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[
"AN EXISTENCE RESULT FOR DISCRETE DISLOCATION DYNAMICS IN THREE DIMENSIONS",
"AN EXISTENCE RESULT FOR DISCRETE DISLOCATION DYNAMICS IN THREE DIMENSIONS"
] | [
"Thomas Hudson "
] | [] | [] | We present a mathematical framework within which Discrete Dislocation Dynamics in three dimensions is well-posed. By considering smooth distributions of slip, we derive a regularised energy for curved dislocations, and rigorously derive the Peach-Koehler force on the dislocation network via an inner variation. We propose a dissipative evolution law which is cast as a generalised gradient flow, and using a discrete-in-time approximation scheme, existence and regularity results are obtained for the evolution, up until the first time at which an infinite density of dislocation lines forms.1991 Mathematics Subject Classification. 35Q74, 74N05, 37N15. | null | [
"https://arxiv.org/pdf/1806.00304v1.pdf"
] | 119,642,378 | 1806.00304 | 076c0e3e4349ffe2b1b09736f5749b0a42c17ae2 |
AN EXISTENCE RESULT FOR DISCRETE DISLOCATION DYNAMICS IN THREE DIMENSIONS
Thomas Hudson
AN EXISTENCE RESULT FOR DISCRETE DISLOCATION DYNAMICS IN THREE DIMENSIONS
We present a mathematical framework within which Discrete Dislocation Dynamics in three dimensions is well-posed. By considering smooth distributions of slip, we derive a regularised energy for curved dislocations, and rigorously derive the Peach-Koehler force on the dislocation network via an inner variation. We propose a dissipative evolution law which is cast as a generalised gradient flow, and using a discrete-in-time approximation scheme, existence and regularity results are obtained for the evolution, up until the first time at which an infinite density of dislocation lines forms.1991 Mathematics Subject Classification. 35Q74, 74N05, 37N15.
INTRODUCTION
In crystalline materials, plastic behaviour is characterised by the generation of slip, which is the process by which the planes of the material's lattice structure are reordered. As the material deforms, slip is propagated via the motion of dislocations, which are topological line defects found in regions where the lattice mismatch required for slip to occur is most concentrated [45,50]. From the very beginnings of the study of dislocations [63,69,76], linear continuum theories have been used to model these defects with great success [9,14,18,19,60,67,79], despite the fact that unphysical singularities are induced in the stress, strain and energy density fields at the dislocation lines. These singularities are a signature of the breakdown of the assumption that the material behaves as a continuum close to dislocation lines, and although no continuum theory is able to accurately capture the properties of dislocation cores, continuum approaches have nevertheless been highly successful at capturing bulk behaviour. Indeed, in a series of recent mathematical works it has been rigorously demonstrated that linear elastoplasticity theory provides an excellent prediction of the strain caused by defects on the atomistic scale [16,34,46,48,49].
Since dislocation motion determines the plastic behaviour of crystalline materials, one of the principal aims of studying these objects is to understand the physical laws which govern their microscopic motion, and consequently, to obtain an accurate description of the evolution of crystal plasticity on a macroscopic scale. To that end, two broad approaches to modelling dislocation motion have developed. Phase field models consider a continuous distribution of dislocations, an approach which has its roots in the classic Peierls-Nabarro model [62,68]; modern examples of such modelling approaches include [43,51,80]. A variety of mathematical results concerning models in this class have been obtained, including well-posedness [10,22], long-time asymptotics [64][65][66], and homogenization results [28,39,40,55,56]. While these models have good mathematical structure, they are typically limited to considering only one family of slip planes at once; moreover, simulating phase field models on a microscopic scale is computationally intensive, since a high resolution mesh is required to accurately resolve the level sets corresponding to individual dislocations.
The second approach is Discrete Dislocation Dynamics (DDD), in which dislocations lines are described as curves within the crystal, and are driven by the action of the Peach-Koehler force [67]. The fact that the dislocations alone are tracked in this approach has the advantage of drastically reducing the computational complexity in comparison with phase field approaches, and as such, DDD has been used as a simulation technique for studying plasticity since the early 1990s [5,6,8,17,33,44,77]. While a significant mathematical literature has developed which considers one-and two-dimensional DDD models for the motion of straight dislocations [1,2,12,13,15,23,24,46,47,78], few mathematical results concerning DDD in a three-dimensional setting exist to date, and in part, this appears to be due to the lack of a clear mathematical statement of what the evolution problem for DDD should be in this setting. This paper therefore seeks to bring together many of the ideas already present in the literature in laying out a well-posed mathematical formulation of DDD which is both general enough to encompass evolution problems similar to those considered by Materials Scientists and Engineers simulating DDD in practice, and mathematically concrete enough to allow the development of further results concerning dislocations in three dimensions. In particular, we hope to open paths towards a deeper mathematical understanding of the numerical schemes used to simulate DDD in practice.
1.1. A regularised theory of DDD in three dimensions. As mentioned above, we seek to develop a well-posed mathematical theory of DDD in three dimensions. In order to be physically-relevant, practical for computation and amenable to mathematical analysis, we make several requirements of this theory:
(1) Dislocations should be curved and satisfy the physically-necessary condition that they are the boundaries of regions of slip. (2) The stress, strain and internal energy density induced in the material by the presence of dislocations are required to be non-singular. (3) The energy of and Peach-Koehler force on a configuration of dislocations should take on explicit expressions in terms of an integral kernel which are computable with a quantifiable error. (4) The underlying material is assumed to be linearly elastic, but need not be isotropic. (5) Dislocation motion should dissipate internal energy.
The first of these conditions is a kinematic requirement for a theory of dislocations to make sense. In common with several recent mathematical works [27,29,73,74], this condition is encoded by describing dislocations as closed 1-currents, which may be viewed closed oriented Lipschitz curves satisfying certain topological constraints.
In order to satisfy the second condition, we construct a regularised version of the classical linear theory via a similar approach to that used in [20,29]: Starting from a regularised distribution of slip and using the ideas of Mura [61], we derive an expression for the internal energy as a double integral which depends on the boundary of the slip surfaces alone. As this expression depends only on the boundaries of the surfaces over which slip has occurred, which correspond exactly to the dislocations in the material, this energy is furthermore consistent with the first requirement made above. Using the Fourier analytic ideas of [9], we study the energy, providing a computable expression for the integral kernel without requiring an explicit expression of the elastic Green's function, which renders the theory broad enough to satisfy both the third and fourth conditions. By performing an inner variation of the energy with respect to the positions of dislocations, we are able to rigorously derive an expression for the Peach-Koehler force.
To fulfil the final condition, we choose to formulate an evolution law for DDD as a generalised gradient flow [4] by prescribing a dissipation potential expressed in terms of the velocity field perpendicular to the dislocation line. This framework enables us to prove our main result, that the evolution problem is well-posed.
1.2. Outline. §1.3 provides a record of the notation used throughout the paper for the reader's convenience, and §2 is devoted to an exposition of the main results of the paper, which are Theorem 1, providing the properties of the regularised energy; Theorem 2, which provides properties of the configurational force on dislocations; and Theorem 3, which asserts the well-posedness of the model for DDD considered here.
The subsequent sections of the paper are then devoted to proving these results. §3 describes the regularisation procedure applied to dislocations and derivation of the explicit representation of the energy due to dislocations; §4 derives the Peach-Koehler (or configurational) force of a dislocation by inner variation; and §5 prove existence, uniqueness, and regularity results for the evolution.
As mentioned above, we consider configurations of slip and dislocations as integral currents [32,35,36,57], since these are the correct mathematical objects to describe the topological restrictions on dislocations [7,27,29,73,74]. Currents generalise the notion of distributions [37] to a geometric setting, and while the theory of these objects can be forbidding, in our setting, the reader should always have in mind surfaces and curves. For convenience, Appendix A recalls the definitions and basic theory related to these objects that is used here.
1.3. Notation. The following notational conventions will be used throughout the paper.
1.3.1. Tensors.
• Important tensors which are fixed throughout (rather than variables) are generally denoted using sans serif fonts, e.g. K ε . • Subscript indices always refer to components in Cartesian coordinates, e.g. f abc .
• Subscript indices appearing after a comma denote partial derivatives: e.g. f i,j = ∂fi ∂xj . • The Einstein summation convention is used throughout, so repeated indices within an expression are always summed, e.g.
a ijik b k = 3 i,k=1 a ijik b k . • Tensor products of vectors are denoted a ⊗ b.
• ∧ denotes the usual alternating product acting on vectors and covectors (for further details, see §A.1). • The dot products between vectors in R 3 is denoted v · τ .
• I ∈ R 3×3 always denotes the identity matrix.
• A ∈ R 3×3×3 denotes the alternating tensor, which satisfies
A ijk = +1 (ijk)
is an even permutation of (123) −1 (ijk) is an odd permutation of (123), 0
otherwise.
• C ∈ R 3×3×3×3 denotes an elasticity tensor, which satisfies the major symmetry C abcd = C cdab , the minor symmetries C abcd = C bacd = C abdc , and a Legendre-Hadamard condition, i.e. there exists
c 0 > 0 such that C abcd v a k b v c k d ≥ c 0 |v| 2 |k| 2 for all v, k ∈ R 3 .
• G : R 3 → R 3×3 denotes the elastic Green's function, i.e. the fundamental solution of the differential operator −C abcd u c,db , which solves
(1) − C αjkl G βk,lj = I αβ δ 0
in the sense of distributions. • G ε := G * ϕ ε denotes a regularised version of the elastic Green's function, solving (2) − C αjkl G ε βk,lj = I αβ ϕ ε in the sense of distributions, where ϕ ε is a smooth, positive, radially symmetric function satisfying R 3 ϕ ε dx = 1.
Sets, currents, measures and integration.
• B r (s) denotes the closed ball of radius r, centred at s ∈ R 3 . • Σ will denote a 2-rectifiable subset of R 3 (i.e. a generalised surface) and Γ will denote a 1rectifiable subset of R 3 (i.e. a generalised curve). • T and S will denote currents, and ∂ is the usual boundary operator.
• I 1 (R 3 ; L ) and I 2 (R 3 ; L ) denote space of integral 1-and 2-currents with multiplicities in a given lattice L ⊂ R 3 . • S ¬ A denotes the restriction of a current S to A. • F # S denotes the pushforward of the current S by F , i.e. the current corresponding to the image F (S). • M(S) denotes the mass of a current S, defined in (4).
• Θ(S) denotes the maximal mass ratio of a current S, defined in (6).
• H m denotes the m-dimensional Hausdorff measure on R 3 . • A f (t)dµ(t) denotes the Lebesgue integral of a Borel measurable function f with respect to the measure µ restricted to a Borel-measurable set A. For some additional details on the basic theory of currents, see Appendix A.
Functions and function spaces.
• Lebesgue, Sobolev and Hölder spaces are all given standard notation, i.e. L p , H k , C k , as are their norms.
• The C 0,γ Hölder seminorm is denoted [F ] γ . • The space of smooth m-forms is denoted D m (R 3 ) (see A.2 for a full definition). • We setḢ 1 (R 3 ) := {∇u ∈ L 2 (R 3 )}, which is equipped with the seminorm u → ∇u L 2 (R 3 ) .
• The space of bounded linear operators mapping a Banach space X to a Banach space Y is denotes L(X, Y ). • The mth Frechet derivative of a function f at a point x is denoted D m f , and its (multilinear)
action on vector v 1 , . . . , v m ∈ R 3 is denoted D m f (x)[v 1 , . . . , v m ].
• The pullback of a function G by F is denoted F # G (see §A.4 for a full definition).
• The identity mapping on R 3 is denoted id.
• The Fourier transform is of a function f is denoted f , and (for f ∈ L 1 (R 3 )) we use the definition
f (k) := R 3 f (x)e −i(k,x) dx, so that f (x) = 1 (2π) 3 R 3 f (k)e i(k,x) dk.
• For numbers, a bar denotes complex conjugation, i.e. if z = x+iy with x, y ∈ R, thenz = x−iy.
• The duality relation between a vector space and its dual is denoted with angular brackets, e.g.
T, φ , e * i , e j . • Inner products are denoted with parentheses, e.g. (u, v).
MAIN RESULTS
Modelling slip and dislocations as currents.
Dislocations are usually modelled as being described by [45,50] • their position, a curve Γ ⊂ R 3 , • their orientation, fixed by defining a tangent field τ : Γ → Λ 1 R 3 , and • their topological 'charge', known as the Burgers vector b ∈ R 3 .
Very often, dislocations are presented as simply being described by the quantities described above, but there is a further topological restriction on possible dislocation configurations, which arises since dislocations must always be the boundary of a region of slip [7,27]. In crystal plasticity, the term slip refers to a displacement across a surface inside a crystal such that the lattice matches perfectly on either side of it: as a consequence, slip is characterised by a lattice vector.
To illustrate the action of slip, suppose B ∈ R 3×3 is an invertible matrix, define a fixed lattice L := BZ 3 ⊂ R 3 which describes the structure of the material considered which satisfies the property that the shortest non-zero lattice vector is of length 1, i.e. min{|b| : b ∈ L \ {0}} = 1. Suppose also that Σ ⊂ R 3 is a compactly-supported oriented surface with a normal field ν across which a slip has occurred. The plastic distortion corresponding to a slip vector b ∈ L across Σ is then the strain field
(3) z = b ⊗ ν H 2 ¬ Σ,
where H 2 ¬ Σ is the 2-dimensional Hausdorff measure restricted to Σ. This description of z as a plastic distortion is motivated by the following remark, which is illustrated in Figure 1: suppose that s ∈ R 3 \ ∂Σ, and δ > 0 is taken such that B δ (s) is disjoint from ∂Σ.
Assuming that Σ ∩ B δ (s) is sufficiently regular, we may partition B δ (s) = Σ ∩ B δ (s) ∪ Σ + ∪ Σ − , where Σ + and Σ − are disjoint open sets. Define the BV vector field u 0 : B δ (s) → R 3 such that u 0 (x) = b x ∈ Σ + , 0 x ∈ Σ − , for which Du 0 = b ⊗ ν H 2 ¬ Σ ∩ B δ (s) .
The function u 0 represents a jump in the displacement of b across the surface Σ ∩ B δ (s), and we can perform this construction for any s / ∈ ∂Σ; however, the same construction fails for s ∈ ∂Σ, which indicates that z has non-trivial distributional curl concentrated on ∂Σ; indeed, the fact that z therefore cannot be globally represented as a gradient is precisely the reason that z is a plastic distortion.
Since the slip z is concentrated on a two-dimensional set Σ, following [27,29], it is natural to define an associated vector-valued integral 2-current T , for which
T, φ := Σ ν, φ b dH 2 for any φ ∈ D 2 (R 3 ).
The space of such 2-currents will be denoted I 2 (R 3 ; L ), and can be endowed with an additive structure, which arises by taking the union of the support, and the sum of the corresponding fields b. If T is an integral current, then its boundary ∂T is also an integral current, and has a consistently oriented tangent field τ : ∂Σ → R 3 ; the equivalent of Stokes Theorem then implies that
∂Σ τ, φ b dH 1 = ∂T, φ = T, dφ = Σ ν, dφ b dH 2 for all φ ∈ D 1 (R 3 ).
The integral 1-current ∂T encodes a configuration of dislocations supported on Γ = ∂Σ, with Burgers vector b : Γ → L , and line direction fixed by the tangent field τ . In analogue with the previous case, the space of all 1-currents taking values in L is denoted I 1 (R 3 ; L ); we note that Theorem 2.5 of [27] describes the structure of I 1 (R 3 ; L ). We call the vector-valued current T a slip configuration, and S = ∂T the corresponding dislocation configuration. It is clear that there are many possible slip configurations corresponding to the same dislocation configuration, since there are many surfaces with the same boundary. The space of admissible dislocation configurations is then defined to be A := S ∈ I 1 (R 3 ; L ) S = ∂T for some T ∈ I 2 (R 3 ; L ) s.t.
T, φ = Σ ν, φ b dH 2 for all φ ∈ D 2 (R 3 ) .
It is straightforward to check that A is an additive subspace of I 1 (R 3 ; L ), since ∂ 2 T = 0 for any T ∈ I 2 (R 3 ; L ).
The choice to require that each S ∈ A is the boundary of a 2-current naturally encodes the fact that dislocation must be the boundary of a region of slip, and this requirement is equivalent to that dislocation configurations satisfy a 'divergence-free' condition, as discussed in [27], which is in turn equivalent to the principle of the conservation of Burgers vector (see [45,50,62]). The additive structure of integral currents allows for the description of complicated configurations of dislocations via a superposition of corresponding elementary currents.
We now define two useful functions which quantify aspects of the geometry of dislocation configurations. The mass of S ∈ I 1 (R 3 ; L ) is defined to be
(4) M(S) := sup | S, φ | φ ∈ D 3 (R 3 ) with |φ(x)| ≤ 1 for all x ∈ R 3 .
In the case where S ∈ A is characterised as above, this is equivalent to the formula which corresponds physically to the total length of dislocation, weighted by the Burgers vector. In analogy with §9.2 in [57], we also define the mass ratio for S ∈ A as
(6) Θ(S) := sup M S ¬ B r (s) r s ∈ supp(S), r > 0 ;
here B r (s) is the closed ball of radius r > 0 centred at s ∈ R 3 , and S ¬ A means the restriction of a current S to a set A. The mass ratio should be viewed as a way of measuring the maximal spatial density of a current; for currents of fixed mass, Θ(S) can be arbitrarily large (see Figure 2 for an explanation). Note however that as long as S = 0, then it follows that Θ(S) ≥ 1: If S is supported on Γ, then the definition of the one-dimensional density of S at s ∈ Γ, defined in analogy with the definition in §9.2 of [57], must satisfy
Functions defined on dislocation configurations.
In order to formulate an appropriate setting in which to consider DDD, we will need to consider fields which are defined on dislocations themselves. As such, we will consider functions which are differentiable 'along' a dislocation configuration S. This section lays out the basic definitions we use, which allow us to state our main results. Suppose that S ∈ A , with corresponding 1-rectifiable set Γ, Burgers vector b : Γ → L and tangent field τ . Consider g : Γ → R 3 which is measurable with respect to H 1 ¬ Γ. We will say that ∇ τ g : Γ → R 3 , a H 1 ¬ Γ-measurable function is a weak derivative of g along S if, for any C 1 function f :
R 3 → R 3 , we have Γ f · ∇ τ g dH 1 = − Γ Df [τ ] · g dH 1 ,
where Df ∈ L(R 3 ; R 3 ) denotes the Frechet derivative of f . We note that if G ∈ C 1 (R 3 , R 3 ) and g := G| Γ , ∇ τ g = DG[τ ]; the fact that ∂S = 0, i.e. S has no boundary, ensures that no 'boundary terms' are required in the definition above. It is straightforward to show that weak derivatives defined in this way are unique when they exist by using the fact that Γ is expressible as a union of images of R under Lipschitz maps, pulling back, and applying the Fundamental Lemma of the Calculus of Variations on R.
For 1 ≤ p < +∞, in analogy with the usual definitions, we set
L p (S; R N ) := g : C → R N g p < +∞ with norm g L p := Γ |g| p dH 1 1/p , L ∞ (S, R N ) := g : Γ → R N g L ∞ < +∞ with norm g L ∞ := ess sup |g(s)| s ∈ Γ},
where the essential supremum in the latter definition is taken up to H 1 null sets. We also define the space
H 1 (S; R 3 ) := g ∈ L 2 (S; R 3 ) ∇ τ g ∈ L 2 (S; R 3 ) ,
which has a real Hilbert space structure when endowed with the inner product
(f, g) = Γ f · g + ∇ τ f · ∇ τ g dH 1 .
As usual, we will denote the dual space of H 1 (S; R N ) as H 1 (S; R N ) * .
2.3. Energy of regularised slip distributions. For any slip configuration T ∈ I 2 (R 3 ; L ), we now define a regularised energy by a procedure similar to that considered in §2.4.2 of [29], distributing slip about the set Σ on which T is supported by mollifying. Physically-speaking, our choice to smooth distributions of slip can be justified by noting that a definition of the lattice plane over which slip has occurred cannot be given with a precision greater than that of a single lattice spacing, and similarly, the position of a dislocation core cannot be ascertained to a precision of less than a few lattice spacings. As such, regularising defines the lengthscale at which linear elasticity is invalid, and as stated in §1.1, this choice has the convenient mathematical benefit that all fields considered are non-singular, in common with the reality on the atomistic scale.
To this end, we suppose that ϕ 1 ∈ C ∞ (R 3 ) is a function which is rapidly decreasing (in the Schwartz sense), is radially symmetric, and satisfies R 3 ϕ 1 (x) dx = 1, such as the Gaussian
(8) ϕ 1 (x) = 1 (2π) 3/2 exp − 1 2 |x| 2 .
For any ε > 0, we then define ϕ ε (x) := ε −3 ϕ 1 (x/ε). If T ∈ I 2 (R 3 ; L ) is a slip configuration supported on Σ, with corresponding normal field ν, and slip b : Σ → L , in analogy with the plastic deformation considered in (3), we define a smoothed plastic distortion z ε T :
R 3 → R 3×3 to be (9) z ε T (x) := Σ b(s) ⊗ ν(s)ϕ ε (x − s)dH 2 (s).
Here H 2 is the 2-dimensional Hausdorff measure (i.e. the surface area measure on Σ), and the field z ε T is well-defined and smooth since ϕ ε is assumed to be C ∞ . Following the ideas of Kröner and Mura [52,53,61], we suppose that the system equilibriates elastically in response to this plastic strain. Using the additive decomposition of the strain, the total distortion β ε is assumed to take the form
β ε = z ε T + Du ε ,
where u ε : R 3 → R 3 , and recalling the definition ofḢ 1 (R 3 ) given in §1.3, the energy at equilibrium due to a configuration of slip described by T ∈ I 2 (R 3 ; L ) is therefore
(10) E ε (T ) := min u ε ∈Ḣ 1 (R 3 ) I(z ε T + Du ε ),
where the total internal energy, I :
L 2 (R 3×3 ) → R, is defined to be (11) I(β) := R 3 1 2 β : C β dx = R 3 1 2 C ijkl β ij β kl dx.
The fundamental insight in the work of Kröner and Mura is that while dislocations must be the boundary of a region of slip, the precise surface over which slip has occurred is irrelevant to the energy and mechanical response of the system, i.e. although (10) appears to depend on T , in fact it only depends upon ∂T . This is compatible with the experimental observation that dislocations moving through a crystal leave no trace of having passed, since the crystal 'heals' perfectly after slip occurs. Using these insights, we prove the following theorem which encodes the dependence of the energy on ∂T alone, and moreover provides an expression for the internal energy given directly in terms of an integral over the dislocation configuration itself. Theorem 1. If T ∈ I 2 (R 3 ; L ) is compactly supported, the energy E ε defined in (10) depends only on S = ∂T , so that we may define Φ ε :
A → R with Φ ε (S) := E ε (T ) for any S ∈ A where S = ∂T.
Further, if T ∈ I 3 2 and S = ∂T ∈ A respectively take the form T, φ = Σ ν, φ b dH 2 for φ ∈ D 2 (R 3 ) and S, η = Γ τ, η b dH 1 for η ∈ D 1 (R 3 ), then the energy functionals E ε (T ) and Φ ε (S) may be written as
E ε (T ) = Σ×Σ 1 2 J ε abcd (s − t)b a (s)ν b (s)b c (t)ν d (t)d(H 2 ⊗ H 2 )(s, t) (12a) Φ ε (S) = Γ×Γ 1 2 K ε abcd (s − t)b a (s)τ b (s)b c (t)τ d (t)d(H 1 ⊗ H 1 )(s, t), (12b) where the kernels J ε , K ε : R 3 → R 3×3×3×3 are defined to be J ε kmgr (s) := R 3 C abcd A bpl C ijkl G ε ai,jn (x − s)A pmn A dqh C ef gh G ε ce,f s (x)A qrs dx, (13a) K ε kpgq (s) := R 3 C abcd A bpl C ijkl G ε ai,j (x − s)A dqh C ef gh G ε ce,f (x) dx, (13b)
where A is the alternating tensor as defined in §1.3, and G ε := G * ϕ ε , i.e. the convolution of the elastic Green's function with ϕ ε . Moreover, J ε and K ε satisfy the following properties.
(1) J ε abcd (s) = J ε cdab (s) = J ε abcd (−s) and K ε abcd (s) = K ε cdab (s) = K ε abcd (−s) for any s ∈ R 3 ; (2) J ε and K ε are smooth. (3) For any m ∈ N, 0 ≤ j ≤ m, and vectors v 1 , . . . , v j ∈ S 2 , there exists a constant C m,j such that for all s ∈ R 3 , D m K ε (s) : v 1 , . . . , v j , s |s| , . . . , s |s| ≤ C m,j ε 2m+2 + ε 2j |s| 2m+2−2j .
The proof of this theorem is given in §3, and proceeds by verifying that E ε (T ) as given in (10) is well-defined, before characterising the solution of the minimisation problem by applying the elastic Green's function and the ideas used to derive Mura's formula [61]. Ideas from [9] (closely related to ideas used in proving Theorem 4.1 in [29]) are then used to derive a Fourier characterisation, which allow us to deduce the asserted properties of the kernels J ε and K ε . We note at this stage that the definition of the regularised energy E ε clearly depends upon the precise choice of ϕ ε , which particularly influences the behaviour of segments of dislocation line which are at a distance on the order of ε apart. As argued in the introduction, this is exactly the scale at which linear elasticity theory fails to be valid, but the results of [29] demonstrate that, to leading order as ε → 0, the Γ-limit of the energy is independent of the choice of regularisation. While this theory (and indeed no continuum theory) can therefore accurately capture the behaviour of dislocations on the lattice scale, we can hope to accurately capture the behaviour of dislocation loops which are 'large' relative to the interatomic distance. Furthermore, to capture additional knowledge about the crystal structure considered, the choice to make ϕ 1 radially symmetric and independent of b and ν could be relaxed, allowing for the modelling of different 'widths' of plane over which slip occurs, although we do not pursue this choice here.
The Peach-Koehler force.
The power of the expression for the energy given in (12b) is that it allows us to take explicit variations of the energy, and thereby to derive the equivalent of the configurational Peach-Koehler force in this model. Since we are varying the set on which the Burgers vector b and line direction τ are defined, an appropriate notion of variation is that of inner variation (see Chapter 3 of [41]) or variation of the reference state (as described in §2.1.5 of [42]). To construct an inner variation, we use the notion of pushforward (defined in §A.4): if g ∈ C 0,1 (R 3 ; R 3 ), we define the inner variation of Φ ε at S ∈ A in the direction g to be the linear functional (14) DΦ ε (S), g :
= d dδ Φ ε (id + δg) # S δ=0
where id : R 3 → R 3 is the identity mapping id(x) := x. The result of taking this variation and properties of the resulting functional are encoded in the following theorem, which provides an expression of the variation as a field defined on the dislocation configuration, as well as a uniform bound and a form of continuity result under deformation.
Theorem 2.
If S ∈ A , the inner variation of the energy at S is given by
(15) DΦ ε (S), g = − Γ f PK i (s, S)g i (s)dH 1 (s), where f PK (s, S) := G(s, S) ∧ τ (s) and G k (s, S) := Γ A klm K ε alcd,m (s − t)b a (s)b c (t)τ d (t)dH 1 (t). If S = ∂T , where T ∈ I 2 (R 3 ; L )
is supported on Σ with Burgers vector b : Σ → L and normal field ν, G can alternatively be written
(16) G k (s, S) = − Σ A def A klm K ε alcd,mb (s − t)b a (s)b c (t)ν f (t)dH 2 (t).
Moreover, recalling the definitions of M and Θ given respectively in (4) and (6), we have the bound
(17) f PK (S) L ∞ ≤ C ε b L ∞ Θ(S) log 1 + 2 M(S) ε Θ(S) ,
and if g : S → R 3 is a Lipschitz map and F := id + g, then
(18) F # f PK (F # S) − f PK (S) L ∞ ≤ 1 + CM(S) ∇ τ g L ∞ + CM(S) g L ∞ ,
where C is a constant independent of v and S.
A proof of this Theorem 2 is given in §4. The main achievement of this result is in obtaining the bound (17), which is crucial to proving the long-time existence result for the formulation of DDD considered in the following sections. This bound is a significant improvement over the more naïve estimate f PK (S) L ∞ ≤ C M(S), which can be deduced directly from the fact that K ε and DK ε are uniformly bounded, proved in Theorem 1. Instead, the estimate is obtained by a rearrangement argument, using the limited geometric information Θ(S) provides to guarantee this weaker dependence on the mass. The bound (18) provides a form of continuity for the Peach-Koehler force, demonstrating that under a Lipschitz variation of S which is close to the identity, the Peach-Koehler force of the new configuration F # S is close to that obtained on the initial configuration. Since these fields are defined on different sets, in order to measure the difference we must pull back this new field onto the initial configuration.
Dislocation mobility.
It is widely believed that moving dislocations dissipate energy via phonon radiation (see for example §3.5 in [50], and §7-7 in [45]): the movement of a dislocation through a 'rough' landscape of local minima generates high-frequency lattice waves which radiate away as heat, leading to drag. When modelling dislocation motion at low to moderate strain rates, it is typically assumed that drag dominates inertial effects, which are therefore neglected. This assumption has been supported by microscopic simulations of dislocation motion (see for example the discussion in §4.5 and §10.2 of [17], and [11,25,26]). Neglecting inertia inevitably entails that dislocation motion (and hence plastic distortion) is modelled as a dissipation-dominated process; as such, a natural mathematical framework for modelling dislocation motion is that of a generalised gradient flow [4].
At low temperatures, the process of slip is dominated by glide, a process which allows dislocations to move while conserving lattice volume [7,45,50]. Requiring that dislocations undergo glide motion only is equivalent to requiring that slip can only evolve on planes which contain b. The evolution of slip in directions parallel to the Burgers vector is called climb, and requires mass transport via point defect diffusion; at low temperatures this is a much slower process than that of glide.
To model these phenomena in DDD simulations, various constitutive assumptions on dislocation mobility are available; for various examples, see [8,21,50]. Usually, the velocity v of a segment of dislocation is related to the configurational force on the dislocation line via a mobility function M, which depends locally on the Burgers vector b, the dislocation orientation τ , and the Peach-Koehler force f PK : v = M(b, τ, f PK ).
Such mobilities are informed by molecular dynamics simulations or experiment, and are generally linear, power laws, or possibly include some frictional threshold before the onset of dislocation motion, mimicking the Peierls barrier. In common with many geometric evolution problems, it is usually In order to define a generalised gradient flow framework which encompasses mobilities of the type referred to above, we restrict ourselves to considering mobilities which take the form
M(b, τ, f ) := −∇ f ψ * (b, τ, −f ),
where ψ * is a entropy production which is convex in f , describing the rate at which entropy is produced by the force f . Requiring the existence of ψ * is not particularly restrictive, as it ensures that energy is conserved in a closed system, and to the author's knowledge, all mobility laws for DDD used in practice take this form.
When an entropy production is defined, it is natural to define a conjugate dissipation potential ψ, which describes the rate at which energy is lost through the variation of a dislocation configuration, and is given as the Legendre-Fenchel transform of ψ * , i.e. (19) ψ
(b, τ, v) := sup f ∈R 3 v, f − ψ * (b, τ, f ) .
The frictional force resulting from a velocity v is then −∇ψ(v); this follows from the fact that for convex conjugate functions,
(20) v = −∇ψ * (−f ) if and only if f = −∇ψ(−v) if and only if f, v = ψ(−v) + ψ * (−f ).
As an example, in the case of a linear relationship between configurational force and velocity, v = B(b, τ )f , it is straightforward to check that we may define
ψ * (f ) = 1 2 f, B(b, τ )f and ψ(v) = 1 2 v, B(b, τ ) −1 v , where B −1 denotes the matrix inverse of B.
In practice, we find that requiring that the dissipation potential is uniquely a function of the dislocation velocity as in (19) is insufficient to guarantee a well-defined evolution. As an illustration, consider the possible scenario in Figure 3. While the dislocation is initially smooth, and f PK is smooth up until the final time, v = B(b, τ )f PK loses regularity exactly as a jump in τ develops. At the final time, the dynamics could require v to be discontinuous, resulting in segments of dislocation 'ripping apart', breaking the physical requirement that dislocations are the boundary of regions of slip, and leading to a blow-up of the evolution.
To avoid this possibility, in the following section, we introduce assumptions requiring that the energy dissipated by dislocation motion also depends on the rate of change of the tangent to the dislocation.
2.6. Dissipation potential. Motivated by the discussion in §2.5, we suppose that the velocity of a dislocation is described by a field along its length, v, and assume that the dissipation potential for a dislocation configuration S supported on Γ ⊂ R 3 with a velocity field v : Γ → R 3 is expressed as
Ψ(S, v) = Γ 1 2 ∇ τ v · A(b, τ )∇ τ v + ψ(b, τ, v) dH 1
where, ∇ τ v denotes the weak derivative of the velocity along the dislocation line as defined in §2.2, and as usual, τ : Γ → S 2 is the tangent field to S. We make the following constitutive assumptions on A and ψ.
(C 1 ) A : L × S 2 → R 3×3 is a matrix-valued function which is symmetric and strictly positive definite everywhere, and there exists α > 0 such that
w · A(b, τ )w ≥ α|w| 2 for any (b, τ, w) ∈ L \ {0} × S 2 × R 3 . (C 2 ) ψ : L × S 2 × R 3 → [0, +∞] is a positive function which is strictly convex in its third argument, satisfying ψ(b, τ, 0) = 0 for any (b, τ ) ∈ L × S 2 . Moreover, ψ(b, τ, v) = +∞ if v · τ = 0. (R) For any b ∈ L , τ → A(b, τ ) is smooth and bounded on S 2 , and (τ, v) → ψ(b, τ, v) is smooth on the set {v · τ ∈ S 2 × R 3 : v · τ = 0}. (G) There exists β > 0 such that ψ(b, τ, v) ≥ 1 2 β|v| 2 for all (b, τ, f ) ∈ L × S 2 × R 3 .
As remarked in the previous section, the assumptions above are broad enough to encode a wide variety of modelling assumptions made when modelling dislocation dynamics:
• The convexity assumptions (C 1 ) and (C 2 ) imply that dissipation potential is always positive and increases as the dislocation velocity or rate of bending increases, and no change in the energy dissipated is produced if dislocations do not deform or translate within the material. The fact that ψ(b, τ, v) = +∞ if v · τ = 0 enforces the requirement that meaningful dislocation velocity fields must be locally perpendicular to the dislocation line. • The regularity assumption (R) ensures that energy dissipation rate varies smoothly with the velocity and orientation of the dislocation line. • The growth assumption (G) is a technical assumption, and is not particularly restrictive, however relaxing it would make some aspects of our analysis more technical. On the other hand, the choice to make the dissipation depend upon ∇ τ v as well as v appears to be a novel addition to DDD. It does not seem to be unreasonable to require that energy is dissipated by the bending or stretching of dislocation lines, and it this additional term gives us control of the bending rate, ruling out the generation of singularities similar to those shown in the figure and allowing us to prove that the evolution is well-posed. It would be of great interest to understand whether this term is indeed physically-justified via a future computational study of a realistic model, or indeed whether this additional dependence can be mathematically removed via (for example) a vanishing viscosity argument.
As an indicative example of a constitutive relation satisfying these assumptions which is closely related to a dislocation mobility which is already present in the literature, we may define
Ψ(S, v) = S 1 2 α|∇ τ v| 2 + ψ(b, τ, v) dH 1 (s),
so that A(b, τ ) = α I, and where we set
ψ(b, τ, v) = 1 2 v T B † (b, τ )v v · τ = 0 +∞ v · τ = 0, with B(b, τ ) = |b ∧ τ | 2 B 2 eg + (b · τ ) 2 B 2 s − 1 2 P(τ )b ⊗ P(τ )b |b ∧ τ | 2 + B 2 ec |b ∧ τ | 2 + B 2 s (b · τ ) 2 (b ∧ τ ) ⊗ (b ∧ τ ) |b| 2 |b ∧ τ | 2 .
Here, B † denotes the Moore-Penrose pseudo-inverse of B; in this case, this is simply the matrix which has the same eigenspaces as B and inverts any non-zero eigenvalues. B eg , B ec , B s > 0 are all mobility parameters describing dissipative timescales resulting from various different modes of motion (bending and stretching, glide of edge dislocations, climb of edge dislocations and glide of screw dislocations), and α −1 > 0 is the energy dissipation rate per unit additional area swept out per unit length of dislocation.
Defining ψ * (b, τ, f ) = sup{v · f − ψ(b, τ, v) | v ∈ R 3 }, i.e. the Legendre-Fenchel transform of ψ, we find that ψ * (b, τ, f ) = 1 2 f, B(b, τ )f , and we note that v = B(b, τ )f = −∇ψ * (b, τ, −f )
is the mobility relation for dislocations in BCC defined in equation (10.40) of [17]. We therefore see that if α = 0, the model as defined above reduces to that considered in [17]. We remark that as a consequence of assumption (C 2 ), we have the following characterisation of the subgradient of ψ:
(21) ∂ v ψ(b, τ, v) = ∅ v · τ = 0, {D ⊥ τ ψ(b, τ, v)} v · τ = 0,
where D ⊥ τ ψ(b, τ, v) means the gradient of ψ taken in directions perpendicular to τ .
2.7. Evolution problem. When viewed as a function defined on H 1 (S; R 3 ), Ψ(S, ·) has a well-defined subdifferential with respect to v,
∂ v Ψ(S, v) ⊂ H 1 (S, R 3 ) * , and recalling the definition of D ⊥ τ ψ made above, ξ ∈ ∂ v Ψ(S, v) implies that ξ, w = S ∇ τ v · A(b, τ )∇ τ w + D ⊥ τ ψ(b, τ, v) · w dH 1 as long as v · τ = 0 H 1 -almost everywhere on S (otherwise, ∂ v Ψ(S, v) = ∅).
As shorthand for the formula above, we will write
∂ v Ψ(S, v) = −div τ A(b, τ )∇ τ v + D ⊥ τ ψ(b, τ, v)
. This expression corresponds to the frictional force induced on the dislocation configuration by moving according to the velocity field v. In the overdamped regime where inertial effects are neglected, these frictional forces balance with the configurational forces, so that v must satisfy
(22) − div τ A(b, τ )∇ τ v + D ⊥ τ ψ(b, τ, v) = f PK in H 1 (S; R 3 ) * .
It is straightforward to check that, since Ψ is convex on H 1 (S; R 3 ), an equivalent criterion is to require that v solves the minimisation problem
(23) v ∈ argmin v∈H 1 (S;R 3 ) Ψ(S, v) + DΦ ε (S), v = argmin v∈H 1 (S;R 3 ) Ψ(S, v) − (f PK , v) L 2 .
The force balance equation (22) is Eulerian in nature: it must be satisfied on the dislocations themselves as they deform. For the purpose of proving existence of solutions, we now cast an alternative Lagrangian formulation for a solution of DDD: we suppose that the position of points on the dislocation line at time t is expressed as a function of time and position on the dislocation line at the initial time. In the language of geometry, this idea is expressed as a pushforward by U (recall §A.4), where U : [0, T ] × S 0 → R 3 . Writing U (t) to denote the mapping at time t, we require that U (0) = id on S 0 , and the 'trajectory' of currents is then
S t := U (0) # S 0 for all t ∈ [0, T ].
In this formulation, we see that U directly identifies the family of currents {S t } ⊂ A . Thanks to §4.1.14 of [35], S t is well-defined as a current as long as U t is a Lipschitz map on S 0 . We will find that an appropriate 'energy space' in which to seek to prove the existence of U is
H 1 ([0, T ]; H 1 (S 0 ; R 3 )), i.e. the space of L 2 Bochner-integrable functions from [0, T ] in H 1 (S 0 ; R 3 )
with L 2 Bochner-integrable weak derivatives; for further detail on the definition of such spaces, see for example Chapter 7 of [72], and we denote the weak time derivative of U (t) asU (t).
A priori, there is no guarantee that a generic U (t) ∈ H 1 (S 0 ; R 3 ) is Lipschitz, and therefore no guarantee that U (t) # S 0 is a current. In order to ensure S t ∈ A for all time, we additionally require that U (t) ∈ C 0,1 (S 0 ; R 3 ). We will say that a pair (U, {v t } t∈[0,T ] ) form a solution of DDD as formulated in (22)
if U ∈ H 1 ([0, T ]; H 1 (S 0 ; R 3 )), U (t) ∈ C 0,1 (S 0 ; R 3 ) and v t ∈ H 1 (U (t) # S 0 ; R 3 ) for almost every t ∈ [0, T ], and(24)∂ v Ψ U (t) # S 0 , v t f PK U (t) # S 0 in H 1 U (t) # S 0 ; R 3 * and U (t) # v t =U (t) in H 1 (S 0 ; R 3 )
for almost every t ∈ [0, T ], where the definitions of pushforward U (t) # and pullback U (t) # are given in §A. 4. Now that we have given a definition of what it means to be a solution to DDD, we have the following well-posedness result.
Theorem 3. If S 0 ∈ A is a finite union of C 1,γ curves with γ ∈ (0, 1], and satisfies Θ(S 0 ) < +∞, then there exists a unique solution (U, {v t }) satisfying (24) for t ∈ [0, T ] where
T := sup t ∈ R : Θ(U (s) # S 0 ) < +∞ for all s ≤ t . Moreover, U ∈ C 0 [0, T ]; C 1 (S 0 ; R 3 ) .
We note that a strength of this result is that it allows for dislocation collision, since there is no requirement that U is invertible, although as currently phrased in a Lagrangian form, it is not clear that dislocations satisfy the correct evolution if they merge. Indeed, in practical simulations of DDD, dislocations are remeshed exactly as dislocation segments approach separation distances of O(ε), as discussed in §10.4 of [17]. It would be of great interest to understand how best to correctly incorporate this phenomenon into a mathematical theory of DDD in future. Generically, we expect Θ(U (t) # S 0 ) < +∞ for all time, since the contrary would require a concentration of internal energy on a small set; while at present we are unable to rule out the possibility of this occurring, it would be interesting in future to confirm existence for all time for at least a large class of initial data.
Conclusion.
We have provided a framework in which to study a regularised form of DDD in three dimensions, inspired by various ideas in both the Engineering and Mathematics literature [9,20,27,29,61,74]. Computable integral formulae for the energy of and configurational forces on a general dislocation configuration were derived, and bounds on the configurational force which depend weakly on the overall length of dislocation were obtained. A gradient flow formalism in which to study DDD was proposed, and within this framework, we obtained a well-posedness result for the evolution up until the first time an infinite density of dislocations develops.
It is hoped that the energetic framework developed here is sufficiently general to open the way to new upscaling results such as those in [31,38,58,59,70,75] in a three-dimensional setting, to build upon the results of [29] in the case where the regularisation lengthscale ε tends to zero, and to allow the mathematical study of the numerical schemes used in practical implementations of DDD.
THE ENERGY OF DISLOCATIONS
Our aim in this section is to prove Theorem 1, providing a characterisation of the energy of dislocations.
3.1. Minimisation problem. Our first step towards proving Theorem 1 is to characterise the solution to the minimisation problem (10).
Lemma 4.
Assuming that C satisfies a Legendre-Hadamard condition, T ∈ I 2 (R 3 ; L ) and z ε T is as defined in (9), the variational problem in (10) has a unique solution, which is smooth and satisfies the equation
(25)
− C ijkl u ε k,lj = C ipqr (z ε T ) qr,p , in the sense of distributions. Moreover, the solution may be represented as
(26) u ε α (x) = R 3 C ijkl G αi,j (x − y)(z ε T ) kl (y)dy = Σ C ijkl G ε αi,j (x − y)b k (s)ν l (s)dH 2 (s),
where G is the elastic Green's function, which, recalling §1.3, is the distributional solution of (1), and the function G ε := G * ϕ ε is a regularised version of G, solving (2).
Proof. The existence of a minimiser follows from the fact that by Theorem 5.25 in [30], any quadratic function is quasiconvex if and only if it is rank-one convex, and in this case rank-one convexity of the integrand is straightforward to check, following from the assumption that C satisfies a Legendre-Hadamard condition. As a consequence, I is weakly lower semicontinuous onḢ 1 (R 3 ) and unique minimisers exist in this space; computing the Frechet derivative of I in the same space shows that the solution satisfies the equation (25) in the sense of distributions. Convolving the distributional equation (1) with f β and contracting the index β, we see that
−C αjkl G βk * f β,lj = f α .
Setting f α = C αpqr (z ε T ) qr,p , we find that u ε α = C βpqr G βα * (z ε T ) qr,p . Integrating by parts to move the derivative with respect to x j onto G, then using the fact that G is a symmetric tensor, i.e. G ij = G ji for any i, j ∈ {1, 2, 3}, we obtain the first equality in (26). The second equality follows by using Fubini's theorem to deduce that
G * ϕ ε * b ⊗ νH 2 ¬ Σ = G * ϕ ε * b ⊗ νH 2 ¬ Σ = G ε * b ⊗ νH 2 ¬ Σ .
We remark that while Lemma 4 establishes that the problem (10) has a unique solution u ε , it does not guarantee the positivity of the energy I(z ε + Du ε ), since we cannot say anything about the sign of I(z ε ) unless z ε is a gradient. We also note that under appropriate growth and quasiconvexity conditions, the existence of u ε can be ensured if the stored energy density takes a more general nonlinear form; related ideas are discussed in [58,59,75] in a two-dimensional setting.
Energy.
A key feature of the linear theory we consider is that it allows the derivation of a representation formula for the distortion due to a configuration of slip, which leads to the following result, allowing us to provide an explicit integral formula for the internal energy (10).
Lemma 5.
If T ∈ I 2 (R 3 ; L ), supported on Σ, with slip vector b and normal field ν, the energy E ε defined in (10) may be represented
(27) E ε (T ) = R 3 Σ×Σ 1 2 C abcd A bpl C ijkl b k (s)G ε ai,jn (x − s)A pmn ν m (s) × A dqh C ef gh b g (t)G ε ce,f s (x − t)A qrs ν r (t) d(H 2 ⊗ H 2 )(s, t) dx where A is the alternating tensor as defined in §1.3. Furthermore, (27) depends only on ∂T , which entails that Φ ε : A → R with Φ ε (S) := E ε (T ) for any S ∈ A where S = ∂T
is well-defined, and if S = ∂T is supported on Γ, with Burgers vector b and tangent field τ , Φ ε (S) may be expressed as
(28) Φ ε (S) = R 3 Γ×Γ 1 2 C abcd A bpl C ijkl b k (s)G ε ai,j (x − s)τ p (s) × A dqh C ef gh b g (t)G ε ce,f (x − t)τ q (t) d(H 1 ⊗ H 1 )(s, t) dx.
Proof. Since T is fixed during this proof, throughout, we write z ε in place of z ε T to keep notation as concise as possible. Applying (26), we write the elastic distortion β ε as (29) β ε ab = u ε a,b + z ε ab = C ijkl G ai,j * z ε kl,b + z ε ab . Using the definition of the elastic Green's function given in (1) with a change of indices, integration by parts, and the major symmetry of the elasticity tensor, z ε ab = I ka δ 0 * z ε kb = −C klij G ai,jl * z ε kb = −C klij G ai,jl * z ε kb,l = −C ijkl G ai,j * z ε kb,l . Substituting this representation into (29) in place of the latter term, we obtain
β ε ab (x) = R 3 C ijkl G ai,j (x − y)z ε kl,b (y) − G ai,j (x − y)z ε kb,l (y) dy, = R 3 C ijkl G ai,j (x − y)[I lm I bn − I bm I ln ]z ε km,n (y)dy, = R 3 A plb C ijkl G ai,j (x − y)A pmn z ε km,n (y)dy,
where we have used the elementary tensor identity A pmn A plb = I lm I bn − I bm I ln . Using the definition of z ε given in (9), we have
β ε ab (x) = R 3 A plb C ijkl b k G ai,j (x − y) Σ A pmn ν m (s)ϕ ε ,n (y − s)dH 2 (s) dy.
where τ is the tangent vector field on ∂Σ. If x / ∈ ∂Σ, Fubini's theorem applies to the above integral representation, so using the definition of G ε given in §1.3 and applying Stokes' Theorem, we find that
β ε ab (x) = Σ A bpl C ijkl b k G ε ai,jn (x − s)A pmn ν m (s)dH 2 (s) = ∂Σ A bpl C ijkl b k G ε ai,j (x − s)τ p (s)dH 1 (s),
where τ : ∂Σ → S 2 is the tangent vector field on ∂Σ. Substituting the former expression of β ε into the definition of I stated in (11) gives (27). Similarly, substituting the latter expression into (11) allows us to directly deduce that Φ ε is well-defined and has the expression given in (28).
Kernel representation.
Inspecting the formulae for E ε and Φ ε given in Lemma 5, we note that both expressions can be regarded as a convolution integral against an interaction kernel; this form allows us to use ideas from [9] to prove the following result, which, when combined with Lemma 4 and Lemma 5, completes the proof of Theorem 1.
Lemma 6.
If T ∈ I 2 (R 3 ; L ) is supported on Σ, with slip vector b and normal field ν, and S = ∂T is supported on Γ with Burgers vector b and tangent field τ , the energy functionals E ε (T ) and Φ ε (S) may be expressed as
E ε (T ) = Σ×Σ 1 2 J ε abcd (s − t)b a (s)ν b (s)b c (t)ν d (t)d(H 2 ⊗ H 2 )(s, t) Φ ε (S) = Γ×Γ 1 2 K ε abcd (s − t)b a (s)τ b (s)b c (t)τ d (t)d(H 1 ⊗ H 1 )(s, t),
where the kernels J ε , K ε :
R 3 → R 3×3×3×3 are defined to be J ε kmgr (s) := R 3 C abcd A bpl C ijkl G ε ai,jn (x − s)A pmn A dqh C ef gh G ε ce,f s (x)A qrs dx, K ε kpgq (s) := R 3 C abcd A bpl C ijkl G ε ai,j (x − s)A dqh C ef gh G ε ce,f (x) dx,
and satisfy the following properties.
(1) J ε abcd (s) = J ε cdab (s) = J ε abcd (−s) and K ε abcd (s) = K ε cdab (s) = K ε abcd (−s) for any s ∈ R 3 ; (2) J ε and K ε are smooth.
(3) For any m ∈ N, 0 ≤ j ≤ m, and vectors v 1 , . . . , v j ∈ S 2 , there exists a constant C m,j such that for all s ∈ R 3 ,
D m K ε (s) : v 1 , . . . , v j , s |s| , . . . , s |s| ≤ C m,j ε 2m+2 + ε 2j |s| 2(m−j)+2 .
Proof. We divide the proof into a series of steps, corresponding to each of the assertions made.
Kernel representation. The existence of the kernels is a straightforward consequence of applying Fubini's theorem to the expressions (27) and (28); J ε (s) and K ε (s) are finite for any s ∈ R 3 since the elastic Green's function satisfies the standard properties that |G ij,k (x)| |x| −2 and |G ij,kl (x)| |x| −3 , and therefore there exist constants C ε such that G ε ij,k (x) ≤ C ε min 1, |x| −2 and G ε ij,kl (x) ≤ C ε min 1, |x| −3 , and it follows that G ε ij,k and G ε ij,kl are in L 2 (R 3 ), and hence the integrals in (13) converge for any s ∈ R 3 .
Fourier characterisation of kernels. To prove that the kernels J ε and K ε satisfy the stated properties, we use a characterisation via the Fourier transform. Applying the Fourier transform to the definition of G ε given in §1.3, we obtain −C abcd G ε ec,db (k) = C abcd k b k d G ε ec (k) = I ae ϕ ε (k).
Define the 2-tensor D(k) ac := C abcd k b k d , and its algebraic inverse D(k) −1 , satisfying the relation D(k) −1 ab D(k) bc = I ac . D(k) −1 is well-defined for k = 0, since the Legendre-Hadamard condition on C entails that D(k) is strictly positive definite in this case, and it follows that
G ε ec (k) = D(k) −1 ec ϕ ε (k) and G ε ab,c (k) = −ik c D(k) −1 ab ϕ ε (k).
Since ϕ ε was assumed to be smooth and rapidly-decreasing, the same holds for ϕ ε . Moreover, D(k) −1 ab k c is −1-homogeneous in k, i.e. (30) D(λk) −1 ab λk c = λ −1 D(k) −1 ab k c , for any λ = 0. Using this observation and applying Plancherel's theorem to the definitions in (13),
J ε kmgr (s) = R 3 C abcd A bpl C ijkl A pmn A dqh C ef gh A qrs G ε ai,jn (k) G ε ce,f s (k)e −ik·s dk = R 3 C abcd A bpl C ijkl A pmn A dqh C ef gh A qrs k j k n k f k s D(k) −1 ai D(k) −1 ce ϕ ε (k) 2 e −ik·s dk, K ε abcd (s) = R 3 C ef gh C aijk C clmn A f ib A hld G ε ej,k (k) G ε gm,n (k)e −ik·s dk = − R 3 C ef gh C aijk C clmn A f ib A hld k k k n D(k) −1 ej D(k) −1 gm ϕ ε (k) 2 e −ik·s dk.
As ϕ ε is assumed to be radially symmetric, it follows that ϕ ε is also radially-symmetric, and therefore k k k n D(k) −1 ej D(k) −1 gm | ϕ ε (k)| 2 is even in k. Using the latter observation, and setting r = |k| and decomposing k = rz for some z ∈ S 2 = {z ∈ R 3 | |z| = 1}, we transform to polar coordinates, and use (30) and the evenness of ϕ ε to obtain
K ε abcd (s) = − R 3 C ef gh C aijk C clmn A f ib A hld z k z n D(z) −1 ej D(z) −1 gm r −2 ϕ ε (rz) 2 cos(rz · s)dk = − S 2 C ef gh C aijk C clmn A f ib A hld z k z n D(z) −1 ej D(z) −1 gm 1 2 +∞ −∞ ϕ ε (rz) 2 e irz·s dr dH 2 (z).
Standard properties of the Fourier transform imply that ϕ ε (k) = ϕ 1 (εk), so applying this relation and changing variable, we find
+∞ −∞ ϕ ε (rz) 2 e irz·s dr = 1 ε +∞ −∞ ϕ 1 (rz) 2 e irz·s/ε dr.
Now, for any ε > 0, we define η ε : R → R to be
η ε (t) := 1 ε +∞ −∞ ϕ 1 (re 1 ) 2 e irt/ε dr = η 1 (t/ε)/ε.
It is straightforward to show that this function is rapidly-decreasing, a property it inherits from ϕ ε . In summary, we have shown that
(31) K ε abcd (s) = − S 2 1 2 C ef gh C aijk C clmn A f ib A hld z k z n D(z) −1 ej D(z) −1 gm η ε (z · s) dH 2 (z).
Performing a similar computation for J ε , we obtain
J ε kmgr (s) = S 2 1 2 C abcd C ijkl C ef gh A bpl A pmn A dqh A qrs z j z n z f z s D(z) −1 ai D(z) −1 ce × ∞ −∞ r 2 ϕ ε (rz) 2 e irz·s dr dH 2 (z).
Considering the inner integral and performing a change of variable,
∞ −∞ r 2 η ε (rz) 2 e irt dr = 1 ε 3 +∞ −∞ r 2 η 1 (re 1 ) 2 e irt/ε dr = −(η ε ) (t); hence J ε kmgr (s) = S 2 1 2 C abcd C ijkl C ef gh A bpl A pmn A dqh A qrs z j z n z f z s D(z) −1 ai D(z) −1 ce (η ε ) (z · s) dH 2 (z).
By applying a series of tensor identities, this representation can be reduced to
(32) J ε kmgr (s) = S 2 1 2 C kmgr − C abgr D(z) −1 ai C ijkm z b z j (η ε ) (z · s) dH 2 (z).
Kernel properties. The symmetry and smoothness properties asserted in (1) and (2) follow directly from the representations (31) and (32), noting that η ε is smooth by construction.
Next, we note that as η 1 ∈ C ∞ (R) is a rapidly-decreasing function, there exist constants C m > 0 for m ∈ N, independent of ε, such that
(33) (η ε ) (m) (r) ≤ C m ε m+1 for all r ∈ R.
Moreover, since C satisfies a Legendre-Hadamard condition and D(k) is strictly positive definite, it follows that there exist M and M such that for all z ∈ S 2
(34) C ef gh C aijk C clmn A f ib A hld z k z n D(z) −1 ej D(z) −1 gm ≤ M and C kmgr − C abgr D(z) −1 ai C ijkm z b z j ≤ M .
Applying the bounds (33) and (34) to the representations (31) and (32), we obtain
(35) D m K ε abcd (s) ≤ 4πC m M ε m+1 and D m J ε abcd (s) ≤ 4πC m+2 M ε m+3
for all s ∈ R 3 and m ∈ N.
Further, taking derivatives of (31), applying the resulting multilinear operator to the collection of vectors v 1 , . . . , v j , s . . . , s, where |v i | = 1, and using (34) once more, we find that
D m K ε (s) : v 1 , . . . , v j , s, . . . , s ≤ M S 2 |z · s| m−j (η ε ) (m) (z · s) dz.
Expressing this upper bound using polar coordinates on S 2 with inclination θ measured relative to an axis parallel to s, and subsequently changing variable to t = |s| ε cos θ, we have
(36) D m K ε (s) v 1 , . . . , v j , s, . . . , s ≤ 2πM π 0 (η 1 ) (m) |s| cos θ ε |s| m−j |cos θ| m−j sin θ ε m+1 dθ = 2πM ε j |s| |s|/ε −|s|/ε |t| m−j (η 1 ) (m) (t) dt ≤ 2πM ε j |s| ∞ −∞ |t| m−j (η 1 ) (m) (t) dt,
where the integral in this upper bound is finite since η 1 is a rapidly-decreasing function. Dividing by |s| m−j , and combining with (35) completes the proof of assertion (3).
We make the following remarks concerning the Fourier representations of the kernels given in formulae (31) and (32):
• In the case where ϕ ε is a Gaussian, as in the example provided in (8), we may explicitly compute η ε as used in the proof above, giving
(37) η ε (t) = 8π 7/2 ε exp − t 2 4ε 2 .
More generally, for the purpose of computation we may choose ϕ ε in order to obtain a convenient expression for η ε . • Combining a convenient choice for η ε with the representations (31) and (32) suggests that the kernels J ε and K ε may be efficiently computed numerically, since these expressions require integration of a smooth function over the unit sphere, and this can be accurately approximated in practice with relatively few quadrature points. Moreover, these expressions are amenable to asymptotic analysis in the case where |s| ε, which should allow for the implementation of explicit expressions to speed-up computation.
DEFORMING DISLOCATIONS AND THE PEACH-KOEHLER FORCE
Theorem 1 established a representation of the elastic energy induced in a material due to the presence of dislocations. In this section, we prove Theorem 2, computing the configurational or Peach-Koehler force induced on a dislocation configuration, and demonstrating its properties. 4.1. The Peach-Koehler force. The first step towards proving Theorem 2 is to establish the expressions (15) and (16).
Lemma 7.
If S ∈ A , the inner variation of Φ ε , defined in (28), is given by
DΦ ε (S), g = − Γ f PK (S) · g dH 1 (s), where f PK (s, S) := G(s, S) ∧ τ (s) and G k (s, S) := Γ A klm K ε alcd,m (s − t)b a (s)b c (t)τ d (t)dH 1 (t).
Moreover, if S = ∂T , where T ∈ I 2 (R 3 ; L ) is supported on Σ with slip vector b and normal field ν, G can alternatively be written
G k (s, S) = Σ A def A klm K ε alcd,mb (s − t)b a (s)b c (t)ν f (t)dH 2 (t).
Proof. Given g ∈ C 1 (R 3 ; R 3 ), we set h δ := id + δg. Pushing forward, we find that
Φ ε (h δ # S) = Γ×Γ 1 2 K ε abcd s − t + δ(g(s) − g(t)) b a (s) τ b (s) + δ∇ τ g b (s) b c (t) × τ d (t) + δ∇ τ g d (t) d(H 1 ⊗ H 1 )(s, t).
Applying the definition (14), we differentiate and set δ = 0, we find
DΦ ε (S), g = Γ×Γ 1 2 K ε abcd,e (s − t)(g e (s) − g e (t))b a (s)τ b (s)b c (t)τ d (t) + 1 2 K ε abcd (s − t)b a (s)∇ τ g b (s)b c (t)τ d (t) + 1 2 K ε abcd (s − t)b a (s)τ b (s)b c (t)∇ τ g d (t) d(H 1 ⊗ H 1 )(s, t).
Applying the symmetries of K ε asserted in Lemma 6, this formula reduces to
DΦ ε (S), g = Γ×Γ K ε abcd,e (s − t)g e (s)b a (s)τ b (s)b c (t)τ d (t) + K ε abcd (s − t)b a (s)∇ τ g b (s)b c (t)τ d (t) d(H 1 ⊗ H 1 )(s, t).
Since S ∈ A satisfies ∂S = 0, i.e. S is formed of closed loops, we may integrate by parts in the variable s, passing a derivative from g onto K ε in the second term, which yields
DΦ ε (S), g = Γ×Γ K ε abcd,e (s − t)g e (s)b a (s)τ b (s)b c (t)τ d (t) − K ε abcd,e (s − t)b a (s)g b (s)τ e (s)b c (t)τ d (t) d(H 1 ⊗ H 1 )(s, t)
Now, applying the tensor identity A ijk A klm = I il I jm − I im I jl , and the definition of G(s, S), we obtain the first result. To obtain the latter expression, we simply apply Stokes' Theorem.
In view of this result, we make two remarks:
• Without additional regularity assumptions on S, we note that f PK is generically only in L ∞ (S), since the tangent field τ on a Lipschitz curve need not be continuous. • More generally, the fact that f PK is the product of a smooth kernel with components of b (which is locally constant on S) and the tangent field τ , entails that the regularity of f PK at a point is dictated by the regularity of the tangent field τ at the same point. This point is one we will return to in §5 when formulating a dynamical theory. • If g and S are assumed to be more regular, it is possible to compute higher-order variations of the energy in a similar way. However, it should be noted that some care is required if variations are made in different directions, since the order in which variations are taken will matter in general.
4.2.
Bounds on the Peach-Koehler force. The second crucial step in proving Theorem 2 is to establish (17), which is encoded in the following result.
Lemma 8. If S ∈ A is an admissible dislocation configuration with Burgers vector b, then the Peach-Koehler force satisfies the uniform bound
f PK (S) L ∞ ≤ C ε b L ∞ Θ(S) log 1 + 2 M(S) ε Θ(S) ,
where C > 0 is a coefficient independent of S and ε, b L ∞ is the maximum Burgers vector, and the mass M(S) and mass ratio Θ(S) were respectively defined in (4) and (6).
Proof. As a consequence of assertion (3) in Lemma 6, we have that
DK ε (s) ≤ C ε ε 2 + |s| 2 ,
with C independent of ε, and therefore the expression given for f PK in (15) directly implies that
f PK (s, S) ≤ b L ∞ Γ DK ε (s − t) |b(t)|dH 1 (t) ≤ C ε b L ∞ Γ |b(t)| ε 2 + |s − t| 2 dH 1 (t),
where Γ is the support of S. This upper bound may now be recast in the following way: Define the function µ :
R + → R + to be µ(r) := M S ¬ B r (s) .
This function is clearly monotonically increasing in r, satisfies µ(0) = 0, and since S ∈ A is compactlysupported, there must exist R ≥ 0 for which (38) µ(r) = M(S) whenever r ≥ R.
As a consequence of these facts, µ is a function of bounded variation (see §3.2 of [3]), and has a weak derivative, µ , which may in general be a measure. Using the definition of µ(r), we have Γ |b(t)| We note that as a consequence of the observation made in (38), ρ(m) = +∞ for m > M(S). Changing variable by setting ρ(m) = r, and since by definition, m = µ(ρ(m)), so that 1 = µ (ρ(m))ρ (m), we have
ε 2 + |s − t| 2 dH 1 (t) = ∞ 0 1 √ ε 2 + r 2 dµ (r).∞ 0 1 √ ε 2 + r 2 dµ (r) = M(S) 0 1 ε 2 + ρ(m) 2 dm.
To estimate this integral, we use the definition of Θ(S) given in (6) to find that µ(r) ≤ Θ(S)r for all r ≥ 0, and hence
ρ(m) ≥ m Θ(S)
for all m ≥ 0.
It follows that the latter integral may be bounded above by
M(S) 0 1 ε 2 + ρ(m) 2 dm ≤ M(S) 0 1 ε 2 + m 2 /Θ(S) 2 dm = Θ(S) log M(S) ε Θ(S) + 1 + M(S) 2 ε 2 Θ(S) 2 ≤ Θ(S) log 1 + 2 M(S) ε Θ(S) ,
which directly entails the stated result.
As discussed in §2, this estimate, while better than simply using the fact that DK ε is globally bounded, does not take into account much detail of the geometric structure, nor the fact that the Peach-Koehler force may be cast as either a line or surface integral (see the result of Lemma 7). It may be of interest for future applications to improve this bound in order to take better account of additional geometric features of a given dislocation configuration.
As a direct consequence of (17), we obtain the following L 2 bound directly via Hölder's inequality.
Corollary 9.
We have the following bound on the Peach-Koehler force:
(39) f PK L 2 ≤ C ε b L ∞ M(S) 1/2 Θ(S) log 1 + 2 M(S) ε Θ(S) .
Both L ∞ and L 2 bounds, (17) and (39), will be important for the proof of Theorem 3.
Continuity of the Peach-Koehler force.
To complete the proof of Theorem 2, we establish (18).
Lemma 10.
If g : S → R 3 is a Lipschitz map and F := id + g, then
F # f PK (F # S) − f PK (S) L ∞ ≤ 1 + CM(S) ∇ τ g L ∞ + CM(S) g L ∞ ,
where C is a constant independent of g and S.
Proof. We recall that
(40) f PK (s, S) = Γ A klm K ε alcd,m (s − t)b a (s)b c (t)τ d (t)dH 1 (t) ∧ τ (s),
and so
(41) f PK F (s), F # S = Γ A klm K ε alcd,m F (s) − F (t) b a (s)b c (t)DF (t)[τ (t)] d dH 1 (t) ∧ DF (s)[τ (s)].
Taking the difference between these formulae, applying the triangle inequality and using the fact that
DF (s)[τ (s)] − τ (s) ≤ ∇ τ g L ∞
and since by assertion (3) of Lemma 6, D 2 K ε is uniformly bounded, we may Taylor expand to obtain
DK ε s − t + g(s) − g(t) − DK ε (s − t) = D 2 K ε s − t + θ g(s) − g(t) g(s) − g(t) ≤ C g L ∞ .
Taking the difference between (40) and (41), and applying the triangle inequality along with the latter estimates, we directly deduce the result.
EVOLUTION PROBLEM AND EXISTENCE RESULTS
We now prove Theorem 3. Our basic strategy for doing so follows a fairly standard scheme. We carry out the following steps:
(1) Construct a family of approximate solutions.
(2) Derive bounds on the approximate solutions which are independent of the approximation.
(3) Use these bounds and a compactness result to extract a convergent approximating sequence. (4) Prove that the approximating sequence satisfies (24) in the limit, and verify the solution is unique.
Each of these steps is carried out in turn over the course of the following sections.
Approximation scheme.
In order to prove existence of a dynamical evolution, we set up an approximation scheme, which may be viewed as an explicit Euler scheme for the gradient flow dynamics. Fixing T > 0 and a sequence of times 0 = t 0 < t 1 < . . . < t K = T, and set δt i := t i+1 − t i , for each i = 0, . . . , K − 1, we will say that (U, {v i } K−1 i=0 ) form an approximate solution to DDD if U (t) = U (t i ) # (id + (t − t i )v i ) for all t ∈ (t i , t i+1 ] and i = 0, . . . , K − 1, and v i satisfies
(42) v i ∈ argmin v∈H 1 (S i ;R 3 ) Ψ(S i , v) − f PK (S i ), v L 2 (S i ) , where S i+1 := (id + δt i v i ) # S i for each i = 0, . . . , K − 1.
We will prove that each of the minimisation problems (42) is well-posed, and given sufficient regularity of S i , v i is regular. The following lemma encodes the first of these results. Lemma 11. If S i ∈ A , then then the minimisation problem in (42) has a unique solution v i ∈ H 1 (S i ; R 3 ), which satisfies
(43) ∂ v Ψ(S i , v i ) f PK (S i ) in H 1 (S i , R 3 ) * and the bound (44) v i H 1 ≤ C M(S i ) 1/2 Θ(S i ) b L ∞ ε min(α, β) log 1 + 2 M(S i ) ε Θ(S i ) .
Proof. First, setting v = 0 demonstrates that the functional which we seek to minimise is finite for some v ∈ H 1 (S i ; R 3 ). Assumptions (C 1 ) and (G) entail that
(45) Ψ(S i , v) ≥ S 1 2 α|∇ τ v| 2 + 1 2 β|v| 2 dH 1 ≥ 1 2 γ v 2 H 1 for any v ∈ H 1 (S i , R 3 ),
where γ = min(α, β). Using the bound for f PK derived in Corollary 9, we also have
(46) S i f PK · v dH 1 ≤ v L 2 C ε b L ∞ M(S) 1/2 Θ(S) log 1 + 2 M(S) ε Θ(S) .
Combining (45) and (46) and using Young's inequality in the usual way, we find that
Ψ(S i , v) − (f PK (S i ), v) L 2 ≥ 1 4 γ v 2 H 1 − C 2 b 2 L ∞ M(S)Θ(S) 2 2γε 2 log 1 + 2 M(S) ε Θ(S) 2
which implies that the functional which we seek to minimise is coercive. As Ψ is strictly convex, and as DΦ ε (S i ) is a bounded linear functional and is therefore also convex, it follows that the map v → Ψ(S i , v) − (f PK (S i ), v) L 2 is weakly lower semicontinuous. A standard application of the Direct Method of the Calculus of Variations therefore implies existence of v i , and strict convexity entails that v i is unique.
To establish (43), we note that convexity of v → Ψ(S i , v) + DΦ ε (S i ), v implies that the subdifferential at the minimum must contain 0, and therefore, by the characterisation of ∂ v ψ given in (21), it follows that
(47) S i ∇ τ v i · A(b, τ )∇ τ w + D ⊥ τ ψ(b, τ, v i ) − f PK · w dH 1 = 0 for any w ∈ H 1 (S i ; R 3 ).
Standard properties of convex functions entail that
ψ(b, τ, 0) ≥ ψ(b, τ, v) − ξ · v for any ξ ∈ ∂ψ(b, τ, v), so since ψ(b, τ, 0) = 0 by (C 2 ), it follows that D ⊥ τ ψ(b, τ, v) · v ≥ ψ(b, τ, v) ≥ 1 2 β|v| 2 for any v such that v · τ = 0,
where we have applied (G). Setting w = v i in (47) and bounding the left-hand side below as in (45), we thereby obtain
1 2 γ v i 2 H 1 ≤ f PK (S i ) L 2 v i L 2 ≤ f PK (S i ) L 2 v i H 1 .
The bound (44) then follows directly from the estimate established in Corollary 9.
Considering the details of the proof above, we make two remarks:
• Estimate (44) hinges upon the L 2 bound on f PK made in Corollary 9, which in turn relies upon the L ∞ bound (17) proved in Lemma 6. Any improvement of (17) would therefore entail an improved bound on v i . • The choice to assume quadratic growth of ψ in (G) allows us to directly obtain an H 1 estimate on v i ; if a weaker growth condition was assumed, we would need to apply a Poincaré-type inequality to obtain a similar estimate. Since such an inequality would inevitably depend upon M(S), this would render some aspects of the arguments which follow more technical.
As a consequence of (44), we may apply the Cauchy-Schwarz inequality to show the following corollary.
Corollary 12. The solution to the minimisation problem (42) satisfies
(48) ∇ τ v i 1 ≤ M(S i ) 1/2 ∇ τ v i 2 ≤ C M(S i )Θ(S i ) b L ∞ ε min(α, β) log 1 + 2 M(S i ) ε Θ(S i ) .
Estimate (48) will be important later, as it provides a control on the maximal growth rate of M(S i ).
5.2.
Properties of approximate solutions. Now that the existence of v i : S i → R 3 has been established, we wish to define S i+1 as the pushforward of S i under the mapping δU i (s) := id(s) + δt i v i (s).
At present, we have only established that v i is in H 1 (S i ; R 3 ); this entails that δU i is continuous as a mapping from S i to R 3 , but in order to be sure that S i+1 ∈ A , we must show that δU i is Lipschitz, and hence we must develop a regularity theory for v i . The following result establishes several crucial properties of v i .
Lemma 13.
If S i is a finite union of C k,γ curves with k ≥ 1 and 0 < γ ≤ 1, then the solution v i to the minimisation problem (42) is C k,γ , and moreover we have the bounds
v i L ∞ ≤ C 1 + 2 M(S i )Θ(S i ) b L ∞ ε min(α, β) log 1 + 2 M(S i ) ε Θ(S i ) (49) ∇ τ v i L ∞ ≤ C ε α b L ∞ 1 + 1 + 2 M(S i ) min(α, β) M(S i )Θ(S i ) log 1 + 2 M(S i ) ε Θ(S i ) (50) [∇ τ v] γ ≤ M(S) 1−γ + [τ ] γ M(S) C ε α b L ∞ 1 + 1 + 2 M(S) min(α, β) Θ(S) log 1 + 2 M(S) ε Θ(S) .(51)
To prove this result, we pull back to a flat domain, recast the resulting equation as an ODE system, and then use the properties assumed of A and ψ along with some elementary integral bounds.
Proof. We first prove regularity, then proceed to obtain the stated estimates. Since they are fixed throughout this proof, we suppress superscripts, writing v and S in place of v i and S i .
Regularity. Since S is assumed to be a union of C k,γ curves, there exists a C k,γ diffeomorphism g which maps the interval (−a, a) to a neighbourhood of s ∈ S. Without loss of generality, we may assume g is an arc-length parametrisation, so g = g # τ on (−a, a), and therefore |g | = 1 since τ = 1. Defining V : (−a, a) → R 3 to be the pullback of v by g, i.e. V := g # v, we find it has weak derivative
g # ∇ τ v = V .
We recall that the equation satisfied by v on S is
−div τ A(b, τ )∇ τ v + D ⊥ τ ψ(b, τ
, v) = f PK , so defining B = g # b and F = g # f PK and 'pulling back' the equation, we find that V : (−a, a) → R 3 must satisfy (52) − A B(r), g (r) V (r) + D ⊥ τ ψ B(r), g (r), V (r) = F (r) almost-everywhere on (−a, a). As remarked at the end of §4.1, the fact that S is assumed to by C k gamma implies that F ∈ C k−1,γ (−a, a); R 3 .
We define the auxiliary function σ := g # A b, τ ∇ τ v , so that V = A(B, g ) −1 σ. Solving (52) for V entails that V and σ must satisfy the system of equations
(53) V (r) = A B(r), g (r) −1 σ(r) σ (r) = D ξ ψ B(r), g (r), V (r) − F (r).
Now, since v ∈ H 1 (S; R 3 ) ⊂ C 0, 1 2 (S; R 3 ), and g ∈ C 1,γ (−a, a); S ⊂ C 0,1 (−a, a); S , it follows that V = v # g ∈ C 0, 1 2 (−a, a); R 3 . The regularity assumptions on ψ and the second equation therefore entail that σ ∈ C 1,η (−a, a); R 3 , where η = min{ 1 2 , γ}, and the regularity assumptions on A applied to the first equation entail that V ∈ C 1,γ (−a, a); R 3 . Bootstrapping, we ultimately find that V, σ ∈ C k,γ (−a, a); R 3 . This local argument entails v ∈ C k,γ (S; R 3 ) via a finite covering of S, which is possible since S ∈ A is a finite union of C k,γ curves.
Uniform bound. Now, taking the inner product between V and V and then integrating and applying the Cauchy-Schwarz inequality, we obtain
1 2 |v(s 1 )| 2 − 1 2 |v(s 0 )| 2 = g(s1) g(s0) V (r) · V (r) dr ≤ V L 2 V L 2 = v L 2 ∇ τ v L 2 .
Integrating with respect to s 0 , dividing by M(S), and using (44), we find that
(54) |v(s 1 )| 2 ≤ 2 v L 2 ∇ τ v L 2 + v 2 L 2 M(S) , ≤ C 2 1 + 2 M(S) Θ(S) 2 b 2 L ∞ ε 2 min(α, β) 2 log 1 + 2 M(S) ε Θ(S) 2 .
which leads directly to (49).
Uniform gradient bound. If V and σ solve (53) on (−a, a), then by integrating the second equation, we find that
(55) |σ(r 1 ) − σ(r 0 )| ≤ C F L ∞ + V L ∞ |r 1 − r 0 | so σ ∈ C 0,1 (−a, a); R 3 )
. Moreover, we may use (49) and the definition of F as a pullback of f PK to find
(56) σ L ∞ ≤ C F L ∞ + v L ∞ 1 2 M(S) ≤ C ε b L ∞ 1 + 1 + 2 M(S) min(α, β) M(S)Θ(S) log 1 + 2 M(S) ε Θ(S)
Using the definition of σ, and noting that (C 1 ) implies that A −1 L ∞ ≤ α −1 , we therefore obtain
∇ τ v L ∞ ≤ C ε α b L ∞ 1 + 1 + 2 M(S) min(α, β) M(S)Θ(S) log 1 + 2 M(S) ε Θ(S) ,
which is (50).
Uniform bound on Hölder seminorm. Applying the assumption (R) to deduce that
A −1 (b, τ 1 )−A −1 (b, τ 2 ) ≤ L|τ 1 − τ 2 | for some L, we have |V (r 1 ) − V (r 0 )| = A B(r 1 ), g (r 1 ) −1 σ(r 1 ) − A B(r 0 ), g (r 0 ) −1 σ(r 0 ) ≤ A B(r 1 ), g (r 1 ) −1 σ(r 1 ) − A B(r 1 ), g (r 1 ) −1 σ(r 0 ) + A B(r 1 ), g (r 1 ) −1 σ(r 0 ) − A B(r 1 ), g (r 0 ) −1 σ(r 0 ) ≤ α −1 |σ(r 1 ) − σ(r 0 )| + L g (r 1 ) − g (r 0 ) σ L ∞ ≤ α −1 [σ] γ + L [g ] γ σ L ∞ |r 1 − r 0 | γ
Using (55), we find that
[σ] γ ≤ C ε b L ∞ 1 + 1 + 2 M(S) min(α, β) M(S) 1−γ Θ(S) log 1 + 2 M(S) ε Θ(S) ,
and estimating the other term using (56), we obtain (51).
Lemma 13 guarantees the spatial regularity of any approximate solution U (t) defined via the procedure prescribed in (42), and therefore ensures that such approximate solutions are well-defined. Our next step will be to prove that approximate solutions converge as max i {δt i } → 0, and that the limit satisfies (24). 5.3. Convergence of approximate solutions. Our approach to proving convergence is via compactness; this requires us to prove appropriate uniform a priori bounds on approximate solutions, which will subsequently allow us to employ the Arzelà-Ascoli theorem. We remark that all bounds on approximate solutions derived thus far depend upon M(S) and Θ(S), and therefore it is these quantities we must bound; the following lemma therefore establishes a bound on the growth of M(U (t) # S).
Lemma 14.
If S 0 ∈ A with Θ(S 0 ) < +∞, then for any ρ > Θ(S 0 ) and M > M(S 0 ), and all δ > 0 sufficiently small, there exists T (M, ρ, δ) > 0 such that any approximate solution of DDD (in the sense
described in §11) with max i {δt i } ≤ δ satisfies M(U (t) # S 0 ) ≤ M and Θ(U (t) # S 0 ) ≤ ρ for all t ∈ [0, T ].
Proof. The proof is divided into first obtaining a uniform bound on the mass growth, then using this bound to guarantee a bound on the growth of the mass ratio.
Uniform mass bound. Our first step is to establish a uniform bound on the mass. By definition,
U (t) # S 0 = (id + (t − t i )v i ) # S i for t ∈ (t i , t i+1 )
, and so (48) implies
M(U (t) # S 0 ) = M (id + (t − t i )v i ) # S i = τ i + (t − t i )∇ τ v i L 1 (S i ) ≤ M(S i ) + (t − t i ) CΘ(S i ) M(S i ) b L ∞ ε min(α, β) log 1 + 2 M(S i ) ε Θ(S i ) .
Estimating log |1 + x| ≤ x for x ≥ 0, and employing the comparison principle for ODEs, we find that M(U (t) # S 0 ) must be bounded above by the solution to
m (t) = 2 C b L ∞ min(α, β) m(t) 2 ε 2 m(0) = M(S 0 ).
Since this is a separable ODE, we may check that
(57) m(t) = 1 M(S 0 ) − 2 C t b L ∞ ε 2 min(α, β) −1
for all t ∈ [0, T ], and therefore it follows that
M(U (t) # S 0 ) ≤ M for all t ≤ T (M ) = C 1 M − M(S 0 ) M M(S 0 ) ,
where C 1 is a constant depending on b L ∞ , ε and min(α, β).
Uniform mass ratio bound. Suppose that U (t) is an approximate solution on [0, T (M )], with max i {δt i } ≤ δ ≤ T (M ). The bounds (49) and (50) entail that for t small enough, the map δU i (t) :
[t i , t i+1 ]×S i → R 3 defined by δU i (t) := id + (t − t i )v i is satisfies |δU i (t, s 1 ) − δU i (t, s 0 )| |s 1 − s 0 | ≤ 1 + (t − t i ) C 2 Θ(S i ) log 1 + 2 M ε Θ(S i )
for any s 0 , s 1 ∈ S i for some constant C 2 > 0. Moreover, since Θ(S i ) ≥ 1 for any S i = ∅, as observed in (7), we may bound the logarithmic terms by log |1 + 2M/ε|, finding that
(58) |δU t i (s 1 ) − δU t i (s 0 )| |s 1 − s 0 | ≤ 1 + C 3 (t − t i )Θ(S i ) .
for some C 3 > 0.
Next, recalling an argument made in §3.2.17 of [35], supposing that F : S → R 3 satisfies
|F (s 1 ) − F (s 0 )| |s 1 − s 0 | ≤ λ for all s 0 , s 1 ∈ B r (s),
for some λ > 1, it follows that
F # S ∩ B r F (s) ⊂ F # S ∩ B λr (s) ,
and therefore
H 1 F # S ∩ B r F (s) r ≤ λ 2 H 1 S ∩ B λr (s) λ r .
Noting the connection between the Hausdorff measure of the support of an integral current and its mass made in (5), we may take suprema over r ≥ 0 and s ∈ R 3 to obtain
(59) Θ(F # S) ≤ λ 2 Θ(S).
In particular, δU i (t) satisfies the conditions on F with
λ = 1 + C 3 (t − t i ) Θ(S i ),
and so we find that
Θ δU i (t) # S i − Θ(S i ) t − t i ≤ C 3 2 + C 3 δ Θ(S i ) Θ(S i ) 2 .
Employing the comparison principle for ODEs once more, and using the fact that, by construction,
Θ(U (t) # S 0 ) = Θ(δU i (t) # S i ), we find that Θ(U (t) # S 0 ) is bounded above by the solution to ρ (t) ≤ C 3 2 + C 3 δρ(t) ρ(t) 2 , with ρ(0) = Θ(S 0 ).
Again, being a separable ODE, we may integrate to find
1 2C 3 Θ(S 0 ) − δ 4 log C 3 δ + 2 Θ(S 0 ) − 1 2C 3 ρ(t) + δ 4 log C 3 δ + 2 ρ(t) = t,
and thereby we see that for any
t ≤ T (ρ, M, δ) = ρ − Θ(S 0 ) 2C 3 (M )Θ(S 0 ) + δ 4 log C 3 (M )δ + 2/ρ C 3 (M )δ + 2/Θ(S 0 )
, the result holds.
The result of Lemma 14 entails that, given an upper limit on the mass and mass ratio, and a maximum step size, there exists an infinite family of approximate solutions on some time interval [0, T ] with T > 0. This is a crucial result which allows us to establish existence by compactness in the following lemma.
Lemma 15. Suppose S 0 ∈ A is C 1,γ for some γ ∈ (0, 1] and fix ρ and M such that M(S 0 ) < M < +∞ and Θ(S 0 ) < ρ < +∞; then there exists T > 0 such that there exists a unique C 0 [0, T ]; C 1 (S 0 ; R 3 ) solution to (24), and moreover M(U (t) # S 0 ) ≤ M and Θ(U (t) # S 0 ) ≤ ρ for all t ∈ [0, T ].
Proof. In Lemma 14, we established the existence of T such that if U n : [0, T ] × R 3 is an approximation solution of DDD with max i {δt i n } ≤ δ, then M U n (t) # S 0 ≤ M and Θ U n (t) # S 0 ≤ ρ. Therefore, taking a sequence of such solutions with max i {δt i n } → 0, we wish to show that this sequence contains a convergent subsequence via an application of the Arzelà-Ascoli Theorem (or equivalently, the compactness properties of Hölder spaces).
Compactness. When combined with the result of Lemma 14, the bounds established in Lemma 13 entail that U n (t) C 1,γ ≤ C(M, ρ, δ) for all n ∈ N;
in turn, this entails that
(60) U t1 n − U t0 n C 1,γ ≤ t1 t0 U n (t) C 1,γ dt ≤ C(M, ρ, δ) |t 1 − t 0 |.
Setting t 0 = 0, it follows that U n : [0, T ]×S 0 → R 3 are uniformly bounded in C 0,1 ([0, T ]; C 1,γ (S 0 ; R 3 )), from which it follows that the sequence of approximate solutions is compact in C 0 [0, T ]; C 1 (S 0 ; R 3 ) . We may therefore extract a convergent subsequence, which we do not relabel; we denote the limit U ∞ . Moreover, the same results also imply that U n (t) # v t n is uniformly bounded in L 2 [0, T ]; H 1 (S 0 ; R 3 ) . It follows that we may extract a further subsequence (which again, we do not label) such that U n (t) # v t n converges weakly in H 1 (S 0 ; R 3 ) for almost every t ∈ [0, T ]. Now that we have demonstrated the existence of a candidate limit, we must demonstrate that it solves (24) Convergence of dissipation potential. By virtue of the fact that U n → U ∞ in C 0,1 [0, T ]; C 1 (S 0 ) ,
U n (t) # τ n (t) = ∇ τ U n (t) → ∇ τ U ∞ (t) = U ∞ (t) # τ ∞ (t), uniformly for t ∈ [0, T ] as n → ∞,
where τ n (t) : U n (t) # S 0 → S 2 and τ ∞ (t) : U ∞ (t) # S 0 → S 2 are tangent fields. Recalling the definition made in (42), by construction, approximate solutions satisfy
∂ v Ψ U n (t i n ) # S 0 , U n (t i n ) #U n (t i n ) = f PK (U n (t i n ) # S 0 ) for each t i n . Applying assumption (R), we have A U n (t) # b, U n (t) # τ n → A U ∞ (t) # b, U ∞ (t) # τ ∞ in L ∞ (S 0 ; R 3×3 )
uniformly in t as n → ∞, and therefore if V n is a sequence of functions in H 1 (S;
R 3 ) such that ∇ τ V n → ∇ τ V ∞ weakly in L 2 (S 0 ; R 3 ), we have (61) lim inf n→∞ S 0 1 2 A U n (t) # b, U n (t) # τ n (t) : [∇ τ V n , ∇ τ V n ] dH 1 ≥ S 0 1 2 A U ∞ (t) # b, U ∞ (t) # τ ∞ (t) : [∇ τ V ∞ , ∇ τ V ∞ ] dH 1 .
Furthermore, the convexity and regularity assumptions (C 2 ) and (R) imply that if V n ∈ L 2 (S 0 ; R 3 ) is a sequence of functions such that V n → V ∞ weakly in L 2 (S 0 ; R 3 ) as n → ∞, then
(62) lim inf n→∞ S 0 ψ U n (t) # b, U n (t) # τ n (t), V n dH 1 ≥ S 0 ψ U ∞ (t) # b, U ∞ (t) # τ ∞ (t), V ∞ dH 1 .
Together, (61), (62) and the fact that U n (t) # v t n converges weakly in
L 2 ([0, T ]; H 1 (S 0 ; R 3 )) imply that (63) lim inf n→∞ U n (t) # Ψ U n (t) # S 0 ; v n ≥ U ∞ (t) # Ψ U ∞ (t) # S 0 ; v ∞ for almost every t ∈ [0, T ].
Convergence of Peach-Koehler force. Applying Lemma 10 and the fact that
U n (t) → U ∞ (t) in C 1 (S 0 ; R 3 )
uniformly in t, we find that
(64) U n (t) # f PK U n (t) # S 0 → U ∞ (t) # f PK U ∞ (t) # S 0 in L ∞ (S 0 ; R 3 ) as n → ∞ uniformly in t. Since U n (t) # v t n → U ∞ (t) # v t ∞ weakly in L 2 [0, T ]; H 1 (S 0 ; R 3 )
, we further obtain that
(65) U n (t) # f PK U n (t) # S 0 , U n (t) # v t n L 2 → U ∞ (t) # f PK U ∞ (t) # S 0 , U ∞ (t) # v t ∞ L 2
as n → ∞ for almost every t ∈ [0, T ].
Convergence of dissipation potential. Since Ψ(S, ·) is a strictly convex functional defined on H 1 (S; R 3 ), it follows that it has a convex conjugate Ψ * (S, ·) : H 1 (S; R 3 ) * → R, defined to be
Ψ * (S, ξ) = sup ξ, v − Ψ(S, v) : v ∈ H 1 (S; R 3 ) .
Employing this definition, we obtain
lim n→∞ U n (t) # Ψ * (U n (t) # S 0 , ξ) = lim n→∞ sup ξ, v − U n (t) # Ψ(U n (t) # S 0 , v) : v ∈ H 1 (S; R 3 ) = sup{ ξ, v − lim n→∞ U n (t) # Ψ(U n (t) # S 0 , v) : v ∈ H 1 (S; R 3 ) = sup ξ, v − U ∞ (t) # Ψ(U ∞ (t) # S 0 , v) : v ∈ H 1 (S; R 3 ) = U ∞ (t) # Ψ * (U ∞ (t) # S 0 , ξ).
Moreover, since (64) holds, it follows that
U n (t) # DΨ ε (U n (t) # S 0 ) → U ∞ (t) # DΨ ε (U ∞ (t) # S 0 ) in H 1 (S 0 ; R 3 ) * ,
and we conclude that
(66) U n (t) # Ψ U n (t) # S 0 , U n (t) # f PK (U n (t) # S 0 ) → U ∞ (t) # Ψ U ∞ (t) # S, U ∞ (t) # f PK U ∞ (t) # S 0 as n → ∞.
Conclusion. Finally, by using standard properties of the Legendre-Fenchel transform [4,54,71], we note that (24) is equivalent to requiring that
U (t) # Ψ U (t) # S 0 , v t + U (t) # Ψ * U (t) # S 0 , −DΨ ε U (t) # S 0 − DΨ ε U (t) # S 0 ), v t = 0
for almost every t ∈ [0, T ]. By considering this expression with U n (t) in place of U (t), and v t n in place of v t , we may combine (63), (65) and (66), to pass to the limit, demonstrating that
0 = T 0 U ∞ (t) # Ψ U ∞ (t) # S 0 ; v t ∞ + U ∞ (t) # Ψ U ∞ (t) # S 0 , U ∞ (t) # f PK (U ∞ (t) # S 0 ) + U ∞ (t) # f PK U ∞ (t) # S 0 , U ∞ (t) # v t ∞ L 2 dt.
This entails that, for almost every t, we have
∂ v Ψ U ∞ (t) # S 0 ; v t ∞ f PK U ∞ (t) # S 0 ,
and since U ∞ (t) # v t ∞ = lim n→∞ U n (t) # v t n = lim n→∞Un (t) =U ∞ (t), we have proved that U ∞ solves (24).
To demonstrate uniqueness of the limit, we note that assumptions (C 1 ) and (C 2 ) entail that the functional Ψ(S, v) is strictly convex on H 1 (S; R 3 ), and therefore a standard argument guarantees that the above limit procedure is independent of the subsequence chosen. 5.4. Conclusion of the proof. Now that we have proved existence for a finite time, we show that we may extend the solution to a possibly infinite time by demonstrating that the total mass of the dislocation configuration is bounded as long as the mass ratio remains bounded.
Lemma 16.
If U : [0, T ] × S 0 → R 3 is a solution of DDD in the sense described in (24), such that Θ(U (t) # S 0 ) ≤ ρ for all t ∈ [0, T ], then we have the uniform bound
M(U (t) # S 0 ) ≤ min M(S 0 ) 1 − C 1 (b, ε) M(S 0 ) T , ε 2 1 + 2 M(S 0 ) ε exp(C2(b,ε,ρ)T ) , where C 1 (b, ε) = 2 C b L ∞ ε 2 min(α, β) and C 2 (b, ε, ρ) = C ρ b L ∞ ε min(α, β) .
Proof. Using the fact that v t must satisfy (48), and by assumption, 1 ≤ Θ(U (t) # S 0 ) ≤ ρ, we have
d dt M(U (t) # S 0 ) ≤ Cρ M(U (t) # S 0 ) b L ∞ ε min(α, β) log 1 + 2 M(U (t) # S 0 ) ε .
Employing the comparison principle for ODEs, it follows that for all t ∈ [0, T ], M(U (t) # S 0 ) must be bounded above by the solution to m (t) = C ρ b L ∞ 2 min(α, β) 1 + 2m(t) ε log 1 + 2 m(t) ε , m(0) = M(S 0 ).
Integrating, we find that
log 1 + 2 m(t) ε = log 1 + 2 M(S 0 ) ε exp C t ρ b L ∞ ε min(α, β)
for all t ∈ [0, T ], which, when combined with (57) which was used to bound the mass in the proof of Lemma 14 now immediately yields the bound stated.
With this result in place, we now deduce the following result, which complete the proof of Theorem 3. This appendix recalls various definitions from the theory of currents, as described in Chapters 1 and 4 of both [35] and [57].
A.1. Vectors and covectors. Recall that the usual exterior product ∧ is multilinear and alternating, i.e. it satisfies (u + λv) ∧ w = u ∧ w + λ(v ∧ w) and u ∧ v = −v ∧ u.
Suppose that e 1 , . . . , e n form an orthonormal basis of R n . The space of m-vectors Λ m R n is then the span Λ m R n := span e i1 ∧ · · · ∧ e im i j ∈ {1, . . . , n}, i 1 < . . . < i m ;
m-vectors should be thought of as describing oriented m-dimensional subspaces of R n . The space of m-covectors, denoted Λ m R n , is the space of linear functions on Λ m R n , i.e. Λ m R n := [Λ m R n ] * . Λ m R n may be identified with Λ m (R n ) * , the span of m-fold wedge products of dual vectors e * i , defined to satisfy e * i , e j = I ij . We may define an inner product, (·, ·) and corresponding norm, |u| = (u, u) 1/2 on Λ m R n , which makes e i1 ∧ · · · ∧ e im with i 1 < · · · < i m an orthonormal basis for the space (see §1.7 of [35]). A similar inner product can be constructed on Λ m R n , which makes e * i1 ∧ · · · ∧ e * im with i 1 < · · · < i m an orthonormal basis for this space.
While the above definitions are general, throughout this work we will exclusively consider n = 3, and m = 1 or m = 2, since these are the cases of interest for the modelling of dislocations. In this case, we have the isometric isomorphisms
R 3 ∼ = Λ 2 R 3 ∼ = Λ 1 R 3 ∼ = Λ 2 R 3 ∼ = Λ 1 R 3 .
In particular, it should be noted that the identification of Λ 2 R 3 with R 3 corresponds to identifying u ∧ v with the usual vector cross product on R 3 .
A.2.
Forms. An m-form is a function φ : R n → Λ m R n . Using the basis of Λ m R n discussed in §A.1, any such function may be expressed as
φ(x) = i1<···<im f i1...im (x) e * i1 ∧ . . . ∧ e * im .
The exterior derivative dφ is the (m + 1)-form defined via
dφ(x) = i1<···<im im+1 f i1...im,im+1 e * im+1
∧ (e * 1 ∧ · · · ∧ e * m ).
For any open set U ⊆ R n , we define the vector space of smooth m-forms which are compactlysupported in U to be D m (U ) := φ : U → Λ m R n φ is C ∞ with compact support in U .
A.3. Currents. A current is a generalisation of the notion of a distribution [37]; whereas distributions act on scalar-valued functions, currents instead act on the spaces D m (R n ). In particular, an m-dimensional current (or m-current) T is a linear functional which acts on D m (R n ), and we denote the action of a current on φ ∈ D m (R n ) to be T, φ ∈ R. The boundary of an m-dimensional current is the (m − 1)-current ∂T , defined to be ∂T, φ = T, dφ for all φ ∈ D m (R n ).
The support of a current is defined to be the closed set supp(T ) := R n \ U ⊂ R n U is open and T, φ = 0 for all φ ∈ D m (U ) .
We recall that Σ ⊂ R n is m-rectifiable if it may be expressed as a countable union of images of bounded subsets of R m under Lipschitz maps, and an m-current is rectifiable if there exists an m-rectifiable set Σ ⊂ R n , a Borel measurable function τ : Σ → Λ m R n with |τ | = 1 on Σ, and a Borel measurable µ : Σ → N such that
T, φ = Σ τ, φ µ dH m ,
where H m denotes the m-dimensional Hausdorff measure on R n . This representation demonstrates that rectifiable m-currents generalise the elementary vector calculus notion of integrals over mdimensional subsets of R n . An m-dimensional rectifiable current is an integral current if both T and ∂T are rectifiable currents. We will denote the space of integral m-currents as I m , and we exclusively consider currents in these classes.
A.4. Pushforward and pullback. Given a Lipschitz map f : R m → R n , we recall that f has a Frechet derivative Df (x) ∈ L(R m ; R n ) for almost every x ∈ R m , where L(R m ; R n ) is the space of bounded linear operators mapping R m to R n .
Following the definitions in §4.3A of [57] (which in turn follow the definitions given in §4.1.6 and §4.1.7 of [35]), we define the pushforward of a simple m-vector v 1 ∧ · · · ∧ v m ∈ Λ m R n under a linear map A ∈ L(R m , R n ) to be
Λ m (A)[v 1 ∧ · · · ∧ v m ] := A[v 1 ] ∧ · · · ∧ A[v m ].
Noting that Λ m R n is spanned by simple m-vectors, this definition can be extended linearly to apply to general m-vectors ξ ∈ Λ m R n . In particular, we recall that the key property of the pushforward is that Λ m Df (x) gives the transformation of tangent space of a manifold under the map f .
The pullback of an m-form φ ∈ D m (R n ) by a Lipschitz map f , denoted f # φ, is defined to be
ξ, f # φ(x) = Λ m Df (x) [ξ], φ f (x) .
Note that φ is evaluated on the target space f (R m ) ⊆ R n . This definition allows us to define the pushforward of a rectifiable current by duality as f # T, φ := T, f # φ .
FIGURE 1 .
1An illustration of the surface Σ discussed in §2.3.
( 5 )
5M(S) = Γ |b| dH 1 ,
FIGURE 2 .
2An example of S ∈ A with large Θ(S). The ball in grey contains a large proportion of the mass. By 'curling' the current tightly, the mass can be contained in an arbitrarily small ball, which may increase Θ(S) while leaving M(S) fixed.
FIGURE 3 .
3Development of a corner in finite time if dissipation potential depends on v alone. Arrows reflect instantaneous velocity of the dislocation. assumed that dislocations have velocity only in normal directions, i.e. M(b, τ, f ), τ = 0 for any b, τ and f .
Corollary 17 .
17If S 0 ∈ A satisfies Θ(S 0 ) < +∞, then there exists a unique solution (U, {v t }) of (24) up until T := sup t ∈ R : Θ(U (s) # S 0 ) < +∞ for all s ≤ t . Proof. For any ρ > Θ(S 0 ), and M > M(S 0 ) Lemma 15 establishes the existence of a unique solution up until the first time T (M, ρ) at which either M(U (T ) # S 0 ) = M or Θ(U (T ) # S 0 ) = ρ, and we note that this existence time is bounded below by a function which is monotone in both M and ρ. Lemma 16 entails that in fact M(U (t) # S 0 ) is finite for any t ≤ T * (ρ), where T * is the first time at which Θ(U (T * ) # S 0 ) = ρ, and thus the maximal existence time is independent of the mass constraint. Letting ρ → ∞, we obtain the result.
Funding.
This work was supported by Early Career Fellowship entitled 'A Mathematical Study of Discrete Dislocation Dynamics' (ECF-2016-526), awarded by the Leverhulme Trust. APPENDIX A. VECTORS, FORMS AND CURRENTS
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| [] |
[
"Large Thermoelectric Power Factor in One-Dimensional Telluride Nb 4 SiTe 4 and Substituted Compounds",
"Large Thermoelectric Power Factor in One-Dimensional Telluride Nb 4 SiTe 4 and Substituted Compounds"
] | [
"Yoshihiko Okamoto \nDepartment of Applied Physics\nNagoya University\n464-8603NagoyaJapan\n\nInstitute for Advanced Research\nNagoya University\n464-8601NagoyaJapan\n",
"Taichi Wada \nDepartment of Applied Physics\nNagoya University\n464-8603NagoyaJapan\n",
"Youichi Yamakawa \nDepartment of Physics\nNagoya University\n464-8602NagoyaJapan\n\nInstitute for Advanced Research\nNagoya University\n464-8601NagoyaJapan\n",
"Takumi Inohara \nDepartment of Applied Physics\nNagoya University\n464-8603NagoyaJapan\n",
"Koshi Takenaka \nDepartment of Applied Physics\nNagoya University\n464-8603NagoyaJapan\n"
] | [
"Department of Applied Physics\nNagoya University\n464-8603NagoyaJapan",
"Institute for Advanced Research\nNagoya University\n464-8601NagoyaJapan",
"Department of Applied Physics\nNagoya University\n464-8603NagoyaJapan",
"Department of Physics\nNagoya University\n464-8602NagoyaJapan",
"Institute for Advanced Research\nNagoya University\n464-8601NagoyaJapan",
"Department of Applied Physics\nNagoya University\n464-8603NagoyaJapan",
"Department of Applied Physics\nNagoya University\n464-8603NagoyaJapan"
] | [] | We found that whisker crystals of Mo-doped Nb 4 SiTe 4 show high thermoelectric performances at low temperatures, indicated by the largest power factor of ~70 W cm 1 K 2 at 230-300 K, much larger than those of Bi 2 Te 3 -based practical materials. This power factor is smaller than the maximum value in the 5d analogue Ta 4 SiTe 4 , but is comparable to that with a similar doping level. First principles calculation results suggest that the difference in thermoelectric performances between Nb and Ta compounds is caused by the much smaller band gap in Nb 4 SiTe 4 than that in Ta 4 SiTe 4 , due to the weaker spin-orbit coupling in the former. We also demonstrated that the solid solution of Nb 4 SiTe 4 and Ta 4 SiTe 4 shows a large power factor, indicating that their combination is promising as a practical thermoelectric material, as in the case of Bi 2 Te 3 and Sb 2 Te 3 . These results advance our understanding of the mechanism of high thermoelectric performances in this one-dimensional telluride system, as well as indicating the high potential of this system as a practical thermoelectric material for low temperature applications. | 10.1063/1.5023427 | [
"https://arxiv.org/pdf/1804.09867v1.pdf"
] | 55,215,210 | 1804.09867 | d54bb4c162dc6a24cdb0ae8cce2958475c17f136 |
Large Thermoelectric Power Factor in One-Dimensional Telluride Nb 4 SiTe 4 and Substituted Compounds
Yoshihiko Okamoto
Department of Applied Physics
Nagoya University
464-8603NagoyaJapan
Institute for Advanced Research
Nagoya University
464-8601NagoyaJapan
Taichi Wada
Department of Applied Physics
Nagoya University
464-8603NagoyaJapan
Youichi Yamakawa
Department of Physics
Nagoya University
464-8602NagoyaJapan
Institute for Advanced Research
Nagoya University
464-8601NagoyaJapan
Takumi Inohara
Department of Applied Physics
Nagoya University
464-8603NagoyaJapan
Koshi Takenaka
Department of Applied Physics
Nagoya University
464-8603NagoyaJapan
Large Thermoelectric Power Factor in One-Dimensional Telluride Nb 4 SiTe 4 and Substituted Compounds
1
We found that whisker crystals of Mo-doped Nb 4 SiTe 4 show high thermoelectric performances at low temperatures, indicated by the largest power factor of ~70 W cm 1 K 2 at 230-300 K, much larger than those of Bi 2 Te 3 -based practical materials. This power factor is smaller than the maximum value in the 5d analogue Ta 4 SiTe 4 , but is comparable to that with a similar doping level. First principles calculation results suggest that the difference in thermoelectric performances between Nb and Ta compounds is caused by the much smaller band gap in Nb 4 SiTe 4 than that in Ta 4 SiTe 4 , due to the weaker spin-orbit coupling in the former. We also demonstrated that the solid solution of Nb 4 SiTe 4 and Ta 4 SiTe 4 shows a large power factor, indicating that their combination is promising as a practical thermoelectric material, as in the case of Bi 2 Te 3 and Sb 2 Te 3 . These results advance our understanding of the mechanism of high thermoelectric performances in this one-dimensional telluride system, as well as indicating the high potential of this system as a practical thermoelectric material for low temperature applications.
Recently, whisker crystals of the one-dimensional telluride Ta 4 SiTe 4 and its substituted compounds were found to show high thermoelectric performances at low temperatures. 1 Their whisker form is typically several mm long and several m in diameter, reflecting the strongly one-dimensional crystal structure comprising Ta 4 SiTe 4 chains. 2,3 The electrical resistivity and thermoelectric power S data measured along the whiskers, i.e. parallel to the Ta 4 SiTe 4 chains, indicate that the power factor P = S 2 / of the chemically-doped Ta 4 SiTe 4 whiskers are significantly larger than those of practical thermoelectric materials in a wide temperature region of 50300 K. 1 Undoped Ta 4 SiTe 4 shows a very large and negative thermoelectric power of |S| ~ 400 V K 1 at 100200 K, while maintaining a small of ~2 m cm. These |S| and yield P = 80 W cm 1 K 2 at the optimum temperature of ~130 K, which is almost twice as large as those of Bi 2 Te 3 -based materials at room temperature. This P is strongly enhanced by electron doping. (Ta 1x Mo x ) 4 SiTe 4 with x = 0.0010.002 shows P = 170 W cm 1 K 2 at 220280 K. These results indicate that Ta 4 SiTe 4 is promising for low temperature applications of thermoelectric conversion, which have never been put to practical use, such as local cooling of electronic devices well below room temperature and power generation utilizing the cold heat of liquefied natural gas. _________________________ a) Electronic mail: [email protected] For the practical use of Ta 4 SiTe 4 as a low-temperature thermoelectric material, it is necessary to study the thermoelectric properties of a sister compound of Ta 4 SiTe 4 with the same crystal structure and electron configuration. Comparing the thermoelectric properties of Ta 4 SiTe 4 to those of a sister compound will help us to understand the physics underlying the high thermoelectric performance in this system. Moreover, forming a solid solution or superlattice structure with sister compounds can improve the thermoelectric performances. In the Bi 2 Te 3 -based materials, for example, the thermoelectric performance is optimized by controlling the electronic state by forming a solid solution with the sister compounds Sb 2 Te 3 and Bi 2 Se 3 . A Bi 2 Te 3 /Sb 2 Te 3 superlattice thin film was reported to show a very large dimensionless figure of merit ZT = 2.4. 4 In this letter, we focus on Nb 4 SiTe 4 as a sister compound of Ta 4 SiTe 4 . Nb 4 SiTe 4 was first synthesized by Badding et al. and was reported to have the same crystal structure as that of Ta 4 SiTe 4 , as shown in Fig. 1(a), which is orthorhombic with the space group Pbam. 5 The of a Nb 4 SiTe 4 single crystal along the chain direction was reported to be ~3 m cm at room temperature. 6 The decreases with decreasing temperature from 260 to 50 K and exhibits a local minimum of 1.5 m cm at ~50 K. 1,5,6 We prepared a series of whisker crystals of chemically doped Nb 4 SiTe 4 and measured their and S. The Mo-doped whisker crystals were found to show the largest P of ~70 W cm 1 K 2 at 230-300 K, far exceeding the practical level. This P is comparable to those of Mo-doped Ta 4 SiTe 4 with similar doping levels but is smaller than the maximum value in Ta 4 SiTe 4 . The difference of thermoelectric performance of the Nb and Ta compounds might be understood by using first principles calculation. We also demonstrated that the Nb 4 SiTe 4 -Ta 4 SiTe 4 solid solution also shows a large P comparable to those of the end members, suggesting that Nb 4 SiTe 4 can be utilized as a sister compound of Ta 4 SiTe 4 , as in the case of Sb 2 Te 3 and Bi 2 Se 3 for Bi 2 Te 3 .
The whisker crystals of Nb 4 SiTe 4 , (Nb 1x Mo x ) 4 SiTe 4 (x 0.05), Nb 4 Si(Te 0.95 Sb 0.05 ) 4 , and (Ta 0.5 Nb 0.5 ) 4 SiTe 4 were synthesized by crystal growth in a vapor phase. A stoichiometric amount of elemental powders and 100% excess of Si powder were mixed and sealed in an evacuated quartz tube with 20 mg of TeCl 4 powder. The tube was heated to and kept at 873 K for 24 h, 1423 K for 96 h, and then furnace cooled to room temperature. The typical size of the whisker crystals is several mm long and several m in diameter, as shown in Fig. 1(b). A sintered sample of Nb 4 SiTe 4 was prepared by a solid-state reaction method with the same procedure for Ta 4 SiTe 4 . 1 Sample characterization was performed by powder X-ray diffraction analysis for pulverized whisker crystals with Cu K radiation at room temperature using a RINT-2100 diffractometer (RIGAKU). We confirmed that the series of whisker crystals are a single phase. In this study, nominal compositions are used to represent the chemical compositions of the whisker crystals. The electrical resistivity and thermoelectric power measurements between 5 and 300 K were performed using a PPMS (Quantum Design). Thermal conductivity was measured by a standard four-contact method. First principles calculations for Nb 4 SiTe 4 were performed using the WIEN2k code. 7 The Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA) is used for the exchange-correlation potential. 8 The self-consistent calculation is done for 9 5 20 k-mesh. The numbers of atoms contained in the unit cell are 16, 4, and 16 for Nb, Si, and Te, respectively. The radii of the muffin-tin spheres are R MT (Nb) = R MT (Te) = 2.5a 0 and R MT (Si) = 2.11a 0 , where a 0 is the Bohr radius (R MT (Ta) = R MT (Te) = 2.5a 0 and R MT (Si) = 1.99a 0 in the calculation of Ta 4 SiTe 4 ). We also set R MT K max = 7.0, G max = 12, and core separation energy as 6.0 Ry. Experimental structural parameters are used for the calculations.
Figures 1(c) and 1(d) show the temperature dependences of S and of whisker crystals of Nb 4 SiTe 4 , (Nb 1x Mo x ) 4 SiTe 4 (x 0.05), and Nb 4 Si(Te 0.95 Sb 0.05 ) 4 measured along the c axis, respectively. The S of undoped Nb 4 SiTe 4 is negative, indicating that the electron carriers are dominant. The |S| of undoped Nb 4 SiTe 4 is 150 V K 1 at 300 K and gradually increases with decreasing temperature, reaching a maximum value of 220 V K 1 at ~150 K, comparable to the |S| of the Bi 2 Te 3 -based material at room temperature. Below 150 K, the |S| decreases toward zero with decreasing temperature. As seen in Fig. 1(c), |S| of Nb 4 SiTe 4 is about half of that of Ta 4 SiTe 4 , although the thermoelectric powers of Nb 4 SiTe 4 and Ta 4 SiTe 4 have the same sign and similar temperature dependences. The of the undoped Nb 4 SiTe 4 is 1 m cm at 300 K and increases to 2 m cm at ~150 K with decreasing temperature, followed by a weak decrease below this temperature. Below ~70 K, gradually increases again. As seen in Fig. 1(d), the values The thermoelectric powers of a series of (Nb 1x Mo x ) 4 SiTe 4 whisker crystals show a systematic change with Mo content. All measured samples have negative S values, as shown in Fig. 1(c), indicative of n-type. The peak temperature, where S shows a maximum value, increases with increasing Mo content from 140 K in the undoped sample. The samples with x 0.02 show d|S|/dT > 0 in all temperature region below room temperature. |S| systematically decreases with increasing x, except that the maximum value of |S| of the x = 0.001 sample is slightly larger than that of the undoped one, consistent with the fact that Mo substitution to the Ta site is electron doping, which is expected to increase the electron carriers. The x = 0.02 and 0.05 samples show |S| = 150 and 80 V K 1 at 300 K, respectively, which are almost the same as those of (Ta 1x Mo x ) 4 SiTe 4 with the same x values. 1 Mo doping tends to decrease , as shown in Fig. 1(d), reflecting the increase of electron carriers by electron doping. The of the x = 0.001 sample shows a broad peak at ~180 K, larger than that of the undoped one above 70 K. The x 0.005 samples show d/dT > 0 in all temperature region below room temperature, suggestive of the presence of metallic conduction in them. The of the x = 0.05 sample, which is most heavily doped in this study, is larger than those of x = 0.03 and 0.04, which might be due to the larger atomic disorder in the x = 0.05 sample caused by the Mo doping.
T (K) (d) x = 0.001, x = 0.005 x = 0.01, x = 0.02 x = 0.03, x = 0.04 x = 0.
Nb 4 Si(Te 0.95 Sb 0.05 ) 4 whisker crystal also shows negative S below 300 K, as shown in Fig. 1(c). Its |S| and are larger than those of the undoped one, respectively, probably reflecting the decrease of electron carriers by hole doping. The |S| of Nb 4 Si(Te 0.95 Sb 0.05 ) 4 reaches 350 V K 1 at 100-130 K, which is much larger than that of Nb 4 SiTe 4 and comparable to that of Ta 4 SiTe 4 . The of the Nb 4 Si(Te 0.95 Sb 0.05 ) 4 sample is about an order of magnitude larger than that of Nb 4 SiTe 4 , although their temperature dependences are similar. Figure 2(a) shows the power factor P = S 2 / of the whisker crystals of Nb 4 SiTe 4 , (Nb 1x Mo x ) 4 SiTe 4 (x 0.05), and Nb 4 Si(Te 0.95 Sb 0.05 ) 4 . The undoped Nb 4 SiTe 4 shows P ~ 20 W cm 1 K 2 between 100 and 300 K and a maximum value of P max = 22 W cm 1 K 2 at the optimum temperature of T max = 150 K. The P values tend to increase with increasing x, reaching 70 W cm 1 K 2 above 230 K at x = 0.04. This power factor is smaller than P max = 170 W cm 1 K 2 for the Mo-doped Ta 4 SiTe 4 , as shown in Fig. 2(b), but is larger than P max = 35 W cm 1 K 2 of the Bi 2 Te 3 -based materials, 9,10 indicating that Nb 4 SiTe 4 is a strong candidate for a low-temperature thermoelectric material same as Ta 4 SiTe 4 . As seen in Fig. 2(b), the P values of the Nb compounds are optimized at x = 0.030.04, which is significantly larger than x < 0.01 for the Ta compounds. Moreover, the x dependence of T max of the Nb compounds is more gradual than that of the Ta ones, as shown in Fig. 2(c). These results suggest that the thermoelectric performances of the Nb compounds are less sensitive to the chemical doping, i.e., more easily optimized than those of the Ta compounds.
We note the dimensionless figure of merit ZT = PT/( el + lat ) of Mo-doped Nb 4 SiTe 4 , where el and lat are the conduction electron and phonon contributions of thermal conductivity, respectively. ZT is directly related to the thermoelectric energy conversion efficiency. Although thermal conductivity of the whisker crystals prepared in this study cannot be measured due to their very thin shape, we estimated the upper limit of ZT of them by using el of the whisker crystals, obtained by applying Wiedemann-Franz law to the data shown in Fig. 1(d), and lat of the Nb 4 SiTe 4 sintered sample. As shown in Fig. 3, thermal conductivity of a Nb 4 SiTe 4 sintered sample is 11 mW cm 1 K 1 at room temperature, almost all of which is the phonon contribution, considering of the sintered sample. The upper limit of ZT for x = 0.03 at room temperature, where P is optimized as shown in Fig. 2(b), is estimated to be 0.5, which is smaller than 2.2 for (Ta 0.999 Mo 0.001 ) 4 SiTe 4 at 250 K. 1 Thus, the whisker crystals of the Nb compounds were found to show the large P and upper limit of ZT, although they are smaller than the maximum values for the Ta compounds. The difference of P values between the Nb and Ta compounds depends on the kind and content of the dopant ions. In the undoped and Sb-doped cases, where electron carrier density is expected to be lower than the Mo-doped case, the P of the Nb compounds are much smaller than those of Ta ones, as shown in Fig. 2(b). In contrast, heavily Mo-doped Nb 4 SiTe 4 shows large P comparable to that of Mo-doped Ta 4 SiTe 4 . Both (Nb 1x Mo x ) 4 SiTe 4 and (Ta 1x Mo x ) 4 SiTe 4 with x 0.01 exhibit d/dT > 0 and dS/dT > 0 below room temperature, different from the undoped Nb 4 SiTe 4 and Ta 4 SiTe 4 , probably reflecting the larger electron carrier density. 1 They also show similar |S| values, giving rise to similar P values, although there is some difference due to the different values.
Here, we discuss the doping dependences of the thermoelectric properties of the Nb-and Ta-based whisker crystals in the light of their electronic structures. Figures 4(a) and 4(b) show the first principles calculation results of Nb 4 SiTe 4 with and without spin-orbit coupling, respectively. The electronic structures of Nb 4 SiTe 4 are similar to those of Ta 4 SiTe 4 , reflecting the same crystal structure and electron configuration. 1 The band dispersions along the k x and k y directions are almost flat and much weaker than that along the k z direction. This strongly one-dimensional electronic structure would be one of the important factors to realize high thermoelectric performances in Nb 4 SiTe 4 and Ta 4 SiTe 4 . 11, 12 We also confirmed that the Mo doping to The difference of thermoelectric performances between the Nb and Ta compounds seems to be caused by a much smaller spin-orbit gap in Nb 4 SiTe 4 . When the spin-orbit coupling is switched off, the energy bands cross on the Z line at around E F in both Nb 4 SiTe 4 and Ta 4 SiTe 4 . In Ta 4 SiTe 4 , the strong spin-orbit coupling of Ta 5d orbitals gives rise to an energy gap of = 0.1 eV near the band crossing points. 1 In contrast, the spin-orbit gap of ~ 0.02 eV in Nb 4 SiTe 4 is much smaller than that of Ta 4 SiTe 4 . These values are probably related to the different behaviors of of the undoped Nb 4 SiTe 4 and Ta 4 SiTe 4 below 50 K mentioned above. We believe that the size of also plays an important role in the thermoelectric performances, such as the significantly different P max between the Nb and Ta compounds in the left side of Fig. 4(b), where the electron carrier densities are lower than those of the right side. In contrast, the S and of the heavily Mo-doped Nb 4 SiTe 4 and Ta 4 SiTe 4 show simple metallic behaviors, suggesting that the E F of them is located well above the bottom of the conduction band due to the electron doping. In this case, fine structures of the energy bands at around E F , such as the size of , seem to have little effect on the thermoelectric properties and result in similar thermoelectric performances in the Nb and Ta compounds. Finally we report the thermoelectric properties of a Nb 4 SiTe 4 -Ta 4 SiTe 4 solid solution sample. Figures 5(a) and 5(b) show the temperature dependences of S and of (Ta 0.5 Nb 0.5 ) 4 SiTe 4 , Nb 4 SiTe 4 , and Ta 4 SiTe 4 whisker crystals, respectively. The S of (Ta 0.5 Nb 0.5 ) 4 SiTe 4 is negative and located between those of Nb 4 SiTe 4 and Ta 4 SiTe 4 . The |S| of (Ta 0.5 Nb 0.5 ) 4 SiTe 4 shows a maximum value of 310 V K 1 at 180 K, indicating that the solid-solution sample has a sufficient |S| as a thermoelectric material. The of the solid-solution sample is almost the same as those of Nb 4 SiTe 4 and Ta 4 SiTe 4 above ~100 K. Below this temperature, the of the solid-solution sample is much smaller than that of Ta 4 SiTe 4 , but is comparable to that of Nb 4 SiTe 4 . This may reflect the presence of a significant contribution of Nb 4d orbitals near the E F in the solid solution samples, which is expected to give rise to a very small spin-orbit gap same as in Nb 4 SiTe 4 . The P max of the solid solution sample is 47 W cm 1 K 2 at T max = 160 K, which exceeds the practical level, suggesting that the combination of Nb 4 SiTe 4 and Ta 4 SiTe 4 is promising as a practical thermoelectric material, same as the Bi 2 Te 3 -Sb 2 Te 3 and Si-Ge alloys.
In conclusion, we have studied the thermoelectric properties of whisker crystals of Nb 4 SiTe 4 , a 4d analogue of Ta 4 SiTe 4 , and its substituted compounds and found that Mo-doped Nb 4 SiTe 4 shows high thermoelectric performances at low temperatures. The power factor of the undoped Nb 4 SiTe 4 is smaller than the maximum value of Ta 4 SiTe 4 , while those of (Nb 1x Mo x ) 4 SiTe 4 with x 0.01 are very large, as represented by 70 W cm 1 K 2 for x = 0.030.04 at 230-300 K, comparable to those in the Ta compounds. The electronic structure of Nb 4 SiTe 4 is similar to that of Ta 4 SiTe 4 in that a small spin-orbit gap opens in the strongly one-dimensional electronic bands, probably yielding the high thermoelectric performances in both systems. Alternatively, the difference of thermoelectric performances between Nb and Ta compounds with low electron carrier density might be related to the significantly smaller spin-orbit gap in Nb 4 SiTe 4 than that in Ta 4 SiTe 4 , suggesting that the spin-orbit gap of ~0.1 eV in Ta 4 SiTe 4 plays an important role in the colossal power factor in the lightly Mo doped Ta 4 SiTe 4 . Moreover, we found that not only the Nb compounds but also the solid solution of the Nb and Ta compounds shows a high thermoelectric performance exceeding those of practical thermoelectric materials, indicating that the combination of Nb 4 SiTe 4 and Ta 4 SiTe 4 is promising to realize practical-level thermoelectric conversion at low temperatures.
FIG. 1 .
1(a) Crystal structure of Nb 4 SiTe 4 . The orthorhombic unit cell is indicated by broken lines. (b) A stereomicroscopic image of whisker crystals of Mo-doped Nb 4 SiTe 4 . The bar in the image indicates 50 m. Temperature dependences of (c) thermoelectric power and (d) electrical resistivity of the whisker crystals of Nb 4 SiTe 4 , (Nb 1x Mo x ) 4 SiTe 4 (x 0.05), and Nb 4 Si(Te 0.95 Sb 0.05 ) 4 measured along the c axis. The data of Ta 4 SiTe 4 are also shown as a reference. 1
FIG. 3 .
3Temperature dependences of thermal conductivity and electrical resistivity of a Nb 4 SiTe 4 sintered sample.
Nb 4
4SiTe 4 results in the upward shift of the chemical potential (5% Mo doping gives ~0.2 eV shift) without a significant change of the band structure by the band calculations using the virtual crystal approximation. 13 FIG. 4. Electronic structures of Nb 4 SiTe 4 without (a) and with (b) spin-orbit coupling. Red and green represent the contributions of the d-orbital of Nb and p-orbital of Te, respectively. The broken lines indicate those of Ta 4 SiTe 4 . 1 The Fermi level is set to 0 eV. The first Brillouin zone is shown in the right panel of (a).
FIG. 5 .
5Temperature dependences of thermoelectric power (a) and electrical resistivity (b) of the whisker crystals of (Ta 0.5 Nb 0.5 ) 4 SiTe 4 , Nb 4 SiTe 4 , and Ta 4 SiTe 4 measured along the c axis.
Si(Te 0.95 Sb 0.05 )4 along the c axis. (b) Maximum power factor P max below 300 K. (c) Optimum temperature T max , at which the power factor shows a maximum value P max , below 300 K. The open symbols indicate T max = 300 K.of of Nb 4 SiTe 4 and Ta 4 SiTe 4 are almost the same above ~70 K, while they are significantly different below this temperature, where the increase of of Nb 4 SiTe 4 is much weaker than that of Ta 4 SiTe 4 .1,5,6 05
Nb 4 SiTe 4
Ta 4 SiTe 4
-500
-400
-300
-200
-100
0
S
(µV
K
-1
)
(c)
Nb 4 Si(Te 0.95 Sb 0.05 ) 4
Nb 4 SiTe 4
Ta 4 SiTe 4
(Nb 1-x Mo x ) 4 SiTe 4
FIG. 2. (a) Temperature dependence of power factor of the
whisker crystals of Nb 4 SiTe 4 , (Nb 1x Mo x ) 4 SiTe 4 (x 0.05), and
Nb 4
ACKNOWLEDGMENTSWe are grateful to Y. Yoshikawa for his help with experiments and A. Yamakage for helpful discussion. This work was partly supported by JSPS KAKENHI (Grant Nos. 16K13664, 16H03848, and 16H01072), the Murata Science Foundation, and the Tatematsu Foundation.
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| [] |
[
"Discovering Dynamic Salient Regions for Spatio-Temporal Graph Neural Networks",
"Discovering Dynamic Salient Regions for Spatio-Temporal Graph Neural Networks"
] | [
"Iulia Duta \nInstitute of Mathematics of the Romanian Academy University \"Politehnica\" of Bucharest\nBitdefenderRomania, Romania\n",
"Andrei Nicolicioiu [email protected] \nInstitute of Mathematics of the Romanian Academy University \"Politehnica\" of Bucharest\nBitdefenderRomania, Romania\n",
"Marius Leordeanu Bitdefender \nInstitute of Mathematics of the Romanian Academy University \"Politehnica\" of Bucharest\nBitdefenderRomania, Romania\n",
"Romania \nInstitute of Mathematics of the Romanian Academy University \"Politehnica\" of Bucharest\nBitdefenderRomania, Romania\n"
] | [
"Institute of Mathematics of the Romanian Academy University \"Politehnica\" of Bucharest\nBitdefenderRomania, Romania",
"Institute of Mathematics of the Romanian Academy University \"Politehnica\" of Bucharest\nBitdefenderRomania, Romania",
"Institute of Mathematics of the Romanian Academy University \"Politehnica\" of Bucharest\nBitdefenderRomania, Romania",
"Institute of Mathematics of the Romanian Academy University \"Politehnica\" of Bucharest\nBitdefenderRomania, Romania"
] | [] | Graph Neural Networks are perfectly suited to capture latent interactions between various entities in the spatio-temporal domain (e.g. videos). However, when an explicit structure is not available, it is not obvious what atomic elements should be represented as nodes. Current works generally use pre-trained object detectors or fixed, predefined regions to extract graph nodes. Improving upon this, our proposed model learns nodes that dynamically attach to well-delimited salient regions, which are relevant for a higher-level task, without using any object-level supervision. Constructing these localized, adaptive nodes gives our model inductive bias towards object-centric representations and we show that it discovers regions that are well correlated with objects in the video. In extensive ablation studies and experiments on two challenging datasets, we show superior performance to previous graph neural networks models for video classification.We propose a method that learns to discover salient regions, well-delimited in space and time, that are useful for modeling interactions between various entities. Such entities could be single objects, parts or groups of objects that perform together a simple action. Each node learns to associate by itself to such salient regions, thus the message passing between nodes is able to model object interactions more effectively. For humans, representing objects is a core knowledge system[8]and to emphasize them in our model, we predict salient regions [9] that give a strong inductive bias towards modeling them.Our method, Dynamic Salient Regions Graph Neural Network (DyReg-GNN) improves the relational processing of videos by learning to discover salient regions that are relevant for the current scene and task. Note that the model learns to predict regions only from the weak supervision given by the high-level video classification loss, without supervision at the region level. Our experiments convincingly show that the regions discovered are well correlated with the objects present in the video, confirming the intuition that action recognition should be strongly related to salient region discovery. The capacity to discover such regions makes DyReg-GNN an excellent candidate model for tackling tasks requiring spatio-temporal reasoning.Our main contributions are summarised as follow:1. We propose a novel method to augment spatio-temporal GNNs by an additional capability:that of learning to create localized nodes suited for spatial reasoning, that adapt to the input. 2. The salient regions discovery enhance the relational processing for high-level video classification tasks: creating GNN nodes from predicted regions obtains superior performance compared to both using pre-trained object detectors or fixed regions 3. Our model leads to unsupervised salient regions discovery, a novelty in the realm of GNNs: it predicts such regions in videos, with only weak supervision at the video class level. We show that regions discovered are well correlated with actual physical object instances.Related workGraph Neural Networks in Vision. GNNs have been recently used in many domains where the data has a non-uniform structure[10,11,12,13]. In vision tasks, it is important to model the relations between different entities appearing in the scene[14,15]and GNNs have strong inductive biases towards relations[16,17], thus they are perfectly suited for modeling interactions between visual instances. Since an explicit structure is not available in the video, it is of critical importance to establish what atomic elements should be represented as graph nodes. As our main contribution revolves around the creation of nodes, we analyse other recent GNN methods regarding the type of information that each node represents, and group them into two categories, semantic and spatial.The approaches of[4,18,19,20,21,22]capture the purely semantic interactions by reasoning over global graph nodes, each one receiving information from all the points in the input, regardless of spatio-temporal position. In [4] the nodes assignments are predicted from the input, while in [18] the associations between input and nodes are made by a soft clusterization. The work of [22] discovers different representation groups by using an iterative clusterization based on self-attention similarity.The downside of these semantic approaches is that individual instances, especially those belonging to the same category, are not distinguished in the graph processing. This information is essential in tasks such as capturing human-object interactions, instance segmentation or tracking.Alternatively, multiple methods, including ours, favour modeling instance interactions by defining spatial nodes associated with certain locations. We distinguish between them by how they extract the nodes from spatial location: as fixed regions or points[23,24], or detected object boxes[25,26,27,28,29]. The method [5] creates nodes from every point in 2D convolutional features maps, while Non-Local [30] uses self-attention [31] between all spatio-temporal positions to capture distant interactions. Further, [3] extract nodes from larger fixed regions at different scales and processes them recurrently. Recent methods based on Transformer [32, 6, 33] also model the interactions between fixed locations on a grid using self-attention. In [7], nodes are created from object boxes extracted by an external detector and are processed using two different graph structures, one given by location and one given by nodes similarity. A related approach is used in [27] in a streaming setting while [34] learns to hop over unnecessary frames. Hybrid approaches use nodes corresponding to points and object features [35, 36] or propagate over both semantic and spatial nodes [37, 38, 39]. | null | [
"https://arxiv.org/pdf/2009.08427v3.pdf"
] | 244,921,149 | 2009.08427 | 6a4a7de57115012a2c044a0c05df3e3a3208bd5e |
Discovering Dynamic Salient Regions for Spatio-Temporal Graph Neural Networks
Iulia Duta
Institute of Mathematics of the Romanian Academy University "Politehnica" of Bucharest
BitdefenderRomania, Romania
Andrei Nicolicioiu [email protected]
Institute of Mathematics of the Romanian Academy University "Politehnica" of Bucharest
BitdefenderRomania, Romania
Marius Leordeanu Bitdefender
Institute of Mathematics of the Romanian Academy University "Politehnica" of Bucharest
BitdefenderRomania, Romania
Romania
Institute of Mathematics of the Romanian Academy University "Politehnica" of Bucharest
BitdefenderRomania, Romania
Discovering Dynamic Salient Regions for Spatio-Temporal Graph Neural Networks
Code is available at https://github.com/bit-ml/DyReg-GNN
Graph Neural Networks are perfectly suited to capture latent interactions between various entities in the spatio-temporal domain (e.g. videos). However, when an explicit structure is not available, it is not obvious what atomic elements should be represented as nodes. Current works generally use pre-trained object detectors or fixed, predefined regions to extract graph nodes. Improving upon this, our proposed model learns nodes that dynamically attach to well-delimited salient regions, which are relevant for a higher-level task, without using any object-level supervision. Constructing these localized, adaptive nodes gives our model inductive bias towards object-centric representations and we show that it discovers regions that are well correlated with objects in the video. In extensive ablation studies and experiments on two challenging datasets, we show superior performance to previous graph neural networks models for video classification.We propose a method that learns to discover salient regions, well-delimited in space and time, that are useful for modeling interactions between various entities. Such entities could be single objects, parts or groups of objects that perform together a simple action. Each node learns to associate by itself to such salient regions, thus the message passing between nodes is able to model object interactions more effectively. For humans, representing objects is a core knowledge system[8]and to emphasize them in our model, we predict salient regions [9] that give a strong inductive bias towards modeling them.Our method, Dynamic Salient Regions Graph Neural Network (DyReg-GNN) improves the relational processing of videos by learning to discover salient regions that are relevant for the current scene and task. Note that the model learns to predict regions only from the weak supervision given by the high-level video classification loss, without supervision at the region level. Our experiments convincingly show that the regions discovered are well correlated with the objects present in the video, confirming the intuition that action recognition should be strongly related to salient region discovery. The capacity to discover such regions makes DyReg-GNN an excellent candidate model for tackling tasks requiring spatio-temporal reasoning.Our main contributions are summarised as follow:1. We propose a novel method to augment spatio-temporal GNNs by an additional capability:that of learning to create localized nodes suited for spatial reasoning, that adapt to the input. 2. The salient regions discovery enhance the relational processing for high-level video classification tasks: creating GNN nodes from predicted regions obtains superior performance compared to both using pre-trained object detectors or fixed regions 3. Our model leads to unsupervised salient regions discovery, a novelty in the realm of GNNs: it predicts such regions in videos, with only weak supervision at the video class level. We show that regions discovered are well correlated with actual physical object instances.Related workGraph Neural Networks in Vision. GNNs have been recently used in many domains where the data has a non-uniform structure[10,11,12,13]. In vision tasks, it is important to model the relations between different entities appearing in the scene[14,15]and GNNs have strong inductive biases towards relations[16,17], thus they are perfectly suited for modeling interactions between visual instances. Since an explicit structure is not available in the video, it is of critical importance to establish what atomic elements should be represented as graph nodes. As our main contribution revolves around the creation of nodes, we analyse other recent GNN methods regarding the type of information that each node represents, and group them into two categories, semantic and spatial.The approaches of[4,18,19,20,21,22]capture the purely semantic interactions by reasoning over global graph nodes, each one receiving information from all the points in the input, regardless of spatio-temporal position. In [4] the nodes assignments are predicted from the input, while in [18] the associations between input and nodes are made by a soft clusterization. The work of [22] discovers different representation groups by using an iterative clusterization based on self-attention similarity.The downside of these semantic approaches is that individual instances, especially those belonging to the same category, are not distinguished in the graph processing. This information is essential in tasks such as capturing human-object interactions, instance segmentation or tracking.Alternatively, multiple methods, including ours, favour modeling instance interactions by defining spatial nodes associated with certain locations. We distinguish between them by how they extract the nodes from spatial location: as fixed regions or points[23,24], or detected object boxes[25,26,27,28,29]. The method [5] creates nodes from every point in 2D convolutional features maps, while Non-Local [30] uses self-attention [31] between all spatio-temporal positions to capture distant interactions. Further, [3] extract nodes from larger fixed regions at different scales and processes them recurrently. Recent methods based on Transformer [32, 6, 33] also model the interactions between fixed locations on a grid using self-attention. In [7], nodes are created from object boxes extracted by an external detector and are processed using two different graph structures, one given by location and one given by nodes similarity. A related approach is used in [27] in a streaming setting while [34] learns to hop over unnecessary frames. Hybrid approaches use nodes corresponding to points and object features [35, 36] or propagate over both semantic and spatial nodes [37, 38, 39].
Introduction
Spatio-temporal data, and videos, in particular, are characterised by an abundance of events that require complex reasoning to be understood. In such data, entities or classes exist at multiple scales and in different contexts in space and time, starting from lower-level physical objects, which are well localized in space and moving towards higher-level concepts which define complex interactions. We need a representation that captures such spatio-temporal interactions at different level of granularity, depending on the current scene and the requirements of the task. Classical convolutional nets address spatio-temporal processing in a simple and rigid manner, determined only by fixed local receptive fields [1]. Alternatively, space-time graph neural nets [2,3] offer a more powerful and flexible approach modeling complex short and long-range interactions between visual entities.
In this paper, we propose a novel method to enhance vision Graph Neural Networks (GNNs) by an additional capability, missing from any other previous works. That is, to have nodes that are constructed for spatial reasoning and can adapt to the current input. Prior works are limited to having either nodes attached to semantic attention maps [4] or attached to fixed locations such as grids [5,3,6]. Moreover, unlike works that require external object detectors [7] our method relies on a learnable mechanism to adapt to the current input. Figure 1: (Left) DyReg-GNN extracts localized node useful for relational processing of videos. For each node i, from the features X t , we predict params o i denoting the location and size of a region. They define a kernel K i , used to extract the localized features v i from the corresponding region of X t . We process the nodes with a spatio-temporal GNN and project each nodev i into its initial location. (Right) B) Node Region Generation: Functions f and {g i } generate the regions params o i ; f extracts a latent representation shared between nodes, while each g i has different params for each node i. C) Node Features Extraction: Each o i creates a kernel that is used in a differentiable pooling w.r.t. o i . This allows us to optimize the generation of these regions' params from the final classification loss, resulting in an unsupervised discovery of salient regions.
However, methods that rely on external modules trained on additional data, such as object detectors, are too dependent on the module's performance. They are unable to adapt to the current problem, being limited to the set of pre-defined annotations designed for another task. Differently, our module is optimized to discover regions useful for the current task, using only the video classification signal.
Recently, the method [40] uses multiple position-aware nodes that take into account the spatial structure. This makes it more suitable for capturing instances, but the nodes have associated a static learned location, where each one is biased towards a specific position regardless of the input. On the other hand, we dynamically assign a location for each node, based on the input, making the method more flexible to adapt to new scenes.
Dynamic Networks. Several works use second-order computations by dynamically predicting different parts of their model from the input, instead of directly optimising parameters. Our work is related to STN [41] that aggregates features by interpolating from an area given by a predicted global transformation and to the differentiable pooling used in some object detectors [42,43,44]. The method [45], replaces the parameters in a standard convolution with weights predicted from the input, resulting in a dynamically generated filter. Deformable convolutions [46,47] predict, based on the input, an offset for each position in the convolutional kernel. Similar, [48] use the same idea of predicting offsets but in a graph formulation. The common topic of these methods is to predict dynamically a support for all points in a convolutional operation while we dynamically generate the input for a set of nodes designed to process high-level interactions. Related ideas, involving high-level processing of a small set of powerful modules, is also highlighted in [49] and [40].
Unsupervised Object Representations. There is an entire area of work devoted to extracting representations centered on objects [50] in a fully unsupervised setting [51,52,53,22]. They are successful in leveraging a reconstruction task to decompose the scene into objects, for synthetic images. In [54] it is shown that representations learned from unsupervised decomposition are also helpful in relational reasoning tasks. Methods for generating unsupervised keypoints or entities [55,56,57,58] have been generally used in synthetic setting. The method [55] generates keypoints from real images of people and faces but they use an image reconstruction objective that could not be aligned with the downsteam task. Our goal is to relate spatio-temporal entities, but without enforcing a clear decomposition of the scene into objects. This allows us to use a simpler but effective method that learns from classification supervision of real-world videos and obtain representations that are correlated to objects. Activity Recognition. Video classification has been influenced by methods designed for 2D images [59,60,61,62]. More powerful 3D convolutional networks have been later proposed [63], while other methods factorise the 3D convolutions [64,65,66] bringing both computational speed and accuracy. Methods like TSM [67] and [68] showed that a simple shift in the convolutional features results in improved accuracy at a low computational budget.
Dynamic Salient Regions GNNs
We investigate how to create node representations that are useful for modeling visual interactions between various entities in space and time using GNNs. Our proposed Dynamic Salient Regions GNN model (DyReg-GNN) learns to dynamically assign each node to a certain interesting region. By dynamic, we mean that we have a fixed number N of regions that change their position and size according to the input at each time step. The regions assigned to each of the N nodes can change from one moment of time to the next depending on their saliency.
The main architecture of our DyReg-GNN model is illustrated in Figure 1. Our model receives feature volume X ∈ R T ×H×W ×C and at each time step t we predict the location and size of N regions. From these regions, a differentiable pooling operation creates graph nodes that are processed by a GNN and then are projected to their initial position. This module can be inserted at any intermediate level in a standard convolutional model.
Node Region Generation
We want to attend only to a few most relevant entities in the scene, thus a small number of nodes are used in DyReg-GNN (in our experiments N = 9) and it is crucial to assign them to the most salient regions. The number of nodes is a hyperparameter that we choose such that it exceeds the expected number of relevant entities in the scene, to increase the robustness of the model. Thus, we propose a global processing (shown in Figure 1 B) that aggregates the entire input features to produce regions defined by parameters indicating their location (∆x, ∆y) and size (w, h).
To generate N salient regions, we process the input X t using position-aware functions f and {g i } i∈1,N that retain spatial information. Nodes should be consistent across time, thus we generate their regions in the same way at all time steps, by sharing in time the parameters of f and {g i }. The function f is a convolutional network that highlights the important regions from the input.
M t = f (X t ) ∈ R H ×W ×C(1)
For each node i, we generate a latent representation of its associated region using the {g i } functions. Each g i has the same architecture, but different parameters for each node and could be instantiated as a fully connected network or as global pooling enriched with spatial positional information. We generate the node regions from a global view to make the decision as informed as possible.
m i,t = g i (M t ) ∈ R C , ∀i ∈ 1, N(2)
Each of the N latent representations is processed independently, with a GRU [69] recurrent network (shared between nodes), to take into account the past regions' representations.
z i,t = GRU(z i,t−1 ,m i,t ) ∈ R C , ∀i ∈ 1, N(3)
At each time step, the final parameters are obtained by a linear projection W o ∈ R C ×4 , transformed by a function α to control the initialisation of the position and size (e.g. regions would start at reference points either in the center of the frame or arranged on a grid). For more details about how to set the transformation α we refer to the Supplemental Materials.
o i,t = (∆x i,t , ∆y i,t , w i,t , h i,t ) = α(W o z i,t ) ∈ R 4(4)
Node Features Extraction
The following operations are applied independently at each time step thus, in the current subsection, we ignore the time index for clarity. We extract the features corresponding to each region i using a differentiable pooling w.r.t. the predicted region parameters o i . All the input spatial locations p ∈ R 2 are interpolated according to the kernel function K (i) (p) as presented in Figure 1 C.
We present the operation for a single axis since the kernel is separable, acting in the same way on both axes:
K (i) (p x , p y ) = k (i) x (p x )k (i) y (p y ) ∈ R(5)
We define the center of the estimated region c i,x + ∆x i , where c i,x is a fixed reference point for node i (located in the frame's center or arranged on a grid). The values of the kernel decrease with the distance to the center and is non-zero up to a maximal distance of w i , where w i and ∆x i are the predicted parameters from Eq. 4.
k (i) x (p x ) = max(0, w i − |c i,x + ∆x i − p x |)(6)
For each time step t, node i is created by interpolating all points in the input X t using the kernel function. By modifying (∆x i , ∆y i ) the network controls the location of the regions, while (h i , w i ) parameters indicate their size.
v i = W px=1 H py=1 K (i) (p x , p y )x px,py ∈ R C(7)
Setting w i = 1 leads to standard bilinear interpolation, but optimising it allows the model to adapt region's size and we observe that larger ones result in a more stable optimisation (see node size ablations from Supp. Material).
The position of the region associated with each node should be taken into account. It helps the relational processing by providing an identity for the node and is also useful in tasks that require positional information. We achieve this by computing a positional embedding for each node i using a linear projection of the kernel K i into the same space as the feature vector v i and summing them.
Key Properties. By construction, the nodes in our method are localized, meaning that they are clearly associated with a location: they pool information from clearly delimited area in space and they maintain position information from the positional embedding. These two aspects could be helpful in tasks involving spatio-temporal reasoning.
The dynamic aspect refers to the key capability of adapting the region's position and size according to the saliency of the input at each time step. This is done by predicting the regions from the input with the operations from equations (1-4).
An essential aspect of this method is that the final classification loss is differentiable with respect to regions' parameters as the gradients are passing from the nodes outputs v i through the kernels k i to the parameters w i and ∆x i . This allows us to learn regions from the final loss, without direct supervision for the region generation. Thus the method has more flexibility in learning relevant regions as appropriate for the task.
Graph Processing
For processing the nodes' features, different spatio-temporal GNNs could be used. Generally, they follow a framework [12] of sending messages between connected nodes, aggregating [70,71] and updating them.
The specific message-passing mechanism is not the focus of the current work, thus we follow a general formulation similar to [3] for recurrent spatio-temporal graph processing. It uses two different stages: one happening between all the nodes at a single time step and the other one updating each node across time. For each time step t, we send messages between each pair of two nodes, computed as an MLP (with shared parameters) and aggregates them using a dot product attention
a(v i , v j ) ∈ R. v i,t = N j=1 a(v j,t , v i,t )MLP([v j,t ; v i,t ]) ∈ R C(8)
We incorporate temporal information through a shared recurrent function across time, applied independently for each node.v
i,t+1 = GRU(v i,t , v i,t ) ∈ R C(9)
The GRU output represents the updated nodes' features and the two steps are repeated K = 3 times. Table 1: Results on val. set of Smt-Smt-V2 showing the importance of salient regions discovery. We compare our predicted (unsupervised) regions to fixed grid regions or boxes given by an object detector using the same GNN model. The mean L 2 distance between the regions and gt. objects proves that DyReG-GNN has regions correlated with objects, while also having superior accuracy and efficiency.
Graph Re-Mapping
To use our method as a module inside any backbone, we produce an output with the same shape as the convolutional input X t ∈ R H×W ×C . The resulting features of each node are sent to all locations in the input according to the weights used in the initial pooling from Section 3.2.
y px,py,t = N i=1 K (i) t (p x , p y )v i,t ∈ R C(10)
Experimental Analysis
While much effort is put into the creation of different video datasets used in the literature, such as Kinetics [63] or Charades [72], it has been argued [73] that they contain biases that make them solvable without complex spatio-temporal reasoning. CATER [73] is proposed to alleviate this, but it is too small (5500 videos) and still has biases that make the last few frames sufficient for good performance [34]. We test our model on two video classification datasets that seem to offer the best advantages, being large enough and requiring abilities to model complex interactions. We evaluate on real-world datasets, Something-Something-V1&V2 [74], while we also test on a variant of the SyncMNIST [3] dataset that is challenging and requires spatio-temporal reasoning, while allowing fast experimentation. The code for our method can be found in our repository 2 .
Human-Object Interactions Experiments
Something-Something-V1&V2 [74] datasets classify scenes involving human-object complex interactions. They consist of 86K / 169K training videos and 11K / 25K validation videos, having 174 classes. Unless otherwise specified, all experiments on Something-Something datasets use TSM-ResNet-50 [67] as a backbone and we add instances of our module at multiple stages.
Studying the Importance of Salient Regions Discovery. We test the importance of the dynamic regions for GNNs vision methods by training models where we replace the predicted regions with the same number of fixed regions on a grid (GNN + Fixed Regions) or boxes (GNN + Detector) as given by a Faster R-CNN [75] trained on MSCOCO [76].
The detector based model has comparable results to the one with fixed regions, seemingly being unable to fully benefit from the correctly identified objects. The relative weaker performance of this model could be due to the fact that the pre-trained detector is not well aligned to the actual salient regions that are relevant for the classification problem.
On the other hand, this weakness is not applicable for DyReg-GNN that learns suitable regions for the current task and it obtains the best performance as seen in Table 1. Not only that it does not require object annotations, but it is also more computationally efficient. Running the detector on a video of Overall, our method, with unsupervised regions obtains superior performance in terms of accuracy and computational efficiency representing a suitable choice for relational processing of a video.
Object-centric representations. The nodes represent the core processing units and their localization enforces a clear decision on what specific regions to focus on while completely ignoring the rest, as a form of hard attention. Different from other works [77], our hard attention formulation is differentiable.
To better understand what elements influence the model predictions, we could inspect the predicted kernels, thus introducing another layer of interpretability to the model, on top of the capabilities offered by the convolutional backbone. Visualisations of our nodes' regions reveal that generally, they cover the objects in the scene. For example, in the first row of Figure 3 the nodes are placed around the phone in the first frames and then separate into two groups, one for the phone one for the hand.
The localized nodes make our model capable of discovering salient regions, leading to object-centric node representations. We quantify this capacity by measuring the mean L 2 distance (normalised to the size of the input) between the predicted regions and ground-truth (gt.) objects given by [28]. The metric is completely defined in the Supp. Materials. We observed that the score improves during the learning process (it reaches 0.129 starting from 0.201), although the model is not optimized for this task. This suggests that the model actually learns object-centric representations.
In Table 1 we also compare the final L 2 distance of our best DyReg-GNN model to an object detector and to fixed grid regions. Although our method is not designed and supervised to find object regions, we observe that it is able to predict locations that are fairly close to gt. objects. The L 2 distance is similar to the one obtained by an external model (0.129 vs 0.125), trained especially for detecting objects.
We observe that learning the regions' size is important for the stability of the optimisation and thus for the final performance (see Tab.5 and Supp. Material -Regions' Size section). However, the predicted size is not as well aligned with the size of the true objects. This gives us a hint that for the action classification task it is important to have good region locations, but their size is less relevant. We leave a more thoroughly investigation for futures work.
These experiments prove that the high-level classification task is well inter-related with the discovery of salient regions and that, in turn, these regions improve the relational processing in the recognition task. First, we show that DyReg-GNN's region obtain superior accuracy and efficiency than other methods of extracting nodes and second, these regions are well correlated to gt. object locations.
Comparison to recent methods. DyReg-GNN can be used with any convolutional model and we show that it consistently boosts the performance of multiple backbones (Table 2). We compare to recent methods from the literature in Table 3 and Table 4. Our method improves the accuracy over the TSM-ResNet50 backbone on both Smt-Smt-V1 and Smt-Smt-V2 by 1.5% and 1.4% respectively and achieves competitive results. Compared to all the other graph based methods we obtain superior results, showing that our discovery of dynamic regions is effective for space-time relational processing.
Implementation Details Unless otherwise specified, we use TSM-ResNet50 (pre-trained on Ima-geNet [84]) as our backbone and add instances of our module in the last three stages. To benefit from ImageNet pre-training, we add our graph module as a residual connection. We noticed that models using multiple graphs have problems learning to adapt the regions from certain layers. We fix this by training models containing a single graph at each single considered stage, as the optimisation process is smoother for a single module, and distill their learned offsets into the bigger model. The distillation is done for the first 10% of the training iterations to kick-start the optimization process and then continue the learning process using only the video classification signal.
In all experiments we follow the training setting of [67], using 16 frames resized to have the shorter side of size 256, and randomly sample a crop of size 224 × 224. For the evaluations, we follow the setting in [67] of taking 3 spatial crops of size 256 × 256 with 2 temporal samplings and averaging their results. For training, we use SGD optimizer with learning rate 0.001 and momentum 0.9, using a total batch-size of 10, trained on two GPUs. We decrease the learning rate by a factor of 10 three times when the optimisation reaches a plateau.
Synthetic Experiments
SyncMNIST is a synthetic dataset involving digits that move on a black background, some in a random manner, while some move synchronously. The task is to identify the digits that move in the same way. We use a harder variant of the dataset (MultiSyncMNIST), where the videos could include multiple digits of the same class. The challenge consists in finding useful entities and model their relationships while being able to distinguish between instances of the same class. Each video contain 5 digits and the goal is to find the smallest and the largest digit class among the subset that moves in the same way. This results in a video classification task with 56 classes. The dataset contains 600k training videos and 10k validation videos with 10 frames each.
Studying the Importance of Dynamic Nodes. We validate our assumption that the nodes should be dynamic, meaning that their regions position and size should be adapted according to the input at each time step. We investigate (Table. 5) different types of localized nodes, each adapting to the input to a varying degree, and show the benefits of our design choices. We experiment with variants of our model, all having the same backbone (2D ResNet-18 [85]), the same graph processing and same pre-determined number of regions, but we constrain the node regions in different ways.
Fixed Model extracts node features from regions arranged on a grid, with a fix location and size.
Static Model investigates the importance of dynamic regions by optimising regions based on the whole dataset but do not take into account the current input. Effectively, the features z i from Eq. 4 become learnable parameters.
Constant-Time Model has regions adapted to the current video but they do not change in time.
DyReg-GNN Model predicts regions defined by location and size, and we can either pre-determine a fixed size for all the regions (Position-Only Model) or directly predict it from the input as in our complete model (DyReg-GNN Model).
These experiments ( Table 5), show that the fixed region approach (Fixed Model) achieves the worst results, slightly improving when the regions are allowed to change according to the learned statistics of the dataset (Static model). Adapting to the input is shown to be beneficial, the performance improving even when the regions are invariant in time (Constant-Time Model), and further more when predicting different regions at every time steps (Position-Only). The best performance is achieved when both the location and the size of the regions are dynamically predicted from the input (DyReg-GNN).
In Figure 2 we show examples of the kernels obtained for each of these models. We observe that the Static Model's kernels are learned to be arranged uniformly on a grid, to cover all possible movements in the scene, while the Constant-Time Model's kernels are adapted for each video such that they cover the main area where the digits move in the current video. The full DyReg-GNN Model learns to reduce the size of its regions and we observe that they closely follow the movement of the digits.
The previous experiments show that performance increases when the model becomes more dynamic, proving that our model benefits from nodes that are adapted to a higher degree to the current input. We show (1st row) the center of all the N regions as predicted by DyReg-GNN (each color for a node). Each node region (last 2 rows) corresponds to a zone from the latent conv features pooled by a node. Studying the Importance of Localized Nodes. We argue that nodes should pool information from different locations according to the input, such that the extracted features correspond to meaningful entities. Depending on the goal, we could balance between semantic nodes globally extracted from all spatial positions or localized (spatial) nodes that are obtained from well-delimited regions.
Semantic Model creates nodes similar to [4,19] where each node extracts features from all the spatial locations and could represent a semantic concept. Each node is extracted by a global average pooling where the weights at every position p are directly predicted from the input features at that location. Practically, we replace the spatially delimited kernel used in our model with this global attention map.
A major downside of this approach is that it does not distinguish between positions with the same features, making it harder to reason about different instances. Figure 2.C shows the attention map of a single node and we observe that it has equally high activations for both instances of the same digit, thus making it hard to distinguish between them.
This limitation does not exist in our DyReg-GNN model, as it predicts localized nodes that favour the modeling of instances. For comparison, we use two variants with a different number of parameters and show that they clearly outperform the semantic model (Table 6). These experiments prove that in cases that involve spatial reasoning of entities, such as the current task, DyReg-GNN is a perfect choice, showing its benefits for spatio-temporal modeling.
Implementation details. All models share the ResNet-18 backbone with 3 stages, where the graph receives the features from the second stage and sends its output to the third stage. We use N = 9 graph nodes and repeat the graph propagation for three iterations. In our main model, f from Eq. 1 is a small convolutional network while g is a fully connected layer. For the lighter model that implements g as a global pooling enriched with spatial positional information, we refer to the Supp. Materials. The graph offsets are initialized such that all the nodes' regions start in the center of the frame. In all experiments, we use SGD optimizer with learning rate 0.001 and momentum 0.9, trained on a single GPU.
Key Results. In the previous section, we experimentally validated that: 1. DyReg-GNN consistently improves multiple backbones ( (Table 5) and 3. these regions are preferable to fixed regions or external object detectors for space-time GNNs (Table 1); 4. predicted nodes correspond to salient regions ( Fig. 2-3) and are well correlated with objects (Table 1).
Conclusions
We propose Dynamic Salient Regions Graph Neural Networks (DyReg-GNN), a relational model for processing spatio-temporal data (videos), that augments visual GNNs by learning to predict localized nodes, adapted for the current scene. This novel method enhances the relational processing of spatio-temporal GNNs and we experimentally prove that it is superior to having nodes anchored in fixed predefined regions or linked to external pre-trained object detectors. Although we do not use region level supervision, the learning dynamics of high-level classification produces salient regions that are well correlated with object instances. We believe that our method of learning dynamic, localized nodes is a valuable direction that could lead to further advances to the growing number of powerful relational models in spatio-temporal domains.
Appendix: Discovering Dynamic Salient Regions for Spatio-Temporal Graph Neural Networks
In this Appendix we present an impact statement, discuss some limitations of the method and then we provide more technical details about DyReg-GNN model and include some additional visualisations and ablation studies.
Section A presents some views on the broader impact of this work.
Section B identifies some limitations of the methods.
Section C presents more details about how the regions are generated.
Section D shows a qualitative analysis of the regions predicted by our model. Section E shows additional ablation studies in the synthetic setting in relation to the number of nodes, the regions size, the importance of recurrence when generating the nodes and comparisons to using ground-truth boxes or other baselines.
Section F presents some training details, describe the metric used to measure the correlation between our regions and the existing objects in the scene and have a runtime analysis of our proposed module.
We provide our full code as supplementary material and we will release it online upon the paper publication. Beside this Appendix, we also provide some videos, visualising the regions discovered by our DyReg-GNN model.
A Broader Impact
We research novel methods that would improve current general models for spatio-temporal processing.
Our goal is to investigate models that emphasize a small number of relevant nodes having the potential to be more explainable and that could lead to more interpretable reasoning. Although this is not fully realised in this paper, we believe that this work is a good step in this direction. Our model enhances any convolutional backbone for video processing and thus inherits the benefits and also the possible harms brought by such models.
When developing our model, we used a synthetic dataset of moving digits and a public dataset for human-object interactions. Our model is kept generic, with no parts specially designed for these tasks. The models trained on these datasets have no obvious direct real application, as the first one is a toy dataset and the second one has restrictive classes meant only to evaluate the capabilities of the models. But developing better models for video understanding leads to more effective applications. On one hand, it could lead to better applications helping visually impaired people navigate the world and on the other hand it could lead to stricter automatic surveillance of workers. In order for ML technology to have a positive broader impact, more discussions between different actors in society should be conducted leading to the development of guidelines and practices.
The proposed work does not rely on using object detectors and only uses video level supervision. Object detectors have a predefined list of objects, that would not be sufficient for many practical cases leading to biases in the system. Moreover, this way we eliminate a possible source of biases coming from the object-level annotations.
B Limitations
By design, DyReg-GNN uses a fixed number of nodes, that we treat as a hyperparameter. This way the model is forced to produce the same number of regions regardless of the complexity of the scene. From simpler scenes, the model learns to group the nodes in overlapping regions, creating redundancy. On the other hand, more complex scenes have an increased number of relevant regions, tending to require distinct regions. This could lead to a discrepancy that would increase the difficulty of the optimisation process. Changes in scene's complexity could be also observed in a single video when the scene suffer major changes in time. For example when elements appear, disappear or are occluded from view, the number of regions predicted by the model remains the same and it is harder to properly model all the elements. Ideally, we want a system that adapts to the complexity of the scene by dynamically predicting the number of nodes. This is a challenging task that requires additional investigations and we leave it for future work.
Preliminary experiments reveals that our method requires a relative large amounts of data to be properly trained. This seems not to be an issue for Something-Something dataset that has 80k-160k training videos, but could be an issue for smaller datasets. On MultiSyncMNIST we could train models with high accuracy on 10% of the whole dataset of 600k videos. But when using only 1% (6k videos) of the data, the predicted regions would not change during training. Given the size of the recent video dataset, this is not a big limitation.
C Node Region Generation
The goal of this sub-module is to generate the regions that correspond to salient zones in the input. We achieve this by processing the input globally with position-aware functions f and {g i }.
Function f . We use f function to aggregate local information from larger regions in the input while preserving sufficient positional information. The input X t ∈ R H ×W ×C is first projected into a lower dimension C since this representation should only encode saliency without the need to precisely model visual elements. Then we increase the receptive field by applying two conv layers, followed by a transposed conv and then a final conv layer. This results in a feature map M t = f (X t ) ∈ R H ×W ×C . Depending on the backbone and the stage where the graph is added H, W have different values and we adapt the hyperparameters of the convolutional layers such that H and W are not smaller than 6. For example, in the synthetic experiments f reduces the input from R 16×16×32 to R 7×7×16 .
Functions {g i }. For each node i we use g i to extract a global latent representation from which we predict the corresponding region parameters. We present two variant of g i function, a larger and more precise one and a smaller, more computational efficient one.
For the bigger one, we use a simple fully connected layer of size C × (H * W * C ) that takes the whole M t and produces a vector of size C. This way g i could distinguish and model the spatial locations of the H × W grid.
The second approach consists in a weighted global average pooling for each node i. The weight associated to each location p is predicted directly from the input M t,p by a 1 × 1 convolution. But this results in a translation-invariant function g i that losses the location information. We alleviate this by adding to each of the H × W location a positional embedding similar to the one used in [31]. This approach predicts regions of slightly poorer quality as the location information is not perfectly encoded in the positional embeddings. For a lighter model, such as the one presented in Table 6 of the main paper we could use the second approach for the {g i } functions and also skip the f processing.
Constraints Equation 4 in the main paper could be expended as:
o i = (∆x i , ∆ỹ i ,w i ,h i ) = γ W o z i ∈ R 4 (11) o i = α(õ i )
To constrain the model to predict valid image regions and also to start from regions with favourable position and size, we apply non-linear functions for each component o i = α(õ i ) . We design the non-linearities such that w i , h i > 0 and ∆x i + C x ∈ [0, W ] and , ∆y i + C y ∈ [0, H], where C is a fixed reference point. In experiments, all nodes share the same constant C , representing the center of the image. In these visualisations the order of the regions is manually selected. In the left column we show the regions at initialisation, and in the right column we present the mean regions as predicted by our learned DyReg-GNN model. Here, we keep the size of the regions fixed, each node has a preferred location in space and assigns salient regions around it. This behaviour is learned by the model to break the symmetry of the nodes.
h = ehh init w = eww init(12)∆x = W 2 tanh ∆x + arctanh ( 2C x W − 1) + W 2 − C x(13)∆y = H 2 tanh ∆ỹ + arctanh( 2C y H − 1) + H 2 − C y(14)
By initialising γ = 0 we obtain h = h init , w = w init and ∆y = ∆x = 0. This means that all regions are initialized centered in the reference point C and start with the predefined size. By default we set h init = H 6 , w init = W 6 .
D Visualising the nodes' regions
The region associated with each node is clearly delimited in space and we can easily visualize them. We train a model on Something-Something-V2 dataset of human-object interactions and in Figure.4 we show its predicted nodes' regions for two videos from the dataset and one out-of-distribution video. Generally the nodes follow relevant regions in the input. We note that the visualisation of the regions is only an approximation of the actual regions that send information to the graph nodes. Each node pools info from a low-resolution region in the latent convolutional features, that corresponds to the high-resolution visualized region. But, the actual area that contributes to each node is actually larger, due to the receptive field of the convolutional network. Moreover, the backbone also contains temporal processing (e.g. in the form of temporal feature shifting in the case of TSM or 3D conv for I3D) such that each node receives information from adjacent time steps. Thus, we expect some misalignment in the visualizations both in space and time.
To better understand how each node attends to the input, we compute the average of its associated regions over the entire evaluation dataset (see Figure. 5). We observe that the regions are initialized in the center of the image and, after training, each node learns to attend to regions around a specific location. For each video, a node predicts a different region, according to the input, but it is situated mostly around a certain part of the image. This behaviour is learned by the model to break the symmetry of the nodes and be able to create an implicit matching between relevant parts of the input and the nodes.
E.2 Ablation: Number of Nodes
We investigate the effect on the performance of the number of nodes for different environments, of varying difficulty. We conduct experiments varying the complexity of MultiSyncMNIST dataset, by changing the number of moving digit (D ∈ {3, 5, 9}). As expected, we observe (in Table 7) that for good performance, it is necessary to set a number of nodes that exceeds the number of relevant entities in the scene.
E.3 Ablation: Regions' Size
In this subsection, we conduct experiments to investigate the effect of the size of the node regions on the final performance. Each node pools information from latent convolutional features of size H × W = 16 × 16. We fix the size of each region to H λ where H = 16 and λ ∈ {6, 7, 8, 11, 16} and show the results of the corresponding models in Table 11. Setting λ = 8 corresponds to regions having approximately the expected values of the regions predicted by the full DyReg-GNN model. We note that the model is relatively robust to reasonable choices of size but the best performance is achieved when the size of each region is dynamically predicted from the input. We also note that by setting λ = H = 16 we arrive at the standard bilinear interpolation kernel. This setting leads us to a model that is more unstable in training than the others and obtains poorer results. There are two probable reasons for this. First, the regions cover a small area thus they must be more precise to cover small entities while also being unable to cover large entities in their entirely. Second, the gradients used to update the region parameters are noisier for small regions. This is because, the gradients of the offsets depend on the features of the predicted regions, and for gradients of the offsets to be informative it means that the features in the regions should also be relevant for the final prediction. Smaller regions have a smaller chance of achieving this.
E.4 Ablation: Comparison to Keypoints Extractor
We conduct an experiment to compare our dynamic way of generating nodes with a previous method [55] that detects keypoints from images. For a fair comparison, we replace the part in our model that predicts the locations of the node regions with a method similar their encoder. The rest of our model would remain the same and will be learned in the same way by the video classification loss. We chose the architecture such that the number of parameters remains the same. As in the original paper, this method predicts the positions but keeps the shape fixed, thus we compare it with our module that has fixed shape, although it obtains poorer results than the full model. We report the results in Table 10. Note that the dynamic regions obtained using the keypoints method (denoted as Keypoints) improve over the fixed-regions approach, reinforcing the idea that dynamic regions are helpful for relational processing. However, our DyReg-GNN models obtain better results both when the size of the regions is fixed and especially when the size is also predicted.
E.5 Ablation: Importance of Recurrence for Region Generation
We conduct an experiment (Table 9) on the MultiSyncMNIST dataset, where we omit the GRU from Eq. 3, thus predicting the regions at each time step only from the features of frame. The performance drops from (DyReg-GNN) to (DyReg-GNN without GRU). This experiment suggests that the temporal modeling in region generation is important for good performance.
E.6 Ablation: Ground-Truth Boxes
To evaluate the quality of our proposed regions on MultiSyncMNIST, we train our model using ground-truth (gt.) boxes instead of generated regions. As this task is defined by the exact movements of digits, the gt. boxes represents the ideal regions for the relational model, giving an upper bound for our method. This oracle model obtains 97.30% accuracy, while the DyReg-GNN model obtains 95.09%. Comparing to the other baselines in the main paper, our DyReg-GNN model obtains closer results to the oracle model, proving the utility of the node generation.
E.7 Comparison to other baselines
We compare our method to additional baselines by replacing our entire DyReg-GNN module with two other models, as seen in Table 7. The first baseline (R18+Conv-LSTM) consists in a convolutional encoder that reduces the spatial dimensions, a shared LSTM applied independently on each spatial position followed by a convolutional decoder. The second baseline (R18-NL) consists in a Non-Local [30] network. Both modules are applied over the same ResNet 18 backbone and have the same number of parameters as DyReg-GNN. DyReg-GNN surpasses the other baselines and the difference in performance is more significant in the hardest setting. When training models with multiple DyReg-GNN modules, we observe that only the regions of the last module behaves well, thus a single graph module is effectively used. To alleviate this problem we train models with a single graph at different stages, and use their region predictions to distill the larger model, for the first 10% of the training iterations. This kick-starts the learning of all graph modules, improving the overall results, as seen in Table 8.
F.2 Implementation Details for Detector Experiment
In the main paper, for the comparison with the regions extracted using object detectors (Section 4.1), we use a Faster RCNN ResNet-50 FPN detector 3 pre-trained on MSCOCO dataset. We extract the top-9 detected boxes based on the confidence score and temporally match them using the hungarian algorithm to maximize the IoU between boxes at consecutive time steps.
F.3 Object-centric metric
To quantify to what degree the nodes cover existing ground-truth objects in the scene, we propose the following metric. We measure the distance between the center of the predicted regions and the center of the gt. objects. For each node region in each frame, we compute the minimum L 2 distance to all gt. object bounding boxes and average all of them.
Dist p = 1 N F F f =1 N i=1 min j |C i + ∆ i − B j | 2(15)
Vice versa we compute for each gt. box the minimum L 2 distance to all predicted regions and average all of them.
Dist r = 1 N B F F f =1 N B j=1 min i |C i + ∆ i − B j | 2(16)
In the previous equations, F is the number of frames in the whole dataset, N the number of nodes, N B the number of objects in the current frame, C i + ∆ i is the center of i-th node's region and B j the center of the j-th object in the current frame and we average over the whole dataset.
The first score (representing precision) ensures that all the predicted regions are close to real objects, while the second (recall) ensures that all the objects are close to at least one predicted region. To balance them, we present as our final score their harmonic mean.
F.4 Runtime Analysis
We compute the number of operations, measured in FLOPS, the parameters and the inference time for our model. We evaluate videos of size 224 × 224 in batches of 16 on a single NVIDIA GTX 1080 Ti GPU. TSM backbone, TSM + DyReg-GNN-r4, and DyReg-GNN-r3-4-5 run at 35.7, 34.8, and 32.7 videos per second respectively, showing that our DyReg-GNN module does not add a large overhead over the backbone. In Table 12, we compare in terms of number of parameters and operations against other current standard models used in video processing. Note that the I3D-based models uses 32 frames but for our method, the number of operations increases linearly with the number of frames so it is easy to make a fair comparison. The I3D+NL+GCN model counts also the parameters and the operations of the detector module used to extract object boxes. This is characteristic to all the relational models where the nodes are extracted using object detectors. Contrary to this approach, our method has a smaller total complexity by directly predicting salient regions instead of using precise object proposals given by external models.
Figure 2 :
2Nodes' regions on MultiSyncMNIST for 3 frames. a) Static Model, ignoring the input, learns a regular grid; b) Constant-Time predicts the same regions for all time steps, covering the movement in the video; c)The attention map of a single node of Semantic that can't distinguish between different instances of the same digit; d) DyReg-GNN generally follows the digits locations at each time steps while also adapting the regions' size.
Figure 3 :
3Nodes' regions on Smt-Smt-V2.
Figure 4 :Figure 5 :
45Visualisations of salient regions associated with each node, as predicted by our DyReg-GNN model on videos from Smt-Smt-v2 dataset (Left and Center) and on out-of-distribution realworld videos (Right). Each node learns to move to different relevant regions in the input. For each video, we show the centers corresponding to all the nodes and, for a better visualisation, a subset of the predicted regions. Due to the receptive field of the backbone, the nodes are actually influenced by larger regions in the initial input. Visualisation of the average locations of the salient regions associated with DyReg-GNN's nodes, computed over the validation set of (a) MultiSyncMNIST and (b) Smt-Smt-v2.
on[3] we create MultiSyncMNIST. It consists of 10 frames videos of size 128 × 128, where MNIST digits move on a black background. Each video has 5 moving digits and a subset of them moves synchronously. Different from the original version, each video could contain multiple instances of the same digit class and any subset can move in the same way. This is done to make it more difficult to distinguish between multiple visual instances. The goal is to detect the smallest and largest digit class among the subset of synchronous digits with each pair of two digits forming a label. In total, we have 55 possible pairs of two digits, and adding a class for videos without synchronous digits results in a 56-way classification task. For example, if a video contains the digits: {2, 4, 6, 7, 7} and the subset {4, 6, 7} is moving in the same way, it has the label associated with the pair: {4, 7}. The dataset contains 600k training videos and 10k validation videos.
Table 2 :
2Consistentimprovements
over different backbones on the vali-
dation set of Smt-Smt-V1 using central
crop evaluation.
Model
Acc (%)
TSM-R18
33.7
TSM-R18 + DyReg-GNN 35.6 (↑ 1.9)
I3D-R50
44.0
I3D-R50 + DyReg-GNN
45.4 (↑ 1.4)
TSM-R50
47.2
TSM-R50 + DyReg-GNN 48.8 (↑ 1.6)
Table 3 :
3Results on val. set of Smt-Smt-V1.Our
Table 4 :
4Results on val. set of Smt-Smt-V2., size 224 × 224 would add 39.7 GFLOPS on its own, comparing to the 1.6G of three DyReg-GNN modules, from which 0.2G represents the regions prediction.in comparisons to recent works. DyReg-GNN
improves the TSM-ResNet50 backbone when
using either one (r4) or three (r3-4-5) modules
of graph processing and it obtains top results.
Model
BB
Top 1 Top 5
TRG [82]
R50
59.8
87.4
GST [78]
R50
62.6
87.9
v-DP [83]
D121 62.9
88.0
SmallBig [79]
R50
63.8
88.9
STM [80]
R50
64.2
89.8
MSNet [81]
R50
64.7
89.4
TSM [67]
R50
63.4
88.5
TSM+DyReg-r4
R50
64.3
88.9
TSM+DyReg-r3-4-5 R50
64.8
89.4
Table 5 :
5Ablationof dynamic nodes on MultiSyncM-
NIST. It is crucial to have regions that adapt based on the
input (Dynamic), both their position (Pos.) and size at
each time step.
Model
Optimise
Time
Dynamic
Acc
Pos.
Varying Pos. Size
Fixed
78.85
Static
81.48
Ct-Time
86.77
Pos-Only
93.41
DyReg-GNN
95.09
Table 6 :
6Semanticvs spatial nodes on
MultiSyncMNIST. The localized (spa-
tial) node regions of DyReg-GNN are
better suited than semantic nodes' maps
obtained by the Semantic Model.
Model
Params (M)
Acc
ResNet-18
2.79
52.29
Fixed
2.82
78.85
Semantic
2.85
82.41
DyReg-GNN-Lite
2.83
91.43
DyReg-GNN
3.08
95.09
Table 2 )
2obtaining competitive results (Table 3, 4); 2. learned dynamic
Table 7 :
7Results on MultiSyncMNIST when varying the number of nodes on datasets of different complexity (with increasing number of moving digits). It is crucial that the number of nodes exceeds the number of important entities in the scene. In the bottom of the table, we also show two additional baselines with the same number of parameters.Model
Dataset
# digits(D)
D=3 D=5 D=9
DyReg 5 Nodes
98.2 89.6 64.4
DyReg 9 Nodes
98.1 95.1 79.3
DyReg 16 Nodes
98.4 95.6 83.0
R18 + Conv-LSTM -
89.1 50.5
R18 + NL
-
93.2 67.5
Table 8 :
8Results on Smt-Smt-V2 val. set, using a single 224 × 224 central crop. We observe that DyReg-GNN models improve over the TSM backbone and that it is important to have the kickstart given by the distillation to learn multiple dynamic graph modules.Model
Top 1 Top 5
TSM
61.1
86.5
DyReg-GNN r3-4-5
62.1
87.4
DyReg-GNN r3-4-5 Distill 62.8
87.7
Table 9 :
9Ablation on MultiSyncMNIST for showing the importance of recurrence for predicting the regions.Model
Accuracy
DyReg-GNN without GRU
91.91
DyReg-GNN
95.09
Table 10 :
10Comparison to Keypoints-based method on MultiSyncMNIST.Model
Acc
ResNet-18
52.29
Fixed
78.85
Keypoints
90.60
DyReg-GNN -Pos-Only 93.41
DyReg-GNN
95.09
Table 11 :
11Experiments on MultiSyncMNIST investigating the size of the learned regions. The best performance is obtained when the size is dynamically predicted while the worst is given by a model with the regions kept at the minimum value, corresponding to the standard bilinear interpolation kernel.Table 12: Comparison in terms of the number of operations and parameters for a single video of size 224 × 224. Comparing to assigning nodes to boxes from external detectors (as in I3D+NL+GCN), our module has a smaller computational overhead. F Human-Object Interactions F.1 Distillation for kick-starting the optimisationLearnable
(Full)
Fix
λ = 6
Fix
λ = 7
Fix
λ = 8
Fix
λ = 11
Fix
bilinear
95.09
93.41 94.11 94.04
94.03
90.99
Model
Frames FLOPS Params
I3D [63]
32
153.0G 28.0M
I3D+NL [30]
32
168.0G 35.3M
I3D+NL+GCN [7]
32
303.0G 62.2M
STM [80]
16
66.5G
24.0M
TSM [67]
16
65.8G
23.9M
TSM + Fixed GNN r4
16
66.3G
24.9M
TSM + DyReg-GNN r4
16
66.4G
25.7M
TSM + Fixed r3-4-5
16
67.2G
26.1M
TSM + DyReg-GNN r3-4-5
16
67.4G
28.7M
https://github.com/bit-ml/DyReg-GNN
https://github.com/facebookresearch/detectron2
Acknowledgment We would like to thank Florin Brad, Elena Burceanu and Florin Gogianu for their valuable feedback and discussions of this work. This work has been supported in part by Bitdefender and UEFISCDI, through projects EEA-RO-2018-0496 and PN-III-P4-ID-PCE-2020-2819.
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[
"Isospin violating dark matter in Stückelberg portal scenarios",
"Isospin violating dark matter in Stückelberg portal scenarios"
] | [
"Víctor Martín Lozano \nInstituto de Física Teórica UAM/CSIC\nUniversidad Autónoma de Madrid\nE-28049Cantoblanco, MadridSpain\n\nDepartamento de Física Teórica\nUniversidad Autónoma de Madrid\nE-28049Cantoblanco, MadridSpain\n",
"Miguel Peiró \nInstituto de Física Teórica UAM/CSIC\nUniversidad Autónoma de Madrid\nE-28049Cantoblanco, MadridSpain\n\nDepartamento de Física Teórica\nUniversidad Autónoma de Madrid\nE-28049Cantoblanco, MadridSpain\n",
"Pablo Soler \nDepartment of Physics\nUniversity of Wisconsin\n1150 University Avenue53706MadisonWIUSA\n"
] | [
"Instituto de Física Teórica UAM/CSIC\nUniversidad Autónoma de Madrid\nE-28049Cantoblanco, MadridSpain",
"Departamento de Física Teórica\nUniversidad Autónoma de Madrid\nE-28049Cantoblanco, MadridSpain",
"Instituto de Física Teórica UAM/CSIC\nUniversidad Autónoma de Madrid\nE-28049Cantoblanco, MadridSpain",
"Departamento de Física Teórica\nUniversidad Autónoma de Madrid\nE-28049Cantoblanco, MadridSpain",
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] | [] | Hidden sector scenarios in which dark matter (DM) interacts with the Standard Model matter fields through the exchange of massive Z bosons are well motivated by certain string theory constructions. In this work, we thoroughly study the phenomenological aspects of such scenarios and find that they present a clear and testable consequence for direct DM searches. We show that such string motivated Stückelberg portals naturally lead to isospin violating interactions of DM particles with nuclei. We find that the relations between the DM coupling to neutrons and protons for both, spin-independent (fn/fp) and spin-dependent (an/ap) interactions, are very flexible depending on the charges of the quarks under the extra U (1) gauge groups. We show that within this construction these ratios are generically different from ±1 (i.e. different couplings to protons and neutrons) leading to a potentially measurable distinction from other popular portals. Finally, we incorporate bounds from searches for dijet and dilepton resonances at the LHC as well as LUX bounds on the elastic scattering of DM off nucleons to determine the experimentally allowed values of fn/fp and an/ap. | 10.1007/jhep04(2015)175 | [
"https://arxiv.org/pdf/1503.01780v1.pdf"
] | 51,694,176 | 1503.01780 | 019bf7b75517c0406d13843eba7bfee688fbc454 |
Isospin violating dark matter in Stückelberg portal scenarios
Víctor Martín Lozano
Instituto de Física Teórica UAM/CSIC
Universidad Autónoma de Madrid
E-28049Cantoblanco, MadridSpain
Departamento de Física Teórica
Universidad Autónoma de Madrid
E-28049Cantoblanco, MadridSpain
Miguel Peiró
Instituto de Física Teórica UAM/CSIC
Universidad Autónoma de Madrid
E-28049Cantoblanco, MadridSpain
Departamento de Física Teórica
Universidad Autónoma de Madrid
E-28049Cantoblanco, MadridSpain
Pablo Soler
Department of Physics
University of Wisconsin
1150 University Avenue53706MadisonWIUSA
Isospin violating dark matter in Stückelberg portal scenarios
Hidden sector scenarios in which dark matter (DM) interacts with the Standard Model matter fields through the exchange of massive Z bosons are well motivated by certain string theory constructions. In this work, we thoroughly study the phenomenological aspects of such scenarios and find that they present a clear and testable consequence for direct DM searches. We show that such string motivated Stückelberg portals naturally lead to isospin violating interactions of DM particles with nuclei. We find that the relations between the DM coupling to neutrons and protons for both, spin-independent (fn/fp) and spin-dependent (an/ap) interactions, are very flexible depending on the charges of the quarks under the extra U (1) gauge groups. We show that within this construction these ratios are generically different from ±1 (i.e. different couplings to protons and neutrons) leading to a potentially measurable distinction from other popular portals. Finally, we incorporate bounds from searches for dijet and dilepton resonances at the LHC as well as LUX bounds on the elastic scattering of DM off nucleons to determine the experimentally allowed values of fn/fp and an/ap.
Introduction
Uncovering the properties of Dark Matter (DM) and, in particular, its possible non gravitational interactions with visible matter is one of the greatest challenges of modern physics, and is accordingly the object of important experimental and theoretical efforts.
A common theoretical framework for DM studies is the hidden sector scenario. In its minimal form, visible matter resides in a sector of the theory that hosts the Standard Model (SM) gauge and matter content (or simple extensions thereof), while DM resides in a hidden sector, with its own gauge and matter content, but is otherwise neutral under the SM group.
Within such a framework, several mechanisms have been proposed to mediate non-gravitational interactions between the different sectors, usually referred to as portals [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Among them, perhaps the most popular is the Higgs portal [1] in which the SM Higgs boson has renormalizable couplings to scalar fields of the hidden sector. This kind of construction leads to important phenomenological consequences such as the contribution of hidden final states to the branching fraction of the Higgs [17,18]. Another popular kind of portal is the Z portal. In this scenario, a hidden sector communicates with the SM via a gauge boson, provided that the SM is enlarged with an extra abelian gauge group [19]. The phenomenology of these constructions is very rich, and ranges from colliders to direct and indirect searches of DM [20][21][22][23][24][25][26].
Particularly important for direct detection experiments are the isospin violation properties of the interactions of DM particles with nuclei induced by different portals. For example, the Higgs portal (at least in its simplest form) automatically predicts isospin preserving interactions. A similar rigidity in the pattern of isospin interactions is present in many other portals. Previous works have shown that a deviation from these patterns would require the presence of several mediators whose contributions to the cross section interfere, and hence, generate an amount of isospin violation potentially tunable [22,[27][28][29].
It is the purpose of this work to show that, in contrast, U (1) extensions of the SM with Stückelberg Z bosons acting as portals naturally accommodate rich patterns of isospin violating interactions. The latter, in turn, provide a clear and testable phenomenological consequence of such models. Extra abelian gauge factors are among the most common extensions of the SM [19], and also among the best motivated from string theory, where massive extra U (1) gauge bosons appear ubiquitously (for reviews see e.g. [30][31][32][33][34][35]). In fact, when one tries to implement a visible sector with the SM gauge group in, say, intersecting brane models, one generically obtains not SU (3) c × SU (2) L × U (1) Y , but rather U (3) c × U (2) L × U (1) p which contains several extra abelian factors (including the centers of U (3) c and U (2) L ).
The models we will consider along this work are based on this type of string constructions. The symmetry structure of this scenario can be represented schematically in the following form,
SU (3) c ×SU (2) L ×U (1) n v × U (1) m h ×G h (1.1) Ψv Ψ h
where the U (1) n v are n abelian gauge factors to which the visible matter fields Ψ v couple. All of the corresponding gauge bosons acquire a mass through the Stückelberg mechanism, except for a particular linear combination of them that corresponds to hypercharge and remains massless (in the phase of unbroken electroweak symmetry). U (1) m h are m abelian gauge factors (some of which could be massless) to which only hidden matter Ψ h couples, and G h represents the semi-simple part of the hidden gauge group.
As mentioned, these type of scenarios can fairly easily be implemented in models of intersecting D6-brane of type IIA string theory. Intuitively, each sector consists of several intersecting stacks of branes wrapping 3-cycles of a six-dimensional compactification space. Each stack hosts a U (N ) gauge factor and chiral matter arises at the brane intersections. Different sectors arise from brane stacks that do not intersect each other and can hence be separated in the internal space (see Fig. 1).
The extra abelian gauge bosons of Eq. (1.1) can provide a portal into the hidden sector in two different ways. The most thoroughly studied is a small kinetic mixing of a light hidden gauge boson with the visible massless photon [36][37][38][39][40][41][42][43][44]. This generates an effective coupling of DM with visible fields which is proportional to their electric charge, and hence, the DM particles only couple to protons and do not couple to neutrons. From the point of view of direct DM searches, this is very important, since the elastic scattering of DM off nucleons only receives contributions from protons. This is, the ratio between the coupling of DM to neutrons and protons vanishes, f n /f p = 0.
} } Compact 6d space Minkowski 4d q A v A h q φ ψ ψ q q A h A v
The second mechanism, which will be the main subject of this paper, was pointed out in Refs. [8,9] (see also [45][46][47][48]). It results from the mixing of massive U (1)s of the visible sector with U (1)s of the hidden one. Despite living in different sectors, the U (1) gauge bosons A n v and A m h often have Stückelberg couplings to the same axions, e.g. RR closed string axions in type II string models. As a consequence the resulting mass matrix can be highly non-diagonal. The 'physical' Z eigenstates obtained upon diagonalization of the kinetic and mass matrices are largely mixed combinations of A n v and A m h and hence couple simultaneously to both, visible and hidden, matter sectors. This mass mixing is a tree-level effect that provides an effective portal into hidden sectors, provided the associated Z bosons are light enough.
Despite the potentially complex gauge and matter structure of the hidden sector, it seems reasonable to assume that it hosts a Dirac fermion ψ that plays the role of DM in the Universe. The stability of these particles is easily guaranteed by the perturbatively conserved U (1) m h symmetries or by non-perturbatively exact discrete subgroups thereof, or simply because they are the lightest particles of the whole sector. In any case, their interaction with the SM fermions will be driven by the exchange of a Z boson. For DM direct detection experiments the leading interaction of the elastic scattering of ψ with quarks is depicted in the right panel of Fig. 1. Following this reasoning, it is clear that the charges of the SM fermions under the U (1) n v groups that mix with the hidden sector will determine the prospects for detecting ψ in these experiments.
In this work we study the phenomenology of a class of scenarios of this kind that can be embeded into well known string theory constructions. In particular, we focus on the isospin violation character of the DM interactions with protons and neutrons induced by the Z bosons. As we will see, isospin violation could distinguish these Stückelberg portal models from other popular setups, such as the Higgs portal or the Z-mediation scenarios. This work is organized as follows. In section 2 we review the general theoretical framework that underlies our models. We describe the mixing mechanism that generates an off-diagonal mass matrix for the U (1) gauge bosons and study general properties the eigenstates of such matrix, which are the physical Z fields that communicate the hidden and the visible sectors. We also discuss how the general form of the effective Lagrangian arises from certain string compactifications with intersecting D6-branes. In section 3 we take a well known class of such string models and determine the SM couplings to the lightest Z mediator in terms of a few mixing parameters. In section 4 we study the isospin violation properties of the DM-nucleon interactions in these constructions in terms of these parameters, and compare them to those arising in other popular scenarios. In section 5 we incorporate to our analysis bounds from direct detection (LUX) and collider searches (LHC) for six benchmark points in the parameter space of the model. Finally, we give some concluding remarks in section 6.
Effective Lagrangian and Z eigenstates
In this section we review the general constructions of Refs. [8,9] which describe the mixing mechanism of massive U (1) gauge bosons from different sectors, the so-called Stückelberg portal. We begin with a discussion in terms of the effective field theory, and describe later on the string implementation of such setup.
Non-diagonal U (1) mass matrix
The abelian sector of the construction sketched in Eq. (1.1) can generically be described by the Lagrangian
L = − 1 4 F T · f · F − 1 2 A T · M 2 · A + α ψ α i∂ / + Q T α · A / ψ α (2.2)
where the vector A T = (A 1 . . . A n+m ) encodes all the U (1) gauge bosons of the system, with field strength F = d A. In this normalization the gauge coupling constants are absorbed in the kinetic matrix f . In hidden sector scenarios, the charge vectors Q α of a given matter field ψ α will have nonzero entries only for one of the sectors (either visible or hidden), while the kinetic and mass matrices f and M can have off-diagonal entries that mix both sectors. We are interested in particular in the mixings induced by the mass matrix M .
The mass terms for Abelian gauge bosons A can be generated by either the Higgs or the Stückelberg mechanisms. In both cases the crucial term in the Lagrangian is the coupling of A to a set of pseudoscalar periodic fields φ i ∼ φ i + 2π whose covariant kinetic terms read
L M = − 1 2 G ij ∂φ i − k i a A a ∂φ i − k j b A b . (2.3)
Here, G ij corresponds to a positive-definite kinetic matrix (the metric in the space of φ i fields), which in our conventions has dimension of (mass) 2 . The factors k i a encode the non-linear gauge transformations
A → A + d Λ =⇒ φ i → φ i + k i a Λ a . (2.4)
The statement that the gauge symmetry group is compact (U (1) rather than R) implies that the transformations must be periodic Λ ∼ Λ + 2π, and hence that the k i a factors (as well as the matter charges Q α of Eq. (2.2)) must be quantized. In fact, under the appropriate normalization they can be assumed to be integers, k i a , Q a α ⊂ Z. In the case of the Higgs mechanism, the axion-like fields φ i are identified with the phases of Higgs fields H i , and the k i a factors are simply the charges of the latter under the U (1) a groups. It is not surprising that these are integer quantities. What is perhaps less obvious is that even for axions not related to a Higgs fields, one can still associate U (1) integer "charges" that determine their gauge transformations.
The U (1) gauge bosons get a mass by absorbing the axions, φ i , through the Lagrangian (2.3). After gauge fixing the U (1) symmetries, the mass term of equation (2.2) is generated, and the corresponding mass matrix takes the form
M 2 = K T · G · K . (2.5)
It is easy to see that this matrix can be highly non-diagonal and have off-diagonal entries that mix hidden and visible sectors. This can happen with particular strength if the mixing is induced by the integer matrix K of axionic charges.
The dynamical origin of the mixing is the simultaneous coupling of vector bosons from different sectors to the same axions (see figure 1). As a toy model, consider two U (1) gauge bosons, a visible A v and a hidden A h , that couple to an axion with charges +1 and -1, respectively, i.e. K = +1 −1 , whose kinetic matrix is G = m 2 . The resulting mass matrix of the U (1) bosons would read M 2 = m 2 1 −1 −1 1 . The resulting physical eigenstates are obviously highly mixed combinations of A v and A h that hence couple with similar strength to both sectors.
In the following we generalize this simple example to the case where several gauge bosons mix with each other by absorbing several axions with mixed charges.
Diagonalization and eigenstates
In order to study the properties of the system described by the Lagrangian of Eq. (2.2), it is convenient to move to a basis in which the gauge bosons have a canonical kinetic term and a diagonal mass matrix. The former can be obtained by a linear transformation:
A ≡ Λ · A (2.6)
such that Λ T · f · Λ = 1. In the case with no kinetic mixing, i.e. f = diag(g −2 1 , . . . , g −2 N ), the transformation matrix is simply Λ = diag(g 1 , . . . , g N ). For the moment we need not assume such simplification, and we work with a general kinetic matrix f .
The Lagrangian in terms of the transformed bosons A reads
L = − 1 4 F 2 − 1 2 A T · Λ T · M 2 · Λ · A + ψ i∂ / + Q T ψ · Λ · A / ψ . (2.7)
Notice that with this new normalization, what appears in the matter coupling to the gauge boson is no longer just the charges Q, but products of these and coupling constants g (and possible kinetic mixing parameters).
We need now an orthogonal transformation O that diagonalizes the mass matrixM 2 ≡ Λ T · M 2 · Λ. That is, we need to find a basis of orthonormal eigenvectors
M 2 · v i = m 2 i v i =⇒ O = ( v 1 v 2 . . . v N ) . (2.8) Conveniently, we define v i ≡ Λ · v i .
The transformation A ≡ O · A brings the Lagrangian to a standard form with canonical kinetic term and diagonal mass matrix:
L = − 1 4 F 2 i − 1 2 m 2 i A 2 i + α ψ α i∂ / + g T α · A / ψ α (2.9)
The coupling of a vector A i to the matter field ψ α is given by a linear combination of the original charges: g
(i) α = Q T α · v (i) . (2.10)
Notice the important fact that, for massless eigenvectors, v i are precisely the zero eigenvectors of the original mass matrix M 2 , i.e. they satisfy K · v i = 0. Since the entries of the matrix K are integer numbers, the entries of the massless eigenvectors v i will be also integers, up to an overall normalization factor. The corresponding gauge bosons will be massless, have quantized charges, and if the form of the matrix K is appropriate, will couple exclusively to one sector of the theory. They are hence perfect candidates to play the role of the SM hypercharge.
These last remarks do not apply to massive eigenstates, for which M 2 · v i = α i v i . Generically, given the non-diagonal character of the mass matrix M 2 , all of the entries of the massive eigenvectors v i will be non-zero and of the same order. The physical massive gauge bosons A i will be hence a linear combination of both visible and hidden bosons and they will act as portals into hidden sectors.
Before concluding this section let us write down an important condition on the vectors
v i . The orthogonality of the transformation matrix O of Eq. (2.8) translates into the condition v i T · v j = δ ij =⇒ v i T · f · v j = δ ij . (2.11)
We will have to take this condition into account in the phenomenological analysis carried out in the following sections.
The string theory interpretation
As mentioned in the introduction, one nice feature of the Stückelberg portal is that it finds a natural implementation in string theory, and a particularly intuitive one in models of intersecting D-branes. A detailed study and explicit examples in the setup of toroidal orientifolds of type IIA string theory can be found in the original references [8,9]. Here, we briefly describe where the different fields and couplings arise in such models (for general reviews on these type of string compactifications, see e.g. [30][31][32][33]).
In type IIA orientifold compactifications, gauge bosons arise from open strings living on D6branes that span the four non-compact dimensions, and wrap three-cycles of the six dimensional compactification space X 6 (usually a Calabi-Yau manifold). A stack of N overlapping such branes usually hosts a gauge group U (N ) ∼ = SU (N ) × U (1). Chiral matter fields arise at the intersections of two stacks. Hence, in order to obtain hidden sector scenarios, one has to choose carefully the cycles wrapped by the branes to make sure that stacks from different sectors do not intersect with each other.
The abelian gauge bosons living in such stacks couple not only to open strings, but also to closed strings which include the graviton, and also Ramond-Ramond (RR) axions that arise from the reduc-tion of RR three forms along three-cycles of X 6 . Being associated to closed strings that propagate in the bulk of the compactification, it is natural to consider that such RR axions couple to gauge fields from different sectors. These couplings are of the Stückelberg type given in Eq. (2.3) and generate masses for the gauge bosons. The charge matrix K is determined by the wrapping numbers of the branes around odd cycles of X 6 (odd with respect to the orientifold projection), and can be engineered in such a way that the mass matrix is highly non-diagonal.
The matrix G that also enters in the formula for the mass matrix M 2 and is identified with the complex structure moduli space metric of the compactification space X 6 , times a string scale factor M 2 s . Unfortunately, except for the simplest compactifications, this metric is unknown. Nevertheless, as long as some RR axion has non-zero charges under U (1) groups from different sectors, the mixing induced by the mass matrix M 2 is expected to be strong and results in physical Z bosons that couple visible and hidden sectors.
The final ingredient in the Lagrangian of Eq. (2.2) is the kinetic matrix f . At tree level, this matrix is diagonal f = diag(g −2 1 , . . . , g −2 N )
, with the couplings determined by the volume of the cycles wrapped by the corresponding branes. Loop corrections can generate off-diagonal terms that produce small kinetic mixings among different U (1)s.
The fate of the U (1) gauge bosons in this type of models is to gain a mass of the order of the string scale, suppressed by the square of the gauge coupling factor, m Z ∝ g 2 M 2 s . This is expected to be very large in a broad class of string constructions. Nevertheless, several mechanisms have been proposed to lower the Z masses, including large volume and anisotropic compactifications, or eigenvalue repulsion effects [8,41,49]. The conclusion is that, although not generic, Z masses at scales as low as the TeV, or even smaller, can be achieved in several setups.
At energy scales much lower than the Z boson masses, the corresponding U (1) symmetries become effectively global. They are in fact perturbatively exact symmetries of the effective Lagrangian, and they are broken only by highly suppressed non-perturbative effects [50][51][52]. Therefore, the U (1) symmetries that extend the visible sector gauge group in realistic D-brane constructions should find an interpretation in terms of known approximate global symmetries of the SM, such as Baryon or Lepton number.
Interestingly, these extra U (1) groups are generically anomalous symmetries of the SM. It is well known, however, that these anomalies are cancelled by a generalized Green-Schwarz mechanism, in which the RR axions, φ i , and the Stückelberg couplings of Eq. (2.3) play a crucial role. Although the gauge bosons associated to such anomalous U (1)s are not considered too frequently in the phenomenological literature, they are a key (and in fact most often unavoidable) ingredient of realistic constructions with open strings.
In the following, we take the Stückelberg portal string constructions we have described in this section as a motivation, and study some of their phenomenological consequences.
SM fermion couplings to Z
As in Refs. [8,9], we focus on visible sectors realized as in [53], the so-called Madrid quivers, which provide some of the simplest realistic models of intersecting D6-branes. In order to reproduce the SM one introduces four stacks of branes yielding a U
(3) A × U (2) B × U (1) C × U (1) D visible gauge group.
The intersection numbers of these branes are chosen in such a way that the model reproduces the SM chiral spectrum and is free of anomalies (with anomalies of extra U (1) factors cancelled by the Green-Schwarz mechanism).
In Table 1 the charges of the SM particles under the four visible U (1) factors are presented. These charges can be interpreted in terms of known global symmetries of the SM. In particular, Q A and Q D are proportional to baryon and lepton number, respectively. With these charge assignments, the hypercharge corresponds to the linear combination
Q Y = 1 6 (Q A − 3Q C + 3Q D ) . (3.12)
One has to make sure that such a combination remains as a massless gauge symmetry of the system (before electroweak symmetry breaking), i.e. that it corresponds to a zero eigenstate of the mass matrix M 2 . Following the discussion below Eq. (2.10), one has to make sure hence that the matrix of axionic charges K has an eigenvector v Y = (1, 0, −3, 3; 0, . . . , 0) with zero eigenvalue. The first entries of this vector correspond to the visible sector, and the latter to the hidden one, so that hypercharge couples exclusively to visible matter. In fact, this condition can be implemented in type II string constructiones by simple topological requirements on the wrapping numbers of the visible branes. Therefore, according to Eq. (2.10), the hypercharge coupling to a matter field ψ α reads
g Y α = e Q Y α = e 6 (Q αA − 3Q αC + 3Q αD ) . (3.13)
In general, the remaining three visible U (1) gauge bosons acquire masses by the Stückelberg mechanism, and as stressed in the previous section, they can have strong mass mixing with hidden U (1) bosons. In this work we are interested in the phenomenology induced by the lightest of the resulting physical Z bosons whose contribution to the DM interaction with SM particles is dominant. Since we cannot know the explicit form of the mass matrix M 2 for generic string compactifications (in particular because of the lack of control of the G matrix that enters the Lagrangian) we will simply parametrise the couplings of the lightest Z boson to the matter fields ψ α by a linear combination
g Z α = a Q αA + b Q αB + c Q αC + d Q αD + m i=1 h i Q (h) αi (3.14)
where we have included the contributions from hidden U (1) factors. The parameters a, b, c, d and h i are precisely the entries of the vector v Z = (a, b, c, d; h 1 , . . .) of Eq. (2.10). For massive Z bosons these are continuous parameters and as already stressed, they are all generically different from zero. Furthermore, notice that, by definition, v Z ≡ Λ · v Z , where v Z is a normalized vector. Since at tree level Λ = diag(g 1 , . . . , g N ), one can see that the parameters a, b, c, d and h i will be proportional to the original gauge coupling constants, and hence perturbative.
The parameters a, b, c and d, which in turn determine the effective couplings of visible matter to DM, are nevertheless, not completely arbitrary. On the one hand, they must be orthogonal to the hypercharge assignment in the sense of Eq. (2.11). Neglecting possible kinetic mixing effects, i.e. taking f = diag (g −2 a . . . g −2 d . . .), the orthogonality condition reads a g 2 Table 1: SM spectrum and U (1) i charges in the four stack models of Ref. [53]. Anomaly cancellation requires the three quark families to be divided into two Q L doublets and two antidoublets q L of U (2) B i.e. they differ in their U (1) B charge. We assign the up and down quarks to the antidoublets.
a + 3b g 2 c − 3d g 2 d = 0 . (3.15) Matter field Q A Q B Q C Q D Y Q L 1 -1 0 0 1/6 q L 1 1 0 0 1/6 U R -1 0 1 0 -2/3 D R -1 0 -1 0 1/3 L 0 -1 0 -1 -1/2 E R 0 0 -1 1 1 N R 0 0 1 1 0
On the other hand, the vectors v (i) must be properly normalized, yielding
g 2 Y 36 1 g 2 a + 9 g 2 c + 9 g 2 d = 1 , (3.16) a 2 g 2 a + b 2 g 2 b + c 2 g 2 c + d 2 g 2 d + m i=1 h 2 i g 2 hi = 1 ,(3.17)
for the Z and Z respectively. Notice that in the second expression the factor m i=1 h 2 i /g 2 hi encodes all the possible interactions of the Z with matter living in the hidden sector. Given the potential complexity of this sector, which we will not fully specify in this work, Eq. (3.17) reduces to a bound on the visible sector couplings
a 2 g 2 a + b 2 g 2 b + c 2 g 2 c + d 2 g 2 d < 1 . (3.18)
Furthermore, the couplings g i can be related to the SM gauge coupling constants by means of the following relations [49,53]:
g 2 a = g 2 3 6 , g 2 b = g 2 2 4 , 1 g 2 a + 9 g 2 c + 9 g 2 d = 36g −2 Y ,(3.19)
where g 3 and g 2 refers to the SU (3) QCD and SU (2) L coupling constants, respectively. These relations arise from the fact that U (1) A and U (1) B are just the center of the groups from which the SU (3) QCD and SU (2) L gauge factors of the SM arise. 1
Now we have all the necessary information to build the couplings of the Z to the SM particles. In virtue of Eq. (3.14) and table 1, the left and right handed (first and second family) of quarks have the following couplings, 20) which can be used to define the vectorial coupling as the sum of the left and right components, 21) and the axial coupling as the difference,
g Z u L = (a + b) , g Z u R = (−a + c) , g Z d L = (a + b) , g Z d R = (−a − c) ,(3.C V u = g Z u L + g Z u R = (b + c) , C V d = g Z d L + g Z d R = (b − c) ,(3.C A u = g Z u L − g Z u R = (2a + b − c) , C A d = g Z d L − g Z d R = (2a + b + c) . (3.22)
Similarly, according to Table 1, for the third family of quarks the vectorial couplings are given by
C V t = (−b − c) , C V b = (−b + c) ,(3.23)
whereas the axial couplings are given by
C A t = (2a − b − c) , C A b = (2a − b + c) . (3.24)
Finally, for the three families of leptons the vector and axial couplings can be written as
C V = (−b − c) , C A = (−b + c − 2d) ,(3.25)
respectively. Note that in all cases, the vectorial couplings are independent of a as well as of d, as was to be expected from the aforementioned interpretation of the charges Q A and Q D in terms of baryon and lepton number. The axial couplings, on the other hand do depend on a and d. This fact will have a remarkable impact on the LHC bounds as we will see later.
Isospin violation from the Stückelberg mechanism
As we have seen previously, the different charges of the SM particles under the U (3) A × U (2) B × U (1) C × U (1) D visible gauge group, together with the mixing of the corresponding abelian bosons, gave rise to very generic vector and axial couplings to the Z boson. As a consequence, a DM particle living in the hidden sector, ψ, will couple to each SM fermion through the Z in a different manner. This fact can be translated into a different coupling strength of ψ to protons and neutrons, and thus, to a rather flexible amount of isospin violation f n /f p (a n /a p ). This is very important from the point of view of DM direct detection experiments [54].
Direct detection experiments are based on the elastic scattering of DM particles off nucleons inside an underground detector which shields it from cosmic rays. These experiments are tremendously sensitive to the recoil energy released by a nucleus of the target material when a DM particle hits it. Since the interaction between the nucleon and the DM particle occurs in the non relativistic limit (the relative velocity of the system in the lab frame is of the order of hundreds of km/s), the energy deposited in the detector after the collision is very small, of the order O(10) keV. Depending on the nature of the DM particles, and the mediator of its interaction with quarks, there exist many different operators that contribute to this interaction. For a Dirac fermion DM with a Z gauge boson mediator, its interactions with quarks can be divided into the so-called spin-independent (SI) interactions, arising from scalar and vector interactions with quarks, and spin-dependent (SD) interactions that originate from axial-vector interactions. Let us now analyse either cases separately.
SI interactions
The spin independent contribution to the total cross section of the DM-nucleus elastic scattering arises from scalar and vector couplings. For an interaction mediated by a vector boson exchange, the effective Lagrangian for the interaction of ψ with nucleons (protons (p) and neutrons (n)) can be written as,
L V SI = f p (ψγ µ ψ)(pγ µ p) + f n (ψγ µ ψ)(nγ µ n),(4.26)
where f p and f n are the vector couplings of ψ to the protons and neutrons, respectively. These quantities depend on the nucleon quark content. For a vector interaction the only quarks that play a role are those of the valence (up and down), while for a scalar interaction the sea quarks are also important for the entire process. Since the up and down quarks are not present in the proton and neutron in the same fraction, one can express f p and f n as follows [55],
f p = 2b u + b d , f n = b u + 2b d ,(4.27)
where b u and b d are the effective vector couplings of the up and down quarks to the DM particles. 2 After integrating out the Z boson, these couplings can be easily written as,
b (u,d) = hC V (u,d) 2m 2 Z ,(4.28)
with h being the coupling strength of the Z boson to ψ, and m Z the mass of the lightest Z boson.
Using now the expressions for the vector couplings of the Z to up and down quarks, given in Eq. (3.21), it is straightforward to deduce that,
b u = hC V u 2m 2 Z = h 2m 2 Z (b + c) , b d = hC V d 2m 2 Z = h 2m 2 Z (b − c) .
(4.29)
These two expressions make obvious that in this framework the ratio between the coupling of ψ to protons and neutrons i.e. the amount of isospin violation f n /f p , according to Eq. (4.27), is given by
f n f p = (3b − c) (3b + c) = (3b/c − 1) (3b/c + 1) . (4.30)
Interestingly, the the total amount of isospin violation depends exclusively on the ratio between the parameters b and c which, as mentioned before, are continuous and different from zero, generating a ratio f n /f p different from ±1. This is a consequence of the introduction of U (1) gauge groups in the visible sector to reproduce the global symmetries of the SM. In particular, the parameter b corresponds to a chiral U (1) symmetry of the Peccei-Quinn type, with mixed SU (3) anomalies; while c is related precisely to the weak isospin symmetry U (1) C [53]. Isospin violation and chirality are the key properties why these new groups generate a general isospin violation in the currents related to the Z interaction.
In Figure 2 (left panel) the quantity f n /f p is shown as a function of b/c according to Eq. (4.30). We have shown some noteworthy theoretical benchmark values of this ratio as well, like Z mediation and dark photon scenarios, f n /f p = −13.3 and f n /f p = 0, respectively. The value of f n /f p ≈ −0.7 is the so-called Xe-phobic dark matter scenario to which Xe-based detectors are poorly sensitive 3 . Interestingly, we notice that our construction naturally generates isospin violating couplings f n /f p = 1 for any value of the parameters b and c. These parameters are expected to be of the same order, |b/c| ∼ O(1), which defines a region in which the value of f n /f p is subject to important changes (for values around b/c = −1/3). This precisely highlights the flexibility in the isospin violation patterns found in these constructions.
All this together can be taken as a clear and testable prediction of this kind of constructions. It also would be distinguishable from other hidden DM scenarios. For instance, if the portal between the visible and the hidden sector occurs via a Higgs boson, the value of f n /f p would be generally 1, since the Higgs boson can not differentiate chiralities of the quarks. 4 It is worth noting that, although the type of constructions we are considering, based on the visible gauge group U (3) A ×U (2) B ×U (1) C ×U (1) D , lead to a flexible amount of isospin violation (generically f n /f p = ±1), 5 there is a well known class of alternative type II string models in which the gauge group U (2) B is replaced by U Sp(2) B ∼ = SU (2) B [57]. In such models, the U (1) B factor, which was crucial in our discussion, is absent. One could realise the Stückelberg portal scenario in such constructions, and follow steps similar as the ones we have taken here. The only difference one would find is that the parameter b would be identically zero, and hence that the DM interactions with the nucleons would automatically satisfy f n /f p ≡ −1.
SD interactions
Let us now move to consider the case of SD interactions. As we have mentioned above, these interactions arise from the axial-vector couplings of DM to protons and neutrons, and thus, occur when the DM particles have a spin different from zero. In terms of the effective Lagrangian we can write, L SD = a p (ψγ µ γ 5 ψ)(pγ µ γ 5 p) + a n (ψγ µ γ 5 ψ)(nγ µ γ 5 n), (4.31) where the parameters a p(n) are the couplings of DM to protons (neutrons), and can be expressed in the following way [58],
a p = q=u,d,s α A q √ 2G F ∆ p q = h 2 √ 2G F m 2 Z C A u ∆ p u + C A d (∆ p d + ∆ p s ) , (4.32) a n = q=u,d,s α A q √ 2G F ∆ n q = h 2 √ 2G F m 2 Z C A u ∆ n u + C A d (∆ n d + ∆ n s ) , (4.33)
where α A q is the effective axial coupling of DM to quarks and G F denotes the Fermi coupling constant. Operators for axial-vector interactions in the nucleon are related to those involving quarks through the quantities ∆ p(n) q , which relate the spin of the nucleon to the operator p(n)|qγ µ γ 5 q|p(n) . For these we have taken the values from Ref. [59]. Now, we can take the ratio between the coupling to protons and neutrons, which gives a n a p =
∆ n u + 2a/c+b/c+1 2a/c+b/c−1 (∆ n d + ∆ n s ) ∆ p u + 2a/c+b/c+1 2a/c+b/c−1 (∆ p d + ∆ p s )
.
(4.34)
As one can see from the previous expression, unlike for f n /f p , this ratio also depends on a/c not only on b/c, and hence, there is one more degree of freedom respect to the SI case.
In Figure 2 (right panel), the ratio a n /a p is depicted as a function of b/c according to Eq. (4.34) for different values of the ratio a/c. In the limit of a/c → ∞ (dot dashed line), we find the case of a n /a p = 1, similar to the case of the SI interactions in the limit b/c → ∞. While, for the cases a/c → 0 (solid line) and a/c = 1 (dashed line), the values of a n /a p are generally different from ±1.
Notice that in this case one can also define the Xe-phobic scenario for a n /a p . However, it depends 5 In the models we discuss, the values fn/fp = ±1 can only be reached in the limits b/c → 0 and b/c → ∞, which although not excluded, are not particularly preferred. This provides a remarkable and potentially measurable distinction of these constructions from other portals. on the ratio between the zero momentum expectation values of the spin for protons and neutrons in xenon which are of the order of O(10 −2 ) (using the latest calculations [60]), and for simplicity it is not included in Figure 2.
Finally, in order to rearrange the results for both SI and SD interactions, in Figure 3 we show the plane a/c versus b/c. As we have seen before, these two ratios determine the amount of isospin violation in DM interactions for both the SI and SD contributions. On the one hand, the dashed vertical lines represent some values of f n /f p , which are independent of a/c, as in Figure 2. On the other hand, the solid lines denote some values for a n /a p . Remarkably, in the region shown, where the values of a, b and c are in general of the same order, the DM interactions are isospin violating in both types of interactions. Furthermore, we see that very high values of the neutron component (with respect to the proton component) can be reached, although, the variation of either f n /f p or a n /a p is very abrupt in this region (see also Figure 2). This is important for direct detection experiments that use target materials in which the ratio between the neutron and proton contribution is significantly different than one. For instance, in Xe-based detectors such as LUX, the SD component is dominated by the neutron scatterings due to the dominance of the neutrons in the total spin of the 129 Xe and 131 Xe isotopes.
Isospin violating DM in light of the LHC and LUX results
As we have shown previously, this kind of constructions generally predict isospin violating DM. The relations between the proton and neutron contributions for SI and SD interactions depend on the Table 2: Input parameters for each BM point.
couplings a, b and c, and more specifically, in their relations.
According to Eqs. (3.21)-(3.25), all couplings of the Z to SM fermions can be written in terms of the four parameters a , b , c , d. In light of this, it is obvious that certain combinations of these parameters will affect the predicted values of some constrained experimental observables. Furthermore, as pointed out in Section 3, there are some constraints on these parameters that come from the building of the Z and Z bosons in this model. This section is aimed at exploring the impact that these constraints have on the allowed values of a/c and b/c, and hence, on the experimentally allowed values of f n /f p and a n /a p . Needless to say, these regions will depend on certain assumptions on the DM mass and its coupling h, the Z mass, d/c, and c, and for this reason we will concentrate on six representative benchmark (BM) points. The values used for each of these parameters are shown in Table 2.
It is legitimate to ask whether the mass scales that appear in such BMs can arise in consistent string compactifications. As we have already mentioned at the end of section 2.3, Z masses of the order of the TeV, although not generic, can be achieved in several ways without much difficulty. On the other hand, notice that the DM particles are charged, often chirally, not only under U (1) hidden groups, but also under non-abelian factors, i.e. the G h in Eq. (1.1). Therefore, the mass of the field ψ is related to possible strong coupling dynamics and symmetry breaking patterns (e.g. a hidden Higgs mechanism) of the hidden non-abelian gauge sector. In this sense, it is quite natural to consider DM masses in the GeV-TeV range, at least as natural as having visible sectors reproducing the masses of the SM particles.
Given the potential complexity of the matter and gauge structure of the hidden sector, it seems reasonable to asume that there could be some mechanisms, either thermal or non-thermal, to account for the relic abundance of ψ other than annihilation through the Z channel. This highlights the dependence of the DM abundance on the particular details of the hidden sector dynamics, which we want to keep as generic as possible. Nevertheless, it is worth mentioning that, in general, annihilation cross sections through the Z channel are lower than the thermal value, and thus indirect detection bounds on the annihilation cross section are generally far from our predictions.
The only contributions to the phenomenology of the model that do not depend on any further assumption on the hidden sector are direct DM searches and LHC searches for resonances 6 . The former only depends on the coupling of ψ to quarks by the exchange of a Z boson (see Figure 1), while the latter depends on the coupling of Z to SM particles (quarks and leptons) and the coupling h. In the following, we will determine the experimentally allowed regions of a/c and b/c in the six BM points shown in Table 2 taking into account the limits from LUX and the LHC.
LUX and LHC limits
The recent null results of the LUX collaboration [63] have placed a very stringent upper limit on the elastic scattering of DM off protons, reducing significantly the parameter space allowed in many theories that provide DM candidates. This limit has been extracted by assuming a scalar DM candidate (zero SD contribution) and f n /f p = 1, which are the typical assumptions that the collaborations use in order to compare their results within a unified framework. However, this prevents us from using this result directly, since none of these two assumptions hold for the DM candidate analysed in this work. Therefore, in order to implement this bound properly we have simulated the LUX experiment, and we have calculated for a given point of the parameter space if it is allowed at 90% C.L. using the Yellin's maximum gap method [64]. To such end, we calculate the predicted total number of events in LUX considering the SI and SD components and computing the f n /f p , a n /a p ratios. To calculate the 90% C.L. exclusion using the maximum gap method, we consider that LUX experiment has observed zero candidate events in the signal region 7 .
To calculate the total number of expected signal events in a Xe-based detector we have followed the prescription of Ref. [65] in the S1 range 2-30 PE for an exposure of 10065 kg days, using the acceptance shown in the bottom of Fig. 1 of Ref. [63] plus an extra 1/2 factor to account for the 50% of nuclear recoil acceptance. We use the S1 single PE resolution to be σ P M T = 0.37 PE [66], a 14% of photon detection efficiency, and the absolute scintillation efficiency digitized from Ref. [63]. For the DM speed distribution, we use the standard isothermal Maxwellian velocity distribution, with v 0 = 220 km/s, v esc = 544 km/s, ρ 0 = 0.3 GeV/cm 3 and v e = 245 km/s, as the one used by the LUX collaboration [63]. As pointed out in Ref. [67] the effect of the form factors can also induce important differences in the expected number of events. In this work we use the Helm factor for the SI component and the SD structure functions given in Ref. [60] for the SD component.
To show explicitly the dependence of the SI and SD elastic scattering cross sections on the parameters of the model, namely, on the ratios a/c and b/c, let us write them as,
σ SI p = 4 π µ 2 p f 2 p = µ 2 p h 2 πm 4 Z (3b + c) 2 , (5.35) σ SD p = 24G 2 F π µ 2 p a 2 p = 3µ 2 p h 2 πm 4 Z [(2a + b − c)∆ p u + (2a + b + c)(∆ p d + ∆ p s )] 2 . (5.36)
Notice that in order to calculate the neutron contributions one has to multiply by (f n /f p ) 2 the SI component and by (a n /a p ) 2 the SD component, whose expressions are given in Eqs. (4.30) and (4.34). Let us mention at this point the existing relation between the SI and the SD elastic cross sections. From the previous equations, and the corresponding neutron counterparts, one can easily see that the contribution from the SD cross section to the total number of expected events dominates if |a/c| |b/c| and |a/c| 1. However, for a given of c the ratio a/c cannot be arbitrarily large due to the normalization Eq. (3.18). In fact, it can be shown that for the SD component to be dominant in LUX for the range of m ψ considered and when |b/c| < 5 (the region shown in the figures) then a/c 100. Using the values of c shown in Table 2, such high values of a/c do not satisfy the Eq. (3.18), and hence, they are not considered.
The production and the subsequent decay of a Z boson into SM particles might leave distinctive signal of new physics that can be searched at colliders, and in particular at the LHC. The ATLAS detector at the LHC searched for high mass resonances decaying into a µ + µ − or an e + e − pair for energies above the Z pole mass, at a center of mass energy √ s = 8 TeV and luminosities of 20.5 fb −1 and 20.3 fb −1 for dimuons and dielectrons resonances, respectively [68]. These results are consistent with the SM predictions allowing to place an upper limit on the signal cross section times the corresponding branching fraction of the process pp → Z → µ + µ − (e + e − ). 8 There are also searches for dijet resonances and monojets plus missing energy that receive additional contributions from the presence of a Z boson, both at the LHC and Tevatron colliders, and hence, they can be used to place constraints on this kind of models as well [69][70][71].
In the model presented here, the coupling of the Z boson to leptons and quarks contributes to the appearance of dimuon, dielectron and dijet resonances, and thus, these searches can constraint the parameter space. In order to include these bounds to determine which regions are allowed in light of these searches, we have followed the approach given in Ref. [72]. In the narrow width approximation, the dilepton production in proton-proton collisions mediated by the Z can be written as,
σ l + l − 1 3 q dL qq dm 2 Z × σ(qq → Z ) × BR(Z → l + l − ) ,(5.37)
where dL qq /dm 2 Z denotes the parton luminosities, σ(qq → Z ) is the peak cross section for the Z boson, and BR(Z → l + l − ) is the branching ratio for the Z decaying into a lepton pair. A close inspection of the previous expression reveals that there is a part which only depends on the model parameters, and the remaining part that only depends on the kinematics of the process. Hence, it can be factorized as,
σ l + l − = π 48s W Z s, m 2 Z × BR(Z → l + l − ) ,(5.38)
where the function W Z is given by:
W Z = q=u,d,c,s c q ω q s, m 2 Z . (5.39)
The coefficients c q are the sums of the squares of the vector and axial couplings, (C V q ) 2 + (C A q ) 2 , to the corresponding quarks. Notice that we do not include the contributions from the bottom and top quarks, since they can be safely neglected in the production process. In this limit, provided that the first and second quark families share the same charges under the U (5.40)
In the previous expression we have reabsorbed a factor 2 in the definition of the ω functions. This factor corresponds to the sum of the up and charm quarks contribution to the up component and, in the same way, for the down and strange quarks for the down component.
Using the equations (5.38) and (5.40) one can easily write the production cross section of a dilepton pair mediated by the Z in proton proton collisions at leading order (LO) as,
σ LO l + l − = c upωup s, m 2 Z + c downωdown s, m 2 Z × BR(Z → l + l − ) ,(5.41)
whereω up,down = (π/48s)ω up,down . To extract the functionsω at √ s = 8 TeV we have benefited from CalcHEP 3.6.22 [73] using the parton distribution functions CTEQ6L to be consistent with the LHC analysis [68]. Furthermore, in order to include Next-to-LO effects, we have used the K-factor given in Ref. [74]. Remarkably, this approach can be used to calculate not just the bounds for dilepton but also for dijet resonances just by substituting BR(Z → l + l − ) (valid for dilepton in the previous expressions) by BR(Z → qq), where a sum over all quarks (except for the top quark [71]) must be performed. Finally, to include properly the dijet resonance searches, the cross section times the branching fraction must be multiplied by a factor A = 0.6 which accounts for the efficiency of the detector [71].
Before moving to the phenomenological analysis of the BM points, let us write the partial widths of the Z boson decay into SM particles and ψ as a function of the model parameters as, smaller value of h makes the Z boson less invisible, which is translated into an increase of both, its production cross section and its branching ratio into SM particles. Remarkably, LHC limits rule out a big portion of the parameter space allowed by LUX, including the Xe-phobic value of f n /f p , and it leaves only a small region allowed corresponding to positive values of a/c and −2 b/c −1.
b/c b/c 5 − 4 − 3 − 2 − 1 − 0 1 2 3 4 5 a/c 4 − 2 −
Interestingly, the allowed regions for both BMs represent isospin violating DM scenarios in which the neutron contribution of the SI component might be much higher than the corresponding proton component but both are generally small in order to evade LUX bounds. For the SD component in BM1, the values of a n /a p are not restricted while for BM1a, the allowed region encodes a n /a p generally larger than one. In conclusion, there exists an outstanding complementarity between LHC and direct detection searches for these BM points. While LUX is more stringent than the LHC for negative values of b/c, the LHC is more constraining for positive ones, and for BM1a also for negative a/c, which highlights the power of combining different experiments in the search for new physics.
Let us move now to BM2 and BM2a. These BM points entail a much heavier DM candidate with respect to the previous ones, now m ψ = 500 GeV, and a Z boson of 3 TeV, heavier than before as well. In this region of DM masses, direct detection experiments start to lose their sensitivity very rapidly, so we have increased the DM coupling h in order for the LUX limit to play a role. Besides, by augmenting d/c we have increased the decay width of the Z boson into leptons, which makes dilepton constraints more stringent. In figure 5 the plane a/c-b/c is depicted for these BM points. Notice that in this case dijet searches at the LHC are shown as dark blue regions with oval-like shapes and are specially important in the upper left corner of the BM2 case.
As in the previous BMs, LUX limits are very stringent in this case, specially for BM2, and again are independent of a/c (dominated by SI interactions). LUX rules out the zone |b/c| 0.4 for BM2 and |b/c| 1.7 for BM2a, since for the latter the values of both c and h are smaller. For BM2, the b/c figure 6 for BM3 (left panel) and for BM3a (right panel). The choice of the parameters is such that for BM3, LUX limits are not very constraining, while for BM3a the increase of c and h makes LUX very restrictive. However, for the latter a new constraint, very strong, has appeared. The grey area denotes a forbidden region because it does not satisfy Eq. (3.18). This is a consequence of the value of c in this case, which is the bigger of all BMs.
Since the Z boson cannot decay into DM particles in BM3 and BM3a, the branching ratios into SM particles are increased, and therefore, we expect LHC limits to constrain very severly. Notably, for BM3 dijet bounds dominate the region −3 b/c −1. The value used for d/c in these BMs makes that for b/c relatively small the Z boson behaves as leptophobic, which results in a decrease in sensitivity of the dilepton searches. As soon as |b/c| increases this behaviour disappears and dilepton bounds are dominant over the dijet ones. In most of the region allowed the SI elastic scattering cross section is dominated by neutrons, except for the region close to b/c ≈ 1. The ratio a n /a p allowed is very similar to those in the previous BM points.
For BM3a, shown in the right panel of figure 6, we find that only a very small region is allowed. The region extending from b/c ≈ −1 up to b/c ≈ 0, and from a/c ≈ 0 to a/c ≈ 1. From a point of view of complementarity, this region is exceptionally exemplifying since it is delimited by all the searches. The upper and lower regions are bounded by dijet searches, the left by LUX and the right by dilepton searches. This is a consequence of increasing c while keeping the ratio d/c constant. In this case, the SI cross section is dominated by neutrons and the SD proton cross section is similar to the neutron component but with a n /a p ≈ −1.
Conclusions
In this article, we have performed a thorough study of phenomenological features of hidden sector scenarios with Stückelberg Z portals that arise as low energy effective actions of certain type II string compactifications with intersecting branes. For our purposes, the crucial property of these constructions is the unavoidable extension of the SM gauge group by several ('anomalous') abelian gauge bosons which gain a mass and can mix with analogous bosons from hidden sectors.
Many interesting phenomenological properties of such setups are determined by the charges of the SM spectrum under the extra U (1)s of the visible sector, together with a handful of mixing parameters (a, b, c, d). The possible choices for the charges are rather scarce, due to the necessary identification of these symmetries with approximate global symmetries of the SM. We have focussed on a particular gauge structure, the Madrid models that arises in a large class of intersecting brane constructions. Some other configurations are possible, and they could be studied in analogy. We believe, nevertheless, that our analysis covers a significant portion of the landscape of semi-realistic brane models.
Once the extra visible U (1) bosons mix with those from the hidden sectors, the lightest Z mass eigenstate generates the dominant interactions between DM and SM fermions. A particularly appealing and characteristic feature of such models, is the natural appearance of rich patterns of isospin violating DM interactions, which contrasts with other simple portals traditionally considered in the literature. We have explored the prospects for f n /f p and a n /a p in six different BM points of the parameter space of these constructions, incorporating LHC and LUX bounds showing that in general values of these ratios tend to be dominated by the neutron contribution. Target materials with more sensitivity to neutron interactions are thus very suitable to explore these scenarios.
Generically, this setup provides isospin violating couplings both in the SI and SD interactions. We have confronted our prospects with LUX and LHC bounds for a set of BM points. By using our own simulation of the LUX experiment, we have performed a check of the exclusion regions for each point using the maximum gap method. This has allowed us to analyse consistently a general scenario with SI and SD (proton and neutron contributions) interactions as well as in general cases of isospin violating couplings of DM. For the LHC we have calculated, for each point of the parameter space, the production cross section of a Z boson times the branching ratio of a specific decay. With this, we have included ATLAS searches for dilepton (e + e − and µ + µ − ) and dijet resonances. Remarkably, all regions experimentally allowed entail much higher neutron than proton cross sections for the SI interactions while for the SD the situation is less constrained.
The findings of this work open the door to generic scenarios in which the signals in direct detection experiments can be dominated by neutrons. Moreover, we show that the existing complementarity between LHC searches and direct detection experiments is specially relevant to disentangle the couplings of the Z boson to SM particles. It is gratifying to see how, not only different experimental strategies, but also phenomenological and fundamental theoretical input can be combined into a single framework to shed some light into the possible properties of the so far elusive nature of dark matter.
Figure 1 :
1Left: Schematic representation of a hidden sector scenario (1.1) with intersecting D-branes. Green and red branes do not intersect each other and hence host different sectors. Right: Diagram contributing to the elastic scattering of hidden sector DM, ψ, off quarks. The mediator of this interaction is a mixing of different string axions, φ, and the vectors Av and A h .
Figure 2 :
2Left: Amount of isospin violation for SI interactions, fn/fp, as a function of b/c (solid line). Some representative values of fn/fp are shown as horizontal dashed lines. Right: Ratio between the coupling of DM to neutrons and protons, an/ap, for the SD interactions as a function of b/c. For a/c we have taken different limits, a/c → 0 (solid line), a/c → ±∞ (dot dashed line), and a/c = 1 (dashed line).
Figure 3 :
3Ratio between the coupling of DM to neutrons and protons for the SD interactions as a function of b/c. For a/c we have take different limits, a/c → 0 (black line), a/c → ∞ (blue line), and a/c = 1 (gray line).
( 3 )
3A × U (2) B × U (1) C × U (1) D gauge symmetry group (see Section 3), the function W Z can be written as a sum of the up and down doublet components of the quarks as W Z = c up ω up s, m 2 Z + c down ω down s, m 2 Z .
Figure 5 :
5Same as figure 4 but for BM2 (left) and BM2a (right). In this case the exclusion region from dijet resonances at the LHC is shown in dark blue.
Note that these relations should be evaluated at the compactification scale. The running of the coupling constants from this scale to the electroweak scale, at which isospin violating properties of DM are defined, can be simply reabsorbed into the definition of the parameters a, b, c and d.
Not to be confused with the coupling b associated with the U (1) B symmetry.
This is a consequence of the ratio between the number of protons and neutrons in xenon isotopes4 In type II 2HDM for tan β ≈ 1 there can be deviations[56].
Hadronic decays[22,61] and the muon anomalous magnetic moment[55,62] do not depend either on any further assumption and can affect the allowed values of parameters a, b and c. However, we have checked that these constraints are not competitive with LUX and LHC in the region of the parameter space considered in this work.
Actually, LUX observed one candidate event that was marginally close to the background region in the log 10 (S2/S1) − S1 plane. Thus our result of the exclusion is closer to the actual LUX limit when considering zero observed events.
Although these results can be used to place constraints on other models of new physics, we are interested in its application for the search of a Z boson.
AcknowledgmentsThe authors are grateful to D. G. Cerdeño48)where l and ν refer to the three families of leptons and neutrinos, respectively. These expressions and the SM couplings of the Z , given in Section 3, allow us to evaluate the LHC bounds as a function of the parameters a/c and b/c.ResultsLet us start analysing BM1 and BM1a. These BM points correspond to a low mass dark matter candidate, with a mass of 50 GeV and a Z boson of 1 TeV. Infigure 4we show the plane a/c-b/c with some values of the ratios f n /f p and a n /a p for BM1 (left panel) and BM1a (right panel). We have superimposed the 90% C.L. LUX exclusion region (shown in red) that rules out high values of |b/c|, while in blue we show the exclusion regions from the LHC searches for e + e − (light blue) and µ + µ − resonances (darker blue). As we have anticipated previously, the LUX limit does not depend on the specific value of a/c since in this region of the parameter space the SI contribution of the elastic scattering dominates over the SD one. For BM1, LUX excludes the regions b/c −0.9 and b/c 1.0, which correspond to the regions in which the proton and neutron components of the SI elastic cross section are similar, |f n /f p | ≈ 1. To understand this behaviour note that the proton contribution given in Eq. (5.35) decreases very fast around b/c = −1/3, faster than f n /f p (due to f 2 p ). This means that, although in the allowed region the neutron contribution to the SI cross section dominates with values of f n /f p that can be very large (see also left panel ofFigure 2), it also decreases, and thus, the LUX limit weakens. For BM1a, since the value of h has been decreased respect to BM1, the coupling of ψ to the Z also diminishes and then the LUX limits are able to constrain much less parameter space, namely, it rules out the region |b/c| 1.9.Unlike direct detection limits, LHC bounds depend on the value of a/c. First of all, we show that for BM1 when |a/c| 2, both e + e − and µ + µ − bounds are less stringent. This can be understood from Eqs. reason for this behaviour is the same as before: in the region not excluded, although the neutron contribution is much higher than the proton contribution, both cross sections are small. LHC limits from dilepton resonances are now very well differentiated and more stringent as a consequence of the increase of d/c (respect to BM1 and BM1a). The difference between e + e − and µ + µ − channels is more notably and comes from the different sensitivity of the ATLAS detector to these channels at this Z mass, since its coupling to each of these leptons is identical. Finally, dijet resonance searches appear in these cases as more constraining than dileptons and LUX in a small region of the parameter space (the upper left corner in the left panel offigure 5). The shape of this constraint is due to the squares of the couplings to quarks, involved either in the production mechanism or in the subsequent decay of the Z . This can be understood as a leptophobic behaviour of the Z in this region of BM2, while we have not found such feature in BM2a due to the increase of d/c which makes the Z more leptophilic.To end with these BM points, as it is shown inFigure 5, there is only a tiny region allowed for BM2, while for BM2a the region is considerably bigger. In terms of isospin violation in the SI interactions, it corresponds to neutron dominance as in the previous cases. Remarkably, the Xe-phobic scenario (f n /f p = −0.7) remains allowed by both LHC and LUX in the two BMs analysed. For the SD interactions, the ratio a n /a p is found to range between 1 and -10, approximately, and thus, it can be concluded that in general all interactions in direct detection experiments would be dominated by neutrons. The complementarity between direct DM searches and the LHC now takes a new shape. LUX rules out the values of b/c stronger than LHC in all cases, however, the LHC is able to constrain high values of |a/c|. Surprisingly, this complementarity is able to delimit the allowed portions of the parameter space so strongly that the we have obtained closed regions.To end with the analysis, we study two BMs in which the decay of the Z into DM particles is kinematically forbidden, BM3 and BM3a, unlike for BM1(a) and BM2(a). Our results are shown in
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| [] |
[
"Mott transition in the triangular lattice Hubbard model: a dynamical cluster approximation study",
"Mott transition in the triangular lattice Hubbard model: a dynamical cluster approximation study"
] | [
"Hung T Dang \nInstitute for Theoretical Solid State Physics\nJARA-FIT and JARA-HPC\nRWTH Aachen University\n52056AachenGermany\n",
"Xiao Yan Xu \nBeijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n",
"Kuang-Shing Chen \nInstitut für Theoretische Physik und Astrophysik\nUniversität Würzburg\nAm HublandD-97074WürzburgGermany\n",
"Zi Yang Meng \nBeijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n",
"Stefan Wessel \nInstitute for Theoretical Solid State Physics\nJARA-FIT and JARA-HPC\nRWTH Aachen University\n52056AachenGermany\n"
] | [
"Institute for Theoretical Solid State Physics\nJARA-FIT and JARA-HPC\nRWTH Aachen University\n52056AachenGermany",
"Beijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"Institut für Theoretische Physik und Astrophysik\nUniversität Würzburg\nAm HublandD-97074WürzburgGermany",
"Beijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"Institute for Theoretical Solid State Physics\nJARA-FIT and JARA-HPC\nRWTH Aachen University\n52056AachenGermany"
] | [] | Based on dynamical cluster approximation (DCA) quantum Monte Carlo simulations, we study the interaction-driven Mott metal-insulator transition (MIT) in the half-filled Hubbard model on the anisotropic two-dimensional triangular lattice, where the degree of frustration is varied between the unfrustrated case and the fully frustrated, isotropic triangular lattice. Upon increasing the DCA cluster size, we analyze the evolution of the MIT phase boundary as a function of frustration in the phase diagram spanned by the interaction strength and temperature, and provide a quantitative description of the MIT phase boundary in the triangular lattice Hubbard model. Qualitative differences in the phase boundary between the unfrustrated and fully frustrated cases are exhibited. In particular, a change in the sign of the phase boundary slope is observed, which via an impurity cluster eigenstate analysis, may be related to a change in the nature of the insulating state. We discuss our findings within the scenario that the triangular lattice electron system might exhibit a quantum critical Mott MIT with a possible quantum spin liquid insulating state, such as considered for the organic charge transfer salts κ-(BEDT-TTF)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2. | 10.1103/physrevb.91.155101 | [
"https://arxiv.org/pdf/1411.7698v2.pdf"
] | 118,444,537 | 1411.7698 | d734c105857d7a576ceeec86184ab6db3cbc78cd |
Mott transition in the triangular lattice Hubbard model: a dynamical cluster approximation study
(Dated: December 1, 2014)
Hung T Dang
Institute for Theoretical Solid State Physics
JARA-FIT and JARA-HPC
RWTH Aachen University
52056AachenGermany
Xiao Yan Xu
Beijing National Laboratory for Condensed Matter Physics
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
Kuang-Shing Chen
Institut für Theoretische Physik und Astrophysik
Universität Würzburg
Am HublandD-97074WürzburgGermany
Zi Yang Meng
Beijing National Laboratory for Condensed Matter Physics
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
Stefan Wessel
Institute for Theoretical Solid State Physics
JARA-FIT and JARA-HPC
RWTH Aachen University
52056AachenGermany
Mott transition in the triangular lattice Hubbard model: a dynamical cluster approximation study
(Dated: December 1, 2014)numbers: 7127+a7110Fd7130+h
Based on dynamical cluster approximation (DCA) quantum Monte Carlo simulations, we study the interaction-driven Mott metal-insulator transition (MIT) in the half-filled Hubbard model on the anisotropic two-dimensional triangular lattice, where the degree of frustration is varied between the unfrustrated case and the fully frustrated, isotropic triangular lattice. Upon increasing the DCA cluster size, we analyze the evolution of the MIT phase boundary as a function of frustration in the phase diagram spanned by the interaction strength and temperature, and provide a quantitative description of the MIT phase boundary in the triangular lattice Hubbard model. Qualitative differences in the phase boundary between the unfrustrated and fully frustrated cases are exhibited. In particular, a change in the sign of the phase boundary slope is observed, which via an impurity cluster eigenstate analysis, may be related to a change in the nature of the insulating state. We discuss our findings within the scenario that the triangular lattice electron system might exhibit a quantum critical Mott MIT with a possible quantum spin liquid insulating state, such as considered for the organic charge transfer salts κ-(BEDT-TTF)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2.
I. INTRODUCTION
The Mott metal-insulator transition (MIT) [1,2] is a fundamental phenomenon in the field of strongly correlated quantum many-body physics and quantum criticality [3,4]. Starting from the early case of NiO [5], the full Mott MIT (temperature versus pressure) phase diagram has later been constructed e.g. for vanadium oxide V 2 O 3 [6], and critical scaling near the end-point of the Mott MIT phase boundary has been observed in conductance measurements [7].
The discovery of strongly correlated organic materials opened new horizons for exploring the Mott MIT. These compounds consist of quasi-two-dimensional triangular lattices of organic charge transfer salts [8][9][10][11][12][13] such as κ-(BEDT-TTF) 2 Cu 2 (CN) 3 and EtMe 3 Sb[Pd(dmit) 2 ] 2 . By varying hydrostatic pressure, these systems exhibit a MIT with an insulating state without any detectable symmetry breaking, possibly exhibiting a quantum spin liquid state [14]. The phase diagram of κ-(BEDT-TTF) 2 Cu 2 (CN) 3 in addition exhibits a superconductivity regime separating the Fermi liquid and the Mott insulating phase [15]. The interplay between geometric frustration and strong electronic interactions is considered crucial for the possibility of the quantum spin liquid phase and unconventional critical behavior exhibited at the MIT itself [16].
On the theoretical side, dynamical mean-field theory (DMFT) [17] and its cluster extensions [18,19] have been established as powerful tools for the study of strongly correlated materials. Over the past decades, DMFT has been used intensively to investigate the Hubbard model [20], the "simplest nontrivial" model for correlated electrons. Along with other numerical approaches, DMFT has revealed the MIT phase diagram of the unfrustrated square lattice Hubbard model at half-filling [17,[21][22][23][24], as well as of extended Hubbard models for multi-orbital systems, resembling transitionmetal oxides [25][26][27][28][29][30]. DMFT studies have furthermore been performed for electron systems with a frustrated lattice geometry. Results for diagonally frustrated twodimensional square as well as triangular lattices have been obtained from DMFT studies with exact diagonalization [31,32], Hirsch-Fye quantum Monte Carlo (QMC) [33], or continuous-time QMC cluster impurity solvers [34][35][36]. With frustration, not only is the position of the van-Hove singularity shifted, and thus antiferromagnetic fluctuations are suppressed at half-filling, several other important aspects such as the re-entrance to insulating behavior [13,33,37] or unconventional superconductivity upon hole doping [36,38] have been discussed. However, due to a severe minus-sign problem in the QMC simulations, a systematic cluster DMFT study of the evolution of the MIT phase diagram from the unfrustrated limit to the strongly frustrated triangular lattice is still lacking, especially for the evolution of the MIT phase boundary upon increasing the cluster size at low temperature, which may encode changes in the MIT physics.
Here, we aim at providing such a systematic study by employing cluster DMFT simulations based on the dynamical cluster approximation (DCA) with continuous-time QMC as cluster solvers [39] to analyze the evolution of the MIT phase boundary in the anisotropic twodimensional triangular lattice Hubbard model. We focus on the low-temperature regime, for which our results reveal a qualitative difference in the phase boundary between the unfrustrated and the fully frustrated triangular lattice, suggesting a connection between the MIT on the frustrated triangular lattice and the possible emergence of a spin liquid phase in a regime, where conventional magnetic order is suppressed by the interplay of geometric frustration and electronic correlations.
The rest of the paper is organized as follows: Section II introduces the model and numerical methods employed in this study. In Sec. III, the MIT phase boundaries for different degrees of frustration and cluster sizes are presented. In Sec. IV, we discuss the changes in the phase boundary as the degree of frustration increases from several physical perspectives. Section V contains our conclusion and outlook. Finally, an appendix provides quantitative details on the sign problem in our QMC simulations.
II. MODEL AND METHODS
A. Model
We consider the half-filled one-band Hubbard model on the triangular lattice, described by the Hamiltonian
H = − ij σ t ij c † iσ c jσ + U i n i↑ n i↓ ,(1)
where ij denotes nearest neighbor sites and the hopping parameter t ij equals either t or t , as sketched in Fig. 1(a), with the hopping t along the x-direction varied from 0 to t. The unfrustrated limit t = 0 is topologically equivalent to a square lattice, while t = t is the fully frustrated triangular lattice case. U is the onsite Coulomb repulsion, which is varied across the MIT boundary.
B. Dynamical cluster approximation
To treat strong correlation effects, we employ the DCA [18], a cluster extension of the single-site DMFT [17], using continuous-time QMC as cluster impurity solvers [39]. Within this approach, the original lattice model is mapped onto a periodic cluster of size N c , dynamically embedded in a self-consistently determined mean-field bath. Spatial correlations inside the cluster are treated explicitly while those at larger distances are described on the mean-field level. Temporal correlations, essential for quantum criticality, are treated explicitly by the QMC solver both on the cluster as well as in the bath.
In DCA, the Brillouin zone (BZ) is divided into N c momentum patches, each of which is generally denoted as K, as shown in Fig. 1 (b)-(d) for N c = 3, 4, 6, respectively. The cluster self-energy is used to approximate the lattice self-energy, i.e. Σ(k, ω)
(a) (b) (c) (d)= Σ K (ω) if k ∈ K.
The DCA self-consistency condition is achieved when the cluster Green's function G K (iω n ) is equal to the coarse-grained lattice Green's function, where the latter is obtained from the coarse-graining process with the cluster self-energy Σ K (ω) viā
G K (iω n ) = N c N k∈K 1 iω n + µ − k − Σ K (iω n ) ,(2)
and the bare dispersion for the anisotropic triangular lattice is given by
k = −2t cos(k x ) − 4t cos √ 3 2 k y cos k x 2 .(3)
An arbitrary initial guess for the bare Green's function, usually the non-interacting one, is taken as the DCA input for the cluster impurity solver. The cluster self-energy is obtained after solving the impurity model, containing information about the correlations. It is then used to computeḠ K (iω n ) based on Eq. (2). The bare Green's function serving as the input for the next iteration is obtained via the Dyson equation as (Ḡ −1 K (iω n ) + Σ K (iω n )) −1 . Once convergence of the selfconsistent calculation is achieved, a final iteration is necessary to measure physical quantities.
The bottleneck of the DCA calculations resides in solving the cluster problem. Depending on the employed cluster solver, the computational effort scales either exponentially with the cluster size or to the cube of the inverse temperature times cluster size [39]. Especially for frustrated systems, the QMC minus-sign problem makes the calculations challenging [see Appendix A]. To obtain good-quality results, we employ two different continuoustime QMC cluster solvers in a complementary manner: for low temperatures we employ the hybridization expansion solver [CT-HYB] [40,41] while for large cluster sizes we use the interaction expansion solver [CT-INT] [42]. In particular, CT-HYB is used for N c = 1, 3 and 4 DCA simulations, for temperatures as low as T = 0.025t and t varying from 0 to t. CT-INT is used for N c = 6 DCA simulations for the isotropic triangular lattice (t = t), as well as for the unfrustrated case (t = 0) with N c = 16, extending thus beyond previous low-temperature results [23] with N c = 8 for the square lattice.
C. Total energy and entropy
Following Refs. [17,[43][44][45][46], we extract the system's entropy (per site) based on a thermodynamic integration of the total energy (per site), E(β), where β = 1/T denotes the inverse temperature. At half-filling and for β = 0, the entropy per site equals S(β = 0) = log 4 [17]. For β > 0, the entropy is then calculated as
S(β) = S(β = 0) + βE(β) − β 0 E(β )dβ .(4)
The total energy E(β) is obtained from the DCA measurements as follows: the kinetic energy is
ij σ t ij c † iσ c jσ = kσ k c † kσ c kσ , where c † kσ c kσ = 1 β iωn e iωn0 + iω n + µ − k − Σ K (iω n ) ,
while the interaction energy U i n i↑ n i↓ is calculated from the cluster double occupancy.
D. Determining the metallic/insulating state
We prefer to study the MIT based on directly accessible Matsubara frequency DCA data. The Mott MIT can be determined from the Matsubara Green's function by considering the expression
G(iω n ) = A(ω)dω iω n − ω ≈ −P A(ω)dω ω − iπA(ω = 0),(5)
where P denotes the principle value. In this expression, ω n is assumed to be small so that the Sokhotski-Plemelj
identity 1/(x − iγ) = P (1/x) + iπδ(x) (
where γ is an infinitesimal quantity) applies. Hence, the spectral weight for each momentum sector K at the Fermi level is given as A K (ω = 0) ≈ −Im G K (iω 0 )/π, with ω 0 = πT the lowest Matsubara frequency available at temperature T . Thus, in order to assess, if an energy gap opens at the Fermi level, one may consult the value of Im G(iω 0 ). However, since it is difficult to draw conclusions from a single data point, in practice we consider Im G(iω n ) at several low Matsubara frequencies and then extract the low-frequency tendency. As demonstrated in Fig. 2, a momentum sector is in a metallic state if Im G(iω n ) approaches a finite value as ω n → 0, whereas if it bends to zero, the sector is insulating. The critical value U c is estimated from the two closest U -values at which the sector changes from metallic to insulating behavior. We have verified that this approach, directly based on the DCA measurements, is consistent with the observation of the spectral function as obtained after an analytic continuation (not shown). The direct approach performs well at low temperatures such that the approximation in Eq. (5) is appropriate. Accordingly, in our study, we mainly focus on the low-temperature region, T ≤ 0.1t.
0.0 0.5 1.0 1.5 ω n (t) 0.5 0.4 0.3 0.2 0.1 0.0 Im G(iω n ) U =7.70t U =7.80t U =7.90t U =7.95t U =8.00t U =8.10t
We note that the Green's function does not usually behave in the same way for different momentum sectors. Therefore the general idea is to check all the momentum sectors for the MIT. However, in practice, the Γ sector [see Fig. 1] lies completely within the Fermi surface; it is always insulating and thus can be safely disregarded. For N c ≤ 4, both the K and M sector [ Fig. 1(b), (c)] contains part of the Fermi surface, they behave similarly and have the same U c for the MIT, thus we require to consider only one of these momentum sectors. For N c ≥ 6, there appears a sector-selective Mott transition [23], and therefore we require to consider all the cluster momentum points close to the Fermi surface [see Fig. 1
III. METAL-INSULATOR TRANSITION AND THE PHASE DIAGRAMS
In this section, we determine the MIT phase diagrams from DCA calculations upon varying the impurity cluster size N c . Although differences in the phase boundaries arise for different cluster sizes, a consistent picture of the phase boundary emerges as the cluster size increases.
Single-site DMFT is known to be exact in infinite dimensions [17] where only local correlations are present. However, for low-dimensional systems such as the triangular lattice (d = 2) considered in this study, spatial correlation effects are important, and need to be taken into consideration. In Fig. 3, we show how the shapes of the DCA phase diagrams crucially change when extending beyond single-site DMFT (N c = 1) to N c = 4 DCA calculations at t = 0 and t = t. Similar to Refs. [17,21], for each panel of Fig. 3, two curves U c1 (T ) and U c2 (T ) are shown that separate stable metallic (left) and insulating (right) regions, while in the intermediate coexistence region both metallic and insulating solutions are available, depending on the initial condition chosen when iterating the self-consistency loop.
When extending beyond N c = 1 DMFT to N c = 4 DCA, we find qualitative changes both in the shape of the phase boundaries and in the numerical values of the transition lines U c (T ). The effects of non-local correlations, that account for these changes, thus depend strongly on the degree of frustration. Considering the unfrustrated case, t = 0, our results in Fig. 3(a), (c) are compatible with previous results in the literature [21,22]. The sub-stantial decrease of the U c values from N c = 1 to N c = 4 DCA for the t = 0 case relates to the fact that antiferromagnetic (AFM) nesting vectors at the Fermi surface are included in the momentum patches of the N c = 4 cluster, such that AFM fluctuations are taken into account, causing a Slater mechanism to open a gap [1,22,24]. The change in the slope of the phase boundaries from negative [N c = 1 DMFT - Fig. 3(a)] to positive [N c = 4 DCA - Fig. 3(c)] is related to the change of the entropy in the insulating state: while within N c = 1 DMFT free local spins reside on the impurity site, N c = 4 DCA exhibits a cluster singlet state within the insulating phase [21].
Moreover, because of the perfect nesting, the U c value in the unfrustrated case is known to approach to 0 as N c → ∞, i.e. antiferromagnetic fluctuations open a gap at arbitrarily weak, finite U > 0 [24,47,48]. This is reflected by a decrease of U c by a factor of 2 between the N c = 1 and N c = 4 calculations [ Fig. 3(a),(c)]. However, upon going from N c = 4 to 8, U c increases, which is due to a momentum-sector-selective MIT [23]. For N c = 8, at T = 0.05t, momentum sectors K = (0, π) and (π, 0) open the gap first at U s ≈ 5.6t, while sector K = ( π 2 , π 2 ) needs a larger U c2 ≈ 6.4t to open the gap, and hence U c2 (at which all the momentum sectors are gapped) for N c = 8 is larger than the U c2 value for N c = 4 where there is no sector-selective MIT [22,23]. To clarify the situation, we performed DCA simulations with an even larger cluster size, N c = 16, for the t = 0 case at low temperature as T = 0.05t, and obtain U s ≈ 3.75t for momentum sectors K = (0, π) and (π, 0), and U c2 ≈ 5.2t for the momentum sector K = ( π 2 , π 2 ). Both values clearly fall below the corresponding values for N c = 8 [23]. We thus believe that, as long as the cluster size N c is large enough to observe the momentum-sector-selective MIT, the U c value for the transition decreases monotonically as N c increases.
The problem becomes more difficult, once frustration is introduced into the system by turning on finite values of t . Figure 3(b) shows that the phase diagrams for the fully frustrated system (t = t) are already different from the unfrustrated case on the single-site DMFT level. In particular, the frustrated case exhibits larger values of U c2 (namely, ∼ 10.2t for t = 0 versus ∼ 12.2t for t = t), the U c1 (T ) curve exhibits a different slope, and the coexistence region is considerately smaller, e.g. less than half the t = 0 value at T = 0.05t. In single-site DMFT, the only contribution to these difference is the shape of the free system's DOS. As shown in the insets of Fig. 3(a), and (b), the DOS in the t = 0 case exhibits a van Hove singularity right at the Fermi level, while it is shifted towards the upper band edge for t = t. In addition, the electronic bandwidth W in the frustrated case is slightly larger (W t =0 = 8t; W t =t = 9t). The free t = 0 electron system is thus more susceptible towards interaction effects, while the t = t case can reside within a more stable Fermi liquid state. These appear to be the main explanations for the differences seen between the two cases within single-site DMFT calculations. Considering larger-cluster DCA calculations, in which short-range correlations are accounted for, our N c = 4 DCA results exhibit for t = 0 an opposite slope for the U c2 (T ) lines compared to the single-site DMFT result [ Fig. 3(c)] and for t = t a coexistence region significantly larger than that in DMFT [ Fig. 3(d)]. Comparing the t = 0 and t = t cases, the qualitatively different U c2 (T ) slopes at N c = 4 can be understood from an increased cluster ground state degeneracy at t = t, as discussed in Sec. IV. We also notice that the decrease in the U c -range when going from N c = 1 to N c = 4 DCA for t = t is weaker than for the t = 0 case. This is in accord with the fact that the perfect AFM nesting in the unfrustrated case is no longer maintained as t → t.
To arrive at a consistent picture for the triangular lattice in the presence of finite cluster size effects, we collect in Fig. 4 the MIT phase diagrams for t = t, as obtained for several different cluster sizes N c = 1, 3, 4 and 6. We find that going beyond N c = 1 DMFT, the phase boundaries converge more rapidly than in the unfrustrated square lattice case. The shapes of the coexistence regime for N c = 3 and N c = 4 DCA are already similar, while the U c values converge quickly, e.g. U c ∼ 8.2t at T = 0.1t for N c = 3, 4 and 6. We note that for N c = 6 DCA, due to the large computational expense and the sign problem, we cannot go to lower temperatures. However, for the temperatures under investigation, the U c2 range appears well converged and in particular the negative slope of curves are consistently determined. This enhanced convergence shows that, unlike the unfrustrated case, where the AFM fluctuations dominate and open a gap at any U > 0 [24,47,48], in the strongly frustrated system, the magnetic fluctuations compete strongly with other interaction channels, such as charge and pairing, possibly giving rise to non-trivial physics such as a quan- tum spin liquid state in the insulating side.
Finally, we consider the evolution of the phase boundary upon varying the frustration by tuning t from 0 to t. Here, we restrict to N c = 4 DCA calculations due to the high computational cost of larger-cluster calculations. Figure 5 shows the evolution of the U c2 (T ) phase boundary from N c = 4 DCA, when the degree of frustration is increased. The phase boundary has clear positive slope at t = 0 and negative slope at t = t. For intermediate values of t , the slope changes gradually and at a critical ratio of t c /t between 0.8 and 0.9, it becomes nearly vertical. Similar slope changes has been found by Liebsch et al. [32]. The nearly vertical phase boundary was not observed in their calculations at t ≈ 0.8t, which may be due to the different numerical method (namely N c = 4 cellular DMFT with an exact-diagonalization solver) employed in Ref. [32]. Nevertheless, based on their results at t = 0.8t and t = t, the vertical behavior can appear for 0.8t < t < t.
Even though our calculations are restricted to the paramagnetic mean-field solution, the negative slope observed in Fig. 4 as well as the changes of the phase boundary and the vertical t c line in Fig. 5 imply important physics. As discussed further below, we expect that as long as the U c2 (T ) phase boundary exhibits a positive slope (such as at t = 0), AFM fluctuations still render the system unstable towards an AFM Mott insulator. However, when different fluctuations (spin, charge, even pairing) start to compete with each other, the system may enter a more exotic, paramagnetic Mott insulator without spontaneous symmetry breaking, such as a quantum spin liquid. We anticipate such behavior is related to the change of the slope in the phase boundary line for t beyond t c .
Other works, based on different methods [31,[49][50][51], also predict a t c around 0.8 to 0.85t for the spin liquid phase, which thus would be consistent with our finding.
IV. UNDERSTANDING THE PHASE DIAGRAMS
In this section, we fathom several aspects of the N c = 4 DCA simulations in order to obtain a more quantitative understanding of the numerical findings. In particular, we analyze the exact eigenstates of a N c = 4 real-space periodic cluster (without bath), the corresponding CT-HYB eigenstate visiting probabilities within the DCA simulations, and estimate the entropy and the (free) energy of the DCA system.
A. Exact diagonalization of the impurity cluster
We start by analyzing the exact low-energy eigenstates of the N c = 4 impurity cluster, obtained via exact diagonalization. Figure 6(a) shows the dependence of the four lowest energy states on the t /t ratio. Since the cluster is periodic, the hoppings are arranged equivalent to a 2 × 2 square lattice with diagonal frustration [cf. the inset of Fig. 6 (a)]. The low-energy states are illustrated in Fig. 6 (b): The ground state S 0 is composed out of a resonance between states with two singlets forming along nearest neighbor sites. The excited level T 0 has three degenerate states, corresponding to a triplet along with a singlet connecting nearest neighbor sites. As shown in Fig. 6 (a), S 0 and T 0 are the two lowest energy states over a wide range of t (0 ≤ t < 0.8). The energy levels S 1 and T 1 , on the other hand, consists of singlet and triplet states involving next-nearest neighbor sites with hopping t . The S 1 level reaches closer to the ground state only when t /t increases, in particular beyond 0.8.
This rather simple analysis turns out to be useful for understanding the behavior of the MIT phase diagrams obtained from the N c = 4 DCA simulations. Increasing the t /t ratio enhances the number of states at low energy: the ground state manifold changes from a single singlet to two-fold degenerate singlet states, and also the first excited level turns from three-to nine-fold degenerate triplet states when varying t from 0 to t. The higher degeneracy leads to a larger entropy of the insulating state at t = t than at t = 0, and hence the insulating state is more stable at high temperature [50]. This explains the negative slope of the U c2 (T ) MIT phase boundary; namely, as temperature decreases, the system goes through a MIT from an insulating to a metallic state in the fully frustrated system, whereas in the unfrustrated case, it is the opposite. Moreover, the changes in the singlet configurations in the ground state manifold suggests that the AFM ground state [captured by the state S 0 in Fig. 6 (b)] of the unfrustrated case will be increasingly competed by the S 1 state which instead exhibits diagonal (Color online) (a) Exact diagonalization results for the periodic Nc = 4 cluster for the evolution of the four lowest energy levels as t increases from 0 to t at half-filling. The inset shows the real-space cluster with the corresponding hopping amplitudes indicated. (b) Illustration of the four lowest energy levels S0, S1, T0 and T1. An ellipse denotes a singlet state (| ↑↓ − | ↓↑ )/ √ 2 among the two enclosed sites; a rectangle denotes the set of three degenerate triplet states, | ↑↑ ,| ↓↓ and (| ↑↓ + | ↓↑ )/ √ 2. Thus, while the S0 and S1 levels are non-degenerate, the degeneracy is three for the T0 and six for the T1 level.
singlets connecting next-nearest neighbor sites.
B. CT-HYB statistics
Besides the capability of going to low temperatures and large interaction strength, a further advantage of CT-HYB cluster solver in DCA calculations is the possibility to measure the visiting probabilities of the cluster eigenstates during the QMC simulations [41]. Analyzing such data provides valuable insight into the system's behavior at the MIT.
Based on the exact diagonalization results in Fig. 6, we measured the visiting probabilities for each cluster eigenstate at the metallic and insulating states closest to the U c2 (T ) phase boundary for the N c = 4 DCA simulations. Figure 7 shows the dependence of the eigenstate visiting probabilities on the t /t ratio at T = 0.05t. We find that S 0 is the most visited state for all values of t . The visiting probability of S 1 on the other hand vanishes for t = 0 but increases and eventually takes on the same value as for S 0 in the fully frustrated case. The other high-energy states (the triplets T 0 , T 1 and all other states) have smaller visiting probabilities, and we thus plot only the sum of all remaining states as the "residual" curve in Fig. 7. There are two t values which deserve further attention in Fig. 7: (i) beyond t ≈ 0.8t, the weight for the state S 1 starts increasing, marking the onset of the role of frustration, (ii) for the insulating solutions [ Fig. 7(a)], the state S 1 exceeds the residual weight beyond t ≈ 0.9t. A value of t /t between 0.8 and 0.9 determining the importance of frustration is consistent with t c ≈ 0.8t for the change of the phase boundary slope [cf. Fig. 5]. We thus believe that this range contains the critical t value beyond which the AFM order is suppressed, allowing for a paramagnetic Mott insulating phase [31,[49][50][51].
C. Entropy and free energy
To further understand the MIT phase boundary slope, we also calculate the entropy and the corresponding free energy of the system. The entropy is obtained from Eq. (4), which requires the temperature dependence of the total energy and hence is computationally rather expensive. A less accurate approach is to approximate the entropy of the system by the impurity cluster entropy, which is calculated as S imp = −Tr(ρ imp logρ imp ), wherê ρ imp denotes the density matrix of the impurity cluster and is easily accessed in the CT-HYB solver.
We show in Fig. 8(a), (b) both types of entropy as functions of temperature at t = 0 and t = t. For the unfrustrated case [panel (a)], there are only results available for the metallic phase because there is almost no coexistence region at U = 5t [ Fig. 3(c)]. However, there are both metallic and insulating curves available for the t = t case [panel (b)] due to the wider coexistence region. The impurity entropy S imp is seen to overestimate the lattice entropy by a constant shift, except for the metallic case at low temperatures for t = t, where a sizeable jump appears in S imp . Nevertheless, over a wide range of parameters, the impurity entropy can be usefully employed for a (fast) estimation of the lattice entropy, if the constant shift is known.
The observed increase in both the total energy as well as the entropy, when the temperature T decreases for the case t = t [ Fig. 8(b), (d)] with a metallic initial state, implies that the final metallic state in this case may be metastable. The lower free energy of the insulating state suggests that the ground state in this regime be more likely insulating. Therefore, the first order transition line inside the coexistence regime may be located well to the left of the U c2 (T ) curve.
V. CONCLUSIONS
We have investigated the Mott MIT for the Hubbard model on the triangular lattice using DCA quantum Monte Carlo simulations. Controlling the degree of frustration by varying the hopping ratio t /t, we studied the evolution of the MIT phase boundary as the geometric frustration increases. Several cluster sizes have been used to convey a consistent picture. We furthermore analyzed the changes of the phase boundary from several perspectives: in the periodic cluster limit, by measuring the visiting probabilities of cluster eigenstates in the QMC simulations, and based on entropy and free energy considerations.
Several differences in the MIT phase diagrams between the unfrustrated (t = 0) and fully frustrated (t = t) cases are exposed. A central difference is the change of the U c2 (T ) phase boundary slope from positive at t = 0 to negative at t = t. From our results for various values of the ratio t /t, we estimated a critical value t c between 0.8t and 0.9t for which the phase boundary is close to vertical. By analyzing the cluster eigenstates and the QMC statistics, t c relates to the formation of the S 1 singlet along the direction of the hopping t , which eventually becomes degenerate with the S 0 singlet along the hopping t direction. The increase in the ground state degeneracy in the frustrated case weakens antiferromagnetic fluctuations considerably when t > t c , such that various short-range fluctuations compete. In essence, this may lead to a paramagnetic Mott insulator, which is possibly a quantum spin liquid state, as suggested in [31,[49][50][51]. On the other hand, the total energy and entropy results for the triangular lattice case suggest that the first order MIT phase boundary is located away from the DCA estimated U c2 (T ) line.
Implications of these results for the physics of frustrated electronic systems would be that observing such an anticipated change in the phase boundary slope can be a signal for the appearance of exotic Mott insulating phases. Several experimental phase diagrams for organic materials [13,15,37] show that the MIT phase boundary varies in a similar manner as our results when frustration is enhanced.
To improve the numerical results, we suggest to employ larger-cluster DMFT calculations with cluster sizes being multiples of three such that the triangular lattice frustration is maintained explicitly. Therefore, cellular DMFT [52] may be a more appropriate scheme, if one could find ways to overcome the minus-sign problem that plaques large-cluster-size and low-temperature simulations. In addition, calculations with symmetry breaking mean-field baths would be useful for specifying magnetic transitions and could provide a more direct prediction for the location of the loss of magnetic order in the insulating state. frustration implies a larger degeneracy on the impurity cluster, thus increasing the chance for swapping impurity electrons in the simulations. Fig. 9(b) shows the temperature dependence of the sign for the isotropic case (t = t). Interestingly, sgn is seen to decrease along with temperature only for T ≤ 0.067t, while it increases below this temperature. This however does not contradict the fact that low temperature worsens the average sign; instead this dip-feature signals the MIT, which occurs at T ∼ 0.067t. The insulating state exhibits a smaller sign due to its closeness to the atomic limit, and the visiting probability is distributed over fewer impurity eigenstates, such that there is a larger chance for sign changes in the QMC probability distribution.
FIG. 1 .
1(Color online) (a): Anisotropic triangular lattice with nearest neighbor hopping t along x direction and t along the other directions. (b)-(d): Brillouin zone (BZ) and momentum patches with Nc = 3, 4, and 6 clusters employed in the DCA simulations. Each patch is represented by the corresponding cluster momentum (solid dots -magenta online).
FIG. 2 .
2(Color online) Imaginary part of the Green's function in the K = M sector [cf. Fig. 1(c)] at the three lowest Matsubara frequencies, calculated for t = 0.9t at T = 0.1t within Nc = 4 DCA. The value of Uc for the MIT is estimated between 7.90t and 7.95t.
site: t =tFIG. 3. (Color online) MIT phase diagrams calculated using Nc = 1 DMFT (the first row) and Nc = 4 DCA (the second row) for the unfrustrated (t = 0) and fully frustrated (t = t) triangular lattice. The coexistence regime is bound by the Uc1(T ) (black squares) and Uc2(T ) (red circles) phase boundaries. The leftmost (rightmost) side of the phase diagrams is metallic (insulating). The insets in panels (a) and (b) show the non-interacting system's DOS filled up to the half-filled level, for t = 0 and t = t, respectively.
online) MIT phase boundaries for t = t from DCA calculations with different impurity cluster sizes (Nc = 1, 3, 4 and 6). For Nc = 6, only the Uc2(T ) phase boundary is available.
online) Evolution of the Uc2(T ) phase boundary as t increases from 0 to t within Nc = 4 DCA calculations. To ease comparison of the slopes for different values of t , the U values are shifted by U0 = Uc2(T = 0.1t) = 5.15t, 6.05t, 7.25t, 7.475t, 7.925t and 8.25t for t = 0, 0.5t, 0.75t, 0.8t, 0.9t and t, respectively.
FIG. 6. (Color online) (a) Exact diagonalization results for the periodic Nc = 4 cluster for the evolution of the four lowest energy levels as t increases from 0 to t at half-filling. The inset shows the real-space cluster with the corresponding hopping amplitudes indicated. (b) Illustration of the four lowest energy levels S0, S1, T0 and T1. An ellipse denotes a singlet state (| ↑↓ − | ↓↑ )/ √ 2 among the two enclosed sites; a rectangle denotes the set of three degenerate triplet states, | ↑↑ ,| ↓↓ and (| ↑↓ + | ↓↑ )/ √ 2. Thus, while the S0 and S1 levels are non-degenerate, the degeneracy is three for the T0 and six for the T1 level.
FIG. 7 .
7(Color online) Dependence of the cluster eigenstates visiting probabilities on t /t. The S0 and S1 states are illustrated inFig. 6(b), while the residual weight is defined as the the sum of the visiting probabilities for all remaining states. The data shown are taken from calculations at T = 0.05t, with U chosen at the insulating [panel (a)] or metallic [panel (b)] states closest to the Uc2(T ) phase boundary. The corresponding pairs of U/t in use are (4.6, 4.8), (5.7, 5.8), (7.1, 7.2), (7.4, 7.45), (7.95, 8.0) and (8.4, 8.45) for t = 0, 0.5t, 0.75t, 0.8t, 0.9t and t, respectively.
FIG. 8 .
8(Color online) Temperature dependence of the entropy S, total energy E, and the free energy F per lattice site. The left (right) column corresponds to t = 0 (t = t). Panels (a) and (b) show the lattice entropy S lat and the impurity entropy Simp. Panels (c) and (d) show the total energy E(T ) and the free energy F (T ). The labels (M) and (I) denote the initial conditions as metallic and insulating states, respectively. Each pair of panels (a,b) and (c,d) shares their legends.
FIG. 9 .
9(a) Dependence of the average sign, sgn , on the degree of frustration, the t /t ratio, at T = 0.067t for U values chosen close to the phase boundary on the insulating side. (b) Temperature dependence of sgn for t = t and U = 8.35t.
(d)] for the MIT.
ACKNOWLEDGMENTSWe thank V. Dobrosavljevic and K. Kanoda for valuable discussions on the nature of the Mott transition and organic materials and also acknowledge discussions with A. Georges, E. Gull, A. Liebsch, A. J. Millis, T. Pruschke and Chuck-Hou Yee. X.Y.X. and Z.Y.M. are supported by the National Thousand-Young-Talents Program of China, their computations were preformed on TianHe-1A, the National Supercomputer Center in Tianjin, China. K.S.C. is supported by the DFG under the grant number AS120/6-2 (Forschergruppe FOR 1162). H.T.D. and S.W. acknowledge support from the DFG within projects FOR 1807, RTG 1995, and WE3649/3-1, as well as the allocation of computing time at JSC Jülich and RWTH Aachen University through JARA-HPC. We used the code for the CT-HYB solvers[40]from the TRIQS project[53], from ALPS-2.0[54,55]and the one written by P. Werner and E. Gull, based on the ALPS-1.3 library[56].Appendix A: The sign problemIn QMC simulations of fermionic systems, the sign problem is the main challenge that prohibits the study of large-scale systems and low temperatures. In fact, the sign problem belongs to the class of NP-hard problems[57]. Swapping two fermions causes a change of sign in the overall wavefunction, thus when constructing the QMC probability distribution, it is possible that the probability is not positive definite, which can lead to measurements with large statistical errors[57,58]. In many cases, the average sign of the probability distribution, sgn , decreases exponentially as the inverse temperature β = 1/T or the system size increases[39,57], causing severely large errors in QMC measurements.While DMFT is an appropriate approximation to avoid the sign problem, the issue cannot be completely resolved; in some cases such as for frustrated systems examined within this study, the sign problem in the impurity solver becomes severe. Therefore, for N c = 4 DCA calculations using the CT-HYB solver, our temperatures are restricted above T = 0.025t for the fully frustrated case (t = t ). Since the complexity of the CT-HYB solver increases exponentially with the cluster size, given limited computational resources, it is not possible to extend this solver to larger cluster sizes. The CT-INT solver was used instead for the N c = 6 DCA calculations, which is exposed more to the sign problem, thus in these DCA calculations we only considered temperatures larger than T = 0.07t.To quantify the sign problem in this study, we show inFig. 9the average sign, sgn , that we observe in our N c = 4 DCA simulations. At intermediate temperatures, T = 0.067t, the average sign decreases as the t /t ratio increases towards 1 [panel (a)]. Frustration is the main cause for the decrease of sgn , as a higher degree of
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| [] |
[
"Automatic end-to-end De-identification: Is high accuracy the only metric?",
"Automatic end-to-end De-identification: Is high accuracy the only metric?"
] | [
"Vithya Yogarajan \nDepartment of Computer Science\nThe University of Waikato\nHamiltonNew Zealand\n",
"Bernhard Pfahringer \nDepartment of Computer Science\nThe University of Waikato\nHamiltonNew Zealand\n",
"Michael Mayo \nDepartment of Computer Science\nThe University of Waikato\nHamiltonNew Zealand\n"
] | [
"Department of Computer Science\nThe University of Waikato\nHamiltonNew Zealand",
"Department of Computer Science\nThe University of Waikato\nHamiltonNew Zealand",
"Department of Computer Science\nThe University of Waikato\nHamiltonNew Zealand"
] | [] | De-identification of electronic health records (EHR) is a vital step towards advancing health informatics research and maximising the use of available data. It is a two-step process where step one is the identification of protected health information (PHI), and step two is replacing such PHI with surrogates. Despite the recent advances in automatic de-identification of EHR, significant obstacles remain if the abundant health data available are to be used to the full potential. Accuracy in de-identification could be considered a necessary, but not sufficient condition for the use of EHR without individual patient consent. We present here a comprehensive review of the progress to date, both the impressive successes in achieving high accuracy and the significant risks and challenges that remain. To best of our knowledge, this is the first paper to present a complete picture of end-to-end automatic deidentification. We review 18 recently published automatic de-identification systems -designed to de-identify EHR in the form of free text-to show the advancements made in improving the overall accuracy of the system, and in identifying individual PHI. We argue that despite the improvements in accuracy there remain challenges in surrogate generation and replacements of identified PHIs, and the risks posed to patient protection and privacy. | 10.1080/08839514.2020.1718343 | [
"https://arxiv.org/pdf/1901.10583v1.pdf"
] | 59,413,754 | 1901.10583 | 4e9f83aab2b1eab3183530b0597ca2f7b18406df |
Automatic end-to-end De-identification: Is high accuracy the only metric?
Vithya Yogarajan
Department of Computer Science
The University of Waikato
HamiltonNew Zealand
Bernhard Pfahringer
Department of Computer Science
The University of Waikato
HamiltonNew Zealand
Michael Mayo
Department of Computer Science
The University of Waikato
HamiltonNew Zealand
Automatic end-to-end De-identification: Is high accuracy the only metric?
ARTICLE HISTORY Compiled January 31, 2019automatic de-identificationpatient privacyelectronic health recordsfree textaccuracychallengesriskssurrogate generationre-identificationusabilityreview
De-identification of electronic health records (EHR) is a vital step towards advancing health informatics research and maximising the use of available data. It is a two-step process where step one is the identification of protected health information (PHI), and step two is replacing such PHI with surrogates. Despite the recent advances in automatic de-identification of EHR, significant obstacles remain if the abundant health data available are to be used to the full potential. Accuracy in de-identification could be considered a necessary, but not sufficient condition for the use of EHR without individual patient consent. We present here a comprehensive review of the progress to date, both the impressive successes in achieving high accuracy and the significant risks and challenges that remain. To best of our knowledge, this is the first paper to present a complete picture of end-to-end automatic deidentification. We review 18 recently published automatic de-identification systems -designed to de-identify EHR in the form of free text-to show the advancements made in improving the overall accuracy of the system, and in identifying individual PHI. We argue that despite the improvements in accuracy there remain challenges in surrogate generation and replacements of identified PHIs, and the risks posed to patient protection and privacy.
Introduction
The application of machine learning research using EHR has the potential to revolutionise health care. There is an abundance of health data available and maximising the utility of this data will result in improving health care, especially in patient care, medical outcomes, surgical outcomes, risk prediction, clinical decision support and medical diagnosis.
Use of patient data typically requires individual patient consent. For research, without individual consent, the data must be de-identified such that the patient's identity or privacy is not breached. Obtaining individual patient consent for massive datasets is time-consuming and is a challenging task. Hence there is a great interest in automating the de-identification process such that EHR can be used in research to improve the health care and quality of patient care without compromising the identity of the patient. CONTACT Vithya Yogarajan. Email: [email protected] There is growing interest internationally in applying big data techniques to electronic health records. However, privacy laws in many jurisdictions -including New Zealand's Health Information Privacy Code and the United States Health Insurance Portability and Accountability Act (HIPAA)-require accurate de-identification of medical documents (such as discharge summaries and electronic health records) before they can be shared outside of their originating institutions.
The sharing of records is crucial for advancing health research. For example, the 2014 Heart Disease Risk Factors Challenge involved participating research groups attempting to predict heart disease risk factors in diabetic patients from longitudinal clinical narratives. As noted above, such a challenge would not have been possible under United States law if the narratives (1,304 medical records from 296 diabetic patients) were not de-identified first. In this case, the records were de-identified manually by multiple medical professionals. Since most institutions will not be able to afford the costs of manual de-identification, automating the process is crucial therefore for sharing data and advancing health research.
We present our findings in three main groups: achievements, challenges, and risks, associated with generating an automatic de-identification. Achievements of automatic de-identification primarily focus on identification of PHI in EHR. Challenges are associated with the surrogate generation and replacement. Risks outline the issues relating to re-identification and medical correctness and usability of de-identified data.
The rest of the paper is structured such that a brief background on de-identification is presented in section 2, achievements of recent de-identification systems in section 3, challenges in section 4, risks in section 5 and finally a discussion in section 6.
Background on De-identification
De-identification is a two-step process where PHI is identified in EHR and replaced with suitable surrogates such that patient privacy and confidentiality is not at risk. Figure 1 provides a detailed example of de-identification of EHR, where original patient discharge notes are de-identified. This figure also outlines the two-step de-identification process. It is important to note that step two requires the use of appropriate surrogates to replace the original PHI and hence automating surrogate generation is a vital step in creating a longitudinal narrative end-to-end automatic de-identification system. Although EHR is in the form of tabular structures (i.e. tables), free-form narratives, and images, this study focuses on medical data in the free form longitudinal text.
De-identification should be considered a means of satisfying rather than circumventing the legal and ethical requirements created to protect patient privacy across the world. Individual countries have different requirements, for example HIPAA in the United States of America (Garfinkel (2015); Stubbs, Uzuner, Kotfila, Goldstein, and Szolovits (2015a); Yogarajan, Mayo, and Pfahringer (2018b)), the European Union's new General Data Protection Regulation (GDPR) (Brasher (2018) ;Polonetsky, Tene, and Finch (2016)), and New Zealand's own health information privacy code (Health & Disability Commissioner (2009); Office of the Privacy Commissioner (2013); Yogarajan, Mayo, and Pfahringer (2018a)). HIPAA is arguably the gold standard, and both HIPAA's regulations on Expert Determination and Safe Harbor are used as the standard benchmarks for de-identification of EHR in the form of free text. We use HIPAA's Safe Harbor guidelines as the basis of assessing the accuracy of the de-identification systems (for details on HIPAA's Safe Harbor and the 18 categories see Yogarajan et al. (2018b)). A superior de-identification system will not only meet legal requirements but will also help build societal consent by assuring the public that their privacy and medical data will be protected. This consent is vital if large-scale research involving medical records is to be accepted in the same way as, for example, Statistics New Zealand's Integrated Data Infrastructure. Acceptance of the latter is arguably in part due to measures were taken by Statistics New Zealand to de-identify data (Ragupathy and Yogarajan (2018); Statistics New Zealand (2016)).
Achievements
In the recent years there has been a substantial development in natural language processing tasks, including de-identification, primarily due to the development in deep learning (Dalianis (2018); Goldberg (2017)). Improving accuracy of de-identification of EHR -step 1 from Figure 1 -has been the primary focus of research in this field, and several de-identification systems have achieved remarkable success. The main reason for such development is the EHR de-identification competitions (Kumar, Stubbs, Shaw, and Uzuner (2015); Stubbs, Filannino, and Uzuner (2017);Stubbs, Kotfila, Xu, and Uzuner (2015); Uzuner (2015c, 2017); Uzuner and Stubbs (2015)). For a complete review of these competitions and significance see Yogarajan et al. (2018b). It is important to note that these competitions provide open access data which allowed the research to grow rapidly. In addition to these competition datasets, the MIMIC dataset (Goldberger, Amaral, Glass, Hausdorff, Ivanov, Mark, Mietus, Moody, Peng, and Stanley (2000); Johnson, Pollard, Shen, Li-wei, Feng, Ghassemi, Moody, Szolovits, Celi, and Mark (2016)) is another open access dataset that has been used to develop de-identification systems. Table 1. De-identification systems summary. Machine learning indicates systems that uses machine learning techniques only. Hybrid systems indicates systems that used a combination of machine learning techniques and hand crafted rules.
Architecture
De-identification system Machine learning S1 (Zhao, Zhang, Ma, and Li (2018)), S2 (Chen, Cullen, and Godwin (2015)) S3 (Dernoncourt, Lee, Uzuner, and Szolovits (2017)), S4 (Yadav, Ekbal, Saha, Pathak, and Bhattacharyya (2017)), S5 ), S6 ) Hybrid S7 (Yang and Garibaldi (2015)) S8 (Liu, Tang, Wang, and Chen (2017)) S9 (Lee, Dernoncourt, Uzuner, and Szolovits (2016)) S10 (Dehghan, Kovacevic, Karystianis, Keane, and Nenadic (2015)) S11 (Yang and Garibaldi (2015)) S12 (He, Guan, Cheng, Cen, and Hua (2015)) S13 (Liu, Chen, Tang, Wang, Chen, Li, Wang, Deng, and Zhu (2015)) S14 (Phuong and Chau (2016)) S15 (Bui, Wyatt, and Cimino (2017a)) S16 (Jiang, Zhao, He, Guan, and Jiang (2017)) S17 (Lee, Wu, Zhang, Xu, Xu, and Roberts (2017)) S18 (Shweta, Kumar, Ekbal, Saha, and Bhattacharyya (2016)) In this section, we outline the most significant achievement of automating end-toend de-identification system: improving accuracy. It has been argued that as far as de-identification is concerned, perfection cannot be achieved; however, 95% accuracy is considered to be the rule of thumb and universally accepted value ; ). We use 18 de-identification systems, as outlined in Table 1, to show that several of these systems have achieved an overall F-measure of ≥ 0.95. Also, we outline the fact that these systems have also identified the majority of the HIPAA PHI with the F-measure of ≥ 0.95. These achievements have been a significant milestone in automating end-to-end de-identification of EHR and has been a significant breakthrough in this area of research. This section will be structured such that a brief overview of the datasets will be provided. This is followed by an outline of the systems that obtained an overall Fmeasure of ≥ 0.95, and also a summary of systems that recorded F-measure of ≥ 0.95 for individual PHIs. The techniques and datasets used for these results are also outlined.
Overview of Datasets
In this section, we provide a quick overview of the most commonly used datasets by the 18 de-identification systems outlined in Table 1. The most commonly used datasets were introduced by the following three competitions: the 2006 Informatics for Integrating Biology and the Bedside (i2b2) competition (Uzuner, Luo, and Szolovits (2007)); the 2014 i2b2/UTHealth shared task ; Stubbs and Uzuner (2015c)); and the 2016 Centers of Excellence in Genomic Science (CEGS) and Neuropsychiatric Genome-Scale and RDOC Individualized Domain (N-GRID) shared task ; ).
The dataset for the 2006 competition included 889 unannotated discharge summaries, also used for smoking challenges, manually broken into sentences and tokenised. The dataset for the i2b2/UTHealth shared task 2014 included 2 -5 records for each patient over a fixed period and was obtained from two large academic tertiary hospitals: Massachusetts General Hospital (MGH), and Brigham and Women's Hospital (BWH) ?. It includes 296 diabetics patients with 1304 longitudinal medical records and contains three cohorts based on the diagnosis of coronary artery disease (CAD) (Kumar et al. (2015); ; Stubbs and Uzuner (2015a,c)).
The 2016 CEGS N-GRID shared task used psychiatric data, making it the first ever competition to use psychiatric intake records ; ). The data for the 2016 competition reflected the records "as is" ; Uzuner, Stubbs, and Filannino (2017)): the state at which data was received from the sources. Unlike other medical data, such as that of the 2014 challenge, psychiatric data contains an abundance of information related to the patients such as places lived, jobs held, children's ages, hobbies, traumatic events, patients' relatives' relationship information, and pet names. This makes it a much more significant challenge to deidentify (Bui, Wyatt, and Cimino (2017b); ).
MIMIC III is one the most extensive publicly available database (Goldberger et al. (2000); Johnson et al. (2016)). It contains health records of approximately sixty thousand admissions of patients in critical care units. The database includes information such as demographics, laboratory test results, procedures, medications, and physician notes.
Also, there were other data used by individual systems such as Chinese data by S1 and Dutch data by Menger, Scheepers, van Wijk, and Spruit (2018). Although we do not describe these systems in this paper, it is important to note that these systems also presented with high F-measure. Table 2 presents the de-identification systems that recorded an overall F-measure of ≥ 0.95. Each entry also outlines the datasets used to obtain such results. The highest recorded overall F-measure was obtained by S3 and S9 using MIMIC III dataset. One possible reason for such high accuracy obtained using MIMIC III dataset could be due to the duplicates created by cut and paste (Gabriel, Shenoy, Kuo, McAuley, and Hsu (2018)). The i2b2 2014 is the most commonly used dataset. It is important to point out S7 as the best performing system from the actual i2b2 2014 competition. As shown in Table 2 there has been a substantial improvement in F-measure since the 2014 competition. Unfortunately, this might partly be due to overfitting of the now known and freely available test set. Table 3 provides an outline of the techniques used by the de-identification systems in Table 2. Machine learning only systems favour deep learning approaches. Hybrid systems with the incorporation of handcrafted rules and dictionary-based approaches are also used by a couple of the de-identification systems to achieve high F-measure.
Overall F-measure of de-identification system
F-measure of individual PHIs
In this section, we provide an overview of the systems that recorded F-measure ≥ 0.95 for individual HIPAA PHIs. Where the F-measure was < 0.95, the highest recorded score is presented. We also provide some possible issues relating to the PHIs that have lower F-measure. This section also provides an overview of the i2b2 PHIs. These are the additional PHIs introduced by the i2b2 2014 and 2016 competitions ; Stubbs and Uzuner (2015a,c)). Although legally, as per HIPAA rules, there are no requirements for these additional PHIs to be de-identified, the competition organisers argue that these extra PHIs provide more security over re-identification of data. Since i2b2 2014 and 2016 datasets are most commonly used in the advancement of de-identification research, we feel it is vital to also present the successes in these additional PHIs. Table 4 provides an overview of the systems that achieved high F-measures. It also outlines the datasets that were used to obtain such results. Except for Fax and Device all other PHIs have obtained an F-measure of ≥ 0.95. This is an incredible achievement and a significant improvement to the results obtained in i2b2 competitions (Yogarajan et al. (2018b)). Although Fax and Device recorded < 0.95 F-measure, it is important to note that only a very few instances (< 10) were found in the datasets for both of these PHIs. This makes improving the accuracy using machine learning approaches Table 5 provides an overview of the techniques used to obtain the F-measures presented in Table 4. As observed in Table 3 there is a clear increase in deep learning methods. With a combination of handcrafted rules, de-identification systems have achieved high F-measure for the majority of the PHI. In several cases, hand-crafted rules only also achieve high F-measure. Good examples are License and Email, where regular expressions work very well. Table 6 provides an overview of F-measures for i2b2 introduced extra PHIs. These PHIs are not part of the legal requirement as per HIPAA regulations but are additional security for ensuring that patient privacy and confidentiality are maintained. Compared to the recorded results in the i2b2 2014 and 2016 competitions, there is a substantial increase in the F-measure. Clearly, Organisation, Location-others, Profession and Country are the PHIs yet to reach the 0.95 F-measure. These were also the PHIs that recorded a very low F-measure in both competitions (see Yogarajan et al. (2018b) for details). The main issue with Country and Organisation is that the data is very sparse. Location-others only occurs in thirteen instances in the dataset. The sparsity of the data and the very low frequencies of same values make achieving higher F-measures very hard. However, there is still an improvement in results compared to that recorded in the competitions.
DATE
Date CRF + Rules + Dictionary (S12); Bi-LSTM (S7); CRF + Rules + Keywords (S11); Hidden Markov model (HMM-DP) (S2); CRF + Rules (S10, S14, S15, S17); LSTM (S16); RNN (S3, S18); LSTM + Rules (S9); CRF (S4) NAME All names RNN (S3, S18) AGE Age Bi-LSTM (S7); CRF + Rules (S14, S15, S17); LSTM (S16); RNN (S1, S3, S18) CONTACT Phone CRF + Rules + Keywords (S11); RNN (S3); CRF + Rules (S14, S17); LSTM + Rules (S9) Fax HMM-DP (S2) Email
Rules (S10, S11); HMM-DP (S2) ID Medicalrecords CRF + Rules + Keywords (S11); LSTM + Rules (S9) IDNUM RNN (S3) Device HMM-DP (S2); RNN (S3) License
Rules (S17) LOCATION all RNN (S3); LSTM + Rules (S9)
In summary
This section showed the achievements in automating de-identification research, with substantial improvement in F-measure of identifying PHI in the overall systems and individual PHIs (notably the HIPAA required PHIs).
Challenges
The biggest challenge in automating end-to-end de-identification is surrogate generation and surrogate replacement (step 2 in Figure 1). At first sight, this appears to be superficially simple when compared to step 1. However, when one considers it in detail, there are many complex subtleties associated with the surrogate generation and surrogate replacement. Unlike the research towards increasing accuracy in identifying PHI, as seen in section 3, this is an area where very little research progress has been made. There have been only a few papers published in the recent years regarding surrogate generation and surrogate replacement for the de-identification problem, with the schema developed in the 2014 i2b2 competitions being the prominent one to date ; Stubbs, Uzuner, Kotfila, Goldstein, and Szolovits (2015b)). PHI are categorised into explicit identifiers and quasi-identifiers. Explicit identifiers such as name, phone number and social security number are directly linked to a patient. Quasi-identifiers such as age, gender, race and zip code are not directly connected to a patient but can be linked to external data sources and consequently be used to identify a patient, hence posing the same risk to patient privacy as explicit identifiers.
In this section, we present examples of standard practices used in surrogate replacement and challenges faced. Automation in the surrogate generation is arguably still a very challenging and unsolved problem. Table 4, but were introduced by i2b2 competitions as additional categories ; ). It also provides the techniques used to achieve these F-measures. Table 7 provides an outline of some PHIs and the standard practices used in a surrogate generation while creating de-identified data. Although all of these practices are based on hand-crafted rules and pre-compiled tables, there was also a need to do a manual check after the data is de-identified. This is to ensure that medical correctness, readability and consistency are maintained across the health data. Table 7 also indicates where manual checking after de-identification was required. Surrogates need to maintain the same form as the original, and where possible same internal temporal and co-reference relationships. Also, as illustrated in Figure 1 semantic links must be maintained, for example between LOCATION and PROFESSION. It is important to note that it is relatively easy to create surrogates randomly and maintain co-references for PHONE, FAX, URLs and ID ; ). Any ambiguous words appearing as part of a name, medical term or acronym were replaced using a set of hand-crafted rules (Pantazos et al. (2017)). This is primarily because, in medicine, it is common to have diseases, signs and symptoms being named after the person first describing it. One such example is "Aaron" which can refer to a name of a person, or be part of a medical term: Aaron sign referring the pain felt in the epigastrium. Table 7. Common practices used in surrogate generation and replacement of PHI as outlined in (Johnson et al. (2016);Pantazos, Lauesen, and Lippert (2017); ; Table 7 provided an overview of techniques used in the surrogate generation and replacement. However, there are many practical issues with some of these rules and techniques which creates challenges in maintaining medical correctness and usability of de-identified data in health advancement research. Moreover, it is important to note in most cases there was a need to manually check the surrogate replaced data to ensure consistency and accuracy is maintained across patient data. When DATE is changed to just the year or randomly changed it removes inferrable information such as the "season" which could result in missing any pandemic outbreak (Li and Qin (2017)). There is a need to maintain the semantic link between the LO-CATION and DATE to ensure such information is not missed. Also, for PHIs DATE and AGE, medical correctness is a major issue. Birth dates have to be transformed such that the patient age is in a similar age range. Otherwise diagnosis patterns will become inapplicable. For example, a 20-year-old de-identified to be a 60-year-old will cause issues in medical diagnosis.
PHI categories
Issues and Challenges due to Surrogate replacement of PHI
When LOCATION such as zip code is replaced by random zip codes (even from a pre-compiled list), geographical information is distorted. For example, a patient living in a high socioeconomic area being moved to low decile area, or vice versa, will result in relevant information about the living conditions and life expectancy changing. This could mislead patient diagnosis, or miss vital information in patient care. In addition to socioeconomic issues relating to LOCATION, there is also ethnicity information. For example, in New Zealand, there are parts of the country, such as Northland, where there is known to be a higher population of New Zealand's indigenous Maori people.
If everyone from Northland is moved to another LOCATION or spread across several LOCATIONS, the ethnicity information is also lost in the de-identified data. It is very challenging to ensure such information is not lost without introducing systemic bias towards a sub-population, e.g. Maori people in the New Zealand example above.
With NAME, if the patient's name, for example, "John", is replaced by "Jack", then there is a need to ensure all of his medical records reflect this change. For instance, in addition to the free text data that was replaced, the change must also be made consistently across all of his longitudinal data, including but not restricted to his structured data and medical images. In addition, the name change should also reflect correctly on his family's medical records, i.e. his wife's records and his childrens. This does allow the consistency and medical correctness to be maintained in de-identified data (Pantazos et al. (2017)). The need to maintain consistency and medical correctness makes automating de-identification a very challenging task and does require manual checks and inputs (Pantazos et al. (2017); ). Also, in order to maintain readability, a patient name must be replaced by a new name that looks real and consistency should also be taken into consideration. For example, the frequency of the name in a database needs to be consistent. A rare name occurring more frequently will not look real.
One of the many challenges faced in de-identifying medical, free text data is ambiguous words. In many cases, it is challenging to differentiate between an everyday word, medical word and part of the patient name. This may result in errors with surrogate replacement where for instance a medical term is replaced by a person's name surrogate.
In summary
This section presented common practices used in surrogate generation and replacement, where most of the techniques rely on hand-crafted rules and pre-compiled tables. We outline some of the important challenges faced in this step of de-identification and argue that automation in surrogacy is still an open question with many obstacles to overcome.
Risks
De-identified data in addition to protecting patient privacy should also meet the following standards: medical correctness, readability and consistency across data (Pantazos et al. (2017)). Risks around de-identification of health data can be classified into two main areas: the risk of re-identification and the risk of losing usability, medical correctness and consistency across data. In this section, we provide a brief overview of these two areas.
Re-identification
Re-identification is a process where a person's identity is identified from the deidentified data. This does result in a serious breach of patient privacy and confidentiality. Explicit identifiers such as person's name and address can be considered obvious identifiers. However, even quasi-identifiers can result in re-identification of a person (Johnson, De Freitas, Glicksberg, Bobe, and Dudley (2018); Li and Qin (2017);Sweeney (2002)). There have been many examples of such occurrences where quasi-identifiers have been matched with external resources to identify patients. For example, it was proven that attributes such as gender, date of birth and zip code could be matched with external sources such as voting data to identify a patient (Li and Qin (2017); Sweeney (2002)). Also, other examples demonstrate that a combination of a small subset of quasi-identifiers, with or without other medical data, may even be enough to identify the individual patient and pose serious threat to patient privacy (El Emam, Jabbouri, Sams, Drouet, and Power (2006); Mayo and Yogarajan (2019)).
In addition to explicit identifiers and quasi-identifiers, there are also the sensitive attributes such as psychiatric diseases, HIV, and cancer, which patients are not willing to be associated with. Due to the specific nature of these sensitive attributes and the need for special care facility these attributes when combined with other identifiers makes re-identification of a patient much more feasible (Gkoulalas-Divanis, Loukides, and Sun (2014)).
The risk of re-identification is real and can lead to serious breaches of patient privacy and confidentiality. While designing an automatic de-identification system, it is important to consider the re-identification risk and take appropriate measures to minimise such risk. Also, there needs to be transparency in acknowledging such concerns. The main questions when it comes to re-identification are:
• what is the accepted level of risk with re-identification?
• who makes that decision, the de-identification system designers, the users or the patients?
There is no easy or correct answer to these questions, but they still need to be considered when designing a de-identification system. There is a need for human input in making such decisions and deciding the boundaries of acceptable risk associated with de-identification of a medical system.
Medical Correctness and Usability of De-identified Data
Maintaining medical correctness, consistency, readability and usability of data is a difficult problem and the risks associated with this are usually overlooked. Compared to de-identification of structured data, unstructured free text is very challenging. It contains medical information about a patient that needs to be preserved for medical correctness. However, it also contains personal details such as name, phone number, family members names and other personal identifying items. Although the accuracy of identifying PHI in such data has improved considerably, ensuring these PHI are replaced with appropriate surrogates, and medical correctness maintained, is an open question. This poses a great risk in using de-identified data for machine learning based health research, as the de-identified health records may compromise the accuracy and outcome of the resulting model. For example, accidentally replacing a word that resembles a name but is not a name (maybe an abbreviation for a disease, or a disease name itself) can result in readability and medical correctness errors (Pantazos et al. (2017)). The hope is that the original data and the de-identified data of a particular problem will result in the same outcome. However, there is no clear evidence that it does. In reality, the only way to check if it does or does not, is by building models for both versions of the data and comparing them.
Many of the surrogate replacements use randomised identifiers. However, in such cases, the readability and consistency are compromised (Pantazos et al. (2017)). Unless manually checked there is no guarantee these randomly replaced PHI makes much sense in the context and provide useful data outcomes.
Another significant risk is accidentally confusing patients. Lets say you have two patients in the same age range, both named Anne Smith, but one presenting with cancer and the other one with cardiovascular issues. Ensuring these two are kept separated across all of their data, especially longitudinal data, can be very hard. This will require using several PHI to match the person's identity. However, in this case, there is an increased risk of re-identification. This poses a question around confidentiality vs verifiability, and as a result, increases the risk. This problem cannot be readily solved by using unique identifiers (such as NHI numbers, date of birth or tax numbers) to match narratives, as automated de-identification systems by design prune such deidentifiers. Furthermore, HIPPAs Safe Harbor provision mandates the removal of such unique identifiers (Garfinkel (2015); ). Similarly, New Zealands Privacy Act and Health Information Privacy Code set strict limits on the assignment and use of unique identifiers (Health & Disability Commissioner (2009); Office of the Privacy Commissioner (2013)).
In Summary
This area outlined the two main risks associated with de-identification: the risk of re-identification and the risk of losing usability, medical correctness and consistency across data. Minimising the risk posed to patient privacy and confidentiality is vital. The risk of re-identification must be considered a severe threat when designing a deidentification system. The de-identified data must also maintain medical correctness, readability and consistency. The advancement of health research using de-identified data does rely on the usability of data and medical correctness of data. There is a need for a manual check to ensure that the de-identified data resembles the original data.
Discussion
To best of our knowledge, this is the first paper to present a complete picture of end-toend automatic de-identification of medical narratives. Noticeably, the majority of the research is done on improving the accuracy of PHI identification in the overall system and of individual PHI. We acknowledge the need for such research, and despite the recent advancements in this area, mainly due to the use of deep learning in natural language processing tasks, there is more room for improvement. At this stage the minimum requirement of 95% F-measure has been met by several systems, but this is only the minimum requirement. There is room for higher F-measures. Also, it will be nice to take these systems to the next level, where in addition to the open access data they use other sources of data. It will be interesting to see the adaptability of these systems.
One of the big downfalls to these systems is that they do not outline the surrogacy generation aspect of de-identification. However, de-identification is not just about identifying the PHI, but is also replacing the identified PHI with appropriate surrogates. As mentioned earlier there is very little research done in this area, and clearly, there are many challenges yet to be overcome. Also, most of the current practices are data specific and use hand-made rules and pre-compiled tables. This is far from achieving full automation in the de-identification problem, and there is an explicit acknowledgement of the need for manual checks after surrogate replacement. We encourage for more research in this area, where the priority is to address some of the challenges outlined in this paper and also to eliminate the need for manual checks.
The importance of automatic de-identification in advancing health research cannot be emphasised enough. However, there is still a need to be aware of the risks associated with designing such systems. This will ensure that risk to patient privacy and confidentiality is minimised while advancing the field of medicine through maximising the potential of EHR with the use of machine learning techniques. Also, it is vital that the medical correctness, consistency, readability and usability of data are all maintained such that the resulting de-identified data provides the parallel output to that of the original data. It must be pointed out that high accuracy of de-identification is directly proportional to medical correctness. It does become harder to maintain the medical correctness and usability of data when achieving high accuracy becomes the focus. Hence, de-identification of data does become a balancing act where barriers associated with risk and benefits must both be considered. This is another area where there needs to be more research done in proving that de-identified data is providing the same outcomes as the original data.
The challenges and risks associated with de-identification have opened new avenues of research in finding alternatives. One has to ask the question: "What if proper de-identification is impossible?". Guinney and Saez-Rodriguez (2018) proposes an alternative idea for sharing confidential data called "model to data" where the flow of information between data generators and modellers is reversed. Another idea presented by Vepakomma, Gupta, Swedish, and Raskar (2018) proposes a deep learning model which excludes the need to share raw patient data or labels. These are merely examples of alternatives, and are just the beginning of possibly solving the problem of sharing and using EHR without the risk to patient privacy and confidentiality.
Figure 1 .
1Example of an end-to-end de-identification process.
Table 2 .
2Table 3. De-identification systems with overall F-measure ≥ 0.95.De-identification systems with overall F-measure ≥ 0.95.
De-identification System
F-measure
i2b2 2006 i2b2 2014 i2b2 2016 MIMIC III
S1
0.9879
0.9805
S3
0.9785
0.9923
S4
0.9746
S5
0.9800
0.9600
S6
0.9770
S7
0.9573
S8
0.9511
S9
0.9926
De-identification Techniques
System
S1
Recurrent neural network (RNN) + statistical text skeleton approach.
S3
RNN
S4
Conditional Random Field (CRF)
S5
Transfer Learning
S6
Artificial neural networks (ANNs)
S7
CRF + Rule based + Dictionary based
S8
CRF + Rule based
S9
Long Short Term Memories (LSTMs) + human-engineered features
Table 4 .
4F-measure ≥ 0.95 for HIPAA categories for de-identification. On occasions where F-measure was not ≥ 0.95, the highest score is presented. CONTACT: URL and IP address; ID: BioID, Healthplan, Social Security no, and Vehicle licence plate no; Face photo; and Any other unique code are PHIs that were not present in any of the dataset, hence not included here.PHI categories Sub-categories F-measure Reference
Dataset
(HIPAA)
DATE
Date
≥ 0.95
S2, S3, S4, S8, S10 i2b2 2014
S11, S12, S13, S18
≥ 0.95
S3, S9
MIMIC
≥ 0.95
S14
i2b2 2006
≥ 0.95
S15, S16, S17, S8
i2b2 2016
NAME
All names
≥ 0.95
S3, S18
i2b2 2014
≥ 0.95
S3
MIMIC
AGE
Age
≥ 0.95
S1, S3, S8, S18
i2b2 2014
≥ 0.95
S3
MIMIC
≥ 0.95
S14
i2b2 2006
≥ 0.95
S15, S16, S17, S8
i2b2 2016
CONTACT
Phone
≥ 0.95
S3, S11
i2b2 2014
≥ 0.95
S9
MIMIC
≥ 0.95
S14
i2b2 2006
≥ 0.95
S17
i2b2 2016
Fax
0.80
S2
i2b2 2014
Email
≥ 0.95
S2, S10, S11
i2b2 2014
ID
Medicalrecords ≥ 0.95
S11, S12
i2b2 2014
≥ 0.95
S9
MIMIC
IDNUM
≥ 0.95
S3
i2b2 2014
Device
0.80
S2, S3
i2b2 2014
License
≥ 0.95
S17
i2b2 2016
LOCATION
all
≥ 0.95
S3
i2b2 2014
≥ 0.95
S9
MIMIC
very hard.
Table 5 .
5Techniques used for the F-measures presented inTable 4for HIPAA PHI categories.PHI categories Sub-categories Techniques
(HIPAA)
Table 6 .
6The best F-measure for i2b2 extra categories for de-identification. This table includes categories not included in
). DATE and AGE Option 1: date shifting where all elements Yes of dates (i.e. day, month and year) are shifted forward by the same random number. Option 2: distorted identifier table is used where date, month were changed but year was kept the same.PHI
Surrogate generation techniques
Manual check
PHONE, FAX,
Randomly created surrogates.
-
URLs, ID
EMAIL address Manual replacement
Yes
NAME
Option 1: permutation tables with existing
-
identifiers are mapped to new ones with
similar frequency of occurrence.
Option 2: Mapping between letters, maintaining
name type and sex.
LOCATION
Random selection of surrogates from pre-compiled
Yes
list or permutation tables ensuring the type of
location is matched.
PROFESSION
hand-crafted rules to select from pre-compiled list.
Yes
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| [] |
[
"The Kinematic Richness of Star Clusters -I. Isolated Spherical Models with Primordial Anisotropy",
"The Kinematic Richness of Star Clusters -I. Isolated Spherical Models with Primordial Anisotropy"
] | [
"Philip G Breen \nSchool of Mathematics and Maxwell Institute for Mathematical Sciences\nUniversity of Edinburgh\nKings BuildingsEH9 3FDEdinburghUK\n",
"Anna Lisa Varri \nInstitute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK\n",
"Douglas C Heggie \nSchool of Mathematics and Maxwell Institute for Mathematical Sciences\nUniversity of Edinburgh\nKings BuildingsEH9 3FDEdinburghUK\n"
] | [
"School of Mathematics and Maxwell Institute for Mathematical Sciences\nUniversity of Edinburgh\nKings BuildingsEH9 3FDEdinburghUK",
"Institute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK",
"School of Mathematics and Maxwell Institute for Mathematical Sciences\nUniversity of Edinburgh\nKings BuildingsEH9 3FDEdinburghUK"
] | [
"MNRAS"
] | We investigate the dynamical evolution of isolated equal-mass star cluster models by means of direct N-body simulations, primarily focusing on the effects of the presence of primordial anisotropy in the velocity space. We found evidence of the existence of a monotonic relationship between the moment of core collapse and the amount and flavour of anisotropy in the stellar system. Specifically, equilibria characterised by the same initial structural properties (Plummer density profile) and with different degrees of tangentially-biased (radially-biased) anisotropy, reach core collapse earlier (later) than isotropic models. We interpret this result in light of an accelerated (delayed) phase of the early evolution of collisional stellar systems ("anisotropic-response"), which we have characterised both in terms of the evolution of the velocity moments and of a fluid model of two-body relaxation. For the case of the most tangentially anisotropic model the initial phase of evolution involves a catastrophic collapse of the inner part of the system which continues until an isotropic velocity distribution is reached. This study represents a first step towards a comprehensive investigation of the role played by kinematic richness in the long-term dynamical evolution of collisional systems. | 10.1093/mnras/stx1750 | [
"https://arxiv.org/pdf/1707.02792v1.pdf"
] | 119,388,228 | 1707.02792 | f4f941646f00aadf62a66b79357b1ef5a1165159 |
The Kinematic Richness of Star Clusters -I. Isolated Spherical Models with Primordial Anisotropy
2017
Philip G Breen
School of Mathematics and Maxwell Institute for Mathematical Sciences
University of Edinburgh
Kings BuildingsEH9 3FDEdinburghUK
Anna Lisa Varri
Institute for Astronomy
University of Edinburgh
Royal Observatory
Blackford HillEH9 3HJEdinburghUK
Douglas C Heggie
School of Mathematics and Maxwell Institute for Mathematical Sciences
University of Edinburgh
Kings BuildingsEH9 3FDEdinburghUK
The Kinematic Richness of Star Clusters -I. Isolated Spherical Models with Primordial Anisotropy
MNRAS
0002017Accepted XXX. Received YYY; in original form ZZZPreprint 12 February 2018 Compiled using MNRAS L A T E X style file v3.0galaxies: star clusters: general -Galaxy: globular clusters: general - methods: numerical
We investigate the dynamical evolution of isolated equal-mass star cluster models by means of direct N-body simulations, primarily focusing on the effects of the presence of primordial anisotropy in the velocity space. We found evidence of the existence of a monotonic relationship between the moment of core collapse and the amount and flavour of anisotropy in the stellar system. Specifically, equilibria characterised by the same initial structural properties (Plummer density profile) and with different degrees of tangentially-biased (radially-biased) anisotropy, reach core collapse earlier (later) than isotropic models. We interpret this result in light of an accelerated (delayed) phase of the early evolution of collisional stellar systems ("anisotropic-response"), which we have characterised both in terms of the evolution of the velocity moments and of a fluid model of two-body relaxation. For the case of the most tangentially anisotropic model the initial phase of evolution involves a catastrophic collapse of the inner part of the system which continues until an isotropic velocity distribution is reached. This study represents a first step towards a comprehensive investigation of the role played by kinematic richness in the long-term dynamical evolution of collisional systems.
INTRODUCTION
The study of the structure and dynamical evolution of collisional stellar systems is often pursued under a relatively stringent set of simplifying assumptions, such as isotropy in the velocity space and the absence of ordered motions (e.g. rotation), which limit significantly the opportunity to explore the kinematic richness of star clusters. As a zerothorder picture, two-body relaxation will certainly bring any dense, collisional system towards a state of approximate thermodynamical equilibrium, which, in phase space, may be characterised in terms of a quasi-Maxwellian distribution function. None the less, the often unexplored kinematic complexity may offer a refreshingly new and fertile degree of freedom which will enrich our fundamental understanding of the formation and dynamical evolution of this class of stellar systems. This is precisely the goal of the present study, which will be devoted to an exploration of the role of primordial (i.e., attributed to the initial conditions) anisotropy in the velocity space in the dynamical evolution of star clusters.
In this respect, possible deviations from isotropy in the E-mail: [email protected] velocity space may be explored along two complementary lines of investigations. On the one hand, after many decades of progressively more realistic numerical simulations, it is well known that the dynamical evolution of collisional stellar systems, as driven by internal and external processes, may significantly affect the properties of their three-dimensional velocity space, generating, for instance, a variable degree of "evolutionary" anisotropy (e.g., see Baumgardt & Makino 2003 and other references mentioned in the next paragraph).
On the other hand, we might also ask whether, especially for star clusters characterised by long relaxation times, any signature of their formation process may actually be preserved in phase space, in the form of "primordial" anisotropy (e.g., see Vesperini et al 2014 and the discussion in the second part of this Section).
In the first case, pioneering numerical experiments have recognised that anisotropy is indeed a natural outcome of star cluster dynamical evolution, especially when the system is in isolation (Hénon 1971;Spitzer & Shapiro 1972). Spitzer and collaborators have shown that isolated globular clusters during their evolution develop a structure composed by two distinct regions: an isotropic core, and a radially anisotropic halo of stars, resulting from the scattering of stars from the centre, preferentially on radial orbits. Idealised models of isolated star clusters based on gaseous methods and linear perturbation theory have later confirmed that isolated systems tend to become progressively more anisotropic in their outer regions (see Bettwieser & Spurzem 1986;Spurzem 1991;Louis & Spurzem 1991). Such a behaviour has been noted also by Giersz & Heggie (1994) in the context of the early exploration of the statistics of N-body simulations and by Takahashi (1995) in two-integral Fokker-Planck models.
This picture has been subsequently extended to the inclusion in the models of the presence of an external tidal field, the effect of which is typically to curb the degree of radial anisotropy developed, most likely as a result of a preferential loss of stars on radial orbits and the general mass loss, which progressively exposes deeper and therefore more isotropic shells of the stellar system (Giersz & Heggie 1997). Nevertheless, the anisotropy profile remains radially-biased at intermediate radii, while it becomes isotropic (or even mildly tangential, as illustrated by Takahashi et al. 1997;Baumgardt & Makino 2003) in the outer regions. Given these two limiting cases, it should not come as a surprise that the degree of anisotropy developed in a collisional system strongly depends on the strength of the tidal field in which it is evolving (Tiongco et al. 2016), with a crucial role played by the population of potential escapers residing in the system (Claydon et al. 2017).
On the side of distribution function-based models, these ideas have inspired the construction of a variety of equilibria, often defined as a direct generalisation of lowered isothermal models, in which the second integral of the motion is considered to be the specific angular momentum, either introduced via a simple exponential dependence (see Michie 1963, Davoust 1977, and, more recently, Gieles & Zocchi 2015), or by more complex prescriptions (e.g., see Merritt 1985, Richstone & Tremaine 1984, Dejonghe 1987, Dehnen 1993. Some of these models have been successfully applied to the interpretation of both observational data (e.g., see Gunn & Griffin 1979) and numerical simulations (Sollima et al. 2015;Zocchi et al. 2016).
As for the investigation of "primordial" anisotropy, i.e. any deviation from velocity isotropy associated with the formation and early dynamical evolution phase, it might very well be that, especially for particularly rich and initially underfilling star clusters, their outer structure is not too far from that of bright elliptical galaxies for which violent relaxation is thought to have acted primarily to make the inner system quasi-relaxed, while the outer parts are only partially relaxed and therefore progressively more dominated by radially-biased anisotropic pressure (e.g., see van Albada 1982). Some numerical experiments of violent relaxation have also been conducted in the presence of an external tidal field (Vesperini et al 2014) and it has been shown that the configurations emerging at the end of such a cold collapse phase are characterised by a distinctive radial variation of the velocity anisotropy and can acquire significant internal differential rotation.
This line of argument motivated the construction of several families of dynamical models, to represent the final state of numerical simulations of the violent relaxation process (e.g., see Stiavelli & Bertin 1985;Trenti et al. 2005). Some of these models have been successfully applied to the study of a sample of Galactic globular clusters under different re-laxation conditions (Zocchi et al. 2012), and recently modified to include an appropriate truncation in phase space, to heuristically mimic the effects of the external tidal field (de Vita et al. 2016).
We emphasise that, both in the case of evolutionary and primordial signatures, most of the attention has been devoted to the case of radially-biased anisotropy, also because of the interest in the regime in which collisionless equilibria may become unstable to the formation of a central bar, due to radial orbit instability (e.g., see Polyachenko & Shukhman 1981;Fridman & Polyachenko 1984;Polyachenko & Shukhman 2015). In turn, the corresponding regime of strong tangential anisotropy, and possible dynamical instabilities associated with it, has been only marginally explored (e.g, see Barnes et al. 1986;Weinberg 1991), partly because equilibria with tangential anisotropy are rare (e.g., see An & Evans 2006).
The exploration of the effects of kinematic complexity on the dynamical evolution of star clusters, especially in the form of deviations from velocity isotropy, is a particularly timely endeavour in light of two main reasons. First, from the observational point of view, it is now finally possible to access the full phase space of selected Galactic globular clusters and characterise their three-dimensional velocity space in terms of anisotropy profile and velocity maps, thanks to the synergy between the astrometric measurements conducted in the central regions of several globulars with Hubble Space Telescope (Watkins et al. 2015) and those which will soon available from Gaia, especially in their periphery (Pancino et al. 2017).
Second, from the theoretical point of view, it is of paramount importance to fill the gap between the highly complex end state predicted by numerical simulations of star formation in a clustered environment and the extremely simplified initial conditions that are usually adopted to study the subsequent long-term dynamical evolution of star clusters. In this respect, some recent hydrodynamical simulations of the formation of young massive star clusters have emphasised the importance of considering the kinematic dimension of protoclusters (e.g., see especially Hennebelle 2016 andMapelli 2017), also in view of the surprising morphological and kinematic richness that has emerged over the past few years in observational studies of young and intermediate age star clusters (e.g., see Hénault-Brunet et al. 2012;Kuhn et al. 2015;Vicente et al. 2016).
In addition, a deeper understanding of the kinematic evolution of star clusters represents an essential step towards a satisfactory solution of topical open problems regarding the phase space characterisation of their elusive multiple stellar populations (see Richer et al 2013 andBellini et al 2014 for the first observational assessments of the difference in velocity anisotropy between different stellar populations in Galactic globulars) and the existence of their putative intermediate mass black holes (see Zocchi et al. 2017 for a recent study of the effect of velocity anisotropy of star clusters on the dynamical mass estimate of a central compact object).
Before approaching the role played by this emerging kinematic complexity in any realistic setting, it is imperative to explore its effects on the gravothermodynamics of simple collisional stellar systems. For this reason, here we start from the case of spherical isolated systems and we consider a se- ries of anisotropic equilibria which are characterised by the same spatial distribution of mass, but different properties of their three-dimensional velocity space. This point is indeed essential, as it has been shown that the structural properties of a stellar system do shape its subsequent collisional evolution (e.g., see Quinlan 1996), especially the moment of core collapse, which will be the primary focus of our investigation (for an earlier study of the evolution of radially anisotropic systems towards core collapse, see Magliocchetti et al. 1998). The second, complementary part of this investigation will be devoted to the study of the role of primordial rotation (Breen, Varri, Heggie, in preparation). The paper is structured as follows. In Section 2 we describe the distribution function-based models we use to define our initial conditions, characterised by spherical symmetry and anisotropy in the velocity space. In Section 3 we present the results of a series of N-body simulations which have been performed to explore the role of anisotropy in the dynamical evolution of collisional systems in isolation. In Section 4 we outline the theory used to explain the early evolution of the N-body models and we discuss possible generalisations of our study. Finally, we present our conclusions in Section 5.
METHOD AND INITIAL CONDITIONS
Unless otherwise stated, the quantities presented in all figures and table included in this paper are given in Hénon units (Hénon 1971). A notable exception are the distribution functions defined in the present Section, in which a notation such that G = M = a = 1 has been adopted (Dejonghe 1987), where M is the total mass and a is the Plummer scale radius. Where appropriate, we have also adopted the definition of the initial half-mass relaxation time (t rh,i ) as presented in equation ( The inner gradient increases with decreasing values of q and becomes positive (i.e. with a temperature inversion) after q = −3/2. The units system, and the colour scheme and line style for the first four models, are consistent with those adopted in Fig. 1.
Anisotropic Plummer models
The initial conditions of the models presented in this study are realisations of Dejonghe (1987) anisotropic Plummer models. The distribution function is given by:
F q (E, L) = 3Γ(6 − q) 2(2π) 5 2 Γ(q/2) E 7 2 −q H 0, 1 2 q, 9 2 − q, 1; L 2 2E (1)
where E is energy, L angular momentum, q is the parameter which controls the amount of anisotropy (and lies in the range −∞ < q < 2), Γ is the Gamma function and H is a function, which may be expressed in terms of the hypergeometric function 2 F 1 . For
x ≤ 1 H(a, b, c, d; x) = Γ(a + b) Γ(c − a)Γ(a + d) x a 2 F 1 (a + b, 1 + a − c; a + d; x) (2)
and for x > 1
H(a, b, c, d; x) = Γ(a + b) Γ(d − b)Γ(b + c) 1 x b 2 F 1 a + b, 1 + b − d; b + c; 1 x .(3)
Depending on the value of q, the resulting models are characterised by isotropy (q = 0), tangential (q < 0) or radial (q > 0) anisotropy. For these equilibria, the anisotropy parameter β has the convenient form
β(r) = 1 − σ 2 t 2σ 2 r = q 2 r 2 1 + r 2 ,(4)
where σ t and σ r are the rms tangential (two-dimensional) and radial velocity components, respectively. For reference, the β profiles for some of the models considered in this study are illustrated in Fig. 1. We note that models with different values of q are characterized by different energy distributions (for a complete description, see Dejonghe 1987, sections 3.3 and 3.4). Nevertheless the total energy and the half-mass relaxation time of all models are the same (in Hénon units).
Discrete numerical realisations of the selected models within the family described above were generated using a similar technique as in Aarseth, Hénon and Wielen (1974), adapted for the properties of the anisotropic distribution functions under consideration 1 .
Radially-biased models
The most radially anisotropic model in the Dejonghe series is the q = 2 model. This configuration is the same as one of the Osipkov-Merritt models (Osipkov 1979;Merritt 1985), i.e. the one with anisotropy radius r a = 1. The distribution function of the radially anisotropic Osipkov-Merritt Plummer model may be defined as
f (Q) = √ 2 378π 3 Gσ(0) −Q σ 2 (0) 7 2 1 − 1 r a + 63 4r 2 a −Q σ 2 (0) −2(5)
where Q = E + J 2 /(2r 2 a ) and σ(0) is the central value of the (scalar, i.e. one-dimensional) velocity dispersion. Here the amount of radial anisotropy grows with decreasing r a . The definition given in Equation 5 is valid for r a > 0.75; for lower values of r a the distribution function becomes negative. In principle, the Osipkov-Merritt models can be used to extend the Dejonghe series to configurations characterised by a higher level of radial anisotropy (i.e., beyond the limiting case of such series, as set by q = 2).
For the purpose of this study, we have considered the Dejonghe model with q = 2 (r a = 1), and, in order to explore the stability threshold of these equilibria with respect to radial orbit instability, we have also explored the evolution of the Osipkov-Merritt configuration with r a = 0.75. A discussion of such threshold is presented in Section 4.1.
Tangentially-biased models
Models with negative values of the parameter q are characterised by tangential anisotropy; for the purpose of the current investigation we have considered the cases with q = −2, −6, −16, −∞; all configurations (N = 8192) appear to be dynamically stable. Merritt (1985) also considered tangential anisotropic distribution functions based on the parameter Q − = E − J 2 /(2r 2 a ) (called "Type II"). However, in order to obtain distributions functions which are easily invertible from the density profile, only the portion of the domain such that r < r a should be considered. In order to extend the models beyond r a , Merritt (1985) offered two solutions: "Type IIa" is such that the density distribution is completed by adding circular orbits outside r a and "Type IIb" by adding orbits that Table 1. Properties of the N-body simulations. From left to right, the columns report the values of the q parameter of the Dejonghe (1987) anisotropic Plummer model, the number of particles N , the number of realisations performed N r , the average core collapse time t c c (with the standard deviation, in units of t r h, i ), the difference in collapse time compared to the isotropic model ∆t c c , and twice the ratio of the kinetic energy of radial motions T r to the kinetic energy of transverse motions T ⊥ . Note that the standard deviation quoted for model q = −6 is most likely underestimated due to low number statistics. correspond to the condition of constant Q − . Both methods result in a discontinuity in the tangential velocity dispersion at r a (cf Merritt 1985). Due to the additional complications that arise for r > r a , in the present investigation we will not consider tangential anisotropic Osipkov-Merritt models further.
q N N r t c c ∆t c c 2 T r /T ⊥
Einstein Sphere
For q → −∞, the model representing the limiting case in the series of tangentially anisotropic Plummer equilibria consists exclusively of stars on circular orbits. This configuration is referred to as "Einstein Sphere" (Fridman & Polyachenko 1984) as it was first studied by Einstein (1939) in the context of an exploration of a class of spherically-symmetric solutions of the field equation in General Relativity. Such a class represented a static cluster of collisionless gravitating particles, as a tool to investigate the physical significance of the Schwarzschild singularity. It is straightforward to construct Einstein Spheres for any spherically symmetric density distribution. These models have the unusual feature that σ 2 → 0 as r → 0. As q decreases (in the Dejonghe 1987 sequence), the gradient of the velocity dispersion at small radii increases, eventually becoming positive (at q = −3/2). The radial profiles of the velocity dispersion calculated for a selection of models are presented in Fig. 2. One may wonder if there is a radial analogue of the Einstein Sphere, i.e. a Plummer model consisting of entirely radial orbits. It can be shown that models with a finite central density can not be constructed by means exclusively of radial orbits (Bouvier & Janin 1968;Richstone & Tremaine 1984).
Numerical simulations
Starting from the initial conditions described in the previous Section, we have performed a series of direct N-body simulations to explore the long-term evolution of the anisotropic Table 1. A simple piecewise linear interpolation is illustrated as a dashed line, to guide the eye. equilibria under consideration. Each model has been studied by means of four independent numerical realisations 2 . All models used stars of equal mass with no primordial binaries and no stellar evolution. The properties of the simulations are presented in Table 1. The simulations were conducted by means of the direct summation code NBODY6 3 (Aarseth 2003;Nitadori and Aarseth 2012), enabled for use with Graphical Processing Units (GPU). We have evolved the models in isolation, without any prescription for removing particles, until they all reached core collapse (as identified from the evolution of the core radius, see Fig. 4).
RESULTS
Dependence of core collapse on anisotropy
The core collapse times (t cc ) of the anisotropic models under consideration are presented in Table 1. There is a clear dependence of the core collapse time on the amount of anisotropy in the system, as expressed by the parameter q or 2 T r /T ⊥ . Specifically, we found that the model characterised by radial anisotropy (q = 2) requires a longer time to reach core collapse, compared to the standard isotropic case; tangentially anisotropic models require a shorter time, 2 This refers to the Dejonghe models listed in Table 1. Only one realisation of the Osipkov-Merritt model with r a = 0.75 was computed 3 NBODY6 is publicly available for download from www.ast.cam.ac.uk/∼sverre/web/pages/nbody.htm. NBODY6 comes with an option to generate an isotropic Plummer model. However, this option imposes a cut-off radius, which we have not adopted when generating our initial conditions. The amount of mass outside the cut off is approximately 1.5%. When the cut off is included it appears to increase the core collapse time by roughly 50 Hénon time units. with the most tangential case (q = −∞, Einstein Sphere) being the absolute fastest to collapse. On the basis of the series of simulations we have performed, we therefore conclude that, in the presence of the same initial structural properties, there appears to exist a monotonic relationship between the amount (and flavour) of primordial anisotropy and the time of core collapse for spherical, isolated collisional systems. The relationship between 2 T r /T ⊥ and the collapse time is shown in Fig 3. It is essential to emphasise that this relationship is likely to depend on the initial structural properties of the models and the spatial distribution of their primordial anisotropy, and the evidence we present for it is limited to the anisotropic Plummer models under consideration. None the less, it suggests that one may certainly consider exploring such a two-dimensional parameter space also for other families of anisotropic equilibria. Some considerations regarding the effect of the spatial distribution of the anisotropy are presented in Section 4.5.
The evolution of the core radius (r c ) of all the models is shown in Fig. 4. Different numerical realisations of the same anisotropic equilibrium are depicted with the same colour, to offer a visual representation of the spread of the core collapse times (see also the standard deviations reported in Table 1). The small spread in values for q = −6 is likely due to the small sample size (four realisations).
If we rescale the time variable so that core collapse occurs at t = 0 (i.e. t = t − t cc ), as performed in Fig 5, it may be noticed that all the models evolve in a similar way once the inner part of the system becomes approximately isotropic (for an evaluation of the time evolution of the level of anisotropy in the models, see Fig. 7).
An assessment of the energy of the innermost 100 particles is offered in Fig. 6. In order to make the energy evaluation more efficient, we calculated the potential by assum- ing spherical symmetry and using the method described by Hénon (1971), then we simply summed the energy of the innermost 100 particles (i.e. E = m φ(r i ) − 0.5v 2 i ), where m is the individual stellar mass. As in the case of the evolution of the core radius, once a condition of isotropy in the central regions is approached, the models tend to evolve on nearly the same track in the energy space.
We wish to note that none of the models listed in Table 1 show signs of radial orbit instability, even though the model with q = 2 has a value 2T r /T ⊥ = 1.96, which sits at the upper limit of the critical range (1.7 ± 0.25) found by Polyachenko & Shukhman (1981). In Section 4.1 we extend this series to explore one example of the more radially anisotropic Osipkov-Merritt configurations, which are indeed unstable. Our attention to the emergence of possible signs of radial orbit instability is motivated by the fact that such a process, by determining the presence of a central bar, would significantly modify the mass distribution of the system, therefore introducing a crucial element of structural difference which would affect the validity of our comparative analysis of the time of core collapse.
Early evolution: the anisotropic response
For a fixed spatial distribution of mass, as we change the amount of anisotropy in the system we are also changing the velocity dispersion profile, as illustrated in Fig. 2. The central velocity dispersion decreases with decreasing q and a temperature inversion develops below q = −3/2. In the case of gravothermal oscillations (e.g. see Bettwieser & Sugimoto 1984), a temperature inversion is usually associated with a phase of expansion. In such a case, it determines an evolutionary condition such that the heat flows from the hotter outer regions to the cooler inner regions, causing the system to expand. Interestingly, in the current case, we observe the opposite behaviour, where the inner regions actually collapse where E = m φ(r i ) − 0.5v 2 i and φ is calculated assuming spherical symmetry and using the same method as described by Hénon (1971). The models approach and then follow a similar evolutionary track in energy space. One realization is shown for each value of q. The curves are ordered as in Fig.5. faster with increasing negative temperature gradient. Below we will interpret this behaviour as a collapse determined by the "conversion" of the tangentially-biased motions (as measured by σ 2 t ) into radial ones (σ 2 r ) as the system relaxes towards an isotropic velocity distribution. For the case of radial models we predict the opposite effect.
By increasing the initial amount of tangential anisotropy in the system, there is a more pronounced collapse in the initial phase of evolution which can be clearly seen in Fig. 4. As we will argue in Sections 4.2 and 4.3, in which we provide a quantitative interpretation of this behaviour in terms of the evolution of the fluid moments, we expect the inner parts of the system to lose support from the tangentially-biased motions as the system relaxes towards an isotropic velocity distribution. For the most tangential case in our series this process results in a catastrophic collapse, as the inner regions rapidly evolve towards an isotropic velocity distribution. The reason for this is the very short central relaxation time, which is caused by the low central velocity dispersion (see Fig. 2).
To support our interpretation, we have studied the evolution of the anisotropy in the central regions of the models by inspecting the behaviour of the value of 2 T r /T ⊥ within the 10% and 50% Lagrangian radii (see Fig. 7). Tangentially anisotropic systems rapidly evolve so that the mass enclosed within the 10% Lagrangian radius becomes isotropic (within about 10%) in about 0.5 t rh,i , and similarly within the halfmass radius, in about 5-6 t rh,i .
As for the initially radial model (q = 2), while the mass within the 10% Lagrangian radius becomes approximately isotropic in about 0.5-1 t rh,i , the region within the halfmass radius remains radially anisotropic for longer (i.e., until about 10 t rh,i ); such a behaviour is probably due to the contribution of high energy particles (on more radial orbits) passing within the half mass radius on their way to and from the halo. These particles would take a longer time to relax compared to the more bound particles. . Evolution of the second moments of the spatial distribution I ii ( j m j x 2 i , i = 1, 2, 3) within the half mass radius of the Dejonghe q = 0, 2 models and the Osipkov-Merritt Plummer model with r a = 0.75. The latter model provides the outlying curves at the top and bottom. The q = 2 model is identical to the Osipkov-Merritt Plummer model with r a = 1.00. We find that the q = 2 models are stable and the r a = 0.75 model is unstable to radial orbit instability.
To further characterise the initial phase of evolution of the anisotropic systems under consideration, we have also analysed the behaviour of the radial profiles of the velocity dispersion, anisotropy parameter β and density; for brevity, in Fig. 8 we report exclusively the evolution of the model with q = −16. In this case, the initially negative gradient of the velocity dispersion is rapidly reduced and completely disappears in about 1 t rh,i . Simultaneously, the amount of tangential anisotropy is significantly reduced, especially in the intermediate regions. Finally, the density profile in the outskirts of the model is virtually unchanged, though there is a noticeable steepening of the slope towards the centre in the first time interval.
DISCUSSION
Radial Orbit Instability
By inspecting the behaviour of the radial modes and the quadrupole moment of equilibria evolved by means of a mean-field N-body code, Dejonghe & Merritt (1988) found that anisotropic Osipkov-Merritt Plummer models with r a 1.1 4 were unstable to the radial orbit instability. On the basis of our numerical simulations performed with a direct Nbody code, we report that configurations with r a = 1.0 (i.e. q = 2) are stable and those with r a = 0.75 are unstable, as visible from the time evolution of the second moments of the spatial distribution illustrated in Fig. 9.
We tend to attribute this slight variation in the value of the threshold for stability to the intrinsic differences between collisional and collisionless codes (see the last paragraph of this subsection), although it is worth mentioning that a number of subsequent investigations, performed on anisotropic configurations characterised by different density distributions, also reported revised stability thresholds. In particular, by monitoring the evolution of the axis ratio of models evolved with a 'self-consistent field' N-body code, Meza & Zamorano (1997) found a stability limit of 2T r /T ⊥ = 2.31 ± 0.27 for selected configurations within the family of spherical anisotropic models proposed by Dehnen (1993). If we assume that the critical value is the same for other models, this result implies that their stability boundary is below r a = 1 (q = 2) and above r a = 0.75 (where 2T r /T ⊥ ≈ 2.4 for our models), which is consistent with our results. Interestingly, a more recent numerical study performed by Trenti & Bertin (2006) specifically examined the dependence of the occurrence of the radial orbit instability on the shape of the anisotropy profile in the central region of a stellar system; they identified stable states with 2T r /T ⊥ ≈ 2.75, as emerging at the end of the process of cold collapse from homogeneous initial conditions. The core collapse time for the r a = 0.75 model is t cc = 2140 ± 30, which is consistent with the r a = 1 (q = 2) model in Table 1 and in excess of the collapse time of the reference isotropic model. From Fig. 9 we can see that the r a = 0.75 model gradually returns to a spherical configuration within the half mass radius, after approximately 3 t rh,i . We wish to note that a detailed investigation of the effects of the radial orbit instability on the time of core collapse is outside the scope of the present study.
Since relaxation can affect the amount of anisotropy in significant ways, this could change the stability boundary of the radial orbit instability. As can be seen in Fig 7, in the collisional setting the more radial models initially evolve towards a more isotropic configuration. This could cause a model which appears to be unstable when simulated with a collisionless code to become stable when simulated with a collisional code. It is perhaps the reason that we find the r a = 1.0 model stable against the radial orbit instability, even though it has previously been found to be unstable (see above). This may imply a relaxation time dependency of the stability boundary. A similar conclusion (i.e. that enhanced collisionality may tend to suppress the radial orbit instability) was reached by Smirnov et al (2017) for anisotropic disk systems.
A theoretical interpretation based on hydrostatic equations
We attempt to provide an interpretation of the early evolution of the anisotropic Plummer models by using the radial Jeans equation. Assuming spherical symmetry, the radial Jeans equation in spherical coordinates can be written as
∂v r ∂t = − 1 ρ ∂(ρσ 2 r ) ∂r − (2σ 2 r − σ 2 t ) r − ∂Φ ∂r .(6)
Under some simplifying assumptions we can predict how the system would respond as it relaxes towards an isotropic velocity distribution. Specifically, we assume that ρ and σ 2 tot (where σ 2 tot = σ 2 r + σ 2 t ) remain constant and that, after one central relaxation time t r (0), σ 2 r becomes σ 2 r + t r (0) t r (r) ( 1 3 σ 2 tot − σ 2 r ) and σ 2 t is simply σ 2 t = σ 2 tot − σ 2 r . Using this result we can evaluate the right-hand side of the radial Jeans equation, Eq. (6). We have performed this calculation numerically for different values of q and plotted the results in Fig 10. This theory predicts that the central regions of a stellar system will experience an acceleration; the sign of such an acceleration term depends on the flavour of the anisotropy, with radially-biased anisotropic models experiencing a positive (outward) acceleration and tangentially-biased anisotropic systems a negative (inward) one. Such a behaviour is consistent with the accelerated (delayed) early phase of evolution characterising the tangential (radial) models which we have considered in Section 3.
A theoretical interpretation based on a fluid model of two-body relaxation
The foregoing theoretical interpretation fails for the isotropic model (which contracts from t = 0). Also, as stated in the assumptions, it is based on the idea that anisotropy evolves without any change in ρ and σ tot ; this gives rise to a significant departure from hydrostatic balance, and a significant acceleration. In fact, ρ, σ tot and σ r all change simultaneously, leaving a small hydrostatic imbalance which gives rise to a mean flow on a time scale of the relaxation time. The following theory recognises this simultaneous evolution, and predicts the initial rate of evolution of central quantities, including the isotropic case. It does not, however, say anything about the radial profile of this evolution, which would require a full numerical solution of the equations we are about to discuss. Louis & Spurzem (1991) describe a gas model of a spherical, anisotropic star cluster, in which the evolution is described in terms of the density ρ, and radial and transverse velocity dispersions σ 2 r , σ 2 t (or equivalent variables), which are functions of radius r and time t. Relaxation drives fluxes of mass and of radial and transverse kinetic energy, which are described by three radial velocities u, v r and v t . While Louis & Spurzem (1991) of the full spatial structure of a system, in the present application we confine our attention to the initial rate of change of one of the central properties, which can be determined without any numerical integration of the partial differential equations of the model. The central property on which we focus is the local specific entropy. Details are given in Appendix A, where it is shown that, for initial conditions of the anisotropic Plummer models studied in the present paper, its initial rate of change is given by the equation
d dt ln σ 3 ρ = − 10λ(3 + 2q) 6 − q √ π log N N GM a 3 1/2 ,(7)
where a is the scale radius of the Plummer model. Incidentally, exactly the same result (except for a change to the dimensionless coefficient) could be obtained from the isotropic model of Lynden-Bell & Eggleton (1980). As those authors mention, the expression on the left of Equation (7) is essentially the rate of change of the specific entropy, s, except for its sign.
For an isotropic Plummer model, we set q = 0, and see that s < 0. Thus either σ decreases or ρ increases initially. Nbody models show that an isotropic Plummer model actually does both initially, though after a small adjustment σ begins to increase. For anisotropic models there are two differences to notice. First, the initial rate of change of s changes to s > 0 for q < −3/2. (Note that this happens to be the value at which a temperature inversion near the centre of the model first appears (as q decreases); see Sec.2.1.3.) Thus, either σ increases or ρ decreases, or both. Second, for large negative q (strongly tangentially anisotropic initial models) the time scale of the initial evolution of s becomes very short. The reason for this is the small central velocity dispersion, which leads to a short relaxation time.
The evidence based on our N-body simulations is illustrated in Fig. 11. Two points at least are clear, despite the sampling fluctuations. The first is that the most ex-tremely tangentially anisotropic model (q = −∞) exhibits the most rapid early evolution, and the second is that its initial rate of change is positive. While these observations are consistent with the predictions we have made on the basis of Equation (7), we readily concede that there is no sign that the transition from positive to negative rate of change takes place near q = −3/2, as predicted by the model.
It might be possible to improve this theory by computing the initial rate of change of the central value of the density function f (r, v), which is also closely related to the central specific entropy. The rate of change can, in principle, be evaluated using an anisotropic form of the Fokker-Planck equation, along the lines of the isotropic calculations presented in Heggie & Stevenson (1988, section 7).
Some considerations on the nature of the phenomenon
One may wonder if the behaviour in models with extreme tangential anisotropy is an instability in the same sense as, for example, the gravothermal instability. Mathematically, an instability in a dynamical system is a property of an equilibrium. While the models considered here are equilibria in the collisionless sense (i.e. they satisfy the collisionless Boltzmann equation), they are not collisional equilibria. For collisional systems the only true equilibria of finite mass are highly artificial systems contained within a perfectly reflecting wall (e.g. Antonov 1962). We note that the models commonly used in stellar dynamics are not collisional equilibria, therefore, strictly speaking, the term instability does not apply to any behaviour associated with core collapse, recognised as a collision-induced phenomenon. However, since in the case of collisional systems contained within a spherical boundary (which do have equilibria), the gravothermal instability is a genuine instability, and since it is fundamentally the same physical mechanism which drives core collapse also in models without strictly defined equilibria, the terminology seems none the less appropriate. Since there does not appear to be an artificial case where the highly tangential models are collisional equilibria (as is the case for gravothermal instability), describing this behaviour as an instability does not seem appropriate One may still ask the question whether the phenomenon of rapid early collapse in models with extreme tangentially biased anisotropy is collisional (as we assumed in Sections 4.2 and 4.3) or collisionless. To clarify this we carried out one simulation for the case q = −∞ which was designed exactly as in Sec.2.2 except that N = 32768. Up to time at least 100 Hénon units in the larger model, the evolution of the Lagrangian radii, up to at least the 10% mass fraction, is almost indistinguishable (within fluctuations) from that in our typical 8k model when the time axis is scaled by the initial relaxation time. This leaves no room for doubt that the very rapid early collapse is indeed a collisional phenomenon, though it does not prove that the system is stable in the collisionless sense. Unfortunately the literature on collisionless stability of systems with circular orbits does not seem to be conclusive (for the Dejonghe model with q = −∞). The uniform sphere is stable (Mikhailovskii et al 1971;Fridman & Polyachenko 1984), which might have been thought relevant to dynamics close to the centre of a model with a Plummer density distribution. On the other hand Wein-berg (1991), drawing partly on the discussion in Polyachenko (1987), mentions an expectation that any model populated by nearly circular orbits should be unstable, at least in extreme or pathological cases. Whatever the theoretical expectation, however, there is no sign of collisionless instability in our extreme models, if only because it is swamped by collisional behaviour taking place on the the very small central two-body relaxation time.
Application to other anisotropic models
The dynamical response of an anisotropic stellar system, especially the time scale of its evolution towards a condition of isotropy in the velocity space, depends in principle on the spatial distribution of the primordial anisotropy. In the present study, we have intentionally limited our investigation to the case of configurations characterised by a Plummer density distribution, generalised by Dejonghe (1987) to allow anisotropy in the velocity space. We have chosen such a family because of the simplicity of the radial profile of the anisotropy parameter (see Equation 4), which, at the half-mass radius, is such that β(r h )= 2 −5/3 q.
We emphasise that the early phase of the "anisotropic response" has direct implications on the time of core collapse: in cases of high tangential anisotropy, for instance, it quickly takes the model a long way towards core collapse (Fig.5). Therefore, the detailed properties of the relationship between the time of core collapse and the initial strength of the primordial anisotropy (see Fig. 3) strictly depend on the initial structure of the three-dimensional velocity space. None the less, on the basis of the theoretical interpretations provided in Sections 4.2 and 4.3, we expect the general behaviour (i.e., acceleration or delay in the presence of tangential or radial primordial anisotropy, respectively) to apply to a wide range of spherically symmetric, anisotropic equilibria. The interpretation based on hydrostatic equations presented in Section 4.2 could easily be repeated for a different set of anisotropic models.
Primordial vs. evolutionary anisotropy
A number of studies have noted that initially isotropic models develop in their outer regions a significant degree of anisotropy of the velocity space, especially after core collapse, whether in isolation or in the presence of a tidal field (e.g., see Giersz & Heggie 1994;Zocchi et al. 2016;Tiongco et al. 2016). This is usually attributed to dynamical scattering near the time of core collapse and interactions with binary stars afterwards.
In order to explore the transition from the regime of primordial anisotropy to that of evolutionary anisotropy, we have measured the indicator 2 T r /T ⊥ in a shell around the location of the half-mass radius, as depicted in Fig 12. All models eventually develop radial anisotropy, to a degree which depends on the flavour and strength of the primordial anisotropy (i.e., the model developing the highest [lowest] value of evolutionary radial anisotropy is the one characterised by the strongest primordial radial [tangential] anisotropy). Though it might seem obvious that a model with greater primordial radial anisotropy will show greater radial anisotropy at later times, it is not quite that sim- 2 T r T q = 2 q = 0 q = -2 q = -6 q = -16 q = -∞ Figure 12. Exploration of the transition between primordial and evolutionary anisotropy, with plots of the time evolution of 2 T r /T ⊥ in a shell around r h for the models illustrated in Fig. 4 (One realization is shown for each value of q). All models eventually become radially anisotropic, to a degree which depends on the flavour and strength of the primordial anisotropy (see Section 4.6 for details).
ple. Since the tangentially-biased models undergo core collapse earlier in their evolution, they could, in principle, have had more time to build up the radial anisotropy. In fact, we believe that the amount of post-core-collapse anisotropy in the intermediate and outer regions of the models should be interpreted as the result of the cooperation or competition between the growth of the genuine evolutionary radial anisotropy and the permanence of the primordial anisotropy, which may still be preserved in the outer portions of the system, as they are characterised by a longer relaxation time.
In any case, we argue that the physical origin of the evolutionary radial anisotropy is qualitatively the same as in the isotropic case (i.e., ejection of stars from the core to the halo preferentially on radial orbits).
CONCLUSIONS
We have studied the effect of primordial anisotropy in a series of equilibrium models characterised by the same spatial distribution of mass, defined as in the spherical isotropic Plummer model, the same initial half-mass relaxation time, but varying anisotropy. The main result of our investigation is that there is a clear dependence of the core collapse time on the amount of velocity anisotropy of the stellar system. Specifically, we report that models characterised by radial anisotropy require a longer time to reach core collapse, compared to the reference isotropic case; tangentially anisotropic models require a shorter time, with the most tangential case (Einstein Sphere) being the absolute fastest to collapse. On the basis of the series of N-body simulations we have performed, we therefore conclude that, in the presence of the same initial structural properties, there appears to exist a monotonic relationship between the amount of primordial anisotropy and the time of core collapse for spherical, isolated collisional systems. The effect is large, the core collapse time for the most tangentially anisotropic models be-ing three times smaller than for the most radially anisotropic models we studied, in which the ratio of the kinetic energies in radial and transverse motions is approximately twice the value in the standard (isotropic) Plummer model (Fig.3). We interpret this behaviour as resulting from the early phase of collisional evolution during which the initially anisotropic configurations rapidly evolve towards a condition of isotropy in the velocity space. From a numerical study of the radial component of the Jeans equations (under some simplifying assumptions) we expect the inner parts of a tangentially anisotropic system to lose support from the tangentially-biased motions as the system relaxes towards an isotropic velocity distribution. For the case of radial models we predict the opposite effect, therefore the systems tend to expand as they evolve towards isotropy. For the case of models which are highly tangentially anisotropic (e.g. Einstein Sphere), the initial evolution involves a catastrophic collapse, because of the very short central relaxation time. Once the models become isotropic in the central regions, they all evolve along a similar evolutionary track, which may be identified also in the energy space.
Figure 1 .
1Radial profiles of the anisotropy parameter β(r) of four representativeDejonghe (1987) anisotropic Plummer models, corresponding to selected values of the main parameter q (for definition, see Sect. 2.1). The half-mass radius is r h = (2 2/3 − 1) −1/2 ≈ 1.3. The units are such that G = M = a = 1. Note that curves with the same absolute value of q are symmetric by reflection across the x-axis, therefore the spatial distribution of their anisotropy is equivalent (with opposite sign).
14.13) ofHeggie & Hut (2003).
Figure 2 .
2Velocity dispersion profile of selectedDejonghe (1987) anisotropic Plummer models with q = −∞, −16, −6, −2, 0 and 2.
Figure 3 .
3Time of core collapse of all models (normalised to the average core collapse time of the isotropic Plummer model) t c c /t c c, i s o expressed as a function of the initial value of twice the ratio of the radial to tangential kinetic energy 2 T r /T ⊥ . The error bars are calculated from the standard deviations presented in
Figure 4 .
4Time evolution of the core radius of selected anisotropic Plummer models with q = −∞, −16, −6, −2, 0, 2. Time is expressed in units of the initial half-mass relaxation time (t r h, i ≈ 112 Hénon units for all models). For q < 0 the models are tangentially anisotropic, for q = 0 isotropic and for q > 0 radially anisotropic (see Section 2). The four different realisations of each model are depicted with the same colour. The six groups of curves correspond to the six values of q, which increases from the group at the extreme left to the group at the extreme right.
Figure 5 .
5Time evolution of the value of the core radius of the anisotropic Plummer models illustrated inFig. 4, though only one realization is shown for each value of q.. The time variable is scaled so that core collapse happens at time t = 0 (i.e. t = t − t c c ). The abscissa at the starting point of a curve is a decreasing function of q. The evolution of r c becomes similar once the inner part of the system reaches a nearly isotropic velocity distribution.
Figure 6 .
6A measure of energy of the innermost 100 particles,
Figure 7 .Figure 8 .Figure 9
789Time evolution of the velocity anisotropy in the central regions of the models illustrated in Fig. 5. From top to bottom, the curves are in order of decreasing q, except where they overlap. Top: 2 T r /T ⊥ within the half-mass radius with the time scaled so that t c c = 0. Middle: Same as Top without shifting time scale. Bottom: 2 T r /T ⊥ within the 10% Lagrangian radius. The velocity distribution of the mass within the 10% and 50% Lagrangian radii becomes approximately isotropic for all models after 50 and 700 Hénon units, respectively (t r h, i ≈ 112 Hénon units for all models). Evolution of kinematics and structure of one realisation of the model with q = −16. Top: velocity dispersion (σ 2 ) profile. Middle: anisotropy profile (β). Bottom: mass density profile (ρ). The black arrows show the direction of evolution and the • marks the position of r h . The profiles are 20 Hénon units apart and cover the range [0, 100] Hénon units. For the construction of the β and σ 2 profiles, the bins contain 500 particles, but 250 particles for the ρ profiles. The value of β in the innermost bin at t = 0 is ≈ −0.7.
Figure 10 .
10The predicted response of selected anisotropic Plummer models (in Hénon units), see Section 4.2 for details.
Figure 11 .
11Early evolution of the central specific entropy in the anisotropic Plummer models illustrated inFig. 4. The initial value of the specific entropy is an increasing function of q (one realization is shown for each value of q). The velocity dispersion and density are computed from the 500 innermost particles. Time is expressed in Hénon units.
MNRAS 000, 1-13(2017)
In the isotropic case the magnitude of the velocity (v) is sampled from a one-dimensional distribution ∝ f (r, v)v 2 , while here, in the anisotropic case, the magnitudes of the tangential and radial velocities (v t , v r ) have been sampled from a joint probability distribution ∝ f (r, v t , v r )v t . For acceptance-rejection sampling a bound on the maximum value of the f (r, v t , v r )v t has to be known at each radius. This is calculated numerically in advance of sampling on a radial grid. During sampling the bound at a particular r is calculated by interpolation.
In this subsection the units of r a are those used in Sec.2.1.1, i.e. such that the scale radius of the Plummer density profile is unity.
ACKNOWLEDGEMENTSWe thank an anonymous referee for several comments including pertinent references. We are grateful to Mark Gieles, Ralf Klessen, Rainer Spurzem, Enrico Vesperini, and Alice Zocchi for interesting conversations about the role of anisotropy in the dynamical evolution of star clusters. All authors acknowledge support from the Leverhulme Trust (Research Project Grant, RPG-2015-408), and ALV also from the EU Horizon 2020 program (Marie Sklodowska-Curie Fellowship, MSCA-IF-EF-RI 658088).APPENDIX A: EVOLUTION DESCRIBED BY AN ANISOTROPIC GASEOUS MODEL OF TWO-BODY RELAXATIONIn this Appendix we describe details of the derivation of an equation which is the focus of Section 4.3. As mentioned there the equation is based on a model for the evolution of an anisotropic spherical system under the action of twobody relaxation, which drives fluxes of radial and transverse kinetic energy with velocities v r , v t , respectively. The model is due toLouis & Spurzem (1991). Actually, these authors describe two variants of the model, one of which (their Model B) would lead to v r and v t differing by a term of order r at small radii in an anisotropic Plummer model. As those authors note, such behaviour is incompatible with a finite central density, and so we adopt their Model A, in which v r = v t = v, where v is the velocity of transport of kinetic energy.Much as in the isotropic gas model ofLynden-Bell & Eggleton (1980), the Lagrangian fluxes are taken to be proportional to temperature gradients, and Louis & Spurzemwhere λ is a dimensionless constant,and T is a relaxation time defined byin which m is the stellar mass. Note also that, in this account of the model, we have adopted from the present paper the definition of σ 2 t as the mean square two-dimensional transverse velocity.In the anisotropic Plummer model, the velocity dispersions are given byDejonghe (1987, equations 22a, b), whence we find that, at small radius, we havethough we have reinstated G, the cluster mass (M) and the scale radius of the Plummer model (R), which Dejonghe scaled away. (We avoid use of the more common a, which is reserved for anisotropy in the model, and is mentioned below.) Hence we find from Equations (A1), (A2) and(A3)thatwhere we have made use of a standard expression for the value of ρ at the centre of a Plummer model. In order to evaluate the effect of these fluxes on central quantities, we substitute them into two of the four moment equations of the model(Louis & Spurzem 1991, equations 43-46). The first of these is mass continuity, i.e.∂ ρ ∂tWe suppose that u = u r + O(r 3 ), where u is its central rderivative, and so find the obvious result thatto lowest order in r, and the variables ρ, ρ also denote central values.Next comes the hydrostatic equation. This is equivalent to our Equation(6), except that the acceleration is absent. In this form it is satisfied by the anisotropic Plummer model, and it gives no information on the rate of change of any central quantity.The last two moment equations are flux balance equations for p and the product pa, whereis the pressure and a is a measure of the anisotropy. Only the first equation is useful, as a = 0 to order r near the centre. The first equation, however, is easily handled like the continuity equation, givingto lowest order in r.A suitable linear combination of Equations (A8) and (A10), with use of Equation (A9), givesand so with Equations (A6) and (A3) we find thatwhere we have again used the standard expression for the central density in a Plummer model, and Equations (A2), (A4) and (A5) for the central value of σ. This is the equation which, with one change of notation, is discussed in Section 4.3.This paper has been typeset from a T E X/L A T E X file prepared by the author.
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| [] |
[
"The mathematical role of time and space-time in classical physics",
"The mathematical role of time and space-time in classical physics"
] | [
"Newton C A Da Costa ",
"Adonai S Sant'anna ",
"\nDepartment of Mathematics\nResearch Group on Logic and Foundations. Institute for Advanced Studies\nUniversity of São Paulo. Av. Prof. Luciano Gualberto, trav. J\n374. 05655-010São Paulo SPBrazil\n",
"\nFederal University of Paraná\nC. P. 01908181531-990CuritibaBrazil\n"
] | [
"Department of Mathematics\nResearch Group on Logic and Foundations. Institute for Advanced Studies\nUniversity of São Paulo. Av. Prof. Luciano Gualberto, trav. J\n374. 05655-010São Paulo SPBrazil",
"Federal University of Paraná\nC. P. 01908181531-990CuritibaBrazil"
] | [] | We use Padoa's principle of independence of primitive symbols in axiomatic systems in order to discuss the mathematical role of time and space-time in some classical physical theories. We show that time is eliminable in Newtonian mechanics and that space-time is also dispensable in Hamiltonian mechanics, Maxwell's electromagnetic theory, the Dirac electron, classical gauge fields, and general relativity. | 10.1023/a:1015561807144 | [
"https://arxiv.org/pdf/gr-qc/0102107v2.pdf"
] | 16,168,248 | gr-qc/0102107 | 19dc9937f3cb0bb280178edf36538a6ea71431c4 |
The mathematical role of time and space-time in classical physics
24 Apr 2001 August 7, 2021
Newton C A Da Costa
Adonai S Sant'anna
Department of Mathematics
Research Group on Logic and Foundations. Institute for Advanced Studies
University of São Paulo. Av. Prof. Luciano Gualberto, trav. J
374. 05655-010São Paulo SPBrazil
Federal University of Paraná
C. P. 01908181531-990CuritibaBrazil
The mathematical role of time and space-time in classical physics
24 Apr 2001 August 7, 2021arXiv:gr-qc/0102107v2
We use Padoa's principle of independence of primitive symbols in axiomatic systems in order to discuss the mathematical role of time and space-time in some classical physical theories. We show that time is eliminable in Newtonian mechanics and that space-time is also dispensable in Hamiltonian mechanics, Maxwell's electromagnetic theory, the Dirac electron, classical gauge fields, and general relativity.
Introduction
In a series of papers G. Jaroszkiewicz and K. Norton have developed a quite interesting research program where the continuum space-time is replaced by a discrete space-time in classical and quantum particle mechanics as well as in classical and quantum field theories [12,13,19,20]. Many other authors have developed similar ideas [1,15,14] in the last years. One of us has recently proposed the use of category theory in order to guarantee the consistency between discrete space-time and the continuum space-time picture for classical particle mechanics [23]. Nevertheless, one of the pioneers in the discretization of space-time was the Japanese physicist T. Tati [31,32]. Tati began his scientific career working with S. Tomonaga. Both of them were concerned with the divergences in quantum field theory. Tomonaga followed the way of renormalization theory. Tati tried the elimination of space-time in quantum theory in order to develop a theory of finite degree of freedom and avoid the divergences. Tomonaga won the Nobel prize. Tati's ideas about the elimination (by means of discretization) of space-time in physics were forgotten. If the reader takes a look on Web of Science, it will be easy to see that Tati's papers about non-space-time physics have no citation at all.
This research program of discretization confirms two things: (i) space-time effectivelly plays a very fundamental role in physics, at least from the intuitive point of view; and (ii) discretization is a deep way to change the usual concepts about space-time. Discrete space-time seems to be more appropriate from the physical point of view than continuum space-time, at least in some cases. In a recent paper, e.g., C. Vafa [33] says:
Many ideas in physics sound like basic objects in number theory. For example, electrical charge comes in quanta, and do not take continuous values; matter comes in quanta, called particles, and do not come in continuous values; etc. In some sense quantum theory is a bending of physics towards number theory. However, deep facts of number theory play no role in questions of quantum mechanics. In fact it is very surprising, in my view, that this area of mathematics has found very little application in physics so far. In particular we do not know of any fundamental physical theories that are based on deep facts in number theory.
Yet, as a scientific prophet, Vafa predicts that in this century ...we will witness deep applications of number theory in fundamental physics.
W. M. Stuckey presents another perspective [25,26,27]. According to him, the usual notion of differentiable manifold as a model of space-time is not appropriate in quantum physics. He does not appeal to any discretization program. He proposes a "pregeometric reduction of transtemporal objects" in order to cope with non-locality and quantum gravity.
On the other hand, there is an insightful paper by U. Mohrhoff [18] that presents a novel interpretation of quantum mechanics, where objective probabilities are assigned to counterfactuals and are calculated on the basis of all relevant facts, including those that are still in the future. According to this proposal, the intuitive distinction between here and there, past and future, has nothing to do with any physical reality:
The world is built on facts, and its spatiotemporal properties are supervenient on the facts. To understand how EPR [Einstein-Podolsky-Rosen Gedanken experiment] correlations are possible, one needs to understand that, in and of itself, physical space is undifferentiated. At a fundamental level, "here" and "there" are the same place. If EPR correlations require a medium, this identity is the medium.
Mohrhoff's ideas about space-time may inspire us for a new perspective about spacetime, quite different from discretization. Tati believed that discretization of space-time is a manner to eliminate it from physics. But in this paper we discuss a different point of view, more radical, in a sense.
We note, in a very general framework, that time and space-time are dispensable in some of the main classical physical theories. This corroborates the intuitive idea given above that space-time properties are supervenient on physical facts.
Our results are simple, almost trivial, from the mathematical point of view, but, as we shall show in future papers, they are philosophically relevant.
Axiomatization of Physical Theories
The presence of mathematics in physics is grounded on the belief that there are mathematical patterns lurking in the dynamics of physical phenomena and natural laws. This is one of the most fundamental, powerful and 'holy' beliefs in science [24]. On the other hand, mathematical concepts cannot, in principle, be considered as a faithful picture of the physical world. There is no length measurement or elapsed time interval, e.g., that actually corresponds to a real number, since measurements do have serious limitations, among others, of precision. This is one justification for the discretization of space-time in physics, according to our discussion in the Introduction. Nevertheless, the hypothesis of the continuum space-time is largely used in physical theories. Although mathematics is a science conceived by the human mind of mathematicians, it has been used to send men to the Moon, to make computers, and to facilitate our lives.
The axiomatic method is the soul of mathematics, the synthesis of the scientific method, and a powerful tool for the scientific philosopher. Rationality, in formal sciences, consists, in particular, in the correct, implicit or explict, use of the axiomatic method.
Roughly speaking, the axiomatic method begins with the definition of an appropriate language. In this language we have (i) the vocabulary, which is a set of symbols, refered to as primitive symbols; (ii) the grammar rules, usually refered to as an effective procedure to distinguish the well-formed formulas from other arbitrary sequences of symbols; (iii) a specific set of well-formed formulas called axioms; and (iv) some rules of inference which allow us to prove theorems from the axioms. This is possible because the rules of inference establish the relations of logical consequence among well-formed formulas.
Primitive concepts are necessary since we cannot define everything. The reason for the very notion of well-formed formulas is to distinguish between meaningful and meaningless expressions. The main purpose of the axioms is to settle up how the primitive concepts are related to each other in a list of assumptions, in order to derive other sentences and bits of information about these primitive concepts by means of the inference rules.
It is not clear, from the historical point of view, if the axiomatic method was born as a philosophical or as mathematical subject. But it was used for the first time in ancient Greece, mainly represented by the famous book Elements by Euclide.
One of the main goals of the axiomatic method in physics is the investigation of the logical foundations of physical theories. Following ideas delineated by Hilbert (1900) in his famous Matematische Probleme, a basic problem may be described as follows [11]:
The investigations of the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part: first of all, the theory of probability and mechanics.
...
If geommetry is to serve as a model for the treatment of physical axioms, we shall try first by a small number of axioms to include as large a class as of physical phenomena, and then by adjoining new axioms to arrive gradually at the more special theories. At the same time Lie's principle of subdivision can perhaps be derived from profound theory of infinite transformation groups. The mathematician will have also to take account not only of those theories coming near to reality, but also, as in geometry, of all logically possible theories. He must be always alert to obtain a complete survey of all conclusions derivable from the system of axioms assumed.
Further, the mathematician has the duty to test exactly in each instance whether the new axioms are compatible with the previous ones. The physicist, as his theories develop, often finds himself forced by the results of his experiments to make new hypotheses, while he depends, with respect to the compatibility of the new hypotheses with the old axioms, solely upon these experiments or upon a certain physical intuition, a practice which in the rigorously logical building up of a theory is not admissible. The desired proof of the compatibility of all assumptions seems to me also of importance, because the effort to obtain each proof always forces us most effectually to an exact formulation of the axioms.
We believe that this kind of investigation may drive us to a better understanding of the actual role of fundamental concepts, and even fundamental principles, in mechanics and physics in general.
In Euclidian geometry, e.g., it is possible to eliminate or even replace one or more postulates by other non-equivalent sentences in order to get non-Euclidian or non-Paschian geometries. Analogously it is possible to derive other set theories such that the axiom of choice, the continuum hypothesis, or Martin's axiom are no longer valid sentences. D. Hilbert considered that this sort of theoretical richness should be achieved in physics by means of a precise use of the axiomatic method.
One important question in connection with the axiomatic method is to determine if a primitive concept of a given axiomatic system is definable or not on the basis of the remaining primitive concepts. For example: Usually force is a primitive concept in classical mechanics. But can we define "force"? The answer depends on the axiomatic system that we employ in order to formulate classical mechanics (see, for example [22]). In the next section we show how Padoa's principle may be used to prove the independence of primitive concepts.
Padoa's Principle of Independence of Primitive Symbols
In an axiomatic system S a primitive term or concept c is definable by means of the remaining primitive ones if there is an appropriate formula, provable in the system, that fixes the meaning of c in function of the other primitive terms of S. This formulation of definability is not rigorous but is enough here. When c is not definable in S, it is said to be independent of the the other primitive terms.
There is a method, introduced by A. Padoa [21], which can be employed to show the independence of concepts. In fact, Padoa's method gives a necessary and sufficient condition for independence [2,28,30].
In order to present Padoa's method, some preliminary remarks are necessary. Loosely speaking, if we are working in set theory, as our basic theory, an axiomatic system S characterizes a species of mathematical structures in the sense of Bourbaki [3]. Actually there is a close relationship between Bourbaki's species of structures and Suppes predicates [29]; for details see [5]. On the other hand, if our underlying logic is higher-order logic (type theory), S determines a usual higher-order structure [4]. In the first case, our language is the first order language of set theory, and, in the second, it is the language of (some) type theory. Tarski showed that Padoa's method is valid in the second case [30], and Beth that it is applicable in the first [2].
From the point of view of applications of the axiomatic method, for example in the foundations of physics, it is easier to assume that our mathematical systems and structures are contructed in set theory [5].
A simplified and sufficiently rigorous formulation of the method, adapted to our exposition, is described in the next paragraphs.
Let S be an axiomatic system whose primitive concepts are c 1 , c 2 , ..., c n . One of these concepts, say c i , is independent from the remaining if and only if there are two models of S in which c 1 , ..., c i−1 , c i+1 , ..., c n have the same interpretation, but the interpretations of c i in such models are different.
Of course a model of S is a set-theoretical structure in which all axioms of S are true, according to the interpretation of its primitive terms [17].
It is important to recall that, according to the theory of definition [28], a definition should satisfy the criterion of eliminability. That means that a defined symbol should always be eliminable from any formula of the theory.
In the sequel we apply Padoa's method to some physical theories (i.e., axiomatic systems), in order to prove that time and space-time are eliminable (or dispensable) from some physical theories.
McKinsey-Sugar-Suppes System of Particle Mechanics
This section is essentially based on the axiomatization of classical particle mechanics due to P. Suppes [28], which is a variant of the formulation by J. C. C. McKinsey, A. C. Sugar and P. Suppes [16]. We call this McKinsey-Sugar-Suppes system of classical particle mechanics and abbreviate this terminology as MSS system. MSS system has six primitive notions: P , T , m, s, f , and g. P and T are sets, m is a real-valued unary function defined on P , s and g are vector-valued functions defined on the Cartesian product P × T , and f is a vector-valued function defined on the Cartesian product P × P × T . Intuitivelly, P corresponds to the set of particles and T is to be physically interpreted as a set of real numbers measuring elapsed times (in terms of some unit of time, and measured from some origin of time). m(p) is to be interpreted as the numerical value of the mass of p ∈ P . s p (t), where t ∈ T , is a 3-dimensional vector which is to be physically interpreted as the position of particle p at instant t. f (p, q, t), where p, q ∈ P , corresponds to the internal force that particle q exerts over p, at instant t. And finally, the function g(p, t) is to be understood as the external force acting on particle p at instant t.
Next we present the axiomatic formulation for MSS system: Definition 1 P = P, T, s, m, f , g is a MSS system if and only if the following axioms are satisfied:
P1 P is a non-empty, finite set.
P2 T is an interval of real numbers.
P3 If p ∈ P and t ∈ T , then s p (t) is a 3-dimensional vector (s p (t) ∈ ℜ 3 ) such that
d 2 sp(t) dt 2
exists.
P4 If p ∈ P , then m(p) is a positive real number.
P5 If p, q ∈ P and t ∈ T , then f (p, q, t) = −f (q, p, t).
P6 If p, q ∈ P and t ∈ T , then [s
p (t), f (p, q, t)] = −[s q (t), f (q, p, t)].
P7 If p, q ∈ P and t ∈ T , then m(p)
d 2 sp(t) dt 2 = q∈P f (p, q, t) + g(p, t).
The brackets [,] in axiom P6 denote external product. Axiom P5 corresponds to a weak version of Newton's Third Law: to every force there is always a counterforce. Axioms P6 and P5, correspond to the strong version of Newton's Third Law. Axiom P6 establishes that the direction of force and counterforce is the direction of the line defined by the coordinates of particles p and q.
Axiom P7 corresponds to Newton's Second Law.
Definition 2 Let P = P, T, s, m, f , g be a MSS system, let P ′ be a non-empty subset of P , let s ′ , g ′ , and m ′ be, respectively, the restrictions of functions s, g, and m with their first arguments restricted to P ′ , and let f ′ be the restriction of f with its first two arguments restricted to P ′ . Then
P ′ = P ′ , T, s ′ , m ′ , f ′ , g ′ is a subsystem of P if ∀p, q ∈ P ′ and ∀t ∈ T , m ′ (p) d 2 s ′ p (t) dt 2 = q∈P ′ f ′ (p, q, t) + g ′ (p, t).
(1)
Theorem 1 Every subsystem of a MSS system is again a MSS system. Definition 4 A MSS system is isolated if and only if for every p ∈ P and t ∈ T , g(p, t) = 0, 0, 0 .
Theorem 2 If P = P, T, s, m, f , g and P ′ = P ′ , T ′ , s ′ , m ′ , f ′ , g ′
are two equivalent systems of particle mechanics, then for every p ∈ P and t ∈ T
q∈P f (p, q, t) + g(p, t) = q∈P ′ f ′ (p, q, t) + g ′ (p, t).
The embedding theorem is the following:
Theorem 3 Every MSS system is equivalent to a subsystem of an isolated system of particle mechanics.
The next theorem can easily be proved by Padoa's method:
Theorem 4 Mass and internal force are each independent of the remaining primitive notions of MSS system.
In the next paragraph follows a sketch of the proof of this last theorem. Consider, for example, two interpretations (A and B) for MSS system, as it follows: In interpretation A we have P = {1, 2}, all internal forces and external forces are null, T is the real interval [0, 1], s p (t) is a constant c for all p and for all t, and m 1 = m 2 = 1; in interpretation B we have P = {1, 2}, all internal forces and external forces are null, T is the real interval [0, 1], s p (t) is the same constant c above for all p and for all t, and m 1 = m 2 = 2. Since s p (t) is a given constant, then all accelerations are null. We can easily see that (i) these two interpretations are models of MSS (in the sense that they satisfy all the axioms of MSS); and (ii) these two models are the same, up to the interpretation of the masses m 1 and m 2 . Then, according to Padoa's principle, mass is an independent concept and, so, is not definable. The same argument may used for internal force.
According to Suppes [28]:
Some authors have proposed that we convert the second law [of Newton], that is, P7, into a definition of the total force acting on a particle. [...] It prohibits within the axiomatic framework any analysis of the internal and external forces acting on a particle. That is, if all notions of force are eliminated as primitive and P7 is used as a definition, then the notions of internal and external force are not definable within the given axiomatic framework.
The next theorem is rather important for our discussion on the dependence of time with respect to the remaining primitive concepts.
Theorem 5 Time is definable from the remaining primitive concepts of MSS system. Proof: According to Padoa's Principle, the primitive concept T in MSS system is independent from the remaining primitive concepts (mass, position, internal force, and external force) iff there are two models of MSS system such that T has two interpretations and the remaining primitive symbols have the same interpretation. But these two interpretations are not possible, since position s, internal force f , and external force g are functions whose domains depend on T . If we change the interpretation of T , then we will change the interpretation of three other primitive concepts, namely, s, f , and g. So, time is not independent and hence it can be defined.✷
The reader will note that our proof does not show how to define time. But that is not necessary if we want to show that time is dispensable in MSS system, since a definition does satisfy the criterion of eliminability.
Theorem 6
The differential equation given in axiom P7 is autonomous, i.e., it does not depend on an independent variable t of time.
Proof: Straightforward, since the parameter time t is definable (theorem 5) by means of position and forces.✷ What is the epistemological meaning of the definability of T ? It is usual to say that one of the main goals of classical mechanics is to make predictions. But a prediction refers to the notion of future and, therefore, to the very notion of time. If time is dispensable, then one question remains: what is the main goal of classical particle mechanics, at least from the point of view of MSS system? It seems to us that the main goal of classical particle mechanics is to describe the physical state of particles. If we give differential equations 2 (with respect to t) for f and g, then the physical state of a particle p, in this framework, is the set of points (s p (t), f (p, q, t), g p (t)) (2) in the phase space of position versus internal force versus external force, given initial conditions. By initial conditions we mean a specific point in the phase space. Since the phase space does not make any reference to the parameter t, then the set of points given above is equivalent to a set of points (a curve)
(s p , f (p, q), g p )(3)
given a specific point (initial condition) in the phase space.
Other Physical Theories
One of us has worked with F. A. Doria in a series of papers concerning the axiomatic foundations of physical theories. Some of these papers are [6,7,8,9]. In [8] there is a unified treatment, by means of axiomatic method, for the mathematical structures underlying Hamiltonian mechanics, Maxwell's electromagnetic theory, the Dirac electron, classical gauge fields, and general relativity. This unified treatment is given by the following definition.
Definition 5 The species of structures (à la Bourbaki) of a classical physical theory is given by the 9-tuple Σ = M, G, P, F , A, I, G, B, ▽ϕ = ι where 1. M is a finite-dimensional smooth real manifold endowed with a Riemannian metric (associated to space-time) and G is a finite-dimensional Lie group (associated to transformations among coordinate systems).
2. P is a given principal fiber bundle P (M, G) over M with Lie group G.
3. F , A, and I are cross-sections of bundles associated to P (M, G), which correspond, respectively, to the field space, potential space, and current or source space. 5. ▽ϕ = ι is a Dirac-like equation, where ϕ ∈ F is a field associated to its corresponding potential by means of a field equation and ι ∈ I is a current. Such a differential equation is subject to boundary or initial conditions B.
We are obviously omitting some important details, which can be found in [8]. The unified picture of classical gauge field equations by means of Dirac equation may be found in [10]. Our point, in this section, is to discuss the mathematical role of space-time in classical physical theories as presented in definition 5.
Theorem 7 Space-time is dispensable in a classical physical theory, in the sense of definition (5).
Proof: Padoa's Principle says that the primitive concept M in Σ is independent from the remaining primitive concepts iff there are two models of Σ such that M has two interpretations and all the other primitive symbols have the same interpretation. But these two interpretations are not possible, since all the remaining concepts, except G, depend on space-time M . Any change of interpretation related to M will imply a change of interpretation of P , F , A, I, G, B, and ▽ϕ = ι. Therefore, space-time is not independent and hence it can be defined. So, according to the criterion of eliminability of definitions, space-time is dispensable.✷ As said in the Introduction, we intend to discuss the philosophical consequences of our results in future works. But one fact seems to be very clear here: since our definition of classical physical theory refers to classical field theories, this last theorem says that the concept of field is more fundamental than the notion of space-time. If this result is not satisfactory from the intuitive point of view, then our mathematical framework for classical field theories should be changed.
Acknowledgments
1 Definition 3
13Two MSS systems P = P, T, s, m, f , g and P ′ = P ′ , T ′ , s ′ , m ′ , f ′ , g ′ are equivalent if and only if P = P ′ , T = T ′ , s = s ′ , and m = m ′ .
4 .
4G ⊆ Diff(M ) ⊗ G ′ is the symmetry group, where Diff(M ) is the group of diffeomorphisms of M and G is the group of gauge transformations of the principal fiber bundle P (M, G).
In the original MSS system it is presented another definition for subsystem, where this theorem is not valid.
These differential equations will be necessarily autonomous, by means of theorem 6.
We acknowledge with thanks some suggestions and criticisms made by Dr. Ulrich Mohrhoff and Dr. Mark Stuckey in connection with a previous version of this text. The idea of the definability of time in MSS system was originated from questions raised by two students (Humberto R. R. Quoirin and Tomas K. Breuckmann) of one of us (ASS) into the context of seminars on the logical foundations of physics at Federal University of Paraná.
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| [] |
[
"First-order directional ordering transition in the three-dimensional compass model",
"First-order directional ordering transition in the three-dimensional compass model"
] | [
"Max H Gerlach \nInstitut für Theoretische Physik and Centre for Theoretical Sciences (NTZ)\nUniversität Leipzig\nPostfach 100 92004009LeipzigGermany\n\nInstitut für Theoretische Physik\nUniversität zu Köln\nZülpicher Str. 7750937KölnGermany\n",
"Wolfhard Janke \nInstitut für Theoretische Physik and Centre for Theoretical Sciences (NTZ)\nUniversität Leipzig\nPostfach 100 92004009LeipzigGermany\n"
] | [
"Institut für Theoretische Physik and Centre for Theoretical Sciences (NTZ)\nUniversität Leipzig\nPostfach 100 92004009LeipzigGermany",
"Institut für Theoretische Physik\nUniversität zu Köln\nZülpicher Str. 7750937KölnGermany",
"Institut für Theoretische Physik and Centre for Theoretical Sciences (NTZ)\nUniversität Leipzig\nPostfach 100 92004009LeipzigGermany"
] | [] | We study the low-temperature properties of the classical three-dimensional compass or t 2g orbital model on simple-cubic lattices by means of comprehensive large-scale Monte Carlo simulations. Our numerical results give evidence for a directionally ordered phase that is reached via a first-order transition at the temperature T 0 = 0.098328(3)J/k B . To obtain our results we employ local and cluster update algorithms, parallel tempering and multiple histogram reweighting as well as model-specific screw-periodic boundary conditions, which help counteract severe finite-size effects. | 10.1103/physrevb.91.045119 | [
"https://arxiv.org/pdf/1406.1750v3.pdf"
] | 118,436,992 | 1406.1750 | 3a3bd9ba7717f8d24296207c1a12c8d21e1134c5 |
First-order directional ordering transition in the three-dimensional compass model
Max H Gerlach
Institut für Theoretische Physik and Centre for Theoretical Sciences (NTZ)
Universität Leipzig
Postfach 100 92004009LeipzigGermany
Institut für Theoretische Physik
Universität zu Köln
Zülpicher Str. 7750937KölnGermany
Wolfhard Janke
Institut für Theoretische Physik and Centre for Theoretical Sciences (NTZ)
Universität Leipzig
Postfach 100 92004009LeipzigGermany
First-order directional ordering transition in the three-dimensional compass model
(Dated: June 9, 2014)numbers: 0570Fh7510Hk7540Mg
We study the low-temperature properties of the classical three-dimensional compass or t 2g orbital model on simple-cubic lattices by means of comprehensive large-scale Monte Carlo simulations. Our numerical results give evidence for a directionally ordered phase that is reached via a first-order transition at the temperature T 0 = 0.098328(3)J/k B . To obtain our results we employ local and cluster update algorithms, parallel tempering and multiple histogram reweighting as well as model-specific screw-periodic boundary conditions, which help counteract severe finite-size effects.
I. INTRODUCTION
The compass model 1 is a generic model for orbitalorbital interactions in certain Mott insulators such as various transition-metal compounds. In systems with partially filled orbital 3d shells it provides a heuristic description for the coupling of t 2g orbitals. If their interaction is dominated by the Kugel-Khomskii superexchange mechanism, the quantum compass model is realized, while the phonon-mediated Jahn-Teller effect gives rise to the classical compass model. 2,3 Beyond the rich physics of orbital order in recent years the quantum compass model has received increased attention because it provides an alternative route to realize qubits that are shielded from decoherence via so-called topological protection. 4,5 In this context the model is realized in the form of arrays of superconducting Josephson junctions, which have already been implemented successfully in experiments. 6 While the compass model is closely related to the wellstudied O(n) and Heisenberg lattice spin models with nearestneighbor interactions, it differs from these in a fundamental aspect: It features an inherent coupling of real space symmetry, realized by the point group of the lattice, to the symmetry of the interactions encoded in the Hamiltonian. The resulting competition of exchange couplings along the different lattice axes prevents a conventional magnetization-like ordered phase, but still allows for long-ranged, essentially onedimensional directional ordering. 7 The peculiar symmetries of the compass model lead to a high degree of degeneracy in its ground states, 8 similarly to other orbital models. 1 Typically such a degeneracy suppresses order for T = 0, while at low, but finite temperatures an ordered phase may still be realized through an order-by-disorder 9,10 mechanism, where certain system configurations are favored entropically. For both the classical and the quantum variation of the compass model in two dimensions (2D) earlier Monte Carlo studies have indeed established the realization of a directionally ordered phase at low temperatures, which is reached by a continuous thermal phase transition in the 2D Ising universality class. [11][12][13] Beyond that, the case of the three-dimensional (3D) compass model remains particularly interesting as it may be significant for the microscopic description of materials in the reach of experimental research. For the 3D quantum compass model high-temperature series expansions have not shown any sign of a finite-temperature phase transition, while the continuous transition could be confirmed for the 2D quantum compass model. 14 The purpose of this paper is to shed more light on the low-temperature properties of the compass model in three dimensions. We present an extensive Monte Carlo study that provides evidence for a first-order phase transition from a high-temperature disordered phase into a directionally ordered phase. While simulations of the quantum model are plagued by a negative-sign problem and hence are infeasible on reasonably sized lattices, we can study the classical variation of the 3D compass model without prohibitive computational cost. Nevertheless, a considerable methodological effort is required to obtain quantitative results for two reasons: The model features very strong finite-size effects that must be treated carefully and long autocorrelation times near the transition point would make it hard to collect sufficient statistics with only a naive Monte Carlo sampling scheme.
The main part of this work is organized as follows: In section II we formally introduce the model and discuss some of its properties. The specific numerical methods we apply are then described in section III before we present our results in section IV. We close in section V with conclusions and an outlook.
II. THE MODEL
In d spatial dimensions the compass model is defined on a simple-hypercubic lattice of size N = L d by the Hamiltonian
H = − d k=1 N i=1 J k s k i s k i+k .(1)
Here s k i is the k-th component of a spin s i at lattice site i. J k is a coupling constant depending on the lattice direction k. The nearest neighbor of site i in the k-th direction is indicated by i +k. In the classical compass model the constituent spins are represented by vectors on the unit hypersphere in d-dimensional space: s i ∈ S d−1 . Two spins on sites neighboring in direction k only interact in their k-th components.
Note that Eq. (1) could be separated into d independent onedimensional Hamiltonians, if the directions were not coupled by the constraint |s i | = 1.
In this paper we limit the discussion to equal coupling constants in every direction: J k ≡ J. The Hamiltonian of the three-dimensional model on a cubic lattice of size N = L 3 then reads
H (3D) = −J N i=1 s x i s x i+x + s y i s y i+ŷ + s z i s z i+ẑ ,(2)
where the spins s i ∈ S 2 can be parametrized by azimuthal and polar angles θ i ∈ [0, π] and ϕ i ∈ [0, 2π):
s i = s(θ i , ϕ i ) = s x i s y i s z i = cos ϕ i sin θ i sin ϕ i sin θ i cos θ i .(3)
In this work we choose a coupling constant of J > 0 corresponding to ferromagnetic interactions. The classical compass model is obtained by taking the limit of large spin S of the quantum mechanical compass model, where the spins would be represented by S = 1/2 operators s i = 2 (σ x , σ y , σ z ) with the Pauli matrices σ k .
The compass model in Eq. (1) has a high number of ground states. To begin with, any constant spin configuration is a ground state. Beyond that, the model exhibits a number of discrete symmetries, which lead to a macroscopic degeneracy of every energetic state, including but not limited to the ground state. 1,8 Most importantly for d = 3 with open or periodic boundary conditions, Eq. (2) is invariant under a reflection of all spins on any line of sites parallel to one of the lattice axes across the orthogonal plane, which leads to a 2 3L 2 -fold degeneracy. As a consequence of these gauge-like symmetries conventional magnetic order is prohibited at any temperature: 7 m = | 1 N i s i | ≡ 0. However, quantities such as s k i s k i+k are invariant under these symmetries and a special type of directional or "nematic" ordering is not precluded. One can construct order parameters that measure directional ordering characterized by long-rang correlations in the direction of fluctuations in spin and lattice spaces, even though magnetic ordering is absent. This type of order is realized by linear spin alignment parallel to the lattice axes so that nearest-neighbor bonds carrying the lowest energy are oriented mostly along one specific direction as illustrated in Fig. 1. It is not obvious to which degree the ground-state degeneracy translates into the number of distinct directionally ordered phases at low finite temperature.
III. NUMERICAL METHODS
A. Observables
We now turn to our numerical simulations of Eq. (2) carried out at various inverse temperatures β = 1/k B T and first discuss the quantities we measure. By On each face of the cube the averaged projection to the orthogonal direction of all spins at sites in one column above that face is given color-coded. While in the high-temperature snapshot at β = 4/J no order can be recognized, there is a strong tendency towards linear alignment of the spins in the ±ẑ-directions in the lowtemperature snapshot at β = 20/J. with k = x, y, z we denote the total bond energy along the k-th lattice axis. Our basic observable is then the total energy
E = E x + E y + E z(4)
with the corresponding heat capacity
C = ∂E ∂T = k B β 2 E 2 − E 2 .(5)
In previous studies an order parameter for directional ordering in the two-dimensional model has been defined by the energy excess in one of the lattice directions compared to the other direction. [11][12][13] Here we consider a three-dimensional extension
D = 1 N (E z − E y ) 2 + (E y − E x ) 2 + (E z − E x ) 2 .(6)
To help with the analysis of the directional ordering phase transition and its finite-size scaling we also consider quantities derived from D: the susceptibility χ and the Binder parameter Q 2 , which are defined as
χ = N D 2 − D 2 , Q 2 = 1 − 1 3 D 4 D 2 2 .(7)
B. Screw-periodic boundary conditions
In most cases simulations of statistical models are carried out on finite lattices with the topology of a torus, i.e., with periodic boundary conditions. The assumption is that compared to open or fixed boundary conditions this choice minimizes finite-size surface effects, which become irrelevant in the thermodynamic limit.
In previous studies of the two-dimensional classical compass model, however, periodic boundary conditions have not turned out to be an ideal choice. In the directionally ordered low-temperature phase the spins form essentially onedimensional chains with decoupled rows and columns of spins on the square lattice. With periodic boundary conditions the spins tend to form closed aligned loops along the boundaries of a finite lattice. Such excitations are particularly stable against thermal fluctuations. In their studies Mishra et al. have noticed such an effect spoiling the finite-size scaling with periodic boundary conditions 11 and suggested that the reason may lie in the existence of a one-dimensional magnetic correlation length ξ 1D which exceeds the linear system size L at low temperatures. Wenzel et al. have confirmed this claim. 13 As a solution the authors of Ref. 11 have adopted special fluctuating or annealed boundary conditions. Here the signs of the coupling constants on the bonds at the lattice boundaries are allowed to fluctuate thermally. In this way, onedimensional chains are effectively broken up. While one can assume that the influence of these dL d−1 fluctuating bonds becomes unimportant in the thermodynamic limit as N = L d → ∞, this choice still constitutes a considerable modification of the model and no good finite-size scaling theory is available for this type of boundary conditions.
As an alternative the authors of Ref. 13 have proposed screw-periodic boundary conditions, which are a particular deformation of the torus topology of regular periodic boundary conditions. We generalize their definition to three dimensions to obtain boundary conditions that interconnect lines of spins along any of the principal lattice directions. Explicitly, the nearest neighbors of a site i = (x, y, z) in directionsx,ŷ,ẑ are specified as follows:
(x, y, z) +x = (x + 1, y, z), if x < L − 1, (0, y, [z + S ] mod L), if x = L − 1, (x, y, z) +ŷ = (x, y + 1, z), if y < L − 1, ([x + S ] mod L, 0, z), if y = L − 1,(8)(x, y, z) +ẑ = (x, y, z + 1), if z < L − 1, (x, [y + S ] mod L, 0), if z = L − 1.
Here the screw length S is a parameter that can be varied. If S is taken as one of the distinct divisors of L, each plane of the lattice can be subdivided into S groups of sites or "loops" in each in-plane directionk, which are linked as pairs of neighbors along that direction. With S = 0 or S = L regular periodic boundary conditions are recovered. With S = 1 there are only single loops for each direction in a plane. The power of screw-periodic boundary conditions lies in the fact that with a sufficiently low choice of S , the loop length exceeds the magnetic correlation length ξ 1D already for small L. Hence, linearly aligned excitations are broken up more easily than with regular periodic boundary conditions. Besides that the screwperiodic boundary conditions reduce the number of discrete symmetries in the compass model and the energetic degeneracy of its configurations such that the leading degeneracy factor mentioned in the end of section II is lowered from 2 3L 2 to 2 3L .
We have found that also for the three-dimensional model regular periodic boundary conditions lead to poor finite-size scaling results. Moreover, the simple definition (6) of the order parameter D is disadvantageous with these boundary conditions because it assigns different values to configurations which differ by planar rotations, but really show an equal degree of order. To remedy both problems we use screwperiodic boundary conditions according to the definition (8) with a choice of S = 1.
The choice of these boundary conditions is not expected to have an influence on the thermodynamic limit. They have also been successfully applied for other purposes, e.g., for the controlled formation of tilted interfaces between ordered domains in the Ising model. 15
C. Monte Carlo methods
In the following section we outline the Monte Carlo algorithms applied in our simulations.
Fundamentally we use the standard Metropolis algorithm 16 for local single-spin updates. In one lattice sweep new orientations are proposed in sequential order for the spins at all sites. The direction of the new spin vector is chosen randomly from a uniform distribution over the surface area of a spherical cap centered around the original vector. During thermalization we adjust the opening angle of this spherical cap in such a way that an average acceptance ratio of 50% is realized at each temperature.
To reduce autocorrelation times we additionally use the one-dimensional version of the Wolff cluster update 17 introduced earlier for the 2D compass model 13 in a direct extension to the 3D model. This update exploits one of the discrete symmetries of the Hamiltonian, which is left invariant if a line of neighboring spins along one of the lattice directions is reflected about the plane orthogonal to that direction. To construct a cluster first a random starting site i and a lattice directionk ∈ {x,ŷ,ẑ} are chosen, then neighboring sites in directions ±k are added to the cluster with probability
P i,i±k (s i , s i±k ) = 1 − exp min 0, 2βJs k i s k i±k .(9)
This step is iterated with i taking the place of the newly adjoined site until no further sites are added. All spins in the strictly one-dimensional cluster constructed in this way are then flipped at the same time. Due to the restricted set of possible reflection planes, this update is not ergodic on its own, but must be used in combination with local spin updates. In our simulations 3L cluster updates in randomly chosen direc-tions are followed by N = L 3 local updates and we count this combination as one Monte Carlo sweep.
To further reduce autocorrelation times and improve statistics we combine these canonical algorithms with a paralleltempering scheme. 18,19 Different replicas of the system are simulated simultaneously at various inverse temperatures β k . We propose exchanges of system configurations between replicas at adjunct temperature points every 100 sweeps. The range of simulation temperatures is chosen according to the scheme of constant entropy increase, 20 which clusters the temperature points close to a phase transition and thus eases diffusion in temperature space, which has been valuable for the simulations on large lattices.
From the measurements taken in the various replicas we obtain time series of the observables D and E at various discrete inverse temperatures β k . Making use of multiple histogram reweighting techniques 21 these observables as well as the derived quantities χ, Q 2 and C can be estimated also at arbitrary intermediate temperatures from the optimally combined simulation data. We limit discretization errors by computing per-sample weighting factors from the density of states and reweighting observable time series directly. 22 By applying Brent's algorithm for minimization 23 we can precisely determine extremal temperature locations and values of χ, Q 2 and C or other quantities which are useful to characterize the finite-size scaling behavior at a phase transition. Estimates of the statistical uncertainties of these quantities are obtained by performing this procedure on jackknife resampled data sets. 24,25
IV. RESULTS
We now present the results we obtain in our Monte Carlo simulations that employ the methods presented in the previous section. The 3D compass model is simulated with screwperiodic boundary conditions with S = 1 on simple-cubic lattices of sizes N = L 3 with L ∈ {8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48}. In each case from 32 to 64 replicas are used in the parallel-tempering scheme. For the smallest lattice inverse temperatures βJ range in {4, . . . , 20}, while for the largest lattice βJ is chosen from {9.5, . . . , 11.5}. Simulations are performed for at least some 10 7 and up to 3.8 × 10 7 Monte Carlo sweeps on the largest lattice after an equilibration phase, typically one-tenth of that length.
For all lattice sizes we observe clear indications of a thermal phase transition around βJ ≈ 10 in the behavior of the order parameter D, which approaches zero in the high-temperature regime (low β) and a finite value D > 0, which characterizes directional ordering, at low temperatures. The two phases are visualized in Fig. 1. Note that up to thermal fluctuations we find all spins in the ordered finite-temperature phase to be aligned with some of the lattice axes even though the ground states of the compass model are not restricted to have such an orientation. Apparently fluctuations around these coaxial configurations are favored through an order-by-disorder mechanism.
The smoothed jump of the order parameter curve D(β) in the temperature region close to the transition point on different lattice sizes can be seen in Fig. 2(a). The transition is accompanied by peaks of the susceptibility χ in Fig. 2(b) and minima of the Binder parameter Q 2 in Fig. 2(c). On the larger lattices also bends in the curves of the normalized energy E(β)/N can be seen in the same temperature region in Fig. 3(a) together with peaks of the specific heat capacity C(β)/N in Fig. 3(b).
Close to the transition we furthermore find signs for phase coexistence, which is realized in histograms of the order parameter D with two peaks: one corresponding to a more disordered and one to a more ordered phase. By combining our reweighting and optimization algorithms, we can precisely estimate the inverse temperatures β D eqH (L), where the two peaks of the probability density P(D) have equal height. The estimates for P(D) at all lattice sizes are shown in Fig. 4. The double-peak structure is already present in the smallest system studied here with L = 8, but from L = 16 to L = 28 the relative suppression at the center of the probability distributions successively goes down and up to L = 24 the two peaks move closer together. Then, starting from L = 32, the behavior changes again: The dip between the two peaks grows with L and also their separation no longer shrinks. Moreover, from L = 36 on there are also double-peak structures in the histograms of the energy E. See Fig. 5 for the distributions P(E) measured at the corresponding inverse temperatures β E eqH (L). Table I lists the estimated values of β χ max (L), χ max (L), β C max (L), C max (L)/N, β Q 2 min (L), Q 2,min (L), β D eqH (L) and β E eqH (L) for all studied lattice sizes L. The signs for phase-coexistence at the transition temperature and the minima of the Binder parameter hint at a first-order phase transition in the thermody-namic limit. In the following we study finite-size scaling relations for the measured quantities to further support or rebut this claim. Even with the application of special screw-periodic boundary conditions finite-size effects appear to be rather severe with an irregular behavior for L ≤ 32.
A. Transition temperature
With β C max (L), β χ max (L), β Q 2 min (L), β D eqH (L) and β E eqH (L) there are various possible definitions of a lattice-size dependent inverse pseudo-transition temperature β * (L). For a discussion of the canonical finite-size scaling at a first-order transition see Ref. 26 and references therein. The inverse pseudo-transition temperatures are expected to have a displacement from the true infinite-volume transition point β 0 which to leading order scales proportionally to the reciprocal system size 1/L 3 :
β * (L) = β 0 + c * L 3 + · · · .(10)
We test this scaling relation for all definitions of β * (L) given above by performing least-squares fits of the β * (L) to Eq. (10) for various ranges of lattice sizes. The results are given in Ta
for L ≥ 24 with χ 2 dof = 1.12. This corresponds to a transition temperature
T 0 = 0.098328(3)J/k B .(12)
The scaling is also visualized in Fig. 6. While it is possible to consider additional terms with higher powers of 1/L 3 or exponential corrections 26 in the scaling law (10), this also leads to a higher number of free parameters and in this case does not improve the quality of the fits. We note that with periodic boundary conditions it may occur that the exponential degeneracy of ground states survives partially also at low, but finite temperatures, leading effectively to a macroscopic degeneracy of distinct ordered states separated from each other by free-energy barriers. This can be understood as a number of ordered phases q that is not constant, but grows exponentially as a function of the system size. It has recently been understood 27 that in such a case a modified scaling law
β * (L) = β 0 + c * ln q L 3 + · · ·(13)
needs to be applied, which predicts a transmuted leading system-size dependence. An advantage of our choice of screw-periodic boundary conditions is that such degeneracies are mostly lifted. In contrast to the gonihedric plaquette model studied in Ref. 27 we do not know about any rigorous calculations of this T > 0 degeneracy for the 3D compass model with periodic boundary conditions, but assuming a degeneracy ln q ∝ L 2 the displacement of β * (L) from the true transition point β 0 would be proportional to 1/L rather than to 1/L 3 . Due to very strong finite-size effects we cannot give a full discussion of the asymptotic scaling behavior with periodic boundary conditions at this point. Our (less extensive) data for this case is compatible with the modified ansatz, but does not allow to discriminate between the two options. We have also checked modified scaling relations corresponding to ln q ∝ L 2 and ln q ∝ L for the case of screw-periodic boundary conditions and have found here no compelling numerical evidence against the conventional 1/L 3 law as reported above.
L β χ max J χ max /J 2 β C max J C max /k B N β Q 2 min J Q 2,min /J 2 β D eqH J P D max /P D min β E eqH J P E max /P E
B. Interface tension
On lattices of size L 3 the suppression of the minimum between the two peaks of the probability distribution of the energy or the order parameter at a first-order phase transition is expected to grow exponentially with L 2 :
P max (L)/P min (L) ∝ e 2βσL 2 .(14)
Configurations corresponding to P min (L) are in a mixture of the ordered and the disordered phases with interfaces that contribute an excess free energy of 2σL 2 , where the freeenergy density σ is the interface tension. 26 We compute lattice size dependent estimates of the reduced interface tension σ(L) = βσ(L) from the double-peaked probability distributions P(D) at β D eqH (L) or P(E) at β E eqH (L) with the relation Table I to estimate the infinite-volume transition point β 0 by relations of the form β * (L) = β 0 + c * /L 3 . Here n is the number of included data points ranging from the smallest considered lattice size L min up to the largest L max = 48. χ 2 dof = χ 2 /(n − 2) is a measure to help with the estimation of the validity of the fit. The best fits are marked bold for each type of pseudo-transition temperature. Table I for L ≥ 16 together with the best fits from Table II, where P max (L)/P min (L) is the ratio of the estimated probabilities in the peak and in the dip as taken from Table I. In Fig. 7 the results are plotted over 1/L 2 for L ≥ 28, which excludes the irregular behavior for the small lattices. While the reduced interface tension does not yet reach its asymptotic constant value on the lattice sizes studied here,σ(L) grows with L and does not appear to vanish in the limit of large systems, which otherwise would be an argument against the first-order nature of the transition. From the available data an approximate infinite-volume value ofσ ≈ 3×10 −4 can be anticipated.
σ(L) = 1 2L 2 ln P max (L) P min (L) ,(15)L min n β χ,max 0 J χ 2 dof β C,max 0 J χ 2 dof β Q 2 ,min 0 J χ 2 dof β eqH D,0 J χ 2 dof β eqH E,
V. SUMMARY AND CONCLUSIONS
In this paper we have presented an extensive Monte Carlo investigation of the classical compass model on the simplecubic lattice. Our results show that directional ordering is present in a low-temperature phase, which is reached via a thermal first-order transition from a disordered hightemperature phase. By a detailed finite-size scaling analysis we could determine a precise estimate of the transition temperature T 0 = 0.098328(3)J/k B . This value agrees with the one mentioned in an earlier publication, 28 but the hightemperature series expansions presented in Ref. 14 could not identify this phase transition. First-order transitions are generally difficult to detect by these techniques, in particular when no low-temperature series are available.
The recently discovered (and for the gonihedric plaquette model numerically confirmed) influence of a macroscopic degeneracy of the low-temperature phases on the leading finitesize scaling behavior of first-order phase transitions 27 renews the interest in a precise characterization of the ground-state and low-temperature degeneracies of the compass model. A rigorous treatment along the lines of Refs. 29-31 for the closely related 120 • model and the gonihedric model is beyond the scope of the present paper focusing on an accurate determination of the first-order character of the phase transition, but would certainly be a worthwhile project for future studies, especially with a view on the "order-by-disorder" mechanism.
Due to the negative-sign problem, quantum Monte Carlo simulations of the 3D compass model are out of reach. However, while additional quantum fluctuations may destroy directional ordering at low temperatures in the quantum model, from Ginzburg-Landau theory one generally expects the na-ture of the phase transition to be the same in the quantum model as in the classical model. Symmetry considerations for the nematic-like type of order parameter of the t 2g compass model support the expectation of a continuous transition in 2D and a first-order transition in 3D, just as observed in the Monte Carlo simulations. Taken together, we firmly anticipate a first-order phase transition to occur also in the quantum compass model and look forward to experimental studies of directional ordering in non-low dimensional samples.
) β = 20/J: D = 1.028J FIG. 1. (Color online) Shown are two typical example spin configurations of the L = 16 system from (a) the disordered hightemperature phase and (b) the directionally ordered low-temperature phase.
online) Monte Carlo data for (a) the order parameter D, (b) its susceptibility χ and (c) the Binder parameter Q 2 . For clarity the inverse temperature range is limited to a region around the transition point and only selected lattice sizes are included in the plots. Markers with error bars are estimates from single-temperature time series. Continuous lines are from the multiple histogram analysis with faint surrounding lines indicating the 1σ-margin of statistical uncertainty.
3. (Color online) Monte Carlo data for (a) the energy per site E/N and (b) the specific heat capacity C/N. For clarity the inverse temperature range is limited to a region around the transition point and only selected lattice sizes are included in the plots. Markers with error bars are estimates from single-temperature time series. Continuous lines are from the multiple histogram analysis with faint surrounding lines indicating the 1σ-margin of statistical uncertainty.
ble II. Fits of good quality can be made based on all possible definitions with the limitation that we only have very few data points for the histogram-based temperature definitions, where the regular behavior sets in at large lattice sizes. The different estimates of the inverse transition temperature β 0 and their statistical uncertainties are in good agreement with each other. This supports the proposed first-order nature of the transition. The best result is found from the β C max (L) data, which yields β 0 = 10.1700(3)/J
FIG. 4 .
4(Color online) Histograms of the order parameter D for different lattice sizes L at the inverse temperatures β D eqH (L), where a doublepeak structure with equal peak height is obtained. (a) The two peaks hinting at phase coexistence can be made out clearly for small lattices.(b) For medium sized lattices with L < 32 the central dip shrinks with growing L. (c) For L ≥ 32 the suppression between the peaks grows with growing L. TABLE I. Lattice-size dependent inverse pseudo-transition temperatures. Listed are the inverse temperature locations β χ max (L), β C max (L) and β Q 2 min (L) of the extrema of the susceptibility, specific heat and Binder parameter together with the extreme values χ max (L), C max (L)/N and Q 2,min (L) as well as the inverse temperatures β D eqH (L) and β E eqH (L) where the histograms of the order parameter D or the energy E have two peaks of equal height together with the ratios of the estimated probabilities P max (L)/P min (L) at the highest peak and at the lowest point in the dip.
FIG. 5 .
5(Color online) Histograms of the energy per site E/N for various lattice sizes L at the inverse temperatures β E eqH (L), where a double-peak structure with equal peak height is obtained for L > 32. For L = 32 and smaller lattices no double-peak distribution can be found at any temperature. The L = 32 histogram in the plot is shown only for comparison and is taken at a temperature close to that of the online) Finite-size scaling of inverse pseudotransition temperatures from
which allow to extrapolate the infinite-volume transition point β 0 . online) Reduced interface tensionsσ(L) calculated from P(D) histograms at β D eqH (L) and from P(E) histograms at β E eqH (L) plotted over 1/L 2 for L ≥ 28.
TABLE II .
IIResults of least-squares fits of the inverse pseudo-transition temperatures β * (L) taken from
ACKNOWLEDGMENTSWe thank C. Hamer, J. Oitmaa and W. Selke for useful discussions initiating this work, as well as A. Rosch and S. Trebst for further helpful conversations. Partial support by the Deutsche Forschungsgemeinschaft (DFG) through Graduate School GSC185 "BuildMoNa" and the Deutsch-Französische Hochschule (DFH-UFA) through the binational German-French Graduate School under grant number CDFA-02-07 is gratefully acknowledged.
. * [email protected] † Janke@itp, uni-leipzig.de* [email protected] † [email protected]; http://www.physik.uni-leipzig.de/cqt. html
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| [] |
[
"One-out-of-two Quantum Oblivious Transfer based on Nonorthogonal States OPEN",
"One-out-of-two Quantum Oblivious Transfer based on Nonorthogonal States OPEN"
] | [
"Yao-Hsin Chou [email protected] \nDepartment of Computer Science and Information Engineering\nNational Chi Nan University\n54561PuliTaiwan\n",
"Guo-Jyun Zeng \nDepartment of Computer Science and Information Engineering\nNational Chi Nan University\n54561PuliTaiwan\n",
"Shu-Yu Kuo \nDepartment of Computer Science and Information Engineering\nNational Chi Nan University\n54561PuliTaiwan\n"
] | [
"Department of Computer Science and Information Engineering\nNational Chi Nan University\n54561PuliTaiwan",
"Department of Computer Science and Information Engineering\nNational Chi Nan University\n54561PuliTaiwan",
"Department of Computer Science and Information Engineering\nNational Chi Nan University\n54561PuliTaiwan"
] | [
"Scientific REPORTS |"
] | This research proposes the first one-out-of-two quantum oblivious transfer (QOT) scheme that does not have a two-level structure and is not subject to Lo's no-go theorem. Instead, the proposed scheme is a simple and efficient approach based on nonorthogonal states. The nonorthogonality causes one of a pair of messages to be unable to be measured to achieve the irreversible goal of discarding a message, resulting in a one-out-of-two selection effect. The proposed QOT protocol is therefore built directly on quantum resources rather than on a two-level structure in which two classical keys must first be created using quantum resources (all-or-nothing QOT) and then a one-out-of-two protocol is built from there. Furthermore, the proposed protocol allows Alice and Bob to test each other's loyalty by comparing measurement results. In addition, the relationship with the no-go theorem is discussed in detail; this relationship is often overlooked in other studies. A security analysis demonstrates that the proposed protocol is secure against both external and internal attacks. In addition, an efficiency analysis shows that the proposed protocol is more efficient than other, two-level-structured protocols.Oblivious transfer (OT) is an important branch of cryptography with many useful and important applications, such as secure computation, bit commitment, remote coin-flipping, and digital contract signing, for which OT protocols are the cryptographic primitives. The two most commonly used OT protocols are the all-or-nothing protocol and the one-out-of-two protocol. All-or-nothing OT was first introduced by Rabin 1 in 1981. In the all-or-nothing OT protocol, a sender Alice wants to send a secret message, m ∈ {0, 1}, to a receiver Bob who has only a 50% probability of receiving m. He will either learn the message m with 100% reliability or learn nothing about m. At the end of all-or-nothing OT, Alice remains oblivious as to whether Bob received the message m. Following the proposal of this protocol, Even et al. 2 presented one-out-of-two OT (or it can be abbreviated as 1-2 OT), in which Alice transfers two messages, m 0 and m 1 , to Bob, and he can choose only one of them and will have no idea what the other message is. When the one-out-of-two OT protocol is complete, Alice learns nothing about which message Bob selected. In 1988, Crépeau 3 presented a method for building a one-out-of-two OT protocol by using p-all-or-nothing OT, in which the receiver has a probability p of receiving the message m, called Crépeau's reduction. The receiver builds two key sets to represent his choice, key 0 and key 1 , one of which he learns with 100% certainty and the other of which he learns with 0% certainty. Based on Bob's choice j ∈ {0, 1}, he asks Alice to encrypt her messages m 0 and m 1 using key j and key j , where = ⇒ j key 0 0 or = ⇒ j key 1 1 . Then, Bob can receive m j under this two-level-structured method.Classical OT protocols are almost all based on the RSA cryptosystem 4 . However, Shor showed that a quantum algorithm 5 can be used to break the RSA cryptosystem in polynomial time, which means that such protocols may be unsafe against quantum algorithms. In 1984, Bennett and Brassard proposed the first quantum key distribution protocol 6 , called BB84, thereby initiating the study of quantum cryptography. Researchers later showed that BB84 is unconditionally secure 7-10 both in theory and in implementation by achieving a one-time pad. The security of quantum cryptography is based on physical laws, unlike that of classical cryptography, which is based on mathematical complexity. This physical basis allows quantum cryptography to easily achieve many goals that were difficult or unthinkable in the past, including unconditional security.Since the proposal of BB84 6 , researchers have been designing quantum oblivious transfer (QOT) protocols using quantum properties. Crépeau and Kilian 11 proposed the first all-or-nothing QOT scheme in 1988, and Bennett et al.12proposed the first one-out-of-two QOT scheme protected by a quantum error-correcting code in 1992. In 1994, Crépeau 13 presented a one-out-of-two QOT scheme based on quantum bit commitment (QBC), which guarantees security under the assumption that Bob cannot delay the quantum measurement. In 1995, Yao 14 further proved that this protocol is secure against coherent measurement if QBC is secure. However, in 1997, Lo 15 doubted that all one-sided two-party computations (in which two parties must input i and j to calculate a function f(i, j) but only one of the two parties is allowed to learn the result) may be insecure, including one-out-of-two QOT (the function f in one-out-of-two QOT is a selector). This was called Lo's no-go theorem, and because of the computational equivalent 3,12 to two OTs, this theorem has caused extreme difficulties in the development of QOT research.Recent studies have, however, proposed various methods of avoiding Lo's no-go theorem. In 2002, Shimizu and Imoto 16 presented an interesting communication method analogous to one-out-of-two QOT with a 50% probability of completing the communication. They 17 then improved the security of their protocol against entangled pair attacks in 2003. Moreover, in 2006, He and Wang 18 proposed a secure all-or-nothing QOT scheme using four entangled states, which, as a result, was no longer subject to Lo's no-go theorem 15 . Consequently, He claimed that Lo's no-go theorem 15 did not truly cover all QOT conditions. Thereafter, He 19 demonstrated that a one-out-of-two QOT scheme built on all-or-nothing QOT protocol using Crépeau's reduction 3 also is not subject to Lo's no-go theorem 15 . The key is that the receiver inputs his choice before the sender inputs her messages m 0 and m 1 , causing the functions f of the one-out-of-two protocol and Lo's no-go theorem 15 to be different.Following He's proof 19 , researchers have been designing new one-out-of-two QOT schemes 19 . In 2007, Wei Yang et al. 20 presented a one-out-of-two QOT scheme using tripartite entangled states based on He's proof19and also showed that this scheme is not covered by the cheating strategy of Lo's no-go theorem 15 . Li Yang 21 presented an all-or-nothing QOT scheme using nonorthogonal states, similar to B92 22 , and used it as a basis for constructing a one-out-of-two QOT scheme in 2013. Subsequently, Yu-Guang Yang and his research team, as part of a research effort that began in 2014, have proposed several QOT protocols. They have been testing various schemes for building one-out-of-two QOT protocols using He's proof 19 . In 2014, they 23 proposed all-or-nothing and one-out-of-two QOT protocols based on an untrusted third party. In 2015, they 24 developed an all-or-nothing QOT protocol by analyzing the probability of the qubit state distribution, which led them to propose a method of testing the loyalty of the sender and then to build a one-out-of-two QOT protocol on this basis. They 25 also designed a one-out-of-two QOT scheme with a two-level structure using BB84 6 and reduced it to B92 22 for an all-or-nothing QOT scheme. In addition, they 26 attempted to use Bell states to achieve the same effect as B92 22 for one-out-of-two QOT. Furthermore, in 2017, they 27 proposed a method of using any two nonorthogonal states by cooperatively measuring the qubit sequence and then built a one-out-of-n QOT scheme using this method.However, these protocols 21,23-27 all have two-level structures, in which two classical keys are created using an all-or-nothing QOT protocol and then a one-out-of-two QOT protocol is built on top. The two-level structure is clearly inefficient, because many quantum resources are consumed for all-or-nothing QOT instead of being used to transfer the message. In addition, this structure reduces the elasticity and diversity of protocol design because such designs can only follow He's proof 19 with minor revisions to the details of the all-or-nothing QOT scheme. In our opinion, He's proof 19 not only revealed a different function f, which is not subject to Lo's no-go theorem 15 , based on a two-level structure but also provided a new approach in the sense that if any protocol can achieve the same effect as that of f in He's proof 19 , then it is also covered by He's proof 19 . In this work, the first one-out-of-two QOT protocol is proposed that is directly based only on the properties of quantum resources, namely, nonorthogonal states, rather than a two-level structure, while also being covered by He's proof 19 . The key to our protocol is that Bob's choice is made before Alice inputs her messages m 0 and m 1 . The property of nonorthogonality ensures that one of the two messages cannot be measured and thus maintains obliviousness, thereby achieving the same effect as that of f in He's proof 19 . Therefore, our protocol is not only secure (and not subject to Lo's no-go theorem) but can achieve greater efficiency than protocols 21,23-27 that are based on a two-level structure.ResultsThis section consists of six subsections, including the preliminaries, the basic idea of our protocol, the proposed protocol itself, its relationship with Lo's no-go theorem 15 and He's proof 19 , and its security and efficiency analyses. The preliminaries introduce the properties of quantum machines and define some notation. Then, the basic idea of the proposed protocol is introduced before the details of the protocol itself, which are described in the subsequent section. Moreover, the relationship among Lo's no-go theorem 15 , He's proof 19 and the proposed protocol is discussed in the subsection titled "Resisting Lo's cheating strategy 15 ". Finally, security and efficiency analyses are presented in the last two subsections.Preliminaries. This subsection introduces the basic definitions of concepts relevant to quantum machines, such as quantum bits, superposition, entanglement, gates, and operations, as well as some properties of quantum machines. | 10.1038/s41598-018-32838-9 | null | 53,105,427 | 1707.06062 | 84a0da1eab1c4272ff0bf0c53da4890762af9da9 |
One-out-of-two Quantum Oblivious Transfer based on Nonorthogonal States OPEN
2018
Yao-Hsin Chou [email protected]
Department of Computer Science and Information Engineering
National Chi Nan University
54561PuliTaiwan
Guo-Jyun Zeng
Department of Computer Science and Information Engineering
National Chi Nan University
54561PuliTaiwan
Shu-Yu Kuo
Department of Computer Science and Information Engineering
National Chi Nan University
54561PuliTaiwan
One-out-of-two Quantum Oblivious Transfer based on Nonorthogonal States OPEN
Scientific REPORTS |
815927201810.1038/s41598-018-32838-9Received: 10 October 2017 Accepted: 14 August 20181 www.nature.com/scientificreports Correspondence and requests for materials should be addressed to Y.-H.C. ( Corrected: Author Correction
This research proposes the first one-out-of-two quantum oblivious transfer (QOT) scheme that does not have a two-level structure and is not subject to Lo's no-go theorem. Instead, the proposed scheme is a simple and efficient approach based on nonorthogonal states. The nonorthogonality causes one of a pair of messages to be unable to be measured to achieve the irreversible goal of discarding a message, resulting in a one-out-of-two selection effect. The proposed QOT protocol is therefore built directly on quantum resources rather than on a two-level structure in which two classical keys must first be created using quantum resources (all-or-nothing QOT) and then a one-out-of-two protocol is built from there. Furthermore, the proposed protocol allows Alice and Bob to test each other's loyalty by comparing measurement results. In addition, the relationship with the no-go theorem is discussed in detail; this relationship is often overlooked in other studies. A security analysis demonstrates that the proposed protocol is secure against both external and internal attacks. In addition, an efficiency analysis shows that the proposed protocol is more efficient than other, two-level-structured protocols.Oblivious transfer (OT) is an important branch of cryptography with many useful and important applications, such as secure computation, bit commitment, remote coin-flipping, and digital contract signing, for which OT protocols are the cryptographic primitives. The two most commonly used OT protocols are the all-or-nothing protocol and the one-out-of-two protocol. All-or-nothing OT was first introduced by Rabin 1 in 1981. In the all-or-nothing OT protocol, a sender Alice wants to send a secret message, m ∈ {0, 1}, to a receiver Bob who has only a 50% probability of receiving m. He will either learn the message m with 100% reliability or learn nothing about m. At the end of all-or-nothing OT, Alice remains oblivious as to whether Bob received the message m. Following the proposal of this protocol, Even et al. 2 presented one-out-of-two OT (or it can be abbreviated as 1-2 OT), in which Alice transfers two messages, m 0 and m 1 , to Bob, and he can choose only one of them and will have no idea what the other message is. When the one-out-of-two OT protocol is complete, Alice learns nothing about which message Bob selected. In 1988, Crépeau 3 presented a method for building a one-out-of-two OT protocol by using p-all-or-nothing OT, in which the receiver has a probability p of receiving the message m, called Crépeau's reduction. The receiver builds two key sets to represent his choice, key 0 and key 1 , one of which he learns with 100% certainty and the other of which he learns with 0% certainty. Based on Bob's choice j ∈ {0, 1}, he asks Alice to encrypt her messages m 0 and m 1 using key j and key j , where = ⇒ j key 0 0 or = ⇒ j key 1 1 . Then, Bob can receive m j under this two-level-structured method.Classical OT protocols are almost all based on the RSA cryptosystem 4 . However, Shor showed that a quantum algorithm 5 can be used to break the RSA cryptosystem in polynomial time, which means that such protocols may be unsafe against quantum algorithms. In 1984, Bennett and Brassard proposed the first quantum key distribution protocol 6 , called BB84, thereby initiating the study of quantum cryptography. Researchers later showed that BB84 is unconditionally secure 7-10 both in theory and in implementation by achieving a one-time pad. The security of quantum cryptography is based on physical laws, unlike that of classical cryptography, which is based on mathematical complexity. This physical basis allows quantum cryptography to easily achieve many goals that were difficult or unthinkable in the past, including unconditional security.Since the proposal of BB84 6 , researchers have been designing quantum oblivious transfer (QOT) protocols using quantum properties. Crépeau and Kilian 11 proposed the first all-or-nothing QOT scheme in 1988, and Bennett et al.12proposed the first one-out-of-two QOT scheme protected by a quantum error-correcting code in 1992. In 1994, Crépeau 13 presented a one-out-of-two QOT scheme based on quantum bit commitment (QBC), which guarantees security under the assumption that Bob cannot delay the quantum measurement. In 1995, Yao 14 further proved that this protocol is secure against coherent measurement if QBC is secure. However, in 1997, Lo 15 doubted that all one-sided two-party computations (in which two parties must input i and j to calculate a function f(i, j) but only one of the two parties is allowed to learn the result) may be insecure, including one-out-of-two QOT (the function f in one-out-of-two QOT is a selector). This was called Lo's no-go theorem, and because of the computational equivalent 3,12 to two OTs, this theorem has caused extreme difficulties in the development of QOT research.Recent studies have, however, proposed various methods of avoiding Lo's no-go theorem. In 2002, Shimizu and Imoto 16 presented an interesting communication method analogous to one-out-of-two QOT with a 50% probability of completing the communication. They 17 then improved the security of their protocol against entangled pair attacks in 2003. Moreover, in 2006, He and Wang 18 proposed a secure all-or-nothing QOT scheme using four entangled states, which, as a result, was no longer subject to Lo's no-go theorem 15 . Consequently, He claimed that Lo's no-go theorem 15 did not truly cover all QOT conditions. Thereafter, He 19 demonstrated that a one-out-of-two QOT scheme built on all-or-nothing QOT protocol using Crépeau's reduction 3 also is not subject to Lo's no-go theorem 15 . The key is that the receiver inputs his choice before the sender inputs her messages m 0 and m 1 , causing the functions f of the one-out-of-two protocol and Lo's no-go theorem 15 to be different.Following He's proof 19 , researchers have been designing new one-out-of-two QOT schemes 19 . In 2007, Wei Yang et al. 20 presented a one-out-of-two QOT scheme using tripartite entangled states based on He's proof19and also showed that this scheme is not covered by the cheating strategy of Lo's no-go theorem 15 . Li Yang 21 presented an all-or-nothing QOT scheme using nonorthogonal states, similar to B92 22 , and used it as a basis for constructing a one-out-of-two QOT scheme in 2013. Subsequently, Yu-Guang Yang and his research team, as part of a research effort that began in 2014, have proposed several QOT protocols. They have been testing various schemes for building one-out-of-two QOT protocols using He's proof 19 . In 2014, they 23 proposed all-or-nothing and one-out-of-two QOT protocols based on an untrusted third party. In 2015, they 24 developed an all-or-nothing QOT protocol by analyzing the probability of the qubit state distribution, which led them to propose a method of testing the loyalty of the sender and then to build a one-out-of-two QOT protocol on this basis. They 25 also designed a one-out-of-two QOT scheme with a two-level structure using BB84 6 and reduced it to B92 22 for an all-or-nothing QOT scheme. In addition, they 26 attempted to use Bell states to achieve the same effect as B92 22 for one-out-of-two QOT. Furthermore, in 2017, they 27 proposed a method of using any two nonorthogonal states by cooperatively measuring the qubit sequence and then built a one-out-of-n QOT scheme using this method.However, these protocols 21,23-27 all have two-level structures, in which two classical keys are created using an all-or-nothing QOT protocol and then a one-out-of-two QOT protocol is built on top. The two-level structure is clearly inefficient, because many quantum resources are consumed for all-or-nothing QOT instead of being used to transfer the message. In addition, this structure reduces the elasticity and diversity of protocol design because such designs can only follow He's proof 19 with minor revisions to the details of the all-or-nothing QOT scheme. In our opinion, He's proof 19 not only revealed a different function f, which is not subject to Lo's no-go theorem 15 , based on a two-level structure but also provided a new approach in the sense that if any protocol can achieve the same effect as that of f in He's proof 19 , then it is also covered by He's proof 19 . In this work, the first one-out-of-two QOT protocol is proposed that is directly based only on the properties of quantum resources, namely, nonorthogonal states, rather than a two-level structure, while also being covered by He's proof 19 . The key to our protocol is that Bob's choice is made before Alice inputs her messages m 0 and m 1 . The property of nonorthogonality ensures that one of the two messages cannot be measured and thus maintains obliviousness, thereby achieving the same effect as that of f in He's proof 19 . Therefore, our protocol is not only secure (and not subject to Lo's no-go theorem) but can achieve greater efficiency than protocols 21,23-27 that are based on a two-level structure.ResultsThis section consists of six subsections, including the preliminaries, the basic idea of our protocol, the proposed protocol itself, its relationship with Lo's no-go theorem 15 and He's proof 19 , and its security and efficiency analyses. The preliminaries introduce the properties of quantum machines and define some notation. Then, the basic idea of the proposed protocol is introduced before the details of the protocol itself, which are described in the subsequent section. Moreover, the relationship among Lo's no-go theorem 15 , He's proof 19 and the proposed protocol is discussed in the subsection titled "Resisting Lo's cheating strategy 15 ". Finally, security and efficiency analyses are presented in the last two subsections.Preliminaries. This subsection introduces the basic definitions of concepts relevant to quantum machines, such as quantum bits, superposition, entanglement, gates, and operations, as well as some properties of quantum machines.
Quantum bit. The classical information carrier is called a "bit". The quantum information carrier is called a "quantum bit", or a "qubit". A qubit collapses to certain states of a basis when it is measured. Two bases are commonly used: the Z-basis and the X-basis. The Z-basis is defined as . A basis is also an orthonormal set.
Superposition. Superposition refers to the phenomenon that a qubit can simultaneously exist in both the |0〉 and |1〉 states; i.e., φ α β = + 0 1, meaning that |φ〉 will collapse to |0〉 and |1〉 with probabilities of α 2 and β 2 , respectively. In addition, the state |−〉 is also considered to be a superposition in the Z-basis. It has a probability of = of collapsing to |1〉.
Entanglement. Another important property, entanglement is the phenomenon that qubits cannot exist singly.
There are four common entangled states, called Bell states, as shown in Eq. 1. For example, when a state |Φ + 〉 is measured, as in Eq. 1, the result may be either |00〉 12 or |11〉 12 , where the subscript indicates the qubit order. As a result, in this case, it is possible to immediately learn the states of two qubits when only one is measured. Einstein referred to this as "spooky action at a distance". Operations I and Z cannot be distinguished in the Z-basis, and Y and Z cannot be distinguished in the X-basis. A single qubit cannot be observed using all four operations, which means that some information is ignored; this is a key element of the proposed protocol. For example, after a Y gate, the state |0〉 becomes −|1〉; i.e., a result of |1〉 will be obtained when the qubit is measured. This negative amplitude is called a global phase and cannot be measured. Another important gate is the Hadamard gate, as described in Eq. 2, also called the H gate. The H gate can be used to convert between two different bases (the Z-basis and the X-basis). For example, after an H gate, the state |0〉 (|+〉) H becomes |+〉 (|0〉). Table 1 The basic idea. This subsection introduces the basic idea underlying encoding and decoding in the proposed protocol. In this study, the four operations "I", "X", "Y" and "Z" represent four messages "00", "10", "11" and "01", respectively, for encoding. Each message can be mapped to m 0 and m 1 , which represent Alice's two messages. Because the properties of the two different bases (shown in Table 2) cause a negative amplitude to be unable to be measured (see the last final state in the left-hand part of Table 2), one of the two messages cannot be measured, and which one depends on the basis in which they are prepared (the receiver's choice). For example, suppose that Bob prepares the state |0〉 and performs either I or H in accordance with his choice, j 0 or j 1 , in order to learn the content of either m 0 or m 1 , respectively. In this way, Bob inputs his choice first, and the initial state |0〉 will be either |0〉 or |+〉, depending on his choice. The results are shown in Table 2; after Alice's operation, if Bob's choice is j 0 (his initial state is |0〉), he learns m 0 unambiguously (the bold text in the left-hand part of Table 2); otherwise, he learns m 1 (the bold text in the right-hand part of Table 2) unambiguously. As a result, one of the two messages is automatically discarded, thereby achieving the requirements of one-out-of-two QOT.
The proposed protocol. As seen from the basic idea presented above, the operations I and H can be regarded as representing Bob's intentions regarding his choice; this makes the proposed protocol similar to B92 22 , which has been proven unconditionally secure both in theory and in implementation 28,29 , meaning that no one can perfectly identify all states of the qubits without any information from their creator. Another key property is that some operations cannot be distinguished in some bases, which means that it is not possible to identify all operations from a single qubit. The proposed protocol allows Alice and Bob to test each other's loyalty, because they can check whether the initial and final states are correct. In other words, if they want to lie to each other, it will create errors, which can be discovered when they test each other. Let us give a simple example at the end of every step of the protocol without channel checking. The proposed protocol consists of 7 steps as follows:
States |0〉 |1〉 |+〉 |−〉 Operations I |0〉 |1〉 |+〉 |−〉 X |1〉 |0〉 |+〉 −|−〉 Y −|1〉 |0〉 |−〉 −|+〉 Z |0〉 −|1〉 |−〉 |+〉(0, 0) I |0〉 |0〉 (0, 0) I |+〉 |+〉 (0, 1) Z |0〉 (1, 0) X |+〉 (1, 0) X |1〉 (1, 1) Y |−〉 (1, 1) Y −|1〉 (0, 1) Z |−〉
Step 1. Bob creates a qubit sequence in accordance with his choice intentions j 0 and j 1 , which correspond to the states |0〉 and |+〉, respectively. The necessary I and H gates can be considered as equivalent to his choice intentions in this stage. This sequence must be longer than the OT sequence, which contains all received message qubits as well as qubits for channel checking and for testing Bob's loyalty. In addition, the channel checking and loyalty testing states are different; the former, also called decoy qubits, belong to {|0〉, |1〉, |+〉, |−〉}, and the latter belong to {|0〉, |+〉}. If N denotes the minimum length (at which Bob will receive N messages), M is the number of channel checking qubits, and K is the number of loyalty testing qubits, then the total length of the QOT sequence is N + M + 2K. Bob randomly prepares M qubits from {|0〉, |1〉, |+〉, |−〉} (each qubit is independent) and inserts them into his sequence. Subsequently, he also inserts his N and 2K candidate choice intentions ({|0〉, |+〉}) into his sequence and then sends the sequence to Alice. Let us give a simple example to describe the proposed one-out-of-two QOT protocol without channel checking, suppose that Bob prepares two qubits in |0+〉 12 (N) to represent his choices and an additional two qubits in |0+〉 34 (2K) for loyalty testing. He then sends these four qubits to Alice.
Step 2. Once she receives the sequence from Bob, Alice first checks the channel for an eavesdropper (Eve) and then tests Bob's loyalty. First, she asks Bob to publish the bases and states that he has created. If the error rate is higher than a given threshold, then an Eve is present on the channel, and Alice and Bob abort their communication; otherwise, Alice goes on to test Bob's loyalty. She discards the qubits for channel checking and then randomly selects several positions and requests that Bob publish his bases. If different results, i.e., ∉ {|0〉, |+〉}, are measured and the error rate is higher than the given error rate, then Bob is considered dishonest, and she aborts this communication; otherwise, she proceeds to the next step. Following the above example, once Alice receives the ordered sequence |0+0+〉 1234 , she randomly chooses a qubit for loyalty testing. Suppose that Alice's random choice is qubit 4; then, she asks Bob to publish the basis of qubit 4, measures it, and compares the published and measured results. If the error rate is higher than the threshold, then Bob is considered as dishonest; after that, qubit 4 is discarded.
Step 3. Since the loyalty test may disturb the order of Bob's choice intentions, Bob must ask Alice to reorder the qubits. It is for this purpose that 2K additional qubits are initially provided to prevent vacancies in the list of choice intentions. In this step, the sequence after reordering represents Bob's real choices. Following the above example, after the loyalty test, Bob asks Alice to reorder the remaining qubits in the order 21, and the states become |+0〉 21 , with qubit 3 discarded. The resulting state |+0〉 21 represents Bob's choices, j 1 and j 0 , respectively.
Step 4. Alice now inputs her secret messages m 0 and m 1 through the I, X, Y and Z operations, corresponding to the combinations "00", "10", "11" and "01", respectively. Following the above example, Alice performs Z and X in accordance with her messages "01 12 " and "10 34 ", where the subscripts indicate Alice's classical bit order, on qubits 2 and 1, respectively. This converts the state |+0〉 21 into |−1〉 21 .
Step 5. Alice then randomly inserts decoy qubits from {|0〉, |1〉, |+〉, |−〉} into the sequence for channel checking and sends the sequence to Bob.
Step 6. When Bob receives the sequence from Alice, he asks Alice to publish the positions and states of the decoy qubits. If the error rate is higher than the channel error rate, they abort this communication and return to step 1. Otherwise, Bob learns the contents of the classical messages by measuring the qubits with the bases he prepared. Following the above example, Bob performs X-and Z-basis measurements to learn the second and first classical messages, "1" and "1" (01 12 and 10 34 , where the subscripts indicate Alice's classical bit order), respectively.
Step 7. At the end of this protocol, where Bob has to test Alice's loyalty to prevent Alice from cheating, she can learn Bob's choice with the probability 25%, or 29.3% by POVM. Bob chooses some random positions and asks Alice to publish her operations. Bob performs the operations according to Alice's announcement, in order to recover those qubits into {|0〉, |+〉}. If the error rate of loyalty testing is higher than the threshold, then Alice is considered as dishonest. Following the above example, Bob asks Alice to publish operation (X), which performs on qubit 1; then, Bob performs X on qubit 1 to recover the state into |0〉.
Resisting Lo's cheating strategy.
Lo's no-go theorem 15 provides a cheating strategy for learning all messages in a one-sided two-party secure computation, which doubts that one-out-of-two QOT is insecure. The key to Lo's cheating strategy 15 is requirement (A-i), namely, that "Bob learns f(i(m 0 , m 1 ), j) unambiguously" (the i(m 0 , m 1 ) represents a pair of messages in a one-out-of-two QOT scenario), which leads to a probability of 100% that the selected state will collapse. In addition, the result is obtained after reversible operations. As a result, choices can be made repeatedly to learn all messages, as shown in Eq. 3, where U j 0 , U j 1 , G and |φ〉 represent two different selected operations, any unitary operation for inputting two messages and any quantum state, respectively. Therefore, once Bob learns the content of a message, he can recover the state G × |φ〉, which is the state after Alice's input, by applying the selected operation U j k and its inverse operation ⁎ U j k , where k ∈ {|0〉, |1〉}. In this way, Bob can change his choice and learn the contents of all messages by repeating the above process.
φ φ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⁎ ⁎ U U G U U G (3) j j j j 0 0 1 1 Scientific REPORTS | (2018) 8:15927 | DOI:10.1038/s41598-018-32838-9
Definition B corresponds to the one-out-of-two QOT scenario covered by Lo's proof 15 . Obviously, definition B is a special case of definition A. Definition B describes the case in which Alice inputs her messages first and then Bob inputs his choices. The important point here is that if Bob inputs his choices first and Alice subsequently input her messages, as in definition C (the proposed protocol), this scenario is not equivalent to the function considered in Lo's proof 15 . This is because the function becomes f(i(m 0 , m 1 , j), j) when Bob inputs his choices first, and f(i(m 0 , m 1 , j 0 ), j 1 ) is meaningless with respect to f(i(m 0 , m 1 , j), j). Therefore, Bob cannot change i from i(m 0 , m 1 , j 0 ) to i(m 0 , m 1 , j 1 ) without Alice's help. Following from the above relation, in the proposed protocol (definition C), the result after Bob's and Alice's actions can be expressed as
⋅ ⋅ ⋅ | ≠ ⋅ ⋅ ⋅ | * * ⟩ ⟩ U G U U G U 0 0 (4) j j j j 0 0 1 1 Here, = U I j 0 , = U H j 1
, and G ∈ {I, X, Y, Z}. Eq. 4 shows that Bob cannot invert the qubit state without possessing information about G. Therefore, Bob cannot perform Lo's cheating strategy 15 . As a result, Bob cannot reverse the effects of his inputs without Alice's help. The condition of Equation 4 shows that the proposed function f(i(m 0 , m 1 , j), j) is similar to that of He's proof 19,30 . This proof shows that the order of input of the choices and messages may change the function f, which means that this protocol is not subject to Lo's no-go theorem. In addition, He 30 has extended the concept of his proof 19 to the general case; if Alice and Bob interact with each other and Bob cannot eliminate the effects of his operations independently, then the interaction is covered by He's proof 19 and resists Lo's cheating strategy 15 .
Security Analysis
Two security conditions are considered in this study: security against external and internal attacks. External attacks involve an eavesdropper, Eve, attempting to steal messages without being detected. Internal attacks involve either Alice or Bob attempting to steal the other's secret information; i.e., Alice wants to learn Bob's choices, or Bob wants to learn the contents of both of Alice's messages.
External Attack. Alice and Bob must ensure that the communication channel between them is secure, because without channel checking or reduced frequency 31 , Eve will be able to illicitly eavesdrop on their messages. In the proposed protocol, several single qubits ∈ {|0〉, |1〉, |+〉, |−〉} are randomly inserted into the transmitted sequence as decoy qubits for channel checking, as described in steps 1 and 5 of the protocol. The positions and states of these qubits are then published and measured to check whether an Eve is present. If the measured results obtained with the same bases are different and the error rate is higher than the channel error rate, then an Eve is present. Two common external attack strategies are the intercept-and-resend attack and the entangling attack. They are discussed below.
Intercept-and-resend attack. Eve intercepts all qubits during transmission when the sender sends the qubit sequence to the receiver, measures them to obtain the message contents, and then resends those qubits to the receiver. This action should disturb the states of the qubits, including the decoy qubits, because Eve does not know which bases have been prepared by Alice and Bob. According to the detection rate of BB84 6 , each qubit has a probability of 1 4 of detecting Eve's presence, and the detection rate increases with an increasing number of decoy qubits M. As a result, the security level can be assessed based on the detection rate by legal agents, ξ 1 , as expressed in Eq. 5. Entangling attack. Eve may instead use a different method that does not disturb the qubit states, namely, the entangling attack. In this attack, she intercepts the transmitted sequence, prepares an ancillary qubit |E〉, and performs a unitary operation U e on the intercepted qubit to entangle it with her qubit |E〉 during transmission. The unitary operation U e is defined as shown in Eq. 6, where |e 00 〉, |e 01 〉, |e 10 〉, and |e 11 〉 are four states determined by the unitary operation U e , + = a b 1 2 2 , and + = c d 1 2 2 . If Eve wishes to avoid detection, the operation U e must satisfy a = d = 1, b = c = 0, and |e 00 〉 = |e 11 〉, and as a result, the proposed protocol ensures that no information can be obtained in this way. Internal Attack. Internal attacks involve the legal agents Alice and Bob attempting to steal each other's secret information; i.e., Alice wants to learn Bob's choices, or Bob wants to learn the contents of all messages sent by Alice. Therefore, two conditions must be discussed, namely, Alice's and Bob's cheating strategies.
ξ = − 1 3 4(5)= + = + + = + + + = + + + + + − − + − − = + − − = + + − − + − − − + U E a e b1 2 ( ) 1 2 ( ) ( ) 1 2 ( 0 1 0 1 ) 1 2 ( ) 1 2 ( )(6)
Alice's cheating strategy. There are two conditions to be discussed. The first condition is that Alice has no ability of entanglement. In this condition, Alice only has the ability to perform a single qubit gate such as {I, X, Y, Z, H} etc., and she has 25% or 29.3% chance to learn Bob's choices; however, this kind of attack can be always detected in our protocol. The second condition is that Alice has the ability of entanglement. In this condition, Alice has the ability to perform two or more qubit gates, which leads to diverse attacks. However, Bob is also required to have the ability of entanglement to resist attacks from Alice, and a dishonest Alice will be detected by the discussion below.
Alice has no ability of entanglement. A dishonest Alice can learn 25% of Bob's choices, as in B92 22 , because a measurement in the incorrect basis can yield incorrect measurement results that nevertheless help Alice to determine Bob's initial state. For example, if Bob sends the state 0 to Alice, she has a probability of 1 2 of using the incorrect basis (X-basis), and when she does so, the incorrect state (|−〉) will be obtained with a probability of 1 2 , resulting in a total probability of = . 1 2 In other word, the remaining 70.7% of Bob's choices will be unknown, which means that Alice should randomly create several state ∈ {|0〉, |1〉, |+〉, |−〉} to send to Bob. However, she cannot know Bob's final measurement results, as he does not publish any information about the bases. In other words, for each bit, he will be unable to correctly decrypt with a probability of − . × × = . at the Alice's loyalty testing stage, which will make him aware of Alice's dishonesty with where ξ 2 can be decided by the user through the number of qubit D for loyalty testing. Therefore, Bob can detect that Alice is cheating. If Alice does not use POVM, the total detection rate is − . × × = . with a single qubit. Indeed, while the detection rate dropped by 18.75% − 17.675% = 1.075% with POVM, it does not change the number of particles too much.
ξ = − − . 1 (1 0 17675) ,(9)
For a simple example regarding the detection rate with a remaining qubit, which Alice randomly prepared, Alice prepares a qubit in state |0〉 and guesses Bob's choice. In this case, she can only publish operation {I, Z} to escape this testing, and there are two branches: 1. Bob uses Z-basis as his choice; in this case, Alice can always escape the testing; 2. Bob uses X-basis as his choice; in this case, Bob has a 50% to get |+〉 or |−〉. When he gets |+〉, the operation Z cannot restore the state |+〉 to |+〉. Otherwise, when he gets |−〉, operation I cannot restore the state |−〉 to |+〉. Therefore, Bob always has a probability to detect Alice's dishonesty. Alice has the ability of entanglement. A dishonest Alice can prepare Bell states in |Φ + 〉 AB to perform a teleportation attack. In this way, she can pass Alice's loyalty testing, and then, learn Bob's choices with 25% or 29.3% chance without being detected. For a simple example to explain the teleportation attack, in step 3, the qubit from Bob after Bob's loyalty testing is called |ϕ〉 C . In step 4, instead of inputting her secret message into qubit C, Alice creates a Bell state in Φ + AB , distributes qubit B to Bob to replace qubit C, and holds qubit A. After that, Alice performs a Bell measurement (a controlled-not gate and a Hadamard gate, which can transfer four Bell states |Φ + 〉, |Φ − 〉, |Ψ + 〉 and |Ψ − 〉 into |00〉, |10〉, |01〉 and |11〉, respectively), BM for short, on qubit C and A, and publishes one of four operations I, X, Y or Z as her secret message according to BM results |00〉 CA , |01〉 CA , |11〉 CA and |10〉 CA , respectively. Bob can then perform one of four operations I, Z, X or Y to recover state |0〉 B or |+〉 B according to a result that Alice published. As shown in Eqs 10 and 11, Bob can always recover the qubit state |0〉 B or |+〉 B , because the BM results |00〉 CA , |10〉 CA , |01〉 CA and |11〉 CA can always match operations I, Z, X and Y, respectively. and sends qubit A 1 to Alice. Under normal conditions, we can know that Alice will do honest behavior. After Alice performs an operation in {I, X, Y, Z} on qubit A 1 , the entangled state will be Eq. 12. We can divide those states into two bases (Eqs 13 and 14), called IY basis and XZ basis, respectively, of which the IY/XZ basis can perfectly distinguish the states after I A 1 /X A 1 and Y A 1 /Z A 1 . Obviously, these two bases are not orthogonal. That is to say, Bob will measure qubits A 1 and B 1 with one of two bases {IY, XZ} according to Alice's operations to check Alice's loyalty.
⊗ + =⇒ = ⊗ + ⊗ + ⊗ + ⊗ 0 1 2 (00 11+ + ⇒ → + + = + + → − − = − + − → + + = + + → + − = + − I Y X Z 1 2 (00 1= − − − − IY basis 1/ 2 1/2 0 1/2 , 0 1/2 1/ 2 1/2 , 1/2 0 1/2 1/ 2 , 1/2 1/ 2 1/2 0 (13) = − − − XZ basis 0 1/2 1/ 2 1/2 , 1/ 2 1/2 0 1/2 , 1/2 1/ 2 1/2 0 , 1/2 0 1/2 1/ 2(14)
However, Alice may be dishonest and perform a teleportation attack. Alice creates an entangled state in Φ + AB and sends qubit B to Bob for the teleportation attack. Alice will perform the BM on qubit A 1 and A, which leads the entire system to be Eq. 15. As a result, we can determine the probabilities of the four qubit states in Eq. in the XZ basis, where the bold numbers are the probabilities of dishonest Alice evading detection. After the above discussion of the defense strategy, we can determine the probability of Alice's average escape detection of each entangled qubits pair is Eq. 20, and security level ξ 3 is as given in Eq. 21, where F is the number of detected entangled qubit pairs. Therefore, Alice and Bob can decide security level ξ 3 , and whether they will continue the protocol according to the detection result. In summary, if the dishonest Alice only has the ability to perform single-qubit operations, then follow this protocol Bob can always have a probability to detect dishonesty one. Moreover, if the dishonest Alice can prepare Bell states or perform teleportation attacks, Alice's cheating becomes more and more difficult because she has to have the technology to store the qubits received from Bob. However, such a long-term quantum storage technology is still a technical challenge and an open issue today. Even though when the long-term quantum storage technology can be built, the protocol still can intentionally delay the operation time between step 4 and step 7 to prevent these attacks. . . . Bob's cheating strategy. A dishonest Bob can prepare entangled qubits of the form |Φ + 〉, as given in Eq. 1. In this way, he can learn the contents of all messages from Alice, with the results shown in Eq. 22, where the subscripts represent the qubit order; i.e., he can perfectly identify which operation Alice performed on qubit 1. However, only two states, |0〉 and |+〉, can be measured in the proposed protocol. Alice randomly selects K positions and asks Bob to publish the bases he prepared in step 2. If different measurement results are given, i.e., ∉ {|0〉, |+〉}, Bob is dishonest. The detection rate by legal agents, or the security level ξ 4 , is as given in Eq. 23.
+ + ⊗ + = ⇒ = + + + − + + + + + + − 1 2 (00 1. − − − − = . − . . − . = ⇒ === . . . . − . . − . − − − − − = . . − . . = ⇒ === . . . . − − − = . . − . . = ⇒ === . . . . − − − − = . . . − . = ⇒ === . . . .ξ = − 1 1 2 (23) K 4
Efficiency Analysis. This section presents a performance comparison of the proposed protocol with three modern two-level-structured one-out-of-two QOT protocols 21,24,25 based on Crépeau's reduction 3 . The protocols of Wei Yang et al. 20 and Yu-Guang Yang et al. 23,27 are not considered because that of Wei Yang et al. may not work, whereas the first protocol of Yu-Guang Yang et al. 23 involves an untrusted third party, and the second 27 is a one-out-of-n QOT protocol for which the resource consumption for one-out-of-two QOT is similar to that of Li Yang's protocol 21 . The three protocols considered for comparison 21,24,25 have two-level structures in which one-out-of-two OT is built on all-or-nothing QOT. However, the probability p of all-or-nothing QOT (where p is the probability of the unambiguous key) is not always 50%. Significant quantum resources are required to build two classical keys (one unambiguous and the other unknown) using all-or-nothing QOT for one-out-of-two OT.
In addition, every transmission should include decoy qubits for channel checking. Some protocols may need many transmissions, and many decoy qubits, to complete all-or-nothing QOT. For fairness, the security level ξ 1 is ensured to be at least 99.9999% by using 50 decoy qubits for each transmission. Then, the most important indicators are the conversion efficiency between two OT protocols and the number of transmissions (which indirectly affects the number of decoy qubits). The total cost of each protocol, as calculated under the requirement that R message bits are received, is given in Table 3. Here, only the quantum cost without loyalty testing is considered; for fairness, we do not include the cost for loyalty testing because two of the other one-out-of-two QOT protocols 21,24 do not consider any loyalty testing, which means that the sender and receiver may not truly trust each other, whereas the loyalty testing of the third protocol 24 requires the consumption of a large number of qubits, making the quantum cost difficult to calculate. Detailed descriptions of the protocols considered for comparison 21,24,25 are given below.
Yang's protocol. This protocol 21 uses the B92 22 protocol as the all-or-nothing QOT protocol on which it is based. Therefore, it requires four qubits on average to obtain an unambiguous key and only one transmission. However, the cited study focused more on the bit-commitment protocol than on the OT protocol, with no further security analysis of the QOT protocol or strategies for detecting eavesdroppers. Therefore, no strategy was provided for loyalty testing between Alice and Bob. In addition, the number of decoy qubits is computed as described above because the original detection strategy of B92 22 is less efficient. Therefore, the same detection strategy with decoy qubits is used. As a result, the total quantum cost is 4 × R + 50. YYLSZ protocol. This protocol 24 also requires at least four qubits on average to obtain an unambiguous key using its all-or-nothing QOT strategy, i.e., = p 1 4 . In the all-or-nothing QOT protocol, Alice first sends a sequence to Bob, and Bob then sends it back after his measurement, which requires two transmissions. After this, Alice can test Bob's loyalty by observing the probability of occurrence of states |+〉 and |−〉. Note that this strategy is based on the law of large number and will consume a large number of qubits. By contrast, Bob cannot really test Alice's loyalty; he is only able to detect errors in the later application of the one-out-of-two OT protocol. The overall cost of the protocol is 4 × R + 2 × 50.
YSW protocol. This protocol 25 reduces BB84 6 to B92 22 . It uses the BB84 6 strategy and the publication of additional state information to allow Bob to generate unambiguous keys as in B92 22 . It also requires four qubits for the generation of an unambiguous key. In addition, it requires one transmission to complete all-or-nothing QOT. However, it does not include a loyalty testing method for the all-or-nothing QOT stage. Errors may be detected at the application level. The overall cost of the protocol is 4 × R + 50.
Proposed protocol. The proposed protocol is based directly on quantum resources and consumes one qubit for each received bit. The proposed protocol is more efficient than the others 21,24,25 , with a probability p of 1 4 . The proposed protocol requires two transmissions between Alice and Bob. The overall cost of the proposed protocol is R + 2 × 50.
As seen from Table 3, the proposed protocol is the most efficient among all of the compared protocols 21,24,25 ; this is also shown in Fig. 1. This result demonstrates that building a one-out-of-two QOT protocol directly is more efficient than Crépeau's reduction 3 , which requires a two-level structure.
Discussion
In conclusion, there three important contributions of the proposed method. First, to the best of our knowledge, the proposed protocol is the first one-out-of-two QOT protocol to be designed directly based on quantum properties without relying on all-or-nothing QOT, and it has been proven to be secure and not subject to Lo's no-go theorem 15 . A simple and efficient one-out-of-two QOT protocol with single nonorthogonal qubits, in which one of the two messages is discarded automatically, has been successfully developed based on the most basic properties of quantum machines. Second, the proposed protocol can effectively prevent both external and internal attacks, as proven by a detailed security analysis. Regarding internal attacks, an important feature of the proposed protocol is that loyalty testing is applied to provide security against internal attacks by a dishonest Alice or Bob; the dishonest one can always be detected at the loyalty testing stages with safety parameters ξ 2 , ξ 3 and ξ 4 , respectively. Finally, the proposed protocol has a lower cost and is more efficient than many traditional protocols based on a two-level structure. In addition, as this protocol uses only a single qubit, it is easily implemented.
Data Availability
No datasets were generated or analysed during the current study.
Quantum gates and operations. Moreover, unitary operations (UU * = U * U = I) are regarded as gates in quantum computers. There are four common operations, represented by operators called Pauli matrices, which are denoted by {I, X, Y, Z}, as shown in Eq. 2.
shows the states {|0〉, |1〉, |+〉, |−〉} after the Pauli operations {I, X, Y, Z}.
•
Definition A: ideal one-sided two-party secure computation (A-i) Bob learns f(i, j) unambiguously. (A-ii) Alice learns nothing about j and f(i, j). (A-iii) Bob learns nothing about i more than what logically follows from the values of j and f(i, j). • Definition B: one-out-of-two OT (Lo's no-go theorem 15 ) (B-i) Alice inputs i, which is a pair of messages (m 0 , m 1 ). (B-ii) Bob inputs j = 0 or 1. (B-iii) At the end of the protocol, Bob learns the content of message m j but not of the other message m j ; i.e., the protocol is an ideal one-sided two-party secure computation, with f(m 0 , m 1 , j = 0) = m 0 and f(m 0 , m 1 , j = 1) = m 1 . (B-iv) Alice does not know which m j Bob received.
•
Definition C: the proposed protocol (C-i) Bob inputs j = 0 or 1 to change the qubit state to {|0〉, |+〉} (Z-or X-basis) in accordance with his choice intention. (C-ii) Alice inputs her messages m 0 and m 1 using {I, X, Y, Z}. (C-iii) Bob learns the content of either m 0 or m 1 using the basis (Z-or X-basis) he prepared.
M
Scientific REPORTS | (2018) 8:15927 | DOI:10.1038/s41598-018-32838-9
Scientific
REPORTS | (2018) 8:15927 | DOI:10.1038/s41598-018-32838-9
15 after the BM by the IY basis and XZ basis are shown in Eqs 16 and 17, where Eqs 16 and 17 are the probabilities
F 3 Scientific
3REPORTS | (2018) 8:15927 | DOI:10.1038/s41598-018-32838-9
Figure 1 .
1Illustration of the comparison results. Scientific REPORTS | (2018) 8:15927 | DOI:10.1038/s41598-018-32838-9
Table 1 .
1All results after {I, X, Y, Z}(m 0 , m 1 )
Alice's
Operation
Bob's Initial
State (j 0 )
Final
State
(m 0 , m 1 )
Alice's
Operation
Bob's Initial
State (j 1 )
Final
State
Table 2 .
2The relationship between the qubit states and the encoding.
. In fact, there is a 29.3% chance that Alice will learn Bob's choices with POVM {E 1 , E 2 , I − E 1 − E 2 } on Bob's qubit, where1
2
1
2
≡
+
≡
+
− − .
E
E
2
1
2
1 1 and
2
1
2
(7)
1
2
Then, she can unambiguously distinguish states {|0〉, |+〉} with probability
= +
+ =
+
≈ . .
E
E
0 0
1
2 (1
2)
29 3%
(8)
Scientific REPORTS | (2018) 8:15927 | DOI:10.1038/s41598-018-32838-9
Table 3 .
3Performance comparison of three modern 1-2 QOT protocols21,24,25 with the proposed protocol.Quantum Resources for One Message Bit: Number of quantum resources consumed for each received bit, without decoy qubits. b Number of Transmissions: Number of transmissions for one sequence. c Decoy Qubits: Number of decoy qubits, considering the number of transmissions. d Total Cost: The total average quantum resource consumption for R received bits.a
Scientific REPORTS | (2018) 8:15927 | DOI:10.1038/s41598-018-32838-9
© The Author(s) 2018
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| [] |
[
"THE REACHABILITY PROBLEM FOR AFFINE FUNCTIONS ON THE INTEGERS",
"THE REACHABILITY PROBLEM FOR AFFINE FUNCTIONS ON THE INTEGERS"
] | [
"Daniel Fremont "
] | [] | [] | We consider the problem of determining, given x, y ∈ Z k and a finite set F of affine functions on Z k , whether y is reachable from x by applying the functions F . We also consider the analogous problem over N k . These problems are known to be undecidable for k ≥ 2. We give 2-EXPTIME algorithms for both problems in the remaining case k = 1. The exact complexities remain open, although we show a simple NP lower bound.Date: December 15, 2012. | null | [
"https://arxiv.org/pdf/1304.2639v1.pdf"
] | 16,536,738 | 1304.2639 | 9de62fec0f761fe2d784f0e4552e1ac958f75194 |
THE REACHABILITY PROBLEM FOR AFFINE FUNCTIONS ON THE INTEGERS
1 Apr 2013
Daniel Fremont
THE REACHABILITY PROBLEM FOR AFFINE FUNCTIONS ON THE INTEGERS
1 Apr 2013
We consider the problem of determining, given x, y ∈ Z k and a finite set F of affine functions on Z k , whether y is reachable from x by applying the functions F . We also consider the analogous problem over N k . These problems are known to be undecidable for k ≥ 2. We give 2-EXPTIME algorithms for both problems in the remaining case k = 1. The exact complexities remain open, although we show a simple NP lower bound.Date: December 15, 2012.
Introduction
Many dynamical systems with simple evolution rules nevertheless exhibit unpredictable long-term behavior. Multidimensional systems in particular can easily be so complex that questions like state reachability are undecidable. For instance, this is the case for states in [0, 1] 2 evolving under a piecewise-linear function [1]. There are even simpler nondeterministic examples, such as states in Q 2 under a finite set of affine functions [2]. For both systems, having states with two coordinates whose evolution is not independent is essential to the undecidability proofs. Generally, while it is often not difficult to prove undecidability for systems with sufficiently high dimension, determining if and when the transition to decidability occurs at lower dimensions is harder. In particular, it is not known whether the reachability problem is decidable for nondeterministic affine evolution on Q.
In this paper, we consider the simpler problem of reachability under nondeterministic affine evolution on Z: given x, y ∈ Z and a finite set F of affine functions f i (z) = a i z + b i with a i , b i ∈ Z, determine whether y is reachable from x by applying functions in F . The generalizations to Z n are undecidable for all n ≥ 2, as implicitly shown in [3] (and a little more clearly in [4,Section 4.9]). We prove that the remaining case, n = 1, is decidable, giving a 2-EXPTIME algorithm for it in Section 2. We also consider the version of this problem with evolution over N: in this problem, the functions f i can still have negative coefficients, but may not be applied if they would yield a negative result. In Section 3 we give a 2-EXPTIME algorithm for this problem by modifying our algorithm for the case over Z. Finally, in Section 4 we show that the problems over Z and N are both NP-hard.
Affine reachability over Z
We begin by defining some notation that we will use throughout this paper.
Definition. If S is a set of affine functions on Z, we will call any function of the form G = s K • · · · • s 1 for some K ∈ N with each s i ∈ S an S-composition. We will want to discuss the individual functions s i which appear in an S-composition, so G is formally the tuple (s 1 , . . . , s K ), but we will often view S-compositions as functions without further comment. The orbit of G applied to the argument x is the set of values {x, s 1 (x), (s 2 • s 1 )(x), . . . , (s K •· · · •s 1 )(x)}. We write x S − → y via G to indicate that G is an S-composition such that G(x) = y, and x S − → y to assert the existence of such a G.
In this notation, the affine reachability problem over Z is to determine, given x, y ∈ Z and a finite set F of affine functions f i (z) = a i z + b i with a i , b i ∈ Z, whether x F − → y. The problem takes several qualitatively different forms depending on the values of the linear coefficients a i . The simplest nontrivial case is when they all satisfy |a i | > 1.
Lemma 1.
There is an EXPTIME algorithm to decide, given any x, y ∈ Z and a finite set F of functions f i (z) = a i z + b i with a i , b i ∈ Z and satisfying |a i | > 1, whether x F − → y.
Proof. Outside of some finite interval, for instance [−Q, Q] with Q = 1+max |b i |, each function f i strictly increases absolute value. Putting R = max{Q, |y|}, for any F -composition G and z ∈ Z with |z| > R we have |G(z)| > |z| > |y| and thus G(z) = y. This means that all preimages of y under F -compositions must lie in the finite interval I = [−R, R]. Create a directed graph D with a vertex for each integer z ∈ I, and add edges from z to f i (z) for each i satisfying f i (z) ∈ I. Since every preimage of y under an F -composition lies in I, we have x F − → y if and only if x ∈ I and there is a path in D from x to y. We can determine whether such a path exists in exponential time using graph search, since D has exponentially-many vertices (at most linearly-many in the values of b i and y).
Remark. As will be important later, with a small modification of this algorithm we can handle the presence of one f j with a j = −1, so that f j (z) = −z + b j . The only change necessary is to broaden I to the interval I ′ = [min(−R, −R + b j ), max(R, R + b j )]. Then f −1 j = f j maps I ′ onto itself, so all preimages of y under F -compositions are in I ′ , and the argument goes through as above.
However, when a function in F is of the form g(z) = z + k this method breaks down, because the preimages of y under F -compositions are no longer bounded. This also happens if there are two functions of the form f (z) = −z + b, since then their composition is of the form g(z) = z + k. Fortunately, functions like g can contribute to an F -composition in basically only one way.
Lemma 2. For any set S of functions f i (z) = a i z + b i with a i , b i ∈ Z for 1 ≤ i ≤ N and function f 0 (z) = z + k, put F = S ∪ {f 0 }. Then for any F -composition G, we have: (a) If G = f e 0 • · · · • f e j • f n 0 • f e j+1 • · · · • f e K , then G(z) = H(z) + ank where H = f e 0 • · · · • f e K and a = a e 0 . . . a e j .
(b) G(z) = H(z)+ak for some S-composition H and a ∈ Z. If a i > 0 for every function f i appearing in G, then a ≥ 0.
Proof. (a) We have (f i • f n 0 )(z) = a i (z + nk) + b i = (a i z + b i ) + a i nk = (f a i n 0 • f i )(z), so G = f e 0 • · · · • f e j • f n 0 • f e j+1 • · · · • f e K = f e 0 • · · · • f e j−1 • f ae j n 0 • f e j • · · · • f e K .
Repeating this j more times gives G = f
ae 0 ...ae j n 0 • f e 0 • · · · • f e K = H(z) + ank with H = f e 0 • · · · • f e K and a = a e 0 . . . a e j . (b) Apply the previous result once for each instance of f 0 in G, giving G = f c 0 0 • · · · • f c L 0 • f e 0 • · · · • f e K for some c 0 , .
. . c L ∈ Z which are products of the coefficients a i . Then G(z) = H(z) + ak with H the S-composition H = f e 0 • · · · • f e K and a = c 0 + · · · + c L . If a i > 0 for each f i appearing in G, then c i > 0 and so a ≥ 0 (we could have a = 0 if there were no instances of f 0 in G).
We now have several cases, based on which S-compositions G satisfy x S − → y (mod k) via G.
Lemma 3.
For any x, y ∈ Z, finite set S of functions f i (z) = a i z + b i with a i , b i ∈ Z and a i = 0, and function g(z) = z + k with k ∈ Z and k = 0, put F = S ∪ {g} and
G = {G : x S − → y (mod k) via G}.
Then the following are true: Proof. (A) By Lemma 2b, any F -composition can be written as an S-composition plus a multiple of k. If G = ∅, then no S-composition can reach y (mod k) from x, and therefore neither can any F -composition. (B) If some G ∈ G satisfies sgn(G(x) − y) = sgn(k), then either sgn(G(x) − y) = 0 or sgn(G(x) − y) = −sgn(k). If the first of these is true, then G(x) = y and so
(A) If G = ∅, then x F − → y. (B) If some G ∈ G satisfies sgn(G(x) − y) = sgn(k), then x F − → y. (C) If some G ∈ G with G = f e 0 • · · · • f e K has a e j < 0 for some j, then x F − → y.x F − → y via G. Otherwise, there is some n ∈ N such that G(x) − y = −nk, so y = G(x) + nk = (g n • G)(x). Putting G ′ = g n • G, we have x F − → y via G ′ . (C) If some G ∈ G with G = f e 0 • · · · • f e K
has a e j < 0 for some j, take the smallest such j. Defining G ′ = f e 0 • · · · f e j • g n • f e j+1 • · · · • f e K , by Lemma 2a we have G ′ (z) = G(z) + ank with a = a e 0 a e 1 · · · a e j , and because a e k > 0 for all k < j by our choice of j, we have a < 0. We may assume that case B does not hold, since otherwise we have x F − → y immediately as shown above. Then we have sgn(G(x)− y) = sgn(k), and so sgn(G(x) − y) = −sgn(ak). Therefore with n sufficiently large we have sgn(G ′ (x) − y) = sgn(G(x) − y + ank) = −sgn(G(x) − y) = −sgn(k). Then as in case
B, we have x F − → y via G ′′ = g m • G ′ for some m ∈ N.
(D) Suppose that x F − → y via some H. Then by Lemma 2b we have H = g a • G for some S-composition G and a ∈ Z. Now G(x) = H(x) − ak ≡ H(x) = y (mod k), so G ∈ G. Since case B does not hold, we have sgn(G(x) − y) = sgn(k). Since case C does not hold, we have a ≥ 0, again by Lemma 2b. If a = 0, then G(x) = H(x) = y, so sgn(G(x) − y) = 0 = sgn(k) and case B holds, contrary to our assumption. So a > 0, and thus sgn(G(x) − y) = sgn(k) = sgn(ak). But this is impossible, since
G(x) − y = H(x) − y − ak = −ak. So we cannot have x F − → y.
To test cases (A), (B), and (C), we use the following algorithm.
f i (z) = a i z + b i with a i , b i ∈ Z and a i = 0 for 1 ≤ i ≤ N , put G = {G : x F − → y (mod k) via G}.
There is a 2-EXPTIME algorithm such that:
(1) If G = ∅, the algorithm returns Empty.
(2) If there is some G ∈ G with G = f e 0 •· · ·•f e K and a e j < 0 for some j, the algorithm returns Negative. The flags Empty and Negative are enough for us to detect cases A and C of Lemma 3. If k < 0, the value V = sup {G(x) : G ∈ G} allows us to recognize case B, because this case holds if and only if V ≥ y. If k > 0 we need the value V ′ = inf {G(x) : G ∈ G} instead, since then case B holds if and only if V ′ ≤ y. The modifications to the algorithm of Lemma 4 required to make it compute V ′ instead of V are simple and obvious (just exchanging "increases" with "decreases" in several places, etc.), so we omit them. Now we prove Lemma 4, assuming a couple of auxiliary lemmas (Lemmas 5 and 6) which we will return to afterwards. Use graph search to determine if there is such a path, and return Empty if not. Since D has |k| vertices, this search takes exponential time.
If there are paths from x (mod k) to y (mod k), we need to analyze all of them to see which ones yield the largest final value. We can conveniently describe the paths using regular expressions. Consider D to be a deterministic finite automaton, where an input symbol e ∈ {1, . . . , N } causes the edge corresponding to applying f e to be followed. Let the initial state be x (mod k), and the only accepting state be y (mod k). If s = e 1 . . . e K is a sequence of input symbols, we write P s = f e K • · · · • f e 1 (note the order!), and then D accepts s if and only if P s (x) ≡ y (mod k). Now we convert D into a regular expression R with the same language L(R). We write concatenation multiplicatively, use | to denote union/alternation, and use ǫ and ∅ as the symbols for the empty string and the empty language respectively. Because D has exponentially-many vertices (and a linearly-sized input alphabet), R has at most doublyexponential size |R|, and the conversion from D to R can be done in time at most polynomial in |R| (see [5,6]). We store R as a tree, with literals, ǫ, or ∅ at the leaf nodes and operators at the other nodes. The "length" |R| in this representation is just the total number of nodes.
Next, reduce R to not include the symbol ∅ by repeatedly passing through R applying the identities E|∅ = E, E∅ = ∅, and ∅ * = ǫ for any expression E. Each pass takes time linear in the length of R and strictly decreases its length, so there can be at most |R| passes and the total time taken is O(|R| 2 ). Afterwards, if the symbol ∅ appears in R it must not be operated on by any operator, since otherwise one of the identities above would apply. Therefore ∅ can appear in R only if R = ∅, but since L(R) is nonempty (because we returned Empty above if so) this is not the case. So R does not contain the symbol ∅.
Now if R contains a literal corresponding to a function f i with a i < 0 (which can
obviously be determined in O(|R|) time), then x F − → y (mod k) via an F -composition which includes f i , so we return Negative. Otherwise, we convert R into disjunctive normal form S 1 |S 2 | · · · |S M
where each S i has no union operations, by iteratively applying the identities (α|β) * = (α * β * ) * , α(β|γ) = αβ|αγ, and (α|β)γ = αγ|βγ. Each identity either decreases the number of unions or moves one closer to the topmost level, so this process will also take time polynomial in |R|.
We say that a regular expression E is reduced if it contains only literals appearing in R and has no ∅ symbols or union operations. The reductions we have done above ensure that any expression produced by concatenating subexpressions of the clauses S i is reduced. Since every literal in R corresponds to a function f i with a i > 0 (because we would have returned Negative otherwise), and the composition of two linear polynomials with positive linear coefficients has a positive linear coefficient, for any reduced expression E and any s ∈ L(E), P s has a positive linear coefficient -this will be important in a moment. We will want to refer to those F -compositions which are generated by reduced expressions, and to decrease the proliferation of symbols, we will say that an F -composition G matches the expression E if there is some s ∈ L(E) such that G = P s . Then G consists precisely of those F -compositions which match R. Now, given some z ∈ Z and a reduced expression E, define I(z, E) to mean that ∃s ∈ L(E) : P s (z) > z. In words, I(z, E) is true if and only if there is some F -composition matching E which increases z. If L(E) is finite, computing I(z, E) is only a matter of testing various cases -the difficulty is handling expressions with stars. Fortunately, we can reduce I to its values on expressions with fewer stars using the identity I(z, ℓα * β) ⇐⇒ I(P ℓ (z), α) ∨ I(z, ℓβ), which we will prove in Lemma 5. This then allows us to compute I(z, E) recursively in polynomial time, as we will show in Lemma 6. Now we are ready to return to the main problem. For each clause S i , we want to find the supremum V i of the possible values x is mapped to by any F -composition matching S i . To do this, we keep track of the supremum of the values x is mapped to by Fcompositions which match progressively-longer prefixes of S i . Write S i = T 1 . . . T K by flattening out concatenations, so that each T j is either a literal or a starred subexpression. Let x
(i) j = sup {P s (x) : s ∈ L(T 1 . . . T j )} for 1 ≤ j ≤ K. Clearly V i = x (i)
K , and we put x (i) 0 = x (since the largest possible value reachable after applying no functions is the
starting value x). For j ≥ 1, we calculate x (i) j in terms of x (i)
j−1 as follows. If T j is a literal, then any F -composition matching T 1 . . . T j must be of the form P T j • q where q is an Fcomposition matching T 1 . . . T j−1 . Since by definition the largest possible value of q(x) is x (i) j−1 , and P T j has a positive linear coefficient, the largest possible value of (P
T j • q)(x) is P T j (x (i) j−1 ). Thus x (i) j = P T j (x (i) j−1 ). If T j is a starred subexpression instead, T j = α * , we compute I(x (i) j−1 , α). If this is true, then some F -composition p matching α increases x (i) j−1 ,
and because p has a positive linear coefficient it must increase all values larger than x
(i) j−1 . So we can increase x (i)
j−1 as much as we want by repeatedly applying p, and thus x
(i) j = ∞. If I(x (i) j−1 , α) is false, then no F -composition matching α increases x (i) j−1 , so x (i) j = x (i) j−1
(since α * is matched by P ǫ , which leaves x Now put V = max V i . Because the union of the languages of each clause S i is the language of R, V is the largest value reachable using F -compositions matching R (or ∞ if arbitrarily large values are reachable). Therefore V = sup {G(x) : G ∈ G}, the desired value, and we return it.
As mentioned above, the first stage of this algorithm takes exponential time, and all subsequent stages take time at most polynomial in |R|. Since |R| is at most doublyexponential in the input length, the entire algorithm takes at most doubly-exponential time.
Lemma 4 depends on our ability to efficiently calculate I(z, E). As was mentioned above, the key to doing this is to reduce I(z, E) to values of I on smaller subexpressions, as made possible by the following lemma.
Lemma 5. If z ∈ Z, α and β are reduced regular expressions, and ℓ is a sequence of literals, the following are true:
(1) I(z, ℓα * β) ⇐⇒ I(z, ℓβ) ∨ I(P ℓ (z), α) (2) I(z, ℓα * ) ⇐⇒ I(z, ℓ) ∨ I(P ℓ (z), α)
(3) I(z, α * β) ⇐⇒ I(z, α) ∨ I(z, β) (4) I(z, α * ) ⇐⇒ I(z, α)
Proof.
(1) I(z, ℓβ) implies I(z, ℓα * β) because ℓβ matches ℓα * β. If I(P ℓ (z), α) holds, then there is an F -composition p matching α which increases P ℓ (z). Because p has a positive linear coefficient as observed above (since α is reduced), p must increase anything greater than P ℓ (z), and thus repeated applications of p can increase P ℓ (z) as much as desired. Therefore for any particular F -composition q matching β, there is some n ∈ N such that q • p n • P ℓ increases z (since β is reduced and so q also has a positive linear coefficient). Since q • p n • P ℓ matches ℓα * β, I(z, ℓα * β) holds.
Suppose neither I(z, ℓβ) nor I(P ℓ (z), α) hold. Any F -composition matching ℓα * β is of the form q • p • P ℓ where q matches β and p is a composition of F -compositions matching α. Since by our assumption no polynomials matching α increase P ℓ (z), and these have positive linear coefficients, they cannot increase anything smaller than P ℓ (z). Thus no composition of F -compositions matching α increases P ℓ (z), and so p does not increase P ℓ (z). Now, since q•P ℓ does not increase z by assumption (since it matches ℓβ), and q has a positive linear coefficient, q•p•P ℓ does not increase z either. Thus no F -composition matching ℓα * β increases z, and I(z, ℓα * β) does not hold.
(2) Put β = ǫ in (1) and use L(ℓα * ǫ) = L(ℓα * ) and L(ℓǫ) = L(ℓ).
(3) Put ℓ = ǫ in (1) and use L(ǫβ) = L(β) and P ǫ (z) = z. (4) Put β = ǫ in (3), use L(α * ǫ) = L(α * ), and note that I(z, ǫ) is clearly false.
These relationships allow us to give a straightforward recursive algorithm to compute I. Lemma 6. There is a P algorithm to compute I(z, E) for any z ∈ Z and reduced regular expression E.
Proof. First, note that if an expression α is a sequence of literals, then only P α matches it, and we may determine I(z, α) = (P α (z) > z) directly by evaluating P α (z). Now we compute I(z, E) recursively, breaking into cases based on the topmost operator or symbol of E:
• E = ǫ: I(z, E) is clearly false.
• E is a literal: As noted above, I(z, E) = (P E (z) > z) may be directly calculated.
• E = F * : By part 4 of Lemma 5, I(z, E) = I(z, F * ) = I(z, F ).
• E = F G: By flattening as necessary, we may write E = F 1 F 2 · · · F K for some K ∈ N with K ≥ 2 and where none of the subexpressions F i are concatenations. If any of the subexpressions F i are ǫ, we simply drop them and renumber appropriately -this obviously leaves L(E) fixed. Thus we may assume that each subexpression F i is either a starred subexpression or a literal. If each one is a literal, then E is a sequence of literals and as noted above I(z, E) can be computed directly. Otherwise, find the smallest j such that F j = α * is a starred subexpression. There are several cases: -j = 1: Then E = α * β where β = F 2 · · · F K , so by part 3 of Lemma 5 we have I(z, E) = I(z, α * β) = I(z, α) ∨ I(z, β). -j = K: Then E = ℓα * where ℓ = F 1 · · · F K−1 is a sequence of literals, so by part 2 of Lemma 5 we have I(z, E) = I(z, ℓα * ) = I(z, ℓ) ∨ I(P ℓ (z), α). -1 < j < K: Then E = ℓα * β where ℓ = F 1 · · · F j−1 is a sequence of literals and β = F j+1 · · · F K , so by part 1 of Lemma 5 we have I(z, E) = I(z, ℓα * β) = I(z, ℓβ) ∨ I(P ℓ (z), α). In each case I(z, E) is either directly computable, equivalent to I(x, F ) for some x ∈ Z and F a proper subexpression of E, or equivalent to I(x, F ) ∨ I(y, G) for some x, y ∈ Z, F a proper subexpression or concatenation of disjoint subexpressions of E, and G a subexpression of E disjoint from F . Because I(z, E) is always reduced to values of I on strictly shorter expressions, the tree of recursive calls has at most O(|E|) levels. Since I(z, E) is always reduced to values of I on disjoint expressions made up of subexpressions of E, each level of the tree can have at most O(|E|) calls. Thus the entire tree has at most O(|E| 2 ) calls with polynomial computation each, so I(z, E) may be computed in polynomial time.
We can now give an algorithm to solve the affine reachability problem over Z in full generality.
Theorem 1. There is a 2-EXPTIME algorithm to decide, given any x, y ∈ Z and a finite set
F of functions f i (z) = a i z + b i with a i , b i ∈ Z, whether x F − → y.
Proof. There are several cases:
(1) For some j, a j = 0: Clearly, x (5) For some j, k with j = k, a j = a k = −1: (3), so it also takes doubly-exponential time. Finally, cases (2) and (1) make one and two recursive calls respectively, each with one less affine function. Thus in total the algorithm will make at most exponentially-many recursive calls (this can easily be reduced to linearly-many by improving the handling of case (1), but this does not decrease the worst-case runtime), with at most a doubly-exponential amount of computation each. Therefore the algorithm runs in at most doubly-exponential time.
Define g = f j • f k . Clearly x F − → y if and only if x F ∪{g} − −−− → y. But g(z) = (f j • f k )(z) = −(−z + b k ) + b j = z + (b j − b k ), and since b j = b k (since f j and f k are distinct functions), g(z) = z + c for some c ∈ Z with c = 0. Recursively solve x
Affine reachability over N
We can also consider the affine reachability problem over N. Much of the analysis is the same, so we will only write out in full detail the considerations which are new. The main difference from the version over Z is that now we cannot apply any functions which would yield a negative result.
Definition. An F -composition is valid with respect to its argument if every integer in its orbit for the given argument is nonnegative. Often the argument of the composition will be clear from context, in which case we will simply say that the composition is valid. We write x F − → + y to indicate that there is a valid F -composition G such that G(x) = y.
With this definition, the affine reachability problem over N is to determine, given x, y ∈ N and a finite set F of functions f i (z) = a i z + b i with a i , b i ∈ Z, whether x F − → + y. As before, there are various cases depending on the values of the linear coefficients a i . The case where they all satisfy |a i | > 1 is still simple.
Lemma 7.
There is an EXPTIME algorithm to decide, given any x, y ∈ N and a finite set Remark. This algorithm also works with any number of functions of the form g(z) = z + k with k > 0, since these strictly increase absolute value on N \ I, and so preimages of y under valid compositions including them must lie in I.
F of functions f i (z) = a i z + b i with a i , b i ∈ Z and satisfying |a i | > 1, whether x F − → + y.
The algorithm of Lemma 7 cannot handle functions of the form g(z) = z − k with k > 0. Fortunately, as before we can reduce problems with this type of function to "modular" problems without them, using the following (much simpler) analog of Lemma 3. We assume for now that all a i > 0, and show how to handle other cases later.
Lemma 8. For any x, y ∈ N, a set S of functions f i (z) = a i z + b i with a i , b i ∈ Z and a i > 0 for 1 ≤ i ≤ N , and function g(z) = z − k with k > 0, put F = S ∪ {g}. Then
x F − → + y if and only if x S − → + z ≡ y (mod k) for some z ≥ y.
Proof. Suppose x F − → + y via G. By Lemma 2b, G = g a • H with a ≥ 0 and H being G with all instances of g removed. Since G(x) = y, we have H(x) ≡ y (mod k). Now note that g always decreases its argument, and since every a i is positive, each f i maps larger inputs to larger outputs. Therefore removing an instance of g from G can only increase the values of integers in its orbit. Since G is valid this means H must be as well, and since G(x) = y this means H(x) ≥ y. Therefore x
S − → + H(x) ≡ y (mod k) with H(x) ≥ y.
Conversely, suppose x S − → + z ≡ y (mod k) via H with z ≥ y. Then (g n • H)(x) = z − nk = y for some n ∈ N. Putting G = g n • H, G is valid because H is, and so we have
x F − → + y via G.
The algorithm of Lemma 4 is almost exactly what we need to test the condition in Lemma 8, since it computes the largest z ≡ y (mod k) reachable by S-compositions. However, we must consider only those reachable by valid S-compositions, and so need to modify the algorithm. This is not hard to do.
Lemma 9. Given any x, y ∈ N, k ∈ Z with k = 0, and a set F of functions
f i (z) = a i z + b i with a i , b i ∈ Z and a i > 0 for 1 ≤ i ≤ N , put G = {G : x F − → + y (mod k) via G}.
There is a 2-EXPTIME algorithm which returns Empty if G = ∅, and otherwise returns sup {G(x) : G ∈ G}.
Proof. As in the algorithm of Lemma 4, construct the graph D and search it to determine if x F − → y (mod k) via any F -composition, not necessarily a valid one. If not, return Empty. Otherwise, consider D to be a finite automaton as before, and convert it into a reduced regular expression R in disjunctive normal form R = S 1 | . . . |S M .
Given some z ∈ N and ℓ a sequence of literals which appear in R, we define V (z, ℓ) to mean that P ℓ is a valid F -composition with respect to z. Now for any z ∈ N and reduced expression E, define I ′ (z, E) to mean ∃s ∈ L(E) : V (z, P s ) ∧ (P s (z) > z). In words, this means that there is some F -composition valid with respect to z which matches E and increases z (this is just the analog of I(z, E) from Lemma 4, but restricted to only valid compositions). By an extension of Lemma 5 which will prove momentarily, Lemma 10, we have that I ′ (z, ℓα * β) ⇐⇒ V (z, ℓ) ∧ (I ′ (P ℓ (z), α) ∨ I ′ (z, ℓβ)). Using this we may compute I ′ (z, E) in polynomial time with the analog of the algorithm of Lemma 6, described shortly in Lemma 11. Now we continue as in the algorithm of Lemma 4, writing S i = T 1 . . . T K and defining x If we discarded every S i , then no valid F -compositions match R, so G = ∅ and we return Empty. Otherwise, if V is the largest of the values V i (at least one of these is defined since we did not discard every S i ) then V = sup {G(x) : G ∈ G} and we return it. As in Lemma 4, this algorithm takes time polynomial in |R|, and thus takes at most doubly-exponential time.
Now we prove the analog of Lemma 5 for I ′ .
Lemma 10. If z ∈ N, α and β are reduced regular expressions, and ℓ is a sequence of literals, the following are true:
(
1) I ′ (z, ℓα * β) ⇐⇒ V (z, ℓ) ∧ (I ′ (z, ℓβ) ∨ I ′ (P ℓ (z), α)) (2) I ′ (z, ℓα * ) ⇐⇒ V (z, ℓ) ∧ (I ′ (z, ℓ) ∨ I ′ (P ℓ (z), α)) (3) I ′ (z, α * β) ⇐⇒ I ′ (z, α) ∨ I ′ (z, β) (4) I ′ (z, α * ) ⇐⇒ I ′ (z, α)
Proof.
(1) I ′ (z, ℓβ) implies I ′ (z, ℓα * β) because ℓβ matches ℓα * β. If I ′ (P ℓ (z), α) holds, then there is an F -composition p matching α which is valid with respect to and increases P ℓ (z). Because p has a positive linear coefficient (since we assumed all a i > 0), p must increase anything greater than P ℓ (z), and thus repeated applications of p can increase P ℓ (z) as much as desired. Therefore for any particular F -composition q matching β, there is some n ∈ N such that q • p n is valid with respect to and increases P ℓ (z), and q • p n • P ℓ increases z. If we also have V (z, P ℓ ), then P ℓ is valid with respect to z, and then q • p n • P ℓ is valid with respect to and increases z. Then, since q • p n • P ℓ matches ℓα * β, I ′ (z, ℓα * β) holds.
Suppose V (z, ℓ) does not hold. Then no F -composition p matching ℓα * β can be valid with respect to z, since the first part of p must be P ℓ , which is not valid with respect to z and thus will cause some integers in the orbit of p to be negative. So I ′ (z, ℓα * β) does not hold.
Suppose that neither I ′ (z, ℓβ) nor I ′ (P ℓ (z), α) hold. Any valid F -composition matching ℓα * β is of the form q•p•P ℓ where q matches β and p is a composition of Fcompositions matching α. Since by our assumption valid F -compositions matching α do not increase P ℓ (z), and these have positive linear coefficients, they do not increase anything smaller than P ℓ (z). Thus no valid composition of F -compositions matching α increases P ℓ (z), and so p does not increase P ℓ (z). Therefore because q is valid with respect to (p•P ℓ )(z), q •P ℓ is valid with respect to z, and since I ′ (z, ℓβ) does not hold, q • P ℓ does not increase z. Because q has a positive linear coefficient, q • p • P ℓ does not increase z either. Thus no valid F -composition matching ℓα * β increases z, and I(z, ℓα * β) does not hold.
(2) Put β = ǫ in (1) and use L(ℓα * ǫ) = L(ℓα * ) and L(ℓǫ) = L(ℓ).
(3) Put ℓ = ǫ in (1) and use L(ǫβ) = L(β) and P ǫ (z) = z. (4) Put β = ǫ in (3), use L(α * ǫ) = L(α * ), and note that I ′ (z, ǫ) is clearly false.
As before, these relationships yield a recursive algorithm for computing I ′ :
Lemma 11. There is a P algorithm to compute I ′ (z, E) for any z ∈ N and reduced regular expression E.
Proof. The algorithm is identical to that of Lemma 6, except that in addition to reducing I ′ (z, E) to values of I ′ on shorter expressions, the identities in Lemma 10 also require evaluations of V (z, ℓ). Clearly V (z, ℓ) can be computed in polynomial time, by simply calculating the entire orbit of P ℓ and checking that every integer in it is nonnegative. Adding a single computation of V at each recursive call in the algorithm of Lemma 6 multiplies its runtime by a polynomial factor, so the overall algorithm still runs in polynomial time.
Now we can give a general algorithm for the affine reachability problem over N.
Theorem 2.
There is a 2-EXPTIME algorithm to decide, given any x, y ∈ N and a finite set
F of functions f i (z) = a i z + b i with a i , b i ∈ Z, whether x F − → + y.
Proof. There are several cases:
(1) For some j, a j = 0: As in Theorem 1, recursively determine x Lemma 7 (which works in this case as noted in the remark), and return the result. The analysis of the runtime is similar to that given in Theorem 1, except for case 2. The graph D created in that case has exponentially-many vertices (linear in the value of b j ), so each invocation of the algorithm makes at most exponentially-many recursive calls. There are at most a linear number of levels, since each recursive call has one less affine function than its parent. Thus there are exponentially-many recursive calls in total. The work done in each call takes at most doubly-exponential time (since the algorithm in Lemma 9 can take this long), so the overall running time of the algorithm is at most doubly-exponential.
A Lower Bound
While it may be hoped that there are vastly more efficient algorithms for the affine reachability problems than the 2-EXPTIME methods we have given here, the following theorem shows that polynomial-time algorithms are unlikely.
Theorem 3. The affine reachability problems over Z and N are NP-hard.
Proof. We give a reduction from the Integer Knapsack Problem (IKP), which is to determine, given w 1 , . . . , w N , C ∈ N, whether there are x 1 , . . . , x N ∈ N such that i w i x i = C. This problem is known to be NP-complete [7]. For a given instance of the IKP, w 1 , . . . , w N , C ∈ N, let the set F consist of the affine functions f i (z) = z +w i for 1 ≤ i ≤ N . If there exist x 1 , . . . , x N ∈ N such that i w i x i = C, then (f x 1 1 • · · · • f x N N )(0) = i w i x i = C, so 0 F − → C. Since the functions f i all commute, if 0 F − → C then C = (f x 1 1 • · · · • f x N N )(0) = i w i x i for some x 1 , . . . , x N ∈ N. Thus 0 F − → C if and only if the IKP instance is solvable. Computing F given an IKP instance can obviously be done in polynomial time, so this gives a polynomial-time many-one reduction from IKP to the affine reachability problem over Z, showing that the latter is NP-hard. The reduction to affine reachability over N is exactly the same, since all F -compositions are valid.
Conclusion
We gave 2-EXPTIME algorithms for the affine reachability problems over Z and N, and showed that they are NP-hard. Beyond improving these upper and lower bounds, a natural generalization that would be interesting to consider is if integer or integer-valued polynomials are allowed instead of just affine functions. Also, the original problem which this paper treated a special case of, namely reachability for affine evolution over Q, remains open. This provides another clear direction for future work.
(
D) If none of the above cases hold, then x F − → y.
Lemma 4 .
4Given any x, y, k ∈ Z with k = 0 and a set F of functions
( 3 )
3Otherwise, the algorithm returns sup {G(x) : G ∈ G}.
Proof of Lemma 4 .
4First we check if there is any F -composition mapping x to y (mod k). Create a directed graph D with a vertex for each congruence class mod k. Add edges indicating which classes are mapped to which under each f i . Then there is a path in D from the congruence class of x to the congruence class of y if and only if x F − → y (mod k).
K
= V i . There are O(|S i |) intermediate values x (i) j which need to be computed, and each one requires at most one call to I on an expression of size at most O(|S i |). Thus we can compute each V i in time polynomial in |S i |, and all the values V i together in time polynomial in |R|.
determine each of these and return true if and only if at least one is true. (2) For some j, a j = 1 and b j = 0: Clearly x F − → y if and only if x F \{f j } −−−−→ y (since f j is the identity), so determine this recursively and return the result. (3) For some j, a j = 1 and b j = 0: Assume for now thatb j < 0 (we will handle b j > 0 momentarily). Run the algorithm of Lemma 4 on F \ {f j } with k = b j .If it returns Empty, then case A of Lemma 3 holds, so return false. If it returns Negative, then case C holds, so return true. Otherwise, the algorithm returnsV = sup {G(x) : x F \{f j } −−−−→ y (mod b j ) via G}. Sincenow either case B or case D of Lemma 3 holds, x F − → y if and only if sgn(V − y) = sgn(b j ) = −1. So we return true if and only if V ≥ y. If b j was in fact positive, we use the variant of the algorithm of Lemma 4 which computes V ′ = inf {G(x) : x F \{f j } −−−−→ y (mod b j ) via G}. By Lemma 3 again we have x F − → y if and only if sgn(V ′ − y) = sgn(b j ) = +1, so we return true if and only if V ′ ≤ y. (4) For some j, a j = −1, and |a i | > 1 for all i = j: Use the algorithm of Lemma 1, modified as in the remark to handle f j (z) = −z + b j .
F
∪{g} − −−− → 0 using case (3) and return the result. (6) Otherwise, |a i | > 1 for all i: Use the algorithm of Lemma 1. Cases (4) and (6) invoke the algorithm of Lemma 1 and take exponential time. Case (3) invokes the algorithm of Lemma 4 and takes doubly-exponential time. Case (5) makes a recursive call which always uses case
Proof. Use the algorithm of Lemma 1, but with the intervalI = [0, R] instead of [−R, R].The argument in the proof of Lemma 1 goes through as before, since all preimages of y under valid F -compositions must lie in I.
sup {P s (x) : s ∈ L(T 1 . . . T j )}. We calculate the values x (i) j in the same way as before, except using I ′ in place of I when dealing with starred subexpressions T j . This ensures that only F -compositions which are valid with respect to x T j is a starred subexpression. When T j is a literal, we put x 0. If so, then P T j is not valid with respect to x (i) j−1 , and thus no F -composition matching S i can be valid with respect to x. So we discard S i and move on. Otherwise again x(i) j can be obtained from x (i) j−1 using a valid F -composition. Then if we compute x (i) K without discarding S i , the value x (i) K can be obtained from x using a valid F -composition, and so V i = x (i) K is the supremum of possible values x is mapped to by any valid F -composition matching S i .
FF
\{f \{f j } −−−−→ y and return true if and only if at least one holds. (2) For some j, a j < 0: There are only a finite number of z ∈ N which f j can be applied to without giving a negative result: for example, they are all in [0, b j ]. Create a directed graph D with a vertex for each of these, as well as vertices for x and y if not already present. Add edges indicating how f j maps these values to each other. Use this algorithm recursively on S = F \ {f j } to add edges corresponding to mappings by all possible valid S-compositions. Then there is a path in D from x to y if and only if x F − → + y. Use graph search to test if such a path exists, and return the result. (3) For some j, a j = 1 and b j = 0: As in Theorem 1, recursively solve x F \{f j } −−−−→ y and return the result. (4) For some j, a j = 1 and b j < 0: Run the algorithm in Lemma 9 on F \ {f j } with k = b j . If it returns Empty, then by Lemma 8 we cannot have x F − → + y and we return false. Otherwise the algorithm returns V = sup {G(x) : x F − → + y (mod k) via G}, and again by Lemma 8 we have x F − → + y if and only if V ≥ y. So return true if and only if V ≥ y. (5) Otherwise, for all i we have a i ≥ 1, and if a i = 1 then b i > 0: Use the algorithm of
E-mail address: [email protected]
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| [] |
[
"A REFINEMENT OF STRONG MULTIPLICITY ONE FOR SPECTRA OF HYPERBOLIC MANIFOLDS",
"A REFINEMENT OF STRONG MULTIPLICITY ONE FOR SPECTRA OF HYPERBOLIC MANIFOLDS"
] | [
"Dubi Kelmer "
] | [] | [] | Let M 1 and M 2 denote two compact hyperbolic manifolds. Assume that the multiplicities of eigenvalues of the Laplacian acting on L 2 (M 1 ) and L 2 (M 2 ) (respectively, multiplicities of lengths of closed geodesics in M 1 and M 2 ) are the same, except for a possibly infinite exceptional set of eigenvalues (respectively lengths). We define a notion of density for the exceptional set and show that if it is below a certain threshold, the two manifolds must be iso-spectral.introduction By a hyperbolic manifold in this paper we refer to a quotient of a real, complex, quaternionic, or octonionic hyperbolic space. Specifically, let G denote a connected semisimple Lie group of real rank one, K ⊂ G a maximal compact subgroup and H = G/K the corresponding symmetric space endowed with the hyperbolic Riemannian metric coming from the Killing form on g = Lie(G). Then for any uniform torsion free lattice Γ ⊂ G, the locally symmetric space M = Γ\H has the structure of a compact hyperbolic manifold. By the spectra of M, that we also call the spectra of Γ, we refer to any one of the following notions described below.The Laplace spectrum is the set of eigenvalues of the Laplace-Beltrami operator acting on L 2 (M) listed with multiplicities. The representation spectrum is the set of π ∈Ĝ occurring in the right regular representation of G on L 2 (Γ\G) listed with multiplicities. HereĜ denotes the unitary dual of G, that is, the set of equivalence classes of unitary representations of G. We note that the representation spectrum determines the Laplace spectrum as the multiplicity of each Laplace eigenvalue is given by the multiplicity of a corresponding representation. We say that two lattices are Laplace equivalent (or iso-spectral) if they have the same Laplace spectrum, and we say that they are representation equivalent if they have the same representation spectrum. | 10.1090/s0002-9947-2014-06102-3 | [
"https://arxiv.org/pdf/1108.2977v2.pdf"
] | 119,652,653 | 1108.2977 | 0846af7f8ddc2331edb194a96cf1b08722cb1acd |
A REFINEMENT OF STRONG MULTIPLICITY ONE FOR SPECTRA OF HYPERBOLIC MANIFOLDS
9 Sep 2011
Dubi Kelmer
A REFINEMENT OF STRONG MULTIPLICITY ONE FOR SPECTRA OF HYPERBOLIC MANIFOLDS
9 Sep 2011
Let M 1 and M 2 denote two compact hyperbolic manifolds. Assume that the multiplicities of eigenvalues of the Laplacian acting on L 2 (M 1 ) and L 2 (M 2 ) (respectively, multiplicities of lengths of closed geodesics in M 1 and M 2 ) are the same, except for a possibly infinite exceptional set of eigenvalues (respectively lengths). We define a notion of density for the exceptional set and show that if it is below a certain threshold, the two manifolds must be iso-spectral.introduction By a hyperbolic manifold in this paper we refer to a quotient of a real, complex, quaternionic, or octonionic hyperbolic space. Specifically, let G denote a connected semisimple Lie group of real rank one, K ⊂ G a maximal compact subgroup and H = G/K the corresponding symmetric space endowed with the hyperbolic Riemannian metric coming from the Killing form on g = Lie(G). Then for any uniform torsion free lattice Γ ⊂ G, the locally symmetric space M = Γ\H has the structure of a compact hyperbolic manifold. By the spectra of M, that we also call the spectra of Γ, we refer to any one of the following notions described below.The Laplace spectrum is the set of eigenvalues of the Laplace-Beltrami operator acting on L 2 (M) listed with multiplicities. The representation spectrum is the set of π ∈Ĝ occurring in the right regular representation of G on L 2 (Γ\G) listed with multiplicities. HereĜ denotes the unitary dual of G, that is, the set of equivalence classes of unitary representations of G. We note that the representation spectrum determines the Laplace spectrum as the multiplicity of each Laplace eigenvalue is given by the multiplicity of a corresponding representation. We say that two lattices are Laplace equivalent (or iso-spectral) if they have the same Laplace spectrum, and we say that they are representation equivalent if they have the same representation spectrum.
The length spectrum (respectively primitive length spectrum) is the set of lengths of closed geodesics (respectively primitive closed geodesics) in M listed with multiplicities. The complex length spectrum is the set of pairs, length and holonomy class, of all closed geodesics listed with multiplicities, where the holonomy class of a closed geodesic is the conjugacy class in SO(d − 1), d = dim(M) obtained by parallel transporting tangent vectors along the geodesic. We say that two spaces are length equivalent if they have the same length spectrum, and that they are complex length equivalent if they have the same complex length spectrum.
When M is a hyperbolic surface, Huber [Hu59] showed that the Laplace spectrum and the length spectrum determine each other (and they also determine the representation and complex length spectra). In higher dimensions the complex length spectrum and the representation spectrum still determine each other (see e.g. [Sa02]), however, the relation between the Laplace spectrum and the length spectrum is more mysterious.
In [Ga77], Gangolli generalized Huber's result to all hyperbolic manifolds, however, this generalization involves a different notion of length spectrum, we will call the 1-length spectrum, assigning to each length a certain real positive number (see remark 1.1). His result implies in particular that the Laplace spectrum of a hyperbolic manifold determines the length set (this is in fact true for all negatively curved manifolds [DG75]). However, the question of whether it determines the multiplicities, and the converse question of whether the length spectrum determines the Laplace spectrum were left open.
In [BR10], Bhagwat and Rajan showed that if two lattices satisfy that all but finitely many Laplace eigenvalues (respectively representations) have the same multiplicities, then they are Laplace equivalent (respectively, representation equivalent 1 ). In [BR11] they studied the length spectrum of real hyperbolic manifolds of even dimension, showing that if two lattices have the same multiplicities for all but finitely many lengths, then they are length equivalent. Similar results were previously shown by Elstrodt, Grunewald, and Mennicke in [EGM,Theorem 3.3] for the Laplace spectrum and 1-length spectrum of hyperbolic 3-manifolds. Elstrodt, Grunewald, and Mennicke then asked if it is possible to prove the same result when the finite set is replaced with an infinite set of sufficiently small density in some suitable sense.
In this paper we give a positive answer to the question of Elstrodt, Grunewald, and Mennicke, for the Laplace spectrum, representation spectrum, 1-length spectrum, and length spectrum. On the way, we also answer a question of Bhagwat and Rajan [BR11] and show that the Laplace spectrum of any compact hyperbolic manifold is completely determined by its length spectrum (cf. [Di89] for a similar result for the spectrum of the Laplacian on forms). The second problem of whether the Laplace spectrum determines the multiplicities in the length spectrum remains open.
Remark 0.1. The results of Bhagwat and Rajan can be thought of as an analogue of the strong multiplicity one theorem for cusp forms. This theorem states that if f and g are two Hecke new-forms for which the eigenvalues of the Hecke operators are equal at all but finitely many primes, then they are equal at all primes, and f = g; see [La76,p. 125]. (The analogy is only with the first part of the theorem, as there are examples of iso-spectral but not isometric hyperbolic manifolds; cf. [Su85,Vi80].) Continuing with this analogy, our result can be compared to Ramakrishnan's refinement [Ra94], stating that the finite set of primes in the strong multiplicity one theorem can be replaced with an infinite set, as long as it has Dirichlet density less than 1/8.
In order to describe our results we need to introduce some more notation. For any uniform lattice Γ ⊂ G and any π ∈Ĝ we denote by m Γ (π) the multiplicity of π in L 2 (Γ\G). For every ℓ ∈ (0, ∞) we denote by m Γ (ℓ) (respectively m o Γ (ℓ)) the number of closed geodesics (respectively primitive closed geodesics) of length ℓ. We also denote by ls Γ (respectively ls o Γ ) the set of lengths of (primitive) closed geodesics in M = Γ\H. Let G = NAK be an Iwasawa decomposition of G and let M denote the centralizer of A in K. For any σ ∈M let π σ,ν , ν ∈ iR and π σ,ν , ν ∈ (0, ρ) denote the unitary principle and complementary series inĜ where ρ = ρ G denotes half the sum of the positive roots for (G, A) (see section 1.1 for more details).
For any two lattices Γ 1 , Γ 2 ⊆ G and every σ ∈M we define the following spectral density function to measure the difference between the (principal part) of the representation spectrum.
(0.1) D σ (Γ 1 , Γ 2 ; T ) = ν∈iR, |ν|≤T |m Γ 1 (π σ,ν ) − m Γ 2 (π σ,ν )|.
We think of the asymptotic growth rate of these functions as T → ∞ as describing the "density" of places for which the multiplicities are different. We note that this captures more information as it also takes into account by how much the multiplicities differ. Using this notion we prove a refinement of [BR10, Theorems 1.1 and 1.2] for real rank one groups.
Theorem 1. Let G denote a real rank one group and Γ 1 , Γ 2 ⊆ G two uniform lattices without torsion.
(1) Let 1 ∈M denote the trivial representation. If
lim T →∞ D 1 (Γ 1 , Γ 2 ; T ) T = 0,
then Γ 1 and Γ 2 are Laplace equivalent.
(2) Let B ⊆M be a finite set. If
lim T →∞ D σ (Γ 1 , Γ 2 ; T ) T = 0, ∀σ ∈ B,
then Γ 1 and Γ 2 are representation equivalent.
Remark 0.2. The finite set B above can also be replaced by an infinite set satisfying a certain sparsity condition. Moreover, when M = SO(2), SO(3) or SU(2) we can naturally identifyM with N and this condition is then essentially that B is of density zero (see section 2.3).
For any two lattices Γ 1 , Γ 2 ⊆ G we also define a length density function
(0.2) D L (Γ 1 , Γ 2 ; T ) = ℓ≤T |m o Γ 1 (ℓ) − m o Γ 2 (ℓ)|,
where the sum is over ℓ ∈ ls o Γ 1 ∪ ls o Γ 2 . Our first results on the length spectrum is of a slightly different nature than the results in [BR10,BR11], in the sense that we start with (partial data) on the length spectrum and retrieve the Laplace spectrum.
Theorem 2. Let G denote a real rank one group and Γ 1 , Γ 2 ⊆ G two uniform lattices without torsion. If D L (Γ 1 , Γ 2 ; T ) e αT with α < ρ G , then Γ 1 and Γ 2 are Laplace equivalent.
Corollary. If two compact hyperbolic manifolds are length equivalent then they are Laplace equivalent.
For hyperbolic surfaces the Laplace spectrum determines the length spectrum, hence, for surfaces the threshold in Theorem 2 also implies that the two lattices are length equivalent.
Remark 0.3. It is interesting to compare this to the result of Buser [Bu92,Theorem 10.1.4] showing that there is a constant c(g, ǫ) such that if two hyperbolic surfaces of genus g and injectivity radius ≥ ǫ have the same multiplicities for all lengths ≤ c(g, ǫ), then they are length equivalent.
In higher dimensions, we do not know if the Laplace spectrum determines the length spectrum. Nevertheless, with the exception of the odd dimensional real hyperbolic spaces, if we impose a smaller threshold for the growth rate we are able to recover the length spectrum directly. Specifically, for each of the rank one groups we define the threshold
(0.3) α 0 (G) = 0 G = SO 0 (2n + 1, 1), n ∈ N 1/2 G = SO 0 (2n + 2, 1), n ∈ N 1 otherwise
With this threshold we can prove a refinement of [BR11, Theorem 1].
Theorem 3. Let G denote a real rank one group and Γ 1 , Γ 2 ⊆ G two uniform lattices without torsion. If D L (Γ 1 , Γ 2 ; T ) e αT for some α < α 0 (G), then Γ 1 and Γ 2 are length equivalent.
For odd dimensional real hyperbolic space the threshold α 0 = 0 so the statement is empty. In this case, even assuming that m o Γ 1 (ℓ) = m o Γ 2 (ℓ) for all but finitely many values of ℓ we were not able to prove that Γ 1 and Γ 2 are length equivalent. However, from Theorem 2 we know that they must be Laplace equivalent and hence must have the same length set and the same volume. Using this fact we can show Theorem 4. Let Γ 1 , Γ 2 ⊆ SO 0 (2n + 1, 1) denote two uniform lattices without torsion. If m o Γ 1 (ℓ) = m o Γ 2 (ℓ) for all ℓ ∈ {ℓ 1 , . . . ℓ k } then ℓ 1 , . . . , ℓ k are rationally dependent. In particular, if k = 1 then Γ 1 and Γ 2 are length equivalent.
We now discuss the sharpness of our thresholds for the density functions. Regarding Theorems 3 and 4, we note that there are no known examples of hyperbolic manifolds that are Laplace equivalent but not representation equivalent. It is thus possible that the correct threshold is actually the same as in Theorem 2.
For Theorem 2 we recall the Prime Geodesic Theorem [Ga77,Ma69], stating that
ℓ≤T m o Γ (ℓ) ∼ Li(e 2ρT ) ∼ e 2ρT 2ρT .
Our threshold is thus roughly the square root of the trivial bound
D L (Γ 1 , Γ 2 ; T ) ≤ 2Li(e 2ρTD L (Γ 1 , Γ 2 ; T ) Li(e 2ρT ) ≤ ǫ.
Remark 0.4. We note that the co-volumes of the lattices we use in the proof go to infinity as ǫ → 0. It is thus still possible that a positive density threshold can hold under the additional assumption that the volumes are uniformly bounded. See section 5.6 for a similar phenomenon in the analogous context of arithmetically equivalent number fields.
For the representation spectrum we suspect that our threshold is not optimal. We recall that the Weyl law for the principal spectrum is
|ν|≤T m Γ (π σ,ν ) ∼ C dim(σ)vol(Γ\G)T d ,
with d = dim(G/K) and C an explicit constant depending on G (see [MV83]). Consequently, the trivial bound for D σ (Γ 1 , Γ 2 ; T ) is O(T d ) and the correct threshold could very well be
lim T →∞ D σ (Γ 1 , Γ 2 ; T ) T d = 0,
or even a positive density threshold of the form D σ (Γ 1 , Γ 2 ; T ) ≤ cT d with c a sufficiently small constant. We note that the first condition implies the two lattices at least have the same co-volume. We conclude this introduction with a brief outline of the paper. In Section 1 we introduce some notation and recall some basic results on the spectral theory of symmetric spaces. In section 2 we give the proof of Theorem 1 using the Selberg trace formula. Our proof is similar to the original proof of [EGM,Theorem 3.3]; the new ingredient which allows us to improve on their result is the use of more general test functions in the trace formula (instead of just the heat trace). In section 3 we develop a new trace formula in which the geometric side involves the length spectrum directly. The price we have to pay is that on the spectral side, in addition to the Laplace spectrum, we have contribution from other representations occurring in L 2 (Γ\G). In section 4 we show that, by using suitable test functions in the trace formula, we can isolate the contribution of each of those representations. This enables us to prove Theorems 2,3, and 4. Finally, in section 5 we recall the construction in [LMNR] of lattices with the same length sets and use it to prove Theorem 5. plaining their construction of spaces with the same length sets. I also thank Peter Sarnak and Masato Wakayama for clarifying a few points regarding the trace formula. This work was partially supported by NSF grant DMS-1001640.
Notation and preliminaries
We write A B or A = O(B) to indicate that A ≤ cB for some constant c. We also write A ≍ B to indicate that A B A. We write A(T ) ∼ B(T ) if A(T )/B(T ) → 1 and A(T ) = o(B(T )) if A(T )/B(T ) → 0 as T → ∞.
1.1. Basic structure on symmetric spaces. Let G denote a connected semisimple Lie group of real rank one, K ⊂ G a maximal compact subgroup and H = G/K the corresponding symmetric space. That is G = SO 0 (n + 1, 1), SU(n + 1, 1), Sp(n, 1) or F II and H is real, complex, quaternionic, or octonionic hyperbolic space respectively.
Fix an Iwasawa decomposition G = NAK and let g = n⊕a⊕k denote the corresponding decomposition of the Lie algebra g. Let M, M * ⊆ K denote the centralizer and normalizer of A in K respectively. Let W = W (G, A) = M * /M denote the baby Weyl group; since we assume that dim(a) = 1 then |W | = 2 and we write W = {1, w}. We denote by Σ = Σ(G, A) the set of restricted roots for the pair (G, A) and by Σ + the set of positive restricted roots, then either Σ + = {α} or Σ + = {α, 2α}. Let ρ = ρ G denote half the sum of the positive roots, that is, ρ = (dim(n 1 ) + 2 dim(n 2 ))α 2 where n = n 1 ⊕ n 2 is the decomposition into the root spaces of α 1 and α 2 respectively. We fix (once and for all) an element H ∈ a with α(H) = 1 and for any t ∈ R we denote by a t = exp(tH) ∈ A. We can identify the dual spaces a * = R and a * C = C via ν = ν(H). With this identification ρ = dim(n 1 )+2 dim(n 2 ) 2 .
1.2. The unitary dual. LetĜ,K, andM denote the unitary duals of G, K, and M respectively. For any σ ∈M we denote by π σ,ν , ν ∈ iR the principal series representations and by π σ,ν , ν ∈ (0, ρ) the complementary series representations.
The action of M * on M (by conjugation) induces an action of the Weyl group W = {1, w} onM . We note that under this action π σ,iν = π wσ,−iν and that these are the only pairs of equivalent principal series representation. We say that a representation σ ∈M is ramified if σ = wσ and unramified otherwise, and we recall that there are unramified σ ∈M only when G = SO 0 (2m + 1, 1).
We denote byĜ c the set of equivalence classes of the principle and complementary series representations, and by S =Ĝ \Ĝ c (that is, S is the set of equivalence classes of discrete series representation, limits of discrete series, and Langland's quotients).
For any π ∈Ĝ we denote by χ π the infinitesimal character of π. We denote by Ω G the Casimir operators of G. Since G is of rank one the character χ π is uniquely determined by its value on Ω G . For π = π σ,ν ∈ G c we have that χ σ,ν (Ω G ) = ν 2 − ρ 2 + χ σ (Ω M ) where Ω M denotes the Casimir operator of M (appropriately normalized). For any Λ ∈ C we denote byĜ(Λ) the (finite set) of representations with χ π (Ω G ) = Λ and by S(Λ) = S ∩Ĝ(Λ).
1.3. Closed geodesics and conjugacy classes. For any γ ∈ Γ we denote by [γ] ∈ Γ # its conjugacy class. Since M is of negative curvature, there is a natural correspondence between conjugacy classes in Γ = π 1 (M), free homotopy classes of closed curves in M, and (oriented) closed geodesics.
We say that an element γ ∈ Γ is primitive if it cannot be written as γ = δ j for some other δ ∈ Γ; note that this only depends on the Γconjugacy class. To any [γ] ∈ Γ # , we define the primitivity index j(γ) as the unique j ∈ N such that γ = δ j with δ ∈ Γ primitive. Under the above correspondence, a closed geodesic is primitive if and only if the corresponding conjugacy class is primitive. Moreover, the primitivity index j(γ) is the number of times the geodesic wraps around itself.
Any hyperbolic γ ∈ Γ is conjugated in G to an element m γ a ℓγ ∈ MA + where A + = {a t |t > 0}. Here ℓ γ is uniquely determined by [γ] and m γ is determined up to conjugacy in M. The pair (ℓ γ , [m γ ]) is then precisely the length and holonomy of the closed geodesic corresponding to [γ].
1.4. The σ-length and representation spectra. To any irreducible representation σ ∈M we attach two function L Γ,σ : R + → R and m Γ,σ : (0, ρ) ∪ iR → N, we call the σ-length spectrum and σ-representation spectrum respectively. These two functions are closely related via the Selberg trace formula.
The σ-length spectrum is defined by
L Γ,σ (ℓ) = [γ]∈Γ # ℓγ =ℓ χ σ (m γ )ℓ 2j(γ)D(γ) ,
where χ σ is the character of σ and
(1.1) D(γ) = e ρℓγ | det((Ad(m γ a ℓγ ) −1 − I) |n )|,
is the Weyl discriminant. When σ is unramified we also define L ± Γ,σ (ℓ) = L Γ,σ (ℓ) ± L Γ,wσ (ℓ). Remark 1.1. The definition of the length spectrum of a hyperbolic 3manifold given in [EGM,Definition 3.1] coincides with what we call the 1-length spectrum, that is, the σ-length spectrum for σ = 1 the trivial representation.
In order to define the σ-representation spectrum we fix a virtual representation η = ⊕ τ ∈K a τ τ with (a τ ∈ Z almost all zeros) such that η | M = σ (respectively σ + wσ if σ is unramified). Let Λ σ,ν = χ σ,ν (Ω G ) and let S =Ĝ \Ĝ c . For π ∈ S we denote by α Γ (π) the corrected multiplicity given by
α Γ (π) = m Γ (ω) − vol(Γ\G)d ω π = ω in discrete series m Γ (π) otherwise ,
with d ω the formal degree of ω. The σ-representation spectrum is given by
(1.2) m Γ,σ (ν) = m Γ (π σ,ν ) + π∈S(Λσ,ν ) α Γ (π)[π | K ; η], where [π | K ; η] = τ a τ [π | K ; τ ].
For σ unramified we also define m ± Γ (ν) by
(1.3) m + Γ,σ (ν) = m Γ (π σ,ν ) + m Γ (π wσ,ν ) + π∈S(Λσ,ν ) m Γ (π)[π | K ; η].
and (1.4) m − Γ,σ (ν) = m Γ (π σ,ν ) − m Γ (π wσ,ν ) We note that by [Mi82, Theorem 1.2] the σ-representation spectrum does not depend on the choice of η. Also, since representations π ∈ S have a minimal K-type (see [Kn86, Chapter XV]), then for any fixed η, there are only finitely many π ∈ S for which [π | K ; η] = 0. In particular, for σ ∈M fixed, m Γ,σ (ν) = m Γ (π σ,ν ) for all but finitely many values of ν. Moreover, for σ = 1 trivial, m Γ,1 (ν) = m Γ (π 1,ν ) for all ν. For any σ ∈M we define the σ-spectral set as
S Γ,σ = {ν ∈ iR + ∪ (0, ρ)| m Γ,σ (ν) = 0}.
1.5. Trace formula attached to σ. Building on the Selberg trace formula developed in [Wal, War], Sarnak and Wakayama [SW99] derived a trace formula attached to each irreducible representation σ ∈M . They derived this formula in general for a (possibly) nonuniform lattice. We will write it down only for the simpler case of a uniform lattice without torsion. The derivation in this case is much simpler as there are no contribution from continuous spectrum or unipotent elements and the treatment of the multiplicities of discrete series is straight forward.
Remark 1.2. For the case of a uniform lattice this formula, with a special test function coming from the fundamental solution to the heat equational, was already derived in [Mi82,MV83]. We note that, in addition to the treatment of non-uniform lattices, another new features in [SW99] which is crucial for our application is the use of general test functions. See also [Di89] for a similar trace formula.
For Γ ⊆ G a uniform lattice without torsion the trace formula corresponding to σ ∈M takes the following form (see [SW99, Theorem 2 and Theorem 6.5] 2 :
• For σ ∈M ramified, for any even g ∈ C ∞ c (R),
ν k ∈S Γ,σ m Γ,σ (ν k )ĝ(iν k ) = vol(Γ\G) Rĝ (ν)µ σ (ν)dν (1.5) + ℓ∈ls Γ g(ℓ)L Γ,σ (ℓ)
whereĝ denotes the Fourier transform of g and µ σ (ν)dν is the Plancherel measure.
• For σ ∈M unramified, for any even g ∈ C ∞ c (R) we have the same formula but with m + Γ,σ , L + Γ,σ and µ + σ = 2µ σ . In addition, for any odd g ∈ C ∞ c (R) we have
ν k ∈s Γ,σ m − Γ (π σ,ν k )ĝ(iν k ) = ℓ∈ls Γ g(ℓ)L − Γ,σ (ℓ). (1.6)
Remark 1.3. For any virtual representation η = a σ σ of M, we can define m Γ,η , L Γ,η and µ η as the corresponding weighted sums. With this convention, the above trace formula holds for any virtual representation and not just the irreducible representations.
Proof of Theorem 1
Let G denote a fixed semisimple group of real rank one and Γ 1 , Γ 2 ⊂ G two uniform lattices without torsion. Throughout this section we will keep the two lattices fixed and suppress them from the notation. In particular, we will denote ∆m
σ = m Γ 1 ,σ −m Γ 2 ,σ , ∆L σ = L Γ 1 ,σ −L Γ 2 ,σ , ∆V = vol(Γ 1 \G) − vol(Γ 2 \G), ls o = ls o Γ 1 ∪ ls o Γ 2 and S σ = S Γ 1 ,σ ∪ S Γ 2 ,σ .
2.1. Density results for a fixed σ. As a first step, for each fixed σ ∈M we will use the trace formula to relate the σ-representation spectrum to the σ-length spectrum. In particular, we show that we can recover the σ-length spectrum from the σ-representation spectrum and vise versa, and moreover, to do that all we need is to know one of them up to an error of density zero.
Proposition 2.1. For any fixed σ ∈M if lim T →∞ 1 T ν∈iR |ν|<T |∆m σ (ν)| = 0,
then vol(Γ 1 \G) = vol(Γ 2 \G) and ∆L σ (ℓ) = 0 for all ℓ.
Proof. Assume first that σ is ramified. The equality of the volumes follows from Weyl's law for the principal series (see [MV83, Corollary 1]). It remains to show the equality for the σ-length spectrum. To do this we will use the trace formula with an appropriate test function.
Fix g ∈ C ∞ c (R) even and supported on [−1, 1] with g(0) = 1. Fix ℓ 0 ∈ (0, ∞) and let g T (ℓ) = g(T (ℓ − ℓ 0 )) + g(T (ℓ + ℓ 0 )) for T a large parameter. Taking the difference of the trace formulas for Γ 1 and Γ 2 applied to g T we get
(2.1) ν k ∈i(0,ρ)∪R ∆m σ (iν k )ĝ T (ν k ) = ℓ∈ls o g T (ℓ)∆L σ (ℓ).
Note that the term involving the volume cancels out.
Since the function g T is supported on
[ℓ 0 − 1 T , ℓ 0 + 1 T ]∪[−ℓ 0 − 1 T , −ℓ 0 + 1 T ]
, for T sufficiently large the right hand side of (2.1) is given by ∆L σ (ℓ 0 )(1 + g(2T ℓ 0 )) which converges to ∆L σ (ℓ 0 ) as T → ∞.
On the other hand,ĝ T (ν) = 1 Tĝ ( ν T )2 cos(νℓ 0 ) is bounded by 2 T |ĝ( ν T )| for ν ∈ R and by 2 Tĝ ( ν T ) cosh(ρℓ 0 ) for ν ∈ i(0, ρ). We can thus bound,
| ν k ∈i(0,ρ)∪R ∆m σ (iν k )ĝ T (ν k )| 1 T ν k ∈i(0,ρ) |∆m σ (iν k )| + 1 T ν k ∈R |∆m σ (iν k )||ĝ( ν k T )|.
The first (finite) sum goes to zero in the limit and for the second sum, using the fast decay ofĝ(ν)
1 1+|ν| 3 for ν ∈ R we can bound 1 T ν k ∈R |∆m σ (iν k )||ĝ( ν k T )| = = 1 T ∞ j=1 |ν k |∈[T (j−1),T j] |∆m σ (iν k )||ĝ( ν k T )| ∞ j=1 1 j 2 1 jT |ν k |≤jT |∆m σ (iν k )| .
From the assumption that 1 T |ν k |≤T |∆m σ (iν k )| → 0 as T → ∞ it is not hard to see that the above sum also goes to zero in the limit. Comparing this with the right hand side, we get that |∆L σ (ℓ 0 )| = 0.
When σ is unramified, we recall that m σ (ν) = m(π σ,ν ) = m(π wσ,−ν ) for all but finitely many values of ν. Consequently, we have that also
lim T →∞ 1 T ν∈iR |ν|<T |∆m ± σ (ν)| = 0.
Using the same argument we get that ∆L σ + ∆L wσ = 0, while a similar argument with an odd test function gives ∆L σ − ∆L wσ = 0, implying that ∆L σ = ∆L wσ = 0.
Remark 2.1. We note that the last argument using the trace formula with an odd test function is only needed when H is real hyperbolic space with dim(H) ≡ 1 (mod 4). Indeed, when dim(H) ≡ 3 (mod 4) any ramified σ ∈M satisfies χ σ (m) = χ wσ (m −1 ). Consequently, since ℓ γ = ℓ γ −1 and m γ −1 = m −1 γ , we get that in this case L Γ,σ (ℓ) = L Γ,wσ (ℓ) automatically. We note however that this is not the case when dim H ≡ 1 (mod 4). Here, χ σ (m) = χ wσ (m −1 ) and there is no obvious reason to suspect that L Γ,σ = L Γ,wσ in this case.
Proposition 2.2. For any ramified σ ∈M , if lim T →∞ 1 T ℓ∈ls o ℓ≤T |∆L σ (ℓ)| = 0, then ∆m σ (ν) = 0 for all ν ∈ (0, ρ) ∪ iR.
When σ is unramified we have the same result with ∆m + σ (ν) and ∆L + σ instead. Proof. We will write down the proof for ramified σ, the proof in the unramified case is identical.
We first show that ∆m σ (ν) = 0 for all ν ∈ (0, ρ). Indeed, if not let ν 0 ∈ (0, ρ) denote the largest element for which ∆m σ (ν 0 ) = 0 and consider the test function
g T (ν) = ν 0 sin(T ν)ĝ(ν) ν sinh(T ν 0 )ĝ(iν 0 ) ,
with g ∈ C ∞ c (R) even, supported on [−1, 1] with g(0) = 1 and satisfies thatĝ(iν 0 ) = 0. Taking the difference of the trace formulas for Γ 1 and Γ 2 we get
ν k ∈i(0,ρ)∪R ∆m σ (iν k )ĝ T (ν k ) = ∆V Rĝ T (ν)µ σ (ν)dν + ℓ∈L Γ g T (ℓ)∆L σ (ℓ) For any ν ∈ (0, ρ) with ν < ν 0 we have thatĝ T (iν) e T (ν 0 −ν) and g T (iν 0 ) = 1. For ν ∈ (0, 1) we can bound |ĝ T (ν)| T sinh(T ν 0 ) and for ν ∈ [1, ∞) we have |ĝ T (ν)| g(ν)
sinh(T ν 0 ) . Consequently we get that as T → ∞ the left hand side of the trace formula converges to ∆m σ (ν 0 ).
To estimate the right hand side, we can bound the integral
| Rĝ T (ν)µ σ (ν)dν| T sinh(ν 0 T ) 1 0 |ĝ(ν)|µ σ (ν)dν + 1 sinh(ν 0 T ) ∞ 1 |ĝ(ν)|µ σ (ν)dν T + 1 sinh(ν 0 T ) .
For the sum, note that g T is given by a convolution
g T (ℓ) = ν 0 sinh(T ν 0 )ĝ(iν 0 ) g * 1 1 T , where 1 1 T is the indicator function of [−T, T ]. In particular, g T is supported on [−T − 1, T + 1] and satisfies |g T (ℓ)| 1 sinh(ν 0 T ) implying that | ℓ j g T (ℓ j )∆L σ (ℓ j )| 1 sinh(ν 0 T ) ℓ j ≤T +1 |∆L σ (ℓ j )| T sinh(ν 0 T ) .
So as T → ∞ the right hand side goes to zero implying that ∆m σ (ν 0 ) = 0 as well.
Next we show that ∆m σ (iν) = 0 for all ν ∈ R. For this we fix ν 0 ∈ [0, ∞) and consider the function g T (ℓ) = 1 T g( ℓ T )2 cos(ℓν 0 ), so that g T (ν) =ĝ(T (ν − ν 0 )) +ĝ(T (ν + ν 0 )). Since we already showed that ∆m σ (ν) = 0 for ν ∈ (0, ρ) the difference of the trace formulas takes the form
ν k ∈R ∆m σ (iν k )ĝ T (ν k ) = ∆V Rĝ T (ν)µ σ (ν)dν + ℓ j g T,x (ℓ j )∆L σ (ℓ j )
A simple change of variables shows that the integral Rĝ T (ν)µ σ (ν)dν goes to zero as T → ∞. The sum
| ℓ j g T (ℓ j )∆L σ (ℓ j )| 1 T ℓ j ≤T |∆L σ (ℓ j )| (2.2)
also goes to zero by our assumption.
Next we estimate the left hand side. Let δ > 0 be such that
m Γ 1 ,σ (iν k ) = m Γ 2 ,σ (iν k ) = 0 for all ν k ∈ [ν 0 − δ, ν 0 + δ] \ {ν 0 }.
The left hand side of the trace formula can be written as
∆m σ (iν 0 )ĝ T (ν 0 ) + ν k ∈R\{ν 0 } ∆m σ (iν k )ĝ T (ν k )
The first term is ∆m σ (iν 0 )(1+o(1)) as T → ∞. For the rest of the sum we can bound |∆m σ | ≤ m Γ 1 ,σ + m Γ 2 ,σ and for each of the two lattices we have
ν k ∈R\{ν 0 } m Γ,σ (iν k )|ĝ T (ν k )| = |ν k −ν 0 |∈(δ,1) m Γ,σ (iν k )|ĝ T (ν k )| + ∞ j=1 |ν k −ν 0 |∈[j,j+1) m Γ,σ (iν k )|ĝ T (ν k )| Using the fast decay ofĝ we can bound |ĝ T (ν)| N 1 T j N for |ν − ν 0 | > j and |ĝ T (ν)| 1 T δ for |ν − ν 0 | ∈ (δ, 1) implying the bound ν k ∈R\{ν 0 } m Γ,σ (iν k )|ĝ T (ν k )| N 1 T |ν k −ν 0 |∈[δ,1] m Γ,σ (iν k ) + ∞ j=1 1 j N |ν k −ν 0 |∈[j,j+1] m Γ,σ (iν k ) .
Since for N sufficiently large (depending only on the dimension) the sum
∞ j=1 1 j N |ν k −ν 0 |∈[j,j+1] m Γ,σ (iν k )
converges we get that the left hand side of the formula converges to ∆m σ (iν 0 ), and since the right hand side goes to zero we have ∆m σ (iν 0 ) = 0.
Since m Γ,σ (ν) = m Γ (π σ,ν ) for all but finitely many values of ν we have that
ν∈iR |ν|≤T |∆m σ (ν)| = D σ (Γ 1 , Γ 2 ; T ) + O(1).
Thus, combining Propositions 2.1 and 2.2 we get Theorem 6. For any fixed σ ∈M the following are equivalent
(1) lim T →∞
Dσ(Γ 1 ,Γ 2 ;T ) T = 0. (2) lim T →∞ 1 T ℓ j ≤T |L Γ 1 ,σ (ℓ j ) − L Γ 2 ,σ (ℓ j )| = 0.
(3) Γ 1 and Γ 2 are σ-length equivalent, σ-representation equivalent and vol(Γ 1 \G) = vol(Γ 2 \G).
2.2.
Proof of Theorem 1. The first part is a special case of Theorem 6 with σ = 1 (recall that m Γ,1 (ν) = m Γ (π 1,ν ) is the multiplicity of the eigenvalue ρ 2 −ν 2 ). We also note that the second condition in Theorem 6 with σ = 1 gives an analogous result for the 1-length spectrum. For the second part, let B ⊂M be a finite set and assume that Dσ(Γ 1 ,Γ 2 ;T ) T → 0 for all σ ∈M \ B. From Theorem 6 we get that vol(Γ 1 \G) = vol(Γ 2 \G) and L Γ 1 ,σ = L Γ 2 ,σ for all σ ∈M \ B. We will show that Γ 1 and Γ 2 have the same complex length spectrum and are hence representation equivalent.
Fix an arbitrary length ℓ 0 ∈ (0, ∞) and holonomy m 0 ∈ M. Since there are only finitely many geodesics of length ℓ 0 we can find
B ⊂ M # a small open neighborhood of [m 0 ] satisfying that {[m γ ]|ℓ γ = ℓ 0 } ∩ B ⊆ {[m 0 ]}. Let F ∈ C ∞ (M) denote a smooth class function supported on B satisfying that F (m 0 ) = 1 and that F (σ) = M F (m)χ σ (m)dµ(m) = 0,
for all σ ∈ B. Since these are finitely many conditions we can clearly find such a function. We then have an expansion
F = σ∈MF (σ)χ σ , with σ∈M |F (σ)| < ∞,
absolutely converges andF (σ) = 0 for all σ ∈ B.
Next let g ∈ C ∞ c (R) be even and supported on a small enough neighborhood of ℓ 0 such that ls Γ 1 ∪ ls Γ 2 intersects its support at {ℓ 0 } and satisfy g(ℓ 0 ) = ℓ −1 0 D(m 0 a ℓ 0 ). We then have
[γ],ℓγ=ℓ 0 mγ =m 0 1 j(γ) = ℓ∈LS Γ g(ℓ)ℓ [γ] ℓγ =ℓ F (m γ ) j(γ)D(γ)
.
Expand F (m γ ) and change the order of summation (note that all series converges absolutely) to get
[γ],ℓγ=ℓ 0 mγ =m 0 1 j(γ) = 2 σ∈MF (σ) ℓ∈LS Γ g(ℓ)L Γ,σ (ℓ).
Since the right hand side is the same for Γ = Γ 1 and Γ = Γ 2 then so is the left hand side, implying that
[γ]∈Γ # 1 (ℓγ ,mγ )=(ℓ 0 ,m 0 ) 1 j Γ 1 (γ) = [γ]∈Γ # 2 (ℓγ ,mγ )=(ℓ 0 ,m 0 ) 1 j Γ 2 (γ) .
Since this is true for any pair (ℓ 0 , m 0 ) we get that the complex length spectrum is the same.
2.3. Further refinement. We note that the above proof will still work if we replace the finite set B with an infinite set, as long as it is sufficiently sparse so that for any small neighborhood B ⊂ M # we can find a smooth class function F supported on B withF (σ) = 0 for all σ ∈ B.
When M = SO(2) we can identifyM with Z where χ σn (θ) = e inθ while for M = SO(3) or SU(2) we can identifyM with N where χ σn (θ) = sin(nθ)
sin(θ) . For these cases the following lemma shows that we can take the set B to be any set satisfying that
(2.3) #{n ∈ B : |n| ≤ T } T α , with α < 1. Lemma 2.3. Let B ⊂ Z satisfy (2.3).
Then for any δ > 0 and
θ 0 ∈ [0, 2π) there is f ∈ C ∞ (Z/2πZ) supported on (θ 0 − δ, θ 0 + δ) with f (θ 0 ) = 1 andf (n) = 0 for all n ∈ B.
Proof. Since the Fourier transform of f (θ − θ 0 ) vanishes together with the Fourier transform of f (θ) we may assume that θ 0 = 0. Also, since we may always renormalize, it is enough to show that there is f ∈ C ∞ c (R) not identically zero that is supported on (−δ, δ) withf (n) = 0 for n ∈ B.
Let β ∈ (α, 1) and let f 1 ∈ C ∞ c (R) be smooth and supported on (−δ/2, δ/2) with Fourier transform satisfyingf 1 (ξ) e −C|ξ| β for ξ ∈ R. We note that a compactly supported function with this decay exists for any β ∈ (0, 1) (see e.g. [Au06, Lemma 6]) and we can make the support as small as we want by dilation. The support of f 1 implies thatf 1 can be extended to an entire function on the complex plane of exponential growth |f 1 (z)| exp(δ|z|/2).
Next consider the function defined by the infinite product
f 2 (z) = n∈B (1 − z n ).
The condition (2.3) implies that the product converges (uniformly on compacta) to an entire function with exponential growth |f 2 (z)| e C|z| α . This function clearly vanishes on B (but it is not the Fourier transform of a compactly supported function). We now definef (z) =f 1 (z)f 2 (z), thenf is entire of exponential growth |f (z)| e δ|z|/2+C|z| α e δ|z| , and it still vanishes on B. Moreover, on the real line it decays like
|f (ξ)| e C|ξ| α −|ξ| β N 1 |ξ| N .
Consequently, the Paley Wiener Theorem (see [Ru66,Theorem 19.3]) implies thatf is the Fourier transform of a smooth function f that is supported on (−δ, δ).
Alternating trace formula
In the proof of Theorem 1 we used the fact that the σ-representation spectrum appearing in the trace formula is essentially given by the multiplicities of the principal series representations. The relation between the σ-length spectrum and the length spectrum is not that straight forward (even for trivial σ). In this section we develop a new formula (which is an alternating sum of trace formulas corresponding to certain virtual representations of M) where the geometric side involves the length spectrum directly. Precisely we show Theorem 7. Let G denote a real rank one group and Γ ⊂ G a uniform lattice without torsion. Let m = ρ G − α 0 (G) ∈ N, then there are m + 1 virtual representations η 0 , η 1 , . . . , η m of M, with η 0 the trivial representation such that for any even g ∈ C ∞ c (R) we have
1 2 ℓ∈pls Γ ℓm o Γ (ℓ) ∞ j=1 g(jℓ) ψ(jℓ) = m q=0 (−1) q ν k ∈S Γ,σ m Γ,ηq (ν k )ĝ q (iν k ) − vol(Γ\G) Rĝ q (ν)µ ηq (ν)dν ,
whereĝ q (ν) =ĝ(ν + i(m−q)) +ĝ(ν + i(q −m)), and the weight function ψ is given by
(3.1) ψ(ℓ) = 1 G = SO 0 (2n + 1, 1) 2 sinh( ℓ 2 ) G = SO 0 (2n + 2, 1) 2 sinh(ℓ) otherwise
Proof. In order to derive this alternating formula we expand the Weyl discriminant (1.1) as a sum of characters of representations of M. Specifically, we will show in Proposition 3.5 below that
(3.2) D(γ) = ψ(ℓ γ ) m q=0 (−1) q (e (m−q)ℓγ + e (q−m)ℓγ )χ ηq (m γ ),
with η 0 , . . . , η m certain virtual representations of M with η 0 trivial. Now, since any γ ∈ Γ can be written in a unique way as γ = δ j with δ ∈ Γ primitive, we have
ℓ∈ls o Γ ℓm o Γ (ℓ) ∞ j=1 g(jℓ) ψ(jℓ) = ℓ∈ls Γ g(ℓ) ψ(ℓ) [γ] ℓγ=ℓ ℓ j(γ) = ℓ∈ls Γ g(ℓ) ψ(ℓ) [γ] ℓγ=ℓ ℓD(γ) j(γ)D(γ)
Plugging in the expansion for D(γ) = D(γ) from (3.2) in the numerator we get
ℓ∈ls o Γ ℓm o Γ (ℓ) ∞ j=1 g(jℓ) ψ(jℓ) = m q=0 (−1) q ℓ∈ls Γ g q (ℓ) [γ] ℓγ =ℓ ℓχ ηq (m γ ) j(γ)D(γ) = 2 m q=0 (−1) q ℓ∈ls Γ g q (ℓ)L Γ,ηq (m γ )
where g q (ℓ) = g(ℓ)(e (m−q)ℓ + e (q−m)ℓ ). We conclude the proof by applying the trace formula attached to each of the virtual representation η q separately.
Remark 3.1. For the odd dimensional orthogonal groups the representations η 1 , . . . , η m are actual irreducible representation. For the other groups, it is also possible to obtain a similar formula using irreducible representations instead of virtual representations by using appropriate linear combinations of theĝ q 's.
The rest of this section will be devoted to the proof of the expansion (3.2) for the Weyl discriminant.
3.1. The Adjoint representation. In order to expand the Weyl discriminant
D(γ) = e ρℓγ | det((Ad(m γ a ℓγ ) −1 − I) |n )|,
as a sum of characters we first need to understand Ad(MA) and in particular its restriction to n 1 and n 2 . Denote by n 1 = dim n 1 and n 2 = dim n 2 . From the definition of n 1 and n 2 , for a ℓ = exp(ℓH) ∈ A we have that Ad(a ℓ ) |n 1 = e ℓ I and Ad(a ℓ ) |n 2 = e 2ℓ I. The following proposition describes the action of Ad(M). Proof. For the orthogonal group this is clear. For the unitary group G = SU(n + 1, 1), we write its Lie algebra as
g = a b b * d |a ∈ Mat n+1 (C), b ∈ C n+1 , a * = −a, d = −Tr(a) ,
and the Lie algebras in the Cartan decomposition g = p⊕k are given by
p = 0 b b * 0 ∈ g , and k = a 0 0 d ∈ g . We write any b ∈ C n+1 as b = v c
with v ∈ C n and c ∈ C. Then a ⊂ p is the real subspace
a = 0 b b * 0 |b = 0 c
, c ∈ R and the root spaces n 1 and n 2 are given by
n 1 = X v = 0 v v −v * 0 0 v * 0 0 v ∈ C n ,
of dimension n 1 = 2n and
n 2 = 0 0 0 0 −d d 0 −d d d ∈ C, d + d * = 0 ,
which is one dimensional. We get that M, the centralizer of A in K, is given by
M = u 0 0 0 x 0 0 0 x u ∈ U(n), x ∈ U(1), x 2 det(u) = 1 .
A simple computation then shows that the adjoint action of M is trivial on n 2 and that Ad(m)X v = mX v m −1 = X x * uv . Fixing the standard basis X e j , X ie j , j = 1, . . . , n for n 1 (recalling the natural inclusion U(n) ⊆ SO(2n)) we get a homomorphism from M to SO(2n). For G = Sp(n, 1) repeating the same arguments replacing C with the quaternions H we get
n 1 = X v = 0 v v −v * 0 0 v * 0 0 v ∈ H n−1 ,
of dimension n 1 = 4(n − 1),
n 2 = Y d = 0 0 0 0 −d d 0 −d d d ∈ H, d + d * = 0 ,
of dimension n 2 = 3, and
M = u 0 0 0 x 0 0 0 x u ∈ Sp(n − 1), x ∈ Sp(1) det(u)x 2 = 1 .
On n 1 we have Ad(m)X v = X uvx * and fixing a suitable basis for H n ∼ = C 2n ∼ = R 4n gives a homomorphism from M to SO(4(n − 1)). On n 2 the action is given by Ad(m)Y d = Y xdx * and choosing a suitable basis this gives a homomorphism into SO(3). Finally, for G = F II we have that n 1 is 8-dimensional and n 2 is 7-dimensional. The smallest irreducible representation of M = Spin (7) is the orthogonal representation which is 7-dimensional and the only 8-dimensional irreducible representation is the spin representation. Let t denote a maximal commutative subspace of m so that h = t ⊕ a is a Cartan algebra for g. Since the root spaces for h are one dimensional, any subspace of n j on which ad(t) acts trivially is at most one dimensional. Consequently, the action of ad(m) is not trivial on n 1 and n 2 , and hence the restriction of Ad(M) to n 2 gives the orthogonal representation and the restriction to n 1 is either the spin representation or a direct sum of the orthogonal and trivial representation. Finally, we note that the latter cannot hold as it would imply that the center of M acts trivially on n.
Remark 3.2. For the unitary and symplectic groups the homomorphism ι 1 : M → SO(n 1 ) is not the standard inclusion U(n) ⊂ SO(2n) (respectively, Sp(n − 1) ⊆ SO(4(n − 1)). In particular, when n = 2 this homomorphism has a nontrivial kernel. The homomorphism ι 2 is trivial for the unitary group and factors through the natural isomorphism Sp(1) ∼ = SO(3) for the symplectic group.
Corollary 3.1. The Weyl discriminant can be factored as D(γ) = | det(e −ℓγ /2 I n 1 − e ℓγ /2 ι 1 (m γ ))|| det(e −ℓγ I n 2 − e ℓγ ι 2 (m γ ))|.
3.2.
Representations of the orthogonal groups. Next, we want to expand each one of these determinants as a linear combination of irreducible characters. To do this we recall some facts about the representation theory of the orthogonal groups that we will need. We refer to [Kn02, Chapter V] for background and more details on representation theory of compact groups.
The irreducible representations of the orthogonal group are parameterized by their highest weights: When n = 2m the possible highest weighs are so(2m) = { j a j e j : a 1 ≥ . . . ≥ a n−1 ≥ |a n |, a i − a j ∈ Z, 2a j ∈ Z}, and when n = 2m + 1 they are so(2m + 1) = { j a j e j : a 1 ≥ . . . ≥ a n ≥ 0, a i − a j ∈ Z, 2a j ∈ Z}.
There is a one to one correspondence between these highest weights and the irreducible representations of the simply connected Lie group Spin(n); a representation of Spin(n) factors through a representation of SO(n) if and only if all the coefficient are integral.
For any θ ∈ (Z/2πZ) m let
u θ = R(θ 1 ) . . . R(θ m ) ∈ SO(2m), with R(θ j ) = cos(θ j ) sin(θ j )
− sin(θ j ) cos(θ j ) . When n = 2m is even, the map θ → u θ is an isomorphism of (Z/2πZ) m with the maximal torus of SO(2m). When n = 2m+ 1 is odd this isomorphism is given by θ →ũ θ = u θ 0 0 1 . For any weight λ, let χ λ denote the character of the irreducible representation of highest weight λ (that we may think of as a function on (Z/2πZ) m by restriction to the maximal torus). We recall the Weyl character formula:
(3.3) χ λ (θ) = 1 D(θ) w∈W sgn(w)ξ w(λ+δ) (θ),
where ξ λ (θ) = exp(i j a j θ j ), W is the Weyl group of SO(n), D is the Weyl discriminant of SO(n) given by
D(θ) = ξ δ (θ) α∈∆ + (1 − ξ −α (θ)) = w∈W sgn(w)ξ wδ (θ),
and δ = 1 2 α∈∆ + α is half the sum of the positive roots. For any fixed ℓ ∈ R consider the function F n (ℓ, ·) : SO(n) → R, F (ℓ, u) = | det(e −ℓ/2 I n − e ℓ/2 u)|. This is clearly a class function on SO(n) and we can write it as a linear combination of irreducible characters.
Proposition 3.2. When n = 2m + 1 is odd
(3.4) F n (ℓ, u) = 2 sinh( ℓ 2 ) m q=0 (−1) q m−q k=q−m e kℓ χ τq (u)
where τ q denotes the irreducible representation of SO(2m+1) of highest weight e 1 + . . . + e q When n = 2m is even where σ q , 0 ≤ q < m denotes the irreducible representation of SO(2m) of highest weight e 1 + . . . + e q , σ m = σ + m ⊕ σ − m with σ ± m of highest weight e 1 + . . . + e m−1 ± e m . In the second line, χ τ is evaluated at u ∈ SO(2m) via the natural inclusion of SO(2m) ⊆ SO(2m + 1).
Proof. We first treat the even case. Evaluating F 2m (ℓ, ·) at u θ we get
F 2m (ℓ, u θ ) = | det(e −ℓ/2 I 2m + e ℓ/2 u θ )| = m j=1 2 cosh(ℓ) − 2 cos(θ j ) = m k=0 (−1) k (2 cosh(ℓ)) m−k S m,k (θ) where (3.6) S m,k (θ) = 1≤j 1 ≤...≤j k ≤m k i=1 2 cos(θ j i ) .
We can think of the functions S m,k as class functions on SO(2m) and write them as a linear combination of irreducible characters. To do this, for any n, k ∈ N with 2k ≤ n we define (3.7)
N 0 (n, k) = #{(j 1 , . . . , j k ) ∈ {1, . . . , n − 1} k |j i+1 ≥ j i + 2}
(3.8) N(n, k) = 1 k = 0 N 0 (n, k) + N 0 (n − 2, k − 1) k ≥ 1
We show in Lemma 3.4 below that
(3.9) S m,k = [k/2] j=0 (−1) j N(m + 2j − k, j)χ σ k−2j .
Plugging this in the above expression we get
F 2m (ℓ, u θ ) = m k=0 (−1) k (2 cosh(ℓ)) m−k S m,k = m k=0 [k/2] j=0 (−1) k+j (2 cosh(ℓ)) m−k N(m + 2j − k, j)χ σ k−2j = (−1) m m k=0 [k/2] j=0 (−1) j−k (2 cosh(ℓ)) k−2j N(k, j) χ σ m−k
The result now follows from the identity
[k/2] j=0 (−1) j (2 cosh(ℓ)) k−2j N(k, j) = 2 cosh(kℓ),
which can be easily proved by induction from the recursion relation
(3.10) N(n, k) = N(n − 1, k) + N(n − 2, k − 1), ∀ 1 ≤ k ≤ n − 1 2 ,
together with the simple observation that N(2k, k) = 2 for all k ≥ 1. The second line of (3.5) is a consequence of the relation
(3.11) χ τ k = χ σ 0 k = 0 χ σ k + χ σ k−1 1 ≤ k ≤ m .
which follows from the decomposition of the restriction of τ k to SO(2m) into irreducible representations of SO(2m).
Now for the odd case, evaluating F 2m+1 (ℓ, ·) onũ θ = u θ 0 0 1 gives
F 2m+1 (ℓ,ũ θ ) = | det(e −ℓ/2 I 2m+1 − e ℓ/2ũ θ )| = 2 sinh(ℓ/2) m j=1 2 cosh(ℓ) − 2 cos(θ j )
= 2 sinh(ℓ/2)F 2m (ℓ, u θ ), and the result follows directly from the even case. We note that using (3.11) we can also write
F 2m+1 (ℓ,ũ θ ) = 2 sinh(ℓ/2) m q=0 (−1) q (e (m−q)ℓ + e (q−m)ℓ )χ ηq (u)
with η q the virtual representation of SO(2m + 1) whose restriction to SO(2m) is σ q .
We still need to prove the identity (3.9) and the recursion (3.10).
Lemma 3.3. The combinatorial terms N(n, k) defined in (3.8) satisfy
N(n, k) = N(n − 1, k) + N(n − 2, k − 1),
for all 1 ≤ k ≤ n−1 2 . Proof. In the definition of N 0 (n, k), for any choice of j 1 ∈ {1, . . . , n − 2k + 1} there are N 0 (n − j − 1, k − 1) choices for j 2 , . . . , j k , leading to the recursion relation N 0 (n, k) = n−2k+1 j=1 N 0 (n − j − 1, k − 1). Now, for k = 1, N(n, 1) = n = n − 1 + 1 = N(n − 1, 1) + N(n − 2, 0) is trivial and for k ≥ 2 substituting N(n, k) = N 0 (n, k) + N 0 (n − 1, k − 2) we get
N(n, k) = n−2k+1 j=1 N(n − j − 1, k − 1) = n−2 j=2(k−1) N(j, k − 1), implying that N(n, k) − N(n − 1, k) = N(n − 2, k − 1).
Lemma 3.4. The functions S m,k defined in (3.6) satisfy
S m,k = [k/2] j=0 (−1) j N(m + 2j − k, j)χ σ k−2j ,
(as class functions on SO(2m)).
Proof. To simplify notation we denote by χ k = χ σ k and χ ± m = χ σ ± m . Let W denote the Weyl group of SO(2m), which acts on the weights e 1 , . . . , e m by permutations and even sign changes. We recall the Weyl character formula
χ k (u) = 1 D(u) w∈W sgn(w)ξ w(δ+e 1 +...+e k ) (u),
where ξ e j (u θ ) = e iθ j , δ = m j=1 (m − j)e j is half the sum of the positive roots, and D(u) = w∈W sgn(w)ξ wδ (u) denotes the Weyl discriminant of SO(2m).
We first prove the formula for k < m. Let
W + = {w ∈ W |sgn(w) = 1}
denote the subgroup of positive elements and observe that we can write
S m,k = 1 N k s∈W + ξ s(e 1 +...+e k ) ,
where N k = #{w ∈ W + |w(e 1 + . . . + e k ) = e 1 + . . . + e k }.
Multiplying by the Weyl discriminant we get
DS m,k = 1 N k w∈W sgn(w)ξ wδ s∈W + ξ s(e 1 +...+e k ) = λ∈E k w∈W sgn(w)ξ w(δ+λ)
where E k = {±e j 1 ± . . . ± e j k |1 ≤ j 1 < . . . < j k ≤ m}. To get this last equality, for each λ ∈ E k and w ∈ W we collected together the N k elements s ∈ W + satisfying that s(e 1 + . . . + e k ) = wλ. We can write any λ ∈ E k as λ = m j=1 µ j e j with µ j ∈ {−1, 0, 1} (with k nonzero entries). We note that if µ j+1 = µ j + 1 for some 0 ≤ j ≤ m − 1 or µ j+2 = µ j + 2 for some 0 ≤ j ≤ m − 2 then the transposition w j,j+1 (respectively w j,j+2 ) of e j and e j+1 (respectively e j+2 ) sends δ + λ to itself. This implies that for any such weight λ the sum w∈W sgn(w)ξ w(δ+λ) = 0. Consequently, the only weights λ ∈ E k that contribute to the sum are of the form
(3.12) λ = e 1 + . . . + e k−2ℓ − e j 1 + e j 1 +1 + . . . − e j ℓ + e j ℓ +1 , for some 0 ≤ ℓ ≤ k 2 where j 1 , . . . , j ℓ satisfy k − 2ℓ + 1 ≤ j 1 < j 1 + 1 < j 2 < . . . < j ℓ ≤ m − 1 or of the form (3.13) λ = e 1 + . . . + e k−2ℓ − e j 1 + e j 1 +1 + . . . − e j ℓ − e j ℓ +1 ,
for some 0 ≤ ℓ ≤ k 2 where j 1 , . . . , j ℓ satisfy k − 2ℓ + 1 ≤ j 1 < j 1 + 1 < j 2 < . . . < j ℓ = m − 1.
For each λ satisfying (3.12) or (3.13) let w λ denote the composition of the transpositions w j i ,j i +1 , i = 1, . . . , ℓ. We then have that sgn(w λ ) = (−1) ℓ and w λ (δ + λ) = δ + e 1 + . . . + e k−2ℓ , implying that 1 D w∈W ξ w(ρ+λ) = (−1) ℓ χ k−2ℓ . Now, for each 1 ≤ ℓ ≤ k 2 there are precisely N 0 (m − k + 2ℓ, ℓ) weights λ ∈ E k satisfying (3.12) and N 0 (m − k + 2ℓ − 2, ℓ − 1) weights satisfying (3.13). We can thus conclude that for any k < m
S m,k = [k/2] ℓ=0 (−1) ℓ N(m + 2ℓ − k, ℓ)χ k−2ℓ
For k = m we need to make small adjustments to the argument as there is no w ∈ W such that s(e 1 + . . . + e m ) = e 1 + . . . + e m−1 − e m . Let W 0 denote the group of all sign changes (this is not a subgroup of the Weyl group as we allow odd sign changes as well). We can write where E m = {±e 1 ± . . . ± e m } as before.
In this case, we get a contribution from all λ ∈ E m satisfying (3.12) and (3.13), but also from the weight λ = e 1 + . . . + e m−1 − e m , giving the formula
S m,m = χ − m + χ + m + [m/2] ℓ=1 (−1) ℓ N(2ℓ, ℓ)χ m−2ℓ = [m/2] ℓ=0 (−1) ℓ N(2ℓ, ℓ)χ m−2ℓ
3.3. The Weyl discriminant. Combining these results we get the following expression for the Weyl discriminant. Proof. We prove it separately for the orthogonal, unitary, symplectic and exceptional groups.
Orthogonal groups: For G = SO 0 (n + 1, 1) we have that ρ = n 2 and m = [ n 2 ]. The formula follows immediately from Corollary 3.1 and Proposition 3.2 where η q = σ q when n = 2m, and it is the virtual representation of SO(2m + 1) whose restriction to SO(2m) is σ q when n = 2m + 1.
Unitary groups: For G = SU(n + 1, 1) we have ρ = n + 1 and m = n. Here, Corollary 3.1 implies that D(γ) = F 2n (ℓ γ , ι 1 (m γ ))|2 sinh(ℓ γ )|, with ι 1 : M → SO(2n) as in Proposition 3.1. The result follows from Proposition 3.2 with η q = σ q • ι 1 .
Symplectic groups: For G = Sp(n, 1) we have that ρ = 2n + 1, m = 2n and the formula in Corollary 3.1 is
D(γ) = F 4(n−1) (ℓ γ , ι 1 (m γ ))F 3 (2ℓ γ , ι 2 (m γ )),
where ι 1 : M → SO(4n − 4) and ι 2 : M → SO(3) are as in Proposition 3.1. Let σ 0 , . . . , σ 2n−2 and τ 0 , τ 1 denote the representations of SO(4n − 4) and SO(3) appearing in (3.5) and (3.4). Consider the representations of M = Sp(n − 1) given byσ q = σ q • ι 1 andτ q = τ q • ι 2 . Using the expansions (3.4) and (3.5) for F 3 (2ℓ, ·) and F 4(n−1) (ℓ, ·) we get
D(γ) = 2 sinh(ℓ γ ) m−2 q=0 (−1) q (e (m−q)ℓγ + e (q−m)ℓγ )χσ q (m γ ) + m−2 q=0 (−1) q (e (m−4−q)ℓγ + e (q−m+4)ℓγ )χσ q (m γ ) + m−2 q=0 (−1) q (e (m−2−q)ℓγ + e (q−m+2)ℓγ )χσ q (m γ ) − m−2 q=0 (−1) q (e (m−2−q)ℓγ + e (q−m−2)ℓγ )χσ q ⊗τ 1 (m γ ) = 2 sinh(ℓ γ ) m q=0 (−1) q (e (m−q)ℓγ + e (q−m)ℓγ )χ ηq (m γ )
with η 0 the trivial representation and η q the virtual representation
η q = σ 1 q = 1 σ q +σ q−2 −σ q−2 ⊗τ 1 2 ≤ q < 4 σ q +σ q−2 +σ q−4 −σ q−2 ⊗τ 1 4 ≤ q ≤ m − 2 2σ m−3 +σ m−5 −σ m−3 ⊗τ 1 q = m − 1 σ m−4 +σ m−2 −σ m−2 ⊗τ 1 q = m
Exceptional group: For G = F II we have ρ = 11, m = 10 and we can write
D(γ) = F 8 (ℓ γ , ι 1 (m γ ))F 7 (2ℓ γ , ι 2 (m γ )),
where ι 1 : M → GL 8 (C) and ι 2 : M → SO(7) are the spin and orthogonal representations of M = Spin(7). Note that the function F 8 (ℓ, ·) is actually a class function on GL 8 (C) so this makes sense. However, ι 1 (m γ ) is not in the orthogonal group so we can't use Proposition 3.2 directly. Nevertheless, since the weights of the spin representations are {sλ| s ∈ W 0 = {±1} 3 } with λ = 1 2 (e 1 + e 2 + e 3 ) we have F 8 (ℓ, ι 1 (·)) = s∈W 0 (e −ℓ/2 − e ℓ/2 ξ sλ ) = 2 cosh(4ℓ) − 2 cosh(3ℓ) s∈W 0 ξ sλ + 2 cosh(2ℓ)
{s 1 ,s 2 }⊆W 0 ξ (s 1 +s 2 )λ − 2 cosh(ℓ)
{s 1 ,s 2 ,s 3 }⊆W 0 ξ (s 1 +s 2 +s 3 )λ + {s 1 ,...,s 4 }⊆W 0 ξ (s 1 +s 2 +s 3 +s 4 )λ
Using similar augments to the ones used in the proof of Lemma 3.4 we identify each of the above sums as a linear combination of irreducible characters of Spin(7) to get F 8 (ℓ, ι 1 (·)) = 2 cosh(4ℓ) − 2 cosh(3ℓ)χ λ + 2 cosh(2ℓ)(χ e 1 +e 2 + χ e 1 + 1) −2 cosh(ℓ)(χ λ + χ λ+e 1 ) + 2(χ 2e 1 + χ e 1 +e 2 +e 3 + χ e 1 +e 2 + χ e 1 + 1) For F 7 (2ℓ γ , ι 2 (m γ )) we can use Proposition 3.2 directly to get
F 7 (2ℓ γ , ι 2 (m γ )) = 2 sinh(ℓ) 3 q=0 3−q k=q−3 e 2kℓγ χ τ k (m γ ).
Multiplying the two expansions and collecting together terms with the same coefficients as we did for the symplectic groups we get that indeed D(γ) = 2 sinh(ℓ γ ) 10 q=0 (e (10−q)ℓγ + e (q−10)ℓγ )χ ηq (m γ ), with η 0 trivial and η q , q = 1, . . . , 10 appropriate virtual representations of M.
Proof of Theorems 2, 3 and 4
We can now use the alternating trace formula to relate the length spectrum with some combination of the representation spectrum. The remarkable point is that we can actually retrieve each one of the multiplicities m Γ,ηq appearing in the formula (rather than just their combination). The key to retrieving the individual multiplicities is the fact that when dilating the functionsĝ q (T ν) they grow exponentially in T and different values of q correspond to different rates of exponential growth. Theorems 2, 3 and 4 will all follow from the following result.
Theorem 8. Let G be as above and Γ 1 , Γ 2 ⊆ G two uniform torsion free lattices. Assume that D L (Γ 1 , Γ 2 ; T ) satisfies
(4.1) D L (Γ 1 , Γ 2 ; T ) e αT ,
for some α ∈ [0, ρ). Then m Γ 1 ,ηq = m Γ 2 ,ηq for all q < ρ − α.
Proof. We retain the notation m = ρ − α 0 with α 0 ∈ {0, 1 2 , 1} (depending on G). We note that the weight function ψ in the alternating trace formula satisfies ψ(ℓ) ≍ e α 0 ℓ for large ℓ > 0.
Assume by contradiction that m Γ 1 ,ηq = m Γ 2 ,ηq for some q < ρ − α and let q 0 denote smallest such q. We first show that the bound (4.1) implies that an appropriate grouping together of the complementary series must cancel. To do this we consider the difference between the alternating trace formulas for Γ 1 and Γ 2 ,
1 2 ℓ∈ls o ℓ∆m o (ℓ) ∞ j=1 g(jℓ) ψ(jℓ) = m q=0 (−1) q+1 ∆V Rĝ q (ν)µ ηq (ν)dν + m q=0 (−1) q ν k ∈R ∆m ηq (iν k )ĝ q (ν k ) + m q=0 (−1) q ν k ∈(0,ρ) ∆m ηq (ν k )ĝ q (iν k )
We can rewrite the third sum as m q=0 (−1) q ν k ∈(0,ρ)
∆m ηq (ν k )ĝ q (iν k ) = m q=0 (−1) q ν k ∈(0,ρ) ∆m ηq (ν k )(ĝ(i(ν k + q)) +ĝ(i(ν k − q))) = x kĝ (ix k ) m q=0 (−1) q (∆m ηq (x k − q) + ∆m ηq (x k − q)) = x kĝ (ix k )C(x k ) where C(x) = m q=0 (−1) q (∆m ηq (x − q) + ∆m ηq (x − q)).
Note that C(x) = 0 only on a finite (possibly empty) set which is contained in (−m, ρ + m). We show that in fact C(x) = 0 for all |x| > m − q 0 . Otherwise, let x 0 ∈ (−m, ρ + m) with |x 0 | > m − q 0 be such that C(x 0 ) = 0 and that C(x) = 0 for all |x| > |x 0 | (we can find such a point as C(x) = 0 on a finite set).
Since we assume that q 0 < m + α 0 − α and that |x 0 | > m − q 0 , we get that α − |x 0 | < α 0 and we can find some c ∈ [0, 1) such that α − |x 0 | < α 0 c < α 0 (where c = 0 iff α 0 = 0). We now estimate the two sides of the alternating trace formula with the test function
g T (ℓ) = x 0ĝ (ix 0 ) sinh(T x 0 ) (g * 1 1 cT,T )(ℓ),
where 1 1 cT,T denotes the indicator function of
[−T, −cT ] ∪ [cT, T ].
Using the fact that |g T (ℓ)| e −T |x 0 | for ℓ ∈ [cT − 1, T + 1] and g T (ℓ) = 0 for ℓ ∈ [cT − 1, T + 1] we can bound the left hand side of the trace formula by some constant times
e −T |x 0 | ℓ≤T ℓ|∆m o (ℓ)| T /ℓ k= cT ℓ e −α 0 kℓ e −T (|x 0 |+α 0 c) ℓ≤T |∆m o (ℓ)| e −T (|x 0 |+α 0 c−α) ,
which goes to zero as T → ∞. For the right side of the trace formula, for any ν ∈ R we can bound
|ĝ T,q (ν)| T e T (m−q−|x 0 |) |ĝ q (ν)|
implying that for all q ≥ q 0 as T → ∞ ν k ∈R |∆m ηq (iν k )||ĝ T,q (ν k )| → 0.
If q 0 > 0 then the minimality of q 0 implies that ∆m η 0 = 0, and since η 0 = 1 is the trivial representation we get that ∆V = 0 (e.g., from Weyl's law). In the case when q 0 = 0, then m − |x 0 | < 0 and the bound |ĝ T,q (ν)| T e T (m−|x 0 |) |ĝ q (ν)| implies that Rĝ T,q (ν)µ q (ν)dν → 0. Consequently, the only part of the right hand side of the formula that does not vanish in the limit is the finite sum |x k |≤|x 0 |ĝ T (ix k )C(x k ) which converges to C(x 0 ). Since the left hand side of the formula goes to zero we get that C(x 0 ) = 0 in contradiction.
Next we show that ∆m σq 0 (iν) = 0 for ν ∈ R. To do this we fix ν 0 ∈ (0, ∞) and estimate both sides of the alternating trace formula with the test function
g T (ℓ) = g( ℓ T ) cos(ℓν 0 ) Tĝ(iT (m − q 0 )) ,
where g ∈ C ∞ (R) is even and supported on (−1, −c) ∪ (c, 1) with c ∈ [0, 1) satisfying that q 0 − m + α < cα 0 < α 0 (again c = 0 iff α 0 = 0). We note that since g is supported on (−1, −c) ∪ (c, 1) we can estimate e c ′ T ĝ(iT ) e T for any c < c ′ < 1.
For the left hand side, using (4.1) together with the bound
|g T (ℓ)| |g( ℓ T )| Tĝ(iT (m − q 0 )) c ′ e −T (m−q 0 )c ′ T ,
for any c ′ ∈ (c, 1) and the fact that g(ℓ) = 0 for ℓ ∈ (−c, c), the same argument as above gives a bound of O(e −T ((m−q 0 )c ′ +α 0 c−α) ). Since q 0 < m − α + α 0 c we can choose c ′ sufficiently close to one to insure that the left hand side goes to zero as T → ∞.
Next we estimate the right hand side. For
x ∈ (−m, ρ + m), if |x| < m − q 0 , then |ĝ T (ix)| =ĝ (T (ix + ν 0 )) +ĝ(T (ix − ν 0 )) g(iT (m − q 0 )) e T (|x|−c ′ (m−q 0 )) → 0, and if |x| = m − q 0 then |ĝ T (ix)| =ĝ (T (i(m − q 0 ) + ν 0 )) +ĝ(T (i(m − q) − ν 0 )) g(iT (m − q 0 )) 1 T ν 0 → 0.
Since we already showed that C(x) = 0 for |x| > m − q 0 we get that the contribution of the complementary series ĝ T (ix k )C(x k ) → 0 as T → ∞.
For ν ∈ R we can estimate |ĝ T,q (ν)| N e −T (q−q 0 ) 1+||ν|−ν 0 | N for all q ≥ q 0 , while if ν = ν 0 thenĝ T,q (ν 0 ) → 1 as T → ∞. Consequently, the right hand side of the trace formula converges to (−1) q 0 ∆m ηq 0 (iν 0 ), and comparing to the left hand side we conclude that ∆m ηq 0 (iν 0 ) = 0.
We have thus shown that ∆m σq 0 (iν) = 0 for all ν ∈ R \ {0}. Consequently, Theorem 6 implies that ∆L ηq 0 = 0 and hence ∆m ηq 0 (ν) = 0 also for ν ∈ [0, ρ).
4.1.
Proof of Theorem 2. Assume that D L (Γ 1 , Γ 2 ; T ) e αT , with 0 ≤ α < ρ. Then Theorem 8 implies that m Γ 1 ,η 0 = m Γ 2 ,η 0 . But η 0 is the trivial representation of M and hence m Γ 1 (π 1,ν ) = m Γ 2 (π 1,ν ) for all ν ∈ (0, ρ) ∪ iR.
4.2.
Proof of Theorem 3. When G/K is not an odd dimensional real hyperbolic space the threshold α 0 > 0. Assume that ℓ≤T |∆L o (ℓ)| e αT , with 0 < α < α 0 . Then Theorem 8 implies that m Γ 1 ,ηq = m Γ 2 ,ηq for q = 0, . . . , m. Since these are all the representation appearing in the alternating trace formula we get that for any even test function
g ∈ C ∞ c (R) ℓ∈ls o ℓ∆m o (ℓ) ∞ j=1 g(jℓ) ψ(jℓ) = 0.
But this can only happen if ∆m o (ℓ) = 0 for all ℓ ∈ R (otherwise, taking g supported around the smallest ℓ where ∆m o (ℓ) = 0 will give a contradiction).
4.3.
Proof of Theorem 4. Let G = SO 0 (2m + 1, 1) and Γ 1 , Γ 2 ⊆ G two uniform torsion free lattices. Assume that m o Γ 1 (ℓ) = m o Γ 2 (ℓ) for all ℓ ∈ R except perhaps a finite set {ℓ 1 , . . . , ℓ k }. We thus have that D L (Γ 1 , Γ 2 ; T ) = O(1) and Theorem 8 implies that ∆m σq (ν) = 0 for 0 ≤ q ≤ m − 1. This also implies that vol(Γ 1 \G) = vol(Γ 2 \G) and hence the alternating trace formula takes the form
1 2 k j=1 ℓ j ∆m o (ℓ j ) ∞ q=1 g(qℓ j ) = (−1) m ν∈Sσ m ∆m σm (ν)ĝ(iν) (4.2)
Using Poisson summation, for each fixed ℓ j we can substitute ∞ q=1 g(qℓ j ) = 1 2 q∈Z g(qℓ j ) − g(0)
= 1 2 ( q∈Zĝ (2πq/ℓ j ) ℓ j − g(0) = ∞ q=0ĝ (2πq/ℓ j ) ℓ j − g(0) 2
Plugging this back into (4.2) replacing g(0) = 2 ∞ 0ĝ (ν)dν we get
1 2 k j=1 ∞ q=0 ∆m o (ℓ j )ĝ(2πq/ℓ j ) − 1 2 k j=1 ℓ j ∆m o (ℓ j ) ∞ 0ĝ (ν)dν = (−1) m ν∈Sσ m ∆m σm (ν)ĝ(iν).
This holds for any even holomorphic functionĝ of uniform exponential type. Since these test functions are dense in C[0, ∞) we get an equality of the corresponding measures
k j=1 ∞ q=0 ∆m o (ℓ j )δ(x − 2πq/ℓ j ) − k j=1 ℓ j ∆m o (ℓ j ) dν = (−1) m ν∈Sσ m 2∆m σm (ν)δ(x − iν).
Since there is no continuous measure on the right hand side we must have that k j=1 ℓ j ∆m o (ℓ j ) = 0 so that {ℓ 1 , . . . , ℓ k } are indeed linearly dependent over Q.
Moreover, this formula gives additional information on the spectral parameters having different multiplicities. Specifically, we get that ∆m σm (ν) = 0 only when ν ∈ iR + satisfies e νℓ j = 1 for some j = 1, . . . , k (that is, when iν = 2πq ℓ j ) in which case
2∆m σm (ν) = (−1) m k j=1 e νℓ j =1
∆m o (ℓ j ).
Spaces with similar length spectrum
In order to prove Theorem 5 (which excludes the possibility of a positive density threshold in Theorem 2) we need to find pairs of lattices for which the length spectrum is as similar as possible. To do this, we borrow the examples constructed by Leininger, McReynolds, Neumann, and Reid [LMNR] of non iso-spectral manifolds having the same length sets. Their construction is based on a modification of the Sunada method [Su85], for constructing non isometric manifolds having the same length spectrum. We start by recalling the Sunada method and in particular his formula for multiplicities of the length spectrum of a finite cover.
Splitting of primitive geodesics in covers.
Let M = Γ\H be as above, let Γ 0 ⊆ Γ denote a finite index subgroup and let M 0 → M the corresponding finite cover. We recall the analogy between the splitting of prime ideals in number field extensions and splitting of primitive geodesics in finite covers.
Let p denote a primitive conjugacy class in Γ (corresponding to a primitive geodesic in M). We say that a primitive conjugacy class P in Γ 0 lies above p (denoted by P|p) if there is some γ ∈ p and a natural number f such that q = [γ f ] Γ 0 . We call f the degree of P and denote f = deg(P) (this does not depend on the choice of representative γ ∈ p). We note that if P 1 , . . . , P r are all the primitive classes above p, then r j=1 f j = [Γ : Γ 0 ]. Assume further that Γ 0 is normal in Γ. Let P 1 , . . . , P r denote all the primitive conjugacy classes in Γ 0 lying above p. Then the group A = Γ/Γ 0 acts naturally on {P 1 , . . . , P r } by permutation. To any class P|p we attache an element σ P ∈ A which generates the stabilizer of P in A (specifically, if P = [γ f ] Γ 0 for some γ ∈ p then σ P is the class of γ in A = Γ/Γ 0 ). We note that the conjugacy class of σ P in A depends only on p and we denote it by σ p .
The analogy with splitting of prime ideals in extensions of number fields is evident from our choice of notation: primitive classes correspond to prime ideals, (normal) covers to (normal) field extensions, and the element σ P and class σ p to the Frobenius element and Frobenius class in the Galois group.
5.2. The Sunada construction. We recall the general setup for the Sunada construction. Let M = Γ\H as above, let Γ 0 ⊂ Γ denote a finite index normal subgroup and let A = Γ/Γ 0 . Let B ⊂ A denote a subgroup and let Γ 0 ⊂ Γ B ⊂ Γ such that Γ B /Γ 0 = B. Fix a primitive class p in Γ and a class P|p in Γ 0 . Let q 1 , . . . , q r denote all the classes in Γ B above p and let f j = deg(q j ).
The length spectrum for Γ B can be recovered from information on Γ and the data on the splitting degrees as follows:
(5.1) m o Γ B (ℓ) = n d=1χ B (σ m p ) = f j |m f j = d|m dA B (p, d),
and using Möbius inversion we get
(5.4) A B (p, m) = 1 m d|m µ(m/d)χ B (σ d p ),
where µ(m) is the Möbius function.
Since the value of χ B (a) is determined by |[a] ∩ B|, we see that if B 1 , B 2 ⊆ A satisfy that |[a] ∩ B 1 | = |[a] ∩ B 2 | for all a ∈ A then Γ B 1 and Γ B 2 are length equivalent. In fact, this condition implies that the two spaces are also representation equivalent (see [Su85]).
Lattices with the same length sets. Leininger McReynolds
Neumann and Reid showed in [LMNR] that, in the same setup, if we assume the weaker condition that |[a] ∩ B 1 | = 0 if and only if |[a]∩B 2 | = 0 for any a ∈ A, then Γ B 1 and Γ B 2 have the same length sets. Moreover, under additional assumptions on the groups A, B 1 , B 2 (see [LMNR,Definition 2.2]) one can guarantee that the primitive length sets are the same. In the same paper they constructed explicit examples of triples B 1 , B 2 ⊆ A satisfying these conditions and showed that these groups can be realized as quotients of lattices in any of the rank one groups.
We briefly recall their examples. Let p ∈ Z denote a prime number and assume that it is inert in the imaginary quadratic extension K = Q(i). In this case we have that the ideal (p) = pO is a prime ideal in O = Z[i] and the quotient O/pO = F q is the finite field with q = p 2 elements. We then have the following inclusion of exact sequences
1 → V p → PSL 2 (Z/p 2 Z) → PSL 2 (F p ) → 1 ↓ ↓ ↓ 1 → V q → PSL 2 (O/p 2 O) → PSL 2 (F q ) → 1 , where V p ∼ = sl 2 (F p ) = {X ∈ Mat 2 (F p )|Tr(X) = 0} via sl 2 (F p ) ∋ X → I + pX ∈ V p ,
and and similarly V q ∼ = sl 2 (F q ). Inside sl 2 (F p ) ∼ = F 3 p we have the plane
R ⊥ (F p ) = {(
x y −y −x ) ∈ sl 2 (F p )|x, y ∈ F p }. As shown in [LMNR], in any rank one group G there is a uniform lattice Γ that surjects onto A = PSL 2 (O/p 2 O) for infinitely many values of p. In this case, the lattices Γ B 1 and Γ B 2 , corresponding to B 1 = V p and B 2 = {I + pX|X ∈ R ⊥ (F p )}, have the same primitive length sets. We now take a closer look at this example and analyze the difference between the multiplicities. 5.4. Computing the splitting types. Let Γ denote a uniform lattice in a rank one group G and assume that there is a surjection
ϕ : Γ ։ A = SL 2 (O/p 2 O).
Consider the following finite index subgroup: Γ 0 = ker(ϕ), Γ Vq = ϕ −1 (V q ) and Γ j = ϕ −1 (B j ), j = 1, 2 where B 1 = V p = {I + pX|X ∈ sl 2 (F p )} and B 2 = {I + pX|X ∈ R ⊥ (F p )} as above. We note that Γ Vq and Γ 0 are both normal subgroups of Γ.
To get hold of the splitting types of primitive classes p in Γ we need to count how many elements lie in the intersection of each conjugacy class with each of the above subgroups. Since V q is normal in PSL 2 (O/p 2 O) a conjugacy class intersects V q if and only if it is contained there. Furthermore, since V q is commutative, the action of PSL 2 (O/p 2 O) on V q by conjugation factors through the quotient PSL 2 (F q ). The following lemma describes the PSL 2 (F q ) conjugacy classes in V q = sl 2 (F q ).
Lemma 5.1. [LMNR, Lemma 4.2] For each Q ∈ F * q the set C Q (F q ) = {X ∈ sl 2 (F q )| det(X) = −Q}
is a conjugacy class. In addition to these classes and the trivial class there are two nilpotent classes with representatives ( 0 0 1 0 ) and ( 0 0 ε 0 ) with ε ∈ F * q not a square. Moreover, for any X ∈ sl 2 (F q ) the size of its conjugacy class is given by
|[X]| = 1 X = 0 q(q + 1) X ∈ C Q , Q ∈ (F * q ) 2 q(q − 1) X ∈ C Q , Q ∈ F * q \ (F * q ) 2 q 2 −1 2 X is nilpotent.
We now compute the size of the intersection of each of these classes with V p ∼ = sl 2 (F p ) and R ⊥ (F p ).
Lemma 5.2. For any X ∈ sl 2 (F q ) we have
|[X] ∩ sl 2 (F p )| = 0 X ∈ C Q (F q ), Q ∈ F * q \ F * p 1 X = 0 p(p + 1) X ∈ C Q (F q ), Q ∈ (F * p ) 2 p(p − 1) X ∈ C Q (F q ), Q ∈ F * p \ (F * p ) 2 p 2 −1 2 X is nilpotent |[X] ∩ R ⊥ (F p )| = 0 X ∈ C Q (F q ), Q ∈ F * q \ F * p 1 X = 0 p − 1 otherwise
Proof. The first part follows from the previous lemma together with the observation that C Q (F q ) ∩ sl 2 (F p ) = C Q (F p ) and that the intersection of a nilpotent conjugacy class with sl 2 (F p ) is the corresponding SL 2 (F p ) conjugacy class.
For the second part, note that
C Q (F q ) ∩ R ⊥ (F p ) = {(
x y −y −x ) |x, y ∈ F p , x 2 − y 2 = Q}, which is of size p − 1 for any Q ∈ F * p . The only nontrivial elements in R ⊥ not yet accounted for are the ones with zero determinant. There are 2(p − 1) such elements and since X is conjugated to ( 0 0 1 0 ) if and only if εX is conjugated to ( 0 0 ε 0 ) we have exactly p − 1 elements in each of the remaining two conjugacy classes.
Fix a primitive conjugacy class p in Γ, a primitive class P in Γ 0 above p, and let σ P ∈ A denote the "Frobenius" element. (That is, σ P = ϕ(γ) for γ ∈ p satisfying that [γ deg(P) ] Γ 0 = P). Since Γ Vq is normal in Γ, there is a natural number d 0 = d 0 (p) such that any γ ∈ p satisfies that γ d 0 is primitive in Γ Vq . Consequently, we have that σ d P ∈ V q if and only if d 0 |d and that A Vq (p, d) = |A| |Vq| δ d,d 0 . We now compute the splitting types for B 1 and B 2 .
Lemma 5.3. For j = 1, 2. If σ d 0 p = 1 then A B j (p, d) = |A| d 0 |B j | δ d,d 0 . Otherwise
A B j (p, d) = χ B j (σ d 0 p ) d 0 δ d,d 0 + ( |A| |B j | − χ B j (σ d 0 p )) pd 0 δ d,pd 0 .
Proof. Since σ d p ∈ V q if and only if d 0 |d we get that χ B j (σ d p ) = 0 unless d = kd 0 for some k ∈ N, and hence A B j (p, d) = 0 unless d = md 0 for some m ∈ N.
If σ d 0 = 1 then σ kd 0 p = 1 for all k ∈ N, and hence χ B j (σ kd 0 p ) = |A| |B j | for all k ∈ N. Using (5.4) we get that in this case
A B j (p, md 0 ) = 1 md 0 k|m µ( m k )χ B j (σ kd 0 p ) = |A| d 0 |B j | δ m,1 .
When σ d 0 P = 1 we write σ d 0 p = I + pX 0 with X 0 ∈ sl 2 (F q ) nontrivial. In this case, σ kd 0 p = (I + pX 0 ) k = I + pkX 0 . In particular, if p|k then σ kd 0 p = 1 and again χ B j (σ kd 0 p ) = |A| |B j | . For k prime to p, if X 0 ∈ C Q then kX 0 ∈ C k 2 Q and if X ∼ ( 0 0 c 0 ) then kX ∼ ( 0 0 kc 0 ). Thus, the sizes of the conjugacy classes of [kX 0 ] and [kX 0 ] ∩ B j don't depend on k so χ B j (σ kd 0 p ) = χ B j (σ d 0 p ). Using (5.4) we get
A B j (p, d 0 m) = 1 d 0 m k|m µ( m k )χ B j (σ kd 0 p ) = χ B j (σ d 0 p ) d 0 m k|m (k,p)=1 µ( m k ) + |A| |B j |d 0 m k|m p|k µ( m k ).
When (m, p) = 1, the second sum is empty and the first sum is k|m µ( m k ) = δ m,1 . When m = pm with (m, p) = 1, the second sum is δm ,1 and the first sum is
k|pm (k,p)=1 µ( m k ) = − k|m µ(m k ) = −δm ,1 .
Finally, when m = p km with k > 1, both sums vanish. We thus get that in any case where X 0 = 0 we have
A B j (p, d 0 m) = δ m,1 d 0 χ B j (σ d 0 p ) + δ m,p pd 0 ( |A| |B j | − χ B j (σ d 0 p )).
5.5. Proof of Theorem 5. Theorem 5 follows from the following proposition with p a sufficiently large inert prime.
Proposition 5.4. Let Γ 2 ⊆ Γ 1 ⊆ Γ be as above. Then there is α < 2ρ such that D L (Γ 1 , Γ 2 ; T ) ≤ 4Li(e 2ρT ) p + 1 + O(e αT ).
Proof. We separate the primitive classes in Γ into 5 types as follows. Let d 0 = d 0 (p) as above and write σ d 0 P = I + pX 0 with X 0 ∈ sl 2 (F q ). We then say that p is of trivial type if X 0 = 0, and of nilpotent type if X 0 is nilpotent, and we say that p is of irregular type, quadratic type, or non-quadratic type if X 0 ∈ C Q (F q ) with Q ∈ F * q \ F * p , Q ∈ (F * p ) 2 or Q ∈ F * p \ (F * p ) 2 respectively (recall that F * p ⊂ (F * q ) 2 ). Note that even though X 0 depends on the choice of P above p, this characterization depends only on p.
Consider the partial counting functions m tr , m ir , m qr , m nq and m ni each given by m I (ℓ, d) = #{p of type I|d 0 (p) = d and ℓ p = ℓ/d}. Let K/Q denote a finite field extension and O K the corresponding ring of integers. For any prime number p ∈ Z we have a decomposition pO K = p e 1 1 · · · p er r with p j ⊆ O K the prime ideals dividing p and e j the ramification degrees (which are all one except for the ramified primes dividing the discriminant). Each quotient O K /p j is a finite field of order p f j where f j denotes the inertia degree of p j . The splitting type of p in K is then given by the numbers A K (p, d) = #{j|f j = d} for d = 1, . . . , n = [K : Q].
Let K 1 , K 2 denote two number fields. We say that a prime number p ∈ Z have the same splitting types in K 1 and K 2 if A K 1 (p, d) = A K 2 (p, d) for all d = 1, . . . , n. In [Pe77, Theorem 1], Perlis showed that if all but finitely many primes have the same splitting types in K 1 and K 2 , then all primes have the same splitting types. In this case the fields have the same Dedekind Zeta functions, but they are not (necessarily) isomorphic; such fields are called arithmetically equivalent fields.
The finite set of exceptions in Perlis's theorem can be replaced with an infinite set of density zero. A positive density threshold that is independent of the number fields probably does not hold. Nevertheless, it is possible to give a positive density threshold which depends on the degree of the smallest Galois field L/Q containing K 1 and K 2 . Specifically, one can show Proposition 5.5. In the above setting, for any 1 ≤ d ≤ n if the set P bad (d) = {p unramified|A K 1 (p, d) = A K 2 (p, d)} has Dirichlet density smaller then 1/|Gal(L/Q)| then P bad (d) = ∅. In particular, if P bad = ∪ n d=1 P bad (d) has density below 1/|Gal(L/Q)| then K 1 and K 2 are arithmetically equivalent.
Proof. The proof is a direct result of Chebotarev's Density Theorem combined with the following observation. For any prime p denote by σ p the Frobenius conjugacy class in Gal(L/Q). Denote by A = Gal(L/Q) and by B j = Gal(L/K j ). Then (using the same argument as in section 5.2) the condition A K 1 (p, d) = A K 2 (p, d) is equivalent to the condition
Proposition 3. 1 .
1For the orthogonal group SO 0 (n+ 1, 1), M ∼ = SO(n) and the restriction of Ad(M) to n = n 1 is the natural isomorphism. For the unitary and symplectic groups the restriction of Ad(M) to n j , j = 1, 2 gives a homomorphism ι j : M → SO(n j ). For G = F II, M = Spin(7) and the restriction of Ad(M) to n 1 and n 2 is the 8-dimensional spin representation and the 7-dimensional orthogonal representation respectively.
e
kℓ χ τq (u)
S m,m = s∈W 0 ξ s(e 1 +...+em) ,
Proposition 3. 5 .
5The Weyl discriminant can be written as(3.14) D(γ) = ψ(ℓ γ ) m q=0 (−1) q (e (m−q)ℓγ + e (q−m)ℓγ )χ ηq (m γ ),with m = ρ − α 0 (G), η 0 , . . . , η m virtual representations of M with η 0 trivial, and ψ(ℓ) the weight function in (3.1).
B j is, as before, the character of the permutation representation of A on A/B j . Consider the set S bad (d) = {a ∈ A| m|d µ(d/m)χ B 1 (a m ) = m|d µ(d/m)χ B 2 (a m )}.
). We note that this is analogous to the threshold in the result of Soundararajan[So04] on strong multiplicity one for the Selberg class. On the other hand, the analogy with the density 1/8 result for cusp forms may lead to the suspicion that the correct threshold should be of the form cLi(e 2ρT ) with some c < 2 a sufficiently small constant. Though we do not know if our threshold of e ρT is sharp, we show that such a positive density threshold cannot hold. Specifically, borrowing examples from[LMNR] we showTheorem 5. For every ǫ > 0 there are non iso-spectral Γ 1 , Γ 2 ⊂ G
satisfying
lim sup
T →∞
ℓp=ℓ/d A B (p, d), where n = [Γ : Γ 0 ], the second sum is over all primitive classes in Γ of length ℓ/d and (5.2) A B (p, d) = #{q : deg(q) = d and q|p}, encodes the splitting type of p. The information on the splitting type encoded in A B (p, d) can be recovered from information on the finite groups A and B. To do this, let χ B denote the character of the representation of A given by the action of A on co-sets in A/B, that is where [a] denotes the conjugacy class of a in A. We then have that (see [Su85, Proof of Theorem 2])(5.3)
χ B (a) =
|Z A (a)||([a] ∩ B)|
|B|
=
|[a] ∩ B|
|[a] ∩ A|
[A : B],
The result on the representation spectrum is proved for any semisimple group not necessarily of rank one.
We corrected here a few typos from the formulas in[SW99] and in particular the mistake in [SW99, Theorem 6.5 (2)] for an odd test function.
Acknowledgments. I thank Ben McReynolds and Alan Reid for ex-where I is one of the types: trivial, irregular, quadratic, non-quadratic, and nilpotent. For the non-trivial types we define additional counting functionsm I (ℓ, d) = #{p of type I|pd 0 (p) = d and ℓ p = ℓ/d}.Let n = |A/B 1 | = p 3 |PSL 2 (F p )| = p 4 (p 2 −1) 2 . Using Lemma 5.2 we can compute χ B j (σ d 0 p ) for each type of p and using Lemma 5.3 we get thatUsing (5.1) for m o Γ j (ℓ) and summing separately over classes of each type we get thatFrom these formulas one can directly see that, indeed, Γ 1 and Γ 2 have the same primitive length sets. Moreover, we get the following formula for the difference of the multiplicities.We can also separate the sum over all classes in Γ of length≤ T into the different types; consequently, if we define π I (T ) andπ I (T ) respectively by This estimate together with (5.5) implies that 2 p 3 +p π qr (T ) = (p − 1)π tr (T ) + p 3 −p 2 −p+1On the other hand, we can bound we get that indeed5.6. Comparison to splitting of primes in number fields.In the examples constructed above, the volumes of the two non iso-spectral spaces grow with p. It is thus possible that a positive density threshold can hold under the additional assumption that the volume is uniformly bounded. To illustrate this we will look at a similar situation in the analogous question regarding the splitting of prime ideals in number fields.This set is invariant under conjugation in A and satisfies thatNow, the Chebotarev density theorem implies that P bad (d) has Dirichlet density |S(d)|/|A|. Thus if the density of P bad (d) is smaller then 1/|A| then S bad (d) = ∅ implying that P bad (d) = ∅ as claimed.Remark 5.1. It follows from the proof that if P bad (1) = ∅ then also S bad (1) = ∅ implying that χ B 1 = χ B 2 and hence that S bad (d) = ∅ for all d. Consequently, we see that to prove arithmetic equivalence one does not need to consider the full splitting type but only the number of prime ideals of inertia degree one dividing each prime.
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301 Carney Hall, Boston College Chestnut Hill, MA 02467 E-mail address: dubi. Department of [email protected] of Mathematics, 301 Carney Hall, Boston College Chest- nut Hill, MA 02467 E-mail address: [email protected]
| [] |
[
"TOWARDS THE ONTOLOGY WEB SEARCH ENGINE",
"TOWARDS THE ONTOLOGY WEB SEARCH ENGINE"
] | [
"Olegs Verhodubs [email protected] "
] | [] | [] | The project of the Ontology Web Search Engine is presented in this paper. The main purpose of this paper is to develop such a project that can be easily implemented. Ontology Web Search Engine is software to look for and index ontologies in the Web. OWL (Web Ontology Languages) ontologies are meant, and they are necessary for the functioning of the SWES (Semantic Web Expert System). SWES is an expert system that will use found ontologies from the Web, generating rules from them, and will supplement its knowledge base with these generated rules. It is expected that the SWES will serve as a universal expert system for the average user. | null | [
"https://arxiv.org/pdf/1505.00755v1.pdf"
] | 11,466,464 | 1505.00755 | 2e2e51ee920723bef7735aebdedd3b629dea32f3 |
TOWARDS THE ONTOLOGY WEB SEARCH ENGINE
Olegs Verhodubs [email protected]
TOWARDS THE ONTOLOGY WEB SEARCH ENGINE
Ontology Web Search EngineSearch EngineCrawlerIndexerSemantic Web
The project of the Ontology Web Search Engine is presented in this paper. The main purpose of this paper is to develop such a project that can be easily implemented. Ontology Web Search Engine is software to look for and index ontologies in the Web. OWL (Web Ontology Languages) ontologies are meant, and they are necessary for the functioning of the SWES (Semantic Web Expert System). SWES is an expert system that will use found ontologies from the Web, generating rules from them, and will supplement its knowledge base with these generated rules. It is expected that the SWES will serve as a universal expert system for the average user.
I. INTRODUCTION
The technological development of the Web during the last few decades has provided us with more information than we can comprehend or manage effectively [1]. Typical uses of the Web involve seeking and making use of information, searching for and getting in touch with other people, reviewing catalogs of online stores and ordering products by filling out forms, and viewing adult material. Keyword-based search engines such as YAHOO, GOOGLE and others are the main tools for using the Web, and they provide with links to relevant pages in the Web. Despite improvements in search engine technology, the difficulties remain essentially the same [2]. Firstly, relevant pages, retrieved by search engines, are useless, if they are distributed among a large number of mildly relevant or irrelevant pages. Secondly, relevant pages may not be retrieved by search engines at all. In truth, it is a rare phenomenon that happens with modern search engines. Thirdly, the pages, retrieved by search engines, are sensitive to vocabulary. Initial keywords do not provide with the results we want, because relevant pages use different terminology from the original query. Finally, retrieved results are single web pages and hence it is necessary manually extract the partial information and put it together [2]. This requires the person, who will browse retrieved pages and extract the information he is looking for. The need of a new approach to manage information is beyond doubt. Development of a SWES (Semantic Web Expert System) is an attempt to change this situation for the better, and this development is the main goal of the research. The SWES is a new expert system, which is based on the Semantic Web technologies [3]. It is assumed that the SWES will use OWL (Web Ontology Languages) ontologies, found in the Web, to generate rules, to supplement the SWES knowledge with these rules, and to reason, based on the rules from the SWES knowledge base and user interaction. The tasks of OWL ontology merging [4], rule generation from OWL ontologies [5], [6] as well as the task of the Jena framework adaptation for fuzzy reasoning had already been investigated and realized. It is expected that potential of the SWES will surpass potential of existing keywordbased search engines, when the task of web page transformation to OWL ontology will be investigated and solved. This task is for the future research. The main purpose of this paper is to develop Ontology Web Search Engine project. This project will describe a search engine for searching and indexing OWL ontologies in the Web. The project is realizable to be integrated in the SWES. Implemented and integrated in the SWES Ontology Web Search Engine would allow the user to utilize all available OWL ontologies in the Web. Thus, the SWES would use knowledge from the Web. This paper is divided into sections as follows. The next section gives an overview of existing ontology search opportunities in the Web. Section III presents several ontology search strategies in the Web. The following section describes the project of the Ontology Web Search Engine, which will be implemented in the near future. The last section presents the conclusions of this work.
II. RELATED WORK
The term "Semantic Web" was coined by Tim Berners-Lee more than 10 years ago [7]. Since then, this term came into use, and rapidly filled with content. Throughout this time the Semantic Web technologies were actively developed, that was why it was not surprising that the task of ontology search in the Web had already been studied and even implemented. There are several implemented ontology web search engines and let us look at them one by one. SWOOGLE was the first search engine for the Semantic Web, and the main contributor of SWOOGLE was Li Ding [8]. SWOOGLE is positioned as a crawler-based indexing and retrieval system for the Semantic Web documents including RDF (Resource Description Framework) and OWL [9]. SWOOGLE consists of four major components SWD (Semantic Web Document) discovery, metadata creation, data analysis and interface. The SWD discovery component is responsible to discover the potential SWDs throughout the Web and keep up-to-date information about SWDs. The metadata creation component caches a snapshot of a SWD and generates objective metadata about SWDs in both syntax level and semantic level. The data analysis component uses the cached SWDs and the created metadata to derive analytical reports. The interface component focuses on providing data service to the Semantic Web community. The architecture of SWOOGLE is data centric and extensible: different components work on different tasks independently [9]. SWOOGLE is realized as a web page, and it is available at http://swoogle.umbc.edu. SWOOGLE is used by means of querying with keywords, after which the SWDs matching those keywords are returned in ranked order. WATSON is one more search engine for the Semantic Web [10]. This search engine is available at http://watson.kmi.open.ac.uk/WatsonWUI/. WATSON is called the gateway for the Semantic Web, which has been guided by the requirements of Semantic Web applications and by lessons learnt from previous systems [11]. The goal of this gateway is to provide an efficient access point to the online ontologies and semantic data. WATSON collects the available semantic content on the Web, analyzes it to extract useful metadata and indexes and also implements efficient query facilities to access the data [11]. In order to support these three tasks, WATSON has been designed around three core activities, each corresponding to a "layer" of its architecture. These layers are the following [12]:
The ontology crawling and discovery layer; it collects the online available semantic content by exploring ontology-based links. The validation and analysis layer; it is core to the architecture and ensures that data about the quality of the collected semantic information is computed, stored and indexed. The query and navigation layer; it grants access to the indexed data through a variety of mechanisms that allow exploring its various semantic features.
WATSON is similar to usual web or desktop search systems in the sense that it is based on keyword search [13]. Automatic independent OWL ontology web search strategy implies developing your own ontology web search engine. This strategy is the most difficult, but it has its own advantages. Of course, the SWES can be tested by means of several ontologies, which are manually found in the Web, but such a Semantic Web Expert System would not be complete. Having its own ontology web search engine, transforms the SWES from a research project into the system that is useful for the end user, and this is the first advantage of the strategy. External ontology web search services are always less reliable than exactly the same in terms of functionality, but built-in services. This is so, because any external services do not depend on our will, but the will of third-party developers is unpredictable, especially in the long term. External to the SWES ontology web search services, provided by SWOOGLE and WATSON, are not an exception. Thus, the SWES reliability increase, achieved by means of the SWES part reliability increase that is the SWES reliability increase, achieved by means of its own OWL ontology web search engine development is the second advantage of the strategy. The third advantage is that own developments are more modifiable, which is essential, if the SWES will continue to evolve. So, automatic independent OWL ontology web search strategy is preferable and therefore it will be implemented in Ontology Web Search Engine project, which will be described in the next section in detail.
IV. ONTOLOGY WEB SEARCH ENGINE PROJECT
Search engine technology keeps up with the growth of the Web. In 2000 there were more than 3200 search engines in the Web [18]. The number of documents, indexed by web search engines, is increasing from year to year. If WWWW (World Wide Web Worm) had an index of 110,000 web pages in 1994, then in 1997 search engines claimed to index from 2 million to 100 million web documents [19]. Nowadays GOOGLE, which is leading web search engine, has an index of over 30 trillion web pages [20]. Despite the differences of web search engines, the typical structure of a web search engine comprises four essential modules [18]:
crawler, an indexer, a query engine, a page repository.
Crawlers are computer programs that browse the Web [18]. They use a starting set of URLs and retrieve (that is copy and store) the content on the Web pages specified by the URLs. The crawlers extract URLs appearing in the retrieved web pages and visit some or all of these URLs, thus repeating the retrieval process.
The indexer takes all the words from each document in the page repository and records the URL, where each word occurred [18]. The result is a very large database, which provides the URLs that point to pages, where a given word occurs. The database may also contain other structural information such as links between documents, incoming URLs to these documents, formatting aspects of the documents, and location of terms with respect to other terms .
The web query engine receives the search requests from users [18]. It takes the query submitted by the user, splits the query into terms, and also searches these terms in the database, which is built by the indexer. The web query engine then retrieves the documents that match the terms within the query and returns these documents to the user .
The page repository stores the Web content that was retrieved during the crawling process [18].
The web search engine typically ranks documents before presenting them to the user. The ranking process is calculating a similarity score between the query and each document. The higher the similarity score, the higher the ranking for a particular document [18].
Semantic Web search engines are very similar to traditional web search engines. For example, such a Semantic Web search engine as WATSON performs the same activities as traditional web search engines [13]:
collecting the Semantic Web content, extracting useful metadata and indexing it, implementing efficient query facilities to access the metadata.
On this basis, it is possible to state that the main difference between traditional web search engines and Semantic Web search engines is in the content they work with. If traditional web search engines work primarily with documents in HTML, then Semantic web search engines work with semantic metadata, mainly contained in RDF and OWL documents. Otherwise, these web search engines are completely identical. Hence, it is necessary to develop its own crawler, indexer, query engine and page repository (ontology repository) for the OWL Ontology Web Search Engine.
The crawler for the Ontology Web Search Engine has to browse the Web to find OWL ontology URL's. There are two main strategies to accomplish the task of OWL ontology URL finding in the Web. The first strategy is to browse OWL ontologies from the Web to find the URL's of other OWL ontologies. This strategy is premature, because analysis shows that ontologies are not so common and not so qualitatively designed to use them for this task. However it should be noted that this strategy may be auxiliary or may be reserved for future use. The second strategy is to browse HTML pages from the Web to find the URL's of OWL ontologies and to find the URLs of other HTML pages to continue browsing the Web. This strategy is preferred for use, because analysis shows that the URL's of ontologies are basically presented on the web pages. The fact that web pages contain the URL's of other web pages has been known for a long time.
There are two ways to implement the crawler for the Ontology Web Search Engine. The first way is to develop such a crawler using one of programming languages. The second one is to adapt one of existing crawlers to our needs. Existing crawlers may be divided into open sources crawlers and others. WebEater, Heritrix, JSpider, Java Web Crawler, Web-Harvest, Crawler4j, Nutch and so on are examples of open sources crawlers. It is necessary to identify crawler requirements in order to choose the way of crawler future implementation in the Ontology Web Search Engine. So, it would be preferable that the crawler would be written in Java, because the SWES is developing in Java. This crawler should have advanced functionality to be able to set the initial URL, from which the web crawling will start, to set the number of crawling web pages, to set the depth of crawling, to set the politeness of crawling that is how fast the crawler will apply to the new URL. Of course, if existing crawler is chosen, then it should be well documented.
It is necessary to utilize the tool for work with OWL ontologies in order to realize the indexer for the Ontology Web Search Engine. Jena [21] can serve as such a tool. This choice is due to several reasons. Firstly, Jena is an open source Semantic Web framework. Secondly, Jena is utilized by means of Java programming language, and this programming language is chosen for the SWES. Thirdly, Jena is used in other SWES subsystems, and it works well. Finally and this is the main reason, Jena gives an opportunity to access to all elements (classes, properties, relations and so on) of the OWL ontology. This opportunity allows storing the information of the ontology elements with the purpose to access to them later. Except the tool for work with OWL ontologies, it is necessary to realize something like database, which would have all necessary information about indexed ontology, in order to implement the indexer. There are two ways for this purpose. The first way is to develop such a database manually. The second way is to exploit one of existing indexers. Lucene, Nutch, Solr, Sphinx are just some of them. The task of developing or choosing the indexer for the Ontology Web Search Engine is the task for the future, however at the same time it is necessary to think about a query engine, performing this task.
The query engine is a subsystem of the Ontology Web Search Engine for searching the OWL ontology URLs, based on the user query. This means that the user types several keywords and the query engine returns the OWL ontology URLs, where the typed keywords appear. The query engine is closely related to the indexer, and actually the query engine looks for the ontology URLs in the database that is supplemented by the indexer.
The page repository of the Ontology Web Search Engine is storage, where the indexed OWL ontologies are located. Physically this storage will be a directory in the computer hard disc, where OWL ontologies will be collected. Such storage will give an opportunity to access to needed OWL ontology as quickly, as necessary.
Speaking about the whole structure of the Ontology Web Search Engine, it is necessary to note that this structure represents the set of the Ontology Web Search Engine functional parts, which are programmatically independent from each other. This means that there are several separate computer programs, which execute different tasks in the area of OWL ontology search in the Web. Thus, the structure of the Ontology Web Search Engine may look like this ( Fig.1.):
Ontology Web Search Engine
The system of Ontology Web Search Engine consists of three computer programs ("Web Crawler", Ontology Indexer", "Query Engine") and three storages of information ("URL Repository", "Indexes", "Ontology Repository"). "Web Crawler" and "Ontology Indexer" are independent computer programs, but "Query Engine" is a software module that will be the part of the Semantic Web Expert System. "URL Repository" will contain the URL's of the OWL ontologies. "Ontology Repository" will contain OWL ontologies, and "Indexes" will contain the information about found OWL ontologies namely URL, size, classes, properties and relations (Fig. 1). "Query Engine" (as the part of the SWES) is started, when "Indexes" and "Ontology Repository" storages are filled with the data. This occurs, when "Web Crawler" and "Ontology indexer" have already worked out. So, "Web Crawler" should be run first. It browses the Web in order to find the URL's of OWL ontologies. The URL's are stored in the "URL Repository", whenever they are found. "Web Crawler" will run until it finds a number of ontologies or it browses a number of web pages, as defined by the user. After the "Web Crawler" finishes running and the "URL Repository" is filled with the URLs, it is possible to run the "Ontology Indexer". The "Ontology Indexer" takes the URL's from the "URL Repository" in order, downloads the ontology at this URL, extracts the ontology components (properties, classes, relations), stores these information in the "Indexes" repository and also stores the ontology entirely in the "Ontology Repository". The "Ontology Indexer" runs until it processes all the URL's from the "URL Repository". The "Query Engine" software module works as follows. It receives the keywords, typed by the user, and looks for the matches in the "Indexes" repository. After the matches are found, the OWL ontology and its URL are identified. This is necessary for the further work of the SWES. In turn, the SWES will merge found ontologies, forming single one-domain ontology. Rules will be generated from this single ontology, and they will be used for reasoning, based on communication with the user.
V. CONCLUSION
The paper has described the project of Ontology Web Search Engine. The project aims to develop a Web Search Engine for searching and indexing OWL ontologies in the Web. This is necessary for the Semantic Web Expert System, which is expected to use OWL ontologies for its work [3]. In the course of the project description, the achievements of search engine technology have been overviewed, the typical structure of a web search engine has been described, ontology web search strategies has been identified, related works in the area of ontology search have been overviewed and also the structure of the Ontology Web Search Engine has been presented. This structure has been described in detail in order to have an opportunity to realize the project as soon, as it becomes possible.
It seems that OWL ontologies cannot displace regular web pages completely. At least it can be argued that this does not happen in the near future. Indeed, this is really difficult task, taking into account the distributed nature of the Web. Such a situation is in contradiction with the stated purpose of the SWES that is to be a universal expert system, which can use all the knowledge of the Web. Therefore it is necessary to be able to use the knowledge, generated from regular We pages, to remove this contradiction. One way to do this is to learn how to transform a regular web page, which, for example, is created in HTML, to OWL ontology. In turn, the process of knowledge (rule) generation from OWL ontology is a technical task, which has already been developed in [5], [6], [22]. So, the transformation of a regular web page to OWL ontology is one of the tasks to be developed.
is utilized to implement this strategy. The uses of the Semantic Web search engines are preferable, because such systems are specially aimed at finding ontologies unlike the uses of the Web search engines. The use of existing web search engines gives an opportunity to get OWL ontology URLs (Uniform Resource Locator), to browse these ontologies and to download them, if it is necessary.Automatic non-independent OWL ontology web search strategy implies searching OWL ontologies using built-in services of existing web search engines. SWOOGLE engine provides search services using REST (Representational State Transfer) interface [14]. It is possible to compose the query in an HTTP (Hypertext Transfer Protocol) GET query and retrieve the result as a dynamic web page encoded in RDF/XML. In turn, WATSON engine provides two types of open web services/APIs [15]: The Java/SOAP API, a complete Java API based on a number of SOAP/WSDL (Simple Object Access Protocol/ Web Services Description Language) services, providing complex search, querying and exploration mechanisms. The REST API, which provides a subset of the functionalities of WATSON through simple HTTP-based access, and giving back results in XML or JSON (JavaScript Object Notation), so that it can be, used easily in any language in particular Javascript.WATSON provides three Web SOAP-based services to allow developers to programmatically access the semantic content (semantic data, ontologies)[16]. These web services have the associated API, which allows developing lightweight Semantic Web application. The WATSON REST API corresponds to a set of services simply accessible through HTTP calls[17]. For instance, it is possible to type certain URL in the Web browser to obtain the URIs (Uniform Resource Identifier) of all the semantic documents containing specific words.In this regard, WATSON and SWOOGLE are alike.
A number of other Semantic Web search engines are known. Among them are such search
engines as Falcons, Sindice, Semantic Web Search, SWSE (Semantic Web Search Engine) and
others [10]. But they will not be discussed here for several reasons. Firstly, some of them are not
available and cannot be tested (Falcons, Semantic Web Search, SWSE). Secondly, other
Semantic Web search engines as Sindice do not work as effectively as it is necessary.
III. ONTOLOGY WEB SEARCH STRATEGIES
OWL ontologies are the main resource for the functioning of the SWES [5], [6]. In principle
there is no difference how ontologies are supplied to the SWES. The main thing is that they
would be, and it is desirable that they would be different. Different ontologies mean here that
they are from different domains. OWL ontology web search strategies can be divided into the
following categories:
manual,
automatic non-independent,
automatic independent.
Manual OWL ontology web search strategy implies searching OWL ontologies using existing
web search engines such as GOOGLE, YANDEX, SWOOGLE, WATSON and so on. There is
no difference, what kind of a web search engine (a web search engine or a Semantic Web search
engine)
Fig. 1. The structure of the Ontology Web Search Engine.The Web
Query
Engine
Web Crawler
URL
Repository
Ontology Indexer
Ontology
Repository
Indexes
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| [] |
[
"An implicit boundary integral method for computing electric potential of macromolecules in solvent",
"An implicit boundary integral method for computing electric potential of macromolecules in solvent"
] | [
"Yimin Zhong ",
"Kui Ren ",
"Richard Tsai "
] | [] | [] | A numerical method using implicit surface representations is proposed to solve the linearized Poisson-Boltzmann equations that arise in mathematical models for the electrostatics of molecules in solvent. The proposed method used an implicit boundary integral formulation to derived a linear system defined on Cartesian nodes in a narrowband surrounding the closed surface that separate the molecule and the solvent. The needed implicit surfaces is constructed from the given atomic description of the molecules, by a sequence of standard level set algorithms. A fast multipole method is applied to accelerate the solution of the linear system. A few numerical studies involving some standard test cases are presented and compared to other existing results.Key words. Poisson-Boltzmann equation, implicit boundary integral method, level set method, fast multipole method, electrostatics, implicit solvent model. | 10.1016/j.jcp.2018.01.021 | [
"https://arxiv.org/pdf/1709.08070v2.pdf"
] | 4,429,606 | 1709.08070 | c80c20b4dd21fafd75a265b398ee971c451836da |
An implicit boundary integral method for computing electric potential of macromolecules in solvent
October 2, 2017
Yimin Zhong
Kui Ren
Richard Tsai
An implicit boundary integral method for computing electric potential of macromolecules in solvent
October 2, 2017AMS subject classifications 2010 45A0565R2065N8078M1692E10
A numerical method using implicit surface representations is proposed to solve the linearized Poisson-Boltzmann equations that arise in mathematical models for the electrostatics of molecules in solvent. The proposed method used an implicit boundary integral formulation to derived a linear system defined on Cartesian nodes in a narrowband surrounding the closed surface that separate the molecule and the solvent. The needed implicit surfaces is constructed from the given atomic description of the molecules, by a sequence of standard level set algorithms. A fast multipole method is applied to accelerate the solution of the linear system. A few numerical studies involving some standard test cases are presented and compared to other existing results.Key words. Poisson-Boltzmann equation, implicit boundary integral method, level set method, fast multipole method, electrostatics, implicit solvent model.
which solvent molecules are treated explicitly and implicit solvent models in which the solvent is represented as a continuous medium. While explicit solvent models are believed to be more accurate, they are computationally intractable when modeling large systems. Implicit models are therefore often an alternative for large simulations. The Poisson-Boltzmann model is one of the popular implicit solvent models in which the solvent is treated as a continuous high-dielectric medium [3,4,5,6,7,9,10,12,14,17,19,20,22,23,25,29,31,32,34,37,42,49,50,52,55,65,66,75,76,82,84].
To introduce the Poisson-Boltzmann model, let us assume that the macromolecule has N c atoms centered at {z j } Nc j=1 , with radii {r j } Nc j=1 and charge number {q j } Nc j=1 respectively. Let Γ be the closed surface that separates the region occupied by the macromolecule and the rest of the space. The typical choice of Γ is the so-called solvent excluded surface, which is defined as the boundary of the region outside the macromolecule which is accessible by a probe sphere with some small radius, say ρ 0 ; see Figure 1 for an illustration. We use Ω to denote the region surrounded by Γ that includes the macromolecule.
We use a single function ψ to denote the electric potential inside and outside of Ω. In the Poisson-Boltzmann model, ψ solves the Poisson's equation for point changes inside Ω, that is,
−∇ · (ε I ∇ψ(x)) = Nc k=1 q k δ(x − z k ), in Ω
where ε I denotes the dielectric constant in Ω. Outside Ω, that is in the solvent that exclude the interface Γ, ψ solves the Poisson's equation for a continuous distribution of charges that models the effect of the solvent, that is,
−∇ · ( E ∇ψ) = ρ B (T, x, ψ(x)), in R 3 \ Ω
where ε E denotes the dielectric constant of the solvent, which often has much higher value than that of the macromolecule, ε E ε I . The source term ρ B is a nonlinear function coming from the Boltzmann distribution with T denoting the temperature of the system. More precisely, for solvent containing m ionic species,
ρ B (T, x, ψ(x)) := e c m i=1 c iqi e −ecq i ψ(x)/k B T , x ∈ R 3 \ Ω
where c i ,q i are the concentration and charge of the ith ionic species, e c is the electron charge, k B is the Boltzmann constant, and T is the absolute temperature.
The nonlinear term ρ B (T, x, ψ) in the Poisson-Boltzmann system poses significant challenges in the computational solution of the system. In many practical applications, it is replaced by the linear function −κ 2 T ψ(x) where the parameterκ T = 8πe 2 I k B T is called the Debye-Huckel screening parameter with κ B , e, and I being the Boltzmann constant, the unit charge, and the ionic strength respectively. This leads to the linearized Poisson-Boltzmann equation (PBE) for the electrostatic potential ψ. It takes the following form −∇ · (ε I ∇ψ(x)) = Nc k=1 q k δ(x − z k ), in Ω,
−∇ · (ε E ∇ψ(x)) = −κ 2 T ψ(x), in Ω c , ψ(x) |Γ + = ψ(x) |Γ − , on Γ, ε E ∂ψ ∂n |Γ + = ε I ∂ψ ∂n |Γ − , on Γ,
|x|ψ(x) → 0, |x| 2 |∇ψ(x)| → 0, as |x| → ∞.
(1)
Here the operator ∂/∂n ≡ n(x) · ∇ denotes the usual partial derivative at x ∈ Γ in the outward normal direction n(x) (pointing from Ω outward). The usual continuity conditions, continuity of the potential and the flux across Γ, are assumed, and the radiation condition, which requires ψ decay to zero far away from the macromolecule, is needed to ensure the uniqueness of solutions to the Poisson-Boltzmann equation. See e.g. [3,7,10,17,31,32,49,52,66,75,76].
Computational solution of the Poisson-Boltzmann equation (1) in practically relevant configurations turns out to be quite challenging. Different types of numerical methods, including for instance finite difference methods [6,16,42,33,57,61], finite element methods [3,39,77,78,79,80], boundary element methods [1,2,15,44,51,53,54,81], and many more hybrid or specialized methods [13,74] have been developed; see [52] for the recent surveys on the subject. Each method has its own advantages and disadvantages. Finite difference methods are easy to implement. They are the methods used in many existing software packages [16,42,33,57,61]. However, finite difference methods all require structure grids which put restrictions on the geometry of the macromolecule domain Ω. Finite element methods provide more flexibility with the geometry. However, like the finite difference methods, they often suffer from issues such as large memory storage requirement and low solution speed when dealing with large problems. Moreover, both finite difference and finite element methods need to truncate the domain in some way so the radiation condition is not satisfied exactly. Boundary element methods are based on integral formulations of the Poisson-Boltzmann equation. They require only the discretization of the macromolecule surface, i.e. Γ, not the macromolecule and solvent domains. The radiation condition is usually exactly, although implicitly, integrated into the integral form to be solved. However, the matrix systems resulted from boundary element formulations are often dense. Efficient acceleration, for instance preconditioning, techniques are needed to accelerate the solution of such dense systems.
In this work, we propose a fast numerical methods for solving the interface/boundary value problem of the linearized Poisson-Boltzmann equation (1). The method is derived from the implicit boundary integral formulation [46] of (1) and relies on some of the classical level set algorithms [63,64] for computing the implicit interfaces and the needed geometrical information. All the involved computational procedures are defined on an underlying uniform Cartesian grid. Thus the proposed method inherit mosts of the flexibilities of a level set algorithm. On the other hand, since the method is derived from a boundary integral for-mulation of (1), it treats the interface conditions and far field conditions in a less involved fashion compared to the standard level set algorithms for similar problems. As such type of implicit boundary integral approaches are relatively new, we describe in detail how to setup a linear system and where a fast multipole method can be used for acceleration of the common matrix-vector multiplications in the resulting linear system. We demonstrate in our simulations involving non-trivial molecules defined by tens of thousands atoms that standard "kernel-independent" fast multipole methods [30] can be used easily and effectively as in a standard boundary integral method.
The rest of this paper is organized as follows. We first introduce in Section 2 the implicit boundary integral formulation of the linearized Poisson-Boltzmann system (1). We then present the details of the implementation of the method in Section 3. In Section 4, we present some numerical simulation results to demonstrate the performance of the algorithm. Concluding remarks are then offered in Section 5.
The implicit boundary integral formulation
The numerical method we develop in this work is based on a boundary integral formulation of the linearized Poisson-Boltzmann equation that is developed in [44].
Boundary integral formulation
Throughout the rest of the paper, all the coefficients involved in the equations are assumed to be constant, i.e. independent of the spatial variable. We define κ =κ T / √ E , and introduce the fundamental solutions
G 0 (x, y) = 1 4π|x − y| and G κ (x, y) = e −κ|x−y| 4π|x − y|
to the Laplace equation and the one with the linear lower order term −κ 2 T ψ in (1). Following the standard way of deriving boundary integral equations, we apply Green's theorem to the system formed by (i) the first equation in (1) and the equation for G 0 , and (ii) the second equation in (1) and the equation for G κ , taking into account the interface and the radiation conditions. A careful routine calculation leads to the following boundary integral equations for the potential ψ and its normal derivative ψ n ≡ ∂ψ/∂n on Γ:
1 2 ψ(x) + Γ ∂G 0 (x, y) ∂n(y) ψ(y) − G 0 (x, y)ψ n (y) dy = Nc k=1 q k I G 0 (x, z k ), 1 2 ψ(x) − Γ ∂G κ (x, y) ∂n(y) ψ(y) − I E G κ (x, y)ψ n (y) dy = 0.(2)
This system of boundary integral equations is the starting point of many existing numerical algorithms for the linearized Poisson-Boltzmann equation.
In our algorithm, we adopt the integral formulation proposed in [44]. This formulation reads:
1 2 1 + E I ψ(x) + Γ ∂G 0 (x, y) ∂n(y) − E I ∂G κ (x, y) ∂n(y) ψ(y)dy − Γ (G 0 (x, y) − G κ (x, y)) ψ n (y)dy = Nc k=1 q k I G 0 (x, z k ),(3)1 2 1 + I E ψ n (x) + Γ ∂ 2 G 0 (x, y) ∂n(x)∂n(y) − ∂ 2 G κ (x, y) ∂n(x)∂n(y) ψ(y)dy − Γ ∂G 0 (x, y) ∂n(x) − I E ∂G κ (x, y) ∂n(x) ψ n (y)dy = Nc k=1 q k I ∂G 0 (x, z k ) ∂n(x) .(4)
The first equation in this formulation, (3), is simply the linear combination of the two equations in (2), while the second equation in this formulation, (4), is nothing but the linear combination of the derivatives of the two equations in (2). It is shown in [44] that the potentially hypersingular integral in (4), involving the second derivatives of G 0 and G κ is actually integrable on Γ, thanks to the fact that
∂ 2 G 0 (x, y) ∂n(x)∂n(y) − ∂ 2 G κ (x, y) ∂n(x)∂n(y) ∼ O(|x − y| −1 ), |x − y| → 0.
Moreover, when κ = 0, (3) is decoupled from (4), and the latter provides an explicit formula for evaluating ∂ψ/∂n using ψ.
The main benefit of the formulation (3)-(4) is that it typically leads to, after discretization, linear systems with smaller condition numbers than the formulation in (2). The typical boundary element methods for this system (and others) require careful triangulation of the interface Γ; see e.g. [1,2,15,44,51,54,81]. In the next subsection, we describe our method to discretize the boundary integral system (3) and (4) on a subset of a uniform Cartesian grid nodes in a narrowband surrounding Γ, without the need to parameterize Γ.
Implicit boundary integral method
Let the interface Γ be a closed surface (in two or three dimension) that is smooth enough so that the distance function to Γ is differentiable in a neighborhood around it. Let d Γ denote the signed distance function to Γ that takes the negative sign for points inside the region enclosed by Γ, and Γ denote the set of points whose distance to Γ is smaller than . An implicit boundary integral formulation of a surface integral defined on Γ is derived by projecting points in Γ onto their closest points on Γ. With the distance function to Γ, the projection operator can be evaluated by
P Γ x := x − d Γ (x)∇d Γ (x).(5)
When is smaller than the maximum principal curvatures of Γ, the closest point projection is well-defined in Γ ; see Figure 1 for an illustration.
An implicit boundary integral method (IBIM) [46] is built upon the following identity:
I Γ [f ] := Γ f (x)ds(x) = Γε f (P Γ x)δ ε (d Γ (x))J(x)dx,(6)
which reveals the equivalence between the surface integral and its extension into a volume integral. We shall call the integral over Γ ε an implicit boundary integral. Some additional quantities are needed for relating the integral to the geometry of Γ; these include 1.
The extension of f (x) as a constant along the normal of Γ at x.
2. The Jacobian J(x) which accounts for the change of variables between Γ and the parallel surface that passes through x.
3. A weight function, δ compactly supported on [−ε, ε] satisfying ε −ε δ ε (η)dη = 1.(7)
In R 3 , The Jacobian J takes the explicit forms
J(x) = 1 − d Γ (x)∆d Γ (x) + d Γ (x) 2 ∇d, ∇ 2 d Γ ∇d Γ .(8)
It can be further related to the products of the singular values of the Jacobian matrix of P Γ , which provides an alternative, and in some cases easier way, for the computation of J. See [47]. It is shown in [48] that if the weight function has more than two vanishing moments, one may replace the Jacobian by J ≡ 1 while keeping the equality in (6) valid, even for piecewise smooth surfaces containing corners and creases.
Numerically we approximate the implicit boundary integral by embedding the computational domain Ω into rectangle U = [a, b] n , and subdivide U into uniform grid U h = hZ n with grid size h = (b − a)/N along each coordinate direction and x i at each grid point. And we approximate the integral by
I Γ [f ] ≈ S h Γ [f ] := x i ∈Γε f (x * i )δ ε (d Γ (x i ))J(x i )h n (9) where x * i = x i − d Γ (x i )∇d Γ (x i ) is the projection of x i onto Γ.
Thus, a typical second kind integral equation of the form
g(x) = λβ(x) + Γ K(x, y)β(y)ds(y), x ∈ Γ,(10)
can be approximated on U h using the IBIM formulation. One would derive a linear system for the unknown functionρ defined on the grid nodes in Γ :
g(P Γ x i ) = λβ(x i ) + h n y j ∈Γ ∩U h K(P Γ x i , P Γ y j )β(y)δ (d Γ (y j ))J(y j ), x i ∈ Γ ∩ U h ,(11)
with the property that as h → 0
β(x i ) −→ β(P Γ x i ), ∀x i ∈ Γ ∩ U h ;
i.e. the solution to the linear system (11) converges to the "constant along the surface normal extension" of the solution of (10); see more discussions in [18,46].
In the context of this paper, equations (3) and (4) will be discretized into
1 2 λ 1ψ (x i ) + h 3 j K 11 (x i , y j )ω jψ (y j ) − h 3 j K 12 (x i , y j )ω jψn (y j ) = g 1 (x i ), 1 2 λ 2ψn (x i ) + h 3 j K 21 (x i , y j )ω jψ (y j ) − h 3 j K 22 (x i , y j )ω jψn (y j ) = g 2 (x i ),(12)
where
λ 1 = 1 2 1 + E I , λ 2 = 1 2 1 + I E , ω j := J(y j )δ (d Γ (y j )),(13)g 1 (x i ) := Nc k=1 q k I G 0 (x i , z k ), g 2 (x i ) := Nc k=1 q k I ∂G 0 (x i , z k ) ∂n(x i ) ,
and K 11 , K 12 , K 21 , K 22 are respectively the regularized versions of
∂G 0 (P Γ x, P Γ y) ∂n(P Γ y) − E I ∂G κ (P Γ x, P Γ y) ∂n(P Γ y) , G 0 (P Γ x, P Γ y) − G κ (P Γ x, P Γ y), ∂ 2 G 0 (P Γ x, P Γ y) ∂n(P Γ x)∂n(P Γ y) − ∂ 2 G κ (P Γ x, P Γ y) ∂n(P Γ x)∂n(P Γ y) , and ∂G 0 (P Γ x, P Γ y) ∂n(P Γ x) − I E ∂G κ (P Γ x, P Γ y) ∂n(P Γ x) .
A simple regularization that we used in our numerical implementation is described below in the next subsection.
This formulation provides a convenient computational approach for computing boundary integrals, where the boundary is naturally defined implicitly, as a level set of a continuous function, and is difficult to parameterized. The geometrical information about the boundary is restricted to the computation of the Jacobian J and the closest point extension of the integrand f -both of which can be approximated easily by simple finite differencing applied to the distance function d Γ (x) at grid point x i within Γ ε . Furthermore, the smoothness of the weight function δ , along with the smoothness of the integrant will allow for higher order in h approximation of I[f ] by simple Riemann sum S h Γ [f ], see for example the discussion in [47].
Regularization of the kernels
While all the kernels (the Greens functions and the particular linear combinations of them) that appeared in (3)-(4) are formally integrable, additional treatment for the singularities Figure 1: A view of the "solvent excluded surface" for the molecule 2aid is shown by the green surface, and half of Γ is shown here by the space bounded between the red and the green surface.
is needed in the numerical computation when x and y are close. Typically, the additional treatment corresponds to either local change of variables so that in the new variables the singularities do not exist or mesh refinement for control of numerical error amplification (particularly for Nystrom methods). The proposed simple discretization of the Implicit Boundary Integral formulation on uniform Cartesian grid can be viewed as an extreme case of Nystrom method, in which no mesh refinement is involved. Therefore we need to regularized the kernels analytically and locally only when P Γ x and P Γ y are sufficiently close with respect to the grid spacing.
In the following, for brevity of the displayed formulas, let x * := P Γ x, y * := P Γ y and
K θ (x, y) := ∂G 0 (x * , y * ) ∂n(y * ) − θ ∂G κ (x * , y * ) ∂n(y * ) , θ ∈ R.
The regularization that we will use is defined by
K reg θ (x, y) = K θ (x), if |x * − y * | P < τ, K θ (x, y), otherwise,(14)
where |x * − y * | P is the distance between projections of x * and y * onto the tangent plane at
x * . K θ (x) is the average of K θ (x, ·) defined as K θ (x) = 1 V (x * ; τ ) V (x;τ ) K θ (x, z)ds(z),(15)
where V (x * ; τ ) is the disc of radius τ in the tangent plane of Γ at x * .
Thus,
K 11 (x, y) := K reg θ (x, y), with θ = E I , K θ (x) = 0,(16)K 22 (x, y) := K reg θ (x, y), with θ = I E , K θ (x) = 0,(17)
Similarly, the averages of G 0 − G κ and
∂ 2 G 0 ∂n(x * )∂n(y * ) − ∂ 2 Gκ
∂n(x * )∂n(y * ) are computed and we define:
K 12 (x, y) = e −κτ −1+κτ 2πκτ 2 , if |x * − y * | P < τ, G 0 (x * , y * ) − G κ (x * , y * ), otherwise,(18)K 21 (x, y) = 0, if |x * − y * | P < τ, ∂ 2 G 0 (x * ,y * ) ∂n(x * )∂n(y * ) − ∂ 2 Gκ(x * ,y * ) ∂n(x * )∂n(y * ) , otherwise.(19)
Finally, we refer the readers to [18] for a recent approach for dealing with hypersingular integrals via extrapolation.
The proposed algorithm
The proposed algorithm consists of a few stages which are outlined below:
1. Preparation of the signed distance function to the "solvent excluded surface" on a uniform Cartesian grid.
This includes initial definition of an initial level set function (Section 3.1), followed by an "inward" eikonal flow of the level set function (Section 3.1.1). After the eikonal flow, we apply a step removing from the implicit surface the interior cavities which are not accessible to solvent (Section 3.1.2). See Figure 2 for an illustration of this process and the various surfaces involved in it.
Finally we smooth out the resulting level set function by a level set reinitialization step (3.1.3). At the end of this stage, one shall obtain the signed distance function to the "solvent excluded surface" on which the Poisson-Boltzmann BIE is solved. The constructed signed distance function has the same sign as the function F , defined in (20) that is used to defined the van der Waals surface, at the prescribed molecule centers.
Preparation of the linear system.
This involves computation of geometrical information, including the closest point mapping and the Jacobian (Section 3.2).
3. Solution of the linear system via GMRES with a fast multipole acceleration for the matrix-vector multiplication (Section 3.2).
At the end of this stage, one obtained the densityψ defined on the grid nodes lying in Γ . This density function will be used in evaluation of the polarization energy.
Evaluation of surface area and polarization energy. (Sections 3.4 and 3.5)
We shall assume that Γ lies in a cubic region U and we shall discretize U by the uniform Cartesian grid hZ 3 U and denote the grid by U h . All computations will be performed on functions defined on U h . For both level set stages, the gradient ∇φ is computed with commonly used routines: i.e. the third order total variation diminishing Runge-Kutta scheme (TVD RK3) [70] for time discretization, and Godunov Hamiltonian [68] for the eikonal terms ±|∇φ| with the fifth order WENO discretization [43] approximating ∇φ. We refer the readers to the book [63] and [72] for more detailed discussions and references. We have also arranged our codes to be openly available on GitHUB. 1 ; 0 Figure 2: The construction of the "solvent excluded surface" (SES) of a fictitious molecule defined by five atoms. The final SES is shown as the solid curve on the right plot. The van der Waals surface corresponding to the molecule is shown by the blue curve. The brown curve is the "solvent accessible surface", from which an inward eikonal flow will shrink it by a distance of ρ 0 to arrived at the pink curves (solid and dashed). The dashed pink curve on the right plot shows that boundary of the cavity enclosed by the molecules. It is removed from our computation.
Creating a signed distance function to the solvent accessible surface
From molecules description the van der Waals surface, Γ vdW , is defined as the zero level set of
F (x) = inf k ( x − z k − r k ) ,(20)
where z k and r k denote respectively the coordinates of the molecule centers and their radii.
From the van der Waals surface, we shall define the so-called solvent excluded surface, Γ, as the zero level set of a continuous function φ SES . φ SES is computed by a simple inward eikonal flows, starting from an initial condition involving F , and is followed by a few iterations of the standard level set reinitialization steps. See Figure 2 for an illustration of this procedure.
Inward eikonal flow
The van der Waals surface is extended outwards for a radius ρ 0 to defined the so-called "solvent accessible surface", which can be conveniently defined as the zero level set of φ SAS :
φ SAS (x) = F (x) − ρ 0 .(21)
The inward eikonal flow will smooth out the corners in the van der Waal surface, while keeping most of its smooth parts unchanged. For 0 < t ≤ ρ 0 , we solve the following equation:
∂φ SAS (x, t) ∂t − ∇φ SAS = 0, x ∈ U, φ SAS (x, 0) = φ SAS (x),(22)
with zero Neumann boundary conditions.
Cavities removal
The zero level set ofφ SAS may contain some pieces of surfaces that isolate cavities that are believed to be void of solvent. Figure 3 provides an example of such cavities in a protein that we used for computation. The cavity removal step uses a simple sweep to remove (the boundaries of) these regions and create a level set function, φ SES , that describes only the exterior, closed and connected surface -the solvent excluded surface:
φ SAS (x, ρ 0 ) −→ φ SES (x).
1. Identify C that contained the cavities. This can be done by a "peeling" process: by marching the boundary of the computational domain inwards. The first layer of the surfaces defined by the zero level set ofφ SAS is defined to be the "solvent excluded surface". We could therefore remove the remaining portion ofφ SAS 's zero level sets, which are regarded corresponding to the cavities. From this process, one can easily compute a characteristic function supported on C .
2. Remove the cavity region by modifying the values ofφ SAS in C :
φ SES (x) := φ SAS (x), x / ∈ C , − ,
x ∈ C . Figure 3: Cavity in protein 1A63. The gray surface is the "solvent excluded surface" used for computing the electric potential for protein 1A63. The regions enclosed by the red surfaces are the cavities to be removed.
Reinitialization
The kinks on solvent accessible surface (SAS) will lower the accuracy in the computation for φ SES (x, ρ 0 ). In addition, the cavity removal step may introduce small jump discontinuities near the removed cavity. We perform several iterations of the standard level set reinitialization to improve the equivalence of the computed φ SES (x, ρ 0 ) and the signed distance function to Γ (which φ SES is supposed to be). The reinitialization equation is defined as
∂φ SES (x, t) ∂t + sgn h (φ SES (·, ρ 0 ))(|∇φ SES | − 1) = 0, φ SES (x, 0) = φ SES (x, ρ 0 ),(23)
where the smoothed-out signum function is defined as
sgn h (φ) = φ φ 2 + h 2 .(24)
Suppose that the reinitialization equation is solve until t = t n , i.e.φ SES (x, t n ) is our approximation to the signed distance function d Γ (x), we shall define Γ by
Γ := {x ∈ R d : − <φ SES (x, t n ) < }.(25)
The smoothness of the signum function sgn h may influence the efficiency and effectiveness of the reinitialization procedure. In our simulations, with the regularized signum function defined in (24), it suffices to solve (23) for O( ) amount of time. With ∆t = C 0 h, and = C 1 h, we run constant number of time step for reinitialization, independent of h. We refer the readers to [21] for some more detailed discussion on reinitialization of level set functions and (closest point) extension of functions from Γ to Γ .
Projections and weights
We locate all grid point x i ∈ U h satisfying that |φ SES (x i )| < ε and compute projections
x * i ∈ Γ by x * i = x i −φ SES (x i , t n )∇φ SES (x i , t n ).(26)
∇φ SES (x i , t n ) can be approximated either by standard central differencing or by the WENO routines. More precisely, on each grid node for each Cartesian coordinate direction, WENO returns two approximations of ∇ψ, say p − and p + , which are generalizations of the standard forward and backward finite differences of ψ. In our numerical simulations, we use
∇φ SES ≈ p − + p + 2 .
For weight function δ ε , we adopt following cosine function with vanishing first moment,
δ ε (η) = 1 2 1 + cos ηπ ε , |η| < , 0, |η| ≥ 0.(27)
For smooth integrands, the above weight function provides at least second order in h convergence, since the sum S h Γ [·] is equivalent to the Trapezoidal rule defined on uniform Cartesian grids. Formally, without loosing orders of accuracy, we can simply take zeroth order approximation of the Jacobian, i.e. we shall set J(x) ≡ 1 in our computation.
Fast linear solvers
Equations (12)- (13) in Section 2.2, together with the regularization of the kernels described in Section 2.2.1, one arrives at the final linear system:
Λp + KWp = g,(28)
with p denoting the vector containing bothψ(x j ) andψ n (x j ),
Λ := 1 2 1 + E I I 0 0 1 + I E I ,
and W is a diagonal matrix defined by the weights ω j := J(y j )δ (d Γ (y j )) as defined in Section 2.2. We solve this system by a standard GMRES algorithm. In the GMRES algorithm, we use the black-box fast multipole method (BBFMM) [30] to accelerate the multiplication of the operator K to any vector. In particular, the solution of the diagonal part of (28) is used as a preconditioner. This means that the GMRES algorithm starts with the particular initial condition:
p (0) := (Λ + D) −1 g,
where D := 0 0 0 0 coming from the regularization of the kernels.
Computing surface area
In our IBIM approach, evaluation of the surface area of Γ is computed by S h Γ [f ] defined in (9) with f ≡ 1.
Computing the polarization energy
The polarization energy G pol of the system is given by
G pol = 1 2 Nc k=1 q k ψ rxn (z k )(29)
where ψ rxn (z k ) is computed by evaluating the following boundary integral at the center of atom k, z k :
ψ rxn (z) = Γ E I ∂G κ (z, y) ∂n(y) − ∂G 0 (z, y) ∂n(y) ψ(y) + (G 0 (z, y) − G κ (z, y)) ∂ψ ∂n (y) ds(y).
In our IBIM approach, evaluation of this integral is computed by
S h Γ [f (z, ·)] defined in (9) with f (z, y) := E I ∂G κ (z, P Γ y) ∂n(y) − ∂G 0 (z, P Γ y) ∂n(y) ψ (y)+(G 0 (z, P Γ y)−G κ (z, P Γ y)) ∂ψ ∂n (y). (30)
Numerical experiments
We now perform some numerical experiments using the computational algorithm we developed. In all the numerical simulations, we set the dielectric parameters I = 1.0, E = 80.0 and Debye-Hückel constant κ = 0.1257Å −1 . We use the following parameters for the implicit boundary integral method:
h denotes the grid spacing in the uniform Cartesian grids, ≡ 2h denotes the width for the narrowband Γ , τ = h or h/2 denotes the regularization parameter used in K 11 , K 12 , K 21 , K 22 .
We set the tolerance in the GMRES algorithm to be 10 −5 , and use 4th order Chebyshev polynomials in the BBFMM preconditioner to achieve tolerance 10 −4 there. Most of the numerical experiments are performed on a desktop with quad-core [email protected], 16GB RAM. The computations involving more than one million unknowns are performed on a Linux computer with sufficient memory; for convenience in comparison, the timings presented in the tables below for simulations performed on this computer are scaled according to the clock speed and processor differences between the two computers. We put an * sign next to the scaled CPU timings in the Tables.
In Section 4.1, we first compare the surface areas computed by our method to the ones computed by a published algorithm. In all later subsections, we present simulations of our algorithms with molecules of different sizes. In certain examples, we compare our computational results to the available published data. Particularly, we perform simulations on more realistic benchmark macromolecules taken from the RCSB Protein Data Bank (PDB) [11], and add missing heavy atoms through software PDB2PQR [26]. The atom charges and radius parameters that we will be using in our simulations are all generated through force field CHARMM [16]. The number of atoms reported in each subsection below correspond to the number in the respective pqr file of each molecule.
Here are some general remarks on the numerical simulations using our algorithm. As we shall see from the numerical accuracy study in Section 4.2, the foremost bottleneck of the proposed algorithm is the low order regularization for the singular integrals. However, regularization is essential, and smaller amount of it (smaller values of τ ) leads to systems which require more GMRES iterations unless the the grid spacing h is sufficiently small. See, for instance, the simulations presented in Tables 4, 5 and 6. Despite the regularization issue, the boundary integrals can be computed very accurately if wider (with respect to the grid spacing h) and the full expression of the Jacobian J are used. However, wider implies larger dense linear system to be solved. Most of the reported computation times are spent on the evaluation of the matrix-vector multiplications. Thus, in the simulations presented below, we choose a regime in which is narrow but sufficient in practice for the adopted simple quadrature to resolve the surface geometry. Finally, regarding to how small h should be for a given molecule and probe size ρ 0 :
h ≈ min j=1,··· ,N {ρ 0 , r j , 2 }/7 < where
is the minimal distance between "different parts of the surface" (think of the thin part of a dumb bell). We shall see in the following examples, that our algorithm seems to perform well even the discretized system is slightly outside of the above regime.
Molecular SES surface area
We compare the performance of our algorithm for calculating surface areas of different proteins with that of the MSMS (Michel Sanner's Molecular Surface) algorithm developed in [69]. For MSMS algorithm, the probe radius is set to be ρ 0 = 1.4Å and the density parameter is to be 1.0 for mesh generation. We use the online implementation by the High-Performance Computing at the NIH group [38] to produce the data for MSMS. In Table 1, we compare results from our method to these from MSMS for seven different proteins on a grid of size 128 3 . We observe that the surface areas computed by our algorithm are quite close to the MSMS's approximate values in general. Since the MSMS results are only approximations to the true values, we did not attempt to tune algorithmic parameters of our method to obtain results that are even closer to the MSMS results.
The single ion model
We start with the single ion model developed in [45] to benchmark the solution accuracy of our numerical algorithm. We use three different relative errors, between exact and numeri-
solution error = Γ |ψ(x) − ψ * (x)| 2 + | ∂ψ(x) ∂n − ∂ψ * (x) ∂n | 2 Γ |ψ * (x)| 2 + | ∂ψ * (x) ∂n | 2 , area error = |A − A * | A * , energy error = |G pol − G * pol | G * pol .(31)
For a single atom with radius r and charge q, the solution to the Poisson-Boltzmann equation is given as [45]
ψ * (x) = q 4π I |x| + q 4πr 1 E (1 + κr) − 1 I , if |x| < r qe −κ(|x|−r) 4π (1 + κr)|x| , otherwise(32)
We can therefore compute the associated polarization energy
G * pol = q 2 8πr 1 E (1 + κr) − 1 I ,(33)
using the fact that the surface is a sphere with area A * = 4πr 2 . We set the atom's radius to be r = 1Å and assigned charge to be q = 1e c .
We performed simulations under different mesh and IBIM parameters. The results are summarized in Table 2. Our method converges in very small numbers (usually 3 ∼ 4) of iterations. This benchmark calculation shows that our numerical algorithms can indeed achieve similar solution accuracies to those achieved by other algorithms developed recently [2,15].
Protein 1A63
In this numerical example, we compute the polarization energy for protein 1A63, the E.Coli Rho factor, of the Protein Data Bank. The protein has 2065 atoms with different radii. The information on the locations and radii of the atoms are all available in [11]. In Figure 4 we plot the potentialψ on the constructed "solvent excluded surface", computed on two different grids, 128 3 (left) and 256 3 (right). Further computational results are tabulated in Table 4. The computed values of the polarization energy G pol can be compared to the existing estimations, G TABI pol = −2374.64 kcal/mol from the treecode-based boundary integral solver TABI [32] and G APBS pol = −2350.58 kcal/mol from the finite difference solver APBS [73].
Protein 2AID
Here we compute the polarization energy for protein 2AID, a non-peptide inhibitor complexed with HIV-1 protease. This protein has 3130 atoms. In Figure 5 we plot the potentialψ on the constructed "solvent excluded surface" of this protein, computed on two different grids. Further computational results are tabulated in Table 4.
Protein 1F15
In this example, we compute the polarization energy for protein 1F15, the cucumber mosaic virus. The protein has 8494 atoms. In Figure 6 we plot the potentialψ on the constructed "solvent excluded surface", computed on two different grids. Further computational results are tabulated in Table 5.
Protein 1A2K
In this numerical example, we compute the polarization energy for protein 1A2K, the GTPase RAN-NTF2 complex. The protein has 13627 atoms. In Figure 7 we plot the potentialψ on the constructed "solvent excluded surface", computed on two different grids. Further computational results are tabulated in Table 6. In this example, we compute the polarization energy for proteasome from thermoplasma acidophilum (PDB id: 1PMA) with 93017 atoms. In Figure 8 we plot the potentialψ on the constructed "solvent excluded surface", computed on two different grids. Further computational results are tabulated in Table 4.7.
Concluding remarks
We present in this paper a new numerical method for solving the boundary value problem of the linearized Poisson-Boltzmann equation, which is widely used to model the electric potential for macromolecules in solvent. Our new method relies on the standard level set method [63,64] for preparing the distance function to the "molecular surface". Contrary to the typical level set method, in which some partial differential equations are discretized with some suitable boundary conditions, ours involved the solution of an integral equation which is derived from an implicit boundary integral formulation [46]. Similar to the typical level set methods, and contrary to the typical boundary integral methods, the proposed method involve computation only with functions defined on uniform Cartesian grids. Our numerical simulations show that in addition to the flexibility that comes from the level set methods, the proposed method can be as computationally efficient as other boundary integral based algorithms. We show by our numerical simulations that the solutions of the resulting linear systems can be accelerated easily by some existing fast multipole methods. Furthermore, the other stages of the proposed algorithm rely on widely available explicit solvers and can be trivially parallelized.
There are several possible improvements that could be investigated in the future. First of all, the quadrature for the implicit boundary integral formulation can be improved to increase the order of accuracy. This includes improvement of the regularization of the kernel singularities and the use of full expression of the Jacobian J. One may also consider different grid geometries, as the underlying mathematical formulation do not require uniform Cartesian grids. For example, the adaptive oct-tree structure used in [37] or radial basis functions may be considered.
As all the presented simulations were computed on two moderate desktop computers, the reported results show the potential of the proposed method for molecular dynamics simulations involving very large molecules.
Finally, let us mention that the numerical method we proposed here can be generalized to solve many similar model problems for electrostatics in related areas of electrochemistry [41,58,60,35,36].
Figure 4 :
4The electrostatic potential on the surface of the PDB-1A63 protein. Left: on grid 128 3 . Right: on grid 256 3 .
Figure 5 :
5The electrostatic potential on the surface of protein 2AID for two grids of size 128 3 (left) and 256 3 (right).
Figure 6 :
6The electrostatic potential on the surface of protein 1F15 for two different grids of sizes 128 3 (left) and 256 3 (right).
Figure 7 :
7The electrostatic potential on the surface of protein 1A2K for two different grids of sizes 128 3 (left) and 256 3 (right).
Figure 8 :
8The electrostatic potential on molecular surface for the proteasome from thermoplasma acidophilum (PDB id: 1PMA). Left: the potential computed on a 128 3 grid. Right: the potential computed on a 512 3 grid.
Table 1 :
1Comparison of area between our method and MSMS for seven different proteins from the RCSB Protein Data Bank. The middle column is calculated by our method.Protein id Area Area (MSMS)
4INS
4732
4761
1HJE
825
801
1A2B
7540
7936
1PPE
7979
8340
2AID
8061
8304
1F15
22000
22725
1A63
6583
6659
cally represented quantities, to measure the quality of numerical solutions. They are defined
as:
Table 2 :
2Benchmarking errors in solution of the single ion model.grid size
h(Å)
/h D.O.F. GMRES solution error area error energy error
64 3
3.91E−1 1
22,756
4
1.11E−02
1.24E−03
1.21E−02
128 3
1.95E−1 1
91,564
3
6.90E−03
2.88E−04
5.68E−03
256 3
9.77E−2 1 366,868
3
3.57E−03
5.01E−05
2.90E−03
64 3
3.91E−1 0.5
22,756
2
5.93E−03
1.24E−03
5.98E−03
128 3
1.95E−1 0.5
91,564
4
3.45E−03
2.88E−04
2.49E−03
256 3
9.77E−2 0.5 366,868
3
2.51E−03
5.01E−05
1.31E−03
Table 3 :
3Numerical results on protein 1A63 under different algorithmic parameters.grid size
h(Å)
τ /h D.O.F. GMRES G pol (kcal/mol) CPU (s) area (Å
2 )
128 3
6.11E−1
1
71,597
12
-2392.22
606.8
6583
256 3
3.05E−1
1
293,627
13
-2366.40
3041
6801
128 3
6.11E−1 0.5
71,597
13
-2345.41
772.3
6583
256 3
3.05E−1 0.5 293,627
14
-2347.74
3808
6801
Table 4 :
4Numerical results on protein 2AID under different algorithmic parameters.grid size
h(Å)
τ /h D.O.F. GMRES G pol (kcal/mol) CPU (s) area (Å
2 )
128 3
5.80E−1
1
97,108
13
-2318.69
940.7
8061
256 3
2.90E−1
1
397,930
14
-2321.72
4521
8335
128 3
5.80E−1 0.5
97,108
24
-2282.14
1745
8601
256 3
2.90E−1 0.5 397,930
15
-2306.70
4906
8335
Table 5 :
5Numerical results on protein 1F15 under different algorithmic parameters.grid size
h(Å)
τ /h
D.O.F. GMRES G pol (kcal/mol) CPU (s) area (Å
2 )
128 3
7.72E−1
1
147,463
17
-7770.00
2586
22000
256 3
3.86E−1
1
613,726
21
-7818.83
12357
22847
512 3
1.93E−1
1
2,497,309
25
-7891.05
38068*
23238
128 3
7.72E−1 0.5
147,463
31
-7682.67
4667
22000
256 3
3.86E−1 0.5
613,726
26
-7774.76
14129
22847
512 3
1.93E−1 0.5 2,497,309
29
-7875.86
43906*
23238
Table 6 :
6Numerical results on protein 1A2K under different algorithmic parameters.grid size
h(Å)
τ /h
D.O.F. GMRES G pol (kcal/mol) CPU (s) area (Å
2 )
128 3
1.08E+0
1
99,783
12
-7190.38
1032
29497
256 3
5.42E−1
1
429,451
17
-8902.62
6120
31501
384 3
3.61E−1
1
983,418
17
-8920.93
11948
32047
512 3
2.71E−1
1
1,765,673
18
-8963.12
21247*
32364
128 3
1.08E+0 0.5
99,783
21
-9004.68
2309
29497
256 3
5.42E−1 0.5
429,451
43
-8789.45
15528
31501
384 3
3.61E−1 0.5
983,418
40
-8859.47
26030
32047
512 3
2.71E−1 0.5 1,765,673
23
-8921.64
25465*
32364
4.7 Protein: 1PMA
Table 7 :
7Protein 1PMA. IBIM's result of relative error in polarization energy and total run time w.r.t different grid sizes.grid size
h(Å)
τ /h
D.O.F. GMRES G pol (kcal/mol) CPU (s)
area (Å
2 )
128 3
1.67E+0
1
222,478
15
-15544.35
3360
1.6014E+5
256 3
8.35E−1
1
1,026,938
18
-46865.78
11784* 1.8034E+5
384 3
5.56E−1
1
2,429,367
20
-50071.90
35773* 1.8851E+5
512 3
4.17E−1
1
4,418,314
22
-50144.63
59067* 1.9262E+5
https://github.com/GaZ3ll3/ibim-levelset
AcknowledgmentsZhong and Ren are partially supported by the National Science Foundation through grants DMS-1321018 and DMS-1620473. Tsai is partially supported by the National Science Foundation through grants DMS-1318975 and DMS-1720171, and Army Research Office Grant No. W911NF-12-1-0519. We would like to thank Professor Chandrajit Bajaj for motivating conversations on topics related to this paper.
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| [
"https://github.com/GaZ3ll3/ibim-levelset"
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[
"Magnetohydrodynamic motion of a two-fluid plasma",
"Magnetohydrodynamic motion of a two-fluid plasma"
] | [
"J W Burby \nCourant Institute of Mathematical Sciences\n10012New YorkNew YorkUSA\n"
] | [
"Courant Institute of Mathematical Sciences\n10012New YorkNew YorkUSA"
] | [] | The two-fluid Maxwell system couples frictionless electron and ion fluids via Maxwell's equations. When the frequencies of light waves, Langmuir waves, and single-particle cyclotron motion are scaled to be asymptotically large, the two-fluidMaxwell system becomes a fast-slow dynamical system. This fast-slow system admits a formally-exact single-fluid closure that may be computed systematically with any desired order of accuracy through the use of a functional partial differential equation. In the leading order approximation, the closure reproduces magnetohydrodynamics (MHD). Higher order truncations of the closure give an infinite hierarchy of extended MHD models that allow for arbitrary mass ratio, as well as perturbative deviations from charge neutrality. The closure is interpreted geometrically as an invariant slow manifold in the infinite-dimensional two-fluid phase space, on which two-fluid motions are free of high-frequency oscillations. This perspective shows that the full closure inherits a Hamiltonian structure from two-fluid theory. By employing infinite-dimensional Lie transforms, the Poisson bracket for the all-orders closure may be obtained in closed form. Thus, conservative truncations of the single-fluid closure may be obtained by simply truncating the single-fluid Hamiltonian. Moreover, the closed-form expression for the all-orders bracket gives explicit expressions for a number of the full closure's conservation laws. Notably, the full closure, as well as any of its Hamiltonian truncations, admits a pair of independent circulation invariants. | 10.1063/1.4994068 | null | 119,349,050 | 1705.02654 | b8e23b6a3de48720627758752a8f27a43c94ebbf |
Magnetohydrodynamic motion of a two-fluid plasma
7 May 2017
J W Burby
Courant Institute of Mathematical Sciences
10012New YorkNew YorkUSA
Magnetohydrodynamic motion of a two-fluid plasma
7 May 2017(Dated: 9 May 2017)
The two-fluid Maxwell system couples frictionless electron and ion fluids via Maxwell's equations. When the frequencies of light waves, Langmuir waves, and single-particle cyclotron motion are scaled to be asymptotically large, the two-fluidMaxwell system becomes a fast-slow dynamical system. This fast-slow system admits a formally-exact single-fluid closure that may be computed systematically with any desired order of accuracy through the use of a functional partial differential equation. In the leading order approximation, the closure reproduces magnetohydrodynamics (MHD). Higher order truncations of the closure give an infinite hierarchy of extended MHD models that allow for arbitrary mass ratio, as well as perturbative deviations from charge neutrality. The closure is interpreted geometrically as an invariant slow manifold in the infinite-dimensional two-fluid phase space, on which two-fluid motions are free of high-frequency oscillations. This perspective shows that the full closure inherits a Hamiltonian structure from two-fluid theory. By employing infinite-dimensional Lie transforms, the Poisson bracket for the all-orders closure may be obtained in closed form. Thus, conservative truncations of the single-fluid closure may be obtained by simply truncating the single-fluid Hamiltonian. Moreover, the closed-form expression for the all-orders bracket gives explicit expressions for a number of the full closure's conservation laws. Notably, the full closure, as well as any of its Hamiltonian truncations, admits a pair of independent circulation invariants.
I. INTRODUCTION
Ideal magnetohydrodynamics 1 (MHD) is a well-known reduced plasma model that treats a plasma as a single conducting fluid. Because real plasma is made up of a large collection of discrete particles, it is natural to wonder how such a single-fluid model could have any predictive capability. This challenging problem has been addressed on numerous occasions 2 , and most assessments conclude that MHD does a good job of predicting plasma equilibrium and stability at "large-scales." However, this answer is unsatisfactory for a variety of reasons.
Most importantly, it does not tell us why MHD works, only that it does. In contrast, there must be some physical mechanism that enables a many-particle-field system to exhibit MHD behavior. Previous investigations provide only vague suggestions of what this mechanism might be.
The purpose of this article is to study an important part of this mechanism. Before elaborating further, however, the sense in which a "part" of the mechanism can even be discussed is worthy of explanation. One possible way to exhibit a mechanism for emergent MHD behavior in a many-particle system is to first show and explain emergent multi-fluid behavior, and then explain how MHD emerges from multi-fluid dynamics. This approach naturally breaks the mechanism into two parts, and this article will discuss (the simpler) one, namely the submechanism by which MHD behavior emerges from the dynamics of multiple charged fluids. Of course, if there is no submechanism for multi-fluid dynamics to emerge in a many-particle model, then the discussion contained in this article would be neither novel nor useful. Therefore the assumption that multi-fluid dynamics can indeed be found within many-particle dynamics will be tacitly assumed henceforth.
Roughly speaking, the dynamical content that is missing from MHD consists of rapidly oscillating modes, including Langmuir waves and light waves. Therefore one tempting explanation for the emergence of MHD motion in a two-fluid plasma is the effective damping of these rapidly varying modes. Even though the ideal two-fluid-Maxwell system does not include collisional dissipation, this damping mechanism may still be present as a result of phase mixing 3 . A second possible explanation may be that the rapidly oscillating modes do not damp, but instead are effectively averaged out. If this explanation is valid, it would be especially interesting because it would suggest that there must be some kind of ponderomotive forcing 4 of the MHD state variables by the rapidly oscillating modes that has not been calculated previously. Finally, there is at least one other possible explanation. Perhaps there are special initial configurations of a two-fluid plasma that do not excite the rapidly oscillating modes at all. In other words, it may be that Langmuir waves and light waves are neither damped, nor averaged-out, but instead fail to be excited in the first place. While all of these possible explanations for emergent MHD behavior may be interesting, interrelated, or perhaps mutually independent, the ensuing discussion and analysis will focus on the third possible mechanism, which is convenient to refer to as "lazy high-frequency modes."
An oversimple caricature of lazy high-frequency modes consists of two pendula placed in a room, one much longer than the other. The most general motion of these pendula (assuming small amplitude oscillations, for simplicity) involves each pendulum swaying at its respective characteristic frequency, and therefore involves a pair of disparate time scales.
However, there are special "slow" motions of the system wherein the short pendulum is motionless, meaning that the fast time scale in the problem is not present. Note that these motions are characterized by special initial conditions that allow only the long pendulum to be displaced from its equilibrium location in phase space. Also note that the slow dynamics is governed by Newton's Second Law applied to just the long pendulum, which is a dynamical system whose dimension is less than that of the total system. Here the short pendulum is the analogue of the rapidly-varying modes in the two-fluid model, while the long pendulum represents MHD motion. Although the special "slow" initial conditions are obvious in this toy problem, the same cannot be said of two-fluid dynamics. There, all of the modes are coupled nonlinearly, and so it is not clear that slow initial conditions even exist, let alone possess a simple parameterization.
In order to argue that slow initial conditions for two-fluid dynamics do exist, this article will deduce three technical results: (a) an asymptotic expansion for the set of slow two-fluid initial conditions, (b) an asymptotic expansion of the reduced dynamical equations that govern slow dynamics, and (c) the variational and Hamiltonian structures underlying the slow dynamics, which are naturally inherited from the corresponding structures underlying two-fluid dynamics. Interestingly, the mathematical tools that will lead to these results are powerful enough to provide a simple, closed-form expression for the slow dynamics' Poisson bracket.
Modulo delicate issues related to convergence of the asymptotic expansions (see Section V for a discussion of this point), these results will show: (1) that there is a collection of slow initial conditions for the two-fluid system of equations that is parameterized by the MHD phase space, (2) that the reduced equations governing the slow dynamics are equivalent to an extended MHD model with low-order truncations that reproduce ideal MHD and Hall MHD, and (3) that the Hall MHD Poisson bracket governs the slow dynamics to all orders. Taken together, (1) and (2) imply that lazy high-frequency modes may indeed be a plausible mechanism for MHD-like motion of a two-fluid plasma. Moreover, (3) implies that the problem of developing dissipation-free approximations of this MHD-like motion reduces to finding an approximate expression for the slow dynamics' Hamiltonian functional. This is a desireable feature for an extended MHD theory to have, for instance, when using such a model to study collisionless reconnection.
II. TWO-FLUID DYNAMICS: SCALING AND VARIATIONAL
PRINCIPLE
The asymptotically-scaled ideal two-fluid-Maxwell system is given by
m i n i (∂ t u i + u i · ∇u i ) = −∇p i (1) − 1 ǫ Z i q e n(E + u i × B) m e (Z i n i + ǫ 2 ǫ o q −1 e ∇ · E)(∂ t u e + u e · ∇u e ) = (2) −∇p e + 1 ǫ q e (Z i n i + ǫ 2 ǫ o q −1 e ∇ · E)(E + u e × B) ∂ t n i + ∇ · (n i u i ) = 0 (3) ∇ × B = 1 ǫ µ o q e Z i n i (u e − u i ) (4) +ǫ µ o ǫ o (∂ t E + u e ∇ · E) ∇ × E = −∂ t B(5)
where n i is the ion number density, u σ is the velocity of species σ, p σ is the species-σ partial pressure, B is the divergence-free magnetic field, E is the electric field, m σ is the species-σ mass, Z i is the ionic atomic number, q e is (minus) the elementary unit of charge, and µ o , ǫ o are the usual MKS vacuum permeability and permittivity. I will assume that the spatial domain is the 3-torus; non-periodic boundary conditions will require a separate analysis.
Upon setting the mass ratio m e /m i = ν, it is also useful to introduce the scalar fields
v 2 A = 1 1 + νZ i µ −1 o |B| 2 m i n i ω 2 p = (1 + νZ i ) q 2 e Z i n i ǫ o m e (6) ω ci = − q e Z i |B| m i ,
which represent the (squared) speed of Alfvén waves, the (squared) frequency of Langmuir oscillations, and the frequency of ion cyclotron motion.
In their order of appearance, the equations comprising the two-fluid Maxwell system express the conservation of momentum for ions and electrons, the conservation of ion number, the Ampére-Maxwell Law, and Faraday's Law. For the sake of simplicity, I have assumed a barotropic equation of state for both electrons and ions
p i = p i (n i ) (7) p e = p e (Z i n i + ǫ 2 ǫ o q −1 e ∇ · E),(8)
but most of the ensuing discussion would only be modified superficially upon adopting an equation of state that accounts for entropic dynamics. Notice that the electron number density does not appear explicitly in the two-fluid Maxwell system as it is written above. This is accomplished by using Gauss' Law to eliminate the electron number density in favor of the electric field and the ion number density. The two-fluid state is therefore given by the tuple of fields Z = (n i , u i , B, u e , E), and the two-fluid-Maxwell system may be regarded as a first-order ODE on Z-space, i.e. Z-space is the (infinite-dimensional) two-fluid-Maxwell phase space. Note that if the electron number density were not eliminated, the two-fluid
Maxwell system would instead take the form of a differential-algebraic system on a slightly larger space. Because I will evetually use some ideas from dynamical systems theory to analyze two-fluid dynamics, this would be a technical inconvenience.
The unscaled (ǫ = 1) two-fluid-Maxwell system, which is perhaps more familiar than the scaled version, may be transformed into the scaled two-fluid-Maxwell system by making the simple substitutions
q e → 1 ǫ q e (9) ǫ o → ǫǫ o .(10)
Formally, rescaling the elementary charge and vacuum permittivity is equivalent to working with dimensionless variables and then adopting the "drift-kinetic ordering"
ω ω ci ∼ ρ i L ∼ v 2 the c 2 ∼ ǫ (11) β ∼ m e m i ∼ u o v the ∼ u E v the ∼ 1,(12)
where v the is the electron thermal velocity,
ρ i = m e /m i v the /ω ci is the ion gyroradius, c = √ µ o ǫ o −1 is the speed of light, β = µ o m i n i v 2 thi /B 2 o is the plasma β, u o is the characteristic flow speed, u E = E o /B o is the characteristic E × B speed,
and ω, L represent the characteristic time and length scales for the slow dynamics. In this article, the dimensional MKS unit system will be adopted instead of the natural dimensionless unit system implied by the drift kinetic ordering. Physical expressions with the correct units may therefore always be recovered by setting ǫ = 1. Physically, the drift kinetic ordering implies that the observation time scale is much longer than the ion cyclotron period, while the observation length scale is much longer than the ion gyroradius. Thus, the usual assumptions underlying guiding center theory are valid in the drift kinetic ordering. The physical interpretation of the remainder of the drift kinetic ordering is: non-relativistic electrons ( v 2 the c 2 ∼ ǫ), arbitrary mass ratio and plasma beta (β ∼ me m i ∼ 1), bulk flow speed comparable to E × B-speed, and E × B-speed comparable to the electron thermal speed ( uo v the ∼ u E v the ∼ 1). This scaling will be leveraged in what follows to find asymptotic expansions for both the special slow two-fluid configurations as well as the reduced evolution law for the slow dynamics.
An important property of the scaled two-fluid-Maxwell system is that it may be derived from a phase space variational principle. The phase space Lagrangian is given by
L = (Z i n i + ǫ 2 ǫ o q −1 e ∇ · E)(m e u e + ǫ −1 q e A) · v e d 3 x + n i (m i u i − ǫ −1 Z i q e A) · v i d 3 x − ǫǫ o E ·Ȧ d 3 x − H,(13)
where the Hamiltonian functional is given by
H = 1 2 m i n i |u i | 2 d 3 x + n i U i (n i ) d 3 x + 1 2 m e (Z i n i + ǫ 2 ǫ o q −1 e ∇ · E)|u e | 2 d 3 x + (Z i n i + ǫ 2 ǫ o q −1 e ∇ · E) U e (Z i n i + ǫ 2 ǫ o q −1 e ∇ · E) d 3 x + 1 2 ǫǫ o |E| 2 d 3 x + 1 2 µ −1 o |B| 2 d 3 x.(14)
The functions U e (n e ) and U i (n i ) are the internal energy densities for electrons and ions respectively. They are determined up to unimportant additive constants by the thermodynamic identities
p σ (n σ ) = n 2 σ U ′ σ (n σ ).(15)
When computing the Euler-Lagrange equations associated with L, variations are taken with
respect to (n i , u i , A, u e , E, v i , v eδn i = −∇ · (n i ξ i )(16)δv i =ξ i + v i · ∇ξ i − ξ i · ∇v i(17)
δv e =ξ e + v e · ∇ξ e − ξ e · ∇v e .
III. PARAMETERIZATION AND TIME EVOLUTION OF SLOW INITIAL
CONDITIONS
Define a slow initial configuration of a two-fluid plasma as an initial condition for the scaled two-fluid-Maxwell system that is O(1) and whose time derivative is O(1). The condition on the time derivative is reasonable because the drift-kinetic ordering involves equating the dynamical time scale with a time scale that is long when compared with the cyclotron period. The question this section will address is "what are the slow initial configurations, and how do they evolve in time?" Satisfying answers will be obtained in an asymptotic sense.
Consider first two basic mathematical properties of the slow initial conditions.
property 1: If Z is a slow initial state, then the state Z(T ) obtained by letting Z evolve for T seconds according to the two-fluid-Maxwell system is also a slow initial state, regardless of the value of T . To see this, first observe that since the path Z(t) is free of rapid oscillations, so too is the path Z ′ (t) = Z(t + T ). But, by the uniqueness of solutions to ordinary differential equations (e.g. the two-fluid-Maxwell system), Z ′ (t) is precisely the solution of the two-fluid-Maxwell system with initial condition Z(T ). In other words, Z(T ), regarded as an initial condition, produces a slow evolution. This proves the claim.
remark: Property 1 is a statement pertaining to the whole set of slow initial conditions. In the context of dynamical systems theory 6 , the collection of slow initial conditions would be referred to as an invariant set. More generally, given a dynamical system on some phase space, an invariant set is a subset of phase space with the property that points in the subset stay inside the subset under dynamical evolution.
property 2: As ǫ → 0, the set of slow initial conditions, S ǫ (which is a subset of Z-space), must contain
S 0 = {(n i , u i , B, u e , E) | u e = u i , E = −u i × B}.(19)
In order to verify that this is true, suppose that Z ǫ = (n iǫ , u iǫ , B ǫ , u eǫ , E ǫ ) is a slow solution of the scaled two-fluid-Maxwell system. Because the solution is slow, the time derivative of Z ǫ must be O(1). Therefore the terms in Eqs. (1)-(5) that are multiplied by ǫ −1 must individually vanish as ǫ → 0. In particular,
0 = E 0 + u i0 × B 0(20)0 = E 0 + u e0 × B 0 (21) 0 = u e0 − u i0 .(22)
The only solution of this system of equations is E 0 = −u i0 × B 0 , u e0 = u i0 . Thus, Z 0 must be contained in the set S 0 . This verifies the claim.
remark: Physically speaking, the set S 0 consists of two-fluid states that are current neutral 7
(u e = u i ) and that satisfy the ideal Ohm's Law (E = −u i × B). Charge neutrality is also enforced because the electron number density
Z i n iǫ + ǫ 2 ǫ o q −1 e ∇ · E ǫ → Z i n i0 as ǫ → 0.
Thus, property 2 already provides some evidence that the slow initial conditions in the two-fluid-Maxwell system must be related to MHD. Property 2 also suggests that slow solutions of the two-fluid-Maxwell system have the property that the electric field and the electron fluid velocity are slaved to the MHD state variables (n i , u i , B), i.e. the former are expressed as functions of the latter. Such slaving relations, which may also be thought of as defining a closure, are commonplace in the theory of slow manifold reduction 8 and geometric singular perturbation theory 9 , which forms the theoretical basis underlying the discussion in this section.
Taken together, properties 1 and 2 indicate that a reasonable approach to finding the slow initial conditions is to look for an invariant subset of Z-space, S ǫ , of the form
S ǫ ={(n i , u i , B, u e , E) | u e = u * eǫ (n i , u i , B) E = E * ǫ (n i , u i , B)},(23)
where u * eǫ and E * ǫ are undetermined O(1) functions of the MHD state variables (n i , u i , B). Keeping in line with the remark below property 2, I will refer to u * eǫ and E * ǫ as the slaving functions of the slow manifold S ǫ . Admittedly, Eq. (23) is nothing more than an ansatz for the set of slow initial conditions. In particular, it is not at all obvious that S ǫ needs to exist.
Moreover, even if S ǫ does exist, that would not imply that S ǫ contains all of the slow initial conditions -in fact it is not obvious at this stage that S ǫ contains any slow initial conditions whatsoever! Nevertheless, (23) will prove to be a good ansatz for two reasons. First, it will turn out that the slaving functions have unique asymptotic expansions in powers of ǫ.
Second, it will be possible to show formally that the dynamics of two-fluid states in S ǫ are indeed slow.
By extending the argument supporting property 2, I will now derive a functional partial differential equation satisfied by the slaving functions in Eq. (23). Suppose that Z ǫ = (n iǫ , u iǫ , B ǫ , u eǫ , E ǫ ) is a solution of the scaled two-fluid-Maxwell system contained in S ǫ .
Then the electric field and the electron fluid velocity must satisfy the slaving relations
u eǫ = u * eǫ (n iǫ , u iǫ , B ǫ ) (24) E ǫ = E * ǫ (n iǫ , u iǫ , B ǫ ),(25)
and Z ǫ must be a solution of the scaled two-fluid-Maxwell equations (1)- (5). An interesting consequence of these two constraints is that the electron momentum equation and the Ampére-Maxwell Law may be written in a manner that does not involve any time derivatives. To see this, first note that the slaving relations imply the partial time derivatives of E ǫ and u eǫ are given by
∂ t u eǫ = D n i u * eǫ [∂ t n iǫ ] + D u i u * eǫ [∂ t u iǫ ] + D B u * eǫ [∂ t B ǫ ](26)∂ t E ǫ = D n i E * ǫ [∂ t n iǫ ] + D u i E * ǫ [∂ t u iǫ ] + D B E * ǫ [∂ t B ǫ ].(27)
Here the symbol D denotes Fréchet derivative (see Appendix A if unfamiliar with the Fréchet derivative). Next use the ion continuity equation, the ion momentum equation, and Faraday's Law to eliminate the partial time derivatives from the right-hand-sides of Eqs. (26) and (27), and
∂ t u eǫ = − D n i u * eǫ [∇ · (n iǫ u iǫ )] + D u i u * eǫ [u iE (n iǫ , u iǫ )] − D B u * eǫ [∇ × E ǫ ] − 1 ǫ q e m e νZ i D u i u * eǫ [E ǫ + u iǫ × B ǫ ](28)∂ t E ǫ = −D n i E * ǫ [∇ · (n iǫ u iǫ )] + D u i E * ǫ [u iE (n iǫ , u iǫ )] − D B E * ǫ [∇ × E ǫ ] − 1 ǫ q e m e νZ i D u i E * ǫ [E ǫ + u iǫ × B ǫ ](29)Here ν = m e /m i is the mass ratio andu iE (n i , u i ) = −u i · ∇u i − (m i n i ) −1 ∇p i (n i )E * ǫ (n i , u i , B): 1 ǫ q e m e νZ i D u i u * eǫ [E * ǫ + u i × B] + 1 ǫ q e m e (E * ǫ + u * eǫ × B) = u * eǫ · ∇u * eǫ + D u i u * eǫ [u iE ] + (m e n * eǫ ) −1 ∇p e (n * eǫ ) − D n i u * eǫ [∇ · (n i u i )] − D B u * eǫ [∇ × E * ǫ ](30)1 ǫ µ o q e Z i n i (u * eǫ − u i ) = ∇ × B + µ o ǫ o q e m e νZ i D u i E * ǫ [E * ǫ + u i × B] − ǫµ o ǫ o D u i E * ǫ [u iE ] + u * eǫ ∇ · E * ǫ − D n i E * ǫ [∇ · (n i u i )] − D B E * ǫ [∇ × E * ǫ ] ,(31)
where n * eǫ = Z i n i + ǫ 2 ǫ o q −1 e ∇ · E * ǫ is shorthand notation for the electron number density. I will refer to this system of functional PDE as the invariance equations.
In general the invariance equations, which are nonlinear and involve both functional and ordinary derivatives, are hopelessly difficult to solve. However, if they admit a solution that is smooth in ǫ and O(1) as ǫ → 0, then this solution has the unique asymptotic expansion
u * eǫ = u * e0 + ǫu * e1 + ǫ 2 u * e2 + . . . (32) E * ǫ = E * 0 + ǫE * 1 + ǫ 2 E * 2 + . . . ,(33)
where the coefficients u * ek , E * k are most readily obtained by substituting the asymptotic expansions into the invariance equations and then solving order by order. For instance, the leading-order invariance equations (O(ǫ −1 ) as written) are
q e m e νZ i D u i u * e0 [E * 0 + u i × B] + q e m e (E * 0 + u * e0 × B) = 0 (34) µ o q e Z i n i (u * e0 − u i ) = 0,(35)
which have the unique solution
u * e0 = u i(36)E * 0 = −u i × B,(37)
representing current neutrality and ideal Ohm's law. In general, the n'th order invariance equation determines uniquely the n'th order terms in the asymptotic expansions (32) and
(33). In particular, the O(1) invariance equations lead to
u * e1 = µ −1 o ∇ × B q e Z i n i (38) E * 1 = − µ −1 o (∇ × B) × B − ∇(p e (Z i n i ) − νZ i p i (n i )) q e Z i n i (1 + νZ i ) ,(39)
while the O(ǫ) invariance equations gives
u * e2 = − ρ MHD u i + ǫ oĖMHD q e Z i n i (40) E * 2 = (ρ MHD u i + ǫ oĖMHD ) × B (1 + νZ i )q e Z i n i + c 2 ω 2 p (∇ × B) · ∇u i + (u i · ∇)∇ × B + (∇ × B)n −1 i ∇ · (n i u i ) + ∇ × ∇ × (u i × B)(41)
where
ρ MHD = − ǫ o ∇ · (u i × B)(42)E MHD = − − u i · ∇u i + −∇(p i (n i ) + p e (Z i n i )) + µ −1 o (∇ × B) × B m i n i (1 + νZ i ) × B − u i × ∇ × (u i × B)(43)
is shorthand notation for the charge density and displacement current given by the ideal MHD model. The higher-order coefficients rapidly become very complicated, but they may be efficiently computed if desired by solving the invariance equations iteratively using a computer algebra system. Now that the slaving functions have been determined, the time evolution of two-fluid states that are contained in the slow manifold S ǫ is easy to determine. Suppose Z(t) ∈ S ǫ is two-fluid trajectory contained in the slow manifold. Combining the ion momentum equation, the ion continuity equation, Faraday's Law, and the slaving relations then implies
m i n i (∂ t u t + u i · ∇u i ) = −∇p i (n i ) − 1 ǫ q e Z i n i (E * ǫ (n i , u i , B) + u i × B)(44)∂ t n i + ∇ · (n i u i ) = 0 (45) ∂ t B = −∇ × E * ǫ (n i , u i , B),(46)
which clearly gives a closed system of equations that determine the time evolution of the MHD state (n i , u i , B). Now, with the time evolution of the MHD state determined, the time evolution of the entire two-fluid state is also determined because two fluid states contained in
S ǫ have the form (n i , u i , B, u * eǫ , E * ǫ )
. It is therefore sensible to refer to the system (44)-(46) as the slow two-fluid equations. Observe that the slaving function E * ǫ appears in the slow two-fluid equations while u * eǫ does not.
The following facts pertaining to two-fluid configurations that begin in the slow manifold S ǫ may now be inferred.
fact 1: Two-fluid states that begin on the slow manifold S ǫ remain on the slow manifold and evolve on the slow (O(1)) timescale. This fact follows from the construction of S ǫ , which guaranteed S ǫ is an invariant set, and from the expressions for the first few terms in the asymptotic expansion for E * ǫ . Indeed, by substituting the first two terms in the asymptotic expansion of E * ǫ into Eq. (44), it is simple to verify that the leading-order truncation of the slow two-fluid equations is given by
(1 + νZ i )m i n i (∂ t u i + u i · ∇u i ) = −∇(p i (n i ) + p e (Z i n i )) + µ −1 o (∇ × B) × B(47)∂ t n i + ∇ · (n i u i ) = 0 (48) ∂ t B = ∇ × (u i × B).(49)
This shows that the time derivatives Interestingly, the first correction to the leading-order slow two-fluid equations is
(∂ t n i , ∂ t u i , ∂ t E) are each O(1).(1 + νZ i )m i n i (∂ t u i + u i · ∇u i ) = −∇(p i (n i ) + p e (Z i n i )) +[µ −1 o (∇ × B) − ǫρ MHD u i − ǫǫ oĖMHD ] × B −ǫ(1 + νZ i )q e Z i n i c 2 ω 2 p (∇ × B) · ∇u i + (u i · ∇)∇ × B +(∇ × B)n −1 i ∇ · (n i u i ) + ∇ × ∇ × (u i × B)(50)∂ t n i + ∇ · (n i u i ) = 0 (51) ∂ t B = ∇ × u i × B + ǫµ −1 o (∇ × B) × B (1 + νZ i )q e Z i n i ,(52)
which is a generalization of Hall MHD 10,11 . The reason for the additional terms relative to ordinary Hall MHD is the following. In the ordinary Hall theory, deviations from charge neutrality, the displacement current, and electron inertia are completely ignored. Formally There is one set of properties possessed by the full slow two-fluid system that can be understood in a rather complete sense. These special properties pertain to the Hamiltonian structure of the slow dynamics. This section will show that the Poisson bracket governing slow two-fluid dynamics may be obtained in closed form. Using this expression, it will be possible to deduce further closed form expressions for some of the conservation laws possessed by the all-orders slow two-fluid system. In addition, this result will lead to a convenient and practically useful method for obtaining conservative truncations of the slow two-fluid system. Because the data consisting of the Poisson bracket and the Hamiltonian functional completely determines the slow two-fluid equations, the problem of truncating while preserving the conservative properties of the slow dynamics is reduced to truncating the Hamiltonian.
In order to uncover the slow two-fluid system's Hamiltonian structure, it is easiest to start from the phase space variational principle governing the full two-fluid-Maxwell system, which is embodied by the phase space Lagrangian (13). Given a solution of the two-fluid Maxwell system, the phase space variational principle states that an arbitrary variation of the action S = L dt around that solution vanishes, δS = 0. In particular, if the solution is contained within the slow manifold S ǫ , any variation of the action that does not leave the slow manifold vanishes. In other words, the action S * obtained by restricting S to curves that are contained in S ǫ has as critical points solutions of the slow two-fluid equations. This implies that the slow two-fluid system inherits a phase space variational principle from the full two-fluid Maxwell system. This slow two-fluid variational principle represents the first crucial step toward obtaining a closed for expression for the slow two-fluid Poisson bracket.
Explicitly, the slow two-fluid action is given by S * = L * dt, where the slow two-fluid Lagrangian is, in accordance with the previous paragraph, given by
L * = (Z i n i + ǫ 2 ǫ o q −1 e ∇ · E * ǫ )(m e u * eǫ + ǫ −1 q e A) · v e d 3 x + n i (m i u i − ǫ −1 Z i q e A) · v i d 3 x − ǫǫ o E * ǫ ·Ȧ d 3 x − H * ,(53)
and the slow two-fluid Hamiltonian is given by
H * = 1 2 m i n i |u i | 2 d 3 x + 1 2 m e (Z i n i + ǫ 2 ǫ o q −1 e ∇ · E * ǫ )|u * eǫ | 2 d 3 x + n i U i (n i ) d 3 x + (Z i n i + ǫ 2 ǫ o q −1 e ∇ · E * ǫ ) U e (Z i n i + ǫ 2 ǫ o q −1 e ∇ · E * ǫ ) d 3 x + 1 2 ǫǫ o |E * ǫ | 2 d 3 x + 1 2 µ −1 o |B| 2 d 3 x.(54)
Notice that in these expressions the velocity u e has been replaced with the slaving function u * eǫ while the velocity v e has not. The reason for this is that the constraint imposed by restricting to the slow manifold S ǫ involves only Eulerian quantities. The Lagrangian configuration maps for both electrons and ions are completely unconstrained on the slow manifold. Of course, one may worry that if the velocity variable u e is constrained, then there might have to be a corresponding non-holonomic constraint on the electron configuration map. This is faulty reasoning because here we are working with a phase space Lagrangian.
The paths that we vary in the action S do not generally satisfy u e = v e , even though this relationship must hold along any solution of the Euler-Lagrange equations.
Using the phase space Lagrangian L * it is possible in principle to identify the Hamiltonian structure underlying the slow two-fluid dynamics. In fact, the Poisson bracket associated with L * is encoded in the part of L * that is linear in the velocities (v i , v e ,Ȧ,u i ) -the so-called "symplectic part" of L * (see Section II.C in Ref. 12). Littlejohn 13,14 gives a lucid discussion of this point, albeit in a finite-dimensional context, in his seminal work on the Hamiltonian formulation of guiding center motion. However, the situtation is more complicated than it seems to be at first glance. Because the symplectic part of L * involves the slaving functions E * ǫ , u * eǫ , the Poisson bracket associated with L * is very complicated. In fact, as a result of the slaving functions being known only as formal infinite series in ǫ, the Poisson bracket must also be a formal infinite series in ǫ. Thus, while the Hamiltonian structure for the slow dynamics is just in reach, the sought-after closed-form expression for the Poisson bracket still lies waiting on the other side of a theoretical chasm that has yet to be crossed.
A closed-form expression for the slow dynamics' Poisson bracket may be obtained by employing some ideas from symplectic geometry. The specific ideas that are necessary are described well by Littlejohn in Ref. 15. From Littlejohn's work, a procedure may be extracted that greatly simplifies the slow system's Poisson bracket. The specific details of this procedure, being perhaps more abstract than the present discussion requires, are contained in Appendix B. However, the essential idea behind the procedure is simple to describe and to understand. The goal is merely to find a near-identity transformation of the slow manifold S ǫ such that the linear terms in the transformed Lagrangian truncate at a finite order in ǫ. If such a transformation can be found, it would lead to an easily-computable closed-form expression for the Poisson bracket. The only price of going in this direction, aside from the labor involved, would be a somewhat more complicated Hamiltonian. As explained in the Appendix, existence of a transformation with the desired properties is ensured by a straightforward application of a geometric tool known as Moser's trick (see Ch.
in Ref. 16).
One transformation that leads to an O(1) truncation of the symplectic part of L * is given by
n i = 1 + ǫ 2 νZ i 1 + νZ i ∇ · (u × B) q e Z i n n + O(ǫ 3 ) (55) u i = u + ǫ(ε −1 − 1 − bb) · v 2 A c 2Ẽ × B |B| 2 + νZ i 1 + νZ iJ q e Z i n + O(ǫ 3 ) (56) A = A − ǫ 2 m i q e Z i (1 − νZ i ) v 2 A c 2Ẽ × B |B| 2 + νZ i 1 + νZ iJ q e Z i n + 1 1 + νZ i ∇ A · ǫ o u × B q e Z i n + O(ǫ 3 ),(57)
where
ε = 1 + ǫ c 2 v 2 A (1 − bb) (58) E = −u × B + ǫE * 1 (59) J = µ −1 o ∇ × B − ǫρ MHD u − ǫǫ oĖMHD ,(60)
and
v 2 A , E * 1 ,Ė MHD , ρ MHD are merely v 2 A , E * 1 ,Ė MHD ,= νZ i 1 + νZ i mnu · v e d 3 x + 1 ǫ q e Z i nA · v e d 3 x + 1 1 + νZ i mnu · v i d 3 x − 1 ǫ q e Z i nA · v i d 3 x −H * (61)
where the transformed Hamiltonian is given byH
* =H * 0 + ǫH * 1 + ǫ 2H * 2 + O(ǫ 3 ), with H * 0 = 1 2 mn|u| 2 d 3 x + n U i (n) d 3 x + Z i n U e (Z i n) d 3 x + 1 2 µ −1 o |B| 2 d 3 x (62) H * 1 = − 1 2 mn v 2 A c 2 |u ⊥ | 2 (63) H * 2 = 1 2 mn v 4 A c 4 |u ⊥ | 2 d 3 x − (1 − νZ i ) mn v 2 A c 2 u ⊥ · µ −1 o ∇ × B (1 + νZ i )q e Z i n d 3 x − 1 2 c 2 ω 2 p µ −1 o |∇ × B| 2 d 3 x + mn v 2 A c 2 u ⊥ · u De + νZ i u Di 1 + νZ i d 3 x.(64)
Here the convenient shorthand notations u ⊥ = (1 − bb) · u and m = (1 + νZ i )m i have been introduced, as well as the so-called diamagnetic drift velocities
u Di = ∇p i (n) × B q e Z i n|B| 2 (65) u De = −∇p e (Z i n) × B q e Z i n|B| 2 .(66)
It is important to emphasize here that even though the displayed expressions for the transformation (n i , u i , A) → (n, u, A) contain only finitely-many terms, the full transformation contains infinitely-many terms. As explained in Appendix B, all of these terms may be calculated in a systematic manner, and they ensure that the symplectic part of the transformed phase space Lagrangian is displayed entirely in Eq. (61); higher-order corrections to the symplectic part are zero in the "nice" coordinate system on S ǫ . It is also worth mentioning that the transformation calculated here is not the only one that leads to a simplified symplectic part in the phase space Lagrangian.
The Poisson bracket associated withL * may be found using standard techniques. In particular, the computation may be done by inverting the Lagrange tensor 17 associated with the symplectic part ofL * as is done for various kinetic systems in Ref. 18. Proceeding in this manner is useful because it ensures that the Poisson bracket satisfies the Jacobi identity; direct verification of the Jacobi identity as in Ref. 19 is not necessary. I will merely report the result of this calculation here. Given two arbitrary functionals G(n, u, B) and H(n, u, B), their Poisson bracket is given by
{G, H} = 1 m ∇ δG δn · δH δu − ∇ δH δn · δG δu d 3 x + 1 mn B · δG δu × ∇ × δH δB − δH δu × ∇ × δG δB d 3 x(67)+ 1 mn ∇ × u · δG δu × δH δu d 3 x −ǫ 1 mn B · |B| ω ce + |B| ω ci ∇ × δG δB × ∇ × δH δB d 3 x +ǫ 2 1 mn ∇ × u · νZ i |B| 2 ω 2 ci ∇ × δG δB × ∇ × δH δB d 3 x(68)
Because this expression does not involve any infinite series, it represents the main result that was meant to be obtained in this section.
With a closed-form expression for the slow system's Poisson bracket in hand, it is now possible to deduce some general properties of the slow two-fluid system that are independent of truncation order. First and foremost, it is now clear that the slow dynamics possess Hamiltonian truncations with any desired level of accuracy. Such Hamiltonian truncations are obtained by using the full Poisson bracket in Eq. (67) while retaining only finitely-many terms in the expansion of the transformed HamiltonianH * =H * 0 + ǫH * 1 + ǫ 2H * 2 + . . . . These truncations of the slow two-fluid system are superior to naive truncations performed at the level of the equations of motion because they ensure that artificial dissipation is not introduced by truncation. An obvious manifestation of this fact is that any Hamiltonian truncation of the slow dynamics will automatically conserve the truncated Hamiltonian exactly.
This follows from the antisymmetry of the Poisson bracket. All Hamiltonian truncations of the slow two-fluid system also possess less obvious conservation laws. Notably, they all possess a pair of circulation invariants, which I will now describe.
For the sake of describing the circulation invariants precisely, let H =H * 0 + ǫH * 1 + ǫ 2H * 2 + · · · +H * n denote the n'th order truncation of the transformed Hamiltonian, where n is an arbitrary non-negative integer. The first circulation invariant, which is associated with motion of the ions, is given by
C i = ℓ i A · dx − ǫ m i q e Z i ℓ i u · dx.(69)
Here ℓ i is a closed loop that moves with the velocity
v * i = 1 mn δH δu − ǫ νZ i 1 + νZ i 1 q e Z i n ∇ × δH δB .(70)
As the notation suggests, the velocity v * i is approximately equal to the ion fluid velocity. It is well-known that the barotropic two-fluid Maxwell system has an analogous circulation invariant, which is equivalent to the circulation of the ion canonical momentum. Actually, the invariant C i is the same quantity, merely restricted to the slow manifold. The second circulation invariant, which is associated with motion of electrons, is given by
C e = ℓe A · dx + ǫνZ i m i q e Z i ℓe u e · dx.(71)
Here ℓ e is an arbitrary (not necessarily closed) curve that moves with the velocity v * e = 1 mn
δH δu + ǫ 1 1 + νZ i 1 q e Z i n ∇ × δH δB .(72)
Just like C i , C e may be interpreted as an invariant inherited from the two-fluid-Maxwell system.
V. DISCUSSION
This article has put forth a new conceptual framework for understanding MHD and its relationship with ideal two-fluid theory. The key insight that underlies this new perspective is that the two-fluid theory, when appropriately scaled, admits a formally-exact single-fluid closure. At leading order, the closure reproduces ideal MHD. At higher orders, the closure leads to new or modified exteneded MHD models. Notably, the latter allow for arbitrary mass ratio as well as perturbative deviations from exact charge neutrality. after making the identifications
u = V B = B * mn = ρ µ o m i n c 2 ω 2 pe = d 2 e √ µ o m i n c ω pi − νZ i c ω pe = d i .
The reason that the d i in Ref. 19
|F (ψ + δψ) − F (ψ) − DF (ψ)[δψ]| V |δψ| W = 0. (A1)
Of course, there is no guarantee that DF (ψ) exists. If it does, it is unique and it is said
that F is Fréchet differentiable at ψ.
The most basic property satisfied by the Fréchet derivative is the chain rule. If F : W → V and G : V → U, then
D(G • F )(ψ)[δψ] = DG(F (ψ))[DF (ψ)[δψ]],(A2)
and it is now necessary to be mindful that DG is evaluated at F (ψ) and DF is evaluated at ψ. A useful consequence of the Chain rule is that the Fréchet derivative may be computed The basic idea that enables a systematic computation of the transformation used in Section IV is best understood in terms of differential forms. Essentially the same idea is described in the proof of Darboux's theorem in Ref. 16. Let Θ = θ + δθ be a 1-form on a manifold M and suppose that ω = −dθ is non-degenerate. Non-degeneracy means that the mapping X → ι X ω, where X is a vector field on M, is injective. Suppose further that δθ ≪ θ. Under these hypotheses, it is possible to find a near-identity transformation Φ of
where the right-hand-side represents a time-ordered exponential of the vector field G t . Because G t must be small, Eq. (B4) provides an asymptotic expansion for the transformed Hamiltonian.
Next, expand the 1-form Θ in Eq. (B5) in a power series Θ = ǫ −1 Θ −1 + Θ 0 + ǫΘ 1 + ǫ 2 Θ 2 + . . . ,
and apply the transformation Φ. Modulo unimportant exact differentials, the 1-form Θ transforms intoΘ = ǫ −1Θ −1 +Θ 0 + ǫΘ 1 + ǫ 2Θ 2 + . . . ,
whereΘ
−1 = Θ −1 (B14) Θ 0 = Θ 0 − ι G 1 dΘ −1 (B15) Θ 1 = Θ 1 − ι G 1 dΘ 0 − ι G 2 dΘ −1 (B16) Θ 2 = Θ 2 − ι G 2 dΘ 0 − ι G 3 dΘ −1 − ι G 1 dΘ 1 + 1 2 (ι G 1 d) 2 Θ 0 (B17) Θ 3 = . . . .(B18)
The vector fields G k , which are usually referred to as Lie generators, are finally determined by demanding that the transformed 1-form truncates at finite-order in ǫ. There is a huge amount of freedom in this step. The prescription for finding the particular transformation that lead to the simple transformed 1-form found in Eq. (61) is given by
ι G 1 dΘ −1 = 0 (B19) Θ k = 0 k > 0.(B20)
These equations may be solved order-by-order in order to determine each of the G k . The first two Lie generators turn out to be G e 1 = 0 (B21)
G i 1 = 0 (B22) G A 1 = 0 (B23) G u 1 = v 2 A c 2 u i⊥ + νZ i 1 + νZ i µ −1 o ∇ × B q e Z i n i ,(B24)
and G e 2 = −
1 1 + νZ i ǫ o u i × B q e Z i n i (B25) G i 2 = νZ i 1 + νZ i ǫ o u i × B q e Z i n i (B26) G A 2 = 1 1 + νZ i ∇ A · ǫ o u i × B q e Z i n i − m i q e Z i (1 − νZ i ) v 2 A c 2 u i × B − νZ i 1 + νZ i µ −1 o ∇ × B q e Z i n i (B27) G u 2 = − 1 2 v 2 A c 2 G u 1⊥ + v 2 A c 2 E * 1 × B |B| 2 − νZ i 1 + νZ i ǫ oĖMHD + ρ MHD u i q e Z i n i .(B28)
Here ξ i,e are arbitrary vector fields defined on the fluid domain. They represent Eulerian displacements of the ion and electron fluids that are generated by variations of the Lagrangian trajectories. While Ref. 5 goes into much more detail on this point, it is worth mentioning that the constrained variations of (n i , v i , v e ) are actually consequences of unconstrained variations of the Lagrangian coordinates of electron and ion fluid parcels.
is the time derivative of the ion velocity in the absence of electromagnetic forces ("E" stands for Euler equation). Because the only temporal derivatives that appear in the electron momentum equation and the Ampére-Maxwell Law are ∂ t u e and ∂ t E, these manipulations suffice to eliminate all of the time derivatives from these equations. Moreover, because Z ǫ is an arbitrary solution contained in the slow manifold S ǫ , the time-derivative-free forms of the electron momentum equation and the Ampére-Maxwell Law may be read as the following system of functional partial differential equations for the unknown functionals u * eǫ (n i , u i , B)
fact 2 :
2The slow two-fluid equations are equivalent to a formally-exact extended MHD model. That slow dynamics extends ideal MHD is immediately apparent from Eqs. (47)-(49), which of course reproduce the ideal MHD model (with a renormalized ion mass).
this corresponds to enforcing ǫ o = 0 and ν = 0. However, the analysis here makes the weaker assumption ǫ o = O(ǫ) and ν = O(1), which allows for perturbative deviations from exact charge neutrality, as well as perturbative contributions to the transverse electric field and the full effects of finite electron inertia. This explains the extra terms in Eqs. (50)-(52). If a subsidiary ordering were introduced, or if I had instead assumed ǫ o = O(ǫ k ), ν = O(ǫ) with k > 1, Eqs. (50)-(52) would be identical to Hall MHD. In particular, all terms proportional to ǫ in the momentum equation would vanish. However, seeing as the ordering used in this analysis is less strict than the conventional ordering, it is entirely possible that the generalized Hall MHD equations given in Eqs. (50)-(52) are more accurate than conventional Hall MHD, especially in situations where deviations from charge neutrality are moderately important.IV. THE HAMILTONIAN STRUCTURE GOVERNING SLOWTWO-FLUID DYNAMICSThe perturbative solution of the invariance equations (30)-(31) presented in the previous section gives suggestive evidence that one possible mechanism for the emergence of MHD behavior within the two-fluid model is the existence of a slow manifold. However, one drawback of the perturbative solution is that high-order contributions to the slaving functions E * ǫ , u * eǫ , which define the slow manifold, are extremely difficult to calculate. This makes it difficult to distinguish between properties that the slow two-fluid equations genuinely possess, and properties that only particular truncations of the slow two-fluid equations possess.For instance, while the leading-order truncation of the slow two-fluid equations (i.e. MHD)gives a system of PDE that is first-order in both space and time, the Hall MHD truncation at next-to-leading order involves second-order derivatives in space. It is therefore entirely unclear what category of PDE the slow two-fluid equations fall into. In fact it is possible, if not likely, that the appearance of high-order space derivatives in high-order truncations of the slow two-fluid equations is merely an artifact of unwittingly expanding nonlocal operators in powers of ǫ.
ρ MHD evaluated using the transformed variables n, u, B = ∇ × A; recall Eqs.(6), (39), (42), and (43). The transformed Lagrangian is given byL *
enforce strict charge neutrality; the closure here only enforces charge neutrality at the leading (MHD) order).The full (i.e. all-orders) single-fluid closure is given as the solution of a certain functional PDE, Eqs. (30)-(31), which can be solved in an asymptotic sense without essential difficulty.Moreover, the solution of the functional PDE has an appealing geometric interpretation as an invariant submanifold in the infinite-dimensional two-fluid phase space, known as a slow manifold. Two-fluid motions that begin within the slow manifold remain within the slow manifold as time passes. These special solutions of the two-fluid system evolve as if they were governed by a small perturbation of the ideal MHD system. They are free of high-frequency oscillations that otherwise generally occur at the cyclotron and plasma frequency time scales. One mechanism by which a two-fluid plasma may exhibit emergent MHD behavior is therefore initialization of the plasma so that its mechanical state lies within the slow manifold.A deep corollary of the geometric interpretation of the single-fluid closure is that the closure actually inherits a Hamiltonian structure from the two-fluid model. Using infinitedimensional Lie transforms, this Hamiltonian structure, which is encoded in the form of a Poisson bracket, may be found in closed form. Therefore, strictly dissipation-free truncations of the single-fluid closure may be obtained by simply truncating an asymptotic expansion for the single-fluid Hamiltonian. Because the single-fluid Hamiltonian is the restriction of the two-fluid Hamiltonian to the slow manifold, an asymptotic expansion of the single-fluid Hamiltonian is readily computable. All such Hamiltonian truncations possess a pair of interesting independent circulation invariants, in addition to the expected invariants associated with space-and time-translation symmetry. This last fact shows that the observations in Ref. 20 pertaining to pairs of circulation invariants in some extended MHD models apply much more generally. It is insightful to compare and contrast the results in this article with previous work on ideal and extended MHD. First consider Ref. 21, where a two-fluid action was used in conjuction with asymptotic methods to deduce action principles for a selection of previously-known extened MHD models. The heart and soul of the method used in the present work to obtain the single-fluid Hamiltonian structure may be traced back to this Reference. To be more precise, the present work demonstrates that the basic principle underlying Ref. 21, namely that of applying asymptotic methods directly to a two-fluid variational principle, may be extended to identify the Hamiltonian structure of the all-orders single-fluid closure of the two-fluid system. The extension involves, amongst some other technical details, employing a two-fluid phase-space variational principle instead of a configuration space variational principle, as well as applying infinite-dimensional Lie transforms to find a closed-form expression for the all-orders Poisson bracket. There are several notable differences between the present work and that of Ref. 21. Most importantly, while the present work studies an all-orders single-fluid closure, Ref. 21 is not concerned with establishing all-orders results. As was explained earlier in Section IV, studying properties of the all-orders theory is important in order to distinguish between phenomena genuinely present in the full single-fluid closure (such as circulation invariants) and phenomena that appear only in particular low-order truncations of the closure (such as the impossibility of magnetic reconnection). On the other hand, given that the primary goal of Ref. 21 was to establish new results on previously-established extended MHD models, there was no particular need for an all-orders theory in Ref. 21.Next consider Ref.19, which identifies a Poisson bracket structure for the strictly-neutral two-fluid system derived byLüst. 22 It is conceptually satisfying to observe that the all-orders Poisson bracket given here in Eq. (67) is equivalent to the bracket given in Eq. (29) of Ref.19
example, let u and B be square-integrable vector fields on R 3 and set F (u, B) = u × B. It is not difficult to show that F takes values in the space of integrable vector fields on R 3 . The domain and range spaces for F are therefore each Banach spaces with obvious norms. The Fréchet derivative of F with respect to u isD u F (u, B)[δu] = d dǫ 0 (u + ǫδu) × B = δu × B.(A5)Appendix B: Simplifying the symplectic part of a phase-space Lagrangian
M 1 HL
1that transforms the 2-form Ω = ω + δω = −dθ − dδθ into the two-form ω. The trick is to express the inverse of transformation as the λ = 1 flow map (here λ is being used as the time parameter for the flow map in order to distinguish it from the physical time t) associated with a time-dependent vector field G λ . Let F λ be the flow map. Now choose G λ in such a way thatF * λ (ω + λδω) = ω (B1) for each λ ∈ [0, 1]. This task may be accomplished if G λ is chosen to be the unique solution of ι G λ (ω + λδω) = δθ. (B2)Note that a solution of (B2) is guaranteed to exist (at least in finite dimensions) by virtueof three key facts: (1) ω is non-degenerate, (2) λδω ≪ ω for λ ∈ [0, 1], and (3) the set of non-degenerate two-forms is open in the space of all 2-forms. Now set Φ = F −1 1 . When this transformation is applied to the manifold M, the 2-form Ω transforms into the 2-form Ω = Φ * Ω = F * 1 Ω = F * 1 (ω + δω) = ω, (B3) which proves the claim. When the 2-form Ω represents the symplectic structure of a Hamiltonian system on M with Hamiltonian H, the transformation Φ transforms the Poisson bracket into the Poisson bracket associated with the 2-form ω. Moreover, the Hamiltonian is transformed intō H = Φ * H = F * = Gλ L G λ H dλ dλ + . . . ,
must be identified with a quantity that is not merely proportional to c/ω pi is that Ref.19 assumes that the mass ratio is small, while in this work geometric interpretation of MHD. Perhaps the first question that naturally raises is that of convergence of the perturbative solution of the functional PDE (30)-(31). By summing all of the terms in the asymptotic solution of this equation, does the result converge and represent a truly-(not just formally-) exact singlefluid closure? If the series does not converge, does that mean that an exact single-fluid closure does not exist? I conjecture that the likely answers to these questions are "no" and "yes, but there may as well be one." This conjecture is based on what happens in finite dimensions. In the finite-dimensional setting[25][26][27] (actually Ref.25 handles systems with finitely-many slow variables and infinitely-many fast variables!), it has been shown that when the PDE defining a slow manifold in a Hamiltonian system can be solved perturbatively, there actually exists a so-called almost-invariant set that is approximated well by truncations of the slow manifold.When solutions start within the almost invariant set, they remain within the almost invariant set, and therefore close to the truncated slow manifold, for exponentially-long periods of time.Extension of these finite-dimensional results to infinite dimensions is certainly a non-trivial task, but one that would have deep implications for the behavior of solutions to a variety of physical models, in plasma physics and elsewhere. Actually, some work in this direction has already been carried out. In Ref.28, Vanneste uses exponential asymptotics to show that while the 3D Boussinesq equations do not admit an exact quasigeostrophic closure, particular solutions that begin on an approximate slow manifold only deviate from the slow manifold by an exponentially-small amount. A second question to ask is "what about the two-fluid states that do not lie within the slow manifold?" This is an important question because plasmas that live on the slow manifold are extremely lucky; most two-fluid states are not contained within the slow manifold. While this is a difficult question, intuition suggests that the most general motion of a two-fluid plasma bears very little resemblance to the predictions given by MHD. However, there is reason to be hopeful that two-fluid plasmas that begin near, but not exactly on the slow manifold still feel the influence of the MHD model. It is not difficult to show that twofluid dynamics near the slow manifold generally decompose into a slow drift along the slow manifold and a rapid oscillation transverse to the slow manifold. In the regime ρ/L ≪ 1, the transverse oscillations describe the dynamics of the Langmuir oscillation and light waves (with the well-known modification to the free-space dispersion relation). By generalizing the oscillation center theory of Refs. 4 and 29 to infinite dimensions, it may be possible to compute the ponderomotive effect of the transverse oscillations on the drift motion along the slow manifold. If this can be done within the context of the Hamiltonian formalism, there will be an adiabatically-invariant field related to the wave action of light waves and Langmuir waves. This adiabatic invariant will couple the averaged transverse dynamics to the MHD-like single-fluid closure via some kind of effective potential. This topic will be thoroughly explored in future work.VI. ACKNOWLEDGEMENTS discussion see Ref.16.) In order to define DF , it is necessary to introduce the technical assumption that W and V are complete normed spaces, i.e. Banach spaces. The norms on W and V will be denoted |·| W and |·| V . Like F , DF is a (possibly nonlinear) functional of ψ.However, the value of DF at ψ, DF (ψ), is not an element of V . Instead DF (ψ) is a linear operator that maps W into V . We may write DF (ψ) : W → V . This particular operator is defined as the linearization of the map F about the point ψ, i.e. if δψ ∈ W represents an infinitesimal displacement of the field ψ, then DF (ψ) is the unique linear operator suchthe mass ratio ν = O(1). Apparently the bracket underlying Lüst's model has a significance
that extends beyond the strictly-neutral two-fluid system. Moreover, invoking the results
of Ref. 20, it must therefore be true that there is a simple relationship between the all-
orders bracket and the bracket 23 for Hall MHD (see Ref. 24 for the first bracket accounting
for Hall physics, albeit in a reduced setting). Indeed, Ref. 20 shows that there is a simple
transformation that maps Lúst's bracket into the Hall MHD bracket. It turns out that the
appropriate transformation for the all-orders model is A →Ā, wherē
A = A + ǫ
νZ i
q e Z i
m i u.
Looking ahead, it is interesting to consider questions that one would not have asked
without this article's newfound phase-space-that
lim
δψ→0
whenever the derivatives on the right-hand-side exist. When context suggests where theFréchet derivative is to be evaluated, it is convenient to suppress the nonlinear argument of DF , i.e. DF [δψ] may sometimes be written instead of DF (ψ)[δψ]. Using this convention, the chain rule may be written as D(G • F )[δψ] = DG[DF [δψ]],
Appendix A: Fréchet derivativesGiven a vector space W of (possibly vector-valued) fields ψ ∈ W , it is natural to wonder how (possibly nonlinear) functionals F : W → V that take values in a vector space V may be differentiated. One natural answer is provided by the Fréchet derivative, DF . (For more This general theory reproduces the results of Section IV when M is taken as the space of tuples (g i , g e , u, A), with g i,e the ion and electron fluid configuration maps, and the 1-form Θ is given byHerev i,e =ġ i,e • g −1 i,e . This explains, conceptually at least, why it is possible to find a coordinate transformation of the slow manifold that leads to a truncated expression for the transformed Poisson bracket.For the sake of actually performing the computation of a transformation that simplifies 1-form Θ, instead of applying Eq. (B2) directly, it is helpful to formulate the problem in terms of iterated transformations as in Ref.15. The remainder of this appendix will explain this computation-oriented approach to find the simplifying transformation.Instead of representing the transformation Φ as the λ = 1 flow of a time-dependent vector field, instead introduce the ansatzwhere G 1 , G 2 , . . . are time-independent vector fields on the space of tuples (g i , g e , u, A).Such vector fields, which are defined on an infinite-dimensional space, may be represented as a tuple of vector fields on configuration space, (G i k , G e k , G u k , G A k ), where each entry in the tuple is a functional of (g i , g e , u, A) that takes values in the space of vector fields on configuration space. In particular, in terms of the component vector fields, the ODE d dλ (g i , g e , u, A) = G k (g i , g e , u, A) (B7) may be written
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| [] |
[
"Linear divisibility sequences and Salem numbers",
"Linear divisibility sequences and Salem numbers"
] | [
"Marco Abrate [email protected] \nDepartment of Mathematics\nUniversity of Turin\nVia Carlo Alberto 1010122TurinITALY\n",
"Stefano Barbero [email protected] \nDepartment of Mathematics\nUniversity of Turin\nVia Carlo Alberto 1010122TurinITALY\n",
"Umberto Cerruti [email protected] \nDepartment of Mathematics\nUniversity of Turin\nVia Carlo Alberto 1010122TurinITALY\n",
"Nadir Murru [email protected] \nDepartment of Mathematics\nUniversity of Turin\nVia Carlo Alberto 1010122TurinITALY\n"
] | [
"Department of Mathematics\nUniversity of Turin\nVia Carlo Alberto 1010122TurinITALY",
"Department of Mathematics\nUniversity of Turin\nVia Carlo Alberto 1010122TurinITALY",
"Department of Mathematics\nUniversity of Turin\nVia Carlo Alberto 1010122TurinITALY",
"Department of Mathematics\nUniversity of Turin\nVia Carlo Alberto 1010122TurinITALY"
] | [] | We study linear divisibility sequences of order 4, providing a characterization by means of their characteristic polynomials and finding their factorization as a product of linear divisibility sequences of order 2. Moreover, we show a new interesting connection between linear divisibility sequences and Salem numbers. Specifically, we generate linear divisibility sequences of order 4 by means of Salem numbers modulo 1. | 10.5486/pmd.2017.7814 | [
"https://arxiv.org/pdf/1709.01995v1.pdf"
] | 119,126,768 | 1709.01995 | 7cb083df5ca35885f441e67505f091f1133f9102 |
Linear divisibility sequences and Salem numbers
6 Sep 2017
Marco Abrate [email protected]
Department of Mathematics
University of Turin
Via Carlo Alberto 1010122TurinITALY
Stefano Barbero [email protected]
Department of Mathematics
University of Turin
Via Carlo Alberto 1010122TurinITALY
Umberto Cerruti [email protected]
Department of Mathematics
University of Turin
Via Carlo Alberto 1010122TurinITALY
Nadir Murru [email protected]
Department of Mathematics
University of Turin
Via Carlo Alberto 1010122TurinITALY
Linear divisibility sequences and Salem numbers
6 Sep 2017
We study linear divisibility sequences of order 4, providing a characterization by means of their characteristic polynomials and finding their factorization as a product of linear divisibility sequences of order 2. Moreover, we show a new interesting connection between linear divisibility sequences and Salem numbers. Specifically, we generate linear divisibility sequences of order 4 by means of Salem numbers modulo 1.
Introduction
A sequence a = (a n ) ∞ n=0 is a divisibility sequence if m|n implies a m |a n . Divisibility sequences that satisfy a linear recurrence relation are particularly studied. A classic example of linear divisibility sequence is the Fibonacci sequence. During the years linear divisibility sequences of order 2 have been deeply studied, see, e.g., [12] and [15]. Hall [11] studied divisibility sequences of order 3 and Bezivin et al. [4] have obtained more general results. Divisibility sequences are very interesting for their beautiful properties. For example, many studies can be found about their connection with elliptic curves [20], [13]. Further results on divisibility sequences can be found, e.g, in [9] where Cornelissen and Reynolds investigate matrix divisibility sequences, and in [23] where Horak and Skula characterize the second-order strong divisibility sequences.
Recently, linear divisibility sequences of order 4 have been deeply examined. In particular, Williams and Guy [21], [22] introduced and studied a class of linear divisibility sequences of order 4 that extends the Lehmer-Lucas theory for divisibility sequences of order 2. In section 2, we consider these sequences proving that all (non degenerate) divisibility sequences of order 4 have characteristic polynomial equals to the characteristic polynomial of sequences of Williams and Guy. Moreover, we provide all factorizations of divisibility sequences of order 4 into the product of divisibility sequences of order 2.
In section 3, we generate linear divisibility sequences of order 4 by means of powers of Salem numbers. This result is particularly intriguing, since connections between Salem numbers and divisibility sequences have been never highlighted. Moreover, the construction of divisibility sequences by means of powers of algebraic integers is an interesting research field that have been recently developed [19].
2 Standard linear divisibility sequences Definition 1. Given a ring R, a sequence a = (a n ) ∞ n=0 over R is a divisibility sequence if m|n ⇒ a m |a n .
Conventionally, we will consider a 0 = 0.
In the following, we will deal with linear divisibility sequences (LDSs), i.e., divisibility sequences that satisfy a linear recurrence. Classic LDSs are the Lucas sequences, i.e., the linear recurrence sequences with characteristic polynomial x 2 −hx+k and initial conditions 0, 1.
In [21] and [22], the authors introduced and studied some linear divisibility sequences of order 4. We recall these sequences in the following definition.
Definition 2. Let us consider linear recurrence sequences of order 4 over Z with characteristic polynomial
x 4 − px 3 + (q + 2r)x 2 − prx + r 2
and initial conditions 0, 1, p, p 2 − q − 3r.
We say that these sequences are standard LDSs of order 4 and we call the previous polynomial as standard polynomial.
In the next theorem, we prove that the product of two LDSs of order 2 is a standard LDS of order 4. First, we need the following lemma proved in [8].
Lemma 1. Let a = (a n ) ∞ n=0 and b = (b n ) ∞ n=0 be linear recurrence sequences with characteristic polynomials f (x) and g(x), respectively. The sequence ab = (a n b n ) ∞ n=0 is a linear recurrence sequence that recurs with f (x) ⊗ g(x), the characteristic polynomial of the matrix F ⊗ G (Kronecker product of matrices), where F and G are the companion matrices of f (x) and g(x), respectively. Remark 1. The previous lemma can be also stated as follows. Let a = (a n ) ∞ n=0 and b = (b n ) ∞ n=0 be linear recurrence sequences whose characteristic polynomials have roots α 1 , ..., α s and β 1 , ..., β t , respectively. Then, the sequence c = (c n ) ∞ n=0 = (a n b n ) ∞ n=0 is also a linear recurrence sequence whose characteristic polynomial has roots γ 1 , ..., γ st , where (γ 1 , ..., γ st ) = (α 1 , ..., α s ) ⊗ (β 1 , ..., β t ), Theorem 1. Let a = (a n ) ∞ n=0 and b = (b n ) ∞ n=0 be LDSs of order 2 with characteristic polynomials x 2 − h 1 x + k 1 , x 2 − h 2 x + k 2 , respectively, and initial conditions 0, 1. The sequence ab = (a n b n ) ∞ n=0 is a standard LDS of order 4 with initial conditions 0, 1,
h 1 h 2 , (h 2 1 − k 1 )(h 2 2 − k 2 ).
Proof. Since a and b are LDSs, it immediately follows that ab is a divisibility sequence and by Lemma 1, we know that it is a linear recurrence sequence of order 4 whose characteristic polynomial is
x 4 − h 1 h 2 x 3 + (k 1 h 2 1 − k 2 h 2 1 + 2k 1 k 2 )x 2 + h 1 k 1 h 2 k 2 x + k 2 1 k 2 2 . By Definition 2, ab is a standard LDS for p = h 1 h 2 , q = h 2 1 k 2 + k 1 (h 2 2 − 4k 2 ), r = k 1 k 2 .
The initial conditions can be directly calculated.
Moreover, we prove that all the LDSs of order 4 have characteristic polynomial equals to the characteristic polynomial of standard LDSs.
Theorem 2. Let a = (a n ) ∞ n=0 be a non degenerate LDS of order 4 with a 0 = 0 and a 1 = 1, then its characteristic polynomial is
x 4 − px 3 + (q + 2r)x 2 − prx + r 2(1)
for some p, q, r.
Proof. Let us suppose that the characteristic polynomial of a has distinct roots in order to avoid degenerate sequences, i.e., ratio of roots are not roots of unity. Let α, β, γ, δ be these roots.
The sequence a is a divisor of the sequence
b = (b n ) ∞ n=0 , where b n = α n − β n α − β · α n − γ n α − γ · α n − δ n α − δ · β n − γ n β − γ · β n − δ n β − δ · γ n − δ n γ − δ .
See [2] and [4]. In other words, there exist a sequence c = (c n ) ∞ n=0 such that b n = a n c n , for any index n.
By Lemma 1 and Remark 1, the sequence p can be written as the product of six Lucas sequences with characteristic polynomials having roots (α, β), (α, γ), (α, δ), (β, γ), (β, δ), (γ, δ), respectively. Thus, the roots of the characteristic polynomial of p are the entries of the following vector of length 64:
B = (α, β) ⊗ (α, γ) ⊗ (α, δ) ⊗ (β, γ) ⊗ (β, δ) ⊗ (γ, δ),
where all the roots appear with the due multiplicity. We can write the vector B as
B = (B 1 , B 2 , B 3 , B 4 ), where • B 1 = (α 3 , α 2 δ) ⊗ (β, γ) ⊗ (β, δ) ⊗ (γ, δ), • B 2 = (α 2 , αδ) ⊗ (βγ, γ 2 ) ⊗ (β, δ) ⊗ (γ, δ), • B 3 = (α 2 β, αβδ) ⊗ (β, γ) ⊗ (β, δ) ⊗ (γ, δ), • B 4 = (αβγ, βγδ) ⊗ (β, γ) ⊗ (β, δ) ⊗ (γ, δ).
Moreover, B = A ⊗ C, where C is a vector whose components are the roots of the characteristic polynomial of c and
A = (ω 1 , ω 2 , ω 3 , ω 4 ),
with (ω 1 , ω 2 , ω 3 , ω 4 ) a certain permutation of (α, β, γ, δ). Thus, we can write
B = (ω 1 C, ω 2 C, ω 3 C, ω 4 C),
i.e., B 1 , B 2 , B 3 , and B 4 are multiple of C. Considering
C = (α 2 , αδ) ⊗ (β, γ) ⊗ (β, δ) ⊗ (γ, δ), we have B 1 = αC, B 2 = γC, B 3 = βC, B 4 = δ · βγ αδ C.
Thus, we have ω 1 = α, ω 2 = γ, ω 3 = β and ω 4 mus be equals to δ, i.e., we must have αδ = βγ, but this is equivalent to say that the characteristic polynomial of a must be of the form (1). Now, we see that any standard LDS can be factorized as a product of two LDS of order 2 over C .
Definition 3. Given the sequences (u n ) +∞ n=0 , (v n ) +∞ n=0 , (s n ) +∞ n=0 , (t n ) +∞ n=0 over a ring R, we say that the product sequences (u n v n ) +∞ n=0 and (s n t n ) +∞ n=0 are equivalent if u n = λ n−1 s n , v n = λ 1−n t n where λ ∈ R is a unit.
Theorem 3. Let a = (a n ) ∞ n=0 be a standard LDS over Z, then a n = b n c n , for all n ≥ 0, where b = (b n ) ∞ n=0 and c = (c n ) ∞ n=0 are LDSs of order 2 over C with initial conditions 0, 1 and characteristic polynomials
x 2 − q + 4r + 2p √ r ± q + 4r − 2p √ r 2 √ r x + 1 x 2 − q + 4r + 2p √ r ∓ q + 4r − 2p √ r 2
x + r when p = 0. Moreover when p = 0 and q + 4r = 0, q = 0 (to avoid degenerate cases) we have the two possible families of characteristic polynomials for b and c given by
x 2 + 1 x 2 − √ q + 4rx + r , x 2 + 1 x 2 − √ qx − r
These are all the families of not equivalent factorizations of a over C.
Proof. We want to factorize a standard polynomial into the Kronecker product of two polynomials of degree 2, i.e., we want to find h 1 , h 2 , k 1 , k 2 such that
(x 2 − h 1 x + k 1 ) ⊗ (x 2 − h 2 x + k 2 ) = x 4 − px 3 + (q + 2r)x 2 − px + r 2 .
Let us observe that the characteristic polynomial of a must have distinct non zero roots in order to guarantee that a is a LDS of order 4 . Let γ 1 , γ 2 and σ 1 , σ 2 be the roots of
x 2 − h 1 x + k 1 and x 2 − h 2 x + k 2 , respectively. We have (γ 1 + γ 2 )(σ 1 + σ 2 ) = p (γ 2 1 + γ 2 2 )σ 1 σ 2 + γ 1 γ 2 (σ 1 + σ 2 ) 2 = q + 2r γ 1 γ 2 σ 1 σ 2 (γ 1 + γ 2 )(σ 1 + σ 2 ) = pr (γ 1 γ 2 σ 1 σ 2 ) 2 = r 2(2)
When p = 0 these conditions are equivalent to the system
k 1 k 2 = r h 1 h 2 = p h 2 1 k 2 + h 2 2 k 1 = q + 4r(3)
which is a particular case of
k 1 k 2 = A h 1 h 2 = B h 2 1 k 2 + h 2 2 k 1 = C where A =
0 since we suppose that the standard polynomial has not zero roots. Thus, we can obtain
A h 2 1 k 1 2 − C h 2 1 k 1 + B 2 = 0 from which we have h 1 = ± k 1 C + 2B √ A ± C − 2B √ A 2 √ A and h 2 = ± C + 2B √ A ∓ C − 2B √ A 2 √ k 1 .
Thus solutions of system 3 are
h 1 = ± √ k 1 q + 4r + 2p √ r ± q + 4r − 2p √ r 2 √ r h 2 = ± q + 4r + 2p √ r ∓ q + 4r − 2p √ r 2 √ k 1 k 2 = r k 1 .
Let us pose
λ = ± √ k 1 , s = q + 4r + 2p √ r ± q + 4r − 2p √ r 2 √ r ,,s = q + 4r + 2p √ r ∓ q + 4r − 2p √ r 2 .
Thus, considering solutions of system 3, we have
x 2 − h 1 x + k 1 = x 2 − sλx + λ 2 and x 2 − h 2 x + k 2 = x 2 −s λ x + r λ 2
, whose roots are
γ 1,2 = λ s ± √ s 2 − 4 2 , σ 1,2 = 1 λ s ± √s 2 − 4 2 .
In this case, we have u n = λ n−1 b n and v n = λ 1−n c n , where b and c are Lucas sequences with characteristic polynomials x 2 − sx + 1 and x 2 −sx + r, respectively. When p = 0 in conditions (2) we may suppose γ 1 + γ 2 = h 1 = 0 and find the two systems
h 1 = 0 k 1 k 2 = r h 2 2 k 1 = p + 4r , h 1 = 0 k 1 k 2 = −r h 2 2 k 1 = p with respective solutions h 1 = 0 h 2 = ± p+4r k 1 k 2 = r k 1 , h 1 = 0 h 2 = ± p k 1 k 2 = − r k 1
which give, with analogous considerations as in the case p = 0, with λ = ± √ k 1 , the two families of characteristic polynomials for b and c related to this case.
Remark 2. It would be interesting to find when previous factorizations determine sequences in Z or Z[i].
In the next section, we see a new connection between LDS of order 4 and Salem numbers.
Construction of linear divisibility sequences by means of Salem numbers of order 4
The Salem numbers have been introduced in 1944 by Raphael Salem [18] and they are closely related to the Pisot numbers [17]. There are several results regarding Pisot numbers and recurrence sequences [5], [6], [7]. In the following, we relate Salem numbers and LDS. There are many equivalent definitions of Salem numbers, here we report the following one.
Definition 4. A Salem number is an algebraic integer τ > 1 of degree d ≥ 4 such that all the conjugate elements belong to the unitary circle, unless τ and τ −1 .
In the following, we work on Salem numbers of degree 4, which can be characterized as follows (see [3], pag. 81). Proposition 1. The Salem numbers of degree 4 are all the real roots τ > 1, of the following polynomials with integer coefficients
x 4 + tx 3 + cx 2 − tx + 1 where 2(t − 1) < c < −2(t + 1).
It is immediate to see that previous polynomials are standard polynomials for p = −t, q = −2 + c, r = 1.
Definition 5. We call Salem standard polynomials the polynomials
x 4 − px 3 + (q + 2)x 2 − px + 1 with 2(−p − 1) < 2 + q < −2(−p + 1).
The study of the distribution modulo 1 of the powers of a given real number greater than 1 is a rich and classic research field (see, e.g, [14]). In the following, we use the same notation of [3] (pag. 61).
Definition 6. Given a real number α, let E(α) be the nearest integer to α, i.e., α = E(α) + ǫ(α) where ǫ(α) ∈ [− 1 2 , 1 2 ] is called α modulo 1. In the original work of Salem [18], he proved that if α is a Pisot number, then α n modulo 1 tends to zero and if α is a Salem number, then α n modulo 1 is dense in the unit interval. Further results on the distribution modulo 1 of the Salem numbers can be found, e.g., in [1] and [24]. Moreover, integer and fractional parts of Pisot and Salem numbers have been studied, e.g., in [10] and [25].
Let R ⊆ C be a ring and α ∈ R with α ∈ R * , then the sequence (α n ) ∞ n=0 is clearly a LDS. Given a couple of irrational numbers λ and α, it is interesting to study when the sequence (E(λα n )) ∞ n=0 is a LDS. Example 1. If we consider 1 √ 5 and the golden mean φ, it is well-known that
E 1 √ 5 φ n = F n ,
where F n is the n-th Fibonacci number, consequently we get a LDS.
Let g(x) be a Salem standard polynomial, g(x) has real roots α > 1, α −1 and complex roots γ, γ −1 with norm 1. Let (u n ) ∞ n=0 be a standard LDS with characteristic polynomial g(x). By the Binet formula, there exist λ, λ 1 , λ 2 , λ 3 such that u n = λα n + λ 1 α −n + λ 2 γ n + λ 3 γ −n .
Since |u n − λα n | ≥ |λ 1 α −n | + |λ 2 | + |λ 3 |, for all ǫ > 0, with n sufficiently large, we have |u n − λα n | ≥ ǫ + |λ 2 | + |λ 3 |.
Thus, if |λ 2 | + |λ 3 | < 1 2 , there exists n 0 such that u n = E(λα n ), ∀n > n 0 and if |λ 1 α −1 | + |λ 2 | + |λ 3 | < 1 2 , then u n = E(λα n ), ∀n ≥ 1.
An interesting case is given by the Salem standard polynomial
x 4 − tx 3 + tx 2 − tx + 1
for t ≥ 6. In this case, we have the Salem numbers
α = 1 4 t + (t − 4)t + 8 + √ 2 t(t + (t − 4)t + 8 − 2) − 4 and λ = 1 (t − 4)t + 8 .
Thus, we can determine infinitely many LDSs generated by powers of a Salem number, specifically the sequences These sequences appear to be new, since they are not listed in OEIS [16]. Moreover, as a consequence, we have the following property on Salem numbers, i.e.,
(θ n (t)) ∞ n=1 = E(λα n ),
d|n ⇒ E(λα d )|E(λα n ).
Finally, in the following proposition we characterize all the Salem standard polynomials that produces LDSs of this kind. Proposition 2. With the above notation, if |λ 1 α −1 | + |λ 2 | + |λ 3 | < 1 2 , then the integer coefficients p, q of g(x) must satisfy the following inequalities 2 ≤ p ≤ 8, −4 − 2p < q < p 4 + 8p 3 − 160p − 400 4p 2 + 32p + 64 p > 8, −4 − 2p < q < −4 + 2p
Proof. The real root α > 1 of g(x) can be written as
α = 1 4 p + p 2 − 4q + (p + p 2 − 4q) 2 − 16 .
Moreover, by the Binet formula λ = λ 1 = αγ (α − γ)(αγ − 1)
, λ 2 = λ 3 = − αγ (α − γ)(αγ − 1) .
Thus, from |λ 1 α −1 | + |λ 2 | + |λ 3 | < 1 2 we get |(α − γ)(αγ − 1)| > 2α + 2.
Posing γ = a + ib, with some calculations we find α 4 − 4aα 3 + 2(2a 2 − 7)α 2 − 4(a + 4)α − 3 > 0 from which we have α > 2 + a + (a + 2) 2 + 1 since −1 < a < 1 and α > 1. Using the explicit expression of α and that a =
∀t ≥ 6 ∈ Z For example, when t = 6 we have the LDS 1, 6, 29, 144, 725, 3654, 18409, ... when t = 7, we have 1, 7, 41, 245, 8897, 53621, ...
2 ≤ p ≤ 8, −4 − 2p < q < p 4 + 8p 3 − 160p − 400 4p 2 + 32p + 64 p > 8, −4 − 2p < q < −4 + 2pp−
√
p 2 −4q
4
,
we finally obtain
1
4
(p+ −16 + (−p − p 2 − 4q) 2 + p 2 − 4q) > 2+
p
4
+ 1 +
1
16
(8 + p − p 2 − 4q) 2 −
1
4
p 2 − 4q,
whose solutions are
.
Salem numbers and distribution modulo 1. S Akiyama, Y Tanigawa, Publ. Math. Debrecen. 64S. Akiyama, Y. Tanigawa, Salem numbers and distribution modulo 1, Publ. Math. Debrecen, Vol. 64, 329-341, 2004.
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R-algebras of linear recurrent sequences. U Cerruti, F Vaccarino, Journal of Algebra. 1751U. Cerruti, F. Vaccarino, R-algebras of linear recurrent sequences, Journal of Algebra, Vol. 175, No. 1, 332-338, 1995.
Matrix divisibility sequences. G Cornelissen, J Reynolds, Acta Arithmetica. 156G. Cornelissen, J. Reynolds, Matrix divisibility sequences, Acta Arithmetica, Vol. 156, 2012
Integer parts of powers of Pisot and Salem numbers. A Dubickas, Archiv der Mathematik. 29A. Dubickas, Integer parts of powers of Pisot and Salem numbers, Archiv der Math- ematik, Vol. 29, 252-257, 2002.
Divisibility sequences of third order. M Hall, Amer. J. Math. 58M. Hall, Divisibility sequences of third order, Amer. J. Math., Vol. 58, 577-584, 1936.
A characterization of the second-order strong divisibility sequences. P Horak, L Skula, The Fibonacci Quarterly. 23P. Horak, L. Skula, A characterization of the second-order strong divisibility se- quences, The Fibonacci Quarterly, Vol. 23, 126-132, 1979.
Elliptic divisibility sequences over certain curves. P Ingram, Journal of Number Theory. 1232P. Ingram, Elliptic divisibility sequences over certain curves, Journal of Number The- ory, Vol. 123, No. 2, 473-486, 2007.
The theory of asymptotic distribution modulo one. J F Koksma, Compositio Mathematica. 16, J. F. Koksma, The theory of asymptotic distribution modulo one, Compositio Math- ematica, Vol. 16, 1-22, 1964.
Characterization of second-order strong divisibility sequences of polynomials. M Norflet, The Fibonacci Quarterly. 432M. Norflet, Characterization of second-order strong divisibility sequences of polyno- mials, The Fibonacci Quarterly, Vol. 43, No. 2, 166-169, 2005.
The On-Line Encyclopedia of Integer Sequences. The On-Line Encyclopedia of Integer Sequences, published electronically at http://oeis.org.
La répartition modulo 1 et nombres algébriques. C Pisot, Ann. Sc. Norm. Super. Pisa, II, Ser. 7C. Pisot, La répartition modulo 1 et nombres algébriques, Ann. Sc. Norm. Super. Pisa, II, Ser. 7, 205-248, 1938.
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Divisibility sequences and powers of algebraic integers. J H Silverman, Documenta Math., Extra. CoatesJ. H. Silverman, Divisibility sequences and powers of algebraic integers, Documenta Math., Extra Volume Coates, 711-727, 2006.
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Some fourth-order linear divisibility sequences. H C Williams, R K Guy, International Journal of Number Theory. 75H. C. Williams, R. K. Guy, Some fourth-order linear divisibility sequences, Interna- tional Journal of Number Theory, Vol. 7, No. 5, 1255-1277, 2011.
Some monoapparitic fourth-order linear divisibility sequences. H C Williams, R K Guy, Integers: The Electronic Journal of Combinatorial Number Theory. 12H. C. Williams, R. K. Guy, Some monoapparitic fourth-order linear divisibility se- quences, Integers: The Electronic Journal of Combinatorial Number Theory, Vol. 12A, Article A17, 2012.
A matrix approach for divisibility properties of the generalized Fibonacci sequences, Discrete Dynamics in Nature and Society. A Yalciner, ID 829535A. Yalciner, A matrix approach for divisibility properties of the generalized Fibonacci sequences, Discrete Dynamics in Nature and Society, Article ID 829535, 2013.
An arithmetical property of powers of Salem numbers. T Zaimi, Journal of Number Theory. 120T. Zaimi, An arithmetical property of powers of Salem numbers, Journal of Number Theory, Vol. 120, 179-191, 2006.
On integer and fractional parts of powers of Salem numbers. T Zaimi, Archiv der Mathematik. 87T. Zaimi, On integer and fractional parts of powers of Salem numbers, Archiv der Mathematik, Vol. 87, 124-128, 2006.
| [] |
[
"A Brief History of Context",
"A Brief History of Context"
] | [
"Kaiyu Wan \nComputer Science Department\nEast China Normal University Shanghai\n200China\n"
] | [
"Computer Science Department\nEast China Normal University Shanghai\n200China"
] | [
"IJCSI International Journal of Computer Science Issues"
] | Context is a rich concept and is an elusive concept to define. The concept of context has been studied by philosophers, linguists, psychologists, and recently by computer scientists. Within each research community the term context was interpreted in a certain way that is well-suited for their goals, however no attempt was made to define context. In many areas of research in computer science, notably on web-based services, human-computer interaction (HCI), ubiquitous computing applications, and context-aware systems there is a need to provide a formal operational definition of context. In this brief survey an account of the early work on context, as well as the recent work on many working definitions of context, context modeling, and a formalization of context are given. An attempt is made to unify the different context models within the formalization. A brief commentary on the usefulness of the formalization in the development of context-aware and dependable systems is included. | null | [
"https://arxiv.org/pdf/0912.1838v1.pdf"
] | 8,260,699 | 0912.1838 | 30709aa6e02f3d49d3b3313fc726a024dc388e9c |
A Brief History of Context
2009
Kaiyu Wan
Computer Science Department
East China Normal University Shanghai
200China
A Brief History of Context
IJCSI International Journal of Computer Science Issues
62200933ContextContext TheoryContext-Awareness
Context is a rich concept and is an elusive concept to define. The concept of context has been studied by philosophers, linguists, psychologists, and recently by computer scientists. Within each research community the term context was interpreted in a certain way that is well-suited for their goals, however no attempt was made to define context. In many areas of research in computer science, notably on web-based services, human-computer interaction (HCI), ubiquitous computing applications, and context-aware systems there is a need to provide a formal operational definition of context. In this brief survey an account of the early work on context, as well as the recent work on many working definitions of context, context modeling, and a formalization of context are given. An attempt is made to unify the different context models within the formalization. A brief commentary on the usefulness of the formalization in the development of context-aware and dependable systems is included.
Introduction
According to the Oxford English Dictionary (OED), context denotes "the circumstances that form the setting for an event". To emphasize a common social usage of the word context OED includes the quotation [12] "I wish honorable gentlemen would have the fairness of what I did say, and not pick out detached words". Although the word context has been used for a long time in many scientific descriptions, literary essays, and in philosophical discourses, its meaning was always left to the reader's understanding. In one of the earlier papers, Clark and Carlson [11] state that Context has become a favorite word in the vocabulary of cognitive psychologists and that it has appeared in the titles of a vast number of articles. They then complain that the denotation of the word has become murkier as its uses have been extended in many directions and deliver the now widespread opinion that context has become some sort of "conceptual garbage can". That context has changed now. The importance of context in information retrieval, knowledge representation, reasoning in AI, and analysis of computer programs have been recognized and there is a serious effort to make a precise technical working definition of the notion of context. More recently, the importance of context was picked up by researchers in many areas of computer science, most importantly those working in Human-Computer Interaction (HCI), semantic web, and trustworthy systems. This intense interest has produced many operational definitions of context, but almost all of them are either informal or use ad hoc notation. We review in this paper the different types of notations and interpretations used for context. The review is classified into Context in Logic, Context in Languages, and Context in Systems. This classification and review are not exhaustive. It is used mainly to trace the historical progression of the systematic study of context in different, but related, areas.
Structure and Interpretation
The word "context" is derived from the Latin words con (meaning "together") and texere (meaning "to weave"). The raw meaning of it is therefore "weaving together". A circumstance is a weaving together of many types of entities. Thus, in describing a context we must define a finite set of entities, a finite set of properties for each entity, and the inter-weaving of the properties. As an example, the setting for a "seminar event" is the weaving together of the entities speaker, topic, audience, time, location and their properties such as name and affiliation for the speaker, title and abstract for the topic, size and status for audience, clocktime for time, and buildingaddress and room-number for locality. We need to associate with each property a value from its domain and bind each entity with the instantiated properties in order to describe the context of seminar. The choice of entities, the choice of properties, and the notation used for binding them are crucial for system development. This choice for context definition has the effect of narrowing down the possible interpretations of declared policies and constraints for system development. Context description also eliminates ambiguities. It should be possible to define contexts in programming languages independent of how it should be used. For example, if a context is defined by locality and time, then many events may happen in a specific context, and each event may produce a different experience in one context. Therefore, the structural definition of a context is only part of its specification, its other part being the semantics of the world associated with the context. The world may be defined by a set of states in programming or by a set of logical formulas in a formal specification. Motivated by a need to specify context as a first class citizen in languages and systems, a formal representation of it was developed in [45].
Context in Logic
In this section we review the study of context in logic as a formal object and reasoning. We review intensional logic and some variations of propositional and predicate logic in which context has been embedded as first class citizens.
Intensional Logic
Intensional Logic [14,42], a family of mathematical formal systems that permits expressions whose value depends on hidden context, came into being from research in natural language understanding. According to Carnap [9], the real meaning of a natural language expression whose truth-value depends on the context in which it is uttered is its intension. The extension of that expression is its actual truth-value in the different possible contexts of utterance, where this expression can be evaluated. Basically, intensional logics add dimensions to logical expressions, and non-intensional logics can be viewed as constant in all possible dimensions, i.e. their valuation does not vary according to their context of utterance. Intensional operators are defined to navigate in the context space. In order to navigate, some dimension tags (or indexes) are required to provide placeholders along dimensions. These dimension tags, along with the dimension names they belong to, are used to define the context for evaluating intensional expressions. Example 1 E: Beijing is now the capital of China. This expression is intensional because the truth value of this expression depends on the context in which it is evaluated. The intensional natural language operator in this expression is now, which refers to the time dimension. Today it is certainly true, but there existed time points in the past when China had a different capital. For example, before 1949, the capital of China was NanJing. Those different values (i.e. True or False) along different time points are extensions of this expression. In other words, the evaluation of the above expression is time-dependent. A natural extension is to consider expressions that depend on more than one dimension, such as time, space, audience, and so on. Example 2 The meaning of the expression: E: the overseas indexes during this period close 10% below their highs can be interpreted when the possible worlds spanned by the dimensions overseas and period are defined. The Table below gives a possible extension of the expression E when the periods are months in a given year, and overseas stock markets are Amsterdam, Brussels, Frankfurt, and London. By varying the year we get 3dimensional extension.
Ja Fe Mr Ap Ma Jn Jl Au Se Oc No De Amsterda F F F F T T T T T F F F Brussels F F F T T T T T T F F F Frankfurt F F T T T T T T T T F F London F T T T T T T T T T T T
Formalizing Context in AI
Contexts in AI were introduced by Weyhrauch (1980) [51] and subsequently developed by McCarthy and Buva c (1998) [28] and Giunchiglia (1993) [17]. Surveys of the formalizations and the usage of contexts can be found in Sharma (1995) [38], Akman and Surav (1996) [3]. Context serves an important purpose in AI and Intelligent Information Processing (IIP). The classic example of the earliest IIP that failed to meet safety criteria is MYCIN [39]. It was observed by McCarthy [2]. MYCIN system advises physicians on treating bacterial infections of the blood and meningitis. When MYCIN was first introduced context was not part of system's query processing phase. When it was given the query "what is the treatment for Chlorae Vibrio" it recommended "two weeks of tetracycline" treatment.
What it failed to inform the physician was that a massive dehydration during the course of the treatment would occur. While the administration of tetracycline would cure the bacteria, the patient would die long before that due to diarrhea. Here is an instance where the context of correct usage was not given to the physician, which ultimately made the system unsafe and hence not trustworthy. A contextual MYCIN will explicitly state the context for correct administration of medications. In AI context is formalized using propositional or predicate logic. Contexts are abstract objects (representation free) and are first-class citizens. Consequently, contexts are freely used in logical formulas, without explicitly defining contexts. In some sense, in the logical approach a context itself is defined by a set of formulas that are true by the truth assignments in that context. In [28] McCarthy, who introduced a logical framework in AI for studying context, gave three reasons justifying his approach.
Axiomatization: The use of contexts simplifies axiomatizations. Axioms from one context can be lifted to more general contexts. Vocabulary and Interpretation: Contexts allow the use of specific vocabulary and information. Terms that are used in one context have particular meaning, which they will not have in general. Building AI Systems: A hierarchy of AI systems can be built be transcending from one context to another.
According to Giunchiglia (1993) [17], the notion of context formalizes the idea of localization of knowledge and reasoning. Intuitively speaking, a context is a set of facts (expressed in a suitable language, usually different for each different set of facts) used locally to prove a given goal, plus the inference routines used to reason about them (which can be different for different sets of facts). A context encodes a perspective about the world. It is a partial perspective as the complete description of the world is given by the set of all the contexts. It is an approximate perspective, in the sense described in McCarthy (1979) [27], as we never describe the world in full detail. Finally, different contexts, in general, are not independent of one another as the different perspectives are about the same world, and, as a consequence, the facts in a context are related to the facts in other contexts.
The work in Giunchiglia and Serafini (1994) provides a logic, called Multi Language Systems (ML Systems), formalizing the principles of reasoning with contexts informally described in Giunchiglia (1993). In ML systems, contexts are formalized using multiple distinct languages, each language being associated with its own theory (a set of formulas closed under a set of inference rules). Relations among different contexts are formalized using bridge rules, namely inference rules with premises and consequences in distinct languages. Recently, Ghidini and Giunchiglia (2001) proposed Local Models Semantics (LMS) as a model-theoretic framework for contextual reasoning, and use ML systems to axiomatize many important classes of LMS. From a conceptual point of view, Ghidini and Giunchiglia argued that contextual reasoning can be analyzed as the result of the interaction of two very general principles: the principle of locality (reasoning always happens in a context); and the principle of compatibility (there can be relationships between reasoning processes in different contexts). In other words, contextual reasoning is the result of the (constrained) interaction between distinct local structures. A good survey of context formalization in AI and a comparison between different formalizations can be found in [2]. According to this exposition, context is either treated within some logical framework or within situation theory. Both approaches deal with abstract contexts and focus only on contextual reasoning.
Context in Languages
We review the role of context in intensional programming languages (IPL) and in ¸λ calculus.
Formalizing Context in AI
The intensional programming paradigm has its foundations on intensional logic. It retains two aspects from intensional logic: first, at the syntactic level, are contextswitching operators, called intensional operators; second, at the semantic level, is the use of possible world semantics. By making difference between intension and extension, IPL provides two different levels for programming. On the higher level, it allows to represent/express problems in a declarative manner; on the lower level, it solves problems without loss of accuracy.
IPL deals with streams of entities which could be numbers, or strings of characters, or any computable structure. These streams are first class objects in intensional languages and functions can be applied to these streams. Because of the infinite nature of IPL, it is especially appropriate for describing the behavior of systems that change with time or physical phenomena that depend on more than one parameter (such as time, space, temperature, etc). It is also an appropriate language for use in business applications that generate data streams, or textual streams, or media streams. Examples include stock market transactions and credit card transactions which are mostly data streams of records where each record contains information on a transaction, call center transactions that generate textual streams of conversations, and multi-media streams that are generated by cable companies to distribute movies on demand. The streams are processed by accessing certain semantic units and interpreting it in different contexts.
There is no notion of type in an IPL. The operators on the stream contents are assumed to be given when one writes the stream functions. The natural logical view of a stream is an infinite sequence, and in writing programs one does not worry about the physical representations of stream contents. This abstraction enables one to understand an IPL program from the statements in it, without any reference to its implementation. The computational model for IPLs is known as eduction. That is, an implementation starts computing the first element that satisfies a given context, then the second, and so on. A context for expression evaluation, as informally understood in Example 2, is described by a set of dimensions (attributes) and a finality (goal). The finality is domain-dependent and is chosen so that a finite set of dimensions would suffice to realize that goal. For example in processing call center streams understanding a conversation may be the finality, and the attributes may be a set of key words chosen in advance to meet the goal. As another example, in processing streams of user interactions with web, the finality may be understanding user patterns and the attributes may be ActivityLocation, ActivityDuration, and VolumeofDataTransfer. Both finality and the attributes defining a context are implicitly used in evaluating the extensions from a stream.
Lambda Calculus
In programming languages, context is a meta concept: static context introduces constants, definitions, and constraints, and dynamic context processes the executable information for evaluating expressions. In [35] context is introduced in the lambda calculus and an argument is made for introducing context as first class objects in programming languages. Their motivation for introducing context in the theory of lambda calculus is to develop a programming language with first-class contexts that has advanced programming features for manipulating open terms. We are motivated along similar lines for introducing context in Lucid. However there are significant differences in the semantics of context between the two approaches.
A context in the lambda calculus is defined as a term with a"hole" in it. The hole in a context can be filled with a term which may involve free variables. To avoid inconsistent hole filling within the scope of lambda binding the holes are labeled, hole abstraction, and context application are separated. In our theory context plays two roles: one role is as a reference to an item in a multi-dimensional stream, and the other role is as a descriptor of situations at which expressions are evaluated. A stream of contexts may be constructed and a context expression may be evaluated at a context. In Lucx expressions and contexts exist independently. A context may be defined without any regard to any specific expression and hence it may be used to evaluate different Lucx expressions. Similarly, an expression can be evaluated at different contexts. It is possible to define context dependent expressions in Lucx. Such expressions may be evaluated at a context distinct from any other context used in its definition. We can define nested contexts, and dependent contexts. These features offer a variety of flexible ways to programming different applications.
Lucid and Lucx
Lucid was originally invented as a Program Verification Language by Ashcroft and Wadge [1]. And later it evolved into a dataflow language [52]. The basic intensional operators are first, next, and fby. The four operators derived from the basic ones are wvr, asa, upon, and prev, where wvr stands for whenever, asa stands for as soon as, upon stands for advances upon, and prev stands for previous. Lucid is a stream (i.e. infinite entity) manipulation language. All the above operators are applied to streams to produce new streams. The definitions of these operators [30] are shown as follows Definition 1 If X = (x0 , x1 , … , xi,… ) and Y = (y0 , y1 , … , yi,…), then
(1) f i r s t X = (x0 , x0 , … , x0 , … )
(2) next X = ( x1 , x2 , … , xi+ 1 , … )
(3) X fby Y = (x0 , y0 , y1 , … , y i -1 , … ) (4) X wvr Y = if f i r s t Y then X Fby (nextX wvr next Y) Else (next X wvr next Y) (5) X asa Y = f i r s t (X wvr Y) (6) X upon Y = X fby (if f i r s t Y then (next X upon next Y) else (X upon next Y)) (7) prev X = X@(#1) 2
Example 3 illustrates the definitions on a stream A whose elements are integers, and a stream B whose elements are boolean. In a boolean stream the symbols 1 and 0 indicate true and false respectively. The symbol nil indicates an undefined value.
Example 3 :
A = 1 2 3 4 5 B = 0 0 1 0 1 first A = 1 1 1 1 1 next A = 2 3 4 5 prev A = nil 1 2 3 4 5 AfbyB = 1 0 0 1 0 1 A wvr B = 3 5 A asa B = 3 3 3 A upon B 1 1 1 3 3 5
With the operators defined above, Lucid only allows sequential access into streams. That is, the (i + 1)th element in a stream is only computed once the ith element has been computed. To enable subcomputations to take place in arbitrary dimensions and all indexical operators to be parameterized by one or several dimensions, two basic intensional operators are added. One is intensional navigation (@.d), which allows the values of a stream to vary along the dimension d. Another is intensional query (#.d), which refers to the current position (i.e. tag value) along the dimension d. This way, it is possible to access streams randomly. The major distinction between contexts in AI and in IPL is that in the former case they are rich objects that are not completely expressible and in the later case they are implicitly expressible. Hence it is possible to write an expression in Lucid whose evaluation is contextdependent. However, a context in the current version of Lucid can not be explicitly manipulated. This restricts the ability of Lucid to be an effective programming language for programming diverse applications. So we have extended Lucid by adding the capability to explicitly manipulate contexts. This is achieved by introducing context as a first class object in the language. That is, contexts can be declared, assigned values, used in expressions, and passed as function parameters. The language thus extended, is called as Lucx [45] (Lucid extended with contexts)(the x is used as the x in TeX). Thus, the rationale for introducing context in Lucid is quite analogous to the introduction of context to enrich knowledge base in AI. However, our notion of context differs significantly from McCarthy's. In our study context is both finite and concrete. It is finite in the sense that only a finite number of dimensions are allowed in defining a context. However it does not impose any limitation on handling infinite streams, because with every dimension an infinite tag set is introduced in the language. A full account of context-based evaluation of expressions in Lucx is given in [45].
Context in Systems
Context-aware adaptation is regarded as the most important feature for pervasive and ubiquitous services [50,20,25]. Web services [26] and mobile computing applications [53,24] immensely benefit with a formal context model. It is in this context that we review the role of context. Context modeling and context-dependent interpretive actions are important in HCI [13,15,49]. However in all these works context is not formalized. In this section, after we review context formalism, we explain how our formal definition provides a rigorous platform for developing context-aware systems.
Formalizing Context
We formalize context as a typed relation, a set of ordered pairs of (d, x) where d is a dimension, Td is the type of d and x : Td.
Context Operators
In this section, context operators are discussed. A context being a relation we borrow the notation and meaning of those relational operators that are available in mathematics. Rest of them we define, using set theory notation. Using these context operators contexts can be managed dynamically and flexibly. The syntax of context expressions are also formally defined. In order to evaluate context expression correctly, precedence rules for context operators are provided as well.
Context operators are : override ⊕ , difference , choice | , conjunction I , disjunction U , undirected range , directed range , projection ↓ , hiding ↑ , substitution / , comparison =, ⊆ , ⊇ . The difference , conjunction I , disjunction U , and comparison =, ⊆ , ⊇ operators are set operators. The rest of the operators are explained and formally defined below.
Definition 3
Override ⊕ This operator takes two contexts c1∈ G, and c2∈ S and returns a context c ∈G, which is the result of the conflict -free union of c1 and c2, as defined below:
_ ⊕ _ : G × S → G, c= c1 ⊕ c2 ={m | ( m ∈ c1 ∧ dim m (m) ∉ dim(c2)) ∨ m∈ c2}
Context Expression
Informally, a context expression is an expression involving context variables and context operators. Let c be a context variable, and D be a set of dimensions. A formal syntax for context expression C is shown in Figure 1(left column). A context expression that satisfies those syntactic rules is a well-formed context expression.
In order to provide a precise meaning for a context expression, we define the precedence rules for all the operators. Figure 1(right column) shows the operator precedence from the highest (top row) to the lowest (bottom row). Parentheses will be used to override this precedence when needed. Operators having the same precedence will be applied from left to right.
Context Set Operators
In Lucx we avoid higher-order sets of contexts, and allow only sets of simple contexts. Hereafter, by "set of contexts" we refer only to "set of simple contexts". There are two kinds of such operators: lifting operators, and relational operators.
_ ↑ _ : Ρ S × Ρ DIM → Ρ S s'=s ↑ D={c ↑ D|c ∈ s}
Definition 13 Substitution. This operator produces a set of contexts s, sP S, for a given set of contexts s, s P S, a dimension and a tag value belonging to that dimension:
_/_ : Ρ S × (DIM × U) → Ρ S s'= s /<d',t'>={c/<d',t'>|c ∈ s}
Definition 14
Choice. This operator accepts two sets of contexts s1, s2 and non-deterministically returns one of them. The definition s= s1|s2 implies that s is either s1 or s2.
_|_ : Ρ S × Ρ S → Ρ S
Definition 15
Override. For every pair of context sets s1, s2, s1, s2 ∈ P S this operator returns a set of contexts s, s ∈ P S, where every context c ∈ s is computed as c1 ⊕ c2, c1 ∈ s1, c2 ∈ s2.
_ ⊕ _ : Ρ S × Ρ S → Ρ S s = s1 ⊕ s2 ={ c1 ⊕ c2| c1 ∈ s1 ∧ c2 ∈ s2}
Definition 16 Difference. For every pair of context sets s1, s2, s1, s2 ∈ P S this operator returns a set of contexts s, s ∈ P S, where every context c ∈ s is computed as c1 c2, c1 ∈ s1, c2 ∈ s2.
_ _ : Ρ S × Ρ S → Ρ S s= s1 s2 ={c1 c2| c1 ∈ s1, c2 ∈ s2}
Lifting the undirected range and directed range to sets of contexts will produce higher-order sets. So, we do not define lifting for these two operators. However, since the results of applying these two operators are sets of contexts, the lifting operators can be applied to the results.
Relational Operators
We define the three relational operations (join), (intersection), and (union) for sets of contexts. In the following definitions, c denotes a context, si ∈ P S and ∆ i=U c' ∈ si dim(c').
Definition 17
Join.
_ _ : Ρ S × Ρ S → Ρ S s=s1 s2 ={ c1 ∪ c2|c1 ∈ s1 ∧ c2 ∈ s2 ∧ c1 ↓ ∆ 3 = c2 ↓ ∆ 3}, where ∆ 3 = ∆ 1 ∩ ∆ 2.
Definition 18
Intersection.
_ _ : Ρ S × Ρ S → Ρ S s=s1 s2 ={ c1 ∩ c2|c1∈c2 ∧ c2∈ s2}.
We can prove that s= s1 s2 =(s1
s2) ↓ ∆ 3, where ∆ 3= ∆ 1 ∩ ∆ 2.
Definition 19 Union.
_ _ : Ρ S × Ρ S → Ρ S s= s1 s2 is computed as follows:
∆ 1 = U c ∈ s 1 dim(c), ∆ 2 = U c ∈ s 2 dim(c), and ∆ 3 = ∆ 1 ∩ ∆ 2 1. Compute X1:X1={c i ∪ cj ↑ ∆ 3 |c i ∈s 1 ∧ c j ∈ s 2 } 2. Compute X2: X2 ={c j ∪ ci ↑ ∆ 3 |ci ∈s 1 ∧ c j ∈ s 2 } 3. The result is : s= X1 U X2.
Earlier we have shown that the results of ci cj and ci cj are sets of contexts. So the relational operators (join), (intersection), and (union) can also be applied to the expressions ci cj and ci cj, where ci and cj are contexts.
Context Set Expressions
Informally, a context set expression is an expression involving sets of contexts and context set operators. Let s ranges over a set of contexts, S over a context set expression and D over a dimension set. A formal syntax for context set expression S is shown in Figure 2(left column).
Fig. 2 Formal Syntax of Context Set Expressions and Precedence
Rules for Context Set Operators.
In order to precisely calculate a context set expression, we define the precedence rules for the context set operators. These are shown in Figure 2(right column) (from the highest precedence at the row to the lowest precedence in the bottom row). Parentheses will be used to override this precedence when needed. Operators having the same precedence will be applied from left to right.
Box Notation
In many applications it is of special interest to consider a set of contexts, all of which have the same dimension set and the tags corresponding to the dimensions in each context satisfy a given constraint. We use the notation Box to denote such a set when the dimension set is ∆ ={d1,…, dk} ⊂ DIM and p is a logical expression.
Note that in p, we are allowing the dimensions as variables, denoting the current tags. That is, if p(d1, d2) = d1 < d2, it means the current tag of d1 is less than the current tag of d2 in the context that has dimensions d1 and d2. It is easy to show that anything defined by the Box notation is a Box.
Using Context Formalism in System Development
The two key terms in the study of context-aware systems are context and awareness. Awareness is of two kinds. One kind is the internal monitoring of the system, called selfawareness or internal awareness. System contexts are dynamic and consequently self-awareness varies from context to context. The other kind is the external monitoring of the system, called external-awareness. External awareness, also known as perception, is normally achieved through sensors and other stimuli, say from users or other system elements. External contexts change as and when the system environment changes, and such changes cause changes to external awareness. The system must use the knowledge it gained from its perception, apply it to the changing internals, and react by either triggering an internal state change or providing an external service. Hence, we must use a context formalism in which both self-awareness and externalawareness can be represented and reasoned about. Using context calculus we can compute dynamically different contexts, combine external and internal context, and calculate an internal context corresponding to an observed external context. Without the formalism such calculations can only be done in an ad hoc manner.
Context calculus has been implemented in C#. This context toolkit is portable and can be used as a plug-in for any contextaware application development. The component-based architecture given in [43] illustrates our approach to using context formalism for developing context-aware systems. Such an approach can be adapted to any context-aware application, including service-oriented systems [47], web services [48], and trustworthy systems [46].
Informally, a context C with a hole in it, written C[ • ], will become the term C[M] when the hole is filled with the term M. The formal way of writing this in calculus is M' M , where the term M' abstracts the hole in the context. The term M that abstracts the hole labeled X itself is written as X.M'. For example, the context C[ • ] = ( x.[ • ] +y)3 is represented by the term M'= ( X( x.X + y)3). The term obtained by filling the hole in M' with x+ z is written ( X.( x.X+y)3) (x+z).
Example 4 illustrates the definitions of these two operators on two streams A and B along the time dimension.
Definition 2
2Let DIM denote the set of all possible dimensions, and T = {Td |d ∈DIM} be the set of types associated with the dimensions. A context c is a finite relation {f(d, x) | d ∈ DIM ∧ x : Td }. The degree of the context c is |dom c|. The empty relation corresponds to Null context. The degree of Null context is 0. A context having only one (dimension, tag) pair is called a micro context. Let G denote the set of contexts over {DIM, T}. The set of micro contexts is M = {c| c∈ G; | c|= 1}. The set of simple contexts is S = {c| c ∈ G, c is a partial function}. Clearly, a simple context c of degree 1 is a micro context. A context which is not simple is referred to a non-simple context. The basic functions dim and tag are to extract the set of dimensions and the values associated with the dimensions in a context. That is, if c = f<d1,x1>,…,<dk,, xk>}, then we may write c = {mi |mi = <di.,xi>}, dim(c) = {d1, d2,…dk,}, and tag(c) = {x1, x2,… xk }. For the tuple (d, x) in a micro context c we use the functions dimm and tagm to extract the tuple components: dimm(c) = d and tagm(c) = x.
Definition 4 Definition 8 Definition 9
489Choice | This operator accepts a finite number of c1ck of contexts and non-deterministically return one of the cis. The definition c= c1|c2|…,|ck implies that c is one of the ci, where 1 ≤ i ≤ k: _|_ : G × G × … × G → G, Definition 5 Projection. This operator takes a context c∈ G and a set of dimensions D ⊂ DIM as arguments and filters only those micro contexts in c that have their dimensions in set D._ ↓ _ : G × D → G c ↓ D = {m| m ∈c ∧ dim m (m) ∈ D}.Hiding. This operator enables a set of dimensions D to be applied on a context c ∈ G to remove all the micro context s in c whose dimensions are in D:_ ↑ _ : G × D → G , c ↑ D={m| m∈ c ∧ dim m (m)Substitution. This operator takes a general context and a simple context as arguments and produces a context which is the result of replacing a sub-context of the general context with a sub-context of the simple context if their domains are equal._/_ : G × S → G , c/s = ( c ↑ dim s) U (s ↓ dim c) Undirected range.This operator takes two contexts c1, c2 ∈ G as arguments and returns a set of simple contexts. The tag set U is assumed to be totally ordered. We give a constructive definition here._ _ : G × G → Ρ SSteps for constructing the final result are shown as follows: 1. Let S' be the set of simple contexts, which is the result of ( c1 c2).2. For each pair of m1∈ c1, m2∈ c2, and dim m (m1) = dim m (m2 ), do the following: Y={Y1,Y2,…Yp}, Where Yi(i = 1,…,p), are the sets of micro context s constructed in Step 2. Define for Yi ∈ Y, first(Yi) ={dim m (m) | m ∈ Yi}, and, second(Yi) ={tag m (m)| m ∈ Yi}. If there exists Yi, Yj∈ Y such that first(Yi) = first(Yj), for i ≠ j, we replace the sets Yi and Yj by their union Yi U Yj, and repeat this process until the first(Yi)s for Yi Y are distinct 4. For Yi ∈ Y, construct the set Z of contexts: Z={{(first(Y1),x1), (first(Y2), x2),…, (first(Yp), xp)}| (x1, x2, …, xp) ∈ ∏ p i=1 second(Yi))}. 5. Define: Xc1 = c1 ↑ U Yi ∈Y first(Yi). 6. Define: Xc2 = c2 ↑ U Yi ∈Y first(Yi).. 7. Construct S': S'= {{z} U X c1 U X c2 | z ∈Z}. Basically, the result consists of three parts: 1. For each pair m1∈ c1, m2∈ c2 which shares the same dimension, constructs a set Yi, this is done in step 2 and step 3. The result of union the set Yi, done in step 4, consists of the first part : Z. 2. All the other micro context s of c1 which have different dimensions consists of the second part all the other micro context s of c2 which have different dimensions consists of the third part : Xc2. Example 8 : Let c1 ={(e, 3), (d, 1)}, c2 ={(e, 1),(d, 3)}, c3 ={(e, 3)}, c4 ={(f, 4)}, c5 ={(e, 1), (f, 4)} then c1 c2={{(e,1),(d,1)}, Directed Range. This operator takes two contexts c1 ∈ G and c2 ∈ S and returns a set of contexts: _ _ : G × S → Ρ G We change only Step 2 of the method described for the undirected range(Page 7) to obtain the result: (a) Define a = tag m (m1), b = tag m (m2 ), if tag m (m1) < tag m (m2), else ignore the pair m1, m2. (b) Define the subrange t ba = a..b. Example 9 : Let c1 = {(d,1)}, c2 = {(d,3), (f,4)}, then c1 c2 = c1 c2 ={(d
Fig. 1 Formal
1Syntax of Context Expressions and Precedence Rules for Context Operators. Example 11 Given a well -formed context expression c3 ↑ D ⊕ c1| c2, where c1 ={(x,3),(y,4),(z,5)}, c2 = {(y,5)}, and c3 ={(x,5),(y,6),(w,5), D={w}, the evaluation steps are shown as follows: [Step1]. c3 ↑ D={(x,5),(y,6)}[Definition 6, Page 6] [Step2]. c1|c2 = c1 or c2 [Page 6] [Step3]. Suppose in Step2, c1 is chosen, c3 ↑ D ⊕ c1 ={(x,3),(y,4),(z,5)} [Definition 3, Page 6] else if c2 is chosen, c3 ↑ D ⊕ c2 ={(x,5),(y,5)}[Definition 3, Page 6]
4.3.1 Lifting OperatorsDefinition 11 Projection. For s ∈ Ρ S, and D ⊆ DIM. The projection operator constructs a set of contexts s' ∈ Ρ S, where s' is obtained by projecting each context from s on to the dimension set D. _ ↓ _ : Ρ S × Ρ DIM → Ρ S s'=s ↓ D={c ↓ D|c ∈ s} Definition 12 Hiding. For s ∈ Ρ S, and D ⊆ DIM. The hiding operator constructs s' ∈ S2, where s' is obtained by hiding each context in s on the dimension set D.
Table 1 :
1Example 1
A formal definition follows: Definition 20 A Box set (or a Box for short) is a set of simple contexts with the same domain. Let φ ≠ {d1, …, dk} ⊆ DIM be a set of dimensions and p be an expression in which the di (1 ≤ i ≤ k) may occur as variables. Then Box[d1, …, dk | p] = {c∈ S | dim(c) ={d1, …, dk} and p is true when, for each I, di is assigned the value c(di)}. The dimension ∆ (b) of an nonempty box b is the dimension of any (all) its elements. The set of Box sets (or Boxes for short) are all sets of simple contexts all of which have the same domain.
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| [] |
[
"Detecting isotropic density and nematic fluctuations using ultrafast coherent phonon spectroscopy",
"Detecting isotropic density and nematic fluctuations using ultrafast coherent phonon spectroscopy"
] | [
"Chandan Setty \nDepartment of Physics\nUniversity of Illinois at Urbana-Champaign\nUrbanaIllinoisUSA\n",
"Kridsanaphong Limtragool \nDepartment of Physics\nUniversity of Illinois at Urbana-Champaign\nUrbanaIllinoisUSA\n",
"Byron Freelon \nDepartment of Physics and Astronomy\nUniversity of Louisville\n40208LouisvilleKY\n",
"Philip W Phillips \nDepartment of Physics\nUniversity of Illinois at Urbana-Champaign\nUrbanaIllinoisUSA\n"
] | [
"Department of Physics\nUniversity of Illinois at Urbana-Champaign\nUrbanaIllinoisUSA",
"Department of Physics\nUniversity of Illinois at Urbana-Champaign\nUrbanaIllinoisUSA",
"Department of Physics and Astronomy\nUniversity of Louisville\n40208LouisvilleKY",
"Department of Physics\nUniversity of Illinois at Urbana-Champaign\nUrbanaIllinoisUSA"
] | [] | We propose a theoretical framework for the detection of order parameter fluctuations in three dimensions using ultrafast coherent phonon spectroscopy. We focus our attention on long wavelength charge density fluctuations (plasmons), and charged nematic fluctuations where the direction of the propagation vector is fixed perpendicular to the plane of anisotropy. By treating phonons and light classically and decoupling interactions to integrate out the fermionic degrees of freedom, we arrive at an effective theory of order parameter fluctuations about the spatially uniform saddlepoint solution. We find that, due to the (k 2x − k 2 y )(B1g) symmetry of the form factor appearing in the vertex, nematic fluctuations couple to light only at fourth order, unlike isotropic density fluctuations which couple at second order. Hence, to lowest order, the interaction between electrons and the electromagnetic field contributes a driving force for plasmon oscillations while it provides a frequency shift for nematic fluctuations. From the resulting coupled harmonic oscillator equations of motion, we argue that ultrafast coherent phonon spectroscopy could be a useful tool to extract and analyze various electronic properties of interest such as the frequency of the collective mode and the coupling between electrons and phonons. Specific experiments are proposed on the normal state of FeSe to observe the frequency shift predicted here resulting directly from orbital ordering (nematic) fluctuations. * Corresponding author: [email protected] [1] A. Larkin and A. Varlamov, Theory of fluctuations in superconductors (Clarendon Press, 2005). [2] P. D. De Reotier and A. Yaouanc, Journal of Physics: Condensed Matter 9, 9113 (1997). | 10.1103/physrevb.99.245157 | [
"https://arxiv.org/pdf/1804.11351v1.pdf"
] | 119,273,826 | 1804.11351 | a1d040fd2fa226c282bffa9c7f2da624c7c69dae |
Detecting isotropic density and nematic fluctuations using ultrafast coherent phonon spectroscopy
Chandan Setty
Department of Physics
University of Illinois at Urbana-Champaign
UrbanaIllinoisUSA
Kridsanaphong Limtragool
Department of Physics
University of Illinois at Urbana-Champaign
UrbanaIllinoisUSA
Byron Freelon
Department of Physics and Astronomy
University of Louisville
40208LouisvilleKY
Philip W Phillips
Department of Physics
University of Illinois at Urbana-Champaign
UrbanaIllinoisUSA
Detecting isotropic density and nematic fluctuations using ultrafast coherent phonon spectroscopy
We propose a theoretical framework for the detection of order parameter fluctuations in three dimensions using ultrafast coherent phonon spectroscopy. We focus our attention on long wavelength charge density fluctuations (plasmons), and charged nematic fluctuations where the direction of the propagation vector is fixed perpendicular to the plane of anisotropy. By treating phonons and light classically and decoupling interactions to integrate out the fermionic degrees of freedom, we arrive at an effective theory of order parameter fluctuations about the spatially uniform saddlepoint solution. We find that, due to the (k 2x − k 2 y )(B1g) symmetry of the form factor appearing in the vertex, nematic fluctuations couple to light only at fourth order, unlike isotropic density fluctuations which couple at second order. Hence, to lowest order, the interaction between electrons and the electromagnetic field contributes a driving force for plasmon oscillations while it provides a frequency shift for nematic fluctuations. From the resulting coupled harmonic oscillator equations of motion, we argue that ultrafast coherent phonon spectroscopy could be a useful tool to extract and analyze various electronic properties of interest such as the frequency of the collective mode and the coupling between electrons and phonons. Specific experiments are proposed on the normal state of FeSe to observe the frequency shift predicted here resulting directly from orbital ordering (nematic) fluctuations. * Corresponding author: [email protected] [1] A. Larkin and A. Varlamov, Theory of fluctuations in superconductors (Clarendon Press, 2005). [2] P. D. De Reotier and A. Yaouanc, Journal of Physics: Condensed Matter 9, 9113 (1997).
INTRODUCTION
Order parameter fluctuations have thermodynamic and transport signatures [1], and can also be detectable by local probes such as Muon Spin Resonance (µSR) [2, 3], Nuclear Magnetic/Quadrupole Resonance (NMR/NQR) [4,5] and Scanning Tunneling Microscopy (STM) [1]. Useful as they may be, all of these measurements are only indirect probes of fluctuations since they are insensitive to both spatial and temporal information. So far, spin and charge order fluctuations have been best probed in the frequency domain through neutron scattering [6,7] and inelastic X-ray scattering [8] or momentum resolved electron energy loss spectroscopy [9] respectively, as they contain both spatial and dynamical properties. More recently, Raman scattering in the frequency domain has found increased utility due to its ability to probe orbital and charge nematic fluctuations [10][11][12] for certain polarization geometries.
While the aforementioned frequency domain measurements have already provided a wealth of information on order parameter fluctuations in a variety of materials, the corresponding time-domain measurements are only now being explored with considerable success [13]. Time domain measurements have the added advantage of observing fluctuating modes directly, provided the temporal resolution is at least equal to the inverse fluctuation scale of the boson. They also enable a direct extraction of the frequency and lifetime of the fluctuating mode, and as we will see below, the coupling to other modes such as phonons. Evidence of charge density wave fluctuations [14][15][16][17][18][19][20] as well as indirect indications of spin fluctuations [21][22][23] have been reported by several groups.
While amplitude modes of the charge density wave seem to be long-lived, the phase modes for a finite momentum vector are overdamped [14] and as a result are barely observable. Hence, the presence of long-lived fluctuating modes is an essential prerequisite for the observation of oscillations in ultrafast time domain spectroscopy. This statement appears facile at first sight−the life-time of the fluctuating mode is independent of whether the measurement is done in time or frequency domain, and decay rates measured in one domain would be expected to carry over to the other. This extension, however, is a non-trivial problem since time domain spectroscopy is inherently out of equilibrium. Therefore, well-known equilibrium results such as the decay rate-self energy relationship and Matthiesen's rule do not generally continue to hold in a non-equilibrium setting [24,25] except in the weak-probe limit.
The use of coherent phonons in ultrafast spectroscopy to elucidate the electronic ground states in solids has been well documented [26][27][28][29][30][31]. Typically, in a pumpprobe measurement, the pump pulse excites multiple fluctuating modes as the excitation energies lie in a similar energy window. Eventually these modes become coupled to one another, and hence one must disentangle the electronic information from the data [32]. This is especially true for the high temperature superconductors since the typical energies of a coherent optical phonon is roughly 5 THz which significantly overlaps with the excitation energies of nematic and spin fluctuations. In these materials, coherent phonon oscillations have been generated [33] and studied both experimentally [34][35][36] and theoretically [37] for different ground states. However, in spite of the immensely significant role played by arXiv:1804.11351v1 [cond-mat.str-el] 30 Apr 2018
(a) (b) (c) A 0 q (d) A i q FIG. 1.
Feynman diagrams contributing to the effective action for the fluctuating order parameter. The thin solid lines denote free electron Green functions. The dashed, dotted and wavy lines denote the plasmon fluctuating order, phonons and the gauge field respectively. The scalar A 0 q and vector potential A i q couple with the electrons with their respective vertices. We have ignored diagrams coupling phonons and the gauge field.
fluctuations in the phenomenology of such materials, few theoretical studies have shed light on the role of fluctuating modes on the oscillations, and no general theoretical framework has been laid.
In this work, we provide such a framework by the theoretical principles for probing electronic fluctuating modes through ultrafast coherent phonon spectroscopy via the interaction between electronic and lattice modes. Our focus will be on two long wavelength collective modes: plasmons and nematic fluctuations where the direction of the propagation vector is fixed perpendicular to the plane of anisotropy. We treat phonons and the electromagnetic field classically, Hubbard-Stratonovichize the interactions, and integrate out the fermionic degrees of freedom to obtain an effective theory of order parameter fluctuations about the uniform saddle-point solution in the disordered phase. Using perturbation theory, we derive the equations of motion for phonons and the collective mode at zero momentum, resulting in a set of coupled differential equations. We find that, due to the (k 2
x − k 2 y )(B 1g ) symmetry of the form factor appearing in the vertex, nematic fluctuations couple to light only at fourth order, unlike isotropic density fluctuations which couple at second order. Hence, to lowest order, the interaction between electrons and the electromagnetic field contributes a driving force for plasmon oscillations while it provides a frequency shift for nematic fluctuations. Finally, we solve the coupled equations of motion and show how one can extract various electronic properties of interest such as the frequency of the collective mode and the coupling between electrons and phonons.
PLASMONS
We begin by recalling the dynamics of zero momentum density fluctuations in three dimensions. The action for electrons with long range Coulomb interactions is given by
S e ψ , ψ = β 0 dτ d 3 rψ ∂ τ +p 2 2m − µ ψ + 1 2 d 3 r d 3 r ψψ e 2 |r − r | ψ ψ (1)
where we have suppressed the indices in the electron Grassmann variables,ψ ≡ψ σ (r, τ ), and similarly for the primed quantities. Here, µ is the chemical potential and β is the inverse temperature. Using standard many body techniques [38], one can arrive at the effective action for the density order parameter fluctuations (σ q ) about the saddle-point solution σ q = 0 given by (in the limit
v f |q| ω 1, v f is the Fermi velocity) S e [σ] = q 1 2 σ q ω 2 − ω 2 p − 3k 2 f ω 2 p m 2 ω 2 |q| 2 σ −q ,(2)
where q denotes a collective variable, q ≡ (iq n , q), containing the Matsubara frequency iq n and momentum q,
ω p = 4πne 2 m
is the plasmon frequency, n is the electron density and k f is the Fermi momentum. We have also used the same notation σ q for the fluctuating field. Note that an expansion in v f |q| ω 1 is possible due to the low energy plasmon gap. For a spatially uniform fluctuating order, the equations of motion for S e [σ] yield a simple harmonic oscillator with frequency ω p . In the presence of an external gauge field and phonons, the action becomes modified to include minimal coupling between electrons and the electromagnetic field, and the electron-phonon interaction. The total action is given by
S = S e ψ , ψ, A + S ph [P, Q] + S e−ph ψ , ψ, Q S e = d 3 rdτ ψ ∂ τ − ieA 0 ψ + 1 2m (∇ + ieA)ψ (∇ − ieA) ψ − µψψ + 1 2 d 3 r d 3 r ψψ e 2 |r − r | ψ ψ (3) S e−ph = i d 3 rψψV (r − R i ).(4)
Here A i are the components of the vector potential A, V (r − R i ) is the potential coupling the lattice to the electrons, and S ph [P, Q] is the lattice part of the action which, in momentum space, can be treated as a collection of harmonic oscillators of the form S ph [P, Q] = 1 2 q P 2 q + Ω 2 q Q 2 q (Ω q is the frequency of mode Q q with canonical momentum P q ). Henceforth, we will also ignore any interaction between phonons and light. Decoupling the quartic interaction and introducing the Hubbard-Stratonovich field φ(r), we rewrite the action as
S = β 0 dτ d 3 r 1 8π (∂ r φ) 2 +ψ G −1 0 +Ĉ(r) ψ ,(5)
whereĈ
≡ − ieA 0 + ie 2m (A · ∇ + ∇ · A) + e 2 |A| 2 2m +ieφ(r) + i V (r − R i ) ,(6)
and the non-interacting Green function is given by
G −1 0 = ∂ τ − ∇ 2 2m − µ.
We next treat the electromagnetic field and lattice vibrations classically [27,28], integrate out the fermions, and find the saddle-point solution. We will work in the limit where electrons are only weakly coupled to phonons and the electromagnetic field, and hence the saddle-saddle-pointpoint solution is not drastically affected. To the zeroth order approximation, this means that we can perform a perturbation expansion for the fluctuations about the original saddle-point solution φ q = 0. The lowest order terms in the fluctuation expansion contribute to the equations of motion for the phonons and the order parameter field trivially because the terms proportional to the gauge fields are independent of the electron density or lattice fluctuations, and the term proportional to the electron-lattice interaction potential only shifts the zero of the oscillations and can thus be ignored. The second-order terms in the expansion which couple the density fluctuation to light are dictated by gauge invariance and, in the limit of
v f |q| ω 1, is given by (in terms of the rescaled function σ q = 1 √ 4π |q| ω φ q ) S σ−A = iω 2 p √ 4πm q A 0 q |q| ω σ −q + A i q σ q . (7)
In the same limits, the coupling between the density fluctuations to phonons yields
S σ−ph = ieN 0 v 2 f 3 √ 4π q ξ q σ q |q| ω Q q ,(8)
where N 0 is the density of states at the Fermi level and ξ q is the matrix element for electron-phonon interactions in momentum space. The form of ξ q is important for the discussions to follow. As we are interested in the zeromomentum transfer limit, the coupling between the order parameter and phonons for a single band goes to zero unless the matrix element diverges as ξ q ∼ 1 |q| . Such a form of the matrix element typically occurs for electrons interacting with lattice displacements through long-range Coulomb interactions. In a multi-band case, zero momentum coupling between electrons and the lattice is allowed even for a constant matrix element. This is because there is a always a non-zero probability amplitude for electrons to be scattered to a different band with zero momentum transfer. Finally, to remain consistent with our previous assumptions of confining ourselves to a minimal model, we ignore the coupling between phonons and the electromagnetic field. This will have no effect on the order at which order-parameter fluctuations contribute Under these assumptions, one can collect all the terms contributing to the total action (Eqs 2, 7 and 8 along with S ph ), and determine the equations of motion for σ q and Q q in the limit of zero momentum transfer (|q| → 0, we denote the variables in this limit as Q 0 and σ 0 ). This leads to a system of coupled differential equations,
σ 0 (t) + ω 2 p σ 0 (t) = iγ 2 A z 0 (t) + iβ 2 Q 0 (t) (9) Q 0 (t) + Ω 2 0 Q 0 (t) = iβ 2 σ 0 (t),(10)
where the double dots denote second derivatives, γ 2 = ω 2 p √ 4πm , β 2 is the coefficient of the 1/|q| factor in the electron-phonon coupling matrix element times
eN0v 2 f 3 √ 4π ,
and Ω 0 is the frequency of the zero momentum optical phonon. We have also taken the vector potential (the zero momentum component of which is denoted by
A i 0 ≡ A i |q|=0 )
to point along the z direction. In deriving Eqs 9 and 10, we have assumed that close to zero momentum, the frequency of the plasmon is approximately a constant at ω ω p . These equations of motions can be solved for Q 0 (t) for a Gaussian pulse with a small width τ (compared to the inverse frequencies of the individual modes) centered around t = 0, and we obtain for t > 0
Q 0 (t) = e −ω 2 1 τ 2 sinω 1 t ω 1 (ω 2 1 − ω 2 2 ) − e −ω 2 2 τ 2 sinω 2 t ω 2 (ω 2 1 − ω 2 2 )
.
We have chosen the initial conditions such that the amplitude and velocity of the individual modes are zero at t = −∞. The individual frequencies of the oscillation are given by
ω 2 1 = 1 2 Ω 2 0 + ω 2 p − ω 2(12)ω 2 2 = 1 2 Ω 2 0 + ω 2 p + ω 2 ,(13)
where we have defined ω 2 ≡ (Ω 2 0 − ω 2 p ) 2 − 4β 2 . Thus, the normalized change in reflectivity follows (approximately in phase) the modulations caused by the phonon oscillations with the new frequencies ω 1,2 . These two frequencies can be extracted experimentally from the reflectivity oscillations and, with prior knowledge of the optical phonon frequency (Ω 0 , which can be measured from a region of the phase diagram where the fluctuations are small), the frequency of the collective mode (ω p ) and the strength of the electron-phonon coupling (β 2 ) can be determined.
For the simpler case of plasmon oscillations we study in this section, the assumption that the frequency is approximately constant (ω ω p ) in the plasmon-phonon coupling (close to zero momentum) can be lifted without much difficulty. In Fourier space, the coupled differential equations of motion for σ 0 (ω) and Q 0 (ω) are given by form factor k 2 x − k 2 y . Note that we have decomposed the electron phonon interaction in the B1g channel as well, and again ignored diagrams coupling phonons to the gauge field,.
(again in the limit
v f |q| ω 1) −ω 2 σ 0 (ω) + ω 2 p σ 0 (ω) = iγ 2 A z 0 (ω) + i β 2 Q 0 (ω) ω ,(14)
−ω 2 Q 0 (ω) + Ω 2 0 Q 0 (ω) = −iβ
2 σ 0 (ω) ω .(15)
We have introduced a new electron-phonon coupling constantβ 2 to be consistent with the units used previously. The effect of retaining the frequency dependence in the electron-phonon coupling is to force the characteristic polynomial determining the pole structure in Q 0 (ω) to be of third order. Hence, an additional frequency appears superposed on the coherent phonon oscillations. These frequencies, denotedω i , can be evaluated exactly in the limit of small electron-phonon couplinḡ
ω 2 1 β 4 Ω 2 0 ω 2 p (16) ω 2 2 Ω 2 0 + β 4 Ω 2 0 (Ω 2 0 − ω 2 p ) (17) ω 2 3 ω 2 p + β 4 ω 2 p (ω 2 p − Ω 2 0 ) .(18)
Once the three frequencies above are extracted from experiment, Ω 0 , ω p and β 2 can be evaluated without any prior knowledge of the bare optical phonon frequency.
NEMATIC FLUCTUATIONS
As an example of an order parameter fluctuation that couples differently to the electromagnetic field, we consider fluctuations above the nematic ordering transition (in the isotropic phase). Since we are interested in a nematic collective mode in the zero-momentum limit, just as in the case of plasmons, we require that the mode be gapped at |q| → 0. This is ensured by the presence of long range interactions in three dimensions that can be decomposed into various irreducible representations of the underlying point group. In the nematic channel for a square lattice, the decomposition will result in a k 2
x − k 2 y (B 1g ) form factor in the nematic susceptibility [10,11]. For simplicity, we will confine ourselves to the case where q is taken to zero along the axis perpendicular to the anisotropy; although this is not the most general treatment of the problem, our analysis can be easily extended to the case where the momentum transfers are in the plane of the anisotropy. To see that the nematic fluctuations are gapped at zero momentum transfer, we have to evaluate the nematic susceptibility [11]
A i q A i q (a) A i q A i q (b) A 0 q A 0 q (c) A 0 q A 0 q (d)χ n 0 (q, iω m ) = 1 V k f 2 k,q n f ( k+q ) − n f ( k ) iω m + k+q − k ,(19)
where 2f k,q = (k 2 x − k 2 y ) + (k x + q x ) 2 − (k y + q y ) 2 , V is the volume and k is the dispersion. Similar to the Lindhard function, χ n 0 (q, iω m ) ∼ q 2 for |q|v f ω, and hence, long range interactions yield a gap in the fluctuation spectrum. Moreover, due to the long-range character of the interactions, even though we work in the continuum limit, the nematic collective mode (like the plasmons) is undamped. This is in contrast with known results [39,40] where, in the presence of screening, damping effects dominate in the isotropic phase, and hence, collective mode oscillations become effectively indetectable. In addition, recent experiments [41,42] in the high temperature superconducting pnictides detect a well defined collective mode as the temperature is lowered, indicating that long-range interactions or lattice effects could be important in keeping the modes sharp and detectable by femto-second spectroscopy.
To write the action for the nematic fluctuations, we replace the interaction term in Eq 1 with (see [39])
S n e = 1 2 d 3 (r,r')ψψ F (r − r )F (∂ 2
x,x , ∂ 2 y,y )ψ ψ, (20) where F (r − r ) contains the long-range part of the interaction andF (∂ 2 (x,x ) , ∂ 2 (y,y ) ) contains the B 1g form factors. Unlike Ref [39], however, we have chosen our interaction to be long-ranged. We now follow the same procedure outlined in the previous section for plasmons. After performing the Hubbard-Stratonovich transformation and expanding in powers of fluctuations about the uniform saddle point, the second-order term (see Fig 2(a)) contributing to the effective action for the nematic fluctuation order parameter σ n q (the superscript n denotes a nematic order parameter fluctuation) is given by (in the limit
v f |q| ω 1)) S n e [σ n ] = q 1 2 σ n q ω 2 − ω 2 n − v 2 f ω 2 n |q| 2 ω 2 σ n −q ,(21)
where ω n is the nematic gap given by ω n = 8πe 2 N n v 2 f ,
N n = mk 5 f (2π) 2 16 105
, and k f is the Fermi wave vector. In the same limit, the coupling between phonons and σ n q is evaluated as (see Fig 2(
b)) S n σ−ph = ie √ 4πv 2 f N n q ξ q σ n −q |q| ω Q q . (22)
Note that to obtain Eq 22, we decomposed the electronphonon interaction in the B 1g channel as well, introducing two (k 2 x − k 2 y ) form factors, and thus keeping the coupling between the nematic fluctuations and phonons nonzero.
The interaction between electrons and light (and thus between σ n q and the gauge field), however, is constrained by minimal coupling, and cannot be decomposed into B 1g lattice form factors. Thus at second order, we can only have one factor of (k 2
x − k 2 y ) whose average value vanishes in the isotropic phase. The lowest non-zero contribution to the σ n q − A q coupling obtains from fourth order terms (see Fig 3; the third order triangle diagrams vanish since they are antisymmetric in the internal momenta). There are two classes of fourth-order terms−those where the fields σ n q and A i,0 q alternate on the vertices and those where they appear together (their various permutations yield the same result). These diagrams are shown in Figs 3(a), (c) and Figs 3(b),(d) respectively, and have different contributions to the effective action. For simplicity, we will choose a gauge where A 0 q is zero and, as discussed before, align the vector potential along the zaxis. For small momentum transfers, the diagrams can be cast in the following form
S n σ−A = q1..3 A z q1 σ n q2 σ n q3 A z −q1−q2−q3 ω q2 b G 1 (ω q k ) − c G 2 (ω q k ) ω q3 + σ n q1 A z q2 σ n q3 A z −q1−q2−q3 ω q1 b K(ω q k ) ω q3 .(23)
Here G i (ω q k ) and K(ω q k ) are rational functions of the Matsubara frequencies ω q k with a pole like structure, whose exact forms are bulky and less enlightening. The first and second terms are obtained from the two classes of diagrams Figs 3(a) and (b) respectively (since we have set A 0 q = 0, (c) and (d) do not contribute). The constants b = 2 525π 2 and c = 2 2205π 2 are obtained from angular integrals of the loop momenta. To simplify Eq 23, we assume spatially uniform fluctuations and gauge fields, and that the fluctuations have approximately a constant frequency ∼ ω n as we did for plasmons near q → 0 (this approximation cannot be made for the frequency of the gauge field since it fails to conserve energy and typically the pump pulse is broad). Under these approximations, the functions G i (ω q k ) and K(ω q k ) (which are now functions of only the gauge frequencies) acquire poles at ω qi = ±ω n , ±2ω n . Fourier transforming into time domain, simplifying using the Dirac delta functions, and performing standard contour integrals, we rewrite the contribution from the nematic-gauge field coupling term as S n σ−A = dt σ n 0 (t) 2 A z 0 (t)f (iω n t).
Here the function f is an oscillatory function of ω n t, and depends on the strength of the external pulse field (note that one factor of A z 0 appears in f (iω n t) after performing the Fourier integrals). Thus, unlike the case of plasmons, due to the lowest-order coupling between σ n 0 −A q being quadratic in σ n 0 (t), the effect of the fourth-order term is to change the frequency of the collective mode. Collating all the terms in Eqs 21, 22, 23 (along with S ph ), and re-writing them in time domain, we find that the equations of motion for σ n 0 (t) and Q 0 (t) are determined bÿ σ n 0 (t) + ω 2 n − γ 2 A z 0 (t)f (iω n t) σ n 0 (t) = iβ 2 Q 0 (t) (24) Q 0 (t) + Ω 2 0 Q 0 (t) = iβ 2 σ n 0 (t). (25) Here β 2 is given by the product of e √ 4πv 2 f N n and the coefficient of 1/|q| in the electron-phonon matrix ele- ment and γ 2 is the effective coupling between σ n 0 (t) − A z 0 that depends on the strength of the external field, and is treated as a parameter that can be obtained experimentally. The Eqns 25, 25 can be solved for non-zero initial amplitudes and zero initial velocities for t < 0 (pulse is incident at t = 0). When the frequency of the nematic mode is similar to that of the coherent phonons (ω n ∼ Ω 0 , which is typically the case experimentally), one obtains a beating pattern in the phonon oscillations (and hence the change in reflectivity) as shown in Fig 4. With knowledge of the frequency of the pure coherent phonon oscillations Ω 0 , the electronic collective mode frequency, ω n , and the coupling to phonons, β 2 , can be extracted in the weak field limit. The effect of γ 2 , then, is to change the period of the beating pattering by varying the effective frequency of the nematic collective mode. This is depicted in Fig 4 for a simple co-sinusoidal form of f (iω n t), with a pulse having a Gaussian electric field profile. Hence, the coupling between light and nematic collective modes (γ 2 ) can be obtained by changing the intensity of light and observing the shift in the beating pattern.
EXPERIMENTS IN FeSe
The absence of (static) ordered phases in the iron chalcogenide compound FeSe makes it an ideal playground for probing fluctuations. In order to utilize the results and test the predictions presented here, several ultrafast techniques could be employed e.g., time-resolved (tr ) x-ray diffraction (XRD), ultrafast electron diffraction (UED) or tr -optical spectroscopy. To be specific, we suggest that an ultrafast XRD experiment applied to FeSe on a substrate might be an excellent test case. If a femto-second optical pump signal, with THz energies, is used to excite coherent optical phonons, a subsequent femto-second X-ray pulse could serve to probe coherent phonon spectral behavior through Bragg peak oscillations. Hence, it should be possible to extract the electron-phonon coupling strength and the nematic collective mode frequency. In this experiment, the nematic fluctuations should be observed as a shift in the optical phonon oscillations. We note that the results of this experimental proposal can be compared to the recent pump-probe measurements [36] which employed both tr XRD and tr -ARPES to determine the electron-phonon coupling strength in FeSe.
SUMMARY
In summary, we described a minimal model that can be used to detect electronic order parameter fluctuations using ultrafast coherent phonon spectroscopy in three dimensions. We focused on two particular order parameter fluctuations in the zero momentum limit−collective (isotropic) modes in the charge density (plasmons), and charge nematic collective modes when the wave vector is taken to zero along a direction perpendicular to the plane of anisotropy (on a square lattice). After performing a perturbation expansion in the fluctuations while treating the electromagnetic field and phonons classically, we derived effective actions for the coupling of the respective collective modes with phonons and the electromagnetic field. Unlike plasmons which couple to light at quadratic order, we found that, due to the B 1g form factor in the nematic susceptibility, nematic modes couple to light only at quartic order. Hence, to lowest order, the coupling between electrons and the electromagnetic field contributes a driving force for plasmons but a frequency shift for nematic fluctuations. We finally determined the equations of motion for the individual collective modes and phonons, and demonstrated how one can extract useful electronic information such as the frequency of the collective mode and electron-phonon/light couplings. Our work provides a first basic theoretical framework for the detection of collective modes using ultrafast coherent phonon spectroscopy, and bears a special relevance to recent time-resolved experiments that have become increasingly popular probes of exotic quantum matter.
FIG. 2 .
2Feynman diagrams contributing to the effective action at second order for the nematic fluctuations. The thin solid lines denote free electron Green functions. The zig-zag and dotted lines denote the nematic fluctuating order and phonons respectively. The black dots each contribute a B1g
FIG. 3 .
3Feynman diagrams contributing to the effective action at fourth order for the nematic fluctuations. The thin solid lines denote free electron Green functions and the zig-zag lines denote the nematic fluctuating order. The black dots each contribute a B1g form factor k 2x −k 2 y . The scalar A 0 q and vector potential A i q couple with the electrons with their respective vertices.
FIG. 4 .
4Oscillations of the q = 0 phonon mode as a function of the delay time t > 0. (Left) For zero coupling (γ 2 = 0) between nematic fluctuations and light and (Right) non-zero coupling γ 2 = 0.5. We have chosen a pulse with a gaussian electric field profile, with a phonon frequency of Ω0 = 1.26T Hz, nematic mode ωn = 1.41T Hz and β 2 = 0.1T Hz. The beat period is increased due to the coupling of nematic fluctuations with light.
Acknowledgments: We acknowledge support from Center for Emergent Superconductivity, a DOE Energy Frontier Research Center, Grant No. DE-AC0298CH1088. We also thank the NSF DMR-1461952 for partial funding of this project.
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| [] |
[
"Existence of inter coupled structural, electronic and magnetic states in Sm 2 NiMnO 6 double perovskite",
"Existence of inter coupled structural, electronic and magnetic states in Sm 2 NiMnO 6 double perovskite"
] | [
"S Majumder \nUGC DAE Consortium for Scientific Research\n452001IndoreIndia\n",
"M Tripathi \nUGC DAE Consortium for Scientific Research\n452001IndoreIndia\n",
". O De Souza \nElettra Sicrotrone Trieste S.C.p.A\nSS 14-km 163.534149BasovizzaItaly\n",
"A Sagdeo \nHXAL\nSUS\nRaja Ramanna Centre for Advanced Technology\n452013IndoreIndia\n\nHomi Bhabha National Institute\nAnushakti nagar400 094MumbaiIndia\n",
"L Olivi \nElettra Sicrotrone Trieste S.C.p.A\nSS 14-km 163.534149BasovizzaItaly\n",
"M N Singh \nHXAL\nSUS\nRaja Ramanna Centre for Advanced Technology\n452013IndoreIndia\n",
"S Pal \nHXAL\nSUS\nRaja Ramanna Centre for Advanced Technology\n452013IndoreIndia\n",
"R J Choudhary \nUGC DAE Consortium for Scientific Research\n452001IndoreIndia\n",
"D M Phase \nUGC DAE Consortium for Scientific Research\n452001IndoreIndia\n"
] | [
"UGC DAE Consortium for Scientific Research\n452001IndoreIndia",
"UGC DAE Consortium for Scientific Research\n452001IndoreIndia",
"Elettra Sicrotrone Trieste S.C.p.A\nSS 14-km 163.534149BasovizzaItaly",
"HXAL\nSUS\nRaja Ramanna Centre for Advanced Technology\n452013IndoreIndia",
"Homi Bhabha National Institute\nAnushakti nagar400 094MumbaiIndia",
"Elettra Sicrotrone Trieste S.C.p.A\nSS 14-km 163.534149BasovizzaItaly",
"HXAL\nSUS\nRaja Ramanna Centre for Advanced Technology\n452013IndoreIndia",
"HXAL\nSUS\nRaja Ramanna Centre for Advanced Technology\n452013IndoreIndia",
"UGC DAE Consortium for Scientific Research\n452001IndoreIndia",
"UGC DAE Consortium for Scientific Research\n452001IndoreIndia"
] | [] | Coupling between different interactions allows to control physical aspects in multifunctional materials by perturbing any of the degrees of freedom. Here, we aim to probe the correlation among structural, electronic and magnetic observables of Sm 2 NiMnO 6 ferromagnetic insulator double perovskite. Our employed methodology includes thermal evolution of synchrotron X-ray diffraction, near edge and extended edge hard X-ray absorption spectroscopy and bulk magnetometry. The magnetic ordering in SNMO adopts two transitions, at T C =159.6K due to ferromagnetic arrangement of Ni-Mn sublattice and at T d =34.1K because of anti-parallel alignment of polarized Sm paramagnetic moments with respect to Ni-Mn network. The global as well as local crystal structure of SNMO undergoes isostructural transitions across T C and T d , observed by means of temperature dependent variation in Ni/Mn-O, Ni-Mn bonding characters and super exchange angle in Ni-O-Mn linkage. Hybridization between Ni, Mn 3d, O 2p electronic states is also modified in the vicinity of magnetic transition. On the other hand, the signature of Ni/Mn anti-site disorders are evidenced from local structure and magnetization analysis. The change in crystal environments governs the magnetic response by imposing alteration in metal -ligand orbital overlap. Utilizing these complimentary probes we have found that structural, electronic and magnetic states are inter-coupled in SNMO which makes it a potential platform for technological usage. | null | [
"https://arxiv.org/pdf/2205.00537v1.pdf"
] | 248,495,977 | 2205.00537 | 3045c09a2bbb71af8579bf9f4fd440364eddb7c1 |
Existence of inter coupled structural, electronic and magnetic states in Sm 2 NiMnO 6 double perovskite
S Majumder
UGC DAE Consortium for Scientific Research
452001IndoreIndia
M Tripathi
UGC DAE Consortium for Scientific Research
452001IndoreIndia
. O De Souza
Elettra Sicrotrone Trieste S.C.p.A
SS 14-km 163.534149BasovizzaItaly
A Sagdeo
HXAL
SUS
Raja Ramanna Centre for Advanced Technology
452013IndoreIndia
Homi Bhabha National Institute
Anushakti nagar400 094MumbaiIndia
L Olivi
Elettra Sicrotrone Trieste S.C.p.A
SS 14-km 163.534149BasovizzaItaly
M N Singh
HXAL
SUS
Raja Ramanna Centre for Advanced Technology
452013IndoreIndia
S Pal
HXAL
SUS
Raja Ramanna Centre for Advanced Technology
452013IndoreIndia
R J Choudhary
UGC DAE Consortium for Scientific Research
452001IndoreIndia
D M Phase
UGC DAE Consortium for Scientific Research
452001IndoreIndia
Existence of inter coupled structural, electronic and magnetic states in Sm 2 NiMnO 6 double perovskite
Coupling between different interactions allows to control physical aspects in multifunctional materials by perturbing any of the degrees of freedom. Here, we aim to probe the correlation among structural, electronic and magnetic observables of Sm 2 NiMnO 6 ferromagnetic insulator double perovskite. Our employed methodology includes thermal evolution of synchrotron X-ray diffraction, near edge and extended edge hard X-ray absorption spectroscopy and bulk magnetometry. The magnetic ordering in SNMO adopts two transitions, at T C =159.6K due to ferromagnetic arrangement of Ni-Mn sublattice and at T d =34.1K because of anti-parallel alignment of polarized Sm paramagnetic moments with respect to Ni-Mn network. The global as well as local crystal structure of SNMO undergoes isostructural transitions across T C and T d , observed by means of temperature dependent variation in Ni/Mn-O, Ni-Mn bonding characters and super exchange angle in Ni-O-Mn linkage. Hybridization between Ni, Mn 3d, O 2p electronic states is also modified in the vicinity of magnetic transition. On the other hand, the signature of Ni/Mn anti-site disorders are evidenced from local structure and magnetization analysis. The change in crystal environments governs the magnetic response by imposing alteration in metal -ligand orbital overlap. Utilizing these complimentary probes we have found that structural, electronic and magnetic states are inter-coupled in SNMO which makes it a potential platform for technological usage.
INTRODUCTION
In materials science there has been always a quest for the strong correlation between charge, spin and lattice degrees of freedom to have additional control over functional arXiv:2205.00537v1 [cond-mat.mtrl-sci] 1 May 2022 aspects of the system [1]. Double perovskites with Ni and Mn as B-site cations A 2 NiMnO 6 (ANMO), have huge potential in new generation quantum electronics owing to rare tunable ferromagnetic insulating ground state [2,3,4,5,6]. In a previous theoretical study it is predicted that LNMO (A=La) holds magnetic order dependent electronic properties [7]. Later, the evidences of magneto-dielectric, magneto-phononic and magneto-elastic correlations are found in LNMO [2,8,9]. On the other hand, based on first principal calculations it is argued that ANMO system may have structural strain driven multiferroic characters [10] which are also experimentally verified in strained LNMO phase [11]. Aforementioned reports indicate that there is an interplay in between charge, spin and lattice, which governs the intriguing properties in ANMO family. However, understanding of these possible inter-coupled multifarious interactions needs comprehensive experimental insights, which is still elusive.
Ground state calculations on ANMO family have predicted that for different Asite ions, the magnetic ordering can transform from ferromagnetic (for A=La) to E type antiferromagnetic (for A=Y and In) [12,13]. The experimental magnetic structure analysis reveals that ANMO for A=La, Nd, Sm and Y ions, it is collinear ferromagnetic [3,14,15]; for A=Tb, Ho, Er and Tm ions, it is canted ferrimagnetic [16]; for A=In, it is incommensurate antiferromagnetic [17] and for A=Sc, it is collinear antiferromagnetic [18]. These studies point out that magnetic ordering in ANMO is highly sensitive to chemical pressure on the lattice structure exerted by different Asite ion sizes. At room temperature, depending upon the cation arrangement there are two possible crystal symmetry for ANMO system, monoclinic P2 1 /n and orthorhombic Pbnm [4,14,19]. In monoclinic structure Ni/Mn ions arrange in ideal rock-salt fashion at alternate octahedral center sites. Whereas, a random occupation between Ni/Mn ions are formed for orthorhombic case. However in practical crystal system, completely ordered or disordered situation is difficult to achieve due to comparable ionic sizes of Ni and Mn species [4,5,6]. Even in highly ordered system, some fraction of disorder is present owing to anti-phase boundary formation [20]. Different ordered and disordered coordinations have drastic role in stabilizing the magnetic phase of the system [4,5,6]. The temperature effect on local and global crystal environments of ANMO is yet to be explored. On the other hand, in previous studies different interpretations about the charge states of Ni and Mn ions are found. For instance, according to Goodenough et al. [21] the magnetic exchange in LNMO (for A=La) is in between Ni 3+ and Mn 3+ ions whereas, Blasse et al. [22] claims for Ni 2+ and Mn 4+ magnetic interaction. Further more, for perovskite nikelets and manganites, temperature driven valence state transition is reported in literature [23,24]. So, it is necessary to check the possibility of such temperature induced valence transition in ANMO system. Moreover, the thermal evolution of crystal and electronic structures holds the key to interpret the unique ferromagnetic-insulating ground state in ANMO family of materials.
Despite several research endeavors, whether the structural, electronic and magnetic observables are inter-coupled in multifuctional ANMO double perovskite or not, is still an open question. In this backdrop we aim to investigate the thermal evolution of local and global structure, electronic state and magnetic ordering in Sm 2 NiMnO 6 (SNMO) system. Double perovskite with Sm at A-site is chosen as a representative platform of ANMO family owing to unique mixed multiplet state of rare earth Sm 3+ ion which can be perturbed by crystal field, exchange field and/or thermal energy [25]. Here, we have utilized a comprehensive analysis of temperature dependent synchrotron radiation X-ray diffraction, hard X-ray absorption spectroscopy in near edge and extended edge regions and bulk magnetometric measurements. The findings of present work establish that in SNMO structural, electronic and magnetic observables are correlated with each other and hence physical properties can be tuned by manipulating any of these degrees of freedom.
EXPERIMENTAL DETAILS
SNMO polycrystalline bulk sample was synthesized by solid state reaction process following the recipe as described in Ref. [4]. The temperature dependence on global crystal structure was investigated by recording powder X-ray diffraction (PXRD) scans in temperature range of 5K T 300K, utilizing synchrotron radiation source at angle dispersive X-ray diffraction (ADXRD) beamline (BL-12, Indus-II, RRCAT, Indore, India). The PXRD profiles were collected using a mar345 image plate detector and obtained two-dimensional patterns were integrated to 2θ scales using the FIT2D program [26]. Incident X-ray wavelength was calibrated by measuring data for reference LaB 6 sample and the estimated value was found to be λ=0.71949Å. Structural refinements of PXRD data were carried out employing FULLPROF software package [27] wherein for background simulations WINPLOTR was used and for visualization of obtained structure VESTA was used. Elemental valence state were probed by Ni and Mn K X-ray absorption near edge spectroscopy (XANES) performed within 13K T 300K. The temperature dependency of local structure was examined by Ni K edge extended X-ray absorption fine structure (EXAFS) spectroscopy measured in temperature range of 13K T 300K. The occurrence of Sm L 3 edge (6716eV) close to Mn K edge (6539eV) limits the EXAFS study for Mn case. XANES and EXAFS experiments were carried out in transmission mode using ionization chamber detectors (Oxford Instruments) and hard X-ray radiation source at XAFS beamline (11.1R, Elettra-Sincrotrone, Trieste, Italy). Incident beam photon energy calibration was done by recording data for Ni and Mn reference metal foils. The intrinsic energy resolution energy resolution ∆E/E was ∼1×10 −4 across the explored energy regime. To obtain normalized absorption, standard normalization process were applied on XANES spectra using ATHENA program [28]. EXAFS spectra were colloected several times to check data reproducibility and merged to have better statistics. After normalization of data, standard background subtraction process were followed to extract k-space (χ(k)) EXAFS signal utilizing AUTOBK algorithm [29] in ATHENA package. In order to obtain EXAFS oscillation in Rspace (χ(R)), Fourier transformations of the k-space signal (within selected k range) were computed. Local structural analysis of EXAFS data were carried out using ARTEMIS software package wherein to generate theoretical standers, ATOMS and FEFF6 programs [30,31] were implemented. For each coordination shell (assume i number of shells) the average coordination distance (R i ) and corresponding mean-square relative displacement factor (σ 2 i ) were computed from standard EXAFS expression [31], wherein all partial contributions from individual scattering paths (i) were summed over to simulate model EXAFS spectra. The dc magnetic response were investigated employing MPMS 7-Tesla SQUID-VSM (Quantum Design Inc., USA) system. Prior to measurement the sample was warmed well above corresponding magnetic transition temperatures to erase previous (if any) magnetic history and standard de-Gaussing protocol was applied to nullify the trapped (if any) magnetic field in the magnetometer superconducting coil. Estimated average magnetic moment sensitivity was ∼10 −8 emu.
RESULTS AND DISCUSSION
The temperature dependent magnetization M(T), measured in typical zero field cooled (ZFC) warming and field cooled warming (FCW) protocols, in presence of applied dc magnetic field µ 0 H=100Oe, are presented in Fig. 1(a). Upon cooling from room temperature, SNMO undergoes two distinct magnetic transitions nomenclatured as, (i) T C , the onset in M(T) and (ii) T d , the downturn in M(T). T C , evaluated from the inflection point at FCW M(T) curve (Inset of Fig. 1(a)), is found to be T C =159.6K. According to GK [32] rule, the virtual hopping of electron from half filled Ni 2+ e g orbital to empty Mn 4+ e g orbital in 180 o geometry should be ferromagnetic in nature. On the other hand, if magnetic exchange is also possible in between Ni e g and Mn t 2g orbitals, it will be antiferromagnetic. However, in previous theoretical studies on prototype LNMO (for A=La) system, it is observed that t 2g electrons are more localized than e g , the magnetic exchange is governed by Ni-Mn e g orbital overlap through intermediate O p orbital and ferromagnetic ground state is energetically more favorable than the antiferromagnetic case [33,34]. Neutron powder diffraction (NPD) measurements below T C on SNMO system reveal long range colinear ferromagnetic ordering of Ni-Mn sublattice moments in F x F z configurations at cation ordered structures [4]. Therefore, observed magnetic transition at T=T C is attributed to Ni-O-Mn superexchange interaction driven transformation of paramagnetic to ferromagnetic phase in SNMO. The inverse susceptibility 1/χ(T) experimental curve starts deviating from Curie-Weiss fitting behaviour at ∼100K above T C (Inset of Fig. 1(a)) value, indicating presence of short scale interactions even in paramagnetic state. This is because of cation disorder (Ni-O-Ni) related short range magnetic interactions in the system (discussed later). In past studies conflicting explanations are found regarding the low temperature transition at T=T d . For example, to discuss the origin of downturn behavior it is proposed that there may be, (i) strong magnetocrystalline anisotropy due to spin-orbit coupling between rare earth -transition metal network [35] or (ii) rare earth long range magnetic ordering [19] or (iii) reentrant transition to spin glass phase [36]. The thermal evolution of microscopic magnetic structure divulges the internal field polarization of paramagnetic Sm moments opposite to ordered Ni-Mn network in the vicinity of T=T d [4]. Unlike T C , the transition temperature T d varies with different measuring field strengths due to the competition between internal field and external field acting on the sublattice moments [4,5]. T d , estimated from the temperature point at which low field (µ 0 H=25Oe) FCW M(T) curve (data not shown here) achieves maximum moment value, is found to be T d =34.1K. The isothermal magnetization measured as a function of applied magnetic field M(H) at T=10K is shown in Insets (i, ii) of Fig. 1(b). M(H) curve does not attain saturation even at µ 0 H=70kOe of applied field. This is possibly because of opposite arrangement between Sm paramgnetic momments and Ni-Mn ordered ferromagnetic moments or the presence of Ni/Mn cation disorder (discussed later) mediated short scale antiferromagnetic interactions in predominant ferromagnetic ordered host matrix. Thermal evolution of coercive field H C and moment M 70kOe are illustrated in Fig. 1(b). Both of H C and M 70kOe behaviors depict abrupt change in the vicinity of T C and T d . Aforementioned magnetic transition temperatures as obtained from bulk magnetometric experiments have well correspondence with our previous microscopic magnetic structure analysis on SNMO system [4].
In order to investigate the global structural evolution across observed magnetic transitions, temperature dependent X-ray diffractograms are measured as displayed in Fig. 2(a). Starting from T=300K down to T=5K, no new Bragg's reflection is observed in PXRD patterns, confirming that crystal symmetry remains the same within the investigated temperature regime. PXRD pattern at T=5K along with Rietveld simulation for monoclinic P2 1 /n structure are depicted in Fig. 2(b). Goodness of fitting indicators for T=5K data analysis are: R p =3.11, R wp =3.81, R exp =4.94, χ 2 =0.593. These statistical factors suggest reasonably good agreement between experimentally observed and simulated patterns. The observance of (011) superstructure reflection in NPD profile confirms dominating Ni, Mn cation ordering at respectively 2c, 2d sites of SNMO monoclinic P2 1 /n (SG 14) lattice [4]. Point to be noted here that it is difficult to resolve such superstructure peak by PXRD measurements owing to nearly equal ionic radius of Ni and Mn species in SNMO. The temperature dependency of a, b, c lattice parameters and corresponding change in unit cell volume are illustrated in Figs. 3 (a-d), respectively. At high temperature typical thermal expansion of cell is observed. With lowering temperature, distinguishable abrupt changes from monotonic decreasing trends are found specifically across the magnetic phase transition temperature T=T d , indicating the presence of iso-structural transitions coupled with magnetic ordering in SNMO. As the magnetic exchange interactions depend on the orbital overlap in transition metal -ligand linkage, we have explored the variation of Ni/Mn-O, Ni-Mn average bond lengths and N i − O − M n average bond angle as a function of temperature, which are shown in Figs. 3(a-d), respectively. Bond lengths and bond angel exhibit anomalous behavior in the vicinity of magnetic transition at T d . The co-occurrence of structural and magnetic transitions indicate coupling between crystal structure and magnetic structure in SNMO.
To check if there is any temperature driven valence state transition of constituent elements liable for the magnetic anomalies in SNMO, temperature dependent X-ray absorption measurements at Ni and Mn K edges are carried out. Both Ni and Mn K edge XANES show characteristic pre-edge and white line features, as displayed in Figs. 4(a, b). The K absorption edge is identified as metal 1s to empty p band electronic transition [37]. In general, with increasing valency of absorbing specie, the band edge position of XANES spectra shifts towards higher energy side [38]. Here edge energy is estimated from the first inflection point in XANES spectra. Comparing the Ni/Mn K XANES spectra for SNMO sample and Mn 4+ (MnO 2 ), Mn 3+ (Mn 2 O 3 ), Ni 2+ (NiO) and Ni 3+ (Ni 2 O 3 ) standard references (data not shown here) at room temperature, it is confirmed that Ni and Mn have chemical valency in between 2+, 3+ and 3+, 4+, respectively [4]. Noteworthily, both Ni and Mn absorption band edges do not show any apparent shift with respect to measuring temperature variation (Figs. 4(a, b)), suggesting that these mixed valence nature remains unaltered across the magnetic transitions.
The pre-edge structures in transition metal K XANES is possibly originated due to electric quadrupole or / and dipole transitions from metal 1s to empty 3d states [37]. In centrosymmetric structure, 1s → 3d dipole excitation is forbidden. Any deviation from centrosymmetry may introduce metal 3d -4p hybridization through ligand involvement and then dipole transition will be weakly allowed [37]. However, the irreducible representation analysis using group theory calculation predicts that in octahedral system metal 3d -4p orbital mixing is not allowed [39]. Point to be noted here that SNMO has centrosymmetric structure (SG: P2 1 /n) where Ni/Mn absorbents are ocahedrally surrounded by O ligands. Therefore, pre-edge structures observed here, are assigned to electric quadrupole transitions from Ni/Mn 1s to 3d states. It is important to mention that owing to significant overlapping between transition metal 3d -O 2p orbitals, XANES K pre-edge also contains information about the metal -ligand hybridization [37]. In a previous report [40], it was observed that with varying Mn-O separations in MnO 4 tetrahedron, the pre-edge structure of Mn K edge absorption changes, which can be understood by considering modification in metal-ligand hybridization due to change in coordination distances. The area under the curve of Ni, Mn pre-edge features are estimated using arc-tangent background function and Gaussian peak shapes as displayed by Insets of Figs. 5(a, b). The thermal evolution of obtained integrated intensity of Ni, Mn pre-edge curves show anomalous behavior close to T=T C and T d , as illustrated in Figs. 5(a, b). These results reveal temperature driven changes in Ni/Mn-O hybridization.
To explore the local structural particularities such as coordination environment, short range disorder etc. present in the crystal system, temperature dependent extended X-ray absorption measurements are carried out at Ni K edge. Figures 6(a, b) present temperature dependent k 2 -weighted k-space oscillations (k 2 χ(k)) and corresponding modulus of Fourier transformed R-space spectra (|χ(R)|) respectively, for SNMO Ni K edge EXAFS. To probe the Ni/Mn cation disorder present in the system, we have performed quantitative fitting analysis on |χ(R)|. The theoretical model is generated using cell parameter values as obtained from Rietveld refinement of PXRD data [4]. The fitting range is confined to 1Å To account for Ni/Mn anti-site disorder in the fitting model, two different structures, both having same core absorber as Ni atom, are considered, as illustrated in Figs. 7(a, b). In ideal ordered configuration, all the next B-sites are filled with Mn atoms (Fig. 7(a)). Whereas, in ideal disordered case, all the next B-sites are occupied with Ni atoms (Fig. 7(b)). The simulated EXAFS spectra have a weighted convolution from these two types of cells [41]. Ni-Mn and Ni-Ni shell coordination numbers are refined to evaluate the fractional weights of each structures. The concentration of antisite disorder is quantified in terms of the probability of encountering disordered bond configurations Q ASD , defined as [42], obtained by analyzing Ni K EXAFS spectra for NiO. As s 2 0 is chemically transferable, same value can be used for SNMO case also. Equal energy shift (∆E 0 ) is used for all coordination shells and for all temperatures to avoid any fictitious artifact due to correlation with coordination distances. During refinement process, the average coordination distances (R) and mean-square relative displacement (MSRD) factors (σ 2 ) are freed to refine. In all fitting, the total coordination numbers (N) are kept fixed at corresponding crystallographic values, N O =6, N Sm =8 and N B−site =6. This should eliminate spurious correlation with σ 2 . Some preliminary trial refinements suggest that same σ 2 can be shared for Ni-Mn/Ni scattering paths as they are highly correlated. The quality of fitting is monitored by R, factor defined as,
Q XAS ASD = N N i−N i /N B−site N B−site = N N i−M n + N N i−N i(1)R = i [Re(χ d (R i ) − χ t (R i )) 2 + Im(χ d (R i ) − χ t (R i )) 2 ] i [Re(χ d (R i )) 2 + Im(χ d (R i )) 2 ](2)
where, χ d and χ t correspond to the experimental and theoretical χ(R) values, respectively. The best fit along with experimentally observed EXAFS pattern measured at T=13K are displayed by k 2 .χ(k) and corresponding Fourier transform of k 2 .χ(k), as illustrated in Figs. 8(a, b), respectively. Here, the partial contribution from different coordination shells used in model analysis are vertically shifted to have better visualization. The goodness of fitting indicator R having value 0.009 suggests reasonable agreement between experimental and simulated profiles. The estimated antiside disorder concentration Q ASD is found to be 3.6(3)%. Obtained Q ASD value is in well agreement with NPD analysis [4]. Therefore, EXAFS modeling confirms the presence of Ni/Mn cation ordered and disordered structures in SNMO lattice. Such local anti-site disorder lefts its imprint in magnetization behaviors by introducing short scale antiferromagnetic interaction in background of long range ferromagnetic ordered SNMO lattice [4,5,6].
The thermal evolution of R and σ 2 obtained from EXAFS fitting at different temperature values, are displayed in Figs. 9(a, b), respectively. For all coordination shells, σ 2 show increasing trend with rising measurement temperatures, indicates more thermal agitation at elevated temperatures. The temperature dependencies of σ 2 can be described considering correlated Einstein model where vibrations in atomic bonds are assumed as harmonic oscillators with single frequency ω E , known as the Einstein frequency [43]. According to Einstein model σ 2 (T ) can be expressed as [43,44],
σ 2 (T ) = σ 2 s + 2µω E coth ω E 2k B T , ω E k B = θ E(3)
where, σ 2 s and µ represent static bond disorder contribution and reduced mass of atomic bond, respectively. θ E , known as the Einstein temperature, measures atomic bond stiffness. Obtained θ E values reduce as
θ N i−O E (∼872K) > θ N i−Sm E (∼779K) > θ N i−M n/N i E (∼124K)
, suggesting decrease in phonon activation energies with increasing coordination distances [44]. Among all possible bonding linked with Ni core, the largest phonon activation energy for Ni-O atom pairs indicates highest rigidity of NiO 6 octahedra [43]. Noteworthily, experimentally obtained σ 2 (T ) for Ni-O bond deviates from the modeled behavior at T=T C by approximately 29% which is larger than statistical uncertainty and hence is not associated to any artifact. Therefore, observed change in σ 2 (T ) N i−O points out the temperature driven anomalous variation of bond disorder across the magnetic transition.
The local and global crystal structure analysis utilizing temperature dependent EXAFS and PXRD measurements reveal presence of isostructural changes in SNMO. The variation in crystal environment drives modification in metal -ligand hybridization which is evidenced from electronic structure studies using thermal evolution of XANES results. Alteration in lattice structure and electronic energy landscape is liable for the observed magnetic transitions in SNMO. In order to probe the electro-magnetic correlation in the SNMO system, we have measured the isothermal magnetization in the presence and absence of applied electric filed, as depicted in Fig. 10. × 100, is found to be 7.3% at T = 5 K and E = 53 V/mm. Therefore, our findings present conclusive evidences of the coupled structural, electronic and magnetic phases in SNMO double perovskite. Such couplings are desirable from device application view point owing to have additional degrees of freedom to control functional aspects of the system.
CONCLUSION
In summary, we have explored temperature dependency on structural, electronic and magnetic properties of Sm 2 NiMnO 6 double perovskite. From bulk magnetization measurements it is observed that SNMO encounters two magnetic transitions, attributed to ferromagnetic ordering of Ni-Mn network and polarazation of Sm paramagnetic moments opposite to Ni-Mn sublattice, at T C =159.6K and T d =34.1K, respectively. SNMO crystallizes in P2 1 /n monoclinic structure with dominating Ni/Mn cation ordered phases. X-ray diffraction results exhibit isostructural change in global crystal structure in the vicinity of T C and T d , whereas the lattice symmetry remains the same within investigated temperature regime (5K T 300K). Moreover, the average bond distances and bond angle between Ni, Mn cations via intermediate O anion are observed to show abrupt variation across magnetic transition, particularly near T=T d . The local coordination environment studies using Ni K edge extended X-ray absorption spectroscopy reveal deviation of Ni-O bonding feature from expected Einstein model behavior particularly at T C . Furthermore, existence of Ni/Mn anti-site disorder is confirmed which influences the bulk magnetic characters. Absence of temperature dependent valence state transition of Ni, Mn ions is confirmed from X-ray absorption near edge spectroscopy. However, changes in Ni, Mn 3d -O 2p hybridization around T C and T d are evidenced. The temperature driven alteration in local as well as global structures provokes modification in energy landscape which eventually results in variation of magnetic observable of SNMO system. Combining these results, we have demonstrated a ferromagnetic insulator platform where structural, electronic and magnetic properties are coupled with each other. We hope present work will have huge relevance on tuning functional aspects of all prototype double perovsite system by manipulating any of the aforementioned degrees of freedom.
Figure 1 .
1(a): Magnetization as a function of temperature M(T), measured in zero filed cooled warming and field cooled warming cycles with applied magnetic field of µ 0 H=100Oe. Inset shows (left panel): first order temperature derivative of magnetization dM(T)/dT (violet curve) and (right panel): inverse susceptibility 1/χ(T) observed data (pink curve) along with Curie-Weiss fitting (olive line). Thermal evolution of (b left panel): coercivity and (b right panel): magnetic moment recorded with µ 0 H=70kOe. Inset (i): Isothermal magnetization as a function of magnetic field M(H), acquired at T=10K. Inset (ii): Enlarged view across low field region of T=10K M(H) curve. The vertical dashed and dotted lines correspond to magnetic transition temperatures T C and T d , respectively.
Figure 2 .
2(a): Temperature dependency of powder X-ray 2θ diffractograms. (b): Representative Rietveld fitting of PXRD 2θ scan measured at T=5K showing observed (red open circles), calculated (black solid line) and difference (blue solid line) patterns along with Bragg's positions (green vertical bars).
Figure 3 .
3The thermal evolution of unit cell parameters (a): a, (b): b, (c): c, (d): volume, atomic bond distances (e): Ni-O, (f): Mn-O, (g): Ni-Mn and (d): average N i − O − M n bond angle obtained from PXRD analysis.
Figure 4 .
4Temperature dependent X-ray absorption spectra in near edge region measured across (a): Ni and (b): Mn K edges.
Figure 5 .
5Temperature dependency of pre-edge feature integrated intensities for (a): Ni and (b): Mn K X-ray absorption spectra. Insets show representative fittings of pre-edge structures recorded at T=300K for Ni and Mn K absorption, respectively.
Figure 6 .
6The temperature dependencies of extended X-ray absorption (a): k 2weighted k-space signals and (b): corresponding modulus of Fourier transformed Rspace oscillations.
R 4 .
42Å in R-space and 2.5Å −1 k 12Å −1 in k-space. Within this selected region, EXAFS signal is contributed from photoelectron scattering by O anions at nearest neighbor octahedral sites, Sm cations at second nearest neighbor sites and Mn/Ni cations at next B-sites connected with core absorber Ni ion through intermediate O ion. The most appropriate single scattering and multiple scattering paths are employed in fitting model to adequately generate the experimental spectral behavior. Although single scattering in Ni-O, Ni-Sm and Ni-Mn/Ni linkages have dominating contribution, the multiple scattering (forward triangle geometry) effects through Ni-O-Mn/Ni can not be ignored in analysis.
where, N N i−M n , N N i−N i are the coordination numbers for ordered and disordered structures respectively and N B−site is the total coordination number of all (Mn/Ni) B-site configurations.In fitting model the amplitude reduction factor (s 2 0 ) is fixed at 0.84. This value is
Figure 7 .
7Schematic representations of cation (a): ordered (Ni-O-Mn) and (b): disordered (Ni-O-Ni) unit cells showing a-b, b-c crystallographic planes. Green, Blue, Red and Grey spheres are for Sm, Ni, Mn and O ions, respectively.
Figure 8 .
8Representative fitting of extended X-ray absorption (a): k 2 -weighted kspace signal and (b): corresponding Fourier transformed R-space oscillation recorder at T=13K. Spectral contributions of different coordination shells accounted in fitting model, are shifted in vertical axes to have clear visualizations.
Figure 9 .
9Thermal evolution of (a): coordination distances and (b): corresponding bond disorder parameters obtained from EXAFS analysis. In (b) geometric symbols are for MSRD values estimated from EXAFS studies and solid lines are for the Einstein model behaviors.
Figure 10 .
10Magnetic isotherms at T = 5 K in absence and presence of applied electric field bias of 53 V/mm. Inset: Enlarged view across the low field region of the M(H) curve.
The strength of electro-magnetic coupling, estimated as (M (0)−M (E)) M (0)
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| [] |
[
"Modelling realistic microgels in an explicit solvent",
"Modelling realistic microgels in an explicit solvent"
] | [
"F Camerin \nCNR-ISC\nUos Sapienza, Piazzale A. Moro, 200185RomaItaly\n\nDipartimento di Scienze di Base e Applicate per l'Ingegneria\nSapienza Università di Roma\nvia A. Scarpa, 1400161RomaItaly\n",
"N Gnan \nCNR-ISC\nUos Sapienza, Piazzale A. Moro, 200185RomaItaly\n\nDipartimento di Fisica\nSapienza Università di Roma\nPiazzale A. Moro, 200185RomaItaly\n",
"L Rovigatti \nCNR-ISC\nUos Sapienza, Piazzale A. Moro, 200185RomaItaly\n\nDipartimento di Fisica\nSapienza Università di Roma\nPiazzale A. Moro, 200185RomaItaly\n",
"E Zaccarelli [email protected] \nCNR-ISC\nUos Sapienza, Piazzale A. Moro, 200185RomaItaly\n\nDipartimento di Fisica\nSapienza Università di Roma\nPiazzale A. Moro, 200185RomaItaly\n"
] | [
"CNR-ISC\nUos Sapienza, Piazzale A. Moro, 200185RomaItaly",
"Dipartimento di Scienze di Base e Applicate per l'Ingegneria\nSapienza Università di Roma\nvia A. Scarpa, 1400161RomaItaly",
"CNR-ISC\nUos Sapienza, Piazzale A. Moro, 200185RomaItaly",
"Dipartimento di Fisica\nSapienza Università di Roma\nPiazzale A. Moro, 200185RomaItaly",
"CNR-ISC\nUos Sapienza, Piazzale A. Moro, 200185RomaItaly",
"Dipartimento di Fisica\nSapienza Università di Roma\nPiazzale A. Moro, 200185RomaItaly",
"CNR-ISC\nUos Sapienza, Piazzale A. Moro, 200185RomaItaly",
"Dipartimento di Fisica\nSapienza Università di Roma\nPiazzale A. Moro, 200185RomaItaly"
] | [] | Thermoresponsive microgels are polymeric colloidal networks that can change their size in response to a temperature variation. This peculiar feature is driven by the nature of the solvent-polymer interactions, which triggers the so-called volume phase transition from a swollen to a collapsed state above a characteristic temperature. Recently, an advanced modelling protocol to assemble realistic, disordered microgels has been shown to reproduce experimental swelling behavior and form factors. In the original framework, the solvent was taken into account in an implicit way, condensing solvent-polymer interactions in an effective attraction between monomers. To go one step further, in this work we perform simulations of realistic microgels in an explicit solvent. We identify a suitable model which fully captures the main features of the implicit model and further provides information on the solvent uptake by the interior of the microgel network and on its role in the collapse kinetics. These results pave the way for addressing problems where solvent effects are dominant, such as the case of microgels at liquid-liquid interfaces. | 10.1038/s41598-018-32642-5 | [
"https://arxiv.org/pdf/1807.07893v2.pdf"
] | 52,846,118 | 1807.07893 | 74f44a40a5248c0e89d3e72294d772107ac90edb |
Modelling realistic microgels in an explicit solvent
F Camerin
CNR-ISC
Uos Sapienza, Piazzale A. Moro, 200185RomaItaly
Dipartimento di Scienze di Base e Applicate per l'Ingegneria
Sapienza Università di Roma
via A. Scarpa, 1400161RomaItaly
N Gnan
CNR-ISC
Uos Sapienza, Piazzale A. Moro, 200185RomaItaly
Dipartimento di Fisica
Sapienza Università di Roma
Piazzale A. Moro, 200185RomaItaly
L Rovigatti
CNR-ISC
Uos Sapienza, Piazzale A. Moro, 200185RomaItaly
Dipartimento di Fisica
Sapienza Università di Roma
Piazzale A. Moro, 200185RomaItaly
E Zaccarelli [email protected]
CNR-ISC
Uos Sapienza, Piazzale A. Moro, 200185RomaItaly
Dipartimento di Fisica
Sapienza Università di Roma
Piazzale A. Moro, 200185RomaItaly
Modelling realistic microgels in an explicit solvent
Thermoresponsive microgels are polymeric colloidal networks that can change their size in response to a temperature variation. This peculiar feature is driven by the nature of the solvent-polymer interactions, which triggers the so-called volume phase transition from a swollen to a collapsed state above a characteristic temperature. Recently, an advanced modelling protocol to assemble realistic, disordered microgels has been shown to reproduce experimental swelling behavior and form factors. In the original framework, the solvent was taken into account in an implicit way, condensing solvent-polymer interactions in an effective attraction between monomers. To go one step further, in this work we perform simulations of realistic microgels in an explicit solvent. We identify a suitable model which fully captures the main features of the implicit model and further provides information on the solvent uptake by the interior of the microgel network and on its role in the collapse kinetics. These results pave the way for addressing problems where solvent effects are dominant, such as the case of microgels at liquid-liquid interfaces.
Introduction
In recent years microgels -colloidal-scale polymer networks -have emerged as a popular model system in condensed matter physics 1 thanks to their colloid/polymer duality 2 . The combination of colloidal properties and responsiveness to external stimuli is the key for their appeal for both applications and fundamental science 3 . Among microgels, the most widely studied are those based on Poly(N-isopropylacrylamide) (PNIPAM), a thermoresponsive polymer able to swell and deswell reversibly as a result of temperature changes. When PNIPAM chains are crosslinked with bisacrylamide (BIS), microgel particles can be prepared in a range of sizes of 10 − 1000 nm by standard synthesis methods 4 , and even reach much larger scale (up to 100 µm) with microfluidic techniques 5 . These particles undergo a Volume Phase Transition (VPT) in water at a temperature of ≈ 305K, from a swollen state at low temperatures to a collapsed one at high temperatures. This swelling-deswelling transition is fully reversible and can be exploited to tune the size of the particles in situ. The VPT is completely controlled by the polymer-solvent interactions, echoing the coil-to-globule transition of linear PNIPAM chains in water 6 . As a matter of fact, the role of water is highly relevant, as the VPT originates from changes in the hydrophilic/hydrophobic character of the interactions of the polymer with the solvent upon temperature variations.
Experimental work on microgels has enormously increased in the last couple of decades, and a comparison of experimental data with theory has been possible thanks to the use of the classical Flory-Rehner theory of swelling 7 . On the other hand, microgel simulations have been less abundant due to the complex, multi-scale nature of the particles. So far, most efforts have relied on the use of unrealistic networks, often based on ordered, diamond-like topologies, in which all strands are of equal length [8][9][10][11][12] . Only a few of these approaches have explicitly considered the role of the solvent [12][13][14] .
Recently, we have introduced a novel method to synthesize realistic microgel particles in silico through the assembly of fully-bonded, disordered networks with arbitrary topology 15,16 . In this approach we initially consider the self-assembly of a mixture of patchy particles, respectively bivalent and tetravalent, to mimic monomers and crosslinkers. To retain a spherically-shaped network, the mixture is confined within a sphere of a given radius. Fully-bonded configurations are obtained by introducing a swapping mechanism that makes it possible to equilibrate the system even at the strong attractions required to maximize the bonding. In this protocol there are two parameters controlling the topology of the resulting network: the concentration of crosslinkers and the confinement radius. Thus, more compact and homogeneous networks are obtained in presence of a large number of crosslinkers and/or for a tight confinement, while looser and more heterogeneous microgels can be produced with a smaller amount of crosslinkers and a very weak confinement. A thorough discussion on how the internal structure of the microgels depends on these parameters can be found in Refs. 15,16 .
Once the network is assembled, we replace the patchy (reversible) interactions with permanent bonds by adopting the classical Kremer-Grest bead-spring model for polymers 17 to preserve the network topology throughout the course of the simulation. In order to reproduce the swelling behavior, it is possible to incorporate in the model an attractive potential that has been shown to capture the variation in polymer-water interactions upon changing temperature. With this approach, the solvent is implicit and the solvophobic potential accounts for it within the thermodynamic properties of the system in an effective way. This implicit solvent model was shown to be able to faithfully reproduce swelling data of individual microgels measured with Dynamic Light Scattering experiments 15 . Even though the use of an explicit versus an implicit solvent model 18,19 should give identical results in terms of equilibrium properties, there are a number of features that cannot be correctly captured and/or described by an implicit model. In particular, the kinetics of swelling and deswelling will depend on the presence of the solvent and on how it is modelled. Besides that, there are situations of fundamental and practical interest in which an explicit solvent will dramatically affect the picture. For instance, to model a system at a liquid-liquid interface, it is necessary to take into account the presence of the two different media in order to capture effects related to the surface tension 20,21 .
In order to be able to handle these situations, here we take the implicit-solvent model of Refs. 15,16 and extend it by developing an explicit solvent description that accurately predicts the swelling behavior of microgel particles. We use the swelling properties exhibited by the implicit solvent model, which has been shown to faithfully reproduce the experimental results, as reference data to calibrate the explicit-solvent parametrisation. By comparing the swelling ratio as a function of temperature and the microgel density profile and form factor with and without solvent, we are able to discriminate among different solvent models and choose the explicit description that works best. In particular, we intend to model a generic solvent that ensures that the key properties of microgel colloids are accurately reproduced rather than to provide a systematic and exhaustive study on the influence of the system parameters on the properties of the particle. We further test the robustness of our approach by repeating the analysis for microgels generated with different topologies and confinement radii. Once established our explicit model, we first look at the arrangement of the solvent inside the microgel across the volume phase transition, and then study the kinetics of the deswelling. Overall, our results open up the possibility to obtain more and more realistic descriptions of microgels, thanks to which it will be possible to tackle exciting problems in which the explicit role of the solvent plays a crucial role [21][22][23][24] . We start by discussing the swelling behavior of microgels in the presence of an explicit solvent as compared to the reference case of the implicit model V α , Fig. 1(a) (see Methods), discussed in Ref. 15 . To this aim, we perform simulations of an individual microgel assembled with a rather loose topology (using a confining radius Z = 25σ ) in different solvents. In particular, we make a comparison between "atomistic" and coarse-grained solvent representations by employing Molecular Dynamics (MD) and Dissipative Particle Dynamics (DPD) simulations, respectively (see Methods). In the former type of approach, we first need to adjust the solvent-solvent interactions, for which the most natural choice is to use a Lennard-Jones (LJ) potential. Next, we address the choice of the monomer-solvent (ms) interactions: these are crucial to describe the swelling transition, because they control the contraction or extension of the polymers chains in the solvent environment. To discriminate between different models and identify the best possible candidate, we explore multiple ms potentials and compare them to the implicit solvent case. The choice of the solvent density allows to tune the pressure exerted by the solvent on the polymer network thus determining the swelling range of the microgel particle, as discussed below.
Results
Similarly to solvent-solvent interactions, a straightforward choice for the monomer-solvent ones is the LJ potential 25 where, by varying the energy minimum ε ms , we control the polymer-solvent affinity. In this way, we obtain the swelling curve reported in Figure 1(b), where the radius of gyration of the microgel R g is shown as a function of ε ms : by decreasing this parameter (with respect to solvent-solvent interaction, which sets the energy scale), the polymer-solvent interactions are less favoured than solvent-solvent ones, giving rise to a reduction of the microgel size. However, an unphysical increase of R g is observed for ε ms → 0: under this condition, both terms in the LJ potential go to zero, i.e. the microgel feels neither attraction nor repulsion with the solvent. Consequently, the network relaxes as the external pressure on the polymer network vanishes, and the microgel swells again, maximizing its configurational entropy.
Such behaviour clearly indicates the unsuitability of the LJ potential to mimic the solvent-monomer interactions. Consequently, the next step is to use a potential in which the attractive term can be tuned arbitrarily without affecting the short-range repulsion. To this aim, we adopt the V λ model, defined in Eq. (3), where the repulsion remains unchanged while the attractive contribution, controlled by the parameter λ , is varied. The swelling behavior of the microgel obtained with this model is reported in Fig. 1(c,d) for two representative solvent densities. The swollen-to-collapsed transition is well reproduced in both cases.
So far, we have assessed the "atomistic" type of solvent. We further examine the possibility to use a coarse-grained solvent by means of DPD simulations, which correctly reproduce hydrodynamic interactions at long times 26 . In order to establish a meaningful comparison with the implicit solvent case and avoid unphysical crossing of the chains, we retain the bead-spring model for monomer-monomer interactions and we apply the DPD treatment only to monomer-solvent and solvent-solvent interactions.
The results of DPD simulations, for the parameters specified in Methods, are reported in Fig. 1(e). In this case, the VPT transition is modulated by the monomer-solvent repulsion quantified by the parameter a ms in Eq. (4): for small values of a ms the microgel is swollen, while it contracts when a ms increases. We notice that R g is systematically larger at comparable swelling for MD-solvents than for DPD results, which, on the other hand, quantitatively reproduce the values obtained in the implicit solvent description. This is due to the softness of the DPD interactions which, contrarily to the MD treatment, do not introduce significant solvent-monomer excluded volume effects, thereby not affecting the microgel size.
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In order to establish a correspondence between different models, we rescale the explicit solvent data onto the implicit one, V α where α is the solvophobic parameter (see Methods). Figure 2(a) shows the normalized R g /R max g , where R max g is the value of the gyration radius at maximum swelling, as a function of the effective swelling parameter χ eff . The latter corresponds to the solvophobic parameter α of the implicit solvent simulations. We report the comparison for the two cases where the agreement is found to be fully satisfactory for all χ eff , namely the DPD and MD V λ models. Of the latter, we consider only the case with the highest solvent density, ρ = 0.87, since deviations with respect to the implicit solvent case are observed with lower densities: the swelling range of the microgel would be shortened, as can be observed in Figure 1(c). Thus, it appears that, while V λ is definitely superior to the simple LJ potential to model the VPT of the microgel, the density of the solvent particles is a key parameter in tuning the details of the transition: a lower density will have a smaller effect on the microgel, resulting in a more limited contraction with respect to the implicit solvent model. From now on we will discard the LJ potential and we will refer to MD simulations as those performed with the V λ interaction. A similar effect can be obtained in DPD simulations by changing the cutoff radius and the interaction parameters of the conservative force, which represents the length scale in DPD and the size of the solvent beads (see Methods).
To verify the robustness of our protocol, we now repeat the above analysis on a microgel configuration assembled with a smaller confinement radius, Z = 15σ . Fig. 2(b) reports the swelling behaviour of the more compact microgel for the DPD and MD models at the optimal solvent density identified above. Together with the data, we also report snapshots of the two microgels (insets) in their maximally swollen state, showcasing the very different topology of the networks. The good agreement between the rescaled swelling curves for both studied microgel configurations allows us to conclude that the developed models are robust and both can faithfully reproduce the swelling behavior observed with the implicit model 15 . Fig. 2(c.I-c.III) further highlights the arrangement of solvent particles inside the microgel for MD simulations at different values of χ eff across the VPT. The microgel remains very permeable to the solvent even close to the transition temperature, finally expelling it only in the fully compact state. In the next sections we focus on MD and DPD to study the effects of the solvent on the microgel structural features and on the kinetics of the volume phase transition.
Structural features of a loose microgel in an explicit solvent
We now discuss the structural features of the microgel at relatively large confinement, corresponding to the swelling curve in Fig. 2(a). First, we show results for the density profile of the microgel in Fig. 3, for several values of the swelling parameter across the VPT for both MD and DPD simulations. We find that, in general, both solvent models yield density profiles that are very similar to the implicit solvent case. This is particularly true for the swollen states, where the typical core-corona structure of the microgels is clearly distinguishable. Under these conditions, DPD simulations are even more accurate than MD ones in reproducing the results of the implicit model. When χ eff increases and the microgel becomes more compact, the difference between the three models becomes more evident. Specifically, as the microgel collapses MD simulations produces lower density profiles in the core region with respect to the implicit-solvent case at the same χ eff , while the DPD model generates more compact structures.
r[s] V a DPD MD c eff » 0.1 c eff » 0.4 c eff » 0.6 c eff » 1.4 c eff » 1.0 c eff » 0.7 a b c f e dV a DPD MD c eff » 0.1 c eff » 1.4 c eff » 0.6 c eff » 1.0 c eff » 0.4 c eff » 0.7 a b c f e d
We notice that low density profiles exhibit a non-flat behavior in the inner core region of the microgel. These inhomogeneities, that are stronger for smaller microgels, can be removed out by averaging over independent topologies 15 . Here we do not perform such an average because we aim to compare the behavior of a given microgel configuration with and without solvent. Beyond the VPT the oscillations are suppressed by the higher density, and hence the profiles are much flatter within the core.
While density profiles provide real-space information on the microgel structure, they are not easily accessible in experiments, except for very recent super-resolution microscopy investigations 27,28 . Instead, they can be indirectly obtained from fitting the form factors to the fuzzy sphere model 29 . The form factors P(q) can be measured by small angle neutron or x-ray scattering experiments. Thus, in contrast to density profiles, numerical P(q) can be used to make a direct comparison with experiments, without having to rely on fits to specific models. Indeed, while the fuzzy-sphere model correctly describes the core-corona structure, it does not take into account the presence of dangling chains in the outer corona shell 15,30 . We thus directly evaluate the form factors of the microgel across the VPT and present them in Fig. 4 as a function of wavevector q for the same values of swelling parameters used in Fig. 3. We find that the use of an explicit solvent does not considerably alter the form factors with respect to the implicit solvent case for all values of the swelling parameters. As χ eff increases and the solvent quality decreases, P(q) shows an increasing number of oscillations which become more and more pronounced. Furthermore, the position of the first peak, which is related to the microgel overall size, shifts to larger and larger wavevectors, indicating the shrinking of the microgel. However, a subtle difference is present between the two types of employed models: while DPD results are perfectly superimposed to the implicit solvent case for all χ eff , the MD results are found to be always shifted to a slightly smaller q-value with respect to them. This is a reflection of the overall microgel size, which is a bit larger for MD explicit-solvent simulations with respect to DPD and implicit solvent, due to stronger excluded volume effects, as evident from Fig. 1. We further notice that at relatively large wavevectors (qσ 1) the MD form factor systematically overestimates the DPD and implicit-solvent ones for intermediate and large values of χ eff . However, all curves superimpose again at qσ ∼ 7, where a small peak is found, independently of the swelling parameter value. The latter corresponds to the monomer-monomer nearest-neighbour peak and is a feature associated to the excluded-volume interactions included in the bead-spring model for polymers and to the finite size of the simulated microgel. Indeed, for larger and larger microgel size, this peak would become more and more separated from the first one, allowing for a larger number of oscillations. In experiments, such a peak is not generally noticeable because of the soft intrinsic nature of the monomers. Thus, it is a limitation of the present modelling, which on very small length scales becomes inaccurate.
We now turn to analyze the solvent density profile ρ s inside the microgel. The normalized profile ρ s /ρ s,bulk , where ρ s,bulk is the bulk solvent density, is shown in Fig. 5 as a function of the distance from the center of mass of the microgel. Clearly, the distribution reflects, as a mirrored image, the one of the microgel monomers. Indeed, when the core of the microgels becomes
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denser and denser, more and more solvent gets expelled. It is interesting to note that, beyond the VPT and except for the very collapsed states, a significant fraction of solvent is retained within the polymer network, even well inside the core region. At the VPT, which takes place at χ eff ∼ 0.6, the density of the solvent inside the core is larger than 50% of the bulk value. Finally, we notice that there seems to be a consistent trend of the MD solvent to be more excluded from the network region with respect to the DPD results, again a feature associated to the larger excluded volume of the MD model. However, the two models yield qualitatively very similar results and reinforce the common view that microgels, despite their inhomogeneous structure and dense core region, retain 90% of water in their swollen configuration and still contain a large amount of water well beyond VPT, in qualitative agreement with the experiments results of Ref. 31 normalized with respect to the bulk solvent density ρ s,bulk , as a function of the distance r from the center of mass of the microgel. Circles and triangles refer to MD and DPD solvent, respectively. Each sub-panel refers to a different swelling state according to Fig. 2(a).
Results for a more confined microgel
We now repeat the above structural analysis for a more compact microgel topology obtained with a smaller confining radius (Z = 15σ ), whose swelling curve was reported in Fig.2(b). The density profiles of the microgel are reported in Fig. 6(a-c) for a few selected values of the swelling parameter and again for both MD and DPD explicit solvents. We find that the DPD model reproduces very well the implicit-solvent data, particularly for the more swollen conditions. When χ eff increases, the DPD monomer density in the core is slightly larger than for the implicit case. However, the corona profiles of the two microgel representations are identical. On the other hand, the MD solvent results underestimate the microgel density profile in the core and also display a different corona profile for all χ eff . If compared to the findings for the looser microgel configuration (Figure 3), the DPD solvent model behaves similarly for both types of networks and well reproduces the implicit model data in all cases. By contrast, the MD results present systematic differences with respect to the other two sets of data making the agreement not completely satisfactory. This is a consequence of the "atomistic" treatment of the solvent, which interacts via excluded volume with the polymer. Especially for compact microgels, when excluded volume becomes more and more relevant, these assumptions in the model may become unrealistic. Thus, while for looser networks both MD and DPD explicit solvents provide a good description of the microgel, for more compact microgels the DPD model has definitely the upper hand. This is also shown in the behavior of the solvent density profiles reported in Fig. 6(d-f). Again we find that the MD solvent is much more excluded from the interior of the microgels at all χ eff . On the other hand, we see that, notwithstanding the relative higher compactness of this microgel, a significant amount of solvent remains inside the core in the swollen states, being roughly 60% of its bulk value close to the VPT, in agreement with what found for the less confined microgel configuration and with experimental estimates 31 .
The form factors, shown in Fig. 6(g-i), further confirm that DPD results are in good agreement with the implicit model ones. However, the MD outcomes display a clear shift in the peak position which is much more evident than for the looser configuration (see Fig. 4). In addition, we observe an excess of signal, highlighted in the insets of Fig. 6(g,h), at qσ ∼ 3.0 in swollen conditions, which is absent in the DPD and implicit solvent simulations. This difference occurs at a length that is roughly twice that of the monomer-monomer peak, thus being associated to monomers that are ∼ 2σ apart, i.e. with a solvent particle in between them. Such a feature is smeared out at increasing χ eff , when the microgel collapses and monomer-monomer interactions become dominant. We notice that the excess signal is not observed for the looser microgel as, at the same χ eff value, excluded volume interactions are far less important. Overall, this further shows that the MD model, while still acceptable for not too dense and open microgels, becomes more inaccurate for rather compact ones.
Collapse kinetics
After having established the explicit solvent models and having analyzed the properties of microgel and solvent particles in equilibrium for different values of the swelling parameters, we now turn our attention to the kinetics of collapse of the microgel in the presence of the solvent. Employing the same approach adopted in Refs. [32][33][34][35] for linear polymers, we start from a swollen microgel in a loose configuration and perform a sudden quench to a different state. In particular, we examine two final states whose value of χ eff correspond to an almost fully collapsed state (χ eff ∼ 1.1) and to a state close to the VPT (χ eff ∼ 0.7). We then assess whether the collapse transition is affected by the presence of the solvent by comparing the kinetics of the implicit-solvent model with that obtained using MD and DPD ones. Figure 7(a.I-II) shows the time evolution of the radius of gyration of the microgel for the three different types of simulations at two different χ eff . In all cases the curves reach at long time the same value of R g but, in these simulation conditions, the time taken to equilibrate is different, being faster in implicit solvent simulations compared to those of DPD and MD (the slowest). All curves display a sharp one-step collapse with no trapping phenomena in metastable states. This is qualitatively in agreement with experiments in which microgels with a similar core-corona structure to ours are subjected to an abrupt temperature jump from low (swollen state) to high temperature (globular state) 36 . In order to highlight the role of the solvent, we perform a cluster analysis to identify how the microgel structure evolves during the collapse. To this aim, we detect clusters of non-bonded monomers (see Methods), and calculate their size distribution for state points having the same R g but simulated with different models. Remarkably, we find the same cluster distribution for both implicit and DPD solvent, indicating that the solvent plays no significant role on the folding dynamics of the microgel, as shown in Fig. 7(b). To visualize the restructuring of the microgel following the instantaneous decrease in the solvent quality, snapshots of the microgel are reported in Fig. 7(c.I-III) for three different times. The microgel, while shrinking, first reorganizes by grouping monomers into small clusters (panel c.I). Each cluster is connected to the others via single or multiple links so that the structure, at an intermediate shrinking stage, displays a large number of holes and becomes increasingly inhomogeneous. As the shrinking proceeds, the clusters start to merge, becoming larger and larger in size and joining the main network (panels c.II-III). Finally, at long times, all non-bonded monomers are connected and only a single cluster is left. We stress that this pattern is also found for the implicit model simulations and for the more confined microgel (not shown). These results strongly indicate that the solvent plays a minor influence on the structure of the microgel during the collapse transition. Indeed, at each swelling stage, the microgel has a similar structure regardless of the solvent employed, suggesting that deswelling occurs via the same sequence of transient states. It would be interesting to compare these findings with more accurate solvent treatments such as Multi-Particle-Collision-Dynamics simulations 13,14 .
Discussion
The tunable swelling of the microgel particles has been, since their discovery, one of the most relevant features of these colloids. Indeed, the opportunity to tune the particle volume fraction without changing their number density, but only the temperature, is a formidable advantage for experimental investigations. However, this poses a computational challenge in choosing a suitable model that best describes their swelling-deswelling transition. The recent assembly of realistic microgel networks in Ref. 15 correctly reproduces experimental density profiles and form factors through an implicit solvent treatment. However, the inclusion of the solvent grants additional information, such as the uptake of solvent within the polymer network or surface tension effects. For these reasons, in this work we have compared the implicit solvent results to explicit solvent ones by employing two common approaches to simulations that allows for an atomistic and a coarse-grained approach, namely MD and DPD. We found that we can reproduce the implicit solvent swelling behavior by tuning the monomer-solvent interaction potentials after having adjusted the solvent density. This stems from the fact that, when the solvent is treated explicitly, the external pressure exerted by the solvent needs to be adjusted. In DPD simulations, the same effect can be obtained by regulating the cut-off radius.
We considered two microgels differing in the degree of compactness, which can be obtained by different synthesis protocol 37 and/or by varying the number of crosslinkers. We found that, particularly when the network is denser, excluded volume interactions play a relevant role in the description of the microgel. Indeed, in the full MD simulations an additional peak in the structure appears at small length scales. At the same time, the internal density profile of the microgel is also affected, resulting in a less dense core and a modified corona behavior, which is more significant in the collapsed state. Despite reducing the size of the solvent may solve excluded volume issues, doing so would dramatically increase the number of particles required to observe the same swelling behavior, as the box size is fixed by the dimensions of the microgel particle.
By contrast, DPD results better describe the implicit model ones for both microgel density profiles and form factors, at all swelling conditions. Furthermore, the DPD model reproduces the behaviour of the radius of gyration of the implicit model at different swelling conditions in an almost quantitative fashion. We have also investigated to what extent the solvent penetrates into the microgel, and we found that in the MD simulations much less solvent is present in the interior of the network, whereas DPD results seem more realistic in comparison to experimental estimates. Indeed, we find that, in the swollen state, the network is completely hydrated, retaining more than 90% of the solvent (with respect to the bulk density) in the core of the microgel. Even above the VPT the microgel contains a large fraction of solvent, which is finally excluded only at very large χ eff 1.4, amounting to temperatures 60 • C according to the mapping established in Ref. 15 for PNIPAM microgels.
We also examined the collapse kinetics and assessed how the presence of the solvent affects it. We observed that, in the conditions we performed simulations, a slowing down of the collapse dynamics occurring for the more structured solvent (MD simulations) and to a smaller extent for the coarse-grained solvent (DPD simulations) with respect to the implicit simulations. However, we also found that the system, when compared at the same swelling degree (quantified by the radius of gyration of the microgel), always presents a similar structure, regardless of the model. In particular, at first the network becomes rather inhomogenous, with regions where monomers have clustered together and empty regions. Later on, the clusters merge together and become larger and larger, until the collapse is complete and the microgel is essentially a fully folded network. Such transient behavior, featured by the appearing of crumples, has also been observed previously in simulations 12,33,34 . The similarity between these results with those found for an implicit solvent treatment suggests that hydrodynamic interactions do not play a major role in the swelling-deswelling transition, which is instead mainly controlled by the quality of polymer-solvent interactions.
In summary, in this work we have established that DPD simulations with a coarse-grained solvent constitute the most suitable method to include explicitly a generic solvent in the simulation of a microgel colloid. Even though a partially satisfactory description can be also obtained with the use of an MD solvent, the atomistic description allows for the presence of significant excluded volume interactions that brings unphysical features in the model. On the other hand, DPD simulations do show a full agreement with the implicit model and provides a realistic description of the solvent arrangement within the network. Thus, our model of realistically assembled microgels in DPD explicit solvent opens up the possibility to tackle those phenomena where the physical presence of the solvent is crucial. In particular, our model may serve as a starting point to numerically investigate the so-called "Mickering" emulsions 38 and in the fascinating case of microgels at fluid-fluid interfaces [21][22][23][24]39 . Finally, a realistic description of how the solvent is trapped within the polymer network may lead to future advances in the field of drug delivery and controlled-release, and can provide further insights into its mechanism 40,41 .
Methods
Microgel assembly. The starting configuration of a microgel particle is prepared as in Ref. 15 . First, we produce a fully-bonded disordered network by self-assembling a binary mixture of bi-and tetravalent patchy particles with an applied spherical confinement. Once the network has assembled, we replace the patchy interactions with the classical bead-spring model for polymers 17
V WCA (r) = 4ε σ r 12 − σ r 6 + ε if r ≤ 2 1 6 σ 0 otherwise ; V FENE (r) = −εk F R 2 0 ln(1 − ( r R 0 σ ) 2 ) if r < R 0 σ(1)
with k F = 15 an adimensional spring constant and R 0 = 1.5 the maximum extension value of the bond. Non-bonded monomers only experience a repulsive WCA potential. Regarding units, lengths are given in units of σ , which corresponds to the diameter of a monomer of unit mass m, energy in units of ε and time in units of mσ 2 /ε. In this work, we build networks of N ∼ 5000 monomers confined within a sphere of two different radii Z, namely Z = 15 σ and Z = 25 σ . The change in Z allows to vary the topology of the network, which becomes more compact for small Z and looser (and with more dangling ends) for larger Z 15,16 . The number of crosslinks is fixed to 3.2% of the total number of monomers.
Implicit solvent. The implicit solvent is modeled through the addition of an attractive potential V α that acts between all monomers, either bonded or non-bonded, of the microgel 44,45 :
V α (r) = −εα if r ≤ 2 1 6 σ 1 2 αε cos δ r σ 2 + β − 1 if 2 1 6 < r ≤ R 0 σ 0. otherwise(2)
where δ = π(2.25 − 2 1/3 ) −1 and β = 2π − 2.25δ 44 . The potential is modulated by the parameter α, which controls the solvophobicity of the monomers and plays the role of an effective temperature. For α = 0, monomers do not experience any attraction and good solvent conditions are reproduced while, by increasing α up to 1.5, the monomers become fully attractive, mimicking bad solvent conditions. Previous analysis has shown that the volume phase transition takes place at α ∼ 0.6 15 . MD simulations are performed at constant reduced temperature T * = k B T /ε = 1 (where k B is the Boltzmann constant) using the Nosè-Hoover thermostat and a timestep t * = 0.002.
Adding an explicit solvent in MD simulations. We take a configuration of the microgel assembled as described above and perform MD simulations in the presence of a varying number of additional spheres that mimic the solvent particles which, for efficiency reasons, have also a diameter σ . The number of solvent particles varies between 2.5 × Here ε is the same as the one used in the WCA of monomer-monomer interactions.
The choice of the monomer-solvent (ms) interactions is crucial in order to implement the solvophobic effect, giving rise to the volume phase transition of the microgel. In this respect, we test different approaches. Our first choice is to employ again the LJ potential in which its depth ε ms is varied, so that the attractive contribution can be tuned. A weaker attraction would thus give rise to a more repulsive monomer-solvent interaction that should cause the shrinking of the microgel. However, as explained by analyzing the swelling curves in the Results section, a decrease of this parameter causes the repulsive barrier to be less efficient, giving rise to unphysical consequences on the swelling behavior. To fix this problem, we consider a λ -dependent Lennard-Jones potential 47 , V λ (r), defined as:
V λ (r) = V WCA − ελ if r ≤ 2 1 6 σ 4ελ σ r 12 − σ r 6 otherwise (3)
where ε is the same as for the monomer-monomer (Eq. (2)) and LJ solvent-solvent interactions, while λ plays the role of an inverse temperature (analogue to the inverse of α in the implicit solvent model). Indeed, for large values of λ there is an attractive contribution between a monomer and a solvent particle, mimicking good solvent conditions, while for λ = 0, the WCA potential is recovered and monomer-solvent interactions are purely repulsive. The potential is truncated and shifted at 2.5σ . The advantage of using such a potential with respect to the simple LJ interactions is that it allows to alter the monomer-solvent interactions, and thus the "quality" of the solvent, without affecting the excluded-volume part; this remains encoded in the V WCA term and it does not depend on λ .
Adding an explicit solvent in DPD simulations. DPD is a mesoscale simulation technique 26, 48 that treats solvent particles as coarse-grained beads and is able to describe hydrodynamic interactions through a momentum-conserving thermostat. In DPD simulations, particles i and j interact by three pairwise additive forces: a conservative force F C i j , a dissipative force F D i j and a random force F R i j where
F C i j = a i j (1 − r i j /R c )r i j if r i j < R c , 0 otherwise ; F D i j = −γw D (r i j )(r i j · v i j )r i j ; F R i j = σ R w R (r i j )θ (∆t) −1/2r i j .(4)
Here r i j = r i − r j with r i the position of particle i, r i j = |r i j |,r i j = r i j /r i j , v i j = v i − v j with v i the velocity of particle i, w D (r i j ) and w R (r i j ) are weight functions, θ is a Gaussian random number with zero mean and unit variance and γ is the friction coefficient (here γ = 4.0); to ensure that the correct Boltzmann distribution is achieved at equilibrium, w D (r i j ) = [w R (r i j )] 2 and σ 2 R = 2γk B T . The interaction region for the dissipative force is defined in the same way as for the conservative force, i.e. w D (r i j ) = 1 − r i j /R c . We refer the reader to Ref. 26 for more details on the DPD simulation technique. Although the DPD protocol is often applied to all the species that are present in the simulation, here we make use of DPD only for the solvent-solvent and the solvent-monomer interactions, while keeping the microgel model unaltered. This hybrid technique allows to simulate a "fast" solvent by retaining important features of the microgel particle, such as the topology and the excluded volume interactions among monomers, already investigated in the implicit solvent case 15 . In DPD simulations, we fix the solvent number density at ρ = 0.73, using N = 2.5 × 10 5 particles in a simulation box of side 70σ , and we tune the interaction parameters and the radius of the solvent beads until the swelling curve of the implicit solvent model is reproduced, in this case at r c = 1.75σ . The same curve may be found by using different combination of these parameters in the limit in which the size of the solvent bead is comparable with that of the microgel monomer. All DPD simulations are also performed with LAMMPS at T * = 1.0 using the velocity-Verlet algorithm to integrate the equations of motion; the DPD thermostat is applied to the solvent particles only. Moreover, the center of mass of the microgel is fixed in the center of the simulation box as in MD simulations. The monomer-solvent interaction parameters a ms i j , that for simplicity we call a ms , plays the role of an effective temperature and, depending on its value, controls the volume phase transition of the polymer network. The repulsion coefficient for solvent-solvent interactions is fixed at a ss i j σ = a ss σ = 25ε.
10/13
Rescaling of swelling curves. The swelling degree of the microgel is expressed via the ratio between the absolute value of the gyration radius and its maximum value, obtained in good solvent conditions. The gyration radius is computed as R g = [N −1 ∑ N i (r i − r CM ) 2 ] 1/2 , where r CM indicates the position of the microgel center of mass. Since each model depends on a different "swelling parameter", that we call generically χ eff , we scale all curves onto the implicit model one, using α as the reference swelling parameter. For those explicit solvent models where a small value of the swelling parameter corresponds to a collapsed state of the microgel, i.e. V LJ and V λ , the scale has to be inverted. In order to properly rescale the x axes onto each other for two curves A and B, we consider two points on the first (x A 1 and x A 2 ) and on the second curve (x B 1 and x B 2 ), respectively. The rescaled x-coordinate is calculated using the following relationship: x new = x − x A ∆x B /∆x A + x B , where x i = 0.5(x i 1 + x i 2 ) and ∆x i = x i 1 − x i 2 with i = A, B. Form factors. The microgel form factor P(q) is calculated as P(q) = 1 N ∑ i j exp (−iq · r i j ) where the angular brackets indicate an average over different equilibrium configurations of the same microgel and over different orientations of the wavevector q. Cluster analysis in the kinetics of deswelling. To investigate the structural changes during the transient kinetics of deswelling, we define clusters within the microgel that are formed by non-bonded monomers only: two such monomers belong to the same cluster when their distance is smaller than 1.2σ , which roughly corresponds to the first peak of the radial distribution function. The cluster size distribution n(s) of clusters of size s is calculated by averaging over six independent configurations of the microgel.
Figure 1 .
1Microgel swelling curves. Radius of gyration R g across the VPT transition for (a) the implicit model, V α ; (b) the explicit LJ solvent with LJ monomer-solvent interactions at a solvent density ρ s = 0.729; (c,d) explicit solvent with V λ monomer-solvent interactions at ρ s = 0.729 and ρ s = 0.875, respectively; (e) DPD simulations where the microgel is modeled as a bead-spring polymer network. All curves report the gyration radius R g as a function of the parameter controlling the solvophobic interactions in each model: (a) α, (b) ε ms , (c-d) λ and (e) a ms .
Figure 2 .
2Effect of microgel topology and solvent arrangement. Swelling curves for the implicit-(full line) and explicit-solvent models that best reproduce the swelling behavior, namely MD simulations with V λ at ρ = 0.87 (dashed lines) and DPD simulations (dotted lines) for (a) a loose microgel (Z = 25σ ) and (b) a more compact microgel (Z = 15σ ). Corresponding microgel snaphots are also shown. Symbols refer to state points in explicit solvent simulations (MD: circles, DPD: triangles) for which further analysis is provided in the next sections, whereas similar colors/shapes refer to similar swelling degrees between the two explicit solvent models. Panels (c.I-c.III) display a central slab of the simulation box for three different values of χ eff , respectively corresponding to the swollen state (c.I), a state very close to the VPT (c.II) and the collapsed state (c.III). The arrangement of solvent (blue spheres) within/around the polymer network (red spheres) depends on χ eff . For visual clarity, only half of the solvent particles are shown.
Figure 3 .
3Density profiles for a loose microgel configuration across the VPT. Monomer radial density profile ρ m (r) for a Z = 25σ microgel as a function of the distance r from its center of mass. Full lines refer to the implicit-solvent model, while symbols are used for MD (circles) and DPD (triangles) simulations. Each sub-panel refers to a different swelling state as inFig. 2(a).
Figure 4 .
4Microgel form factors for a loose microgel across the VPT. P(q) as a function of qσ . Full lines refer to the implicit-solvent model, while symbols are used for MD (circles) and DPD (triangles). Each sub-panel refers to a different swelling state according toFig. 2(a).
Figure 5 .
5Solvent density profiles for a loose microgel configuration across the VPT. We show the solvent density profile ρ s
Figure 6 .
6Microgel density profiles, solvent density profiles and form factors for a compact microgel across the VPT. (a-c) microgel density profiles ρ m as a function of the distance r from the center of mass of the microgel; (d-f) solvent density profiles ρ s normalized with respect to the solvent bulk density ρ s,bulk as a function of r; (g-i) microgel form factors as a function of the wavenumber. Data are reported for a swollen state (χ eff = 0.1), a state close to the VPT (χ eff = 0.6) and a compact state (χ eff = 1.0). Full lines refer to the implicit solvent (V α ), while symbols are used for DPD (triangles) and MD (circles). The insets in panels g and h show an enlargement of the high wavevector region where solvent-monomer excluded volume interactions induce an excess of signal for the MD data.
Figure 7 .
7Collapse kinetics. Radius of gyration R g as a function of time for a loose microgel (Z = 25σ ) for χ eff = 1.1 (a.I) and 0.7 (a.II) for implicit (V α , full line), MD (dashed and dotted lines) and DPD solvents (dashed lines); (b) cluster size distribution n(s) for R g = 14.9 (indicated as III in a.I) for implicit and DPD solvents. In order to improve statistics data are averaged over six different microgels configurations; (c.I-III) simulation snapshots for state points I-III (circles in a.I). Clusters are highlighted by different colors according to their size N c (as indicated in the color bar). Light grey monomers are either found in small clusters (N c < 10) or belong to the main network (N c > 100).
, in which bonded monomers interact via the sum of a Weeks-Chandler-Andersen 42 (WCA), V WCA (r), and a Finite-Extensible-Nonlinear-Elastic 43, 44 (FENE), V FENE (r), potentials:
10 5 and 3 × 10 5 in a simulation box of size L = 70 σ , yielding solvent number densities 0.729 ≤ ρ s ≤ 0.845, for which the LJ solvent is in the fluid regime. Lower densities would bring the LJ solvent to phase separate, while higher ρ s would lead to a crystallization of the solvent particles. All MD simulations with explicit solvent are performed with the LAMMPS simulation package 46 at T * = 1 making use of the Nosè-Hoover thermostat and a timestep t * = 0.002. The center of mass of the microgel is fixed in the center of the simulation box. To model solvent-solvent interactions we use a Lennard-Jones (LJ) potential, V LJ (r)
AcknowledgementsWe acknowledge support from the European Research Council (ERC Consolidator Grant 681597, MIMIC).Author contributions statementFC, NG, LR and EZ performed simulations, analyzed results and wrote the paper.Additional informationThe authors declare no competing interests.
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| [] |
[
"POTTS MODELS WITH A DEFECT LINE",
"POTTS MODELS WITH A DEFECT LINE"
] | [
"Sébastien Ott ",
"Yvan Velenik "
] | [] | [] | We provide a detailed analysis of the correlation length in the direction parallel to a line of modified coupling constants in the ferromagnetic Potts model on Z d at temperatures T > Tc. We also describe how a line of weakened bonds pins the interface of the Potts model on Z 2 below its critical temperature. These results are obtained by extending the analysis in[13]from Bernoulli percolation to FK-percolation of arbitrary parameter q ≥ 1.1.1. Correlation length of the Potts model on Z d above T c . Thanks to the self-duality of the 2d Ising model, the problems analyzed in [1, 2] admit equivalent reformulations in terms of the inverse correlation length of a 2d Ising model above its critical temperature, in the presence of a line along which the coupling constants are modified. Such an analysis, based on exact computations, was undertaken by McCoy and Perk [20], independently of the previously mentioned works and at the same time. An advantage of this dual version is that it admits immediate generalizations to higher-dimensional lattices. In this section, we investigate this 1 arXiv:1706.09130v4 [math-ph] 18 May 2018 2 SÉBASTIEN OTT AND YVAN VELENIK problem in the more general case of Potts models on Z d . The low temperature setting for the Potts model on Z 2 will be discussed in Section 1.2.GivenLet Ω d q = {1, . . . , q} Z d be the set of configurations of the q-state Potts model on Z d . Given Λ Z d (that is, Λ ⊂ Z d and finite) and J ≥ 0, we associate to ω ∈ Ω d q the energywhere the coupling constants (J i,j ) i,j∈Z d are given byPOTTS MODELS WITHA DEFECT LINE 3 J c ξ β J ξ β (J) Figure 1.1. The graph of ξ β (J) for the two-dimensional Ising model at β = .75β c , as computed in [20]. As is proved for general Potts models in Theorem 1.2, J c = 1 in this case.(iii) J → ξ β (J) is Lipschitz-continuous and nonincreasing.(iv) There exist c + , c − > 0, depending on β, q and d, such thatIt follows in particular thatis well defined, for any β < β c , and satisfies ∞ > J c ≥ 1. (SeeFigure 1.1 for an illustration in the case of the two-dimensional Ising model.)Remark 1.1. The word "longitudinal" above refers to the fact that we consider the correlation length in a direction parallel to the defect line. One could, in a similar fashion, define the transverse correlation length, by replacing e 1 by e 2 in the definition. However, it is not difficult to show that this quantity always coincides with the corresponding quantity in the homogeneous model.Our next result provides information on the value of J c : Theorem 1.2. J c = 1 when d = 2 or d = 3, but J c > 1 when d ≥ 4.When J > J c , more precise information is available.Theorem 1.3. The following properties hold, for any β < β c :(i) J → ξ β (J) is real-analytic and strictly decreasing on (J c , ∞).(ii) When d = 2, there exist c + , c − , > 0, depending on β, q and d, such that, for all J ∈ (J c , J c + ),(iii) When d = 3, there exist c + , c − , > 0, depending on β, q and d, such that, for all J ∈ (J c , J c + ), e −c+/(J−Jc) ≥ ξ β (J c ) − ξ β (J) ≥ e −c−/(J−Jc) .(iv) For all J > J c , there exists C β,J = C β,J (q, d) > 0 such that, as n → ∞, µ β,J (ω 0 = ω n e1 ) = 1 q + C β,J e −ξ β (J)n (1 + o(1)).The behavior in the last statement should be contrasted with (1). | 10.1007/s00220-018-3197-6 | [
"https://arxiv.org/pdf/1706.09130v4.pdf"
] | 119,732,557 | 1706.09130 | 65c26f0c9acd9086b85fcca559ad631fae68226b |
POTTS MODELS WITH A DEFECT LINE
Sébastien Ott
Yvan Velenik
POTTS MODELS WITH A DEFECT LINE
We provide a detailed analysis of the correlation length in the direction parallel to a line of modified coupling constants in the ferromagnetic Potts model on Z d at temperatures T > Tc. We also describe how a line of weakened bonds pins the interface of the Potts model on Z 2 below its critical temperature. These results are obtained by extending the analysis in[13]from Bernoulli percolation to FK-percolation of arbitrary parameter q ≥ 1.1.1. Correlation length of the Potts model on Z d above T c . Thanks to the self-duality of the 2d Ising model, the problems analyzed in [1, 2] admit equivalent reformulations in terms of the inverse correlation length of a 2d Ising model above its critical temperature, in the presence of a line along which the coupling constants are modified. Such an analysis, based on exact computations, was undertaken by McCoy and Perk [20], independently of the previously mentioned works and at the same time. An advantage of this dual version is that it admits immediate generalizations to higher-dimensional lattices. In this section, we investigate this 1 arXiv:1706.09130v4 [math-ph] 18 May 2018 2 SÉBASTIEN OTT AND YVAN VELENIK problem in the more general case of Potts models on Z d . The low temperature setting for the Potts model on Z 2 will be discussed in Section 1.2.GivenLet Ω d q = {1, . . . , q} Z d be the set of configurations of the q-state Potts model on Z d . Given Λ Z d (that is, Λ ⊂ Z d and finite) and J ≥ 0, we associate to ω ∈ Ω d q the energywhere the coupling constants (J i,j ) i,j∈Z d are given byPOTTS MODELS WITHA DEFECT LINE 3 J c ξ β J ξ β (J) Figure 1.1. The graph of ξ β (J) for the two-dimensional Ising model at β = .75β c , as computed in [20]. As is proved for general Potts models in Theorem 1.2, J c = 1 in this case.(iii) J → ξ β (J) is Lipschitz-continuous and nonincreasing.(iv) There exist c + , c − > 0, depending on β, q and d, such thatIt follows in particular thatis well defined, for any β < β c , and satisfies ∞ > J c ≥ 1. (SeeFigure 1.1 for an illustration in the case of the two-dimensional Ising model.)Remark 1.1. The word "longitudinal" above refers to the fact that we consider the correlation length in a direction parallel to the defect line. One could, in a similar fashion, define the transverse correlation length, by replacing e 1 by e 2 in the definition. However, it is not difficult to show that this quantity always coincides with the corresponding quantity in the homogeneous model.Our next result provides information on the value of J c : Theorem 1.2. J c = 1 when d = 2 or d = 3, but J c > 1 when d ≥ 4.When J > J c , more precise information is available.Theorem 1.3. The following properties hold, for any β < β c :(i) J → ξ β (J) is real-analytic and strictly decreasing on (J c , ∞).(ii) When d = 2, there exist c + , c − , > 0, depending on β, q and d, such that, for all J ∈ (J c , J c + ),(iii) When d = 3, there exist c + , c − , > 0, depending on β, q and d, such that, for all J ∈ (J c , J c + ), e −c+/(J−Jc) ≥ ξ β (J c ) − ξ β (J) ≥ e −c−/(J−Jc) .(iv) For all J > J c , there exists C β,J = C β,J (q, d) > 0 such that, as n → ∞, µ β,J (ω 0 = ω n e1 ) = 1 q + C β,J e −ξ β (J)n (1 + o(1)).The behavior in the last statement should be contrasted with (1).
Introduction and results
In 1980-81, Abraham published two papers [1,2] on the effect of a row of modified coupling constants on the interface of the two-dimensional Ising model, discussing what would later be recognized as pinning and wetting transitions. Being based on exact computations, these results provided precise information but little understanding on the underlying mechanisms. The desire to obtain a better understanding immediately led to an an intense activity (see [22,3,6,7,19,21] for some examples published in 1981 and [12] for a well-known early review). In all these papers, the same problems were tackled in the much simpler setting of effective interface models: basically, modeling the interface as the trajectory of some random walk in suitable potentials. This approach provided not only a better understanding, but also allowed to consider various generalizations: one-dimensional paths in higher dimension (modeling a polymer, for example), higher-dimensional interfaces, random potentials, etc. Note that there is still interest in such issues in the physics community (see, for example, [9] for a recent exact approach, based on more sophisticated field theoretical techniques). The analysis of effective models has also generated a lot of interest among mathematical physicists and probabilists: see, for instance, [23,14] for reviews. In the meantime, new techniques to analyze nonperturbatively various lattice spin systems have been developed [4,5], making it potentially possible to import back the results about effective interface models to the "genuine" spin systems that originally motivated their analysis. This is precisely the purpose of the present paper, in which we provide a detailed description of the longitudinal correlation length of the Potts model on Z d above the critical temperature in the presence of a line of modified coupling constants, as well as an analysis of the pinning of a Potts interface by a line of defects in the two-dimensional model below its critical temperature. (More generally, our results apply to all randomcluster models with parameter q ≥ 1.) The results we obtain are in full agreement with the predictions by effective models.
The Gibbs measure in Λ Z d , with boundary condition η ∈ Ω d q and at inverse temperature β = 1/T , is the probability measure on Ω d q given by
µ η Λ;β,J (ω) = Z η Λ;β,J −1 e −βH Λ;J (ω) if ω i = η i ∀i ∈ Λ, 0 otherwise.
Finally, the associated infinite-volume Gibbs measures are all probability measures µ on Ω d q satisfying µ( · | F Λ c )(η) = µ η Λ;β,J ( · ) ∀Λ Z d for µ-almost every η ∈ Ω d q . Here, F Λ c is the σ-algebra generated by the random variables (ω i ) i ∈Λ .
We first recall a few results concerning the homogeneous model, in which J = 1. In this case, it is well-known that, for any d ≥ 2, there exists β c = β c (d) ∈ (0, ∞) such that there is a unique infinite-volume Gibbs measure when β < β c = 1/T c , but infinitely many infinite-volume Gibbs measures when β > β c . Assume that β < β c and denote by µ β the (unique) infinite-volume Gibbs measure. Then, the inverse correlation length is positive [11]:
ξ β = lim n→∞ − 1 n log µ β (ω 0 = ω n e1 ) − 1 q > 0.
More precisely, the following Ornstein-Zernike asymptotics hold [5]: there exists C β = C β (q, d) > 0 such that, as n → ∞, µ β (ω 0 = ω n e1 ) = 1 q + C β n (d−1)/2 e −ξ β n (1 + o(1)).
(1)
Let us now consider general values of J ≥ 0. We still assume that β < β c (with the β c defined above). It turns out 1 that there is still a unique infinite-volume Gibbs measure in this case, which we denote by µ β,J . We define the longitudinal inverse correlation length as follows: for any x ∈ Z d ,
ξ β (J) = lim n→∞ − 1 n log µ β;J (ω x = ω x+n e1 ) − 1 q .(2)
We first claim that Theorem 1.1. For any β < β c , the following properties hold: (i) The limit in (2) exists and is independent of x.
(ii) ξ β (J) > 0 for all J ≥ 0. 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 the functions Γ + n and Γ − n ; note that the interface is contained in the regions they delimit (hatched area).
1.2.
Pinning of the interface of the 2d Potts model below T c . We now restrict our attention to the lattice Z 2 . Let
L * = {(x, 0), (x, 1)} ∈ E 2 : x ∈ Z .
We now consider the q-state Potts model on Z 2 with coupling constants (J i,j ) i,j∈Z 2 given by
J i,j = 1 if{i, j} ∈ E 2 \ L * , J if {i, j} ∈ L * , 0 otherwise.
Let Λ n = {−n, . . . , n} × {−n + 1, . . . , n} and let η Dob ∈ Ω 2 q be the Dobrushin-type boundary condition defined by
η Dob i = 1 if i ⊥ ≥ 1, 2 if i ⊥ ≤ 0.
We denote by µ ± n;β,J the Gibbs measure in Λ n with boundary condition η Dob at inverse temperature β.
In the remainder of this section, we assume that β > β c . In that case, there is long-range order and it is convenient to describe configurations in terms of their Peierls contours. First, given {i, j} ∈ E 2 , denote by {i, j} * = {x ∈ R 2 : x − i ∞ = x − j ∞ = 1 2 } the dual edge separating i and j. The contours of a configuration ω ∈ Ω 2 q are the maximal connected components of {i,j}∈E 2 : ωi =ωj {i, j} * .
When ω i = η Dob i for all i ∈ Λ n , there is a unique unbounded contour. We call its intersection with [−n − 1 2 , n + 1 2 ] × R the interface and denote it by Γ n . Note that Γ n is a two-dimensional object, but with a macroscopic extension only along the first coordinate axis and an (essentially) bounded width, as we explain now.
Consider first the homogeneous case J = 1. It can then be shown [5] that, under µ ± n;β,1 , the interface has a width of order log n. Namely, for each x ∈ [−n− 1 2 , n+ 1 2 ], define Γ + n (x) = max{y ∈ R : (x, y) ∈ Γ n }, Γ − n (x) = min{y ∈ R : (x, y) ∈ Γ n }. Then, there exists C(β, q) such that lim n→∞ µ ± n;β,1 max x (Γ + n (x) − Γ − n (x)) ≤ C(β) log n = 1. Moreover, under diffusive scaling, the interface weakly converges to a Brownian bridge [5]: for any β > β c , there exists κ β > 0 such that, as n → ∞,
1 √ n Γ + n (n ·) ⇒ κ β B · ,
where (B t ) −1≤t≤1 denotes the standard Brownian bridge on [−1, 1].
The main result of this section is that, whenever J < 1, the interface ceases to behave diffusively and instead localizes along the defect line: Note that, under diffusive scaling, the limit is then identically 0: an arbitrary weakening of the coupling constants along L * pins the interface. Actually, the claim in the theorem will follow from a detailed description of the structure of the interface (see Theorem 7.2), which provides a much stronger claim than what is stated above. In particular, the width of the interface is typically bounded, with only rare deformations of order log n. (In fact, Theorem 1.4 will essentially be a corollary of Item (iv) of Theorem 1.3.)
Before closing this introduction, let us briefly mention that although we restricted our attention to a defect along a line of the lattice, this is by no means necessary. Straightforward adaptation of our arguments would allow the analysis, for example, of a defect along the lattice approximation of any line with rational "slope", or other periodic structures. Similarly, the restriction to nearest-neighbor interactions is only necessary for the statement of Theorem 1.4 (the proof of which relies on duality); for the other claims, any finite-range, translation-invariant, reflectionsymmetric interaction would do. • Determine the sharp asymptotics of the 2-point function when J ≤ J c . Only the case J = 1 has been treated in complete generality up to now. For the two-dimensional Ising model, the asymptotic behavior was explicitly computed in [20] and found to be of the form (1)) when J < J c = 1. Note the exponent of the prefactor, which is not of the usual Ornstein-Zernike form. Again, by analogy with what happens in effective models (see [14,Theorem 2.2]), we expect the prefactor to be of order n −(d−1)/2 when d ≥ 4 and n −1 (log n) −2 when d = 3.
µ β,J (ω 0 = ω n e1 ) = 1 2 + C β n 3/2 e −ξ β n (1 + o
• Closely related to the previous problem, determine the scaling limit of the interface in the two-dimensional model when J < J c . We expect the latter to be given by a Brownian excursion after diffusive scaling, as a consequence of entropic repulsion away from L. This is fully compatible with the exponent in the prefactor mentioned in the previous point.
Moreover, there are a number of natural generalizations, to which we plan to return in future works:
• What happens when the defect is located along the boundary of the system? In dimension 2, this amounts to studying the wetting problem for the Potts model. • What happens when the defect is of dimension d ∈ (1, d)? Note that, in this case, the system may display long-range order along the defect even when the bulk is disordered. In particular, the longitudinal inverse correlation length vanishes for finite values of J. • Is it possible to adapt some of the technology used to deal with pinning of a random walk by a disordered potential to cover the case of random (quenched, ferromagnetic) coupling constants along the defect?
2. Random cluster representation, notations and strategy of the proof
In this section, we introduce a few notations which will be recurrent throughout this article, we recall briefly the random-cluster (or Fortuin Kastelyn) representation of the Potts model and we give a short outline of the proofs of the theorems of Section 1.
2.1. Random-cluster representation of the Potts model. The Potts model on a finite graph G = (V G , E G ) can be mapped to a percolation model defined on {0, 1} E G (identifying the value 1 with the presence of an edge and the value 0 with its absence) in the following way. For any edge configuration ω ⊂ E G , we denote by κ(ω) the number of connected components in (V G , ω). Writing x = (x e ) e∈E G , with x e = e βJe − 1 for each e ∈ E G , we associate to ω ⊂ E G the probability
P x,q (ω) = (Z x,q ) −1 q κ(ω) e∈ω x e ,
where Z x,q = ω⊂E G q κ(ω) e∈ω x e . The corresponding expectation will be denoted by E x,q . We say that an edge e with ω e = 1 is open and denote by |ω| or o(ω) the number of open edges. We say that u, v ∈ G are connected, which we write u ↔ v, if they lie in the same connected component. For A ⊂ E G , denote by ω A the configuration ω restricted to A and, for e ∈ E G , by ω \e the configuration ω E G \{e} .
The random-cluster measures with q ≥ 1 enjoy the following properties.
Finite energy: For any e ∈ E G and any configuration ω \e ,
x e x e + q ≤ P x,q (ω e = 1|ω \e ) ≤ x e x e + 1 .
Positive association: Let f, g be two nondecreasing functions (w.r.t. the partial order induced by 0 ≤ 1 on {0, 1} E G ). Then the FKG inequality holds:
E x,q [f g] ≥ E x,q [f ] E x,q [g]
. Stochastic monotonicity: Assume that x e ≤ y e for all e ∈ E G . Then P x,q P y,q . The random-cluster model does not enjoy the usual spatial Markov property but an analogue can be used: for Λ ⊂ G, the random-cluster measure in Λ with boundary condition ω G\Λ depends only on the connectivity properties of the vertices in the inner boundary of Λ, thus a boundary condition is a partition of those vertices (every set of the partition is a connected component). In particular, the measure with wired boundary condition (denoted P w x,q,Λ ) is obtained by setting ω G\Λ ≡ 1, while the measure with free boundary condition (P f x,q,Λ ) is obtained using ω G\Λ ≡ 0. Stochastic monotonicity then implies that these two measures are extremal with respect to stochastic ordering.
In the sequel, we will work with the random-cluster measure on Z d induced by
the weights x e = x = e β − 1 if e ∈ L c x = e βJ − 1 if e ∈ L
. We denote the corresponding law P x ;
it corresponds to the random-cluster measure associated with the Potts measure described in the previous section. In particular, the 2-point correlation function of the Potts model can be rewritten as (see, for example, [17, (1.16)
]) µ β;J (σ u = σ v ) = 1 q + q − 1 q P x (u ↔ v).(3)
From this, it immediately follows that the inverse correlation length ξ β (J) is equal to
ξ x = lim n→∞ − 1 n log P x (0 ↔ n e 1 ).(4)
We will write P ≡ P x and E ≡ E x for the law and expectation of the homogeneous model; the corresponding measure in a finite volume Λ Z d with boundary condition # ∈ {f, w} will be denoted P # Λ . Everywhere in the analysis below, except in the proof of Theorem 1.4, we will implicitly assume that x < x c = e βc − 1 and q ≥ 1 are fixed and we will thus omit them from the notation.
We also write ξ ≡ ξ x for the corresponding inverse correlation length. The following exponential decay of connectivities under P, established in [11], plays a crucial role in our analysis.
Lemma 2.1. Let x < x c .
Then there exists ν 1 > 0 such that, for n large enough,
P w Λn (0 ↔ Λ c n ) ≤ e −ν1n .
We will prove all the results of Section 1 in the random-cluster representation. They can then be translated straightforwardly to the Potts model language via (3).
Remark 2.1. Since x < x c , we can always work in large but finite boxes. Indeed, for any event A depending on a finite number of edges, we can find a finite box Λ ⊂ Z d such that
1 2 P f Λ (A) ≤ P(A) ≤ 2P f Λ (A)
. This will be done in several instances for technical reasons, but we will keep the same notation as for the infinite-volume measure for readability purposes. The choice of boundary condition does not matter, thanks to the uniqueness of the infinite volume measure in the sub-critical regime.
2.2. Notations. For u, v ∈ Z d , we denote d(u, v) = v − u 1 the graph distance between u and v; for A, B ⊂ V (Z d ), we set d(A, B) = min u∈A,v∈B d(u, v). For a < b ∈ R, the notation L [a,b] denotes the subgraph of Z d : a, b × {0 d−1 } where 0 d is the origin of Z d and a, b = [a, b] ∩ Z; we also use the notations L <n ≡ L (−∞,n−1] and L >n ≡ L [n+1,∞) . Let A ⊂ Z d . We denote u A
↔ v the event in which u ↔ v using only edges originally present in A. We will use the following notion of boundaries: ∂A = {i ∈ A : ∃j ∈ A c , j ∼ i} and ∂ ext A = {i ∈ A c : ∃j ∈ A, j ∼ i}. We will also use the notation ∂A to denote the set of edges having exactly one endpoint in A.
Sums of the form b i=a for a, b not integers are to be understood as the corresponding sums with a, b replaced by the appropriate integers; for example, if this notation is used in the course of proving an upper bound, and the summand is nonnegative, then b i=a = b i= a (taking integer part would not change our estimates, so we chose not to write them explicitly for readability purposes).
In the following proofs, we will say that a quantity f r (K) is o K (1) if the following is true: for every m > 0, one can find r > 0 and K 0 > 0 such that, for every K > K 0 , f r (K) ≤ K −m (the quantities r, K will make sense later and the notation will become clear from the context; we define this here for easy reference, since this appears in several places in the following sections).
We will also use the notation θ 1 = min e∈Z d min ω \e P x (ω e = 1|ω \e ) = x x+q ∧ x
x +q and θ 0 = min e∈Z d min ω \e P x (ω e = 0|ω \e ) = 1 1+(x∨x ) . Finally, all constants appearing in the proofs below depend a priori on q, x and d, but this will not be mentioned explicitly every time.
For a set E ⊂ E G and a random-cluster configuration ω, we write o E (ω) for the number of open edges of ω in E.
Given x ∈ Z d and a random-cluster configuration ω, we denote by C x = C x (ω) the cluster of x in ω.
2.3.
Outline of the paper. In the next section, we provide the proof of Theorem 1.1. In the process, we introduce some tools and calculations that will reappear in the proofs of Theorems 1.2, 1.3 and 1.4.
The procedure leading to the main claims is as follows: in Section 4, we reinterpret long connections in the homogeneous model in terms of a random walk with i.i.d. increments. This is done combining the coarse-graining procedure of [5] with a variant of the construction of [8] (see the comments at the beginning of Appendix C), which is described in a self-contained way in Appendix C. The statement of Theorem 1.2 and the second and third points of Theorem 1.3 follow, on the one hand, by studying a pinning problem for the random walk obtained in Section 4 (see Section 5) and, on the other hand, by an energy/entropy argument induced by the Russo-like formula described in Appendix B.2 (see Section 6). Finally, the first and fourth points of Theorem 1.3 are established in Section 7 by studying the localization of the random walk trajectory in a small neighborhood of L via a coarse-graining argument. The claim of Theorem 1.4 follows from the same analysis combined with self-duality, as explained in Section 7.6.
Basic properties and estimates
In this section, we prove Theorem 1.1. We assume throughout that x < x c . Using the correspondence described in the previous section, it is sufficient to establish the following lemma. Lemma 3.1 (Basic properties of ξ x ). The limit in (4) exists and defines a function ξ x with the following properties. a) ∀u ∈ Z d ,
lim n→∞ − 1 n log P x (u ↔ u + n e 1 ) = ξ x . Moreover, ∀u, v ∈ Z d , P x (u ↔ v) ≤ e −ξ x |v1−u1| .(5)
b) ξ x = ξ for all x ≤ x and ξ x < ξ for x sufficiently large. c) x → ξ x is locally Lipschitz continuous, nonincreasing on [0, ∞] and strictly positive for
x ∈ [0, ∞). d) There exist c + , c − > 0 such that c− x ≤ ξ x ≤ c+ x for x large enough. In particular, there exists x c ∈ [x, ∞) such that ξ x = ξ for all x ≤ x c and ξ x < ξ for all x > x c .
Remark 3.1. We actually prove something stronger than strict positivity of ξ x : we show that there exists c > 0 such that, for all n,
P w x ,Λn (0 ↔ ∂Λ n ) ≤ e −cn .(6)
By stochastic monotonicity, this implies the same bound for any boundary condition.
Proof. • The existence and the first part of a) are shown using Fekete's lemma. We first prove existence of ξ u x = lim n→∞ − 1 n log P x (u ↔ u + n e 1 ). Define π n = log P x (u ↔ u + n e 1 ). We see that (−π n ) n is a subadditive sequence: by FKG and translation invariance in the e 1 -direction,
π n+m = log P x (u ↔ u + (n + m) e 1 )
≥ log P x (u ↔ u + n e 1 ↔ u + (n + m) e 1 ) ≥ log P x (u ↔ u + n e 1 )P x (u + n e 1 ↔ u + (n + m) e 1 ) = log P x (u ↔ u + n e 1 ) + log P x (u ↔ u + m e 1 ) = π n + π m .
Fekete's lemma then implies that ξ u x = lim n→∞ −πn n = inf n −πn n exists; in particular, P x (u ↔ u + n e 1 ) ≤ e −ξ u x n .
(7) To prove that ξ u x = ξ 0 x ≡ ξ x , just observe that, for all u ∈ Z d , P x (u ↔ u + n e 1 ) ≥ P x (u ↔ (u 1 , 0 d−1 ) ↔ (u 1 + n, 0 d−1 ) ↔ u + n e 1 ) ≥ θ 2d(u,L) 1 P x (0 ↔ n e 1 )
, and therefore ξ u x ≤ ξ x . The same argument, exchanging the role of 0 and u, yields the reverse inequality.
• The second part of a) follows from
P x (u ↔ v) 2 ≤ P x (u ↔ v ↔ū v ) ≤ P x (u ↔ū v ) ≤ e −ξ x 2|v1−u1| ,
whereū v denotes the point obtained from u by a reflection through the hyperplane orthogonal to L containing v. The last inequality is a direct consequence of the bound (7) and the identity ξ u x = ξ x . • The monotonicity of x → ξ x follows from the stochastic domination P x 1 P x 2 when x 1 ≥ x 2 . • To get the first point of item b), we fix x ≤ x and work in a finite volume (see Remark 2.1). We will use a coupling Φ(ω, η) between P and P x satisfying (we denote C L (ω) the connected component of the line L in ω):
(i) ω ∼ P and η ∼ P x ; (ii) ω ≥ η; (iii) ω = η outside of C L (ω).
A sketch of the construction of such a coupling (as well as references) is provided in Appendix A. Choosing 1 > α > β > 1/2 and setting [j] = n α e 2 + j e 1 , we have
P x (0 ↔ n e 1 ) ≥ θ 2n α 1 P x ([0] ↔ [n]) ≥ θ 2n α 1 n 1−β i=1 P x ([(i − 1)n β ] ↔ [in β ]) = θ 2n α 1 P x ([0] ↔ [n β ]) n 1−β . Let ∆ = {u ∈ Z d : d(u, L) ≤ n α /2} and Λ n (u) = {v ∈ Z d : u − v ∞ ≤ n}.
Then,
P x ([0] ↔ [n β ]) ≥ Φ(1 [0]↔[n β ] (η)1 {C [0] (ω)∩L=∅} ) = Φ(1 [0]↔[n β ] (ω)1 {C [0] (ω)∩L=∅} ) = P([0] ↔ [n β ])P(L ↔ / [0] | [0] ↔ [n β ]) = P([0] ↔ [n β ])P(L ↔ / [0] | [0] ↔ [n β ], L <−n ↔ / [0], L >2n ↔ / [0]) ×P(L <−n ↔ / [0], L >2n ↔ / [0] | [0] ↔ [n β ]) ≥ C n β(d−1)/2 e −ξn β 1 − P w ∆ (L [−n,2n] ↔ ∂∆) × 1 − P([0] ↔ ∂Λ n ([0]) | [0] ↔ [n β ]) ≥ C n β(d−1)/2 e −ξn β (1 − 3ne −cn α /2 )(1 − n dβ e −cn )
. Together with P x (0 ↔ n e 1 ) ≤ P(0 ↔ n e 1 ) when x ≤ x, we get the result.
• For the second point of b), notice first that, for any edge e ∈ L, P x (ω e = 1 | ω \e ) ≥ 1/(1 + q x ) uniformly on ω \e . Therefore, by opening all edges from L [0,n] ,
P x (0 ↔ n e 1 ) ≥ e − log(1+ q x )n ≥ e − q x n .
Choosing x such that q x < ξ, the result follows. Moreover, we obtain that ξ x ≤ q x , which corresponds to one side of item d).
• We now prove a variant of Lemma 2.1, establishing exponential decay of connectivities uniformly over boundary conditions under the measure P x .
Lemma 3.2. Assume that x < x c . Then, for any x ≥ 0, there exists a constant ν 2 = ν 2 (x, x , q, d) > 0 such that
P w x ,Λn(u) (u ↔ ∂Λ n (u)) ≤ e −ν2n(8)
uniformly over u ∈ Z d .
Proof. First observe that the claim is an immediate consequence of FKG and Lemma 2.1 when x ≤ x. We thus assume from now on that x > x. Let us write P w x ,Λn(u) (u ↔ ∂Λ n (u)) = P w x ,Λn(u) (u ↔ ∂Λ n (u), u ↔ / L) + P w x ,Λn(u) (u ↔ ∂Λ n (u), u ↔ L) (9) and treat separately the two terms in the right-hand side. For the first term, we rely again on the coupling Φ(ω, η) between P w x,Λn(u) and P w x ,Λn(u) as above: so that the claim follows again from Lemma 2.1. Let us finally consider the second term in the right-hand side of (9). The proof in this case relies on a coarse-graining procedure similar to the one used in [5]. Fix a scale K and a number r (both of which will be later chosen sufficiently large, independently of n) and define 2K . We first coarse-grain the connected components of F using the following algorithm:
P w x ,Λn(u) (u ↔ ∂Λ n (u), u ↔ / L) = Φ(u η ↔ ∂Λ n (u), u η ↔ / L) = Φ(u ω ↔ ∂Λ n (u), u η ↔ / L) ≤ Φ(u ω ↔ ∂Λ n (u)) = P(u ↔ ∂Λ n (u)), ∆ ∆ 2K L u Λ n∆ k (v) = −k, k d + v, ∆(v) = ∆ K (v), where K = K + r log(K). Given a set of vertices A ⊂ Z d , we write [A] k = v∈A ∆ k (v). Set Λ ≡ Λ n+2K (u) and Λ n ≡ Λ n (u). Let F = {i ∈ Λ n \ [L] 2K : i ↔ [L] 2K } and D = ∂ ext [L] 2K ∩ Λ n . Note that u ∈ F ∪ [L]
Algorithm 1: Coarse-graining procedure
Set V = ∅, n = 1; while ∃v ∈ F ∩ D such that v F \[V ] K ←−−−→ ∂∆ K (v) do
Let v be the smallest such vertex and add it to V ; set V n = {v} and E n = ∅;
while ∃w ∈ F ∩ ∂ ext [V ] K such that w F \[V ] K ←−−−→ ∂∆ K (w) do
Let w be the smallest such vertex and add it to V and to V n ; Let w ∈ V be the smallest vertex such that w ∈ ∂ ext [w] K and add the edge {w , w} to E n ; end Set T n = (V n , E n ); Update n → n + 1; end This algorithm yields a (possibly empty) family of trees T 1 = (V 1 , E 1 ), . . . , T M = (V M , E M ), possessing the following properties: (i) the root of each T k belongs to
D; (ii) every edge {w, w } ∈ E k , 1 ≤ k ≤ M , satisfies w ∈ ∂ ext [w] K ; (iii) all connected components of F \ M i=1 [V i ] K have ( ∞ -)diameter at most 2K.
In view of the property (i), it is convenient to relabel the trees according to the position of their root. Namely, for any v ∈ D, we denote by T v = (V v , E v ) the (possibly empty) tree with root at v obtained using the above algorithm.
Denote by C K the number of vertices in ∂∆. The number of possible configurations of the tree T v , with fixed root v, is at most equal to the number of trees with branching number C K , which is in turn at most e c1 log(C K )|Vv| by an argument due to Kesten (see [16,Section 4.2]). Therefore, by Lemmas 2.1 and B.1 (which can be applied provided we choose r large enough), the probability that the algorithm yields a given collection of trees (T v ) v∈D with total number of vertices N is bounded above by e −ν1N K e c1 log(C K )N ≤ e − 1 2 ν1N K once K is chosen large enough. Therefore, for any ρ > 0, there exists K 0 > 0 and c > 0 such that, for all K ≥ K 0 ,
P w x ,Λn v∈D |V v | ≥ ρn/K ≤ N ≥ρn/K v ≥0,v∈D: v v =N e − 1 2 ν1K v v ≤ N ≥ρn/K e − 1 4 ν1KN ∞ =0 e − 1 4 ν1K |D| ≤ e −cρn ,(10)
for all n large enough. This immediately implies that, whenever u connects to a side H of Λ n with H ∩ L = ∅, the desired exponential decay follows, since, in that case, v∈D |V v | ≥ n/(2K).
It only remains to take care of connexions to the two sides of Λ n intersecting L; by symmetry, it suffices to consider the side with largest e 1 component, which we denote by H. Let us split Λ n into slices (see Figure 3.2). Define
S i = (i − 1)3K, i3K × Z d−1 + u − n e 1 ∩ Λ n (u), i = 1, 2, . . . , 2n/3K, and set B i = S i ∩ [L] 2K . We say that the box B i is covered if S i−1 F ← → S i+1
and uncovered otherwise. Observe that, by property (iii) above, v∈D |V v | cannot be smaller than the number of covered boxes. Denoting by B + uncov = {n/3K ≤ i ≤ 2n/3K : B i is uncovered} the indices of all the uncovered boxes "on the right of" u, it thus follows from (10) that there exists c > 0 such that, for all large n,
P w x ,Λn u ↔ H, u ↔ L ≤ P w x ,Λn u ↔ H, |B + uncov | ≥ n/6K + e −cn .
The proof will be complete once we prove that the first term in the right-hand side decays exponentially with n. Let us decompose
P w x ,Λn u ↔ H, |B + uncov | ≥ n/6K = A: |A|≥n/6K P w x ,Λn u ↔ H | B + uncov = A P w x ,Λn B + uncov = A .
Observe now that, in order for u to be connected to H, it is necessary that none of the boxes B i , i ∈ B + uncov is empty, in the sense of all the edges inside of it being closed. Clearly, B uncov only depends on the state of the edges in Λ n \ [L] 2K . Since the probability that all the edges inside an uncovered box B i are closed is bounded below by θ 2d|Bi| 0 > 0, uniformly in the state of all the other edges, we conclude that
P w x ,Λn u ↔ H | B + uncov = A ≤ P w x ,Λn B i not empty, for all i ∈ B + uncov | B + uncov = A ≤ 1 − θ |Bi| 0
n/6K , and the conclusion follows.
Remark 3.2. Note that, using a standard coupling argument, (8) implies that there is a unique infinite-volume random-cluster measure for any x ≥ 0. Since there is a.s. no infinite cluster under this measure, we conclude from the Edwards-Sokal coupling that there is a unique infinite-volume Potts measure for any finite value of J.
• We can now prove the other half of item d). Notice that the same procedure as in the previous point ensures that, on the event 0 ↔ n e 1 , we can find K ≡
K(x, d), c 2 ≡ c 2 (K, d, x) (uniformly in x ) such that at least half of the boxes B i are uncovered with P x -probability at least 1 − e −c2n .
Then, by finite energy, we can find ≡ (K, d, x) > 0 and c 3 ≡ c 3 (K, d, x) such that at least n/K boxes contain an edge in L that is pivotal for 0 ↔ n e 1 with P x -probability at least 1 − e −c3n (again, both and c 3 do not depend on x ). Denote this event B . Then, proceeding as before,
P x (0 ↔ n e 1 ) ≤ P x (B , 0 ↔ n e 1 ) + e −c3n ≤ x 1 + x n K + e −c3n ≤ e − log(1+ 1 x ) n K (1 + e −(c3−log(1+ 1 x ) n K ) )
.
Choosing x such that c 3 − log 1 + 1 x K > 0, we obtain ξ x = lim n→∞ − 1 n log P x (0 ↔ n e 1 ) ≥ log 1 + 1 x K ≥ 2K 1 x ,
for x large enough. • To prove continuity, we work again in large but finite boxes (following Remark 2.1). We start with a small computation (which will be used again in Section 5). Let x 1 ≤ x 2 and write λ = log(x 2 /x 1 ). Then,
ξ x 1 − ξ x 2 = lim n→∞ 1 n log P x 2 (0 ↔ n e 1 ) P x 1 (0 ↔ n e 1 ) = lim n→∞ 1 n log E x 1 e λo L 0 ↔ n e 1 E x 1 e λo L .
Now, we partition the numerator in the logarithm w.r.t. the cluster of 0:
E x 1 e λo L 0 ↔ n e 1 E x 1 e λo L = C 0,n e1 1 E x 1 e λo L P x 1 C 0 = C 0 ↔ n e 1 ×E x 1 e λo L C 0 = C ≤ C 0,n e1 P x 1 C 0 = C 0 ↔ n e 1 E x 1 e λo L\C ∂C closed e λ|C∩L| ×E x 1 e λo L\C ∂C closed = C 0,n e1 P x 1 C 0 = C 0 ↔ n e 1 e λ|C∩L| = E x 1 e λ|C0∩L| 0 ↔ n e 1 .(11)
Partitioning w.r.t. the leftmost and rightmost point of C 0 ∩ L (denoted L and R), we then obtain
E x 1 e λ|C0∩L| 0 ↔ n e 1 ≤ 0 l=−∞ ∞ r=n e λ(r−l) P x 1 (L = l, R = r | 0 ↔ n e 1 ) ≤ e λ3n + e λ3n ∞ l=0 ∞ r=0 e λ(r+l) P x 1 (0 ↔ (3n + l + r) e 1 ) P x 1 (0 ↔ n e 1 ) ≤ e λ3n 1 + ∞ l=0 ∞ r=0 e λ(r+l) e −ξ x 1 (3n+l+r) e ξ x 1 (1+o(1))n ≤ e λ3n 1 + e −ξ x 1 n ∞ l=0 ∞ r=0 e −(ξ x 1 −λ)(l+r) .
Note that λ < ξ x 1 when x 2 is close enough to x 1 (since ξ x 1 > 0). In this case, the last double sum converges and we get
ξ x 1 − ξ x 2 ≤ lim n→∞ 1 n log e λ3n (1 + e −ξ x 1 n C x 2 −x 1 ) ≤ 3 log(x 2 ) − log(x 1 ) = x 2 x 1 3 s ds ≤ 3 x 1 (x 2 − x 1 ).
Random Walk representation
In this section, we explain how one can couple the cluster C 0 under P(· | 0 ↔ n e 1 ) (remember that P denotes the homogeneous (that is, when x = x) random-cluster measure on Z d ) with a directed random walk on Z d for all x < x c . This coupling will allow us to analyze in detail the large-scale properties of C 0 . The construction is based on the decomposition of the cluster into irreducible pieces, as described in [5], and on the arguments exposed in Appendix C that explain how to get rid of the dependency between the irreducible pieces. The exposition is not self-contained and its goal is mostly to setup notations and remind the reader of the main steps of the construction. A reader not familiar with [5] should refer to that work for details and additional explanations.
We start with a brief description of the coarse-graining in [5]. As we only consider the e 1 direction, the construction simplifies slightly. Let us first introduce the geometric objects required for the coarse-graining procedure. Let 0 < ψ ≤ π/2 and let
Y ψ = {u ∈ Z d : u, e 1 ≥ u 2 cos(ψ/2)}
be the cone of angular aperture ψ and axis direction e 1 ; we will usually omit ψ from the notation and simply write Y . We also set Y = −Y and introduce the "diamonds"
D(v 1 , v 2 ) = (v 1 + Y ) ∩ (v 2 + Y ). Let us say that v ∈ C 0 is a cone-point if C 0 ⊂ (v + Y ) ∪ (v + Y ).
We introduce three families of clusters:
• B L is the set of all clusters C such that: 0 ∈ C; C has a cone-point v such that C ⊂ v + Y ; C possesses no other cone-point with nonnegative e 1 -coordinate.
• B R is the set of all clusters C such that: 0 ∈ C; C has a cone-point v such that C ⊂ v + Y ; C possesses no other cone-point with nonpositive e 1 -coordinate. • A is the set of all clusters C such that: C possesses exactly two cone-points, 0 and v ∈ Y , and C ⊂ D(0, v). (Note that the single-vertex cluster {0} belongs to both B L and B R .) We define a displacement application D from each of these three sets into Y by setting (v is the vertex appearing in the previous definitions): A cluster γ b ∈ B L can be naturally concatenated with m ≥ 0 clusters γ 1 , . . . , γ m ∈ A by first translating each γ k by D(γ b ) + D(γ 1 ) + · · · + D(γ k−1 ). We can then also concatenate a cluster γ f ∈ B R by first translating it by
D(C) = v when C ∈ B L ∪ A and D(C) = −v when C ∈ B R . v v 0 v 0 0D(γ b )+ m k=1 D(γ k )+D(γ f ). The resulting object is denoted γ b γ 1 · · · γ m γ f . Let v 1 , v 2 , . . . , v m+1 be all the cone-points of C 0 with e 1 -coordinate in {0, .
. . , n}. We assume that they are ordered according to increasing e 1 coordinates. (We also assume that m ≥ 1, since this will occur with high probability, as explained below.) These vertices induce a decomposition of C 0 into a string of m irreducible components (belonging to A) and two boundary-components (belonging, respectively, to B L and B R ):
C 0 = γ b γ 1 · · · γ m γ f . Note that all the pieces are unambiguously identified after inverting the translations due to the concatenation, except for γ f . The latter ambiguity disappears if we impose that As shown in [5], there exists ν 3 > 0 and ν 4 > 0 such that the number of irreducible pieces is at least ν 4 n with P( · | 0 ↔ n e 1 )-probability at least 1 − e −ν3n . In particular,
D(γ b , γ 1 , . . . , γ m , γ f ) ≡ D(γ b ) + k D(γ k ) + D(γ f ) = n e 1 . n e 1 v m+1 0 v 2 v 1P(0 ↔ n e 1 ) = (1 + o(1)) m≥ν2n γ b ,γ1,...,γm,γ f : D(γ b ,γ1,...,γm,γ f )=n e1 P(C 0 = γ b γ 1 · · · γ m γ f ) = (1 + o(1)) m≥ν2n γ b ,γ1,...,γm,γ f : D(γ b ,γ1,...,γm,γ f )=n e1 P(Γ b ∩ Γ 1 ∩ · · · ∩ Γ m ∩ Γ f ),(12)
where the percolation events Γ b , Γ k and Γ f are defined as follows. Let γ k be the translate of γ k obtained after the concatenation operation and denote by v k , v k+1 the corresponding cone-points. We set
W( γ k ) = {all edges of γ k are open}, ∂ γ k = (∂ γ k ) \ {v k − e 1 , v k }, {v k+1 , v k+1 + e 1 } , N ( γ k ) = {all edges of∂ γ k are closed}, Γ k = W( γ k ) ∩ N ( γ k ); the definitions of Γ b , Γ f are completely similar.
In order to apply the results of Appendix C, let us reformulate the above in the language of Appendix C (see the latter for details). Let us write D 1 (γ) = D(γ) · e 1 and set
Ψ(γ b ) = e ξD1(γ b ) P(Γ b ), Ψ m (γ b γ 1 · · · γ m γ f ) = e ξD1(γ b ,γ1,...,γm,γ f ) P(Γ b ∩ Γ 1 ∩ . . . ∩ Γ m ∩ Γ f ), Ψ(γ k | γ b γ 1 · · · γ k−1 ) = e ξD1(γ k ) P(Γ k | Γ b ∩ Γ 1 ∩ . . . ∩ Γ k−1 ), Ψ(γ f | γ b γ 1 · · · γ m ) = e ξD1(γ f ) P(Γ f | Γ b ∩ Γ 1 ∩ . . . ∩ Γ m ).
Of course, with these definitions, we have
Ψ m (γ b γ 1 · · · γ m γ f ) = Ψ(γ b ) m k=1 Ψ(γ k | γ b γ 1 · · · γ k−1 ) Ψ(γ f | γ b γ 1 · · · γ m ),
as desired. Moreover, the required properties are satisfied. To shorten notation, we simply write γ m = (γ b , γ 1 , . . . , γ m , γ f ). We first assume that s > 1. In this case, it follows from (12) (1)) . Let us now assume that s < 1. Since, by FKG, P(0 ↔ (n, x)) ≤ e −ξn for any x ∈ Z d−1 and n ≥ 1, it again follows from (12) that
that m≥0 s m γ m Ψ m (γ m ) ≥ n≥1 s ν4n n m=ν4n γ m : D(γ)=n e1 Ψ m (γ m ) ≥ C n≥1 s ν4n e ξn P(0 ↔ n e 1 ) = +∞, since P(0 ↔ n e 1 ) = e −ξn(1+o∞ > n≥1 s n x∈Z d−1 : (n,x)∈Y e ξn P(0 ↔ (n, x)) ≥ n≥1 s n n m=ν2n γ m : D1(γ)=n Ψ m (γ m ) ≥ m≥1 s m/ν4 m/ν4 n=m γ m : D1(γ)=n Ψ m (γ m ) ≥ C m≥1 s m/ν4 γ m Ψ m (γ m ).
We conclude that the radius of convergence of z → m≥1 z m γ m Ψ m (γ m ) is equal to 1, which establishes (H3).
In view of the above, we can import the results of Appendix C.1 to the present
context. Let S = n≥0 A n , S * = n≥1 A n , B L = {b L x : b L ∈ B L , x ∈ S} and B R = {x b R : b R ∈ B R , x ∈ S}.
One can then define (see (32)) two finite, positive measures ρ L and ρ R on B L and B R respectively, and a probability measure p on S * .
To any family γ = (γ b , γ 1 , . . . , γ m , γ f ), with m ≥ 1, we can associate uniquely a cluster of 0 (not necessarily containing n e 1 ), with cone-points v 1 , . . . , v m+1 (more precisely: γ b is not translated, while the other ones are concatenated as explained above). Any subset
x = {x 1 , . . . , x k+1 } ⊂ {v 1 , . . . , v m+1 } induces a decompositioñ γ = (γ b ,γ 1 , .
. . ,γ k ,γ f ) by concatenating irreducible pieces not separated by conepoints in x. We then introduce a (positive, finite) measure on triples (γ, x, y), with y ∈ Y , by settingQ
(γ, x, y) = 1 {D(γ)=y} ρ L (γ b ) ρ R (γ f ) k i=1 p(γ i ).
By Lemma C.1 and Theorem C.4, there exists c > 0 such that, for any bounded function f of the cluster C 0 ,
Q (f, D(γ) = n e 1 ) − e ξn E(f 1 {0↔n e1} ) ≤ e −cn ,(13)
for all n large enough.
Given k distinct vertices x 1 , . . . , x k such that x 1 ∈ Y , x k ∈ n e 1 + Y and x i+1 ∈ x i + Y for 1 ≤ i < k, and an additional vertex y ∈ x k + Y , we can writê Q(x, y) =ρ L (x 1 )ρ R (y − x k ) k−1 i=1p (x i+1 − x i ),
whereρ L ,ρ R andp are the push-forwards of ρ L , ρ R and p by the displacement map D. By Theorem C.4, these measures have exponential tails: there exist c > 0 and c p > 0 such that, for any x ∈ Y ,
ρ L (x) ≤ e −c x ,ρ R (x) ≤ e −c x ,p(x) ≤ e −cp x ,(14)
for all x large enough. Moreover,
ρ L (0) > 0,ρ R (0) > 0,(15)
as follows from Lemma C.3 and Remark C.2. Let us denote by P u and E u the distribution and expectation associated to the random walk (S ) ≥0 , starting at u ∈ Z d with transition probabilities given byp.
Write X i = S i − S i−1 for the increments of S. Let also R(v) = {∃ ≥ 0 : S = v} and set P u,v = P u ( · | R(v)) and E u,v = E u [ · | R(v)].
As a direct consequence of (13), observe that
e ξn P(0 ↔ n e 1 ) = u∈Y v∈n e1+Y ρ L (u)ρ R (n e 1 − v) P u (R(v)) + e −cn ,
for some c > 0. By the local limit theorem (see [18]), uniformly in u, v such that u , n e 1 − v ≤ n 1/2−α (for some fixed α > 0),
P u (R(v)) P 0 (R(n e 1 )) = 1 + o(1),(16)
and
P 0 (R(n e 1 )) = (1 + o(1)) c 4 n −(d−1)/2 .(17)
In particular, we obtain the Ornstein-Zernike asymptotics:
e ξn P(0 ↔ n e 1 ) = (1 + o(1)) c 5 n −(d−1)/2 ,(18)
with
c 5 = c 4 u∈Y v∈n e1+Y ρ L (u)ρ R (n e 1 − v). 5. Upper bound on ξ x − ξ x
In this section, we prove the upper bounds in Items (ii) and (iii) of Theorem 1.3, as well the d ≥ 4 part of Theorem 1.2. The argument in this section is a variant of the argument in [13], which applied to the case of Bernoulli percolation.
We work once more in large but finite volumes (as explained in Remark 2.1). In view of Theorem 1.1, we can assume, without loss of generality, that x ≥ x. In particular, λ = log(x /x) ≥ 0. By (11), we have the upper bound (13), there exist λ 0 > 0 and c > 0 such that, for any λ < λ 0 ,
E e λo L 0 ↔ n e 1 E e λo L ≤ E e λ|L∩C0| 0 ↔ n e 1 .
By Lemma 3.2 and
e ξn E e λ|L∩C0| 1 {0↔n e1} ≤ e ξn E e λ|L∩C0| 1 {0↔n e1,|L∩C0|≤c n} + e −cn ≤Q(e λ|L∩C0| , D(γ) = n e 1 ) + e −cn .
In particular, using (18) and (17),
E e λ|L∩C0| | 0 ↔ n e 1 ≤ c 6Q (e λ|L∩C0| | D(γ) = n e 1 ) + e −cn ,
for some c > 0. Then,
|L ∩ C 0 | = |γ b ∩ L| + |γ f ∩ L| + M i=1 |γ i ∩ L| ≤ |γ b ∩ L| + |γ f ∩ L| + M i=1 |D(S i−1 , S i ) ∩ L| ≤ |γ b ∩ L| + |γ f ∩ L| + M i=1 X i 1 {X i ≥|S ⊥ i−1 |} ,
where the last inequality relies on the fact that the angle of the "diamonds" is at most π/2 and the fact that a step cannot cross the line if its parallel component is smaller than the distance between its starting point and the line. We get
Q(e λ|L∩C0| | D(γ) = n e 1 ) ≤ C u,v ρ λ b (u)ρ λ f (n e 1 − v)E u,v e λ M i=1 X i 1 {X i ≥|S ⊥ i−1 |} ,
where ρ λ b (y) and ρ λ f (y) decay exponentially in y , provided that λ be small enough. In particular, we can restrict the sum to the pairs u, v with |u|, |v| ≤ n 1/2−α and
we have u,v ρ λ b (u)ρ λ f (v) < ∞.
At this stage, notice that the problem has been reduced to the analysis of a variant of the random-walk pinning problem. Then, for any m 0 ≥ 1, we can write
E u,v e λ M i=1 X i 1 {X i ≥|S ⊥ i−1 |} = n m=1 E u,v e λ m i=1 X i 1 {X i ≥|S ⊥ i−1 |} , M = m ≤ e λn P u,v (M < m 0 ) + n sup m0≤m≤n A u,v (m),(19)
with
A u,v (m) = E u,v e λ m i=1 X i 1 {X i ≥|S ⊥ i−1 |} ≤ O(n (d−1)/2 ) m k=1 1,..., k i=m E u k i=1 (e λX t i − 1) 1 {X t i ≥|S ⊥ t i −1 |} ≤ O(n (d−1)/2 ) m k=1 λ k 1,..., k i=m E u k i=1 e λX t i X ti 1 {X t i ≥|S ⊥ t i −1 |} , where t i = i j=1 j .
The first inequality is obtained using (17), writing
m i=1 e λX i 1 {X i ≥|S ⊥ i−1 |} = m i=1 (e λX i − 1)1 {X i ≥|S ⊥ i−1 |} + 1
and expanding the product. The second inequality is obtained using e λx − 1 = e λx (1 − e −λx ) ≤ e λx λx. Now, we use the Markov property and the local limit theorem in dimension d − 1 to get that, for all j,
E u e λX t j X tj 1 {X t j ≥|S ⊥ t j −1 |} S tj−1 , X tj = X tj e λX t j P St j−1 |S ⊥ tj −1 | ≤ X tj ≤ X tj e λX t j c 7 (X tj ) d−1 −(d−1)/2 j . As P u (X i > a) ≤ e −cpa , we obtain E u e λX t j X tj 1 {X t j ≥|S ⊥ t j −1 |} S tj−1 ≤ c 8 −(d−1)/2 j a≥1 e −cpa a d e λa ≡ c 9 −(d−1)/2 j ,
with c 9 < ∞ provided that λ < c p . Therefore, defining
A(m) = m k=1 (c 9 λ) k 1 ,..., k j =m k j=1 −(d−1)/2 j ,(20)
we get A u,v (m) ≤ c 10 n (d−1)/2 A(m).
In dimension 4 and larger, we bound A(m) uniformly over m by ignoring the constraint j = m:
A(m) ≤ ∞ k=1 c 9 λ ∞ =1 −(d−1)/2 k ,
which is convergent for λ > 0 small enough. Using (19) with m 0 = 1, this implies that E e λ|L∩C0| 0 ↔ n e 1 ≤ c 11 n (d+1)/2 , which in turn yields ξ ≤ ξ x . Since, ξ ≥ ξ x always holds, we conclude that ξ x = ξ for λ > 0 small enough, and thus that x c > x in dimension 4 and larger. This proves the d ≥ 4 part of Theorem 1.2.
In dimension 2 and 3, we get a diverging (in m) upper bound on A(m). Consider the generating function associated to the sequence (A(m)) m≥1 and define B(z):
A(z) = ∞ m=1 A(m)z m , B(z) = c 9 λ ∞ =1 −(d−1)/2 z .
Using (20), we have the relation
A(z) = k≥1 B(z) k .
Note that B is increasing on R + . Let f (λ) > 0 be the unique number such that B(e −f (λ) ) = 1. Since A(e −2f (λ) ) < ∞, we conclude that A(m) ≤ e 2f (λ)m for all large enough m. Now, using (19) with m 0 = c 12 n, c 12 > 0 small enough, and taking λ sufficiently small, we obtain, when n is large,
E e λ|L∩C0| 0 ↔ n e 1 ≤ c 13 (1 + O(n d−1
2 )e 2f (λ)n ).
It then follows from Theorem A.2 in [14] that, as λ ↓ 0, f (λ) behaves as
f (λ) = c 14 λ 2 (1 + o(1)) when d = 2, exp −c 15 /λ(1 + o(1)) when d = 3.
This proves the upper bounds in Items (ii) and (iii) of Theorem 1.3.
Lower bound on
ξ x c − ξ x when d = 2, 3
In this section, we prove the lower bounds in Items (ii) and (iii) of Theorem 1.3, which will then also imply the d ∈ {2, 3} part of Theorem 1.2.
For technical reasons, we work with large but finite systems (see Remark 2.1). The proof is based on an energy-entropy argument induced by
P x (0 ↔ n e 1 ) P(0 ↔ n e 1 ) ≥ P x (M δ ) P(0 ↔ n e 1 ) ≥ P x (M δ ) P(M δ ) P(M δ |0 ↔ n e 1 ),
where δ ∈ (0, 1) is arbitrary and M δ is the event {there exists an open path γ ∈ Γ δ } with Γ δ the set of self-avoiding paths from 0 to n e 1 with at least δn cone-points on L [0,n] (cone-points for the path itself, not the cluster of 0). The analysis below applies to arbitrary values of the parameter δ. A specific choice will be made at the end of the section.
Lemma 6.1. Let A be an increasing event and take x > x. Then,
P x (A) P(A) ≥ exp x x 1 s(1 + s) e∈L [0,n] P s (e ∈ Piv A | A) ds .
Proof. This is a straightforward application of Lemma B.2 and of the following consequence of the FKG inequality: P x (A) ≥ P This inequality allows us to control the "energy" part. Lemma 6.2. There exists ρ > 0, depending on x, such that
P x (M δ ) P(M δ ) ≥ exp (x − x) x (1 + x ) ρδn .
Proof. Notice that M δ is increasing. We can thus use Lemma 6.1:
P x (M δ ) P(M δ ) ≥ exp x x 1 s(1 + s) e∈L [0,n] P s (e ∈ Piv M δ | M δ ) ds ≥ exp 1 x (1 + x ) x x e∈L [0,n]
P s (e ∈ Piv 0↔n e1 | M δ ) ds , the last inequality following from the fact that Piv 0↔n e1 ⊂ Piv M δ on M δ . The claim will thus follow if we can prove that
e∈L [0,n] P s (e ∈ Piv 0↔n e1 | M δ ) ≥ ρδn for some ρ > 0, uniformly in s ∈ [x, x ].
Fix an arbitrary total order on Γ δ . For γ ∈ Γ δ , define B γ = {γ is the first path in Γ δ realizing M δ }. Then,
e∈L [0,n] P s (e ∈ Piv 0↔n e1 | M δ ) = e∈L [0,n] γ∈Γ δ P s (B γ | M δ ) P s (e ∈ Piv 0↔n e1 | B γ ) ≥ γ∈Γ δ e∈C(γ) P s (B γ | M δ ) P s (e ∈ Piv 0↔n e1 , B γ | γ open) P s (B γ | γ open) ≥ γ∈Γ δ P s (B γ | M δ ) e∈C(γ) P s (e ∈ Piv 0↔n e1 | γ open),
where C(γ) is the set of edges in γ ∩ L [0,n] having a cone-point of γ as an endpoint. The last inequality uses FKG: on the one hand, the measure P s (· | γ open) is a random-cluster measure on the complement of γ with wired boundary condition on γ, and is thus positively associated; on the other hand, B γ and {e ∈ Piv 0↔n e1 }, for e ∈ γ, are positively correlated as they are decreasing events on configurations in which the edges of γ are open.
We are thus left with showing that P s (e ∈ Piv 0↔n e1 | γ open) ≥ ρ uniformly over γ ∈ Γ δ , e ∈ C(γ) and s ∈ [x, x ].
Fix K ≥ K 0 large enough. Consider the cone Y ψ , as introduced in Section 4. Write Y ψ (u) = u + (Y ψ ∪ Y ψ ).
Let ∆ K = [−K, K] d (with a slight abuse of notation, ∆ K will denote the sites and/or the edges of ∆ K ∩ Z d ) and u * ψ = ∂(u + Y * ψ ) for * ∈ { , } and u ∈ Z d . Write v for the end-point of e which is a cone-point for γ (the right one if both are). Then, writing∆
K (v) = (v + ∆ K ) \ Y ψ (v), P s (e ∈ Piv 0↔n e1 | γ open) ≥ P s (e ∈ Piv 0↔n e1 ,∆ K (v) closed | γ open) ≥ ρ K 1 − P s (v ψ Y ψ (v) c ← −−− → v ψ | γ open,∆ K (v) closed) (21) by finite energy (ρ K depends polynomially on K). γ Y ψ (v) ∆ K (v) v v + ∆ KP s (v ψ Y ψ (v) c ← −−− → v ψ | γ open,∆ K (v) closed) ≤ ∞ r, =K u L ∈v ψ (u L )1=v1− u R ∈v ψ (u R )1=v1+r P w Y ψ (v) c (u L Y ψ (v) c ← −−− → u R |∆ K (v) closed) ≤ ∞ r, =K e −c1(r+ ) 2r tan (ψ)/2) d−2 (2 tan (ψ)/2 d−2 ≤ e −2c1K ∞ k=0 e −c1k P K,d,ψ (k) ≤ C K e −c2K ,
where P K,d,ψ (k) is a polynomial in k of degree at most 2d − 3 and C K is a constant depending polynomially on K; Lemma 2.1 was used to derive the second inequality. We take K large enough for the right-hand side to be at most 1/2. Plugging this into (21), we get the desired result with ρ = 1 2 ρ K and thus the initial claim. Let us now consider the "entropy" term P(M δ | 0 ↔ n e 1 ). Lemma 6.3. There exist c 1 , c 2 > 0, depending on x, such that, for small enough δ > 0,
P(M δ | 0 ↔ n e 1 ) ≥ e −c1δ 2 n when d = 2, e −c2(δ/ log(δ))n when d = 3.
Proof. Proceeding as in the previous section, we work with the measureQ and the random-walk S = (S ⊥ , S ) associated to C 0 (with increments X i = (X ⊥ i , X i )). Notice that, every time S steps on L, the corresponding point is a cone-point for any open path in Γ δ . Let C δ be the event {S hits L at least δn times}. Define the sequence of hitting times of L: τ 0 = 0 and τ k = inf{m > τ k−1 : S m ∈ L} for k ≥ 1. Using (15), we can restrict to the case whereγ b andγ f are reduced to {0}, respectively, and writing R n = R(n e 1 ), we get P(M δ | 0 ↔ n e 1 ) ≥ c P 0 (C δ | R n ).
From (17), we get that, for all sufficiently large n and k ≤ n/2, P0(R n−k ) P0(Rn) ≥ c for some c > 0. Then, using the strong Markov property, and δ * = δ4E[X 1 ], P 0 (S τ δn ≤ n/2) ≥ P 0 (S τ δn ≤ n/2, N n/2 ≥n) ≥ P 0 (L ⊥ (n) ≥ δn, N n/2 ≥n) ≥ P 0 (L ⊥ (n) ≥ δ * n ) − P 0 n k=1 X k > n/2
P 0 (C δ | R n ) = P 0 (S τ δn ≤ n | R n ) ≥ n/2 k=0 P 0 (S τ δn = k) P 0 (R n−k ) P 0 (R n ) ≥ c P 0 (S τ δn ≤ n/2).≥ P 0 (L ⊥ (n) ≥ δ * n ) − e −cn ,
where we used an elementary large deviation estimate for a sum of independent random variables in the last line. Finally, the event {L ⊥ (n) ≥ δ * n } depends only on S ⊥ which is a random walk with i.i.d. increments in Z d−1 , and thus (see Corollary B.3 in [13]):
P 0 (L ⊥ (n) ≥ δ * n ) ≥ e −cδ 2 * n when d = 2, e −c(δ * /| log δ * |)n when d = 3. Now, choosing δ to be δ = C x (x − x) when d = 2, exp −C x /(x − x) when d = 3,
with C x , C x large enough (observe that x (1 + x ) > x(1 + x)/2 when x − x is sufficiently small), in Lemmas 6.2 and 6.3, we obtain the lower bounds stated in Items (ii) and (iii) of Theorem 1.3, which also implies the d ∈ {2, 3} part of Theorem 1.2.
Coarse-graining procedure and advanced properties
In this section, we prove the first and last items of Theorem 1.3, as well as Theorem 1.4. Namely, we show that, for x > x c , the connectivity function along the e 1 axis has pure exponential decay and that x → ξ x is real analytic and strictly decreasing on (x c , ∞). This will be done with the help of a coarse-graining procedure similar to the one we used in Section 3. Given a set of vertices A ⊂ Z d , define A = u∈A ∆(u). Given B ⊂ Z d , we denote by C 0 | B the connected component of 0 in C 0 ∩ B. We coarse-grain C 0 using the following algorithm:
Algorithm 2: Coarse-graining procedure of C 0 v 0 = 0; V = {v 0 } and E = ∅; i = 0; A = {v ∈ ∂V ∩ C 0 | V : v ∆(v)\V ←−−−→ ∂∆(v)}; while A = ∅ do i = i + 1;
Set v i to be the smallest site of A w.r.t. the lexicographical order;
Set e i = (v * , v i ), where v * is the smallest vertex in V among those closest to v i ; Update V = V ∪ {v i } and E = E ∪ {e i }; Update A = {v ∈ ∂V ∩ C 0 | V : v ∆(v)\V ←−−−→ ∂∆(v)}; end T 0 = (V, E);
Adapting the definitions introduced in the coarse-graining argument used in the proof of Item c) of Lemma 3.1 to our new boxes, denote by C K the number of vertices in ∂∆. As already used there, the number of trees T with N vertices that can be obtained via Algorithm 2 is at most e c16 log(C K )N . We say that v ∈ Z d is L-free if ∆(v) ∩ L = ∅; otherwise, we call it an L-vertex. The next lemma will give us control on the probability to see a specific tree T . Proof. First, notice that whenever an L-free vertex is created, a connection as described in Lemma B.1 is induced forcing a cost e −ξK uniformly over the tree constructed so far (we fix r large enough to be able to apply Lemma B.1). When an L-vertex is created, two things can happen. The first possibility is that a crossing (in the easy direction) of a box −K, K × 0, K d−1 at distance at least r log(K) from the line L is induced (see Figure 7.2), costing e −ξK (1 + o K (1)) ≤ e −ξ x K (1 + o K (1)) by Lemma B.1. The second possibility is that the vertex is connected to a side of ∆ crossed by L. The procedure used in the proof of Lemma B.1 together with (5) and (6) yield probability e −ξ x K (1 + o K (1)) for such a crossing (uniformly over the tree constructed so far). Therefore, as τ = ξ − ξ x > 0 when x > x c , we have (for T containing m L-free vertices and N L-vertices):
P x (T 0 = T ) ≤ e −τ Km e −ξ x K(N +m) (1 + o K (1)) m+N .
Using this, we argue similarly as in the proof of Item c) of Lemma 3.1. Remark that, up to a term of order e −2ξ x n(1+o(1)) , we can restrict connections 0 ↔ n e 1 to those not connecting to Z <−n × Z d−1 or to Z >2n × Z d−1 . Then, for any ρ ∈ (0, 1),
2K L K r log K≤ ∞ m=ρn/K c17n/K N =1∨n/K−m e −τ Km e −ξ x K(N +m) e c d log(K)(m+N ) ≤ e − K K ξ x n C 1 n K e −(τ ρK/K−c18 log(K)/K)n .
Thus, for any ρ ∈ (0, 1), we can find K 0 ≡ K 0 (ρ, τ ) such that, for K ≥ K 0 , there exists ν(ρ) > 0 depending on ρ, x and x such that P x (T 0 contains at least ρn/K L-free vert., 0 ↔ n e 1 ) ≤ e − K K ξ x n e −ν(ρ)n .
7.2. Renewal on L. We now use the coarse-graining of the previous section to show that, under P x ( · | 0 ↔ n e 1 ), C 0 possesses a number of cone-points on L of order n when x > x c (as defined in Section 4). For convenience, we look at cones Y (and diamonds) having angular aperture π/2. Theorem 7.2. When x > x c , there exist ρ cp ≡ ρ cp (x ) ∈ (0, 1) and ν 5 > 0 such that P x C 0 contains at least ρ cp n cone-points on L 0 ↔ n e 1 ≥ 1 − e −ν5n .
Proof. Start by observing that C 0 is included in a K-neighborhood of T 0 . Then, define the shade sh(v) of a point v by
sh(v) = L [− v ⊥ +v , v ⊥ +v ] .
This corresponds to the portion of L that cannot contain cone-points of C 0 as soon as v ∈ C 0 . In the same fashion, define the shade of v ∈ T 0 to be the union over u ∈ [∆(v)] K (the K neighborhood of ∆(v)) of the shade of u. Finally, define the shade of T 0 as the union of the shades of the L-free vertices of T 0 . We will show that, with high probability, this shade does not cover a substantial proportion of L; then we will use a finite-energy argument to show that a positive fraction of the unshaded points are cone-points of C 0 . A first observation is that there exists c 19 not depending on K, such that the size of the shade of T 0 is at most
c 19 K#{L-free vertices of T 0 }.
This is proved by induction on the number of L-free vertices of T 0 . The first one is at distance at most 5K from the line and the inequality thus holds by definition of the shade. Then, adding an L-free vertex either adds the same shade size to the total shade (the vertex is far from the existing ones) or it increases the shade size by at most c 20 K, for some constant c 20 < 10, (see figure 7.3). 3K (which is included in the shade of the L-free vertices of T 0 ). We have, using (22),
S i = (i − 1)7K, i7K × Z d−1 and B i = S i ∩ [L] 3K , where [L] 3K is the 3K-neighborhood of L. We will say that B i is illuminated if L [(i−1)7K+3K,i7K−3K] is not included in the shade of C 0 \ [L]P x #{illuminated B i } ≥ n 14K ≥ 1 − P x |sh(T 0 )| > n 2 ≥ 1 − P x c 19 #{L-free vertices of T 0 } > n 2 ≥ 1 − e − K K ξ x n e −ν6n .
Thus, as P x (0 ↔ n e 1 ) ≥ e −ξ x n(1+o(1)) ,
P x #{illuminated B i } ≥ n 14K 0 ↔ n e 1 ≥ 1 − e −ν7n ,(23)
for some ν 7 > 0. Noticing that the number of B i is n 7K , this implies that at least half the boxes are illuminated with high probability. Now, we describe a surgery procedure creating a cone-point on L from an illuminated B i and bound its cost uniformly over the rest of the cluster of 0. 7.3. Pure exponential decay when x > x c . We are now in position to prove the last item of Theorem 1.3. This will be done in the same fashion as in Section 4, except that the "random walk" will here be pinned to the line, replacing the powerlaw correction present in (18) by a constant (which is related to the frequency of occurrence of cone-points on the line). We work here with cones of angular aperture π/2. As in Section 4, let w 1 , . . . , w m be the cone-points of C 0 lying on L (by Theorem 7.2, m is typically of order n). Let
ζ i = C 0 ∩ w i · e 1 , w i+1 · e 1 × Z d−1
define the cone-confined irreducible components of C 0 , and let ζ b and ζ f be the two components of C 0 \ (ζ 1 ∪ ζ 2 ∪ ... ∪ ζ m−1 ) containing respectively 0 (backwardirreducible) and n e 1 (forward-irreducible); they can possibly be reduced to a single vertex. All definitions of Section 4 extend with almost no modification to the irreducible components ζ. In particular, we can define percolation events
Ξ b , Ξ 1 , . . . , Ξ m−1 , Ξ f associated with ζ b , ζ 1 , . . . , ζ m−1 , ζ f so that P x (C 0 = ζ b ζ 1 · · · ζ m−1 ζ f ) = P x (Ξ b )P x (Ξ f | Ξ b , . . . , Ξ m−1 ) m−1 i=1 P x (Ξ i | Ξ b , . . . , Ξ i−1 ).
Then, for u, v ≥ 1, we can define
ρ b (u) = e ξ x u ζ b 0 D(ζ b )=u P x (Ξ b ) and ρ f (v) = e ξ x v ζ f 0 D(ζ f )=v P x (Ξ f ).
By Theorem 7.2, they satisfy
ρ b (u) ≤ e −ν5u and ρ f (v) ≤ e −ν5v .
Again, all the properties listed in Proposition 4.1 hold in the present setting (with essentially the same proof). This allows us to proceed as in Section 4 in order to "couple" C 0 with a random walk S on Z >0 with i.i.d. increments X i in Z >0 having exponential tails. We denote its law and expectation by Q . The measures associated to the boundaries pieces will be denoted byρ b ,ρ f ; they have exponential tails. The arguments leading to (18) yield in the present setting
e ξ x n P x (0 ↔ n e 1 ) = u,vρ b (u)ρ f (v) Q ∃t > 0 : t i=1 X i = n − v − u + e −cn .
We can clearly restrict the sum to u, v < n/4. The conclusion then follows from the Renewal Theorem and Theorem 7.2, since they imply that Q ∃t > 0 :
t i=1 X i = n − v − u n→∞ − −−− → 1 Q (X 1 ) > 0,
uniformly in u, v < n/4.
7.4
. ξ x is strictly decreasing when x > x c . As discussed in Remark 2.1, we can find a sequence (a n ) n≥1 of large enough numbers such that ξ x = lim n→∞ ξ
(n) x , where ξ (n) x = − 1 n log P f x ,Λa n (0 ↔ n e 1 ).
We can then bound d dx ξ (n) x using Lemma B.2:
d dx ξ (n) x ≤ − 1 n 1 x (x + 1) e∈L P f
x ,Λa n (e ∈ Piv 0↔n e1 | 0 ↔ n e 1 ).
By Theorem 7.2, the number of cone-points can be assumed to grow linearly with n. As every cone-point induces at least one pivotal edge for 0 ↔ n e 1 , we can find a positive constant such that d dx ξ
(n) x ≤ −c 21 uniformly in n. Thus, x → ξ x is strictly decreasing. Indeed, for x c < x 1 < x 2 < ∞, ξ x 2 − ξ x 1 = lim n→∞ ξ (n) x 2 − ξ (n) x 1 = lim n→∞ x 2 x 1 d ds ξ (n) s ds ≤ −c 22 (x 2 − x 1 ).
Notice that the constant depends on x 2 .
7.5. Analyticity of x → ξ x for x > x c .
For any x 0 > x c , we are going to prove analyticity of ξ x for x in a neighbourhood of x 0 . Let us thus fix x 0 > x c . We first make the following two assumptions, which will be proved at the end of the section:
Claim 7.1. ξ x can be obtained as the limit of − 1 n log P Assuming this, we can rewrite lim n→∞ 1 n log P (n)
x (0 ↔ n e 1 ) = lim
n→∞ 1 n log E (n) x 0 x x 0 o L [0,n] 1 {0↔n e1} E (n) x 0 x x 0 o L [0,n] = lim n→∞ 1 n log e −(f x +ξ x 0 )n E (n) x 0 x x 0 o L [0,n] 0 ↔ n e 1 .
The same construction as in the previous subsection (coarse-graining and finite energy), together with the strict monotonicity of ξ x on (x c , ∞), guarantee that there exists 0 > 0 such that, for any x in a neighbourhood of x 0 , we have might not lie on L [0,n] ). As before, the length of these irreducible components has exponential tails. We obtain
e −(f x +ξ x 0 )n E (n) x 0 x x 0 o L [0,n] 0 ↔ n e 1 = e −(f x +ξ x 0 )n E (n) x 0 x x 0 o L [0,E (n) x 0 x x 0 o L [0,n] 1 {Cp(L [0,n] )> 0 n} 0 ↔ n e 1 = E (n) x 0 x x 0 o D b ∩L [0,n] x x 0 o D f ∩L [0,n] m i=1 x x 0 o D i ∩L 1 {Cp(L [0,n] )> 0n} 0 ↔ n e 1 . (24)
Proceeding as in Sections 4 and 7.3, we can partition ϑ b , ϑ 1 , . . . , ϑ m , ϑ f into finite strings of irreducible piecesθ b ,θ 1 , . . . ,θ k−1 ,θ f , and construct a probability measure Q on the irreducible componentsθ i and two finite measures p L , p R onθ b and θ f , respectively. All three measures have exponential tails, so that, up to an error of order e −cn with c > 0 uniform in x in a small neighbourhood of x 0 , (24) becomes
C n k=1 0+···+ k =n p L ( 0 )p R ( k ) k−1 i=1 Q x x 0 oD i ∩L |D i | = i Q(|D i | = i ),(25)
where the C comes from the pure exponential decay behaviour of P x 0 (0 ↔ n e 1 ) and the associated conditioning, and where
p L/R ( ) = p L/R x x 0 oD b/f ∩L [0,n] 1 {|D b/f |= } , are exponentially decaying in for x in a small neighbourhood of x 0 . Notice that Q x x 0 oD i ∩L |D i | = i is a polynomial in x of degree at most i . Denote q = Q(|D 1 | = ) (which is exponentially decaying in ) and p (x ) = Q x x 0 oD 1 ∩L |D 1 | = e −(f x +ξ x 0 ) ; observe that p (x )
is an analytic function of x . Define α 0 (x ) = 1 and
α n (x ) = n k=1 1+···+ k =n k i=1 Q x x oD i ∩L |D i | = i Q(|D i | = i )e −(f x +ξ x 0 ) i = n k=1 1+···+ k =n k i=1 q i p i (x ).(26)Denote d n = e −(f x +ξ x 0 )n E (n) x 0 x x 0 o L [0,n]
0 ↔ n e 1 and D(z) = n≥1 d n z n .
By definition, the radius of convergence of D is e ξ x . Then, consider the generating functions
A(t) = ∞ n=0 α n (x )e tn and B(t) = ∞ =1 q p (x )e t .
By (24) and (25), the radius of convergence of A is given by the one of D(e t ), so that the radius of convergence of A(t) is equal to ξ x . Moreover, notice that B(t) converges for all t < t 0 for some t 0 > ξ x as q is exponentially decaying in . Then, (26) implies that
A(t) = 1 1 − B(t)
.
Thus B(ξ x ) = 1, which provides an implicit expression for ξ x . Defining
Φ(w, z) = ∞ =1 q p (w) e z ,
analyticity follows from solving Φ(w, z) = 1 for z in a neighborhood of (x , ξ x ). Analyticity of Φ(w, z) close to (x , ξ x ) follows from the exponential decay property of q , and ∂Φ ∂z (x ,ξ x ) = 0 from a direct computation. The claim follows using the (analytic version of the) implicit function theorem.
Proof of Claim 7.1. To simplify notations, we will write j ≡ j e 1 when the meaning is clear from the context. First notice that P (n)
x (0 ↔ n) ≤ P x (0 ↔ n) by mono- tonicity, since x ≥ x. Thus ξ x ≤ − lim n→∞ 1 n log P (n)
x (0 ↔ n). To obtain the reverse inequality, we partition, for M > 0,
P x (0 ↔ n) = P x (0 ↔ n, 0 ↔ (L <−M ∪ L >n+M )) + P x (0 ↔ n, 0 ↔ / (L <−M ∪ L >n+M )). (27)
Now, we bound separately the two terms in the RHS.
P x (0 ↔ n, 0 ↔(L <−M ∪ L >n+M )) ≤ k<−M P x (k ↔ 0 ↔ n) + k>n+M P x (0 ↔ n ↔ k) ≤ 2 ∞ k=0 P x (0 ↔ n + M + k) ≤ 2e −ξ x n e −ξ x M 1 1 − e −ξ x ≤ CP x (0 ↔ n)e −ξ x M 1 1 − e −ξ x ≡ C x e −M ξ x P x (0 ↔ n),(28)
where the first inequality is a union bound, the second uses invariance under translation, the third is by Lemma 3.1, Item a), and the fourth follows from Theorem 1.3, Item (iv) with C ≥ 0 not depending on n. Then,
P x (0 ↔ n, 0 ↔ / (L <−M ∪ L >n+M )) = = C 0,n C∩(L <−M ∪L >n+M )=∅ P x (∂C 0 closed)P x (C open | ∂C 0 closed) ≤ C 0,n C∩(L <−M ∪L >n+M )=∅ P (−M,n+M ) x (∂C 0 closed)P (−M,n+M ) x (C open | ∂C 0 closed) ≤ P (−M,n+M ) x (0 ↔ n) = E x x o L [−M,n+M ] 1 {0↔n} E x x o L [−M,n+M ] ≤ x x 2M P (n) x (0 ↔ n),(29)
where P
P x (0 ↔ n) ≤ x x 2M 1 1 − C x e −M ξ x P (n) x (0 ↔ n) ≤ 2 x x 2M P (n) x (0 ↔ n),
implying ξ x ≥ − lim n→∞ 1 n log P (n)
x (0 ↔ n e 1 ).
Proof of Claim 7.2. The proof will be done using the same line of ideas as described above for the analyticity of ξ x . First notice that, for x ≥ x 0 ,
E (n) x 0 x x o L [0,n+m] ≥ E (n) x 0 x x o L [0,m] E (n) x 0 x x o L [0,n] ,
by FKG, as o L [0,n] is an increasing function, and by translation invariance of E (for x < x 0 , the reverse inequality holds). Thus, existence of f x follows by Fekete's Lemma. Analyticity of f x follows the same lines as ξ x : the same representation of L [0,n] cluster holds under E (n) x 0
, and the rest of the argument carries out in the same (in fact, simpler) fashion as in the ξ x case. 7.6. Interface localization. Theorem 1.4 is an essentially immediate corollary of the analysis leading to Theorem 7.2 and classical tools for the analysis of the random-cluster model (see [17]): the Edwards-Sokal coupling and the coupling between the high-and low-temperature random-cluster measures on Z 2 . It is enough to make the following observations:
• Whenever {i, j} ∈ E 2 is such that {i, j} * is part of the interface, the edge {i, j} is closed in the random-cluster configuration associated to the Potts configuration by the Edwards-Sokal coupling. Note that this randomcluster model has wired boundary condition and a constraint that {i ∈ Z 2 : i ⊥ > 0} \ Λ n must no be connected (in Λ n ) to {i ∈ Z 2 : i ⊥ ≤ 0} \ Λ n . • By the standard coupling between the random-cluster model on Z 2 and its dual (which has parameters p * < p c , J * > 1 and the same value of q), the latter has free boundary condition and is conditioned on the two dual vertices (−n − 1 2 , 1 2 ) and (n + 1 2 , 1 2 ) being connected. Let us denote by C n the corresponding cluster.
• By the above, the Potts interface is a subset of the (dual) cluster C n .
• The analysis leading to Theorem 7.2 can be repeated essentially verbatim, the fact that one is working in a finite system having no incidence. • This implies that the cone-points of C n are also cone-points of the Potts interface, from which the desired result follows immediately.
L L Figure 7.6. The FK representation of low temperature Potts model with Dobrushin boundary condition (the top is conditioned to not intersect the bottom) and the corresponding high temperature dual FK configuration (where the two points are conditioned to be connected).
Appendix A. Couplings
We sketch here the proofs of the existence of some couplings used in the paper. Similar construction (with more details) can be found, for example, in [15] and [10].
Lemma A.1. Let G be a finite graph and let P x,q be the random-cluster measure with edges weights (x e ) e∈E G and cluster weight q on G. Then, for any e ∈ E G , there exists a coupling (ω, η) ∼ Φ of P x,q (· | ω e = 1) and P x,q (· | ω e = 0) such that (i) ω ∼ P x,q (· | ω e = 1) and η ∼ P x,q (· | η e = 0),
(ii) Φ(ω ≥ η) = 1.
Proof. This lemma is standard and follows from a Markov chain argument: start from ω (0) ≥ η (0) and perform a heat bath dynamic simultaneously on the two configurations. Having constructed ω (n−1) , η (n−1) , construct ω (n) , η (n) in the following way: select an edge f uniformly at random from E G ; resample its state in ω (n−1) according to P x,q (· | ω g = ω (n−1) g ∀g / ∈ {f, e}, ω e = 1) to obtain ω (n) ; resample its state in η (n−1) according to P x,q (· | η g = η (n−1) g ∀g / ∈ {f, e}, η e = 0) to obtain η (n) . The two dynamics can be coupled so that for every n, the law of ω (n+1) dominates the law of η (n+1) . Letting n → ∞, this gives the desired coupling.
Lemma A.2. Let G be a finite graph, let P x,q be the random cluster measure with edges weights (x e ) e∈E G and cluster weight q on G and, for E ⊂ E G , let P y,q be the random cluster measure with edges weights y e = x e if e / ∈ E y e < x e if e ∈ E , and cluster weight q. Then, there exists a coupling (ω, η) ∼ Φ of P x,q and P y,q such that (i) ω ∼ P x,q and η ∼ P y,q ,
(ii) Φ(ω ≥ η) = 1, (iii) Φ(ω| (C E (ω)) c = η| (C E (ω)) c ) = 1.
Proof. This coupling is slightly more involved and is done via an exploration process. Fix an arbitrary ordering of E G . We will explore the configurations by exploring the cluster of E. Denote C (n) E (ω) the cluster of E in ω (that is, the union of the clusters of the endpoints of the edges in E) restricted to the explored edges after step n (it always contains the endpoints of the edges in E) and let ∂C In this way, when an edge is open in η, it is also open in ω. Observe that, once the cluster of E in ω is explored, its boundary will be closed in both configurations. We can thus sample the remaining edges in both configurations according to P f x,q;(C E (ω)∪∂C E (ω)) c , so the two agree outside of C E (ω).
Appendix B. Basic results in FK percolation B.1. A decoupling inequality. The following lemma is inspired by an analogous claim in [5].
Lemma B.1. Let R > 0 and let A be an increasing event depending only on edges in a finite set
D ⊂ Z d . Define D R = i∈D (i + −R, R d ).
Then, for all R large enough,
P w D R (A) P(A) ≤ 1 − |∂D R/2 ||∂D R |e −ν1R/2 −1 .
Proof. Notice first that, for u ∈ ∂D R/2 and v ∈ ∂D R , the distance between u and v is at least R/2. Then, partitioning according to whether the event ∂D R ↔ ∂D R/2 occurs, we get
P w D R (A) ≤ P w D R (A) P w D R \D (∂D R ↔ ∂D R/2 ) + P(A) ≤ P w D R (A)|∂D R/2 ||∂D R |e −ν1R/2 + P(A)
, where we used monotonicity in volume and in boundary conditions for the first inequality and Lemma 2.1 for the second one.
B.2.
A Russo-like formula. There exist various extensions of the Russo formula from Bernoulli percolation to FK percolation. However, we will need the following version, which we did not find in the literature. Recall that an edge e is pivotal for an event A in a configuration ω if the value of 1 {ω∈A} depends on the value of ω e . Denote Piv A (ω) the set of edges pivotal for A in ω.
Lemma B.2. Let P x,q be the random-cluster measure on a finite graph G, with weights (x e ) e∈E G and q ≥ 1. Let E ⊂ E G a collection of edges in G. Denote by P s
x,q the random-cluster measure obtained by modifying the weights x by setting x e = s, ∀e ∈ E. Then, for s 2 > s 1 and any nondecreasing event A, we have
P s2 x,q (A) P s1 x,q (A) ≥ exp s2 s1 1 s(1 + s) e∈E P s x,q (e ∈ Piv A | A)ds . Proof. First, we compute d ds log P s x,q (A) = 1 sP s x,q (A) Cov s x,q (1 A , o E ) = 1 sP s x,q (A) e∈E P s x,q (ω e = 1) P s x,q (A | ω e = 1) − P s x,q (A) = 1 sP s x,q (A) e∈E P s x,q (ω e = 1)P s x,q (ω e = 0) × P s x,q (A | ω e = 1) − P s x,q (A | ω e = 0) . (30)
Consider a coupling (ω, η) ∼ Φ of P s x,q (· | ω e = 1) and P s x,q (· | ω e = 0) such that ω ≥ η and ω ∼ P s x,q (· | ω e = 1) and η ∼ P s x,q (· | ω e = 0). Then compute
P s x,q (A | ω e = 1) − P s x,q (A | ω e = 0) = Φ(1 A (ω) − 1 A (η)) = Φ(1 A (ω)1 A c (η)) ≥ Φ(1 {e∈Piv A } (ω))
= P s x,q (e ∈ Piv A | ω e = 1) = P s x,q (e ∈ Piv A , ω e = 1) P s x,q (ω e = 1) = P s x,q (e ∈ Piv A , A) P s x,q (ω e = 1)
, where we used, in the second line, that A is increasing and ω ≥ η, so that η ∈ A =⇒ ω ∈ A and ω ∈ A c =⇒ η ∈ A c and thus 1 A (ω) − 1 A (η) = 1 A (ω)1 A c (η); we have also used the fact that e ∈ Piv A (ω) =⇒ ω ∈ A and η ∈ A c for the inequality. Plugging this into (30) gives
d ds log P s x,q (A) ≥ 1 sP s (A) e∈E P s x,q (ω e = 0)P s x,q (e ∈ Piv A , A) = 1 s e∈E P s x,q (ω e = 0)P s x,q (e ∈ Piv A | A) ≥ 1 s(1 + s) e∈E P s x,q (e ∈ Piv A | A) ,
where the last inequality follows from finite energy of P s x,q . Integrating both sides between s 1 and s 2 and taking the exponential leads to the desired inequality.
Appendix C. Renewal for long-range memory process
The goal of this appendix is to present a way to factorize measures on sequences with exponential mixing. The procedure employed is a representation of the mixing property as a memory-percolation picture. The ideas used here are inspired from the construction done in [8], but our set-up being a bit different (we deal with general kernels instead of probability kernels and we need "finite volume" estimates rather than estimates on the stationary measure), the results from [8] do not immediately apply, so we provide here a self-contained exposition.
C.1. Setting, Notations and Definition. We will work with A an alphabet (finite or countable), and B L , B R two sets containing ∅ (finite or countable). The objects of study will be measures on sequences of the form
(b L , x 1 , x 2 , . . . , x n , b R ) ∈ B L × A n × B R .
We will say and assume:
• elements of A are called letters, sequences (or concatenation) of letters are called words; • A does not contain words;
• for x ∈ A n , denote |x| = n the length of the word x.
As we work with sequences, it will be useful to have a few operations on them. We first define the concatenation operation.
Definition C.1. For x = (. . . , x k−1 , x k ) a right-finite sequence and y = (y l , y l+1 , . . . ) a left-finite sequence, the concatenation of x and y is the sequence x y = (. . . , x k−1 , x k , y l , y l+1 , . . . ).
By convention, the labels of the new sequence will be chosen to be consistent with the labels of x:
(x y) i = x i if i ≤ k y l+i−k−1 if i > k .
Elements of A will be considered as one-element sequences for concatenation. We then define the extraction operation.
Definition C.2. For k ≤ l ≤ m ≤ n ∈ Z and x = (. . . , x k , . . . , x l , . . . , x m , . . . , x n , . . . ) a sequence, the (l, m)-extraction of x is the sequence x m l = (x l , x l+1 , . . . , x m−1 , x m ). We will use the following notations:
• S = n≥0 A n the set of finite sequences (A 0 = {∅}), and S * = n≥1 A n the set of non-empty finite sequences; • S + = A Z ≥0 , resp. S − = A Z<0 , the set of right-infinite, resp. left-infinite, sequences;
• B L = {b L x : b L ∈ B L , x ∈ S} and B R = {x b R : b R ∈ B R , x ∈ S}.
In all this Appendix, when not explicitly said otherwise, b L , b R , x will always denote elements of B L , B R , S * and x = (x 1 , . . . , x n ).
We will consider measures Ψ n on B L × S n × B R that are given by a kernel Ψ : B L × (A ∪ B R ) → R + and a weight function Ψ : B L → R + (for simplicity, we denote both by the same letter...). Namely, writing
Ψ(b, s) ≡ Ψ(s | b),
we assume that
Ψ n (b L x b R ) = Ψ(b L ) n k=1 Ψ(x k | b L x k−1 1 ) Ψ(b R | b L x n 1 )
To lighten the notations, we will sometimes write
Ψ(b L x n 1 ) = Ψ(b L ) n i=1 Ψ(x i | b L x i−1 1 ).
We will make the following additional assumptions on Ψ: (H1) uniform summability: there exists K < ∞ such that
s∈A∪B R/L Ψ(s | b) ≤ K
for all b ∈ B L/R ; (H2) ratio exponential mixing: there exist L 0 , c > 0 such that
1 − Ψ(z|b 2 x) Ψ(z|b 1 x) ≤ e −c|x|
for all n ≥ L 0 , z, x = (x 1 , . . . , x n ) ∈ S, b 1 , b 2 ∈ B L ; (H3) sub-exponential decay (or growth) of the mass:
lim n→∞ 1 n log(µ n ) = 0, where µ n = b L ,b R x∈A n Ψ n (b L x b R ). (H4) there exist s ∈ A and 0 > 0 such that inf l∈B L Ψ(s | l) ≥ 0 .
C.2. The Memory Percolation Picture. A stick-percolation configuration on an interval J ⊂ R is a partitioning of J into disjoints open intervals (called clusters) and their endpoints (called cuts). Given a stick-percolation configuration ω, denote by Cuts(ω) the set of its cuts. We will consider stick-percolation configurations induced by functions I : Z → Z ≥0 via the following procedure: to every k ∈ Z, associate the (open) interval (k −I k − 1 2 , k + 1 2 ). Then, the connected components of the union of those intervals give the clusters of the stick-percolation configuration, while the complement of this union gives the set of cuts. We will say that an edge e k = (k, k + 1) is a cut if k + 1 2 is. This definition extends straightforwardly for stick-percolation configurations on finite subsets of Z. With this in hand, we augment each sequence b L x b R , x ∈ A n , with a stick-percolation realization on [0, n + 1] ∩ Z. This will be done with the help of a memory threshold sequence (following [8]). Let x = (x 1 , . . . , x n ) ∈ S * , b ∈ B L and define are nondecreasing sequences in k for any s, b and x. One can thus consider the "covered mass at depth k":
a 0 (s | b x) ≡ a 0 (s | * ) = inf l∈B L Ψ(s | l), a k (s | b x) ≡ a k (s | x n n−k+1 ) = inf l∈B L Ψ(s | l x n n−k+1 ), a k (b x) = s∈A a k (s | b x n 1 ), if 0 ≤ k ≤ n,∆ 0 (s | b x) = a 0 (s | * ), ∆ k (s | b x) = a k (s | b x) − a k−1 (s | b x), ∆ k (b x) = s∈A ∆ k (s | b x).
All these definitions are for s ∈ A, but they extend straightforwardly to the case where s is replaced by b ∈ B R . Observe that Assumption (H4) is equivalent to the existence of 0 > 0 such that a 0 ≥ 0 . Now, noticing that
Ψ(s | b x n 1 ) = a n+1 (s | b x n 1 ) = a 0 (s | b x n 1 ) + n k=0 a k+1 (s | b x n 1 ) − a k (s | b x n 1 ) = n+1 k=0 ∆ k (s | b x n 1 ), one can write Ψ n (b L x b R ) = Ψ(b L ) n k=1 Ψ(x k | b L x k−1 1 ) Ψ(b R | b L x n 1 ) = Ψ(b L ) n k=1 k i=0 ∆ i (x k | b L x k−1 1 ) n+1 i=0 ∆ i (b R | b L x n 1 ) = Ψ(b L ) I∈In ∆ In+1 (b R | b L x n 1 ) n k=1 ∆ I k (x k | b x k−1 1 ),
where I n = I : {0, 1, . . . , n + 1} → Z ≥0 : I k ≡ I(k) ≤ k . Now enters the memory-percolation picture: I can be seen as the realization of a stick-percolation. In this way, each I can be associated to a cluster set that we will represent as the sequence of the lengths of its clusters:
C n = n+1 k=0 (l 0 , . . . , l k ) : l i ≥ 1, k i=0 l i = n + 2 .
We will write I ∼ (l 0 , . . . , l k ) if the cuts of the stick-percolation configuration induced by I are (l 0 − 1, l 0 ), (l 0 + l 1 − 1, l 0 + l 1 ), . . . , (l 0 + · · · + l k−1 − 1, l 0 + · · · + l k−1 ). One can thus see Ψ n as a measure on (B L A n B R ) × C n (equipped with the discrete sigma-algebra):
Ψ n (b L x b R , (l 0 , l 1 , . . . , l k )) = Ψ(b L ) I∈In I∼(l0,...,l k ) ∆ In+1 (b R | b L x n 1 ) n i=1 ∆ Ii (x i | b L x i−1 1 ). (31)
Notice that, for a given cluster realization (l 0 , . . . , l k ), the value of the weight ∆ Ii (x i | b L x i−1 1 ) for a given i is independent of the value of x j for j < i − I i . It is this essential property that will be exploited in our analysis.
b L x 1 x 2 x 3 x 4 x 5 x 6 x 7
x n−1 x n b R x n−2 ... Define
ρ L (b L x n 1 ) = Ψ(b L ) I∼(n+1) n k=1 ∆ I k (x k | b L x k−1 1 ) ρ R (x n 1 b R ), = I∼(1,n+1) ∆ In+1 (b R | x n 1 ) n k=1 ∆ I k (x k | x k−1 1 ) p(x n 1 ), = I∼(1,n) n k=1 ∆ I k (x k | x k−1 1
).
(32)
These are obviously nonnegative measures on, respectively, B L × S,S × B R and S. Moreover, denoting M + 1 the (variable) number of clusters in the percolation configuration, and defining A = {M ≥ 1, l 0 + l M < n + 2}, we have
Ψ n (b L x b R , A) = n+1 L=2 l≥1,r≥1 r+l=L Ψ n (b L x b R , l 0 = l, l M = r) = n+1 r+l=L=2 r,l≥1 n+2−L k=1 l1,...,l k ≥1 li=n+2−L Ψ(b L ) I∼(l) l−1 i=1 ∆ Ii (x i | b L x i−1 1 ) × × k j=1 I∼(1,lj ) lj i=1 ∆ Ii x l+l1···+lj−1+i x l+l1+···+lj−1+i−1 l+l1···+lj−1+1 × × I∼(1,r) ∆ Ir b R x l+···+l k +r−1 l+···+l k +1 r−1 i=1 ∆ Ii x l+···+l k +i x l+···+l k +i−1 l+···+l k +1 = n+1 r+l=L=2 r,l≥1 n+2−L k=1 l1,...,l k ≥1 i li=n+2−L ρ L b L x l−1 1 k j=1 p x l+···+lj−1+lj l+···+lj−1+1 × × ρ R x l+···+l k +r−1 l+···+l k +1 b R .(33)
C.3. Decoupling of Random Sequences. We now present a factorisation result for weakly coupled measures. We always see Ψ n as a measure on (B L ×A n ×B R )×I n (with the discrete σ-algebra) as the percolation picture is induced by the memory values (I ∈ I n ) and thus all weights that we consider can be expressed as sums of weight of elements in (B L × A n × B R ) × I n .
The idea being to approximate Ψ n by a factorized measure, we introduce the product measure P = p Z>0 , X = (X 1 , X 2 , . . . ) and B L/R sequences "sampled" from P, ρ L/R (one can just think as if they were random variables, and look at them as a convenient way of defining certain sets). Then define R L = {∃k ≥ 1 :
k i=1 |X i | = L} and Ξ = ρ L × P × ρ R , (34) Ξ n ( · ) = Ξ( · , |B L | + |B R | < n, R n−|B L |−|B R | ),(35)
where Ξ n is understood as a measure on (B L , A n , B R ) × C n . Percolation estimates and the construction described in the previous section allow one to prove the following result.
Lemma C.1. Let (Ψ n ) n and Ψ be as described in Section C.1 and such that Conditions (H1), (H2), (H3) and (H4) are satisfied. Let ρ L , ρ R , p be defined by (32). Then, p is a probability measure and (i) there exist C 2 , c 23 > 0 such that
ρ R (|B R | = l) ∨ ρ L (|B L | = l) ∨ p(|X| = l) ≤ C 2 e −c23l ;
(ii) there exist C 3 , c 24 > 0 such that, for any f :
B L × A n × B R → R bounded, f (y)Ψ n (y) − f (y)Ξ n (y) ≤ f ∞ C 3 e −c24n ,
where the sum is over y ∈ B L × A n × B R .
Proof. We start by showing Item (i), as the second point follows from it and (33).
To lighten notations, we will use the notation s n k ≡ (s i ) n i=k for any kind of sequence (not just words) and write s n k =s n k instead of: s i =s i for i = k, k +1, ..., n. First observe that, using (H2) and the definition of a n , a n (s | x n 1 )
Ψ(s | b x n 1 ) − 1 = 1 − a n (s | x n 1 ) Ψ(s | b x n 1 ) ≤ e −cn ,(36)
uniformly in b ∈ B L and s ∈ A (or s ∈ B R ) whenever n ≥ L 0 . Thus, for any b ∈ B L and x ∈ A n , n ≥ L 0 ,
s∈A Ψ(s | b x n 1 ) − a n (s | x n 1 ) ≤ e −cn s∈A Ψ(s | b x n 1 ).
This and the fact that a n ≥ a 0 ≥ 0 imply that s∈A a n (s | x n 1 )
s∈A Ψ(s | b x n 1 ) ≥ 1 − e −cn if n ≥ L 0 , 0 K if n < L 0 .(37)
We can then use (36) to obtain a "uniform" exponential decay estimate on I k : for L 0 ≤ l < k,
Ψ n (I k ≤ l, I n+1 k+1 = r n+1 k+1 ) = = b L ,b R ,x n 1 Ψ(b L x k−1 1 )a l (x k | b L x k−1 1 ) n i=k+1 ∆ ri (x i | b L x i−1 1 ) ≥ b L ,b R ,x n 1 Ψ(b L x k 1 )(1 − e −cl ) n i=k+1 ∆ ri (x i | b L x i−1 1 ) = (1 − e −cl )Ψ n (I n+1 k+1 = r n+1 k+1 ).(38)
For l ≤ L 0 , the same computation and (37) gives Ψ n (I k ≤ l, I n+1 k+1 = r n+1 k+1 ) ≥ 0 Ψ n (I n+1 k+1 = r n+1 k+1 ). Reformulating, one has
Ψ n (I k > l, I n+1 k+1 = r n+1 k+1 ) Ψ n (I n+1 k+1 = r n+1 k+1 ) ≤ e −cl if l ≥ L 0 , 1 − 0 K if l < L 0 .(39)
For 0 < i ≤ j ≤ n + 1, let X [i,j] = max{i − m + I m : i ≤ m ≤ j} be "the distance reached by [i, j]"; note that it is nonnegative. Doing (almost) the same computation as in (38), one obtains the following Claim C.1. There exist c > 0, L 1 ≥ 0 such that,
Ψ n (X [i,j] > l, I n+1 j+1 = r n+1 j+1 ) Ψ n (I n+1 j+1 = r n+1 j+1 ) ≤ e −cl if l ≥ L 1 , 1 − 0 K L1 if l < L 1 ,
uniformly in i, j and r n+1 j+1 . In particular, there exist C ≥ 0, c > 0 such that:
Ψ n (X [i,j] > l, I n+1 j+1 = r n+1 j+1 ) Ψ n (I n+1 j+1 = r n+1 j+1 ) ≤ Ce −cl .
We will also need a uniform cut estimate.
Claim C.2. There exists > 0 such that
Ψ n (X [i,j] = 0, I n+1 j+1 = r n+1 j+1 ) Ψ n (I n+1 j+1 = r n+1 j+1 ) ≥ ,
uniformly in i, j and r n+1 j+1 .
Proof. Proceeding as in (38),
Ψ n (X [i,j] = 0, I n+1 j+1 = r n+1 j+1 ) = Ψ n (I i+k ≤ k, 0 ≤ k ≤ j − i, I n+1 j+1 = r n+1 j+1 ) ≥ 0 K L0 j−i k=L0 (1 − e −ck )Ψ n (I n+1 j+1 = r n+1 j+1 ) ≥ Ψ n (I n+1 j+1 = r n+1 j+1 ) 0 K L0 ∞ k=1
(1 − e −ck ) = Ψ n (I n+1 j+1 = r n+1 j+1 ), since the infinite product converges.
We now use Claims C.1 and C.2 to implement an exploration argument which will imply that having no cuts in a long interval carries an exponentially small measure. This in turn implies exponential decay of ρ L , ρ R and p (Item (i)). Fix l, n large enough and m ∈ [0, n
+ 1] ∩ Z such that [m − l, m] ⊂ [0, n + 1]. Define D m (l) = {[m − l, m] ∩ Cuts = ∅}.
We want to prove the following Claim C.3. There exist L 2 ≥ 0 and c > 0 such that, for all l ≥ L 2 and n ≥ m ≥ l,
Ψ n (D m (l)) ≤ e −cl .
Proof. The idea is the following: look at the furthest point reached by [m, n + 1]; call it i 1 . With measure at least , e i1−1 is a cut. If not, look at the furthest point reached by [i 1 , n + 1], and so on and so forth. To make this precise, we introduce
Y 1 = X [m,n+1] , Y 2 = X [m−Y1,m−1] , Y 3 = X [m−Y1−Y2,m−Y1−1] , . . . , S k = k i=1 Y i , so that Y k = X [m−S k−1 ,m−S k−2 −1] , T = min{k : S k ≥ l}.
All these quantities are functions of the memory configuration I. Now, for any 1 > δ > 0, Ψ n (D m (l)) = Ψ n (D m (l), T ≥ δl) + Ψ n (D m (l), T < δl)
≤ (1 − ) δl + Ψ n (T < δl),
via the uniformity in Claim C.2. Finally, for t > 0,
Ψ n e tS δl = ∞ k1,...,k δl =0 e t δl i=1 ki Ψ n (Y 1 = k 1 , ..., Y δl = k δl ) ≤ ∞ k1,...,k δl =0 e t δl i=1 ki δl i=1 Ce −cki = C 1 − e −c/2 δl ,
for any t ≤ c/2, where we used the uniform exponential decay property of the Y i s (Claim C.1) in the inequality. This gives:
Ψ n (T < δl) = Ψ n (S δl > l) ≤ e −cl
for some c > 0 and δ small enough, via the application of the exponential version of Markov inequality.
With (33), Claim C.3 implies exponential decay of ρ L , ρ R and p (item (i)), as well as the bound Ψ n (A) ≤ e −cn (where A is defined just above (33)).
To conclude the proof of Lemma C.1, we must still establish Item (ii) and show that p is indeed a probability measure. We start with the latter. As p is a positive measure, we have to prove that x∈S * p(x) = 1. This will be done using a standard renewal argument. We will need the weights
ρ L n = b L ∈B L ,x∈A n−1 ρ L (b L x), ρ R n = b R ∈B R ,x∈A n−1 ρ L (x b R ), ν n = x∈A n p(x),
and the associated generating functions (recall
µ n = b L ,b R x∈A n Ψ n (b L x b R ) from (H3)) A(z) = ∞ n=1 µ n z n , B(z) = ∞ n=1 ν n z n , C L (z) = ∞ n=0 ρ L n z n , C R (z) = ∞ n=0 ρ R n z n . Since x∈S * p(x) = ∞ n=1 x∈A n p(x) = ∞ n=1 ν n = B(1),
we only need to show that B(1) = 1. We will deduce this from a functional equation satisfied by the previously introduced generating functions:
A(z) = ∞ n=1 z n b L ,b R x∈A n Ψ n (b L x b R ) = ∞ n=1 z n Ψ n (A c ) + ∞ n=1 z n b L ,b R x∈A n Ψ n (b L x b R , A) = g A c (z) + ∞ k=1 ∞ n=k l,r,l1,...,l k ≥1 l+r+ li=n+2 ρ L l−1 z l−1 ρ R r−1 z r−1 k i=1 ν li z li = g A c (z) + ∞ k=1 C L (z)B(z) k C R (z) = g A c (z) − C L (z)C R (z) + C L (z)C R (z) 1 − B(z) .(40)
Now, denoting r g Ac , r A , r B , r C L and r C R the radii of convergence of, respectively, g Ac , A, B, C L and C R , (i) and the exponential decay of Ψ n (A c ) imply that
r g Ac > 1, r B > 1, r C L > 1, r C R > 1.
Furthermore, Properties (H3) implies that
r A = 1.
Together with (40), this yields B(r A ) = 1 and thus B(1) = 1. Since all notations and estimates are provided here, we prove a few technical points which are not directly useful in this paper but which might be of use in later investigations.
Lemma C.2. Under the assumption of Lemma C.1, there exists > 0 such that
Ψ n (b L x k 1 , (k, k + 1) ∈ Cuts) Ψ n (b L x k 1 ) ≥ ,
uniformly in b L , k ≥ 0 and x k 1 . Proof. This follows form the same computation as in the argument leading to Claim C.2 in the proof of Lemma C.1. Proof. By definition of p, p((s 0 )) = a 0 (s 0 ) = inf b∈B L Ψ(s 0 | b) ≥ δ.
Before ending this sub-section, we observe two facts about the boundary pieces:
Remark C.1.
• The same argument as in Lemma C.3 gives the same result for p replaced by ρ R and b R ∈ B R instead of s 0 ∈ A. • If Ψ(b L ) ≥ δ for some b L , then ρ L (b L ) ≥ δ (using the definition of ρ L ).
C.4. Application to Random Walks with Exponentially Decaying Memory. In order to avoid confusion, we continue to use | · | for the length of a sequence and use · for the norm in R d . We now apply results of the previous section to the setting where B L , S, B R come equipped with a displacement application: i.e. a function V : B L ∪ B R ∪ S → Z d . This naturally induces an applicationṼ from B L ×S n ×B R to the space of trajectories of n+2-steps random walk in Z d : denoting
V (x n 1 ) = n i=1 V (x i ) (and similarly for b L x b R , etc.), V u (b L x n 1 b R ) = u, u + V (b L ), u + V (b L x 1 1 ), . . . , u + V (b L x n 1 b R ) ,
where u ∈ Z d is the starting point of the trajectory. We continue to denote Ψ n the push-forward of Ψ n byṼ . It will be convenient to denotẽ
S = (S m ) n+1 m=0 ≡S(u, b L x n 1 b R ) =Ṽ u (b L x n 1 b R ).
In turn,Ṽ induces an applicationV from (B L × S n × B R , C n ) to the trajectories of random walk with ≤ n + 2 steps via (denote y n+2
1 = b L x n 1 b R )
V u (y, (l 0 , . . . , l k )) = (u, u + V (y l0 1 ), u + V (y l0+l1
Ψ n S = 0 d , (v L , v ⊥ L ), (v 1 , v ⊥ 1 ), . . . , (v R , v ⊥ R ) = Ψ n S = 0 d , (v L , −v ⊥ L ), (v 1 , −v ⊥ 1 ), . . . , (v R , −v ⊥ R ) .
Recall that S * = ∞ k=1 A n and see the beginning of Section C.3 for the definitions of P, X, R L , B L/R , Ξ and Ξ n . Denote the push-forward of Ξ (resp. Ξ n ) by Ξ traj (resp. Ξ traj n ). Notice that, by construction, Ξ traj n and Ψ traj n are both measures on Traj n = n k=1 (Z d ) 2+k . Recall ρ L , ρ R and p defined in the previous section. We use the same notations for their push-forward. The goal of this section is to prove the following Theorem C.4. If Ψ satisfies hypotheses (H1), (H2), (H3) and (H4), and Property (P1), then p is a probability measure on Z d and:
(1) there exist c > 0, C ≥ 0 such that, for all n and any bounded f : Traj n → R,
v∈Traj n f (v)Ψ traj n (v) − v∈Traj n f (v)Ξ traj n (v) ≤ f ∞ Ce −cn .
(2) Let X be a random variable with law p and write X = V (X) ∈ Z d (and define similarly B L , B R from B L , B R ). There exist c > 0, C ≥ 0 such that p( X = l) ≤ Ce −cl and ρ L/R ( B L/R = l) ≤ Ce −cl .
Given S 0 ∈ Z d and an i.i.d. sequence X 1 = V (X 1 ), X 2 = V (X 2 ), . . . (with X i ∼ p), denote
S k = S 0 + k i=1 X i
and write S = (S k ) k≥0 and S ⊥ = (S ⊥ k ) k≥0 . Under (P2) we have: • if (P3) is satisfied, S ⊥ and S are aperiodic, • if (P3) and (P4) are satisfied, S ⊥ is also irreducible and S can attain every k > S 0 with positive probability. • If (P5) is satisfied, p(X ⊥ 1 = u) = p(X ⊥ 1 = −u) and P (0,u) ∃n : S n = (L, v) = P (0,v) ∃n : S n = (L, u) , where P (k,u) denotes the law of S for S 0 = (k, u).
Before starting the proof, we make a small remark on the boundary conditions:
Remark C.2. By Theorem C.4, for n large enough, Ψ n ( · , b L = (0, 0), b R = (0, 0)) ≥ 1 2 Ξ( · , B L = (0, 0), B R = (0, 0)). If Ψ((0, 0)) ≥ δ 1 > 0 and a 0 ((0, 0)) ≥ δ 2 > 0, then one can apply Remark C.1 to obtain: Ψ n (·, b L = (0, 0), b R = (0, 0)) ≥ δ 1 δ 2 2 P (0,0) ·, ∃k : S k = (n, 0) .
Proof. Using hypotheses (H1), (H2), (H3) and (H4), we can deduce from Lemma C.1 that • ρ L , ρ R are positive measures on S * with finite total mass and satisfying ρ L/R (|B L/R | = l) ≤ Ce −cl ;
• p is a probability measure on S * satisfying p(|X| = l) ≤ Ce −c26l ;
• Item 1 holds. We thus only need to deduce exponential decay in · from exponential decay in |·|. Let us denote expectation under p by E. Exponential decay in · follows from Claim C.4. There exists t 0 > 0 such that E e t X < ∞, for all t ≤ t 0 .
Proof. We can assume, without loss of generality, that t ≥ 0. In that case, provided that e 2tm0 + e −(c 25 −2t)m 0 1−e −(c 25 −2t) e −c26 < 1, which is true for t ∈ [0, t 0 ) for some t 0 = t 0 (m 0 ) > 0, once m 0 is chosen large enough.
E e t X ≤ ∞ n=1 E e t n i=1 V (Xi) 1 {|X|=n} ≤ ∞ n=1 E e 2t n i=1 Xi E 1 {|X|=n} ,(41)
The previous argument extends easily to obtain exponential decay in · under ρ L/R . We now turn to the additional properties. The aperiodicity of S ⊥ follows immediately from (P3), since the latter gives p(X ⊥ = 0) ≥ 1 > 0. The aperiodicity of S is done identically and so is the irreducibility of S ⊥ under (P4).
The symmetry is slightly less obvious. Start by observing that, by definition of a k (s|b L x n 1 ) and ∆ k (s|b L x n 1 ), (P5) implies that a k ((v , v ⊥ ) | b L x n 1 ) = s∈A:
V (s)=(v ,v ⊥ ) a k (s | b L x n 1 ) = s∈A: V (s)=(v ,−v ⊥ ) a k (s | b L x n 1 ) = a k ((v , −v ⊥ ) | b L x n 1 ),
and, thus,
∆ k ((v , v ⊥ )|b L x n 1 ) = ∆ k ((v , −v ⊥ )|b L x n 1 )
. Using this in the expansion of p V (X) = (v , v ⊥ ) as k v1+···+v k =v xi:V (xi)=vi p(X = (x 1 , . . . , x k )), and using the definition of p, one straightforwardly obtains
p V (X) = (v , v ⊥ ) = p V (X) = (v , −v ⊥ ) .
Finally, for L > 0 and u, v ∈ Z d−1 , P (0,u) ∃n : S n = (L, v) = P (0,v) ∃n : S n = (L, u) follows by summing over possible trajectories and applying the previous symmetry result.
Figure 1 . 2 .
12Left: The interface of the 2d Potts model. Right:
Figure 1 . 3 .
13Interface in the 2d 4-state Potts model. Left: J = 1. Right: J = 1/2. The same inverse temperature β > β c is used in both cases.
Theorem 1 . 4 .
14For any β > β c and any 0 ≤ J < 1, there exists C β,J = C β,J n (x) ≤ C β,J log n, min x Γ − n (x) ≥ −C β,J log n = 1.
1. 3 .
3Open problems. In view of the results presented above, there remain a few interesting open problems: • Determine the behavior of ξ β (J) in the neighborhood of J c in dimensions d ≥ 4. By analogy with the results for effective models (see [14, Theorem 2.1]), we conjecture that the qualitative behavior of ξ β (J c ) − ξ β (J) as J ↓ J c is as follows: Θ((J − J c ) 2 ) when d = 4, Θ((J − J c )/| log(J − J c )|) when d = 5 and Θ(J − J c ) when d ≥ 6.
Figure 3 . 1 .
31Coarse-graining of F .
Figure 3 . 2 .
32Splitting of [L] 2K into boxes. The four covered boxes are darker (only the relevant clusters of F are drawn).
Figure 4 . 1 .
41Example of clusters in B L , B R and A. The corresponding values D are depicted as vectors.
Figure 4 . 2 .
42The decomposition of the common cluster of 0 and n e 1 into irreducible pieces.
Proposition 4 . 1 .
41Properties (H1), (H2), (H3), (H4), (P1), (P2), (P3), (P4) and (P5) of Appendix C all hold in the present setting. Proof. (H1), (H2) and (P1) are direct consequences of [5, Theorem 2.2]. (P2) and (P5) are obvious. (H4) and (P3) hold by finite energy, since the edge {(0, 0 d−1 ), (1, 0 d−1 )} belongs to A; we can argue similarly for (P4). Let us check (H3).
random-cluster measure with probabilities modified to x only on L [0,n] .
Figure 6 . 1 .
61Left: The set Y ψ (v). Right: The type of connections we want to prevent. Now,
Now, denote by N n/2 = max{k ≤ n : S k ≤ n/2} the number of steps before exiting[0, n 2 e 1 ] × Z d−1 and by L ⊥ ( ) = #{0 ≤ i ≤ : S ⊥ i = 0} the local time at 0 of S ⊥ up to time . Writingn = n 4E[X 1 ]
7. 1 .
1Coarse-graining. We first describe our coarse-graining procedure. Fix a scale K and a number r (both to be chosen later, independent of n) and define∆(v) = −K, K × −2K − 3r log(K), 2K + 3r log(K) d−1 + v, ∆(v) = −K, K × −2K − 4r log(K), 2K + 4r log(K) d−1 + v,where K = K + r log(K).
Figure 7 . 1 .
71Elementary piece of the coarse-graining.
Lemma 7 . 1 .
71The probability of a given tree T with m L-free vertices and N Lvertices satisfiesP x (T 0 = T ) ≤ e −ξKm e −ξ x KN (1 + o K (1)) m+N .
Figure 7 . 2 .
72Connection to the boundary of ∆. we have (with c 17 > 0 a constant depending on the dimension): P x (T 0 contains at least ρn/K L-free vert., 0 ↔ n e 1 ) (T 0 contains m L-free v. and N L-vert.)
Figure 7 . 3 .
73Evolution of T 0 shade. Now, we split 0, n × Z d−1 into slices. For i = 1, 2, . . . , n/7K, define
Given the restriction of a configuration ω| [L] c 3K outside [L] 3K such that B i is illuminated, let v L and v R be the leftmost and rightmost vertices of L [(i−1)7K+3K,i7K−3K] not shaded by C 0 ∩ [L] c 3K . Notice that, by definition of the shade, the whole segment [v L , v R ] is not in the shade of C 0 ∩ [L] c 3K . Denote by A v L ,v R the event that all edges of [v L , v R ] are open, all edges inside (v L − Y) ∩ B i and (v R + Y) ∩ B i are open and the remaining edges of B i are closed (see Figure 7.4).
Figure 7 . 4 .
74Local surgery procedure to create cone-points. The edges between v L and v R and those in the shaded regions are all open, those in the white regions are all closed. By finite energy, min ω| [L] c 3K P x (A v L ,v R | 0 ↔ n e 1 ) ≥ θ > 0 where the minimum is taken over configurations such that B i is illuminated. Thus, a positive density of illuminated B i 's contain cone-points and, by (23), a positive density of B i 's are illuminated, thereby completing the proof of Theorem 7.2.
Figure 7 . 5 .
75The original process based on the cone-points of C 0 on the line L, together with the associated process of independent pieces.
xo
(0 ↔ n e 1 ) (where P (n) x denotes the measure with only L [0,n] weights modified). L [0,n] exists and is analytic in x in a small neighbourhood of x 0 .
n] 1 {Cp(L [0,n] )> 0 n} 0 ↔ n e 1 (1 + o(1)), where Cp(L [0,n] ) is the number of cone-points for the cluster of L [0,n] (the union of all the clusters containing at least one vertex of L [0,n] ) lying on L [0,n] . Let ϑ b , ϑ 1 , .. . , ϑ m , ϑ f be the (random) sequence of diamonds-confined L [0,n] -clusters (that is, the irreducible, in the sense of Section 7.3, cone-confined pieces of the cluster of L [0,n] ). We write D 1 , . . . , D m the sequence of diamonds containing ϑ 1 , . . . , ϑ m and |D i | ≥ 1 the e 1 displacement (or length) of D i . We stress at this point that ϑ i contains the whole information about all the clusters touching L ∩ D i . We also denote D b , D f the diamonds containing ϑ b , ϑ f (their left (resp. right) endpoints
measure with modified weights on L [−M,n+M ] . The first inequality is by monotonicity (and P x (· | ∂C 0 closed) = P (−M,n+M ) x (· | ∂C 0 closed) as the interior of ∂C 0 does not contain edges from L <−M or L >n+M ). Putting together (27),(28), (29), and choosing M large enough so that C x e −M ξ x ≤ 1 2 we obtain:
E
(ω). At step n, sample the smallest edge e n ∈ ∂C (n−1) E (ω) as follows: sample U n ∼ Unif([0, 1]) and set ω en = 1 {Un≤P x,q (· | ωe 1 ,...,ωe n−1 )} and η en = 1 {Un≤P y,q (· | ηe 1 ,...,ηe n−1 )} .
Figure C. 1 .
1Left stick-percolation configuration and the cut process.
and a k (s | b x) = a ∞ (s | b x) = Ψ(s | b x) for k > n. Under Assumption (H1), all those numbers are in [0, K] for all k and a k (s | b x), a k (b x)
Figure C. 2 .
2Sequence with memory.
Finally, we prove Item (ii). For f : B L × A n × B R → R bounded, f (y)Ψ n (y) − f (y)Ξ n (y) ≤ f (y)Ψ n (y, A) − f (y)Ξ n (y) + f ∞ Ψ n (A c ) ≤ f ∞ Ce −cnby exponential decay of Ψ n (A c ) (implied by Claim C.3) and equation (33).
Lemma C. 3 .
3If there exist δ > 0 and an element s 0 ∈ A such thatΨ(s 0 | b) ≥ δ,for all b ∈ B L , then p X = (s 0 ) ≥ δ.
S
= (S m ) k+1 m=0 ≡S(u, b L x n 1 b R , (l 0 , . . . , l k )) =V u (b L x n 1 b R , (l 0 , . . . , l k )).The goal of this section is to give properties of the push-forward measure of Ψ n bȳ V , denoted Ψ traj n , under hypotheses (H1), (H2), (H3) and (H4) and some additional properties, namely:(P1) there exist C, c > 0 such thatΨ( V (x) = l | b) ≤ Ce −cl uniformly in b (in particular, there exists c 25 > 0 such that Ψ( V (x) = l | b) ≤ e −c25l for l large enough); (P2) directedness: V (x) · e 1 > 0 for all x ∈ B L ∪ B R ∪ S. In this case, it makes sense to distinguish the displacement along the first coordinate from the others and to denoteS k = (S k ,S ⊥ k ) ∈ Z × Z d−1 , and similarly forS; (P3) aperiodicity: there exists s 0 ∈ S with V (s 0 ) = e 1 such that Ψ(s | b) ≥ 1 > 0 uniformly over b ∈ B; (P4) irreducibility: there exist r > 0 and s i ∈ S, i = 1, . . . , d − 1, with V (s i ) = (r, q i ) such that Ψ(s i | b) ≥ 2 > 0 uniformly over b ∈ B, where (q i ) j = 1 {i=j} , j = 1, . . . , d − 1; (P5) trajectory symmetry (under (P2), starting point u = 0):
,...,n} A={a1,...,a k }e 2tm0(n−k) E e 2t k i=1 Xa i 1 { Xa i ≥m0,i=1,..enough m 0 (the first inequality holds for any m 0 > 0, but we need m 0 to be large enough to use (P1) (uniform exponential decay of the steps) in the second inequality). Plugging this into (
This follows, for example, from our analysis below; see Remark 3.2.
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
The authors thank the two referees for their careful reading and their questions and comments that substantially improved the readability of this work. They also acknowledge the support of the Swiss National Science Foundation through the NCCR SwissMAP.
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CH-1211 Genève, Switzerland E-mail address: [email protected] Section de Mathématiques. CH-1211GenèveSection de Mathématiques, Université de Genève ; Université de GenèveSwitzerland E-mail address: [email protected] de Mathématiques, Université de Genève, CH-1211 Genève, Switzerland E-mail address: [email protected] Section de Mathématiques, Université de Genève, CH-1211 Genève, Switzerland E-mail address: [email protected]
| [] |
[
"Quantum fields in Bianchi type I spacetimes. The Kasner metrc",
"Quantum fields in Bianchi type I spacetimes. The Kasner metrc"
] | [
"Jerzy Matyjasek \nInstitute of Physics\nMaria Curie-Sk lodowska University pl. Marii Curie-Sk lodowskiej 120-031LublinPoland\n"
] | [
"Institute of Physics\nMaria Curie-Sk lodowska University pl. Marii Curie-Sk lodowskiej 120-031LublinPoland"
] | [] | Vacuum polarization of the quantized massive fields in Bianchi type I spacetime is investigated from the point of view of the adiabatic approximation and the Schwinger-DeWitt method. It is shown that both approaches give the same results that can be used in construction of the trace of the stress-energy tensor of the conformally coupled fields. The stress-energy tensor is calculated in the Bianchi type I spacetime and the back reaction of the quantized fields upon the Kasner geometry is studied. A special emphasis is put on the problem of isotropization, studied with the aid of the directional Hubble parameters.Similarities with the quantum corrected interior of the Schwarzschild black hole is briefly discussed. | 10.1103/physrevd.98.104054 | [
"https://arxiv.org/pdf/1811.00993v1.pdf"
] | 119,021,214 | 1811.00993 | 50917409ed5c0e1c48581c4bd37fa7533af29b2a |
Quantum fields in Bianchi type I spacetimes. The Kasner metrc
Jerzy Matyjasek
Institute of Physics
Maria Curie-Sk lodowska University pl. Marii Curie-Sk lodowskiej 120-031LublinPoland
Quantum fields in Bianchi type I spacetimes. The Kasner metrc
PACS numbers: 0462+v,0470-s
Vacuum polarization of the quantized massive fields in Bianchi type I spacetime is investigated from the point of view of the adiabatic approximation and the Schwinger-DeWitt method. It is shown that both approaches give the same results that can be used in construction of the trace of the stress-energy tensor of the conformally coupled fields. The stress-energy tensor is calculated in the Bianchi type I spacetime and the back reaction of the quantized fields upon the Kasner geometry is studied. A special emphasis is put on the problem of isotropization, studied with the aid of the directional Hubble parameters.Similarities with the quantum corrected interior of the Schwarzschild black hole is briefly discussed.
I. INTRODUCTION
In this paper we shall consider quantized massive fields in homogeneous anisotropic cosmological Bianchi type I models described by the line element
ds 2 = −dt 2 + a 2 (t)dx 2 + b 2 (t)dy 2 + c 2 (t)dz 2 ,(1)
where a, b and c, the directional scale factors, are functions of time. A special emphasis will be put on the Kasner spacetime [1][2][3]
ds 2 = −dt 2 + t 2p 1 dx 2 + t 2p 2 dy 2 + t 2p 3 dz 2 ,(2)
where the parameters p 1 , p 2 and p 3 satisfy
p 1 + p 2 + p 3 = p 2 1 + p 2 2 + p 2 3 = 1.(3)
These conditions define a Kasner plane and a Kasner sphere (see Fig 1). < p 1 < 0 < p 2 < 2/3 < p 3 < 1 (4) and the comoving volume element expands in two directions and compresses in one. In what follows however, we shall take p 1 as an independent parameter and express all remaining quantities in terms of it. It can be done easily since
p 3 = 1 − p 1 − p 2(5)
and
p 2 = 1 2 1 − p 1 − 1 + 2p 1 − 3p 2 1(6)
for the lower branch, and
p 2 = 1 2 1 − p 1 + 1 + 2p 1 − 3p 2 1(7)
for the upper branch. This terminology is self-explanatory if we consider the parameter p 2 as a function of p 1 (see Fig. 2). (For different parametrizations see Ref. [4]). There is an interesting relation between the Kasner metric and the metric that describes the closest vicinity of the Schwarzschild central singularity. Indeed, it can be shown that the Schwarzschild interior approaches the Kasner metric with, say, p 1 = −1/3 and p 2 = p 3 = 2/3 as r goes to 0 (see e.g. Ref. [5][6][7]).
The physical content of the quantum field theory in curved background is encoded in the regularized stress-energy tensor, T b a , and to certain extend in the field fluctuation, φ 2 . In this formulation the spacetime is treated classically whereas the fields propagating on it are quantized.
Even with this simplified approach the semiclassical theory should be able to describe quite a number of interesting phenomena, such as vacuum polarization, particle creation and the influence of the quantized field upon the background geometry [8][9][10][11]. Moreover, one expects that the results obtained within the framework of the quantum field theory in curved background remain accurate as long as the quantum gravity effects are negligible.
Ideally, the stress-energy tensor should depend functionally on a general metric or at least on a wide class of metrics and be related to the non-local one-loop effective action, W R , in a standard way. Unfortunately, such calculations are very hard (if not impossible) in practice. Indeed, the solutions of the field equations are not expressible in terms of the known special functions, the formal products of the operator valued distributions have to be regularized and the (perturbative) series are divergent. All this makes the exact analytical calculations practically impossible and to circumvent these problems one is forced either to refer to the numerical methods or to make use of some approximations.
In this note we shall follow the latter approach and make use of the local one-loop effective action constructed within the framework of the Schwinger-DeWitt approximation [12][13][14], (see also Ref. [15]). In cosmology, however, there is another powerful approach to the problem, namely the adiabatic approximation [16][17][18][19][20][21][22][23][24][25][26][27] and closely related n-wave method [15,[28][29][30]. For the Robertson-Walker spacetime it has been shown that regardless of the chosen method, the results of the calculations are identical. It has been demonstrated that the Schwinger-DeWitt approach and the adiabatic method give precisely the same result in this context for 4 ≤ D ≤ 8. (See [31][32][33]). Building on this we expect that a similar correspondence also appears in the anisotropic homogeneous cosmologies. Although we do not attempt to perform the full calculations of the stress-energy tensor within the framework of the adiabatic approach and show that the both methods yield the same result, here we will solve somewhat simpler problem and demonstrate that this equality holds for the vacuum polarization, φ 2 , of the massive scalar field with arbitrary curvature coupling in the Bianchi type I cosmology. Specifically, it will be shown that the leading and the next-to-leading term of the approximate vacuum polarization calculated within the framework of the adiabatic approximation are precisely the same as the analogous terms calculated with the aid of the Schwinger-DeWitt approach. Interestingly, using our vacuum polarization results, we will be able to calculate the trace of the stress-energy tensor of the conformally coupled massive scalar field.
The paper is organized as follows. In section II we construct the leading and the next-toleading term of the approximate vacuum polarization of the massive scalar field in the Bianchi type I spacetime. In section III the stress-energy tensor of the scalar, spinor and vector fields in the Kasner spacetime is calculated and discussed, whereas in Sec. IV we study the back reaction of the quantized fields upon the background geometry. To the best of our knowledge the results of Sections II-IV are essentially new. Throughout the paper the natural units are chosen and we follow the Misner, Thorne and Wheeler conventions.
II. VACUUM POLARIZATION
In this section we will be concerned with the neutral massive scalar field
− 2φ + (m 2 + ξR)φ = 0,(8)
with the arbitrary curvature coupling, ξ, in the anisotropic Bianchi type I specetime. Our main task is to construct the vacuum polarization.
A. Adiabatic approximation
To simplify calculations we will introduce a new time coordinate [28,34]
η = t V −1/3 dt ,(9)
where V = abc, and redefine the field putting
f = V 1/3 φ.(10)
The solution of the transformed equation can be written in the form
f = 1 (2π) 3/2 d 3 k A k f k (η)e ikx + A † k f k (η)e −ikx ,(11)
where f k (η) satisfies
f k (η) + Ω 2 + Q + Q 1 f k (η) = 0(12)
with
Ω 2 = V 2/3 m 2 + k 2 1 a 2 + k 2 2 b 2 + k 2 3 c 2 ,(13)Q = 1 3 1 3 − ξ a a − b b 2 + a a − c c 2 + b b − c c 2(14)
and
Q 1 = 2 ξ − 1 6 a a + b b + c c .(15)
The A k obey the standard commutation relations
[A k , A k ] = [A † k , A † k ] = 0(16)
and
[A k , A † k ] = δ(k − k ),(17)
provided the functions f k satisfy the Wronskian condition
f k (η)f k (η) − f k (η)f k (η) = i.(18)
The ground state is defined by the relation A k |0 = 0 (19) and the formal (divergent) expression for the vacuum polarization has the following simple form
φ 2 = 1 (2π) 3 V 2/3 d 3 k|f k | 2 .(20)
Now, we demand that the positive frequency functions f k can be written in the form
f k = 1 (2V 1/3 W ) 1/2 exp −i η V 1/3 (s)W (s)ds ,(21)
where W (η) satisfies the following differential equation
Ω 2 + Q + Q 1 − V 2/3 W 2 + 7V 2 36V 2 + V W 6V W + 3W 2 4W 2 − V 6V − W 2W = 0.(22)
The solution of this equation can be constructed iteratively assuming that the functions W (η) (we have omitted the subscript k) can be expanded as
W = ω 0 + ω 2 + ω 4 + ...(23)
with the zeroth-order solution given by
ω 0 = m 2 + k 2 1 a 2 + k 2 2 b 2 + k 2 3 c 2 1/2 .(24)
Now, in order to make our calculations more systematic and transparent, we shall introduce a dimensionless parameter ε that will help to determine the adiabatic order of the complicated expressions:
d dη → ε d dη and W = j=0 ε 2j ω 2j ,(25)
and, after the substitution of (23) into (22), we collect the terms with the like powers of ε. It should be noted that ω 2 is of the second adiabatic order, ω 4 is of the fourth adiabatic order, and so forth. Solving the chain of the algebraic equations of ascending complexity and substituting the thus obtained W into the formal expression for the vacuum polarization (20) one obtains
φ 2 = 1 2(2π) 2 V d 3 k ω 0 − ε 2 2(2π) 2 V d 3 k ω 2 ω 2 0 + ε 4 2(2π) 2 V d 3 k ω 2 2 ω 3 0 − ω 4 ω 2 0 − ε 6 2(2π) 2 V d 3 k ω 3 2 ω 4 0 − 2 ω 2 ω 4 ω 3 0 + ω 6 ω 2 0 + ...(26)
The first integral is divergent whereas the second one contains the terms that are divergent. Starting from the third term in the right hand side of the formal expression for the vacuum polarization the integrals are finite and their computation in the anisotropic case presents no problems. Now, following the standard prescription [8] we subtract from the formal φ 2 the first two terms, i.e., we subtract all the terms of a given adiabatic order if at least one of them is divergent.
Thus far our analysis has been formal. Now, let us investigate when the adopted method can
give well defined functions ω n . In order to make our analysis more precise let us introduce the directional Hubble parameters
H a = a a V 1/3 , H b = b b V 1/3 and H c = c c V 1/3(27)
and observe that ω 0 is much smaller than ω 2 provided H i /m 1 where i = a, b, c. Moreover, in this regime one has
a (n) m n aV n/3 ∼ H a m n ,(28)
where a (n) denotes n-th derivative of a(η). Since similar relations for the remaining directional scale factors b and c hold, the magnitude of the terms at the n-th adiabatic order are therefore equal to
H a m α H b m β H c m γ ,(29)
with α + β + γ = n, where α, β, γ are nonnegative integers. One expects that in this regime the particle creation can be safely ignored.
Although the ε 4 -term (i.e., the leading term) looks innocent it leads to the quite complicated final result. Indeed, integration over angles yields 723 terms of the type
F (a, b, c; ξ) dp p q (m 2 + p 2 ) r ,(30)
where
p 2 = k 2 1 a 2 + k 2 2 b 2 + k 2 3 c 2(31)
and the functions F (a, b, c; ξ) are constructed from a, b, c and their derivatives. The number of derivatives in each term at each adiabatic order is constant. Finally, after integration over p one obtains 135 terms. Similarly, the next-to-leading term (i.e. the sixth adiabatic order) consists of 761 terms.
As the vacuum polarization of the massive scalar field with the arbitrary coupling in a general Bianchi type I spacetime is rather complicated we will confine ourselves to the minimal and conformal couplings only. 1 Integrating over p we find
φ 2 ξ=1/6 = φ 2 ξ=0 + ∆,(32)
where
κ φ 2 ξ=0 = − 2a (4) 15a + 2a (3) b 45ab + 2a (3) c 45ac + 13a b 45ab + 8a b c 45abc − a b 2 15ab 2 − a c 2 15ac 2 + 4a 2 9a 2 − 14a b c 2 135abc 2 − 14a 3 b 135a 3 b − 2a 2 b 2 135a 2 b 2 − 14a 3 c 135a 3 c + 10a 4 27a 4 + 4a (3) a 9a 2 + 4a a c 45a 2 c − 16a 2 a 15a 3 + 4b b c 45b 2 c + cycl,(33)κ∆ = a (4) 9a − a (3) b 27ab − a (3) c 27ac − 8a b 27ab − 4a b c 27abc + 2a b 2 27ab 2 + 2a c 2 27ac 2 − 10a 2 27a 2 + 2a 2 b c 27a 2 bc + 8a 3 b 81a 3 b + a 2 b 2 27a 2 b 2 + 8a 3 c 81a 3 c − 25a 4 81a 4 − 10a (3) a 27a 2 − a a c 9a 2 c + 8a 2 a 9a 3 − b b c 9b 2 c + cycl,(34)
κ = 32π 2 m 2 V 4/3 and cycl denotes the terms that should be added after performing cyclic transfor-
mations {a(η), b(η), c(η)} → {b(η), c(η), a(η)} → {c(η), a(η), b(η)}.
The next-to-leading term is too complicated to be presented here. Finally observe that the thus constructed vacuum polarization can easily be expressed as a function of t by a simple transformation of the time coordinate.
Now, let us return to the Kasner spacetime and calculate the first two terms of the vacuum polarization. Because of the simplicity of the metric the result is independent of the coupling constant and reads
φ 2 = 1 180π 2 m 2 t 4 p 2 1 − p 3 1 + 1 16π 2 m 4 t 6 − 8 35 p 2 1 + 8 35 p 3 1 + 1 315 p 4 1 − 2 315 p 5 1 + 1 315 p 6 1 .(35)
As we shall see the second term in the right-hand-side of the above equation (multiplied by m 2 )
is precisely the the main approximation of the trace of the stress-energy tensor of the conformally coupled massive field taken with the minus sign.
B. Schwinger-DeWitt approximation
As is well known the regularized vacuum polarization of the quantized massive scalar field in a large mass limit can be constructed within the framework of the Schwinger-DeWitt method.
Subtracting the terms that are divergent in the coincidence limit of the Schwinger-DeWitt approximation to the Green function, the φ 2 can be written as a series
φ 2 = 1 16π 2 m 2 k=2 a k (m 2 ) k−1 (k − 2)!,(36)
where a k are the coincidence limit of the Hadamard-DeWitt coefficients. The usual criterion for the validity of the approximation is that the Compton length associated with the massive fields is smaller that the characteristic radius of curvature of the spacetime. Note, that one can safely use the conditions H i /m 1 in this regard. The first two coefficients have the form
a 2 = 1 180 R abcd R abcd − 1 180 R ab R ab + 1 6 1 5 − ξ R a ;a + 1 2 1 6 − ξ 2 R 2 ,(37)
and
a 3 = 1 7! b (0) 3 + 1 360 b (ξ) 3 ,(38)
where
b (0) 3 = 35 9 R 3 + 17R ;a R ;a − 2R ab;c R ab;c − 4R ab;c R ac;b + 9R abcd;e R abcd;e −8R c ab;c R ab − 14 3 RR ab R ab + 24R b ab;c R ac − 208 9 R ab R a c R bc + 64 3 R ab R cd R acbd + 16 3 R ab R a cde R becd + 80 9 R abcd R a c e f R bedf + 14 3 RR abcd R abcd + 28RR a ;a + 18R a b ;a b + 12R abcd e ;e R abcd + 44 9 R abcd R ab ef R cdef (39) and b (ξ) 3 = −5R 3 ξ + 30R 3 ξ 2 − 60R 3 ξ 3 − 12ξR ;a R ;a + 30ξ 2 R ;a R ;a − 22RξR a ;a −6ξR a b ;a b − 4ξR ;ab R ab + 2RξR ab R ab − 2RξR abcd R abcd +60Rξ 2 R a ;a .(40)
It can be shown that calculating Hadamard-DeWitt coefficients a 2 and a 3 for the anisotropic Bianchi type I spacetime described by the line element
ds 2 = −V 2/3 dη 2 + a 2 (η)dx 2 + b 2 (η)dy 2 + c 2 (η)dz 2 ,(41)
one obtains for the vacuum polarization precisely the same results as in the adiabatic method. The main approximation a 2 /16π 2 m 2 equals the fourth-order adiabatic term and a 3 /16π 2 m 4 is the same as the sixth-order adiabatic term. This one-to-one correspondence should hold for the higher-order terms.
C. Trace of the stress-energy tensor
At first sight it seems that the calculations reported in this section have nothing to do with the stress-energy tensor. However, for the conformally coupled fields there is an interesting relation between the trace of the stress-energy tensor and the vacuum polarization [35]. Indeed, provided
ξ = 1/6 one has T a a = a 2 16π 2 − m 2 φ 2 .(42)
Since the leading behavior of the vacuum polarization is proportional to the trace anomaly term,
i.e., a 2 /(4π) 2 , the main approximation of the trace of the stress-energy tensor is simply the nextto-leading term of the vacuum polarization taken with the minus sign. Of course it equals also the minus sixth-order term calculated within the framework of the adiabatic approximation. In the next section we shall demonstrate, among other things, that the trace of the stress-energy tensor calculated from the one-loop effective action is given precisely by (42).
III. STRESS-ENERGY TENSOR OF QUANTIZED MASSIVE FIELDS
The one-loop effective action of the quantized fields in curved spacetime is nonlocal and describes both particle creation and the vacuum polarization. However, when the mass of the field is sufficiently large, the creation of the real particles is suppressed and the effective action becomes local and is determined by the geometry. To be more precise consider a test field of the mass m and the associated Compton length λ C in a spacetime with the characteristic radius of curvature L. One expects that if λ C /L 1 the vacuum polarization part of the effective action dominate, making the expansion in inverse powers of m 2 possible. Suppose that these assumptions are satisfied, then the effective action is given by the Schwinger-DeWitt expansion [13,[36][37][38]]
W R = 1 32π 2 ∞ n=3 (n − 3)! (m 2 ) n−2 d 4 x √ g Tra n ,(43)
where a n are the coincidence limit of the Hadamard-DeWitt coefficients and Tr is a supertrace operator. Inspections of Eq. (43) shows that the main approximation requires knowledge of the fourth Hadamard-DeWitt coefficients a 3 . Their exact form is known for the vector, spinor and scalar fields, satisfying respectively
(δ a b 2 − ∇ b ∇ a − R a b − δ a b m 2 )φ (1) a = 0,(44)(γ a ∇ a + m)φ (1/2) = 0(45)
and Eq. (8), where γ a are the Dirac matrices. In the main approximation, the effective action of the quantized scalar, spinor and vector fields, after discarding total divergences and expressing the final result in the basis of the curvature invariants, can be written as [38] W (1)
ren = 1 192π 2 m 2 d 4 xg 1/2 α (s) 1 R2R + α (s) 2 R ab 2R ab + α (s) 3 R 3 + α (s) 4 RR ab R ab +α (s) 5 RR abcd R abcd + α (s) 6 R a b R b c R c a + α (s) 7 R ab R cd R a b c d + α (s) 8 R ab R a ecd R becd +α (s) 9 R ab cd R eh ab R cd eh + α (s) 10 R a b c d R e s a b R c d e s = 1 192π 2 m 2 d 4 xg 1/2 10 i α i I i ,(46)
where the numerical coefficients α i are given in a Table I Sometimes it is more efficient to adopt a less ambitious approach and instead of using general formulas of Refs. [40] focus on the effective action for a given line element and make use of the Lagrange-Euler equations. Below we shall briefly discuss how it can be achieved for the Kasner metric. First, observe that we can solve a more general problem and calculate components of the stress-energy tensor of the quantized fields in a general Bianchi type I spacetime with g 00 = −f (t).
(Henceforth, for calculational convenience, we slightly abuse our notation and put g 11 = a(t), g 22 = b(t) and g 33 = c(t)). Since the effective action is invariant under the cyclic transformation
{a(t), b(t), c(t)} → {b(t), c(t), a(t)} → {c(t), a(t), b(t)}(48)
it suffices to calculate T 0 0 and T 1 1 . The remaining components can be obtained using this chain of transformation in T 1 1 . Indeed, under the action of the cyclic transformation (48) one has
T 1 1 → T 2 2 → T 3 3 .(49)
For the first spatial component the Lagrange-Euler equations give
T 1 1 = a 96π 2 m 2 √ −g ∂L ∂a + n k=1 (−1) k+1 d k dt k ∂L ∂a (k) ,(50)
where L is the Lagrange function density, n is the maximal order of derivatives of the function a(t) and a (k) = d k a/dt k . The component T 0 0 can be constructed in a similar way. One can also start with the simplified line element with f (t) = 1 and using (50) calculate only T 1 1 . The remaining spatial components can be obtained making use of the cyclic transformation whereas the time component of the stress-energy tensor can be constructed from the ∇ a T a b = 0, which in the case at hand reduces to
T 0 0 ȧ 2a +ḃ 2a +ḃ 2a − T 1 1ȧ 2a − T 2 2ḃ 2b − T 3 3ċ 2a +Ṫ 0 0 = 0.(51)
The integration constant should be put to zero at the end of the calculations. Since the Ricci tensor (and Ricci scalar) vanish for the Kasner metric the calculations can be substantially simplified from the very beginning. For example, the nonvanishing terms in δI 5 /δg ab and δI 8 /δg ab come solely from from the variations of the Ricci tensor. Regardless of the choice of method the obtained results must be, of course, the same.
The stress-energy tensor of the quantized massive fields in the general Bianchi type I spacetime is very complicated and for obvious reasons it will be not presented here. We only remark that the number of terms in T 0 0 with the coefficients c i unspecified is 1431 whereas the number of terms in the each spatial component of the stress-energy tensor is 1709. For a(t) = b(t) = c(t) the result reduces to the well-known T b a obtained in the spatially flat Robertson-Walker spacetime. Although the general stress-energy tensor in the Bianchi type I spacetime is quite complicated, the final result for the Kasner metric is not. Indeed, because of spatial homogeneity there are massive simplifications and the resulting T b a consists of small number of terms and has a simple form that can schematically be written as follows:
T (i)b a = 1 96π 2 m 2 t 6 diag[T (i) 0 , T (i) 1 , T (i) 2 , T (i) 3 ] b a (52) where each T (i)
k is a sixth-order polynomial of p 1 and i refers either to the upper branch or the lower branch in the parameter space. Since there is no danger of confusion the spin index is omitted.
A. Massive scalar fields
First let us consider the quantized massive scalar field.
Making the substitution
{a(t), b(t), c(t)} → {t 2p 1 , t 2p 2 , t 2p 3 } in
where β = 1 + 2p 1 − 3p 2 1 . Inspection of the above formulas reveals some interesting general features: (i) for any allowable p 1 both T 0 0 and T 1 1 does not depend on the branch, (ii) the differences appear only for the remaining spatial components and they are related to the change of the sign of β, (iii) despite the dependence of the stress-energy tensor on the branch there are only four independent components as T (u)2 2 = T (l) 3 3 and T (u) 3 3 = T (l)2 2 , that is in concord with the symmetries of the background geometry, (iv) only I 5 , I 8 , I 9 and I 10 contribute to the final result, and, consequently, (v) the stress-energy tensor depends linearly on ξ. One expects, that except (v) the features (i)-(iv) are independent of the spin of the quantized field. The results of the calculations for the massive scalar field that are plotted in Figs. 3-6 reveal quite complicated (oscillatory) behavior of the T i on the (p 1 , ξ)-space. Here we shall focus on ξ = 0 (the minimal coupling) and ξ = 1/6 (the conformal coupling), i.e., we restrict ourselves to its physical values. First observe that the components of the stress-energy tensor are either negative or positive, and they vanish for the degenerate configurations for which one of the p i equals 1. The basic properties ale listed in the tables II and III. As the functions T i have a simple structure
ξ = 0 T 0 0 ≤ 0 T 1 1 ≥ 0 T 2 2 ≥ 0 T 3 3 ≥ 0 ξ = 1/6 T 0 0 ≤ 0 T 1 1 ≥ 0 T 2 2 ≥ 0 T 3 3 ≥ 0T i (p 1 ) = p 2 1 (p 1 − 1)W 3 (p 1 ),(57)
where W 3 (p 1 ) is a third-order polynomial, the first local extremum of the stress-energy tensor is always at p 1 = 0, whereas location of the second one (on the lower branch) is tabulated in Table III.
For the upper branch the results for T 2 2 and T 3 3 should be interchanged. It should be noted that for the conformal coupling the second extremum is always at p 1 = 2/3, i.e., for the degenerate configuration.
35
.
The results for the lower branch can be obtained, as before, form the conditions T effects are more pronounced for the spinor field. A more detailed analysis shows that the stressenergy tensor has two local extrema: one of them is at p 1 = 0 and the location of the other is given in Table IV. Finally observe that for the lower branch at p 1 = 2/3 the components T
IV. BACK REACTION ON THE METRIC
Although the stress-energy tensor is interesting in its own right it has much wider applications.
Most importantly, it can be regarded as the source term of the semiclassical Einstein field equations
R ab − 1 2 Rg ab + Λg ab = 8π T (cl)b a + T b a ,(66)
plotted as a function of p 1 . T i are defined as T i = 96π 2 m 2 t 6 T i i (no summation) and s = 1.
where T
(cl)b a
is the classical part of the total stress-energy tensor. The resulting system of differential equations has to be solved self-consistently for the quantum-corrected metric. To simplify our discussion we shall assume that the cosmological constant and the coupling parameters k 1 and k 2 in the quadratic part of the total action functional
d 4 x √ g k 1 R ab R ab + k 2 R 2(67)
vanish after the renormalization. Unfortunately, because of the technical complexity of the problem it is practically impossible to find the solution of the equations without referring to approximations or numerics. The exact self-consistent solutions exist only for simple geometries with a high degree of symmetry. Moreover, for the stress-energy tensor obtained from the effective action (46) there is a real danger that some classes of solution of the semiclassical equations would be non-physical. It is because of the appearance of the higher-order derivatives in the equations. Because of that our strategy (that is in concord with the philosophy of the effective lagrangians) is as follows. Since the modifications of the classical spacetime caused by the quantum effects are expected to be small, the natural approach to the problem is to solve the semiclassical equations perturbatively. If both the classical and the quantum parts of the total stress-energy tensor depend functionally on the metric, the equations to be solved have the following form
G ab [g] = 8π T (cl)b a [g] + εT b a [g] ,(68)
with
g ab = g (0) ab + ε∆g ab ,(69)
where ∆g ab is a first-order correction to the metric and to keep control of the order of terms in complicated series expansions, we have introduced once again the dimensionless parameter ε.
Focusing on the first two term of the expansion one has
G ab = G (0) ab + ε∆G ab .(70)
Of course, one expects that the quantized fields acting upon the classical Kasner spacetime deform it, i.e., the quantum corrected metric is still of the Bianchi type I type, but it is not the Kasner metric any more (See however Ref. [41]). Having this in mind we assume that each metric potential a(t), b(t) and c(t) can be expand as the classical background plus a correction. Since we are interested in the corrections to the classical vacuum solution we put T
(cl)b a
where ∆G b a is given by the linear in ε part of G b a , is more complicated. However, before going further it is worthwhile to briefly discuss our general strategy. Following Ref. [42] let us assume that for t < t 0 (t 0 t P l ) the stress-energy of the quantum fields vanishes. The modes with the frequencies satisfyingω k (t 0 ) > t −1 0 are in the adiabatic regime for t > t 0 and the creation of the particles is exponentially damped. On the other hand, for the modes satisfyingω k (t 0 ) < t −1 0 we have creation, although it can be made small taking sufficiently massive fields. In what follows the particle creation will be ignored.
The general solution (ψ 1 (t), ψ 2 (t)) depends on three integration constants. The fourth one must be equated to zero on the account of the covariant conservation of the stress-energy tensor. Since T b a (t) = 0 for t ≤ t 0 we have the Kasner metric tensor and its derivative (a left-hand derivative at t 0 ) in that region. Consequently one is left with a simple solution
ψ 1 (t) = 1 15 T 1 − 1 10 T 2 1 t 4(80)
and
ψ 2 (t) = − T 1 12t 4(81)
with ψ 2 (t) = ψ 3 (t). On the other hand, for a general configuration one has
ψ i = B i t 4 ,(82)
where B i for the upper branch have the form
B 1 = T 1 3p 2 1 − 2p 1 + 15 2880πm 2 + T 2 3(β − 9) − 3p 2 1 + (3β − 10)p 1 5760πm 2 − T 3 3(β + 9) + 3p 2 1 + (3β + 10)p 1 5760πm 2 ,(83)B 2 = T 1 [p 1 (14 − 3p 1 + 3β) − 5(7 + β)] 5760πm 2 + T 2 [31 + p 1 (2 − 3p 1 − 3β) + β] 5760πm 2 + T 3 [p 1 (3p 1 − 2) − 4(4 + β)] 2880πm 2(84)
and
B 3 = T 1 [p 1 (14 − 3p 1 − 3β) + 5(β − 7)] 5760πm 2 − T 2 [p 1 (3p 1 − 2) + 4(β − 4)] 2880πm 2 − T 3 [31 − β + p1(2 − 3p 1 + 3β)] 5760πm 2 . (85) 1 .(88)
As the result has a general structure H ij = H ij is bigger or smaller than 1. Before we discuss the general case let us analyze the degenerate configuration (−1/3, 2/3, 2/3).
For the massive scalar field the sign of the perturbation δH ab depends the coupling constant ξ.
Indeed, when ξ < 47/216 the perturbation is positive and the vacuum polarization isotropizes background spacetime. It shold be noted that both minimally and conformally coupled fields make the background spacetime more isotropic. Moreover, it is precisely the same inequality that should hold for the coupling constant of the massive scalar field to make the interior of the Schwarzschild black hole more isotropic [5]. It becomes even more interesting when we realize that for the Schwarzschild black hole the degenerate Kasner metric is approached asymptotically only in the closest vicinity of the singularity. For the spinor field δH ab is always positive whereas for the vector fields it is always negative. Once again a similar behavior is observed for the quantum corrected interior of the Schwarzschild spacetime. Now, let us return to the general case. We shall analyze the influence of the minimally and conformally coupled massive scalar fields on the anisotropy. Here we describe only the minimally coupled fields since a similar qualitative behavior of H (0) ij and δH ij can be observed for the conformal coupling. On the lower branch (excluding configurations of the type (0, 0, 1)) the ratio H (0) ab is always negative, H (0) bc is positive for p 1 < 0 and negative for p 1 > 0, and finally H (0) ca is negative for p 1 < 0 and positive for p 1 > 0. On the other hand, δH ab is positive for p 1 < 0 and negative for p 1 > 0.
Further, δH bc is always positive, whereas δH ca is negative for p 1 < 2/3 and positive for p 1 > 2/3.
A similar analysis carried out for the upper branch shows that H The quantum correction δH ab is positive for p 1 < 2/3 and negative for p 1 > 2/3, δH bc is always negative, and, finally, δH ca is negative for p 1 < 0 and positive for p 1 > 0.
All this can be stated succinctly in the following way: roughly speaking, for the upper branch, the quantum effects tend to increase anisotropy in (x, y)-directions for p 1 < 2/3 and decrease for p 1 > 2/3. The anisotropy is always decreased by the vacuum polarization in (y, z)-directions and in (x, z)-directions the anisotropy is strengthened for p 1 < 0 and damped for p 1 > 0. On the other hand, for the lower branch the behavior of δH 12 is qualitatively similar to δH 23 on the upper branch, whereas δH 23 is qualitatively similar to δH 12 . The qualitative behavior of H 31 is identical on both branches. Finally observe that for the corrections generated by the spinor and vector fields one has a similar equivalence. More specifically, analysis of δH 12 for the massive vectors shows that the anisotropy always increases, whereas that of δH 23 increases for p 1 < 0 and decreases for p 1 > 0. δH 31 leads to decreasing anisotropy for p 1 < 0 and to increasing for p 1 > 0. The appropriate results for the massive spinors field are opposite, i.e., 'increase' should be replaced by 'decrease' and vice-versa.
V. FINAL REMARKS
In this paper we have calculated the vacuum polarization, φ 2 , of the massive scalar field in the Bianchi type I spacetime within the framework of the Schwinger-DeWitt method and the adiabatic approximation. It has been demonstrated that both methods yield the same result. We expect that a similar equality will hold for the stress-energy tensors. Although we have verified this only for the trace of the stress-energy tensor of the conformally coupled scalar field, we believe that the demonstration of this equality in a general case is conceptually easy but quite involved computationally. Building on this we have calculated the stress-energy tensor of the scalar, spinor and vector fields in the Bianchi type I spacetime making use the Schwinger-DeWitt one-loop effective action and checked the influence of the quantized fields upon the Kasner spacetime. The special emphasis has been put on the problem of isotropization of the background geometry. It should be emphasized once again that being local the Schwinger-DeWitt technique does not take particle creation into account. It is therefore possible that the actual influence of the quantized fields, e.g., calculated numerically, will be more pronounced [5]. On the other hand however, we expect that if the conditions H i /m 1 hold our results should provide a reasonable approximation.
Finally observe that the semiclassical Einstein equations with the right hand side given by the stress-energy tensor of the quantized fields constructed from the one-loop effective action (46) may be treated as the theory with higher curvature terms. Theories of this type are currently actively investigated (see e.g. Refs. [43][44][45] and references therein).
The Kasner metric is a solution of the vacuum Einstein field equations, and, because of its simplicity, it is also a solution of the equations of the quadratic gravity. The Kasner conditions exclude the possibility that all three exponents are equal, however, there are configurations for which two of them are the same. We shall call these configurations degenerate. The Kasner solution with p 1 = −1/3 and p 2 = p 3 = 2/3 arXiv:1811.00993v1 [gr-qc] 2 Nov 2018 has rotational symmetry. The choice of p i in the form (1, 0, 0) defines the flat Kasner metric In the nondegenerate case one can order the parameters p i as follows − 1 3
FIG. 1 :
1The Kasner sphere and the Kasner plane in the parameter space. FIG. 2: Two branches of the allowable parameters in the (p 1 , p 2 ) space. The parameter p 3 can be obtained from Eq. 5. The branch points represent degenerate configurations (−1/3, 2/3, 2/3) and (1, 0, 0).
the general formulas in the Bianchi type I spacetime, after some algebra, one obtains (for the lower (l) and upper (u) branches)
: The extrema of the components of the stress-energy tensor of the massive scalar field. The calculations have been carried out for the lower branch. FIG. 3: T 1 = 96π 2 m 2 t 6 T 0 0 plotted as a function of p 1 and ξ.
FIG. 4 :
4T 1 = 96π 2 m 2 t 6 T 1 1 plotted as a function of p 1 and ξ.
16 FIG. 5 :
165T 2 = 96π 2 m 2 t 6 T 2 2 plotted as a function of p 1 and ξ.
FIG. 6 :
6T 3 = 96π 2 m 2 t 6 T 3 3 plotted as a function of p 1 and ξ.B. Massive spinor asnd vector fieldsSimilar calculations can be carried out for the massive fields of higher spin. First, let us consider the spinor field. The stress-energy tensor in the Kasner spacetime has a
.
The components of T b a are still of the form given by Eq. (52) and the functions T i are plotted in Figs. 7 and 8. A comparison of the results shows that the components of the stress-energy tensor change their sign with a change of spin and vanish for the degenerate configurations of the type (1, 0, 0). The energy density ρ = −T 0 0 is nonnegative for the spinor field whereas it is negative (or zero) for the vector field. Moreover, the quantum FIG. 7: The functions T 0 (dashed curve), T 1 (dotted curve), T 2 (dot-dashed curve) and T 3 (solid curve) plotted as a function of p 1 . T i are defined as T i = 96π 2 m 2 t 6 T i i (no summation) and s = 1/2.
.
On the other hand,at p 1 = −1/3, (the left branch point) T 2 2 = T 3 3 .This behavior is, of course, expected as we have (2/3, −1/3, 2/3)-configuration in the first case,
summation over a).
( 0 )
0ij + δH ij we shall call H(0) ij the classical part and δH ij its correction. First, consider the zeroth-order effects: if the H (0) ij is positive the spacetime is expanding or contracting in the both spacetime directions, moreover, if H (0) ij = 1 then the evolution is isotropic. On the other hand, if the sign is negative then the spacetime is expanding in one direction and contracting in the other. From this one sees that the influence of the quantum fields depends not only on the relative signs of the classical Hubble parameters and their corrections, but also if H (0)
is positive for p 1 < 0 and negative for p 1 > 0, and H (0) ca is always negative.
TABLE I :
IThe coefficients α (s) i for the massive scalar with arbitrary curvature coupling ξ , spinor, and vector field can be calculated form the standard relation and consists of the purely geometric terms constructed from the Riemann tensor, its covariant derivatives and contractions. Each I i , after variations with respect to the metric tensor, leads to the covariantly conserved quantity. The type of the quantum field enters through the spindependent coefficients α i . The resulting stress-energy tensor is a linear combination of almost 100T ab =
2
√ g
δW
(1)
ren
δg ab
(47)
local geometric terms. (Their actual number depends on the simplification strategies and identities
satisfied by the Riemann tensor used during the calculation). The general formulas describing the
stress-energy tensor have been given in Refs. [39, 40].
TABLE II :
IIThe sign of the components of the stress-energy tensor of the massive scalar field in the Kasner spacetime. The calculations have been carried out for the lower branch.
TABLE IV :
IVThe extrema of the components of the stress-energy tensor of the massive spinor and vector fields. The calculations have been carried out for the lower branch
The general results can be obtained on request from the author.
AcknowledgmentsThe author would like to thank Darek Tryniecki for discussions.= 0, and, consequently, the metric, with a little prescience, can be expanded asb(t) = t 2p 2 (1 + εψ 2 (t)) ,Now, expanding the semiclassical Einstein field equations in the powers of ε and retaining the first two terms in the Einstein tensor, one hasThe solution of the zeroth-order equations is the Kasner metric, whereas the system of first the order equationsFor a given spin, T i are defined as in Eqs. (52). The results for the lower branch can be obtained by putting β → −β and taking T i appropriate for that branch. With a little effort one can check that the functions ψ 1 , ψ 2 and ψ 3 satisfy Eq (74) and this completes the solution of the first order semiclassical equations.Now we try to answer the natural question if the quantum effects dampen or strengthen the anisotropy[42]. As its natural measure let us take the ratios of the directional Hubble parameters of the quantum-corrected spacetime. To the first order in ε one has
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| [] |
[
"Manipulating antiferromagnets with magnetic fields: ratchet motion of multiple domain walls induced by asymmetric field pulses",
"Manipulating antiferromagnets with magnetic fields: ratchet motion of multiple domain walls induced by asymmetric field pulses"
] | [
"O Gomonay ",
"M Kläui ",
"J Sinova ",
"\nInstitut für Physik\nJohannes Gutenberg Universität Mainz\nD-55099MainzGermany\n",
"\nInstitut für Physik\nNational Technical University of Ukraine \"KPI\"\n03056KyivUkraine\n",
"\nInstitut für Physik\nJohannes Gutenberg Universität Mainz\nD-55099MainzGermany\n",
"\nInstitute of Physics, Academy of Sciences of the Czech Republic\nJohannes Gutenberg Universität Mainz\nCukrovarnicka 10D-55099, 162 00Mainz, Praha 6Germany, Czech Republic\n"
] | [
"Institut für Physik\nJohannes Gutenberg Universität Mainz\nD-55099MainzGermany",
"Institut für Physik\nNational Technical University of Ukraine \"KPI\"\n03056KyivUkraine",
"Institut für Physik\nJohannes Gutenberg Universität Mainz\nD-55099MainzGermany",
"Institute of Physics, Academy of Sciences of the Czech Republic\nJohannes Gutenberg Universität Mainz\nCukrovarnicka 10D-55099, 162 00Mainz, Praha 6Germany, Czech Republic"
] | [] | Future applications of antiferromagnets (AFs) in many spintronics devices rely on the precise manipulation of domain walls. The conventional approach using static magnetic fields is inefficient due to the low susceptibility of AFs. Recently proposed electrical manipulation with spin-orbit torques is restricted to metals with a specific crystal structure. Here we propose an alternative, broadly applicable approach: using asymmetric magnetic field pulses to induce controlled ratchet motion of AF domain walls. The efficiency of this approach is based on three peculiarities of AF dynamics. First, a time-dependent magnetic field couples with an AF order parameter stronger than a static magnetic field, which leads to higher mobility of the domain walls. Second, the rate of change of the magnetic field couples with the spatial variation of the AF order parameter inside the domain and this enables synchronous motion of multiple domain walls with the same structure. Third, tailored asymmetric field pulses in combination with static friction can prevent backward motion of domain walls and thus lead to the desired controlled ratchet effect. The proposed use of an external field, rather than internal spin-orbit torques, avoids any restrictions on size, conductivity, and crystal structure of the AF material. We believe that our approach paves a way for the development of new AF-based devices based on controlled motion of AF domain walls.Antiferromagnets (AFs) are considered perspective materials for spintronic applications: they exhibit fast magnetic dynamics with excitations in the THz range, are fundamentally insensitive to external magnetic fields, and produce no stray fields. 1-3 One of the further advantages of AFs, important for fast switching between different states, is related with the motion of domain walls (DWs). In contrast to ferromagnets (FM), the dynamics of AF DWs shows no Walker breakdown. Thus, the DW velocity is only limited by the group velocity of spin waves, which is of the order of tens of km/s (e.g., 40 km/s for NiO). This is orders of magnitude larger than the typical velocities in FM, where the Walker breakdown limits the achievable velocities, and also larger than velocities in synthetic AFs.4However, the manipulation of AF DWs faces significant difficulties. In particular, 180 • AF domains are indistiguishable even in the presence of a constant homogeneous magnetic field. So, in contrast to FMs, an applied external field cannot move the 180 • AF DWs at all. In addition, coupling between the external magnetic field and the AF order parameter (Néel vector) is suppressed due to the strong exchange coupling between the magnetic sublattices. In this case typical values of fields necessary to produce any noticeable shift of the DW are of the order of the spin-flop field and range 1-10 T.5Recently the possibility to move DW in an AF with the help of a staggered Néel spin-orbit torque was demon-strated in Ref. 6. While this mechanism can be very effective, its application is restricted to metals that have a broken local inversion symmetry, which the vast majority of the AF systems do not have. Furthermore, manipulation using regular spin-orbit torques has been shown to be restricted to specific DW types, sample geometry and AF spin structure configuration, which narrows the applicability of these torques.7Finally, recent calculations predict that temperature gradients can move the AF DWs in metals and isolators as well.8,9However, manipulation of the DWs using this mechanism is restricted to one-directional motion and is yet to be observed. Hence, at present there is no broadly applicable approach to manipulate AF DWs.In this Letter we develop such a broadly applicable approach to manipulate AF DWs based on the use of asymmetric magnetic field pulses. We show that this approach is highly efficient for devices as it enables to attain high DW mobilities, to induce synchronous motion of multiple DWs and to control the DW displacement through a ratchet effect.We compare the dynamics of AF DWs induced by static and by time-dependent magnetic fields and show that a time-dependent field produces a larger effective force than its static counterpart. This difference originates from the strong exchange field which reduces the magnetic static susceptibility. Our results show that the force produced by the rate of change of the magnetic field arXiv:1608.05967v1 [cond-mat.mtrl-sci] | 10.1063/1.4964272 | [
"https://arxiv.org/pdf/1608.05967v1.pdf"
] | 118,427,539 | 1608.05967 | 29fadda2be5ba370b1ef38cdac33a28c075ef520 |
Manipulating antiferromagnets with magnetic fields: ratchet motion of multiple domain walls induced by asymmetric field pulses
21 Aug 2016
O Gomonay
M Kläui
J Sinova
Institut für Physik
Johannes Gutenberg Universität Mainz
D-55099MainzGermany
Institut für Physik
National Technical University of Ukraine "KPI"
03056KyivUkraine
Institut für Physik
Johannes Gutenberg Universität Mainz
D-55099MainzGermany
Institute of Physics, Academy of Sciences of the Czech Republic
Johannes Gutenberg Universität Mainz
Cukrovarnicka 10D-55099, 162 00Mainz, Praha 6Germany, Czech Republic
Manipulating antiferromagnets with magnetic fields: ratchet motion of multiple domain walls induced by asymmetric field pulses
21 Aug 2016
Future applications of antiferromagnets (AFs) in many spintronics devices rely on the precise manipulation of domain walls. The conventional approach using static magnetic fields is inefficient due to the low susceptibility of AFs. Recently proposed electrical manipulation with spin-orbit torques is restricted to metals with a specific crystal structure. Here we propose an alternative, broadly applicable approach: using asymmetric magnetic field pulses to induce controlled ratchet motion of AF domain walls. The efficiency of this approach is based on three peculiarities of AF dynamics. First, a time-dependent magnetic field couples with an AF order parameter stronger than a static magnetic field, which leads to higher mobility of the domain walls. Second, the rate of change of the magnetic field couples with the spatial variation of the AF order parameter inside the domain and this enables synchronous motion of multiple domain walls with the same structure. Third, tailored asymmetric field pulses in combination with static friction can prevent backward motion of domain walls and thus lead to the desired controlled ratchet effect. The proposed use of an external field, rather than internal spin-orbit torques, avoids any restrictions on size, conductivity, and crystal structure of the AF material. We believe that our approach paves a way for the development of new AF-based devices based on controlled motion of AF domain walls.Antiferromagnets (AFs) are considered perspective materials for spintronic applications: they exhibit fast magnetic dynamics with excitations in the THz range, are fundamentally insensitive to external magnetic fields, and produce no stray fields. 1-3 One of the further advantages of AFs, important for fast switching between different states, is related with the motion of domain walls (DWs). In contrast to ferromagnets (FM), the dynamics of AF DWs shows no Walker breakdown. Thus, the DW velocity is only limited by the group velocity of spin waves, which is of the order of tens of km/s (e.g., 40 km/s for NiO). This is orders of magnitude larger than the typical velocities in FM, where the Walker breakdown limits the achievable velocities, and also larger than velocities in synthetic AFs.4However, the manipulation of AF DWs faces significant difficulties. In particular, 180 • AF domains are indistiguishable even in the presence of a constant homogeneous magnetic field. So, in contrast to FMs, an applied external field cannot move the 180 • AF DWs at all. In addition, coupling between the external magnetic field and the AF order parameter (Néel vector) is suppressed due to the strong exchange coupling between the magnetic sublattices. In this case typical values of fields necessary to produce any noticeable shift of the DW are of the order of the spin-flop field and range 1-10 T.5Recently the possibility to move DW in an AF with the help of a staggered Néel spin-orbit torque was demon-strated in Ref. 6. While this mechanism can be very effective, its application is restricted to metals that have a broken local inversion symmetry, which the vast majority of the AF systems do not have. Furthermore, manipulation using regular spin-orbit torques has been shown to be restricted to specific DW types, sample geometry and AF spin structure configuration, which narrows the applicability of these torques.7Finally, recent calculations predict that temperature gradients can move the AF DWs in metals and isolators as well.8,9However, manipulation of the DWs using this mechanism is restricted to one-directional motion and is yet to be observed. Hence, at present there is no broadly applicable approach to manipulate AF DWs.In this Letter we develop such a broadly applicable approach to manipulate AF DWs based on the use of asymmetric magnetic field pulses. We show that this approach is highly efficient for devices as it enables to attain high DW mobilities, to induce synchronous motion of multiple DWs and to control the DW displacement through a ratchet effect.We compare the dynamics of AF DWs induced by static and by time-dependent magnetic fields and show that a time-dependent field produces a larger effective force than its static counterpart. This difference originates from the strong exchange field which reduces the magnetic static susceptibility. Our results show that the force produced by the rate of change of the magnetic field arXiv:1608.05967v1 [cond-mat.mtrl-sci]
Future applications of antiferromagnets (AFs) in many spintronics devices rely on the precise manipulation of domain walls. The conventional approach using static magnetic fields is inefficient due to the low susceptibility of AFs. Recently proposed electrical manipulation with spin-orbit torques is restricted to metals with a specific crystal structure. Here we propose an alternative, broadly applicable approach: using asymmetric magnetic field pulses to induce controlled ratchet motion of AF domain walls. The efficiency of this approach is based on three peculiarities of AF dynamics. First, a time-dependent magnetic field couples with an AF order parameter stronger than a static magnetic field, which leads to higher mobility of the domain walls. Second, the rate of change of the magnetic field couples with the spatial variation of the AF order parameter inside the domain and this enables synchronous motion of multiple domain walls with the same structure. Third, tailored asymmetric field pulses in combination with static friction can prevent backward motion of domain walls and thus lead to the desired controlled ratchet effect. The proposed use of an external field, rather than internal spin-orbit torques, avoids any restrictions on size, conductivity, and crystal structure of the AF material. We believe that our approach paves a way for the development of new AF-based devices based on controlled motion of AF domain walls. Antiferromagnets (AFs) are considered perspective materials for spintronic applications: they exhibit fast magnetic dynamics with excitations in the THz range, are fundamentally insensitive to external magnetic fields, and produce no stray fields. 1-3 One of the further advantages of AFs, important for fast switching between different states, is related with the motion of domain walls (DWs). In contrast to ferromagnets (FM), the dynamics of AF DWs shows no Walker breakdown. Thus, the DW velocity is only limited by the group velocity of spin waves, which is of the order of tens of km/s (e.g., 40 km/s for NiO). This is orders of magnitude larger than the typical velocities in FM, where the Walker breakdown limits the achievable velocities, and also larger than velocities in synthetic AFs. 4 However, the manipulation of AF DWs faces significant difficulties. In particular, 180 • AF domains are indistiguishable even in the presence of a constant homogeneous magnetic field. So, in contrast to FMs, an applied external field cannot move the 180 • AF DWs at all. In addition, coupling between the external magnetic field and the AF order parameter (Néel vector) is suppressed due to the strong exchange coupling between the magnetic sublattices. In this case typical values of fields necessary to produce any noticeable shift of the DW are of the order of the spin-flop field and range 1-10 T. 5 Recently the possibility to move DW in an AF with the help of a staggered Néel spin-orbit torque was demon-strated in Ref. 6. While this mechanism can be very effective, its application is restricted to metals that have a broken local inversion symmetry, which the vast majority of the AF systems do not have. Furthermore, manipulation using regular spin-orbit torques has been shown to be restricted to specific DW types, sample geometry and AF spin structure configuration, which narrows the applicability of these torques. 7 Finally, recent calculations predict that temperature gradients can move the AF DWs in metals and isolators as well. 8,9 However, manipulation of the DWs using this mechanism is restricted to one-directional motion and is yet to be observed. Hence, at present there is no broadly applicable approach to manipulate AF DWs.
In this Letter we develop such a broadly applicable approach to manipulate AF DWs based on the use of asymmetric magnetic field pulses. We show that this approach is highly efficient for devices as it enables to attain high DW mobilities, to induce synchronous motion of multiple DWs and to control the DW displacement through a ratchet effect.
We compare the dynamics of AF DWs induced by static and by time-dependent magnetic fields and show that a time-dependent field produces a larger effective force than its static counterpart. This difference originates from the strong exchange field which reduces the magnetic static susceptibility. Our results show that the force produced by the rate of change of the magnetic field arXiv:1608.05967v1 [cond-mat.mtrl-sci] 21 Aug 2016 will move DWs with similar structure in the same direction. In contrast, the force produced by a static magnetic field is independent of the DW structure and induces a shrinking and disappearance of unfavourable domains. We find the conditions for ratchet-like motion by calculating the critical rate of magnetic field that overcomes a static friction force. We also propose an optimal configuration to implement controlled DW motion for the archetypical AFs Mn 2 Au and NiO. We consider the generic case of a compensated AF with two magnetic sublattices with magnetizations M 1 and The magnetic structure of an AF texture can be explicitly decribed in terms of the AF (Néel) vector L = M 1 − M 2 which is considered as a field variable, L(r, t). The closed equations of motion for AF vector 10-12 have the following form: 13
M 2 (|M 1 | = |M 2 | = M s /2).L × L − c 2 ∆L + γ 2 H ex M s ∂w AF ∂L = T − γα G H ex L ×L.
(1) In Eqs. (1) we introduced the magnon velocity c which coincides with the limiting velocity for the DW motion, γ is the gyromagnetic ratio, and w an is the density of magnetic anisotropy energy which depends upon the crystal structure. The effective field H ex parametrizes the exchange coupling between the magnetic sublattices. The last term in the r.h.s. of Eq. (1) describes viscous damping parametrized by the Gilbert constant α G .
The vector T in the r.h.s. of Eqs. (1) describes the effective forces (torques) induced by the external magnetic field H:
T = γL ×Ḣ × L − 2γL(HL) − γ 2 L × H (L · H) . (2)
In many practical cases the shape of the moving DW does not change or changes slightly. So, the DW can be considered as a point particle, whose dynamics is described by only two vectors: the generalized momentum P and its canonically conjugated coordinate R (position of the DW center). The dynamics of Eq. (1) can then be reduced to a standard equation for a point mass: 11
dP dt = −γα G H ex P + F,(3)
where F is the resulting external force, and the first term in the r.h.s. is analogous to viscous damping with relaxation time τ relax = 1/(γα G H ex ). Equation (3) is derived from the original Eq. (1) in the following way. First, we define the DW momentum P as an integral of motion related with homogeneity of space:
P j = − 1 γ 2 M s H ex L (0) ∂ j L (0) dV, j = x, y, z.(4)
Here L (0) (r, t) is a solution of Eq. (1) in the absence of a field (T = 0) and damping (α G = 0). Second, we assume that L(t, r) = L (0) (t, r − R). Finally, calculating explicitly the time derivative of Eq. (4) and taking into account Eq. (1) we obtain Eq. (3). Among the forces, acting on the DW, we specify three types, essential for our consideration, F = F pond +F diss + F fric . The first one is the ponderomotive force
F pond = nS 2M s H ex (L 2 H) 2 − (L 1 H) 2 .(5)
It stems from the difference in energy density between the left (Néel vector L 1 ) and right (Néel vector L 2 ) AF domains and is directed along the normal to the DW plane, n (see, e.g. Fig. 1). The ponderomotive force is proportional to the square of the magnetic field. Its value is weakened due to the strong exchange coupling between the magnetic sublattices. In addition, this force is insensitive to the structure of the DW itself and acts equally on the Bloch-like and Néel-like DW. The second force is dissipative and it is given by
F diss · n = − 1 γM s H ex Ḣ · L (0) × (n · ∇)L (0) dV.(6)
It is induced by the time-dependent component of the magnetic field and is sensitive to the relative orientation of the the external field and AF vectors inside the DW, i.e. the DW structure. This force is maximal if the magnetic field is perpendicular to the (L 1 , L 2 ) plane, see, e.g. Fig. 2. In spite of the small factor 1/H ex , the dissipative force can be larger than F pond , especially for high frequencies. Moreover, in contrast to F pond , the dissipative force can move 180 • domain walls. Lastly, the third force is a friction force, F fric . It is related to the magnetic defect distribution within the crystal and pinning strength of the defects. We consider this force as a static friction force which defines a threshold for the dynamics. This force is sample-dependent and can be estimated from the coercitivity. In the calculations below we take it to be 10% of the spin-flop field.
The important difference between the ponderomotive and dissipative forces is illustrated in Fig. 1, where we consider a stripe AF domain structure. The ponderomotive force is directed from the favourable domain (with lower energy density) to the unfavourable. So, in a stripe structure adjacent DWs move in opposite directions, thus shrinking the fraction of unfavourable domains. Contrary to this, the orientation of the dissipative force depends upon the AF DW structure. So, all the DWs with the same chirality move in the same direction. Thus, the application of a time-dependent field provides an effective tool for manipulating AF-domains in an AF-based race-track type memory.
We next analyse the dynamics of AF DWs through Eq. (3). We consider the simple case of a tetragonal AF (e.g. Mn 2 Au) and 90 • domain structure with orthogonal AF vectors in neighboring domains, L 1 ⊥ L 2 . In this case the optimal orientation of the static magnetic field, H dc , which produces the ponderomotive force, is parallel to one of the Néel vectors, e.g. H dc L 1 (see Fig. 2). On the other hand, the optimal orientation of the time dependent field, H ac , is related to the DW type.
In a thin film the AF vectors inside the DW rotate within the film plane and the DW is of a Néel type. For this case, the most efficient H ac is perpendicular to the film plane (Fig. 2a). In a bulk sample, a Bloch wall is also possible and the most efficient H ac is perpendicular to the DW plane (Fig. 2b). In both geometries H ac ⊥ H dc and thus the time-dependent component does not contribute to the ponderomotive force.
Although H dc and H ac fields have different orientations, the corresponding forces, F pond and F diss , are both parallel to the DW normal.
The time dependent component of the magnetic field allows one to manipulate the AF DW motion in a very effective way. To illustrate this fact, we start from the constant (time-independent) forces produced by H dc and steadily increasing/decreasing H ac . The velocity of the steady motion is
v steady = c (πḢ ac /γ + H 2 dc )/(2H ex ) α 2 G H an H ex + (πḢ ac /γ − H 2 dc ) 2 /(2H ex ) 2 ,(7)
as can be obtained from Eq. (3). Here H an is the anisotropy field.
Contributions of the time-dependent and the static component to v steady are compared in Fig. 3. For the calculations we use field values H ex =1400 T, H an =30 mT typical for AFs with high Néel temperature (like Mn 2 Au 14,15 and NiO 16 ). We set the AF magnon velocity c = 30 km/s. As the damping parameters of metals and insulators are different, we take α G = 10 −4 for insulating NiO 17 and α G = 10 −3 for metalic Mn 2 Au. These values correspond to relaxation times τ relax = 50 ps and 5 ps, respectively. The friction force per unit DW area is taken 9 N/m 2 which corresponds to an effective coercive field of 0.1 T. Fig. 3 shows that the mobility (= dv/dH) of the DW in an ac field is much higher than in the static field and an amplitude value H ac =1 T is enough to reach the limiting velocity. However, a practical fast increase of the magnetic field is only possible on short time scales and up to a limited amplitude of H ac . These facts exclude monotonously varying H ac as a useful tool for DW manipulation.
A more experimentally realistic alternating (cos-like) field H ac ∝ cos(ωt) can only induce oscillations of the DW with zero permanent displacement by drift. The green line in Fig.4 (left axis) shows the displacement of the DW induced by a symmetric field pulse. For all pulses we have taken the time between rise and fall times to be 700 ps and field amplitude H ac =10 mT. During the rising edge and falling edge periods of the pulses, the DW moves in opposite directions with exactly the same velocity ( Fig.4(right axis)), resulting in zero displacement.
Nonzero displacement can be achieved with an asymmetric pulse, as illustrated by the magenta (fall time 100 ps) and blue (fall time 200 ps) lines in Fig.4. The corresponding asymmetry of the velocity during the raising and falling intervals ( Fig.4 (b)) is due to the frictional force. Friction sets a threshold for the DW depinning and prevents DW motion for small field ratesḢ ac . As a result, the velocity of backward motion diminishes with increasing fall time. At some critical value of fall time the backward motion of the DW is blocked (blue lines in Fig.4) and the displacement of the DW is maximal.
The maximal DW displacement during the pulse depends upon the relation between risetime τ raise and relaxation time of the DW, τ relax . For a given material (fixed relaxation time) the optimal rising time is close to τ relax . For a given experimental technique (fixed raise time) a longer relaxation time is preferable (magenta vs blue lines). Note, that a small relaxation time is typical for the metallic systems like Mn 2 Au, while a large τ relax is more typical for insulators like NiO.
Although the displacement of a DW during one pulse is limited by the internal damping, the friction, and the attainable pulse parameters, a DW can be moved to any distance by a periodic set of pulses, as shown in Fig. 5 (b). The average velocity of such rachet-like motion (in this example is 0.44 m/s), can be controlled by a proper choice of the pulse duration and the interval between the pulses. To attain maximal velocity, the time between rise and fall times should be minimized (white range in Fig. 5). We also note that this type of ratchet force is different from its counterpart in FM materials, where an oscillating motion is induced instead. 18 In summary, we exploit the use of asymmetric field pulses to displace AF DWs. We ascertain that asymmetric sawtooth-shaped pulses of the magnetic field in combination with the natural defect-induced static friction enable unidirectional controlled rachet-like motion of an AF DW. This mechanism is broadly applicable to many different types of AF materials and can induced synchronous motion of multiple domain walls as required for applications.
We acknowledge support from Humboldt Foundation, the EU (Wall PEOPLE-
FIG. 1 .
1Evolution of the stripe AF domain structure induced by a constant, H dc , (a) and a time-dependent, Hac, (b) magnetic field. Black arrows show the directions of the ponderomotive, F pond , and the dissipative, F diss , forces. The grey dashed lines in (b) mark the previous position of the DWs.
The metallic Mn 2 Au and the isolating NiO are good examples of such AFs.
FIG. 2 .
2Sample with a single Néel (a) or Bloch (b) domain wall and optimal orientation of static (H dc ) and timedependent (Hac) field. Curved arrows show the direction in which the AF vector rotates within the DW.
FIG. 3 .
3Velocity of a 90 • AF DW vs static field (magenta line) and an amplitude of steadily increasing time-dependent field (blue line) Hac(t) = H (0) ac t/τraise calculated according to Eq. (7) for Mn2Au, τraise = 5 ps. The vertical dashed line separates regions with linear and log-scale.
FIG. 4 .
4Time dependence of DW displacement (left axis, solid lines) and velocity (right axis, dashed lines) induced by field pulses with different fall times. τraise =50 ps. Fall time is 50 ps (green line), 100 ps (magenta line), and 200 ps (blue line). Relaxation time τ relax =50 ps. The grey dotted line shows the pulse shape with exponential rise/fall ∝ exp(−t/τ ). The rising/falling interval is shown with shaded area.
FIG. 5 .
52013-ITN 608031; MultiRev ERC-2014-PoC 665672) as well as the Center of Innovative and Emerging Materials at Johannes Gutenberg Rachet-like displacement (magenta line) of the DW under the sawtooth-shaped pulses (grey dotted line). Blue line shows the effective displacement as a function of time, average velocity being 0.44 m/s. τ relax = τraise=50 ps. The fall time is 100 ps. University Mainz, the Graduate School of Excellence Materials Science in Mainz (CSC 266), the DFG (in particular SFB TRR 173 Spin+X), the Ministry of Education of the Czech Republic Grant No. LM2011026, and from the Grant Agency of the Czech Republic Grant no. 14-37427
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"Pablo Diaz \nDepartment of Physics and Astronomy\nUniversity of Lethbridge\n4401 University DriveT1K 3M4LethbridgeAlbertaCanada\n",
"Saurya Das \nDepartment of Physics and Astronomy\nUniversity of Lethbridge\n4401 University DriveT1K 3M4LethbridgeAlbertaCanada\n",
"Mark Walton \nDepartment of Physics and Astronomy\nUniversity of Lethbridge\n4401 University DriveT1K 3M4LethbridgeAlbertaCanada\n",
"Pablo Diaz \nDepartment of Physics and Astronomy\nUniversity of Lethbridge\n4401 University DriveT1K 3M4LethbridgeAlbertaCanada\n",
"Saurya Das \nDepartment of Physics and Astronomy\nUniversity of Lethbridge\n4401 University DriveT1K 3M4LethbridgeAlbertaCanada\n",
"Mark Walton \nDepartment of Physics and Astronomy\nUniversity of Lethbridge\n4401 University DriveT1K 3M4LethbridgeAlbertaCanada\n"
] | [
"Department of Physics and Astronomy\nUniversity of Lethbridge\n4401 University DriveT1K 3M4LethbridgeAlbertaCanada",
"Department of Physics and Astronomy\nUniversity of Lethbridge\n4401 University DriveT1K 3M4LethbridgeAlbertaCanada",
"Department of Physics and Astronomy\nUniversity of Lethbridge\n4401 University DriveT1K 3M4LethbridgeAlbertaCanada",
"Department of Physics and Astronomy\nUniversity of Lethbridge\n4401 University DriveT1K 3M4LethbridgeAlbertaCanada",
"Department of Physics and Astronomy\nUniversity of Lethbridge\n4401 University DriveT1K 3M4LethbridgeAlbertaCanada",
"Department of Physics and Astronomy\nUniversity of Lethbridge\n4401 University DriveT1K 3M4LethbridgeAlbertaCanada"
] | [] | With the aim of investigating the relation between gravity and non-locality at the classical level, we study a bilocal scalar field model. Bilocality introduces new (internal) degrees of freedom that seem to reproduce gravity. We show that the equations of motion of the massless branch of the free bilocal model match those of linearized gravity. We also discuss higher orders of perturbation theory, where there is self-interaction in both gravity and the bilocal field sectors. | null | [
"https://arxiv.org/pdf/1609.08631v4.pdf"
] | 118,513,905 | 1609.08631 | fa1fe42062db32cca25343d1d9e8130e05ee6a77 |
Gravity and non-locality
October 5, 2016 4 Oct 2016
Pablo Diaz
Department of Physics and Astronomy
University of Lethbridge
4401 University DriveT1K 3M4LethbridgeAlbertaCanada
Saurya Das
Department of Physics and Astronomy
University of Lethbridge
4401 University DriveT1K 3M4LethbridgeAlbertaCanada
Mark Walton
Department of Physics and Astronomy
University of Lethbridge
4401 University DriveT1K 3M4LethbridgeAlbertaCanada
Gravity and non-locality
October 5, 2016 4 Oct 2016Gravitybilocal fieldnon-localityperturbative gravitygravita- tional waves
With the aim of investigating the relation between gravity and non-locality at the classical level, we study a bilocal scalar field model. Bilocality introduces new (internal) degrees of freedom that seem to reproduce gravity. We show that the equations of motion of the massless branch of the free bilocal model match those of linearized gravity. We also discuss higher orders of perturbation theory, where there is self-interaction in both gravity and the bilocal field sectors.
Introduction
It is believed that resolving two spacetime points is impossible when they are sufficiently close to each other. A simple argument goes as follows: resolving two nearby points amounts to probing that region of spacetime with particles of wavelength of the order of the distance we want to resolve. As we consider closer points the wavelength must be shorter, with more and more energetic probes. This process cannot go on forever.
According to general relativity, if enough radiation is aimed into a region, the concentration of energy warps spacetime and the region becomes a black hole. The event horizon of the black hole prevents us from resolving any points beyond it [1].
The theoretical impossibility of resolving arbitrarily small distances may indicate a fundamental non-locality. In other words, semiclasical gravity suggests non-locality at a fundamental level. This motivates us to ask the reverse question: does non-locality imply, or accommodate, gravity in a natural way? An affirmative answer to this question would provide a novel, non-local framework in which general relativity is an effective, emergent, low-energy theory. Note that the effective nature of gravity has been conjectured for a long time [2].
We investigate this possibility in the current article. Although we do not offer a formal proof, we provide significant evidence that non-locality naturally gives rise to gravity. Specifically, we study a simple model of a bilocal scalar field, and show how bilocality opens up new (internal) degrees of freedom that can be matched exactly to those of a spin-two field, at least to first order in perturbation theory. That is, we show that the free bilocal model encodes gravity waves, precursors to gravitons in a quantized theory. This is the strongest result of the paper. We then sketch how this match should hold in subsequent orders, where there is self-interaction in gravity and in the bilocal field.
It came as a surprise when it was discovered that string theory was a theory of gravity [13,14]. Some would argue that this is a sign that string theory is the theory of everything. However, if the claim we make in this paper is true, that is if gravity comes along with non-locality, the discovery would be less surprising, since strings are extended (and so non-local) objects.
Bilocal fields were first studied by Yukawa in a series of papers [5]. The physical motivation was to study mesonic excitations. The two points the field depends on were the location of the quarks. As he insisted, the bilocal field has to be genuine, in the sense that it cannot be decomposed into local field quantities. The use of bilocal and trilocal field models to explain confinement in hadrons became popular in the next couple of decades [3,6,7], until the success of quantum chromodynamics made them fade away.
In this work we revisit the bilocal models in the context of gravity. This echoes the history of string theory, first conceived as a description of the strong interaction before becoming a theory of gravity, and everything else.
Bilocal fields have also been reconsidered in the context of the AdS/CFT correspondence (in the higher-spin version) by Jevicki et al. during the last decade [8,9]. It is surprising how bilocal fields in CFT help to reconstruct the AdS bulk. Bilocality plays a crucial role in the reconstruction, and is not just a matter of convenience. In fact some of these works, including [9], inspired us to examine bilocality in the context of gravity in a simple setting.
The bilocal model that we use to compare with gravity is perhaps the simplest version of a bilocal model, in the sense of involving no more than two derivatives. It was extensively used in the realm of the strong interaction, as mentioned before. A comprehensive study of it can be found in [3]. Since the bilocal field was aimed to describe mesons, only massive solutions were considered. In this work we study the massless solutions of that model. To our knowledge, this is the first study of the massless sector.
The pleasant surprise is that the most general massless solution of the free bilocal model is nothing but gravity waves. The match that we show in this paper is at the level of equations of motion and solutions: massless solutions of the bilocal model match one-to-one with gravity waves. It is known that there is no such a thing as a solution in gravity without fixing the gauge. Thus, in the process of comparing solutions we are forced to lose covariance.
The paper is organized as follows. Section 2 is a brief review of linearized gravity. In section 3 we discuss the bilocal field model that we will use later. Massive solutions of that model are presented in section 3.1. They are not new and we will not actually use them for our purposes, but they motivate our ansatz for the massless solutions. Section 3.2 describes the massless solutions, which constitute the main results of this paper.
They seem at first sight not well behaved at infinity, a fact that would render them physically inadmissible. However, it is possible to construct solutions everywhere finite and with Gaussian decay. This is shown in section 3.2.1.
Section 4 is devoted to present the precise match between the solutions. The first order, when both gravity and the bilocal model are not self-interacting, matches exactly.
We see how we get gravitational waves from the bilocal field model in section 4.1. We claim that the match should hold at all orders, so in 4 we sketch the way in which they should be compared. Basically, the inclusion of self-interacting terms in the gravity side should generalize the free bilocal model to the one described by equations (4.15) and (4.16), for suitable potentials.
Before the concluding section we also present a short section 4.3, where we interpret some features of the model and we speculate about the meaning of the bilocal field.
Although this article is focused on classical theories, the bilocal model may be quantized. Therefore in section 4.3, we discuss an interpretation of the bilocal field as measuring probability amplitudes associated with identifying points. We also consider in this section the way the bilocal field should couple to other (local) matter fields.
Linearized gravity
First order equations of Einstein gravity are well-known and lead to the gravitational wave equations in the appropriate gauge. Linearized gravity equations are of the form
− h µν + h α ν ,µα + h α µ ,να − h ,µν − h αβ ,αβ η µν + η µν h = 16πGT µν . (2.1)
In this work we will consider only vacuum solutions like in (4.11), so T µν = 0. Upon taking trace-reversed variablesh
µν = h µν − 1 2 hη µν ,µν = A µν e iP x ,(2.5)
where P µ is a null vector, the gauge can be completely fixed leading to the familiar form
A = 0 0 0 0 0 A + A × 0 0 A × −A + 0 0 0 0 0 . (2.6)
The two remaining degrees of freedom, i.e. the two polarizations, are thus associated to
A + and A × .
Bilocal field model
The bilocal field models proposed by Yukawa [5] in the early 50's to describe mesons and confinement were extensively used, with slight modifications, in the following two decades. Yukawa's original model was improved by lowering the order of the differential equations, while the main features of the solutions remained the same. We will use the simplest model, which still has bilocality. It is described in detail in [3]. Here we offer a summary of it. Massive solutions of the model are reviewed in section 3.1 and massless solutions are found and described in 3.2.
The basic object of a bilocal model is the bilocal field Φ(x, y), which as indicated, depends on two points in spacetime as opposed to a single point as in local theories. It is common to work with the more physical coordinates given by the combinations
X µ = 1 2 (x µ + y µ ), r µ = x µ − y µ ,
which are referred to as centre of mass and relative coordinates, respectively. The field in the new coordinates will be called Φ(X, r), with the definition Φ(X, r) =Φ(x, y).
We will work in four dimensions, and assume µ = 0, 1, 2, 3. As is customary, we consider the bilocal field symmetric under the exchange x ↔ y. That is, we considerΦ(x, y) = Φ(y, x) or, equivalently,
Φ(X, r) = Φ(X, −r). (3.1)
This assumption is made for simplicity and ensures that the physics is invariant under the permutation of the two spacetime points determining a bilocal field.
We see that the effect of the bilocal field is to bring in new degrees of freedom, namely the dependence of the field on the relative coordinates. We will show that it is precisely in those degrees of freedom that gravity is encoded.
The model we are considering has two equations:
− 4U 0 + 4 r − 4α 4 r 2 Φ(X, r) = 0 (3.2) ∂ ∂X µ ∂ ∂r µ − α 2 r µ Φ(X, r) = 0, (3.3)
where is the d'Alembertian associated with the coordinates X µ , while r is the d'Alembertian associated with the coordinates r µ . Parameter α has dimensions of mass and U 0 has dimensions of mass 2 and are a priori arbitrary.
Equation (3.2) is dynamical whereas equation (3.3) is a constraint. In [3] the reader
can find motivations for constraint (3.3). The basic idea is that it serves to integrate the time mode of the relative dimension. In fact, it identifies the proper time for the two different points x and y, which in turn defines an action at a distance. The constraint (3.3) turns out to be crucial for the model to be well-defined and bilocal although, as commented in [3], different constraints than (3.3) are possible and have been used in the literature (see, for instance, [7]).
Equations (3.2) and (3.3) define a free system on the coordinates X. There are two branches of solutions for these equations, as we show in the following.
Massive solutions
Massive solutions of (3.2) and (3.3) where studied in detail in [3], and read
Φ n 1 n 2 n 3 ,P (X, r) = N n 1 n 2 n 3 e iP X exp − α 2 2 R µν r µ r ν i=1,2,3 H n i (αr i ),(3.4)
where N n 1 n 2 n 3 is a normalization constant that will be irrelevant for our discussion, H n i stands for Hermite polynomials, with n i = 0, 1, 2, 3... and P is the momentum four-vector of the center of mass. The symmetric tensor R µν needs to be
R µν = η µν + 2 P µ P ν M 2 , P 2 = −M 2 , (3.5)
where M 2 or the "rest mass" of the system is related to the parameters of the model as
M 2 = 4U 0 + 8α 2 (n − 1), n = i=1,2,3 n i .
We see that we can always boost the system to the centre of mass frame, where P = (M, 0). From (3.9) we can easily extract the basic properties
R µν | P =(M,0) = δ µν , R µ µ = 2, R µν R ν σ = η µσ , P µ (R µν + η µν ) = 0,
from which (3.4) are verified to be solutions of (3.2) and (3.3).
Massless solutions
There is another branch of solutions of (3.2) and (3.3) which we call "massless solutions", for which P 2 = −M 2 = 0. These massless solutions are new. It is likely that they were not studied before because they are irrelevant to the description of (massive) hadrons.
A null vector P µ can generally be written as P = K(1, n), where n is a threedimensional spatial vector such that δ ij n i n j = 1. Without loss of generality, we will choose from now on the unit vector n parallel to the z-axis, so P = (K, 0, 0, K).
The limit M → 0 is ill-defined in (3.5). This is related to the fact that a timelike vector cannot be boosted to a null vector. For this reason, "massive" and "massless"
solutions are not connected, and one cannot pass from one set to the other.
For massless solutions, we start with the ansatz
Φ λ,P (X, r) = N λ e iP X exp − α 2 2 R µν r µ r ν f λ (r) ,(3.6)
similar to (3.4). λ is a label (or a collection of labels) for the excited states, and N λ is the corresponding normalization constant, which is irrelevant for our discussion.
We search for solutions of (3.2) and (3.3) of the form (3.6). Although Φ is a scalar, spin-0 field, its nonlocality permits the Gaussian term involving the symmetric tensor R µν contracted with the relative coordinates. The magic of bilocality seems to reside in the fact the scalar field develops an internal structure that involves a symmetric two tensor, whose dynamics will match with those of linearized gravity in the following section.
First of all, since P µ is a null vector, it is clear that the ansatz (3.6) is a solution of the equation
Φ λ,P (X, r) = 0.
Therefore, in order for (3.6) to be a massless solution of (3.2) it must fulfill
− U 0 + r − α 4 r 2 Φ λ,P (X, r) = 0. (3.7)
The trivial solution R µ,ν = η µν is a solution of (3.3) and (3.7). This isolated solution comes along with U 0 = −4α 2 . Apart from the trivial case there are no other solutions of (3.3) and (3.7) unless we restrict ourselves to the hyperplane defined by
P µ r µ = 0, in which case, equation (3.3) forces P µ R µν = 0. Now, (3.7) requires that relations R iν R ν j = δ ij , i, j = 1, 2 (3.8)
hold. The matrix R µν has the following structure:
R = 0 0 0 0 0 a b 0 0 b d 0 0 0 0 0 . (3.9)
There are only two possibilities: either b = 0 and a = d = 1, which is the trivial solution (the identity in the subspace spanned by {r 1 , r 2 }), or
R = 1 det(−B) 0 0 0 0 0 a b 0 0 b −a 0 0 0 0 0 , B = a b b −a ,(3.10)
with a, b arbitrary real numbers. These are the solutions we will study.
Solutions (3.10) are traceless, for this reason U 0 must be 0 in (3.7). The linear
equation for functions f λ is − 2α 2 r σ R σµ ∂ µ + r f λ (r) = 0. (3.11)
The ground state corresponds to constant f λ , which we will put to 1, since functions will eventually be normalized. The f λ for the excited states are nontrivial polynomials involving the parameters a and b. Although we have not studied the general solutions of (3.11), the first excited state has been found to correspond to f (r) = α 2 (−br 2 1 + br 2 2 + 2ar 1 r 2 ).
Next we discuss about physically admissible solutions, which must vanish when r → ∞. In the context of quantum mechanics, admissible solutions are normalizable.
Classically too, solutions that do not decay at infinity are problematic, since we do not expect the effects of non-locality to be relevant for distant points. Having well-behaved solutions at infinity requires that the matrix R be positive definite, that is,
r µ r ν R µν > 0. (3.12)
The trivial solution is the only one that satisfies it, although, as said above, this solution leaves us with no degrees of freedom and, for that reason, it is of no interest for us.
Solutions (3.10) are not square integrable. Actually, taking (3.10) we see that
r µ r ν R µν = ar 2 1 − ar 2 2 + 2br 1 r 2 ,(3.13)
which make (3.6) blow up, upon assuming a > 0, when r 2 → ∞. These solutions should be physically inadmissible for this reason. We will see in the next subsection how to construct solutions that behave well at infinity.
Admissible solutions
In this section we will see how to construct physically admissible solutions of the form (3.6) defined on the whole (r 1 , r 2 )-plane. Let us focus on the ground state, so f λ = 1.
We will call Φ (a,b) (X, r) the ground state functions with R µν as in (3.10).
As said above, the sign of r µ r ν R µν tells us when solutions are well-behaved at infinity.
In fact, r µ r ν R µν > 0 guarantees that the functions decay as Gaussians. Without loss of generality we take a, b ≥ 0 from now on. Analyzing the sign of r µ r ν R µν in (3.13) we see that
r µ r ν R µν > 0 −→ r 1 > cr 2 , c = −b a + 1 + b 2 a 2 .
The constant c takes values in the interval (0, 1), so for the wedge r 1 ≥ r 2 ≥ 0 the sign of r µ r ν R µν is always positive, and the solutions Φ (a,b) (X, r) are physically admissible in this wedge. Let us notice that, for the same reason, the function Φ (−a,b) (X, r) makes the product r µ r ν R µν always positive in the wedge r 2 ≥ r 1 ≥ 0. One is tempted to consider the solution defined as Φ (a,b) (X, r) if r 1 ≥ r 2 ≥ 0, and as Φ (−a,b) (X, r) when r 2 ≥ r 1 ≥ 0, so it will behave well at infinity in the entire first quadrant. However we have to make sure that solutions are continuous and with continuous derivatives all over the plane, and this choice would not have continuous derivatives on the line r 1 = r 2 . We should impose continuity of the solutions and of the directional derivative ∂ r 2 − ∂ r 1 of the solutions along the line r 1 = r 2 . It is easy to see that the solution
Φ (a,b) (X, r) − Φ (0,b) (X, r), r 1 > r 2 ≥ 0 −Φ (−a,b) (X, r) + Φ (0,b) (X, r), r 2 ≥ r 1 ≥ 0
is continous and has continous derivatives everywhere, so it is a well behaved function defined on the first quadrant. Next, we notice that functions Φ (−a,−b) (X, r) decay as Gaussians in the wedge |r 1 | < r 2 and r 1 ≤ 0, and again we can find the way of gluing it together properly along the line r 1 = 0. This can be done till we complete the circle and the solutions are defined on the entire (r 1 , r 2 )-plane. So, for any a, b ≥ 0 we find a global solution ,0) , |r 1 | > |r 2 | and r 1 · r 2 ≤ 0.
Φ G (a,b) (X, r) = Φ I (a,b) ≡ Φ (a,b) , |r 1 | ≥ |r 2 | and r 1 · r 2 > 0, Φ II (a,b) ≡ −Φ (−a,b) + 2Φ (0,b) , |r 2 | > |r 1 | and r 1 · r 2 ≥ 0, Φ III (a,b) ≡ Φ (−a,−b) − 2Φ (0,−b) − 2Φ (−a,0) + 4, |r 2 | ≥ |r 1 | and r 1 · r 2 < 0, Φ IV (a,b) ≡ −Φ (a,−b) + 2Φ (a
(3.14)
Solutions (3.14) are shown in the figure and are everywhere regular.
E r 1 r 1 T r 1 = r 2 r 1 = r 2 d d d d d d d d d d d d d d d d d d d d d d r 1 = −r 2 r 1 = −r 2 r 2 r 2 Φ I (a,b) Φ I (a,b) Φ II (a,b) Φ II (a,b) Φ III (a,b) Φ III (a,b) Φ IV (a,b) Φ IV (a,b)
The global solution as we glue the 8 wedges. Note the symmetry Φ G (a,b) (X, r) = Φ G (a,b) (X, −r).
Gravity from a bilocal scalar field
We now show how to obtain gravity from the bilocal model, perturbatively. For the first order we will see that the match is perfect: massless solutions of the free bilocal model match gravitons. Next, we will show how to extend the bilocal model (3.2) and (3.3) perturbatively and how it should be compared with perturbative gravity, to higher orders.
Let us write our bilocal field in the center of mass and relative coordinates (X, r)
given by (3) and Taylor expand it on the coordinate r around r = 0. Remember that Φ(X, r) is even in the relative coordinates. We have Φ(X, r) = φ(X) + H µν r µ r ν + D µνσρ r µ r ν r σ r ρ · · · , (4.1)
with the identification
φ(X) ≡ Φ(X, 0) H µν (X) ≡ 1 2 ∂ µ ∂ ν Φ(X, r) r=0 D µνσρ (X) ≡ 1 4! ∂ µ ∂ ν ∂ σ ∂ ρ Φ(X, r) r=0 . . . (4.2)
where the partial derivatives are related to the coordinates r. This is the way of seeing that the bilocal scalar field is equivalent to a unique tower of local higher spin fields.
We will ignore the role of the higher spin fields in this work and focus on the lowest contribution H µν of the expansion (4.1), which makes contact with gravity.
First order
Equations (3.2) and (3.3) will be considered first order in perturbation theory, which will be carried out with the parameter κ B . Inserting where c is a constant, we recover in (4.4) the Lorentz gauge condition of linear gravity.
Note that fields H µν are functions of the center of mass coordinates X, which are to be associated with spacetime coordinates in gravity. For example, using the definitions (4.2) in (4.5) for the case in which Φ(X, r) is in the ground state of (3.6) makes
h µν = −cα 2 R µν e iP X .
It is natural to assign the value c = −1/α 2 , so we recover the usual shape of the first order solutions in gravity.
Again, using the ansatz (3.6) we immediately see that
h µν = 0,b √ a 2 + b 2 ←→ A × , a √ a 2 + b 2 ←→ A + .
A question arises from (4.6). What is the use of the full operator (3.2)? We remind that for massless solutions we have Φ(X, r) = 0 which, combined with (3.2), leads to There is no gauge freedom in our bilocal model. An easy way to see this is to realize that if R µν leads to a solution of the model, then R µν + ∂ (µ ξ ν) does not. This is consistent with the statement that the match with gravity happens in a specific gauge, which is precisely the gauge where gravitational waves are found.
subgroup of the Poincaré group that stabilizes a null 4-momentum. The group E(2) has three generators, which can be visualized by their actions on the 2-plane: one of them generates rotations, and the other two generate translations on the plane. It is known that out of the three generators of E(2) the two associated with translations produce gauge transformations [11,12] Another nice surprise is the π/4-symmetry pattern of the solutions Φ G (a,b) (X, r) as can be seen in (3.14). Notice that in the bilocal model there is not a natural concept of helicity, or we could say that its helicity is 0, as it is for scalar fields. The π/4 pattern of the global solution is forced by physical requirements of admissibility, nothing else. The internal space "mimics" a spin-2 field this way, and it reproduces the well-known π/4 pattern of the "cross" and "plus" polarizations of gravitational waves.
In summary, once we fix the appropriate gauge in gravity, the theories match perfectly, at least to linear order in perturbation theory.
Second and higher orders in perturbation theory
In this subsection, we sketch how both theories, gravity and bilocal model, should be compared at second and higher orders in the perturbative expansion. First, we review the perturbative expansion as usually performed in gravity.
Perturbative gravity
In perturbation theory one assumes the existence of a one-parameter family of solutions g(κ) µν to Einstein's equations
R µν − 1 2 g µν R + Λg µν = 8πGT µν . (4.7)
The perturbation equations get generated as the Einstein's equations are expanded in powers of κ and the terms with equal powers are equated. In this paper, for simplicity, we will consider pure gravity with no Cosmological Constant, so
G µν ≡ R µν − 1 2 g µν R = 0. (4.8)
Now, both the solutions and the operator G µν are to be expanded in powers of κ. Any new solution of the κ-family 1 is expanded around flat metric as g µν (κ) = η µν + κh (1) µν + κ 2 h (2) µν + · · · , (4.9)
where h (i) µν are order i contributions to the solution g µν (κ) and are found as solutions of some (linear) differential equations when solved in ascending order. To find out the tower of equations that h (i) µν solves, we expand the Einstein's tensor. This is done by writing
G µν [η ab + κh ab ] = G (0) µν [η ab ] + κG (1) µν [h ab ] + κ 2 G (2) µν [h ab ] + · · · .
(4.10)
A general order operator is them computed as
G (n) µν [h ab ] = 1 n! d n dκ n G µν [η ab + κh ab ] κ=0 .
Performing expansions (4.9) and (4.10) and equating equal powers of κ we get the tower of equations
0 = G (0) µν [η ab ] 0 = G (1) µν [h (1) ab ], (4.11) 0 = G (1) µν [h (2) ab ] + G (2) µν [h (1)
ab ], (4.12) . . . (4.13) To find h (2) ab one must solve (4.12) with the linear solutions h (1) ab found by solving the first order. G (1) is a linear differential operator. The operator G (2) [h] produces 24 terms which are schematically either type h∂ 2 h or type (∂h) 2 with different index contractions.
The usefulness of the perturbative method is that computing h (1) is just solving linear differential equations. Then, one computes G (2) [h (1) ] by inserting the obtained solutions.
So G (2) [h (1) ] is a known function and then equation (4.12) is just a set of linear differential equations for h (2) . The solution will seed the third order perturbation equations, and so on. So, at end of the day, one can find approximate solutions to Einstein equations by solving, iteratively, sets of linear differential equations.
Perturbative bilocal model
As it is customary in perturbation theory, we expand the solutions in powers of κ B , so 3) . . . (4.14)
Φ = Φ (1) + κ B Φ (2) + κ 2 B Φ (
1 The parameter κ is actually dimensionful. By consistency with the non-perturbative treatment it is found to be κ = √ 16πG, where G is the Newton constant. (4.15) and
+ 4 r − 4α 4 r 2 Φ = V (Φ),(3.3) by ∂ ∂X µ ∂ ∂r µ − α 2 r µ Φ(X, r) = W (Φ),(4.16)
where we have already take into account that U 0 = 0 for massless solutions. V (Φ) and W (Φ) will be expanded in powers of a coupling κ B with the correct dimensions 2 . So, after plugging (4.14) into (4.15) and (4.16) we get a tower of equations similar to (4.13), an equation for any power of κ B . The claim we make in this paper is that after suitable choices of the potentials V (Φ) and W (Φ), solutions of both systems should match order by order in κ, κ B being a function of κ. We have already checked it out for the first order, for which linearized gravity solutions, that is, solutions of (4.11), have been proven to match those of linear (4.15) and (4.16), when their RHS is 0. We will now write
+ 4 r − 4α 4 r 2 Φ (1) (X, r) = 0,(4.17)
and
∂ ∂X µ ∂ ∂r µ − α 2 r µ Φ (1) (X, r) = 0
as first order equations.
The natural second order contribution for bilocal theories is a term like
κ B Φ (x, z)Φ(z, y)dz, (4.18)
which is a function of x and y, and so of X and r. The second order equation is obtained by equating terms proportional to κ B in (4.15) and (4.16). For instance, from equation (4.15) we will have the κ-second order equation
+ 4 r − 4α 4 r 2 Φ (2) (X, r) = Φ (1) (x, z)Φ (1) (z, y)dz,(4.19)
where Φ (1) are solutions of (4.17). The functions Φ (2) (X, r) lead, after derivation, to functions H (2) µν (X) which will be identified with second order contributions in gravity h (2) .
The identification we claim is nontrivial. In gravity, the seed functions h (1) enter in the second order equations with their derivatives, whereas in the bilocal model they get integrated, as in (4.19). It is the special properties of Gaussians with respect to derivation and integration what are expected to make it possible.
Physical interpretation of the bilocal model
The bilocal model defined by equations (3.2) and (3.3) is probably the simplest model that accounts for genuine bilocality (with an action at a distance defined by (3.3)), at short range. For the last purpose it is crucial that, in the relative coordinates, we have
( r − α 4 r 2 )Φ(X, r) = 0,
with the quadratic term that produces Gaussian solutions. As we saw in Section 3.2, this quadratic term is also responsible for the solutions to match term-by-term with gravitational waves. Therefore, all in all, it seems that forcing short range non-locality in the bilocal model inevitably gives rise to graviton-like solutions.
The parameter α in our bilocal field model measures non-locality. The limit
lim α→∞ α √ 2π e − 1 2 α 2 x 2 = δ(x),
implies that the relative dimensions vanish for large α. So, one is left with a local theory in the limit α → ∞.
Let us speculate about the physical meaning of the bilocal field itself. First of all, if we declare that an observer cannot, by any means, resolve very small distances, an "honest" theory of gravity should take this limitation into account. This means that very close points should be identified since they are indistinguishable. Of course, this restriction should vanish as we consider well-separated points. We will assign 3 to a pair of points (x, y) a probability P (x = y) that decays with the distance between x and y.
The easiest probability distribution that makes the job with just one parameter, α, is the Gaussian
P (x = y) = e − α 2 2 d 2 (x,y) ,
which decays exponentially for distances greater than 1/α. We suggest that such a probability is encoded in the bilocal field, once it is quantized. Along these lines, we could also interpret the bilocal fieldΦ(x, y) as the probability amplitude of identifying points x and y when we try to resolve them.
On the other hand, the identification of spacetime points at small distances will result in a complex topology at those scales and, in turn, they should modify the usual commutator relations of matter fields. For instance, for scalar field φ one should replace
[φ(x), φ † (y)] ∝ δ(x − y) → [φ(x), φ † (y)] ∝ P (x = y),
which ensures that for delta distributions we recover the usual commutator relations.
This will apply to all matter fields as it should, since gravity should couple to everything.
Conclusion
In this paper we have investigated the relation between gravity and non-locality at the classical level. Non-locality has been implemented with a simple bilocal scalar field model. Bilocality introduces new (internal) degrees of freedom that can accommodate gravity. Specifically, we have shown that the massless branch of solutions of the free scalar field match exactly those of linearized gravity. To our knowledge, this is the first time that massless solutions of this model have been studied. We have then shown how to proceed in order to match solutions in higher orders in perturbation theory, where there is self-interaction in both gravity and the bilocal field.
Although we do not provide a formal proof, we offer strong evidence for the emergence of gravity from non-locality. The claim is that full gravity can be obtained from interacting bilocal models as defined by equations (4.15) and (4.16). If this is correct, then we believe that the analysis of this paper goes beyond the bilocal field to general non-local fields. A non-local theory that is effectively local at large scales is expected to be effectively bilocal to leading order, much as the dipole term dominates a multipole expansion.
The classification of non-local theories can be made by considering the number of points that define the field. Therefore we have bilocal, trilocal..., and when the field depends on an infinite number of points, one can obtain classical string theory [4]. From the point of view of locality, bilocal theories are specially interesting since they encode departures from locality in the simplest form. We think that they deserve more attention.
There are many lines along which this work can be extended. From the classical point of view it would be interesting to test the match of gravity and bilocal field in second and higher orders [15]. It is known that one can obtain a full theory of gravity in a unique way starting from linearized gravity and imposing reasonable consistency conditions [16,17]. It would be interesting to know what these conditions translate into in the bilocal model. This, in the end, would tell us about the full shape of potentials V (Φ) and W (Φ) in equations (4.15) and (4.16).
The background we have used as the starting point for the perturbative expansion is the Minkowski metric. It should not be hard to adapt the model to accommodate a cosmological constant, so that the expansions would be performed around de Sitter or anti-de Sitter backgrounds.
If solutions match it is natural to think that there should be some equivalence at the level of actions [18]. This should be useful for subsequent quantization. It would also shed light on the way general covariance enters on the bilocal field side. In the same spirit, gauge invariant variables could be considered. Gauge invariant variables [19] have been proven to be extremely useful for computations in perturbation theory. It would be interesting to see how this formalism is realized in a bilocal model. It can potentially provide a gauge invariant picture.
One of the initial motivations of this work was to find an appropriate setup for quantizing gravity. It is believed that the bilocal theories are, in general finite. The absence of divergence is essentially due to the existence of a fundamental length. In this work, this length scale is α, which measures non-locality. Once it is understood how bilocal fields encode gravity, it seems possible to keep track of the "induced" metric field after quantizing the bilocal field. We hope that this points the way toward an eventual finite theory of quantum gravity.
Global solutions ( 3 .
314) are essentially the only choice to assemble a well-behaved solution. Because our free model is linear, linear superpositions lead to other global solutions. A general solution can be written asΦ G A (X, r) = a,b∈ + A(a, b)Φ G (a,b) (X, r) da db,(3.15)for any real square-integrable function A(a, b).
tower of equations as we equate powers of r. Those equations involve all the higher spin fields although in this article we will focus on spin 2. The lowest power of rin (3.3) leads to ∂ ∂X µ (2H µν − α 2 η µν φ)r ν = 0 ∀r −→ ∂ ∂X µ (2H µν − α 2 η µν φ) = 0. (4.3)If we take the ansatz (3.6), for which P µ r µ since (3.3) is a constraint, it is natural to associate it with a constraint in gravity. With the identification cH µν ←→h µν , (4.5)
as the dynamical equation of linear gravity in the Lorentz gauge. More specifically, solutions of the form (3.6) with (3.10), and so solutions (3.14), lead to (the complete fixed gauge) gravitational waves upon the identifications
(3. 7 )
7. Equation (3.7) is not dynamical with respect to coordinates X, so it must be seen as a constraint. Actually, as explain in subsection 3.2, it is equation (3.7) which constraints the shape of R µν in the solutions to be of the form (3.10) which, in the end, is to be associated with the gravitons in the fully fixed gauge: (2.5) with (2.6). So, all in all, the dynamics of the relative coordinates in the bilocal field model are seen as constraints from the gravity point of view.
for the electromagnetic field and gravity. The only one which does not affect the gauge is the generator of rotations. Since there is no gauge symmetry in the bilocal model the solutions can only involve SO(2) ⊂ E(2), as seen from the gravity point of view. The other two generators must act trivially, as it happens in the usual gauge of gravitational waves. This is a consistency check since, as we saw in section 3.2, massless solutions (3.10) are general for the bilocal model.
At first sight it could seem strange to plug a potential W (Φ) in (4.16), since equation (4.16) is a constraint. Actually, (4.16) matches the Lorentz gauge condition for the free case. However, this constraint may change in gravity for higher orders of perturbation, as noticed by Wald[17]. So the presence of a nonzero W (Φ) for higher orders in perturbation theory must be taken into account accordingly in the bilocal model.
Note that to make this claim it is essential that the symmetry Φ(X, r) = Φ(X, −r), assumed at the begining, holds.
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AdS4/CFT3 Construction from Collective Fields. R De Mello Koch, A Jevicki, K Jin, J P Rodrigues, Phys.Rev. 8325006R. de Mello Koch, A. Jevicki, K. Jin, J. P. Rodrigues, AdS4/CFT3 Construction from Collective Fields, Phys.Rev. D83 (2011) 025006.
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| [] |
[
"Transformations of q-boson and q-fermion algebras",
"Transformations of q-boson and q-fermion algebras"
] | [
"P Narayana Swamy \nPhysics Professor Emeritus\nSouthern Illinois University\n62026EdwardsvilleILUSA\n"
] | [
"Physics Professor Emeritus\nSouthern Illinois University\n62026EdwardsvilleILUSA"
] | [] | We investigate the algebras satisfied by q-deformed boson and fermion oscillators, in particular the transformations of the algebra from one form to another. Based on a specific algebra proposed in recent literature, we show that the algebra of deformed fermions can be transformed to that of undeformed standard fermions. Furthermore we also show that the algebra of q-deformed fermions can be transformed to that of undeformed standard bosons. a electronic address: [email protected] PACS numbers: 05.30.-d; 05.20.-y; 05.70.-a Typeset using REVT E X | 10.1142/s0217984901002671 | [
"https://arxiv.org/pdf/quant-ph/0109062v1.pdf"
] | 18,384,087 | quant-ph/0109062 | 5f4059f01b6ad0336005eceba19d3cd45bbd69dd |
Transformations of q-boson and q-fermion algebras
arXiv:quant-ph/0109062v1 13 Sep 2001
P Narayana Swamy
Physics Professor Emeritus
Southern Illinois University
62026EdwardsvilleILUSA
Transformations of q-boson and q-fermion algebras
arXiv:quant-ph/0109062v1 13 Sep 2001
We investigate the algebras satisfied by q-deformed boson and fermion oscillators, in particular the transformations of the algebra from one form to another. Based on a specific algebra proposed in recent literature, we show that the algebra of deformed fermions can be transformed to that of undeformed standard fermions. Furthermore we also show that the algebra of q-deformed fermions can be transformed to that of undeformed standard bosons. a electronic address: [email protected] PACS numbers: 05.30.-d; 05.20.-y; 05.70.-a Typeset using REVT E X
The study of quantum groups have led to theories of q-deformed boson and fermion oscillators based on a deformation [1] of the standard algebra of boson and fermion creation and annihilation operators. It has recently been shown that the thermostatistics of q-deformed bosons can be realized in a self-consistent manner if the statistical mechanics of such systems is developed by using the q-calculus based on Jackson derivatives [2]. Recent interest has focussed on the proposition [3,4] that the q-deformation of fermion oscillators may or may not reduce to the algebra of ordinary fermions obeying Pauli principle depending on the particular algebra that is postulated. Chaichian et al [5] have pointed out the importance of transformations which transform such algebras from one form to another, while investigating the connection between q-oscillator statistics and fractional statistics. Specifically Chaichian et al argue that there is a special formulation of the q-deformed fermion algebra which cannot be reduced to the algebra of ordinary fermions in this manner. In view of the importance of such transformations which map one kind of algebra to another, it is quite important to investigate the nature of such transformations in the most general context.
In this note we study the general properties of such transformations generated by operator functions of the number operator and conclude that transformations from deformed fermions to ordinary fermions or ordinary bosons is possible.
Let us begin with the standard boson algebra defined by the commutation relations
a 0 a † 0 − a † 0 a 0 = 1, [N, a 0 ] = −a 0 , [N, a † 0 ] = a † 0 , [ a 0 , a 0 ] = [ a † 0 , a † 0 ] = 0 ,(1)
where the number operator is N = a † 0 a 0 . We have suppressed the particle index for simplicity. The q-deformed bosons [1] can be introduced by the algebra
aa † − qa † a = q −N , [N, a] = −a, [N, a † ] = a † ,(2)
where q is a positive real number, N is the q-boson number operator and a, a † , are the annihilation, creation operators respectively. The number operator is given by
N = 1 2 ln q ln aa † + q −1 a † a aa † + qa † a(3)
and depends on q. We note that N = a † a only in the limit q → 1 when the algebra in Eq. (2) reduces to that in Eq.(1). The Fock representation of this algebra for the eigenstates |n of the number operator is enabled by the construction
|n = (a † ) n [n]! |0 , a|0 = 0, N|n = n|n .(4)[x] = q x − q −x q − q −1 .(5)
In this Fock space, the following relations are easily obtained: a † a = [N], aa † = [N + 1] . It is of importance to discuss possible transformations on the creation and annihilation operators which can transform the algebra from one form to another. For instance [5], the transformation defined by
a 1 = q N/2 a, a † 1 = a † q N/2 ,(6)
introduces new operators which satisfy the algebra
a 1 a † 1 − q 2 a † 1 a 1 = 1.(7)
In this representation, we find a † 1 a 1 = q N −1 [N] and a 1 a † 1 = q N [N + 1]. We shall now consider a general transformation of the q-boson algebra by introducing
A = f 1 2 a , a † = a † f 1 2 ,(8)
where f = f (N) can be taken to be a general function of N. In order to study such transformations in a general context, let us first develop a useful construction, a lemma.
Consider an operator f (N) which is a function of the number operator. We can establish the following well-known fundamental relation [3,4] , which we shall refer to as the commutation property:
f (N)a = af (N − 1), f (N)a † = a † f (N + 1),(9)
which follows from Eq. (2)for any polynomial f (N). Eq. (7) is then easily established using this construction. Later we shall prove this commutation property for a general class of functions.
Let us now consider a different transformation
a 2 = f 1 2 a, a † 2 = a † f 1 2 ,
where f stands for a real polynomial function f (N), satisfying the commutation property. We may now ask under what conditions the transformed operators satisfy the algebra a 2 a † 2 − a † 2 a 2 = 1. We see by using the commutation property that this requires the necessary condition
[N + 1]f (N) − [N]f (N − 1) = 1.(10)
One solution is f (N) = (N + 1)/[N + 1]. Consequently, the transformation
a = [N + 1] N + 1 1 2 a 2 , a † = a † 2 [N + 1] N + 1 1 2(11)
transforms it to the undeformed standard boson algebra
a 2 a † 2 − a † 2 a 2 = 1, [N, a 2 ] = −a 2 , [N, a † 2 ] = a † 2 .(12)
This is identical with the algebra of a 0 , a † 0 , Eq.(1). That the q-deformation of bosons defined by Eq.(2) can be transformed away in this manner is a known result [5].
Since such transformations are quite interesting and useful in the analysis of q-deformed boson and fermion oscillators, we must investigate the general validity of Eq.(9) in more detail. In what follows, a referring to Eq.(9), represents a generic annihilation operator obeying [N, a] = −a, [N, a † ] = a † , and could represent either the q-boson algebra or the q-deformed fermion algebra. First, the commutation property is clearly valid for N, as is evident from Eq.(2). The proof then extends to N 2 or for any monomial N r and hence for a polynomial function of N. It is easy to verify the property for the case when f = 1/N and then for any negative power of N. We can then extend the proof to the case when f (N) = q N by observing that q N a = e N ln q a = 1 + N ln q + N 2 2! (ln q) 2 + · · · a (13)
= a 1 + (N − 1) ln q + (N − 1) 2 2! (ln q) 2 + · · · = aq N −1 .(14)
Similarly, the property is valid for q −N . In all these cases, it can be easily proved that the transformed operators a, a
(−1) N q N f = e iπN e N ln q f = 1 + N(ln q + iπ) + N 2 2! (ln q + iπ) 2 + · · · f (16) = f (−1) N −1 q N −1 .(17)
This enables us to extend the validity of the commutation property for [N] F , the q-deformed fermionic basic number introduced in ref. [5] and hence to operators such as [N +1] F /(N +1). We have thus proved the validity of the commutation property, Eq.(9), for a large class of operator functions of the number operator N. We shall now proceed to derive some interesting consequences of such transformations. We begin with the algebra of the standard fermions defined by
b 0 b † 0 + b † 0 b 0 = 1, [N, b 0 ] = −b, [N, b † 0 ] = b † 0 , { b 0 , b 0 } = { b † 0 , b † 0 } = 0 ,(18)
where the fermion number is given by N = b † 0 b 0 . It follows from the above relations that b 2 0 = 0, (b † 0 ) 2 = 0 and thus Pauli exclusion principle is contained in the algebra. Next let us proceed to investigate the q-deformation theory based on the q-fermion algebra [5] by introducing
bb † + qb † b = q −N , [N, b] = −b , [N, b † ] = b † ,(19)
where N is the number operator of the q-fermions. The q-fermionic basic number is defined by
[x] F = q −x − (−1) x q x q + q −1 ,(20)
which is quite different from the basic number, Eq.(5), introduced in the theory of q-bosons. The eigenstates of the number operator in this theory are
|n = (b † ) n [n] F ! .(21)
This algebra plays a major role in the theory [3,5] of generalized q-deformed fermions, whose characteristic feature is that states with more than one fermion are allowed, thus going beyond the Pauli exclusion principle: b 2 = 0, (b † ) 2 = 0 is realized only in the limit q → 1 and not true generally when q = 1. We then have
b † b = [N] F , bb † = [N + 1] F .(22)
We now ask if we can transform this algebra to the algebra of ordinary undeformed fermions. For this purpose, consider the transformations
b → b 1 = T (N)b, b † → b † 1 = b † T (N) .(23)
If we choose
T (N) = 1 2[N + 1] F 1 2 ,(24)
then the algebra reduces to
b 1 b † 1 + b † 1 b 1 = 1 ,b 1 b † 1 + b † 1 b 1 = 1, [N, b 1 ] = −b 1 , [N, b † 1 ] = b † 1 .(25)
We can prove that this algebra is consistent with the exclusion principle: explicitly,
{b 1 , b 1 } = {T b, T b} = T {b, T }b − T T {b, b} + {T, T }bb − T {T, b}b .(26)
The first and the last terms cancel and the right hand side then vanishes identically. This algebra is then no different from the one in Eq.(18). Consequently we have shown that there is a transformation which can transform the deformation away, reducing the non-exclusion algebra to an algebra with exclusion, hence our conclusion is different from that of ref. [5]. We can prove another interesting result: the deformed q-fermion algebra can be transformed into ordinary undeformed boson algebra, thus implying a q-fermion-boson transmutation. The desired transformation is
b → a 2 = N + 1 [N + 1] F b, b † → a † 2 = b † N + 1 [N + 1] F(27)
is such a transformation. It can be verified explicitly that the operators a 2 , a † 2 satisfy the relations
a 2 a † 2 − a † 2 a 2 = 1, [N, a 2 ] = −a 2 , [N, a † 2 ] = a † 2(28)
and we also find a 2 a † 2 = N + 1 and a † 2 a 2 = N. This is thus identical with the algebra of ordinary bosons as in Eq.(1).
The transformation which leads to the undeformed boson algebra is very intriguing. We shall conclude by pointing out that the transformed operators a 2 , a † 2 satisfies all the expected properties. We know that the boson operators must obey [a 2 , a 2 ] = 0, [a † 2 , a † 2 ] = 0. We can prove this explicitly as follows. From Eq.
(27) with a 2 = T b = (N + 1)/[N + 1] F 1 2 b, we immediately obtain [a 2 , a 2 ] = [T b, T b] (29) = T {b, T }b − T T {b, b} + {T, T }bb − T {T, b}b .(30)
The first and fourth terms cancel out and we find
[a 2 , a 2 ] = −T 2 {b, b} + {T, T }bb = −2T 2 bb + 2T 2 bb ≡ 0.(31)
We do not require b 2 = 0 and thus the above result is true for any q. Similarly a † 2 commutes with itself. Finally the following results can be obtained immediately:
[N, a 2 ] = [a † 2 a 2 , a 2 ] = −a 2 , [N, a † 2 ] = [a † 2 a 2 , a † 2 ] = a † 2(32)
Hence we have shown that all of the familiar standard properties of the boson algebra for a 2 , a † 2 follow while the operators b, b † satisfy the q-deformed fermion algebra, without requiring b 2 = 0, (b † ) 2 .
In conclusion, we may state that the algebra of deformed fermions can be transformed to that of undeformed standard fermions as well as to that of undeformed standard bosons. This indicates a q-fermi-q-bose transmutation. It would be worthwhile to investigate whether these transformations would allow an interpolation between Fermi and Bose distributions. It is interesting to note that Vokos et al have mentioned the possibility of q-fermions behaving like q-bosons [6] and Lutzenko et al have demonstrated [7] the behavior of q-bosons as quasifermions.
ACKNOWLEDGMENTS
It is a pleasure to thank Andrea Lavagno for many useful discussions and for interest in this work.
† satisfy the commutation relations [N, a] = −a and [N, a † ] = a † . For instance, in the case of the transformation used in Eq.(11), due to Eq.turn our attention to more general functions. The commutation property is true for the basic number [N] since it contains monomials. Alternatively we can readily verify it for this case since [N + 1]a = aa † a = a[N]. This can be readily extended to power functions such as [N] r and to fractional powers of the basic number such as [N] 1 2 and [N] − 1 2 and hence functions of the form [N +1]/(N +1). We can next extend the proof to the operator function (−1) N q N by observing
which is the algebra of undeformed standard fermions. Next we obtain [N, b 1 ] = [N, T (N)b] = T [N, b] = −b 1 and similarly [N, b † 1 ] = b † 1 . Consequently the transformed operators b 1 , b † 1 satisfy
. L Biedenharn, J. Phys. A. 22873L.Biedenharn, J. Phys. A 22, L873 (1989);
. A Macfarlane, J. Phys. A. 224581A. Macfarlane, J. Phys. A 22, 4581 (1989).
See also all the references cited here. A Lavagno, P. Narayana Swamy, Phys. Rev. 611218A. Lavagno and P. Narayana Swamy, Phys. Rev. E61, 1218 (2000). See also all the references cited here.
. R Parthasarathy, K S Viswanathan, J. Phys. 24613R. Parthasarathy and K.S. Viswanathan, J. Phys. A24, 613 (1991);
. K S Viswanathan, R Parthasarathy, R Jagannathan, J. Phys. 25335K.S. Viswanathan, R. Parthasarathy and R. Jagannathan, J. Phys. A25, L335 (1992).
. S Jing, J Xu, J. Phys. 24891S. Jing and J. Xu, J. Phys. A24,L891 (1991).
See also all the references cited here. M Chaichian, R Gonzalez Felipe, C Montonen, J. Phys. 264017M. Chaichian, R. Gonzalez Felipe and C. Montonen, J. Phys. A26, 4017(1993). See also all the references cited here.
. S Vokos, C Zachos, Mod. Phys. Lett. 9S. Vokos and C.Zachos, Mod. Phys. Lett. A9, 1-9 (1994).
. I Lutzenko, A Zhedanov, Phys.Rev. 5097I. Lutzenko and A. Zhedanov, Phys.Rev. E50, 97 (1994)
| [] |
[
"SYMBOLIC DYNAMICS OF ODD DISCONTINUOUS BIMODAL MAPS",
"SYMBOLIC DYNAMICS OF ODD DISCONTINUOUS BIMODAL MAPS"
] | [
"Henrique M Oliveira "
] | [] | [] | Iterations of odd piecewise continuous maps with two discontinuities, i.e., symmetric discontinuous bimodal maps, are studied. Symbolic dynamics is introduced. The tools of kneading theory are used to study the homology of the discrete dynamical systems generated by the iterations of that type of maps. When there is a Markov matrix, the spectral radius of this matrix is the inverse of the least root of the kneading determinant. | 10.12785/amis/oliveira | [
"https://arxiv.org/pdf/1503.01593v2.pdf"
] | 56,332,047 | 1503.01593 | 43bffbad5edaf392139bf6df3878f0382b3a3887 |
SYMBOLIC DYNAMICS OF ODD DISCONTINUOUS BIMODAL MAPS
5 Mar 2015
Henrique M Oliveira
SYMBOLIC DYNAMICS OF ODD DISCONTINUOUS BIMODAL MAPS
5 Mar 2015
Iterations of odd piecewise continuous maps with two discontinuities, i.e., symmetric discontinuous bimodal maps, are studied. Symbolic dynamics is introduced. The tools of kneading theory are used to study the homology of the discrete dynamical systems generated by the iterations of that type of maps. When there is a Markov matrix, the spectral radius of this matrix is the inverse of the least root of the kneading determinant.
Introduction
In this paper we apply techniques of Markov partitions and kneading theory to the study of iterates of discontinuous maps of the interval (or the real line) in itself. We show that these systems can be studied with a proper framework, which is related to kneading theory and Markov matrices.
We cite, as examples of discontinuous one dimensional cases, the Lorenz maps, Newton maps, circle and tree maps, see [1,2,3,7] among other literature.
In [9] Lampreia and Sousa Ramos studied symbolic dynamics of continuous bimodal maps on the compact interval. Using similar techniques, we study in this paper the case of symmetric (odd) discontinuous maps in the real line or some suitable interval with two discontinuity points and three maximal intervals (laps) of continuity, which are as well maximal intervals of monotonicity. We call to this type of mapping a symmetric bimodal discontinuous map because of the existence of exactly three laps as in the continuous bimodal case.
In section two, we introduce the notation, the main definitions and revision of basic results. We include as well, the notions of symbolic dynamics, kneading theory and Markov partitions. We relate these concepts with lap growth number. We tried to define with great detail all the concepts presented. Since good definitions are essential for the constructive proof of the main result, which is actually done along the full length of the paper, section two is relatively long.
In section three, we present the main result of the paper, i.e., the spectral radius of the Markov matrix is the inverse of the least root of the kneading determinant for that kind of maps. We point out that the introduction of the linear operator µ in section three is one of the main ideas of this paper along with the matrix Θ relating the kneading and the Markov data. The linear transformation µ, representing the symmetry of this type of non-continuous maps, is completely different of its continuous counterpart [9]. We think that the proof of the result can be instructive giving methods that can be applied to other non-continuous mappings.
1.1. Motivation. The iterates of the complex tangent family λ tan z, introduced in [5] and [8], when the parameter λ = iβ is pure imaginary and the initial condition x 0 is a real number can be identified with the iterates of the real alternating map [f 1,β , f 2,β ] [4] in the real line
x 1 = f 1,β (x 0 ) = β tan (x 0 )
x 2 = f 2,β (x 1 ) = −β tanh (x 1 )
x 3 = f 1,β (x 2 ) x 4 = f 1,β (x 3 ) .
. . .
The composition map g β is
(1) g β : R → R x → −β tanh (β tan (x)) ,
which can be interpreted as the second return map to the real axis for the mapping λ tan (z). Knowing x 0 and g β , we obtain all the even iterates of the original system. To obtain the odd iterates knowing the even iterates is easy
x 2n+1 = β tan (x 2n ) .
The geometric behavior of the maps g β in this family depends on the parameter β. The map is periodic and the real line is mapped on the interval I = (−β, β). We restrict the map only to the interval I. When π 2 < β < 3π 2 the maps g β have two discontinuities. The study of the real projection of the complex tangent map is a good clue to the dynamics in the complex plane, similarly to the case of quadratic maps.
In this paper, we center our study on the symbolic dynamics of the iterates of maps F with the same geometrical properties of g β . Considering that the tangent family was an initial motivation and a good example, we point out that the results are independent on the choice of the family.
(1) F is odd F (x) = −F (−x) (2) F is piecewise continuous having two discontinuities c 1 < c 2 , c 1 = −c 2 ,
where lim x→c ± i F (x) = ±a, and lim x→±a F (x) = ±b, where b is a real number.
(3) F is decreasing in every interval of continuity (−a, c 1 ), (c 1 , c 2 ) and (c 2 , a).
Example 2.2. The family of maps g β defined in (1) is a family of bimodal symmetric discontinuous maps.
Definition 2.1 applies to maps with infinite jumps at c 1 and c 2 as we can see in the next example. Actually, as we see in the next example, any such map is smoothly conjugated to a map with finite jumps via a diffeomorphism.
Example 2.3. Consider u : R → R such that u (x) = −1 if x ≤ − 1 2 0 if − 1 2 < x < 1 2 1 if 1 2 ≤ x , the family G α : R → R x → x 4x 2 −1 − α u (x)
is a family of bimodal symmetric discontinuous maps with b = −α, c 1 = − 1 2 , c 2 = 1 2 and a = +∞. Any map in this family satisfies definition 2.1 and is smoothly conjugated to a map with finite jumps using for instance the diffeomorphism h (x) = arctan (x) such that
G α (x) = h • G α • h −1 (x) , x ∈ − π 2 , π 2 .
The map G α can be prolonged by continuity to the endpoints ± π 2 of the interval, since lim
x→± π 2 h • G α • h −1 (x) = lim x→±∞ h • G α (x) = ∓ arctan (α) .
Symbolic dynamics.
For sake of completeness and readability we introduce here briefly notions well known like orbit, periodic orbit and symbolic itinerary among other concepts, see for instance [6].
Definition 2.4. We define the orbit of a real point x 0 as a sequence of numbers
O (x 0 ) = {x j } j=0,1,... such that x j = F j (x 0 ) where F j is the j-th composition of F with itself.
Definition 2.5. Any point x is periodic with period n > 0 if the condition F n (x) = x is fulfilled with n minimal.
Because of condition 1 in definition 2.1, the orbit of any point x is symmetric relative to the orbit of −x. To avoid ambiguities in the definition of the orbit of the pre discontinuity points we adopt the convention that F (c 1 ) = F c − 1 = −a and F (c 2 ) = F c + 2 = +a. Definition 2.6. [11] Consider the alphabet A = {L, A, M, B, R} the address A (x) of a real point x is defined such that
A (x) = L if x < c 1 A if x = c 1 M if c 1 < x < c 2 B if c 2 = x R if c 2 < x .
We can apply this function to an orbit of a given real point x 0 , we associate to that orbit one infinite symbolic sequence.
Definition 2.7. Consider the sequence of symbols in A
It (x 0 ) = A (x 0 ) A (x 1 ) A (x 2 ) ...A (x n ) ...
this infinite sequence is the symbolic itinerary of x 0 .
The orbit O (−a) is x (1) j : x (1) j = F j (−a) , j = 0, 1, . . . . The orbit of O (+a) is x (2) j : x (2) j = F j (+a) , j = 0, 1, . . . , with F (+a) = b.
Definition 2.8. [11] Kneading sequences and kneading pairs. The kneading sequences are defined as the symbolic itineraries of the orbits of a and −a. The kneading pair is the ordered pair formed by these two symbolic sequences Definition 2.11. [11] Let A N denote the set of all sequences written with the alphabet A. We define an ordering ≺ on the set A N such that: given two symbolic sequences P = P 0 P 1 P 2 ... and Q = Q 0 Q 1 Q 2 ... let n be the first integer such that P n = Q n . Denote by S = S 0 S 1 S 2 ...S n−1 the common first subsequence of both P and Q. Then, we say that P ≺ Q if P n ≺ Q n and ρ (S) = +1 or Q n ≺ R n if ρ (S) = −1. If no such n exists then P = Q.
This ordering is originated by the fact that when x < y then It (x) It (y).
To state the rules of admissibility the shift operator σ will be used, defined as usual.
Definition 2.12. Shift operator σ. The shift operator is defined
σ (P 0 P 1 P 2 ...) = P 1 P 2 ....
When we have a finite sequence S the shift operator acts such that
σ (S 0 S 1 S 2 ...S n−1 ) = S 1 S 2 ...S n−1 S 0 .
The orbit of +a has the symbolic itinerary It (+a). The sequence It (+a) is maximal (resp. It (−a) is minimal) in the ordering defined in this section. Maximal in the sense that every shift of the sequence It (+a) is less or equal than It (+a). Every orbit with initial condition x 0 is symmetric to the orbit with initial condition −x 0 . Thus any orbit beginning by +a is accompanied by a symmetric orbit started by −a. Therefore, we shall focus the admissibility rules for kneading sequences only on the itineraries with the first symbol R (corresponding to +a).
Definition 2.13. [11] Operator τ. The operator τ :
A N → A N is defined such that τ L = R, τ A = B, τ M = M, τ B = A, τ R = L.
Given a sequence Q = Q 0 Q 1 Q 2 ..., τ acts such that
τ Q = τ Q 0 τ Q 1 τ Q 2 ...
The operator τ interchanges the symbols L and R, letting the symbols M unchanged. For instance τ ((RLM R)
∞ ) = (LRM L) ∞ .
Proposition 2.14. [11] It (x 0 ) = τ It (−x 0 ).
Proof. Is a direct consequence of condition 1 in definition 2.1. Given any itinerary of +a denoted by S, the corresponding itinerary of −a is τ S. The kneading pair is (S, τ S). To know the kneading sequence S, corresponding to the orbit of +a, is to know the kneading pair. By some abuse of notation, sometimes (mainly in the examples) we use only the kneading sequence S instead of the kneading pair.
Definition 2.15. [11] Admissibility rules: Let S be a given sequence of symbols and (S, τ S) be a pair of sequences. (S, τ S) is a kneading pair and S is a kneading sequence, if S satisfies the admissibility condition: τ S σ k S S, for every integer k. The set of the admissible sequences is denoted by Σ ⊂ A N . Definition 2.16. Given a finite sequence P with length p, the sequence S = P ∞ is called a p-periodic sequence.
We will work sometimes only with P instead of P ∞ when there is no danger of confusion.
Definition 2.17. [11] A bistable periodic orbit contains both the orbit of +a and the orbit of −a. Any bistable orbit has an itinerary S = P ∞ = (Qτ Q) ∞ or shortly P = Qτ Q As a consequence of the previous definition bistable orbits and associated symbolic itineraries must have even period.
2.3.
Kneading theory. In [10] were introduced the concepts of invariant coordinate, kneading increments, kneading matrix and kneading determinant. We will use the definitions of the cited work with the convenient adaptations for the discontinuous case. We present here a brief exposition of the results obtained applying kneading theory to this type of maps.
Definition 2.18. Invariant coordinate of an initial condition θ x0 (t). Is defined using the sequence X = X 0 X 1 X 2 . . . = It (x 0 ). Is the formal power series
θ x0 (t) = +∞ k=0 (−1) k X k t k .
With the notation θ c ±
i (t) = lim x→c ± i θ x (t)
, for each discontinuity point, the kneading increment is defined.
Definition 2.19. Kneading increment and kneading matrix. The kneading increment is
ν i (t) = θ c + i (t) − θ c − i (t) .
This quantity is a formal power series measuring the discontinuity. After collecting the terms associated to each symbol, and remarking that, in this case, c − 1 corresponds to L, c + 1 corresponds to M , c − 2 corresponds to M and c + 2 corresponds to R, the decomposition
ν i (t) = N i1 (t) L + N i2 (t) M + N i3 (t) R is obtained. The kneading matrix is N = N 11 (t) N 12 (t) N 13 (t) N 21 (t) N 22 (t) N 23 (t) .
Definition 2.20. Kneading determinant. Omitting the j-th column of the kneading matrix we compute the determinant D j . The kneading determinant is
D (t) = (−1) j+1 D j (1 + t) .
The denominator in the kneading determinant results from the fact that F is decreasing in the three intervals where this map is defined [10]. Note that
D 1 = −D 2 = D 3 . Definition 2.21. [11] Given a sequence X = X 1 X 2 . . . we define a function Φ : A −→ {−1, 0, 1}, such that Φ (X i ) = −1 0 1 if X i = L, A if X i = M if X i = R, B . Definition 2.22. [11] Given a sequence X = X 1 X 2 . . . from A N we define a formal power series u (t), such that u (t) = +∞ k=1 (−1) k Φ (X k ) t k . When X is finite with length p we define the formal polynomial u p (t) = p k=1 (−1) k Φ (X k ) t k .
Let S = S 1 S 2 . . . ∈ Σ be a kneading sequence with the kneading pair (S, τ S), then the kneading determinant is given by.
(2) D (t) = 1 + 2u (t) t + 1 .
When S = P ∞ is p-periodic, the expression of the kneading determinant simplifies into D (t) (t + 1) = 1 + 2up(t) 1−(−1) p t p . When the kneading sequence S is bistable: S = P ∞ with P = Qτ Q of period p = 2q, with the associated kneading pair (S, τ S) = (Qτ Q, (τ Q) Q) the kneading determinant is 1 +
(3) ρ = lim n→∞ n ℓ (F n ).
Remark 2.25. [11] The growth number of F can be computed using the relation
ρ = 1 t 0 ,
where t 0 is the least root in the unit interval of the kneading determinant D (t). The proof is provided defining the power series Λ (t) = n≥1 ℓ (F n ) t n−1 , where each coefficient is the lap number of the iterate F n . This new power series is closely related to the kneading determinant because of the relation
Λ (t) = 1 t(1−t 2 )D(t) − 1 t .
Example 2.26. The kneading sequence (RM R) ∞ corresponds to the kneading determinant
D (t) = 1−2t−t 3 (t+1)(1+t 3 )
, which is realized for instance by g β (x) with β approximately 3.1588 or by G α (from example 2.3) with α = 1 4 √ 5 − 1 . We obtain Λ (t) = 3 + 7t + 17t 2 + 39t 3 + 87t 4 + 193t 5 + . . ., in this case ℓ (F ) = 3, ℓ F 2 = 7, ℓ F 3 = 17 . . .
It will be an interesting work to see if the usual relationship between the topological entropy and growth number still remains valid in the case of discontinuous maps.
2.5. Markov partition. Whenever we can define Markov matrices, the method of Markov transition matrices in the case of continuous maps is an equivalent approach to the computation the roots of the kneading determinant. To each p-periodic kneading pair we associate a Markov transition matrix, see [9] and related references on that paper. Now, denote by
x (2) j = F j c + 2 , j = 0, 1, . . . , p − 1, x (1) j = F j c − 1 , j = 0, 1, .
. . , p − 1, the orbits of the discontinuity points. An ordered sequence (z k ) k=1,...,2p is obtained reordering the elements x (m) j , m = 1, 2, and getting a partition
I k = (z k , z k+1 ) with k = 1, . . . , 2p − 1.
The discontinuity points are present in the above partition. We call z k1 = c 1 and z k2 = c 2 . To compute the Markov matrix note that I k1−1 = z k1−1 , c − 1 and I k1 = c + 1 , z k1+1 and similarly with the two intervals adjacent to the discontinuity point c 2 . With this precision made, the Markov transition matrix can be defined.
ψ ij = 1 if I j ⊂ F (I i ) , 0 otherwise.
In [9] the relationship between Markov partitions and kneading theory is explained for bimodal continuous maps. It is also presented the proof of the equality of the reciprocal of t 0 and the spectral radius of the matrix Ψ. In this paper we will prove the same equivalence of definitions in the case of bimodal symmetric discontinuous maps.
In the next example we obtain this equivalence for a particular case, computing directly both the kneading determinant and the characteristic polynomial of the Markov matrix.
Example 2.28. The kneading pair
((RM R) ∞ , (LM L) ∞ )
corresponds to a pair of orbits satisfying
(4) x(1)1 < x(1)0 = c 1 < x (2) 2 < x (1) 2 < x(2)0 = c 2 < x(2)
1 , renaming the elements of the partition we get
z 1 = x (1) 1 , z 2 = x (1) 0 , z 3 = x (2) 2 , z 4 = x (1) 2 , z 5 = x (2) 0 , z 6 = x (2) 1 . The Markov matrix is Ψ = 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 0 0 0 0 1 1 1 .
The smallest solution t 0 of the equation
det (I − Ψt) = 1 − t + t 2 1 − 2t − t 3 = 0
in the unit interval gives the growth number ρ = 1 t0 = 1 0.4534 = 2.2056, exactly the same root obtained with the kneading determinant of the example 2.26.
Relation between Markov partition and the orbits of the discontinuity points. Giving the p-periodic orbits
O c + 2 = x (1) j : x (1) j = F j c + 2 , j = 0, 1, . . . , p − 1 and O c − 1 = x (2) j : x (2) j = F j c − 1 , j = 0, 1, . . . , p − 1 , we define the vector y = y 1 . . . y p y p+1 . . . y 2p = x (2) 0 . . . x (2) p−1 x (1) 0 . . . x (1) p−1 .
Let z be the vector {z i } i=1,...,2p where z i−1 < z i < z i+1 are the ordered elements of y. There is a 2p × 2p permutation matrix π such that z = πy.
k = σ k−1 (τ S) . Naturally, we have x (1) k = τ x (2) k .
To each z j corresponds the symbolic itinerary w j = It (z j ). We define v j = It (y j ).
1 = RM B = v 2 , x (2) 2 = M BR = v 3 , x (2) 0 = BRM = v 1 , x (1) 1 = LM A = v 5 , x (1) 2 = M AL = v 6 , and x (1) 0 = ALM = v 4 . We get w 1 = v 5 , w 2 = v 4 , w 3 = v 3 , w 6 = v 4 , w 5 = v 1 and w 6 = v 2 . The matrix π is π = (2)(1 + t) D (t) = 1 + 2 −t − t 2 + t 4 1 − t 4 = 1 − 2t − 2t 2 + t 4 1 − t 4 .
The kneading pair is
((RLM B) ∞ , (LRM A) ∞ ), x (2) 1 = RLM B = v 2 , x (2) 2 = LM BR = v 3 , x (2) 3 = M BRL = v 4 , x (2) 0 = BRLM = v 1 , x (1) 1 = LRM A = v 6 , x (1) 2 = RM AL = v 7 , x (1) 3 = M ALR = v 8 and x (1) 0 = ALRM = v 5 . We get w 1 = v 6 , w 2 = v 3 , w 3 = v 5 , w 4 = v 4 , w 5 = v 8 , w 6 = v 1 , w 7 = v 7 and w 8 = v 2 . The Markov matrix is Ψ = . The matrix π is π = .
The matrix π can also be used to reorder the shifts of the kneading sequences, giving the vector v = {v 1 , . . . , v 2p } and the vector w = {w 1 , . . . , w 2p }, we have w = πv.
Main Result
3.1. The Markov and kneading endomorphism in spaces of chain complexes. Let C 0 be the vector space of the 0-chains spanned by the shifts of the kneading sequences {v j } j=1,...,2p this space is isomorphic of the space of the 0-chains spanned by the points of the orbit {y j } j=1,...,2p . The space π (C 0 ) is spanned by {w k } k=1,...,2p which is isomorphic to the space of the 0-chains spanned by {z j } j=1,...,2p . Let C 1 be the space of the 1-chains spanned by {I k } k=1,...,2p−1 , isomorphic to the linear space of the 1-chains spanned by {I ′ k } k=1,...,2p−1 where I ′ k is the set of all the admissible sequences w: w k w w k+1 . In what follows we identify I ′ k with I k and use the same symbol both for sequences and intervals and call both the linear transformations and the corresponding matrix representations by the same letters.
The border of a 1-chain is obtained using the linear transformation ∂ :
C 1 → D 0 such that ∂ (I k ) = w k+1 − w k , ∂ (C 1 ) = D 0 where D 0 is spanned by {w k+1 − w k } k=1,...,2p−1 .
It is clear that D 0 ⊂ π (C 0 ).
We define the linear transformation ∂ s :
C 1 → D 0 such that ∂ s (I k ) = ∂ (1 − τ ) I k = ∂ (I k − I 2p−k ) = ∂ (I k ) − ∂ (I 2p−k ) .
The image of I k by ∂ s is
∂ s (I k ) = w k+1 − w k − (w 2p+1−k − w 2p−k )
and is an element of D 0 . We can define another linear transformation that acts on π (C 0 ) with matrix representation
µ = −1 1 0 · · · 0 1 −1 0 −1 1 · · · 1 −1 0 . . . . . . . . . . . . . . . . . . . . . 0 1 −1 · · · −1 1 0 1 −1 0 · · · 0 −1 1 .
where µ ij = δ i+1,j − δ i,j − δ 2p+1−i,j + δ 2p−i,j , i = 1, . . . , 2p − 1, j = 1, . . . , 2p, and δ is the Kronecker delta symbol. This linear transformation represents the order relation of the points of the real line and the symmetry of the original mapping. It is immediate from the above definitions that Image (∂ s ) = B 0 is a proper subspace of D 0 such that D 0 = B 0 ⊕ B, where B has dimension one, and B 0 is isomorphic to µπ (C 0 ). We define η = µπ and the endomorphism ω acting on C 0 with matrix representation
ω = σ 0 0 σ ,
where σ is the shift operator with matrix representation p × p
σ = 0 1 · · · 0 0 0 0 · · · 0 0 . . . . . . . . . . . . . . . 0 0 · · · 0 1 1 0 · · · 0 0 .
Let α be the endomorphism induced in B 0 by the rotation ω in C 0 which results from the commutativity of the diagram η ∂s
C 0 −→ B 0 ←− C 1 ω ↓ ↓ α ↓ α C 0 −→ B 0 ←− C 1 η ∂s .
It is easy to see that α = −Ψ, where Ψ is the Markov matrix. Note that ηω = αη. Every entry in the matrix α is non-positive, because the images of the intervals are obtained by the images of the boundary points and F is reverse order in any of each interval of continuity (lap).
η = 1 −1 0 1 −1 0 −1 0 1 −1 0 1 0 0 0 0 0 0 1 0 −1 1 0 −1 −1 1 0 −1 1 0 and ηω = 0 1 −1 0 1 −1 1 −1 0 1 −1 0 0 0 0 0 0 0 −1 1 0 −1 1 0 0 −1 1 0 −1 1 ,
which is precisely −Ψη.
3.2.
Matrix Θ. Giving the right p-periodic kneading sequence S, the symbolic itinerary of the right discontinuity point is σ p−1 S = S 0 S 1 S 2 . . . = S 0 S. We construct the vector
s (S) = Φ (S 0 ) Φ (S 1 ) . . . Φ (S p−1 ) Φ (τ S 0 ) Φ (τ S 1 ) . . . Φ (τ S p−1 ) = 1 Φ (S 1 ) . . . Φ (S p−1 ) −1 Φ (τ S 1 ) . . . Φ (τ S p−1 ) ,
where Φ was defined in definition 2.21. When applied to the other kneading sequence τ S the vector takes the form s (τ S) = σ p s (S) .
Let Γ be a square matrix which columns 1 and p + 1 are s and σ p s, respectively, and the other elements are zeros. Now, we introduce the matrices γ = Γ − I and Θ = γω. The matrix Θ has the form
0 0 0 ... 0 0 0 −Φ(S0) 0 ... 0 0 0 Φ(S1) −1 ... 0 0 0 −Φ(S1) 0 ... 0 0 0 Φ(S2) 0 ... 0 0 0 −Φ(S2) 0 ... 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 Φ(Sp−2) 0 ... 0 −1 0 −Φ(Sp−2) 0 ... 0 0 −1 Φ(Sp−1) 0 ... 0 0 0 −Φ(Sp−1) 0 ... 0 0 0 −Φ(S0) 0 ... 0 0 0 0 0 ... 0 0 0 −Φ(S1) 0 ... 0 0 0 Φ(S1) −1 ... 0 0 0 −Φ(S2) 0 ... 0 0 0 Φ(S2) 0 ...0 −Φ(Sp−2) 0 ... 0 0 0 Φ(Sp−2) 0 ... 0 −1 0 −Φ(Sp−1) 0 ... 0 0 −1 Φ(Sp−1) 0 ... 0 0 .
Example 3.2. With the same kneading sequences of the example 2.29, we have
γ = 0 0 0 −1 0 0 1 −1 0 −1 0 0 0 0 −1 0 0 0 −1 0 0 0 0 0 −1 0 0 1 −1 0 0 0 0 0 0 −1 and Θ = 0 0 0 0 −1 0 0 1 −1 0 −1 0 −1 0 0 0 0 0 0 −1 0 0 0 0 0 −1 0 0 1 −1 0 0 0 −1 0 0 .
Proposition 3.3. The following diagram commutes
η C 0 −→ B 0 γ ↓ ↓ −I C 0 −→ B 0 η .
Proof. We must show that ηγ = −η or η (Γ − I) = −η, in other words that ηΓ = 0, or that s (S) , s (τ S) ∈ kernel (η). But η = µπ, and π reorders s (S) in terms of the order of the real line, giving
πs (S) = ν (I (z 1 )) ν (I (z 2 )) . . . ν (I (z 2p−1 )) ν (I (z 2p )) ,
knowing that I (z j ) = τ I (z 2p−j ) , j = 1, . . . p, we have ν (I (z j )) = −ν (I (z 2p−j )). It is obvious that µπs (S) = µπs (τ S) = 0.
Theorem 3.4. The characteristic polynomial of the matrix Θ = γω is
P Θ (t) = det (I − tΘ) = (1 − (−1) p t p ) 2 (1 + t) D (t) ,
where D (t) is the kneading determinant.
Proof. The determinant of the matrix I − tΘ is We first multiply the row 1 by −t and add the result to the row p, we do the same with the rows p + 1 and 2p. Then we develop the determinant by the columns 1 and p getting a (2p − 1) × (2p − 1) determinant:
1 0 0 . . . 0 0 0 tΦ (S 0 ) 0 . . . 0 0 0 1 − tΦ (S 1 ) t . . . 0 0 0 tΦ (S 1 ) 0 . . . 0 0 0 −tΦ (S 2 ) 1 . . . 0 0 0 tΦ (S 2 ) 0 . . .0 −tΦ (S p−2 ) 0 . . . 1 t 0 tΦ (S p−2 ) 0 . . . 0 0 t −tΦ (S p−1 ) 0 . . . 0 1 0 tΦ (S p−1 ) 0 . . . 0 0 0 tΦ (S 0 ) 0 . . . 0 0 1 0 0 . . . 0 0 0 tΦ (S 1 ) 0 . . . 0 0 0 1 − tΦ (S 1 ) t . . . 0 0 0 tΦ (S 2 ) 0 . . . 0 0 0 −tΦ (S 2 ) 1 . . .1−tΦ(S1) t ... 0 0 tΦ(S1) 0 ... 0 0 −tΦ(S2) 1 ... 0 0 tΦ(S2) 0 ... 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . −tΦ(Sp−2) 0 ... 1 t tΦ(Sp−2) 0 ... 0 0 −tΦ(Sp−1) 0 ... 0 1 tΦ(Sp−1)−t 2 Φ(S0) 0 ... 0 0 tΦ(S1) 0 ... 0 0 1−tΦ(S1) t ... 0 0 tΦ(S2) 0 ... 0 0 −tΦ(S2) 1 ... 0tΦ(Sp−2) 0 ... 0 0 −tΦ(Sp−2) 0 ... 1 t tΦ(Sp−1)−t 2 Φ(S0) 0 ... 0 0 −tΦ(Sp−1) 0 ... 0 1
, then we multiply the row p − 1 by −t and add the result to the row p − 2. We do the same with the last two rows. Then we develop the determinant by the columns p − 1 and the last one getting a (2p − 2) × (2p − 2) determinant: , where r 1 (t) = −tΦ (S p−2 ) + t 2 Φ (S p−1 ) and r 2 (t) = tΦ (S p−2 ) − t 2 Φ (S p−1 ) + t 3 Φ (S 0 ). Repeating this reducing process we get a 2 × 2 determinant
1−tΦ(S1) t ... 0 tΦ(S1) 0 ... 0 −tΦ(S2) 1 ... 0 tΦ(S2) 0 ...1 − p−1 k=1 (−1) k t k Φ (S k ) − p k=1 (−1) k t k Φ (S k ) − p k=1 (−1) k t k Φ (S k ) 1 − p−1 k=1 (−1) k t k Φ (S k ) .
Remembering that
u p (t) = p k=1 (−1) k t k Φ (S k ) , S 0 = S p = B and (−1) p t p Φ (B) = (−1) p t p ,
the previous determinant is equal to
(1 − (−1) p t p ) + u p (t) −u p (t) −u p (t) (1 − (−1) p t p ) + u p (t) ,
which gives P Θ (t) = (1 − (−1) p t p ) 2 1 + 2 u p (t) 1 − (−1) p t p and this is precisely
P Θ (t) = (1 − (−1) p t p ) 2 (1 + t) D (t) ,
as desired.
Example 3.5. We use the kneading sequences of the example 2.30 to illustrate this last result, the matrix Θ is
0 0 0 0 0 −1 0 0 0 1 −1 0 0 −1 0 0 0 −1 0 −1 0 1 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 −1 0 0 0 1 −1 0 0 1 0 0 0 −1 0 −1 0 0 0 0 −1 0 0 0 ,
with characteristic polynomial
1 − t 4 1 − 2t − 2t 2 + t 4 ,
which agrees with the value of the kneading determinant (1 + t) D (t) = 1−2t−2t 2 +t 4 1−t 4
.
We have now all the ingredients to state the main results of this paper.
Theorem 3.6. The following diagram commutes
η C 0 −→ B 0 ↓ Θ ↓ Ψ C 0 −→ B 0 η .
and P Θ (t) = (1 + t) det (I − tΨ).
Proof. Noticing that Θ = γω and Ψ = −α the result is only a direct consequence of
η C 0 −→ B 0 ↓ω η ↓α C 0 −→ B 0 ↓ γ η ↓ −I C 0 −→ B 0
conjugated with the fact that the two rows in the next diagram
inj η 0 −→ B −→ C 0 −→ B 0 −→ 0 ↓ −I inj ↓ Θ η ↓ Ψ 0 −→ B −→ C 0 −→ B 0 −→ 0
are exact sequences, where inj is the natural embedding. We think that the results of this work can be extended to general discontinuous maps with finite number of discontinuities. That is a natural extension of this work.
Bimodal symmetric discontinuous map. Definition 2.1. Bimodal symmetric discontinuous map of type (−, −, −). Let I = (−a, a) be a real interval (where a can be +∞) and F : I →I, such that:
(
It (a) ,It (−a)) .
Definition 2 . 9 .
29Order relation in A. The order on A is naturally induced from the order in the real axis L ≺ A ≺ M ≺ B ≺ R. Definition 2.10. [11] Parity function ρ (S). Given any finite sequence S with length p, ρ (S) is such that ρ (S) = (−1) p .
Growth number. The kneading determinant is essential in the computation of the growth number of laps. Definition 2.23. Lap number ℓ (F n ) is the number of maximal intervals of continuity of each composition of F with itself. Definition 2.24. The growth number is defined
Definition 2 . 27 .
227The Markov transition matrix Ψ = [ψ ij ] is defined by the rule:
S 1 S 2 . . . = S is the kneading sequence of +a. Let x τ S is the kneading sequence of −a. It is also clear that x σ p−1 (τ S). To each k = 1, . . . , p corresponds a symbolic sequence x(2) k = σ k−1 (S) . To each k = 1 . . . , p,corresponds another sequencex(1)
Example 2 . 29 .
229Given the kneading sequence (RM B) ∞ , equivalent to (RM R) ∞ already used before. The kneading pair is ((RM B) ∞ , (LM A) ∞ ), x
.
Given the kneading sequence (RLM B) ∞ , the kneading determinant D (t) is such that
Example 3. 1 .
1With the matrices of the examples 2.29 we have
(S p−2 ) 0 . . . 0 0 0 −tΦ (S p−2 ) 0 . . . 1 t 0 tΦ (S p−1 ) 0 . . . 0 0 t −tΦ (S p−1 ) 0 . . . 0 1
Corollary 3 . 7 .
37The inverse of the least root in the unit interval of the periodic kneading determinant is the spectral radius of the Markov matrix.Proof. Is an immediate consequence of the relation(1 − (−1) p t p ) 2 D (t) = det (I − tΨ) ,obtained in the last theorem.
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AcknowledgementThe author thanks some precious comments of the referee that simplified and improved a great deal the scope and clarity of this work. Namely, the possibility of infinite jumps at the discontinuity points. The author was partially funded by FCT/Portugal through project PEst-OE/EEI/LA0009/2013.
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. 1049-001 LisboaPortugal E-mail address: [email protected]. Center for Mathematical Analysis, Geometry and Dynamical Systems, Mathematics Department, Instituto Superior Técnico, Universidade de LisboaAv. Rovisco Pais. ptCenter for Mathematical Analysis, Geometry and Dynamical Systems, Mathematics Department, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal E-mail address: [email protected]
| [] |
[
"The Hyper-Cortex of Human Collective-Intelligence Systems",
"The Hyper-Cortex of Human Collective-Intelligence Systems"
] | [
"Marko A Rodriguez [email protected] \nComputer Science Department\nUniversity of California at Santa Cruz and Center for Evolution\nComplexity\n\nCognition Vrije Universiteit Brussel\nBelgium\n"
] | [
"Computer Science Department\nUniversity of California at Santa Cruz and Center for Evolution\nComplexity",
"Cognition Vrije Universiteit Brussel\nBelgium"
] | [] | Individual-intelligence research, from a neurological perspective, describes the cortex as a medium for performing conceptual abstraction and specification. This idea has been used to explain how motor-cortex regions responsible for different behavioral modalities such as writing and speaking can express the same general concept represented in the cortex. For example, the concept of a dog, abstractly represented in the higher-layers of the cortex, can either be written or spoken about depending on the context. Abstract models in the higher-layers propagate activation patterns down the cortical hierarchy to the desired region of the motor-cortex for worldly implementation. In this paper, the individual-intelligence framework is expanded to incorporate collective-intelligence within a hyper-cortical construct. This hyper-cortex is a multi-layered network used to represent abstract collective concepts. This collective-intelligence framework plays an important role in understanding how collective-intelligence systems can be engineered to handle collective problem-solving. To conclude the paper, five common problems in the scientific community are solved using an artificial hyper-cortex generated from digital-library metadata. | null | [
"https://arxiv.org/pdf/cs/0506024v3.pdf"
] | 14,100,895 | cs/0506024 | fbf5f3a52398123b7bab1185722862835c2fe988 |
The Hyper-Cortex of Human Collective-Intelligence Systems
Marko A Rodriguez [email protected]
Computer Science Department
University of California at Santa Cruz and Center for Evolution
Complexity
Cognition Vrije Universiteit Brussel
Belgium
The Hyper-Cortex of Human Collective-Intelligence Systems
ECCO Working Paper 2004-06
Individual-intelligence research, from a neurological perspective, describes the cortex as a medium for performing conceptual abstraction and specification. This idea has been used to explain how motor-cortex regions responsible for different behavioral modalities such as writing and speaking can express the same general concept represented in the cortex. For example, the concept of a dog, abstractly represented in the higher-layers of the cortex, can either be written or spoken about depending on the context. Abstract models in the higher-layers propagate activation patterns down the cortical hierarchy to the desired region of the motor-cortex for worldly implementation. In this paper, the individual-intelligence framework is expanded to incorporate collective-intelligence within a hyper-cortical construct. This hyper-cortex is a multi-layered network used to represent abstract collective concepts. This collective-intelligence framework plays an important role in understanding how collective-intelligence systems can be engineered to handle collective problem-solving. To conclude the paper, five common problems in the scientific community are solved using an artificial hyper-cortex generated from digital-library metadata.
Introduction
Research published by Jeff Hawkins and Sandra Blakeslee [1] has provided the foundation for many of the ideas in this paper. Hawkins and Blakeslee define the human cortex as a predictive system that patternmatches the current sensory experience to an analogous concept in memory. Multiple subjectively similar events are mapped to a single abstract model. These lossy models are called invariant representations. Invariant representations in the cortex may be associated with a specific learnt behavior. In such cases, if the associated behavior proved successful in the past, then the individual executes the behavior relative to current context. In other instances, the invariant model may trigger a continuous cascade of associated memories-other invariant representations. In such situations, the individual performs multiple internal transformations of the input signal in the process of conceptual thinking. In short, the cortex is a memory structure that associates present stimuli with previously held concepts about the world. Pattern-matching will be shown to be the cortical function required for individual problem-solving.
The paper then extends the individual cortex model to encapsulate a socially-embedded hyper-cortex 1 residing outside the individual, existing within the collective. The layers of a hyper-cortex represent information which is unable to be fully contained within any single individual. These hyper-cortical layers are created by the aggregate interactions of multiple individuals over time. In order to represent abstract collective concepts, a hierarchy of layers can cluster similar lower layer regions into subjectively related ideas. During specific problem-solving instances, an individual who is unable to locate an appropriate invariant solution within their cortex can use the collective's hyper-cortex to derive a solution. In effect, a hyper-cortex functions as a pattern-matching infrastructure-mapping user realized problems to collectively-derived solutions.
Outlining the paper's structure, the initial section (Section 2) explains the general individual-intelligence framework described by Hawkins and Blakeslee [1] and further formalized more concretely by other researchers in the neurosciences. Section 3 will extend this framework into the domain of collectiveintelligence. In Sections 4, an artificial hyper-cortex is generated from digital-library metadata, and in Section 5, the digital-library hyper-cortex is used to solve five problems commonly encountered within the scientific community. 1. locating references for an initial idea: the hyper-cortex correlates a collection of related manuscripts with an individual's rough and unformulated idea.
2. finding potential collaborators for an idea: given a model of the specific domain of the individual's initial idea, the hyper-cortex will suggest fit collaborators. 3. determining a journal for paper submission: once the collaborating team has formalized the idea into a written manuscript, a journal suitable for the paper will be recommended by the hyper-cortex. 4. locating peer-reviewers to review a paper: journal editors can use the hyper-cortex to find qualified referees to review the submitted manuscript. 5. distribute the accepted paper to appropriate members in the community: the hyper-cortex creates a plausible mapping between the newly published paper and potentially interested scientists in the field.
Individual-Intelligence in the Cortex
Before delving into this paper's collective-intelligence framework, discussion of the human cortex's role in individual-intelligence is necessary. Individual problem-solving will be defined as the cortical act of matching a given problem's context to potential solutions previously existing in memory (Subsections 2a,2b&2c). The section concludes with a real-world example of an individual solving a problematic and unexpected event-an event that can't be represented by the cortex-by utilizing another individual's memory of the solution (Subsection 2d) .
2a. Mapping Sensory Information to Experiential Memories
Any individual's sensory input is represented within their cortex as a series of activation patterns situated within space and/or time. Propagating sensory patterns associate themselves with an individual's stored subjective categorization of the external world. These internal categories are called invariant representations and are formed as the individual's cortex organizes the countless, different but relatable everyday experiences into abstract generalizations [3]. Invariant models possess a one-to-many compression ratio with their continuously fluxing sensory equivalents. That is, many experiences are mapped by the individual's cortex to one invariant representation. The term 'invariant' expresses the idea that, even under various transformations of the input signal (i.e. rotations, translations, scalings, etc.) the same abstract model can represent all transformations of the signal [4]. The invariant representation's modeling ability is impervious to slight variations of the input signal, and is therefore an approximation of many related, yet different occurrences. Due to this compression functionality, invariant representations form the foundation of conceptual thinking, the process of generalization and abstraction.
A more complex concept, one relying on multiple sensory modalities, is elicited when the multiple sensory input signals ascend the cortical hierarchy and converge on one single invariant model. This is how the smell, sight, and sound derived from a person can unite to activate the individual's pre-existing concept of a well known friend [5]. Furthermore, even without a visual signal from a friend's physical presence, an auditory signal-e.g. a friend's voice heard from behind-will prime the individual to expect seeing their friend upon turning around. This example illustrates that the human cortex can use subsets of an invariant model to predict future events-anticipation [6]. Over time, as more sensory information related to the invariant model of the friend is made available, the individual will further affirm their subjective belief that the friend co-exists with them in physical reality. Invariant representations, stimulated by their analogous sensory information, can trigger a cascade of memories, which lead to predictions and behaviors. Within this framework, it is shown that predictions and behaviors are one in the same phenomenon [1]. Predictions and behaviors are activation patterns of invariant models, descending down the hierarchy to either; prime sensory cells to anticipate the next instant's experience, or elicit certain behaviors attempting to increase the predictability of the individual's environment. With any sensory input the individual's interpretation of the signal will be unavoidably associated with their previously held invariant concepts about their world. An individual thinks and acts with a continuous and innate reliance on their internal memories. If and when expectations-derived from memory-are not met, invariant models will be altered. Thereby the human cortex organizes a diverse and seemingly unpredictable world into a more consistent subjective experience. Invariant representations change throughout the individual's life as unexpected experiences adversely associate with inappropriate internal models [4]. A constant process of mapping the present and the past into predictable and more palatable cortical models of reality is at the root of intelligent problem-solving.
2b. Hierarchical and Lateral Projections within the Cortex
The hierarchical structure of the human cortex is composed of interconnecting layers of neural tissue. Neurons on the higher levels of the cortex's structural hierarchy have greater invariance to sensory signal fluctuations than do those located on lower levels [7]. A cluster of lower-layered neurons, each of which project to a single higher-layer neuron, is called the invariant neuron's receptive field ( Figure 1). Only the lower-level layers are responsive to small deviations of the sensory signal (i.e. light contrast, lines, and basic shapes). When considering a conceptually stable object experienced in the world, the activation pattern in the receptive field of the object's invariant model varies significantly through time. The invariant model of that object, on the other hand, maintains a constant firing pattern. Therefore, an object can move through the individual's visual scene and still be recognized as the same object. Within the layers of the cortex, neurons are also connected with horizontal projections. Horizontal projections within a cortical layer can produce a broadening or contraction of a sensory signal's potential categorization [8]. Partial excitation of a receptive field can spread over the entire receptive field, causing the invariant neuron of that receptive field to fire. For example, an object can still be recognized even if it is partially hidden from view. On the other hand, inhibitory projections to remote receptive fields can dampen the probability of the remote receptive field stimulating its respective invariant representation [9]. When a receptive field inhibits the activation of a neighboring receptive field, the sensory experience will be contained to one abstract interpretation. Lateral inhibition has been studied extensively in the visual system [10]. Without lateral inhibition the same sensory experience would be mapped to multiple unrelated concepts. Without lateral excitation sensory experiences would seem too varied and inconsistent to be grouped above and beyond the lower-layer's excitation pattern. A fine balance between these two algorithms provides the human with a chaotic-stable experience of reality.
2c. Problem-Solving Sensory Perturbations
A problem, in its most basic form, is any sensory input-or perturbation. When sensory information enters through the individual's sensory modalities, the cortex begins to categorize the information into concepts of higher and higher order. A cascade of action potentials throughout the cortex causes the individual to either contain the signal internally as a rationalization in thought, or externally as a behavior that alters the external environment. This resolution of sensory perturbations is called problem-solving. Signal resolution can happen through priming or through action. In priming, the current sensory pattern can cause an invariant model to prime its receptive field to expect a particular pattern at the next instant. If the expectation is met, then the receptive field is fully excited and thus the invariant model neuron is activated again. Other times, expectations are not met and the primed cortical region does not receive enough excitation from the sensory signal to elicit the experience of the concept. In this case, the individual has a parallel experience of reality-one is their memory-based expectation and the other is their sensory-based experience. When a discrepancy between expectation and experience occurs, a larger problem for the individual exists. In such cases, the sensory signal continues to elicit activations laterally and vertically through the cortex until an appropriate model of the signal is located. For example, a cortex that has been habituated to a constant sensory signal through overexposure is unperturbed by continuous exposure to that same signal. The signal causes a higher-level invariant representation to continuously prime the low-level cortical neurons to expect that same signal at the next time step. If that expectation is met, then the signal is easily resolved as an instance of the invariant representation. In this case, the future is constantly being accurately predicted and therefore the problem-solving environment is simple-the individual contains the appropriate internal model to resolve the sensory perturbation. If at any time in the future the overexposed signal is altered in such a way that the primed receptive field of the invariant representation isn't activated, then the cortex's expectation is not met. In not meeting the prediction, a rationalization of the altered signal's properties must be deduced. Sometimes the signal can be resolved in the head and other times it requires the active manipulation of the environment-which in turn begins the process again. The human cortex is constantly problem-solving by changing its internal structure and its external world to ensure accurate predictions through time. The human cortex is constantly reaching for equilibrium by through constant adaptation of its internal structure in order to accurately represent its external world.
2d. Individual-Intelligence Problem-Solving
Let's apply this general framework to a simple individual-intelligence problem-solving scenario. Suppose an individual is working in his or her office. As the individual scans the room, each item in the office streams through the individual's visual system via a spatial-temporal pattern of activation across their retina (Figure 2a-P). If the human's neural-network is well adapted to the office then the sensory pattern is easily mapped to an invariant representation-the concept of their room and the standard items it contains. Nothing is out of the ordinary since the signal matches the primed invariant model (Figure 2a-S). In this example, the activation pattern further habituates the neural-network to the external world and stimulates no active behavior in the world. The individual's predictions of the world are true and the individual is unconscious of any problem.
Figure 2a&b: internal and external problem-solving resolution. (P) problem-model and (S) for solution-model
Now if at some point in time a new item is placed into this individual's office then the input pattern is not so effectively mapped to the invariant representation of the office (Figure 2b-P). Though some aspects of the office model will fire (i.e. that it's the same office, same chair, same desk, etc.), the other aspects of the signal concerning the new item will have to find resolution elsewhere in the cortex. To handle this perturbation, the cortex performs a series of transformation on the unresolved portion of the input signal until an appropriate understanding of the new items existence is realized. For this example, a potential abstract solution is that someone in the house must have placed the new item in the room. In gross simplification, this abstract solution propagates an activation pattern down the cortical hierarchy to the individual's motor-cortex. The activation of the motor-cortex prompts the individual to leave the office and ask others in the house about the new item (Figure 2b-S). The behavior output of this complex problem-solving process is the physical act of questioning everyone for a solution. This is the beginning of collective intelligence.
Many of the ideas outlined in this section lack strong boundaries between them. For instance, memories, invariant representations, internal models, concepts, and habituated cortical-regions are all essentially the same idea. When discussing the diversity of expression allowed by the cortex through its fundamental properties of a layered network, specific ideas are difficult to contain.
Collective-Intelligence in the Hyper-Cortex
The individual-intelligence framework presented in the previous section was a mix of cybernetic concepts and ideas outlined by Hawkins and Blakeslee. The depth of their book has not been done justice as only those ideas pertaining to the goal of this paper have been outlined. It is recommended that the reader read their book for a greater understanding of this simple yet effective model of human intelligence. This section will now generalize the individual-intelligence model to represent collective-intelligence within a hyper-cortex. This hyper-cortex maintains the highest layers of a societal problem-solving hierarchy.
3a. Memory in a Hyper-Cortex
As stated in the previous section, the cortex can be generically described as a memory system. Each experience contributes to the adaptation of the cortical structure-habituating the cortical structure to accurately associate its sensory experiences with stable container concepts. These concepts allow for the many-to-one relationship seen when the ever flux experience of the external world is mapped to a conceptually consistent internal explanation. It is only through habituation (learning) and analogy (thinking) that this stability emerges. External to the individual, within the domain of the collective, there also exists such networks of habituation and analogy. The learning web system formulated in [11] is a collectively created network of associated concepts. As individuals traverse the word-network, linking from concept to concept, they strengthen old associations and consequently weaken others. The learning web is a network which continuously adapts its structure to learn the collective preference of its users. This collectively created word-network has been called a collective mental-map [11]. What other such maps exist in the world today? The World Wide Web is a network of related web pages, tied by their associations within domains subjectively determined by their authors. Unlike the cortical structure of the human brain, the word-network and the World Wide Web are both flat hierarchies of interconnected nodes-these networks have no distinguishable layers of invariant representations. With an explicit hierarchical network abstract models of collectively derived associations can be created. One such layered network can be generated for the scientific community. As scientist go about their research, a multi-layered network consisting of scientists, their papers, and their journals can be derived from digital-library metadata. The individuals, their papers, and their journals form the three layers of the hyper-cortex. Each layer interconnects its nodes with horizontal projections: authors are associated in thought by their coauthored papers, papers are connected to one another by their citations, and journals and conference proceedings are related by their domain. As one moves up the scientific community's hyper-cortical hierarchy the breadth of the invariant representations become more pronounced. Authors, as general problem-solvers of their domain, are lossy models of the specific problems facing the world-many problem's map to one author. Written papers are lossy models of the knowledge contained within their authors and referenced authors-many authors map to one paper. And finally, journals and conference proceedings are lossy models of the papers they contain-many papers map to one journal. The further up the abstraction layer, the more general the concept, the more invariant the representation, and the more information lost to the necessity of compression. It is within the domain of scientific publications that an explicit, well documented, hierarchical hyper-cortex has already begun to emerge. And it is one of the goals of this paper to extract this information to construct a collective problem-solving hyper-cortex.
3b. Collective Attention as the Flow of Information-Molecules
At any moment, the collective mind is thinking about a particular topic. This collectively generated topic can be broad, spanning, what seems to the individual, a confusion of unrelated ideas. In a momentary span of attention there exists a collective subjective experience of the collective's external world. The collective is currently thinking about its environment; internalizing the activation patterns of its individual's current problems to continuously transform them into associated models within and between the layers of its hypercortex. This continuous spread of hyper-cortical activation is the collective thought-an expression of collective problem-solving.
Millions upon millions of individuals are currently moving through the World Wide Web-looking up web pages that relate to their questions, and, in turn, finding solutions to their problems. At no point in time has the World Wide Web been viewed in its entirety by every member of the collective. With billions upon billions of web pages and only millions upon millions of users, at any one moment only a small percentage of the web is actually being viewed. The perceived subset of the web is the focus of the collective attention-a pattern of activation representing the populations currently problem-solving activity [12]. Collective thought then is the evolution of this pattern through time as people link between pages and use search engines to jump between disparate ideas. The individuals, or information molecules, through their aggregate linking, create the fluidity the collective mind's experience [13]. These molecules have a goal, a history, and a method to their behavior. Their environment has a structure that constricts the absolute mobility of their movements. Each information molecule is a unique cortical problem-solving system. How do all these information molecules contribute to the collective problem-solving process? How does the interaction of all the individuals generate a solution to a problem greater than those experienced by their own individuality?
In the collective-mental map presented in [11], these information molecules habituate the hyper-cortex network through repeated selection of ingrained paths-thus restricting the future behaviors of other information molecules. This is a form of stigmergy. A single information molecule affects the probability of another molecules path by altering the external environment. In this system, each word-node in the network has a list of other word-nodes that an individual can link to. This list is ranked according to how many other individuals took the same path-popular paths are higher on the list. The ranked list of each node primes future individuals to choose well ingrained paths and thus restricts the probability of their choices. The World Wide Web works in a similar fashion. Web page authors alter the probability of an information molecule's next step via web links. Search engines do the same with their page ranking algorithms. In the scientific community journals have impact ratings, manuscripts have references, and digital library search engines have discipline categories. These structural biases cause the collective mind to focus on popular regions of the collective mental-map, which, in turn, further incites a realization of the world within the terms of this focus. This is the pattern-match of the current world perception to a popular invariant representation stored in the hyper-cortical memory. The continuous flow of the collective's attention within trend regions of the hyper-cortex can further increase the probability of future activity in that area of the network. Future experiences of the collective have a higher probability of being modeled by these trend representations. This is why terms such as 'complex systems', 'scale-free networks', and 'multi-agent systems' are the abstract models that the collective currently uses to represent their problems and solutions. With virtually unlimited space on the web and in the federated digital libraries, old ideas are not forgotten; they are simply not attended too. They exist structurally, but the probability of an information molecule reaching these ideas is low. Therefore, outdated invariant representations within the hyper-cortex are not tied to newer experiences. Where, on the other hand, the invariant representations of the collective's attention are the epicenter of novelty and neuronal growth within the collective mind.
3c. Problem Abstraction and Solution Specification
As stated before, when an individual experiences their environment, the cortex automatically categorizes the sensory pattern in relation to its invariant representations. In order to move that information outside their head and into a medium that is transparent to others within the collective, that information must be explicitly represented. Within the scientific community, this explicit representation is a written manuscript-a paper formalizing a solution to a perceived problem in the environment. This is the highestlevel of abstract problem-solving that the single individual can accomplish. The individual, upon completing a paper, can store this novel solution model in hyper-cortex's paper layer. In the future, new problems of the community can potentially, after a series of transformation, return the individual's written manuscript as a potentially usable solution model. An individual perceives their world, abstracts it, and represents it within the community. The community references it, organizes it, and provides the means for another individual to use that model for finding their solutions. This model is already in use by the scientific community's current non-tangible cortical architecture. Individuals and their collaborators perceive problems in the environment and then solve them. These solutions are represented as papers. These papers are categorized in journals and conference proceedings. Other individuals within the domain subscribe to these journals. Subscribing individuals read the focused papers of these broad ranging journals. The information in these papers alters the reader's internal mental-map-causing them to make new associations between disparate ideas or simply reinforcing old ideas. These new invariant representations become the focus of their attention and therefore the means by which they perceive the new problems of their environment. And so within the human and collective cortices, the problems of one individual, through the process of abstraction and specification, become the solutions of another (Figure 3).
The scientific community's publication process has existed for many years, but only in the last 15 years has the community moved their work into the computational domain-into the digital-library domain. This explicit representation of the community provides the necessary medium to algorithmically optimize the publication process-creating a more friction-less scientific community. The next two sections will create a hyper-cortex from the metadata contained in digital-library systems and then describes the way the hypercortex solves five common problems in the scientific community.
Constructing the Scientific Communities Hyper-Cortex
The scientific community's hyper-cortex has been made more explicit due in large part to the extensive use of digital-libraries. Many institutions use digital-libraries to electronically publish the pre-prints of their research findings. Digital-library technology not only stores digital copies of written manuscripts, it also maintains a record of each paper's associated metadata. Manuscript metadata such as authoring scientists, cited papers, and publishing journals make it possible to algorithmically generate the scientific community's hyper-cortex. This section will describe how to utilize digital-library metadata to construct the three layers of the hyper-cortex and interconnect them into a problem-solving hierarchy.
4a. Digital-Library Metadata Review
The most important metadata specification in use to date is the OAI 2 -metadata standard [14]. The OAI-PMH is the associated protocol used to harvest OAI-metadata from digital-libraries [15]. Initially this standard was used exclusively for searching, but recent research has recommended the development of more advanced services [16]. Before discussing how the OAI-metadata can be used to generate a digital representation of a hyper-cortex, the OAI <record> container will be reviewed. A <record> in the OAI specification contains all the metadata for a specific repository item-a submitted paper. The randomly chosen CiteSeer [17] record below is represented within the Dublin Core (<dc:>) and CiteSeer (<oai_citeseer:>) namespaces. For the sake of brevity, multiple tags that do not pertain to the goals of this paper have been left out. Also the abstract description and the references section at the end of the record have been shortened. Important tags that will be referenced in the following subsections are highlighted with bold font and have the ∆# notation near the right hand margin. <record> <header> <identifier>oai:CiteSeerPSU:99914</identifier> <datestamp>1998-10-05</datestamp> </header> <metadata> <oai_citeseer:oai_citeseer xmlns:oai_citeseer="http://copper.ist.psu.edu/oai/oai_citeseer/" xmlns:dc ="http://purl.org/dc/elements/1.1/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://copper.ist.psu.edu/oai/oai_citeseer/ http://copper.ist.psu.edu/oai/oai_citeseer.xsd "> <dc:title>The Asymptotics of Waiting Times Between Stationary Processes, Allowing Distortion</dc:title> <oai_citeseer:author name="Amir Dembo" /> ∆1 <oai_citeseer:author name="Ioannis Kontoyiannis" /> ∆1 <dc:description>this paper is to extend these asymptotic…</dc:description> <dc:identifier>http://citeseer.ist.psu.edu/99914.html</dc:identifier> <dc:source>http://www.stat.purdue.edu/people/yiannis/PAPERS/ms.ps.gz</dc:source> <oai_citeseer:relation type="References"> ∆2a <oai_citeseer:uri>oai:CiteSeerPSU:143950</oai_citeseer:uri> </oai_citeseer:relation> <oai_citeseer:relation type="References"> ∆2a <oai_citeseer:uri>oai:CiteSeerPSU:50737</oai_citeseer:uri> </oai_citeseer:relation> <oai_citeseer:relation type="Is Referenced By"> ∆2b <oai_citeseer:uri>oai:CiteSeerPSU:308347</oai_citeseer:uri> </oai_citeseer:relation> <oai_citeseer:relation type="Is Referenced By"> ∆2b <oai_citeseer:uri>oai:CiteSeerPSU:204535</oai_citeseer:uri> </oai_citeseer:relation> . . . <dc:rights>unrestricted</dc:rights> <dc:publisher>Annals of Applied Probability</dc:publisher> ∆3 </oai_citeseer:oai_citeseer> </metadata> </record>
4b. The Three Layers of the Hyper-Cortex and their Lateral Projections
The hyper-cortex described by this paper has three layers-an author layer, a paper layer, and a journal/proceedings layer. Each layer is represented using a co-authorship network [18] or co-citation network [19], a citation network [20] or co-citation network, and a domain network, respectively. The goal of each network layer is to relate their nodes according to some homophilic property-some quality of relatedness. For purposes of this hyper-cortex, the goal is to connect the nodes of each layer according to similar research ideas.
Co-authorship networks define the past collaborations of authors in the community. When two authors publish a paper together, an edge between them in the co-authorship network is created. It is assumed that two co-authors are similar in research ideas if they have collaborated on a project together. The strength of this relationship can be determined many ways. For one, the more two authors publish together, the greater their connection strength. If there are multiple co-authors on a particular paper, then the strength between any two co-authors is inversely proportional to the total amount of co-authors of the paper [18]. If an edge value between two authors already exists, then this new value is simply added. The lateral connections within the co-authorship network have the semantic meaning: 'has co-authored with'. An OAI <record> contains the necessary information to build a co-authorship network-∆1. In terms of the paper in the previous subsection, given that there are two authors, and assuming that no other papers have been co-authored by these two authors, each directions edge weight is 1.0. Co-citation data provide another means for determining author similarity. If many papers in the field co-cite two authors, then there is a high degree of similarity between the research ideas of these two authors [19] [21]. The process to create a co-citation network is more difficult than a co-authorship network, but it is possible using both the ∆1 and ∆2a entries in the OAI metadata record. The next layer above the co-authorship layer is the paper layer. The paper layer is created using a citation network which is different than a co-citation network. A co-citation network will argue that two papers are similar if some other third paper references both of them, while a citation network states that two papers are related if either one of the two papers references the other. Directed edges in a citation network represent a reference from one paper to another. The edge strength for a particular citation link is inversely proportional to the amount of references contained in the referencing paper.
edge weight is inversely proportional to the amount of reference (n) in the paper
In a citation network, two directed edges connect the papers-feedforward and feedback. A feedforward projection states that the first paper cites the second paper-∆2a. A feedback projection states that the second paper has been cited by the first paper-∆2b. The two outgoing and the two incoming edges of the paper in the previous subsection each have a weight of 0.5. Co-citation is another method for constructing networks of related manuscripts. To determine author similarity by co-citation data, if two papers are cited together by some third paper, then there similarity increases.
Finally, riding atop the paper layer is the journal and conference proceedings layer, but for brevity's sake, this layer will simply be called the journal layer. The journal layer is represented by a domain network. There currently exists no research in this area and it is expected that the edges and associated strengths of the domain network can be created algorithmically by doing word analysis of papers contained within the journals or by using the user base to create a collective mental-map similar to the one described in [11]. The difficulty of defining this layer is that journals have issues and conferences have years. It may be possible to represent the journal layer into two sub-layers. The lower of the two layers represents the issue of the journal or the year of the conference, and the higher layer represents the overall journal or conference. Even with this complication, there still exist many papers in the digital-library archives that have not been published in a journal or conference proceeding-pre-prints. It is for this reason that this paper, though not discussing this idea in-depth, promotes the concept of a virtual journal [22]. The basic idea is that scientists in the community are able group electronic papers into virtual journals as they see appropriate-and store this information as a record in the digital-library archive. Scientists in the community can then subscribe to those electronic journals most related to their research. If this technology is brought to the forefront then journal similarity is determined by the amount of papers two virtual journals share in common.
n e j i = ,
edge weight is equal to the amount of papers(n) the two virtual journals share in common
Layer Network Type Feedfoward Projections Feedback Projections ∆1-author co-authorship 'co-authored with this author' 'co-authored with this author' ∆2-paper citation 'cites this paper' 'cited by this paper' ∆3-journal domain 'similar to this journal' 'similar to this journal'
4c. The Three Layers of the Hyper-Cortex and their Vertical Projections
The previous subsection described the three layers of the scientific community's hyper-cortex and the meaning of their lateral projections. The cortical hierarchy is created by interconnecting the three layers with vertical projections. The author layer projects up the hierarchy with edges that represent that that author has written that paper. The paper layer projects down to the author layer stating that that paper has been written by that author. The paper layer then projects up to the journal layer stating that that paper has been published in that journal-∆3. Finally, the journal layer projects down the hierarchy stating that that journal contains that paper. In Figure 4, upward projections are on the left hand side of the object and downward projections are on the right hand side. The imbalance of projections was done to preserve the clarity of the diagram.
Layer Projections Down
Projections Up ∆1-author n/a 'wrote this paper' ∆2-paper 'written by this author' 'published in this journal' ∆3-journal 'contains this paper' n/a With today's current digital-library technology, layer's 1 and 2 are easily constructed. What is not so obvious is the lateral and horizontal projections within and between layer 3. As stated before, it is the hope that the ideas in this paper will stimulate the development of a virtual journal record in the OAIspecification. In next section, a specific hyper-cortex pattern-matching algorithm is used to solve five common problems in scientific community.
Problem-Solving in the Scientific Community's Hyper-Cortex
Collective problem-solving, like in individual problem-solving, is a pattern-matching function. Given a model of the environment (the problem)-an author, a paper, a journal, or some collection of the three-the hyper-cortex will then perform a series of transformations to the problem in order to derive a collection of potential solutions. The basic idea is to continuously map the input signal within and between layers of the hyper-cortex until some desired format emerges.
Each problem of the following subsections is represented graphically as a three layered hyper-cortex. A set of input nodes, or problem-model, is represented as a collection of either a P+ or P-nodes. P+ nodes refer to nodes that are the initial sources of positive or excitatory energy. P-nodes are nodes that are the initial sources of negative or inhibitory energy. Depending on the context of the problem, some energy should be excitatory and others inhibitory. The output nodes, or S nodes, represent the solution-model and are determined at the end of the problem-solving process. Energy is contained within discrete information molecules or particles. Each time step, a particle takes a particular an outgoing edge of its current node depending on a probability derived by normalizing all outgoing edge's weight to 1.0. With each time step, the energy content of the node decays according to a decay scalar parameter-a value between 0.0 and 1.0. A 1.0 decay scalar means that at each time step 100 percent of a particle's energy content remains and therefore the total energy in the network will never decrease. A decay scalar of 0.0 means that after one time step all energy in the particles will decay to 0.0. Whenever a node receives an incoming particle it adds the energy content of that particle to its current energy content-for negative energy, the nodes total energy is decreased. It is important to remember that for each problem-solving process each node keeps a history of how much energy has passed through it. The particle dissemination algorithm continues until the energy content of all the particles has reach 0.0-until the decay scalar has completely reduced each particle's energy content. The set of all nodes which have an energy value greater than 0.0 is considered the solution-model. The solution-model can be trimmed using an energy threshold value if the S node set is too large. In the scientific community's hyper-cortex there can be multiple S node sets: S A , S P , and S J . The various S sets are solution-models in author, paper, and journal format. The actual solution to the problem is dependant upon the observers desired solution format-which is dependant on the problem's context.
The following subsections take the reader through five stages of the paper writing process. The community's hyper-cortex can support a scientist through the development of an initial idea to the distribution of the idea's manuscript to interested members in the collective. Each of the five major problem-solving endeavors are possible via problem/solution pattern-matching.
5a. Realizing References
It is the goal of any scientist to publish papers. A scientist must use their internal cognitive faculties and the past and present ideas of their colleagues to generate novel solutions to the problems facing their discipline. Every paper starts with some initial idea that grows into a formalized model ready for reading by their colleagues. In many cases that idea is triggered by some paper they read. From here, this novel idea must be nurtured by and tied to other papers in the domain. So, from an initial idea, sparked by some keystone paper, the scientist must find the niche for this new, soon to be formalized, idea. The keystone paper of the paper currently being read was the book by Hawkins and Blakeslee [1]. From there many papers from the discipline of neuroscience, collective-intelligence research, and digital-library technology were sought to give the paper context within the community.
For this individual scientist's problem, the P P + set represents the keystone paper. This is the only sensory information the hyper-cortex has and therefore this small set is the hyper-cortex's problem-model. In order to trigger the cascade of energy through the hyper-cortex, the P P + paper is given a collection of positive energy particles. These particles propagate within the paper layer and between the author and journal layers incrementing the energy of all the nodes they pass through. The process continues until all the energy of all the particles has decayed to 0.0. At the end of the particle dissemination process there is an S A , S P , and an S J set. Since the scientist is concerned with finding potential references for the new idea, only the S P set is reviewed. If too many papers are contained in S P , the scientist can increase the energy threshold value until only the amount of papers desired is returned. For the example in Figure 5, the S P set is reduced to two nodes. At this point a scientist has initially found a paper that has stimulated an unformulated novel idea in their mind. The scientist has used that paper as a source for finding other related papers to further his understanding of the problem domain. This process of finding related papers is not a new idea. The hypercortex has always existed in the world-being partially created as individuals navigate the web of relations between authors, documents, and journals. The difference is that now the process is algorithmically performed on an explicitly represented hyper-cortex. If no explicit hyper-cortex existed, then the scientist would follow the references of the keystone paper. Review the journals that these papers are published in. Look through the curriculum vitae of the keystone paper's author for other papers. So on and so forth. This would have been a manual process, but because of the collectively generated hyper-cortex, this process can be expressed outside the individual. The scientist utilizes this collective mental-map to match his problem-find papers related to the keystone paper-to his solution-a collection of related papers worth reading.
5b. Realizing Collaborators
Now that the scientist has a collection of papers that are representative of his initial idea he may be interested in finding potential collaborators. A way to make this possible via the hyper-cortex is to find a collection of authors that are related to that collection of papers. The intuitive notion of this idea is that individuals whom are related in thought to the papers derived in subsection 5a are more than likely going to understand the author's vague notion of an idea and, in turn, be able to provide assistance in formalizing the idea more clearly. To determine collaborators, the input set of the hyper-cortex, P P +, is the solution set, S A , from subsection 5a. These positive energy particles propagate through the hyper-cortex-up to the journal layer and down to the author layer. Since the scientist is interested in finding collaborating authors the scientist reviews the S A solution-model once the energy content of all the particles has decayed to 0.0. The collection of authors in S A can then be contacted for discussion about these ideas and a potential collaborative paper writing relationship may form.
5c. Realizing Journals to Publish In
After finding papers to reference and other scientists to discuss the initial vague idea with, the original scientists and his collaborators have written a paper that contains a novel idea that they feel is worth publishing in a journal. In order to find a good arena to publish their work in, the authors must find a journal that is related to the now formalized paper. There could be many ways to organize this problemmodel, but for the example demonstrated in Figure 7, the problem-model is the set of nodes in both the paper and author layer. P A + is the set of all collaborating authors on the new paper. Next, in the paper layer, the P P + set is the set of all papers referenced by the new unpublished paper. After the initial distribution of all positive energy, the collective thought process runs until the energy content of all the particles has decayed to 0.0. Since the scientists are interested in a solution that refers to a potential journal to publish in, the scientists set the energy threshold value to the value of the highest energy node in the journal layer. This trims the S J set to one node-the journal to submit the paper to. The intuitive idea is to search the hyper-cortex for journals that have published the papers of the cited articles or of individuals whom are related in thought to the collaborating scientists.
5d. Realizing Peer-Reviewers
With the paper written and submitted it is up to the journal to determine the quality of the work. This is accomplished by means of the peer-review process. The editors of the journal locate individuals within the community for whom they believe to be experts in the paper's domain. The paper is then distributed to these referees for review. Sometimes the paper is accepted as is, other times it's returned for revision, and still other times, the paper is rejected. It is up to the reviewers to determine whether the paper is deemed of high enough quality to allow others in the community to use the work in their future publications. Finding experts in the field can be a laborious task of manual effort. The hyper-cortex can solve this problem by matching an unpublished paper to a set of expert individuals in the author layer. The unpublished paper is abstractly represented by the P P + set as all its references since it's representative node hasn't been placed in the hyper-cortex-since no invariant representation of the paper exists. Secondly, a P J + set can be created with one node-the reviewing journal. The reason for stimulating the reviewing journal is because the reviewing journal may want to search the web of authors whom have published in their journal before knowing that they understand the quality requirements of the journal. Finally, and of most interest is the P A -set. This is the set of authors whom have written the paper. Since an author cannot be a referee to their own paper, and since it is advisable that past collaborators of those authors also not be referees, these nodes should be given negative energy to inhibit the activation of themselves and the individuals around them in the co-authorship network. For an in depth look at this process please refer to [23]. In [23] the algorithm was carried out solely in the author layer by utilizing the authors of the referenced paper (instead of the referenced papers themselves) as the problem-model. Furthermore, what is of interest is that the set S A can have an energy threshold value of 0.0. By doing this, every individual in the output set S A is preserved. Each member in that set then has some proportion of the total network energy relative to the others in the collective. This varying degree of energy can be used to determine the individual's decision-making influence in the review process-where more expert individuals, with regards to the paper in question, should have more say as to the quality of the paper. This idea has been proposed as a means of augmenting digital-library technology to support the peer-review process and is further developed in [23].
5e. Realizing Readers
Finally the paper has been written, submitted, and accepted. It is the role of the journal to find the largest distribution base for the paper. Journals, that are not fee-based, can aid the author and the community by matching the paper to a set of readers who may be interested in the newly published paper. This is done by creating a P P + set of all the referenced papers of the newly published paper and creating a P A + set of the authors of the paper. The particle dissemination algorithm runs and the solution set, S A , is the set of all individuals in the author layer whom received an amount of energy greater than 0.0. If these individuals have provided email address with the digital-library system, then these authors can be automatically contacted with an email containing the title of the paper, its abstract, and a link to it in the digital-library repository. Authors can increase the energy threshold value of their author node if they wish to receive only those papers that are most pertinent to them. P/S process Problem-Model Solution-Model References P P + (related keystone paper) S P (related papers) Collaborators P P + (related papers) S A (potential collaborators) Journals P P + (related papers) · P A + (co-authors) S J (submitting journal) Peer-reviewers P J + (journal) · P P + (related papers) · P A -(co-authors) S A (potential reviewers) Readers P P + (related papers) · P A + (co-authors) S A (interested readers) Table 3: the problem-model and solution-models of 5 hyper-cortical problem-solving processes A single hyper-cortical structure can provide the functionality to perform many complex problem-solving tasks for the community (Table 3). By testing various parameters such as the amount of particles to distribute, the energy threshold values, and the multiplicative properties of particle interaction, these basic ideas can be finely tuned to meet the needs of the user. There are still many other problem's that can be developed that have yet to be realized by this paper. With future research into the hyper-cortex architecture and its particle-flow problem-solving process, digital-library's can be augmented above and beyond simple manuscript searching.
Conclusion
What has emerged in the scientific community is computational medium that supports collective problemsolving. Each individual stigmergetically contributes to the evolution of this structure and therefore the potential paths of its users. The collective brain continues to grow at a phenomenal rate as more and more individuals publish their work into the federated OAI digital-libraries. How will such a hyper-cortex affect the collective-attention? How does the hyper-cortex promote the self-organization of the relationships of the community members and their work? A well engineered hyper-cortex is the foundation for a 'frictionless' community [24]. What this means is that information will get to where it needs to go when it needs to get there-finely tuning the community by meditating its information flow and member relations. This is in line with the goals of the Open Archives Initiative. A hyper-cortically supported scientific community is a self-organizing entity that constantly derives solutions to its problems by matching its present state with its past realizations via the use of its artificial neural-network.
This paper has provided a glimpse into a collective-intelligence paradigm extended from work done within the individual-intelligence research domain. Other such hyper-cortical structures can be created. A movement away from flat networks is needed if more abstract problem-solving is to be possible. Other collectively generated social-networks can make use of this research to facilitate a new form of humancollective problem-solving unseen in today's current technology.
Figure 1 :
1the receptive field of the abstract neuron (black) is a cluster of connected lower-layer neurons
Figure 3 :
3abstracting a problem outside the individual and implementing the solution by another
increases according to how many co-authors (n) are co-authoring a new paper
Figure 4 :
4digital-library metadata can create the vertical and horizontal projections of the scientific community's hyper-cortex
Figure 5 :
5PP+ (keystone paper) → SP (related papers)
Figure 6 :
6PP+ (related papers) → SA (collaborators)
Figure 7 :
7PP+ (referenced papers) · PA+ (collaborators) → SJ (submitting journal)
Figure 8 :
8PJ+ (reviewing journal) · PP+ (referenced papers) · PA-(collaborators) → SA (reviewers)
Figure 9 :
9PA+ (paper authors) · PP+ (referenced papers) → SA (interested readers)
Table 1 :
1the semantic meanings of the feedfoward and feedback lateral projections within the hyper-cortical layers
Table 2 :
2the semantic meanings of the horizontal and vertical projections in the hyper-cortex
Discussed by Pierre Lévy[2] but originally coined by Pierre Teilhard de Chardin
Open Archives Initiative (http://www.openarchives.org/) "develops and promotes interoperability standards that aim to facilitate the efficient dissemination of content."
This paper's production is due in large part to funding from the U.S. Department of Education's GAANN Fellowship and the Fonds voor Wetenschappelijk Onderzoek -Vlaanderen. A special thanks goes out to Francis Heylighen and Carlos Gershenson for extended talks on this topics. This paper's production is due in large part to funding from the U.S. Department of Education's GAANN Fellowship and the Fonds voor Wetenschappelijk Onderzoek -Vlaanderen. A special thanks goes out to Francis Heylighen and Carlos Gershenson for extended talks on this topics.
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| [] |
[
"Multi-Particle Spectral Properties in the Transverse Field Ising Model by Continuous Unitary Transformations",
"Multi-Particle Spectral Properties in the Transverse Field Ising Model by Continuous Unitary Transformations"
] | [
"Benedikt Fauseweh \nLehrstuhl für Theoretische Physik I\nTU Dortmund\nOtto-Hahn Straße 444221DortmundGermany\n",
"Götz S Uhrig \nLehrstuhl für Theoretische Physik I\nTU Dortmund\nOtto-Hahn Straße 444221DortmundGermany\n"
] | [
"Lehrstuhl für Theoretische Physik I\nTU Dortmund\nOtto-Hahn Straße 444221DortmundGermany",
"Lehrstuhl für Theoretische Physik I\nTU Dortmund\nOtto-Hahn Straße 444221DortmundGermany"
] | [] | The one-dimensional transverse field Ising model is solved by continuous unitary transformations in the high-field limit. A high accuracy is reached due to the closure of the relevant algebra of operators which we call string operators. The closure is related to the possibility to map the model by Jordan-Wigner transformation to non-interacting fermions. But it is proven without referring to this mapping. The effective model derived by the continuous unitary transformations is used to compute the contributions of one, two, and three elementary excitations to the diagonal dynamic structure factors. The three-particle contributions have so far not been addressed analytically, except close to the quantum critical point. | 10.1103/physrevb.87.184406 | [
"https://arxiv.org/pdf/1302.0230v3.pdf"
] | 118,454,229 | 1302.0230 | f4bae168a6fd04cefabcc79002716fcd7bd668d1 |
Multi-Particle Spectral Properties in the Transverse Field Ising Model by Continuous Unitary Transformations
10 May 2013 (Dated: May 14, 2013)
Benedikt Fauseweh
Lehrstuhl für Theoretische Physik I
TU Dortmund
Otto-Hahn Straße 444221DortmundGermany
Götz S Uhrig
Lehrstuhl für Theoretische Physik I
TU Dortmund
Otto-Hahn Straße 444221DortmundGermany
Multi-Particle Spectral Properties in the Transverse Field Ising Model by Continuous Unitary Transformations
10 May 2013 (Dated: May 14, 2013)arXiv:1302.0230v3 [cond-mat.str-el]numbers: 7540Gb7510Pq7530Ds0230Mv
The one-dimensional transverse field Ising model is solved by continuous unitary transformations in the high-field limit. A high accuracy is reached due to the closure of the relevant algebra of operators which we call string operators. The closure is related to the possibility to map the model by Jordan-Wigner transformation to non-interacting fermions. But it is proven without referring to this mapping. The effective model derived by the continuous unitary transformations is used to compute the contributions of one, two, and three elementary excitations to the diagonal dynamic structure factors. The three-particle contributions have so far not been addressed analytically, except close to the quantum critical point.
I. INTRODUCTION
Understanding strong quantum fluctuations continues to be a formidable challenge. Where two (or more) phases compete and are separated by a continuous quantum phase transition 1 , i.e., at zero temperature, such fluctuations are particularly strong. Generically, such a quantum phase transition is signaled by the decay of elementary excitations. Far away from the phase transition, spectroscopic probes show dominant sharp δ-peaks at low energies which result from stable elementary excitations, so-called quasi-particles. But on approaching the phase transition, the spectral weight in the dominant quasiparticle peak is reduced further and further and shifted to contributions of more quasi-particle. Multi-particle spectra with considerable weight are an important signature of dominant quantum fluctuations in general. The vanishing of the single quasi-particle peaks at zero temperature is a smoking gun for a quantum phase transition in particular.
In this context, the transverse field Ising model (TFIM) 2 is a popular generic model describing magnetic excitations and displaying a quantum phase transition between the disordered phase in the limit of dominating transverse field and the ordered phase in the limit of dominating longitudinal Ising coupling. Due to its relative simplicity, the TFIM provides a convenient test case in the development of theoretical approaches 3,4 . This is particularly true for the one dimensional case, for which fermionization by a Jordan-Wigner transformation 5 yields an exact solution [6][7][8] .
The calculation of dynamical correlations in the TFIM is an active field of research. While transverse correlations can be treated in terms of fermions 8,9 , longitudinal correlations require different approaches, because of their non-locality in the fermionic picture. Based on an equation of motion for the longitudinal correlations 10 there has been a series of papers investigating the scaling region around the critical point Ref. 11-16. In 2009, Perk and Au-Yang computed results for the time-dependent longitudinal correlation functions by solving the coupled differential equations and complementing them with longtime asymptotics 17 . But so far no momentum and frequency resolved analysis has been performed, which also applies away from the scaling regime.
Quantum magnets in the vicinity of quantum phase transitions are dominated by strong quantum fluctuations. Quantum fluctuations are strongly favored by low-dimensionality and by frustration. Theoretically, a particularly clear sign of dominant quantum fluctuations is the fractionalization of elementary excitations. For instance, conventional spin waves (magnons) split into two spinons in antiferromagnetic Heisenberg chains [18][19][20][21] . Before complete fractionalization occurs, the spectral weight, as observed in inelastic neutron scattering, shifts from the channel of a single elementary excitation to the channel where two and more elementary excitations are created 22,23 . Thus, also the experimental focus is directed more and more to continua formed by more than one excitation, see for instance Refs. 19,20,24,and 25. In view of the above considerations, the present article pursues two goals in a study of the one-dimensional (1D) TFIM. First, we show how the special algebraic structure ('string algebra') of the operators occurring in the 1D TFIM enables its solution by a continuous unitary transformation (CUT) in the high-field phase to very high accuracy. Upon completion of our calculations, we learned that this algebra was introduced and used before in an algebraic solution of the TFIM 26 .
This algebra paves the way to treat a larger class of models of which the Hamilton operators can be expressed by operators belonging to the string algebra, for instance XY models in transverse fields. Second, we compute the three-particle contributions to the diagonal dynamic structure factors (DSF) in the CUT framework in the high-field phase. To our knowledge, this is the first time that these subdominant contributions are computed, except in the scaling region around the quantum phase transition. Thereby, interesting predictions for future ex-perimental studies are provided.
In Sect. II, the model and known exact results are recalled. In the following section, the continuous unitary transformations (CUTs) are briefly reviewed. Sect. IV is devoted to the algebra of string operators which are subsequently employed. Sects. V and VI comprise our static and dynamic results, respectively, while the conclusions are drawn in Sect. VII.
II. MODEL AND EXACT RESULTS
The Hamiltonian for the transverse field Ising model (TFIM) reads
H TFIM = Γ 2 i σ z i + J 4 i σ x i σ x i+1 ,(1)
where the σ α are the Pauli matrices and the sum i runs over all lattice sites. We normalized the distance between two sites to one. The transverse field strength is given by the parameter Γ while J denotes the strength of the antiferromagnetic coupling between two adjacent sites. The antiferromagnetic exchange can be converted to a ferromagnetic exchange J → −J by a π rotation around S z i for every second site i. This translates to a shift of π in momentum space.
The model has a quantum phase transition at J = 2Γ and it is self dual 1 . Similar to Ref. 1, we introduce the parameter x = J/2Γ. The starting point for the CUT calculations is J = 0. Hence, an elementary excitation is given by a single spin flip. For finite J the energy of these excitations becomes momentum dependent. We will refer to these excitations as quasi-particles. We expect that the perturbative ansatz breaks down once we reach the critical value J = 2Γ. Hence we focus on the static and dynamic properties for J < 2Γ.
The model was solved exactly by Pfeuty in 1970 7 , based on the work by Lieb et al. 27 and Niemeijer 6 . Pfeuty's solution uses the Jordan-Wigner transformation 5 to map the Hamiltonian in Eq. (1) to a chain of free fermions which is diagonalized by a Bogoliubov transformation 28 . This approach yields the exact expression for the ground state energy per site
E 0 N Γ = − 1 2π π 0 ω(q)dq,(2)
where ω(q) denotes the dimensionless dispersion
ω(q) = 1 + x 2 − 2x cos(q).(3)
The dispersion with dimension is given by Γω(q). From the dispersion we can easily extract the energy gap of the lowest lying excitations
∆ Γ = |1 − x| .(4)
Another interesting quantity worked out by Pfeuty is the transverse magnetization
M z = 1 N i g| σ z i |g = 1 π π 0 1 + x cos(q) ω(q) dq,(5)
where |g denotes the ground state of the TFIM. In the following sections we will compare our results with these exact expressions in order to validate the CUT approach. Beside the static properties stated above, dynamic properties are important in order to explain experimental results. Although the TFIM is analytically integrable, the evaluation of longitudinal dynamic correlations remains a very difficult task which requires considerable numerics, see for instance Refs. 17 or 29. In the fermionic picture, this is due to the non-locality of the Jordan-Wigner transformation.
One important quantity in the study of spin systems is the dynamic structure factor (DSF)
S αβ (ω, Q) = 1 N ∞ −∞ dt 2π l,l ′ e iωt e −iQ(l−l ′ ) σ α l (t)σ β l ′ ,(6)
where α, β ∈ {x, y, z}. Here Q denotes the total momentum and ω the frequency. The DSF is directly linked to the differential cross section in inelastic scattering experiments, see for instance Ref. 30. Due to the symmetry σ x i → −σ x i of H TFIM no correlations occur for α = x, β = z and α = y, β = z and vice versa, but for α = x, β = y and for α = β the DSF may and will obtain finite values.
In the following, we focus on the diagonal part of the DSF, i.e., α = β. For α = z exact expressions are known 8,9,31 , because the observable σ z remains local in the Jordan-Wigner representation of the TFIM. At zero temperature case, this expression reads
S zz (Q, ω) = π −π dk 1 [1 − f (Q, k 1 )] δ(ω − ω(k 1 − Q/2) − ω(k 1 + Q/2)), (7a) with f (Q, k 1 ) = Γ + J 2 cos(k 1 − Q/2) Γ + J 2 cos(k 1 + Q/2) ω(k 1 − Q/2)ω(k 1 + Q/2) .(7b)
It consists of a spectral density of scattering states of two elementary excitations with total momentum Q. For α = x and α = y, only the one-particle contributions have been calculated by Hamer et al. in 2006 32 . They used series expansion techniques to propose the expressions
S xx 1 (Q) = 1 − x 2 1 4 ω(Q) ,(8a)S yy 1 (Q) = 1 − x 2 1 4 ω(Q)(8b)
for the one-particle contribution to the equal-time structure factor. By comparing their results to correlation functions in the two-dimensional classical Ising model, see Ref. 33 and 34, they could show that the expressions above are indeed exact. Hence the full one-particle structure factor is given by
S αα 1 (Q, ω) = S αα 1 (Q)δ(ω − ω(Q)).(9)
For higher-particle contributions to the longitudinal DSF much less is known. In 1978, Vaidya and Tracy computed exact expressions for the longitudinal correlation functions in the anisotropic XY model in the time domain 35 . They evaluated the resulting expression in frequency space up to the three-particle contributions. But their results are limited to the scaling region at low energies, very close to the critical point. Furthermore Müller and Shrock calculated frequency-integrated wave number dependent susceptibilities for the TFIM at the critical point in Refs. 15 and 16. Our results will be complementary to theirs.
III. CONTINUOUS UNITARY TRANSFORMATIONS
We use the method of continuous unitary transformations (CUT) to derive effective models which allow for an easier evaluation of static ground state properties and dynamic correlation functions. The idea of CUT was first introduced by Wegner 36 and independently by G lazek and Wilson 37,38 .
The concept of CUT is to systematically finda unitary transformation that maps the Hamiltonian to a diagonal representation. One introduces a family of unitary transformations U (l) depending differentiably on a parameter l ∈ R + . The unitary transformation is characterized by its anti-Hermitian generator η(l) = (∂ l U (l))U † (l)= −η † (l). Then, a short calculation yields the flow equation
∂ l H(l) = [η(l), H(l)](10)
for the l-dependent Hamiltonian H(l). In general, it represents a system of coupled differential equations for the prefactors of all operators appearing in H(l). We refer to it as the differential equation system (DES). Without further truncation, the DES generically comprises an infinite number of variables. In practice, various truncation schemes help to keep the DES finite. For l → ∞ the Hamiltonian acquires its final form and it is denoted as the effective Hamiltonian
H eff = H(l) l=∞ = U (∞)HU † (∞).(11)
The convergence for ℓ → ∞ is assumed; it cannot be proven generally for infinite dimensional quantum systems because it depends on the specific form of the generator as well as on the employed truncation scheme.
Note that observables O also need to be transformed to effective observables by the same unitary transformation. This results in the flow equation for observables
∂ l O(l) = [η(l), O(l)](12)
which yields the effective observable O eff for l → ∞. The generator characterizes the CUT and the flow of the Hamiltonian. Thus, the choice of the generator is an important issue and it still represents an active field of research, cf. Refs. 4, 36, 39-42. In this paper we use the (quasi-)particle conserving (pc) generator, which was first proposed by Mielke 39 in the context of banded matrices and independently by Knetter and Uhrig 40 for manybody problems. By 'quasi-particle' we mean the elementary excitation. The pc generator directly aims at these quasi-particles. The goal is to eliminate terms that do not conserve the number Q of quasi-particles
H eff , Q = 0.(13)
The pc generator is given in matrix representation in the eigenbasis of Q by
η pc,ij (l) = sgn(q i − q j )h ij (l),(14)
where q i denotes the eigenvalues of the operator Q. An equivalent description of the pc generator can be given by decomposing the Hamiltonian into parts that create, H + (l), conserve, H 0 (l), and annihilate, H − (l), quasi-particles. Then the Hamilonian reads H(l) = H + (l) + H 0 (l) + H − (l) (15) and the quasi-particle conserving generator
η pc = H + (l) − H − (l).(16)
The convergence of the flow induced by this generator is proven for finite-dimensional systems; extensions to infinite systems are also available 43 . The pc generator preserves the blockband diagonal structure, i.e., the maximum number of particles created or annihilated does not change during the flow 39,40,44 . The CUT method consists of two basic steps. The commutator in Eq. (10) needs to be calculated, followed by the integration of the resulting flow equation. The latter can easily be done with standard numerical integration algorithms or even analytically.
In general, commuting H with η creates new types of terms which were originally not part of the Hamiltonian. For systems in the thermodynamic limit, all sorts of new terms may arise connecting more and more sites over larger and larger distances. In a numerical calculation we cannot treat an infinite number of operators, hence we have to restrict ourselves to operators which are physically relevant. In this paper we use the previously introduced directly evaluated enhanced perturbative CUT (deepCUT) 45 . The idea of deepCUT is to truncate operators and contributions to the DES according to their effects in powers of a small expansion parameter x. Roughly speaking, the order n in x is the truncation criterion. More precisely, a certain contribution to the DES is kept if it affects the targeted quantities (here: ground state energy and one-particle dispersion) in order m ≤ n in x. Details can be found in Ref. 45.
Thus we write our initial Hamiltonian in the form
H = H 0 + xV(17)
where H 0 describes the unperturbed Hamiltonian and V represents a perturbation. We expand the operators in the basis {A i } which is chosen such that the effective Hamiltonian can be computed exactly 45 up to order n in the parameter x. Then the flowing Hamiltonian can be denoted as
H(l) = i h i (l)A i(18)
where the prefactors h i (l) depend on the flow parameter l. For the generator we choose the same operator basis with the same prefactors
η(l) = i η i (l)A i = i h i (l)η [A i ](19)
where η [·] is a superoperator applying the generator scheme. For the pc generator, η
[A i ] = A i if A i creates more quasiparticles than it annihilates, η [A i ] = −A i if A i annihilates∂ l h i (l) = j,k D ijk h j (l)h k (l).(20)
We call the D ijk ∈ C the contributions to the DES. They are obtained in a perturbative calculation up to order n by calculating the commutator in Eq. (10) and expanding the results in the chosen operator basis. Note that the numerically evaluated DES also comprises powers in x beyond the order n 45 .
IV. STRING OPERATORS
In the previous section, we explained how continuous unitary transformations are applied in a general context. Here we specify the approach for the transverse field Ising model. For the TFIM in the high-field limit, we use the the state with all spins down
|0 = |· · · ↓ j−1 ↓ j ↓ j+1 · · · ,(21)
as the reference state, i.e., as the vacuum of elementary excitations. This corresponds to the strong field limit Γ → ∞ in the TFIM, which is also the starting point for a perturbative approach in the parameter x = J/(2Γ). An elementary excitations, i.e., a quasi-particle, is created by the spin-flip operator σ + l . We denote this state by
|l = σ + l |0 .(22)
It is obvious that no two excitations can be present at the same site so that the quasi-particles behave like hardcore bosons. But multi-particle states can straightforwardly be created by flipping spins at different sites. These ideas suggest a basis of operators consisting of monomials made from the local operators
σ + j , σ − j , σ + j σ − j , 1 ,(23)
where σ + j stands for particle creation, σ − j for particle annihilation, and σ + j σ − j counts whether a particle is present
at site j (σ + j σ − j = 1) or not (σ + j σ − j = 0)
. This approach is in line with the general structure explained in Ref. 46; we call it henceforth the multi-particle representation.
The number of such monomials grows exponentially with the number of sites which are non-trivially involved because at any site a quasi-particle may be created or annihilated or simply counted. For each additional site occurring in the course of the iterated commutations, the number of operators to be tracked grows roughly by a factor of 4 (neglecting reductions due to symmetry effects). This is a major drawback if one aims at higher orders. Therefore, we introduce a simpler modified operator basis, which we call string algebra, which is more appropriate for the TFIM, see also Ref. 26. We stress that the possibility to introduce a string algebra is connected to the Jordan-Wigner representation of the Hamiltonian in terms of non-interacting fermions.
The string algebra consists of operators which are given by the following product of Pauli operators
T φǫ n := j σ φ j j+n−1 k=j+1 σ z k σ ǫ j+n (24a) = j σ φ j σ z j+1 σ z j+2 · · · σ z j+n−1 σ ǫ j+n ,(24b)
with {φ, ǫ} ∈ {+, −} and n ∈ N. Each string operator consists of a product of adjacent σ z operators, framed by spin flip creation-and/or annihilation operators. The σ z operators form the string between the pair of spin flip operators. We refer to n as the spatial range of an operator. In contrast to the local set of operators used in the multi-particle representation (23) the string algebra is more transparently expressed by the set
σ + j , σ − j , σ z j , 1 .(25)
The key point of the string algebra is that excitations or annihilations of quasi-particles occur only at the end points of the string. Thus, for given end points, there are only four string operators to be considered. If excitations or annihilation could occur anywhere along the string one would have exponential growth of the number of operators. In Eq. (24), we defined the translationally invariant form of string operators. When dealing with local observables, it is also useful to introduce local string operators
O φǫ j,n := σ φ j j+n−1 k=j+1 σ z k σ ǫ j+n (26a) = σ φ j σ z j+1 σ z j+2 · · · σ ǫ j+n ,(26b)
with {φ, ǫ} ∈ {+, −} and n ∈ N . Note that a translationally invariant string operator is given by the sum of local string operators. It is also useful to define a string operator of range 0 consisting of a single σ z matrix.
T 0 := j σ z j ,(27a)O j,0 := σ z j .(27b)
For n = 1, the definitions (24) and (26) correspond to a normal hopping term or pair creation or annihilation operator. These operators cannot be distinguished from operators in the multi-particle representation 46 . For n > 1, the situation is different. For example, in the case n = 2, φ = + and ǫ = − can be re-expressed in multi-particle representation by
T +− 2 = j σ + j σ z j+1 σ − j+2 (28a) = j σ + j (2σ + j+1 σ − j+1 − 1)σ − j+2 (28b) = j 2σ + j σ + j+1 σ − j+1 σ − j+2 − σ + j σ − j+2 ,(28c)
which is the sum of a quartic interaction term and a hopping term, because we re-expressed σ z j+1 = 2σ + j+1 σ − j+1 −1. This simple example illustrates the computational advantage of the string algebra. If we tracked all operators in multi-particle representation, a single string operator of range n would require 2 n−1 multi-particle operators, clarifying the previous statement on the exponential growth of the number of such monomials. Therefore, if a model can be diagonalized within the string algebra, it is highly advantageous to describe all operators in terms of string operators.
With the above definitions, the Hamiltonian of the transverse field Ising model is formulated in terms of string operators
H TFIM = Γ 2 j σ z j + J 4 j σ + j σ − j+1 + σ + j σ + j+1 + h.c. (29a) = Γ 2 T 0 + J 4 T +− 1 + T −+ 1 + T ++ 1 + T −− 1 .(29b)
Next, we study the action of hopping terms on single particle-states
T −+ n |l = j σ − j σ z j+1 σ z j+2 · · · σ + j+n |l (30a) = j δ l,j σ z j+1 σ z j+2 · · · σ + j+n |0 (30b) = (−1) n−1 |l + n .(30c)
In the second line we used the property σ − j |l = δ l,j |0 and we know that σ z j |0 = − |0 , which yields the final result. If there is only one quasi-particle in the system, the only difference to conventional hopping is the factor (−1) n−1 . For subspaces with more quasi-particles, we have to take into account that there may be particles on the sites l + 1, l + 2 . . . l + n − 1. They modify the exponent of (−1) and can thus change the sign of the resulting state.
In order to apply the deepCUT to the TFIM, we have to calculate the contributions to the DES in Eq. (20). Thus, we calculate the commutator between two operators of the Hamiltonian H and the generator η. In Appendix A we show that the string algebra is closed under such commutations. This means that the commutator of two string operators can again be written as a linear combination of string operators. The closure of the string algebra allows us to set up the flow equation in very high order because the number of operators to be tracked grows only linearly for the Hamiltonian. For local observables within the string algebra it grows quadratically which is still a moderate growth. This is the key observation of the present article.
Explicitly calculating all distinct commutators of string operators, see Appendix A, allows us to determine all contributions to the DES analytically. In this way, we calculate the flow equation up to infinite order in x. We parametrize the Hamiltonian
H TFIM (l) = t 0 (l)T 0 + ∞ n=1 t +− n (l) T +− n + h.c. + ∞ n=1 t ++ n (l) T ++ n + h.c. ,(31)
and the generator for the CUT
η(l) = ∞ n=1 t ++ n (l) T ++ n − h.c. .(32)
In Appendix B we derive the flow equation for the pref-actors t 0 , t +− n , t ++ n . It reads
∂ l t 0 = 2 ∞ n=1 t ++ n 2 ,(33a)∂ l t +− m = 2 k+l=m k,l=1 t ++ k t ++ l − 2 |k−l|=m k,l=1 t ++ k t ++ l , (33b) ∂ l t ++ m = −4t ++ m t 0 + 2 |k−l|=m k,l=1 sgn(k − l)t ++ k t +− l + 2 k+l=m k,l=1 t ++ k t +− l ,(33c)
with m ∈ N. Note that this is a differential equation with an infinite number of variables which grows, however, only linearly in the spatial range. This result is remarkable considering the fact that it would require tremendously more flow parameters if we formulated the problem in multi-particle representation. Thus the string algebra allows us to evaluate the Hamiltonian transformation up to very high orders, which is especially important on approaching the quantum critical point x = J/2Γ = 1.
V. STATIC RESULTS
In this section we evaluate and present static results for the transverse field Ising model. The expression 'static' refers to time-independent properties. We treat the ground state energy per site in Sect. V A, the magnetization in Sect. V B and the momentum-integrated spectral weight in Sect. V C and the momentum-resolved static structure factor in Sect. V D.
A. Ground state energy
Due to the only linearly growing number of string operators, we are able to obtain the ground state energy per site up to order 256. Higher orders do not improve the results significantly so that we restrict ourselves to orders up to 256. Figure 1 compares the exact result for the ground state energy per site to CUT results in various orders in x. As expected, the accuracy increases for increasing order. Even close to the critical point the CUT result of order 128 and the exact results can barely be separated. The inset shows the deviations from the exact results. On the logarithmic scale, straight lines indicate power laws for these deviations as expected in a perturbatively controlled approach. We checked that the slopes of the lines correspond to the order of calculation by fitting the deviations to power laws, see Tab. I.
From the inset we also read off that the deepCUT in order 128 calculates the ground state energy per site correctly to the fifth digit, even at the critical point. The
J [Γ] -9 -8 -7 -6 -5 -4 -3 -2 -1 -0.2 -0.1 0 log 10 (∆ E0 ΓN ) log 10 ( J 2Γ )
Order 3 Order 9 Order 128 Exact
B. Transverse magnetization
Next, we examine the transverse magnetization
M z = − 1 N j g| σ z j |g .(34)
Here |g denotes the ground state of the Hamiltonian which is mapped to the zero-particle state by the CUT. Note that the expression 'transverse' refers to the direction of the external field, which is the z-axis in our model. In the limit J → 0, all spins are aligned along the external field, so that M z = 1 holds. For J > 0, the spins are perturbed by the antiferromagnetic interaction which reduces the transverse magnetization. To obtain M z , we transform the observable σ z j by the continuous unitary transformation to an effective observable. The operator of the transverse magneti-
Due to this identity and the fact that the string algebra is closed under commutation, we know that the final effective observable can be written as a linear combination of string operators
Due to this simple form of the observable very high orders can be reached again. The transverse magnetization calculated by the CUT is compared to the exact result in Fig. 2. As expected the results improve upon increasing order. The largest error occurs at the critical point where the transverse magnetization displays a singularity.
C. Spectral weight
In this section, we discuss the CUT results for the momentum-integrated quasi-particle weight in the two diagonal channels S xx and S yy . We use the CUT framework to calculate effective observables which renders an easy evaluation for the spectral weight possible in various quasi-particle channels, see also Ref. 47. The total spectral weight can be split according to
I αα = 1 N l σ α l σ α l = I 1 + I 2 + I 3 + . . . ,(38)
where I n denotes the weight in the channel with n quasiparticles in the system. Introducing the CUT framework results in
I αα n = 0| σ α n,eff σ α, † n,eff |0 ,(39)
where σ α n,eff denotes the part of the effective observable which annihilates n quasi-particles. Since σ x and σ y create an odd number of spin flips, i.e., quasi-particles, and since the generator of the CUT preserves the parity of an observable, S xx and S yy consist of 1, 3, 5, . . . quasiparticle contributions. With the help of the sum rule I αα = 1, we can also check if our results are still valid for large values of x.
We emphasize that the local observables σ x l and σ y l transform into non-local operators under the Jordan-Wigner transformation which act on an infinite number of sites. Therefore, no easy analysis of these observables is possible in fermionic terms. An explicit evaluation requires either analytical mappings which enable an evaluation in the scaling region 35 or extensive numerics in terms of Pfaffians whose dimensions grow linearly with the spatial range of the correlation 29 .
The problem of an infinite number of operators is avoided in the string operator basis (25). But the calculation remains cumbersome because the observables are not part of the string algebra and thus the structure of the effective observables is more complicated. This complicated structure prevents us from achieving very high orders, because the number of representatives to be tracked grows exponentially on increasing order. We are able to obtain results up to order 38. Then the computational effort reaches its limit in the present implementation because the contributions to the differential equation system take more than 8 GB of memory and the number of operators to track is larger than 7 million.
First, we address S xx as function of J for which results are depicted in Fig. 3. The CUT results are compared to the exact results from Ref. 32. The one-particle contribution shows a very sharp drop for J → 2Γ with a singularity at the critical point. The CUT agrees very well with the exact results as long as the order of calculation is below the correlation length. Recall that the order is proportional to the range of the physical processes included in the calculation. For the calculation of effective observables within the string algebra this was no problem because large orders > 100 could be achieved. But for the longitudinal correlations, we obtain only order 38 so that particularly sharp edges such as the one in the one-particle spectral weight are not captured. But the agreement improves on increasing order.
The spectral weight of the three-particle channel increases on approaching the critical point. Hence, spectral weight is transferred from the one-particle channel Figure 3. (Color Online) One-particle and three-particle spectral weight as function of the parameter J. Comparison of the exact expression for the one-particle weight (8a) with the CUT results. to higher quasi-particle channels. The one-and threeparticle terms are the dominant contributions to the total spectral weight for the parameters investigated. But for J > 1.9, the sum rule starts being violated in the CUT calculation. We attribute this violation to the calculation in finite order. It appears that the CUT calculation overestimates the one-particle contributions close to the critical point.
Next, we investigate S yy . This correlation is depicted in Fig. 4 in comparison to the exact result. Again, the one-particle contributions also vanish for J → 2Γ. But the edge at the critical point is by far not as sharp as in S xx because more spectral weight is transferred to higher quasi-particle spaces for lower parameters J. The sum rule is again violated for J > 1.9Γ due to finite order errors.
D. Equal-time structure factor
The momentum-resolved equal-time structure factor contains more information so that it is another interesting quantity
S αα (Q) = 1 N l,l ′ e −iQ(l−l ′ ) σ α l σ α l ′ .(40)
For a single quasi-particle it is directly connected to the full DSF by Eq. (9) because there is no mixing between different quasi-particle spaces 22 . Our first focus is S xx 1 (Q). Within the CUT framework, this quantity can be computed from the effective observable σ x j,eff by Fourier transformation of the terms that exactly create one particle S αα n (Q) = 0| σ α n,eff (−Q)σ α, † n,eff (Q) |0 where n stands for the number of quasi-particles involved and α may take the values x or y. In Fig. 5 we compare the CUT results to the exact expression (8a) for the parameters J = Γ, J = 1.5Γ and J = 1.9Γ. The agreement is very impressive though it worsens upon approaching the critical point. For J = Γ the DSF is essentially converged and the absolute errors are below 10 −10 Γ −1 . For J = 1.9Γ the error is below 10 −3 Γ −1 for Q < π/2. For Q > π/2 the absolute error rises up to 10 −1 Γ −1 .
A closer analysis reveals that the largest absolute error occurs for all parameters at the wave vector Q = π. The DSF diverges at this point for J → 2Γ. The relative error (not shown in the graphs) remains fairly constant over the whole Brillouin zone. Consequently our results differ from the exact ones found by Hamer et al. 32 by only about 1% even close to the critical point at J = 1.9Γ.
Finally, we consider S yy 1 (Q). In Fig. 6 we compare the CUT results to the exact expression for the parameters J = Γ, J = 1.5Γ and J = 1.9Γ. For J = Γ the DSF is essentially converged and the absolute errors are below 10 −10 Γ −1 . This changes for rising parameter J. For J = 1.9Γ the error is below 10 −2 Γ −1 for Q < π/2. For Q > π/2 the absolute error remains below 10 −3 Γ −1 . In contrast to the S xx channel, the lowest absolute error is located in the S yy channel at Q ≈ π. This is can be easily understood because S yy 1 (Q = π) constitutes a local minimum for all parameters J. Again, the relative error (not shown) remains essentially constant over the whole Brillouin zone.
VI. DYNAMIC PROPERTIES
In this section, we evaluate and present the dynamic results for the transverse field Ising model. Here, 'dynamic' refers to frequency dependent quantities. We deal with the dispersion in Sect. VI A and with the DSF in general in Sect. VI B and its different channels in Sects. VI C (S zz ), VI D (S xx ) and VI E (S yy ).
The general DSF is an important quantity because it is directly measurable in scattering experiments. Furthermore, dynamic correlations strongly depend on the model under study and often exhibit features which reveal the microscopic interactions in the Hamiltonian. Despite the fact that the TFIM is integrable, the calculation of dynamic correlations remains a difficult and complex problem.
A. Dispersion
As before we were able to reach order 256 for the CUT calculation of the Hamiltonian. In particular, we obtain the hopping matrix elements up to a range of 256. For small parameters J, a low order calculation is sufficient to achieve a good agreement with the exact result. Closer to the critical point this changes distinctly, see Fig. 7, which makes higher order calculations necessary. For J = 1.9Γ the absolute error of the result in order 6 is below 0.01Γ for q < π/2 and below 0.1Γ for q > π/2. For the order 32 result, it is below 10 −4 Γ for q < π/2 and below 10 −3 Γ for q > π/2. For J = 2Γ, the error of the order 32 result is below 10 −3 Γ for q < π/2 and below 10 −1 Γ for q > π/2. For the order 256 result, it is below 10 −5 Γ for q < π/2 and below 10 −3 Γ for q > π/2. This behavior is expected because the excitations become more and more dispersive on increasing J. Consequently, hopping processes over larger and larger distances become more important. To include these physical processes we need higher orders because the maximum range we can describe directly corresponds to the order of calculation (for lattice constant equal to unity). Directly at the critical point the energy gap closes and the correlation length diverges concomitantly.
The calculation of the dispersion is worst in the vicinity of the critical wave vector q = π. We stress, however, that the value directly at q = π, the energy gap of the TFIM, is calculated exactly up to numerical errors below 10 −10 Γ. This is an accidental result because the energy gap happens to be a linear function of J so that it is captured correctly by any perturbative approach in linear order and beyond, compare Eq. (4).
B. Dynamic structure factor
The DSF at T = 0 is linked to the imaginary part of the retarded Green function by the fluctuation-dissipation theorem at zero temperature 48
S αα (ω, Q) = − 1 π ImG αα (ω, Q).(42)
At T = 0, it is useful to write this Green function for ω > 0 as a resolvent (43) where E 0 is the ground state energy and
G αα (ω, Q) = g|σ α (−Q) 1 ω − (H(Q) − E 0 ) + i0 + σ α (Q)|gσ α (Q) = 1 √ N l e iQl σ α l(44)
is the Fourier transformed spin operator σ α l . In the CUT framework, we replace all operators by the effective operators and the ground state by the zeroparticle state, i.e., the vacuum of quasi-particles
G αα (ω, Q) = 0|S α eff (−Q) 1 ω − (H eff (Q) − E 0 ) + i0 + S α eff (Q)|0 . (45)
We evaluate the resolvent in Eq. (45) by means of a Lanczos tridiagonalization yielding a continued fraction representation of the resolvent 49,50
G αα (ω, Q) = b 2 0 ω − a 0 − b 2 1 ω−a1− b 2 2 . . . ,(46)
where the coefficients a n and b n are the matrix elements of the tridiagonal matrix of the effective Hamiltonian. We refer the reader to Appendices C and D where we explicitly calculate S α eff (Q)|0 as well as the action of the effective Hamiltonian for the Lanczos tridiagonalization. The continued fraction is terminated by a standard square-root terminator. This is appropriate for square root singularities at the band-edges.
Another piece of information that can be obtained from the sequences {a n } and {b n } are the exponents α and β of the band-edge singularities, see Fig. 8. They are directly connected to the asymptotics 49
a n = a ∞ + b ∞ β 2 − α 2 2n 2 + O 1 n 3 (47a) b n = b ∞ + b ∞ 1 − 2α 2 − 2β 2 8n 2 + O 1 n 3 .(47b)
These relations allow us to obtain the band-edge singularities up to their signs by fitting to the continued fraction coefficients. For two massive hardcore particles without interaction and with finite range hopping in one dimension the bandedge singularities are known to be α = β = 1/2, see for instance Refs. 51 and 52. We expect this behavior also to be true in the case of the TFIM because there is no interaction in the exact solution. In this case the relations (47a) and (47b) yield
f (n) = C + D n 2 (48) 0 0 ω A ω B Intensity ω (ω − ω A ) α (ω B − ω) βa n = a ∞ + O 1 n 3 (49a) b n = b ∞ + O 1 n 3 .(49b)
We confirm this behavior in the two-particle case of the S zz channel in Sect. VI C.
C. S zz channel
Because the observable σ z stays local in the Jordan-Wigner representation of the TFIM the DSF in the S zz channel can be obtained analytically, see Eq. (7a). The DSF in the zz channel results from the two-particle continuum. Even for large parameters J, no weight is shifted towards four or more particle spaces. All dynamics induced by this observable is captured in the two-particle sector.
We emphasize that this fact holds as well in in the CUT treatment formulated in terms of the string operator algebra. The corresponding local operator σ z j = O j,0 is element of the string algebra so that its effective observable after the CUT consists of a linear combination of string operators where the maximum range n is limited by the order of the calculation. We stress again that the operators O j+d,0 and O +− j+d,n do not create any excitations, while the operators O ++ j+d,n create exactly two excitations. Thus, the vector S α eff (Q)|0 is only element of the zero-and of the two-particle Hilbert space. The same holds in the fermionic picture, where the operator σ z i is a local density term which at most creates two fermionic excitations after the Bogoliubov diagonalization.
σ z j,eff = d o j+d O j+d,0(50)
Concomitantly, very high orders can be reached also in the transformation of the local observable. Because we transform a non translational-invariant operator, we have to consider the positions j +d and the starting site j so that the number of terms increases quadratically with the order. But we are still able to achieve an order of 128.
False color plots of the DSF obtained in this way are shown in Fig. 9 in order 128. The two-particle continuum is depicted in dependence of the total momentum Q and the energy ω. The overall intensity rises for larger parameters J. This stems from the decrease of the transverse magnetization which induces a shift of spectral weight from the zero-particle channel to the two-particle channel. Furthermore, we see that for small parameters J/(2Γ) most of the weight is concentrated in the region Q < π/2 while the opposite behavior occurs for larger parameters x. Note the singularity inside the continuum on the right side of the Brillouin zone that separates two regions with low and high spectral weight, see the case J = 1.9Γ. Knowing the exact expression (7a) we can explain this singularity as van Hove singularity in the two-particle density-of-states. The two-particle energy ω(Q/2 + q) + ω(Q/2 − q) displays a local maximum besides the global extrema as function of q, if Q and J are large enough.
We want to investigate more profoundly how the CUT calculation differs from the exact calculation by examining the DSF for fixed parameters J and total momentum Q. Fig. 10 shows S zz (ω, Q) for J = 1.5Γ and J = 1.9Γ and for the momenta Q = 0, Q = π/2 and Q = π calculated by the CUT in order 128 in comparison to the exact result. Note the excellent agreement for all parameters and momenta. The form of the DSF is very close to a half ellipse for low values of J because the continued fraction coefficients converge very quickly towards their final values a ∞ and b ∞ . For large J more spectral weight is concentrated at the lower band-edge which we attribute to a complex interplay between momentum and energy conservation. For the parameters J = 1.9 and Q = π/2, one clearly sees the singularity inside the continuum of the DSF which is the above mentioned van Hove singularity from a local maximum.
A detailed analysis shows that the error is lower in the middle of the continuum than at the band-edge singularities. This is expected due to the strong change of the DSF at the edges. On average, the error is below 10 −6 Γ −1 even for large parameters J. At the band-edges the error can rise up to 10 −3 Γ −1 . We presume that the errors are The maximum range for the Lanczos algorithm is dmax = 1000 sites, the continued fraction was evaluated to a depth of 50 and then terminated by the square root terminator. The color indicates the spectral density, see legend to the right. The upper and lower edge of the two-particle continuum are indicated by white lines. mainly due to inaccuracies in the Lanczos tridiagonalization and due to the limited maximum range in the transformation of the observable by the CUT. Nonetheless the errors are still very small and justify our approach.
Next, we investigate how the continued fraction coefficients approach their limiting values. Thereby, we estimate the exponents of the band-edge singularities according to Eqs. (47a) and (47b). The continued fraction coefficients for the case J = 1.5Γ and total momenta Q = 0 and Q = π/2 are shown in Fig. 11. The coefficients for the case Q = 0 approach their limit exponentially. Therefore we know by Eqs. (47a) and (47b) that both exponents take the value 1/2. Exact Q = 0 Order 128 Q = 0 Exact Q = π/2 Order 128 Q = π/2
Exact Q = π Order 128 Q = π J = 1.5Γ J = 1.9Γ Figure 10. (Color online) DSF S zz (ω, Q) for the parameters J = 1.5Γ (left) and J = 1.9Γ (right) for three total momenta Q. The maximum range for the Lanczos algorithm is dmax = 4000 sites, the continued fraction was evaluated to a depth of 100 and then terminated by the square root terminator. n (a n − a ∞ )Q = 0 For the case Q = π/2, the coefficients do not converge so rapidly. We fit them for this case versus 1/n 2 to check if they display terms in O(1/n 2 ). Both coefficients oscillate around the final value which can not be described by Eqs. (47a) and (47b). This again indicates that the exponents at the band-edges are 1/2. We also checked other momenta and they support the assumption that all exponents are 1/2 for the two-particle case as it has to be according to the fermionic analytical results. Thus, these findings corroborate the validity of our approach and analysis.
(b n − b ∞ )Q = 0 (a n − a ∞ )Q = π/2 (b n − b ∞ )Q = π/2
D. S xx channel
In Ref. 32 Hamer et al. derived an analytic expression for the one-particle contribution for the longitudinal structure factors. To our knowledge no data is available in the literature for higher quasi-particle contributions Q α β 0 2.5 ± 0.3 1.0 ± 0.2 π/2 3.0 ± 0.2 2.7 ± 0.2 π 3.0 ± 0.2 2.8 ± 0.1 Table II. Exponents for the band-edge singularities of S xx 3 (ω, Q) for J = 1.5Γ. The errors are determined from the fits using the Levenberg-Marquardt algorithm 56,57 . away from the scaling region 35 . Here our approach is able to provide complementary quantitative knowledge.
Similar to the two-particle case S zz (ω, Q), the threeparticle case S xx 3 (ω, Q) consists of a continuum of states. We are limited to a maximum order 38 due to the complicated structure of the local observable σ x which is not part of the closed string algebra. Overview plots for the DSF obtained by the CUT are found in Fig. 12. In these plots the three-particle continuum is depicted in dependence of total momentum Q and the energy ω.
The total weight rises on increasing J because spectral weight is transferred from the one-particle sector to the higher quasi-particle channels. The same qualitative behavior is observed for dimerized spin chains and spin ladders, and related systems [53][54][55] . In addition, we notice that most of the spectral weight is concentrated at momenta Q < π/2 for small parameters J. The weight slowly shifts for growing J similar to the S zz case. For J = 1.9Γ, most of the spectral weight is concentrated rather strongly at the lower band-edge of the continuum. The same tendency was found in the S zz case as well. Still the shape of the DSF in the S xx case differs strongly from a semi-ellipse in contrast to the S zz DSF.
A more quantitative investigation is shown in Fig. 13 where S xx 3 is plotted for J = 1.9Γ and momenta Q = 0, Q = π/2 and Q = π. It is confirmed that most of the spectral weight is concentrated at the lower bandedge for large values of J. For Q = π, a strong wiggling occurs which is to be attributed to the errors due to the calculation in finite order.
Next, we investigate the band-edge singularities in the three-particle case S xx 3 (ω, Q). As in the two-particle case, we use the relations (47a) and (47a) by fitting a 1/n 2 power law to the continued fraction coefficients. An example is shown in Fig. 14 In contrast to the two-particle case, no exponential approach towards the limit values occurs. For all momenta, both a n and b n show a behavior proportional to 1/n 2 . We stress that the O(1/n 3 ) terms are significant up to 1/n 2 ≈ 0.0002 ⇒ n ≈ 70. The values for the band-edge singularities obtained from the fits are shown in Tab. II.
For Q = π and Q = π/2, both exponents are close to 3 while for Q = 0 the exponents differ and we deduce that α = 2.5 and β = 1 holds. We stress, however, that the obtained exponents may still be affected by rather large numerical errors. In Ref. 58, a general expression for the multi-particle band-edge singularities is derived for a simple one-dimensional model of hardcore bosons for n > 1,
, but close to the extrema of the dispersion. For the three-particle case, this yields S n ∝ ω 3 which agrees well with our results for Q = π and Q = π/2, but differs for Q = 0. The discrepancy in the latter case may be due to the more complex structure of the dispersion in the effective Hamiltonian for the TFIM which includes longer range hopping processes. Order 38 Q = 0 Order 38 Q = π/2
Order 38 Q = π Figure 13. (Color online) DSF S xx 3 (ω, Q) for the parameter J = 1.9Γ for three chosen total momenta Q. The maximum range for the Lanczos algorithm is dmax = 300 sites, the continued fraction was evaluated to a depth of 100 and then terminated by the square root terminator.
E. S yy channel
As in the S xx channel, the S yy channel consists of 1, 3, 5, . . . particle contributions. Overview plots for the three-particle DSF obtained by the CUT in order 38 are found in Fig. 15. In these plots, the three-particle continuum is plotted in dependence of the total momentum Q and the energy ω. For small J, the S yy channel looks similar to the S xx channel. The only difference is the absolute weight because the three-particle continuum in the S yy channel gains weight sooner,i.e., for smaller J/(2Γ), than in the S xx channel. For higher values of J, there are already qualitative differences between the S xx and the S yy channel. Most of the spectral weight is still concentrated at the lower band-edge of the continuum. No spectral weight is gained in the region of the critical wave vector Q = π which constitutes a major difference to the S xx channel, see Fig. 12.
Scans of S yy 3 at fixed Q are shown in Fig. 16 for J = 1.9Γ and momenta Q = 0, Q = π/2 and Q = π. For Q ≤ π/2, most of the spectral weight is concentrated at the lower band-edge. This changes distinctly for Q ≥ π/2, especially for Q ≈ π. Here spectral weight is spread Order 38 Q = 0 Order 38 Q = π/2
Order 38 Q = π Figure 16. (Color online) DSF S yy 3 (ω, Q) for the parameter J = 1.9Γ for three chosen total momenta Q. The maximum range for the Lanczos algorithm is dmax = 300 sites, the continued fraction was evaluated to a depth of 100 and then terminated by the square root terminator.
Q α β 0 2.9 ± 0.1 1.0 ± 0.1 π/2 2.9 ± 0.2 2.7 ± 0.2 π 2.9 ± 0.1 2.8 ± 0.2 rather equally over frequency space. We also observe some wiggling which is due to finite order errors.
The values for the band-edge singularities obtained by fits as described before in the channels S zz and S xx 3 are given in Tab. III. They mostly equal those obtained for in the S xx channel within numerical errors. Only the case Q = 0 differs. We deduce that α = 3 and β = 1 holds for S yy 3 generally.
VII. CONCLUSION
Summarizing, we showed that the one-dimensional transverse field Ising model (1D TFIM) in the high field limit can be expressed in terms of string operators which form an algebra which is closed under commutation, which agrees with the previous finding by Jha and Valatin 26 . This property allowed us to solve the 1D TFIM in the high field limit to very high accuracy by continuous unitary transformations without resorting to the Jordan-Wigner transformation to non-interacting fermions. Note that the solution provided formally also covers the low field limit due to the duality of the model.
The only remaining restriction in the presented solution is the truncation in a given order n in the ratio x = J/(2Γ) of the Ising coupling J to the field strength Γ. But due to the closure of the string algebra the num-ber of terms to be tracked grows only linearly in the order n so that very high orders up to n = 256 can be achieved. Thus, accurate results for all practical purposes could be obtained. The order corresponds directly to the range of physical processes which are included if the lattice spacing is set to unity. We employed the recently developed deepCUT approach which is perturbatively correct in the targeted order and provides a robust extrapolation beyond this order 45 .
High orders are accessible for the Hamiltonian and all observables which belong to the string algebra. They cannot be obtained for observables which do not belong to the string algebra such as the longitudinal spin components. For this reason, the longitudinal components could be unitarily transformed only up to order 38.
We gauged the results in computing various static properties such as the ground state energy, the transverse magnetization, and the one-particle contribution to the equal time structure factors S αα 1 (Q) for α = x and α = y. The first two quantities can be compared directly to the analytically accessible results via Jordan-Wigner transformation. The one-particle contribution to the equal time structure factors has been conjectured by Hamer and co-workers by series expansions and strongly underlined by the mapping to a two-dimensional classical Ising model for which the exact results are known 32 .
Similarly, we computed dynamic quantities such as the one-particle dispersion and the momentum-and frequency-resolved diagonal dynamic structure factors. The dispersion and the transverse structure factor can again be gauged against the analytical result obtained in terms of non-interacting fermions. The longitudinal structure factors are much more difficult to address because their excitation operators are highly non-local in terms of the non-interacting fermions. While the oneparticle contribution can be derived from the static oneparticle structure factor and the exactly known dispersion there are no results for the next important threeparticle contributions for general coupling x ≤ 1. Only in the scaling region around the quantum phase transition results for the three-particle contributions in frequency domain exist 35 . Our results are reliable further away from the scaling region so that they are complementary to the existing information. The equation of motion approach pursued by Perk and Au-Yang 17 provides information on the correlations in the time and real space domain. But so far no analysis with frequency and momentum resolution has been performed.
The presented theoretical three-particle data for the static and the dynamic structure factor provides predictions where in momentum and frequency space one can expect significant three-particle signal. This information may guide future experimental searches for many-particle contributions. Concretely, our results show that the S yy channel is considerably better suited for such searches than the S xx channel. In the S xx channel the single particle contributions dominates over the multi-particle contributions except very close to the quantum phase transition.
Moreover, we found that the spectral weight in the three-particle dynamic structure factors is concentrated close to the lower band-edge if the parameters are such that the system is not too far away from criticality. Further away from criticality the main response is rather featureless and hardly displays a dependence on the total momentum Q. Then the spectral weight is still concentrated close to the lower bandedge around Q = 0, while it is spread out in the middle of the band around Q = π.
Our approach can be pursued further for all onedimensional models of which the Hamilton operators can be expressed within the string algebra. Further investigations for other response functions are possible as well.
ACKNOWLEDGMENTS
We thank Nils Drescher, Mohsen Hafez, Frederik Keim, Holger Krull, and Joachim Stolze for useful discussions and Jacques Perk for bringing the series of papers based on his equation of motion to our attention. We acknowledge financial support of the Helmholtz Virtual Institute "New states of matter and their excitations". 1 √ N r,d0,d1,j e iQr · s d0,d1 eff,j |r + d 0 , r + d 0 + d 1 ,
with d 1 > 0 must be considered. The sum over j addresses all operators that create an excitation at r + d 0 and another at r + d 0 + d 1 which are different in their content of factors σ z i at various sites. The index j is used to distinguish them. In contrast, in a strict multi-particle representation there would be only one operator. Shifting the exponent by d 0 + d 1 /2, the center of mass, results in the expression
where we have introduced |Q, d 1 which is the Fourier transformation of a two-particle state with distance d 1 . For the three-particle structure factor the state
where we introduced |Q, d 1 , d 2 , which is the Fourier transformation of a three-particle state with distance d 1 between the first two particles and distance d 2 between the second two particles.
Figure 1 .
1(Color online) Ground state energy per site as function of J. Comparison of the exact result to CUT results in various orders of x = J/(2Γ). The inset shows the absolut difference between the exact result and the CUT calculation on a logarithmic scale. The critical point is located at log 10 J 2Γ = 0. calculation in order 256 is not shown in the graphs because it would be indistinguishable from the other curves. It improves the result in order 128 at the critical point by about one digit.
Figure 2 .
2(Color Online) Transverse magnetization as a function of J. Comparison of the exact result with the CUT calculation. The inset shows a close view on the critical point J = 2Γ.
the vacuum expectation value. But they can not be omitted during the flow of the observable. The transverse magnetization after the CUT reads M z = o 0 (∞).
Figure 4 .
4(Color online) One-particle and three-particle spectral weight as function of the parameter J. Comparison of the exact expression for the one-particle weight (8b) with the CUT results.
Figure 5 .
5(Color Online) One-particle equal time structure factor S xx 1 (Q) for the parameters J = Γ, J = 1.5Γ and J = 1.9Γ. Comparison of the exact expression (8a) for the oneparticle weight with the CUT results.
Figure 6 .
6(Color online) One-particle equal time structure factor S yy 1 (Q) for the parameters J = Γ, J = 1.5Γ and J = 1.9Γ. Comparison of the exact expression (8b) with the CUT results.
Figure 7 .
7(Color online) Energy dispersion for J = 2.0Γ (top) and J = 1.9Γ (bottom). Comparison of the the CUT calculations to the exact results.
Figure 8 .
8Qualitative sketch of the band-edge singularities in the DSF.
Figure 9 .
9(Color online) The DSF S zz (ω, Q) for the parameters J = Γ (top), J = 1.5Γ (center) and J = 1.9Γ (bottom).
Figure 11 .
11(Color online) Absolute difference between the continued fraction coefficients and their final values for the case J = 1.5Γ and total momenta Q = 0 and Q = π/2.
Figure 12 .
12(Color online) The DSF S xx 3 (ω, Q) for the parameters J = Γ (top), J = 1.5Γ (center) and J = 1.9Γ (bottom). The maximum range for the Lanczos algorithm is dmax = 100 sites, the continued fraction was evaluated to a depth of 50 and then terminated by the square root terminator. The color indicates the spectral density, see legend to the right. The dispersion is indicated by the white solid line. The upper and lower edge of the three-particle continuum are indicated by white dashed lines. All results are computed in order38. with nearest-neighbor hopping. The result reads
Figure 14 .
14(Color online) Continued fraction coefficients for the case J = 1.5Γ and total momentum Q = 0. The upper panel shows the coefficients an and the lower panel shows the coefficients bn. The limit values are indicated by horizontal lines. The red/green lines indicated linear fits in 1/n 2 ; note the scale of the x-axis.
Figure 15 .
15(Color online) DSF S yy 3 (ω, Q) for the parameters J = Γ (top), J = 1.5Γ (center) and J = 1.9Γ (bottom). The maximum range for the Lanczos algorithm is dmax = 100 sites, the continued fraction was evaluated to a depth of 50 and then terminated by the square root terminator. The color indicates the spectral density, see legend to the right. The dispersion is indicated by the white solid line. The upper and lower edge of the three-particle continuum are indicated by white dashed lines. All results are computed in order 38.
eff,j (Q) |Q, d 1 ,
Table I. Exponents of the power laws for the deviation of the ground state energy obtained by numerical fits.Order Exponent Fitting Error
9
10
± 3
16
21
± 4
32
33
± 3
64
72
± 6
128
132
± 5
-0.64
-0.62
-0.6
-0.58
-0.56
-0.54
-0.52
-0.5
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
E0
N [Γ]
J [Γ]
-0.64
-0.62
-0.6
-0.58
-0.56
-0.54
-0.52
-0.5
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
E0
N [Γ]
Table III .
IIIExponents for the Band Edge Singularities for S yy 3 (ω, Q) for J = 1.5Γ. The errors are determined from the fits using the Levenberg-Marquardt algorithm 56,57 .
· |r + d 0 , r + d 0 + d 1 , r + d 0 + d 1 + d 2 ,(C4)with d 1 > 0 and d 2 > 0 must be considered. Shifting the exponent by d 0 + 2d 1 /3 + d 2 /3, the center of mass, results in the expression iQ(r+d0+2d1/3+d2/3) |r + d 0 , r + d 0 + d 1 , r + d 0 + d 1 + d 21
√
N r,d0,d1,d2,j
e iQr s d0,d1,d2
eff,j
S eff
3
0 (Q) |0 =
d0,d1,d2,j
e −iQ(d0+2d1/3+d2/3) s d0,d1,d2
eff,j
·
1
√
N r
e :=|Q,d1,d2
,
(C5a)
=
d0,d1,d2,j
e −iQ(d0+2d1/3+d2/3) s d0,d1,d2
eff,j
:=s
d 0 ,d 1 ,d 2
eff ,j
(Q)
|Q, d 1 , d 2 ,
(C5b)
=
d0,d1,d2,j
s d0,d1,d2
eff,j
(Q) |Q, d 1 , d 2 ,
t 0 (∞)T 0 |Q, d 1 , d 2 = t 0 (∞)(−N + 6) |Q, d 1 , d 2 ,(D4a)= (E 0 + 6t 0 (∞)) |Q, d 1 , d 2 . (D4b)Next, we analyze the action of the operator T +− n
Appendix A: Closure of the string algebraHere we show that the string algebra is closed under commutation. This means that the commutator of two string operators can again be written as a linear combination of string operators. First, we show that on a chain two string operators commute if neither of their start-/end-operators are on the same site. Without loss of generality this means 0 = O φǫ j,n , O χξ l,m , l > j, l + m < j + n, (A1a) 0 = O φǫ j,n , O χξ l,m , l > j, l + m > j + n,for φ, ǫ, χ, ξ ∈ {+, −}. The last commutator (A1c) is zero because operators acting on completely different sites always commute in a bosonic algebra. A simple calculation yields for the first commutator (A1a)using [σ z , σ χ ] = χ2σ χ , σ z σ χ = χσ χ and σ χ σ z = −χσ χ , if all operators act on the same site. Analogous calculations yield that (A1b) holds as well.The remaining contributions consist of commutators where either the start-and/or end-operators are on the same site. The start-and/or end-operators on the same site of two string operators need to be different because otherwise the identities σ + σ + = σ − σ − = 0 imply a vanishing result. Explicit calculations yield for the nonvanishing commutatorswith m < n andExplicit calculations for the translationally invariant string operators yield the following set of commutator relations for the case n, m ∈ N + , n < mand for the case n = mwhich are all linear combinations of string operators. Contributions with n > m are also included by exchange of the arguments in the commutators. This concludes the derivation of the closure of the string algebra.Appendix B: Proof of infinite order flow equationTo prove the expression (33) we proceed in two steps. First, we show that all kinds of string operators of arbitrary range will be created during the flow. Next we show which contributions occur in the DES.Our starting point for step one is the Hamiltonian of the TFIM in string operators in Eq.(31). By induction we show that once we have a complete set of operators of maximum range n, T 0 , T ±± 1 , T ±± 2 , . . . T ±± n we can create a new complete set of operators of range n + 1 by commutation with a string pair-creation-operator,Thereby, we created the string operators of range n+1. Because the Hamiltonian in Eq. (31) already comprises a complete set of range one we can deduce that all ranges n ∈ N + will be created during the flow. Hence, we can conclude for the generator of the TFIM(B2)For step two we consider the relations in Eq. (A5) and Eq. (A6). We start with the contributions to the operator T 0 . Such contributions are created only in the case m = n. For a given range n there are two contributions from the commutatorsboth with prefactor one. Note that T ++ n and T −− n have the same prefactor up to a sign due to hermiticity/antihermiticity. Finally, these considerations yieldand for |k − l| = mwith prefactor one. Note that the operator T −+ m have the same prefactor as T +− m . These calculations yieldLast we consider the operator T ++ m and T −− m , respectively. They are created by three different kinds of commutators. For k + l = mwhere the sign function stems from the different signs in the cases [T ++ n , T +− m ] and [T ++ m , T +− n ] in Eq. (A5). Finally, the third case is given byNow we can write down the complete flow equation forwhich concludes our derivation for the flow equation for infinite order.Appendix C: Calculation of S α eff (Q)|0We start from Eq.(45). To apply the Lanczos algorithm we need to calculateWe split the vector into its components of different particle number. For the two-particle structure factor the state S eff 2 0 (Q) |0 = Appendix D: Action of the effective HamiltonianTo apply the Lanczos algorithm we need to know the action of the effective Hamiltonian on the two-and threeparticle states, calculated in App. C. We stress that after the CUT there are no terms that violate particle-number conservation. We analyze the action of the operators T 0 , T +− n , T −+ n separately. Starting with the simple operator T 0 yieldsNext, we analyze the action of the operator T +−and of the operator T −+Note the different signs of the second and third term due to the properties of the string operator.Similarly to the two-particle state we examine the action of the effective Hamiltonian on the three-particle state. The simple operator T 0 yields
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| [] |
[
"Berry phase theory of Dzyaloshinskii-Moriya interaction and spin-orbit torques",
"Berry phase theory of Dzyaloshinskii-Moriya interaction and spin-orbit torques"
] | [
"Frank Freimuth \nPeter Grünberg Institut and Institute for Advanced Simulation\nForschungszentrum Jülich and JARA\n52425JülichGermany\n",
"Stefan Blügel \nPeter Grünberg Institut and Institute for Advanced Simulation\nForschungszentrum Jülich and JARA\n52425JülichGermany\n",
"Yuriy Mokrousov \nPeter Grünberg Institut and Institute for Advanced Simulation\nForschungszentrum Jülich and JARA\n52425JülichGermany\n"
] | [
"Peter Grünberg Institut and Institute for Advanced Simulation\nForschungszentrum Jülich and JARA\n52425JülichGermany",
"Peter Grünberg Institut and Institute for Advanced Simulation\nForschungszentrum Jülich and JARA\n52425JülichGermany",
"Peter Grünberg Institut and Institute for Advanced Simulation\nForschungszentrum Jülich and JARA\n52425JülichGermany"
] | [] | Recent experiments on current-induced domain wall motion in chiral magnets suggest important contributions both from spin-orbit torques (SOTs) and from the Dzyaloshinskii-Moriya interaction (DMI). We derive a Berry phase expression for the DMI and show that within this Berry phase theory DMI and SOTs are intimately related, in a way formally analogous to the relation between orbital magnetization (OM) and anomalous Hall effect (AHE). We introduce the concept of the twist torque moment, which probes the internal twist of wave packets in chiral magnets in a similar way like the orbital moment probes the wave packet's internal self rotation. We propose to interpret the Berry phase theory of DMI as a theory of spiralization in analogy to the modern theory of OM. We show that the twist torque moment and the spiralization together give rise to a Berry phase governing the response of the SOT to thermal gradients, in analogy to the intrinsic anomalous Nernst effect. The Berry phase theory of DMI is computationally very efficient because it only needs the electronic structure of the collinear magnetic system as input. As an application of the formalism we compute the DMI in Pt/Co, Pt/Co/O and Pt/Co/Al magnetic trilayers and show that the DMI is highly anisotropic in these systems. | 10.1088/0953-8984/26/10/104202 | [
"https://arxiv.org/pdf/1308.5983v1.pdf"
] | 21,831,686 | 1308.5983 | 7c0e62a0b4c56e5810933663b1b6d744fb0a7087 |
Berry phase theory of Dzyaloshinskii-Moriya interaction and spin-orbit torques
27 Aug 2013
Frank Freimuth
Peter Grünberg Institut and Institute for Advanced Simulation
Forschungszentrum Jülich and JARA
52425JülichGermany
Stefan Blügel
Peter Grünberg Institut and Institute for Advanced Simulation
Forschungszentrum Jülich and JARA
52425JülichGermany
Yuriy Mokrousov
Peter Grünberg Institut and Institute for Advanced Simulation
Forschungszentrum Jülich and JARA
52425JülichGermany
Berry phase theory of Dzyaloshinskii-Moriya interaction and spin-orbit torques
27 Aug 2013
Recent experiments on current-induced domain wall motion in chiral magnets suggest important contributions both from spin-orbit torques (SOTs) and from the Dzyaloshinskii-Moriya interaction (DMI). We derive a Berry phase expression for the DMI and show that within this Berry phase theory DMI and SOTs are intimately related, in a way formally analogous to the relation between orbital magnetization (OM) and anomalous Hall effect (AHE). We introduce the concept of the twist torque moment, which probes the internal twist of wave packets in chiral magnets in a similar way like the orbital moment probes the wave packet's internal self rotation. We propose to interpret the Berry phase theory of DMI as a theory of spiralization in analogy to the modern theory of OM. We show that the twist torque moment and the spiralization together give rise to a Berry phase governing the response of the SOT to thermal gradients, in analogy to the intrinsic anomalous Nernst effect. The Berry phase theory of DMI is computationally very efficient because it only needs the electronic structure of the collinear magnetic system as input. As an application of the formalism we compute the DMI in Pt/Co, Pt/Co/O and Pt/Co/Al magnetic trilayers and show that the DMI is highly anisotropic in these systems.
Broken inversion symmetry in chiral magnets, such as B20 compounds, (Ga,Mn)As and asymmetric bior trilayers opens new perspectives for current-induced magnetization control via so-called spin-orbit torques (SOTs) [1][2][3][4][5]. Notably, magnetization switching by SOTs in single collinear ferromagnetic layers has been demonstrated experimentally [6,7]. In addition to SOTs also the Dzyaloshinskii-Moriya interaction (DMI) arises from the interplay of broken inversion symmetry and spinorbit interaction (SOI) in magnetic systems [8,9]. Recent experiments and simulations suggest that both SOTs and DMI substantially influence current-induced domain-wall motion in chiral magnets [10][11][12] and that their combination may lead to a very efficient coupling of domain-wall motion to the applied current. Additionally, relations between SOTs and DMI have been proposed theoretically based on model calculations [13].
Expanding the micromagnetic free energy density F (r) at position r in terms of gradients ∂n/∂r j of magnetization directionn(r), we obtain in first order of the gradients
F (1) (r) = j D j (n(r)) · n(r) × ∂n(r) ∂r j ,(1)
where the Dzyaloshiskii vectors D j (n) will generally depend on magnetization directionn(r). Within ab initio density-functional theory (DFT) methods, DMI is often computed by adding SOI perturbatively to spirals with finite wave vectors q and extracting D j from the q-linear term in the dispersion E(q) [14][15][16]. Alternative methods for the calculation of DMI are based on multiplescattering theory [17,18] or a tight-binding representation of the electronic structure [19].
In the present work we develop a Berry phase theory of DMI. Our approach is based on expanding the free energy in terms of small gradients of magnetization direction within quantum mechanical perturbation theory. Formally, our Berry phase theory closely resembles the quantum theory of OM. It drastically reduces the computational burden, because it allows for calculating the DMI based on the collinear electronic structure. Additionally, the dependence of D j (n) on magnetization directionn is readily available, which is an advantage for general magnetic structures whenever D j (n) is strongly anisotropic. Moreover, the relationship between SOT and DMI becomes visible within the Berry phase theory. It turns out that DMI and SOT are related in a similar way like OM and AHE are related. We introduce the concept of a twist torque moment of a given band, which turns out to be analogous to the orbital moment of a band in OM theory and propose to interpret DMI as a spiralization, i.e., a twist torque moment per volume, analogous to the concept of magnetization as a magnetic moment per volume. We investigate how thermal gradients and gradients in the chemical potential can give rise to SOTs and find that both the twist torque moment and the DMI spiralization contribute to the SOT driven by statistical forces. Thus, DMI and SOT, the two effects which can make the coupling between domain wall motion and applied current highly efficient, are obtained from a common basis within the Berry phase theory. Finally, we apply our new method to the calculation of DMI in Pt/Co, Pt/Co/O and Pt/Co/Al magnetic thin trilayer films and find that DMI is strongly anisotropic in these systems.
In the following we derive an expression for the calculation of D j from the collinear electronic structure. We setn = (sin(γ), 0, cos(γ)) and consider small sinusoidal spatial oscillations of the angle γ around zero, i.e., γ(r) = η sin(q·r), where η is the smallness parameter. Up to first order in η we haven(r) = (η sin(q·r), 0, 1). Insert-ing this oscillating magnetization direction into Eq. (1), we obtain a spatially oscillating free energy density
F (1) (r) = η j D j (ê z ) ·ê y q j cos(q · r),(2)
whereê y andê z are unit vectors along the y and z directions, respectively. In order to extract D j from this oscillating free energy density, we multiply by cos(q · r) and integrate over the volume V :
lim q→0 2 V η ∂ ∂q j V F (1) (r) cos(q · r)d 3 r = D j (ê z ) ·ê y . (3)
The exchange interaction term in the LDA Hamiltonian is given by µ B B xc (r)n(r) · σ, where µ B is Bohr's magneton, σ is the vector of Pauli spin matrices, and B xc (r) is the exchange field. The oscillations of the magnetization directionn(r) perturb the wave functions of the collinear system. To first order in the smallness parameter η the perturbation operator is given by δV (r) = µ B B xc (r)σ x η sin(q · r). Below, we will first evaluate
K (1) = 2 N kn f kn × × ℜ V (ψ kn (r)) * (H 0 − µN )δψ kn (r) cos(q · r)d 3 r,(4)
where ψ kn (r) = e ik·r u kn (r) is the unperturbed Bloch function of band n at k-point k, δψ kn (r) is the change of ψ kn (r) within first order perturbation theory due to the perturbation δV (r), f kn = f (E kn ) with Fermi function f and E kn the unperturbed band energy, H 0 is the unperturbed LDA Hamiltonian of the collinear system, µ the chemical potential, N the particle number operator, and N the number of k points.
Based on the relation ∂ ∂β (βF ) = E − µN between free energy and grand-canonical energy, where β = (k B T ) −1 , we can relate K (1) and D j as follows:
lim q→0 2 V η ∂ ∂q j K (1) = ∂ ∂β (βD j (ê z ) ·ê y ).(5)
The evaluation of Eq. (4) is very similar to the derivation of the quantum theory of OM [20]. Using the firstorder perturbation theory expression for δψ kn (r) and switching from Bloch functions ψ kn (r) to their lattice periodic parts u kn (r) we obtain
K (1) = µ B η 4N knm [E kn + E k+qm − 2µ] [f kn − f k+qm ] × × ℑ u kn |u k+qm u k+qm |B xc (r)σ x |u kn E kn − E k+qm .(6)
Differentiating with respect to q j , taking the limit q → 0 as prescribed by Eq. (5), and generalizing to arbi-trary directionn we arrive at the expression
∂ ∂β (βD j (n) ·ê i ) = N V k m =n [E kn − µ] [f kn − f km ] × × ℑ u kn |T i |u km u km |v j (k)|u kn (E km − E kn ) 2 + N V k m =n [E kn − µ] f ′ (E kn )× × ℑ u kn |T i |u km u km |v j (k)|u kn E km − E kn ,(7)
where v j (k) = 1 ∂ ∂kj [e −ik·r H 0 e ik·r ] is the j-component of the velocity operator in crystal momentum representation, and the torque operator at position r is given by T (r) = m × B xc (r) in terms of the spin magnetic moment operator m = −µ B σ and the exchange field B xc (r) and T i is its i component.
Integrating Eq. (7) and defining D ij (n) = D j (n) ·ê i yields
D ij = 1 N V kn f kn A knij + 1 β ln[1 + e −β(E kn −µ) ]B knij ,(8)where A knij = m =n ℑ u kn |T i |u km u km |v j (k)|u kn E km − E kn(9)
and
B knij = −2 m =n ℑ u kn |T i |u km u km |v j (k)|u kn (E km − E kn ) 2 .
(10) Using
T = ∂H 0 ∂θê φ − 1 sin θ ∂H 0 ∂φê θ ,(11)
where θ and φ specifyn in spherical coordinates, i.e., n = (sin θ cos φ, sin θ sin φ, cos θ), andê θ = ∂n/∂θ,ê φ = (1/ sin θ)∂n/∂φ, we obtain alternative expressions of A kn and B kn in terms of derivatives of the wave functions with respect to both crystal momentum k and the spherical coordinates of the magnetization direction:
A knij = −ê i · ê φ ℑ ∂u kn ∂θ (E kn − H 0 (k)) ∂u kn ∂k j −ê θ 1 sin θ ℑ ∂u kn ∂φ (E kn − H 0 (k)) ∂u kn ∂k j ,(12)
and
B knij = −2ê i · ê φ ℑ ∂u kn ∂θ ∂u kn ∂k j −ê θ 1 sin θ ℑ ∂u kn ∂φ ∂u kn ∂k j ,(13)
where H 0 (k) = e −ik·r H 0 e ik·r is the crystal momentum representation of the Hamiltonian H 0 . We note that the special case of θ = 0, in which case sin θ = 0 in the numerators, is obtained from these equations by considering a finite θ and then taking the limit of θ → 0. The case of θ = π is treated similarly. Alternatively, one may avoid the problems at θ = 0 and θ = π by choosing the polar axis such that θ is between 0 and π. At zero temperature Eq. (8) becomes
D ij = 1 N V kn f kn [A knij − (E kn − µ)B knij ] = −ê i N V · ê φ ℑ ∂u kn ∂θ (2µ − E kn − H 0 (k)) ∂u kn ∂k j −ê θ sin θ ℑ ∂u kn ∂φ (2µ − E kn − H 0 (k)) ∂u kn ∂k j .
(14) Comparing Eq. (8) with the quantum theory of OM [20,21] one finds strong formal analogies, where B knij corresponds to the Berry curvature i ∇ k u kn | × |∇ k u kn and A knij corresponds to the orbital moment
(e/2 )i ∇ k u kn | × [E kn − H 0 ]|∇ k u kn of state n.
Therefore, we define the twist torque moments of state n by A knij . In perfect analogy to the theory of OM the DMI at zero temperature given by Eq. (14) is not simply the sum of the twist torque moments of all occupied states divided by the volume but there is a Berry curvature correction due to B knij . In order to counterpart the formal analogies on the level of terminology, it is tempting to call D ij the DMI spiralization.
Considering finite systems rather than infinite periodic crystals we can further substantiate the analogies between OM and DMI spiralization. For finite systems, the orbital magnetic moment p orb can be expressed via the moment r × j(r) of the current density j(r) as
p orb = 1 2 n f n d 3 r(ψ n (r)) * r × j(r)ψ n (r).(15)
Since the position operator r is not compatible with periodic boundary conditions, it is replaced either by the velocity operator or by derivatives with respect to crystal momentum in the theory of OM for infinite systems whenever the formalism is based on the Bloch-periodic eigenfunctions of the Hamiltonian [22][23][24][25]. In this situation the OM is expressed in terms of Berry phases. Likewise, it is natural to interpret DMI in finite systems in terms of the moments r i T j (r) of the torque, i.e.,
D ij = 1 V n f n d 3 r(ψ n (r)) * r j T i (r)ψ n (r),(16)
because if the magnetization rotates by an angle δΦ around the axis s, the associated energy change is T ·sδΦ. Thus, in this picture of DMI the free energy change due to magnetization gradients arises from the asymmetry of the torque, which can be quantified by its moments. However, like in the case of the orbital moment, an expression of DMI involving the position operator cannot be used for infinite systems with periodic boundary conditions and the correct theory has instead of the position operator either velocity operators (see Eq. (9) and Eq. (10)) or derivatives with respect to crystal momentum (see Eq. (12) and Eq. (13)). While Thonhauser et al. [23] and Shi et al. [20] derived the expressions of the OM quantum mechanically rigorously for Bloch electrons in crystalline solids, exactly the same expressions have been obtained from semiclassical wave-packet dynamics [21]. At first glance astonishing, this agreement results from the semiclassical theory getting exact as the length scale of the perturbation goes to infinity [20]. This limit q → 0 is also taken explicitly in our definition of the DMI spiralization in Eq. (5). Thus, the expressions obtained for DMI within the semiclassical formalism have to reproduce our results. It will become clear in the following that developing the semiclassical picture of SOT and DMI is quite rewarding.
To get started semiclassically, we define the twist torque moment of a wave packet |W kn constructed from band n with average crystal momentum k by
A W knij = W kn |(r j − r W knj )T i |W kn ,(17)
where r W knj = W kn |r j |W kn are the coordinates of the center of |W kn . In Ref. [26] the details of the construction of |W kn from Bloch functions are given and a wave packet formalism is presented there which allows for rewriting wave packet expectation values in terms of Berry phase expressions. In the case of the twist torque moment we obtain
A W knij = A knij .(18)
Thus, our previous expression for the twist torque moment of band n in Eq. (12) is equivalent to the expression of the twist torque moment of the wave packet constructed from band n if the crystal momentum k in Eq. (12) is identified with the mean wave vector of the wave packet. In the theory of SOT and DMI the twist torque moment plays the same role as the orbital moment does in the theory of AHE and OM. It is associated with the internal twist of the wave packet which locally prefers a noncollinear spiral magnetic structure such that any enforcement of magnetic collinearity leads to torques countering this collinearity, the moment of which is the twist torque moment. Inclusion of the SOT into the picture marks the next stage of our semiclassical expedition. If an external electric field E is applied to the system, a torque T arises, which is given within linear response by T = tE, where t is the torkance tensor. As we have shown recently [27], the intrinsic even torkance t even ij (n) = (t ij (n)+t ij (−n))/2 can be expressed in terms of the Berry curvature Eq. (13) as
t even ij (n) = − e N kn f kn B knij ,(19)
where e > 0 is the elementary positive charge. Thus, the electron in band n with crystal momentum k exerts the torque
T kn = ∂E kn ∂θê φ − 1 sin θ ∂E kn ∂φê θ − ij eB knijêi E j (20)
on the magnetization. The first two terms are simply the expectation value of the torque operator Eq. (11) and give rise to the magnetocrystalline anisotropy energy. The last term is the even SOT. This equation should be compared to the wave packet's semiclassical equation of motion
dr W kn dt = 1 ∂E kn ∂k + e E × Ω kn ,(21)
where the first term is the group velocity and the last term is the anomalous velocity due to the Berry curvature Ω kn which gives rise to the AHE. We find that the even SOT, related to the Berry curvature B knij , is analogous to the AHE, related to the Berry curvature Ω kn . In metallic systems the application of an external electric field leads to additional responses of the system besides the ones due to these Berry curvature terms, because the Fermi sphere is shifted and the states are occupied according to a nonequilibrium distribution function. Evaluated for such a nonequilibrium distribution, the first two terms in Eq. (20) yield the odd torque T odd ij (n) = (T ij (n) − T ij (−n))/2 [27], while the first term in Eq. (21) gives rise to the normal electrical transport current.
According to the Einstein relation, a gradient in the chemical potential ∇µ has the same effects as an applied electric field E = ∇µ/e. However, in the absence of an applied electric field the rightmost terms in Eq. (20) and in Eq. (21) vanish. Therefore, the question arises how the Berry phases enter the theory when statistical rather than mechanical forces drive the electrons. Xiao et al. [21] have shown that in order to solve this puzzle in the case of Eq. (21), it is important to distinguish between local currents and transport currents and to take the wave packet's finite spread into account in the equation for the local current. The derivation of a local torque in analogy to the local current of Xiao et al. is rather straightforward. Due to the twist torque moment gradients of the temperature or of the chemical potential lead to a correction term for the local torque, which is given by
δT loc i = 1 N kn j ∂ ∂r j f kn (r)A knij ,(22)
where f kn (r) depends on r due to gradients in temperature or in chemical potential. Thus, a Berry phase term, namely A knij , manifests itself in the local torque whenever temperature or chemical potential are inhomogeneous across the sample, because the twist torque moment couples to these gradients, whereby it is partly converted into a torque. Next, we have to subtract the gradients of the DMI spiralization to obtain the measurable torque. This step is analogous to the subtraction of the curl of magnetization from the local current density in the work of Xiao et al. and leads to the following correction term to the measurable torque in the presence of gradients of T or µ:
δT i = − 1 N knj B knij ∂ ∂r j 1 β ln[1 + e −β(E kn −µ) ].(23)
From Eq. (23) one easily obtains the torque due to a chemical potential gradient ∇µ:
δT i = − e N knj f kn B knij 1 e ∂µ ∂r j = j t even ij 1 e ∂µ ∂r j ,(24)
showing that the Einstein relation is satisfied. The torque due to a temperature gradient can be written as
δT i = j 1 e ∂T ∂r j dE df kn dµ t even ij (E)| T =0 E − µ T ,(25)
where t even ij (E)| T =0 is the even torkance at zero temperature with Fermi energy set to E. Thus, analogously to the intrinsic anomalous Nernst effect [21], the proper definition of local and measurable torques introduces the Berry phases into the response to thermal gradients.
We proceed to formulate DMI in terms of Green functions. Thereby, our objective is twofold. First, the Green function formulation will allow us to connect DMI and SOT from a different perspective. Second, we expect that a Green function theory for DMI will sometimes be favorable, e.g. for the investigation of DMI in disordered systems. Using the residue theorem we can prove the identity
ℑ dEf (E) 1 E − E q + i0 + 1 (E − E p + i0 + ) 2 = = π [f (E p ) − f (E q )] (E p − E q ) 2 + f ′ (E p ) E q − E p ,(26)
which allows us to rewrite Eq. (7) in terms of Green functions as follows:
∂ ∂β (βD ij ) = −1 2πV 2 ℑ dEf (E)× ×Tr T i G R (E)v j G R (E)(H 0 − µ)G R (E) −T i G R (E)(H 0 − µ)G R (E)v j G R (E) ,(27)
where G R (E) = [E − H 0 + i0 + ] −1 is the retarded Green function. Making use of
0 = ℑ dEf (E)× ×Tr T i G R (E)v j G R (E)(H 0 − E)G R (E) −T i G R (E)(H 0 − E)G R (E)v j G R (E)(28)
we can simplify Eq. (27) into the form
∂ ∂β (βD ij ) = 1 hV ℜ dEf (E)(E − µ)× × Tr T i G R (E)v j dG R (E) dE − T i dG R (E) dE v j G R (E) .(29)
This result can be directly compared with the SOT torkance t ij , which is a sum of three contributions [27], i.e., t ij = t
I(a) ij + t I(b) ij + t II ij , where t I(a) ij = − e h dE df (E) dE Tr T i G R (E)v j G A (E) t I(b) ij = e h dE df (E) dE ℜTr T i G R (E)v j G R (E) t II ij = e h dEf (E) ℜTr T i G R (E)v j dG R (E) dE − T i dG R (E) dE v j G R (E) ,(30)
with G A (E) the advanced Green function. Clearly, ∂ ∂β (βD ij ) differs from t II ij only by an additional factor (E − µ)/e in the integrand as well as a factor 1/V which can be left away whenever energy per unit cell is the desired unit of energy density. It is interesting that only part of the torkance, namely only t II ij , is related to the D ij . But in fact also for the torkance itself, t II ij plays a special role, because in contrast to the other two terms it is a Fermi sea integral. Moreover, its complex version -i.e., without taking the real part -can be analytically continued into the upper half complex plane. Additionally, t II ij contributes only to the even SOT torkance but not to the odd one. Since DMI is a ground state property then so is t II ij . The complete SOT torkance is generally not a ground state property but a transport one, in particular the intraband contribution to the odd torkance involves the relaxation times [27].
Defining the energy-resolved torkance ϑ ij (E) at T = 0 as
ϑ ij (E) = e h δ(E − µ) Tr T i G R (E)v j G A (E) − e h δ(E − µ)ℜTr T i G R (E)v j G R (E) + e h θ(µ − E)ℜTr T i G R (E)v j dG R (E) dE − T i dG R (E) dE v j G R (E) ,(31)
we may write at T = 0:
t ij = dEϑ ij (E) D ij = 1 eV dE(E − µ)ϑ ij (E).(32)
Since δ(E − µ)(E − µ) = 0, the first two lines in Eq. (31) do not contribute to D ij . Eq. (32) suggests that t ij and D ij will generally behave similarly, in particular from the symmetry point of view. Doing a symmetry analysis for ϑ ij is sufficient to determine the symmetries of both t ij and D ij . In systems with strongly anisotropic SOT [28] we expect also the DMI to be anisotropic as a consequence of Eq. (32). Furthermore, sign and magnitude of t ij and D ij will often be correlated.
We turn now to the computational aspects of the Berry phase formalism of DMI spiralisation. Recently, a Wannier function based method for the calculation of the orbital moment directly from the Berry phase expressions has been presented [29]. Conceptually, the direct evaluation of observables from their Berry phase representation (e.g. Eq. (12) and Eq. (13)) is very appealing and appears to be advantageous over the use of the corresponding Kubo formula expressions (e.g. Eq. (9) and Eq. (10)). However, in the case of DMI, the curvatures involve also angular derivatives. They are similar to phase-space Berry phases, or mixed real-space momentum-space Berry phases [30]. This opens an interesting practical aspect for the generalization of the Wannier function concept to higher dimensions by performing Fourier transforms not only with respect to crystal momentum, but also with respect to other parameters of the Hamiltonian, such as the magnetic structure parameters. In order to compute the DMI in disordered systems, Eq. (29) should generally be suitable. Since Eq. (29) involves only the retarded Green function, the energy integration can be performed in the upper half complex plane, analogously to the calculation of the charge density from the Green function in Green function based DFT codes. Irrespective of whether the explicit Berry phase based or the wave function based or the Green function based version of the DMI theory is used, one will in most cases achieve computational speed-ups compared to extracting DMI from spin spiral calculations due to several reasons: The symmetry of the collinear system is higher than the symmetry of the noncollinear spin spiral system. If spin-orbit interaction is treated within second variation, the computational time demand is dominated by the diagonalization of the collinear Hamiltonians, which can be faster by a factor of 4 compared to diagonalizing the noncollinear Hamiltonian. In systems with anisotropic DMI sampling the DMI vector with the Berry phase method is more efficient than reconstructing the information from spiral calculations which by construction average over the various magnetization directions comprising the spiral.
As an application of the Berry phase method we compute DMI in Pt/Co, Pt/Co/O and Pt/Co/Al thin films composed of 10 atomic layers of Pt(111), 3 atomic layers of hcp Co and one additional atomic layer of O or Al. Our Pt/Co, Pt/Co/O and Pt/Co/Al thin films are realistic models of trilayer structures such as Pt/Co/AlO x and Pt/Co/MgO currently studied extensively experimentally due to SOT and due to the combination of SOT and DMI to allow for highly efficient current induced domain wall motion. The computational details of the DFT electronic structure calculations are given in Ref. [27], in which the authors studied the SOT in these systems. We constructed maximally localized Wannier functions from the relativistic first-principles Bloch functions in order to evaluate Eq. (14), Eq. (9) and Eq. (10) computationally efficiently by making use of the Wannier interpolation technique [31][32][33]. Forn perpendicular to the films (i.e., along z direction), the computed DMI spiralizations D yx = −D xy of Pt/Co, Pt/Co/O and Pt/Co/Al are respectively 11.3, 15.0 and 20.7 meVÅ per unit cell (one unit cell contains 3 Co atoms). The in-plane unit cell area is 6.65Å 2 . Interestingly, it has been found that the OM converges much faster with respect to the density of the k-mesh than the AHE, even though both are computed from similar Berry phase expressions [29]. We report an analogous observation for the DMI spiralization: Using uniform 32x32, 64x64, 128x128 and 512x512 k-meshes we obtain spiralizations D yx of 12.8, 11.8, 12.2, and 11.3 meVÅ per unit cell for Pt/Co. This suggests the option of doing quick estimates of D ij using coarse k grids. Such estimates could also be done without Wannier interpolation directly within the first principles codes. Computing D yx for various directions ofn, we find the DMI to depend strongly on the direction ofn. E.g. forn along x direction, D yx is smaller by a factor of 3 than for n in perpendicular to film direction in the case of the Pt/Co/Al film. Anisotropies of this order of magnitude are typical of transport coefficients in non-cubic crystals. Inclusion of such anisotropy terms of the DMI into the micromagnetic energy functionals used for simulation of current-induced domain-wall motion in chiral magnets is therefore expected to affect results on the quantitative level.
In conclusion we showed that DMI can be formulated in terms of a Berry phase theory. We derived this Berry phase theory by expanding the free energy functional within rigorous quantum mechanical perturbation theory in terms of gradients of magnetization direction. Formally, our Berry phase theory closely resembles the quantum theory of OM and drastically reduces the computational burden, because it allows for calculating the DMI based on the collinear electronic structure. We worked out the analogies between OM and DMI and showed that the orbital moment of a band is counterparted by a twist torque moment within the Berry phase DMI theory. This twist torque moment is of the same fundamental impor-tance for the intrinsic even SOT driven by thermal gradients like the orbital moment of a band is for the intrinsic anomalous Nernst effect. We investigated the formal relations between DMI and SOT and found them to be the same as those between OM and AHE. We propose to interpret DMI as a spiralization, i.e., a twist torque moment per volume, in analogy to the magnetization, which can be interpreted as a magnetic moment per volume. Besides formulating the DMI explicitly as a Berry phase theory we also derived various equivalent alternative expressions which can be conveniently implemented within first principles DFT codes, including expressions in terms of Green functions, which allow the computation of DMI in disordered systems. As a practical application of the formalism we computed the DMI in Pt/Co, Pt/Co/O and Pt/Co/Al thin films. We found the DMI to be strongly anisotropic in these systems.
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| [] |
[
"DEGENERATE SL n : REPRESENTATIONS AND FLAG VARIETIES",
"DEGENERATE SL n : REPRESENTATIONS AND FLAG VARIETIES"
] | [
"Evgeny Feigin "
] | [] | [] | The degenerate Lie group is a semidirect product of the Borel subgroup with the normal abelian unipotent subgroup. We introduce a class of the highest weight representations of the degenerate group of type A, generalizing the PBW-graded representations of the classical group. Following the classical construction of the flag varieties, we consider the closures of the orbits of the abelian unipotent subgroup in the projectivizations of the representations. We show that the degenerate flag varieties F a n and their desingularizations Rn can be obtained via this construction. We prove that the coordinate ring of Rn is isomorphic to the direct sum of duals of the highest weight representations of the degenerate group. In the end, we state several conjectures on the structure of the highest weight representations. | null | [
"https://arxiv.org/pdf/1202.5848v2.pdf"
] | 119,124,890 | 1202.5848 | 6ff143159652697d9ac2792aad1a27a4ec82228c |
DEGENERATE SL n : REPRESENTATIONS AND FLAG VARIETIES
28 Feb 2012
Evgeny Feigin
DEGENERATE SL n : REPRESENTATIONS AND FLAG VARIETIES
28 Feb 2012arXiv:1202.5848v2 [math.RT]
The degenerate Lie group is a semidirect product of the Borel subgroup with the normal abelian unipotent subgroup. We introduce a class of the highest weight representations of the degenerate group of type A, generalizing the PBW-graded representations of the classical group. Following the classical construction of the flag varieties, we consider the closures of the orbits of the abelian unipotent subgroup in the projectivizations of the representations. We show that the degenerate flag varieties F a n and their desingularizations Rn can be obtained via this construction. We prove that the coordinate ring of Rn is isomorphic to the direct sum of duals of the highest weight representations of the degenerate group. In the end, we state several conjectures on the structure of the highest weight representations.
Introduction
Let g be a simple Lie algebra with the Borel subalgebra b and let G, B be the corresponding groups. The irreducible highest weight representation of g play fundamental role in the algebraic and geometric Lie theory. In particular, the generalized flag varieties for G can be realized as G-orbits inside the projectivizations of these modules. Let g a and G a be the degenerate Lie algebra and the degenerate Lie group (see [Fe1], [Fe2], [FF], [FFiL], [PY], [Y]). The Lie algebra is a sum of the subalgebra b and of the abelian ideal g/b with the adjoint action of b on g/b. The Lie group is the semidirect product of the Borel subgroup with the normal abelian unipotent subgroup exp(g/b). In this paper we are concerned with the following question: what are the analogues of the finite-dimensional representations of g and of the flag varieties in the degenerate situation?
In this paper we only study the type A case, so from now on g = sl n and G = SL n . Recall that in this case the fundamental representations V ω k are labeled by a number k = 1, . . . , n−1 and one has V ω k ≃ Λ k (C n ). Now let λ = n−1 i=1 m i ω i be a dominant weight, m i are non-negative integers. Then the corresponding highest weight representation V λ sits inside the tensor product of V ω i 's, where each factor appears exactly m i times. The image of the embedding is nothing but the g-span of the tensor product of highest weight vectors. It has been shown in [FFoL1], [FFoL2], [Fe1] that the degenerate Lie algebra g a naturally acts on the PBW-graded representations V a λ (the associated graded to V λ with respect to the PBW filtration). In addition, V a λ 1 still sits inside the tensor product of V a ω i 's in the same way as V λ in the tensor product of V ω i 's. The natural question is: are there natural representations of g a different from V a λ ? It turns out that this question is important for the study of the PBW-degeneration of flag varieties, see [Fe1], [Fe2], [FF]. Let us recall basic steps here.
Let n − be the nilpotent subalgebra such that g = b ⊕ n − . We denote by N − the corresponding unipotent subgroup of G. Let (N − ) a ≃ exp(g/b) be the abelian unipotent subgroup of G a . This group is isomorphic to the product of dim n − copies of the group G a -the additive group of the field. Let v λ ∈ V λ be a highest weight vector. Recall that the generalized flag variety F λ is defined as the G-orbit of the highest weight line Cv λ in P(V λ ) (see [FH], [Fu], [K]). The corresponding degenerate flag variety is defined as the closure of the (N − ) a -orbit of Cv λ in P(V a λ ) (see [Fe1], [Fe2]). For general λ, this is a normal singular projective variety enjoying explicit description in terms of linear algebra. Let us restrict here to the case of regular dominant λ (i.e. all m i are positive). Then all such flag varieties are isomorphic and we denote the corresponding variety by F a n . F a n can be explicitly realized as the variety of collections (V i ) n−1 i=1 of subspaces of an n-dimensional space subject to certain conditions. We note that the entries V i correspond to fundamental weights ω i .
In [FF] a desingularization R n for F a n was constructed in terms of linear algebra. These R n are smooth projective algebraic varieties, which are Bott towers, i.e. can be constructed as succesive fibrations with fibers P 1 . A point in R n is a collection (V i,j ) with 1 ≤ i ≤ j ≤ n − 1. This suggests that the "fundamental" representations of the Lie algebra g a are in one-to-one correspondence with the set of positive (non-necessarily simple) roots. In fact, it turns out that to each positive root α one can attach a representation M α . These representations are highest weight in a sense that M α is generated from a highest weight vector v α by the action of the symmetric algebra of n − (for example, for simple roots α i one has M α i ≃ V a ω i ). In addition, R n can be embedded into the product of P(M α ) (over all positive roots α) as the closure of the (N − ) a -orbit through the product of the highest weight lines. Similar to the classical situation, fundamental representations give us a way to construct a large class of g a -modules. Namely, given a collection of non-negative integers m = (m α ), α -positive root, we consider the g a -module
M m = S • (n − ) · v m ⊂ M ⊗mα α , v m = v ⊗mα α .
In particular, if m α = 0 for non-simple α, then M m ≃ V a λ for λ = m α i ω i (α i are simple roots). We give more examples of such modules and formulate several conjectures concerning the structure off M m .
By definition, for any two collections m 1 and m 2 , we have a g a -equivariant embedding M m 1 +m 2 ⊂ M m 1 ⊗ M m 2 , v m 1 +m 2 → v m 1 ⊗ v m 2 . Dualizing, we obtain an algebra m M * m . Our main theorem is as follows:
Theorem 0.1. The coordinate ring of R n is isomorphic to m M * m . Our main tool is the explicit form of the Plücker-type relations in the coordinate ring of R n .
Finally we note that one can naturally attach to a module M m the corresponding "flag variety". Namely, let F(M m ) ⊂ P(M m ) be the closure of the orbit (N − ) a · Cv m .These are the so-called G a -varieties (see [A], [HT]). We note that if all m α are positive, then F(M m ) ≃ R n and if all m α but m α i vanish, then F(M m ) ≃ F a n . Our paper is organized as follows: In Section 1 we settle notation and recall main definitions and constructions. In Section 2 we state our results and provide examples. Section 3 is devoted to the proofs and in Section 4 several conjectures are stated.
1. Notation and main objects 1.1. Classical story. Let g be a simple Lie algebra with the Cartan decomposition g = n ⊕ h ⊕ n − . Let b = n ⊕ h be the Borel subalgebra. We denote by Φ + the set of positive roots of g and by α 1 , . . . , α l ∈ Φ + the set of simple roots. We sometimes write α > 0 instead of α ∈ Φ + . For α > 0 we denote by f α ∈ n − a weight −α element. Thus we have n − = α>0 Cf α . Let G, B, N , N − and T be the Lie groups corresponding to the Lie algebras g, b, n, n − and h.
Let ω 1 , . . . , ω l be the fundamental weights. The fundamental weights and simple roots are orthogonal with respect to the Killing form (·, ·) on h * :
(ω i , α j ) = δ i,j . A dominant integral weight λ is given by l i=1 m i ω i , m i ∈ Z ≥0
. For a dominant integral λ let V λ be the finite-dimensional irreducible highest weight g-module with highest weight λ and a highest weight vector v λ such that nv λ = 0, hv λ = λ(h)v α (h ∈ h) and V λ = U(n − )v λ .
The (generalized) flag varieties for G are defined as quotient G/P by the parabolic subgroups. These varieties play crucial role in the geometric representation theory. An important feature of the flag varieties is that they can be naturally embedded into the projectivization of the highest weight modules. Namely, let λ be a dominant weight such that the stabilizer of the line [v λ ] in G is equal to P . Here and below for a vector v in a vector space V we denote by [v] ∈ P(V ) the line spanned by v. Then one gets the embedding G/P ⊂ P(V λ ) as the G-orbit of the highest weight line [v λ ]. For a dominant weight λ we denote by F λ ⊂ P(V λ ) the orbit G[v λ ] of the highest weight line. These are smooth projective algebraic varieties. It is clear that F λ ≃ F µ if and only if for all i (λ, ω i ) = 0 is equivalent to (µ, ω i ) = 0.
1.2. Degenerate version. Let g a be the degenerate Lie algebra defined as a direct sum b ⊕ g/b of the Borel subalgebra b and abelian ideal g/b (see [Fe1], [Fe2]). The algebra b acts on g/b via the adjoint action. We denote the space g/b by (n − ) a (a is for abelian). Let (N − ) a = exp(n − ) a be the abelian Lie group, which is nothing but the product of dim n − copies of the group G a -the additive group of the field. Let G a be the semidirect product B ⋉ (N − ) a of the subgroup B and of the normal abelian subgroup (N − ) a (the action of B on (N − ) a is induced by the action of B on n − by conjugation).
Similar to the classical situation, we say that a g a -module M is a highest weight module if there exists v ∈ M such that nv = 0, the line [v] is h-stable and M = S(n − )v, where S(n − ) denotes the symmetric algebra of n − , which is isomorphic to the polynomial ring C[f α ] α>0 . It is clear that the highest weight g a -modules are in one-to-one correspondence with the b-invariant ideals in C[f α ] α>0 . Namely, M defines the annihilating ideal AnnM ⊂ C[f α ] α>0 and a b-invariant ideal I produces the g a -module C[f α ] α>0 /I. In [FFoL1], [FFoL2] the modules V a λ were studied for dominant integral λ. The module V a λ is the associated graded module to V λ with respect to the PBW filtration. We note that all V a λ are highest weight modules. However, these are not all highest weight g a -modules. Below we study a wider class of representations for g = sl n (still not exhaustive). We note that all the highest weight g a -modules which show up so far, are graded compatibly with with the grading by the total degree on C[f α ] α>0 . Summarizing, we put forward the following definition:
Definition 1.1. A g a -module M = s≥0 M (s)
is called a graded highest weight module if the following holds:
• M (0) = Cv, nv = 0, hv ⊂ M (0), • M = C[f α ] α>0 v, • f α M (s) ⊂ M (s + 1), α > 0.
We note that the definition implies that bM (s) ⊂ M (s). In what follows we say that elements of M (s) have PBW-degree s. Now we want to define degenerate version of the flag varieties. It is not reasonable to take the quotient of G a now (for example, G a /B is simply an affine space). Instead we use the highest weight representations. Let M be a graded highest weight g a -module with a highest weight vector v. Then we put forward the following definition:
Definition 1.2. The variety F(M ) is the closure of the (N − ) a -orbit of the highest weight line:
F(M ) = (N − ) a · [v] ⊂ P(M ).
The degenerate flag varieties F a λ are isomorphic to F(V a λ ). We note that the group (N − ) a is isomorphic to the product of dim n − copies of the group G a -the additive group of the field. Therefore, all the varieties F(M ) are the so-called G a -varieties, i.e. the compactifications of the abelian unipotent group (see [A], [HT]).
1.3. The type A case. From now on g = sl n and G = SL n . In this case the set of positive roots Φ + consists of the roots
α i,j = α i + α i+1 + · · · + α j , 1 ≤ i ≤ j ≤ n − 1.
In what follows we use the shorthand notation f i,j for f α i,j . The following proposition is proved in [FFoL1] (• denotes the adjoint action of b on S(n − ) ≃ S(g/b)).
Proposition 1.3. For a dominant weight λ the module V a λ is isomorphic to the quotient S(n − )/I λ , where I λ is the ideal generated by the subspace U(b) • span(f (β,λ)+1 β , β ∈ Ψ + ).
It is proved in [Fe2] that the degenerate flag varieties enjoy the following explicit realization. For simplicity, we describe the case of the complete flags here. So let λ be a regular dominant weight, i.e. (λ, ω i ) > 0 for all i. Fix an n-dimensional vector space W and a basis w 1 , . . . , w n of W . The operators f i,j act on W by the standard formula f i,j w k = δ i,k w j+1 . Let pr k : W → W be a projection along the k-th basis vector, i.e. pr k w k = 0 and pr k w j = w j for j = k. All the flag varieties F a λ are isomorphic (λ is regular) and we denote this variety by F a n . Then F a n consists of collections (V 1
, . . . , V n−1 ) of subspaces of W such that dim V i = i and pr i+1 V i ⊂ V i+1 .
The varieties F a n are singular (starting from n = 3) projective algebraic varieties, naturally embedded into the product of Grassmannians. A desingularization R n for F a n was constructed in [FF] (see also [FFiL] for the symplectic version). Namely, R n consists of collections (V i,j ) 1≤i≤j≤n−1 of subspaces of W subject to the following conditions:
• dim V i,j = i, V i,j ⊂ span(w 1 , . . . , w i , w j+1 , . . . , w n ), • V i,j ⊂ V i+1,j , 1 ≤ i < j ≤ n − 1, • pr j+1 V i,j ⊂ V i,j+1 , 1 ≤ i ≤ j < n − 1.
One can show that the variety R n is a Bott tower, i.e. there exists a tower
R n = R n (0) → R n (1) → R n (2) → · · · → R n (n(n − 1)/2) = pt such that each map R n (l) → R n (l + 1) is a P 1 -fibration. Hence, R n is smooth. The surjection R n → F a n is given by (V i,j ) 1≤i≤j≤n−1 → (V i,i ) n i=1 . Let W i,j = span(w 1 , . . . , w i , w j+1 , . . . , w n ). Then from the definition of R n we obtain (1.1) R n ⊂ 1≤i≤j≤n−1 P(Λ i W i,j ).
Representations and coordinate rings
In this section we state our results and provide examples. The proofs are given in the next section. Throughout the section g = sl n .
2.1. Representations. For a positive root α = α i,j let M α be the g amodule defined as the quotient C[f β ] β>0 /I α , where I α is the ideal generated by the subspace
U (b) • f 2 β , β ≥ α; f β , β ≥ α . (Here • denotes the adjoint action). We denote the highest weight vector (the image of 1) in M α by v α . Example 2.1. Let α = α 1 + · · · + α n−1 be the highest root. Then M α is two-dimensional space with a basis v α , f α v α . All the elements f β , β = α as well as b ⊂ g a , act trivially on M α . Lemma 2.2. If α is a simple root α i , then M α i ≃ V a ω i as g a -modules. Proof. Follows from Proposition 1.3. Lemma 2.3. Let α = α i,j . Then M α ≃ Λ i W i,j .
Proof. Let us define the structure of g a -module on Λ i W i,j in the following way. Consider V a ω i . Then as a vector space this module is isomorphic to Λ i W . It is easy to see that the subspace U spanned by the vectors
w l 1 ∧ · · · ∧ w l i , ∃r : i < l r ≤ j is g a -invariant. Now clearly we can identify the quotient of Λ i W/U with Λ i W i,j .
In addition, this quotient is isomorphic to M α i,j accordiing to Proposition 1.3.
Remark 2.4. 1). We note that M α = s≥0 M α (s) (see Definition 1.1) and M α (s) is spanned by the vectors w l 1 ∧ · · · ∧ w l i , l a ∈ {1, . . . , i, j + 1, . . . , n}, #{a : l a > j} = s.
2). The operators f a,b act trivially on M α i,j unless a ≤ i ≤ j ≤ b.
The modules M α i,j play a role of the fundamental representations for the Lie algebra g a . We now define a larger class of representations "generated" by the fundamental ones. Let m = (m i,j ) i≤j be a collection of non-negative integers.
Here and in what follows
we write i ≤ j (or a ≤ b) instead of 1 ≤ i ≤ j ≤ n − 1 (or 1 ≤ a ≤ b ≤ n − 1) if it does not lead to any confusion.
We put forward the following definition
Definition 2.5. The g a -module M m is the top (Cartan) component in the tensor product of modules M α i,j , i.e.: M m = S(n − ) · i≤j v ⊗m i,j α i,j ⊂ i≤j M ⊗m i,j α i,j .
We denote the highest weight vector of M m , which is the tensor product of the vectors v α i,j , by v m .
Example 2.6. Let m i,j = 0 unless i = j. Then according to [FFoL1] we have k 1 ≤ 1, k 1 + k 2 ≤ 2, . . . , k 1 + · · · + k n−1 ≤ n − 1.
M m ≃ V a λ , where λ = n−1 i=1 m i,i ω i .
Moreover, the monomials satisfying conditions above are linearly independent. We note that the number of collections (k 1 , . . . , k n−1 ) subject to the restrictions (2.1) is equal to the Catalan number C n . In addition, the dimensions of the homogeneous components dim M m (s) are given by the entries of the Catalan triangle (see e.g. http://mathworld.wolfram.com/CatalansTriangle.html).
The following Lemma explains the importance of the above defined modules. We say that a collection m is regular if m i,j > 0 for all i ≤ j.
Proposition 2.8. For any regular m, the variety R n is embedded into the projective space P(M m ) as the closure of the SL a n -orbit or (N − ) a -orbit through the highest weight line:
R n = SL a n · [v m ] = (N − ) a · [v m ] ⊂ P(M m ). Proof.
It suffices to show that our proposition holds if all m i,j = 1. We note that the orbits SL a n · [v m ] and (N − ) a · [v m ] coincide and are embedded into the product of Grassmannians i≤j Gr i (W i,j ). Recall that we have fixed a basis w 1 , . . . , w n in W . For a collection L = (l 1 , . . . , l i ) ⊂ {1, . . . , i, j + 1, . . . , n} and an element V i,j ∈ Gr i (W i,j ), let X L (V i,j ) be the Plücker coordinate of V i,j . We claim that the orbit (N − ) a [v m ] consists exactly of the collections of subspaces (V i,j ) i≤j ∈ R n such that X 1,...,i (V i,j ) = 0 (see Lemma 2.11 and Lemma 2.12 of [FF]). This implies the proposition.
In what follows we consider R n as being embedded into P(M m ) with m i,j = 1 for all i ≤ j, or, equivalently, being embedded into α>0 P(M α ). We denote such a collection by 1.
Coordinate ring. We denote by X
(i,j) L = X (i,j)
l 1 ,...,l i the dual (Plücker) coordinates in W i,j with respect to w l 1 ∧· · ·∧w l i , where 1 ≤ l 1 < · · · < l i ≤ n and L ⊂ {1, . . . , i, j + 1, . . . , n}. For a permutation σ ∈ S n we set
X (i,j) l σ(1) ,...,l σ(i) = (−1) σ X (i,j) l 1 ,...,l i .
The PBW-degree of a variable X (i,j) L is defined as the number of entries l a such that l a > j.
Remark 2.9. The variables of the PBW-degree zero are of the form X Let us give explicit description of the SL a n -orbit through the highest weight line in M 1 . For a collection of complex numbers (c a,b ) a≤b consider the element exp(
a≤b c a,b f a,b ) ∈ (N − ) a .
Let L = (1 ≤ l 1 < · · · < l i ≤ n) and let d be a number such that l d ≤ i < l d+1 . We consider the numbers 1 ≤ r 1 < · · · < r i−d ≤ i such that
(−1) d p=1 (lp−p) det(c ra,l d+b −1 ) i−d a,b=1 . Proof.
We note that f a,b w c = δ a,c w b+1 . Therefore in W i,j one has: exp( a≤b c a,b f a,b )w 1 ∧ · · · ∧ w i = (w 1 + c 1,j w j+1 + · · · + c 1,n−1 w n ) ∧ · · · ∧ (w i + c i,j w j+1 + · · · + c i,n w n ). Let CX n be the ring of multi-homogeneous polynomials in variables X (i,j) L , 1 ≤ i ≤ j ≤ n − 1, L = (l 1 , . . . , l i ). In other words, CX n is the coordinate ring of the product of projective spaces P(W i,j ), 1 ≤ i ≤ j ≤ n − 1. We fix the decomposition
CX n = m∈Z n(n−1)/2 CX m ,
where CX m is the space of polynomials of total degree m i,j in all variables X (i,j) L . For an ideal I ⊂ CX n we denote by X(I) ⊂ i≤j P(Λ i W i,j ) the variety of zeros of I and for a subvariety X in the product of projective spaces we denote by I(X) the ideal of multi-homogeneous polynomials vanishing on X.
Our goal is to find the ideal I(R n ) in CX n of multi-homogeneous polynomials vanishing on R n . Then the coordinate ring Q n of R n is given by
Q n = CX n /I(R n ).
We first describe the analogues of the Plücker relations contained in I(R n ).
The relations are labeled by the following data:
• two pairs (i 1 , j 1 ), (i 2 , j 2 ) with i 1 ≥ i 2 and 1 ≤ i 1 ≤ j 1 ≤ n − 1, 1 ≤ i 2 ≤ j 2 ≤ n − 1; • an integer k satisfying 1 ≤ k ≤ i 2 ;
• two collections of indices L = (l 1 , . . . , l i 1 ), P = (p 1 , . . . , p i 2 ) with L ⊂ {1, . . . , i 1 , j 1 + 1, . . . , n}, P ⊂ {1, . . . , i 2 , j 2 + 1, . . . , n}.
We denote the relation labeled by the data above by R (i 1 ,j 1 ),(i 2 ,j 2 );k L,J
. This is a polynomial in the ring CX n given by the following formulas:
First, let i 1 = i 2 and j 1 ≥ j 2 . Then
R (i 1 ,j 1 ),(i 2 ,j 2 );k L,P = X (i 1 ,j 1 ) L X (i 2 ,j 2 ) P − 1≤r 1 <···<r k ≤n X (i 1 ,j 1 ) L ′ X (i 2 ,j 2 ) P ′ ,
where the summation rums over r i satisfying l r i ∈ {1, . . . , i 2 , j 1 + 1, . . . , n} and for a term X
(i 1 ,j 1 ) L ′ X (i 2 ,j 2 ) P ′
corresponding to the sequence (r 1 , . . . , r k ) one has L ′ = (l 1 , . . . , l r 1 −1 , p 1 , l r 1 +1 , . . . , l r k −1 , p k , l r k +1 , . . . , l i 1 ), (2.3) P ′ = (l r 1 , . . . , l r k , p k+1 , . . . , p i ).
(2.4) Second, let j 1 ≤ j 2 . Then we take the relations as above with an additional restriction L ⊂ {1, . . . , i 1 , j 2 + 1, . . . , n}.
Remark 2.12. We note that the initial term X l 1 ,...,l i ] generated by the relations R (i,j),(i,j);k L,P is exactly the ideal defining the Grassmann variety Gr i (W i,j ).
Proposition 2.16. The polynomials R (i 1 ,j 1 ),(i 2 ,j 2 );k L,P vanish on R n .
Let J n ⊂ CX n be the ideal generated by the Plücker relations above. We will prove the following theorem.
Theorem 2.17. The ideal I(R n ) is the saturation of J n . More precisely, for any F ∈ I(R n ) there exist non-negative integers
N i,j , i ≤ j, such that i≤j X (i,j) 1,...,i N i,j F ∈ J n .
We put forward the following conjecture.
Conjecture 2.18. The ideal J n is prime, i.e. I(R n ) = J n .
Proofs.
3.1. The ideal.
Proposition 3.1. J n ⊂ I(R n ), i.e. all the relations R (i 1 ,j 1 ),(i 2 ,j 2 ),k L,P vanish on R n .
Proof. We recall that the Sylvester identity (see e.g. [Fu], section 8.1, Lemma 2) states that for any two s by s matrices M and N and a number k ≤ s one has
det M det N − 1≤r 1 <···<r k ≤s det M ′ det N ′ = 0,
where the matrices M ′ and N ′ are obtained from M and N by interchanging the first k columns of N with the columns r 1 , . . . , r k of M (preserving the order of columns). Combining the Sylvester identity with Lemma 2.10 we obtain our proposition.
Lemma 3.2. The relations R (i,j),(i,j);k L,P , R (i+1,j),(i+1,j);k L,P and R (i,j),(i+1,j);k L,P with all possible L and P define the variety of pairs (V i,j
, V i+1,j ) ⊂ Gr i (W i,j )× Gr i+1 (W i+1,j ) satisfying V i,j ⊂ V i+1,j .
Proof. First, the relations R (i,j),(i,j);k L,P and R (i+1,j),(i+1,j);k L,P cut out the product of Grassmann varieties Gr i (W i,j ) × Gr i+1 (W i+1,j ) inside the product of projective spaces P(Λ i W i,j )×P(Λ i+1 W i+1,j ). Second, the relations R (i,j),(i+1,j);k L,P are exactly the classical Plücker relations saying that the first subspace has to be a subset of the second. with arbitrary L and P define the variety of pairs (V i,j
, V i,j+1 ) ⊂ Gr i (W i,j )× Gr i (W i,j+1 ) satisfying pr j+1 V i,j ⊂ V i,j+1 .
Proof. Similar to the proof of the lemma above. The only difference that the relations R (i,j),(i,j+1);k I,P are degenerate Plücker relations (see [Fe1]).
Proposition 3.4. We have R n = X(J n ), i.e. the common zeroes of J n give R n .
Proof. We need to prove that if all the relations R vanish at a point x ∈ × i≤j P(M α i,j ), then x ∈ R n . Follows from Lemmas 3.2 and 3.3.
The following proposition is of the key importance for us.
Proposition 3.5. For an element F ∈ CX n there exist numbers N i,j ∈ Z ≥0 , i ≤ j and another polynomial G ∈ CX n such that
i≤j (X (i,j) 1,...,i ) N i,j F − G ∈ J n
and G depends only on the PBW-degree zero or one variables X (i,j) 1,...,i , i ≤ j and X (i,j) 1,...,r−1,r+1,...,i,m , 1 ≤ r ≤ i ≤ j < m, In addition, G can be chosen in such a way that if G depends on a variable X (i,j) 1,...,r−1,r+1,...,i,m for some i, j then it does not depend on all variables of the form X (i 1 ,j 1 ) 1,...,r−1,r+1,...,i 1 ,m for (i 1 , j 1 ) = (i, j).
Proof. Recall the PBW-degree of a variable X (i,j) L given by the number of l a ∈ L such that l a > j. Let X (i,j) L be a variable of PBW-degree greater than one with r / ∈ L for some r ≤ i. We consider the relation R (i,j),(i,j);1 L,(r,1,...,r−1,r+1,...,i) . One has
(−1) r−1 R (i,j),(i,j);1 L,(1,...,i) = X (i,j) L X (i,j) (1,...,i) − a: la>j X (i,j) (l 1 ,...,l a−1 ,r,l a+1 ,...,l i ) X (i,j)
(1,...,r−1,la,r+1,...,i) .
We note that the PBW-degree of each variable X (i,j) (l 1 ,...,l a−1 ,r,l a+1 ,...,l i ) is one less than that of X (i,j) L . Hence, by decreasing induction, we obtain the first statement of our proposition. Now assume that a polynomial G depends on the variables X (i 1 ,j 1 ) 1,...,r−1,r+1,...,i 1 ,m and X (i 2 ,j 2 ) 1,...,r−1,r+1,...,i 2 ,m for two different pairs (i 1 , j 1 ) and (i 2 , j 2 ). Let i 1 ≥ i 2 . We consider the relation (−1) r−1 R (i 1 ,j 1 ),(i 2 ,j 2 );1 (1,...,r−1,r+1,...,i 1 ,m),(r,1,...,r−1,r+1,...,i 2 ) = X (i 1 ,j 1 ) 1,...,r−1,r+1,...,i 1 ,m X
(i 2 ,j 2 ) 1,...,i 2 − (−1) i 2 +i 1 X (i 1 ,j 1 ) 1,...,i 1 X (i 2 ,j 2 )
1,...,r−1,r+1,...,i 2 ,m .
Using these relations we can get rid of the variable X (i 1 ,j 1 ) 1,...,r−1,r+1,...,i 1 ,m (multiplying by X (i 2 ,j 2 ) 1,...,i 2 ).
3.2. Representation of Q n . Let Q n = CX n /I(R n ) be the coordinate ring of R n . Consider a polynomial algebra A n in variables T i,j , 1 ≤ i ≤ j ≤ n − 1 and Z i,j , 1 ≤ i ≤ j ≤ n − 1. We define a homomorphism Ψ : CX n → A in the following way. Let X (i,j) L be a variable with 1 ≤ l 1 < · · · < l d ≤ i ≤ j < l d+1 < · · · < l i (thus the PBW-degree of this variable is i − d). We define i × i matrix M by
(3.1) M a,b = (−1) la−a , if b = a ≤ d, l a ∈ L Z a,l b −1 , if a / ∈ L, a ≤ i, b > d, 0, otherwise.
We define the homomorphism of polynomial algebras by the formula
(3.2) X (i,j)
l 1 ,...,l i → T i,j det M. Proposition 3.6. The kernel of Ψ coincides with I(R n ).
Proof.
Recall the open dense cell U ⊂ R n , which is the (N − ) a -orbit through the highest weight line. We note that I(R n ) = I(U ). Now the embedding
R n ⊂ i≤j P(Λ i W i,j )
is defined using the same determinants as in (3.1) (see Lemma (2.10)). Hence a polynomial in CX n vanishes on U if and only if belongs to the kernel of Ψ (we note that the variables T i,j guarantee the multi-homogeneity).
Corollary 3.7. We have an embedding Q n → A n defined by (3.2).
Corollary 3.8. One has the decomposition
Q n = m∈Z n(n−1)/2 ≥0 Q m ,
where Q m consists of polynomials of total degree m i,j in variables X (i,j) L . We note that this decomposition is very similar to the one for the classical flag varieties, where the coordinate ring decomposes into the direct sum of dual irreducible g-modules. Thus it is very natural to ask about the properties of the spaces Q m .
3.3. Cocyclicity. Our goal in this subsection is to prove that there exists a natural action of the Lie algebra (n − ) a on Q n such that each Q m is cocycilc with respect to the action of (n − ) a with a cocyclic vector
v * m = i≤j (X (i,j) 1,...,i ) m i,j .
Let m be a collection such that for some i, j we have m i,j = 1 and m c,d = 0 for all (c, d) = (i, j) (these are analogues of the fundamental weights). We denote the corresponding homogeneneous component Q m by Q i,j .
We first define the structure of (n − ) a -module on Q i,j . Note that the space Q i,j has a basis X (i,j) L with L = (1 ≤ l 1 < · · · < l i ≤ n) and L ⊂ {1, . . . , i, j + 1, . . . , n}. We identify Q i,j with the dual space Λ i (W i,j ) by setting X (i,j) L w P = δ L,P for any P = (1 ≤ p 1 < · · · < p i ≤ n), where w P = w p 1 ∧ · · · ∧ w p i . This endows Q i,j with the structure of g a -module,
Q * i,j ≃ Λ i (W i,j ). In particular, one has f a,b X (i,j) L = −X (i,j)
l 1 ,...,l r−1 ,a,l r+1 ,...,l i , if a ≤ i ≤ j < b and l r = b + 1, 0, otherwise.
(Recall that for σ ∈ S i we have X σL = (−1) σ X L ). We note that Q i,j is coclycic with respect to the action of (n − ) a and X (i,j) 1,...,i is a cocyclic vector. Now, let us define the structure of (n − ) a -module on all Q m . By definition, there exists a surjective map i≤j Q ⊗m i,j i,j → Q m . We have a natural structure of (n − ) a -module on the tensor product and hence we only need to show that I(R n ) is (n − ) a -invariant. We first prove this for J n .
Lemma 3.9. The ideal J n is invariant with respect to the (n − ) a -action defined above.
Proof. It suffices to prove that for any element f a,b ∈ (n − ) a one has
f a,b R (i 1 ,j 1 ),(i 2 ,j 2 );k L,P ∈ J n .
But it is easy to see that f a,b R (i 1 ,j 1 ),(i 2 ,j 2 );k L,P is again (different) generalized Plücker relation from our list.
Theorem 3.10. The ideal I(R n ) is (n − ) a -invariant and it is a saturation of J n .
Proof. We first prove the second statement. Using Proposition 3.5 we find numbers N i,j and a polynomial G such that
(3.3) i≤j (X (i,j) 1,...,i ) N i,j F − G ∈ J n
and G depends on the variables of PBW-degree at most 1. In addition, only one variable of the form X (i,j) 1,...,r−1,r+1,...,i,m shows up in G for any pair r < m. We claim that if F ∈ I(R n ) then G = 0. In fact, we choose pairs r, m and i, j with r ≤ i ≤ j < m such that G depends on X (i,j) 1,...,r−1,r+1,...,a,m . Let G = p≥0 X (i,j) 1,...,r−1,r+1,...,i,m p G p be the decomposition with respect to powers of X (i,j) 1,...,r−1,r+1,...,i,m (i.e. G p are independent of X (i,j) 1,...,r−1,r+1,...,i,m ). Proposition 3.6 gives that ΨG = 0. We note that ΨX (i,j) 1,...,r−1,r+1,...,i,m = (−1) i−r T i,j Z r,m−1 . Therefore, the variable Z r,m−1 comes only from X (i,j) 1,...,r−1,r+1,...,i,m . We conclude that all G p = 0. Continuing this procedure, we arrive at G = 0. Therefore, (3.3) now reads as i≤j (X (i,j) 1,...,i ) N i,j F ∈ J n and thus I(R n ) is the saturation of J n (see [H], Exercise 5.10).
In order to prove the first statement of the proposition we note that for all a, b, i, j one has f a,b X (i,j) 1,...,i = 0. Since J n is g a -invariant, we obtain
i≤j (X (i,j) 1,...,i ) N i,j (f a,b F ) ∈ J n ⊂ I(R n ). Now since I(R n ) is simple, we obtain f a,b F ∈ I(R n ).
Thanks to the theorem above, each Q m carries the structure of (n − ) amodule. Our next goal is to prove that it is cocyclic with cocyclic vector being the product of X (i,j) 1,...,i , i.e for any nontrivial F ∈ Q m one has
i≤j (X (i,j) 1,...,i ) m i,j ∈ C[f a,b ] a≤b F.
Theorem 3.11. Q m is cocyclic with a coclycic vector i≤j (X
(i,j) 1,...,i ) m i,j .
Proof. Let us denote by v * m the image of i≤j (X (i,j) 1,...,i ) m i,j in Q n . Let F ∈ CX n and F / ∈ I(R n ). We use Proposition 3.5 and write
i≤j (X (i,j) 1,...,i ) N i,j F − G ∈ J n
for some numbers N i,j and a polynomial G. Since F / ∈ I(R n ) we have G = 0. Suppose that a term i≤j (X and the result is proportional (with a non-zero coefficient) to the product of powers of X (i,j) 1,...,i . Hence we arrive at the following identity: there exist numbers N i,j , p(r, m) such that
r<m f p(r,m) r,m i≤j (X (i,j) 1,...,i ) N i,j F = const · i≤j (X (i,j) 1,...,i ) N i,j +m i,j .
Since all f a,b annihilate X and hence ( r<m f p(r,m) r,m )F = v * m in Q n . Corollary 3.12. The modules Q m are cocyclic with respect to the abelian algebra (n − ) a with cocyclic vectors v * m . One has the isomorphism of (n − ) amodules Q * m ≃ M m . Proof. It is easy to see that the (n − ) a -modules Q * i,j and M i,j are isomorphic. Since both families Q * m and M m can be constructed inductively via the (n − ) a -embeddings M m 1 +m 2 ⊂ M m 1 ⊗ M m 2 , Q * m 1 +m 2 ⊂ Q * m 1 ⊗ Q * m 2 , our corollary follows.
Conjectures and further directions.
In this section we collect conjectures concerning geometric and algebraic objects described in the present paper.
For the readers convenience we first recall the conjecture about the Plücker relations already stated in the previous section. Recall the ideal J n generated by the (generalized) Plücker relations.
Conjecture 4.1. The ideals J n and I(R n ) coincide.
One way to prove such kind of statements is to find a set of linearly independent monomials in Q n = CX n /I(R n ) such that these monomials form a spanning set for CX n /J n (recall J n ⊂ I(R n )). This is usually done in terms of some kind of (semi)standard tableaux (see [Fu], [Fe1]). It is interesting to describe the corresponding combinatorics in our situation.
The following conjecture provides a presentation of the modules M m in terms of generators and relations. It generalizes similar statements from [FFoL1], where the case of m supported on the diagonal was considered (m i,j = 0 unless i = j).
Conjecture 4.2. The following g a -module M m is isomorphic to the quotient C[f i,j ] i≤j /I m , where I m is the ideal generated by the subspace We also conjecture that the monomial basis of [FFoL1], [V] can be extended to our case. Namely, let S m be the subset of Z n(n−1)/2 ≥0 consisting of collections s = (s α ) α>0 such that for any Dyck path p starting at α i and ending at α j one has β∈p s β ≤ i≤k≤l≤j m k,l . We note that two conjectures above are closely related. Namely, it is easy to show that there is a surjection C[f i,j ] i≤j /I m → M m (because the relations of the left hand side hold in M m ). In addition we have the following lemma:
Lemma 4.4. The vectors {f s , s ∈ S m } span the right hand side of (4.1).
Proof. The proof is very similar to the one in [FFoL1].
Hence in order to prove two conjectures above it suffices to show that the vectors {f s v m , s ∈ S m } are linearly independent. Computer experiments support this conjecture, but we are not able to prove it so far. The main difference with the case of [FFoL1] is that the the Minkowsky sum S m 1 +S m 2 is no longer equal to S m 1 +m 2 for general m 1 and m 2 .
Example 2. 7 .
7Let m 1,j = 1 for j = 1, 2, . . . , n − 1 and m i,j = 0 if i > 1. Then f i,j v m = 0 unless i = 1 and it is easy to see that f
...,i . The variables of PBW-degree one are of the form X (i,j) 1,...,r−1,r+1,...,i,m for some r ≤ i ≤ j < m.
{r 1 , . . . , r i−d } ∪ {l 1 , . . . , l d } = {1, . . . , i}. Lemma 2.10. For L ⊂ {1, . . . , i, j + 1, . . . , n}, the Plücker coordinate X (i,j) L of the point exp( a≤b c a,b f a,b )[v 1 ] is given by
Example 2 . 11 .
211Let d = i−1, i.e. l i−1 ≤ i and l i > j. Let {r} = {1, . . . , i}\L. i−r c r,l i −1 .
..
in the relations if and only if {p 1 , . . . , p k } ⊂ {1, . . . , i 2 , j 1 + 1, . . . , n}.Example 2.13. Let n = 3. Then CX 3 the ring of multi-homogeneous polynomials in three groups of variables Example 2.14. Let n = 4. Then the only relation containing variables X Example 2.15. Fix a pair 1 ≤ i ≤ j ≤ n − 1. Then the ideal in C[X(i,j)
...,r−1,r+1,...,i(r,m),m ) p(r,m) with some indices i(r, m) ≤ j(r, m) and powers p(r, m) shows up in G with a non-zero coefficient. Then since f a,b X ...,r−1,r+1,...,i(r,m),m ) p(r,m)
...,i ) N i,j r<m f p(r,m) ...,i ) N i,j +m i,j ∈ I(R n ). ...,i ) m i,j ∈ I(R n )
Conjecture 4 . 3 .
43The elements {f s v m , s ∈ S m } form a basis of M m .
AcknowledgmentsThis work was partially supported by the Russian President Grant MK-3312.2012
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O Yakimova, arXiv:1202.3009One-parameter contractions of Lie-Poisson brackets. Moscow, RussiaNational Research University Higher School of EconomicsEvgeny Feigin: Department of Mathematics. Vavilova str. 7, 117312. and Tamm Theory Division, Lebedev Physics Institute E-mail address: [email protected], One-parameter contractions of Lie-Poisson brackets, arXiv:1202.3009. Evgeny Feigin: Department of Mathematics, National Research University Higher School of Economics, Vavilova str. 7, 117312, Moscow, Russia, and Tamm Theory Division, Lebedev Physics Institute E-mail address: [email protected]
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"C Tortora \nAstronomical Observatory of Capodimonte -INAF\nvia Moiariello 16I-80131NapoliItaly\n",
"G Longo \nDepartment of Physics\nUniversity Federico II\nvia Cinthia 680126NapoliItaly\n",
"N R Napolitano \nAstronomical Observatory of Capodimonte -INAF\nvia Moiariello 16I-80131NapoliItaly\n",
"M Radovich \nAstronomical Observatory of Padua\nvicolo dell'Osservatorio 5I-35122PadovaItaly\n",
"F La Barbera \nAstronomical Observatory of Capodimonte -INAF\nvia Moiariello 16I-80131NapoliItaly\n",
"M Capaccioli \nVSTCen -INAF\nvia Moiariello 16I-80131NapoliItaly\n",
"J T A De Jong \nLeiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands\n",
"F Getman \nAstronomical Observatory of Capodimonte -INAF\nvia Moiariello 16I-80131NapoliItaly\n",
"A Grado \nAstronomical Observatory of Capodimonte -INAF\nvia Moiariello 16I-80131NapoliItaly\n",
"M Paolillo \nDepartment of Physics\nUniversity Federico II\nvia Cinthia 680126NapoliItaly\n"
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"Astronomical Observatory of Capodimonte -INAF\nvia Moiariello 16I-80131NapoliItaly",
"Astronomical Observatory of Capodimonte -INAF\nvia Moiariello 16I-80131NapoliItaly",
"Astronomical Observatory of Capodimonte -INAF\nvia Moiariello 16I-80131NapoliItaly",
"Department of Physics\nUniversity Federico II\nvia Cinthia 680126NapoliItaly",
"Astronomical Observatory of Capodimonte -INAF\nvia Moiariello 16I-80131NapoliItaly",
"Astronomical Observatory of Padua\nvicolo dell'Osservatorio 5I-35122PadovaItaly",
"Astronomical Observatory of Capodimonte -INAF\nvia Moiariello 16I-80131NapoliItaly",
"VSTCen -INAF\nvia Moiariello 16I-80131NapoliItaly",
"Leiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands",
"Astronomical Observatory of Capodimonte -INAF\nvia Moiariello 16I-80131NapoliItaly",
"Astronomical Observatory of Capodimonte -INAF\nvia Moiariello 16I-80131NapoliItaly",
"Department of Physics\nUniversity Federico II\nvia Cinthia 680126NapoliItaly"
] | [
"Mon. Not. R. Astron. Soc"
] | We estimated photometric redshifts (z phot ) for more than 1.1 million galaxies of the ESO Public Kilo-Degree Survey (KiDS) Data Release 2. KiDS is an optical wide-field imaging survey carried out with the VLT Survey Telescope (VST) and the OmegaCAM camera, which aims at tackling open questions in cosmology and galaxy evolution, such as the origin of dark energy and the channel of galaxy mass growth. We present a catalogue of photometric redshifts obtained using the Multi Layer Perceptron with Quasi Newton Algorithm (MLPQNA) model, provided within the framework of the DAta Mining and Exploration Web Application REsource (DAMEWARE). These photometric redshifts are based on a spectroscopic knowledge base which was obtained by merging spectroscopic datasets from GAMA (Galaxy And Mass Assembly) data release 2 and SDSS-III data release 9. The overall 1σ uncertainty on ∆z = (z spec − z phot )/(1 + z spec ) is ∼ 0.03, with a very small average bias of ∼ 0.001, a NMAD of ∼ 0.02 and a fraction of catastrophic outliers (|∆z| > 0.15) of ∼ 0.4%. | 10.1093/mnras/stv1496 | [
"https://arxiv.org/pdf/1507.00754v3.pdf"
] | 118,703,429 | 1507.00754 | 726fd3092271dcad430603f6a422367d6fe081a9 |
Machine Learning based photometric redshifts for the KiDS ESO DR2 galaxies
30 Jul 2015. 2015. July 31. 2015
S Cavuoti
Astronomical Observatory of Capodimonte -INAF
via Moiariello 16I-80131NapoliItaly
M Brescia
Astronomical Observatory of Capodimonte -INAF
via Moiariello 16I-80131NapoliItaly
C Tortora
Astronomical Observatory of Capodimonte -INAF
via Moiariello 16I-80131NapoliItaly
G Longo
Department of Physics
University Federico II
via Cinthia 680126NapoliItaly
N R Napolitano
Astronomical Observatory of Capodimonte -INAF
via Moiariello 16I-80131NapoliItaly
M Radovich
Astronomical Observatory of Padua
vicolo dell'Osservatorio 5I-35122PadovaItaly
F La Barbera
Astronomical Observatory of Capodimonte -INAF
via Moiariello 16I-80131NapoliItaly
M Capaccioli
VSTCen -INAF
via Moiariello 16I-80131NapoliItaly
J T A De Jong
Leiden Observatory
Leiden University
P.O. Box 95132300 RALeidenThe Netherlands
F Getman
Astronomical Observatory of Capodimonte -INAF
via Moiariello 16I-80131NapoliItaly
A Grado
Astronomical Observatory of Capodimonte -INAF
via Moiariello 16I-80131NapoliItaly
M Paolillo
Department of Physics
University Federico II
via Cinthia 680126NapoliItaly
Machine Learning based photometric redshifts for the KiDS ESO DR2 galaxies
Mon. Not. R. Astron. Soc
00030 Jul 2015. 2015. July 31. 2015Accepted 2015 July 03; Received 2015 July 02; in original form 2015 May 05(MN L A T E X style file v2.2)techniques: photometric -galaxies: distances and redshifts -galaxies: photome- try
We estimated photometric redshifts (z phot ) for more than 1.1 million galaxies of the ESO Public Kilo-Degree Survey (KiDS) Data Release 2. KiDS is an optical wide-field imaging survey carried out with the VLT Survey Telescope (VST) and the OmegaCAM camera, which aims at tackling open questions in cosmology and galaxy evolution, such as the origin of dark energy and the channel of galaxy mass growth. We present a catalogue of photometric redshifts obtained using the Multi Layer Perceptron with Quasi Newton Algorithm (MLPQNA) model, provided within the framework of the DAta Mining and Exploration Web Application REsource (DAMEWARE). These photometric redshifts are based on a spectroscopic knowledge base which was obtained by merging spectroscopic datasets from GAMA (Galaxy And Mass Assembly) data release 2 and SDSS-III data release 9. The overall 1σ uncertainty on ∆z = (z spec − z phot )/(1 + z spec ) is ∼ 0.03, with a very small average bias of ∼ 0.001, a NMAD of ∼ 0.02 and a fraction of catastrophic outliers (|∆z| > 0.15) of ∼ 0.4%.
INTRODUCTION
Photometric redshifts (z phot ) derived from multi-band digital surveys are crucial to a variety of cosmological applications (Scranton et al. 2005;Myers et al. 2006;Hennawi et al. 2006;Giannantonio et al. 2008). In the last years a plethora of methods has been developed to estimate z phot (cf. Hildebrandt et al. 2010), but the advent of a new generation of photometric surveys (to quote just a few, Pan-STARRS : Kaiser 2004;Euclid: Laureijs et al. 2011;KiDS 1 : de Jong et al. 2013) demands for higher accuracy (Brescia et al. 2014b).
The evaluation of photometric redshifts requires the mapping of the photometric space into the spectroscopic redshift space. All methods, one way or the other, require the use of a Knowledge Base (KB) consisting in a set of templates, and differ mainly in the following aspects: (i) the way in which the KB is constructed (spectroscopic redshifts or, rather, empirically or theoretically derived spectral energy distributions or SEDs), and (ii) the adopted interpolation/fitting algorithm. Methods based on the interpolation of a spectroscopic KB are usually labeled as empirical.
Many different implementations of empirical methods exist and we shall recall just a few: i) polynomial fitting (Connolly et al. 1995); ii) nearest neighbors (Csabai et al. 2003); iii) neural networks (D'Abrusco et al. 2007;Yéche et al. 2010 and references therein); iv) support vector machines (Wadadekar 2005); v) regression trees (Carliles et al. 2010); vi) gaussian processes (Way & Srivastava 2006;Bonfield et al. 2010), and vii) diffusion maps (Freeman et al. 2009).
In this paper we discuss the derivation of photometric redshifts for the galaxies in the Kilo-Degree Survey (KiDS) data release 2 (de Jong et al. 2015). KiDS is an ESO public survey, based on the VLT Survey Telescope (Capaccioli & Schipani 2011) with the OmegaCAM camera (Kuijken 2011), that will image 1500 square degrees in four filters (u, g, r, i), in single epochs per filter. The high spatial resolution of VST (0.2 ′′ /pixel), the photometric depth and area covered make it a front-edge tool for weak gravitational lensing and galaxy evolution studies. The measurement of unbiased and high-quality z phot is a crucial step to pursue many of the scientific goals which motivated the KiDS survey (de Jong et al. 2015).
We present the z phot for a sample of ∼ 1.1 million galaxies.
These redshifts were derived with the Multi Layer Perceptron with Quasi Newton Algorithm (MLPQNA) method, described in detail elsewhere (Brescia et al. , 2013, hence we refer to those articles for all the mathematical and technical details. Recently, this method has been also used to derive a catalogue of z phot for the entire SDSS-DR9 (Brescia et al. 2014b). In the PHAT1 contest (Hildebrandt et al. 2010), which blindly compared most existing methods to estimate z phot on a very limited KB (∼ 500 objects only), the MLPQNA method proved to be one of the best empirical methods to date . It is however worth noticing that in the PHAT1 contest, MLPQNA did not perform as well as many SED fitting methods, due to the very limited base of knowledge available. This situation reverses when significantly larger KB's properly sampling the photometric parameter space become available (Brescia et al. 2013(Brescia et al. , 2014b.
The MLPQNA model is publicly available in the DAta Mining & Exploration Web Application REsource infrastructure (DAME-WARE; Brescia et al. 2014a) and has also been implemented in the PhotoRaptor service package (Cavuoti et al. 2015).
The paper is organized as follows. In Sect. 2 we present the dataset, while in Sect. 3 the experiments and related outcome are described and discussed. In Sect. 4 we give a description of the resulting photometric redshift catalogue. Finally, in Sect. 5 we draw our conclusions and future prospects.
THE DATA
The sample of galaxies for which we provide z phot was extracted from the second data release of the ESO Public Kilo-Degree Survey (KiDS-ESO-DR2). A detailed description of all the steps followed to extract the catalogues is given in de Jong et al. (2015). KiDS is a wide-area optical imaging survey in the four filters (u, g, r, i), carried out with the VLT Survey Telescope (VST) and the Omega-CAM camera. The KiDS observation strategy consists of a standard diagonal dithering pattern (5 dithers in g, r, i and 4 in u-band) in order to minimize the effect of the inter-CCD gaps in the Omega-CAM science array. Therefore the final footprint of each single tile is slightly larger than the nominal 1 square degree (de Jong et al. 2015).
The data processing procedure used is based on the Astro-WISE (AW) optical pipeline (McFarland et al. 2013). After the first basic data reduction steps (such as cross-talk, de-biasing and overscan correction, flat-fielding, illumination correction, de-fringing, pixel masking, satellite-track removal and background subtraction), the pipeline performs photometric and astrometric calibrations.
Source extraction is based on a task provided in the AW environment, KiDS-CAT (de Jong et al. 2015), based on algorithms developed for the software 2DPHOT (La Barbera et al. 2008). KiDS-CAT automatically performs a seeing assessment of the image, using best-quality stars in the image, and subsequently optimizes the configuration files of SExtractor (Bertin & Arnouts 1996) to perform the source extraction in the individual bands. In this process, besides the photometric flag provided by SExtractor, detected sources are also flagged according to their proximity to star spikes and haloes (IMAFLAGS_ISO flag), which are identified in the KiDS images through a dedicated masking procedure (de Jong et al. 2015, see also Huang et al. in preparation).
In order to derive our photometric redshifts, we used the multiband source catalogues, which rely on the double-image mode of SExtractor. These catalogues are based on source detection in the r-band images. While magnitudes are measured in all filters, the Star-Galaxy separation, as well as the source positional and shape parameters, are based on the r-band data only. The choice of the r-band as a reference is motivated by the fact that it is observed under the best seeing conditions (∼ 0.7 ′′ seeing FWHM, on average), and therefore it typically has the best image quality, thus providing the most reliable source positions and shapes. Aperture photometry in the four bands within several aperture radii, together with MAG_AUTO, shape parameters and flags, are available from SExtractor and KiDS-CAT. In the final catalogue, in order to maximize the sample with z phot estimates available, we have retained ∼ 10 7 sources with r-band SExtractor flag FLAGS _r < 4 and rejected ∼ 2 × 10 5 objects having close and bright companion sources, affected by bad pixels or originally blended with other objects (see Bertin & Arnouts 1996 for a detailed description of extraction flags). The limiting magnitudes of KiDS catalogues 2 at the 1σ level are:
• MAGAP_4_u = 25.17 • MAGAP_6_u = 24.74 • MAGAP_4_g = 26.03 • MAGAP_6_g = 25.61 • MAGAP_4_r = 25.89 • MAGAP_6_r = 25.44 • MAGAP_4_i = 24.53 • MAGAP_6_i = 24.06
KiDS DR2 contains 148 tiles observed in all filters during the first two years of operations. From the original catalogue of ∼ 18 millions of sources, the Star/Galaxy separation leaves ∼ 10 million galaxies, of which ∼ 6 million have null IMAFLAGS_ISO in all the filters, i.e. they are observed in unmasked regions. Out of these, we succeeded in estimating z phot for 1, 142, 992 sources (see Sec. 4 for details).
In order to build the needed spectroscopic knowledge base, the KiDS galaxy sample was matched with two independent spectroscopic surveys: GAMA (Galaxy And Mass Assembly) and SDSS (Sloan Digital Sky Survey). The final spectroscopic sample was obtained by merging data from GAMA data release 2 (112k new redshifts in the first three years, Driver et al. 2011, Liske et al. in prep.) and SDSS-III data release 9 (Ahn et al. 2012(Ahn et al. , 2014Bolton et al. 2012;Chen et al. 2012). The redshift distribution of the mixed catalogue is shown in Fig. 1.
GAMA observes galaxies out to z = 0.5 and r < 19.8 (r-band petrosian magnitude), by reaching a spectroscopic completeness of 98% for the main survey targets. It provides also information about the quality of the redshift determination by using the probabilistically defined normalised redshift quality scale nQ. The redshifts with nQ > 2 are the most reliable (Driver et al. 2011;Hopkins et al. 2013).
For the SDSS-III we used the low z (LOWZ) and constant mass (CMASS) samples of the Baryon Oscillation Sky Survey (BOSS). The BOSS project aims to obtain spectra (redshifts) for 1.5 million luminous galaxies up to z ∼ 0.7. The LOWZ sample consists of galaxies with 0.15 < z < 0.4 with colors similar to those of luminous red galaxies (LRGs) at z 0.4. Objects were selected by applying suitable cuts on magnitudes and colors with the aim of extending the SDSS LRG sample towards fainter magnitudes/higher redshifts (see e.g. Ahn et al. 2012;Bolton et al. 2012).
The CMASS sample contains three times more galaxies than the LOWZ sample, and it was designed to select galaxies in the range 0.4 < z < 0.8. The rest-frame color distribution of the CMASS sample is significantly broader than that of the LRG one, thus CMASS contains a nearly complete sample of massive galaxies down to log M ⋆ /M ⊙ ∼ 11.2. The faintest galaxies are at r = 19.5 in the LOWZ sample and i = 19.9 in the CMASS one.
Our spectroscopic sample is therefore dominated by galaxies from GAMA (46, 603 vs. 1, 618 from SDSS) at low-z (z 0.4), while SDSS galaxies dominate the higher redshift regime (out to z ∼ 0.7), with r < 22.
EXPERIMENTS AND DISCUSSION
Dealing with machine learning supervised methods, it is common practice to select and use the available KB to build a minimum of three disjoint data subsets: (i) a data set to train the method looking for the correlation hidden in the photometric information among the input features necessary to perform the regression (known as training set); (ii) a validation set to be used to check and verify the training performance against a loss of generalization capabilities (a phenomenon also known as overfitting); (iii) finally, a test set needed to blindly evaluate the overall performances of the model with data samples never submitted to the model before.
In this work, the validation process was embedded into the training phase, by applying the standard leave-one-out k-fold cross validation mechanism (Geisser 1975). We would like to stress that none of the objects included in the training (and validation) sample were included in the test sample and only the test data were used to generate the statistics and scatter plots.
We created training and test samples with relative sizes of 60% (36, 222 objects) and 40% (24, 150 objects) by randomly drawing without replacement from the KB. The histogram in Fig. 1 shows the distribution of the KB as a function of the z spec in both the training and test sets, while in Fig. 2 the distribution of z spec and z phot in the blind test set is shown. As it can be seen, the three distributions are in almost perfect agreement.
The results were evaluated using a standard set of statistical indicators applied to the quantity ∆z = zspec −z phot 1+zspec :
• the bias, defined as the mean value of the residuals ∆z;
• the standard deviation (σ) of the residuals; • the normalized median absolute deviation or N MAD of the residuals, defined as N MAD(∆z) = 1.48 × Median (|∆z|).
As input photometric parameters (or features) we used the MAGAP_4 and MAGAP_6 aperture magnitudes (u, g, r, i), a choice which was based on our past experience, since almost always this combination lead to the best performances (Brescia et al. 2013(Brescia et al. , 2014b. However, it needs to be emphasized that an improvement in the performances of a machine learning method can be expected from an exhaustive exploration of the parameter space through feature selection (cf. Polsterer et al. 2014). This approach, however, is usually too much demanding in terms of computing time.
MLPQNA z phot are in excellent agreement with z spec , as we show in Figs. 3 and 4, where the results of the experiment are summarized. The upper panel of Fig. 3 shows the predicted photometric redshift estimates versus the spectroscopic redshift values for all objects in the blind test set. In the lower panel of Fig. 3 the z spec is plotted vs. the residuals ∆z. The underpopulated redshift bins, visible in Fig. 3, reflect the distribution of the spectroscopic sample which is less populated at redshift ∼ 0.22 and ∼ 0.42 (see Fig. 1 and Fig. 2).
In Fig. 4 we show the distribution of residuals which has a kurtosis of 1.8 and a skewness of 7.07 × 10 −16 , i.e. a leptokurtic and symmetric distribution, as already found in the SDSS-DR9 case by applying the same method (Brescia et al. 2014b). In other words, the distribution reveals an over-density of objects in its central region (i.e. objects with a small error), which also reflects on the very low percentage of outliers and a low NMAD value (see below).
Overall, we find a bias of 9.9 × 10 −4 , a standard deviation of 0.0305 and a NMAD of 0.021. The σ 68 (i.e. the range in which falls the 68% of the residuals) is 0.022, smaller than the standard deviation, as it has to be expected from a leptokurtic distribution. Moreover, our method leads to a very small fraction of outliers, i.e. less than 0.39% and 3.30% using the |∆z| > 0.15 and |∆z| > 2σ criteria, respectively. If we refer to the sample of objects with I MAFLAGS _IS O = 0, the bias, standard deviation and NMAD become 0.00072, 0.0288 and 0.0207, respectively. While the fraction of outliers is of 0.32% and 3.26%. Thus, although the present approach is quite immune to systematic effects in photometry, we find a small improvement in the statistics when the sources in the masked regions are removed from the analysis.
THE PHOTOMETRIC CATALOGUE
To produce the final z phot catalogue, we initially considered the multi-source KiDS catalogue, i.e. sources detected in r-band and measured in all KiDS bands. However, it is important to underline that all empirical z phot prediction methods suffer from a poor capability to extrapolate outside the range of distributions imposed by the training sample. In literature, several approaches have been proposed to extend the applicability test of empirical methods outside the boundaries of the parameter space properly sampled by the spectroscopic KB (c.f. Vanzella et al. 2004;Hoyle et al. 2015). While useful in some cases, this artificial augmentation of the KB introduces a further level of complexity and leads to statistical biases which are difficult to evaluate and control.
In the available spectroscopic KB we found that ∼ 99% of the KB objects falls within the following region of the parameter space: Hence, to produce the final z phot catalogue, we have removed all the objects that do not match the above criteria in more than one band. The choice to retain objects with only one band not matching the above criteria was dictated by the need to maximize the number of objects with a redshift estimate and supported by the well tested robustness of the MLPQNA method against non detections or missing data . In Table 1 we report the statistical indicators evaluated for two groups of objects: those having all data points falling within the above region (clean objects) and those (contaminated objects) with only one band falling outside of it. It appears evident that for a one-band failure there is only a small decrease of performance.
However, in order to keep track of this effect, we include a z phot
CONCLUSIONS
In this work we applied the MLPQNA neural network to the ESO KiDS DR2 photometric galaxy data, using a knowledge base derived from the SDSS and GAMA spectroscopic samples, to produce a catalogue of photometric redshifts based on optical photometric data only. We obtained an overall 1σ uncertainty on ∆z = (z spec − z phot )/(1 + z spec ) of 0.0305 with a very small average bias of 9.9 × 10 −4 , a low N MAD of 0.021, and a low fraction of outliers (0.39% above the standard limit of 0.15).
The trained network was then used to process all galaxies in the data set that populate a parameter space similar to that defined by the SDSS+GAMA spectroscopic sample, producing z phot esti-mates for about 1.1 million KiDS galaxies. The catalogue will be made available on CDS VizieR facility.
Deriving photometric redshifts is an essential task when dealing with large samples of galaxies, such as that expected from the KiDS photometric survey. These redshifts are currently being used by the Kids collaboration for a variety of studies regarding the evolution of galaxy stellar masses, integrated colours, colour gradients and the structural parameters with redshift (Napolitano et al. in prep.). The characterization of how completeness and biases of the photo-z catalogue affect the final scientific goals is therefore postponed to later works. This types of studies will allow us to better constrain the processes leading to the (mass) growth of galaxies in the last half of the current age of the universe.
Figure 1 .Figure 2 .
12Spectroscopic redshift distribution of objects included in the training set (black line) and test set (gray line) normalized to the splitting rate. Redshift distribution of objects included in the blind test set, spectroscopic (black line) and photometric (gray line).
Figure 3 .
3Upper panel: spectroscopic versus photometric redshifts for objects of the blind test set. Lower panel: spectroscopic redshift versus (z specz phot )/(1+z spec ) for the same objects. 2013) under grant agreement n. 267251 Astronomy Fellowships in Italy (AstroFIt). This work was partially funded by the MIUR PRIN Cosmology with Euclid. MB acknowledges the support by the PRIN-INAF 2014 Glittering kaleidoscopes in the sky: the multifaceted nature and role of Galaxy Clusters.
Table 1. Statistical indicators computed for two different subsets of the blind test set. The clean set includes only data for which the photometry falls within the limits listed in Sec. 4. The contaminated subset includes the objects which fall outside those limits in only one band. quality flag in the catalogue, set to 1 for best quality (i.e. clean) and 0 for worse quality (i.e. contaminated) objects.The final z phot catalogue consists of 1, 142, 992 objects (699, 155 objects have all I MAFLAGS _IS O = 0 and 710, 127 with best quality).Subset
|bias|
σ
NMAD
Outliers %
Outliers %
|∆z| > 0.15
|∆z| > 2σ
clean
0.0011
0.0303
0.0212
0.38
3.13
contaminated
0.0003
0.0339
0.0223
0.47
5.80
We use the MAGAP_4 and MAGAP_6 magnitudes, measured within circular apertures of 4 ′′ and 6 ′′ of diameter, respectively. These magnitudes are provided within the produced z phot catalogue.
ACKNOWLEDGEMENTSThe authors would like to thank the anonymous referee for extremely valuable comments and suggestions. Based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 177.A-3016, 177.A-3017 and 177.A-3018, and on data products produced byTar
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| [] |
[
"Rotation-and temperature-dependence of stellar latitudinal differential rotation ⋆",
"Rotation-and temperature-dependence of stellar latitudinal differential rotation ⋆"
] | [
"A Reiners [email protected] \nAstronomy Department\nUniversity of California\n521 Campbell Hall94720BerkeleyCA\n\nHamburger Sternwarte\nUniversität Hamburg\nGojenbergsweg 11221029HamburgGermany\n"
] | [
"Astronomy Department\nUniversity of California\n521 Campbell Hall94720BerkeleyCA",
"Hamburger Sternwarte\nUniversität Hamburg\nGojenbergsweg 11221029HamburgGermany"
] | [] | More than 600 high resolution spectra of stars with spectral type F and later were obtained in order to search for signatures of differential rotation in line profiles. In 147 stars the rotation law could be measured, with 28 of them found to be differentially rotating. Comparison to rotation laws in stars of spectral type A reveals that differential rotation sets in at the convection boundary in the HR-diagram; no star that is significantly hotter than the convection boundary exhibits the signatures of differential rotation. Four late A-/early F-type stars close to the convection boundary and at v sin i ≈ 100 km s −1 show extraordinarily strong absolute shear at short rotation periods around one day. It is suggested that this is due to their small convection zone depth and that it is connected to a narrow range in surface velocity; the four stars are very similar in T eff and v sin i. Detection frequencies of differential rotation α = ∆Ω/Ω > 0 were analyzed in stars with varying temperature and rotation velocity. Measurable differential rotation is more frequent in late-type stars and slow rotators. The strength of absolute shear, ∆Ω, and differential rotation α are examined as functions of the stellar effective temperature and rotation period. The highest values of ∆Ω are found at rotation periods between two and three days. In slower rotators, the strongest absolute shear at a given rotation rate ∆Ω max is given approximately by ∆Ω max ∝ P −1 , i.e., α max ≈ const. In faster rotators, both α max and ∆Ω max diminish less rapidly. A comparison with differential rotation measurements in stars of later spectral type shows that F-stars exhibit stronger shear than cooler stars do and the upper boundary in absolute shear ∆Ω with temperature is consistent with the temperature-scaling law found in Doppler Imaging measurements. | 10.1051/0004-6361:20053911 | [
"https://arxiv.org/pdf/astro-ph/0509399v2.pdf"
] | 8,642,707 | astro-ph/0509399 | a6618d925b866f542a9a6fa51a92b2866f8c4599 |
Rotation-and temperature-dependence of stellar latitudinal differential rotation ⋆
August 11, 2018 August 11, 2018
A Reiners [email protected]
Astronomy Department
University of California
521 Campbell Hall94720BerkeleyCA
Hamburger Sternwarte
Universität Hamburg
Gojenbergsweg 11221029HamburgGermany
Rotation-and temperature-dependence of stellar latitudinal differential rotation ⋆
August 11, 2018 August 11, 2018arXiv:astro-ph/0509399v2 30 Sep 2005 Astronomy & Astrophysics manuscript no. 3911 (DOI: will be inserted by hand later)Stars: activity -Stars: late-type -Stars: rotation -Stars: individual: HD 6869HD 60 555HD 109 238HD 307 938Cl* IC 4665 V 102
More than 600 high resolution spectra of stars with spectral type F and later were obtained in order to search for signatures of differential rotation in line profiles. In 147 stars the rotation law could be measured, with 28 of them found to be differentially rotating. Comparison to rotation laws in stars of spectral type A reveals that differential rotation sets in at the convection boundary in the HR-diagram; no star that is significantly hotter than the convection boundary exhibits the signatures of differential rotation. Four late A-/early F-type stars close to the convection boundary and at v sin i ≈ 100 km s −1 show extraordinarily strong absolute shear at short rotation periods around one day. It is suggested that this is due to their small convection zone depth and that it is connected to a narrow range in surface velocity; the four stars are very similar in T eff and v sin i. Detection frequencies of differential rotation α = ∆Ω/Ω > 0 were analyzed in stars with varying temperature and rotation velocity. Measurable differential rotation is more frequent in late-type stars and slow rotators. The strength of absolute shear, ∆Ω, and differential rotation α are examined as functions of the stellar effective temperature and rotation period. The highest values of ∆Ω are found at rotation periods between two and three days. In slower rotators, the strongest absolute shear at a given rotation rate ∆Ω max is given approximately by ∆Ω max ∝ P −1 , i.e., α max ≈ const. In faster rotators, both α max and ∆Ω max diminish less rapidly. A comparison with differential rotation measurements in stars of later spectral type shows that F-stars exhibit stronger shear than cooler stars do and the upper boundary in absolute shear ∆Ω with temperature is consistent with the temperature-scaling law found in Doppler Imaging measurements.
Introduction
Stellar rotation rates range from those too slow to be detected by Doppler broadening up to rates at which centrifugal forces become comparable to surface gravity. Surface magnetic fields, either fossil or generated by some type of magnetic dynamo, can couple to ionized plasma and brake a star's rotation via a magneto-thermal wind. Magnetic braking is observed in stars with deep convective envelopes where magnetic dynamo processes can efficiently maintain strong magnetic fields. As a consequence, field stars of a spectral type later than F are generally slow rotators with surface velocities below 10 km s −1 .
Magnetic braking requires the existence of a magnetic field, which also causes the plethora of all the effects found in stellar ⋆ Based on observations carried out at the European Southern Observatory, Paranal and La Silla, PIDs 68. D-0181, 69.D-0015, 71.D-0127, 72.D-0159, 73.D-0139, and on observations collected at the Centro Astronómico Hispano Alemán (CAHA) at Calar Alto, operated jointly by the Max-Planck Institut für Astronomie and the Instituto de Astrofísica de Andalucía (CSIC) ⋆⋆ Marie Curie International Outgoing Fellow magnetic activity. While early magnetic braking may be due to fossil fields amplified during contraction of the protostellar cloud, magnetic activity at later phases requires a mechanism that maintains magnetic fields on longer timescales. In the Sun, a magnetic dynamo located at the interface between convective envelope and radiative core is driven by radial differential rotation. This dynamo has been identified as the main source of magnetic fields, although there is growing evidence that it is not the only source of magnetic field generation (e.g., Schrijver & Zwaan, 2000) and that magnetic fields also exist in fully convective stars (Johns-Krull & Valenti, 1996). The solartype dynamo, however, has the potential to generate magnetic fields in all non-fully convective stars, as long as they show a convective envelope, and magnetic fields have been observed in a variety of slowly rotating stars (e.g., Marcy, 1984;Gray, 1984;Solanki, 1991).
Calculations of stellar rotation laws, which describe angular velocity as a function of radius and latitude, have been carried out by Kitchatinov & Rüdiger (1999) and Küker & Rüdiger (2005) for different equatorial angular veloc-ities. Kitchatinov & Rüdiger (1999) investigated rotation laws in a G2-and a K5-dwarf. They expect stronger latitudinal differential rotation in slower rotators, and their G2-dwarf model exhibits stronger differential rotation than the K5-dwarf does. Küker & Rüdiger (2005) calculate a solar-like model, as well as a model of an F8 main sequence star. They also come to the conclusion that differential rotation is stronger in the hotter model; the maximum differential rotation in the F8 star is roughly twice as strong as in the G2 star for the same viscosity parameter. The calculated dependence of horizontal shear on rotation rate, however, does not show a monotonic slope but has a maximum that occurs near 10 d in the F8 type star and around 25 d in the solar-type star. The strength of differential rotation depends on the choice of the viscosity parameter, which is not well constrained, but the trends in temperature and rotation are unaffected by that choice.
Observational confirmation of solar radial and latitudinal differential rotation comes from helioseismological studies that provide a detailed picture of the differentially rotating outer convection zone (e.g., Schou, 1998). Such seismological studies are not yet available for any other star. Asteroseismological missions, like COROT and Kepler, may open a new window on stellar differential rotation, but its data quality may provide only a very limited picture in the near future (Gizon & Solanki, 2004). In the case of the Sun, radial differential rotation manifests itself in latitudinal differential rotation that can be observed at the stellar surface, but all stars except the Sun are at distances where their surfaces cannot be adequately resolved. With the advent of large optical interferometers, more may be learned from observations of spatially resolved stellar surfaces (Domiciano de Souza et al., 2004). For now, we have to rely on indirect methods to measure the stellar rotation law.
Photometric programs that search for stellar differential rotation assume that starspots emerge at various latitudes with different rotation rates, as observed on the Sun. Hall (1991) and Donahue et al. (1996) measured photometric rotation periods and interpreted seasonal variations as the effect of differential rotation on migrating spots. Although these techniques are comparable to the successful measurements of solar differential rotation through sunspots (Balthasar et al., 1986), they still rely on a number of assumptions, e.g., the spot lifetime being longer than the observational sequence, an assumption difficult to test in stars other than the Sun (cf. Wolter et al., 2005). Photometric measurements report lower limits for differential rotation on the order of 10% of the rotation velocity (i.e. the equator rotating 10% faster than the polar regions), and differential rotation is reported to be stronger in slower rotators.
Doppler Imaging (DI) has been extensively used to determine latitudinal differential rotation where the derived maps are constructed from time-series of high-resolution spectra. Differential rotation can then be extracted from comparing two surface maps taken with time separation of a few rotation periods (e.g., Donati & Collier Cameron, 1997;Wolter et al., 2005). Doppler maps can also be constructed that incorporate differential rotation during the inversion algorithm (Petit et al., 2002). Detections of differential rotation through DI have recently been compiled by Barnes et al. (2005), who also analyze dependence on stellar rotation and temperature, and then compare them to results obtained from other techniques. Their results will be discussed in Sect. 6.
The technique employed in this paper is to search for latitudinal differential rotation in the shape of stellar absorption line profiles. This method is applicable only to stars not dominated by spots. From a single exposure, latitudinal solar-like differential rotation -i.e. the equator rotating faster than polar regions -can be derived by measuring its unambiguous fingerprints in the Fourier domain. The foundations of the Fourier transform method (FTM) were laid by Gray (1977Gray ( , 1998, and first models were done by Bruning (1981). A detailed description of the fingerprints of solar-like differential rotation and the first successful detections are given in Reiners & Schmitt (2002a, 2003a. The FTM is limited to moderately rapid rotators (see Sect. 2), but the big advantage of this method is that latitudinal differential rotation can be measured from a single exposure. This allows the analysis of a large sample of stars with a comparably small amount of telescope time. Reiners & Schmitt (2003a) report on differential rotation in ten out of a sample of 32 stars of spectral types F0-G0 with projected rotation velocities 12 km s −1 < v sin i < 50 km s −1 . Reiners & Schmitt (2003b) investigated a sample of 70 rapid rotators with v sin i > 45 km s −1 and found a much lower fraction of differential rotators. Differential rotation has also been sought in A-stars that have no deep convective envelopes. Reiners & Royer (2004) report on three objects out of 76 in the range A0-F1, 60 km s −1 < v sin i < 150 km s −1 , which show signatures of differential rotation.
In the cited works, differential rotation is investigated in stars of limited spectral types and rotation velocities. In this paper, I aim to investigate all measurements of differential rotation from FTM, add new observations, and finally compare them to results from DI and theoretical predictions. Currently, more than 600 stars were observed during the course of this project, and in 147 of them the rotation law could be measured successfully.
Differential rotation in line profiles
Latitudinal differential rotation has a characteristic fingerprint in the shape of the rotational broadening that appears in each spectral line. Since all other line broadening mechanisms, like turbulence, thermal, and pressure broadening, etc., also affect the shape of spectral lines, the effects of differential rotation are very subtle. Signatures of differential rotation and a recipe for measuring them are presented in Reiners & Schmitt (2002a). Since the signatures are so small, and spectral line blending is a serious issue even in stars of moderate rotation rates at v sin i ≈ 20 km s −1 , single lines are not measured, but instead a total broadening function is constructed from many lines of similar intrinsic shape. This process typically involves 15 lines in slow rotators observed at very high resolution (Sect. 3), and 300 lines in rapid rotators observed at lower resolution. In order to derive a unique broadening function at the required precision, high data quality is required. Detailed information about demands on data quality can be found in Reiners & Schmitt (2003a,b).
Interpretation of the profile's shape with rotational broadening requires that the line profiles are not affected by starspots, stellar winds, spectroscopic multiplicity, etc.. The Fourier transform method is therefore limited to unspotted single stars with projected rotation velocities v sin i > ∼ 12 km s −1 , and an upper limit at v eq ≈ 200 km s −1 is set by gravitational darkening. More specific information on the influence of starspots, of very rapid rotation, and some examples of detected signatures of differential rotation in line profiles can be found in Reiners (2003), and Reiners & Schmitt (2002a, 2003a.
From the derived broadening profile, the rotation law is determined by measuring the first two zeros of the profile's Fourier transform, q 1 and q 2 . In sufficiently rapid rotators, those are direct indicators of rotational broadening since other broadening mechanisms (for example turbulence or instrumental broadening) do not show zeros at such low frequencies. This is also the reason only stars with v sin i > ∼ 12 km s −1 can be studied; in slower rotators, line broadening is dominated by turbulence and the zeros due to rotation cannot be measured. In stars spinning fast enough to be analyzed with FTM, approximation of net broadening by convolutions is also justified. The important point in choosing the Fourier domain for profile analysis is that convolutions become multiplications in Fourier space. Thus, the fingerprints of rotational broadening are directly visible in the observed broadening profile's Fourier transform, and the spectra do not have to be corrected for instrumental or for any other line broadening, as long as the targets' rotation dominates the important frequency range.
In the following, the stellar rotation law will be approximated in analogy to the solar case. Differential rotation is expressed in terms of the variable α, with Ω the angular velocity and l the latitude. The rotation law is approximated by Ω(l) = Ω Equator · α sin 2 (l),
(1)
α = Ω Equator − Ω Pole Ω Equator = ∆Ω Ω .
(2) Solar-like differential rotation is characterized by α > 0 (α ⊙ ≈ 0.2). Sometimes ∆Ω is called absolute differential rotation. To avoid confusion with relative differential rotation α = ∆Ω/Ω, I will refer to ∆Ω as absolute shear and to α as differential rotation. ∆Ω is given in units of rad d −1 . As shown in Reiners & Schmitt (2003a), the parameter α/ √ sin i, with i the inclination of the stellar rotation axis, can be directly obtained from the ratio of the Fourier transform's first two zeros, q 2 /q 1 . Thus, determination of the stellar rotation law in Eq. 1 from a single stellar spectrum is a straightforward exercise.
Variables to describe differential rotation
The role of differential rotation especially for magnetic field generation in rapid rotators is not understood well. In the case of the Sun, we know that its magnetic cycle is driven, at least in part, by radial differential rotation, which itself is reflected in latitudinal differential rotation. How (and if) magnetic dynamos depend on the strength of differential rotation in terms of α or on absolute shear ∆Ω has not yet been empirically tested. Furthermore, different observing techniques measure different quantities, and authors express their results on stellar rotation laws in different variables. A variable frequently used is the lap time, i.e. the time it takes the equator to lap the polar regions (or vice versa), which is essentially the reciprocal of the shear ∆Ω. Differential rotation α = ∆Ω/Ω is the shear divided by the angular velocity. Expressing rotation velocity in terms of rotation period P essentially means the reciprocal of angular velocity Ω, but using ∆P instead of ∆Ω introduces several problems, since one has to consider whether P denotes equatorial or polar rotation period, the latter being larger with solar-like differential rotation (cp. Reiners & Schmitt, 2003b). In this paper, I will express the rotation law in terms of α and ∆Ω, in order to search for correlations with rotation velocity. The quantity measured by the FTM is α with a fixed observational threshold of α min ≈ 0.05 thereby limiting the detection of deviations to rigid rotation. The uncertainty in α measured by FTM is approximately δα = 0.1. The perhaps more intuitive parameter for the physical consequences of differential rotation, however, is the shear ∆Ω = αΩ, where Ω has to be obtained from v sin i and the radius. When analyzing the rotation law in terms of ∆Ω, one has to keep in mind that it can never exceed angular velocity itself, if polar and equatorial regions are not allowed to rotate in opposite directions. In other words, differential rotation cannot be larger than 100%.
Observational data
Data for this study have been compiled from observations carried out at different telescopes. Observations of field stars with projected rotation velocities higher than v sin i = 45 km s −1 have been carried out with FEROS on the 1.5m telescope (R = 48 000), ESO, La Silla, or with FOCES (R = 40 000) on the 2.2m at CAHA. Slower rotating field stars were observed at higher resolution at the CES with the 3.6m telescope, ESO, La Silla (R = 220 000). Additionally, observations in open clusters were carried out with the multi-object facility FLAMES feeding VLT's optical high-resolution spectrograph UVES at a resolution of R = 47 000. Details of the spectra and instruments used are given in Table 1. Parts of the data were published in Reiners & Schmitt (2003a,b) and Reiners et al. (2005). For more information about observations and data reduction, the reader is referred to these papers. Ten of the cluster targets (FLAMES/UVES), as well as 34 FOCES, targets were not reported on in former publications 1 .
More than 600 stars were observed for this project during the last four years. For this analysis, I selected the 147 stars for which the ratio of the Fourier transform's zeros q 2 /q 1 -the tracer of the rotation law -is measured with a precision better than 0.1, i.e. better than 6% in case of the typical value of q 2 /q 1 = 1.76. These 147 stars exhibit broadening functions that are (i) symmetric (to avoid contamination by starspots), and (ii) reveal rotation velocities between 1 It should be noted that the FLAMES/UVES targets observed in open cluster fields are not necessarily cluster members, as was shown for example for Cl* IC 4665 V 102 in Reiners et al. (2005). A detailed investigation of cluster membership goes beyond the scope of this analysis; rotation velocity and spectral type make them sufficiently comparable to the (probable) field stars in this context. Table 1. Data used for this analysis. Spectra were taken with different high-resolution spectrographs. Detection of differential rotation requires higher resolution in slower rotators. The minimum rotation velocities required for determining the rotation law are given in column 4. From the 147 spectra from which differential rotation could be determined, 28 stars show signatures of differential rotation v sin i = 12 km s −1 and 150 km s −1 , the latter being an arbitrary threshold in order to minimize the amount of gravity-darkened stars with v eq ≫ 200 km s −1 (Reiners, 2003). Depending on the spectral resolution, the minimum rotation velocity is higher than v sin i = 12 km s −1 (CES) or 45 km s −1 (FEROS, FOCES, UVES, see also Table 1). More than 450 stars have not been used in this analysis. Their broadening functions show a whole variety of broadening profiles. Many are slow rotators, although some were reported as rapid rotators in earlier catalogues (cf. Reiners & Schmitt, 2003b). A large number of spectroscopic binaries (or even triples) were found, and another large part of the sample shows spectra that are apparently distorted by starspots or other mechanisms that cause the spectra to appear asymmetric. For analyzing the rotation law from the broadening profile alone, these spectra are useless and are not considered in this work. They may be promising targets for DI techniques or other science; a catalogue of broadening profiles is in preparation. Reiners & Royer (2004) measured the rotation law in 78 stars of spectral type A including a few early F-type stars. This sample will only be incorporated in the HR-diagram in Sect. 6.1. As will be shown there, rotation laws in A-stars are fundamentally different from those in F-stars. For this reason, I will not incorporate the A-stars in the sample analyzed for rotation-and temperature-dependencies of differential rotation. One of the stars observed during this project has recently been studied with DI, with comparison of the results given in the Appendix in Sect. A.
Accuracy of rotation law measurements
Projected rotation velocities v sin i were derived from the first zero q 1 in the Fourier transform as explained in Reiners & Schmitt (2003b). As mentioned there, the precision of this measurement is usually < 1 km s −1 , but simulations revealed a systematic uncertainty of ∼ 5% in v sin i. Thus, I chose the maximum of the uncertainty in the intrinsic measurement and the 5% limit as the error on v sin i.
The typical uncertainty for the determined values of relative differential rotation in terms of α = ∆Ω/Ω is δα ≈ 0.1 (Reiners & Schmitt, 2003a). In slow rotators, this uncertainty is dominated by the noise level due to comparably sparse sampling of the profile even at very high resolution. In fast rotators, the profile (i.e., the zeros of the profile's Fourier transform) can be measured with very high precision. Here the uncertainty in α stems from poor knowledge of the limb darkening parameter ǫ. For the case of a linear limb darkening law, the value of the measured ratio of the Fourier transform's first two zeros, q 1 and q 2 , is in the range 1.72 < q 2 /q 1 < 1.85 (Dravins et al., 1990). With the very conservative estimate that ǫ is essentially unknown (0.0 < ǫ < 1.0), every star with 1.72 < q 2 /q 1 < 1.85 in this analysis was considered a rigid rotator, and q 2 /q 1 < 1.72 was interpreted as solar-like differential rotation with the equator rotating faster than the polar regions. From the measured value of q 2 /q 1 , the parameter α/ √ sin i was determined as explained above.
Stars can exhibit ratios of q 2 /q 1 > 1.85 as well, but this value also does not agree with rigid rotation of a homogenous stellar surface. In the investigated sample of 147 stars, eight (5%) exhibit a ratio of q 2 /q 1 > 1.85. In contrast to solar-like differential rotation with the equator rotating more rapidly than polar regions, this case may be due to anti solar-like differential rotation with polar regions rotating faster than the equator. On the other hand, it can also be caused by a cool polar cap, which is expected to occur in rapidly rotating stars (Schrijver & Title, 2001). The lower flux emerging from a cool polar cap makes the line center shallow and has the same signature as anti solarlike differential rotation. While anti solar-like differential rotation cannot be distinguished from a cool polar cap by investigating the rotation profile, the existence of a cool polar cap in rapidly rotating F-stars seems much more plausible than anti solar-like differential rotation. Differentiation between the two cases can only be achieved by measuring differential rotation independent of the line shape. For the scope of this paper, however, I will interpret stars with q 2 /q 1 > 1.85 as rigid rotators with a cool polar cap.
Rotation velocity and spectral type
The observations carried out with FEROS, FOCES, UVES, and the CES are very homogeneous in terms of data quality and the investigated wavelength region, and only measurements that fulfill the criterion δα < 0.1 are considered. The targets have similar spectral types, but they do not form a statistically unbiased sample. The quality requirements discussed above make it difficult to analyze stars in a well-defined sample. This sample certainly is severely biased by observational and systematic effects. The most important bias is probably due to rotational Table 2. braking, by connecting spectral type with rotational velocity in the presumably old field stars. Later spectral types suffer from stronger magnetic braking and are expected to be generally slower than earlier spectral types. Projected rotation velocities v sin i of the sample stars are plotted versus color B − V in Fig. 1, while differentially rotating stars (i.e. stars with α > 0) are indicated by filled circles and will be discussed in detail in Sect. 6. A clear dependence of v sin i on color is apparent, as expected. In Fig. 1, the sample is divided arbitrarily into four subsamples with projected rotation velocities that are higher (lower) than v sin i = 70 km s −1 , and color redder (bluer) than B − V = 0.4. The total numbers of stars in each subsample and the numbers of differential rotators are given in Table 2. Most targets occupy regions I and IV. The scarcity in region II is due to the early F-type stars not being subject to strong magnetic braking; since the sample mainly consists of field stars, most stars later than B − V = 0.4 have been decelerated into region IV and do not appear in region III. Thus, slower rotating stars generally have a later spectral type in the sample. This implies that the effects of temperature and rotation velocity on the fraction of differentially rotating stars are degenerate in this sample; slow rotation implies late spectral type. It is therefore not possible to uniquely distinguish between the effects of rotation and spectral type in the investigated sample. This degeneracy has to be kept in mind when trying to interpret rotation-and temperature-dependencies of differential rotation in the following chapters.
Results
As a first result of the profile analysis, I present v sin i and the measurement of the rotation law for all 147 objects in Appendix B. Fields stars (FEROS, FOCES, and CES observations) are compiled in Table B.1, FLAMES/UVES targets in Table B.2.
In the sample of 147 stars for which the rotation law was measured, 28 (19%) exhibit signatures of solar-like differential rotation (q 2 /q 1 < 1.72). CES-, FEROS-, and FOCESsamples contain field stars of spectral type later than F0, with the majority of them brighter than V = 6 mag. Stars in fields of open clusters contained in the UVES-sample probably are mostly younger than the field stars, and all of them should have reached the main sequence (Stahler & Palla, 2004).
Differential rotation in the HR-diagram
All currently available measurements of the stellar rotation law from rotation profile shape (i.e, 147 stars from this sample and 78 stars from Reiners & Royer, 2004) are plotted in an HR-diagram in Fig. 2. For the field and A-stars, effective temperature T eff and bolometric magnitude M bol were derived from uvbyβ photometry taken from Hauck & Mermilliod (1998) using the program UVBYBETA published by Moon (1985). For T eff a new calibration by Napiwotzki et al. (1993) based on the grids of Moon & Dworetsky (1985) was used, and the statistical error of the temperature determination is about ∆T eff = 150 K. For three of the field stars, no uvbyβ photometry is available; all three are rigid rotators, so no value of T eff was calculated to avoid inconsistencies. For the cluster stars, no uvbyβ data is available, so radius and temperature are estimated from B−V color using zero-age main sequence (ZAMS) polynomials taken from Gray (1976), i.e., they are assumed to be young. For them, M bol is calculated from
M bol = 42.36 − 10 log T eff − 5 log R/R ⊙ ,(3)
with R ⊙ the solar radius. To get absolute V-magnitudes M V from bolometric magnitudes M bol , the bolometric correction calibrated from Hayes (1978) was applied. In Fig. 2, stars are plotted as open or filled circles that indicate rigid (α = 0) or differential rotation (α > 0), respectively. Circle sizes display projected rotation velocities v sin i as explained in the figure. The 147 stars given in Tables B.1 and B.2 are plotted as black symbols, while stars from Reiners & Royer (2004) are plotted as grey symbols. Evolutionary main sequence tracks for M = 1.2, 1.4, 1.6, 2.0, and 3.0 M ⊙ and the ZAMS from Siess et al. (2000) are also shown. Near the interface of A-stars and F-stars, the "granulation boundary" from Gray & Nagel (1989) is indicated with dashed lines. This is the region where line bisectors measured in slow rotators show a reversal. For dwarfs and subgiants, the "granulation boundary" coincides with theoretical calculations of the "convection boundary" and thus can be identified as the region where deep envelope convection disappears (cf. Gray & Nagel, 1989). The exact region of this boundary is not defined well and may depend on factors other than temperature and luminosity. HR-diagram of all currently available measurements of latitudinal differential rotation from FTM. Stars analyzed in this work are plotted in black, and stars from the A-star sample in Reiners & Royer (2004) in grey. Differential rotators are indicated by filled circles. Symbol size represents ranges of projected rotation velocity v sin i as explained in the figure. A typical error bar is given in the upper right corner. Evolutionary tracks and the ZAMS from Siess et al. (2000) are overplotted. Dashed lines mark the area of the granulation boundary according to Gray & Nagel (1989), while no deep convection zones are expected on the hot side of the boundary. Downward arrows indicate the four stars from Table 3 (see text).
The stars shown in Fig. 2 cover a wide range in temperature on both sides of the convection boundary. Because field stars are investigated, many stars have evolved away from the ZAMS. Rotation velocity as indicated by symbol size follows the well-known behavior of magnetic braking; late type stars are generally slowed down and exhibit slower rotation velocities. The striking fact visible in Fig. 2 is that none of the differential rotators detected with FTM lies significantly on the hot side of the convection boundary -latitudinal differential rotation has only been detected in stars believed to possess a deep convective envelope.
The largest group of differential rotators can be found near the ZAMS at all temperatures that are cooler than the convection boundary. The convection boundary itself is also populated by differential rotators. A few others can be found far away from the ZAMS (at masses > ∼ 1.6 M ⊙ ) to the right end of the main sequence tracks. There is a hint that differential rotators are lacking in the region between these few rotators and those near the ZAMS and again between them and those near the convection boundary, although this region is occupied by rigidly rotating stars (i.e., stars with differential rotation weaker than the observational threshold). However, it is not clear from the available sample whether temperature (and evolutionary stage) or rotation velocity may be the important parameter in determining differential rotation in F-type stars (see Sect. 5).
Differentially rotating A-stars at the convection boundary
Among all the differentially rotating stars shown as filled circles in Fig. 2, the four stars with the strongest shear ∆Ω are found very close to the convection boundary. This group of rapidly rotating late A-/early F-type stars is listed in Table 3 2 . 2 It should be noted that the only star in this group, that lies on the cool side of the convection boundary, is Cl* IC 4665 V 102. Temperature and luminosity were estimated from B − V color using ZAMS polynomials. Cl* IC 4665 V 102, however, is not a member of the cluster IC 4665 (Reiners et al., 2005), hence its position in the HR-diagram is most uncertain. It may have a much higher absolute luminosity that shifts it even closer towards the convection boundary. Table 3. Four stars with extreme values of differential rotation ∆Ω. It is speculated that these objects represent a special class of targets (marked with downward arrows in Fig. 2) in a narrow region of rotation velocity and effective temperatures near the convection boundary
Star Sp v sin i ∆Ω sin i T eff M V [km s −1 ] [rad d −1 ] [K]
HD 6869 A9 95 ± 5 1 2.3 ± 1.5 1 7390 3.01 HD 60555 A6 109 ± 5 1 1.3 ± 0.7 1 7145 1.49 HD 109238 F0 103 ± 4 1 1.3 ± 0.6 1 7184 1.31 IC 4665 V 102 A9 105 ± 12 2 3.6 ± 0.8 2 7136 2.93 1 Reiners & Royer (2004) 2 Reiners et al. (2005) All four exhibit very similar effective temperatures around T eff = 7200 K, putting them into the region where convective envelopes are extremely shallow. The four stars show remarkably similar projected rotation velocities all within 10% of v sin i = 100 km s −1 , and all four have rotation periods shorter than P = 2 d. Two of them, HD 6869 and Cl* IC 4665 V 102, exhibit a shear that is as strong as ∆Ω > 2 rad d −1 at rotation periods that are shorter than one day. This contrasts the 34 other stars that show comparably small rotation periods but have different temperatures or surface velocities. None of the 34 other rapid rotators exhibits a shear ∆Ω sin i in excess of 0.7 rad d −1 , i.e. a factor of three weaker than the two mentioned above, and only three of the 34 show ∆Ω > 0 at all.
The fact that the four strongest differential rotators are found at hot temperatures and high rotation velocities contradicts the general trend that differential rotation is more common in slowly rotating cool stars, which will be discussed in the following sections. All four are very close to the convection boundary or even on its blue side, meaning extremely shallow convective envelopes. This leads to the suspicion that the mechanism responsible for the strong shear is different from the one that drives the shear in stars with deeper convection zones. This is supported by the observation that all four stars also exhibit very similar (and comparably large) surface velocities of about 100 km s −1 . It is thus suggested that the strong surface shear in the four stars in Table 3 is not comparable to the rest of the sample, but is facilitated by the high surface velocity in a particularly shallow convective envelope. Such a mechanism could be supported by eigenmodes that are comparable to pulsational instabilities, for example in δ Scu stars, but further investigation is beyond the scope of this paper.
The fractional amount of differential rotators
In this section, the hypothesis that the fraction of differentially rotating stars with α > 0 is independent of rotation velocity v sin i and color B − V will be tested. The actual strength of differential rotation is not taken into account, but will be investigated in the following sections.
In Fig. 1, the whole sample of 147 stars has been divided into the four arbitrary regions mentioned in Sect. 5 (dividing at v sin i = 70 km s −1 and B − V = 0.4). The number of stars and differential rotators (α > 0, for this sample this means stars with α above the observational threshold of α min ≈ 0.05), and the percentage of differential rotators is given in Table 2. The percentage of differentially rotating stars among slow rotators in regions II and IV is higher than it is among rapid rotators in regions I and III, respectively; a trend towards a higher fraction of differential rotators at slower rotation velocity is visible in both color regimes. The same is true for the percentage of differentially rotating stars among late-type stars in regions III and IV. A comparison to earlier type stars among regions I and II, yields a higher fraction of differentially rotating stars towards later spectral type. I tested the hypothesis that subsamples I-IV are drawn from the same distribution with a total mean of 19 % differential rotators. Samples II and III are consistent with this hypothesis (5 and 4 expected differential rotators, respectively). For samples I and IV, the hypothesis can be rejected at a 99 % level (99.3 %, 11 expected, and 99.6 %, 8 expected for samples I and IV, respectively). Thus, the fraction of differential rotators with α larger than the observational threshold of α min ≈ 0.05which does not depend on v sin i or color -is not constant. It is larger in slower rotators and stars of a later spectral type.
The distribution of differential rotators in v sin i and B − V is investigated further in Fig. 3. The upper panel shows the total number of stars divided into five bins in B − V (left), and seven bins in v sin i (right). The lower panel of Fig. 3 displays the fraction of differential rotators (α > ∼ 0.05) in the respective color/rotation bins with 2σ-errors. For example, 48% of the 25 stars with projected rotation velocities v sin i between 10 and 30 km s −1 show signatures of differential rotation. Although some bins are sparsely populated and have large errors, the change from color B − V = 0.4 to B − V = 0.5 and the transition from slow rotators to stars with v sin i > 30 km s −1 are significant. The trends indicated in Table 2 stand out in the lower panel of Fig. 3. In this sample, profiles with α > ∼ 0.05 are more frequent in slow rotators, which implies that they are more frequent in stars of later spectral type (due to the sample bias, cf. Sect. 5).
Differential rotation and rotation period
In this and the following sections, I will investigate the strength of differential rotation and especially the maximum strength of differential rotation at different rotation rates. I discuss differential rotation in terms of α = ∆Ω/Ω in Sect. 6.4.1, and analyze the shear ∆Ω in Sect. 6.4.2. Although both quantities essentially have the same meaning, it is instructive to look at both of them separately. Rotation speed will be expressed as a function of equatorial rotation period instead of rotation velocity. The rotation period itself is not measured for the majority of stars, so I will use P/ sin i instead, as calculated from measured v sin i and the radius according to Eq. 3.
In the left and right panels of Fig. 4, values of differential rotation α and absolute shear ∆Ω are plotted against projected rotation period P/ sin i, respectively. From the 147 stars 4. Differential rotation vs. projected rotation period P/ sin i. Left: Differential rotation α = ∆Ω/Ω; right: Absolute shear ∆Ω (note the logarithmic scale on the ordinate). Field F-stars are indicated by filled circles, open circles display stars from cluster observations. Crosses mark the three differential rotators from Reiners & Royer (2004). Short dashed lines are models from Küker & Rüdiger (2005) for an F8-star with a viscosity parameter of c ν = 0.15. The four stars with extraordinarily strong shear ∆Ω were discussed in Sect. 3, a different mechanism is suggested to cause their strong shear. For the remaining stars, the upper envelopes are indicated qualitatively by the long dashed lines. In the right panel, dotted lines show regions of constant differential rotation α = 0.1 (approximate sensitivity of FTM) and α = 1.0 (maximum if no counter-rotation of the polar regions is allowed for).
in Table 1, only the 28 stars with signatures of differential rotation are shown, while the other 119 objects populate the α = ∆Ω/Ω = 0 region at 0.5 d < P/ sin i < 11.0 d. Field stars from this sample with available uvbyβ measurements are plotted as filled circles, while the four cluster targets for which the ZAMS-age has been assumed are plotted as open circles. In both panels, the three differentially rotating stars from Reiners & Royer (2004), which have been discussed in Sect. 6.2, are marked as crosses.
In the left and right panels of Fig. 4, the long dashed lines qualitatively indicate the upper envelope of α/ √ sin i and ∆Ω √ sin i, respectively. No fit was attempted, so the lines should only guide the eye to clarify what will be discussed in the next sections.
Differential rotation α
The advantage of using α is that it is measured directly and its detection does not depend on rotation period, hence radius, while measuring ∆Ω does. Differential rotation α is smaller than 0.45 for all rotation periods. While a minimum threshold of (α/ √ sin i) min ≈ 0.05 applies, the observational technique has no limitations towards high values of α. Thus, the highest detected value of differential rotation (α/ √ sin i) max ≈ 0.45 is not due to limitations of the FTM.
At rotation periods between two and ten days, the targets populate the whole region 0 < α/ √ sin i < 0.45, while the slower targets could not be analyzed due to the limitations of the FTM (cp. Sect 2). Among the rapid rotators with projected rotation periods that are shorter than two days, the upper envelope shows a clear decline among the F-stars (filled circles in the left panel of Fig. 4). Except for the group of A-stars discussed in Sect. 6.2 (listed in Table 3), no star with a projected rotation period less than two days shows α > 0.2, and no star with P/ sin i < 1 d shows α > 0.1. Neglecting those four stars, the maximum value in differential rotation, (α/ √ sin i) max , grows from virtually zero at P = 0.5 d to (α/ √ sin i) max ≈ 0.45 in stars slower than P = 2 d. In slower rotators, (α/ √ sin i) max remains approximately constant.
Absolute shear ∆Ω
Absolute shear (∆Ω √ sin i) is shown in the right panel of Fig. 4. Since observed values of both α and Ω depend on inclination i, the observed absolute shear is
∆Ω obs = α obs Ω obs = α/ √ sin i Ω sin i = ∆Ω √ sin i.(4)
In this and the following sections, I omit the factor √ sin i for readability. Note that in the case of small inclination angles, the value of ∆Ω can be larger than ∆Ω √ sin i, while α can be smaller than α/ √ sin i. In the right panel of Fig. 4, the two dotted lines indicate the slopes of constant differential rotation α = ∆Ω/Ω. The upper line is for α = 1.0, i.e., the maximum differential rotation possible regardless of the observation technique used, and the lower line shows α = 0.05, the approximate minimum threshold for the FTM, as explained above.
As expected from Sect. 6.4.1, the F-stars form a relatively smooth upper envelope in the maximum absolute shear ∆Ω max , as observed at different rotation rates. The slowest rotators exhibit low values of ∆Ω max ≈ 0.2 rad d −1 at P ≈ 10 d. ∆Ω max grows towards a faster rotation rate with a maximum between two and three days before it diminishes slightly with more rapid rotation. The strongest differential rotation occurs at rotation periods P between two and three days at a magnitude of ∆Ω ≈ 1 rad d −1 (i.e., lapping times on the order of 10 d). At higher rotation rates in the range 0.5 d < P < 2 d, the maximum shear has a slope of roughly ∆Ω ∝ P +0.4 . This slope, however, is not constrained well due to the large uncertainties and sparse sampling. The data are also consistent with a plateau at ∆Ω ≈ 0.7 rad d −1 for 0.5 < P < 3 d. At slower rotation, right from the maximum, the slope is approximately ∆Ω ∝ P −1 . Küker & Rüdiger (2005) recently calculated ∆Ω in an F8 star for different rotation periods. Their results for ∆Ω(P) (with a viscosity coefficient c ν = 0.15) are displayed qualitatively in both panels of Fig. 4 as a short-dashed line (Fig. 6 in Küker & Rüdiger, 2005). One of their results is that in their model ∆Ω does not follow a single scaling relation for all periods (as was approximately the case in the calculations by Kitchatinov & Rüdiger, 1999), but that a maximum shear arises at a rotation period of about P = 7 d in the case of the modeled F8-star. Comparison of their calculations (right panel of Fig. 4) to the upper envelope suggested in this work (longdashed line) still shows a large quantitative discrepancy. The qualitative slopes of both curves, however, are in reasonable agreement with each other. The theoretical curve was calculated for an F8-star. Küker & Rüdiger (2005) also show ∆Ω(P) for a solar-type star, where ∆Ω(P) is essentially moved towards higher rotation periods and lesser shear; i.e. the short-dashed curve in the right panel of Fig. 4 moves to the lower right for later spectral types. Although earlier spectral types are not calculated by Küker & Rüdiger (2005), it can be expected that ∆Ω(P) will shift towards higher shear and shorter rotation periods in stars of an even earlier spectral type. Since most stars investigated in this sample are earlier than F8, this would suggest qualitative consistency between theoretical curves and the slope of ∆Ω max shown here.
Comparing different techniques
Differential rotation measurements are now available from a variety of observational techniques (see Sect. 1), comparison of results from techniques becomes possible. However, such a comparison has to be carried out with great care. Photometrically measured periods, for example, are only sensitive to latitudes covered by spots; they reflect only parts of the rotation law and are always lower limits. Furthermore, temperature has been shown to be the dominating factor for the strength of differential rotation (Kitchatinov & Rüdiger, 1999;Küker & Rüdiger, 2005;Barnes et al., 2005), which has to be taken into account when analyzing the rotation dependence of differential rotation.
In the past, analyses of relations between rotation and differential rotation have generally assumed a monotonic scaling relation between period P (or angular velocity Ω = 2π/P) and ∆Ω. Such a relation was expected from calculations by Kitchatinov & Rüdiger (1999). As mentioned above, Küker & Rüdiger (2005) recently presented new calculations showing that the ∆Ω vs. Ω-relation may have a temperature dependent maximum.
Searching for dependence on angular velocity, Barnes et al. (2005) recently compiled data from differential rotation measurements from DI, photometric monitoring, and FTM. Fitting a single power law to the compiled data, they derive ∆Ω ∝ Ω +0.15±0.10 , which is compared to the case of a G2 dwarf calculated in Kitchatinov & Rüdiger (1999). From the latter, they cite the theoretical G2 dwarf relation as ∆Ω ∝ Ω +0.15 , and claim agreement to their fit. Although the work of Kitchatinov & Rüdiger (1999) has been superseded by Küker & Rüdiger (2005), it should be mentioned that Kitchatinov & Rüdiger (1999) report ∆Ω ∝ Ω −0.15 , implying stronger shear for slower rotation instead of weaker (note that Küker & Rüdiger, 2005, also report a negative exponent for their solar-like star model at periods less than 20 d). In fact, the large scatter in the compilation of all measurements from different techniques (Fig. 3 in Barnes et al., 2005) and the severe bias due to systematic uncertainties (like the lack of small values ∆Ω in the sample measured with FTM) leads to any conclusion about the period dependence from such a heteroge- Fig. 5. Differential rotation ∆Ω with effective temperature T eff for sample stars (symbols as in Fig. 4). Big circles indicate large rotation velocity. Stars analyzed with DI are indicated as squares, and the fit to DI targets from Barnes et al. (2005) is overplotted. F-stars with the strongest differential rotation are consistent with the fit.
neous sample very uncertain. The new data in this paper does not significantly improve this situation and no analysis that improves upon the one performed by Barnes et al. (2005) can be expected.
Although the constantly growing amount of differential rotation measurements provides a relatively large sample, the results from analyzing all measurements as one sample do not yet provide convincing evidence for a unique rotation dependence of differential rotation over the whole range of rotation periods.
Upper bound of differential rotation and effective temperature
Effective temperature is the second parameter after rotation to govern stellar differential rotation, since convection zone depth, as well as convection velocity, are very sensitive to T eff . Küker & Rüdiger (2005) and Kitchatinov & Rüdiger (1999) report stronger differential rotation with higher effective temperatures when comparing a solar-type star to stars of spectral types F8 and K5. The differential rotators in the sample of F-stars investigated here span a range in effective temperature between 6000 K and 7150 K. As a result, analysis of temperature effects on differential rotation is limited by the small range of targets in T eff , and is biased by the large range in rotation periods, as discussed above. Thus, I limit the analysis of temperature dependence to a comparison to differential rotation in stars that are significantly cooler than T eff = 6000 K.
Considering stars with effective temperatures in the range 3400 K < T eff < 6000 K, Barnes et al. (2005) found a powerlaw dependence in their measurements of absolute shear ∆Ω on stellar surface temperature. Their result is compared to the sample of this work in Fig. 5. Absolute shear follows the trend expected by Küker & Rüdiger (2005) and Kitchatinov & Rüdiger (1999) with a stronger surface shear at higher effective temperatures. Although a large scatter is observed in the F-star measurements, they qualitatively follow this trend and connect to cooler stars at roughly the expected values. In addition, the rotation rate of the F-stars is indicated by symbol size, larger symbols displaying higher rotation rate. Stars exhibiting a shear in excess of the expected rate for their temperature tend to show very rapid rotation. Thus, the investigated F-stars agree qualitatively with the temperature fit derived by Barnes et al. (2005). A more quantitative analysis is complicated by the large scatter among measurements of absolute shear, which is visible also in the sample of Barnes et al. (2005). Again, it should be noted that the values plotted for the F-star sample only display stars for which signatures of differential rotation have been measured. A large number of stars populate the region of weaker surface shear or rigid rotation (α = ∆Ω = 0), and the temperature law applies only to the strongest differential rotators.
Conclusions
The sample of stars with rotation laws measured from spectral broadening profiles with the FTM is constantly growing. In this work, 44 new observations of stars of spectral type F and later were added to the results from former publications. Currently, rotation laws have been analyzed in a homogeneous data set of 147 stars of spectral type F and later, and in a second data set in 78 stars of spectral type A. Among all these observations covering the temperature range between 5600 K and 10 000 K (including A-stars from Reiners & Royer, 2004), 31 stars exhibit signatures of solar-like differential rotation. Only three of them are of spectral type A. In the HR-diagram, differential rotators appear near and on the cool side of the convection boundary. No differentially rotating star hotter than 7400 K is known, and it is obvious that the signatures of solar-like differential rotation are closely connected to the existence of deep convective envelopes.
Most differential rotators can be found near to the ZAMS at young ages, but due to the limited sample and severe selection effects, this needs confirmation from a less biased sample. Particularly, a number of slower rotators with temperatures around T eff = 6600 K are needed.
Four differential rotators are very close to the convection boundary. All four show extraordinarily strong absolute shear and exhibit projected rotation velocities within 10% around v sin i = 100 km s −1 . It is suggested that these stars form a group of objects in which rotation velocity and convection zone depth facilitate very strong absolute shear and that the mechanism causing the shear is different from the later F-type stars.
Among the F-stars, differential rotation occurs in the whole range of temperatures and rotation rates. The sample of 147 stars of spectral type F and later was investigated for the dependence of differential rotation on rotation and temperature. 28 of them (19%) exhibit signatures of differential rotation. The distribution of differential rotation was approached with two different strategies: (i) investigation of the fraction of stars exhibiting differential rotation (α > 0); and (ii) analysis of the maximum α and maximum ∆Ω as a function of rotation period and temperature. The first approach reflects the typical values of <α> and <∆Ω> at any given temperature and rotation rate, while the second focuses on the question how strong the abso-lute shear can possibly be in such stars. Due to the large uncertainties in the measurements and the high minimum threshold in differential rotation, the mean values of α and ∆Ω are not particularly meaningful. Furthermore, it is not clear whether a smooth transition from stars exhibiting strong differential rotation to "rigidly" rotating stars (α < 0.05) or a distinction between these two groups exists, and the mentioned approaches were preferred for the analysis.
In the sample, hotter stars generally rotate more rapidly, and effects due to rotation velocity and temperature cannot be disentangled. The distribution of differential rotators depends on color and/or rotation rate, and the fraction of stars with differential rotation (α > 0) increases with cooler temperature and/or slower rotation. It is not clear what this means to the mean shear, <∆Ω>. For example, it is not inconsistent with <∆Ω> being constant at all rotation rates. In this case, <α>=<∆Ω/Ω> would be smaller in more rapidly rotating stars, and thus a lower fraction of stars would exhibit differential rotation α above the observational threshold, as is observed.
On the other hand, the maximum observed values of differential rotation, α max , and of the absolute shear, ∆Ω max , do vary depending on the rotation rate. The strongest absolute shear of ∆Ω ≈ 1 rad d −1 is found at rotation periods between two and three days with significantly smaller values in slower rotators. The more rapidly rotating stars show a slight decrease in absolute shear as well, although the sparse data are also consistent with a plateau at ∆Ω ≈ 0.7 rad d −1 for 0.5 < P < 3 d. A maximum in differential rotation ∆Ω has recently been predicted by Küker & Rüdiger (2005) for an F8-star, although of lesser strength and at a slower rotation rate, a difference that may in parts be due to their later spectral type.
The investigated sample does not cover a wide range in effective temperature since only very few late-type field stars rotate fast enough for the method applied. Although temperature is expected to strongly influence the strength of differential rotation, the large range in rotation rate and the connection between rotation rate and temperature in the sample makes conclusions about temperature effects insecure. The results were compared to differential rotation measurements in cooler stars and found in qualitative agreement with an extrapolation of the empirical temperature dependence that Barnes et al. (2005) found when analyzing a sample of differentially rotation measurements done with DI.
The implications of the relations discussed here for stellar magnetic activity and the nature of the dynamo working in F-type stars still remain unclear from an observational point of view. Naively, one would expect stronger magnetic activity to occur in stars with stronger differential rotation among groups of comparable temperature or rotation rate. Those stars for which X-ray measurements are available, however, do not yet exhibit such a trend, but a meaningful investigation is hampered by the limited amount of data points available (especially for comparable temperature or rotation velocity). It has been shown that qualitative conclusions can be derived from the currently available measurements of stellar rotation laws, but a more detailed investigation of the consequences on the dynamo operating in F-type stars has to wait until a statistically betterdefined sample of stars is available. , 1996, ApJ, 459, 95 Wolter, U., Schmitt, J.H.M.M., &andVan Wyk, F., 2005, A&A, 436, 261 Appendix A: The first direct comparison to Doppler Imaging Recently, Marsden et al. (2005) have presented Doppler Images of HD 307 938, a young active G dwarf in IC 2602. This star was also observed for this project in the FLAMES/UVES campaign, and I report on the rotation law in Table B.2. Marsden et al. (2005) took a time series over four nights detecting spectroscopic variability. Their data is contaminated by a significant amount of sunlight reflected by the moon, which they have carefully removed before constructing Doppler Images. The result of Doppler Imaging is that HD 307 938 has a cool polar cap extending down to ∼ 60 • latitude, and v sin i = 92 ± 0.5 km −1 , which excellently corresponds to v sin i = 93.7 ± 4.7 km s −1 , as derived here (on comments about uncertainties in v sin i, see Sect. 6). They also report marginal differential rotation of ∆Ω = 0.025 ± 0.015 rad d −1 with a 1σ error. From this result, one expects a value of q 2 /q 1 that is only marginally less than 1.76, the value for rigid rotation and the best guess for the limb darkening parameter. The cool polar spot found on HD 307 938 influences q 2 /q 1 as well, as it enlarges q 2 /q 1 much more than the small deviation from rigid rotation does; q 2 /q 1 is thus expected to be larger than 1.76. The spectrum secured for the analysis presented here is not contaminated by sunlight, and the broadening function derived is shown in Fig. A.1. The profile does not reveal large asymmetry, although a spot may be visible around v ≈ −30 km s −1 . The profile is fully consistent with the broadening functions presented in Marsden et al. (2005), who were able to show temporal variations in the profile from lower quality data. The profile parameter determined from FTM is q 2 /q 1 = 1.78 ± 0.01, indicating no signs of solar-like differential rotation that is large enough to be detected with this method. However, the fact that q 2 /q 1 is slightly larger than 1.76 supports the idea that a cool spot occupies the polar caps. Thus, the finding of marginal differential rotation and a cool polar spot is consistent with the result derived from FTM (besides the good consistency in v sin i). This is the first time that a direct comparison of the results is possible, since the majority of Doppler Imaging targets usually show spot signatures that are stronger than what can be dealt with using FTM.
Fig. 1 .
1Projected rotation velocity plotted vs. B − V for the whole sample of 147 stars. Differential rotators are plotted as filled circles. Contents of subsamples I-IV are given in
Fig. 2 .
2Fig. 2. HR-diagram of all currently available measurements of latitudinal differential rotation from FTM. Stars analyzed in this work are plotted in black, and stars from the A-star sample in Reiners & Royer (2004) in grey. Differential rotators are indicated by filled circles. Symbol size represents ranges of projected rotation velocity v sin i as explained in the figure. A typical error bar is given in the upper right corner. Evolutionary tracks and the ZAMS from Siess et al. (2000) are overplotted. Dashed lines mark the area of the granulation boundary according to Gray & Nagel (1989), while no deep convection zones are expected on the hot side of the boundary. Downward arrows indicate the four stars from Table 3 (see text).
Fig. 3 .
3Upper panel: Histograms of the sample in B − V (left column) and v sin i (right column). Lower panel: Fraction of stars with strong relative differential rotation in the bins plotted in the upper panel. 2σ-errors are overplotted.
Fig.
Fig. 4. Differential rotation vs. projected rotation period P/ sin i. Left: Differential rotation α = ∆Ω/Ω; right: Absolute shear ∆Ω (note the logarithmic scale on the ordinate). Field F-stars are indicated by filled circles, open circles display stars from cluster observations. Crosses mark the three differential rotators from Reiners & Royer (2004). Short dashed lines are models from Küker & Rüdiger (2005) for an F8-star with a viscosity parameter of c ν = 0.15. The four stars with extraordinarily strong shear ∆Ω were discussed in Sect. 3, a different mechanism is suggested to cause their strong shear. For the remaining stars, the upper envelopes are indicated qualitatively by the long dashed lines. In the right panel, dotted lines show regions of constant differential rotation α = 0.1 (approximate sensitivity of FTM) and α = 1.0 (maximum if no counter-rotation of the polar regions is allowed for).
Fig. A. 1 .
1Broadening function of HD 307 938. This object has also been studied with Doppler Imaging by Marsden et al. (2005). Stahler, S.W., & Palla, F., 2004, The Formation of Stars, Wiley-VCH Johns-Krull, C.M., & Valenti, J.A.
Table 2 .
2Amount of differential rotators, total number of stars, and percentage of differential rotators for each subsample ofFig. 1Subsample # of diff. rotators total # of stars percentageI
2
57
4%
II
4
24
17%
III
5
21
24%
IV
17
45
38%
Table B
B.1. Field Stars with measured differential rotation. sin i [km s −1 ] [km s −1 ] rad d −1 rad d −1Star
HR
v sin i
δ v sin i q 2 /q 1 δq 2 /q 1
∆Ω
δ∆Ω
T eff
M V
P/ K
mag
d
HD 432
21
71.0
3.6
1.78
0.03
0.00
6763 0.96
3.08
HD 4089
187
23.5
1.2
1.82
0.02
0.00
6161 2.75
5.01
HD 4247
197
42.7
2.1
1.77
0.04
0.00
6825 3.08
1.90
HD 4757
230
93.8
4.7
1.73
0.03
0.00
6706 1.34
2.00
HD 6706
329
46.0
2.3
1.83
0.06
0.00
6551 2.77
2.21
HD 6903
339
87.7
4.4
1.69
0.02
0.54
0.534
6273 3.38
0.96
HD 15524
728
59.8
3.0
1.81
0.08
0.00
6592 2.44
1.95
HD 17094
813
45.1
2.3
1.49
0.02
0.86
0.231
7141 2.17
2.50
HD 17206
818
25.6
1.3
1.70
0.02
0.15
0.184
6371 3.68
2.78
HD 18256
869
17.2
0.9
1.71
0.04
0.05
0.121
6332 2.93
5.91
HD 22001 1083
13.2
0.7
1.83
0.07
0.00
6601 2.98
6.88
HD 22701 1107
55.0
2.8
1.83
0.03
0.00
6610 2.70
1.88
HD 23754 1173
13.8
0.7
1.87
0.05
0.00
6518 2.98
6.75
HD 24357 1201
65.8
3.3
1.82
0.05
0.00
6895 2.83
1.35
HD 25457 1249
18.0
0.9
1.71
0.02
0.09
0.152
6333 4.07
3.35
HD 25621 1257
16.7
0.8
1.73
0.03
0.00
6091 2.26
9.02
HD 27459 1356
78.3
3.9
1.78
0.05
0.00
7642 2.15
1.29
HD 28677 1432
134.8
6.7
1.77
0.03
0.00
6981 2.76
0.67
HD 28704 1434
88.1
4.4
1.79
0.03
0.00
6672 2.43
1.30
HD 29875 1502
47.8
2.4
1.82
0.06
0.00
7080 3.39
1.37
HD 29992 1503
97.5
4.9
1.75
0.04
0.00
6742 2.64
1.04
HD 30034 1507
103.6
5.2
1.82
0.06
0.00
7484 2.34
0.92
HD 30652 1543
17.3
0.9
1.78
0.03
0.00
6408 3.59
4.23
HD 33167 1668
47.5
2.4
1.82
0.03
0.00
6493 2.10
2.97
HD 35296 1780
15.9
0.8
1.75
0.02
0.00
6060 4.11
4.09
HD 37147 1905
109.9
5.5
1.77
0.07
0.00
7621 2.25
0.88
HD 41074 2132
87.8
4.4
1.78
0.06
0.00
6912 2.45
1.20
HD 43386 2241
19.5
1.0
1.83
0.03
0.00
6512 3.50
3.77
HD 44497 2287
89.4
4.5
1.79
0.03
0.00
7010 1.79
1.56
HD 46273 2384
106.9
5.3
1.74
0.05
0.00
6674 2.10
1.25
HD 48737 2484
66.1
3.3
1.78
0.04
0.00
6496 2.32
1.93
HD 51199 2590
91.7
4.6
1.77
0.04
0.00
6730 2.06
1.45
HD 55052 2706
81.8
4.1
1.80
0.04
0.00
6668 0.63
3.21
HD 56986 2777
129.7
6.5
1.75
0.08
0.00
6837 1.95
1.05
HD 57927 2816
89.5
4.5
1.75
0.02
0.00
6772 1.40
1.99
HD 58579 2837
147.1
7.4
1.81
0.03
0.00
7053 1.69
0.98
HD 58946 2852
59.0
3.0
1.76
0.07
0.00
6892 2.79
1.54
HD 60111 2887
117.5
5.9
1.80
0.05
0.00
7181 3.01
0.65
HD 61035 2926
124.2
6.2
1.80
0.03
0.00
6986 3.09
0.62
HD 61110 2930
91.1
4.6
1.78
0.04
0.00
6575 1.51
1.98
HD 62952 3015
127.5
6.4
1.80
0.08
0.00
6933 1.53
1.26
HD 64685 3087
47.6
2.4
1.88
0.03
0.00
6838 3.04
1.73
HD 67483 3184
52.4
4.1
1.50
0.07
0.69
0.279
6209 2.07
3.01
HD 69548 3254
53.9
2.7
1.77
0.05
0.00
6705 3.53
1.27
HD 70958 3297
45.5
2.3
1.74
0.05
0.00
6230 3.53
1.76
HD 70958 3297
46.1
2.3
1.75
0.01
0.00
6230 3.53
1.74
HD 72943 3394
56.8
2.8
1.70
0.06
0.17
0.361
6897 1.81
2.51
HD 75486 3505
128.1
6.4
1.79
0.02
0.00
6993 1.22
1.42
HD 76143 3537
83.0
4.2
1.79
0.03
0.00
6579 1.58
2.10
HD 76582 3565
90.5
4.5
1.80
0.05
0.00
7884 2.58
0.87
HD 77370 3598
60.4
3.0
1.71
0.03
0.20
0.408
6609 2.97
1.51
HD 77601 3603
140.7
7.0
1.82
0.02
0.00
6421 0.44
2.21
HD 79940 3684
117.2
5.9
1.78
0.03
0.00
6397 0.88
2.18
HD 81997 3759
30.4
1.5
1.73
0.01
0.00
6471 3.28
2.72
HD 82554 3795
129.7
6.5
1.63
0.05
0.66
0.460
6272 1.32
1.68
HD 83287 3829
102.5
5.1
1.77
0.08
0.00
7815 2.69
0.74
HD 83962 3859
140.3
7.0
1.72
0.08
0.00
6507 1.53
1.30
HD 84607 3879
93.1
4.7
1.79
0.01
0.00
7000 1.78
1.51
HD 88215 3991
97.5
4.9
1.78
0.05
0.00
6823 3.04
0.85
HD 89254 4042
63.5
3.2
1.81
0.02
0.00
7173 2.42
1.57
Table B .1. continued
Bsin i [km s −1 ] [km s −1 ] rad d −1 rad d −1Star
HR
v sin i
δ v sin i q 2 /q 1 δq 2 /q 1
∆Ω
δ∆Ω
T eff
M V
P/ K
mag
d
HD 89449 4054
17.3
1.7
1.44
0.04
0.45
0.127
6398 3.05
5.44
HD 89569 4061
12.2
0.7
1.57
0.02
0.21
0.070
6290 3.10
7.83
HD 89571 4062
133.9
6.7
1.79
0.07
0.00
HD 90089 4084
56.2
2.8
1.70
0.09
0.35
0.987
6674 3.60
1.19
HD 90589 4102
51.6
2.6
1.93
0.04
0.00
6794 2.95
1.68
HD 96202 4314
93.4
4.7
1.77
0.05
0.00
6747 2.34
1.25
HD 99329 4410
137.9
6.9
1.73
0.04
0.00
6990 2.66
0.68
HD100563 4455
13.5
0.7
1.67
0.04
0.14
0.134
6489 3.61
5.24
HD105452 4623
23.5
1.2
1.59
0.02
0.40
0.151
6839 2.98
3.60
HD106022 4642
77.2
3.9
1.86
0.06
0.00
6651 2.28
1.60
HD107326 4694
132.2
6.6
1.80
0.05
0.00
7107 2.08
0.90
HD108722 4753
97.0
4.8
1.78
0.04
0.00
6490 1.71
1.74
HD109085 4775
60.0
3.0
1.75
0.05
0.00
6813 3.13
1.32
HD109141 4776
135.7
6.8
1.79
0.03
0.00
6881 2.84
0.66
HD110385 4827
105.2
5.3
1.75
0.01
0.00
6717 1.71
1.49
HD110834 4843
133.3
6.7
1.73
0.06
0.00
6244 0.99
1.93
HD111812 4883
63.0
3.2
1.78
0.05
0.00
5623 2.87
2.22
HD112429 4916
119.6
6.0
1.77
0.03
0.00
7126 2.97
0.66
HD114378 4968
19.9
1.0
1.76
0.01
0.00
6324 3.82
3.40
HD115810 5025
99.2
5.0
1.82
0.06
0.00
7185 1.95
1.24
HD116568 5050
36.8
1.8
1.73
0.01
0.00
6485 3.20
2.32
HD118889 5138
140.6
7.0
1.82
0.06
0.00
6951 2.40
0.76
HD119756 5168
63.9
3.2
1.78
0.03
0.00
6809 3.06
1.29
HD120136 5185
15.6
1.0
1.57
0.04
0.31
0.134
6437 3.38
5.13
HD121370 5235
13.5
1.3
1.46
0.03
0.21
0.059
6024 2.36
0.90
HD122066 5257
40.6
2.0
1.81
0.01
0.00
6395 2.18
3.45
HD124780 5337
70.7
3.5
1.80
0.04
0.00
7204 2.21
1.54
HD124850 5338
15.0
0.8
1.91
0.04
0.00
6075 2.85
7.71
HD127739 5434
55.8
2.8
1.81
0.01
0.00
6787 2.20
2.20
HD127821 5436
55.6
2.8
1.75
0.04
0.00
6601 3.76
1.14
HD129153 5473
105.7
5.3
1.78
0.06
0.00
7693 2.70
0.73
HD129502 5487
47.0
2.4
1.80
0.03
0.00
6695 2.94
1.91
HD129926 5497
112.5
5.6
1.74
0.03
0.00
6048 3.91
0.64
HD132052 5570
113.2
5.7
1.77
0.06
0.00
6964 2.19
1.03
HD136359 5700
20.3
1.0
1.78
0.01
0.00
6296 3.02
4.86
HD136751 5716
72.7
3.6
1.77
0.02
0.00
6810 2.31
1.60
HD138917 5788
85.8
5.4
1.99
0.06
0.00
HD139225 5804
104.4
5.2
1.76
0.03
0.00
6960 2.50
0.97
HD139664 5825
71.6
3.6
1.77
0.05
0.00
6681 3.57
0.94
HD142908 5936
75.8
3.8
1.80
0.05
0.00
6849 2.33
1.50
HD143466 5960
141.3
7.1
1.77
0.05
0.00
7235 2.20
0.77
HD147365 6091
72.5
3.6
1.77
0.06
0.00
6657 3.46
0.99
HD147449 6093
76.4
3.8
1.79
0.04
0.00
6973 2.70
1.21
HD147449 6093
77.1
3.9
1.80
0.03
0.00
6973 2.70
1.20
HD148048 6116
84.8
4.2
1.78
0.05
0.00
6731 2.43
1.32
HD150557 6205
61.8
3.1
1.83
0.04
0.00
6959 2.01
2.06
HD151613 6237
47.5
2.4
1.84
0.07
0.00
6630 2.80
2.06
HD155103 6377
57.9
2.9
1.81
0.05
0.00
7150 2.55
1.63
HD156295 6421
107.4
5.4
1.79
0.07
0.00
7818 2.56
0.75
HD160915 6595
12.4
0.6
1.60
0.03
0.23
0.101
6356 3.58
6.07
HD164259 6710
69.3
3.5
1.75
0.04
0.00
6704 2.70
1.44
HD165373 6754
79.9
4.0
1.74
0.03
0.00
6976 2.08
1.54
HD171834 6987
71.3
3.6
1.77
0.04
0.00
6622 2.50
1.58
HD173417 7044
53.9
2.7
1.83
0.04
0.00
6780 1.81
2.74
HD173667 7061
18.0
2.0
1.40
0.02
0.44
0.104
6363 2.78
5.99
HD175317 7126
17.1
0.9
1.58
0.04
0.30
0.136
6563 3.11
5.07
HD175824 7154
53.7
2.7
1.69
0.03
0.15
0.179
6232 1.72
3.43
HD182640 7377
87.3
4.4
1.67
0.06
0.62
0.753
7016 2.46
1.17
HD185124 7460
87.0
4.4
1.71
0.03
0.30
0.604
6680 2.98
1.02
HD186005 7489
149.9
7.5
1.79
0.04
0.00
6988 1.68
0.98
Table B .1. continued
BStarHRv sin i δ v sin i q 2 /q 1 δq 2 /q 1 ∆Ω δ∆Ω T eff M V P/ sin i [km s −1 ] [km s −1 ] rad d −1 rad d −1Table B.2. Stars in cluster fields, FLAMES/UVES observations sin i [km s −1 ] [km s −1 ] rad d −1 rad d −1K
mag
d
HD187532 7553
77.5
3.9
1.75
0.03
0.00
6788 3.34
0.94
HD189245 7631
72.6
3.6
1.74
0.03
0.00
6259 4.06
0.85
HD190004 7657
136.1
6.8
1.80
0.04
0.00
6974 2.47
0.76
HD197692 7936
41.7
2.1
1.62
0.02
0.65
0.286
6587 3.33
1.86
HD199260 8013
13.7
0.7
1.79
0.03
0.00
6213 4.18
4.36
HD201636 8099
58.8
2.9
1.86
0.03
0.00
6700 2.20
2.15
HD203925 8198
70.7
3.5
1.80
0.03
0.00
6845 1.15
2.77
HD205289 8245
57.5
2.9
1.74
0.03
0.00
6525 3.24
1.44
HD206043 8276
134.0
6.7
1.74
0.04
0.00
7092 2.87
0.62
HD207958 8351
69.3
3.5
1.84
0.03
0.00
6747 2.97
1.26
HD210302 8447
13.6
0.7
1.72
0.04
0.00
6465 3.52
5.46
HD210459 8454
147.4
7.4
1.76
0.03
0.00
6376 0.21
2.89
HD213051 8558
48.4
2.4
1.85
0.08
0.00
HD213845 8592
35.7
1.8
1.71
0.02
0.14
0.234
6551 3.35
2.18
HD219693 8859
19.9
1.0
1.81
0.01
0.00
6461 2.82
5.14
HD220657 8905
73.4
3.7
1.74
0.03
0.00
5801 2.41
2.15
Star
Cluster
v sin i
δ v sin i q 2 /q 1 δq 2 /q 1
∆Ω
δ∆Ω
T eff
M V
P/ K
mag
d
NGC 6475 69
NGC 6475
94.9
4.7
1.76
0.01
0.00
6822 3.27
0.75
NGC 6475 41
NGC 6475
80.1
4.0
1.71
0.01
0.41
0.497
6346 3.86
0.76
BD+20 2161
Praesepe
72.7
3.6
1.76
0.01
0.00
6864 3.23
0.99
BD+20 2170
Praesepe
94.4
4.7
1.71
0.01
0.46
0.557
6484 3.69
0.68
Cl* NGC 2632 KW 230
Praesepe
78.9
3.9
1.75
0.01
0.00
6628 3.51
0.85
Cl* IC 2391 L 33
IC 2391
81.1
4.1
1.74
0.01
0.00
6414 3.78
0.77
HD 307938
IC 2602
93.7
4.7
1.78
0.01
0.00
5743 4.64
0.51
HD 308012
IC 2602
45.7
2.3
1.70
0.01
0.36
0.306
6147 4.12
1.23
Cl* IC 4665 V 69
IC 4665
45.1
2.3
1.74
0.01
0.00
6180 4.08
1.26
Cl* IC 4665 V 102
IC 4665
105.0
12.0
1.39
0.01
3.62
0.776
7136 2.93
0.73
Cl* IC 4665 V 97
IC 4665
52.0
2.6
1.90
0.01
0.00
6628 3.51
1.29
A. Reiners: Differential rotation with temperature and rotation
A. Reiners: Differential rotation with temperature and rotation
Acknowledgements. I am thankful to G. Basri for carefully reading the manuscript and for very helpful discussions, and to J. Schmitt for valuable comments on an earlier version of the manuscript. I thank the referee, Dr. John Barnes, for a careful and very constructive report. A.R. has received research funding from the European Commission's Sixth Framework Programme as an Outgoing International Fellow (MOIF-CT-2004-002544).Appendix B: Tables of stars with measured differential rotation.
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| [] |
[
"EUROPEAN ORGANISATION FOR NUCLEAR RESEARCH (CERN)",
"EUROPEAN ORGANISATION FOR NUCLEAR RESEARCH (CERN)"
] | [] | [] | [] | Measurement of W ± -boson and Z-boson production cross-sections in p p collisions at √ s = 2.76 TeV with the ATLAS detectorThe ATLAS CollaborationThe production cross-sections for W ± and Z bosons are measured using ATLAS data corresponding to an integrated luminosity of 4.0 pb −1 collected at a centre-of-mass energy √ s = 2.76 TeV. The decay channels W → ν and Z → are used, where can be an electron or a muon. The cross-sections are presented for a fiducial region defined by the detector acceptance and are also extrapolated to the full phase space for the total inclusive production cross-section. The combined (average) total inclusive cross-sections for the electron and muon channels are: σ tot W + → ν = 2312 ± 26 (stat.) ± 27 (syst.) ± 72 (lumi.) ± 30 (extr.) pb, σ tot W − → ν = 1399 ± 21 (stat.) ± 17 (syst.) ± 43 (lumi.) ± 21 (extr.) pb, σ tot Z→ = 323.4 ± 9.8 (stat.) ± 5.0 (syst.) ± 10.0 (lumi.) ± 5.5(extr.) pb.Measured ratios and asymmetries constructed using these cross-sections are also presented. These observables benefit from full or partial cancellation of many systematic uncertainties that are correlated between the different measurements. | 10.1140/epjc/s10052-019-7399-7 | [
"https://arxiv.org/pdf/1907.03567v2.pdf"
] | 195,833,389 | 1907.03567 | 401826ad88375a7ad0beb6d4ede9687ed97dd2d5 |
EUROPEAN ORGANISATION FOR NUCLEAR RESEARCH (CERN)
9th July 2019 8 Jul 2019
EUROPEAN ORGANISATION FOR NUCLEAR RESEARCH (CERN)
9th July 2019 8 Jul 2019Submitted to: Eur. Phys. J. C Reproduction of this article or parts of it is allowed as specified in the CC-BY-4.0 license.
Measurement of W ± -boson and Z-boson production cross-sections in p p collisions at √ s = 2.76 TeV with the ATLAS detectorThe ATLAS CollaborationThe production cross-sections for W ± and Z bosons are measured using ATLAS data corresponding to an integrated luminosity of 4.0 pb −1 collected at a centre-of-mass energy √ s = 2.76 TeV. The decay channels W → ν and Z → are used, where can be an electron or a muon. The cross-sections are presented for a fiducial region defined by the detector acceptance and are also extrapolated to the full phase space for the total inclusive production cross-section. The combined (average) total inclusive cross-sections for the electron and muon channels are: σ tot W + → ν = 2312 ± 26 (stat.) ± 27 (syst.) ± 72 (lumi.) ± 30 (extr.) pb, σ tot W − → ν = 1399 ± 21 (stat.) ± 17 (syst.) ± 43 (lumi.) ± 21 (extr.) pb, σ tot Z→ = 323.4 ± 9.8 (stat.) ± 5.0 (syst.) ± 10.0 (lumi.) ± 5.5(extr.) pb.Measured ratios and asymmetries constructed using these cross-sections are also presented. These observables benefit from full or partial cancellation of many systematic uncertainties that are correlated between the different measurements.
Introduction
The processes that produce W and Z bosons1 in pp collisions via Drell-Yan annihilation are two of the simplest at hadron colliders to describe theoretically. At lowest order in quantum chromodynamics (QCD), W-boson production proceeds via qq → W and Z-boson production via qq → Z. Therefore, precision measurements of these production cross-sections yield important information about the parton distribution functions (PDFs) for quarks inside the proton. Factorisation theory allows PDFs to be treated separately from the perturbative QCD high-scale collision calculation as functions of the event energy scale, Q, and the momentum fraction of the parton, x, for each parton flavour. Usually PDFs are defined for a particular starting scale Q 0 and can be evolved to other scales via the DGLAP equations [1][2][3][4]. Measurements of on-shell W/Z-boson production probe the PDFs in a range of Q 2 that lies close to m 2 W /Z . The range of x that is probed depends on the centre-of-mass energy, √ s, of the protons and the rapidity coverage of the detector. Each measurement of these production cross-sections at a new value of √ s thus provides information complementary to previous measurements. The combinations of initial partons participating in the production processes of W + ,W − , and Z bosons are different, so each process provides complementary information about the products of different quark PDFs. This paper presents the first measurements of the production cross-sections for W + , W − and Z bosons in pp collisions at √ s = 2.76 TeV. The data were collected by the ATLAS detector at the Large Hadron Collider (LHC) [5] in 2013 and correspond to an integrated luminosity of 4.0 pb −1 . To provide further sensitivity to PDFs, and to reduce the systematic uncertainty in the predictions, ratios of these cross-sections and the charge asymmetry for W-boson production are also presented. The measurements are performed for leptonic (electron or muon) decays of the W and Z bosons, in a defined fiducial region, and also extrapolated to the total cross-section.
Previous measurements of the W-boson and Z-boson production cross-sections in pp collisons at the LHC were performed by the ATLAS and CMS Collaborations at √ s = 5.02 TeV [6], 7 TeV [7, 8], 8 TeV [9,10] and 13 TeV [11,12], and by the PHENIX and STAR Collaborations at the RHIC at √ s = 500 GeV [13,14] and 510 GeV [15]. This is the first measurement at 2.76 TeV. Other measurements of these processes were performed in pp collisons at
ATLAS detector
The ATLAS detector [24] at the LHC covers nearly the entire solid angle around the collision point. It consists of an inner tracking detector surrounded by a thin superconducting solenoid, electromagnetic (EM) and hadronic calorimeters, and a muon spectrometer (MS) incorporating three large superconducting toroid magnets. The inner-detector system (ID) is immersed in a 2 T axial magnetic field and provides charged-particle tracking in the pseudorapidity range |η| < 2.5.2
The high-granularity silicon pixel detector covers the vertex region and typically provides three measurements per track. It is followed by the silicon microstrip tracker, which usually provides eight measurements from eight strip layers. These silicon detectors are complemented by the transition radiation tracker (TRT), which enables radially extended track reconstruction up to |η| = 2.0. The TRT also provides electron identification information based on the fraction of hits (typically 30 in total) above a higher energy-deposit threshold associated with the presence of transition radiation.
The calorimeter system covers the pseudorapidity range |η| < 4.9. Within the region |η| < 3.2, EM calorimetry is provided by barrel and endcap high-granularity lead/liquid-argon (LAr) sampling calorimeters, with an additional thin LAr presampler covering |η| < 1.8 that is used to correct for energy loss in material upstream of the calorimeters. Hadronic calorimetry in this region is provided by the steel/scintillator-tile calorimeter, segmented into three barrel structures with |η| < 1.7, and two copper/LAr hadronic endcap calorimeters. The solid angle coverage is completed with forward copper/LAr and tungsten/LAr calorimeter modules optimised for EM and hadronic measurements, respectively.
The muon spectrometer comprises separate trigger and high-precision tracking chambers measuring the deflection of muons in a magnetic field generated by superconducting air-core toroids. The precision chamber system covers the region |η| < 2.7 with three layers of monitored drift tubes, complemented by cathode strip chambers in the forward region, where the backgrounds are highest. The muon trigger system covers the range |η| < 2.4 with resistive plate chambers in the barrel and thin gap chambers in the endcap regions.
The ATLAS detector selected events using a three-level trigger system [25]. The first-level trigger is implemented in hardware and used a subset of detector information to reduce the event rate to a design 2 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the z-axis along the beam pipe. The x-axis points from the IP to the centre of the LHC ring, and the y-axis points upwards. Cylindrical coordinates (r, φ) are used in the transverse plane, φ being the azimuthal angle around the z-axis. The pseudorapidity is defined in terms of the polar angle θ as η = − ln tan(θ/2). Angular distance is measured in units of ∆R ≡ (∆η) 2 + (∆φ) 2 .
value of at most 75 kHz. This was followed by two software-based triggers that together reduced the event rate to about 200 Hz.
Data and simulation samples
The data used in this measurement were collected in February 2013 during a period when proton beams at the LHC were collided at a centre-of-mass energy of 2.76 TeV. During this running period a typical value of the instantaneous luminosity was 1 × 10 32 cm −2 s −1 , significantly lower than in 7, 8 and 13 TeV data-taking conditions. The typical value of the mean number of collisions per proton bunch crossing (pile-up) µ was 0.3. Only data from stable collisions when the ATLAS detector was fully operational are used, yielding a data sample corresponding to an integrated luminosity of 4.0 pb −1 .
Samples of Monte Carlo (MC) simulated events are used to estimate the signals from W-boson and Z-boson production, and the backgrounds containing prompt leptons: electroweak-diboson production and top-quark pair (tt) production. Background contributions arising from multijet events that do not contain prompt leptons are estimated directly from data, with simulated events used to cross-check these estimations in the muon channel.
Production of single W and Z bosons was simulated using P -B v1 r1556 [26][27][28][29]. The parton showering was performed using P 8.17 [30]. The PDF set used for the simulation was CT10 [31], and the parton shower parameter values were those of the AU2 tune [32]. Additional quantum electrodynamics (QED) emissions from electroweak (EW) vertices and charged leptons were simulated using P ++ v3.52 [33]. Additional samples of simulated W-boson events generated with S 2.1 [34] are used to estimate uncertainties arising from the choice of event generator model. In these S samples, simulation of W-boson production in association with up to two additional partons was performed at next-to-leading order (NLO) in QCD while production of W bosons in association with three or four additional partons was performed at leading order (LO) in QCD. The sample cross-sections were normalised to next-to-next-to-leading-order (NNLO) QCD predictions for the total cross-sections described in Section 8.
P
-B v1 r2330 was used to generate tt samples [35]. These samples had parton showering performed using P 6.428 [36] with parameters corresponding to the Perugia2011C tune [37]. The CT10 PDF set was used. Additional QED final-state radiative corrections were applied using P ++ v3.52 and τ-lepton decays were performed using T v25feb06 [38]. Single production of top quarks is a negligible contribution to this analysis, compared with tt production, so no such samples were generated.
Production of two massive electroweak bosons (WW, Z Z, W Z) was simulated using H 6.5 [39], with multiparton interactions modelled using J 4.13 [40]. The CTEQ6L1 PDF set [41] and AUET2 tune [42] were used for these samples.
Multijet production containing heavy-flavour final states, arising from the production of bb or cc pairs, were simulated using P 8.186. The CTEQ6L1 PDF set and AU2 tune were used. Events were required to contain an electron or muon with transverse momentum p T > 10 GeV and |η| < 2.8.
The detector response to generated events was simulated by passing the events through a model of the ATLAS detector [43] based on G 4 [44]. Additional minimum-bias events generated using P 8.17 and the A2 set of tuned parameters, were overlaid in such a way that the distribution of µ for
Process
Generator
Generator QCD precision Signal Samples W → ν P -B +P 8 NLO Z → + − P -B +P 8 NLO Background Samples W → τν P -B +P 8 NLO Z → τ + τ − P -B +P 8 NLO tt P -B +P 6 NLO WW H LO Z Z H LO W Z H LO bb P 8 LO cc P 8 LO
simulated events reproduced that in the real data. The resulting events were then passed through the same reconstruction software as the real data.
The simulated samples used for the baseline analysis are summarised in Table 1, which shows the generator used for each process together with the order in QCD at which they were generated.
Event selection
This section describes the selection of events consistent with the production of W bosons or Z bosons. The W-boson selection requires events to contain a single charged lepton and large missing transverse momentum. The Z-boson selection requires events to contain two charged leptons with opposite charge and the same flavour.
Events were selected by triggers that required at least one charged electron (muon) with p T > 15 GeV (10 GeV). These thresholds yield an event sample with a uniform efficiency as a function of the E T and p T requirements used subsequently to select the final event sample. The hard-scatter vertex, defined as the vertex with highest sum of squared track transverse momenta (for tracks with p T > 400 MeV), is required to have at least three associated tracks.
Electrons are reconstructed from clusters of energy in the EM calorimeter that are matched to a track reconstructed in the ID. The electron is required to have p T > 20 GeV and |η| < 2.4 (excluding the transition region between barrel and endcap calorimeters of 1.37 < |η| < 1.52). Each electron must satisfy a set of identification criteria designed to suppress misidentified photons or jets. Electrons are required to satisfy the medium selection, following the definition provided in Ref. [45]. This includes requirements on the shower shape in the EM calorimeter, the leakage of the shower into the hadronic calorimeter, the number of hits measured along the track in the ID, and the quality of the cluster-track matching. A Gaussian sum filter [46] algorithm is used to re-fit the tracks and improve the estimated electron track parameters. To suppress background from misidentified objects such as jets, the electron is required to be isolated using calorimeter-based criteria. The sum of the transverse energies of clusters lying within a cone of size ∆R = 0.2 around the centroid of the electron cluster and excluding the core3 must be less than 10% of the electron p T .
Muon candidates are reconstructed by combining tracks reconstructed in the ID with tracks reconstructed in the MS [47]. They are required to have p T > 20 GeV and |η| < 2.4. The muon candidates are also required to be isolated, by requiring that the scalar sum of the p T of additional tracks within a cone of size ∆R = 0.4 around the muon is less than 80% of the muon p T .
The missing transverse momentum vector [48] (E miss T ) is calculated as the negative vector sum of the transverse momenta of electrons and muons, and of the transverse momentum of the recoil. The magnitude of this vector is denoted by E miss T . The recoil vector is obtained by summing the transverse momenta of all clusters of energy measured in the calorimeter, excluding those within ∆R = 0.2 of the lepton candidate. The momentum vector of each cluster is determined by the magnitude and coordinates of the energy deposits; the cluster is assumed to be massless. Cluster energies are initially measured assuming that the energy deposition occurs only through EM interactions, and are then corrected for the different calorimeter responses to hadrons and electromagnetically interacting particles, for losses due to dead material, and for energy that is not captured by the clustering process [49]. The definition of the recoil does not make use of reconstructed jets, to avoid threshold effects. The procedure used to calibrate the recoil closely follows that used in the recent ATLAS measurement of the W-boson mass [50], first correcting the modelling of the overall recoil in simulation and then applying corrections for residual differences in the recoil response and resolution that are derived from Z-boson data and transferred to the W-boson sample.
The W-boson selection requires events to contain exactly one lepton (electron or muon) candidate and have E miss T > 25 GeV. The lepton must match a lepton candidate that met the trigger criteria. The transverse mass, m T , of the W-boson candidate in the event is calculated using the lepton candidate and
E miss T according to m T = 2p T E miss T (1 − cos(φ − φ E miss T )
). The transverse mass in W-boson production events is expected to exhibit a Jacobian peak around the W-boson mass. Thus, requiring that m T > 40 GeV suppresses background processes. After these requirements there are 3914 events in the W → e + ν channel, 2209 events in the W → e −ν channel, 4365 events in the W → µ + ν channel, and 2460 events in the W → µ −ν channel.
The Z-boson selection requires events to contain exactly two lepton candidates with the same flavour and opposite charge. At least one lepton must match a lepton candidate that met the trigger criteria. Background processes are suppressed by requiring that the invariant mass of the lepton pair satisfies 66 < m < 116 GeV. After these requirements there are 430 events in the Z → e + e − channel, and 646 events in the Z → µ + µ − channel.
Background estimation
The background processes that contribute to the sample of events passing the W-boson and Z-boson selections can be separated into two categories: those estimated from MC simulation and theoretical calculations, and those estimated directly from data. The main backgrounds that contribute to the event sample passing the W-boson selection are processes with a τ-lepton decaying into an electron or muon plus neutrinos, leptonic Z-boson decays where only one lepton is reconstructed, and multijet processes.
The main background contribution to the event sample passing the Z-boson selection is production of two massive electroweak bosons.
The backgrounds arising from W → τν, Z → + − , diboson production, and tt production are estimated from the simulated samples described in Section 3. Predictions of the backgrounds to the W-boson and Z-boson production measurements arising from multijet production suffer from large theoretical uncertainties, and therefore the contribution to this background in the W-boson measurement is estimated from data. This is achieved by constructing a shape template for the background using a discriminating variable in a control region and then performing a template fit to the same distribution in the signal region to extract the background contribution. The choice of template variable is motivated by the difference between signal and background and by the available number of events. Previous ATLAS measurements at 7 TeV [7] and 13 TeV [12] found that multijet production makes a background contribution of less than 0.1% for Z-boson measurements; this is therefore neglected.
Electron candidates in multijet background events are typically misidentified candidates produced when jets mimic the signature of an electron, for example when a neutral pion and a charged pion overlap in the detector. Additional candidates can arise from 'non-prompt' electrons produced when a photon converts, and in decays of heavy-flavour hadrons. To construct a control region for the multijet template, a selection is used that differs from the W-boson selection described in Section 4 in only two respects: the medium electron identification criteria are inverted (while keeping the looser identification criteria) and the E miss T requirement is removed. By construction, this control region is statistically independent of the W-boson signal region. A template for the shape of the multijet background in the E miss T distribution is then obtained from that distribution in the control region after subtraction of expected contributions from the signal and other backgrounds determined using MC samples. The normalisation of the multijet background template in the signal region is extracted by performing a χ 2 fit of the E miss T distribution (applying all signal criteria except the requirement on E miss T ) to a sum of the templates for the multijet background, the signal, and all other backgrounds. The normalisation of the signal is allowed to vary freely in the fit as is the multijet background; however, the other backgrounds are only allowed to vary from their expected values by up to 5%, corresponding to the largest level of variation in predicted electroweak-boson production cross-sections obtained from varying the choice of PDF. The normalisation from this fit can then be used together with the inverted selection to construct multijet background distributions in any other variable that is not correlated with the electron identification criteria.
Muon candidates in multijet background events are typically 'non-prompt' muons produced in the decays of hadrons. The multijet background contribution to the W → µν selection is estimated by using the same method as described for the W → eν selection. In this case the control region is defined by inverting the isolation requirement and removing the requirement on m T . The distribution used for the fits is m T .
The overall number of multijet background events is estimated from a fit to the total W-boson sample. Fits to the separate W + -boson and W − -boson samples are used in the evaluation of the systematic uncertainties, as described in Section 7. The final estimated multijet contributions are 30 ± 11 events for W → e + ν and W → e − ν and 2.5 ± 1.9 events for W + → µ + ν and W − → µ − ν. The relative contribution of the multijet events (1%) is lower than in 13 TeV (4%) and 7 TeV (3%) data. This is in agreement with expectations for this lower pile-up running, where the resolution in E miss T is improved compared to the higher pile-up running.
Correction for detector effects
The measurements in this paper are performed within specific fiducial regions and extrapolated to the total W-boson or Z-boson phase space. The fiducial regions are defined by the kinematic and geometric selection criteria given in Table 2; in simulations these are applied at the generator level before the emission of QED final-state radiation from the decay lepton(s) (QED Born level).
The fiducial W-boson/Z-boson production cross-section is obtained from the number of observed events meeting the selection criteria after background contributions are subtracted, N sig W, Z , using the following formula:
σ fid W, Z→ ν, = N sig W, Z C W, Z · L int ,
where L int is the total integrated luminosity of the data samples used for the analysis. The factor C W, Z is the ratio of the number of generated events that satisfy the final selection criteria after event reconstruction to the number of generated events within the fiducial region. It includes the efficiency for triggering, reconstruction and identification of W, Z → ν, + − events falling within the acceptance. The different components of the efficiency are calculated using a mixture of MC simulation and measurements from data.
The total W-boson and Z−boson production cross-sections are obtained using the following formula:
σ tot W, Z→ ν, ≡ σ tot × B(W, Z → ν, ) = N sig W, Z A W, Z · C W, Z · L int .
The factor B(W, Z → ν, ) is the per-lepton branching fraction of the vector boson. The factor A W, Z is the acceptance for W/Z-boson events being studied. It is defined as the fraction of generated events that satisfy the fiducial requirements. This acceptance is determined using MC signal samples, corrected to the generator QED Born level, and is used to extrapolate the measured cross-section in the fiducial region to the full phase space. The central values of A W, Z are around 0.6 for these measurements, compared with 0.5 at √ s = 7 TeV and 0.4 at √ s = 13 TeV, so the fiducial region is closer to the full phase space in this measurement than for those at higher centre-of-mass energies. This is due to a combination of higher p T thresholds for leptons in other measurements, and more-central production of vector bosons at lower
Systematic uncertainties
The systematic uncertainty in the electron reconstruction and identification efficiency is estimated using the tag-and-probe method in 8 TeV data [45,51] and extrapolated to the 2.76 TeV dataset. The extrapolation procedure results in absolute increases of ±2%, due to uncertainties in the effect of the differing pileup conditions in 2.76 TeV data relative to the 8 TeV data. Transverse-momentum-dependent isolation corrections, calculated with the tag-and-probe method in 2.76 TeV data, are very close to 1, so the systematic W-boson fiducial region Z-boson fiducial region p T > 20 GeV p
+,− T > 20 GeV |η | < 2.4 |η +,− | < 2.4 E miss T > 25 GeV 66 < m + − < 116 GeV m T > 40 GeV
uncertainty in the electron isolation requirement is set to the size of the correction itself, that is ±1% for low p T and ±0.3% for higher p T . The electron energy scale has associated statistical uncertainties and systematic uncertainties arising from a possible bias in the calibration method, the choice of generator, the presampler energy scale, and imperfect knowledge of the material in front of the EM calorimeter [52]. The total energy-scale uncertainty is calculated as the sum in quadrature of these components.
Systematic uncertainties associated with the muon momentum can be divided into three major independent categories: momentum resolution of the MS track, momentum resolution of the ID track, and an overall scale uncertainty. The total momentum scale/resolution uncertainty is the sum in quadrature of these components. An η-independent uncertainty of approximately ±1.1% in the muon trigger efficiency, determined using the tag-and-probe method [47] in 2.76 TeV data, is taken into account. Furthermore, a p Tand ηdependent uncertainty in the identification and reconstruction efficiencies of approximately ±0.3 %, derived using the tag-and-probe method on 8 TeV data is applied. The uncertainty in the p T -dependent isolation correction in the muon channel, calculated with the tag-and-probe method in 2.76 TeV data, is about ±0.6% for low p T and ±0.5% for higher p T .
The luminosity uncertainty for the 2.76 TeV data is ±3.1%. This is determined, following the same methodology as was used for the 7 TeV data recorded in 2011 [53], from a calibration of the luminosity scale derived from beam-separation scans performed during the 2.76 TeV operation of the LHC in 2013.
Systematic uncertainties in the E miss T arising from the smearing and bias corrections applied to obtain satisfactory modelling of the recoil [48] affect the C W factors in the W → ν measurement, and are taken into account.
Uncertainties arising from the choice of PDF set are evaluated using the error sets of the initial CT10 PDF set (at 90% confidence level (CL)) and from comparison with the results obtained using the central PDF sets from ABKM09 [54], NNPDF23 [55], and ATLAS-epWZ12 [56]. The effect of this uncertainty on A W + (A W − ) is estimated to be ±1.0% (1.2%), and the effect on A Z is estimated to be ±1.4%. The effect on C W, Z is between ±0.05% and ±0.4% depending on the channel.
A summary of the systematic uncertainties in the C W, Z factors is shown in Table 3. The muon trigger, and electron reconstruction and identification uncertainties are dominant.
Uncertainties arising from the choice of event generator and parton shower models are estimated by comparing results obtained when using S 2.1 signal samples instead of the (nominal) P -B +P 8. The effect of this uncertainty on A W, Z is estimated to be ±0.9%.
The systematic uncertainty in the multijet background estimation can be divided into several components: the normalisation uncertainty from the χ 2 fit, the uncertainty in the modelling of electroweak processses by simulated samples in the fitted region, uncertainty from fit bias due to binning choice, and uncertainty from template shape. The scale normalisation uncertainty from the χ 2 fit is approximately ±13% for the W → eν channel. This uncertainty is neglected in the W → µν channel where the template bias is dominant. The mismodelling uncertainty is estimated by comparison of the fit results for + and − , and for the combined ± candidates. The central value used is 0.5N ± with the uncertainties N + − 0.5N ± and N − − 0.5N ± , where N + is the fitted number of + background events, N − is the fitted number of − and N ± is the fitted total number of ± background events. In the W → eν channel this leads to an uncertainty of ±28% in the multijet background. In the W → µν channel the multijet template normalisation is derived from the fit in the small-m T region, where electroweak contributions are negligible and there are many data events, and this source of systematic error is found to be negligible. The fit-bias uncertainty arising from the choice of bin width is estimated by repeating the fit with different binnings. This component is negligible in the W → µν case and ±15% in the W → eν case. The uncertainty due to a potential bias from template choice is estimated by employing different template selections. For the W → eν channel, different inverted-isolation criteria were investigated. The overall differences are considered negligible. For the W → µν channel, template variations were estimated from fits that use bb + cc MC samples as the multijet templates, leading to an uncertainty of ±75%; this is the largest uncertainty in the multijet background in the W → µν channel.
Combining results and building ratios or asymmetries of results require a model for the correlations of particular systematic uncertainties between different measurements. Correlations arise mostly due to the fact that electrons, muons, and the recoil are reconstructed identically in the different measurements. Further correlations occur due to similarities in the analysis methodology such as the methods of signal and background estimation.
The systematic uncertainties from the electroweak background estimations are treated as uncorrelated between the W-boson and Z-boson measurements, and fully correlated among different flavour decay channels of the W and Z boson. The top-quark background is treated as fully correlated across all W-boson and Z-boson decay channels. The multijet background and recoil-related systematic uncertainties are also treated as fully correlated between all four W-boson decay channels despite there being an expected uncorrelated component, since the statistical uncertainty is dominant in this case.
The systematic uncertainties due to the choice of PDF are treated as fully correlated between all W-boson and Z-boson channels. The uncertainties in electron and muon selection, reconstruction and efficiency are treated as fully correlated between all W-boson and Z-boson channels.
A simplified form of the correlation model with the grouped list of the sources of systematic errors is presented in Table 4.
Source
Muon channel Electron channel
Z W + W − Z W + W − Muon trigger A A A - - - Muon reconstruction/ID A A A - - - Muon energy scale/resolution A A A - - - Muon isolation A A A - - - Electron trigger - - - A * A * A * Electron reconstruction/ID - - - A A A Electron energy scale/resolution - - - A A A Electron isolation - - - A A A Recoil related - A A - A A EW background A B B A B B Top-quark background A A A A A A Multijet background - A A - A A PDF A A A A A A
Results
The numbers of events passing the event selections described in Section 4 are presented in Table 5, together with the estimated background contributions described in Section 5. The distribution of m T for W → ν candidate events is shown in Figure 1, compared with the expected distribution for signal plus backgrounds, where the signal is normalised to the NNLO QCD prediction. Similarly, Figure 2 shows the distribution of m for Z → + − candidate events compared with the expectations for signal. In this case, the background contributions are not shown, because they would not be visible in the figure if included.
The measured fiducial (σ fid ) and total (σ tot ) cross-sections in the electron and muon channels are presented separately in Table 6. For these measurements, the dominant contribution to the systematic uncertainty arises from the luminosity determination.
The results obtained from the electron and muon final states are consistent. The fiducial measurements from electron and muon final states are combined following the procedure described in Ref. [57] and the result is extrapolated to the full phase space to obtain the total cross-section. The total W-boson cross-section is calculated by summing the separate W + and W − cross-sections. The results are shown in Table 7.
Theoretical predictions of the fiducial and total cross-sections are computed for comparison with the measured cross-sections using D 1.5 [58] and F 3.1 [59][60][61][62], which provide calculations at NNLO in the strong-coupling constant, O(α 2 s ), including the boson decays into leptons ( + ν, −ν or + − ) with full spin correlations, finite width and interference effects. These calculations allow kinematic Table 6: Results of the fiducial and total cross-sections measurements of the W + -boson, W − -boson, and Z-boson production cross-sections in the electron and muon channels. The cross-sections are shown with their statistical, systematic and luminosity uncertainties (and extrapolation uncertainty for total cross-section).
Value ± stat. ± syst. ± lumi. (± extr.) Value ± stat. ± syst. ± lumi. (± extr.) [65]. The following input parameters are taken from the Particle Data Group's Review of Particle Properties 2014 edition [66]: the Fermi constant, the masses and widths of W and Z bosons as well as the elements of the CKM matrix. The cross-sections for vector bosons decaying into these leptonic final states are calculated such that they match the definition of the measured cross-sections in the data. Thus, from complete NLO EW corrections, the following components are included: virtual QED and weak corrections, real initial-state radiation (ISR), and interference between ISR and real final-state radiation (FSR) [67]. The calculated effect of these corrections on the cross-sections is (−0.26 ± 0.02)% for σ fid W + , (−0.21 ± 0.03)% for σ fid W − , and (−0.25 ± 0.12)% for σ fid Z . D is used for the central values of the predictions while F is used for the PDF variations and all other systematic variations such as QCD scale and α s . The predictions are calculated Table 7: Combined fiducial and total cross-section measurements for W + -boson, W − -boson and Z-boson production. The cross-sections are shown with their statistical, systematic and luminosity uncertainties (and extrapolation uncertainty for total cross-section).
W + → eν W + → µν σ fid W + [pb] 1416 ± 24 ± 36 ± 44 1438 ± 23 ± 19 ± 45 σ tot W + [pb] 2284 ± 38 ± 58 ± 71 (±30) 2319 ± 36 ± 30 ± 72 (±30) W − → eν W − → µν σ fid W − [
Value ± stat. ± syst. ± lumi. (± extr.) Value ± stat. ± syst. ± lumi. (± extr.)
W + → ν W − → ν σ fid W [pb]
1433 ± 16 ± 17 ± 44 798 ± 12 ± 10 ± 25
σ tot W [pb] 2312 ± 26 ± 27 ± 72 (±30) 1399 ± 21 ± 17 ± 43 (±21) W → ν σ fid W [pb] 2231 ± 20 ± 26 ± 69 σ tot W [pb] 3711 ± 34 ± 43 ± 115 (±51) Z → σ fid Z [pb] 203.7 ± 6.2 ± 3.2 ± 6.3 σ tot Z [pb]
323.4 ± 9.8 ± 5.0 ± 10.0 (±5.5)
using the CT14nnlo [68], NNPDF3.1 [69], MMHT14nnlo68cl [70], ABMP16 [71], HERAPDF2.0 [72], and ATLAS-epWZ12nnlo PDF sets. The dynamic scale, m , and fixed scale, m W , are used as the nominal renormalisation, µ R , and factorisation, µ F , scales for Z and W predictions, respectively.
Theoretical uncertainties in the predictions are also derived from the following sources:
PDF: these uncertainties are evaluated from the variations of the NNLO PDFs according to the recommended procedure for each PDF set. A table with all PDF uncertainties and their central values is shown in Appendix A; the PDF uncertainty from CT14nnlo was rescaled from 90% CL to 68% CL.
Scales: the scale uncertainties are defined by the envelope of the variations in which the scales are changed by factors of two subject to the constraint 0.5 ≤ µ R /µ F ≤ 2.
α s : the uncertainty due to α s was estimated by varying the value of α s used in the CT14nnlo PDF set by ±0.001, corresponding to a 68% CL variation.
The statistical uncertainties in these theoretical predictions are negligible. The numerical values of the predictions for the CT14nnlo PDF set are presented in Table 8. The predictions for the acceptance factor A W, Z can differ by a few percent from those derived from simulated signal samples, this may be due to a poorer description of production of low p T W-bosons by the fixed-order calculations. The predictions are shown in comparison with the combined W-boson and Z-boson production measurements, and with results from pp and pp collisions at other centre-of-mass energies in Figure 3. A comparison of the measurements with predictions from various different PDF sets is presented in Figures 4 and 5. Overall there is good agreement.
Taking ratios of measurements leads to results that have significantly reduced systematic uncertainties due to full or partial cancellation of correlated systematic uncertainties, as discussed in Section 7. The ratios of the fiducial cross-sections for W-boson and Z-boson production are presented, together with the ratio for W + -boson and W − -boson production, in Figure 6. It can be seen that the predictions from the different PDF sets are mostly in good agreement with the measurements. There is a slight (less than two standard deviations) tension between the data and the prediction using the ABMP16 PDF set. The measured values of the ratios are:
R W /Z = 10.95 ± 0.35 (stat.) ± 0.10 (syst.); R W + /W − = 1.797 ± 0.034 (stat.) ± 0.009 (syst.).
The measurement of the ratio R W + /W − is sensitive to the u v and d v valence quark distributions, while the ratio R W /Z can place constraints on the strange quark distributions. A common alternative way of presenting this information is in terms of the charge asymmetry, A , in W-boson production:
A = σ fid W + − σ fid W − σ fid W + + σ fid W − .
This observable also benefits from the cancellation of systematic uncertainties in the same way as the cross-section ratios. The measured value is:
A = 0.285 ± 0.009(stat.) ± 0.002(syst.).
The ratio of measured cross-sections in the electron and muon decay channels provides a test of lepton universality in W-boson decays. The measured ratios are: Table 7. The inner shaded band represents the statistical uncertainty only, the outer band corresponds to the experimental uncertainty (including the luminosity uncertainty). The theory predictions are given with the corresponding PDF (total) uncertainty shown by inner (outer) error bar. Table 7. The inner shaded band represents the statistical uncertainty only, the outer band corresponds to the experimental uncertainty (including the luminosity uncertainty). The theory predictions are given with the corresponding PDF (total) uncertainty shown by inner (outer) error bar. Figure 6: The measured ratio of fiducial cross-sections for (a) W-boson production to Z-boson production, (b) W + -boson production to W − -boson production. The measurements are compared with theoretical predictions at NNLO in QCD based on a selection of different PDF sets. The inner shaded band corresponds to statistical uncertainty while the outer band shows statistical and systematic uncertainties added in quadrature. The theory predictions are given with the corresponding PDF (total) uncertainty shown by inner (outer) error bar.
R W + = σ fid W + →e + ν σ fid W + →µ + ν = 0.985 ± 0.023 (stat.) ± 0.028 (syst.) R W − = σ fid W − →e −ν σ fid W − →µ −ν = 0.+ W (pp) CT14nnlo - W ATLAS ν l → ATLAS / CMS W / ν + l → + ATLAS / CMS W / ν - l → - ATLAS / CMS W / ν (l/e) → CDF W / ν ) µ (e/ → D0 W / ν l → UA1 W ν e → UA2 W ν ) - /e + (e → ± Phenix W / ν ) - µ / + µ ( → ± Phenix W / ν ) - /e + (e → ± Star W / (a) [TeV] s 1 10 ll) [nb] → * γ B(Z/ × * γ Z/ σ 2 − 10 1 − 10 1 ) CT14nnlo p * (p γ Z/ * (pp) CT14nnlo γ Z/ ATLAS ll → * γ ATLAS Z/ ll → * γ CMS Z/ µ µ ee/ → * γ CDF Z/ / ee → * γ D0 Z/ ee → * γ UA1 Z/ µ µ → * γ UA1 Z/ ee → * γ UA2 Z/ ee → * γ Star Z/ (b)R Z = σ fid Z→e + e − σ fid Z→µ + µ − = 0.96 ± 0.06 (stat.) ± 0.05 (syst.)
These results lie within one standard deviation of the Standard Model prediction and previous measurements by ATLAS.
Conclusion
This paper presents measurements of the W → ν and Z → production cross-sections based on about 12 400 W-boson and 1100 Z-boson candidates, after subtracting background events, reconstructed from √ s = 2.76 TeV proton-proton collision data recorded by the ATLAS detector at the LHC, corresponding to integrated luminosity of 4.0 pb −1 . The total inclusive W-boson production cross-sections for the combined electron and muon channels are σ tot W + → ν = 2312 ± 26 (stat.) ± 27 (syst.) ± 72 (lumi.) ± 30 (extr.) pb, σ tot W − → ν = 1399 ± 21 (stat.) ± 17 (syst.) ± 43 (lumi.) ± 21 (extr.) pb, and the total inclusive Z-boson cross-section in the combined electron and muon channels is: σ tot Z→ = 323.4 ± 9.8 (stat.) ± 5.0 (syst.) ± 10.0 (lumi.) ± 5.5(extr.) pb.
The results obtained, and the ratios and charge asymmetries constructed from them, are in agreement with theoretical calculations based on NNLO QCD.
A Theoretical predictions
This appendix presents the theoretical predictions used for comparison with the measurements in the main body of the paper. Table 9 shows the predictions using the MMHT14nnlo68cl, NNPDF31_nnlo_as_0118, ATLASepWZ12, HERAPDF2.0, and ABMP16 PDF sets with associated PDF uncertainties.
The ATLAS Collaboration
The values of C W are approximately 0.67 for the W → eν channels and 0.75 for the W → µν channels. The values of C Z are 0.55 for the Z → e + e − channel and 0.79 for Z → µ + µ − . The C W, Z values are a little higher than for previous measurements at √ s = 7 TeV and √ s = 13 TeV.
Figure 1 :
1The distribution of m T for W → ν candidate events. The expected signal, normalised to the NNLO theoretical predictions, is shown as an unfilled histogram on top of the stacked background predictions. Backgrounds that do not originate from W production are grouped together into the 'Others' histogram. Systematic uncertainties for the signal and background distributions are combined in the shaded band. Systematic uncertainties from the measurement of the integrated luminosity are not included. The lower panel shows the ratio of the data to the prediction.
Figure 2 :
2The distribution of m for Z → + − candidate events. The expected signal, normalised to the NNLO theoretical predictions, is shown as an unfilled histogram. Systematic uncertainties for the signal and background distributions are combined in the shaded band. Systematic uncertainties from the measurement of the integrated luminosity are not included. The background distributions are neglected here, but would not be visible if included. The lower panel shows the ratio of the data to the prediction.
Figure 3 :
3The measured values of (a) σ W × B(W → ν) for W + bosons, W − bosons and their sum and (b) σ Z/γ * × B(Z/γ * → ) for proton-proton and proton-antiproton collisions as a function of √ s. Data points at the same √ s are staggered to improve readibility. All data points are shown together with their total uncertainty. The theoretical calculations are performed at NNLO in QCD using D 1.5 and F 3.1 as described in the text. The theoretical uncertainties are not shown.
Figure 4 :
4NNLO predictions for the fiducial cross-section (a) σ fid W + and (b) σ fid W − for the six PDFs CT14nnlo, MMHT2014, NNPDF3.1, ATLASepWZ12, ABMP16 and, HERApdf2.0 compared with the measured fiducial cross-section as given in
Figure 5 :
5NNLO predictions for the fiducial cross-sections (a) σ fid W and (b) σ fid Z for the six PDFs CT14nnlo, MMHT2014, NNPDF3.1, ATLASepWZ12, ABMP16 and HERApdf2.0 compared with the measured fiducial cross-section as given in
Table 1 :
1Summary of the baseline simulated samples used.
Table 2 :
2Summary of the selection criteria that define the measured fiducial regions.
Table 3 :
3Relative systematic uncertainties (%) in the correction factors C W, Z in different channels.δC/C[%]W + →e + ν W − →e − ν Z→e + e − W + →µ + ν W − →µ − ν Z→µ + µ −Lepton trigger
0.14
0.13
< 0.01
1.07
1.07
0.03
Lepton reconstr. and ident.
2.31
2.33
4.55
0.30
0.32
0.62
Lepton isolation
0.71
0.71
1.41
0.51
0.51
1.01
Lepton scale and resolution
0.44
0.43
0.34
0.05
0.05
0.04
Recoil scale and resolution
0.25
0.20
-
0.22
0.22
-
PDF
0.22
0.29
0.11
0.11
0.20
0.06
MC statistical uncertainty
0.24
0.31
0.30
0.24
0.34
0.43
Total
2.5
2.5
4.8
1.3
1.3
1.3
Table 4 :
4The correlation model for the grouped systematic uncertainties for the measurements of W-boson and Z-boson production. The entries in different rows are uncorrelated with each other. Entries in a row with the same letter are fully correlated. Entries in a row with a starred letter are mostly correlated with the entries with the same letter (most of the individual sources of uncertainties within a group are taken as correlated). Entries with different letters in a row are either fully or mostly uncorrelated with each other.
Table 5 :
5The numbers of observed candidate events with the estimated numbers of selected electroweak (EW) plus top, and multijet background events, together with their total uncertainty. In addition, the number of background-subtracted signal events is shown with the first uncertainty given being statistical and the second uncertainty being the total systematic uncertainty, obtained by summing in quadrature the EW+top and multijet uncertainties. Uncertainties shown as ±0.0 have a magnitude less than 0.05.Measurement Observed Background Background Background-subtracted
Channel
candidates (EW + top)
(Multijet)
data N
sig
W
W + → e + ν
3914
108 ± 6
30 ± 11
3776 ± 63 ± 12
W − → e −ν
2209
74.2 ± 3.3
30 ± 11
2105 ± 47 ± 12
W + → µ + ν
4365
152 ± 7
2.5 ± 1.9
4210 ± 66 ± 7
W − → µ −ν
2460
108 ± 4
2.5 ± 1.9
2350 ± 50 ± 5
Z → e + e −
430
1.3 ± 0.0
-
428.7 ± 20.7 ± 0.0
Z → µ + µ −
646
1.6 ± 0.1
-
644.4 ± 25.4 ± 0.1
bosons. The calculation was done in the G µ EW schemepb]
789 ± 18 ± 20 ± 25
799 ± 17 ± 11 ± 25
σ tot
W − [pb]
1385 ± 31 ± 36 ± 43 (±21)
1402 ± 30 ± 19 ± 44 (±21)
Z → ee
Z → µµ
σ fid
Z [pb]
197.6 ± 9.6 ± 9.5 ± 6.1
205.6 ± 8.1 ± 2.6 ± 6.4
σ tot
Z [pb]
313.6 ± 15.2 ± 15.0 ± 9.7 (±5.3)
326.3 ± 12.9 ± 4.1 ± 10.1 (±5.5)
requirements to be implemented for direct comparison with experimental data. The procedure used follows
that used for the previous ATLAS measurement at
√
s = 7 TeV [7].
Corrections for NLO EW effects are calculated with F
3.1 for the Z bosons and with S
[63, 64]
for the W
Table 8 :
8The predictions, using the CT14nnlo PDF set, for the cross-sections measured.The calculations are
NNLO QCD, inner uncert.: PDF only) NNLO QCD, inner uncert.: PDF only)σ
ATLAS
-1
= 2.76 TeV, 4.0 pb
s
total uncertainty
±
Data
stat. uncertainty
±
Data
ABMP16
CT14nnlo
NNPDF3.1
MMHT14nnlo68CL
ATLAS-epWZ12nnlo
HERAPDF2.0nnlo
(Z
fid
σ
/
±
W
fid
σ
=
W/Z
R
(a)
1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9
-
W
fid
σ
/
+
W
fid
σ
ATLAS
-1
= 2.76 TeV, 4.0 pb
s
total uncertainty
±
Data
stat. uncertainty
±
Data
ABMP16
CT14nnlo
NNPDF3.1
MMHT14nnlo68CL
ATLAS-epWZ12nnlo
HERAPDF2.0nnlo
(-
W
fid
σ
/
+
W
fid
σ
=
-
/W
+
W
R
(b)
Table 9 :
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Quantity
MMHT14 NNPDF31 ATLASepWZ12 HERAPDF20 ABMP16
σ fid
W +
1397 +29
−30
1428 +24
−24
1375 +34
−30
1429 +91
−49
1397 +14
−14
σ fid
W −
773 +17
−20
778 +14
−14
784 +19
−19
806 +31
−21
746 +9
−9
σ fid
Z
199 +4
−4
203 +4
−4
199 +4
−4
199 +11
−5
198.6 +2.0
−2.0
σ tot
W +
2138 +43
−45
2271 +36
−36
2086 +54
−47
2140 +140
−70
2214 +21
−21
σ tot
W −
1295 +28
−33
1330 +22
−22
1296 +48
−29
1338 +52
−32
1283 +16
−16
σ tot
Z
308 +6
−6
313 +5
−5
308 +6
−5
312 +16
−7
305.7 +3.0
−3.0
In this paper it is implicit that Z boson refers to Z/γ * bosons.
The core of the shower is the contribution within ∆η × ∆φ = 0.125 × 0.175 around the cluster barycentre.
AcknowledgementsWe thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently.[73].Appendix
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Arling 46 , A.J. Armbruster 36 , L.J. Armitage 92 , A. Armstrong 171 , O. Arnaez 167 , H. Arnold 120 , A. Artamonov 111,* , G. Artoni 135 , S. Artz 99 , S. Asai 163 , N. Asbah 59 , E.M. Asimakopoulou 172 , L. Asquith 156 , K. Assamagan 29 , R. Astalos 28a , R.J. Atkin 33a , M. Atkinson 173 , N.B. Atlay 151 , H. Atmani 132 , K. Augsten 142 , G. Avolio 36 , R. Avramidou 60a , M.K. Ayoub 15a , A.M. Azoulay 168b , G. Azuelos 109,ax , M.J. Baca 21 , H. Bachacou 145 , K. Bachas 67a,67b , M. Backes 135 , F. Backman 45a,45b , P. Bagnaia 72a,72b , M. Bahmani 84 , H. Bahrasemani 152 , A.J. Bailey 174 , V.R. Bailey 173 , J.T. Baines 144 , M. Bajic 40 , C. Bakalis 10 , O.K. Baker 183 , P.J. Bakker 120 , D. Bakshi Gupta 8 , S. Balaji 157 , E.M. Baldin 122b,122a , P. Balek 180 , F. Balli 145 , W.K. Balunas 135 , J. Balz 99 , E. Banas 84 , A. Bandyopadhyay 24 , Sw. Banerjee 181,i , A.A.E. Bannoura 182 , L. Barak 161 , W.M. Barbe 38 , E.L. Barberio 104 , D. Barberis 55b,55a , M. Barbero 101 , T. 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Brickwedde 99 , D.L. Briglin 21 , D. Britton 57 , D. Britzger 115 , I. Brock 24 , R. Brock 106 , G. Brooijmans 39 , W.K. Brooks 147b , E. Brost 121 , J.H Broughton 21 , P.A. Bruckman de Renstrom 84 , D. Bruncko 28b , A. Bruni 23b , G. Bruni 23b , L.S. Bruni 120 , S. Bruno 73a,73b , B.H. Brunt 32 , M. Bruschi 23b , N. Bruscino 139 , P. Bryant 37 , L. Bryngemark 96 , T. Buanes 17 , Q. Buat 36 , P. Buchholz 151 , A.G. Buckley 57 , I.A. Budagov 79 , M.K. Bugge 134 , F. Bührer 52 , O. Bulekov 112 , T.J. Burch 121 , S. Burdin 90 , C.D. Burgard 120 , A.M. Burger 129 , B. Burghgrave 8 , K. Burka 84 , J.T.P. Burr 46 , J.C. Burzynski 102 , V. Büscher 99 , E. Buschmann 53 , P.J. Bussey 57 , J.M. Butler 25 , C.M. Buttar 57 , J.M. Butterworth 94 , P. Butti 36 , W. Buttinger 36 , A. Buzatu 158 , A.R. Buzykaev 122b,122a , G. Cabras 23b,23a , S. Cabrera Urbán 174 , D. Caforio 56 , H. Cai 173 , V.M.M. Cairo 153 , O. Cakir 4a , N. Calace 36 , P. Calafiura 18 , A. Calandri 101 , G. Calderini 136 , P. Calfayan 65 , G. Callea 57 , L.P. Caloba 80b , S. Calvente Lopez 98 , D. Calvet 38 , S. Calvet 38 , T.P. Calvet 155 , M. Calvetti 71a,71b , R. Camacho Toro 136 , S. Camarda 36 , D. Camarero Munoz 98 , P. Camarri 73a,73b , D. Cameron 134 , R. Caminal Armadans 102 , C. Camincher 36 , S. Campana 36 , M. Campanelli 94 , A. Camplani 40 , A. Campoverde 151 , V. Canale 69a,69b , A. Canesse 103 , M. Cano Bret 60c , J. Cantero 129 , T. Cao 161 , Y. Cao 173 , M.D.M. Capeans Garrido 36 , M. Capua 41b,41a , R. Cardarelli 73a , F.C. Cardillo 149 , G. Carducci 41b,41a , I. Carli 143 , T. Carli 36 , G. Carlino 69a , B.T. Carlson 139 , L. Carminati 68a,68b , R.M.D. Carney 45a,45b , S. Caron 119 , E. Carquin 147b , S. Carrá 46 , J.W.S. Carter 167 , M.P. Casado 14,e , A.F. Casha 167 , D.W. Casper 171 , R. Castelijn 120 , F.L. Castillo 174 , V. Castillo Gimenez 174 , N.F. Castro 140a,140e , A. Catinaccio 36 , J.R. Catmore 134 , A. Cattai 36 , J. Caudron 24 , V. Cavaliere 29 , E. Cavallaro 14 , M. Cavalli-Sforza 14 , V. Cavasinni 71a,71b , E. Celebi 12b , F. Ceradini 74a,74b , L. Cerda Alberich 174 , K. Cerny 130 , A.S. Cerqueira 80a , A. Cerri 156 , L. Cerrito 73a,73b , F. Cerutti 18 , A. Cervelli 23b,23a , S.A. Cetin 12b , D. Chakraborty 121 , S.K. Chan 59 , W.S. Chan 120 , W.Y. Chan 90 , J.D. Chapman 32 , B. Chargeishvili 159b , D.G. Charlton 21 , T.P. Charman 92 , C.C. Chau 34 , S. Che 126 , A. Chegwidden 106 , S. Chekanov 6 , S.V. Chekulaev 168a , G.A. Chelkov 79,aw , M.A. Chelstowska 36 , B. Chen 78 , C. Chen 60a , C.H. Chen 78 , H. Chen 29 , J. Chen 60a , J. Chen 39 , S. Chen 137 , S.J. Chen 15c , X. Chen 15b,av , Y. Chen 82 , Y-H. Chen 46 , H.C. Cheng 63a , H.J. Cheng 15a,15d , A. Cheplakov 79 , E. Cheremushkina 123 , R. Cherkaoui El Moursli 35e , E. Cheu 7 , K. Cheung 64 , T.J.A. Chevalérias 145 , L. Chevalier 145 , V. Chiarella 51 , G. Chiarelli 71a , G. Chiodini 67a , A.S. Chisholm 36,21 , A. Chitan 27b , I. Chiu 163 , Y.H. Chiu 176 , M.V. Chizhov 79 , K. Choi 65 , A.R. Chomont 72a,72b , S. Chouridou 162 , Y.S. Chow 120 , M.C. Chu 63a , X. Chu 15a , J. Chudoba 141 , A.J. Chuinard 103 , J.J. Chwastowski 84 , L. Chytka 130 , K.M. Ciesla 84 , D. Cinca 47 , V. Cindro 91 , I.A. Cioară 27b , A. Ciocio 18 , F. Cirotto 69a,69b , Z.H. Citron 180 , M. Citterio 68a , D.A. Ciubotaru 27b , B.M. Ciungu 167 , A. Clark 54 , M.R. Clark 39 , P.J. Clark 50 , C. Clement 45a,45b , Y. Coadou 101 , M. Cobal 66a,66c , A. Coccaro 55b , J. Cochran 78 , H. Cohen 161 , A.E.C. Coimbra 36 , L. Colasurdo 119 , B. Cole 39 , A.P. Colijn 120 , J. Collot 58 , P. Conde Muiño 140a,f , E. Coniavitis 52 , S.H. Connell 33b , I.A. Connelly 57 , S. Constantinescu 27b , F. Conventi 69a,ay , A.M. Cooper-Sarkar 135 , F. Cormier 175 , K.J.R. Cormier 167 , L.D. Corpe 94 , M. Corradi 72a,72b , E.E. Corrigan 96 , F. Corriveau 103,ad , A. Cortes-Gonzalez 36 , M.J. Costa 174 , F. Costanza 5 , D. Costanzo 149 , G. Cowan 93 , J.W. Cowley 32 , J. Crane 100 , K. Cranmer 124 , S.J. Crawley 57 , R.A. Creager 137 , S. Crépé-Renaudin 58 , F. Crescioli 136 , M. Cristinziani 24 , V. Croft 120 , G. Crosetti 41b,41a , A. Cueto 5 , T. Cuhadar Donszelmann 149 , A.R. Cukierman 153 , S. Czekierda 84 , P. Czodrowski 36 ,
Huseynov 79,af , J. Huston 106 , J. Huth 59 , R. Hyneman 105 , S. Hyrych 28a , G. Iacobucci 54 , G. Iakovidis 29 , I. Ibragimov 151 , L. Iconomidou-Fayard 132 , Z. Idrissi 35e , P.I. Iengo 36 , R. Ignazzi 40 , O. Igonkina 120,z , R. Iguchi 163 , T. Iizawa 54 , Y. Ikegami 81 , M. Ikeno 81 , D. Iliadis 162 , N. Ilic 119 , F. Iltzsche 48 , G. Introzzi 70a,70b , M. Iodice 74a , K. Iordanidou 168a , V. Ippolito 72a,72b , M.F. Isacson 172 , M. Ishino 163 , M. Ishitsuka 165 , W. Islam 129 , C. Issever 135 , S. Istin 160 , F. Ito 169 , J.M. Iturbe Ponce 63a , R. Iuppa 75a,75b , A. Ivina 180 , H. Iwasaki 81. E F Kay, N. Jeong ; E. Kneringer 76 , E.B.F.G. KnoopsSotiropoulou 71a,71b , S. Sottocornola 70a,70b , R. Soualah 66a,66c,g , A.M. Soukharev 122b,122a , D. South 46 , S. Spagnolo 67a,67b , M. Spalla 115 , M. Spangenberg 178 , F. Spanò 93 , D. Sperlich 52 , T.M. Spieker 61a , R. Spighi 23b , G. Spigo 36 , M. Spina 156 , D.P. Spiteri 57 , M. Spousta 143 , A. Stabile 68a,68b , B.L. Stamas 121 , R. Stamen 61a , M. Stamenkovic 120 , E. Stanecka 84 , R.W. Stanek 6 , B. Stanislaus 135 , M.M. Stanitzki. A.L. Steinhebel 131 , B. Stelzer 152 , H.J. Stelzer 139 , O. Stelzer-Chilton 168a , H. Stenzel 56 , T.J. Stevenson 156 , G.A. Stewart 36 , M.C. Stockton 36 , G. Stoicea 27b , M. Stolarski 140a , P. Stolte 53 , S. Stonjek 115 , A. Straessner 48 , J. Strandberg 154 , S. Strandberg 45a,45b , M. Strauss 128 , P. Strizenec 28b , R. Ströhmer 177 , D.M. Strom 131 , R. Stroynowski 42 , A. Strubig 50 , S.A. Stucci 29 , B. Stugu 17 , J. Stupak 128 , N.A. Styles 46 , D. Su 153 , S. Suchek 61a , V.V. Sulin 110 , M.J. Sullivan 90 , D.M.S. Sultan 54 , S. Sultansoy 4c , T. Sumida 85 , S. Sun 105 , X. Sun 3 , K. Suruliz 156 , C.J.E. Suster 157 , M.R. Sutton 156 , S. Suzuki 81 , M. Svatos 141 , M. Swiatlowski 37 , S.P. Swift 2 , T. Swirski 177 , A. Sydorenko 99 , I. Sykora 28a , M. Sykora 143 , T. Sykora 143 , D. Ta 99 , K. Tackmann 46,y , J. Taenzer 161 , A. Taffard 171 , R. Tafirout 168a , H. Takai 29 , R. Takashima 86 , K. Takeda 82 , T. Takeshita 150 , E.P. Takeva 50 , Y. Takubo 81 , M. Talby 101 , A.A. Talyshev 122b,122a , N.M. Tamir 161 , J. Tanaka 163 , M. Tanaka 165 , R. Tanaka 132 , S. Tapia Araya 173 , S. Tapprogge 99 , A. Tarek Abouelfadl Mohamed 136 , S. Tarem 160 , G. Tarna 27b,c , G.F. Tartarelli 68a , P. Tas 143 , M. Tasevsky 141 , T. Tashiro 85 , E. Tassi 41b,41a , A. Tavares Delgado 140a,140b , Y. Tayalati 35e , A.J. Taylor 50 , G.N. Taylor 104 , W. Taylor 168b , A.S. Tee 89 , R. Teixeira De Lima 153Y. Smirnov; t , O. Smirnova; G.H. Stark; J. Stark 58 , S.H Stark34J. Walder. R. Walker 114 , S.D. Walker 93 , W. Walkowiak 151 , V. Wallangen 45a,45b , A.M. Wang 59 , C. Wang 60b , F. Wang 181 , H. Wang 18 , H. Wang 3 , J. Wang 157 , J. Wang 61b , P. Wang 42 , Q. Wang 128 , R.-J. Wang 99 , R. Wang 60a , R. Wang 6 , S.M. Wang 158 , W.T. Wang 60a , W. Wang 15c,ae , W.X. Wang 60a,ae , Y. Wang 60a,am , Z. Wang 60c , C. Wanotayaroj 46 , A. Warburton 103 , C.P. Ward 32 , D.R. Wardrope 94 , N. Warrack 57 , A. Washbrook 50 , A.T. Watson 21 , M.F. Watson 21 , G. Watts 148 , B.M. Waugh 94 , A.F. Webb 11 , S. Webb 99 , C. Weber 183 , M.S. Weber 20 , S.A. Weber 34 , S.M. Weber 61a , A.R. Weidberg 135 , J. Weingarten 47 , M. Weirich 99 , C. Weiser 52 , P.S. Wells 36 , T. Wenaus 29 , T. Wengler 36 , S. Wenig 36 , N. Wermes 24 , M.D. Werner 78 , M. Wessels 61a , T.D. Weston 20 , K. Whalen 131 , N.L. Whallon 148 , A.M. Wharton 89 , A.S. White 105 , A. White 8 , M.J. White 1 , D. Whiteson 171 , B.W. Whitmore 89 , F.J. Wickens 144 , W. Wiedenmann 181 , M. Wielers 144 , N. Wieseotte 99 , C. Wiglesworth 40 , L.A.M. Wiik-Fuchs 52 , F. Wilk 100 , H.G. Wilkens 36 , L.J. Wilkins 93 , H.H. Williams 137 , S. Williams 32 , C. Willis 106 , S. Willocq 102 , J.A. Wilson 21 , I. Wingerter-Seez 5 , E. Winkels 156 , F. Winklmeier 131 , O.J. Winston 156 , B.T. Winter 52 , M. Wittgen 153 , M. Wobisch 95 , A. Wolf 99 , T.M.H. Wolf 120 , R. Wolff 101 , R.W. Wölker 135 , J. Wollrath 52 , M.W. Wolter 84 , H. Wolters 140a,140c , V.W.S. Wong 175 , N.L. Woods 146 , S.D. Worm 21 , B.K. Wosiek 84 , K.W. Woźniak 84 , K. Wraight 57 , S.L. Wu 181 , X. Wu 54 , Y. Wu 60a , T.R. Wyatt 100 , B.M. Wynne 50 , S. Xella 40 , Z. Xi 105 , L. Xia 178 , D. Xu 15a , H. Xu 60a,c , L. Xu 29 , T. Xu 145 , W. Xu 105 , Z. Xu 60b , Z. Xu 153 , B. Yabsley 157 , S. Yacoob 33a , K. Yajima 133 , D.P. Yallup 94 , D. Yamaguchi 165 , Y. Yamaguchi 165 , A. Yamamoto 81 , T. Yamanaka 163 , F. Yamane 82 , M. Yamatani 163 , T. Yamazaki 163 , Y. Yamazaki 82 , Z. Yan 25 , H.J. Yang 60c,60d , H.T. Yang 18 , S. Yang 77 , X. Yang 60b,58 , Y. Yang. Y. Yasu 81 , E. Yatsenko 60c,60d , J. Ye 42 , S. Ye 29 , I. Yeletskikh 79 , M.R. Yexley 89 , E. Yigitbasi 25 , K. Yorita 179 , K. Yoshihara 137 , C.J.S. Young 36 , C. Young 153 , J. Yu 78 , R. Yuan 60b , X. Yue 61a , S.P.Y. Yuen 24 , B. Zabinski 84 , G. Zacharis 10 , E. Zaffaroni 54 , J. Zahreddine 136 , A.M. Zaitsev 123,ao , T. Zakareishvili 159b , N. Zakharchuk 34 , S. Zambito 59 , D. Zanzi 36 , D.R. Zaripovas 57 , S.V. Zeißner 47 , C. Zeitnitz 182 , G. Zemaityte 135 , J.C. Zeng 173 , O. Zenin 123 , D. Zerwas 132 , M. Zgubič 135 , D.F. Zhang 15b , F. Zhang 181 , G. Zhang 60a , G. Zhang 15b , H. Zhang 15c , J. Zhang 6 , L. Zhang 15c , L. Zhang 60a , M. Zhang 173 , R. Zhang 60a , R. Zhang 24 , X. Zhang 60b , Y. Zhang 15a,15d , Z. Zhang 63a , Z. Zhang 132 , P. Zhao 49 , Y. Zhao 60b , Z. Zhao 60a , A. Zhemchugov 79 , Z. Zheng 105 , D. Zhong 173 , B. Zhou 105 , C. Zhou 181 , M.S. Zhou 15a,15d , M. Zhou 155 , N. Zhou 60c , Y. Zhou 7 , C.G. Zhu 60b , H.L. Zhu 60a , H. Zhu 15a , J. Zhu 105 , Y. Zhu 60a , X. Zhuang 15a , K. Zhukov 110 , V. Zhulanov 122b,122a , D. Zieminska 65 , N.I. Zimine 79 , S. Zimmermann 52 , Z. Zinonos 115 , M. Ziolkowski 151 , G. Zobernig 181 , A. Zoccoli 23b,23a , K. Zoch 53 , T.G. Zorbas 149 , R. Zou 37 , L. Zwalinski 36M.J. Da Cunha Sargedas De Sousa 60b , J.V. Da Fonseca Pinto 80b , C. Da Via 100 , W. Dabrowski 83a , T. Dado 28a , S. Dahbi 35e , T. Dai 105 , C. Dallapiccola 102 , M. Dam 40 , G. D'amen 23b,23a , V. D'Amico 74a,74b , J. Damp 99 , J.R. Dandoy 137 , M.F. Daneri 30 , N.P. Dang 181 , N.D Dann 100 , M. Danninger 175 , V. Dao 36 , G. Darbo 55b , O. Dartsi 5 , A. Dattagupta 131 , T. Daubney 46 , S. D'Auria 68a,68b , W. Davey 24 , C. David 46 , T. Davidek 143 , D.R. Davis 49 , I. Dawson 149 , K. De 8 , R. De Asmundis 69a , M. De Beurs 120 , S. De Castro 23b,23a , S. De Cecco 72a,72b , N. De Groot 119 , P. de Jong 120 , H. De la Torre 106 , A. De Maria 15c , D. De Pedis 72a , A. De Salvo 72a , U. De Sanctis 73a,73b , M. De Santis 73a,73b , A. De Santo 156 , K. De Vasconcelos Corga 101 , J.B. De Vivie De Regie 132 , C. Debenedetti 146 , D.V. Dedovich 79 , A.M. Deiana 42 , M. Del Gaudio 41b,41a , J. Del Peso 98 , Y. Delabat Diaz 46 , D. Delgove 132 , F. Deliot 145,q , C.M. Delitzsch 7 , M. Della Pietra 69a,69b , D. Della Volpe 54 , A. Dell'Acqua 36 , L. Dell'Asta 73a,73b , M. Delmastro 5 , C. Delporte 132 , P.A. Delsart 58 , D.A. DeMarco 167 , S. Demers 183 , M. Demichev 79 , G. Demontigny 109 , S.P. Denisov 123 , D. Denysiuk 120 , L. D'Eramo 136 , D. Derendarz 84 , J.E. Derkaoui 35d , F. Derue 136 , P. Dervan 90 , K. Desch 24 , C. Deterre 46 , K. Dette 167 , C. Deutsch 24 , M.R. Devesa 30 , P.O. Deviveiros 36 , A. Dewhurst 144 , S. Dhaliwal 26 , F.A. Di Bello 54 , A. Di Ciaccio 73a,73b , L. Di Ciaccio 5 , W.K. Di Clemente 137 , C. Di Donato 69a,69b , A. Di Girolamo 36 , G. Di Gregorio 71a,71b , B. Di Micco 74a,74b , R. Di Nardo 102 , K.F. Di Petrillo 59 , R. Di Sipio 167 , D. Di Valentino 34 , C. Diaconu 101 , F.A. Dias 40 , T. Dias Do Vale 140a , M.A. Diaz 147a , J. Dickinson 18 , E.B. Diehl 105 , J. Dietrich 19 , S. Díez Cornell 46 , A. Dimitrievska 18 , W. Ding 15b , J. Dingfelder 24 , F. Dittus 36 , F. Djama 101 , T. Djobava 159b , J.I. Djuvsland 17 , M.A.B. Do Vale 80c , M. Dobre 27b , D. Dodsworth 26 , C. Doglioni 96 , J. Dolejsi 143 , Z. Dolezal 143 , M. Donadelli 80d , J. Donini 38 , A. D'onofrio 92 , M. D'Onofrio 90 , J. Dopke 144 , A. Doria 69a , M.T. Dova 88 , A.T. Doyle 57 , E. Drechsler 152 , E. Dreyer 152 , T. Dreyer 53 , A.S. Drobac 170 , Y. Duan 60b , F. Dubinin 110 , M. Dubovsky 28a , A. Dubreuil 54 , E. Duchovni 180 , G. Duckeck 114 , A. Ducourthial 136 , O.A. Ducu 109 , D. Duda 115 , A. Dudarev 36 , A.C. Dudder 99 , E.M. Duffield 18 , L. Duflot 132 , M. Dührssen 36 , C. Dülsen 182 , M. Dumancic 180 , A.E. Dumitriu 27b , A.K. Duncan 57 , M. Dunford 61a , A. Duperrin 101 , H. Duran Yildiz 4a , M. Düren 56 , A. Durglishvili 159b , D. Duschinger 48 , B. Dutta 46 , D. Duvnjak 1 , G.I. Dyckes 137 , M. Dyndal 36 , S. Dysch 100 , B.S. Dziedzic 84 , K.M. Ecker 115 , R.C. Edgar 105 , T. Eifert 36 , G. Eigen 17 , K. Einsweiler 18 , T. Ekelof 172 , H. El Jarrari 35e , M. El Kacimi 35c , R. El Kosseifi 101 , V. Ellajosyula 172 , M. Ellert 172 , F. Ellinghaus 182 , A.A. Elliot 92 , N. Ellis 36 , J. Elmsheuser 29 , M. Elsing 36 , D. Emeliyanov 144 , A. Emerman 39 , Y. Enari 163 , J.S. Ennis 178 , M.B. Epland 49 , J. Erdmann 47 , A. Ereditato 20 , M. Errenst 36 , M. Escalier 132 , C. Escobar 174 , O. Estrada Pastor 174 , E. Etzion 161 , H. Evans 65 , A. Ezhilov 138 , F. Fabbri 57 , L. Fabbri 23b,23a , V. Fabiani 119 , G. Facini 94 , R.M. Faisca Rodrigues Pereira 140a , R.M. Fakhrutdinov 123 , S. Falciano 72a , P.J. Falke 5 , S. Falke 5 , J. Faltova 143 , Y. Fang 15a , Y. Fang 15a , G. Fanourakis 44 , M. Fanti 68a,68b , A. Farbin 8 , A. Farilla 74a , E.M. Farina 70a,70b , T. Farooque 106 , S. Farrell 18 , S.M. Farrington 178 , P. Farthouat 36 , F. Fassi 35e , P. Fassnacht 36 , D. Fassouliotis 9 , M. Faucci Giannelli 50 , W.J. Fawcett 32 , L. Fayard 132 , O.L. Fedin 138,o , W. Fedorko 175 , M. Feickert 42 , S. Feigl 134 , L. Feligioni 101 , A. Fell 149 , C. Feng 60b , E.J. Feng 36 , M. Feng 49 , M.J. Fenton 57 , A.B. Fenyuk 123 , J. Ferrando 46 , A. Ferrante 173 , A. Ferrari 172 , P. Ferrari 120 , R. Ferrari 70a , D.E. Ferreira de Lima 61b , A. Ferrer 174 , D. Ferrere 54 , C. Ferretti 105 , F. Fiedler 99 , A. Filipčič 91 , F. Filthaut 119 , K.D. Finelli 25 , M.C.N. Fiolhais 140a , L. Fiorini 174 , F. Fischer 114 , W.C. Fisher 106 , I. Fleck 151 , P. Fleischmann 105 , R.R.M. Fletcher 137 , T. Flick 182 , B.M. Flierl 114 , L.F. Flores 137 , L.R. Flores Castillo 63a , F.M. Follega 75a,75b , N. Fomin 17 , J.H. Foo 167 , G.T. Forcolin 75a,75b , A. Formica 145 , F.A. Förster 14 , A.C. Forti 100 , A.G. Foster 21 , M.G. Foti 135 , D. Fournier 132 , H. Fox 89 , P. Francavilla 71a,71b , S. Francescato 72a,72b , M. Franchini 23b,23a , S. Franchino 61a , D. Francis 36 , L. Franconi 20 , M. Franklin 59 , A.N. Fray 92 , B. Freund 109 , W.S. Freund 80b , E.M. Freundlich 47 , D.C. Frizzell 128 , D. Froidevaux 36 , J.A. Frost 135 , C. Fukunaga 164 , E. Fullana Torregrosa 174 , E. Fumagalli 55b,55a , T. Fusayasu 116 , J. Fuster 174 , A. Gabrielli 23b,23a , A. Gabrielli 18 , G.P. Gach 83a , S. Gadatsch 54 , P. Gadow 115 , G. Gagliardi 55b,55a , L.G. Gagnon 109 , C. Galea 27b , B. Galhardo 140a , G.E. Gallardo 135 , E.J. Gallas 135 , B.J. Gallop 144 , P. Gallus 142 , G. Galster 40 , R. Gamboa Goni 92 , K.K. Gan 126 , S. Ganguly 180 , J. Gao 60a , Y. Gao 90 , Y.S. Gao 31,l , C. García 174 , J.E. García Navarro 174 , J.A. García Pascual 15a , C. Garcia-Argos 52 , M. Garcia-Sciveres 18 , R.W. Gardner 37 , N. Garelli 153 , S. Gargiulo 52 , V. Garonne 134 , A. Gaudiello 55b,55a , G. Gaudio 70a , I.L. Gavrilenko 110 , A. Gavrilyuk 111 , C. Gay 175 , G. Gaycken 24 , E.N. Gazis 10 , A.A. Geanta 27b , C.N.P. Gee 144 , J. Geisen 53 , M. Geisen 99 , M.P. Geisler 61a , C. Gemme 55b , M.H. Genest 58 , C. Geng 105 , S. Gentile 72a,72b , S. George 93 , T. Geralis 44 , L.O. Gerlach 53 , P. Gessinger-Befurt 99 , G. Gessner 47 , S. Ghasemi 151 , M. Ghasemi Bostanabad 176 , M. Ghneimat 24 , A. Ghosh 77 , B. Giacobbe 23b , S. Giagu 72a,72b , N. Giangiacomi 23b,23a , P. Giannetti 71a , A. Giannini 69a,69b , S.M. Gibson 93 , M. Gignac 146 , D. Gillberg 34 , G. Gilles 182 , D.M. Gingrich 3,ax , M.P. Giordani 66a,66c , F.M. Giorgi 23b , P.F. Giraud 145 , G. Giugliarelli 66a,66c , D. Giugni 68a , F. Giuli 73a,73b , S. Gkaitatzis 162 , I. Gkialas 9,h , E.L. Gkougkousis 14 , P. Gkountoumis 10 , L.K. Gladilin 113 , C. Glasman 98 , J. Glatzer 14 , P.C.F. Glaysher 46 , A. Glazov 46 , M. Goblirsch-Kolb 26 , S. Goldfarb 104 , T. Golling 54 , D. Golubkov 123 , A. Gomes 140a,140b , R. Goncalves Gama 53 , R. Gonçalo 140a,140b , G. Gonella 52 , L. Gonella 21 , A. Gongadze 79 , F. Gonnella 21 , J.L. Gonski 59 , S. González de la Hoz 174 , S. Gonzalez-Sevilla 54 , G.R. Gonzalvo Rodriguez 174 , L. Goossens 36 , P.A. Gorbounov 111 , H.A. Gordon 29 , B. Gorini 36 , E. Gorini 67a,67b , A. Gorišek 91 , A.T. Goshaw 49 , C. Gössling 47 , M.I. Gostkin 79 , C.A. Gottardo 24 , M. Gouighri 35b , D. Goujdami 35c , A.G. Goussiou 148 , N. Govender 33b,a , C. Goy 5 , E. Gozani 160 , I. Grabowska-Bold 83a , E.C. Graham 90 , J. Gramling 171 , E. Gramstad 134 , S. Grancagnolo 19 , M. Grandi 156 , V. Gratchev 138 , P.M. Gravila 27f , F.G. Gravili 67a,67b , C. Gray 57 , H.M. Gray 18 , C. Grefe 24 , K. Gregersen 96 , I.M. Gregor 46 , P. Grenier 153 , K. Grevtsov 46 , C. Grieco 14 , N.A. Grieser 128 , J. Griffiths 8 , A.A. Grillo 146 , K. Grimm 31,k , S. Grinstein 14,x , J.-F. Grivaz 132 , S. Groh 99 , E. Gross 180 , J. Grosse-Knetter 53 , Z.J. Grout 94 , C. Grud 105 , A. Grummer 118 , L. Guan 105 , W. Guan 181 , J. Guenther 36 , A. Guerguichon 132 , F. Guescini 115 , D. Guest 171 , R. Gugel 52 , T. Guillemin 5 , S. Guindon 36 , U. Gul 57 , J. Guo 60c , W. Guo 105 , Y. Guo 60a,s , Z. Guo 101 , R. Gupta 46 , S. Gurbuz 12c , G. Gustavino 128 , P. Gutierrez 128 , C. Gutschow 94 , C. Guyot 145 , M.P. Guzik 83a , C. Gwenlan 135 , C.B. Gwilliam 90 , A. Haas 124 , C. Haber 18 , H.K. Hadavand 8 , N. Haddad 35e , A. Hadef 60a , S. Hageböck 36 , M. Hagihara 169 , M. Haleem 177 , J. Haley 129 , G. Halladjian 106 , G.D. Hallewell 101 , K. Hamacher 182 , P. Hamal 130 , K. Hamano 176 , H. Hamdaoui 35e , G.N. Hamity 149 , K. Han 60a,ak , L. Han 60a , S. Han 15a,15d , K. Hanagaki 81,v , M. Hance 146 , D.M. Handl 114 , B. Haney 137 , R. Hankache 136 , P. Hanke 61a , E. Hansen 96 , J.B. Hansen 40 , J.D. Hansen 40 , M.C. Hansen 24 , P.H. Hansen 40 , E.C. Hanson 100 , K. Hara 169 , A.S. Hard 181 , T. Harenberg 182 , S. Harkusha 107 , P.F. Harrison 178 , N.M. Hartmann 114 , Y. Hasegawa 150 , A. Hasib 50 , S. Hassani 145 , S. Haug 20 , R. Hauser 106 , L.B. Havener 39 , M. Havranek 142 , C.M. Hawkes 21 , R.J. Hawkings 36 , D. Hayden 106 , C. Hayes 155 , R.L. Hayes 175 , C.P. Hays 135 , J.M. Hays 92 , H.S. Hayward 90 , S.J. Haywood 144 , F. He 60a , M.P. Heath 50 , V. Hedberg 96 , L. Heelan 8 , S. Heer 24 , K.K. Heidegger 52 , W.D. Heidorn 78 , J. Heilman 34 , S. Heim 46 , T. Heim 18 , B. Heinemann 46,as , J.J. Heinrich 131 , L. Heinrich 36 , C. Heinz 56 , J. Hejbal 141 , L. Helary 61b , A. Held 175 , S. Hellesund 134 , C.M. Helling 146 , S. Hellman 45a,45b , C. Helsens 36 , R.C.W. Henderson 89 , Y. Heng 181 , S. Henkelmann 175 , A.M. Henriques Correia 36 , G.H. Herbert 19 , H. Herde 26 , V. Herget 177 , Y. Hernández Jiménez 33c , H. Herr 99 , M.G. Herrmann 114 , T. Herrmann 48 , G. Herten 52 , R. Hertenberger 114 , L. Hervas 36 , T.C. Herwig 137 , G.G. Hesketh 94 , N.P. Hessey 168a , A. Higashida 163 , S. Higashino 81 , E. Higón-Rodriguez 174 , K. Hildebrand 37 , E. Hill 176 , J.C. Hill 32 , K.K. Hill 29 , K.H. Hiller 46 , S.J. Hillier 21 , M. Hils 48 , I. Hinchliffe 18 , F. Hinterkeuser 24 , M. Hirose 133 , S. Hirose 52 , D. Hirschbuehl 182 , B. Hiti 91 , O. Hladik 141 , D.R. Hlaluku 33c , X. Hoad 50 , J. Hobbs 155 , N. Hod 180 , M.C. Hodgkinson 149 , A. Hoecker 36 , F. Hoenig 114 , D. Hohn 52 , D. Hohov 132 , T.R. Holmes 37 , M. Holzbock 114 , L.B.A.H Hommels 32 , S. Honda 169 , T. Honda 81 , T.M. Hong 139 , A. Hönle 115 , B.H. Hooberman 173 , W.H. Hopkins 6 , Y. Horii 117 , P. Horn 48 , L.A. Horyn 37 , J-Y. Hostachy 58 , A. Hostiuc 148 , S. Hou 158 , A. Hoummada 35a , J. Howarth 100 , J. Hoya 88 , M. Hrabovsky 130 , J. Hrdinka 76 , I. Hristova 19 , J. Hrivnac 132 , A. Hrynevich 108 , T. Hryn'ova 5 , P.J. Hsu 64 , S.-C. Hsu 148 , Q. Hu 29 , S. Hu 60c , Y. Huang 15a , Z. Hubacek 142 , F. Hubaut 101 , M. Huebner 24 , F. Huegging 24 , T.B. Huffman 135 , M. Huhtinen 36 , R.F.H. Hunter 34 , P. Huo 155 , A.M. Hupe 34 , N. Huseynov 79,af , J. Huston 106 , J. Huth 59 , R. Hyneman 105 , S. Hyrych 28a , G. Iacobucci 54 , G. Iakovidis 29 , I. Ibragimov 151 , L. Iconomidou-Fayard 132 , Z. Idrissi 35e , P.I. Iengo 36 , R. Ignazzi 40 , O. Igonkina 120,z , R. Iguchi 163 , T. Iizawa 54 , Y. Ikegami 81 , M. Ikeno 81 , D. Iliadis 162 , N. Ilic 119 , F. Iltzsche 48 , G. Introzzi 70a,70b , M. Iodice 74a , K. Iordanidou 168a , V. Ippolito 72a,72b , M.F. Isacson 172 , M. Ishino 163 , M. Ishitsuka 165 , W. Islam 129 , C. Issever 135 , S. Istin 160 , F. Ito 169 , J.M. Iturbe Ponce 63a , R. Iuppa 75a,75b , A. Ivina 180 , H. Iwasaki 81 , J.M. Izen 43 , V. Izzo 69a , P. Jacka 141 , P. Jackson 1 , R.M. Jacobs 24 , B.P. Jaeger 152 , V. Jain 2 , G. Jäkel 182 , K.B. Jakobi 99 , K. Jakobs 52 , S. Jakobsen 76 , T. Jakoubek 141 , J. Jamieson 57 , K.W. Janas 83a , R. Jansky 54 , J. Janssen 24 , M. Janus 53 , P.A. Janus 83a , G. Jarlskog 96 , N. Javadov 79,af , T. Javůrek 36 , M. Javurkova 52 , F. Jeanneau 145 , L. Jeanty 131 , J. Jejelava 159a,ag , A. Jelinskas 178 , P. Jenni 52,b , J. Jeong 46 , N. Jeong 46 , S. Jézéquel 5 , H. Ji 181 , J. Jia 155 , H. Jiang 78 , Y. Jiang 60a , Z. Jiang 153,p , S. Jiggins 52 , F.A. Jimenez Morales 38 , J. Jimenez Pena 174 , S. Jin 15c , A. Jinaru 27b , O. Jinnouchi 165 , H. Jivan 33c , P. Johansson 149 , K.A. Johns 7 , C.A. Johnson 65 , K. Jon-And 45a,45b , R.W.L. Jones 89 , S.D. Jones 156 , S. Jones 7 , T.J. Jones 90 , J. Jongmanns 61a , P.M. Jorge 140a , J. Jovicevic 36 , X. Ju 18 , J.J. Junggeburth 115 , A. Juste Rozas 14,x , A. Kaczmarska 84 , M. Kado 72a,72b , H. Kagan 126 , M. Kagan 153 , C. Kahra 99 , T. Kaji 179 , E. Kajomovitz 160 , C.W. Kalderon 96 , A. Kaluza 99 , A. Kamenshchikov 123 , L. Kanjir 91 , Y. Kano 163 , V.A. Kantserov 112 , J. Kanzaki 81 , L.S. Kaplan 181 , D. Kar 33c , M.J. Kareem 168b , E. Karentzos 10 , S.N. Karpov 79 , Z.M. Karpova 79 , V. Kartvelishvili 89 , A.N. Karyukhin 123 , L. Kashif 181 , R.D. Kass 126 , A. Kastanas 45a,45b , Y. Kataoka 163 , C. Kato 60d,60c , J. Katzy 46 , K. Kawade 82 , K. Kawagoe 87 , T. Kawaguchi 117 , T. Kawamoto 163 , G. Kawamura 53 , E.F. Kay 176 , V.F. Kazanin 122b,122a , R. Keeler 176 , R. Kehoe 42 , J.S. Keller 34 , E. Kellermann 96 , D. Kelsey 156 , J.J. Kempster 21 , J. Kendrick 21 , O. Kepka 141 , S. Kersten 182 , B.P. Kerševan 91 , S. Ketabchi Haghighat 167 , M. Khader 173 , F. Khalil-Zada 13 , M.K. Khandoga 145 , A. Khanov 129 , A.G. Kharlamov 122b,122a , T. Kharlamova 122b,122a , E.E. Khoda 175 , A. Khodinov 166 , T.J. Khoo 54 , E. Khramov 79 , J. Khubua 159b , S. Kido 82 , M. Kiehn 54 , C.R. Kilby 93 , Y.K. Kim 37 , N. Kimura 66a,66c , O.M. Kind 19 , B.T. King 90,* , D. Kirchmeier 48 , J. Kirk 144 , A.E. Kiryunin 115 , T. Kishimoto 163 , D.P. Kisliuk 167 , V. Kitali 46 , O. Kivernyk 5 , E. Kladiva 28b,* , T. Klapdor-Kleingrothaus 52 , M. Klassen 61a , M.H. Klein 105 , M. Klein 90 , U. Klein 90 , K. Kleinknecht 99 , P. Klimek 121 , A. Klimentov 29 , T. Klingl 24 , T. Klioutchnikova 36 , F.F. Klitzner 114 , P. Kluit 120 , S. Kluth 115 , E. Kneringer 76 , E.B.F.G. Knoops 101 , A. Knue 52 , D. Kobayashi 87 , T. Kobayashi 163 , M. Kobel 48 , M. Kocian 153 , P. Kodys 143 , P.T. Koenig 24 , T. Koffas 34 , N.M. Köhler 115 , T. Koi 153 , M. Kolb 61b , I. Koletsou 5 , T. Komarek 130 , T. Kondo 81 , N. Kondrashova 60c , K. Köneke 52 , A.C. König 119 , T. Kono 125 , R. Konoplich 124,an , V. Konstantinides 94 , N. Konstantinidis 94 , B. Konya 96 , R. Kopeliansky 65 , S. Koperny 83a , K. Korcyl 84 , K. Kordas 162 , G. Koren 161 , A. Korn 94 , I. Korolkov 14 , E.V. Korolkova 149 , N. Korotkova 113 , O. Kortner 115 , S. Kortner 115 , T. Kosek 143 , V.V. Kostyukhin 24 , A. Kotwal 49 , A. Koulouris 10 , A. Kourkoumeli-Charalampidi 70a,70b , C. Kourkoumelis 9 , E. Kourlitis 149 , V. Kouskoura 29 , A.B. Kowalewska 84 , R. Kowalewski 176 , C. Kozakai 163 , W. Kozanecki 145 , A.S. Kozhin 123 , V.A. Kramarenko 113 , G. Kramberger 91 , D. Krasnopevtsev 60a , M.W. Krasny 136 , A. Krasznahorkay 36 , D. Krauss 115 , J.A. Kremer 83a , J. Kretzschmar 90 , P. Krieger 167 , F. Krieter 114 , A. Krishnan 61b , K. Krizka 18 , K. Kroeninger 47 , H. Kroha 115 , J. Kroll 141 , J. Kroll 137 , J. Krstic 16 , U. Kruchonak 79 , H. Krüger 24 , N. Krumnack 78 , M.C. Kruse 49 , J.A. Krzysiak 84 , T. Kubota 104 , S. Kuday 4b , J.T. Kuechler 46 , S. Kuehn 36 , A. Kugel 61a , T. Kuhl 46 , V. Kukhtin 79 , R. Kukla 101 , Y. Kulchitsky 107,aj , S. Kuleshov 147b , Y.P. Kulinich 173 , M. Kuna 58 , T. Kunigo 85 , A. Kupco 141 , T. Kupfer 47 , O. Kuprash 52 , H. Kurashige 82 , L.L. Kurchaninov 168a , Y.A. Kurochkin 107 , A. Kurova 112 , M.G. Kurth 15a,15d , E.S. Kuwertz 36 , M. Kuze 165 , A.K. Kvam 148 , J. Kvita 130 , T. Kwan 103 , A. La Rosa 115 , L. La Rotonda 41b,41a , F. La Ruffa 41b,41a , C. Lacasta 174 , F. Lacava 72a,72b , D.P.J. Lack 100 , H. Lacker 19 , D. Lacour 136 , E. Ladygin 79 , R. Lafaye 5 , B. Laforge 136 , T. Lagouri 33c , S. Lai 53 , S. Lammers 65 , W. Lampl 7 , C. Lampoudis 162 , E. Lançon 29 , U. Landgraf 52 , M.P.J. Landon 92 , M.C. Lanfermann 54 , V.S. Lang 46 , J.C. Lange 53 , R.J. Langenberg 36 , A.J. Lankford 171 , F. Lanni 29 , K. Lantzsch 24 , A. Lanza 70a , A. Lapertosa 55b,55a , S. Laplace 136 , J.F. Laporte 145 , T. Lari 68a , F. Lasagni Manghi 23b,23a , M. Lassnig 36 , T.S. Lau 63a , A. Laudrain 132 , A. Laurier 34 , M. Lavorgna 69a,69b , M. Lazzaroni 68a,68b , B. Le 104 , O. 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Lund-Jensen 154 , M.S. Lutz 102 , D. Lynn 29 , R. Lysak 141 , E. Lytken 96 , F. Lyu 15a , V. Lyubushkin 79 , T. Lyubushkina 79 , H. Ma 29 , L.L. Ma 60b , Y. Ma 60b , G. Maccarrone 51 , A. Macchiolo 115 , C.M. Macdonald 149 , J. Machado Miguens 137 , D. Madaffari 174 , R. Madar 38 , W.F. Mader 48 , N. Madysa 48 , J. Maeda 82 , K. Maekawa 163 , S. Maeland 17 , T. Maeno 29 , M. Maerker 48 , A.S. Maevskiy 113 , V. Magerl 52 , N. Magini 78 , D.J. Mahon 39 , C. Maidantchik 80b , T. Maier 114 , A. Maio 140a,140b,140d , O. Majersky 28a , S. Majewski 131 , Y. Makida 81 , N. Makovec 132 , B. Malaescu 136 , Pa. Malecki 84 , V.P. Maleev 138 , F. Malek 58 , U. Mallik 77 , D. Malon 6 , C. Malone 32 , S. Maltezos 10 , S. Malyukov 36 , J. Mamuzic 174 , G. Mancini 51 , I. Mandić 91 , L. Manhaes de Andrade Filho 80a , I.M. Maniatis 162 , J. Manjarres Ramos 48 , K.H. Mankinen 96 , A. Mann 114 , A. Manousos 76 , B. Mansoulie 145 , I. Manthos 162 , S. Manzoni 120 , A. Marantis 162 , G. Marceca 30 , L. Marchese 135 , G. Marchiori 136 , M. Marcisovsky 141 , C. Marcon 96 , C.A. Marin Tobon 36 , M. Marjanovic 38 , F. Marroquim 80b , Z. Marshall 18 , M.U.F Martensson 172 , S. Marti-Garcia 174 , C.B. Martin 126 , T.A. Martin 178 , V.J. Martin 50 , B. Martin dit Latour 17 , L. Martinelli 74a,74b , M. Martinez 14,x , V.I. Martinez Outschoorn 102 , S. Martin-Haugh 144 , V.S. Martoiu 27b , A.C. Martyniuk 94 , A. Marzin 36 , S.R. Maschek 115 , L. Masetti 99 , T. Mashimo 163 , R. Mashinistov 110 , J. Masik 100 , A.L. Maslennikov 122b,122a , L.H. Mason 104 , L. Massa 73a,73b , P. Massarotti 69a,69b , P. Mastrandrea 71a,71b , A. Mastroberardino 41b,41a , T. Masubuchi 163 , A. Matic 114 , P. Mättig 24 , J. Maurer 27b , B. Maček 91 , S.J. Maxfield 90 , D.A. Maximov 122b,122a , R. Mazini 158 , I. Maznas 162 , S.M. Mazza 146 , S.P. Mc Kee 105 , T.G. McCarthy 115 , L.I. McClymont 94 , W.P. McCormack 18 , E.F. McDonald 104 , J.A. Mcfayden 36 , M.A. McKay 42 , K.D. McLean 176 , S.J. McMahon 144 , P.C. 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Minashvili 159b , A.I. Mincer 124 , B. Mindur 83a , M. Mineev 79 , Y. Minegishi 163 , Y. Ming 181 , L.M. Mir 14 , A. Mirto 67a,67b , K.P. Mistry 137 , T. Mitani 179 , J. Mitrevski 114 , V.A. Mitsou 174 , M. Mittal 60c , A. Miucci 20 , P.S. Miyagawa 149 , A. Mizukami 81 , J.U. Mjörnmark 96 , T. Mkrtchyan 184 , M. Mlynarikova 143 , T. Moa 45a,45b , K. Mochizuki 109 , P. Mogg 52 , S. Mohapatra 39 , R. Moles-Valls 24 , M.C. Mondragon 106 , K. Mönig 46 , J. Monk 40 , E. Monnier 101 , A. Montalbano 152 , J. Montejo Berlingen 36 , M. Montella 94 , F. Monticelli 88 , S. Monzani 68a , N. Morange 132 , D. Moreno 22 , M. Moreno Llácer 36 , C. Moreno Martinez 14 , P. Morettini 55b , M. Morgenstern 120 , S. Morgenstern 48 , D. Mori 152 , M. Morii 59 , M. Morinaga 179 , V. Morisbak 134 , A.K. Morley 36 , G. Mornacchi 36 , A.P. Morris 94 , L. Morvaj 155 , P. Moschovakos 36 , B. Moser 120 , M. Mosidze 159b , T. Moskalets 145 , H.J. Moss 149 , J. Moss 31,m , K. Motohashi 165 , E. Mountricha 36 , E.J.W. Moyse 102 , S. Muanza 101 , J. Mueller 139 , R.S.P. Mueller 114 , D. Muenstermann 89 , G.A. Mullier 96 , J.L. Munoz Martinez 14 , F.J. Munoz Sanchez 100 , P. Murin 28b , W.J. Murray 178,144 , A. Murrone 68a,68b , M. Muškinja 18 , C. Mwewa 33a , A.G. Myagkov 123,ao , J. Myers 131 , M. Myska 142 , B.P. Nachman 18 , O. Nackenhorst 47 , A.Nag Nag 48 , K. Nagai 135 , K. Nagano 81 , Y. Nagasaka 62 , M. Nagel 52 , E. Nagy 101 , A.M. Nairz 36 , Y. Nakahama 117 , K. Nakamura 81 , T. Nakamura 163 , I. Nakano 127 , H. Nanjo 133 , F. Napolitano 61a , R.F. Naranjo Garcia 46 , R. Narayan 42 , D.I. Narrias Villar 61a , I. Naryshkin 138 , T. Naumann 46 , G. Navarro 22 , H.A. Neal 105,* , P.Y. Nechaeva 110 , F. Nechansky 46 , T.J. Neep 21 , A. Negri 70a,70b , M. Negrini 23b , C. Nellist 53 , M.E. Nelson 135 , S. Nemecek 141 , P. Nemethy 124 , M. Nessi 36,d , M.S. Neubauer 173 , M. Neumann 182 , P.R. Newman 21 , T.Y. Ng 63c , Y.S. Ng 19 , Y.W.Y. Ng 171 , H.D.N. Nguyen 101 , T. Nguyen Manh 109 , E. Nibigira 38 , R.B. Nickerson 135 , R. Nicolaidou 145 , D.S. Nielsen 40 , J. Nielsen 146 , N. Nikiforou 11 , V. Nikolaenko 123,ao , I. Nikolic-Audit 136 , K. Nikolopoulos 21 , P. Nilsson 29 , H.R. Nindhito 54 , Y. Ninomiya 81 , A. Nisati 72a , N. Nishu 60c , R. Nisius 115 , I. Nitsche 47 , T. Nitta 179 , T. Nobe 163 , Y. Noguchi 85 , I. Nomidis 136 , M.A. Nomura 29 , M. Nordberg 36 , N. Norjoharuddeen 135 , T. Novak 91 , O. Novgorodova 48 , R. Novotny 142 , L. Nozka 130 , K. Ntekas 171 , E. Nurse 94 , F.G. Oakham 34,ax , H. Oberlack 115 , J. Ocariz 136 , A. Ochi 82 , I. Ochoa 39 , J.P. Ochoa-Ricoux 147a , K. O'Connor 26 , S. Oda 87 , S. Odaka 81 , S. Oerdek 53 , A. Ogrodnik 83a , A. Oh 100 , S.H. Oh 49 , C.C. Ohm 154 , H. Oide 55b,55a , M.L. Ojeda 167 , H. Okawa 169 , Y. Okazaki 85 , Y. Okumura 163 , T. Okuyama 81 , A. Olariu 27b , L.F. Oleiro Seabra 140a , S.A. Olivares Pino 147a , D. Oliveira Damazio 29 , J.L. Oliver 1 , M.J.R. Olsson 171 , A. Olszewski 84 , J. Olszowska 84 , D.C. O'Neil 152 , A. Onofre 140a,140e , K. Onogi 117 , P.U.E. Onyisi 11 , H. Oppen 134 , M.J. Oreglia 37 , G.E. Orellana 88 , Y. Oren 161 , D. Orestano 74a,74b , N. Orlando 14 , R.S. Orr 167 , V. O'Shea 57 , R. Ospanov 60a , G. Otero y Garzon 30 , H. Otono 87 , M. Ouchrif 35d , J. Ouellette 29 , F. Ould-Saada 134 , A. Ouraou 145 , Q. Ouyang 15a , M. Owen 57 , R.E. Owen 21 , V.E. Ozcan 12c , N. Ozturk 8 , J. Pacalt 130 , H.A. Pacey 32 , K. Pachal 49 , A. Pacheco Pages 14 , C. Padilla Aranda 14 , S. Pagan Griso 18 , M. Paganini 183 , G. Palacino 65 , S. Palazzo 50 , S. Palestini 36 , M. Palka 83b , D. Pallin 38 , I. Panagoulias 10 , C.E. Pandini 36 , J.G. Panduro Vazquez 93 , P. Pani 46 , G. Panizzo 66a,66c , L. Paolozzi 54 , C. Papadatos 109 , K. Papageorgiou 9,h , A. Paramonov 6 , D. Paredes Hernandez 63b , S.R. Paredes Saenz 135 , B. Parida 166 , T.H. Park 167 , A.J. Parker 89 , M.A. Parker 32 , F. Parodi 55b,55a , E.W.P. Parrish 121 , J.A. Parsons 39 , U. Parzefall 52 , L. Pascual Dominguez 136 , V.R. Pascuzzi 167 , J.M.P. Pasner 146 , E. Pasqualucci 72a , S. Passaggio 55b , F. Pastore 93 , P. Pasuwan 45a,45b , S. Pataraia 99 , J.R. Pater 100 , A. Pathak 181 , T. Pauly 36 , B. Pearson 115 , M. Pedersen 134 , L. Pedraza Diaz 119 , R. Pedro 140a , T. Peiffer 53 , S.V. Peleganchuk 122b,122a , O. Penc 141 , H. Peng 60a , B.S. Peralva 80a , M.M. Perego 132 , A.P. Pereira Peixoto 140a , D.V. Perepelitsa 29 , F. Peri 19 , L. Perini 68a,68b , H. Pernegger 36 , S. Perrella 69a,69b , K. Peters 46 , R.F.Y. Peters 100 , B.A. Petersen 36 , T.C. Petersen 40 , E. Petit 101 , A. Petridis 1 , C. Petridou 162 , P. Petroff 132 , M. Petrov 135 , F. Petrucci 74a,74b , M. Pettee 183 , N.E. Pettersson 102 , K. Petukhova 143 , A. Peyaud 145 , R. Pezoa 147b , L. Pezzotti 70a,70b , T. Pham 104 , F.H. Phillips 106 , P.W. Phillips 144 , M.W. Phipps 173 , G. Piacquadio 155 , E. Pianori 18 , A. Picazio 102 , R.H. Pickles 100 , R. Piegaia 30 , D. Pietreanu 27b , J.E. Pilcher 37 , A.D. Pilkington 100 , M. Pinamonti 73a,73b , J.L. Pinfold 3 , M. Pitt 180 , L. Pizzimento 73a,73b , M.-A. Pleier 29 , V. Pleskot 143 , E. Plotnikova 79 , D. Pluth 78 , P. Podberezko 122b,122a , R. Poettgen 96 , R. Poggi 54 , L. Poggioli 132 , I. Pogrebnyak 106 , D. Pohl 24 , I. Pokharel 53 , G. Polesello 70a , A. Poley 18 , A. Policicchio 72a,72b , R. Polifka 143 , A. Polini 23b , C.S. Pollard 46 , V. Polychronakos 29 , D. Ponomarenko 112 , L. Pontecorvo 36 , S. Popa 27a , G.A. Popeneciu 27d , D.M. Portillo Quintero 58 , S. Pospisil 142 , K. Potamianos 46 , I.N. Potrap 79 , C.J. Potter 32 , H. Potti 11 , T. Poulsen 96 , J. Poveda 36 , T.D. Powell 149 , G. Pownall 46 , M.E. Pozo Astigarraga 36 , P. Pralavorio 101 , S. Prell 78 , D. Price 100 , M. Primavera 67a , S. Prince 103 , M.L. Proffitt 148 , N. Proklova 112 , K. Prokofiev 63c , F. Prokoshin 79 , S. Protopopescu 29 , J. Proudfoot 6 , M. Przybycien 83a , D. Pudzha 138 , A. Puri 173 , P. Puzo 132 , J. Qian 105 , Y. Qin 100 , A. Quadt 53 , M. Queitsch-Maitland 46 , A. Qureshi 1 , P. Rados 104 , F. Ragusa 68a,68b , G. Rahal 97 , J.A. Raine 54 , S. Rajagopalan 29 , A. Ramirez Morales 92 , K. Ran 15a,15d , T. Rashid 132 , S. Raspopov 5 , M.G. Ratti 68a,68b , D.M. Rauch 46 , F. Rauscher 114 , S. Rave 99 , B. Ravina 149 , I. Ravinovich 180 , J.H. Rawling 100 , M. Raymond 36 , A.L. Read 134 , N.P. Readioff 58 , M. Reale 67a,67b , D.M. Rebuzzi 70a,70b , A. Redelbach 177 , G. Redlinger 29 , K. Reeves 43 , L. Rehnisch 19 , J. Reichert 137 , D. Reikher 161 , A. Reiss 99 , A. Rej 151 , C. Rembser 36 , M. Renda 27b , M. Rescigno 72a , S. Resconi 68a , E.D. Resseguie 137 , S. Rettie 175 , E. Reynolds 21 , O.L. Rezanova 122b,122a , P. Reznicek 143 , E. Ricci 75a,75b , R. Richter 115 , S. Richter 46 , E. Richter-Was 83b , O. Ricken 24 , M. Ridel 136 , P. Rieck 115 , C.J. Riegel 182 , O. Rifki 46 , M. Rijssenbeek 155 , A. Rimoldi 70a,70b , M. Rimoldi 46 , L. Rinaldi 23b , G. Ripellino 154 , B. Ristić 89 , E. Ritsch 36 , I. Riu 14 , J.C. Rivera Vergara 176 , F. Rizatdinova 129 , E. Rizvi 92 , C. Rizzi 36 , R.T. Roberts 100 , S.H. Robertson 103,ad , M. Robin 46 , D. Robinson 32 , J.E.M. Robinson 46 , C.M. Robles Gajardo 147b , A. Robson 57 , E. Rocco 99 , C. Roda 71a,71b , S. Rodriguez Bosca 174 , A. Rodriguez Perez 14 , D. Rodriguez Rodriguez 174 , A.M. Rodríguez Vera 168b , S. Roe 36 , O. Røhne 134 , R. Röhrig 115 , C.P.A. Roland 65 , J. Roloff 59 , A. Romaniouk 112 , M. Romano 23b,23a , N. Rompotis 90 , M. Ronzani 124 , L. Roos 136 , S. Rosati 72a , K. Rosbach 52 , G. Rosin 102 , B.J. Rosser 137 , E. Rossi 46 , E. Rossi 74a,74b , E. Rossi 69a,69b , L.P. Rossi 55b , L. Rossini 68a,68b , R. Rosten 14 , M. Rotaru 27b , J. Rothberg 148 , D. Rousseau 132 , G. Rovelli 70a,70b , D. Roy 33c , A. Rozanov 101 , Y. Rozen 160 , X. Ruan 33c , F. Rubbo 153 , F. Rühr 52 , A. Ruiz-Martinez 174 , A. Rummler 36 , Z. Rurikova 52 , N.A. Rusakovich 79 , H.L. Russell 103 , L. Rustige 38,47 , J.P. Rutherfoord 7 , E.M. Rüttinger 46,j , Y.F. Ryabov 138 , M. Rybar 39 , G. Rybkin 132 , A. Ryzhov 123 , G.F. Rzehorz 53 , P. Sabatini 53 , G. Sabato 120 , S. Sacerdoti 132 , H.F-W. Sadrozinski 146 , R. Sadykov 79 , F. Safai Tehrani 72a , B. Safarzadeh Samani 156 , P. Saha 121 , S. Saha 103 , M. Sahinsoy 61a , A. Sahu 182 , M. Saimpert 46 , M. Saito 163 , T. Saito 163 , H. Sakamoto 163 , A. Sakharov 124,an , D. Salamani 54 , G. Salamanna 74a,74b , J.E. Salazar Loyola 147b , P.H. Sales De Bruin 172 , A. Salnikov 153 , J. Salt 174 , D. Salvatore 41b,41a , F. Salvatore 156 , A. Salvucci 63a,63b,63c , A. Salzburger 36 , J. Samarati 36 , D. Sammel 52 , D. Sampsonidis 162 , D. Sampsonidou 162 , J. Sánchez 174 , A. Sanchez Pineda 66a,66c , H. Sandaker 134 , C.O. Sander 46 , I.G. Sanderswood 89 , M. Sandhoff 182 , C. Sandoval 22 , D.P.C. Sankey 144 , M. Sannino 55b,55a , Y. Sano 117 , A. Sansoni 51 , C. Santoni 38 , H. Santos 140a,140b , S.N. Santpur 18 , A. Santra 174 , A. Sapronov 79 , J.G. Saraiva 140a,140d , O. Sasaki 81 , K. Sato 169 , E. Sauvan 5 , P. Savard 167,ax , N. Savic 115 , R. Sawada 163 , C. Sawyer 144 , L. Sawyer 95,al , C. Sbarra 23b , A. Sbrizzi 23a , T. Scanlon 94 , J. Schaarschmidt 148 , P. Schacht 115 , B.M. Schachtner 114 , D. Schaefer 37 , L. Schaefer 137 , J. Schaeffer 99 , S. Schaepe 36 , U. Schäfer 99 , A.C. Schaffer 132 , D. Schaile 114 , R.D. Schamberger 155 , N. Scharmberg 100 , V.A. Schegelsky 138 , D. Scheirich 143 , F. Schenck 19 , M. Schernau 171 , C. Schiavi 55b,55a , S. Schier 146 , L.K. Schildgen 24 , Z.M. Schillaci 26 , E.J. Schioppa 36 , M. Schioppa 41b,41a , K.E. Schleicher 52 , S. Schlenker 36 , K.R. Schmidt-Sommerfeld 115 , K. Schmieden 36 , C. Schmitt 99 , S. Schmitt 46 , S. Schmitz 99 , J.C. Schmoeckel 46 , U. Schnoor 52 , L. Schoeffel 145 , A. Schoening 61b , P.G. Scholer 52 , E. Schopf 135 , M. Schott 99 , J.F.P. Schouwenberg 119 , J. Schovancova 36 , S. Schramm 54 , F. Schroeder 182 , A. Schulte 99 , H-C. 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Shimogama 179 , M. Shimojima 116 , I.P.J. Shipsey 135 , S. Shirabe 87 , M. Shiyakova 79,aa , J. Shlomi 180 , A. Shmeleva 110 , M.J. Shochet 37 , S. Shojaii 104 , D.R. Shope 128 , S. Shrestha 126 , E. Shulga 180 , P. Sicho 141 , A.M. Sickles 173 , P.E. Sidebo 154 , E. Sideras Haddad 33c , O. Sidiropoulou 36 , A. Sidoti 23b,23a , F. Siegert 48 , Dj. Sijacki 16 , M. Silva Jr. 181 , M.V. Silva Oliveira 80a , S.B. Silverstein 45a , S. Simion 132 , E. Simioni 99 , R. Simoniello 99 , S. Simsek 12b , P. Sinervo 167 , V. Sinetckii 113,110 , N.B. Sinev 131 , M. Sioli 23b,23a , I. Siral 105 , S.Yu. Sivoklokov 113 , J. Sjölin 45a,45b , E. Skorda 96 , P. Skubic 128 , M. Slawinska 84 , K. Sliwa 170 , R. Slovak 143 , V. Smakhtin 180 , B.H. Smart 144 , J. Smiesko 28a , N. Smirnov 112 , S.Yu. Smirnov 112 , Y. Smirnov 112 , L.N. Smirnova 113,t , O. Smirnova 96 , J.W. Smith 53 , M. Smizanska 89 , K. Smolek 142 , A. Smykiewicz 84 , A.A. Snesarev 110 , H.L. Snoek 120 , I.M. Snyder 131 , S. Snyder 29 , R. Sobie 176,ad , A.M. Soffa 171 , A. Soffer 161 , A. Søgaard 50 , F. Sohns 53 , C.A. Solans Sanchez 36 , E.Yu. Soldatov 112 , U. Soldevila 174 , A.A. Solodkov 123 , A. Soloshenko 79 , O.V. Solovyanov 123 , V. Solovyev 138 , P. Sommer 149 , H. Son 170 , W. Song 144 , W.Y. Song 168b , A. Sopczak 142 , F. Sopkova 28b , C.L. Sotiropoulou 71a,71b , S. Sottocornola 70a,70b , R. Soualah 66a,66c,g , A.M. Soukharev 122b,122a , D. South 46 , S. Spagnolo 67a,67b , M. Spalla 115 , M. Spangenberg 178 , F. Spanò 93 , D. Sperlich 52 , T.M. Spieker 61a , R. Spighi 23b , G. Spigo 36 , M. Spina 156 , D.P. Spiteri 57 , M. Spousta 143 , A. Stabile 68a,68b , B.L. Stamas 121 , R. Stamen 61a , M. Stamenkovic 120 , E. Stanecka 84 , R.W. Stanek 6 , B. Stanislaus 135 , M.M. Stanitzki 46 , M. Stankaityte 135 , B. Stapf 120 , E.A. Starchenko 123 , G.H. Stark 146 , J. Stark 58 , S.H Stark 40 , P. Staroba 141 , P. Starovoitov 61a , S. Stärz 103 , R. Staszewski 84 , G. Stavropoulos 44 , M. Stegler 46 , P. Steinberg 29 , A.L. Steinhebel 131 , B. Stelzer 152 , H.J. Stelzer 139 , O. Stelzer-Chilton 168a , H. Stenzel 56 , T.J. Stevenson 156 , G.A. Stewart 36 , M.C. Stockton 36 , G. Stoicea 27b , M. Stolarski 140a , P. Stolte 53 , S. Stonjek 115 , A. Straessner 48 , J. Strandberg 154 , S. Strandberg 45a,45b , M. Strauss 128 , P. Strizenec 28b , R. Ströhmer 177 , D.M. Strom 131 , R. Stroynowski 42 , A. Strubig 50 , S.A. Stucci 29 , B. Stugu 17 , J. Stupak 128 , N.A. Styles 46 , D. Su 153 , S. Suchek 61a , V.V. Sulin 110 , M.J. Sullivan 90 , D.M.S. Sultan 54 , S. Sultansoy 4c , T. Sumida 85 , S. Sun 105 , X. Sun 3 , K. Suruliz 156 , C.J.E. Suster 157 , M.R. Sutton 156 , S. Suzuki 81 , M. Svatos 141 , M. Swiatlowski 37 , S.P. Swift 2 , T. Swirski 177 , A. Sydorenko 99 , I. Sykora 28a , M. Sykora 143 , T. Sykora 143 , D. Ta 99 , K. Tackmann 46,y , J. Taenzer 161 , A. Taffard 171 , R. Tafirout 168a , H. Takai 29 , R. Takashima 86 , K. Takeda 82 , T. Takeshita 150 , E.P. Takeva 50 , Y. Takubo 81 , M. Talby 101 , A.A. Talyshev 122b,122a , N.M. Tamir 161 , J. Tanaka 163 , M. Tanaka 165 , R. Tanaka 132 , S. Tapia Araya 173 , S. Tapprogge 99 , A. Tarek Abouelfadl Mohamed 136 , S. Tarem 160 , G. Tarna 27b,c , G.F. Tartarelli 68a , P. Tas 143 , M. Tasevsky 141 , T. Tashiro 85 , E. Tassi 41b,41a , A. Tavares Delgado 140a,140b , Y. Tayalati 35e , A.J. Taylor 50 , G.N. Taylor 104 , W. Taylor 168b , A.S. Tee 89 , R. Teixeira De Lima 153 , P. Teixeira-Dias 93 , H. Ten Kate 36 , J.J. Teoh 120 , S. Terada 81 , K. Terashi 163 , J. Terron 98 , S. Terzo 14 , M. Testa 51 , R.J. Teuscher 167,ad , S.J. Thais 183 , T. Theveneaux-Pelzer 46 , F. Thiele 40 , D.W. Thomas 93 , J.O. Thomas 42 , J.P. Thomas 21 , A.S. Thompson 57 , P.D. Thompson 21 , L.A. Thomsen 183 , E. Thomson 137 , Y. Tian 39 , R.E. Ticse Torres 53 , V.O. Tikhomirov 110,ap , Yu.A. Tikhonov 122b,122a , S. Timoshenko 112 , P. Tipton 183 , S. Tisserant 101 , K. Todome 23b,23a , S. Todorova-Nova 5 , S. Todt 48 , J. Tojo 87 , S. Tokár 28a , K. Tokushuku 81 , E. Tolley 126 , K.G. Tomiwa 33c , M. Tomoto 117 , L. Tompkins 153,p , K. Toms 118 , B. Tong 59 , P. Tornambe 102 , E. Torrence 131 , H. Torres 48 , E. Torró Pastor 148 , C. Tosciri 135 , J. Toth 101,ab , D.R. Tovey 149 , A. Traeet 17 , C.J. Treado 124 , T. Trefzger 177 , F. Tresoldi 156 , A. Tricoli 29 , I.M. Trigger 168a , S. Trincaz-Duvoid 136 , W. Trischuk 167 , B. Trocmé 58 , A. Trofymov 132 , C. Troncon 68a , M. Trovatelli 176 , F. Trovato 156 , L. Truong 33b , M. Trzebinski 84 , A. Trzupek 84 , F. Tsai 46 , J.C-L. Tseng 135 , P.V. Tsiareshka 107,aj , A. Tsirigotis 162 , N. Tsirintanis 9 , V. Tsiskaridze 155 , E.G. Tskhadadze 159a , M. Tsopoulou 162 , I.I. Tsukerman 111 , V. Tsulaia 18 , S. Tsuno 81 , D. Tsybychev 155 , Y. Tu 63b , A. Tudorache 27b , V. Tudorache 27b , T.T. Tulbure 27a , A.N. Tuna 59 , S. Turchikhin 79 , D. Turgeman 180 , I. Turk Cakir 4b,u , R.J. Turner 21 , R.T. Turra 68a , P.M. Tuts 39 , S Tzamarias 162 , E. Tzovara 99 , G. Ucchielli 47 , K. Uchida 163 , I. Ueda 81 , M. Ughetto 45a,45b , F. Ukegawa 169 , G. Unal 36 , A. Undrus 29 , G. Unel 171 , F.C. Ungaro 104 , Y. Unno 81 , K. Uno 163 , J. Urban 28b , P. Urquijo 104 , G. Usai 8 , J. Usui 81 , Z. Uysal 12d , L. Vacavant 101 , V. Vacek 142 , B. Vachon 103 , K.O.H. Vadla 134 , A. Vaidya 94 , C. Valderanis 114 , E. Valdes Santurio 45a,45b , M. Valente 54 , S. Valentinetti 23b,23a , A. Valero 174 , L. Valéry 46 , R.A. Vallance 21 , A. Vallier 36 , J.A. Valls Ferrer 174 , T.R. Van Daalen 14 , P. Van Gemmeren 6 , I. Van Vulpen 120 , M. Vanadia 73a,73b , W. Vandelli 36 , A. Vaniachine 166 , D. Vannicola 72a,72b , R. Vari 72a , E.W. Varnes 7 , C. Varni 55b,55a , T. Varol 42 , D. Varouchas 132 , K.E. Varvell 157 , M.E. Vasile 27b , G.A. Vasquez 176 , J.G. Vasquez 183 , F. Vazeille 38 , D. Vazquez Furelos 14 , T. Vazquez Schroeder 36 , J. Veatch 53 , V. Vecchio 74a,74b , M.J. Veen 120 , L.M. Veloce 167 , F. Veloso 140a,140c , S. Veneziano 72a , A. Ventura 67a,67b , N. Venturi 36 , A. Verbytskyi 115 , V. Vercesi 70a , M. Verducci 74a,74b , C.M. Vergel Infante 78 , C. Vergis 24 , W. Verkerke 120 , A.T. Vermeulen 120 , J.C. Vermeulen 120 , M.C. Vetterli 152,ax , N. Viaux Maira 147b , M. Vicente Barreto Pinto 54 , T. Vickey 149 , O.E. Vickey Boeriu 149 , G.H.A. Viehhauser 135 , L. Vigani 135 , M. Villa 23b,23a , M. Villaplana Perez 68a,68b , E. Vilucchi 51 , M.G. Vincter 34 , V.B. Vinogradov 79 , A. Vishwakarma 46 , C. Vittori 23b,23a , I. Vivarelli 156 , M. Vogel 182 , P. Vokac 142 , S.E. von Buddenbrock 33c , E. Von Toerne 24 , V. Vorobel 143 , K. Vorobev 112 , M. Vos 174 , J.H. Vossebeld 90 , M. Vozak 100 , N. Vranjes 16 , M. Vranjes Milosavljevic 16 , V. Vrba 142 , M. Vreeswijk 120 , T. Šfiligoj 91 , R. Vuillermet 36 , I. Vukotic 37 , T. Ženiš 28a , L. Živković 16 , P. Wagner 24 , W. Wagner 182 , J. Wagner-Kuhr 114 , H. Wahlberg 88 , K. Wakamiya 82 , V.M. Walbrecht 115 , J. Walder 89 , R. Walker 114 , S.D. Walker 93 , W. Walkowiak 151 , V. Wallangen 45a,45b , A.M. Wang 59 , C. Wang 60b , F. Wang 181 , H. Wang 18 , H. Wang 3 , J. Wang 157 , J. Wang 61b , P. Wang 42 , Q. Wang 128 , R.-J. Wang 99 , R. Wang 60a , R. Wang 6 , S.M. Wang 158 , W.T. Wang 60a , W. Wang 15c,ae , W.X. Wang 60a,ae , Y. Wang 60a,am , Z. Wang 60c , C. Wanotayaroj 46 , A. Warburton 103 , C.P. Ward 32 , D.R. Wardrope 94 , N. Warrack 57 , A. Washbrook 50 , A.T. Watson 21 , M.F. Watson 21 , G. Watts 148 , B.M. Waugh 94 , A.F. Webb 11 , S. Webb 99 , C. Weber 183 , M.S. Weber 20 , S.A. Weber 34 , S.M. Weber 61a , A.R. Weidberg 135 , J. Weingarten 47 , M. Weirich 99 , C. Weiser 52 , P.S. Wells 36 , T. Wenaus 29 , T. Wengler 36 , S. Wenig 36 , N. Wermes 24 , M.D. Werner 78 , M. Wessels 61a , T.D. Weston 20 , K. Whalen 131 , N.L. Whallon 148 , A.M. Wharton 89 , A.S. White 105 , A. White 8 , M.J. White 1 , D. Whiteson 171 , B.W. Whitmore 89 , F.J. Wickens 144 , W. Wiedenmann 181 , M. Wielers 144 , N. Wieseotte 99 , C. Wiglesworth 40 , L.A.M. Wiik-Fuchs 52 , F. Wilk 100 , H.G. Wilkens 36 , L.J. Wilkins 93 , H.H. Williams 137 , S. Williams 32 , C. Willis 106 , S. Willocq 102 , J.A. Wilson 21 , I. Wingerter-Seez 5 , E. Winkels 156 , F. Winklmeier 131 , O.J. Winston 156 , B.T. Winter 52 , M. Wittgen 153 , M. Wobisch 95 , A. Wolf 99 , T.M.H. Wolf 120 , R. Wolff 101 , R.W. Wölker 135 , J. Wollrath 52 , M.W. Wolter 84 , H. Wolters 140a,140c , V.W.S. Wong 175 , N.L. Woods 146 , S.D. Worm 21 , B.K. Wosiek 84 , K.W. Woźniak 84 , K. Wraight 57 , S.L. Wu 181 , X. Wu 54 , Y. Wu 60a , T.R. Wyatt 100 , B.M. Wynne 50 , S. Xella 40 , Z. Xi 105 , L. Xia 178 , D. Xu 15a , H. Xu 60a,c , L. Xu 29 , T. Xu 145 , W. Xu 105 , Z. Xu 60b , Z. Xu 153 , B. Yabsley 157 , S. Yacoob 33a , K. Yajima 133 , D.P. Yallup 94 , D. Yamaguchi 165 , Y. Yamaguchi 165 , A. Yamamoto 81 , T. Yamanaka 163 , F. Yamane 82 , M. Yamatani 163 , T. Yamazaki 163 , Y. Yamazaki 82 , Z. Yan 25 , H.J. Yang 60c,60d , H.T. Yang 18 , S. Yang 77 , X. Yang 60b,58 , Y. Yang 163 , W-M. Yao 18 , Y.C. Yap 46 , Y. Yasu 81 , E. Yatsenko 60c,60d , J. Ye 42 , S. Ye 29 , I. Yeletskikh 79 , M.R. Yexley 89 , E. Yigitbasi 25 , K. Yorita 179 , K. Yoshihara 137 , C.J.S. Young 36 , C. Young 153 , J. Yu 78 , R. Yuan 60b , X. Yue 61a , S.P.Y. Yuen 24 , B. Zabinski 84 , G. Zacharis 10 , E. Zaffaroni 54 , J. Zahreddine 136 , A.M. Zaitsev 123,ao , T. Zakareishvili 159b , N. Zakharchuk 34 , S. Zambito 59 , D. Zanzi 36 , D.R. Zaripovas 57 , S.V. Zeißner 47 , C. Zeitnitz 182 , G. Zemaityte 135 , J.C. Zeng 173 , O. Zenin 123 , D. Zerwas 132 , M. Zgubič 135 , D.F. Zhang 15b , F. Zhang 181 , G. Zhang 60a , G. Zhang 15b , H. Zhang 15c , J. Zhang 6 , L. Zhang 15c , L. Zhang 60a , M. Zhang 173 , R. Zhang 60a , R. Zhang 24 , X. Zhang 60b , Y. Zhang 15a,15d , Z. Zhang 63a , Z. Zhang 132 , P. Zhao 49 , Y. Zhao 60b , Z. Zhao 60a , A. Zhemchugov 79 , Z. Zheng 105 , D. Zhong 173 , B. Zhou 105 , C. Zhou 181 , M.S. Zhou 15a,15d , M. Zhou 155 , N. Zhou 60c , Y. Zhou 7 , C.G. Zhu 60b , H.L. Zhu 60a , H. Zhu 15a , J. Zhu 105 , Y. Zhu 60a , X. Zhuang 15a , K. Zhukov 110 , V. Zhulanov 122b,122a , D. Zieminska 65 , N.I. Zimine 79 , S. Zimmermann 52 , Z. Zinonos 115 , M. Ziolkowski 151 , G. Zobernig 181 , A. Zoccoli 23b,23a , K. Zoch 53 , T.G. Zorbas 149 , R. Zou 37 , L. Zwalinski 36 .
. Physics Department, SUNY Albany. Physics Department, SUNY Albany, Albany NY; United States of America.
. Canada, Canada.
. Université Lapp, Grenoble Alpes, FranceUniversité Savoie Mont Blanc, CNRS/IN2P3, AnnecyLAPP, Université Grenoble Alpes, Université Savoie Mont Blanc, CNRS/IN2P3, Annecy; France.
. Physics Department. National and Kapodistrian University ofPhysics Department, National and Kapodistrian University of Athens, Athens; Greece.
. Physics Department. GreeceNational Technical University ofPhysics Department, National Technical University of Athens, Zografou; Greece.
. Azerbaijan Academy of Sciences. AzerbaijanInstitute of PhysicsInstitute of Physics, Azerbaijan Academy of Sciences, Baku; Azerbaijan.
. Spain, Chinese Academy of Sciences. 15Institute of High Energy Physics ; b) Physics Department, Tsinghua University, Beijing; (c) Department of Physics, Nanjing University, Nanjing; (d) University of Chinese Academy of Science (UCAS), Beijing; ChinaSpain. 15(a) Institute of High Energy Physics, Chinese Academy of Sciences, Beijing; (b) Physics Department, Tsinghua University, Beijing; (c) Department of Physics, Nanjing University, Nanjing; (d) University of Chinese Academy of Science (UCAS), Beijing; China.
. Physics Division. Lawrence Berkeley National Laboratory and University of CaliforniaPhysics Division, Lawrence Berkeley National Laboratory and University of California, Berkeley CA;
United States of America. United States of America.
Bogota; Colombia. 23(a) INFN Bologna and Universita' di Bologna, Dipartimento di Fisica; (b) INFN Sezione di Bologna. Facultad de Ciencias y Centro de Investigaciónes, Universidad Antonio Nariño,ItalyFacultad de Ciencias y Centro de Investigaciónes, Universidad Antonio Nariño, Bogota; Colombia. 23(a) INFN Bologna and Universita' di Bologna, Dipartimento di Fisica; (b) INFN Sezione di Bologna; Italy.
. Physikalisches Institut. UniversitätPhysikalisches Institut, Universität Bonn, Bonn; Germany.
Iasi; (d) National Institute for Research and Development of Isotopic and Molecular Technologies. Physics and Informatics. Transilvania University of Brasov, Brasov; (b) Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest; (c) Department of Physics, Alexandru Ioan Cuza University of Iasi, ; (e) University Politehnica Bucharest, Bucharest; ( f ) West University in ; Comenius University, Bratislava; (b) Department of Subnuclear Physics, Institute of Experimental Physics of the Slovak Academy of SciencesPhysics Department. Slovak RepublicTransilvania University of Brasov, Brasov; (b) Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest; (c) Department of Physics, Alexandru Ioan Cuza University of Iasi, Iasi; (d) National Institute for Research and Development of Isotopic and Molecular Technologies, Physics Department, Cluj-Napoca; (e) University Politehnica Bucharest, Bucharest; ( f ) West University in Timisoara, Timisoara; Romania. 28(a) Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava; (b) Department of Subnuclear Physics, Institute of Experimental Physics of the Slovak Academy of Sciences, Kosice; Slovak Republic.
. Física Departamento De, Buenos AiresUniversidad de Buenos AiresDepartamento de Física, Universidad de Buenos Aires, Buenos Aires;
. Argentina , Argentina.
Casablanca; (b) Faculté des Sciences, Université Ibn-Tofail, Kénitra; (c) Faculté des Sciences Semlalia, Université Cadi Ayyad, LPHEA-Marrakech; (d) Faculté des Sciences, Université Mohamed Premier and LPTPM, Oujda; (e) Faculté des sciences. Canada, 35Faculté des Sciences Ain Chock, Réseau Universitaire de Physique des Hautes Energies -Université Hassan II, ; Université Mohammed V, Rabat; MoroccoCanada. 35(a) Faculté des Sciences Ain Chock, Réseau Universitaire de Physique des Hautes Energies -Université Hassan II, Casablanca; (b) Faculté des Sciences, Université Ibn-Tofail, Kénitra; (c) Faculté des Sciences Semlalia, Université Cadi Ayyad, LPHEA-Marrakech; (d) Faculté des Sciences, Université Mohamed Premier and LPTPM, Oujda; (e) Faculté des sciences, Université Mohammed V, Rabat; Morocco.
. Geneva ; Cern, Switzerland, CERN, Geneva; Switzerland.
. LPC. LPC, Université Clermont Auvergne, CNRS/IN2P3, Clermont-Ferrand;
. France, France.
41(a) Dipartimento di Fisica, Università della Calabria, Rende; (b) INFN Gruppo Collegato di Cosenza, Laboratori Nazionali di Frascati. Niels Bohr Institute, University of Copenhagen, Copenhagen; DenmarkItalyNiels Bohr Institute, University of Copenhagen, Copenhagen; Denmark. 41(a) Dipartimento di Fisica, Università della Calabria, Rende; (b) INFN Gruppo Collegato di Cosenza, Laboratori Nazionali di Frascati; Italy.
. Greece, Greece.
. Deutsches Elektronen-Synchrotron DESY. Deutsches Elektronen-Synchrotron DESY, Hamburg and Zeuthen; Germany.
. Infn E Laboratori Nazionali Di Frascati, ; Frascati, Italy, INFN e Laboratori Nazionali di Frascati, Frascati; Italy.
. Physikalisches Institut, Albert-Ludwigs-Universität , Freiburg, Freiburg; GermanyPhysikalisches Institut, Albert-Ludwigs-Universität Freiburg, Freiburg; Germany.
(a) Dipartimento di Fisica, Università di Genova, Genova; (b) INFN Sezione di Genova. GenèveDépartement de Physique Nucléaire et Corpusculaire, Université de GenèveSwitzerland. 55. ItalyDépartement de Physique Nucléaire et Corpusculaire, Université de Genève, Genève; Switzerland. 55(a) Dipartimento di Fisica, Università di Genova, Genova; (b) INFN Sezione di Genova; Italy.
. Physics and Astronomy. United KingdomSUPA -School of Physics and Astronomy, University of Glasgow, Glasgow; United Kingdom.
. Université Lpsc, Grenoble Alpes, Cnrs/In2p3, Inp Grenoble, ; Grenoble, France, LPSC, Université Grenoble Alpes, CNRS/IN2P3, Grenoble INP, Grenoble; France.
60(a) Department of Modern Physics and State Key Laboratory of Particle Detection and Electronics, University of Science and Technology of China, Hefei; (b) Institute of Frontier and Interdisciplinary Science and Key Laboratory of Particle Physics and Particle Irradiation (MOE). Physics and Astronomy. Laboratory for Particle Physics and Cosmology, Harvard University, Cambridge MA; United States of AmericaLaboratory for Particle Physics and Cosmology, Harvard University, Cambridge MA; United States of America. 60(a) Department of Modern Physics and State Key Laboratory of Particle Detection and Electronics, University of Science and Technology of China, Hefei; (b) Institute of Frontier and Interdisciplinary Science and Key Laboratory of Particle Physics and Particle Irradiation (MOE), Shandong University, Qingdao; (c) School of Physics and Astronomy, Shanghai Jiao Tong University, KLPPAC-MoE, SKLPPC, Shanghai; (d) Tsung-Dao Lee Institute, Shanghai; China.
. Kirchhoff-Institut Für Physik, Heidelberg, Heidelberg; GermanyRuprecht-Karls-Universität Heidelberg, Heidelberg; (b) Physikalisches Institut, Ruprecht-Karls-UniversitätKirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Heidelberg; (b) Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg; Germany.
Kong; (c) Department of Physics and Institute for Advanced Study. N T Shatin, Hong Kong, Hong. Faculty of Applied Information Science, Hiroshima Institute of Technology, Hiroshima; Japan. 63(a) Department of Physics, Chinese University of Hong Kong ; b) Department of Physics, University of Hong Kong ; Hong Kong University of Science and Technology, Clear Water BayFaculty of Applied Information Science, Hiroshima Institute of Technology, Hiroshima; Japan. 63(a) Department of Physics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong; (b) Department of Physics, University of Hong Kong, Hong Kong; (c) Department of Physics and Institute for Advanced Study, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong; China.
Sezione di Trieste, Udine; (b) ICTP, Trieste; (c) Dipartimento Politecnico di Ingegneria e Architettura. Infn Gruppo Collegato Di Udine, UdineUniversità di UdineItalyINFN Gruppo Collegato di Udine, Sezione di Trieste, Udine; (b) ICTP, Trieste; (c) Dipartimento Politecnico di Ingegneria e Architettura, Università di Udine, Udine; Italy.
Università del Salento, Lecce; Italy. 68(a) INFN Sezione di Milano; (b) Dipartimento di Fisica, Università di Milano, Milano; Italy. 69(a) INFN Sezione di Napoli; (b) Dipartimento di Fisica. INFN Sezione di Lecce(b) Dipartimento di Matematica e FisicaNapoliUniversità di NapoliItalyINFN Sezione di Lecce; (b) Dipartimento di Matematica e Fisica, Università del Salento, Lecce; Italy. 68(a) INFN Sezione di Milano; (b) Dipartimento di Fisica, Università di Milano, Milano; Italy. 69(a) INFN Sezione di Napoli; (b) Dipartimento di Fisica, Università di Napoli, Napoli; Italy.
Università di Pavia, Pavia; Italy. 71(a) INFN Sezione di Pisa; (b) Dipartimento di Fisica E. Fermi, Università di Pisa, Pisa; Italy. 72(a) INFN Sezione di Roma; (b) Dipartimento di Fisica. Infn Sezione Di Pavia ; B) Dipartimento Di Fisica, Sapienza Università di Roma. ItalyINFN Sezione di Pavia; (b) Dipartimento di Fisica, Università di Pavia, Pavia; Italy. 71(a) INFN Sezione di Pisa; (b) Dipartimento di Fisica E. Fermi, Università di Pisa, Pisa; Italy. 72(a) INFN Sezione di Roma; (b) Dipartimento di Fisica, Sapienza Università di Roma, Roma; Italy.
Roma Tor Vergata; (b) Dipartimento di Fisica. Di Infn Sezione, RomaUniversità di Roma Tor VergataINFN Sezione di Roma Tor Vergata; (b) Dipartimento di Fisica, Università di Roma Tor Vergata, Roma;
. Italy, Italy.
Tre; (b) Dipartimento di Matematica e Fisica. Di Infn Sezione, Roma. ItalyINFN Sezione di Roma Tre; (b) Dipartimento di Matematica e Fisica, Università Roma Tre, Roma; Italy.
INFN-TIFPA; (b) Università degli Studi di Trento. Trento; ItalyINFN-TIFPA; (b) Università degli Studi di Trento, Trento; Italy.
Juiz de Fora; (b) Universidade Federal do. Rio De Janeiro COPPE/EE/IF, Rio de Janeiro; São PauloJoint Institute for Nuclear Research, Dubna; Russia. 80(a) Departamento de Engenharia Elétrica, Universidade Federal de Juiz de Fora (UFJF) ; Universidade Federal de São João del Rei (UFSJ), São João del Rei; (d) Instituto de Física, Universidade de São PauloJoint Institute for Nuclear Research, Dubna; Russia. 80(a) Departamento de Engenharia Elétrica, Universidade Federal de Juiz de Fora (UFJF), Juiz de Fora; (b) Universidade Federal do Rio De Janeiro COPPE/EE/IF, Rio de Janeiro; (c) Universidade Federal de São João del Rei (UFSJ), São João del Rei; (d) Instituto de Física, Universidade de São Paulo, São Paulo;
. Brazil, Brazil.
Energy Accelerator Research Organization, Tsukuba. High Kek, JapanKEK, High Energy Accelerator Research Organization, Tsukuba; Japan.
. Graduate School of Science. JapanGraduate School of Science, Kobe University, Kobe; Japan.
. Nuclear Physics Polish Academy of Sciences. PolandInstitute of Nuclear Physics Polish Academy of Sciences, Krakow; Poland.
. Japan, Japan.
. School of Physics and Astronomy. Queen Mary University of LondonUnited KingdomSchool of Physics and Astronomy, Queen Mary University of London, London; United Kingdom.
. Lunds Fysiska Institutionen, Universitet, ; Lund, Sweden, Fysiska institutionen, Lunds universitet, Lund; Sweden.
Centre de Calcul de l'Institut National de Physique Nucléaire et de Physique des Particules (IN2P3). FranceVilleurbanneCentre de Calcul de l'Institut National de Physique Nucléaire et de Physique des Particules (IN2P3), Villeurbanne; France.
Departamento de Física Teorica C-15 and CIAFF. Universidad Autónoma de Madrid, Madrid; SpainDepartamento de Física Teorica C-15 and CIAFF, Universidad Autónoma de Madrid, Madrid; Spain.
. School of Physics and Astronomy. United KingdomSchool of Physics and Astronomy, University of Manchester, Manchester; United Kingdom.
. Aix-Marseille Cppm, Université, Cnrs/In2p3, ; Marseille, France, CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille; France.
. Canada, Canada.
Stepanov Institute of Physics. B I , National Academy of Sciences of Belarus, Minsk; BelarusB.I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Minsk; Belarus.
. Montreal QC. Group of Particle Physics, University of MontrealGroup of Particle Physics, University of Montreal, Montreal QC;
. Canada, Canada.
Lebedev Physical Institute of the Russian Academy of Sciences. P , Moscow; RussiaP.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow; Russia.
. D V , MoscowSkobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State UniversityD.V. Skobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State University, Moscow;
. Russia, Russia.
Max-Planck, Institut für Physik. München; GermanyWerner-Heisenberg-InstitutMax-Planck-Institut für Physik (Werner-Heisenberg-Institut), München; Germany.
. Nagasaki Institute of Applied Science. JapanNagasaki Institute of Applied Science, Nagasaki; Japan.
. Netherlands, Netherlands.
. Ochanomizu University, Otsuka, Bunkyo-Ku, ; Tokyo, Japan, Ochanomizu University, Otsuka, Bunkyo-ku, Tokyo; Japan.
. L Homer, Dodge Department of Physics and Astronomy, University of Oklahoma, Norman OK; United States of AmericaHomer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman OK; United States of America.
. Université Lal, Paris-Sud, FranceParis-Saclay, OrsayCNRS/IN2P3, UniversitéLAL, Université Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay; France.
. Graduate School of Science. Graduate School of Science, Osaka University, Osaka; Japan.
. Sorbonne Lpnhe, Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3, Paris; FranceLPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3, Paris; France.
140(a) Laboratório de Instrumentação e Física Experimental de Partículas -LIP; (b) Departamento de Física, Faculdade de Ciências. LisboaDepartment of Physics and Astronomy, University of Pittsburgh, Pittsburgh PA; United States of America ; Universidade de Lisboa, Lisboa; (c) Departamento de Física, Universidade de Coimbra, Coimbra; (d) Centro de Física Nuclear da Universidade de Lisboa(e) Departamento deDepartment of Physics and Astronomy, University of Pittsburgh, Pittsburgh PA; United States of America. 140(a) Laboratório de Instrumentação e Física Experimental de Partículas -LIP; (b) Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Lisboa; (c) Departamento de Física, Universidade de Coimbra, Coimbra; (d) Centro de Física Nuclear da Universidade de Lisboa, Lisboa; (e) Departamento de
Granada (Spain); (g) Dep Física and CEFITEC of Faculdade de Ciências e Tecnologia. Física, Universidade do Minho, Braga; ( f ) Universidad de Granada ; Universidade Nova de Lisboa, Caparica; PortugalFísica, Universidade do Minho, Braga; ( f ) Universidad de Granada, Granada (Spain); (g) Dep Física and CEFITEC of Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Caparica; Portugal.
. Physics of the Czech Academy of Sciences. Institute of Physics of the Czech Academy of Sciences, Prague; Czech Republic.
. Faculty of Mathematics and Physics. Charles UniversityCharles University, Faculty of Mathematics and Physics, Prague; Czech Republic.
. Cea Irfu, Université Paris-Saclay, Gif-sur-YvetteIRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette;
. France, France.
147(a) Departamento de Física. United States of America. Santa Cruz Institute for Particle Physics, University of California Santa Cruz ; Pontificia Universidad Católica de Chile, Santiago; (b) Departamento de Física, Universidad Técnica FedericoSanta Cruz Institute for Particle Physics, University of California Santa Cruz, Santa Cruz CA; United States of America. 147(a) Departamento de Física, Pontificia Universidad Católica de Chile, Santiago; (b) Departamento de Física, Universidad Técnica Federico Santa María, Valparaíso; Chile.
. Canada, Canada.
. Physics Department, Royal Institute of Technology. Physics Department, Royal Institute of Technology, Stockholm; Sweden.
. E Andronikashvili, Tbilisi; GeorgiaInstitute of Physics, Iv. Javakhishvili Tbilisi State University, Tbilisi; (b) High Energy Physics Institute, Tbilisi State UniversityE. Andronikashvili Institute of Physics, Iv. Javakhishvili Tbilisi State University, Tbilisi; (b) High Energy Physics Institute, Tbilisi State University, Tbilisi; Georgia.
. Beverly Raymond, Sackler School of Physics and Astronomy. Tel Aviv University, Tel Aviv; IsraelRaymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv; Israel.
. Graduate School of Science and Technology. Graduate School of Science and Technology, Tokyo Metropolitan University, Tokyo; Japan.
. Canada, Canada.
. Vancouver Triumf, Bc, Toronto ONb) Department of Physics and Astronomy, York UniversityTRIUMF, Vancouver BC; (b) Department of Physics and Astronomy, York University, Toronto ON;
. Canada, Canada.
Division of Physics and Tomonaga Center for the History of the Universe, Faculty of Pure and Applied Sciences. University of Tsukuba, TsukubaJapanDivision of Physics and Tomonaga Center for the History of the Universe, Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba; Japan.
. Física Instituto De, IFICCorpuscular, IFICCentro Mixto Universidad de Valencia -CSIC, Valencia; SpainInstituto de Física Corpuscular (IFIC), Centro Mixto Universidad de Valencia -CSIC, Valencia; Spain.
. Canada, Canada.
Armenia. a Also at Centre for High Performance Computing, CSIR Campus. Yerevan Physics Institute, Yerevan; Rosebank, Cape Town; Aix-Marseille Université, CNRS/IN2P3, Marseille; GenèveFrance. d Also at Département de Physique Nucléaire et Corpusculaire, Université de GenèveSouth Africa. b Also at CERN, Geneva; Switzerland. c Also at CPPMYerevan Physics Institute, Yerevan; Armenia. a Also at Centre for High Performance Computing, CSIR Campus, Rosebank, Cape Town; South Africa. b Also at CERN, Geneva; Switzerland. c Also at CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille; France. d Also at Département de Physique Nucléaire et Corpusculaire, Université de Genève, Genève;
East Bay; United States of America. l Also at Department of Physics. Wigner Research Centre for Physics. Switzerland. e Also at Departament de Fisica de la Universitat Autonoma de Barcelona, Barcelona; Spain. f Also at Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Lisboa; Portugal. g Also at Department of Applied Physics and Astronomy, University of Sharjah, Sharjah; United Arab Emirates. h Also at Department of Financial and Management Engineering, University of the Aegean, Chios; Greece. i Also at Department of Physics and Astronomy, University of Louisville, Louisville, KY; United States of America. j Also at Department of Physics and Astronomy, University of Sheffield, Sheffield; United Kingdom. k Also at Department of Physics, California State University ; California State University ; California State University, Sacramento; United States of America. n Also at Department of Physics, King's College London, London; United Kingdom. o Also at Department of Physics, St. Petersburg State Polytechnical University, St. Petersburg; Russia. p Also at Department of Physics, Stanford University, Stanford CA; United States of America. q Also at Department of Physics, University of Adelaide, Adelaide; Australia. r Also at Department of Physics, University of Fribourg ; University of Michigan ; Lomonosov Moscow State University, Moscow; Russia. u Also at Giresun University, Faculty of Engineering, Giresun; Turkey. v Also at Graduate School of Science, Osaka University, Osaka; Japan. w Also at Hellenic Open University, Patras; Greece. x Also at Institucio Catalana de Recerca i Estudis Avancats, ICREA, Barcelona; Spain. y Also at Institut für Experimentalphysik, Universität Hamburg, Hamburg; Germany. z Also at Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen/NikhefUnited States of America. m Also at Department of Physics. Switzerland. s Also at Department of Physics. Netherlands. aa Also at Institute for Nuclear Research and Nuclear Energy (INRNE) of the Bulgarian Academy of Sciences. Bulgaria. ab Also at Institute for Particle and Nuclear PhysicsSwitzerland. e Also at Departament de Fisica de la Universitat Autonoma de Barcelona, Barcelona; Spain. f Also at Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Lisboa; Portugal. g Also at Department of Applied Physics and Astronomy, University of Sharjah, Sharjah; United Arab Emirates. h Also at Department of Financial and Management Engineering, University of the Aegean, Chios; Greece. i Also at Department of Physics and Astronomy, University of Louisville, Louisville, KY; United States of America. j Also at Department of Physics and Astronomy, University of Sheffield, Sheffield; United Kingdom. k Also at Department of Physics, California State University, East Bay; United States of America. l Also at Department of Physics, California State University, Fresno; United States of America. m Also at Department of Physics, California State University, Sacramento; United States of America. n Also at Department of Physics, King's College London, London; United Kingdom. o Also at Department of Physics, St. Petersburg State Polytechnical University, St. Petersburg; Russia. p Also at Department of Physics, Stanford University, Stanford CA; United States of America. q Also at Department of Physics, University of Adelaide, Adelaide; Australia. r Also at Department of Physics, University of Fribourg, Fribourg; Switzerland. s Also at Department of Physics, University of Michigan, Ann Arbor MI; United States of America. t Also at Faculty of Physics, M.V. Lomonosov Moscow State University, Moscow; Russia. u Also at Giresun University, Faculty of Engineering, Giresun; Turkey. v Also at Graduate School of Science, Osaka University, Osaka; Japan. w Also at Hellenic Open University, Patras; Greece. x Also at Institucio Catalana de Recerca i Estudis Avancats, ICREA, Barcelona; Spain. y Also at Institut für Experimentalphysik, Universität Hamburg, Hamburg; Germany. z Also at Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen/Nikhef, Nijmegen; Netherlands. aa Also at Institute for Nuclear Research and Nuclear Energy (INRNE) of the Bulgarian Academy of Sciences, Sofia; Bulgaria. ab Also at Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Budapest;
Also at Institute of High Energy Physics. Hungary, Chinese Academy of Sciences. China. ad Also at Institute of Particle Physics (IPPHungary. ac Also at Institute of High Energy Physics, Chinese Academy of Sciences, Beijing; China. ad Also at Institute of Particle Physics (IPP);
United States of America. am Also at LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3, Paris; France. an Also at Manhattan College. United States of America. av Also at The Collaborative Innovation Center of Quantum Matter (CICQM). Ruston LA; New York NY; Guangzhou; The City College of New York, New York NY; Beijing; Dolgoprudny; Vancouver BC; NapoliCanada. ae Also at Institute of Physics, Academia Sinica, Taipei; Taiwan. a f Also at Institute of Physics, Azerbaijan Academy of Sciences, Baku; Azerbaijan. ag Also at Institute of Theoretical Physics, Ilia State University, Tbilisi; Georgia. ah Also at Instituto de Fisica Teorica ; IFT-UAM/CSIC, Madrid; Spain. ai Also at Istanbul University, Dept. of Physics, Istanbul; Turkey. a j Also at Joint Institute for Nuclear Research, Dubna; Russia. ak Also at LAL, Université Paris-Sud ; CNRS/IN2P3, Université Paris-Saclay, Orsay; France. al Also at Louisiana Tech University ; United States of America. ao Also at Moscow Institute of Physics and Technology State University, Dolgoprudny; Russia. ap Also at National Research Nuclear University MEPhI, Moscow; Russia. aq Also at Physics Department, An-Najah National University, Nablus; Palestine. ar Also at Physics Dept, University of South Africa, ; Sun Yat-sen University ; China. aw Also at Tomsk State University, Tomsk, and Moscow Institute of Physics and Technology State University ; Canada. ay Also at Universita di Napoli ParthenopePretoria; South Africa. as Also at Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Freiburg; Germany. at Also at School of Physics. China. au Also at. Russia. ax Also at TRIUMF. Italy. * DeceasedCanada. ae Also at Institute of Physics, Academia Sinica, Taipei; Taiwan. a f Also at Institute of Physics, Azerbaijan Academy of Sciences, Baku; Azerbaijan. ag Also at Institute of Theoretical Physics, Ilia State University, Tbilisi; Georgia. ah Also at Instituto de Fisica Teorica, IFT-UAM/CSIC, Madrid; Spain. ai Also at Istanbul University, Dept. of Physics, Istanbul; Turkey. a j Also at Joint Institute for Nuclear Research, Dubna; Russia. ak Also at LAL, Université Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay; France. al Also at Louisiana Tech University, Ruston LA; United States of America. am Also at LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3, Paris; France. an Also at Manhattan College, New York NY; United States of America. ao Also at Moscow Institute of Physics and Technology State University, Dolgoprudny; Russia. ap Also at National Research Nuclear University MEPhI, Moscow; Russia. aq Also at Physics Department, An-Najah National University, Nablus; Palestine. ar Also at Physics Dept, University of South Africa, Pretoria; South Africa. as Also at Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Freiburg; Germany. at Also at School of Physics, Sun Yat-sen University, Guangzhou; China. au Also at The City College of New York, New York NY; United States of America. av Also at The Collaborative Innovation Center of Quantum Matter (CICQM), Beijing; China. aw Also at Tomsk State University, Tomsk, and Moscow Institute of Physics and Technology State University, Dolgoprudny; Russia. ax Also at TRIUMF, Vancouver BC; Canada. ay Also at Universita di Napoli Parthenope, Napoli; Italy. * Deceased
| [] |
[
"Chiral Edge Currents For ac Driven Skyrmions In Confined Pinning Geometries",
"Chiral Edge Currents For ac Driven Skyrmions In Confined Pinning Geometries"
] | [
"C Reichhardt \nTheoretical Division and Center for Nonlinear Studies\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n",
"C J Olson Reichhardt \nTheoretical Division and Center for Nonlinear Studies\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA\n"
] | [
"Theoretical Division and Center for Nonlinear Studies\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA",
"Theoretical Division and Center for Nonlinear Studies\nLos Alamos National Laboratory\n87545Los AlamosNew MexicoUSA"
] | [] | We show that ac driven skyrmion lattices in a weak pinning channel confined by regions of strong pinning exhibit edge transport carried by skipping orbits while skyrmions in the bulk of the channel undergo localized orbits with no net transport. The magnitude of the edge currents can be controlled by varying the amplitude and frequency of the ac drive or by changing the ratio of the Magnus force to the damping term. We identify a localized phase in which the orbits are small and edge transport is absent, an edge transport regime, and a fluctuating regime that appears when the ac drive is strong enough to dynamically disorder the skyrmion lattice. We also find that in some cases, multiple rows of skyrmions participate in the transport due to a drag effect from the skyrmionskyrmion interactions. The edge currents are robust for finite disorder and should be a general feature of skyrmions interacting with confined geometries or inhomogeneous disorder under an ac drive. We show that similar effects can occur for skyrmion lattices at interfaces or along domain boundaries for multiple coexisting skyrmion species. The edge current effect provides a new method to control skyrmion motion, and we discuss the connection of these results with recent studies on the emergence of edge currents in chiral active matter systems and gyroscopic metamaterials. | 10.1103/physrevb.100.174414 | [
"https://arxiv.org/pdf/1909.12884v1.pdf"
] | 203,593,528 | 1909.12884 | 5b0b2177a141d8354a5ad07b22435b82ea31a1a0 |
Chiral Edge Currents For ac Driven Skyrmions In Confined Pinning Geometries
C Reichhardt
Theoretical Division and Center for Nonlinear Studies
Los Alamos National Laboratory
87545Los AlamosNew MexicoUSA
C J Olson Reichhardt
Theoretical Division and Center for Nonlinear Studies
Los Alamos National Laboratory
87545Los AlamosNew MexicoUSA
Chiral Edge Currents For ac Driven Skyrmions In Confined Pinning Geometries
(Dated: May 2, 2022)
We show that ac driven skyrmion lattices in a weak pinning channel confined by regions of strong pinning exhibit edge transport carried by skipping orbits while skyrmions in the bulk of the channel undergo localized orbits with no net transport. The magnitude of the edge currents can be controlled by varying the amplitude and frequency of the ac drive or by changing the ratio of the Magnus force to the damping term. We identify a localized phase in which the orbits are small and edge transport is absent, an edge transport regime, and a fluctuating regime that appears when the ac drive is strong enough to dynamically disorder the skyrmion lattice. We also find that in some cases, multiple rows of skyrmions participate in the transport due to a drag effect from the skyrmionskyrmion interactions. The edge currents are robust for finite disorder and should be a general feature of skyrmions interacting with confined geometries or inhomogeneous disorder under an ac drive. We show that similar effects can occur for skyrmion lattices at interfaces or along domain boundaries for multiple coexisting skyrmion species. The edge current effect provides a new method to control skyrmion motion, and we discuss the connection of these results with recent studies on the emergence of edge currents in chiral active matter systems and gyroscopic metamaterials.
I. INTRODUCTION
A paradigmatic example of a system that exhibits edge states or edge currents in confinement is electrons in a magnetic field undergoing cyclotron motion. Here, bulk electrons follow closed circular orbits while charges near the boundaries enter skipping orbits, and the direction of the resulting current is determined by the chirality of the cyclotron motion [1][2][3][4][5]. Edge currents also arise in cold atom systems [6], chiral active matter spinners [7][8][9][10], coupled gyroscopes [11][12][13][14], and colloids placed on periodic magnetic substrates to create colloidal topological insulators [15,16] in analogy to electronic topological insulators [17]. In all of these systems, there is some form of periodic orbit with a particular chirality as well as some type of boundary or interface. Edge currents can also arise for magnetic skyrmions, which have dynamics that are similar in many ways to those of electrons in a magnetic field. Skyrmions can exhibit complex cyclotron orbits with a fixed chirality, and in certain sample geometries, the skyrmions interact with some form of boundary or interface [18,19].
Skyrmions are particle like textures that were originally proposed to appear in magnets in 1989 [20]. Magnetic skyrmion lattices were experimentally observed in 2009 using neutron scattering [21] and were directly imaged with Lorentz microscopy [22]. Since these initial observations, an increasing variety of materials have been identified which support skyrmions, including systems in which skyrmions are stable at room temperature [23][24][25][26][27]. When skyrmions are set into motion by an applied current [28][29][30][31][32][33][34], they exhibit depinning and sliding phases similar to those found for vortices in type-II superconductors and Wigner crystals [35]. Due to their size scale and the fact that they can be moved easily with a current, skyrmions are also candidates for possible memory and computing applications [36][37][38][39], and the understanding of skyrmion dynamics on the individual and collective level will be integral to the creation of such devices. Although skyrmions have many similarities to other particle-based systems that exhibit depinning, they also have several distinct properties, the most prominent of which is the domination of skyrmion dynamics by the Magnus force. Skyrmion motion has many similarities to the dynamics of electrons in a magnetic field; however, the skyrmions can also experience significant damping, and the ratio of the Magnus force to the damping force depends on the material parameters and can produce different dynamical effects [18,19,27,28,40]. The Magnus force generates a skyrmion velocity component that is perpendicular to the net external force acting on the skyrmion. One consequence of a finite Magnus term is that the skyrmions move at an angle with respect to an applied drive that is known as the skyrmion Hall angle [18,19,21]. In the absence of quenched disorder or pinning, the skyrmion Hall angle is constant; however, when quenched disorder is present, the skyrmion Hall angle becomes drive or velocity dependent, starting at a value of nearly zero just above the depinning threshold and approaching the disorder-free limit at higher drives [41][42][43][44][45][46][47][48]. The Magnus force also causes the skyrmions to exhibit cyclotron or spiraling motion when they are in a confining potential or interacting with a pinning site [41,[49][50][51][52][53][54][55][56]. Skyrmions can perform circular orbits under biharmonic drives [57], oscillating fields [58,59], and in certain types of driven bilayer systems [60]. In numerical studies of skyrmions with one-dimensional (1D) periodic and asymmetric substrate arrays, skyrmions under ac, dc, and combined ac and dc drives exhibited complex periodic closed and running orbits [61][62][63].
Since skyrmions can undergo chiral motion when interacting with disorder or subjected to a drive, it is natural to look for edge currents in the presence of a confining geometry when the skyrmions are driven or perturbed in arXiv:1909.12884v1 [cond-mat.mes-hall] 27 Sep 2019 some way. In this paper, we examine skyrmion lattices in a system containing a pin-free channel surrounded by strong pinning, with an initially uniform distribution of skyrmions in the entire sample. Under an ac drive, the skyrmions in the pin-free channel undergo periodic motion. If there were no confinement, the periodic orbits would take the form of 1D paths oriented at an angle to the applied drive; however, when confinement is present, the skyrmions near the edges of the pin-free channel follow circular or elliptical orbits with a chirality that is determined by the sign of the Magnus force. These orbits interact with the edge potential created by the pinned skyrmions and become skipping orbits that generate edge currents moving to the right or left, depending on the side of the pin-free channel at which they appear. The skyrmions in the bulk of the pin-free channel remain localized. We find slip phenomena for the skyrmions participating in the edge transport, and the effectiveness of the edge currents depends on the ratio of the Magnus force to the damping force as well as on the amplitude and frequency of the ac drive and on the skyrmion density.
We identify three regimes of behavior. For ac drives with high frequency or low amplitude, we observe a localized phase in which no edge transport occurs but the orbits at the edges of the pin-free channel are much more circular than the orbits in the bulk of the pin-free channel. We also find a regime of edge transport in which the skyrmions in the pin-free channel form a lattice, as well as a liquid regime in which the ac drive is large enough to melt the skyrmion lattice dynamically. Within the liquid regime, near the transition to the lattice regime, edge transport can still appear in which skyrmions at the edge of the pin-free channel can be transported over some distance before exchanging with skyrmions in the bulk of the pin-free channel; however, deeper within the liquid regime, the edge transport is lost. In the limit of zero Magnus force, there is no edge transport and the orbits are 1D throughout the sample. The edge transport we observe is a collective effect, and it does not appear in the low density limit.
We also find that the width and type of orbit along the edge of the pin-free channel depend on the direction of the applied ac drive. When the drive is parallel to the channel, pronounced edge transport appears, while a drive that is perpendicular to the channel generally produces less edge transport since the orbits become 1D and are oriented parallel to the edge of the channel. In regimes where strong edge transport occurs, the moving skyrmions along the edge can drag the adjacent rows of skyrmions, causing these rows to exhibit a drift of reduced magnitude. These effects are robust for varying amounts of disorder, and in some cases, a disorder-free system that shows no edge transport can develop edge currents when disorder is introduced. In the low density regime the edge transport is generally weak or absent, but we find that commensurate-incommensurate effects between 1D rows of skyrmions can create currents flow-ing in the bulk of the pin-free channel instead of along its edges. Such effects occur when the edge row and an adjacent row contain different numbers of skyrmions, producing a skyrmion gear motion. Finally, we show that edge currents should be a general feature that appears at any type of skyrmion interface, such as along the domain wall separating two different species of skyrmions. These effects should arise for confining geometries under ac drives in samples with inhomogeneous pinning, edge roughness, grain boundaries, or twin boundaries. Our results provide a new method for transporting skyrmions that avoids the skyrmion Hall effect found for dc driven skyrmions.
The paper is organized as follows. In Section II we discuss the system and the simulation method. In section III, we describe the conditions under which edge currents arise and demonstrate the different dynamical regimes. Section IV shows the effect of applying the ac driving along different directions, while in Section V we examine the role of disorder. In Section VI we vary the skyrmion density and explore skyrmion pumping produced by commensurate-incommensurate effects, while in section VII we discuss our results, describe geometries in which the edge currents could arise, and show that edge currents can occur even for pin-free systems at an interface between different skyrmion species. In Section VIII we summarize our results.
II. SIMULATION
We consider a two-dimensional (2D) system of size L × L with periodic boundary conditions in the x and y-directions. Half of the sample contains a square array of pinning sites, and the other half of the sample consists of a pin-free channel aligned with the x direction. The system contains N sk skyrmions and N p pinning sites. The skyrmion density is n sk = N sk /L 2 and the matching density at which the number of skyrmions would equal the number of pins in a sample with uniform pinning is n φ = 2N p /L 2 . The initial positions of the skyrmions are obtained using simulated annealing. The skyrmionskyrmion interactions are repulsive, and in the absence of pinning, the skyrmions form a uniform triangular lattice. After annealing, we apply an ac drive parallel or perpendicular to the pin-free channel. In Fig. 1(a) we illustrate the skyrmion locations and trajectories in a pin-free system under an ac drive applied along the x-direction. Here the skyrmions form a triangular lattice and execute 1D orbits oriented at an angle with respect to the drive direction given by the skyrmion Hall angle θ int sk . In Fig. 1(b), half of the sample contains a square pinning lattice and a pin-free channel is oriented along the x direction. Here, when an ac drive is applied, the skyrmions in the pinning sites remain immobile and produce a confining potential for the skyrmions in the pin-free channel. Within the pin-free channel, the skyrmions form a triangular lattice but no longer undergo the strictly 1D motion of Fig. 1(a). Instead, the skyrmions follow circular or elliptical orbits, with a continuously translating or edge current appearing along the edges of the channel, as highlighted by the arrows. The skyrmions in the bulk of the pin-free channel perform localized orbits. In this case, the skyrmion orbits are counterclockwise, so the skyrmions on the bottom edge of the pin-free channel are moving in the −x direction while those on the top edge of the channel are translating in the +x direction.
We model the skyrmions using a particle based approach for skyrmions interacting with pinning as employed previously [40,41,[61][62][63][64].
The motion of skyrmion i is governed by the following equation of motion:
α d v i + α mẑ × v i = F ss i + F sp i + F ac i .(1)
The first term on the right is the repulsive skyrmionskyrmion interaction force F i = N j =i K 1 (r ij )r ij , where r ij = |r i − r j | is the distance between skyrmions i and j and K 1 (r) is the modified Bessel function which decreases exponentially at large r. The pinning force is given by F sp i and the pinning sites are modeled as parabolic traps of maximum strength F p and radius r p . In this work we set F p sufficiently large that skyrmions within the pinning sites remain immobile for all the parameters we consider. Additionally, r p is small enough that each pinning site captures at most one skyrmion. We focus on the regime in which there are twice as many skyrmions as pinning sites, so that the pin-free and pinned regions contain equal numbers of skyrmions.
An ac driving force F ac is applied in either the x or y direction, F ac = A sin(ωt)(x,ŷ). The term α d is the damping constant which aligns the velocities in the direction of the net force while α m is the coefficient of the Magnus term which aligns the velocity perpendicular to the net force. The dynamics can be characterized by the ratio α m /α d , and in the absence of pinning the skyrmions move at an angle with respect to a dc drive known the intrinsic skyrmion Hall angle θ int sk = arctan(α m /α d ). When α m = 0, the system is in the overdamped limit. The initial positions of the skyrmions are obtained by starting from a high temperature liquid state and cooling to T = 0.0 in order to obtain an overall skyrmion density that is roughly constant in both the pinned and unpinned regions.
III. EDGE TRANSPORT AND DYNAMICAL REGIMES
We first illustrate that skyrmions translate along the edges of the pin-free region. In Fig. 2 Fig. 2. The skyrmion on the top edge of the pin-free channel moves a distance of δx = 30 in the positive x direction during the same time period, and the skyrmion at the center of the channel has no net displacement. The skyrmions that are undergoing a net translation do not exhibit completely periodic motion but occasionally become localized for a period of time when phase slips occur, producing the disorder in the orbits found in Fig. 3(b). Since the lattice constant of the pinning sites is a = 1.56, if the skyrmion lattice was perfectly ordered and the skyrmion orbit perfectly matched the pinning lattice size scale to produce an ideal locking between the cyclic orbit and the periodic drive, a total displacement of |δx ideal | = 110 would occur during the course of n = 70 cycles. The size of the elliptical orbit on the edges of the pin-free channel depends on the frequency and amplitude of the ac drive as well as on the value of α m /α d , so the efficiency |δx|/|δx ideal | of the transport current depends on these parameters.
In Fig. 4 we plot the skyrmion trajectories for the system from Fig. 3 at different values of the ac drive amplitude A and frequency ω. with A = 0.075 at higher frequencies of ω = 6 × 10 −5 and ω = 1 × 10 −4 , respectively. Edge currents are still present when ω = 6×10 −5 in Fig. 4(a), but in Fig. 4(b) at ω = 1 × 10 −4 , the edge transport is lost, with skyrmions at the edge of the pin-free channel executing circular orbits and skyrmions in the bulk of the pin-free channel following 1D trajectories. In general, as the ac drive frequency increases, the size of the skyrmion orbits shrinks, but edge transport only occurs when the orbits are large enough that they either overlap with each other or are wider than the period of the confining potential. Figure 4(c) shows a sample with ω = 3.75×10 −5 at a smaller A = 0.025, where the orbits are small enough that the edge currents are lost. In Fig. 4(d) at ω = 3.75×10 −5 and A = 0.15, the orbits are large enough that the skyrmions in the pin-free channel become strongly disordered and form a dynamical liquid or fluctuating state. In this fluctuating regime, it is still possible for edge transport to occur when skyrmions near the edges of the channel translate for some distance before exchanging with a skyrmion in the bulk of the pin-free channel. Thus, we observe both a crystal with edge transport and a liquid with edge transport. For larger values of A, the skyrmions become more disordered and the edge transport is strongly re- duced or absent, and when A is sufficiently large, the skyrmions at the pinning sites begin to depin and the distinction between skyrmions in the pin-free channel and those in the pinned region is destroyed. In Fig. 5 we plot the average drift distance d = N −1 edge N edge i=1 |δx i | for the N edge skyrmions on the edges of the pin-free channel during a time period of n = 20 ac drive cycles. Figure 5(a,b) shows d versus A and ω, respectively, for the system in Fig. 4 at α m /α d = 15. In both cases there is an optimum drive parameter that maximizes the edge transport, while for low A or high ω, the edge skyrmions are localized. In Fig. 5(a), when A > 0.115 the skyrmion lattice starts to become disordered and forms a liquid state. Figure 5(b) indicates that the edge transport is more efficient at lower frequencies since fewer phase slips occur; however, the orbits also become increasingly one-dimensional as ω decreases, so at small values of ω the edge transport is reduced. In Fig. 5(c,d) we plot d versus A and ω for a system with α m /α d = 5.0, where similar behavior appears.
In Fig. 6 we illustrate the skyrmion trajectories for varied α m /α d . In the overdamped limit of α m /α d = 0.0, shown in Fig. 6(a) for A = 0.075 and ω = 3.75 × 10 −5 , there are no edge currents and the skyrmions in the pin-free channel move in strictly 1D paths aligned with the x-direction. Here the orbits are large enough that they overlap. In Fig. 6(b), the orbits for a system with α m /α d = 0.5, A = 0.025, and ω = 1.5 × 10 −4 are elliptical but there are still no edge currents. The sample with α m /α d = 5.0, A = 0.075, and ω = 3.75 × 10 −5 illustrated in Fig. 6(c) has strong edge transport as well as transport of skyrmions that are up to three rows from the edge of the pin-free channel. This occurs when the edge transport is sufficiently strong that it exerts a drag effect on the adjacent rows, which pick up a net motion at a reduced velocity. In Fig. 7 we illustrate the x-position of individual skyrmions in the top, second from top, and third from top rows in the pin-free channel for the system in Fig. 6(c). Each of these rows has a net transport in the positive x direction, but the magnitude of the transport decreases as the rows become further from the edge of the channel. The forth row from the top of the channel has no net drift. Similarly, at the bottom of the pinfree channel (not shown) the three rows closest to the channel edge are moving in the −x direction. We have also found regimes in which only two rows are moving as well as regimes in which only the edgemost row is moving. Figure 6(d) shows the skyrmion trajectories for a sample with α m /α d = 25, ω = 3.75 × 10 −5 , and A = 0.075. The orbits are smaller than those that appear for the same ac drive parameters at α m /α d = 15 or 10 since the skyrmion orbits become more curved when the Magnus force is larger. When the skyrmion orbits are smaller, the edge currents are reduced in magnitude. In Fig. 8 we plot d versus α m /α d for an ac drive with ω = 3.75 × 10 −5 and A = 0.075. Edge currents are absent when α m /α d is small. For 2.5 < α m /α d < 17.5, multiple rows participate in the edge transport, while for α m /α d ≥ 17.5, only the outer row of the pin-free channel has a net transport. The overall shape of d versus α m /α d depends strongly on the values of ω and A, and if these quantities are too small, d = 0.0.
By varying A, ω, and α m /α d , we identify three regimes of behavior: a localized lattice phase with no edge transport, an edge transport phase in which the skyrmions remain in a lattice structure, and a disordered or fluctuating state. In Fig. 9 we construct a dynamic phase diagram as a function of A versus α m /α d for a system with fixed ω = 3.75 × 10 −5 , highlighting the localized phase, lattice edge transport phase, and fluctuating liquid phase. As α m /α d approaches zero, the localized phase grows in extent. In the overdamped limit, the skyrmions form a triangular lattice that moves elastically back and forth in the x-direction with no net transport, as illustrated in Fig. 6(a). Figure 9 indicates that there is a tendency for the localized region to grow at large values of α m /α d due to the shrinking of the skyrmion orbit size with increasing Magnus force. Within the fluctuating regime, there is still some transport of skyrmions along the edges at smaller values of A; however, for larger A the system becomes more liquid like and the edge transport vanishes. We can construct a similar phase diagram for fixed A and varied ω (not shown), where we find that at high drive frequencies, the system enters the localized regime.
IV. AC DRIVING IN THE PERPENDICULAR DIRECTION
Up until now we have considered a driving force applied along the x-direction, parallel to the pin-free channel. In this orientation, the Magnus force generated by the drive produces a skyrmion velocity component aligned mostly along the y-direction. If the ac drive is instead applied along the y-direction, perpendicular to the pin-free channel, the Magnus-induced velocity is mostly aligned with the x-direction, and as a result, for large α m /α d , the skyrmion trajectories become nearly 1D along the x-direction and the edge currents are absent. In Fig. 10(a) we illustrate the skyrmion trajectories in the absence of pinning for a system with an ac drive applied along the y-direction at A = 0.075, ω = 3.75×10 −5 , and α m /α d = 10. Here the skyrmions follow 1D paths that are aligned mostly in the x-direction. Figure 10(b) shows the same system with pinning present in the overdamped limit of α m /α d = 0.0. The skyrmions in the pin-free channel move in 1D paths aligned in the ydirection and there is no edge transport. In Fig. 10(c) at α m /α d = 10, the skyrmions move in mostly 1D paths aligned with the x-direction, and edge transport is absent. In general, for driving in the y-direction we only observe edge transport for low values of α m /α d and low ac drive frequencies ω. The edge transport is the most efficient when the system is near the transition to the fluctuating state. In Fig. 10(d) at A = 0.075, α m /α d = 2.0, and ω = 5×10 −6 , just past the transition to the fluctuating state, we highlight the trajectory of a single skyrmion that undergoes edge transport when it is near the edge of the pin-free channel. The skyrmion eventually wanders off into the bulk and becomes trapped. In Fig. 11(a,b) we plot d versus ω and d versus α m /α d , respectively, for the system in Fig. 10(d). The frequency and amplitude dependence of the edge currents is similar to that found for driving in the x-direction.
Here there are no edge currents for α m /α d > 3.5 since the orbits become too one-dimensional.
V. EFFECTS OF DISORDER
We next consider the addition of disorder, achieved by applying random offsets to the pinning site locations in both the x and y directions. The offsets are uniformly distributed and have a maximum size of δr, which is always less than 0.5a, where a is the pinning lattice constant. Depending on the parameters, we find that the disorder can enhance or decrease the edge transport. In Fig. 12(a) we illustrate the skyrmion positions and trajectories for a system with δr = 0.5, A = 0.075, ω = 1.025 × 10 −4 , and α m /α d = 15. At this value of ω, the ordered system has no edge transport, as shown in Fig. 5(b). In the presence of sufficiently large disorder, however, edge transport occurs in the two edgemost rows. Figure 12(b) shows the same sample at a higher ac drive frequency of ω = 1.5 × 10 −4 , where the orbits are small enough that the edge currents are lost even in the presence of disorder.
In other cases, the addition of disorder reduces the edge currents and produces intermittent or chaotic motion in which the edge current drops to zero for a period time before becoming finite again. In Fig. 13 we show an example of this behavior in a sample with ω = 3.75 × 10 −5 , A = 0.075, δr = 0.5, and α m /α d = 10, which exhibits pronounced edge transport when δr = 0.0. The plot of the x position of a single skyrmion versus ac cycle number n shows that there are several time intervals during which the skyrmion becomes localized. The edge skyrmions trace the same path during each ac cycle in the localized interval, but this path is chaotic, allowing the skyrmions eventually to jump back into a translating orbit.
In Fig. 14(a) we plot d versus δr for the system in For larger disorder, however, the transport is reduced due to the pinning of the edge current by the disorder. In Fig. 14(b), we show d versus δr for the system in Fig. 13 where the addition of disorder decreases the edge transport. We find similar effects for driving in the y-direction.
VI. DENSITY AND SKYRMION PUMPING EFFECTS
To determine the effects of changing the skyrmion density, we study both the case in which the pinning density is held fixed and the skyrmion density is varied, as well as the case in which the ratio of the number of pins to the number of skyrmions is held fixed at N p /N sk = 1/2 but the matching density n φ of the system is changed. In Fig. 15(a) we plot the skyrmion trajectories at α m /α d = 10, A = 0.075, and ω = 3.75 × 10 −5 at a skyrmion density of n sk = 0.1 and a fixed matching den- sity of n φ = 0.4. The skyrmion density is low enough that no edge transport occurs. Figure 15(b) shows the same system at n sk = 0.5 where there is strong edge transport, and Fig. 15(c) illustrates the trajectories at n sk = 1.0, where the edge transport is reduced and there is an increase in the motion of interstitial skyrmions within the pinned region. In Fig. 15(d), the matching density is increased to n φ = 1.0 and the skyrmion density is n sk = 1.0. Here, at A = 0.025 and ω = 3.75 × 10 −5 , edge currents are present. In Fig. 16(a) we plot d versus n sk for the system in Fig. 15 with a fixed matching density of n φ = 0.4. For n sk < 0.15, there is no edge current, while for 0.4 < n sk < 0.8, pronounced edge transport appears which falls off when n sk > 0.8. The decrease in the edge transport with increasing n sk at higher densities occurs because the number of interstitial skyrmions in the pinned region is increasing, contributing a fluctuating component of increasing magnitude to the edge potential. In addition, at higher skyrmion densities, the skyrmion orbits become compressed due to the decrease in the skyrmion lattice constant. This effect is similar to what we find for reduced ac amplitude or higher ac drive frequencies, both of which suppress the edge transport. The peak value in d is produced by a resonance effect in which the width of the skyrmion orbits along the channel edge locks to the periodicity of the confining potential created by the pinned skyrmions. In Fig. 17 we plot the x position of a single skyrmion on the lower edge of the pin-free region in the system from Fig. 15(a,b,c) with n φ = 0.4 at three different skyrmion densities. At n sk = 0.1, the skyrmion undergoes a brief transient motion before becoming localized. When n sk = 0.5, the skyrmion translates rapidly, moving nearly one pinning lattice constant per ac cycle, giving a transport that is close to optimal. At n sk = 1.0, the edge current is strongly reduced.
In Fig. 16(b) we plot d versus the matching density n φ in samples with n sk = n φ for the same parameters as in Fig. 16(a). Here, a finite edge current appears only when n φ > 0.3, and the edge transport remains robust as n φ is further increased. This result indicates that edge currents should be a general effect which can be observed whenever collective interactions between the skyrmions are important.
Although we do not find any edge transport when n φ < 0.3, we observe a different type of translating orbit that we call a skyrmion pump effect. In Fig. 18, we illustrate the pump effect in a sample with n φ = 0.1, n sk = 0.1, A = 0.075, ω = 3.75 × 10 −5 , and α m /α d = 10.0. The skyrmion trajectories, obtained over a time period of n = 250 cycles, indicate that the skyrmions in the second row from the top of the pin-free channel are undergoing transport, while the top row remains localized. The pump effect flow is much slower than the edge transport motion found at higher n φ and it is also oriented in the opposite direction, with skyrmions mov- ing in the negative x direction instead of in the positive x direction. In Fig. 19 we plot the x position versus ac drive cycle number n for a skyrmion in the second row from the top of the pin-free channel in the system from Fig. 18. The skyrmion is moving in the negative x-direction and exhibits x position oscillations which are much smaller than those found for edge transport. For comparison, we also plot the x position of a skyrmion in the top row of the pin-free channel, which becomes localized after an initial transient motion. The skyrmion pump effect arises when the number of skyrmions in the top row of the pin-free channel is incommensurate with the number of skyrmions in the adjacent row. Here, the top row contains 11 skyrmions while the second row from the top contains 12 skyrmions. In contrast, both of the bottom two rows in the pin-free channel contain 11 skyrmions. The incommensuration causes the skyrmions in the top row to act as an effective gear that gradually translates the skyrmions in the second row from the top. If the number of skyrmions in the adjacent row is smaller rather than larger than the number of skyrmions in the top row (10 skyrmions instead of 12), the pump flow direction is reversed. We observe the pump effect when 0.05 < n φ < 0.2 in samples with n sk = n φ , and it is always associated with an incommensuration between the edge row and an adjacent row. For small densities n φ ≤ 0.05, the skyrmions are so far apart that their interactions become unimportant and the pump effect disappears. We note that a similar pump effect could also occur at higher densities n φ ≥ 0.2 where edge transport appears; however, since the pump effect is very weak, it is difficult to detect in the presence of the much stronger edge transport.
VII. DISCUSSION
Edge states have previously been proposed to occur in skyrmion systems, such as for frustrated magnets [65] where a dc current can induce motion on the edges of the sample; however, this effect is very different from the edge transport we propose here and it occurs due to a different mechanism. Chiral magnonic edge states for antiferromagnetic skyrmion crystals have also been proposed [66], but these differ from the edge current and skyrmion transport that we consider here. We note that our results do show similarities with recent studies of edge transport in chiral active matter or active spinning systems, where directed transport can occur near the edge of the sample or between regions of spinners of opposite chirality [7]. There are still several differences in that the chiral active spinning particles always undergo circular motion, whereas in the absence of confinement, skyrmions subjected to an ac drive move along 1D paths, with circular orbits appearing only due to the presence of a confining potential or similar quenched disorder. The pinning potential we consider is produced by pinned skyrmions; however, it is possible to create other types of confining potentials, such as by using nanoscale pinning sites that repel skyrmions, by modifying the materials properties along the edges or in stripe patterns [67,68], or by using nanowires with rough edges [69].
Beyond nanostructured geometries, our work suggests that skyrmion edge transport could also be observed at the interface between lattices of two different skyrmion species, between skyrmions with different chiralities [70], or along grain boundaries or regions of coexisting skyrmion and helical states [71][72][73][74]. In Fig. 20(a) we illustrate an example of an edge state that appears in a pin-free system at the interface between different skyrmion species. We consider a system with α m /α d = 10, ω = 1.25 × 10 −4 , A = 0.075, and n sk = 0.2 where half of the skyrmions have a Magnus term that is opposite in sign to the other half of the skyrmions. The skyrmions of opposite sign are placed in a band aligned with the x direction. Under an ac drive applied in the x-direction, elliptical orbits appear along the domain boundaries separating the two species, while in the bulk of each domain, the skyrmions follow 1D trajectories. In Fig. 20(b) we show the same system at n sk = 0.4, where clear edge currents emerge at the boundaries between the species.
Other methods of introducing confining boundaries in the sample include using inhomogeneous pinning strength so that skyrmions in one region of the sample are more strongly (or less strongly) pinned than skyrmions in adjacent regions. It would be interesting to examine the effects of confinement on different types of skyrmions, such as antiferromagnetic skyrmions [75][76][77]. If the Magnus force is absent, the skyrmion dynamics would be con- sistent with what is found in an overdamped system and the edge transport would be absent. There are also other possible systems that could exhibit edge currents, such as the trochoidal motion of skyrmions in certain bilayer systems [60] as well as the dynamics of anti-skyrmion lattices [77], meron lattices [78], or polar skyrmions [79].
In this work we considered only a single ac drive applied either parallel or perpendicular to the pin-free channel. It is possible, however, to apply more complex ac drives such as F ac = A sin(ωt)x + A cos(ωt)ŷ. A circular ac drive of this type applied to an underdamped system can create commentate-incommensurate transitions and localized or delocalized states in the presence of periodic or random disorder [80,81]. Addition of an asymmetry in either the substrate or the circular orbit itself can produce directed motion in the overdamped system [82][83][84][85], so a rich array of phenomena is expected to appear for skyrmions under multiple ac drives. For example, ratcheting skyrmion motion was observed in work on biharmonic drives [57]. In Fig. 21(a) we show the trajectories for a skyrmion system with n φ = 0.1, n sk = n φ , α m /α d = 10, A = 0.075, and ω = 3.75 × 10 −5 under a circular ac drive. In this case there is no edge current; however, the skyrmions in the bulk of the pin-free channel follow orbits that are much more circular than the orbits of the skyrmions at the edges of the channel, in contrast to what we find for a linear ac drive. In Fig. 21(b), the same system at a matching density of n = 0.2 exhibits edge currents, with particularly rapid transport occurring along the lower edge of the pin-free channel.
Studies of gyroscopic metamaterials [11,13,14] showed that perturbations migrate to the edge of the sample and propagate around the edge in one direction. In our system, for very weak damping or small α m , it would be possible to perturb skyrmions in the bulk of the pinfree channel and obtain motion that is localized on the edges of the channel, producing a transient current which eventually damps away. Since the skyrmion lattice has many similarities to gyroscopic metamaterials, other effects observed in the latter system could be relevant for skyrmions in confinement, such as the odd viscosity and odd elasticity found in driven chiral matter [86][87][88].
It would also be possible to use an edge current to create a device. For example, in a race track geometry filled with a skyrmion lattice, an ac drive could cause a signal to propagate along the edge of the race track. One advantage of this mode of operation is that the propagating skyrmions will exhibit a skyrmion Hall angle equal to zero, eliminating the problem of having skyrmions escape from the edges of the track as they move. Our results suggest that the propagation speed of the signal will not monotonically increase with increasing ac frequency, but will instead drop to zero at high frequencies. In our work we have considered only rigid skyrmions, but in continuum systems, additional edge modes could arise such as the prorogation of breathing modes or perturbations of the internal modes of the skyrmions.
VIII. SUMMARY
In summary, we have shown that for a skyrmion lattice in a confining pinned geometry, application of an ac drive combined with the intrinsic Magnus force produces circular orbits of skyrmions near the edge of the pin-free channel. This can generate an edge current of skyrmions in which the direction of transport is controlled by the sign of the Magnus force and the orientation of the edge. Simultaneously, skyrmions in the bulk of the pin-free channel are localized and follow closed periodic orbits. The magnitude of the edge current depends on the ratio of the Magnus force to the damping term as well as on the amplitude and frequency of the ac drive. We identify three dynamic phases: a localized state in which no edge transport occurs, a lattice state with edge transport, and a disordered fluctuating state that appears for high ac drive amplitude. In regimes where the edge transport is strong, the transport can extend beyond the edges and can involve one or two additional rows of skyrmions adjacent to the edgemost row. In the overdamped limit, the skyrmion orbits become one-dimensional and the edge transport is lost. The edge currents are the most pronounced when the ac drive is parallel to the pin-free channel since the Magnus force induces motion perpendicular to the driving direction. A parallel drive pushes the skyrmions against the edges of the confining potential and induces the circular orbits that are necessary to generate the edge transport. In contrast, when the ac drive is applied perpendicular to the pin-free channel, the motion of the skyrmions is more one dimensional, the coupling to the confining potential is reduced, and the edge transport is diminished. We show that these results are robust against disorder and that in some cases the addition of disorder can induce edge transport. For lower skyrmion densities, we observe a skyrmion pump effect produced by an incommensuration between the number of skyrmions in the row at the edge of the pin-free channel and the number of skyrmions in the adjacent pin-free row. The emergence of edge currents should be a generic feature of driven or excited skyrmion lattices that are subjected to confinement or that contain an interface, and as an example we demonstrate that an ac drive can induce edge transport along the domain boundary separating two different species of skyrmions.
ACKNOWLEDGMENTS
This work was supported by the US Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the U. S. Department of Energy (Contract No. 892333218NCA000001).
. 1. (a) Skyrmion positions (dots) and trajectories (lines) for a system without any pinning at a skyrmion density of n sk = 0.2 and a matching density of n φ = 0.2 for αm/α d = 10.0 under an x-direction ac drive with amplitude A = 0.075 and frequency ω = 3.75 × 10 −5 . The skyrmions form a triangular lattice and move at an angle of θ = 84 • with respect to the driving direction. (b) The same for a sample in which half of the system is filled with a square array of pinning sites (open circles). The skyrmions in the pin-free channel form a triangular lattice and execute closed counterclockwise orbits, while the skyrmions in the pinned region remain immobile. Edge transport occurs along the boundaries of the pin-free channel, as indicated by the arrows.
we plot the x position versus ac cycle number n for individual skyrmions on the bottom edge, top edge, and center of the pinfree channel in a system with α m /α d = 15, n sk = 0.4, n φ = 0.4, A = 0.075, and ω = 3.75 × 10 −5 . All three skyrmions start at an x position near x = 29. In Fig. 3(a) position vs ac cycle number n for individual skyrmions at the bottom edge of the pin-free channel (blue), the top edge of the pin-free channel (red), and in the center of the pin-free channel (green) in a sample with x direction driving, αm/α d = 10, A = 0.075, ω = 3.75 × 10 −5 , skyrmion density n sk = 0.4, and matching density n φ = 0.4. The skyrmion on the top edge moves in the positive x direction, the skyrmion on the bottom edge moves in the negative x direction, and the skyrmion in the center remains localized.The trajectories of these three skyrmions are illustrated inFig. 3(b). The jumps between x = 0 and x = 36 occur when the skyrmion crosses the periodic boundary.we illustrate the skyrmion positions, pinning site locations, and trajectories for all skyrmions from the system inFig. 2, while inFig. 3(b)we trace the trajectories of only the three skyrmions shown inFig. 2. The skyrmion on the bottom edge of the pin-free channel moves in the −x direction, undergoing a displacement of δx = −45 during the course of 70 ac cycles, as shown in
Figure 4 FIG. 3 .
43Skyrmion positions (dots), all skyrmion trajectories (lines), and pinning site locations (open circles) for the system in Fig. 2 with x direction ac driving illustrating the edge transport. Here αm/α d = 10, A = 0.075, ω = 3.75 × 10 −5 , and n sk = n φ = 0.4. (b) Image of the same system with the trajectories of only the three skyrmions shown in Fig. 2 plotted. Red: skyrmion at the top of pin-free channel; green: skyrmion in the center of pin-free channel; blue: skyrmion at the bottom of the pin-free channel.
FIG. 4 .
4Skyrmion positions (dots), trajectories (lines), and pinning site locations (open circles) for the system in Fig. 2 at different x direction ac drive frequencies and amplitudes, where αm/α d = 10 and n sk = n φ = 0.4. (a) For ω = 6 × 10 −5 and A = 0.075, the orbits are smaller but edge transport persists. (b) At ω = 1 × 10 −4 and A = 0.075, the orbits are small enough that no edge transport occurs. (c) At ω = 3.75 × 10 −5 and A = 0.025, there is also no edge transport due to the small size of the orbits. (d) At ω = 3.75×10 −5 and A = 0.15, the system is in a disordered or fluctuating state.
FIG. 5 .
5The average drift distance d for the edge skyrmions during 20 ac drive cycles for the system in Fig. 4(a) with x direction driving and n sk = n φ = 0.4. (a) d vs A for a sample with αm/α d = 15 at ω = 3.75 × 10 −5 . (b) d vs ω for the system in (a) at A = 0.075. (c) d vs A for a sample with αm/α d = 5.0 at ω = 3.75 × 10 −5 . (d) d vs ω for the system in (c) at A = 0.075.
FIG. 6 .
6Skyrmion positions (dots), trajectories (lines), and pinning site locations (open circles) for varied ratios of the Magnus force to the damping term in samples with x direction driving and n sk = n φ = 0.4. (a) At A = 0.075 and ω = 3.75 × 10 −5 for αm/α d = 0.0 in the overdamped limit, there is no edge transport and the skyrmions move in strictly 1D orbits. (b) At A = 0.025 and ω = 1.5×10 −4 for αm/α d = 0.5, there is no edge transport and the skyrmions move in elliptical paths. (c) At A = 0.075 and ω = 3.75 × 10 −5 for αm/α d = 5, there are multiple rows participating in the edge transport. (d) At A = 0.075 and ω = 3.75 × 10 −5 for αm/α d = 20, the edge transport involves only the edgemost rows and the skyrmion orbits are smaller.
FIG. 7 .
7x position vs ac cycle number n for individual skyrmions in the first (blue), second (red), and third (green) rows from the top of the pin-free channel for the sample inFig. 6(c) with x direction driving at n sk = n φ = 0.4, αm/α d = 5.0, ω = 3.75 × 10 −5 , and A = 0.075. Multiple rows are moving but the transport is reduced for rows that are further from the edge of the pin-free channel.
FIG. 8 .
8d vs αm/α d for an x direction ac drive with ω = 3.75 × 10 −5 and A = 0.075, showing that d goes to zero in the overdamped limit of αm/α d = 0.0. Here n sk = n φ = 0.4.
FIG. 9 .
9Dynamic phase diagram as a function of A vs αm/α d for fixed ω = 3.75 × 10 −5 and x direction driving in samples with n sk = n φ = 0.4. Squares (yellow): localized state; triangles (green): crystal state with edge currents; circles (red): disordered fluctuating state.
10. Skyrmion positions (dots), trajectories (lines), and pinning site locations (open circles) for systems with n sk = n φ = 0.4 in which the ac drive is applied in the y-direction. (a) A sample with no pinning at A = 0.075, ω = 3.75 × 10 −5 and αm/α d = 10. The trajectories are 1D in nature and are aligned mostly along the x-direction. (b) The same sample with pinning at αm/α d = 0.0 for A = 0.075 and ω = 3.75 × 10 −5 , where there are no edge currents. (c) At αm/α d = 10.0 for A = 0.075 and ω = 3.75 × 10 −5 , edge currents appear. (d) A sample with αm/α d = 2.0, A = 0.075, and ω = 5 × 10 −6 in the fluctuating state where the trajectory of only a single skyrmion is plotted. This skyrmion undergoes edge transport before becoming trapped in the bulk of the pin-free channel.
11. (a) d vs ω at αm/α d = 2 and (b) d vs αm/α d at ω = 5 × 10 −6 for the system inFig. 10(d)with y direction driving, n sk = n φ = 0.4, and A = 0.075.
FIG. 12 .
12Skyrmion positions (dots), trajectories (lines), and pinning site locations (open circles) for samples in which disorder has been added in the form of random offsets δr of the pinning site locations. Here, the driving is in the x direction and n sk = n φ = 0.4. (a) At ω = 1.025 × 10 −4 , A = 0.075, αm/α d = 15, and δr = 0.5, there is disorder-induced edge transport. (b) In the same system at ω = 1.5 × 10 −4 , there is no edge transport.
Fig. 12 (
12a), where we find an increase in the edge current for intermediate values of δr.
.x position vs ac cycle number n for a single skyrmion in a system with x direction driving, n sk = n φ = 0.4, ω = 3.75 × 10 −5 , A = 0.075, δr = 0.5, and αm/α d = 10. The disorder causes the skyrmion to become localized for extended periods of time.FIG. 14. d vs δr for the system in Fig. 12(a) with x direction driving showing that addition of disorder can induce an edge current. Here n sk = n φ = 0.4, ω = 1.025 × 10 −4 , A = 0.075, and αm/α d = 15. (b) d vs δr for the system in Fig. 13 at n sk = n φ = 0.4, ω = 3.75 × 10 −5 , A = 0.075, and αm/α d = 10. where addition of disorder reduces the edge current.
15. Skyrmion positions (dots), trajectories (lines), and pinning site locations (open circles) for a system with x direction driving at A = 0.075, ω = 3.75 × 10 −5 , and αm/α d = 10 for fixed matching density n φ = 0.4 and varied skyrmion density. (a) At n sk = 0.1, there is no edge current. (b) At n sk = 0.5, there is strong edge transport. (c) At n sk = 1.0, there is considerable disorder in the trajectories and the edge transport is reduced. (d) A system with a matching density of n φ = 1.0 and skyrmion density n sk = 1.0 for A = 0.25, ω = 3.75 × 10 −5 , and αm/α d = 10, showing edge transport.
FIG. 16. (a) d vs n sk for a system with a fixed matching density of n φ = 0.4 under x direction driving at ω = 3.75 × 10 −5 , A = 0.075, and αm/α d = 10. The edge currents are lost at low skyrmion density and diminish with increasing n sk at high skyrmion density. (b) d vs the matching density n φ in samples with n sk = n φ , ω = 3.75 × 10 −5 , A = 0.075, and αm/α d = 10, showing that the edge transport is robust over a wide range of system densities. 17. x positions vs ac cycle number n for an individual skyrmion on the lower edge of the pin-free channel in the system fromFig. 15(a,b,c) with x direction driving at A = 0.075, ω = 3.75 × 10 −5 , αm/α d = 10, and n φ = 0.4. When n sk = 0.1 (blue), the skyrmion is localized. For n sk = 0.5 (red) there is strong edge transport, while at n sk = 1.0 (green) the edge transport is reduced.
18. Skyrmion positions (dots), trajectories (lines), and pinning site locations (open circles) for a system with x direction driving and a matching density of n φ = 0.1 at n sk = 0.1, A = 0.075, ω = 3.75 × 10 −5 , and αm/α d = 10. A different type of current, termed a skyrmion pump effect, occurs in the second row from the top of the pin-free channel. The skyrmion pump effect appears due to an incommensuration between this second row and the row at the upper edge of the pin-free channel. The skyrmions are propagating in the negative x-direction, opposite to the edge current motion that appears for higher n φ .
19. x position vs ac cycle number n for a skyrmion in the second row from the top of the pin-free channel (blue) and a skyrmion in the top row (green) for the system inFig. 18with x direction driving at n sk = n φ = 0.1, A = 0.075, ω = 3.75 × 10 −5 , and αm/α d = 10. The pumped skyrmion is moving in the −x direction, which is opposite to the direction of motion found for edge transport of the top row at higher n φ .
FIG. 20 .
20Skyrmion positions (dots) and trajectories (lines) in a system with x direction driving and no pinning containing two different skyrmion species (red and green) that have Magnus terms of the opposite sign. (a) At n sk = 0.2, αm/α d = 10, A = 0.075, and ω = 1.25×10 −4 , the largest circular orbits appear along the boundary between the two species. (b) For the same parameters but at n sk = 0.4, edge currents are flowing along the domain walls separating the two species.
FIG. 21 .
21Skyrmion positions (dots), trajectories (lines), and pinning site locations (open circles) for a system with a circular ac drive applied along the x direction. (a) At αm/α d = 10, A = 0.075, ω = 3.75 × 10 −5 , n φ = 0.1, and n sk = n φ , the orbits are localized. (b) In the same system at n φ = 0.2 and n sk = n φ , edge currents are present.
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| [] |
[
"Building an irreversible Carnot-like heat engine with an overdamped harmonic oscillator A. Prados",
"Building an irreversible Carnot-like heat engine with an overdamped harmonic oscillator A. Prados"
] | [
"Carlos A Plata ",
"David Guéry-Odelin ",
"Emmanuel Trizac ",
"\nDipartimento di Fisica e Astronomia \"Galileo Galilei\", INFN\nLaboratoire Collisions, Agrégats, Réactivité, IRSAMC\nUniversità di Padova\nVia Marzolo 835131PadovaItaly\n",
"\nUniversité de Toulouse\nCNRS\nUPS\nToulouseFrance\n",
"\nFísica Teórica\nUniversité Paris-Saclay\nCNRS, LPTMS\n91405OrsayFrance\n",
"\nUniversidad de Sevilla\nApartado de Correos 1065, E41080SevillaSpain\n"
] | [
"Dipartimento di Fisica e Astronomia \"Galileo Galilei\", INFN\nLaboratoire Collisions, Agrégats, Réactivité, IRSAMC\nUniversità di Padova\nVia Marzolo 835131PadovaItaly",
"Université de Toulouse\nCNRS\nUPS\nToulouseFrance",
"Física Teórica\nUniversité Paris-Saclay\nCNRS, LPTMS\n91405OrsayFrance",
"Universidad de Sevilla\nApartado de Correos 1065, E41080SevillaSpain"
] | [] | We analyse non-equilibrium Carnot-like cycles built with a colloidal particle in a harmonic trap, which is immersed in a fluid that acts as a heat bath. Our analysis is carried out in the overdamped regime. The cycle comprises four branches: two isothermal processes and two locally adiabatic ones. In the latter, both the temperature of the bath and the stiffness of the harmonic trap vary in time, but in such a way that the average heat vanishes for all times. All branches are swept at a finite rate and, therefore, the corresponding processes are irreversible, not quasi-static. Specifically, we are interested in optimising the heat engine to deliver the maximum power and characterising the corresponding values of the physical parameters. The efficiency at maximum power is shown to be very close to the Curzon-Ahlborn bound over the whole range of the ratio of temperatures of the two thermal baths, pointing to the near optimality of the proposed protocol. | 10.1088/1742-5468/abb0e1 | [
"https://arxiv.org/pdf/2006.09426v1.pdf"
] | 219,721,515 | 2006.09426 | bda7e3c38507743ce98dee36edbed5a10eddf39e |
Building an irreversible Carnot-like heat engine with an overdamped harmonic oscillator A. Prados
Carlos A Plata
David Guéry-Odelin
Emmanuel Trizac
Dipartimento di Fisica e Astronomia "Galileo Galilei", INFN
Laboratoire Collisions, Agrégats, Réactivité, IRSAMC
Università di Padova
Via Marzolo 835131PadovaItaly
Université de Toulouse
CNRS
UPS
ToulouseFrance
Física Teórica
Université Paris-Saclay
CNRS, LPTMS
91405OrsayFrance
Universidad de Sevilla
Apartado de Correos 1065, E41080SevillaSpain
Building an irreversible Carnot-like heat engine with an overdamped harmonic oscillator A. Prados
We analyse non-equilibrium Carnot-like cycles built with a colloidal particle in a harmonic trap, which is immersed in a fluid that acts as a heat bath. Our analysis is carried out in the overdamped regime. The cycle comprises four branches: two isothermal processes and two locally adiabatic ones. In the latter, both the temperature of the bath and the stiffness of the harmonic trap vary in time, but in such a way that the average heat vanishes for all times. All branches are swept at a finite rate and, therefore, the corresponding processes are irreversible, not quasi-static. Specifically, we are interested in optimising the heat engine to deliver the maximum power and characterising the corresponding values of the physical parameters. The efficiency at maximum power is shown to be very close to the Curzon-Ahlborn bound over the whole range of the ratio of temperatures of the two thermal baths, pointing to the near optimality of the proposed protocol.
Introduction
The investigation of heat engines is a pillar of classical thermodynamics [1]. The practical interest of the conversion of thermal energy into mechanical work led to unravel the laws of thermodynamics. These laws have been well formulated since the 19th century for macroscopic systems, for which fluctuations are negligible. In this context, the Carnot heat engine has played a major role: the Carnot cycle comprises two isothermal and two adiabatic branches, which are swept in a quasi-static, reversible, way. This reversible Carnot heat engine maximises the efficiency but the infinite time operation entails that the delivered power vanishes. In the adiabatic branches, the system is thermally isolated from the bath and there is no heat exchange-moreover, reversibility implies that there is no entropy variation either.
The extension of thermodynamic results to mesoscopic systems, where fluctuations are of paramount importance, is not straightforward ; stochastic thermodynamics has been developed to this end [2][3][4], the focus of which lies on non-equilibrium dynamics. In recent years, researchers have looked into the possibility of speeding up the relaxation of physical systems between two given equilibrium states [5][6][7][8][9][10][11]. The "engineered swift equilibration" (ESE) techniques that emerged, which have also been referred to as "shortcut to isothermality" are essentially equivalent and can be viewed as the counterpart in the classical realm of the "shortcuts to adiabaticity" (STA) developed in quantum systems [12]. Both STA and ESE processes make it possible to connect given initial and target states in a time that is much shorter than the natural characteristic relaxation time of the system at hand. To avoid confusion, it is perhaps worthwhile pointing that "adiabaticity" in "STA" refers to a slow variation, not to he absence of heat transfer, at variance with terminology to be used below.
STA and ESE techniques make it possible to build heat engines that connect equilibrium states in a finite time, i.e. in an irreversible way. Therefore, the irreversible counterparts of the classical heat engines can be constructed at the mesoscopic level. In fact, our main goal is building an irreversible version of the Carnot heat engine with a colloidal particle in a harmonic trap of stiffness k, immersed in a fluid at equilibrium with temperature T . This system is relevant from both the theoretical and experimental standpoints. However, a difficulty arises: for mesoscopic systems, it is impossible to completely decouple the system from the heat bath to thermally isolate it-the interaction between a Brownian particle and the fluid in which it is immersed cannot be switched off. Moreover, zero heat and no entropy increment are not equivalent for finite time processes.
The above discussion entails that the definition of adiabatic-in the thermodynamical sense-process is far from trivial at the mesoscale. Notwithstanding, very recently, finite-time adiabatic processes have been characterised for a wide class of mesoscopic systems [13], in the overdamped description of the dynamics. In these processes, the average heat vanishes for all times but there is entropy creation, as imposed by the second principle. We employ these finite-time adiabatic processes to build the corresponding adiabatic branches of the irreversible Carnot engine. Therefore, our approach differs from other recent attempts to construct an irreversible Carnot engine [14][15][16][17][18], the limitations of which are discussed in what follows. Specifically, we focus on the respective definitions of "adiabaticity". In Ref. [14], working in the overdamped regime, the term adiabatic has been employed for a process in which the bath temperature T is instantaneously changed, while the configurational distribution is frozen. However, as already noted by the authors of that work, neither heat nor the entropy increment vanishes in such a process, which are thus non-adiabatic, because of the kinetic contri-bution thereto. In Refs. [15,17,18], the adiabatic branches are constructed by changing both the temperature of the bath and the stiffness k of the trap but keeping the ratio T 2 /k constant, which is obtained in the underdamped description. Nevertheless, the condition T 2 /k = const. has been shown to correspond to isoentropic processes only in the quasi-static limit [15,19], so such a process is not adiabatic either for finite time operation. ‡ A completely different approach is proposed in Ref. [16]. Therein, the oscillator follows a Hamiltonian dynamics and is completely decoupled from the heat bath during the adiabatic branches, a procedure that cannot be implemented with a Brownian particle immersed in a fluid.
The performance of a heat engine is characterised by its efficiency and power. The maximum efficiency achievable operating between a hot bath at temperature T h and a cold bath at temperature T c is the well-known Carnot efficiency η C = 1−T c /T h . However, it is only reached for infinite time operation, which makes the power vanish. The cycle must be swept in a finite time to yield a nonzero power output. This acceleration of the process entails a non-equilibrium dynamics and reduces the reachable efficiency. The study of efficiency at maximum power is a classical problem associated with the field of finite-time thermodynamics [20][21][22][23][24][25]. Curzon and Ahlborn derived that the efficiency at maximum power for a macroscopically endoreversible heat engine is given by
η CA = 1 − T c /T h [20].
There is no general proof ensuring that the efficiency of any arbitrary heat engine at maximum power is bounded by the Curzon-Ahlborn value. Nevertheless, myriads of different studies hint at the existence of some universal properties, connected to the Curzon-Ahlborn bound, for the efficiency at maximum power. Specifically, it has been proven that in the limit of small relative temperature difference, the two first terms in the expansion of the efficiency at maximum power in the Carnot efficiency are universal [26][27][28]. This finding is completely consistent with the results for the efficiency at maximum power in different stochastic heat engines constructed either with a Brownian particle [14], a Feynman ratchet [29], or a quantum dot [30].
Another main objective of our work is the optimisation of the irreversible Carnot engine. Specifically, in connection with the discussion above, we are interested in looking into the optimisation in a sense to be specified soon below, of the delivered power and its associated efficiency. In this regard, the optimal protocols for isothermal and the adiabatic branches, which have been explicitly worked out recently [13,14,31,32], play a crucial role. It appears that work should be minimised in the isothermal processes [14,31,32], whereas the connection time is minimised in the adiabatic ones [13].
The rest of the paper is organised as follows. In section 2, we introduce the model system with which we construct our heat engine: a Brownian particle moving in a harmonic trap. Special attention is paid to its energetics. Section 3 is devoted to putting forward the optimal protocols for both the isothermal and adiabatic branches. ‡ Reference [13], although working in the overdamped description, incorporates the kinetic contribution to the energy balance. Therein, the ratio T 2 /k has been shown to be a non-decreasing function of time for finite-time adiabatic processes, being constant only in the quasi-static limit.
These protocols allow us to build the Carnot-like cycle, which is analysed in section 4. The efficiency at maximum power is thoroughly investigated in section 5. In section 6, the main conclusions of our work are presented. Finally, we refer to the Supplementary Material for some further technical details, which complement the main text.
The model system
Definition
We consider a one-dimensional (1D) overdamped harmonic oscillator of stiffness k in contact with a thermal bath at temperature T . A Brownian particle, confined by optical tweezers, provides an accurate realisation. The stochastic dynamics of the system may be modelled at either the Langevin or the Fokker-Planck levels of description. The average variance of the oscillator x 2 , which is given by
λ d x 2 dt = −2k x 2 + 2k B T,(1)
being k B the Boltzmann constant and λ the friction coefficient. Physically, we are considering that the harmonic oscillator is immersed in a certain fluid that plays the role of the heat bath, which provides the values of the temperature T and the friction coefficient λ. Throughout our work, we take λ as constant §, time-independent, but we assume both the stiffness of the oscillator k and the bath temperature T to be externally controlled. While the time control of trap stiffness is now routinely achieved experimentally, we refer to Refs. [4,33] for the time control of temperature. At any time t, the state of the system is characterised by the state-point (k, x 2 , T ). Equilibrium states fulfil the equation of state x 2 eq = k B T /k. The above relation, which has been obtained by making the rhs of equation (1) vanish, defines the equilibrium surface in the (k, x 2 , T ) three-dimensional space. We want to describe the energetics of this system at the average level. Thus we define the average energy
E = 1 2 k x 2 + 1 2 k B T,(2)
where we have taken into account that the velocity variable is always at equilibrium in the overdamped limit. The equilibrium value of the energy is then E eq = k B T . Let us now consider a process starting from a certain state A and ending in another state B. Work and heat are defined by the relations [2]
W AB = 1 2 B A x 2 dk,(3)Q AB = 1 2 B A k d x 2 + k B dT = 1 2 B A k d x 2 + k B 2 (T B − T A ) ,(4)
where T A and T B are the temperature values for the initial and final states, A and B, respectively. Thus, the first law of thermodynamics reads ∆E ≡ E B −E A = W AB +Q AB .
We have used the following sign convention: for W, Q > 0 energy is transferred from the environment to the system, whereas for W, Q < 0 energy is transferred from the system to the environment, irrespective of the "kind" of energy involved. Therefore, in order to consider a heat engine, we are interested in cycles with a negative total work.
Non-dimensional variables
First of all, we introduce dimensionless variables as follows: we divide the stiffness and the temperature by their respective initial values, κ = k/k 0 , θ = T /T 0 , and the variance by its equilibrium value at the initial temperature, y = x 2 / x 2 eq,0 = k 0 x 2 /(k B T 0 ). Then, we have that y(t = 0) = 1 if the system starts from an equilibrium state. Second, a dimensionless time is defined as s = k 0 t/λ. With the above definitions, the evolution of the system in non-dimensional variables is governed by
dy ds = −2κy + 2θ,(5)
where the equilibrium surface (or equation of state) reads,
κy eq = θ.(6)
Regarding the energetics, we introduce the dimensionless energy by dividing E by the equilibrium value at the initial time, k B T 0 . Consistently, non-dimensional work and heat are defined with the same energy unit, that is,
E = 1 2 κy + 1 2 θ,(7)W AB = 1 2 B A y dκ, Q AB = 1 2 B A κ dy + 1 2 (θ B − θ A ) .(8)
The first law reads ∆E ≡ E B − E A = W AB + Q AB , and the equilibrium value of the energy is E eq = θ.
In dimensionless variables, the state of the system is characterised by the statepoint (κ, y, θ) at any time s. For our purposes, it is useful to consider the movement of the projection of the state-point onto the (κ, y) plane. In particular, the work W, as given by equation (8), is proportional to the area below the curve (κ(s), y(s)) swept by the system as time increases.
Building blocks of the cycle: Isothermal and adiabatic processes
Herein, we aim at building an irreversible heat engine with the above described overdamped harmonic oscillator. Our heat engine operates cyclically between a "hot" source, at dimensionless temperature θ h , and a "cold" source, at temperature θ c < θ h . Specifically, the non-equilibrium cycle comprises four different processes: two isothermal ones, at temperatures θ h and θ c , and two locally adiabatic ones that connect the isotherms. No heat is exchanged in average during these locally adiabatic processes at all times, as described below. This is the usual use of the term adiabatic in equilibrium thermodynamics, in which adiabatic is employed for a process in which the system is thermally insulated from the environment.
In each cycle, the engine takes energy from the hot reservoir as heat, Q h > 0, and performs work, that is, W < 0. Therefore, the projection of the state-point onto the (κ, y) plane sweeps a certain closed curve (κ(s), y(s)), which characterises the considered cycle, in the counterclockwise direction. In the light of the above, isothermal and adiabatic processes can be considered as the building blocks for our irreversible heat engine. In the following, we summarise some results obtained in previous studies for isothermal [14,32] and adiabatic processes [13].
Isothermal processes
We consider two kinds of isothermal processes at temperature θ: quasi-static and optimal. In both of them, the initial and final states characterised by (κ A , y B ) and (κ A , y B ), respectively, correspond to equilibrium situations. Therefore,
κ A y A = κ B y B = θ.
First, we deal with the quasi-static case. Therein, κ is slowly tuned in such a way that the system sweeps the equilibrium curve y(s) = θ/κ(s) in the (κ, y) plane. Therefrom,
W = θ 2 ln κ B κ A , Q = −W, ∆E = 0, E B = E A = θ.(9)
Of course, this quasi-static process takes an infinite time. Second, we look into the optimal process for a given finite time s f . Therein, we are interested in the process for which the work performed by an external agent on the system is minimum, or in other words, we look for the maximum work produced by the system. The evolution of the variance in the optimal process is [14,32].
y(s) = √ y A + ( √ y B − √ y A ) s s f 2 .(10)
From now, tilde denotes optimality in some sense: either for the profiles or for the values of the physical quantities or parameters. Note thatỹ(s) is continuous in the
whole interval [0, s f ].
The optimal evolution for the stiffness is obtained from the evolution equation (5) in the open interval (0, s f ),
κ(s) = θ y(s) − 1 2 d ds lnỹ(s), 0 < s < s f .(11)
We recall that the stiffness is discontinuous at both the initial and final times,κ(s = 0) = κ A ,κ(s = s f ) = κ B . In this problem, the elastic constant κ(s) plays the role of the "control" function in optimal control theory [34,35]. Similar discontinuities in the "control" function have been repeatedly found in stochastic thermodynamics [6,14,31,32,[36][37][38].
This is a consequence of the corresponding "Lagrangian" being linear in the "velocities" [39], which is sometimes called the Miele problem [40].
The optimal values of work and heat can also be readily calculated. The results are
W = θ 2 ln κ B κ A + θ s f 1 √ κ B − 1 √ κ A 2 , Q = − W,(12)
Of course, in this isothermal process there is no energy change between the initial and final states ∆E = 0, E B = E A = θ. Note, however, that the energy of the system does change in the intermediate times, E(s) = θ for 0 < s < s f because we are dealing with a non-equilibrium process and y(s) = θ/κ(s), as expressed by equation (11).
Adiabatic processes
Now we turn our attention to adiabatic processes, there is no heat transfer at any point of the system trajectory. Therefore, bearing in mind equation (8) we have that the infinitesimal heat vanishes, i.e.
dQ ≡ κ dy + dθ = 0,(13)
Note that temperature becomes a function of time that goes from θ A to θ B in adiabatic processes. Similarly to the case of isothermal processes, we only consider adiabatic processes connecting two equilibrium states and then
κ A y A = θ A , κ B y B = θ B .
The energetics of adiabatic processes is quite simple. The energy change is given by the change in temperature,
E A = θ A , E B = θ B , ∆E = θ B − θ A .
Since there is no heat exchange, Q = 0, work coincides with the energy change, W = θ B − θ A . The above expressions for energy, heat and work apply for any adiabatic process, regardless of its duration, and therefore are valid for both quasi-static and non-equilibrium processes. Nevertheless, the equivalence between adiabatic and isoentropic processes occurs only in the quasi-static limit. It is in the non-equilibrium case that we deviate from the proposals in Refs. [15,17].
Again we consider two kinds of processes: quasi-static and optimal. First, in the quasi-static case, κ and θ are tuned in an infinitely slow way to allow the system sweep the equilibrium curve (6). Combining equations (6) and (13), one gets
y(s) = y A θ A θ(s) = y A κ A κ(s) .(14)
Second, we investigate optimal adiabatic processes. Here, optimal means something different from the sense we used in the previous section. As already said above, the work value is fixed by the initial and target states and thus cannot be optimised. However, two arbitrary states cannot be connected by an adiabatic transformation, the following inequality
θ(s) θ A ≥ y(s) y A −1 ,(15)
holds for all times [13]. Therefore, for the initial and final times,
θ B θ A ≥ y B y A −1 or, equivalenty, θ B θ A 2 ≥ κ B κ A(16)
must be fulfilled. The equality in equations (15) and (16) corresponds to the quasi-static case (14). There exists a minimum time to carry out an adiabatic process [13], namelỹ
s f = (y B − y A ) 2 2 (y B θ B − y A θ A )
.
This minimum time is reached for a protocol in which the variance and the temperature evolve according tõ
y(s) = y A + (y B − y A ) s s f ,θ(s) = y A θ A + (y B θ B − y A θ A ) s s f y A + (y B − y A ) s s f ,(18)
which are valid in the whole interval [0,s f ]. Therefore, bothỹ(s) andθ(s) are continuous functions of time, including the initial and final times. The stiffness is given bỹ
κ(s) = − dỹ(s) ds −1 dθ(s) ds , 0 < s <s f .(19)andκ(s = 0) = κ A ,κ(s =s f ) = κ B .
The discontinuity at the boundaries of κ(s) does not break the adiabatic character of the process: there is no instantaneous heat transfer at the initial and/or final times. Since both the variance y and the temperature θ are continuous at the boundaries, the integration of the differential of heat, as defined in equation (13), between s = 0 and s = 0 + (or between s =s − f and s =s f ) vanishes. On the contrary, there is an instantaneous contribution to the work at both boundaries.
Irreversible Carnot-like heat engine
The aim of this work is to study a (stochastic) thermodynamic cycle comprising the following processes: (i) Isothermal expansion starting from (κ A , y A , θ A ) up to (κ B , y B , θ B = θ A ) in contact with a hot bath at temperature θ A , (ii) adiabatic expansion starting from (κ B , y B , θ B = θ A ) up to (κ C , y C , θ C ), (iii) isothermal compression starting from (κ C , y C , θ C ) up to (κ D , y D , θ D = θ C ) in contact with a cold bath at temperature θ C , and (iv) adiabatic compression going from (κ D , y D , θ D = θ C ) to (κ A , y A , θ A ). We always choose the normalisation constants (units) such that (κ A , y A , θ A ) = (1, 1, 1).
As a consequence of the above processes being isothermal/adiabatic, we have the following general identities,
W AB = −Q AB , W BC = E C − E B = θ C − θ A , Q BC = 0, W CD = −Q CD , W DA = E A − E D = θ A − θ C = −W BC , Q DA = 0.
We focus on a heat engine, that is, a device that extracts heat from the hot bath and performs work, i.e.
Q AB = −W AB > 0, W AB + W BC +W CD + W DA = W AB +W CD < 0.(20)
The efficiency of such a device is defined by
η ≡ − (W AB + W CD ) Q AB = 1 − W CD Q AB < 1,(21)(a) κ y θ A 1 1 1 B χ χ −1 1 C ν 2 χ ν −1 χ −1 ν D ν 2 ν −1 ν (b) κ y θ A 1 1 1 B χ χ −1 1 C cν 2 χ c −1 ν −1 χ −1 ν D dν 2 d −1 ν −1 ν
whereas the power that delivers is given by
P ≡ − (W AB + W CD ) s AB + s BC + s CD + s DA ,(22)
where s AB is the time employed for going from A to B, and so on.
Quasi-static case
First, we concentrate on the quasi-static limit, that is, we consider a Carnot cycle in which the harmonic oscillator is always at equilibrium. In principle, we must give 12 numbers to characterise the four operating points of the cycle (A, B, C, D), but we have the following constraints: (i) due to normalisation, state A is given, The cycle is thus completely characterised by the temperature ratio ν and the compression ratio along the first isotherm χ. The values of the state variables (κ, y, θ) at the operating points of the cycle are collected in panel (a) of Table 1. Note that the isotherm condition implies that y B /y A = κ A /κ B = χ −1 , so the parameter χ certainly gives the compression ratio along the first isotherm. Hereafter, to keep our wording simpler, we call χ the compression ratio.
The efficiency of a Carnot cycle
η C = 1 − θ C θ A = 1 − ν,(23)
is well known and can be derived for any system without the knowledge of its state equation through entropic considerations [1]. Here, it can also be explicitly checked by calculating work and heat over the branches of the cycle. The power delivered by this engine is zero, because the processes are quasi-static and thus involve an infinite time.
Irreversible Carnot-like cycle at finite speed
Now we consider a similar cycle, being the only difference that the processes are carried out in a finite time and are thus irreversible. The adiabaticity of the second and third process impose two inequalities, as expressed by equation (16). Therefore, we have that
θ 2 C θ 2 B ≥ κ C κ B , θ 2 A θ 2 D ≥ κ A κ D ,(24)
which become equalities only for reversible processes, as those in the previous section. Thus, we need two additional parameters to define the cycle unambiguously, specifically we choose to introduce
c = κ C ν −2 χ −1 ≤ 1, d = κ D ν −2 ≥ 1,(25)
which assure that equation (24) is fulfilled. In panel (b) of Table 1, we summarise the values of the state variables (κ, y, θ) at the operating points of this non-equilibrium cycle. A comparative plot of the reversible and irreversible Carnot engines is shown in figure 1.
In the following, we focus our attention on the maximisation of the power delivered by the engine. Therefore, we build the heat engine that operates at maximum power for fixed operating points (A, B, C, D) or, equivalently, for given values of (ν, χ, c, d). We approach the problem of the maximisation of the power defined in equation (22) in a stepwise manner. As discussed in detail below, the main idea is that the global maximum of P can be obtained as the maximum of maximums, that is, we start by maximising with respect to some parameters keeping the remainder fixed. Afterwards, this maximum can be in turn be maximised with respect to the previously fixed parameters. For instance, maximisation can be performed for a given set (ν, χ, c and d), or in a more global fashion, specifying ν only.
Maximising equation (22) implies to take the shortest possible adiabatic protocols and the minimal work for the isothermal processes. This is readily understood as follows.
The only dependence on the adiabatic protocols come from s BC and s DA , so they have to be minimum in order to give the maximum value for P. With respect to the isothermal processes, for fixed values of s AB and s CD , we have to maximise the respective work values −W AB and −W CD , that is, minimise W AB and W CD . Therefore, we end up with the optimal processes, either isothermal or adiabatic, discussed in section 3. Making use of equations (10)-(11) for the isothermal processes and equations (17)- (19) for the adiabatic ones, we get
W AB = 1 2 ln χ + 1 s AB 1 √ χ − 1 2 , Q AB = −W AB ,(26)W CD = − ν 2 ln cχ d + 1 νs CD 1 √ cχ − 1 √ d 2 , Q CD = −W CD ,(27)s BC = (1 − cν) 2 2cχν 2 (1 − c) ,s DA = (dν − 1) 2 2dν 2 (d − 1)(28)
It is worth commenting some points before proceeding further. On the one hand, W CD is always positive and thus Q CD is negative; isothermal compression work has to be done on the system and heat is always transferred from the device to the cold bath. On the other hand, in the isothermal expansion, W AB is negative for large enough s AB , but W AB becomes positive if we intend to compress the system too fast: we have to exert work on the system in that case and moreover Q AB becomes negative and heat is transferred from the system to the hot bath. Therefore, we are not interested here in these too fast isothermal expansions because we would not be building a heat engine in that case. Below we show that this poses no problem because (i) the optimal values AB yields a negative value of W AB and (ii) the optimal values CD makes that W CD < −W AB , that is, W CD + W AB < 0. Thus, the heat engine conditions are met.
Let us build on the ideas above. We must impose the inequalities (20) to have a heat engine. In particular, these inequalities should hold when s AB and s CD go to infinity (reversible isotherms). It is useful to introduce the definitions (29) where we have taken into account that χ < 1, c ≤ 1, d ≥ 1, and
W 1 ≡ lim s AB →∞ W AB = 1 2 ln χ < 0, W 2 ≡ lim s CD →∞ W CD = − ν 2 ln cχ d > 0,W ∞ ≡ W 1 + W 2 = 1 2 ln χ − ν 2 ln cχ d < 0.(30)
Although W 1 coincides with the value of the work over the first isotherm in the fully reversible engine, neither W 2 nor W ∞ does because they depend on c and d. The negativeness of W ∞ leads to the constraint
c d > χ 1−ν ν .(31)
Strictly speaking, this constraint has been shown to hold only in the limit as s AB , s CD → ∞, but below we prove that it also holds for finite-time operation. Therefore, we just have to maximise the power
P = ν−1 2 ln χ + ν 2 ln c d − 1 s AB 1 √ χ − 1 2 − 1 νs CD 1 √ d − 1 √ cχ 2 s AB +s BC + s CD +s DA ,(32)
with respect to s AB and s CD , by imposing that the partial derivatives of P with respect to s AB and s CD vanish for the optimal durations of the isothermal processes. ¶ In order to write the expressions fors AB ands CD , it is convenient to introduce the parameters
∆ 1 = √ y B − √ y A = 1 √ χ − 1 > 0,(33)∆ 2 = √ y D − √ y C = 1 √ ν 1 √ d − 1 √ cχ < 0,(34)
which measure the expansion and compression of the system in the first and second isotherms, respectively, and
σ = 1 + (s BC +s DA ) (−W ∞ ) (∆ 1 − ∆ 2 ) 2 > 1.(35)
As a function of these parameters, we can write now that
s AB = ∆ 1 (∆ 1 − ∆ 2 )(1 + σ) −W ∞ ,s CD = −∆ 2 (∆ 1 − ∆ 2 )(1 + σ) −W ∞ .(36)
The condition W ∞ < 0 ensures the positivity of the optimal times. Using the above definitions, we can write the work values for the optimal durations of the isothermal processes as
W AB = W 1 + ∆ 2 1 s AB = −W 1 ∆ 2 − W 2 ∆ 1 + W 1 σ(∆ 1 − ∆ 2 ) (∆ 1 − ∆ 2 )(1 + σ) < 0,(37)W CD = W 2 + ∆ 2 2 s CD = W 1 ∆ 2 + W 2 ∆ 1 + W 2 σ(∆ 1 − ∆ 2 ) (∆ 1 − ∆ 2 )(1 + σ) > 0.(38)
By combining the expressions above, the total work in the cycle with the optimal durations is found to be
W AB + W CD = W ∞ − 1 1 + σ W ∞ = σ 1 + σ W ∞ < 0(39)
The signs of W AB and W AB + W CD show that we have, in fact, a "good" engine. Moreover, we get a physical interpretation for the parameter σ: it measures the deviation of the total irreversible work from the value for infinitely slow isothermal processes W ∞ . In the limit as σ → ∞, we have that W AB + W CD → W ∞ . We have found the optimal values of the times for the isothermal and adiabatic protocols, for given values of the parameters (ν, χ, c, d) that univocally define the operating points of our irreversible Carnot-like heat engine. As a function of these parameters, the optimal power is thus given by
P = −W ∞ − ∆ 2 1 s AB − ∆ 2 2 s CD s AB +s BC +s CD +s DA = −W ∞ σ 1+σ s AB +s BC +s CD +s DA ,(40)
Later, we address the issue of optimising the cycle further, by looking for the maximum of the P as a function of c, d and χ for a fixed value of the temperature ratio ν. ¶ Note that, since they do not depend on s AB and s CD , we have not substituted explicitly the values ofs BC ands DA , given by equation (28), so as not to clutter the expression.
Efficiency at maximum power and the Curzon-Ahlborn bound
Maximal power at fixed temperature and compression ratios
Let us look into the the efficiency of the maximum power cycle,
η = − W AB + W CD Q AB = 1 + W CD W AB ,(41)
which depends on (ν, χ, c, d). To keep our notation simple, either for P in equation (40) orη in equation (41), we do not write explicitly the parameters which they depend on. This choice also applies to the remainder of the paper. Making use of equation (37), we can rewriteη as
η = 1 − ν η C + (ν − 1) (W 1 ∆ 2 + W 2 ∆ 1 ) − (W 2 + W 1 ν) σ (∆ 1 − ∆ 2 ) W 1 ∆ 2 + W 2 ∆ 1 − W 1 σ (∆ 1 − ∆ 2 )(42)
All the terms in the denominator are clearly positive, whereas all the terms in the numerator are negative by taking into account that W 2 + W 1 ν = −(ν/2) ln(c/d) > 0. Therefore,η < η C : the efficiency is always below the Carnot bound, as expected.
On the other hand, the comparison with the Curzon-Ahlborn bound [14,20,41,42]
η CA = 1 − √ ν(43)
requires a more detailed analysis. Let us investigate two different cases. First, we consider values of the ratio c/d such that χ −1+ν −1 < c/d < χ −1+ν −1/2 , which entails that
W 2 + √ νW 1 > 0. for which √ ν − 1 <0 (W 1 ∆ 2 >0 + W 2 ∆ 1 >0 ) − (W 2 + W 1 √ ν) >0 σ (∆ 1 − ∆ 2 ) >0 < 0.(44)
In this region, a manipulation similar to the one done for showing that η < η C gives
η = 1− √ ν+ ( √ ν − 1) (W 1 ∆ 2 + W 2 ∆ 1 ) − (W 2 + W 1 √ ν) σ (∆ 1 − ∆ 2 ) W 1 ∆ 2 + W 2 ∆ 1 − W 1 σ (∆ 1 − ∆ 2 )
. (45) Again, the denominator and the numerator are positive and negative respectively, which leads to the inequalityη < η CA . Nevertheless, for the complementary case, χ −1+ν −1/2 < c/d < 1, we can no longer assure that the Curzon-Ahlborn is an upper bound. Indeed, in the double limit as (c, d) → (1, 1), we have thats BC ands DA diverge for fixed ν < 1. In that limit, not only do the adiabatic processes become quasi-static but also the isothermal ones, recovering the quasi-static Carnot engine introduced in section 4.1, with optimal efficiency lim (c,d)→(1,1)η = η C . Because of continuity, we can always find values of c and d, given a value of χ, such that the efficiency of our optimal heat engine is arbitrarily close to the Carnot value and thus greater than the Curzon-Ahlborn bound. However, it has to be taken into account that the optimal power for this case is very small, because the denominator in equation (40) diverges. In section 1 of the Supplementary Material, we consider the leading order ofη andP. To illustrate the above results, we present in figure 2 a density plot of the optimal power, equation (40), and the corresponding efficiency, equation (41), as a function of c ν−1 √ ν c, above which we know that η < η CA . Below the aforementioned line, we cannot assure thatη < η CA and in the limit as (c, d) → (1, 1) we know thatη → η C . The curve over whichη = η CA , which departs from the hypotenuse vertices (open squares) of this second triangle and is fully contained within it, has been evaluated numerically and plotted (dashed line) along with the point of delivery of maximum power (circle).
There are several implications that can be drawn from this analysis. First, along all the sides of the delimiting triangle, the maximum power is zero because some of the optimal times diverge. Second, as a consequence of the previous point and the positiveness of P, there always appears a maximum of the optimal power as a function of (c, d) (for fixed ν and χ), at a certain pointc,d. Third, the numerical estimate for this maximum is very close to the dotted line, at whichη = η CA . This last observation is especially robust for either small χ or large ν, as can be seen in section 2 of the Supplementary Material, in which analogous plots for different couples of values (ν, χ) are presented.
Maximal power for fixed temperature ratio ν
The numerical analysis shown in figure 2 suggests that studying further the maximum power that can be achieved for fixed values of ν and χ, that is, as a function of c and d, may be illuminating. This is a meaningful physical question: recall that the reversible Carnot engine is completely determined by these two parameters. Moreover, its efficiency η C does not depend on the compression ratio χ, which makes interesting even a further maximisation in the compression ratio χ.
It is possible to address this problem by maximising again the optimal power in equation (40) with respect to c and d, and finally with respect to χ. Doing so analytically is not feasible since it involves transcendental equations. Nevertheless, a systematic asymptotic analysis can be carried out for ν → 1. In this regime, the main idea is to expand all the physical quantities in powers of η C = 1 − ν. In order to avoid cluttering the information flow with the technicalities of the asymptotic analysis, we present the detailed calculation in sections 3 and 4 of the Supplementary Material. Therein, it is shown that the expansions of P andη in the Carnot efficiency up to order η 4 C and η 3 C , respectively, are
P = η 2 C 16 − η 5/2 C 8 + 5 48 η 3 C − 11 144 η 7/2 C + 937 17280 η 4 C + O(η 9/2 C ),(46)η = η C 2 + η 2 C 8 + η 3 C 32 + O(η 7/2 C ).(47)
We recall that the expansion of the Curzon-Ahlborn efficiency is
η CA = η C 2 + η 2 C 8 + η 3 C 16 + O(η 4 C ),(48)
Similarly to the situation reported in Refs. [14,30] the first two terms in the expansion ofη in powers of η C coincide with those in η CA and the deviation occurs in the third term, of the order of O(η 3 C ). The obtained efficiency at maximum power is smaller than the Curzon-Ahlborn bound, similarly to the situation found in Ref. [14]. + In figure 3, we plot the efficiency at maximum power as a function of ν. Power has been numerically maximised over c, d and χ. The obtained efficiencyη is compared with (i) the Curzon-Ahlborn bound, (ii) the efficiency for the engine with instantaneous "adiabatic" branches developed in [14], η (I) ref = 2η C /(4 − η C ), and (iii) the efficiency obtained for large dissipation in the recent proposal, using a fast forward approach [18], to build a Carnot-like engine, η
(II) ref = (1 − ν)(1 + √ ν)/ [2 + √ ν(1 + ν)] ≤ η (I) ref .
It is clearly observed thatη ≥ η (I) ref for all ν, with the difference between them increasing as ν decreases. Moreover, the closeness between the efficiency of our engine at maximum power and the Curzon-Ahlborn bound goes beyond our expectations based on the asymptotic analysis, holding not only within the limit ν → 1 but also for the whole range of ν. Specifically, the relative deviation between our numerical values for efficiency at maximum power and the Curzon-Ahlborn bound always remains under 2%. Therefore, + See equations (24) and (25) in that paper. However, the reverse situation has also been found, see for instance equation (20) in Ref. [30]. Efficiency at maximum power as a function of the temperature ratio ν. The value obtained for the efficiency, once that the optimisation of the power is numerically performed for the rest of parameters, is almost indistinguishable from the Curzon-Ahlborn bound η CA . Our construction develops a better efficiency compared with those shown in Refs. [14] and [18].
our novel irreversible Carnot-like heat engine is certainly a very efficient one at maximum power.
Conclusions
In this work, we have put forward an irreversible Carnot-like heat engine. Our model system is a Brownian particle trapped in a harmonic potential, in the overdamped regime. The adiabatic branches of the proposed cycle are truly adiabatic in the classical thermodynamic sense: at every point thereof, there is no heat exchange with the thermal bath. Of course, the heat exchange vanishes in average: it is impossible to completely decouple the colloidal particle from the surrounding fluid. Therefore, our locally adiabatic branches contrast with the approach followed in other works, in which the system has a non-vanishing heat exchange in the "adiabatic" parts of the cycle [4,14,17,19].
The cycle of the reversible Carnot heat engine is completely characterised by the temperature ratio ν and the compression ratio χ. For our irreversible counterpart of the Carnot heat engine, we need two more parameters in order to fully characterise the four operating points of the cycle: the adiabatic condition imposes restrictions on-but does not univocally define-the operating points.
We have thoroughly studied the performance of the Carnot-like heat engine at maximum power. We have adopted a step-by-step optimisation approach. First, the maximum power is shown to be obtained for the optimal protocols for both isothermalmaximum work [14,31,32]-and adiabatic-minimum duration [13]-branches. In a second step, we have optimised the power over the duration of the isothermal processes. These two stages of the optimisation have been carried out for fixed operation points in the state space (κ, y, T )-(stiffness, variance of position, temperature). Finally, we have maximised the power over the operation points by just fixing the temperature ratio ν.
The efficiency at maximum power for our heat engine is very close to the Curzon-Ahlborn bound. This behaviour is predicted by an asymptotic analysis for ν → 1. Nevertheless, we have numerically shown that this result remarkably holds for the whole range of temperature ratios, well beyond the asymptotic prediction. This implies that our cycle is a close to optimal choice for building an efficient mesoscopic heat engine, as compared with the theoretical predictions for other constructions [14,18]. Possible venue for future work includes the study of fluctuations, beyond the mean scenario reported here [43,44].
(κ A , y A , θ A ) = (1, 1, 1) (3 constraints), (ii) points (B, C, D) are equilibrium states (3 constraints), (iii) two isothermal relations A-B and C-D (2 constraints), and (iv) two adiabatic relations B-C and D-A (2 constraints). So, we need only 12 − 3 − 3 − 2 − 2 = 2 variables to univocally define the quasi-static cycle.
Figure 1 .
1(a) Projection of the movement of the state-point onto the (κ, y) plane for a reversible Carnot engine. (b) Projection of the movement of the state-point onto the (κ, y) plane for an irreversible Carnot-like engine. Specifically, we have used the parameter values ν = 0.6, χ = 0.6, c = 0.96, d = 1.03 and the corresponding optimal protocols discussed in the text. In both plots, red lines correspond to the isothermal processes and green lines to the adiabatic ones. The dashed segments mark the jumps in the stiffness at the initial and final points of each of the four branches of the cycle.
Figure 2 .
2Density plots of the optimal power (left) and its corresponding efficiency (right) in the (c, d) plane. The curves whereη = η CA (dashed line), with its initial and final points (open squares) over the axes d = 1 and c = 1, respectively, and the point at which the maximum power (circle) is reached, are displayed in both panels. We have taken ν = 0.75 and χ = 0.5. and d. Specifically, we consider given values of the temperature ratio ν = 0.75 and the compression ratio χ = 0.5. The constraint (31) entails that the meaningful region in the plane (c, d) is a right triangle of vertices (c min = χ 1−ν ν , 1), (1, 1) and (1, d max = c −1 min ). Within this region, we can define another right triangle with the right angle in the same vertex and the hypotenuse given by the line d = χ √
Figure 3 .
3Figure 3. Efficiency at maximum power as a function of the temperature ratio ν. The value obtained for the efficiency, once that the optimisation of the power is numerically performed for the rest of parameters, is almost indistinguishable from the Curzon-Ahlborn bound η CA . Our construction develops a better efficiency compared with those shown in Refs. [14] and [18].
Table 1 .
1Operating points of the Carnot engines. Panels (a) and (b) correspond to the reversible and irreversible versions, respectively.
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| [] |
[
"Cryptomorphisms for abstract rigidity matroids",
"Cryptomorphisms for abstract rigidity matroids"
] | [
"Emanuele Delucchi ",
"Tim Lindemann "
] | [] | [] | This note contributes to the structure theory of abstract rigidity matroids in general dimension. In the spirit of classical matroid theory, we prove several cryptomorphic characterizations of abstract rigidity matroids (in terms of circuits, cocircuits, bases, hyperplanes). Moreover, the study of hyperplanes in abstract rigidity matroids leads us to state (and support with significant evidence) a conjecture about characterizing the class of abstract rigidity matroids by means of certain "prescribed substructures". We then prove a recursive version of this conjecture. 1 | 10.1007/978-3-319-31580-5_8 | [
"https://arxiv.org/pdf/1503.03795v1.pdf"
] | 119,754,252 | 1503.03795 | d6cb996770c2deb1dec266388c98528825d76ea9 |
Cryptomorphisms for abstract rigidity matroids
12 Mar 2015 March 13, 2015
Emanuele Delucchi
Tim Lindemann
Cryptomorphisms for abstract rigidity matroids
12 Mar 2015 March 13, 2015
This note contributes to the structure theory of abstract rigidity matroids in general dimension. In the spirit of classical matroid theory, we prove several cryptomorphic characterizations of abstract rigidity matroids (in terms of circuits, cocircuits, bases, hyperplanes). Moreover, the study of hyperplanes in abstract rigidity matroids leads us to state (and support with significant evidence) a conjecture about characterizing the class of abstract rigidity matroids by means of certain "prescribed substructures". We then prove a recursive version of this conjecture. 1
Introduction
Rigidity matroids are combinatorial structures introduced by Graver in [3] which, roughly speaking, model some aspects of the theory of rigid bar-link framework, whose combinatorial study goes back to work of Laman [5].
More precisely, a framework is a (finite) graph (V, E) together with a straight-line embedding into R d , and it is called rigid if the only continuous motions of the embedding of the vertices which fixes the distance of adjacent vertices are composites of rotations and translations. Testing rigidity of a framework involves thus checking for nontrivial solutions of a system of linear equations, every equation corresponding to an edge of the graph. Therefore, the rows of this matrix define a matroid on the ground set E. This matroid depends of course on the chosen embedding, but it always satisfies some abstract properties (usually defined in terms of the closure operator, see Definition 1.8). An abstract rigidity matroid is any matroid defined on the set of edges of a graph which satisfies these additional properties. For example, the 'usual' cycle matroid of the given graph is in fact an abstract rigidity matroid, but abstract rigidity matroids are a much bigger class. For a comprehensive introduction to the combinatorial study of rigidity of frameworks we point to the book of Graver, Servatius and Servatius [2].
Here, we approach the subject from the point of view of pure matroid theory, which we briefly introduce in Section 1, and focus on two main subjects.
First, the fact that classical matroid theory features a web of equivalent definitions -so-called cryptomorphic approaches, see Remark 1 -has been recognized as one of its main theoretical strength since at least the seminal work of Crapo and Rota [1]. In Section 2, we develop the theory of abstract rigidity matroids in a corresponding way and derive different, equivalent characterizations, each reflecting one of the classical approaches to matroid theory.
Second, we focus on structural aspects and ask whether being an abstract rigidity matroid can be identified as a "structural" matroid property, e.g., whether the class of abstract rigidity matroids can be characterized through some "prescribed" substructures. This is a quite standard organizational process in graph theory and matroid theory: in particular, prominent open questions and important results about characterizing certain classes of matroids through "excluded minors" abound. In Section 3, we prove a characterization of abstract rigidity matroids in terms of "prescribed classes of hyperplanes" in all restrictions (Theorem 3.8) and conjecture a nonrecursive characterization of abstract rigidity matroids via the requirement of the existence of a prescribed class of hyperplanes. We offer some stringent evidence towards this conjecture (e.g., Propositions 3.2 and 3.3) as well as in support of the general significance of the suggested substructures (e.g. through Proposition 3.4).
Review
In this section we will review some basic definitions and results about abstract rigidity matroids. In particular, we will state Viet-Hang Nguyens's combinatorial characterization of abstract rigidity matroids (Proposition 1.16). We will assume the reader familiar with the basics of matroid theory, and suggest Oxley's textbook [7] for an introduction to the subject.
Graphs
We consider a graph to be a pair (V, E) consisting of a finite set V and a set E of two-element subsets of V (thus, our graphs will not have loops nor parallel edges). Given two vertices u, v ∈ V we will often use uv as a shorthand for {u, v}. For any finite set W we will let K(W ) ∶= {uv u, v ∈ W, u ≠ v} so that (W, K(W )) is the complete graph on the vertex set W . Given a natural number n we will use the notation K n to refer to any (and thus every)
K(W ) with W = n. For any E ⊆ K(V ) let V (E) ∶= {u ∈ V uw ∈ E for some w ∈ V }; V (E) is called the support of the edge set E.
In the following, we will often consider graphs on a fixed vertex set and will then, if no confusion can occur, refer to edge-sets as "graphs".
• For v ∈ V let star(v) ∶= {vu ∈ K(V ) u ≠ v}}.
Sets of the form star(v) with m − 1 arbitrary edges deleted are simply called "vertex stars minus m − 1 edges".
• For a set V ′ ⊆ V such that V ′ = m let bigstar(V ′ ) ∶= K(V ) K(V V ′ ).
Thus, the set bigstar(V ′ ) is the edge-set of the graph where every
vertex v 0 not in V ′ is attached to K(V ′ ) by the family of edges {v 0 v ′ v ′ ∈ V ′ } (one edge to every vertex in V ′ ). V = 7, m = 2 V = 6, m = 3 V = 8, m = 4
Matroids
Matroid theory finds its origins in the attempt, by Hassler Whitney, to define combinatorial structures abstracting some properties of linear independency in vectorspaces. For instance, it is an easy check that the set of bases of a vectorspace (say, over a finite field) satisfies Definition 1.1 below. Our goal in this introductory paragraph is to define matroids and some of the related terminology, and to explain what matroid theorists mean by cryptomorphism (see Remark 1). Indeed the word may sound unusual, but the concept is one that is both useful in applications and -most importantly for us here -as a theoretical feature which was singled out as one of the main aspects of interest of matroid theory ever since at least Crapo and Rota's seminal treaty [1]. Definition 1.1. Let S be a finite set and let B be a collection of subsets of S which fulfills
(i) B ≠ ∅. (ii) If B 1 , B 2 ∈ B, then B 1 = B 2 . (iii) For all B 1 , B 2 ∈ B, x ∈ B 1 B 2 , there exists y ∈ B 2 B 1 , such that (B 1 {x}) ∪ {y} ∈ B.
Then the pair M = (S, B) is called a matroid and B is called the collection of bases of M. A set I ⊆ S is called independent if there is a basis B ∈ B such that I ⊆ B, otherwise dependent. An inclusion-minimal dependent subset of S is called a circuit of M. A maximal set which does not contain a basis is called a hyperplane.
The closure of a set A ⊆ S, denoted by σ B (A), is the intersection of all hyperplanes containing A (if no such hyperplane exist, the closure is defined as A).
A direct consequence of the basis axioms is that the sets of complements of bases does also fulfill these axioms; the matroid M * ∶= (S, B * ) is called dual to M, and (M * ) * = M. If a subset of S is a circuit (or a hyperplane, or a basis etc) of M * one says that it is a cocircuit (resp. cohyperplane, cobasis) of M.
We will have use for the following two standard facts, whose proof can be found e.g. in [7]. (i) If C is a circuit and C ′ is cocircuit, then C ∩ C ′ ≠ 1. (iv) If x, y ∈ S and x ∈ σ(A ∪ {y}), then y ∈ σ(A ∪ {x}).
Then σ(⋅) is called a matroid closure operator, and any set A ⊆ S is called
closed (or flat) if σ(A) = A.
The next theorem is basic and can be found e.g. in [7, Chapter 1] Remark 1 (Cryptomorphisms). In the parlance of matroid theory Theorem 1.4 is referred to as a cryptomorphism between the definition of matroids in terms of bases and the definition in term of closure operator. In fact, just like in Definition 1.3, one can isolate some distinguishing properties of the family of circuits (or of cocircuits, or hyperplanes, etc.) of a matroid and prove that any family of sets satisfying those formal properties can be obtained as the set of circuits (or... etc.) of a matroid defined e.g. as in Definition 1.1.
We close this short presentation of matroids by defining two more concepts we will have use for later. As a reference we point, again, to [7].
rk(X) ∶= max {B ∩ X B ∈ B} ,
i.e., the size of the biggest independent set contained in X.
Abstract rigidity matroids
We now are ready to introduce the main character of this note. As these structures are less classical than graphs of matroids, we will go into some more detail. Notice that the ground set of an abstract rigidity matroid is the set of edges of a complete graph and, although perhaps not evident from our abstract point of view, 'rigidity' of a set of edges is meant to be related to (and indeed comes from) the concept of rigidity of a bar-andjoints framework in m-space. The reader will perhaps find a useful intuition in thinking of taking the closure of a certain set of elements (i.e., edges) of an abstract rigidity matroid as of increasing the given set by all edges whose presence would not change the degree of rigidity of the given set.
Definition 1.8. Let V be a finite set and let A = (K(V ), σ) be a matroid on K(V ) with closure operator σ. A set E ⊆ K(V ) is then called rigid (with respect to A) if σ(E) = K(V (E)). Let m ∈ N >0 ; the matroid A is called a m-dimensional abstract rigidity matroid if C1. if E, F ⊆ K(V ) and V (E) ∩ V (F ) < m, then σ(E ∪ F ) ⊆ K(V (E)) ∪ K(V (F )) C2. if E, F ⊆ K(V ) are rigid and V (E) ∩ V (F ) ≥ m, then E ∪ F is rigid.
Roughly speaking condition (C1) says that edge-sets which do not share enough common vertices cannot unite to a rigid set, while (C2) says that rigid sets which are connected through enough vertices form a rigid union.
Note that (C1) also states that σ(E) ⊆ K(V (E)) for all E ⊆ K(V ).
Remark 2. Recall the notation of Section 1.1 and notice that, if A is an m-dimensional rigidity matroid on K(V ), every K(V ′ ) with V ′ = m + 2 is a circuit and every vertex star minus m − 1 edges is a cocircuit of A.
Remark 3. For the sake of simplicity, if not otherwise specified, throughout the text we will consider m-dimensional rigidity matroids on K(V ) where m ∈ N >0 and V ≥ m + 1 be fixed. Indeed one easily checks that every abstract rigidity matroid on K(V ) with V ≤ m is trivial in the sense that every edge-set would be an independent set in such a matroid.
Definition 1.9. An edge-set is said to fulfill Laman's condition in dimension m, if for all F ⊆ E with V (F ) ≥ m, we have F ≤ m V (F ) − m+1 2 .
In fact every independent set in a m-dimensional abstract rigidity matroid fulfills Laman's condition.
Lemma 1.10. [2, Lemma 2.5.6.] Let U ⊆ V and let A be a m-dimensional abstract rigidity matroid on K(V ). Then r(K(U )) = U 2 if U ≤ m + 1 m U − m+1 2 if U ≥ m + 1
Therefore every m-dimensional abstract rigidity matroid on K(V ) is of rank m V − m+1 2 , and rigid edge-sets on at least m + 1 vertices have at least m V − m+1 2 edges. Definition 1.11. An edge-set which is both rigid and independent is called isostatic.
A result in rigidity theory states that 2-isostatic sets are at least 2-vertex connected [2, Exercise 4.7.]. Our Proposition 3.4 will prove that in fact every rigid set in a m-dimensional rigidity matroid is m-vertex connected.
Note that every rigid edge-set must have an isostatic subset, so rigidity of a graph can be seen as related to connectivity. This is emphasized by the following result: Lemma 1.12 (Theorem 3.11.8. of [2]). There is only one 1-dimensional abstract rigidity matroid on K(V ), the cycle matroid on K(V ).
However, for m > 1, there can be many different m-dimensional abstract rigidity matroids on K(V ). This makes the theory interesting, and is investigated in more detail for example in [2] and [8].
Here we will only briefly recall some of the structural aspects of abstract rigidity matroids in general dimension, especially as related to the problem of characterizations alternative to Definition 1.8 (e.g., towards "cryptomorphisms" -cf. Remark 1)
Definition 1.13. Let E ⊆ K(V ) be an edge-set, let v 1 , . . . v k ∈ V (E) and w ∈ V V (E). Then the set E ∪ {v 1 w, . . . v k w} is called a k-valent-0-extension of E.
Intuitively, we are here attaching a vertex to a given edgeset by means of exacly k new edges. (i) no circuit of A contains a vertex of valence less than m + 1.
(ii) for k ≤ m, every k-valent-0-extension of an independent edge-set is independent.
(iii) for k ≥ m, every k-valent-0-extension of a rigid edge-set is rigid if (C2) holds in A (with respect to m).
Proof. For any
V ′ ⊆ V with V ′ = m, bigstar(V ′ ) is the set K(V ′ ) with every vertex of V V ′ attached to it via m-valent-0-extensions. Note that K(V ′ )
is independent and rigid, because it is a complete graph on less than m + 2 vertices. Hence Lemma 1.14 implies that bigstar(V ′ ) is a basis. ∎
The problem of finding alternative characterizations of abstract rigidity matroids has been raised in the literature, and we conclude this short review with the 2010 result of Nguyen which gave an answer to this problem. This will be the starting point for our considerations. (ii) all K m+2 are circuits of A.
(iii) r(K(V )) = m V − m+1 2 .
Axiomatizations of rigidity matroids
In this section we will derive several cryptomorphic definitions of abstract rigidity matroids. Our starting point will be Proposition 1.16, which states conditions on the circuits, coircuits and on the rank function. We strive for characterizations fitting into the classical reformulations of matroid theory -in particular: cocircuits, hyperplanes, bases, circuits. We start by removing from Proposition 1.16 the reference to rank and circuits. Proof. First of all let A be an m-dimensional abstract rigidity matroid. We have already seen in Remark 2 that all vertex stars minus m − 1 edges are cocircuits in abstract rigidity matroids, so (D2) holds. We prove (D1) by contraposition, and so let C ⊆ K(V ) be a set with V (C) ≤ V − m. If C is empty, then it is independent in any matroid and thus not a cocircuit. Let then C ≠ ∅. Thus, C contains an edge with endpoints, say, v 1 , v 2 ∈ V (C). Because of the constraint on the cardinality of V (C), we can choose v 3 , . . . , v m+2 ∉ V (C).
By Proposition 1.16.(ii), K({v 1 , . . . , v m+2 }) is a circuit in A which intersects the set C in the single edge v 1 v 2 . Therefore, C cannot be a cocircuit by Proposition 1.2. Now let A be a matroid satisfying (D1) and (D2). By Proposition 1.16, to prove that A is an abstract rigidity matroid, we only need to verify that r(A) = m V − m+1 2 . So choose an m-element subset V ′ of V . Since, by (D2), all vertex stars minus m − 1 edges are cocircuits, bigstar(V ′ ) is independent by Lemma 1.14. Moreover, its complement is K(V V ′ ) -a complete graph on V −m vertices -and thus coindependent. So bigstar(V ′ ) is also a spanning set, hence a basis, and we can use it to show that the rank of A has the correct value, as follows.
r(A) = r(bigstar(V ′ )) = K(V ′ ) + m( V − m) = m 2 + m( V − m) = m V + m 2 − m 2 = m V − m + 1 2 ∎
We now state hyperplane axioms for abstract rigidity matroids, as we will be concerned extensively with hyperplanes in the next section. They are easily derived from the cocircuit axioms by complementation. H2. for all v ∈ V , the sets of the form ∆ A v where A = m−1 (i.e., m−1-valent 0-extensions of K(V v)) are hyperplanes.
Definition 2.2. For v ∈ V and A ⊆ V {v} let ∆ A v ∶= K({v} ∪ A) ∪ K(V {v}).
We now move to a characterization of abstract rigidity matroids in terms of bases. To prove (B2) notice that, whenever V ′ = m, bigstar(V ′ ) is a m-valent-0-extensions of some K m , thus independent. Moreover, such a bigstar(V ′ ) cannot be contained in any hyperplane, because m of its vertices have valence V − 1, thus it violates (H2).
We now turn to the reverse direction and let B fulfill (B1) and (B2). We will prove that, then, the set of hyperplanes of the given matroid satisfies For the reverse implication, let C satisfy (Z1) and (Z2), and let B denote the set of bases of A. To prove that (B2) holds, choose then any subset V ′ ⊆ V with V ′ = m, and consider the set bigstar(V ′ ). Every edge xy not in bigstar(V ′ ) is contained in the circuit K(V ′ ∪ {x, y}) and thus the closure of bigstar(V ′ ) is K(V ). Also, bigstar(V ′ ) does not contain any circuit: by (Z1), all vertices of V V ′ have valence m, and so any circuit in bigstar(V ′ ) must be a subset of K(V ′ ), which itself is independent because it only contains vertices of valence m − 1. Summarizing, there is no circuit in bigstar(V ′ ) and so it is independent, thus a basis and (B2) holds. In order to prove (B1), suppose by way of contradiction that there is a basis B with a vertex v of valence less than m. Then, there is a vertex w ∈ V such that vw ∉ B. But B ∪ {vw} must be dependent: thus it contains a circuit C which must contain the vw. In particular, v has at most valence m in C, contradicting (Z1). Hence (B1) holds. ∎
Hyperplanes of abstract rigidity matroids
All axiomatizations given in Section 2 share a similar structure: they require some "prescribed substructure" (e.g., conditions (H2), (B2), (Z2), (D2)) and impose some condition on the valency of vertices (e.g., (H1), (B1), (Z1), (D1)). From a structural point of view one can ask whether it is possible to characterize anstract rigidity matroids from a "prescribed substructure" point of view alone. This question turns out to be intriguing and not trivial. We will investigate it from the vantage point of hyperplanes, where the "prescribed substructures" are hyperplanes of the form ∆ A v for A = m−1 (see Definition 2.2). We suggest to look at a bigger family of "prescribed substructures", namely the following.
H m ∶= {H = K(V 1 ) ∪ K(V 2 ) V i ⊈ V j for i ≠ j, V 1 ∪ V 2 = V, V 1 ∩ V 2 = m − 1}
The family H m consists of pairs of sets of edges of complete graphs which intersect in a complete graph on m − 1 vertices. Figure 3 shows how this sets may look like. Evidence towards an affirmative answer to this question is given in the next two propositions.
Proposition 3.2.
Let A m be a m-dimensional abstract rigidity matroid on K(V ) with hyperplanes H. Then H m ⊆ H. Thus
Proof. Let H ∈ H m such that H = K(V 1 )∪K(V 2 ) with V 1 , V 2 ⊆ V , V 1 ∪V 2 = V , V 1 ∩ V 2 = m − 1 and V i ⊈ V j for i ≠ j. We have: σ(H) = σ(K(V 1 ) ∪ K(V 2 )) (C1) ⊆ K(V (K(V 1 ))) ∪ K(V (K(V 2 ))) = K(V 1 ) ∪ K(V 2 ) = H ⊆ σ(H).σ(H) = H ≠ K(V ). Let V ′ ∶= V 1 ∩ V 2 and let e ∈ K(V ) H, e = v 1 v 2 . Then w.l.o.g. v i ∈ V i (i = 1, 2), v 1 , v 2 ∉ V ′ . Note that K(V ′ ∪ {v 1 , v 2 })
is as a complete edge-set rigid, as is K(V 1 ). In elements of H 2 (V ) and x is not in their union. By the hyperplane axioms for matroids, the set on the right has to be subset of another hyperplane which is at least 2-vertex-connected, hence not an element of H 2 (V ).
addition V (K(V ′ ∪ {v 1 , v 2 })) ∩ V 1 = V ′ ∪ {v 1 } = m, so by (C2), also their union K(V ′ ∪ {v 1 , v 2 }) ∪ K(V 1 ) is rigid, as is K(V ′ ∪ {v 1 , v 2 }) ∪ K(V 2 ).
Proposition 3.3.
Let A be matroid on K(V ) with hyperplanes H and H m ⊆ H. Then its closure-operator fulfills (C1).
Proof. Let E, F ⊆ K(V ) such that V (E) ∩ V (F ) < m. For given v ∈ V ,
the complement of star(v) minus m − 1 arbitrary edges is a hyperplane. So we are able to find hyperplanes which intersect into K(V {v}). This means that for all v ∈ V , the set K(V {v}) is closed in A and thus is every complete edgeset. Hence σ(E) ⊆ K(V (E)) and σ(F ) ⊆ K(V (F )), so we may assume w.l.o.g. that V (E) ⊈ V (F ) and vice versa. Now we are able to choose
V 1 , V 2 ⊆ V such that K(V 1 ) ∪ K(V 2 ) is an element of H m -a hyperplane - and (K(V 1 ) ∪ K(V 2 )) ∩ (K(V (E ∪ F ))) = K(V (E)) ∪ K(V (F )).
Because the sets which intersect lefthandside are closed in A, K(V (E)) ∪ K(V (F )) must also be closed.
It contains E ∪ F , so σ(E ∪ F ) ⊆ K(V (E)) ∪ K(V (F )). ∎
We conclude this section with one more token of the advantage of considering the set H m : the following proposition gives an immediate proof, for all dimension, of a strenghtening of the problem posed in [2,Exercise 4.7.] for dimension 2. Proof. Otherwise choose a vertex-set U ⊆ V (E) such that U < m and deleting U out of (V (E), E) would leave a disconnected graph. Like in the proof before, we are able to find
V 1 , V 2 ⊆ V such that K(V 1 ) ∪ K(V 2 ) is a hyperplane in H m , E ⊆ K(V 1 ) ∪ K(V 2 ) and U ⊆ V 1 ∩ V 2 .
We can construct V 1 , V 2 in such a way, that at least one vertex v 1 of one component of the graph after deleting U lies in V 1 V 2 , while another vertex v 2 of another such component lies in
V 2 V 1 . This implies that v 1 , v 2 ∈ V (E) but v 1 v 2 ∉ K(V 1 ) ∪ K(V 2 ). Therefore v 1 v 2 ∉ σ(E) = K(V (E)
). This is a contradiction. ∎
An inductive characterization
The last part of our study moves form the observation that certain restrictions of rigidity matroids are again rigidity matroids. Proof. We check condition (ii) and (iii) of Proposition 1.16. Condition (ii) is clearly inherited from A (and possibly empty) and Lemma 1.10 implies (iii). ∎
The following definition identifies the "prescribed substructures" through which we will be able to give an intrinsic characterization of abstract rigidity matroids.
H (1) m ∶= {K(V 1 ) ∪ K(V 2 ) ∈ H m V 1 = m} = {∆ A v A = m − 1}
The fact that the subfamily H m ⊆ H(A)can be already seen from Lemma 1.14, whose premise can be rephrased as "H (1) m ⊆ H(A)". Moreover, the same requirement is equivalent to condition (i) in Proposition 1.16 -hence, answering our question requires a detailed study of the rank function of matroids on
K(V ) for which H (1)
m is a subset of the hyperplane set. Our main result in this section is the following. Remark 5. The proof will actually work also with milder assumptions: it is enough to assume that there exists an enumeration v 1 , . . . , v k of the vertices such that H The proof of the theorem relies on the following two general lemmas. For the second it will be enough to prove that r A (F j ) = r 0 − j. First notice that, for every j > 1, F j covers F j−1 . In fact, by definition F 0 covers F 1 and, for j > 0, so we see that the cardinality F j F j−1 = 1. Therefore clearly there can be no G with F j ⊊ G ⊊ F j+1 . The chain F m ⊊ . . . ⊊ F 0 is therefore saturated, and its length thus equals the difference between the rank of F 0 and the rank of F m . ∎ Proof. Suppose by way of contradiction that K(V ′ ) contains a circuit C, let uv be an element of this circuit and A ∶= (V ′ {v, u}) ∪ v 0 , where v 0 is any element of V V ′ (nonempty for cardinality reasons). Then, ∆ A v is closed in A and contains all of K(V ′ ) except uv, a contradiction to the fact that uv is in a circuit of K(V ′ ). ∎ Proof of Theorem 3.8. One direction ("left to right") is proved by Lemma 3.6. For the other direction let V 1 ⊆ V 2 ⊆ . . . ⊆ V n be a filtration of V such that V i = i for all i. notice that the hereditarity assumption allows us to apply Lemma 3.9 recursively to get that
r(K(V i )) = r(K(V )) − m( V − V i )
for all i ≥ m, and with Lemma 3.
Figure 1 :
1Examples of some sets bigstar(V ′ ).
Lemma 1 . 2 .
12Let M be a matroid on a finite set S.
(
ii) H is a hyperplane if and only if S H is a cocircuit.(iii) F ⊆ S is closed if and only if, for every circuit C, C F ≤ 1 implies C ⊆ F .An easy check shows that any function of the form σ B (⋅) satisfies the following definition.
Definition 1 . 3 .
13Let S be a finite set and let σ ∶ 2 S → 2 S be a function such that for all A, B ⊆ S:(i) A ⊆ σ(A).
(ii) A ⊂ B implies σ(A) ⊆ σ(B).
Theorem 1 . 4 .
14The function B ↦ σ B (⋅) is a bijection between the set of families B satisfying Definition 1.1 and the set of functions σ(⋅) satisfying Definition 1.3.
Definition 1 . 5 .
15Let M be a matroid on the set S and consider T ⊆ S. The restriction of M to T , written M[T ], is the matroid with ground set T and closure operator defined for each X ⊆ T as σ(X) ∶= σ M (X) ∩ T . Definition 1.6. Let M = (S, B) be a matroid. The rank of any X ⊆ S is
Lemma 1 . 7 .
17Let M = (S, σ) be a matroid, and T ⊆ S be a closed set of M. Consider any unrefinable chain T = T 0 ⊊ T 1 ⊊ . . . ⊊ T j = S of closed sets. Then rk(S) − rk(T ) = j.
Figure 2 :
2An example of a 2-valent-0-extension Lemma 1.14 (Lemma 0.1 of [4]). Let A be a matroid on K(V ) such that all vertex stars minus m − 1 edges are cocircuits. Then:
Corollary 1. 15 .
15For any V ′ ⊆ V with V ′ = m, bigstar(V ′ )is a basis in any abstract rigidity matroid of dimension m.
Proposition 1 .
116. [6, Theorem 2.2.] [4, Theorem 0.2] Let m ∈ N and let V be a finite set withV ≥ m + 1. Furthermore, let A be a matroid on K(V ).Then, A is a m-dimensional abstract rigidity matroid if and only if any two of the following conditions hold:(i) all vertex stars minus m − 1 edges are cocircuits of A.
Let A be a matroid on K(V ). Then A is a m-dimensional abstract rigidity matroid if and only if D1. all cocircuits have strictly more than V − m vertices and D2. all vertex stars minus m − 1 edges are cocircuits.
Theorem 2. 3 .
3Let A be a matroid on K(V ) with hyperplanes H. Then A is a m-dimensional abstract rigidity matroid if and only if H fulfills: H1. if H ∈ H then at most m − 1 vertices in V (H) have valence V − 1.
Let A be a matroid on K(V ) with bases B. Then A is a m-dimensional abstract rigidity matroid if and only if B fulfills: B1. if B ∈ B then V (B) = V and every vertex in B has at least valence m,B2. for all V ′ ⊆ V with V ′ = m, the set bigstar(V ′ ) is a basis.Proof. Suppose A is a m-dimensional abstract rigidity matroid and there is a basis B of A with a vertex v of degree less than m. Let N (v) denote the neighbors of v in the graph induced by B, and choose some A ∈ star(v) with A = m − 1 and N (v) ⊆ A. Then, B ⊆ ∆ A v , which is a hyperplane by (H2) and should then not have maximal rank. Thus no such basis exists and (B1) holds.
(H1) and (H2). First, by (B2) every set of edges inducing a graph with m or more vertices of valence V − 1 contains a basis: thus no hyperplane has more than m − 1 vertices of valence V − 1 and (H1) holds. To prove (H2) consider any v ∈ V and any A ⊆ V {v} with A = m − 1, and let H ∶= ∆ A v . Note that, by (B1), H cannot contain a basis because v has degree m − 1 in H. Moreover, any edge e not in H must have v as an endpoint. So, by (B2) H ∪ {e} contains a basis. We conclude that H is a maximal set which does not contain a basis -i.e., a hyperplane -and (H2) holds. ∎ We close this section with circuit axioms. Theorem 2.5. Let A be a matroid on K(V ) with set of circuits C. Then A is a mdimensional abstract rigidity matroid if and only if C fulfills: Z1. if C ∈ C, no vertex of V (C) has valence less than m + 1. Z2. all K m+2 are circuits. Proof. Let A be a m-dimensional abstract rigidity matroid. Then, Proposition 1.16.(ii) gives directly (Z2). Proposition 1.16.(i) says that all vertex stars minus m − 1 edges are cocircuits of A and so, by Lemma 1.14.(i), (Z1) holds.
Definition 3 . 1 .
31Given a vertex set V and any m ≥ 1, define
Question 1 .
1Is every matroid A on the ground set K(V ) which satisfies H m ⊆ H(A) an m-dimensional abstract rigidity matroid?
1 Figure 3 :
13V = 10, m = 3 c) V = 11, m = 2 d) V = 10, m = 3 e) V = 8, m = 2 f) V = 8, m = Examples of elements of H m .
.
The union of these sets is H ∪ e andσ(H∪{e}) = σ((K(V ′ ∪{v 1 , v 2 })∪K(V 1 ))∪K((V ′ ∪{v 1 , v 2 })∪K(V 2 ))) Notice that,in general, the inclusion H m ⊆ H(A) is strict. The figure below illustrates this. The edge-sets on the left and in the middle are
Figure 4 :
4An example where V = 6 and m = 2.
Let A be a matroid on K(V ), with hyperplanes H and H m ⊆ H. Furthermore let E be a rigid edgeset (i.e. σ(E) = K(V (E))). Then (V (E), E) is at least m-vertex-connected.
Definition 3. 5 .
5Let A be an m-dimensional abstract rigidity matroid on the set K(V ). Given X ⊆ V , let A[X] denote the restriction of A to the set K(X).
Lemma 3. 6 .
6Let A be an m-dimensional abstract rigidity matroid on V . For every X ⊆ V , X ≥ m + 1, A[X] is an m-dimensional rigidity matroid.
Theorem 3 . 8 .
38Let V ≥ m + 1 and A a matroid on K(V ). The matroid A is an m-dimensional abstract rigidity matroid if and only if, for all X ⊆ V , H (1) m (K(X)) ⊆ H(A[X]).
2
(K({v 1 , . . . , v j })) ⊆ H(A[{v 1 , . . . , v j }]) for all j = 2, . . . k.
Lemma 3. 9 .
9Let m ≥ 2, V ≥ m + 1 and let A be a matroid on the ground set K(V ). Choose any v 0 ∈ V .• If all sets in H(1) m are closed in A, K(V v 0 ) is closed in A. • If H (1) m ⊆ H(A), the rank of the flat K(V v 0 ) in A is r(K(V )) − m. Proof. Choose v 0 ∈ V and let A ∶= V v 0 . Choose a 1 , . . . , a m ∈ A and let A 0 ∶= {a 1 , . . . , a m) }. Moreover, set A j ∶= {v 0 } ∪ A 0 a j .For j = 1, . . . mMoreover, set F 0 ∶= K(V ) and let r 0 = r A (F 0 ). As intersections of hyperplanes, all F j are flats, so we have a chainF m ⊊ F m−1 ⊊ . . . ⊊ F 0in the lattice of flats of A. Since K(V v 0 ) = F m , we have proved the forst claim.
F
j = K(A) ∪ K({v 0 , a j+1 , . . . , a m })
Lemma 3 . 10 .
310Let m ≥ 2, V ≥ m + 1 and let A be a matroid on the groundset K(V ). If all sets in H (1) m are closed in A, K(V ′ ) is independent in A for every V ′ ⊆ V with V ′ = m.
10 we have r(K(V m ) = m 2 . Thus we can write m 2 = r(K(V )) − m( V − m) hence r(K(V )) = m V − m 2 + (m 2 − m) 2 = m V − m+ 1 2 and we conclude with Proposition 1.16. ∎ Corollary 3.11. Question 1 is now equivalent to the following Question 2. Let m ∈ N >0 and V a set with V ≥ m+1. Is it true that, for any matroid A on the vertex set K(V ), H m ⊆ H(A) implies H (1) m ⊆ H(A[X]) for all X with X ≥ m + 1?
H Henry, Gian-Carlo Crapo, Rota, On the foundations of combinatorial theory: Combinatorial geometries. The M.I.T. Press. Cambridge, Mass.-Londonpreliminary editionHenry H. Crapo and Gian-Carlo Rota. On the foundations of combina- torial theory: Combinatorial geometries. The M.I.T. Press, Cambridge, Mass.-London, preliminary edition, 1970.
Combinatorial rigidity. Jack Graver, Brigitte Servatius, Herman Servatius, Graduate Studies in Mathematics. American Mathematical Society. 2Jack Graver, Brigitte Servatius, and Herman Servatius. Combinatorial rigidity, volume 2 of Graduate Studies in Mathematics. American Math- ematical Society, Providence, RI, 1993.
Rigidity matroids. Jack E Graver, SIAM J. Discrete Math. 43Jack E. Graver. Rigidity matroids. SIAM J. Discrete Math., 4(3):355- 368, 1991.
Abstract rigidity in m-space. Jack E Graver, Brigitte Servatius, Herman Servatius, Jerusalem combinatorics '93. Providence, RIAmer. Math. Soc178Jack E. Graver, Brigitte Servatius, and Herman Servatius. Abstract rigidity in m-space. In Jerusalem combinatorics '93, volume 178 of Con- temp. Math., pages 145-151. Amer. Math. Soc., Providence, RI, 1994.
On graphs and rigidity of plane skeletal structures. G Laman, J. Engrg. Math. 4G. Laman. On graphs and rigidity of plane skeletal structures. J. Engrg. Math., 4:331-340, 1970.
On abstract rigidity matroids. Viet-Hang Nguyen, SIAM J. Discrete Math. 242Viet-Hang Nguyen. On abstract rigidity matroids. SIAM J. Discrete Math., 24(2):363-369, 2010.
Matroid theory. James Oxley, Oxford Graduate Texts in Mathematics. 21Oxford University Presssecond editionJames Oxley. Matroid theory, volume 21 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, second edition, 2011.
Abstract and generic rigidity in the plane. Sachin Patkar, Brigitte Servatius, K V Subrahmanyam, J. Combin. Theory Ser. B. 621Sachin Patkar, Brigitte Servatius, and K. V. Subrahmanyam. Abstract and generic rigidity in the plane. J. Combin. Theory Ser. B, 62(1):107- 113, 1994.
| [] |
[
"Lx, 25.20.Lj, 25.75.-q * Talk at the session of Russian Academy of Sciences",
"Lx, 25.20.Lj, 25.75.-q * Talk at the session of Russian Academy of Sciences"
] | [
"S M Kiselev \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n"
] | [
"Institute for Theoretical and Experimental Physics\nMoscowRussia"
] | [
"ITEP"
] | Using the extrapolation of existing data estimations of prompt photon production at FAIR energies have been made. At y = y c.m. the rapidity density of prompt photons with p t > 1.5 GeV/c per central Au+Au event at 25 AGeV is estimated as ∼ 10 −4 . With the planed beam intensity 10 9 per second and 1% interaction probability, for 10% of most central events one can expect the prompt photon rate ∼ 10 2 photons per second.Direct photons from the hadron scenario of ion collisions generated by the Hadron-String-Dynamics (HSD) transport approach with implemented meson scatterings πρ → πγ, ππ → ργ have been analyzed. Photons from short-living resonances (e.g. ω → π 0 γ) decaying during the dense phase of the collision should be considered as direct photons. They contribute significantly in the direct photon spectrum at p t = 0.5 − 1 GeV/c. At the FAIR energy 25 AGeV in Au+Au central collisions the HSD generator predicts, as a lower estimate, γ direct /γ π 0 ≃ 0.5% in the region p t = 0.5 − 1 GeV/c. At p t = 1.5 − 2 GeV/c γ prompt /γ π 0 ≃ 2%.Thermal direct photons have been evaluated with the Bjorken Hydro-Dynamics (BHD) model. The BHD spectra differ strongly from the HSD predictions. The direct photon spectrum is very sensitive to the initial temperature parameter T 0 of the model. The 10 MeV increase in the T 0 value leads to ∼ 2 times higher photon yield. | 10.1134/s1063778809030168 | [
"https://arxiv.org/pdf/0801.1425v1.pdf"
] | 15,499,691 | 0801.1425 | f88f8414789a245da17839c28a6de685dda86a03 |
Lx, 25.20.Lj, 25.75.-q * Talk at the session of Russian Academy of Sciences
Moscow, 26 -30 November 2007
S M Kiselev
Institute for Theoretical and Experimental Physics
MoscowRussia
Lx, 25.20.Lj, 25.75.-q * Talk at the session of Russian Academy of Sciences
ITEP
Moscow, 26 -30 November 20071
Using the extrapolation of existing data estimations of prompt photon production at FAIR energies have been made. At y = y c.m. the rapidity density of prompt photons with p t > 1.5 GeV/c per central Au+Au event at 25 AGeV is estimated as ∼ 10 −4 . With the planed beam intensity 10 9 per second and 1% interaction probability, for 10% of most central events one can expect the prompt photon rate ∼ 10 2 photons per second.Direct photons from the hadron scenario of ion collisions generated by the Hadron-String-Dynamics (HSD) transport approach with implemented meson scatterings πρ → πγ, ππ → ργ have been analyzed. Photons from short-living resonances (e.g. ω → π 0 γ) decaying during the dense phase of the collision should be considered as direct photons. They contribute significantly in the direct photon spectrum at p t = 0.5 − 1 GeV/c. At the FAIR energy 25 AGeV in Au+Au central collisions the HSD generator predicts, as a lower estimate, γ direct /γ π 0 ≃ 0.5% in the region p t = 0.5 − 1 GeV/c. At p t = 1.5 − 2 GeV/c γ prompt /γ π 0 ≃ 2%.Thermal direct photons have been evaluated with the Bjorken Hydro-Dynamics (BHD) model. The BHD spectra differ strongly from the HSD predictions. The direct photon spectrum is very sensitive to the initial temperature parameter T 0 of the model. The 10 MeV increase in the T 0 value leads to ∼ 2 times higher photon yield.
Introduction
The FAIR (Facility for Antiproton and Ion Research) [1] accelerators will provide heavy ion beams up to Uranium at beam energies ranging from 2 -45 AGeV (for Z/A=0.5) and up to 35 AGeV for Z/A=0.4. The maximum proton beam energy is 90 GeV. The planed ion beam intensity is 10 9 per second.
The CBM (Compressed Baryonic Matter) [2] detector will have good possibilities for vertex reconstruction, tracking and identification of particles (hadrons, leptons and photons). Though direct photons are of great interest for the research program of the CBM experiment, a feasibility study has not been done yet.
During high-energy heavy-ion collisions direct photons, defined as photons not from particle decays, have very little interaction with the surrounding medium and are therefore not altered by rescattering. Therefore, they provide a very interesting probe and convey unique and unperturbed information on the all stages of the collision (see, e.g. the review [3]).
On the quark-gluon level the main processes are Compton scattering qg → qγ and annihilation qq → gγ. Photons from initial hard NN collisions are named prompt photons and are the main source at large p t . On the hadron level the main source of direct photons is meson rescatterings: πρ → πγ, ππ → ργ, πK → K * γ, Kρ → Kγ, ... First two channels are most important.
Direct photons from a thermalized quark-gluon or hadron system, if any, are named thermal photons and are the main source at low p t . Most of theoretical predictions for direct photons assume the local thermalization, evaluate photon production rates from the equilibrated quark-gluon or hadron system which then are convoluted with space-time evolution of the system.
Of cause, during the nucleus-nucleus collision there is a non-equilibrium stage. It can be described by the transport approach free from the local thermalization assumption. However, most of existing transport codes do not include the hadronic source (meson rescatterings) of direct photons [4].
Identification of direct photons in heavy-ion collisions is a very difficult experimental task, especially at low transverse momentum, because of the large background from decay photons, mostly from π 0 decays. First direct photons, γ direct /γ measured ≃ 20% at p t > 1.5 GeV/c, have been extracted by the WA98 collaboration at SPS [5]. The PHENIX collaboration at RHIC using a novel analysis technique decreased a systematic error to 2-3% and revealed direct photons since p t > 1 GeV/c, γ direct /γ measured ≃ 10% at p t = 1 -2 GeV/c [6].
Here we would like to estimate for the FAIR energies: prompt photons using the extrapolation of existing p + p → γX data and photon contribution from hadron sources exploiting as the microscopic Hadron-String-Dynamics (HSD) transport approach [7] and the Bjorken Hydro-Dynamics (BHD) [8].
Existing p + p(p) → γX data [9] cover the energy range √ s = 20 − 1800 GeV.
Thus, The CBM experiment at the FAIR accelerator can fill the gap √ s < 14
GeV.
At transverse momentum x t = 2p t / √ s > 0.1 cross sections can be fitted in the central rapidity region by the formula Ed 3 σ pp /d 3 p = 575( √ s) 3.3 /p 9.14 t pb/GeV 2 [10]. Using this fit one can estimate the prompt photon spectrum in nucleus-nucleus, A+B, collisions at the impact parameter b:
Ed 3 N AB (b)/d 3 p = Ed 3 σ pp /d 3 p · AB · T AB (b)
, where the nuclear overlapping function is defined as
T AB (b) = N coll (b)/σ pp in , where N coll (b)
is the average number of binary NN collisions. Nuclear effects (Cronin, shadowing) are ignored in this approach. Au+Au events (N coll =650) at 25 AGeV one can expect ∼ 10 −4 prompt photons with p t > 1.5 GeV/c. At the beam intensity 10 9 per second and 1% interaction probability, for 10% of most central collisions ∼ 10 2 prompt photons per second are expected. PYTHIA [11] simulations agree reasonably with the data extrapolation to FAIR energies [4]. For the prompt photon cross section PYTHIA predicts: σ ≈ 2 · 10 −4 mb, 85% from the process gq → γq and 15% from qq → γg.
γ π → ρ π HSD, γ ρ → π π HSD, >1 0 ρ / ρ , γ 0 π → ω HSD, HSD, sum γ prompt
Rescattering and decay photons from HSD
Cross sections for meson rescatterings πρ → πγ and ππ → ργ evaluated in the theory of the π and ρ meson gas [12] have been prepared by the ITEP group [4] and implemented by E.Bratkovskaya into the HSD code. The cross sections diverge at a threshold but averaging over the spectral function of the ρ meson solves the problem. Fig. 2 shows comparison of HSD predictions with the data of the WA98 collaboration at the SPS energy [5]. Besides the meson rescatterings, photons from decays of short-living resonances (e.g. ω → π 0 γ) are also taken into account. In the case when the life time of a resonance is less than characteristic time of the nucleus-nucleus collision it is difficult to reconstruct the resonance because the decay hadron (π 0 ) can reinteract with surrounding medium especially if this medium is dense. About 10% of ω → π 0 γ decays take place during the dense nuclear matter stage when ρ/ρ 0 > 1 (ρ 0 is the normal nuclear density). We will assume that it will not be possible to reconstruct these decays.
From Fig. 2 one can see that at p t =0.5 -1 GeV/c the decay photon contribution is comparable with the contribution from πρ → πγ. The model prediction is ∼ 10 times lower then one can expect from the experimental data. There are other sources of direct photons as on the hadron (Kρ → Kγ, ...) and the quarkgluon (qg → qγ, qq → gγ) levels which are so far not taken into account in the HSD transport code. Thus the HSD results can be considered as a lower estimate of data. Photon rapidity densities dN/dy at y = y c.m. are 0.38, 0.26 and 0.17 for πρ, ππ and ω respectively. On the same plot the prompt photon estimation ( see the section 2) is also presented by a line at p t > 1.5 GeV/c. Fig. 3 demonstrates predictions of the HSD code at the FAIR energy 25 AGeV for Au+Au central collisions. As at the SPS energy here at p t = 0.5 -1 GeV/c one can observe significant contribution of photons from the decays ω → π 0 γ in the dense matter. On the same plot the photon spectrum from π 0 decays is also shown. In the region p t = 0.5 − 1 GeV/c γ direct /γ π 0 ≃ 0.5%. As have been 2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 mentioned it is a lower estimate. Rapidity densities dN/dy at y = y c.m. are 0.10, 0.12 and 0.09 for πρ, ππ and ω respectively. Average multiplicities per event are 0.31, 0.32 and 0.25 for πρ, ππ and ω respectively. At high p t =1.5 -2 GeV/c the part of direct photons is higher γ prompt /γ π 0 ≃ 2%.
γ π → ρ π BHD, γ π → ρ π HSD, γ ρ → π π BHD, γ ρ → π π HSD, γ 0 π → ω BHD, >1 0 ρ / ρ , γ 0 π → ω HSD,
Direct photons from BHD
In the Bjorken hydrodynamics [8] it is assumed that during the ion-ion collision the system is mainly expanding in beam direction in a boost-invariant way. Natural variables are the proper time τ = √ t 2 − z 2 and rapidity. Thermodynamical variables (pressure, temperature, ...) do not depend on the rapidity but are functions of τ , e.g. T = T 0 (τ 0 /τ ) 1/3 for an ideal ultrarelativistic gas. Main initial parameters are the proper time τ 0 and temperature T 0 . Viscosity and conductivity effects are neglected. For this simple space-time evolution one can evaluate simple formula for photon spectrum with the photon emission rate as input (the section 2.2.2 of [3]). Photon yield is proportional to ∼ τ 2 0 . Though the Bjorken scenario is expected to be valid for ultrarelativistic energies (RHIC, LHC) one can assume it can be used for estimations at SPS and FAIR energies. The parameterizations presented in [13] have been used for the thermal photon emission rates of the channels πρ → πγ and ππ → ργ. The direct photon WA98 data can be reproduced at p t > 1.5 GeV/c by the BHD model with the parameters τ 0 = 1 fm and T 0 = 235 MeV [14]. Fig. 4 shows BHD predictions with the parameters τ 0 = 1 fm, T 0 = 180 MeV at the FAIR energy 25 AGeV for central Au+Au events side by side with the HSD results discussed before. The BHD spectra are broader than ones in the HSD transport approach. The main reason is the local termalization assumption used in the BHD model. γ direct /γ π 0 ≃ 0.5% and 3% at p t = 1 and 1.5 GeV/c respectively. Fig. 5 demonstrates a sensitivity of the direct photon spectrum to the initial parameter T 0 . The 10 MeV increase in the T 0 value leads to 1.5 and 2.5 times higher photon yield at p t = 0.1 GeV/c and 2 GeV/c respectively.
Using the extrapolation of existing data estimations of prompt photon production at FAIR energies have been made. At y = y c.m. the rapidity density of prompt photons with p t > 1.5 GeV/c per central Au+Au event at 25 AGeV is estimated as ∼ 10 −4 . With the planed beam intensity 10 9 per second and 1% interaction probability, for 10% of most central events one can expect the prompt photon rate ∼ 10 2 photons per second.
Direct photons from the hadron scenario of ion collisions generated by the HSD transport approach with implemented meson scatterings πρ → πγ, ππ → ργ have been analyzed. Photons from short-living resonances (e.g. ω → π 0 γ) decaying during the dense phase of the collision should be considered as direct photons. They contribute significantly in the direct photon spectrum at p t = 0.5 -1 GeV/c. At the FAIR energy 25 AGeV in Au+Au central collisions the HSD generator predicts, as a lower estimate, γ direct /γ π 0 ≃ 0.5% in the region p t = 0.5 − 1 GeV/c. At p t = 1.5 − 2 GeV/c γ prompt /γ π 0 ≃ 2%.
Thermal direct photons have been evaluated with the BHD model. The BHD spectra differ strongly from the HSD predictions. The direct photon spectrum is very sensitive to the initial temperature parameter T 0 of the model. The 10 MeV increase in the T 0 value leads to ∼ 2 times higher photon yield.
This work was partially supported by the Russian Foundation for Basic Research, grant number 06-08-01555 and Federal agency of Russia for atomic energy (Rosatom).
Figure 1 :
1Rapidity density at y = y c.m. of prompt photons with p t > p 0 t at FAIR energies.
Figure 2 :
2The invariant direct photon spectra for central Pb+Pb events at 158 AGeV in the central rapidity region |y − y c.m. | < 0.5.
Figure 3 :
3The invariant photon spectra predicted by the HSD generator for central (b=0.5 fm) Au+Au events at 25 AGeV in the central rapidity region |y − y c.m. | < 0.5.
Figure 4 :
4Comparison of the BHD (τ 0 = 1 fm, T 0 = 180 MeV) and HSD invariant photon spectra for central Au+Au events at 25 AGeV in the central rapidity region.
Figure 5 :
5Direct photon spectrum in the central rapidity regionpredicted by the BHD model for central Au+Au collisions at 25 AGeV for different values of the initial parameter T 0 .
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| [] |
[] | [
"Valter Moretti [email protected] \nDepartment of Mathematics\nTrento University\nI-38050Povo (TN)Italy\n",
"Valter Moretti [email protected] \nDepartment of Mathematics\nTrento University\nI-38050Povo (TN)Italy\n"
] | [
"Department of Mathematics\nTrento University\nI-38050Povo (TN)Italy",
"Department of Mathematics\nTrento University\nI-38050Povo (TN)Italy"
] | [] | This is a quick review on some technology concerning the local zeta function applied to Quantum Field Theory in curved static (thermal) spacetime to regularize the stress energy tensor and the field fluctuations. | 10.1007/978-3-642-19760-4_30 | [
"https://arxiv.org/pdf/1010.1630v1.pdf"
] | 119,272,418 | 1010.1630 | 3dd29d2941b495bcf08a5600dcafca690342660f |
8 Oct 2010 October 2010
Valter Moretti [email protected]
Department of Mathematics
Trento University
I-38050Povo (TN)Italy
8 Oct 2010 October 2010Local ζ-functions, stress-energy tensor, field fluctua-tions, and all that, in curved static spacetime Dedicated to Prof. Emilio Elizalde on the occasion of his 60th birthday
This is a quick review on some technology concerning the local zeta function applied to Quantum Field Theory in curved static (thermal) spacetime to regularize the stress energy tensor and the field fluctuations.
1 Quasifree QFT in curved static manifolds, Euclidean approach ζ-function technique.
1.1 The ζ-function determinant. Suppose we are given a n×n positive-definite Hermitian matrix A with eigenvalues 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ n . One can define the complex-valued function
ζ(s|A) = n j=1 λ −s j ,(1)
where s ∈ I C. (Notice that λ −s j is well-defined since λ j > 0.) By direct inspection one proves that:
det A = e − dζ(s|A) ds | s=0 .(2)
This trivial result can be generalized to provide a useful definition of the determinant of an operator working in an infinite-dimensional Hilbert space. To this end, focus on a non-negative self-adjoint operator A whose spectrum is discrete and each eigenspace has a finite dimension, and consider the series with s ∈ I C (the prime on the sum henceforth means that any possible null eigenvalues is omitted)
ζ(s|A) := j ′ λ −s .(3)
Looking at (2), the idea [Ha77] is to define, once again.
det A = e − dζ(s|A) ds | s=0 ,
where now the function ζ on the right-hand side is, in the general case, the analytic continuation of the function defined by the series (3) in its convergence domain, since the series may diverge at s = 0 -and this is the standard situation in the infinite-dimensional case! -provided that the analytic extension really reaches a neignorhood of the point s = 0. The interesting fact is that this procedure truly works in physically relevant cases, related to QFT in curved spacetime, producing meaningful results as we go to discuss in the following section.
1.2 Thermal QFT in static spacetimes. A smooth globally hyperbolic spacetime (M, g) is said to be static if it admits a (local) time-like Killing vector field ∂ t normal to a smooth spacelike Cauchy surface Σ. Consequently, there are (local) coordinate frames (
x 0 , x 1 , x 2 , x 3 ) ≡ (t, x)
where g 0i = 0 (i = 1, 2, 3) and ∂ t g µν = 0 and x are local coordinates on Σ. Though the results we are going to present may be generalize to higher spin fields, we henceforth stick to the case of a real scalar field φ propagating in M and satisfying an equation of motion of the form
P φ = 0 ,(4)
where P := −∇ µ ∇ µ + V , V being a smooth scalar field like
V (x) := ξR + m 2 + V ′ (x) .(5)
We also assume that V ′ satisfies ∂ t V ′ = 0 so that the space of solutions of (4) is invariant under t-displacements. Furthermore ξ ∈ IR, is a constant, R is the scalar curvature and m 2 the squared mass of the particles associated to the field. The domain of P is the space of real-valued C ∞ functions compactly supported Cauchy data on Σ. In the quasifree case, a straightforward way to define a QFT consistes of the assignment of a suitable Green function of the operator P [FR87], in particular the Feynman propagator G F (x, x ′ ) or, equivalently, the Wightman functions W ± (x, x ′ ). Then the GNS theorem (e.g. see [KW91]) allows one to construct a corresponding Fock realization of the theory. In a globally hyperbolic static spacetime it is possible to chose t-invariant Green functions. In that case, in static coordinates, one performs the Wick rotation obtaining the Euclidean formulation of the same QFT. This means that (locally) one can pass from the Lorentzian manifold (M, g) to a Riemannian manifold (M E , g (E) ) by the analytic continuation t → iτ where t, τ ∈ IR. This defines a (local) Killing vector ∂ τ in the Riemannian manifold M E and a corresponding (local) "static" coordinate frame (τ, x) therein. As is well-known [FR87], when the orbits of the Euclidean time τ are closed with period β, T = 1/β has to be interpreted as the temperature of the quantum state because the Wightman two-point function of the associated quasifree state satisfy the KMS condition at the inverse temperature β. In this approach, the Feynman propagator G F (t − t ′ , x, x ′ ) determines -and (generally speaking [FR87]) it is completely determined by -a proper Green function (in the spectral theory sense) S β (τ − τ ′ , x, x ′ ) of a corresponding self-adjoint extension A of the operator
A ′ := −∇ (E) a ∇ (E)a + V ( x) : C ∞ 0 (M E ) → L 2 (M E , dµ g (E) ) .(6)
S β (τ − τ ′ , x, x ′ ) is periodic with period β in the τ − τ ′ entry and it is said the Schwinger function. As a matter of fact, S β turns out to be the integral kernel of A −1 when A > 0.
The partition function of the quantum state associated to S β is the functional integral evaluated over the field configurations periodic with period β in the Euclidean time
Z β = Dφ e −S E [φ] ,(7)
the Euclidean action
S E being (dµ g (E) := g (E) d 4 x) S E [φ] = 1 2 M dµ g (E) (x) φ(x)(Aφ)(x) .(8)
Thus, extending the analogous result for finite dimensional Gaussian integral, one has
Z β = det A µ 2 −1/2 ,(9)
where µ is a mass scale which is necessary for dimensional reasons. To give a sensitive interpretation of that determinant, the idea [Ha77] is to try to exploit (2).
If M E is a D-dimensional Riemannian compact manifold and A ′ is bounded below by some constant b ≥ 0, A ′ admits the Friedrichs self-adjoint extension A which is also bounded below by the same bound of A ′ , moreover the spectrum of A is discrete and each eigenspace has a finite dimension. Then, as we said, one can consider the series
ζ(s|A/µ 2 ) := j ′ λ j µ 2 −s .(10)
Remarkably [Ha77,BCEMZ03], in the given hypotheses, the series above converges for Re s > D/2 and it is possible to continue the right-hand side above into a meromorphic function of s which is regular at s = 0. Following (2) and taking the presence of µ into account, we define:
Z β := e 1 2 d ds | s=0 ζ(s|A/µ 2 ) ,(11)
where the function ζ on the right-hand side is the analytic continuation of that defined in (10). It is possible the define the ζ function in terms of the heat kernel of the operator A, K(t, x, y|A) [BCEMZ03]. This is the smooth integral kernel of the (Hilbert-Schmidt, trace-class) operators e −tA , t > 0. One has, for Re s > D/2,
ζ(s|A/µ 2 ) = M dµ g (E) (x) +∞ 0 dt µ 2s t s−1 Γ(s) [K(t, x, x|A) − P 0 (x, x|A)] ,(12)
P (x, y|A) is the integral kernel of the projector on the null-eigenvalues eigenspace of A.
When M E is not compact, the spectrum of A may included a continuous-spectrum part, however, it is still possible to generalize the definitions and the results above considering suitable integrals on the spectrum of A, provided A is strictly positive (e.g, see [Wa79]). Another very useful tool is the local ζ function that can be defined in two differen,t however equivalent, ways [Wa79, Mo98, BCEMZ03]:
ζ(s, x|A/µ 2 ) = +∞ 0 dt µ 2s t s−1 Γ(s) [K(t, x, x|A) − P 0 (x, x|A)] ,(13)
and, φ j being the smooth eigenvector of the eigenvalue λ j ,
ζ(s, x|A/µ 2 ) = j ′ λ j µ 2 −s φ j (x)φ * j (x) .(14)
Both the integral and the series converges for Re s > D/2 and the local zeta function enjoys the same analyticity properties of the integrated ζ function. For future convenience it is also useful to define, in the sense of the analytic continuation,
ζ(s, x, y|A/µ 2 ) = +∞ 0 dt µ 2s t s−1 Γ(s) [K(t, x, y|A) − P 0 (x, y|A)](15)
(see [Mo98,Mo99] for the properties of this off-diagonal ζ-function). In the framework of the ζ-function regularization framework, the effective Lagrangian is defined as
L(x|A) µ 2 := 1 2 d ds | s=0 ζ(s, x|A/µ 2 ) ,(16)
and thus, in a thermal theory, Z β = e −S β where S β = dµ g L βµ 2 . A result which generalizes to any dimension an earlier results by Wald [Wa79] is the following [Mo98]. The above-defined effective Lagrangian can be computed by a point-splitting procedure: For D even
L(y|A) µ 2 = lim x→y − +∞ 0 dt 2t K(t, x, y|A) − a D/2 (x, y) 2(4π) D/2 ln µ 2 σ(x, y) 2 + D/2−1 j=0 ( D 2 − j − 1)! a j (x, y|A) 2(4π) D/2 2 σ(x, y) D/2−j − 2γ a D/2 (y, y) 2(4π) D/2 ,(17)
for D odd (notice that µ disappears)
L(y|A) µ 2 = lim x→y − +∞ 0 dt 2t K(t, x, y|A) − 2 σ(x, y) a (D−1)/2 (x, y) 2(4π) D/2 + (D−3)/2 j=0 (D − 2j − 2)!! 2 (D+1)/2−j a j (x, y|A) 2(4π) D/2 2 σ(x, y) D/2−j .(18)
Above, σ(x, y) is one half the square of the geodesical distance of x from y and the coefficients a j are the well-known off-diagonal coefficients of the small-t expansion of the heat-kernel. These coefficients, in spite of their non symmetric definition, turns out to by invariant when interchanging x and y [Mo99, Mo00].
2 Stress-energy tensor and field fluctuations 2.1 Generalizations of the local ζ function technique. Physically relevant quantities are the (quantum) field fluctuation and the averaged (quantum) stress tensor, respectively:
< φ 2 (x) > = δ δJ(x) | J≡0 ln Dφ e −S E + dµ g (E) φ 2 J ,(19)< T ab (x) > = 2 g (E) (x) δ δg (E)ab (x) ln Dφ e −S E [g (E) ] .(20)
A standard method to compute them is the so-called point-splitting procedure [
g (E) (x) < T ab (x) > " = "2 δ δg (E)ab (x) ln Z β " = " δ δg (E)ab (x) d ds | s=0 ζ(s|A/µ 2 ) " = " δ δg (E)ab (x) d ds | s=0 j ′ λ j µ 2 −s " = " d ds | s=0 µ −2s j ′ δλ −s j δg (E)ab (x) .(21)
Following this route, one define the ζ-regularized (or renormalized) stress tensor as
< T ab (x|A) > µ 2 := 1 2 d ds | s=0 Z ab (s, x|A/µ 2 ) ,(22)
where, in the sense of the analytic continuation of the left-hand side
Z ab (s, x|A/µ 2 ) := 2 j ′ µ −2s δλ −s j δg ab (x) .(23)
The difficult problem is now twofold: how to compute the functional derivative in the righthand side of (23) and whether or not the series in the right-hand side of (23) defines an analytic function of s in a neighborhood of s = 0. We have the result [Mo97, Mo99]:
Theorem 1. If M E is compact, A ≥ 0 and µ 2 > 0, then Z ab (s, x|A/µ 2 ) is well-defined and is a C ∞ function of x which is also meromorphic in s ∈ I C. In particular, it is analytic in a neighborhood of s = 0.
The result above has been checked even in several noncompact manifolds (containing singularities) [Mo97,BCEMZ03]. In that case, the summation in the right-hand side of (23) has to be replaced by a suitable spectral integration. The series in the right-hand side of (23) can be explicitly computed as [Mo97,Mo99]:
s j ′ 2 µ 2 λ j µ 2 −s−1 T ab [φ j , φ * j ](x) + g ab (x) λ j µ 2 −s , T ab [φ j , φ * j ](x)
being the classical stress tensor evaluated on the modes of A (see [Mo97,Mo99,BCEMZ03] for details). The series converges for Re s > 3D/2 + 2. It is similarly possible to define a ζ-function regularizing the field fluctuation [IM98,Mo98]:
< φ 2 (x|A) > µ 2 := d ds | s=0 Φ(s, x|A/µ 2 ) ,
where Φ(s, x|A/µ 2 ) := s µ 2 ζ(s + 1, x|A/µ 2 ) .
The properties of these functions have been studied in [IM98,Mo98] and several applications on concrete cases are considered (e.g. cosmic-string spacetime and homogeneous spacetimes). In particular, in [Mo98], the problem of the change of the parameter m 2 in the field fluctuations has been studied.
2.2 Physically meaningfulness of the procedures. We are now interested in the physical meaningfulness of the presented regularization techniques. The following general results strongly suggest that it is the case [Mo97, Mo99, Mo03].
Theorem 2. If M E is compact, A ≥ 0 and µ 2 > 0, and the averaged quantities above are those defined above in terms of local ζ-function regularization, then (a) < T ab (x|A) > µ 2 defines a C ∞ symmetric tensorial field.
(b) Similarly to the classical result,
∇ b < T bc (x|A) > µ 2 = − 1 2 < φ 2 (x|A) > µ 2 ∇ c V ′ (x) .(25)
(c) Concerning the trace of the stress tensor, it is naturally decomposed in the classical and the known quantum anomalous part
g ab (x) < T ab (x|A) > µ 2 = ξ D − ξ 4ξ D − 1 ∆ − m 2 − V ′ (x) < φ 2 (x|A) > µ 2 + δ D a D/2 (x, x|A) (4π) D/2 − P 0 (x, x|A) ,(26)
where δ D = 0 if D is odd and δ D = 1 if D is even,
ξ D = (D − 2)/[4(D − 1)]. (d) for any α > 0 < T ab (x|A) > αµ 2 =< T ab (x|A) > µ 2 +t ab (x) ln α ,(27)
where, the form of t ab (x) which depends on the geometry only and is in agreement with Wald's axioms [Wa94], has been given in [Mo99,Mo03]. (e) In the case ∂ 0 = ∂ τ is a global Killing vector, the manifold admits periodicity β along the lines tangent to ∂ 0 and M remains smooth (near any fixed points of the Killing orbits) fixing arbitrarily β in a neighborhood and, finally, Σ is a global surface everywhere normal to ∂ 0 , then
∂ ∂β ln Z(β) µ 2 = Σ d x g( x) < T 0 0 (x, β|A) > µ 2 .(28)
Another general achievement regards the possibility to recover the Lorentzian theory from the Euclidean one [Mo99]:
Theorem 3. Let M E be compact, A ≥ 0, µ 2 > 0. Also assume that M E is static with global Killing time ∂ τ and (orthogonal) global spatial section Σ and finally, 1, 2, 3, ..., D − 1 where the averaged quantities above are those defined above in terms of local ζ-function regularization and coordinates τ ≡ x 0 , x ∈ Σ are employed.
∂ τ V ′ ≡ 0. Then (a) ∂ τ < φ 2 (x|A) > µ 2 ≡ 0; (b) ∂ τ L(x|A) µ 2 ≡ 0; (c) ∂ τ < T ab (x|A) > µ 2 ≡ 0; (d) < T 0i (x|A) > µ 2 ≡ 0 for i =
These properties allow one to continue the Euclidean considered quantities into imaginary values of the coordinate τ → it obtaining real functions of the Lorentzian time t. Some of the properties above (regarding Thm.1, Thm. 2, Thm.3) have been found to be valid in some noncompact manifolds too (Rindler spacetime, cosmic string spacetime, Einstein's open spacetime, H N spaces, Gödel spacetime, BTZ spacetime) [Mo97,IM98,Ca98,Ra98,Ra98b,Ra99,Ra05,BMVZ98,RF02,SS04,AMR05]. In particular, the presented theory has been successfully exploited to compute the quantum back reaction on the three-dimensional BTZ metric [BMVZ98] in the case of the singular ground state containing a naked singularity. A semiclassical implementation of the cosmic censorship conjecture has been found in that case.
where G(x, y) is the symmtric part the two-point Wightman function of the considered quantum state or, in Euclidean approach, the corresponding Schwinger function. H(x, y) is the Hadamard parametrix which depends on the local geometry only and takes the short-distance singularity into account. H(x, y) is represented in terms of a truncated series of functions of σ(x, y). The operator D ab (x, y) is a bi-tensorial operator obtained by "splitting" the argument of the classical expression of the stress tensor (see [Mo99,Mo03]). Finally Q(y) is a scalar obtained by imposing several physical conditions (essentially, the appearance of the conformal anomaly, the conservation of the stress tensor and the triviality of the Minkowskian limit) [Wa94] in the lefthand side of (30) (see [BD82,Fu91,Wa94,Mo99] for details). More recently, in the framework of Lorentzian generally locally covariant algebraic quantum field theory in curved spacetime, it has established [Mo03] that Q can be omitted, redefining the classical stress-energy tensor, and thus D ab (x, y), into a way that it does not affect the classical expression of T µν when computed on solutions of the equations of motion, improving the point-splitting procedure. See [Hac10] where that point-splitting procedure is discussed and applied especially to cosmology. In geodesically convex neighborhoods:
H µ (x, y) = L j=0 u j (x, y)σ(x, y) j (4π) D/2 (σ(x, y)/2) D/2−1 + δ D M j=0 v j (x, y)σ(x, y) j ln µ 2 σ(x, y) 2 + δ D N j=0 w j (x, y)σ(x, y) j .
L, M, N are fixed integers (see [Mo99,Mo03] for details), δ D = 0 if D is odd and δ D = 1 otherwise. The coefficients u j and v j are smooth functions of (x, y) which are completely determined by the local geometry. The coefficients w j are determined once one has fixed w 0 , and they are omitted [Mo03] when dropping Q. Dealing with Euclidean approaches, it is possible to explicitly compute u j and v j in terms of heat-kernel coefficients [Mo98,Mo99]. One has the following result [Mo98,Mo99].
Theorem 4. If M E is compact, A ≥ 0 and µ 2 > 0, and the averaged quantities in the left-hand side below are those defined above in terms of local ζ-function regularization, then
< φ 2 (y|A) > µ 2 = lim x→y G(x, y) − H µ ′ (x, y) ,(32)
< T ab (y|A) > µ 2 = lim x→y D ab (x, y) G(x, y) − H µ ′ (x, y) + g ab (y)Q(y) ,
where G(x, y) = ζ(1, x, y|A/µ 2 ) given in (15), H µ ′ is completely determined by (31) with the requirement w 0 (x, y) := − a D/2−1 (x, y|A) (4π) D/2 [2γ + ln µ ′ 2 ] ,
and the term Q is found to be Q(y) = 1 D −P 0 (y, y|A) + δ D a D/2 (y, y|A) (4π) D/2 .
If D ab is defined in order to drop Q in the right-hand side of (33), H µ 2 is determined by fixing w 0 (x, y) = 0. The scales µ and µ ′ satisfies µ = cµ ′ for some constant c > 0.
2. 3 .
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| [] |
[
"Percolation on hypergraphs with four-edges",
"Percolation on hypergraphs with four-edges"
] | [
"Ojan Khatib \nDepartment of Physics\nUniversity of Michigan\n48109-1040Ann ArborMIUSA\n",
"Damavandi ",
"Robert M Ziff [email protected] \nCenter for the Study of Complex Systems\nDepartment of Chemical Engineering\nUniversity of Michigan\n48109-2136Ann ArborMIUSA\n"
] | [
"Department of Physics\nUniversity of Michigan\n48109-1040Ann ArborMIUSA",
"Center for the Study of Complex Systems\nDepartment of Chemical Engineering\nUniversity of Michigan\n48109-2136Ann ArborMIUSA"
] | [] | We study percolation on self-dual hypergraphs that contain hyperedges with four bounding vertices, or "four-edges", using three different generators, each containing bonds or sites with three distinct probabilities p, r, and t connecting the four vertices. We find explicit values of these probabilities that satisfy the selfduality conditions discussed by Bollobás and Riordan. This demonstrates that explicit solutions of the self-duality conditions can be found using generators containing bonds and sites with independent probabilities. These solutions also provide new examples of lattices where exact percolation critical points are known. One of the generators exhibits three distinct criticality solutions (p, r, t). We carry out Monte-Carlo simulations of two of the generators on two different hypergraphs to confirm the critical values. For the case of the hypergraph and uniform generator studied by Wierman et al., we also determine the threshold p = 0.441374 ± 0.000001, which falls within the tight bounds that they derived. Furthermore, we consider a generator in which all or none of the vertices can connect, and find a soluble inhomogeneous percolation system that interpolates between site percolation on the union-jack lattice and bond percolation on the square lattice.Abstract. Supplementary material. | 10.1088/1751-8113/48/40/405004 | [
"https://arxiv.org/pdf/1506.06125v2.pdf"
] | 118,481,075 | 1506.06125 | 1ed7d0027bb325ee6f8300e17e9549321d53a9a2 |
Percolation on hypergraphs with four-edges
August 2015 17 Sep 2015
Ojan Khatib
Department of Physics
University of Michigan
48109-1040Ann ArborMIUSA
Damavandi
Robert M Ziff [email protected]
Center for the Study of Complex Systems
Department of Chemical Engineering
University of Michigan
48109-2136Ann ArborMIUSA
Percolation on hypergraphs with four-edges
August 2015 17 Sep 2015Percolation on hypergraphs with four-edges 2
We study percolation on self-dual hypergraphs that contain hyperedges with four bounding vertices, or "four-edges", using three different generators, each containing bonds or sites with three distinct probabilities p, r, and t connecting the four vertices. We find explicit values of these probabilities that satisfy the selfduality conditions discussed by Bollobás and Riordan. This demonstrates that explicit solutions of the self-duality conditions can be found using generators containing bonds and sites with independent probabilities. These solutions also provide new examples of lattices where exact percolation critical points are known. One of the generators exhibits three distinct criticality solutions (p, r, t). We carry out Monte-Carlo simulations of two of the generators on two different hypergraphs to confirm the critical values. For the case of the hypergraph and uniform generator studied by Wierman et al., we also determine the threshold p = 0.441374 ± 0.000001, which falls within the tight bounds that they derived. Furthermore, we consider a generator in which all or none of the vertices can connect, and find a soluble inhomogeneous percolation system that interpolates between site percolation on the union-jack lattice and bond percolation on the square lattice.Abstract. Supplementary material.
Introduction
Percolation is a fundamental model in both statistical physics and mathematics [1,2,3,4,5,6]. It is concerned with the formation of long-range connectivity which occurs when the occupation of sites or bonds exceeds a critical threshold. Finding the exact threshold for different lattices is one of the primary goals in the study of percolation. Precise thresholds are necessary for applying percolation models to real systems, and for studying the behavior of systems near the critical point. For many years, exact values of threshold had been known for only a handful of lattices (e.g., square, triangular, honeycomb, kagome, and bow-tie lattices, for site and/or bond percolation) [7,8,9]. Recently, percolation thresholds have been found for a broad class of lattices that can be represented in the form of three-hypergraphs self-dual under the triangle-triangle transformation [10,11]. In a hypergraph representation, the hyperedges (in the shape of triangles, squares, etc.) can represent any collection of bonds and internal sites, including correlated sites and bonds; all that matters is the connection probabilities between the boundary vertices. For a triangular generator, there is a single nontrivial self-duality condition which gives a unique percolation threshold [12,10,13,14] Prob(all vertices connect) = Prob(none connect) (1) and this has led to many exactly soluble lattices when applied on self-dual threehypergraphs [10]. However, several important lattices correspond to hypergraphs with edges of four vertices (four-edges) and cannot be represented on a three-hypergraph, and it is desirable to find thresholds for such lattices as well. Bollobás and Riordan [14] discussed the self-duality conditions for self-dual four-hypergraphs, and recently Wierman et al. [15] used those conditions and a stochastic ordering method to find bounds for a 16-bond uniform probability generator on a certain four-hypergraph. The purpose of this paper is to find explicit self-dual four-vertex generators and to confirm with Monte-Carlo simulation that when they are placed in a self-dual hypergraph containing four-edges, the system is indeed at the threshold. In contrast to the case of hypergraphs with triangular generators, the conditions for a square do not reduce to one single equation but to three as we will see below; therefore, we allow for three different probabilities to ensure that the set of equations have nontrivial solutions. It will turn out that the number of solutions depends on the generator we choose. We study the hypergraph introduced by Bollobás and Riordan (hypergraph A) shown in Fig. 1 along with another hypergraph based upon the (3 2 ,4,3,4) Archimedean lattice, which we call hypergraph B, shown in Fig. 2. We consider in detail two generators, one with 12 bonds (generator I), and one with 16 bonds introduced by Wierman et al. [15] (generator II), both containing three distinct bond probabilities p, r and t as shown in Fig. 3. We verify the criticality of these solutions numerically. We also study the uniform case p = r = t for generator II on hypergraph A numerically and find that the threshold falls well within the bounds found in [15]. Furthermore, a variant of generator I (which we call generator III) is considered by replacing the inner square in Fig. 3a with a single site with probability t as shown in Fig. 3c. We do not however, perform Monte Carlo simulations for this case as this generator is not the primary focus of this paper. Hypergraph A is a four-uniform self-dual hypergraph-i.e., consists of only fouredges connecting boundary vertices in a self-dual configuration, as shown in Fig. 1. In constructing the dual, a vertex is put in each empty polygon (or face of the hypergraph), and a dual hyperedge is drawn around each hyperedge. Hypergraph B is a bit different from hypergraph A in that it also contains two-edges (ordinary bonds). These bonds have probabilities p 1 and 1 − p 1 in a manner that leaves the hypergraph self-dual (Fig. 2a). Because the hypergraph is self-dual, the value of p 1 does not matter at the critical point, as long as the generators in the four-edges satisfy the duality conditions. We can therefore manipulate p 1 and nothing should change. A variant of this hypergraph which only involves four-edges is obtained by setting p 1 = 0 (Fig. 2b). We note that another approach to finding thresholds for some lattices has recently been put forth by Grimmett and Manolescu [16], for lattices that can be represented in an isoradial form in which each polygon can be inscribed in a circle of equal radius. This method can be used to find a geometrical proof [17] of Wu's criticality condition for the square checkerboard lattice [18], however, it cannot be used to study the types of square lattices considered here. We also note that methods have recently been developed to find approximate thresholds for many lattices to extraordinarily high precision [19,20].
In the following sections, we discuss the definitions (Section 2), the general selfduality conditions (Section 3), the derivation of the explicit critical points of the two generators (Section 4), Monte-Carlo simulations (Section 5), the analysis of the critical manifold for generator I (Section 6), the derivation of a generalized union-jack lattice for site percolation (Section 7), and conclusions (Section 8). Explicit polynomials for the generators are given in the Supplementary Material.
Definitions
Consider a generator G with four boundary vertices A, B, C, and D. Denote the dual generator by G * . Any configuration (i.e., designation of internal edges as open or closed) on G determines a partition of the boundary vertices into clusters of vertices that are connected by edges. We use the following notation defined in [11]: connected vertices are grouped into clusters, and distinct clusters are separated by a vertical bar. For instance, P G [AB|CD] means that A and B are connected, C and D are also connected, but the two sets are not connected to each other. We also introduce the quantities P n for isotropic systems defined as follows:
P 1 =P 6 = two unconnected pairs = P G [AB|CD] or P G [AD|BC](2)
Note that we cannot have P G [AC|BD]; otherwise the graph will be non-planar. Normalization requires
P 1 + 4P 2 + 4P 3 + P 4 + 2P 5 + 2P 6 = 1(3)
Self-duality conditions
By the duality relationship between G and G * , an edge in G * crossing an open edge in G cannot be open. Immediately it follows that P G 4). Assuming that at the critical point, G and G * should have the same probability of connection between the vertices, we get the first self-duality condition which needs to be satisfied:
[ABCD] = P G * [A * |B * |C * |D * ] (Fig.P 4 (p, r, t) = P 1 (p, r, t)(4)
analogous to (1) for the triangular case. However, here we get two additional conditions:
P 2 (p, r, t) = P 3 (p, r, t)(5)
P 5 (p, r, t) = P 6 (p, r, t)
as given in [14]. Figures 5 and 6 illustrate these two additional duality conditions. The three relations yield three nontrivial equations, unlike in the case of hypergraphs with triangular generators where there is only one nontrivial equation (1 for criticality. It is for this reason that we chose three distinct bond probabilities (p, r, t) within the generators instead of only one uniform probability p.
Derivation of the critical points
To find the critical points, we need explicit expressions for the various P n (p, r, t). For the two generators, the six probabilities defined in (2) will be of the form
P n (p, s, t) = N i=0 4 j=0 4 k=0 c (n) ijk p i q N −i r j s 4−j t k u 4−k(7)
where q = 1 − p, s = 1 − r, and u = 1 − t, and N = 4 for generator I and N = 8 for generator II. To find the c (n) ijk , we used the method of exact enumeration, as follows: Go through every possibility of bond placement (there are 2 12 and 2 16 possibilities for generators I and II respectively); calculate the probability of percolation for each bond configuration noting that each configuration corresponds to a unique set of {i, j, k}; then count all the possible configurations that give the same probability of percolation (i.e., the same {i, j, k}); also keep track of the vertex connectivity for each configuration using standard cluster algorithms. The last two steps identify c Having found the probabilities, we then solve (4), (5) and (6) numerically to find the self-dual points. For generator II we find a single physically meaningful solution: As we will discuss in Sec. 6, these three points lie on a manifold in (p, r, t) space. By slightly changing the values we can find the other points on the critical manifold, which now depends upon the hypergraph being considered. We find for a range of probabilities on the critical manifold away from these three exact solutions, the duality relations are very closely (but not exactly) satisfied.
We also look at generator III (a variant of generator I), in which the vertices on the inner square are correlated in such a way that either all of them are connected with probability t or none are connected with probability 1 − t, i.e. the inner square in generator I is squeezed into a site that is open with probability t or closed with probability 1 − t (Fig. 3). In this case we find a single physical solution:
p = 0.19738202171710149917476592797778709312 r = 0.44960772662591992436180558214955545825 (12) t = 0.99956609784920984836488846372795998038
Interestingly, the solution for t is very close to 1 but not exactly 1. If t = 1 were a solution, then we would have a two-probability generator that satisfies the three duality relations. Likewise, a generator with four outside bonds and two crossing diagonal bonds does not satisfy the duality conditions. The values of p, r, and t above are closest to the third solution to generator I (11), but still quite different (especially t).
For comparison, we present P i (p, r, t) for the solutions given in equations (8)(9)(10)(11)(12) in Table 1. We observe that the values in any of the columns are very close to each other. In fact if we plot for example P i vs. P j they nearly fall on a straight line. In Sec. 6 we give a plot of P 2 vs. P 1 . Table 1. The values of P i (p, r, t) for the self-dual points given in (8)(9)(10)(11)(12).
generator P 1 (= P 4 ) P 2 (= P 3 ) P 5 (= P 6 ) I (
Monte-Carlo simulations
We carried out Monte-Carlo simulations using a Leath-type of growth algorithm on a lattice of size 16384 × 16384; clusters were grown up to a size cutoff 2 20 = 1048576, guaranteeing that the boundaries of the system were never reached. This allowed us to find an unbiased estimate of P ≥s = the probability that a point belongs to a cluster whose size is greater than s. According to scaling theory, we have at p c , P ≥s ∼ s 2−τ where in 2d, τ − 2 = 5/91, and for p close to p c , we have
s τ −2 P ≥s ∼ af (b(p − p c )s σ ) ≈ A + B(p − p c )s σ(13)
where f (z) is the universal scaling function, and the last term above follows from a Taylor series expansion of f (z), where a, b, A, and B are non-universal constants which are specific to the system being studied, while σ = 36/91 and f (z) are universal. Thus a plot of s τ −2 P ≥s vs. (p − p c )s σ yields a straight line (for large s) with a slope proportional to p − p c . When p = p c , s τ −2 P ≥s is constant, apart from deviations for small s where scaling is not valid.
Generator I on hypergraph A
In this case, because we have two parallel bonds where the square generators touch, it was convenient to replace those two bonds having probability p by one bond having probability p = 2p − p 2 [9], as shown in Fig. 7. Figure 8 shows the results for the simulation of 250 000 samples on a lattice of 16384×16384 for generator I on hypergraph A, at the predicted critical point of the first solution (9) and with values of p equal to 0.0001 above and below the critical value 0.07878785 . . .. It can be seen that (9) corresponds to a critical point within high numerical accuracy (at least 10 −5 ). Likewise, we verified that the other two solutions are also at the critical point within this error.
Generator II on hypergraph A
We also confirmed the criticality of the self-dual point (8) uniform probabilities with this generator and hypergraph, and found the critical value p = r = t = 0.441374 ± 0.000001, which falls well within the bounds given by Wierman et al. [15]: ‡ 0.424072 ≤ p c ≤ 0.463661 (14) This is a non-self-dual percolation threshold, specific to the uniform generator II on hypergraph A. At this value of p, the connection probabilities are P 1 = 0.174841, P 2 = 0.0645187, P 3 = 0.0786097, P 4 = 0.153594, P 5 = 0.0375026, and P 6 = 0.0120231. The duality conditions (4-6) are far from being satisfied by these values, and the values of the P i are far from those of the other generators listed in Table 1.
To find this result we simulated up to 10 8 samples for each value of p using a smaller cutoff of 2 15 = 32768. Our Monte-Carlo results are plotted in Fig. 9. Here we show an alternate way of analyzing the data, where we plot s τ −2 P ≥s vs. s −Ω where Ω = 72/91 [21,22] represents the finite-size scaling for smaller s. At p c , we have s τ −2 P ≥s = A + Cs −Ω , so at p c our plot should yield a straight line. This form of plotting the data is useful when p is very close to p c .
Generator I on hypergraph B
Here, we have the two additional bonds, one with probability p 1 and the other with probability p 2 = 1 − p 1 . However, since we are looking at self-dual square generators, ‡ We mention that there appears to be a minor calculation error in [15]. In section 4.2, which concerns the model considered here, the authors give the fourth probability as 0.432569051787763. However, using Mathematica to solve the fifth equation in their Table 4,
P p [C 1 ] + P p [C 2 ] + P p [C 3 ] = P p [C 4 ] + P p [C 5 ] + P p [C 6
], we find instead p = 0.4444066898118085. (The fifth probability they give is the solution to the fourth equation in Table 4.) However, because this value falls within their other values, it does not alter the final bounds (14).
which do not depend on p 1 , we expect that by changing p 1 nothing changes, and this is exactly what is observed. We performed the simulation for four different values of p 1 (0.00, 0.30, 0.50, and 0.85) and for the three self-dual points given in equations (9-11), the system was found to be critical in all of these cases within numerical accuracy.
Generator II on hypergraph B
We verified that the self-dual point (8), is the critical point for the four values of p 1 used above, consistent with our expectations. We also tested the uniform probability critical point (p = r = t = 0.441374) that we found for generator II hypergraph A, and observed that in this case, as expected, p 1 does matter. The system is at criticality only for the two points p 1 ≈ 0.7665 and p 1 ≈ 1 − 0.7665 = 0.2335.
Further analysis of critical manifolds
For generator I, we further studied the self-dual relations. We independently evaluated each of the three conditions given in eqns. (4), (5) and (6) and found p in terms of r and t. In this way, we found three equations for p(r, t), one for each duality condition:
P 4 (p, r, t) = P 1 (p, r, t) ⇒ p (1) (r, t)(15)P 2 (p, r, t) = P 3 (p, r, t) ⇒ p (2) (r, t)(16)P 5 (p, r, t) = P 6 (p, r, t) ⇒ p (3) (r, t)(17)
Figures 10, 11 and 12 show the manifolds for the three self-duality conditions. These three manifolds intersect at three points-precisely our self-dual points. To see the intersection of these surfaces, we set pairs of the p (n) (r, t) equations equal to each other and find t as a function of r. This gives us three non-independent relations t(r):
p (1) (r, t) = p (2) (r, t) ⇒ t (1) (r)(18)p (1) (r, t) = p (3) (r, t) ⇒ t (2) (r)(19)p (2) (r, t) = p (3) (r, t) ⇒ t (3) (r)(20)
where t (1) (r) is satisfied if both self-duality conditions (4) and (5) are satisfied, and so forth. The interesting fact is that if we plot the curves of t (1) (r), t (2) (r) and t (3) (r), they are very nearly identical in a wide region (Fig. 13), which implies that the points along these curves should be quite close to self-duality. Finally, we plot their differences in Fig. 14, which shows clearly that the curves cross at the three self-dual points. The curves remain within 0.00001 of each other over a wide range of r, meaning that we "almost" have a manifold of thresholds. Motivated by the results given in Table 1, we find P i (p, r, t) along the curves in Fig. 13. P 2 vs. P 1 is plotted in Fig. 15. Also on the same plot, we show the values for the self-dual points of generator I (where the three curves cross) along with generator III, which is a variant of generator I. It can be seen that the points lie almost on a straight line; we have no explanation for this surprising result. The point (P 1 , P 2 ) for generator II falls on the same approximate line, however, it is not shown on Fig. 15 as it is farther away from other points on the plot. Similar nearly linear curves can be obtained for other P i vs. P j . Figure 15. Plot of P 2 (p, r, t) vs. P 1 (p, r, t) for the points t (i) (r) along the curves in Fig. 13. The circles mark the values of the self-dual probabilities. These are indeed at the crossing points of the curves. The black square shows the probabilities for generator III, which falls nearly on the curve determined by t (3) (r).
The same analysis was not possible for generator II because the p (n) (r, t)'s were found to be singular at many points, and solutions for t (n) (r) could not be found.
Results for site percolation
By choosing a generator in which either all or none of the four vertices are connected together, we can generate critical site percolation systems. (For the three-hypergraph case where the hypergraph is a regular array of triangles, this procedure yields simply the system of site percolation on the triangular lattice.) Thus, we assume P 2 = P 3 = P 5 = P 6 = 0, and the self-duality conditions are satisfied if P 1 = P 4 = 1/2. Applying this to hypergraph A, we find site percolation on the triangular lattice once again (Fig. 16). In Fig. 17 we consider a stretched version of hypergraph A, which is also self-dual, and applying the all-or-none generator to this system yields another lattice with a site threshold of 1/2, but with non-planar bonds.
Applying the all-or-none generator to hypergraph B, we find site percolation on the union-jack lattice with non-uniform probabilities p 1 , (1 − p 1 ) and 1/2, as shown in Fig. 18. In fact, as shown in that figure, we can generalize this further by having a hypergraph with alternating dual hyperedges, where the blue hyperedge is the dual to the red hyperedge, and the system is still self-dual. This yields a generalization to a union-jack lattice with probabilities p 1 and p 2 as shown in Fig. 18b. This is an interesting system that interpolates continuously between the uniform union-jack lattice (p 1 = p 2 = 1/2) and the covering lattice or line graph for bond percolation on the square lattice (p 1 = 0 or 1).
Other lattices
The finding of an inhomogeneous site percolation model motivated us to look for other soluble cases of inhomogenous site percolation models to compare with. Here we list several of them. For comparison, we can think of our union-jack system as having sites with probabilities p 1 , p 2 , p 3 , and p 4 , with p 1 + p 3 = 1 and p 2 + p 4 = 1, implying the condition (
1 − p 1 − p 3 )(1 − p 2 − p 4 ) = 0 or 1 − p 1 − p 2 − p 3 − p 4 + p 1 p 2 + p 1 p 4 + p 2 p 3 + p 3 p 4 = 0(21)
For inhomogeneous site percolation on the kagome lattice-the covering lattice of the honeycomb lattice-Sykes and Essam [8] showed that the threshold is given by
1 − p 1 p 2 − p 1 p 3 − p 2 p 3 + p 1 p 2 p 3 = 0(22)
In the limit p 3 → 1, this gives p 1 + p 2 = 1, in which case the system becomes the nonplanar square-lattice covering lattice, similar to what happens to our union-jack lattice when p 1 → 0 or 1. We can find the site threshold for the inhomogeneous martini lattice [23] by considering a martini generator with the three outer bonds having probability p 1 , p 2 Figure 18. Hypergraph B and its equivalent site percolation system. (a) We replace the bonds with probability p 1 and 1 − p 1 with green and orange sites respectively. More generally, we can also have hyperedges with alternate probability p 2 and 1 − p 2 , a configuration which we demonstrate by blue and red squares respectively. (b) generalized site percolation on the union-jack lattice with site probabilities p 1 , 1 − p 1 , p 2 , and 1 − p 2 . and p 3 , and triangular part correlated with all vertices connected with probability p 4 and disconnected with probability 1 − p 4 . The covering lattice to this is the martini site lattice, and using (1) we find that the threshold is given by
1 − p 1 p 2 p 4 − p 1 p 3 p 4 − p 2 p 3 p 4 + p 1 p 2 p 3 p 4 = 0(23)
For the covering lattice of the inhomogeneous martini lattice, shown in Fig. 19b, the site threshold is [24,10,25,26] 1 − p 1 p 2 r 3 − p 2 p 3 r 1 − p 1 p 3 r 2 − p 1 p 2 r 1 r 2 − p 1 p 3 r 1 r 3 − p 2 p 3 r 2 r 3 + p 1 p 2 p 3 r 1 r 2 + p 1 p 2 p 3 r 1 r 3 + p 1 p 2 p 3 r 2 r 3 + p 1 p 2 r 1 r 2 r 3 + p 1 p 3 r 1 r 2 r 3 + p 2 p 3 r 1 r 2 r 3 − 2p 1 p 2 p 3 r 1 r 2 r 3 = 0
which interpolates between site percolation on the inhomogeneous triangular covering lattice (by setting all p's equal to 1) and site percolation on the inhomogeneous kagome lattice (setting all r's equal to 1). While there evidently exists several examples of site thresholds for non-uniform systems, none of these yield the union-jack lattice in any limit, and consequently do not yield our result given above.
Conclusions
The self-duality and criticality of a hypergraph with four-edges is determined by the duality conditions (4), (5) and (6) given by Bollobás and Riordan. While it is easy Figure 19. (a) The inhomogeneous martini lattice. The outer star of bonds of the generator has probabilities p 1 , p 2 and p 3 , and the inner triangle bonds have probabilities r 1 , r 2 and r 3 . The covering lattice transformation is also shown on the top generator where each bond is replaced with a site having the same probability. Two sites are connected by a dashed line if the corresponding bonds are connected. (b) the equivalent covering lattice, which is also called the (1 × 1) : (2 × 2) subnet lattice [25].
to construct generators with correlated bonds that satisfy these conditions, it is not obvious that one can do it with independently occupied bonds, or independent bonds and internal sites. In this paper we showed that this can indeed be done, making use of three different four-vertex generators (I, II, and III). We imposed the self-duality conditions and found the percolation thresholds for these three generators, as given in (8)(9)(10)(11)(12). These critical probabilities are independent of the hypergraph, as long as the hypergraph is self-dual. Then we carried out Monte-Carlo simulations for generators I and II on the two self-dual hypergraphs (A and B) and observed these critical probabilities indeed give the percolation threshold for the corresponding generator, independent of the irrelevant p 1 for hypergraph B. Furthermore, we found for generator II on hypergraph A the homogeneous threshold p = r = t = 0.441374 ± 0.000001, in agreement with the bounds (14) found by Wierman et al. [15]. Because the uniform generator is not self-dual, this homogeneous threshold is specific to that hypergraph (A), and can result in criticality in hypergraph B only if we choose p 1 ≈ 0.7665 or p 1 ≈ 0.2335. We observe that although for generator I there are three self-dual critical points, there are also infinitely many points very close to being critical. This can be seen from Fig. 13, 14 and 15.
Using the all-or-none generator, we can find models for site percolation based upon the hypergraphs. With this generator on hypergraph B we found a non-uniform union-jack model that interpolates between the uniform union-jack lattice and bond percolation on the square lattice, providing a range of simple fully-triangulated systems that maybe useful for criticality studies. For example, one can see how the lattice metric factor [27] or the number of clusters per site [28] varies as one goes from one system to the other.
This work shows that while four-hypergraphs are perhaps more restricted than the case of three-hypergraphs (where many self-dual configurations and also uniformprobability generators can easily be found), there are still exact systems with independent bond and site occupancies that can be constructed. It would be interesting to find other examples of self-dual four-hypergraphs and study other generators than the ones considered here, perhaps some that satisfy the duality conditions with two or one distinct probability.
Exact connection polynomials
Following are exact enumeration polynomials for P 1 (p, r, t) through P 6 (p, r, t), defined in Eq. (2) of the paper, for generators I, II, and III, where q = 1 − p, s = 1 − r and u = 1 − t:
9.1. Generator I P 1 = 4q 4 rs 3 t 4 + q 4 s 4 t 4 + 16q 4 rs 3 t 3 u + 4q 4 s 4 t 3 u + 20q 4 r 2 s 2 t 2 u 2 + 24q 4 rs 3 t 2 u 2 + 6q 4 s 4 t 2 u 2 + 8q 4 r 3 stu 3 +20q 4 r 2 s 2 tu 3 +16q 4 rs 3 tu 3 +4q 4 s 4 tu 3 +q 4 r 4 u 4 +4q 4 r 3 su 4 +6q 4 r 2 s 2 u 4 +4q 4 rs 3 u 4 + q 4 s 4 u 4 P 2 = pq 3 r 2 s 2 t 4 +q 4 r 2 s 2 t 4 +4pq 3 rs 3 t 4 +pq 3 s 4 t 4 +4pq 3 r 2 s 2 t 3 u+4q 4 r 2 s 2 t 3 u+16pq 3 rs 3 t 3 u+ 4pq 3 s 4 t 3 u + 4pq 3 r 3 st 2 u 2 + 4q 4 r 3 st 2 u 2 + 23pq 3 r 2 s 2 t 2 u 2 + 3q 4 r 2 s 2 t 2 u 2 + 24pq 3 rs 3 t 2 u 2 + 6pq 3 s 4 t 2 u 2 + pq 3 r 4 tu 3 + q 4 r 4 tu 3 + 10pq 3 r 3 stu 3 + 2q 4 r 3 stu 3 + 21pq 3 r 2 s 2 tu 3 + q 4 r 2 s 2 tu 3 + 16pq 3 rs 3 tu 3 + 4pq 3 s 4 tu 3 + pq 3 r 4 u 4 + 4pq 3 r 3 su 4 + 6pq 3 r 2 s 2 u 4 + 4pq 3 rs 3 u 4 + pq 3 s 4 u 4 P 3 = p 2 q 2 r 3 st 4 + 2pq 3 r 3 st 4 + q 4 r 3 st 4 + 3p 2 q 2 r 2 s 2 t 4 + 4pq 3 r 2 s 2 t 4 + 4p 2 q 2 rs 3 t 4 + p 2 q 2 s 4 t 4 + 4p 2 q 2 r 3 st 3 u + 8pq 3 r 3 st 3 u + 4q 4 r 3 st 3 u + 12p 2 q 2 r 2 s 2 t 3 u + 16pq 3 r 2 s 2 t 3 u + 16p 2 q 2 rs 3 t 3 u + 4p 2 q 2 s 4 t 3 u+p 2 q 2 r 4 t 2 u 2 +2pq 3 r 4 t 2 u 2 +q 4 r 4 t 2 u 2 +11p 2 q 2 r 3 st 2 u 2 +14pq 3 r 3 st 2 u 2 +q 4 r 3 st 2 u 2 + 28p 2 q 2 r 2 s 2 t 2 u 2 + 10pq 3 r 2 s 2 t 2 u 2 + 24p 2 q 2 rs 3 t 2 u 2 + 6p 2 q 2 s 4 t 2 u 2 + 2p 2 q 2 r 4 tu 3 + 2pq 3 r 4 tu 3 + 12p 2 q 2 r 3 stu 3 + 4pq 3 r 3 stu 3 + 22p 2 q 2 r 2 s 2 tu 3 + 2pq 3 r 2 s 2 tu 3 + 16p 2 q 2 rs 3 tu 3 + 4p 2 q 2 s 4 tu 3 + p 2 q 2 r 4 u 4 + 4p 2 q 2 r 3 su 4 + 6p 2 q 2 r 2 s 2 u 4 + 4p 2 q 2 rs 3 u 4 + p 2 q 2 s 4 u 4 P 4 = p 4 r 4 t 4 +4p 3 qr 4 t 4 +6p 2 q 2 r 4 t 4 +4pq 3 r 4 t 4 +q 4 r 4 t 4 +4p 4 r 3 st 4 +16p 3 qr 3 st 4 +20p 2 q 2 r 3 st 4 + 8pq 3 r 3 st 4 + 6p 4 r 2 s 2 t 4 + 24p 3 qr 2 s 2 t 4 + 20p 2 q 2 r 2 s 2 t 4 + 4p 4 rs 3 t 4 + 16p 3 qrs 3 t 4 + p 4 s 4 t 4 + 4p 3 qs 4 t 4 + 4p 4 r 4 t 3 u + 16p 3 qr 4 t 3 u + 24p 2 q 2 r 4 t 3 u + 16pq 3 r 4 t 3 u + 4q 4 r 4 t 3 u + 16p 4 r 3 st 3 u + 64p 3 qr 3 st 3 u + 80p 2 q 2 r 3 st 3 u + 32pq 3 r 3 st 3 u + 24p 4 r 2 s 2 t 3 u + 96p 3 qr 2 s 2 t 3 u + 80p 2 q 2 r 2 s 2 t 3 u + 16p 4 rs 3 t 3 u+64p 3 qrs 3 t 3 u+4p 4 s 4 t 3 u+16p 3 qs 4 t 3 u+6p 4 r 4 t 2 u 2 +24p 3 qr 4 t 2 u 2 +30p 2 q 2 r 4 t 2 u 2 + 12pq 3 r 4 t 2 u 2 + 24p 4 r 3 st 2 u 2 + 96p 3 qr 3 st 2 u 2 + 84p 2 q 2 r 3 st 2 u 2 + 8pq 3 r 3 st 2 u 2 + 36p 4 r 2 s 2 t 2 u 2 + 144p 3 qr 2 s 2 t 2 u 2 + 52p 2 q 2 r 2 s 2 t 2 u 2 + 24p 4 rs 3 t 2 u 2 + 96p 3 qrs 3 t 2 u 2 + 6p 4 s 4 t 2 u 2 + 24p 3 qs 4 t 2 u 2 + 4p 4 r 4 tu 3 + 16p 3 qr 4 tu 3 + 12p 2 q 2 r 4 tu 3 + 16p 4 r 3 stu 3 + 64p 3 qr 3 stu 3 + 24p 2 q 2 r 3 stu 3 + 24p 4 r 2 s 2 tu 3 + 96p 3 qr 2 s 2 tu 3 + 12p 2 q 2 r 2 s 2 tu 3 + 16p 4 rs 3 tu 3 + 64p 3 qrs 3 tu 3 + 4p 4 s 4 tu 3 + 16p 3 qs 4 tu 3 + p 4 r 4 u 4 + 4p 3 qr 4 u 4 + 4p 4 r 3 su 4 + 16p 3 qr 3 su 4 + 6p 4 r 2 s 2 u 4 + 24p 3 qr 2 s 2 u 4 + 4p 4 rs 3 u 4 + 16p 3 qrs 3 u 4 + p 4 s 4 u 4 + 4p 3 qs 4 u 4 P 5 = q 4 r 2 s 2 t 4 + 4q 4 r 2 s 2 t 3 u + 2q 4 r 3 st 2 u 2 + 2q 4 r 2 s 2 t 2 u 2 P 6 = 2p 2 q 2 r 2 s 2 t 4 + 2pq 3 r 2 s 2 t 4 + 4p 2 q 2 rs 3 t 4 + p 2 q 2 s 4 t 4 + 8p 2 q 2 r 2 s 2 t 3 u + 8pq 3 r 2 s 2 t 3 u + 16p 2 q 2 rs 3 t 3 u + 4p 2 q 2 s 4 t 3 u + p 2 q 2 r 4 t 2 u 2 + 2pq 3 r 4 t 2 u 2 + q 4 r 4 t 2 u 2 + 8p 2 q 2 r 3 st 2 u 2 + 8pq 3 r 3 st 2 u 2 + 26p 2 q 2 r 2 s 2 t 2 u 2 + 6pq 3 r 2 s 2 t 2 u 2 + 24p 2 q 2 rs 3 t 2 u 2 + 6p 2 q 2 s 4 t 2 u 2 + 2p 2 q 2 r 4 tu 3 + 2pq 3 r 4 tu 3 + 12p 2 q 2 r 3 stu 3 + 4pq 3 r 3 stu 3 + 22p 2 q 2 r 2 s 2 tu 3 + 2pq 3 r 2 s 2 tu 3 + 16p 2 q 2 rs 3 tu 3 + 4p 2 q 2 s 4 tu 3 + p 2 q 2 r 4 u 4 + 4p 2 q 2 r 3 su 4 + 6p 2 q 2 r 2 s 2 u 4 + 4p 2 q 2 rs 3 u 4 + p 2 q 2 s 4 u 4 9.2. Generator II P 1 = 4p 2 q 6 r 4 t 4 + 8pq 7 r 4 t 4 + q 8 r 4 t 4 + 16p 2 q 6 r 3 st 4 + 32pq 7 r 3 st 4 + 4q 8 r 3 st 4 + 24p 2 q 6 r 2 s 2 t 4 + 48pq 7 r 2 s 2 t 4 + 6q 8 r 2 s 2 t 4 + 16p 2 q 6 rs 3 t 4 + 32pq 7 rs 3 t 4 + 4q 8 rs 3 t 4 + 4p 2 q 6 s 4 t 4 + 8pq 7 s 4 t 4 + q 8 s 4 t 4 + 16p 2 q 6 r 4 t 3 u + 32pq 7 r 4 t 3 u + 4q 8 r 4 t 3 u + 64p 2 q 6 r 3 st 3 u + 128pq 7 r 3 st 3 u + 16q 8 r 3 st 3 u+16p 3 q 5 r 2 s 2 t 3 u+136p 2 q 6 r 2 s 2 t 3 u+192pq 7 r 2 s 2 t 3 u+24q 8 r 2 s 2 t 3 u+32p 3 q 5 rs 3 t 3 u+ 144p 2 q 6 rs 3 t 3 u + 128pq 7 rs 3 t 3 u + 16q 8 rs 3 t 3 u + 16p 3 q 5 s 4 t 3 u + 56p 2 q 6 s 4 t 3 u + 32pq 7 s 4 t 3 u + 4q 8 s 4 t 3 u + 24p 2 q 6 r 4 t 2 u 2 + 48pq 7 r 4 t 2 u 2 + 6q 8 r 4 t 2 u 2 + 96p 2 q 6 r 3 st 2 u 2 + 192pq 7 r 3 st 2 u 2 + 24q 8 r 3 st 2 u 2 + 4p 4 q 4 r 2 s 2 t 2 u 2 + 80p 3 q 5 r 2 s 2 t 2 u 2 + 320p 2 q 6 r 2 s 2 t 2 u 2 + 288pq 7 r 2 s 2 t 2 u 2 + 36q 8 r 2 s 2 t 2 u 2 + 20p 4 q 4 rs 3 t 2 u 2 + 176p 3 q 5 rs 3 t 2 u 2 + 380p 2 q 6 rs 3 t 2 u 2 + 192pq 7 rs 3 t 2 u 2 + 24q 8 rs 3 t 2 u 2 + 16p 4 q 4 s 4 t 2 u 2 + 112p 3 q 5 s 4 t 2 u 2 + 124p 2 q 6 s 4 t 2 u 2 + 48pq 7 s 4 t 2 u 2 + 6q 8 s 4 t 2 u 2 + 16p 2 q 6 r 4 tu 3 + 32pq 7 r 4 tu 3 + 4q 8 r 4 tu 3 + 64p 2 q 6 r 3 stu 3 + 128pq 7 r 3 stu 3 + 16q 8 r 3 stu 3 + 8p 4 q 4 r 2 s 2 tu 3 + 128p 3 q 5 r 2 s 2 tu 3 + 368p 2 q 6 r 2 s 2 tu 3 + 192pq 7 r 2 s 2 tu 3 + 24q 8 r 2 s 2 tu 3 + 64p 4 q 4 rs 3 tu 3 +320p 3 q 5 rs 3 tu 3 +336p 2 q 6 rs 3 tu 3 +128pq 7 rs 3 tu 3 +16q 8 rs 3 tu 3 +64p 4 q 4 s 4 tu 3 + 128p 3 q 5 s 4 tu 3 + 96p 2 q 6 s 4 tu 3 + 32pq 7 s 4 tu 3 + 4q 8 s 4 tu 3 + 4p 2 q 6 r 4 u 4 + 8pq 7 r 4 u 4 + q 8 r 4 u 4 + 16p 2 q 6 r 3 su 4 + 32pq 7 r 3 su 4 + 4q 8 r 3 su 4 + 2p 4 q 4 r 2 s 2 u 4 + 32p 3 q 5 r 2 s 2 u 4 + 92p 2 q 6 r 2 s 2 u 4 + 48pq 7 r 2 s 2 u 4 + 6q 8 r 2 s 2 u 4 + 16p 4 q 4 rs 3 u 4 + 80p 3 q 5 rs 3 u 4 + 84p 2 q 6 rs 3 u 4 + 32pq 7 rs 3 u 4 + 4q 8 rs 3 u 4 + 16p 4 q 4 s 4 u 4 + 32p 3 q 5 s 4 u 4 + 24p 2 q 6 s 4 u 4 + 8pq 7 s 4 u 4 + q 8 s 4 u 4 P 2 = p 4 q 4 r 4 t 4 + 4p 3 q 5 r 4 t 4 + 4p 2 q 6 r 4 t 4 + 4p 4 q 4 r 3 st 4 + 16p 3 q 5 r 3 st 4 + 16p 2 q 6 r 3 st 4 + 6p 4 q 4 r 2 s 2 t 4 + 24p 3 q 5 r 2 s 2 t 4 + 24p 2 q 6 r 2 s 2 t 4 + 4p 4 q 4 rs 3 t 4 + 16p 3 q 5 rs 3 t 4 + 16p 2 q 6 rs 3 t 4 + p 4 q 4 s 4 t 4 +4p 3 q 5 s 4 t 4 +4p 2 q 6 s 4 t 4 +4p 4 q 4 r 4 t 3 u+16p 3 q 5 r 4 t 3 u+16p 2 q 6 r 4 t 3 u+16p 4 q 4 r 3 st 3 u+ 64p 3 q 5 r 3 st 3 u + 64p 2 q 6 r 3 st 3 u + 2p 5 q 3 r 2 s 2 t 3 u + 36p 4 q 4 r 2 s 2 t 3 u + 110p 3 q 5 r 2 s 2 t 3 u + 90p 2 q 6 r 2 s 2 t 3 u + 4p 5 q 3 rs 3 t 3 u + 40p 4 q 4 rs 3 t 3 u + 92p 3 q 5 rs 3 t 3 u + 52p 2 q 6 rs 3 t 3 u + 2p 5 q 3 s 4 t 3 u + 16p 4 q 4 s 4 t 3 u + 30p 3 q 5 s 4 t 3 u + 10p 2 q 6 s 4 t 3 u + 6p 4 q 4 r 4 t 2 u 2 + 24p 3 q 5 r 4 t 2 u 2 + 24p 2 q 6 r 4 t 2 u 2 + 24p 4 q 4 r 3 st 2 u 2 + 96p 3 q 5 r 3 st 2 u 2 + 96p 2 q 6 r 3 st 2 u 2 + 10p 5 q 3 r 2 s 2 t 2 u 2 + 92p 4 q 4 r 2 s 2 t 2 u 2 + 206p 3 q 5 r 2 s 2 t 2 u 2 + 118p 2 q 6 r 2 s 2 t 2 u 2 + 24p 5 q 3 rs 3 t 2 u 2 + 130p 4 q 4 rs 3 t 2 u 2 + 186p 3 q 5 rs 3 t 2 u 2 + 54p 2 q 6 rs 3 t 2 u 2 + 14p 5 q 3 s 4 t 2 u 2 + 59p 4 q 4 s 4 t 2 u 2 + 44p 3 q 5 s 4 t 2 u 2 + 9p 2 q 6 s 4 t 2 u 2 + 4p 4 q 4 r 4 tu 3 + 16p 3 q 5 r 4 tu 3 +16p 2 q 6 r 4 tu 3 +16p 4 q 4 r 3 stu 3 +64p 3 q 5 r 3 stu 3 +64p 2 q 6 r 3 stu 3 +16p 5 q 3 r 2 s 2 tu 3 + 112p 4 q 4 r 2 s 2 tu 3 + 192p 3 q 5 r 2 s 2 tu 3 + 56p 2 q 6 r 2 s 2 tu 3 + 48p 5 q 3 rs 3 tu 3 + 168p 4 q 4 rs 3 tu 3 + 120p 3 q 5 rs 3 tu 3 + 24p 2 q 6 rs 3 tu 3 + 32p 5 q 3 s 4 tu 3 + 48p 4 q 4 s 4 tu 3 + 24p 3 q 5 s 4 tu 3 + 4p 2 q 6 s 4 tu 3 + p 4 q 4 r 4 u 4 +4p 3 q 5 r 4 u 4 +4p 2 q 6 r 4 u 4 +4p 4 q 4 r 3 su 4 +16p 3 q 5 r 3 su 4 +16p 2 q 6 r 3 su 4 +4p 5 q 3 r 2 s 2 u 4 + 28p 4 q 4 r 2 s 2 u 4 + 48p 3 q 5 r 2 s 2 u 4 + 14p 2 q 6 r 2 s 2 u 4 + 12p 5 q 3 rs 3 u 4 + 42p 4 q 4 rs 3 u 4 + 30p 3 q 5 rs 3 u 4 + 6p 2 q 6 rs 3 u 4 + 8p 5 q 3 s 4 u 4 + 12p 4 q 4 s 4 u 4 + 6p 3 q 5 s 4 u 4 + p 2 q 6 s 4 u 4 P 3 = p 6 q 2 r 4 t 4 + 6p 5 q 3 r 4 t 4 + 12p 4 q 4 r 4 t 4 + 8p 3 q 5 r 4 t 4 + 4p 6 q 2 r 3 st 4 + 24p 5 q 3 r 3 st 4 + 48p 4 q 4 r 3 st 4 + 32p 3 q 5 r 3 st 4 + 6p 6 q 2 r 2 s 2 t 4 + 36p 5 q 3 r 2 s 2 t 4 + 72p 4 q 4 r 2 s 2 t 4 + 48p 3 q 5 r 2 s 2 t 4 + 4p 6 q 2 rs 3 t 4 + 24p 5 q 3 rs 3 t 4 + 48p 4 q 4 rs 3 t 4 + 32p 3 q 5 rs 3 t 4 + p 6 q 2 s 4 t 4 + 6p 5 q 3 s 4 t 4 + 12p 4 q 4 s 4 t 4 + 8p 3 q 5 s 4 t 4 + 4p 6 q 2 r 4 t 3 u + 24p 5 q 3 r 4 t 3 u + 48p 4 q 4 r 4 t 3 u + 32p 3 q 5 r 4 t 3 u + 16p 6 q 2 r 3 st 3 u + 96p 5 q 3 r 3 st 3 u + 192p 4 q 4 r 3 st 3 u + 128p 3 q 5 r 3 st 3 u + 26p 6 q 2 r 2 s 2 t 3 u + 150p 5 q 3 r 2 s 2 t 3 u + 282p 4 q 4 r 2 s 2 t 3 u + 172p 3 q 5 r 2 s 2 t 3 u + 20p 6 q 2 rs 3 t 3 u + 108p 5 q 3 rs 3 t 3 u + 180p 4 q 4 rs 3 t 3 u + 88p 3 q 5 rs 3 t 3 u + 6p 6 q 2 s 4 t 3 u + 30p 5 q 3 s 4 t 3 u + 42p 4 q 4 s 4 t 3 u + 12p 3 q 5 s 4 t 3 u + 6p 6 q 2 r 4 t 2 u 2 + 36p 5 q 3 r 4 t 2 u 2 + 72p 4 q 4 r 4 t 2 u 2 + 48p 3 q 5 r 4 t 2 u 2 + 24p 6 q 2 r 3 st 2 u 2 + 144p 5 q 3 r 3 st 2 u 2 + 288p 4 q 4 r 3 st 2 u 2 + 192p 3 q 5 r 3 st 2 u 2 + 44p 6 q 2 r 2 s 2 t 2 u 2 + 238p 5 q 3 r 2 s 2 t 2 u 2 + 400p 4 q 4 r 2 s 2 t 2 u 2 + 200p 3 q 5 r 2 s 2 t 2 u 2 + 40p 6 q 2 rs 3 t 2 u 2 + 178p 5 q 3 rs 3 t 2 u 2 + 218p 4 q 4 rs 3 t 2 u 2 + 54p 3 q 5 rs 3 t 2 u 2 + 14p 6 q 2 s 4 t 2 u 2 + 48p 5 q 3 s 4 t 2 u 2 + 28p 4 q 4 s 4 t 2 u 2 + 4p 3 q 5 s 4 t 2 u 2 + 4p 6 q 2 r 4 tu 3 + 24p 5 q 3 r 4 tu 3 + 48p 4 q 4 r 4 tu 3 +32p 3 q 5 r 4 tu 3 +16p 6 q 2 r 3 stu 3 +96p 5 q 3 r 3 stu 3 +192p 4 q 4 r 3 stu 3 +128p 3 q 5 r 3 stu 3 + 36p 6 q 2 r 2 s 2 tu 3 + 176p 5 q 3 r 2 s 2 tu 3 + 236p 4 q 4 r 2 s 2 tu 3 + 56p 3 q 5 r 2 s 2 tu 3 + 40p 6 q 2 rs 3 tu 3 + 120p 5 q 3 rs 3 tu 3 + 64p 4 q 4 rs 3 tu 3 + 8p 3 q 5 rs 3 tu 3 + 16p 6 q 2 s 4 tu 3 + 16p 5 q 3 s 4 tu 3 + 4p 4 q 4 s 4 tu 3 + p 6 q 2 r 4 u 4 +6p 5 q 3 r 4 u 4 +12p 4 q 4 r 4 u 4 +8p 3 q 5 r 4 u 4 +4p 6 q 2 r 3 su 4 +24p 5 q 3 r 3 su 4 +48p 4 q 4 r 3 su 4 + 32p 3 q 5 r 3 su 4 + 9p 6 q 2 r 2 s 2 u 4 + 44p 5 q 3 r 2 s 2 u 4 + 59p 4 q 4 r 2 s 2 u 4 + 14p 3 q 5 r 2 s 2 u 4 + 10p 6 q 2 rs 3 u 4 + 30p 5 q 3 rs 3 u 4 + 16p 4 q 4 rs 3 u 4 + 2p 3 q 5 rs 3 u 4 + 4p 6 q 2 s 4 u 4 + 4p 5 q 3 s 4 u 4 + p 4 q 4 s 4 u 4 P 4 = p 8 r 4 t 4 + 8p 7 qr 4 t 4 + 24p 6 q 2 r 4 t 4 + 32p 5 q 3 r 4 t 4 + 16p 4 q 4 r 4 t 4 + 4p 8 r 3 st 4 + 32p 7 qr 3 st 4 + 96p 6 q 2 r 3 st 4 + 128p 5 q 3 r 3 st 4 + 64p 4 q 4 r 3 st 4 + 6p 8 r 2 s 2 t 4 + 48p 7 qr 2 s 2 t 4 + 144p 6 q 2 r 2 s 2 t 4 + 192p 5 q 3 r 2 s 2 t 4 + 96p 4 q 4 r 2 s 2 t 4 + 4p 8 rs 3 t 4 + 32p 7 qrs 3 t 4 + 96p 6 q 2 rs 3 t 4 + 128p 5 q 3 rs 3 t 4 + 64p 4 q 4 rs 3 t 4 + p 8 s 4 t 4 + 8p 7 qs 4 t 4 + 24p 6 q 2 s 4 t 4 + 32p 5 q 3 s 4 t 4 + 16p 4 q 4 s 4 t 4 + 4p 8 r 4 t 3 u + 32p 7 qr 4 t 3 u + 96p 6 q 2 r 4 t 3 u + 128p 5 q 3 r 4 t 3 u + 64p 4 q 4 r 4 t 3 u + 16p 8 r 3 st 3 u + 128p 7 qr 3 st 3 u + 384p 6 q 2 r 3 st 3 u + 512p 5 q 3 r 3 st 3 u + 256p 4 q 4 r 3 st 3 u + 24p 8 r 2 s 2 t 3 u + 192p 7 qr 2 s 2 t 3 u + 564p 6 q 2 r 2 s 2 t 3 u + 720p 5 q 3 r 2 s 2 t 3 u + 336p 4 q 4 r 2 s 2 t 3 u + 16p 8 rs 3 t 3 u + 128p 7 qrs 3 t 3 u + 360p 6 q 2 rs 3 t 3 u + 416p 5 q 3 rs 3 t 3 u + 160p 4 q 4 rs 3 t 3 u + 4p 8 s 4 t 3 u + 32p 7 qs 4 t 3 u + 84p 6 q 2 s 4 t 3 u + 80p 5 q 3 s 4 t 3 u + 16p 4 q 4 s 4 t 3 u + 6p 8 r 4 t 2 u 2 + 48p 7 qr 4 t 2 u 2 + 144p 6 q 2 r 4 t 2 u 2 + 192p 5 q 3 r 4 t 2 u 2 + 96p 4 q 4 r 4 t 2 u 2 + 24p 8 r 3 st 2 u 2 + 192p 7 qr 3 st 2 u 2 + 576p 6 q 2 r 3 st 2 u 2 + 768p 5 q 3 r 3 st 2 u 2 + 384p 4 q 4 r 3 st 2 u 2 + 36p 8 r 2 s 2 t 2 u 2 + 288p 7 qr 2 s 2 t 2 u 2 + 812p 6 q 2 r 2 s 2 t 2 u 2 + 944p 5 q 3 r 2 s 2 t 2 u 2 + 368p 4 q 4 r 2 s 2 t 2 u 2 + 24p 8 rs 3 t 2 u 2 + 192p 7 qrs 3 t 2 u 2 + 472p 6 q 2 rs 3 t 2 u 2 + 400p 5 q 3 rs 3 t 2 u 2 + 64p 4 q 4 rs 3 t 2 u 2 + 6p 8 s 4 t 2 u 2 + 48p 7 qs 4 t 2 u 2 + 92p 6 q 2 s 4 t 2 u 2 + 32p 5 q 3 s 4 t 2 u 2 + 2p 4 q 4 s 4 t 2 u 2 + 4p 8 r 4 tu 3 + 32p 7 qr 4 tu 3 + 96p 6 q 2 r 4 tu 3 + 128p 5 q 3 r 4 tu 3 + 64p 4 q 4 r 4 tu 3 + 16p 8 r 3 stu 3 + 128p 7 qr 3 stu 3 + 384p 6 q 2 r 3 stu 3 + 512p 5 q 3 r 3 stu 3 + 256p 4 q 4 r 3 stu 3 + 24p 8 r 2 s 2 tu 3 + 192p 7 qr 2 s 2 tu 3 + 496p 6 q 2 r 2 s 2 tu 3 + 448p 5 q 3 r 2 s 2 tu 3 + 64p 4 q 4 r 2 s 2 tu 3 + 16p 8 rs 3 tu 3 + 128p 7 qrs 3 tu 3 + 224p 6 q 2 rs 3 tu 3 + 64p 5 q 3 rs 3 tu 3 + 4p 8 s 4 tu 3 + 32p 7 qs 4 tu 3 + 16p 6 q 2 s 4 tu 3 + p 8 r 4 u 4 + 8p 7 qr 4 u 4 + 24p 6 q 2 r 4 u 4 + 32p 5 q 3 r 4 u 4 + 16p 4 q 4 r 4 u 4 + 4p 8 r 3 su 4 + 32p 7 qr 3 su 4 + 96p 6 q 2 r 3 su 4 + 128p 5 q 3 r 3 su 4 + 64p 4 q 4 r 3 su 4 + 6p 8 r 2 s 2 u 4 + 48p 7 qr 2 s 2 u 4 + 124p 6 q 2 r 2 s 2 u 4 + 112p 5 q 3 r 2 s 2 u 4 + 16p 4 q 4 r 2 s 2 u 4 + 4p 8 rs 3 u 4 + 32p 7 qrs 3 u 4 + 56p 6 q 2 rs 3 u 4 + 16p 5 q 3 rs 3 u 4 + p 8 s 4 u 4 + 8p 7 qs 4 u 4 + 4p 6 q 2 s 4 u 4 P 5 = p 4 q 4 r 4 t 4 + 4p 3 q 5 r 4 t 4 + 4p 2 q 6 r 4 t 4 + 4p 4 q 4 r 3 st 4 + 16p 3 q 5 r 3 st 4 + 16p 2 q 6 r 3 st 4 + 6p 4 q 4 r 2 s 2 t 4 + 24p 3 q 5 r 2 s 2 t 4 + 24p 2 q 6 r 2 s 2 t 4 + 4p 4 q 4 rs 3 t 4 + 16p 3 q 5 rs 3 t 4 + 16p 2 q 6 rs 3 t 4 + p 4 q 4 s 4 t 4 + 4p 3 q 5 s 4 t 4 + 4p 2 q 6 s 4 t 4 + 4p 4 q 4 r 4 t 3 u + 16p 3 q 5 r 4 t 3 u + 16p 2 q 6 r 4 t 3 u + 16p 4 q 4 r 3 st 3 u + 64p 3 q 5 r 3 st 3 u + 64p 2 q 6 r 3 st 3 u + 28p 4 q 4 r 2 s 2 t 3 u + 100p 3 q 5 r 2 s 2 t 3 u + 88p 2 q 6 r 2 s 2 t 3 u + 24p 4 q 4 rs 3 t 3 u + 72p 3 q 5 rs 3 t 3 u + 48p 2 q 6 rs 3 t 3 u + 8p 4 q 4 s 4 t 3 u + 20p 3 q 5 s 4 t 3 u + 8p 2 q 6 s 4 t 3 u + 6p 4 q 4 r 4 t 2 u 2 + 24p 3 q 5 r 4 t 2 u 2 + 24p 2 q 6 r 4 t 2 u 2 + 24p 4 q 4 r 3 st 2 u 2 + 96p 3 q 5 r 3 st 2 u 2 + 96p 2 q 6 r 3 st 2 u 2 + 50p 4 q 4 r 2 s 2 t 2 u 2 + 156p 3 q 5 r 2 s 2 t 2 u 2 + 108p 2 q 6 r 2 s 2 t 2 u 2 + 46p 4 q 4 rs 3 t 2 u 2 + 104p 3 q 5 rs 3 t 2 u 2 +38p 2 q 6 rs 3 t 2 u 2 +16p 4 q 4 s 4 t 2 u 2 +16p 3 q 5 s 4 t 2 u 2 +4p 2 q 6 s 4 t 2 u 2 +4p 4 q 4 r 4 tu 3 + 16p 3 q 5 r 4 tu 3 +16p 2 q 6 r 4 tu 3 +16p 4 q 4 r 3 stu 3 +64p 3 q 5 r 3 stu 3 +64p 2 q 6 r 3 stu 3 +44p 4 q 4 r 2 s 2 tu 3 + 112p 3 q 5 r 2 s 2 tu 3 + 40p 2 q 6 r 2 s 2 tu 3 + 32p 4 q 4 rs 3 tu 3 + 32p 3 q 5 rs 3 tu 3 + 8p 2 q 6 rs 3 tu 3 + p 4 q 4 r 4 u 4 + 4p 3 q 5 r 4 u 4 + 4p 2 q 6 r 4 u 4 + 4p 4 q 4 r 3 su 4 + 16p 3 q 5 r 3 su 4 + 16p 2 q 6 r 3 su 4 + 11p 4 q 4 r 2 s 2 u 4 + 28p 3 q 5 r 2 s 2 u 4 + 10p 2 q 6 r 2 s 2 u 4 + 8p 4 q 4 rs 3 u 4 + 8p 3 q 5 rs 3 u 4 + 2p 2 q 6 rs 3 u 4 P 6 = 2p 6 q 2 r 2 s 2 t 3 u + 8p 5 q 3 r 2 s 2 t 3 u + 8p 4 q 4 r 2 s 2 t 3 u + 4p 6 q 2 rs 3 t 3 u + 16p 5 q 3 rs 3 t 3 u + 16p 4 q 4 rs 3 t 3 u + 2p 6 q 2 s 4 t 3 u + 8p 5 q 3 s 4 t 3 u + 8p 4 q 4 s 4 t 3 u + 10p 6 q 2 r 2 s 2 t 2 u 2 + 40p 5 q 3 r 2 s 2 t 2 u 2 + 40p 4 q 4 r 2 s 2 t 2 u 2 + 20p 6 q 2 rs 3 t 2 u 2 + 68p 5 q 3 rs 3 t 2 u 2 + 56p 4 q 4 rs 3 t 2 u 2 + 10p 6 q 2 s 4 t 2 u 2 + 28p 5 q 3 s 4 t 2 u 2 + 11p 4 q 4 s 4 t 2 u 2 + 16p 6 q 2 r 2 s 2 tu 3 + 64p 5 q 3 r 2 s 2 tu 3 + 64p 4 q 4 r 2 s 2 tu 3 + 32p 6 q 2 rs 3 tu 3 + 80p 5 q 3 rs 3 tu 3 + 32p 4 q 4 rs 3 tu 3 + 16p 6 q 2 s 4 tu 3 + 16p 5 q 3 s 4 tu 3 + 4p 4 q 4 s 4 tu 3 + 4p 6 q 2 r 2 s 2 u 4 + 16p 5 q 3 r 2 s 2 u 4 + 16p 4 q 4 r 2 s 2 u 4 + 8p 6 q 2 rs 3 u 4 + 20p 5 q 3 rs 3 u 4 + 8p 4 q 4 rs 3 u 4 + 4p 6 q 2 s 4 u 4 + 4p 5 q 3 s 4 u 4 + p 4 q 4 s 4 u 4
Generator III
For generator III, the probabilities can be found by hand by going through the different configurations of the outside bonds first, resulting in the first entry. The second entry is the expansion which was verified by exact enumeration: P 1 = q 4 (t (4rs 3 + s 4 ) + u) = 4q 4 rs 3 t + q 4 s 4 t + q 4 u P 2 = pq 3 (t (r 2 s 2 + 4rs 3 + s 4 ) + u)+q 4 r 2 s 2 t = pq 3 r 2 s 2 t+q 4 r 2 s 2 t+4pq 3 rs 3 t+pq 3 s 4 t+pq 3 u P 3 = p 2 q 2 (t (rs 3 + s) + u) + 2pq 3 rs (1 − s 2 ) t + q 4 r 3 st = p 2 q 2 r 3 st + 2pq 3 r 3 st + q 4 r 3 st + 3p 2 q 2 r 2 s 2 t + 4pq 3 r 2 s 2 t + 4p 2 q 2 rs 3 t + p 2 q 2 s 4 t + p 2 q 2 u P 4 = p 4 + 4p 3 q + 4p 2 q 2 r (1 − s 3 ) t + 2p 2 q 2 (1 − s 2 ) 2 t + 4pq 3 r 2 (1 − s 2 ) t + q 4 r 4 t = p 4 r 4 t+4p 3 qr 4 t+6p 2 q 2 r 4 t+4pq 3 r 4 t+q 4 r 4 t+4p 4 r 3 st+16p 3 qr 3 st+20p 2 q 2 r 3 st+8pq 3 r 3 st+ 6p 4 r 2 s 2 t + 24p 3 qr 2 s 2 t + 20p 2 q 2 r 2 s 2 t + 4p 4 rs 3 t + 16p 3 qrs 3 t + p 4 s 4 t + 4p 3 qs 4 t + p 4 u + 4p 3 qu P 5 = q 4 r 2 s 2 t P 6 = p 2 q 2 (t (2r 2 s 2 + 4rs 3 + s 4 ) + u) + 2pq 3 r 2 s 2 t = 2p 2 q 2 r 2 s 2 t + 2pq 3 r 2 s 2 t + 4p 2 q 2 rs 3 t + p 2 q 2 s 4 t + p 2 q 2 u
Figure 1 .
1(a) Hypergraph A, and (b) showing the self-duality.
Figure 2 .
2(a) Hypergraph B with four-edges (squares) and single bonds of probability p 1 and 1 − p 1 . (b) Hypergraph B with probability p 1 set to zero.
Figure 3 .
3(a) Generator I (b) Generator II (c) Generator III
Figure 4 .
4Solid lines represent a four-hyperedge with boundary vertices A, B, C, and D. Dashed lines represent the dual hyperedge, with its boundary vertices A * , B * , C * , and D * .
Figure 5 .Figure 6 .
56If A, B, and D are connected in G (a case of P 3 ), only A * and B * can be connected in G * (a case of P 2 ). If A and C are connected diagonally in G (a case of P 5 ), then B * and C * , and A * and D * can be connected in G * (a case of P 6 ).
for a given n and {i, j, k}. The results of these calculations are given in the Supplementary Material.
First
generator I, we find three distinct solutions:
Figure 7 .Figure 8 .
78of generator II on hypergraph A by a Monte-Carlo simulation of 150 000 samples, and find results similar toFig. 8but will not be shown here. In addition, we carried out extensive simulations for the case of Replacing the two central parallel bonds (dashed lines) having probability p by one bond having effective probability p = 2p − p 2 , necessary when Generator I on hypergraph A is used. The dotted lines represent the interior of the generator with probabilities defined inFig. 3a. Plot of s τ −2 P ≥s vs. s σ for the Monte-Carlo simulation of generator I on hypergraph A, for the first self-dual point, eqn. (9) (middle), for p + 0.0001 and the same r and t as in (9) (upper), and for p − 0.0001 and again the same r and t (lower). The horizontal plot for the first case confirms that the system is at the critical point with those probabilities.
Figure 9 .
9Plot of s τ −2 P ≥s vs. s −Ω with Ω = 72/91 for Monte-Carlo results of the uniform probability generator II p = r = t on hypergraph A. The dashed line represents the linear behavior which is expected for p = p c , and suggests p c = 0.441374±0.000001. Some representative error bars are shown.
Figure 10 .
10p (1) (r, t) satisfying P 4 = P 1 . The column at the corner of the plot is the artifact of the plotting program caused by a singularity.
Figure 11 .Figure 12 .
1112p (2) (r, t) satisfying P 2 = P 3 . p (3) (r, t) satisfying P 5 = P 6 .
Figure 13 .Figure 14 .
1314Plot of the curves of t (1) (r), t (2) (r) and t (3) (r). There are actually three different curves here but they are not distinguishable in this figure. Plot of the difference between pairs of curves of t (n) (r) ofFig. 13. The curves cross exactly at self-dual points(9)(10)(11).
Figure 16 .
16If each hyperedge is replaced by a site in hypergraph A, we get a triangular site percolation system.
Figure 17 .
17Replacing all hyperedges by a site in this self-dual hypergraph created by "stretching out" hypergraph A yields another system with site percolation threshold of p = 1/2, but with non-planar crossing bonds (on the left). The average coordination number of the site lattice is six.
none of the vertices connected = P G [A|B|C|D] P 2 = only two nearest vertices connected = P G [AB|C|D] or P G [BC|A|D] or P G [CD|A|B] or P G [DA|B|C] P 3 = three out of four vertices connected = P G [ABC|D] or P G [ABD|C] or P G [ACD|B] or P G [BCD|A] P 4 = all four vertices connected = P G [ABCD] P 5 = only two vertices connected diagonally = P G [AC|B|D] or P G [BD|A|C]
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| [] |
[
"On the Maximum Density of Graphs with Good Edge-Labellings",
"On the Maximum Density of Graphs with Good Edge-Labellings"
] | [
"Abbas Mehrabian ",
"Dieter Mitsche ",
"Pawe ",
"Pra Lat "
] | [] | [] | A good edge-labelling of a simple, finite graph is a labelling of its edges with real numbers such that, for every ordered pair of vertices (u, v), there is at most one nondecreasing path from u to v. In this paper we prove that any graph on n vertices that admits a good edge-labelling has at most n log 2 (n)/2 edges, and that this bound is tight for infinitely many values of n. Thus we significantly improve on the previously best known bounds. The main tool of the proof is a combinatorial lemma which might be of independent interest. For every n we also construct an n-vertex graph that admits a good edge-labelling and has n log 2 (n)/2 − O(n) edges. | null | [
"https://arxiv.org/pdf/1211.2641v1.pdf"
] | 119,173,245 | 1211.2641 | a75f43a19650809c4ea7a3accf3bef859addc3cb |
On the Maximum Density of Graphs with Good Edge-Labellings
12 Nov 2012
Abbas Mehrabian
Dieter Mitsche
Pawe
Pra Lat
On the Maximum Density of Graphs with Good Edge-Labellings
12 Nov 2012
A good edge-labelling of a simple, finite graph is a labelling of its edges with real numbers such that, for every ordered pair of vertices (u, v), there is at most one nondecreasing path from u to v. In this paper we prove that any graph on n vertices that admits a good edge-labelling has at most n log 2 (n)/2 edges, and that this bound is tight for infinitely many values of n. Thus we significantly improve on the previously best known bounds. The main tool of the proof is a combinatorial lemma which might be of independent interest. For every n we also construct an n-vertex graph that admits a good edge-labelling and has n log 2 (n)/2 − O(n) edges.
these observations they concluded that if n is a power of two, then
f (n) ≥ n 2 log 2 (n) ,
and that for all n,
f (n) ≤ n √ n √ 2 + O n 4/3 .
The first author of this paper proved that any good graph whose maximum degree is within a constant factor of its average degree (in particular, any good regular graph) has at most n 1+o(1) edges-see [5] for more details.
Before we state the main result of this paper, we need one more definition. Let b(n) be the function that counts the total number of 1's in the binary expansions of all integers from 0 up to n − 1. This function was studied in [4]. Our main result is the following theorem.
Theorem 1. For all positive integers n,
n 2 log 2 3n 4 ≤ b(n) ≤ f (n) ≤ n 2 log 2 (n) .
It follows that the asymptotic value of f (n) is n log 2 (n)/2−O(n). Note that Theorem 1 implies that any good graph on n vertices has at most n log 2 (n)/2 edges, significantly improving the previously known upper bounds. Moreover, this bound is tight if n is a power of two. We also give an explicit construction of a good graph with n vertices and b(n) edges for every n.
The Proofs
This section is devoted to prove the main result, Theorem 1.
The upper bound
For a graph G, an edge-labelling φ : E(G) → R, and an integer t ≥ 0, a nice t-walk from v 0 to v t is a sequence v 0 v 1 . . . v t of vertices such that v i−1 v i is an edge for 1 ≤ i ≤ t, and
v i−1 = v i+1 and φ(v i−1 v i ) ≤ φ(v i v i+1 ) for 1 ≤ i ≤ t − 1.
We call v t the last vertex of the walk. When t does not play a role, we simply refer to a nice walk. The existence of a self-intersecting nice walk implies that the edge-labelling is not good:
let v 0 v 1 . . . v t be a shortest such walk with v 0 = v t . Then there are two nondecreasing paths v 0 v 1 . . . v t−1 and v 0 v t−1 from v 0 to v t−1 .
Thus if for some pair of distinct vertices (u, v) there are two nice walks from u to v, then the labelling is not good. Also, if for some vertex v, there is a nice t-walk from v to v with t > 0, then the labelling is not good. Consequently, if the total number of nice walks is larger than 2 n 2 + n = n 2 , then the labelling is not good. The following lemma will be very useful. Lemma 1. Let G and H be graphs with good edge-labellings on disjoint vertex sets. Then if we add a matching between the vertices of G and H (i.e., add a set of edges, such that each added edge has an endpoint in either of V (G) and V (H), and every vertex in V (G) ∪ V (H) is incident to at most one added edge), then the resulting graph is good.
Proof. Consider a good edge-labelling of G and H, and let M be a number greater than all existing labels. Then label the matching edges with M, M + 1, M + 2, etc. It is not hard to verify that the resulting edge-labelling is still good.
Corollary 2. We have f (1) = 0 and for all n > 1, f (n) ≥ max f (n 1 ) + f (n 2 ) + min{n 1 , n 2 } : 1 ≤ n 1 , 1 ≤ n 2 , n 1 + n 2 = n .
The proof of the upper bound in Theorem 1 relies on the analysis of a one-player game, which is defined next. The player, who will be called Alice henceforth, starts with n sheets of paper, on each of which a positive integer is written. In every step, Alice does an operation as follows. She chooses any two sheets. Assume that the numbers written on them are a and b. She erases these numbers, and writes a + b on both sheets. We denote the move by a pair (a, b). The configuration of the game is a multiset of size n, containing the numbers written on the sheets, in which the multiplicity of number x equals the number of sheets on which x is written. Notice that there may be multiple pairs of sheets with numbers a and b written on them, and we treat choosing any such pair as the same move, since they all result in the same configuration (that is, all moves are isomorphic). Note also that the moves (a, b) and (b, a) have the same effect.
Clearly, after playing the move (a, b), the sum of the numbers is increased by a + b. The aim of the game is to keep the sum of the numbers smaller than a certain threshold. Let S be the starting configuration of the game, namely, a multiset of size n containing the numbers initially written on the sheets, and let k ≥ 0 be an integer. We denote by opt(S, k) the smallest sum Alice can get after doing k operations. An intuitively good-looking strategy is the following: in each step, choose two sheets with the smallest numbers. We call this the greedy strategy, and show that it is indeed an optimal strategy. Specifically, we prove the following theorem, which may be of independent interest. Theorem 2. For any starting configuration S and any nonnegative integer k, if Alice plays the greedy strategy, then the sum of the numbers after k moves equals opt(S, k).
Before proving Theorem 2, we show how this implies our upper bound.
Proof of the upper bound of Theorem 1. Let G be a graph with n vertices and m > n log 2 (n)/2 edges. We need to show that G does not have a good edge-labelling. Consider an arbitrary edge-labelling φ : E(G) → R. Enumerate the edges of G as e 1 , e 2 , . . . , e m such that
φ(e 1 ) ≤ φ(e 2 ) ≤ · · · ≤ φ(e m ) .
We may assume that the inequalities are strict. Indeed, if some label L appears p > 1 times, we can assign the labels L, L + 1, . . . , L + (p − 1) to the edges originally labelled L, and increase by p the label of edges with original label larger than L. It is easy to see that the modified edge-labelling is still good, and by repeatedly applying this operation all ties are broken. . , e m one by one, in this order. Fix an i with 1 ≤ i ≤ m. Let u and v be the endpoints of e i . After adding the edge e i , for any t, any nice t-walk with last vertex u (respectively, v) in G i−1 can be extended via e i to a nice (t + 1)-walk with last vertex v (respectively, u) in G i . So, we have a
(i) u = a (i) v = a (i−1) u + a (i−1) v and a (i) w = a (i−1) w for w / ∈ {u, v}.
Thus we are in the same setting as the one-player game described before, with starting
configuration S = {1, 1, . . . , 1}, so we have v∈V (G) a (m) v ≥ opt(S, m) .
Hence, in order to prove that φ is not a good edge-labelling, it is sufficient to show that opt(S, m) > n 2 .
Let m 0 be the largest number for which opt(S, m 0 ) ≤ n 2 , and let α = ⌊log 2 (n)⌋. First, assume that n is even. By Theorem 2, we may assume that Alice plays according to the greedy strategy. The smallest number on the sheets is initially 1, and is doubled after every n/2 moves. Hence after αn/2 moves, the smallest number becomes 2 α , so the sum of the numbers would be 2 α n. In every subsequent move, the sum is increased by 2 α+1 , so Alice can play at most (n 2 − 2 α n)/2 α+1 more moves before the sum of the numbers becomes greater than n 2 . Consequently,
m 0 ≤ α n 2 + n(n − 2 α ) 2 α+1 . Now, define h(x) := log 2 (x) − x + 1.
Then h is concave in [1,2] and h(1) = h(2) = 0, which implies that h(x) ≥ 0 for all x ∈ [1,2]. In particular, for x 0 = n/2 α , we have
n − 2 α 2 α = x 0 − 1 ≤ log 2 (x 0 ) = log 2 n 2 α = log 2 (n) − α .
Therefore,
m 0 ≤ n 2 α + n 2 n − 2 α 2 α ≤ n 2 log 2 (n) < m ,
which completes the proof.
Finally, assume that n is odd. Since 2n is even, we have f (2n) ≤ n log 2 (2n) = n log 2 (n) + n .
On the other hand, by Corollary 2, f (2n) ≥ 2f (n) + n .
Combining these inequalities gives
f (n) ≤ n 2 log 2 (n) ,
completing the proof of the lemma.
The rest of this section is devoted to prove Theorem 2. Let S = {s 1 , s 2 , . . . , s n } be the starting configuration of the game. Note that after each step, the sum of the numbers is of the form n i=1 c i s i for some positive integers {c i } n i=1 . The sequence {c i } n i=1 depends only on the sheets Alice chooses in each step, and does not depend on {s i } n i=1 . We say that (c 1 , c 2 , . . . , c n ) is k-feasible if after k steps Alice can actually get a sum of the form n i=1 c i s i . For example, (1, 1, . . . , 1) is the only 0-feasible n-tuple, and there exist n 2 different 1-feasible n-tuples, one of them being (2, 2, 1, 1, . . . , 1). Notice that for any permutation π of {1, 2, . . . , n}, if (c 1 , c 2 , . . . , c n ) is k-feasible, then so is (c π(1) , c π(2) , . . . , c π(n) ), since Alice can first permute the sheets according to the permutation π, and then apply the same strategy as before.
For multisets S and T of size n, we write S ≤ T if for all k ≥ 0 we have opt(S, k) ≤ opt(T, k). Note that if we can arrange the elements of S and T as S = {s 1 , s 2 , . . . , s n } and T = {t 1 , t 2 , . . . , t n } such that s i ≤ t i holds for all 1 ≤ i ≤ n, then S ≤ T .
First, we make two useful observations. Lemma 3. Let k ≥ 2 and assume that the starting configuration is
S = {a, b, c, x 1 , x 2 , . . . , x m },
where a ≤ b ≤ c. Also suppose that either there is an optimal k-step strategy in which Alice plays (b, c) and (a, b + c) in the first and second steps, or there is an optimal k-step strategy in which Alice plays (a, c) and (b, a + c) in the first and second steps. Then, there exists an optimal k-strategy in which Alice plays (a, b) and (c, a + b) in the first and second steps.
Proof. In the first case, the configuration after the second step is
U = {b + c, a + b + c, a + b + c, x 1 , x 2 , . . . , x m }.
In the second case, the configuration after the second step is
V = {a + c, a + b + c, a + b + c, x 1 , x 2 , . . . , x m }.
In the third case, the configuration after the second step is
T = {a + b, a + b + c, a + b + c, x 1 , x 2 , . . . , x m }.
Since a ≤ b ≤ c, we have T ≤ U and T ≤ V , and therefore opt(T, k − 2) ≤ opt(U, k − 2) and opt(T, k − 2) ≤ opt(V, k − 2). Hence, the strategy starting with moves (a, b) and (c, a + b) is also an optimal k-step strategy. Proof. Let k ≥ 0. We first show that opt(S, k) ≤ opt(T, k). Let w 1 , w 2 , . . . , w n be such that
opt(T, k) = m i=1 w i x i + w m+1 (a + c) + w m+2 (a + c) + w m+3 (b + d) + w m+4 (b + d) .
Since (w 1 , w 2 , . . . , w m , w m+1 , w m+2 , w m+3 , w m+4 ) and (w 1 , w 2 , . . . , w m , w m+3 , w m+4 , w m+1 , w m+2 ) are k-feasible, we have Similarly, we show that opt(S, k) ≤ opt(U, k). Let w 1 , w 2 , . . . , w n be such that
opt(S, k) ≤ m i=1 w i x i + w m+1 (a + b) + w m+2 (a + b) + w m+3 (c + d) + w m+4 (c + d) = opt(T, k) + (w m+1 + w m+2 − w m+3 − w m+4 )(b − c) , and opt(S, k) ≤ m i=1 w i x i + w m+3 (a + b) + w m+4 (a + b) + w m+1 (c + d) + w m+2 (c + d) = opt(T, k) + (w m+1 + w m+2 − w m+3 − w m+4 )(d − a) . Now, if w m+1 + w m+2 − w m+3 − w m+4 ≥ 0,opt(U, k) = m i=1 w i x i + w m+1 (a + d) + w m+2 (a + d) + w m+3 (b + c) + w m+4 (b + c) .
As before, let us notice that both (w 1 , w 2 , . . . , w m , w m+1 , w m+2 , w m+3 , w m+4 ) as well as (w 1 , w 2 , . . . , w m , w m+3 , w m+4 , w m+1 , w m+2 ) are k-feasible, hence we have
opt(S, k) ≤ m i=1 w i x i + w m+1 (a + b) + w m+2 (a + b) + w m+3 (c + d) + w m+4 (c + d) = opt(U, k) + (w m+1 + w m+2 − w m+3 − w m+4 )(b − d) , and opt(S, k) ≤ m i=1 w i x i + w m+3 (a + b) + w m+4 (a + b) + w m+1 (c + d) + w m+2 (c + d) = opt(U, k) + (w m+1 + w m+2 − w m+3 − w m+4 )(c − a) .
Now, if w m+1 + w m+2 − w m+3 − w m+4 ≥ 0, then the first inequality gives opt(S, k) ≤ opt(U, k), and otherwise, the second inequality gives opt(S, k) ≤ opt(U, k).
Lemma 4 immediately implies the following corollary.
Corollary 5. Let k ≥ 2 and let a ≤ b ≤ c ≤ d be four numbers in the starting configuration. Suppose that either there is an optimal k-step strategy in which Alice plays (a, c) and (b, d) in the first two steps, or there is an optimal k-step strategy in which Alice plays (a, d) and (b, c) in the first two steps. Then there exists an optimal k-step strategy in which Alice plays (a, b) and (c, d) in the first and second steps.
The following lemma finishes the proof of Theorem 2.
Lemma 6. Let k ≥ 0, and let S = {s 1 , s 2 , . . . , s n } be the starting configuration, arranged such that s 1 ≤ s 2 ≤ · · · ≤ s n . Then there exists an optimal k-step strategy with the first move (s 1 , s 2 ).
Proof. We prove the lemma by induction on k. The induction base is obvious for k = 0 and easy for k = 1, so assume that k ≥ 2. Consider an optimal k-step strategy. Assume that the first move is (s i , s j ), where 1 ≤ i < j ≤ n. If i = 1 and j = 2, then we are done. Otherwise, there are 5 cases to consider:
{i, j} = {1, 3}. There are two subcases:
1. If s 1 + s 3 ≤ s 4 , then by the induction hypothesis, for the configuration that arose after the first step, there is an optimal (k − 1)-step strategy in which Alice's first move is (s 2 , s 1 + s 3 ). That is, there is an optimal k-step strategy for the initial configuration in which Alice plays (s 1 , s 3 ) and (s 2 , s 1 + s 3 ) in the first two steps. Thus, by Lemma 3 there also exists an optimal k-step strategy (for the initial configuration) with first move (s 1 , s 2 ).
2. If s 1 + s 3 > s 4 , then by the induction hypothesis, there is an optimal strategy with second move (s 2 , s 4 ). Then, by Lemma 4, there exists an optimal strategy with first move (s 1 , s 2 ).
{i, j} = {2, 3}. As before, there are two subcases:
1. If s 2 + s 3 ≤ s 4 , then by the induction hypothesis, there is an optimal strategy with second move (s 1 , s 2 + s 3 ). As before, by Lemma 3, there exists an optimal strategy with first move (s 1 , s 2 ).
2. If s 2 + s 3 > s 4 , then by the induction hypothesis, there is an optimal strategy with second move (s 1 , s 4 ). Then, by Lemma 4, there exists an optimal strategy with first move (s 1 , s 2 ).
i = 1 and j > 3. By the induction hypothesis, there is an optimal strategy with second move (s 2 , s 3 ), and thus by Lemma 4, there exists an optimal strategy with first move (s 1 , s 2 ).
i = 2 and j > 3. By the induction hypothesis, there is an optimal strategy with second move (s 1 , s 3 ), and again, Lemma 4 implies that there exists an optimal strategy with first move (s 1 , s 2 ).
i > 2 and j > 3. By the induction hypothesis, there is an optimal strategy with second move (s 1 , s 2 ). Swapping the first and second moves gives an optimal strategy with first move (s 1 , s 2 ).
The lower bound
In this section we prove the lower bound in Theorem 1. Recall that b(n) is equal to the total number of 1's in the binary expansions of all integers from 0 up to n − 1. It is known [4] that b(1) = 0 and b(n) satisfies the recursive formula b(n) = max{b(n 1 ) + b(n 2 ) + min{n 1 , n 2 } : 1 ≤ n 1 , 1 ≤ n 2 , n 1 + n 2 = n} , and the lower bound in Theorem 1 follows by using induction and applying Corollary 2. Moreover, McIlroy [4] proved that b(n) ≥ n log 2 3 4 n /2. For every n we also give an explicit construction of a good graph with n vertices and b(n) edges. It is easy to see that b(n) equals the number of edges in the graph G n with vertex set {0, 1, . . . , n − 1}, and with vertices i and j being adjacent if the binary expansions of i and j differ in exactly one digit. This graph is an induced subgraph of the ⌈log 2 (n)⌉-dimensional hypercube graph. It can be shown by induction and Lemma 1 that the hypercube graph is good, which implies that G n is also good (since the restriction of a good edge-labelling for the supergraph to the edges of the subgraph is a good edgelabelling for the subgraph). Hence G n is a good graph with n vertices and b(n) edges.
Concluding Remarks
We proved that any n-vertex graph with a good edge-labelling has at most n log 2 (n)/2 edges, and for every n we constructed a good n-vertex graph with n log 2 (n)/2 − O(n) edges. Thus we proved f (n) = n log 2 (n)/2 − O(n). One can try to investigate the second order term of the function f (n). Perhaps it is the case that our construction is best possible; that is, in fact f (n) = b(n)?
It would be interesting to further investigate the connection between having a good edge-labelling and other parameters of the graph; in particular, the length of the shortest cycle (known as the girth) of the graph (see, e.g., [3]). Araújo et al. [1] proved that any planar graph with girth at least 6 has a good edge-labelling, and asked whether 6 can be replaced with 5 in this result. The first author [5] proved that any graph with maximum degree ∆ and girth at least 40∆ is good. This does not seem to be tight, and improving the dependence on ∆ is an interesting research direction.
Let us denote by G i the subgraph of G induced by {e 1 , e 2 , . . . , e i }. For each vertex v and 0 ≤ i ≤ m, let a (i) v be the number of nice walks with last vertex v in G i . Clearly, a (0) v = 1 for all vertices v. Suppose the graph is initially empty and we add the edges e 1 , e 2 , . .
Lemma 4 .
4Let x 1 , x 2 , . . . , x m , a, b, c, d be positive integers with a ≤ b ≤ c ≤ d. Define S, T , and U as follows: S := {x 1 , x 2 , . . . , x m , a + b, a + b, c + d, c + d} ; T := {x 1 , x 2 , . . . , x m , a + c, a + c, b + d, b + d} ; U := {x 1 , x 2 , . . . , x m , a + d, a + d, b + c, b + c} .Then, we have S ≤ T and S ≤ U.
then the first inequality gives opt(S, k) ≤ opt(T, k), and otherwise, the second inequality gives opt(S, k) ≤ opt(T, k).
Good edge-labelling of graphs. J Araújo, N Cohen, F Giroire, F Havet, Discrete Appl. Math. 160J. Araújo, N. Cohen, F. Giroire, and F. Havet, Good edge-labelling of graphs, Discrete Appl. Math. (2012), Vol. 160, Issue 18, pp. 2502-2513.
Directed acyclic graphs with unique path property. J.-C Bermond, M Cosnard, S Pérennes, RR-6932INRIA. Technical ReportJ.-C. Bermond, M. Cosnard, and S. Pérennes, Directed acyclic graphs with unique path property, Technical Report RR-6932, INRIA, May 2009.
Good edge-labelings and graphs with girth at least five, preprint. M Bode, B Farzad, D Theis, arXiv:1109.1125M. Bode, B. Farzad, and D. Theis, Good edge-labelings and graphs with girth at least five, preprint, 2011 (available on arXiv:1109.1125).
The number of 1's in binary integers: bounds and extremal properties. M Mcilroy, SIAM J. Comput. 3M. McIlroy, The number of 1's in binary integers: bounds and extremal properties, SIAM J. Comput. (1974), Vol. 3, pp. 255-261.
On the density of nearly regular graphs with a good edge-labelling. A Mehrabian, SIAM J. Discrete Math. 263A. Mehrabian, On the density of nearly regular graphs with a good edge-labelling, SIAM J. Discrete Math. (2012), Vol. 26, No. 3, pp. 1265-1268.
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"Emmanuel Lhuillier \nSorbonne Université\nCNRS\nNanoSciences de Paris\nINSP\nF-75005ParisInstitutFrance\n"
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] | [] | Nanocrystals are often described as an interesting strategy for the design of low-cost optoelectronic devices especially in the infrared range. However the driving materials reaching infrared absorption are generally heavy metalcontaining (Pb and Hg) with a high toxicity. An alternative strategy to achieve infrared transition is the use of doped semiconductors presenting intraband or plasmonic transition in the short, mid and long-wave infrared. This strategy may offer more flexibility regarding the range of possible candidate materials. In particular, significant progresses have been achieved for the synthesis of doped oxides and for the control of their doping magnitude. Among them, tin doped indium oxide (ITO) is the one providing the broadest spectral tunability. Here we test the potential of such ITO nanoparticles for photoconduction in the infrared. We demonstrate that In2O3 nanoparticles presents an intraband absorption in the mid infrared range which is transformed into a plasmonic feature as doping is introduced. We have determined the cross section associated with the plasmonic transition to be in the 1-3x10 -13 cm 2 range. We have observed that the nanocrystals can be made conductive and photoconductive due to a ligand exchange using a short carboxylic acid, leading to a dark conduction with n-type character. We bring further evidence that the observed photoresponse in the infrared is the result of a bolometric effect. | null | [
"https://arxiv.org/pdf/1903.10295v1.pdf"
] | 85,500,507 | 1903.10295 | 9212ea316beac21fb4279d0e09019e39fbf8daf1 |
Transport and Phototransport in ITO Nanocrystals with Short to Long-Wave Infrared Absorption
Junling Qu
Sorbonne Université
CNRS
NanoSciences de Paris
INSP
F-75005ParisInstitutFrance
Clément Livache
Sorbonne Université
CNRS
NanoSciences de Paris
INSP
F-75005ParisInstitutFrance
Laboratoire de Physique et d'Etude des Matériaux
ESPCI-Paris
PSL Research University
Sorbonne
Université UPMC Univ Paris 06
CNRS
10 rue Vauquelin75005ParisFrance
Bertille Martinez
Sorbonne Université
CNRS
NanoSciences de Paris
INSP
F-75005ParisInstitutFrance
Laboratoire de Physique et d'Etude des Matériaux
ESPCI-Paris
PSL Research University
Sorbonne
Université UPMC Univ Paris 06
CNRS
10 rue Vauquelin75005ParisFrance
Charlie Gréboval
Sorbonne Université
CNRS
NanoSciences de Paris
INSP
F-75005ParisInstitutFrance
Audrey Chu
Sorbonne Université
CNRS
NanoSciences de Paris
INSP
F-75005ParisInstitutFrance
Elisa Meriggio
Sorbonne Université
CNRS
NanoSciences de Paris
INSP
F-75005ParisInstitutFrance
Julien Ramade
Sorbonne Université
CNRS
NanoSciences de Paris
INSP
F-75005ParisInstitutFrance
Hervé Cruguel
Sorbonne Université
CNRS
NanoSciences de Paris
INSP
F-75005ParisInstitutFrance
Zhen Xiang
Xu
Laboratoire de Physique et d'Etude des Matériaux
ESPCI-Paris
PSL Research University
Sorbonne
Université UPMC Univ Paris 06
CNRS
10 rue Vauquelin75005ParisFrance
Anna Proust
Sorbonne Université
CNRS
Institut Parisien de Chimie Moléculaire
IPCM
F-75005ParisFrance
Florence Volatron
Sorbonne Université
CNRS
Institut Parisien de Chimie Moléculaire
IPCM
F-75005ParisFrance
Grégory Cabailh
Sorbonne Université
CNRS
NanoSciences de Paris
INSP
F-75005ParisInstitutFrance
Nicolas Goubet
Sorbonne Université
CNRS
NanoSciences de Paris
INSP
F-75005ParisInstitutFrance
Laboratoire de Physique et d'Etude des Matériaux
ESPCI-Paris
PSL Research University
Sorbonne
Université UPMC Univ Paris 06
CNRS
10 rue Vauquelin75005ParisFrance
Interactions et Spectroscopies
Laboratoire de la Molécule aux Nano-objets ; Réactivité
Sorbonne Université
CNRS
MONARIS
F-75005ParisFrance
Emmanuel Lhuillier
Sorbonne Université
CNRS
NanoSciences de Paris
INSP
F-75005ParisInstitutFrance
Transport and Phototransport in ITO Nanocrystals with Short to Long-Wave Infrared Absorption
Oxide nanocrystalstransportphotoconductionplasmon
Nanocrystals are often described as an interesting strategy for the design of low-cost optoelectronic devices especially in the infrared range. However the driving materials reaching infrared absorption are generally heavy metalcontaining (Pb and Hg) with a high toxicity. An alternative strategy to achieve infrared transition is the use of doped semiconductors presenting intraband or plasmonic transition in the short, mid and long-wave infrared. This strategy may offer more flexibility regarding the range of possible candidate materials. In particular, significant progresses have been achieved for the synthesis of doped oxides and for the control of their doping magnitude. Among them, tin doped indium oxide (ITO) is the one providing the broadest spectral tunability. Here we test the potential of such ITO nanoparticles for photoconduction in the infrared. We demonstrate that In2O3 nanoparticles presents an intraband absorption in the mid infrared range which is transformed into a plasmonic feature as doping is introduced. We have determined the cross section associated with the plasmonic transition to be in the 1-3x10 -13 cm 2 range. We have observed that the nanocrystals can be made conductive and photoconductive due to a ligand exchange using a short carboxylic acid, leading to a dark conduction with n-type character. We bring further evidence that the observed photoresponse in the infrared is the result of a bolometric effect.
INTRODUCTION
Over the recent years, huge progresses 1,2,3,4 have been made in the field of infrared (IR) optoelectronics using colloidal quantum dots as a low-cost alternative to historical semiconductor technologies (InGaAs, InSb, HgCdTe…). In the visible range, InP 5,6 has appeared as viable alternative strategy to Cd-based nanocrystals for light emission. However, the infrared range is still driven by toxic compounds and more particularly lead 7,8 and mercury chalcogenides 9,10 .
Since materials above arise a toxicological concern for mass-market applications, it is of utmost importance to screen alternative materials with a reduced toxicity and an optical absorption in the mid infrared. However, in practice narrow band gap is often found with heavy atoms which are more likely to be toxic. In this sense, doped semiconductors presenting intraband transition in the range of wavelength of interest need to be considered. Mercury chalcogenides Self-doped nanocrystals 11 present a tunable intraband absorption 12,13,14 in the 3 to 60 µm range 15 . Ag2Se 16,17,18 , which is heavy metal-free, presents a very similar absorption spectrum to HgSe, but its performance for photodetection remains far weaker than its Hg-containing counterpart 19 .
Degenerately doped oxides 20,21,22 such as Ga 23 , Al 24 and In 25 doped ZnO, indium tin oxide (ITO) 26,27,28,29 , tungsten oxide 30 and cadmium oxide have seen prompt advances in terms of synthetic maturity over the recent years. Compared with the doping of II-VI and IV-VI materials which remains difficult and mostly induced through surface effects, the doping of oxides seems more straightforward and tunable. Like bulk semiconductors, the introduction of heteroatoms leads to the generation of free carriers, resulting in plasmonic mid infrared absorption and carrier conduction. 31,32 The optical absorption of such doped oxides has been extensively discussed, in particular to understand the correlation between the plasmonic absorption linewidth 33 and dopant position. 34,35,36,37 On the application side, such doped oxides have been driven by their use in the field of smart windows, 38 sensing 39 and catalysis. 40 However, their photoconductive performance remains nearly unexplored. In a semiconductor, the photocurrent is proportional to two key material parameters which are material absorption and photocarrier lifetime. For interband and intraband transitions, the absorption cross section per nanoparticle is quite weak, typically in the 10 -15 cm 2 range. 41,42 This implies that for an intraband transition from a lightly doped semiconductor (1 carrier per dot), only one electron is active every 10 3 -10 4 atoms. On the other extreme, metal with 1 active carrier per atom can reach high cross section 43 (10 -12 cm 2 per particle). The two type of transitions also differ by the fact that intraband absorption is quantum confined (ie size depedent) while plasmonic absorption is a bulk like property. In a degenerately doped semiconductor, the situation is typically intermediate, but the cross section is higher than that of marginally doped materials. The drawback of plasmonic absorption comes from its shorter lifetime compared with exciton which may limit the benefit of a higher absorption. Thus, the exact potential of oxide nanocrystals for IRphotoconduction remains to be determined. In addition, several open questions have to be tackled such as the exact nature of the transition in the low doping regime and the IR photoconduction potential. In particular, the electrical nature (photoconduction vs bolometric) of the response under IR illumination needs to be revealed. In this paper, we probe the transport properties of ITO nanoparticles both in dark condition and under illumination to answer the previously mentioned questions. It is also of utmost importance to quantize the IR photoresponse performance of these ITO nanocrystals with the one obtained using undoped or barely doped excitonic semiconductors.
DISCUSSION
We start by synthetizing a series of ITO nanocrystals with various levels of doping from 0 (In2O3) to 15% of Sn content using a previously reported method. 31 The nanocrystal solution, see Figure 1 a, shows a color switching from colorless for undoped material to blue for doped material. The nanoparticles are of similar size (≈12 nm) with a near sphere shape for all doping levels, as revealed by transmission electron microscopy (TEM), see Figure 1 b and c and S1. The Xray diffraction, see figure S2, confirms the bixbyite structure of the In2O3 lattice. These nanoparticles present an absorption peak in the infrared as well as a band edge in the UV range, see Figure 1d. Both transitions blueshift with the increase of Sn content. The band edge transition shift is consistent with the bleach of the lowest energy state as the conduction band state get filled, see figure S3. The infrared peak shifts from 9.3 µm (≈148 meV) for the undoped material to 2 µm (≈614 meV) for the 10% doped particles, see Table 1. We observe that above 10% the transition barely blueshifts, but strongly broadens due to an increase of electron-electron scattering. 37 In the following, we will focus more specifically on undoped (In2O3), 1.7% Sn and 10% Sn doped ITO nanoparticles. The infrared peak has been assigned to plasmonic absorption, which allows us to determine the carrier density. From the Drude model 44 36 , ≈ 2 the dielectric constant of the environment and the plasmon damping constant. The latter is estimated as the linewidth of the plasmon transition, i.e. the full width at half maximum (FWHM) of the transition. The average scattering time can be estimated from = ħ/ , with ħ the reduced Planck constant. The value of has been estimated to be ≈ 5±1 fs, see Figure 1e. The plasma frequency directly relates to the carrier density (n) through the relation 2 = 2 0 * with e the proton charge, 0 the vacuum dielectric constant and m* the ITO conduction band effective mass, here taken equal to 0.4m0 45,34 with m0 the electron rest mass. We estimate the carrier density to be in the 10 20 -10 21 cm -3 range, which corresponds to 250 (resp. 900) free electrons in the case of the 1.7% doped ITO nanoparticles (resp. 10%), see Figure 1f. In the case of the undoped material, this corresponds typically to 100 electrons per nanocrystal, however the validity of the Drude model is questionable in this case, as discussed later in the text.
We also have determined the absorption cross-section of the nanoparticles as a function of doping, see Figure 1g, Table 1 and Methods. Doped nanoparticles have a cross section in the 1-3x10 -13 cm 2 range. While the cross section increases quasi linearly with doping for doped nanoparticles, the undoped nanoparticles present a much weaker absorption with a cross section of only 4x10 -15 cm 2 . This is a first indication that the number of free carriers in undoped nanoparticles can be much lower than the one predicted by the Drude model.
Transport measurements are used to probe the carrier density. To ensure a strong interparticle coupling in the film, a ligand exchange step is required. While thiols have been extensively used for Cd, Hg or Pb based nanocrystals, the hard base nature of In 3+ leads to a poor affinity for thiol. As a result, we choose a hard acid, according to Pearson's theory, such as acetic acid to replace the initial oleic acid ligands. The transport properties of the ITO nanocrystals under dark condition are probed using an electrolyte field effect transistor configuration, see a scheme of the setup in Figure 2a. Electrolyte gating, here made by dissolving LiClO4 in a polyethylene glycol matrix, ensures a large gate capacitance which is critical to modulate the carrier density of degenerately doped material. In addition, it allows air operation and gating of thick films 46 . Thin films of ITO and undoped In2O3 nanoparticle only present n-type conduction (i.e. rise of conduction under positive gate bias), see Figure 2b and Figure S4. No evidence for hole transport has been observed, which is consistent with the wide band gap nature of ITO. 46 We nevertheless observe a clear reduction of the current modulation (on/off ratio) with the Sn content increase as well as a shift of the threshold voltage toward more negative potentials, see Figure 2c. Both are consistent with the increase of doping leading to a Fermi level deeper within the conduction band. Around room temperature, the transport is thermally activated, see Figure 2d. An activation energy of almost 100 meV in the case of In2O3 is extracted from the Arrhenius fit of the I-T curve. This value is weaker for doped samples at 30±6 meV, independent of the doping level, see the inset of Figure 2d. The latter value can be related to the charging energy (EC) of the nanoparticle where Ec is given by = 2 2 with C the self-capacitance of the nanoparticle. C can be estimated from = 2 0 with 0 the vacuum permittivity, the material dielectric constant (4) and d the nanoparticle diameter (12 nm), which leads to 29 meV. In this sense, the driving energy of the hopping transport in the array of degenerately doped semiconductors is the same one as for metallic nanoparticles. 47 While dark conduction in ITO nanoparticle film has already been explored, 31,32,48 their photoconductive properties remain mostly unexplored. We choose to study their photoconductive properties by exciting them at the band edge (UV irradiation) or by directly exciting their plasmonic feature in the near-IR for 10% doped and in the mid-IR for the 1.7% doped nanoparticles. In practice, we use four light sources to resonantly excite each of these transitions, see Figure 3a. c. Current as a function of time while a thin film In2O3 nanoparticles is exposed to a pulse of UV light. d. Current modulation (ratio of the photocurrent over the dark current) as function of the Sn content of an ITO thin film exposed to a short illumination of UV light.
Under UV excitation, a large current modulation is observed, see Figure 3 and S5. A striking feature relative to the excitation of the band edge is the slow time response of the photocurrent, see Figure 3c. The decay time can be as long as 1h, which is the signature of the involvement of deep traps in the photoconductive process. Similar memory effect has already been observed for ZnO 49,50 nanoparticles and are now used as a strategy to activate the conduction of electron transport layer 51 in a solar cell device. The magnitude of the photoresponse tends to be reduced while the Sn content is increased. Again, the undoped nanoparticles present a significantly larger modulation than all doped nanoparticles, see Figure 3d, due to a much lower dark current. This observation combined with the weaker cross section of this material confirms that the free carrier density in the undoped material is low and certainly much weaker than the one determined using a Drude model. This suggests that the infrared peak observed for In2O3 nanoparticles has an intraband character, rather than plasmonic. Once the doping is introduced, this transition acquires a more collective character as it has been proposed in the case of self-doped HgS quantum dots. 52 To support this claim, we graft polyoxometalates (POM) on the surface of the nanoparticles. It was recently demonstrated by Martinez et al 53 that POM can be used as agents to tune the doping magnitude of HgSe nanocrystals. The POMs used are a strongly oxidized (+VI) form of tungsten, and then behave as electron attractors. Once POMs are grafted on the surface of degenerately n-doped nanoparticles, they can strip electrons from the nanoparticles 53 . This strongly impacts the electron radial distribution 36 (Figure 4a) thanks to a post synthetic process. For doped forms of ITO, the grafting of the POMs has a severe effect of the linewidth of the plasmonic transition, see Figure 4b.
Initially ITO behaves as a Jellium 37 (Figure 4a), meaning that positive charges resulting from Sn dopants are fixed while free electrons are homogeneously spread all over the nanoparticle. After the POM grafting, the nanoparticle surface is electron deficient, which leaves positive impurities unscreened on the surface. This tends to enhance the electron impurity scattering and leads to the observed increase of the spectrum linewidth.
On the other hand, for In2O3, the infrared peak is neither shifted nor broaden, see c. In other words, the IR spectrum of the In2O3 nanoparticles with grafted POMs is just the sum of the absorption of the two materials. This reflects the fact that this transition is determined by the density of states of the semiconductor, rather than the electron gas.
Figure 4 a. Scheme of a non-functionalized nanoparticle where positive charges coming from Sn dopants are randomly dispersed in the nanocrystal, leading to a homogeneous distribution for the electron gas (in grey). In presence of electron attracting groups at the nanoparticle surface (here polyoxometalates), surface is electronically depleted and leave some of the positive charges unscreened. b. Infrared spectra of 1.7% doped ITO with and without POM grafted on the surface. c. Infrared spectrum of the In2O3 (bottom part), pristine POM (central part) and from the In2O3 nanoparticles grafted with POM (top part, scatter). The latter signal is well fitted (red curve) by an addition of the signal from the independent POM and In2O3 nanoparticles.
There are several significant consequences to this observation beyond the obvious facts that undoped nanoparticles have lower conductivity and weaker absorption. First, reducing the doping and even the residual material nonstoichiometry will not allow to redshift the transition of ITO toward wavelengths longer than 9 µm. This may be a limitation for detection application in the low energy part of the 8-12 µm atmospheric transparency window. Secondly, if tuning of the intraband transition of In2O3 needs to be investigated, alternative paths such as change of confinement or material alloying need to be considered.
In the last part of the paper, we aim to discuss the potential of ITO nanocrystals as photoconductive material in the infrared range. Three ranges of detection have been considered : long-wave infrared for In2O3 nanoparticles, midwave infrared for 1.7% doped ITO and short-wave infrared for 10% doped ITO, see Figure 3a. Samples are cooled down to reduce their thermally activated carrier density, see Figure 5a, d and g. The responsivity achieved by the film of In2O3 nanocrystals reaches 40 µA/W in the long wavelength range, 4 µA/W in the mid infrared (1.7% doped ITO at 4.4 µm) and 14 µA/W at 1.55 µm for the 10 % Sn ITO.
The In2O3 nanoparticles present a dramatically slow photoresponse with turn-on and turn-off time close to 10 min, Figure 5b. All doped nanoparticles have a similar time responses, with a turn-on time above 10 s and a turn-off time of ≈1 min typically, see Figure 5e and h. Even if their dynamics are shorter than the photoconductive dynamics at the band edge, they remain extremely slow compared with other IR active colloidal materials in the same range of wavelengths. For example, with HgTe quantum dots, time response shorter than the µs are commonly reported. 54,55 This long time response actually suggests that the current modulation results from a bolometric effect. In other words, the change of carrier density induced by the light (i.e. the photocurrent) is negligible compared to the increase of thermally activated carrier density (i.e. the light-induced change of the dark current). We can conclude that the benefit of a stronger absorption brought by the plasmonic absorption is strongly balanced by the short lifetime of the photocarrier. Thus, to take full benefit of the larger plasmonic absorption, a fast extraction of the hot electrons will have to be implemented. Regarding the operating temperature of such films, we measured that the photocurrent remains a marginal modulation of the dark current, see Figure 5c, f and i. Only the undoped material achieve BLIP (background limited performances, defined here as the temperature when dark current becomes smaller that the photocurrent) operation at T<50K, see Figure 5c. This is the result of the large doping, required to achieve plasmonic absorption, but which comes at the price of a large dark current.
Figure 5 I-V curves for a thin film of (a) In2O3 nanoparticles, (d) 1.7% doped ITO nanoparticles and (g) 10% doped ITO nanoparticles at various temperatures ranging from room temperature down to 25 K. The inset in (a) is the I-V curves at 25 K under dark condition and under illumination by a blackbody source (TBB=982°C). Current as a function of time of a thin film of (b) In2O3 nanoparticles, (e) 1.7% doped ITO nanoparticles and (h) 10% doped ITO nanoparticles while exposed to (b) a pulse illumination by a blackbody source, (e) a short illumination of a quantum cascade laser operating at 4.4 µm and (h) a short illumination
resulting from 1.55µm laser diode. Ratio of the photocurrent over the dark current as a function of the temperature for a thin film of (c) In2O3 nanoparticles exposed to the excitation from a blackbody source, (f) 1.7% doped ITO nanoparticles exposed to the excitation from a 4.4 µm QCL and (i) 10 % doped ITO nanoparticles exposed to the excitation from a 1.55 µm laser diode.
CONCLUSION
To summarize, we have tested the potential of ITO nanocrystals for infrared photoconduction. We have determined that the undoped In2O3 nanoparticles present an intraband absorption at 8-9 µm which transforms into a plasmonic transition as the Sn doping is introduced. The intraband character of this transition will make the tuning of the absorption toward longer wavelength more challenging. Doped nanoparticles of ITO present a cross section of a few 10 -13 cm 2 per nanoparticle, corresponding to an absorption coefficient above 10 5 cm -1 . The material presents n-type conduction with a weak temperature dependence. Photoconduction in the infrared range can be obtained at low temperature and results from a bolometric effect. The responsivity achieved by thin films of ITO nanoparticle are in the few to few tens of µA/W -1 range, which is not yet competitive with other materials such as PbS and HgTe. Future development will have to integrate a strategy to take advantage of the short living hot electrons.
METHODS
List of chemicals
Indium(III) acetate (Sigma-Aldrich, 99.99 % ), tin(IV) acetate ( Alfa Aesar), oleic acid (Sigma-Aldrich), olyel alcohol (Alfa Aesar, tech. 80-85%), ethanol absolute anhydrous (Carlo Erba, 99.9%), Chloroform (Carlo Erba), n-hexane (Carlo Erba), n-octane (SDS, 99%), acetic acid (Sigma-Aldrich, ≥ 99%), N,N-dimethylformamide (DMF, Sigma Aldrich), triethyloxonium tetrafluoroborate (Et3O + BF4 -, Sigma Aldrich), lithium perchlorate (LiClO4, Sigma-Aldrich, 98%), polyethylene glycol (PEG, Mw=6 kg/mol) were used as received.
Synthesis of indium oxide and Sn-doped indium oxide nanocrystals:
In this study, the nanocrystals were synthesized with a two-step slow-injection method established by Jansons et al. 35 with minor modifications. In one three-neck flask, a desired composition of indium acetate and tin (IV) acetate (5 mmol in total) was mixed with oleic acid (10 mL). The mixture was heated at 150 °C under Ar until all the powders are fully dissolved (typically 1 hour). The obtained metal oleate solution (0.5M) was light yellow in color. In another flask, 13 mL of oleyl alcohol was heated to 305 °C under Ar. Then 1 mL of the as-prepared metal oleate solution was injected to the oleyl alcohol bath using a syringepump at a rate of 0.2 mg/mL. After the injection, the reaction was baked at 305 °C under Ar for 20 min before cooled down with air flux. The obtained nanocrystals were precipitated with ethanol and redispersed in chloroform for 3 times and finally dispersed in chloroform for storage. The nominal doping of Sn in nanocrystal was determined by the Sn/(In+Sn) ratio in the mixed precursor. In this manner, we synthesized a series of nanocrystals with 0% (In2O3), 1.7%, 3%, 5% and 10% of Sn doping. 56 POM grafting on ITO nanoparticles: We used a two-step ligand exchange method with the removal of organic ligands followed by the functionalization with POM. Firstly 1 mL of NCs in hexane (10 mg/mL) was mixed with 1 mL of triethyloxonium tetrafluoroborate in DMF (20 mg/mL) vigorously with vortex and sonication. After 5 min, the NCs were transferred from nonpolar phase (hexane) to the polar one (DMF) on the bottom, due to the stripping of their organic ligands. Then, the bare NCs in DMF were precipitated with toluene and redissolved in 1 mL of POM in DMF (20 mg/mL) for functionalization. After complete mixing, the POM-capped NCs were precipitated with toluene to remove the POM excess and were redispersed in DMF for tests.
POM-COOH (TBA)3[PW11O39{O(SiC2H4COOH)2}] synthesis
Ligand exchange for transport measurements: To improve electron transport, the long organic ligands of nanocrystals were replaced by short ones using an on-film ligand exchange method. In more details, we first drop-casted the nanocrystal solution onto a substrate. Once the substrate was dried, it was dipped into a solution of acetic acid in ethanol (0.5 wt%) for 60 s then in pure ethanol for 30 s. The substrate was then annealed at 250 °C for 15 min to finish one round of ligand exchange. The procedure was repeated once before transport measurements.
Material characterization: Absorption spectra are acquired using a Jasco V730 spectrometer for the UV visible part, while a Thermo Fischer IS50 Fourrier transform infrared spectrometer in ATR configuration is used in the IR range. For the determination of the absorption cross section, a film of nanoparticles has been deposited onto a double side polished Si wafer. Its thickness is determined using a Dektak 150 profilometer. The absorption coefficient is given by · 10 · with A the absorption, t the film thickness, and f the film volume fraction taken equal to 0.64 which correspond to a randomly close pack film. The cross section per particle is simply obtained by multiplying the absorption coefficient by the nanoparticle volume, assuming a spherical shape. For Transmission electron microscopy, we used a JEOL 2010. Si/SiO2 substrate for electrodes: The surface of a Si/SiO2 wafer (400 nm oxide layer) is cleaned by sonication in acetone. The wafer is rinsed with isopropanol and finally cleaned using an O2 plasma. AZ 5214E resist is spin-coated and baked at 110°C for 90 s. The substrate is exposed under UV through a pattern mask for 2 s. The film is further baked at 125°C for 2 min to invert the resist. Then a 40 s flood exposure is performed. The resist is developed using a bath of AZ 326 for 32 s, before being rinsed in pure water. We then deposit a 5 nm chromium layer and 80 nm gold layer using a thermal evaporator. The lift-off is performed by dipping the film in acetone for 1 h. The electrodes are finally rinsed using isopropanol and dried by an air flow. The electrodes are 2.5 mm long and spaced by 20 µm. These electrodes are used for DC measurements (IV curves and transistor measurements).
Electrolyte gating: For electrolyte gating, we first mix in a glove box 0.5 g of LiClO4 with 2.3 g of PEG (MW = 6 kg.mol -1 ). The vial is heated at 170°C on a hot plate for 2 h until the solution gets clear. To use the electrolyte, the solution is warmed around 100°C and brushed on the top of the ITO nanoparticle film.
Electrical measurements DC transport: The sample is connected to a Keithley 2634b which applied bias and measured current. For measurement under illumination three sources has been used: a UV flash light at 365 nm, a 1.55µm laser diode and a quantum cascade laser operating at 4.4 µm. For measurement as a function of temperature, the sample is mounted on the cold finger of a cryostat and the sample is biased using a Keithley 2634b. The current is acquired while the temperature of the sample is cooled down.
Transistor measurements: The sample is connected to a Keithley 2634b which sets the drain bias (VDS = 20 mV for doped ITO, VDS = 200 mV for In2O3), controls the gate bias (VGS) between -2 and +2 V with a step of 1 mV and measures the associated currents IDS and IGS. All measurement are conducted in room condition (temperature and pressure).
Supporting Informations
The Supporting Information is available free of charge on the ACS Publications website at DOI: Size determination of nanoparticles, X-ray diffractogram, Tauc plot, process and effect of POM grafting, field effect transistor measurements and photoconduction under UV illumination
Figure 1a .
1aImage of ITO nanoparticles dispersed in CHCl3 with various levels of Sn content. b. Transmission electron microscopy image of the 1.7% Sn ITO nanocrystals. c. High-resolution TEM image of the 1.7% Sn ITO nanocrystals. d. Infrared and UV-visible spectra of ITO nanocrystals with various levels of doping. e. Full width at half maximum (FWHM) of the infrared absorption and the estimated average electron scattering time τ (ℏ/ ) as a function of the Sn content. f. Absorption cross section per nanoparticle and film absorption coefficient as a function of the Sn content. g. Estimated carrier density from the plasmon peak energy per unit of volume and per nanoparticle as a function of Sn content.
Figure 2a .
2aScheme of an electrolyte gated transistor with a thin film of ITO nanocrystals as canal. b. Transfer curve (drain and gate current as a function of the applied gate voltage) for a transistor whose canal is made of In2O3 nanocrystals. c. On/off ratio and threshold voltage as a function of the Sn content for a transistor made of ITO nanocrystals thin film. d. Current as a function of temperature for ITO nanocrystal thin films with various Sn contents. The inset provides the fitted activation energy for the doped samples assuming an Arrhenius fit of the high temperature part of the I-T curve.
Figure 3a .
3aBand diagram of undoped, 1.7% Sn and 10 % Sn doped ITO nanocrystals. b. I-V curve of In2O3 nanoparticles under dark condition and under UV illumination.
P
: K7[PW11O39] (0.64 g, 0.2 mmol) was dissolved in a water/acetonitrile mixture (30 mL, 1:2). A 1 M HCl aqueous solution was added drop by drop until an apparent pH equals to 3. The solution was cooled in an ice bath and the Si(OH)3(CH2)2COONa (0.476 mL, 0.8 mmol) was inserted. The 1 M HCl solution was added drop by drop again to reach pHapp=2. After an overnight reaction, TBABr (0.26 g, 0.8 mmol) was added and the solution concentrated with a rotary evaporator to make precipitate the product. The oily compound obtained was dissolved in the minimum of acetonitrile then precipitated again with an excess of ether. A sticky solid was recovered by centrifugation and washed thoroughly with ether to obtain a white powder (0.6 g, 82%). The dried compound was found to be partially deprotonated (cf. IR and EA analysis) with the exact general formula(TBA)NMR (121 MHz, CD3CN) δ (ppm) -12.28 ; IR (KBr pellet) : δ =2963 (s), 2935 (m), 2874 (w), 1710 (s), 1623 (w), 1483 (s), 1471 (s), 1420 (w), 1381 (m), 1112 (vs), 1064 (vs), 1052 (s), 1036 (s), 964 (vs), 870 (vs), 824 (vs) MS (ESI-), m/z (%) : calcd for W11PSi2O44C6H10 : 965.41 [M] 3-; found : 965.42 (100) ; calcd for W11PSi2O44C22H46N : 1569.26 [M+TBA] 2-; found : 1569.
The grids were prepared by a drop cast of dilute solution of nanocrystals dispersed in hexane and degassed overnight under secondary vacuum. X-ray diffraction pattern is obtained by drop casting a solution of nanocrystals on a Si wafer. The diffractometer is a Philips X'Pert, based on the emission of the Cu Kα line operated at 40 kV and 40 mA current.
Table 1
1Band edge energy, energy of the infrared absorption peak and absorption cross section associated with the infrared transition for three Sn content of ITO nanoparticles.Sn content Band edge
energy (eV)
Infrared peak
energy (meV)
Infrared cross
section (x10 -15 cm 2 )
0 %
3.8
148 (8.3 µm)
4
1.7 %
4
315 (3.95 µm)
90
10 %
4.3
614 (2.02 µm)
220
ACKNOWLEDEGMENTSWe thank C. Delerue for valuable discussion on the effect of the charge distribution. EL thanks the support ERC starting grant blackQD (grant n° 756225). We acknowledge the use of clean-room facilities from the "Centrale de Proximité Paris-Centre". This work has been supported by the Region Ile-de-France in the framework of DIM Nano-K (grant dopQD). This work was supported by French state funds managed by the ANR within the Investissements d'Avenir programme under reference ANR-11-IDEX-0004-02, and more specifically within the framework of the Cluster of Excellence MATISSE and also by the grant Nanodose and IPER-Nano2. JQ thanks the Chinese Scholar council for PhD grant while NG and JR thank Nexdot for post doctorate funding.
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Polyanions et polycations; Monographies de chimie minérale. P Souchay, ParisSouchay, P. Polyanions et polycations; Monographies de chimie minérale; Gauthier-Villars: Paris, 1963.
| [] |
[] | [
"\nFaculty of Mathematics and Physics\nDepartment of Algebra\nCharles University\nSokolovská 83186 75Praha 8Czech Republic\n"
] | [
"Faculty of Mathematics and Physics\nDepartment of Algebra\nCharles University\nSokolovská 83186 75Praha 8Czech Republic"
] | [] | Following the paper [7], we investigate in ZFC the following compactness question: for which unountable cardinals κ, an arbitrary nonempty system S of homogeneous Z-linear equations is nontrivially solvable in Z provided that each its nonempty subsystem of cardinality < κ is nontrivially solvable in Z?Date: December 18, 2018. 2010 Mathematics Subject Classification. 08A45, 13C10 (primary) 20K30, 03E35, 03E55 (secondary). | null | [
"https://arxiv.org/pdf/1812.06958v1.pdf"
] | 119,703,596 | 1812.06958 | 28f96e38f0aab6e37b572db159a87867a6917869 |
17 Dec 2018
Faculty of Mathematics and Physics
Department of Algebra
Charles University
Sokolovská 83186 75Praha 8Czech Republic
17 Dec 2018arXiv:1812.06958v1 [math.LO] ON THE NONTRIVIAL SOLVABILITY OF SYSTEMS OF HOMOGENEOUS LINEAR EQUATIONS OVER Z IN ZFC JANŠAROCHand phrases homogeneous Z-linear equationκ-free groupLω 1 ω -compact cardinal
Following the paper [7], we investigate in ZFC the following compactness question: for which unountable cardinals κ, an arbitrary nonempty system S of homogeneous Z-linear equations is nontrivially solvable in Z provided that each its nonempty subsystem of cardinality < κ is nontrivially solvable in Z?Date: December 18, 2018. 2010 Mathematics Subject Classification. 08A45, 13C10 (primary) 20K30, 03E35, 03E55 (secondary).
Introduction and preliminaries
In what follows, group means always an abelian group, i.e. a Z-module. We say that a system S of homogeneous Z-linear equations with a set X = {x i | i ∈ I} of variables is nontrivially solvable in a group H if there exists a mapping f : X → H \ {0} such that, whenever j∈J a j x j = 0 is an equation from S (where J is a finite subset of I and a j ∈ Z for each j ∈ J), then j∈J a j f (x j ) = 0 holds in H.
This notion of nontriviality is a little bit unusual. If we assume instead that the mapping f goes to H and it is not constantly zero on all x ∈ X that appear in the system S, we say that the system S is weakly nontrivially solvable in H. More natural as it might me, this weaker notion has got one significant disadvantage: unlike with nontrivial solvability, if a system S is weakly nontrivially solvable and T is a nonempty subsystem of S, then T need not be weakly nontrivially solvable. Notice also that an empty system S is (weakly) nontrivially solvable by definition.
Motivated by the work in [7], our aim is to characterize the class S (WS, resp.) of all infinite cardinals κ such that any system S of homogeneous Z-linear equations is nontrivially (weakly nontrivially, resp.) solvable in Z provided that each subsystem T ⊆ S of cardinality < κ is nontrivially (weakly nontrivially, resp.) solvable in Z. In [7,Section 2.2], the authors present several well-known examples of countable S which show in ZF that ℵ 0 ∈ S ∪ WS. They also discuss various interesting related questions in ZF: among other things, they provide a model of ZF without choice where ℵ 1 ∈ S while they note that the result is not known in ZFC.
In this short note, we use κ-free groups with trivial dual to show that ZFC actually proves ℵ n ∈ S for each n ∈ ω. Moreover, it is consistent with ZFC that S = WS = ∅. On the other hand, we are able to prove that κ ∈ WS ∩ S whenever there exits a regular L ω1ω -compact cardinal below κ (see Corollary 2.2 and Theorem 3.2). A detailed discussion is contained in Section 4.
For an unexplained terminology, we recommend, for instance, the very wellwritten extensive book [4].
The case of S
Recall that, given an uncountable cardinal ν, we say that a cardinal κ is L νωcompact if every κ-complete filter on any set I can be extended to a ν-complete ultrafilter. Observe that a cardinal µ is L νω -compact whenever there exists an L νω -compact cardinal λ such that λ ≤ µ. This is obviously a large cardinal notion for the existence of an L νω -compact cardinal implies the existence of a measurable cardinal.
The following result is a classic one. We give a proof for the reader's convenience.
Proposition 2.1. Let λ be a regular L νω -compact cardinal and Z a structure in a first order language L with |Z| < ν. Then a system S consisting of first order formulas in variables from a set X is realized in Z provided that each its subsystem T of cardinality < λ is realized in Z.
Proof. First, let E denote the set Z X of all mappings from X to Z. By the assumption, for each T ∈ [S] <λ , there exists e ∈ E such that Z |= ϕ[e] for each ϕ ∈ T . Let F be the filter on E generated by the sets
E T = {e ∈ E | Z |= ϕ[e] for all ϕ ∈ T }.
Since λ is regular, we see that F is a λ-complete filter. Let G denote an extension of F to a ν-complete ultrafilter.
For each (x, z) ∈ X × Z, put E x,z = {e ∈ E | e(x) = z} and define f ∈ Z X by the assignment f (x) = z ⇔ E x,z ∈ G. We can do it like that since the ultrafilter G picks, for each fixed x ∈ X, exactly one element from the disjoint partition E = z∈Z E x,z ; recall that |Z| < ν. Now let ϕ ∈ S be arbitrary and x 1 , . . . , x n be variables freely occuring in ϕ.
Then ∅ = E {ϕ} ∩ n i=1 E xi,f (xi) ∈ G
, and so f ∈ E {ϕ} . We conclude that S is realized in Z using the evaluation f . Corollary 2.2. Let κ be a cardinal and λ ≤ κ a regular L ω1ω -cardinal. Then every nonempty system S of homogeneous Z-linear equations in variables from a set X is nontrivially solvable in Z whenever each its nonempty subsystem of cardinality < κ is nontrivially solvable in Z. In other words κ ∈ S.
Proof. In the system S replace each equation ψ in variables x 1 , . . . , x n ∈ X by the formula ψ & n i=1 x i = 0 and use Proposition 2.1. Before we turn our attention to the negative part, we need one preparatory lemma which holds in the general context of R-modules over a (commutative) noetherian domain. Recall, that, for a module M ∈ Mod-R and an ordinal number σ, an increasing chain M = (M α | α ≤ σ) of submodules of M is called a filtration of M if M 0 = 0, M β = α<β M α whenever β ≤ σ is a limit ordinal, and M σ = M .
Lemma 2.3. Let R be an infinite noetherian domain, M a free R-module of rank µ ≥ ℵ 0 , and M = (M α | α ≤ σ) be a filtration of M where, for all α < σ, M α+1 = M α + a α with a α ∈ M \ M α . For each α < σ, let z α ∈ R be arbitrary.
Then there is a homomorphism ψ : M → R such that ψ(a α ) = z α for all α < σ.
Proof. First, assume that µ = ℵ 0 . Let {g n | n < ω} be a set of free generators of M . For each α < σ, we express a α as n∈Iα b nα g n , where I α is a finite subset of ω and b nα ∈ R \ {0} for every n ∈ I α .
Using the fact that a free R-module of finite rank is noetherian, we infer that, for each n < ω, the set A n = {α < σ | I α ⊆ {0, 1, . . . , n}} is finite. Note that σ = n<ω A n . On the free generators of M , we recursively construct a homomorphism ψ : M → R as follows:
Let ψ(g 0 ) be arbitrary such that, for each α ∈ A 0 , b 0α ψ(g 0 ) = z α . There is always an applicable choice by the hypothesis on R. Assume that n > 0, ψ(g n−1 ) is defined, and ψ(a α ) = z α for each α ∈ A n−1 .
We define ψ(g n ) arbitrarily in such a way that, for each α ∈ A n \ A n−1 , we have
b nα ψ(g n ) = z α − k∈Iα\{n} b kα ψ(g k ).
This is possible, since A n \ A n−1 is finite, b nα = 0 for each α from this set, and R is an infinite domain. It immediately follows that ψ(a α ) = z α for each α ∈ A n . Now, let µ be an uncountable cardinal. Again, let {g β | β < µ} be a set of free generators of M , and put
G B = g β | β ∈ B for all B ⊆ µ.
We use ideas from [6, Section 7.1]. First, we set A α = a α ≤ M . We say that
a subset S of the ordinal σ is 'closed' if every α ∈ S satisfies M α ∩ A α ⊆ β∈S,β<α A β .
Notice that any ordinal α ≤ σ is a 'closed' subset of σ. For a 'closed' subset S, we define M (S) = α∈S A α . The results from [6, Section 7.1] give us the following:
(1) For a system (S i | i ∈ I) of 'closed' subsets, i∈I S i and i∈I S i is 'closed' as well.
(2) For S, S ′ 'closed' subsets of σ, we have S ⊆ S ′ ⇐⇒ M (S) ⊆ M (S ′ ). (3) Let S be a 'closed' subset of σ and X be a countable subset of M . Then there is a 'closed' subset S ′ such that M (S) ∪ X ⊆ M (S ′ ) and |S ′ \ S| < ℵ 1 .
Using the properties listed above, we are going to construct a filtration N =
(M (S α ) | α ≤ µ) of M such that, for each α < µ, a) S α is 'closed', b) S α+1 \ S α is countable, and c) there exists B α ⊆ µ such that G Bα = M (S α ) and α ⊆ B α .
We proceed by the transfinite recursion, starting with S 0 = B 0 = ∅. Let S α and B α be defined and α < µ. Then |S α | + |B α | < µ (using b) and c)).
Let B 0 ⊇ B α ∪{α} be any subset of µ with |B 0 \ B α | = ℵ 0 . By (3), we find S 0 ⊇ S α such that M (S 0 ) ⊇ G B 0 and |S 0 \ S α | < ℵ 1 . Assuming B n , S n are defined for n < ω, we can find B n+1 ⊇ B n with |B n+1 \ B n | < ℵ 1 such that G B n+1 ⊇ M (S n ), and S n+1 ⊇ S n with |S n+1 \ S n | < ℵ 1 such that M (S n+1 ) ⊇ G B n+1 . Put S α+1 = n<ω S n and B α+1 = n<ω B n .
This completes the isolated step. In limit steps, we simply take unions. Since M (S µ ) = M , we have S µ = σ by (2). Now, for each α < µ, we have the countable sets C α = B α+1 \ B α and T α = S α+1 \ S α , and the canonical projection π α : M (S α+1 ) → G Cα . Let τ be the ordinal type of (T α , <), and fix an order-preserving bijection i : τ → T α .
Since S α ∪ (S α+1 ∩ β) is 'closed' for any β ≤ σ by (1), the part (2) yields that the chain (N β | β ≤ τ ) of modules defined as N β = M (S α ∪ (S α+1 ∩ i(β))), for β < τ , and N τ = M (S α+1 ) is strictly increasing. Notice that N 0 = M (S α ).
If we putN β = π α [N β ] for all β ≤ τ , it follows that the strictly increasing chain (N β | β ≤ τ ) is a filtration of the free module G Cα of countable rank. Moreover, for each β < τ , we haveN β+1 =N β + π α (a i(β) ) .
Finally, we recursively define the homomorphism ψ : M → R. Let α < µ and assume that ψ ↾ G Bα is constructed with the property ψ(a γ ) = z γ for all γ ∈ S α . By the already proven part for µ = ℵ 0 , we can define ψ ↾ G Cα in such a way that ψ(π α (a γ )) = z γ − ψ(a γ − π α (a γ )) for all γ ∈ T α ; observe that the right-hand side of the inequality is already defined since a γ − π α (a γ ) ∈ G Bα . We immediately get ψ(a γ ) = z γ for all γ ∈ S α+1 .
For the negative part, we start with an uncountable cardinal κ and a κ-free group G with the trivial dual, i.e. with the property G * := Hom(G, Z) = 0; here, κ-free means that any < κ-generated subgroup of G is free. We will discuss the existence of such groups, as well as the question whether G can be taken with |G| = κ, later on. Firstly, we show how the existence of such G implies that κ ∈ S.
Let us denote by λ the cardinality of G and express G as a quotient F/K where F is a free group of rank λ. Notice that λ ≥ κ. Let π : F → F/K denote the canonical projection and let {e α | α < λ} be a set of free generators of the group F . For each A ⊆ λ, let F A denote the subgroup of F generated by {e α | α ∈ A}. We can w.l.o.g. assume that Im(π ↾ F β ) Im(π ↾ F β+1 ) for each ordinal β < λ.
( * )
The group K is also free of rank λ. If it had a smaller rank, G would have possessed a free direct summand-a contradiction with G * = 0. Let {k β | β < λ} denote a set of (free) generators of the group K. Consider the uncountable set
S = α∈J β a αβ x α = 0 | β < λ, J β ∈ [λ] <ω , (∀α ∈ J β )(a αβ ∈ Z), α∈J β
a αβ e α = k β of homogeneous Z-linear equations with the set {x α | α < λ} of variables. We will show that this is the desired counterexample.
First of all, S does not have even a weakly nontrivial solution in Z. Indeed, any such solution would define a nonzero homomorphism ψ from F to Z which is zero on K. Hence ψ would provide for a nonzero homomorphism from G to Z, a contradiction.
On the other hand, we can show Proposition 2.4. Any system T ⊆ S with cardinality < κ has a nontrivial solution in Z.
Proof. Let A ∈ [λ] <κ be an infinite set such that whenever x α appears in an equation from T then α ∈ A. Put M = Im(π ↾ F A ).
Since G is κ-free, M is a free group (of infinite rank). Let σ denote the ordinal type of A and fix an order-preserving bijection i : σ → A. For each α ≤ σ, set M α = π(e i(β) ) | β < α . Then (M α | α ≤ σ) is a filtration of M such that M α+1 = M α + π(e i(α) ) where π(e i(α) ) ∈ M α for all α < σ (using ( * )).
Applying Lemma 2.3 with R = Z and z γ = 0 for all γ < σ, we obtain a homomorphism ψ : M → Z such that ψ(π(e α )) = 0 for all α ∈ A. The assignment x α → ψ(π(e α )), α ∈ A, is the desired nontrivial solution of the system T in Z.
Corollary 2.5. Let κ be an uncountable cardinal. If there exists a κ-free group G with G * = 0, then κ ∈ S ∪ WS.
The case of WS
For the weaker notion of nontrivial solvability, we have the following general result. (1) There exists a regular cardinal λ ≤ κ which is L ω1ω -compact.
(2) There is a regular cardinal λ ≤ κ such that each group A ∈ Ker Hom(−, Z) is the sum of its subgroups of cardinality < λ which are contained in Ker Hom(−, Z). Proof. The equivalence of (1) and (2) follows directly from [1,Corollary 5.4]. Let us show that (2) is equivalent to (3). To this end, we are going to use the following two-way translation.
Given any system S = {k j = 0 | j ∈ J} of homogeneous Z-linear equations with the set X of variables, we can build a group A = F/K where F is freely generated by the elements of the set X and K is generated by the set {k j | j ∈ J}. Then Hom(A, Z) = 0 if and only if S has no weakly nontrivial solution in Z. On the other hand, for a given group A and its presentation F/K where F is freely generated by a set X, the same equivalence holds for the system S = {k j = 0 | j ∈ J} of homogeneous Z-linear equations where {k j | j ∈ J} is a fixed set of generators of K expressed as Z-linear combinations of elements from the set X.
Proving (2) =⇒ (3), we start with a system S and a set C ∈ [J] <κ . Consider the group A constructed for S as in the previous paragraph, and let Y 0 denote the set of all the elements from X appearing in equations k j = 0, j ∈ C.
Let µ ≥ λ be a regular uncountable cardinal such that |C| < µ ≤ κ. Since Ker Hom(−, Z) is closed under direct sums and quotients, and µ is regular, there exists, by (2), G 0 ∈ Ker Hom(−, Z) such that G 0 is a subgroup of A, |G 0 | < µ and
Y 0 + K := {y + K | y ∈ Y 0 } ⊆ G 0 . Now, take any Y 1 ∈ [X] <µ , Y 0 ⊆ Y 1 such that: (a) G 0 is contained in the subgroup of A generated by Y 1 + K. (b) There exists C 0 ∈ [J] <µ such that Y 0 ∩ K is contained in the subgroup of K generated by {k j | j ∈ C 0 }, and Y 1 contains all the elements from X appearing in equations k j = 0, j ∈ C 0 .
For this Y 1 , we obtain, using (2), a subgroup G 1 of A with |G 1 | < µ, and so on.
After ω steps, we have the group G = n<ω G n ∈ Ker Hom(−, Z) generated by Y +K where Y = n<ω Y n ∈ [X] <µ . By the construction, we have also G = y+K | y ∈ Y ∼ = Y / k j | j ∈ n<ω C n . Finally, we put T = {k j = 0 | j ∈ n<ω C n }. Now, let us prove the implication ¬(1) =⇒ ¬(3). First, assume that κ is not L ω1ω -compact. Following [1, Theorem 5.3] and its proof, we start with A = Z I /F where F is a κ-complete filter on I which cannot be extended to an ω 1 -complete ultrafilter. From the latter part, it follows that Hom(A, Z) = 0. The κ-completeness of F , on the other hand, assures that any subgroup of A of cardinality < κ can be embedded into Z I .
Consider a system S of homogeneous Z-linear equations associated to the group A presented as F/K where F is freely generated by a set X. We can w. l. o. g. assume that no x ∈ X is contained in K. Let C ∈ [J] <κ be non-empty. We shall show that the system {k j = 0 | j ∈ C} has weakly nontrivial solution in Z.
As in the proof of the other implication, we can possibly enlarge C to some D ⊆ J such that |D| ≤ |C| + ℵ 0 and-denoting by Y the set of all the elements from X appearing in equations k j = 0, j ∈ D-y + K | y ∈ Y ∼ = Y / k j | j ∈ D . Let us denote the latter group by H and fix an embedding i : H → Z I (which exists since |H| < κ).
Let y ∈ Y be any element appearing in (one of the) equations k j = 0, j ∈ C. Since i(y + K) = 0 there is a projection π : Z I → Z such that πi(y + K) = 0. The assignment x → πi(x + K) defines the desired weakly nontrivial solution of the system {k j = 0 | j ∈ C} in Z.
It remains to tackle the possibility that κ is the least L ω1ω -compact cardinal and κ is singular. We know by [2] that γ = cf (κ) is greater than or equal to the first measurable cardinal in this case. Let (κ α | α < γ) be an increasing sequence of cardinals < κ converging to κ.
Consider the group A = α<γ A α where, for each α < γ, A α ∈ Ker Hom(−, Z) is not a sum of its subgroups of cardinality < κ α which belong to Ker Hom(−, Z). Assume, for the sake of contradiction, that (3) holds for the system S of homogeneous Z-linear equations associated to the group A (more precisely, to its presentation F/K).
By the definition of A, there exists, for each α < γ, an element a α ∈ A such that a α is not contained in any subgroup H of A of cardinality < κ α with the property Hom(H, Z) = 0.
We know that there is C 0 ∈ [J] <κ and Y 0 ⊆ X consisting of the elements from X appearing in the equations k j = 0, j ∈ C 0 such that
{a α | α < γ} ⊆ y + K | y ∈ Y 0 ∼ = Y 0 / k j | j ∈ C 0 .
For this C 0 , we obtain a corresponding T 0 ∈ [J] <κ using (3). We continue by finding C 1 ∈ [J] <κ and Y 1 ∈ [X] <κ such that T 0 ⊆ C 1 , Y 0 ⊆ Y 1 and y + K | y ∈ Y 1 ∼ = Y 1 / k j | j ∈ C 1 , and so forth.
Put T = n<ω T n = n<ω C n and Y = n<ω Y n . The system {k j = 0 | j ∈ T } has cardinality < κ (since γ is uncountable) and it has no weakly nontrivial solution in Z. Whence the subgroup H = y + K | y ∈ Y ∼ = Y / k j | j ∈ T of A belongs to Ker Hom(−, Z). However, this is impossible since a α ∈ H for α < γ satisfying |H| < κ α .
In the proof above, we have actually showed a little bit more. In fact, we have the following Theorem 3.2. Let κ be a cardinal, and assume that κ is not at the same time singular and the least L ω1ω -compact cardinal. The following conditions are equivalent:
(1) κ is L ω1ω -compact.
(2) Any nonempty system S of homogeneous Z-linear equations has a weakly nontrivial solution in Z provided that each its nonempty subsystem of cardinality < κ has one. In other words, κ ∈ WS.
Proof. The implication '(1) =⇒ (2)
Concluding remarks
The problem of existence of κ-free groups with trivial dual turns out to be rather delicate. Under the assumption V = L (even a much weaker one), there are κ-free groups with trivial dual for any uncountable cardinal κ. Moreover, if κ is regular and not weakly compact, then the groups can be constructed of cardinality κ, see [3]. If κ is singular or weakly compact, then κ-free implies κ + -free. For more information on the topic, we refer to [4,Chapter VII]. Anyway, we have S = WS = ∅ under V = L by Corollary 2.5.
In [5], Göbel and Shelah show in ZFC that ℵ n -free groups with cardinality n and trivial dual exist for all 0 < n < ω. This is further generalized in [8] 1 , where Shelah proves in ZFC the existence of κ-free groups with trivial dual for any uncountable κ < ℵ ω1·ω . On the other hand, he also shows (modulo the existence of a supercompact cardinal) that it is relatively consistent with ZFC that there is no ℵ ω1·ω -free group with trivial dual. By Corollary 2.5, we thus know in ZFC that κ ∈ S for κ < ℵ ω1·ω . However, we do not know what happens for larger cardinals κ since the existence of a κ-free group with trivial dual is just a sufficient condition for κ ∈ S. We have only the upper bound given by Corollary 2.2. It might still be possible that S = WS where for the latter class, we have a decent description in Theorem 3.2. As shown in [1], relative to the existence of a supercompact cardinal, there are models of ZFC where the smallest L ω1ω -compact cardinal κ is singular. In this only case, we cannot resolve the question whether κ ∈ WS although we conjecture that this is not the case, which would readily imply that at least WS ⊆ S always holds.
A possible direction for further research is to investigate further what more can be proved in ZFC about the class S.
Proposition 3 . 1 .
31Let κ be an uncountable cardinal. The following conditions are equivalent:
( 3 )
3For any nonempty system S of homogeneous Z-linear equations such that S has no weakly nontrivial solution in Z, and any C ∈ [S] <κ , there exists T ∈ [S] <κ such that C ⊆ T and T has no weakly nontrivial solution in Z.
' follows immediately from '(1) =⇒ (3)' in Proposition 3.1. The other implication then from the first part of the proof of '¬(1) =⇒ ¬(3)' in Proposition 3.1.
Very heavy in content. Unpublished outside arXiv.org so far.
Acknowledgement : I would like to thank Petr Glivický for a fruitful discussion about compactness problems in set theory.
J Bagaria, M Magidor, Group radicals and strongly compact cardinals. 366J. Bagaria, M. Magidor, Group radicals and strongly compact cardinals, Trans. Amer. Math. Soc. 366 (2014), no. 1, 1857-1877.
On ω 1 -strongly compact cardinals. J Bagaria, M Magidor, J. Symb. Log. 791J. Bagaria, M. Magidor, On ω 1 -strongly compact cardinals, J. Symb. Log. 79 (2014), no. 1, 266-278.
Every cotorsion-free ring is an endomorphism ring. M Dugas, R Göbel, Proc. London Math. Soc. 45M. Dugas, R. Göbel, Every cotorsion-free ring is an endomorphism ring, Proc. London Math. Soc. 45 (1982), 319-336.
P C Eklof, A H Mekler, Almost Free Modules. North-Holland, New YorkRevised ed.P. C. Eklof, A. H. Mekler, Almost Free Modules, Revised ed., North-Holland, New York 2002.
ℵn-free modules with trivial dual. R Göbel, S Shelah, Res. Math. 54R. Göbel, S. Shelah, ℵn-free modules with trivial dual, Res. Math. 54 (2009), 53-64.
Approximations and Endomorphism Algebras of Modules. R Göbel, J Trlifaj, de Gruyter Expositions in Mathematics. 412nd revised and extended editionR. Göbel, J. Trlifaj, Approximations and Endomorphism Algebras of Modules, de Gruyter Expositions in Mathematics 41, 2nd revised and extended edition, Berlin-Boston 2012.
On the Solvability of Systems of Linear Equations over the Ring Z of Integers. H Herrlich, E Tachtsis, Comment. Math. Univ. Carolin. 582H. Herrlich, E. Tachtsis, On the Solvability of Systems of Linear Equations over the Ring Z of Integers, Comment. Math. Univ. Carolin. 58 (2017), no. 2, 241-260.
Quite free complicated abelian group, pcf and black boxes. S Shelah, Sh:1028, arxiv.org/abs/1404.2775S. Shelah, Quite free complicated abelian group, pcf and black boxes, Sh:1028, arxiv.org/abs/1404.2775.
| [] |
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"Magnetization and spin dynamics of a Cr-based magnetic cluster: Cr 7 Ni",
"Magnetization and spin dynamics of a Cr-based magnetic cluster: Cr 7 Ni"
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"A Bianchi \nDipartimento di Fisica\nUniversità di Parma\nViale Usberti 7/AI-43100ParmaItaly; and S\n",
"S Carretta \nDipartimento di Fisica\nUniversità di Parma\nViale Usberti 7/AI-43100ParmaItaly; and S\n",
"P Santini \nDipartimento di Fisica\nUniversità di Parma\nViale Usberti 7/AI-43100ParmaItaly; and S\n",
"G Amoretti \nDipartimento di Fisica\nUniversità di Parma\nViale Usberti 7/AI-43100ParmaItaly; and S\n",
"Y Furukawa \nDivision of Physics\nHokkaido University\n060-0810SapporoJapan\n",
"K Kiuchi \nDivision of Physics\nHokkaido University\n060-0810SapporoJapan\n",
"Y Ajiro \nCNR-INFM\nI-41100ModenaItaly\n\nDepartment of Chemistry\nKyoto University\n606-8502KyotoJapan\n\nCREST Japan Science and Technology (JST)\n\n\nCNR-INFM\nI-41100ModenaItaly\n",
"Y Narumi \nThe Institute for Solid State Physics\nThe University of Tokyo\n277-8581KashiwaChibaJapan\n",
"K Kindo \nThe Institute for Solid State Physics\nThe University of Tokyo\n277-8581KashiwaChibaJapan\n",
"J Lago \nDipartimento di Fisica \"A. Volta\"\nUniversità di Pavia\nCNR-INFM\nVia Bassi 6I-27100PaviaItaly\n",
"E Micotti \nDipartimento di Fisica \"A. Volta\"\nUniversità di Pavia\nCNR-INFM\nVia Bassi 6I-27100PaviaItaly\n",
"P Arosio \nIstituto di Fisologia Generale e Chimica Biologica\nUniversità di Milano\nI-20134MilanoItaly\n\nCNR-INFM\nI-27100PaviaItaly; and S\n",
"A Lascialfari \nDipartimento di Fisica \"A. Volta\"\nUniversità di Pavia\nCNR-INFM\nVia Bassi 6I-27100PaviaItaly\n\nIstituto di Fisologia Generale e Chimica Biologica\nUniversità di Milano\nI-20134MilanoItaly\n\nCNR-INFM\nI-27100PaviaItaly; and S\n",
"F Borsa \nDipartimento di Fisica \"A. Volta\"\nUniversità di Pavia\nCNR-INFM\nVia Bassi 6I-27100PaviaItaly\n",
"G Timco \nSchool of Chemistry\nUniversity of Manchester\nOxford RoadM13 9PLManchesterUnited Kingdom\n",
"R E P Winpenny \nSchool of Chemistry\nUniversity of Manchester\nOxford RoadM13 9PLManchesterUnited Kingdom\n"
] | [
"Dipartimento di Fisica\nUniversità di Parma\nViale Usberti 7/AI-43100ParmaItaly; and S",
"Dipartimento di Fisica\nUniversità di Parma\nViale Usberti 7/AI-43100ParmaItaly; and S",
"Dipartimento di Fisica\nUniversità di Parma\nViale Usberti 7/AI-43100ParmaItaly; and S",
"Dipartimento di Fisica\nUniversità di Parma\nViale Usberti 7/AI-43100ParmaItaly; and S",
"Division of Physics\nHokkaido University\n060-0810SapporoJapan",
"Division of Physics\nHokkaido University\n060-0810SapporoJapan",
"CNR-INFM\nI-41100ModenaItaly",
"Department of Chemistry\nKyoto University\n606-8502KyotoJapan",
"CREST Japan Science and Technology (JST)\n",
"CNR-INFM\nI-41100ModenaItaly",
"The Institute for Solid State Physics\nThe University of Tokyo\n277-8581KashiwaChibaJapan",
"The Institute for Solid State Physics\nThe University of Tokyo\n277-8581KashiwaChibaJapan",
"Dipartimento di Fisica \"A. Volta\"\nUniversità di Pavia\nCNR-INFM\nVia Bassi 6I-27100PaviaItaly",
"Dipartimento di Fisica \"A. Volta\"\nUniversità di Pavia\nCNR-INFM\nVia Bassi 6I-27100PaviaItaly",
"Istituto di Fisologia Generale e Chimica Biologica\nUniversità di Milano\nI-20134MilanoItaly",
"CNR-INFM\nI-27100PaviaItaly; and S",
"Dipartimento di Fisica \"A. Volta\"\nUniversità di Pavia\nCNR-INFM\nVia Bassi 6I-27100PaviaItaly",
"Istituto di Fisologia Generale e Chimica Biologica\nUniversità di Milano\nI-20134MilanoItaly",
"CNR-INFM\nI-27100PaviaItaly; and S",
"Dipartimento di Fisica \"A. Volta\"\nUniversità di Pavia\nCNR-INFM\nVia Bassi 6I-27100PaviaItaly",
"School of Chemistry\nUniversity of Manchester\nOxford RoadM13 9PLManchesterUnited Kingdom",
"School of Chemistry\nUniversity of Manchester\nOxford RoadM13 9PLManchesterUnited Kingdom"
] | [] | We study the magnetization and the spin dynamics of the Cr7Ni ring-shaped magnetic cluster. Measurements of the magnetization at high pulsed fields and low temperature are compared to calculations and show that the spin Hamiltonian approach provides a good description of Cr7Ni magnetic molecule. In addition, the phonon-induced relaxation dynamics of molecular observables has been investigated. By assuming the spin-phonon coupling to take place through the modulation of the local crystal fields, it is possible to evaluate the decay of fluctuations of two generic molecular observables. The nuclear spin-lattice relaxation rate 1/T1 directly probes such fluctuations, and allows to determine the magnetoelastic coupling strength. | 10.1016/j.jmmm.2009.03.006 | [
"https://arxiv.org/pdf/0808.2621v1.pdf"
] | 118,673,647 | 0808.2621 | a40751ca633737be95fd234326762df9c979ff3c |
Magnetization and spin dynamics of a Cr-based magnetic cluster: Cr 7 Ni
19 Aug 2008 (Dated: August 19, 2008)
A Bianchi
Dipartimento di Fisica
Università di Parma
Viale Usberti 7/AI-43100ParmaItaly; and S
S Carretta
Dipartimento di Fisica
Università di Parma
Viale Usberti 7/AI-43100ParmaItaly; and S
P Santini
Dipartimento di Fisica
Università di Parma
Viale Usberti 7/AI-43100ParmaItaly; and S
G Amoretti
Dipartimento di Fisica
Università di Parma
Viale Usberti 7/AI-43100ParmaItaly; and S
Y Furukawa
Division of Physics
Hokkaido University
060-0810SapporoJapan
K Kiuchi
Division of Physics
Hokkaido University
060-0810SapporoJapan
Y Ajiro
CNR-INFM
I-41100ModenaItaly
Department of Chemistry
Kyoto University
606-8502KyotoJapan
CREST Japan Science and Technology (JST)
CNR-INFM
I-41100ModenaItaly
Y Narumi
The Institute for Solid State Physics
The University of Tokyo
277-8581KashiwaChibaJapan
K Kindo
The Institute for Solid State Physics
The University of Tokyo
277-8581KashiwaChibaJapan
J Lago
Dipartimento di Fisica "A. Volta"
Università di Pavia
CNR-INFM
Via Bassi 6I-27100PaviaItaly
E Micotti
Dipartimento di Fisica "A. Volta"
Università di Pavia
CNR-INFM
Via Bassi 6I-27100PaviaItaly
P Arosio
Istituto di Fisologia Generale e Chimica Biologica
Università di Milano
I-20134MilanoItaly
CNR-INFM
I-27100PaviaItaly; and S
A Lascialfari
Dipartimento di Fisica "A. Volta"
Università di Pavia
CNR-INFM
Via Bassi 6I-27100PaviaItaly
Istituto di Fisologia Generale e Chimica Biologica
Università di Milano
I-20134MilanoItaly
CNR-INFM
I-27100PaviaItaly; and S
F Borsa
Dipartimento di Fisica "A. Volta"
Università di Pavia
CNR-INFM
Via Bassi 6I-27100PaviaItaly
G Timco
School of Chemistry
University of Manchester
Oxford RoadM13 9PLManchesterUnited Kingdom
R E P Winpenny
School of Chemistry
University of Manchester
Oxford RoadM13 9PLManchesterUnited Kingdom
Magnetization and spin dynamics of a Cr-based magnetic cluster: Cr 7 Ni
19 Aug 2008 (Dated: August 19, 2008)
We study the magnetization and the spin dynamics of the Cr7Ni ring-shaped magnetic cluster. Measurements of the magnetization at high pulsed fields and low temperature are compared to calculations and show that the spin Hamiltonian approach provides a good description of Cr7Ni magnetic molecule. In addition, the phonon-induced relaxation dynamics of molecular observables has been investigated. By assuming the spin-phonon coupling to take place through the modulation of the local crystal fields, it is possible to evaluate the decay of fluctuations of two generic molecular observables. The nuclear spin-lattice relaxation rate 1/T1 directly probes such fluctuations, and allows to determine the magnetoelastic coupling strength.
I. INTRODUCTION
The great efforts devoted to the synthesis and investigation of nanosize magnetic molecules are motivated both by interests in fundamental physics and by the envisaged technological applications. For instance, some of these systems have shown phenomena such as quantum tunneling of magnetization between quasi-degenerate levels, slow relaxation at low T , and revealed to be promising for high density information storage and quantum computing [1,2,3]. The magnetic core of molecular magnets is constituted by transition metal ions sorrounded by an organic shell which prevents intramolecular magnetic interactions. As a result, the microscopic properties of these nanoscale clusters can be investigated by means of bulk samples. Among these systems, there are homonuclear antiferromagnetic (AF) ring-shaped molecules formed by n transition metal ions in an almost planar ring. In particular, in even membered rings the dominant AF exchange interactions lead to a singlet S T = 0 ground state and the energy spectrum is characterized by rotational bands, with the lowest-lying levels approximately following the so-called Landé's rule [1]. In this paper we study the magnetization and the phonon-induced relaxation in the heterometallic ring Cr 7 Ni. This compound derives from the even membered AF ring Cr 8 and thus provides a opportunity of a deeper insight in the role of topology in the static and dynamical quantum properties of magnetic wheels [6,7]. Cr 7 Ni compound is obtained by the chemical substitution of a Cr 3+ ion with a Ni 2+ ion in the structure of the Cr 8 ring. This leads to a new molecular system formed by an odd number of unpaired electrons with dominant AF nearest neighbour interactions as inferred by susceptibility measurements [4]. The resulting S T = 1/2 ground state has been shown to be suitable to encode a qubit [5].
II. MAGNETIZATION
The magnetic molecule has been theoretically analyzed within a spin Hamiltonian approach, with the Hamiltonian given by:
H = i>j J ij s(i) · s(j) + i d i s 2 z (i) − s i (s i + 1)/3 + i>j s(i) · D ij · s(j) − µ B i g i H · s(i),(1)
where s(i) is the spin operator of the the ith ion in the molecule (s(i)=3/2 for Cr 3+ ions, and s(i)=1 for the Ni 2+ ion). The first term of the above equation is the dominant nearest neighbour Heisenberg exchange interaction. The second and the third terms describe the uniaxial local crystal fields and anisotropic intracluster spin-spin interactions respectively (with the z axis assumed perpendicular to the plane of the ring). The last term represents the Zeeman coupling with an external field H. The parameters of the above Hamiltonian were determined by inelastic neutron scattering (INS) experiments [8,9]. In order to corroborate the microscopic description of the Cr 7 Ni from INS data, a detailed study of high field magnetization is very powerful. In fact, with high pulsed fields up to almost 60T, spin multiplets not accessible to a standard INS experiment can be explored. In Fig.1a the magnetization curve as a function of the magnetic field H is reported. A clear staircase structure with plateaus at ≈ 1µ B , ≈ 3µ B and odd multiples of µ B reflects the change in the ground state due to the external field at the level anticrossing fields H n . An hysteresis of the measured magnetization curves has been observed. The effect arises from the non-equilibrium condition due to the high pulsed magnetic field with a few millisecond duration [7] and has been discussed in terms of phonon bottleneck effects and magnetic Foehn effects [12]. There is a very good agreement between the measured and calculated magnetization curves. This is clearly visible in Fig.1b where the positions of the main peaks of the calculated and measured dM/dH matches correctly. The smaller peaks in the experimental dM/dH are due to level anticrossings between excited energy levels. The effects are caused by the non-equilibrium exeperimental conditions and are not included in equilibrium calculations reported in Fig.1 [7]. These results confirm that the microscopic picture derived from INS experiments [8,9] perfectly holds even at very high applied magnetic fields.
III. SPIN DYNAMICS
A major obstacle to the proposed technological applications of magnetic molecules is constituted by phononinduced relaxation. In fact, molecular observables, e.g. the magnetization, are deeply affected by the interaction of the spins with other degrees of freedom such as phonons [10]. Here we investigate the molecular spin-spin correlations through an approach based on a density matrix theory [10]. The irreversible evolution of the density matrixρ(t) can be determined through the secular approximation and focusing on time scales detectable by low-frequency techiques such as NMR. Within this theoretical framework, a general expression for the quasielastic part of the Fourier transform of cross correlation functions is given by [10,13]:
S AB (ω) = m,n=1,N p (eq) m (B mm − B eq ) ×(A nn − Â eq )Re 1 iω − W nm ,(2)
where N is the dimension of the Hilbert spin space of the molecule, p (eq) m is the equilibrium population of the mth level and B mm = m|B |m , |m being the mth eigenstate of the spin Hamiltonian, while W is the so called rate-matrix. The mn element W mn of W represents the probability per unit time of a transition between eigenstates |m and |n induced by the interaction of the spins with phonons. By assuming that spin-bath interaction takes place mostly through modulation of local crystal fields, the rate matrix can be calculated on the basis of the eigenstates of molecular spin Hamiltonian by firstorder perturbation theory. With the choice of a spherical magnetoelastic (ME) coupling [14] the transition rates W mn are given by:
W mn = γπ 2 ∆ 3 mn n(∆ mn ) i,j=1,N q1,q2=x,y,z m| O q1,q2 (s i ) |n × n| O q1,q2 (s j ) |m ,(3)
with n(x) = (e β x − 1) −1 , ∆ mn = (E m − E n )/ the gap between the eigenstates |m and |n of the molecule. In the last equation O q1,q2 (s i ) = (s q1,i s q2,i + s q2,i s q1,i )/2 are quadrupolar operators [14]. Finally, γ represents the spin-phonon coupling strength, which can be determined by comparing the theoretical results with experimental data. In fact, the nuclear spin-lattice relaxation rate 1/T 1 probes the fluctuations of molecular observables, thus giving information on the relaxation dynamics [10]. Exploiting the Moriya formula [15], the proton NMR 1/T 1 can be evaluated in absolute units using as inputs the positions of the Cr and Ni ions and of the hydrogens of the molecule: . This equation shows that the spectrum of fluctuations of M is given by a sum of N Lorentzians, each with characteristic frequency λ i , given by the eigenvalues of −W . For a wide range of H and T in these systems only a single relaxation frequency λ 0 significantly contribute to S Sz,Sz (ω, T, H). As a result, if the dominant frequency λ 0 intersects the Larmor angular frequency, i.e. when λ 0 (T 0 ) = ω L , at the temperature T 0 the proton NMR 1/T 1 shows a sharp peak [10,11]. Being an heterometallic ring, this explanation does not hold for Cr 7 Ni and Eq.(4) has to be used. Nevertheless, our calculations show that a peak in the reduced 1/(T 1 χT ) occurs in agreement with experimental data (see Fig.2). By fitting the observed peak position we have obtained γ = 0.8 × 10 −7 THz −2 .
1 T 1 = i,j=1,N q,q ′ =x,y,z α qq ′ ij S s q i ,s q ′ j (ω L ) + S s q i ,s q ′ j (−ω L ) ,(4)
IV. CONCLUSIONS
A magnetization study of the heteronuclear antiferromagnetic ring-shaped nanomagnet Cr 7 Ni has been performed. A clear step-wise increase of magnetization with increasing field is observed. The very good agreement of high field magnetization measurements up to almost 60T with calculation shows the spin Hamiltonian approach to be suitable even at very high fields. The relaxation dynamics of the compound has been investigated by the proton nuclear-spin relaxation rate 1/T 1 . Our calculations are in very good quantitative agreement with experimental data.
FIG. 1 :
1(Color online) Magnetization curve of Cr7Ni (top) and derivative dM/dH (bottom) at T=1.3K. Red (dark gray) and black lines represent the down and up experimental magnetic field processes respectively. The dashed blue lines represent the theoretical calculation with the following parameters: JCr−Cr=16.9K, JCr−Ni=19.6K, dCr=-0.3K, dNi=-4K, gCr=1.98, gNi=2.2.
online) Experimental data (scatters) and calculations (lines) of reduced proton NMR 1/(T1χT ) for different values of the applied field along z (parallel to the ring axis). where the S s q i ,s q ′ j (ω L ) are the Fourier transforms of the cross correlation functions from Eq. (2) calculated at the Larmor angular frequency ω L , while the α qq ′ ij are geometric coefficients of the hyperfine dipolar interaction between magnetic ions and protons probed by NMR. The occurence of a peak in the proton NMR 1/T 1 has been clearly explained in homonuclear ringshaped molecules with small anisotropy such as Cr 8 [10]. In fact, in this case 1/T 1 ∝ S Sz,Sz (ω L ), where S Sz,Sz (ω, H, T ) is the Fourier transform of the au-tocorrelation function of M [10]: S Sz,Sz (ω, T, H) = i=1,N A(λ i , T, H)λ i (T, H)/[λ i (T, H) 2 +ω 2 ]
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| [] |
[
"The stability and properties of high-buckled two-dimensional tin and lead",
"The stability and properties of high-buckled two-dimensional tin and lead"
] | [
"Pablo Rivero \nDepartment of Physics\nUniversity of Arkansas. Fayetteville AR\n72701USA\n",
"Jia-An Yan \nDepartment of Physics, Astronomy and Geosciences\nTowson University. Towson\n21252MDUSA\n",
"Víctor M García-Suárez \nDepartamento de Física and Centro de Investigación en Nanociencia y Nanotecnología\nUniversidad de Oviedo\n\n",
"Jaime Ferrer \nDepartamento de Física and Centro de Investigación en Nanociencia y Nanotecnología\nUniversidad de Oviedo\n\n",
"Salvador Barraza-Lopez \nDepartment of Physics\nUniversity of Arkansas. Fayetteville AR\n72701USA\n"
] | [
"Department of Physics\nUniversity of Arkansas. Fayetteville AR\n72701USA",
"Department of Physics, Astronomy and Geosciences\nTowson University. Towson\n21252MDUSA",
"Departamento de Física and Centro de Investigación en Nanociencia y Nanotecnología\nUniversidad de Oviedo\n",
"Departamento de Física and Centro de Investigación en Nanociencia y Nanotecnología\nUniversidad de Oviedo\n",
"Department of Physics\nUniversity of Arkansas. Fayetteville AR\n72701USA"
] | [] | In realizing practical non-trivial topological electronic phases stable structures need to be determined first. Tin and lead do stabilize an optimal two-dimensional high-buckled phase -a hexagonalclose packed bilayer structure with nine-fold atomic coordination-and they do not stabilize topological fullerenes, as demonstrated by energetics, phonon dispersion curves, and the structural optimization of finite-size samples. The high-buckled phases are metallic due to their high atomic coordination. The optimal structure of fluorinated tin lacks three-fold symmetry and it stabilizes small samples too. It develops two oblate conical valleys on the first Brillouin zone coupling valley, sublattice, and spin degrees of freedom with a novel τzσxsx term, thus making it a new 2D platform for valleytronics. | 10.1103/physrevb.90.241408 | [
"https://arxiv.org/pdf/1411.5702v1.pdf"
] | 53,136,110 | 1411.5702 | 5dd50b3b9abf1eb572f52d6c6dc7f3131c211144 |
The stability and properties of high-buckled two-dimensional tin and lead
Pablo Rivero
Department of Physics
University of Arkansas. Fayetteville AR
72701USA
Jia-An Yan
Department of Physics, Astronomy and Geosciences
Towson University. Towson
21252MDUSA
Víctor M García-Suárez
Departamento de Física and Centro de Investigación en Nanociencia y Nanotecnología
Universidad de Oviedo
Jaime Ferrer
Departamento de Física and Centro de Investigación en Nanociencia y Nanotecnología
Universidad de Oviedo
Salvador Barraza-Lopez
Department of Physics
University of Arkansas. Fayetteville AR
72701USA
The stability and properties of high-buckled two-dimensional tin and lead
(Dated: November 24, 2014)
In realizing practical non-trivial topological electronic phases stable structures need to be determined first. Tin and lead do stabilize an optimal two-dimensional high-buckled phase -a hexagonalclose packed bilayer structure with nine-fold atomic coordination-and they do not stabilize topological fullerenes, as demonstrated by energetics, phonon dispersion curves, and the structural optimization of finite-size samples. The high-buckled phases are metallic due to their high atomic coordination. The optimal structure of fluorinated tin lacks three-fold symmetry and it stabilizes small samples too. It develops two oblate conical valleys on the first Brillouin zone coupling valley, sublattice, and spin degrees of freedom with a novel τzσxsx term, thus making it a new 2D platform for valleytronics.
Introduction.-Carbon forms two-dimensional (2D) layers with a hexagonal lattice [1,2] and silicon, germanium [3], AlAs, AlSb, GaP, InP, GaAs, InAs, GaSb, InSb [4], phosphorus [5], and tin [6][7][8] are all predicted to form stable low-buckled (LB) hexagonal 2D layers. High-buckled (HB) 2D phases cannot occur for carbon, silicon, nor germanium [3,9]. Can tin and lead stabilize the HB phase?
Proceeding by direct analogy to silicene and germanene [3], known studies of the electronic properties of 2D tin [6][7][8]10] are performed under the implicit assumption that the HB phase is not viable. In addition, the guess structures and the electronic gaps in Ref. [8] had been previously reported [7]. Contrary to common assumption, the HB 2D structures of heavy column-IV elements tin and lead are stable and lower in energy than their LB counterparts, thus representing the true optimal structures of these two-dimensional systems. The structural stability of HB tin and HB lead will have fundamental consequences for the practical realization of substrate-free non-trivial topological phases based from these elements.
The optimal phase of 2D fluorinated stanene is not analogous to tetrahedrally-coordinated graphane [11] as it was postulated in Refs. [7,8]. Studies of 2D fluorinated tin dismiss the existence of bulk crystalline fluorinated phases stable at room temperature. There is no indication for tetrahedral coordination of tin atoms in bulk fluorinated tin [12] and tetrahedral coordination [7,8] does not yield the most stable 2D fluorinated tin either.
We uncover six metastable fluorinated phases for 2D tin, the graphane-like phase [6][7][8] being one of them. Consistent with the literature in bulk flourinated tin [12,13], we demonstrate that two tilted F atoms mediate the interaction among two Sn atoms in the optimal 2D structure. This stable optimal phase displays two gapped * Electronic address: [email protected] oblate Dirac cones on the first Brillouin zone where valley τ , pseudospin σ, and spin s couple as τ z σ x s x [14].
Unlike known 2D materials with a hexagonal lattice in which three valleys with momentum directions separated by 120 o rotations are related due to threefold symmetry [15][16][17][18][19], the optimal 2D flourinated tin leads to strictly two valleys due to its reduced structural symmetry. This allows an unprecedented specificity in coupling three quantum degrees of freedom around the Fermi energy: the valley, the crystal momentum including direction, and the electronic spin.
The results here provided invite to look closely into 2D materials postulated for their remarkable electronic properties that may not realize ground-state, optimal struc- [3] and helps in confirming the stability of HB tin and HB lead unequivocally: The stanene sample (LB tin) becomes thick and amorphous and the initial LB lead sample turns into HB lead. (d) 2D tin and lead do not realize topological fullerenes. tures [7,8].
The HB phase is more favorable than the LB phase with increasing atomic number.-The energetics of column-IV 2D materials on Fig. 1(a) were obtained with the PBE exchange-correlation potential [20] on a version of the SIESTA code [21,22] that includes a self-consistent spinorbit interaction (SOI) [23]. Our basis sets are of doublezeta plus-polarization size [24]. The trends in Fig. 1 remain regardless of the inclusion of SOI, and were crosschecked with VASP calculations [25,26].
The lattice constant at the HB energy minima a HB is equal to 3.418Å for tin, and a HB =3.604Å for lead. These values become 3.413 and 3.575Å, respectively, when the SOI is included in calculations. These strikingly stable HB structures have not been reported before; lattice parameters in the literature [6][7][8]10] Germanium (with atomic number Z = 32) cannot form a HB phase, even though the energy minima of the optimized HB phase is lower than the local minima at the optimal LB phase already [ Fig. 1(a)] [3]. This HB minima becomes sizeable deeper and the energy barriers separating these phases become shallower with increasing atomic number. Figure 1(a) invites to ponder whether HB tin and HB lead are stable. In answering this question we address the atomistic coordination of HB phases first. The optimal HB structure is a hexagonal close-packed bilayer.-HB phases were represented as three-fold coordinated [3], but the relative height ∆z among atoms in complementary sublattices A and B increases as the lattice constant a 0 is compressed, so the distance a AB = a 2 0 /3 + ∆z 2 among atoms belonging in complementary sublattices increases towards a 0 . Indeed, a AB = a 0 / √ 3 for a planar hexagonal unit cell -dashed horizontal line on Fig. 1 Fig. 1(b) [27]. Numerical results yield a AB 0.95a HB (solid vertical line on Fig. 1(b)). Thus, six atoms are a distance a HB apart on a triangular lattice, and three atoms belonging on complementary sublattices are separated by a AB 0.95a HB , leading to the nine-fold coordinated HCP bilayer structure [28,29] on Fig. 1(c). A transition among low-and high-buckled structures occurs around a 0 1.2a HB on Fig. 1(b). HB tin and HB lead are stable.-We show in Figs. 2(a-b) phonon dispersion curves for HB tin and lead [30]. The effect of SOI is small, thus justifying the trends without SOI shown on Fig. 1(a-b) [31]. Similar dispersions were obtained using the Quantum Espresso code [32]. The lack of significant negative energies indicates that HB tin and HB lead are indeed stable: The Chemistry of Si and Ge does not translate to Sn and Pb because with increasing atomic number the s−orbital lowers its energy with respect to p−orbital, thus reducing the s − p hybridization.
(b)-but an ideal hexagonal close-packed (HCP) structure has ∆z = √ 2a 0 / √ 3 yielding a AB = a 0 -solid horizontal line on
The ultimate test of relative stability is a structural optimization of small 2D flakes with initial HB or LB conformations [ Fig. 2(c)] where the lines joining atoms reveal their atomistic coordination. The finite-size HB structures have 122 atoms; the LB structures have eight additional atoms (red dots on the LB initial structure) so that all edge atoms are two-fold coordinated. We set a stringent force tolerance cutoff of at least 0.01 eV/Å.
The LB Si and LB Ge samples (subplots ii and iv) show crumpling originating out from the boundaries yet the hexagonal lattice remains visible around the center of mass after the force relaxation [3]. On the other hand, the amorphous shape and the random-looking atomistic coordination of HB Si and HB Ge (subplots i and iii) indicate that these phases are unstable [3]. Confirming the structural stability inferred from phonon dispersion curves, HB Sn and HB Pb do stabilize on finite-size samples: Starting from an ideal HB phase, the optimized Sn structure retains the HB coordination within the area highlighted by an oval (Fig. 2(c), subplot v). The finite LB Sn sample, on the other hand, crumples upon optimization (Fig. 2(c), subplot vi). In fact, the region highlighted by the tiny oval on subplot vi in Fig. 2(c) displays the local coordination expected of a HB phase already. Similar conclusions would be reached in Ref. [10] when periodic constraints are removed.
−1 0 1 Γ Κ Μ Γ Γ Κ Μ Γ E−E F (eV) (b) (a) without SOI with SOI
Haldane's honeycomb model has been studied in closed geometries [33] and one of the many candidates for its practical realization is LB tin (stanene). Unfortunately, a fullerene-like Sn 60 is not stable [ Fig. 2(d)] so tin and lead are no-go elements for topological fullerenes. Based on Fig. 1(a) Viable electronic materials require stable structures. Tin and lead films have been created experimentally [38][39][40] and structural aspects must be addressed diligently to realize two-dimensional materials with a strong SOI. Electronic properties of HB tin and lead.-Graphene, silicene and germanene are three-fold coordinated and have a conical dispersion around the K−points with small gaps due to SOI [14,[34][35][36][37]. The nine-fold-coordinated 2D HB structures display no conduction gaps [ Fig. 3]. Bulk limits.-HCP bilayers could be cleaved out of HCP or FCC bulk structures. Lead forms a FCC structure with interatomic distances of 3.614Å, which compare favorably with a HB = 3.575 and make HB lead stable.
Tin stabilizes a tetragonal structure (β−tin [41,42]) and a diamond structure (α−tin [43]). The β−phase is higher in energy than the α−phase by E β − E α = 0.58 eV/atom. Every atom on β−tin has four neighbors at 3.11Å, two neighbors at 3.26Å, and four neighbors 3.87 A apart: these ten atoms are 3.44Å apart on average. On the nine-fold coordinated HB tin a HB = 3.42Å and a AB = 3.281, having an atomistic coordination comparable to bulk β−tin. The α−tin phase has four neighbors 2.89Å apart, which compares well to a Sn−Sn = 2.85Å for a 2D LB structure. Importantly, in two-dimensions energetics switch and the HB phase -compatible with bulk β−tin-is more stable than LB tin -compatible with Fig. 1(a)]. Fluorinated 2D tin.-The phase space for decorated 2D tin is larger than originally anticipated [ Fig. 4(a)]: The graphane-like phase [7,8] realizes the metastable minima labeled 6 that turns into phase 4 upon in-plane compression. Placement of F atoms directly on top of/under Sn atoms results on two dissociated triangular Sn lattices bonded on opposite sides by F atoms (structures 2/1).
In the optimal structure, 7, four-fold coordinated Sn atoms form a sequence of parallel zig-zag one-dimensional chains with two fluorine atoms mediating interactions among neighboring Sn chains. The structure is realized on a triangular lattice with a 0 = 5.230Å [ Fig. 4(b)]. The Wigner-Seitz unit cell is within the dotted area in Fig. 4(b); the symmetry axes are shown as well. A similar "bridging" fluorine coordination is realized on bulk tin(II) fluoride (e.g., Fig. 2 in Ref. [12]).
Bulk Tin(II) fluoride is highly stable at room temperature and can be found in household products. Structural stability of optimal 2D tin is probed with phonon dispersion calculations [ Fig. 4(c)] along the high-symmetry lines shown in Fig. 5(a). The phonon frequency range is comparable with that of graphene, and it is one order- of-magnitude larger than those in Fig. 2(a,b). As an additional successful check, small-size flakes were subjected to a successful structural optimization [Fig 4(d)]. The peculiar coupling of quantum degrees of freedom on this system may encourage experimental routes towards the synthesis of 2D fluorinated tin. The stability of its parent 3D compound at room temperature [12,13] invites experimental investigations of potential viability in 2D.
Γ Κ 2 Μ 1 Μ 2 Κ 1 k y (A −1 ) k x (A −1 ) (a) (b) V 1 V 2 V 2 V 1 conduction valence −1 0 1 V 1 +M 2 V 1 V 1 -M 2 V 1 −1 0 1 2 E−E F (eV) Γ Κ 1 Κ 2 Μ 1 Μ 2 Γ −0.1 0.0 0.1 Κ 1 Γ without SOI with SOI model without SOI E−E F (eV) (d) Valence-1 Sublattice A Sublattice B Conduction Conduction+1 Valence V 1 (e)
The first Brillouin zone in Fig. 5(a) shows a top view of the conduction band and the high-symmetry points in momentum space. As seen in Fig. 5(b), the arrangement of parallel 1D Sn wires gives rise to an electronic structure with only two anisotropic Dirac cones on the First Brillouin zone located away from the K-points at positions V 1 and V 2 = ±0.85K 1 , respectively. From now on we identify the x−axis with the line joining tin atoms across fluorine bridges. The Fermi velocity is close in magnitude to that of graphene and it is anisotropic: v F y = 5.4 × 10 5 m/s [ Fig. 5(c)], and v F x = 2.1 × 10 5 m/s [ Fig. 5(d)] and a 2∆ = 0.02 eV gap opens due to SOI, five times larger than the intrinsic gap due to SOI in graphene [44]. Phase 6 transitions from a topological insulator to a trivial insulator [8], but the electronic structure of the optimal phase remains robust under larger isotropic strain.
The electronic dispersion in Fig. 5(b-d) can be understood in terms of a 2 × 2 π−electron tight-binding Hamiltonian [45] in which an effective coupling t is set among the tin atoms originally linked by fluorine bridges [thin bonds on Fig. 4(b)], and t is the coupling among actual Sn-Sn atoms [thick bonds on Fig. 4(b)]. Using interatomic distances among Sn atoms from Table I we obtain the blue dashed lines in Fig. 5(c,d) with t = 0.8 eV and t = v F x v F y t which reproduce first-principles results. To account for SOI, we realize an oblate low-energy Dirac-Hamiltonian at the vicinity of the V 1,2 points. The relevant subspace is four-dimensional at any given valley, and the task is to reproduce the spin texture displayed in Fig. 5(e) where spin projects onto the +x or the −x directions while leaving the sublattice (pseudospin) degree of freedom unpolarized. The numerical results on Fig. 5(e) are consistent with a coupling τ z σ x s x . Indeed, eigenvectors of τ z σ x s x in Table II project spins onto the −x, +x, +x, −x axis parallel to the Sn-F bonds, inverting signs at each valley and lacking sublattice polarization, consistently with ab-initio data [ Fig. 5(e)]. Thus, the low-energy dynamics is given by:
H = −i Ψ † (v F x τ z σ x ∂ x + v F y σ y ∂ y )Ψ + Ψ † (∆τ z σ x s x )Ψ.
An unprecedented specific coupling of momentumincluding direction-with spin oriented alongx and valley degrees of freedom is thus realized by the second term in previous equation. The valley degree of freedom can be addressed by a bias along the V 1 −V 2 axis that breaks inversion symmetry. Similarly, a magnetic field along thex axis will break time-reversal symmetry, locking the valley and crystal momentum direction at the V 1 , V 2 points. The dynamics invites the use of 2D fluorinated tin for valleytronic applications.
We demonstrated the structural stability of HB tin and HB lead and discussed their electronic properties, showed that tin and lead are not viable routes towards topological fullerenes, and discovered the structural, valley, sublattice and spin properties of optimal fluorinated stanene. We are grateful to G. Montambaux, M. Kindermann and L. Bellaiche, and acknowledge the Arkansas Biosciences Institute (P.R. and S.B.L.); the Faculty Development and Research Committee, grant OSPR 140269 and the FCSM Fisher General Endowment at Towson Uni-
PACS numbers: 73.22-f, 71.70.Ej, 68.55.at
are ∼140% larger. Normalization of a 0 in terms of a HB in Figs. 1(a) facilitates an unified display of energetics regardless of atomic species. The vertical dashed line in Figs. 1(a) and 1(b) at about a 0 1.2a HB highlights the lattice constant a 0 for which energy barriers separating the LB and the HB phases become largest.
FIG. 3 :
3(Color online.) Electronic dispersion for (a) HB tin and (b) HB lead. The nine-fold atomic coordination of the HB phases is behind the metallic electronic dispersion.
HB lead is extremely stable: It stabilizes finite HB samples with no change in atomic coordination [Fig. 2(c), subplot vii] and turns a LB structure onto a HB-coordinated one [Fig. 2(c), subplot viii].
FIG. 4 :
4(Color online.) (a) Phases of 2D fluorinated tin; structures shown to the right. (b) Symmetries of the most stable structure (7), depicting triangular (dashed) and Wigner-Seitz (within dotted perimeter) unit cells, the two symmetry axes, and the two Sn sublattices A and B. Structural stability is demonstrated by (c) phonon dispersion curves and (d) the structural stabilization of a finite-size sample. TABLE I: Basis vectors for fluorinated stanene (a0 = 5.23Å). Sn: (0.000, 0.000, 0.000)a0, (0.583, 0.336, −0.221)a0 F: (0.216, 0.124, −0.348)a0, (0.367, 0.212, 0.128)a0 α−tin-by E LB − E HB = 0.25 eV/atom [c.f.
II: Eigenvectors of τzσxsx. |sx; ± are eigenstates of sx, and |A , |B are eigenstates of the pseudospin operator.
online.) (a) Conduction band on the first Brillouin zone, highlighting high-symmetry points and locations of valleys V1 and V2 away from the K-and K'-points. (b) The two valleys on the Brillouin zone arise from the two-fold symmetry of the atomic structure. (c-d) Band structures along high-symmetry lines, including a two-band tight-binding fit. (e) Spin texture resolved over valley (τ ), energy, and sublattice (σ) degrees of freedom. (The spin projection onto the z−axis is of the order of 1% at most.)
at the high-buckled energy minimum; the structure transitions to a low-buckled phase at roughly 1.2aHB. (c) The high-buckled structure is a HCP bilayer.1.0
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Phonon dispersions E ph (k) for HB tin and lead demonstrate their structural stability; SOI does not change phonon dispersions dramatically. (c) The structural optimization of finite samples reflects previous findings for Si and GearXiv:1411.5702v1 [cond-mat.mes-hall] 20 Nov 2014
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); the Spanish MICINN, grant FIS2012-34858, and European Commission FP7 ITN "MOLE-SCO," grant number 606728. fellowship RYC-2010-06053Computations were carried out at Arkansas and TACC (XSEDE TG-PHY090002). J.A.Y.); and a Ramón y Cajalversity (J.A.Y.); the Spanish MICINN, grant FIS2012- 34858, and European Commission FP7 ITN "MOLE- SCO," grant number 606728 (V.M.G.S. and J.F.); and a Ramón y Cajal fellowship RYC-2010-06053 (V.M.G.S.). Computations were carried out at Arkansas and TACC (XSEDE TG-PHY090002).
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| [] |
[
"Deep kernel processes",
"Deep kernel processes"
] | [
"Laurence Aitchison ",
"Adam X Yang ",
"Sebastian Ober "
] | [] | [] | We define deep kernel processes in which positive definite Gram matrices are progressively transformed by nonlinear kernel functions and by sampling from (inverse) Wishart distributions. Remarkably, we find that deep Gaussian processes (DGPs), Bayesian neural networks (BNNs), infinite BNNs, and infinite BNNs with bottlenecks can all be written as deep kernel processes. For DGPs the equivalence arises because the Gram matrix formed by the inner product of features is Wishart distributed, and as we show, standard isotropic kernels can be written entirely in terms of this Gram matrix -we do not need knowledge of the underlying features. We define a tractable deep kernel process, the deep inverse Wishart process, and give a doubly-stochastic inducing-point variational inference scheme that operates on the Gram matrices, not on the features, as in DGPs. We show that the deep inverse Wishart process gives superior performance to DGPs and infinite BNNs on fully-connected baselines. 1 | null | [
"https://arxiv.org/pdf/2010.01590v2.pdf"
] | 222,133,058 | 2010.01590 | 438d0a7a85e5958c5bf877679666c0d3cc09bcdf |
Deep kernel processes
Laurence Aitchison
Adam X Yang
Sebastian Ober
Deep kernel processes
We define deep kernel processes in which positive definite Gram matrices are progressively transformed by nonlinear kernel functions and by sampling from (inverse) Wishart distributions. Remarkably, we find that deep Gaussian processes (DGPs), Bayesian neural networks (BNNs), infinite BNNs, and infinite BNNs with bottlenecks can all be written as deep kernel processes. For DGPs the equivalence arises because the Gram matrix formed by the inner product of features is Wishart distributed, and as we show, standard isotropic kernels can be written entirely in terms of this Gram matrix -we do not need knowledge of the underlying features. We define a tractable deep kernel process, the deep inverse Wishart process, and give a doubly-stochastic inducing-point variational inference scheme that operates on the Gram matrices, not on the features, as in DGPs. We show that the deep inverse Wishart process gives superior performance to DGPs and infinite BNNs on fully-connected baselines. 1
Introduction
The deep learning revolution has shown us that effective performance on difficult tasks such as image classification (Krizhevsky et al., 2012) requires deep models with flexible lower-layers that learn task-dependent representations. Here, we consider whether these insights from the neural network literature can be applied to purely kernel-based methods. (Note that we do not consider deep Gaussian processes or DGPs to be "fully kernel-based" as they use a feature-based representation in intermediate layers).
Importantly, deep kernel methods (e.g. Cho & Saul, 2009) 1 Reference implementation at: github.com/LaurenceA/ bayesfunc already exist. In these methods, which are closely related to infinite Bayesian neural networks (Lee et al., 2017;Matthews et al., 2018;Garriga-Alonso et al., 2018;Novak et al., 2018), we take an initial kernel (usually the dot product of the input features) and perform a series of deterministic, parameter-free transformations to obtain an output kernel that we use in e.g. a support vector machine or Gaussian process. However, the deterministic, parameter-free nature of the transformation from input to output kernel means that they lack the capability to learn a top-layer representation, which is believed to be crucial for the effectiveness of deep methods (Aitchison, 2019).
Contributions
1. We propose deep kernel processes (DKPs), which combine nonlinear transformations of the kernel, as in Cho & Saul (2009) with a flexible learned representation by exploiting a Wishart or inverse Wishart process (Dawid, 1981;Shah et al., 2014).
This complexity means that common variational approximate posteriors can give a very poor approximation to the true posterior. In contrast, the Gram matrices in DKPs are invariant to permutations/rotations of the weights/features and thus have much simpler true posteriors which are more easily captured by variational approximate posteriors. Second, in DIWPs the "width" parameter is learnable, and in the limit of infinite width gives a series of deterministic kernel transformations, as in an infinite neural network. This gives DIWPs the ability to learn on a layer-by-layer basis where a deterministic kernel transformation is appropriate, or where more flexibility in the kernel is needed.
Background
We briefly revise Wishart and inverse Wishart distributions. The Wishart distribution is a generalization of the gamma distribution that is defined over positive semidefinite matrices. Suppose that we have a collection of P -dimensional random variables x i with i ∈ {1, . . . , N } such that
x i iid ∼ N (0, V) ,(1)N i=1 x i x T i = S ∼ W (V, N )(2)
has Wishart distribution with scale matrix V and N degrees of freedom. When N > P − 1, the density is,
W (S; V, N ) = |S| (N −P −1)/2 e − Tr(V −1 S/2) 2 N P |V|Γ P N 2(3)
where Γ P is the multivariate gamma function. Further, the inverse, S −1 has inverse Wishart distribution, W −1 V −1 , N . The inverse Wishart is defined only for N > P − 1 and also has closed-form density. Finally, we note that the Wishart distribution has mean N V while the inverse Wishart has mean V −1 /(N −P −1) (for N > P +1).
Deep kernel processes
We define a kernel process to be a set of distributions over positive definite matrices of different sizes, that are consistent under marginalisation (Dawid, 1981;Shah et al., 2014). The two most common kernel processes are the Wishart process and inverse Wishart process, which we write in a slightly unusual form to ensure their expectation is K. We take G and G to be finite dimensional marginals of the underlying Wishart and inverse Wishart process,
G ∼ W (K /N, N ) , (4a) G * ∼ W (K * /N, N ) ,(4b)
G ∼ W −1 (δK , δ+(P +1)) ,
G * ∼ W −1 (δK * , δ+(P * +1)) ,
and where we explicitly give the consistent marginal distributions over K * , G * and G * which are P * × P * principal submatrices of the P × P matrices K, G and G dropping the same rows and columns. In the inverse-Wishart distribution, δ is a positive parameter that can be understood as controlling the degree of variability, with larger values for δ implying smaller variability in G .
We define a deep kernel process by analogy with a DGP, as a composition of kernel processes, and show in App. A that under sensible assumptions any such composition is itself a kernel process. 2
DGPs with isotropic kernels are deep Wishart processes
We consider deep GPs of the form (Fig. 1 top) with X ∈ R P ×N0 , where P is the number of input points and N 0 is the number of features in the input.
K = 1 N0 XX T for = 1, K (G −1 ) for ∈ {2, . . . , L+1},(5a)P (F |K ) = N λ=1 N f λ ; 0, K ,(5b)G = 1 N F F T .(5c)
Here, F ∈ R P ×N are the N hidden features in layer ; λ indexes hidden features so f λ is a single column of F , representing the value of the λth feature for all training inputs. Note that K(·) is a function that takes a Gram matrix and returns a kernel matrix, whereas K is a (possibly random) variable representing the kernel matrix at layer . Note, we have restricted ourselves to kernels that can be written as functions of the Gram matrix, G , and do not require the full set of activations, F . As we describe later, this is not too restrictive, as it includes amongst others all isotropic kernels (i.e. those that can be written as a function of the distance between points Williams & Rasmussen, 2006). Note that we have a number of choices as to how to initialize the kernel in Eq. (5a). The current choice just uses a linear dotproduct kernel, rather than immediately applying the kernel function K. This is both to ensure exact equivalence with infinite NNs with bottlenecks (App. C.3) and also to highlight an interesting interpretation of this layer as Bayesian inference over generalised lengthscale hyperparameters in the squared-exponential kernel (App. B e.g. Lalchand & Rasmussen, 2020).
For DGP regression, the outputs, Y, are most commonly given by a likelihood that can be written in terms of the output features, F L+1 . For instance, for regression, the 2 Note that we leave the question of the full Kolmogorov extension theorem (Kolmogorov, 1933) for matrices to future work: for our purposes, it is sufficient to work with very large but ultimately finite input spaces as in practice, the input vectors are represented by elements of the finite set of 32-bit or 64-bit floating-point numbers (Sterbenz, 1974).
X K 1 F 1 G 1 K 2 F 2 G 2 K 3 F 3 Y X G 1 G 2 F 3 Y Layer 1
Layer 2 Output Layer distribution of the λth output feature column could be
P (y λ |F L+1 ) = N y λ ; f L+1 λ , σ 2 I ,(6)
alternatively, we could use a classification likelihood,
P (y|F L+1 ) = Categorical y; softmax F L+1 λ .(7)
Importantly, our methods can be used with any likelihood with a known probability density function.
The generative process for the Gram matrices, G , consists of generating samples from a Gaussian distribution (Eq. 5b), and taking their product with themselves transposed (Eq. 5c). This exactly matches the generative process for a Wishart distribution (Eq. 1), so we can write the Gram matrices, G , directly in terms of the kernel, without needing to sample features ( Fig. 1 bottom),
P (G 1 |X) = W 1 N1 1 N0 XX T , N 1 ,(8a)P (G |G −1 ) = W (K (G −1 ) /N , N ) ,(8b)P (F L+1 |G L ) = N L+1 λ=1 N f L+1 λ ; 0, K (G L ) . (8c)
Except at the output, the model is phrased entirely in terms of positive-definite kernels and Gram matrices, and is consistent under marginalisation (assuming a valid kernel function) and is thus a DKP. At a high level, the model can be understood as alternatively sampling a Gram matrix (introducing flexibility in the representation), and nonlinearly transforming the Gram matrix using a kernel (Fig. 2).
This highlights a particularly simple interpretation of the DKP as an autoregressive process. In a standard autoregressive process, we might propagate the current vector, x t , through a deterministic function, f (x t ), and add zero-mean Gaussian noise, ξ,
x t+1 = f (x t ) + σ 2 ξ such that E [x t+1 |x t ] = f (x t ) .(9)
By analogy, the next Gram matrix has expectation centered on a deterministic transformation of the previous Gram matrix,
E [G |G −1 ] = K (G −1 ) ,(10)
so G can be written as this expectation plus a zero-mean random variable, Ξ , that can be interpreted as noise,
G = K (G −1 ) + Ξ .(11)
Note that Ξ is not in general positive definite, and may not have an analytically tractable distribution. This noise decreases as N increases,
V G ij = V Ξ ij (12) = 1 N K 2 ij (G −1 ) + K 2 ii (G −1 )K 2 jj (G −1 )
. Notably, as N tends to infinity, the Wishart samples converge on their expectation, and the noise disappears, leaving us with a series of deterministic transformations of the Gram matrix. Therefore, we can understand a deep kernel process as alternatively adding "noise" to the kernel by sampling e.g. a Wishart or inverse Wishart distribution (G 2 and G 3 in Fig. 2) and computing a nonlinear transformation of the kernel (K(G 2 ) and K(G 3 ) in Fig. 2) Remember that we are restricted to kernels that can be written as a function of the Gram matrix,
K = K (G ) = K features (F ) , K ij = k F i,: , F j,: .(13)
where K features (·) takes a matrix of features, F , and returns the kernel matrix, K , and k is the usual kernel function, which takes two feature vectors (rows of F ) and returns an element of the kernel matrix. This does not include all possible kernels because it is not possible to recover the features from the Gram matrix. In particular, the Gram matrix is invariant to unitary transformations of the features: the Gram matrix is the same for F and F = UF where U is a unitary matrix, such that UU T = I,
G = 1 N F F T = 1 N F U U T F T = 1 N F F T . (14)
Superficially, this might seem very limiting -leaving us only with dot-product kernels (Williams & Rasmussen, 2006) such as,
k(f , f ) = f · f + σ 2 .(15)
However, in reality, a far broader range of kernels fit within this class. Importantly, isotropic or radial basis function kernels including the squared exponential and Matern depend 0 100 200 input index Visualisations of a single prior sample of the kernels and Gram matrices as they pass through the network. We use 1D, equally spaced inputs with a squared exponential kernel. As we transition K(G −1 ) → G , we add "noise" by sampling from a Wishart (top) or an inverse Wishart (bottom). As we transition from G to K(G ), we deterministically transform the Gram matrix using a squared-exponential kernel.
K(G 1 ) G 2 K(G 2 ) G 3 K(G 3 )
only on the squared distance between points, R, (Williams & Rasmussen, 2006)
k(f , f ) = k (R) , R = |f − f | 2 .(16)
These kernels can be written as a function of G, because the matrix of squared distances, R, can be computed from G,
R ij = 1 N N λ=1 F iλ − F jλ 2 = 1 N N λ=1 F iλ 2 − 2F iλ F jλ + F jλ 2 = G ii − 2G ij + G jj .(17)
Variational inference in deep kernel processes
A key part of the motivation for developing deep kernel processes was that the posteriors over weights in a BNN or over features in a deep GP are extremely complex and multimodal, with a large number of symmetries that are not captured by standard approximate posteriors (MacKay, 1992;Moore, 2016;Pourzanjani et al., 2017). For instance, in the Appendix we show that there are permutation symmetries in the prior and posteriors over weights in BNNs (App. D.1) and rotational symmetries in the prior and posterior over features in deep GPs with isotropic kernels (App. D.2). The inability to capture these symmetries in standard variational posteriors may introduce biases in the parameters inferred by variational inference, because the variational bound is not uniformly tight across the state-space (Turner & Sahani, 2011). Gram matrices are invariant to permutations or rotations of the features, so we can sidestep these complex posterior symmetries by working with the Gram matrices as the random variables in variational inference. However, variational inference in deep Wishart processes (equivalent to DGPs Sec. 4.1 and infinite NNs with bottlenecks App. C.3) is difficult because the approximate posterior we would like to use, the non-central Wishart (App. E), has a probability density function that is prohibitively costly and complex to evaluate in the inner loop of a deep learning model (Koev & Edelman, 2006). Instead, we consider an inverse Wishart process prior, for which the inverse Wishart itself makes a good choice of approximate posterior.
The deep inverse Wishart processes
By analogy with Eq. (8), we define a deep inverse Wishart processes (DIWPs). However, the inverse Wishart process introduces a new difficulty: that at the input layer, 1 N0 XX T may be singular if there are more datapoints than features. Instead of attempting to use a singular Wishart distribution over G 1 , which would be complex and difficult to work with (Bodnar & Okhrin, 2008;Bodnar et al., 2016), we instead define an approximate posterior over the full-rank N 0 × N 0 matrix, Ω, and use G 1 = 1 N0 XΩX T ∈ R P ×P .
P (Ω) = W −1 (δ 1 I, δ 1 +N 0 +1) ,(18)(with G 1 = 1 N0 XΩX T , ) P G ∈{2...L} |G −1 = W −1 (δ K (G −1 ) , P +1+δ ) , P (F L+1 |G L ) = N L+1 λ=1 N f L+1 λ ; 0, K (G L ) ,
remembering that X ∈ R P ×N0 , G ∈ R P ×P and F ∈ R P ×N L+1 .
Critically, the distributions in Eq. (18) are consistent under marginalisation as long as δ is held constant (Dawid, 1981), with P taken to be the number of input points, or equivalently the size of K −1 . Further, the deep inverse Wishart process retains the interpretation as a deterministic trans-formation of the kernel plus noise because the expectation is,
E [G |G −1 ] = δ K (G −1 ) (P + 1 + δ ) − (P + 1) = K (G −1 ) .(19)
The resulting inverse Wishart process does not have a direct interpretation as e.g. a deep GP, but does have more appealing properties for variational inference, as it is always full-rank and allows independent control over the approximate posterior mean and variance. Finally, it is important to note that Wishart and inverse Wishart distributions do not differ as much as one might expect; the standard Wishart and standard inverse Wishart distributions have isotropic distributions over the eigenvectors so they only differ in terms of their distributions over eigenvalues, and these are often quite similar, especially if we consider a Wishart model with ResNet-like structure (App. H).
An approximate posterior for the deep inverse Wishart process
Choosing an appropriate and effective form for variational approximate posteriors is usually a difficult research problem. Here, we take inspiration from Ober & Aitchison (2020) by exploiting the fact that the inverse-Wishart distribution is the conjugate prior for the covariance matrix of a multivariate Gaussian. In particular, if we consider an inverse-Wishart prior over Σ ∈ R P ×P with mean Σ 0 , which forms the covariance of Gaussian-distributed matrix, V ∈ R P ×P , consisting of columns v λ , then the posterior over Σ is also inverse-Wishart,
P (Σ) = W −1 (Σ; δΣ 0 , P +1+δ) ,(20a)P (V|Σ) = N V λ=1 N (v λ ; 0, Σ) ,(20b)P (Σ|V) = W −1 δΣ 0 + VV T , P +1+δ+N V . (20c)
Inspired by this exact posterior that is available in simple models, we choose the approximate posterior in our model to be,
Q Ω = W −1 δ 1 I + V 1 V T 1 , δ 1 +γ 1 +N 0 +1 , (with G 1 = 1 N0 XΩX T , ) Q G |G −1 = W −1 δ K (G −1 ) +V V T , δ +γ +P +1 , Q F L+1 |G L = N L+1 λ=1 N f L+1 λ ; Σ λ Λ λ v λ , Σ λ , where Σ λ = K −1 (G L ) + Λ λ −1 ,(21)
and where V 1 is a learned N 0 × N 0 matrix, {V } L =2 are P × P learned matrices and γ are learned non-negative real numbers. For more details about the input layer, see App. F. At the output layer, we use the global inducing approximate posterior for DGPs from Ober & Aitchison (2020), with learned parameters being vectors, v λ , and positive definite matrices, Λ λ (see App. G).
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N L+1 λ=1 and {Λ λ } N L+1
λ=1 at the output. In all our experiments, we optimize all five parameters {δ , V , γ } L =1 and ({v λ , Λ λ } N L+1 λ=1 ), and in addition, for inducing-point methods, we also optimize a single set of "global" inducing inputs, X i ∈ R Pi×N0 , which are defined only at the input layer.
Doubly stochastic inducing-point variational inference in deep inverse Wishart processes
For efficient inference in high-dimensional problems, we take inspiration from the DGP literature (Salimbeni & Deisenroth, 2017) by considering doubly-stochastic inducing-point deep inverse Wishart processes. We begin by decomposing all variables into inducing and training (or
test) points X t ∈ R Pt×N0 , X = X i X t F L+1 = F L+1 i F L+1 t G = G ii G it G ti G tt(22)
where e.g. G ii is P i × P i and G it is P i × P t where P i is the number of inducing points, and P t is the number of testing/training points. Note that Ω does not decompose as it is N 0 × N 0 . The full ELBO including latent variables for all the inducing and training points is,
L= E log P (Y|F L+1 ) +log P Ω, {G } L =2 , F L+1 |X Q Ω, {G } L =2 , F L+1 |X(23)
where the expectation is taken over
Q Ω, {G } L =2 , F L+1 |X .
The prior is given by combining all terms in Eq. (18) for both inducing and test/train inputs,
P Ω, {G } L =2 , F L+1 |X = P (Ω) L =2 P (G |G −1 ) P (F L+1 |G L ) ,(24)
where the X-dependence enters on the right because G 1 = 1 N0 XΩX T . Taking inspiration from Salimbeni & Deisenroth (2017), the full approximate posterior is the product of an approximate posterior over inducing points and the conditional prior for train/test points,
Q Ω, {G } L =2 , F L+1 |X = (25) Q Ω, {G ii } L =2 , F L+1 i |X i P {G it } L =2 , {G tt } L =2 , F L+1 t |Ω, {G ii } L =2 , F L+1 i , X
and the prior can be written in the same form,
P Ω, {G } L =2 , F L+1 |X = (26) P Ω, {G ii } L =2 , F L+1 i |X i P {G it } L =2 , {G tt } L =2 , F L+1 t |Ω, {G ii } L =2 , F L+1 i , X
To obtain the full ELBO, we substitute Eqs. (25) and (26) into Eq.(23), the conditional prior terms cancel,
L= E log P (Y|F L+1 ) +log P Ω, {G ii } L =2 , F L+1 |X Q Ω, {G ii } L =2 , F L+1 |X(27)
where,
P Ω,{G ii } L =2 , F L+1 i |X i = (28) P (Ω) L =2 P G ii |G −1 ii P F L+1 i |G L ii , Q Ω,{G ii } L =2 , F L+1 i |X i = (29) Q (Ω) L =2 Q G ii |G −1 ii Q F L+1 i |G L ii .
Importantly, the first term in the ELBO (Eq. 27 is a summation across test/train datapoints, and the second term depends only on the inducing points, so as in Salimbeni & Deisenroth (2017) we can compute unbiased estimates of the expectation by taking only a minibatch of datapoints, and we never need to compute the density of the conditional prior (Eq. 30), we only need to be able to sample it.
Finally, to sample the test/training points, conditioned on the inducing points, we need to sample,
P {G it , G tt } L =2 , F L+1 t |Ω, {G ii } L =2 , F L+1 i , X = P F L+1 t |F L+1 i , G L L =2 P G it , G tt |G ii , G −1 .(30)
The first distribution, P F L+1 t |F L+1
i , G L , is a multivariate Gaussian, and can be evaluated using methods from the GP literature (Williams & Rasmussen, 2006;Salimbeni & Deisenroth, 2017). The difficulties arise for the inverse Wishart terms, P G it , G tt |G ii , G −1 . To sample this distribution, note that samples from the joint over inducing and train/test locations can be written,
G ii G it G ti G tt ∼ W −1 Ψ ii Ψ it Ψ ti Ψ tt , δ + P i + P t + 1 , Ψ ii Ψ it Ψ ti Ψ tt = δ K (G −1 ) ,(31)
and where P i is the number of inducing inputs, and P t is the number of train/test inputs. Defining the Schur complements,
G tt·i = G tt − G ti G ii −1 G it ,(32)Ψ tt·i = Ψ tt − Ψ ti Ψ −1 ii Ψ it .(33)
We know that G tt·i and G ii −1 G it have distribution, (Eaton, 1983)
G tt·i G ii , G −1 ∼ W −1 (Ψ tt·i , δ +P i +P t +1) , (34a) G it G tt·i , G ii , G −1 ∼ MN G ii Ψ −1 ii Ψ it , G ii Ψ −1 ii G ii , G tt·i ,(34b)
where MN is the matrix normal. Now, G it and G tt , can be recovered by algebraic manipulation. Finally, because of the doubly stochastic form for the objective, we do not need to sample multiple of jointly consistent samples for test points; instead, (and as in DGPs Salimbeni & Deisenroth, 2017) we can independently sample each test point (App. I), which dramatically reduces computational complexity.
The full algorithm is given in Alg. 1, where the P and Q distributions for Ω and for inducing points are given by Eq. (18) and (21)
Computational complexity
As in non-deep GPs, the complexity is O(P 3 ) for time and O(P 2 ) for space for standard DKPs (the O(P 3 ) time dependencies emerge e.g. because of inverses and determinants required for the inverse Wishart distributions). For DSVI, there is a P 3 i time and P 2 i space term for the inducing points, because the computations for inducing points are exactly the same as in the non-DSVI case. As we can treat each test/train point independently (App. I), the complexity for test/training points must scale linearly with P t , and this term has P 2 i time scaling, e.g. due to the matrix products in Eq. (32). Thus, the overall complexity for DSVI is O(P 3 i + P 2 i P t ) for time and O(P 2 i + P i P t ) for space which is exactly the same as non-deep inducing GPs. Thus, and exactly as in non-deep inducing-GPs, by using a small number of inducing points, we are able to convert a cubic dependence on the number of input points into a linear dependence, which gives considerably better scaling.
Surprisingly, this is substantially better than standard DGPs. In standard DGPs, we allow the approximate posterior covariance for each feature to differ (Salimbeni & Deisenroth, 2017), in which case, we are in essence doing standard inducing-GP inference over N hidden features, which gives complexity of O(N P 3 i + N P 2 i P t ) for time and O(N P 2 i + N P i P t ) for space (Salimbeni & Deisenroth, 2017). It is possible to improve this complexity by restricting the approximate posterior to have the same covariance for each point (but this restriction harms performance).
Algorithm 1 Computing predictions/ELBO for one batch
P parameters: {δ } L =1 . Q parameters: {V , γ } L =1 , ({v λ , Λ λ } N L+1 λ=1 ), X i . Inputs: X t ; Targets: Y combine inducing and test/train inputs X = X i X t sample first Gram matrix and update ELBO Ω ∼ Q (Ω) L = log P (Ω) − log Q (Ω) G 1 = 1 N0 XΩX T for in {2, . . . , L} do sample inducing Gram matrix and update ELBO G ii ∼ Q G ii |G −1 ii L ← L + log P G ii |G −1 ii − log Q G ii |G −1 ii sample full Gram matrix from conditional prior Ψ = δ K(G −1 ) Ψ tt·i = Ψ tt − Ψ ti Ψ −1 ii Ψ it G tt·i ∼ W Ψ tt·i , δ +P i +P t +1 G it ∼ MN G ii Ψ −1 ii Ψ it , G ii Ψ −1 ii G ii , G tt·i G tt = G tt·i + Ψ ti Ψ −1 ii Ψ it . G = G ii G it G ti G tt end for sample GP inducing outputs and update ELBO F L+1 i ∼ Q F L+1 i |G L ii L ← L + log P F L+1 i |G L ii − log Q F L+1 i |G L ii sample GP predictions conditioned on inducing points F L+1 t ∼ Q F L+1 t |G L , F L+1 i add likelihood to ELBO L ← L + log P Y|F L+1 t
Results
We began by comparing the performance of our deep inverse Wishart process (DIWP) against infinite Bayesian neural networks (known as the neural network Gaussian process or NNGP) and DGPs. To ensure sensible comparisons against the NNGP, we used a ReLU kernel in all models (Cho & Saul, 2009). For all models, we used three layers (two hidden layers and one output layer), with three applications of the kernel. In each case, we used a learned bias and scale for each input feature, and trained for 8000 gradient steps with the Adam optimizer with 100 inducing points, a learning rate of 10 −2 for the first 4000 steps and 10 −3 for the final 4000 steps. For evaluation, we used approximate posterior 100 samples, and for each training step we used 10 approximate posterior samples in the smaller datasets (boston, concrete, energy, wine, yacht), and 1 in the larger datasets.
We found that DIWP usually gives better predictive performance and (and when it does not, the differences are very small; Table 1). We expected DIWP to be better than (or the same as) the NNGP as the NNGP was a special case of our DIWP (sending δ → ∞ sends the variance of the inverse Wishart to zero, so the model becomes equivalent to the NNGP). We found that the DGP performs poorly in comparison to DIWP and NNGPs, and even to past baselines on all datasets except protein (which is by far the largest). This is because we used a plain feedforward architecture for all models. In contrast, Salimbeni & Deisenroth (2017) found that good performance (or even convergence) with DGPs on UCI datasets required a complex GP-prior inspired by skip connections. Here, we used simple feedforward architectures, both to ensure a fair comparison to the other models, and to avoid the need for an architecture search. In addition, the inverse Wishart process is implicitly able to learn the network "width", δ , whereas in the DGPs, the width is fixed to be equal to the number of input features, following standard practice in the literature (e.g. Salimbeni & Deisenroth, 2017).
Next, we considered fully-connected networks for small image classification datasets (MNIST and CIFAR-10; Table 2). We used the same models as in the previous section, with the omission of learned bias and scaling of the inputs. Note that we do not expect these methods to perform well relative to standard methods (e.g. CNNs) for these datasets, as we are using fully-connected networks with only 100 inducing points (whereas e.g. work in the NNGP literature uses the full 60, 000 × 60, 000 covariance matrix). Nonetheless, as the architectures are carefully matched, it provides another opportunity to compare the performance of DIWPs, NNGPs and DGPs. Again, we found that DIWP usually gave statistically significant gains in predictive performance (except for CIFAR-10 test-log-likelihood, where DIWP lagged by only 0.01). Importantly, DIWP gives very large improvements in the ELBO, with gains of 0.09 against DGPs for MNIST and 0.08 for CIFAR-10 (Table 2). For MNIST, remember that the ELBO must be negative (because both the log-likelihood for classification and the KL-divergence term give negative contributions), so the change from −0.301 to −0.214 represents a dramatic improvement.
Related work
Our first contribution was the observation that DGPs with isotropic kernels can be written as deep Wishart processes as the kernel depends only on the Gram matrix. We then gave similar observations for neural networks (App. C.1), infinite neural networks (App. C.2) and infinite network with bottlenecks (App. C.3, also see Aitchison, 2019). These observations motivated us to consider the deep inverse Wishart process prior, which is a novel combination of two preexisting elements: nonlinear transformations of the kernel (e.g. Cho & Saul, 2009) and inverse Wishart priors over kernels (e.g. Shah et al., 2014). Deep nonlinear transformations Table 1. Performance measured as predictive log-likelihood for a three-layer (two hidden layer) DGP, NNGP and DIWP on UCI benchmark tasks. We have consider relu and squared exponential kernels. Errors are quoted as two standard errors in the difference between that method and the best performing method, as in a paired t-test. This is to account for the shared variability that arises due to the use of different test/train splits in the data (20 splits for all but protein, where 5 splits are used Gal & Ghahramani, 2015) some splits are harder for all models, and some splits are easier. Because we consider these differences, errors for the best measure are implicitly included in errors for other measures, and we cannot provide a comparable error for the best method itself. is not yet in widespread use, it allows us to make a distinction that is very useful in our context. In particular, a generalised Wishart process is a distribution over infinitely many finite-dimensional marginally Wishart matrices. For instance, these might represent the noise in a dynamical system. In that case, there would in principle be infinitely covariance matrices, one for each state-space location or time-point (Wilson & Ghahramani, 2010;Heaukulani & van der Wilk, 2019;Jorgensen et al., 2020). In contrast, kernel processes (Dawid, 1981;Bru, 1991) are distributions over a single infinite dimensional matrix. We stack these kernel process to form a (non-genearlised) deep kernel pro- Heaukulani & van der Wilk, 2019), but it is not possible to optimize the degrees of freedom and the posterior over these features usually has rotational symmetries (App. D.2) that are not captured by standard variational posteriors. In contrast, we give a novel doubly-stochastic variational inducing point inference method that operates purely on Gram matrices and thus avoids needing to capture these symmetries.
Conclusions
We proposed deep kernel processes which combine non-
A. DKPs are kernel processes
We define a kernel process to be a distribution over positive (semi) definite matrices, K(K), parameterised by a postive (semi) definite matrix, K ∈ R P ×P . For instance, we could take,
K(K) = W (K, N ) or K(K) = W −1 (K, δ + (P + 1)) ,(35)
where N is a positive integer and δ is a positive real number. A kernel process is defined by consistency under marginalisation and row/column exchangeability. Consistency under marginalisation implies that if we define K * and G * as principle submatrices of K and G, dropping the same rows and columns, then G being distributed according to a kernel process implies that G * is distributed according to that same kernel process,
G ∼ K(K) implies G * ∼ K(K * ).(36)
row/column exchangeability means,
G ∼ K(K) implies G σ ∼ K(K σ )(37)
where σ is a permutation of the rows/columns. Note that both the Wishart and inverse Wishart as defined in Eq. (35) are consistent under marginalisation and are row/column exchangeable.
A deep kernel process, D, is the composition of two (or more) underlying kernel processes, K 1 and K 2 ,
G ∼ K 1 (K), H ∼ K 2 (G), (38a) H ∼ D(K).(38b)
We define K * , G * and H * as principle submatrices of K, G and H respectively, dropping the same rows and columns, and again, K σ , G σ and H σ are those matrices with the rows and columns permuted. To establish that D is consistent under marginalisation, we use the consistency under marginalisation of K 1 and K 2
G * ∼ K 1 (K * ), H * ∼ K 2 (G * ),(39a)
and the definition of the D as the composition of K 1 and K 2 (Eq. 38)
H * ∼ D(K * ).(39b)
Likewise, to establish row/column exchangeability, we use row/column exchangeability of K 1 and K 2 ,
G σ ∼ K 1 (K σ ), H σ ∼ K 2 (G σ ),(40a)
and the definition of the D as the composition of K 1 and K 2 (Eq. 38)
H σ ∼ D(K σ ).(40b)
The deep kernel process D is thus consistent under marginalisation and has exchangeable rows/columns, and hence a deep kernel process is indeed itself a kernel process.
Further, note that we can consider K to be a deterministic distribution that gives mass to only a single G. In that case, K can be thought of as a deterministic function which must satisfy a corresponding consistency property,
G = K(K), G * = K(K * ), G σ = K(K σ ),(41)
and this is indeed satisfied by all deterministic transformations of kernels considered here. In practical terms, as long as G is always a valid kernel, it is sufficient for the elements of G i =j to depend only on K ij , K ii and K jj and for G ii to depend only on K jj , which is satisfied by e.g. the squared exponential kernel (Eq. 17) and by the ReLU kernel (Cho & Saul, 2009).
B. The first layer of our deep GP as Bayesian inference over a generalised lengthscale
In our deep GP architecture, we first sample F 1 ∈ R P ×N1 from a Gaussian with covariance K 0 = 1 N0 XX T (Eq. 5a). This might seem odd, as the usual deep GP involves passing the input, X ∈ R P ×N0 , directly to the kernel function. However, in the standard deep GP framework, the kernel (e.g. a squared exponential kernel) has lengthscale hyperparameters which can be inferred using Bayesian inference. In particular,
k param ( 1 √ N0 x i , 1 √ N0 x j ) = exp − 1 2N0 (x i − x j ) Ω (x i − x j ) T .(42)
where k param is a new squared exponential kernel that explicitly includes hyperparmeters Ω ∈ R N0×N0 , and where x i is the ith row of X. Typically, in deep GPs, the parameter, Ω, is diagonal, and the diagonal elements correspond to the inverse square of the lengthscale, l i , (i.e. Ω ii = 1/l 2 i ). However, in many cases it may be useful to have a non-diagonal scaling. For instance, we could use,
Ω ∼ W 1 N1 I, N 1 ,(43)
which corresponds to,
Ω = WW T , where W iλ ∼ N 0, 1 N1 , W ∈ R N0×N1 .(44)
Under our approach, we sample F = F 1 from Eq. (5b), so F can be written as,
F = XW, f i = x i W,(45)
where f i is the ith row of F. Putting this into a squared exponential kernel without a lengthscale parameter,
k( 1 √ N0 f i , 1 √ N0 f j ) = exp − 1 2N0 (f i − f j ) (f i − f j ) T , = exp − 1 2N0 (x i W − x j W) (x i W − x j W) T , = exp − 1 2N0 (x i − x j ) WW T (x i − x j ) T , = exp − 1 2N0 (x i − x j ) Ω (x i − x j ) T , = k param ( 1 √ N0 x i , 1 √ N0 x j ).(46)
We find that a parameter-free squared exponential kernel applied to F is equivalent to a squared-exponential kernel with generalised lengthscale hyperparameters applied to the input.
C. BNNs as deep kernel processes
Here we show that standard, finite BNNs, infinite BNNs and infinite BNNs with bottlenecks can be understood as deep kernel processes.
C.1. Stanard finite BNNs (and general DGPs)
Standard, finite BNNs are deep kernel processes, albeit ones which do not admit an analytic expression for the probability density. In particular, the prior for a standard Bayesian neural network (Fig. 3 top) is,
P (W ) = N λ=1 N w λ ; 0, I/N −1 , W ∈ R N −1 ×N , (47a) F = XW 1 for = 1, φ (F −1 ) W otherwise, F ∈ R P ×N ,(47b)
where w λ is the λth column of W . In the neural-network case, φ is a pointwise nonlinearity such as a ReLU. Integrating out the weights, the features, F , become Gaussian distributed, as they depend linearly on the Gaussian distributed weights, W ,
P (F |F −1 ) = N λ=1 N f λ ; 0, K = P (F |K ) ,(48)X W 1 F 1 W 2 F 2 W 3 F 3 W 4 F 4 Y X K 1 F 1 K 2 F 2 K 3 F 3 K 4 F 4 Y X K 1 K 2 K 3 K 4 F 4 Y
Layer 1 Layer 2 Layer 3 Output Layer Figure 3. A series of generative models for a standard, finite BNN. Top. The standard model, with features, F , and weights W (Eq. 47).
Middle. Integrating out the weights, the distribution over features becomes Gaussian (Eq. 50), and we explicitly introduce the kernel, K , as a latent variable. Bottom. Integrating out the activations, F , gives a deep kernel process, albeit one where the distributions P (K |K −1 ) cannot be written down analytically, but where the expectation, E [K |K −1 ] is known (Eq. 51).
where
K = 1 N −1 φ(F −1 )φ T (F −1 ).(49)
Crucially, F depends on the previous layer activities, F −1 only through the kernel, K . As such, we could write a generative model as (Fig. 3 middle),
K = 1 N0 XX T for = 1, 1 N −1 φ(F −1 )φ T (F −1 ) otherwise, (50a) P (F |K ) = N λ=1 N f λ ; 0, K ,(50b)
where we have explicitly included the kernel, K , as a latent variable. This form highlights that BNNs are deep GPs, in the sense that F λ are Gaussian, with a kernel that depends on the activations from the previous layer. Indeed note that any deep GP (i.e. including those with kernels that cannot be written as a function of the Gram matrix) as a kernel, K , is by definition a matrix that can be written as the outer product of a potentially infinite number of features, φ(F ) where we allow φ to be a much richer class of functions than the usual pointwise nonlinearities (Hofmann et al., 2008). We might now try to follow the approach we took above for deep GPs, and consider a Wishart-distributed Gram matrix, G = 1 N F F T . However, for BNNs we encounter an issue: we are not able to compute the kernel, K just using the Gram matrix, G : we need the full set of features, F . Instead, we need an alternative approach to show that a neural network is a deep kernel process. In particular, after integrating out the weights, the resulting distribution is chain-structured (Fig. 3 middle), so in principle we can integrate out F to obtain a distribution over K conditioned on K −1 , giving the DKP model in Fig. 3 (bottom),
P (K |K −1 ) = dF −1 δ D K − 1 N φ(F −1 )φ T (F −1 ) P (F −1 |K −1 ) ,(51)
where P (F −1 |K −1 ) is given by Eq. (50b) and δ D is the Dirac-delta function consisting of a point-mass at zero. Using this integral to write out the generative process only in terms of K gives the deep kernel process in Fig. 3 (bottom). While this distribution exists in principle, it cannot be evaluated analytically. But we can explicitly evaluate the expected value of K given K −1 using results from Cho & Saul (2009). In particular, we take Eq. 50a, write out the matrix-multiplication explicitly as a series of vector outer products, and note that as f λ is IID across , the empirical average is equal to the expectation of a single term, which is computed by Cho & Saul (2009),
E [K +1 |K ] = 1 N N λ=1 E φ(f λ )φ T (f λ )|K = E φ(f λ )φ T (f λ )|K , = df λ N f λ ; 0, K φ(f λ )φ T (f λ ) ≡ K(K ).(52)
Finally, we define this expectation to be K(K ) in the case of NNs.
C.2. Infinite NNs
We have found that for standard finite neural networks, we were not able to compute the distribution over K conditioned on K −1 (Eq. (51)). To resolve this issue, one approach is to consider the limit of an infinitely wide neural network. In this limit, the K becomes a deterministic function of K −1 , as K can be written as the average of N IID outer products, and as N grows to infinity, the law of large numbers tells us that the average becomes equal to its expectation,
lim N →∞ K +1 = lim N →∞ 1 N N λ=1 φ(f λ )φ T (f λ ) = E φ(f λ )φ T (f λ )|K = K(K ).(53)
C.3. Infinite NNs with bottlenecks
In infinite NNs, the kernel is deterministic, meaning that there is no flexibility/variability, and hence no capability for representation learning (Aitchison, 2019). Here, we consider infinite networks with bottlenecks that combine the tractability of infinite networks with the flexibility of finite networks (Aitchison, 2019). The trick is to separate flexible, finite linear "bottlenecks" from infinite-width nonlinearities. We keep the nonlinearity infinite in order to ensure that the output kernel is deterministic and can be computed using results from Cho & Saul (2009). In particular, we use finite-width F ∈ R P ×N and infinite width F ∈ R P ×M , (we send M to infinity while leaving N finite),
P (W ) = N λ=1 N w λ ; 0, I/M −1 M 0 = N 0 ,(54a)F = XW if = 1, φ(F −1 )W otherwise,(54b)P (M ) = M λ=1 N m λ ; 0, I/N , (54c) F = F M .(54d)
This generative process is given graphically in Fig. 4 (top).
Integrating over the expansion weights, M ∈ R N ×M , and the bottleneck weights, W ∈ R M −1 ×N , the generative model ( Fig. 4 second row) can be rewritten,
K = 1 N0 XX T for = 1, 1 M −1 φ F −1 φ T F −1 otherwise,(55a)P (F |K ) = N λ=1 N f λ ; 0, K ,(55b)G = 1 N F F T ,(55c)P (F |G ) = M λ=1 N f λ ; 0, G .(55d)
Remembering that K +1 is the empirical mean of M IID terms, as M → ∞ it converges on its expectation
lim M →∞ K +1 = lim M →∞ 1 M N λ=1 φ f λ φ T f λ = E φ(f λ )φ T (f λ )|G = K(G ).(56)
and we define the limit to be K(G ). Note if we use standard (e.g. ReLU) nonlinearities, we can use results from Cho & Saul (2009) to compute K(G ). Thus, we get the following generative process,
K = 1 N0 XX T for = 1, K(G −1 ) otherwise,(57a)P (G ) = W G ; 1 N K , N .(57b)
Finally, eliminating the deterministic kernels, K , from the model, we obtain exactly the deep GP generative model in Eq. 8 (Fig. C.3 fourth row).
D. Standard approximate posteriors over features and weights fail to capture symmetries
We have shown that it is possible to represent DGPs and a variety of NNs as deep kernel processes. Here, we argue that standard deep GP approximate posteriors are seriously flawed, and that working with deep kernel processes may alleviate these flaws. Last row. Eliminating all deterministic random variables, we get a model equivalent to that for DGPs (Fig. 1 bottom).
X W 1 F 1 M 1 F 1 W 2 F 2 M 2 F 2 W 3 F 3 Y X K 1 F 1 G 1 F 1 K 2 F 2 G 2 F 2 K 3 F 3 Y X K 1 G 1 K 2 G 2 K 3 F 3 Y X G 1 G 2 F 3 Y
Layer 1 Layer 2 Output Layer
In particular, we show that the true DGP posterior has rotational symmetries and that the true BNN posterior has permutation symmetries that are not captured by standard variational posteriors.
D.1. Permutation symmetries in DNNs posteriors over weights
Permutation symmetries in neural network posteriors were known in classical work on Bayesian neural networks (e.g. MacKay, 1992). Here, we spell out the argument in full. Taking P to be a permutation matrix (i.e. a unitary matrix with PP T = I with one 1 in every row and column), we have,
φ(F)P = φ(FP).(58)
i.e. permuting the input to a nonlinearity is equivalent to permuting its output. Expanding two steps of the recursion defined by Eq. (47b),
F = φ(φ(F −2 )W −1 )W ,(59)
multiplying by the identity,
F = φ(φ(F −2 )W −1 )PP T W ,(60)
where P ∈ R N −1 ×N −1 , applying Eq. (58)
F = φ(φ(F −2 )W −1 P)P T W ,(61)
defining permuted weights,
W −1 = W −1 P, W = P T W ,(62)
the output is the same under the original or permuted weights,
F = φ(φ(F −2 )W −1 )W = φ(φ(F −2 )W −1 )W .(63)
Introducing a different perturbation between every pair of layers we get a more general symmetry,
W 1 = W 1 P 1 ,(64a)W = P T −1 W P for ∈ {2, . . . , L}, (64b) W L+1 = P L W L+1 ,(64c)
where P ∈ R N −1 ×N −1 . As the output of the neural network is the same under any of these permutations the likelihoods for original and permuted weights are equal, P (Y|X, W 1 , . . . , W L+1 ) = P Y|X, W 1 , . . . , W L+1 ,
and as the prior over elements within a weight matrix is IID Gaussian (Eq. 47a), the prior probability density is equal under original and permuted weights,
P (W 1 , . . . , W L+1 ) = P W 1 , . . . , W L+1 .(66)
Thus, the joint probability is invariant to permutations,
P (Y|X, W 1 , . . . , W L+1 ) P (W 1 , . . . , W L+1 ) = P Y|X, W 1 , . . . , W L+1 P W 1 , . . . , W L+1 ,(67)
and applying Bayes theorem, the posterior is invariant to permutations,
P (W 1 , . . . , W L+1 |Y, X) = P W 1 , . . . , W L+1 |Y, X .(68)
Due in part to these permutation symmetries, the posterior distribution over weights is extremely complex and multimodal. Importantly, it is not possible to capture these symmetries using standard variational posteriors over weights, such as factorised posteriors, but it is not necessary to capture these symmetries if we work with Gram matrices and kernels, which are invariant to permutations (and other unitary transformations; Eq. 14).
D.2. Rotational symmetries in deep GP posteriors
To show that deep GP posteriors are invariant to unitary transformations, U ∈ R N ×N , where U U T = I, we define transformed features, F ,
F = F U .(69)
To evaluate P F |F −1 , we begin by substituting for F −1 ,
P F |F −1 = N λ=1 N f λ ; 0, K 1 N −1 F −1 F T −1 ,(70)= N λ=1 N f λ ; 0, K 1 N −1 F −1 U −1 U T −1 F T −1 ,(71)= N λ=1 N f λ ; 0, K 1 N −1 F −1 F T −1 ,(72)
= P (F |F −1 ) .
To evaluate P (F |F −1 ), we substitute for F in the explicit form for the multivariate Gaussian probability density,
P (F |F −1 ) = − 1 2 Tr F T K −1 −1 F + const,(74)= − 1 2 Tr K −1 −1 F F T + const,(75)= − 1 2 Tr K −1 −1 F U U T F T + const,(76)= − 1 2 Tr K −1 −1 F F T + const,(77)= P (F |F −1 ) .(78)
where
K −1 = K 1 N −1 F −1 F T −1
, and the constant depends only on F −1 . Combining these derivations, each of these conditionals is invariant to rotations of F and F −1 ,
P F |F −1 = P (F |F −1 ) = P (F |F −1 ) .(79)
The same argument can straightforwardly be extended to the inputs, P (F 1 |X),
P (F 1 |X) = P (F 1 |X) ,(80)
and to the final probability density, for output activations, F L+1 which is not invariant to permutations,
P (F L+1 |F L ) = P (F L+1 |F L ) ,(81)
Therefore, we have,
P (F 1 , . . . , F L , F L+1 , Y|X) = P (Y|F L+1 ) P (F L+1 |F L ) L =2 P F |F −1 P (F 1 |X) ,(82)= P (Y|F L+1 ) P (F L+1 |F L ) L =2 P (F |F −1 ) P (F 1 |X) ,(83)= P (F 1 , . . . , F L , F L+1 , Y|X) .(84)
Therefore, applying Bayes theorem the posterior is invariant to rotations,
P (F 1 , . . . , F L , F L+1 |X, Y) = P (F 1 , . . . , F L , F L+1 |X, Y) .(85)
Importantly, these posterior symmetries are not captured by standard variational posteriors with non-zero means (e.g.
Salimbeni & Deisenroth, 2017).
D.3. The true posterior over features in a DGP has zero mean
We can use symmetry to show that the posterior of F has zero mean. We begin by writing the expectation as an integral,
E [F |F −1 , F +1 ] = dF F P (F =F|F −1 , F +1 ) .(86)
Changing variables in the integral to F = −F, and noting that the absolute value of the Jacobian is 1, we have
= dF (−F ) P (F = (−F ) |F −1 , F +1 ) ,(87)
using the symmetry of the posterior,
= dF (−F ) P (F =F |F −1 , F +1 ) ,(88)= − E [F |F −1 , F +1 ] ,(89)
the expectation is equal to minus itself, so it must be zero
E [F |F −1 , F +1 ] = 0.(90)
E. Difficulties with VI in deep Wishart processes
The deep Wishart generative process is well-defined as long as we admit nonsingular Wishart distributions (Uhlig, 1994;Srivastava et al., 2003). The issue comes when we try to form a variational approximate posterior over low-rank positive definite matrices. This is typically the case because the number of datapoints, P is usually far larger than the number of features. In particular, the only convenient distribution over low-rank positive semidefinite matrices is the Wishart itself,
Q (G ) = W G ; 1 N Ψ, N .(91)
However, a key feature of most variational approximate posteriors is the ability to increase and decrease the variance, independent of other properties such as the mean, and in our case the rank of the matrix. For a Wishart, the mean and variance are given by,
E Q(G ) [G ] = Ψ,(92)V Q(G ) G ij = 1 N Ψ 2 ij + Ψ ii Ψ jj .(93)
Initially, this may look fine: we can increase or decrease the variance by changing N . However, remember that N is the degrees of freedom, which controls the rank of the matrix, G . As such, N is fixed by the prior: the prior and approximate posterior must define distributions over matrices of the same rank. And once N is fixed, we no longer have independent control over the variance.
To go about resolving this issue, we need to find a distribution over low-rank matrices with independent control of the mean and variance. The natural approach is to use a non-central Wishart, defined as the outer product of Gaussian-distributed vectors with non-zero means. While this distribution is easy to sample from and does give independent control over the rank, mean and variance, its probability density is prohibitively costly and complex to evaluate (Koev & Edelman, 2006).
F. Singular (inverse) Wishart processes at the input layer
In almost all cases of interest, our the kernel functions K(G) return full-rank matrices, so we can use standard (inverse) Wishart distributions, which assume that the input matrix is full-rank. However, this is not true at the input layer as K 0 = 1 N0 XX T will often be low-rank. This requires us to use singular (inverse) Wishart distributions which in general are difficult to work with (Uhlig, 1994;Srivastava et al., 2003;Bodnar & Okhrin, 2008;Bodnar et al., 2016). As such, instead we exploit knowledge of the input features to work with a smaller, full-rank matrix, Ω ∈ R N0×N0 , where, remember, N 0 is the number of input features in X. For a deep Wishart process,
1 N0 XΩX T = G 1 ∼ W 1 N1 K 0 , N 1 , where Ω ∼ W 1 N1 I, N 1 ,(94)
and for a deep inverse Wishart process,
1 N0 XΩX T = G 1 ∼ W −1 (δ 1 K 0 , δ 1 + P + 1) , where Ω ∼ W −1 (δ 1 I, δ 1 + N 0 + 1) .(95)
Now, we are able to use the full-rank matrix, Ω rather than the low-rank matrix, G 1 as the random variable for variational inference. For the approximate posterior over Ω, in a deep inverse Wishart process, we use
Q (Ω) = W −1 δ 1 I + V 1 V T 1 , δ 1 + γ 1 + (N 0 + 1) .(96)
Note in the usual case where there are fewer inducing points than input features, then the matrix K 0 will be full-rank, and we can work with G 1 as the random variable as usual.
G. Approximate posteriors over output features
To define approximate posteriors over inducing outputs, we are inspired by global inducing point methods (Ober & Aitchison, 2020). In particular, we take the approximate posterior to be the prior, multiplied by a "pseudo-likelihood",
Q (F L+1 |G L ) ∝ P (F L+1 |G L ) N L+1 λ=1 N v λ ; f L+1 λ , Λ −1 λ .(97)
This is valid both for global inducing inputs and (for small datasets) training inputs, and the key thing to remember is that in either case, for any given input (e.g. an MNIST handwritten 2), there is a desired output (e.g. the class-label "2"), and the top-layer global inducing outputs, v λ , express these desired outcomes. Substituting for the prior,
Q (F L+1 |G L ) ∝ N L+1 λ=1 N f L+1 λ ; 0, K(G L ) N v λ ; f L+1 λ , Λ −1 λ ,(98)
and computing this value gives the approximate posterior in the main text (Eq. 21).
H. Using eigenvalues to compare deep Wishart, deep residual Wishart and inverse Wishart priors
One might be concerned that the deep inverse Wishart processes in which we can easily perform inference are different to the deep Wishart processes corresponding to BNNs (Sec. C.1) and infinite NNs with bottlenecks (App. C.3). To address these concerns, we begin by noting that the (inverse) Wishart priors can be written in terms of samples from the standard (inverse) Wishart where K = LL T such that,
G = LΩL T , G = LΩ L T ,(99)Ω ∼ W 1 N I, N , Ω ∼ W −1 (N I, λN ) ,(100)G ∼ W 1 N K, N , G ∼ W −1 (N K, λN ) .(101)
Note that as the standard Wishart and inverse Wishart have uniform distributions over the eigenvectors (Shah et al., 2014), they differ only in the distribution over eigenvalues of Ω and Ω . We plotted the eigenvalue histogram for samples from a Wishart distribution with N = P = 2000 (Fig. 5 top left). This corresponds to an IID Gaussian prior over weights, with 2000 features in the input and output layers. Notably, there are many very small eigenvalues, which are undesirable as they eliminate information present in the input. To eliminate these very small eigenvalues, a common approach is to use a ResNet-inspired architecture (which is done even in the deep GP literature, e.g. Salimbeni & Deisenroth, 2017). To understand the eigenvalues in a residual layer, we define a ResW distribution by taking the outer product of a weight matrix with itself,
WW T = Ω ∼ ResW (N, α) ,(102)
where the weight matrix is IID Gaussian, plus the identity matrix, with the identity matrix weighted as α,
W = 1 √ 1+α 2 1 N ξ + αI , ξ i,λ ∼ N (0, 1) .(103)
With α = 1, there are still many very small eigenvalues, but these disappear as α increases. We compared these distributions to inverse Wishart distributions (Fig. 5 bottom) with varying degrees of freedom. For all degrees of freedom, we found that inverse Wishart distributions do not produce very small eigenvalues, which would eliminate information. As such, these eigenvalue distributions resemble those for ResW with α larger than 1.
I. Doubly stochastic variational inference in deep inverse Wishart processes
Due to the doubly stochastic results in Sec. 5.3, we only need to compute the conditional distribution over a single test/train point (we do not need the joint distribution over a number of test points). As such, we can decompose G and Ψ as, . Samples from a one-layer (top) and a two-layer (bottom) deep IW process prior (Eq. 18). On the far left, we have included a set of samples from a GP with the same kernel, for comparison. This GP is equivalent to sending δ0 → ∞ in the one-layer deep IW process and additionally sending δ1 → ∞ in the two-layer deep IW process. All of the deep IW process panels use the same squared-exponential kernel with bandwidth 1. and δ0 = δ1 = 0. For each panel, we draw a single sample of the top-layer Gram matrix, GL, then draw multiple GP-distributed functions, conditioned on that Gram matrix.
G = G ii g T it g it g tt , Ψ = Ψ ii ψ T it ψ it ψ tt ,(104)
where G ii , Ψ ii ∈ R Pi×Pi , g it ∈ R Pi×1 and ψ it ∈ R Pi×1 are column-vectors, and g tt and ψ tt are scalars. Taking the results in Eq. (34) to the univariate case,
g tt·i = g tt − g T it G ii −1 g it , ψ tt·i = ψ tt − ψ T it Ψ −1 ii ψ it .(105)
As g tt·i is univariate, its distribution becomes Inverse Gamma, g tt·i |G ii , G −1 ∼ InverseGamma α = 1 2 (δ + P t + P i + 1) , β = 1 2 ψ tt·i .
As g it is a vector rather than a matrix, its distribution becomes Gaussian,
G ii −1 g it |g tt·i , G ii , G −1 ∼ N Ψ −1 ii ψ it , g tt·i Ψ −1 ii .(107)
J. Samples from the 1D prior and approximate posterior First, we drew samples from a one-layer (top) and two-layer (bottom) deep inverse Wishart process, with a squaredexponential kernel (Fig. 6). We found considerable differences in the function family corresponding to different prior samples of the top-layer Gram matrix, G L (panels). While differences across function classes in a one-layer IW process can be understood as equivalent to doing inference over a prior on the lengthscale, this is not true of the two-layer process, and to emphasise this, the panels for two-layer samples all have the same first layer sample (equivalent to choosing a lengthscale), but different samples from the Gram matrix at the second layer. The two-layer deep IW process panels use the same, fixed input layer, so variability in the function class arises only from sampling G 2 .
Next, we exploited kernel flexibilities in IW processes by training a one-layer deep IW model with a fixed kernel bandwidth on data generated from various bandwidths. The first row in Figure 7 shows posterior samples from one-layer deep IW processes trained on different datasets. For each panel, we first sampled five full G 1 matrices using Eq.(34a) and (34b). Then for each G 1 , we use Gaussian conditioning to get a posterior distribution on testing locations and drew one sample from the posterior plotted as a single line. Remarkably, these posterior samples exhibited wiggling behaviours that were consistent with training data even outside the training range, which highlighted the additional kernel flexibility in IW processes. On the other hand, when model bandwidth was fixed, samples from vanilla GPs with fixed bandwidth in the second row displayed almost identical shapes outside the training range across different sets of training data. . The additional flexibility in a one-layer deep IW process can be used to capture mismatch in the kernel. We plot five posterior function samples from trained IW processes in the first row, and samples from trained GPs below. We generate different sets of training data from a GP with different kernel bandwidths (0.5, 1, 2, 5, 10) across columns, while we keep the kernel bandwidth in all models being 1.
K. Why we care about the ELBO
While we have shown that DIWP offers some benefits in predictive performance, it gives much more dramatic improvements in the ELBO. While we might think that predictive performance is the only goal, there are two reasons to believe that the ELBO itself is also an important metric. First, the ELBO is very closely related to PAC-Bayesian generalisation bounds (e.g. Germain et al., 2016). In particular, the bounds are generally written as the average training log-likelihood, plus the KL-divergence between the approximate posterior over parameters and the prior. This mirrors the standard form for the ELBO,
L = E Q(z)
[log P (x|z)] − D KL (Q (z) || P (z)) ,
where x is all the data (here, the inputs, X and outputs, Y), and z are all the latent variables. Remarkably, Germain et al. (e.g. 2016) present a bound on the test-log-likelihood that is exactly the ELBO per data point, up to additive constants. As such, in certain circumstances, optimizing the ELBO is equivalent to optimizing a PAC-Bayes bound on the test-log-likelihood. Similar results are available in Rivasplata et al. (2019). Second, we can write down an alternative form for the ELBO as the model evidence, minus the KL-divergence between the approximate and true posterior, L = log P (x) − D KL (Q (z) || P (z|x)) ≤ log P (x) .
As such, for a fixed generative model, and hence a fixed value of the model evidence, log P (x), the ELBO measures the closeness of the variational approximate posterior, Q (z) and the true posterior, P (z|x). As we are trying to perform Bayesian inference, our goal should be to make the approximate posterior as close as possible to the true posterior. If, for instance, we can set Q (z) to give better predictive performance, but be further from the true posterior, then that is fine in certain settings, but not when the goal is inference. Obviously, it is desirable for the true and approximate posterior to be as close as possible, which corresponds to larger values of L (indeed, when the approximate posterior equals the true posterior, the KL-divergence is zero, and L = log P (x) ).
L. Differences with Shah et al. (2014)
For a one-layer deep inverse Wishart process, using our definition in Eq. (18)
K 0 = 1 N0 XX T ,(110a)
P (G 1 |K 0 ) = W −1 (δ 1 K 0 , δ 1 + (P + 1)) , (110b) P (y λ |K 1 ) = N (y λ ; 0, K (G 1 )) .
(110c) Importantly, we do the nonlinear kernel transformation after sampling the inverse Wishart, so the inverse-Wishart sample acts as a generalised lengthscale hyperparameter (App. B), and hence dramatically changes the function family.
In contrast, for Shah et al. (2014), the nonlinear kernel is computed before, the inverse Wishart is sampled, and the inverse Wishart sample is used directly as the covariance for the Gaussian,
K 0 = K 1 N0 XX T ,(111a)
P (G 1 |K 0 ) = W −1 (δ 1 K 0 , δ 1 + (P + 1)) , (111b) P (y λ |K 1 ) = N (y λ ; 0, G 1 ) .
This difference in ordering, and in particular, the lack of a nonlinear kernel transformation between the inverse-Wishart and the output is why Shah et al. (2014) were able to find trivial results in their model (that it is equivalent to multiplying the covariance by a random scale).
Figure 1 .
1Generative models for two layer (L = 2) deep GPs. (Top) Generative model for a deep GP, with a kernel that depends on the Gram matrix, and with Gaussian-distributed features. (Bottom) Integrating out the features, the Gram matrices become Wishart distributed.
Figure 2 .
2Figure 2. Visualisations of a single prior sample of the kernels and Gram matrices as they pass through the network. We use 1D, equally spaced inputs with a squared exponential kernel. As we transition K(G −1 ) → G , we add "noise" by sampling from a Wishart (top) or an inverse Wishart (bottom). As we transition from G to K(G ), we deterministically transform the Gram matrix using a squared-exponential kernel.
Figure 4 .
4A series of generative models for an infinite network with bottlenecks. First row. The standard model. Second row. Integrating out the weights. Third row. Integrating out the features, the Gram matrices are Wishart-distributed, and the kernels are deterministic.
Figure 5 .
5Eigenvalue histograms for a single sample from the labelled distribution, with N = 2000.
Figure 6
6Figure 6. Samples from a one-layer (top) and a two-layer (bottom) deep IW process prior (Eq. 18). On the far left, we have included a set of samples from a GP with the same kernel, for comparison. This GP is equivalent to sending δ0 → ∞ in the one-layer deep IW process and additionally sending δ1 → ∞ in the two-layer deep IW process. All of the deep IW process panels use the same squared-exponential kernel with bandwidth 1. and δ0 = δ1 = 0. For each panel, we draw a single sample of the top-layer Gram matrix, GL, then draw multiple GP-distributed functions, conditioned on that Gram matrix.
Figure 7
7Figure 7. The additional flexibility in a one-layer deep IW process can be used to capture mismatch in the kernel. We plot five posterior function samples from trained IW processes in the first row, and samples from trained GPs below. We generate different sets of training data from a GP with different kernel bandwidths (0.5, 1, 2, 5, 10) across columns, while we keep the kernel bandwidth in all models being 1.
1 Department of Computer Science, Bristol, BS8 1UB, UK 2 Department of Engineering, Cambridge, CB2 1PZ, UK. Correspondence to: Laurence Aitchison <[email protected]>. Proceedings of the 38 th International Conference on Machine Learning, PMLR 139, 2021. Copyright 2021 by the author(s).
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| [] |
[
"3×3 transfer matrix modelling",
"3×3 transfer matrix modelling"
] | [
"Matteo Cherchi \nVTT -Technical Research Centre of Finland\n\n"
] | [
"VTT -Technical Research Centre of Finland\n"
] | [] | Unlike common devices based on ring resonators, the structure inFig. 1.a involves not only 2×2 couplers but also a 3×3 coupler, which means that a 3×3 transfer matrix approach is required to model the system. To the best of our knowledge, no such a model has been developed before. The only model available in the literature is based on a clever recursive 2×2 transfer matrix model[1], which requires lengthy calculations that depend on the chosen boundary conditions and on the particular geometry chosen. The scope of this document is instead to show how to generalize the standard 2×2 transfer matrix approach to cover any system with 3×3 couplers, and calculate the transfer matrix of any complicated system just as a product of simple 3×3 matrices.Fig. 1 (a) Schematic of the structure under study, (b) relevant section to be modelled, and (c) topologically equivalent circuit divided in 5 sections to build a suitable transfer matrix model.We start by noticing that the relevant part of the system to be modelled is simply the one inFig. 1(b), being the double input just a matter of boundary conditions. We then suitably divide the system in a topologically equivalent cascade of 3-arm sections, including two sections with 2×2 couplers and one uncoupled waveguide, two sections with three uncoupled arms, and one section where all three waveguides are coupled to each other, as shown inFig. 1.c. All sections are characterized by a forward propagating field and two backward propagating fields and . In particular we want to determine the transfer matrix linking the field on the right-hand side to the left-hand side of the system, such that (a) (b) (c) | null | [
"https://arxiv.org/pdf/1803.06208v2.pdf"
] | 246,430,590 | 1803.06208 | 86b99fc1dbb2eaf05b542a7ff60c0391718a1cc4 |
3×3 transfer matrix modelling
Matteo Cherchi
VTT -Technical Research Centre of Finland
3×3 transfer matrix modelling
1
Unlike common devices based on ring resonators, the structure inFig. 1.a involves not only 2×2 couplers but also a 3×3 coupler, which means that a 3×3 transfer matrix approach is required to model the system. To the best of our knowledge, no such a model has been developed before. The only model available in the literature is based on a clever recursive 2×2 transfer matrix model[1], which requires lengthy calculations that depend on the chosen boundary conditions and on the particular geometry chosen. The scope of this document is instead to show how to generalize the standard 2×2 transfer matrix approach to cover any system with 3×3 couplers, and calculate the transfer matrix of any complicated system just as a product of simple 3×3 matrices.Fig. 1 (a) Schematic of the structure under study, (b) relevant section to be modelled, and (c) topologically equivalent circuit divided in 5 sections to build a suitable transfer matrix model.We start by noticing that the relevant part of the system to be modelled is simply the one inFig. 1(b), being the double input just a matter of boundary conditions. We then suitably divide the system in a topologically equivalent cascade of 3-arm sections, including two sections with 2×2 couplers and one uncoupled waveguide, two sections with three uncoupled arms, and one section where all three waveguides are coupled to each other, as shown inFig. 1.c. All sections are characterized by a forward propagating field and two backward propagating fields and . In particular we want to determine the transfer matrix linking the field on the right-hand side to the left-hand side of the system, such that (a) (b) (c)
3×3 transfer matrix modelling
Matteo Cherchi, VTT -Technical Research Centre of Finland Unlike common devices based on ring resonators, the structure in Fig. 1.a involves not only 2×2 couplers but also a 3×3 coupler, which means that a 3×3 transfer matrix approach is required to model the system. To the best of our knowledge, no such a model has been developed before. The only model available in the literature is based on a clever recursive 2×2 transfer matrix model [1], which requires lengthy calculations that depend on the chosen boundary conditions and on the particular geometry chosen. The scope of this document is instead to show how to generalize the standard 2×2 transfer matrix approach to cover any system with 3×3 couplers, and calculate the transfer matrix of any complicated system just as a product of simple 3×3 matrices. We start by noticing that the relevant part of the system to be modelled is simply the one in Fig. 1(b), being the double input just a matter of boundary conditions. We then suitably divide the system in a topologically equivalent cascade of 3-arm sections, including two sections with 2×2 couplers and one uncoupled waveguide, two sections with three uncoupled arms, and one section where all three waveguides are coupled to each other, as shown in Fig. 1.c. All sections are characterized by a forward propagating field and two backward propagating fields and . In particular we want to determine the transfer matrix linking the field on the right-hand side to the left-hand side of the system, such that
(a) (b) (c) = tot ,(1)
where the vectors ( = , ) are defined as
= ( ).(2)
From the different sections in Fig. 1.c, we can see that the transfer matrix is in fact a product of five matrices:
tot = 2 2 1 1 ,(3)
where the matrices account for the 2×2 coupling, the propagate the uncoupled waveguides and the matrix models the 3×3 coupling.
Matrices
can be easily inferred from the 2×2 model, and can be written, without any loss of generality, as
= ( 0 0 0 1/ 0 0 0 1 ) ,(4)
where the phase change of can be arbitrarily set to zero, while ≡ /2 are the phases accumulated by the and fields propagating with propagation constant (that can have also non-negligible imaginary part, to account for propagation losses) in half the ring length .
Similarly the matrices can be written as
= 1 ( −1 * 0 − 1 0 0 0 ) ,(5)
where the element ( ) 33 = 1 imposes again zero phase change to , without any loss of generality (the asterisk stands for complex conjugate). The top-left 2×2 submatrix is nothing but the standard transfer matrix of a coupler [2][3][4] (or, equivalently, of a mirror) with field transmission and field reflection .
The derivation of matrix requires instead some more effort. It is actually instructive to recall how the transfer matrix of the 2×2 coupler is derived from the scattering matrix [5] in the 2×2 case, so to follow a similar approach for the 3×3 case. In the 2×2 case the scattering matrix = ( ) links input and output of the coupler (see Fig. 2.a) as follows:
= ,(6)
where input and output vectors ( = , ) are defined as = ( 1 2 ).
We want to derive the 2×2 transfer matrix linking left-and right-hand side of the coupler (Fig. 2.b) as follows
= ,(8)
where the vectors ( = , ) are defined as = ( ).
From the linear system of equations corresponding to Eq. (6), using the identities = 1 , = 1 , = 2 , and = 2 , it is straightforward to calculate as
where | | is the determinant of the matrix . Given the standard form of the scattering matrix :
= ( * ) ,(11)
and assuming a lossless coupler (| | = 1), leads to
= 1 ( −1 * − 1 ) .(12)
is also unitary, with | | = 1. For synchronous couplers:
(a) (b) (c) (d) { = cos( ) = sin( ) ,(13)
where is the coupling coefficient and is the effective length of the coupler. We can also extend these relations to asynchronous couplers. Assuming waveguides with propagation constant differing by , reflection and transmission coefficient can be written as follows [6]:
{ = cos( ) + sin( )cos ( ) = sin( ) sin ( ) ,(14)
where ≡ √ 2 + ( /2) 2 and cos ≡ − /(2 ) (which implies sin ≡ / ).
We want now to apply the same strategy to derive the 3×3 transfer matrix of the tri-coupler in Fig. 2.d from the 3×3 scattering matrix of Fig. 2.c. In this case the following identities hold:
= 1 , = 1 , = 2 , = 3 ,
where ≡ |( ) ≠ , ≠ | are the so called (i,j)minors of the matrix , i.e. the determinants of the submatrices obtained by eliminating the i-th row and the j-th column.
At this point, we just miss the scattering matrix of the coupler in Fig. 2.c, which we could not find anywhere in the literature. To calculate the matrix we follow the same approach used in reference [6], i.e. starting from the differential equations of the coupled mode theory in space [7,8], found in the existing literature on tricouplers [9,10]. We will focus on asynchronous couplers where the two external waveguides are identical, whereas the middle waveguide is different, being the difference of propagation constants. Therefore, assuming nearest neighbour coupling only and no losses, the system of differential equation is simply:
{ 1 = 2 + 2 1 2 = ( 1 + 3 ) − 2 2 3 = 2 + 2 3 ,(16)
and by defining the symmetric and anti-symmetric combinations ≡ 1
√2
( 1 + 3 ) and
≡ 1 √2 ( 1 − 3 )
they reduce two the 2×2 case, being already an eigenmode of the system:
{ 2 = √2 − 2 2 = √2 2 + 2 = 0 .(17)
Here denotes the coupling coefficient between two adjacent waveguides. If we introduce the input and output vectors Φ ( = , )
Φ = ( 2 ) ,(18)
the solution can be written as Φ = ̅ Φ [6], where ̅ is the scattering matrix in the rotated basis { 2 , , } ̅ ≡ ( 0 * 0 0 0 1 ) .
(
and we have defined the reflection and transmission coefficients
{ = cos( ) + sin( )cos ( ) = sin( ) sin ( ) ,(20)
being the effective length of the couplers, ≡ √2 2 + ( /2) 2 and cos ≡ − /(2 ) (which implies sin ≡ √2 / ). Synchronous couplers are just the limiting case for = 0, leading to = cos(√2 ) and = sin(√2 ).
Going back in the original basis { 1 , 2 , 3 } can be easily done through the matrix of basis change
= 1 √2 ( 1 0 1 0 √2 0 1 0 −1 ) ,(21)
which happens to be an involutory matrix, i.e. such that = −1 . The scattering matrix takes the form
= ̅ = 1 2 ( 1 + √2 −(1 − ) √2 2 * √2 −(1 − ) √2 1 + ) ,(22)
which is the unitary matrix describing the coupler of Fig. 2.c. From Eq. (15) we can finally write the transfer matrix for Fig. 2.d, that is the missing matrix from Fig. 1.c:
= 1 1 − * ( −2 1 + * − √2 −(1 + * ) 2 * − √2 √2 − √2 −(1 − )
) .
Noticeably, is unitary, with | | = − (1 − ) (1 − * ) ⁄ . This result completes the 3×3 transfer matrix model of the coupled ring resonators.
One last step is still necessary to impose the boundary conditions of Fig. 1.a and calculate the unknowns , , and , assuming input fields = 1 √2 ⁄ and = √2 ⁄ , where for the sake of generality we are assuming a relative phase difference between the two inputs. By explicitly writing the system of linear equations = tot , in terms of the matrix components tot = ( ) and of the fields components, it can be easily calculated:
where ≡ |( ) ≠ , ≠ | indicate once again the (i,j)minors of the matrix tot .
The same way it is possible to treat any other boundary conditions. For example the response of the system when light is launched in one input only can be easily calculated as: (25)
Fig. 1
1(a) Schematic of the structure under study, (b) relevant section to be modelled, and (c) topologically equivalent circuit divided in 5 sections to build a suitable transfer matrix model.
Fig. 2 (
2a) Scattering matrix model of a 2×2 coupler, (b) transfer matrix model for the same coupler, (c) scattering matrix model of a 3×3 coupler, and (d) transfer matrix model for the same coupler.
Wavelength-selective switching using double-ring resonators coupled by a three-waveguide directional coupler. L Zhou, R Soref, J Chen, Opt. Express. 23L. Zhou, R. Soref, and J. Chen, "Wavelength-selective switching using double-ring resonators coupled by a three-waveguide directional coupler," Opt. Express 23, 13488-13498 (2015).
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Bloch analysis of finite periodic microring chains. M Cherchi, Appl. Phys. B. 80M. Cherchi, "Bloch analysis of finite periodic microring chains," Appl. Phys. B 80, 109-113 (2004).
H A Haus, Waves and Fields in Optoelectronics. Prentice Hall, IncorporatedH. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Incorporated, 1984).
Wavelength-Flattened Directional Couplers: A Geometrical Approach. M Cherchi, Appl. Opt. 427141M. Cherchi, "Wavelength-Flattened Directional Couplers: A Geometrical Approach," Appl. Opt. 42, 7141 (2003).
Coupled-mode theory. H A Haus, W Huang, Proc. IEEE 79. IEEE 79H. A. Haus and W. Huang, "Coupled-mode theory," Proc. IEEE 79, 1505-1518 (1991).
Coupled-mode theory for optical waveguides: an overview. W.-P Huang, J. Opt. Soc. Am. A. 11W.-P. Huang, "Coupled-mode theory for optical waveguides: an overview," J. Opt. Soc. Am. A 11, 963- 983 (1994).
The coupling of modes in three dielectric slab waveguides. K Iwasaki, S Kurazono, K Itakura, Electron. Commun. Jpn. 58K. Iwasaki, S. Kurazono, and K. Itakura, "The coupling of modes in three dielectric slab waveguides," Electron. Commun. Jpn. 58, 100-108 (1975).
Three-waveguide couplers for improved sampling and filtering. H Haus, C Fonstad, IEEE J. Quantum Electron. 17H. Haus and C. Fonstad, "Three-waveguide couplers for improved sampling and filtering," IEEE J. Quantum Electron. 17, 2321-2325 (1981).
| [] |
[
"Families of Quantum Fingerprinting Protocols",
"Families of Quantum Fingerprinting Protocols"
] | [
"Benjamin Lovitz ",
"Norbert Lütkenhaus ",
"\nInstitute for Quantum Computing\nDepartment of Physics and Astronomy\nInstitute for Quantum Computing\nDepartment of Physics and Astronomy\nUniversity of Waterloo\n200 University Ave WN2L 3G1WaterlooONCanada\n",
"\n3G1, and Perimeter Institute for Theoretical Physics\nUniversity of Waterloo\n200 University Ave W, Waterloo, 31 Caroline St N, WaterlooN2L, N2L 2Y5ON, Canada, ON, Canada\n"
] | [
"Institute for Quantum Computing\nDepartment of Physics and Astronomy\nInstitute for Quantum Computing\nDepartment of Physics and Astronomy\nUniversity of Waterloo\n200 University Ave WN2L 3G1WaterlooONCanada",
"3G1, and Perimeter Institute for Theoretical Physics\nUniversity of Waterloo\n200 University Ave W, Waterloo, 31 Caroline St N, WaterlooN2L, N2L 2Y5ON, Canada, ON, Canada"
] | [] | We introduce several families of quantum fingerprinting protocols to evaluate the equality function on two n-bit strings in the simultaneous message passing model. The original quantum fingerprinting protocol uses a tensor product of a small number of O(log n)-qubit high dimensional signals [1], whereas a recently-proposed optical protocol uses a tensor product of O(n) single-qubit signals, while maintaining the O(log n) information leakage of the original protocol [2]. We find a family of protocols which interpolate between the original and optical protocols while maintaining the O(log n) information leakage, thus demonstrating a trade-off between the number of signals sent and the dimension of each signal.There has been interest in experimental realization of the recently-proposed optical protocol using coherent states[3,4], but as the required number of laser pulses grows linearly with the input size n, eventual challenges for the long-time stability of experimental set-ups arise. We find a coherent state protocol which reduces the number of signals by a factor 1/2 while also reducing the information leakage. Our reduction makes use of a simple modulation scheme in optical phase space, and we find that more complex modulation schemes are not advantageous. Using a similar technique, we improve a recently-proposed coherent state protocol for evaluating the Euclidean distance between two real unit vectors [5] by reducing the number of signals by a factor 1/2 and also reducing the information leakage. | 10.1103/physreva.97.032340 | [
"https://arxiv.org/pdf/1712.02895v1.pdf"
] | 73,589,990 | 1712.02895 | ecf98bb92d8ea92632f8f788a2138855d1443215 |
Families of Quantum Fingerprinting Protocols
Benjamin Lovitz
Norbert Lütkenhaus
Institute for Quantum Computing
Department of Physics and Astronomy
Institute for Quantum Computing
Department of Physics and Astronomy
University of Waterloo
200 University Ave WN2L 3G1WaterlooONCanada
3G1, and Perimeter Institute for Theoretical Physics
University of Waterloo
200 University Ave W, Waterloo, 31 Caroline St N, WaterlooN2L, N2L 2Y5ON, Canada, ON, Canada
Families of Quantum Fingerprinting Protocols
(Dated: December 11, 2017)
We introduce several families of quantum fingerprinting protocols to evaluate the equality function on two n-bit strings in the simultaneous message passing model. The original quantum fingerprinting protocol uses a tensor product of a small number of O(log n)-qubit high dimensional signals [1], whereas a recently-proposed optical protocol uses a tensor product of O(n) single-qubit signals, while maintaining the O(log n) information leakage of the original protocol [2]. We find a family of protocols which interpolate between the original and optical protocols while maintaining the O(log n) information leakage, thus demonstrating a trade-off between the number of signals sent and the dimension of each signal.There has been interest in experimental realization of the recently-proposed optical protocol using coherent states[3,4], but as the required number of laser pulses grows linearly with the input size n, eventual challenges for the long-time stability of experimental set-ups arise. We find a coherent state protocol which reduces the number of signals by a factor 1/2 while also reducing the information leakage. Our reduction makes use of a simple modulation scheme in optical phase space, and we find that more complex modulation schemes are not advantageous. Using a similar technique, we improve a recently-proposed coherent state protocol for evaluating the Euclidean distance between two real unit vectors [5] by reducing the number of signals by a factor 1/2 and also reducing the information leakage.
I. INTRODUCTION
In this work we introduce several families of equality protocols in the simultaneous message passing model. In this model, two parties (Alice and Bob) receive inputs x, y ∈ {0, 1} n respectively, conditioned on which they each send classical or quantum states to the referee, who performs a measurement to determine the output of some function f (x, y). We consider the case in which f is the equality function. We are interested in protocols which minimize the information leakage, that is, the amount of information the referee learns about the parties' inputs. Using classical states, the information leakage is lower bounded by Ω( √ n) [6]. In contrast, there exist protocols using quantum states with information leakage O(log n) [1,2].
The original quantum fingerprinting protocol uses O(log n)-qubit highly entangled signals and a controlledswap measurement [1]. A more recent and experimentally realizable "optical" protocol uses a tensor product of O(n) binary phase-modulated laser pulses, which can be represented as qubits, and a beamsplitter comparison measurement [2]. In this work, we find a family of protocols which interpolate between these two, exhibiting a trade-off between the number of signals sent and the dimension of each signal. We show that this family of protocols has information leakage O(log n).
The optical protocol of [2] has been implemented using co- * [email protected] † [email protected] herent states [3,4], but as the required number of laser pulses grows linearly with the input size n, large inputs become difficult to handle due to the limited long-time stability of experimental set-ups. We reduce the number of signals by a factor 1/2, while also reducing the information leakage, by utilizing the imaginary component of the phase space representation of coherent states. We find several natural generalizations of this protocol which further reduce the number of signals, but find numerical evidence that the information leakage of these protocols is higher in both the ideal and experimental settings, even under an abstract, optimal measurement performed by the referee.
Using a similar technique, we also reduce the number of signals and information leakage of a recently-proposed coherent state protocol for evaluating the Euclidean distance between two real unit vectors [5], and find a similar protocol for complex unit vectors.
In addition, in Appendix A we prove a tangentially related result that a simple beamsplitter measurement similar to that used in [7] achieves optimal unambiguous state comparison (USC) between any two coherent states of equal amplitude and opposite phase when the states are given with equal a priori probabilities. Optimal USC of two states given with equal a priori probabilities was first solved in [8] and generalized to arbitrary a priori probabilities in [9]. A method to realize the optimal USC of two single-photon states prepared with arbitrary a priori probabilities is proposed in [10], but to our knowledge optimal USC has not yet been experimentally realized. The beamsplitter scheme has the advantage of being experimentally realizable, with the drawback of being suboptimal for not-equal a priori probabilities.
The main text is organized as follows. In Section II we interpolate between the original and existing optical equality protocols. In Section III we introduce our coherent state protocols which improve on the existing optical equality and Euclidean distance protocols, and find numerical evidence that some natural generalizations to coherent state equality protocols using fewer signals have higher information leakage. In Section IV we derive the information leakage bounds we have used for our protocols.
II. INTERPOLATION
The original equality protocol of [1] uses a small number of O(log n)-qubit signals, whereas an existing optical equality protocol of [2] uses a tensor product of O(n) binary phasemodulated coherent state signals, which can be represented as single qubits (a notion formalized in Section II A). In this section we interpolate between the original and optical protocols, thus demonstrating a trade-off between the number of signals sent and the dimension of each signal. In Sections II A and II B we introduce slight adaptations to the existing protocols which are more natural candidates for the interpolation, and in Section II C we interpolate between these adaptations.
Before proceeding, we outline a general protocol framework which we will use for all equality protocols that we consider. First, Alice and Bob receive inputs x, y ∈ {0, 1} n respectively, conditioned on which they send pure states |ψ x , |ψ y to the referee which are sufficiently distinguishable when x = y. The referee then performs a comparison measurement on |ψ xy := |ψ x |ψ y and outputs either Equal or NotEqual. We define the error probability of the protocol as the worst case error probability over all x, y ∈ {0, 1} n . In the ideal setting, the error probability of every protocol is one-sided: if the inputs are equal the referee will always output Equal. In every protocol, the states |ψ x are product vectors. We refer to individual tensor factors of |ψ x as signals, and to the entire object |ψ x as a state. For many protocols that we consider, the states will contain multiple copies of identical signals.
To make the states sufficiently distinguishable to the referee when x = y, Alice and Bob map their inputs x, y to codewords E(x), E(y) ∈ {0, 1} m of an error-correcting code E characterized by some minimum distance. They then encode these codewords into states |ψ x , |ψ y whose overlap is a decreasing function of the distance between codewords. The code E is chosen to have constant minimum distance mδ and constant rate, which we will see ensures that the states used in each protocol are sufficiently distinguishable to the referee and give rise to information leakage O(log n).
A. Adaptation of existing optical equality protocol
Now we review the existing optical equality protocol of [2] (in the ideal setting) and propose a slight adaptation which is a more natural candidate for the interpolation. In the existing optical protocol, the j-th signal is one of two coherent states depending on the j-th codeletter of the codeword
E(x) ∈ {0, 1} m [2] |α x EQ,1 = m j=1 (−1) E(x) j α √ m j .(1)
For each index j, the referee interferes the j−th pair of signals received from Alice and Bob in a beamsplitter, and measures the dark port with a single photon threshold detector, obtaining one of two outcomes: "dark port detection" or "no dark port detection". The referee outputs NotEqual if at least one outcome "dark port detection" occurs. On input |β a |β b , outcome "no dark port detection" occurs with probability
| β a , β b | = e − 1 2 |β a −β b | 2 .(2)
It follows that the error probability given different inputs x = y is equal to | α x , α y |, and the error probability given equal inputs is zero. The worst case error probability occurs when the codewords differ by minimum distance mδ bits, and is equal to exp[−2 |α| 2 δ ], which is brought to within any ε > 0 through appropriate choice of α.
In the existing optical protocol, the set of two possible coherent states for each signal span a two-dimensional space, and thus can be written in a basis as (i.e. they are isometrically equivalent to) two qubits |q ε 0 , |q ε 1 with inner product determined by ε. The beamsplitter measurement can also be converted to this basis (see Appendix A). By the invariance of entropy under isometries, the information leakage (defined in Section IV) is equal for protocols using states that are equal up to change of basis.
In contrast to the existing optical protocol, the original protocol of [1] attains the desired error probability ε by sending multiple copies of signals which are fixed independent of ε. To facilitate our interpolation between these two protocols, we adapt the optical protocol to more closely resemble the original by fixing qubits |q 0 , |q 1 independent of ε, and sending identical copies of each qubit signal until ε is attained (we refer to each individual copy as a signal). In the coherent state basis, this corresponds to fixing the amplitude of each signal independent of ε (any two qubits can be written in a basis as coherent states). Define the repetition number r as the number of copies sent to attain ε. The referee uses the qubit-basis beamsplitter measurement described in Appendix A on each signal, and outputs NotEqual if any outcome "dark port detection" occurs.
Remarkably, as the probability (2) of "no dark port detection" is given by the overlap between the input pair of signals, it follows that if ε is attained with equality in both the existing optical protocol and our adapted protocol, then the states used in each protocol are equal up to a change of basis. Specifically, the set of signals {| α √ m , | −α √ m } used in the existing optical protocol are equal up to a change of basis to the set of unit vectors {|q 0 ⊗r , |q 1 ⊗r } containing the r copies of each signal used in our adapted protocol. Indeed, the worst case error probability is given by
ε = | q 0 , q 1 | mδ r = | −α √ m , α √ m | mδ , so | q 0 , q 1 | r = | −α √ m , α
√ m |, which completes the proof by Property 3 of [11] that two sets of unit vectors {|v a ∈ H v } a∈Z , {|w a ∈ H w } a∈Z are equal up to change of basis if and only if there exist real numbers θ a , a ∈ Z such that v a , v b = e i(θ a −θ b ) w a , w b for all a, b ∈ Z. Note that the choice of coherent state amplitude α √ m for the optical protocol thus corresponds to the choice α √ rm for our adapted optical protocol in the coherent state basis.
B. Adaptation of original equality protocol
Here we describe the original quantum fingerprinting protocol of [1], and propose a slight adaptation which is a more natural candidate for the interpolation. Similarly to the adapted optical protocol, in the original protocol Alice and Bob send identical copies of the signals
|ψ (m) x = 1 √ m m ∑ i=1 |i |E(x) i(3)
until the desired error probability ε is attained. On each pair of signals |ψ (m) xy
:= |ψ (m) x |ψ (m) y
sent by Alice and Bob, the referee performs a controlled-swap measurement (effectively a projective measurement onto the symmetric and anti-symmetric subspaces), which returns outcome "antisymmetric" with probability 1
2 (1 − W)|ψ (m) xy 2 ,
where W is the operator which swaps the state of Alice and Bob's systems. The referee outputs NotEqual if any outcome "antisymmetric" occurs. The worst case error probability occurs when the codewords differ by minimum distance mδ bits, and is given by (1 − δ (1 − δ 2 )) r for repetition number r [1]. In our adapted original protocol, the signals used are the same but the referee instead performs a direct sum of the controlled-swap measurement with the qubit-basis beamsplitter measurement. We use this measurement for the interpolation because on one end, in the adapted optical protocol, it converges to the beamsplitter measurement (see Section II C); and on the other end, in the adapted original protocol, it closely resembles (and slightly improves on) the controlledswap measurement.
Operationally, the measurement in the adapted original protocol proceeds as follows. First, the referee performs a nondestructive measurement on |ψ (m) xy with two measurement operators, the first (second) of which projects onto the subspace containing the first (second) term of the decomposition
|ψ (m) xy = 1 m m ∑ i=1 |i |E(x) i |i |E(y) i + 1 m m ∑ l,h=1 l =h |l |E(x) l |h |E(y) h .(4)
If the non-destructive measurement projects onto the first subspace, the referee performs a measurement which applies the identity matrix to the first and third registers, and to the twoqubit space contained in the second and fourth registers it applies the measurement operators M d and M nd , implicitly defined in Appendix A, which correspond to the measurement outcomes "dark port detection" and "no dark port detection", respectively. If the measurement projects onto the second subspace, the referee performs the controlled-swap measurement on the entire state. The referee outputs NotEqual if any outcome "dark port detection" or "anti-symmetric" occur. In Appendix B 1 we show that this protocol has worst case error probability
Pr I m (Err) = 1 − δ 1 − δ 2 + 1 2m r ,(5)
a minor improvement over the original protocol.
In Appendix A we show that the qubit-basis beamsplitter measurement on the two-qubit space contained in the second and fourth registers can be further decomposed as a direct sum of the controlled-swap measurement with an unambiguous state discrimination measurement. Thus, the full adapted measurement on the larger Hilbert space can also be decomposed in this way.
C. Interpolation between adapted protocols
Now we find a family of protocols which interpolates between the adapted protocols described in Sections II A and II B, thus demonstrating a trade-off between the number of signals sent and the dimension of each signal. In the interpolation protocol with block size k, blocks of k bits of E(x) are encoded into each signal:
|ψ (k) x = m/k j=1 1 √ k ∑ i∈I[ j,k] |i q (k) E(x) i ∈ (C k ⊗ C 2 ) ⊗ m k ,(6)
where |q
|ψ (1) x = m j=1 | j q (1) E(x) j ,(7)
which reproduces Eq. (1) written in the qubit basis and appended with basis vectors | j . For each index j = 1, . . . , m/k , the referee measures the jth pair of signals received from Alice and Bob as follows. As in the adapted original protocol, they perform a direct sum of the qubit-basis beamsplitter measurement and the controlledswap measurement on the first and second terms of the decomposition 1
k ∑ i∈I[ j,k] |i q (k) E(x) i |i q (k) E(y) i + 1 k ∑ l,h∈I[ j,k] l =h |l q (k) E(x) l |h q (k) E(y) h ,(8)
and decide NotEqual on any outcome "dark port detection" or "anti-symmetric". For k = m this measurement converges to the measurement used in the adapted original protocol, and for k = 1 the second term of the decomposition (8) disappears and this measurement converges to the qubit-basis beamsplitter measurement. The worst case error probability of this measurement is derived in Appendix B 1.
As noted in the previous section, this measurement can equivalently be decomposed as a direct sum of the controlledswap measurement with an unambiguous state discrimination measurement.
As shown in Eq. (6), each signal contains k qubit states, each chosen from the set {|q
(k) 0 , |q (k)
1 }. We will choose these qubits to satisfy q (k) 0 , q (k) 1 = 1 − k/m, which converges to the adapted orginal and optical protocols for k = m and k = 1 respectively, and which has information leakage O(log n) (see Section IV A). For a given block size k and repetition number r, our interpolation uses r m/k signals, each of dimension 2k, exhibiting a trade-off between the number of signals sent and the dimension of each signal.
III. OPTICAL PROTOCOLS
In this section we consider several families of optical coherent state simultaneous message passing model protocols which reduce the number of signals below that of the existing protocols. In Section III A we introduce optical protocols for equality and Euclidean distance which reduce the number of signals by a factor 1/2 and reduce the information leakage below that of the existing optical protocols of [2] and [5]. In Section III B we introduce two families of optical equality protocols which further reduce the number of signals, but find numerical evidence that they increase the information leakage in both the ideal and experimental settings, even under an abstract, optimal measurement performed by the referee.
A. Improved optical protocols for equality and Euclidean distance
In our improved optical equality protocol, two bits of E(x) are encoded into each signal by utilizing the imaginary component of the phase space representation of coherent states, which reduces the number of signals by a factor 1/2. Codeletters 01/10 are encoded into phases ±i, and codeletters 00/11 are encoded into phases ±1, as shown in Figure 1. Explicitly, the parties send the states
|α x EQ,2 = m j=1 odd (−1) E(x) j · (i) E(x) j ⊕E(x) j+1 α m/2 j .(9)
The referee uses the same beamsplitter measurement as in the existing optical protocol: she interferes pairs of signals in a beamsplitter, measures the dark port with a single photon threshold detector, and decides NotEqual if at least one outcome "dark port detection" occurs. As before, the desired error probability ε is attained through appropriate choice of α.
The above states have the same total mean photon number |α| 2 , and give rise to the same error probability, as the existing optical protocol. To show the second statement, recall that the probability of "no dark port detection" depends only on the squared distance between the amplitudes of the incoming pair of coherent state signals (2). For pairs of codeletters (E(x) j , E(x) j+1 ) and (E(y) j , E(y) j+1 ) which differ by one bit this quantity is given by w 2 = |α| 2 /m as in the existing optical protocol (see Figure 1), and for pairs which differ by two bits this quantity is 2w 2 , which gives rise to the same probability of "no dark port detection" as the existing optical protocol (see Appendix B 2). By the information leakage bound ∼ O(|α| 2 log m k ) (where m k is the number of signals) derived in Appendix C, our improved protocol has lower information leakage than the existing protocol. It can be shown that this statement also holds under the stronger bound derived in Section IV B for |α| 2 m k using standard approximation techniques. Below we will refer to this protocol, including its use of the beamsplitter measurement, as the two-bit protocol.
Re[β ]
Im[β ]β 1 = α/ √ m; for k = 2, β 2 = α/ m/2; and in general, β k = µ k /(m/k),
where µ k is the total mean photon number of the total state.
We improve the existing optical Euclidean distance protocol of [5] in similar fashion. In the existing protocol, Alice and Bob receive real unit vectors u, v ∈ R s respectively and prepare the states
|α u ED,1 := s j=1 |u j α j .(10)
The referee interferes each pair of signals received from Alice and Bob in a beamsplitter and measures both output ports with single photon threshold detectors. The quantity u − v 2 is a function of |α| 2 and the expected difference between the number of detections observed in the two output ports, so using Chernoff bounds the referee can estimate u − v 2 to within an additive constant ε with probability at least 1 − δ by repeating the protocol O(log(1/δ )/ε 2 ) times [5]. As in our improved equality protocol, our improved Euclidean distance protocol utilizes the imaginary component of the phase space representation of coherent states to reduce the number of signals by a factor 1/2. Alice and Bob prepare the states
|α u ED,2 := s j=1 odd |(u j + iu j+1 )α j ,(11)
and the referee uses the same measurement as before. These states have the same total mean photon number |α| 2 , and it can be shown using nearly identical analysis to [5] that the measurement outcome statistics are also the same. Thus, as before, our protocol has lower information leakage than the existing protocol under the information leakage bound of Appendix C. Alternatively, one can adapt the existing protocol to evaluate the Euclidean distance between two complex unit vectors u, v ∈ C s using the same measurement and the states (10).
B. Families of optical equality protocols
In this section we introduce two families of optical coherent state equality protocols which further reduce the number of signals. In Section III B 1 we find numerical evidence that these protocols have higher information leakage than the twobit protocol in the ideal setting, even under the optimal measurement. In Section III B 2 we find numerical evidence of the same behaviour under realistic experimental imperfections.
Ideal setting
We first describe our families of optical equality protocols in the ideal setting. In the ring (lattice) protocol with block size k, blocks of k bits of E(x) are encoded into one of 2 k coherent state signals arranged in a ring (lattice) in phase space using a Gray code, as shown in Figure 1. The size of the ring (lattice) is determined by the desired error probability ε, and is held constant for all signals.
The ring (lattice) Gray code is a mapping from k-bit strings to a ring (lattice) of coherent state signals which satisfies the property that all nearest neighbour signals differ in exactly one bit [12][13][14]. Note that in the ring encoding, each coherent state signal has two nearest neighbours, while in the lattice encoding a given coherent state signal can have as many as four nearest neighbours. We have chosen this code so that a greater number of bit differences between two k-bit blocks of E(x) and E(y) corresponds to greater distinguishability between the two coherent state signals. In Appendix B 2 we prove that this is the optimal encoding of k-bit blocks of binary codewords for all k = 2, 3, 4, and that it outperforms an analogous encoding of q−ary codewords. We consider two different measurements performed by the referee. The beamsplitter measurement proceeds identically to that of Section II A: the referee interferes pairs of signals in a beamsplitter and measures the dark port with a single photon threshold detector. She decides NotEqual if at least one outcome "dark port detection" occurs. Recall that the error probability under the beamsplitter measurement is one-sided, i.e. there is zero error for equal inputs. We also consider the optimal one-sided error measurement, which is described in Appendix B 3, and is shown to have error probability lower bounded by the square of the error probability of the beamsplitter measurement.
In Figure 2 we plot the information leakage of the ring encoding under the bound of Section IV B, compared to the classical information leakage lower bound of [6]. We have optimized over δ and assumed the code E saturates the Gilbert-Varshamov bound [15][16][17]
n m = 1 − h(δ ).(12)
For k = 1, 2, 3 we plot the information leakage under the beamsplitter measurement, and for k = 4, 5, 6 we lower bound the information leakage under the optimal measurement using the quadratic bound on the error probability derived in Appendix B 3. We see that the one-bit and two-bit protocols have the lowest information leakage (they are numerically indistinguishable in this parameter regime). We have observed the same result for ε in the range [10 −5 , 10 −2 ] in both the ring and lattice protocols. Before continuing on to consider experimental imperfections, we argue that for fixed block size k, both the ring and lattice protocols have asymptotic information leakage O(log n) in the ideal setting. In Appendix C we show that any simultaneous message passing model protocol which uses m k coherent state signals and fixed maximum total mean photon number µ max,k has information leakage ∼ O(µ max,k log m k ). Note that m k = m/k = O(n) by the fact that E is a constant rate code. Furthermore, in Appendix B 2 we show that any fixed error probability is attained with µ max,k constant in n.
Together, these results imply that the information leakage is O(log n).
Experimental imperfections
In the experimental setting, for larger block sizes the states have fewer signals, so dark counts have less effect on the error probability. In this section we find numerical evidence that despite this effect, the two-bit protocol still outperforms the ring protocols with block size k > 2 in the experimental setting, and speculate that the same result holds for the lattice protocols.
The experimental ring protocol uses the same states and beamsplitter measurement as in the ideal setting. However, to account for transmittivity η the initial total mean photon number µ k is rescaled to µ k /η, and to account for dark count probability p dark per signal the referee uses a different criteria to decide Equal or NotEqual. The referee decides NotEqual if the number of outcomes "dark port detection" exceeds a threshold value T k which is chosen to minimize the worst case error probability over all inputs x, y ∈ {0, 1} n (which is no longer one-sided when p dark > 0). This technique is based on that introduced in [3] for experimental implementation of the existing optical protocol. Now we determine the optimal threshold value T k . Define a random variable D E,k for the number of outcomes "dark port detection" given equal inputs. Define D D,k identically, but for different inputs which have the lowest expected number of outcomes "dark port detection". For a given threshold value T k , the worst case error probability is then given by max{Pr(D E,k ≥ T k ), Pr(D D,k < T k )}. As the first (second) element is monotonically decreasing (increasing) with T k , it follows that the optimal threshold value T k satsifies
Pr(D E,k ≥ T k ) = Pr(D D,k < T k ),(13)
which is also the worst case error probability of the protocol under this choice. Note that Eq. (13) may not attain exact equality due to the fact the threshold value must be an integer, so both probabilities are step functions. In our considered parameter regime, it can be shown that the number of clicks are well-approximated by binomial dis-
tributions D D,k ∼ Bin(m/k, p D,k ) and D E,k ∼ Bin(m/k, p E,k ), where p D,k = (1 − (kδ − kδ )) 1 − e −|β k | 2 1−cos 2π 2 k kδ + (kδ − kδ ) 1 − e −|β k | 2 1−cos 2π 2 k ( kδ +1) + p dark p E,k = p dark(14)
for all k = 1, . . . , 6 under the Gray code (see Appendix B 2), where β k = µ k /(m/k) is the amplitude of each coherent state signal. We have used this approximation to calculate T k and the corresponding worst case error probability (13).
In Figure 3 we plot the information leakage of the ring protocols under the bound of Section IV B for realistic experimental imperfections, compared to the classical information leakage lower bound of [6]. For given values of m, k, δ , and p dark we have chosen the signal amplitude β k to attain the desired error probability ε under the approximation (14). As before, we have optimized over δ and assumed the code E saturates the Gilbert-Varshamov bound (12). This plot uses worst case error probability ε = 0.01, transmittivity η = 0.3, and dark count probability per signal p dark = 7.3 × 10 −11 . We include a plot of of the existing optical protocol with interferometric visibility 99% for reference. We observe the same hierarchy as the ideal setting, but as dark counts have less effect for protocols sending fewer signals, the k = 2 (two-bit) protocol now has visibly lower information leakage than the k = 1 protocol. We have also observed the same hierarchy for p dark in the range [0, 10 −9 ] and ε in the range [10 −5 , 10 −2 ]. We speculate that the same behaviour holds for the lattice protocol under experimental imperfections.
IV. INFORMATION LEAKAGE
In this section we bound the information leakage of the protocols we have considered. The quantum information leakage of a simultaneous message passing model protocol Π in which, conditioned on input x, Alice (Bob) sends state σ x to the referee, is given by [18] QIL(Π) = max
P∈Pr(X×Y ) I(XY : AB) ρ ,(15)
where
ρ = ∑ x,y∈{0,1} n P(x, y)|xy xy| XY ⊗ σ A x ⊗ σ B y .
For every protocol we consider, σ x = |ψ x ψ x | is pure, which implies H(AB|XY ) = 0, so Eq. (15) reduces to
QIL(Π) = max P∈Pr(X×Y ) H(AB) ρ .(16)
To bound the information leakage, we will find an orthonormal basis for the Hilbert space used in each protocol such that the expectation values of |ψ x ψ x | w.r.t. this basis form a fixed probability vector Λ for all x ∈ {0, 1} n . The Schur-Horn theorem [19,20] then implies ρ AB Λ ⊗2 , so H(AB) ≤ H(Λ ⊗2 ) (see, e.g. [21]), which implies QIL(Π) ≤ 2H(Λ) (17) by the additivity of entropy under tensor product. While not all simultaneous message passing model protocols will have such a basis, we do find such a basis for both the interpolation and optical ring protocol families, and use this technique to bound their information leakage as O(log n). In Appendix C we bound the information leakage of the optical lattice protocol family as O(log n) (for which we were unable to find such a basis). Our Appendix C bound also holds for the optical ring protocols, but we have found numerically that our bound of this section is 15 − 40% lower in the considered parameter regime. This follows intuitively from the fact that the bound in this section takes into account the particular form of the states used, whereas the Appendix C bound simply projects onto a neighbourhood of the total mean photon number.
A. Information leakage for family of interpolation protocols
Here we bound the information leakage for the family of interpolation protocols under the choice of qubit overlap q As a corollary, for block size k being given by any function of n such that k(n) ∈ [m] for each n, the family of protocols which uses the interpolation family with block size k(n) for each n has information leakage O(log n). For example, k could be held to a fixed constant as in the existing optical protocol, or the ratio k/m could be held fixed as in the original protocol.
Proof. We prove that the information leakage is upper bounded by C r log n for some C > 0. In Appendix B 1 we show that for fixed error probability ε, the repetition number r is fixed, completing the proof.
Note that the states
|φ (k) x = m/k j=1 1 − p k |00 + p k 2k ∑ i∈I[ j,k] |i |0 +(−1) E(x) i |1 for p k = 1 − | q (k) 0 , q(k)
1 | reproduce the overlap structure of the states |ψ (k)
x , so by Property 3 of [11] (reviewed in Section II A) they are equal up to a change of basis. For all x ∈ {0, 1} n , the diagonal entries of |φ
(k) x φ (k)
x | are given by
Λ ⊗ m/k I,k = 1 − p k , p k 2k , . . . , p k 2k ⊗ m/k .(18)
Thus, by Eq. (17) the information leakage of the interpolation protocol with block size k under the choice p k = k/m is upper bounded by
QIL(Π I k ) ≤ 2 m/k rH(Λ I,k ) = 2 m/k r (1 − k/m) log 1 1 − k/m + (k/m) log (2m) ≤ 2r(2 + (1 + k/m) log(2m)).(19)
As E is a constant rate code, then m is linear in n, so this quantity is upper bounded by C r log n for some fixed constant C > 0.
B. Information leakage for family of optical ring protocols
Here we bound the information leakage for the family of optical ring protocols described in Section III B 1. We use a similar technique as in the information leakage bound for the family of interpolation protocols, and write the states in a basis for which the diagonal entries form a fixed probability vector Λ.
For block size k and total mean photon number µ k , each signal consists of one of 2 k coherent states equally spaced on a ring with amplitude β k = µ k /(m/k). Formally, each signal is contained in the set
S k = ω j β k ω j β k , j = 0, . . . , 2 k − 1 ,(20)
where ω = exp 2πi 2 k .
As ω jl = ω jn for all n ≡ l mod 2 k , then the states in S k are equal to
ω j β k = e −|β k | 2 2 2 k −1 ∑ l=0 ω jl ∑ h≡l mod 2 k β h k √ h! |h(22)
for j = 0, . . . , 2 k − 1, which can be written in a basis as
2 k −1 ∑ l=0 ω jl λ l |l(23)
for
λ l = e −|β k | 2 2 ∑ h≡l mod 2 k |β k | 2h h! .(24)
In this basis, the diagonal entries of each state in S k form the probability vector Λ O,k = (λ 2 0 , . . . , λ 2 2 k −1 ).
Writing each signal of the states used in the optical ring protocol in this basis, for all x ∈ {0, 1} n the diagonal entries of the transformed states are given by Λ ⊗m/k O,k . By Eq. (17) and additivity of entropy under tensor product, the information leakage of the optical ring protocol Π O k with block size k is upper bounded by
QIL(Π O k ) ≤ 2m k H(Λ O,k ).(25)
This bound is straightforward to calculate in practice, and was used to produce Figures 2 and 3. We have found numerically that this bound gives an advantage of 15-40% over the bound of Appendix C in our considered parameter regime. This bound can also be used to prove the asymptotic information leakage O(log n) for any fixed block size k using standard techniques.
V. CONCLUSION
In this work we developed several families of quantum fingerprinting protocols. One family demonstrates a trade-off between the number of signals sent and the dimension of each signal. The other families use coherent state signals arranged in a ring and lattice in phase space, thus catering to optical implementations. We found that one such coherent state protocol reduced the number of signals from the existing coherent state protocol by a factor 1/2, while also reducing the information leakage. Although the other protocols in the ring and lattice families use even fewer signals, we found convincing numerical evidence that they have higher information leakage under the bounds we have used. It would be interesting to investigate whether some other family of coherent state protocols might further reduce the number of signals while maintaining or reducing the information leakage, and whether our information leakage bounds could be improved.
We also proved that a simple beampslitter measurement similar to that used in [7] performs optimal unambiguous state comparison between two coherent states given with equal a priori probabilities. It would be interesting to explore whether a similar measurement could be used to perform optimal unambiguous state comparison of coherent states given with arbitrary a priori probabilities.
VI. ACKNOWLEDGEMENTS
We thank Konrad Banaszek for his insight on the q−ary encoding described in Appendix B 2, which helped us improve the presentation of these results. We thank Dave Touchette for his insight on the information leakage bound of Appendix C 2. We thank Juan Miguel Arrazola, Richard Cleve, David Lovitz, and John Watrous for helpful discussions. This research was supported in part by NSERC, Industry Canada and ARL CDQI program. The Institute for Quantum Computing and the Perimeter Institute are supported in part by the Government of Canada and the Province of Ontario.
VII. APPENDIX Appendix A: Beamsplitter measurement in qubit basis, and optimal unambiguous state comparison with a beamsplitter
Here we explicitly write any set of two coherent states {|β 0 , |β 1 } in a basis as qubits, and construct a measurement in this basis (based on a measurement introduced in [8]) which reproduces the outcome probabilities of the beamsplitter measurement described in Section II A. Using a result of [8] we then find that a beamsplitter measurement with single photon threshold detectors placed at both output ports performs optimal unambiguous state comparison on {|β , |−β } for any complex number β when the states are given with equal a priori probabilities.
We begin by writing {|β 0 , |β 1 } in a basis as qubits. Let β = 1 2 (β 0 − β 1 ) and
|q a = e −|β | 2 2 cos h(|β | 2 ) |0 +(−1) a sin h(|β | 2 ) |1 = 1 − p |0 +(−1) a √ p |1 (A1) for p = exp[− |β | 2 ] sinh(|β | 2 )
. It is straightforward to verify that | q 0 , q 1 | = | β 0 , β 1 |, so by Property 3 of [11] (reviewed in Section II A), the sets {|q 0 , |q 1 } and {|β 0 , |β 1 } are equal up to a change of basis. Now we review a measurement (introduced in [8]) in the qubit basis which reproduces the outcome probabilities of the beamsplitter measurement described in Section II A, i.e. on input |q a |q b for a, b ∈ {0, 1}, it outputs "different" (previously, "dark port detection") with probability 1 − | q a , q b |, and fails to output "different" with probability | q a , q b |.
The measurement is a direct sum of an unambiguous state discrimination measurement and a controlled-swap measurement on the first and second terms of the decomposition
|q a |q b =(1 − p) |00 +(−1) a+b p |11 + p(1 − p)[(−1) b |01 +(−1) a |10 ]. (A2)
Operationally, a non-destructive measurement is first performed with two measurement operators, the first (second) of which projects onto the subspace containing the first (second) term of the decomposition (A2). If it projects onto the first term, the remaining state takes one of two forms depending on the equality of a and b. In this case, an unambiguous state discrimination (USD) measurement is performed to distinguish between these two states. In particular, the USD measurement is performed which is optimal for the case in which each state is given with equal a priori probabilities [22]. Of course, this choice is sub-optimal for different a priori probabilities, but we make it because it gives rise to a measurement which reproduces the outcome probabilities of the beamsplitter measurement. We refer to the two unambiguous outcomes of this measurement as "plus" and "minus" corresponding to the two possible states of the first term of Eq. (A2), and the inconclusive outcome as "?". If the non-destructive measurement projects onto the second term of Eq. (A2), a controlledswap measurement is performed (which effectively projects onto the symmetric and anti-symmetric subspaces). Outcomes "minus" and "anti-symmetric" are mapped to a single outcome "different" which unambiguously determines a = b, and occurs with probability 1 − | q a , q b | [8], thus reproducing the outcome probabilities of the beamsplitter measurement.
Following [8], we map outcomes "plus" and "symmetric" to a single outcome "same" which unambiguously determines a = b with probability 1 − q a , q b for b = 1 ⊕ b [8]. In [8] it is shown that when the states are given with equal a priori probabilities, this measurement performs optimal unambiguous comparison between the cases a = b and a = b, i.e. it minimizes the probability of an inconclusive outcome "?".
On input |(−1) a β (−1) b β to a beamsplitter with single photon threshold detectors placed at both the light and dark ports, the dark port registers a detection with probability 1 − | (−1) a β , (−1) b β | and the light port registers a detection with probability 1 − | (−1) a β , (−1) b β |, which are identical to the outcome probabilities of the above optimal unambiguous state comparison measurement in the qubit basis. Thus, this beamsplitter measurement performs optimal unambiguous state comparison on {|β , |−β } when the states are given with equal a priori probabilities.
Appendix B: Error analysis
In Appendix B 1 we calculate the worst case error probabilities for the family of interpolation protocols, which we use in Section IV A to show that they have information leakage O(log n).
In Appendix B 2 we calculate the worst case error probabilities for the family of optical ring protocols described in Section III B, which we use in Section IV B and Appendix C to show that they have information leakage O(log n). We also prove optimality of these protocols over several similar protocols which use coherent state signals arranged in a ring in phase space, and state without proof analogous results for optical lattice protocols.
In Appendix B 3 we determine the optimal one-sided error measurement for any simultaneous message passing model equality protocol, and show that the worst case error probability is lower bounded by the square of the error probability of the beamsplitter measurement. We use these results in Section III to show numerically that larger block sizes k do not outperform the two-bit protocol, even under the optimal measurement.
For most protocols that we consider, each signal encodes several bits of E(x) ∈ {0, 1} m . For different inputs x = y ∈ {0, 1} n , the error probability depends on the distribution of the bit differences between the codewords across the signals. For a given code, the worst case distribution over the particular 2 n codewords could be difficult to calculate. Instead, for most arguments we make the following simplifying assumption.
Remark 1.
In calculating the worst case error probability of a protocol, we assume that the code is uncharacterized apart from its minimum distance, and take the worst case over all strings E A = E B in the output space of the code which differ by at least the minimum distance.
Error analysis for family of interpolation protocols
Here we calculate the worst case error probability of the interpolation protocol with block size k for any choice of positive qubit overlap q
(k) 0 , q (k) 1
≥ 0. Similar results hold for arbitrary overlap, but we make this assumption because it simplifies the algebra. We then show that under the choice q (k) 0 , q (k) 1 = 1 − k/m the error probability is upper bounded by 2 −δ r , for repetition number r denoting the number of identical copies of |ψ (k) x sent. This implies that for a given desired error probability ε, the repetition number can be fixed independent of n. We use this fact in the proof of Proposition 1 that the information leakage of the interpolation is O(log n).
As mentioned in Remark 1, the worst case is taken over all strings E A = E B ∈ {0, 1} m which differ by minimum distance mδ bits. To determine the worst case over this set, we first calculate the probability that the referee's measurement on the j-th pair of signals of |ψ (k) xy returns "dark port detection" or "anti-symmetric" under the assumption that E A,i = E B,i for d indices i ∈ I[ j, k]. Recall that the qubit-basis beamsplitter measurement is performed on the first term of the decomposition (8), and the controlled-swap measurement is performed on the second term.
Recall that on input |q
(k) 0 |q (k)
1 , the qubit-basis beamsplitter measurement outputs "dark port detection" with probabil-
ity 1 − q (k) 0 , q (k) 1 (see Eq.
(2) and Appendix A). Thus, when d bits differ the qubit-basis beamsplitter measurement on the first term of the decomposition (8) outputs "dark port detection" ("D") with probability
Pr I k ("D"|d bits differ) = d k 2 1 − q (k) 0 , q (k) 1 .(B1)
Recall that on input |ψ , the controlled-swap outputs "antisymmetric" with probability 1 2 (1 − W) |ψ 2 , where W is the operator which swaps Alice and Bob's systems. It follows that the controlled-swap on the second term of the decomposition (8) outputs "anti-symmetric" with probability
Pr I k ("A-S"|d) = 1 4k 2 ∑ i,l∈I[ j,k] i =l |q (k) E(x) i |q (k) E(y) l − |q (k) E(y) i |q (k) E(x) l 2 = 1 2k 2 k(k − 1) − ∑ i,l∈I[ j,k] i =l q (k) E(x) i , q (k) E(y) i q (k) E(y) l , q (k) E(x) l = 1 2k 2 k(k − 1) − (k − d)(k − d − 1) − 2d(k − d) q (k) 0 , q (k) 1 −d(d − 1) q (k) 0 , q (k) 1 2
The first and second equalities are straightforward. The third follows from the fact that (k − d)(k − d − 1) terms in the sum are equal to one, 2d(k − d) terms have one bit different, and d(d − 1) terms have two bits different. Let "no detection" ("ND") denote the event that neither "dark port detection" nor "anti-symmetric" occur for a given signal. After simplification, the probability Pr I k ("ND"|d) of outcome "no detection" when the two k-bit blocks differ by d bits is given by Pr I k ("ND"|d) = 1 − Pr I k ("D"|d) − Pr I k ("A-S"|d)
= 1 − d k p k 1 − (d − 1)p k 2k ,(B2)
where
p k = 1 − q (k) 0 , q (k)
1 . For a given distribution of bit differences d 1 , . . . , d m/k beetween two codewords among the m/k blocks, the total error probability for repetition number r is given by
Pr I k (Err|d 1 , . . . , d m/k ) = m/k ∏ i=1 Pr I k ("ND"|d i ) r ,(B3)
which reproduces Eq. (5) for p k = 1 and k = m in the worst case d = mδ , as expected. Now we prove that the worst case error probability for each block size k occurs when all bit differences between the codewords are consolidated into the fewest number of signals. Indeed, by straightforward calculation it can be shown that for all 0 ≤ p k ≤ 1, for every pair of integers 1 ≤ d, c ≤ k − 1 such that d ≥ c, Pr I k ("ND"|d)Pr I k ("ND"|c) ≤ Pr I k ("ND"|d + 1)Pr I k ("ND"|c − 1), which proves the claim. Thus, the worst case error probability is given by
Pr I k (Err) = 1 − p k 1 − (k − 1)p k 2k mδ k r 1 − p k t k 1 − (t − 1)p k 2k r (B4)
for t the remainder given by mδ
k = mδ k + t k .
In what follows, we show that under the choice p k = k/m the worst case error proability is upper bounded by a constant to the power r. This implies that any fixed error ε can be attained with fixed repetition number. We use this fact in Proposition 1 to show that the information leakage of the interpolation under this choice is O(log n).
Under the choice p k = k/m, the worst case error probability is upper bounded by 2 −δ r :
Pr I k (Err|p k = k/m) = 1 − k m 1 − (k − 1) 2m mδ k r 1 − t m 1 − (t − 1) 2m r ≤ 1 − k m 1 − k 2m mδ k r 1 − t m 1 − t 2m r ≤ 1 − k m 1 − k 2m mδ k r ≤ 2 −δ r(B5)
for all 1 ≤ k ≤ m. The equality follows from simplification of Eq. (B4). The first inequality is straightforward, and the second and third both follow from the fact that the function
[1 − x(1 − x 2 )]
1/x is strictly increasing with x for all 0 < x < 1.
Error analysis for family of optical protocols
Here we derive the worst case error probability of the optical ring protocol described in Section III B with block size k, which we use in Section IV B and Appendix C to bound the information leakage as O(log n). We also prove that for all k = 1, . . . , 4 the ring Gray code is an optimal encoding of kbit blocks of binary codewords into coherent states arranged in a ring in phase space. We then show that the ring Gray code outperforms an alternative which uses q−ary codewords. We state without proof analogous results for the optical lattice protocols.
Lemma 1. For any positive integer k, the ideal ring protocol described in Section III B 1 with block size k, using the Gray code, states of total mean photon number µ k , and the beamsplitter measurement described in Section II A, satisfies the following.
The worst case error probability Pr O k (Err) is upper bounded by
exp − µ k 1 − (1 − (kδ − kδ )) cos 2π kδ 2 k − (kδ − kδ ) cos 2π( kδ + 1) 2 k . (B6)
Furthermore, if the worst case error probability is taken over all codewords E A = E B ∈ {0, 1} m which differ by at least the minimum distance mδ bits (as described in Remark 1), this bound is attained with equality for all 0 ≤ δ ≤ 3/k.
As a corollary, since every binary code with more than two codewords has minimum distance δ ≤ 1/2, then for all k = 1, . . . , 6, the worst case error probability Pr O k (Err) is given by Eq. (B6) with equality in the ideal setting.
Proof. The error probability of the beamsplitter measurement is given by the probability that "no dark port detection" ("ND") occurs for every pair of signals. The worst case error probability depends on the worst case distribution of the bit differences between the codewords across the signals. To upper bound the worst case error probability, we first bound the probability Pr O k ("ND"|d) of "ND" when a pair of k-bit blocks differ by d bits. As a property of the Gray code, nearestneighbour coherent state signals in phase space correspond to blocks which differ by one bit. Thus, for any pair of blocks which differ by d bits, the corresponding pair of coherent state signals must be spaced at least d steps apart on the ring. It follows that Pr O k ("ND"|d) is upper-bounded by the probability of "ND" when the pair of coherent state signals are spaced d steps apart on the ring, which is given by
Pr O k ("ND"|d) ≤ exp − |β k | 2 1 − cos 2πd 2 k ,(B7)
which follows from straightforward calculation using equation 2 (recall β k = µ k /(m/k) is the signal amplitude). For d = 1, 2, 3 under the Gray code, Eq. (B7) is satisfied with equality (see Figure 1 or [12]). Now we upper bound the worst case error probability using the bound (B7) for each signal. For a given distribution of bit differences d 1 , . . . , d m/k among the blocks, the error probability is given by
Pr O k (Err|d 1 , . . . , d m/k ) = m/k ∏ i=1 Pr O k ("ND"|d i ) ≤ exp −µ k + |β k | 2 m/k ∑ i=1 cos 2πd i 2 k . (B8)
Note that cos 2πd 2 k + cos
2πc 2 k ≤ cos 2π 2 k d + c 2 + cos 2π 2 k d + c 2
for all 1 ≤ d, c ≤ k. This can be proven by direct calculation for k = 3, and for k ≥ 4 it follows from the fact that the function cos(2πν) + cos(2π(a − ν)) is strictly decreasing with ν whenever 0 ≤ ν < a − ν < 1 4 , along with the fact that k 2 k ≤ 1 4 . It follows that the righthand side of Eq. (B8) is maximized for strings E A = E B ∈ {0, 1} m such that the bit differences d 1 , . . . , d m/k are evenly distributed among the m/k blocks and differ by minimum distance mδ bits. In this case, m k (kδ − kδ ) pairs of signals differ by kδ + 1 bits, and m k (1 − (kδ − kδ )) pairs differ by kδ bits, giving rise to the bound (B6). Below, we refer to this case of inputs as adjacent.
If δ ≤ 3/k, then mδ ≤ 3m/k, so when the mδ bit differences are evenly distributed among the m/k blocks, d i ≤ 3 for all i = 1, . . . , m/k. As mentioned above, under the Gray code such inputs satisfy Eq. (B7) with equality for every signal, and thus also satisfy Eq. (B6) with equality.
In the experimental setting, it can be shown using similar techniques to those used in the proof of Lemma B6 that adjacent inputs give rise to the lowest expected number of outcomes "dark port detection" in our considered parameter regime. The approximation 14 to the behaviour of the random variable D D,k follows using standard techniques. Now we prove statements of optimality for the Gray code in the optical ring protocols. Analogous results also hold for the lattice protocols.
Proposition 2. In all protocols considered below, assume the referee uses the beamsplitter measurement described in Section II A, and that the worst case error probability is taken over all codewords E A = E B ∈ {0, 1} m which differ by at least the minimum distance mδ bits, as described in Remark 1.
For any positive integer k, the ideal ring protocol described in Section III B 1 with block size k, using states of total mean photon number µ k , satisfies the following: For all 0 ≤ δ ≤ 2/k, the Gray code minimizes the worst case error probability over all encodings of k-bit blocks of binary codewords into a ring of equally spaced coherent state signals in phase space.
As a corollary, since every binary code with more than two codewords has minimum distance δ ≤ 1/2, the Gray code is optimal for all k = 1, . . . , 4.
Proof. We prove optimality of any encoding in which nearest neighbour coherent state signals differ by one bit (of which the Gray code is an example). First we show that for any such encoding, the worst case error probability is given by Eq. (B6) with equality. For any such encoding, any pair of coherent state signals which are second-nearest neighbours correspond to a pair of blocks which differ by two bits, and thus satisfy Eq. (B7) with equality. By the same arguments used to prove Lemma 1, it follows that the worst case error probability for all δ ≤ 2/k under any such encoding is given by Eq. (B6) with equality.
Now we show that any encoding for which there exist nearest-neighbour coherent state signals which differ by d ≥ 2 bits has worst case error probability greater than Eq. (B6). The case assumptions δ ≤ 2/k and d ≥ 2 imply mδ ≤ d m/k. Thus, there exist codewords which differ in mδ bits and for which the bit differences d 1 , . . . , d m/k are distributed among the m/k blocks such that d i = d for mδ /d indices i, and all other bit differences are zero. Furthermore, the codewords can be chosen so that for every index i satisfying d i = d , the corresponding pair of coherent state signals are nearest neighbours in phase space. Such codewords clearly give rise to error probability greater than Eq. (B6). Here we have assumed d divides mδ , but similar techniques can be used to prove the same statement in the case when d does not divide mδ .
We have shown that under certain conditions, the Gray code is an optimal encoding of binary codewords into a ring. We now consider another family of optical ring protocols which map q-ary codewords into q equally-spaced nodes on a ring. Under the assumption that all codes saturate the Gilbert -Varshamov bound, we show that the Gray coding of binary codewords outperforms this family for all q powers of two. An analogous result holds for the lattice protocols.
Proposition 3. In all protocols considered below, assume all codes saturate the Gilbert-Varshamov bound. Furthermore, assume that the referee uses the beamsplitter measurement described in Section II A, and that the worst case error probability is taken over all codewords E A = E B in the output space of the code which differ by at least the minimum distance of the code, as described in Remark 1.
Let q be any power of two. For all ε > 0, for any q-ary ring protocol described above which attains error probability ε with total mean photon number µ q and m q signals, the ring protocol described in Section III B with block size k = log q and Gray coding can attain the same error probability ε with the same total mean photon number µ q and fewer than m q signals.
As a corollary, because the ring protocol with block size k = log q uses the same total mean photon number and fewer signals than the q-ary ring protocol, then it has lower information leakage under the bound derived in Appendix C. It can also be shown that this statement holds under the stronger bound derived in Section IV B when n µ q using standard approximation techniques.
Proof. Let m q denote the length of the q−ary code (i.e. the number of "qits" in the code), and let δ q m q denote its minimum distance (i.e. the minimum number of differing qits between codewords). It is easy to see that the worst case error probability occurs when the codewords E A = E B ∈ [q] m q differ by δ q m q qits, and every pair of differing qits correspond to nearest-neighbour coherent state signals; and is given by
Pr q (Err) = exp −µ q δ q 1 − cos 2π q . (B9)
As the q−ary code saturates the (q−ary) Gilbert-Varshamov bound, the quantity δ q satisfies
n m q = log q − δ q log(q − 1) − h(δ q )(B10)
and 0 ≤ δ q < 1 − 1/q [17]. Note that Eq. (B10) is the ratio of n to the number of signals used. Now consider the ring protocol with block size k = log q using the Gray code, the same total mean photon number, the beamsplitter measurement, and minimum distance
δ = δ q log q . (B11) Note that δ < 1 − 1 q log q < 2 log q . (B12)
The first inequality follows from δ q < 1 − 1/q and the second is straightforward. By Lemma 1, the inequality (B12) implies that the worst case error probability is given by Eq. (B9) with equality. As the code saturates the Gilbert-Varshamov bound, the ratio of n to the number signals is given by
n m/ log q = (1 − h(δ q / log q)) log q.(B13)
For all 0 ≤ δ q < 1 − 1/q, we now show that the righthand side of Eq. (B10) is no greater than the righthand side of Eq. (B13), which implies that this protocol sends fewer signals than the q-ary ring protocol, completing the proof. After substituting log q = k and δ q = kδ , the desired inequality becomes
h(δ ) δ − h(kδ ) kδ ≤ log(2 k − 1) (B14)
for all 0 ≤ δ ≤ (1 − 2 −k )/k. Using standard calculus techniques, it can be shown that the lefthand side of Eq. (B14) is a strictly increasing function of δ . Thus, the inequality need only be shown for δ = (1 − 2 −k )/k, which is proven using standard techniques.
Optimal measurement
Here we derive the optimal one-sided error measurement for any simultaneous message passing model equality protocol. We then show that the error probability of the optimal measurement is lower bounded by the square of the error probability of the beamsplitter measurement. We use these results in Section III to show numerically that the optical protocols with block size k > 2 do not outperform the two-bit protocol, even under the optimal measurement.
Consider a general setting in which the referee receives a state σ AB z , where z is contained in one of two sets EQ or NEQ. The referee then wishes to determine whether z is contained EQ or NEQ under the constraint that if z ∈ EQ they never err. It is straightforward to show that the measurement {Π EQ , 1 − Π EQ } (the projection onto the space spanned by the image of the states σ AB z∈EQ and its orthogonal complement) minimizes the worst case error probability of this task. Proposition 4. In the setting described above, for
EQ = {(x, x) : x ∈ {0, 1} n } NEQ = {(x, y) : x = y ∈ {0, 1} n }, (B15) if σ AB (x,y) = |ψ x ψ x | ⊗ |ψ y ψ y | (B16)
for all x, y ∈ {0, 1} n , then the error probability on input x = y ∈ {0, 1} n is lower bounded by ψ x , ψ y 2 .
As a corollary, since the error probability of the beamsplitter measurement in the optical protocols is given by | ψ x , ψ y | on coherent state inputs (see Eq. (2) and the subsequent discussion), then the error probability of the optimal measurement is lower bounded by the square of the error probability of the beamsplitter measurement.
Proof. The error probability is lower bounded by
ψ x ψ y Π EQ ψ x ψ y ≥ 2 ψ x , ψ y 2 1 + ψ x , ψ y 2 ≥ ψ x , ψ y 2 .
The second inequality is straightforward and the first is derived by considering only the first two terms of the decomposition
Π EQ = |ψ x ψ x ψ x ψ x | + |φ φ | + . . . ,(B17)
where |φ is the normalized component of ψ y ψ y orthogonal to |ψ x ψ x . mean photon number lying in a fixed range [µ min , µ max ] for all x, y ∈ {0, 1} n . In Section III B 1 we use these results to bound the information leakage of the families of optical ring and lattice protocols described as O(log n).
In Appendix C 1 we give a practical bound on the information leakage using a continuity bound on entropy. Due to the dimension dependence of the continuity bound, this bound does not give the desired O(log m k ) limiting behaviour, but has the advantage of being straightforward to calculate in practice. In Appendix C 2 we bound the asymptotic behaviour as O(log m k ).
Practical information leakage bound
Here we use an extension of Theorem 1 of [2] and a continuity bound on entropy to bound the information leakage of any simultaneous message passing model protocol satisfying the conditions outlined above.
By Eq. (16), the information leakage is equal to the entropy of ρ AB , maximized over prior distributions P ∈ Pr(X ×Y ). We use the Fannes-Audenaert inequality H(ρ AB ) ≤ H(Γ AB ) + γ log dim(AB) + h(γ),
which bounds H(ρ AB ) in terms of H(Γ AB ), γ = 1 2 ρ AB − Γ AB 1 , and dim(AB) for any state Γ AB [23,24]. We choose Γ AB = Π 0 ρ AB Π 0 /Tr(Π 0 ρ AB ), where Π 0 projects onto a "typical subspace" of ρ, given by the set of Fock states of total photon number lying within some radius ∆ ∈ N of the interval [µ min , µ max ], as in [2]. By straightforward extension of Theorem 1 of [2], ψ xy Π 0 ψ xy ≥ 1 − max 0, e −µ min eµ min
µ min − ∆ µ min −∆ − e −µ max eµ max µ max + ∆ µ max +∆ ,(C2)
and log dim(Π 0 ) ≤ (µ max + ∆) log(µ max + ∆ + m k − 1) + log(µ max − µ min + 2∆ + 1). (C3)
We choose any ε > 0 (which can be optimized over), and fix ∆ such that ψ xy Π 0 ψ xy ≥ 1 − ε . We now bound the quantities H(Γ AB ), γ, and dim(AB) for a given choice of ε . First, by the dimension bound on entropy, H(Γ AB ) is upper bounded by Eq. (C3). Second, γ ≤ 1 2 ∑
x,y∈{0,1} n P(x, y) |ψ xy ψ xy | − Π 0 |ψ xy ψ xy |Π 0 Tr(Π 0 ρ AB ) 1 ≤ ∑
x,y∈{0,1} n P(x, y) 1 − ψ xy Π 0 ψ xy
2 ≤ 1 − (1 − ε ) 2 < √ 2ε (C4)
where the first inequality is the triangle inequality, the second is the Fuchs-van de Graaf inequality along with Tr(Π 0 ρ AB ) ≤ 1, the third is our parameter choice ensuring that ψ xy Π 0 ψ xy ≥ 1 − ε , and the fourth is straightforward. Third, as there are 2 2n states |ψ xy , they span at most a 2 2n -dimensional space. Combining these bounds, the Fannes-Audenaert inequality gives
H(ρ AB ) ≤ log dim(Π 0 ) + 2n √ 2ε + h( √ 2ε ).(C5)
The quantity (C5) holds for any distribution P, and thus upper bounds the information leakage. Although this bound is easily calculable in practice, it is linear in n. For all optical protocols we consider, n is linear in m k , so this bound does not give the desired O(log m k ) asymptotic behaviour.
Information leakage asymptotic analysis
Here we prove the O(log m k ) asymptotic information leakage of any simultaneous message passing model protocol satisfying the conditions outlined above. By Eq. (16), the information leakage is equal to the entropy of ρ AB , maximized over prior distributions P ∈ Pr(X × Y ). We bound H(ρ AB ) as O(log m k ) by projecting onto Fock states lying within telescoping neighbourhoods ∆ 0 , ∆ 1 , . . . of [µ min , µ max ].
In general, consider any projective measurement {Π 0 , Π 1 . . . , } (with possibly infinitely many measurement operators), and define an isometry Pr(C 1 = j) log dim(Π j ),
where the first equality follows from the fact that isometries preserve entropy and the second equality follows from the chain rule and H(C 1 C 2 ) = H(C 1 ). The first inequality follows from strong subadditivity, and the second inequality follows from the dimension bound on quantum entropy. For a fixed positive integer ∆, let ∆ 0 = {N ≥ 0 : µ min − ∆ ≤ N ≤ µ max + ∆} ∆ j = {N ≥ 0 : j∆ + 1 ≤ µ min − N ≤ ( j + 1)∆ or j∆ + 1 ≤ N − µ max ≤ ( j + 1)∆} (C8)
for each j = 0, 1, . . . . Let Π j be the projection onto the space of Fock states with total photon number lying in the set ∆ j . Then the set {Π 0 , Π 1 , . . . } forms a measurement. We now show that Pr(C 1 = j) decreases exponentially with j and log dim(Π j ) is O(log m k ) for each j (with prefactors not growing too quickly with j) to bound Eq. (C7) as O(log m k ). Using similar techniques to those used to prove Theorem 1 of [2], it can be shown that Pr(C 1 = j) ≤ e −µ max eµ max µ max + j∆
µ max + j∆ (C9)
for all j = 0, 1, . . . under the simplifying assumption ∆ > µ max , and log dim(Π j ) ≤ (µ max + ( j + 1)∆) log(µ max + ( j + 1)∆ + m k − 1) + log(∆) for all j = 1, 2, . . .
It is straightforward to show that under these bounds the infinite sum appearing in the second term of Eq. (C7) converges and is O(log m k ).
It is also straightforward to show that H(C 1 ) is no greater than the entropy of the µ max Poisson distribution, which is finite and constant in m k (in fact, it is well-approximated by 1 2 log(2πeµ max ) when µ max 1 [25]). Thus, the asymptotic information leakage is O(log m k ).
. The set I[ j, k] indexes the j-th block of k bits, i.e. it is the set of integers in the range [( j − 1)k + 1, jk]. If k does not divide m the remaining qubits in the final signal are set to |q (k) 0 . As before, Alice and Bob send identical copies of |ψ (k) x until the desired error probability ε is attained. Note that Eq. (6) converges to Eq. (3) for k = m, and for k = 1 it converges to
FIG. 1 :
1Gray coding of k−bit blocks into a ring of coherent states in phase space for k = 1 (blue), k = 2 (blue and red combined), and k = 3 (blue, red and teal combined). Here, β is the coherent state amplitude of each signal. For k = 1,
FIG. 2 :
2Quantum information leakage (QIL), measured in bits, as a function of the input size n for our ring protocols to attain error probability ε = 0.01 in the ideal setting.
FIG. 3 :
3Quantum information leakage (QIL), measured in bits, as a function of the input size n for our ring protocols to attain error probability ε = 0.01 under transmittivity η = 0.3 and dark count probability p dark = 7.3 × 10 −11 . Existing optical protocol with interferometric visibility 99% included for reference.
− k/m as O(log n): Proposition 1. For any fixed error probability ε > 0, there exists a constant C ≥ 0 such that for every positive integer n the following holds: for all k = 1, . . . , m (where m is the length of the codewords E(x) ∈ {0, 1} m ), the information leakage of the interpolation protocol with block size k and qubit overlap q − k/m is no greater than C log(n).
⊗ | j ⊗ | j ∈ U(AB, ABC 1 C 2 ). AB) ρ = H(ABC 1 C 2 ) V ρV † = H(C 1 ) + H(AB|C 1 C 2 ) ≤ H(C 1 ) + H(AB|C 1 )
Appendix C: Information leakage for families of optical ring and lattice protocols.Here we bound the information leakage of any pure state simultaneous message passing model protocol in which every state ψ xy is a tensor product of m k coherent states with total
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| [] |
[
"QKD-based quantum private query without a failure probability",
"QKD-based quantum private query without a failure probability"
] | [
"Liu Bin \nState Key Laboratory of Networking and Switching Technology\nBeijing University of Posts and Telecommunications\n100876BeijingChina\n\nCollege of Computer Science\nChongqing University\n400044ChongqingChina\n",
"Gao Fei \nState Key Laboratory of Networking and Switching Technology\nBeijing University of Posts and Telecommunications\n100876BeijingChina\n",
"Wei Huang \nState Key Laboratory of Networking and Switching Technology\nBeijing University of Posts and Telecommunications\n100876BeijingChina\n",
"Wen Qiaoyan \nState Key Laboratory of Networking and Switching Technology\nBeijing University of Posts and Telecommunications\n100876BeijingChina\n"
] | [
"State Key Laboratory of Networking and Switching Technology\nBeijing University of Posts and Telecommunications\n100876BeijingChina",
"College of Computer Science\nChongqing University\n400044ChongqingChina",
"State Key Laboratory of Networking and Switching Technology\nBeijing University of Posts and Telecommunications\n100876BeijingChina",
"State Key Laboratory of Networking and Switching Technology\nBeijing University of Posts and Telecommunications\n100876BeijingChina",
"State Key Laboratory of Networking and Switching Technology\nBeijing University of Posts and Telecommunications\n100876BeijingChina"
] | [
"Article . SCIENCE CHINA Physics, Mechanics & Astronomy"
] | In this paper, we present a quantum-key-distribution (QKD)-based quantum private query (QPQ) protocol utilizing single-photon signal of multiple optical pulses. It maintains the advantages of the QKD-based QPQ, i.e., easy to implement and loss tolerant. In addition, different from the situations in the previous QKD-based QPQ protocols, in our protocol, the number of the items an honest user will obtain is always one and the failure probability is always zero. This characteristic not only improves the stability (in the sense that, ignoring the noise and the attack, the protocol would always succeed), but also benefits the privacy of the database (since the database will no more reveal additional secrets to the honest users). Furthermore, for the user's privacy, the proposed protocol is cheat sensitive, and for security of the database, we obtain an upper bound for the leaked information of the database in theory. quantum private query, quantum key distribution, single-photon signal PACS number(s): 03.67.Dd, 03.65.Ud Citation: Liu B, Gao F, Huang W, et al. QKD-based quantum private query without a failure probability. | 10.1007/s11433-015-5714-3 | [
"https://arxiv.org/pdf/1511.05267v1.pdf"
] | 45,165,339 | 1511.05267 | 50f748fac44aea80d9a64f75a257e60db33bdfab |
QKD-based quantum private query without a failure probability
2015. 100301. 2015
Liu Bin
State Key Laboratory of Networking and Switching Technology
Beijing University of Posts and Telecommunications
100876BeijingChina
College of Computer Science
Chongqing University
400044ChongqingChina
Gao Fei
State Key Laboratory of Networking and Switching Technology
Beijing University of Posts and Telecommunications
100876BeijingChina
Wei Huang
State Key Laboratory of Networking and Switching Technology
Beijing University of Posts and Telecommunications
100876BeijingChina
Wen Qiaoyan
State Key Laboratory of Networking and Switching Technology
Beijing University of Posts and Telecommunications
100876BeijingChina
QKD-based quantum private query without a failure probability
Article . SCIENCE CHINA Physics, Mechanics & Astronomy
10101003012015. 100301. 201510.1007/s11433-015-5714-3Received June 2, 2015; accepted June 18, 2015
In this paper, we present a quantum-key-distribution (QKD)-based quantum private query (QPQ) protocol utilizing single-photon signal of multiple optical pulses. It maintains the advantages of the QKD-based QPQ, i.e., easy to implement and loss tolerant. In addition, different from the situations in the previous QKD-based QPQ protocols, in our protocol, the number of the items an honest user will obtain is always one and the failure probability is always zero. This characteristic not only improves the stability (in the sense that, ignoring the noise and the attack, the protocol would always succeed), but also benefits the privacy of the database (since the database will no more reveal additional secrets to the honest users). Furthermore, for the user's privacy, the proposed protocol is cheat sensitive, and for security of the database, we obtain an upper bound for the leaked information of the database in theory. quantum private query, quantum key distribution, single-photon signal PACS number(s): 03.67.Dd, 03.65.Ud Citation: Liu B, Gao F, Huang W, et al. QKD-based quantum private query without a failure probability.
Introduction
The applications of quantum mechanics have achieved great success in mutual trusted private communication, for which quantum key distribution (QKD) can allow two parties who trust each other to generate a secret key over a public channel with "unconditional" security [1][2][3]. In the last three decades, mutual trusted private communication has been fully developed both in theory [1][2][3][4][5][6][7][8][9][10][11][12] and in experiment [13,14]. While in some cryptographic communications, we need not only protect the security of the transmitted message against the outside adversaries, but also the communicators' individual privacies against each other [15][16][17][18]. An interesting example is the problem of private query to a database, where Alice (the user) wants to get an item in Bob's database, which is composed of a quantity of secret messages. In this scenario, Alice does not want Bob to know which item she is interested in and Bob does not want Alice to get more items of his database. Symmetrically private information retrieval (SPIR) [19] protocols are designed for such circumstance. It ensures the privacies of both Alice and Bob. That is, after the communication, Alice can correctly obtain precisely one item (i.e., the one Alice wants to get) from Bob's database and simultaneously, Bob does not know the address Alice has retrieved.
Quantum private query (QPQ) is the quantum scheme for SPIR problem. Similar to the two-party quantum secure computations, the task of SPIR cannot be achieved ideally even in its quantum version [20]. More practically, the aim of quantum private query is generally loosened into that Alice can elicit a few more items than the ideal requirement (i.e, just one item) in the database, and Bob's attack will be discovered with a non-zero probability by Alice if he tries to obtain the address Alice is retrieving (equivalently, Alice's privacy is guaranteed in the sense of cheat-sensitive instead of an ideal one).
In 2008 V. Giovannetti et al. proposed the first quantum private query protocol [21][22][23], where the database is represented by an unitary operation (i.e. oracle operation) and it is performed on the two coming query states. In 2011 L. Olejnik presented an improved protocol where only one query state is needed [24]. Compared with the previous (classical) SPIR schemes, the above two protocols not only exhibit advantage in security (i.e. security based on fundamental physical principles instead of assumptions on computational complexity) but also display an exponential reduction in both communication complexity and running-time computational complexity.
Though the above two protocols have significant advantages in theory, they are difficult to implement because when large database is concerned, the dimension of the oracle operation will be very high. To solve this problem, M. Jakobi et al. gave a new private query protocol based on quantum oblivious key transfer (QOKT) [25].This type of QPQ protocol is called QKD-based QPQ [25] or QOKT-based QPQ [26]. The previous QKD-based QPQ protocols can be generally divided into the following three Stages.
(1) States transfer (Raw oblivious key distribution). Alice and Bob generate a raw oblivious key satisfying: (a) Bob knows the whole key; (b) Alice only knows part of it; (c) Bob does not know which bits Alice knows.
(2) Post-processing (Oblivious key dilution). Alice and Bob "dilute" the raw oblivious key into a final oblivious key satisfying: (a) Bob knows the whole key; (b) Alice knows only one (or a little more than one) bit; (c) Bob does not know which bit Alice knows. Note that the key dilution stage in all the previous QKD-based QPQ protocols would fail with a small probability and if it happens, Alice and Bob have to restart the whole protocol. And generally, to make the above failure probability smaller, the expectation of the number of the bits Alice gains is always larger than one, which is actually not in Bob's favor.
(3) Private Query. Alice claims a shift to the final key according to the item she wants to query. Then Bob encrypts his database by the shifted final oblivious key and sends the whole encrypted database to Alice. Thus Alice will know exactly the bit she is interested in.
Compared with the previous QPQ protocols, the above type of QPQ protocol, i.e., QKD-based QPQ, is easy to be realized. More concretely, it can be easily generalized to large database, and it is loss tolerant. Therefore, as a practical model, QKD-based QPQ is very attractive and has become a research hotspot. Different manners to distribute the raw key were given in Refs. [27][28][29][30][31][32][33], and different methods for postprocessing were presented in Refs. [34,35] and analyzed by us in Ref. [26]. What's more, recently, people start to study the error-correction scheme for the QOKT [26,36].
Inspired by a recently proposed QKD protocol [37] utilizing single-photon signal of multiple optical pulses, we propose a QPQ protocol here. As a QKD-based QPQ protocol, it maintains the original advantages, i.e., easy to implement and loss tolerant. (Note that the single-photon signal of multiple optical pulses sources have good experimental foundation, for instance the multi-mode W-state has been generated and precisely characterised [38,39]). Besides, different from the situation in the previous QKD-based QPQ protocols, the number of the bits an honest user (Alice) knows in the initially generated oblivious key (i.e., the raw oblivious key in the previous protocols) is always one. Therefore, no more key dilution process is required here, and consequently there is no more failure probability for our protocol. This characteristic not only improves the stability, in the sense that, ignoring the noise and the attack, the protocol would always succeed, but also benefits the privacy of the database since the database owner (Bob) will no more reveal additional secrets to the honest users (Alice). Moreover, in all the previous QKD-based QPQ protocols, only some specific attacks to the database are analyzed. Differently, here we calculate an upper bound for the leaked information of the database in the proposed protocol in theory. What's more, like all the previous QPQ protocols (not only the QKD-based ones), our protocol is also cheat sensitive, in the sense that if Bob tries to steal the information on which item Alice is querying, he will not give Alice the item precisely and, therefore, Alice could detect Bob's cheating by the incorrect data value.
The rest of this paper is organized as follows. The protocol is presented in next section. In section 3, we analyze both the database security and the user security of the proposed protocol. And a short conclusion is given in section 4.
Quantum Private Query Utilizing Single-Photon Signal
Bit sharing process utilizing single-photon signal of multiple pulses
Recently, utilizing a single-photon state of L optical pulses, Sasaki et al. proposed a QKD protocol without monitoring signal disturbance, which has high tolerance of bit errors. Here we introduce a variant of the bit sharing process in Sasaki et al. protocol, where the realization condition is a little more idealized than that of the original version.
a Bob generates a L-length binary string S and a singlephoton state of L optical pulses |Ψ 0 , where the only photon might be equiprobably in each of the pulses.
Here,
S = s 0 s 1 · · · s L−1 ,(1)
where s i = 0 or 1, i = 0, 1, · · · , L−1; and
|Ψ 0 = 1 √ L L−1 k=0 |k ,(2)
where the photon is in the k-th pulse for state |k − 1 .
Then according to S , Bob modulates the phase of each pulse and the state changes into
|Ψ S = 1 √ L L−1 k=0 (−1) s k |k .(3)
Then Bob sends |Ψ S to Alice.
b Alice generates a random value r ∈ {1, 2, · · · L−1} in advance. When she receives |Ψ S , she first splits each pulse by a half beam-splitter and then, according to r, she changes the order of the pulses in one of the light paths into
r, r + 1, · · · , L − 1, 0, 1, · · · , r − 1,(4)
while the order of the pulses in the other light path remains 0, 1, · · · , r − 1, r, r + 1, · · · , L − 1.
Then she superposes the pulses in the two light paths in order. According to the measurement result (i.e., in which pair of pulses he denotes an photon and the interference result), she will get one of the following values (if no eavesdropping and noise exist)
s i⊕ L r ⊕ s i ,(6)
where i = 0, 1, · · · L−1, ⊕ denotes summation modulo 2 and ⊕ L denotes summation modulo L.
c If Alice gets the interference result of the (i 0 +1)-th pair of the pulses (i.e., the value of s i 0 ⊕ L r ⊕ s i 0 ,), she publishes i 0 and r. Alice and Bob agree on that the shared key bit in this round is just
s i 0 ⊕ L r ⊕ s i 0 .
We denote the processes presented above as Protocol I. The more concrete content about Sasaki et al. QKD protocol, for example the security analysis and the implementation scheme in practice, please refer to the Ref. [37]. Here we will not dwell on it since this paper only covers this novel coding scheme described above.
The proposed oblivious key distribution protocol
Utilizing single-photon signal of multiple pulses, we can distribute a oblivious key which is very suitable for the QPQ problem. Our QPQ protocol (denoted as Protocol II) for an N-element database utilizing single-photon signal of multiple optical pulses is as follows.
1 Bob generates a single-photon state of N+1 optical pulses |Ψ S according to an (N+1)-bit binary string S (just like the state in Eq. 3),
S = s 0 , s 1 , s 2 , · · · , s N ,(7)
and
|Ψ S = 1 √ N + 1 N k=0 (−1) s k |k ,(8)
where s k is randomly 0 or 1, and k=0, 1, 2, · · · N. Then he sends |Ψ S to Alice in sequence.
2 Alice generates a random value r ∈ {1, 2, · · · N}, and then she uses the interference circuits in Fig. 1 (just as what Alice does in Step b in Protocol I) to randomly get one of the following values
s j ⊕ s j⊞r ,(9)
where j = 0, 1, · · · N, ⊕ denotes summation modulo 2 and ⊞ denotes summation modulo N+1. For ease of description, we suppose the value Alice finally gets is
s t ⊕ s t⊞r .(10)
Note that Alice got the interference result of the (t+1)th pair of the pulses in two paths. (for all pulses)
D1 D0 r r+1 r+1 r+1 · · · N 0 · · · r−3 r−2 r−1 0 1 2 3 · · · N−r N−r+1 · · · N−2 N−1 N
The interference process Figure 1 Alice's information extraction process: After splitting the pulses of |Ψ S , Alice sets up two circuits in the two light paths according to r, respectively. One, Circuit 1, is with the length of N + 1 pulses and this circuit only works for the first r pulses, i.e., the rest pulses will take shortcut without Circuit 1. The other, Circuit 2, is with the length of r pulses and all pulses in this light path will go through it. Thus, Alice can get interference results of the two-way pulses in the way shown in the bottom half of the figure.
3 Alice publishes t and they agree on the N-bit key
s t ⊕ s 0 , s t ⊕ s 1 , · · · , s t ⊕ s t−1 , s t ⊕ s t+1 , · · · , s t ⊕ s N−1 , s t ⊕ s N .(11)
Note that Alice knows s t ⊕ s t⊞r .
4 According to the position of the bit Alice knows in the N-bit key above and the position of the item she wants to query, Alice claims a shift to the key. For example, if Alice knows the j-th bit in the key and wants to retrieve the i-th item in Bob's database, she declares a shift value s= j−i so that Bob can shift his key by s.
5 Bob shifts the key according to the value Alice declared. Then he encrypts his database by the shifted oblivious key and sends the encrypted database to Alice.
6 Alice decodes the item she wants to query by the shifted oblivious key.
In Protocol II, Alice always knows precisely one bit in the oblivious key, therefore, the protocol would always succeed if no attack and noise exist, and Bob no more needs to leak additional information about his database to the honest Alice.
In addition, to verify Bob's honesty, Alice could periodically check the validity of the received data in other database, or query to the same item repeatedly and compare the outcomes [24]. Another way, similar to the first QPQ protocol [21], they would perform Protocol II twice at one query process, denoted as Protocol II ′ , where Alice is allowed to query the same item twice to check Bob's honesty. In fact, the two ways to verify Bob's honesty above are both based on that if Bob gains information on the bit position Alice knows in the key, he will lose information on the bit value she has recorded [25]. And the latter (Protocol II ′ ) appears more active since Alice might detect Bob's dishonest action immediately, however, it leaks more information on the database to the dishonest Alice.
Correctness of the proposed protocol
Now we analyze how each single-photon signal of multiple optical pulses Bob sends to Alice evolves in the whole processes of Protocol II. The initial state is as that in Eq. 8. When the signal passed Alice's first half beam-splitter BS 1 , it changed into
1 √ 2(N + 1) N k=0 (−1) s k |k |0 + i N k=0 (−1) s k |k |1 ,(12)
where the state |0 represents the photon is in the transmission path and |1 represents the photon is in the reflection path. After passing the two delay circuits and the two reflectors, it becomes
i √ 2(N + 1) N k=0 (−1) s k |k |0 + i N k=0 (−1) s k⊞r |k |1 . (13)
After the two shifted pulses passes through the second half beam-splitter BS 2 , the state becomes
i 2 √ (N + 1) N k=0 [(−1) s k − (−1) s i,k⊞r ] |k |1 ′ +i N k=0 [(−1) s k + (−1) s k⊞r ] |k |0 ′ ,(14)
where the state |0 ′ represents the photon is in the light path leading to the detector D 0 (i.e., the transmission-reflection path and also the reflection-transmission path), and |1 ′ represents the photon is in the light path leading to the detector D 0 (the transmission-transmission path and also the reflectionreflection path). We can see that if s k =s k⊞r , Alice might detect the photon in D 0 but never detect it in D 1 , and correspondingly if s k s k⊞r , she might detect the photon in D 1 but never detect it in D 0 . Therefore, once Alice detect the photon at (t+1)-th pair of the pulses, she can know the value of s t ⊕ s t⊞r .
Security analysis of the proposed protocol
Database security
It is difficult to figure out a specific attack strategy for Alice to steal more information of Bob's database. Here we calculate an upper bound of a dishonest Alice's information about Bob's database by the Holevo bound. That is
H(A : Database) H(A : S ) (15) S ( 1 2 N+1 2 N+1 k=1 |Ψ k Ψ k |) − 1 2 N+1 2 N+1 k=1 S (|Ψ k Ψ k |)
Here, S (ρ) represents the Von Neumann entropy of the state ρ, H(A : Database) denotes the amount of information that Alice can gain about Bob's database, H(A : S ) denotes the amount of information that Alice can gain about Bob's binary strings S , and each |Ψ k denotes one of the 2 N+1 different scenarios of the states in Eq. 8. And we have
S ( 1 2 N+1 2 N+1 k=1 |Ψ k Ψ k |) = S (I N+1 ) = log(N + 1),(16)S (|Ψ k Ψ k |) = 0.(17)
Therefore,
H(A : Database) H(A : S ) log(N + 1).(18)
(Obviously, H(A : Database) 2 log(N + 1) for Protocol II ′ .) This upper bound is lower than that of the previous QKDbased QPQ protocols when a very large database is considered. For example in the QPQ protocol in Refs. [25,27], the average information Alice can get for each bit in the final key can be described by a constant δ if Alice stores the qubits corresponding to the same bit in the final key and performs collective measurements to them (which is just a specific attack strategy and there might be some more powerfull attacks, for example, to perform collective measurements to the whole received photons together), and consequently she can get Nδ information about Bob's database in total. And when N become larger, the advantage of our protocol's bound B 1 log(N + 1) would be highlighted comparing with B 2 Nδ in Ref.s [25,27].
User privacy
Just as in the previous QPQ protocols [21,24,25,27], the security of user's privacy in our protocols is based on that if Bob gets some effective information on which item Alice queries, he might give Alice the wrong value of the queried one, and Alice would detect Bob's attack by the incorrect item. Firstly, we consider that Bob sends Alice a pure state of single photon with N + 1 pulses as follows
|Ψ A = N k=0 a k |k ,(19)
where k |a k | 2 = 1. (The signals with 2 or more photons will be detect by Alice's interference circuits.) When the above signal passes the two half beam-splitters, it becomes
1 2 N k=0 (a k + a k⊞r ) |k 0 ′ + i(a k − a k⊞r ) |k 1 ′ .(20)
Thus, if Alice chooses r, she will get the photon at the (k+1)th pair of pulses with the probability
1 2 (|a k | 2 + |a k⊞r | 2 ).(21)
The joint probability distribution of different r and k is
1 2N (|a k | 2 + |a k⊞r | 2 ) r,k ,(22)
where r=1, 2, · · · , N and k=0, 1, 2, · · · , N. And the minimum probability of Bob giving a wrong value to Alice is
P min = N k=0 min |a k + a k⊞r | 2 4N , |a k − a k⊞r | 2 4N .(23)
Obviously, |Ψ A should be prepared with the condition a i = ±a j for all i and j, if Bob wants to give Alice the item she is querying precisely. Ignoring the overall phase, it is just the legal state |Ψ S in Eq. 8. Now we consider that Bob prepares an entangled state and sends part of it to Alice. To give Alice the correct item, Bob must know every phase difference s t ⊕ s j where j = 0, 1, . . . , N, since r is generated randomly by Alice and j could be any of the natural numbers no bigger than N except t. Therefore, the state he sends to Alice must in the following form
2 N i=1 λ i |Λ i B |Ψ S i A ,(24)
where i |λ i | 2 = 1,
|Ψ S i = 1 √ N + 1 |0 + N k=1 (−1) s i,k |k ,(25)
Ψ S i |Ψ S j 1 for i j, and Λ i |Λ j = σ i j . In the above equation, each value of i is corresponding to one of the 2 N different states |Ψ S i . Note that, ignoring the overall phase, |Ψ S i is the same with
|Ψ ′ S i = 1 √ N + 1 −|0 + N k=1 (−1) s i,k ⊕1 |k .(26)
Then Bob sends Alice the second half of the state in Eq. 24, i.e., system A, and keeps the first, i.e., system B. After Alice has measured A, he can measure B in bases {|Λ i } N i=1 to get the value of S i , according to which he can give the correct value of the item she wants to query. However, by doing this, he will get no information about which item Alice has queried.
Or he can make other measurements on system B to steal the above position information, but thus he cannot get the whole information about S i and, therefore, he cannot give Alice the correct value all the time. So our protocol is also cheat sensitive just as the previous ones.
For example, Bob may prepare the entangled state s 0 ,s 1 ,...,s N
|s 0 0 |s 1 1 . . . |s N N N k=0 (−1) s k |k A (N + 1) × 2 N ,(27)
where Bob keeps the first N+1 systems {0, 1, . . . , N} and sends system A to Alice. When Alice has measured system A and published part of the measurement results t in Step 2 and 3, the whole system for Bob collapsed into a mixture state with the ensembles of pure states
{ 1 2 , |ϕ + t,t⊞r i t, i t⊞r |+ i ⊗ |t0 ′ A ; 1 2 , |ψ + t,t⊞r i t, i t⊞r |+ i ⊗ |t1 ′ A },(28)
where,
|+ = 1 √ 2 (|0 + |1 ) ,(29)|ϕ + = 1 √ 2 (|00 + |11 ) ,(30)|ψ + = 1 √ 2 (|01 + |10 ) .(31)
If Bob measures the first N+1 system in Z basis, he can give the correct item to Alice. While he can measure them in X basis, and if he gets the result |− =1/ √ 2(|0 −|1 ) in system t⊞r, with the probability 1/2, he would get Alice's privacy. Thus, he will totally lose the information on the bit value of the key. And Alice would detect Bob's cheating with the probability 1/2.
Summary and conclusion
In this paper, utilizing the new encoding style by single photon for multiple optical pulses, we proposed a QKD-based QPQ protocol, which, we think, has the following advantages.
• It maintains the advantages of the QKD-based QPQ, i.e., easy to implement and loss tolerant.
• The failure probability is always 0. The proposed protocol is more reliable compared with the previous QKD-based QPQ protocols, in the sense that, ignoring the noise and the attack, it would always succeed.
• The number of items an honest user will obtain is always 1. Therefore, it is more reasonable compared with the previous QKD-based QPQ protocols, since Bob will no more reveal unexpected secrets to the honest Alice.
For the security of the database, we have calculated an upper bound for the leaked information of the database in theory. This upper bound is lower than the previous QKD-based QPQ protocols, however, it is larger than that of the ones in Refs. [21,24]. And correspondingly, the implementation difficulty of our protocols is also between them. That is, Bob does not need to perform a collective unitary operation on a very large quantum system, however, the single photon signal with N+1 pulses is more difficult to prepare than the single photons. For the privacy of the user, our protocols are also cheat sensitive as the previous ones, since once Bob tries to get the address of the item which Alice is querying, he cannot give the correct item to Alice precisely and Alice could detect Bob's cheating by the incorrect data value.
In conclusion, the proposed protocol is the first QKDbased QPQ protocol without the process of the oblivious key dilution, and, therefore, it is the first QKD-based one with no failure probability and no information reveal for the database when the user is honest. Therefore, we think the proposed protocol initiates a new branch of QKD-based QPQ. And we hope that the contents of this paper would be both interesting and instructive to the researchers who are working on quantum private query and the related fields.
Length: N+1. (only for the first r pulses) Circuit2. Length: r.
This work is supported by NSFC (Grant Nos. 61272057, 61170270), Beijing Higher Education Young Elite Teacher Project (Grant Nos. YETP0475, YETP0477), BUPT Excellent Ph.D. Students Foundation (Grant CX201442).
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| [] |
[
"On Prediction and Tolerance Intervals for Dynamic Treatment Regimes",
"On Prediction and Tolerance Intervals for Dynamic Treatment Regimes"
] | [
"Daniel J Lizotte \nDepartments of Computer Science and Epidemiology & Biostatistics\nDepartment of Epidemiology & Biostatistics\nThe University of Western Ontario\nLondonOntarioCanada\n",
"Arezoo Tahmasebi \nThe University of Western Ontario\nLondonOntarioCanada\n"
] | [
"Departments of Computer Science and Epidemiology & Biostatistics\nDepartment of Epidemiology & Biostatistics\nThe University of Western Ontario\nLondonOntarioCanada",
"The University of Western Ontario\nLondonOntarioCanada"
] | [] | We develop and evaluate tolerance interval methods for dynamic treatment regimes (DTRs) that can provide more detailed prognostic information to patients who will follow an estimated optimal regime. Although the problem of constructing confidence intervals for DTRs has been extensively studied, prediction and tolerance intervals have received little attention. We begin by reviewing in detail different interval estimation and prediction methods and then adapting them to the DTR setting. We illustrate some of the challenges associated with tolerance interval estimation stemming from the fact that we do not typically have data that were generated from the estimated optimal regime. We give an extensive empirical evaluation of the methods and discussed several practical aspects of method choice, and we present an example application using data from a clinical trial. Finally, we discuss future directions within this important emerging area of DTR research. | 10.1177/0962280217708662 | [
"https://arxiv.org/pdf/1704.07453v1.pdf"
] | 45,659,519 | 1704.07453 | 5d7bf26925300bc8a0c5857e835643545a7d339d |
On Prediction and Tolerance Intervals for Dynamic Treatment Regimes
April 26, 2017
Daniel J Lizotte
Departments of Computer Science and Epidemiology & Biostatistics
Department of Epidemiology & Biostatistics
The University of Western Ontario
LondonOntarioCanada
Arezoo Tahmasebi
The University of Western Ontario
LondonOntarioCanada
On Prediction and Tolerance Intervals for Dynamic Treatment Regimes
April 26, 2017
We develop and evaluate tolerance interval methods for dynamic treatment regimes (DTRs) that can provide more detailed prognostic information to patients who will follow an estimated optimal regime. Although the problem of constructing confidence intervals for DTRs has been extensively studied, prediction and tolerance intervals have received little attention. We begin by reviewing in detail different interval estimation and prediction methods and then adapting them to the DTR setting. We illustrate some of the challenges associated with tolerance interval estimation stemming from the fact that we do not typically have data that were generated from the estimated optimal regime. We give an extensive empirical evaluation of the methods and discussed several practical aspects of method choice, and we present an example application using data from a clinical trial. Finally, we discuss future directions within this important emerging area of DTR research.
Introduction
Dynamic Treatment Regimes (DTRs), also known as adaptive treatment strategies or treatment policies, are a key tool for providing data-driven sequential decision-making support. A DTR is a sequence of decision functions that take up-to-date patient information as input and produce a recommended treatment. Thus, a DTR is a mathematical representation of the sequential decision-making process. Using this representation, we can use previously collected decision-making data to estimate an "optimal" DTR, where optimality is most often defined in terms of expected outcome. That is, a DTR is optimal if it produces the best outcome, on average, over a patient population. We will use this definition of optimality throughout our work.
Each decision in an optimal DTR is made in the service of achieving maximal expected outcome. However, the outcome of any particular individual under an optimal regime may vary widely from this expectation. Indeed, DTRs have been applied in many very challenging areas of medicine, including psychiatry, cancer, and HIV, where patient outcomes are known to be highly variable, or, equivalently from our perspective, difficult to predict.
It is with this variability in mind that we consider different methods for assessing the variability in individual outcomes under a given DTR. Our objective is to quantify for the decision-maker not our certainty about the expectation of outcomes, but rather our uncertainty about what the observed outcome might be for a particular patient.
We begin by formally defining DTRs, and we review point and interval estimation techniques for relevant parameters of the optimal DTR. We then review definitions and existing methods for confidence intervals, 1 prediction intervals, and tolerance intervals. Following this background, we formally describe our problem of interest in the context of using DTRs to provide decision support.
We will see that the main technical challenge associated with constructing tolerance intervals for DTRs stems from not having a sample drawn from the correct distribution. Thus, our methods will use re-weighting and re-sampling to allow us to apply existing tolerance interval methods in this setting. To help illustrate the technical challenge, we first describe a naïve strategy for constructing tolerance intervals whose performance is poor, and we then present two novel strategies for constructing valid tolerance intervals for the response under a given dynamic treatment regime. We present an empirical evaluation of the methods, and we conclude by discussing their implications and directions for future work.
Background
In the following, we review basic concepts pertaining to DTRs, the estimation of optimal regimes, and concepts and issues surrounding interval estimation and prediction.
Dynamic Treatment Regimes
DTRs are a mathematical formalism meant to capture the decision-making cycle of information gathering, followed by treatment choice, followed by outcome evaluation. They have been defined at different levels of generality by many authors [Schulte et al., 2014, Laber et al., 2014b,a, Lizotte et al., 2012, Nahum-Shani et al., 2012a,b, Lizotte et al., 2010, Shortreed et al., 2011. Here, we focus on regimes with two decision points; thus for this work we consider a DTR to be a sequence of two functions (π 1 , π 2 ) which map up-to-date patient information at the first and second decision points, respectively, to distributions over the space of available treatments at each decision point. We represent the information (covariates) about a given patient at point t by s t , which we view as a realization of a random variable S t . Similarly, we denote the chosen treatment (action) by a t , which is a realization of A t . For a patient who follows a DTR (π 1 , π 2 ), we will have A 1 ∼ π 1 (s 1 ) and A 2 ∼ π 2 (s 1 , a 1 , s 2 ). We let y be the observed outcome or reward attained by a patient after following a regime, and we follow the convention that larger values of y are preferable. For a patient following a given regime, we observe (s 1 , a 1 , s 2 , a 2 , y), the trajectory for that patient.
Trajectory data may come from various observational and experimental sources, for example from Sequential Multiple Assignment Randomized Trials (SMARTs) [Nahum-Shani et al., 2012a,b, Collins et al., 2014. A SMART is an experimental design under which patients follow a DTR that applies randomly assigned treatments. We will call such a DTR an exploration DTR or exploration policy. The goal of running a SMART is analogous to that of running a pragmatic randomized controlled trial-to evaluate the comparative effectiveness of different treatment options in an unbiased way. This comparative effectiveness information can then be used to estimate an optimal DTR. An optimal DTR is a pair of decision functions (π 1 , π 2 ) that maximize E[Y |S 1 , A 1 , S 2 , A 2 ; π 1 , π 2 ] where A t ∼ π t (S t ). Thus, an optimal DTR produces maximal expected outcome when applied to a population of patients. In this work, we focus on the setting where the exploration DTR is stochastic, but the candidate optimal DTRs under consideration are deterministic.
Q-learning
Several methods are available for estimating an optimal DTR from data collected under an exploration DTR. Here, we review one such method called Q-learning [Schulte et al., 2014, Huang et al., 2015. Q-learning works by estimating Q functions (Q for "quality") that predict expected outcome given current covariates and treatment choice. In our 2-decision point setting, we have
Q 2 (s 1 , a 1 , s 2 , a 2 ) = E[Y |S 1 =s 1 , A 1 =a 1 , S 2 =s 2 , A 2 =a 2 ].
Note that unlike the expectation in the previous section which averages over patients, Q 2 gives the expectation of Y conditioned on particular patient observations and treatment choices. The definition of Q 2 implies an optimal decision function π * 2 (s 1 , a 1 , s 2 ) = arg max a 2 Q 2 (s 1 , a 1 , s 2 , a 2 ). Q 2 can be estimated using any 2 regression method. Having obtained an estimateQ 2 of Q 2 , our estimate of the optimal second decision function isπ * 2 (s 1 , a 1 , s 2 ) = arg max a 2Q 2 (s 1 , a 1 , s 2 , a 2 ). The optimal Q-function for the first decision point produces the conditional mean of Y given S 1 and A 1 and given that the optimal decision function π * 2 is used at the second decision point. In Q-learning, we estimate Q 1 byQ
1 (s 1 , a 1 ) ≈ E[max a 2Q 2 (s 1 , a 1 , S 2 , a 2 )|S 1 =s 1 , A 1 =a 1 ]
where the expectation is over S 2 conditioned on S 1 and A 1 . The quantity max a 2Q 2 (s 1 , a 1 , S 2 , a 2 ) is sometimes called the pseudooutcome, and is denotedỹ. In order to estimate Q 1 , we compute the pseudooutcome for each trajectory in our dataset, and then regress them on S 1 and A 1 to estimate Q 1 . Again, any regression method can be used to estimate Q 1 , in principle. Our corresponding estimate of the optimal first decision function is thenπ * 1 (s 1 ) = arg max a 1Q 1 (s 1 , a 1 ), and our estimate of the optimal DTR is (π * 1 ,π * 2 ). Note that this DTR is deterministic.
We focus on Q-learning in this work, but several other methods are available for estimating optimal DTRs, including A-learning [Blatt et al., 2004, Schulte et al., 2014, the closely-related g-estimation [Moodie, 2009, Orellana et al., 2010, Barrett et al., 2014, and direct policy search Laber, 2014, Zhao et al., 2015].
Interval Estimation
For consistency, in the following we use ys to represent observed outcomes, and xs to represent covariates, even in non-regression settings.
Confidence Intervals
A confidence interval ( c , u c ) with level 1 − α for a parameter θ is a functional of a dataset Y = {y 1 , ..., y n } of realizations of a random variable Y , with the property that
Pr[θ ∈ ( c , u c )] ≥ 1 − α.
(1)
The probability statement (1) is over datasets containing i.i.d. samples of Y . The goal of a confidence interval is to provide confidence information about the estimated location of an underlying distributional parameter.
Though not our main focus, confidence intervals are by far the most well-known class of interval estimates, and they are closely related to the prediction and tolerance intervals we will develop and investigate.
Prediction Intervals
A prediction interval ( p , u p ) with level 1 − α is a functional of a dataset Y = {y 1 , ..., y n } of realizations of a random variable Y , with the property that
Pr[Y new ∈ ( p , u p )] ≥ 1 − α.(2)
Here, Y new represents a single future observation that was not contained in the original data Y. The goal of a prediction interval is to provide confidence information about where this new observation might fall. However, we note, as others have [Vardeman, 1992], that there is often confusion surrounding the probability statement (2). In particular, the statement is over the joint distribution of Y 1 , ..., Y n , Y new . A prediction interval formed from a dataset traps one additional observation with probability 1−α. It offers no guarantees about trapping more than one additional observation, and indeed no guarantees regarding our confidence in the content of an interval, that is, of the quantity
F Y (u p ) − F Y ( p ) where F Y is the cumulative distribution function of Y .
(For example, a prediction interval that has content 1.0 half the time and content 0.9 half the time has property (2) for α = 0.05, as does an interval that always has content 0.95.)
The well-known normal-theory prediction interval for Y [Neter, John and Wasserman, William and Kutner, 1989] is given by
( p , u p ) N =ȳ ± t α/2;n−1σY 1 + 1 n(3)
whereȳ is the sample mean,σ Y the sample standard deviation, and t α/2;n−1 is the α/2 quantile of a tdistribution with n − 1 degrees of freedom. Note that the validity of (3) is predicated on normality of Y , regardless of sample size. The corresponding prediction interval for Y |X=x in the linear regression setting on p parameters is
( p , u p ) N =ŷ ± t α/2;n−pσY |X=x 1 + x T (X T X) −1 x(4)
where x represents the location of a new sample,ŷ is the prediction of E[Y |X=x],σ Y |X=x is the sample standard deviation of the residuals, X is the design matrix for the regression, and t α/2;n−p is the α/2 quantile of a t-distribution with n − p degrees of freedom. Equation (4) is predicated on the normality of Y |X = x and on homoscedasticity of the residuals.
Tolerance Intervals
A tolerance interval ( t , u t ) with level 1 − α and content γ is also a functional of a dataset Y = {y 1 , ..., y n }.
It has the property that
Pr[F Y (u t ) − F Y ( t ) ≥ γ] ≥ 1 − α.(5)
where F Y is the cumulative distribution function of Y . Thus, a tolerance interval formed from a dataset traps at least γ of the probability content of Y with probability 1 − α, where the 1 − α probability is over datasets.
One well-known normal theory approximate tolerance interval for Y with confidence 1 − α and content γ is given by Krishnamoorthy, Kalimuthu and Mathew [2009] as
( t , u t ) N =ȳ ∓σ Y (n − 1)χ 2 γ;1,1/n χ 2 α;n−1 (6)
whereȳ is the sample mean, χ 2 γ;1,1/n is the γ quantile of a non-central χ 2 distribution with 1 degree of freedom and noncentrality parameter 1/n, and χ 2 α;n−1 is the α quantile of a χ 2 with n − 1 degrees of freedom. The corresponding tolerance interval for Y |X=x in the linear regression setting on p parameters is [Young, 2013] ( t , u t ) N =ŷ ∓σ Y |X=x (n − p)χ 2 γ;1,1/n * χ 2 α;n−p
whereŷ is the prediction of E[Y |X=x],σ Y |X=x is the sample standard deviation of the residuals, n * = σ 2 Y |X=x /σ 2 y is Wallis' "effective number of observations" (1951), andσŷ is the standard error ofŷ|X=x. Again, validity of (6) and (7) is predicated on the normality of Y and Y |X=x, respectively; (7) is also predicated on homoscedasticity. Wilks [1941] proposed a non-parametric tolerance interval that assumes only continuity of F Y . The interval is given by the sample values corresponding to the minimum and maximum ranks r for which
(1 − F Beta (γ; n − 2r + 1, 2r)) > 1 − α(8)
where F Beta is the beta cumulative distribution function. Thus, the interval is constructed simply by truncating the sample to the ranks satisfying (8), and then taking the minimum and maximum of the truncated sample to be the lower and upper limits of the tolerance interval, respectively.
DTRs for Decision Support
DTRs are an ideal formalism for providing data-driven decision support. The most basic approach to providing decision support would be to estimate an optimal DTR from SMART data, and then provide the estimated DTR (π * 1 ,π * 2 ) to a decision maker, perhaps as a computer-based tool that produces the estimated optimal treatment by using current patient information as input to the previously estimated DTR.
Early in the development of DTRs it was recognized that this approach is problematic because it provides no confidence information about our recommendations. Just as we would not recommend one treatment over another if no statistically significant difference were obtained from a standard randomized controlled trial (RCT), neither should we recommend a single treatment in a DTR if in fact the alternatives are not known to be inferior with high confidence. This led to the development of confidence interval methods for the difference in mean expected outcome under different treatment choices within a regime [Chakraborty et al., 2010, 2013, Chakraborty and Moodie, 2013, Laber et al., 2014b, Chakraborty et al., 2014.
Such intervals can give us confidence that if we do recommend a single treatment, that treatment will provide a better outcome, in expectation over patients. However, they do not provide any information about what the range of possible outcomes might actually be for an individual patient. In particular, large SMARTs with 100s to 1000s of patients may discover statistically significant differences in mean outcome even when the effect sizes are small to moderate and variance in outcomes is still substantial. If this is the case, it may be better to avoid recommending a single treatment, or at least to provide more nuanced information about what the patient's experience is likely to be under the different treatment options.
In this work, we consider tolerance intervals as one method for providing this information. For a patient with S 1 = s 1 at the first decision point, rather than recommending treatmentπ * 1 (s 1 ) (even if it is statistically significantly better than the alternative in terms of mean outcome) we would present tolerance intervals for the outcome Y under each possible action, and allow the decision-maker (or the patient-clinician dyad, in the context of patient-centred care [Barry and Edgman-Levitan, 2012]) to decide on treatment based on the range of probable outcomes indicated by the intervals. For each interval, we condition on the observed s 1 , the hypothetical a 1 , and the estimated optimal regimeπ * 2 for the second stage. Thus, we will construct tolerance intervals for Y |S 1 = s 1 , A 1 = a 1 ; π 2 =π * 2 , marginal over S 2 (whose distribution is governed by S 1 and A 1 ) and A 2 (whose distribution is governed by A 1 , S 1 , S 2 and π 2 .) To do so, we will adapt several standard methods because typically we do not have observations drawn from this distribution. This is because, as we noted above, data from SMART studies and similar sources are generated according to an exploration DTR (π 0 1 , π 0 2 ), rather than according to an estimated optimal DTR (π * 1 ,π * 2 ).
Aside: Non-regularity
It is well-known that many kinds of inference on the parameters of an estimated optimal dynamic treatment regime, including confidence intervals, are plagued by issues of non-regularity [Laber et al., 2014b]. Briefly, non-regularity is a result of the sampling distributions of corresponding estimators changing abruptly as a function of the true underlying parameters. It can lead to bias in estimates and anti-conservatism in inference. In dynamic treatment regimes, non-regularity occurs and inference is problematic when two or more treatments produce (nearly) the same mean optimal outcome. In this work, we will not specifically develop methods that are robust to non-regularity. This is because even in the absence of non-regularity, i.e. when optimal Q values are well-separated from sub-optimal ones, there is significant variability in the performance of "standard" tolerance interval methods that is worthy of exploration and analysis. We will return to this point in the Discussion.
Methods
We now detail our strategies for constructing tolerance intervals for Y |S 1 =s 1 , A 1 =a 1 ; π 2 =π * 2 . As we mentioned above, the fundamental challenge of constructing intervals for this quantity is that in general we do not have samples drawn from this distribution-otherwise, we could use off-the-shelf tolerance interval methods. Note that we can use off-the-shelf methods for tolerance intervals for Y |S 2 =s 2 , A 2 =a 2 , because there is no need to account for future decision-making in that case; thus our work focuses on the first decision point. We begin by presenting a naïve approach to constructing tolerance intervals that helps illustrate the main technical challenge to be addressed, and then we present our two proposed strategies: inverse probability weighting, and residual borrowing.
Naïve Q-Learning Tolerance Intervals
Standard Q-learning involves estimating Q 1 (s 1 , a 1 ), which predicts the expected Y under the optimal regime. However, it does so using the pseudooutcomeỸ = max a 2Q 2 (s 1 , a 1 , s 2 , a 2 ) as the regression target, rather than the observed Y . Since the pseudooutcome targets are themselves predicted conditional means of Y , they carry no variance information about Y |S 2 , S 1 , A 1 under the estimated optimal policy, even among trajectories that (by chance) followed the estimated optimal policy. To see this, suppose that we had several trajectories, all of which had the same s 1 , a 1 , s 2 , a 2 , and all of whom happened to follow the estimated optimal policy. Even though their observed outcomes y might have all been different, simply due to unexplained (but still important) variation in Y , they would all be assigned the same pseudooutcome value, and the sample variance of the pseudooutcomes in this group is zero.
This observation highlights the key aspect of Q-learning and related methods that precludes direct estimation of variability in Y . Dynamic programming methods for estimating conditional means of sequential outcomes can "throw away" residual variance without negative repercussions when backing up values, essentially because of the law of total expectation. The benefit of this approach is a reduction in the variance of Q estimates by allowing the use of the entire dataset of trajectories for estimating Q-functions for earlier decision points. The drawback is that such methods cannot directly estimate other distributional properties of Y , including variance and higher-order moments, quantiles, and so on.
If most of the variability in Y were explained by S 2 and A 2 -that is, if the variance of Y |S 2 , A 2 were nearly zero-we might be able to construct approximate tolerance intervals for Y by constructing parametric tolerance intervals for the pseudooutcome, for example using (7). In the case of a saturated model with discrete S 1 and A 1 , we could construct non-parametric tolerance intervals for each pattern of (s 1 , a 1 ) using the pseudooutcome with (8). However, as expected, will see in our empirical results that this approach is not very effective if in fact the variance of Y |S 2 , A 2 is not near zero.
Inverse Probability Weighting
One approach to obtaining variance information about Y underπ * 2 is to select from our dataset only those trajectories whose second-stage treatment matches whatπ * 2 would have assigned, i.e., the trajectories (s 1 , a 1 , s 2 , a 2 , y) for which a 2 =π * 2 (s 1 , a 1 , s 2 ). This subset contains all of the trajectories that have positive probability under the estimated DTR.
Consider a joint distribution over S 2 , A 2 , Π, A 0 2 , A * 2 , M, Y conditioned on S 1 and A 1 . (All statements in the remainder of this subsection are implicitly conditioned on S 1 and A 1 ; explicitly maintaining this is too cumbersome.) Here, A 0 2 is the action chosen by π 0 2 , and A * 2 is the action chosen byπ * 2 , which is assumed to be deterministic given S 2 . Let M (for match 1 ) be 1 if A * 2 = A 0 2 , or 0 otherwise. Let Π be binary, and define
A 2 such that A 2 = A 0 2 if Π = 0 and A 2 = A * 2 if Π = 1.
The dependencies among all of these variables are illustrated in Figure 1 using a directed graphical model [Koller and Friedman, 2009].
The distribution of Y among matched trajectories is governed by
Y |Π = 0, M = 1. The distribution of Y among trajectories gathered usingπ * 2 is Y |Π = 1. Note that while the distribution of Y |S 2 , A 2 , Π=0, M =1 is identical to the distribution of Y |S 2 , A 2 , Π=1 due to the conditional independence structure, the distribution of Y |Π = 0, M = 1 may be different from Y |Π = 1 if there is dependence of M on S 2 .
We describe this phenomenon using the following lemma.
S 2 A 0 2 A * 2 M A 2 Y Π Figure 1: Graphical model depicting the dependence structure of S 2 , A 0 2 , A * 2 , A 2 , Π, M, Y |S 1 , A 1 .
Note that the structure is the same for all values of S 1 and A 1 .
Lemma 1. Let S 2 , A 0 2 , A * 2 , A 2 , Π, M, Y be defined as above, and assume Pr(S 2 ) > 0 =⇒ Pr(S 2 |M =1) > 0. Then Pr(Y |Π=1) = S2 Pr(S 2 ) Pr(S 2 |M =1) Pr(Y, S 2 |Π=0, M =1).
Proof. In the following, we abuse notation by allowing Pr to represent a probability or a density, as appropriate, and we allow to indicate a sum or an integral. The message in any case remains the same. First we note that
Pr(Y |Π=0, M =1) = S2,A2 Pr(Y, S 2 , A 2 |Π=0, M =1) = S2,A2 Pr(Y |S 2 , A 2 , Π=0, M =1)· Pr(A 2 |S 2 , Π=0, M =1)· Pr(S 2 |Π=0, M =1) .(9)
The data generating distribution underπ * 2 is
Pr(Y |Π=1) = S2,A2 Pr(Y, S 2 , A 2 |Π=1) = S2,A2 Pr(Y |S 2 , A 2 , Π=1) Pr(S 2 , A 2 |Π=1) = S2,A2 Pr(Y |S 2 , A 2 , Π=0, M =1) Pr(S 2 , A 2 |Π=1)(10)
where the last step follows from conditional independence of Y and (Π, M ) given S 2 and A 2 . Furthermore,
Pr(S 2 , A 2 |Π=1) = Pr(A 2 |S 2 , Π=1) Pr(S 2 |Π=1) = Pr(A 2 |S 2 , Π=0, M =1) Pr(S 2 |Π=1) = Pr(A 2 |S 2 , Π=0, M =1) Pr(S 2 |Π=0)(11)
where the second step follows because A * 2 is deterministic given S 2 2 and from the definition of Π and M , and the third step follows from independence of S 2 and Π. By combining (10) and (11) and comparing with (9), we obtain
Pr(Y |Π=1) = S2,A2 Pr(Y |S 2 , A 2 , Π=0, M =1)· Pr(A 2 |S 2 , Π=0, M =1)· Pr(S 2 |Π=0) = S2 Pr(Y |S 2 , Π=0, M =1)· Pr(S 2 |Π=0) = S2 Pr(S 2 |Π=0) Pr(S 2 |Π=0, M =1) Pr(Y, S 2 |Π=0, M =1) = S2 Pr(S 2 ) Pr(S 2 |M =1) Pr(Y, S 2 |Π=0, M =1)
where the final step is by independence of S 2 and Π.
Corollary 1. If S 2 and M are independent, then Y |Π=0, M =1 has the same distribution as Y |Π=1.
Proof. Follows immediately from Lemma 1.
To achieve independence of S 2 and M , we could ensure during data collection that A 0 2 is independent of S 2 , which in turn can be achieved by equal randomization independent of S 2 . This is common, but not universal, in SMART designs [Collins et al., 2014].
If A 0 2 |S 2 =s 2 ∼ Bernoulli(θ 0 ) and A * 2 |S 2 =s 2 ∼ Bernoulli(θ * s2 ), then Pr(M =1|S 2 =s 2 ) = θ 0 θ * s2 + (1 − θ 0 )(1 − θ * s2 ) = 1 − θ 0 + θ * s2 (2θ 0 − 1).
Hence, if θ 0 = 0.5, then Pr(M =1|S 2 ) = Pr(M =1) = 0.5, and Pr(S 2 |M =1) = Pr(S 2 ). Using this subset of trajectories whose s 2 matchesπ * 2 (s 2 ), we can regress Y on S 1 and A 1 to construct tolerance intervals using (7), or, as above, we can construct non-parametric tolerance intervals for each pattern of (s 1 , a 1 ) using (8).
Dependence of M on S 2 is problematic because of the effect of S 2 on Y . When M depends on S 2 , conditioning on M can affect the distribution of Y through S 2 , meaning that the distribution of Y |S 1 , A 1 , Π=0, M =1 we estimate by collecting data under π 0 2 is not what we would have obtained had we collected data underπ * 2 and ignored (i.e. marginalized over) M . To correct the problem of the distribution of S 2 |S 1 , A 1 among the matched trajectories, we employ inverse probability weighting. To do so, we construct a propensity score model, not for the probability of treatment, but for the probability of following the estimated optimal DTR, i.e. Pr(M =1|S 2 , S 1 , A 1 ). Using this model, we can then re-weight the trajectories so that the distribution of S 2 |M =1, S 1 , A 1 matches the distribution of S 2 |S 1 , A 1 as well as possible. The weight function is therefore
w(s 1 , a 1 , s 2 ) = Pr(S 2 =s 2 |S 1 =s 1 , A 1 =a 1 ) Pr(S 2 =s 2 |S 1 =s 1 , A 1 =a 1 , M =1) .(12)
These are sometimes known as importance weights. We note that in causal inference, importance weights are sometimes used to adjust for an association between the probability of receiving treatment and the observed outcome. Here, they are used to adjust for an association between the probability of following the estimated optimal policy and the observed outcome through the variable S 2 . Note that estimating the two densities in (12) separately is not necessary to estimate the function w; it can be estimated using any density ratio estimation method. Logistic regression is one common approach but many others are available. In related weighting methods for causal inference, practitioners have found that a flexible model for w is often preferable to a simpler one [Ghosh, 2011].
To use the weighted data for building tolerance intervals, we must adapt existing methods for use with the weights. To build normal-theory regression tolerance intervals using the weighted data, we first estimate y|X=x using weighted least squares. We then use the resulting mean estimate, together with a weight-based sandwich estimate ofσŷ to construct the tolerance interval as per (7). To build non-parametric tolerance intervals, we obtain weighted estimates of the ranks obtained by linear interpolation of the weighted empirical distribution [Harrell et al., 2015]. We then construct the Wilks interval as per (8). Figure 2 shows the empirical results of applying weighted tolerance intervals in a simple scenario. Our goal here is to verify that the weighting scheme can counteract some of the dependence on M . (We will evaluate them more fully in the next section.) The data are drawn from a two-variable generative model with M ∼ Bernoulli(0.5) and Y |M =m ∼ N (µ m , σ m ). Our goal is to produce a tolerance interval for Y , marginal over M , using only data for which M =1. The sample size for M = 1 was n = 500, and the weights were computed analytically. Parameters for Y |M =0 were fixed at µ 0 = 0 and σ 0 = 1. Parameters for Y |M =1 were varied to illustrate how performance of the weighted tolerance intervals changed as the distribution of Y |M =1 deviated from the marginal distribution of Y . The top row of heatmaps shows the coverage of each method, that is, the proportion of times out of 1000 Monte Carlo replicates for which the computed tolerance interval had at least γ = 0.9 probability content. The confidence level 1 − α was set to 0.95; in the plot, Monte Carlo coverages that are not statistically significantly different from 0.95 are coloured pure white. Over-coverage is coloured blue, and under-coverage is coloured orange. The second row plots the average width of the tolerance intervals, normalized by the width of the optimal tolerance interval constructed from the true quantiles of Y , with unit relative width coloured white.
Methods beginning with U are unweighted, and methods beginning with W are weighted. Methods containing NP are nonparametric, and those without NP are normal-theory. (Table 1 gives the complete key to the method names.) Note that except when µ 1 = 0 and σ 1 = 1, Y is nonnormal. As one would expect, performance when µ 1 = 0, σ 1 = 1 is very good across all methods; in this case, Pr(Y |M = 1) = Pr(Y ), and weighting is not needed. When µ 1 is near zero and/or σ 2 is larger than σ 0 , most of the mass of Pr(Y |M = 1) overlaps the mass of Pr(Y ), and all intervals tend to over-cover. This is indicated by the blue regions in the upper-left corner of the coverage plots, and is larger in the weighed methods than the unweighted Weighted Normal-theory TI methods. Conversely, when Pr(Y |M = 1) does adequately overlap the mass of Pr(Y ) because µ 1 is farther from 0 and/or σ 1 is less than σ 0 , we see undercoverage indicated by the orange in the lower-right of the plots. Again, this is mitigated by weighting. The non-parametric methods provide better coverage than the normal-theory methods; this is not surprising since Y is not normal in most cases. The width plots verify that the weighted methods bring the extreme widths observed from the unweighted methods closer to optimal. This example verifies that the weighted methods we propose can substantially reduce over-and undercoverage in cases where there is mismatch between the observed distribution and the distribution of interest. However, they cannot eliminate it entirely when the distributions of Y and Y |M =1 are very different. This is to be expected; estimating say the mean of one distribution using an importance-weighted sample is challenging in practice. Estimating the tails of that distribution is even more challenging. Nonetheless, there is value in the weighted approach, and we will explore it further in the DTR setting in the next section.
Residual Borrowing
We now present a different approach to ensuring that our analysis captures the joint distribution Y, S 2 |S 1 , A 1 correctly, and hence captures variability in Y |S 1 , A 1 correctly when we marginalize over S 2 . To do so, we return to the Q-learning approach, which estimates E[Y |S 2 , A 2 ] using regression. As discussed above, the pseudooutcomeỹ for each trajectory represents our best estimate of E[Y |S 1 =s 1 , A 1 =a 1 , S 2 =s 2 , A 2 ] when A 2 ∼ π * 2 (s 1 , a 1 , s 2 ). This estimate is available for all trajectories in our dataset, including those for which M =1. Rather than naïvely constructing tolerance intervals based on the regression ofỸ on S 1 and A 1 , we create a new pseudooutcomey for each point: For trajectories with m=1, we sety = y. For trajectories with m = 0, we sety =ỹ + , where ∼ E, and E is an estimate of the distribution of the residuals among trajectories with M = 1. We call this procedure residual borrowing. We then construct tolerance intervals using the regression ofy on S 1 and A 1 .
Unlike theỹ, they retain information about the distribution of Y |S 2 , A 2 . Furthermore, since we use all of the trajectories in our original dataset, our empirical distribution of S 2 |S 1 , A 1 is representative of the true generative model. The distribution E could be the empirical distribution of the appropriate residuals, or it could be a smoothed estimate, e.g., a kernel density estimate. In our simulations, we found that a smoothed estimate works better than sampling from the empirical distribution.
Empirical Results
We now present results of six tolerance interval methods, which are listed in Table 1, using a simulation study. Our goals are to: 1. verify that inverse probability weighted methods can succeed where the unweighted methods fail, and test their limits; and, 2. to assess the difference in performance between the inverse probability weighted methods and the residual borrowing methods. Note that we do not include results from the naïve method as it performs very poorly.
The generative model from the study is taken from Schulte et al. [2014], with modifications. We begin by reviewing that model and discussing our modifications to it; we then present and discuss the performance of our methods.
Generative Model
The generative model has 2 decision points. S 1 is binary, A 1 is binary, S 2 is continuous, A 2 is binary, and Y is continuous. The generative model under the exploration DTR is given by
S 1 ∼ Bernouli(0.5) A 0 1 |S 1 =s 1 ∼ Bernoulli{expit( ξ φ {φ 0 10 + φ 0 11 s 1 })} S 2 |S 1 =s 1 , A 1 =a 1 ∼ Normal(δ 0 10 + δ 0 11 s 1 + δ 0 12 a 1 + δ 0 13 s 1 a 1 , 2) A 0 2 |S 1 =s 1 , S 2 =s 2 , A 1 =a 1 ∼ Bernoulli{expit( ξ φ {φ 0 20 + φ 0 21 s 1 + φ 0 22 a 1 + φ 0 23 s 2 + φ 0 24 a 1 s 2 + φ 0 25 s 2 2 })} Y |S 1 =s 1 , S 2 =s 2 , A 1 =a 1 , A 2 =a 2 ∼ Ydist {µ Y (s 1 , s 2 , a 1 , a 2 ), σ 2 ε }
µ Y (s 1 , s 2 , a 1 a 2 ) = β 0 20 + β 0 21 s 1 + β 0 22 a 1 + β 0 23 s 1 a 1 + β 0 24 s 2 + β 0 25 s 2 2 + a 2 ξ ψ (ψ 0 20 + ψ 0 21 a 1 + ψ 0 22 s 2 )
Here, expit(x) = e x /(e x + 1). The original model is indexed by
φ 0 1 = (φ 0 10 , φ 0 10 ) = (0.3, −0.5) δ 0 1 = (δ 0 10 , δ 0 11 , δ 0 12 , δ 0 13 ) = (0, 0.5, −0.75, 0.25) φ 0 2 = (φ 0 20 , φ 0 21 , φ 0 22 , φ 0 23 , φ 0 24 , φ 0 25 ) = (0, 0.5, 0.1, −1, −0.1, 0) β 0 2 = (β 0 20 , β 0 21 , β 0 22 , β 0 23 , β 0 24 , β 0 25 ) = (3, 0, 0.1, −0.5, −0.5, 0) ψ 0 2 = (ψ 0 20 , ψ 0 21 , ψ 0 22 ) = (1, 0.25, 0.5)
to which we have added four parameters: ξ ψ is a factor multiplying ψ 0 2 , its default is 1; ξ φ is a factor multiplying φ 0 2 , its default is 1; Ydist(µ, σ 2 ε ) gives the conditional distribution of Y with given mean and variance; its default is the normal distribution and the default σ 2 ε is 10. We have emphasized these parameters by displaying them in boxes.
Our parameter ξ φ allows us to control the degree to which state information influences treatment selection under the exploration (data-gathering) DTR. For ξ φ = 1, we have the original exploration used by Schulte et al., and for ξ φ = 0, we have uniform randomization over treatments independent of state and previous treatment. ξ ψ allows us to control the effect of treatment A 2 on Y . For ξ ψ = 1 we have the treatment effect specified by Schulte et al., and for ξ ψ = 0 we have no treatment effect at the second stage. Ydist allows us to control the shape of the error distribution to see its effect on the tolerance interval methods; Schulte et al. used a normal error, but we will explore heavier and lighter-tailed errors while holding variance constant.
This family of generative models allows us to explore what happens to the performance of tolerance interval methods when we have dependence of S 2 on A 2 during the generating process. While most of the SMART studies we are aware of use a simple randomization strategy where the distribution of A 2 does not depend on S 2 (which is the case here when e.g. ξ φ = 0, giving a simple 50:50 randomization strategy), we expect that more studies akin to "adaptive trials" with state-dependent randomization will become attractive in the future.
Based on the function 3 µ Y which determines the expected value of Y |S 1 , A 1 , S 2 , A 2 , we can immediately see that the optimal second stage decision function is π * 2 (a 1 , s 2 ) = arg max a 2 a 2 ξ ψ (ψ 0 20 + ψ 0 21 a 1 + ψ 0 22 s 2 ) = I{ξ ψ (ψ 0 20 + ψ 0 21 a 1 + ψ 0 22 s 2 ) > 0}.
3 Denoted m by Schulte et al.
11
Working Model
Our working model for Q 2 is Q 2 (s 1 , a 1 , s 2 , a 2 ; β 2 , ψ 2 ) = β 20 + β 21 s 1 + β 22 a 1 + β 23 s 1 a 1 + β 24 s 2 + β 25 s 2 2 + a 2 (ψ 20 + ψ 21 a 1 + ψ 22 s 2 ) (13)
Having computed least squares estimatesβ 2 andψ 2 , our estimate of the optimal second-stage decision function isπ * 2 (s 1 , a 1 ) = I{ψ 20 +ψ 21 a 1 +ψ 0 22 s 2 > 0} (14) and the pseudooutcome for the ith trajectory is y i =β 20 +β 21 s 1i +β 22 a 1i +β 23 s 1i a 1i +β 24 s 2i +β 25 s 2 2i + |ψ 20 +ψ 21 a 1i +ψ 0 22 s 2i | + .
Our working model for Q 1 is the saturated model Q 1 (s 1 , a 1 ; β 1 , ψ 1 ) = β 10 + β 21 s 1 + a 1 (ψ 10 + ψ 11 s 1 ).
Having computed least squares estimatesβ 2 andψ 2 by regressing the pseudooutcomes on s 1 and a 1 , our estimate of the optimal first-stage decision function would be 4 π * 1 (s 1 , a 1 ) = I{ψ 10 +ψ 11 a 1 > 0}.
Tolerance Intervals
In many studies of DTR methods, the focus is on point and interval estimates of the optimal stage 1 decision parameters [Chakraborty et al., 2010, 2013, Chakraborty and Moodie, 2013, Laber et al., 2014b, Chakraborty et al., 2014. In this work, we will investigate methods for constructing tolerance intervals for
Y |S 1 =0, A 1 =0;π * 2 Y |S 1 =0, A 1 =1;π * 2 Y |S 1 =1, A 1 =0;π * 2 Y |S 1 =1, A 1 =1;π * 2 .
Note that our goal is to construct tolerance intervals for Y under the estimated optimal regime rather than under the optimal regime. The reason for this is pragmatic: we assume that it is the estimated optimal regime that would be deployed in future to support decision-making. We begin by estimatingπ * 2 using the working models (13,14). We then compute the pseudooutcomeỹ i for each trajectory, and the match indicator m i = I{π * 2 (s 1i , a 1i , s 2i ) = a 2i }.
Unweighted Methods
To construct the unweighted normal-theory TIs, we regress y on s 1 and a 1 according to working model (15) but using only trajectories with m = 1. We then apply (7) to construct the four tolerance intervals.
To construct the unweighted nonparametric TIs, we divide the trajectories with m = 1 into four mutually exclusive groups according to their (s 1 , a 1 ) values. We then construct the four tolerance intervals by applying the Wilks method (8) to each group.
Weighted Methods
To construct the weights, we first form kernel density estimatesf E (s 2 ; a 1 , a 1 , m=1) for S 2 |S 1 =s 1 , A 1 =a 1 , M =1 andf E (s 2 ; s 1 , a 1 ) for S 2 |S 1 =s 1 , A 1 =a 1 . The weight for a trajectory with index i that has m = 1 is then given by
w i =f E (s 2i ; s 1i , a 1i ) f E (s 2i ; s 1i , a 1i , m = 1) .(17)
While logistic regression might be viewed as a more obvious choice for this task, we found that its attendant monotonicity assumptions were often violated, and that the pair of kernel density estimates were the simplest way to produce a more flexible model in this low-dimensional setting.
To construct the weighted normal-theory TIs, as above we compute a weighted regression of y on s 1 and a 1 according to working model (15) but using only trajectories with m = 1. We then apply (7) to construct the four tolerance intervals; in this case, we use the sandwich estimate [Huber, 1967, White, 1980 with the weights to computeσ Y |X=x . This makes the method somewhat more robust.
To construct the unweighted nonparametric TIs, we divide the trajectories with m = 1 into four mutually exclusive groups according to their (s 1 , a 1 ) values. We then construct the four tolerance intervals by applying our weighted modification of the Wilks method (8) to each group.
Residual Borrowing
For the residual borrowing methods, within each (s 1 , a 1 ) group, we first form a kernel density estimatê f R (r; s 1 , a 1 ) using the residuals y i −ỹ i among the trajectories with m = 1. We then sety i = y i for each trajectory with m i = 1, and sampley i from the kernel density estimate for trajectories with m i = 0. We then either regressy i using the working models to create the regression tolerance intervals, or we again divide up the data according to s 1 and a 1 to construct non-parametric tolerance intervals.
Results
Using the foregoing generative model, working models, and tolerance interval methods, we ran a suite of simulations to investigate performance. Experiments varied by ξ φ , ξ ψ , σ 2 ε , and Ydist, for a total of 1, 089 different experimental settings. Both ξ φ and ξ ψ were varied from 0 to 1 in 0.1 increments, and σ 2 ε took values in {10, 1, 0.1}. We examined settings with Ydist as normal, uniform, and t with 3 degrees of freedom, each scaled to have the appropriate σ 2 ε . For each setting, we drew 1000 simulated datasets each of size n = 1000, computed tolerance intervals using each of the six methods, and evaluated their content, that is, what proportion of Y was captured by each interval, and their relative width, given by (u t − t )/h * , where h * is the width of the optimal tolerance interval computed using the γ/2 and 1 − γ/2 quantiles of the true distribution. For all experiments, we set 1 − α = 0.95 and γ = 0.9. All kernel density estimates were one-dimensional, and used the default optimal bandwidth. All experimental code was written in R [R Core Team, 2015], and is publicly available. Figures 5, 6, and 7 display the results of all of our experiments as heatmaps using the same approach as Figure 2. Monte Carlo coverages that are not statistically significantly different from 0.95 are coloured pure white, Over-coverage is coloured blue, and under-coverage is coloured orange. The second row of each subplot gives the average width of the tolerance intervals. Figure 5a contains the original model setting proposed by Schulte et al. [2014] in the upper-right corners of its heatmaps. In this setting, the weighted and unweighted normal-theory tolerance intervals undercover slightly, while the weighted and unweighted non-parametric methods overcover, and are much wider. The residual-borrowing methods perform best in this setting, with the normal-theory residual-borrowing intervals achieving near-nominal coverage with modest width. There is relatively little variation in coverage and width across ξ φ and ξ ψ in this setting, we believe because the noise level is quite high relative to the effect of a 2 even when ξ ψ = 1. In Figure 5b, Ydist was chosen to be a t-distribution with 3 degrees of freedom, scaled to have variance σ 2 ε = 10 and shifted by µ Y . In this heavy-tailed setting, it is the non-parametric residual-borrowing method that slightly undercovers, while the other methods overcover somewhat. As in the normal case, the weighted and unweighted nonparametric methods are very wide. Figure 5c uses a scaled and shifted uniform distribution for Ydist, again maintaining σ 2 ε = 10. In this light-tailed setting, in contrast to Figure 5b, it is the normal-theory intervals which tend to be wide, while the non-parametric ones are narrower. The residual-borrowing intervals are wide as well. All intervals achieve nominal or greater coverage in this setting.
We see a striking change as we examine the lower-noise settings in Figure 6, which have σ 2 ε = 1. Here, we start to see dependence of performance on ξ ψ and ξ φ . As in Figure 5a, in Figure 6a we see the normal theory intervals undercovering, although we now see a definite trend that worsens as ξ φ increases, and as Figure 3: Bias in the estimated value of the optimal policy. This is a surrogate measure of non-regularity; note that maximal bias occurs when ψ, which controls the effect of A 2 , is small. Phi Factor (ξ φ in the text) controls the effect of covariates on the exploration DTR, and Psi Factor (ξ ψ in the text) controls the effect of A 2 on the conditional mean of Y . 14 ξ ψ decreases. We also see this trend among the non-parametric methods, which range from overcovering to undercovering as we move across ξ φ and ξ ψ . Overall, we see the greatest coverage when the effect of A 2 is quite strong (topmost rows), or if the dependence of A 0 2 on S 2 is weak (leftmost rows.) As we discussed earlier, when ξ φ = 0 (leftmost columns) there is no dependence of M on S2, and thus weighting is unnecessary. Furthermore, we not only obtain a uniform probability of M = 1 across S 2 , but also a uniform probability of A 0 2 across S 2 . This uniformity likely leads to improved estimates of Y |S 2 , A 2 , and in turn to better coverage of the tolerance intervals. The decrease in performance for low ξ ψ may be due to non-regularity: when ξ ψ = 0, there is in fact no effect of A 2 on Y . However, assuming continuity of the appropriate distributions, our estimatedψ 2 will be nonzero almost always, and our plug-in estimate of the value ofπ * 2 will be positive almost always. Definingâ * 2i =π * 2 (s 1i , a 1i , s 2i ), the empirical bias in the value ofπ * 2 is iâ * 2i (ψ 20 +ψ 21 a 1i +ψ 22 s 2i )−â * 2i ξ ψ (ψ 0 20 + ψ 0 21 a 1i + ψ 0 22 s 2i ). Figure 3 shows the average empirical bias in our estimate of the average value of usingπ * 2 , as a function of ξ φ and ξ ψ . We can see that the bias is concentrated at the bottom of the plots, near ξ ψ = 0. This is precisely where there is more than one nearly-optimal action and non-regularity is known to be a problem.
We see the problems worsen in Figure 7, where we set σ 2 = 0.1. We hypothesise that this is because proportionately even more of the variability in Y is attributable to variability in S 2 , and accurate estimation of Y |S 2 , A 2 becomes that much more important. All of the matched subset methods have severe undercoverage for large values of φ and low values of ψ. Weighted methods mitigate this. The residual-borrowing methods achieve much better coverage, but at the cost of much wider intervals.
Discussion
Based on our simulation study experiments, we believe that designing the exploration DTR to have uniform randomization over actions is highly beneficial for estimating tolerance intervals. When this is the case, all methods gave reasonable results in almost all scenarios. Some knowledge of the error distribution may help choose a method that will result in reasonable widths. If uniform exploration is not possible, the residualborrowing methods appear to be the most robust to undercoverage, followed by the weighted methods, followed by the unweighted methods. That said, it would be prudent to perform a simulation study under a scenario "close" to the analysis at hand if possible; to facilitate this we have released our R code [R Core Team, 2015] so that researchers and practitioners can explore other scenarios.
6 Example: STAR*D We present an example of the application of the TI methods we have described to real-world clinical trial data. The Sequenced Treatment Alternatives to Relieve Depression (STAR*D) study followed an initial population of 4041 patients as they were treated using different antidepressant medications and cognitive behavioural therapy [Rush et al., 2004]. There were a total of three decision points at which randomisation took place, with different treatment options available at each one. Outcomes were measured using the clinician-rated Quick Inventory of Depressive Symptomatology [Rush et al., 2003]. We will examine two such decision points corresponding to Level 2 and Level 3 of the study, which will correspond to the first and second decision points in our analysis.
We construct tolerance intervals for STAR*D at Level 2 (our decision point 1), having estimated a Q function and estimated optimal policy for Level 3 (our decision point 2.) We use exactly the same Q-learning working model and estimation procedure as Schulte et al. [2014] to developπ * 2 and the pseudooutcomes; we refer the interested reader to their work for more details. In summary, the state variables we use are up-todate QIDS measures of patient symptoms, and the outcomes we use are based on later QIDS measurements that have been negated so that higher values are preferable. At decision point 1, we elect to use a binary state Figure 4: Tolerance intervals for STAR*D at Level 2. The six TI methods used previously are applied to the data, using the choice to switch or augment treatment as A 1 , and letting S 1 be an indicator variable for previous QIDS slope being greater than the median. In this setting, higher outcomes are preferable, but higher QIDS scores (and slopes) indicate worse symptoms.
variable indicating whether the previous slope in QIDS score for a patient is greater than the median. Higher QIDS scores indicate worse symptom levels, so this state variable effectively identifies patients whose disease status is worsening most quickly. At both decision points, the treatment choice is whether to "augment" the current medication with another, or to "switch" to another medication altogether. We applied the six TI methods described previously to the data, using the choice to switch or augment treatment as A 1 , and letting S 1 be an indicator variable for QIDS slope being greater than the median slope. We see that generally the intervals are quite wide, and that there is severe overlap of TIs for different treatments. This reflects the high variance and low treatment effect we observe in this data. However, the intervals do capture prognostic information: the intervals for S 1 = "Yes" (indicating severely worsening symptoms) are wider, with a decreased lower bound indicating that such patients may have poorer outcomes relative to those with more stable symptoms prior to the decision point. The maximum attainable outcome in this problem is 0, since QIDS cannot go below 0. We note that the parametric TI methods can produce upper bounds greater than 0 and lower bounds that appear to be a bit optimistic. Hence, we suggest that one of the non-parametric methods would be a sensible choice for STAR*D.
Conclusion
We have developed and evaluated tolerance interval methods for dynamic treatment regimes that can provide more detailed prognostic information to patients who will follow an estimated optimal regime. We began by reviewing in detail different interval estimation and prediction methods and then adapting them to the DTR setting. We illustrated some of the challenges associated with tolerance interval estimation stemming from the fact that we do not typically have data that were generated from the estimated optimal regime. We gave an extensive empirical evaluation of the methods and discussed several practical aspects of method choice. We demonstrated the methods using data from a pragmatic clinical trial. We now take the opportunity to discuss future directions of research on tolerance intervals for dynamic treatment regimes.
Future Directions
Our work lays the foundation for extending tolerance interval methods for dynamic treatment regimes in several different directions.
The normal theory TI methods we employed used an estimate of the residual distribution that is pooled over S 1 and A 1 . The non-parametric methods estimated the residual distributions separately for the different discrete S 1 , A 1 . A compromise solution that partially shares residual information across different configurations of (S 1 , A 1 ), perhaps in a data-driven, adaptive fashion, may provide improved performance and wider applicability. (Note that the non-parametric methods we described are not applicable if S 1 is continuous.)
We have treated DTRs with two decision points, but in general we would like to have tolerance intervals for multiple decision points. Such methods would potentially have to address uncertainty stemming from "parameter sharing," across time points. It is known [Chakraborty et al., 2016] that the effects of model misspecification and non-regularity can compound in the multiple decision point setting, and the impact of this on tolerance intervals is not yet known.
While we assumed a single outcome measure Y throughout our work, several methods have been described for estimating DTRs in the presence of multiple outcomes [Lizotte et al., 2012, Laber et al., 2014a, Lizotte and Laber, 2015. Joint tolerance intervals/tolerance regions for this setting would be equally important as they are in the standard, single-outcome setting.
We observed some problems associated with biased estimates of the value of the estimated policy, which is caused by non-regularity. The problem of non-regularity in optimal DTR estimation has been addressed in the confidence interval setting using different approaches, including pre-testing [Laber et al., 2014b] and shrinkage [Chakraborty et al., 2010. We have not explicitly incorporated either of these ideas in the methods we presented; doing so may lead to methods that are more robust to small or zero treatment effects at the second stage yet do not pay a high cost in terms of width.
Fernholz and Gillespie [2001] have presented a method to re-calibrate tolerance intervals using the bootstrap. They propose a bootstrap method to estimate the content γ of a given tolerance interval-first they construct a tolerance interval with nominal (or "requested") content γ, but then they use the bootstrap to estimate what the actual content. This could potentially be used to identify when tolerance methods fail on dynamic treatment regimes, or they may be used simply to give more accurate confidence information to the decision maker. For example, we may attempt to construct a tolerance interval for γ = 0.9, but if it turns out that the actual content is 0.85, the interval may still be useful if the decision-maker is made aware of this fact. Future work to adapt the calibration procedure could prove promising.
Finally, a Bayesian approach to the predictive estimation problem may prove fruitful in some settings. Saarela et al. [2015] have laid groundwork for this direction of research.
Figure 2 :
2Comparison of coverage and width of inverse probability weighted tolerance interval methods. Axes represent a space of simple generative models. Lighter colouration indicates better performance.
Figure 5 :
5Coverage and average Relative Width for all methods and σ 2 ε = 10. Phi Factor (ξ φ in the text) controls the effect of covariates on the exploration DTR, and Psi Factor (ξ ψ in the text) controls the effect of A 2 on the conditional mean of Y . 21
Figure 6 :
6Coverage and average Relative Width for all methods and σ 2 ε = 1. Phi Factor (ξ φ in the text) controls the effect of covariates on the exploration DTR, and Psi Factor (ξ ψ in the text) controls the effect of A 2 on the conditional mean of Y . 22
Figure 7 :
7Coverage and average Relative Width for all methods and σ 2 ε = .1. Phi Factor (ξ φ in the text) controls the effect of covariates on the exploration DTR, and Psi Factor (ξ ψ in the text) controls the effect of A 2 on the conditional mean of Y . 23
Table 1 :
1Plot Acronyms for Tolerance IntervalsRBQNPTI Residual-Borrowing Non-parametric TI
RBQTI
Residual-Borrowing Normal-theory TI
UNPTI
Unweighted Non-parametric TI
UTI
Unweighted Normal-theory TI
WNPTI
Weighted Non-parametric TI
WTI
Note we are not matching trajectories with other trajectories-we are identifying trajectories whose action matches a DTR of interest.
This assumption is critical: if A * 0 |S 2 is not deterministic, the relationship between Y |Π=1 and Y |Π=0, M =1 is more complicated.
Schulte et al. [2014] give the true optimal values of β 1 and ψ 1 as a function of the other model parameters.
AcknowledgementsThis work was supported by the Natural Sciences and Engineering Research Council of Canada. Data used in the preparation of this article were obtained from the limited access datasets distributed from the NIH-supported "Sequenced Treatment Alternatives to Relieve Depression" (STAR*D). STAR*D focused on non-psychotic major depressive disorder in adults seen in outpatient settings. The primary purpose of this research study was to determine which treatments work best if the first treatment with medication does not produce an acceptable response. The study was supported by NIMH Contract #N01MH90003 to the University of Texas Southwestern Medical Center. The ClinicalTrials.gov identifier is NCT00021528.
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| [] |
[
"A CHOICE-FREE CARDINAL EQUALITY",
"A CHOICE-FREE CARDINAL EQUALITY"
] | [
"Guozhen Shen "
] | [] | [] | For a cardinal a, let fin(a) be the cardinality of the set of all finite subsets of a set which is of cardinality a. It is proved without the aid of the axiom of choice that for all infinite cardinals a and all natural numbers n, 2 fin(a) n = 2 [fin(a)] n . On the other hand, it is proved that the following statement is consistent with ZF: there exists an infinite cardinal a such that 2 fin(a) < 2 fin(a) 2 < 2 fin(a) 3 < · · · < 2 fin(fin(a)) .2 fin(fin(a)) = 2 fin(fin(fin(a))) = 2 fin(fin(fin(fin(a)))) = · · · .2010 Mathematics Subject Classification. Primary 03E10, 03E25. | 10.1215/00294527-2021-0028 | [
"https://arxiv.org/pdf/1912.12435v1.pdf"
] | 209,515,520 | 1912.12435 | 4fde2b3bf385978583c8ce7e2d4670dc3ae17ec9 |
A CHOICE-FREE CARDINAL EQUALITY
28 Dec 2019
Guozhen Shen
A CHOICE-FREE CARDINAL EQUALITY
28 Dec 2019
For a cardinal a, let fin(a) be the cardinality of the set of all finite subsets of a set which is of cardinality a. It is proved without the aid of the axiom of choice that for all infinite cardinals a and all natural numbers n, 2 fin(a) n = 2 [fin(a)] n . On the other hand, it is proved that the following statement is consistent with ZF: there exists an infinite cardinal a such that 2 fin(a) < 2 fin(a) 2 < 2 fin(a) 3 < · · · < 2 fin(fin(a)) .2 fin(fin(a)) = 2 fin(fin(fin(a))) = 2 fin(fin(fin(fin(a)))) = · · · .2010 Mathematics Subject Classification. Primary 03E10, 03E25.
Introduction
For a cardinal a, let fin(a) be the cardinality of the set of all finite subsets of a set which is of cardinality a. The axiom of choice implies that fin(a) = a for any infinite cardinal a. However, in the absence of the axiom of choice, this is no longer the case. In fact, in the ordered Mostowski model (cf. [2, pp. 198-202]), the cardinality a of the set of atoms satisfies fin(a) < [fin(a)] 2 < fin(a) 2 < [fin(a)] 3 < fin(a) 3 < · · · < fin(fin(a)) < fin(fin(fin(a))) < · · · < ℵ 0 · fin(a). (1) It is natural to ask which relationships between the powers of the cardinals in (1) for an arbitrary infinite cardinal a can be proved without the aid of the axiom of choice.
The first result of this kind is Läuchli's lemma (cf. [3] or [2,Lemma 5.27]), which states that for all infinite cardinals a, 2 ℵ 0 ·fin(a) = 2 fin(a) .
Läuchli's lemma implies that, in the ordered Mostowski model, the powers of the cardinals in (1) are all equal, where a is the cardinality of the set of atoms.
In this paper, we give a complete answer to the above question. We first prove in ZF that for all infinite cardinals a, Then, as our main result, we prove in ZF that for all infinite cardinals a and all natural numbers n, 2 fin(a) n = 2 [fin(a)] n .
Finally, we prove that the following statement is consistent with ZF: there exists an infinite cardinal a such that 2 fin(a) < 2 fin(a) 2 < 2 fin(a) 3 < · · · < 2 fin(fin(a)) .
Basic notions and facts
Throughout this paper, we shall work in ZF. In this section, we indicate briefly our use of some terminology and notation. The cardinality of x, which we denote by |x|, is the least ordinal α equinumerous to x, if x is well-orderable, and the set of all sets y of least rank which are equinumerous to x, otherwise. We shall use lower case German letters a, b for cardinals.
For a function f , we shall use
dom(f ) for the domain of f , ran(f ) for the range of f , f [x] for the image of x under f , f −1 [x]
for the inverse image of x under f , and f ↾x for the restriction of f to x. For functions f and g, we use g • f for the composition of g and f .
Definition 2.1. Let x, y be arbitrary sets, let a = |x|, and let b = |y|.
(1) x y means that there exists an injection from x into y; a b means that x y. (2) x * y means that there exists a surjection from a subset of y onto x; a * b means that x * y. (3) a b (a * b) denotes the negation of a b (a * b). (4) a < b means that a b and b a. (5) a = * b means that a * b and b * a.
It follows from the Cantor-Bernstein theorem that if a b and b a then a = b. Clearly, if a b then a * b, and if a * b then 2 a 2 b . Thus a = * b implies that 2 a = 2 b . Definition 2.2. Let x, y be arbitrary sets, let a = |x|, and let b = |y|.
(1) x y is the set of all functions from y into x; a b = |x y |.
(2) x y is the set of all injections from y into x; a b = |x y |.
(3) [x] y is the set of all subsets of x which have the same cardinality as y;
[a] b = |[x] y |. (4) seq(x) = n∈ω x n ; seq(a) = | seq(x)|.
(5) seq 1-1 (x) = n∈ω x n ; seq 1-1 (a) = | seq 1-1 (x)|. (6) fin(x) = n∈ω [x] n ; fin(a) = | fin(x)|.
Below we list some basic properties of these cardinals. We first note that fin(a) * seq 1-1 (a) seq(a). Fact 2.3. For all cardinals a, seq 1-1 (a) fin(fin(a)).
Proof. For every set x, the function f defined on seq 1-1 (x) given by f (t) = {t[n] | n dom(t)} is an injection from seq 1-1 (x) into fin(fin(x)). Proof. Let x be an infinite set. Let p be a bijection from ω × ω onto ω such that n p(m, n) for any m, n ∈ ω. Let f be the function defined on seq 1-1 (x) given by
f (t) = (m, t↾n),
where m, n ∈ ω are such that dom(t) = p(m, n). It is easy to see that f is a surjection from seq 1-1 (x) onto ω × seq 1-1 (x). Proposition 2.7. For all infinite cardinals a, seq 1-1 (a) = * fin(fin(a)) = * fin(fin(fin(a))) = * · · · = * seq(a).
Proof. Immediately follows from Fact 2.3 and Lemmata 2.4, 2.5 and 2.6. Corollary 2.8. For all infinite cardinals a, 2 seq 1-1 (a) = 2 fin(fin(a)) = 2 fin(fin(fin(a))) = · · · = 2 seq(a) .
Proof. Immediately follows from Proposition 2.7.
The following lemma will be used in Section 4. Lemma 2.9. For all cardinals a and all n ∈ ω, a 2 n fin(a) n+1 .
Proof. Let x be an arbitrary set and let n ∈ ω. Let f be the function defined on x ℘(n) such that for all t ∈ x ℘(n) , f (t) is the function on n + 1 given by
f (t)(k) = {t(∅)},
if k = n; {t(a) | a ⊆ n and k ∈ a}, otherwise.
Clearly, ran(f ) ⊆ fin(x) n+1 . It is easy to verify that for all t ∈ x ℘(n) , t is the function defined on ℘(n) given by
t(a) = f (t)(n), if a = ∅; k∈a f (t)(k) \ k∈n\a f (t)(k) , otherwise. Hence, f is an injection from x ℘(n) into fin(x) n+1 .
The main theorem
In this section, we prove our main result which states that for all infinite cardinals a and all natural numbers n,
2 fin(a) n = 2 [fin(a)] n .
The main idea of the proof is originally from [3].
Fix an arbitrary infinite set A and a non-zero natural number n. For a finite sequence x 1 , . . . , x n of length n, we write x = x 1 , . . . , x n for short. For finite sequences x = x 1 , . . . , x n and y = y 1 , . . . , y n , we introduce the following abbreviations: x ⊑ y means that x i ⊆ y i for any i = 1, . . . , n;
x ⊏ y means that x ⊑ y but x = y; x ⊔ y denotes the finite sequence x 1 ∪ y 1 , . . . , x n ∪ y n ; x ⊓ y denotes the finite sequence x 1 ∩ y 1 , . . . , x n ∩ y n ; ∅ denotes the finite sequence ∅, . . . , ∅ of length n. For an operator H and an m ∈ ω, we write H (m) (X) for H(H(· · · H(X) · · · )) (m times), and if m = 0 then H (0) (X) is X itself.
Definition 3.1. For all natural numbers k 1 , . . . , k n and l 1 , . . . , l n such that k i l i for any i = 1, . . . , n, we introduce the following three functions:
(1) F n, k, l is the function defined on ℘([A] k 1 × · · · × [A] kn ) given by F n, k, l (X) = y ∈ [A] l 1 × · · · × [A] ln x ⊑ y for some x ∈ X ; (2) G n, k, l is the function defined on ℘([A] k 1 × · · · × [A] kn ) given by G n, k, l (X) = x ∈ [A] k 1 × · · · × [A] kn for all y ∈ [A] l 1 × · · · × [A] ln if x ⊑ y then y ∈ F n, k, l (X) ; (3) H n, k, l is the function defined on ℘([A] k 1 × · · · × [A] kn ) given by H n, k, l (X) = G n, k, l (X) \ X.
The proof of the following fact is easy and will be omitted.
Fact 3.2. Let k 1 , . . . , k n and l 1 , . . . , l n be natural numbers such that k i l i for any i = 1, . . . , n.
(i) If X ⊆ Y ⊆ [A] k 1 × · · · × [A] kn then F n, k, l (X) ⊆ F n, k, l (Y ). (ii) If X ⊆ [A] k 1 × · · · × [A] kn then X ⊆ G n, k, l (X). (iii) If X ⊆ Y ⊆ [A] k 1 × · · · × [A] kn then G n, k, l (X) ⊆ G n, k, l (Y ). (iv) If X ⊆ [A] k 1 × · · · × [A] kn then G n, k, l (G n, k, l (X)) = G n, k, l (X). (v) If X ⊆ [A] k 1 × · · · × [A] kn then F n, k, l (G n, k, l (X)) = F n, k, l (X). (vi) F n, k, l is injective on {X ⊆ [A] k 1 × · · · × [A] kn | G n, k, l (X) = X}. (vii) If X ⊆ [A] k 1 × · · · × [A] kn and m ∈ ω then H (m) n, k, l (X) = G n, k, l (H (m) n, k, l (X)) \ H (m+1) n, k, l (X). (viii) Let l ′ 1 , . . . , l ′ n be natural numbers such that l i l ′ i for any i = 1, . . . , n. If X ⊆ [A] k 1 × · · · × [A] kn then G n, k, l (X) ⊆ G n, k, l ′ (X), and hence G n, k, l ′ (X) = X implies that G n, k, l (X) = X.
The key step of our proof is the following lemma.
i = 1, . . . , n, if X ⊆ [A] k 1 × · · · × [A] kn then H (k 1 +···+kn+1) n, k, l (X) = ∅.
Before we prove Lemma 3.3, we use it to prove our main theorem. Proof. Let A be an infinite set such that |A| = a. The case n = 0 is obvious. So assume that n is a non-zero natural number. For all natural numbers k 1 , . . . , k n , m, let s( k, m) be the finite sequence
p k 1 1 · · · p kn n p m n+1 p i n+2 1 i n
where p j is the j-th prime number, and let t( k) = s( k, k 1 + · · · + k n ).
For all X ⊆ fin(A) n and all natural numbers k 1 , . . . , k n , m, we define
X k = X ∩ ([A] k 1 × · · · × [A] kn ); Y k,m = G n, k,t( k) (H (m) n, k,t( k) (X k )); Z k,m = F n, k,s( k,m) (Y k,m ).
Notice that for any finite sequence x = x 1 , . . . , x n , ran( x) = {x 1 , . . . , x n }. Now, let Φ be the function defined on ℘(fin(A) n ) given by Φ(X) = ran( y) ∃k 1 , . . . , k n , m ∈ ω m k 1 + · · · + k n and y ∈ Z k,m .
We claim that Φ is an injection from ℘(fin(A) n ) into ℘([fin(A)] n ).
Let X ⊆ fin(A) n . For all y = y 1 , . . . , y n ∈ Z k,m , it is easy to see that |y i | = p k 1 1 · · · p kn n p m n+1 p i n+2 for any i = 1, . . . , n, and thus |y 1 | < · · · < |y n |, which implies that ran( y) ∈ [fin(A)] n . Hence Φ(X) ⊆ [fin(A)] n . Moreover, X is uniquely determined by Φ(X) in the following way:
First, for all natural numbers k 1 , . . . , k n , m such that m k 1 + · · · + k n , Z k,m is uniquely determined by Φ(X):
Z k,m = y ∈ [A] l 1 × · · · × [A] ln ran( y) ∈ Φ(X) ,
where l i = p k 1 1 · · · p kn n p m n+1 p i n+2 for any i = 1, . . . , n. Then, for all natural numbers k 1 , . . . , k n , m such that m k 1 + · · · + k n , by Fact 3.2(iv)(vi)(viii), Y k,m is the unique subset of
[A] k 1 × · · · × [A] kn such that G n, k,t( k) (Y k,m ) = Y k,m and F n, k,s( k,m) (Y k,m ) = Z k,m , which implies that Y k,m is uniquely determined by Φ(X).
Now, for all natural numbers k 1 , . . . , k n , it follows from Fact 3.2(vii) and Lemma 3.3 that
X k = Y k,0 \ (Y k,1 \ (· · · (Y k,k 1 +···+kn−1 \ Y k,k 1 +···+kn ) · · · )),
and thus X k is uniquely determined by Φ(X).
Finally, since
X = k 1 ,...,kn∈ω X k ,
it follows that X is also uniquely determined by Φ(X). Hence, Φ is an injection from ℘(fin(A) n ) into ℘([fin(A)] n ), and thus 2 fin(a) n 2 [fin(a)] n . Since [fin(a)] n * fin(a) n , it follows that 2 [fin(a)] n 2 fin(a) n , and thus 2 fin(a) n = 2 [fin(a)] n follows from the Cantor-Bernstein theorem.
We still have to prove Lemma 3.3. To this end, we need the following version of Ramsey's theorem, whose proof will be omitted.
Lemma 3.5. Let n be a non-zero natural number. There exists a function R defined on ω n × (ω \ {0}) × ω such that for all natural numbers j 1 , . . . , j n , c, r with c > 0 and all finite sets S 1 , . . . , S n , Y 1 , . . . , Y c , if |S i | R(j 1 , . . . , j n , c, r) for any i = 1, . . . , n and
[S 1 ] j 1 × · · · × [S n ] jn = Y 1 ∪ · · · ∪ Y c , then for each i = 1, . . . , n there exist a T i ∈ [S i ] r such that [T 1 ] j 1 × · · · × [T n ] jn ⊆ Y d for some d = 1, . . . , c.
Proof of Lemma 3.3. Let A be an arbitrary infinite set and n a non-zero natural number. Let k 1 , . . . , k n and l 1 , . . . , l n be natural numbers such that k i l i for any i = 1, . . . , n. Since in this proof the natural numbers n, k 1 , . . . , k n , l 1 , . . . , l n are fixed, we shall omit the subscripts in F n, k, l , G n, k, l and H n, k, l for convenience.
Consider the following two formulae: φ(X, x, y): X ⊆ [A] k 1 × · · · × [A] kn and x, y ∈ fin(A) n are such that |x i | k i for any i = 1, . . . , n, such that x ⊓ y = ∅, and such that x ⊔ z ∈ X for any z ∈ [y 1 ] k 1 −|x 1 | × · · · × [y n ] kn−|xn| . ψ(X, x): For all r ∈ ω there exists a y ∈ ([A] r ) n such that φ(X, x, y).
We claim that for all X ⊆ [A] k 1 × · · · × [A] kn and all x ∈ fin(A) n , if ψ(H(X), x) then ψ(X, u) for some u ⊏ x.
(
Once we prove (2), we finish the proof of Lemma 3.3 as follows. Assume towards a contradiction that X ⊆ [A] k 1 × · · · × [A] kn and there exists an x ∈ H (k 1 +···+kn+1) (X). It is obvious that ψ(H (k 1 +···+kn+1) (X), x). Now, by repeatedly applying (2), we get a descending sequence
x ⊐ u 1 ⊐ · · · ⊐ u k 1 +···+kn+1 , which is absurd, since x ∈ [A] k 1 × · · · × [A] kn .
Now, let us prove (2). Let X ⊆ [A] k 1 × · · · × [A] kn and let x ∈ fin(A) n be such that ψ(H(X), x). It suffices to prove that
∀r l 1 + · · · + l n ∃ u ⊏ x ∃ y ∈ ([A] r ) n φ(X, u, y),(3)
since then there must be a u ⊏ x such that for infinitely many r ∈ ω there exists a y ∈ ([A] r ) n such that φ(X, u, y), and for this u we have ψ(X, u). We prove (3) as follows. Let r l 1 + · · · + l n . Let R be the function whose existence is asserted by Lemma 3.5. We define r ′ = max{R(j 1 , . . . , j n , 2, r) | j i k i for any i = 1, . . . , n}; r ′′ = R(l 1 − |x 1 |, . . . , l n − |x n |, 2 |x 1 |+···+|xn| , r ′ ).
Since ψ(H(X), x), we can find an S = S 1 , . . . ,
S n ∈ ([A] r ′′ ) n such that φ(H(X), x, S). Notice that x ⊓ S = ∅. For each u ⊑ x, let Y u = w ∈ [S 1 ] l 1 −|x 1 | × · · · × [S n ] ln−|xn| u ⊔ v ∈ X for some v ⊑ w .
We claim that
[S 1 ] l 1 −|x 1 | × · · · × [S n ] ln−|xn| = {Y u | u ⊑ x}. (4) Let w ∈ [S 1 ] l 1 −|x 1 | × · · · × [S n ] ln−|xn| . Take a z ∈ [S 1 ] k 1 −|x 1 | × · · · × [S n ] kn−|xn| such that z ⊑ w.
Then it follows from φ(H(X), x, S) that x ⊔ z ∈ H(X), and thus x ⊔ z ∈ G(X). Since x ⊔ z ⊑ x ⊔ w ∈ [A] l 1 × · · · × [A] ln , it follows that x ⊔ w ∈ F (X), and hence a ⊑ x ⊔ w for some a ∈ X. Now, if we take u = a ⊓ x and v = a ⊓ w, then we have u ⊔ v = a ∈ X and hence w ∈ Y u . By (4) and Lemma 3.5, we can find a u = u 1 , . . . , u n ⊑ x such that for each i = 1, . . . , n there exist a T i ∈ [S i ] r ′ such that
[T 1 ] l 1 −|x 1 | × · · · × [T n ] ln−|xn| ⊆ Y u . (5) Let Z = v ∈ [T 1 ] k 1 −|u 1 | × · · · × [T n ] kn−|un| u ⊔ v ∈ X .
Since |T i | = r ′ R(k 1 −|u 1 |, . . . , k n −|u n |, 2, r) for any i = 1, . . . , n, it follows from Lemma 3.5 that we can find a y = y 1 , . . . , y n such that y i ∈ [T i ] r for any i = 1, . . . , n, and such that either
[y 1 ] k 1 −|u 1 | × · · · × [y n ] kn−|un| ⊆ Z (6) or ([y 1 ] k 1 −|u 1 | × · · · × [y n ] kn−|un| ) ∩ Z = ∅.(7)
We claim that (7) is impossible. Since |y i | = r l i l i − |x i | for any i = 1, . . . , n, there is a w ∈ [y 1 ] l 1 −|x 1 | × · · · × [y n ] ln−|xn| , and thus it follows from (5) that w ∈ Y u , which implies that u ⊔ v ∈ X for some v ⊑ w and such a v is in ([y 1 ] k 1 −|u 1 | × · · · × [y n ] kn−|un| ) ∩ Z. Therefore (6) must hold, from which φ(X, u, y) follows.
It remains to show that u = x. Since φ(H(X), x, S) and y ⊑ S, it follows that φ(H(X), x, y). If u = x, then we also have φ(X, x, y), which is impossible: Since |y i | = r l i k i k i − |x i | for any i = 1, . . . , n, there is a z ∈ [y 1 ] k 1 −|x 1 | × · · · × [y n ] kn−|xn| , and for such a z, we cannot have both x ⊔ z ∈ H(X) and x ⊔ z ∈ X.
Consistency results
In this section, we establish some consistency results by the method of permutation models. Permutation models are not models of ZF; they are models of ZFA (the Zermelo-Fraenkel set theory with atoms). Nevertheless, they indirectly give, via the Jech-Sochor theorem (cf. [2, Theorem 17.2]), models of ZF.
For our purpose, we only consider the basic Fraenkel model V F (cf. [2, pp. 195-196]). The set A of atoms of V F is denumerable, and x ∈ V F if and only if x ⊆ V F and x has a finite support, that is, a set B ∈ fin(A) such that every permutation of A fixing B pointwise also fixes x.
Lemma 4.1. Let A be the set of atoms of V F and let a = |A|. In V F , Proof. Let n ∈ ω. We claim that in V F , 2 a 2 n 2 fin(a) n .
Assume towards a contradiction that there exists an injection f ∈ V F from ℘(A 2 n ) into ℘(fin(A) n ). Let B be a finite support of f . Take an arbitrary C ∈ [A \ B] 2 n +1 and a u ∈ C 2 n . We say that a permutation π of A is even (odd ) if π moves only elements of C and can be written as a product of an even (odd) number of transpositions. It is well-known that a permutation of A cannot be both even and odd. Now, let E = {π(u) | π is an even permutation of A}, and let O = {σ(u) | σ is an odd permutation of A}.
Clearly, {E, O} is a partition of C 2 n , for all even permutations π of A we have π(E) = E, and for all odd permutations σ of A we have σ(E) = O. Now, let us consider f (E). For each t ∈ f (E), let ∼ t be the equivalence relation on C such that for all a, b ∈ C,
a ∼ t b if and only if ∀k < n a ∈ t(k) ↔ b ∈ t(k) .
For all even permutations π of A, since B is a finite support of f , it follows that π(f ) = f , and thus π(f (E)) = f (E). For all odd permutations σ of A and all t ∈ f (E), since |C/∼ t | 2 n and |C| = 2 n + 1, there are a, b ∈ C such that a = b and a ∼ t b, and therefore the transposition τ that swaps a and b fixes t, which implies that σ(t) = (σ • τ )(t) ∈ f (E) since σ • τ is even. Hence, for all odd permutations σ of A, σ(f (E)) = f (E), which implies that f (O) = f (σ(E)) = σ(f (E)) = f (E), contradicting the injectivity of f . Now, it follows from Lemma 2.9 that a 2 n fin(a) n+1 , and therefore 2 a 2 n 2 fin(a) n+1 , which implies that 2 fin(a) n < 2 fin(a) n+1 by (8). It follows from Theorem 3.4 that 2 fin(a) n = 2 [fin(a)] n 2 fin(fin(a)) . Hence 2 fin(a) < 2 fin(a) 2 < 2 fin(a) 3 < · · · < 2 fin(fin(a)) . Now the following proposition immediately follows from Lemma 4.1 and the Jech-Sochor theorem.
Proposition 4.2. The following statement is consistent with ZF: there is an infinite cardinal a such that 2 fin(a) < 2 fin(a) 2 < 2 fin(a) 3 < · · · < 2 fin(fin(a)) .
It is natural to wonder whether the conclusion of Theorem 3.4 can be strengthened to fin(a) n * [fin(a)] n . We shall give a negative answer to this question. The case n = 1 of the following lemma is proved in [5]. Lemma 4.3. Let A be the set of atoms of V F . In V F , for every n ∈ ω, fin(A) n is dually Dedekind finite; that is, every surjection from fin(A) n onto fin(A) n is injective.
Proof. Let n ∈ ω. Take an arbitrary surjection f ∈ V F from fin(A) n onto fin(A) n . In order to prove the injectivity of f , it suffices to show that for all t ∈ fin(A) n there is an m > 0 such that f (m) (t) = t.
(9)
Let B be a finite support of f . For each t ∈ fin(A) n , let ∼ t be the equivalence relation on A \ B such that for all a, b ∈ A \ B,
a ∼ t b if and only if ∀k < n a ∈ t(k) ↔ b ∈ t(k) .
Let ⊑ be the preorder on fin(A) n , such that for all t, u ∈ fin(A) n ,
t ⊑ u if and only if ∼ u ⊆ ∼ t .
Claim 4.4. There is an l ∈ ω such that every ⊑-chain without repetition must have length less than l.
Proof of Claim 4.4. We first prove that for all u ∈ fin(A) n ,
t ∈ fin(A) n ∼ t = ∼ u 2 (|B|+2 n )·n .(10)
Let u ∈ fin(A) n . Let g be the function defined on fin(A) n such that for all t ∈ fin(A) n , g(t) is the function on n given by
g(t)(k) = t(k) ∩ B, w ∈ (A \ B)/∼ u w ⊆ t(k) .
Clearly, ran(g) ⊆ ℘(B) × ℘((A \ B)/∼ u ) n . It is also easy to see that
g↾{t ∈ fin(A) n | ∼ t = ∼ u } is injective. Since |(A \ B)/∼ u | 2 n , we have t ∈ fin(A) n ∼ t = ∼ u ℘(B) × ℘((A \ B)/∼ u ) n 2 (|B|+2 n )·n .
For each t ∈ fin(A) n , let k t = |(A \ B)/∼ t |. Clearly, for all t, u ∈ fin(A) n such that t ⊑ u, we have 0 < k t k u 2 n , and if k t = k u then ∼ t = ∼ u . Thus, by (10), every ⊑-chain without repetition must have length less than or equal to 2 (|B|+2 n )·n · 2 n . Now, it suffices to take l = 2 (|B|+2 n +1)·n + 1. Proof of Claim 4.5. Assume towards a contradiction that ∼ u ∼ f (u) for some u ∈ fin(A) n . Let a, b ∈ A \ B be such that a ∼ u b but not a ∼ f (u) b. Clearly a = b. Let τ be the transposition that swaps a and b. Then τ (u) = u but τ (f (u)) = f (u), contradicting that B is a finite support of f . We prove (9) as follows. Let t ∈ fin(A) n . By Claim 4.4, there is an l ∈ ω such that every ⊑-chain without repetition must have length less than l. Let h be a function from l into fin(A) n , such that h(0) = t and for all i < l if i + 1 < l then h(i) = f (h(i + 1)). Such an h exists since f is surjective. Clearly, for all i < l, f (i) (h(i)) = t. By Claim 4.5, h is a ⊑-chain, and since the length of h is l, we can find i, j < l such that i < j and h(i) = h(j). Now, if we take m = j − i, then we have m > 0 and f (m) (t) = f (j−i) (t) = f (j−i) (f (i) (h(i))) = f (j) (h(j)) = t. Now the following proposition immediately follows from Lemma 4.3 and the Jech-Sochor theorem.
Proposition 4.6. The following statement is consistent with ZF: there is an infinite set A such that fin(A) n is dually Dedekind finite for any n ∈ ω. Proof. Notice that for all infinite sets A and all natural numbers n 2, there exists a non-injective surjection from fin(A) n onto [fin(A)] n . Hence, this corollary follows from Proposition 4.6.
We conclude this paper with two open problems. Question 4.8. Is it provable in ZF that 2 2 fin(a) = 2 2 fin(fin(a)) for any infinite cardinal a?
Notice that Proposition 4.2 shows that 2 fin(a) = 2 fin(fin(a)) cannot be proved in ZF for an arbitrary infinite cardinal a. Question 4.9. Does ZF prove that 2 2 a = 2 2 a+1 for any infinite cardinal a?
Notice that for all Dedekind finite cardinals a we have a < a + 1, and for all power Dedekind finite cardinals a (i.e., cardinals a such that 2 a is Dedekind finite) we have 2 a < 2 a+1 . Question 4.9 is asked in [3] (cf. also [2, p. 132]). Notice that, in [3], Läuchli proves in ZF that for all infinite cardinals a, 2 2 a = 2 2 a +1 .
Acknowledgements. I would like to give thanks to Professor Qi Feng for his advice and encouragement during the preparation of this paper.
Lemma 2. 4 .
4For all non-zero cardinals a, seq(seq(a)) = seq(a). Proof. Cf. [1, Lemma 2]. Lemma 2.5. For all non-zero cardinals a, seq(a) = ℵ 0 · seq 1-1 (a). Proof. Cf. [4, Lemma 2.22].Lemma 2.6. For all infinite cardinals a, ℵ 0 · seq 1-1 (a) * seq 1-1 (a).
Lemma 3 . 3 .
33For all natural numbers k 1 , . . . , k n and l 1 , . . . , l n such that k i l i for any
Theorem 3 . 4 .
34For all infinite cardinals a and all natural numbers n, 2 fin(a) n = 2 [fin(a)] n .
Claim 4 . 5 .
45For all u ∈ fin(A) n we have f (u) ⊑ u.
Corollary 4 . 7 .
47The following statement is consistent with ZF: there exists an infinite cardinal a such that fin(a) n * [fin(a)] n for any n 2.
Generalized idempotence in cardinal arithmetic. E Ellentuck, Fund. Math. 58E. Ellentuck, Generalized idempotence in cardinal arithmetic, Fund. Math. 58 (1966), 241-258.
L Halbeisen, Combinatorial Set Theory: With a Gentle Introduction to Forcing. ChamSpringer2nd ed.L. Halbeisen, Combinatorial Set Theory: With a Gentle Introduction to Forcing, 2nd ed., Springer Monogr. Math., Springer, Cham, 2017.
Ein Beitrag zur Kardinalzahlarithmetik ohne Auswahlaxiom. H Läuchli, Z. Math. Log. Grundl. Math. 7H. Läuchli, Ein Beitrag zur Kardinalzahlarithmetik ohne Auswahlaxiom, Z. Math. Log. Grundl. Math. 7 (1961), 141-145.
Factorials of infinite cardinals in ZF, Part I: ZF results. G Shen, J Yuan, 10.1017/jsl.2019.74J. Symb. Log. to appearG. Shen and J. Yuan, Factorials of infinite cardinals in ZF, Part I: ZF results, J. Symb. Log. (2019), to appear. https://doi.org/10.1017/jsl.2019.74
People's Republic of China School of Mathematical Sciences. J Truss, Institute of Mathematics, Academy of Mathematics and Systems Science. 84University of Chinese Academy of SciencesClasses of Dedekind finite cardinals. People's Republic of China E-mail address: [email protected]. Truss, Classes of Dedekind finite cardinals, Fund. Math. 84 (1974), 187-208. Institute of Mathematics, Academy of Mathematics and Systems Sci- ence, Chinese Academy of Sciences, Beijing 100190, People's Republic of China School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China E-mail address: [email protected]
| [] |
[
"Evidence for Direct and Indirect Gap in FeSi from Electron Tunneling Spectroscopy",
"Evidence for Direct and Indirect Gap in FeSi from Electron Tunneling Spectroscopy"
] | [
"M Bat'ková \nInstitute of Experimental Physics\nSlovak Academy of Sciences\nWatsonova 47040 01KošiceSlovakia\n",
"I Bat'ko \nInstitute of Experimental Physics\nSlovak Academy of Sciences\nWatsonova 47040 01KošiceSlovakia\n",
"M Mihalik \nInstitute of Experimental Physics\nSlovak Academy of Sciences\nWatsonova 47040 01KošiceSlovakia\n"
] | [
"Institute of Experimental Physics\nSlovak Academy of Sciences\nWatsonova 47040 01KošiceSlovakia",
"Institute of Experimental Physics\nSlovak Academy of Sciences\nWatsonova 47040 01KošiceSlovakia",
"Institute of Experimental Physics\nSlovak Academy of Sciences\nWatsonova 47040 01KošiceSlovakia"
] | [] | We report electron tunneling spectroscopy studies on single crystalline FeSi sample performed for the case of homogeneous tunnel junction (TJ) contacts and for the case of counter electrode made from Pt-Rh alloy. Our results reveal that while the tunneling spectroscopy in the configuration with Pt-Rh tip is preferably sensitive to the d-partial density of states (DOS) and to the indirect energy gap, the FeSi-FeSi type of TJ yields the spectroscopic information on the c-partial DOS and on the direct gap in FeSi. | 10.1016/j.ssc.2006.11.014 | [
"https://arxiv.org/pdf/2010.10838v1.pdf"
] | 93,126,976 | 2010.10838 | 5217a4f9d81d9b6d2a4445dada56bca2f1cb53f1 |
Evidence for Direct and Indirect Gap in FeSi from Electron Tunneling Spectroscopy
21 Oct 2020
M Bat'ková
Institute of Experimental Physics
Slovak Academy of Sciences
Watsonova 47040 01KošiceSlovakia
I Bat'ko
Institute of Experimental Physics
Slovak Academy of Sciences
Watsonova 47040 01KošiceSlovakia
M Mihalik
Institute of Experimental Physics
Slovak Academy of Sciences
Watsonova 47040 01KošiceSlovakia
Evidence for Direct and Indirect Gap in FeSi from Electron Tunneling Spectroscopy
21 Oct 2020(Dated: October 22, 2020)numbers: 7127+a7128+d7530Mb7340Gk
We report electron tunneling spectroscopy studies on single crystalline FeSi sample performed for the case of homogeneous tunnel junction (TJ) contacts and for the case of counter electrode made from Pt-Rh alloy. Our results reveal that while the tunneling spectroscopy in the configuration with Pt-Rh tip is preferably sensitive to the d-partial density of states (DOS) and to the indirect energy gap, the FeSi-FeSi type of TJ yields the spectroscopic information on the c-partial DOS and on the direct gap in FeSi.
I. INTRODUCTION
Cubic compound FeSi is well known by the unusual physical properties arising from its unconventional band structure. Despite the fact that FeSi is a d-transition metal compound, it shares certain characteristic features with rare earth hybridization gap semiconductors such as SmB 6 , YbB 12 or Ce 3 Bi 4 Pt 3 , in which the hybridization of f-and conduction electrons is believed to take place. As a rule, FeSi is placed into the same special group of heavy electron systems with a narrow gap, named "Kondo insulators" or "Kondo semiconductors" 1,2 .
According to transport and optical studies, at high temperatures FeSi can be characterized as a dirty metal 3,4 . As the temperature falls to around 300 K, it shows a metal to semiconductor crossover 5 and behaves as a narrow gap semiconductor below ∼100 K [5][6][7] . A value of the activation energy ∆ differs from about 50 meV to ∼110 meV upon the used experimental method [4][5][6][7][8][9][10] . At the lowest temperatures FeSi has a metallic character 3,7,11 .
Several different theoretical approaches have been suggested to explain striking properties and an origin of the energy gap in FeSi [12][13][14][15] . A simple physical picture assumes that instead of f-electrons in the rare earth semiconductors, a set of rather localized d-orbitals having a strong on-site repulsion, hybridizes in FeSi with a broad itinerant conduction band of noninteracting celectrons forming in this way two bands separated by the gap 1,5,15 . According to the theory of Rozenberg, Kotliar and Kajueter 15 , the d-electron partial DOS shows opening of an indirect gap, ∆ ind , accompanied by the formation of strong and narrow quasiparticle bands visible as two symmetric peaks on either side of the Fermi level. A direct gap, ∆ dir , in c-electron partial DOS is merely a dip without peaks in the framework of this model 15 . Moreover it is expected that ∆ ind << ∆ dir 15 . In accordance with number of published experimental results, it appears that the magnetic gap probed by magnetic susceptibility 4-7 and optical reflectivity 6,7,10 measurements is the larger direct gap, and that the transport gap reflected in resistivity 4-7,10 , point contact spectroscopy 6,10 and tunneling spectroscopy 8 measurements is the smaller indirect gap. In this paper we show for the first time that tunneling spectroscopy is sensitive to both electron subsystems, and so that it can probe not only the indirect but also the direct gap in FeSi.
II. EXPERIMENT
The samples used for experiments were pieces of the same FeSi single crystal grown from the melt by the "triarc" Czochralski technique, for which magnetic, transport, photo-emission, point-contact spectroscopy, as well as infrared studies were reported previously 6,10,16 . The ratio of low temperature to room temperature resistivity, ρ(5 K)/ρ(300 K) = 5×10 3 is an indication for high quality sample 10 . In addition, the electron probe microanalysis did not reveal a presence of any second phase 6 . As reported before, the value of the transport gap obtained from resistivity data is ∆ t ∼56 meV, while the magnetic susceptibility studies indicate the gap of ∆ m ∼95 meV or ∼103 meV, depending on the fitting method used 6 .
The tunneling measurements were performed by the scanning tunneling spectroscopy approach using mechanically controlled TJs in two different configurations: (i) with Pt-Rh tip as a counter electrode and, for the first time as we know, (ii) with the tip made from a piece of the same FeSi single crystal as the studied sample. The Pt-Rh tip was prepared from commercially available Pt 0.9 Rh 0.1 wire. Differential conductance, dI/dV , was numerically calculated from the measured currentvoltage characteristics. It should be mentioned that due to the formation of an oxide layer on the FeSi surface, the TJ electrodes were in mechanical contact, separated just by this native insulating surface layer.
III. RESULTS AND DISCUSSION
Typical differential conductance curves taken with Pt-Rh tip at 4.2 K and 300 K are shown in Fig. 1. As temperature was lowered from 300 K to 4.2 K, in the region of |V | < ∼300 mV the differential conductance increased except the vicinity of the zero-bias, where a strong and sharp dip was formed, with the minimum of (almost) the same value as the zero bias conductance at 300 K. (Here should be noted that there is a relatively strong dependence of the zero bias conductance on the width of the interval used for calculation of numerical derivation. We used the widest possible interval still not increasing the zero bias conductance; this is the reason for a bigger error of dI/dV curve at 4.2 K in Fig. 1.) Exemplary spectra of the FeSi-FeSi configuration are depicted in Fig. 2. The conductances observed at 300 K seem to be very similar to those measured out with Pt-Rh tip. On the other hand, the spectra taken at 4.2 K exhibit qualitatively different features, as they show a visible decrease of the differential conductance under the values of the 300 K curve in the whole energy interval corresponding to |V | < ∼300 mV. The zero-bias conductance shows the greatest decrease, but at 4.2 K it remains still finite. According to Hammers 17 , the differential conductance dI/dV in the tip-sample configuration can be expressed by
dI dV = ρ s (r, eV )ρ t (r, 0)T (eV, eV, r) + eV 0 ρ s (r, E)ρ t (r, ±eV , ∓E) dT (E, eV, r) dV dE,(1)
where ρ s (r, E) and ρ t (r, E) are the state densities of the sample and the tip, respectively, at location r and at the energy increases monotonically with V and contributes by a smoothly varying "background", on which the spectroscopic information is superimposed 17 . Because of the smooth and monotonic increase, a structure in dI/dV as a function of V usually can be assigned to changes in the state density via the first term of equation (1), thus permitting the DOS to be determined as a function of energy at any particular location on the surface 17 . Analogously, if DOS changes due to a variance of temperature, the change in DOS at constant energy E = eV is correspondingly detected as the change of dI/dV at the voltage V . The change of the DOS due to the temperature variance from T 1 to T 2 can be then inferred from the difference dI/dV (T 2 ) − dI/dV (T 1 ). (Of course, in such a case the effects due to temperature smoothing are neglected. Because of the energy scale, which is a few thousand Kelvins in our case, such an approximation seems to be acceptable.) Fig. 3 shows the difference between the dI/dV curves taken at 300 K and 4.2 K for both TJ configurations. The curve at the top, corresponding to the contacts with Pt-Rh tip, shows two almost symmetrical local maxima at both sides of the Fermi level with estimated peak-to-peak distance of about 110 meV, separated by the dip with the minimum of practically the zero-value at the zero-bias. Because of the temperature independent DOS of Pt-Rh tip, the observed influence of the temperature on the tunnel spectra should reflect changes in the DOS of FeSi only. In spite of the fact that the obtained curve can not be without a renormalization quantitatively related to the DOS, it clearly indicates a formation of two symmetrically placed peaks at gap edges. (According to the finite values of the differential conductance in all the studied cases, it would be probably more appropriate to speak about the direct and the indirect "pseudogap" in FeSi. Nevertheless, for the purposes of this paper we will use the term "gap".) The difference curve for FeSi-FeSi configuration at the bottom of Fig. 3, is negative in the whole region of |V | < ∼300 mV with the strong dip centered at the zero bias. Such a behavior can be attributed to a decrease of the quasiparticle DOS around the Fermi level by falling the temperature from the room one to 4.2 K. As follows from the subsequent discussion, we have associated the curve at the top and at the bottom of Fig. 3 with the temperature change of the d-and the c-partial DOS, respectively. As emphasized by Tromp 18 , because of the determination of the tunneling current by the tunneling transmission probability and by the DOS of both electrodes, in addition to a finite DOS there must be a significant overlap between the corresponding sample and tip wave functions 18 . If the sample has a large DOS, but these states do not overlap with the tip, they are inaccessible in the tunneling experiment 18 .
In the case of the heterogeneous TJs with the metallic Pt-Rh tip, a sufficient overlap of the rather spatially extended wave function of conduction electrons of the tip with that of the correlated c-and d-electrons in FeSi can be expected. Because of much larger and peaked dpartial DOS a major tunnel current will originate from delectrons. In fact, the shape of the curve at the top of Fig. 3 strongly resembles the experimental results of other authors 19 and the situation in d-partial DOS proposed by the theory of Rozenberg, Kotliar and Kajueter 15 . Therefore, in agreement with Fäth and coworkers 8 , we conclude that the tunneling spectroscopy measurements on the tunnel junctions with the counter electrode from the metal without electron correlation effects probe preferably the much larger and peaked d-partial DOS in FeSi.
In replacing the Pt-Rh tip by FeSi tip, the relatively slowly decaying wave function of conduction electrons of Pt-Rh tip is then replaced by the wave functions of correlated c-and d-electrons of FeSi. Due to the localized nature of d -electrons their wave function is less spatially extended than one of c-electrons, so it is reasonable to expect that an overlap between the c-states will be dominating here. In spite of the high density of d-states, the tunneling process in FeSi-FeSi type of TJ seems to be predominantly governed by the lower-density c-electrons, as the effect of the overlap between the c-states seems to be superior to the effect of high density of d-states. In addition to this, the shape of the curve at the bottom of Fig. 3 resembles the theoretically predicted change of the c-partial DOS due to temperature variation in the region of the direct gap 15 . Based on the given arguments we conclude that the FeSi-FeSi tunneling configuration yields preferably the spectroscopic information on c-partial DOS.
According to the previous discussion, the dip observed in TJs with Pt-Rh tip is associated with the indirect gap. Although the shape of the dip is modified due to Shottky barrier effect 8 , its width should still correlate with the width of the indirect gap. As ∆ ind is associated with transport measurements 7,15 , it is to be expected that the width of this dip will be comparable with the transport gap value ∆ t = 56 meV previously derived for the studied sample from the resistivity data 6 . In Fig. 3, where ∆ t is correspondingly indicated, a definite correlation between ∆ t and the width of the dip can be seen. On the other hand, the magnetic gap is related to the direct gap ∆ dir 7,15 . The width of the magnetic gap of our sample ∆ m ∼ = 95 (or 103 meV, depending on the fitting procedure used) 6 is larger than ∆ t in accordance with the theoretical predictions 15 . The comparison of ∆ m with the width of the dip visible in the curve at the bottom of Fig. 3 requires taking into account that in FeSi-FeSi type of TJ there are gap structures at both sides of the TJ. So that (analogously like it is in superconductor-insulatorsuperconductor TJs), the decrease of the differential conductance should develop in the energy interval of 2∆ m . Doubled values of ∆ m are indicated in Fig. 3. Although the determination of the dip width is not straightforward, it can be seen that the indicated values of 2∆ m lie in the crossover between the dip region and the background region, and so, it can be said that the width of the dip region correlates with the ∆ m . The observed good correspondence between our tunneling data and the earlier results of the transport and magnetic studies, provides a further support for the interpretation of the tunneling data given above.
IV. CONCLUSIONS
Our tunneling spectroscopy studies of FeSi have revealed that two types of electron subsystems and two different energy gaps are present in FeSi. Depending on the used type of the counter electrode, the lowering of temperature from 300 K to 4.2 K causes either the formation of the peaks in the partial DOS, which we associate with the properties of d-electrons and with the indirect gap; or decrease of the partial DOS at and in the vicinity of the Fermi level, what we assign to the direct gap formation in the c-partial DOS. The obtained results strongly support the theoretical model of Rozenberg, Kotliar and Kajueter 15 and correlate well with the independent evaluation of the transport and the magnetic gap of our FeSi sample. Moreover, our work shows that the type of spectroscopic information obtained by tunneling spectroscopy can be "selected" by the type of counter electrode used. For the more detailed information on the role of the counter electrode material and on the properties of correlated electron subsystems in FeSi, additional experiments are going to be done at various temperatures, utilizing several types of counter electrode materials.
FIG. 1 :
1Typical tunnel spectra of FeSi taken with Pt-Rh tip at 4.2 K and 300 K.
FIG. 2 :
2E, measured with respect to their individual Fermi levels; T (E, eV, r) is the tunneling transmission probability for electrons with energy E at applied bias voltage V . The upper and the lower signs in equation (1) correspond to a positive and a negative sample bias, respectively 17 . At any fixed location, T (E, eV, r) Typical spectra of the FeSi-FeSi tunnel junction at 4.2 K and 300 K.
FIG. 3 :
3Change of the differential conductance after cooling down from 300 K to 4.2 K for Pt-Rh tip (top) and for FeSi-FeSi tunnel junction (bottom).
* Electronic address: [email protected]. * Electronic address: [email protected]
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| [] |
[] | [
"Michael Bateman "
] | [] | [] | We prove L p , p ∈ (1, ∞) estimates on the Hilbert transform along a one variable vector field acting on functions with frequency support in an annulus. Estimates when p > 2 were proved by Lacey and Li in[4]. This paper also contains key technical ingredients for a companion paper[3]with Christoph Thiele in which L p estimates are established for the full Hilbert transform. | 10.4171/rmi/748 | [
"https://arxiv.org/pdf/1109.6395v1.pdf"
] | 54,742,239 | 1109.6395 | 0231d7983c21eb5ad40f9ee7a09c759136532107 |
29 Sep 2011
Michael Bateman 29 Sep 2011SINGLE ANNULUS L p ESTIMATES FOR HILBERT TRANSFORMS ALONG VECTOR FIELDS
We prove L p , p ∈ (1, ∞) estimates on the Hilbert transform along a one variable vector field acting on functions with frequency support in an annulus. Estimates when p > 2 were proved by Lacey and Li in[4]. This paper also contains key technical ingredients for a companion paper[3]with Christoph Thiele in which L p estimates are established for the full Hilbert transform.
Acknowledgments
The author thanks Ciprian Demeter and Christoph Thiele for helpful discussions, and especially Christoph Thiele for making many comments on an early version of this paper. The author also thanks Francesco Di Plinio for pointing out an important typo. The author was partially supported by NSF grant DMS-0902490.
Introduction
Let v be a nonvanishing vector field that depends one variable, i.e., v : R 2 → R 2 \{0} and v(x 1 , x 2 ) = v(x 1 ). In this paper we prove L p estimates on the Hilbert transform along v precomposed with frequency restriction to an almost-annular region. More specifically, define
H v f (x) = p.v. f (x − tv(x)) t dt.
Because of the structure of the Hilbert kernel, the magnitude of v is irrelevant, provided it is nonzero. For this reason we may assume that v(x 1 , x 2 ) = (1, u(x 1 )). We will further assume that the slope of v is bounded by 1. This will be helpful for some technical reasons in this paper, but our main interest is in the action of H v on arbitrary functions (i.e., those not necessarily having frequency support in an annulus); in this more general case, the operator is invariant under dilations in the vertical variable. See [3] for more on the symmetries of this problem. This invariance allows us to assume, in that case, that the slope of v is bounded by 1. (This is mostly a technical convenience, that allows us to think of rectangles and parallelograms as being the same kind of objects.) Since this general problem is the primary motivation for this paper, we adopt the restriction on the slope here as well. The general problem is addressed in a companion paper with Christoph Thiele [3]. This paper is logically prior to the other, and is therefore self-contained. Fix w ≥ 0, and define τ to be the trapezoid with corners (− 1 w , 1 w ), ( 1 w , 1 w ), (− 2 w , 2 w ), and ( 2 w , 2 w ). Also define Π τ f (ξ) = 1 τ (ξ)f (ξ).
Here we prove the following Theorem 1. Let v be a vector field depending on one variable with slope bounded by 1. Let p ∈ (1, ∞). Then
||(H v • Π τ )f || p ||Π τ f || p .
We remark that the estimate in this theorem is independent of the parameter w in the definition of τ , which comes as no surprise given the dilation invariance of the problem. Further, the restriction to a trapezoid specifically is nothing to take seriously. Using the assumption on the slope of the vector field we can already assume suppf lies in a two-ended cone near the vertical axis, because H v acts trivially on functions with support outside this cone. More precisely, iff is support in a cone close to the horizontal axis, then we have with the constant vector field (1,0):
H v f (x, y) = H (1,0) f (x, y) ,(1.1)
because H (1,0) is a multiplier corresponding to right and left half-planes. But H (1,0) is trivially bounded, justifying our claim. Finally, a trapezoid is the restriction of the cone to a horizontal frequency band. We could have equally well stated the theorem for functions with support in the full band, and reduced it to the trapezoidal case. Alternatively, we could have worked with an annular region, or an annular region intersected with a cone. Our methods work equally well in these cases. We chose the horizontal band (rather than an annulus) because of the special structure of one-variable vector fields, but for other vector fields an annular region may be more appropriate.
Perhaps the biggest contribution of this paper (aside from its applicability to [3]) is a more streamlined and mechanized collection of two-dimensional time-frequency tools. Building heavily on important earlier work of Lacey-Li (see [4] and [5]), we clarify the relationship between the density-related maximal operators (see Lemma 20) and the more classical time-frequency tools. Specifically, a key sublemma in [1], combined with this more efficient understanding, allows us to obtain the full range of exponents p ∈ (1, ∞) here. Further, although the results are stated only for one-variable vector fields, it is clear how to combine a maximal theorem for a different vector field with the methods of this paper. We should remark that time-frequency analysis in two-dimensions is rather less-well-developed than in one-dimension, with work of Lacey-Li being the only natural precursor to this paper. We therefore strove to make the paper self-contained and to include proofs of a number of lemmas that are standard in one-dimension, but whose proofs in the two-dimensional situation do not seem to appear in the literature.
Related work.
Study of such problems is motivated by the obvious connection to the problem of estimating the Hilbert transform on functions that have not been Fourier-localized. Stein, for example, conjectured that if v is Lipschitz, then H v (or rather, a truncated version of it) is a bounded operator on L 2 . We note that when v depends on only one variable, the L 2 boundedness of H v is a rather immediate consequence of Carleson's theorem, as shown in [5]. Stein's conjecture is the singular integral variant of Zygmund's well-known conjecture on the differentiation of Lipschitz vector fields. For a fuller history, see [5]. More recently, Thiele and the author proved a range of L p estimates on the full Hilbert transform along a one variable vector field, using some key lemmas from the present paper. It is known that the operator H v is related to the return-times theorem from ergodic theory; see [3] for more on this connection.
We remark that the operator C is quite similar to Carleson's operator (i.e., the maximal Fourier partial sum operator). The argument in [4] is also quite similar to the Lacey-Thiele proof of Carleson's theorem (see [6]). The argument here draws on ingredients from [4], but obtaining L p estimates for p < 2 in this situation requires more effort, partly because the relevant maximal operators are more complicated, but also because making use of the maximal theory is more complicated. In the 1-D situation, exceptional sets are unions of intervals; nothing so simple is the case here.
Theorem 1 was proved for arbitrary vector fields when p > 2 by Lacey and Li in [4]. (In fact, they proved a weak L 2 result.) The same authors, in [5], introduced a method for obtaining L p , p < 2, estimates on H v • Π τ when a certain maximal theorem is available for the vector field v in question. (The story is a bit technical: they proved a theorem contingent on the existence of this certain maximal theorem in the case of truncated Hilbert kernels. However the method had little to do with the truncation of the kernel, allowing us to extend it here.) The author proved such an L p maximal theorem when v depends on one variable in [1]. Given this result, it is not surprising that the method from [5] yields a result for some p < 2, but the value of p obtained from the method in [5] seemed far from sharp. (At the very least, the method seemed nonsharp. Of course, this was not important for the authors there.) It was clear, for example that new ideas would be required to even reach p close to 3 2 . The author recently improved the estimates in this maximal theorem to (essentially) best possible in [2]. Because of this, the author decided to investigate the precise range of p for which Theorem 1 holds.
1.2. New ideas. The novelties in this paper that allow us to obtain the full range of p claimed in Theorem 1 are a simplification of the approach in [5], and a more efficient appeal to the maximal theorems.
We elaborate a bit more on these points for readers already familiar with the argument in [5].
Regarding the first point: In [5], tiles are sorted into trees via standard density and orthogonality (size) lemmas. An important additional observation made in [5] is that if T is a collection of trees such that for each T ∈ T the "size" of T is about σ and the "density" of the top of T is about δ, then we can control T ∈T |top(T )| by using an appropriate maximal theorem. Their argument, however, requires an additional twist to handle trees with large size whose tiles have density ∼ δ, but whose tops have density much less than δ. Here we use an organization of the tiles that admits a more straightforward argument. This organization is carried out in Section 8, which contains more discussion as well.
Regarding the second point: A rather simple observation allows us to appeal to a key ingredient in the proof of the maximal theorem, rather than the theorem itself. This strengthens estimates on T ∈T |top(T )| for trees as mentioned in the last paragraph. This observation allows us to obtain the full range of p. This observation uses the proof of [1], and hence does not even take advantage of the sharp L p estimates on the maximal operator obtained in [2]. See Lemma 20.
1.3. Organization of paper. Readers familiar with time-frequency analysis, having a bit of faith, and wanting an executive summary should follow this outline: Skip to the definition of the model operator in Section 2.4. Then (possibly after skimming Section 3 to review essentially standard definitions,) read Sections 4, 5, and 8. Those wanting to check the numerology should also read Section 6. A comprehensive outline is below.
In Section 2, we reduce the theorem to an analogous one for a model operator.
In Section 3, we present some key definitions needed for the organization of our set of tiles. (Recall that the operators in question are model sums over tiles.)
In Section 4, we make the main decomposition of the collection of tiles and state several key estimates that follow from the decomposition.
In Section 5, we state the main lemmas needed to prove the estimates stated in Section 4.
In Section 6, we balance these various estimates to prove the main theorem. There is no serious content here.
In Section 7, we prove the density lemma, which estimates T ∈T |top(T )| for certain collections T by using elementary covering ideas.
In Section 8, we prove the maximal estimate, which controls T ∈T |top(T )| for certain collections T by using more sophisticated techniques in combination with L p and BM O-type estimates on a square function related to the "projection" operator associated to trees.
In Section 9, we compare the size of a tree to its intersection with the function in the definition of size.
In Section 10, we prove the tree lemma, which controls the contribution to the model sum from one tree. The proof mirrors that of the (more) classical one-dimensional tree lemma, with a small bit of extra work required to handle two-dimensional tail terms.
In Section 11, we prove the size lemma, which estimates T ∈T |top(T )| for certain collections T by using orthogonality.
In Section 12, we prove a refined Bessel inequality that allows us to control tail terms in the size and tree lemmas, as well as in the proof of localized L p estimates for the square function mentioned above.
In Section 13, we prove localized (to the top of a tree) L p estimates for a square function associated to a tree. Once again, we follow a relatively standard argument and appeal to the refined Bessel inequality to handle some two-dimensional technicalities.
In Section 14, we prove that higher L p norms of the square function are controlled by lower L p ones by using standard BM O techniques.
In the appendix, we recall the proof in [4] of the L p , p > 2 case of our main theorem.
Reductions
In this section we reduce the L p estimates in Theorem 1 to restricted weak-type estimates on a model operator. The model operator should look familiar to readers familiar with developments in time-frequency analysis from the last ten to fifteen years: it is a sum over "tiles" of wave packets. The model operator arises from decomposing (1) the Hilbert kernel 1 t into (smoothly cutoff) dyadic intervals on the frequency side; for technical reasons we make these annuli rather thin, resulting in two summation indices for the Hilbert kernel. In fact, we actually decompose the projection operator onto positive frequencies, and write the Hilbert transform as a linear combination of this operator and the identity operator.
(2) given any integer l ≥ 0,f on τ into ∼ 2 l pieces; again, the "∼" here comes from another summation introduced to provide strict orthogonality between the various pieces.
l (t) = ψ (0) (2 −l t). Now define ψ (0) = l∈Z ψ (0)
l . By appropriately defining ψ (i) 0 with similarly sized support, and defining ψ
l (t) = ψ (i) 0 (2 −l t),
we can construct a partition of unity for R + ; i.e.
1 (0,∞) = 99 i=0 ψ (i) .
This gives us the Hilbert kernel as a linear combination of 100 model kernels and the identity. More precisely, let
H (i) l g(x, y) = ψ (i) l (t)g(x − t, y − tu(x))dt.
Then writing I for the identity operator,
c 1 H • Π τ f (x, y) + I • Π τ f (x, y) = c 2 l∈Z 99 i=0 H (i) l • Π τ f (x, y).
By the triangle inequality, we have
||H • Π τ f || p ||I • Π τ f || p + 99 i=0 ||H (i) • Π τ f || p ,
where
H (i) = l H (i)
l . We note that H l • Π τ f = 0 for l ≤ log 1 w + c because of the Fourier support of the kernel of the operator H l .
2.2.
Discretizing the function. We next focus on discretizing the function f . For l ≥ 0, we write D l to denote the collection of dyadic intervals of length 2 −l contained in [−2, 2]. Fix a smooth positive function β : R → R such that β(x) = 1 for x ∈ [−1, 1] and such that β(x) = 0 when |x| ≥ 2. Also assume that √ β is a smooth function. This point will become relevant for the definition of ϕ immediately before Lemma 2. Now fix an integer c (whose exact value is unimportant) and for each ω ∈ D l , define
β ω (x) = β(2 l+c (x − c ω 1 )),
where ω 1 is the right half of ω, and c ω 1 is the center of ω 1 . Define
β l (x) = ω∈D l β ω (x).
Note that
β l (x + 2 −l ) = β l (x) for x ∈ [−2, 2 − 2 −l ]. Now define γ l (x) = 1 2 1 −1 β l (x + t)dt.
Because of the local periodicity mentioned above, we have that γ l (x) is constant for x ∈ [−1, 1]; say γ l (x) = δ, where δ is a constant independent of l.
Hence 1 δ γ l (x)1 [−1,1] (x) = 1 [−1,1] (x)
.
Define yet another multiplierβ : R → R with support in [ 1 2 , 5 2 ], andβ(x) = 1 for x ∈ [1,2]. Just as γ l is an average over translates of β l , so each H (i) is an average of model operators. We define the corresponding multipliers on R 2 :
m ω (ξ, η) =β(η)β ω ( ξ η ) m l,t (ξ, η) =β(η)β l (t + ξ η ) m l (ξ, η) =β(η)γ l ( ξ η ).
We know that for each l m l (ξ, η)1 τ (ξ, η) = 1 τ (ξ, η)
for (ξ, η) ∈ τ . Note that for each i,
||H (i) (Π τ • f )|| p = || l (H (i) l • Π τ )( 1 δ m l * f )|| p = || 1 2 1 −1 l (H (i) l • Π τ )( 1 δ m l,t * f )dt|| p ≤ 1 2 1 −1 || l (H (i) l • Π τ )( 1 δ m l,t * f )|| p dt,
so it is enough to consider the discretized projections m l,t . In what follows, we will assume, without loss of generality, that t = 0 = i and omit the dependence on t and i.
2.3.
Constructing the tiles. For each ω ∈ D with l ≥ 0, let U ω be a partition of R 2 by parallelograms of width w and length w |ω| whose long side has slope θ, where tan θ = c(ω) and where c(ω) is the center of the interval ω, and whose projection onto the x-axis is a dyadic interval. We remark that l < 0 need not be considered. (See the remark immediately prior to Section 2.2. Note that the index l plays a slightly different role there.) Briefly, the parts of the Hilbert kernel whose frequency support is outside the interval [− 1 w , 1 w ] ⊆ R ((i.e., ψ l for l < log 1 w ) have no interaction with our function f whose frequency support is contained in the annulus of radius 1 w . Finally, let U = ω∈D U ω . If s ∈ U ω , we will write ω s := ω.
An element of U is called a "tile". The following lemma, stated in essentially this form in [4], allows us to further discretize our operator into a sum over tiles. Let R ω denote an element of U ω containing the origin. Suppose ϕ ω is such that | ϕ ω | 2 = m ω . Note that ϕ ω is smooth, by our assumption on the function β mentioned above. Further, each region
{(ξ, η) : ξ η ∈ ω, η ∈ [1, 2]}
can be obtained by a linear transformation of the trapezoid with corners (−1, 1), (1, 1), (−2, 2), (2,2), which ensures that the functions ϕ ω , with ω ∈ D := ∪ l≥0 D l , satisfy uniform decay conditions. To see this, consider the transformations Note that the functions m ω are L 1 normalized, so the functions ϕ s are L 2 normalized. Lemma 2. Using notation above, we have
f * m ω (x) = lim N →∞ 1 4N 2 [−N,N ] 2 s∈Uω
f, ϕ s (p + ·) ϕ s (p + x)dp.
Proof. We compute directly:
f * m ω (x) = z∈R 2 f (z) p∈R 2 ϕ ω (p)ϕ ω (p + x − z)dpdz = z∈R 2 f (z) s∈Uω p∈s ϕ ω (p + z)ϕ ω (p + x)dpdz = s∈Uω p∈s z∈R 2 f (z)ϕ ω (p + z)dzϕ ω (p + x)dp = s∈Uω p∈s f, ϕ ω (p + ·) ϕ ω (p + x)dp = s∈Uω 1 |R ω | p∈Rω f, ϕ s (p + ·) ϕ s (p + x)dp = lim N →∞ 1 4N 2 [−N,N ] 2 s∈Uω
f, ϕ s (p + ·) ϕ s (p + x)dp.
To see the last equality, note that the integrand is periodic in p, and the error (which arises from the fact that [−N, N ] 2 will not exactly agree with the boundaries of the tiles s) goes to zero as N → ∞.
This lemma allows us to conclude (using the dominated convergence theorem) that
H l (f * m ω )(x) = lim N →∞ 1 4N 2 [−N,N ] 2 H l s∈Uω f, ϕ s (p + ·) ϕ s (p + x) dp.
This allows us to restrict attention to the model operator that we define shortly. Define
ψ s = ψ log(length(s))
and
φ s (x 1 , x 2 ) = ψ s (t)ϕ s (x 1 − t, x 2 − tv(x))dt.
We record the following fact for use in the proof of the tree lemma in Section 10.2.2.
Lemma 3. We have φ s (x) = 0 unless v(x) ∈ ω s,2 .
Proof. Use Plancherel's theorem and the Fourier supports of ψ s and ϕ s .
2.4. The model operator. We can finally define our model operator:
Cf = s∈U f, ϕ s φ s .
For readers following the executive summary: ϕ s is a standard wave packet adapted to the tile s, and φ s is the appropriate scale of the Hilbert transform acting on ϕ s . A good mental shortcut is to imagine φ s (x) = ϕ s (x)1 ω s,2 (u(x)), an expression quite similar to one appearing in the Lacey-Thiele proof of Carleson's theorem. By Lemma 2, each operator H (i) is an average of models of the form C. Hence it is enough to prove the following theorem.
Theorem 4. With C defined immediately above, and p ∈ (1, ∞), we have
||Cf || p ||f || p . (2.1)
By appealing to restricted weak-type interpolation, it suffices to prove
| C1 F , 1 E | |E| 1− 1 p |F | 1 p
for arbitrary E, F ⊆ R 2 and p ∈ (1, ∞). Of course by the triangle inequality it suffices to prove the following inequality:
s∈S | 1 F , ϕ s 1 E , φ s | |E| 1− 1 p |F | 1 p (2.2)
for any p ∈ (1, ∞), any E, F ⊆ R 2 , and any finite S ⊆ U . This is our task for the rest of the paper. Lacey and Li have already proved this estimate for arbitrary vector fields when p ≥ 2. We discuss this proof in the appendix. Note that for p ≤ 2, we have
|E| 1− 1 p |F | 1 p = |E| 1 2 |F | 1 2 |F | |E| 1 p − 1 2 |E| 1 2 |F | 1 2 whenever |F | |E| because 1 p − 1 2 > 0.
Hence our estimate is already proved when |F | |E|, so we restrict attention to the case |F | ≤ c|E| for some small constant c.
Key definitions
Definition 5. Given a parallelogram R, we write CR to denote the parallelogram with the same center as R but dilated by a factor of C.
Definition 6. Given two parallelograms R 1 and R 2 in U , we will write
R 1 ≤ R 2 whenever R 1 ⊆ CR 2 and ω R 2 ⊆ ω R 1 .
Recall that ω R is defined in Section 2.3. The exact value of C in the last definitions is not important: 10 is enough. We need that if R 1 ∩ R 2 = ∅ and
ω R 1 ⊆ ω R 2 , then R 2 ≤ R 1 . Definition 7. A tree is a collection T of parallelograms with a top parallel- ogram, denoted top(T ), with top(T ) ∈ U , such that for all s ∈ T , we have s ≤ top(T ). A tree T is a j-tree if ω top(T ) ∩ ω s,j = ∅.
Given a tree T , we will write T j to denote the maximal j-tree contained in T .
Recall that ω s,1 is the right half of ω s and ω s,2 is the left half. The following definitions will help us organize our collections of tiles. Recall that our vector field v is defined on a set E; this set plays a role in the definitions of dense and dense below. Similarly, the definitions of size depends on our other set F .
For x ∈ R 2 , let χ(x) = 1 1+|x| 100 . For any parallelogram s, let χ (p)
s be an L p normalized version of χ adapted to the parallelogram s.
Definition 8. Define the following for a parallelogram s and a collection of parallelograms S:
E s = {(x, y) ∈ E : u(x) ∈ ω s } dense(s) = Es χ (1) s dense(s) = sup s ′ ≥s,s ′ ∈U dense(s ′ ) size(S) = sup 1-trees T ⊆S 1 |top(T )| s∈T | 1 F , ϕ s | 2 1 2 .
We remark that the function χ is needed for density since the wave packets ϕ s have Schwartz tails. See the proofs of the tree and density lemmas. The extra technicality involved in defining dense (as opposed to just dense) is needed for our proof of the tree lemma (just as it is in the one-dimensional theory of [6]). The cost is rather high: a density estimate (see Estimate 12 below) is still easily obtainable, but the maximal estimate becomes much more difficult to prove. If dense(s) were equal to dense(s) for every tile s, then the tops of the trees constructed in Section 4 are already prepared for an application of maximal technology. Unfortunately this is not the case, and this difficulty prompts our consideration of the collections R j in Section 8. See also the delicate sorting algorithm in Lacey-Li [5], where the authors wrestle with the same issue.
Organization
In this section we carry out the main decomposition of the collection of tiles. We sort a given collection of tiles into subsets of tiles of approximately constant density, and further into trees of approximately constant size. The relevance of trees is shown in the following:
Lemma 9 (Tree lemma). Let T be a tree. Suppose dense(T ) ≤ δ. Suppose size(T ) ≤ σ. Then s∈T | 1 F , ϕ s 1 E , φ s | δσ|top(T )|.
This is the "Tree Lemma" from [4], which is the 2-D version of the same in [6]. We prove it in Section 10. It reduces (2.2) to proving for each 0 < ǫ < 1
δ σ T ∈T δ,σ δσ|top(T )| |F | 1−ǫ |E| ǫ .
We can already prove this with the Estimates 11, 12, 13 (appearing in the next lemma) and some bookkeeping -this is carried out in Section 6.
Lemma 10 (Organizational Lemma). Let S be a finite collection of tiles. Then there exist a partition of S into trees T δ,σ where δ, σ are dyadic with δ 1, (i.e., S = δ,σ T ∈T δ,σ T ) such that the following estimates hold:
Estimate 11. [Orthogonality] T ∈T δ,σ |top(T )| |F | σ 2 .
Estimate 12.
[Density]
T ∈T δ,σ |top(T )| |E| δ .
Estimate 13.
[Maximal] For any ǫ > 0,
T ∈T δ,σ |top(T )| |F | 1−ǫ |E| ǫ δσ 1+ǫ .
Remark 14. In fact we can take σ 1, which we need (and prove) in the appendix.
In the remainder of this section we construct the collections of trees T δ,σ . In the following sections we prove the estimates above. Estimate 11 follows from the construction of the trees T δ,σ , and the proof of the standard size lemma; we give a proof in Section 11. We prove Estimates 12 and 13 in Section 8. We remark that we make these claims about the same family of trees. This is in contrast to [6], [4], [5], in which the argument has the form "There exists a family T size such that S δ = ∪ T ∈T size T and such that the size estimate holds for the collection T size ; further there is a (potentially different!) family T density such that S δ = ∪ T ∈T density T and such that the density estimate holds for the collection T density ."
First, we sort the tiles by density: Let
S δ = {s ∈ S : dense(s) ∈ ( 1 2 δ, δ]}
for dyadic δ. By the definition of dense, we need only consider δ ≤ ||χ|| 1 1. We next sort each collection S δ into families of trees with comparable size. The following algorithm is a slight variant of the sorting algorithm used in [6] and in [4]. We want to ensure that top(T ) ∈ T for each tree T in our construction. There are some small technicalities that arise in the 2-D situation due to the non-transitivity of the relation "≤". Without loss of generality, we may assume our collection of tiles S is finite, so we know there exists σ max such that size(S) ≤ σ max for every T ⊆ S δ . This gives us a starting point for the following lemma.
Lemma 15. Let S be a collection of tiles satisfying size(S) < σ. Then there exists a disjoint collection of trees T σ such that for all T ∈ T σ , we have top(T ) ∈ T , and
size S \ T ∈Tσ T < σ 2 .
Finally, we have the estimate
T ∈Tσ |top(T )| |F | σ 2 , (4.1)
where here F is the set used in the definition of size.
Remark 16. Having top(T ) ∈ T will be helpful in Section 8. See in particular the construction of the rectangles R T and the collections T R .
Proof. Initialize
ST OCK = S T σ = ∅.
In the following scheme we write C to denote the constant used in the definition of tree (see Definition 7), which we assume is somewhat large.
While there is a 1-tree T ⊆ ST OCK with 1 top(T ) s∈T | 1 f , ϕ s | 2 ≥ σ C and with top(T ) ∈ T , choose T with c(ω top(T )
) most clockwise, letT be the maximal tree with top equal to top(T ), and update
ST OCK := ST OCK \T T σ := T σ ∪ {T }.
(Again, we write c(ω top(T ) ) to denote the center of ω top(T ) .)
Remark 17. We remark that our choice of c(ω top(T ) ) most clockwise will be used in the proof of Estimate 4.1 in Section 11. See specifically Claim 34.
When no such trees remain, we have the collection of trees T σ described in the statement of the lemma. By construction we see that top(T ) ∈T and that size(T ) ≥ σ C for eachT ∈ T σ . The estimate (4.1) follows rather standard arguments; we present the proof in Section 11. It remains to prove the following:
Claim 18. size (ST OCK) < σ 2 .
Consider a tree T ⊆ ST OCK. Without loss of generality, T is a 1-tree (since the definition of size only takes into consideration 1-tree subtrees of T anyway). We will partition T into a collection T T of subtrees of T , each of which contains its top, as follows: Initialize
P AN T RY := T T max := ∅.
While P AN T RY is nonempty, choose a tile t of maximal length in P AN T RY , let T t be the maximal subset of P AN T RY such that s ≤ t for s ∈ T t , and update
P AN T RY := P AN T RY \ T t T max := T max ∪ {t}.
It is clear that this construction exhausts all of T ; i.e., eventually P AN T RY becomes empty. Since the tiles t ∈ T max all satisfy ω top(T ) ⊆ ω t , and since each is maximal with respect to "≤", we know these tiles are pairwise disjoint. On the other hand, they are all contained in Ctop(T ), and t = top
(T t ), so t∈Tmax |top(T t )| ≤ C|top(T )|.
Further, since each tree T t for t ∈ T max contains its top, we know
1 top(T ) s∈T | 1 f , ϕ s | 2 ≤ σ C ,
for otherwise T t would have been selected and put into T σ . Hence
s∈T | f, ϕ s | 2 = t∈Tmax s∈Tt | f, ϕ s | 2 ≤ t∈Tmax |top(T t )| σ 2 C 2 ≤ σ 2 |top(T )| C .
This implies
size(T ) ≤ σ √ C ,
which proves the claim provided C ≥ 4.
By applying the lemma iteratively to each collection S δ , we obtain collections S δ,σ and T δ,σ such that
S δ,σ = T ∈T δ,σ T
where the union is disjoint, such that dense(s) ∼ δ for s ∈ S δ,σ , and such that
size(T ) ∼ σ ∼ 1 top(T ) s∈T | 1 f , ϕ s | 2
for T ∈ T δ,σ . This proves Lemma 10, except for Estimates 12 and 13. Note that Estimate 11 follows from (4.1).
Main Lemmas
Here we present the main lemmas needed to prove Estimates 12 and 13.
Lemma 19. Suppose R is a collection of pairwise incomparable (under "≤") parallelograms of uniform width such that dense(R) ≥ δ for R ∈ R. Then R∈R |R| |E| δ .
Lemma 19 is nothing more than the Density Lemma from [6] with straightforward modifications for the 2-D setting.
Lemma 20. Suppose R is a collection of pairwise incomparable (under "≤") parallelograms of uniform width such that for each R ∈ R, we have
|E ∩ u −1 (ω R ) ∩ R| |R| ≥ δ (5.1) and 1 |R| R 1 F ≥ λ. (5.2)
Then for each ǫ > 0,
R∈R |R| |F | δλ 1+ǫ .
The proof of Lemma 20 is contained in Section 3 of [1]. More specifically, see estimate (3.10) on page 959, as well as the construction of the collection of parallelograms called R 1 there. Note that this last lemma requires an assumption of the form
1 |R| R 1 F > λ;
on the other hand, our assumption on T ∈ T δ,σ is that size(T ) σ and
1 |top(T )| s∈T 1 | 1 F , ϕ s | 2 1 2 σ,
where T 1 is the maximal 1-tree in T . The following lemma shows that the second kind of fact implies the first without much loss:
Lemma 21. Let F ⊆ R 2 . Suppose T is a tree with size(T ) σ and 1 |top(T )| s∈T 1 | 1 F , ϕ s | 2 1 2 σ,
where T 1 is the maximal 1-tree in T . Then for any ǫ > 0,
|σ −ǫ top(T ) ∩ F | |σ −ǫ top(T )| σ 1+ǫ .
Lemma 21 is proved in Section 9; it follows from L p and BM O-type estimates on a square function related to the notion of size.
Estimate 13 deserves more prominent mention. An estimate in this spirit was proved in [5]. However here we have much better dependence on the parameter δ due to a rather simple observation. The argument in [5] follows essentially the argument of the density lemma, with an appeal to a maximal theorem to control |{M δ 1 F > λ}|. In our case of a vector field depending on only one variable, the relevant maximal operator was studied by the author in [1], [2]. However this approach is inefficient. Instead of combining the density argument with a maximal function estimate (each of which costs in terms of 1 δ ), we appeal to an argument made in [1], which directly estimates
R∈R |R| |F | δλ 1+ǫ
for any ǫ > 0. In fact, this estimate was established en route to a covering lemma which implies the maximal theorem. Interestingly, the improved L 2 estimates established in [2], which interpolate to give improved L p estimates, are unhelpful in this setting, precisely because they are estimates on the operator norm, rather than on a sum like the one appearing immediately above.
Balancing the estimates
In this section we carry out some computations which allow us to prove (2.2), and hence the main theorem. We now estimate
δ σ T ∈T δ,σ δσ|top(T )|.
We have two cases. Recall that E and F are sets with |F | ≤ |E|.
6.1. Case 1: δ ≥ |F | |E| . A quick computation shows that (up to additive O(ǫ) terms in the exponents)
• the maximal estimate is more efficient when σ ≥ |F | |E| • the density lemma is more efficient when σ ≤ |F | |E| .
Remark 22. The maximal estimate is more effective than the size estimate for δ ≥ |F | |E| and σ close to |F | |E| . Without this, we would not be able to obtain L p estimates for any p < 2.
For the first range, with δ fixed, we have for any ǫ > 0
σ≥ |F | |E| T ∈T δ,σ δσ|top(T )| σ≥ |F | |E| δσ |F | 1−ǫ |E| ǫ δσ 1+ǫ = |F | 1−ǫ |E| ǫ σ≥ |F | |E| 1 σ ǫ ∼ |F | 1−2ǫ |E| 2ǫ .
Summing this over dyadic 1 δ ≥ |F | |E| gives us a total of |F | 1−3ǫ |E| 3ǫ . For the second range, with δ fixed, we have
|F | |E| ≥σ T ∈T δ,σ δσ|top(T )| |F | |E| ≥σ δσ |E| δ = |F | |E| ≥σ σ|E| ∼ |F |.
Once again, summing this over dyadic 1 δ ≥ |F | |E| gives us a total of |F | 1−ǫ |E| ǫ .
6.2. Case 2: δ ≤ |F | |E| .
In this case, the size and density estimates alone will be enough for us. A quick computation shows that
• The size estimate is most efficient when σ ≥ δ |F | |E| • The density estimate is most efficient when σ ≤ δ |F | |E| . We decompose our sum over σ into these two ranges. For the first range, we have
σ≥ δ |F | |E| δσ |F | σ 2 = |F |δ σ≥ δ |F | |E| 1 σ |F ||E|δ.
Summing over δ ≤ |F | |E| gives us a total of |F | |F | 1−ǫ |E| ǫ , since |F | ≤ |E|. For the second range, we have
σ≤ δ |F | |E| δσ |E| δ ∼ |E| σ≤ δ |F | |E| σ ∼ |F ||E|δ.
Once again, summing over δ ≤ |F | |E| gives us a total of
|F | |F | 1−ǫ |E| ǫ , since |F | ≤ |E|.
This completes the proof of the main estimate (2.2) modulo the proofs of the lemmas, which are given in the following sections.
Density lemma
In this section we prove Lemma 19. Let R be as in the hypotheses of the lemma. For k = 0, 1, 2, . . . , let R k be the collection of R ∈ R such that
|u −1 (ω R ) ∩ 2 k R ∩ E| ≥ 1 100 δ2 20k |2 k R|,
and such that k is the least integer with this property.
Note R = ∪ k R k , since if R ∈ R but R ∈ ∪ k R k , then dense(R) ≤ E R χ (1) R ≤ ∞ k=0 |u −1 (ω R ) ∩ 2 k R ∩ E|2 −100k 1 |R| ≤ 1 100 δ |R| ∞ k=0 2 25k |R|2 −100k ≤ δ 50 .
We now run an iterative selection procedure to find a subset of R k such that the parallelograms 2 k R are disjoint: Initialize
ST OCK = R k R k = ∅.
While ST OCK = ∅, choose R with maximal length, let
A R = {R ′ ∈ ST OCK : 2 k R ′ ∩ 2 k R = ∅ and ω R ′ ∩ ω R = ∅},
and update
ST OCK : = R k \ A R R k = R k ∪ {R}.
Note that the parallelograms in A R are pairwise disjoint by the pairwise incomparability of parallelograms in R, and because ω R ′ ∩ ω R = ∅ for R ′ ∈ A R . Hence, using the definition of R k , we have
R∈R k |R| = R∈ R k R ′ ∈A R |R ′ | 2 2k R∈ R k |R| 2 2k 2 −20k 1 δ R∈ R k |u −1 (ω R ) ∩ 2 k R ∩ E| 2 −18k 1 δ |E|,
where in the last inequality we have used the fact that the parallelograms 2 k R are pairwise incomparable, and that ω R = ω 2 k R , so that the sets {u −1 (ω R ) ∩ 2 k R} are disjoint. Finally, we sum over k to obtain the result.
Proofs of maximal and density estimates
We now look more closely at the collections T δ,σ . For the remainder of this section we regard δ and σ as fixed. Notation in this section is understood to depend on both δ and σ. (So, for example, T = T δ,σ .) We begin by isolating a collection of tiles with density δ. First, let
R = {R ∈ U : dense(R) ∼ δ}.
We now find a maximal subset ofR whose elements are pairwise incomparable. Initialize:
ST OCK =R R = ∅. While ST OCK = ∅, choose R of maximal length in ST OCK. Define A R = {R ′ ∈ ST OCK : R ′ ≤ R},
and update
ST OCK = ST OCK \ A R R = R ∪ {R}.
When the loop terminates, elements of R are pairwise incomparable (under ≤), and R is maximal with respect to this property.
Remark 23. Recall that for T ∈ T , dense(top(T )) ∼ δ, but maybe dense(top(T )) is much less than δ. This makes the maximal Lemma 20 unavailable to us. Note that several ingredients are required, and top(T ) may lack the dense required. The work in this section goes to organizing the trees in such a way that we can legitimately appeal to Lemma 20.
Next we associate to each tree T ∈ T a parallelogram R T ∈ R. This requires a few steps. Note that for each s ∈ ∪ T ∈T T , we have dense(s) ∼ δ. By Lemma 15, we know that top(T ) ∈ T for each T ∈ T . Hence dense(top(T )) ∼ δ. This means there exists a parallelogramR ∈R such that dense(R) ∼ δ and such that top(T ) ≤R. (This is the reason why it is convenient to have top(T ) ∈ t.) Further, for eachR ∈R, there is R ∈ R (again, possibly not unique) such thatR ≤ R. Hence we may assign to each T ∈ T some R ∈ R, and there isR such that top(T ) ≤R ≤ R. (Of course there may be more than one R to choose from for each T ; choose one!) Call this parallelogram R T . Now for each R ∈ R, define
T R = {T ∈ T : R T = R}.
By construction,
T = ∪ R∈R T R .
Our goal now is to control
R∈R T ∈T R |top(T )|.
First, we'll show that for all R ∈ R,
T ∈T R |top(T )| |R|.
The collection {top(T ) : T ∈ T R } need not be pairwise disjoint, but we do have the following satisfactory substitute.
Claim 24. There exists T R ⊆ T R such that {top(T ) : T ∈ T R } is pairwise disjoint and such that We stop when ST OCK is empty. By construction, the tops of the trees in T R are pairwise disjoint. Now we show that
T ∈T R |top(T )| T ∈T R |top(T )|. Proof. Initialize ST OCK = T R T R = ∅.T ′ ∈A T |top(T ′ )| ≤ C ′ |top(T )|.
With this we'll know that
T ∈T R |top(T )| = T ∈T R T ′ ∈A T |top(T ′ )| ≤ C ′ T ∈T R |top(T )|. Suppose not. Define S = ∪ T ′ ∈A T T ′ 1 ,
where for a tree T , define T 1 to be the maximal 1-tree contained in T . We claim S can be partitioned into a small number of trees S j , j = 1, . . . , 10C 2 , with each a 1-tree. To see that they are 1-trees, suppose s ∈ T ′ ∈ A T . Then ω s,2 ⊇ ω top(T ′ ) ⊇ ω top(T ) , so ω s,1 ∩ ω top(T ) = ∅. To see that we only need a few trees, just note that for each T ′ ∈ A T , top(T ′ ) ⊆ C(top(T )). Then since each s ∈ T ′ satisfies s ⊆ C(top(T ′ )), we know that S can be partitioned into ∼ C 2 subtrees S j by considering (possibly overlapping) tiles in C 2 top(T ) of height w and length the same as length of top(T ). Hence
10C 2 j=1 s∈S j | f, ϕ s | 2 ≥ T ′ ∈A T s∈T ′ 1 | f, ϕ s | 2 ≥ 1 4 T ′ ∈A T σ 2 |top(T ′ )| ≥ σ 2 C ′ 4 |top(T )|
Provided C ′ is taken large enough (with respect to a universal constant C mentioned in Section 3), one of the trees S j satisfies size(S j ) ≥ 10σ, which is impossible since the trees T ∈ T R were chosen from a collection with size less than σ. This proves the second claim about T R .
8.1. Proof of the density estimate. We are already in position to prove Estimate 12. Note that the collection R constructed above is of pairwise incomparable parallelograms of uniform width and dense ∼ δ. Hence the previous claim, together with Lemma 19, implies
R∈R T ∈T R |top(T )| R∈R |R| |E| δ .
8.2. Proof of the maximal estimate. The proof of Estimate 13 is a bit more involved. For the rest of this section, fix ǫ > 0. The first key step is to sort the parallelograms in R by how heavily they are covered by the trees in T R . Specifically: for integers j ≥ 0, define
R j = {R ∈ R : T ∈T R |top(T )| ∼ 2 −j |R|}.
Since our goal is to control
R∈R T ∈T R |top(T )| ∼ j R∈R j T ∈T R |top(T )| ∼ j 2 −j R∈R j |R|,
it is enough to estimate R∈R j |R| with suitable dependence on j.
In order to apply maximal technology (in the form of Lemma 20), we must find parallelograms R that heavily intersect F , and that also contain a large subset on which v points in the direction of R. Because of the Schwartz tails in the definition of dense, we do not know that each R ∈ R j satisfies
|u −1 (ω R ) ∩ E ∩ R| δ|R|.
Rather, we know that
|u −1 (ω R ) ∩ E ∩ 2 k R| 2 20k δ|R| (8.1)
for some integer k ≥ 0, as in Section 7. Define R j,k to be the set of R ∈ R j such that condition (8.1) holds for R but such that it does not hold with any smaller k. Similarly, we cannot conclude that R itself intersects F heavily. Recall that Lemma 21 guarantees that F intersects σ −ǫ top(T ) heavily, whenever T ∈ T δ,σ ; we cannot however, conclude that F intersects top(T ) itself. This causes some minor differences in the treatment of the cases 2 k ≥ σ −ǫ and 2 k ≤ σ −ǫ that the reader should not take too seriously. It suffices then to control sums like R∈R j,k |R| with suitable dependence on k and j.
8.2.1. Case1: 2 k ≥ σ −ǫ . We want to apply Lemma 20 to the collection R j,k . The defining condition of R j,k gives us the kind of information needed by the hypothesis (5.1). The following claim gives us the kind of information needed by the hypothesis (5.2).
Claim 25. For R ∈ R j,k |F ∩ 2 k R| |2 k R| 2 −j σ 1+3ǫ σ −ǫ 2 k 2 .
We postpone the proof of the claim until the end of this section. With the claim, the only ingredient still needed to apply Lemma 20 is the pairwise incomparability of the parallelograms in question. We arrange this with the usual type of sorting algorithm. Initialize
ST OCK = R j,k R j,k = ∅. While ST OCK = ∅, choose R with maximal length, let A R = {R ′ ∈ ST OCK : 2 k R ′ ∩ 2 k R = ∅ and ω R ′ ∩ ω R = ∅},
and update
ST OCK : = R j,k \ A R R j,k = R j,k ∪ {R}.
(Note ω R = ω CR for any C.) Since the parallelograms R ′ ∈ A R are pairwise incomparable, we know they are in fact disjoint (see earlier in Section 8 for a similar argument), so
R ′ ∈A R |R ′ | |2 k R|. Hence j k R∈R j,k T ∈T R |top(T )| j k R∈R j,k 2 −j |R| j k R∈ R j,k R ′ ∈A R 2 −j |R ′ | j k R∈ R j,k 2 −j |2 k R|.
We now focus our attention on
R∈ R j,k 2 −j |2 k R|.
Claim 25 together with the defining condition for parallelograms in R j,k allows us to apply Lemma 20, with "δ" in (5.1) being 2 20k δ and "λ" in (5.2) being 2 −j 2 −2k σ 1+O(ǫ) , as in Claim 25. The huge gain in k from (8.1) allows us to sum the contributions from the various R j,k . More specifically, Lemma 20 yields
R∈ R j,k |2 k R| 1 2 20k δ |F | (σ 1+ǫ 2 −2k 2 −j ) 1+ǫ
This obviously sums in k to prove
R∈R j T ∈T R |top(T )| R∈R j 2 −j |R| 1 δ 2 ǫj |F | (σ 1+ǫ ) 1+ǫ ;
this estimate is effective for small j. Estimate 12 tells us that for any j,
R∈R j T ∈T R |top(T )| R∈R j 2 −j |R| 2 −j |E| δ ;
this estimate is effective for large j. It remains to balance these two estimates:
j≥0 R∈R j T ∈T R |top(T )| = j≤log |E|σ |F | R∈R j 2 −j |R| + j≥log |E|σ |F | R∈R j 2 −j |R| j≤log |E|σ |F | 2 ǫj |F | δσ (1+ǫ) 2 + j≥log |E|σ |F | 2 −j |E| δ |F | 1−ǫ |E| ǫ δσ (1+ǫ) 2 |F | 1−5ǫ |E| 5ǫ δσ 1+5ǫ ,
Remark 26. Of course the first sum above is empty when σ ≤ |F | |E| ; in this case we recover the density estimate. Recalling Section 6, we see that in this range of σ we have no need for the maximal estimate anyway.
This completes the proof of the maximal estimate, except for the proof of Claim 25, which we turn to now.
Proof of Claim 25. For each T ∈ T R , Lemma 21 tells us that
|σ −ǫ top(T ) ∩ F | |σ −ǫ top(T )| ≥ σ 1+ǫ .
One minor technical problem is that the parallelograms σ −ǫ top(T ) might not be disjoint. But since all parallelograms {top(T ) : T ∈ T R } have (essentially) the same orientation, we may use a standard covering argument to select a subset T R of T R such that
{σ −ǫ top(T )} T ∈ T R
is pairwise disjoint, and such that
| T ∈ T R σ −ǫ top(T )| | T ∈T R σ −ǫ top(T )|. Hence |F ∩ Cσ −ǫ R| | T ∈ T R σ −ǫ top(T ) ∩ F | = T ∈ T R |σ −ǫ top(T ) ∩ F | by disjointness σ 1+ǫ T ∈ T R |σ −ǫ top(T )| by Lemma 21 σ 1+ǫ | T ∈ T R σ −ǫ top(T )| σ 1+ǫ | T ∈T R σ −ǫ top(T )| σ 1+ǫ | T ∈T R top(T )| σ 1+ǫ T ∈T R |top(T )| by disjointness σ 1+ǫ T ∈T R |top(T )| by Claim 24 σ 1+ǫ 2 −j |R| by definition of R j .
This finishes the proof of Claim 25.
8.2.2. Case 2: 2 k ≤ σ −ǫ . This section is very similar to the previous section. As in the last section, we verify the hypotheses of Lemma 20 for a suitable collection.
We consider all of these collections R j,k together. Let
R j,small = 0≤k≤log σ −ǫ R j,k .
Now we sort the tiles as before: Initialize
ST OCK = R j,small R j,small = ∅.
While ST OCK = ∅, choose R with maximal length, let
A R = {R ′ ∈ ST OCK : σ −ǫ R ′ ∩ σ −ǫ R = ∅ and ω R ′ ∩ ω R = ∅},
and update
ST OCK : = R small \ A R R j,small = R j,small ∪ {R}.
As before, we have
R∈R j,small |R| ≤ R∈ R j,small R ′ ∈A R |R ′ | ≤ R∈ R j,small |σ −ǫ R|.
We again note several properties of the parallelograms in R j,small . First, they are pairwise incomparable. Second, they satisfy the estimate
|σ −ǫ R ∩ E ∩ u −1 (ω σ −ǫ R )| |σ −ǫ R| σ 2ǫ δ.
This gives us the density estimate
R∈ R j,small |σ −ǫ R| |E| σ 2ǫ δ ,(8.|σ −ǫ R ∩ F | |σ −ǫ R| 2 −j σ 1+ǫ .
So by Lemma 20, we have
R∈ R j,small |σ −ǫ R| |F | δ (2 −j σ 1+ǫ ) 1+ǫ . (8.3)
As before, we split the sum into large and small j and use (8.2) and (8.3), respectively:
j≥0 R∈R j,small T ∈T R |top(T )| = j≤log |E|σ |F | R∈R j,small 2 −j |R| + j≥log |E|σ |F | R∈R j,small 2 −j |R| j≤log |E|σ |F | 2 ǫj |F | δσ (1+ǫ) 2 + j≥log |E|σ |F | 2 −j |E| σ 2ǫ δ |F | 1−5ǫ |E| 5ǫ δσ 1+5ǫ
, which is what we needed, since ǫ is arbitrary.
Large size implies large intersection with F
Remark 27. The title of the section is technically a bit misleading, since size(T ) is actually the supremum over all subtrees of T of an l 2 -type norm; nevertheless, the trees obtained through the selection procedure in Section 4 all satisfy the property that the full tree (essentially) achieves this supremum.
To prove Lemma 21, we need the following notation. For a fixed 1-tree T , define the operator
∆(f ) = s∈T | f, ϕ s | 2 1 s |s| 1 2 .
We need the following facts about ∆.
Lemma 28. For any N > 0, we have
||∆f || p ||f β N,T || p for p ∈ (1, ∞), where β N (x 1 , x 2 ) = 1 1 + |x 1 | N + |x 2 | N ,
and β N,T is an L ∞ -normalized version of β N adapted to top(T ). The implicit constant depends on N but not on T .
We prove Lemma 28 in Section 13. Of course proving ||∆f || 2 ||f || 2 is straightforward; indeed, it is an easy special case of Lemma 36. The work is in inserting the smooth cutoff β N , which is the point of Lemma 36, and moving below L 2 . Second,
Lemma 29. ||∆f || 2 1 |top(T )| 1 2 Ctop(T ) ∆f,
provided that T satisfies the following uniform size estimate:
sup 1-trees T ′ ⊆T 1 |top(T ′ )| s∈T ′ | f, ϕ s | 2 1 2 1 |top(T )| s∈T | f, ϕ s | 2 1 2 .
The condition in the last lemma is the one mentioned in the remark at the beginning of this section. We prove Lemma 29 in Section 14. The point of these lemmas is that ||∆f || 2 is closely related to size(T ). Indeed,
||∆f || 2 2 = s∈T | f, ϕ s | 2 .
On the other hand, we want information about |F ∩ top(T )| (or possibly |F ∩ M top(T )| for a dilate M top(T ) of top(T ), which is actually what we will obtain below), which is much more closely related to ||∆f || p for p close to 1, as we see below. Combining these two lemmas and Hölder's inequality gives us
1 |top(T )| s∈T | f, ϕ s | 2 1 2 = 1 |top(T )| 1 2 ||∆f || 2 1 |top(T )| Ctop(T ) ∆f 1 |top(T )| (∆f ) 1+ǫ 1 1+ǫ 1 |top(T )| (f β N,T ) 1+ǫ 1 1+ǫ . 1 |top(T )| s∈T | f, ϕ s | 2 1 2 ∼ σ gives us for any N , σ 1+ǫ |top(T )| 1 F (β N,T ) 1+ǫ |σ −ǫ top(T ) ∩ F | + σ (N −2)ǫ |σ −ǫ top(T )|
This proves Lemma 21 since N can be chosen arbitrarily large with respect to ǫ.
Proof of Tree Lemma
In this section we present a proof of Lemma 9. Recall that we have a fixed tree T in mind. For notational convenience we assume that the slope of the long side of top(T ) is zero. We write π 1 (E), π 2 (E) to denote the vertical, horizontal (respectively) projections of a set E. Of course the width of every tile in T is a fixed number w. Let J 1 be a partition of R (the horizontal axis) into dyadic intervals such that 3J × R does not contain any tile s ∈ T , and such that J is maximal with respect to this property. Now let J 2 be a partition of R (the vertical axis) into intervals of width 1 3 |π 2 (top(T ))|. Let P =
J 1 ∈J 1 J 2 ∈J 2 J 1 × J 2 .
This is a partition of R 2 . The parallelograms P ∈ P are the smallest relevant parallelograms for this tree. The parallelograms P ∈ P with π 1 (P ) far away from top(T ) are defined so as to still be able to take advantage of the density estimate for tiles in T . Now for each P ∈ P we split the operator L into two pieces, one corresponding to tiles with larger x-projection than P , the other to tiles with smaller x-projection than P : let
T + P = {s ∈ T : |π 1 (s)| > |π 1 (P )|} T − P = {s ∈ T : |π 1 (s)| ≤ |π 1 (P )|} L + P = s∈T + P f, ϕ s φ s 1 E L − P = s∈T − P f, ϕ s φ s 1 E .
Note that for appropriate ǫ s with |ǫ s | = 1, we have
s∈T | f, ϕ s φ s 1 E | = s∈T ǫ s f, ϕ s φ s 1 E = s∈T ǫ s f, ϕ s φ s 1 E = P ∈P P s∈T ǫ s f, ϕ s φ s 1 E = P ∈P P L − P + P ∈P P L + P . (10.1)
The main term will come from parallelograms P ∈ P close to top(T ); estimates on parallelograms P away from top(T ) will come with a decay factor. To make things more precise, define for k ≥ 1,
P 0 = {P ∈ P : dist(π 2 (P ), π 2 (top(T ))) |π 2 (top(T ))| ≤ 1} P k = {P ∈ P : dist(π 2 (P ), π 2 (top(T ))) |π 2 (top(T ))| ∈ (2 k−1 , 2 k ]|}.
We focus first on the first term in (10.1). To control it we need only spatial decay in both the horizontal and vertical directions.
10.1. Small tiles. For notational convenience, we further consider for l ≥ 1,
P k,0 = {P ∈ P k : dist(π 1 (P ), π 1 (top(T ))) |π 1 (top(T ))| ≤ 1}, P k,l = {P ∈ P k : dist(π 1 (P ), π 1 (top(T ))) |π 1 (top(T ))| ∈ (2 l−1 , 2 l ]}.
We divide the sum in the definition of L − P into pieces according to how large the tiles are. Specifically, let
T j = {s ∈ T − P : |s| = 2 −j |top(T )|}.
The reason for this is that since the tiles s ∈ T − P are shorter than P , their frequency intervals can be much larger than that of P , meaning we lose control on |P ∩ supp(L − P )|. We use the extra decay from Schwartz tails to compensate for this. The upper bound of size(T ) ≤ σ implies that for individual tiles s ∈ T we have | f, ϕ s | ≤ σ|s|
1 2 . Hence | s∈T j f, ϕ s φ s 1 E | s∈T j σχ (∞) s σ2 −N k m≥2 j+l m −N σ2 −N k 2 −Nj 2 2 −Nl 2 .
But note that since dense(s) δ, we have
δ Es χ (1) s ≥ 2 −100(k+j+l) |P ∩ supp( s∈T j f, ϕ s φ s 1 E )| |P | ,
This last estimate follows from considering the distance between s and P relative to the length of s. Hence for any P ∈ P k,l , we have
P |L − P | ≤ j≥0 P | s∈T j f, ϕ s φ s 1 E )| σ j≥0 2 −N k 2 −Nj 2 2 −Nl 2 |P ∩ supp( s∈T j f, ϕ s φ s 1 E | δ|P |σ2 −10(l+k)
Summing over k, l and P gives us
P ∈P P |L − P | l≥0 k≥0 P ∈P k,l P |L − P | l≥0 k≥0 P ∈P k,l σδ|P |2 −10k 2 −10l σδ|top(T )|,
with the primary contribution coming from P near top(T ) as usual.
10.2. Large tiles. We start by remarking that sorting with respect to horizontal distance from T (i.e., using the index l, as in the previous subsection) is unnecessary in this subsection. For if P ∈ P k,l with l ≥ C, then T + P is empty, because |Π 1 (P )| > |Π 1 (top(T ))|. This fact will appear several times in what follows. Next, we show that the term under consideration in this section has small support. Precisely:
Claim 30. For P ∈ P k , L + P 1 E is supported on a set of size δ|P |2 100k . The factor 2 100k arises from the tail in the definition of dense and the fact that P is away from top(T ). Fortunately, the decay in the functions ϕ s for s ∈ T is even greater when P is away from top(T ).
Proof. It is convenient to proceed by contradiction. Assume L + P 1 E has much larger support than δ|P |2 100k . By the construction of P , we know that there is some s ∈ T such that s ⊆ C2 k P . But this implies there is R of the same dimensions as P , but located spatially over T , with ω R ⊆ ω s and such that dense(R) ≥ 100δ, say. Since this implies s ≤ R, we have contradicted the assumption that dense(s) ≤ δ.
We now turn our attention to the second term in (10.1). Recall the definitions of 1-trees and 2-trees. Clearly for every s ∈ T , either ω s,1 ∩ ω top(T ) = ∅ or ω s,2 ∩ ω top(T ) = ∅, so our tree T can be partitioned as T = T 1 ∪ T 2 , where T j is a j-tree. Let (T + P ) j = T + P ∩ T j for j = 1, 2. Of course (T + P ) j is still a j-tree. We treat the two cases separately.
10.2.1. The 2-tree case. This case is a bit easier to handle because of the location of the support of the function φ s . More to the point: Since T 2 is a 2-tree, if there exists x such that φ s (x)φ t (x) = 0 for s, t ∈ T 2 , then |s| = |t|. This follows from the fact that φ s (x) = 0 unless v(x) ∈ ω s,2 , together with the fact that ω s,1 ⊇ ω top(T ) , and similarly for t. (This was mentioned near the definition of φ s in Section 2.) Further, we know that for any tile s ∈ T , we have | f, ϕ s | ≤ σ|s| 1 2 by the size estimate for T . Combining these observations with Claim 30 and the rapid decay of φ s in the vertical direction gives us for P ∈ P k that
P s∈(T + P ) 2 f, ϕ s φ s 1 E σδ2 −10k |P |,
since the integrand is uniformly bounded by σ2 −200k . As mentioned earlier, if |π 1 (s)| ≥ |π 1 (P )|, then π 1 (P ) ⊆ Cπ 1 (top(T )). Hence
k P ∈P k P s∈(T + P ) 2 f, ϕ s φ s 1 E δσ|top(T )|.
This completes the estimate for T 2 .
10.2.2. The 1-tree case. In this case we appeal to orthogonality in the form of the Bessel inequality in Lemma 36. For parallelograms P ∈ P whose vertical component is large, we need the decay factor from Lemma 36. We first introduce some extra functions associated to the tiles: let
α s (x) = ψ s (t)ϕ s (x 1 − t, x 2 )dt.
The difference between α s and φ s is that the vector field v makes no explicit appearance in the definition of α s ; rather, the integral is taken over a horizontal line for every x. In φ s , however, the integral is taken over an almost horizontal line, where the precise definition of almost depends on the length of s. (The line is horizontal because we assumed that the slope of the long side of top(T ) is zero. In the general case it is parallel to top(T ).) We have the obvious equality
P s∈(T + P ) 1 ǫ s f, ϕ s φ s 1 E = P s∈(T + P ) 1 ǫ s f, ϕ s α s 1 E + P s∈(T + P ) 1 ǫ s f, ϕ s (φ s − α s )1 E .
This decomposition allows us to reduce our problem to proving the following two claims:
Claim 31. For each P ∈ P,
P s∈(T + P ) 1 ǫ s f, ϕ s α s 1 E δ j≥0 2 −N j 1 |2 j P | 2 j P | s∈T 1 f, ϕ s α s |.
Claim 32. For P ∈ P k ,
s∈(T + P ) 1 ǫ s f, ϕ s (φ s − α s )1 E 2 −200k σ.
Notice that supp α s ⊆ supp ϕ s , since
α s (ξ) = ψ s (t)e −2πitξ 1 ϕ s (ξ)dt.
This will allow us to prove orthogonality statements about the α s later in the proof. For example, From this we can conclude that
|| s∈T 1 ǫ s f, ϕ s α s || 2 2 s∈T 1 | f, ϕ s | 2 ,(10.2)
because the fact stated above about the Fourier support of the functions α s allows us to prove this inequality in the same way we prove the Bessel inequality in Section 12: expand the square, and notice that α s , α t = 0 unless |s| = |t|.
Again we remark that if T + P is nonempty, then π 1 (P ) ⊆ Cπ 1 (topT ). Hence in the summation below we can ignore dependence on the parameter l used in the last section. Given these claims, together with Claim 30, we control the first term in (10.1) by
P ∈P P L + P P ∈P P ǫ s s∈(T + P ) 1 f, ϕ s α s 1 E + P ∈P P ǫ s s∈(T + P ) 1 f, ϕ s (φ s − α s )1 E k P ∈P k δ P j≥0 2 −N j 1 |2 j P | 2 j P | s∈T 1 f, ϕ s α s | + k P ∈P k 2 −200k σ|P ∩ supp(L + P )|.
Note that the second term in the last display is controlled by Claim 30. For P ∈ P k , it is convenient to split the function s∈T 1 f, ϕ s α s into two pieces, using the identity
1 R 2 = 1 D k−5 + 1 (D k−5 ) c , where D k = {(x, y) : |y| 2 k |π 2 (top(T ))|}.
In other words, D k is horizontal strip of width ∼ 2 k |π 2 (top(T ))|. (Obvious modifications can be made in the case k ≤ 5.) For the first piece-the one closer to top(T )-we can use the fact that the tile P is far from top(T ) together with the decay in j to obtain good control. For the second piecethe one away from top(T ) -we can take advantage of the decay in the wave packets associated to tiles in T in the form of the Bessel inequality in Lemma 36. We focus first on the term close to top(T ):
k P ∈P k δ P j≥0 2 −N j 1 |2 j P | 2 j P | s∈T 1 f, ϕ s α s 1 D k−5 | = k P ∈P k δ P j≥k 2 −N j 1 |2 j P | 2 j P | s∈T 1 f, ϕ s α s 1 D k−5 | k P ∈P k δ2 −N k P M (| s∈T 1 f, ϕ s α s 1 D k−5 |) = δ ∪ C l=0 ∪ P ∈P k,l P 2 −N k M (| s∈T 1 f, ϕ s α s 1 D k−5 |) δ2 −N k ∪ C l=0 ∪ P ∈P k,l P 1 2 | s∈T 1 f, ϕ s α s 1 D k−5 | 2 1 2 .
This nearly finishes the proof for the first term, since we may estimate this L 2 norm by using orthogonality in the x-variable just as in the proof of Lemma 36 below. (Readers uncomfortable with this should look to the proof of Lemma 36.) Specifically, we have
| s∈T 1 f, ϕ s α s 1 D k−5 | 2 = s∈T 1 s ′ ∈T 1 f, ϕ s f, ϕ s ′ D k α s α s ′ s∈T 1 | ϕ s , f | 2 s ′ : |s|=|s ′ | |α s α s ′ | s∈T 1 | ϕ s , f | 2 σ 2 |top(T )|.
We have used symmetry and the x-orthogonality in the first inequality above. This finishes the proof for the first term. To control the second term (the one away from top(T )), we can appeal directly to a Bessel-type inequality. Here we use such an inequality for the functions α s rather than the functions ϕ s , just as in the estimate above, but we also obtain significant decay in k just as in Lemma 36. The proof is identical to the proof of Lemma 36. Hence
k C l=0 P ∈P k,l δ P j≥0 2 −N j 1 |2 j P | 2 j P | s∈T 1 f, ϕ s α s 1 (D k−5 ) c | δ ∪ C l=0 ∪ P ∈P k,l P M (| s∈T 1 f, ϕ s α s 1 (D k−5 ) c |) δ ∪ C l=0 ∪ P ∈P k,l P 1 2 | s∈T 1 f, ϕ s α s 1 (D k−5 ) c | 2 1 2 δ2 k |top(T )| 1 2 (σ 2 2 −100k |top(T )|) 1 2 2 −10k δσ|top(T )|,
which is what we want.
Proof of Claim 31. Recall that we are considering a point x ∈ P for some parallelogram P , and we consider the sum
s∈T 1 : |π 1 (s)|>|π 1 (P )| f, ϕ s φ s (x).
The restriction in the summation already implies that for any x, there is m(x) such that all tiles s who make an appearance in the sum above satisfy |π 1 (s)| ≥ m(x). Further, since we know that u(x) ∈ ω s,2 , we also have M (x) such that all tiles s who make an appearance in the sum above satisfy |π 1 (s)| ≤ M (x). Both of these claims are reversible, so
{s ∈ T 1 : |π 1 (s)| > |π 1 (P )|} = {s ∈ T : m(x) ≤ L(s) ≤ M (x)}.
Hence it is our goal to estimate
s∈T : m(x)≤L(s)≤M (x) f, ϕ s α s .
Denote by k a Schwartz function such that suppk ⊆ [−1 − 1 100 , 1 + 1 100 ] 2 , and such thatk(ξ) = 1 for ξ ∈ [−1, 1] 2 . further denote by k r the function obtained by adapting k to the rectangle [ −1 r , 1 r ] × [ −1 w , 1 w ]; i.e., let k r (x, y) = k( x r , y w ). With this definition, we know for any N (which appears in the last line of the computation below) f, ϕ s (φ s (x) − α s (x))1 ω s,2 (u(x)).
s∈T 1 : m(x)≤L(s)≤M (x) f, ϕ s α s = s∈T 1 : m(x)≤L(s) f, ϕ s α s − s∈T 1 : L(s)>M (x) f, ϕ s α s = ( s∈T 1 f, ϕ s α s ) * k m(x) − ( s∈T 1 f, ϕ s α s ) * k M (x) ≤ j≥0 2 −N j 1 |2 j P | 2 j P | s∈T 1 f, ϕ s α s |.
To do this we first estimate |φ s − α s |. By definition, we have
|φ s (x) − α s (x)| ≤ |ψ s (t)||ϕ s (x 1 − t, x 2 − tu(x)) − ϕ(x 1 − t, x 2 )|dt.
To compute the difference in the integrand, estimating the following quantity will be helpful:
⋆ := sup z∈[0,tu(x)] ∂ ∂x 2 ϕ s (x 1 − t, x 2 − z).
Fix an integer j ≥ 1 and consider |t| ∼ 2 j |π 1 (s)|. If (x 1 , x 2 ) ∈ 2 j+10 s, then ⋆ χ (2) s (x 1 , x 2 ). If (x 1 , x 2 ) ∈ 2 j+10 s, then ⋆ 1. We also have that ψ s (t) 1 2 Nj |s| for any N . Analogous facts hold when j = 0 and |t| ≤ |π 1 (s)|. Let I j = {t : |t| ∼ 2 j |π 1 (s)|} for j ≥ 1 and I 0 = {t : |t| ≤ |π 1 (s)|}. Combining these observations gives us for (x 1 , x 2 ) ∈ 2 j+10 s that
|φ s (x) − α s (x)| j≥0 I j 1 2 N j |s| 2 j |π 1 (s)| |u(x)| w χ (2) s (x 1 , x 2 )dt |π 1 (s)| |u(x)| w χ (2) s (x 1 , x 2 ). If (x 1 , x 2 ) ∈ 2 j+10 s, then we have ⋆ 2 100j χ (2) s , so |φ s (x) − α s (x)| j≥0 I j 1 2 −N j |s| 2 j |π 1 (s)| |u(x)| w dt |π 1 (s)| |u(x)| w χ (2) s (x 1 , x 2 ).
Since u(x) ∈ ω s,2 for all s ∈ T 1 , we know u(x) ≤ w |π 1 (s)| . Combining this with the fact that | f, ϕ s | σ|s| 1 2 and the estimate immediately above, we have
| m(x)≤|π 1 (s)|≤M (x) f, ϕ s (φ s − α s )| ≤ |π 1 (s)|≤ w u(x) σ|s| 1 2 |u(x)| |π 1 (s)| w χ (2) s (x 1 , x 2 ) σχ (∞) top(T ) (x 1 , x 2 ),
which is what we claimed.
Proof of size estimate
In this section we write f = 1 F ; note that we do not use the fact that f is a characteristic function. As with the tree lemma, there are small modifications required from the one-dimensional situation to handle Schwartz tails in the vertical direction. We use the Bessel inequality from Lemma 36 to do this. First we note that by assumption,
σ 2 T ∈T |top(T )| T ∈T s∈T | f, ϕ s | 2 = f T s f, ϕ s ϕ s ≤ ||f || 2 || T s f, ϕ s ϕ s || 2 .
It is enough to prove
|| T s f, ϕ s ϕ s || 2 ≤ σ T ∈T |top(T )|.
By expanding the square and using symmetry, we have
|| T ∈T s∈T f, ϕ s ϕ s || 2 2 = T ∈T T ′ ∈T s∈T ′ s ′ ∈T ′ f, ϕ s f, ϕ s ′ ϕ s , ϕ s ′ T ∈T s∈T T ′ ∈T s ′ ∈T ′ : |s ′ |=|s| | f, ϕ s f, ϕ s ′ ϕ s , ϕ s ′ | + T ∈T s∈T T ′ ∈T s ′ ∈T ′ : |s ′ |<|s| f, ϕ s f, ϕ s ′ ϕ s , ϕ s ′ = B + C.
Note that
{s ′ : |s ′ | = |s| and ω s ∩ ω s ′ = ∅} partitions R 2 , so |s ′ |=|s| | ϕ s , ϕ s ′ | ∼ 1.
Hence we can estimate the first term, using symmetry again, by
B T ∈T s∈T T ′ ∈T s ′ ∈T : |s ′ |=|s| | f, ϕ s | 2 | ϕ s , ϕ s ′ | T ∈T s∈T | f, ϕ s | 2 ∼ σ 2 T ∈T |top(T )|.
Now we look at the second term C. By Cauchy-Schwarz, we have
C ≤ T ∈T s∈T | f, ϕ s | 2 1 2 s∈T T ′ ∈T ′ s ′ ∈T ′ : |s ′ |<|s| ϕ s , ϕ s ′ f, ϕ s ′ 2 1 2 T ∈T σ|top(T )| 1 2 D(T ) 1 2 where D(T ) = s∈T T ′ ∈T ′ s ′ ∈T ′ : |s ′ |<|s| ϕ s , ϕ s ′ f, ϕ s ′ 2 .
It remains to analyze D(T ) for a tree T ∈ T . We claim that the set of tiles over which the inner sum ranges is actually independent of s. More specifically, define
A = {s ′ ∈ T ′ =T,T ′ ∈T
T ′ : ω s,1 ∩ ω s ′ ,1 = ∅ and |s ′ | < |s| for some s ∈ T }.
Then
Claim 33. For each s ∈ T ,
T ′ ∈T s ′ ∈T ′ : |s ′ |<|s| ϕ s , ϕ s ′ f, ϕ s ′ = s ′ ∈A ϕ s , ϕ s ′ f, ϕ s ′ .
Proof. It is clear from the definition of A that the summation on the left is over a set of tiles that is contained in A. So suppose s ′ ∈ A; by definition of A, this gives uss ∈ T such that |s ′ | < |s| and such that ωs ,1 ∩ ω s ′ ,1 = ∅. This last condition guarantees that ω s ′ ,1 ⊇ ω T . If |s| ≥ |s|, then of course |s| > |s ′ | and ω s,1 ∩ ω s ′ ,1 = ∅, so that in fact the tile s ′ appears in the summation on the left hand side of the claim. If |s| < |s| and |s| > |s ′ | then we are done as before. So assume |s| ≤ |s ′ | < |s|. In this case ω s,1 ∩ω s ′ ,1 = ∅, which implies that ϕ s , ϕ s ′ = 0, finishing the proof of the claim.
Now for a collection of tiles C, define
F (C) = t∈C f, ϕ t ϕ t .
With this notation, we have
D(T ) = s∈T | ϕ s , F (A) | 2 .
Before we proceed, we mention a key disjointness property of tiles in A.
Claim 34. Tiles in A are pairwise disjoint.
Proof. Suppose t, t ′ ∈ A. Then there are s, s ′ ∈ T such that ω t,2 ⊇ ω s ⊇ ω top(T ) and such that ω t ′ ,2 ⊇ ω s ′ ⊇ ω top(T ′ ) . Hence ω t,2 ∩ ω t ′ ,2 = ∅, we may assume without loss of generality that ω t,2 ⊆ ω t ′ ,2 , i.e., that |t ′ | ≤ |t|. This means the tree T * containing t was selected before the tree containing t ′ . Finally, note that t and t ′ cannot belong to the same 1-tree, since ω t,2 ⊆ ω t ′ ,2 . If t ∩ t ′ = ∅, then in fact t ′ ⊆ V (top(T * )), and hence t ′ was included in the maximal tree T * containing the 1-tree T * ; see the selection algorithm in Consider a function g ∈ L 2 , and remember that | f, ϕ s | σ |s|. Then using the claim above about disjointness of tiles s ∈ A k , we have
F (A k )g1 R k−3 = s∈A k f, ϕ s ϕ s (x)1 R k−3 (x)g s∈A k 2 −10k σχ (∞) s (x)g 2 −10k σ s∈A k s M g ≤ 2 −10k σ s∈A k s M g ≤ 2 −10k σ |R k | 1 2 ||g|| 2 ≤ 2 −10k σ(2 2k |top(T )|) 1 2 ||g|| 2 ,
which implies that
I k ||1 R k−3 F (A k )|| 2 2 σ 2 2 −4k |top(T )|.
This proves (11.1) for I k .
To estimate II k , we need only estimate ||F (A k )|| 2 and apply Lemma 36. We do this just as above: let g be such that ||g|| 2 = 1. Then
F (A k )g ≤ | s∈A k f, ϕ s ϕ s g| | s∈A k σχ (∞) s g| σ ∪ s∈A k s M g σ|R k | 1 2 . So ||F (A k )|| 2 2 σ 2 | ∪ A k | σ 2 2 2k |top(T )|.
Hence by Lemma 36,
II k 2 −10k ||F (A k )|| 2 2 σ 2 2 −8k |top(T )|.
Summing in k proves D(T ) σ 2 |top(T )|, which finishes the proof.
Localized Bessel inequality
In this section we prove a Bessel inequality for 1-trees with functions supported away from the top of the tree. Specifically:
Lemma 36. Let T be a 1-tree. For k ≥ 1, let R k = 2 k top(T ). For k ≥ 1, let Ω k = R k \ R k−1 . Define Ω 0 = top(T ). Then for any N > 0, s∈T | f 1 Ω k , ϕ s | 2 2 −N k ||f 1 Ω k || 2 2 .
Remark 37. For a classical 1-dimensional tree, this can be proved by using the extreme spatial decay of the wave packets ϕ s , s ∈ T , away from top(T ). We use this in conjunction with orthogonality in the x-variable to handle interactions of functions ϕ s , ϕ s ′ horizontally close to the tree, where tail estimates do not improve for shorter tiles in the tree. This is the reason for the decomposition of Ω k into B k and C k in the proof below.
Proof. For notational convenience, we will assume that the parallelogram top(T ) is centered at the origin, has width 1, and has sides parallel to the coordinate axes. First note that
s∈T | f 1 Ω k , ϕ s | 2 = s∈T | f 1 B k , ϕ s | 2 s∈T | f 1 C k , ϕ s | 2 =: B + C, where B k = {(x, y) ∈ Ω k : |y| ≥ 2 k } C k = Ω k \ B k .
To estimate B we will need to use orthogonality in the horizontal variable. To estimate C we will need only spatial decay, as in the one-dimensional case.
Note that by Cauchy-Schwarz
B 2 = B k f s∈T f 1 B k , ϕ s ϕ s ≤ ||f 1 Ω k || 2 s∈T s∈T ′ |y|≥2 k x∈R f 1 B k , ϕ s f 1 B k , ϕ s ′ ϕ s (x, y)ϕ s ′ (x, y)dxdy 1 2 .
Also note that if |s| = |s ′ |, then for every y, we have
x ϕ s (x, y)ϕ s ′ (x, y) = 0.
This follows from the definition of the wave packets ϕ s ; specifically, note that π 1 (supp(φ s )) ∩ π 1 (supp(φ s ′ )) = ∅ whenever ω s,1 ∩ ω s,2 = ∅, which happens whenever s, s ′ are in the same 1-tree and |s| = |s ′ |. By symmetry we may estimate | f 1 Ω k , ϕ s f 1 Ω k , ϕ s ′ | ≤ | f 1 Ω k , ϕ s | 2 , which gives us
s∈T s ′ ∈T |y|≥2 k x f 1 B k , ϕ s f 1 Ω k , ϕ s ′ ϕ s (x, y)ϕ s ′ (x, y) ≤ s∈T s ′ ∈T : |s|=|s ′ | | f 1 B k ϕ s | 2 |y|≥2 k x |ϕ s ||ϕ s ′ |.
But note that
s ′ ∈T : |s|=|s ′ | |y|≥2 k x |ϕ s ||ϕ s ′ | ≤ 2 −N k ,
because the prototype ϕ is Schwartz, s ∈ T , and Ω k is far away from top(T ). Hence
B 2 − N 2 k ||f 1 Ω k || 2 s∈T | f 1 Ω k ϕ s | 2 1 2 .
We now estimate C. Define
T j = {s ∈ T : |s| = 2 −j |top(T )|}.
Note that if s ∈ T j , then | f 1 C k , ϕ s | 2 − N 2 k−50j ||f 1 Ω k || 2 by Cauchy-Schwarz and the fact that ||ϕ s 1 c k || 2 2 − N 2 k−50j . This last claim follows from the fact that ϕ s is highly localized to top(T ), and because C k is far away from top(T ) horizontally. (Of course we could not make the same argument for B because we can do no better than ||ϕ s 1 B k || 2 2 −N k for s ∈ T j ; i.e., there is no decay in the parameter j.) This is already enough:
C ≤ j≥0 s∈T j | f 1 Ω k ϕ s | 2 2 − N 2 k ||f 1 Ω k || 2 ,
which finishes the proof of the lemma.
Square function estimates
In this section we prove Lemma 28. The proof is similar to the standard proof of L p boundedness for the analogous one-dimensional square function, with a few tweaks to handle the two-dimensionality. For notational convenience we will assume, without loss of generality, that the tree T has top that is axis parallel and centered at the origin. Proving the lemma with the spatial localization requires us to decompose ∆ spatially as follows. For
k ≥ 1, define the set Ω k = 2 k top(T ) \ 2 k−1 top(T ). For k = 0, define Ω k = top(T ). Now define ∆ k (f ) = s∈T | f, 1 Ω k ϕ s | 2 1 s |s| 1 2 .
By Minkowski's inequality, we have
∆f (x) = s∈T | f, k 1 Ω k ϕ s | 2 1 s |s| 1 2 ≤ k ∆ k f (x)
pointwise, so again by Minkowski's inequality we have
||∆f || p ≤ k ||∆ k f || p .
We will prove that for any N ,
||∆ k f || p 2 −N k ||1 Ω k f || p . (13.1)
With this, we can use Hölder's inequality to see that for any N , we have
||∆f || p k 2 −N k ||1 Ω k f || p k 2 −N k Ω k |f | p 1 p |β N,T f | p 1 p ,
where β N,T is the function defined in the statement of Lemma 28, which finishes the proof of Lemma 28. It remains to prove (13.1). Note that Lemma 36 is exactly this when p = 2. By interpolation, it is enough to prove the following weak type estimate:
|{∆ k f > λ}| 2 2k ||f || 1 λ 1 . By dividing the function f into 2 2k pieces, we may assume the support of f is contained in a translate of top(T ). With this assumption, it is enough to prove for such f that |{∆ k f > λ}| ||f || 1 λ 1 . Our argument proceeds more or less by the usual path of Calderon-Zygmund decomposition.
Denote by R k the rectangle with same center and length as R but 2 k times the height. Let B be the collection of maximal rectangles of width w taken from the collection such that 1
|R k | R k |f | > 2 5k λ,
and for each R ∈ B, let R ′ = π 1 (R) × π 2 (Ctop(T )). Then let B = {R ′ : R ∈ B}. We can see already that R∈ B |R| ≤ R∈B |R| ||f || 1 λ . This follows from the weak (1,1) inequality for the Hardy-Littlewood maximal function, which holds for rectangles of fixed width: if we write, for k ≥ 0,
B k = {R ∈ B : 1 |R k | R k |f | > 2 5k λ}, then we have R∈ B |R| k≥0 2 k ||f || 1 2 5k λ ||f || 1 λ . For each (x, y) ∈ R, let b(x, y) = f (x, y) − 1 |π 1 (R)| π 1 (R) f (z, y)dz.
Note that by definition we have for each y ∈ π 2 (top(T )) that
π 1 (R) b(x, y)dx = 0.
We also have the following helpful fact:
Claim 38. For each y ∈ π 2 (Ctop(T )), we have 1
|π 1 (R)| π 1 (R) f (z, y)dz ≤ Cλ.
Proof of Claim. Note that f is supported in the annulus of width 1 w . Let k be a function such that k(ξ) = 1 for ξ ∈ [−4w, 4w]. Then
f (x, y) = f (x, w)k(y − w)dw, so 1 |π 1 (R)| π 1 (R) |f (z, y)|dz = 1 |π 1 (R)| π 1 (R) | f (z, w)k(y − w)dw|dz
Because k rapidly decays away from a rectangle of height w, if we denote by R k the rectangle with same center and length as R but 2 k times the height, then 1
|π 1 (R)| π 1 (R) | f (z, w)k(y − w)dw|dz 1 |π 1 (R)| π 1 (R) k 1 2 k 2 k 2 k f (z, w)2 −10k dwdz ≤ λ,
where the last inequality is by assumption on R.
With this claim, we define g(x, y) = f (x, y) for (x, y) ∈ R∈B and g(x, y) = 1
|π 1 (R)| π 1 (R) f (z, y)dz for (x, y) ∈ R ∈ B.
Note that by the claim we have g(x, y) λ for (x, y) ∈ R. Further, for almost every (x, y) ∈ R∈B R such that g(x, y) = f (x, y) >> λ, there exists a horizontal line segment L through (x, y) such that 1 |L| L f >> λ, which implies there is a rectangle of width w containing (x, y) on which the average of f is larger than λ, contradicting our assumption that (x, y) ∈ R∈B R. Hence g λ almost everywhere.
To see the purpose of including the rectangles 5CR ′ in the exceptional set (rather than a small dilate of R itself), consider a rectangle R north of the tree T , and a mean zero function h supported on R. Analysis of (5CR) c ∆h is a bit more complicated than in the one-dimensional case because the collection {ϕ s } s∈T has no orthogonality in the vertical direction. However by excluding R ′ , we need only consider small tiles s supported away from the vertical translate of 5CR, allowing us to take advantage of the spatial decay (in the horizontal variable) of the functions ϕ s .
With this modification, the proof now proceeds as expected: Use the fact that |g| λ, together with the L 2 estimate on ∆ to see |{∆ k g > λ}| |g| 2 λ 2 ||f || 1 λ .
Additionally, by the Chebyshev and triangle inequalities, together with sublinearity of ∆ k , we have
|{x ∈ E : ∆ k ( R b R ) > λ}| ≤ 1 λ R (5CR ′ ) c |∆ k (b R )|.
To finish the proof we show that for each R ∈ B, we have
(5CR ′ ) c |∆ k (b R )| |b R |,(13.2)
which will give us that |{x ∈ E : ∆(
R b R ) > λ}| 1 λ R |b R | R |R| ||f || 1 λ .
Once again, to prove (13.2), we essentially follow the one-dimensional argument, dealing with a few extra nuisances along the way. A reader having trouble seeing through the technicalities should note that all of the computations below are essentially the same as in the one-dimensional case. The problem is understanding why the present situation is essentially the same as the one-dimensional case. More specifically, to prove (13.2), it is convenient to make a few simplifying (and valid) assumptions. For each parallelogram s ∈ T definẽ s = π 1 (s) × Cπ 2 (top(T )).
Since s ⊆s, it is clear that if we definẽ
∆ k f = s∈T | f 1 Ω k , ϕ s | 2 1s |s| 1 2 ,(13.3)
then ∆ k f ≤∆ k f pointwise. For each s ∈ T , we know that π 1 (s) is contained in the union of two dyadic intervalss L ands R each of size π 1 (s). Further, because the set of tiles of a given size and orientation partition R 2 (i.e., for each ω ∈ D, we have R∈Uω R = R 2 ; see the definitions in Section 2), and because |π 1 (s)| ≥ |π 2 (s)| we know that for any dyadic interval I, there are 1 tiles s ∈ T such that I = π 1 (s L ) or I = π 1 (s R ). All of this allows us to assume (possibly after dividing T into ∼ 1 pieces) that the tiles s are parameterized by dyadic intervals, and that for each x ∈ Ctop(T ), and each dyadic interval I, there is at most one s ∈ T such that x ∈s and π 1 (s) = I.
To prove (13.2), we split the sum inside ∆f into two pieces, one over tiles whose vertical projection is smaller than the length of R, and the other over tiles whose vertical projection is larger than the length of R. We begin by controlling the sum over smaller tiles. Note that the dominant term in both cases comes from tiles such that |π 1 (s)| ∼ |π 1 (R)|. In the integral below, we need only consider x ∈ Ctop(T ) such that π 1 (x) ∈ π 1 (5CR). This allows us to prove the desired estimate using spatial decay alone. Further, since 1s(x) is constant on vertical segments projecting to π 2 (Ctop(T )), we have ||b R || 1 |π 1 (R)| t∈R : |t|≥5|π 1 (R)| 1 | t |π 1 (R)| | 5 dt ||b R || 1 .
We emphasize that the integral in the second-to-last line is one-dimensional. It remains to control the sum over the tiles with vertical projection larger than |π 1 (R)|. This requires using the mean-zero-along-horizontal-line-segments property of the function b R . Note that for any smooth function h, we have b R , h = y∈π 2 (R) x∈π 1 (R) b R (x, y)h(x, y)dxdy ≤ y∈π 2 (R) x∈π 1 (R) |b R (x, y)||h(x, y) − h(c π 1 (R) , y)|dxdy.
Our goal is to apply this to the wave packets ϕ s . Specifically, we will show Proof. We must deal with a small technicality here: the tiles s need not be precisely axis parallel, but fortunately they are close. Precisely, we have that the vertical component (when using the coordinate frame of s) of (x, y) − (c π 1 (R) , y) is less than w|π 1 (R)| |π 1 (s)| . Of course we have the horizontal component (when using the coordinate frame of s) of (x, y) − (c R , y) is less than |π 1 (R)|. Further, we know that This completes the proof of (13.2) and thus the proof of Lemma 28.
14. BM O type estimates for the square function
In this section we prove Lemma 29. As in the previous section, we consider the related operator∆. See (13.3) for the definition, as well as the discussion immediately following the definition for several simplifying assumptions that we make. To prove the Lemma, we prove the following key claim. Here, and in the rest of the proof, we write σ = size(T ); note that we also have
σ ∼ 1 |top(T )| s∈T | f, ϕ s | 2 1 2 .
As in the last section, we consider a slightly modified version of ∆: definẽ ∆f = s∈T | f, ϕ s | 2 1s |s| 1 2 where the rectangless are defined immediately above (13.3).
Claim 40.
|{∆f > σn}| 2 −n 2 |{∆f > σ}|.
(Of course we do not need the full exponential-squared decay, but we do have it.)
With the Claim, we are almost done: For larger δ, we use the estimate size 1:
Claim 42. If the function in the definition of size(T ) is called f , then size(T ) ||f || ∞ .
Of course we are using f = 1 F , which proves that here size(T ) 1.
Proof. For k ≥ 1, define
Ω 0 = top(T ) Ω k = 2 k top(T ) \ 2 k−1 top(T ).
We need only note that for any 1-tree T , by Lemma 36, Combining these two estimates proves (15.1) since |E| ≤ |F |.
s∈T | f, ϕ s | 2 1 2 ≤ k s∈T | 1 Ω k f, ϕ s | 2 1 2 k 2 −N k ||1 Ω k f || 2 2 ||f || 2 ∞ |top(T )| since |Ω k | 2 2k |top(T )|.
(ǫ + λ), 2M ), (2M (−ǫ + λ), 2M ), which is precisely the area of support for ϕ ω when M , ǫ, and λ are chosen appropriately. Define ϕ s (p) = |s|ϕ ω (p − c(s)).
While
ST OCK = ∅, choose T ∈ ST OCK such that top(T ) is of maximal length. Then define A T = {T ′ ∈ ST OCK : top(T ′ ) ∩ top(T ) = ∅}, and update ST OCK := ST OCK \ A T T R := T R ∪ {T }.
Proof of Claim 32 .
32By the argument at the beginning of the proof of Claim 31, it suffices to estimate s∈T : m(x)≤|π 1 (s)|≤M (x)
s (x, y) − ϕ s (c(π 1 (R)), yields, writing Γ = Ktop(T ) ∩ (5CR ′ ) c ,
2.1. Discretizing the kernel. In this section we decompose the operator H • Π τ into a sum of model operators.We begin by selecting a Schwartz function ψ
(0)
0 such that ψ
(0)
0 is supported
on [ 98
100 , 102
100 ] and equal to 1 on [ 99
100 , 101
100 ] . Let ψ
(0)
With this, we see thatσ 2 |top(T )| ∼ ||∆f || 2which is what we need. It remains to prove the claim.Proof of Claim 40 . Of course to prove the claim it is enough to show that |{∆f > √ nσ}| 2 −n |{∆f > σ}|, and this is equivalent to showing |{(∆f ) 2 > nσ 2 }| 2 −n |{(∆f ) 2 > σ 2 }|, |E|.||∆f || 2
2
{∆f ≤σ}
(∆f ) 2 +
n
∞
n=1
(σn) 2 |{∆f > nσ}|
{∆f ≤σ}
(∆f ) 2 +
n
∞
n=1
(σn) 2 |2 −n 2 {∆f > σ}|
{∆f ≤σ}
(∆f ) 2 + σ 2 |{∆f > σ}|
σ
{∆f ≤σ}∆
f + σ
{∆f >σ}∆
f
= σ
∆ f.
2
σ
∆ f,
which proves that
||∆f || 2 ∼ σ|top(T )|
1
2
1
|top(T )|
1
2
∆ f,
This proves the claim.Hence
δ≥ |E|
|F |
σ≤1
δσ
|E|
δ
|E| log
|F |
|E|
.
Michael Bateman, UCLA E-mail address: [email protected]
Section 4 for construction of this tree T * . Hence the tiles in A are pairwise disjoint.We now introduce some more notation to sort the tiles in A according to how far they are from top(T ). For k > 1, let R k = 2 k top(T ). Let R 0 = top(T ). Then letWe will use the spatial localization of the tiles s ∈ T to top(T ) to obtain the desired decay in k. We haveFirst we estimate I k . For x ∈ R k−3 and s ∈ A k , we haveWe now estimate ||1 R k−3 F (A k )|| 2 by duality. We make one small observation as a preliminary:Claim 35. If M is the strong maximal operator, thenWe remark that each s ∈ A is essentially pointed in the direction of T , so the strong maximal operator is appropriate here.Proof.which can be shown in a rather straightforward manner following the proof of the John-Nirenberg inequality. Recall that for each dyadic I we have an associated tile in T , which we call s(I). For notational convenience, define for intervals I, KWe first note that for any K, if I is a maximal interval on whichthen we knowWe begin by defining a collection of intervals I 0 :Then having defined I n−1 , define for any K ∈ I n−1We remark that for any K,To see this we only need to use Chebyshev and the estimate on size(T ):where the last inequality is due to the estimate on size(T ). Similarly,Putting together all K in I n−1 gives us that In this appendix we briefly discuss the proof of Theorem 1 for p > 2, which is essentially the proof in[4].Following the tree decomposition of Section 4 and the remarks in Section 5, we need to showThis time we care most about p close to ∞. We may assume |E| ≤ |F | because if |E| > |F | then we may apply the previous arguments for the case p ≤ 2. We emphasize here that there is no circularity. Both the argument in this section (in which we assume |E| ≤ |F | ) and the argument in the bulk of the paper (in which we assume |E| ≥ |F |) work when p = 2. Hence the p = 2 case of the estimate in (4) is established for arbitrary E, F . This allows us to assume |E| ≤ |F | in this section, where p ≥ 2, and allows us to assume |E| ≥ |F | in the earlier part of the paper, where p ≤ 2. By Estimates 11 and 12 it suffices to provefor p ≥ 2.The following simple estimate will be helpful:Claim 41. For any δ, we haveProof. We need only observe that the two terms in the minimum are equal when σ = δ |F | |E| and split the sum over σ accordingly. We split the sum (15.1) in δ into two pieces, with the dividing line being δ = |E| |F | . For smaller δ, we use Claim 41 above:
L p estimates for maximal averages along one-variable vector fields in R 2. Michael Bateman, Proc. Amer. Math. Soc. 137Cited on 3, 4, 5, 15, 16.Bateman, Michael. L p estimates for maximal averages along one-variable vector fields in R 2 , Proc. Amer. Math. Soc. 137, (2009), 955-963. (Cited on 3, 4, 5, 15, 16.)
Maximal averages along a planar vector field depending on one variable. Michael Bateman, to appear in Trans. AMS, preprint available at. Cited on 4, 5, 16.Bateman, Michael. Maximal averages along a planar vector field depend- ing on one variable, to appear in Trans. AMS, preprint available at http://lanl.arxiv.org/abs/1104.2859 (Cited on 4, 5, 16.)
L p estimates for the Hilbert transform along a one variable vector field. Michael Bateman, Christoph Thiele, Cited on 1, 2, 3, 4.Bateman, Michael, and Thiele, Christoph. L p estimates for the Hilbert transform along a one variable vector field , preprint available at http://www.math.ucla.edu/ thiele/papers/BTmar2011.pdf (Cited on 1, 2, 3, 4.)
Maximal Theorems for the Directional Hilbert Transform on the Plane Trans. Michael Lacey, Xiaochun Li, Amer. Math. Soc. 358Cited on 1, 3, 4, 6, 8, 12, 13, 49.Lacey, Michael, and Xiaochun Li. Maximal Theorems for the Directional Hilbert Transform on the Plane Trans. Amer. Math. Soc. 358 (2006), 4099-4117. (Cited on 1, 3, 4, 6, 8, 12, 13, 49.)
Michael Lacey, Xiaochun Li, Conjecture of EM Stein on the Hilbert Transform on Vector Fields Memoirs of the AMS 205. Cited on 3, 4, 11, 12, 16.Lacey, Michael, and Xiaochun Li. On a Conjecture of EM Stein on the Hilbert Trans- form on Vector Fields Memoirs of the AMS 205 (2010), no. 965. (Cited on 3, 4, 11, 12, 16.)
A proof of boundedness of the Carleson operator. Michael Lacey, Christoph Thiele, Mathematical Research Letters. 74Cited on 4, 11, 12, 13, 15.Lacey, Michael, and Thiele, Christoph. A proof of boundedness of the Carleson oper- ator Mathematical Research Letters vol. 7 no. 4, (2000), pp 361-370. (Cited on 4, 11, 12, 13, 15.)
| [] |
[
"Three-flavour neutrino oscillation update",
"Three-flavour neutrino oscillation update"
] | [
"Thomas Schwetz [email protected] \nPhysics Departement\nTheory Division\nCERN CH-1211Geneva 23Switzerland\n",
"Mariam Tórtola [email protected] \nDepartamento de Física and CFTP\nInstituto Superior Técnico Av\nRovisco Pais 11049-001LisboaPortugal\n",
"José W F Valle [email protected] \nAHEP Group\nInstituto de Física Corpuscular -C.S.I.C./Universitat de València\nEdificio Institutos de Paterna\nApt 22085E-46071ValenciaSpain\n"
] | [
"Physics Departement\nTheory Division\nCERN CH-1211Geneva 23Switzerland",
"Departamento de Física and CFTP\nInstituto Superior Técnico Av\nRovisco Pais 11049-001LisboaPortugal",
"AHEP Group\nInstituto de Física Corpuscular -C.S.I.C./Universitat de València\nEdificio Institutos de Paterna\nApt 22085E-46071ValenciaSpain"
] | [] | We review the present status of three-flavour neutrino oscillations, taking into account the latest available neutrino oscillation data presented at the Neutrino 2008 Conference. This includes the data released this summer by the MINOS collaboration, the data of the neutral current counter phase of the SNO solar neutrino experiment, as well as the latest KamLAND and Borexino data. We give the updated determinations of the leading 'solar' and 'atmospheric' oscillation parameters. We find from global data that the mixing angle θ 13 is consistent with zero within 0.9σ and we derive an upper bound of sin 2 θ 13 ≤ 0.035 (0.056) at 90% CL (3σ). | 10.1088/1367-2630/10/11/113011 | [
"https://arxiv.org/pdf/0808.2016v3.pdf"
] | 119,192,940 | 0808.2016 | f51b5aa0b2f749d4beb14a2623c3f78233ea0d03 |
Three-flavour neutrino oscillation update
11 Feb 2010
Thomas Schwetz [email protected]
Physics Departement
Theory Division
CERN CH-1211Geneva 23Switzerland
Mariam Tórtola [email protected]
Departamento de Física and CFTP
Instituto Superior Técnico Av
Rovisco Pais 11049-001LisboaPortugal
José W F Valle [email protected]
AHEP Group
Instituto de Física Corpuscular -C.S.I.C./Universitat de València
Edificio Institutos de Paterna
Apt 22085E-46071ValenciaSpain
Three-flavour neutrino oscillation update
11 Feb 2010Neutrino mass and mixingsolar and atmospheric neutrinosreactor and accelerator neutrinos PACS numbers: 2665+t, 1315+g, 1460Pq, 9555Vj
We review the present status of three-flavour neutrino oscillations, taking into account the latest available neutrino oscillation data presented at the Neutrino 2008 Conference. This includes the data released this summer by the MINOS collaboration, the data of the neutral current counter phase of the SNO solar neutrino experiment, as well as the latest KamLAND and Borexino data. We give the updated determinations of the leading 'solar' and 'atmospheric' oscillation parameters. We find from global data that the mixing angle θ 13 is consistent with zero within 0.9σ and we derive an upper bound of sin 2 θ 13 ≤ 0.035 (0.056) at 90% CL (3σ).
Introduction
Thanks to the synergy amongst a variety of experiments involving solar and atmospheric neutrinos, as well as man-made neutrinos at nuclear power plants and accelerators [1] neutrino physics has undergone a revolution over the last decade or so. The long-soughtfor phenomenon of neutrino oscillations has been finally established, demonstrating that neutrino flavor states (ν e , ν µ , ν τ ) are indeed quantum superpositions of states (ν 1 , ν 2 , ν 3 ) with definite masses m i [2]. The simplest unitary form of the lepton mixing matrix relating flavor and mass eigenstate neutrinos is given in terms of three mixing angles (θ 12 , θ 13 , θ 23 ) and three CP-violating phases, only one of which affects (conventional) neutrino oscillations [3]. Here we consider only the effect of the mixing angles in current oscillation experiments, the sensitivity to CP violation effects remains an open challenge for future experiments [4,5]. Together with the mass splitting parameters ∆m 2 21 ≡ m 2 2 − m 2 1 and ∆m 2 31 ≡ m 2 3 − m 2 1 the angles θ 12 , θ 23 are rather well determined by the oscillation data. In contrast, so far only upper bounds can be placed upon θ 13 , mainly following from the null results of the short-baseline CHOOZ reactor experiment [6] with some effect also from solar and KamLAND data, especially at low ∆m 2 31 values [7]. Here we present an update of the three-flavour oscillation analyses of Refs. [7] and [8]. This new analysis includes the data released this summer by the MINOS collaboration [9], the data from the neutral current counter phase of the SNO experiment (SNO-NCD) [10], the latest KamLAND [11] and Borexino [12] data, as well as the results of a recent re-analysis of the Gallex/GNO solar neutrino data presented at the Neutrino 2008 conference [13]. In Section 2 we discuss the status of the parameters relevant for the leading oscillation modes in solar and atmospheric neutrinos. In Section 3 we present the updated limits on θ 13 and discuss the recent claims for possible hints in favour of a non-zero value made in Refs. [14,15,16]. We summarize in Section 4.
The leading 'solar' and 'atmospheric' oscillation parameters
Let us first discuss the status of the solar parameters θ 12 and ∆m 2 21 . The latest data release from the KamLAND reactor experiment [11] has increased the exposure almost fourfold over previous results [17] to 2.44 × 10 32 proton·yr due to longer lifetime and an enlarged fiducial volume, corresponding to a total exposure of 2881 ton·yr. Apart from the increased statistics also systematic uncertainties have been improved: Thanks to the full volume calibration the error on the fiducial mass has been reduced from 4.7% to 1.8%. Details of our KamLAND analysis are described in appendix A of Ref. [8]. We use the data binned in equal bins in 1/E to make optimal use of spectral information, we take into account the (small) matter effect and carefully include various systematics following Ref. [18]. As previously, we restrict the analysis to the prompt energy range above 2.6 MeV where the contributions from geo-neutrinos [19] as well as backgrounds are small and the selection efficiency is roughly constant [11]. In that energy range 1549 reactor neutrinos events and a background of 63 events are expected without oscillations, whereas the observed number of events is 985 [20].
The Sudbury Neutrino Observatory (SNO) has released the data of its last phase, where the neutrons produced in the neutrino NC interaction with deuterium are detected mainly by an array of 3 He proportional counters to measure the rate of neutralcurrent interactions in heavy water and precisely determine the total active boron solar neutrino flux, yielding the result 5.54 +0. 33 −0.31 (stat) +0.36 −0.34 (syst) × 10 6 cm −2 s −1 [10]. The independent 3 He neutral current detectors (NCD) provide a measurement of the neutral current flux uncorrelated with the charged current rate from solar ν e , different from the statistical CC/NC separation of previous SNO phases. Since the total NC rate receives contributions from the NCD as well as from the PMTs (as previously) a small (anti-) correlation between CC and NC remains. Following Ref. [16] we assume a correlation of ρ = −0.15. In our SNO analysis we add the new data on the CC and NC fluxes to the previous results [21] assuming no correlation between the NCD phase and the previous salt phase, see Ref. [7] for further details. The main impact of the new SNO data is due to the lower value for the observed CC/NC ratio, (φ CC /φ NC ) NCD = 0.301 ± 0.033 [10], compared to the previous value (φ CC /φ NC ) salt = 0.34 ±0.038 [21]. Since for 8 B neutrinos φ CC /φ NC ≈ P ee ≈ sin 2 θ 12 , adding the new data point on this ratio with the lower value leads to a stronger upper bound on sin 2 θ 12 .
We also include the direct measurement of the 7 Be solar neutrino signal rate performed by the Borexino collaboration [12]. They report an interaction rate of the 0.862 MeV 7 Be neutrinos of 49±3(stat)±4(syst) counts/(day·100 ton). This measurement constitutes the first direct determination of the survival probability for solar ν e in the transition region between matter-enhanced and vacuum-driven oscillations. The survival probability of 0.862 MeV 7 Be neutrinos is determined to be P 7 Be,obs ee = 0.56 ± 0.1. We find that with present errors Borexino plays no significant role in the determination of neutrino oscillation parameters. Apart from the fact that the uncertainty on the survival probability is about a factor 3 larger than e.g., the uncertainty on the SNO CC/NC ratio measurement, it turns out that the observed value for P ee quoted above practically coincides (within 0.1σ) with the prediction at the best fit point: P 7 Be,pred ee = 0.55. The new data from SNO and Borexino are combined with the global data on solar neutrinos [22,23,24,25], where we take into account the results of a recent re-analysis of the Gallex data yielding a combined Gallex and GNO rate of 67.6±4.0±3.2 SNU [13].
The numerical χ 2 profiles shown in Fig. 1 have to very good accuracy the Gaussian shape χ 2 = (x − x best ) 2 /σ 2 , when the different σ for upper an lower branches are used as given in Eq.
(1). Spectral information from KamLAND data leads to an accurate determination of ∆m 2 21 with the remarkable precision of 8% at 3σ, defined Figure 1. Determination of the leading "solar" oscillation parameters from the interplay of data from artificial and natural neutrino sources. We show χ 2 -profiles and allowed regions at 90% and 99.73% CL (2 dof) for solar and KamLAND, as well as the 99.73% CL region for the combined analysis. The dot, star and diamond indicate the best fit points of solar data, KamLAND and global data, respectively. We minimise with respect to ∆m 2 31 , θ 23 and θ 13 , including always atmospheric, MINOS, K2K and CHOOZ data.
as (x upper − x lower )/(x upper + x lower ). We find that the main limitation for the ∆m 2 21 measurement comes from the uncertainty on the energy scale in KamLAND of 1.5%. KamLAND data start also to contribute to the lower bound on sin 2 θ 12 , whereas the upper bound is dominated by solar data, most importantly by the CC/NC solar neutrino rate measured by SNO. The SNO-NCD measurement reduces the 3σ upper bound on sin 2 θ 12 from 0.40 [8] to 0.37.
Let us now move to the discussion of the status of the leading atmospheric parameters θ 23 and ∆m 2 31 . The Main Injector Neutrino Oscillation Search experiment (MINOS) has reported new results on ν µ disappearance with a baseline of 735 km based on a two-year exposure from the Fermilab NuMI beam. Their data, recorded between May 2005 and July 2007 correspond to a total of 3.36×10 20 protons on target (POT) [9], more than doubling the POT with respect to MINOS run I [26], and increasing the Figure 2. Determination of the leading "atmospheric" oscillation parameters from the interplay of data from artificial and natural neutrino sources. We show χ 2 -profiles and allowed regions at 90% and 99.73% CL (2 dof) for atmospheric and MINOS, as well as the 99.73% CL region for the combined analysis (including also K2K). The dot, star and diamond indicate the best fit points of atmospheric data, MINOS and global data, respectively. We minimise with respect to ∆m 2 21 , θ 12 and θ 13 , including always solar, KamLAND, and CHOOZ data. exposure used in the latest version of Ref. [8] by about 34%. The latest data confirm the energy dependent disappearance of ν µ , showing significantly less events than expected in the case of no oscillations in the energy range 6 GeV, whereas the high energy part of the spectrum is consistent with the no oscillation expectation. We include this result in our analysis by fitting the event spectrum given in Fig. 2 of Ref. [9]. Current MINOS data largely supersedes the pioneering K2K measurement [27] which by now gives only a very minor contribution to the ∆m 2 31 measurement. We combine the long-baseline accelerator data with atmospheric neutrino measurements from Super-Kamiokande [28], using the results of Ref. [8], see references therein for details. In this analysis sub-leading effects of ∆m 2 21 in atmospheric data are neglected, but effects of θ 13 are included, in a similar spirit as in Ref. [29]. Fig. 2 illustrates how the determination of the leading atmospheric oscillation parameters θ 23 and |∆m 2 31 | emerges from the complementarity of atmospheric and accelerator neutrino data. We find the following best fit points and 1σ errors:
sin 2 θ 23 = 0.50 +0.07 −0.06 , |∆m 2 31 | = 2.40 +0.12 −0.11 × 10 −3 eV 2 .(2)
The determination of |∆m 2 31 | is dominated by spectral data from the MINOS longbaseline ν µ disappearance experiment, where the sign of ∆m 2 31 (i.e., the neutrino mass hierarchy) is undetermined by present data. The measurement of the mixing angle θ 23 is still largely dominated by atmospheric neutrino data from Super-Kamiokande with a best fit point at maximal mixing. Small deviations from maximal mixing due to subleading three-flavour effects have been found in Refs. [30,31], see, however, also Ref. [32] for a preliminary analysis of Super-Kamiokande. A comparison of these subtle effects can be found in Ref. [33]. At present deviations from maximality are not statistically significant.
Status of θ 13
The third mixing angle θ 13 would characterize the magnitude of CP violation in neutrino oscillations. Together with the determination of the neutrino mass spectrum hierarchy (i.e., the sign of ∆m 2 31 ) it constitutes a major open challenge for any future investigation of neutrino oscillations [4,5]. Fig. 3 summarizes the information on θ 13 from present data. Similar to the case of the leading oscillation parameters, also the bound on θ 13 emerges from an interplay of different data sets, as illustrated in the left panel of Fig. 3. An important contribution to the bound comes, of course, from the CHOOZ reactor experiment [6] combined with the determination of |∆m 2 31 | from atmospheric and long-baseline experiments. Due to a complementarity of low and high energy solar neutrino data, as well as solar and KamLAND data, one finds that also solar+KamLAND provide a non-trivial constraint on θ 13 , see e.g., Refs. [7,8] [34]. We obtain at 90% CL (3σ) the following limits ‡:
sin 2 θ 13 ≤ 0.060 (0.089) (solar+KamLAND) 0.027 (0.058) (CHOOZ+atm+K2K+MINOS) 0.035 (0.056) (global data)(3)
In the global analysis we find a slight weakening of the upper bound on sin 2 θ 13 due to the new data from 0.04 (see Ref. [33] or v5 of [8]) to 0.056 at 3σ. The reason for this is two-fold. First, the shift of the allowed range for |∆m 2 31 | to lower values due to the new MINOS data implies a slightly weaker constraint on sin 2 θ 13 (cf. Fig. 3, left), and second, the combination of solar and new KamLAND data prefers a slightly non-zero value of sin 2 θ 13 which, though not statistically significant, also results in a weaker constraint in the global fit (cf. Fig. 3, right).
As has been noted in Ref. [16] the slight downward shift of the SNO CC/NC ratio due to the SNO-NCD data leads to a "hint" for a non-zero value of θ 13 . From the combination of solar and KamLAND data we find a best fit value of sin 2 θ 13 = 0.03 with ∆χ 2 = 2.2 for θ 13 = 0 which corresponds to a 1.5σ effect (86% CL). We illustrate the interplay of solar and KamLAND data in the left panel of Fig. 4. The survival probability in KamLAND is given by
P KamL ee ≈ cos 4 θ 13 1 − sin 2 2θ 12 sin 2 ∆m 2 21 L 4E ,(4)
leading to an anti-correlation of sin 2 θ 13 and sin 2 θ 12 [8], see also [14,34]. In contrast, for solar neutrinos one has P solar ee ≈ cos 4 θ 13 1 − 1 2 sin 2 2θ 12 low energies cos 4 θ 13 sin 2 θ 12 high energies .
Eq. (5) shows a similar anti-correlation as in KamLAND for the vacuum oscillations of low energy solar neutrinos. For the high energy part of the spectrum, which undergoes the adiabatic MSW conversion inside the sun and which is subject to the SNO CC/NC measurement, a positive correlation of sin 2 θ 13 and sin 2 θ 12 emerges. As visible from Fig. 4 (left) and as discussed already in Refs. [8,34], this complementarity leads to a non-trivial constraint on θ 13 and it allows to understand the hint for a non-zero value of θ 13 , which helps to reconcile the slightly different best fit points for θ 12 for solar and KamLAND separately [14,16]. This trend was visible already in pre-SNO-NCD data, though with a significance of only 0.8σ, see Fig. 4 (right) showing the present result together with our previous one from v6 of [8].
Let us briefly comment on a possible additional hint for a non-zero θ 13 from atmospheric neutrino data [15,30]; Refs. [16,30] find from atmospheric+long-baseline+CHOOZ data a 0.9σ hint for a non-zero value: sin 2 θ 13 = 0.012 ± 0.013. In our atmospheric neutrino analysis (neglecting ∆m 2 21 effects) combined with CHOOZ data the best fit occurs for θ 13 = 0 (cf. Fig. 3, right), in agreement with Ref. [29]. Also, in the atmospheric neutrino analysis from Ref. [31] (which does include ∆m 2 21 effects, as Refs. [16,30]) the preference for a non-zero θ 13 is much weaker than the one from [30], with a ∆χ 2 0.2. In our global analysis the hint from solar+KamLAND gets diluted by the constraint coming from atmospheric+CHOOZ data, and we find the global χ 2 minimum at sin 2 θ 13 = 0.01, but with θ 13 = 0 allowed at 0.9σ (∆χ 2 = 0.87). Hence, we conclude that at present there is no significant hint for a non-zero θ 13 . As already stated, the origin of slightly different conclusions of other studies is related with including or neglecting the effect of solar terms in the atmospheric neutrino oscillation analysis, and translates also into a possibly nonmaximal best fit value for θ 23 . Note, however, that all analyses agree within ∆χ 2 values of order 1 and therefore there is no significant disagreement. A critical discussion of the impact of sub-leading effects in atmospheric data on θ 13 and θ 23 as well as a comparison of the results of different groups can be found in Ref. [33].
Before summarizing let us update also the determination of the ratio of the two mass-squared differences,
α ≡ ∆m 2 21 |∆m 2 31 | = 0.032 , 0.027 ≤ α ≤ 0.038 (3σ) ,(6)
which is relevant for the description of CP violation in neutrino oscillations in longbaseline experiments.
Summary
In this work we have provided an update on the status of three-flavour neutrino oscillations, taking into account the latest available world neutrino oscillation data presented at the Neutrino 2008 Conference. Our results are summarized in Figures 1, 2 and 3. Table 1 provides best fit points, 1σ errors, and the allowed intervals at 2 and 3σ for the three-flavour oscillation parameters. Appendix A.1. Updates in the solar neutrino analysis SSM: We consider the recently updated standard solar model from [35]. Among the different models presented in that reference, we use the low metallicity model, labelled as AGSS09, that incorporates the most recent determination of solar abundances [36] as our standard choice. The solar abundances in that model are a bit higher than previous determinations by the same group, alleviating the disagreement with helioseismic data. From the point of view of solar neutrinos, the most important changes with respect to the previous SSM used in our analysis (BS05(OP), with high metallicities [37]) is the 15% and 5% reduction in the Boron and Beryllium fluxes respectively. This is due to the reduced central temperature in the new model with respect to the previous one. Given the condition of fixed solar luminosity, this reduction is compensated by a slight increase in the pp and pep neutrino fluxes. We discuss also the impact of a new SSM with high metallicity, the GS98 model (presented in [35] as well), see also the recent discussion in [38]. [40]. Note that the recently published reanalysis of GALLEX data [41] has been reported already at Neutrino 2008 [13] and was therefore included in the original version of this paper. SNO: In our update we include also the results from the recent joint re-analysis of data from the Phase I and Phase II (the pure D 2 O and salt phases) of the Sudbury Neutrino Observatory (SNO) [42]. In this analysis, an effective electron kinetic energy threshold of 3.5 MeV has been used (Low Energy Threshold Analysis, LETA), and the total flux of 8 B neutrinos has been determined to be [10], one can see that the determination of the total neutrino flux has been improved by about a factor 2. These improvements have been possible thanks mainly to the increased statistics, in particular the NC event sample in the LETA is increased by about 70%, since the previously used higher energy thresholds of 5 MeV in phase I and 5.5 MeV in phase II have cut away a significant portion of the NC events. Furthermore, energy resolution, backgrounds suppression, and systematic uncertainties have been improved. We include the LETA SNO data by fitting the predicted energy-dependent neutrino survival probability and day-night asymmetry in terms of the polynomials given by the SNO collaboration, see Tabs. XXVI and XXVII in [42]. We have checked that our results agree with the analysis including all solar neutrino experiments made by SNO. Note that we have adopted in our present analysis of the SNO-NCD phase data the detailed correlations between the CC, NC and ES neutrino fluxes recently given by the SNO collaboration [43], thereby improving our previous treatment presented in Sec. 2. Fig. A1 shows the impact of the updates in the solar analysis on the determination of the solar parameters. The left panel compares solar and solar+KamLAND allowed regions for the previous and updated analyses, the middle panel illustrates the impact of the SNO LETA analysis, and the right panel shows the effect of changing between the low (AGSS09) and high (GS98) metallicity solar models. We observe that the main changes come from the SNO LETA analysis, whereas the impact of solar metallicity is small. In general changes are rather small, once KamLAND data is added to the solar data, with a small tightening of the lower bound on sin 2 θ 12 . The best fit point value for the solar mixing angle has been shifted to a slightly higher value mainly due to the lower values reported for the total boron neutrino flux, either from the SNO LETA measurements as well as from the updated SSM with low metallicities. The allowed range for ∆m 2 21 of solar only data has been somewhat reduced, however this effect gets completely diluted after combining with KamLAND data. The updated best fit values and allowed ranges for sin 2 θ 12 and ∆m 2 21 can be found in Tab. A1. The impact of the updated solar analysis on θ 13 is illustrated in Fig. A2. Our Figure A1. Impact of the changes in the solar neutrino analysis. In all panels the blue (shaded) regions corresponds to the 3σ regions from solar and solar+KamLAND updated analysis. The regions delimited by the red contour curves correspond to our previous analysis (left), an analysis using the previous high-threshold SNO phase I and II analysis but the same solar model (middle), and an analysis using the high metallicity GS98 instead of our standard low metallicity AGSS09 solar model (right).
analysis of solar + KamLAND data gives sin 2 θ 13 = 0.022 +0.018 −0.015 in excellent agreement with the value obtained by the SNO Collaboration [42], sin 2 θ 13 = 0.0200 +0.0209 −0.0163 . Hence, we obtain a lower best fit value with respect to the one we obtained in our previous analysis (sin 2 θ 13 = 0.03). This is due to the fact that now solar data prefer a somewhat higher value of θ 12 (as KamLAND does), and therefore, a smaller value of θ 13 is required to reconcile solar and KamLAND data, as can be seen by comparing left panels of Figs. 4 and A2. The fact that now solar data prefer a larger value for sin 2 θ 12 results in a stronger bound on θ 13 from the combination of solar + KamLAND data. The allowed solar region in the panel (sin 2 θ 12 , sin 2 θ 13 ) is more shifted to the right (because of the higher θ 12 preferred by the new smaller boron neutrino flux), where the allowed KamLAND region is narrower. At θ 13 = 0 we find ∆χ 2 = 2.2, same value as before.
As stated above, the small improvement in the θ 13 bound is related to the solar model used. For models with higher solar metallicities like GS98, a slightly weaker bound is obtained [38], see Fig. A2 (right). In that case we obtain a slightly larger best fit point, sin 2 θ 13 = 0.027 +0.019 −0.015 and ∆χ 2 = 3.05 at θ 13 = 0.
Appendix A.2. MINOS ν e appearance data In Ref. [44] a search for ν µ → ν e transitions by the MINOS experiment has been presented, based on a 3.14 × 10 20 protons-on-target exposure in the Fermilab NuMI beam. 35 events have been observed in the far detector with a background of 27 ± 5(stat) ± 2(syst) events predicted by the measurements in the near detector. This corresponds to an excess of about 1.5σ which can be interpreted as a weak hint for ν e appearance due to a non-zero θ 13 . We fit the MINOS ν e spectrum by using the GLoBES simulation software [45], where we calibrate our predicted spectrum by using the information given in [46]. A full three-flavour fit is performed taking into account a 7.3% uncertainty on the background normalization (Tab. I of [44]), and a 5% uncertainty on the matter density along the neutrino path.
In the MINOS detector, being optimized for muons, it is rather difficult to identify ν e CC events, since they lead to an electromagnetic shower. NC and ν µ CC events often have a similar signature, and hence lead to a background for the ν e appearance search. Indeed, in Ref. [47] an analysis of "NC events" has been performed, where "NC events" in fact include also ν e CC events due to the similar event topology. Therefore, a possible ν µ → ν e oscillation signal would contribute to the "NC event" sample of [47] and these data can be used to constrain θ 13 . We have performed a fit to the observed spectrum, again using the GLoBES software, by summing the NC events induced from the total neutrino flux with the ν e CC appearance signal due to oscillations. We include a 4% error on the predicted NC spectrum and a 3% error on the ν µ CC induced background (Tab. II of [47]).
In Fig. A2 (right) we show the constraint on sin 2 θ 13 from these MINOS data. The χ 2 has been marginalized with respect to all parameters except θ 13 , where for the solar and atmospheric parameters we imposed Gaussian errors taken from Tab. A1, without including any other information on θ 13 except from MINOS. We show the ∆χ 2 profiles for ν e appearance data and NC data, for a fixed neutrino mass hierarchy. The best fit point is always obtained for the inverted hierarchy (IH, ∆m 2 31 < 0), and in that case in general the constraint on sin 2 θ 13 is weaker, since for IH the matter effect tends to suppress the ν e appearance probability. The ∆χ 2 for normal hierarchy (NH, ∆m 2 31 > 0) is given with respect to the best fit for IH. In the global analysis we also marginalize over the two hierarchies, and hence, the actual information from MINOS comes from the IH. We see from the figure that MINOS ν e appearance data shows a slight preference for a non-zero value of θ 13 , with a best fit point of sin 2 θ 13 = 0.032(0.043) for NH (IH) with ∆χ 2 = 1.8 at sin 2 θ 13 = 0. In contrast, no indication for a non-zero θ 13 comes from the NC data. Furthermore, one observes that NC gives a slightly more constraining upper bound on sin 2 θ 13 than ν e appearance, while both are significantly weaker than the bound from ν µ disappearance data + CHOOZ or solar+KamLAND. Let us mention that the result for the NC analysis strongly depends on the value assumed for the systematic uncertainty, whereas the ν e appearance result is more robust with respect to systematics, being dominated by statistics.
In the global analysis we do not combine the χ 2 's from MINOS ν e and NC data, since presumably the data are not independent and adding them would imply a double counting of the same data. Therefore, we adopt the conservative approach and use only ν e appearance data without the information from NC data in the global analysis. We have checked, however, that adding both MINOS data sets leads to practically the same result in the global fit, both for the "hint" for θ 13 > 0 as well as the global bound, the latter being dominated by other data sets.
The present situation on the mixing angle θ 13 is summarized in Fig. A3. We obtain the following bounds at 90% (3σ) CL: We note a slight tightening of the bounds from solar+KamLAND as well as the global bound, due to the update in the solar analysis, see Appendix A.1, whereas the bound from CHOOZ+atm+K2K+MINOS gets slightly weaker, due to MINOS appearance data. In the global analysis we obtain the following best fit value and 1σ range: This corresponds to a 1.5σ hint for θ 13 > 0 (∆χ 2 = 2.3 at θ 13 = 0). As discussed in sec. 3 above, in our previous analysis the 1.5σ hint for θ 13 > 0 from solar+KamLAND data was diluted after the combination with atmospheric, long-baseline and CHOOZ data, resulting in a combined effect of 0.9σ. Now, thanks to the new MINOS appearance data, we find that the atmospheric + long-baseline + CHOOZ analysis already gives a nonzero best fit value of θ 13 (see Fig. A3), leading to the above global result, eq. A.3.
Finally, let us comment on the possible hint for a non-zero θ 13 from atmospheric data [16,30], as discussed in sec. 3. The possible origin of such a hint has been investigated in Ref. [48] and recently in [38], see also [49]. From these results one may conclude that the statistical relevance of the claimed hint for non-zero θ 13 from atmospheric data depends strongly on the details of the rate calculations and of the χ 2 analysis. Furthermore, the origin of that effect might be traced back to a small excess (at the 1σ level) in the multi-GeV e-like data sample in SK-I, which however, is no longer present in the combined SK-I and SK-II, as well as SK-I+II+III data. Tab. A1 gives an updated summary of the present best fit values and allowed ranges for the three-flavor oscillation parameters.
Fig. 1
1illustrates how the determination of the leading solar oscillation parameters θ 12 and ∆m 2 21 emerges from the complementarity of solar and reactor neutrinos. From the global three-flavour analysis we find (1σ errors) sin 2 θ 12 = 0.304 +0.022 −0.016 , ∆m 2 21 = 7.65 +0.23 −0.20 × 10 −5 eV 2 .
Figure 3 .
3Constraints on sin 2 θ 13 from different parts of the global data.
Figure 4 .
4Left: Allowed regions in the (θ 12 −θ 13 ) plane at 90% and 99.73% CL (2 dof) for solar and KamLAND, as well as the 99.73% CL region for the combined analysis. ∆m2 21 is fixed at its best fit point. The dot, star, and diamond indicate the best fit points of solar, KamLAND, and combined data, respectively. Right: χ 2 profile from solar and KamLAND data with and without the 2008 SNO-NCD data.
φ
N C = 5.140 +0.160 −0.158 (stat) +0.132 −0.117 (syst) × 10 6 cm −2 s −1 . (A.1) Comparing this number with the result obtained in SNO Phase III (the NC detector phase, NCD): 5.54 +0.33 −0.31 (stat) +0.36 −0.34 (syst) × 10 6 cm −2 s −1
Figure A2 .
A2Left (update ofFig. 4, left): Allowed regions in the (θ 12 − θ 13 ) plane at 90% and 99.73% CL (2 dof) for solar and KamLAND, as well as the 99.73% CL region for the combined analysis. Right: ∆χ 2 as a function of sin 2 θ 13 . The blue curves illustrate the impact of the updates in the solar neutrino analysis on the bound from the global solar+KamLAND data. The red curves show the constraint coming from the MINOS ν e appearance (red solid) and NC (red dashed) data, where we show the ∆χ 2 assuming NH (thin) and IH (thick), both with respect to the common minimum, which occurs for IH. The green solid curve corresponds the bound from CHOOZ + atmospheric + K2K + MINOS (disappearance) data.
Figure A3 .
A3The constraint on sin 2 θ 13 from MINOS ν e appearance data, solar + KamLAND data, atmospheric + CHOOZ + K2K + MINOS (disappearance as well as appearance), and the combined global data.
Table 1. Best-fit values with 1σ errors, and 2σ and 3σ intervals (1 d.o.f.) for the three-flavour neutrino oscillation parameters from global data including solar, atmospheric, reactor (KamLAND and CHOOZ) and accelerator (K2K and MINOS) experiments.parameter
best fit
2σ
3σ
∆m 2
21 [10 −5 eV 2 ]
7.65 +0.23
−0.20
7.25-8.11
7.05-8.34
|∆m 2
31 | [10 −3 eV 2 ]
2.40 +0.12
−0.11
2.18-2.64
2.07-2.75
sin 2 θ 12
0.304 +0.022
−0.016
0.27-0.35
0.25-0.37
sin 2 θ 23
0.50 +0.07
−0.06
0.39-0.63
0.36-0.67
sin 2 θ 13
0.01 +0.016
−0.011
≤ 0.040
≤ 0.056
Table A1. Current update of Tab. 1: Best-fit values with 1σ errors, and 2σ and 3σ intervals (1 d.o.f.) for the three-flavour neutrino oscillation parameters from global data including solar, atmospheric, reactor (KamLAND and CHOOZ) and accelerator (K2K and MINOS) experiments.parameter
best fit
2σ
3σ
∆m 2
21 [10 −5 eV 2 ]
7.59 +0.23
−0.18
7.22-8.03
7.03-8.27
|∆m 2
31 | [10 −3 eV 2 ]
2.40 +0.12
−0.11
2.18-2.64
2.07-2.75
sin 2 θ 12
0.318 +0.019
−0.016
0.29-0.36
0.27-0.38
sin 2 θ 23
0.50 +0.07
−0.06
0.39-0.63
0.36-0.67
sin 2 θ 13
0.013 +0.013
−0.009
≤ 0.039
≤ 0.053
‡ Note that the bounds given in Eq. (3) are obtained for 1 dof, whereas inFig. 3 (left)the 90% CL regions for 2 dof are shown.
Acknowledgments. This work was supported by MEC grant FPA2005-01269, by EC Contracts RTN network MRTN-CT-2004-503369 and ILIAS/N6 RII3-CT-2004-506222. We thank Michele Maltoni for collaboration on global fits to neutrino oscillation data.
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| [] |
[
"PROBLEMS IN GROUP THEORY MOTIVATED BY CRYPTOGRAPHY",
"PROBLEMS IN GROUP THEORY MOTIVATED BY CRYPTOGRAPHY"
] | [
"Vladimir Shpilrain "
] | [] | [] | This is a survey of algorithmic problems in group theory, old and new, motivated by applications to cryptography. | null | [
"https://arxiv.org/pdf/1802.07300v2.pdf"
] | 3,655,323 | 1802.07300 | 859fec15a70634d49fb9d58e183099b96403471a |
PROBLEMS IN GROUP THEORY MOTIVATED BY CRYPTOGRAPHY
4 Mar 2018
Vladimir Shpilrain
PROBLEMS IN GROUP THEORY MOTIVATED BY CRYPTOGRAPHY
4 Mar 2018arXiv:1802.07300v2 [math.GR]
This is a survey of algorithmic problems in group theory, old and new, motivated by applications to cryptography.
INTRODUCTION
The object of this survey is to showcase algorithmic problems in group theory motivated by (public key) cryptography.
In the core of most public key cryptographic primitives there is an alleged practical irreversibility of some process, usually referred to as a one-way function with trapdoor, which is a function that is easy to compute in one direction, yet believed to be difficult to compute the inverse function on "most" inputs without special information, called the "trapdoor". For example, the RSA cryptosystem uses the fact that, while it is not hard to compute the product of two large primes, to factor a very large integer into its prime factors appears to be computationally hard. Another, perhaps even more intuitively obvious, example is that of the function f (x) = x 2 . It is rather easy to compute in many reasonable (semi)groups, but the inverse function √ x is much less friendly. This fact is exploited in Rabin's cryptosystem, with the multiplicative semigroup of Z n (n composite) as the platform. In both cases though, it is not immediately clear what the trapdoor is. This is typically the most nontrivial part of a cryptographic scheme.
For a rigorous definition of a one-way function we refer the reader to [71]; here we just say that there should be an efficient (which usually means polynomial-time with respect to the complexity of an input) way to compute this function, but no visible (probabilistic) polynomial-time algorithm for computing the inverse function on "most" inputs.
Before we get to the main subject of this survey, namely problems in combinatorial and computational group theory motivated by cryptography, we recall historically the first public-key cryptographic scheme, the Diffie-Hellman key exchange protocol, to put things in perspective. This is done in Section 2. We note that the platform group for the original Diffie-Hellman protocol was finite cyclic. In Section 2.1, we show how to convert the Diffie-Hellman key exchange protocol to an encryption scheme, known as the ElGamal cryptosystem.
In the subsequent sections, we showcase various problems about infinite non-abelian groups. Complexity of these problems in particular groups has been used in various cryptographic primitives proposed over the last 20 years or so. We mention up front that a significant shift in paradigm motivated by research in cryptography was moving to search versions of decision problems that had been traditionally considered in combinatorial group theory, see e.g. [45,60]. In some cases, decision problems were used in cryptographic primitives (see e.g. [48]) but these occasions are quite rare.
The idea of using the complexity of infinite non-abelian groups in cryptography goes back to Wagner and Magyarik [31] who in 1985 devised a public-key protocol based on the unsolvability of the word problem for finitely presented groups (or so they thought). Their protocol now looks somewhat naive, but it was pioneering. More recently, there has been an increased interest in applications of non-abelian group theory to cryptography initially prompted by the papers [1,26,68].
We note that a separate question of interest that is outside of the scope of this survey is what groups can be used as platforms for cryptographic protocols. We refer the reader to the monographs [44], [45], [15] for relevant discussions and examples; here we just mention that finding a suitable platform (semi)group for one or another cryptographic primitive is a challenging problem. This is currently an active area of research; here we can mention that groups that have been considered in this context include braid groups (more generally, Artin groups), Thompson's group, Grigorchuk's group, small cancellation groups, polycyclic groups, (free) metabelian groups, various groups of matrices, semidirect products, etc.
Here is the list of algorithmic problems that we discuss in this survey. In most cases, we consider search versions of the problems as more relevant to cryptography, but there are notable exceptions.
-The word (decision) problem: Section 5 -The conjugacy problem: Section 3 -The twisted conjugacy problem: Section 3.2 -The decomposition problem: Section 4 -The subgroup intersection problem: Section 4.2 -The factorization problem: Section 4.4 -The isomorphism inversion problem: Section 6 -The subset sum and the knapsack problems: Section 8 -The Post correspondence problem: Section 9 -The hidden subgroup problem: Section 10 Also, in Section 7 we show that using semidirect products of (semi)groups as platforms for a Diffie-Hellman-like key exchange protocol yields various peculiar computational assumptions and, accordingly, peculiar search problems.
In the concluding Section 11, we describe relations between some of the problems discussed in this survey.
THE DIFFIE-HELLMAN KEY EXCHANGE PROTOCOL
The whole area of public-key cryptography started with the seminal paper by Diffie and Hellman [7]. We quote from Wikipedia: "Diffie-Hellman key agreement was invented in 1976 . . . and was the first practical method for establishing a shared secret over an unprotected communications channel." In 2002 [21], Martin Hellman gave credit to Merkle as well: "The system . . . has since become known as Diffie-Hellman key exchange. While that system was first described in a paper by Diffie and me, it is a public-key distribution system, a concept developed by Merkle, and hence should be called 'Diffie-Hellman-Merkle key exchange' if names are to be associated with it. I hope this small pulpit might help in that endeavor to recognize Merkle's equal contribution to the invention of public-key cryptography." U. S. Patent 4,200,770, now expired, describes the algorithm, and credits Diffie, Hellman, and Merkle as inventors.
The simplest, and original, implementation of the protocol uses the multiplicative group Z * p of integers modulo p, where p is prime and g is primitive mod p. A more general description of the protocol uses an arbitrary finite cyclic group.
(1) Alice and Bob agree on a finite cyclic group G and a generating element g in G. We will write the group G multiplicatively. (2) Alice picks a random natural number a and sends g a to Bob.
(3) Bob picks a random natural number b and sends g b to Alice. (4) Alice computes K A = (g b ) a = g ba . (5) Bob computes K B = (g a ) b = g ab .
Since ab = ba (because Z is commutative), both Alice and Bob are now in possession of the same group element K = K A = K B which can serve as the shared secret key.
The protocol is considered secure against eavesdroppers if G and g are chosen properly. The eavesdropper, Eve, must solve the Diffie-Hellman problem (recover g ab from g a and g b ) to obtain the shared secret key. This is currently considered difficult for a "good" choice of parameters (see e.g. [32] for details).
An efficient algorithm to solve the discrete logarithm problem (i.e., recovering a from g and g a ) would obviously solve the Diffie-Hellman problem, making this and many other public-key cryptosystems insecure. However, it is not known whether or not the discrete logarithm problem is equivalent to the Diffie-Hellman problem.
We note that there is a "brute force" method for solving the discrete logarithm problem: the eavesdropper Eve can just go over natural numbers n from 1 up one at a time, compute g n and see whether she has a match with the transmitted element. This will require O(|g|) multiplications, where |g| is the order of g. Since in practical implementations |g| is typically at least 10 300 , this method is considered computationally infeasible.
This raises a question of computational efficiency for legitimate parties: on the surface, it looks like legitimate parties, too, have to perform O(|g|) multiplications to compute g a or g b . However, there is a faster way to compute g a for a particular a by using the "square-andmultiply" algorithm, based on the binary form of a. For example, g 22 = (((g 2 ) 2 ) 2 ) 2 ·(g 2 ) 2 ·g 2 . Thus, to compute g a , one actually needs O(log 2 a) multiplications, which is feasible given the magnitude of a.
2.1. The ElGamal cryptosystem. The ElGamal cryptosystem [9] is a public-key cryptosystem which is based on the Diffie-Hellman key exchange. The ElGamal protocol is used in the free GNU Privacy Guard software, recent versions of PGP, and other cryptosystems. The Digital Signature Algorithm (DSA) is a variant of the ElGamal signature scheme, which should not be confused with the ElGamal encryption protocol that we describe below.
(1) Alice and Bob agree on a finite cyclic group G and a generating element g in G.
(2) Alice (the receiver) picks a random natural number a and publishes c = g a .
(3) Bob (the sender), who wants to send a message m ∈ G (called a "plaintext" in cryptographic lingo) to Alice, picks a random natural number b and sends two elements, m · c b and g b , to Alice. Note that c b = g ab .
(4) Alice recovers m = (m · c b ) · ((g b ) a ) −1 .
A notable feature of the ElGamal encryption is that it is probabilistic, meaning that a single plaintext can be encrypted to many possible ciphertexts.
We also point out that the ElGamal encryption has an average expansion factor of 2, meaning that the ciphertext is about twice as large as the corresponding plaintext.
THE CONJUGACY PROBLEM
Let G be a group with solvable word problem. For w, a ∈ G, the notation w a stands for a −1 wa. Recall that the conjugacy problem (or conjugacy decision problem) for G is: given two elements u, v ∈ G, find out whether there is x ∈ G such that u x = v. On the other hand, the conjugacy search problem (sometimes also called the conjugacy witness problem) is: given two elements a, b ∈ G and the information that u x = v for some x ∈ G, find at least one particular element x like that.
The conjugacy decision problem is of great interest in group theory. In contrast, the conjugacy search problem is of interest in complexity theory, but of little interest in group theory. Indeed, if you know that u is conjugate to v, you can just go over words of the form u x and compare them to v one at a time, until you get a match. (We implicitly use here an obvious fact that a group with solvable conjugacy problem also has solvable word problem.) This straightforward algorithm is at least exponential-time in the length of v, and therefore is considered infeasible for practical purposes.
Thus, if no other algorithm is known for the conjugacy search problem in a group G, it is not unreasonable to claim that x → u x is a one-way function and try to build a (public-key) cryptographic protocol on that. In other words, the assumption here would be that in some groups G, the following problem is computationally hard: given two elements a, b of G and the information that a x = b for some x ∈ G, find at least one particular element x like that. The (alleged) computational hardness of this problem in some particular groups (namely, in braid groups) has been used in several group based cryptosystems, most notably in [1] and [26]. However, after some initial excitement (which has even resulted in naming a new area of "braid group cryptography", see e.g. [6]), it seems now that the conjugacy search problem in a braid group may not provide sufficient level of security; see e.g. [22,42,43] for various attacks.
We start with a simple key exchange protocol, due to Ko, Lee et al. [26], which is modeled on the Diffie-Hellman key exchange protocol, see Section 2.
(1) An element w ∈ G is published.
(2) Alice picks a private a ∈ G and sends w a to Bob.
(3) Bob picks a private b ∈ G and sends w b to Alice. (4) Alice computes K A = (w b ) a = w ba , and Bob computes K B = (w a ) b = w ab .
If a and b are chosen from a pool of commuting elements of the group G, then ab = ba, and therefore, Alice and Bob get a common private key K B = w ab = w ba = K A . Typically, there are two public subgroups A and B of the group G, given by their (finite) generating sets, such that ab = ba for any a ∈ A, b ∈ B.
In the paper [26], the platform group G was the braid group B n which has some natural commuting subgroups. Selecting a suitable platform group for the above protocol is a very nontrivial matter; some requirements on such a group were put forward in [58]:
(P0) The conjugacy (search) problem in the platform group either has to be well studied or can be reduced to a well-known problem (perhaps, in some other area of mathematics). (P1) The word problem in G should have a fast (at most quadratic-time) solution by a deterministic algorithm. Better yet, there should be an efficiently computable "normal form" for elements of G. This is required for an efficient common key extraction by legitimate parties in a key establishment protocol, or for the verification step in an authentication protocol, etc.
(P2) The conjugacy search problem should not have an efficient solution by a deterministic algorithm.
We point out here that proving a group to have (P2) should be extremely difficult, if not impossible. The property (P2) should therefore be considered in conjunction with (P0), i.e., the only realistic evidence of a group G having the property (P2) can be the fact that sufficiently many people have been studying the conjugacy (search) problem in G over a sufficiently long time.
The next property is somewhat informal, but it is of great importance for practical implementations: (P3) There should be a way to disguise elements of G so that it would be impossible to recover x from x −1 wx just by inspection. One way to achieve this is to have a normal form for elements of G, which usually means that there is an algorithm that transforms any input u in , which is a word in the generators of G, to an output u out , which is another word in the generators of G, such that u in = u out in the group G, but this is hard to detect by inspection.
In the absence of a normal form, say if G is just given by means of generators and relators without any additional information about properties of G, then at least some of these relators should be very short to be used in a disguising procedure.
To this one can add that the platform group should not have a linear representation of a small dimension since otherwise, a linear algebra attack might be feasible.
3.1. The Anshel-Anshel-Goldfeld key exchange protocol. In this section, we are going to describe a key establishment protocol from [1] that really stands out because, unlike other protocols based on the (alleged) hardness of the conjugacy search problem, it does not employ any commuting or commutative subgroups of a given platform group and can, in fact, use any non-abelian group with efficiently solvable word problem as the platform. This really makes a difference and gives a big advantage to the protocol of [1] over most protocols in this and the following section. The choice of the platform group G for this protocol is a delicate matter though. In the original paper [1], a braid group was suggested as the platform, but with this platform the protocol was subsequently attacked in several different ways, see e.g. [3], [13], [14], [22], [28], [29], [42], [43], [72]. The search for a good platform group for this protocol still continues. Now we give a description of the AAG protocol. A group G and elements a 1 , ..., a k , b 1 , ..., b m ∈ G are public.
(1) Alice picks a private x ∈ G as a word in a 1 , ..., a k (i.e., x = x(a 1 , ..., a k )) and sends b x 1 , ..., b x m to Bob. (2) Bob picks a private y ∈ G as a word in b 1 , ..., b m and sends a y 1 , ..., a y k to Alice. It may seem that solving the (simultaneous) conjugacy search problem for b x 1 , ..., b x m ; a y 1 , ..., a y k in the group G would allow an adversary to get the secret key K. However, if we look at Step (3) of the protocol, we see that the adversary would have to know either x or y not simply as a word in the generators of the group G, but as a word in a 1 , ..., a k (respectively, as a word in b 1 , ..., b m ); otherwise, he would not be able to compose, say, x y out of a y 1 , ..., a y k . That means the adversary would also have to solve the membership search problem:
Given elements x, a 1 , ..., a k of a group G, find an expression (if it exists) of x as a word in a 1 , ..., a k .
We note that the membership decision problem is to determine whether or not a given x ∈ G belongs to the subgroup of G generated by given a 1 , ..., a k . This problem turns out to be quite hard in many groups. For instance, the membership decision problem in a braid group B n is algorithmically unsolvable if n ≥ 6 because such a braid group contains subgroups isomorphic to F 2 × F 2 (that would be, for example, the subgroup generated by σ 2 1 , σ 2 2 , σ 2 4 , and σ 2 5 , see [5]), where F 2 is the free group of rank 2. In the group F 2 × F 2 , the membership decision problem is algorithmically unsolvable by an old result of Mihailova [34].
We also note that if the adversary finds, say, some
x ′ ∈ G such that b x 1 = b x ′ 1 , ..., b x m = b x ′ m , there is no guarantee that x ′ = x in G. Indeed, if x ′ = c b x, where c b b i = b i c b for all i (in which case we say that c b centralizes b i ), then b x i = b x ′
i for all i, and therefore b x = b x ′ for any element b from the subgroup generated by b 1 , ..., b m ; in particular, y x = y x ′ . Now the problem is that if x ′ (and, similarly, y ′ ) does not belong to the subgroup A generated by a 1 , ..., a k (respectively, to the subgroup B generated by b 1 , ..., b m ), then the adversary may not obtain the correct common secret key K. On the other hand, if x ′ (and, similarly, y ′ ) does belong to the subgroup A (respectively, to the subgroup B), then the adversary will be able to get the correct K even though his x ′ and y ′ may be different from x and y, respectively. Indeed, if
x ′ = c b x, y ′ = c a y, where c b centralizes B and c a centralizes A (elementwise), then (x ′ ) −1 (y ′ ) −1 x ′ y ′ = (c b x) −1 (c a y) −1 c b xc a y = x −1 c −1 b y −1 c −1 a c b xc a y = x −1 y −1 xy = K
because c b commutes with y and with c a (note that c a belongs to the subgroup B, which follows from the assumption y ′ = c a y ∈ B, and, similarly, c b belongs to A), and c a commutes with x. We emphasize that the adversary ends up with the corrrect key K (i.e., K = (x ′ ) −1 (y ′ ) −1 x ′ y ′ = x −1 y −1 xy) if and only if c b commutes with c a . The only visible way to ensure this is to have x ′ ∈ A and y ′ ∈ B. Without verifying at least one of these inclusions, there seems to be no way for the adversary to make sure that he got the correct key.
Therefore, it appears that if the adversary chooses to solve the conjugacy search problem in the group G to recover x and y, he will then have to face either the membership search problem or the membership decision problem; the latter may very well be algorithmically unsolvable in a given group. The bottom line is that the adversary should actually be solving a (probably) more difficult ("subgroup-restricted") version of the conjugacy search problem: Given a group G, a subgroup A ≤ G, and two elements g, h ∈ G, find x ∈ A such that h = x −1 gx, given that at least one such x exists.
3.2.
The twisted conjugacy problem. Let φ, ψ be two fixed automorphisms (more generally, endomorphisms) of a group G. Two elements u, v ∈ G are called (φ, ψ)-double-twisted conjugate if there is an element w ∈ G such that uw φ = w ψ v. When ψ = id, then u and v are called φ-twisted conjugate, while in the case φ = ψ = id, u and v are just usual conjugates of each other.
The twisted (or double twisted) conjugacy problem in G is:
decide whether or not two given elements u, v ∈ G are twisted (double twisted) conjugate in G for a fixed pair of endomorphisms φ, ψ of the group G.
Note that if ψ is an automorphism, then (φ, ψ)-double-twisted conjugacy problem reduces to φψ −1 -twisted conjugacy problem, so in this case it is sufficient to consider just the twisted conjugacy problem. This problem was studied from group-theoretic perspective, see e.g. [73,53,11], and in [64] it was used in an authentication protocol. It is interesting that the research in [73,53] was probably motivated by cryptographic applications, while the authors of [11] arrived at the twisted conjugacy problem motivated by problems in topology.
THE DECOMPOSITION PROBLEM
Another ramification of the conjugacy search problem is the following decomposition search problem:
Given two elements w and w ′ of a group G, find two elements x ∈ A and y ∈ B that would belong to given subsets (usually subgroups) A, B ⊆ G and satisfy x · w · y = w ′ , provided at least one such pair of elements exists.
We note that if in the above problem A = B is a subgroup, then this problem is also known as the double coset problem.
We also note that some x and y satisfying the equality x · w · y = w ′ always exist (e.g. x = 1, y = w −1 w ′ ), so the point is to have them satisfy the conditions x ∈ A and y ∈ B. We therefore will not usually refer to this problem as a subgroup-restricted decomposition search problem because it is always going to be subgroup-restricted; otherwise it does not make much sense. We also note that the most commonly considered special case of the decomposition search problem so far is where A = B.
We are going to show in Section 11 that solving the conjugacy search problem is unnecessary for an adversary to get the common secret key in the Ko-Lee (or any similar) protocol (see our Section 3); it is sufficient to solve a seemingly easier decomposition search problem. This was mentioned, in passing, in the paper [26], but the significance of this observation was downplayed there.
We note that the membership condition x, y ∈ A may not be easy to verify for some subsets A. The authors of [26] do not address this problem; instead they mention, in justice, that if one uses a "brute force" attack by simply going over elements of A one at a time, the above condition will be satisfied automatically. This however may not be the case with other, more practical, attacks.
We also note that the conjugacy search problem is a special case of the decomposition problem where w ′ is conjugate to w and x = y −1 . The claim that the decomposition problem should be easier than the conjugacy search problem is intuitively clear since it is generally easier to solve an equation with two unknowns than a special case of the same equation with just one unknown. We admit however that there might be exceptions to this general rule. Now we give a formal description of a typical protocol based on the decomposition problem. There is a public group G, a public element w ∈ G, and two public subgroups A, B ⊆ G commuting elementwise, i.e., ab = ba for any a ∈ A, b ∈ B.
(1) Alice randomly selects private elements a 1 , a 2 ∈ A. Then she sends the element a 1 wa 2 to Bob. (2) Bob randomly selects private elements b 1 , b 2 ∈ B. Then he sends the element b 1 wb 2 to Alice. (3) Alice computes K A = a 1 b 1 wb 2 a 2 , and Bob computes K B = b 2 a 1 wb 1 a 2 . Since a i b i = b i a i in G, one has K A = K B = K (as an element of G), which is now Alice's and Bob's common secret key.
We now discuss several modifications of the above protocol.
4.1. "Twisted" protocol. This idea is due to Shpilrain and Ushakov [63]; the following modification of the above protocol appears to be more secure (at least for some choices of the platform group) against so-called "length based" attacks (see e.g. [13], [14], [22]), according to computer experiments. Again, there is a public group G and two public subgroups A, B ≤ G commuting elementwise.
(1) Alice randomly selects private elements a 1 ∈ A and b 1 ∈ B. Then she sends the element a 1 wb 1 to Bob. (2) Bob randomly selects private elements b 2 ∈ B and a 2 ∈ A. Then he sends the element b 2 wa 2 to Alice.
(3) Alice computes K A = a 1 b 2 wa 2 b 1 = b 2 a 1 wb 1 a 2 , and Bob computes K B = b 2 a 1 wb 1 a 2 .
Since
a i b i = b i a i in G, one has K A = K B = K (as an element of G),
which is now Alice's and Bob's common secret key.
4.2.
Finding intersection of given subgroups. Another modification of the protocol in Section 4 is also due to Shpilrain and Ushakov [63]. First we give a sketch of the idea. Let G be a group and g ∈ G. Denote by C G (g) the centralizer of g in G, i.e., the set of elements h ∈ G such that hg = gh.
For S = {g 1 , . . . , g k } ⊆ G, C G (g 1 , . . . , g k ) denotes the centralizer of S in G, which is the intersection of the centralizers C G (g i ), i = 1, ..., k.
Now, given a public w ∈ G, Alice privately selects a 1 ∈ G and publishes a subgroup B ⊆ C G (a 1 ) (we tacitly assume here that B can be computed efficiently). Similarly, Bob privately selects b 2 ∈ G and publishes a subgroup A ⊆ C G (b 2 ). Alice then selects a 2 ∈ A and sends w 1 = a 1 wa 2 to Bob, while Bob selects b 1 ∈ B and sends w 2 = b 1 wb 2 to Alice.
Thus, in the first transmission, say, the adversary faces the problem of finding a 1 , a 2 such that w 1 = a 1 wa 2 , where a 2 ∈ A, but there is no explicit indication of where to choose a 1 from. Therefore, before arranging something like a length based attack in this case, the adversary would have to compute generators of the centralizer C G (B) first (because a 1 ∈ C G (B)), which is usually a hard problem by itself since it basically amounts to finding the intersection of the centralizers of individual elements, and finding (the generators of) the intersection of subgroups is a notoriously difficult problem for most groups considered in combinatorial group theory. Now we give a formal description of the protocol from [63]. As usual, there is a public group G, and let w ∈ G be public, too.
(1) Alice chooses an element a 1 ∈ G, chooses a subgroup of C G (a 1 ), and publishes its generators A = {α 1 , . . . , α k }. Bob. (4) Bob chooses a random element b 1 from gp < α 1 , . . . , α k > and sends P B = b 1 wb 2 to Alice.
(5) Alice computes K A = a 1 P B a 2 . (6) Bob computes K B = b 1 P A b 2 . Since a 1 b 1 = b 1 a 1 and a 2 b 2 = b 2 a 2 , we have K = K A = K B , the shared secret key.
We note that in [72], an attack on this protocol was offered (in the case where a braid group is used as the platform), using what the author calls the linear centralizer method. Then, in [3], another method of cryptanalysis (called the algebraic span cryptanalysis) was offered, applicable to platform groups that admit an efficient linear representation. This method yields attacks on various protocols, including the one in this section, if a braid group is used as the platform.
4.3.
Commutative subgroups. Instead of using commuting subgroups A, B ≤ G, one can use commutative subgroups. Thus, suppose A, B ≤ G are two public commutative subgroups (or subsemigroups) of a group G, and let w ∈ G be a public element.
(1) Alice randomly selects private elements a 1 ∈ A, b 1 ∈ B. Then she sends the element a 1 wb 1 to Bob. (2) Bob randomly selects private elements a 2 ∈ A, b 2 ∈ B. Then he sends the element a 2 wb 2 to Alice. Given an element w of a group G and two subgroups A, B ≤ G, find any two elements a ∈ A and b ∈ B that would satisfy a · b = w, provided at least one such pair of elements exists.
(3) Alice computes K A = a 1 a 2 wb 2 b 1 , and Bob computes K B = a 2 a 1 wb 1 b 2 . Since a 1 a 2 = a 2 a 1 and b 1 b 2 = b 2 b 1 in G, one has K A = K B = K (
The following protocol relies in its security on the computational hardness of the factorization search problem. As before, there is a public group G, and two public subgroups A, B ≤ G commuting elementwise, i.e., ab = ba for any a ∈ A, b ∈ B.
(1) Alice randomly selects private elements a 1 ∈ A, b 1 ∈ B. Then she sends the element a 1 b 1 to Bob. (2) Bob randomly selects private elements a 2 ∈ A, b 2 ∈ B. Then he sends the element a 2 b 2 to Alice. (3) Alice computes
K A = b 1 (a 2 b 2 )a 1 = a 2 b 1 a 1 b 2 = a 2 a 1 b 1 b 2 ,
and Bob computes
K B = a 2 (a 1 b 1 )b 2 = a 2 a 1 b 1 b 2 .
Thus, K A = K B = K is now Alice's and Bob's common secret key.
We note that the adversary, Eve, who knows the elements a 1 b 1 and a 2 b 2 , can compute
(a 1 b 1 )(a 2 b 2 ) = a 1 b 1 a 2 b 2 = a 1 a 2 b 1 b 2 and (a 2 b 2 )(a 1 b 1 ) = a 2 a 1 b 2 b 1 , but neither of these prod- ucts is equal to K if a 1 a 2 = a 2 a 1 and b 1 b 2 = b 2 b 1 .
Finally, we point out a decision factorization problem:
Given an element w of a group G and two subgroups A, B ≤ G, find out whether or not there are two elements a ∈ A and b ∈ B such that w = a · b.
This seems to be a new and non-trivial algorithmic problem in group theory, motivated by cryptography.
THE WORD PROBLEM
The word problem "needs no introduction", but it probably makes sense to spell out the word search problem:
Suppose H is a group given by a finite presentation < X ; R > and let F(X ) be the free group with the set X of free generators. Given a group word w in the alphabet X , find a sequence of conjugates of elements from R whose product is equal to w in the free group F(X ).
Long time ago, there was an attempt to use the undecidability of the decision word problem (in some groups) in public key cryptography [31]. This was, in fact, historically the first attempt to employ a hard algorithmic problem from combinatorial group theory in public key cryptography. However, as was pointed out in [4], the problem that is actually used in [31] is not the word problem, but the word choice problem: given g, w 1 , w 2 ∈ G, find out whether g = w 1 or g = w 2 in G, provided one of the two equalities holds. In this problem, both parts are recursively solvable for any recursively presented platform group G because they both are the "yes" parts of the word problem. Therefore, undecidability of the actual word problem in the platform group has no bearing on the security of the encryption scheme in [31].
On the other hand, employing decision problems (as opposed to search problems) in public-key cryptography would allow one to depart from the canonical paradigm and construct cryptographic protocols with new properties, impossible in the canonical model. In particular, such protocols can be secure against some "brute force" attacks by a computationally unbounded adversary. There is a price to pay for that, but the price is reasonable: a legitimate receiver decrypts correctly with probability that can be made very close to 1, but not equal to 1. This idea was implemented in [48], so the exposition below follows that paper.
We assume that the sender (Bob) is given a presentation Γ (published by the receiver Alice) of a group G by generators and defining relators:
Γ = x 1 , x 2 , . . . , x n | r 1 , r 2 , . . . .
No further information about the group G is available to Bob. Bob is instructed to transmit his private bit to Alice by transmitting a word u = u(x 1 , . . . , x n ) equal to 1 in G in place of "1" and a word v = v(x 1 , . . . , x n ) not equal to 1 in G in place of "0". Now we have to specify the algorithms that Bob should use to select his words.
Algorithm "0" (for selecting a word v = v(x 1 , . . . , x n ) not equal to 1 in G) is quite simple: Bob just selects a random word by building it letter-by-letter, selecting each letter uniformly from the set X = {x 1 , . . . , x n , x −1 1 , . . . , x −1 n }. The length of such a word should be a random integer from an interval that Bob selects up front, based on his computational abilities.
Algorithm "1" (for selecting a word u = u(x 1 , . . . , x n ) equal to 1 in G) is slightly more complex. It amounts to applying a random sequence of operations of the following two kinds, starting with the empty word:
(1) Inserting into a random place in the current word a pair hh −1 for a random word h.
(2) Inserting into a random place in the current word a random conjugate g −1 r i g of a random defining relator r i . The length of the resulting word should be in the same range as the length of the output of Algorithm "0", for indistinguishability.
Encryption emulation attack. Now let us see what happens if a computationally un-
bounded adversary uses what is called encryption emulation attack on Bob's encryption. This kind of attack always succeeds against "traditional" encryption protocols where the receiver decrypts correctly with probability exactly 1. The encryption emulation attack is:
For either bit, generate its encryption over and over again, each time with fresh randomness, until the ciphertext to be attacked is obtained. Then the corresponding plaintext is the bit that was encrypted. Thus, the (computationally unbounded) adversary is building up two lists, corresponding to two algorithms above. Our first observation is that the list that corresponds to the Algorithm "0" is useless to the adversary because it is eventually going to contain all words in the alphabet X = {x 1 , . . . , x n , x −1 1 , . . . , x −1 n }. Therefore, the adversary may just as well forget about this list and focus on the other one, that corresponds to the Algorithm "1". Now the situation boils down to the following: if a word w transmitted by Bob appears on the list, then it is equal to 1 in G. If not, then not. The only problem is: how can one conclude that w does not appear on the list if the list is infinite? Of course, there is no infinity in real life, so the list is actually finite because of Bob's computational limitations. Still, at least in theory, the adversary does not know a bound on the size of the list if she does not know Bob's computational limits.
Then, perhaps the adversary can stop at some point and conclude that w = 1 with overwhelming probability, just like Alice does? The point however is that this probability may not at all be as "overwhelming" as the probability of the correct decryption by Alice. Compare:
(1) For Alice to decrypt correctly "with overwhelming probability", the probability P 1 (N) for a random word w of length N not to be equal to 1 should converge to 1 (reasonably fast) as N goes to infinity. (2) For the adversary to decrypt correctly "with overwhelming probability", the probability P 2 (N, f (N)) for a random word w of length N produced by the Algorithm "1" to have a proof of length ≤ f (N) verifying that w = 1, should converge to 1 as N goes to infinity. Here f (N) represents the adversary's computational capabilities; this function can be arbitrary, but fixed.
We see that the functions P 1 (N) and P 2 (N) are of very different nature, and any correlation between them is unlikely. We note that the function P 1 (N) is generally well understood, and in particular, it is known that in any infinite group G, P 1 (N) indeed converges to 1 as N goes to infinity.
On the other hand, functions P 2 (N, f (N)) are more complex; note also that they may depend on a particular algorithm used by Bob to produce words equal to 1. The Algorithm "1" described in this section is very straightforward; there are more delicate algorithms discussed in [45].
Functions P 2 (N, f (N)) are currently subject of active research, and in particular, it appears likely that there are groups in which P 2 (N, f (N)) does not converge to 1 at all, if an algorithm used to produce words equal to 1 is chosen intelligently.
We also note in passing that if in a group G the word problem is recursively unsolvable, then the length of a proof verifying that w = 1 in G is not bounded by any recursive function of the length of w.
Of course, in real life, the adversary may know a bound on the size of the list based on a general idea of what kind of hardware may be available to Bob; but then again, in real life the adversary would be computationally bounded, too. Here we note (again, in passing) that there are groups G with efficiently solvable word problem and words w of length n equal to 1 in G, such that the length of a proof verifying that w = 1 in G is not bounded by any tower of exponents in n, see [51].
Thus, the bottom line is: in theory, the adversary cannot positively identify the bit that Bob has encrypted by a word w if she just uses the "encryption emulation" attack. In fact, such an identification would be equivalent to solving the word problem in G, which would contradict the well-known fact that there are (finitely presented) groups with recursively unsolvable word problem.
It would be nice, of course, if the adversary was unable to positively decrypt using "encryption emulation" attacks even if she did know Bob's computational limitations. This, too, can be arranged, see the following subsection.
5.2.
Encryption: trick and treat. Building on the ideas from the previous subsection and combining them with a simple yet subtle trick, we describe here an encryption protocol from [48] that has the following features:
(F1) Bob encrypts his private bit sequence by a word in a public alphabet X . (F2) Alice (the receiver) decrypts Bob's transmission correctly with probability that can be made arbitrarily close to 1, but not equal to 1. (F3) The adversary, Eve, is assumed to have no bound on the speed of computation or on the storage space. (F4) Eve is assumed to have complete information on the algorithm(s) and hardware that Bob uses for encryption. However, Eve cannot predict outputs of Bob's random numbers generator. (F5) Eve cannot decrypt Bob's bit correctly with probability > 3 4 by emulating Bob's encryption algorithm. This leaves Eve with the only possibility: to attack Alice's decryption algorithm or her algorithm for obtaining public keys, but this is a different story. Here we only discuss the encryption emulation attack, to make a point that this attack can be unsuccessful if the probability of the legitimate decryption is close to 1, but not exactly 1.
Here is the relevant protocol (for encrypting a single bit). (P0) Alice publishes two presentations:
Γ 1 = x 1 , x 2 , . . . , x n | r 1 , r 2 , . . . Γ 2 = x 1 , x 2 , . .
. , x n | s 1 , s 2 , . . . . One of them defines the trivial group, whereas the other one defines an infinite group, but only Alice knows which one is which. Bob is instructed to transmit his private bit to Alice as follows: (P1) In place of "1", Bob transmits a pair of words (w 1 , w 2 ) in the alphabet X = {x 1 , x 2 , . . . , x n , x −1 1 , . . . , x −1 n }, where w 1 is selected randomly, while w 2 is selected to be equal to 1 in the group G 2 defined by Γ 2 (see e.g. Algorithm "1" in the previous section). (P2) In place of "0", Bob transmits a pair of words (w 1 , w 2 ), where w 2 is selected randomly, while w 1 is selected to be equal to 1 in the group G 1 defined by Γ 1 . Under our assumptions (F3), (F4) Eve can identify the word(s) in the transmitted pair which is/are equal to 1 in the corresponding presentation(s), as well as the word, if any, which is not equal to 1. There are the following possibilities:
(
1) w 1 = 1 in G 1 , w 2 = 1 in G 2 ; (2) w 1 = 1 in G 1 , w 2 = 1 in G 2 ; (3) w 1 = 1 in G 1 , w 2 = 1 in G 2 .
It is easy to see that the possibility (1) occurs with probability 1 2 (when Bob wants to transmit "1" and G 1 is trivial, or when Bob wants to transmit "0" and G 2 is trivial). If this possibility occurs, Eve cannot decrypt Bob's bit correctly with probability > 1 2 . Indeed, the only way for Eve to decrypt in this case would be to find out which presentation Γ i defines the trivial group, i.e., she would have to attack Alice's algorithm for obtaining a public key, which would not be part of the encryption emulation attack anymore. Here we just note, in passing, that there are many different ways to construct presentations of the trivial group, some of them involving a lot of random choices. See e.g. [38] for a survey on the subject.
In any case, our claim (F5) was that Eve cannot decrypt Bob's bit correctly with probability > 3 4 by emulating Bob's encryption algorithm, which is obviously true in this scheme since the probability for Eve to decrypt correctly is, in fact, precisely 1 2 · 1 2 + 1 2 · 1 = 3 4 . (Note that Eve decrypts correctly with probability 1 if either of the possibilities (2) or (3) above occurs.)
THE ISOMORPHISM INVERSION PROBLEM
The isomorphism (decision) problem for groups is very well known: suppose two groups are given by their finite presentations in terms of generators and defining relators, then find out whether the groups are isomorphic. The search version of this problem is well known, too: given two finite presentations defining isomorphic groups, find a particular isomorphism between the groups. Now the following problem, of interest in cryptography, is not what was previously considered in combinatorial group theory:
Given two finite presentations defining isomorphic groups, G and H, and an isomorphism ϕ : G → H, find ϕ −1 . Now we describe an encryption scheme whose security is based on the alleged computational hardness of the isomorphism inversion problem. Our idea itself is quite simple: encrypt with a public isomorphism ϕ that is computationally infeasible for the adversary to invert. A legitimate receiver, on the other hand, can efficiently compute ϕ −1 because she knows a factorization of ϕ in a product of "elementary", easily invertible, isomorphisms.
What is interesting to note is that this encryption is homomorphic because ϕ(g 1 g 2 ) = ϕ(g 1 )ϕ(g 2 ) for any g 1 , g 2 ∈ G. The significance of this observation is due to a result of [49]: if the group G is a non-abelian finite simple group, then any homomorphic encryption on G can be converted to a fully homomorphic encryption (FHE) scheme, i.e., encryption that respects not just one but two operations: either boolean AND and OR or arithmetic addition and multiplication.
In summary, a relevant scheme can be built as follows. Given a public presentation of a group G by generators and defining relations, the receiver (Alice) uses a chain of private "elementary" isomorphisms G → H 1 → ... → H k → H, each of which is easily invertible, but the (public) composite isomorphism ϕ : G → H is hard to invert without the knowledge of a factorization in a product of "elementary" ones. (Note that ϕ is published as a map taking the generators of G to words in the generators of H.) Having obtained this way a (private) presentation H, Alice discards some of the defining relations to obtain a public presentation H. Thus, the group H, as well as the group G (which is isomorphic to H), is a homomorphic image of the groupĤ. (Note thatĤ has the same set of generators as H does but has fewer defining relations.) Now the sender (Bob), who wants to encrypt his plaintext g ∈ G, selects an arbitrary word w g (in the generators of G) representing the element g and applies the public isomorphism ϕ to w g to get ϕ(w g ), which is a word in the generators of H (orĤ, since H and H have the same set of generators). He then selects an arbitrary word h g in the generators ofĤ representing the same element ofĤ as ϕ(w g ) does, and this is now his ciphertext: h g = E(g). To decrypt, Alice applies her private map ϕ −1 (which is a map taking the generators ofĤ to words in the generators of G) to h g to get a word w ′ g = ϕ −1 (h g ). This word w ′ g represents the same element of G as w g does because ϕ −1 (h g ) = ϕ −1 (ϕ(w g )) = w g in the group G since both ϕ and ϕ −1 are homomorphisms, and the composition of ϕ and ϕ −1 is the identity map on the group G, i.e., it takes every word in the generators of G to a word representing the same element of G. Thus, Alice decrypts correctly.
We emphasize here that a plaintext is a group element g ∈ G, not a word in the generators of G. This implies, in particular, that there should be some kind of canonical way (a "normal form") of representing elements of G. For example, for elements of an alternating group A m (these groups are finite non-abelian simple groups if m ≥ 5), such a canonical representation can be the standard representation by a permutation of the set {1, . . . , m}. Now we are going to give more details on how one can construct a sequence of "elementary" isomorphisms starting with a given presentation of a group G = x 1 , x 2 , . . . | r 1 , r 2 , . . . . (Here x 1 , x 2 , . . . are generators and r 1 , r 2 , . . . are defining relators). These "elementary" isomorphisms are called Tietze transformations. They are universal in the sense that they can be applied to any (semi)group presentation. Tietze transformations are of the following types: (T1): Introducing a new generator: Replace x 1 , x 2 , . . . | r 1 , r 2 , . . . by y, x 1 , x 2 For each Tietze transformation of the types (T1)-(T3), it is easy to obtain an explicit isomorphism (as a map on generators) and its inverse. For a Tietze transformation of the type (T4), the isomorphism is just the identity map. We would like here to make Tietze transformations of the type (T4) recursive, because a priori it is not clear how Alice can actually implement these transformations. Thus, Alice can use the following recursive version of (T4):
(T4 ′ ) In the set r 1 , r 2 , . . . , replace some r i by one of the: r −1 i , r i r j , r i r −1 j , r j r i , r j r −1 i , x −1 k r i x k , x k r i x −1 k , where j = i, and k is arbitrary. One particularly useful feature of Tietze transformations is that they can break long defining relators into short pieces (of length 3 or 4, say) at the expense of introducing more generators, as illustrated by the following simple example. In this example, we start with a presentation having two relators of length 5 in 3 generators, and end up with a presentation having 4 relators of length 3 and one relator of length 4, in 6 generators. The ∼ = symbol below means "is isomorphic to."
Example 1. G = x 1 , x 2 , x 3 | x 2 1 x 3 2 , x 1 x 2 2 x −1 1 x 3 ∼ = x 1 , x 2 , x 3 , x 4 | x 4 = x 2 1 , x 4 x 3 2 , x 1 x 2 2 x −1 1 x 3 ∼ = x 1 , x 2 , x 3 , x 4 , x 5 | x 5 = x 1 x 2 2 , x 4 = x 2 1 , x 4 x 3 2 , x 5 x −1 1 x 3 ∼ = (now switching x 1 and x 5 -this is (T3)) ∼ = x 1 , x 2 , x 3 , x 4 , x 5 | x 1 = x 5 x 2 2 , x 4 = x 2 5 , x 4 x 3 2 , x 1 x −1 5 x 3 ∼ = x 1 , x 2 , x 3 , x 4 , x 5 , x 6 | x −1 1 x 5 x 2 2 , x −1 4 x 2 5 , x −1 6 x 4 x 2 , x 6 x 2 2 , x 1 x −1 5 x 3 = H.
We note that this procedure of breaking relators into pieces of length 3 increases the total relator length (measured as the sum of the length of all relators) by at most a factor of 2.
Since we need our "elementary" isomorphisms to be also given in the form x i → y i , we note that the isomorphism between the first two presentations above is given by x i → x i , i = 1, 2, 3, and the inverse isomorphism is given by x i → x i , i = 1, 2, 3; x 4 → x 2 1 . By composing elementary isomorphisms, we compute the isomorphism ϕ between the first and the last presentations: ϕ : 3 . By composing the inverses of elementary isomorphisms, we compute ϕ −1 :
x 1 → x 5 , x 2 → x 2 , x 3 → xx 1 → x 1 x 2 2 , x 2 → x 2 , x 3 → x 3 , x 4 → x 2 1 , x 5 → x 1 , x 6 → x 2 1 x 2 .
We see that even in this toy example, recovering ϕ −1 from the public ϕ is not quite trivial without knowing a sequence of intermediate Tietze transformations. Furthermore, if Alice discards, say, two of the relators from the last presentation to get a publicĤ =
x 1 , x 2 , x 3 , x 4 , x 5 , x 6 | x −1 1 x 5 x 2 2 , x 6 x 2 2 , x 1 x −1 5 x 3
, then there is no isomorphism betweenĤ and G whatsoever, and the problem for the adversary is now even less trivial: to find relators completing the public presentationĤ to a presentation H isomorphic to G by way of the public isomorphism ϕ, and then find ϕ −1 .
Moreover, ϕ as a map on the generators of G may not induce an onto homomorphism from G toĤ, and this will deprive the adversary even from the "brute force" attack by looking for a map ψ on the generators ofĤ such that ψ :Ĥ → G is a homomorphism, and ψ(ϕ) is identical on G. If, say, in the example above we discard the relator x 1 x −1 5 x 3 from the final presentation H, then x 4 will not be in the subgroup ofĤ generated by ϕ(x i ), and therefore there cannot possibly be a ψ :Ĥ → G such that ψ(ϕ) is identical on G.
We now describe a homomorphic public key encryption scheme a little more formally. : Key Generation: Let ϕ : G → H be an isomorphism. Alice's public key then consists of ϕ as well as presentations G andĤ, whereĤ is obtained from H by keeping all of the generators but discarding some of the relators. Alice's private key consists of ϕ −1 and H. : Encrypt: Bob's plaintext is g ∈ G. To encrypt, he selects an arbitrary word w g in the generators of G representing the element g and applies the public isomorphism ϕ to w g to get ϕ(w g ), which is a word in the generators of H (orĤ, since H andĤ have the same set of generators). He then selects an arbitrary word h g in the generators of H representing the same element ofĤ as ϕ(w g ) does, and this is now his ciphertext: h g = E(g). : Decrypt: To decrypt, Alice applies her private map ϕ −1 to h g to get a word w ′ g = ϕ −1 (h g ). This word w ′ g represents the same element of G as w g does because ϕ −1 (h g ) = ϕ −1 (ϕ(w g )) = w g in the group G since both ϕ and ϕ −1 are homomorphisms, and the composition of ϕ and ϕ −1 is the identity map on the group G. In the following example, we use the presentations 2 2 , and the isomorphism ϕ : x 1 → x 5 , x 2 → x 2 , x 3 → x 3 from Example 1 to illustrate how encryption works.
G = x 1 , x 2 , x 3 | x 2 1 x 3 2 , x 1 x 2 2 x −1 1 x 3 ,Ĥ = x 1 , x 2 , x 3 , x 4 , x 5 , x 6 | x 1 = x 5 x 2 2 , x 4 = x 2 5 , x 6 = x 4 x 2 , x 6 x
Example 2. Let the plaintext be the element g ∈ G represented by the word x 1 x 2 . Then ϕ(x 1 x 2 ) = x 5 x 2 . Then the word x 5 x 2 is randomized inĤ by using relators ofĤ as well as "trivial" relators x i x −1 i = 1 and x −1 i x i = 1. For example: multiply x 5 x 2 by x 4 x −1 4 to get
x 4 x −1 4 x 5 x 2 .
Then replace x 4 by x 2 5 , according to one of the relators ofĤ, and get x 2 5 x −1 4 x 5 x 2 . Now insert x 6 x −1 6 between x 5 and x 2 to get x 2 5 x −1 4 x 5 x 6 x −1 6 x 2 , and then replace x 6 by x 4 x 2 to get x 2 5 x −1 4 x 5 x 4 x 2 x −1 6 x 2 , which can be used as the encryption E(g). Finally, we note that automorphisms, instead of general isomorphisms, were used in [18] and [37] to build public key cryptographic primitives employing the same general idea of building an automorphism as a composition of elementary ones. In [37], those were automorphisms of a polynomial algebra, while in [18] automorphisms of a tropical algebra were used along the same lines. We also note that "elementary isomorphisms" (i.e., Tietze transformations) are universal in nature and can be adapted to most any algebraic structure, see e.g. [36], [35], and [65].
SEMIDIRECT PRODUCT OF GROUPS AND MORE PECULIAR COMPUTATIONAL
ASSUMPTIONS
Using a semidirect product of (semi)groups as the platform for a very simple key exchange protocol (inspired by the Diffie-Hellman protocol) yields new and sometimes rather peculiar computational assumptions. The exposition in this section follows [20] (see also [24]).
First we recall the definition of a semidirect product:
Definition 1. Let G, H be two groups, let Aut(G) be the group of automorphisms of G, and let ρ : H → Aut(G) be a homomorphism. Then the semidirect product of G and H is the set
Γ = G ⋊ ρ H = {(g, h) : g ∈ G, h ∈ H}
with the group operation given by
(g, h)(g ′ , h ′ ) = (g ρ(h ′ ) · g ′ , h · h ′ ).
Here g ρ(h ′ ) denotes the image of g under the automorphism ρ(h ′ ), and when we write a product h · h ′ of two morphisms, this means that h is applied first.
In this section, we focus on a special case of this construction, where the group H is just a subgroup of the group Aut(G). If H = Aut(G), then the corresponding semidirect product is called the holomorph of the group G. Thus, the holomorph of G, usually denoted by Hol(G), is the set of all pairs (g, φ), where g ∈ G, φ ∈ Aut(G), with the group operation given by
(g, φ) · (g ′ , φ ′ ) = (φ ′ (g) · g ′ , φ · φ ′ ).
It is often more practical to use a subgroup of Aut(G) in this construction, and this is exactly what we do below, where we describe a key exchange protocol that uses (as the platform) an extension of a group G by a cyclic group of automorphisms.
One can also use this construction if G is not necessarily a group, but just a semigroup, and/or consider endomorphisms of G, not necessarily automorphisms. Then the result will be a semigroup Thus, let G be a (semi)group. An element g ∈ G is chosen and made public as well as an arbitrary automorphism φ ∈ Aut(G) (or an arbitrary endomorphism φ ∈ End(G)). Bob chooses a private n ∈ N, while Alice chooses a private m ∈ N. Both Alice and Bob are going to work with elements of the form (g, φ r ), where g ∈ G, r ∈ N. Note that two elements of this form are multiplied as follows: (g, φ r ) · (h, φ s ) = (φ s (g) · h, φ r+s ).
The following is a public key exchange protocol between Alice and Bob.
(1) Alice computes (g, φ) m = (φ m−1 (g) · · · φ 2 (g) · φ(g) · g, φ m ) and sends only the first component of this pair to Bob. Thus, she sends to Bob only the element a = φ m−1 (g) · · · φ 2 (g) · φ(g) · g of the (semi)group G.
(2) Bob computes (g, φ) n = (φ n−1 (g) · · · φ 2 (g) · φ(g) · g, φ n ) and sends only the first component of this pair to Alice. Thus, he sends to Alice only the element b = φ n−1 (g) · · · φ 2 (g) · φ(g) · g of the (semi)group G.
(3) Alice computes (b, x) · (a, φ m ) = (φ m (b) · a, x · φ m ). Her key is now K A = φ m (b) · a.
Note that she does not actually "compute" x · φ m because she does not know the automorphism x = φ n ; recall that it was not transmitted to her. But she does not need it to compute K A .
(4) Bob computes (a, y) · (b, φ n ) = (φ n (a) · b, y · φ n ). His key is now K B = φ n (a) · b. Again, Bob does not actually "compute" y · φ n because he does not know the automorphism y = φ m .
(5) Since (b, x) · (a, φ m ) = (a, y) · (b, φ n ) = (g, φ) m+n , we should have K A = K B = K, the shared secret key.
Remark 1. Note that, in contrast with the original Diffie-Hellman key exchange, correctness
here is based on the equality h m · h n = h n · h m = h m+n rather than on the equality (h m ) n = (h n ) m = h mn . In the original Diffie-Hellman set up, our trick would not work because, if the shared key K was just the product of two openly transmitted elements, then anybody, including the eavesdropper, could compute K.
We note that the general protocol above can be used with any non-commutative group G if φ is selected to be a non-trivial inner automorphism, i.e., conjugation by an element which is not in the center of G. Furthermore, it can be used with any non-commutative semigroup G as well, as long as G has some invertible elements; these can be used to produce inner automorphisms. A typical example of such a semigroup would be a semigroup of matrices over some ring. Now let G be a non-commutative (semi)group and let h ∈ G be an invertible non-central element. Then conjugation by h is a non-identical inner automorphism of G that we denote by ϕ h . We use an extension of the semigroup G by the inner automorphism ϕ h , as described in the beginning of this section. For any element g ∈ G and for any integer k ≥ 1, we have ϕ h (g) = g −1 gh; ϕ k h (g) = h −k gh k . Now our general protocol is specialized in this case as follows.
(1) Alice and Bob agree on a (semi)group G and on public elements g, h ∈ G, where h is an invertible non-central element.
(2) Alice selects a private positive integer m, and Bob selects a private positive integer n.
(3) Alice computes (g, ϕ h ) m = (h −m+1 gh m−1 · · · h −2 gh 2 · h −1 gh · g, ϕ m h ) and sends only the first component of this pair to Bob. Thus, she sends to Bob only the element
A = h −m+1 gh m−1 · · · h −2 gh 2 · h −1 gh · g = h −m (hg) m .
(4) Bob computes (g, ϕ h ) n = (h −n+1 gh n−1 · · · h −2 gh 2 · h −1 gh · g, ϕ n h ) and sends only the first component of this pair to Alice. Thus, he sends to Alice only the element
B = h −n+1 gh n−1 · · · h −2 gh 2 · h −1 gh · g = h −n (hg) n . (5) Alice computes (B, x) · (A, ϕ m h ) = (ϕ m h (B) · A, x · ϕ m h ).
Her key is now K Alice = ϕ m h (B) · A = h −(m+n) (hg) m+n . Note that she does not actually "compute" x · ϕ m h because she does not know the automorphism x = ϕ n h ; recall that it was not transmitted to her. But she does not need it to compute K Alice .
(6) Bob computes (A, y) · (B, ϕ n h ) = (ϕ n h (A) · B, y · ϕ n h ).
His key is now K Bob = ϕ n h (A) · B. Again, Bob does not actually "compute" y · ϕ n h because he does not know the automorphism y = ϕ m h .
K = ϕ m h (B) · A = ϕ n h (A) · B = h −(
m+n) (hg) m+n . Therefore, our security assumption here is that it is computationally hard to retrieve the key K = h −(m+n) (hg) m+n from the quadruple (h, g, h −m (hg) m , h −n (hg) n ). In particular, we have to take care that the elements h and hg do not commute because otherwise, K is just a product of h −m (hg) m and h −n (hg) n . Once again, the problem is:
Given a (semi)group G and elements g, h, h −m (hg) m , and h −n (hg) n of G, find h −(m+n) (hg) m+n . Compare this to the Diffie-Hellman problem from Section 2:
Given a (semi)group G and elements g, g n , and g m of G, find g mn . A weaker security assumption arises if an eavesdropper tries to recover a private exponent from a transmission, i.e., to recover, say, m from h −m (hg) m . A special case of this problem, where h = 1, is the "discrete log" problem, namely: recover m from g and g m . However, the "discrete log" problem is a problem on cyclic, in particular abelian, groups, whereas in the former problem it is essential that g and h do not commute.
By varying the automorphism (or endomorphism) used for an extension of G, one can get many other security assumptions. However, many (semi)groups G just do not have outer (i.e., non-inner) automorphisms, so there is no guarantee that a selected platform (semi)group will have any outer automorphisms. On the other hand, it will have inner automorphisms as long as it has invertible non-central elements.
In conclusion, we note that there is always a concern (as well as in the standard Diffie-Hellman protocol) about the orders of public elements (in our case, about the orders of h and hg): if one of the orders is too small, then a brute force attack may be feasible.
If a group of matrices of small size is chosen as the platform, then the above protocol turns out to be vulnerable to a "linear algebra attack", similar to an attack on Stickel's protocol [70] offered in [59], albeit more sophisticated, see [41], [55], [54]. A composition of conjugating automorphism with a field automorphism was employed in [23], but this automorphism still turned out to be not complex enough to make the protocol withstand a linear algebra attack, see [8], [55]. Selecting a good platform (semi)group for the protocol in this section still remains an open problem.
Finally, we mention another, rather different, proposal [50] of a cryptosystem based on the semidirect product of two groups and yet another, more complex, proposal of a key agreement based on the semidirect product of two monoids [2].
THE SUBSET SUM AND THE KNAPSACK PROBLEMS
As usual, elements of a group G are given as words in the alphabet X ∪ X −1 . We begin with three decision problems:
The subset sum problem (SSP): Given g 1 , . . . , g k , g ∈ G decide if
(1) g = g ε 1 1 . . . g ε k k for some ε 1 , . . . , ε k ∈ {0, 1}.
The knapsack problem (KP): Given g 1 , . . . , g k , g ∈ G decide if
(2) g = g ε 1 1 . . . g ε k k
for some non-negative integers ε 1 , . . . , ε k . The third problem is equivalent to KP in the abelian case, but in general this is a completely different problem:
The Submonoid membership problem (SMP): Given elements g 1 , . . . , g k , g ∈ G decide if g belongs to the submonoid generated by g 1 , . . . , g k in G, i.e., if the following equality holds for some g i 1 , . . . , g i s ∈ {g 1 , . . . , g k }, s ∈ N:
(3) g = g i 1 , . . . , g i s .
The restriction of SMP to the case where the set of generators {g 1 , . . . , g n } is closed under inversion (so that the submonoid is actually a subgroup of G) is a well-known subgroup membership problem, one of the most basic algorithmic problems in group theory.
There are also natural search versions of the decision problems above, where the goal is to find a particular solution to the equations (1), (2), or (3), provided that solutions do exist.
We also mention, in passing, an interesting research avenue explored in [39]: many search problems can be converted to optimization problems asking for an "optimal" (usually meaning "minimal") solution of the corresponding search problem. A well-known example of an optimization problems is the geodesic problem: given a word in the generators of a group G, find a word of minimum length representing the same element of G.
The classical (i.e., not group-theoretical) subset sum problem is one of the very basic NPcomplete problems, so there is extensive related bibliography (see [25]). The SSP problem attracted a lot of extra attention when Merkle and Hellmann designed a public key cryptosystem [33] based on a variation of SSP. That cryptosystem was broken by Shamir in [56], but the interest persists and the ideas survive in numerous new cryptosystems and their variations (see e.g. [47]). Generalizations of knapsack-type cryptosystems to non-commutative groups seem quite promising from the viewpoint of post-quantum cryptography, although relevant cryptographic schemes are yet to be built.
In [39], the authors showed, in particular, that SSP is NP-complete in: (1) the direct sum of countably many copies of the infinite cyclic group Z; (2) free metabelian non-abelian groups of finite rank; (3) wreath product of two finitely generated infinite abelian groups; (4) Thompson's group F; (5) Baumslag-Solitar group BS(m, n) for |m| = |n|, and in many other groups.
In [27], the authors showed that the subset sum problem is polynomial time decidable in every finitely generated virtually nilpotent group but there exists a polycyclic group where this problem is NP-complete. Later in [46], Nikolaev and Ushakov showed that, in fact, every polycyclic non-virtually-nilpotent group has NP-complete subset sum problem.
Also in [27], it was shown that the knapsack problem is undecidable in a direct product of sufficiently many copies of the discrete Heisenberg group (which is nilpotent of class 2). However, for the discrete Heisenberg group itself, the knapsack problem is decidable. Thus, decidability of the knapsack problem is not preserved under direct products. In [12], the effect of free and direct products on the time complexity of the knapsack and related problems was studied further.
THE POST CORRESPONDENCE PROBLEM
The Post correspondence problem PCP(A) for a semigroup (or any other algebraic structure) A is to decide, given two n-tuples u = (u 1 , . . . , u n ) and v = (v 1 , . . . , v n ) of elements of A, if there is a term (called a solution) t(x 1 , . . . , x n ) in the language of A such that t(u 1 , . . . , u n ) = t(v 1 , . . . , v n ) in A. In 1946 Post introduced this problem in the case of free monoids (free semigroups) and proved that it is undecidable [52]. (See [69] for a simpler proof.)
The PCP in groups is closely related to the problem of finding the equalizer E(φ, ψ) of two group homomorphisms φ, ψ : H → G. The equalizer is defined as E(φ, ψ) = {w ∈ H | φ(w) = ψ(w)}. Specifically, PCP in a group G is the same as to decide if the equalizer of a given pair of homomorphisms φ, ψ ∈ Hom(H, G), where H is a free group of finite rank in the variety Var(G) generated by G, is trivial or not. Indeed, in this case every tuple u = (u 1 , . . . , u n ) of elements of G corresponds to a homomorphism φ u from a free group H with a basis x 1 , . . . , x n in the variety Var(G) such that φ u (x 1 ) = u 1 , . . . , φ u (x n ) = u n . The equalizer E(φ u , φ v ) describes all solutions w for the instance (u, v).
There is an interesting variation of the Post correspondence problem for semigroups and groups that we call a non-homogeneous Post correspondence problem, or a general Post correspondence problem GPCP, following [40]: given two tuples u and v of elements in a (semi)group S as above and two extra elements a, b ∈ S, decide if there is a term t(x 1 , . . . , x n ) such that at(u 1 , . . . , u n ) = bt(v 1 , . . . , v n ) in S. Interesting connections between GPCP and the (double) twisted conjugacy problem were reported in [40]. Specifically, it was shown in [40] that the double endomorphism twisted conjugacy problem in a relatively free group in Var(G) is equivalent to GPCP(G), and, in general, the double endomorphism twisted conjugacy problem in G is P-time reducible to GPCP(G).
Another interesting observation made in [40] is that if GPCP is decidable in a group G then there is a uniform algorithm to solve the word problem in every finitely presented (relative to G) quotient of G. Furthermore, since decidability of GPCP in G is inherited by all subgroups of G, decidability of GPCP in G implies the uniform decidability of the word problem in every finitely presented quotient of every subgroup of G.
Examples of groups with undecidable GPCP include free groups and free solvable groups of derived length at least 3 and sufficiently high rank ( [40]). On the other hand, examples of groups where GPCP is decidable in polynomial time include all finitely generated nilpotent groups.
Furthermore, it was shown in [40] that in a free group, the bounded GPCP is NPcomplete. (In the bounded version of GPCP one is looking only for solutions (i.e., the words t(x 1 , . . . , x n )), whose length is bounded by a given number.)
The search version of the Post correspondence problem (or of the bounded version thereof) is to find a solution for a given instance, provided at least one solution exists. As usual, search versions can potentially be used to build cryptographic primitives, although it is not immediately clear how to convert the search version of the (bounded or not) Post correspondence problem to a one-way function with trapdoor.
THE HIDDEN SUBGROUP PROBLEM
Given a group G, a subgroup H ≤ G, and a set X , we say that a function f : G → X hides the subgroup H if for all g 1 , g 2 ∈ G, one has f (g 1 ) = f (g 2 ) if and only if g 1 H = g 2 H for the cosets of H. Equivalently, the function f is constant on the cosets of H, while it is different between the different cosets of H.
The hidden subgroup problem (HSP) is:
Let G be a finite group, X a finite set, and f : G → X a function that hides a subgroup H ≤ G. The function f is given via an oracle, which uses O(log |G| + log |X |) bits. Using information gained from evaluations of f via its oracle, determine a generating set for H.
A special case is where X is a group and f is a group homomorphism, in which case H corresponds to the kernel of f . The importance of the hidden subgroup problem is due to the facts that:
• Shor's polynomial time quantum algorithm for factoring and discrete logarithm problem (as well as several of its extensions) relies on the ability of quantum computers to solve the HSP for finite abelian groups. Both factoring and discrete logarithm problem are of paramount importance for modern commercial cryptography.
• The existence of efficient quantum algorithms for HSP for certain non-abelian groups would imply efficient quantum algorithms for two major problems: the graph isomorphism problem and certain shortest vector problems in lattices. More specifically, an efficient quantum algorithm for the HSP for the symmetric group would give a quantum algorithm for the graph isomorphism, whereas an efficient quantum algorithm for the HSP for the dihedral group would give a quantum algorithm for the shortest vector problem. We refer to [74] for a brief discussion on how the HSP can be generalized to infinite groups.
RELATIONS BETWEEN SOME OF THE PROBLEMS
In this section, we discuss relations between some of the problems described earlier in this survey. In the preceding Sections 8 and 9 we have already pointed out some of the relations, now here are some other relations, through the prizm of cryptographic applications.
We start with the conjugacy search problem (CSP), which was the subject of Section 3, and one of its ramifications, the subgroup-restricted conjugacy search problem:
Given two elements w, h of a group G, a subgroup A ≤ G, and the information that w a = h for some a ∈ A, find at least one particular element a like that. In reference to the Ko-Lee protocol described in Section 3, one of the parties (Alice) transmits w a for some private a ∈ A, and the other party (Bob) transmits w b for some private b ∈ B, where the subgroups A and B commute elementwise, i.e., ab = ba for any a ∈ A, b ∈ B. Now suppose the adversary finds a 1 , a 2 ∈ A such that a 1 wa 2 = a −1 wa and b 1 , b 2 ∈ B such that b 1 wb 2 = b −1 wb. Then the adversary gets a 1 b 1 wb 2 a 2 = a 1 b −1 wba 2 = b −1 a 1 wa 2 b = b −1 a −1 wab = K, the shared secret key.
We emphasize that these a 1 , a 2 and b 1 , b 2 do not have anything to do with the private elements originally selected by Alice or Bob, which simplifies the search substantially. We also point out that, in fact, it is sufficient for the adversary to find just one pair, say, a 1 , a 2 ∈ A, to get the shared secret key:
a 1 (b −1 wb)a 2 = b −1 a 1 wa 2 b = b −1 a −1 wab = K.
In summary, to get the secret key K, the adversary does not have to solve the (subgrouprestricted) conjugacy search problem, but instead, it is sufficient to solve an apparently easier (subgroup-restricted) decomposition search problem, see our Section 4.
Then, one more trick reduces the decomposition search problem to a special case where w = 1, i.e., to the factorization problem, see our Section 4.4. Namely, given w ′ = a · w · b, multiply it on the left by the element w −1 (which is the inverse of the public element w) to get w ′′ = w −1 a · w · b = (w −1 a · w) · b.
Thus, if we denote by A w the subgroup conjugate to A by the (public) element w, the problem for the adversary is now the following factorization search problem:
Given an element w ′ of a group G and two subgroups A w , B ≤ G, find any two elements a ∈ A w and b ∈ B that would satisfy a · b = w ′ , provided at least one such pair of elements exists. Since in the original Ko-Lee protocol one has A = B, this yields the following interesting observation: if in that protocol A is a normal subgroup of G, then A w = A, and the above problem becomes: given w ′ ∈ A, find any two elements a 1 , a 2 ∈ A such that w ′ = a 1 a 2 . This problem is trivial: a 1 here could be any element from A, and then a 2 = a −1 1 w ′ . Therefore, in choosing the platform group G and two commuting subgroups for a protocol described in our Section 3 or Section 4, one has to avoid normal subgroups. This means, in particular, that "artificially" introducing commuting subgroups as, say, direct factors is inappropriate from the security point of view.
At the other extreme, there are malnormal subgroups. A subgroup A ≤ G is called malnormal in G if, for any g ∈ G, A g ∩ A = {1}. We observe that if, in the original Ko-Lee protocol, A is a malnormal subgroup of G, then the decomposition search problem corresponding to that protocol has a unique solution if w / ∈ A. Indeed, suppose w ′ = a 1 · w · a ′ 1 = a 2 · w · a ′ 2 , where a 1 = a 2 , say. Then a −1 2 a 1 w = wa ′ 2 a ′−1 1 , hence w −1 a −1 2 a 1 w = a ′ 2 a ′−1 1 . Since A is malnormal, the element on the left does not belong to A, whereas the one on the right does, a contradiction. This argument shows that, in fact, already if A w ∩ A = {1} for this particular w, then the corresponding decomposition search problem has a unique solution.
Finally, we describe one more trick that reduces, to some extent, the decomposition search problem to the (subgroup-restricted) conjugacy search problem. Suppose we are given w ′ = awb, and we need to recover a ∈ A and b ∈ B, where A and B are two elementwise commuting subgroups of a group G.
Pick any b 1 ∈ B and compute:
[awb, b 1 ] = b −1 w −1 a −1 b −1 1 awbb 1 = b −1 w −1 b −1 1 wbb 1 = (b −1 1 ) wb b 1 = ((b −1 1 ) w ) b b 1 .
Since we know b 1 , we can multiply the result by b −1 1 on the right to get w ′′ = ((b −1 1 ) w ) b . Now the problem becomes: recover b ∈ B from the known w ′′ = ((b −1 1 ) w ) b and (b −1 1 ) w . This is the subgroup-restricted conjugacy search problem. By solving it, one can recover a b ∈ B.
Similarly, to recover an a ∈ A, one picks any a 1 ∈ A and computes:
[(awb) −1 , (a 1 ) −1 ] = awba 1 b −1 w −1 a −1 a −1 1 = awa 1 w −1 a −1 a −1 1 = (a 1 ) w −1 a −1 a −1 1 = ((a 1 ) w −1 ) a −1 a −1 1 . Multiply the result by a 1 on the right to get w ′′ = ((a 1 ) w −1 ) a −1 , so that the problem becomes: recover a ∈ A from the known w ′′ = ((a 1 ) w −1 ) a −1 and (a 1 ) w −1 .
We have to note that, since a solution of the subgroup-restricted conjugacy search problem is not always unique, solving the above two instances of this problem may not necessarily give the right solution of the original decomposition problem. However, any two solutions, call them b ′ and b ′′ , of the first conjugacy search problem differ by an element of the centralizer of (b −1 1 ) w , and this centralizer is unlikely to have a non-trivial intersection with B. A similar computation shows that the same trick reduces the factorization search problem, too, to the subgroup-restricted conjugacy search problem. Suppose we are given w ′ = ab, and we need to recover a ∈ A and b ∈ B, where A and B are two elementwise commuting subgroups of a group G. Pick any b 1 ∈ B and compute
[ab, b 1 ] = b −1 a −1 b −1 1 abb 1 = (b −1 1 ) b b 1 .
Since we know b 1 , we can multiply the result by b −1 1 on the right to get w ′′ = (b −1 1 ) b . This is the subgroup-restricted conjugacy search problem. By solving it, one can recover a b ∈ B.
This same trick can, in fact, be used to attack the subgroup-restricted conjugacy search problem itself. Suppose we are given w ′ = a −1 wa, and we need to recover a ∈ A. Pick any b from the centralizer of A; typically, there is a public subgroup B that commutes with A elementwise; then just pick any b ∈ B. Then compute Multiply the result by b −1 on the right to get (b −w ) a , so the problem now is to recover a ∈ A from (b −w ) a and b −w . This problem might be easier than the original problem because there is flexibility in choosing b ∈ B. In particular, a feasible attack might be to choose several different b ∈ B and try to solve the above conjugacy search problem for each in parallel by using some general method (e.g., a length-based attack). Chances are that the attack will be successful for at least one of the b's.
( 3 )
3Alice computes x(a y 1 , ..., a y k ) = x y = y −1 xy, and Bob computes y(b x 1 , ..., b x m ) = y x = x −1 yx. Alice and Bob then come up with a common private key K = x −1 y −1 xy (called the Commutator commutator of x and y) as follows: Alice multiplies y −1 xy by x −1 on the left, while Bob multiplies x −1 yx by y −1 on the left, and then takes the inverse of the whole thing: (y −1 x −1 yx) −1 = x −1 y −1 xy.
( 2 )
2Bob chooses an element b 2 ∈ G, chooses a subgroup of C G (b 2 ), and publishes its generators B = {β 1 , . . . , β m }. (3) Alice chooses a random element a 2 from gp < β 1 , . . . , β m > and sends P A = a 1 wa 2 to
as an element of G), which is now Alice's and Bob's common secret key. 4.4. The factorization problem. The factorization search problem is a special case of the decomposition search problem:
, . . . | ys −1 , r 1 , r 2 , . . . , where s = s(x 1 , x 2 , . . . ) is an arbitrary element in the generators x 1 , x 2 , . . . . (T2): Canceling a generator (this is the converse of (T1)): If we have a presentation of the form y, x 1 , x 2 , . . . | q, r 1 , r 2 , . . . , where q is of the form ys −1 , and s, r 1 , r 2 , . . . are in the group generated by x 1 , x 2 , . . . , replace this presentation by x 1 , x 2 , . . . | r 1 , r 2 , . . . . (T3): Applying an automorphism: Apply an automorphism of the free group generated by x 1 , x 2 , . . . to all the relators r 1 , r 2 , . . . . (T4): Changing defining relators: Replace the set r 1 , r 2 , . . . of defining relators by another set r ′ 1 , r ′ 2 , . . . with the same normal closure. That means, each of r ′ 1 , r ′ 2 , . . . should belong to the normal subgroup generated by r 1 , r 2 , . . . , and vice versa. Tietze proved (see e.g. [30]) that two groups given by presentations x 1 , x 2 , . . . | r 1 , r 2 , . . . and y 1 , y 2 , . . . | s 1 , s 2 , . . . are isomorphic if and only if one can get from one of the presentations to the other by a sequence of transformations (T1)-(T4).
( 7 )
7Since (B, x) · (A, ϕ m h ) = (A, y) · (B, ϕ n h ) = (M, ϕ h ) m+n ,we should have K Alice = K Bob = K, the shared secret key. Thus, the shared secret key in this protocol is
[w ′ , b] = [a −1 wa, b] = a −1 w −1 ab −1 a −1 wab = a −1 w −1 b −1 wab = (b −w ) a b.
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. The Department Of Mathematics, City, Of, New York, York, NY 10031 E-mail address: [email protected] OF MATHEMATICS, THE CITY COLLEGE OF NEW YORK, NEW YORK, NY 10031 E-mail address: [email protected]
| [] |
[
"Spin effects and compactification",
"Spin effects and compactification"
] | [
"Alexander J Silenko ",
"Oleg V Teryaev ",
"\nBelarus and Bogoliubov Laboratory of Theoretical Physics\nResearch Institute for Nuclear Problems\nBelarusian State University\n220030Minsk\n",
"\nBogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research\nJoint Institute for Nuclear Research\n141980, 141980Dubna, DubnaRussia, Russia\n"
] | [
"Belarus and Bogoliubov Laboratory of Theoretical Physics\nResearch Institute for Nuclear Problems\nBelarusian State University\n220030Minsk",
"Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research\nJoint Institute for Nuclear Research\n141980, 141980Dubna, DubnaRussia, Russia"
] | [] | We consider the dynamics of Dirac particles moving in the curved spaces with one coordinate subjected to compactification and thus interpolating smoothly between three-and two-dimensional spaces. We use the model of compactification, which allows us to perform the exact Foldy-Wouthuysen transformation of the Dirac equation and then to obtain the exact solutions of the equations of motion for momentum and spin in the classical limit. The spin precesses with the variable angular velocity, and a "flick" may appear in the remnant two-dimensional space once or twice during the period. We note an irreversibility in the particle dynamics because the particle can always penetrate from the lower-dimensional region to the higher-dimensional region, but not inversely. | 10.1103/physrevd.89.041501 | [
"https://arxiv.org/pdf/1311.5984v2.pdf"
] | 118,657,757 | 1311.5984 | eb6d83cb89d3d336407c59541941d3333ee14ecb |
Spin effects and compactification
16 Feb 2014
Alexander J Silenko
Oleg V Teryaev
Belarus and Bogoliubov Laboratory of Theoretical Physics
Research Institute for Nuclear Problems
Belarusian State University
220030Minsk
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research
Joint Institute for Nuclear Research
141980, 141980Dubna, DubnaRussia, Russia
Spin effects and compactification
16 Feb 2014numbers: 0420Jb0365Pm1110Ef1125Mj * Electronic address: alsilenko@mailru † Electronic address: teryaev@theorjinrru
We consider the dynamics of Dirac particles moving in the curved spaces with one coordinate subjected to compactification and thus interpolating smoothly between three-and two-dimensional spaces. We use the model of compactification, which allows us to perform the exact Foldy-Wouthuysen transformation of the Dirac equation and then to obtain the exact solutions of the equations of motion for momentum and spin in the classical limit. The spin precesses with the variable angular velocity, and a "flick" may appear in the remnant two-dimensional space once or twice during the period. We note an irreversibility in the particle dynamics because the particle can always penetrate from the lower-dimensional region to the higher-dimensional region, but not inversely.
I. INTRODUCTION
Low-dimensional structures are now under scrutiny in nonperturbative QCD, cosmology, high-energy physics, and condensed matter physics. Properties of particles placed into such structures are usually described by considering quantum theory in two dimensions. However, there is no doubt that real space remains three dimensional, which may lead to qualitative differences in some observables. This especially concerns the particle spin properties, which are crucially different at two and three spatial dimensions (see, e.g., Refs. [1][2][3]). Thus, transition to (2+1)-dimensional spacetimes leads to losses of a significant part of such properties. At the same time, in the two-dimensional space, anyons [4] may appear.
In the present work, we investigate the problem of transformation of the spin properties under the compactification of some spatial dimension. This problem is generally very difficult because the spin dynamics depends on many factors. To extract some common properties, we consider the toy model [5] of the curved space of variable dimensionality smoothly changing from three to two. A great preference of the model used is a possibility to obtain exact quantum-mechanical solutions.
We use the conventional Dirac equation for a consistent description of spin-1/2 particle motion in the curved space and take into account relativistic effects. While such effects are not too important in condensed matter physics (except for graphene), we keep in mind their further applications to the processes at Large Hadron Collider in the case [6,7] of variable (momentum) space dimension. We use the relativistic method [8] of the Foldy-Wouthuysen (FW) transformation [9] to derive exact quantum-mechanical equations of motion and obtain their classical limit.
In this work, we focus our attention on the spin properties. We show that, in contrast to a "naive" estimation, the spin in an effectively two-dimensional space may precess about the noncompactified dimensions and therefore a "flick" may appear in the remnant space once or twice during the period.
Let us start with the following metric proposed by Fiziev [5]: 2 , the primes define derivatives with respect to z, and ρ i are the functions of z. The spatial coordinates vary in the limits −∞ < z < ∞, 0 < Φ 1,2 < 2π.
ds 2 = c 2 dt 2 − ρ 1 (z) 2 dΦ 2 1 − ρ 2 (z) 2 dΦ 2 2 − ρ 3 (z) 2 dz 2 , (2.1) where ρ 3 (z) 2 = 1 + ρ ′ 1 (z) 2 + ρ ′ 2 (z)
We suppose ρ i (z) to be positive. The (3+1)-dimensional manifold defining this metric is a hypersurface in a flat pseudo-Euclidean (5+1)-dimensional space. The tetrad e 0 0 = 1, e j i = δ ij √ g ii allows us to define the local Lorentz (tetrad) frame. This considerably simplifies an analysis of results from possibly using the rescaled Cartesian coordinates dX = ρ 1 (z)dΦ 1 , dY = ρ 2 (z)dΦ 2 , dZ = ρ 3 (z)dz in the neighborhood of any point.
Taking the limit ρ 1 (z) → 0 or the limit ρ 2 (z) → 0 may lead to the reduction of dimension of the physical space from d = 3 to d = 2. We consider the case when the compactification of the e 1 (e 2 ) direction results in the confinement of the particle in a narrow interval of
Φ 1 (Φ 2 ) angles.
The transverse part of the metric (if z is assumed to be a longitudinal coordinate) has the structure of the Clifford torus, which is the product of two unit circles in the fourdimensional Euclidean space:
y 2 1 + y 2 2 = y 2 3 + y 2 4 = 1. (2.2)
The Clifford tori are used for analyzing twisted materials [10] and vesicles [11][12][13]. There is also some qualitative similarity to projection of a tube in a six-dimensional space onto a three-dimensional space, which was used for the construction of the quasicrystals theory [14].
We consider Clifford tori as a toy model of dimensional reduction. [16]). For the metric (2.1), the Hermitian Dirac Hamiltonian was first derived in Ref. [17]. It can be presented in the form
H D = βmc 2 − i c ρ 1 α 1 ∂ ∂Φ 1 − i c ρ 2 α 2 ∂ ∂Φ 2 − i c 2 α 3 1 ρ 3 , ∂ ∂z , (2.3)
where {. . . , . . . } denotes an anticommutator.
We transform this Hamiltonian to the FW representation by the method elaborated in
Ref. [8] which was earlier applied in our previous works [15,16,18]. After the exact FW transformation, we get the result
H F W = β √ a + Σ · b, (2.4) where a = m 2 c 4 + c 2 p 2 1 ρ 2 1 + c 2 p 2 2 ρ 2 2 + c 2 4 1 ρ 3 , p 3 2 , b = b 1 e 1 + b 2 e 2 = c 2 ρ ′ 2 ρ 2 2 ρ 3 p 2 e 1 − c 2 ρ ′ 1 ρ 2 1 ρ 3 p 1 e 2 , (2.5) and (p 1 , p 2 , p 3 ) = −i ∂ ∂Φ 1 , −i ∂ ∂Φ 2 , −i ∂ ∂z is the generalized momentum operator.
Primes denote derivatives with respect to z. The e 1 , e 2 , e 3 vectors form the spatial part of the orthonormal basis defining the local Lorentz (tetrad) frame. For the given timeindependent metric, the operators H F W , p 1 , and p 2 are integrals of motion.
Neglecting a noncommutativity of the a and b operators allows us to omit anticommutators and results in
H F W = β 2 √ a + b + √ a − b + Π · b 2b √ a + b − √ a − b ,(2.6)
where Π = βΣ is the spin polarization operator. It can be proven that extra terms appearing from the above noncommutativity are of order of | /(p z l)| 3 , where p z is the particle momentum and l is the characteristic size of the nonuniformity region of the external field (in the z direction). With this accuracy,
H F W = β √ a − 2 b 2 8a 3/2 + Π · b 2 √ a . (2.7)
The second term proportional to 2 is important even when it is relatively small. This term contributes to the difference between gravitational interactions of spinning and spinless particles and therefore violates the weak equivalence principle. Its importance relative to the main term is defined by the ratio ( b/a) 2 . The weak equivalence principle is also violated by the spin-dependent Mathisson force (see Refs. [15,19] and references therein) defined by the third term in Eq. (2.7). While the third term is usually much bigger than the second one, it vanishes for unpolarized spinning particles. The second term proportional to (Π · b) 2 is always nonzero. An analysis of Eqs. (2.5) and (2.7) leads to the conclusion that this term can be comparable with the main one (proportional to √ a) when l ∼ λ B , where λ B is the de Broglie wavelength. The existence of the term proportional to 2 is not a specific property of the toy model used. The appearance of such terms in the FW Hamiltonians describing a Dirac particle in Riemannian spacetimes was noticed in several works [18,20,21], whereas its relation to the spin-originated effect leading to the violation of the weak equivalence principle was never mentioned.
The equation of spin motion is given by
dΠ dt = Ω × Π, Ω = β b √ a . (2.8)
As a result, the spin rotates relative to e i vectors (i = 1, 2, 3) with the angular velocity Ω. Its motion relative to the Cartesian axes is much more complicated.
It has been proven in Ref. [22] that finding a classical limit of relativistic quantum mechanical equations reduces to the replacement of operators by respective classical quantities when the condition of the Wentzel-Kramers-Brillouin approximation, /|pl| ≪ 1, is satisfied.
It has also been shown that the classical limit of the FW Hamiltonians for Dirac [15,16,18] and scalar [23] particles in Riemannian spacetimes coincides with the corresponding purely classical Hamiltonians.
III. MOTION OF PARTICLE AT VARIABLE DIMENSIONS
Let us first study the motion of the particle by neglecting the influence of the spin onto its trajectory. Since p 1 and p 2 are integrals of motion, they can be replaced with the eigenvalues P 1 and P 2 , respectively. Let us choose the e 1 axis as the compactified dimension and suppose that ρ 1 (z) is a decreasing function (ρ 1 (z) → 0 when z → ∞). We can neglect a dependence of ρ 2 on z, assuming that this function changes much more slowly. We denote initial values of all parameters by additional zero indices and consider the general case when the initial value of the metric component, ρ 10 ≡ ρ 1 (z 0 ), is not small.
The classical limit of the Hamiltonian is given by
H = m 2 c 4 + c 2 P 2 1 ρ 2 1 + c 2 P 2 2 ρ 2 2 + c 2 p 2 3 ρ 2 3 . (3.1)
The possibility of making general conclusions with the special model used is based on the fact that the Hamiltonian of a particle in an arbitrary static spacetime is given by
H = c 2 (m 2 c 2 + g ij p i p j ) g 00 , i, j = 1, 2, 3. (3.2) Equation (3.
2) covers spinless [24] and spinning [15,16] particles in classical gravity as well as the classical limit of the corresponding quantum-mechanical Hamiltonians for scalar [23] and Dirac [15] particles. For spinning particles, the term s · Ω should be added to this
Hamiltonian [15,16]. When the metric is diagonal, g ii = 1/g ii and Eq. To describe the compactification, we can introduce the compactification radius δ so that the "compactification point" z c can be defined by ρ 1 (z c ) = δ. Due to the energy E conservation, the particle can reach this point if
E ≥ m 2 c 4 + c 2 P 2 1 δ 2 + c 2 P 2 2 ρ 2 2 (z c )
.
(3.3)
Note that the decrease of compatification radius δ while E remains finite implies the corresponding decrease of P 1 .
The particle velocity is equal to
v z ≡ dz dt = ∂H ∂p 3 = c 2 p 3 Eρ 2 3 = c sgn (p 3 ) Eρ 3 (z) E 2 − m 2 c 4 − c 2 R(z), R(z) = P 2 1 ρ 2 1 (z) + P 2 2 ρ 2 2 (z)
. A tedious but simple calculation allows us to obtain the longitudinal component of the particle acceleration:
a z ≡ d 2 z dt 2 = − c 4 E 2 ρ 2 3 R ′ 2 + p 2 3 ρ ′ 3 ρ 3 3
.
(3.5)
It is obvious that p 3 (z f ) = 0, R ′ (z f ) ≥ 0 (for monotonic continuously differentiable R(z)),
so that a z (z f ) ≤ 0. Therefore, z f is the turning (if R ′ (z f ) > 0) or attracting (if R ′ (z f ) = 0)
point. For nonmonotonic R(z) there is a possibility of passage to the region z > z f due to possible growth of ρ 2 (z). The particle motion is then limited by the pointz f corresponding to the neglect of the motion in the e 2 direction
E = m 2 c 4 + c 2 P 2 1 ρ 1 (z f ) . (3.6)
The important particular case of Eq. (3.1) corresponds to P 1 = 0. The particle penetrates into the region of the effective dimensional reduction (z → ∞) and does not reverse the direction of its motion.
In this study, as was mentioned above, we consider that the smooth adiabatic transition from the three-dimensional space to the effectively two-dimensional one does not necessarily attribute the physical sense to all intermediate points in particle motion. At the same time, the true change of the dimensionality was discussed in cosmology (see Refs. [7,[25][26][27]) and
in connection with experiments at the LHC (see Refs. [6,7,28,29]). Our analysis can also be applicable at the LHC.
Note also that the motion in the opposite direction of increasing dimension does not impose any conditions for the initial state of the particle. One may say that the region of lower dimension is "repulsive" whereas the region of higher dimension is "attractive", implying a sort of irreversibility in the particle dynamics. This property emerges because of the appearance of ρ 1 in the expression for the Hamiltonian in the denominator. Such a situation is a general one that can be seen from Eq. (3.2) in the case of diagonal metric.
This may give additional support to the hypothesis [25,26] that such a transition from the lower dimensionality to the higher one leaded to the evolution of the Universe.
IV. SPIN EVOLUTION AT VARIABLE DIMENSIONS
In the classical limit, the angular velocity of spin precession is given by Since
Ω = b E = c 2 Eρ 3 P 2 ρ ′ 2 ρ 2 2 e 1 − P 1 ρ ′ 1 ρ 2 1 e 2 .ρ 1 (z f ) = c |P 1 | E 2 − m 2 c 4 − c 2 P 2 2 ρ 2 20 −1/2 ,(4.5)
the total spin turn (z = z f ) is given by
∆ϕ = sgn (P 1 ) · π 2 − arctan P 1 ρ 30 ρ 10 p 30 . (4.6)
The passage of the particle to the region of compactification implies, as was discussed above, the relative smallness of the second term so that the spin rotates by about 90 • .
If P 1 = 0, the spin projection onto the e 1 direction is always conserved. The total spin turn (z = z f ) is given by 2) so that one can expect that qualitative features of spin and momentum dynamics will persist for other compactification-related metrics as well.
∆φ = arctan P 2 ρ 30 ρ 20 p 30 − sgn (P 2 ) · π 2 .
The analysis of particle momentum evolution allows us to describe the motion at the boundary between the regions of space having different dimensions. The passage to the region of lower dimension is more natural in the special case when the generalized momentum in the compactified direction P 1 = 0. At the same time, the transition to the region of higher dimension (considered in Refs. [25,26] as a possible way of the evolution of the Universe)
does not impose the constraints for its initial state, manifesting a sort of irreversibility.
The particle motion (especially near the turning point) is characterized by the three main properties which cannot be naturally explained from the point of view of observer residing in the compactified spacetime: i) a reversion of the direction of motion; ii) a rather quick motion along the compactified direction, which may be seen as a sort of "zitterbewegung"; iii) the appearance of a pseudovector of spin in the compactified (2+1)-dimensional space and its rotation or flickering [when the spin pseudovector crosses the remnant (2+1)-dimensional layer].
The experimental tests of the emerging spin effects may be performed by studies of spin polarizations of Λ (and, probably, also Λ c ) hyperons produced in the high-energy collisions where the compactification [6,7] takes place. This may bear a resemblance to the recently proposed [30] tests of the vorticity in heavy-ion collisions, although a detailed analysis is required.
We can finally conclude that the transition to (2+1)-dimensional spacetime leads to the nontrivial behavior of spin which, generally speaking, cannot be adequately described from the point of view of an observer residing at (2+1) dimensions.
correspond to the two different directions of the longitudinal particle motion.Note that the arrival to the compactification point with zero velocity (z c = z f being the final point of particle trajectory) corresponds to the equality sign in Eq. (3.3).
ds/dt = v z (z)(ds/dz), Eqs.(3.4) and (4.1) define an easily solvable system of first-order homogeneous linear differential equations. Equation (4.1) is rather informative about details of the compactification. Only the Ω 2 component contains parameters of the compactified dimension. Although |P 1 |/|P 2 | ≪ 1, the presence of additional factors does not allow for neglecting Ω 2 as compared with Ω 1 (under the condition that P 1 = 0).When ρ 2 (z) = const, Ω 1 = 0 and the spin rotates about the e 2 axis, the spin projection onto the e 2 e 3 surface, which is the spatial part of the (2+1)-dimensional spacetime, oscillates. The spin appears in this surface only once (in the special case when the cone of spin precession is tangent to this surface) or twice per rotation period. Evidently, the origin of this spin "flickering", as well as the appearance of pseudovector, is completely unexplainable in terms of the two-dimensional space.The model used allows to obtain an exact analytical description of the spin evolution. It is characterized by a change of the angle ϕ defining the direction of the spin in the plane orthogonal to Ω: of spin evolution at the effective dimensional reduction can be solved in a general form. To simplify the analysis, let us consider the case of ρ 2 (z) = ρ 20 = const. In this case, the exact value of the integral
the Dirac fermion dynamics in the curved space model of variable dimension. The advantage of the toy model used is the possibility of performing the exact FW transformation of the Dirac equation and then obtaining the exact solutions of the equations of motion for momentum and spin in the classical limit. At the same time, the obtained Hamiltonian (3.1) is similar to the generic one (3.
The spin can, however, rotate about the e 1 direction if ρ 2 depends on z. In this case, the angle of the spinturn is equal to
∆φ(z) = − arcsin
cP 2
Bρ 2 (z)
+ arcsin
cP 2
Bρ 20
, B =
√
E 2 − m 2 c 4 =
c 2 p 2
30
ρ 2
30
+
c 2 P 2
2
ρ 2
20
. (4.7)
AcknowledgmentsWe are indebted to P.P. Fiziev, V.P. Neznamov, and D.V. Shirkov for stimulating discussions. This work was supported in part by the RFBR (Grants No. 11-02-01538 and 12-02-91526) and BRFFR (Grant No. Φ12D-002).
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| [] |
[
"Electrical and magneto transport in 2D semiconducting MXene Ti 2 CO 2",
"Electrical and magneto transport in 2D semiconducting MXene Ti 2 CO 2"
] | [
"Anup Kumar Mandia ",
"Anna Namitha ",
"Koshi ",
"Bhaskaran Muralidharan ",
"Seung-Cheol Lee ",
"¶ ",
"Satadeep Bhattacharjee [email protected] ",
"\nof Electrical Engineering\nScience and Technology Center (IKST)\n¶Electronic Materials Research Center\nIndian Institute of Technology Bombay\nMumbai-400076560065Powai, Jakkur, BengaluruIndia ‡Indo-Korea, India\n",
"\nKIST\n136-791SeoulSouth Korea\n"
] | [
"of Electrical Engineering\nScience and Technology Center (IKST)\n¶Electronic Materials Research Center\nIndian Institute of Technology Bombay\nMumbai-400076560065Powai, Jakkur, BengaluruIndia ‡Indo-Korea, India",
"KIST\n136-791SeoulSouth Korea"
] | [] | The Hall scattering factor is formulated using Rode's iterative approach to solving the Boltzmann transport equation in such a way that it may be easily computed within the scope of ab-inito calculations. Using this method in conjunction with density functional theory based calculations, we demonstrate that the Hall scattering factor in electron-doped Ti 2 CO 2 varies greatly with temperature and concentration, ranging from 0.2 to around 1.3 for weak magnetic fields. The electrical transport was modelled primarily using three scattering mechanisms: piezoelectric scattering, acoustic scattering, and polar optical phonons. Even though the mobility in this material is primarily limited by acoustic phonons, piezoelectric scattering also plays an important role which was not highlighted earlier. | 10.1039/d2tc01279k | [
"https://arxiv.org/pdf/2203.12234v4.pdf"
] | 247,618,692 | 2203.12234 | fc35a61621244298b6db6e16c7eee1ccd980fbe3 |
Electrical and magneto transport in 2D semiconducting MXene Ti 2 CO 2
Anup Kumar Mandia
Anna Namitha
Koshi
Bhaskaran Muralidharan
Seung-Cheol Lee
¶
Satadeep Bhattacharjee [email protected]
of Electrical Engineering
Science and Technology Center (IKST)
¶Electronic Materials Research Center
Indian Institute of Technology Bombay
Mumbai-400076560065Powai, Jakkur, BengaluruIndia ‡Indo-Korea, India
KIST
136-791SeoulSouth Korea
Electrical and magneto transport in 2D semiconducting MXene Ti 2 CO 2
The Hall scattering factor is formulated using Rode's iterative approach to solving the Boltzmann transport equation in such a way that it may be easily computed within the scope of ab-inito calculations. Using this method in conjunction with density functional theory based calculations, we demonstrate that the Hall scattering factor in electron-doped Ti 2 CO 2 varies greatly with temperature and concentration, ranging from 0.2 to around 1.3 for weak magnetic fields. The electrical transport was modelled primarily using three scattering mechanisms: piezoelectric scattering, acoustic scattering, and polar optical phonons. Even though the mobility in this material is primarily limited by acoustic phonons, piezoelectric scattering also plays an important role which was not highlighted earlier.
Introduction
Since the advent of graphene, 1 two-dimensional (2D) materials have become the focus of intensive research due to their novel electrical, chemical, optical and mechanical properties. 2? -5 Many other 2D materials like h-BN, 6-8 transition metal dichalcogenides (TMDs) [9][10][11][12][13] and Xenes, 14-21 group V graphyne ? have been fabricated and deeply investigated for nanoelectronics application.
Recently, MXene, a new class of 2D transition metal carbides, nitrides and carbonitrides have been synthesized [22][23][24][25][26][27] from M n+1 AX n phases (where "M" represents an early transition metal, "A" represents group IIIA or IVA element and "X" represents C and/or N, and MXenes are generally written as M n+1 X n T x , with T standing for the surface terminating groups. Zha et al. 28 have investigated the mechanical, structural and electronic properties of M 2 CT 2 MXenes and shown that surface functional groups (T = F,OH,O) have considerable impact on the crystal structure of these materials. Oxygen functionalized MXene structure possess smaller lattice parameter and also shows higher mechanical stability as compared to fluorine and hydroxyl functionalized ones. They also have higher thermodynamic stability than fluorine and hydroxyl functionalized ones. [29][30][31] The family of 2D materials has been quickly expanding since the discovery of the first MXene in 2011. Several MXenes have been synthesised to date, and their characteristics have been extensively studied by various researchers. Chemical stability, hydrophilic behaviour, strong electronic conductivity, and outstanding mechanical qualities are among the characteristics they exhibit. Numerous experimental and theoretical works have presented the potential applications of MXene in various fields like energy storage, 30,32-37 gas sen-sors, 38,39 bio-sensors, 40 adsorbents, 31,41 supercapacitor, 42,43 water treatment, 31 biomedical application, 40,44 electromagnetic interference shielding 45,46 and so on. Some of these applications are related to high electronic conductivity and there is a need to explore it. From experimental and theoretical investigations, it is known that the metallic or semiconducting nature of MXenes depend on the surface termination group and a vast majority of the members of this family are metallic. 43,44,[47][48][49] Also, T i 3 C 2 T x is the most conductive one among all the MXenes synthesized. Metallic conductivity of MXenes can be engineered by controlling their surface chemistry or intercalation mechanism. 50 In this work, we work we have performed an comprehensive study of the electronic and magneto-transport in Ti 2 CO 2 using first principles based transport calculations. A key quantity for studying the carrier concentration and drift mobility in a semiconductor is the so called Hall factor, 51,52 which is commonly considered to be equal to one, which implies Hall mobility and the drift mobility are the same. But in reality, in many materials it differs from one which lead to wrong estimation of the carrier density and the drift mobility. In the present work, we formulate the Hall factor in a simple form which can be calculated within the framework of Rode's iterative scheme of solving the Boltzmann transport equation. We apply such a scheme to understand the temperature dependence of the Hall factor in Ti 2 CO 2 with the inputs obtained from DFT based simulations. Furthermore, even though the transport properties of Ti 2 CO 2 have been addressed in the previous studies, the role of piezoelectric scattering was never discussed. We also demonstrate that piezoelectric scattering plays an important role in this material. It should be mentioned here that we have excluded the effect of spin-orbit coupling for the moment. For MXenes containing heavy 4d and 5d transition metals, the relativistic spin-orbit coupling (SOC) affects the electronic structures significantly. 29 There are few known exceptions to it in MXene literature. One such example is, the BB phase of Ti2CF2, which demonstrate multiple Dirac cones and giant spin orbit coupling. The effect of SOC on the electronic structure of pristine/bare Ti2C is investigated by B. Akgenc et al. ? For the minimum energy 1T phase of Ti2C, the effect of SOC is minute whereas it becomes significant for 2H-Ti2C. It modifies dispersion of the bands arising from d-orbitals. We have considered the 1T phase for modeling Ti2CO2 and hence we expect the effect of SOC to be small.
Structural model
We start with the investigation of structural properties of Ti 2 C. Keeping up with transition metal dichalcogenides (TMD) notation, two phases 1T and 2H of MXenes are considered. 54 Both 1T and 2H phases have hexagonal symmetry with C atom sandwiched between two Ti triangular lattices. In the 1T phase, the transition metal atoms are not in line (side view - Figure 1 AFM calculations, we construct 2×1 supercell consisting of four transition metal atoms and the corresponding AFM configurations are given in SI. In AFM1, intralayer coupling is ferromagnetic and interlayer configuration is antiferromagnetic ordering. In the case of AFM2, the intralayer coupling is antiferromagnetic and interlayer ordering is ferromagnetic.
From comparison of total energies, AFM1 ordering is preferred for 1T-Ti 2 C whereas the lowest energy configuration is FM for 2H-Ti 2 C. This is consistent with reported literature.
The magnetic moment of FM configuration of 2H-Ti 2 C is 2 µ B /cell. The optimized lattice constants of 1T-and 2H-Ti 2 C are 3.06 and 3.05 Å respectively. The corresponding thickness of the layer are 2.30 and 2.47 Å respectively and match with previous DFT reports. 55 The 1T phase of Ti 2 C is semiconducting in nature whereas the 2H phase is half-metallic (details are given in SI). Further, we study the structure of oxygen functionalized 1T-Ti 2 C and the most stable configuration is given in Figure 1. Here, we see that the oxygen atom on top lies in line with Ti atom in the lower layer and vice versa. This structure is non-magnetic and belongs to the spacegroup P-3m1 (No. 164). The optimized lattice parameter is 3.03 Å which agrees with other literature. 57 The thickness of the layer (calculated as the distance between the two oxygen atoms) is around 4.45 Å.
ab-initio calculations
Electronic structure calculations are carried out using density functional theory (DFT) implemented in plane wave code, Vienna ab-initio Simulation Package (VASP). For pseudopotentials, the projector augmented wave (PAW) approach is used. The exchange-correlation functional is treated using generalized gradient approximation (GGA) parameterized by Perdew-Burke-Ernzerhof (PBE) formalism. The plane wave cut off energy is set to 500 eV.
The conjugate gradient algorithm is used for structural optimization. The convergence criteria for energy and force are 10 −6 eV and -0.01 eV/Å respectively. A vacuum of thickness 20 Å along the z-direction is employed to avoid interactions between the neighboring layers and a Monkhorst-Pack k-mesh of 17×17×1 is used for Brillouin zone sampling. The DFT-D2 method is used for van der Waals correction. The crystal structures are visualized using VESTA. 53 Phonon spectrum is calculated using VASP in combination with Phonopy software. Here, we employ a supercell of size 4×4×1 and a 3×3×1 k-mesh to determine the dynamical matrix.
Electronic structure and lattice dynamics
The electronic band structure of Ti 2 CO 2 with the corresponding density of states (DOS) is given in Figure 2. It is semiconducting with a band gap of 0.25 eV with the valence band maximum (VBM) and conduction band minimum (CBM) at Γ and M respectively.To determine the dynamical stability, phonon dispersion spectrum of Ti 2 CO 2 is calculated and presented in Figure 2. There are no imaginary frequencies in the phonon spectrum hence ab-initio parameters needed for the transport calculation
The calculated ab-initio parameters obtained using DFT calculations are reported in the Table- Case I: Carriers are in electric field BTE for the electron distribution function f is given by
∂f(k) ∂t + v · ∇ r f + eE · ∇ k f = ∂f ∂t coll ,(1)
where e is the electronic charge, v is the carrier velocity, E is the applied electric field,
eE · ∇ k f = [s(k, k )f (1 − f ) − s(k , k) f (1 − f )]dk ,(2)
where s(k, k ) represents the transition rate of an electron from a state k to a state k .
At lower electric fields, the distribution function is given by [60][61][62]
f (k) = f 0 [ (k)] + g(k)cosθ,(3)
where f 0 [ (k)] is the equilibrium distribution function, g(k) is perturbation in the distribution function, and cos θ is the angle between applied electric field and k. Higher order terms are neglected here, since we are calculating mobility under low electric field conditions. Now perturbation in the distribution function g(k) is required for calculating the low-field transport properties. The perturbation in the distribution function g(k) is given by [60][61][62]
g k,i+1 = S i (g k , i) − v(k)( ∂f ∂z ) − eE ( ∂f ∂k ) S o (k) .(4)
where S i represents in-scattering rates due to the inelastic processes and S o represents the sum of out-scattering rates.
S o = 1 τ in (k) + 1 τ el (k) . Where 1 τ el (k)
is the sum of the momentum relaxation rates of all elastic scattering processes and 1 τ in (k) is the momentum relaxation rate due to the in-elastic processes.
The expression for τ el (k), S i and 1 τ in (k) are given by the following equations
1 τ el (k) = (1 − X)s el (k, k )dk (5) S i (g k , i) = Xg k ,i [s in (k , k)(1 − f ) + s in (k, k )f ]dk (6) 1 τ in (k) = [s in (k, k )(1 − f ) + s in (k , k)f ]dk(7)
where X is the cosine of the angle between the initial and the final wave vectors, s in (k, k ) and s el (k, k ) represents transition rate of an electron from state k to k due to inelastic and elastic scattering mechanisms respectively. Since, S i is function of g(k), thus equation 4 is to be calculated iteratively. 62 In our previous work we have used the same procedure to calculate mobility for ZnSe 59 and CdS. 58 Drift mobility µ is then calculated by the following
expression 59 µ = 1 2E v( )D s ( )g( )d D s ( )f ( )d ,(8)
where D S ( ) represents density of states. The carrier velocity is then calculated directly from the ab-initio band structure by using the following expression v(k) = 1 ∂ ∂k .
From these, we can evaluate the electrical conductivity given as
σ = neµ e t z ,(10)
where n is the electron carrier concentration, t z is the thickness of the Ti 2 CO 2 layers along z-direction which is 4.45Å in this case.
Case II: Carriers are in both electric and magnetic field
A similar method, also introduced by Rode 63 to solve BTE under arbitrary magnetic field.
In this case distribution function is given by 63
f (k) = f 0 [ (k)] + xg(k) + yh(k),(11)
where h(k) represents perturbation in distribution function due to magnetic field, and y is direction cosine from B × E to k, where B is applied magnetic field. Substituting equation 11 in equation 1, we get a pair of coupled equations that can be solved iteratively 63
g i+1 (k) = S i (g i (k)) − eE ( ∂f ∂k ) + βS i (h i (k)) S o (k)(1 + β 2 ) . (12) h i+1 (k) = S i (h i (k)) + β eE ( ∂f ∂k ) − βS i (g i (k)) S o (k)(1 + β 2 )
.
where β = ev(k)B kSo(k) . The above expression shows that the perturbations to the distribution function due to the electric field (g) and magnetic field (h) are coupled to each other through the factor β and the in scattering rates S i . It should be highlighted that such a representation cannot be obtained using standard relaxation time approximation (RTA) and
can only be seen using the current method. The components of conductivity tensor in terms of perturbations are given by
σ xx = e v( )D s ( )g( )d 2E (14) σ xy = e v( )D s ( )h( )d 2E(15)
The Hall coefficient R H , Hall mobility µ H and Hall factor r are respectively calculated by
R H = σ xy B(σ xx σ yy + σ 2 xy )(16)µ H = σ xx (0)|R H | (17) r = µ H µ(18)
where σ xx (0) is value of σ xx in the absence of the magnetic field.
Scattering Mechanisms
Acoustic Scattering
The scattering rates due to the acoustic phonons can be expressed as 57
1 τ j ac (E) = D 2 jA k B T k 2 C j A v(19)
where T is temperature, k is the wave vector, C A is the elastic modulus, is the reduced Planck's constant, D jA is acoustic deformation potential for the j th acoustic mode and k B
is Boltzmann constant. v is the group velocity of the electrons. j ∈ LA,TA,ZA. The energy dependence here enters through the wave-vector k. We have performed an analytical fitting of the lowest conduction band with a six-degree polynomial to get smooth curve for group velocity which help us to get a one by one mapping between the wavevector k and the band energies.
Piezoelectric Scattering
The piezoelectric Scattering rates 64 are calculated as follows,
1 τ pz (E) = 1 τ ac (E) × 1 2 × e 11 e 0 D A 2(20)
where e 11 is piezoelectric constant (unit of C/m), 0 is vacuum permeability.
Polar Optical Phonon (POP) Scattering
The polar optical phonons are the source of inelastic scattering in the system. The inelastic scattering rates from a given k-state are given in terms of S in (in) and S o scattering as in the Eq. 6 and Eq. 7.
The out scattering contribution due to polar optical phonon scattering is given by 65,66
1 τ in (k) = C pop (1 − f 0 (E)) [N V (1 − f 0 (E + ω pop )I + (E) k + v(E + ω pop ) + (N V + 1)(1 − f 0 (E − ω pop )I − (E) k − v(E − ω pop ) ](21)
where
I + (E) = 2π 0 1 q a dθ(22)I − (E) = 2π 0 1 q e dθ (23) q a = k 2 + k + 2 − 2kk + cosθ (24) q e = k 2 + k − 2 − 2kk − cosθ(25)
where θ is angle between initial wave vector k and final wave vector k , k + and k − represents wave vector at energy E + ω and E − ω respectively.
C pop = e 2 ω pop 8π 0 × 1 κ ∞ − 1 κ 0(26)
κ ∞ and κ 0 represents high frequency and low frequency dielectric constant.
The in scattering contribution due to polar optical phonon scattering can be represented by the sum of in-scattering due to absorption and emission of polar optical phonons
S in i (k) = S in a (k) + S in e (k)(27)
where S in a (k) represents in-scattering due to absorption of polar optical phonon from energy E − ω pop to energy E and S in e (k) represents in-scattering due to emission of polar optical phonon from energy E + ω pop to energy E.
S in a (k) = C pop (N V + 1)f 0 (E)J − (E) k − v(E − ω pop )f 0 (E − ω pop ) (28) S in e (k) = C pop (N V )f 0 (E)J + (E) k + v(E + ω pop )f 0 (E + ω pop )(29)N V = 1 exp( ω pop /k B T ) − 1 (30) J + (E) = 2π 0 cosθ q i,e dθ (31) J − (E) = 2π 0 cosθ q i,a dθ (32) q i,a = k − 2 + k 2 − 2kk − cosθ (33) q i,e = k + 2 + k 2 − 2kk + cosθ(34)
While driving the expression for different scattering rate we have replaced term by k m * by group velocity. 58,59 The group velocity will be calculated directly from the DFT band structure. 58,59
Results and Discussion
Let us discuss about electron transport first, before we go on to magnetotransport. Also, in order to fully understand magnetotransport in this material, these results must be comprehended.
Electronic transport
In the Fig.3(a), we show the scattering rates due to the phonons (acoustic and optical) and
due the piezoelectric scattering. It can be seen that the most dominant contribution is due the acoustic phonons, followed by the piezoelectric scattering. The POP scattering has the least effect. Therefore the conductivity or mobility in Ti 2 CO 2 is limited by acoustic phonons.
To understand the nature of the acoustic phonons that limits the conductivity in Ti 2 CO 2 ,
we compute the scattering rates due to individual acoustic phonons which are shown in the Fig.3(b). From this figure, it can be understood that LA phonons are the ones which play key role here. Therefore the two main carrier scattering mechanisms involved here are due to LA phonons and piezoelectric scattering. To realize the trend of mobility with respect to temperature and electron concentration, let us consider Matthiessen's rule
1 µ = 1 µ ac + 1 µ pz + 1 µ pop(35)
where µ denotes total mobility and the suffixes ac, pz, and pop denote the acoustic, piezoelectric, and polar optical contributions, respectively. In the Fig.4 (a) however there is a drastic decrease of the mobility which can be seen from the Fig.4(b) at electron concentration of 1 × 10 14 cm −2 where the mobility drops rapidly. We fit the temperature dependent mobility was through a power law model as follows,
µ = AT −γ(36)
here A is the prefactor and γ is the power law exponent. The exponent depends on the con- The increase in conductivity due to the increase in carrier concentration and (2) the decrease in conductivity (due to the decrease in mobility) with increase in carrier concentration. The latter has a temperature dependence (µ = AT −γ ). As we have shown above the exponent γ has small value (about 0.7) for the carrier concentration of about 10 14 cm −2 , there is a crossing of the conductivity curves that can be seen in the Fig-5 (a) for the concentration 10 13 cm −2 and 10 14 cm −2 .
In the Fig.6, we show the contribution of different components of mobility and their temperature evolution. In the left panel ( Fig.6(a)) we show both piezoelectric as well as phonon contribution (acoustic and optical) to the mobility. According to the Eq. it is understood that LA acoustic mode is the dominant contribution at all the temperature.
However, there is a significant contribution of piezoelectric scattering something that was not addressed by other studies on the same material.
Magnetotransport: The Hall factor
In many situations the Hall factor is assumed to be equal to one, then we have µ H = µ. This is true when R H = 1 ne . The Hall and the drift mobility are same in this case. This is the case when one assumes constant relaxation time and the band is parabolic. However, for a general case, the Hall factor can therefore be expressed as
r = neR H = ne B σ xy (σ xx σ yy + σ 2 xy ) ∼ ne B σ xy σ 2 xx(37)
The above expression results from the fact that σ xy is very small and its square can be neglected in the denominator. Also we consider σ xx ∼ σ yy . Combining Eq.14, Eq.15 and Eq.
37, the Hall factor can be expressed as,
r = n B 2E v( )D s (ε)h(ε)dε [ v( )D s (ε)g(ε)dε] 2(38)
The above formulation shows the Hall factor is directly proportional to the carrier concentration, and the strength of the electric field, while it is inversely proportional to the magnetic field. In our formulation temperature dependence enters via the perturbations to the distribution function g(ε) and h(ε) which in tern depend on the temperature dependent scattering rates. Recently, Macheda et al. have calculated the Hall scattering factor in graphene. 69 They also formulated Hall factor in terms of the solutions of the Boltzmann transport equation. However, they used RTA and the effect of magnetic field on the distribution function was not considered in an explicit way as we have done. In our case we use two perturbations g(ε) and h(ε) which are coupled to each other as can be seen in the equations 12 and 13. It can also seen from those equations that coupling between g(ε) and h(ε) further enhances at larger magnetic field through the factor β. For even small inelastic scattering, the coupling between g(ε) and h(ε) could be considerably large due to effect of magnetic field. In the scenario of larger magnetic fields, we expect our technique to perform better than the one proposed by Macheda et al. 69 It should be noted here that, if we drop the inelastic contribution which means that
S i =0 and S o = 1 τ el ,
Conclusions
In conclusion, we have studied the electrical and magneto-transport properties in electron doped semiconducting MXene Ti 2 CO 2 using a combined approach of Rode's iterative scheme with DFT based methods. The electronic and vibrational properties needed as inputs are obtained from DFT simulations. Hall factor shows large deviation from unity as function of temperature and carrier concentration. At low doping and temperature the Hall factor is as low as 0.2 while at higher temperature and carrier concentration, the Hall factor is bigger than one which also depend on the value of magnetic field considered. This suggest that one has to take precautions while measuring the carrier concentration and drift mobility in such systems.
n=1-3) by selective exfoliation of "A" atoms as the M-A bonds are much weaker than the M-X-M bonds. M n+1 AX n belong to the family of layered compounds with P 6 3 /mmc symmetry. Other possible pathways of synthesizing MXenes are Due to the etching process during synthesis, MXenes are terminated with -O, -OH and -F groups. So, functionalized
(a)). The symmetry group of 1T-Ti 2 C is P − 3m 1 (No. 164). 55,56 For 2H structure, transition metal atoms are in line (side view -Figure 1(c)) or they are stacked on top of each other and has the symmetry group P − 6m 2 (No. 187). From the total energy calculations, 1T phase is lowest in energy for non-magnetic (NM) configuration. To obtain the correct magnetic ground state, we calculate the energies of the different magnetic orderings (ferromagnetic-FM, antiferromagnetic-AFM1 and antiferromagnetic-AFM2). For
1 .
1The acoustic deformation potential as well as elastic moduli are calculated along the out-of-plane (ZA), longitudinal (LA) and transverse (TA) directions using the method described in Zha et al .57 The piezoelectric and dielectric constants (both high and low frequency) are calculated using density functional perturbation theory (DFPT). ?Methodology: Solution of Boltzmann Transport Equation using Rode's iterative methodTransport coefficient calculation are performed by using our tool AMMCR.58,59 Brief methodology of solving the Boltzmann Transport Equation (BTE) is presented below.
f
describes the probability distribution function of carrier in real and momentum space as a function of time, ∂f ∂t coll represents the change in the distribution function with time due to collisions. Under steady state , ∂f(k) ∂t = 0 and spatial homogeneous condition (∇ r f = 0), equation 1 can be written as
centration. With increase in carrier concentration from 10 11 cm −2 to 10 14 cm −2 the exponent changes from 1.4 to 0.7. We fit the mobility data to the Eq.36 within the temperature region 100-700K. The values of the exponents look similar to values seen in standard electron doped semiconductors particularly when we consider the scattering mechanisms are dominated by acoustic phonons.67,68 This explains also the nature of the longitudinal conductivity shown in theFig-5. The conductivity here evolve as a result of competition between two terms:(1)
35 above, the reciprocal nature of the relationship between total mobility and individual components results in the case where the component with the least value is the most significant. From the figure,
in this case we stop at the zeroth iteration (i=0, in Eq.12 and Eq.13). The situation is equivalent of doing calculations within RTA, (equivalent to the work done by Macheda et al ). It is vital to note, however, that unlike compound semiconductors, POP scattering should be almost absent in the case of a free-standing graphene, making the estimate of Macheda et al. quite justifiable. In the Fig.7 (a) we show the Hall-conductivity with respect to the temperature. The value of the magnetic field used is 0.4 T along the z-direction. The figure shows highest Hall conductivity at the electron concentration of 10 13 cm −2 .In the Fig.8(a) we show the variation of the Hall factor with the temperature, while in the Fig.8(b) we show how Hall factor changes with doping concentration. It can be seen that the temperature dependence of the Hall factor is higher for the high doping concentration. The Hall factor varies within the range [0.1-0.5] up to concentration 10 11 − 10 13 cm −2 . The Hall factor rises steeply within the concentration range 10 13 − 10 14 cm −2 . At concentration, 10 14 Hall factor reaches value 1.3 which is comparable to that of bulk Si. The same behaviour can be understood from the heat-map of the Hall scattering factor at various temperatures and concentrations shown in the Fig.9 where the results are shown for two values of the magnetic field 0.4T and 0.8T. It can be seen that the Hall scattering factor is as small as 0.2 for small doping and temperature. Therefore one has be cautions about estimating the carrier concentration and drift mobility in those regions. Use of Hall factor as one may lead to the overestimation of the carrier concentration and underestimation of the drift velocity.
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Figure 1 :
1Side view and top view of (a-b) 1T and (c-d) 2H phases of Ti 2 C and (e-f) Ti 2 CO 2 . Blue, brown and red balls correspond to Ti, C and O respectively.
Figure 2 :
2(a) Band structure of Ti 2 CO 2 with the VBM and CBM denoted by green and red points respectively, (b) projected density of states and (c)phonon dispersion of Ti 2 CO 2 along the high symmetry directions of the Brillouin zone.
Figure 3 :Figure 4 :Figure 5 :Figure 6 :
3456Scattering Rate vs energy at 300 K (a)Variation of mobility with temperature (b) Variation of mobility with doping concentration. (a) Variation of conductivity with temperature (b) Variation of conductivity with doping concentration (a)Contribution of mobility due to polar optical phonons, acoustic phonons and piezoelectric scattering (b) due to LA, TA and ZA acoustic phonon mode with temperature.
Figure 7 :
7Variation of Hall conductivity with (a)temperature and (b) carrier concentration.
Figure 9 :
9Hall scattering factor as a function of temperature and carrier concentration at (a)0.4T and (b)0.8T
Table 1 :
1Material Parameters used for T i 2 CO 2Parameters
Values
PZ constant, e 11 (C/m)
3 × 10 −13
Acoustic deformation potentials, D A (ev) :
D A,LA
8.6
D A,T A
3.5
D A,ZA
0.7
Elastic modulus, C A (N/m)
C A,LA
301.7
C A,T A
391.6
C A,ZA
59.3
Polar optical phonon frequency ω pop (T Hz)
ω pop,LO
3.89
ω pop,T O
3.89
ω pop,HP
8.51
High frequency dielectric constant, κ ∞
23.570798
Low frequency dielectric constant, κ 0
23.6
we show the temperature variation of mobility at different electron concentration. As expected the mobility decrease with increasing electron concentration as well as temperature. The mobility values are not very different for electron concentration within the range [1 × 10 11 cm −2 -1 × 10 13 cm −2 ]
Conflict of interestThe authors declare that they have no conflict of interest.Author contributionsSB conceived the idea. SB and NAK performed the DFT calculations. The transport calculations using AMMCR code were performed by AKM under supervision of SB. The manuscript was written by SB, NAK and AKM. BM and SCL contributed in discussions.
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Akm Acknowledements, BM gratefully acknowledge funding from Indo-Korea Science and Technology Center (IKST). BangaloreAcknowledements AKM and BM gratefully acknowledge funding from Indo-Korea Science and Technology Center (IKST), Bangalore.
| [] |
[
"XOC: Explainable Observer-Classifier for Explainable Binary Decisions",
"XOC: Explainable Observer-Classifier for Explainable Binary Decisions"
] | [
"Stephan Alaniz ",
"Zeynep Akata "
] | [] | [] | When deep neural networks optimize highly complex functions, it is not always obvious how they reach the final decision. Providing explanations would make this decision process more transparent and improve a user's trust towards the machine as they help develop a better understanding of the rationale behind the network's predictions. Here, we present an explainable observer-classifier framework that exposes the steps taken through the model's decision-making process. Instead of assigning a label to an image in a single step, our model makes iterative binary sub-decisions, which reveal a decision tree as a thought process. In addition, our model allows to hierarchically cluster the data and give each binary decision a semantic meaning. The sequence of binary decisions learned by our model imitates human-annotated attributes. On six benchmark datasets with increasing size and granularity, our model outperforms the decision-tree baseline and generates easy-to-understand binary decision sequences explaining the network's predictions. | null | [
"https://arxiv.org/pdf/1902.01780v1.pdf"
] | 59,604,430 | 1902.01780 | 4bc821ff7cc7e4906fe7c4f9f1d8f180a0ef3e18 |
XOC: Explainable Observer-Classifier for Explainable Binary Decisions
Stephan Alaniz
Zeynep Akata
XOC: Explainable Observer-Classifier for Explainable Binary Decisions
When deep neural networks optimize highly complex functions, it is not always obvious how they reach the final decision. Providing explanations would make this decision process more transparent and improve a user's trust towards the machine as they help develop a better understanding of the rationale behind the network's predictions. Here, we present an explainable observer-classifier framework that exposes the steps taken through the model's decision-making process. Instead of assigning a label to an image in a single step, our model makes iterative binary sub-decisions, which reveal a decision tree as a thought process. In addition, our model allows to hierarchically cluster the data and give each binary decision a semantic meaning. The sequence of binary decisions learned by our model imitates human-annotated attributes. On six benchmark datasets with increasing size and granularity, our model outperforms the decision-tree baseline and generates easy-to-understand binary decision sequences explaining the network's predictions.
Introduction
A classification decision made by a deep neural network (DNN) is hard to interpret. However, to spread adoption and create a widespread acceptance by the user, the predictions of a neural network need to be explainable. An explanation can help an end-user to establish trust or help a machinelearning practitioner to understand or debug deep models. We distinguish between two types of explanations: posthoc rationalizations and introspections. Post-hoc rationales are generated by a second neural network that justify the output of the first network, i.e., the decision maker. For instance, a language model explaining the classification decision of a vision model by talking about discriminative features of the object (Hendricks et al., 2016; falls into this category. Post-hoc rationales, however, do not guarantee 1 University of Amsterdam. Correspondence to: Stephan Alaniz <[email protected]>. that the explanation reflects the decision-maker's internal thought process. We focus on introspective explanations revealing which pathway is taken to determine the final decision. Work on introspective explanations includes, e.g., visualizing features (Springenberg et al., 2014;Zhou et al., 2016;Selvaraju et al., 2017), saliency maps (Simonyan et al., 2013), interpretable features (Adel et al., 2018), and modular networks (Andreas et al., 2016). Among those, Andreas et al. (2016) expose a transparent network structure and Adel et al. (2018) give network features a semantic meaning. We combine both of these traits in a single model.
Our Observer-Classifier (OC) framework exposes a decision path in the form of an explainable decision tree. It breaks down the decision process into many small decisions via a two-agent setup (see Figure 1). The first agent, the classifier, does not have access to any visual information about the input. It formulates a question to the second agent, the observer, who uses a perception unit to respond to this question with a yes/no answer. By repeating these questions and their binary answers several times, the classifier obtains information about the input. The communication between the agents is learned end-to-end that allows us to condition the binary answers to be from a set of human-interpretable class-attributes, i.e., Explainable Observer-Classifier (XOC). When our XOC model solves a classification task, we extract explainable binary decisions that lead to the final prediction.
arXiv:1902.01780v1 [cs.LG] 5 Feb 2019
We emphasize that our OC framework uses hard yes/no decisions instead of binary distribution probabilities for the classification procedure. Because of the strict use of binary decisions, the decision trees as well as the resulting hierarchical clustering of the data become easy to interpret.
Furthermore, an additional loss on our binary decisions encourages to mimic attributes that attach a semantic meaning to the binary answers. We propose zero-shot learning as a natural test bed for evaluating the interpretability of the attributes learned by our model. Zero-shot learning relies on transferable side information in order to generalize from classes seen during training time to new classes only seen at test time. We show that our model learns discriminative class-level attributes that perform well in zero-shot learning.
Our contributions are as follows: 1) We propose a twoagent Observer-Classifier framework that learns to make iterative binary decisions to collaboratively solve an imageclassification task. 2) We showcase on six datasets that our model outperforms classic decision trees and qualitatively demonstrate that our model learns attributes about the decision process that lead to explainable decision chains. 3) We propose zero-shot learning as an evaluation metric for interpretability and show that our learned attributes outperform Word2Vec-based semantic embeddings.
Related Work
Our work is related to the combination of deep learning and decision trees, multi-agent communication, and interpretable machine learning.
Decision Trees with Neural Networks. Adaptive Neural Trees (Tanno et al., 2018) directly model the neural network as a decision tree, where each node and edge correspond to one or more modules of the network. Our model is self-adapting through the use of a recurrent network in the classifier that makes a prediction at every node and can be easily rolled out to a greater depth without changing the architecture or number of weights. The prior work closest to ours is the Deep Neural Decision Forest (Kontschieder et al., 2016), which first uses a CNN to determine the routing probabilities on each node and then combines nodes to an ensemble of decision trees that jointly make the prediction. Similarly, in our Explainable Observer, we compute the binary decisions, i.e., router nodes of the tree, once before using them ad hoc. Our method differs in that we focus on explainability by explicitly only considering a hard binary decision at each node while other methods use soft decisions, making a large portion of the tree responsible for the predictions, and, thus, are harder to interpret.
Multi-Agent Communication. Learning to communicate in a multi-agent setting has recently gained interest in the research community mostly due to the emergence of deep reinforcement learning (Foerster et al., 2016;Havrylov & Titov, 2017). Most related to our work, Foerster et al. (2016) propose the use of an agent that composes a message of categorical symbols at once and another agent that uses the information in these messages to solve a referential game. For discrete symbols, they also rely on the Gumbel-softmax estimator, but, in contrast to our model, their focus is not on explainability and their communication is not iterative, i.e., it concludes after one message, which does not allow the fine-grained introspection as in our model.
Explainability. The importance of explanations for an end-user has been studied from the psychological perspective (Lombrozo, 2012), showing that humans use explanations as a guidance for learning and understanding by building inferences and seeking propositions or judgments that enrich their prior knowledge. They usually seek explanations to fill the requested gap depending on prior knowledge and the goal in question.
In support of this trend, explainability has been recently growing as a field in computer vision and machine learning (Hendricks et al., 2016;Park et al., 2018;Andreas et al., 2016;Zintgraf et al., 2017). Following the convention of Park et al. (2018), our focus is on introspective explanations, where a deep network as the decision maker is trained to explain its own decision which is useful in increasing trust for the end user and provides a means to detect errors in the predictions of the model.
Textual explanations are explored by Hendricks et al. (2016) who propose a loss function based on sampling and reinforcement learning that learns to generate sentences that realize a global sentence property, such as class specificity. Andreas et al. (2016) compose collections of jointly trained neural modules into deep networks for question answering by decomposing the questions into their linguistic substructures and using these structures to dynamically instantiate modular networks with reusable components.
As for visual explanations, Zintgraf et al. (2017) propose to apply a prediction-difference analysis to a specific input. Park et al. (2018) utilize a visual-attention module that justifies the predictions of deep networks for visual question answering and activity recognition. Grad-CAM (Selvaraju et al., 2017) uses the gradients of any target concept, e.g., a predicted action, flowing into a convolutional layer to produce a localization map highlighting the important regions in the image that lead to predicting the target concept. Interpretable CNNs (Zhang et al., 2018a) modify the convolutional layer, such that each filter map corresponds to an object part in the image, and a follow-up work (Zhang et al., 2018b) uses a classical decision tree to explain the predictions based on the learned object-part filters.
Our framework is set up as a sequential interaction between two agents to solve an image-classification task by communicating. The first agent, the observer, holds information about the image x ∈ R D , by having access to the actual image or pre-extracted image features z ∈ R Z . The second agent, the classifier, predicts the associated ground-truth class y ∈ Y using only the messages broadcast by the observer, without having direct access to the image.
For a single image, the classifier sets its initial state to its prior belief of the class distributionŷ 0 ∈ R C . In a balanced dataset, this would correspond to a uniform distribution over all the class labels. It then creates a query message h t ∈ R H sent to the observer requesting information about the image. The observer processes the query together with the input image to construct a binary response d t ∈ {0, 1}.
The classifier uses this one bit of information to update its state minimizing the classification error. This constitutes one iteration of the observer-classifier communication. The interaction repeats until a maximum number of steps is reached or until the classifier is confident in its decision.
The classification loss is minimized when the two agents jointly learn to communicate the most important bit of information about the image at each time step, such that the classifier's label prediction improves. We deliberately limit the observer's messages to be binary as clear yes/no answers are easier for humans to interpret than probability values.
Classifier
Main output of the classifier is the classification decision. The decision tree is a byproduct obtained from restricting the inputs of the classifier to be binary. Hence, the binary response sequence of the observer corresponds to one particular path along the decision tree. Since the classifier only takes discrete inputs, we can map out all possible binary sequence paths up to a desired length at test time, which provides us with a binary tree structure. We construct the classifier as an LSTM (Hochreiter & Schmidhuber, 1997), making each node of the decision tree correspond to a hidden state of the LSTM and the classifier's prediction distribution.
We use the hidden state h t ∈ R H of the LSTM as a query message for the observer. Each iteration the classifier receives the observer's binary decision together with the message from the previous time step [d t−1 , h t−1 ] to update its state and, thus, produce h t . The message h t is also used to make the class prediction using an affine transformation followed by a softmax:
y t = softmax(W h t + b)(1)
with W ∈ R |Y|×H , b ∈ R |Y| . Since the primary objective of The classifier sends a query message ht to the observer that the observer combines with image features z to construct a binary response dt using its MLP and the Gumbel-softmax estimator. The response is fed into the classifier in the next time step, such that the predicted class distributionŷ better fits the true class label.
the classifier is to maximize the classification performance, we minimize the cross-entropy loss of the predicted class probabilitiesŷ t and the true class probabilities y:
L CE (y,ŷ t ) = − i y i logŷ t,i .(2)
While decision trees usually employ a classification loss on the leaf nodes, we use the cross-entropy loss also after every binary decision. This results in the loss function:
L = 1 T T t=1 L CE (y,ŷ t )(3)
where we enforce a loss on the current classification distribution after each iteration of the observer-classifier communication loop. This encourages the network to predict the correct class in as few communication steps as possible.
Observer: Binary Decisions
The observer model can have different architectures depending on whether or not certain capabilities are required. We present the General Observer, which optimizes for classification accuracy without using attribute data, as well as the Explainable Observer that introduces an additional attribute loss for human interpretability.
General Observer. Our General Observer architecture is outlined in Figure 2. In this setting, the observer consists of a multi-layer perceptron (MLP) that takes as inputs both the image features and the message from the classifier [z, h t ] and produces the binary decision d t with M LP g : R Z+H → {0, 1}. Since we require the output of M LP g to be a discrete binary signal, its construction needs Figure 3. Explainable Observer-Classifier (XOC) Model: Our explainable observer uses two MLPs to process z and ht separately. From image features zt, it predicts binary attributes, while from the query message ht, it learns to pick an attribute as a response to the classifier. We impose a loss on our learned attributes to match them with human-annotated attributes to make them understandable for the user.
to be differentiable in order to train our model end-to-end. The Gumbel-softmax estimator (Jang et al., 2017;Maddison et al., 2017) allows us to sample from a discrete categorical distribution and use the reparameterization trick to obtain the gradients of this sampling process. To obtain a binary sample with the Gumbel-softmax estimator, we sample g i from a Gumbel distribution and then calculate a continuous relaxation of the categorical distribution
d i = exp((log o i + g i )/τ ) K j=1 exp((log o j + g j )/τ )(4)
where log o is the unnormalized output of our MLP and τ is the temperature that parameterizes the discrete approximation. When τ approaches 0, the output becomes a one-hot vector (binary when K = 2) and otherwise, it is a continuous signal. Popular training strategies include annealing the parameter τ over time or augmenting the Gumbel-softmax with an arg max function that discretizes the activation in the forward pass and a straight-through identity function in the backward pass. We resort to the second strategy since doing so guarantees the response d t from the observer to be always binary during training, which empirically leads to better results.
Explainable Observer. In addition, we develop our Explainable Observer (see Figure 3), where the observer needs to first predict a set of binary attributes for the image independent of the query message of the classifier. The query message h t is then used to decide on the attribute as a re-sponse, which could be seen as the classifier requesting a particular attribute at each communication time step. The perception module is extended with an Attribute MLP, i.e., MLP Attr that outputs a set of binary attributesâ ∈ {0, 1} A with the dimensionality A chosen in advance. Each of these attributes uses the Gumbel-softmax estimator with K = 2 to sample and discretize them as binary features. Once the binary attribute vector is obtained, a Response MLP, i.e., MLP Resp takes the query message from the classifier as input and produces a probability distribution π t ∈ R A over the binary attributes to indicate which one serves as the response to the classifier. The choice distribution π t over the attributes is again discretized using the Gumbel-softmax estimator resulting in a one-hot vector c t of size A. The final binary decision d t =â c t is computed by selecting the position encoded by c t fromâ (denoted by ).
Attribute Loss for Explainable Observer. The classification loss allows to learn both the observer and the classifier in conjunction. Minimizing this loss at each time step is equivalent to finding the binary split of the data that reduces the class-distribution entropy the most. In this regard, it is similar to what is usually referred to as information gain in classical decision trees. However, a split that best separates the data is not always easy to interpret, especially when the features used to do this split result from a non-linear transformation such as, in our case, with the CNN.
We, therefore, propose to align the binary decisions with class-level attributes corresponding to human-interpretable features. We consider attributes to be assigned a binary value, e.g., statements that are either true or false about a particular entity in question, such as "the object is red". Since this information is only needed on the class level, minimal additional supervision is required to obtain this labeled data. As the Explainable Observer predicts a set of A binary attributes, we add a second cross-entropy term for these learned attributes to be consistent with the humanlabeled attributes:
L = 1 T T t=1 L CE (y,ŷ t ) + λL CE (α y,ct ,â ct )(5)
weighted by a hyperparameter λ. α y,ct corresponds to the true attribute label of class y that was chosen by the observer at time step t andâ ct is the matching inferred attribute of the same choice c t . This loss encourages the network to learn attributes that agree with human-annotated attributes while optimizing for classification accuracy. Note that the attribute loss is only imposed on those attributes employed by the model. When λ > 0, our model uses ground-truth attributes in order to give the split a semantic meaning. In this case, we set the number of learned attributes A to the number of ground-truth attributes. However, our model learns to use only a subset of the ground-truth attributes, focusing on AWA2 CUB aPY MNIST CIFAR-10 ImageNet attributes available not available # image 37K 11K 15K 70K 60K 1.2M # class 50 200 24 10 10 1K scale medium small large difficulty coarse fine coarse fine Table 1. A summary of the datasets that we use in terms of the availability of attributes, number of images, and number of classes. As our model tackles image classification, we group the datasets based on scale in terms of classes (medium, small, and large-scale) and based on difficulty (coarse-grained and fine-grained) the ones that minimize the cumulative attribute loss. When λ = 0, i.e., our model does not use any human-annotated attributes, it automatically discovers attributes to be used as side information for zero-shot learning.
Experiments
In this section, we describe our experimental setup, provide quantitative and qualitative results demonstrating the performance of our model and evaluate the attributes learned by our model in zero-shot learning.
Experimental Setup
Here we detail our setup in terms of datasets, side information and experimental settings.
Datasets. We experiment on six datasets (see Table 1). MNIST (Lecun et al., 1998) Side Information. The attributes are collected manually from experts by asking the relevance of an attribute for each class. Since our model does not consider splits on soft probabilities but rather on hard binary decisions, it is beneficial to have binary attribute data. When we train our explainable model with attribute loss, we binarize all the attributes with a threshold at 0.5, i.e., an attribute is present if more than 50% of the annotations agree.
The attribute quality directly affects the zero-shot-learning performance. Hence, evaluating our model on this task shows the effectiveness of our XOC model in attribute prediction. We compare the learned attributes by our model with Word2Vec feature vectors (Mikolov et al., 2013) extracted from Wikipedia articles (Akata et al., 2015).
Setting. For fully supervised learning, we randomly assign 20% of each class as test data for image classification when an official test set is not provided. In all the experiments across the datasets, we randomly separate 10% of the training data as a validation set. The MLPs consist of two layers with a ReLU non-linearity in between. It is beneficial to learn the temperature hyper-parameter τ of the Gumbel-softmax estimator jointly with the network.
Decision-Tree Baseline
We use the classical decision tree as a baseline for evaluating our OC framework. A decision tree is fitted on top of the image features extracted from ResNet (He et al., 2016) using one dimension of the features at a time to make a split, e.g., z 42 < 0.23, until a leaf node only contains samples of the same class or a regularization strategy leads to early stopping. Since no semantic meaning is attached to a split in the tree, this model serves as the decision-tree baseline optimized for classification accuracy.
For the explainable decision-tree baseline, we first train a linear layer on top of the image features to predict the binarized class attributes (instead of the class labels) for each image using a binary cross-entropy loss as in our XOC model. Secondly, we predict attributes for each image and fit a decision tree to determine the class label. We emphasize that the network will predict probabilities per attribute which we binarize before fitting the decision tree. This is important to be comparable with our proposed XOC model that also uses binary attributes and to be in line with the yes/no questions for improved interpretability. If we did not enforce this, the decision tree could, for instance, learn two subsequent splits, such as p("swims") < 0.8 and p("swims") > 0.6, to isolate samples that cover a certain range of probabilities for an attribute, which weakens the explainability. Equivalently to our XOC model, the explainable decision-tree baseline splits on whether an attribute is present or not.
We use the Gini impurity index as splitting criterion because it has a slight computational advantage over entropy-based methods (Raileanu & Stoffel, 2004), such as information gain. We report the outcome with the best validation results after randomized hyperparameter search on regularization parameters, i.e., minimum sample size for splits, minimum reduction in impurity per split.
Image Classification with OC
We compare the classifier of our OC model with the softmax classifier as the non-explainable upper bound and with the classical decision tree as the baseline. We use a simple CNN for MNIST, ResNet-18 for CIFAR-10, and ResNet-152 pretrained on ImageNet and fine-tuned on each of the datasets for the remaining datasets. Figure 4 shows the classification accuracy of our OC model compared with the decision-tree baseline as well as the upper bound for each dataset. During training of the upperbound model, the softmax classifier can update the weights of the CNN perception module. We fix the CNN's weights from the upper-bound model and use it to extract image features for the decision tree and our model. While our model works with constrained single-bit communications to improve explainability, it succeeds in maintaining the same classification accuracy as the upper-bound model on the medium/small-scale and coarse-grained datasets (AWA2, MNIST, CIFAR-10). Furthermore, our model exposes a binary decision-tree structure and hierarchical clustering of the data as additional outcome that improves interpretability of the predictions. For instance, on CIFAR-10, we observe that our OC model separates the animal classes from the vehicles in the first binary decision. While this helps in getting a better understanding on the data, it can also be used to debug failure cases by analyzing at which part in the decision-tree path a mistake occurred.
We also observe that our OC model consistently outperforms the decision-tree baseline across all datasets. On CUB and aPY specifically, the classification accuracy of our model is Figure 5. Our XOC model, where each decision node has a humaninterpretable meaning, is compared to the decision-tree baseline. In all cases, image features come from a state-of-the-art CNN pre-trained on ImageNet (except for MNIST) and fine-tuned on the respective datasets. The attribute loss is weighted with λ = 0.25.
3.5 and 2 times higher, respectively. Fine-grained datasets, such as CUB, are challenging to explain because their classification relies on nuances. Moreover, this makes it hard for non-experts to judge the correctness of the predictions, making explanations particularly important. Thus, achieving good results on CUB helps in addressing this issue. Im-ageNet poses an extreme challenge for both decision-tree approaches due to being a large-scale dataset (1,000 classes and 1,2 million images) that requires significantly bigger trees than the other datasets. In addition to outperforming the decision-tree baseline in terms of accuracy, our OC model scales better with increasing tree size in large-scale datasets. This is due to increasing the tree depth simply translates to incrementing the number of binary decisions, i.e., time steps of the Observer-Classifier communication. Therefore, our model scales linearly with the depth of the tree while the number of weights stays constant as opposed to classical decision trees that grow exponentially with the depth of the tree.
Explainable Image Classification with XOC
We present results of our Explainable Observer-Classifier (XOC) model, where we include the attribute loss to incorporate explainable binary decisions. We set λ to 0.25 across all experiments, as it leads to the best results based on grid search on the validation set.
Comparing our explainable model with the explainable decision-tree baseline in Figure 5, we observe that on AWA2 and aPY, the classification accuracy comes close to the performance of the model without attribute loss. This shows that constraining the model to use attributes as binary decisions does not harm classification accuracy and a discriminative decision tree can be learned either way. On the other hand, the CUB dataset is more challenging. Since the dataset is fine grained, distinguishing between closely related classes require a large number of class attributes, which leads to sparse attribute vectors and an imbalanced decision tree. Nonetheless, our XOC achieves nine-timeshigher accuracy than the decision tree on CUB. It also sig- Figure 6. Learned Explainable Decision Tree on AWA2 using our Explainable Observer-Classifier model and the attribute loss. We show the first decisions of the most likely path for each class and give each decision a human-understandable meaning based on the class-attribute that was used at each node. The tree exposes a transparent overview of how our model comes to its classification conclusions, e.g., it decides to separate meat-eating animals from all other animals in the first step.
nificantly ourperforms the decision tree on AWA2 and aPY. These results show the benefit of our XOC model from the joint optimization of the the classification loss as well as the attribute loss. The decision-tree baseline first optimizes for the best attribute prediction and then builds the tree on top of these features. By doing this process collectively, our model can choose to ignore attributes that are not suitable for good decision-tree splits since predicting these attribute would only constitute to a larger penalty in the attribute loss. We specifically design the attribute loss, such that it only acts on attributes our XOC model uses. Fine-grained datasets, such as CUB, show the difficulty in predicting these attributes, where the decision-tree baseline performs particularly poorly.
Introspective Explanations: Visualizing Decisions
The main benefit of the XOC model is that classification predictions can now be explained with a human-understandable thought process, which we demonstrate qualitatively. Our model reveals its decision-making process by pointing to the tree branch, into which a certain image falls, and the attribute being chosen at each node. We also inspect the learned structure of the decision tree by illustrating the splits from our model on AWA2.
For more examples, we refer to the supplementary material.
In Figure 6, we observe the separation of classes in three decision steps and each decision is associated with a humaninterpretable attribute. The left/right sub-tree indicates that the attribute is present/absent respectively. By visualizing the tree, we can get an explainable overview of the internal decision process of the whole classifier. For instance, the first decision deals with identifying meat-eating animals and those who do not eat meat. With this attribute split, our model effectively separates dogs, bears, cats, big cats, and foxes from all the other animals. These categories get further refined with each binary split building a hierarchical clustering that defines the XOC model's decision structure.
Since the pool of attributes determines the vocabulary of the explanations, it is worth considering different sets of attributes depending on the use case, e.g., if only attributes that refer to visual features are desired.
Explanations are often contrastive . In addition to justifying a positive decision, our model can reason about negative decisions. When images are misclassified, we inspect the point in the tree where the error occurred, exposing detailed information of when the features are mistaken to be from another class.
As an example, in Figure 7, we show the first six decisions for two tiger images. The lower path corresponds to when the model thinks the attribute is present for a given class. Both images follow the same path for five decisions that indicate the animal is a meat-eater, a hunter, a stalker, is strong, and lives near bushes. The last shown decision of whether the animal has stripes is different for the two images. For one of the tigers, our model decides it does not have stripes. Therefore it is ultimately incorrectly classified as a lion, while the other image is correctly classified as a tiger. In addition, our XOC model depicts its current belief of the correct class at any time during the process. This also reveals some critical binary decisions, such as the one shown in the example of whether the animal has stripes or not. In fact, we observe a high probability of "tiger" if the answer is "yes" and a high probability of "not tiger" if the answer is "no". This way, a user inspecting the individual rationals can make a more informed decision on the value of the model's predictions.
Explanations help other tasks: Zero-Shot Learning
Explanations are useful when they enable solving an independent task (Lombrozo, 2012). We argue that zero-shot learning is suitable for this purpose because solving this task requires using interpretable features as side information.
In zero-shot learning, the training and test classes are disjoint. In order to predict the unseen class for a query image, the model needs to do information transfer. Side information, often in a complementary modality, is used as a means of such a transfer as it associates seen and unseen classes.
The most widely used side information comprises humanannotated attributes. In our XOC model, in order not to use any expert annotation we set λ = 0. After obtaining the probabilities of each learned binary attribute via softmax, we stack the attributes in a per-class attribute vector and scale the attribute values to be between -1 and 1. For a particular class, the attributes are averaged over all training set images. We compare against Word2Vec (Mikolov et al., 2013) trained on Wikipedia articles to make a fair comparison with a technique that does not use human annotations as side information for zero-shot learning. Our experiments follow the recommendations of Xian et al. (2017), i.e., we use the same image features and their proposed train-test split. As the zero-shot learning model, we use Structured Joint Embedding (SJE) (Akata et al., 2015), the best zeroshot-learning performance on AWA2 according to Xian et al. (2017).
Our results in Table 2 show that while expert-annotated attributes reach 66.2% accuracy on AWA2, the attributes learned by our XOC model achieve 54.1% accuracy, significantly outperforming Word2Vec with 41.0%. This behavior generalizes to other datasets. On aPY, we observe that our learned attributes (36.0%) come close to the performance of human-annotated attributes (38.0%) and on the CUB dataset, our model (43.6%) outperforms Word2Vec (25.9%) by a large margin. This result is encouraging as it demonstrates that our learned attributes lead to discriminative and interpretable representations that are useful for tackling the challenging task of zero-shot learning. It also shows that our model learns representations more discriminative than the ones Word2Vec extracted from Wikipedia articles.
Another interesting observation is that while the dimensionality of Word2Vec equals 400, our learned attributes reach a significantly better accuracy using considerably fewer attributes: 40 on AWA2, 30 on aPY, and 100 on CUB. The number of attributes learned for CUB is required to be larger as CUB is a fine-grained dataset. These results suggest that the hierarchical clustering from our decision tree carries an interpretable meaning that generalizes to unseen classes.
Conclusion
In this work, we presented a two-agent framework that tackles the image-classification task using human-interpretable binary decisions in the form of a hierarchical process. Trained end-to-end, our model achieves marginally close accuracy as the state-of-the-art non-explainable models while revealing its internal thought process. In addition, our XOC model relates its binary decisions to human-understandable concepts. The hierarchical clustering and explainable binary decisions allow to better understand the iterative predictions of the network as well as help identify failure cases at test time. We propose zero-shot learning as an evaluation metric for evaluating explanations and show promising results validating that our model indeed learns transferable and discriminative binary features across classes.
Figure 1 .
1Our Observer-Classifier Framework. A. For classifying an image into cat, dog, car, plane categories, the observer combines image features with a query message of the classifier at each time step and creates a binary response. The classifier, i.e. an LSTM, uses its hidden state as a query message. At every step, the state of the LSTM is updated with the binary response to improve classification accuracy. B. The corresponding decision making process as the underlying decision tree.
Figure 2 .
2General Observer-Classifier (OC) Model:
Figure 4 .
4Ablation Study: Our OC model (General Observer without any attribute loss) is compared to the decision-tree baseline as well as the non-explainable upper bound (softmax on image features) for image classification. In all cases, image features come from a state-of-the-art CNN pre-trained on ImageNet (except for MNIST) and fine-tuned on the respective datasets.
Explainable Observer-ClassifierDoes it have whiskers?... ...Classifier
t+1
Classifier
Classifier
t
General Observer-Classifier
Gumbel-softmax
MLP_Resp
MLP_Attr
z
Is it an animal?
[
]
_Attr
Gumbel
softmax
.
Response
Selection
t
t+1
NO
Observer
Attr. 1
Attr. n-1 Attr. n
Response/Attribute Index
Probability
Sample
Probability
Sample
Attribute
Prediction
Gumbel
softmax
Y N
Y N
Y N
Y N
1
…
n-1
n
Selection Operator
Probability Sample
Y
N
Y
N
YES
Observer
Classifier
Figure 7. Qualitative Explanation of Classification of Two Tigers in AWA2. We show the binary decisions made for two images of a tiger along with the current label prediction at each step. The lower path corresponds to when the attribute is present for a given class. Both images follow the same path except for the last shown decision of whether the animal has stripes. The one tiger, for which our model decides there are no stripes, becomes ultimately classified incorrectly as a lion.Has stripes
Meateater
Hunter
Stalker
Strong
Not a
meateater
Not a
hunter
Not a
stalker
Not strong
Lives near
bushes
Not living
near bushes
No stripes
side info superv. AWA2 (d) CUB (d)
aPY (d)
att
human 66.2 (85) 47.6 (312) 38.0 (64)
att (ours) learned 54.1 (40) 43.6 (100) 36.0 (30)
w2v
learned 41.0 (400) 25.9 (400) 34.6 (400)
Table 2 .
2Zero-Shot Learning on AWA2, CUB, aPY using SJE. We compare side information (d = features' dimensionality): humanannotated attributes (att), learned attributes by the XOC model (att (ours), λ = 0), learned attributes byWord2Vec (w2v)
Supplementary Material for: XOC: Explainable Observer-Classifier for Explainable Binary DecisionsA. Qualitative Results on aPYWe trained our XOC model on the aPY dataset equivalently as on AWA2 reported in the paper. InFigure 8, we show the explainable decision tree learned by our XOC model. The left/right path of each node indicate the presence/absence of the human-interpretable attribute used to make the decision. InFigure 9, we illustrate a qualitative example of the classification of two images of chairs made by our model. Both follow the upper branch that indicates the chairs are not furry, not a vertical object, not from metal, and do not have a snout. In the last shown step that decides whether or not it is made of wood, the decision is different for the chairs, and the one not made of wood according to our model is ultimately classified incorrectly. This serves as another example of introspection that our model allows to make a more informed decision about the value of the network's prediction.Figure 8. Learned Explainable Decision Tree on aPY using our XOC model and the attribute loss (λ = 0.25). We show the first decisions of the most likely path for each class and give each decision a human-understandable meaning based on the class attribute that was used at each node.Figure 9. Qualitative Explanation of Classification of Two Chairs in APY. We show the binary decisions made for two images of a chair along with the current label prediction at each step. The upper path corresponds to when the attribute is not present for a given class. Both images follow the same path except for the last shown decision of whether the object is made of wood. The one chair for which our model decides it is not made of wood is ultimately incorrectly classified.
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Zero-shot learningthe good, the bad and the ugly. Y Xian, B Schiele, Akata , Z , IEEE Conference on Computer Vision and Pattern Recognition. Xian, Y., Schiele, B., and Akata, Z. Zero-shot learning - the good, the bad and the ugly. In IEEE Conference on Computer Vision and Pattern Recognition, (CVPR), 2017.
Interpretable convolutional neural networks. Q Zhang, Y Nian Wu, S.-C Zhu, IEEE Conference on Computer Vision and Pattern Recognition (CVPR). Zhang, Q., Nian Wu, Y., and Zhu, S.-C. Interpretable con- volutional neural networks. In IEEE Conference on Com- puter Vision and Pattern Recognition (CVPR), 2018a.
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Learning deep features for discriminative localization. B Zhou, A Khosla, À Lapedriza, A Oliva, A Torralba, IEEE Conference on Computer Vision and Pattern Recognition. Zhou, B., Khosla, A., Lapedriza,À., Oliva, A., and Torralba, A. Learning deep features for discriminative localization. In IEEE Conference on Computer Vision and Pattern Recognition, (CVPR), 2016.
Visualizing deep neural network decisions: Prediction difference analysis. L M Zintgraf, T S Cohen, T Adel, M Welling, Zintgraf, L. M., Cohen, T. S., Adel, T., and Welling, M. Visualizing deep neural network decisions: Prediction difference analysis. In ICLR, 2017.
| [] |
[
"SEISMIC NOISE MEASURES FOR UNDERGROUND GRAVITATIONAL WAVE DETECTORS",
"SEISMIC NOISE MEASURES FOR UNDERGROUND GRAVITATIONAL WAVE DETECTORS"
] | [
"Somlai L ",
"Gráczer Z ",
"Lévai P ",
"Vasúth M ",
"Wéber Z ",
"Ván P "
] | [] | [] | The selection of sites for underground gravitational wave detectors based on spectral and cumulative characterisation of the low frequency seismic noise. The evaluation of the collected long term seismological data in the Mátra Gravitational and Geophysical Laboratory revealed several drawbacks of the previously established characteristics. Here we demonstrate the problematic aspects of the recent measures and suggest more robust and more reliable methodology. In particular, we show, that the mode of the data is noisy, sensitive to the discretization and intrinsic averaging, and the rms 2Hz is burdened by irrelevant information and not adapted to the technological changes. Therefore the use of median of the data instead of the mode and also the modification of the frequency limits of the rms is preferable. | 10.1007/s40328-019-00257-5 | [
"https://arxiv.org/pdf/1810.06252v1.pdf"
] | 119,448,021 | 1810.06252 | e794664a9fce3b0f59cf6739653ba522348cd6e4 |
SEISMIC NOISE MEASURES FOR UNDERGROUND GRAVITATIONAL WAVE DETECTORS
Somlai L
Gráczer Z
Lévai P
Vasúth M
Wéber Z
Ván P
SEISMIC NOISE MEASURES FOR UNDERGROUND GRAVITATIONAL WAVE DETECTORS
The selection of sites for underground gravitational wave detectors based on spectral and cumulative characterisation of the low frequency seismic noise. The evaluation of the collected long term seismological data in the Mátra Gravitational and Geophysical Laboratory revealed several drawbacks of the previously established characteristics. Here we demonstrate the problematic aspects of the recent measures and suggest more robust and more reliable methodology. In particular, we show, that the mode of the data is noisy, sensitive to the discretization and intrinsic averaging, and the rms 2Hz is burdened by irrelevant information and not adapted to the technological changes. Therefore the use of median of the data instead of the mode and also the modification of the frequency limits of the rms is preferable.
Introduction
The improved sensitivity of future third generation gravitational wave (GW) detectors requires various technological developments. One of the plans is to optimize the facility for underground operation, in order to reduce the noise between the frequencies from 1 Hz to 10 Hz. According to the related sensitivity calculations the seismic and Newtonian noises represent the most important noise contributions in this frequency range [1,2]. During the preparatory studies of the so-called Einstein Telescope (ET), the European initiative, several short term seismic measurements were performed in various locations [3,4,6]. Based on these studies two performance measures were established: a spectral and a cumulative one. According to the spectral recommendation the average horizontal acceleration Amplitude Spectral Density should be smaller than the A BF limit, (1) A BF = 2 · 10 −8 m/s 2 √ Hz , in the region 1Hz ≤ f ≤ 10Hz.
This is the so-called Black Forest line, named after one of the investigated sites. This spectral criterion corresponds to a cumulative value, the square root of the displacement Power Spectral Density integrated from the Nyquist frequency down to 2 Hz, this is the rms 2Hz and its value for the Black Forest line is 0.1 nm.
In the ET survey the three best sites that fulfilled these requirements are the LSC Canfranc laboratory in Spain (rms 2Hz = 0.070 nm), the Sos Enattos mine in Sardinia, Italy (0.077 nm) and the Gyöngyösoroszi mine in Hungary (0.082 nm and 0.12 nm in depths 400 m and 70 m respectively). The data collection was performed up to a week at most in the various sites. In spite of the similar cumulative rms 2Hz values, the spectra of these sites is far from being uniform: the contributions of civilization noise, oceanic and sea waves appear in different frequency ranges and with different weights.
The Mátra Gravitational and Geophysical Laboratory (MGGL) has been operating since March 2016 with the purpose to evaluate and survey the Mátra mountain range as a possible ET candidate site. The primary goal of the laboratory is to collect seismic noise data for long period and evaluate them from the points of view of ET [7]. The laboratory is located at the coordinates (399 MAMSL, 47 • 52' 42.10178", 19 • 51' 57.77392" OGPSH 2007 (ETRS89)), along a horizontal tunnel of the mine, 1280 m from the entrance, 88 m depth from the surface. It is situated near to the less deep location of the above mentioned former short term measurements and it is prepared for long term data collection in a telemetric operation mode. In the laboratory a Guralp CMG-3T seismometer (hereafter referred as ET1H) was installed and has been operating continuously except shut downs which happens at strongly interfering mine activities (e.g. explosions). There is an ongoing reclamation activity in the mine and therefore the human activity is not negligible in recent years. The regular operation of the mine railway, the continuously working large water pumps in the vicinity of the laboratory and the related technical service and construction activities are producing industrial noise. These instruments will not be present in the future, especially during gravitational wave detection.
The ET related analysis of long term noise data revealed some particular aspects, that are not apparent in short term measurements, and could influence the operational conditions and detection possibilities of GWs in an underground location. Therefore we need to expose these effects for the optimal operation of the detector facilities. These are in particular the presence of various short term seismic disturbances with large amplitudes and the methodology of long term data evaluation.
The short time, large amplitude disturbances are unpredictable, unavoidable and must be left out to obtain reliable estimation of the average low noise level. However, any particular truncation or cutting process generates biases on the spectral and also the cumulative noise measures. To avoid these biases we suggest to use the percentiles of the complete data. The percentiles select the highest and lowest values, this selection is relative, and based on the intrinsic feature of the data set.
Any long term analysis and the evaluation of spectra and rms may require some intermediate averaging over the basic averaging length of the Fourier transformation. For this purpose here we suggest two different averaging steps:
(a) calculate short time averages (STA) to get manageable size of the data sets and to use the optimal time-scales of the planned detector. (b) calculate intermediate -for whole day, night or working periods i.e. natural periodicity of the data -percentiles and analyse the averages of them to study daily, annual, etc. variations. In the following we will call this intermediate or long time averaging as intrinsic averages (INA).
In particular the averaged daily percentiles of the complete data set can be used to estimate the spectral and cumulative variation of the data and the averaged daily median -the 50th percentile -values for the comparison with the Black Forest line. If the data collection period is longer than half year, then it is practical to use INA. The paper is organized as follows. First we shortly survey the evaluation procedure including its pitfalls, like the usage of mode for rms calculation. Then we analyse the effect of averaging on spectral and cumulative measures established by Beker et al [4]. After that the utility of INA is studied and we examine rms values with different frequency interval. Finally we conclude our experiences of calculation process and measurements and suggest further quantities to compare sites.
Data and data analysis
The new seismological data for our recent analysis were collected by a Guralp CMG 3T low noise, broadband seismometer, which is sensitive to ground vibrations with flat velocity response in the frequency range 0,008-50Hz. The self noise of the seismometer is below the New Low Noise Model of Peterson in the region 0.02 Hz to 10 Hz [5]. In this paper we study data collected by one instrument (ET1H). This station was permanently installed in the MGGL. The seismometer is deployed on a concrete pier which is connected to the bedrock. Between the pier and the seismometer a granite plate has been placed. The data collection period for ET1H has been started on 2016-03-01. In this paper we focus on methodology restricting the studied data period from 2017-01-01 until 2017-12-15 (349 days).
In our analysis we followed the data processing method of [4], e.g. the so-called Nuttal-window was applied with 3/4 overlap. In this section we recall the basic definitions. The Power Spectral Density (PSD) for the velocity is defined as
(2) P (v) = 2 f s · N · W |V k | 2 ,
where f s is the sampling rate, N is the length of the analysed data sample, and
W = 1 N N n=1 w[n] 2 with the Nuttall window function w[n]. The coefficients V k = F (w[n] · (v[n] − v )
, represent the Fourier transform F of the deviation of raw velocity data v[n] from its average value v . In our analysis PSDs were calculated with 50 s data samples. The choice of this sample length for Fourier transformation is a compromise between the frequency resolution of the spectra and the detectability of short noisy events. The resulted 0.02Hz resolution seems to be reasonably fine and we can reliably identify less than a second long seismic events. We did not use the advantage of fast Fourier algorithm on the expense of increasing the lowest frequency value 1 . Before further processing, raw data were highpass-filtered with f HP = 0.02 Hz.
Our STA is chosen to be 300 s. As we have mentioned above, the basic Fourier length is influenced by the sampling rate of the instrument. On the other hand for long term data the analysis can be easily adapted to the natural human and industrial noise periods. In the previous studies STA was 1800 s, which is natural with the basic 128 s Fourier length, considering the overlap. With our choice of STA, the comparison of the two analysis is with minimal bias, simply because 6 × 300 s = 1800 s ≈ 14 × 128 s.
The Amplitude Spectral Density (ASD) for the velocity can be calculated from
PSD via A (v) = √ P (v)
. Both amplitude and power spectral densities can be expressed also as either acceleration (a) or displacement (d) by multiplying or dividing by ω = 2 · π · f or the square of it respectively. For example, A (d) = A (v) /ω. Therefore the mentioned ET comparison level, (1), can be transferred easily to other 1 Other instruments, the Trillium seismometer of the previous study, work with 128 Hz sampling rate. Then the 128 s interval is convenient for fast Fourier calculation, but hourly or daily spectra require truncations. It is convenient to characterize sites in terms of acceleration ASD spectra and its variation and also by displacement rms as a single cumulative property. The displacement rms is the square root of the integral of displacement PSD between two frequency values (4) rms
(d) = 1 T N/2+1 k=l P (d) k ,
where l is the cutoff index, T = N fs . The usual choice is 2 Hz for comparing ET candidate sites [4]. The displacement power spectral density of the daily average of 2017-10-22 of ET1H station, East direction is shown on Figure 1. The Black Forest line is the solid straight line, the New Low Noise Model of Peterson (NLNM) [8] is the dashed one. The rms 2Hz is the square of the area of the shaded region.
For the comparison of various spectra of ET sites it is worth to show the Black Forest line, Eq. (1) and to recall the corresponding rms (d) = 0.1nm value at 2Hz:
rms (d) 2Hz = fs/2 2 P (a) BF 1 (2π) 4 1 f 4 df = A BF (2π) 2 fs/2 2 1 f 4 df ≈ A BF (2π) 2 [f −3 /(−3)] ∞ 2 ≈ 0.1nm,(5)
where we considered that the displacement PSD values decrease significantly at higher frequencies and expanded the domain of BF line to the infinity with the same value. See Figure 1 as an illustration, where it is obvious that in some cases higher frequencies can contribute significantly to the rms value. For the particular data shown in Figure 1 the rms 2Hz = 0.209 nm, the rms 2−10Hz = 0.144 nm and their ratio is 69.7%.
In the following only displacement rms will be used so the (d) superscript is omitted. Furthermore, two more rms will be considered: the rms 2−10Hz and the rms 1−10Hz . For the Black Forest line they are:
rms 2−10Hz = 10 2 P (a) BF 1 (2π) 4 1 f 4 df ≈ 0.1nm,(6)rms 1−10Hz = 10 1 P (a) BF 1 (2π) 4 1 f 4 df ≈ 0.29nm.(7)
In Section 5 we will show how both values can specify new information about the site.
Mode vs. median
In Beker et al. [4] the mode of the STA was used to characterize the typical noise level. Here we show that for long term data analysis the mode strongly depends on the discretization of the spectrum. In order to illustrate the differences, we use the same method as Beker [4] to determine modes. Only 7 days of data, between 2017-01-01 and 2017-01-07, was chosen for the recent analysis. The modes are shown together with the 10th, 50th and 90th percentiles of the half-hour averages on Figure 2 between 5 Hz and 7 Hz, with 1 dB and 0.1 dB bins. It is clear the fluctuation of the mode is discretisation dependent and larger than the fluctuation of the median. It is remarkable that the rms 5−7Hz -s are 0.0218nm, 0.0198nm and 0.0211nm for the median, mode 1 dB and 0.1 dB respectively. Our next step is to illustrate the advantage of median when considering different short time averaging (STA) lengths. A "well-behaving" characterization is expected to keep its profile for different STAs, in order to avoid process dependent artifacts. As it was mentioned, the STA is 300s in our case. In Beker's site selection study [4] approximately half-hour (1800s) STA was chosen. The differences between the expected values are illustrated in Figure 3. The mode is noisier than the median. The median is slightly increasing above 5 Hz with increasing STA.
For the median there is no need for noise level discretization and it is not sensitive to noise level distribution. In general it is a more stable quantity. The following simple example demonstrates this. Let us consider the half-hour PSD-s calculated for one-week interval. Then we have about 350 samples. Then a PSD bin with 0.1dB, and an 10dB difference between the 10th, 90th percentiles, we obtain 100 bins for calculating the mode. With a sufficiently uniform noise level distribution at a given frequency only 4 PSD values could define the mode. Then it is understandable, hat with several noise peaks with varying strength the mode can fluctuate violently. On the other hand the median characterizes the best/worst 50 % of the data, it is not sensitive of the form of the distribution and does not require power discretisation. Therefore, in the following analysis the use of median is preferred.
The effect of intrinsic averaging
To study long term -annual and seasonal -seismic noise variation and investigate site properties for the planned detector the use of intrinsic averages can also be necessary. Considering one year of data implies 365 days × 288 ST A ≈ 100 000 data point so the 90th percentile is defined by the 10 000 worst STAs. It could be a problem that one has not get any information about the density distribution: the 90th percentile is defined by just few noisy months or by three hours every day. Therefore intrinsic averaging (INA) is suggested to handle this difficulty and it can also be use to optimize the process whether discretization is omitted or not. The natural periodicity of the noise data indicates the use of daily averaging.
To illustrate it, we defined night period (00:00 -2:00 and 20:00 -24:00 UTC)in order to reduce the effect of human activity and focus on the noise changesand calculated the percentiles with and without INA in Figure 4. As it can be seen the medians have almost the same values for the whole interval, but either 10th or 90th percentiles show slightly different properties of the site. In general the use of intrinsic averaging shows small differences when compared to the evaluation without INA. The spectrum is slightly noisier with INA, therefore we cannot underestimate the noise level using that. On the other hand for large amount of data the analysis and the calculations are more convenient with INA.
The frequency range and cumulative statistics
Beker's original rms 2Hz compare sites with the help of a single parameter 2 , using a particular frequency range from 2Hz to the upper frequency determined by the speed of the data acquisition. However, the noise budget of the low frequency part of ET is more frequency dependent.
The term "seismic noise" covers two different aspects: the "original" seismic noise -the movement of Earth shakes the mirrors -and the Newtonian noise, or gravity gradient -the seismic activity causes perturbation in the local gravity field. The first one can be damped by passive filtering (e.g. by a suspension system) but the second one cannot, thus active filtering is necessary [2,9]. The seismic noise is relevant until 1 − 2Hz and the Newtonian, or gravity gradient noise is relevant above that frequency up ot 7 Hz according to ET low frequency sensitivity budget [3]. The exact values depend on the suspension system and the efficiency of the applied filtering methods.
Therefore it is reasonable to consider the modification of the frequency range for cumulative characterisation of ET sites. There are two aspects that influence our choice. First, the rare very noisy events at high frequencies, above 10Hz, are collected in rms 2Hz . This can be seen in Figure 5. Noises from this region eventuate irrelevant properties of the site, therefore high frequency cutoff is reasonable.
Also the low frequency limit is worth consideration. The recent cutoff at 2Hz was determined by the properties of the planned mirror suspensions. If one expects that mirror technology enables and science requires observations down to 1Hz, then the difference in the spectral properties of various sites must be characterized accordingly. Therefore we suggest to introduce suitable quantities and use also rms 2−10Hz and rms 1−10Hz for further site selection information. The referential Black Forest line values are given in Eqs. (7). To illustrate it in Figure 6 the normalized values of rms-s and they ratio are plotted. The figure indicates that there is a qualitative difference in the noisiness when lower frequencies are considered, otherwise one would expect an approximately constant ratio.
Conclusions
In the previous sections we have studied and characterised the specific aspects of long term low frequency seismological data evaluation in order to find the best site characterisation measures for Einstein Telescope. Our general observation is, that there are several sensitive aspects in the spectral and cumulative characteristics and in their calculation methods. The differences may become significant when the noise spectra are different and also, these performance measures are not the same from the point of view of potential ET requirements.
In order to reduce this sensitivity we suggest the following improvements in site characterisation measures (1) Use median instead of mode. Then we can omit the discretization and therefore its uncertainties and avoid STA sensitivity. Also the mode is unstable if the distribution of the data contains new peaks, a phenomenon observed several times in our data. The use of median provides a selection. (2) Use optimal STAs and INAs. That is advantageous for handling large amount of data. Moreover, the chosen interval length can be related directly to operational conditions and requirements of the low frequency part of the ET 3 . (3) Use both rms 2−10Hz and rms 1−10Hz . The upper limit in the frequency range removes the information from the rms which is irrelevant for the low frequency operation. The 1 − 10Hz frequency range enhances the lower frequency properties of the site. The suggested new rms measures are different because of the mode-median difference and the change of the frequency range. We illustrate the differences in Table 1, where the first row shows the reference values from the Black Forest line and the second row contains the values calculated from the 2017 data (349 days) of the ET1H station in the MGGL. Here the first column is calculated from the mode and the other columns from the median of the data. The mode related and median related values calculated from the same data are different. Usually the median is larger, but not necessarily. A detailed evaluation of the MGGL data is shown in [10]. 3 The planned detection length of GW signals could reach 1-10ks. It could be reasonable to use a more suitable -less than the order of expected detection length -than the 14x128s averaging of Beker et al [4]. Furthermore the STA makes the overlapping much easier to handle. It is also remarkable that the expected duration of gravitational wave signals can be considered already in site selection. For example it may be reasonable to choose the STA periods according to observational requirements. If one expects, that a continuous observation of a minimal length (e.g. 128s for a black hole merger) is suitable, then the percentiles of low level averaging directly characterize the observation capabilities of the particular site.
We have seen that several seemingly minor aspects of the noise measures (e.g. the width of the noise levels in the mode calculations) may introduce different numbers and spectra, emphasizing different properties of the overall noisiness. Long periods are more sensitive to these aspects than short ones.
Acknowledgement
Figure 1 .
1Illustration of rms at a displacement PSD spectrum (blue line) together with the Black Forest line (solid black) and the New Low Noise Model of Peterson (dashed black). The filled area represents rms 2 2Hz .spectral densities, e.g. for the Black Forest line:
Figure 2 .
2In this figure the median (solid blue line) and the modes (dashed and dotted lines) with 1 dB and 0.1 dB bins are compared. The solid black horizontal line represents the Black Forest line and the blue area indicates the 10th-90th percentiles.
Figure 3 .
3Upper figure displays the median with different short time averaging lengths. Lower figure shows the mode of the same data with the same different STA lengths. The Black Forest line is solid black. The data is from the first week of January in 2017.
Figure 4 .
4Here the differences of spectra with and without intrinsic averaging (INA) are shown. Red curves belong to no INA and green ones to the averages of night percentiles -averages of 10th, and 90th percentiles are the upper and lower limits of the shaded area. The median is shown with green and red lines in the middle.
Figure 5 .
5The upper figure displays the daily rms 2Hz (blue) toghether with rms 2−10Hz (red). The lower figure shows the ratio of the rms 2Hz and rms 2−10Hz . The data is calculated from daily averages at the North direction of each day in 2017.
Figure 6 .
6The upper figure displays the normalized rms 2−10Hz (blue) and rms 1−10Hz values (red) of the daily averages in 2017 in the North direction. The lover figure shows the ratio of the same normalized rms 2−10Hz and rms 1−10Hz . In both cases the normalization is made by the Black Forest values.
rms 2Hz (mode) rms 2Hz rms 2−10Hz rms 1−10Hz Black Forest line [nm] Table 1. The various suggested rms values for the Black Forest line and calculated from the 2017 data of the ET1H station.0.1
0.1
0.1
0.29
ET1H 2017 [nm]
0.136
0.153
0.152
0.502
Beker originally defined sigma_ET also to distinguish the distributions of PSDs. In this paper we do not want to explore this quantity but focus only to the rms.
The work was supported by the grants National Research, Development and Innovation Office -NKFIH 116197(116375) NKFIH 124366(124508) and NKFIH 123815. The support of the PHAROS (CA16214) and G2net (CA17137) COST Actions is also acknowledged. The authors thank Géza Huba for the constant support and help, for Zoltán Zimborás and Jan Harms for important remarks. Also the help and support of the Nitrokémia Zrt and GEO-FABER Zrt. is greatly acknowledged.
Einstein telescope site selection: Seismic and gravity gradient noise. Jfj Van Den Brand, M Beker, Doets, D S Hennes, Rabeling, Journal of Physics: Conference Series. IOP Publishing20312076JFJ van den Brand, MG Beker, M Doets, E Hennes, and DS Rabeling. Einstein telescope site selection: Seismic and gravity gradient noise. In Journal of Physics: Conference Series, volume 203, page 012076. IOP Publishing, 2010.
Terrestrial gravity fluctuations. Living reviews in relativity. Jan Harms, 183Jan Harms. Terrestrial gravity fluctuations. Living reviews in relativity, 18(1):3, 2015.
ET-0106C-10Conceptual Design Study. ET Science Team. Einstein gravitational wave TelescopeTechnical ReportET Science Team. Einstein gravitational wave Telescope, Conceptual Design Study. Technical Report ET-0106C-10, June 2011. http://www.et-gw.eu/etdsdocument.
Subterranean ground motion studies for the Einstein Telescope. M G Beker, J Van Den Brand, D S Rabeling, Classical and Quantum Gravity. 32225002M G Beker, J F J van den Brand, and D S Rabeling. Subterranean ground motion studies for the Einstein Telescope. Classical and Quantum Gravity, 32(2):025002, 2015.
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First report of long term measurements of the MGGL laboratory in the Mátra mountain range. G G Barnaföldi, T Bulik, M Cieslar, E Dávid, M Dobróka, E Fenyvesi, Z Gráczer, G Hamar, G Huba, Á Kis, R Kovács, I Lemperger, P Lévai, J Molnár, D Nagy, A Novák, L Oláh, P Pázmándi, D Piri, D Rosinska, L Somlai, T Starecki, M Suchenek, G Surányi, S Szalai, D Varga, M Vasúth, P Ván, B Vásárhelyi, V Wesztergom, Z Wéber, arXiv:1610.07630Classical and Quantum Gravity. 34222017G.G. Barnaföldi, T. Bulik, M. Cieslar, E. Dávid, M. Dobróka, E. Fenyvesi, Z. Gráczer, G. Hamar, G. Huba, Á. Kis, R. Kovács, I. Lemperger, P. Lévai, J. Molnár, D. Nagy, A. Novák, L. Oláh, P. Pázmándi, D. Piri, D. Rosinska, L. Somlai, T. Starecki, M. Suchenek, G. Surányi, S. Szalai, D. Varga, M. Vasúth, P. Ván, B. Vásárhelyi, V. Wesztergom, and Z. Wéber. First report of long term measurements of the MGGL laboratory in the Mátra mountain range. Classical and Quantum Gravity, 34:114001(22), 2017. arXiv: 1610.07630.
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Long term measurements from Matra Gravitational and Geophysical Laboratory. P Ván, under publicationP. Ván, et al. Long term measurements from Matra Gravitational and Geophysical Labora- tory, under publication
| [] |
[
"The environment of graphene probed by electrostatic force microscopy",
"The environment of graphene probed by electrostatic force microscopy"
] | [
"J Moser \nCIN2\nCNM Barcelona\nCampus UABE-08193BellaterraSpain\n",
"A Verdaguer \nICN\nCampus UABE-08193BellaterraSpain\n",
"D Jiménez \nDepartament d'Enginyeria Electrònica\nEscola Tècnica Superior d'Enginyeria\nUniversitat Autònoma de Barcelona\nE-08193BellaterraSpain\n",
"A Barreiro \nCIN2\nCNM Barcelona\nCampus UABE-08193BellaterraSpain\n",
"A Bachtold \nCIN2\nCNM Barcelona\nCampus UABE-08193BellaterraSpain\n"
] | [
"CIN2\nCNM Barcelona\nCampus UABE-08193BellaterraSpain",
"ICN\nCampus UABE-08193BellaterraSpain",
"Departament d'Enginyeria Electrònica\nEscola Tècnica Superior d'Enginyeria\nUniversitat Autònoma de Barcelona\nE-08193BellaterraSpain",
"CIN2\nCNM Barcelona\nCampus UABE-08193BellaterraSpain",
"CIN2\nCNM Barcelona\nCampus UABE-08193BellaterraSpain"
] | [] | We employ electrostatic force microscopy to study the electrostatic environment of graphene sheets prepared with the micro-mechanical exfoliation technique. We detect the electric dipole of residues left from the adhesive tape during graphene preparation, as well as the dipole of water molecules adsorbed on top of graphene. Water molecules form a dipole layer that can generate an electric field as large as ∼ 10 9 V · m −1 . We expect that water molecules can significantly modify the electrical properties of graphene devices. | 10.1063/1.2898501 | [
"https://arxiv.org/pdf/0803.2032v1.pdf"
] | 119,193,676 | 0803.2032 | 57e55f82dd15931b8f13aba42f1690aeb8a7412d |
The environment of graphene probed by electrostatic force microscopy
13 Mar 2008
J Moser
CIN2
CNM Barcelona
Campus UABE-08193BellaterraSpain
A Verdaguer
ICN
Campus UABE-08193BellaterraSpain
D Jiménez
Departament d'Enginyeria Electrònica
Escola Tècnica Superior d'Enginyeria
Universitat Autònoma de Barcelona
E-08193BellaterraSpain
A Barreiro
CIN2
CNM Barcelona
Campus UABE-08193BellaterraSpain
A Bachtold
CIN2
CNM Barcelona
Campus UABE-08193BellaterraSpain
The environment of graphene probed by electrostatic force microscopy
13 Mar 2008(Dated:)arXiv:0803.2032v1 [cond-mat.other]
We employ electrostatic force microscopy to study the electrostatic environment of graphene sheets prepared with the micro-mechanical exfoliation technique. We detect the electric dipole of residues left from the adhesive tape during graphene preparation, as well as the dipole of water molecules adsorbed on top of graphene. Water molecules form a dipole layer that can generate an electric field as large as ∼ 10 9 V · m −1 . We expect that water molecules can significantly modify the electrical properties of graphene devices.
Graphene, a two-dimensional crystal of carbon atoms arranged in a honeycomb lattice, is among the thinnest objects imaginable [1,2]. The structural properties of graphene make it a system of choice to study the physics of Dirac fermions [3,4]; it is also envisioned as a building block for a novel generation of electronic devices. One inherent technological difficulty however remains: because all the atoms are at the surface and are directly exposed to the environment, the electronic properties are easily affected by unwanted adsorbates.
In this Letter, we employ electrostatic force microscopy (EFM) to probe the electrostatic environment of graphene sheets. Two adsorbate species possessing an electric dipole are identified: water molecules and residues left from the adhesive tape during fabrication. Water molecules form a dipole layer on top of graphene that can generate an electric field as large as ∼ 10 9 V · m −1 . Tape residues form an ultra-thin layer nearby the graphene sheets on top of the silicon oxide substrate, which is not detectable using standard topographic atomic force microscopy.
We start by briefly describing our fabrication technique and our experimental setup. Graphene sheets are obtained using the conventional micro-mechanical exfoliation technique [5,6]: a flake of bulk Kish graphite is cleaved repeatedly with an adhesive tape and pressed down onto a silicon wafer coated with 280 nm of thermal silicon oxide. Two types of adhesive tape are used: standard wafer protection tape for microfabrication by ICROS and Magic Tape by 3M. Thin graphene sheets are located optically, imaged by atomic force microscopy, and occasionally examined by Raman spectroscopy to identify single layer specimens. EFM measurements are carried out under various humidity conditions in a constant flow of either dry N 2 gas, or moist N 2 gas produced by bubbling dry N 2 in deionized water.
The EFM technique [7] is well suited to study dipoles on surfaces. Our EFM protocol proceeds in two passes along a given scan line: first the topography is recorded in tapping mode ( Fig. 1(a,b)), then the AFM tip is lifted by a given amount and a bias V dc is applied to the Si backgate and a potential V ac cos(ωt) is applied to the tip. The tip experiences a force whose term at the frequency ω reads ( Fig. 1(c)):
F ω = ∂C ∂z (V dc − ∆φ)V ac(1)
where C and ∆φ are the capacitance and the contact potential difference between the sample and the tip. We measure F ω over a range of V dc ( Fig. 2(a) and Fig. 3(ac)). As in Kelvin probe force microscopy, ∆φ is given by V dc that minimizes F ω . Prior to discussing the measurements, we recall the connection between ∆φ and the presence of dipoles on surfaces. The energy eφ is the energy barrier that an electron has to overcome in order to be extracted from a material to the vacuum. Any electric field existing at the surface of this material, originating e.g. from dipoles, modifies the contact potential φ by an amount χ: φ = W + χ, where W is the contact potential without electric field at the surface. For further insight into the underlying physics of ∆φ, see Refs. [8,9]. In our experiment, the contact potential of the tip is φ tip , and the one of the Si wafer depends on whether the graphene sheet is present or not:
φ graphene Si = W Si + χ graphene (2) φ no graphene Si = W Si + χ no graphene(3)
We first look into the electrostatic traces left by the adhesive tape. In the case of a SiO 2 surface onto which the adhesive tape has been pressed (without graphite), EFM measurements reveal that the contact potential difference ∆φ = φ no graphene Si −φ tip is significantly shifted to negative values (Fig. 2(a)). ∆φ is found to vary between -2 V and -0.5 V depending upon where on the wafer the measurement is carried out. This has to be compared with ∆φ ≃ 0 V for a pristine wafer. Both adhesive tapes that we used yield similar results. This shift in ∆φ suggests that adhesive residues change χ no graphene , which we attribute to the deposition of dipoles on the surface. Note that the shift in ∆φ is stable over long periods of time, which suggests that it is not related to individual charges that would get neutralized, for example, by charged molecules in the environment.
These tape residues do not seem to roughen the surface, as inferred from topography measurements ( Fig. 2(b,c)). Presumably this is why they have not been reported so far, as a standard topography scan fails to detect them. Further studies are needed, especially to find out whether or not these residues are present underneath the graphene sheets. At first sight, they should not, as the SiO 2 surface is masked by the graphene sheets when pressing the tape down onto the wafer. However, micromechanical exfoliation is rather difficult to control, and residues may well lie beneath graphene sheets. The latter situation could have important consequences on the transport properties of graphene devices (see discussion below for water). We now turn to EFM measurements on graphene sheets where we modify the humidity of the environment. Fig. 3 compares measurements on a single graphene layer and on the oxide ∼ 1µm away. The measurement is first performed in dry N 2 with a relative humidity RH of less than 3%, thus corresponding to a submonolayer water coverage on clean SiO 2 (Fig. 3(a)) [10]. Without taking it out of the dry environment, the sample is heated to 160 • C for 1 hour and measured again (Fig. 3(b)). Eventually, moist N 2 is introduced until RH≃ 50% (∼ 3 water monolayers on clean SiO 2 [10]), see Fig. 3(c). ∆φ on the graphene sheet is observed to vary between each step significantly (−2.1 V → −0.7 V → −1.9 V) [11]. By contrast, ∆φ measured on the oxide stays pretty much constant (∼ −2 V), which suggests that φ tip , W , and χ no graphene are not affected much by humidity [12]. Therefore the quantity that appears to be sensitive to water is χ graphene , which changes by ∼ 1.3 V from one step to another in Fig. 3. We attribute this strong variation to water molecules that desorb from and readsorb onto the graphene sheet [13].
The adsorption of water on graphene may appear surprising, as it is well established that graphite is highly hydrophobic [14]. However, it has been reported that water can adsorb on graphite for RH≃ 60% at the surface defects such as steps [15]. It has also been reported that water can adsorb on carbon structures such as carbon nanotubes [16] and on self-assembled monolayers (SAM) of carbon chains [17]. It has been argued that the key ingredient in water adsorption on SAM is the surface roughness. Interestingly, the surface of graphene is also corrugated as it tends to follow the roughness of the substrate [18,19].
Because ∆φ = φ graphene Si − φ tip shifts to more negative values as humidity is raised, water molecules adsorb on average with the oxygen atoms pointing towards the graphene sheet. As such, the layer of water molecules can be described as a dipole layer with the negative charges towards the graphene sheet (see Fig. 3(d)). The electric field is maximum within the dipole layer, and vanishes to zero as e −az as the distance z to the layer is increased (in the particular case where the layer consists of a periodic array of dipoles that are all pointing in the same direction, a is the distance between two neighboring dipoles).
We can obtain a rough estimate for the strength of the electric field E within the water dipole layer. Assuming that the width of the dipole layer is d = 1 nm, we get E ≃ ∆χ graphene /d ≃ 10 9 V · m −1 , where ∆χ graphene ≃ 1 V is the shift in χ graphene from dry to wet environment. As a comparison, we calculate E to be of the order of 10 9 to 10 10 V · m −1 using the water electric dipole 6.2 · 10 −30 Cm, assuming a monolayer of water molecules that are all pointing in the direction perpendicular to the surface with a density from 1 to 10 nm −2 , and modeling the dipole layer as a parallel plate capacitor. The strength of E changes somewhat when considering that the dipoles are not all oriented in the same direction or can flip in the field.
An electric field of 10 9 V · m −1 within the dipole layer corresponds to the strength obtained when applying ≃ 300 V between the Si backgate and a graphene device. It is important to note that the graphene sheet experiences a fraction of this field only, as it lies outside of the dipole layer. Nevertheless, we expect that E will significantly shift the Fermi level of graphene devices [20]. Moreover, this field is likely to be inhomogeneous, resulting in puddles of electrons and holes near the charge neutrality point [21]. Another consequence is that water can screen charge impurities [22] as well as modify electron-electron interactions.
In conclusion, we show that EFM is a powerful tool for the characterization of the electrostatic environment of graphene. Water molecules form a dipole layer on top of graphene that generates a large electric field. We expect water to have a strong influence on the transport properties of graphene devices.
FIG. 1 :
1(color online) (a) Schematic of the EFM setup. (b) In a first pass, the topography of the graphene sheet is recorded. (c) In a second pass, a dc voltage V dc is applied to the Si substrate and an ac voltage Vac = 1 mV is applied to the tip at a frequency set to the resonance frequency of the cantilever. The sample is scanned at a constant height of about 50 nm and the oscillating force is measured.
FIG. 2 :
2(color online) (a) Force term Fω experienced by the AFM tip as a function of V dc on a pristine SiO2 surface and on SiO2 covered with tape residues. (b-c) Topography of a typical graphene sheet. Plots in (c) are sections along the red and blue lines depicted in (b).
FIG. 3 :
3(color online) Force term Fω experienced by the AFM tip as a function of V dc in (a) dry N2; (b) dry N2 after heating the sample to 160 • C for 1 hour in dry N2; (c) moist N2. Data on graphene and on silicon oxide ∼ 1µm away from graphene are shown. The dotted line at V dc = −2 V draws the attention to the rather weak dependence on humidity of Fω(V dc ) measured on the oxide. (d) Water molecules adsorb on average with the oxygen atom pointing towards the graphene sheet, and form a dipole layer with an effective surface charge σ (schematic not on scale)
* Electronic address: [email protected] † Electronic address: adrian. [email protected]* Electronic address: [email protected] † Electronic address: [email protected]
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This is attributed to the graphene sheet that changes the capacitance between the tip and the Si substrate. In principle, it should be possible to extract the dielectric constant ǫG of the graphene sheet. We numerically solved the Poisson equation and found that ∂C/∂z has a very weak dependence on ǫG when ǫG > ∼ 2. As a result. The slope of Fω(V dc ) is different on the oxide and on the graphene sheet. estimating the dielectric constant of graphene is difficultThe slope of Fω(V dc ) is different on the oxide and on the graphene sheet. This is attributed to the graphene sheet that changes the capacitance between the tip and the Si substrate. In principle, it should be possible to extract the dielectric constant ǫG of the graphene sheet. We numerically solved the Poisson equation and found that ∂C/∂z has a very weak dependence on ǫG when ǫG > ∼ 2. As a result, estimating the dielectric constant of graphene is difficult.
Note that χ no graphene varies slightly as humidity is changed. The size and the sign of the variation are consistent with previous work on SiO2 surfaces contaminated with a submonolayer of carbon. 10Note that χ no graphene varies slightly as humidity is changed. The size and the sign of the variation are consis- tent with previous work on SiO2 surfaces contaminated with a submonolayer of carbon [10].
Note that the wafer was exposed to air during manipulation before these measurements, at which point water and other molecules were likely to adsorb on the graphene sheet. Note that the wafer was exposed to air during manipu- lation before these measurements, at which point water and other molecules were likely to adsorb on the graphene sheet.
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| [] |
[
"A Note on the Complexity of Computing the Smallest Four-Coloring of Planar Graphs *",
"A Note on the Complexity of Computing the Smallest Four-Coloring of Planar Graphs *"
] | [
"André Große ",
"Jörg Rothe ",
"Gerd Wechsung ",
"\nInstitut für Informatik Heinrich-Heine\nInstitut für Informatik Friedrich-Schiller-Universität Jena\n07740JenaGermany\n",
"\nInstitut für Informatik Friedrich-Schiller\nUniversität Düsseldorf\n40225DüsseldorfGermany\n",
"\nUniversität Jena\n07740JenaGermany\n"
] | [
"Institut für Informatik Heinrich-Heine\nInstitut für Informatik Friedrich-Schiller-Universität Jena\n07740JenaGermany",
"Institut für Informatik Friedrich-Schiller\nUniversität Düsseldorf\n40225DüsseldorfGermany",
"Universität Jena\n07740JenaGermany"
] | [] | We show that computing the lexicographically first four-coloring for planar graphs is ∆ p 2 -hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem NP-hard, and conclude that it is not self-reducible in the sense of Schnorr, assuming P = NP. We discuss this application to non-self-reducibility and provide a general related result.In this note, we raise Khuller and Vazirani's NP-hardness lower bound for computing the lexicographically smallest four-coloring of a planar graph to ∆ p 2 -hardness. Our result is optimal, since this problem belongs to (the function analog of) the class ∆ p 2 . The class ∆ p 2 = P NP , which belongs to the second level of the polynomial hierarchy [MS72, Sto77], contains exactly the problems solvable in deterministic polynomial time with an NP oracle. Papadimitriou [Pap84] proved that Unique-Optimal-Traveling-Salesperson is ∆ p 2 -complete, and Krentel [Kre88] and Wagner [Wag87] established many more ∆ p 2 -completeness results, including the result that the problem Odd-Max-SAT is ∆ p 2 -complete. The complexity of colorability problems has been studied in a number of papers, see, e.g., [AH77a, AH77b, Sto73, GJS76, Wag87, KV91, Rot03].As mentioned above, if for some problem in P computing the lexicographically smallest solution is hard, then the problem itself cannot be self-reducible in the sense of Schnorr [Sch76, Sch79], unless P = NP. We discuss this application to non-self-reducibility and provide a general related result. In particular, it follows from this result that even a set as simple as Σ * has representations in which it is not self-reducible in Schnorr's sense, unless P = NP. | null | [
"https://arxiv.org/pdf/cs/0106045v2.pdf"
] | 3,206,262 | cs/0106045 | 7b78bc741886cbaf010abdd5eabc7868e0e5314e |
A Note on the Complexity of Computing the Smallest Four-Coloring of Planar Graphs *
7 Feb 2006 February 7, 2006
André Große
Jörg Rothe
Gerd Wechsung
Institut für Informatik Heinrich-Heine
Institut für Informatik Friedrich-Schiller-Universität Jena
07740JenaGermany
Institut für Informatik Friedrich-Schiller
Universität Düsseldorf
40225DüsseldorfGermany
Universität Jena
07740JenaGermany
A Note on the Complexity of Computing the Smallest Four-Coloring of Planar Graphs *
7 Feb 2006 February 7, 2006complexity of smallest solutionsself-reducibilitygraph colorability
We show that computing the lexicographically first four-coloring for planar graphs is ∆ p 2 -hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem NP-hard, and conclude that it is not self-reducible in the sense of Schnorr, assuming P = NP. We discuss this application to non-self-reducibility and provide a general related result.In this note, we raise Khuller and Vazirani's NP-hardness lower bound for computing the lexicographically smallest four-coloring of a planar graph to ∆ p 2 -hardness. Our result is optimal, since this problem belongs to (the function analog of) the class ∆ p 2 . The class ∆ p 2 = P NP , which belongs to the second level of the polynomial hierarchy [MS72, Sto77], contains exactly the problems solvable in deterministic polynomial time with an NP oracle. Papadimitriou [Pap84] proved that Unique-Optimal-Traveling-Salesperson is ∆ p 2 -complete, and Krentel [Kre88] and Wagner [Wag87] established many more ∆ p 2 -completeness results, including the result that the problem Odd-Max-SAT is ∆ p 2 -complete. The complexity of colorability problems has been studied in a number of papers, see, e.g., [AH77a, AH77b, Sto73, GJS76, Wag87, KV91, Rot03].As mentioned above, if for some problem in P computing the lexicographically smallest solution is hard, then the problem itself cannot be self-reducible in the sense of Schnorr [Sch76, Sch79], unless P = NP. We discuss this application to non-self-reducibility and provide a general related result. In particular, it follows from this result that even a set as simple as Σ * has representations in which it is not self-reducible in Schnorr's sense, unless P = NP.
Introduction
Khuller and Vazirani [KV91] proved an NP-hardness lower bound for computing the lexicographically first solutions of the planar graph four colorability problem, which we denote by Pl-4-Color. It follows from their result that, assuming P = NP, the polynomial-time decidable problem Pl-4-Color is not self-reducible in the sense of Schnorr [Sch76,Sch79]. Noting that their result appears to be the first such non-self-reducibility result for problems in P, they proposed as an interesting task to find other problems in P that are not self-reducible under some plausible assumption.
Computing the Smallest Four-Coloring of a Planar Graph
Solving the famous Four Color Conjecture in the affirmative, Appel and Haken [AH77a,AH77b] showed that every planar graph can be colored with no more than four colors. In contrast, for each k ≥ 4, computing the lexicographically first k-coloring of a planar graph is hard: Khuller and Vazirani [KV91] established an NP-hardness lower bound for this problem. We raise their lower bound to ∆ p 2 -hardness. Since the lexicographically smallest k-coloring of a planar graph can be computed in (the function analog of) ∆ p 2 , this improved lower bound is optimal.
Definition 2.1 Let k > 1, and let 0, 1, . . . , k − 1 represent k colors.
• A k-coloring of an undirected graph G = (V, E) is a mapping ψ G : V → {0, 1, . . . , k − 1}.
• A k-coloring ψ G is said to be legal if and only if for each edge {u, v} ∈ E, ψ G (u) = ψ G (v).
• A graph G is said to be k-colorable if and only if there exists a legal k-coloring of G.
• Let Pl-k-Color denote the planar graph k-colorability problem.
Stockmeyer [Sto73] proved that Pl-3-Color is NP-complete, see also Garey et al. [GJS76]. By Appel and Haken's above-mentioned result, every planar graph is four-colorable. Thus, Pl-k-Color is in P for each k ≥ 4.
Definition 2.2 (Khuller and Vazirani [KV91])
Let k > 1, and let the vertex set of a given undirected graph G = (V, E) with n vertices be ordered as V = {v 1 , v 2 , . . . , v n }. Then, every kcoloring ψ G of G can be represented by a string ψ G in {0, 1, . . . , k − 1} n :
ψ G = ψ G (v 1 )ψ G (v 2 ) · · · ψ G (v n ).
Define the lexicographically smallest (legal) k-coloring by
LF Pl-k-Color (G) = min{ψ G | ψ G is a legal k-coloring of G}, if G ∈ Pl-k-Color,
where the minimum is taken with respect to the lexicographic ordering of strings, and define LF Pl-k-Color (G) = 10 n if G ∈ Pl-k-Color.
We now prove our main result.
Theorem 2.3
Computing the lexicographically smallest k-coloring for planar graphs is ∆ p 2 -hard for any k ≥ 4.
Proof. For simplicity, we show this claim only for k = 4. Let ρ 4 be the reduction of Khuller and Vazirani [KV91, Theorem 3.1]. Recall that ρ 4 maps a given planar graph G = (V, E), whose vertices are ordered as V = {v 1 , v 2 , . . . , v m }, to the planar graph H = (U, F ) defined as follows:
• The vertex set of H is ordered as U = {u 1 , u 2 , . . . , u 2m }, where u i is a new vertex and u m+i = v i is an old vertex for each i, 1 ≤ i ≤ m.
• The edge set of H is defined by
F = E ∪ {{u i , u m+i } | 1 ≤ i ≤ m}.
It follows immediately from this construction that
(1) G ∈ Pl-3-Color ⇐⇒ LF Pl-4-Color (ρ 4 (G)) ∈ {0 m w | w ∈ {1, 2, 3} m },
that is, "G ∈ Pl-3-Color?" can be decided by looking at the first m bits of LF Pl-4-Color (H).
We give a reduction from the problem Odd-Min-SAT, which is defined to be the set of all boolean formulas F = F (x 1 , x 2 , . . . , x n ) in conjunctive normal form for which, assuming F is satisfiable, the lexicographically smallest satisfying assignment α : {x 1 , x 2 , . . . , x n } → {1, 2} is "odd," i.e., for which α(x n ) = 1. Here, "1" represents "true," and "2" represents "false."
It is well known that Odd-Min-SAT is ∆ p 2 -complete; Krentel [Kre88] and also Wagner [Wag87] proved the corresponding claim for the dual problem Odd-Max-SAT.
Let F = F (x 1 , x 2 , . . . , x n ) be any given boolean formula. Without loss of generality, we may assume that F is in conjunctive normal form with exactly three literals per clause. Assume that F has z clauses. Let σ be the Stockmeyer reduction from 3-SAT to Pl-3-Color, see Stockmeyer [Sto73] and also Garey et al. [GJS76]. This reduction σ, on input F , yields a graph G = (V, E) with m > n vertices, where m = m(F ) depends on the number n of variables, the number z of clauses, and the structure of F . Note that F 's structure induces a certain number of "crossovers" of edges to guarantee the planarity of G; see [GJS76,Sto73] for details.
Order the vertex set of G as
V = {v 1 , v 2 , . . . , v m } such that (a) for each i, 1 ≤ i ≤ n, v i represents the variable x i , and (b) for each i, n < i ≤ m, v i represents some other vertex of G.
Note that G is a planar graph satisfying that (i) F is satisfiable if and only if G is 3-colorable, using the colors 1, 2, and 3, and (ii) every satisfying assignment α of F corresponds to a 3-coloring ψ α of G such that for each i,
1 ≤ i ≤ n, ψ α (v i ) = α(v i ) ∈ {1, 2}.
The color 3 is used for the other vertices of G.
Now apply the reduction ρ 4 of Khuller and Vazirani to G and obtain a planar graph H = ρ 4 (G) = ρ 4 (σ(F )) that satisfies Equation (1) as described above. It follows immediately from this construction and from Equation (1) that
F ∈ Odd-Min-SAT ⇐⇒ LF Pl-4-Color (ρ 4 (σ(F ))) ∈ {0 m w1y | w ∈ {1, 2} n−1 and y ∈ {1, 2, 3} m−n },
that is, "F ∈ Odd-Min-SAT?" can be decided by looking at the first m bits and at the (m + n)th bit of LF Pl-4-Color (H).
For k > 4, the claim of the theorem follows from an analogous argument that employs in place of ρ 4 the appropriate reduction ρ k from [KV91, Thm. 3.2].
Non-Self-Reducible Sets in P
From their NP-hardness lower bound for computing the lexicographically first four-coloring of planar graphs, Khuller and Vazirani [KV91] conclude that, unless P = NP, the polynomial-time decidable problem Pl-k-Color is not self-reducible for k ≥ 4. The type of (functional) self-reducibility used by Khuller and Vazirani is due to Schnorr [Sch76,Sch79], see also [BD76]. For more background on self-reducibility, see, e.g., [Sel88,JY90,Rot05].
Definition 3.1 (Schnorr [Sch76, Sch79])
• Let Σ and Γ be alphabets with at least two symbols each. Instances of problems are encoded over Σ, and solutions of problems are encoded over Γ. For any set B ⊆ Σ * × Γ * and any polynomial p, the p-projection of B is defined to be the set
proj p (B) = {x ∈ Σ * | (∃y ∈ Γ * ) [|y| ≤ p(|x|) and (x, y) ∈ B]}.
• A partial order ≤ on Σ * is polynomially well-founded and length-bounded if and only if there exists a polynomial q such that (a) every ≤-decreasing chain with maximum element x has at most q(|x|) elements, and (b) for all strings x, y ∈ Σ * , x < y implies |x| ≤ q(|y|).
• Let A = proj p (B) for some set B ⊆ Σ * × Γ * and some polynomial p. The projection A is said to be self-reducible with respect to (B, p) if and only if there exist a polynomial-time computable function g mapping from Σ * × Γ to Σ * and a polynomially well-founded and length-bounded partial order ≤ such that for all strings x ∈ Σ * , for all strings y ∈ Γ * , and for all symbols γ ∈ Γ,
(i) g(x, γ) < x, and (ii) (x, γy) ∈ B ⇐⇒ (g(x, γ), y) ∈ B.
If the pair (B, p) for which A = proj p (B) is clear from the context, we omit the phrase "with respect to (B, p)."
We mention in passing that various other important types of self-reducibility have been studied, such as the self-reducibility defined by Meyer and Paterson [MP79] and the disjunctive selfreducibility studied by Selman [Sel88], Ko [Ko83], and many others. We refer the reader to the excellent survey by Joseph and Young [JY90] for an overview and for pointers to the literature. Note that, in sharp contrast with Schnorr's self-reducibility, every set in P is self-reducible in the sense of Meyer and Paterson [MP79], Ko [Ko83], and Selman [Sel88].
Definition 3.2 Let Σ = {0, 1}. Given any set A in NP with A ⊆ Σ * , there is an associated set B A ⊆ Σ * × Σ * and an associated polynomial p A such that B A is in P and A = proj p A (B A ).
• For any x ∈ Σ * , define the set of solutions for x with respect to B A and p A by
Sol (B A ,p A ) (x) = {y ∈ Σ * | |y| ≤ p A (|x|) and (x, y) ∈ B A }. Note that x ∈ A if and only if Sol (B A ,p A ) (x) = ∅.
• For any x ∈ Σ * , define the lexicographically first solution with respect to B A and p A by
LF (B A ,p A ) (x) = min Sol (B A ,p A ) (x) if x ∈ A bin(2 p(|x|) ) otherwise,
where the minimum is taken with respect to the lexicographic ordering of Σ * , and bin(n) denotes the binary representation of the integer n without leading zeros.
If the pair (B A , p A ) for which A = proj p A (B A ) is clear from the context, we use Sol A (x) and LF A (x) as shorthands for respectively Sol (B A ,p A ) (x) and LF (B A ,p A ) (x).
It is well known that if A is self-reducible then LF A can be computed in polynomial time by prefix search, via suitable queries to the oracle A. Moreover, if A is in P then LF A can even be computed in polynomial time without any oracle queries. It follows that if computing LF A is NPhard then A cannot be self-reducible, assuming P = NP.
Khuller and Vazirani [KV91] propose to prove polynomial-time decidable problems other than Pl-4-Color non-self-reducible, under the assumption P = NP. As Theorem 3.5 below, we provide a general result showing that it is almost trivial to find such problems: For any NP problem A for which LF A is hard to compute, one can define a P-decidable version D of A such that LF D is still hard to compute; hence, D is not self-reducible, assuming P = NP.
To formulate this result, we now define the functional many-one reducibility that was introduced by Vollmer [Vol94] as a stricter reducibility notion than Krentel's metric reducibility [Kre88]. We also define the function class min ·P that was introduced by Hempel and Wechsung [HW00].
Definition 3.3 (Vollmer [Vol94])
Let f and h be functions from Σ * to Σ * .
• We say f is polynomial-time functionally many-one reducible to h (in symbols, f ≤ FP m h) if and only if there exists a polynomial-time computable function g such that for all x ∈ Σ * , f (x) = h(g(x)).
• We say h is ≤ FP m -hard for a function class C if and only if for where ·, · : Σ * × Σ * → Σ * is a standard pairing function. If the set over which the minimum is taken is empty, define by convention f (x) = bin(2 p(|x|) ).
every f ∈ C, f ≤ FP m h. • We say h is ≤ FP m -complete for C if and only if h ∈ C and h is ≤ FP m -hard.
Note that LF A = LF (B,p) is in min ·P for every NP set A and for every representation of A as a p-projection A = proj p (B) of some suitable set B ∈ P and polynomial p.
Theorem 3.5 Let A be any set in NP, and let B and D be sets in P and p be a polynomial such that A = proj p (B) ⊆ D and LF A is ≤ FP m -complete for min ·P. Then, there exist a set C ∈ P and a polynomial q such that D = proj q (C) and computing LF D is ∆ p 2 -hard. Hence, D is not self-reducible with respect to (C, q), assuming P = NP.
Proof. Let A, B, and p be given as in the theorem, where A ⊆ Σ * and B ⊆ Σ * × Σ * and Σ = {0, 1}. Let D be any set in P with A ⊆ D. Define C = B ∪ {(x, bin(2 p(|x|) )) | x ∈ D}, and let q(n) = p(n) + 1 for all n. Note that C ∈ P and D = proj q (C). It also follows that
LF D (x) = LF A (x) if x ∈ D, and LF D (x) = 2 · LF A (x) if x ∈ D.
We now show that computing LF D is as hard as deciding the ∆ p 2 -complete problem Odd-Min-SAT, which was defined in Section 2. Since LF A is ≤ FP m -complete for min ·P, we have LF SAT (F ) = LF A (t(F )) for some polynomial-time computable function t. Hence,
F ∈ Odd-Min-SAT ⇐⇒ LF SAT (F ) ≡ 1 mod 2 ⇐⇒ LF A (t(F )) ≡ 1 mod 2 ⇐⇒ LF D (t(F )) ≡ 1 mod 2.
Thus, one can decide "F ∈ Odd-Min-SAT?" by looking at the last bit of LF D (t(F )).
Corollary 3.6 If P = NP then Σ * has representations in which it is not self-reducible.
Proof. Replacing the set D of Theorem 3.5 by Σ * , it is clear that the hypothesis of the theorem can be satisfied by suitably choosing A, B, and p. It follows that Σ * , unconditionally, has representations in which it is not self-reducible in the sense of Schnorr, unless P = NP.
Conclusions and Open Questions
In Theorem 2.3, we strengthened Khuller and Vazirani's [KV91] lower bound for computing the lexicographically first four-coloring for planar graphs from NP-hardness to ∆ p 2 -hardness. The nonself-reducibility of the Pl-4-Color problem follows immediately from these lower bounds. Khuller and Vazirani [KV91] asked whether similar non-self-reducibility results can be proven for problems in P other than Pl-4-Color, under some plausible assumption such as P = NP. We established as Theorem 3.5 a general result showing that it is almost trivial to find such problems.
This general result subsumes a number of results [Gro99] providing concrete-although somewhat artificial-problems in P that are not self-reducible in Schnorr's sense, unless P = NP. Why are these problems artificial? The reason is that they are P versions of standard NP-complete problems-such as the satisfiability problem, the clique problem, and the knapsack problem-that are defined by (a) encoding directly into each solvable problem instance a trivial solution to this instance, and simultaneously (b) ensuring that computing the smallest solution remains a hard problem by fixing a suitable ordering of the solutions to a given problem instance.
Here are some examples of such problems:
1. (a) P-SAT is the set of pairs F, x i such that F is a boolean formula in conjunctive normal form and x i is a variable occurring in each clause of F in positive form.
(b) Let the variables of a given formula F be ordered as F = F (x 1 , x 2 , . . . , x n ). Just as for the satisfiability problem, a solution to a P-SAT instance I = F, x i is any satisfying assignment ψ I of F . A solution ψ I of I is represented by the string ψ I = ψ I (x 1 )ψ I (x 2 ) · · · ψ I (x n ) from {0, 1} n , where "1" represents "true" and "0" represents "false."
2. (a) P-Clique is the set of pairs G, C such that G = (V, E) is a graph and C ⊆ V is a clique in G.
(b) Let the vertex set of a given graph G = (V, E) be ordered as V = {v 1 , v 2 , . . . , v n }. Just as for the clique problem, a solution to a P-Clique instance I = G, C is any cliquê C ⊆ V that is of size at least ||C||. A solutionĈ of I is represented by the string ψ I = χĈ(v 1 )χĈ (v 2 ) · · · χĈ(v n ) from {0, 1} n , where χĈ denotes the characteristic function ofĈ, i.e., χĈ(v) = 1 if v ∈Ĉ, and χĈ(v) = 0 if v ∈Ĉ.
3. (a) P-Knapsack is the set of tuples U, s, v, k, b such that U is a finite set, s and v are functions mapping from U to the positive integers, and there exists an element u ∈ U satisfying s(u) ≤ b and v(u) ≥ k.
(b) Let the set U of a given P-Knapsack instance I = U, s, v, k, b be ordered as U = {u 1 , u 2 , . . . , u n }. Just as for the knapsack problem, a solution to I is any subsetÛ ⊆ U that satisfies the "knapsack property," i.e., that satisfies the conditions u∈Û s(u) ≤ b and u∈Û v(u) ≥ k.
A solutionÛ of I is represented by the string ψ I = χÛ (v 1 )χÛ (v 2 ) · · · χÛ (v n ) from {0, 1} n .
Note that the lexicographic ordering of strings induces a suitable ordering of the solutions to a given problem instance. For each of the above-defined problems Π ∈ P, computing LF Π can be shown to be NP-hard [Gro99], which implies that Π is non-self-reducible unless P = NP.
Analogously, every standard NP-complete problem yields such an artificial, non-self-reducible problem in P. In contrast, the Pl-4-Color problem is a quite natural problem. Is it possible to prove, under a plausible assumption such as P = NP, the non-self-reducibility of other natural problems in P?
Definition 3. 4 (
4Hempel and Wechsung [HW00]) Define the class min ·P to consist of all functions f for which there exist a set A ∈ P and a polynomial p such that for all x ∈ Σ * , f (x) = min{y ∈ {0, 1} * | |y| ≤ p(|x|) and x, y ∈ A},
Acknowledgments.We thank the anonymous referees of the conference version of this paper for their helpful and insightful comments on the paper. In particular, we thank the referee who suggested an idea that led to Theorem 3.5, which subsumes some results from an earlier draft of this paper.
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| [] |
[
"ON THE PUSH-FORWARDS FOR MOTIVIC COHOMOLOGY THEORIES WITH INVERTIBLE STABLE HOPF ELEMENT",
"ON THE PUSH-FORWARDS FOR MOTIVIC COHOMOLOGY THEORIES WITH INVERTIBLE STABLE HOPF ELEMENT"
] | [
"Alexey Ananyevskiy "
] | [] | [] | We present a geometric construction of push-forward maps along projective morphisms for cohomology theories representable in the stable motivic homotopy category assuming that the element corresponding to the stable Hopf map is inverted in the coefficient ring of the theory. The construction is parallel to the one given by A. Nenashev for derived Witt groups. Along the way we introduce cohomology groups twisted by a formal difference of vector bundles as cohomology groups of a certain Thom space and compute twisted cohomology groups of projective spaces. | 10.1007/s00229-015-0799-6 | [
"https://arxiv.org/pdf/1406.2894v2.pdf"
] | 55,849,507 | 1406.2894 | d59d8d007623989a03d0d126bf9cb4be9a807511 |
ON THE PUSH-FORWARDS FOR MOTIVIC COHOMOLOGY THEORIES WITH INVERTIBLE STABLE HOPF ELEMENT
11 Jun 2014
Alexey Ananyevskiy
ON THE PUSH-FORWARDS FOR MOTIVIC COHOMOLOGY THEORIES WITH INVERTIBLE STABLE HOPF ELEMENT
11 Jun 2014
We present a geometric construction of push-forward maps along projective morphisms for cohomology theories representable in the stable motivic homotopy category assuming that the element corresponding to the stable Hopf map is inverted in the coefficient ring of the theory. The construction is parallel to the one given by A. Nenashev for derived Witt groups. Along the way we introduce cohomology groups twisted by a formal difference of vector bundles as cohomology groups of a certain Thom space and compute twisted cohomology groups of projective spaces.
Introduction
Existence of push-forward maps in a cohomology theory gives a powerful tool that allows to perform various computations and analyze properties of the considered cohomology theory. The best understood algebraic cohomology theories, such as etale cohomology, Chow groups and algebraic K-theory, have push-forward maps for arbitrary projective morphisms. Roughly speaking, oriented cohomology theories (see [PS03,LM07,S07a]) are cohomology theories possessing push-forwards along arbitrary projective morphisms and satisfying certain natural properties. The theory in the oriented setting is quite well-developed: one may obtain a projective bundle theorem and introduce Chern classes of vector bundles [S07a,PS09], study morphisms between such theories and obtain Riemann-Roch type theorems [PS04,S07b], construct a universal oriented cohomology theory [LM07] that allows to perform computations in the universal setting, study the corresponding categories of motives and obtain various motivic decompositions [NZ06], etc.
On the other hand, there are some interesting cohomology theories for which one can not define push-forward maps along arbitrary projective morphisms.
Among the examples are derived Witt groups, hermitian K-theory, oriented Chow groups and Witt cohomology (see [Bal99,Sch10,BM00] for the definitions). A more down-to-earth example is given by choosing an embedding of the base field to R, taking real points of the considered variety and computing singular cohomology with integer coefficients. Another example is given by motivic stable cohomotopy groups S i,j , i.e. by the cohomology theory represented by the spherical spectrum in the motivic stable homotopy category. All these theories have in common that the usual version of projective bundle theorem fails, i.e. A(P n ) ∼ = A(pt)[t]/t n+1 , where A denotes the corresponding cohomology theory. Nevertheless, sometimes one can obtain a certain computation for projective space, for example for derived Witt groups we know that W * (P n k ) ⊕ W * (P n k , O(1)) is a free module over W * (Spec k) of rank two [G01,Wa03,Ne09b]. Based on this computation A. Nenashev defined for derived Witt groups push-forwards along projective morphisms [Ne09a]: for a projective morphism of smooth varieties f : Y → X and a line bundle L over X he defined a homomorphism
f W : W * (Y, f * L ⊗ ω Y ⊗ f * ω −1 X ) → W * +c (X, L), where c = dim X − dim Y .
The twists should agree in the way as it is stated above, for example, for a projection p : P 2 k → Spec k we do not have a pushforward map p W : W * (P 2 k ) → W * −2 (Spec k). It is noteworthy that there is another way to define push-forward maps for derived Witt groups based on Grothendieck duality [CH11] yielding similar homomorphisms.
For a cohomology theory representable in the motivic stable homotopy category there is a general approach to the construction of push-forward maps based on the Atiyah duality, which was settled in the motivic setting by P. Hu and I. Kriz [H05] via geometric methods and by J. Riou [R05] using four functors formalism developed by V. Voevodsky and J. Ayoub. Consider a projective morphism f : Y → X of smooth varieties. Suppose for simplicity that both Y and X are projective. Then we have the dual morphism f ∨ :
Σ ∞ T X ∨ → Σ ∞ T Y ∨ for Σ ∞ T X ∨ = Hom(Σ ∞ T X, S) and Σ ∞ T Y ∨ = Hom(Σ ∞ T Y, S) being the dual spectra. Atiyah duality gives isomorphisms Σ ∞ T X ∨ ∼ = Σ ∞ T T h(−T X ) and Σ ∞ T Y ∨ ∼ = Σ ∞ T T h(−T Y ), where Σ ∞ T T h(−T X ) and Σ ∞ T T h(−T Y )
are suspension spectra of stable normal bundles, i.e. we use the Jouanalou device (see [J73], [We89,§ 4]) replacing varieties by affine ones, consider vector bundles complement to the tangent bundles and take an appropriate shifts of the suspension spectra of the respective Thom spaces. Hence we have the corresponding morphism of the cohomology groups (f ∨ ) A : A * , * (Σ ∞ T T h(−T Y )) → A * , * (Σ ∞ T T h(−T X )). Identifying A * , * (Σ ∞ T T h(−T X )) and A * , * (Σ ∞ T T h(−T Y )) with cohomology of X and Y via appropriate Thom isomorphisms we obtain push-forward maps. In particular, for the derived Witt groups we have Thom isomorphisms [Ne07]
W * (Σ ∞ T T h(−T Y )) ∼ = W * +dim Y (Y, (det T Y ) −1 ) = W * +dim Y (Y, det ω Y ) W * (Σ ∞
T T h(−T X )) ∼ = W * +dim X (X, (det T X ) −1 ) = W * +dim X (X, det ω X ) and that is a geometric reason why we have the twist by ω Y ⊗ f * ω −1 X . In this paper we generalize the construction of projective push-forwards for derived Witt groups given by A. Nenashev [Ne09a] to the case of representable cohomology theories with invertible stable Hopf element η ∈ A −1,−1 (pt) (see Definition 8 for the precise definition of stable Hopf element). Among the examples are Witt cohomology H * (−, W ) and stable cohomotopy groups with inverted stable Hopf element S * , * η (−). Analyzing A. Nenashev's approach one can notice that it is based on the computation of derived Witt groups of projective space, which could be done via induction as in [Ne09b] using the following properties of derived Witt groups:
(1) identification W * (P 2 ) = W * (pt);
(2) Thom isomorphisms W * +n X (E) ∼ = W * (X, det E) for a rank n vector bundle E over a smooth variety X;
PUSH-FORWARDS FOR THEORIES WITH INVERTIBLE STABLE HOPF ELEMENT 3
(3) isomorphisms W * (X, L 1 ⊗ L ⊗2 2 ) ∼ = W * (X, L 1 ). For a general representable cohomology theory A * , * (−) one may rephrase these properties in the following way:
(1) the stable Hopf element η ∈ A −1,−1 (pt) arising from the Hopf map A 2 − 0 → P 1 is invertible (see Definition 8 and Remark 1); (2) A * , * (−) is SL-oriented in the sense of [PW10b,Definition 5.1] (see also [An12]); (3) for every line bundle L over every smooth variety X there are Thom isomorphisms A * +2, * +1 X (L ⊗2 ) ∼ = A * , * (X) (cf. [PW10c, Definition 3.3]). As we show in Theorem 1 only the first property is essential:
Theorem. Let A be a commutative ring T -spectrum and let X be a smooth variety. Then
A * , * η (P 2n X ) ∼ = A * , * η (X), A * , * η (P 2n−1 X ) ∼ = A * , * η (X) ⊕ A * −4n+1, * −2n η (X), where A * , * η (−) = A * , * (−)[η −1 ]
. In order to obtain this theorem we consider the projection
H 2n : (A 2n − {0}, (1, 1, . . . , 1)) → P 2n−1 /P 2n−2 given by H 2n (x 1 , x 2 , . . . , x 2n ) = [x 1 : x 2 : . . . : x 2n ].
It turns out that this projection is, up to canonical isomorphisms, a suspension of the Hopf map, thus induces an isomorphism on cohomology groups with inverted η. This isomorphism allows us to compute cohomology groups of projective spaces by induction. Note that real points of H 2n give a morphism S 2n−1 → S 2n−1 and one can easily see that it corresponds to 2 ∈ π 2n−1 (S 2n−1 ), while real points of the Hopf map give a two-folded covering S 1 → S 1 .
In order to define push-forwards we adopt ideas arising from Atiyah duality and introduce cohomology groups twisted by a vector bundle as shifted cohomology of the Thom space of the vector bundle. Then, using Jouanalou device, we consider cohomology groups of a smooth variety X twisted by a formal difference of vector bundles as shifted cohomology groups of appropriate Thom space. These twisted groups are denoted by A * , * (X; E 1 ⊖ E 2 ). In particular, we have cohomology groups twisted by a complement to the tangent bundle, A * , * (X; ⊖T X ).
It is well-known that one can define push-forwards along closed embeddings using deformation to the normal bundle (see, for example, [PS03]). In our setting, for a closed embedding f : Y → X of codimension c we obtain a push-forward map
f A : A * , * (Y ; ⊖T Y ) → A * +2c, * +c (X; ⊖T X ).
In Theorem 2 we compute cohomology of P n twisted by an arbitrary vector bundle assuming that stable Hopf element is inverted, in particular, we have the following theorem (for the general statement see loc.cit.).
Theorem. Let A be a commutative ring T -spectrum and let X be a smooth variety. Then there is an isomorphism
A * , * η (X; ⊖T X ) (ia) A − −− → A * +4n, * +2n η (P 2n X ; ⊖T P 2n X )
given by push-forward map along the closed embedding
i a : X → P 2n X , i(x) = (x, a)
, where a is a rational point on P 2n .
For the canonical projection p : P 2n X → X we define the push-forward map
p A = (i a ) −1 A : A * +4n, * +2n η (P 2n X ; ⊖T P 2n X ) → A * , * η (X; ⊖T X ) as the isomorphism inverse to (i a ) A .
Then for a projective morphism f of codimension c we define push-forward map
f A : A * , * η (Y ; ⊖T Y ) → A * +2c, * +c η (X; ⊖T X )
representing f as a composition f = p • i of closed embedding and projection and taking push-forwards along these morphisms. This construction generalizes immediately to twisted cohomology groups yielding for a codimension c projective morphism f : Y → X and a formal difference of vector bundles
E 1 ⊖ E 2 over X push-forward map f A : A * , * η (Y ; f * E 1 ⊖ (f * E 2 ⊕ T Y )) → A * +2c, * +c η (X; E 1 ⊖ (E 2 ⊕ T X )).
Using A. Nenashev's constructions (which follow in general the ones introduced in [S07a,PS09]) one can check that this definition does not depend on the choice of p and i and obtain the usual properties of push-forwards: functoriality, projection formula and compatibility with transversal base change. The paper is organized in the following way. In Section 2 we recall some well-known facts about motivic homotopy theory and representable cohomology theories. In the next two sections we introduce cohomology of a smooth variety twisted by a formal difference of vector bundles and check some basic properties, in particular, in Corollary 3 we show that twisted cohomology depend only on the class of the twist in reduced K 0 . In Section 5 we recall the well-known construction of push-forwards along closed embeddings. The main part of the paper is Section 6 where we compute twisted cohomology of projective spaces assuming that stable Hopf element is inverted. In the rest two sections we define push-forwards along projective morphisms and obtain basic properties.
Acknowledgement. The author wishes to thank I. Panin for numerous conversations on the subject. This research is supported by RFBR grants 13-01-00429 and 14-01-31095, by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government grant 11.G34.31.002, by JSC "Gazprom Neft" and by "Dynasty" foundation. The final stage of the research is supported by RSF grant 14-11-00456.
Preliminaries on motivic homotopy theory
In this section we recall some basic definitions and constructions in the nonstable and stable motivic homotopy categories H • (k) and SH(k). We refer the reader to the foundational papers [MV99,V98] for the details.
Let k be a field and let Sm/k be the category of smooth varieties over k.
Definition 1. A motivic space over k is a simplicial presheaf on Sm/k. Every smooth variety X defines an unpointed motivic space Hom Sm/k (−, X) constant in the simplicial direction. We will occasionally write pt for Spec k regarded as a motivic space. We use the injective model structure on the category of the pointed motivic spaces M • (k). Inverting all the weak motivic equivalences in M • (k) (see [MV99]) we obtain the pointed motivic unstable homotopy category H • (k).
Definition 2. For a vector bundle E over a smooth variety X we denote
T h(E) = E/(E − X) ∈ H • (k) the Thom space of E.
We need the next well-known lemma claiming that an action given by an elementary matrix is homotopy trivial. First recall the precise definition of elementary matrix.
Definition 3. Let X be a smooth variety and let ½ n X be a trivialized vector bundle over X. An automorhism α ∈ Aut X (½ n X ) is elementary if the corresponding matrix A ∈ GL n (k[X]) is elementary, i.e. belongs to the subgroup
E n (X) = I n + g(x)e ij | g(x) ∈ k[X], i = j ≤ GL n (k[X])
generated by transvections (shear mappings). Here I n is the identity matrix of size n × n, g(x) is a regular function and e ij is a matrix unit (matrix with 1 at (i, j) and 0 everywhere else).
Recall that every matrix of determinant 1 over a field or over a Euclidean domain is elementary by Gaussian elimination (combined with Euclid's algorithm for Euclidean domains). In particular, every automorphism of ½ n X given by a determinant 1 matrix with coefficients from the base field k, or from the integers is elementary. Moreover, the automorphism of ½ X ⊕ ½ X given by
the matrix s(x) −1 0 0 s(x) with s(x) ∈ k[X] * is elementary since the matrix t −1 0 0 t ∈ GL 2 (k[t, t −1 ]) is elementary.
Lemma 1. Let X be a smooth variety, let E be a vector bundle over X, let ½ n X be a trivialized vector bundle over X, and let α ∈ Aut X (½ n X ) be an elementary automorphism. Then α ⊗ id ∈ Aut X (½ n X ⊗ E) = Aut X (E ⊕n ) induces identity morphisms on T h(E ⊕n ) and P(E ⊕n ) in H • (k).
Proof. The proof does not depend on the vector bundle E, so we assume that E = ½ X . Moreover, the proofs for the Thom space and for the projective bundle are quite the same, so we give the detailed proof only for the first one.
By the definition we know that the matrix A corresponding to α could be represented as a product
A = k t i k j k (g k (x)),
where t i k j k (g k (x)) = I n + g k (x)e i k j k . Put
A(t) = k t i k j k (tg k (x)) ∈ GL n (k[X][t]).
Denote p : X × A 1 → X the canonical projection and let i 0 , i 1 : X → X × A 1 be the embeddings given by i 0 (x) = (x, 0) and i 1 (x) = (x, 1). These maps induce isomorphisms of Thom spaces
p : T h(p * ½ n X ) ≃ − → T h(½ n X ), i 0 , i 1 : T h(½ n X ) ≃ − → T h(p * ½ n X )
.
We have pi 0 = pi 1 = id T h(½ n X ) , hence i 0 = i 1 = p −1 in H • (k). On the other hand A(t) defines an automorphism A(t) : T h(p * ½ n X ) ≃ − → T h(p * ½ n X ) and id T h(½ n X ) = pA(t)i 0 = pA(t)i 1 = α, where α : T h(½ n X ) ≃ − → T h(½ n X ) is the automorphism induced by α. Definition 4. Let T = A 1 /(A 1 − {0}) be the Morel-Voevodsky object. A T - spectrum M [Jar00]
is a sequence of pointed motivic spaces (M 0 , M 1 , M 2 , . . . ) equipped with structural maps σ n : T ∧ M n → M n+1 . A map of T -spectra is a sequence of maps of pointed motivic spaces which is compatible with the structure maps. Inverting the stable motivic weak equivalences as in [Jar00] we obtain the motivic stable homotopy category SH(k).
A pointed motivic space Y gives rise to a suspension T -spectrum Σ ∞ T Y . Set S = Σ ∞ T (pt + ) for the spherical spectrum. Both H • (k) and SH(k) are equipped with symmetric monoidal structures (∧, pt + ) and (∧, S) respectively and
Σ ∞ T : H • (k) → SH(k) is a strict symmetric monoidal functor.
Definition 5. Recall that there are two spheres in M • (k), the simplicial one S 1,0 = S 1 s = ∆ 1 /∂(∆ 1 ) and S 1,1 = (G m , 1). Here we follow the notation and indexing introduced in [MV99, p.111]. For the integers p, q ≥ 0 we write S p+q,q for (S 1 s ) ∧p ∧ (G m , 1) ∧q and Σ p+q,q for the suspension functor − ∧ S p+q,q . This functor becomes invertible in the stable homotopy category SH(k), so we extend the notation to arbitrary integers p, q in an obvious way.
Definition 6. Any T -spectrum A defines a bigraded cohomology theory on the category of pointed motivic spaces. Namely, for a pointed motivic space (Y, y) one sets A p,q (Y, y) = Hom SH(k) (Σ ∞ T (Y, y), Σ p,q A) and A * , * (Y, y) = p,q A p,q (Y, y). For an unpointed motivic space Y we set A p,q (Y ) = A p,q (Y + , +) with A * , * (Y ) defined accordingly. We can regard a smooth variety X as an unpointed motivic space and obtain groups A p,q (X). In case of i−j, j ≥ 0 one has a canonical suspension isomorphism A p,q (Y, y) ∼ = A p+i,q+j (Σ i,j (Y, y)) induced by the shuffling isomorphism S p,q ∧S i,j ∼ = S p+i,q+j . In the motivic homotopy category there is a canonical isomorphism T ∼ = S 2,1 [MV99, Lemma 2.15], we write
· · · → A * , * (X) → A * , * (U 1 ) ⊕ A * , * (U 2 ) → A * , * (U 1 ∩ U 2 ) → A * +1, * (X) → . . .
(4) Cup-product: for a pointed motivic space Y we have a functorial graded ring structure ∪ :
A * , * (Y ) × A * , * (Y ) → A * , * (Y ).
Moreover, let i 1 : Z 1 → X and i 2 : Z 2 → X be closed embeddings of varieties with X being smooth. Then we have a functorial, bilinear and associative cup-product
∪ : A * , * (X/(X − Z 1 )) × A * , * (X/(X − Z 2 )) → A * , * (X/(X − Z 1 ∩ Z 2 )).
In particular, setting Z 1 = X we obtain an A * , * (X)-module structure on A * , * (X/(X − Z 2 )). All the morphisms in the localization sequence are homomorphisms of A * , * (X)-modules. We will sometimes omit ∪ from the notation.
(5) Module structure over stable cohomotopy groups: for every motivic space Y the structure morphism S → A induces a homomorphism of graded rings S * , * (Y ) → A * , * (Y ), which defines an S * , * (pt)-module structure on A * , * (Y ). For a smooth variety X the ring A * , * (X) is a graded S * , * (pt)-algebra via S * , * (pt) → S * , * (X) → A * , * (X), where the first morphism is induced by the projection X → pt.
We end this preliminary section with the definitions related to the stable Hopf map. we may regard H as an element of Hom H•(k) (S 3,2 , S 2,1 ). The stable Hopf element is the unique η ∈ S −1,−1 (pt) such that Σ 3,2 η = Σ ∞ T H, i.e. η is the stabilization of H moved to S −1,−1 (pt) via the canonical isomorphisms. For a commutative ring T -spectrum A we will usually denote by the same letter η the corresponding element η ∈ A −1,−1 (pt) in the coefficient ring.
Definition 9. Let A be a commutative ring T -spectrum and let Y be a pointed motivic space. Denote
A * , * η (Y ) = A * , * (Y ) ⊗ S * , * (pt) (S * , * (pt)[η −1 ])
. One can easily check that A * , * η (−) is a cohomology theory and satisfies properties from Definiton 7. We refer to A * , * η (−) as cohomology theory A with inverted stable Hopf element.
Remark 1. It is quite well-known that η ∈ A −1,−1 (pt) is invertible (i.e. H A is an isomorphism) if and only if the canonical projection p : P 2 → pt induces an isomorphism p A : A * , * (pt) ≃ − → A * , * (P 2 ). It follows from the fact that mapping cone of the Hopf map is equivalent to (P 2 , [0 : 0 : 1]). More precisely, one needs to consider the long exact sequence associated to the zero section embedding
P 1 → O P 1 (1) combined with isomorphisms T h(O P 1 (1)) ∼ = P 2 /(P 2 − P 1 ) ∼ = (P 2 , [0 : 0 : 1]) and isomorphism O P 1 (1) − P 1 ∼ = A 2 − 0, . . . / / A * , * (T h(O P 1 (1))) / / A * , * (P 1 ) / / A * , * (O P 1 (1) − P 1 ) / / . . . . . . / / A * , * (P 2 , [0 : 0 : 1]) ∼ = O O / / A * , * (P 1 ) H A / / A * , * (A 2 − 0) / / ∼ = O O . . .
Cohomology twisted by a vector bundle
In this section we introduce the language of twisted cohomology groups which turns out to be a very convenient tool in our approach to the construction of push-forwards. Roughly speaking, for a vector bundle E over X twisted cohomology groups are defined to be the cohomology groups of E supported on X. For an oriented cohomology theory one can untwist it, and twisted cohomology groups coincide with the ordinary ones, but in general they are quite different. We show that these groups depend only on the class [E] in K 0 (X) and in the next section extend the definition allowing to twist by a formal difference of vector bundles Then we derive some handy properties of these groups.
Definition 10. Let E be a vector bundle of rank n over a smooth variety X and let A be a commutative ring T -spectrum. Denote
A * , * (X; E) = A * +2n, * +n (T h(E))
and refer to it as cohomology groups of X, twisted by E.
Let f : Y → X be a morphism of smooth varieties. Denote
f A = f A E : A * , * (X; E) → A * , * (Y ; f * E) the natural homomorphism induced by the corresponding map of Thom spaces T h(f * E) → T h(E).
We will usually omit the subscript E from the notation.
Let φ : E ′ → E be a monomorphism of vector bundles and put
k = rank E − rank E ′ . Denote φ A : A * , * (X; E) → A * +2k, * +k (X; E ′ ) the natural homomorphism induced by the corresponding map T h(E ′ ) → T h(E). Since φ is a monomorphism then φ(E ′ − X) ⊂ E − X and φ indeed induces a map of Thom spaces.
Definition 11. Let p 1 : E 1 → X and p 2 : E 2 → X be vector bundles over a smooth variety X. There is a product
∪ : A * , * (X; E 1 ) × A * , * (X; E 2 ) → A * , * (X; E 1 ⊕ E 2 )
induced by the usual cup-product
A * , * ((E 1 ⊕ E 2 )/(E 1 ⊕ E 2 − E 2 )) × A * , * ((E 1 ⊕ E 2 )/(E 1 ⊕ E 2 − E 1 )) ∪ − → ∪ − → A * , * ((E 1 ⊕ E 2 )/(E 1 ⊕ E 2 − X)) combined with isomorphisms A * , * ((E 1 ⊕ E 2 )/(E 1 ⊕ E 2 − E 2 )) ∼ = A * , * (E 1 /(E 1 − X)), A * , * ((E 1 ⊕ E 2 )/(E 1 ⊕ E 2 − E 1 )) ∼ = A * , * (E 2 /(E 2 − X))
induced by contractions of corresponding bundles.
Lemma 2. The following functoriality properties hold.
(1) Let Z g − → Y f
− → X be morphisms of smooth varieties and let E be a vector bundle over X. Then the following diagram commutes.
A * , * (X; E) f A ( ( P P P P P P
Then the following diagram commutes.
A * , * (X; E 1 ) × A * , * (X; E 2 ) φ A 1 ×φ A 2 ∪ / / A * , * (X; E 1 ⊕ E 2 ) (φ 1 ⊕φ 2 ) A A * +2k, * +k (X; E ′ 1 ) × A * +2l, * +l (Y ; E ′ 2 ) ∪ / / A * +2(k+l), * +(k+l) (X; E ′ 1 ⊕ E ′ 2 ) Proof.
The results are straightforward since all the homomorphisms are given by pull-backs.
Remark 2. The above notation is inspired by the following observation which I learned from Ivan Panin who attributed it to Charles Walter. Let L be a line bundle over a smooth variety X. Then for derived Witt groups introduced by Paul Balmer [Bal99] one has a canonical isomorphism W * +1 X (L) ∼ = W * (X; L) of derived Witt groups with support (i.e. derived Witt groups of the Thom space T h(L)) and derived Witt groups with the twisted duality [Ne07, Theorem 2.5]. Thus for a general cohomology theory A and a line bundle L over a smooth variety X one can introduce twisted cohomology groups as cohomology groups with support: A(X; L) = A X (L).
Remark 3. For an oriented cohomology theory [PS03] we have canonical Thom isomorphisms A * , * (X; E) ∼ = A * , * (X) functorial in X. For symplectically and SL-oriented cohomology theories [PW10a, PW10b, An12] we have similar Thom isomorphisms for symplectic and special linear vector bundles.
Lemma 3. Let E be a vector bundle over a smooth variety X. Then for the dual vector bundle E ∨ there is a natural isomorphism
A * , * (X; E) ∼ = A * , * (X; E ∨ ).
Proof. Let n be the rank of E and denote by E 0 and (E ∨ ) 0 the complements to the zero section of the corresponding bundles. Consider variety
Y = {(v, f ) ∈ E × X E ∨ | f (v) = 1} .
Natural projections identify Y with affine bundles over E 0 and (E ∨ ) 0 , thus we obtain isomorphisms
A * , * (E 0 ) ∼ = A * , * (Y ) ∼ = A * , * ((E ∨ ) 0 ) of A * , * (X)-algebras. Moreover, projections provide homomorphisms A * −2n, * −n (X; E) → A * , * ((E × X E ∨ )/Y ), A * −2n, * −n (X; E ∨ ) → A * , * ((E × X E ∨ )/Y ).
Then the localization sequences combined with the 5-lemma give the claim
. . . / / A * −2n, * −n (X; E) / / A * , * (X) / / ≃ A * , * (E 0 ) / / ≃ . . . . . . / / A * , * ((E × X E ∨ )/Y ) / / A * , * (E × X E ∨ ) / / A * , * (Y ) / / . . . . . . / / A * −2n, * −n (X; E ∨ ) / / O O A * , * (X) / / ≃ O O A * , * ((E ∨ ) 0 ) / / ≃ O O . . .
Lemma 4. Let E 1 and E 2 be vector bundles over a smooth variety X. Denote p : E 1 − X → X the canonical projection and suppose that there exists some th ∈ A * , * (X; E 2 ) such that
A * , * (X) −∪th − −− → A * , * (X; E 2 ), A * , * (E 1 − X) −∪p A (th) −−−−−→ A * , * (E 1 − X; p * E 2 )
are isomorphisms. Then
A * , * (X; E 1 ) −∪th − −− → A * , * (X; E 1 ⊕ E 2 )
is an isomorphism as well.
Proof. Consider the localization sequence for the zero section X → E 1 and its twisted version:
. . . / / A * , * (X; E 1 ) / / −∪th A * , * (X) / / −∪th ≃ A * , * (E 1 − X) / / −∪p A (th) ≃ . . . . . . / / A * , * (X; E 1 ⊕ E 2 ) / / A * , * (X; E 2 ) / / A * , * (E 1 − X; p * E 2 ) / / . . .
The claim follows via 5-lemma.
Corollary 1. Let E be a vector bundle over a smooth variety X and let A * , * (−) be an SL-oriented cohomology theory in the sense of [PW10b, Definition 5.1] represented by a commutative ring T -spectrum A. Then there is a canonical isomorphism
A * , * (X; E) ∼ = A * , * (X; det E).
Proof. We have canonical trivializations
det(det E ⊕ det E ∨ ) ∼ = ½ X , det(E ⊕ det E ∨ ) ∼ = ½ X ,
hence the above lemma combined with the Remark 3 yields the claim:
A * , * (X; E) ∼ = A * , * (X; E ⊕ det E ∨ ⊕ det E) ∼ = ∼ = A * , * (X; det E ⊕ det E ∨ ⊕ E) ∼ = A * , * (X; det E).
Corollary 2. Let E be a vector bundle over a smooth variety X and let ½ n X be a trivialized vector bundle over X. Then there is a canonical isomorphism
A * , * (X; E) −∪Σ n T 1 − −−− → A * , * (X; E ⊕ ½ n X )
. Moreover, for a vector bundle E ′ over X the following diagram commutes.
A * , * (X; E ′ ) × A * , * (X; E) (id,−∪Σ n T 1) / / ∪ A * , * (X; E ′ ) × A * , * (X; E ⊕ ½ n X ) ∪ A * , * (X; E ′ ⊕ E) −∪Σ n T 1 / / A * , * (X; E ′ ⊕ E ⊕ ½ n X ).
Proof. The first claim follows from the above lemma since the suspension isomorphism is functorial. The second claim follows from the associativity of cup-product.
Corollary 3. Let E 1 and E 2 be vector bundles over a smooth variety X such that
[E 1 ] = [E 2 ] in K 0 (X) = K 0 (X)/(Z[½ X ]). Then A * , * (X; E 1 ) ∼ = A * , * (X; E 2 ).
Proof. Using the Jouanolou device (see [J73], [We89,§4]) one may assume that X is affine. Then for some m and n there exists an isomorphism
E 1 ⊕ ½ n X ∼ = E 2 ⊕ ½ m X
, and the claim follows by Corollary 2:
A * , * (X; E 1 ) ∼ = A * , * (X; E 1 ⊕ ½ n X ) ∼ = A * , * (X; E 2 ⊕ ½ m X ) ∼ = A * , * (X; E 2 ).
Remark 4. Let X be a smooth variety and consider P ∈ K 0 (X). Then the above corollary suggests us to define A * , * (X; P ) in the following way: using the Jouanolou device we may assume X to be affine. Then P = [E] for some vector bundle E over X and we may put A * , * (X; P ) = A * , * (X; E).
The problem is that although all the choices for E such that P = [E]
give isomorphic A * , * (X; P ), but the isomorphisms are not canonical. Moreover, for a morphism f : Y → X we do not have a natural homomorphism
f A : A * , * (X; P ) → A * , * (Y ; f * P ).
In order to obtain a functorial definition one needs to keep track of all the involved vector bundles.
Cohomology twisted by a formal difference of vector bundles
In this section we introduce twists by a formal difference of vector bundles and establish its basic properties. Roughly speaking, in order to define cohomology groups twisted by formal differences we add a trivialized vector bundle of large rank and use the definition from previous section. Keeping track of all the isomorphisms allows us to obtain functoriality. Throughout this section we continuously use Jouanalou device ( [J73], [We89,§4]) if needed, thus for every considered variety X and vector bundle E over X we assume that there is a vector bundle E such that E ⊕ E ∼ = ½ 2n X . Definition 12. Let E ′ and E be vector bundles over a smooth variety X. Applying Jouanalou device ( [J73], [We89,§4]) we may assume that X is affine.
Choose a vector bundle E and an isomorphism θ :
E ⊕ E ∼ = ½ 2n X for some n and put A * , * (E,θ) (X; E ′ ⊖ E) = A * , * (X; E ′ ⊕ E)
. We will show in the next lemma that these groups depend on the choice of (E, θ) up to a canonical isomorphism. We will omit the subscript (E, θ) identifying the groups for all the choices using the canonical isomorphisms and refer to them as cohomology groups of X, twisted by the formal difference of vector bundles E 1 ⊖ E 2 .
Remark 5. For an oriented cohomology theory one has natural isomorphisms A * , * (X; E 1 ⊖ E 2 ) ∼ = A * , * (X) and for an SL-oriented cohomology theory Corollary 1 yields A * , * (X;
E 1 ⊖ E 2 ) ∼ = A * , * (X; det E 1 ⊗ (det E 2 ) −1 ).
Definition 13. Let E ′ and E be vector bundles over a variety X. Consider
(E i , θ i ), i = 1, 2, where E i is a vector bundle over X and θ i : E ⊕ E i ≃ − → ½ 2n i X is an isomorphism of vector bundles. Define canonical isomorphisms Θ (E 2 ,θ 2 ) (E 1 ,θ 1 ) : A * , * (X; E ′ ⊕ E 1 ) ≃ − → A * , * (X; E ′ ⊕ E 2 )
to be given by the following sequence of isomorphisms:
A * , * (X; E ′ ⊕ E 1 ) −∪Σ 2n 2 T 1 A * , * (X; E ′ ⊕ E 2 ) −∪Σ 2n 1 T 1 A * , * (X; E ′ ⊕ E 1 ⊕ ½ 2n 2 X ) (id ⊕θ 2 ) A A * , * (X; E ′ ⊕ E 2 ⊕ ½ 2n 1 X ) (id ⊕θ 1 ) A A * , * (X; E ′ ⊕ E 1 ⊕ E ⊕ E 2 ) (id ⊕τ s (E 1 ,E 2 )) A / / A * , * (X; E ′ ⊕ E 2 ⊕ E ⊕ E 1 )
Here τ s (E 1 , E 2 ) : E 2 ⊕E ⊕E 1 → E 1 ⊕E ⊕E 2 swaps E 2 and E 1 and multiplies the first vector bundle by −1, i.e. τ s is given by the following matrix:
τ s = 0 0 − id E 2 0 id E 0 id E 1 0 0 .
Lemma 5. Let E ′ and E be vector bundles over a smooth variety X.
(1) Let (E, θ) be a vector bundle over X together with an isomorphism of vector bundles θ :
E ⊕ E ≃ − → ½ 2n X . Then Θ (E,θ) (E,θ) = id. (2) Let (E i , θ i ), i = 1, 2, 3, be vector bundles over X with isomorphisms of vector bundles θ i : E ⊕ E i ≃ − → ½ 2n i X . Then Θ (E 3 ,θ 3 ) (E 2 ,θ 2 ) • Θ (E 2 ,θ 2 ) (E 1 ,θ 1 ) = Θ (E 3 ,θ 3 ) (E 1 ,θ 1 ) . (3) Let (E i , θ i ), i = 1, 2, be vector bundles over X together with isomor- phisms of vector bundles θ i : E ⊕ E i ≃ − → ½ 2n i X and let f : Y → X be a morphism of smooth varieties. Denote f * (θ i ) : f * E ⊕ f * E i ≃ − → ½ 2n i Y the
isomorphism induced by f and θ i . Then the following diagram commutes.
A * , * (X; E ′ ⊕ E 1 ) f A Θ (E 2 ,θ 2 ) (E 1 ,θ 1 ) / / A * , * (X; E ′ ⊕ E 2 ) f A A * , * (Y ; f * E ′ ⊕ f * E 1 ) Θ (f * E 2 ,f * (θ 2 )) (f * E 1 ,f * (θ 1 )) / / A * , * (Y ; f * E ′ ⊕ f * E 2 )
Proof.
(1) Permutation τ s (E, E) is given by the matrix
τ s = 0 0 − id E 0 id E 0 id E 0 0 .
Lemma 1 shows that τ s (E, E) induces in H • (k) the identity automorphism of the Thom space, thus (id ⊕τ s (E, E)) A = id.
u u • • • • • • • • • • • • • • • E 1 ⊕ E ⊕ E 2 ⊕ E ⊕ E 3 τ s (E⊕E 2 ,E⊕E 3 ) τ s (E 1 ,E 2 ) / / E 2 ⊕ E ⊕ E 1 ⊕ E ⊕ E 3 τ s (E⊕E 1 ,E⊕E 3 ) E 1 ⊕ E ⊕ E 3 / / τ s (E 1 ,E 3 ) E 1 ⊕ E ⊕ E 3 ⊕ E ⊕ E 2 τ s (E 1 ,E 3 ) E 2 ⊕ E ⊕ E 3 ⊕ E ⊕ E 1 τ s (E 2 ,E 3 ) E 2 ⊕ E ⊕ E 3 τ s (E 2 ,E 3 ) o o E 3 ⊕ E ⊕ E 1 ⊕ E ⊕ E 2 τ s (E⊕E 1 ,E⊕E 2 ) / / E 3 ⊕ E ⊕ E 2 ⊕ E ⊕ E 1 E 3 ⊕ E ⊕ E 1 5 5 • • • • • • • • • • • • • • E 3 o o / / O O E 3 ⊕ E ⊕ E 2 i i | | | | | | | | | | | | | |
We leave to the reader insert E ′ , A * , * , θ A i (Σ n i T 1) and superscripts accordingly. The hexagon in the middle commutes since one way differs from the other by τ s (E, E) which equals to identity in H • (k) by Lemma 1. Triangles and some squares commute by obvious reasons, while the squares involving τ s (E ⊕ E i , E ⊕ E j ) commute since E ⊕ E i and E ⊕ E j are trivial vector bundles of even rank (in particular, Lemma 1 yields that (− id) T h(E⊕E i ) = id in H • (k)). The claim follows.
(3) The third claim is straightforward and follows from Lemma 2.
Remark 6. Our definition of cohomology groups twisted by a formal difference of vector bundles yields that in order to define a morphism involving cohomology groups twisted by a formal difference of vector bundles it is sufficient to treat the case of the twist by a vector bundle and check that the definition agrees with canonical isomorphisms Θ (E 2 ,θ 2 ) (E 1 ,θ 1 ) . Moreover, in order to check that a diagram involving cohomology groups twisted by formal differences of vector bundles commutes it is sufficient to treat the case of the twists by vector bundles.
Definition 14. Let E 1 = E ′ 1 ⊖ E 1 and E 2 = E ′ 2 ⊖ E 2 be formal differences of vector bundles over a smooth variety X and let f : Y → X be a morphism of smooth varieties. Define the direct sum of formal differences of vector bundles and pull-back of formal difference of vector bundles as
E 1 ⊕ E 2 = E ′ 1 ⊕ E ′ 2 ⊖ (E 1 ⊕ E 2 ), f * E = f * E ′ 1 ⊖ f * E ′ 1 . Definition 15. Let E = E ′ ⊖ E
be a formal difference of vector bundles over a smooth variety X. Choose a vector bundle E over X and an isomorphism θ :
E ⊕ E ≃ − → ½ 2n X .
For a morphism of smooth varieties f : Y → X we define pull-back f A as
A * , * (E,θ) (X; E) f A / / A * , * (f * E,f * θ) (Y ; f * E) A * , * (X; E ′ ⊕ E) f A / / A * , * (Y ; f * E ′ ⊕ f * E)
where the bottom map is the pull-back introduced in Definition 10. Lemma 5 (3) yields that this definition respects the canonical isomorphisms, thus we will usually omit the explicit choice of E and θ. For a monomorphism of vector bundles φ : V 1 → V 2 we define pull-back φ A in a similar way:
A * , * (E,θ) (X; E ⊕ V 2 ) φ A / / A * +2k, * +k (E,θ) (X; E ⊕ V 1 ) A * , * (X; E ′ ⊕ V 2 ⊕ E) (id ⊕φ⊕id) A / / A * +2k, * +k (X; E ′ ⊕ V 1 ⊕ E)
where k = rank V 2 − rank V 1 and the bottom map was introduced in Definition 10. One can easily see that this definition respects the canonical isomorphisms, allowing us to omit the explicit choice of E and θ.
Definition 16. Let E 1 = E ′ 1 ⊖ E 1 and E 2 = E ′ 2 ⊖ E 2 be formal differences of vector bundles over a smooth variety X. Choose vector bundles E i , i = 1, 2, over X and isomorphisms θ i :
E i ⊕ E i ≃ − → ½ 2n i X .
Define the cup-product in the following way:
A * , * (E 1 ,θ 1 ) (X; E 1 ) × A * , * (E 2 ,θ 2 ) (X; E 2 ) ∪ / / A * , * (E 1 ⊕E 2 ,θ 1⊕2 ) (X; E 1 ⊕ E 2 ) A * , * (X; E ′ 1 ⊕ E 1 ) × A * , * (X; E ′ 2 ⊕ E 2 ) ∪ + + ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ A * , * (X; E ′ 1 ⊕ E ′ 2 ⊕ E 1 ⊕ E 2 ) A * , * (X; E ′ 1 ⊕ E 1 ⊕ E ′ 2 ⊕ E 2 ) (id ⊕τ (E 1 ,E ′ 2 )⊕id) A O O
where θ 1⊕2 is defined as the composition
θ 1⊕2 : E 1 ⊕ E 2 ⊕ E 1 ⊕ E 2 τ (E 2 ,E 1 ) − −−−− → E 1 ⊕ E 1 ⊕ E 2 ⊕ E 2 θ 1 ⊕θ 2 − −− → ½ 2(n 1 +n 2 ) X ,
τ (E 2 , E 1 ) and τ (E 1 , E ′ 2 ) swap the corresponding vector bundles and the bottom map in the diagram is the cup-product introduced in Definition 11.
A straightforward but rather lengthy computation similar to the one carried out in the proof of Lemma 5 (2) shows that if we choose some other E i , i = 1, 2, and isomorphisms ρ i :
E i ⊕ E i ≃ − → ½ 2k i
X then the following diagram commutes:
A * , * (X; E ′ 1 ⊕ E 1 ⊕ E ′ 2 ⊕ E 2 ) (id ⊕τ (E 1 ,E ′ 2 )⊕id) A A * , * (X; E ′ 1 ⊕ E 1 ) × A * , * (X; E ′ 2 ⊕ E 2 ) ∪ 3 3 ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ Θ ( E 1 ,ρ 1 ) (E 1 ,θ 1 ) ×Θ ( E 2 ,ρ 2 ) (E 2 ,θ 2 ) A * , * (X; E ′ 1 ⊕ E ′ 2 ⊕ E 1 ⊕ E 2 ) Θ ( E 1 ⊕ E 2 ,ρ 1⊕2 ) (E 1 ⊕E 2 ,θ 1⊕2 ) A * , * (X; E ′ 1 ⊕ E 1 ) × A * , * (X; E ′ 2 ⊕ E 2 ) ∪ + + ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ A * , * (X; E ′ 1 ⊕ E ′ 2 ⊕ E 1 ⊕ E 2 ) A * , * (X; E ′ 1 ⊕ E 1 ⊕ E ′ 2 ⊕ E 2 ) (id ⊕τ ( E 1 ,E ′ 2 )⊕id) A
In view of the above compatibility with canonical isomorphisms we will usually omit explicit choices of E i and θ i .
Definition 17. Let Y f − → X and Z g − → X be morphisms of smooth varieties and E = E ′ ⊖ E and F = F ′ ⊖ F be formal differences of vector bundles over Y and Z respectively. Suppose that for every formal difference of vector bundles V = V ′ ⊖ V over X we have homomorphisms of abelian groups
A * , * (Z; g * V ⊕ F ) Φ l V −→ A * , * (Y ; f * V ⊕ E), A * , * (Z; F ⊕ g * V) Φ r V −→ A * , * (Y ; E ⊕ f * V).
We say that Φ l 0 (or Φ r 0 ) is a homomorphism of twisted A * , * (X)-bimodules if for every formal difference of vector bundles V = V ′ ⊖ V the following diagrams commute:
A * , * (X; V) × A * , * (Z; F ) id ×Φ l 0 / / g A ×id A * , * (X; V) × A * , * (Y ; E) f A ×id A * , * (Z; g * V) × A * , * (Z; F ) ∪ A * , * (Y ; f * V) × A * , * (Y ; E) ∪ A * , * (Z; g * V ⊕ F ) Φ l V / / A * , * (Y ; f * V ⊕ E) A * , * (Z; F ) × A * , * (X; V) Φ r 0 ×id / / id ×g A A * , * (Y ; E) × A * , * (X; V) id ×f A A * , * (Z; F ) × A * , * (Y ; g * V) ∪ A * , * (Y ; E) × A * , * (Y ; f * V) ∪ A * , * (Z; F ⊕ g * V) Φ r V / / A * , * (Y ; E ⊕ f * V)
If we have only Φ l V (or Φ r V ) and the first (resp. second) diagram commutes then we say that Φ l 0 (resp. Φ r 0 ) is a homomorphism of left (resp. right) twisted A * , * (X)-modules.
We usually write down explicitly only one homomorphism Φ = Φ l 0 (or Φ r 0 ), since the other homomorphisms are clear from the context. For example, the next lemma yields that for a morphism of smooth varieties f : Y → X, a formal difference of vector bundles E = E ′ ⊖ EX over X and a monomorphism E 1 φ − → E 2 of vector bundles over X with k = rank E 2 − rank E 1 the homomorphisms
f A : A * , * (X; E) → A * , * (Y ; f * E), φ A : A * , * (X; E ′
2 ) → A * +2k, * +k (X; E 1 ) are homomorphisms of twisted A * , * (X)-bimodules (in the notation of the definition we should put Y = Y, Z = X, f = f, g = id, Φ = f A in the first case and Z = Y = X, f = g = id, Φ = φ A in the second one).
Lemma 6. All the properties (1)-(5) from Lemma 2 hold for the twists by formal differences of vector bundles.
Proof. Follows from the definitions and Lemma 2 itself.
Definition 18. Let E ′ be a vector bundle over X and let ψ : E 2 ≃ − → E 1 be an isomorphism of vector bundles over X. Choose a vector bundle E 1 → X and an isomorphism θ :
E 1 ⊕ E 1 ≃ − → ½ 2n X . Define ψ A : A * , * (E1,θ) (X; E ′ ⊖ E 1 ) = A * , * (X; E ′ ⊕ E 1 ) id − → A * , * (X; E ′ ⊕ E 1 ) = A * , * (E1,θ•(ψ,id)) (X; E ′ ⊖ E 2 ).
One easily checks that this homomorphism is compatible with canonical isomorphisms, i.e. for another vector bundle E 2 over X with an isomorphism ρ :
E 1 ⊕ E 2 ≃ − → ½ k X the diagram A * , * (X; E ′ ⊕ E 1 ) id / / Θ (E 2 ,ρ) (E 1 ,θ) A * , * (X; E ′ ⊕ E 1 ) Θ (E 2 ,ρ•(ψ,id)) (E 1 ,θ•(ψ,id)) A * , * (X; E ′ ⊕ E 2 ) id / / A * , * (X; E ′ ⊕ E 2 )
commutes. Thus we may omit explicit choice of (E 1 , θ) and regard ψ A as a homomorphism between A * , * (X; E ′ ⊖E 1 ) and A * , * (X; E ′ ⊖E 2 ). Moreover, one can easily check that ψ A is a homomorphism of twisted A * , * (X)-bimodules,
for isomorphisms E 3 φ − → E 2 ψ − → E 1 we have (ψφ) A = φ A ψ A and for a morphism of smooth varieties f : Y → X we have f A ψ A = (f * (ψ)) A f A .
Note that in general even if E 1 = E 2 this homomorphism is not an identity, but an automorphism given by
A * , * (E 1 ,θ) (X; E ′ ⊖ E 1 ) A * , * (X; E ′ ⊕ E 1 ) Θ (E 1 ,θ) (E 1 ,θ•(ψ,id)) A * , * (E 1 ,θ) (X; E ′ ⊖ E 1 ) A * , * (X; E ′ ⊕ E 1 )
Lemma 7. Let E = E ′ ⊖ E be a formal difference of vector bundles over a smooth variety X and let E 1 be a vector bundle over X. Then there is a canonical isomorphism of left twisted A * , * (X)-modules
A * , * (X; E) ≃ − → A * , * (X; E ⊕ (E 1 ⊖ E 1 )).
This isomorphism is functorial in X.
Proof. Let E 1 be a vector bundle over X and let θ 1 :
E 1 ⊕ E 1 → ½ 2n
X be an isomorphism of vector bundles. Consider an element th(E 1 , θ 1 ) = θ A 1 (Σ 2n T 1) ∈ A * , * (X; E 1 ⊕ E 1 ). One can easily check that for another vector bundle E 1 → X and an isomor-
phism ρ 1 : E 1 ⊕ E 1 → ½ 2k X one has Θ ( E 1 ,ρ 1 ) (E 1 ,θ 1 ) (th(E 1 , θ 1 )) = th( E 1 , ρ 1 ).
Hence this element is compatible with canonical isomorphisms and we may think about it as a certain element th(E 1 ⊖ E 1 ) ∈ A * , * (X; E 1 ⊖ E 1 ). The desired isomorphism is given by the cup-product with th(E 1 ⊖ E 1 ):
A * , * (X; E) ∪th(E 1 ⊖E 1 ) − −−−−−− → A * , * (X; E ⊕ (E 1 ⊖ E 1 )).
This map is an isomorphism since it is induced by the suspension isomorphism and it is clearly an isomorphism of left twisted A * , * (X)-modules. Functoriality follows from the observation that for a morphism of smooth varieties
Y f − → X one has f A (th(E 1 ⊖ E 1 )) = th(f * E 1 ⊖ f * E 1 ).
Lemma 8. Let E = E ′ ⊖ E be a formal difference of vector bundles over a smooth variety X and let
0 → E 1 i − → E 2 p − → E 3 → 0
be an exact sequence of vector bundles over X. Then there are canonical isomorphisms
A * , * (X; E ⊕ E 2 ) ∼ = A * , * (X; E ⊕ E 3 ⊕ E 1 ), A * , * (X; E ⊕ E 3 ) ∼ = A * , * (X; E ⊕ (E 2 ⊖ E 1 )).
The first isomorphism is an isomorphism of twisted A * , * (X)-bimodules and the second one is an isomorphism of left twisted A * , * (X)-modules.
Proof. The second isomorphism follows from the first one via Lemma 7, so we focus on the first one. Recall that in the definition of twisted cohomology groups we use Jouanalou device, thus we may assume that X is affine and the short exact sequence splits, producing a non-canonical isomorphism φ :
E 3 ⊕ E 1 ≃ − → E 2 . It is sufficient to show that isomorphism φ A : A * , * (X; E ⊕ E 2 ) ≃ − → A * , * (X; E ⊕ E 3 ⊕ E 1 )
do not depend on the choice of the splitting.
Consider two splittings
E 1 i * * E 2 j 1 j j p 4 4 E 3 s 1 t t E 1 i * * E 2 j 2 j j p 4 4 E 3 s 2 t t .
They induce isomorphisms
φ 1 = (s 1 , i), φ 2 = (s 2 , i) : E 3 ⊕ E 1 ≃ − → E 2 .
The inverse isomorphisms are given by p j 1 and p j 2 respectively. We need to check that
φ −1 2 φ 1 = ps 1 pi j 2 s 1 j 2 i = id E 3 0 j 2 s 1 id E 1
induces the identity morphism on the twisted cohomology groups. We claim that this map induces the identity morphism in H • (k) on the corresponding Thom space, and it could be shown in the similar way as in the proof of Lemma 1: inserting t in the lower-left entry of the matrix one obtains an explicit A 1 -homotopy.
Corollary 4. Let E = E ′ ⊖ E be a formal difference of vector bundles over a smooth variety X and let be an exact sequence of vector bundles over X. Denote k = rank E 3 −rank E 2 + rank E 1 . Then there is a canonical homomorphism of left twisted A * , * (X)modules A * , * (X; E ⊕ E 3 ) → A * +2k, * +k (X; E ⊕ (E 2 ⊖ E 1 )).
Proof. Lemma 8 yields
A * +2k, * +k (X; E ⊕ (E 2 ⊖ E 1 )) ∼ = A * +2k, * +k (X; E ⊕ E 2 /E 1 ).
There is a monomorphism φ : E 2 /E 1 → E 3 arising from the exact sequence and composing the above isomorphism with φ A we obtain the claim.
Push-forwards along closed embeddings
In this section we recall the geometric part of the well-known construction of push-forwards along closed embeddings for cohomology theories possessing Thom classes (cf. [Ne06,Ne07,PS03,S07a,PS09,PW10a]). In our exposition we follow [Ne07] adapting it to our context. The key ingredient of the presented construction is the homotopy purity theorem [MV99, Section 3, Theorem 2.23].
Definition 19. Let i : Y → X be a closed embedding of smooth varieties with normal bundle N i of rank n. The deformation space D(X, Y ) is obtained as follows.
(1) Consider X × A 1 .
(2) Blow-up it along Y × 0.
(3) Remove the blow-up of X × 0 along Y × 0. This construction produces a smooth variety D(X, Y ) over A 1 . The fiber over 0 is canonically isomorphic to N i . The fiber over 1 is isomorphic to X. We have the corresponding closed embeddings i 0 : N i → D(X, Y ) and i 1 : X → D(X, Y ). There is a closed embedding z : Y × A 1 → D(X, Y ) such that over 0 it coincides with the zero section s : Y → N i of the normal bundle and over 1 it coincides with the closed embedding i : Y → X. Finally, we have a projection p : D(X, Y ) → X.
Thus we have the following homomorphisms of A * , * (X)-modules with module structure given by p A :
A * , * (D(X, Y )/(D(X, Y ) − Y × A 1 )) i A 0 t t ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ i A 1 * * ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ A * , * (N i /(N i − Y )) A * , * (X/(X − Y ))
These homomorphisms are isomorphisms since by [MV99, Proposition 2.24] the corresponding maps are isomorphisms in the pointed motivic homotopy category H • (k):
D(X, Y )/(D(X, Y ) − Y × A 1 ) N i /(N i − Y ) i A 0 ≃ 5 5 ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ X/(X − Y ) i A 1 ≃ i i ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ We set d A i = i A 1 • (i A 0 ) −1 : A * , * (N i /(N i − Y )) ≃ − → A * , * (X/(X − Y ))
to be the deformation to the normal bundle isomorphism.
Definition 20. Let i : Y → X be a closed embedding of smooth varieties with a rank n normal bundle N i . Applying Jouanalou device to Y and X accordingly we may assume them to be affine. More precisely, apply Jouanalou device to X producing an A r -bundle p : X → X with affine X and consider cartesian square
Y j / / q X p Y i / / X Here q : Y → Y is an A r -bundle, Y is affine and there is a canonical isomor- phism of normal bundles q * N i ∼ = N j .
Let E be a rank m vector bundle over X and denote i E : Y → E the composition of i and zero section of E. Note that there is a short exact sequence
0 → N i → N i E → i * E → 0,
choosing a splitting (Y is affine) we obtain an isomorphism
A * , * (Y ; N i E ) ∼ = A * , * (Y ; i * E ⊕ N i )
as in Lemma 8, and this isomorphism do not depend on the splitting. Identify
A * +2(m+n), * +(m+n) (E/(E − Y )) ∼ = A * , * (Y ; N i E ) ∼ = A * , * (Y ; i * E ⊕ N i ),
A * +2m, * +m (E/(E − X)) ∼ = A * , * (X; E) and let z : E/(E − X) → E/(E − Y ) be the quotient map. Then
i E A = z A • d A i E : A * , * (Y ; i * E ⊕ N i ) → A * +2n
, * +n (X; E) is the push-forward map. We will usually omit vector bundle from the notation for the push-forward map and write i A = i E A . One can easily extend the definition of push-forward map from cohomology groups twisted by a vector bundle to cohomology groups twisted by a formal difference of vector bundles E = E ′ ⊖ E: we already used Jouanalou device and all the involved morphisms were pull-backs, which are compatible with canonical isomorphisms Θ ( E,ρ) (E,θ) . Thus we have push-forward maps
i A : A * , * (Y ; i * E ⊕ N i ) → A * +2n, * +n (X; E).
Sometimes it will be more convenient to write the push-forward map in the following way:
i A : A * , * (Y ; i * E ⊖ T Y ) → A * +2n, * +n (X; E ⊖ T X ).
The latter map is quite the same as the former one since N i = i * T X /T Y and varying E we may derive one from the other using Lemma 7 and Lemma 8.
Remark 7. Push-forward i A is clearly a homomorphism of left twisted A * , * (X)modules, i.e. the projection formula holds:
i A (i A (α) ∪ β) = α ∪ i A (β) for α ∈ A * , * (X; E 1 ), β ∈ A * , * (Y ; i * E 2 ⊕ N i ),
where E 1 and E 2 are formal differences of vector bundles over X.
In the next two lemmas we establish basic properties of push-forward maps: compatibility with base change and functoriality.
Lemma 9 (cf. [Ne07, Corollary 3.8]). Consider the following cartesian diagram of smooth varieties
Y ′ i ′ / / f Y X ′ f Y i / / X
with all the morphisms being smooth. Suppose that i and i ′ are closed embeddings of codimension n and m. Denote φ : N i ′ → f * Y N i the induced monomorphism of vector bundles and let E = E ′ ⊖ E be a formal difference of vector bundles over X. Then the following diagram commutes.
A * +2n, * +n (Y ′ ; (if Y ) * E ⊕ N i ′ ) i ′ A / / A * +2(m+n), * +(m+n) (X ′ ; f * E) A * +2m, * +m (Y ′ ; (if Y ) * E ⊕ f * Y N i ) (id⊕φ) A O O A * +2m, * +m (Y ; i * E ⊕ N i ) i A / / f A Y O O A * +2(m+n), * +(m+n) (X; E) f A O O
Proof. There is the following commutative diagram (see [Ne07,Proposition 3.4] for the details).
N i ′ /(N i ′ − Y ′ ) i ′ 0 ≃ / / φ D(X ′ , Y ′ )/(D(X ′ , Y ′ ) − Y ′ × A 1 ) f D X ′ /(X ′ − Y ′ ) i ′ 1 ≃ o o f X ′ f o o f * Y N i /(f * Y N i − Y ′ ) f Y N i /(N i − Y ) i 0 ≃ / / D(X, Y )/(D(X, Y ) − Y × A 1 ) X/(X − Y ) i 1 ≃ o o X o o
Here all the morphisms of quotient spaces are induced by the corresponding morphism of varieties. The claim for E = 0 follows from the commutativity of the diagram. In order to obtain the claim for E = E ′ we may consider a similar diagram with the following modifications: X replaced by E ′ , X ′ replaced by
f * E ′ , N i replaced by N i E ′ and N i ′ replaced by N (i ′ ) f * E ′ .
Finally, the case of E = E ′ ⊖ E formally follows from the above: one needs to consider E = E ′ ⊕ E with E being a complement to E.
A * +2m, * +m (Y ; j * E ⊕ N j ) j A / / A * +2m+2n, * +m+n (X; E) A * , * (Z; (ji) * E ⊕ N i ⊕ i * N j ) i A O O ≃ / / A * , * (Z; (ji) * E ⊕ N ji ) (ji) A O O
Here the bottom isomorphism is the canonical one given by Lemma 8 applied to the short exact sequence
0 → N i → N ji → i * N j → 0.
Proof. Suppose that E = 0 and let D(X, Y, Z) = D(D(X, Z), D(Y, Z)) be the double deformation space. Then there exists the following commutative diagram (see [Ne07,5.2] for the details).
N j /(N j − Y ) i0 ≃ / / i1 D(X, Y )/(D(X, Y ) − Y × A 1 ) X i1 o o i1 D(N j , i * N j )/(D(N j , i * N j ) − Z × A 1 ) / / D(X, Y, Z)/(D(X, Y, Z) − Z × A 2 ) D(X, Z)/(D(X, Z) − Z × A 1 ) o o (N i ⊕ i * N j )/(N i ⊕ i * N j − Z) i0 ≃ O O i0 ≃ / / D(N ji , N i )/(D(N ji , N i ) − Z × A 1 ) O O N ji /(N ji − Z) i0 ≃ O O i1 o o
Here we denote by i 0 and i 1 the morphisms arising from various deformations to the normal bundles and identify the total spaces of the normal bundles for the closed embeddings i * N j → N j and i * N j → N ji with the total space of i * N j ⊕ N i . One can easily see that the left column and top row of the diagram represent i A and j A respectively, the right column represents (ji) A , and the bottom row gives us the canonical isomorphism described in Lemma 8.
We leave to the reader substitute everywhere E ′ for X and make all the necessary modifications in order to obtain the claim for E = E ′ . As usual, the claim for general E = E ′ ⊖ E formally follows from the case of vector bundle: one needs to consider E = E ′ ⊕ E with E being a complement to E.
Twisted cohomology of projective space
This section is the main part of the paper, here we compute twisted cohomology groups A * , * η (P n X ; p * 1 V ⊕ p * 2 E) where E = E 1 ⊖ E 2 is a formal difference of vector bundles over P n , V = V 1 ⊖ V 2 is a formal difference of vector bundles over X, p 1 : P n X → X and p 2 : P n X → P n are canonical projections. In the next two sections this computation will allow us to define projective push-forwards for A * , * η (−). We start with the following easy but useful lemma.
Lemma 11. Let X be a smooth variety, let s ∈ k[X] * be an invertible regular function and consider a morphism f s 2 : X + ∧ T → X + ∧ T given by f s 2 (x, u) = (x, s(x) 2 u). Then Σ ∞ T f s 2 = id in SH(k). Proof. Canonical isomorphism T ∼ = (P 1 , ∞) combined with the splitting Hom SH(k) ((X × P 1 ) + , X + ∧ (P 1 , ∞)) = = Hom SH(k) (X + ∧ (P 1 , ∞), X + ∧ (P 1 , ∞)) ⊕ Hom SH(k) (X + , X + ∧ (P 1 , ∞)) yields that it is sufficient to show that f s 2 = g ∈ Hom H•(k) ((X × P 1 ) + , X + ∧ (P 1 , ∞)),
where f s 2 (x, [u : v]) = (x, [s(x) 2 u : v]) and g(x, [u : v]) = (x, [u : v]). We have g(x, [u : v]) = (x, [u : v]) = (x, [s(x)u : s(x)v]), hence g = f • h with h : (X × P 1 ) + → (X × P 1 ) + , h(x, [u : v]) = (x, [s(x) −1 u, s(x)v]).
By Lemma 1 we have h = id in H • (k) and the claim follows.
Lemma 12. Let X be a smooth variety and let A be a commutative ring T -spectrum. Denote 0 P = [0 : 0 : . . . : 0 : 1] ∈ P 2n−1 , 1 A = (1, 1, . . . , 1) ∈ A 2n − 0.
Then projection H 2n : A 2n − 0 → P 2n−1 given by 0, 1 A )). Proof. The proof is purely geometrical and do not depend on the base, so we omit X from notation. One may smash everything with X + and use the same the reasoning.
H 2n (x 1 , x 2 , . . . , x 2n ) = [x 1 : x 2 : . . . x 2n ] induces an isomorphism H A X,2n : A * , * η (X + ∧ (P 2n−1 /(P 2n−1 − 0 P ))) ≃ − → A * , * η (X + ∧ (A 2n −
Put Y = ((A 2n−2 × (1, 1)) ∪ ((A 2n−2 − 0) × A 2 )) ⊂ A 2n − 0 and consider the following diagram.
T ∧2n−2 ∧ (P 1 , [1 : 1]) ≃ T ∧2n−2 ∧ (A 2 − 0, (1, 1)) i id∧H 2 o o ≃ / / (A 2n − 0)/Y H 2n | | T ∧2n−2 ∧ (P 1 /A 1 ) T ∧2n−2 ∧ ((A 2 − 0)/(A 1 × G m )) id∧H 2 o o H 2n & & ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ (A 2n − 0, 1 A ) ≃ O O H 2n T ∧2n−2 ∧ T ≃ O O T ∧2n−2 ∧ T ∧ G m+ ≃ O O id∧φ t t ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ id∧ψ o o Φ T ∧2n−2 ∧ T T ∧2n−1 j ≃ / / P 2n−1 /(P 2n−1 − 0 P )
Here all the arrows marked with "≃" are induced by tautological embeddings and these maps are isomorphisms by excision or A 1 -contractibilty, using that Y could be contracted to a point in two steps:
A 2n−2 × (1, 1) ∪ (A 2n−2 − 0) × A 2 ∼ A 2n−2 × (1, 1) ∼ 1 A .
Morphism i is also induced by tautological inclusion. We denote by H 2n all the maps induced by H 2n . Morphisms ψ, φ : T ∧ (G m+ , +) → T are given by ψ(x, t) = x/t and φ(x, t) = x/t 2n−1 accordingly. Finally, j is given by j(x 1 , x 2 , . . . , x 2n−1 ) = [x 1 : x 2 : · · · : x 2n−1 : 1]
and Φ is given by Φ(x 1 , x 2 , . . . , x 2n−1 , t) = (x 1 /t, x 2 /t, . . . , x 2n−1 /t).
A straightforward check shows that the diagram commutes except possibly for the lower-left square involving ψ, φ and Φ. Morphisms id∧φ and Φ coincide in H • (k) by Lemma 1, since they differ by an automorphism of T ∧2n−1 ∧G m+ = T h(½ 2n−1 Gm ) given by diagonal matrix diag(t, t, . . . , t, 1/t 2n−2 ). Lemma 11 yields that for
f t 2(n−1) : T ∧ G m+ → T ∧ G m+ , f t 2(n−1) (x, t) = (t 2(n−1) x, t), we have Σ ∞ T ψ = Σ ∞ T (φf t 2(n−1) ) = Σ ∞ T φΣ ∞ T f t 2(n−1) = Σ ∞ T φ and Σ ∞ T ψ = Σ ∞ T φ. Hence the Σ ∞
T -suspension of the diagram commutates. The claim of the lemma follows from the diagram chase combined with the observation that
(Σ 2n−1 T η) A = (id ∧ H 2 ) A : A * , * η (T ∧2n−2 ∧ (P 1 , [1 : 1])) → A * , * η (T ∧2n−2 ∧ (A 2 − 0, (1, 1)))
is an isomorphism since we have inverted η in the coefficients.
Theorem 1. Let A be a commutative ring T -spectrum and let X be a smooth variety. Then (1) projection p : P 2n X → X induces an isomorphism A * , * η (P 2n X ) ∼ = A * , * η (X); (2) projection H 2n induces an isomorphism
H A X,2n : A * , * η (P 2n−1 X ) ≃ − → A * , * η (A 2n X − (0 × X))
, and, together with a choice of a section s :
X → A 2n X −(0 ×X), induces an isomorphism A * , * η (P 2n−1 X ) ∼ = A * , * η (X) ⊕ A * −4n+1, * −2n η (X).
Proof. The proof do not depend on the base, so we omit it from the notation. Not that the first claim is equivalent to the claim that A * , * η (P 2n , x) = 0 for a rational point x ∈ P 2n and the second is equivalent to the claim that H A 2n : A * , * η (P 2n−1 , H 2n (y)) − → A * , * η (A 2n − 0, y) is an isomorphism for a rational point y ∈ A 2n − 0.
Proceed by induction: the case of P 0 is trivial. Denote 1 P k = [1 : 1 : . . . : 1] ∈ P k , 0 P k = [0 : . . . : 0 : 1] ∈ P k , 1 A k = (1, 1, . . . , 1) ∈ A k − 0.
2n → 2n + 1. Consider the long exact sequence associated to the closed embedding (P 2n , 1 P 2n ) → (P 2n+1 , 1 P 2n+1 ) given by x → [1 : x]:
. . . − → A * −1, * η (P 2n , 1 P 2n ) ∂ − → A * , * η (P 2n+1 /P 2n ) r A − → A * , * η (P 2n+1 , 1 P 2n+1 ) − → A * , * η (P 2n , 1 P 2n ) ∂ − → . . .
By the induction assumption we know that A * , * η (P 2n , 1 P 2n ) = 0, thus r A is an isomorphism. Moreover, P 2n+1 − 0 P 2n+1 ∼ = O P 2n (1) is a vector bundle over P 2n yielding A * , * η (P 2n+1 /(P 2n+1 − 0 P 2n+1 )) ∼ = A * , * η (P 2n+1 /P 2n ). Applying Lemma 12 we obtain:
A * , * η (P 2n+1 , 1 P 2n+1 ) ∼ = A * , * η (P 2n+1 /P 2n ) ∼ = A * , * η (P 2n+1 /(P 2n+1 − 0 P 2n+1 )) ∼ = ∼ = A * , * η (A 2n+2 − 0, 1 A 2n+2 ) ∼ = A * −4n−3, * −2n−2 η (pt).
One can easily check that the above composition is given precisely by H A 2n+2 .
2n − 1 → 2n. A similar argument as above shows that the quotient map induces an isomorphism A * , * η (P 2n−1 /P 2n−2 ) ≃ − → A * , * η (P 2n−1 , 1 P 2n−1 ). Hence Lemma 12 gives an isomorphism
H A 2n : A * , * η (P 2n−1 ) ≃ − → A * , * η (A 2n − 0)
. Recall that the complement to the zero section of the tautological line bundle over P 2n−1 is isomorphic to A 2n − 0 and the canonical projection composed with this isomorphism gives a projection A 2n − 0 → P 2n−1 which coincides with H 2n . The long exact sequence
. . . / / A * , * η (T h(O P 2n−1 (1))) / / A * , * η (O P 2n−1 (1)) / / A * , * η (O P 2n−1 (1) 0 ) / / . . . A * , * η (P 2n−1 ) H 2n ≃ / / A * , * η (A 2n − 0) yields A * , * η (T h(O P 2n−1 (1))) = 0. Embed A 2n → P 2n , x → [x : 1]. Excision provides an isomorphism A * , * η (P 2n /A 2n ) ∼ = A * , * η (T h(O P 2n−1
(1))) = 0. A 1 -contractibility of A 2n gives us the claim:
A * , * η (P 2n , 1 P 2n ) ∼ = A * , * η (P 2n /A 2n ) = 0. Corollary 5.
Let A be a commutative ring T -spectrum and let E be a vector bundle of even rank over a smooth variety X. Then canonical projection p : P X (E) → X induces an isomorphism p A : A * , * η (X) ≃ − → A * , * η (P X (E)). Proof. Follows via Mayer-Vietoris long exact sequence.
Corollary 6. Let A be a commutative ring T -spectrum and let V = V ′ ⊖ V be a formal difference of vector bundles over a smooth variety X. Then projection p : P 2n X → X induces an isomorphism p A : A * , * η (X; V) ≃ − → A * , * η (P 2n X ; p * V). Proof. As usual, it is sufficient to deal with the case of V = V ′ , while the general case formally follows. Consider the following diagram consisting of long exact sequences associated to the zero sections of V ′ and p * V ′ .
. . . / / A * −2n, * −n η (X; V ′ ) / / p A A * , * η (X) / / ≃ p A A * , * η (V ′ − X) / / ≃ p A . . . . . . / / A * −2n, * −n η (P 2n X ; p * V ′ ) / / A * , * η (P 2n X ) / / A * , * η (p * V ′ − P 2n X ) / / . . . Note that p * V ′ − P 2n X = (V ′ − X) × P 2n
, so the second and the third arrows are isomorphisms by Theorem 1. The claim follows via 5-lemma.
Theorem 2. Let X be a smooth variety and let A be a commutative ring T -spectrum. Denote p 1 : P k X → X and p 2 : P k X → P k the canonical projections. Consider a formal difference V = V ′ ⊖ V of vector bundles over X, a formal difference E = E ⊖ E ′ of vector bundles over P k of degree d = deg det E − deg det E ′ and a rational point a ∈ P k . Denote i a : X → P k X the closed embedding given by i(x) = (x, a). Then we have the following isomorphisms depending on the parity of d and k.
Ia) If d = 2m − 1, k = 2n − 1 then A * , * η (P k X ; p * 1 V ⊕ p * 2 E) = 0. Ib) If d = 2m, k = 2n − 1 then there is a split short exact sequence A * −2k, * −k η (X; i * a (p * 1 V ⊕ p * 2 E) ⊕ N ia ) (ia) A − −− → A * , * η (P k X ; p * 1 V ⊕ p * 2 E) i A − → A * , * η (X; i * a (p * 1 V ⊕ p * 2 E)). IIa) If d = 2m − 1, k = 2n then A * , * η (P k X ; p * 1 V ⊕ p * 2 E) (ia) A ← −− − A * −2k, * −k η (X; i * a (p * 1 V ⊕ p * 2 E) ⊕ N ia ) is an isomorphism. IIb) If d = 2m, k = 2n then A * , * η (P k X ; p * 1 V ⊕ p * 2 E) i A a − → A * , * η (X; i * a (p * 1 V ⊕ p * 2 E)) is an isomorphism.
Proof. It is sufficient to obtain the claim in the case of E ′ = 0, i.e. when E = E is a vector bundle, and the general case formally follows. We shorten i a to i omitting the index a. The proof do not depend on the base X and formal difference of vector bundles V, so we omit them from notation and suppose that X = pt, V = 0. In case of nontrivial X and V everything is virtually the same except that one should use Corollary 6 in place of Theorem 1.
We will argue by induction on the dimension of projective space. The case of P 0 is clear. Throughout the proof we will continuously use the following long exact sequences associated to a linear embedding r : P l → P m . The complement q : P m − P l → P m is a vector bundle over P m−l−1 and P m /(P m − P l ) is isomorphic to the Thom space T h(O P l (1) ⊕m−l ) via excision. Thus for a vector bundle E over P m we have a long exact sequence · · · → A * −2m+2l, * −m+l
η (P l ; r * E ⊕ O P l (1) ⊕m−l ) r A − → A * , * η (P m ; E) q A − → A * , * η (P m−l−1 ; q * E) → . . .
Ia) for P 1 . Embed P 1 to P 2m . We have a long exact sequence · · · → A * −4m+2, * −2m+1 η (P 1 ; O P 1 (1) ⊕2m−1 ) → A * , * η (P 2m ) → A * , * η (P 2m−2 ) → . . . . Theorem 1 yields that q A is an isomorphism and A * , * η (P 1 ; O P 1 (1) ⊕2m−1 ) = 0. Corollary 3 combined with Lemma 3 and the fact that
K 0 (P 1 ) = Z[½ P 1 ] ⊕ Z[O P 1 (1)]
gives claim Ia) for P 1 .
Ia)⇒ Ib). Consider the long exact sequence associated to the embedding i : P 0 → P 2n−1 .
· · · → A * −4n+2, * −2n+1 η (P 0 ; i * E ⊕ O P 0 (1) ⊕2n−1 ) i A − → A * , * η (P 2n−1 ; E) q A − → A * , * η (P 2n−2 ; q * E) → . . .
Induction assumption yields an isomorphism A * , * η (P 2n−2 ; q * E) ∼ = A * , * η (pt; i * E). Thus we obtain a long exact sequence
· · · → A * −4n+2, * −2n+1 η (pt; i * E ⊕ N i ) i A − → A * , * η (P 2n−1 ; E) i A − → A * , * η (pt; i * E) → . . .
Hence it is sufficient to show that i A is a split surjection. Consider the twisted version of the localization sequence for the complement to the zero section of O P 2n−1 (1) (recall that we denote the canonical projection for the complement by H 2n : A 2n − 0 → P 2n−1 ):
· · · → A * −2, * −1 η (P 2n−1 ; E ⊕ O P 2n−1 (1)) → A * , * η (P 2n−1 ; E) H A 2n − − → A * , * η (A 2n − 0; H * 2n E) → . . .
Assumption Ia) gives A * −2, * −1 η (P 2n−1 ; E ⊕ O P 2n−1 (1)) = 0, thus H A 2n is an isomorphism. Hence it is sufficient to show that
j A : A * , * η (A 2n − 0; H * 2n E) → A * , * η (pt; j * H * 2n E)
is a split surjection for every embedding j : pt → A 2n − 0. It basically follows from Corollary 3 and the fact that K 0 (A 2n − 0) = 0. More precisely, every vector bundle over Y = A 2n − 0 stably trivial, thus we may find some s, t and an isomorphism θ :
H * 2n E ⊕ ½ s Y ∼ = ½ t Y .
Lemma 4 together with suspension isomorphisms gives us commutative diagram
A * , * η (Y ; H * 2n E) ≃ / / j A A * , * η (Y ; H * 2n E ⊕ ½ s Y ) ≃ θ A / / j A A * , * η (Y ; ½ t Y ) j A A * , * η (Y ) ≃ o o j A A * , * η (pt; j * H * 2n E) ≃ / / A * , * η (pt; j * H * 2n E ⊕ ½ s pt ) ≃ j * θ A / / A * , * η (pt; ½ t pt ) A * , * η (pt) ≃ o o
We denote the vertical morphisms by the same letter since all of them are induced by the embedding j. The rightmost vertical morphism is clearly surjective with the splitting given by p A for the projection p : Y → pt. Hence the leftmost vertical morphism is a split surjection and we get the claim. Ia)⇒ IIa). Consider the long exact sequence associated to the embedding i : P 0 → P 2n .
· · · → A * −4n, * −2n η (P 0 ; i * E ⊕ O P 0 (1) ⊕2n ) i A − → A * , * η (P 2n ; E) → A * , * η (P 2n−1 ; q * E) → . . . Assumption Ia) yields that A * , * η (P 2n−1 ; q * E) = 0, thus i A is an isomorphism. Ia)⇒ IIb).
Consider an embedding P 2n−1 → P 2n and the corresponding long exact sequence · · · → A * −2, * −1 η (P 2n−1 ; r * E ⊕ O P 2n−1 (1)) → A * , * η (P 2n ; E) q A − → A * , * η (P 0 ; q * E) → . . . Assumption Ia) yields that A * −2, * −1 η (P 2n−1 ; r * E ⊕ O P 2n−1 (1)) = 0, thus q A is an isomorphism and every choice of a rational point provides a splitting isomorphism.
Ia) for P 2n−1 ⇒ Ia) for P 2n+1 . Embed P 1 to P 2n+1 and consider the long exact sequence · · · → A * −4n, * −2n
η (P 1 ; r * E ⊕ O P 1 (1) ⊕2n ) → A * , * η (P 2n+1 ; E) → A * , * η (P 2n−1 ; q * E) → . . .
By the induction assumption we have A * , * η (P 1 ; r * E ⊕ O P 1 (1) ⊕2n ) = 0 and A * , * η (P 2n−1 ; q * E) = 0, so the claim follows. We sum up our computations in the following form. Note that the isomorphisms in the following Corollary depend on the choice of trivializations of certain bundles. Corollary 7. In notation of Theorem 2 we have the following isomorphisms depending on the parity of d and k.
Ia) d = 2m − 1, k = 2n − 1 : A * , * η (P k X ; p * 1 V ⊕ p * 2 E) = 0. Ib) d = 2m, k = 2n − 1 : A * , * η (P k X ; p * 1 V ⊕ p * 2 E) ∼ = A * , * η (X; V) ⊕ A * −2k, * −k η (X; V)
.
IIa) d = 2m − 1, k = 2n : A * , * η (P k X ; p * 1 V ⊕ p * 2 E) ∼ = A * −2k, * −k η (X; V). IIb) d = 2m, k = 2n : A * , * η (P k X ; p * 1 V ⊕ p * 2 E) ∼ = A * , * η (X; V).
diagram consisting of canonical projections.
X × P m × P n p ′ / / q ′ X × P m q X × P n p / / X Denote r = p ′ q = q ′ p : X × P m × P n → X.
Then the following diagram commutes.
A * , * η (X × P m × P n ; r * V ⊖ T X×P m ×P n ) p ′ A / / q ′ A A * −2n, * −n η (X × P m ; q * V ⊖ T X×P m ) q A A * −2m, * −m η (X × P n ; p * V ⊖ T X×P n ) p A / / A * −2(m+n), * −(m+n) η (X; V ⊖ T X )
Proof. Consider the following diagram.
X × P 2k−1 × P 2l−1 / / X × P 2k × P 2l−1 / / X × P 2l−1 X × P 2k−1 × P 2l / / X × P 2k × P 2l / / X × P 2l X × P 2k−1 / / X × P 2k / / X
Then consider the corresponding diagram of push-forward maps (we omit it since it should contain all the twists occupying a lot of space). In that diagram one may invert push-forwards along projection for projective spaces of even dimension obtaining a diagram consisting of push-forwards along closed embeddings. The latter diagram commutes by Lemma 10, thus the former one commutes as well. Examining these commutative diagrams one obtains the claim of the lemma: the bottom-right square corresponds to the case when both m and n are even, the boundary contour gives claim for both m and n being odd, and the right and bottom halves of the diagram correspond to the cases with one of m and n being odd.
Lemma 15 (cf. [Ne09a,3.4]). Let V = V ′ ⊖ V be a formal difference of vector bundles over a smooth variety X, let f : X ′ → X be a smooth morphisms of smooth varieties and let p : P n X → X be the canonical projection. Consider the following cartesian diagram.
P n
X ′ f ′ / / p ′ P n X ′ p X ′ f / / X Denote r = f ′ p = p ′ f : P n X ′ → X.
Then the following diagram commutes.
A * , * η (P n X ′ ; r * V ⊖ T P n X ′ ) p ′ A A * , * η (P n X ; p * V ⊖ T P n X ) p A (f ′ ) A o o A * −2n, * −n η (X ′ ; f * V ⊖ T X ′ ) A * −2n, * −n η (X; V ⊖ T X ) f A o o
Proof. Consider the following diagrams.
P 2l−1
X ′ f 2 / / j ′ P 2l−1 X j P 2l X ′ f 1 / / p ′ 1 P 2l X p 1 X ′ f / / i ′ T T X i J J A * −2, * −1 η (P 2l−1 X ′ ; r * 2 V ⊖ T P 2l−1 X ′ ) j ′ A A * −2, * −1 η (P 2l−1 X ; p * 2 V ⊖ T P 2l−1 X ) j A f A 2 o o A * , * η (P 2l X ′ ; r * 1 V ⊖ T P 2l X ′ ) (p ′ 1 ) A A * , * η (P 2l X ; p * 1 V ⊖ T P 2l X ) (p 1 ) A f A 1 o o A * −4l, * −2l η (X ′ ; f * V ⊖ T X ′ ) i ′ A R R A * −4l, * −2l η (X; V ⊖ T X ) f A o o i A K K
Here p 1 and p ′ 1 are natural projections, j and j ′ induced by a linear embedding P 2l−1 → P 2l , i and i ′ are given by i(x) = (x, a) and i ′ (x ′ ) = (x ′ , a) for some rational point a ∈ P 2l and both squares in the left diagram are cartesian; in the right diagram p 2 = p 1 j : P 2l−1 X → X is the canonical projection, r 1 = p 1 f 1 = f p ′ 1 and r 2 = f p ′ 1 j ′ = p 1 jf 2 . By Lemma 9 we have f A
1 j A = j ′ A f A 2 and f A 1 i A = i ′ A f A , thus, since (p 1 ) A = i −1 A and (p ′ 1 ) A = (i ′ ) −1
A , the right diagram commutes. The claim of the lemma follows from the commutativity of the lower square in case of n = 2l and from the commutativity of the outer contour in case of n = 2l − 1.
Lemma 16 (cf. [Ne09a, 3.5]). Let V = V ′ ⊖ V be a formal difference of vector bundles over a smooth variety X, let j : P m X → P n X be an embedding induced by a linear embedding P m → P n and let p m : P m X → X and p n : P n X → X be the canonical projections. Then the following diagram commutes.
A * , * η (P m X ; p * m V ⊖ T P m X )
j A / / (pm) A + + ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱
A * +2(n−m), * +(n−m) η (P n X ; p * n V ⊖ T P n X )
(pn) A A * −2m, * −m η (X; V ⊖ T X )
Proof. We give a detailed proof only for the case of m = 2k − 1, n = 2l − 1 since the other cases are quite the same. Consider the following diagrams.
P 2k−1 X j / / j1 P 2l−1 X j2 P 2k Xj / / p 2k ! ! P 2l X p 2lX i 2k O O i 2l P P A * , * η (P 2k−1 X ; p * 2k−1 V ⊖ T P 2k−1 X ) j A / / (j1) A A * +4(l−k), * +2(l−k) η (P 2l−1 X ; p * 2l−1 V ⊖ T P 2l−1 X ) (j2) A A * +2, * +1 η (P 2k X ; p * 2k V ⊖ T P 2k X )j A / / (p 2k ) A + + ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ A * +4(l−k)+2, * +2(l−k)+1 η (P 2l X ; p * 2l V ⊖ T P 2l X ) (p 2l ) A
A * −2(2k−1), * −(2k−1) η (X; V ⊖ T X )
(i 2k ) A h h (i 2l ) A W W
Here p 2k and p 2l are canonical projections, i 2k and i 2l are given by i 2k (x) = (x, a), i 2l (x) = (x,j(a)) for some rational point a ∈ P 2k and j 1 , j 2 ,j are closed embeddings induced by linear embeddings in such a way that the left diagram commutes. In the right diagram we have (j 2 ) A j A =j A (j 1 ) A and (i 2l ) A = j A (i 2k ) A by Lemma 10. We have (p 2k ) A = (i 2k ) −1 A and (p 2l ) A = (i 2l ) −1 A , thus the right diagram commutes, giving us the claim. In the rest cases one should consider similar diagrams.
Lemma 17 (cf. [Ne09a,3.6]). Let V = V ′ ⊖ V be a formal difference of vector bundles over a smooth variety X, let j : Y → X be a codimension m closed embedding of smooth varieties and let p : P n X → X and p ′ : P n Y → Y be canonical projections. Denote j ′ : P n Y → P n X the closed embedding induced by j and put r = jp ′ = j ′ p. Then the following diagram commutes.
A * , * η (P n Y ; r * V ⊖ T P n Y ) j ′ A / / p ′ A A * +2m, * +m η (P n X ; p * V ⊖ T P n X ) p A A * −2n, * −n η (Y ; j * V ⊖ T Y ) j A / / A * +2(m−n), * +(m−n) η (X; V ⊖ T X )
Proof. Suppose that n = 2k − 1. Consider the following diagrams. Here p 2k and p ′ 2k are canonical projections, i and i ′ are given by i(x) = (x, a), i ′ (y) = (y, a) for some rational point a ∈ P 2k ,j is induced by j, and j 1 and j 2 are induced by a linear embedding P 2k−1 → P 2k , so the left diagram commutes. Thus the right diagram commutes by Lemma 10, and the claim follows. In case of n = 2k one should consider only the bottom parts of the diagrams.
P 2k−1 Y j ′ / / j1 P 2k−1 X j2 P 2k Yj / / p ′ 2k P 2k X p 2k Y i ′ T T j / / X i J J A * , * η (P 2k−1 Y ; r * V ⊖ T P 2k−1 Y ) j ′ A / / (j1) A A * +2m, * +m η (P 2k−1 X ; p * V ⊖ T P 2k−1 X ) (j2) A A * +2, * +1 η (P 2k Y ; (jp ′ 2k ) * V ⊖ T P 2k Y )j A / / (p ′ 2k ) A A * +2m+2, * +m+1 η (P 2k X ; p * 2k V ⊖ T P 2k X ) (p 2k ) A
Lemma 18 (cf. [Ne09a, Proposition 3.9]). Let V = V ′ ⊖ V be a formal difference of vector bundles over a smooth variety X and let s : X → P n X be a section of the canonical projection p : P n X → X, i.e. s = (id, f ) for some morphism f : X → P n . Then p A s A = id : A * , * η (X; V ⊖ T X ) → A * , * η (X; V ⊖ T X ). Proof. It is sufficient to show that p A s A (1) = 1 in case of V = T X , the general case follows at once since s A and p A are homomorphisms of left twisted A * , * (X)-modules and projection formula holds. First we obtain the claim for n = 2k, X = P 2k and diagonal section s = ∆ = (id, id), and then derive the general case.
• n = 2k, X = P 2k , s = ∆ = (id, id). Choose a rational point a ∈ P 2k , denote the corresponding embedding i : pt → P 2k and define i a , i ′ a : P 2k → P 2k × P 2k via i a (x) = (x, a) and i ′ a (x) = (a, x). Let p 1 , p 2 : P 2k × P 2k → P 2k be the canonical projections to the first and second factors. Consider the following commutative diagrams.
P 2k ∆ / / ia / / P 2k × P 2k pt i O O i / / P 2k i ′ a O O A * , * η (P 2k ) ∆ A / / (ia) A / / i A A * +4k, * +2k η (P 2k × P 2k ; ⊖p * 2 T P 2k ) (i ′ a ) A
Definition 8 .
8The Hopf map is the morphism of pointed motivic spaces H : (A 2 − {0}, (1, 1)) → (P 1 , [1 : 1]), H(x, y) = [x : y]. Using canonical isomorphisms (A 2 − 0, (1, 1)) ∼ = S 3,2 (see [MV99, Example 2.20]) and (P 1 , [1 : 1]) ∼ = S 2,1 (see [MV99, Lemma 2.15 and Corollary 2.18])
Lemma 10 (cf. [Ne07, Proposition 5.1]). Let Z i − → Y j − → X be closed embeddings of smooth varieties of codimensions m and n. Consider a formal difference E = E ′ ⊖ E of vector bundles over X. Then the following diagram commutes.
Proof. The claim follows from Theorem 2 combined with Corollary 3 and.Push-forwards along projectionsIn this section we use the computation from Theorem 2 to define pushforward maps along canonical projections P n X → X and obtain basic properties of these push-forward maps.Lemma 13 (cf. [Ne09a, Lemma 3.1 (ii)]). Let V = V ′ ⊖ V be a formal difference of vector bundles over a smooth variety X. Consider two linear embeddings j 1 , j 2 : P m → P k and denote by p k : P k X → X and p m : P m X → X the canonical projections. ThenProof. There is an isomorphism α : P k → P k given by a matrix of SL k+1 (k)Definition 21. Let V = V ′ ⊖ V be a formal difference of vector bundles over a smooth variety X and let p : P k X → X be the canonical projection. k = 2n. Choose a rational point a ∈ P k and denote i a : X → P k X the embedding i a (x) = (x, a). Recall that deg T P k is odd, thus Theorem 2 allows us to defineConsider a linear embedding j : P k → P k+1 and denotep : P k+1 X → X the canonical projection. DefineWe refer to p A in both cases as push-forward map along projection p. By Lemma 13 the defined push-forwards p A do not depend on the choice of rational point a and linear embedding j.Remark 8. Note that by Remark 7 push-forward map p A is a homomorphism of left twisted A * , * η (X)-modules in the sense of Definition 17, i.e. the projection formula holds:where V 1 and V 2 are formal differences of vector bundles over X.In the rest part of the section we establish some basic properties of pushforwards along projections following[Ne09a]. We usually provide only some sketches of the proofs since one needs to introduce only slight modifications to the reasoning from loc. cit.Lemma 14 (cf.[Ne09a,3.3]). Let V = V ′ ⊖ V be a formal difference of vector bundles over a smooth variety X. Consider the following commutativeIn the next lemma we show that f A does not depend on the chosen factorization f = p • i.Remark 9. Note that by Remarks 7 and 8 push-forward map f A is a homomorphism of left twisted A * , * η (X)-modules in the sense of Definition 17, i.e. the projection formula holds:for α ∈ A * , * (X; V 1 ), β ∈ A * , * (Y ; f * V 2 ⊖ T Y ), where V 1 and V 2 are formal differences of vector bundles over X.Lemma 19 (cf. [Ne09a, Proposition 4.2]). Let V = V ′ ⊖ V be a formal difference of vector bundles over a smooth variety X and let f : Y → X be a pure codimension m projective morphism of smooth varieties. Then the pushforward mapdoes not depend on the choice of a factorization f = p • i.Proof. We split the proof into two steps. First we deal with a closed embedding f and a non-trivial decomposition f = p • i, and then we turn to the general case.• f = p•i is a closed embedding. It is sufficient to show that f A = p A i A with the first map f A being the push-forward along a closed embedding. Consider the following commutative diagram.Here the square is cartesian and s is the section of p ′ determined by i = f ′ s. Applying Lemmas 10, 17 and 18 we obtainis an arbitrary projective morphism. Consider the following diagram.Here the right square is cartesian and I is the unique morphism making the diagram commutative. The obtained result for a closed embedding together with Lemma 14 yieldsLemma 20 (cf.[Ne09a,4.5]). Let V = V ′ ⊖ V be a formal difference of vector bundles over a smooth variety X and let Z g − → Y f − → X be projective morphisms of smooth varieties of codimensions m and n. Then the following diagram commutes.A * +2(m+n), * +(m+n) η (X; V ⊖ T X )Proof. Decompose g = i 1 • p 1 : Z → Y × P 2k → Y, f = i 2 • p 2 : Y → X × P 2l → X and consider the following diagram.
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E-mail address: alseang@gmail. E-mail address: [email protected]
| [] |
[
"Thermal properties of 3BaO-3TiO 2 -B 2 O 3 glasses",
"Thermal properties of 3BaO-3TiO 2 -B 2 O 3 glasses"
] | [
"Rahul Vaish \nMaterials Research Centre\nIndian Institute of Science\n560 012BangaloreIndia\n",
"K B R Varma \nMaterials Research Centre\nIndian Institute of Science\n560 012BangaloreIndia\n"
] | [
"Materials Research Centre\nIndian Institute of Science\n560 012BangaloreIndia",
"Materials Research Centre\nIndian Institute of Science\n560 012BangaloreIndia"
] | [] | Transparent glasses in the system 3BaO-3TiO 2 -B 2 O 3 (BTBO) were fabricated via the conventional melt-quenching technique. The as-quenched samples were confirmed to be glassy by differential thermal analysis (DTA) technique. The thermal parameters were evaluated using non-isothermal DTA experiments. Kauzmann temperature was found to be 759 K based on heating rate dependent glass transition and crystallization temperatures. Theoretical relation for temperature dependent viscosity was proposed for these glasses and glass-ceramics.2 | null | [
"https://arxiv.org/pdf/1007.2276v1.pdf"
] | 116,892,509 | 1007.2276 | a99056618d728c6bbead24856c3004810f3d857b |
Thermal properties of 3BaO-3TiO 2 -B 2 O 3 glasses
Rahul Vaish
Materials Research Centre
Indian Institute of Science
560 012BangaloreIndia
K B R Varma
Materials Research Centre
Indian Institute of Science
560 012BangaloreIndia
Thermal properties of 3BaO-3TiO 2 -B 2 O 3 glasses
*Corresponding Author; E-Mail : [email protected]; 1
Transparent glasses in the system 3BaO-3TiO 2 -B 2 O 3 (BTBO) were fabricated via the conventional melt-quenching technique. The as-quenched samples were confirmed to be glassy by differential thermal analysis (DTA) technique. The thermal parameters were evaluated using non-isothermal DTA experiments. Kauzmann temperature was found to be 759 K based on heating rate dependent glass transition and crystallization temperatures. Theoretical relation for temperature dependent viscosity was proposed for these glasses and glass-ceramics.2
Abstract:
Transparent glasses in the system 3BaO-3TiO 2 -B 2 O 3 (BTBO) were fabricated via the conventional melt-quenching technique. The as-quenched samples were confirmed to be glassy by differential thermal analysis (DTA) technique. The thermal parameters were evaluated using non-isothermal DTA experiments. Kauzmann temperature was found to be 759 K based on heating rate dependent glass transition and crystallization temperatures. Theoretical relation for temperature dependent viscosity was proposed for these glasses and glass-ceramics.
Introduction:
Glasses embedded with polar crystallites have been of increasing importance because of their multifarious properties that include electro-optic, ferroelectric, pyroelectric, piezoelectric etc. [1][2][3][4]. Indeed the physical properties of these glass-ceramics are dependent upon the size and shape of the crystallites, their volume fraction and connectivity. These parameters could be optimized using a controlled heat-treatment process. We have been fabricating various ferroelectric glass-ceramics from the point of view of exploiting them for various aforementioned device applications.
Borate-based single crystalline materials which include Li 2 B 4 O 7 , LiB 3 O 5 , BaB 2 O 4 , BiB 3 O 6 , etc. were reported to exhibit interesting non-linear optic and related properties [5][6][7][8]. These crystals have wide transmission window, high optical damage threshold, good chemical and mechanical stability. We thought it was worth investigating into the physical properties of the glass-ceramics of these fascinating materials as this route facilitates to accomplish strict control over their microstructural characteristics and hence the physical properties. Ba 3 Ti 3 O 6 (BO 3 ) 2 crystals were reported to possess excellent non-linear optical properties [9,10]. The optical second harmonic generation (SHG) efficiency of these crystals was about 95% of that of LiNbO 3 single crystals [10]. We have been making systematic attempts to fabricate glasses and glass-ceramics in the BaO-TiO 2 -B 2 O 3 system and visualize their physical properties. The composition for the present study is chosen in such a way that one could obtain Ba 3 Ti 3 O 6 (BO 3 ) 2 crystalline phase on crystallization of 3BaO-3TiO 2 -B 2 O 3 glasses. To begin with a detailed analyses of the glass-transition, crystallization kinetics and viscosity behavior of the glasses are in order. This would facilitate the optimization of certain parameters for obtaining transparent glass nano/microcrystals composites. The details pertaining to these studies are reported in the following sections.
Experimental:
Transparent glasses in the composition 3BaO-3TiO 2 -B 2 O 3 (in molar ratio) were fabricated via the conventional melt-quenching technique. For this, BaCO 3 , TiO 2 and H 3 BO 3 were thoroughly mixed and melted in a platinum crucible at 1473K for 1h. The batch weight was 10 gm. Melts were quenched by pouring on a steel plate that was preheated at 423K and pressed with another plate to obtain 1-1.5 mm thick glass plates.
All these samples were annealed at 773 K (5 h) which is well below the glass transition temperature. X-ray powder diffraction (Philips PW1050/37, Cu Ka radiation) study was performed on the as-quenched powdered samples at room temperature to confirm their amorphous nature. The DTA (TA Instruments SDTQ 600) runs were carried out in the 573K -1173K temperature range. For non-isothermal experiments, the glass samples were heated from 573K to 1173K at the rate of 10, 20, 30, 35, 40 and 50K/min. Baseline (empty) and standard (indium) runs were also made in the same temperature range and at the same heating rates. The bulk glass samples of 1 mm thick, weighing 35 mg were taken in a platinum crucible and alumina powder was used as the reference for each experiment. All the experiments were conducted in dry nitrogen ambience.
Results and discussion
The DTA trace that was obtained for the as-quenched glass plates at a heating rate of 10 K/min is shown in Fig. 1. As expected, an endotherm followed by an exotherm were encountered for the as-quenched samples under study. The endotherm at 827 K corresponds to the glass-transition temperature (T g ). A subsequent exotherm that is incident around 920 K is attributed to the crystallization (T cr ) of the BTBO glasses. The X-ray powder diffraction (XRD) pattern of the as-quenched glasses confirms their amorphous nature ( Fig. 2 (a)). The XRD pattern recorded for the sample heat-treated at the crystallization temperature (920 K) is shown in Fig. 2 (b). The Bragg peaks that are observed in this pattern are in good agreement with that reported in the literature for the poly-crystalline sample of Ba 3 Ti 3 O 6 (BO 3 ) 2 [9] obtained by the solid-state reaction route.
The DTA traces that are obtained for the as-quenched glasses at different heating rates (10,20,30,35, 40 and 50K/min) are shown in Fig. 3. The glass transition (T g ) and the crystallization (T cr ) temperatures shift towards higher temperatures with increasing heating rate (as illustrated in Fig. 3), indicating the kinetic nature of the glass transition and the crystallization. The heating rate dependent glass transition temperature is rationalized using fragmentation model [11]. According to this model, viscous behavior is caused by the fragmentation of random networks due to bond breaking. The density of the broken bonds could be obtained from the rate equation which is dependent on the heating rate [11]. The fragmentation would occur when the density of the broken bonds reaches a certain level. The system comprising fragments must exhibit a lower viscosity than that of the continuous random networks, and the viscosity of the fragmentized system decreases as the fragmentation proceeds. Therefore, the glass transition is dependent on the heating rate, as the fragmentation processes are heating rate dependent.
The investigation into the heating rate dependent glass-transition and crystallization could be helpful for the understanding of thermal behavior of BTBO glasses.
The effect of heating rate (a) on T g and T cr has been rationalized using the Lasocka's relation [12]. The relation between the T g , T cr and a could be represented empirically (Lasocka's relation) as follows:
α log g g g B A T + =(1)
and α log cr cr cr
B A T + = (2)
where A g and A cr are the glass transition and crystallization temperatures for the heating rate of 1 K/min. B g and B cr are the constants. The plots of T g and T cr versus log a for BTBO glass-plates along with the theoretical fits (solid lines) are shown in Fig. 4. The above relations (Eqs. 1 and 2) can be written as;
T g (K) = (811 ± 2) + (16 ± 1) log a (K/min)(3)
and T cr (K) = (882 ± 1) + (38 ± 1) log a (K/min)
The crystallization has a stronger dependence on heating rate than that of the glass transition (B cr > B g ). The relative location of T cr with reference to T g in the DSC/DTA trace is a measure of the thermal stability of glasses. The thermal stability (T cr -T g ) of the glasses is a crucial parameter to be noted from their technological applications point of view. Glasses should be sufficiently stable against crystallization. However, one must be able to form nuclei and subsequently grow crystals within the glass matrix on heat treatment inorder to obtain glass-ceramics. The higher values of the T cr -T g delay the nucleation process. The parameter (T cr -T g ) is heating rate dependent and on combining Eqs. 1 and 2, one arrives at:
T cr -T g = (A cr -A g ) + (B cr -B g ) log a(5)
The above relation for BTBO glass-plates can be written as
T cr -T g (K) = (71 ± 2) + (22 ± 1) log a (K/min)(6)
Using the above equation, the values of T cr -T g at various heating rates could be obtained.
It is clear that T cr -T g decreases with decreasing heating rate (Eq. 6) and can be equal (T cr = T g ) at some lower value of heating rate a k , which can be expressed as;
log a k = -[(A cr -A g )/(B cr -B g )](7)
The a k value that is obtained for BTBO glass-plates is 0.0006 K/min. It is to be noted that at this heating rate (a k ), glass is expected to undergo the glass-transition and the crystallization processes simultaneously. This is known as the Kauzmann temperature (T k ) [13][14][15]. The calculated value for T k for the present glass system is 759 K. It is experimentally difficult to observe this ideal glass-transition temperature (T k ) as the heating rate required is extremely low (0.0006 K/min). The estimated time involved is about 531 days to achieve the above temperature (T k ) at that heating rate.
It is of interest to investigate the activation energy of the glass-transition for the BTBO glass samples. The activation energy (E g ) associated with the glass-transition is involved in the molecular motion and rearrangement of atoms. The activation energy for the glasstransition for the present glasses could be evaluated using Kissinger's method [16] and is given by the following relation:
ln (a/T g 2 ) = -E g /RT g + constant
where R is the universal gas constant. The above equation (Eq. 8) was formerly derived for the process of crystallization. However, it is known in the literature [17] that the Kissinger equation could also be used to evaluate the activation energy associated with the glass transition. A plot of ln(a/T g 2 ) versus (1000/T g ) yields a linear relation which is depicted in Fig. 5. The experimental points of the present work along with a linear fit (solid line) to the above relation suggest its validity. From the slope of the straight line, the value of E g is found to be 740 ± 10 kJ/mol. The variation in ln T g 2 with a is insignificant as compared to the change in ln a [18]. Therefore, the eq. 8 could be written as;
ln a = -E g /RT g + constant (9) The value of E g is obtained from the slope of a straight line (linear fit) of ln a vs 1/T g plot.
The plots of ln a vs 1000/T g along with linear fits are shown in Fig. 5
( ) − = o o T T B exp η η(12)
where ? o and B are constants. And T o is the VFT temperature corresponding to the kinetic instability point [22]. Chen [23] showed that the activation energies associated with the glass transition and the crystallization could be associated with those for viscous flow as a function of temperature;
2 2 ) ( ) / 1 ( ) (ln ) ( o T T RBT T d d R T E − = = η(13)
The above equation (Eq. 13) for the glass-transition and the crystallization could be written as;
2 2 ) ( o g g g T T RBT E − =(14)
and for the crystallization,
After solving Eqs. 14 and 15 for B and T o , one would arrive at;
2 * − = g cr g g T T T C E B(16)
and
− − = 1 C T C T T cr g o (17) where 2 = g cr cr g T T E E C(18)
The obtained values for E g and E cr were used to deduce the constants (B and T o ) in the above relations (Eqs. 16 and 17).
The viscosity at various temperatures for the BTBO glass in the supercooled liquid region can be estimated using the Eq. 19. The value of viscosity is expected to increase with the increase in the volume fraction of crystallization. The viscosity expression for the glassceramic of various fractions of crystallization is [24];
( ) n eff mx + = 1 η η(20)
where eff η is the viscosity for the crystallized glass with volume fraction, x and ? is the viscosity of the as-quenched glasses. The constants m and n are dependent on the shape and orientation of the crystallites in the glass matrix. Under the assumption that spherical crystals are randomly oriented in the glasses, Eq. 20 could be written as
( ) 3 1 − − = x eff η η(21)
The above equation is valid only for the value of x lying in the range 0-0.6. One can obtain the viscosities at various temperatures for the BTBO glass-ceramics using the Eqs.
Conclusions:
The glass-transition and crystallization kinetics of 3BaO-3TiO 2 Exo.
Endo.
T (K)
10 K/min 20 K/min 30 K/min 35 K/min 40 K/min 50 K/min This document was created with Win2PDF available at http://www.daneprairie.com. The unregistered version of Win2PDF is for evaluation or non-commercial use only.
For this purpose, the values of T g and T cr at 10 K/min were considered. The values obtained for B and T o are around 5180 K and 676 K, respectively. It is to be noted that the value of T o is less than the that of Kauzmann temperature, (T k .) It is reported in the literature that the relation T k = T o is valid only for fragile glasses whereas for strong glasses T k >T o[22].To establish the theoretical relation for temperature dependence of viscosity, it is considered that a liquid on cooling becomes a glass when the viscosity equals 10 12 poise (?(T g ) = 10 12 poise). The numerical value for ? o could be deduced from Eq. 12 using the values for B, T o and ?(T g ) (10 12 poise). The expression for the viscosity (Eq.12) for the BTBO glasses may be written as;
19 and 21 .
21The plots of viscosity versus temperature for the glasses and glass-ceramics (x=0.3 & 0.6) are depicted in Fig. 7. It is clear from the figure that the viscosity increases with increase in the volume fraction of crystallization and decreases with increase in temperature.
-B 2 O 3 glasses have been studied. The activation energies associated with the glass-transition and crystallization were used to determine the temperature dependent viscosity relation (VFT equation) K. The value obtained for the fragility index indicates that the present composition belongs to a strong glass forming family of oxides. The thermal parameters that are obtained in the present studies are crucial for controlling the crystallization in bulk samples.
Figure captions : Fig. 1 :Fig. 4 :
:14DTA trace for as-quenched 3BaO-3TiO 2 -B 2 O 3 glass-plates Fig. 2: X-ray diffraction patterns for the (a) as-quenched and (T g and T cr versus log a for 3BaO-3TiO 2 -B 2 O 3 glass-plates Fig. 5: ln a and ln (a/T g 2 ) versus 1000/T g Fig. 6: ln (a/T P 2 ) versus 1000/T P plot Fig. 7: ln ? & ln ? eff versus T plots for glasses and glass-ceramics of two different fractions
Fig
Fig. 2
Fig. 3
Fig. 4
Fig
Fig. 6
. The average value of E g obtained from the slope of the linear fit is 735 ± 8 kJ/mol which is in close agreement with that obtained by Kissinger's method. Another important parameter of the glasses is their Fragility index (F) which is a measure of the rate at which the relaxation time decreases with increasing temperature around T g . The glass forming liquids that exhibit an approximately Arrhenius temperature dependence of the viscosity are defined as strong glass formers and those which exhibit a It was known in the literature that kinetically strong-glass forming liquids have low values of F (F≈ 16), while fragile glass forming liquids have higher values of F (F≈ 200).The value of F obtained for the glasses under investigation at the heating rate of 10K/min is 47 ± 2. This value which is well within the above mentioned range indicates that the present system can be considered to belong to a strong glass forming family.Kissinger[16] developed a method for a non-isothermal analysis of the crystallization which is as follows:non-Arrhenius behavior (Vogel Fulcher Tammann (VFT) equation [19,20]) are known as
fragile glass formers. The value of the fragility index (F) could be estimated using the
following relation [21]:
10
ln
g
g
RT
E
F =
(10)
It is essential to have apropri knowledge about the crystallization behavior of the BTBO
glasses for fabricating the glass-ceramics with the desired characteristics. Crystallization
behavior of the BTBO glass samples was studied by non-isothermal methods. For
evaluating the activation energy (E cr ) for crystallization using the variation in T p with a,
p
cr
o
c
p
RT
E
K
R
E
T
−
+
−
=
ln
ln
ln
2
α
(11)
The plots of
)
/
ln(
2
P
T
α
versus 1000/T p for these glasses are shown in Fig. 6. The value of
the activation energy and the frequency constant (K o ) are obtained from the slope (E cr /R)
and intercept (
−
R
E
K
cr
ln
ln 0
) of the linear fit to the experimental points. The value of
E cr is 350 ± 10 kJ/mol. for the glasses understudy. The value for the frequency constant
is about 1.6 X 10 19 sec -1 which indicates the number of attempts per second made by the
nuclei to overcome the energy barrier.
In order to have further insight into the thermal behavior of BTBO glasses, its
temperature-dependent behavior of the viscosity (?) is studied using VFT equation. It is
assumed that the temperature dependent viscosity for the BTBO glasses could be
expressed by VFT relation as follows
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| [] |
[
"On the properties of a single OPLS-UA model curcumin molecule in water, methanol and dimethyl sulfoxide. Molecular dynamics computer simulation results",
"On the properties of a single OPLS-UA model curcumin molecule in water, methanol and dimethyl sulfoxide. Molecular dynamics computer simulation results"
] | [
"T Patsahan \nInstitute for Condensed Matter Physics\nNational Academy of Sciences of Ukraine\n1 Svientsitskii St79011LvivUkraine\n",
"J M Ilnytskyi \nInstitute for Condensed Matter Physics\nNational Academy of Sciences of Ukraine\n1 Svientsitskii St79011LvivUkraine\n",
"O Pizio \nInstituto de Investigaciones en Materiales\nUniversidad Nacional Autónoma de México\nCircuito Exterior04510Cd. MxMéxico\n"
] | [
"Institute for Condensed Matter Physics\nNational Academy of Sciences of Ukraine\n1 Svientsitskii St79011LvivUkraine",
"Institute for Condensed Matter Physics\nNational Academy of Sciences of Ukraine\n1 Svientsitskii St79011LvivUkraine",
"Instituto de Investigaciones en Materiales\nUniversidad Nacional Autónoma de México\nCircuito Exterior04510Cd. MxMéxico"
] | [
"Condensed Matter Physics"
] | The properties of model solutions consisting of a solute -single curcumin molecule in water, methanol and dimethyl sulfoxide solvents have been studied using molecular dynamics (MD) computer simulations in the isobaric-isothermal ensemble. The united atom OPLS force field (OPLS-UA) model for curcumin molecule proposed by us recently [ J. Mol. Liq., 2016, 223, 707] in combination with the SPC/E water, and the OPLS-UA type models for methanol and dimethyl sulfoxide have been applied. We have described changes of the internal structure of the solute molecule induced by different solvent media in very detail. The pair distribution functions between particular fragments of a solute molecule with solvent particles have been analyzed. Statistical features of the hydrogen bonding between different species were explored. Finally, we have obtained a selfdiffusion coefficient of curcumin molecules in three model solvents. | 10.5488/cmp.20.23003 | [
"https://arxiv.org/pdf/1706.07253v1.pdf"
] | 55,476,089 | 1706.07253 | 46d936f685a19b3bab5eab034428f1f6fafacb38 |
On the properties of a single OPLS-UA model curcumin molecule in water, methanol and dimethyl sulfoxide. Molecular dynamics computer simulation results
2017. 23003
T Patsahan
Institute for Condensed Matter Physics
National Academy of Sciences of Ukraine
1 Svientsitskii St79011LvivUkraine
J M Ilnytskyi
Institute for Condensed Matter Physics
National Academy of Sciences of Ukraine
1 Svientsitskii St79011LvivUkraine
O Pizio
Instituto de Investigaciones en Materiales
Universidad Nacional Autónoma de México
Circuito Exterior04510Cd. MxMéxico
On the properties of a single OPLS-UA model curcumin molecule in water, methanol and dimethyl sulfoxide. Molecular dynamics computer simulation results
Condensed Matter Physics
2022017. 2300310.5488/CMP.20.23003Received March 17, 2017, in final form May 4, 2017curcuminunited atom modelmolecular dynamicswatermethanoldimethyl sulfoxide PACS: 0270Ns6120Ja8230Rs8715hp
The properties of model solutions consisting of a solute -single curcumin molecule in water, methanol and dimethyl sulfoxide solvents have been studied using molecular dynamics (MD) computer simulations in the isobaric-isothermal ensemble. The united atom OPLS force field (OPLS-UA) model for curcumin molecule proposed by us recently [ J. Mol. Liq., 2016, 223, 707] in combination with the SPC/E water, and the OPLS-UA type models for methanol and dimethyl sulfoxide have been applied. We have described changes of the internal structure of the solute molecule induced by different solvent media in very detail. The pair distribution functions between particular fragments of a solute molecule with solvent particles have been analyzed. Statistical features of the hydrogen bonding between different species were explored. Finally, we have obtained a selfdiffusion coefficient of curcumin molecules in three model solvents.
Introduction
Curcumin, derived from the root of turmeric Curcuma longa, is well known not only as a spice and natural colorant but also as an important substance in biomedicine due to possible ample applications. Namely, the issues concerning the antioxidant, anti-inflammatory, antiviral and anticancer activity of curcumin have become the subject of numerous experimental studies for the recent few decades, see e.g., recent reviews and references therein [1][2][3][4][5]. We are unable to comment on the problems related to the applications of curcumin-based agents in medicinal chemistry [6] and just wish to mention that experimental research has urged the use of methods of theoretical chemistry in this area, more specifically, methods of quantum chemistry. However, as noted by Wright [7], from a theoretical modelling perspective, there is a bewildering number of variables involved in the experiments. Consequently, there is much room for the development of adequate modelling and for confronting predictions from computer simulations against experimental data.
The studies of liquid systems containing curcumin solutes by using computer simulation techniques have been initiated quite recently [8][9][10][11][12][13][14][15][16][17]. The works [12,13] are of particular interest for the present study. In each of them, the modelling of a curcumin molecule force field has been undertaken starting from quantum chemical calculations within the B3LYP DFT method by using different versions of Gaussian software. Unfortunately, not all the details of modelling were presented in the original papers and in supplementary materials to them. For, example, in [13] only charges for the interaction sites are given whereas all parameters concerning bonds, angles and dihedral angles were omitted. This was one of the reasons that in the previous work from this laboratory we have developed the OPLSunited atom model for enol-tautomer of curcumin molecule and tested it in vacuum and water using classical MD simulations [18]. It has been already mentioned [7] that experiments were performed in a variety of solvents (e.g., chlorobenzene, isopropanol-water mixture, dimethylsulfoxide, acetonitrile) due to marginal solubility of curcumin in pure water. Therefore, in the current study we extend our recent work by considering a single curcumin molecule in pure methanol (MeOH) and pure dimethylsulfoxide (DMSO). It is worth mentioning that specific features of curcumin molecule conformations in liquid DMSO were studied experimentally quite recently [19]. Our principal objective here is to elucidate the similarities and differences in the behavior of this molecule in water, MeOH and DMSO. With this aim we also needed to recalculate some of the results for curcumin in water from the previous study [18] in order to make the time extent of computer simulations for all three systems similar.
Model and technical details of calculations
Simulations of a single curcumin molecule in three solvents, H 2 O, MeOH and DMSO, were performed in the isothermal-isobaric (NPT) ensemble at a temperature of 298.15 K and at 1 bar. In all cases we used GROMACS simulation engine [20] version 4.6.7.
In close similarity to the previous work [18], our attention is restricted to the enol-tautomer of curcumin solely. Its chemical formula is given in figure 1. This tautomer was investigated in the recent computer simulation studies [12,13]. It is a dominant tautomer both in solids and in various solutions as confirmed in various works, see e.g., [19,[21][22][23]. Besides, it is the major phytoconstituent of extracts of Curcuma longa according to [6]. Thus, we are interested in the effects of solvents on a particular tautomer, while diketo-tautomer will be explored elsewhere. A "ball-and-stick" schematic representation of the curcumin molecule is shown in figure 2, with labels denoting the sites (carbon groups, oxygen and hydrogen) of the united atom force field model. All the parameters of the force field are taken from [18], see supporting information file to that article.
For water we used the SPC/E model [24]. For MeOH, the OPLS-UA model was taken according to [25]. However, in contrast to [25], we used the harmonic bond-angle potential for CH 3 -O-H with the force constant from the OPLS-UA database [26]. In the case of DMSO, we used the four-site model with all parameters taken from the OPLS-UA database, except that the improper dihedral angle S-CH 3 -O-CH 3 was added, it was described by the GROMOS set of parameters [27].
The geometric combination rules were used to determine parameters for cross interactions (rule 3 of GROMACS package). To evaluate the contributions of Coulomb interactions the particle mesh Ewald method was used (fourth-order spline interpolation and grid spacing for fast Fourier transform equal to 0.12). The electrostatic and Lennard-Jones cut-off distances were chosen equal to 1.1 nm. The van der Waals tail correction terms to the energy and pressure were taken into account. The lengths of the bonds were constrained by LINCS. For each system, a periodic cubic simulation box was set up with 2000 solvent particles and a single curcumin molecule.
The initial configuration of particles was prepared by placing them randomly in the simulation box using the GROMACS genbox tools. Next, the systems underwent energy minimization to remove the possible overlap of atoms in the starting configuration. This was done by applying the steepest descent algorithm. After minimization we performed equilibration of each system for 50 ps at a temperature of 298.15 K and at 1 bar using a timestep 0.1 fs.
The Berendsen thermostat with the time coupling constant of 0.1 ps and isotropic Berendsen barostat with the time constant of 2 ps were used. We are aware that this thermostat may yield certain inaccuracies in common MD, see, e.g., comments in [28] concerning the application of replica exchange MD to conformational equilibria, and becomes completely unsatisfactory in calculations of fluctuationbased properties. Still, for purposes of the present work, and to keep a consistency with the previous development [18], we have decided to use this thermostat for the moment.
The compressibility parameter was taken equal to 4.5 · 10 −5 bar −1 for water, 1.2 · 10 −4 bar −1 for methanol, similar to [29], and 5.25·10 −5 bar −1 for DMSO. All the properties were collected by performing the averages over 10 consecutive simulation runs of the length of 50 ns. Each run started from the last configuration of the previous one. The timestep in the production runs was chosen to be 1 fs.
Results and discussion
First, we would like to discuss the changes of the internal structure of a single curcumin molecule induced by each solvent under study. Various properties are described in terms of histograms of the probability distributions termed as the distribution functions. For each production run, we select a set of frames with the distance in time equal to 2 ps. The outputs for any property of interest are collected on a set of frames and the resulting histogram is then normalized. Next, the average over a set of runs is performed.
Distributions of dihedral angles
The histograms of probability distributions of the dihedral angles of curcumin in water, MeOH and DMSO are presented in three panels of figure 3. The abbreviated notations for dihedral angles are given in table 1. To begin with, we consider the EnHdih dihedral angle distribution [figure 3 (a)] that describes the orientation of hydrogen atom H15 within the enol group. For all three solvents in question, p(φ) is characterized by the maxima at φ = 0°and φ = ±180°of different height. The maxima located at φ = ±180°indicate that H15 is preferentially directed toward the ketone oxygen (O18), rather than outward the enol group. Thus, the trans-conformation of EnHdih angle is dominant compared to the cis-conformation in all cases. It seems that intramolecular electrostatic interaction between H15 and O18 leads to these trends. However, solvent effect is manifested in the difference of heights of the maxima at characteristic angles. Apparently, the cis-conformation is suppressed, comparing to its transcounterpart in two solvents capable of solubilizing curcumin. By contrast, in water, H15 has enough freedom to accommodate in both conformations. An abundant population of cis-conformation witnesses a possibility of the formation of H-bonds between water oxygens and H15 belonging to the enol group. It is worth noting that for curcumin molecule in vacuum, the only state of EnHdih is trans-conformation [18].
Histograms of probability distribution of dihedral angles of phenol and anisole groups [ figure 3 (b)] practically coincide for three systems studied [the curve describing the influence of MeOH on p(φ) is omitted to make the figure less loaded]. Thus, interpretation of the results is similar for all cases. The
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Phdih dihedrals have a principal maximum at φ = 0°(cis-conformation) due to the hydrogens H4 and H28 directed toward the oxygens O2 and O29, respectively. Trans-conformation for Phdih can happen, but with negligible probability as it follows from the figure. The Andih angle distributions have solely two maxima at φ = ±180°. In the trans-conformation, the methyl groups, C1 and C30, point away from the oxygens O5 and O27, respectively.
Dihedral distributions, B 1 , B 2 and B 3 [figure 3 (c)] characterize the flexibility of the spacer connecting aromatic rings to the central enol group. The changes in the B 1 dihedrals describe the rotation of aromatic rings apart from the spacer, whereas the changes of the B 2 and B 3 are related to the rotation of all the side branches including aromatic rings and the spacer fragments connected to them. The B 2 distribution qualitatively differs from the B 1 and B 3 curves. While the B 1 and B 3 histograms have maxima at φ = 0°a nd φ = ±180°, the B 2 has a maximum around φ = ±180°(trans-conformation). Consequently, there is no rotation around the 11−12 and 19−20 bonds. The same behaviour for the B 2 is observed for a curcumin molecule in water and in DMSO, two curves practically coincide (the MeOH curve is omitted). The difference between the B 1 and B 3 curves in water and in DMSO is better pronounced but small. We are unaware of any direct experimental spectroscopic evidence or computer simulation studies of the relevant properties to estimate the quality of our predictions. Nevertheless, some indirect supporting arguments are given in the following subsection.
Distance and angles distributions
The distribution function of the EnHdih angle discussed in the previous subsection can be interpreted in terms of the changes of the distance between hydrogen, H15, and ketone oxygen, O18, figure 4 (a). Well pronounced maxima at 0.196 nm and 0.375 nm correspond to H15 pointing toward and outward the enol group, respectively. As expected from the behavior of EnHdih [figure 3 (a)], the site H15 prefers to be directed toward the enol group in all solvents considered. The maximum of the O18-H15 distance distribution at 0.196 nm is higher for the system with DMSO solvent, in comparison with MeOH and water. On the other hand, the maximum at 0.375 nm is much lower in magnitude for curcumin-DMSO system compared with two other solvents. Consequently, the trans-conformation of the enol group is most frequent and possibly most stable in thermodynamic sense for curcumin-DMSO solution.
This is a valuable result. There is no experimental evidence supporting our observation. However, in computer simulations of Samanta and Roccatano [12], the H15-O18 distance distribution was discussed. According to their model [12], the H15-O18 distance distribution in water solvent is bimodal with two maxima of almost equal height located at ≈ 0.16 and 0.35 nm. These values do not differ much from what we observe. It seems, however, that the discrepancy of the absolute values of maxima reported in figure 4 of [12] and our data in figure 4 (a) is due to the differences of parameters used in modelling the dihedral angles. Moreover, in the case of MeOH solvent, the second maximum is totally absent in the model of [12] whereas in our distribution it is seen. The trends of the behavior of the distances H4-O2 and H28-O29 have been analyzed by us as well. Histograms of the probability distribution for each of these distances in figure 4 (b) confirm our conclusions concerning the dihedral angles Phdih [cf. figure 3 (b)]. In particular, one can observe a bimodal distribution with two maxima at r = 0.22 nm and 0.36 nm. They correspond to the cis-and trans-conformations. It is clearly seen that the larger distance (trans-conformation) is realized with very low probability in water and DMSO solvents, though in DMSO it is noticeably lower. Therefore, we can conclude that such a rare change of Phdih conformation can hardly have an effect on the overall behavior of the curcumin molecule.
Probability distributions describing the flexibility of a curcumin molecule in three solvents are shown in figure 5 (a) and figure 5 (b). The Left ring-Right ring distance is defined as a distance between centers of aromatic rings, whereas the Center-Left and Center-Right are the distances from the center of the enol ring to the center of the left and right ring, respectively. The center of a ring is chosen as a center of mass of carbons and hydrogens composing it. For the Left-Right distance of curcumin in water presented in figure 5 (a), three well pronounced maxima at 0.90 nm, 1.03 nm and 1.15 nm are observed.
As it was discussed in our previous study [18], this shape is explained by the conformational states available for the B 1 , B 2 and B 3 dihedrals. It was also concluded that only the B 3 dihedrals involving 16-13-12-11 and 20-19-17-16 groups are responsible for the Left-Right distance of a curcumin molecule. Since only three possible combinations of cis-and trans-conformation states of these two groups may occur, the number of the most probable Left-Right distances is three.
The maximum which corresponds to the middle distance 1.03 nm is the most pronounced, and consequently this structure is the most probable, figure 5 (a). The left-hand maximum in the figure corresponding to the distance 0.90 nm is much lower. Finally, the lowest maximum can be found at the distance 1.15 nm. The obtained distances agree well with the results from our previous investigation of curcumin in SPC-E water [18].
In the case of the MeOH solvent [ figure 5 (a)], we again observe three maxima, but of different heights, in comparison with water. Moreover, all of them are slightly shifted towards higher distances. The middle peak becomes higher in a "better" solvent that solubilizes curcumin in contrast to water. Specifically, in the case of MeOH, the middle maximum of the Left-Right distance distribution is higher than in water. It is even higher for curcumin in DMSO compared to the curcumin molecule in MeOH. Similar behavior is observed for the right-hand maximum. By contrast, opposite trends are seen for the left-hand maximum. Finally, for curcumin molecules in DMSO, the height of the right-hand maximum becomes higher than the left-hand one. The Center-Left and Center-Right distance distributions are shown in figure 5 (b) only for the curcumin in water, since for MeOH and DMSO, the results are very close to the system with water. As can be seen, both distributions have a single maximum, approximately of equal heights, just slightly shifted. The Center-Left distance has the most probable value at 0.61 nm whereas the Center-Right most probable distance is slightly higher. The shift can be attributed to the asymmetry of the spacer of the curcumin molecule. We are not aware of the data coming from other models of a curcumin molecule in various solvents. However, a detailed discussion of fine features of the molecular internal structure can be helpful in designing more sophisticated models in future research. Namely, the modification and reparameterization of the dihedral-angle potentials can be attempted along the lines proposed by Kurokawa et al. [30].
Changes of the internal structure of a molecule can be interpreted in terms of the angles between certain fragments of the curcumin molecule immersed in a solvent. To begin with, we explored the trends of the behavior of the angle between planes of aromatic rings. A plane is determined by picking up three sites belonging to a ring, namely C6, C7, C9 for the left-hand ring and C22, C24, C25 for the right-hand ring. The distribution of the angles between the rings, α, has been analysed, but the figure is not shown for the sake of brevity. The probability distributions of α are very similar in all the solvents considered and they almost coincide with the histogram for p(α) presented in figure 10 of [18]. The p(α) distribution has a bimodal shape principally determined by the intrinsic structure of a molecule within the framework of the modelling applied.
Another characteristic angle is β, which is the side-side angle defined via the triplet of sites, C3-C16-C26. Its behavior is given in terms of angular distribution p(β), figure 6 (a). Actually, the p(β) distribution is unimodal with the maximum at ≈ 130°. At a larger value of the angle, there is a shoulder for all three solvents. However, the shape of p(β) in DMSO is very close to that in MeOH. Only in water, a weakly pronounced secondary maximum can be observed at a smaller angle, ≈ 112°. These distributions reflect the bending of curcumin molecules and in some sense should yield information similar to figure 5 (a). However, splitting into three maxima using the distance variable [figure 5 (a)] in terms of the angles is manifested solely in shoulders of the p(β) distribution. Nevertheless, the angular distribution curves indicate that the curcumin molecule is slightly more bent in water compared to DMSO or MeOH. In figure 6 (b), we also show the corresponding distributions for the angle cosine [cos(β)]. It provides conclusions similar to p(β), though in such representation the shoulders become more prominent.
Dipole moment
The dipole moment distribution only marginally changes upon the solvent change within the framework of the model. Therefore, only the histograms of the probability distribution of the magnitude of the dipole moment of a curcumin molecule in water and DMSO are presented in figure 7, the curve for MeOH is omitted.
The dipole moment distribution reflects superposition of the conformational states of a curcumin molecule discussed in terms of dihedral angles and the related properties in the previous subsections. It seems that an essential conformation change is the position of hydrogen site in the enol group. The splitted maximum in the dipole moment distribution p(d) is due to the enol hydrogen, H15, pointing toward ketone-oxygen, O18, or outward the enol group. To get a profounder insight, we have analyzed two distributions obtained from the curcumin trajectories. Namely, we picked up a part of the trajectory when H15 is pointing solely outwards (case A: r OH = 0.375 nm) and the other part when H15 is directed toward (case B: r OH = 0.195 nm) the enol group. Histograms describing p(r) for these trajectories are given in figure 8 (a). The corresponding p(d) functions are presented in figure 8 (b). The p(r) is clearly a unimodal function in each case. However, in the case A, we observe two maxima d = 3.5 D and d = 5.3 D on p(d) coming from distinct conformation states. Additional exploration leads to the conclusion that this bimodality in the present case comes from the behavior of p(β), i.e., from the bending of the spacer. In the case B, the splitting of a maximum part of the curve is less pronounced. From the results above, it seems that "closed" states of the enol ring influence the formation of the left-hand side maximum on the resulting curve in figure 7, or, in other words, contribute to lower values of the dipole moment compared to other conformations.
To summarize, for a curcumin molecule in MeOH and DMSO, we observe a similar behavior of the dipole moment distribution as in water ( figure 7). However, the maxima at the p(d) distribution are found to be a bit more pronounced in MeOH and DMSO. Further studies are necessary to establish the validity of these conclusions for other protic and aprotic solvents.
Pair distribution functions
In order to characterize the surrounding of the curcumin molecule in different solvents, we consider some selected pair distribution functions (PDFs). Here, our focus is on PDFs of solvent molecules around the polar groups of curcumin, since these groups formally yield the strongest attractive interactions between a curcumin molecule and solvent species compared to other groups of a molecule. We pick up the groups where oxygen is present. Taking into account the molecular symmetry of curcumin, some of them are combined into pairs, e.g., for the hydroxyl groups (O5-H4 and O27-H28), we consider the functions H4/H28-OX and O5/O27-OX (X = OW, OMe and OD) each resulting from the averaging of the two PDFs. A similar procedure is used for methoxy groups O2-C1 and O29-C30. Prior to any kind of such averaging, we have checked whether the functions are similar, up to statistical inaccuracy.
As concerns the groups at aromatic rings, one should mention that the PDFs O5/O27-OX are characterized by a sharp first maximum at 0.28 nm for all solvents in question (figure 9, left-hand panel). The first maximum in MeOH and in DMSO is much higher than in water (2.13, 2.60 and 1.44, respectively). The second peak is weakly pronounced both in water (at about 0.47 nm) and in MeOH (at 0.49 nm). Actually, this is a local maximum. Only in the case of DMSO, the second maximum is well manifested but it is not high (the height is 1.17 at 0.57 nm). The ordering of solvent species around hydrogens H4/H28 (H4/H28-OX PDFs) resembles the above description. Finally, for the methoxy groups, it is observed that the first peak of PDF O2/O29-OX in the MeOH solvent is less than unity at the distance 0.30 nm, while in water it is absent at all. In the case of DMSO, the first maximum is well defined at the same distance 0.30 nm, but not as high as in the case of hydroxyl groups.
Our conclusions concerning three left-hand panels are as follows. This part of the curcumin molecule perturbs the solvent density up to two layers at most. Distribution of solvent species around this region of curcumin is heterogenous. The most "dry" part is around the methoxy group. Only the DMSO solvent fills the first layer in the methoxy part. By contrast, the DMSO particles form two layers in the hydroxy group region. Moreover, it seems that there exists a possibility that OD can form hydrogen bonds with H4/H28. These trends become weaker for MeOH. Finally, in terms of adsorption terminology, one can say that water molecules tend to their bulk rather than approach this part of curcumin molecules or "dewet" it. Now, consider the arrangement of solvent molecules around the hydroxyl and ketone groups at the spacer of a curcumin molecule (figure 9, right-hand panel). The first maximum of PDF for O14-OX (at 0.28 nm) in the case of DMSO solvent is found lower than in water and in MeOH, for which the first maxima are observed at about the same distances. The second maximum in DMSO is well manifested at 0.55 nm and is much higher than in water and MeOH. Those are just local maxima on a gradually increasing function that tends to unity. For the PDFs H15-OX in water, the distribution function is similar to H4/H28-OX, but the height of the first maximum is somewhat lower. For MeOH and DMSO solvents, we observe essentially lower first maxima compared to the curves for solvent species around hydroxyl groups at the aromatic rings. However, the picture is different for the second maximum, which is absent in MeOH, but well pronounced in DMSO at 0.58 nm. Finally, the PDFs for the last group on the list of oxygen containing groups of the curcumin molecule (the ketone group) exhibits a "desorption" type arrangement of solvent molecules. The first maximum of PDF in the case of water is less than unity at r = 0.29 nm. In MeOH, it is much lower and in DMSO it is absent. The second maximum in DMSO is weak. The same is observed for the MeOH. The second maximum in water is just a local perturbation.
To summarize these observations, we would like to mention the following. The solvent density around this part of curcumin molecule is perturbed to larger distances, cf. right-hand and left-hand three panels of figure 9. Important events occur not only at the distance of the closest approach for sites and solvent species but in the second layer as well. Actually, the second layer should be included in the discussion of the structure of the surroundings of a curcumin molecule. This central part of the molecules does not exhibit much adsorptive capability, seemingly an interfacial region within this area is richer in hydrogen bonds compared to other parts of the surface of curcumin, while in other parts, the solvent may cover a solute without forming bonds. These issues are touched upon herein below.
From the PDFs presented in figure 9 we have calculated the first coordination numbers using the definition,
n i (r min ) = 4πρ j r min ∫ 0 g i j (r)r 2 dr,(1)
where ρ j is the number density of species j, and r min refers to the minimum of the corresponding pair distribution function. The n(r min ) indicate the quantity of oxygens of a solvent located within the first coordination shell of the corresponding oxygen of a curcumin molecule (table 2). The distances r min are determined from the first minimum of the PDFs. The coordination numbers serve as the evidence that water molecules prefer to be located in the first coordination shells formed by oxygens of the enol ring rather than by the oxygens of the side phenol rings. By contrast, MeOH and DMSO surround mostly the methoxy and hydroxyl groups of the side rings. Weaker trends to observe these molecules in the first coordination shells of the enol and keto groups are evident. Thus, the side rings play an important role in the curcumin solubility in MeOH and DMSO solvent. The arrangements of the solvent molecules around the hydrophobic groups of curcumin are analyzed as well (figure 10). For this purpose, we consider the PDFs of solvent oxygens around several sites at the curcumin molecule, which correspond to the sites C1/C30, C16 and C9/C22. The obtained PDFs C1-OX and C30-OX, C9-OX and C22-OX are combined due to the symmetry of a curcumin molecule. As expected, these PDFs have well pronounced extremal points in the case of DMSO solvent. A less structure is observed for water and MeOH. However, all the solvents have PDFs indicating a rather strong augmented density close to C1 and C30 sites.
The analysis of the solvent arrangements between the two rings in the vicinity of spacer shows that this location is preferably occupied by DMSO molecules, rather than by water or MeOH molecules. This conclusion emerges from the well manifested first maximum of the PDF C16-OX for DMSO at r = 0.34 nm. In the case of water and MeOH, the PDF C16-OX describes "desorptive" trends, the solvent molecules tend to the bulk rather than come close to a curcumin molecule. Similar trends are observed for the PDFs O9-O22.
Spatial distributions
Additional insights into the solvent surrounding the curcumin molecule that are not accessible from the PDFs, follow from the analyses of the spatial density distribution of solvent species. The correspond-
23003-12
ing plots have been built with the use of the VMD software [31] in the form of isosurfaces (figures 11, 12 and 13). The input data for the construction of isosurfaces of the three-dimensional density distribution function (SDF) of solvent molecules are calculated from the simulation trajectories. Each of the trajectories were obtained from the independent 10 ns runs performed in the NVT ensemble and for the fixed positions of sites of the curcumin molecule. The values of number density for the construction of isosurfaces were chosen to make the plots as illustrative as possible. For water oxygen, it is 0.15, whereas for methanol oxygen -0.045 and finally for DMSO oxygen -0.050.
We have considered two specific conformations of curcumin, which correspond to the enol hydrogen pointing toward and outward the ketone group, respectively. In figures 11, 12 and 13 we discern four main regions of solvent molecules involved into curcumin surroundings: (i) two regions with molecules near the side rings, (ii) a region near the enol group and (iii) near the spacer between the rings. It should be noted that these regions can be split into groups. Thus, for water and MeOH, one can observe two distinct groups near each of the side rings in the region (i). This can be attributed to the formation of hydrogen bonds between the corresponding hydroxyl groups and molecules of these solvents. A similar splitting can be seen near the hydroxyl group of enol and the ketone group in the region (ii). A different situation has been found for the DMSO solvent. Since in this case hydrogen bonding is absent, such splitting does not occur. Moreover, it is seen that the region (ii) is affected by the enol group conformation.
In the case of water solvent, the reorientation of water molecules in the groups of the region (ii) is observed due to the change of enol conformation. Finally, in all solvents, certain amount of molecules are concentrated in the region (iii). However, in water and MeOH, the molecules are grouped into beltlike cluster, while the DMSO molecules form several distinct clusters of a rather high density, which corresponds to the first maximum of the PDF C16-OX in figure 10.
Hydrogen bonds
Solvent molecules in the vicinity of curcumin molecule are involved in the hydrogen bond (HB) formation. We have calculated the number of HBs, n hb , between the oxygens or hydrogens of solvent molecules and the groups of curcumin molecule, that contain oxygens (O14, O18, O5/O27, O2/O29). GROMACS software was used for this purpose. In order to identify the hydrogen bonds, we applied a commonly used two-parameter geometric criterium, where donor-acceptor distance should be shorter than 0.35 nm while the angular cut-off is taken equal to 30 degrees. It is worth noting that oxygens of water and MeOH molecules can act both as proton donors and proton acceptors, while oxygen of DMSO can be a hydrogen bond acceptor exclusively.
In figure 14 we present the histograms of the probability distributions of the number of HBs. It is observed that if an oxygen of the solvent acts as acceptor, the distributions for all the solvents studied are close to each other. Moreover, the maxima of these distributions obtained for water and MeOH are practically the same, and they yield a maximum at n hb = 2. The corresponding maximum for DMSO is somewhat higher, but for n hb = 2 as well. On the other hand, when oxygens of a solvent are donors, the results are different. Since the DMSO oxygen cannot be a donor, the corresponding function is not present. The distribution of HBs for water is shifted toward higher values of n hb , and the most probable number of HBs is in the interval 4−5. This distribution has a big dispersion yielding non-zero values of probability density even for n hb equal to 9. On the other hand, for MeOH, the distributions in figure 14 (b) are not very different from that shown in figure 14 (a). For water, the average values of n hb in the donor case are much larger than in the acceptor case (4.56 against 1.72, respectively), while for MeOH, the corresponding values are rather close to each other.
Finally, in figure 14 (c), we present the probability distributions of the total number of the HBs formed between solvent particles and the curcumin molecule, regardless of the role of solvent oxygens (acceptor or donor) in hydrogen bonding. It follows that the most probable number of HBs formed between water and curcumin is equal to 6, for MeOH it is 3 and for DMSO it is equal to 2. We have also calculated the averaged numbers of HBs for curcumin in each of the solvents ( n hb for water as solvent is 6.28. It is almost twice higher than the estimate for MeOH ( n hb = 3.20). At the same time, n hb for MeOH is almost twice higher than the one calculated for DMSO ( n hb = 1.74).
In addition, we have analyzed the number of HBs formed with the particular groups of the curcumin molecule. It is shown in figure 15 that the major contribution to the total number of HBs follows from the oxygens of phenol groups (O5/O27). For water, the most probable number of HBs occurring between the phenol groups and water is found to be equal to 2, for MeOH this value is in the interval 1−2 whereas for DMSO it is always unity.
The probability of the formation of two HBs between distinct water molecules with the enol group (O14) is rather high, but this is not the case for MeOH molecules that prefer to form a single bond. For DMSO, only a single H-bond with the enol group can be formed, although with a rather small probability. Furthermore, hydrogen bonds can be formed between water molecules and ketone group (O18), figure 15 (c). One hydrogen bond can be formed between them with reasonable probability, the formation of two bonds being much less probable. In DMSO, a single hydrogen bond can be formed, again with low probability. The hydrogen bonding between solvent species and the anisole groups is quite a rare event. The average numbers of HBs from our calculations are summarized in table 3.
Self-diffusion coefficient
The dynamic properties of a solute can be strongly affected by the solvent environment. The selfdiffusion coefficient, D self , of a single curcumin molecule in three solvents is estimated from the meansquare displacement (MSD) dependence on time, according to the Einstein relation [32],
D self = 1 6 lim t→∞ d dt |r(τ + t) − r(τ)| 2 ,(2)
where τ is the time origin. This expression is used for each species. First, we have checked the shape of the MSDs and the values for self-diffusion coefficients for pure solvents that follow directly from the GROMACS software [ figure 16 (a)]. Time interval 10−20 ns was used, the curves for MSD are almost linear therein. The obtained results agree reasonably well with the already reported values for various models of solvents considered [33][34][35][36][37]. A similar procedure was then used to obtain the self-diffusion coefficient of curcumin in each solvent [ figure 16 (b)]. The results for a single curcumin molecule follow from the trajectory of the center of mass (COM) that can be conveniently generated by software. Moreover, the COM MSD functions on time coming from each run (50 ns) were collected to yield the average COM MSD curve, which is plotted in the figure. Then, the D self value was evaluated from the slope.
As one can see, the smallest value of D self is found for curcumin in DMSO (0.353 · 10 −5 cm 2 /s). In water, it is somewhat higher (0.423 · 10 −5 cm 2 /s). A much higher value for D self has been obtained for curcumin in MeOH (0.710 · 10 −5 cm 2 /s). We are not aware of the experimental data for the self-diffusion coefficient of curcumin molecule at infinite dilution in the solvents considered. On the other hand, the self-diffusion coefficient of curcumin in water and methanol was reported by Samanta and Roccatano [12] from MD simulation for their curcumin model. Contrary to our results, they obtained D self of curcumin 23003-16 in water higher than in methanol and their absolute values are much higher than ours. This discrepancy can be due to the SPC model used in [12] whereas we used the SPC/E model. Additional calculations are necessary to reach a definite conclusion. Nevertheless, in order to validate our results, we used the procedure based on the Stokes-Einstein equation,
D = k B T 6πη S R h ,(3)
where R h is the hydrodynamic radius and η S is the solvent viscosity. Applying the definition,
1 R h = 1 N 2 N i j 1 |r i − r j | t ,(4)
we have calculated the hydrodynamic radius R h for curcumin in three solvents. The histograms of the probability distributions of R h are shown in figure 17 (a) and the corresponding average values can be found in
R 2 g = 1 N N i=1 |r i − r COM | 2 t ,(5)r COM = 1 N N j=1 r j ,(6)
as it follows from the results in figure 17 (b) given in the form of histograms of probability distributions. (4) exhibit similar trends upon changing the solvent as obtained from the MSDs. However, the values differ quantitatively. One of possible reasons is that equation (3) is approximate and does not take into account the molecular shape. However, we do not expect that the observed trends would be reversed.
Summary and conclusions
To conclude, in this work we have presented a very detailed description of the properties of model solutions consisting of a single curcumin molecule in 2000 molecules of water, MeOH and DMSO solvent. We have evaluated and analyzed the changes of the internal structure of a curcumin molecule in terms of dihedral angles, most important characteristic distances between particular atoms as well as the distances and angles between segments of the molecule. The solvent effect on the value of the dipole moment of a curcumin molecule has been elucidated. An interface between the molecule and solvent surroundings has been characterized in terms of pair distribution functions, coordination numbers and spatial distribution density maps. Quantitative description of the probability of cross hydrogen bonds between atoms of a curcumin molecule and solvent species has been given. Self-diffusion coefficient of curcumin in three solvents has been evaluated and analyzed.
Our principal findings comprise the following points. The intrinsic bending of the molecule already discussed in vacuum [18] is affected by different solvents differently. In water, the molecule avoids contacts with solvent particles and becomes even more bent whereas in MeOH and DMSO, more extended conformations are observed. We believe that this behavior can be attributed to an overall hydrophobicity of the molecule. However, it would be of interest to test this hypothesis using two or more curcumin molecules in different solvents. "Switching" of the enol hydrogen is an important event as well, inward and outward conformations of this atom occur with different frequency upon changing the solvent. This behavior was analyzed in terms of the corresponding dihedral angle and histograms of the distance distribution. A similar effect has been discussed in [12] within the framework of another curcumin model. Both, the bending trends and "switching" of the enol hydrogen contribute to the resulting value of the dipole moment of the molecule in solvent media. Larger values of the magnitude of the dipole moment in comparison with vacuum [18,40] are observed.
The pair distribution functions built for particular atoms of a curcumin molecule with solvent species reflect that the surroundings of the molecule are heterogenous in terms of density, higher density is probable around phenol groups whereas "desorption" type effects are more probable close to enol group. The formation of cross hydrogen bonds is most probable therein. Seemingly, the formation of hydrogen bonds counterarrests hydrophobicity of certain fragments of the molecule, while in water this effect is weak. The solvent density is perturbed in approximately single layer around phenol groups whereas it is more extended around the enol fragment. Important effects can be seen in the second layer therein. Coordination numbers of particular atoms and density distribution maps illustrate a solvent distribution around the molecule at a quantitative and at a qualitative level.
In the final part of this work, our results are given for the self-diffusion coefficient of a curcumin molecule in three solvents. We presented additional arguments to prove that the trends observed upon changing the solvent are correct by performing auxiliary calculations having involved concepts of radius of gyration and hydrodynamic radius.
It is of much interest to extend this study along several lines. For the moment, it seems most important to perform comparisons of the behavior of a single curcumin molecule in other protic and aprotic solvents and extend present calculations to a finite concentration interval for curcumin molecules in the spirit of recent work [13]. Construction of bridges to experimental observations is not just desirable but certainly indispensable for a better understanding of curcumin solutions with simultaneous improvement of the force fields.
Figure 1 .
1Chemical structure of the enol form of curcumin.
Figure 2 .
2(Color online) Schematic representation for the united-atom curcumin model with sites numbering. Carbon groups are shown as dark gray spheres, oxygens -as red spheres, hydrogens -as small light-gray spheres.
Figure 3 .
3(Color online) Histograms of the probability distributions for the dihedral angles of curcumin molecule in different solvents.
Figure 4 .
4(Color online) Histograms of the probability distribution of the H15-O18 (a) and H4-O2 (H28-O29) (b) distances for curcumin in different solvents.
Figure 5 .
5(Color online) Histograms of the probability distribution of the ring-ring (a) and center-ring (b) distances for curcumin in different solvents.
Figure 6 .
6(Color online) Histograms of the probability distributions of angles β (defined in the text) and angle cosine cos(β), in panels (a) and (b), respectively, for the curcumin molecule in different solvents.
Figure 7 .Figure 8 .
78(Color online) A comparison of the probability distribution for the dipole moment d of the curcumin molecule in water and DMSO. (Color online) Histograms of the probability distribution of O14-H15 distance p(r) obtained from different fragments of curcumin trajectory, cases A and B, left-hand panel. The corresponding probability distribution for the dipole moment p(d) of curcumin in water obtained by using these trajectories (right-hand panel).
Figure 9 .
9(Color online) Pair distribution function for the hydrogens and oxygens of curcumin molecule and oxygens (OX) of different solvents.
Figure 10 .
10(Color online) Pair distribution function for the carbons of curcumin molecule and oxygens (OX) of different solvents.
Figure 11 .
11(Color online) Spatial distribution of the density for oxygens (in red) and hydrogens (in blue) of water molecules around a curcumin molecule for the case of the enol hydrogen pointing toward (left-hand panel) and outward (right-hand panel) the enol group.
Figure 12 .
12(Color online) Spatial distribution of the density for oxygens (in red), hydrogens (in blue) and methyl group (transparent cyan) of methanol molecules around a curcumin molecule for the case of the enol hydrogen pointing toward (left-hand panel) and outward (right-hand panel) the enol group.
Figure 13 .
13(Color online) Spatial distribution of the density for oxygens (in red), sulfur (in yellow) and methyl groups (transparent cyan) of DMSO molecules around a curcumin molecule for the case of the enol hydrogen pointing toward (left-hand panel) and outward (right-hand panel) the enol group.
Figure 14 .
14(Color online) Histograms of the probability distributions of the total number of hydrogen bonds (n hb ) between the curcumin molecule and molecules of different solvents.
Figure 15 .
15(Color online) Histograms of the probability distributions of the number of hydrogen bonds (n hb ) formed between the groups containing oxygens O5(O27), O14, O18 and molecules of different solvents.
Figure 16 .
16(Color online) Mean-square displacements of the solvents and of the curcumin molecule immersed in them.
Figure 17 .
17(Color online) Histograms of the probability distributions of hydrodynamic radius (a) and radius of gyration (b) for the curcumin molecule in different solvents.
Table 1 .
1Nomenclature for the groups of dihedral angles of a curcumin molecule. Site numbers are from figure 2.dih. group abbr.
dihedrals
EnHdih
12-13-14-15
Phdih
3-6-5-4
and 26-24-27-28
Andih
6-3-2-1
and 24-26-29-30
B 1
12-11-8-7
and 25-21-20-19
B 2
13-12-11-8 and 21-20-19-17
B 3
16-13-12-11 and 20-19-17-16
23003-3
Table 2 .
2Position of the first minimum of the pair distribution functions and the first coordination numbers. The sites OW, OM and OD correspond to oxygens belonging to water, MeOH and DMSO molecules, respectively.CUR-SOL
r min , nm n(r min )
O14-OW
0.355
3.17
O18-OW
0.390
4.58
O5/O27-OW
0.350
3.28
O2/O29-OW
0.365
2.46
O14-OM
0.336
0.86
O18-OM
0.310
0.35
O5/O27-OM
0.350
1.54
O2/O29-OM
0.365
1.11
O14-OD
0.360
0.58
O5/O27-OD
0.360
0.97
O2/O29-OD
0.370
1.00
23003-11
table 3). The average number of HBs0
1
2
3
4
n hb
0
0.2
0.4
0.6
0.8
p(n
hb
)
water (<n hb >=1.84)
methanol (<n hb >=1.19)
dmso (<n hb >=0.78)
O5 (O27)
(a)
0
1
2
3
4
n hb
0
0.2
0.4
0.6
0.8
p(n
hb )
water (<n hb >=1.41)
methanol (<n hb >=0.56)
dmso (<n hb >=0.18)
O14
(b)
0
1
2
3
4
n hb
0
0.2
0.4
0.6
0.8
p(n
hb
)
water (<n hb >=0.87)
methanol (<n hb >=0.22)
O18
(c)
Table 3 .
3Average number of H-bonds between each of oxygen-containing groups of curcumin and solvent molecules.water
MeOH
DMSO
acceptor donor both acceptor donor both acceptor
O14
0.40
1.01 1.41
0.25
0.31 0.56
0.18
O18
−
0.87 0.87
−
0.22 0.22
−
O5/O27
0.66
1.18 1.84
0.66
0.53 1.19
0.78
O2/O29
−
0.16 0.16
−
0.02 0.02
−
Sum
1.72
4.56 6.28
1.58
1.64 3.20
1.74
table 4 .
4It could be seen that the most probable values for hydrodynamic radius of curcumin0.66
0.68
0.7
0.72
R h (nm)
Table 4 .
4Hydrodynamic radius R h and the radius of gyration R g for a single curcumin molecule in different solvents. The experimental values of solvent viscosities η S were taken from[38] and[39]. The self-diffusion coefficient D was calculated from the Stokes-Einstein equation(3).solventR h , nm R g , nm η S , 10 −3 Pa·s D, 10 −5 cm 2 s −1 D self , 10 −5 cm 2 s −1 practically does not change with the solvent. More pronounced changes are observed for the radius of gyration R g ,water
0.689
0.519
0.890
0.356
0.423
methanol 0.690
0.523
0.545
0.580
0.710
DMSO
0.690
0.525
1.996
0.158
0.353
The average values for R g are included in table 4. The experimental values of viscosity η S of water, methanol and DMSO are given in table 4 as well. We have found that the values for the self-diffusion coefficient calculated from the equation
Властивостi OPLS-UA моделi однiєї молекули куркумiну у водi, метанолi та диметилсульфоксидi. Результати комп'ютерного моделювання методом молекулярної динамiки Т. Пацаган 1 , Я.М. Iльницький 1 , О. Пiзiо 2 1 Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 Львiв, Україна 2 Iнститут матерiалознавства, Нацiональний автономний унiверситет м. Мехiко, Мехiко, Мексика За допомогою методу молекулярної динамiки (МД) проведено комп'ютерне моделювання та дослiджено властивостi однiєї молекули куркумiну у водi, метанолi та диметилсульфоксидi при постiйних температурi та тиску. Для цього використано модель, запропоновану нами нещодавно для куркумiну в рамках силового поля OPLS-UA [ J. Mol. Liq., 2016, 223, 707] та поєднано її iз моделлю води SPC/E та моделями OPLS-UA для метанолу i диметилсульфоксиду. Нами отримано детальний опис змiн у внутрiшнiй структурi розчиненої молекули, якi спричинюються рiзними середовищами розчинника. Проаналiзовано парнi функцiї розподiлу мiж окремими фрагментами молекули куркумiну та молекулами розчинника. Дослiджено статистичнi характеристики водневих зв'язкiв мiж рiзними компонентами. Насамкiнець, отримано коефiцiєнт самодифузiї молекули куркумiну в трьох модельних розчинниках.Ключовi слова: куркумiн, молекулярна динамiка, вода, метанол, диметилсульфоксид
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| [] |
[
"Distinct electronic nematicities between electron and hole underdoped iron pnictides",
"Distinct electronic nematicities between electron and hole underdoped iron pnictides"
] | [
"J J Ying \nHefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China\n",
"X F Wang \nHefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China\n",
"T Wu \nHefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China\n",
"Z J Xiang \nHefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China\n",
"R H Liu \nHefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China\n",
"Y J Yan \nHefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China\n",
"A F Wang ",
"M Zhang \nHefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China\n",
"G J Ye \nHefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China\n",
"P Cheng [email protected] \nHefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China\n",
"J P Hu \nHefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China\n\nDepartment of Physics\nPurdue University\n47907West LafayetteIndianaUSA\n",
"X H Chen \nHefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China\n"
] | [
"Hefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China",
"Hefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China",
"Hefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China",
"Hefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China",
"Hefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China",
"Hefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China",
"Hefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China",
"Hefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China",
"Hefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China",
"Hefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China",
"Department of Physics\nPurdue University\n47907West LafayetteIndianaUSA",
"Hefei National Laboratory for Physical Science at Microscale\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiPeople's Republic of China"
] | [] | We systematically investigated the in-plane resistivity anisotropy of electron-underdoped EuF e2−xCoxAs2 and BaF e2−xCoxAs2, and hole-underdoped Ba1−xKxF e2As2. Large in-plane resistivity anisotropy was found in the former samples, while tiny in-plane resistivity anisotropy was detected in the latter ones. When it is detected, the anisotropy starts above the structural transition temperature and increases smoothly through it. As the temperature is lowered further, the anisotropy takes a dramatic enhancement through the magnetic transition temperature. We found that the anisotropy is universally tied to the presence of non-Fermi liquid T-linear behavior of resistivity. Our results demonstrate that the nematic state is caused by electronic degrees of freedom, and the microscopic orbital involvement in magnetically ordered state must be fundamentally different between the hole and electron doped materials. | 10.1103/physrevlett.107.067001 | [
"https://arxiv.org/pdf/1012.2731v1.pdf"
] | 30,555,330 | 1012.2731 | c300a1badac148f887539747077bc7bd1c60fd35 |
Distinct electronic nematicities between electron and hole underdoped iron pnictides
13 Dec 2010
J J Ying
Hefei National Laboratory for Physical Science at Microscale
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiPeople's Republic of China
X F Wang
Hefei National Laboratory for Physical Science at Microscale
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiPeople's Republic of China
T Wu
Hefei National Laboratory for Physical Science at Microscale
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiPeople's Republic of China
Z J Xiang
Hefei National Laboratory for Physical Science at Microscale
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiPeople's Republic of China
R H Liu
Hefei National Laboratory for Physical Science at Microscale
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiPeople's Republic of China
Y J Yan
Hefei National Laboratory for Physical Science at Microscale
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiPeople's Republic of China
A F Wang
M Zhang
Hefei National Laboratory for Physical Science at Microscale
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiPeople's Republic of China
G J Ye
Hefei National Laboratory for Physical Science at Microscale
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiPeople's Republic of China
P Cheng [email protected]
Hefei National Laboratory for Physical Science at Microscale
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiPeople's Republic of China
J P Hu
Hefei National Laboratory for Physical Science at Microscale
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiPeople's Republic of China
Department of Physics
Purdue University
47907West LafayetteIndianaUSA
X H Chen
Hefei National Laboratory for Physical Science at Microscale
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiPeople's Republic of China
Distinct electronic nematicities between electron and hole underdoped iron pnictides
13 Dec 2010
We systematically investigated the in-plane resistivity anisotropy of electron-underdoped EuF e2−xCoxAs2 and BaF e2−xCoxAs2, and hole-underdoped Ba1−xKxF e2As2. Large in-plane resistivity anisotropy was found in the former samples, while tiny in-plane resistivity anisotropy was detected in the latter ones. When it is detected, the anisotropy starts above the structural transition temperature and increases smoothly through it. As the temperature is lowered further, the anisotropy takes a dramatic enhancement through the magnetic transition temperature. We found that the anisotropy is universally tied to the presence of non-Fermi liquid T-linear behavior of resistivity. Our results demonstrate that the nematic state is caused by electronic degrees of freedom, and the microscopic orbital involvement in magnetically ordered state must be fundamentally different between the hole and electron doped materials.
The newly discovered iron-based high temperature superconductors provide a new family of materials to explore the mechanism of high-T c superconductivity besides high-T c cuprates superconductors [1][2][3][4]. The parent compounds undergo a tetragonal to orthorhombic structure transition and a collinear antiferromagnetic (CAFM) transition.
The structure transition temperature(T S ) is higher than the CAFM transition temperature(T N ) in 1111 system, while for the 122 parent compound T N is almost the same with T S . Superconductivity arises when both transitions are suppressed via electron or hole doping. The origin of the structure transition and CAFM transition is still unclear, it was proposed that antiferromagnetic fluctuation [6,7] or orbital ordering [5,17,18] may play an important role in driving the transitions.
Recent works showed a large in-plane resistivity anisotropy below T S or T N in Co-doped Ba122 system, though the distortion of the orthorhombic structure is less than 1 % [8,9] in the CAFM state. The resistivity is smaller along the antiferromagnetic a direction than along the ferromagnetic b direction which is opposite to our intuitive thinking. Moreover, the resistivity anisotropy grows with doping and approaches its maximum close to the point where superconductivity emerges. Other experiments also reported the existence of anisotropy in many different physical properties in the CAFM state of iron pnictides including magnetic exchange coupling [14], Fermi surface topology [15] and local density distribution [16]. Electronic anisotropies have also been observed in many other materials, including underdoped cuprates, quantum Hall systems and * Corresponding author; Electronic address: [email protected] [12,13]. In cuprates, electronic anisotropy can not be explained by small structural orthorhombicity [10,11] and it has been proposed the large electron anisotropy is due to the emergence of electron nematic phase. This idea has been widely investigated in the other two systems as well. In iron-pnictides, the origin of the the electronic anisotropy is still elusive because of their complicated electronic structures and the presence of many different degrees of freedom.
Sr 3 Ru 2 O 7
In this paper, we systematically investigated the in-plane resistivity anisotropy of electron-underdoped EuF e 2−x Co x As 2 and BaF e 2−x Co x As 2 , and holeunderdoped Ba 1−x K x F e 2 As 2 systems. Large in-plane resistivity behavior was found below T S or T N in EuF e 2−x Co x As 2 and BaF e 2−x Co x As 2 , while in-plane anisotropy is dramatically decreased and barely observable in Ba 1−x K x F e 2 As 2 . The resistivity shows T-linear behavior above the temperature wherever the resistivity anisotropy starts to emerge. In the Ba 1−x K x F e 2 As 2 , no T-linear behavior in the resistivity is observed in the normal state. For EuF e 2−x Co x As 2 , the resistivity behaves very differently at T S and T N . The in-plane anisotropy starts to emerge even above T S and increases smoothly through it. As the temperature is lowered further, it takes a dramatic enhancement through the magnetic phase transition. Therefore, our results provide direct evidence ruling out the possibility that the anisotropy is caused by lattice distortion. The dramatic difference of the anisotropies in the electron and hole doped materials suggests that magnetism must be orbital-selective in a way that the magnetism in the hole underdoped ironpnictides may stem mainly from the d xy orbitals while the magnetism in the electron underdoped ones attributes mainly to the d yz and d xz orbitals.
High quality single crystals with nominal composition EuF e 2−x Co x As 2 (x=0 ,0.067 ,0.1 ,0.225) and Ba 1−x K x F e 2 As 2 (x=0.1 ,0.18) were grown by self-flux method as described elsewhere [19]. Many shinning platelike single crystals can be obtained. EuF e 2 As 2 has not only a CAFM or structural transition around 190 K, but also an antiferromagnetic transition of Eu 2+ ions around 20 K [20]. As Co-doping increases, the SDW transition and structure transition were gradually separated and suppressed like the BaF e 2−x Co x As 2 system. Resistivity reentrance behavior due to the antiferromagnetic ordering of Eu 2+ spins is also observed at low temperature. The detailed in-plane resistivity of EuF e 2−x Co x As 2 can be found elsewhere [21,22]. Ba 0.9 K 0.1 F e 2 As 2 and Ba 0.82 K 0.18 F e 2 As 2 undergoes structural transition and SDW transition at around 128K and 113 K, respectively. It is difficult to directly measure the in-plane resistivity anisotropy because the material naturally forms structure domains below T S . In order to investigate the intrinsic in-plane resistivity anisotropy we developed a mechanical cantilever device that is able to detwin crystals similar to the Ref. 8. Crystals were cut parallel to the orthorhombic a and b axes so that the orthorhombic a(b) direction is perpendicular (parallel) to the applied pressure direction. ρ a (current parallel to a) and ρ b (current parallel to b) were measured on the same sample using standard 4-point configuration. We can get the same result with Ref.8 in Co-doped Ba122 system as shown in Fig.4(d) and (e). Fig.1(a) shows the temperature dependence of in-plane resistivity with the current flowing parallel to the orthorhombic b direction (black) and orthorhombic a direction (red) of the detwined EuF e 2 As 2 sample. Compounds with a small amount of Co-doping show large in-plane anisotropy as displayed in Fig.1(b) and (c) for x=0.067 and x=0.1, respectively. More specifically, we can find a much more obvious upturn of ρ b around T N or T S compared to the twined in-plane resistivity as the blue line shows while ρ a shows no upturn and drops very rapidly with the temperature decreases below T N or T S . The behavior of ρ a is very similar to the EuF e 2 As 2 polycrystalline samples [23]. When the Co-doping increases, the upturn of ρ b at T N or T S becomes much more sharper while the drop of ρ a at T N or T S gradually disappears and turns into a small upturn at the temperature slightly lower than ρ b . For the sample of x=0.225, no in-plane anisotropy was found due to the complete suppression of the structural or magnetic transition as shown in Fig.1(d). The different behavior of ρ a and ρ b observed here is very similar to the ones in other parent or electron underdoped iron-pnictides [8,9].
Further examining our data, we also notice that the structural transition temperature T S and the magnetic transition temperature T N become well separated as the Co-doping increases in EuF e 2−x Co x As 2 . This separation provides us an opportunity to investigate the effects of the structural transition and magnetic transition on the ρ a and ρ b separately. The two transition tempera- tures can be obtained by analyzing the heat capacity of the samples. Treating the heat capacity in the x=0.225 sample as phonon background and subtracting it, we obtain the electronic part of the heat capacity of the x=0.067 and x=0.1 samples, ∆C P . Taking the x=0.067 sample as an example, the ∆C P has two distinct features as a function of temperature: a very sharp peak at 150 K and a broad hump around 157 K. Similar to other isostructure materials, such as BaF e 2−x Co x As 2 [25], we can attribute the sharp peak to the magnetic transition and the broad hump to the structure transition. In the x=0.067 sample, as shown in fig.2(a), ρ a shows a dip like behavior from T S to T N . ρ a starts to drop rapidly at the temperature coincident with T S , and after reaching its minimum value, it shows a very weak upturn at the temperature coincident with T N . ρ b , however, has no observable feature at T S but shows a large upturn around T N . When the sample is not detwinned, the in-plane resistivity ρ ab also shows a large upturn. For the x=0.1 sample, T S is suppressed to 140 K and T N is suppressed to 130 K from the heat capacity measurement as shown in Fig.2(b). ρ b starts to go upward above T S but ρ a does not show any obvious response. The dρ ab /dT curve of the twined ρ ab peaks around T N while the peak of dρ b /dT is higher than T N and the one of dρ a /dT is slightly lower than T N . With increasing the Co doping, the effect of the structure transition on the ρ a is gradually wiped out which is similar to BaF e 2−x Co x As 2 system, while the upturn behavior T N becomes more noticeable.
We characterized the degree of in-plane resistivity anisotropy by the ratio ρ b /ρ a . Fig.3 plane resistivity anisotropy ρ b /ρ a and its related differential curve for x=0.067 and x=0.1 samples, respectively. The amplitude of the anisotropy is greatly increased with Co doping though the magnetic transition and structure transition are suppressed, similar to BaF e 2−x Co x As 2 . The in-plane resistivity anisotropy increases very rapidly at T N and still gradually increases down to 4 K. A very sharp peak can be observed in the differential curve of ρ b /ρ a at the temperature coincident with T N . However, ρ b /ρ a did not show any obvious anomaly at T S . The sharp increase of in-plane resistivity anisotropy at T N indicates that in-plane anisotropy is correlated to the magnetic transition rather than structure transition. It also indicates that the in-plane electron anisotropy is driven by the magnetic fluctuation or other hidden electronic order rather than the small orthorhombic distortion.
(a) and (b) show in-
To understand the common feature of the inplane resistivity anisotropies in EuF e 2−x Co x As 2 and BaF e 2−x Co x As 2 , we plot ρ a and ρ b in an enlarged temperature region as shown in Fig.4(a), (b), (c), (d) and (e). For EuF e 2−x Co x As 2 , it is very clear that the in-plane resistivity anisotropy emerges at temperatures higher than the structure transition temperature as the black arrows indicate. It suggests that fluctuations associated with the resistivity anisotropy must emerge well above the T S . Moreover, T-linear resistivity behaviors appear in a large temperature region above the temperature which ρ a and ρ b begin to show discrepancy. We also observed this kind of feature in Co-doped Ba122 system as shown in Fig.4(d), (e) for BaF e 2 As 2 and BaF e 1.83 Co 0.17 As 2 respectively. This non-Fermi liquid behavior, or the Tlinear behavior has also been observed in other ironpnictides, for example, BaF e 2 As 2−x P x single crystals and SmO 1−x F x F eAs polycrystaline samples [26,27]. It suggests that this behavior might be universal in electron underdoped iron pnictides at high temperature. The Tlinear resistivity behavior is also similar to the one in cuprates. However, in Ba 2−x K x F e 2 As 2 , such a T-linear behavior is rapidly suppressed through K doping. As shown in Fig.4(f) and (g), Ba 0.9 K 0.1 F e 2 As 2 shows very tiny in-plane anisotropy compared with its parent compound, Ba 0.82 K 0.18 F e 2 As 2 almost show no in-plane resistivity anisotropy and meanwhile the normal state resistivity does not follow the T-linear behavior. The correspondence between the anisotropy and T-linear behavior of resistivity presents in all of measured materials. We did not find a single exception. Our above results have important implications for the origin of the nematicity in iron-pnictides. First, our results strongly support the nematic state in iron-pnictides is indeed an electronic nematic state. Our measurements show that the in-plane resistivity anisotropy is closely related to the magnetic transition rather than the structural transition and persists at temperature higher than structural transition, suggesting the existence of nematicity even in tetragonal lattice structure. Second, the distinct anisotropy of resistivity between the hole and electron underdoped materials reveals the hidden interplay between magnetic and orbital degrees of free-dom. Recently, many theories focus on orbital ordering which generates an unequal occupation of d xz and d yz orbitals [28] that breaks the rotational symmetry and causes the lattice distortion [29]. ARPES measurements have provided orbital ordering evidence [15]. Since there is little anisotropy in the hole doped samples in their magnetically ordered states, our results suggest that the resistivity anisotropy is most likely induced by orbital ordering rather than magnetic ordering. The orbital ordering is suppressed rapidly by hole-doping while it is relatively robust to electron-doping. The strong enhancement of the anisotropy around T N in electron doped systems indicates that the magnetic ordering and orbital ordering are intimately connected to each other in these systems. However, our result on hole doped samples suggest the magnetic ordering is not simply a result of orbital ordering. Considering the facts that the dominating orbitals are t 2g and the orbital ordering is most likely due to the d xz and d yz , we can conclude that although the magnetically ordered states in both electron and hole doped materials have an identical ordering wavevector, the two states must differ microscopically from their orbital involvements, namely, the magnetism is orbital selective in a way that the magnetic ordering in hole-doped systems is attributed mostly from the d xy orbital while the d yz and d xz make important contributions to the magnetism in the electron doped ones. This implication can be tested explicitly by ARPES experiments in hole doped materials. Finally, the correspondence between the anisotropy and T-linear behavior suggests the importance of electron-electron correlation in causing the orbital and magnetic ordering. The non-Fermi liquid behavior of T-linear resistivity can be understood by the presence of strong orbital or magnetic fluctuations in all three t 2g orbitals. Consequently, the suppression of the non-fermi liquid behavior by hole-doping indicates the suppression of orbital and magnetic fluctuations in the d xz and d yz ones.
In conclusion, we have measured the in-plane resistivity anisotropy on electron-underdoped EuF e 2−x Co x As 2 and BaF e 2−x Co x As 2 , and hole-underdoped Ba 1−x K x F e 2 As 2 single crystals. Large in-plane resistivity anisotropy was found in EuF e 2−x Co x As 2 which is quite similar to the isostructural BaF e 2−x Co x As 2 system, however anisotropy disappears in hole-doped samples. We identified an universal correspondence between the anisotropy and a T-linear behavior of resistivity at high temperature. The different behavior of the anisotropy at T N and T S rules out the anisotropy is originated from the lattice degree of freedom. The magnetic states in the hole and electron doped systems are significantly different. This work is supported by the Nature Science Foundation of China, Ministry of Science and Technology and by Chinese Academy of Sciences.
online) Temperature dependence of in-plane resistivity with the electric current flow along a direction (red) and b direction (black) respectively for the parent compound (a): EuF e2As2, (b): EuF e1.933Co0.067As2, (c): EuF e1.9Co0.1As2, (d): EuF e1.775Co0.225As2. The twined in-plane resistivity was also shown for comparison(blue line).
FIG. 2 :
2(Color online) Temperature dependence of ρa, ρ b and twined ρ ab , their differential curve and heat capacity around the TS and TN for the sample of (a): x=0.067 and(b): x=0.1. The red dashed line and blue solid line indicated TS and TN respectively.
FIG. 3 :
3(Color online) In-plane resistivity anisotropy ρ b /ρa and its related differential curve for the sample of (a):x=0.067 and (b): x=0.1. The blue solid line indicated TN and red dashed line indicated TS.
FIG. 4 :
4(Color online)The temperature dependence of ρa and ρ b for (a)EuF e2As2, (b): EuF e1.933Co0.067As2, (c): EuF e1.9Co0.1As2, (d): BaF e2As2, (e): BaF e1.83Co0.17As2, (f): Ba0.9K0.1F e2As2 and (g): Ba0.82K0.18F e2As2. The black arrows indicate the temperature where ρa and ρ b begin to show discrepancy. The red line indicates the TS. The blue line is the linear fit of the resistivity above the black arrow indicated temperature.
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| [] |
[
"THREE PROBLEMS FOR THE CLAIRVOYANT DEMON",
"THREE PROBLEMS FOR THE CLAIRVOYANT DEMON"
] | [
"Geoffrey Grimmett "
] | [] | [] | A number of tricky problems in probability are discussed, having in common one or more infinite sequences of coin tosses, and a representation as a problem in dependent percolation. Three of these problems are of 'Winkler' type, that is, they ask about what can be achieved by a clairvoyant demon. | 10.1017/cbo9781139107174.018 | [
"https://arxiv.org/pdf/0903.4749v2.pdf"
] | 3,155,505 | 0903.4749 | da2a77337d81f59dbbde59ef9ed5ad13a7dc5f70 |
THREE PROBLEMS FOR THE CLAIRVOYANT DEMON
28 Jun 2009
Geoffrey Grimmett
THREE PROBLEMS FOR THE CLAIRVOYANT DEMON
28 Jun 2009
A number of tricky problems in probability are discussed, having in common one or more infinite sequences of coin tosses, and a representation as a problem in dependent percolation. Three of these problems are of 'Winkler' type, that is, they ask about what can be achieved by a clairvoyant demon.
Introduction
Probability theory has emerged in recent decades as a crossroads where many sub-disciplines of mathematical science meet and interact. Of the many examples within mathematics, we mention (not in order): analysis, partial differential equations, mathematical physics, measure theory, discrete mathematics, theoretical computer science, and number theory. The International Mathematical Union and the Abel Memorial Fund have recently accorded acclaim to probabilists. This process of recognition by others has been too slow, and would have been slower without the efforts of distinguished mathematicians including John Kingman. JFCK's work looks towards both theory and applications. To single out just two of his theorems: the subadditive ergodic theorem [21,22] is a piece of mathematical perfection that has also proved rather useful in practice; his 'coalescent' [23,24] is a beautiful piece of probability that is now a lynchpin of mathematical genetics. John is also an inspiring and devoted lecturer, who continued to lecture to undergraduates even as the Bristol Vice-Chancellor, and the Director of the Isaac Newton Institute in Cambridge. Indeed, the current author learned his measure and probability from partial attendance at John's course in Oxford in 1970/71.
To misquote Frank Spitzer [34,Chap. 8], we turn to a very down-to-earth problem: consider an infinite sequence of light bulbs. The basic commodity of probability is an infinite sequence of coin tosses. Such a sequence has been studied for so long, and yet there remain 'simple to state' problems that appear very hard. We present some of these problems here. Sections 3-5 are devoted to three famous problems for the so-called clairvoyant demon, a non-human being to whom is revealed the (infinite) realization of the sequence, and who is permitted to plan accordingly for the future.
Each of these problems may be phrased as a geometrical problem of percolation type. The difference with classical percolation [13] lies in the dependence of the site variables. Percolation is reviewed briefly in Section 2. This article ends with two short sections on related problems, namely: other forms of dependent percolation, and the question of 'percolation of words'.
Site percolation
We set the scene by reminding the reader of the classical 'site percolation model' of Broadbent and Hammersley [9]. Consider a countably infinite, con- Let 0 be a given vertex, called the origin, and let θ(p) be the probability that the origin lies in an infinite open self-avoiding path of G. It is clear that θ is non-decreasing in p, and θ(0) = 0, θ(1) = 1. The critical probability is given as
nected graph G = (V, E). To each 'site' v ∈ V we assign a Bernoulli random variable ω(v) with density p. That is, ω = {ω(v) : v ∈ V } isp c = p c (G) := sup{p : θ(p) = 0}.
It is a standard exercise to show that the value of p c does not depend on the choice of origin, but only on the graph G.
One may instead associate the random variables with the edges of the graph, rather than the vertices, in which case the process is termed 'bond percolation'. Percolation is recognised as a fundamental model for a random medium. It is important in probability and statistical physics, and it continues to be the source of beautiful and apparently hard mathematical problems, of which the most outstanding is to prove that θ(p c ) = 0 for the three-dimensional lattice Z 3 . Of the several recent accounts of the percolation model, we mention [13,14].
Most attention has been paid to the case when G is a crystalline lattice in two or more dimensions. The current article is entirely concerned with aspects of two-dimensional percolation, particularly on the square and triangular lattices illustrated in Figure 1. Site percolation on the triangular lattice has featured prominently in the news in recent years, owing to the work of Smirnov, Lawler-Schramm-Werner, and others on the relationship of this model (with p = p c = 1 2 ) to the process of random curves in R 2 termed Schramm-Löwner evolutions (SLE), and particularly the process denoted SLE 6 . See [35]. When G is a directed graph, one may ask about the existence of an infinite open directed path from the origin, in which case the process is referred to as directed (or oriented ) percolation.
Variants of the percolation model are discussed in the following sections, with the emphasis on models with site/bond variables that are dependent.
Clairvoyant scheduling
Let G = (V, E) be a finite connected graph. A symmetric random walk on G is a sequence X = (X k : k = 0, 1, 2, . . . ) of V -valued random variables satisfying, for all v, w ∈ V and k ≥ 0,
P (X k+1 = w | X k = v) = 1 ∆ v if v ∼ w, 0 if v ≁ w,
where ∆ v is the degree of vertex v, and ∼ denotes the adjacency relation of G. Random walks on general graphs have attracted much interest in recent years, see [14, Chap. 1] for example. Let X and Y be independent random walks on G with distinct starting sites x 0 , y 0 , respectively. We think of X (respectively, Y ) as describing the trajectory of a particle labelled X (respectively, Y ) around G. A clairvoyant demon is set the task of keeping the walks apart from one another for all time.
To this end, (s)he is permitted to schedule the walks in such a way that exactly one walker moves at each epoch of time. Thus, the walks may be delayed, but they are required to follow their prescribed trajectories.
More precisely, a schedule is defined as a sequence Z = (Z 1 , Z 2 , . . . ) in the space {X, Y } N , and a given schedule Z is implemented in the following way. From the X and Y trajectories, we construct the rescheduled walks Z(X) and Z(Y ), where:
1. If Z 1 = X, the X-particle takes one step at time 1, and the Y -particle remains stationary. If Z 1 = Y , it is the Y -particle that moves, and the X-particle that remains stationary. Thus,
if Z 1 = X then Z(X) 1 = X 1 , Z(Y ) 1 = Y 0 , if Z 1 = Y then Z(X) 1 = X 0 , Z(Y ) 1 = Y 1 .
2. Assume that, after time k, the X-particle has made r moves and the Y -particle k − r moves, so that Z(X) k = X r and Z(Y ) k = Y k−r .
If Z k+1 = X then Z(X) k+1 = X r+1 , Z(Y ) k+1 = Y k−r , if Z k+1 = Y then Z(X) k+1 = X r , Z(Y ) k+1 = Y k−r+1 .
We call the schedule Z good if Z(X) k = Z(Y ) k for all k ≥ 1, and we say that the demon succeeds if there exists a good schedule Z = Z(X, Y ). (We overlook issues of measurability here.) The probability of success is θ(G) := P(there exists a good schedule), and we ask: for which graphs G is it the case that θ(G) > 0? This question was posed by Peter Winkler (see the discussion in [10,11]). Note that the value of θ(G) is independent of the (distinct) starting points x 0 , y 0 .
The problem takes a simpler form when G is the complete graph on some number, M say, of vertices. In order to simplify it still further, we add a loop to each vertex. Write V = {1, 2, . . . , M}, and θ(M) := θ(G). A random walk on G is now a sequence of independent, identically distributed points in {1, 2, . . . , M}, each with the uniform distribution. It is expected that θ(M) is non-decreasing in M, and it is clear by coupling that θ(kM) ≥ θ(M) for k ≥ 1. Also, it is not too hard to show that θ(3) = 0. This problem has a geometrical formulation of percolation-type. Consider the positive quadrant N 2 = {(i, j) : i, j ≥ 1, 2, . . . } of the square lattice Z 2 . A path is taken to be an infinite sequence (u n , v n ), n ≥ 0, with (u 0 , v 0 ) = (0, 0) such that, for all n ≥ 0,
either (u n+1 , v n+1 ) = (u n + 1, v n ) or (u n+1 , v n+1 ) = (u n , v n + 1).
With X, Y the random walks as above, we declare the vertex (i, j) to be open if X i = Y j . It may be seen that the demon succeeds if and only if there exists a path all of whose vertices are open.
Some discussion of this problem may be found in [11]. The law of the open vertices is 3-wise independent but not 4-wise independent in the sense of Section 6.
The problem becomes significantly easier if paths are allowed to be undirected. For the totally undirected problem, it is proved in [3,36]
Clairvoyant compatibility
Let p ∈ (0, 1), and let X 1 , X 2 , . . . and Y 1 , Y 2 . . . be independent sequences of independent Bernoulli variables with common parameter p. We say that a collision occurs at time n if X n = Y n = 1. The demon is now charged with the removal of collisions, and to this end (s)he is permitted to remove 0s from the sequences.
Let W = {0, 1} N , the set of singly-infinite sequences of 0s and 1s. Each w ∈ W is considered as a word in an alphabet of two letters, and we generally write w n for its nth letter. For w ∈ W, there exists a sequence i(w) = (i(w) 1 , i(w) 2 , . . . ) of non-negative integers such that w = 0 i 1 10 i 2 1 · · · , that is, there are exactly i j = i(w) j zeros between the (j − 1)th and jth appearances of
1. For x, y ∈ W, we write x → y if i(x) j ≥ i(y) j for j ≥ 1. That is, x → y
if and only if y may be obtained from x by the removal of 0s.
Two infinite words v, w are said to be compatible if there exist v ′ and w ′ such that v → v ′ , w → w ′ and v ′ n w ′ n = 0 for all n. For given realizations X, Y , we say that the demon succeeds if X and Y are compatible. Write ψ(p) = P p (X and Y are compatible).
Note that, by a coupling argument, ψ is a non-increasing function.
Question 4.1. For what p is it the case that ψ(p) > 0.
It is easy to see as follows that ψ( 1 2 ) = 0. When p = 1 2 , there exists a.s. an integer N such that
N i=1 X i > 1 2 N, N i=1 Y i > 1 2 N.
With N chosen thus, it is not possible for the demon to prevent a collision in the first N values. By working more carefully, one may see that ψ( 1 2 − ǫ) = 0 for small positive ǫ; see the discussion in [12].
Peter Gács has proved in [12] that ψ(10 −400 ) > 0, and he has noted that there is room for improvement.
Clairvoyant embedding
The clairvoyant demon's third problem stems from work on long-range percolation of words (see Section 7). Let X 1 , X 2 , . . . and Y 1 , Y 2 , . . . be independent sequences of independent Bernoulli variables with parameter 1 2 . Let M ∈ {2, 3, . . . }. The demon's task is to find a monotonic embedding of the X i within the Y j in such a way that the gaps between successive terms are no greater than M.
Let v, w ∈ W. We say that v is M-embeddable in w, and we write v ⊆ M w, if there exists an increasing sequence (m i : i ≥ 1) of positive integers such that
v i = w m i and 1 ≤ m i − m i−1 ≤ M for all i ≥ 1. (We set m 0 = 0.) A similar definition is made for finite words v lying in one of the spaces W n = {0, 1} n , n ≥ 1.
The demon succeeds in the above task if X ⊆ M Y , and we let
ρ(M) = P(X ⊆ M Y ).
It is elementary that ρ(M) is non-decreasing in M. This question is introduced and discussed in [15], and partial but limited results proved. One approach is to estimate the moments of the number N n (w) of M-embeddings of the finite word w = w 1 w 2 · · · w n ∈ W n within the random word Y . It is elementary that E(N n (w)) = (M/2) n for any such w, and it may be shown that
E(N n (X) 2 ) E(N n (X)) 2 ∼ A M c n M as n → ∞,
where A M > 0 and c M > 1 for M ≥ 2. That E(N n (w)) ≡ 1 when M = 2 is strongly suggestive that ρ(2) = 0, and this is part of the next theorem.
Theorem 5.2. [15] We have that ρ(2) = 0, and furthermore, for M = 2,
P(w ⊆ 2 Y ) ≤ P(a n ⊆ 2 Y ) for all w ∈ W n ,(5.3)
where a n = 0101 · · · is the alternating word of length n.
It is immediate that (5.3) implies ρ(2) = 0 on noting that, for any infinite periodic word π, P(π ⊆ M Y ) = 0 for all M ≥ 2. One may estimate such probabilities more exactly through solving appropriate difference equations. Herein lies a health warning for simulators. One knows that, almost surely, a n ⊆ M Y for large n, but one has to look on scales of order 2 2M −1 if one is to observe its extinction with reasonable probability. One may ask about the 'best' and 'worst' words. Inequality (5.3) asserts that an alternating word a n is the most easily embedded word when M = 2. It is not known which word is best when M > 2. Were this a periodic word, it would follow that ρ(M) = 0. Unsurprisingly, the worst word is a constant word c n (of which there are of course two). That is, for all M ≥ 2,
P(w ⊆ M Y ) ≥ P(c n ⊆ M Y ) for all w ∈ W n ,
where, for definiteness, we set c n = 1 n ∈ W n . Let M = 2, so that the mean number E(N n (w)) of embeddings of any word of length n is exactly 1 (as remarked above). A further argument is required to deduce that ρ(2) = 0. Peled [32] has made rigorous the following alternative to that used in the proof of Theorem 5.2. Assume that the word w ∈ W n satisfies w ⊆ 2 Y . For some small c > 0, one may identify (for most embeddings, with high probability) cn positions at which the embedding may be altered, independently of each other. This gives 2 cn possible 'local variations' of the embedding. It may be deduced that the probability of embedding a word w ∈ W n is exponentially small in n, and also ρ(2) = 0.
The sequences X, Y have been taken above with parameter 1 2 . Little changes in a more general setting. Let the two (respective) parameters be p X , p Y ∈ (0, 1). It turns out that the validity of the statement "for all M ≥ 2, P(X ⊆ M Y ) = 0" is independent of the values of p X , p Y . See [15].
A number of easier variations on Question 5.1 spring immediately to mind, of which two are mentioned here.
1. Suppose the gap between the embeddings of X i−1 and X i must be bounded above by some M i . How slow a growth on the M i suffices that the embedding probability be strictly positive? [An elementary bound follows by the Borel-Cantelli lemma.] 2. Suppose that the demon is allowed to look only boundedly into the future. How much clairvoyance may (s)he be allowed without the embedding probability becoming strictly positive? Further questions (and variations thereof) have been proposed by others.
1. In a 'penalised embedding' problem, we are permitted mismatches by paying a (multiplicative) penalty b for each. What is the cost of the 'cheapest' penalised embedding of the first n terms, and what can be said as b → ∞? [Erwin Bolthausen] 2. What can be said if we are required to embed only the 1s? That is, a '1' must be matched to a '1', but a '0' may be matched to either '0' or '1'. [Simon Griffiths] 3. The above problems may be described as embedding Z in Z. In this language, might it be possible to embed Z m in Z n for some m, n ≥ 2? [Ron Peled] Question 5.1 may be expressed as a geometrical problem of percolation type. With X and Y as above, we declare the vertex (i, j) ∈ N 2 to be open if X i = Y j . A path in N 2 is defined as an infinite sequence (u n , v n ), n ≥ 0, of 1. the argument of Peled [32] may be applied to problem (b) with M = 2 to obtain that P(w ⊆ 2 Y ) = 0 for all w ∈ W, 2. problem (e) is easily seen to be trivial. It is, as one might expect, much easier to embed words in two dimensions than in one, and indeed this may be done along a path of Z 2 that is directed in the north-easterly direction. This statement may be made more precise as follows. Let Y = (Y i,j : i, j ≥ 1) be a two-dimensional array of independent Bernoulli variables with parameter p ∈ (0, 1), say. A word v ∈ W is said to be M-embeddable in Y , written v ⊆ M Y , if there exist strictly increasing sequences (m i : i ≥ 1), (n i : i ≥ 1) of positive integers such that
v i = Y m i ,n i and 1 ≤ (m i − m i−1 ) + (n i − n i−1 ) ≤ M for all i ≥ 1. (We set m 0 = n 0 = 0.)
The following answers a question of [29]. Note added at revision: A related result has been discovered independently in [30].
Theorem 5.5. [14] Suppose R ≥ 1 is such that 1 − p R 2 − (1 − p) R 2 > p c ,
the critical probability of directed site percolation on Z 2 . With strictly positive probability, every infinite word w satisfies w ⊆ 5R Y .
The identification of the set of words that are 1-embeddable in the twodimensional array Y , with positive probability, is much harder. This is a problem of percolation of words, and the results to date are summarised in Section 7.
Proof. We use a block argument.
Let R ∈ {2, 3, . . . }. For (i, j) ∈ N 2 , define the block B R (i, j) = ((i − 1)R, iR] × ((j − 1)R, jR] ⊆ N 2 .
On the graph of blocks, we define the (directed) relation B R (i, j) → B R (m, n) if (m, n) is either (i + 1, j + 1) or (i + 1, j + 2). By drawing a picture, one sees that the ensuing directed graph is isomorphic to N 2 directed north-easterly. Note that the L 1 -distance between two vertices lying in adjacent blocks is no more than 5R.
We call a block B R good if it contains at least one 0 and at least one 1. It is trivial that
P p (B R is good) = 1 − p R 2 − (1 − p) R 2 .
If the right side exceeds the critical probability of directed site percolation on Z 2 , then there is a strictly positive probability of an infinite directed path of good blocks in the block graph, beginning at B R (1, 1). Such a path contains 5R-embeddings of all words.
The problem of clairvoyant embedding is connected to a question concerning isometries of random metric spaces discussed in [33]. In broad terms, two metric spaces (S i , µ i ), i = 1, 2, are said to be 'quasi-isometric' (or 'roughisometric') if their metric structure is the same up to multiplicative and additive constants. That is, there exists a mapping T : S 1 → S 2 and positive constants M, D, R such that:
1 M µ 1 (x, y) − D ≤ µ 2 (T (x), T (y)) ≤ Mµ 1 (x, y) + D, x, y ∈ S 1 ,
and, for x 2 ∈ S 2 , there exists x 1 ∈ S 1 with µ 2 (x 2 , T (x 1 )) ≤ R. It has been asked whether two Poisson process on the line, viewed as random sets with metric inherited from R, are quasi-isometric. This question is open at the time of writing. A number of related results are proved in [33], where a history of the problem may be found also. It turns out that the above question is equivalent to the following. Let X = (. . . , X −1 , X 0 , X 1 , . . . ) be a sequence of independent Bernoulli variables with common parameter p X . The sequence X generates a random metric space with points {i : X i = 1} and metric inherited from Z. Is it the case that two independent sequences X and Y generate quasi-isometric metric spaces? A possibly important difference between this problem and clairvoyant embedding is that quasi-isometries of metric subspaces of Z need not be monotone.
Dependent percolation
Whereas there is only one type of independence, there are many types of dependence, too many to be summarised here. We mention just three further types of dependent percolation in this section, of which the first (at least) arises in the context of processes in random environments. In each, the dependence has infinite range, and in this sense these problems have something in common with those treated in Sections 3-5.
For our first example, let X = {X i : i ∈ Z} be independent, identically distributed random variables taking values in [0, 1]. Conditional on X, the vertex (i, j) of Z 2 is declared open with probability X i , and different vertices receive (conditionally) independent states. The ensuing measure possesses a dependence that extends without limit in the vertical direction. If the law µ of X 0 places probability both below and above p c , there exist (almost surely) vertically-unbounded domains that consider themselves subcritical, and others that consider themselves supercritical. It depends on the choice of µ whether or not the process possesses infinite open paths, and necessary and sufficient conditions have proved elusive. The most successful technique for dealing with such problems seems to be the so-called 'multiscale analysis'. This leads to sufficient conditions under which the process is subcritical (respectively, supercritical). See [25,26].
There is a variety of models of physics and applied probability for which the natural random environment is exactly of the above type. Consider, for example, the contact model in d dimensions with recovery rates δ x and infection rates λ e , see [27,28]. Suppose that the environment is randomised through the assumption that the δ x (respectively, λ e ) are independent and identically distributed. The graphical representation of this model may be viewed as a 'vertically directed' percolation model on Z d × [0, ∞), in which the intensities of infections and recoveries are dependent in the vertical direction. See [1,8,31] for example.
Vertical dependence arises naturally in certain models of statistical physics also, of which we present one example. The 'quantum Ising model' on a graph G may be formulated as a problem in stochastic geometry on a product space of the form G × [0, β], where β is the inverse temperature. A fair bit of work has been done on the quantum model in a random environment, that is, when its parameters vary randomly around different vertices/edges of G. The corresponding stochastic model on G × [0, β] has 'vertical dependence' of infinite range. See [7,16].
It is easy to adapt the above structure to provide dependencies in both horizontal and vertical directions, although the ensuing problems may be considered (so far) to have greater mathematical than physical interest. For example, consider bond percolation on Z 2 , in which the states of horizontal edges are correlated thus, and similarly those of vertical edges. A related three-dimensional system has been studied by Jonasson, Mossel and Peres [18]. Draw planes in R 3 orthogonal to the x-axis, such that they intersect the x-axis at points of a Poisson process with given intensity λ. Similarly, draw independent families of planes orthogonal to the yand z-axes. These three families define a 'stretched' copy of Z 3 . An edge of this stretched lattice, of length l, is declared to be open with probability e −l , independently of the states of other edges. It is proved in [18] that, for sufficiently large λ, there exists (a.s.) an infinite open directed percolation cluster that is transient for simple random walk. The method of proof is interesting, proceeding as it does by the method of 'exponential intersection tails' (EIT) of [6]. When combined with an earlier argument of Häggström, this proves the existence of a percolation phase transition for the model.
The method of EIT is invalid in two dimensions, because random walk is recurrent on Z 2 . The corresponding percolation question in two dimensions was answered using different means by Hoffman [17].
In our final example, the dependence comes without geometrical information. Let k ≥ 2, and call a family of random variables k-wise independent if any k-subset is independent. Note that the vertex states in Section 3 are 3-wise independent but not 4-wise independent.
Benjamini, Gurel-Gurevich and Peled [4] have investigated various properties of k-wise independent Bernoulli families, and in particular the following percolation question. Consider the n-box B n = [1, n] d in Z d with d ≥ 2, in which the measure governing the site variables {ω(v) : v ∈ B n } has local density p and is k-wise independent. Let L n be the event that two given opposite faces are connected by an open path in the box. Thus, for large n, the probability of L n under the product measure P p has a sharp threshold around p = p c (Z d ). The problem is to find bounds on the smallest value of k such that the probability of L is close to its value P p (L n ) under product measure.
This question may be formalised as follows. Let Π = Π(n, k, p) be the set of probability measures on {0, 1} Bn that have density p and are k-wise independent. Let ǫ n (p, k) = max P∈Π P(L n ) − min P∈Π P(L n ), and K n (p) = min{k : ǫ n (p, k) ≤ δ},
where for definiteness we may take δ = 0.01 as in [4]. Thus, roughly speaking, K n (p) is a quantification of the amount of independence required in order that, for all P ∈ Π, P(L n ) differs from P p (L n ) by at most δ.
Benjamini, Gurel-Gurevich and Peled have proved, in an ongoing project, that K n (p) ≤ c log n when d = 2 and p = p c (and when d > 2 and p < p c ), for some constant c = c(p, d). They have in addition a lower bound for K n (p) that depends on p, d, and n, and goes to ∞ as n → ∞.
Percolation of words
Recall the set W = {0, 1} N of words in the alphabet comprising the two letters 0, 1. Consider the site percolation process of Section 2 on a countably infinite connected graph G = (V, E), and write ω = {ω(v) : v ∈ V } for the ensuing configuration. Let v ∈ V and let S v be the set of all self-avoiding walks starting at v. Each π ∈ S v is a path v 0 , v 1 , v 2 . . . with v 0 = v. With the path π we associate the word w(π) = ω(v 1 )ω(v 2 ) · · · , and we write W v = {w(π) : π ∈ S v } for the set of words 'visible from v'. The central question of site percolation concerns the probability that W v ∋ 1 ∞ , where 1 ∞ denotes the infinite word 111 · · · . The so-called AB-percolation problem concerns the existence in W v of the infinite alternating word 01010 · · · , see [2].
More generally, for given p, we ask which words lie in the random set W v . Partial answers to this question may be found in three papers [5,19,20] of Kesten and co-authors Benjamini, Sidoravicius, and Zhang, and they are summarised here as follows.
For Z d , with p = 1 2 and d sufficiently large, we have from [5] where the set of words seen includes all periodic words apart from 0 ∞ and 1 ∞ . The measure on W can be taken in (7.1) as any non-trivial product measure. This extends the observation that AB-percolation takes place at p = 1 2 , whereas there is no infinite cluster in the usual site percolation model.
Finally, for the 'close-packed' lattice Z 2 cp obtained from Z 2 by adding both diagonals to each face, P p (W 0 = W) > 0 for 1 − p c < p < p c , with p c = p c (Z 2 ). Moreover, every word is (a.s.) seen along some self-avoiding path in the lattice. See [20].
$
a family of independent, identically distributed random variables taking the values 0 and 1 with respective probabilities 1 − p and p. A vertex v is called open if ω(v) = 1, and closed otherwise.
Figure 1 .
1The square lattice Z 2 and the triangular lattice T, with their dual lattices.
Question 3 . 1 .
31Is it the case that θ(M) > 0 for sufficiently large M? Perhaps θ(4) > 0?
that there exists an infinite open path with strictly positive probability if and only if M ≥ 4.
Question 5. 1 .
1Is it the case that ρ(M) > 0 for sufficiently large M?
For example, v n (M) = P(a n ⊆ M Y ) satisfies v n+1 (M) = (α + (M − 1)β)v n − β(M − 2α)v n−1 , n ≥ 1, (5.4) with boundary conditions v 0 (M) = 1, v 1 (M) = α. Here, α = 1 − 2 −M , β = 2 −M . The characteristic polynomial associated with (5.4) is a quadratic with one root in each of the disjoint intervals (0, Mβ) and (α, 1). The larger root equals 1 − (1 + o(1))2 1−2M for large M, so that, in rough terms v n (M) ≈ (1 − 2 1−2M ) n .
Figure 2 .
2Icons describing a variety of embedding problems. vertices such that: (u 0 , v 0 ) = (0, 0), (u n+1 , v n+1 ) = (u n + 1, v n + d n ), for some d n satisfying 1 ≤ d n ≤ M. It is easily seen that X ⊆ M Y if and only if there exists a path all of whose vertices are open. (We declare (0, 0) to be open.)With this formulation in mind, the above problem may be represented by the icon at the top left ofFigure 2. The further icons of that figure represent examples of problems of similar type. Nothing seems to be known about these except:
(W 0 = W) > 0, and indeed there exists (a.s.) some vertex v for which W v = W. Partial results are obtained for Z d with edge-orientations in increasing coordinate directions.For the triangular lattice T and p = 1 2 , we have from[19] that P 1 v∈V W v contains almost every word = 1, (7.1)that
P 1
2
2
AcknowledgementsThe author acknowledges conversations with his co-authors Tom Liggett and Thomas Richthammer. He profited from discussions with Alexander Holroyd while at the Department of Mathematics at the University of British Columbia, and with Ron Peled and Vladas Sidoravicius while visiting the Institut Henri Poincaré-Centre Emile Borel, both during 2008. This article was written during a visit to the Section de Mathématiques at the University of Geneva, supported by the Swiss National Science Foundation. The author thanks Ron Peled for his comments on a draft.
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| [] |
[
"Some new Stein operators for product distributions",
"Some new Stein operators for product distributions"
] | [
"Robert E Gaunt \nThe University of Manchester\nINRIA and Université de Liège\n\n",
"Guillaume Mijoule \nThe University of Manchester\nINRIA and Université de Liège\n\n",
"Yvik Swan \nThe University of Manchester\nINRIA and Université de Liège\n\n"
] | [
"The University of Manchester\nINRIA and Université de Liège\n",
"The University of Manchester\nINRIA and Université de Liège\n",
"The University of Manchester\nINRIA and Université de Liège\n"
] | [] | We provide a general result for finding Stein operators for the product of two independent random variables whose Stein operators satisfy a certain assumption, extending a recent result of[18]. This framework applies to non-centered normal and non-centered gamma random variables, as well as a general sub-family of the variance-gamma distributions. Curiously, there is an increase in complexity in the Stein operators for products of independent normals as one moves, for example, from centered to non-centered normals. As applications, we give a simple derivation of the characteristic function of the product of independent normals, and provide insight into why the probability density function of this distribution is much more complicated in the non-centered case than the centered case. | 10.1214/19-bjps460 | [
"https://arxiv.org/pdf/1901.11460v1.pdf"
] | 119,283,944 | 1901.11460 | ad65c74e9b7c24bb0a0d099fe7bc5f3b176b0f82 |
Some new Stein operators for product distributions
31 Jan 2019
Robert E Gaunt
The University of Manchester
INRIA and Université de Liège
Guillaume Mijoule
The University of Manchester
INRIA and Université de Liège
Yvik Swan
The University of Manchester
INRIA and Université de Liège
Some new Stein operators for product distributions
31 Jan 2019Stein's methodStein operatorsproduct distributionsproduct of independent normal random variables AMS 2010 Subject Classification: Primary 60F05Secondary 62E15
We provide a general result for finding Stein operators for the product of two independent random variables whose Stein operators satisfy a certain assumption, extending a recent result of[18]. This framework applies to non-centered normal and non-centered gamma random variables, as well as a general sub-family of the variance-gamma distributions. Curiously, there is an increase in complexity in the Stein operators for products of independent normals as one moves, for example, from centered to non-centered normals. As applications, we give a simple derivation of the characteristic function of the product of independent normals, and provide insight into why the probability density function of this distribution is much more complicated in the non-centered case than the centered case.
Introduction
In 1972, Charles Stein [33] introduced a powerful technique for deriving explicit bounds in normal approximations. Shortly after, in 1975, Louis Chen [7] adapted the method to the Poisson distribution, and since then Stein's method has been extended to a wide variety of distributional approximations. For a given target distribution p, the first step in the general procedure is to find a suitable operator A acting on a class of functions F such that IE[Af (X)] = 0 for all f ∈ F, where the random variable X has distribution p. The operator A is called the Stein operator, and for continuous distributions is typically a differential operator; for the N (µ, σ 2 ) distribution, the classical operator is Af (x) = σ 2 f ′ (x)−(x−µ)f (x). This leads to the Stein equation
Af h (x) = h(x) − IEh(X),(1)
where h is a real-valued function. If A is well chosen then, for a given h, the Stein equation (1) can be solved for f h , and one can see from (1) that the problem of estimating the proximity of the distribution of a random variable W of interest to the distribution of the target random variable X, as measured by the quantity |IEh(W ) − IEh(X)|, reduces to one of bounding the expectation |IE[Af h (W )]|. For a detailed account of the method and its numerous applications to normal and non-normal approximations, we refer the reader to the survey [31] and the monographs [5,8,24,34]. In addition to the normal and Poisson distributions, Stein's method has been adapted to many classical distributions, such as the exponential [6], gamma [22] and Laplace [29], as well as quite general families of distributions, such as the Pearson family [32] and variance-gamma distributions [12] and a wide class of distributions satisfying a certain diffusive assumption [11,20]; for an overview see [21]. As such, over the years, a number of techniques have been developed for finding Stein operators for a variety of distributions. These include the density method [34,21,23], the generator method [4,19], the differential equation duality approach [14,21], and probability generating function and characteristic function based approaches of [35] and [2]; useful surveys are given by [21,30]. The corpus of literature concerning Stein operators and their applications is now vast, and it continues growing at a steady pace. Stein operators provide handles on target distributions which are in some sense just as important and natural characteristics of a probability distribution as its moments, its moment generating function, its p.d.f., c.d.f. or even its characteristic function. Finding tractable Stein operators is thus, naturally, an important question.
In this paper we pursue the work begun in [13,16] concerning the following question : "given two independent random variables X and Y with Stein operators A X and A Y , can one find a Stein operator for Z = XY ". More specifically, the present paper is a complement (sequel) to our paper [18] where we developed an algebraic technique for finding Stein operators for products of independent random variables with polynomial Stein operators satisfying a technical condition. Let M (f ) = (x → xf (x)), D(f ) = (x → f ′ (x)) and I be the identity operator. We say that the absolutely continuous random variables X and Y have polynomial Stein operators if they allow Stein operators of the form A = i,j a ij M i D j for a ij some real numbers. The highest value of j such that a ij = 0 is called the degree of the operator. In [18] we provided a method for deriving operators under the technical assumption that # {j − i | a ij = 0} ≤ 2 (see Assumption 3 and Lemma 2.6 of [18] for more details on this condition). For such random variables, Proposition 2.12 of [18] gives a polynomial Stein operator for the product XY . A number of classical random variables have Stein operators which satisfy this assumption, such as the N (0, σ 2 ) Stein equation with Stein operator σ 2 D − M , with others including the gamma, beta, tdistribution, and even some more exotic distributions such as the zero-mean symmetric variance-gamma distribution and PRR distribution of [28]. However, some very natural densities do not satisfy the assumption. In fact, even the non-centered normal distribution does not satisfy this assumption, as its Stein operator σ 2 D + µI − M instead satisfies # {j − i | a ij = 0} = 3. In Proposition 2.1 of this paper, we shall address the natural problem of extending the result of [18] to treat the product of two independent random variables satisfying this new assumption. Here we have only added one level of complexity in the operator; nevertheless, as we will see later on, it is sufficient to include the classical cases of non-centered normal and non-centered gamma, and a more general sub-family of the variance-gamma distributions. Also, as noted in Remark 2.3, the proof technique is novel and seems to be a useful addition to the Stein toolkit for finding Stein operators.
The Stein operators for the products of independent normal random variables are particularly theoretically interesting, and we devote Section 3 to exploring some of their properties. For the case of two independent centered normals a second order Stein operator was obtained by [13], whereas, rather curiously, we find a third order operator for the product of two i.i.d. normals, and a fourth order operator for the product of two independent general normals; see Table 1. (By n-th order Stein operator, we mean that it is a n-th order differential operator.) It is an important and natural question to ask whether our operators have minimal order amongst all Stein operators with polynomial coefficients. We believe this is the case but are unable to prove it. However, in Section 3.1, we are able to provide a brute force approach for verifying this assertion for polynomial coefficients up to a particular order. This brute force approach is very general and in principle can be applied to any polynomial Stein operators. In Section 3.2, we prove that our Stein operators for products of independent normals characterise the distribution. We do this by appealing to a more general result, Proposition 3.2, which treats distributions that are determined by their moments.
For the Stein operator of [13] for the product of two independent standard normal random variables, it was possible to solve the corresponding Stein equation and bound the derivatives of the solution. As a result, [13] was able to derive explicit bounds for product normal approximations. However, it seems to be beyond the scope of existing techniques in the Stein's method literature to solve and then bound the derivatives of the solution to our more complicated third and fourth order Stein equations for products of non-centered normals. It should be noted, though, that there is still great utility to Stein equations even when it is not possible to obtain bounds for the solution. For example, as has been demonstrated
= σ X σ Y ) N (0, σ 2 X ) × N (0, σ 2 Y ) σ 2 (xf ′′ (x) + xf (x)) − xf (x) N (µ, σ 2 X ) × N (µ, σ 2 Y ) σ 3 xf (3) (x) + σ 2 (σ − x)f ′′ (x) − σ(x + (σ + µ 2 ))f ′ (x) + (x − µ 2 )f (x) N (µ X , σ 2 X ) × N (µ Y , σ 2 Y ) σ 4 X σ 4 Y xf (4) (x) + σ 4 X σ 4 Y f (3) (x) − σ 2 X σ 2 Y (2x + µ X µ Y )f ′′ (x) −(σ 2 X σ 2 Y + µ 2 X σ 2 Y + µ 2 Y σ 2 X )f ′ (x) + (x − µ X µ Y )f (x)
in several papers such as [25,26,1,3,2], Stein operators can be used for comparison of probability distributions directly without solving Stein equations. We also stress that Stein operators are also of use in applications beyond proving approximation theorems; for example, in obtaining distributional properties [13,16,18] and have even found surprising applications in the derivation of new definite integrals of special functions [15]. Indeed, in Section 3.3, we use our Stein operators to obtain a simple derivation of the characteristic function of two independent normals, and also provide valuable insight into why there is a dramatic increase in complexity in the probability density function from the centered to non-centered case.
2 New Stein operators for product distributions
A general result
Throughout this paper, we shall make the following assumptions, which were also made in [18]; we refer the reader to that paper for some remarks on these assumptions. Assumption 1. X admits a smooth density p with respect to the Lebesgue measure on IR; this density is defined and non-vanishing on some (possibly unbounded) interval J ⊆ IR.
2. X admits an operator A acting on F which contains the set of smooth functions with compact support C ∞ 0 (IR).
Let P be a real polynomial. Then it is easily proved (by checking it when P is a monomial, then by linearity) that
P (M D)M = M P (M D + I), DP (M D) = P (M D + I)D (2) (recall the notations M (f ) = (x → xf (x)), D(f ) = (x → f ′ (x)) and I the identity operator from the introduction). Now, for a ∈ IR \ {0}, let τ a (f ) = (x → f (ax)). Simple computations show that (see [18, Lemma 2.5]) τ a M = aM τ a , and Dτ a = aτ a D.
This implies that for any real polynomial P ,
τ a P (M D) = P (M D)τ a .
Proposition 2.1. Let X and Y be i.i.d. with common Stein operator of the form
A = M − Q(M D) − P (M D)D
for P, Q two real polynomials. Then, a Stein operator for Z = XY is
A Z = R 1 (M D)D 2 + R 2 (M D)D + R 3 (M D) + M R 4 (M D),(3)
where R 1 (U ) = (P (U )) 2 P (U + 1)(U + 1)Q(U + 2),
R 2 (U ) = −(P (U )) 2 Q(U )(U + 1) − Q(U + 1)P (U )Q 2 (U ), R 3 (U ) = −U Q(U )P (U − 1) − Q(U − 1)(Q(U )) 2 , R 4 (U ) = Q(U − 1). Proof. Let Z = XY and f ∈ F. Denote U = M D. We have IE[Xf (Z)] = IE[M τ Y f (X)] = IE[Q(U )τ Y f (X) + P (U )Dτ Y f (X)] = IE[τ Y Q(U )f (X) + Y τ Y P (U )Df (X)] IE[Xf (Z)] = IE[Q(U )f (Z) + Y P (U )Df (Z)].(4)
Similarly,
IE[Y f (Z)] = IE[Q(U )f (Z) + XP (U )Df (Z)].(5)
Apply (5) to P (U )Df and add up to (4) to get
IE[X(I − P (U )DP (U )D)f (Z)] = IE[(Q(U ) + Q(U )P (U )D)f (Z)],
which is also, using (2),
IE[X(I − P (U + I)P (U )D 2 )f (Z)] = IE[(Q(U ) + Q(U )P (U )D)f (Z)].(6)
Now using (5) and conditioning, we can compute
IE[Zf (Z)] = IE[XIE[Y f (Z) | X]] = IE XQ(U )f (Z) + X 2 P (U )Df (Z) .(7)
We also have
IE[X 2 f (Z)] = IE[M 2 τ Y f (X)] = IE[Q(U )M τ Y f (X) + P (U )DM τ Y f (X)] = IE[M τ Y Q(U + I)f (X) + τ Y P (U )(U + I)f (X)] IE[X 2 f (Z)] = IE[XQ(U + I)f (Z) + P (U )(U + I)f (Z)].
Thus we obtain by (7) IE
[(M − P (U )P (U )(U + I)D)f (Z)] = IE[X (Q(U ) + Q(U + I)P (U )D) f (Z)].(8)
Apply (8) to P (U )Df and add up to (6) applied to Q(U − I)f to obtain
IE[X(Q(U )P (U )D + Q(U − I))f (Z)] = IE[(U P (U − I) − (P (U )) 2 P (U + I)(U + I)D 2 + (Q(U ) + Q(U )P (U )D)Q(U − I))f (Z)].
Apply the preceding equation to Q(U )f and subtract to (8) applied to Q(U − I)f to get the result.
The case that P and Q are polynomials of degree one is important, as it is applicable to non-centered normal and non-centered gamma random variables, as well as a general sub-family of the variance-gamma distributions. To this end, let us define the operator T r := M D + rI. We note that the limit of T r as r → ∞ is ill-defined, but we do have lim r→∞ r −1 T r = I (see [18,Remark 2.3]). Corollary 2.2. Let α, β ∈ IR and a, b ∈ IR ∪ {∞} (if either a or b are set to +∞, then we proceed as described above). Let X, Y be i.i.d. with common Stein operator
A = M − αT a − βT b D.
Then, a Stein operator for Z = XY is
A Z = (M − α 2 T 2 a − β 2 T 2 b T 1 D)(T a−1 − βT b T a+1 D) − 2α 2 βT 2 a T b T a+1 D.(9)
Proof. Set Q(U ) = α(U + a) and P (U ) = β(U + b) in (3). A calculation then verifies that (9) and (3) are equivalent operators in this case (up to a factor −α).
= M − Q X (M D) − P X (M D)D and A Y = M − Q Y (M D) − P Y (M D)D.
We were only able to find a Stein operator for the product XY under the very restrictive condition that
P Y (U )Q X (U )Q X (U + 1) = P X (U )Q Y (U )Q Y (U + 1
Examples
Product of non-centered normals
Assume X and Y are independent standard normal random variables. A Stein operator for X + µ (or Y + µ) is A = D − M + µI. Applying Corollary 2.2 with α = µ, β = 1 and a = b = ∞ gives the following Stein operator for Z = (X + µ)(Y + µ):
A Z = (M − µ 2 I − T 1 D)(I − D) − 2µ 2 D = M D 3 + (I − M )D 2 − (M + (1 + µ 2 )I)D + M − µ 2 I.(10)
(Here, and for the rest of this paper, for ease of exposition we only consider the unit variance case; the extension to general non-zero, finite variances follows from a straightforward rescaling and the resulting Stein operator for the product is given in Table 1.) Note that when µ = 0, the above operator becomes
A Z f (x) = M (D 3 − D 2 − D + I)f (x) + (D 2 − D)f (x) = x(f (3) (x) − f ′′ (x)) + (f ′′ (x) − f ′ (x)) + x(f ′ (x) − f (x)).
Taking
g(x) = f ′ (x) − f (x) then yields A Z f (x) =Ã Z g(x) = xg ′′ (x) + g ′ (x) − xg(x),(11)
which we recognise as the product normal Stein operator that was obtained by [13].
Product of non-centered gammas
Assume X and Y are distributed as a Γ(r, 1), with p.d.f. p(x) = 1 Γ(r) x r−1 e −x , x > 0, and let µ ∈ IR. A Stein operator for X + µ (or Y + µ) is A = T r+µ − µD − M. Corollary 2.2 applied with α = 1, β = −µ, a = r + µ, b = ∞ yields the following fourth-order Stein operator for Z = (X + µ)(Y + µ):
A Z = (M − T 2 r+µ − µ 2 T 1 D)(T r+µ−1 + µT r+µ+1 D) + 2µT 2 r+µ T r+µ+1 D.
Note also that when µ = 0, this operator reduces to (M − T 2 r )T r−1 , which is the product gamma Stein operator of [16] applied to T r−1 f instead of f .
Product of variance-gamma random variables
The variance-gamma distribution with parameters r > 0, θ ∈ IR, σ > 0, µ ∈ IR has p.d.f.
f (x) = 1 σ √ πΓ( r 2 ) e θ σ 2 (x−µ) |x − µ| 2 √ θ 2 + σ 2 r−1 2 K r−1 2 √ θ 2 + σ 2 σ 2 |x − µ| , x ∈ IR,(12)
where the modified Bessel function of the second kind is given by K ν (x) = ∞ 0 e −x cosh(t) cosh(νt) dt, x > 0. If a random variable W has density (12) then we write W ∼ VG(r, θ, σ, µ). A VG(r, θ, σ, 0) Stein operator is given by σ 2 T r D + 2θT r/2 − M (see [12]). Applying Corollary 2.2 with α = 2θ, β = σ 2 , a = r/2, b = r, we get the following Stein operator for the product of two independent VG(r, θ, σ, 0) random variables:
A = (M − 4θ 2 T 2 r/2 − σ 4 T 2 r T 1 D)(T r/2−1 − σ 4 T r T r/2+1 D) − 8θ 2 σ 2 T 2 r/2 T r T r/2+1 D. Note that when θ = 0 we have Af (x) = (M − σ 4 T 2 r T 1 D)(T r/2−1 − σ 4 T r T r/2+1 D)f (x).
Defining g : IR → IR by xg(x) = −(T r/2−1 − σ 4 T r T r/2+1 D)f (x) gives
Ag(x) = (σ 4 T 2 r T 1 D − M )M g(x) = σ 4 T 2 r T 2 1 g(x) − M 2 g(x),
which is in agreement with the product variance-gamma Stein operator given in Section 3.2 of [18]. Lastly, we note that the VG(r, θ, σ, µ) Stein operator of [12], as given by
σ 2 (M − µ)D 2 + (rσ 2 + 2θ(M − µ))D + (rθ − (M − µ))I,
satisfies # {j − i | a ij = 0} = 4 when µ = 0, and therefore one cannot apply Proposition 2.1 or Corollary 2.2 to find a Stein operator for the product of two such random variables.
Product of non-identically distributed non-central normals
By working on a case-by-case basis it is possible to use the proof technique of Proposition 2.1 to find Stein operators for the product of two non-identically distributed random variables, whose Stein operators satisfy the assumptions of the proposition. We find that a Stein operator for the product of independent normals N (µ X , 1) and N (µ Y , 1) is given by
M D 4 + D 3 − (2M + µ X µ Y I)D 2 − (1 + µ 2 X + µ 2 Y )D + M − µ X µ Y I.(13)
Let us now provide a derivation of this Stein operator. Let X and Y be independent standard normal random variables and define Z = (X + µ X )(Y + µ Y ). We will use repeatedly the fact that IE[W g(W )] = IE[g ′ (W )] for W ∼ N (0, 1), as well as conditioning arguments, and we let IE W [] stand for the expectation conditioned on W . Let f : IR → ∞ be four times differentiable and such that IE|Zf (i) (Z)| < ∞ for i = 0, 1, . . . 4 and IE|f (i) (Z)| < ∞ for i = 0, 1, 2, 3, where f (0) ≡ f . Then
IE[Zf (Z)] = IE[(X + µ X )(Y + µ Y )f ((X + µ X )(Y + µ Y ))] = IE[(Y + µ Y )IE Y [Xf ((X + µ X )(Y + µ Y )]] + µ X IE[(Y + µ Y )f (Z)] = IE[(Y + µ Y ) 2 f ′ (Z)] + µ X IE[(Y + µ Y )f (Z)] = IE[Y (Y + µ Y )f ′ (Z)] + µ X IE[(Y + µ Y )f (Z)] + µ Y IE(Y + µ Y )f ′ (Z)] = IE[f ′ (Z)] + IE[(X + µ X )(Y + µ Y )f ′′ (Z)] + µ X IE[(Y + µ Y )f (Z)] + µ Y IE[(Y + µ Y )f ′ (Z)] = (1 + µ 2 Y )IE[f ′ (Z)] + IE[Zf ′′ (Z)] + µ X µ Y IE[f (Z)] + µ X IE[Y f (Z)] + µ Y IE[Y f ′ (Z)].(14)
By again applying a conditioning argument we obtain
IE[Y f (Z)] = IE[(X + µ X )f ′ (Z)] = µ X IE[f ′ (Z)] + IE[Xf ′ (Z)] = µ X IE[f ′ (Z)] + IE[(Y + µ Y )f ′′ (Z)]
(and the same applies to IE[Y f ′ (Z)]). Hence
IE[Zf (Z)] = (1 + µ 2 Y )IE[f ′ (Z)] + IE[Zf ′′ (Z)] + µ X µ Y IE[f (Z)] + µ 2 X IE[f ′ (Z)] + µ X µ Y IE[f ′′ (Z)] + µ X IE[Y f ′′ (Z)] + µ Y µ X IE[f ′′ (Z)] + µ 2 Y IE[f (3) (Z)] + µ Y IE[Y f (3) (Z)] = (1 + µ 2 X + µ 2 Y )IE[f ′ (Z)] + IE[Zf ′′ (Z)] + µ X µ Y IE[f (Z)] + 2µ X µ Y IE[f ′′ (Z)] + µ 2 Y IE[f (3) (Z)] + µ X IE[Y f ′′ (Z)] + µ Y IE[Y f (3) (Z)].(15)
Isolating the expressions depending on Y from equations (14) and (15), we have that
µ X IE[Y f ′′ (Z)] + µ Y IE[Y f (3) (Z)] = IE[(Z − µ X µ Y )f (Z) − (1 + µ 2 X + µ 2 Y )f ′ (Z) − (Z + 2µ X µ Y )f ′′ (Z) − µ 2 Y f (3) (Z)](16)
and
µ X IE[Y f ′′ (Z)] + µ Y IE[Y f (3) (Z)] = IE[(Z − µ X µ Y )f ′′ (Z) − (1 + µ 2 Y )f (3) (Z) − Zf (4) (Z)].(17)
Substract (17) to (16) to get
IE[Zf (4) (Z) + f (3) (Z) − (2Z + µ X µ Y )f ′′ (Z) − (1 + µ 2 X + µ 2 Y )f ′ (Z) + (Z − µ X µ Y )f (Z)] = 0,
from which we deduce that (13) is a Stein operator for Z. ✷ Lastly, we note that applying the third order operator (10)
to f (x) = g ′ (x) + g(x) yields xg (4) (x) + g (3) (x) − (2x + µ 2 )g ′′ (x) − (1 + 2µ 2 )g ′ (x) + (x − µ 2 )g(x),
which we recognise as the Stein operator (13) in the special case µ X = µ Y = µ.
Sums of products of normals
Let us begin by noting a simple result, that has perhaps surprisingly not previously been stated explicitly in the literature. Suppose X, X 1 , . . . , X n are i.i.d., with Stein operator
A X f (x) = m k=0 (a k x+b k )f (k) (x),
where m ≥ 1 and the a k and b k are real-valued constants. Let W = n j=1 X j . Then, by conditioning,
IE[(a 0 W + nb 0 )f (W )] = n j=1 IE IE (a 0 X j + b 0 )f (W ) X 1 , . . . , X j−1 , X j+1 , . . . , X n = − n j=1 IE IE m k=1 (a k X j + b k )f (k) (W ) X 1 , . . . , X j−1 , X j+1 , . . . , X n = −IE m k=1 (a k W + nb k )f (k) (W ) .
Thus, a Stein operator for W is given by
A W f (x) = m k=0 (a k x + nb k )f (k) (x).(18)
This result can be used, for example, to obtain the χ 2 (d) Stein operator T d/2 − 1 2 M from the χ 2 (1) Stein operator T 1/2 − 1 2 M , since all coefficients in this Stein operator are linear. Remark 2.6. Identity (18) actually generalises similar observations for score functions and Stein kernels, for which such an additive stability is well-known, see [26].
Since the coefficients in the Stein operators (10) and (13) are linear, we can use (18) to write down a Stein operator for the sum
W = r i=1 X i Y i , where (X i ) 1≤i≤r ∼ N (µ X , 1) and (Y i ) 1≤i≤r ∼ N (µ Y , 1) are independent. When µ X = µ Y = µ, we have A W = M D 3 + (rI − M )D 2 − (M + r(1 + µ 2 )I)D + M − rµ 2 I,(19)
and when µ X and µ Y are not necessarily equal, we have
A W = M D 4 + D 3 − (2M + rµ X µ Y I)D 2 − r(1 + µ 2 X + µ 2 Y )D + M − rµ X µ Y I.(20)
When µ X = µ Y = 0, the random variable W follows the VG(r, 0, 1, 0) distribution (see [12], Proposition 1.3). Taking g = f ′ − f in (19) (as we did in arriving at (11)), we obtain
A W g(x) = xg ′′ (x) + rg ′ (x) − xg(x),
which we recognise as the VG(r, 0, 1, 0) Stein operator that was obtained in [12].
3 Some results concerning the Stein operators for products of independent normal random variables
On the minimality of the operators
The product operator (9) is at most a seventh order differential operator. However, for particular cases, such as the product of two i.i.d. non-centered normals, the operator reduces to one of lower order, see Section 2.2.1. Whilst we believe that the third order operator (10) is a minimal order polynomial operator, we have no proof of this claim (nor do we have much intuition as to whether the seventh order operator (9) is of minimal order). We believe this question of minimality to be of importance and state it as a conjecture.
Conjecture 3.1. There exists no second order Stein operator (acting on smooth functions with compact support) with polynomial coefficients for the product of two independent non-centered normal random variables.
We have not been able to devise a proof strategy to establish Conjecture 3.1. However, one can use a brute force approach to verify the conjecture for polynomials of fixed order (if the conjecture is true). Such results would be worthwhile in practice, because a third order Stein operator with linear coefficients may be easier to work with in applications than second order Stein operators with polynomial coefficients of degree greater than one.
Let us now demonstrate how the brute force approach can be used to prove that there is no second order Stein operator with linear coefficients for the product of two independent non-centered normals (generalisations are obvious). Let X and Y be independent N (1, 1) random variables and let Z = XY . Suppose that there was such a Stein operator for Z, then it would be of the form
A Z f (x) = 2 j=0 (a 0,j + a 1,j x)f (j) (x), where f (0) ≡ f . Now, if A Z was a Stein operator for Z, we would have IE[A Z f (Z)]
= 0 for all f in some class F that contains the monomials {x k : k ≥ 1}. Taking f (x) = x k , k = 0, 1, . . . , 5, we obtain six equations for six unknowns. Letting µ k denote IEZ k , we have µ 1 = 1, µ 2 = 4, µ 3 = 16, µ 4 = 100, µ 5 = 676 and µ 6 = 5776. This leads to the following system of equations: a 1,0 + a 0,0 = 0 a 1,1 + a 0,1 + 4a 1,0 + a 0,0 = 0 2a 1,2 + 2a 0,2 + 8a 1,1 + 2a 0,1 + 16a 1,0 + 4a 0,0 = 0 24a 1,2 + 6a 0,2 + 48a 1,1 + 12a 0,1 + 100a 1,0 + 16a 0,0 = 0 192a 1,2 + 48a 0,2 + 400a 1,1 + 48a 0,1 + 676a 1,0 + 100a 0,0 = 0 2000a 1,2 + 320a 0,2 + 3380a 1,1 + 500a 0,1 + 5776a 1,0 + 676a 0,0 = 0.
We used Mathematica to compute that the determinant of the matrix corresponding to this system of equations is 783 360 = 0. Therefore, there is a unique solution, which is clearly a 1,2 = · · · = a 0,0 = 0. Thus, there does not exist a second order Stein operator with linear coefficients for Z.
Similarly, one can show that there is no third order Stein operator with linear coefficients for the product of two independent normals with different means. Here we took X ∼ N (1, 1) and Y ∼ N (2, 1), and sought a Stein operator of the form A Z f (x) = 3 j=0 (a 0,j + a 1,j x)f (j) (x). We then used the monomials f (x) = x k , k = 0, 1, . . . , 7, to generate eight linear equations in eight unknowns, and found the determinate of the matrix corresponding to this system of equations to be 10 157 222 707 200 = 0.
Characterisation by the operators
We begin with a simple general result, which perhaps surprisingly has not previously been given in the literature. The proof technique has, however, appeared in the literature; see the proof of Lemma 5.2 of [31] for case of the exponential distribution.
Proposition 3.2. Suppose that the law of the random variable X, supported on I ⊂ IR, is determined by its moments. Let the operator A X = n i=1 p j=1 a i,j M j D i , where a i,j ∈ IR, act on a class of functions F which contains all polynomial functions. Suppose A X is a Stein operator for X: that is, for all f ∈ F,
IE[A X f (X)] = 0. (21) Now, let m = max i,j (j − i) − min i,j (j − i) − 1,
where the maxima and minima are taken over all i, j such that a i,j = 0. Suppose that the first m moments of Y are equal to those of X and that
IE[A X f (Y )] = 0(22)
for all f ∈ F. Then Y has the same law as X.
Proof. We prove that all moments of Y are equal to those of X. As the moments of X determine its law, verifying this proves the Proposition. The monomials {x k : k ≥ 1} are contained in the class F, so applying f (x) = x k , k ≥ m, to (22) yields the recurrence relation
i,j a i,j C k IEY k+j−i = 0, k ≥ m,(23)
where C k = k(k − 1) · · · (k − i + 1) if k − i + 1 > 0 and C k = 0 otherwise. We have that IEY 0 = 1 and we are given that IEY k = IEX k for k = 1, . . . , m. We can then use forward substitution in (23) to (uniquely) obtain all moments of Y . Due to (21), IE[A X f (X)] = 0 for all f ∈ F, and so it follows by the above reasoning that
i,j a i,j C k IEX k+j−i = 0, k ≥ m.
But this is same recurrence relation as (23) and, since IEY k = IEX k for k = 1, . . . , m, it follows that IEY k = IEX k for all k ≥ m as well.
If we have obtained a Stein operator A X for a random variable X, then Proposition 3.2 tells us that the operator characterises the law of X if X is determined by its moments. This characterisation is weaker than those typically found in Stein's method literature, as it involves moment conditions on the random variable Y . This is perhaps not surprising, because the characterisations given in the literature have mostly been found on a case-by-case basis, whereas ours applies to a wide class of distributions.
The distribution of the product of two independent normal distributions is determined by its moments, which can be seen from the existence of its moment generating function M (t) for all |t| < 1; see Section 3.3.1. The following full characterisation of the distribution is thus immediate from Proposition 3.2. Proposition 3.3. (i) Let W be a real-valued random variable whose first three moments are equal to that of the random variable Z = XY , where X ∼ N (µ X , 1) and Y ∼ N (µ Y , 1) are independent. Then W is equal in law to Z if and only if
IE W f (4) (W ) + f (3) (W ) − (2W + µ X µ Y )f ′′ (W ) − (1 + µ 2 X + µ 2 Y )f ′ (W ) + (W − µ X µ Y )f (W ) = 0 (24) for all f ∈ C 4 (IR) such that IE|Zf (j) (Z)| < ∞ for 0 ≤ j ≤ 4, and IE|f (k) (Z)| < ∞ for 0 ≤ k ≤ 3, where f (0) ≡ f .
(ii) Now suppose that µ X = µ Y = µ, and that the first two moments of W are equal to those of Z. Then W is equal in law to Z if and only if
IE W f (3) (W ) + (1 − W )f ′′ (W ) − (W + 1 + µ 2 )f ′ (W ) + (W − µ 2 )f (W ) = 0 for all f ∈ C 3 (IR) such that IE|Zf (j) (Z)| < ∞ for 0 ≤ j ≤ 3, and IE|f (k) (Z)| < ∞ for 0 ≤ k ≤ 2.
Proposition 3.2 can be used to prove that some other Stein operators given in the literature fully characterise the distribution. For example, the Stein operator for the product of n independent Beta random variables obtained by [16] is characterising, since this product is supported on (0, 1) and thus the distribution is determined by its moments.
Applications of the operators
Characteristic function
As the Stein operator (13) has linear coefficients, it turns out to be straightforward to use the characterising equation (24) to find a formula for the characteristic function of the random variable Z = XY , where X ∼ N (µ X , 1) and Y ∼ N (µ Y , 1) are independent.
On taking f (x) = e itx in the characterising equation (24) and setting φ(t) = IE[e itZ ], we deduce that φ(t) satisfies the differential equation
(t 4 + 2t 2 + 1)φ ′ (t) + (−it 3 + µ X µ Y t 2 − (1 + µ 2 X + µ 2 Y )it − µ X µ Y )φ(t) = 0.(25)
(Note that it is the fact that the coefficients of (13) are linear that ensures this is a first order differential equation.) Solving (25) subject to the condition that φ(0) = 1 then gives that
φ(t) = 1 √ 1 + t 2 exp −t(µ 2 X t + µ 2 Y t − 2iµ X µ Y ) 2(1 + t 2 )
.
Similarly, one can write down the following formula for the moment generating function M (t) = IE[e tZ ], which exists for |t| < 1:
M (t) = 1 √ 1 − t 2 exp t(µ 2 X t + µ 2 Y t + 2µ X µ Y ) 2(1 − t 2 )
.
We doubt that these formulas are new, but it is interesting to note that we were able to obtain such a simple proof via the Stein characterisation of the distribution.
Probability density function
Let X ∼ N (µ X , 1) and Y ∼ N (µ Y , 1) be independent, and denote their product by Z = XY . For the case µ X = µ Y = 0, it is a well-known and easy to prove result that the p.d.f. is given by p Z (x) = 1 π K 0 (|x|), x ∈ IR. However, in general, the p.d.f. takes a much more complicated form (see [9])):
p Z (x) = 1 π e −(µ 2 X +µ 2 Y )/2 ∞ n=0 2n m=0 x 2n−m |x| m−n (2n)! 2n m µ m X µ 2n−m Y K m−n (|x|), x ∈ IR.(26)
It is possible to use the Stein operators for the product Z to gain insight into why there is such a dramatic increase in complexity from the zero mean case to non-zero mean case. To see this, we recall a duality result given in Remark 2.7 of [18] (see also Section 4 of that paper for further details). If V admits a smooth density p, which solves the differential equation Bp = 0 with B = i,j b ij M j D i , then a Stein operator for V is given by A = i,j (−1) i b ij D i M j , and similarly given a Stein operator for V one can write down a differential equation satisfied by p. In this manner, we can write down differential equations satisfied by the density p W of the random variable W = r i=1 X i Y i , where the X i and Y i are independent copies of X and Y respectively, using the Stein operators (19) and (20) for this distribution. When µ X = µ Y = µ, we have
xp (3) W (x) + (x + 3 − r)p ′′ W (x) − (x + r(1 + µ 2 ) − 2)p ′ W (x) − (x + 1 − rµ 2 )p W (x) = 0,(27)
and in general xp (4) W (x) + 3p
(3) W (x) − (2x + rµ X µ Y )p ′′ W (x) + (r(1 + µ 2 X + µ 2 Y ) − 4)p ′ W (x) + (x − rµ X µ Y )p W (x) = 0,(28)
and of course setting r = 1 yields differential equations for the p.d.f. of Z. In the special case µ X = µ Y = 0, the density of Z satisfies the modified Bessel differential equation xp ′′ Z (x) + p ′ Z (x) − xp Z (x) = 0. From Section 3.1 and the duality result of [18], we know that there do not exist differential equations for p Z with linear coefficients that have a lower degree than (27) and (28). Moreover, we were unable to transform (27) or (28) into a well-understood class of third or fourth order differential equations, such as the Meijer G-function differential equation (see Section 16.21 of [27]). Therefore, the increase in complexity in the density p Z of Z from the zero mean case to non-zero mean case can be understood from the increase in complexity of the differential equation satisfied by p Z . Also, due to the above reasoning, it seems plausible that the formula (26) cannot be simplified further.
Finally, we note that there is not a severe increase in complexity in the differential equations satisfied by W from the r = 1 case to the general case. To the best of our knowledge, a formula for general r ≥ 1 has not been obtained in the literature, and even if the differential equations (27) and (28) are not ultimately used to derive such a formula, they do indicate that the formula should be at a similar level of complexity to that of (26), and thus provide motivation for obtaining such a formula. We note that such a result would be of interest due to the occurrence of such random variables in, for example, electrical engineering applications, see [36].
Table 1 :
1Stein operators for products of normal random variables.Product P
Stein operator A P f (x) (here we set σ :
Remark 2.3. The proof of Proposition 2.1 involves applying certain equations to test functions of the form Lf , where L is a linear differential operator. This allowed us to cancel certain terms to obtain the Stein operator (3). We consider this technique to be a useful addition to the Stein toolkit for finding Stein operators. Indeed, this approach was recently used by[17] to find Stein operators for the random variables H 3 (Z) and H 4 (Z), where H n is the n-th Hermite polynomial and Z ∼ N (0, 1). In Section 2.2.4, we also use the technique to derive a Stein operator for the product of independent non-centered normals with different means. This more specific setting provides a useful illustration of the technique.Remark 2.4. We attempted to generalise Proposition 2.1 so that X and Y are no longer identically distributed, for which X and Y have Stein operators of the form A X
). This Stein operator also had the unusual feature of not being symmetric in X and Y . In certain simple cases, we can, however, apply the argument used in the proof of Proposition 2.1 to derive a Stein operator for the product of two non-identically distributed random variables; see Section 2.2.4 for an example.Remark 2.5. Note that, whilst the Stein operator for X and Y in Proposition 2.1 satisfies the condition # {j − i | a ij = 0} = 3, the Stein operator (3) for their product satisfies # {j − i | a ij = 0} = 4. Thus, it is not possible to iterate Proposition 2.1 to find a Stein operator for product of three i.i.d. random variables. This is in contrast to the work of[18] which was carried out under the assumption # {j − i | a ij = 0} = 2.
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A bound for the error in the normal approximation to the the distribution of a sum of dependent random variables. C Stein, Proc. Sixth Berkeley Symp. Math. Statis. Prob. (1972). Sixth Berkeley Symp. Math. Statis. Prob. (1972)BerkeleyUniv. California Press2Stein, C. A bound for the error in the normal approximation to the the distribution of a sum of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Statis. Prob. (1972), vol. 2, Univ. California Press, Berkeley, pp. 583-602.
Approximate Computation of Expectations. C Stein, IMS. Stein, C. Approximate Computation of Expectations. IMS, Hayward, California, 1986.
On Stein operators for discrete approximations. N S Upadhye, V Čekanavičius, P Vellaisamy, Bernoulli. 23Upadhye, N. S.,Čekanavičius, V. and Vellaisamy, P. On Stein operators for discrete approximations. Bernoulli 23 (2017), pp. 2828-2859.
Approximating the distribution for sums of products of normal variables. Working paper. R Ware, F Lad, New ZealandDepartment of Mathematics and Statistics, University of CanterburyWare, R. and Lad, F. Approximating the distribution for sums of products of normal variables. Working paper, Department of Mathematics and Statistics, University of Canterbury, New Zealand, 2013.
| [] |
[
"Permutation Orbifolds and Chaos",
"Permutation Orbifolds and Chaos"
] | [
"Alexandre Belin [email protected] \nInstitute for Theoretical Physics\nUniversity of Amsterdam\nScience Park 9041098 XHAmsterdamThe Netherlands\n"
] | [
"Institute for Theoretical Physics\nUniversity of Amsterdam\nScience Park 9041098 XHAmsterdamThe Netherlands"
] | [] | We study out-of-time-ordered correlation functions in permutation orbifolds at large central charge. We show that they do not decay at late times for arbitrary choices of low-dimension operators, indicating that permutation orbifolds are non-chaotic theories. This is in agreement with the fact they are free discrete gauge theories and should be integrable rather than chaotic. We comment on the early-time behaviour of the correlators as well as the deformation to strong coupling. | 10.1007/jhep11(2017)131 | [
"https://arxiv.org/pdf/1705.08451v2.pdf"
] | 119,098,679 | 1705.08451 | 3acab68ddc57e18fa3e331938e8987655abfbe4b |
Permutation Orbifolds and Chaos
28 Nov 2017 November 29, 2017
Alexandre Belin [email protected]
Institute for Theoretical Physics
University of Amsterdam
Science Park 9041098 XHAmsterdamThe Netherlands
Permutation Orbifolds and Chaos
28 Nov 2017 November 29, 2017
We study out-of-time-ordered correlation functions in permutation orbifolds at large central charge. We show that they do not decay at late times for arbitrary choices of low-dimension operators, indicating that permutation orbifolds are non-chaotic theories. This is in agreement with the fact they are free discrete gauge theories and should be integrable rather than chaotic. We comment on the early-time behaviour of the correlators as well as the deformation to strong coupling.
Introduction and Summary
Quantum chaos has recently received a surge of interest in the context of holography, which has led to new insights on thermal physics of quantum gravity and conformal field theories. This program was initiated by a holographic realization of the butterfly effect [1], where it was shown that a small boundary perturbation can have drastic consequences in the bulk provided it happens at sufficiently early times. This effect along with several generalization were then studied in various quantum systems relevant for gravitational physics [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].
In classical physics, the Poisson bracket i {q(t), p(0)} measures the sensitivity of q(t) to the initial conditions and hence diagnoses classical chaos. In analogy, one can consider the commutator
W (t), V (0) ,(1)
which measures the perturbation by V on measurements of W . If the time separation is small, this commutator will be small. However, it can grow quickly if the quantum system is chaotic. In this paper, we will consider a quantity closely connected to this commutator [9], the out-of-time-ordered (OTO) correlation function
F (t) = V (0)W (t)V (0)W (t) β V V β W W β .(2)
The behaviour of this Lorentzian correlation function was argued to be a sharp diagnostic of quantum chaos [9]. In particular, it should decay at late times in chaotic conformal field theories for arbitrary choice of "simple" operators V and W . Simple means that V and W should be a product of an O(1) number of degrees of freedom. For large N theories, one can distinguish two parametrically distinct time-scales. First, there is the dissipation time t d ∼ β which is the characteristic time scale of the exponential decay of two point functions V (0)V (t) β in the thermal state. This will also be the time scale at which typical time-ordered correlation functions reach there late time limits, for example
V (0)V (0)W (t)W (t) ∼ V V W W , for t > t d .(3)
For holographic theories, this time scale will be connected to the behaviour of the quasinormal modes of black holes. Next, there is the time-scale at which the out-oftime-ordered correlation function becomes small in chaotic theories, which is named the scrambling time t s . For large N theories, the scrambling time is parametrically larger than the dissipation time
t s ∼ log N ≫ t d .(4)
A way to characterize the strength of chaos is by looking at the behaviour of F (t) for times between t d and t s . Chaotic theories are expected to exhibit an exponential growth in this regime which takes the form
V (0)W (t)V (0)W (t) β ∼ 1 − 1 N e λ L t ,(5)
where λ L is called the Lyapunov exponent. The first term in the r.h.s. of (5) corresponds to the disconnected part of the four-point function, whereas the second term is the connected contribution. We see that the connected contribution starts out to be small but becomes of the same order as the disconnected part for t ∼ t s . The 1/N hierarchy between connected and disconnected contributions is a result of large N factorization and always remains valid for Euclidean correlators. Chaos can also be viewed as a breakdown of large N factorization for large Lorentzian times. In [9], a bound on the Lyapunov exponent was found to be
λ L ≤ 2π β ,(6)
which is saturated by black holes in Einstein gravity, showing support for the claim that black holes are the fastest scramblers in nature [22,23]. This bound on chaos can also be used to carve out from the space of all CFTs, those that have nice enough properties to have potential Einstein gravity duals [19,24]. We will be interested in the case of two-dimensional CFTs where it was shown that identity block domination of the correlation function (2) yields maximal chaos [6].
In this paper, we will study out-of-time-ordered correlation function in a large class of theories: permutation orbifolds. Starting from a two-dimensional CFT C with central charge c, we can define
C N = C ⊗N G N ,(7)
for G N ⊆ S N , giving an orbifold CFT with central charge Nc. Provided that the group G N is oligomorphic, this will provide a vast landscape of two-dimensional CFTs that have a good large N limit [25][26][27]. Permutation orbifolds should be viewed as free discrete gauge theories and hence provide examples of weakly-coupled CFTs. In this paper, we will show that these theories are not chaotic, in agreement with the fact that they are free gauge theories. We will show this explicitly by considering the OTO four-point function of arbitrary low-dimension operators. Although one could have suspected this outcome on the grounds that orbifold theories are free gauge theories, there are other perspectives from which the result can appear more surprising. First, the result is completely independent of the choice of seed theory C. This means we can pick a seed theory that is chaotic with a spectrum of operators that is, at least in principle, as crazy as we want. We can certainly make most of the operator dimensions irrational. One might suspect that this leaves an important imprint in the orbifold theory. However, our results indicate that it does not. Because the N copies are non-interacting, the details of the seed theory's spectrum are completely washed out in the large N limit. The details of the seed theory will only be important at early times t ∼ O(1) but do not matter at times of order the scrambling time t ∼ log N.
The other reason why one could have suspected the outcome to be different is that permutation orbifolds have been capable of reproducing many universal features of Einstein gravity in the semi-classical limit. This is often tied to the dominance of the identity block in CFT correlation functions. At least for the symmetric group, the orbifolds theories give the same partition function as Einstein gravity [28,29], which also means they correctly reproduce the BTZ black hole entropy. Furthermore, one can check that they satisfy all conditions demanded in [30], which means that finite temperature two-point functions match those calculated in the BTZ background. Finally, other observables involving late-time dynamics such as two point functions in excited states also show qualitative similarities to a theory dual to Einstein gravity [31] (even though there are quantitative differences in terms of time-scales). In some sense, these theories are doing a way better job at mimicking general relativity than what they should be doing. This is probably related to the fact that gravity in three dimensions is very special, while the story is more complicated in higher dimensions [32]. Nevertheless, these facts suggest that several observables of permutation orbifolds might be well approximated by the identity block, and pinning down which ones are and which ones aren't appears to be quite important. In this paper, we will show that the OTO correlator does not fall in this class as it behaves drastically differently from how a theory dual to Einstein gravity would behave.
In some cases (like for example when G N = S N and C is the non-linear sigma model on T 4 ), one can deform the orbifold theory to go to strong coupling. In that case, one expects maximal chaos which means that the Lyapunov exponent should increase as we increase the coupling. We do not perform this calculation but make some general comments on the deformation to strong coupling.
Summary of Results
We will show that the OTO four-point function in permutation orbifolds takes the general form
F (t) = 1 + g 2 ( K, ϕ)f 2 (t) + 1 N g 3 ( K, ϕ)f 3 (t) + g 4 ( K, ϕ)f 4 (t) + ... (8)
where the g i ( K, ϕ) are order one combinatorial factors that depend on the choice of group and the choice of operators for V and W while the f i (t) are related to ipoint functions of the seed theory. The sums are taken over the different i-point functions of the seed theory that can appear. This results follows directly from large N factorization. For times much greater than the dissipation time, we will show that the OTO correlator behaves as
F (t) ≈ 1 + 1 N g 4 ( K, ϕ)f 4 (t) + ... , t ≫ t d ,(9)
namely that the two and three-point function contributions become small and we are simply left with the seed theory four-point functions. If we consider twisted sector operators, f 4 (t) is not directly related to seed theory four-point functions on the plane but rather to some higher genus amplitude. Nevertheless, we can also think of its contribution as coming from four-point functions in a S k orbifold theory with k ∼ O(1). This will be enough to show that permutation orbifolds cannot be chaotic. There are essentially two scenarios. If the seed theory (or Sym k (C) for the twisted sectors) is chaotic, then the four-point function of the seed theory vanishes at lates times. In that case, F (t) ∼ 1 at late times. If the seed theory is not chaotic, its OTO four-point function will stabilizes at some value α ∼ O(1) which means F (t) ∼ 1 + α/N ∼ 1. Again, F (t) does not decay. This result is universal and holds for all oligomorphic permutation orbifolds. Even if the seed theory is chaotic, the effect is washed away once we take N non-interacting copies and orbifold. The orbifolding procedure does not introduce any interaction between the N copies but only projects to G N singlets and hence considerably reduces the number of low-energy states. This result should be a taken as a general feature of free gauges theories in the large N limit.
The paper is organized as follows: in section 2, we discuss the kinematics of the OTO correlation functions and define notation. In section 3, we introduce permutation orbifolds and show how large N factorization arises. In section 4, we study the OTO correlators in permutation orbifolds, considering both untwisted and twisted sector operators. We also discuss the behaviour of the function at early times as well as a possible deformation of the theory to strong coupling.
Out-of-time-ordered Correlators in 2d CFTs
We are interested in calculating out-of-time-ordered correlation functions in two dimensional conformal field theories. In this paper, we will focus on four point functions and consider correlation functions in the thermal state on the infinite line. We will closely follow [6] to set up our convention. 2d CFTs have the nice property that a correlation function in the thermal state on the infinite line can be mapped to usual vacuum expectation values through the map
z(x, t E ) = e 2π β (x+it E ) ,z(x, t E ) = e 2π β (x−it E ) ,(10)
where x, t E label points along the spatial and thermal direction respectively. With this transformation one can easily compute Euclidean correlators by mapping the operators to the plane. The operators transform as
O(x, t) = 2πz β h 2πz β h O(z,z) .(11)
In this paper, we will be interested in computing Lorentzian correlators so the conformal transformation (10) becomes
z(x, t) = e 2π β (x+t) ,z(x, t) = e 2π β (x−t) ,(12)
where t is now Lorentzian time. Note that although z * =z for Euclidean times, it is no longer true in Lorentzian time. However, any Lorentzian correlation function with arbitrary ordering of operators can always be obtained from its Euclidean counterpart upon doing the appropriate analytic continuation. We will describe the procedure shortly. The correlation function we wish to compute is
F (t) ≡ V (0, 0)W (x, t)V (0, 0)W (x, t) β V (0, 0)V (0, 0) β W (x, t)W (x, t) β ,(13)
which by (12) can be mapped to
F (t) = W (z 1 ,z 1 )W (z 2 ,z 2 )V (z 3 ,z 3 )V (z 4 ,z 4 ) W (z 1 ,z 1 )W (z 2 ,z 2 ) V (z 2 ,z 2 )V (z 4 ,z 4 ) .(14)
The positions of the operators are
z 1 = e 2π β (x+t+iǫ 1 )z 1 = e 2π β (x−t−iǫ 1 ) z 2 = e 2π β (x+t+iǫ 2 )z 2 = e 2π β (x−t−iǫ 2 ) z 3 = e 2π β (iǫ 3 )z 3 = e 2π β (−iǫ 3 )(15)z 4 = e 2π β (iǫ 4 )z 4 = e 2π β (−iǫ 4 ) .
The various factors of ǫ i are regulators that are needed to analytically continue the Euclidean correlator to Lorentzian time. The procedure is as follows. We start with finite values of ǫ i at t = 0. This is a Euclidean correlator. We then analytically continue in Lorentzian time by increasing t keeping the ǫ i finite. Finally, one can smear the operators over Lorentzian time and then take the ǫ i → 0. The order in which we take the ǫ i to zero will determine the ordering in Lorentzian time. Similarly to [6], we will omit this final step and keep the ǫ i finite and refer the reader to section 2.4 of [6] for a more detailed discussion. Note that by conformal symmetry, this function is only a function of the cross ratios given by
u = z 12 z 34 z 13 z 24 ,ū =z 12z34 z 13z24 ,(16)
which means
F (t) = F (u,ū) .(17)
If the correlator is purely Euclidean, then the function F (u,ū) is single valued. However, this is no longer true for Lorentzian times and F (u,ū) becomes multivalued: a branch cut stretches from 1 to ∞. One must specify which sheet we do the computation on, which again is related to the choice of ordering for the operators. The ordering given in (13) corresponds to doing the analytic continuation
1 − u → 1 − u , 1 −ū → e 2πi (1 −ū) ,(18)
which circles around the branch pointū = 1 and hence crosses the branch cut. However, nothing happens for the holomorphic cross-ratio u because we have broken the symmetry between u andū by considering Lorentzian times. Note that at late times, the cross-ratios are small and read
u ≈ −e − 2π β (t+x) ǫ * 12 ǫ 34 ,ū ≈ −e − 2π β (t−x) ǫ * 12 ǫ 34 ,(19)
with
ǫ ij = i e 2πi β ǫ i − e 2πi β ǫ j .(20)
We will now turn to the computation of these correlation function for permutation orbifolds.
Permutation Orbifolds
We will consider OTO correlation functions in a particular family of 2d CFTs: permutation orbifolds. Permutation orbifolds give a huge landscape of 2d CFTs at large central. They are built in the following way: consider any 2d CFT C with central charge c; we will call C the seed theory. Now consider the N-fold tensor product
C ⊗N ,(21)
which has central charge Nc. This theory has a global S N symmetry that permutes any of the N copies of C. We may then take a quotient of this product theory by any subgroup of the permutation group G N ⊆ S N . We are thus led to define
C N = C ⊗N G N .(22)
One can define such a theory for any seed theory C and for any group G N , which by taking N large, gives a huge landscape of 2d CFTs with a large central charge and thus a possible semi-classical holographic dual.
It is important to note that not all permutation groups have a well-defined large N limit. For example, the number of states at fixed dimension ∆ may diverge as N → ∞. We will therefore work with a subset of permutation orbifolds, those for which the group G N is oligomorphic [33][34][35]. Oligomorphic means that the group has a finite number of orbits on K-tuples as N → ∞ which in turn gives a finite number of states [25][26][27] of fixed dimensions ∆ as N → ∞. For example, this excludes the cyclic orbifolds Z N but allows group quite smaller than S N such as the wreath product S √ N ≀ S √ N .
Large N factorization
In [27], it was shown that a class of permutation orbifolds satisfy large N factorization. This was shown to be the case for the symmetric group, the wreath product, as well as any democratic group, i .e. groups with orbits of the same size in the large N limit.
In this paper, we will assume that large N factorization holds for all oligomorphic permutation groups. Even if a general proof is still missing, we believe this to be true. As evidence, note that even the cyclic group that is not oligomorphic still satisfies large N factorization. Alternatively, one can consider our results to apply to democratic permutation groups. For simplicity, we will derive the expressions for the symmetric group S N and only reintroduce the factors counting the numbers of orbits when we discuss the OTO correlators. We now review the derivation of large N factorization given in [27] and generalize it to four point functions. For simplicity, we will only consider untwisted sector operators but the generalization to arbitrary twisted sectors follows trivially. The untwisted sector operators can be described in the following way. Consider an ordered K-tuple K of distinct integers, and a K-vector ϕ of states in the seed theory,
φ = φ ( K, ϕ) .(23)
The notation is that the CFT K i is in state ϕ i while all other CFTs are in the vacuum. This is a state of the product theory (21). To obtain a state invariant under the action of G N , we must sum over images of the group. This gives
Φ = g∈G N φ (g. K, ϕ) ,(24)
where g acts only on the vector of integers, namely it permutes which of the N copies are in excited states. Any untwisted sector state of the orbifold theory can be expressed this way. Note that states where a single copy is in a non-trivial state correspond to single-trace operators, whereas those that have multiple excited states give multi-trace operators.
We will now prove that a general 4-point function will have the following schematic structure
Φ 1 Φ 2 Φ 3 Φ 4 N 1 N 2 N 3 N 4 ≈ ϕϕ + N −n 3 /2−n 4 ϕϕϕ n 3 ϕϕϕϕ n 4 ϕϕ ,(25)
where the N i are normalization factors and ϕ 1 ...ϕ k corresponds to a k-point function of the seed theory. The leading term in the 1/N expansion corresponds to the disconnected contribution for single-trace operators. For multi-trace operators, it simply corresponds to the sum over all Wick contractions [36]. It is usefull to consider the following schematic contraction of the different excited factors of the seed theory.
φ 1 : N • • • • • • • • • • • • • • K 1 • • • • • · · · • φ 2 : • • • • • • • • • • • • • • • • • • • · · · • φ 3 : • • • • • • • • • n 123 • • • • • • • • • • · · · • φ 4 : • • • • • n 4 • • • • • • • • • • • • • n 34
• · · · • Each black dot corresponds to an excited states whereas white dots correspond to vacua 1 . The numbers n ij or n ijk tell us the number of 2 or 3 point overlaps of the seed theory and n 4 gives the number of 4 point overlaps. The numbers are not all independent, we have n 12 + n 13 + n 14 + n 123 + n 124 + n 134 + n 4 = K 1 n 12 + n 23 + n 24 + n 123 + n 124 + n 234 + n 4 = K 2 n 13 + n 23 + n 34 + n 123 + n 134 + n 234 + n 4 = K 3 (26) n 14 + n 24 + n 34 + n 124 + n 134 + n 234 + n 4 = K 4 .
Each state in this pictorial representation is accompanied by its own sum over the permutation group. To calculate the correlation function, we only need to keep track of the non-zero contributions, which means we only keep the terms where there are at least two states overlapping because any 1-point function would vanish. We will take the states of the seed to be orthonormal such that
ϕ i ϕ j = δ ij .(27)
We must also take into account the normalization factors N i for the four operators Φ i . The two point function can be shown to be
Φ i Φ i = N!(N − K i )! ,(28)
which gives
N i = N!(N − K i )! .(29)
We are now ready to evaluate the contribution to the 4p-function. Keeping only the N-dependent factors, we obtain the following contributions:
• The sum over the group S 1 N simply gives N! . The vacua of φ 2 also give a contribution of (N − K 3 )!.
• Finally, the only contribution from S 4 N comes from the vacua as all the other contractions are fixed. This gives (N − K 4 )! Adding the normalization factors, we get a contribution of
(N − K 1 )!(N − K 2 )!(N − K 3 )!(N − K 4 )! N!(N − 1/2(K 1 + K 2 + K 3 + K 4 )! + n 4 + n 3 /2) ,(30)
with n 3 = n 123 + n 124 + n 134 + n 234 . Using Stirling's approximation, it is easy to see that we recover the form (25). A term which has n 3 and n 4 three and four-point overlap and will be of order
N −n 3 /2+n 4(31)
We will now proceed to the evaluation of F (t) using the results we just derived.
Out-of-time-ordered Correlators in Permutation Orbifolds
Untwisted Sector Single-trace Operators
We will start with the simplest possible choice of operators: single-trace operators. These will be operators given by
V = N k=1 ϕ k i , W = N k=1 ϕ k j ,(32)
where i and j label operators of the seed theory and the sum over k sums over the N copies. They are symmetric operators invariant under S N . One might wonder wether there are multiple operators of this form for a given φ i if the group is a subgroup of S N , rather than the full symmetric group. This would mean that there are multiple orbits of the group when acting on 1-tuples. While there clearly are examples of oligomorphic permutation groups that have this property (for example S N/2 × S N/2 ), we will not consider them here. These theories would have more than one stress tensor and would hence be peculiar. We will focus on oligomorphic permutation groups who have a single orbit when acting on 1-tuples.
For such a choice of operators, it is easy to see that there cannot be 3-point overlaps hence n 3 = 0. We get
F (t) = 1 + 1 N ϕ i (z 1 ,z 1 )ϕ i (z 2 ,z 2 )ϕ j (z 3 ,z 3 )ϕ j (z 4 ,z 4 ) ϕ i (z 1 ,z 1 )ϕ i (z 2 ,z 2 ) ϕ j (z 2 ,z 2 )ϕ j (z 4 ,z 4 ) .(33)
This shows that the dynamics of the four-point function at late times is completely fixed by the behaviour of the OTO correlation function in the seed theory. It is now easy to see that F (t) cannot become small at late times. To see that, notice that there are essentially two scenarios. First, the seed theory could be chaotic. In that case, its own OTO correlator would vanish at late times. Second, it could be non-chaotic which means its OTO would not vanish at late times and stay of order one. In any event, the OTO correlator of the seed theory can never become O(N). In fact, it cannot even know about the existence of a parameter N. This shows that the OTO of all single-trace operators stays of O(1) at late times. We now turn to multi-trace operators.
Untwisted Sector Multi-trace Operators
For a multi-trace operator, the expression of F (t) will be more complicated. In general it will be of the form
F (t) = 1 + g 2 ( K, ϕ)f 2 (t) + 1 N g 3 ( K, ϕ)f 3 (t) + g 4 ( K, ϕ)f 4 (t) + ...(34)
where f 2 (t), f 3 (t), f 4 (t) are the contribution coming from 2-point, 3-point and 4-point overlaps respectively. The g i are combinatorial factors that depend on the number of seed operators chosen, wether they are all the same or not, and the choice of the group. It can be determined purely from group theory arguments by counting the number of orbits on a given (un)ordered K-tuple. In any case, note that
g i ∼ O(1) ,(35)
as implied by the fact that we are considering oligomorphic permutation groups which by definition have a finite number of orbits as N → ∞. Also note that the sums in (34) run over an O(1) number of possibilities. This results from the fact that we considered V and W to be "simple" operators, made out of an O(1) number of seed theory operators. In particular, V and W must have ∆ ≪ N. We will now analyze the various contributions to (34).
2-point Overlaps
The 2-point overlap captures the disconnected contribution to the four-point function.
For single-trace operators, this term simply gave one. For multi-trace operators, there can be contractions between seed operators in V and seed operators in W if V and W share a same seed theory operator. This means we can have terms of the form
ϕ a (z 1 )ϕ a (z 4 ) ϕ a (z 2 )ϕ a (z 3 ) ϕ a (z 1 )ϕ a (z 2 ) ϕ a (z 3 )ϕ a (z 4 ) = u 1 − u 2ha ū 1 −ū 2ha ϕ a (z 1 )ϕ a (z 3 ) ϕ a (z 2 )ϕ a (z 4 ) ϕ a (z 1 )ϕ a (z 2 ) ϕ a (z 3 )ϕ a (z 4 ) = u 2haū2ha .(36)
One can quickly see that these terms do not do anything interesting upon taking an analytic continuation. Terms of the first type simply gives a phase upon analytic continuation whereas those of the second type do nothing. Furthermore, u,ū → 0 when t ≫ t d which means both types of terms will quickly decay. We now turn to the three-point overlaps.
3-point Overlaps
First, it is important to note that there must be an even number of three point overlaps because the four point function we consider has two pairs of identical operators. It is then easy to see that the most general form of f 3 (t) will be
f 3 (t) = C 2 i u p 1 (h i ) (1 − u) p 2 (h i )ūp 1 (h i ) (1 −ū)p 2 (h i )(37)
where the p i are linear functions of the conformal weights and C 2 i is an ope coefficient of three operators of the seed theory squared. Also, one can easily check that p 1 ,p 1 > 0. This shows that the three-point overlaps do not have interesting analytic continuations and just like the two-point overlaps, they would pick up a simple phase and anyway decay for times much greater than the dissipation time.
4-point Overlaps
The contributions from the 4-point overlap are of course very similar to the case where we had single-trace operators. They take the form
ϕ a (z 1 )ϕ a (z 2 )ϕ b (z 3 )ϕ b (z 4 ) ϕ a (z 1 )ϕ a (z 2 ) ϕ b (z 3 )ϕ b (z 4 ) ,(38)
which again is an OTO correlation function in the seed theory. The only difference with the single-trace operators is that there will be a sum over different OTO correlators in the seed theory, weighted by sum combinatorial factor that depends on the number of different states and the group G N . Nevertheless, this implies that the OTO correlators of the multi-trace operators cannot decrease at late times as it is built out of an O(1) number of seed theory OTO correlators that each may decay or not, but at least can never become large. This closes our analysis of the untwisted sector operators.
We have shown that an arbitrary choice of untwisted sector operators V and W with ∆ ≪ N yields an OTO correlator that cannot decay at late times. It is tempting to conclude that this already proves that permutation orbifolds cannot be chaotic. Note however that the growth of operators in the symmetric product theory is dominated by twisted sector operators [25], which grow as
ρ tw (∆) ≈ e 2π∆ ,(39)
whereas the growth of untwisted sector operators only goes as
ρ untw (∆) ≈ e ∆ log ∆ .(40)
One could then argue that an untwisted sector operator is actually not generic, and that it is perhaps the reason why they do not decay at late times. We will now show that the twisted sector operators behave exactly in the same way.
Twisted Sector Operators
The expression for generic twisted sector operators will still take the form (34). As showed in the previous section, there is no fundamental difference between single-trace and multi-trace operators at this level so we will consider multi-trace operators to stay as general as possible. It is easy to show that the 2-point and 3-point overlaps behave exactly the same way as for the untwisted sector operators. This results from the fact that the only data necessary do derive (36) and (37) was the conformal weights. f 3 (t) also carries an OPE coefficient of the seed theory squared but these are O(1) numbers and do not play an important role. For this reason, no interesting contribution can come from the two and three-point overlaps.
We still need to consider the four-point overlaps. The general structure of these contributions is sketched in [27,[37][38][39] and reads
ϕ a (z 1 )ϕ a (z 2 )ϕ b (z 3 )ϕ b (z 4 ) ϕ a (z 1 )ϕ a (z 2 ) ϕ b (z 3 )ϕ b (z 4 ) ,(41)
where the ϕ a,b are now operators in some S k orbifold theory where k ∼ O(1). For example, if the twisted sector contains a single cycle of length k, the operators will be twisted sector operators in a Z k orbifold theory, which were considered in [21] 2 . But now, the same logic we applied for the untwisted sector operators can be used here. Independently of wether this S k orbifold theory is chaotic or not, its OTO fourpoint function can never become O(N). It would be nice to be able to bound the OTO four-point function, for example by its value at t = 0. This can be done for the spectralform factor which is an analytic continuation of the partition function [40,41]. The bound simply comes from the fact that the partition function is a sum over positive contributions and introducing phases can only decrease its value. Unfortunately, a similar reasoning does not apply to the OTO four-point function as the Euclidean correlator is not a sum over positive contributions. Also, if V and W are null separated then the Lorentzian correlator must diverge. Nevertheless, once we move away from the lightcones and go to late times the correlator (41) still cannot be of order N.
As an example, one can consider the four-point function of two twisted and two untwisted operators in the D1D5 CFT. The two twist operators we will consider are special in that they correspond to the operators that deform the theory towards the strongly coupled regime. The Euclidean correlator was calculated in [39] and it was shown in [19] that it does not vanish at late times after we analytically continue to the OTO setup. A general four-point function of twisted operators in the D1D5 CFT at the orbifold point will be hard to compute but for the reasons mentioned above, we do not expect it to decay at late times.
This closes our discussion of all possible operators of dimension ∆ ≪ N in permutation orbifolds. We have shown that their OTO correlators do not decay at late times, indicating that permutation orbifolds are not chaotic theories. This is in fact expected, since permutation orbifolds correspond to free discrete gauge theories.
The Early Time Behaviour
So far, we have been interested in the behaviour of the OTO correlator at late times (t ∼ t s ) and have simply investigated wether it decays to zero or not. In this sense, permutation orbifolds are universal and all share the same structure: the OTO of generic operators does not decay at late times and stays of order one.
However, one may wonder what happens at earlier times (t ≪ t s ). This is where the universality will break down and the physics will be theory-dependent. In particular, the choice of the seed theory will dictate the dynamics at early times. This dependence on details of the theory was made explicit in [21] where the authors considered the behaviour of the OTO correlator in Z n orbifolds of T 2 . As they show, the answer depends strongly on the compactification radius and wether it is a rational number or not. For irrational compactification radii, they find a polynomial decay. This is directly relevant for the behaviour of the OTO correlator in permutation orbifolds, where this type of behaviour will be relevant at early times (see also [42] for a discussion of OTO correlators in rational CFTs).
In general, if one picks a seed theory that is chaotic, we expect there to be some interesting time-dependence at early times dictated by the physics of the seed theory. Note however that there is no clear notion of a Lyapunov exponent for CFTs with central charge c ∼ O(1) as there is no parametrically large difference between the dissipation time and the scrambling time.
Deformation to Strong Coupling
We know that CFTs dual to weakly-coupled supergravity should be maximally chaotic since black holes in Einstein Gravity saturate the chaos bound [9]. This means that deforming the D1D5 CFT away from the orbifold point should drastically change the behaviour of the OTO correlation function. To see this, one needs to study the orbifold theory deformed by a twist-2 operator. The deformation is δS = α dzdzO(z,z)
where O(z,z) is the exactly marginal operator described in [39,43,44] built from the twist-2 operator. The scaling of the coupling α can be shown to be [38] α ∼ λN 1/2 (43) where λ is the 't Hooft coupling and is fixed in the limit N → ∞. Only even powers of λ will appear in the OTO correlators as we do conformal perturbation theory. This means that the first correction will be of the form
λ 2 N V W V W σσ con V V W W .(44)
It would be very interesting to compute this correction using second-order conformal perturbation theory. Nonetheless, much more work is needed to compute the Lyapunov exponent perturbatively. One would need to resum ladder diagrams along the lines of [13], which in this case means considering four-point function on arbitrary genus Riemann surfaces, integrated over the moduli of the surface. This appears to be a very complicated task and it is not clear to us how to use the ladder diagram re-organization in this context. It would be interesting to attempt this calculation but we leave this for future work.
•
For S 2 N , there are two contributions. First, the excited states of φ 2 that are not contracted with φ 1 can be distributed in any way on the vacua of φ 1 . This gives a contribution of N −K 1 K 2 −n 12 −n 123 −n 124 −n 4 . Then, there is a factor of (N − K 2 )! coming from the permutation of the vacua of φ 2 . • For S 3 N There are also two contributions. First the contractions of 3 and 4 only can be distributed in any way along the vacua not occupied by 1 or 2. This gives a contribution of N −K 1 −n 23 −n 24 −n 234 n 34
In principal, one would have to keep track of the different seed theory states. This would amount to giving different colors to the black dots. This will only keep track of N -independent numbers so we neglect it for simplicity. See[36] for an exact expression in the case of the symmetric group.
It is also possible to view these twisted sector four-point functions as correlation functions of the seed theory but on a complicated Riemann surface, although it will not be particularly helpful here.
AcknowledgementsI would like to thank Fotis Dimitrakopoulos, Guy Gur-Ari, Christoph Keller, Dan Roberts, Gabor Sarosi, Steve Shenker and Ida Zadeh for useful discussions. I am especially grateful to Nathan Benjamin and Ethan Dyer for collaboration at an early stage of this project. I thank the Galileo Galilei Institute for Theoretical Physics (GGI) for the hospitality and INFN for partial support during the completion of this work, within the program New Developments in AdS 3 /CFT 2 Holography. I am supported by the Foundation for Fundamental Research on Matter (FOM). This work is part of the ∆-ITP consortium, a program of the NWO funded by the Dutch Ministry of Education, Culture and Science (OCW).
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| [] |
[
"Finite size scaling of the 5D Ising model with free boundary conditions",
"Finite size scaling of the 5D Ising model with free boundary conditions"
] | [
"P H Lundow \nDepartment of mathematics and mathematical statistics\nUmeå University\nSE-901 87UmeåSweden\n",
"K Markström \nDepartment of mathematics and mathematical statistics\nUmeå University\nSE-901 87UmeåSweden\n"
] | [
"Department of mathematics and mathematical statistics\nUmeå University\nSE-901 87UmeåSweden",
"Department of mathematics and mathematical statistics\nUmeå University\nSE-901 87UmeåSweden"
] | [] | There has been a long running debate on the finite size scaling for the Ising model with free boundary conditions above the upper critical dimension, where the standard picture gives a L 2 scaling for the susceptibility and an alternative theory has promoted a L 5/2 scaling, as would be the case for cyclic boundary. In this paper we present results from simulation of the far largest systems used so far, up to side L = 160 and find that this data clearly supports the standard scaling. Further we present a discussion of why rigorous results for the random-cluster model provides both supports the standard scaling picture and provides a clear explanation of why the scalings for free and cyclic boundary should be different. | 10.1016/j.nuclphysb.2014.10.011 | [
"https://arxiv.org/pdf/1408.5509v1.pdf"
] | 119,205,597 | 1408.5509 | 94cedb3a30c05d9618a85117066203f65de0aae1 |
Finite size scaling of the 5D Ising model with free boundary conditions
23 Aug 2014 (Dated: August 26, 2014)
P H Lundow
Department of mathematics and mathematical statistics
Umeå University
SE-901 87UmeåSweden
K Markström
Department of mathematics and mathematical statistics
Umeå University
SE-901 87UmeåSweden
Finite size scaling of the 5D Ising model with free boundary conditions
23 Aug 2014 (Dated: August 26, 2014)Ising modelfinite-size scalingboundarysusceptibility
There has been a long running debate on the finite size scaling for the Ising model with free boundary conditions above the upper critical dimension, where the standard picture gives a L 2 scaling for the susceptibility and an alternative theory has promoted a L 5/2 scaling, as would be the case for cyclic boundary. In this paper we present results from simulation of the far largest systems used so far, up to side L = 160 and find that this data clearly supports the standard scaling. Further we present a discussion of why rigorous results for the random-cluster model provides both supports the standard scaling picture and provides a clear explanation of why the scalings for free and cyclic boundary should be different.
INTRODUCTION
Above the upper critical dimension d = 4, for the Ising model with nearest-neigbour interaction, the critical exponents assume their mean field values [1,2]; α = 0 (finite specific heat), β = 1/2, γ = 1, ν = 1/2, and the so called hyperscaling law dν = 2 − α fails for d > 4. However, for periodic boundary conditions most finite-size scaling properties near the critical (inverse) temperature K c are well known. For example, the susceptibility behaves as χ ∝ L 5/2 for d = 5, in sharp contrast to L γ/ν for 1 < d < 5, with a logarithmic correction for d = 4. There is plenty of literature on d > 4 for periodic boundary conditions, to name but a few, see e.g. [3][4][5][6][7][8][9].
Much less has been written on the subject of free boundary conditions above the upper critical dimension, but see e.g. [10,11]. As can be seen from the references in those two papers there has been some debate on whether the standard scaling picture, saying that e.g. the susceptibility scales as L 2 for free boundary, holds or whether an alternative theory proposing that it scales as L 5/2 is correct. In [10] the current authors simulated the 5-dimensional model on larger systems than previous authors and found that the data supported the standard scaling picture. In a reply [11] it was again suggested that the alternative picture is correct and that the results of [10] were due to too small systems, dominated by finite size effects stemming from their large boundaries.
The purpose of this paper is two-fold. We have extended the 5-dimensional simulations with free boundary from [10] to much larger systems, up to L = 160, where the boundary vertices make up less than 6.1% of the system. First, using the new data, we give improved estimates of the critical energy, specific heat and several other quantities at the critical point. Second, we compare how well the standard scaling and the alternative theory fit our new large system data, and discuss why, based on mathematical results on the random cluster model, there are good reasons for expecting the standard picture to be the correct one, as the data also suggests.
DEFINITIONS AND DETAILS
For a given graph G the Hamiltonian with interactions of unit strength along the edges is H = − ij S i S j where the sum is taken over the edges ij. As usual K = 1/k B T is the dimensionless inverse temperature and we denote the temperature equilibrium mean by · · · . The susceptibility is defined as χ = N m 2 , where m = (1/N ) i S i is the magnetisation per spin, and the specific heat as C = N U 2 − U 2 , where U = (1/N ) ij S i S j is the energy per spin, and for short we write U = U .
The underlying graph in question is an L×L×L×L×L grid graph with free boundary conditions, or equivalently, the cartesian product of five paths on L vertices.
We have collected data for grid graphs of linear order L = 3, 5,7,11,15,19,23,31,39,47,55,63,72,96,128 and 160. For L = 160 we thus present data for systems on more than 100 billion vertices. States were generated with the Wolff cluster method [12]. Between measurements, clusters were updated until an expected L 5 spins were flipped.
For 3 ≤ L ≤ 63 we kept 64 separate systems at nine couplings K = 0.1139130, 0.1139135, . . . 0.1139170. For L = 72, 96 we used 48 separate systems, 116 for L = 128 and 108 for L = 160. For these larger systems the number of measurements were just short of 30000 for L = 72 down to about 4000 for L = 160. Means and standard errors were estimated by exploiting the separate systems. For the larger systems (L ≥ 72), due to the comparably few measurements, we also used bootstrapping on the entire data set for estimating standard errors. To doublecheck for equilibration problems we compared with subsets of the data after rejecting early measurements.
The different couplings for 3 ≤ L ≤ 63 showed no discernible difference in their scaling behaviour for L ≤ 63. Based on the behaviour for L ≤ 63 we have designated K c = 0.1139150, which is a little higher than what we used in Ref. [10] and marginally lower than that used in e.g. Ref. [11]. The lion's share of sampling were then made at 0.1139150 and for L ≥ 72 we have measured only at K c .
For all sizes we measured magnetisation and energy, storing their moment sums. For 3 ≤ L ≤ 63 we also measured many properties regarding the clusters that were generated and used for consistency checks, and one of them will be shown in the Discussion section.
Geometry and boundary effects
A potentially important issue for systems with free boundary condition is the size of the boundary, and in particular the fraction of vertices on the boundary. Of the N = L 5 vertices in the graph (L − 2) 5 are inner vertices and thus L 5 − (L − 2) 5 vertices sit on the boundary. The fraction of boundary vertices is then 1 − (1 − 2/L) 5 . For L = 8 this means that the boundary constitutes no less than 76% of all vertices. To continue, for L = 16 the boundary's share is 49%, for L = 32 it is 28%, for L = 64 it is 15%, for L = 128 it is 7.6% and for L = 160, our largest system studied here, it makes up 6.1%. So our largest system have a clear minority of their vertices on the boundary.
Another important measure is the number of vertices with a given minimum distance to the boundary. If we consider the cube of side cL around the centre vertex of the cube, i.e. the set of vertices with distance at least cL 2 to the boundary we find that it contains a c 5 fraction of the N vertices. That means that at least 50% of the vertices are at a distance of at most 0.065L from the boundary, for every L. Similarly, the central cube with side L/2 contains just 3.1% of the vertices of the cube.
This means that even in the limit the effect of vertices close to the boundary will always be large, and that the vertices close to the central vertex will also remain atypical for any property which depend both on the distance to the boundary and the majority of the vertices in the cube. In particular we should expect such properties to be bounded from above the corresponding values in the infinite system thermodynamic limit, if they tend to decrease with the distance to the boundary.
ENERGY AND SPECIFIC HEAT
It is known [2] that in the limit ,for d ≥ 5, the specific heat, i.e. the energy variance, is bounded for all temperatures, but the value is not known, and likewise for the critical energy. In Fig. 1 the mean energy U is shown versus 1/L. The leading scaling term is here set to the order 1/L and the correction term to order 1/L 3/2 . This gave by far the most stable coefficients of the fitted curve among the simple exponents. We find that the best fitted curve is 0.675647(3) − 1.013(1) x + 0.395(1) x 3/2 , where x = 1/L. The fit is excellent down to L = 3. The coefficients and their error estimates are here based on the median and interquartile range of the coefficients when deleting one of the data points from the fitting process.
In the inset picture in Fig. 1 we zoom into the plot by showing L(U − U c ) versus 1/L 1/2 together with the line −1.013 + 0.395x. The fit is vey good and hence we conclude that the correction term is of the order 1/L 1/2 . Note that the error bars in both plots are included but they are far too small to be seen at this scale. The limit energy U c = lim L→∞ U(K c , L) = 0.675647 (3) is only marginally larger than the value we gave in [10] which may be explained by the slightly smaller K c . Note that for free boundary conditions the limit is reached from below whereas for periodic boundary conditions U approaches its limit from above, roughly as U(K c , L)−U c ∼ 5.5/L 5/2 [10].
For the specific heat we note that the error bars are noticeable, but this is to be expected. It is not known at which rate C(K c , L) approaches its asymptotic value C c but judging from the excellent line-up of the points in (2)x, where x = 1/L 1/3 . As before the error estimates of the coefficients are based on the variability of a fitted curve after deleting one of the points. We estimate thus that C c = lim L→∞ C(K c , L) = 14.69(1). The inset picture of Fig. 2 zoom into the correction term by plotting L 1/3 (C − C c ) versus 1/L 1/3 together with the constant line −14.93. The error bars now become quite noticeable, especially for the larger L. There is no clear trend upwards or downwards in the data points which suggests that any further corrections to scaling must be truly negligible.
MAGNETISATION AND SUSCEPTIBILITY
The scaling of the modulus of the magnetisation |m| for free boundary conditions is very different from that of periodic boundary conditions. In the first case we find |m| ∝ L −3/2 whereas in the second it is wellknown that |m| ∝ L −5/4 . In the free boundary case we note the need for correction to scaling. Indeed, if we want perfectly fitted curves down to L = 3 we need two correction terms. We have instead chosen to ignore L ≤ 7 and stay with just one correction term for the remaining 13 points. In Fig. 3 we show |m| L 3/2 versus 1/L for 11 ≤ L ≤ 160 together with the curve 0.22958(6) + 1.101(3) x − 1.63(3) x 2 . We test the fit of this curve by zooming into the picture and instead show ( |m| L 3/2 − 0.22958)L which then should be well fitted by the line 1.101 − 1.63 x. As the inset of Fig 3 shows, it is and we conclude that to leading order |m| ∼ 0.22958(6)L −3/2 . However, the error bars for the largest systems are now quite pronounced.
As we mentioned above the susceptibility χ = N m 2 scales to leading order as χ ∝ L 5/2 for periodic boundary conditions but it is not known what the corresponding order is for free boundary conditions. We find here, as in Ref. [10], that χ ∝ L 2 is by far the best scaling rule. In Fig. 4 we show χ/L 2 versus 1/L for 7 ≤ L ≤ 160 together with the line 0.08269(2) + 0.8174 (3) x. The coefficients were determined after excluding L ≤ 5 from the fitting process. Had we included the two smallest systems an extra correction to scaling term would have been required. Again we zoom in and the inset of shows (χ/L 2 − 0.08269)L versus 1/L together with the constant line 0.8174. Though the error bars are quite big for the largest systems the fit is quite acceptable. In short we find χ(K c ) ∼ 0.08269(2)L 2 . We might add that the corresponding expression for periodic boundary is not known exactly but we suggested recently [8] that χ ∼ 1.742L 5/2 .
SUSCEPTIBILITY COMPARED TO L 5/2
It has been suggested [13] that L 2 is in fact not the correct scaling for the susceptibility. The authors of [13] claim that the correct scaling should be L 5/2 , as for the case with cyclic boundary conditions, and that the exponent previously found by us, and other authors, are based on either finite size effects due to too many boundary vertices in small systems, for simulation studies, or incomplete theory. In our previous work the boundary did indeed contain a large fraction of the system's vertices but in our current study this fraction has been reduced to a lower value than in any previous study, including the truncated systems used in [13].
To avoid implicit bias in our scaling of the susceptibility to we can also test the ratio χ/L 5/2 . If the claims of [13] are correct this quantity should converge to a finite non-zero limit, at least for large enough systems, and if the standard scaling is correct it should to leading order converge to 0 as L −0.5 . We make a scaling ansatz c 0 + c 1 x λ1 + c 2 x λ2 , where x = 1/L, and let Mathematica find the five free parameters using a least squares fit, after excluding L = 3, 5. As usual, we let each remaining point be deleted in turn from the fitting data to obtain error bars of the parameters. We find on average the curve 0.0000(4) + 0.085(9) x 0.51(3) + 0.820(7) x 1.51 (3) . Clearly the parameters we estimated above for χ/L 2 falls inside these estimates, though the error bars are a magnitude larger here. Using the middle point values we plot the curve together with the data points in Fig. 5. The data for large systems is clearly consistent with the standard scaling, even for an unrestricted data fitting like this.
FOURTH MOMENT AND KURTOSIS
Our final property of interest is the fourth moment of the magnetisation at K c . We find that the fourth moment of the magnetisation scales as m 4 ∝ L −6 . The general rule would then be |m k | ∝ L −3k/2 whereas the corresponding rule for periodic boundary conditions is |m k | ∝ L −5k/4 . Proceeding in the same manner as before, we plot the normalised fourth moment's behaviour as m 4 L 6 versus 1/L in Fig. 6 together with the estimated polynomial 0.02051(3) + 0.4045 (8) x + 1.989(4) x 2 . We excluded L = 3 from the coefficient estimates. To leading order we thus find m 4 ∼ 0.02051(3)L −6 .
The moment ratio Q = m 4 / m 2 2 , or kurtosis, indicates the shape of the underlying magnetisation distribution. In Fig. 7 we show the kurtosis versus 1/L and the line 3−0.14 x. The error bars are based on the formula for the error of a quotient, d(x/y 2 ). The line is based on the coefficient estimates for m 2 and m 4 above by taking the quotient of their respective series expansions in the standard fashion. Inserting the coefficients and their error estimates gives the limit m 4 / m 2 2 → 3.000(6) which is the characteristic value of a gaussian distribution. Recall that for periodic boundary conditions the kurtosis at K c takes the asymptotic value Γ(1/4) 4 /2π 2 = 2.1884 . . ., see Refs. [7,8]
KURTOSIS AT AN EFFECTIVE CRITICAL POINT
That the kurtosis takes the asymptotical value 3 at K c does of course not mean that the kurtosis converges to 3 for every sequence of temperatures K c (L) that has K c as limit. We exemplify this by reexamining some of our data used in Ref. [10] where we relied on extremely detailed data on a wide temperature range for 4 ≤ L ≤ 20. For L = 4, 6, 8, 10 we also have magnetisation distributions. Let us say that K c (L) is the point where the variance of the modulus magnetisation, i.e.χ = N m 2 − |m| 2 , takes its maximum value. The distribution is here at its widest and on the threshold of breaking up into two parts, see Fig. 8 where we show a scaled distribution at K c (L) for L = 4, 6, 8, 10, in stark contrast to the distribution at K c of Fig. 9. Measuring the kurtosis at this point produces Fig. 10 which shows Q(K c (L), L) versus 1/L and a fitted 2nd degree polynomial which suggests the limit 1.520. The absence of error bars is due to the method by which the original data were produced. However, we expect the error to be smaller than the plotted points. In fact, repeating this exercise for periodic boundary conditions suggests the limit 1.517. There is a distinct possibility that these two limits are in fact the same, but that would require high-resolution data for larger systems to resolve than is at our disposal. In any case this subject falls outside the scope of this paper.
DISCUSSION AND CONCLUSIONS
As we have seen the sampled data for cubes up to side L = 160 agree well with the standard scaling picture for free boundary conditions, and e.g. recent long series expansions [14] also appears to favour this version, nontheless without a rigorous bound for the rate of convergence a simulation study is always open to the claim that it is dominated by finite size effects.
However, the last decade has seen a number of rigorous mathematical results on the behaviour of the randomcluster model which leads us to believe that the standard scaling is indeed the right one. The Fortuin-Kasteleyn random-cluster model has a parameter q which governs the properties of the model. We'll refer the reader to [15] for more details and history. We recall that for q = 1 the model is the standard bond percolation model, for q = 2 it is equivalent to the Ising model, and in the limit q → 0 we get the uniform random spanning tree, or the uniform spanning forest, depending on the parameter p.
For the random spanning tree on the d-dimensional lattice with different boundary conditions Pemantle [16] begun a study which related it to the loop erased random walk and demonstrated a strong dependency on both the dimension and boundary condition. In later papers [17][18][19] these results were refined to show, among other things, that for large enough d that for two points in a grid with side L and free boundary the distance between the two points within the tree scale as L 2 , but that for the torus of side L the distance scales as L d/2 .
Coming to the case q = 1, Aizenman [20] studied the behaviour of the largest crossing clusters, i.e clusters which contain vertices on opposite sides of the box, in percolation on grids in different dimensions. Among other things he conjectured that at the critical point p c , for all large enough dimensions d the largest cluster in the case with free boundary should scale as L 4 , and for the torus, or cyclic boundary, it should scale as L 2d/3 . This conjecture was proven in [21,22], and so we know that for percolation the boundary condition have a large and non-vanishing effect on the size of the largest clusters at the critical point. It is also important to point out that the results from [21,22] are for the scaling exactly at the critical point p c . For other sequences of points converging to p c different scaling behaviours can appear.
The reason for the drastic difference between the torus and free boundary case here is that for large d the clusters in the model become much more spread out than in low dimensions. For d = 2 clusters at p c are always finite and have a boundary which in the scaling limit is very close to a brownian motion [23]. For a finite box of side L, with free boundary the number of crossing clusters, has a finite mean, bounded as L grows, and the probability that there are more than k such clusters is less than exp(−a 1 k 2 ), for some positive constant a 1 [20]. For a box with free boundary in d > 6, the critical dimension for q = 1, the number of crossing clusters is at least a 2 L d−6 , for some constant a 2 [20]. Further, the largest cluster and a positive proportion of the crossing clusters have size proportional to L 4 , independently of d. The clusters are here of much lower dimension relative to that of the lattice and much more tree like in their structure than for d = 2. If we instead consider a torus with d > 6 the large number of these more expanding clusters lead them to connect up with other clusters, which would have been separate in the free boundary case, and this merging leads to maximum clusters that are vastly larger than in the free boundary case.
The current authors expect a picture similar to that for q = 1 to hold for the Ising case q = 2 as well. Since the susceptibility in the Ising model is proportional to the average cluster size in the random-cluster model this leads to a prediction of L 2 as the correct scaling for the free boundary case and L 5/2 for cyclic boundaries, for d = 5. This would also lead us to expect the largest clusters in the free boundary case to scale as L 2 . For our small to medium sized systems we collected the size of the cluster containing the central vertex of our cubes, a property which was also considered in [13] for truncated systems.
In Fig. 11 we show the normalised mean cluster size S 0 /L 2 for 3 ≤ L ≤ 55 as used in [13], together with a linear function estimated to be 0.3505(2) + 0.600 (1)x. The inset shows the zoomed-in version S 0 L −5/4 − 0.3505L 3/4 versus 1/L which then should essentially take the constant value 0.600, the red line. As we can see we have an excellent fit to the prediction that this cluster size should scale as L 2 .
Aizenman's prediction [20] of L 2d/3 as the correct scaling for percolation on the torus, for high d, came from a comparison with the Erdös-Renyi random graph on N vertices, for which the largest connected component at the critical probability, scales as N 2/3 . In [24] we follow this analogy further by comparing the largest cluster for the random-cluster model on 5-dimensional tori with the detailed rigorous results on the random-cluster model for complete graphs from [25], and a good agreement is found.
To conclude, we find that both the data from our simulations and the current mathematical result for the random-cluster model gives good support for the standard scaling picture for the Ising model with free boundary conditions, as well as a framework predicting further properties for the case with cyclic boundary.
on-line) Mean energy U at Kc versus 1/L for L = 7, 11, 15, 19, 23, 31, 39, 47, 55, 63, 72, 96, 128 and 160. The red curve is 0.675647 − 1.013 x + 0.395 x 3/2 where x = 1/L. Error bars are too small to be seen. See text for details. The inset shows the scaled energy L(U − Uc) at Kc versus 1/L 1/2 for the same range of L. Error bars are too small to be seen. The red curve −1.013 + 0.395 x.
Fig. 2it appears C(K c , L)− C c ∝ 1/L 1/3 . This is the only time we see a 1/3 in a scaling exponent for free boundary conditions and we have no theoretical basis for it. InFig. 2we show C(K c , L) versus 1/L 1/3 and the line 14.69(1) − 14.93
160. The red line is 14.69 − 14.93 x where x = 1/L 1/3 . The inset shows the scaled specific heat L 1/3 (C − Cc) at Kc versus 1/L 1/3 for the same range of L. The red line is the constant −14.93.
Fig. 4
4
on-line) Modulus magnetisation |m| L 3/2 at Kc versus 1/L for L = 11, 15, 19, 23, 31, 39, 47, 55, 63, 72, 96, 128 and 160. The red curve is 0.22958 + 1.101 x − 1.63 x 2 , where x = 1/L. The inset shows the scaled magnetisation ( |m| L 3/2 − 0.22958)L at Kc versus 1/L for the same range of L. The red line is 1.101 − 1.63 x.
FIG. 4 :
4(Colour on-line) Normalised susceptibility χ/L 2 at Kc versus 1/L for L = 7, 11, 15, 19, 23, 31, 39, 47, 55, 63, 72, 96, 128 and 160. The red line is 0.08269 + 0.8174 x, where x = 1/L. The inset shows the scaled and normalised susceptibility (χ/L 2 − 0.08269)L at Kc versus 1/L for the same range of L. The red line is the constant 0.8174.
on-line) Normalised susceptibility χ/L 5/2 at Kc versus 1/L for L = 7, 11, 15, 19, 23, 31, 39, 47, 55, 63, 72, 96, 128 and 160. The red curve is 0.085 x 0.51 + 0.82 x 1.51 , where x = 1/L.
on-line) Normalised fourth moment m 4 L 6 at Kc versus 1/L for L = 7, 11, 15, 19, 23, 31, 39, 47, 55, 63, 72, 96, 128 and 160. The red curve is 0.02051 + 0.4045 x + 1.989 x 2 , where x = 1/L.
on-line) Kurtosis m 4 / m 2 2 at Kc versus 1/L for the same range of L as inFig. 6. The red line is 3 − 0.14 x, where x = 1/L.
FIG. 8 :
8(Colour on-line) Scaled magnetisation distributions Pr(m)L 7/2 versus mL 3/2 at Kc(L) (see text) for L = 4, 6, 8 and 10.
FIG. 9 :
9(Colour on-line) Scaled magnetisation distributions Pr(m)L 7/2 versus mL 3/2 at Kc for L = 4, 6, 8 and 10 (increasing at y-axis). Data from Ref.[10].
10: (Colour on-line) Kurtosis Q = m 4 / m 2 2 at Kc(L) (see text) versus 1/L for L = 4, 6, 8, 10, 12, 16, 20. The red curve 1.52 − 0.17 x + 0.56 x 2 , where x = 1/L.
FIG
. 11: (Colour on-line) The normalised average size of the cluster containing the centre vertex, S0 /L 2 , plotted versus 1/L 3/4 for L = 3, 5, 7, 11, 15, 19, 23, 31, 39, 47, 55. The red line is 0.3505 + 0.600x where x = 1/L 3/4 . The inset shows the scaled and normalised cluster size S0 L −5/4 − 0.3505L 3/4 versus 1/L. The red line is the constant 0.600.
ACKNOWLEDGEMENTSThe simulations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at High Performance Computing Center North (HPC2N) and at Chalmers Centre for Computational Science and Engineering (C3SE).
* Electronic address: [email protected] † Electronic address: klas. [email protected]* Electronic address: [email protected] † Electronic address: [email protected]
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| [] |
[
"Relativistic superfluidity and vorticity from the nonlinear Klein-Gordon equation",
"Relativistic superfluidity and vorticity from the nonlinear Klein-Gordon equation"
] | [
"Chi Xiong \nInstitute of Advanced Studies\nNanyang Technological University\n639673Singapore\n",
"Michael R R Good \nInstitute of Advanced Studies\nNanyang Technological University\n639673Singapore\n",
"Yulong Guo \nSchool of Computer Science and Engineering\nSouth China University of Technology\n510641GuangzhouChina\n",
"Xiaopei Liu \nSchool of Computer Engineering\nNanyang Technological University\n639673Singapore\n",
"Kerson Huang \nInstitute of Advanced Studies\nNanyang Technological University\n639673Singapore\n\nPhysics Department\nMassachusetts Institute of Technology\n02139CambridgeMAUSA\n"
] | [
"Institute of Advanced Studies\nNanyang Technological University\n639673Singapore",
"Institute of Advanced Studies\nNanyang Technological University\n639673Singapore",
"School of Computer Science and Engineering\nSouth China University of Technology\n510641GuangzhouChina",
"School of Computer Engineering\nNanyang Technological University\n639673Singapore",
"Institute of Advanced Studies\nNanyang Technological University\n639673Singapore",
"Physics Department\nMassachusetts Institute of Technology\n02139CambridgeMAUSA"
] | [] | We investigate superfluidity, and the mechanism for creation of quantized vortices, in the relativistic regime. The general framework is a nonlinear Klein-Gordon equation in curved spacetime for a complex scalar field, whose phase dynamics gives rise to superfluidity. The mechanisms discussed are local inertial forces (Coriolis and centrifugal), and current-current interaction with an external source. The primary application is to cosmology, but we also discuss the reduction to the non-relativistic nonlinear Schrödinger equation, which is widely used in describing superfluidity and vorticity in liquid helium and cold-trapped atomic gases. | 10.1103/physrevd.90.125019 | [
"https://arxiv.org/pdf/1408.0779v2.pdf"
] | 5,195,244 | 1408.0779 | 3ac1605e1cb60ca24f52ed17c90bf0d5a19b1be5 |
Relativistic superfluidity and vorticity from the nonlinear Klein-Gordon equation
Chi Xiong
Institute of Advanced Studies
Nanyang Technological University
639673Singapore
Michael R R Good
Institute of Advanced Studies
Nanyang Technological University
639673Singapore
Yulong Guo
School of Computer Science and Engineering
South China University of Technology
510641GuangzhouChina
Xiaopei Liu
School of Computer Engineering
Nanyang Technological University
639673Singapore
Kerson Huang
Institute of Advanced Studies
Nanyang Technological University
639673Singapore
Physics Department
Massachusetts Institute of Technology
02139CambridgeMAUSA
Relativistic superfluidity and vorticity from the nonlinear Klein-Gordon equation
relativistic superfluidityquantized vorticitynonlinear Klein-Gordon equation PACS number: 0462+v0375Kk0375Lm
We investigate superfluidity, and the mechanism for creation of quantized vortices, in the relativistic regime. The general framework is a nonlinear Klein-Gordon equation in curved spacetime for a complex scalar field, whose phase dynamics gives rise to superfluidity. The mechanisms discussed are local inertial forces (Coriolis and centrifugal), and current-current interaction with an external source. The primary application is to cosmology, but we also discuss the reduction to the non-relativistic nonlinear Schrödinger equation, which is widely used in describing superfluidity and vorticity in liquid helium and cold-trapped atomic gases.
I. INTRODUCTION
In quantum mechanics, a system of particles is describable by a complex wave function, which has a modulus as well as a phase, and the existence of the quantum phase is an essential distinction between quantum mechanics and classical mechanics. Macroscopic phase coherence (correlation of the quantum phase over macroscopic distances) gives rise to superfluidity and occurs in Bose-Einstein condensates (BEC) in diverse systems:
• Liquid 4 He at low temperatures,
• Cold trapped atomic gases,
• Central region in relativistic heavy-ion collisions,
• The Higgs or Higgs-like fields over cosmological scales,
• Superconducting metals,
• Liquid 3 He at low temperatures,
• Interior of neutron stars.
The last three refer to superconductivity, which can be viewed as the superfluidity of condensed fermion pairs. Ginzburg and Landau [1] propose a general phenomenological theory that describes the phase coherence in terms of a complex scalar field, which is viewed as an order parameter that emerges in a phase transition below a critical temperature. This phase transition is associated with spontaneous breaking of global gauge symmetry, i.e., the invariance of the wave function under a constant change of phase. We adopt such an approach here, and, in view of cosmological applications, begin with a complex scalar field in curved spacetime, and then consider the flat-spacetime and non-relativistic limits. Our treatment is confined to the neighborhood of the absolute zero of temperature.
We denote the complex scalar field by
Φ = Fe iσ ,(1)
which satisfies a nonlinear Klein-Gordon equation (NLKG) of the form Φ + f (Φ) = 0, where the d'Alembertian operator is the generalization of ∂ µ ∂ µ = ∇ 2 − 1 c 2 ∂ 2 ∂t 2 to curvilinear coordinate frames, and f is a nonlinear function, which contains a potential that has a minimum at Φ 0. Thus, there exists a nonzero vacuum field, which breaks global gauge symmetry spontaneously. The superfluid velocity is proportional to the spatial gradient of the phase ∇σ, which has quantized circulation:
C ∇σ · ds =2πn (n = 0, ±1, ±2, . . .) ,(2)
The integral extends over a spatial closed loop C, and the quantization is a consequence of the fact that Φ, and hence the phase factor e iσ , must be a continuous function in space. The contour C encircles a vortex line, which may meander in space but must end on itself, forming a closed loop, or terminate on boundaries. The modulus F must vanish on the vortex line, where it approaches zero continuously, over a characteristic distance, the healing length, as illustrated in Fig.1. Thus the vortex length is in actuality a vortex filament with a finite effective radius. The superfluid velocity v s is a hydrodynamic quantity that obeys certain conservation laws. As discussed below, it has the form v s c = ξ s ∇σ,
where ξ s is a correlation length that is generally spacetime dependent, with ξ s → /mc in the non-relativistic limit, where m is a mass scale. The fact that the modulus F must vanish at arXiv:1408.0779v1 [hep-th] 4 Aug 2014 the vortex center effectively makes the space non-simply connected, i.e., there are closed circuits that cannot be shrunken to zero continuously. This is why we can have ∇ × v s 0, even though it is proportional to a gradient (at least nonrelativistically).
In the large-scale motion of a quantum fluid, as in a classical medium such as the atmosphere or the ocean, vorticity is ubiquitous, being induced through different means in different systems. In liquid helium in the laboratory, they can be created through rotation of the container, or through local heat perturbations. The latter can create quantum turbulence in the form of a vortex tangle. In the cosmos, quantized vortices in the background superfluid can be created by a rotating black hole, a rotating galaxy, or colliding galaxies. The big bang era could witness the creation of quantum turbulence. Our main theme is that all these effects can be traced to a universal mechanism characterized by operator terms in the NLKG that can be associated with the Coriolis and the centrifugal force:
R Coriolis = 2Ω c 2 ∂ 2 ∂φ∂t , R centrifugal = − Ω 2 c 2 ∂ 2 ∂φ 2 ,(4)
where Ω is an angular velocity, and φ the angle of rotation. The angular velocity may be an externally given constant, or it may be a spacetime function arising from dynamics, interaction with external sources, or spacetime geometry. The various threads of our investigation may be summarized by the chart in Fig.2. We begin in Sec.2 with a general formulation of the NLKG in general relativity, and define the superfluid velocity field v s in terms of the phase of the field. This is followed in Sec.3 by a reduction to flat Minkowski spacetime. In Sec.4, we then consider the NLKG in a global rotating frame for orientation, and exhibit the Coriolis force and centripetal force (the top link in Fig.2). This enables us to extract local Coriolis and centripetal terms from the curvedspace equation, by expansion in the local angular velocity, (the left to right link in Fig.2). The local rotating frames associated with the Coriolis force leads to frame-dragging. We then make a non-relativistic reduction in Sec.5, recovering the nonlinear Schrödinger equation, particularly the form used to describe vortex generation in rotating BEC (the link to the bottom square in Fig.2). In Sec.6, we discuss the introduction of an external source via current-current coupling, and note how it can have the same effect as the Coriolis force (the right link in Fig.2). Finally, in Sec.7, we show some results from numerical computations.
II. SUPERFLUIDITY IN CURVED SPACETIME
We start with the NLKG in a general background metric g µν (µ, ν = 0, 1, 2, 3). The action of the complex scalar field is given by where g = det g µν , and V denotes a self-interaction potential that depends only on Φ * Φ. The equation of motion is
S = − dt d 3 x √ −g g µν ∂ µ Φ * ∂ ν Φ + V ,(5)− V Φ = 0,(6)
where
V = dV/d (Φ * Φ), and Φ ≡ ∇ µ ∇ µ Φ = 1 √ −g ∂ µ ( √ −gg µν ∂ ν Φ).(7)
Here, ∇ µ is the covariant derivative defined by
∇ µ ϕ = ∂ µ ϕ, ∇ ν A µ = ∂ ν A µ + Γ µ λν A λ (8)
where ϕ is a scalar, A µ a 4-vector, and Γ µ λν is the affine connection.
In the phase representation Φ = Fe iσ , the real and imaginary parts of (6) lead to the equations:
− V F − F∇ µ σ∇ µ σ = 0, 2∇ µ F∇ µ σ + F∇ µ ∇ µ σ = 0.(9)
The first equation can be rewritten in the form of a relativistic Euler equation
∇ µ F −1 F − V − 2F −2 ∇ λ (F 2 ∇ λ σ ∇ µ σ) = 0.(10)
The second equation is a continuity equation
∇ µ j µ = 0,(11)
with
j µ ≡ F 2 ∇ µ σ = F 2 ∂ µ σ.(12)
There exists a conserved charge
Q = d 3 x √ −g j 0 ,(13)
which satisfies dQ/dt = 0. This follows from the relation
√ −g∇ µ J µ = ∂ µ ( √ −gJ µ ), which can be shown using Γ λ λµ = (−g) −1/2 ∂ µ (−g) 1/2 .
The energy-momentum tensor is defined as
T µν = −2 √ −g δS δg µν ,(14)
leading to
T µν = ∇ µ Φ * ∇ ν Φ + ∇ ν Φ * ∇ µ Φ − g µν ∇ τ Φ * ∇ τ Φ − g µν V,(15)
which is conserved covariantly:
∇ µ T µν = 0.(16)
We turn to the definition of the superfluid velocity v s . The field Φ (x, t) corresponds to an order parameter at absolute zero, and thus represents a pure superfluid. Let τ be the proper time along a timelike worldline of a fluid element whose coordinates are
x µ = c t(τ), x i (τ) ,(17)
where i = 1, 2, 3. The superfluid velocity is the 3-velocity of the fluid element:
v s = dx dt .(18)
We define a 4-velocity
U µ ≡ dx µ dτ = γ (c, v s ) ,(19)
where γ ≡ dt/dτ. From ds 2 = −c 2 dτ 2 = g µν dx µ dx ν , we obtain
γ = −g 00 − 2g 0i v i c − g i j v i v j c 2 −1/2 .(20)
The superfluid density ρ s is defined through
j µ = ρ s U µ .(21)
Comparison with (12) leads to
ρ s = (cγξ s ) −1 F 2 , v s c = ξ s ∇σ (22) where ξ s = ∂ 0 σ −1 .(23)
III. REDUCTION TO MINKOWSKI SPACETIME
In Minkowski metric (−1, 1, 1, 1), the Euler equation and the continuity equation become
∂ µ F −1 F − V − 2F −2 ∂ λ (F 2 ∂ λ σ ∂ µ σ) = 0, ∂ µ (F 2 ∂ µ σ) = 0.(24)
The 4-velocity is given by
U µ = (γc, γv s ) , γ = 1 1 − v 2 s /c 2 .(25)
One can check U µ U µ = −c 2 , which guarantees v s < c. The superfluid density and velocity are given by
ρ s = − σ c 2 γ F 2 , v s c = − ċ σ ∇σ.(26)
In a stationary solution of the form Φ (x, t) = e −iωt χ (x) , we haveσ = −ω. Eq. (26) was first derived in Ref. [6].
IV. SUPERFLUID ROTATION
A. Rotating frame
A superfluid can flow frictionlessly past a wall at low velocities; dissipations can occur only when the velocity exceeds a critical value necessary to excite the system. Similarly, a superfluid contained in a rotating bucket will remain at rest, until the angular frequency of the bucket exceeds a critical value, whereupon a rotational velocity field occurs through the creation of quantized vortices, with vortex lines parallel to the axis of rotation. In a steady state, these vortices form a lattice of specific form. Theoretical treatment of this problem is best done in a coordinate frame rotating with the bucket, for it easily exposes the inertial forces, i.e., the Coriolis force and the centrifugal force, that are responsible for vortex creation. The actual superfluid we are studying may not be contained in a bucket, and we may not be in a rotating frame, but the mechanism for vortex creation may be attributed to a local version of these forces.
Consider a rotating frame with angular velocity Ω 0 about the z axis. In spatial spherical coordinates, the lab frame (t , r , θ , φ ) and the rotating frame (t, r, θ, φ) are related by the transformation
t = t , r = r , θ = θ , φ = φ − Ω 0 t .(27)
The d'Alembertian operator in the rotating frame in flat spacetime is given by
= (0) + R Coriolis + R centrifugal ,(28)where (0) = ∇ 2 − 1 c 2 ∂ 2 ∂t 2 , and R Coriolis = 2Ω 0 c 2 ∂ 2 ∂t∂φ , R centrifugal = − Ω 2 0 c 2 ∂ 2 ∂φ 2 .(29)
Eq. (28) can be applied to the study of any Klein-Gordon equations in the rotation frame with interactions, e.g. a selfinteraction V which gives the NLKG. Note that the coordinate transformation (27) is restricted by the condition Ω 0 R c where R is the radius of system. A general treatment will be given in subsection C.
B. Feynman's relation
Consider N vortices in rotating bucket of radius R and angular velocity Ω. At the wall of the bucket, the superfluid velocity is v s = ΩR. Thus, ∇σ = v s / (cξ s ) = ΩR/ (cξ s ) . From the relation ds · ∇σ = 2πN, we obtain
Ω = πcξ s n v ,(30)
where n v = N/ πR 2 is the number of vortices per unit area. This formulas can give an estimate of the local vortex density when the superfluid flows with varying local angular velocity. In non-relativistic limit ξ s → /mc, we have Ω = (π /m) n v , which is called Feynman's relation.
C. Frame-dragging from spacetime geometry
The transformation to a rotating coordinate frame is equivalent to using the metric in the following line element:
ds 2 0 = −c 2 dt 2 + dr 2 + r 2 sin 2 θdθ 2 + r 2 (dφ + Ω 0 dt) 2 .(31)
The cross term 2Ω 0 r 2 dφdt corresponds to a rotating frame of angular velocity Ω 0 . A spacetime-dependent Ω 0 leads to frame-dragging, i.e., local rotating frames. We give a general treatment of this effect in the following. Consider a stationary axially-symmetric background metric with two Killing vectors ξ a = (∂/∂t) a and ψ a = (∂/∂φ) a , which characterize the stationary condition and the rotational symmetry, respectively. The metric can be parametrized with coordinates {t, u, v, φ} as follows [3]:
ds 2 = −Adt 2 + g uu du 2 + 2g uv dudv + g vv dv 2 + 2Bdφdt + Cdφ 2 ,(32)
where the functions A, B, C are related to the Killing vectors:
A = −g tt = −ξ a ξ a , B = g tφ = ξ a ψ a , C = g φφ = ψ a ψ a .(33)
Frame-dragging arises from the metric components g tφ = B,
g tφ = B AC + B 2 .(34)
In general we should solve the NLKG in the background metric (32) for rotation problems. Here we consider a small B approximation. By expanding the d'Alembertian operator in powers of B, we find a cross term identifiable with the Coriolis force:
2g tφ ∂ 2 ∂t∂φ = 2B AC ∂ 2 ∂t∂φ + O B 3 .(35)
Noting that Ω = −B/C is the coordinate angular velocity of locally non-rotating observers [3], we can write
R Coriolis = 2Ω g tt ∂ 2 ∂t∂φ .(36)
In a similar manner we obtain the centrifugal term
R centrifugal = Ω 2 g tt ∂ 2 ∂φ 2 .(37)
Examples of metrics with frame-dragging are the BTZ metric in 2+1 dimensional spacetime, and the Kerr metric in 3+1 dimensional spacetime, which describe the spacetime curvature around a black hole. Frame-dragging in these metrics arise from the angular velocity of the black hole. Vortex creation due to this purely geometric effect is investigated in [7].
V. NON-RELATIVISTIC LIMIT
A. From NLKG to NLSE
A solution to the NLKG generally contains frequencies of both signs. A large frequency ω 0 of one sign (say, positive) could develop, due to an initial field with large positive charge, or a large potential energy, and the system will approach the non-relativistic limit. We define a wave function Ψ through
Φ (x, t) = e −iω 0 t Ψ (x, t) .(38)
Introducing a mass scale m by putting ω 0 = mc 2 / , we find the equation for Ψ:
i ∂Ψ ∂t = − 2 2m ∇ 2 Ψ + UΨ + 2 2mc 2 ∂ 2 Ψ ∂t 2 ,(39)
where U = 2 /2m V − mc 2 /2 . In the limit c → ∞, we drop the term ∂ 2 Ψ/∂t 2 and obtain the nonlinear Schrödinger equation (NLSE). Let the phase of the non-relativistic wave function be denoted by β (x, t) :
Ψ = |Ψ|e iβ .(40)
The phase of the relativistic field Φ is thus given by
σ = −ω 0 t + β,(41)
Putting
u k ≡ ∂ k σ = ∂ k β,(42)
we obtain, from (24),
∂ t 1 −β/ω 0 F 2 u k + F 2 4ω 0 ∂ k ∂ t ln F 2 + c 2 ω 0 ∂ j (F 2 u j u k ) + ∂ k P = 0, ∂ t 1 −β/ω 0 F 2 + c 2 ω 0 ∂ j F 2 u j = 0,(43)
where the pressure P is defined by
P = 1 2 F 2 V − V − c 2 4ω 0 F 2 ∇ 2 ln F 2 .(44)
The terms in braces are due to self interaction, while the last term is the "quantum pressure" [2]. The non-relativistic superfluid density and velocity can be obtained from (26) by putting ω 0 = mc 2 / , and formally let c → ∞ :
ρ s = m|Ψ| 2 1 + β /mc 2 − v 2 s /c 2 + O c −4 , v s = m ∇β 1 + β /mc 2 + O c −4 .(45)
From (24) we obtain the non-relativistic hydrodynamic equations:
∂ρ s ∂t + ∇ · (ρ s v s ) = 0, ∂ ∂t + v s · ∇ v s = − ∇P ρ s − R.(46)
where R = (∇ · v s + v s · ∇ ln ρ s )v s vanishes for an incompressible, divergenceless fluid.
B. NLSE in rotating frame
We start with the NLKG in a rotating frame:
∇ 2 − 1 c 2 ∂ 2 ∂t 2 − V + R Coriolis + R centrifugal Φ = 0, V = mc 2 − 2mλ |Φ| 2 , R Coriolis = 2Ω c 2 ∂ 2 ∂t∂φ , R centrifugal = − Ω 2 c 2 ∂ 2 ∂φ 2(47)
where we use a specific form of the potential. Going to the non-relativistic limit yields
i ∂Ψ ∂t = − 2 2m ∇ 2 + i Ω ∂ ∂φ + λ |Ψ| 2 Ψ,(48)
which is the NLSE usually used to describe a rotating BEC [2]. The trapping potential can be included as well. The centrifugal term does not appear, because it is − 2 Ω 2 /c 2 ∂ 2 /∂φ 2 , and thus of order c −2 .
VI. CURRENT-CURRENT INTERACTION WITH A SOURCE
The complex scalar field Φ may be coupled to an external source. In Minkowski spacetime, the only Lorentz-invariant interaction in the Lagrangian density is the current-current interaction
L int = −ηJ µ j µ ,(49)
where j µ is the conserved current (12) of the complex scalar field, η a coupling constant, and J µ is the conserved current of the external source. A direct coupling is possible only if the scalar field has multi-components representing internal symmetry. We can write, more explicitly,
L int = −ηF 2 J µ ∂ µ σ,(50)
showing that it is a derivative coupling of the phase of the field. The NLKG now reads
− V Φ − iηJ µ ∂ µ Φ = 0.(51)
Ref. [6] uses an external current of the form
J µ = (ρ, J) , J = ρΩ × r,(52)
to simulate the presence of a galaxy in a cosmic superfluid.
Here, ρ(x) is the density of the galaxy, represented by a Gaussian distribution, and Ω is its angular velocity. In this case, the NLKG takes the form
− V Φ + iηρ ∂Φ ∂t + Ω × r · ∇Φ = 0.(53)
The last term reads, in a cylindrical coordinate system with azimuthal angle φ about the rotation axis,
iηρΩ × r · ∇Φ = iηρΩ ∂Φ ∂φ .(54)
Since ρΩ effectively gives a spatially dependent angular velocity, this term gives rise to frame-dragging due to interactions. We can compare it with a Coriolis term
Ω ∂ 2 Φ ∂t∂φ = −iωΩ ∂Φ ∂φ ,(55)
where we have put ∂Φ/∂t = −iωΦ for a stationary solution.
VII. NUMERICAL COMPUTATIONS
We solve the NLKG with various inertial or interaction terms in (2+1)-and (3+1)-dimensional Minkowski spacetime. Our emphasis here is on vortex states and vortex dynamics. For the effects of gravity and dark matter modeling, see Ref. [6].
The equations are solved by finite difference scheme and spectrum methods are used to decouple the discretization for derivatives ∂ t and ∂ φ in the Coriolis term. Semi-implicit scheme similar to the Crank-Nicolson method is used for the linear terms. The non-linear term is handled explicitly. We impose second order Neumann boundary condition in solving the NLKG in the rotating frame (47) and periodic boundary condition in studying the vortex rings lattice and vortex line reconnection with (53), respectively. Fig.3 shows in (2+1) dimensions the number of vortices in a rotating bucket as a function of the angular velocity and the field frequency. The computational data agrees quite well with the relativistic Feynman relation (30). As examples Fig.4 shows the modulus (left) and the phase (right) plots of the complex scalar field in stationary states with 8 and 63 vortices, respectively. The locations of vortices are indicated by the dots in the modulus plots, corresponding to positions in the phase plots around which the color changes from blue to red, indicating a phase change of 2π. The vortex distribution is mainly determined by the Coriolis term while some nonuniformity is caused by the centrifugal term.
In (3+1) dimensions, we introduce an external current of the form (52) to simulate a rotating "star" immersed in a cosmic superfluid and then solve (53). A vortex-ring solution is found and plotted in Fig.5. It shows that the stars drags the superfluid into local rotation, and there appear vortex rings surrounding the star. As pointed out in Ref. [5,6], the so-called "non-thermal filaments" observed near the center of the Milky Way may be parts of such vortex rings around rotating black holes. Vortex lines in 3D can cross and reconnect. The originally smooth vortex lines become lines with cusps at the reconnection point, and these cusps spring away from each other rapidly, creating two jets of energy in the superfluid. Through repeated reconnections, large vortex rings can be degraded until they become a vortex tangle of fractal dimensions (quantum turbulence). This mechanism has been used in [4,5] to create matter in the cosmos during the big bang era. We can simulate such a reconnection via the NLKG, as shown in Fig.6. Vortex reconnection in the NLSE has been studied in Ref. [8]. For actual photographs of vortex reconnection in superfluid helium, see Ref. [9].
VIII. CONCLUSION
We formulate a framework for investigating superfluidity and the mechanism for creating quantized vorticity in the relativistic regime, based on a nonlinear Klein-Gordon equation for a complex scalar field. This framework is constructed in curved spacetime, then is reduced to flat spacetime and also the non-relativistic limit. Our numerical study focuses on the flat-spacetime cases. It shows that quantized vorticity can be created by local inertial forces (Coriolis and centrifugal) and current-current interaction. Feynman's relation relating the number of vortices to the angular frequency is numerically verified in the relativistic regime. Our numerical solutions to the NLKG exhibit such features as vortex lattices, 3D vortex rings around rotating stars, and vortex reconnections.
FIG. 1 .
1Profile of the modulus of a complex scalar field with a quantized vortex, as obtained from a solution to the nonlinear Klein-Gordon equation. Here, r is the normal distance from the vortex line. The field goes to zero at the vortex, with a characterized healing length. FIG. 2. Chart showing various threads of our investigation. The general theme, shown by the central and the left square, is that the mechanisms for vortex creation are the inertia forces (Coriolis and centrifugal) in rotating frames, generated by external means or by the spacetime metric, or by external sources through a current-current interaction. See Introduction to follow the various threads.
FIG. 3 .
3The number N of vortices in a rotating bucket as function of the angular velocity Ω and the field frequency ω. The straight line represents the relativistic Feynman relation (30), which can be rewritten N = (R 2 /c 2 )ωΩ. Here, R is the radius of the bucket, and ω is the frequency of the the stationary solution to the NLKG.
FIG. 4 .
4Two lattice states with 8 and 63 vortices respectively: The modulus of the field is plotted on the left side and its phase is on the right side. The color map from blue to red (see web version) indicates a phase change of 2π.
FIG. 5 .
5System of 3D vortex rings surrounding a rotating "star", viewed from different perspectives.FIG. 6. Frames 1-4 show the progress of a vortex reconnection, at equal time steps. An initial configuration of a 3D vortex-line array is constructed with the current-current source term, which is then turned off to allow free evolution and reconnection of the vortex lines.
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| [] |
[
"ON NEAR-MARTINGALES AND A CLASS OF ANTICIPATING LINEAR SDES",
"ON NEAR-MARTINGALES AND A CLASS OF ANTICIPATING LINEAR SDES"
] | [
"Hui-Hsiung Kuo ",
"Pujan Shrestha ",
"ANDSudip Sinha ",
"Padmanabhan Sundar "
] | [] | [] | The primary goal of this paper is to prove a near-martingale optional stopping theorem and establish solvability and large deviations for a class of anticipating linear stochastic differential equations. We prove the existence and uniqueness of solutions using two approaches: (1) Ayed-Kuo differential formula using an ansatz, and (2) a novel braiding technique by interpreting the integral in the Skorokhod sense. We establish a Freidlin-Wentzell type large deviations result for solution of such equations. | null | [
"https://arxiv.org/pdf/2204.01932v1.pdf"
] | 247,958,320 | 2204.01932 | a520368ede8c8bcc1e3c0cbd3173226fec1d4da8 |
ON NEAR-MARTINGALES AND A CLASS OF ANTICIPATING LINEAR SDES
Hui-Hsiung Kuo
Pujan Shrestha
ANDSudip Sinha
Padmanabhan Sundar
ON NEAR-MARTINGALES AND A CLASS OF ANTICIPATING LINEAR SDES
arXiv:2204.01932v1 [math.PR] 5 Apr 2022
The primary goal of this paper is to prove a near-martingale optional stopping theorem and establish solvability and large deviations for a class of anticipating linear stochastic differential equations. We prove the existence and uniqueness of solutions using two approaches: (1) Ayed-Kuo differential formula using an ansatz, and (2) a novel braiding technique by interpreting the integral in the Skorokhod sense. We establish a Freidlin-Wentzell type large deviations result for solution of such equations.
Introduction
Anticipating stochastic calculus has been an active and important research area for several years, and lies at the intersection of probability theory and infinite-dimensional analysis. Enlargement of filtration, Malliavin calculus, and white noise theory provide three distinct methodologies to incorporate anticipation (of future) into classical Itô theory of stochastic integration and differential equations.
It is to the credit of Itô who constructed an anticipating stochastic integral in 1976 [6], and laid the foundation for the idea of enlargement of the underlying filtration. Ever since, the method was embraced by several researchers that led to many important works (see articles in [7]). The advent of an integral invented by Skorokhod resulted in an impressive edifice built by Malliavin on stochastic calculus of variations in order to prove Hörmander's hypoellipticity result by stochastic analysis. Malliavin calculus provided a natural basis for the development and study of anticipative stochastic analysis and differential equations. Around the same time, a systematic study of Hida distributions gave rise to white noise theory and a general framework for stochastic calculus.
Malliavin calculus and white noise theory have vast applicability to the theory of stochastic differential equations with anticipation. However, the results obtained by these theories are primarily abstract though general. A more tractable theory was envisaged by Kuo based on a concrete stochastic integral known as the Ayed-Kuo integral [1]. Under less generality, the latter allows one to obtain results under easily understood, verifiable hypotheses.
In this article, we prove some results about stopped near-martingales, which are generalizations of martingales. We then study existence, uniqueness and large deviation principle for linear stochastic differential equations with anticipating initial conditions and drifts. While we rely mostly on the Ayed-Kuo formalism, other theories are minimally used either out of necessity, or to compare and contrast the conclusions of certain results.
The structure of the paper is as follows. In section 2, we introduce the Ayed-Kuo integral. In section 3, study near-martingales. We show that Ayed-Kuo integrals are nearmartingales. We also show that stopped near-martingales are near-martingales, and prove an optional stopping theorem for near-submartingales. In section 4, we study methods for solving anticipating linear stochastic differential equations by interpreting the anticipating stochastic integral from two perspectives. For the Ayed-Kuo formulation, we use the differential formula and an ansatz to derive the solution. For the Skorokhod interpretation, we introduce a novel braiding technique inspired by Trotter's product formula [12]. We show that the solutions coincide when the assumptions are identical. Finally, in section 5, we derive large deviation principles for the solutions of the class of anticipating linear stochastic differential equations studied in section 4. In this paper, we assume t ∈ [0, 1], unless specified otherwise.
The Ayed-Kuo anticipating stochastic calculus
Before we define the Ayed-Kuo integral, we need to define instantly independent processes. A stochastic process φ(t) is called instantly independent with respect to {F t } if for each t ∈ [0, 1], the random variable φ(t) and the σ-field F t are independent. Instantly independent processes are the counterpart of adapted processes in this theory.
We refer to [4, section 2] for a detailed definition of the anticipating stochastic integral. In what follows, we highlight the crucial steps in the definition in a concise manner.
Definition 2.1 ([4, definition 2.3]).
The anticipating integral is defined in following three steps:
(1) Suppose f (t) is an F t -adapted continuous stochastic process and φ(t) is a continuous stochastic processes that is instantly independent with respect to {F t }. Then the stochastic integral of Φ(t) = f (t)φ(t) is defined by
1 0 f (t)φ(t) dW t = lim ∆n →0 n j=1 f (t j−1 ) φ(t j ) (W t j − W t j−1 ),
provided that the limit exists in probability. (2) For any stochastic process of the form
Φ(t) = n i=1 f i (t)φ i (t),
where f i (t) and φ i (t) are given as in step (1), the stochastic integral is defined by
1 0 Φ(t) dW t = n i=1 1 0 f i (t) φ i (t) dW t .
(3) Let Φ(t) be a stochastic process such that there is a sequence (Φ n (t)) ∞ n=1 of stochastic processes of the form in step (2) satisfying (a) 0 Φ n (t) dW t converges in probability as n → ∞. Then the stochastic integral of Φ(t) is defined by
1 0 Φ(t) dW t = lim n→∞ 1 0 Φ n (t) dW t in probability.
This integral is well defined, as demonstrated in [4, lemma 2.1]. In order to use the definition of the integral, we first need to decompose the integrand into a sum of products of adapted and instantly independent parts. The main idea is to then use the left-endpoints of subintervals to evaluate the adapted parts and the right-endpoints of subintervals to evaluate the instantly independent parts.
3. Near-martingales 3.1. Near-martingale property of the Ayed-Kuo integral. Martingales are an extremely important class of processes that are used to model fair games, and hence find applications not only in probability theory, but also in mathematical finance and numerous other fields. Itô's integrals are essentially continuous martingale transforms, and therefore retain the martingale nature of the integrator. Since the Ayed-Kuo integral is an extension of the Itô integral, it is natural to ask if Ayed-Kuo integrals are martingale. Unfortunately, they are not. However, we have a very similar property, which gives rise to the idea of near-martingales.
Definition 3.1. An integrable stochastic process N t is called a near-submartingale with respect to the filtration {F t } if for any s ≤ t, we have E(N t − N s | F s ) ≥ 0 almost surely. It is called a near-martingale if E(N t − N s | F s ) = 0 almost surely.
The following result links martingales and near-martingales. In particular, it says that conditioned near-martingales are martingales. Ayed-Kuo integrals are near-martingales, as stated by this theorem. Theorem 3.3. Let Θ(x, y) be a function that is continuous in both variables such that the stochastic integral,
N t = t a Θ(W s , W b − W s ) dW s , a ≤ t ≤ b,
exists and E| N t | < ∞ for each t in [0, 1]. Furthermore, assume that the family of partial sums
n i=1 Θ(W t i , W 1 − W t i−1 ) W t i − W t i−1 are uniformly integrable. Then N t , a ≤ t ≤ b,E [N t − N s | F s ] =E t s Θ(W v , W b − W v ) dW v | F s =E lim n→∞ n k=1 Θ(W k−1 , W b − W k )∆W k | F s = lim n→∞ n k=1 E [Θ(W k−1 , W b − W k )∆W k | F s ] . (3.1) Consider, H (b) a = σ(F a ∪ G (b) ). Then F s ⊆ F k−1 ⊆ H (k)
k−1 . Using this fact alongside the continuity of Θ in both variables, we have that Θ(
W k−1 , W b − W k ) is H (k) k−1 -measurable. Furthermore, via the independence of the Brownian increments, ∆W k is independent of H (k) k−1 . Thus, E [Θ(W k−1 , W b − W k )∆W k | F s ] = E E Θ(W k−1 , W b − W k )∆W k | H (k) k−1 | F s = E [Θ(W k−1 , W b − W k )E [∆W k ] | F s ] = 0.
Using this result for each k in equation (3.1), we have E [N t − N s | F s ] = 0, and so N t is a near-martingale. Figure 1. A t-dependence plot of the disjoint increments of W . The shaded regions represents the forward and separation σ-field.
H (t k ) t k−1 F s 0 s t k−1 t k t 1 B t k−1 B 1 − B t k B t k − B t k−1
3.2.
Stopped near-martingales. In this section, we show that stopped near-martingales are near-martingales. We also generalize Doob's optional stopping theorem for near-martingales.
Definition 3.4. Let (A n ) ∞
n=0 be an adapted process and (X n ) ∞ n=0 a discrete time nearsubmartingale. Then the processes (Y n ) ∞ n=0 , where Y 0 = 0 and
Y n = (A • X) n = n i=1 A n−1 (X n − X n−1 )
is called the near-martingale transform of X by A.
Near-martingale transforms retain the near-martingale property, as is shown in the following result.
Proposition 3.5.
(1) If X is a near-submartingale and A is a bounded non-negative adapted process, then (A • X) is a near-submartingale.
(2) If X is a near-martingale and A is a bounded adapted process, then (A • X) is a near-martingale. (3) If X and A are both square integrable, then we do not require the boundedness condition in items 1 and 2.
Proof. We only prove item 1 because the rest follow the same process. Let X be a nearsubmartingale and Y = (A • X). Suppose n is an arbitrary time. Note that Y n − Y n−1 = A n−1 (X n − X n−1 ), which is integrable since A is bounded. Using the adaptedness of A, we get
E(Y n − Y n−1 | F n−1 ) = E(A n−1 (X n − X n−1 ) | F n−1 ) = A n−1 E(X n − X n−1 | F n−1 ) ≥ 0,
where the last inequality holds since A is non-negative.
We now show that stopped near-submartingales are near-submartingales.
Theorem 3.6. Suppose X is a discrete time near-submartingale and τ a stopping time.
Then the stopped process X τ defined by X τ n = X τ ∧n is a (discrete time ) near-submartingale.
Proof. Let A n = ½ {n≤τ } , so the process A is bounded, non-negative, and adapted. Now, note
that X τ n − X 0 = X τ ∧n − X 0 = (A • X) n .
Therefore, by proposition 3.5, we get that X τ is a near-submartingale. Now, we show the equivalent result of Doob's optional stopping theorem for discrete time near-submartingales.
Theorem 3.7. Let X be a discrete time near-submartingale. Suppose σ and τ are two bounded stopping times with σ ≤ τ . Then X σ and X τ are integrable, and E(X τ − X σ | F σ ) ≥ 0 almost surely.
Proof. Since σ and τ are bounded, there exists N < ∞ such that σ ≤ τ ≤ N. Let Y be any near-submartingale. Clearly, Y σ is integrable. Suppose B ∈ F σ . Then for any n ≤ N, we have B ∩ {σ = n} ∈ F n , and so
B∩{σ=n} (Y N − Y σ ) d P = B∩{σ=n} (Y N − Y n ) d P ≥ 0. Summing over n, we get B (Y N − Y σ ) d P ≥ 0, and so E(Y N − Y σ | F σ ) ≥ 0. Finally, let Y n = X τ n to get E(X τ N − X τ σ | F σ ) = E(X τ − X σ | F σ ) ≥ 0.
We need the following definition and lemma to prove the result in continuous time.
Definition 3.8. Let (F n ) ∞ n=1 be a decreasing sequence of σ-algebras, and let X = (X n ) ∞ n=1 be a stochastic process. Then the pair (X n , F n ) ∞ n=1 is called a backward near-submartingale if for every n,
(1) X n is integrable and F n -measurable, and
(2) E(X n − X n+1 | F n+1 ) ≥ 0. Lemma 3.9. Let (X n , F n ) ∞ n=1 be a backward near-submartingale with lim n→∞ E(X n ) > −∞.
If X is non-negative for every n, then X is uniformly integrable.
Proof. As n ր ∞, we have E(X n ) ց lim n→∞ E(X n ) = inf n E(X n ) > −∞. Fix ǫ > 0.
By the definition of infimum, there exists a K > 0 such that for any n ≥ K, we have
E(X K ) − lim n→∞ E(X n ) < ǫ.
For any k ≥ n and δ > 0, we have
E |X k | ½ {|X k |>δ} = E X k ½ {X k >δ} + E X k ½ {X k ≥−δ} − E(X k ) . Moreover, since X is a backward near-submartingale, E X k ½ {X k ≥δ} ≤ E X n ½ {X k ≥δ} . There- fore, E |X k | ½ {|X k |>δ} ≤ E X n ½ {X k >δ} + E X n ½ {X k ≥−δ} − (E(X n ) − ǫ) ≤ E |X n | ½ {|X k |>δ} + ǫ.
By Markov's inequality and the non-negativity of X,
P{|X k | > δ} ≤ 1 δ E|X k | = 1 δ E(X k ) ≤ 1 δ E(X 1 ) → 0 as δ → ∞.
This concludes the proof.
We are now ready to prove the near-martingale optional stopping theorem in continuous time.
Theorem 3.10. Let N be a near-submartingale with right-continuous sample paths. Suppose σ and τ are two bounded stopping times with σ ≤ τ . If N is either non-negative or uniformly integrable, then N σ and N τ are integrable, and
E(N τ − N σ | F σ ) ≥ 0 almost surely.
Proof. We use a discretization argument to prove the result. Let T > 0 be a bound for τ . For every n ∈ N, define the discretization function
f n : [0, ∞) → k n : k = 0, . . . , n : x → ⌊2 n x⌋ + 1 2 n ∧ T,(3.2)
and let σ n = f n (σ) and τ n = f n (τ ). Now, for any n and t,
{τ n ≤ t} = {f n (τ ) ∈ [0, t]} = τ ∈ f −1 n [0, t] = τ ∈ f −1 n 0, ⌊2 n t⌋ 2 n ∈ F ⌊2 n t⌋ 2 n ⊆ F t ,
so τ n is a stopping time. Similarly, σ n is a stopping time. Moreover, it can be easily seen that σ n ≤ τ n for every n, and σ n ց σ and τ n ց τ as n ր ∞. Therefore, by theorem 3.7, we get N σn and N τn are integrable, and E(N τn − N σn | F σn ) ≥ 0 almost surely. Furthermore, it is easy to see that F σ = ∞ n=1 F σn ⊆ F σn for any n. Therefore, E(N τn − N σn | F σ ) ≥ 0 almost surely for any n.
If N is non-negative, by construction, (N σn , F σn ) ∞ n=1 is a backward near-submartingale such that N σn ≥ 0 for every n. Therefore, E(N σn ) ց E(N σ ) > −∞ as n ր ∞. Using lemma 3.9, (N σn ) ∞ n=1 is uniformly integrable. Similarly, (N τn ) ∞ n=1 is also uniformly integrable. On the other hand, if N is uniformly integrable, this is trivial.
Using the right continuity of N and the boundedness assumption of σ and τ , we get lim n→∞ N σn = N σ and lim n→∞ N τn = N τ almost surely. Furthermore, the uniform integrability of (N σn ) ∞ n=1 and (N τn ) ∞ n=1 allows us to conclude that N σ and N τ are integrable and that the convergence is also in L 1 , giving us E(N τ − N σ | F σ ) ≥ 0 almost surely.
We highlight the special case of theorem 3.10.
Corollary 3.11. Let N be a non-negative near-martingale with right-continuous sample paths and τ a bounded stopping time. Then N τ is integrable, and E(N τ ) = E(N 0 ) almost surely.
Anticipating linear stochastic differential equations
In our previous works [9,8], we studied linear stochastic differential equations with anticipating initial conditions, where the stochastic integral is in the Ayed-Kuo sense. In [3], the authors gave examples of linear stochastic differential equations with anticipating diffusion coefficient. In this paper, we focus on anticipating drift.
In particular, we shall be concerned about the solution of a class of anticipating linear stochastic differential equations of the form
dX t = σ t X t dW t + f 1 0 γ t dW t X t dt, t ∈ [0, 1], X 0 = ξ, (4.1)
where W t is a Brownian motion, f : R → R is bounded function, ξ a random variable, and σ t is an bounded adapted process such that all integrability conditions are satisfied. We choose this class because we want to study linear stochastic differential equations where the anticipation comes from the drift coefficient being Brownian functionals.
4.1. The Ayed-Kuo sense. We look at an extension of Itô's formula that can account for instantly independent processes. Let X t and Y (t) be stochastic processes of the form
X t = X a + t a g(s) dB(s) + t a h(s) ds, (4.2) Y (t) = Y (b) + b t ξ(s) dB(s) + b t η(s) ds, (4.3)
where g(t), h(t) are adapted (so X t is an Itô process), and ξ(t), η(t) are instantly independent such that Y (t) is also instantly independent.
Theorem 4.1 ([4, theorem 3.2]). Suppose {X (i) t } n i=1 and {Y (t)
j } m j=1 are stochastic processes of the form given by equations (4.2) and (4.3), respectively. Suppose θ(t, x 1 , . . . , x n , y 1 , . . . , y m ) is a real-valued function that is C 1 in t and C 2 in other variables. Then the stochastic differential of θ(t, X
(1) t , . . . , X (n) t , Y (t) 1 , . . . , Y (t) m ) is given by dθ(t, X (1) t , . . . , X (n) t , Y (t) 1 , . . . , Y (t) m ) = θ t dt + n i=1 θ x i dX (i) t + m j=1 θ y j dY (t) j + 1 2 n i,k=1 θ x i x k dX (i) t dX (k) t − 1 2 m j,l=1 θ y j y l dY (t) j dY (t)
l .
The above differential formula allows us to calculate the solutions of anticipating stochastic differential equations. We shall see two instances of its application in section 4.1.
We apply the differential formula to derive a general result for existence of linear stochastic differential equations with anticipating coefficients. 1], and ξ be a random variable independent of the Wiener process W . Moreover, suppose f ∈ C 2 (R) along with f, f ′ , f ′′ ∈ L 1 (R). Then the solution of equation (4.1) in the Ayed-Kuo theory is given by Motivated by this, we define
Theorem 4.2. Suppose σ ∈ L 2 ad ([0, 1] × Ω), γ ∈ L 2 [0,Z t = ξ exp t 0 σ s dW s − 1 2 t 0 σ 2 s ds + t 0 f 1 0 γ u dW u − t s γ u σ u du ds .θ(t, x 1 , x 2 , y) = ξ exp x 1 − 1 2 t 0 σ 2 s ds + t 0 f x 2 + y − t s γ u σ u du ds .
Moreover, let
X (1) t = t 0 σ s dW s (so dX (1) t = σ t dW t ), X (2) t = t 0 γ s dW s (so dX (2) t = γ t dW t ),
and
Y (t) = 1 t γ s dW s (so dY (t) = −γ t dW t ).
Then we can write Z t = θ t, X
t , X
t , Y (t) . For conciseness, we denote F = f 1 0 γ t dW t − t s γ u σ u du , and similarly the derivatives
F ′ = f ′ 1 0 γ t dW t − t s γ u σ u du and F ′′ = f ′′ 1 0 γ t dW t − t s γ u σ u du .
Note that for the 8 derivatives of θ, we have
θ x 1 = θ x 1 x 1 = θ, θ x 2 = θ x 1 x 2 = θ y = θ · t 0 F ′ ds, θ x 2 x 2 = θ yy = θ · t 0 F ′ ds 2 + θ · t 0 F ′′ ds, and θ t = − 1 2 θσ 2 t + θf (x 2 + y) − γ t σ t θ y ,
where we used the Leibniz integral rule and the second line for the last identity.
Since ξ is independent of the Wiener process, by theorem 4.1, we get
dθ = θ t dt + θ x 1 dX (1) t + θ x 2 dX (2) t + θ y dY (t) + 1 2 θ x 1 x 1 dX (1) t 2 + 1 2 θ x 2 x 2 dX (2) t 2 + θ x 1 x 2 dX (1) t dX (2) t − 1 2 θ yy ( dY (t) ) 2 .
Using the relationships between the derivatives of θ and its differential form, we have
dθ = θ t dt + θσ t dW t + ❳ ❳ ❳ ❳ ❳ θ y γ t dW t − ❳ ❳ ❳ ❳ ❳ θγ t dW t + 1 2 θσ 2 t dt + ✟ ✟ ✟ ✟ ✟ 1 2 θ yy γ 2 t dt + θ y γ t σ t dt − ✟ ✟ ✟ ✟ ✟ 1 2 θ yy γ 2 t dt = θ t + 1 2 θσ 2 t + θ y γ t σ t dt + θσ t dW t . Now, θ t + 1 2 θσ 2 t + θ y γ t σ t = − ❅ ❅ ❅ 1 2 θσ 2 t + θf (x 2 + y) − ✘ ✘ ✘ ✘ θ y γ t σ t + ❅ ❅ ❅ 1 2 θσ 2 t + ✘ ✘ ✘ ✘ θ y γ t σ t = θf (x 2 + y),
and so dθ = f (x 2 + y)θ dt + σ t θ dW t .
Since Z t = θ t, X
t , X
t , Y (t) , we get
dZ t = f 1 0 γ s dW s Z t dt + σ t Z t dW t ,
which is exactly equation (4.1).
Theorem 4.1 is an indispensable tool for analyzing anticipating processes. We show another example by finding the stochastic differential equation corresponding to the square of the above solution.
dV t = σ 2 t + f 1 0 γ s dW s + 2γ t σ t t 0 f ′ 1 0 γ u dW u − t s γ u σ u du ds V t dt + 2σ t V t dW t , V 0 = ξ 2 is solved by Z 2 t ,
where Z is given by equation (4.4).
Remark 4.4. An interesting feature is that the derivative of f appears in the stochastic differential equation.
Proof. We follow the exact same strategy as the proof of theorem 4.2. The initial condition is trivially true. Let V t = Z 2 t . Taking the square of both sides of equation (4.4), we get
V t = ξ 2 exp t 0 2σ s dW s − t 0 σ 2 s ds + t 0 2f 1 0 γ u dW u − t s γ u σ u du ds We have V t = θ t, X (1) t , X (2) t , Y (t) , where θ(t, x 1 , x 2 , y) = ξ 2 exp x 1 − t 0 σ 2 s ds + t 0 2f x 2 + y − t s γ u σ u du ds , and X (1) t = t 0 2σ s dW s (so dX (1) t = 2σ t dW t ), X (2) t = t 0 γ s dW s (so dX (2) t = γ t dW t ),
and
Y (t) = 1 t γ s dW s (so dY (t) = −γ t dW t ).
As before, writing
F = f 1 0 γ t dW t − t s γ u σ u du , F ′ = f ′ 1 0 γ t dW t − t s γ u σ u du , and F ′′ = f ′′ 1 0 γ t dW t − t s γ u σ u du , we get θ x 1 = θ x 1 x 1 = θ, θ x 2 = θ x 1 x 2 = θ y = 2θ · t 0 F ′ ds, θ x 2 x 2 = θ yy = θ · t 0 F ′ ds 2 + θ · t 0
F ′′ ds, and θ t = −θσ 2 t + 2θf (x 2 + y) − γ t σ t θ y . Using the general Itô formula (theorem 4.1), we get
dθ =θ t dt + θ x 1 dX (1) t + θ x 2 dX (2) t + θ y dY (t) + 1 2 θ x 1 x 1 dX (1) t 2 + 1 2 θ x 2 x 2 dX (2) t 2 + θ x 1 x 2 dX (1) t dX (2) t − 1 2 θ yy ( dY (t) ) 2 =θ t dt + 2θσ t dW t + ❳ ❳ ❳ ❳ ❳ θ x 2 γ t dW t − ❳ ❳ ❳ ❳ ❳ θγ t dW t + 2θσ 2 t dt + ✟ ✟ ✟ ✟ ✟ ✟ 1 2 θ x 2 x 2 γ 2 t dt + 2θ y γ t σ t dt − ✟ ✟ ✟ ✟ ✟ 1 2 θ yy γ 2 t dt = θ t + 2θσ 2 t + 2θ y γ t σ t dt + 2θσ t dW t = θσ t + 2θf (x 2 + y) + 2γ t σ t θ t 0 F ′ ds dt + 2θσ t dW t . 10
Finally, using V t = θ t, X
t , X (2) t , Y (t) , we get the stochastic differential equation.
4.2.
A novel braiding technique for the Skorokhod sense. In the prior section, we showed the existence of the solution via the Ayed-Kuo differential formula. However, the procedure started with intelligently guessing an ansatz for the solution and applying the differential formula to it. Can a solution be found without this "guessing"? In this section, we use elementary ideas from Malliavin calculus to interpret the stochastic differential equation in the Skorokhod sense. We introduce an iterative "braiding" technique in the spirit of Trotter's product formula [12] that allows us to construct the solution without needing to know the form of the solution. Note that we expect to arrive at the same solution as in section 4.1 since under the definition of the Ayed-Kuo integral using L 2 (Ω) convergence, the Hitsuda-Skorokhod integral and the Ayed-Kuo integrals are equivalent, as shown in [11, theorem 2.3]. In what follows, we briefly introduce some ideas of Malliavin calculus and Skorokhod integral so that we can introduce our braiding technique. A well known extension of the Itô integral is Hitsuda-Skorokhod integral. For this text, we shall introduce the Hitsuda-Skorokhod integral as the adjoint of the Gross-Malliavin derivative. Let us first set up the spaces to operate on. Consider the probability space (Ω, F , P ) where F is the σ-field generated by the Brownian motion. Let H = L 2 [0, 1] be the space of square integrable functions defined on the positive reals. For any h ∈ H, consider the Wiener integral
W (h) = 1 0 h(t) dW t . In particular, if h = ½ [0, 1 2 ] ∈ H then W ½ [0, 1 2 ] = 1 0 ½ [0, 1 2 ] (t) dW t = W1 2 .
This Hilbert space H plays an important role in the definition of the derivative. Let S be the class of smooth random variables such that F ∈ S has the form
F = f (W (h 1 ), W (h 2 ) . . . , W (h n )) , h i ∈ H, i ∈ {1, 2, . . . , n},
where f is a real valued n-dimensional smooth function whose derivatives have at most polynomial growth.
D t F = n i=1 ∂ i f (W (h 1 ), W (h 2 ) . . . , W (h n )) h i (t),
where d i is the derivative with respect to the ith variable.
We denote D 1,2 as the closure of the derivative operator D from L 2 (Ω) to L 2 (Ω; H). In other words, D 1,2 is the completion of the class of smooth Brownian functionals with respect to the inner product
F, G 1,2 = E (F G) + E ( DF, DG H ) .
We now introduce the Skorokhod integral operator δ. (1) The domain of δ is the set of H-valued square integrable random variables u ∈ L 2 (Ω; H) such that for any F ∈ D 1,2 , where c is some constant depending on u.
E( DF, u H ) ≤ c F 2 .
(2) If u belongs to the domain of δ, then δ(u) is the element of L 2 (Ω) characterized by
E(F δu) = E( DF, u H ) .
for any F ∈ D 1,2 .
It is natural to ask about the nature of relationship of these two stochastic integral. While that is an open question, we refer to the following result. ). Let f be an adapted L 2 -continuous stochastic process and φ be an instantly independent L 2 -continuous stochastic process such that the sequence
n i=1 f (t i−1 )φ(t i ) W t i − W t i−1 ,
converges strongly in L 2 (Ω) as ∆ n → 0. Then the limit I(f ψ) equals the Hitsuda-Skorokhod integral δ(f ψ) in Dom(δ). Now, we move on to finding the solution of the linear stochastic differential equation when the anticipating integral is taken in the sense of Skorokhod. First, fix the family of translation on the space of continuous functions starting at the origin in the Cameron-Martin direction given by
(A t (ω)) s = ω s − t∧s 0 σ(u) du and (T t (ω)) s = ω s + t∧s 0 σ(u) du.
We look at an existence result for stochastic differential equations in the Skorokhod sense.
Lemma 4.8. Suppose σ ∈ L 2 [0, 1] and ξ ∈ L p (Ω) for some p > 2. Then the stochastic differential equation
dZ t = σ(t) Z t dW t Z 0 = ξ, (4.5)
has the unique solution given by
Z t = (ξ • A t ) E t . (4.6)
Proof. It is clear that the family {(ξ • A t )E t | t ∈ [0, 1]} is L r (Ω)-bounded for all r < p by Girsanov's theorem and Hölder's inequality. Let G be any smooth random variable. Multiply both sides of equation (4.5) by G. With the process X given by (4.6),
E G t 0 σ(s) Z s dW s = E t 0 σ(s) Z s D s G ds = E ξ t 0 σ(s) (D s G)(T s ) ds (using Girsanov theorem) = E ξ t 0 d ds G(T s ) ds = E(ξ(G(T t ) − G)) = E(ξ(A t ) E t G) − E(ξ G) (again by Girsanov theorem) = E(Z t G) − E(ξ G) .
Thus, a solution of the stochastic equation equation (4.5) is explicitly given by (4.6). Uniqueness follows since the solution of equation (4.5) started at ξ ≡ 0 is identically zero at all times. Now we introduce the braiding technique to solve equation (4.1), where γ ∈ L 2 [0, 1] and f : R → R. To simplify notation, define
I γ = 1 0 γ s dW s , A v u (ω · ) = ω · − (·∧v)∨u u σ(s) ds, E v u = exp v u σ(s) dW s − 1 2 v u σ(s) 2 ds , and g v u = exp [f (I γ ) (v − u)
] . Directly from the definitions above, for any u < v < w, we get the compositions
A w v • A v u = A w u , E v u • A w v = E v u , g v u • A w v = exp [f (I γ • A w v ) (v − u)]
, and the products
E v u · E w v = E w u , and g v u · g w v = g w u .
We suppress the dependence on ω for notational convenience.
Fix t ∈ [0, 1], and consider a sequence of partitions ∆ n = {0 = t 0 < t 1 < · · · < t n = t} of [0, t] such that ∆ n = sup {t i − t i−1 | i ∈ [n]} → 0. On each subinterval, we (1) solve the equation having only the diffusion with the initial condition as the solution of the previous step, and (2) use the solution obtained in step 1 as the initial condition and solve the equation having only the drift. Figure 2. A graphical representation of the braiding technique.
t t 0 = 0 t 1 t 2 t n−1 t n = t 1 Y (1) ξ X (1) Y (2) X (2) Y (n) X (n) Z · · ·
For the first subinterval, the initial condition of step 1 is taken to be ξ. For a visual representation of the idea, see figure 2.
We explicitly demonstrate the process for the first two subintervals. An index (i) in the superscript refers to the ith subinterval. First subinterval.
(1) The stochastic differential equation that we want to solve is
dY (1) u = σ(u)Y (1) u dW u , u ∈ [0, t 1 ], Y (1) 0 = ξ.(1) u = (ξ • A u 0 ) E u 0 , so Y (1) t 1 = (ξ • A t 1 0 ) E t 1 0
on a set Ω 1 , where P(Ω 1 ) = 1. (2) For each ω ∈ Ω 1 , we want to solve the ordinary differential equation
dX (1) u = f (I γ ) X (1) u du, u ∈ [0, t 1 ], X (1) 0 = Y (1) t 1 .
By the existence and uniqueness theorem of ordinary differential equations, the unique solution is given by X
(1) u = Y (1) t 1 g u 0 = (ξ • A t 1 0 ) E t 1 0 g u 0 , and so X (1) t 1 = (ξ • A t 1 0 ) E t 1 0 g t 1 0 . Second subinterval.
(1) The stochastic differential equation that we want to solve is
dY (2) u = σ(u) Y (2) u dW u , u ∈ [t 1 , t 2 ], Y(2)t 1 = X (1) t 1 .t 1 • A u t 1 ) E u t 1 . Now, Y (2) u = (ξ • A t 1 0 ) E t 1 0 g t 1 0 • A u t 1 E u t 1 = (ξ • A t 1 0 • A u t 1 ) E t 1 0 E u t 1 (g t 1 0 • A u t 1 ) = (ξ • A u 0 ) E u 0 (g t 1 0 • A u t 1 ),
where we used the fact that E t 1 0 is invariant under A u t 1 . This is because, by definition,
A u t 1 (ω · ) = ω · − (·∧u)∨t 1 t 1 σ(s) ds. Now, for E t 1 0 , we have t ∈ [0, t 1 ]
. Therefore,
A u t 1 (ω t ) = ω t − t 1 t 1 σ(s) ds = ω t ,
showing the invariance. This gives the motivation behind why we define A as such, and is a key trick in the method. Continuing, we get
Y (2) t 2 = (ξ • A t 2 0 ) E t 2 0 (g t 1 0 • A t 2 t 1 ) on a set Ω 2 ⊆ Ω 1 , where P(Ω 2 ) = 1.
(2) For each ω ∈ Ω 2 , we have the ordinary differential equation
dX (2) u = f (I γ ) X (2) u du, u ∈ [t 1 , t 2 ], X (2) t 1 = Y (2) t 1 .
The unique solution is given by X
(2) u = Y (2) t 1 g u t 1 . Using the definition of Y(2)
t 1 and the fact that A t 2 t 2 is the identity function,
X (2) t 2 = (ξ • A t 2 0 ) E t 2 0 (g t 1 0 • A t 2 t 1 ) (g t 2 t 1 • A t 2 t 2 ) = (ξ • A t 2 0 ) E t 2 0 2 i=1 (g t 2 t 1 • A t 2 t 2 )
It should now become obvious what the pattern is. We prove this using induction in the following lemma. Lemma 4.9. Let ξ ∈ L p (Ω) for some p > 2. Consider the kth subinterval u ∈ [t k−1 , t k ] for any k ∈ [n], and define (1) the stochastic differential equation
dY (k) u = σ(u) Y (k) u dW u , u ∈ [t k−1 , t k ], Y (k) t k−1 = X (k−1)
t k−1 , and (2) the ordinary differential equation
dX (k) u = f (I γ ) X (k) u du, u ∈ [t k−1 , t k ], X (k) t k−1 = Y (k) t k .
Then there exists a set Ω k ⊆ Ω with P(Ω k ) = 1 such that on Ω k , we have
X (k) t k = (ξ • A t k 0 ) E t k 0 k i=1 (g t i t i−1 • A t k t i ).
Proof. Base cases. This is true for k = 1 and k = 2 as shown in the computations above. Induction step. Assume that the result holds for k = m − 1. This means that there exists Ω m−1 with P(Ω m−1 ) = 1 such that on Ω m−1 , we have
X (m−1) t m−1 = (ξ • A t m−1 0 ) E t m−1 0 · m−1 i=1 (g t i t i−1 • A t m−1 t i ).
Using the ideas of computations on the second subinterval, we get that there exists Ω m with P(Ω m ) = 1 such that on Ω m , we have
Y (m) tm = (ξ • A tm 0 ) E tm 0 m−1 i=1 (g t i t i−1 • A tm t i ).
Since A tm tm is the identity function, on Ω m , we have
X (m) tm = Y (m) t m−1 g tm t m−1 = (ξ • A tm 0 ) E tm 0 m i=1 (g t i t i−1 • A tm t i ).
The proof is now complete by mathematical induction.
We are now able to derive a closed form solution of equation (4.1) in the Skorokhod sense. This is the main theorem of the section.
Z t = (ξ • A t 0 ) exp t 0 σ(s) dW s − 1 2 t 0 σ(s) 2 ds + t 0 f 1 0 γ u dW u − t s γ u σ(u) du ds .
Remark 4.11. Note that ξ may depend on the Wiener process.
Proof. Using lemma 4.9, for any t ∈ [0, 1], we have
X (n) t = (ξ • A t 0 ) E t 0 k i=1 (g t i t i−1 • A t t i ). Now, k i=1 (g t i t i−1 • A t t i ) = k i=1 exp f (I γ • A t t i ) (t i − t i−1 ) = exp k i=1 f 1 0 γ u dW u − t 0 γ u σ(u) du ∆t i . 16
Finally, taking n → ∞, we get
Z t = lim n→∞ X (n) t = (ξ • A t 0 ) E t 0 exp t 0 f 1 0 γ u dW u − t 0 γ u σ(u) du ds ,
which exactly equals the proposed solution.
The solution exists almost surely, due to the continuity of the measure. Moreover, the solution is unique. For if not, there are two solutions which disagree for the first time on a particular interval, say the kth interval. Recall that the solutions obtained using Malliavin calculus and also for ordinary differential equations are unique for each interval of time. Therefore, such a disagreement would violate these uniqueness results.
Large deviation principles
The theory of large deviation allow us to find probabilities of rare events that decay exponentially. Our goal is to derive large deviation principles for the solutions of LSDEs that we derived in section 4. But first, we give the formal setting for sample path large deviations. . Let X and Y be two polish spaces, I a rate function on X , and f a continuous function mapping X to Y. Then the following conclusions hold.
(1) For each y ∈ Y,
J(y) = inf I(x) | x ∈ f −1 (y)
is a rate function on Y, (2) If {X n } satisfies large deviation principle on X with rate function I, then {f (X n )} satisfies large deviation principle on Y with rate function J.
When are large deviation principles are conserved? To answer this question, we introduce the idea of superexponential closeness. . For ǫ > 0, let X ǫ and Y ǫ be families of random variables on (Ω, F , P ) that take values in X . Then the families X ǫ and Y ǫ (and their corresponding families of laws) are said to be superexponentially close if
lim ǫ→0 ǫ log P {d(X ǫ , Y ǫ ) > δ} = −∞.
The following theorem says that large deviation principles are preserved for superexponentially close families.
Theorem 5.4 ([2, theorem 4.2.13]). Suppose X ǫ and Y ǫ be superexponentially close families of random variables on (Ω, F , P ). Then X ǫ follows large deviation principle with rate function I if and only if Y ǫ follows large deviation principle with the same rate function I.
Finally, we give an example of large deviation principle. Consider the family of process ( √ ǫW ) ǫ>0 , where a Wiener process W is scaled down by a parameter √ ǫ. As ǫ → 0, we have √ ǫW → 0 almost surely. But at what rate does the convergence happen? This is answered by Schilder's theorem. In order to use the continuity principle (theorem 5.2), we need the following lemma.
∞ ≤L f ( γ ∞ x − y ∞ + |γ(1) − γ(0)| x − y ) ∞ =3L f γ ∞ x − y ∞ ,
which proves the continuity of ψ.
The following result now follows directly from the continuity of θ (lemma 5.6), the continuity principle (theorem 5.2), and Schilder's theorem (theorem 5.5).
1 0
1|Φ n (t) − Φ(t)| 2 dt → 0 almost surely as n → ∞, and (b)
Theorem 3.2 ([5, theorem 2.11]). Let N t be an integrable stochastic process and let M t = E(N t | F t ). Then N t is a near-martingale if and only if M t is a martingale.
is a near-martingale with respect to the filtration generated by Brownian motion given by {F t } Proof. Let s ≤ t and consider a partition, ∆ n , of [s, t] with t 0 = s and t n = t. The definition of the Ayed-Kuo stochastic integral in conjunction with the uniform integrability condition 3 on the partial sums implies
(4. 4 )
4Proof. We show that equation (4.4) solves equation (4.1). The initial condition is trivially verified.Note that equation(4.4) can be written as σ u du ds .
Theorem 4 . 3 .
43Under the condition of theorem 4.2, the stochastic differential equation
Definition 4.5 ([10, definition 1.2.1]). The Gross-Malliavin derivative of a smooth random variable F ∈ S is the real valued random variable given by
denote by δ the adjoint of the operator D. That is, δ is an unbounded operator on L 2 (Ω; H) with values in L 2 (Ω) such that:
Lemma 4.8 gave us the almost sure unique solution Y
Theorem 4 . 10 .
410Suppose σ, γ ∈ L 2 [0, 1], f : R → R, and ξ ∈ L p (Ω) for some p > 2. Then the unique solution of equation (4.1) in the Skorokhod sense is given by
Definition 5 . 1 .
51Let (X , d) be a Polish space and (µ ǫ ) ǫ>0 a sequence of Borel probability measures on X . Suppose I : X → ∞ is a lower semicontinuous functional. Then the sequence (µ ǫ ) ǫ>0 is said to satisfy a large deviation principle on X with rate function I if and only if (1) (upper bound) for every closed set F ⊆ X , lim ǫ→0 ǫ log µ ǫ (F ) ≤ − inf x∈F I(x), (2) (lower bound) and for every open set G ⊆ X , lim ǫ→0 ǫ log µ ǫ (G) ≥ − inf x∈G I(x). The next result states how large deviation principles are transferred under continuous transformations.
0
0The sequence of probability measure {p ǫ } asǫ → 0 follows Large Deviation Principle on C 0 ([0, 1]) with rate function I(f ) ′ (t)| 2 dt if f ∈ H 1 ∞ otherwise.whereH 1 = {f ∈ C 0 ([0, 1]) | f is absolutely continuous and f ′ ∈ L 2 [0, 1]}.5.1. LSDEs with constant initial conditions. Suppose σ and γ are deterministic functions of bounded variation on [0, 1]. Moreover, suppose f ∈ C 2 (R) is Lipschitz continuous along with f, f ′ , f ′′ ∈ L 1 (R). For a fixed κ ∈ R, consider the family of linear stochastic differential equations with parameter ǫ >
Lemma 5. 6 .
6The function θ : C 0 → C κ defined by u dx(u) − ǫ t s γ u σ(u) du ds , is continuous in the topology induced by the canonical supremum norm. ds and exp are continuous transformations, continuity of θ is guaranteed if we prove continuity of φ and ψ. This is what we show below. For φ, we haveφ(x) − φ(y) ∞ x − y ∞ + |σ(t) − σ(0)| x − y ∞ ≤ 3 σ ∞ x − y ∞ , so φ is continuous.For ψ, if L f is the Lipschitz constant for f , we get ψ(x) − ψ(y)
Theorem 5.7. The laws of the solutions Z ǫ κ given by equation(5.2) of the family of stochastic differential equations given by equation (5.1) follow a large deviation principle on (C κ , · ∞ ) with the rate function J(y) = inf I • θ −1 (y) ,(5.3)where θ is as defined in lemma 5.6, and I is the rate function given by theorem 5.5.5.2.LSDEs with random initial conditions. Is it necessary for the family of linear stochastic differential equations equation (5.1) to start at a constant point κ ∈ R in order for it to have a large deviation principle? In this section, we generalize theorem 5.7 and show that we can derive a similar result under a stronger version of exponential equivalence and more restrictive conditions on the functions f , σ, and γ. Suppose σ and γ are deterministic functions of bounded variation on [0, 1]. Moreover, suppose f ∈ C 2 (R) is Lipschitz continuous along with f, f ′ , f ′′ ∈ L 1 (R). Consider the family of linear stochastic differential equations with parameter ǫ > 0 given bywhere ξ ǫ s are random variables independent of the Wiener process W . For each ǫ, just as before, the unique solution to equation (5.4) is given byWe now state a more general large deviation principle.Theorem 5.8. Let κ ∈ R and consider the family of random variables ξ ǫ such that the following holdMoreover, assume that the functions f, f ′ , σ, γ are all bounded. Then the laws of the solutions Z ǫ ξ given by equation (5.5) of the family of stochastic differential equations given by equation(5.4)follow a large deviation principle on (C κ , · ∞ ) with the rate function given by equation(5.3), where θ is as defined in lemma 5.6, and I is the rate function given by theorem 5.5.Then V ǫ satisfies the stochastic differential equation7)20 whose solution is given byLet φ(z) = |z| 2 and let U ǫ = φ(V ǫ ). From theorem 4.3, U ǫ satisfies the integral equationFix δ > 0 and let τ = inf {t ∈ [0, 1] : |V ǫ t | ≥ δ}. Taking expectation of the stopped process U ǫ t∧τ , we getThe second integral on the right-hand side is a near-martingales by theorem 3.3. Suppose f, f ′ , σ, γ are all bounded by some M > 1. Using non-negativity of U ǫ and the nearmartingale optional stopping theorem (corollary 3.11), we getBy Gronwall's inequality, we getis a monotonically increasing non-negative function in |z|, we use Markov's inequality to get21Taking log and multiplying by ǫ, we getFinally, taking limit of ǫ → 0 and using equation (5.6), lim ǫ→0 ǫ log P{|V ǫ τ | ≥ δ} = −∞.This result allows us to say that Z ǫ ξ and Z ǫ κ are exponentially equivalent. Since exponentially equivalent families have the same large deviation principle due to theorem 5.4, Z ǫ ξ follows a large deviation principle with the same rate function given by equation(5.3).
An extension of the Itô integral. Wided Ayed, Hui-Hsiung Kuo, 10.31390/cosa.2.3.05Communications on Stochastic Analysis. 2Wided Ayed and Hui-Hsiung Kuo. "An extension of the Itô integral". In: Communica- tions on Stochastic Analysis 2.3 (2008), pp. 323-333. doi: 10.31390/cosa.2.3.05.
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. Hui-Hsiung Kuo, Baton Rouge, LA 70803, USADepartment of Mathematics, Louisiana State UniversityEmail address: [email protected] Kuo: Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA Email address: [email protected]
. Pujan Shrestha, 57Baton Rouge, LA 70803, USA Email addressDepartment of Mathematics, Louisiana State UniversityPujan Shrestha: Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA Email address: [email protected]
USA Email address: sudipsinha@protonmail. 70803Baton Rouge, LASudip Sinha: Department of Mathematics, Louisiana State UniversitySudip Sinha: Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA Email address: [email protected] URL: https://sites.google.com/view/sudip-sinha
. Padmanabhan Sundar, Baton Rouge, LA 70803, USADepartment of Mathematics, Louisiana State UniversityEmail address: [email protected] Sundar: Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA Email address: [email protected]
| [] |
[
"Robust Sign Language Recognition System Using ToF Depth Cameras",
"Robust Sign Language Recognition System Using ToF Depth Cameras"
] | [
"Morteza Zahedi [email protected] \nDepartment of Computer Engineering\nIT Shahrood University of Technology Shahrood\nIran\n",
"Ali Reza Manashty [email protected] \nDepartment of Computer Engineering\nIT Shahrood University of Technology Shahrood\nIran\n"
] | [
"Department of Computer Engineering\nIT Shahrood University of Technology Shahrood\nIran",
"Department of Computer Engineering\nIT Shahrood University of Technology Shahrood\nIran"
] | [
"World of Computer Science and Information Technology Journal (WCSIT)"
] | Sign language recognition is a difficult task, yet required for many applications in real-time speed. Using RGB cameras for recognition of sign languages is not very successful in practical situations and accurate 3D imaging requires expensive and complex instruments. With introduction of Time-of-Flight (ToF) depth cameras in recent years, it has become easier to scan the environment for accurate, yet fast depth images of the objects without the need of any extra calibrating object. In this paper, a robust system for sign language recognition using ToF depth cameras is presented for converting the recorded signs to a standard and portable XML sign language named SiGML for easy transferring and converting to real-time 3D virtual characters animations. Feature extraction using moments and classification using nearest neighbor classifier are used to track hand gestures and significant result of 100% is achieved for the proposed approach. | null | [
"https://arxiv.org/pdf/1105.0699v1.pdf"
] | 11,702,492 | 1105.0699 | cae8f4f1074a1e854e914640632c720e3c07b1c7 |
Robust Sign Language Recognition System Using ToF Depth Cameras
2011
Morteza Zahedi [email protected]
Department of Computer Engineering
IT Shahrood University of Technology Shahrood
Iran
Ali Reza Manashty [email protected]
Department of Computer Engineering
IT Shahrood University of Technology Shahrood
Iran
Robust Sign Language Recognition System Using ToF Depth Cameras
World of Computer Science and Information Technology Journal (WCSIT)
13201150sign languageTime-of-Flight camerasign recognitionSigMLMomentshand trackingrange cameras
Sign language recognition is a difficult task, yet required for many applications in real-time speed. Using RGB cameras for recognition of sign languages is not very successful in practical situations and accurate 3D imaging requires expensive and complex instruments. With introduction of Time-of-Flight (ToF) depth cameras in recent years, it has become easier to scan the environment for accurate, yet fast depth images of the objects without the need of any extra calibrating object. In this paper, a robust system for sign language recognition using ToF depth cameras is presented for converting the recorded signs to a standard and portable XML sign language named SiGML for easy transferring and converting to real-time 3D virtual characters animations. Feature extraction using moments and classification using nearest neighbor classifier are used to track hand gestures and significant result of 100% is achieved for the proposed approach.
I. INTRODUCTION
Emerging technologies help scientific researches both in speed and accuracy. Sometimes, previous research efforts on a particular area might become completely or partially out of date with the new technology replacing the previous one. Time-of-Flight (ToF) or Range cameras are type of cameras that are able to create distance images from the environment, ranging from few meters to several kilometers. These devices have been shipped just since the first decade of 21th century when the semiconductor products became fast enough to provide the speed needed for processing the data of such cameras. Recently with current technology, the price of these cameras is reduced about twenty times. This efficiency in the price makes it possible to use such a technology in ordinary life of people when they are handy. The new capabilities of such devices make it possible to track gestures and environments from another point of view that reduces the recognition error of previously used RBG cameras and sophisticated 3D sensing systems.
On the other hand, voice is the primary means of communication between human beings; yet, it is not possible to always rely on voice for communication. This disability might be caused from different situations, e.g. people with auditory defects that also lost their ability to speak or just prefer to do so, military situations that voice communication may lead to revealing the stealth of the troops and ordinary situations when signs are preferred to voice communication such as in audio recording studios. These reasons, made a lot of people learn the sign language and create a standard for it and they even created a language for it. American Sign Language (ASL) is one example of a commonly used sign language.
Many projects and efforts are aimed for removing the gap between the signers and regular speakers [1] using regular cameras. These efforts sometimes involve creating new databases or modifying and discussing the older ones hoping for a better recognition rate [2,3]. The use of depth in recognition is mostly limited to stereo imaging and the results are not satisfactory for general public usage [4].
To transfer a sign sentence via mediums to another place, the easiest way is to transfer the video sequence of the recorded sign language over the transmission medium to the destination so that a person on the other side of this communication can interpret the meaning of the sign language. Although this seems very simple, the video transmission requires high bandwidth to be transferred. Due to the explained problem, a standard sign language that can be used as a middle coding for transferring such data is devised and is named SiGML (Signing Gesture Markup Language) [5]. This language is still in use in recent researches [6]. SiGML is an XML language that is defined based on the single movements of the body, from eyes to fingertips. This language that is also compatible with the Hamburg Sign Language Notation System (or HamNoSys, a phonetic transcription system for sign languages) is used in most research studies.
In this paper, we present the use of ToF depth cameras to track different hand movements and after recognition of the single movements, convert them to the appropriate SiGML notation so that it can be easily transferred as a text and then the same recognized action can be shown using 3D virtual characters. Different steps toward such system are organized in this paper as sections so after previous works on the next section, section 3 is about data acquisition and database, section 4 describes the preprocessing steps, section 5 and 6 discuss the feature extraction and classification schemes using nearest neighbor (NN) classifier, respectively, and then the paper is concluded with the results and future works.
II. PREVIOUS WORKS
The need for storing and transferring the sign data makes different sign notation systems to evolve. Stokoe system, HamNoSys and SignWriting are some of those [7]. From the above, The Hamburg Sign Language Notation System, HamNoSys [8], is a -phonetic‖ transcription system, which is a more general form of Stokoe system. It includes more than 200 symbols which are used to transcribe signs in four levels consisting of hand shape, hand configuration, location and movement. The symbols are based on iconic symbols to be easily recognizable. Facial expressions can be represented as well, but their development is not finished yet. HamNoSys is still being improved and extended as the need arises and hence is the most suitable for converting real sign gestures to the equivalent notation.
The Signing Gesture Markup Language, SiGML [5] is based on HamNoSys symbols and represents signs in form of Extensible Markup Language (XML)a simple but flexible format for transferring structured and semi-structured data. There are some tools available to convert signs written in SiGML format to virtual character animations, both as applications in regular operating systems and as applets in web browsers. These tools make SiGML a preferable language for converting the recognized signs to it.
For robust gesture recognition, in most of the efforts, the equipment used in order to capture the videos is usually complex and expensive. For example, authors in [9] use two cameras for recording or authors in [10] use dense ranging images captured by expensive 3D sensors. Others usually use some equipment other than cameras, such as special sensor gloves, like [11] and [12]. The use of regular and efficient ToF cameras which can be easily purchased by individuals is not a habit in sign language recognition. Specially the efforts to use ToF cameras that can obtain the data remotely, accurately and fast for converting the signs to SiGML language (so that it can be easily transferred to be visual again using virtual animated characters) has not been tested before. The latter effort helps people to communicate over networks with lower bandwidth or use short text messages systems on cellular phones to transfer, convert and watch the animated sign language representation of the text message easily.
In this paper, a novel approach is proposed for recognizing the signs that are captured using a ToF depth camera and converting the recognized signs to SiGML language. The signs in this language can be easily replayed using SiGML service player that uses a 3D virtual character animation to replay the recorded signs.
III. DATA ACQUISITION
To acquire data for recognition, a ToF camera similar to Microsoft Kinect for Xbox 360 is used. The output of these cameras is depth sequence of images (video) in gray-scale at about 30 frames per second. The images show the environment in such a way that the nearest objects have more gray-scale intensity and are close to white and farther objects have lower gray-scale intensity and are close to black.
A. Gestures to Capture
The SiGML language has many elements that can be used to define a movement. This language also supports HamNoSys elements to be used. For a robust start, four different movements have been chosen that are to be recognized. These movements are listed below on Table 1. As it can be observed, each movement has a symbol in HamNoSys font and system and has an equivalent in SiGML language too. The point in these four different movements is that they are not started in a particular area of the screen and are not even finished in a predefined area. They can be started anywhere in the screen and be finished anywhere else, so all different possible starting and ending point for these gestures must be included in the database.
B. Database
For creating a suitable and fault-tolerant database, the video of nine different sets of all four previously mentioned movements is captured. This results in 36 single movements. These videos are captured using a ToF camera in 640x480 resolution and maximum of 30 frames per second. The database includes every action in one consecutive video but a single blank action is recorded between every set of actions. Single actions are separated using the two steps described in the preprocessing section.
IV. PREPROCESSING
The raw video data that is captured using the ToF camera needs some processing so that it can be used for classification. As it has been mentioned earlier, the range image frames are gray-scale with the whiter pixels meaning they are nearer to the camera. Most of the gesture recognition problems were that the recognition of the skin and the hand itself needed some noticeable amount of time and complexity. The unavoidable errors caused by lighting and noise made it difficult to track the hand itself. When the output image of the ToF camera is used, without any effect of light, the results have the same Start End
Start End
Start End characteristics in any room lighting condition (because of the nature of the ToF cameras, sunlight defects the received light of such devices).
A. Intensity Filtering
The first preprocessing step applied to the video frames is to threshold the pixels to a specific range, resulting in all images (video frames) with the real-world distance filter applied to. In this research, the pixels with intensity of less than 112 and more than or equal to 128 have been filtered. This means that only a domain of 16 pixel values are selected for the images and the others turned to black. This filtering makes it possible to have only the hand in the screen and have every other useless details removed very fast and accurately. In an automatic manner, these numbers can change automatically depending on the distance of the person but in this research, the distance of the person is changed so that when he adjusts his hand to be visible in the filtered visible range of the camera, the person does not move during the whole database creation period. While having little varying z coordinate, each set of actions has completely different x and y coordinates. Fig. 1 shows the starting and ending frame of a regular gesture video taken from ToF camera after filtering. The images are gray-scale and for convenient, some color spectrum has been used to clarify it. Fig. 2 and Fig. 3 show images of an action that is not performed near to the center of the screen and a defected captured action, respectively. Both the actions shown in Fig. 2 and Fig. 3 are included in the database to show the robustness of the recognition.
B. Action Separation
The second preprocessing step that can also be automatically done later, is to separate single movements in continuous set of actions. This is done by (1) separating each single action frames from all the video frames and (2) selecting the starting and ending image of each sequence. The second step described above is performed due to nature of the actions in the database that is selected for classifying. If a more complex action is going to be recognized, many inner frames other than the starting and ending frame will be needed and some other algorithms and tools such as hidden Markov model can be used.
V. FEATURE EXTRACTION
After the preprocessing step, two images per action are remained, showing the starting and ending frame of that action. Each of these images includes the hand of the signer fully or partially. To classify these patterns correctly, moments are used to get the features from the images:
A. Moments
The features that are selected for classification are based on moments [13]. The two dimensional (p+q)th order moments of the grey-value image with the pixel intensities X(i, j) are defined as following:
I i J j q p pq j i X j i m ) , ((1)
If X(i, j) is the piecewise continuous and it only has nonzero values in the finite part of the two dimensional plane, then the moments of all orders exist and the sequence {m pq } is uniquely determined by X(i, j) and vice versa. The small order moments of the X(i, j) describe the shape of the region. For example m 00 is equal to the area size, and m 01 and m 10 give the x and y coordinates of the center of the gravity [7].
B. Feature Selection
As the criteria for using moments are met in our database images, two features are chosen from the moments of the images: m 01 and m 10 which are called x CoG and y CoG as the x and y coordinates of the center of gravity of the hand image. For every action, there are two images, one for the starting image of the action and the other for the ending image of the same action. So the total number of features for any action is equal to four, two for the center of gravity of the starting image and two for the center of gravity of the ending image.
Nearest neighbor classifier is used for the classification of the actions. First, it must be assured the input features are discriminant enough to result in a good classification. To ensure that, the scatter diagram of the center of gravity points of both the starting and ending images has been plotted separately. As it is visible in the Fig. 4 and Fig. 5, because the starting and ending point of each set of action is not at the center of the screen, each movement could start and end anywhere in the screen so that it may be misinterpreted for another hand movement. Because these features are indiscriminant, instead of using the center of gravities as features, the movement vector of the center of gravity of each action, resulted from the following equation, is used:
(x, y) CoG-Movement Vector = (x CoG-End -x CoG-Start , y CoG-End -y CoG-Start )(2)
In (2), as the resulting vector is two dimensional, the total number of features is reduced by two. The scatter diagram of the center of gravity movement vector calculated above is illustrated in Fig. 6. As it can be easily observed, each action is now scattered in a particular part of the space and is now easily classifiable by nearest neighbor. Each hand movement vector in Fig. 6 is cumulated with its similar hand movement vectors, all in their particular part of the space.
VI. CLASSIFICATION
The feature vector created from the previous step consists of two values for each action. As movement vector of the center of gravity is used, the extracted features are scattered in a discriminant way so that they can be easily classified using the nearest neighbor algorithm. The nearest neighbor classifier used for the classification utilizes Euclidian distance measure to determine the distance between each sample.
B. Classification Results
The permutation of five training sets that can be chosen from nine total samples is equal to 126. The train sets have been changed 126 times so that all possible combination of train and test data can be verified for consistency. In all 126 classifications of hand gesture movements using nearest neighbor classification, 100% classification result is achieved. After successfully classifying the input hand gestures, according to the Table 1, each successful recognized action can be converted to the corresponding SiGML element. Using a template for a SiGML file, the elements are added one after another after classification. The resulting SiGML file becomes ready for being played in the SiGML service player [14]. The SiGML service player can play SiGML files and converts the movements defined in the file to a 3D virtual character animation. An example for resulting animation of the recognized real human gestures that is being played in the SiGML service player is shown in Fig. 7. In this paper, the effective role of ToF depth cameras in making the sign language recognition a faster and more reliable task is presented. The importance of sign language is clear both as a means of communication for disabled people and also as a second language for situations in which communication using voice is not possible.
VII. GESTURE TO SIGML TRANSLATION
The transmission of performed sign languages is important since transferring the input video might be too costly and space consuming. On the other hand, relying on individuals for interpreting the signs is not always possible and efficient. So some sign writing languages are devised to transfer the recognized signs, e.g. SiGML and HamNoSys notation systems. The aim of this paper is to present a way to recognize the input video of performed signs and to convert them in the easiest and most accurate way to SiGML language for easy transferring through the media. The robustness of the features selected for the classification is approved with the result of 100% for the selected gestures.
In near future, the database must be expanded to cover the most useful and common symbols and actions of the sign language to propose a system that is capable of recognizing more complex and handy signs. Different aspects of signing must be considered in such tasks [15]. There is also the need for a model that can help the translation of more advanced sign actions to be an easier job. The automation of distance calibrating and action separation is also in high priority.
Figure 1 .
1The starting and ending frame of a right hand movement after filtering. The center of gravity of the hand is also calculated.
Figure 2 .
2The starting and ending frame of a right hand movement that is not performed near to the center of the screen; after filtering. The center of gravity of the hand is also calculated.
Figure 3 .
3The starting and ending frame of a defected captured image showing hand movement to the top; after filtering. The center of gravity of the hand is also calculated despite most of the hand is not visible.
Figure 4 .
4The scatter diagram of the center of gravity points of the starting image of different actions.The colors of the points that correspond to each action are dark Red, Blue, Yellow and Cyan, for movement to Top, Bottom, Right and Left, respectively.
Figure 5 .
5The scatter diagram of the center of gravity points of the ending image of different actions.The colors of the points that correspond to each action are dark Red, Blue, Yellow and Cyan, for movement to Top, Bottom, Right and Left, respectively.
Figure 6 .
6The scatter diagram of the movement vectors of the center of gravity points of the actions. The colors of the points that correspond to each action are dark Red, Blue, Yellow and Cyan, for movement to Top, Bottom, Right and Left, respectively. The points are scattered in their groups and are discriminant enough for classification using nearest neighbor algorithm.
Figure 7 .
7An example of resulting animation of the recognized real human gestures that is being played in the SiGML service player VIII. CONCLUSION AND FUTURE WORKS
TABLE 1 .
1THE FOUR DIFFERENT HAND MOVEMENTS IN THE DATABASE These elements are special HamNoSys equivalent elements.Movement
Equivalent Symbols
SiGML a
HamNoSys
Unicode
HamNoSys
Symbol
Hand to Right
hammover
E082
Hand to Left
hammovel
E086
Hand to Up
hammoveu
E080
Hand to Down
hammoved
E084
a.
A .
ATraining and Testing From nine sets of four different hand movements, 5 sets are used for training and four sets for testing. This results in 20 reference samples and 16 test samples. Using nearest neighbor, each of 16 test samples are compared with 20 train samples and the train sample with the least distance will determines the test sample's class. Each test sample must be grouped to one of the following four classes: Movement to Right, Movement to Left, Movement to Top and Movement to Bottom.
ACKNOWLEDGMENTWe sincerely thank Zahra Forootan Jahromi for helping us in creating the database and testing the ToF camera in the first place.AUTHORS PROFILEMorteza Zahedi received the B.Sc. degree in computer engineering (hardware) from Amirkabir University of Technology, Iran, in 1996, the M.Sc. in machine intelligence and robotics from University of Tehran, Iran, in 1998 and the Ph.D. degree in manmachine interaction from RWTH-Aachen University, Germany, in 2007, respectively. He is currently an assistant professor in Department of Computer Engineering and IT at Shahrood University of Technology, Shahrood, Iran. He is the Head of Computer Engineering and IT Department. His research interests include pattern recognition, sign language recognition, image processing and machine vision.Ali Reza Manashty is a M.Sc. student of artificial intelligent at Shahrood University of Technology, Shahrood, Iran. He received his B.Sc. degree in software engineering from Razi University, Kermanshah, Iran, in 2010. He has been researching on mobile application design and smart environments especially smart digital houses since 2009. His publications include 6 papers in international journals and conferences and one national conference paper. He has earned several national and international awards regarding mobile applications developed by him or under his supervision and registered 4 national patents. He is also a member of The Elite National Foundation of Iran. His research interests include 3D image processing, pattern recognition, smart enviroments and mobile agents.
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| [] |
[
"Acceleration and ejection of ring vortexes by a convergent flow as a probable mechanism of arising jet components of AGN",
"Acceleration and ejection of ring vortexes by a convergent flow as a probable mechanism of arising jet components of AGN"
] | [
"S A Poslavsky s:[email protected] ",
"E Yu Bannikova [email protected] \nInstitute of Radio Astronomy NAS\nUkraine\n",
"V M Kontorovich \nInstitute of Radio Astronomy NAS\nUkraine\n",
"V N Karazin Kharkov ",
"\nNational University\n\n"
] | [
"Institute of Radio Astronomy NAS\nUkraine",
"Institute of Radio Astronomy NAS\nUkraine",
"National University\n"
] | [] | Exact solutions of two-dimensional hydrodynamics equations for the symmetric configurations of two and four vortices in the presence of an arbitrary flow with a singular point are found. The solutions describe the dynamics of the dipole toroidal vortex in accretion and wind flows in active galactic nuclei. It is shown that the toroidal vortices in a converging (accretion) flow, being compressed along the large radius, are ejected with acceleration along the axis of symmetry of the nucleus, forming the components of two-sided jet. The increment of velocities of the vortices is determined by the monopole component of the flow only. The dipole component of the flow determines the asymmetry of ejections in the case of an asymmetric flow. | null | [
"https://arxiv.org/pdf/1006.5736v1.pdf"
] | 118,171,160 | 1006.5736 | 539c6c93f039d58577bb7b588cac733b32966925 |
Acceleration and ejection of ring vortexes by a convergent flow as a probable mechanism of arising jet components of AGN
29 Jun 2010
S A Poslavsky s:[email protected]
E Yu Bannikova [email protected]
Institute of Radio Astronomy NAS
Ukraine
V M Kontorovich
Institute of Radio Astronomy NAS
Ukraine
V N Karazin Kharkov
National University
Acceleration and ejection of ring vortexes by a convergent flow as a probable mechanism of arising jet components of AGN
29 Jun 2010galaxy -active, jetsvortices -ring, planeflow -accretion
Exact solutions of two-dimensional hydrodynamics equations for the symmetric configurations of two and four vortices in the presence of an arbitrary flow with a singular point are found. The solutions describe the dynamics of the dipole toroidal vortex in accretion and wind flows in active galactic nuclei. It is shown that the toroidal vortices in a converging (accretion) flow, being compressed along the large radius, are ejected with acceleration along the axis of symmetry of the nucleus, forming the components of two-sided jet. The increment of velocities of the vortices is determined by the monopole component of the flow only. The dipole component of the flow determines the asymmetry of ejections in the case of an asymmetric flow.
Introduction
A large number of works (see, for example, the monograph [1]) is devoted to the origin of jets. In the most of them the decisive role is played by a strong magnetic field [2][3][4], or "external", or arising from the development of instabilities in the plasma of the accretion disk. This field serves as a guide for the movement of particles under the action of electromagnetic, centrifugal and gravitational forces, allowing them to move against the gravity and carry away the angular momentum that is necessary for the effective accretion process, which is responsible for the activity of the nucleus (see discussion and references in reviews [5][6][7]). At the same time, the very possibility of the existence of strong magnetic fields in the accretion disks around the black holes is not entirely clear. In this connection, the models of continuous flow without magnetic field are also considered, including those, in which the structure of jets resemble the hydrodynamic tornadoes [8].
The observations, however, show that at small distances from the nucleus (parsec scales for active galactic nuclei (AGN)) the emergence of the individual (including the superluminal) components of the radio jets are observed [9]. In the model we discuss here [10][11][12] some ejections are derived from the kinematics of the interaction of vortices and the exposure of the magnetic field is not required. An important role in this process has the flow that can affect the velocity of ejection. Therewith, greater velocities of the ejected components are attained in a converging (accretion) flow. We regard a system of toroidal vortices [10], surrounding the central part of AGN, the outermost of which is observed as a "obscuring torus". The vortex motion arises in the torus due to twisting by the wind and radiation. Due to the flow symmetry the movement in the torus possesses the dipole character ( Fig. 1 in [10]). In its simplest form, this movement can be represented as the motion of two opposite rotating vortex rings in a radial flow. As is known, the dynamics of a vortex ring can be described as the movement of a pair of point vortices that arise in the cross-section of the ring (torus) by the plane of symmetry. In our case it is a symmetrical system of two or four vortices (or two vortex pairs) in an arbitrary flow with a point singularity.
Setting of a problem
We consider the motion of a system of point vortices on a background flow caused by the stationary singular point, which is placed at the origin. Following [13,14], the stream function can be represented as the sum of two constituents: its regular part ψ reg which describes the background flow and the singular part ψ sing , which describes the point vortices.
The complex potential of the flow can be represented as a series
w reg = C 0 ln z + C 1 z + C 2 z 2 + ... (z = x + iy)(1)
When the flow is symmetric about the axis Ox, all the coefficients C k in (1) are real. Expressing the complex potential through the usual velocity potential ϕ and the stream function ψ according to w = ϕ + iψ and passing on to the polar coordinates z = r · exp (iθ) we obtain the stream function in the form
ψ reg = C 0 θ − C 1 r sin θ − C 2 r 2 sin 2θ − ....(2)
The radial and azimuthal velocity components of the background flow are determined by the conditions
v r = 1 r ∂ψ reg ∂θ , v θ = − ∂ψ reg ∂r .
Combining the monopole at the origin (source or sink), dipole, quadrupole, ... with intensity C 0 , C 1 , C 2 , ... we can obtain a given distribution of the radial velocity component at the circle | z |= R which describes the considered flow.
The system of vortices and the flow has Hamiltonian form.
3. Symmetric motion of two vortex pairs in the flow with singularity of the type "source + quadrupole + ..."
Now we consider the dynamics of a system of two vortex pairs in the stream flow generated by a fixed singular point provided the existence of two axes of symmetry ( fig.1). This case can be interpreted as the motion of a point vortex in the right angle (which sides are impermeable "walls") at the apex of which is placed the referred hydrodynamic singularity. Note that Figure 1: Scheme of the movement of four vortices (two vortex pairs) in a symmetric flow, containing the sink and the quadrupole for C 0 = −1,
C 1 = 0, C 2 = 1, Γ = −4π
the solution of the problem for a symmetric system of four vortices in the absence of the background flow was found in the classical Grobli's work (see [15,16]). In a purely radial flow it admits a Hamiltonian formulation [11] and the exact solution of the dynamic problem [11,12].
The stream function for the discussed flow is represented in the form
ψ reg = C 0 θ − C 2 r 2 sin 2θ − C 4 r 4 sin 4θ − ....(3)
The vortex components of the right pair are arranged symmetrically about the axis Ox in the points (x; y) and (x;-y), and the intrinsic (due to the interaction of vortices and not related to the presence of the background flow) velocity of the vortex in the 1-st quadrant is
− → V sing = Γ 4π 1 y − y x 2 + y 2 ; x x 2 + y 2 − 1 x .
In the case of only one vortex pair without a background flow that corresponds to the known expression
− → V s = (Γ/4πy; 0) ,(4)
where Γ is the intensity of the vortex, x and y are its abscissa and ordinate.
The dynamics of two pairs of vortices in the flow is described by the equationṡ
x = Γ 4π 1 y − y x 2 + y 2 + ∂ψ reg ∂y ;ẏ = x x 2 + y 2 − 1 x − ∂ψ reg ∂x(5)
and the Hamiltonian of the system of 4 vortices is reduced to the form
H = Γ 4π ln xy √ x 2 + y 2 + C 0 arctan y x − C 2 2xy (x 2 + y 2 ) 2 − ....(6)
Accordingly, the equations (5) can be represented aṡ
x = ∂H ∂y ,ẏ = − ∂H ∂x
Equation H = E = const determines the trajectory of vortices. Moreover, if the motion is unbounded, then the vortex pair comes from infinity along one axis (Ox ), exchange their components and goes away to infinity along the other axis (Oy):
y ∞ = exp 4πE Γ , x → +∞; x ∞ = exp 4π(E − C 0 π 2 ) Γ , y → +∞; C 0 = 0, C 1 = 0, C 2 = 0.
For the asymptotic values x ∞ and y ∞ coordinates of the vortex moving in the first quadrant, we have the correlation
y ∞ = x ∞ exp 2π 2 C 0 /Γ ,(7)
which coincides with the result for the purely radial flow 11,12 . This means that the ratio y ∞ /x ∞ of the limit distances between elements of the vortex pairs at infinity and, correspondingly, the ratio of their velocities, are only determined by the intensity of the C 0 of the source (sink) at the origin and do not depend on the other multipole components.
Since the velocity of the translational motion of a vortex pair is inversely proportional to the distance between them (cf. (4)), then we obtain the ratio of the asymptotic values of the velocities of the pairs, coming from infinity V − = Γ/(4πy ∞ ) and going to infinity V + = Γ/(4πx ∞ )
V − /V + = x ∞ /y ∞ = exp −2π 2 C 0 /Γ ,(8)
that coincides with the ratio obtained in [11,12]. Obviously, the vortex pairs go away from the source at a slower velocity, and from the sinkcorresponding to accretion -with a greater velocity than they come in. Thus, the obtained solutions confirm the main result of the work [12] about the acceleration of ejections by the radial accretion flow, and expand it on the general case of a symmetric accretion flow. As we will show below the dipole component of the flow may be responsible for the asymmetry of ejections.
Motion of two vortex pairs in the flow from dipole which axis is the axis of symmetry of the flow
In the case of motion of two vortex pairs in the flow from a dipole with the axis Oy with intensity C 1 (that corresponds to replacement C 1 → iC 1 in (1)) the equations of dynamics of the vortices can be expressed through the coordinates of the two vortices in the right half-plane in the forṁ
x 1 = ∂H ∂y 1ẏ 1 = − ∂H ∂x 1ẋ 2 = − ∂H ∂y 2ẏ 2 = ∂H ∂x 2 ; H = C 1 x 1 x 2 1 + y 2 1 − C 1 x 2 x 2 2 + y 2 2 + Γ 4π ln (x 1 − x 2 ) 2 + (y 1 − y 2 ) 2 − (9) Γ 4π ln (x 1 + x 2 ) 2 + (y 1 − y 2 ) 2 + Γ 4π ln x 1 + Γ 4π ln x 2 .
The vortex moving in the first quadrant has the intensity Γ and coordinates (x 1 ; y 1 ) and the vortex in the fourth quadrant has the intensity -Γ and coordinates (x 2 ; y 2 ). The vortices located in the left half-plane have the parameters -Γ, (−x 1 ; y 1 ) and Γ, (−x 2 ; y 2 ) accordingly. From the first integral H = E = const one can get a relationship between the asymptotic values y ∞ , x 1∞ , x 2∞ : the half of the distance (y ∞ ) between the elements of the vortex pairs coming from infinity along the axis Ox, and the same for the pairs going to infinity along the axis Oy (x 1∞ and x 2∞ )
x 1∞ x 2∞ = y 2 ∞ .(10)
The dipole power occurs implicitly through the difference of asymptotics x 1∞ and x 2∞ . Really, as it is easily seen from the fig.3, the interaction between the vortices gives the same distortion in the x direction for both the 1-st and 2-nd vortices. The difference only arises due to the influence of the flow. In the case when there is also a source (sink) with intensity C 0 in the origin, the relationship (10) becomes
x 1∞ x 2∞ = y 2 ∞ exp − 4π 2 C 0 Γ .(11)
Note that the relation (11) remains valid if there are also the multipoles of the higher orders. It has been taken into account that the initial asymptotics corresponds to the symmetric pair of vortices. In terms of the ring vortices this corresponds to the dipole toroidal vortex of "infinite" radius, which is compressed by the interaction of the ring components. Dipole flow introduces an asymmetry in the motion, and different velocity correspond to vortex rings of different radius thrown in opposite directions. Thus, within the framework of the dipole-toroidal model one can naturally explain both the very appearance of the ejections accelerated by the accretion flow and the observed asymmetry of the ejections in AGNs.
Conclusion
The model of active galactic nuclei proposed in [10] gives the possibility within the framework of hydrodynamics of an ideal incompressible fluid to study the dynamic behavior of toroidal structures and the influence on their movement of accretion-wind flows.
In this paper we considered a simplified plane model, for which the analogue of a vortex ring is a pair of point vortices, which axis coincides with the axis of the ring. The dipole-vortex structure of the torus in the 2D model is represented by the two pairs of vortices with a common axes and the angular momenta of the opposite signs. It takes into account that the symmetric pairs of vortices correspond to the initial asymptotics.
In the terms of the ring vortices that corresponds to initial dipole toroidal vortex of "infinite" radius which is compressed due to interaction of the ring components. In the absence of the background flow this problem resembles the classical problem of the Helmholtz vortex ring interaction with the wall parallel to the plane in which the vortex ring lies. The wall can be replaced by a mirror image of the vortex, and the problem can be reduced to the interaction of oppositely rotating vortex rings. However, in our case the direction of rotation is opposite to the corresponding to approaching the vortex to the wall, which was considered by Helmholtz. (Our choice corresponds to the moving off the vortex from the wall.) In a purely radial flow it admits a Hamiltonian formulation and exact solution of the dynamic problem [11,12].
x y Figure 4: Scheme of the motion of ring vortices in a converging flow, appropriate to the discussed planar analog (symmetrical flow [12]).
In this study the conclusion about acceleration of the ejections by the radial accretion flow is expanded on the general case of two-dimensional flow. It is shown that the monopole component of the flow is only responsible for the acceleration of ejections, and a dipole flow component may be responsible for the asymmetry of bilateral ejections.
Thus, the dipole-toroidal model can naturally explain both the very emergence of ejections, accelerated by the accretion flow in the case of the general character of the flow, and the observed asymmetry of the ejections of active galactic nuclei and quasars.
The text basically corresponds to [17]. We have added some figures and corrected typos in formulas.
For an extended discussion of this subject see [18]. The authors are sincerely grateful to N.N. Kizilova for the useful notes.
Figure 2 :
2Phase portrait for the vortex of the first quadrant, corresponding to the movement of the four vortices in the accretion flow. C 0 = −1, C 1 = 0, C 2 = 1, Γ = −4π. Thin and bold lines correspond to the trajectories of the vortex and the separatrix accordingly. The asymptotic values of coordinates (x ∞ and y ∞ ) correspond to half of the distance between the components of the pairs of vortices at the inlet and outlet of the system.
Figure 3 :
3The scheme of motion of four vortices in the dipole flow which destroys the symmetry relatively of the axis Ox. It is shown the asymmetry of the movement of vortices in such a stream.
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. C A Poslavsky, E Yu, M Bannikova & V, Kontorovich, Astrophysics. 532174C.A. Poslavsky, E.Yu. Bannikova & V.M. Kontorovich, Astrophysics, Vol. 53, No. 2, 174 (2010)
| [] |
[
"Dynamics of the solar magnetic bright points derived from their horizontal motions",
"Dynamics of the solar magnetic bright points derived from their horizontal motions"
] | [
"L P Chitta \nHarvard-Smithsonian Center for Astrophysics\n60 Garden Street MS-1502138CambridgeMAUSA\n\nIndian Institute of Astrophysics\n560 034BangaloreIndia\n",
"A A Van Ballegooijen \nHarvard-Smithsonian Center for Astrophysics\n60 Garden Street MS-1502138CambridgeMAUSA\n",
"L Rouppe Van Der Voort \nInstitute of Theoretical Astrophysics\nUniversity of Oslo\nBlindernP.O. Box 1029NO-0315OsloNorway\n",
"E E Deluca \nHarvard-Smithsonian Center for Astrophysics\n60 Garden Street MS-1502138CambridgeMAUSA\n",
"R Kariyappa \nIndian Institute of Astrophysics\n560 034BangaloreIndia\n"
] | [
"Harvard-Smithsonian Center for Astrophysics\n60 Garden Street MS-1502138CambridgeMAUSA",
"Indian Institute of Astrophysics\n560 034BangaloreIndia",
"Harvard-Smithsonian Center for Astrophysics\n60 Garden Street MS-1502138CambridgeMAUSA",
"Institute of Theoretical Astrophysics\nUniversity of Oslo\nBlindernP.O. Box 1029NO-0315OsloNorway",
"Harvard-Smithsonian Center for Astrophysics\n60 Garden Street MS-1502138CambridgeMAUSA",
"Indian Institute of Astrophysics\n560 034BangaloreIndia"
] | [] | The sub-arcsec bright points (BP) associated with the small scale magnetic fields in the lower solar atmosphere are advected by the evolution of the photospheric granules. We measure various quantities related to the horizontal motions of the BPs observed in two wavelengths, including the velocity auto-correlation function. A 1 hr time sequence of wideband Hα observations conducted at the Swedish 1-m Solar Telescope (SST ), and a 4 hr Hinode G-band time sequence observed with the Solar Optical telescope are used in this work. We follow 97 SST and 212 Hinode BPs with 3800 and 1950 individual velocity measurements respectively. For its high cadence of 5 s as compared to 30 s for Hinode data, we emphasize more on the results from SST data. The BP positional uncertainty achieved by SST is as low as 3 km. The position errors contribute 0.75 km 2 s −2 to the variance of the observed velocities. The raw and corrected velocity measurements in both directions, i.e., (v x , v y ), have Gaussian distributions with standard deviations of (1.32, 1.22) and (1.00, 0.86) km s −1 respectively. The BP motions have correlation times of about 22 − 30 s. We construct the power spectrum of the horizontal motions as a function of frequency, a quantity that is useful and relevant to the studies of generation of Alfvén waves. Photospheric turbulent diffusion at time scales less than 200 s is found to satisfy a power law with an index of 1.59. | 10.1088/0004-637x/752/1/48 | [
"https://arxiv.org/pdf/1204.4362v1.pdf"
] | 118,617,314 | 1204.4362 | a948b2a086a708f8dd7a697ece82a17f9860deef |
Dynamics of the solar magnetic bright points derived from their horizontal motions
19 Apr 2012
L P Chitta
Harvard-Smithsonian Center for Astrophysics
60 Garden Street MS-1502138CambridgeMAUSA
Indian Institute of Astrophysics
560 034BangaloreIndia
A A Van Ballegooijen
Harvard-Smithsonian Center for Astrophysics
60 Garden Street MS-1502138CambridgeMAUSA
L Rouppe Van Der Voort
Institute of Theoretical Astrophysics
University of Oslo
BlindernP.O. Box 1029NO-0315OsloNorway
E E Deluca
Harvard-Smithsonian Center for Astrophysics
60 Garden Street MS-1502138CambridgeMAUSA
R Kariyappa
Indian Institute of Astrophysics
560 034BangaloreIndia
Dynamics of the solar magnetic bright points derived from their horizontal motions
19 Apr 2012Subject headings: Sun: photosphere -Sun: surface magnetism
The sub-arcsec bright points (BP) associated with the small scale magnetic fields in the lower solar atmosphere are advected by the evolution of the photospheric granules. We measure various quantities related to the horizontal motions of the BPs observed in two wavelengths, including the velocity auto-correlation function. A 1 hr time sequence of wideband Hα observations conducted at the Swedish 1-m Solar Telescope (SST ), and a 4 hr Hinode G-band time sequence observed with the Solar Optical telescope are used in this work. We follow 97 SST and 212 Hinode BPs with 3800 and 1950 individual velocity measurements respectively. For its high cadence of 5 s as compared to 30 s for Hinode data, we emphasize more on the results from SST data. The BP positional uncertainty achieved by SST is as low as 3 km. The position errors contribute 0.75 km 2 s −2 to the variance of the observed velocities. The raw and corrected velocity measurements in both directions, i.e., (v x , v y ), have Gaussian distributions with standard deviations of (1.32, 1.22) and (1.00, 0.86) km s −1 respectively. The BP motions have correlation times of about 22 − 30 s. We construct the power spectrum of the horizontal motions as a function of frequency, a quantity that is useful and relevant to the studies of generation of Alfvén waves. Photospheric turbulent diffusion at time scales less than 200 s is found to satisfy a power law with an index of 1.59.
Introduction
The discrete and small scale component of the solar magnetic field is revealed in the high spatial resolution observations of the Sun. Ground based observations (Muller 1983(Muller , 1985Berger et al. 1995) show clusters or a network of many bright points (hereafter BPs) in the inter-granular lanes, with each individual BP having a typical size of 100−150 km. These BPs are known to be kilogauss flux tubes in the small scale magnetic field (SMF), and are extensively used as proxies for such flux tubes (Chapman & Sheeley 1968;Stenflo 1973;Stenflo & Harvey 1985;Title et al. 1987, see de Wijn et al. (2009 for a review on the SMF). High cadence observations and studies show that magnetic BPs are highly dynamic and intermittent in nature, randomly moving in the dark intergranular lanes 1 . These motions are mainly due to the buffeting of granules. The SMF is passively advected to the boundaries of supergranules creating the magnetic network in the photosphere.
Earlier works by several authors have reported mean rms velocities of magnetic elements in the order of a few km s −1 . With the ground based observations of the granules at 5750Å (white light), Muller et al. (1994) have identified many network BPs with turbulent proper motion and a mean speed of 1.4 km s −1 . Berger & Title (1996) have used G-band observations of the photosphere and found that the G-band BPs move in the intergranular lanes at speeds from 0.5 to 5 km s −1 . Berger et al. (1998) observed the flowfield properties of the photosphere by comparing the magnetic network and nonmagnetic quiet Sun. They show that the convective flow structures are smaller and much more chaotic in magnetic region, with a mean speed of 1.47 km s −1 for the tracked magnetic BPs. With the G-band and continuum filtergrams, van Ballegooijen et al. (1998) used an object tracking technique and determined the autocorrelation function describing the temporal variation of the bright point velocity, with a correlation time of about 100 s. Correcting for measurements errors, Nisenson et al. (2003) measured a 0.89 km s −1 rms velocity of BPs. Advances in the ground based observations like rapid high cadence sequences with improved adaptive optics to minimize seeing effects, and also the space based observations at high resolutions, continued to attract many authors to pursue BP motion studies. For example, Utz et al. (2010) used space based Hinode G-band images to measure BP velocities and their lifetime. The BP motions can be used to measure dynamic properties of magnetic flux tubes and their interaction with granular plasma. Photospheric turbulent diffusion is one such dynamical aspect that can be derived consequently from the BP random walk. Manso Sainz et al. (2011) measured a diffusion constant of 195 km 2 s −1 from the BP random walk and their dispersion. Abramenko et al. (2011) studied photospheric diffusion at a cadence of 10 s with high resolution TiO observations of a quiet Sun area. They found a super-diffusion regime, satisfying a power law of diffusion with an index γ = 1.53, which is pronounced in the time intervals 10 − 300 s.
The implications of these magnetic random walk motions have been found very fruitful recently. Such motions are capable of launching magneto-hydrodynamic (MHD) waves (Spruit 1981), which are potential candidates for explaining the high temperatures observed in the solar chromosphere and corona. For example, a three-dimensional MHD model developed by van Ballegooijen et al. (2011) suggests that random motions inside BPs can create Alfvén wave turbulence, which dissipates the waves in a coronal loop (also see Asgari-Targhi & van Ballegooijen 2012). Observations by De Pontieu et al. (2007b); Jess et al. (2009);McIntosh et al. (2011) provide strong evidence that the Alfvénic waves (which are probably generated by the BP motions), have sufficient energy to heat the quiet solar corona. To test theories of chromospheric and coronal heating, more precise measurements of the velocities and power spectra of BP motions are needed. Nisenson et al. (2003) worked on the precise measurements of BP positions, taking into account the measurement errors. The auto-correlations derived by them for the x− and y− components of the BP velocity using high spatial resolution and moderate cadence of 30 s observations gave a correlation time of about 60 s, which is twice the cadence of the observations. This suggests an over-estimation of correlation time and an under-estimation of the rms velocity power, with a significant hidden power in the time scales less than 30 s, and thus warranting for observations at even higher cadence. This is important because, the measured power profile, which is the Fourier transform of the auto-correlation function, gives us an estimation of the velocity amplitudes and energy flux carried by the waves that are generated by the BP motions in various and especially at higher frequencies.
In this study, we use 5 s cadence wideband Hα observations from the Swedish 1-m Solar Telescope to track the BPs and measure their rms velocities. For comparison, we also use a 30 s cadence G-band observational sequence from Solar Optical Telescope onboard Hinode. These independent and complementary results take us closer to what could be the true rms velocity and power profile of the lateral motions of the BPs. The details of the datasets used, analysis procedure, results and their implications are discussed in the following sections.
Datasets
In this study, we have analyzed time sequence of intensity filtergrams with 5 and 30 s cadence. A brief description of the observations is given below. ;van Noort et al. 2005) image restoration method under excellent seeing conditions. The target area is a quiet Sun region away from disk center at (x, y) = (−307 ′′ , −54 ′′ ) and µ = 0.94 (see Figure 1). The time sequence is of one hour duration starting at 13:10 UT. Here we analyze images from the wideband channel of the Solar Optical Universal Polarimeter (SOUP; Title & Rosenberg 1981) which received 10% of the light before the SOUP tunable filter but after the SOUP prefilter (see De Pontieu et al. 2007a, for the optical setup of the instrument). The prefilter was an FWHM=8 A wide interference filter centered on the Hα line. The SOUP filter was tuned to the blue wing of Hα at -450 mÅ but that data is not considered here. On the wideband channel there were 2 cameras (running at 37 frames per sec) positioned as phase-diversity pair -one in focus and one camera 13.5 mm out of focus. the data from the two cameras has been processed with the MOMFBD restoration method in sets of 5 seconds, creating a 5 s cadence time sequence with a total of 720 images. After MOMFBD processing, the restored images were de-rotated to account for the field rotation due to the altazimuth mount of the telescope. Furthermore, the images were aligned using cross-correlation on a large area of the field of view (FOV) . The images were then clipped to 833 × 821 pixels (with 0.065 ′′ per pixel), to keep the common FOV (the CCDs have 1024 × 1024 pixels, some pixels are lost after alignment between focus and defocus cameras).
For a reference direction, the solar north in the SST time sequence is found by aligning an earlier SST observation of that day of a magnetogram of an active region (AR) to a full disk SOHO /MDI magnetogram (the AR was just outside the MDI High Res region). From that comparison, we fix the direction of solar north and disk center (black and white arrows respectively in Figure 1). Though we do not rotate the images to match the solar north during our analysis, the angles are taken into account at a later stage to correct for the projection effects in the velocity measurements.
30 sec data: We use G-band filtergrams observed with Solar Optical Telescope (SOT) on board Hinode (Kosugi et al. 2007;Tsuneta et al. 2008), on 14 April 2007. The observations were made for a duration of 4 hr, with a 30 s cadence in a FOV of 55 ′′ × 55 ′′ ( 0.05 ′′ per pixel; 1024 pixels in both x and y directions), near disk center. The images were processed using standard procedures available in the solarsoft library.
Procedure
In this section, we briefly describe the method of determining the BP positions, and the velocity measurements through the correlation tracking.
BP Positions
We manually select the BPs to estimate their position to a sub-pixel accuracy. We consider the coordinates of maximum intensity of a given BP to be the position of that BP and the method for measuring these positions involves two steps. In the first step, we visually identify a BP and it is selected for analysis for a period during which it is clearly distinguishable from the surrounding granules. On an average, we follow a BP for about 3 − 5 min. The BPs with elongated shapes are not considered for the analysis. Also, we stop following a BP if it is substantially distorted or elongated from its initial shape. Though time consuming, manual selection gives a handle on the validity of the positional accuracy of a BP from frame to frame. At each time step, using a cursor, an approximate location (x ′ app , y ′ app ) of a particular BP is fed to an automated procedure to get its accurate position, which is step two in our method.
Step two is completely an automated procedure. Here, we use a surface interpolation technique to get a precise position of that BP (to a sub-pixel accuracy). Approximate position from the previous step is used to construct a grid of 5 × 5 pixels covering the full BP (with (x ′ app , y ′ app ) as the center of that grid). Now, our procedure fits a 2-D, 4 th degree surface polynomial to that grid (using SF IT , an IDL procedure); interpolates the fit to one-hundredth of a pixel; returns the fine location of its peak (δx ′ , δy ′ ) within that grid and finally stores the accurate position (x ′ BP , y ′ BP ) of that BP (which is the sum of its approximate and fine positions (x ′ app + δx ′ , y ′ app + δy ′ )), for further analysis. Therefore, the position of a BP with index j in a frame i is given by
(x ′ BP , y ′ BP ) j i = (x ′ app + δx ′ , y ′ app + δy ′ ) j i ,(1)
and all the coordinates till this point are relative to the lower left corner of the image.
Reference Frame
Though the positional measurements of BPs as described in Section 3.1 are accurate, they cannot be directly used to measure the velocities as there are artificial velocity sources viz. the instrumental drifts, seeing variations, jittery motions and also the solar rotation, which vectorially add to BP velocities and thus are required to be removed from the analysis. While Hinode (space based) data is not subjected to seeing variations, SST (ground based) data has been corrected for seeing as described in Section 2. Further, we need to correct for instrumental drifts, jitters and solar rotation. Calculating the offsets between the successive images is necessary to remove these artificial velocities. In this section, we describe the method of our cross correlation analysis used to co-align the images.
Cross correlation (C) of two images f (x, y) and g(x, y) is defined as
C = 1 k − 1 x,y (f (x, y) − f )(g(x, y) − g) σ f σ g ,(2)
where, f [g] and σ f [σ g ] are mean value and standard deviation of f (x, y) [g(x, y)] respectively, k is the number of pixels in each image, for normalization. With the above definition of cross correlation, to get the offsets between two images, we need to shift one image with respect to the other (in both x and y directions ) and find at what offsets (independent in x and y) the correlation function attains the maximum value. In general, for shifts of −l to +l, the cross correlation is a 2-D function with 2l + 1 rows and columns. Let l x and l y be the coarse offsets between the two images in x and y directions, respectively, such that the cross correlation reaches its maximum value: max(C) = C(l x , l y ), where −l < l x , l y < l. To get the sub-pixel offsets, the fine offsets (δl x , δl y ) are calculated. The method is similar to finding the fine position of BP by using a 5 × 5 pixel grid but now about (l x , l y ) of C.
Instead of cross correlating every successive image with its previous one, we keep a reference image for about 200 s, i.e., a frame i taken at time t (i t ) is used as a reference for the subsequent frames till t+200 s (i t+200 ) for cross correlation. Therefore the 5 sec (SST ) and the 30 sec ( Hinode) data have about 40 and 7 images respectively in each set. By keeping the last image of a set equal to the first image in its next set, we can co-align different sets. In this way, the accumulation of errors in the offsets can be minimized.
With the above background on co-aligning images to find various drifts, we present the results of drifts found in Hinode data. As an illustration, we divide the full (i.e., 55 ′′ × 55 ′′ × 4 hr) Hinode time sequence into 4 quadrants with 27.5 ′′ × 27.5 ′′ × 4 hr each. Further, we do correlation tracking (as described above by keeping 7 frames per set) on each quadrant separately and plot the results in Figure 2. The four dashed lines in the left and the right panels are the offsets in x− and y− directions respectively, the thick dashed line in each panel is the average of the offsets (i.e., average of four dashed lines), and the solid red curve is the offset obtained by considering the full FOV. Clearly, in each quadrant, the offsets have a trend similar to that of the full FOV (solid red curve) and an additional component of their own. This additional component is probably the real velocity on the Sun due to flows with varying length scales (for example, super-granular, meso-granular and granular) and with flow directions changing over areas of a few tens of arcsec 2 on the Sun.
In this paper we are mainly interested in the dynamics of the BPs relative to their local surroundings, as granulation flows will have a dominant effect on the BP velocities and their variations on short time scales. Hence, we consider a 5 ′′ × 5 ′′ area about the BP as a reference frame for that BP (i.e., keeping the BP in the center of the local area). The cross correlation is performed on this 5 ′′ × 5 ′′ area instead of the full FOV to get the offsets, which are subtracted from the (x ′ BP , y ′ BP ) j i . The BP positions corrected for offsets are now given by
(x ′ BP C , y ′ BP C ) j i = (x ′ BP , y ′ BP ) j i − (l x + δl x , l y + δl y ) j local i ,(3)
where j local represents the local area of BP j .
In the case of the SST data, the observations are off disk center at (−307 ′′ , −54 ′′ ), which corresponds to a heliocentric angle of arccos(0.94). This will introduce a projection effect on the measured horizontal velocities in both x ′ − and y ′ − directions and needs to be corrected. To do this, the coordinate system (x ′ , y ′ ) defined by the original SST observations, is rotated by 45°in the anti-clockwise direction. Now the image plane is oriented in E-W (parallel to equator, new x−) and N-S (new y−) directions. Further, the E-W coordinate is multiplied by a factor of 0.94 −1 . Hence the new coordinate system (x, y) is given by
x = (x ′ cos 45°+ y ′ sin 45°) × 1 0.94 , y = (−x ′ sin 45°+ y ′ cos 45°).(4)
SST BP positions (x ′ BP C , y ′ BP C ) j i , as measured from Equation (3), are remapped to (x BP C , x BP C ) j i , using the above coordinate transformations 2 .
Results
In this section, we present various results in detail giving more emphasis on the SST results. We have selected 97 SST BPs with ∼3800 individual velocity measurements 3 . Figure 3 shows the paths of four individual SST BPs. Some of the BPs move in a relatively smoother path while some exhibit very random motions to the shortest time steps available. BPs drift about a few hundred km in a few minutes. The instantaneous velocity (v x , v y ) j i+1 of a BP is given by (x BP C , y BP C ) j i+1 − (x BP C , y BP C ) j i , multiplied by a factor to convert the units of measured velocity to km s −1 (9.4 in case of SST which is, image scale of SST in km divided by the time cadence in sec). Figure 4 shows the plot of such velocities as a function of time for BP#3 (path of BP#3 is shown in the lower left panel of Figure 3). Usually, the changes in the velocity are gradual in time but, sometimes we do see sudden and large changes in the magnitude and direction of the velocity (for example at 1 min in v x and at 2 min in v y in Figure 4). Note that a large change of velocity of one sign is followed immediately by a change of the opposite sign, so the net change in position is not very large. This suggest that these changes are due to errors in the positional measurements. A position error at one time will affect the velocities in the intervals immediately before and after that time. In the following we will assume that such changes in velocity are due to measurement errors. However, we cannot rule out that some of these changes are due to real motions on the Sun on a time scales less than 5 seconds.
The means and standard deviations of v x and v y are listed in Table 1 (first line). Histograms of the distribution of velocities v x , v y and v = v 2
x + v 2 y , are shown in Figure 5 (panels (a), (b) and (c) respectively). Solid lines in panels (a) and (b) are Gaussian fits to the histograms with raw standard deviations (σ v,r ) of 1.32 and 1.22 km s −1 . A scatter plot of v x against v y is shown in panel (d), which is symmetric in the v−space. However, a small non-zero and positive mean velocity of about 0.2 km s −1 is noticed, suggesting that there is a net BP velocity with respect to the 5 arcsec boxes that we used as reference frames. Values of the mean and rms velocities as determined from the fits are also listed in Table 1 (second line). These distributions are a mix of both true velocities and measurement errors.
We can gain more insight into the the dynamical aspects of the BP motions by studying their 2 Note that the transformations in Equation (4) are only to modify the SST BP positions and in the rest of the paper, we use (x, y) for the remapped (x ′ , y ′ ) of SST and (x, y) of Hinode.
3 Similarly, we have identified 212 Hinode BPs with 1950 individual velocity measurements. observed velocity correlation function c(t), defined as
c xx,n = v j x,i v j x,i+n , c yy,n = v j y,i v j y,i+n(5)c xy,n = v j x,i v j y,i+n ,(6)
where c xx,n , c yy,n are the auto-correlations, and c xy,n is the cross-correlation of v x and v y , n is the index of the delay time. . . . denotes the average over all values of the time index i and BP index j but for a fixed value of n. These results are shown in Figure 6. Top left and right panels are the plots of c xx and c yy respectively. Black curves are for the SST whereas the red curves show the Hinode results for comparison. Both the SST and Hinode results are consistent for delay times < 1 min. However, the Hinode auto-correlations quickly fall to lower values. This is mainly a statistical error, since we do not have a large number of measurements in the case of Hinode. Focusing on periods < 1 min, it is clear from the auto-correlation plots that the core of the Hinode data within ±30 s delay time, which is sampled with three data points is now well resolved with the aid of the SST data. Also, at shorter times, c takes a cusp-like profile. Extrapolating this to delay times of the order of 1 s, we expect to see a steep increase in the rms velocities 4 of the BP motions. The bottom left panel shows the cross-correlation as function of delay time. The SST data show a small but a consistent and overall negative c xy while the Hinode data show a small positive correlation. We suggest that the real cross-correlation c xy ≈ 0, and that the measured values are due to a small number of measurements with high velocities (largely exceeding the rms values). The lower right panel of Figure 6 shows the number of measurements N n used in the correlation analysis for both the SST and Hinode data. To obtain good statistics we collected enough BP measurements to ensure that N n 500 for all bins.
In the rest of the section, we describe the method of estimating the errors in the velocity measurements due to positional uncertainties by analyzing c(t). Following Nisenson et al. (2003), we assume that the errors in the positions are uncorrelated from frame to frame and randomly distributed with a standard deviation of σ p . Since the velocities are computed by taking simple differences between position measurements (see above), the measurement errors increase the observed velocity correlation at n = 0 by ∆ (error), and reduce the correlations at n = ±1 by −∆/2 where ∆ = 2(σ p /δt) 2 and δt is the cadence (see Equation 3 in their paper). We define
∆ n = ∆ when n = 0 − 1 2 ∆ when n = ±1 0 otherwise,(7)
which is valid only with our two-point formula for the velocity. Once ∆ is determined, the rms values (σ v,c ) of the true solar velocities can be measured as σ 2 v,c = σ 2 v,r − ∆.
A previous study using data from the Swedish Vacuum Solar Telescope (van Ballegooijen et al. 1998) assumed c(t) to be a Lorentzian. Here, we clearly see that c(t) differs from a Lorentzian, and it can be fitted with a function C, which is a sum of the true correlation of solar origin (C ′ ) and ∆, given by
C n (∆, τ, κ) = C ′ n (τ, κ) + ∆ n ,(8)
where
C ′ n (τ, κ) = a + b 1 + |tn| τ κ (9)
is a generalized Lorentzian. ∆, τ (correlation time), and κ (exponent) are the free parameters of the fit; a and b are the functions of (∆, τ, κ), which are determined analytically by least square minimization (see Appendix A). We also bring to the notice of the reader that our formula for C is a monotonically decreasing function of t n . However, there is an unexplained increase in the observed c yy beyond ±100 s (panel (b) in Figure 6). To eliminate any spurious results due to this anomaly, we use a maximum delay time of ±105 s to fit c with C by minimizing the sum of the squares of their difference, as defined in Equation (A1).
The top panel in Figure 7 shows the results listing the best fit values of the free parameters (∆, τ, κ), a, and b for a maximum t n of ±105 s. C (black) and C ′ (thin red) are plotted as functions of the delay time over c xx,n (left, symbols), and c yy,n (right, symbols). The value of ∆ where χ 2 has its global minimum is found to be 0.75 km 2 s −2 , for both c xx and c yy . The bottom panel shows the contours of χ 2 as a function of τ and κ at ∆ = 0.75 km 2 s −2 , and the min(χ 2 ) is denoted by plus symbols. Dashed and solid lines are the regions of 1.5 and 2 times the min(χ 2 ) respectively. χ 2 is a well bounded function for κ < 2, confirming a cusp-like profile. The correlation time is 22 − 30 s, which is about 4 − 6 times the time cadence.
Taking into account the variance in errors (i.e., ∆ = 0.75 km 2 s −2 ), we get σ p = 3 km, and the corrected rms velocities (σ v,c ) of v x and v y are now 1.00 and 0.86 km s −1 . These results are plotted as dashed curves in panels (a) and (b) of Figure 5, and the values are tabulated in the last row of Table 1. The corrected distribution of v is shown as a dashed Rayleigh distribution in panel (c). With higher cadence observations, these results can be refined and modified, as (∆, τ, κ) depend on the shape of the core of c. By comparing the SST and Hinode results, we expect that the observed c probably increases rapidly below 5 s and thus changing the set of parameters to some extent.
Summary and Discussion
We studied the proper motions of the BPs using the wideband Hα observations from the SST and the G-band data from Hinode. BPs are manually selected and tracked using 5 ′′ × 5 ′′ areas surrounding them as reference frames. The quality of the SST observations allowed us to measure the BP positions to a sub-pixel accuracy with an uncertainty of only 3 km, which is at least seven times better than the value reported by Nisenson et al. (2003), and comparable to the rms value of 2.7 km due to image jittering reported by Abramenko et al. (2011). They adopted this rms value of 2.7 km as a typical error of calculations of the BP position. We found that the horizontal motions of the BPs in x and y are Gaussian distributions with raw (including the true signal and measurement errors) rms velocities of 1.32 and 1.22 km s −1 , symmetric in v−space, observed at 5 s cadence. The above estimate of the measurement uncertainty is obtained from a detailed analysis of the velocity auto-correlation functions. For this, we fitted the observed c(t) with C, a function of the form shown in Equation (8), and estimated an rms error of about 0.87 km s −1 in v x and v y . The removal of this error makes the v x and v y Gaussians narrower with new standard deviations 1.00 and 0.86 km s −1 (a fractional change of 30%). The total rms velocity (v x and v y combined) is 1.32 km s −1 . The correlation time is found to be in the range of 22 − 30 s.
Following is a brief note and discussion on the additional results we derive from our work. BPs are advected by the photospheric flows. Thus, taking these features as tracers, we can derive the diffusion parameters of the plasma. As BPs usually have life times of the order of minutes, the motion of these features can be used to study the nature of photospheric diffusion at short time scales. The mean squared displacement of BPs (∆r) 2 as function of time is a measure of diffusion. It is suggested in the literature that (∆r) 2 can be approximated as a power law with an index γ (i.e., (∆r) 2 ∼ t γ , see for example Cadavid et al. 1999;Abramenko et al. 2011). In Figure 8, we plot the observed (∆r) 2 (symbols), for the 200 s interval on a log-log scale. Solid line is the least square fit with a slope of 1.59, which is consistent with the value γ = 1.53 found by Abramenko et al. (2011) for quiet sun. Despite the differences in the observations (instruments and observed wavelengths), and analysis methods (identification and tracking of BPs), a close agreement in the independently estimated γ suggests that this is a real solar signal. Both these results assert the presence of super-diffusion (i.e., γ > 1), for time intervals less than 300 s. Since most of the BPs in this study are tracked for only 3 − 4 min, we cannot comment on the diffusion at longer times.
Note that there is a general relationship between the mean squared displacement (∆r) 2 and the velocity auto-correlation function C ′
(∆r) 2 = t 0 v x (t ′ )dt ′ 2 + t 0 v y (t ′ )dt ′ 2 (10) = 2 t 0 t 0 C ′ (t ′′ − t ′ )dt ′ dt ′′ ,(11)
where we assume isotropy of the BP motions (C ′ xx = C ′ yy = C ′ ). For a known auto-correlation or mean squared displacement, the other quantity can be derived using the above relation.
We already saw that the horizontal motions of the BPs yield several important properties of the lower solar atmosphere. One more such important property is the possibility of the generation of Alfvén waves due to these motions. Here we qualitatively estimate and compare the power spectrum of horizontal motions as a function of frequency for two forms of the velocity correlation function 5 : (a) the form C ′ (Equation (9)), obtained in this study, and (b) a Lorentzian function. For case (a) we use a = 0, and also assume that C ′ xx,n = C ′ yy,n , with the parameters b, τ , and κ taking the mean values of x and y. For case (b) we use a modified form of C ′ with κ = 2. The other parameters (a, b, and τ ) are the same as in case (a). Figure 9 shows the power spectra for the two described cases: (a) solid line, and (b) dashed line. We observe that for frequencies exceeding 0.02 Hz (< 50 s), the horizontal motions generally have more power in case (a) as compared to case (b). This highlights the fact that the dynamics of the BPs at short time scales are very important. Therefore, it is highly desirable to do these observations and calculations at very high cadence.
The measurements presented in this paper provide important constraints of models for Alfvén and kink wave generation in solar magnetic flux tubes. As discussed in the Introduction, such waves may play an important role in chromospheric and coronal heating. In the Alfvén wave turbulence model (van Ballegooijen et al. 2011;Asgari-Targhi & van Ballegooijen 2012) it was assumed that the photospheric footpoints of the magnetic field lines are moved about with rms velocity of 1.5 km s −1 , similar to the rms velocity of 1.32 km s −1 found here. However, the models include only the internal motions of a flux tube, whereas the observations refer to the displacements of the flux tube as a whole. Clearly, to make more direct comparisons between models and observations will require imaging with high spatial resolution (< 0.1 arcsec). This may be possible in the future with the Advanced Technology Solar Telescope.
In this work we presented the results of the BP motions, some of their implications and use in the context of photospheric diffusion and coronal wave heating mechanisms. We interpret the location of the intensity maximum of a BP as its position at any given time. This is certainly plausible for time periods when we begin to see the physical motion of a BP as a rigid body due to the action of the convection on the flux tubes. But at timescales shorter than one minute, other interpretations are also plausible: the motions marked by the intensity maxima could be intensity fluctuations in an otherwise static BP. Nevertheless, these fluctuations are manifestations of some disturbances inside the BP, which are equally important and interesting to explore further. Table 1. Properties of the velocity distributions in Figure 5 v
x (km s −1 ) v y (km s −1 ) v x σ(v x ) v y σ(v y )Histogram
A. Determination of a and b
In this section, we briefly describe a method of determining a and b for a set of parameters (∆, τ, κ). We define χ 2 of autocorrelation functions of v x and v y as
χ 2 xx (∆, τ, κ) = N n=−N [c xx,n − C n (∆, τ, κ)] 2 , and χ 2 yy (∆, τ, κ) = N n=−N [c yy,n − C n (∆, τ, κ)] 2 (A1)
where, c xx,n and c yy,n are the observed autocorrelation values of velocities v x and v y and C is a model of the correlation function given by Equation (8). By minimizing the χ 2 with respect to a and b (i.e., ∂χ 2 ∂a = 0 and ∂χ 2 ∂b = 0, separately for x and y), and solving the resulting system of linear equations, we have ∆, τ , and κ obtained by minimizing the χ 2 (see Appendix A), of the observed c (shown as symbols, c xx : left, and c yy : right, also shown as black curves in the top panels of Figure 6), and the modeled correlation function C, for a delay time of ±105 s in steps of 5 s. Thin red curve is the profile of C ′ . Bottom: Contour plots of χ 2 as a function of κ and τ , for a value of ∆ (∆ x = ∆ y = 0.75 km 2 s −2 ), where χ 2 attains the global minimum. Plus symbol is the global minimum of χ 2 , dashed and solid lines are the contours of 1.5min(χ 2 ) and 2min(χ 2 ) respectively. -from a Lorentz profile with same a, b, and τ as in case (a) but with κ = 2 (see text for details).
a = 1 αβ ′ − α ′ β β ′ A − βB (A2) b = 1 αβ ′ − α ′ β αB − α ′ A(
5 sec data: These observations were obtained on 18 June 2006, with the Swedish 1-m Solar Telescope (SST ; Scharmer et al. 2003a) on La Palma, using the adaptive optics system (AO; Scharmer et al. 2003) in combination with the Multi-Object Multi-Frame Blind Deconvolution (MOMFBD
Fig. 1 .Fig. 2 .
12-First image from the time sequence of SST wideband Hα observations at 13:10 UT on 18 June 2006. Black arrow is pointing towards solar north and the white arrow is towards disk center. -Illustration of different offsets seen in the full and partial FOV of Hinode data. Dashed curves in the left (right) panel show the drifts in the x− (y−) direction of the four selected quadrants. Thick dashed profile is the average of four dashed curves. Thick red profile is the drift of the full FOV (see text for details).
Fig. 3 .
3-Examples of the paths of four BPs taken from the SST data. Initial position of each BP is marked with a star. Time shown at the top right corner in each panel is the duration for which respective BP is followed.
Fig. 4 .
4-Velocities v x and v y as a function of time for a typical BP (shown here for BP#3, seeFigure 3for the track of BP#3).
Fig. 5 .
5-Histograms of measured BP velocities:(a) v x , (b) v x , (c) v = v 2 x + v 2y . Solid Gaussians in the top panels are fits to the histograms. Dashed Gaussians in panels (a), (b), and dashed Rayleigh profile in panel (c), are the new distributions of velocities after correcting for the measurement errors (see text for details); (d) shows v x plotted against v y .
Fig. 6 .
6-Correlation functions of BP velocity v x and v y . (a) Observed auto-correlation c xx,n as a function of delay time t (black: SST, red: Hinode). (b) Similar for the observed auto-correlation c yy,n . (c) Cross-correlation c xy,n as function of delay time. (d) Number of measurements per bin used in panels (a), (b), and (c).
Fig. 7 .
7-Top: C (black curve) plotted as a function of delay time with the best fit values of a, b,
Fig. 9 .
9-Power spectrum of the horizontal motions of BPs as a function of frequency derived from auto-correlation function for two cases. Solid line: case (a) -from this study. Dashed line: case (b)
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The correlation at zero-time lag is the variance of the velocity distribution.
Fourier transform of the velocity auto-correlation is the power spectrum.
This preprint was prepared with the AAS L A T E X macros v5.2.
Authors thank the referee for many comments and suggestions that helped in improving the presentation of the manuscript. LPC. is a 2011 − 2012 SAO Pre-Doctoral Fellow at the Harvard-Smithsonian Center for Astrophysics. The Swedish 1-m Solar Telescope is operated on the island of La Palma by the Institute for Solar Physics of the Royal Swedish Academy of Sciences in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. Funding for LPC and EED is provided by NASA contract NNM07AB07C. Hinode is a Japanese mission developed and launched by ISAS/JAXA, collaborating with NAOJ as a domestic partner, NASA and STFC (UK) as international partners. Scientific operation of the Hinode mission is conducted by the Hinode science team organized at ISAS/JAXA. This team mainly consists of scientists from institutes in the partner countries. Support for the post-launch operation is provided by JAXA and NAOJ (Japan), STFC (U.K.), NASA (U.S.A.), ESA, and NSC (Norway). This research has made use of NASA's Astrophysics Data System.
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Mean squared displacement (∆r) 2 as a function of time t on a log-log scale. Solid line is the least square fit of the observations (symbols. with a slope of 1.59Fig. 8.-Mean squared displacement (∆r) 2 as a function of time t on a log-log scale. Solid line is the least square fit of the observations (symbols), with a slope of 1.59.
| [] |
[
"Quality Assessment of Low Light Restored Images: A Subjective Study and an Unsupervised Model",
"Quality Assessment of Low Light Restored Images: A Subjective Study and an Unsupervised Model"
] | [
"Vignesh Kannan [email protected] \nIndian Institute of Science\n\n",
"Sameer Malik [email protected] \nIndian Institute of Science\n\n",
"Rajiv Soundararajan [email protected] \nIndian Institute of Science\n\n"
] | [
"Indian Institute of Science\n",
"Indian Institute of Science\n",
"Indian Institute of Science\n"
] | [] | The quality assessment (QA) of restored low light images is an important tool for benchmarking and improving low light restoration (LLR) algorithms. While several LLR algorithms exist, the subjective perception of the restored images has been much less studied. Challenges in capturing aligned low light and well-lit image pairs and collecting a large number of human opinion scores of quality for training, warrant the design of unsupervised (or opinion unaware) no-reference (NR) QA methods. This work studies the subjective perception of low light restored images and their unsupervised NR QA. Our contributions are two-fold. We first create a dataset of restored low light images using various LLR methods, conduct a subjective QA study and benchmark the performance of existing QA methods. We then present a self-supervised contrastive learning technique to extract distortion aware features from the restored low light images. We show that these features can be effectively used to build an opinion unaware image quality analyzer. Detailed experiments reveal that our unsupervised NR QA model achieves state-of-the-art performance among all such quality measures for low light restored images. | null | [
"https://arxiv.org/pdf/2202.02277v1.pdf"
] | 246,608,128 | 2202.02277 | 81dafff94bd1bd5c714a2cc46705dec6255f09ad |
Quality Assessment of Low Light Restored Images: A Subjective Study and an Unsupervised Model
Vignesh Kannan [email protected]
Indian Institute of Science
Sameer Malik [email protected]
Indian Institute of Science
Rajiv Soundararajan [email protected]
Indian Institute of Science
Quality Assessment of Low Light Restored Images: A Subjective Study and an Unsupervised Model
The quality assessment (QA) of restored low light images is an important tool for benchmarking and improving low light restoration (LLR) algorithms. While several LLR algorithms exist, the subjective perception of the restored images has been much less studied. Challenges in capturing aligned low light and well-lit image pairs and collecting a large number of human opinion scores of quality for training, warrant the design of unsupervised (or opinion unaware) no-reference (NR) QA methods. This work studies the subjective perception of low light restored images and their unsupervised NR QA. Our contributions are two-fold. We first create a dataset of restored low light images using various LLR methods, conduct a subjective QA study and benchmark the performance of existing QA methods. We then present a self-supervised contrastive learning technique to extract distortion aware features from the restored low light images. We show that these features can be effectively used to build an opinion unaware image quality analyzer. Detailed experiments reveal that our unsupervised NR QA model achieves state-of-the-art performance among all such quality measures for low light restored images.
Introduction
The ability to produce good quality images under low ambient lighting conditions is an important feature of modern mobile photography. Low light image restoration involves solving the twin challenges of contrast enhancement and denoising. While several classical approaches based on histogram equalization [2,5,24,67] and retinex theory [16,25] have been studied in literature, the emergence of deep learning techniques [6,26,43,58,64] has evoked a lot of research interest in this area. Despite the existence of a plethora of low light restoration algorithms, the quality assessment (QA) of the restored images has received much less attention. The focus of our work is in the study of perceptual quality assessment of low light restored images.
Although several large scale generic image QA (IQA) databases exist, the subjective quality assessment of low light images is still a nascent research area. The camera image database (CID) [42] consists of a variety of images captured by cameras, including low light distorted images. The natural night-time image database (NNID) [51] consists of a diverse set of low light images captured from three cameras. While the subjective QA of low light image captures has been studied, there is no large-scale study of low light restored images. The distortions that arise while applying contrast enhancement and denoising algorithms on low light images are more diverse and different from images captured under low light conditions. The problems associated with the restoration algorithms themselves lead to several artifacts. In particular, low light restored images often suffer from various combinations of blur, noise, over/under enhancement, poor color saturation, and color casts. Thus there is a need for a detailed subjective study of low light restored images. This would help better compare the performance of low light images restored using different algorithms and advance their design.
On the other hand, the objective assessment of low light restored images is also less explored. While full reference algorithms such as peak signal to noise ratio (PSNR), structural similarity (SSIM) [46] and LPIPS [62] are typically used for assessing the quality of the low light restored images, the requirement for a reference high quality image can be a limitation. Thus, there is a need to design no reference (NR) IQA algorithms. Further, NR IQA algorithms are often designed by training on a dataset of human opinion scores and evaluated on a test set. Since it is cumbersome to collect such human scores, such supervised algorithms are difficult to train on each dataset. Their generalization performance across datasets has also been limited [30]. Thus, there is a need to study unsupervised NR IQA algorithms without any training on human opinion scores.
The NIQE [31] and IL-NIQE [60] indices represent seminal examples of unsupervised NR IQA algorithms. They are designed by extracting natural scene statistics based features and comparing with a corpus of pristine images.
Since, they do not particularly account for the distortions that arise during low light restoration, our experiments reveal that their performance for evaluating low light restored images is limited. This motivates the study and design of unsupervised NR IQA algorithms for low light restored images. Our goal is to design NR IQA algorithms that use a large corpus of low light restored images without training them using human labels.
Our main contributions in addressing the above challenges are as follows:
Dataset for Subjective Assessment of Low Light Restoration. We create a large dataset of 1035 low light restored images obtained from a variety of contrast enhancement, denoising and joint contrast enhancement and denoising algorithms. Our dataset is unique and different from other IQA datasets in terms of the generation of distorted images through low light restoration algorithms. We then conduct an online subjective study involving 88 human subjects with 22,500 ratings.
Performance Benchmarking of IQA Measures on Low Light Restored Images. We conduct a detailed benchmarking of the performance of several popularly used full reference and no reference image quality measures against the subjective scores obtained through our study.
Unsupervised Quality Assessment of Low Light Restored Images. We present a scene wise multi-scale multiview contrastive learning method on a large corpus of low light restored images to learn quality aware features. We then compare these features against features from a corpus of unpaired sharp and colorful high quality images captured under low light conditions to design an unsupervised NR IQA model. We refer to our method as M-SCQALE (Multiscale -Sub-band Contrastive learning for Quality Assessment of Low light Enhancement). We show that M-SCQALE achieves the state-of-the-art performance in terms of correlation with human scores when compared to all other NR unsupervised quality measures.
Our database and code will be made publicly available.
Related work
Datasets for QA of low light restoration: While lowlight restoration datasets such as SID [6], LOL [48], ELD [49] contaning low-light / well-lit pairs exist in literature, there are no QA studies of low light restored images. Xiang et al. [51] create a night time IQA dataset which assesses low light images having camera/device induced distortions. CID [42] is an older database containing fewer images, a more diverse set of distortions but catering to both low light and well lit scenarios. Both the above datasets contain camera captured distortions but do not address images obtained through low light restoration algorithms.
No Reference image quality assessment: NR-IQA methods can broadly be divided into two categories viz. su-pervised (or opinion aware) and unsupervised (or opinion unaware) methods. Classical supervised method involve designs of handcrafted features which are regressed to learn the mapping to the quality labels. Some popular methods include BRISQUE, an NSS based method that extracts features in the spatial domain [30] and CORNIA, a dictionary based feature learning framework [53]. Several deep learning methods have also been successfully used for NR IQA. Zhang et al. [63] propose using a deep bilinear network that works well for both synthetic as well as authentic distortions. Zhu et al. [65] propose a meta-learning based method to learn a quality prior model which is then fine tuned on the target NR-IQA task. Other approaches include learning the distribution of scores instead of a single label [38] and studying the relationships between patches and picture quality [57]. Among methods to assess low light camera captured distortions, Xiang et al. [51] propose a night time IQA metric learnt using brightness and texture features.
Unsupervised methods are much less explored in literature. Mittal et al. [31] propose a completely blind quality analyzer that is based on measuring deviations from statistical regularities found in natural images. Zhang et al. [60] extend this frame to enrich features with more statistical features. Gu et al. [52] design a contrast distortion based metric which combines local as well as global aspects of an image. Nevertheless, unsupervised quality measures have not been specifically studied in the context of low light image restoration.
Self-Supervised feature learning: Self-supervised learning has been studied extensively in literature for image classification, object detection and segmentation. Among these, contrastive learning based methods [4,9,10,39,50] have been very successful. Wu et al. [50] propose an instance discrimination based classification problem and use noise contrastive estimation to learn features. He et al. [18] build large and consistent dynamic dictionaries for unsupervised feature learning. Chen et al. [7] do away with the need for memory banks and specialized architectures by proposing a simple framework for contrastive learning of visually relevant features. Tian et al. [39] present a multiview contrastive learning method such that the mutual information between different views of a scene is maximized. We find that the multiview contrastive learning approach lends itself naturally to isolate the quality aware features.
Dataset and Subjective Study
We create DSLR -Dataset for Subjective assessment of Low light Restoration. DSLR is unique in terms of the application of diverse low light restoration algorithms on low light captured images leading to a wide spectrum of distortion types. It consists of 1035 low light restored images across 80 different scenes. Further we conduct an online subjective study thereby collecting a total of 22,500 quality ratings from 88 subjects.
Dataset
To create the low light restored images, we use lowlight images from two publicly available datasets SID (test set) [6] and ELD [49]. These images are converted from RAW format to sRGB using the python rawpy library. Further we downsample the images to a resolution of 624 × 936 without changing its aspect ratio. We use a combination of 14 contrast enhancement techniques coupled with 3 denoising methods, and also 7 joint contrast enhancement and denoising techniques including deep learning methods to generate a huge corpus of around 27,000 images. A detailed list of techniques used can be found in the supplementary.
Since there is redundancy in the nature of distortions in this large corpus, we select a subset of 1035 images for the subjective study. We perform this selection by first identifying majorly occurring distortion types such as noise, blur, color distortions, over-enhancement, under-enhancement, and poor color saturation. For distorted versions of each scene, we classify the images according to the above distortion types, and select a subset of images for these distortion types spanning the low, medium and high quality ranges. When a restored image suffers from multiple distortion types, we associate it with the dominant distortion in the above selection steps. While finalizing on the set of distortions for a particular scene we make sure that the low light as well as the high exposure version of the scene available in the respective datasets [6,49] is picked. We call the high exposure version of the scene as 'well-lit'. We also include all the images from other distortion types such as halo artifacts, streak noise, fog which were few in number. In Figure 1, we show a low light image and corresponding restored versions suffering from different distortion types.
Subjective study
We conduct an online subjective study owing to the pandemic situation where each subject participates in two sessions of half an hour each spaced 24 hours apart. The study protocols were reviewed and approved by an appropriate committee overseeing human ethics. The human subjects were informed about the purpose of the study and consent obtained about how their data would be used. Neither the study nor the database will reveal personally identifiable information.
Although the study is done online where users rated images under different conditions similar to other crowdsourced studies [11,14,20,37,57], we observe that there is a high level of agreement among the subjects as seen in Section 3.3. A total of 88 subjects took our study leading to around 20 or more ratings per image.
We conduct a single stimulus continuous procedure study, where the subjects were asked to rate an image based on its perceptual quality on a continuous scale from 0-100. The entire study interface was designed and implemented in-house using a Node.js + MySQL backend. Each viewing session was preceded by a training phase so that the subjects get accustomed to the framework as well as get a sense of the range of qualities that the study might span. Within a session, images were shown in a randomized order, taking care that no two images from the same scene occur in succession. The subject was allowed to go to the next image only after selecting a value on the slider. Processing of subjective scores: We first convert raw human scores to 'Z-scores' [36] by removing the mean and dividing by the standard deviation for each subject for each session. We then employ standard outlier rejection procedures described in [1]. A total of 5 subjects were discarded as outliers. Finally the scores from the inlier subjects are linearly rescaled to the range 0-100 to obtain the final MOS scores.
Subjective Study Inferences
MOS Distribution: We show the MOS distribution across the dataset in Figure 2. We see that the MOS values span the entire subjective quality range with more number of medium quality images as desirable in any challenging IQA database.
Inter-subject consistency: We validate the study by randomly splitting the population into two equal halves and compute the median Pearson linear correlation coefficient (PLCC) between the MOS obtained from the two halves across a number of random splits. We find that the median PLCC across 100 such splits is 0.93. We show the scatter plot for one such split in Figure 3. We see that there is a high agreement between two random halves of the population.
Validation using lab controlled study: In order to validate our online study, we also conduct an in-lab study with controlled conditions. We use a 24 inch LED monitor with a screen resolution of 1920x1080. The subjects are positioned at a viewing distance of approximately four times the screen height. The room illumination is 6 lux. These settings are decided based on the recommendations given in [1]. We employ a limited number of 10 subjects in our in-lab study. Each subject participated in a single session of half an hour each and was shown the same set of 166 images from DSLR. The PLCC between the MOS scores computed from the online study vs the in-lab study is 0.89. This validates the authenticity of the online study.
MOS variability across distortions:
The MOS values of all images in our dataset lie in the range [20.36,78.26]. We note that the high exposure shots have MOS scores towards the higher end of the spectrum with a mean of 67.94 6.66. Figure 4 shows box plots for different distortion types. As each comparative distortion type has roughly an equal span over the entire quality range, we draw certain conclusions from the boxplots in Figure 4. We observe that humans find noise more annoying when compared to blur, and over-enhancement more annoying when compared to under-enhancement. We also conduct two sample t-tests to validate the statistical significance of the above conclusions.
Unsupervised NR Low Light Restored IQA
We now present the details of M-SCQALE, our unsupervised NR IQA method for low light restored images. We first present the self-supervised quality feature learning approach followed by the quality computation method.
Contrastive Learning of Quality Features
We present our approach for self-supervised learning of quality features for low light image restoration through multi-view contrastive learning [39]. One of our key contributions is in the choice of the multiple views as relevant for learning quality features. We hypothesize that the joint distribution of quality features among different patches from the same distorted image can be contrasted with patches drawn from different distorted images to learn rich features. In particular, we divide a low light restored image into nonoverlapping patches and choose different patches as multiple views. Secondly, our choice of positive and negative pairs enables the learning of features that are sensitive to quality and less variant to content. In particular, we draw the positive and negative pairs of views from distorted versions of the same scene in a manner that enables the learning of quality relevant features. Here, the only difference among the positive and negative pairs is in the visual quality while the content remains the same. We learn these features in each sub-band of a multi-scale Laplacian pyramid decomposition to mimic the multi-scale processing of human vision [47].
Through our choice of patches in contrastive multi-view coding, the joint distribution of patches drawn from the same image tends to capture global quality features that are shared across the image. We believe that such feature extraction is analogous to how global features are extracted by modeling the natural scene statistics of image sub-bands for QA. Such feature extraction helps obtain a global sense of the image quality which is ultimately predicted as a single number. The global features are still a function of local features obtained through several layers of convolutions and non-linear activations in early stages. Now we formulate our multi-view contrastive learning as shown in Figure 5 and describe it below. For a given scene n ∈ {1, 2, · · · , N }, we choose K distorted versions denoted as {I n1 , I n2 , I n3 , · · · , I nK }. These distorted versions correspond to either the original low light image or its restored versions obtained by applying different contrast enhancement and denoising algorithms. We also include an image captured with high exposure as one of these distorted versions, although it may be of high quality. We now describe the feature learning from the images. The same procedure is repeated for each subband independently.
We define two functions P n 1 (.) and P n 2 (.) where P n 1 (.) crops a patch from the images in the batch {I n1 , I n21 ...I nK } at the exactly same location while P n 2 (.) corresponds to a different non-overlapping patch. For simplicity, we divide the image into vertical or horizontal halves and randomly pick the largest square patch in each half. We note that the patch location is the same across distorted versions of the same scene but can vary for different scenes. Thus, we obtain a total of 2K patches for each scene where View 1 contains {P n 1 (I n1 ), · · · , P n 1 (I nK )} and View 2 contains {P n 2 (I n1 ), · · · , P n 2 (I nK )}. We pick a random patch from View 1 and label it as an anchor. Let P n 1 (I nk ) be the anchor. Corresponding to this anchor, (P n 1 (I nk ), P n 2 (I nk )) is a positive pair of samples since they are drawn from the same image while (P n 1 (I nk ), P n 2 (I nj )) for j = k is a negative pair of samples. Let the feature extraction network be denoted as f (·). Let z (n) k1 = f (P n 1 (I nk )) and z (n) k2 = f (P n 2 (I nk )). Let S(·) denote the normalized cosine similarity between two vectors u and v be defined as
S(u, v) = u T v ||u||||v|| .
The contrastive loss corresponding to anchor P n 1 (I nk ) is obtained as
l(P n 1 (I nk )) = − log exp S(z (n) k1 , z (n) k2 )/τ K j=1 exp S(z (n) k1 , z (n) j2 )/τ ,(1)
where τ is the temperature parameter. We obtain the loss similarly when P n 2 (I nk ) is chosen as the anchor. Thus, the overall loss function for N scenes in a batch is obtained as
L = 1 N K N n=1 K k=1
[l(P n 1 (I nk )) + l(P n 2 (I nk ))] . (2) The feature extraction network parameters are learnt by minimizing L. We employ ResNet-50 as the feature extraction network in our experiments. We learn contrastive features for each subband of a Laplacian pyramid decomposition by feeding the subbands instead of the images to the above contrastive learning framework. We learn network parameters independently for each subband.
Blind Quality Prediction
We design a quality prediction model based on our features obtained through contrastive learning using the framework of NIQE [31]. We extract features from M subbands of an M -level Laplacian pyramid decomposition. Let the features obtained for Subband-m, m ∈ {1, 2, · · · , M } be z m . Let z 0 denote the features obtained from the image directly. We concatenate all these features to obtain (z 0 , z 1 , · · · , z M ). However, the dimension of this resulting feature vector is very large. Thus, we perform principal component analysis (PCA) to reduce the feature dimension to D components for fair comparison with other features. We then use these reduced dimensionality features from the test image to compute quality as follows.
Similar to the NIQE evaluation framework, we take a corpus of pristine image patches of size P × P satisfying the sharpness criterion described in [31]. We select these patches from scenes that do not overlap with any of the test images. Since low light restored images also suffer from color artifacts, we believe that choosing colorful pristine patches can help better assess the quality of the restored images. Thus, among the selected sharp patches, we further compute the colorfulness of each patch using the method described in [17]. We select those patches possessing a colorfulness index greater than a threshold τ c . We obtain the features from the sharp and colorful pristine patches by applying our feature extraction network and subject them to PCA. Let the pristine multivariate Gaussian (MVG) model learnt on the reduced dimensionality features be (µ r , Σ r ). For each test image, we take all P × P patches with an overlap of P/2, extract features, reduce dimensionality and fit an MVG model over them to determine (µ d , Σ d ). Finally we compute the quality score of the test image as
Q = (µ r − µ d ) T Σ r + Σ d 2 −1 (µ r − µ d ) . (3)
Experiments
Performance Evaluation and Benchmarking
We benchmark state of the art full reference (FR) and supervised and unsupervised NR IQA measures on our dataset. Popular FR measures that we benchmark include PSNR, SSIM [46], Feature SIM [61] and Multiscale SSIM [47]. We also benchmark RIQMC [15] which is a contrast distortion based FR measure. Finally we benchmark LPIPS [62], which is a deep learning based full reference measure. We use the high exposure shot of the scene as the reference for computing the full reference quality scores.
We benchmark no reference measures that need to be trained against human opinion scores and refer to them as supervised NR IQA measures. Among non-deep learning based methods, we evaluate BRISQUE [30], PI [3] and Ma et al. [28]. We also compare recent deep learning based NR IQA methods such as MetaIQA [66] and NIMA [38]. Finally, we evaluate a simple benchmark based on pretrained ResNet-50 features obtained after global average pooling using a support vector regression model as in [22].
Among unsupervised IQA measures we compare against NIQE [31] and IL-NIQE [60]. Both these methods require a set of pristine images for comparison and we use the same set as that which is used for our prediction framework as described in the implementation details. We also benchmark NIQMC [52] which is an unsupervised contrast distortion based blind quality index.
Implementation details of M-SCQALE: We learn quality features through contrastive learning using images from a larger corpus of 60,000 distorted images that we create similar to DSLR. In particular, we take 300 scenes from the training set of the SID dataset, different from the test set of SID that is used in creating DSLR. We then apply contrast enhancement and denoising algorithms on these images similar to DSLR. Thus, there is no overlap of scene content between the images on which contrastive learning is performed and DSLR. After feature learning, we use high exposure/pristine images from the SID [6] training set (300 images) and LOL dataset [48] (500 images) to learn the feature statistics for our pristine MVG model.
We choose the number of scales as M = 3 in our experiments and study the performance variation with M in Section 5.2. We use two RTX 2080 Ti GPUs using Pytorch framework for our training. Starting at the image level, we choose the number of scenes as N = 4 and the number of distorted versions K = 10 in a mini-batch. As the resolution of the subbands decreases with scale, based on our GPU memory constraints, we increase N and K by a factor of 2 in each scale. The exact values for each sub-band can be found in the supplementary. We train a ResNet-50 network for each sub-band independently for 110 epochs using Adam optimizer with a learning rate of 0.01. We set the temperature parameter as τ = 0.1 in all our experiments. We use sharpness threshold as τ s = 0.3 times the maximum sharpness and colorfulness threshold τ c = 0.8 times the maximum colorfulness in each image for selecting the pristine patches in our quality prediction framework. We set D = 2048 and P = 96 similar to NIQE [31].
Performance evaluation: We use Spearman's rank order correlation coefficient (SRCC) and Pearson's linear correlation coefficient (PLCC) between the subjective scores and quality predictions to evaluate performance. PLCC is computed after passing the predicted scores through a nonlinearity as described in [35]. For all methods which require training, we randomly split the dataset into 80% for training and 20% for testing such that there is no scene overlap between the train and the test splits. We report the median performance across 100 such splits in Table 1. For fair comparison among all the methods, we evaluate the methods which do not require training also on the 20% test splits and report their median performance.
Performance analysis: We observe that among the FR measures, LPIPS achieves the best performance. Among supervised NR measures, we see that ResNet-50 features that have been pre-trained for image classification can be regressed quite well to predict MOS. This is also not surprising given its stable competitive performance on multiple recent authentically distorted IQA datasets [14,57]. Among the unsupervised NR IQA models, M-SCQALE achieves the state of the art performance. This shows that contrastive learning can be effectively used to learn features that can predict quality in an unsupervised fashion.
Ablations
We evaluate the strength of different components of M-SCQALE. Various experiments in this subsection involve comparing our features and feature learning method with other feature extraction mechanisms. We perform all these comparisons by replacing our features with other features in Equation (3) for unsupervised NR QA. As M-SCQALE is training free, we report the SRCC between the predicted scores and the actual MOS on the entire DSLR dataset.
Other deep features vs. M-SCQALE: We experiment with various self-supervised pretrained ResNet-50 models such as Simclrv2 [9], Mocov2 [10], CMC [39] and Swav [4] to extract features and use them in our quality prediction framework. We also compare with features extracted with a ResNet-50 that has been pre-trained in a supervised fashion for image classification denoted as Sup* ResNet-50. We see from Table 2 that our features outperform these pre-trained features although they have been trained on huge amounts of data and with far higher computing power. Thus our contrastive learning paradigm contributes to this performance difference with other deep features. Further, we also try training the frameworks of Simclr [7] and CMC [39] on our dataset. However, they perform poorer than the pre-trained features, perhaps due to the lack of enough training data. Exact results can be found in the supplementary. Choice of patches from the same scene for M-SCQALE: One of the key aspects of our training method is the way we choose Features SRCC Sup* ResNet-50 0.46 Simclrv2 [9] 0.40 Mocov2 [10] 0.50 CMC [39] 0.41 Swav [4] 0.47 M-SCQALE 0.70 Table 2. Evaluating various self-supervised features. patches for multiple views. In particular, we choose pairs of patches such that the negative pairs come from different distorted versions of the same scene. We believe that this is critical in learning features that relate to quality and not content. To understand the benefit of this aspect, we experiment with a learning method where the negative pairs can come from images of different scenes. We perform this comparison for the features learnt for a single scale (just the image). The first two rows in Table 3 reveal a huge difference in the performance indicating the importance of scene content remaining the same while computing the contrastive loss.
Multiscale subband features: Recall that in M-SCQALE we evaluate the features in different sub-bands of a Laplacian pyramid decomposition and concatenate all these features learnt along with the features extracted from the image. We study the performance variation with the number of scales in the Laplacian pyramid decomposition M and report the performance in Figure 6. M = 0 implies features are extracted only at the image level. M = 1 implies feature extraction at the image level, the first high pass sub-band and the corresponding low pass band and so on for other values of M . We observe that after M = 3, the performance saturates. We also show the comparison of the multi-scale feature extraction method for M = 3 in Table 3.
Role of sharpness and colorfulness: We next analyze the strength of the different criteria we use for patch se- lection from the pristine patches to learn the pristine MVG model. We evaluate the impact of patch selection using both sharpness and colorfulness. We see from Table 3 that selecting colorful patches improves the performance.
Performance analysis on other datasets
We evaluate M-SCQALE on other datasets containing low light images such as CID [42] and NNID [51]. We do not learn features using images from these datasets and use the same model evaluated on DSLR. Thus, there is no training of any form on these datasets. As in [51], we evaluate M-SCQALE on the 79 low-contrast images in CID. NNID contains 2240 images with distortions such as underenhancement, blur and color distortion. We evaluate different unsupervised models on the above datasets in Table 4. We observe that NNID contains images with widely varying resolutions. To be consistent with the resizing step of IL-NIQE, we resize all the images to a resolution of 512 × 512 before evaluating NIQE and M-SCQALE on NNID. This resizing is important especially since these approaches use fixed patch sizes. We see that M-SCQALE achieves the best performance on both these datasets demonstrating its excellent generalization performance.
Visualization of features
We also try to visually interpret the M-SCQALE features using a t-SNE plot. We compute features on all images of DSLR, then reduce dimensionality to 50 dimensions using PCA following [41], and then visualize the features. We show the scatter plot of t-SNE features from different distortion types in Figure 7. We observe that the different distortion types form clearly demarcated clusters. This explains that our contrastively learnt features are able to differentiate images in the distortion space.
Failure cases: We analyze where M-SCQALE fails by comparing the subjective scores and our quality predictions passed through a non-linearity to predict MOS. As shown in Figure 8 (a), M-SCQALE overestimates the quality of over enhanced images. Although our feature learning mechanism is able to identify such distortions, we believe that our unsupervised quality prediction framework still lacks the artillery to carefully quantify such distortions. In another example shown in Figure 8 (b), we observe that our model underestimates the quality of this colorful image. This could be happening because the distribution of colorful pristine patches may not be as colorful as the enhanced image.
Conclusion
Our DSLR database represents one of the first exhaustive QA studies on low light restored images. Our novel unsupervised NR method for QA of low light restored images, M-SCQALE, addresses the twin challenges of the need for a reference image as well as human quality labels for training. In particular, we show that contrastive learning can be a valuable tool for effective learning of quality aware features. Although, we do not train on subjective data, M-SCQALE still achieves very good performance in accordance with human judgements. While the features we learn are able to distinguish distortion types, their unsupervised mapping to perceptual quality can be improved further.
In Table 5, we report the list of LLR algorithms we use to generate images in DSLR. We use a combination of 14 contrast enhancement techniques coupled with 3 denoisers, and also 7 joint contrast enhancement and denoising techniques which include deep learning methods. Note that we use a variety of algorithms belonging to different categories such as histogram equalization based, retinex based and multiscale approaches.
Block Diagram of M-SCQALE
In Figure 9, we show the complete workflow of the our M-SCQALE method for unsupervised NR-QA of restored low light images. Note that NIQE distance refers to Equation (3) in the main paper.
Additional Training Details
Batch sizes
For training different sub-bands in M-SCQALE, we choose the number of scenes N and the number of distorted versions K in a mini-batch as given in Table 6.
Additional Experiment Details
Training Simclr and CMC for Quality Assessment
In the main paper, we reported the performance of using features from pre-trained Simclr and CMC ResNet-50 networks in our quality prediction framework. We remarked there that training these frameworks on the set of low light restored images on which M-SCQALE's contrastive learning framework was trained, results in poorer performance as compared to the pre-trained features. We report the performance in Table 7.
For Simclr, we follow the exact method proposed by Chen et al. [8]. In a minibatch, B data points are randomly Similarly for CMC, we follow the exact method proposed by Tian et al. [40]. During training, Y and DbDr from the same data point are considered as the positive pair, while DbDr channels from other randomly sampled data points are considered as a negative pair (for a given Y).
We use two RTX 2080 Ti GPUs using Pytorch framework for our training. We train a ResNet-50 network for each framework for 110 epochs using Adam optimizer with a learning rate of 0.01. We set the temperature parameter as τ = 0.1. We observe that setting τ = 0.07 for CMC (as in the original CMC paper) gives very poor performance for our task and hence we report the performance with τ = 0.1. We use a batch size of 40 as limited by our memory constraints. We use a patch size of 312 as used in M-SCQALE's image level contrastive learning framework for fair comparison.
Unsupervised IQA with CORNIA:
CORNIA [53] is an example of an unsupervised feature learning algorithm. Typically, CORNIA features are used in a supervised setting, where the features are trained against human opinion scores. However, to compare these features against M-SCQALE's features, we design an unsupervised quality index based on the CORNIA features. We
Contrast Enhancement Techniques
Automatic Color Enhancement (ACE) and its Fast Implementation [13] Efficient Contrast Enhancement Using Adaptive Gamma Correction With Weighting Distribution [21] A Bio-Inspired Multi-Exposure Fusion Framework for Low-light Image Enhancement [54] A New Low-Light Image Enhancement Algorithm Using Camera Response Model [56] Contrast-limited adaptive histogram equalization [67] Contextual and Variational Contrast Enhancement [5] Single Image Haze Removal Using Dark Channel Prior [19] A New Image Contrast Enhancement Algorithm Using Exposure Fusion Framework [55] A fusion-based enhancing method for weakly illuminated images [12] Histogram Equalization Contrast Enhancement Based on Layered Difference Representation of 2D Histograms [24] LIME: Low-Light Image Enhancement via Illumination Map Estimation [16] Naturalness Preserved Image Enhancement Using a Priori Multi-Layer Lightness Statistics [45] Weighted Adaptive Histogram Equalization [2] Joint Contrast Enhancement and Denoising Techniques Fast and Efficient Image Quality Enhancement via Desubpixel Convolutional Neural Networks [44] MBLLEN: Low-Light Image/Video Enhancement Using CNNs [27] LLRNET [29] UNET [34] Learning to See in the Dark [6] Joint Enhancement and Denoising Method via Sequential Decomposition [33] Structure-Revealing Low-Light Image Enhancement Via Robust Retinex Model [25] Denoisers Deep Iterative Down-Up CNN for Image Denoising (DIDN) [59] DIDN pre-trained [59] The Noise Clinic: a Blind Image Denoising Algorithm [23]
Level
N K Image level 4 10 Level 1 high pass sub-band 4 10 Level 2 high pass sub-band 8 20 Level 3 high pass sub-band 16 40 Low pass sub-band 32 40 Table 6. Values of N and K used for different sub-bands.
Method SRCC Simclr 0.13 CMC 0.28 Table 7. Results of training Simclr and CMC for quality assessment.
use CORNIA features instead of M-SCQALE's features in our NIQE [32] based quality prediction framework. In our experiment, we first learn the dictionary in an unsupervised fashion [53] on the set of low light restored images on which M-SCQALE's contrastive learning frame-work was trained. We then use the learnt features from in our quality prediction framework. This gives a median SROCC of 0.25 and a median PLCC of 0.23 on the same 100 test splits as reported in Table 1 in the main paper.
Additional Details of Subjective Study
We recorded various parameters such as screen size, screen resolution, subject distance from screen using a mandatory survey form at the start of the subjective study. We study the role of these parameters in a similar fashion as analyzed by Deepti et al. in [14]. We compute the MOS from disjoint sections of the population separated based on these parameters for 5 randomly sampled images from DSLR. We plot the corresponding MOS values and their associated 95% confidence intervals in Figure 10. We observe that the difference between the mean of the ratings obtained on the five randomly sampled image between disjoint sets of the population separated by either resolution, viewing distance or screen size is not statistically significant. However we do not disregard the plausible influences that these parameters might have on distortion perception based on an analysis of five randomly sampled images. A study that exactly studies the relationship between a subject's Influence of resolution.
Influence of subject viewing distance.
Influence of screen size. Figure 10. Plots illustrating the influence of various parameters on a subject's quality rating on five randomly sampled images from DSLR. perception of quality and the inter-play of these factors is an interesting area for future work, considering that crowdsourced studies are becoming very popular in recent times.
Figure 1 .
1± 5.65. The low light images have MOS scores towards the lower end of the spectrum with a mean of 33.45 ± Low light, well-lit and different restored versions of a low light image showing various types of distortions in the dataset.
Figure 2 .
2Distribution of MOS values across dataset.
Figure 3 .
3Inter-subject consistency between two random halves of the population.
Figure 4 .
4Boxplot of MOS values for different distortions. Here Enh* refers to Enhancement
Figure 5 .
5Illustration of views in our contrastive learning framework.
Figure 6 .
6Performance variation of M-SCQALE for different number of sub-bands.
Figure 7 .Figure 8 .
78Scatter plot of 2-dimensional features obtained by using t-SNE on M-SCQALE features. Failure cases.
Figure 9 .
9Block diagram of M-SCQALE. sampled and their corresponding augmented pairs are generated using the prescribed augmentation procedure resulting in 2B data points. Given a sample and it's augmented version (positive pair), the remaining 2(B − 1) data points in the minibatch are treated as negative examples.
Table 1. Median performance on 100 random train test splits of DSLR dataset. Std refers to the standard deviation in performance.Method
SRCC
PLCC
Median Std Median Std
Full Reference measures
PSNR
0.39
0.05
0.41
0.04
RIQMC [15]
0.16
0.07
0.22
0.06
SSIM [46]
0.66
0.04
0.68
0.03
FSIM [61]
0.72
0.03
0.74
0.03
MS-SSIM [47]
0.72
0.03
0.74
0.03
LPIPS [62]
0.83
0.03
0.82
0.02
Supervised No Reference measures
BRISQUE [30]
0.57
0.04
0.61
0.03
Meta-IQA [65]
0.80
0.01
0.73
0.02
NIMA [38]
0.78
0.03
0.78
0.03
Ma et al. [28]
0.53
0.04
0.60
0.04
PI [3]
0.53
0.04
0.59
0.04
Resnet50 + SVR
0.84
0.03
0.84
0.03
Unsupervised No Reference measures
NIQMC [52]
0.17
0.06
0.22
0.06
NIQE [31]
0.54
0.04
0.56
0.04
IL-NIQE [60]
0.50
0.05
0.49
0.08
M-SCQALE
0.70
0.03
0.71
0.03
Table 5 .
5List of low light restoration algorithms used to generate DSLR.
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"Waad Bouaguel \nLARODEC, ISG\nUniversity of Tunis\n41, rue de la Liberté2000Le BardoTunisie\n",
"Ghazi Bel Mufti \nESSEC\nUniversity of Tunis\n4, rue Abou Zakaria El Hafsi1089MontfleuryTunisie\n"
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"LARODEC, ISG\nUniversity of Tunis\n41, rue de la Liberté2000Le BardoTunisie",
"ESSEC\nUniversity of Tunis\n4, rue Abou Zakaria El Hafsi1089MontfleuryTunisie"
] | [
"IJARAI) International Journal of Advanced Research in Artificial Intelligence"
] | Filter selection techniques are known for their simplicity and efficiency. However this kind of methods doesn't take into consideration the features inter-redundancy. Consequently the un-removed redundant features remain in the final classification model, giving lower generalization performance. In this paper we propose to use a mathematical optimization method that reduces inter-features redundancy and maximize relevance between each feature and the target variable. | 10.14569/ijarai.2012.010502 | [
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An improvement direction for filter selection techniques using information theory measures and quadratic optimization
2012
Waad Bouaguel
LARODEC, ISG
University of Tunis
41, rue de la Liberté2000Le BardoTunisie
Ghazi Bel Mufti
ESSEC
University of Tunis
4, rue Abou Zakaria El Hafsi1089MontfleuryTunisie
An improvement direction for filter selection techniques using information theory measures and quadratic optimization
IJARAI) International Journal of Advanced Research in Artificial Intelligence
152012Feature selectionmRMRQuadratic mutual informationfilter
Filter selection techniques are known for their simplicity and efficiency. However this kind of methods doesn't take into consideration the features inter-redundancy. Consequently the un-removed redundant features remain in the final classification model, giving lower generalization performance. In this paper we propose to use a mathematical optimization method that reduces inter-features redundancy and maximize relevance between each feature and the target variable.
INTRODUCTION
In many classification problems we deal with huge datasets, which likely contain not only many observations, but also a large number of variables. Some variables may be redundant or irrelevant to the classification task. As far as the number of variables increase, the dimensions of data amplify, yielding worse classification performance. In fact with so many irrelevant and redundant features, most classification algorithms suffer from extensive computation time, possible decrease in model accuracy and increase of overfitting risks [17,12]. As a result, it is necessary to perform dimensionality reduction on the original data by removing those irrelevant features.
Two famous special forms of dimensionality reduction exist. The first one is feature extraction, in this category the input data is transformed into a reduced representation set of features, so new attributes are generated from the initial ones. The Second category is feature selection. In this category a subset of the existing features without a transformation is selected for classification task. Generally feature selection is chosen over feature extraction because it conserves all information about the importance of each single feature while in feature extraction the obtained variables are, usually, not interpretable. In this case it is obvious that we will study the feature selection but choosing the most effective feature selection method is not an easy task.
Many empirical studies show that manipulating few variables leads certainly to have reliable and better understandable models without irrelevant, redundant and noisy data [21,20]. Feature selection algorithms can be roughly categorized into the following three types, each with different evaluation criteria [7]: filter model, wrapper model and embedded. According to [18,3,9] a filter method is a pre-selection process in which a subset of features is firstly selected independently of the later applied classifier. Wrapper method on the other hand, uses search techniques to rank the discriminative power of all of the possible feature subsets and evaluate each subsets based on classification accuracy [16], using the classifier that was incorporated in the feature selection process [15,14]. The wrapper model generally performs well, but has high computational cost. Embedded method [20] incorporates the feature selection process in the classifier objective function or algorithm. As result the embedded approach is considered as the natural ability of a classification algorithm; which means that the feature selection take place naturally as a part of classification algorithm. Since the embedded approach is algorithm-specific, it is not an adequate one for our requirement.
Wrappers on other hand have many merits that lie in the interaction between the feature selection and the classifier. Furthermore, in this method, the bias of both feature selection algorithm and learning algorithm are equal as the later is used to assess the goodness of the subset considered. However, the main drawback of these methods is the computational weight. In fact, as the number of features grows, the number of subsets to be evaluated grows exponentially. So, the learning algorithm needs to be called too many times. Therefore, performing a wrapper method becomes very expensive computationally.
According to [2,17] filter methods are often preferable to other selection methods because of their usability with alternative classifiers and their simplicity. Although filter algorithms often score variables separately from each other without considering the inter-feature redundancy, as result they do not always achieve the goal of finding combinations of variables that give the best classification performance [13].Therefore, one common step up for filter methods is to consider dependencies and relevance among variables. mRMR [8] (Minimal-Redundancy-Maximum-Relevance) is an effective approach based on studying the mutual information among features and the target variable; and taking into account the inter-features dependency [19]. This approach selects those features that have highest relevance to the target class with the minimum inter-features redundancy. The mRMR algorithm, selects features greedily.
The new approach proposed in this paper aims to show how using mathematical methods improves current results. We use quadratic programming [1] in this paper, the studied www.ijacsa.thesai.org objective function represents inter-features redundancy through quadratic term and the relationship between each feature and the class label is represented through linear term. This work has the following sections: in section 2 we review studies related to filter methods; and we study the mRMR feature selection approach. In section 3 we propose an advanced approach using mathematical programming and mRMR algorithm background. In section 4 we introduce the used similarity measure. Section 5 is dedicated to empirical results.
II. FILTER METHODS
The processing, of filter methods at most cases can be described as it follows: At first, we must evaluate the features relevance by looking at the intrinsic properties of the data. Then, we compute relevance score for each attribute and we remove ones which have low scores. Finally, the set of kept features forms the input of the classification algorithm. In spite of the numerous advantages of filters, scoring variables separately from each other is a serious limit for this kind of techniques. In fact when variables are scored individually they do not always achieve the object of finding the perfect features combination that lead to the optimal classification performance [13].
Filter methods fail in considering the inter-feature redundancy. In general, filter methods select the top-ranked features. So far, the number of retained features is set by users using experiments. The limit of this ranking approach is that the features could be correlated among themselves. Many studies showed that combining a highly ranked feature for the classification task with another highly ranked feature for the same task often does not give a great feature set for classification. The raison behind this limit is redundancy in the selected feature set; redundancy is caused by the high correlation between features.
The main issue with redundancy is that with many redundant features the final result will not be easy to interpret by business managers because of the complex representation of the target variable characteristics. With numerous mutually highly correlated features the true representative features will be consequently much fewer. According to [8], because features are selected according to their discriminative powers, they do not fully represent the original space covered by the entire dataset. The feature set may correspond to several dominant characteristics of the target variable, but these could still be fine regions of the relevant space which may cause a lack in generalization ability of the feature set.
A. mRMR Algorithm
A step up for filter methods is to consider dissimilarity among features in order to minimize feature redundancy. The set of selected features should be maximally different from each other. Let S denote the subset of features that we are looking for. The minimum redundancy condition is
, ) , ( | | 1 = , , 2 1 1 j i S j x i x x x M S P MinP (1)
where we use M(i, j) to represent similarity between features, and |S |is the number of features in S. In general, minimizing only redundancy is not enough sufficient to have a great performance, so the minimum redundancy criteria should be supplemented by maximizing relevance between the target variable and others explicative variables. To measure the level of discriminant powers of features when they are differentially expressed for different target classes, again a similarity measure ) , ( i x y M is used, between targeted classes y={0,1} and the feature expression i
x . This mesure quantifies the relevance of i x for the classification task. Thus the maximum relevance condition is to maximize the total relevance of all features in S:
). ,
( | | 1 = , 2 2 2 i S i x x y M S P MaxP (2)
Combining criteria such as: maximal relevance with the target variable and minimum redundancy between features is called the minimum redundancy-maximum relevance (mRMR) approach. The mRMR feature set is obtained by optimizing the problems 1 P and 2 P receptively in Eq. (1) and Eq. (2) simultaneously. Optimization of both conditions requires combining them into a single criterion function
. } { 2 1 P P Min (3)
mRMR approach is advantageous of other filter techniques. In fact with this approach we can get a more representative feature set of the target variable which increases the generalization capacity of the chosen feature set. Consistently, mRMR approach gives a smaller feature set which effectively cover the same space as a larger conventional feature set does. mRMR criterion is also another version of MaxDep [19] that chooses a subset of features with both minimum redundancy and maximum relevance.
In spite of the numerous advantages of mRMR approach; given the prohibitive cost of considering all possible subsets of features, the mRMR algorithm selects features greedily, minimizing their redundancy with features chosen in previous steps and maximizing their relevance to the class. A greedy algorithm is an algorithm that follows the problem solving heuristic of making the locally optimal choice at each stage with the hope of finding a global optimum; the problem with this kind of algorithms is that in some cases, a greedy strategy do not always produce an optimal solution, but nonetheless a greedy heuristic may yield locally optimal solutions that approximate a global optimal solution.. On the other hand, this approach treats the two conditions equally important. Although, depending on the learning problem, the two conditions can have different relative purposes in the objective function, so a coefficient balancing the MaxDep and the MinRev criteria should be added to mRMR objective function. To improve the theory of mRMR approach we use in the next section; mathematical knowledge to modify and balance the mRMR objective function and solve it with quadratic programming. www.ijacsa.thesai.org
III. QUADRATIC PROGRAMMING FOR FEATURE SELECTION
A. Problem Statement
The problem of feature selection was addressed by statistics machine learning as well as by other mathematical formulation. Mathematical programming based approaches have been proven to be excellent in terms of classification accuracy for a wide range of applications [5,6]. The used mathematical method is a new quadratic programming formulation. Quadratic optimization process, use an objective function with quadratic and linear terms. Here, the quadratic term presents the similarity among each pair of variables, whereas the linear term captures the correlation of each feature and the target variable.
Assume the classifier learning problem involves N training samples and m variables [20]. A quadratic programming problem aims to minimize a multivariate quadratic function subject to linear constraints as follows:
. 1 = , , 1 = 0 . 2 1 = ) ( 1 = i m i i T T x m i x Subjectto F Q Minf x x x x (4)
where F is an m-dimensional row vector with non-negative entries, describing the coefficients of the linear terms in the objective function. F measures how correlated each feature is with the target class (relevance). Q is an (m x m) symmetric positive semi-definite matrix describing the coefficients of the quadratic terms, and represents the similarity among variables (redundancy). The weight of each feature decision variables are denoted by the m-dimensional column vector x .
We assume that a feasible solution exists and that the constraint region is bounded. When the objective function ) (x f is strictly convex for all feasible points the problem has a unique local minimum which is also the global minimum. The conditions for solving quadratic programming, including the Lagrangian function and the Karush-Kuhn-Tucker conditions are explained in details in [1]. After the quadratic programming optimization problem has been solved, the features with higher weights are better variables to use for subsequent classifier training.
B. Conditions balancing
Depending on the learning problem, the two conditions can have different relative purposes in the objective function.
Therefore, we introduce a scalar parameter as follows:
, ) (1 2 1 = ) ( x x x x T T F Q Minf (5). = F Q Q (6)
IV. BASED INFORMATION THEORY SIMILARITY MEASURE
The information theory approach has proved to be effective in solving many problems. One of these problems is feature selection where information theory basics can be exploited as metrics or as optimization criteria. Such is the case of this paper, where we exploit the mean value of the mutual information between each pair of variables in the subset as metric in order to approximate the similarity among features. Formally, the mutual information of two discrete random variables i x and j x can be defined as:
, ) ( ) ( ) , ( ) , ( = ) , ( j i j i j i S j x S i x j i x p x p x x p log x x p x x I (7)
and of two continuous random variables is denoted as follows:
. ) ( ) ( ) , ( ) , ( = ) , ( j i j i j i j i j i dx dx x p x p x x p log x x p x x I (8) V.
EMPIRICAL STUDY
In general mutual information computation requires estimating density functions for continuous variables. For simplicity, each variable is discretized using Weka 3.7 software [4]. We implemented our approach in R using the quadprog package [10,11]. The studied approach should be able to give good results with any classifier learning algorithms, for simplicity the logistic regression provided by R will be the underlying classifier in all experiments.
The generality of the feature selection problem makes it applicable to a very wide range of domains. We chose in this paper to test the new approach on two real word credit scoring datasets from the UCI Machine Learning Repository.
The first dataset is the German credit data set consists of a set of loans given to a total of 1000 applicants, consisting of 700 examples of creditworthy applicants and 300 examples where credit should not be extended. For each applicant, 20 variables describe credit history, account balances, loan purpose, loan amount, employment status, and personal information. Each sample contains 13 categorical, 3 www.ijacsa.thesai.org continuous, 4 binary features, and 1 binary class feature. The second data set is the Australian credit dataset which is composed by 690 instances where 306 instances are creditworthy and 383 are not. All attributes names and values have been changed to meaningless symbols for confidential reason. Australian dataset present an interesting mixture of continuous features with small numbers of values, and nominal with larger numbers of values. There are also a few missing values.
The aim of this section is to compare classification accuracy achieved with the quadratic approach and others filter techniques. Table I and Table ii show the average classification error rates for the two data sets as a function of the number of features. Accuracy results are obtained with α= 0.511 for German data set and α= 0.489 for Australian data set, which means that an equal tradeoff between relevance and redundancy is best for the two data sets. From Table 1 and Table II it's obvious that using the quadratic approach for variable selection lead to the lowest error rate.
VI. CONCLUSION
This paper has studied a new feature selection method based on mathematical programming; this method is based on the optimization of a quadratic function using the mutual information measure in order to capture the similarity and nonlinear dependencies among data.
that is, features with higher weights are those which have lower similarity coefficients with the rest of features. Every data set has its best choice of the scalar . However, a reasonable choice of must balances the relation between relevance and redundancy. Thus, a good estimation of must be calculated. We know that the relevance and redundancy terms in Equation 6 are balanced whenabove x , Q and F are defined as before and
[0,1]
, if
1
=
, only relevance is considered. On the
opposing, if
0
=
, then only independence between features
is considered F
Q
=
)
(1
, where Q is the estimate of the mean value
of the matrix Q ; and F is the estimate of the mean value of
vector F elements. A practical estimate of is defined as
TABLE I .
IRESULTS SUMMARY FOR GERMAN DATASET, WITH 7SELECTED FEATURES
Test
error
Type I
error
Type II
error
Quadratic
0.231
0.212
0.222
Relief
0.242
0.233
0.287
Information Gain
0.25
0.238
0.312
CFS Feature Set Evaluation
0.254
0.234
0.344
mRMR
0.266
0.25
0.355
MaxRel
0.25
0.238
0.312
TABLE II.
RESULTS SUMMARY FOR AUSTRALIAN DATASET, WITH 6
SELECTED FEATURES
Test
error
Type I
error
Type II
error
Quadratic
0.126
0.155
0.092
Relief
0.130
0.164
0.099
Information Gain
0.127
0.163
0.094
CFS Feature Set Evaluation
0.126
0.165
0.098
mRMR
0.130
0.164
0.099
MaxRel
0.139
0.179
0.101
| P a g e
| P a g e
| P a g e www.ijacsa.thesai.org
ACKNOWLEDGMENTThe authors would like to thank Prof. Mohamed Limam who provides valuable advices, support and guidance. The product of this research paper would not be possible without his help.
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| [] |
[
"Consistent Cosmic Bubble Embeddings",
"Consistent Cosmic Bubble Embeddings"
] | [
"S Shajidul Haque ",
"Bret Underwood ",
"\nDepartment of Mathematics & Applied Mathematics\nDepartment of Physics\nLaboratory for Quantum Gravity & Strings\nUniversity of Cape Town\nSouth Africa\n",
"\nPacific Lutheran University\n98447TacomaWAUSA\n"
] | [
"Department of Mathematics & Applied Mathematics\nDepartment of Physics\nLaboratory for Quantum Gravity & Strings\nUniversity of Cape Town\nSouth Africa",
"Pacific Lutheran University\n98447TacomaWAUSA"
] | [] | The Raychaudhuri equation for null rays is a powerful tool for finding consistent embeddings of cosmological bubbles into a background spacetime in a way that is largely independent of the matter content. We find that spatially flat or positively curved thin wall bubbles surrounded by a cosmological background must have a Hubble expansion that is either contracting or expanding slower than the background, presenting an obstacle for models of local inflation or false vacuum bubble embeddings. Similarly, a cosmological bubble surrounded by Schwarzschild space must either be contracting (for spatially flat and positively curved bubbles) or bounded in size by the apparent horizon, presenting an obstacle for embedding cosmic bubbles into a background Universe. * Electronic address: [email protected] † Electronic address: [email protected] 1 The assumption of an infinitesimally thin bubble wall does not apply to phase transition bubbles formed in new inflation [1]. | 10.1103/physrevd.95.103513 | [
"https://arxiv.org/pdf/1701.07771v3.pdf"
] | 119,010,696 | 1701.07771 | 825abacfe86dc119ec94ac773276a763ab321992 |
Consistent Cosmic Bubble Embeddings
S Shajidul Haque
Bret Underwood
Department of Mathematics & Applied Mathematics
Department of Physics
Laboratory for Quantum Gravity & Strings
University of Cape Town
South Africa
Pacific Lutheran University
98447TacomaWAUSA
Consistent Cosmic Bubble Embeddings
(Dated: June 1, 2017)
The Raychaudhuri equation for null rays is a powerful tool for finding consistent embeddings of cosmological bubbles into a background spacetime in a way that is largely independent of the matter content. We find that spatially flat or positively curved thin wall bubbles surrounded by a cosmological background must have a Hubble expansion that is either contracting or expanding slower than the background, presenting an obstacle for models of local inflation or false vacuum bubble embeddings. Similarly, a cosmological bubble surrounded by Schwarzschild space must either be contracting (for spatially flat and positively curved bubbles) or bounded in size by the apparent horizon, presenting an obstacle for embedding cosmic bubbles into a background Universe. * Electronic address: [email protected] † Electronic address: [email protected] 1 The assumption of an infinitesimally thin bubble wall does not apply to phase transition bubbles formed in new inflation [1].
I. INTRODUCTION
Cosmic bubbles -spherically symmetric homogeneous and isotropic spacetimes surrounded by a spherically symmetric background spacetime -can arise in many interesting applications of General Relativity, including local inflation, eternal inflation, cosmological phase transitions, and gravitational collapse. In these scenarios, spacetime is separated into two separate manifolds M + and M − , with their own distinct metrics, which are typically joined across a "thin wall" hypersurface 1 using the Israel junction conditions [2].
The dynamics of such bubbles can be extremely complicated, depending on the matter content of the background and interior of the bubble as well as the tension on the bubble wall and how it interacts with the surrounding matter. The most well-understood solutions involve simple assumptions, such as the dust-collapse model of Oppenhemier-Snyder [3] or false vacuum de-Sitter bubbles in empty space [4,5]. Some results have also been obtained for interior and exterior spacetimes described by homogeneous and isotropic FLRW spaces [6][7][8]. See also [9][10][11][12][13][14][15][16] for other investigations of cosmic bubble dynamics.
More recently, there has been renewed interest in the onset of local inflation arising from inhomogeneous initial conditions, in which inflation starts in a small patch surrounded by a non-inflating background. Earlier analytical arguments and numerical work suggested that the inflationary patch will not grow unless the initial size of the homogeneous patch is larger than the Hubble length (see e.g. [17][18][19][20] and the recent review [21]). These results have been revisited by recent work [22,23], which use modern numerical relativity codes to study the conditions under which local inflation begins. While these numerical analyses are not necessarily spherically sym-metric, inflation will act to homogenize and isotropize the spacetime inside the bubble, so we can view these models as cosmic bubbles embedded into a larger spacetime.
The dynamics of cosmic bubbles appear to be strongly model-dependent, so it has been difficult to make general statements about their behavior. In this paper, we will consider cosmic bubbles from a different angle, by using the null Raychaudhuri equation to study the consistency of cosmic bubble embeddings. While we will not be able to derive dynamical equations of motion for the bubble wall -which arise from the Israel boundary conditions -we will nonetheless find interesting constraints from the null Raychaudhuri equation on the types of bubble embeddings which are consistent with the propagation of null rays across the bubble wall. The Raychaudhuri equation is particularly powerful because it is independent of specific solutions to the Einstein equations, and has been used before [24,25] to study local inflation. In this paper, we will consider more general bubble and background spacetimes beyond inflation.
In particular, we will be interested in the behavior of the expansion θ of radial, inwardly directed, futureoriented null rays N α . For an affinely-parameterized null tangent vector N α ∇ α N β = 0, the expansion θ = ∇ α N α satisfies the null Raychaudhuri equation:
dθ dλ = − 1 2 θ 2 − |σ| 2 − R αβ N α N β ,(1)
where λ is an affine parameter along the geodesic and σ is the shear tensor. The Raychaudhuri equation (1) arises from the geometric properties of null vectors, and as such is independent of the Einstein equations and their solutions. Imposing the Einstein equations on the last term of (1), we have:
dθ dλ = − 1 2 θ 2 − |σ| 2 − T αβ N α N β .(2)
If we assume that the matter inside and outside of the bubble as well as on the bubble wall obeys the Null En- equation (2) implies that the expansion must be nonincreasing,
ergy Condition (NEC) T αβ N α N β ≥ 0,dθ dλ ≤ 0 .(3)
As a null ray traverses the bubble wall boundary Σ from the background spacetime into the bubble, as shown in Figure 1, the value of θ will, in principle, change. In order for the bubble embedding to be consistent with the Raychaudhuri equation (3), the value of θ must not increase across the wall,
∆θ = θ − − θ + ≤ 0 .(4)
It is common to study cosmic bubbles in the "thin wall" limit in which the boundary is infinitesimally thin, for which (4) must be true when evaluated at the (singular) boundary; however, (4) must also be satisfied for "thick wall" bubbles as well, as a null ray leaves the background and enters the bubble. The Raychaudhuri equation (2) is independent of specific solutions to the Einstein equations, thus (4) must be satisfied for any inwardly directed radial null ray, independent of the details of the matter content and spacetime structure of the background and bubble spacetimes as well as the bubble wall, as long such matter obeys the NEC. As such, our results will be generally applicable to a wide class of bubble embeddings. We will now consider several different scenarios for the background and bubble spacetimes and use (4) to constrain the allowed embeddings.
II. A BUBBLE IN A COSMOLOGICAL BACKGROUND
We will begin by considering a spatially flat homogeneous and isotropic cosmological bubble embedded at the origin of a homogeneous and isotropic cosmological background; non-spatially flat bubble and background spacetimes will be considered in Section II B. Both spacetimes are described by the FLRW metric:
ds 2 ± = −dt 2 ± + a ± (t ± ) 2 dr 2 ± + a ± (t ± ) 2 r 2 ± dΩ 2 ,(5)
where ± refers to the background/bubble spacetimes, respectively. The (timelike) boundary upon which these spacetime regions are joined is the wall of the bubble Σ. The metric must be continuous across the bubble wall 2 , which implies that the coefficient of dΩ 2 in the metric must be continuous across the boundary a + (t + )r
+ | Σ = a − (t − )r − | Σ ≡ R(τ ),
where τ is the proper time of the wall. Since the bubble and background spacetimes have (in principle) different cosmological evolutions, the comoving radial coordinates r ± cannot be continuous across the wall. The "areal radius"r ≡ a(t)r, however, will be continuous across the wall and we will set r + | Σ =r − | Σ = R on the bubble wall.
Radial, inwardly directed, future-oriented null rays in the cosmological spacetimes (5) are described by the (affine) null tangent vector:
N α ± = a −1 ± , −a −2 ± , 0, 0(6)
in either spacetime. The expansion of these rays θ ± = ∇ α N α ± is then
θ ± = 2 a ± H ± − 1 r .(7)
Following [26,27], (7) defines an apparent horizon for the bubble and background spacetimes where θ ± = 0:
r ± AH = H −1 ± .(8)
Null rays that are at an areal distance from the origin that is smaller than the radius of the apparent horizoñ r <r ± AH have negative expansion, as is expected for converging rays. However, null rays that are at a areal distance larger than the radius of the apparent horizoñ r >r ± AH have a positive divergence, indicating that the expansion of space is overcoming the expected geometric convergence of the rays.
A. Crossing the Bubble Wall
In the limit of an infinitesimally thin bubble wall, as a null ray crosses the boundary the radial coordinate is continuousr ± | Σ = R but the scale factor a ± and Hubble expansion rate H ± are not. Thus, the requirement that θ must decrease (4) becomes:
∆θ = 2 a − H − − 1 R − 2 a + H + − 1 R ≤ 0 .(9)
The metrics (5) contain an ambiguity: it is always possible to make a simultaneous rescaling of the scale factor and comoving radial coordinate by a constant a ± → λ ± a ± , r ± → λ −1 ± r ± that leaves the metric and physical distances invariant. We will use this freedom to fix the bubble and background scale factors to be equal to one at the time of the null ray wall crossing t ±,cross for the respective spacetimes only, e.g. a + (t +,cross ) = a − (t −,cross ) = 1. Since the matter content of the background and bubble generically differ, this implies that the scale factors should typically not be the same at any other time. Utilizing this freedom, we can simplify (9) to:
∆θ = θ − − θ + = 2(H − − H + ) ≤ 0 .(10)
This constraint has important implications for the embedding of cosmological bubbles inside cosmological background spacetimes. If the background is expanding but the bubble spacetime is collapsing H − < 0, then we are able to satisfy (10) without any difficulty, indicating that a collapsing bubble inside an expanding background is always an allowed solution. Alternatively, if the background spacetime is collapsing, then (10) indicates that the bubble spacetime cannot be expanding while still satisfying the Raychaudhuri equation. Since this is of little practical cosmological interest for us, however, we will not consider this situation further.
If both the bubble and the background spacetimes are expanding H ± > 0, then (10) is only satisfied for a bubble that has a lower Hubble rate than the background H − < H + , indicating that the energy density of the bubble is smaller than the background ρ − < ρ + . Some simple examples include a true vacuum bubble embedded inside a false vacuum background, or a low temperature bubble inside a higher temperature background. However, (10) fails for a bubble that has a higher Hubble rate than the background H − > H + , including false vacuum or high temperature bubbles. This represents a significant challenge for embedding false vacuum or "hot" bubbles into cosmological backgrounds. This further extends the results of [7,8], which found that the Israel junction conditions do not allow for a spatially flat bubble with H − > H + unless the bubble is superhorizon-sized. We find a stronger result here, which is that a spatially flat bubble with H − > H + is not allowed for any size bubble. We emphasize that this result is independent of the details of the matter content inside and outside of the bubble as well as the tension of the bubble wall, as long as the matter obeys the NEC. The Raychaudhuri equation thus provides a very strong constraint on allowed bubble embeddings.
A limited form of this result was found in [24], which considered an inflationary bubble embedded inside a noninflationary cosmological background with H − > H + . [24] noticed that for inflationary bubbles that are larger than their own apparent horizon but smaller than the apparent horizon of the backgroundr − AH < R <r + AH , an ingoing light ray traversing the bubble wall that starts in the background will begin with negative expansion θ + < 0. However, after traversing the bubble wall boundary, the expansion is now positive θ − > 0 and therefore θ is non-decreasing, in violation of the Raychaudhuri equa-tion. This change in sign of θ can be avoided if the inflationary bubble is larger than both the bubble and background apparent horizons R >r + AH ,r − AH , so that θ ± is positive in both spacetimes [24]. Putting aside the difficulty of establishing such a configuration in a causal way, our result (10) indicates that it is not sufficient for θ ± to simply be positive in both spacetimes, as θ − for the inflationary bubble is still larger than θ + for the background when H − > H + for any size bubble.
The challenge presented by (10) to bubbles with H − > H + is quite basic, and it is compelling to view the failure of (10) as due to the unrealistic assumption that the bubble and background cosmological spacetimes (5) are glued together on an infinitesimally thin wall. Since the gravitating energy enclosed by the bubble is larger than it would be if filled with the background energy density, we expect the energy density of the bubble to locally backreact on the background metric so that it deviates from the pure cosmological form. Indeed, for a pure de-Sitter bubble embedded in a background that is vacuum [5] or dominated by a positive cosmological constant [12], the spacetime is approximately Schwarzschild in the vicinity of the bubble wall. Unfortunately, analytic forms for the backreaction of the bubble energy density are not known when the bubble and background are filled with a generic cosmological fluid.
Nevertheless, we can move beyond our approximation of an infinitesimally thin wall in a generic way by assuming that the bubble wall now has a "thickness" of 2δ, as in Figure 2. We will leave the spacetime geometry inside the thick wall unspecified, as its form likely will include the unknown backreaction of the bubble and tension of the wall. Since we do not know the details of the metric inside of the thick wall, we cannot compute θ inside the wall. Nevertheless, it must still be true that (4) is satisfied for a null ray as it enters and exits the thick wall.
An ingoing null ray enters the thick wall from the background atr = R + δ and leaves the wall into the bubble atr = R − δ, so we have 3 :
∆θ = 2(H − − H + ) − 4δ R 2 − δ 2 ≤ 0 .(11)
In contrast to (10), it is now possible to solve (11) for an expanding bubble with H − > H + . In particular, (11) is satisfied if the thickness of the wall is larger than
δ ≥ 1 2 R 2 ∆H ,(12)
where ∆H = H − − H + , and we assumed 4 R|∆H| 1. It is interesting to see how the presence of a "thick" wall satisfying (12) also evades the argument of [24] described above. In the presence of a thick wall of sizer − AH < R <r + AH satisfying (12), a null ray exits the wall and enters the bubble cosmology at the inner boundary of the wallr =
R − δ. Since H − (R − δ) ≤ H − R 1 − 1 2 H − R 1 − H+ H− ≤ 1− 1 2 1 − H+ H− < 1
, the inner boundary of the wall is smaller than the bubble apparent horizon R−δ < H −1 − =r − AH . Thus, the expansion of the null ray is negative when it enters the cosmological part of the bubble, and Raychaudhuri's equation (11) can be satisfied.
It is important to note that the condition (12) is a necessary condition for the thickness of a wall surrounding a cosmological bubble with H − > H + , but it is not sufficient. Our approach has avoided specifying the behavior of θ inside the "thick" bubble wall, and it must be the case that θ is decreasing inside the thick bubble wall in a way that interpolates between the expansions θ of the exterior and interior spacetimes to satisfy the Raychaudhuri equation. In Section III we consider a cosmological bubble surrounded by a Schwarzschild spacetime, which can serve as one possible model for the spacetime inside the "thick wall." In the next subsection, we generalize our argument to include non-zero spatial curvature for both the bubble and the background, finding that the main results of this section hold for a larger range of sings of the spatial curvature of the bubble and background.
B. Non-zero Spatial Curvature for Bubble and Background
The constraint (10), while independent of the matter content of the bubble and background, applies only to bubble and background spacetimes that are spatially flat. In general, however, the bubble and background can have their own distinct spatial curvature. Indeed, [25] gener-alized the argument of [24] to non-flat spatial sections and found that it is possible to avoid changes of sign of θ ± when crossing the boundary if the bubble is initially negatively curved. In this section, we will generalize our results from Section II A to non-flat spatial geometries by requiring that θ be non-increasing.
Including spatial curvature in a homogeneous and isotropic spacetime, we start with the metrics:
ds 2 ± = −dt 2 ± + a ± (t) 2 dr 2 ± 1 − k ± r 2 ± + r 2 ± dΩ 2 ,(13)
where k ± > 0 (k ± < 0) corresponds to positive (negative) spatial curvature, and k ± = 0 is flat space. Note that (13) still allows for an independent constant rescaling of the scale factors a ± → λ ± a ± as long as the comoving radial coordinates and spatial curvatures are rescaled as well r ± → λ −1 ± r ± , k ± → λ 2 ± k ± . This is only consistent if one does not choose k ± = ±1, as is common in the presence of spatial curvature, which we will avoid.
Radially inward affine null rays in the spacetimes (13) take the form:
N α ± = 1 a ± 1, − 1 − k ± r 2 ± a ± , 0, 0 ,(14)
with corresponding expansion θ ± = ∇ α N α ± ,
θ ± = 2 a ± H ± − 1 − k ± r 2 ± r ,(15)
where again we have used the areal radiusr = a − r − = a + r + , which is continuous across the boundary, even in the presence of spatial curvature. As in flat space, a vanishing expansion for ingoing null rays θ ± = 0 defines an apparent horizon for non-spatially flat FLRW space:
r ± AH = 1 H 2 ± + k ± /a 2 ± .(16)
Forr > r ± AH , the expansion is positive due to the expansion of space.
As the null ray crosses the boundaryr| Σ = R, the requirement from the Raychaudhuri equation (4) that θ must be non-increasing ∆θ = θ − − θ + ≤ 0 becomes:
∆θ = (17) 2 a − H − − 1 − k − r 2 − R − 2 a + H + − 1 − k + r 2 + R = 2(H − − H + ) − 2 R 1 − k − r 2 − − 1 − k + r 2 + ≤ 0 ,
where we again used our rescaling freedom to set a ± = 1 at the wall crossing. It does not seem possible to make general statements about the implications of (17), so we will examine the constraints imposed by (17) for specific cases. The results are summarized in Table I.
In particular, for a background that is either flat or negatively curved k + ≤ 0, we have 1 − k + r 2 + ≥ 1, while a bubble that is either flat or positively curved k − ≥ 0 has 1 − k − r 2 − ≤ 1. Combining these two inequalities with (17), we find
H − − H + ≤ 0 .(18)
This is an analogous constraint as (10) from Section II A: for an expanding bubble and background, only bubbles with a smaller Hubble rate than that of the background are consistent with the Raychaudhuri equation. (Similar arguments as given in Section II A hold for a nonexpanding bubble or background). For a background with positive spatial curvature k + > 0 and positive or flat spatial curvature for the bubble k − ≥ 0, we can rearrange (17) into the form
H − − H + ≤ 1 R 1 − k − r 2 − − 1 − k + r 2 + ≤ 1 R ,(19)
where the last inequality follows because the square roots are bounded for non-negative k ± . This constraint (19) is considerably less stringent than (18) since while an expanding bubble with a Hubble rate smaller than that of the background still satisfies (19), we can now have an expanding bubble with a Hubble rate larger than that of the background, provided that the size of the bubble is smaller than R ≤ (H − − H + ) −1 . If the bubble and the background expansion rates are not too different, this does not amount to too stringent of a constraint. However, if the bubble is expanding much faster than the background H − H + , then the bubble size is bounded above by the bubble's inverse Hubble length R < H −1 − . Similarly, if the background itself is collapsing H + < 0, the size of the bubble is again constrained by a combination of the expansion rates. Alternatively, if the bubble is collapsing, (19) is automatically satisfied, as in the flat space case.
Finally, to enumerate all possibilities we must consider the cases when the bubble is negatively curved k − < 0. In all of these cases, (17) implies a bound on the size of the bubble:
R ≤ 1 + |k − | r 2 − − 1 − k + r 2 + H − − H + ,(20)
(where we took the absolute value of the bubble spatial curvature for clarity). It is interesting to compare the results of Table I with the constraints obtained from the Israel boundary conditions on FLRW spacetimes embedded in FLRW backgrounds from [7,8]. In particular, [8] finds that there are no restrictions on the bubble embedding when the difference in energy density between the background and bubble is positive ρ + > ρ − and larger than the surface energy density S of the bubble wall ρ + − ρ − > 6πGS. However, when this difference is either negative, as when ρ − > ρ + , or smaller than the surface energy density, then it is not possible to satisfy the Israel junction conditions for flat or negatively curved spacetimes k + ≤ 0 (for any value of the bubble spatial curvature) unless the bubble is larger than super-Hubble size R > H −1 + . In contrast, we find a much stronger result since if the background is flat or negatively curved and the bubble is flat or positively curved, the embedding is inconsistent with the NEC and the Raychaudhuri equation unless the bubble expansion rate is smaller than the background H − < H + , for any sized bubble. On the other hand, if the bubble is negatively curved, our constraint (20) appears to put an upper bound on the size of an allowed bubble, rather than the lower bound of [8]. Thus, we see that the constraints imposed by the Raychaudhuri equation are complementary, and in some cases much stronger, than constraints obtained by the Israel boundary conditions. As in Section II A, it is tempting to see the failure of the Raychaudhuri equation for non-spatially flat bubbles as due to a possibly unrealistic assumption that the size of the bubble wall is infinitesimally thin. We can generalize our argument here to include a wall with a thickness of 2δ as in Figure 2 with unspecified geometry, so that a null ray leaves the background and enters the wall at r = R + δ and leaves the wall and enters the bubble at r = R − δ. The expansion θ must be non-increasing from when the null ray leaves the background and enters the bubble wall, ∆θ = θ − |r =R−δ − θ + |r =R+δ (21) leading to the constraint:
Background Bubble k+ > 0 k+ = 0 k+ < 0 k− > 0 H− − H+ ≤ R −1 H− < H+ k− = 0 k− < 0 R(H− − H+) ≤ 1 + |k−| r 2 − − 1 − k+ r 2 +2 a − (t * − ) H − − 1 − k − (R − δ) 2 /a 2 − R − δ (22) − 2 a + (t * + ) H + − 1 − k + (R + δ) 2 /a 2 + R + δ ≤ 0 ,
where t * ± are the times when the null ray crosses the boundary out of (into) the background (bubble) spacetimes. It is difficult to draw generic conclusions from (22). Specific constraints can be obtained in the H + → 0 limit, though since in this limit the background is empty, so we should consider a Schwarzschild background spacetime instead.
FIG. 3: A FLRW bubble surrounded by a Schwarzschild back-
ground can serve as a model for a "vacuole" embedded in a larger expanding background. We study the behavior of radial null rays as they traverse from the Schwarzschild background into the bubble.
III. BUBBLE IN A SCHWARZSCHILD BACKGROUND
We will now consider our cosmological bubble to be surrounded by a background which is vacuum Schwarzschild spacetime. This can serve as a model of a false vacuum bubble in flat space, as in [4,5,28], or as a Schwarzschild "vacuole" embedded in a larger cosmological background, as in Figure 3. Indeed, [15,16] have argued that vacuum bubbles will develop just such a Schwarzschild layer as they evolve in a post-inflationary Universe. This Schwarzschild layer can thus serve as a simple model for the "thick wall" introduced in Section II, or as a background for the bubble in its own right.
We will thus take the background spacetime to be spatially flat Schwarzschild space:
ds 2 S = − 1 − 2GM r S dt 2 S + 1 − 2GM r S −1 dr 2 S +r 2 S dΩ 2 ,(23)
where we will use a subscript "S" for the background to avoid confusion with the FLRW background coordinates from the previous section. A radially ingoing (affine) null ray in this spacetime,
N α S = 1 − 2GM r S −1 , −1, 0, 0 ,(24)
has the expansion:
θ S = ∇ α N α S = − 2 r S ,(25)
which is what we would expect to get by setting H + = 0 in the FLRW background of the previous section. As the null ray traverses the bubble wall, the spacetime changes from the exterior Schwarzschild space to the interior FLRW cosmology (we will consider k − = 0 here):
ds 2 = −dt 2 − + a − (t − ) 2 1 − k − r 2 − dr 2 − + a − (t − ) 2 r 2 − dΩ 2 .(26)
As before, we will find it convenient to work with the areal radiusr − = a − (t − )r − , which is continuous across the boundary of the bubble. A radially ingoing affine null ray for this non-spatially-flat FLRW spacetime takes the form
N α − = 1 a − 1, −a −1 − 1 − k − r 2 − , 0, 0 ,(27)
which has the expansion
θ − = 2 a − H − − 1 − k −r 2 − /a 2 − r − .(28)
In order for the Raychaudhuri equation (4) to be satisfied as the null ray crosses the thin wall boundaryr S | Σ = r − | Σ = R, we must have,
∆θ = θ − − θ S = 2 a − H − + 2 R 1 − 1 − k − R 2 /a 2 − a − ≤ 0 .(29)
Note that since θ − is positive when the size of an expanding bubble is larger than its apparent horizon R >r − AH , while θ S is always negative, the requirement from the Raychaudhuri equation that θ be non-increasing implies that the bubble may never be larger than its own apparent horizon. Thus, an arbitrarily large expanding bubble cannot develop when surrounded by a Schwarzschild spacetime. It is interesting to compare this result to that of [28], which found that an expanding bubble larger than its apparent horizon must have begun in an initial singularity. We find a complementary result here, that such a bubble embedding would not be consistent with the Raychaudhuri equation to begin with.
Utilizing our freedom to rescale the scale factor of the bubble to a(t * − ) = 1 at the time of crossing t * − , the condition (29) becomes
2H − + 2 R 1 − 1 − k − R 2 ≤ 0 .(30)
As in the previous section, we can impose stronger constraints from (30) for specific assumptions about the spatial curvature of the bubble. Assuming a bubble that is either spatially flat or positively curved k − ≥ 0, the second term of (30) is always positive; thus (30) requires that the bubble spacetime be contracting
H − ≤ 0 .(31)
This implies that a flat or positively curved cosmic bubble embedded inside of a Schwarzschild background must be collapsing in order to satisfy the Raychaudhuri equation. This is consistent with the the k − ≥ 0 Oppenheimer-Synder solutions [3] for a ball of collapsing dust, some of the first and most famous solutions of cosmological bubbles embedded in a Schwarzschild spacetime background. While the Oppenheimer-Synder solution is specifically for pressureless dust and zero surface tension, our result (31) does not make any assumptions about the matter content of the bubble or surface tension (as long as it satisfies the NEC), thus generalizing the conditions under which the bubble collapses.
One motivation for considering a background described by a Schwarzschild spacetime was that it could serve as a simple model of a "thick wall" between a cosmological FLRW background and the bubble. Because of this, we have assumed the thin wall approximation for the bubble wall. However, it can be the case that the boundary between the Schwarzschild and bubble spacetimes is also itself thick; we will therefore generalize our results for a thick wall of size 2δ, as in Section II.
First, note that even in the presence of a thick wall, the inner boundary of the wall R − δ must be smaller than the apparent horizon of the bubble, for the same reasons as described below (29), again indicating that the bubble size may not be larger than its apparent horizon for any value of the bubble spatial curvature. Constraints on the minimum thickness of a sub-horizon bubble wall can be obtained from (22) by setting H + = 0 and k + = 0 (since θ S and θ + agree in this liimit). For a spatially flat bubble, the thickness of the bubble wall must be larger than
δ > H −1 − −1 + 1 + R 2 H 2 − ,(32)
in order to satisfy the Raychaudhuri equation. For bubbles much smaller than their apparent horizon size RH − 1, the required thickness δ min is small compared to the size of the bubble δ min /R = 1 2 H − R 1. However, for bubbles that are comparable in size their apparent horizon H − R ∼ 1, the wall thickness is comparable to the size of the bubble itself δ ∼ R, necessitating a different spacetime structure for the entire bubble. Thus, even with a "thick wall" with an unspecified metric, large expanding bubbles embedded into Schwarzschild space are inconsistent with the Raychaudhuri equation.
IV. DISCUSSION
The Raychaudhuri equation requires that the expansion of radially inward null rays must be non-increasing as long as matter obeys the Null Energy Condition (NEC), independent of the Einstein equations. We have used this to study allowed spherically symmetric embeddings of FLRW cosmological bubbles into various background spacetimes.
We found that when the background is FLRW space with non-positive spatial curvature k + ≤ 0 and the bub-ble has non-negative spatial curvature k − ≥ 0, the Raychaudhuri equation constrains the bubble's Hubble expansion rate to be smaller than that of the background's, H − < H + , for any size bubble. (For other combinations of the spatial curvature of the bubble and background, see Table I.) This result has several important implications for cosmological bubbles. In particular, it rules out a broad class of embeddings of false vacuum bubbles into true vacuum backgrounds obtained by joining the spacetimes together across the bubble wall, irrespective of the details of the matter content on the wall as long as the matter obeys the NEC. It also implies a difficulty with embedding local inflation into a larger cosmological background, as first suggested by [24] in a more limited context. In particular, since the Hubble expansion rate in a near-dS inflating bubble is likely to be larger than that of its non-dS background (since the latter will decrease with time while the former does not), a model of local inflation obeying the NEC conflicts with this constraint. This has important implications for recent studies of the onset of inflation from inhomogeneous initial conditions [22,23].
We also considered bubbles surrounded by empty Schwarzschild space, which could serve as a model of a "vacuole" embedded in a larger FLRW background. We found that expanding bubbles must be smaller than their own apparent horizon in order to satisfy the Raychaudhuri equation. This rules out the possibility of embedding an expanding bubble Universe in flat space, extending the result [28] that such a bubble must start from an initial singularity. Further, we show that a bubble of any size with flat or positives spatial curvature cannot be expanding but rather must be contracting, generalizing the collapsing behavior of the Oppenheimer-Synder dust ball solution [3] to any matter content satisfying the NEC.
Our results place strong limits on the embedding of spherically symmetric bubbles into a background spacetime, so we also considered the relaxation of the infinitesimal "thin wall" approximation for the bubble wall. Without specifying the metric within the "thick wall" of the bubble, the Raychaudhuri equation requires that the expansion of the null rays be non-increasing between the outer and inner boundaries of the bubble wall. For a cosmological background, we derived a lower bound on the required thickness of the bubble wall to satisfy the Raychaudhuri equation, and showed that for a spatially flat background and bubble the inner boundary of the bubble is smaller than the apparent horizon of the bubble spacetime, so that again the bubble may not be larger than its own apparent horizon. For a Schwarzschild background the bubble must be much smaller than its own apparent horizon or else the required thickness of the wall is comparable to the size of the bubble itself.
It is possible to avoid the constraints we have derived here by embedding the cosmological bubble in such a way that the bubble is casually disconnected from the background spacetime. In particular, an expanding cosmological bubble tucked behind a wormhole will be casually disconnected from radially ingoing null rays from the background, as in Figure 4. Solutions of this type have been described previously in [4,5,9] for a de-Sitter FIG. 4: One way to evade the constraints on bubble embeddings from the Raychaudhuri equation is for the bubble to become disconnected from the background spacetime by the creation of a wormhole, as shown in this Penrose diagram adapted from [15]. Radially ingoing null rays starting in the background either see the bubble contracting or are casually disconnected from the bubble, while the bubble continues to expand in a "baby Universe".
bubble and Schwarzschild background, [6] for some limited FLRW spacetimes, [10,12] for de-Sitter bubble and background spacetimes, and [15,16] for false-vacuum and domain-wall bubbles embedded in FLRW backgrounds.
It is important to note, however, that the conclusions of [28] still apply, so that bubbles that are bigger than their apparent horizons still must begin in an initial singularity, even if they are tucked behind a wormhole.
Finally, it would be useful to see how robust these arguments are to deviations from homogeneous spherical symmetry in the bubble and the background (see [29] for a discussion of an inhomogeneous background). We have also been assuming that the matter content of the background, bubble interior, and bubble wall obeys the NEC; it is possible to consider cosmological models based on violations of the NEC, either through quantum gravity effects [30], NEC-violating matter fields (see e.g. [31,32] for some common models), or non-minimal coupling [33,34]. We will leave a detailed study of bubbles with these effects for future work.
FIG. 2 :
2A null ray traveling across a "thick wall" boundary between a cosmological background H+ and bubble H− leaves the background atr = R + δ and enters the bubble atr = R − δ.
TABLE I :
IRequiring that the expansion θ of a null ray be non-increasing as the ray traverses the boundary between a background spacetime and bubble spacetime when including spatial curvature leads to different constraints depending on the relative signs of the spatial curvatures of the spaces.
See[7,8] for an analysis of the Israel junction conditions across the boundary.
We have again used the rescaling freedom in the scale factor to set a + = a − = 1 when the ray crosses the respective exterior and interior boundaries.4 If R∆H ∼ O(1), then the wall must be approximately the same size as the bubble itself δ ≥ ∆H ∼ R in order to solve(11), which implies that the bubble spacetime is almost certainly not described by a homogeneous and isotropic cosmological metric.
AcknowledgementsThe authors would like to thank T. Ali and C. Hellaby for useful discussions. We would also like to thank A. Bhattacharyya for conversations on an earlier version of this work. SSH is supported by the Claude Leon Foundation.
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| [] |
[
"AN EXTENSION OF WIENER INTEGRATION WITH THE USE OF OPERATOR THEORY",
"AN EXTENSION OF WIENER INTEGRATION WITH THE USE OF OPERATOR THEORY"
] | [
"Palle E T Jorgensen ",
"Myung-Sin Song "
] | [] | [] | With the use of tensor product of Hilbert space, and a diagonalization procedure from operator theory, we derive an approximation formula for a general class of stochastic integrals. Further we establish a generalized Fourier expansion for these stochastic integrals. In our extension, we circumvent some of the limitations of the more widely used stochastic integral due to Wiener and Ito, i.e., stochastic integration with respect to Brownian motion. Finally we discuss the connection between the two approaches, as well as a priori estimates and applications. | 10.1063/1.3196622 | [
"https://arxiv.org/pdf/0901.0195v1.pdf"
] | 18,904,839 | 0901.0195 | 7526ac1c095804e2a72f7f3cbb8ef274ca3e3deb |
AN EXTENSION OF WIENER INTEGRATION WITH THE USE OF OPERATOR THEORY
1 Jan 2009
Palle E T Jorgensen
Myung-Sin Song
AN EXTENSION OF WIENER INTEGRATION WITH THE USE OF OPERATOR THEORY
1 Jan 2009
With the use of tensor product of Hilbert space, and a diagonalization procedure from operator theory, we derive an approximation formula for a general class of stochastic integrals. Further we establish a generalized Fourier expansion for these stochastic integrals. In our extension, we circumvent some of the limitations of the more widely used stochastic integral due to Wiener and Ito, i.e., stochastic integration with respect to Brownian motion. Finally we discuss the connection between the two approaches, as well as a priori estimates and applications.
Introduction
Recently there has been increase in the number of applications of stochastic integration and stochastic differential equations (SDEs). In addition to the traditional applications in physics and dynamics, stochastic processes have found uses in such areas as option pricing in finance, filtering in signal processing, computations biological models. This fact suggests a need for a widening of the more traditional approach centered about Brownian motion B(t) and Wiener's integral.
Since SDEs are solved with the use of stochastic integrals, we will focus here on integration with respect to a wider class of stochastic processes than has previously been considered. In evaluation of a stochastic integral we deal with the term dB(t) by making use of the basic properties of Brownian motion, such as the fact that B(t) has independent increments. If instead X(t) is an arbitrary stochastic process, it is not at all clear how to make precise a stochastic integration with respect to dX(t). We will develop a method, based on a Karhunen-Loève diagonalization, for doing precisely that.
The theory of stochastic integrals is well developed, see e.g., [Kuo06,IM65]. For many applications, such as the solutions to stochastic differential equations in physics and finance, it is important to have tools for evaluating integrals with respect to dB where B is Brownian motion. The reason for the technical issues involved in the computation of stochastic integrals can be understood this way: A naive approach runs into difficulties, for example because the length of Brownian paths is infinite, and because Brownian paths are discontinous (with probability one). Wiener and Ito offered a, by now, well known way around this difficulty. The idea of Wiener in fact is operator theoretic: It is to establish the value of an integral as a limit that takes place in a Hilbert space of random variables. This is successful because of the existence of an isometry between this Hilbert space on the one hand and a standard L 2 −Lebesgue space on the other.
In this paper we extend this operator theoretic approach to a much wider class of stochastic integrals, i.e., integration with respect to dX where X belongs to a rather general class of stochastic processes. And we give some applications.
In the proof of our theorem we make use of a result from two earlier papers [JS07,JS08] by the coauthors. The idea is again operator theoretic, and it is based on an application of von Neumann's spectral theorem to an integral operator directly associated with the process X under consideration.
While the applications of stochastic integrals to physics (e.g., [BC97], and their interplay with operator theory (e.g., [AL08a])are manifold, the idea of exploring and extending the scope in the present direction appears to be new. The need for such an extension is convincing: For example, physical disturbances or perturbations will typically take you outside the particular path-space framework where the theory was initially developed.
Earlier approaches to stochastic integrals with reproducing kernels include [AL08a, AAL08, AL08b] and operator theory [JM08]; and papers exploring physical ramifications: [BC97, BDSG + 07, Hud07b,Hud07a]. Although the papers cited here with physics applications represent only the tip of an iceberg!
Notation and Definitions
To make precise the operator theoretic tools going into our construction, we must first introduce the ambient Hilbert spaces. Since stochastic integrals take values in a space of random variables, we must specify a fixed probability space Ω, with sigma algebra and probability measure. In the case of Brownian motion, the probability space amounts to the standard construction of Wiener and Kolmogorov: The essential axiom in that case is that all finite samples are jointly Gaussian, but we will consider general stochastic processes, and so we will not make these restricting assumptions on the sample distributions and on the underlying probability space. For more details on this case, see section 4.
The kind of integrals we consider presently are stated precisely in Definition 2.1, eq (2.5) below. I particular, initially we consider only functions of time in the integrant, so f (t)dX t . When the stochastic process X is given, we show (Theorem 3.1) that the corresponding integrals live in a Hilbert space which is a direct sum of standard Lebesgue Hilbert spaces carrying the function f (t). In the case of Brownian motion, we show (Example 4.1) that the direct sum representation then only has one term.
We now list the symbols and the terminology.
• L 2 : an L 2 -space.
• L 2 (R): all L 2 -functions on R.
• J ⊂ R a finite closed interval.
• L 2 (J): L 2 with respect to the Lebesgue measure restricted to J.
• (Ω, S, P ): a fixed probability space.
• Ω: sample space.
• S: some sigma algebra of subsets of Ω.
• P : a probability measure defined on S.
• L 2 (J × Ω, m × P ): the L 2 -space on J × Ω with respect to product measure m × P where m denotes Lebesgue measure.
(1) X t : Ω → R, t ∈ R a stochastic process, (2) X t (ω) := X(t, ω), t ∈ R, ω ∈ R.
(3) E(·) := Ω ·dP : the expectation with respect to (Ω, S, P ).
Restricting Assumptions:
(i) X ∈ L 2 (J × Ω, m × P ) for all finite intervals J ⊂ R. (ii) (s, t) −→ E(X s X t ) is continuous on J × J. (E(X t ) = 0).
(iii) For all J, and all s ∈ J, the function,
(2.1) t −→ E(X s X t )
is of bounded variation. For J ⊂ R fixed, we consider partitions (2.2) π : t 0 < t 1 < · · · < t n−1 < t n =: t, J = [t 0 , t];
and we set
(2.3) |π| := max i (t i+1 − t i ), and ∆t i := t i+1 − t i , 0 ≤ i < n.
If f : R → C is continuous, we set
(2.4) S π (f, X) := n−1 i=0 f (t i )(X ti+1 − X ti ).
Definition 2.1. By a stochastic integral, we mean a limit
(2.5) lim |π|→0 S π (f, X) =: t t0 f (s)dX s
We now turn to questions of existence of this limit for a rather general family of stochastic processes X t ; see (i)-(iii) below.
Statement of the Main Theorem
When the stochastic process X is given, we proved in Theorem 3.1 that the corresponding integrals live in a Hilbert space which is a direct sum of standard Lebesgue Hilbert spaces carrying the function f (t). In the case of Brownian motion, we now show that the direct sum representation then only has one term. Yet the method from section 3 still offers a Fourier decomposition of the Wiener integration.
Theorem 3.1. Let (Ω, S, P ) be given as above, and let X be a stochastic process satisfying conditions (i) − (iii). Let f be given and continous.
(a) Then the stochastic integral t t0 f (s)dX s exists and is in L 2 (Ω, P ). (b) There is a family of bounded variation functions ϕ 1 , ϕ 2 , · · · and numbers λ 1 , λ 2 , · · · satisfying the following conditions:
λ 1 ≥ λ 2 ≥ · · · > 0, λ k → 0 in fact k λ k < ∞, such that (3.1) E( t t0 f (s)dX s 2 ) = ∞ k=1 λ k t t0 f (s)dϕ k (s) 2
where the terms t t0 f (s)dϕ k (s) refer to Stieltjes integration. (c) If an interval J is chosen such that t 0 and t ∈ J, and if f has a weak derivative f ′ in L 2 (J), then the following estimate holds for the RHS in (3.1):
(3.2) RHS ≤ "Const" + λ 1 t t0 |f ′ (s)| 2 ds
where "Const" depends on certain boundary conditions, and where λ 1 is the maximal eigenvaluel see (5.4) below.
In the next corollary, we stay with the assumptions from the theorem; in particular X t is a stochastic process subject to conditions (i)-(iii), and a compact interval J is fixed.
Corollary 3.2. Covariance relations: • E(X s X t ) = ∞ k=1 λ k ϕ k (s)ϕ k (t) • E(X 2 t ) = ∞ k=1 λ k |ϕ k (t)| 2 • Dependency of increments: If s < t < u in J, then E((X t − X s )(X u − X t )) = ∞ k=1 λ k (ϕ k (t)ϕ k (u) − ϕ k (s)ϕ k (u) + ϕ k (s)ϕ k (t) − |ϕ k (t)| 2 ) • E((X t+∆t − X t ) 2 ) = ∞ k=1 λ k |ϕ k (t + ∆t) − ϕ k (t)| 2
An Application
In this section we restrict the setting of Theorem 3.1 to the special case when X = B, i.e., to the special case of integration with respect to Brownian motion. We then work out the eigenfunctions and eigenvalues for the covariance operator. It turns out to be the familiar Fourier basis. Actually there is a choice of bases depending on boundary conditions. A choice of the Dirichlet conditions yields the ONB of the sine functions. We further show that when the eigenvalue expansion is summed (using orthogonality) we then arrive at the familiar Wiener-Ito formula.
Example 4.1. X = B = Brownian motion.
• (Ω, S, P ) Gaussian space;
• Ω a space of functions, S := cylinder sets σ− algebra; the sigma algebra generated by the cylinder-sets.
• J = [0, 1]; • E(X s X t ) = min(s, t) =: s ∧ t; • X t (ω) = ω(t), ω ∈ Ω; • E((X t+∆t − X t ) 2 ) = ∆t.
We now show that the known formula
(4.1) E( t 0 f (s)dB s 2 ) = t 0 |f (s)| 2 ds
follows from the theorem; and in particular from (3.1).
In the case of Brownian motion for the functions ϕ k we may take
(4.2) ϕ k (t) = √ 2 sin(kπt); k = 1, 2, · · · .
Note that
(4.3) ϕ k (0) = ϕ k (1) = 0, ∀k = 1, 2, · · · ; and (4.4) λ k = 1 (kπ) 2 Set t 0 = 0 for simplicity. Note that if (4.5) g(t) := 1 0 t ∧ sf (s)ds, then ( d dt ) 2 g(t) = −f (t),
so the eigenvalue problem (4.6)
1 0 t ∧ sf (s)ds = λf (t)
has the solution given by (4.2) and (4.4). Note further that (4.3) is a choice of boundary conditions. To see that (4.1) follows from (3.1) in the theorem, we proceed as follows; starting with the RHS in (3.1) and using d(sin(kπt)) = −kπ cos(kπt)dt:
∞ k=1 λ k t 0 f (s)dϕ k (s) 2 = ∞ k=1 2 (kπ) 2 t 0 f (s)d sin(kπs) 2 = by(4.3) 2 ∞ k=1 t 0 f (s) cos(kπs)ds 2 = byP arseval ′ sf ormula t 0 |f (s)| 2 ds;
and the desired conclusion follows.
Proof of Theorem 3.1
Here we give the details of proof of theorem 3.1. Since the proof is long, to help the reader our presentation is divided into two parts, A and B.
Part A is an outline of the steps in the proof itself, and part B contains the details arguments making up each part in the proof. Part A begins with the notation and the terminology, introducing an auxiliary selfadjoint operator, its matrix approximations, and its spectral resolution.
Part A.
• Select a fixed interval J := [a, b], a < b.
• From the assumptions on the process (X t ) t∈R note that the operator
(5.1) (T J f )(t) := J E(X t X s )f (s)ds
is compact and selfadjoint in the Hilbert space L 2 (J). • For every partition π : t 0 < t 1 < · · · < t n , t 0 = a, t n = b; the following matrix
(5.2) (M J,π ) i,j := E(X ti X tj )
offers a discrete approximation for the operator T j in (5.1).
• Set (5.3) H(J) := L 2 (J) ⊖ ker(T J ) = {g ∈ L 2 (J) : g, k L 2 = 0, ∀k ∈ ker(T J )}.
Then an application of the spectral theorem to T J yields the following sequence of orthogonal eigenfunctions ϕ 1 , ϕ 2 , · · · in H(J), and numbers λ 1 , λ 2 , · · · ∈ R + such that λ k → 0;
λ 1 ≥ λ 2 ≥ · · · > 0, λ k → 0 such that (5.4) T J ϕ k = λ k ϕ k k = 1, 2, · · ·
orthogonality relations in the t−domain:
(5.5) ϕ j , ϕ k L 2 (J) = J ϕ j ϕ k dm = δ j,k ;
and the closed span of {ϕ k } is H(J). • Set
(5.6) Z k (·) := 1 √ λ k J φ k (t)X t (·)dt,
and note that each Z k , k = 1, 2, · · · is a random variable, Z k ∈ L 2 (Ω, S, P ).
Moreover, a calculation yields (orthogonality relations in the ω−domain:
(5.7) E(Z j Z k ) = δ j,k
• Aside; note that if (X t ) is assumed Gaussian, then each Z k , k = 1, 2, · · · is Gaussian as well. • Karhunen-Loève, or Generalized Fourier Expansion:
In L 2 (J × Ω, m × P ), we have the following pointwise a. e. representation and so from the assumptions (i)-(iii) and eq. (2.1) we conclude that each of the eigenfunctions ϕ k (·) is continuous and of bounded variation. This means that whenever t 0 , t ∈ J, i.e., a ≤ t 0 < t ≤ b, the expression
(5.11) t t0 f (s)dϕ k (s)
is a well defined Stieltjes integral. Morever, if f is assumed of bounded variation,
(5.12) t t0 f dϕ k = [f ϕ k ] t t0 − t t0 ϕ k (s)f ′ (s)ds.
We now turn to the approximation (2.4) from the theorem, and we use the Karhunen-Loève expansion (5.8) in the computation of (5.13) ∆X ti := X ti+1 − X ti for a fixed (chosen) partition π as specified in (2.2). Using condition (3)(ii) in the statement of the theorem, we note that for fixed J, the operator T j in (5.1) is trace-class. From operator theory (Mercer's theorem), we know that (5.14)
trace
(T J ) = J E(X 2 t )dt = ∞ k=1 λ k < ∞
i.e., integration in (5.1) over the diagonal s = t. And so in particular, finiteness of ∞ k=1 λ k follows.
In the study of the operator T J from (5.1) we make use of tools from Hilbert space theory of integral operators. In particular, in the estimate (5.14) we use Mercer's theorem. However in applications to covariance kernels (2.1) one often has stronger properties. It is known that if the kernel in (2.1) is Lipschitz of degree γ with γ > 1 2 in one of the two variables (with the other fixed), then the operator T J in (5.1) will automatically be nuclear. For the literature on this we refer to [Dos93,Küh83,LL52,Sti58]. We further note that this Lipschitz condition is indeed satisfied for the covariance kernel of fractional Brownian motion, see e.g., [IA07].
Set (∆ϕ k ) ti := ϕ k (t + ∆t) − ϕ k (t). Using the Hilbert space L 2 (J × Ω, m × P ) and its tensor-product representation, L 2 (J) ⊗ L 2 (Ω, S, P ), we get
n−1 t=0 f (t i )∆X ti (ω) = by (5.8) n−1 i=0 ∞ k=1 f (t i ) λ k (ϕ k (t i+1 ) − ϕ k (t i ))Z k (ω) = ∞ k=1 ( n−1 i=0 f (t i )(∆ϕ k ) ti ) λ k Z k (ω) 7 and therefore (5.15) E(|S π (f, X)| 2 ) = by (5.7) ∞ k=1 λ k n−1 i=0 f (t i )(∆ϕ k ) ti 2 .
Since f is assumed contions, and each ϕ k is of bounded variation, the following convergence holds:
(5.16) lim |π|→0 n−1 i=0 f (t i )(∆ϕ k ) ti = t t0 f dϕ k .
Now if the function f is satisfying f ∈ L 2 (J) and f ′ ∈ L 2 (J), then we get the following estimate, relying on the boundary representation (5.12) and Parseval, see also (5.5): For the RHS in (5.15) we have; after passing to the limit: which is the desired conclustion in part (c) of the theorem.
Proof. Of Corollary 3.2: The essential point is formular (5.8). However, in substitution of the expression on the RHS in (5.8) we make use of double-orthogonality, i.e., (5.5) and (5.7). Specifically, we have the tensor product representation L 2 (J × Ω, m × P ) = L 2 (J) ⊗ L 2 (Ω), and so X = ∞ k=1 √ λ k ϕ k ⊗ Z k in (5.8) refers to the tensor representation.
Entropy: Optimal Bases
In this section we compare the choice of ONB from section 3 with alternative choices of ONBs. The application of Karhunen-Loève dictates a particular choice of ONB.
Historically, the Karhunen-Loève arose as a tool from the interface of probability theory and information theory; see details with references inside the paper. It has served as a powerful tool in a variety of applications; starting with the problem of separating variables in stochastic processes, say X t ; processes that arise from statistical noise, for example from fractional Brownian motion. Since the initial inception in mathematical statistics, the operator algebraic contents of the arguments have crystallized as follows: starting from the process X t , for simplicity assume zero mean, i.e., E(X t ) = 0; create a correlation matrix T J (s, t) = E(X s X t ). (Strictly speaking, it is not a matrix, but rather an integral kernel. Nonetheless, the matrix terminology has stuck.) The next key analytic step in the Karhunen-Loève method is to then apply the Spectral Theorem from operator theory to a corresponding selfadjoint operator, or to some operator naturally associated with the integral kernel: Hence the name, the Karhunen-Loève Decomposition (KLC). In favorable cases (discrete spectrum), an orthogonal family of functions (ϕ n (t)) in the time variable arise, and a corresponding family of eigenvalues. We take them to be normalized 8 in a suitably chosen square-norm. By integrating the basis functions ϕ n (t) against X t , we get a sequence of random variables Z n . It was the insight of Karhunen-Loève [Loe52] to give general conditions for when this sequene of random variables is independent, and to show that if the initial random process X t is Gaussian, then so are the random variables Z n .
Below, we take advantage of the fact that Hilbert space and operator theory form the common language of both quantum mechanics and of signal/image processing. Recall first that in quantum mechanics, (pure) states as mathematical entities "are" one-dimensional subspaces in complex Hilbert space H, so we may represent them by vectors of norm one. Observables "are" selfadjoint operators in H, and the measurement problem entails von Neumann's spectral theorem applied to the operators.
In signal processing, time-series, or matrices of pixel numbers may similarly be realized by vectors in Hilbert space H. The probability distribution of quantum mechanical observables (state space H) may be represented by choices of orthonormal bases (ONBs) in H in the usual way (see e.g., [Jor06]).In the 1940s, Kari Karhunen ([Kar46], [Kar52]) pioneered the use of spectral theoretic methods in the analysis of time series, and more generally in stochastic processes. It was followed up by papers and books by Michel Loève in the 1950s [Loe52], and in 1965 by R.B. Ash [Ash90]. (Note that this theory precedes the surge in the interest in wavelet bases!)
Parallel problems in quantum mechanics and in signal processing entail the choice of "good" orthonormal bases (ONBs). One particular such ONB goes under the name "the Karhunen-Loève basis." We will show that it is optimal.
Definition 6.1. Let H be a Hilbert space. Let (ψ i ) and (ϕ i ) be orthonormal bases (ONB). If (ψ i ) i∈I is an ONB, we set Q n := the orthogonal projection onto span{ψ 1 , ..., ψ n }.
We now introduce a few facts about operators which will be needed in the paper. In particular we recall Dirac's terminology [Dir47] for rank-one operators in Hilbert space. While there are alternative notation available, Dirac's bra-ket terminology is especially efficient for our present considerations. Dirac's bra-ket and ket-bra notation is is popular in physics, and it is especially convenient in working with rank-one operators and inner products. For example, in the middle term in eq (6.3), the vector u is multiplied by a scalar, the inner product; and the inner product comes about by just merging the two vectors. If (ψ i ) i∈N is an ONB then the projection Q n := proj span{ψ 1 , ..., ψ n } is given by
(6.5) Q n = n i=1 |ψ i ψ i |;
and for each i, |ψ i ψ i | is the projection onto the one-dimensional subspace Cψ i ⊂ H.
Definition 6.4. Suppose X t is a stochastic process indexed by t in a finite interval J, and taking values in L 2 (Ω, P ) for some probability space (Ω, P ). Assume the normalization E(X t ) = 0. Suppose the integral kernel E(X t X s ) can be diagonalized, i.e., suppose that
J E(X t X s )ϕ k (s)ds = λ k ϕ k (t)
with an ONB (ϕ k ) in L 2 (J). If E(X t ) = 0 then
X t (ω) = k λ k φ k (t)Z k (ω), ω ∈ Ω
where E(Z j Z k ) = δ j,k , and E(Z k ) = 0. The ONB (ϕ k ) is called the KL-basis with respect to the stochastic processes {X t : t ∈ J}.
Theorem 6.5. (See [JS07]) The Karhunen-Loève ONB gives the smallest error terms in the approximation to a frame operator.
Proof. Given the operator T J which is trace class and positive semidefinite, we may apply the spectral theorem to it. What results is a discrete spectrum, with the natural order λ 1 ≥ λ 2 ≥ ... and a corresponding ONB (ϕ k ) consisting of eigenvectors, i.e., (6.6) T J ϕ k = λ k ϕ k , k ∈ N called the Karhunen-Loève data. The spectral data may be constructed recursively starting with (6.7)
λ 1 = sup ϕ∈H, ϕ =1 ϕ|T J ϕ = ϕ 1 |T J ϕ 1 and (6.8) λ k+1 = sup ϕ∈H, ϕ =1 ϕ⊥ϕ 1 ,ϕ 2 ,...,ϕ k ϕ|T J ϕ = ϕ k+1 |T J ϕ k+1
Now an application of [ArKa06]; Theorem 4.1 yields
(6.9) n k=1 λ k ≥ tr(Q ψ n T J ) = n k=1 ψ k |T J ψ k for all n,
where Q ψ n is the sequence of projections from (6.5), deriving from some ONB (ψ i ) and arranged such that (6.10) ψ 1 |T J ψ 1 ≥ ψ 2 |T J ψ 2 ≥ ... .
Hence we are comparing ordered sequences of eigenvalues with sequences of diagonal matrix entries.
Finally, we have
tr (T J ) = ∞ k=1 λ k = ∞ k=1 ψ k |T J ψ k < ∞.
The assertion in Theorem 6.5 is the validity of (6.11) E ϕ n ≤ E ψ n for all (ψ i ) ∈ ON B(H), and all n = 1, 2, ...; and moreover, that the infimum on the RHS in (6.11) is attained for the KL-ONB (ϕ k ). But we see that (6.11) is equivalent to the system (6.9) in the Arveson-Kadison theorem.
The Arveson-Kadison theorem is the assertion (6.9) for trace class operators, see e.g., refs [Arv06] and [ArKa06]. That (6.11) is equivalent to (6.9) follows from the definitions.
Our next theorem gives Karhunen-Loève optimality for sequences of entropy numbers. The formula will be used on cut-down versions of an initial operator T J . In some cases only the cut-down might be trace-class. Since the Spectral Theorem applies to T J , the RHS in (6.12) is also (6.13) S(T J ) = − ∞ k=1 λ k log λ k .
For simplicity we normalize such that 1 = trT J = ∞ k=1 λ k , and we introduce the partial sums ψ k |T J ψ k log ψ k |T J ψ k .
Let (ψ i ) ∈ ON B(H), and set d ψ k := ψ k |T J ψ k ; then the inequalities (6.9) take the form: S KL n (T J ) ≤ S ψ n (T J ), n = 1, 2, ... follow. i.e., the KL-data minimizes the sequence of entropy numbers.
k ϕ k (·)Z k (·) L 2 (m×P ) t X s )ϕ k (s)ds = λ k ϕ k (t);
Definition 6 . 2 .
62Let vectors u, v ∈ H. Then (6.1) u|v = inner product ∈ C, (6.2) |u v| = rank-one operator, H → H, where the operator |u v| acts as follows (6.3) |u v|w = |u v|w = v|w u, for all w ∈ H.
Definition 6 . 3 .
63If S and T are bounded operators in H, in B(H), then (6.4) S|u v|T = |Su T * v|
Theorem 6.6. (See [JS07]) The Karhunen-Loève ONB gives the smallest sequence of entropy numbers in the approximation. Proof. We begin by a few facts about entropy of trace-class operators T J . The entropy is defined as (6.12) S(T J ) := −tr(T J log T J ).
1 ≥ d ψ 2 ≥ ...has been chosen.
λ i ), n = 1, 2, ... . Since the RHS in (6.18) is −tr(T J log T J ) = −S KL n (T J ), the desired inequalities (6.19)
Acknowledgment. The present work was motivated by details form a graduate course taught by the first named author on stochastic integration and its applications. We are grateful to the students in the course, especially Mr Yu Xu, for their comments and their inspiration.
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| [] |
[
"Kinetic theory of pattern formation in mixtures of microtubules and molecular motors",
"Kinetic theory of pattern formation in mixtures of microtubules and molecular motors"
] | [
"Ivan Maryshev \nSchool of Biological Sciences\nInstitute of Cell Biology\nCentre for Synthetic and Systems Biology\nMax Born Crescent\nThe University of Edinburgh\nEH9 3BFEdinburghUnited Kingdom\n",
"Davide Marenduzzo \nSchool of Physics and Astronomy\nSUPA\nThe University of Edinburgh\nJames Clerk Maxwell Building, Peter Guthrie Tait RoadEH9 3FDEdinburghUnited Kingdom\n",
"Andrew B Goryachev \nSchool of Biological Sciences\nInstitute of Cell Biology\nCentre for Synthetic and Systems Biology\nMax Born Crescent\nThe University of Edinburgh\nEH9 3BFEdinburghUnited Kingdom\n",
"Alexander Morozov \nSchool of Physics and Astronomy\nSUPA\nThe University of Edinburgh\nJames Clerk Maxwell Building, Peter Guthrie Tait RoadEH9 3FDEdinburghUnited Kingdom\n"
] | [
"School of Biological Sciences\nInstitute of Cell Biology\nCentre for Synthetic and Systems Biology\nMax Born Crescent\nThe University of Edinburgh\nEH9 3BFEdinburghUnited Kingdom",
"School of Physics and Astronomy\nSUPA\nThe University of Edinburgh\nJames Clerk Maxwell Building, Peter Guthrie Tait RoadEH9 3FDEdinburghUnited Kingdom",
"School of Biological Sciences\nInstitute of Cell Biology\nCentre for Synthetic and Systems Biology\nMax Born Crescent\nThe University of Edinburgh\nEH9 3BFEdinburghUnited Kingdom",
"School of Physics and Astronomy\nSUPA\nThe University of Edinburgh\nJames Clerk Maxwell Building, Peter Guthrie Tait RoadEH9 3FDEdinburghUnited Kingdom"
] | [] | In this study we formulate a theoretical approach, based on a Boltzmann-like kinetic equation, to describe pattern formation in two-dimensional mixtures of microtubular filaments and molecular motors. Following the previous work by Aranson and Tsimring [Phys. Rev. E 74, 031915 (2006)] we model the motor-induced reorientation of microtubules as collision rules, and devise a semianalytical method to calculate the corresponding interaction integrals. This procedure yields an infinite hierarchy of kinetic equations that we terminate by employing a well-established closure strategy, developed in the pattern-formation community and based on a power-counting argument. We thus arrive at a closed set of coupled equations for slowly varying local density and orientation of the microtubules, and study its behaviour by performing a linear stability analysis and direct numerical simulations. By comparing our method with the work of Aranson and Tsimring, we assess the validity of the assumptions required to derive their and our theories. We demonstrate that our approximation-free evaluation of the interaction integrals and our choice of a systematic closure strategy result in a rather different dynamical behaviour than was previously reported. Based on our theory, we discuss the ensuing phase diagram and the patterns observed. | 10.1103/physreve.97.022412 | [
"https://arxiv.org/pdf/1711.06240v1.pdf"
] | 3,898,228 | 1711.06240 | eb70b94f7b11431529c61e47e9f2a7c9cd21d615 |
Kinetic theory of pattern formation in mixtures of microtubules and molecular motors
(Dated: November 17, 2017)
Ivan Maryshev
School of Biological Sciences
Institute of Cell Biology
Centre for Synthetic and Systems Biology
Max Born Crescent
The University of Edinburgh
EH9 3BFEdinburghUnited Kingdom
Davide Marenduzzo
School of Physics and Astronomy
SUPA
The University of Edinburgh
James Clerk Maxwell Building, Peter Guthrie Tait RoadEH9 3FDEdinburghUnited Kingdom
Andrew B Goryachev
School of Biological Sciences
Institute of Cell Biology
Centre for Synthetic and Systems Biology
Max Born Crescent
The University of Edinburgh
EH9 3BFEdinburghUnited Kingdom
Alexander Morozov
School of Physics and Astronomy
SUPA
The University of Edinburgh
James Clerk Maxwell Building, Peter Guthrie Tait RoadEH9 3FDEdinburghUnited Kingdom
Kinetic theory of pattern formation in mixtures of microtubules and molecular motors
(Dated: November 17, 2017)
In this study we formulate a theoretical approach, based on a Boltzmann-like kinetic equation, to describe pattern formation in two-dimensional mixtures of microtubular filaments and molecular motors. Following the previous work by Aranson and Tsimring [Phys. Rev. E 74, 031915 (2006)] we model the motor-induced reorientation of microtubules as collision rules, and devise a semianalytical method to calculate the corresponding interaction integrals. This procedure yields an infinite hierarchy of kinetic equations that we terminate by employing a well-established closure strategy, developed in the pattern-formation community and based on a power-counting argument. We thus arrive at a closed set of coupled equations for slowly varying local density and orientation of the microtubules, and study its behaviour by performing a linear stability analysis and direct numerical simulations. By comparing our method with the work of Aranson and Tsimring, we assess the validity of the assumptions required to derive their and our theories. We demonstrate that our approximation-free evaluation of the interaction integrals and our choice of a systematic closure strategy result in a rather different dynamical behaviour than was previously reported. Based on our theory, we discuss the ensuing phase diagram and the patterns observed.
In this study we formulate a theoretical approach, based on a Boltzmann-like kinetic equation, to describe pattern formation in two-dimensional mixtures of microtubular filaments and molecular motors. Following the previous work by Aranson and Tsimring [Phys. Rev. E 74, 031915 (2006)] we model the motor-induced reorientation of microtubules as collision rules, and devise a semianalytical method to calculate the corresponding interaction integrals. This procedure yields an infinite hierarchy of kinetic equations that we terminate by employing a well-established closure strategy, developed in the pattern-formation community and based on a power-counting argument. We thus arrive at a closed set of coupled equations for slowly varying local density and orientation of the microtubules, and study its behaviour by performing a linear stability analysis and direct numerical simulations. By comparing our method with the work of Aranson and Tsimring, we assess the validity of the assumptions required to derive their and our theories. We demonstrate that our approximation-free evaluation of the interaction integrals and our choice of a systematic closure strategy result in a rather different dynamical behaviour than was previously reported. Based on our theory, we discuss the ensuing phase diagram and the patterns observed.
I. INTRODUCTION
Self-organisation of mixtures of biological polymers and molecular motors provides a fascinating manifestation of active matter [1,2]. Microtubules are actively re-oriented by the molecular motors, and can form farfrom-equilibrium global, cell-scale structures, such as the mitotic spindle apparatus [3]. It is believed that different motor types favour formation of distinct patterns: microtubule-sliding motors organise antiparallel bundles, while clustering motors control the formation of spindle poles and asters [3,4].
Despite steady advance in the experimental analysis of such systems [5][6][7][8][9][10][11][12], their theoretical description is stymied by the paucity of approaches able to connect individual microscopic motor-induced interactions of filaments to the macroscopic dynamics at lengthscales relevant to the whole cytoskeleton. Here we build on a kinetic method developed earlier in [13,14] to provide a revised version of the hydrodynamic equations that govern collective behaviour of microtubules in the presence of clustering motors.
Microtubules are long and stiff rod-like biopolymers [15]. Because of the asymmetry of the constituting tubulin subunits, the microtubule filament has intrinsic orientation and distinct ends denoted as '-' and '+'. Molecular motors use chemical energy stored as ATP to move processively along microtubule filaments in one preferred direction. Some motors can bind two filaments simultaneously and, therefore, reorient and translocate them with respect to each other [15,16]. The activity of such * [email protected] bivalent motors generates global order on a scale which is much larger than the length of a single filament.
Spontaneous transitions to various ordered states and patterns has been extensively studied in several in vitro experiments with cell extracts [11,17] and in the reconstituted systems containing mixtures of stabilised microtubules and purified motors. The latter systems recapitulate formation of structures with nematic [8,9] or polar order, e.g., asters and vortices [5][6][7]. This type of systems is considered further in the current contribution.
Multiple approaches had been developed to advance our understanding of the dynamics typical of microtubule-motor mixtures. Besides direct agent-based simulations [6,18], mean field equations have been derived first on the basis of symmetry considerations [19], and then from the detailed microscopic rules of interaction [14,[20][21][22].
In the kinetic approach employed by Aranson and Tsimring [13,14,23], pairwise motor-mediated interactions of microtubules were treated as instantaneous collisions. These authors considered plus-directed clustering motors, which can align and bundle microtubules. Hydrodynamic equations for the two field variables, filament concentration and orientation, were derived by coarsegraining of the corresponding Boltzmann-type equation for the probability distribution function (PDF). Their model successfully recapitulated such phenomena as spontaneous ordering, bundling and formation of asters and vortices.
In this paper we revisit this technique. We demonstrate that using the exact form of the collision rate function, instead of the phenomenological expression suggested in [14], yields the system of equations with significantly distinct "phase diagram". Specifically, we find that in our model instabilities occur in a different order. We also argue that the previously neglected excluded volume effect needs to be considered to prevent density blow-up in the bundling regime. Additionally, we compare the closure of the equation expansion considered in [14] with the more conventional method, which is historically used in Landau-Ginzburg-like equations [24]. The latter approach necessitates introduction of an additional variable, the nematic order parameter or Q-tensor. The introduction of this additional field variable results in a novel instability, not observed in the previous work.
The paper is organised as follows. Our kinetic model is introduced in Section II. In Section III we rederive the hydrodynamic equations obtained in [14] and critically discuss the approximations used in this derivation. We demonstrate that the model exhibits bundling instability and argue for the need to include excluded volume effects. Section IV presents the derivation of such excluded volume terms. In Section V we discuss evaluation of the interaction integrals. Our equations of motion are presented in Section VI: these are derived according to two different types of closure. We perform stability analysis of the equations of motion corresponding to both these closure schemes, present the corresponding phase diagrams, and provide the results of numerical simulations in Section VII. Finally, Section VIII contains a discussion of our results.
II. KINETIC THEORY
The Boltzmann-like kinetic equation
The setup of our problem follows that of [14]. We consider a two-dimensional collection of microtubules, which we treat as slender rigid rods of length L. Since microtubules are polar objects, we describe their local orientation by a vector n that points from the 'minus'-to the 'plus'-end of a microtubule. We introduce a Cartesian coordinate system (x, y), and parametrise the orientation vector by a single angle -i.e., n = (cos φ, sin φ). To describe spatial and orientational inhomogeneities in the system, we introduce the probability distribution function P (r, φ, t), defined in the usual way: P (r, φ, t)drdφ gives the number of microtubules in a small volume of the phase space drdφ which are at position r and possess an orientation given by φ at time t. (1). Two colliding microbutular bundles are re-oriented by the action of the molecular motors to assume a common orientation along the bisector of the original angle between them. Their centre of mass does not move in the process.
Following [14], the time-evolution of the probability distribution function is assumed to be governed by a Boltzmann-like kinetic equation
∂ t P (r, φ) = D r ∂ 2 φ P (r, φ) + ∇ i D ij ∇ j P (r, φ) + dξ π −π dω W r − ξ 2 , φ − ω 2 ; r + ξ 2 , φ + ω 2 × P r − ξ 2 , φ − ω 2 P r + ξ 2 , φ + ω 2 − W (r, φ; r − ξ, φ − ω)P (r, φ) P (r − ξ, φ − ω) ,(1)
where ∇ i = ∂/∂x i , x i are the Cartesian components of r, and we use the Einstein summation convention; from now on we suppress the explicit time-dependence of P for brevity. The first two terms in Eq.(1) describe thermal rotational and translational diffusion of individual microtubule bundles, while the last two terms represent motormediated interactions between microtubule bundles. The first integral in Eq.(1) is a gain term, which accounts for events where two microtubule bundles with different positions and orientations are re-oriented by the motors to assume position and orientation (r, φ). The specific form of this term encodes our assumptions about how motors and microtubules interact; the details of such interactions are summarised in Fig. 1. Again following [14], we assume that after a re-orientation event both bundles align along the bisector of the original angle between them, while their centre of mass does not move in the process.
To motivate the latter choice we note that a motor simultaneously attached to both bundles applies a pair of equal and opposite forces to the system -i.e., it behaves as a force-dipole. Since the total force applied to the centre of mass is zero, its position is conserved. This assumption is in contrast with the work reported in [1,20,21,25] where it was instead argued that directed motion of molecular motors along the microtubules can create a flow in the surrounding fluid that would result in microtubule self-propulsion, and, hence, the position of the centre of mass of two bundles can change during an interaction event. Such effects are rather difficult to quantify in dense suspensions of microtubules that are confined close to a boundary, as is typically the case in experiments, hence we neglect them here. The second integral in Eq.(1) is a loss term, describing the process by which a bundle with the position and orientation (r, φ) leaves that configuration due to an interaction event with another bundle. The rate of both motor-induced processes is given by the function W discussed below. It is finally important to underscore that the pair-wise nature of the interaction terms in Eq.(1) is not related to a dilute-limit assumption, as is often the case in Boltzmann-like kinetic theories, but rather stems from the fact that a molecular motor can only simultaneously attach to two microtubules [15,26].
Long-wavelength expansion
To proceed, we observe that without loss of generality the probability distribution function can be expanded in Fourier harmonics
P (r, φ) = ∞ n=−∞ P n (r)e inφ ,(2)
where P * −n (r) = P n (r), since P is real, and ' * ' denotes complex conjugation. Next, we note that motormediated interactions between microtubules are shortranged, and the integrand in Eq.(1) is non-zero only when |ξ| L, independent of the particular form of the interaction strength W . Since we are interested in patterns that evolve slowly on scales comparable to L, we perform a gradient expansion of P and keep terms up to fourth order. Projecting the resulting equation on the s-th Fourier harmonic yields the following equation,
∂ t P s (r) = −s 2 D r P s (r) + ∇ i D ij ∇ j P (r, φ) s + ∞ n,m=−∞ I (0) nm s P n P m + 1 2 I (1) i,nm s A i,nm + 1 8 I (2) ij,nm s A ij,nm + 1 48 I (3) ijk,nm s A ijk,nm + 1 384 I (4) ijkl,nm s A ijkl,nm + · · · (3) − ∞ n,m=−∞ P n J (0) nm s P m − J (1) i,nm s ∇ i P m + 1 2 J (2) ij,nm s ∇ i ∇ j P m − 1 6 J (3) ijk,nm s ∇ i ∇ j ∇ k P m + 1 24 J (4) ijkl,nm s ∇ i ∇ j ∇ k ∇ l P m + · · · , where (. . . ) s = 1 2π 2π 0 e −isφ (. . . ) ,(4)
and
A i,nm = P n ∇ i P m − P m ∇ i P n , A ij,nm = P n ∇ i ∇ j P m − 2 (∇ i P n ) (∇ j P m ) + P m ∇ i ∇ j P n , A ijk,nm = P n ∇ i ∇ j ∇ k P m − 3 (∇ i P n ) (∇ j ∇ k P m ) + 3 (∇ i ∇ j P n ) (∇ k P m ) − P m ∇ i ∇ j ∇ k P n , A ijkl,nm = P n ∇ i ∇ j ∇ k ∇ l P m − 4 (∇ i P n ) (∇ j ∇ k ∇ l P m ) + 6 (∇ i ∇ j P n ) (∇ k ∇ l P m ) − 4 (∇ i ∇ j ∇ k P n ) (∇ l P m ) + P m ∇ i ∇ j ∇ k ∇ l P n .
In Eq.(3) all P n 's and P m 's are functions of r and t. The interaction integrals are given by
I (0) nm = e i(n+m)φ dξ π −π dω W 1 e i(m−n) ω 2 , I (1) i,nm = e i(n+m)φ dξ π −π dω W 1 e i(m−n) ω 2 ξ i , I (2) ij,nm = e i(n+m)φ dξ π −π dω W 1 e i(m−n) ω 2 ξ i ξ j ,(5)I (3) ijk,nm = e i(n+m)φ dξ π −π dω W 1 e i(m−n) ω 2 ξ i ξ j ξ k , I(4)
ijkl,nm = e i(n+m)φ dξ π −π dω W 1 e i(m−n) ω 2 ξ i ξ j ξ k ξ l , and
J (0) nm = e i(n+m)φ dξ π −π dω W 2 e −imω , J (1) i,nm = e i(n+m)φ dξ π −π dω W 2 e −imω ξ i , J (2) ij,nm = e i(n+m)φ dξ π −π dω W 2 e −imω ξ i ξ j ,(6)J (3) ijk,nm = e i(n+m)φ dξ π −π dω W 2 e −imω ξ i ξ j ξ k , J (4) ijkl,nm = e i(n+m)φ dξ π −π dω W 2 e −imω ξ i ξ j ξ k ξ l ,
where ξ i are the Cartesian components of ξ, and we introduced
W 1 ≡ W r − ξ 2 , φ − ω 2 ; r + ξ 2 , φ + ω 2 , W 2 ≡ W (r, φ; r − ξ, φ − ω).
Eq.(3) comprises an infinite hierarchy of equations for the Fourier harmonics P n (r, t). Its practical application relies on a strategy to reduce the number of relevant fields to just a few harmonics, and on the ability to calculate the interaction integrals for a particular function W . Our approach to both these issues is discussed below in Sections VI A and VI B, and in Section II 4), respectively.
Diffusion terms
The diffusion coefficients in Eq.(1) are approximated by their values for a single rod of length L and diameter d moving in an infinite, three-dimensional fluid with viscosity η [27]
D r = 12 k B T πηL 3 ln(L/d),(7)
and
D ij = D n i (φ)n j (φ) + D ⊥ [δ ij − n i (φ)n j (φ)] ,(8)
where
D ⊥ = k B T 4πηL ln(L/d), D = 2D ⊥ .
Here, T is the temperature of the solution, and k B is the Boltzmann constant. Note that D r is four times larger than the value given by Doi and Edwards [27] due to the difference in our choice of the angular variable (i.e. φ rather than n). Using Eq.(8) in Eq.(3), and projecting onto the s-th Fourier harmonics, leads to the following contributions to the equations of motion
∂ t P s (r) = −s 2 D r P s (r) + D + D ⊥ 2 ∇ 2 P s (r) + D − D ⊥ 4 ∇ 2 x − 2i∇ x ∇ y − ∇ 2 y P s−2 (r) + D − D ⊥ 4 ∇ 2 x + 2i∇ x ∇ y − ∇ 2 y P s+2 (r) + · · · ,(9)
where '· · · ' denotes contributions from the interaction integrals discussed below. Formally, Eqs.(7) and (8) limit the scope of Eq.(3) to rather dilute suspensions far away from liquid-solid or liquid-liquid boundaries, while both assumptions are routinely violated in experiments [2,9,28,29]. As we will see below, the kinetic theory equations that we are going to derive will only depend on the ratios D /D r and D ⊥ /D r that are less sensitive to the local density of other microtubules and proximity of a boundary. We note, however, that a proper study of this effect is outside the scope of this work.
Interaction kernel W
Within our kinetic theory, molecular details of motormicrotubules interactions are encoded at a coarsegrained level in the interaction function W . Its physical interpretation is given by Eq.(1), which identifies W (r 1 , φ 1 ; r 2 , φ 2 ) as a rate at which two microtubular bundles at (r 1 , φ 1 ) and (r 2 , φ 2 ) are displaced and reoriented by molecular motors. These changes in the bundles positions and orientations occur when a molecular motor is attached to both bundles and moves along them. Therefore, a motor-induced re-orientation event can only take place when the shortest distance between the bundles is not larger than the size of the motors. Since the latter is significantly smaller than the length of individual microtubules, or the typical size of the patterns formed by the suspension, see e.g. [9,15], we consider motors to be point-like. Under this assumption, W is non-zero only when the bundles intersect in their original configuration. In turn, this implies that in real systems one of the bundles leaves the xy-plane of the suspension and deviates slightly into the third dimension. Such deviations are small compared either to L or the typical pattern size, hence we will treat such bundles as intersecting in 2D. The intersection condition can be written as plus-end of the microtubule. By taking the cross product of Eq.(10) with either n 1 or n 2 , the contour-length parameters can be found to be
r 1 + n 1 L 2 τ 1 = r 2 + n 2 L 2 τ 2 ,(10)τ 1 = 2 L (r 2 − r 1 ) × n 2 · e z (n 1 × n 2 ) · e z ,(11)τ 2 = 2 L (r 2 − r 1 ) × n 1 · e z (n 1 × n 2 ) · e z ,(12)
where e z is a unit vector perpendicular to the xyplane. Since |τ 1,2 | should be smaller than unity, the intersection condition can equivalently be written as
Θ(1 − |τ 1 |) Θ(1 − |τ 2 |) = 0, where Θ is the Heaviside step function.
Having established the condition for bundle intersection, we turn to modelling their re-orientation rate. Following Aranson and Tsimring [14], we take this rate to be proportional to the local motor density at the intersection point. In the following we assume that the motors are abundant in the solution, and their dynamics of association/dissociation with the microtubules are much faster than the typical pattern-formation time. This was the case in several in-vitro experiments (see [6], for example).
With these assumptions, the motor distribution along individual microtubules instantaneously reaches its equilibrium profile. For plus-directed motors and under similar conditions, the equilibrium motor distribution was measured experimentally [30,31], and it was shown that the motor density stays low and approximately constant in the vicinity of the minus-end of the microtubules, until it rises sharply and saturates at another constant value close to the plus-end. This behaviour is corroborated by 1D non-equilibrium models [14,[30][31][32] that relate this distribution to the formation of traffic jams at the plusend. The equilibrium motor distribution m(τ ), which gives the motor density at the contour length position τ , can therefore be approximated by
m(τ ) = m − + (m + − m − )Θ(τ − τ 0 ),(13)
where m − and m + are the motor densities at the minusand plus-ends, correspondingly, and τ 0 sets the position of the transition between those values; see Fig.2 for details.
The re-orientation rate can finally be written as
W (r 1 , φ 1 ; r 2 , φ 2 ) = G Θ(1 − |τ 1 |) Θ(1 − |τ 2 |) × 1 + Ξ Θ(τ 1 − τ 0 ) + Θ(τ 2 − τ 0 ) ,(14)
where Ξ = (m + − m − )/(2m − ), and τ 1,2 are given by Eqs. (11) and (12). The constant G is proportional to the motor properties, such as its processivity along the microtubules [15,16], and varies with the motor type. However, as we will demonstrate below, G can be removed from the model by a rescaling of the dynamical fields. While its value would be important to map the parameter values used in the equations of motion back to dimensional units, it plays no role in determining the phase diagram of our model. Indeed, the interaction function W depends on two dimensionless parameters, τ 0 and Ξ, where the latter quantifies the mismatch between the motor densities at the two ends of a microtubule. While it would be tempting to ignore this complexity and set Ξ = 0 for simplicity, previous work suggests this to be a crucial ingredient of the theory. As was shown by Aranson and Tsimring [14] for their model, there is no interesting pattern formation taking place in the absence of the motor density mismatch, and only a trivial instability is present in that case (see below). A similar conclusion was reached by Marchetti, Liverpool and co-workers [1,20,21,25], where the analogous parameter was the motor speed anisotropy along a microtubule. We, therefore, consider Ξ = 0 below.
III. APPROXIMATIONS IN THE ARANSON-TSIMRING THEORY
In this section we review the approximations used by Aranson and Tsimring in [14] to evaluate the integrals in Eqs. (5) and (6), and to terminate the infinite hierarchy of coupled equations in Eq.(3). Here, we sketch their argument in some detail as it will be important in the further discussion.
The first step involves replacing the exact interaction kernel W in Eq. (14) with an effective simplified kernel, which is given by
W AT (r 1 , φ 1 ; r 2 , φ 2 ) =G b 2 π exp − (r 1 − r 2 ) 2 b 2 × 1 − β L (r 1 − r 2 ) · (n 1 − n 2 ) ,(15)
where b ∼ L is a lengthscale, andG is a motor-related constant, similar to G in Eq. (14). This expression replaces the complicated spatial and angular dependence of Eq. (14) with a Gaussian cut-off that, essentially, allows any interactions as long as the bundle centres of mass are separated by a typical distance set by b; the term in the brackets can be seen as the first terms of the Fourier expansion of the true angular dependence in Eq. (14). The parameter β is a measure of how anisotropic the motor distribution is along individual microtubules, and is analogous to Ξ in Eq. (14). The obvious benefit of this approximation is that the integrals in Eqs. (5) and (6) can now be evaluated analytically. In [14] it is claimed that while not exact, Eq.(15) retains the main features of Eq. (14). We demonstrate in the next Sections that together with the choice of the parameter b made in [14], the approximation in Eq.(15) leads to a different phase diagram with respect to that obtained when the original kernel Eq. (14) is retained.
The second approximation developed in [14] concerns the way to terminate the infinite hierarchy in Eq.(3). To illustrate this strategy, we neglect spatial variations of the probability distribution and keep only its angular dependence. This approximation implies that the dominant mechanism of the instability in this system should be the appearance of orientational order, while the density fluctuations are assumed to be subdominant. The validity of this approximation will be re-assessed after the same methodology is applied to the full system of equations with both the spatial and angular dependencies included. Using Eq.(15) in Eq. (3), and setting β and the spatial gradients to zero, we obtain
∂ t P s = −s 2 D r P s − 2πGP 0 P s +G ∞ m=−∞ 4 sin π 2 (2m − s) 2m − s P s−m P m . (16)
Keeping only the first three Fourier harmonics in the expansion, this system of equations reads
∂ t P 0 = 0,(17)∂ t P 1 = −D r P 1 +G (8 − 2π) P 0 P 1 − 8 3G P * 1 P 2 , (18) ∂ t P 2 = −4D r P 2 + 2πG P 2 1 − P 0 P 2 ,(19)
where the first equation is the direct consequence of the total probability conservation. The isotropic solution of these equations is given by P 1 = P 2 = 0, while the evolution of small perturbations around this state, δp 1 and δp 2 , is governed by the following equations,
∂ t δp 1 = λ 1 δp 1 , ∂ t δp 2 = λ 2 δp 2 ,(20)
where
λ 1 = −D r +G (8 − 2π) P 0 and λ 2 = −4D r − 2πGP 0 ; here, P 0 is a constant.
The isotropic solution becomes unstable with respect to perturbations δp 1 when λ 1 becomes positive, while perturbations in the second mode, δp 2 , are decaying since λ 2 is always negative. Therefore, close to the instability threshold the dynamics of the second mode P 2 is enslaved to the dynamics of the linearly unstable field P 1 [24], and P 2 can only acquire a non-zero value due to the non-linear forcing by the P 2 1 term in Eq. (19). Thus, P 2 quickly relaxes to the value set by the r.h.s. of Eq.(19)
P 2 = 2πG 4D r + 2πGP 0 P 2 1 .(21)
As can be shown from Eq.(16), the same holds for all higher modes P m with m > 1, where P m ∼ O (P m 1 ). Since close to the instability threshold the saturated value of P 1 is small, all higher harmonics are significantly smaller, and can be neglected. Therefore, the authors of Ref. [14] restricted the infinite hierarchy Eq.(3) to contain only the first three modes, P 0 , P 1 and P 2 , where the latter does not possess its own dynamics but is assumed to be well-approximated by the adiabatically-adjusted value given in Eq. (21), even in the presence of spatial variations and non-zero β.
These approximations allow for Eq.(3) to be converted into a system of partial differential equations for the hydrodynamic (i.e., slowly varying) fields ρ(r) and p(r) defined by the moments of P (r, φ) as follows,
ρ(r) = 2π 0 dφP (r, φ) = 2πP 0 (r),(22)p(r) = 1 2π 2π 0 dφ n(φ)P (r, φ) = P −1 (r) + P 1 (r) 2 , P −1 (r) − P 1 (r) 2i .(23)
Here, ρ(r) is the local density of microtubular bundles, and p(r) is proportional to their local orientation; note that p is not a unit vector.
To render equations dimensionless, time, space and the slow fields ρ and p are scaled by D −1 r , L andGL 2 /D r , respectively. The final dimensionless equations used in [14] read
∂ t ρ = ∇ 2 ρ 32 − B 2 ρ 2 16 − πB 2 H 16 3∇ · p∇ 2 ρ − ρ∇ 2 p + 2∂ i (∂ j ρ∂ j p i − ∂ i ρ∂ j p j ) − 7ρ 0 B 4 256 ∇ 4 ρ,(24)∂ t p = 5 192 ∇ 2 p + 1 96 ∇(∇ · p) + (ρ/ρ cr − 1) p −Ã 0 |p| 2 p − H ∇ρ 2 16π − π − 8 3 p(∇·p) − 8 3 (p · ∇)p + B 2 ρ 0 4π ∇ 2 p,(25)
where ρ 0 is the conserved average density, B = b/L, H = βB 2 , andρ cr = π/(4−π); the constantà 0 = 16π/(3(ρ 0 + 4)) and the corresponding term in Eq.(25) arise from the dimensionless version of Eq. (21). In addition to the approximations developed above, Eqs. (24) and (25) entail some additional assumptions. First, the only non-linear terms (i.e. terms proportional to H) kept in these equations correspond to the lowest order non-zero terms in the gradient expansion (cubic and linear in gradients in the equations for ρ and p, respectively). This is done to ensure that both equations are coupled to each other. Additionally, Eq.(25) contains a series of terms quadratic in the gradient that are linearised around ρ 0 , giving rise to the last term in that equation. This linearisation is justified if there are only small density variations close to the instability threshold. Finally, to ensure the absence of short-wavelengthinstability, the fourth-order terms in the gradient expansion are again linearised around ρ 0 to yield the biharmonic term in Eq. (24).
The analysis presented in the Aranson-Tsimring theory suggests that Eqs. (24) and (25) exhibit two linear instabilities: an isotropic-polar transition at ρ 0 =ρ c , where the system acquires a global polarisation p(r) = const, while ρ(r) = ρ 0 , and the bundling transition at ρ 0 =ρ b ≡ 1/ 4B 2 with p(r) = 0, where the linearised diffusion-like term in Eq.(24) becomes negative indicating the tendency of the system to accumulate disordered microtubular bundles in localised clusters; both instabilities are long-wavelength and set in at the scale of the system size. By setting B such thatρ c <ρ b , Aranson and Tsimring red could show numerically that for ρ c < ρ 0 <ρ b Eqs. (24) and (25) exhibit a disordered quasi steady-state array of vortex and aster-like structures, dominated by vortices, at low H, and by asters, at larger H. For ρ 0 >ρ b , there is a competition between vortices, asters, and disordered high-density clusters at high values of H.
Below we systematically examine the assumptions leading to Eqs. (24) and (25). First, we devise a semianalytical strategy to evaluate the integrals in Eq. (3) with the exact interaction kernel Eq. (14) instead of the effective approximation Eq.(15). We will demonstrate that, as a result, the bundling transition sets in at lower density than the instability towards a globally ordered state, substantially changing the phase diagram. This can already be seen from comparing Eq.(15) with Eq. (14): since L is the only lengthscale that appears in the true interaction kernel, the parameter b of the approximate kernel should only differ from L by a factor of order unity, which impliesρ b ≈ 1/4 < ρ c . Next we note that the terms that appear in Eqs. (24) and (25) were selected on the basis of approximations whose validity is difficult to control a priori: as a result, close to the instability threshold the final equations combine terms which effectively are of different orders. We show how to systematically keep terms of the same order and that this requires modification of the closure given by Eq. (21). Finally, we observe that in the absence of anisotropy in the interaction kernel, i.e. H = 0, Eq.(24) exhibits pathological behaviour for ρ 0 >ρ b , since there are no non-linear terms that can cut-off exponential growth of the linearlyunstable modes. The same problem persists at small values of H, while at large H the non-linear coupling to the polarisation field limits the instability growth, as shown in [14]. To cure this problem, which is more severe wheñ ρ b <ρ cr , here we explicitly account for excluded volume interactions between the microtubular bundles that stabilise the dynamics even in the absence of the polarisation field.
IV. EXCLUDED VOLUME INTERACTIONS
In this Section, we incorporate the excluded volume interactions between microtubular bundles into the dynamic equation for the density. To do so, we start from the Smoluchowski equation, similarly to the work by Ahmadi et al. [21] and Baskaran and Marchetti [33], and then incorporate these terms into the dynamical equations that we derived from the Boltzmann-like Eq.(1). Another approach is to introduce the excluded volume interactions directly in the Boltzmann-like equation [34], but this is more cumbersome. Formally, the two approaches are expected to be equivalent, but note their detailed comparison by Bertin et al. [34].
We begin by introducing the Onsager free energy [35][36][37] for a collection of solid rods in terms of irreducible integrals [38]
F k B T = dr dφ P (r, φ) ln Λ 2 P (r, φ) − 1 − 1 2 drdr dφdφ P (r, φ)P (r , φ )f (r, φ; r , φ ) − 1 6 drdr dr dφdφ dφ P (r, φ)P (r , φ )P (r , φ ) × f (r, φ; r , φ )f (r, φ; r , φ )f (r , φ ; r , φ ) + · · · .(26)
Here, P is the probability distribution function, as in Section II, Λ is the thermal de Broglie wavelength, and f = exp (−U/k B T ) − 1 is the Mayer function. The interaction potential U between two microtubular bundles is infinite, when the bundles cross, and zero otherwise.
In the absence of any external driving, the equilibrium probability distribution is given by [27]
δ δP (r, φ) F k B T − λ dr dφ P (r, φ) = 0,(27)
where δ/δP (r, φ) denotes a functional derivative w.r.t. P (r, φ), and we have introduced a Lagrange multiplier λ to ensure that P (r, φ) satisfies the normalisation condition
dr dφ P (r, φ) = N.(28)
The solution to this equation can formally be written as
P (r, φ) = const × exp − U sc (r, φ) k B T ,(29)
where the self-consistent potential U sc is a functional of the probability distribution function,
U sc (r, φ) k B T = − dr dφ P (r , φ )f (r, φ; r , φ ) − 1 2 dr dr dφ dφ P (r , φ )P (r , φ )
× f (r, φ; r , φ )f (r, φ; r , φ )f (r , φ ; r , φ ) (30) + · · · , and the constant, const, is determined from the normalisation condition, Eq. (28). The two terms in Eq.(30) are the second and third irreducible integrals [38] that correspond to two-bundle and three-bundle interactions, respectively. As was shown by Ahmadi et al. [21], the first term leads to a contribution to the density equation proportional to ∇ 2 ρ 2 . When added to Eq.(24), for example, this contribution can limit the growth of the density fluctuations only for certain values of the parameter B, and in order to avoid this restriction we also include the three-bundle term in Eq. (30), which, as we show below, leads to a contribution proportional to ∇ 2 ρ 3 and provides a stabilisation mechanism for any density and any values of the parameters.
To evaluate the first integral in Eq. (30) we observe that the Mayer function f (r, φ; r , φ ) is only non-zero when a bundle at r with an orientation given by φ intersects the test bundle at r with an orientation given by φ, and in that case f = −1. The first integral, therefore, reduces to
dr dφ P (r , φ ),(31)
integrated over intersecting configurations only. To enumerate such configurations, we use the contour variables introduced in Eq. (10), and write the condition of two bundles intersecting on a plane as
r + n(φ) L 2 τ = r + n (φ ) L 2 τ ,(32)
where τ and τ are the dimensionless positions of the intersection point along the corresponding bundle; see discussion after Eq.(10) for details. When the bundles intersect, Eq.(32) can be used to change integration variables from r to τ and τ , yielding
L 2 4 1 −1 dτ dτ 2π 0 dφ P r + L 2 (n(φ)τ − n (φ )τ ) , φ × |e z · (n(φ) × n (φ )) |,(33)
where the last factor comes from the Jacobian of the transformation of variables. To proceed, we use the Fourier expansion of the probability density function, Eq.(2), in the integral above, and note, as before, that since we are interested in patterns evolving on spatial scales significantly larger than L, we can Taylor expand the Fourier modes P n r + L 2 (n(φ)τ − n (φ )τ ) in gradients of P n (r). The leading contribution to this expansion comes simply from the zeroth order term P n (r) and we use this approximation in this calculation. Additionally, since we are interested in stabilising the dynamics of the density fluctuations, we will only keep P 0 , ignoring all other Fourier harmonics, in the analysis below. The ignored contributions from higher Fourier modes and their spatial gradients have been discussed by Ahmadi et al. [21]. We will argue below that they are subdominant in the regime we are interested in.
Proceeding with the approximations discussed above, the first integral in Eq.(30) becomes
L 2 4 1 −1 dτ dτ 2π 0 dφ ρ(r) 2π |e z · (n(φ) × n (φ )) | = L 2 ρ(r) 2π 2π 0 dφ | sin(φ − φ )| = 2 π L 2 ρ(r).(34)
In a similar fashion, the second term in Eq.(30) can be written in terms of the dimensionless variables τ ij that denote the position along the bundle i of its crossing with the bundle j, where we have numbered the bundles with (r, φ), (r , φ ), and (r , φ ), as bundles 1, 2, and 3, respectively. The conditions of simultaneous intersection of all three bundles is then
r + n(φ) L 2 τ 12 = r + n (φ ) L 2 τ 21 , r + n(φ) L 2 τ 13 = r + n (φ ) L 2 τ 31 ,(35)r + n (φ ) L 2 τ 23 = r + n (φ ) L 2 τ 32 .
As for the two-body interaction term, we use the first two conditions of Eq.(35) to change variables from (r , r ) to (τ 12 , τ 21 , τ 13 , τ 31 ), and use the last condition to ensure that the bundles 2 and 3 cross. This yields for the second term in Eq.(30)
1 2 ρ(r) 2π 2 L 2 6 2π 0 dφ dφ | sin(φ − φ )|| sin(φ − φ )|| sin(φ − φ )| × 1 −1 dτ 12 dτ 21 dτ 13 dτ 31 dτ 23 dτ 32 δ L 2 {n(φ) (τ 12 − τ 13 ) + n (φ ) (τ 23 − τ 21 ) + n (φ ) (τ 31 − τ 32 )} = ρ(r) 2 L 4 8π 4 2π 0 dφ dφ | sin(φ − φ )|| sin(φ − φ )|| sin(φ − φ )| × dk sin(k · n(φ)) k · n(φ) 2 sin(k · n (φ )) k · n (φ ) 2 sin(k · n (φ )) k · n (φ ) 2 ,(36)
where we replaced the two-dimensional Dirac deltafunction by its integral representation [39] δ(a) = 1
(2π) 2 dk e i k·a ,(37)
and performed integration over τ 's. We could not find a closed analytic form for the integral in Eq. (36), and calculated it numerically by replacing the
[−∞, ∞] × [−∞, ∞] integration range for k by [−R, R] × [−R, R]
, with R appropriately large, but finite. We summed up the integrand on a a grid with ∆k x = ∆k y = ∆φ = 0.01 and used R = 60. Since the value of the integral should not depend on the absolute orientation of the first bundle, i.e. on the angle φ, we also averaged the result over φ to increase accuracy. The resulting value is numerically very close to 2π 3 , and we will use that approximation below, for convenience. Finally, to third order in the bundle density the self-consistent potential becomes
U sc (r, φ) k B T = 2 π L 2 ρ(r) + 1 4π L 4 ρ(r) 2 .(38)
The first term in the equation above was calculated by Ahmadi et al. [21], and the three-dimensional version of the second term was discussed by Straley [40]. Diffusion of a rod in an external potential is described by the Smoluchowski equation [21,27]
∂P (r, φ) ∂t = ∇ i D ij ∇ j P (r, φ) + P (r, φ)∇ j U k B T ,(39)
where D ij is given by Eq. (8). Using U sc for the external potential, projecting onto the zeroth Fourier mode, and selecting only the terms containing the density, we arrive at the following contribution of the excluded volume effects to the dynamical equation for the density
∂ t ρ = D + D ⊥ 2π ∇ 2 L 2 ρ(r) 2 + 1 6 L 4 ρ(r) 3 + . . . .(40)
While the ρ 3 term provides stabilisation against the otherwise unbounded growth of the bundling instability, resolving the competition between the ρ 2 and ρ 3 terms numerically requires fine temporal resolutions, as at large timesteps the quadratic term can still lead to a finite-time blow-up due to an insufficient time for the qubic term to curb that growth. We, therefore, introduce a further approximation that allows us to avoid working with small timesteps by re-summing the virial expansion in Eq.(40) as
L 2 ρ(r) 2 + 1 6 L 4 ρ(r) 3 + · · · ≈ L 2 ρ(r) 2 e 1 6 L 2 ρ(r) ,(41)
where we added an infinite number of higher-order terms that mimic the effect of the higher order virial coefficients; their influence is small for sufficiently small densities, and their main function is to safe-guard against very fast growth of local density fluctuations in our numerical simulations presented below. Finally, the contribution of the excluded volume interactions to the density equation is written as
∂ t ρ = D + D ⊥ 2π ∇ 2 L 2 ρ(r) 2 e 1 6 L 2 ρ(r) + . . . ,(42)
where . . . denote the diffusion terms from Eq.9, and the terms originating from the interaction integrals are discussed next.
V. EVALUATION OF INTERACTION INTEGRALS
In this Section we proceed by evaluating the interaction integrals from Eqs. (5) and (6) with the exact kernel Eq. (14). As an example, we calculate the value of I j,nm can be written as
I (1) j,nm s = GL 3 1 2π 2π 0 dφ e i(n+m)φ e −isφ 2π 0 dχ ∞ 0 dζ ζ 2 π −π dωW 1 e i(m−n) ω 2 cos(χ + φ) sin(χ + φ) j ,(43)
where Projection onto the s-th Fourier harmonics yields
W 1 = Θ | sin ω| − 2ζ sin χ − ω 2 Θ | sin ω| − 2ζ sin χ + ω 2 × 1 + Ξ Θ −2ζ sin χ + ω 2 sin ω − τ 0 + Θ −2ζ sin χ − ω 2 sin ω − τ 0 .(44)1 2π 2π 0 e −isφ e i(n+m)φ cos(χ + φ) sin(χ + φ) j dφ = e iχ δ n,s−m−1 + e −iχ δ n,s−m+1 /2 e iχ δ n,s−m−1 − e −iχ δ n,s−m+1 /(2i) j ,(45)
and the spatial components of I ,
where
B (1) s,m =G −1 2π 0 dχ ∞ 0 dζζ 2 π −π dωW 1 e i(2m−s+1) ω 2 e iχ , B (2) s,m =G −1 2π 0 dχ ∞ 0 dζζ 2 π −π dωW 1 e i(2m−s−1) ω 2 e −iχ ,
are functions of Ξ and τ 0 . The structure of Eq.(44) suggests that each of these integrals can be split into two contributions
B (k) m,s (Ξ, τ 0 ) = B (k)iso m,s + Ξ B (k)ani m,s (τ 0 ),(47)
where B
(k)iso m,s is a number associated with the isotropic (i.e. Ξ-independent) part of the kernel Eq.(14), while B (k)ani m,s is a function of τ 0 . We evaluate these contributions numerically, as explained below.
We illustrate our method by calculating B
1,1 . In this case, numerical integration on a grid with ∆ζ = ∆ω = ∆χ = 0.01 gives B
(1)iso 1,1 = 0. We note, however, that in general the isotropic contributions to this and other integrals are not necessarily zero for all values of the indices. To evaluate the anisotropic part B
(1)ani 1,1 (τ 0 ), we perform a similar numerical integration for a range of τ 0 from the interval [−1, 1], and plot the resulting values in Fig.3 (solid circles). We observe that these values are well-approximated by
B (1)ani 1,1 (τ 0 ) = 8 15 1 − τ 2 0 ,(48)
as can be seen from Fig.3 (solid line). We therefore obtain
B (1) 1,1 (Ξ, τ 0 ) = 8 15 Ξ 1 − τ 2 0 .(49)
All other integrals I's and J's, Eqs. (5) and (6), are evaluated in the same way. For all these integrals, the anisotropic contributions are simple polynomials in τ 0 that are readily guessed, while their prefactors and the isotropic contributions are well approximated by ratios of simple integers (see Supplemental Material [41] for detail).
VI. HYDRODYNAMIC EQUATIONS
We now have all the ingredients to formulate our version of the equations of motion for the hydrodynamic fields. As mentioned above, our approach differs from the work of [14] in several important ways, and we will show that this significantly changes the phase diagram of the system. In order to be able to attribute the changes observed to a particular aspect of our theory, we use the following approach. First, we use our values of the interaction integrals calculated with the exact kernel in Eq.(3) combined with a closure strategy employed in [14], see Eq. (21). Then we repeat the same derivation but with a different closure devised to keep only the terms that are relevant in the vicinity of the instability onset. In both cases we add the excluded volume terms to the equation for the density to be able to resolve the dynamics in the presence of a bundling instability, as discussed in Section IV.
A. Aranson-Tsimring Closure
Here, we repeat the derivation from Section III with our values of the interaction integrals. The equations are rendered dimensionless by scaling time, space and the Fourier harmonics of P by D −1 r , L and GL 2 /D r , respectively. In Eq.(3), we keep only the first Fourier harmonics, P 0 , P ±1 , and P ±2 , but drop any gradient of P ±2 . For the second Fourier harmonics, Eq.(3) is an algebraic equation that is solved by P ±2 = A 0 P 2 ±1 , similar to the closure Eq. (21), while for the density and polarisation, Eqs. (22) and (23), we obtain
∂ t ρ = ∇ 2 ρ 32 − (1 + a 3 )ρ 2 48π + 1 32π α∇ 2 ρ 2 e αρ 6 − 91 69120π (1 + a 5 ) ρ 0 ∇ 4 ρ − 1 240 a 4 3∇ · p∇ 2 ρ − ρ∇ 2 p + 2∂ i (∂ j ρ∂ j p i − ∂ i ρ∂ j p j ) ,(50)∂ t p = − p + 5 192 ∇ 2 p + 1 96 ∇(∇ · p) + (1 + a 1 ) 2 3π ρp − 28 15 A 0 |p| 2 p − a 2 ∇ρ 2 32π 2 − 1 20 p(∇ · p) − 9 20 (p · ∇)p + 1 24 ∇ (p · p) + (1 + a 3 ) ρ 0 40π ∇ 2 p + 2 9 ∇(∇ · p) .(51)
Here,
A 0 = 3π 3π (1 + a 1 ) −1 + ρ 0 ,(52)
and
a 1 = Ξ (1 − τ 0 ) , a 2 = Ξ(1 − τ 2 0 ), a 3 = Ξ 1 − τ 0 1 + τ 2 0 /2 ,(53)a 4 = Ξ 1 − τ 2 0 1 + τ 2 0 /2 , a 5 = Ξ 1 − τ 0 1 + τ 2 0 2 /2 .
In Eq.(50), the term proportional to α is the dimensionless version of the excluded volume contribution from Eq. (42), where α = D r /G. This quantity can be understood as a ratio of two timescales, t m /t r , where t m ∼ G −1 is a typical time over which a bundle changes its orientation due to the activity of molecular motors, while t r ∼ D −1 r is a typical re-orientation time due to rotational diffusion. In the absence of motor activity, α becomes very large, and the excluded volume term prevents formation of any significant density fluctuations. In the motor-activity-dominated regime, α is small, and this regime is the focus of the rest of this work.
We also note that apart from the ∇ (p · p) term and the excluded volume contribution, Eqs.(50) and (51) have the same tensorial structure as the equations in the Aranson-Tsimring theory, Eqs. (24) and (25). Perhaps surprisingly, though, the differences in their dependence on the parameters of the kernel, and different numerical prefactors are sufficient to produce a rather different phase diagram, as we discuss in Section VII.
B. Self-consistent closure and Q-tensor
The inherent problem of the previous closure is that it combines terms that are of various orders (equivalently, degrees of smallness) close to the instability threshold. Dropping spatial gradient in the equation for the second Fourier harmonics of P implies that, to obtain a coupled system, the density equation should contain third-order spatial gradients, whilst only first-order terms are sufficient in the polarisation equation. To address this inconsistency, we employ a systematic procedure of deriving hydrodynamic equations that was originally developed for Ginzburg-Landau-like amplitude equations in pattern formation [24] and was recently applied in the context of self-propelled rods [42,43] and microtubule-motor mixtures [44].
Similar to Eq.(25), Eq.(51) suggests that a uniformly polarised state becomes stable above some ρ cr , given by the time-and spatially-independent version of Eq.(51). If we introduce 2 = ρ 0 − ρ cr , balancing the terms in Eq.(51) implies that |p| ∼ , ∇ ∼ , ∂ t ∼ 2 , and the deviation of the density ρ(r, t) from its average value ρ 0 scales as δρ(r, t) ≡ ρ(r, t) − ρ 0 ∼ 2 . Using these scalings we can see that the coupling terms, i.e. the terms proportional to a 4 , in Eq.(50) contain a term proportional to ρ 0 ∇ 2 (∇ · p) ∼ 4 , while the rest of the coupling terms are ∼ 6 . Moreover, this scaling implies that ignoring spatial gradients of P 2 or spatial gradients in the equation for P 2 is not justified as, for example, the term ∇ i p j is of the same order as p i p j , used in the algebraic closure above. Therefore, here we re-derive the equation for P 2 keeping all the terms that are ∼ 2 .
To simplify the notation, we introduce the so-called Q-tensor that is proportional to the second Fourier harmonics of P (r, t),
Q ij (r) = 1 π 2π 0 n i n j − 1 2 δ ij P (r, φ)dφ.(54)
The two independent components of the Q-tensor can be explicitly written as follows,
Q xx (r) = P 2 (r) + P −2 (r) 2 , Q xy (r) = P −2 (r) − P 2 (r) 2i .(55)
Note that Q yy = −Q xx and Q yx = Q xy as the Q-tensor is traceless and symmetric. Keeping the terms proportional to 2 in Eq.(3) for the second harmonics, we obtain
Q ij = 1 1 + 1 3π ρ 0 (1 + a 1 ) (1 + a 1 ) 2p i p j − (p · p)δ ij + a 2 ρ 0 48π ∂ i p j + ∂ j p i − δ ij (∇ · p) ,(56)
which, in the absence of spatial gradients, is the same as the closure used above. Similarly, keeping the leading terms in , which are proportional to 3 and 4 for the first and the zeroth harmonics, respectively, we arrive at the following dynamical equations
∂ t ρ = ∇ 2 ρ 32 − (1 + a 3 )ρ 2 48π + 1 32π α∇ 2 ρ 2 e αρ 6 − 91 69120π (1 + a 5 ) ρ 0 ∇ 4 ρ + π 48 − 1 36 (1 + a 3 )ρ 0 ∂ i ∂ j Q ij + 1 80 a 4 ρ 0 ∇ 2 (∇ · p) ,(57)∂ t p i = −p i + 5 192 ∇ 2 p i + 1 96 ∇ i (∇ · p) + (1 + a 1 ) 2 3π ρ p i − 28 15 Q ij p j − a 2 1 16π 2 ρ 0 ∂ i ρ − 1 20 p i (∇ · p) − 9 20 (p · ∇)p i + 1 24 ∇ i (p · p) + 1 240 ρ 0 ∂ k Q ik + (1 + a 3 ) ρ 0 8 1 5π ∇ 2 p i + 2 45π ∇ i (∇ · p) .(58)
In Eq.(57) we kept several terms that are inconsistent with this approximation scheme. While being of higher order than the rest of the equation, they represent the lowest order terms responsible for a particular effect. Thus, we keep the term that causes the bundling instability and the excluded volume term that saturates it, and we follow Aranson and Tsimring [14] in keeping the bilaplacian term that selects the lengthscale of the bundling instability. This system of equations is the central result of our paper.
VII. RESULTS
In this Section we present analysis of the dynamical behaviour exhibited by the models derived above. For convenience, we will be referring to Eqs.(50) and (51) as the Aranson-Tsimring-closure (ATC) model, and to Eqs.(57) and (58) -as the Q-tensor-closure (QC) model. First, we perform a linear stability analysis of the homogeneous and isotropic base state for both models and determine the regions of the parameter space where nontrivial behaviour can be expected. Then we perform direct numerical simulations of the ATC and QC models in these parts of the parameter space and discuss the resulting patterns.
A. Linear stability analysis
We start by observing that both models support exact solutions in the form of a homogeneous state with ρ(r, t) = ρ 0 and p(r, t) = P, as was already mentioned above. In both cases, the evolution equation for P is given by
∂ t P i = −1 + (1 + a 1 ) 2 3π ρ − 28 15 A 0 P 2 P i ,(59)
where P = |P|. Trivially, P = 0 is always a solution to this equation for any density. For densities larger than
ρ cr = 3π 2(1 + a 1 ) ,(60)
the isotropic solution loses its stability, as Eq.(59) suggests, and another solution sets in with
P = 1 1 + a 1 15 28 ρ 0 ρ cr − 1 ρ 0 2ρ cr + 1 ,(61)
and a random orientation selected through a spontaneous symmetry breaking. We refer to this solution as the globally-ordered state. The homogeneous and isotropic state with ρ(r, t) = ρ 0 and p(r, t) = 0 is also unstable with respect to density fluctuations, as was already mentioned above; there, it was referred to as a bundling instability. Assuming small spatial variations of the density profile ρ(r, t) = ρ 0 + δρ(t)e i(kxx+kyy) and the absence of polarisation fluctuations, the linear dynamics of the density perturbations are given by
∂ t δρ = λ b (k)δρ, where λ b (k) = − 1 32 + (1 + a 3 )ρ 0 24π − αρ 0 192π e αρ 0 6 (12 + αρ 0 ) k 2 − 91 69120π (1 + a 5 ) ρ 0 k 4 ,(62)
and k 2 = k 2 x + k 2 y . For a selected wavevector, density perturbations grow when λ b (k) becomes positive, which can only happen when the coefficient of k 2 is positive, since the prefactor of k 4 is negative for realistic values of τ 0 . Therefore this instability sets in at a critical density ρ b , given by
− 1 32 + (1 + a 3 )ρ b 24π − αρ b 192π e αρ b 6 (12 + αρ b ) = 0,(63)
which in the absence of the excluded volume, α = 0, becomes ρ b = 3π/(4 (1 + a 3 )), similar to the expression obtained in [14]. In Fig.4, we plot the solutions of Eq.(63) as a function of the excluded volume strength α for fixed values of the asymmetry parameter Ξ. For any value of Ξ, there exist two regions of this parameter space. For large values of α there is no bundling instability as strong excluded volume effects preclude growth of any density variations. Instead, for smaller values of α there is a band of density values (the shaded regions in Fig.4), where the bundling instability exists. The upper boundary of this band goes to infinity when α approaches zero.
Since α sets the strength of the excluded volume interactions, which is just a geometric effect not related to motor activity, we fix its value, and treat Ξ and ρ 0 as the control parameters. In Fig.5, we plot the instability boundaries found above in terms of these control parameters. The dotted black line in Fig.5 is the critical density ρ cr , given by Eq.(60), while the solid lines, given by Eq.(63), enclose the region of the bundling instability (shaded regions in Fig.5). As α increases, the bundling instability is pushed towards larger values of Ξ, but is always present. We, therefore, select a representative case of α = 0.35 (blue line and the blue shaded region in Fig.5), and perform direct numerical simulations of the ATC and QC models for a range of densities and fixed motor asymmetry parameter Ξ = 0.2 and Ξ = 0.5. The former case exhibits only the instability towards a globally ordered state, while the latter case has both types of instability. The densities we use in our simulations are denoted by green circles in Fig.5. Finally, we note that a full linear stability analysis (see below) shows that the transition to global order and the bundling instability are the only instabilities of the homogeneous and isotropic state for both models.
B. Direct numerical simulations
To explore the nonlinear behaviour of the ATC and QC models, we perform direct numerical simulations of Eqs.(50) and (51), and of Eqs.(57) and (58) in the parts of the parameter space identified above. We discretise spatial derivatives by second-order finite-differences, and employ a second-order predictor-corrector method for time integration [45,46]. Our computations are performed on square domains 150 × 150 with periodic boundary conditions with spatial resolution ∆h = 0.5, where the unit length is chosen to be the microtubular length (see Section VI for details of our dimensional units); the timestep is set to ∆t = 0.005. Unless explicitly stated, we set α = 0.35 and τ 0 = 0, as discussed above. Below, we present our simulation results in composite images showing simultaneously the local density profile ρ(r) (colour) and the polarisation vector field p(r) (arrows), normalised by its magnitude in the globallyordered state, Eq.(61).
We start by examining the behaviour of the ATC model for Ξ = 0.2 where, according to Fig.5 one should expect a transition to global polar order for sufficiently high densities. For ρ 0 = 2 and 3, there exists no instability of the homogeneous and isotropic state, and any random initial condition in our simulations quickly returns to that state. For densities above the global-instability threshold (black dotted line in Fig.5), we observe rapid formation of a globally oriented state with a large number of defects, as can be seen from Fig.6a for ρ 0 = 5. These defects consist of vortices, inward-pointing asters that correspond to an increase of the local density, and spatially-distributed defects of the opposite topological charge that correspond to the minima of the local density. After sufficiently long simulation times, these defects annihilate leaving behind a uniform, globally polarised state. The same behaviour persists at higher densities, the only difference being that there are now sharper density gradients around topological defects. We also observe that the typical time for all defects to annihilate grows quickly with ρ 0 . In Fig.6b, for instance, we show the final snapshot of a long run for ρ 0 = 12, which continued to coarsen over the course of the whole simulation.
At Ξ = 0.5 the behaviour of the ATC model changes considerably. According to Fig.5, as the density is increased, the bundling instability is the first one to set in. For larger densities, the bundling instability coexists with the globally polarised state, while at yet large densities, one should again expect uniform polar order throughout the system. This scenario is supported by our direct numerical simulations. Below the bundling instability threshold, the system always returns to the homogeneous and isotropic state. At higher densities, we observe the following dynamical structures. For ρ 0 = 5 and ρ 0 = 7, (Figs.7a and 7b, respectively), the bundling instability competes with the emergence of global order, and the ensuing high-density clusters tend to elongate to keep local polarisation aligned. Such elongated clusters often end up in yet-higher-density regions with inwardpointing asters. Even after a long time, the system does not settle into a steady-state; instead its dynamics comprise slow re-arrangements of the high-density clusters, mostly along the direction set by the local polarisation, punctuated by fast re-orientation waves that align locally the polarisation vector with the density gradient. A similar behaviour is observed in simulations with ρ 0 = 3, which is within a narrow range of densities that are below the global instability threshold, but above the bundling instability one. In this case the system first develops clusters of high density dispersed in a low-density background until the local density inside the clusters exceeds the global instability threshold, after which the dynamics resemble its higher-density counterpart discussed above. At yet higher density, above the bundling instability region (ρ 0 = 12, see Fig.7c), the system does not exhibit global order as predicted by the linear stability analysis (Fig.5). Instead it forms high-density clusters, which tend to merge into large-scale structures at very long times, see Fig.7c. Each cluster contains polarisation field in the form of inward-pointing asters. Perhaps, this state may be viewed as an example of microphase separation, as clusters do not coarsen indefinitely but appear to reach a self-limiting size. However, we do not know whether it survives at yet longer simulation times or in larger systems. Now we compare these observations against the results of our direct numerical simulations of the QC model. Since the linear stability properties of the homogeneous and isotropic state are the same for both models, one might expect the QC model to exhibit a dynamical behaviour similar to the ATC one. Surprisingly, the two models are instead substantially different. As for the ATC model, the cases of ρ 0 = 2, with Ξ = 0.2 and 0.5, and of ρ 0 = 3, with Ξ = 0.2, yield no instabilities, and the system returns to the homogeneous and isotropic state.
Above the global instability threshold, the QC model exhibits the same type of dynamics for both Ξ = 0.2 and Ξ = 0.5 (see Fig.8 and Fig.9, respectively). Although visually these structures appear to be similar to the ATC patterns at Ξ = 0. ). While the high-density clusters of the ATC model exhibit slow, largely coarsening-type dynamics with the polarisation quickly adjusting to slowly evolving local density gradients, here the density and polarisation evolve on comparable timescales, never settle down, and appear to be chaotic for any value of Ξ and ρ 0 in Figs.8 and 9. Even in the regions of approximately homogeneous local density, the polarisation field exhibits significant time dependence, suggesting that the globally polarised state is linearly unstable for these parameters.
To validate this statement, we performed a linear stability analysis of the globally polarised state for the ATC and QC models, see Supplemental Material for details [41]. First, this analysis confirms that the homogeneous and isotropic state, P = 0, of both models does not have any other instability than the bundling and globalorder instabilities, discussed above. Next, we observe that while the globally polarised state is always linearly stable for the ATC model, for the QC model there is a range of parameters where it becomes unstable with respect to coupled polarisation and density fluctuations. In Fig.10 we plot the results of both types of linear stability analysis of the QC model. There, the black dotted line and the blue dashed line (both taken from Fig.5) correspond to the instability boundary of the globallyoriented state and the region of the bundling instability, respectively. The solid brown line marks the boundary above which the globally ordered state is linearly unstable. Additionally, within that region there are two possi- ble instability modes. The first one is characterised by a modulation in the density and polarisation along the direction of the global order (magenta shaded region); the second has modulations both perpendicular and parallel to that direction (brown shaded region). We, therefore, speculate that when there is global order (i.e., above or to the right of the dotted black line in Fig.10), the QC model exhibits three types of behaviour that cannot coexist: (i) the tendency to create global orientation with a uniform density profile, (ii) the bundling instability, and (iii) the instability of the global order. The interaction between these three instabilities is what leads to irregular dynamics, as we see in Figs.8 and 9 and Supplemental Movies [41]. As we can see from Fig.10, for ρ 0 > 3, all our simulations (green circles) belong to the unstable region of the parameter space. We therefore performed additional simulations (not shown) of the QC model for Ξ = 0.2 with ρ 0 = 20 and Ξ = 0.5 with ρ 0 = 25, that both lie outside the unstable region (brown line), and confirmed the absence of chaotic-like behaviour at long times. Instead, both systems settled into a globally polarised state interlaid with topological defects, similarly to the case of the ATC model.
The non-trivial dynamics presented above relies on the simultaneous existence of at least two types of instability for the same values of Ξ and ρ 0 . Fig.4 suggests that for moderate values of Ξ, the bundling instability only exists for small values of α. To study the dynamics of both models outside of this regime, we set α = 0.6 and considered Ξ = 0.2 and Ξ = 0.5, as before. The linear stability analysis of the globally polarised state suggests that both the ATC and QC models are linearly stable in that regime, and the only instability threshold is given by Eq.(60). Our simulations confirm that both models exhibit rather simple dynamics, similar to the case of the ATC model with Ξ = 0.2 and α = 0.35: below ρ cr , the system returns to the homogeneous and isotropic state, while above ρ cr , it goes through a series of long-lived topological defects before, eventually, settling into the homogeneous and isotropic state. At the highest density considered, ρ 0 = 12, the system gets trapped into a state with an apparently stable (or long-lived metastable) arrangement of topological defects (see Fig.11). The main difference between the two models, however, is that the inward-pointing asters of the ATC model correspond to local density enhancement, while similar topological defects in the QC model lead to local density minima. The two situations presented above, α = 0.35 and α = 0.6, seem to comprehensively cover the behaviour of the ATC and Q models, and we have not observed any other dynamical structures besides the patterns presented above. As mentioned at the beginning of this Section, we restricted our simulations to a realistic, albeit arbitrary, case of τ 0 = 0. Another value of τ 0 would lead to a quantitative effect on the instability boundaries, while the qualitative behaviour is still the same. This is only the case for 1 + a 5 > 0, as Eq.(62) suggests, which is always true for Ξ < 1. When 1 + a 5 < 0, the bilaplacian terms in Eqs.(50) and (57) do not result in the lengthscale selection for the bundling instability, and a yet higher-order gradient has to be added to the equations in that case.
VIII. DISCUSSION
The main goal of this study was to revisit the kinetic theory of microtubule-motor mixtures originally derived in [14], as well as its coarse-graining into a set of dynamical equations for (slowly-varying) density and orientation fields, Eqs. (24) and (25). We also studied (by linear stability analysis and direct numerical simulations) the resulting equations, and analysed the corresponding pattern formation dynamics.
In particular, we considered the validity of the effective interaction kernel, Eq.(15), used in [14]. To address this issue, we developed a semi-analytical method that allowed us to calculate the interaction integrals, Eqs. (5) and (6), exactly. We also studied the closure relationship, Eq.(21), used in [14], and compared it to a closure method routinely used in Ginzburg-Landautype theories of pattern formation [24,[42][43][44]. We derived two dynamical systems of equations, which we respectively called ATC model and QC model, that utilise our approximation-free values of the interaction integrals, but use various closure relations. While the ATC model uses the same closure as [14], the QC model uses the selfconsistent closure derived in Section VI B. Together with the original equations of Aranson and Tsimring, Eqs. (24) and (25) (which we refer to as the original Aranson-Tsimring model), these models allowed us to assess the importance of each of the assumptions mentioned above.
We used a linear stability analysis and direct numerical simulations to compare these three models. For the parameters of the effective kernel chosen by Aranson and Tsimring [14], the model predicts three types of behaviour: (i) the homogeneous and isotropic state for low densities, (ii) the globally-polarised state with various topological defects for intermediate densities, and (iii) the bundling instability leading to the formation of high-density clusters at high densities. Our analysis with the exact kernel demonstrated that under similar assumptions the order of the phases is different, with the bundling instability often setting in before the globallypolarised state. Therefore, in order to fully resolve the dynamics at late times, the equations of motion should have a physical mechanism that limits the otherwise unchecked growth of the bundling instability. The original Aranson-Tsimring model simply relies on the nonlinear coupling terms (i.e., terms proportional to H) in Eq. (24) to cut the growth of density fluctuations -however this is a viable route only for sufficiently large values of H. To cure this problem we introduced steric repulsion between the microtubular bundles: this has to be calculated up to the third virial coefficient or higher in order to provide a stabilisation mechanism that works for any density. This procedure allowed us to resolve the dynamics of our models in the region of the parameter space where the bundling and global instabilities co-exist. Our main conclusion here is that the usage of the exact kernel significantly alters the positions of the instability boundaries and, unless the exclusion volume parameter α is rather large, the bundling instability co-exists with the global order, leading to patterns absent from the original Aranson-Tsimring model [14]. When the bundling instability is absent, the ATC model exhibits the transition to a globally-polarised state, mediated by a variety of topological defects, similar to the original Aranson-Tsimring model [14].
Additionally, by comparing the ATC and QC models, we concluded that the self-consistent closure employed in the latter model, changes the stability properties of the globally-polarised state in the region of the parameter space where it co-exists with the bundling instability, leading to seemingly chaotic patterns. Also, the topological defects observed for this model in the absence of the bundling instability are of rather different nature than the corresponding defects in the ATC or original Aranson-Tsimring models.
We, therefore, conclude that out of the three sets of equations we compared, the QC model more faithfully reproduces the long-wavelength dynamics of Eq.(3) with Eq. (14). When either the effective kernel Eq.(15) or a closure similar to Eq.(21) is employed, the resulting phase diagram differs significantly from the phase diagram of the QC model. This suggests that it might be of interest to analyse how the results in previous studies on microtubule-motor mixtures such as [23] may be affected by the use of the QC equations of motion.
We would like to point out that there is another additional remarkable difference between the QC and the other two models: in the absence of the motor asymmetry along microtubular bundles -i.e., for Ξ = 0 -the density and the polarisation equations of the Aranson-Tsimring and the ATC models decouple from each other, while this is not the case in the QC model, which still exhibits dynamical, seemingly chaotic patterns, similar to the Ξ = 0 case (not shown).
Finally, we note that as the main goal of this study was to hone the techniques required to derive consistent hydrodynamic equations, we adopted a simple set of collision rules, formulated in Fig.1, that are the basis for Eq.(1), and the expression for the interaction kernel, Eq. (14). Detailed studies of the interactions between microtubular bundles [7,16,47,48] suggest that these assumptions are unlikely to be fully realistic, and will require refinement. Additionally, there is a need to understand the role that potential microtubular selfpropulsion, discussed by Liverpool, Marchetti and coworkers [1,20,21,25], might play in the dynamics of microtubules-molecular motor mixtures. We plan to address some of these questions in our future work.
IX. ACKNOWLEDGEMENT
Discussions with Igor Aranson and Lev Tsimring are kindly acknowledged. AG acknowledges funding from the Biotechnology and Biological Sciences Research Council of UK (BB/P01190X, BB/P006507). DM acknowledges support from ERC CoG 648050 (THREEDCELL-PHYSICS). Additional simulation movies and further research outputs generated for this project can be found at http://dx.doi.org/10.7488/ds/2246.
FIG. 1 .
1Collision rule employed in Eq.
FIG. 2 .
2where the left-and right-hand side of this equation is the position of the intersection point written with respect to the centre of mass (middle point) of either the first or the second bundle, i.e. r 1 or r 2 . The microtubule orientation is given by n i = (cos φ i , sin φ i ), i = 1, 2. Here we have introduced the dimensionless contour lenghts τ 1,2 that parametrise the position along each microtubule: τ = −1 corresponds to the minus-end, and τ = 1 to the Model anisotropic distribution of the molecular motors along a microtubular filament.
the same technical features shared by all other interaction integrals, whose values are given in Supplemental Material [41]. By introducing new variables χ = ψ − φ and ζ = ξ/L, the integral I (1)
FIG. 3. B (1)ani 1,1 as a function of τ0.
FIG. 4 .
4Regions of existence of the bundling instability for τ0 = 0 and various values of Ξ. The solid lines are solutions to Eq.(63), while the shaded regions indicate where the homogeneous and isotropic state is unstable with respect to density fluctuations (the bundling instability). The solid lines can therefore be seen as spinodal lines, and the shaded areas as regions of phase separation.
FIG. 5 .
5The data from Fig.4 replotted as Ξ vs. ρ0 graph for τ0 = 0 and various values of α. The dotted black line is the onset of global order, given by Eq.(60). Green circles indicate point for which we perform direct numerical simulations with the ATC and QC models: τ0 = 0, α = 0.35, Ξ = 0.2 and Ξ = 0.5 with ρ0 = 2, 3, . . . , 12.
FIG. 6 .
6Instantaneous snapshots from the direct numerical simulations of the ATC model with τ0 = 0, α = 0.35, and Ξ = 0.2.
FIG. 7 .
7Same as Fig.6 but with Ξ = 0.5.
FIG. 8 .
8Instantaneous snapshots from the direct numerical simulations of the QC model with τ0 = 0, α = 0.35, and Ξ = 0.2.
FIG. 9 .
9Same as Fig.8 but with Ξ = 0.5.
FIG. 10 .
10Linear stability diagram of the QC model for τ0 = 0 and α = 0.35. As in Fig.5, the dotted black line is the onset of global order, given by Eq.(60), and the dashed blue line delineates the region of the parameter space where the homogeneous and isotropic state exhibits the bundling instability (the same as the solid blue line in Fig.5). The brown solid line indicates the region where a homogeneous, globally-ordered state becomes linearly unstable. Inside this line we also specify the instability mode: magenta-shaded region corresponds to the density and polarisation fluctuations modulated along the direction of the global order, while the brown-shaded region corresponds to modulations both perpendicular and parallel to that direction.(a) ATC model (b) QC model FIG. 11. Comparison between the long-time dynamics of the ATC and QC models with τ0 = 0, α = 0.6, Ξ = 0.5 and ρ0 = 12.
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| [] |
[
"Asymptotic expansions of the solutions of the Cauchy problem for nonlinear parabolic equations",
"Asymptotic expansions of the solutions of the Cauchy problem for nonlinear parabolic equations"
] | [
"Kazuhiro Ishige :[email protected] ",
"Tatsuki Kawakami :[email protected] ",
"\nMathematical Institute\nDepartment of Mathematical Sciences\nTohoku University\n980-8578AobaSendaiJapan\n",
"\nOsaka Prefecture University\n599-8531SakaiJapan\n"
] | [
"Mathematical Institute\nDepartment of Mathematical Sciences\nTohoku University\n980-8578AobaSendaiJapan",
"Osaka Prefecture University\n599-8531SakaiJapan"
] | [] | Let u be a solution of the Cauchy problem for the nonlinear parabolic equationand assume that the solution u behaves like the Gauss kernel as t → ∞. In this paper, under suitable assumptions of the reaction term F and the initial function ϕ, we establish the method of obtaining higher order asymptotic expansions of the solution u as t → ∞. | 10.1007/s11854-013-0038-6 | [
"https://arxiv.org/pdf/1202.1037v1.pdf"
] | 119,635,174 | 1202.1037 | 01c9063d0f797da85a0a9fe8b2dd206d2a480bc6 |
Asymptotic expansions of the solutions of the Cauchy problem for nonlinear parabolic equations
6 Feb 2012
Kazuhiro Ishige :[email protected]
Tatsuki Kawakami :[email protected]
Mathematical Institute
Department of Mathematical Sciences
Tohoku University
980-8578AobaSendaiJapan
Osaka Prefecture University
599-8531SakaiJapan
Asymptotic expansions of the solutions of the Cauchy problem for nonlinear parabolic equations
6 Feb 2012This paper is a generalization of our previous paper [18], and our arguments are applicable to the large class of nonlinear parabolic equations. 2010 Mathematics Subject Classification Numbers. 35B40, 35K15, 35K58.
Let u be a solution of the Cauchy problem for the nonlinear parabolic equationand assume that the solution u behaves like the Gauss kernel as t → ∞. In this paper, under suitable assumptions of the reaction term F and the initial function ϕ, we establish the method of obtaining higher order asymptotic expansions of the solution u as t → ∞.
Introduction
Let u be a unique solution of the Cauchy problem for the nonlinear parabolic equation
(1.1) ∂ t u = ∆u + F (x, t, u, ∇u) in R N × (0, ∞), u(x, 0) = ϕ(x) in R N ,
where N ≥ 1, ∂ t = ∂/∂t, F ∈ C(R N × (0, ∞) × R × R N ), and (1.2) ϕ ∈ L 1 K := φ ∈ L 1 (R N ) :
R N (1 + |x|) K |φ(x)|dx < ∞
for some constant K ≥ 0. Let A > 1 and assume that the solution u satisfies
(C A )
|F (x, t, u(x, t), ∇u(x, t))| ≤ C * (1 + t) −A (|u(x, t)| + (1 + t) 1/2 |∇u(x, t)|)
for almost all (x, t) ∈ R N × (0, ∞), where C * is a constant. Then it can be proved that u ∈ S := v ∈ L ∞ loc (0, ∞ : W 1,∞ (R N )) :
sup t>0 t N/2 v(t) L ∞ (R N ) + t 1/2 ∇v(t) L ∞ (R N ) < ∞ ,
and the solution u behaves like the Gauss kernel as t → ∞, that is,
(1.3) R N
u(x, t)dx converges to a constant M as t → ∞ and
lim t→∞ u(t) − M G(1 + t) L q (R N ) / G(1 + t) L q (R N ) = 0 for any q ∈ [1, ∞], where G(x, t) = (4πt) − N 2 exp − |x| 2 4t
(see Theorem 3.1). We introduce the condition (F A ) on the reaction term F :
(F A )
(i) F (x, t, 0, 0) = 0 for all (x, t) ∈ R N × (0, ∞);
(ii) For any v 1 and v 2 ∈ S, there exists a constant C such that |F (x, t, v 1 (x, t), ∇v 1 (x, t)) − F (x, t, v 2 (x, t), ∇v 2 (x, t))| ≤ C(1 + t) −A (|v 1 (x, t) − v 2 (x, t)| + (1 + t) 1/2 |∇v 1 (x, t) − ∇v 2 (x, t)|)
for almost all (x, t) ∈ R N × (0, ∞).
Condition (F A ) ensures that, if v ∈ S, then v satisfies condition (C A ). In this paper, under these conditions (C A ) and (F A ), we study the large time behavior of the solution u of (1.1), and establish the method of obtaining higher order asymptotic expansions of the solution u as t → ∞.
Consider the Cauchy problem for the semilinear heat equation
(1.4) ∂ t u = ∆u + λ|u| p−1 u in R N × (0, ∞), u(x, 0) = ϕ(x) in R N ,
where N ≥ 1, λ ∈ R, p > 1 + 2/N , and ϕ ∈ L 1 (R N ) ∩ L ∞ (R N ). Under suitable assumptions, Cauchy problem (1.4) has a unique global in time solution, and the large time behavior of the solution has been studied in many papers by various methods (see for example [3], [6], [11]- [18], [20], [23]- [25], [29]- [31], [34], and references therein). In particular, it is known that, if ϕ ∈ L 1 (R N ) ∩ L ∞ (R N ) and ϕ L N(p−1)/2 (R N ) is sufficiently small, then there exists a unique global in time solution of (1.4), satisfying (1.3). In [16] the authors of this paper and Ishiwata studied the large time behavior of the solution of (1.4), and investigated the decay rate of the difference between the solution u satisfying (1.3) and the Gauss kernel (see also [17], [24], [25], [31], and [30,Proposition 20.13]). Subsequently, in [18], improving the arguments in [16], the authors of this paper studied the Cauchy problem for the nonlinear parabolic equations of type
∂ t u = ∆u + F (x, t, u) in R N × (0, ∞),
and gave higher order asymptotic expansions of the solution satisfying (1.3). Their results are applicable to the solution of (1.4), satisfying (1.3). We remark that, if the solution u of (1.4) satisfies (1.3), then there holds
λ|u(x, t)| p−1 u(x, t) ≤ C(1 + t) − N 2 (p−1) |u(x, t)|, (x, t) ∈ R N × (0, ∞)
for some constant C, and conditions (C A ) and (F A ) are satisfied with A = N (p − 1)/2 > 1.
On the other hand, for the Cauchy problem for the nonlinear parabolic equations of type
(1.5) ∂ t u = ∆u + ∇ · F (x, t, u) in R N × (0, ∞),
under suitable assumptions on F and the initial function, there exists a global in time solution satisfying (1.3), and the asymptotics of the solution has been studied in detail by many mathematicians (see for example [1], [2], [4], [5], [7], [8], [10], [22], [27], [28], [32], [33], [35], and references therein). The solution u of the Cauchy problem for (1.5) satisfies
(1.6) R N u(x, t)dx = R N u(x, 0)dx
under suitable integrability conditions on the solution u, and property (1.6) has been used effectively in the study of the asymptotic expansions of the solution of (1.5) in the papers. However the solution of (1.1) does not necessarily have property (1.6), and it seems difficult to apply their arguments to Cauchy problem (1.1) for general nonlinear parabolic equations directly.
This paper is a generalization of our previous paper [18], and the main results of this paper are given in Section 4. In this paper, by using the operator P [K] (t) introduced by [16] (see Section 2.1) we establish the method of obtaining higher order asymptotic expansions of the solution of Cauchy problem (1.1) under conditions (C A ) and (F A ). Furthermore we give decay estimates of the difference between the solution and its asymptotic expansions. Our results can give not only higher order asymptotic expansions of the solutions of general nonlinear parabolic equations systematically but also sharp asymptotic expansions of the solutions for some typical examples of nonlinear parabolic equations. In Section 6 we apply our results to some selected examples of nonlinear parabolic equations including the convection-diffusion equation and the Keller-Segel system of parabolic-parabolic type, and explain the advantage of our results.
The rest of this paper is organized as follows. In Section 2 we give some notation and introduce the operator P [K] (t). Furthermore we recall some properties of the solution of the heat equation and the operator P [K] (t), and give a preliminary lemma on the volume potential (see also Section 7). In Section 3 we give a theorem, which implies that the solution of (1.1) belongs to S and satisfies (1.3) and which ensures the well-definedness of P [K] (t)u(t) and P [K] (t)F (·, t, u(t), ∇u(t)). In Section 4 we state the main results of this paper, and give higher order asymptotic expansions of the solution u of (1.1) under conditions (C A ) and (F A ) with A > 1. Section 5 is devoted to the proof of theorems given in Section 4. In Section 6 we apply our main results to some selected examples of nonlinear parabolic equations. Section 7 is an appendix, and there we prove the Hölder continuity of the gradient of the volume potential.
Notation and preliminary results
In this section we give some notation and the definition of the solution of (1.1). Furthermore we introduce an operator P [K] (t), and recall some preliminary lemmas on the solution of the heat equation and the operator P [K] (t).
Notation and operator P [K] (t)
We introduce some notation.
Let N 0 = N ∪ {0}. For any k ∈ R, let [k] be an integer such that k − 1 < [k] ≤ k. For any multi-index α = (α 1 , · · · , α N ) ∈ N N 0 , we put |α| := N i=1 |α i |, α! := N i=1 α i !, x α := N i=1 x α i i , ∂ α x := ∂ |α| ∂x α 1 1 · · · ∂x α N N , J(α) := {ρ = (ρ 1 , · · · , ρ N ) ∈ N N 0 \ {α} : ρ i ≤ α i for all i = 1, · · · , N }, g α (x, t) := (−1) |α| α! (∂ α x G)(x, 1 + t).
In particular, we write g(x, t) = g 0 (x, t) for simplicity. We denote by e t∆ ϕ the unique bounded solution of the Cauchy problem for the heat equation with the initial function ϕ ∈ L 1 (R N ), that is,
(2.1) (e t∆ ϕ)(x) := R N G(x − ξ, t)ϕ(ξ)dξ.
For any two nonnegative functions f 1 and
f 2 defined in a subset D of [0, ∞), we say f 1 (t) f 2 (t) for all t ∈ D if there exists a positive constant C such that f 1 (t) ≤ Cf 2 (t) for all t ∈ D.
In addition, we say
f 1 (t) ≍ f 2 (t) for all t ∈ D if f 1 (t) f 2 (t) and f 2 (t) f 1 (t) for all t ∈ D.
In what follows, we write
· q = · L q (R N ) , ||| · ||| m = · L 1 (R N ,(1+|x|) m dx)
for simplicity, where q ∈ [1, ∞] and m ≥ 0.
We give the definition of the solution of Cauchy problem (1.1).
Definition 2.1 Let ϕ ∈ L 1 (R N ) and assume F ∈ C(R N × (0, ∞) × R × R N ). Then the function u ∈ L ∞ loc (0, ∞ : W 1,1 (R N )) is said to be a solution of (1.1) if u(x, t) = R N G(x − ξ, t)ϕ(ξ)dξ + t 0 R N G(x − ξ, t − s)F (ξ, s, u(ξ, s), ∇u(ξ, s))dξds holds for almost all (x, t) ∈ R N × (0, ∞).
Let k ∈ N 0 , i ∈ {0, . . . , k}, and t > 0. Next we follow [16] and [18], and introduce a linear operator P i (t) on L 1 k by
(2.2) [P i (t)f ](x) := f (x) − |α|≤i M α (f, t)g α (x, t), where f ∈ L 1 k and M α (f, t) is the constant defined inductively (in α) by (2.3) M 0 (f, t) := R N f (x)dx, M α (f, t) := R N x α f (x)dx if |α| = 1, M α (f, t) := R N x α f (x)dx − ρ∈J(α) M ρ (f, t) R N x α g ρ (x, t)dx if |α| ≥ 2.
Then the operator P i (t) has the following property,
(2.4) R N x α [P i (t)f ](x)dx = 0, |α| ≤ i,
which is a crucial property in our analysis. Here, under the assumption ϕ ∈ L 1 K with K ≥ 0, we apply the operator P [K] (t) to e t∆ ϕ, and obtain
P [K] (t)e t∆ ϕ = e t∆ ϕ − |α|≤[K] M α (e t∆ ϕ, t)g α (x, t) = e t∆ ϕ − |α|≤[K] M α (ϕ, 0)g α (x, t) = e t∆ [P [K] (0)ϕ]
for all t > 0. (See also Lemma 2.3 (ii).) Then, due to property (2.4), we have
(2.5) t N 2 (1− 1 q ) e t∆ ϕ − |α|≤[K] M α (ϕ, 0)g α (t) q = o(t − K 2 ) if K = [K], O(t − K 2 ) if K > [K],
Preliminaries
In this section we recall some preliminary results on the behavior of solutions for the heat equation and the operator P [K] (t). Furthermore we give preliminary lemmas on the volume potential and an integral inequality.
Let α ∈ N N 0 and g α be the function given in Section 2.1. Then, for any j = 0, 1, 2, . . . , there exists a constant C 1 such that
(2.6) |∂ j t ∂ α x G(x, t)| ≤ C 1 t − N+|α|+2j 2 1 + |x| t 1/2 |α|+2j exp − |x| 2 4t
for all (x, t) ∈ R N × (0, ∞). This inequality yields the inequalities
(2.7) g α (t) q (1 + t) − N 2 (1− 1 q )− |α| 2 , R N |x| l |g α (x, t)|dx (1 + t) l−|α| 2 , t > 0,
for any q ∈ [1, ∞] and l ≥ 0. Furthermore, by (2.1) and (2.6) we have:
(G1) For any multi-index α and 1 ≤ p ≤ q ≤ ∞, there exists a constant c |α| , independent of p and q, such that
∂ α x e t∆ ϕ q ≤ c |α| t − N 2 ( 1 p − 1 q )− |α| 2 ϕ p , t > 0.
In particular, there holds e t∆ ϕ q ≤ ϕ q for all t > 0;
(G2) For any l ≥ 0 and δ > 0, there exists a constant C 2 such that
R N |x| l |(e t∆ ϕ)(x)|dx ≤ (1 + δ) R N |x| l |ϕ(x)|dx + C 2 t l 2 R N |ϕ(x)|dx, t > 0
(see also Lemma 2.1 in [16]). This inequality implies that
|||e t∆ ϕ||| l ≤ (1 + δ)|||ϕ||| l + C 3 (1 + t l 2 ) ϕ 1 , t > 0,
for some constant C 3 ;
(G3) For any l ≥ 0, there exists a constant C 4 such that
R N |x| l |∇(e t∆ ϕ)(x)|dx ≤ C 4 t − 1 2 R N |x| l |ϕ(x)|dx + C 4 t l−1 2 R N |ϕ(x)|dx, t > 0.
This inequality implies that
|||∇(e t∆ ϕ)||| l ≤ C 5 t − 1 2 |||ϕ||| l + C 5 t − 1 2 (1 + t l 2 ) ϕ 1 , t > 0,
for some constant C 5 .
Moreover we give one lemma on e t∆ ϕ. See [16, Lemmas 2.2 and 2.5].
Lemma 2.1 Let ϕ ∈ L 1 k with k ≥ 0 and assume
R N x α ϕ(x)dx = 0, |α| ≤ m,
for some integer m ∈ {0, . . . , [k]}. Then there holds the following:
(i) If 0 ≤ m ≤ [k] − 1, for any l ∈ [0, k − m − 1], there exists a constant C 1 such that R N |x| l (e t∆ ϕ)(x) dx ≤ C 1 t − m+1 2 R N |x| m+l+1 |ϕ(x)|dx + t l 2 R N |x| m+1 |ϕ(x)|dx , t > 0; (ii) If m = [k], for any l ∈ [0, k − [k]], there exists a constant C 2 such that R N |x| l (e t∆ ϕ)(x) dx ≤ C 2 t − k−l 2 R N |x| k |ϕ(x)|dx
for all t > 0. In particular, if k = [k], then lim t→∞ t k 2 e t∆ ϕ 1 = 0.
Next we recall the following two lemmas on the operator P k (t
Lemma 2.2 Let K ≥ 0 and f be a measurable function in R N × (0, ∞) such that f (t) ∈ L 1 K
for all t > 0. Then there holds the following: (i) Assume that there exist constants β ≥ 0 and γ ≥ 0 such that
sup t>0 (1 + t) − l 2 +γ t β |||f (t)||| l < ∞
for all l ∈ [0, K]. Then, for any multi-index α with |α| ≤ [K], there exists a constant C 1 such that
|M α (f (t), t)| ≤ C 1 (1 + t) |α| 2 −γ t −β , t > 0. Furthermore sup t>0 t N 2 (1− 1 q )+γ+β P [K] (t)f (t) − f (t) q + (1 + t) − l 2 +γ t β |||P [K] (t)f (t)||| l < ∞
for any l ∈ [0, K] and q ∈ [1, ∞];
(ii) If there exist constants β ′ ≥ 0 and γ ′ ≥ 0 such that
sup t>0 t N 2 (1− 1 q )+γ ′ +β ′ f (t) q + (1 + t) − l 2 +γ ′ t β ′ |||f (t)||| l < ∞ for all l ∈ [0, K] and q ∈ [1, ∞], then t N 2 (1− 1 q )+ j 2 ∇ j t 0 e (t−s)∆ P [K] (s)f (s)ds q t − K 2 t 0 (1 + s) K 2 −γ ′ s −β ′ ds, t > 0,
for any q ∈ [1, ∞] and j = 0, 1.
Lemma 2.3 Let k ≥ 0 and f = f (x, t) ∈ C(R N × (0, ∞)) ∩ L ∞ (R N × (0, ∞)) such that sup 0<τ <t |||f (τ )||| k < ∞ for all t > 0.
Let u be a solution of the Cauchy problem
∂ t u = ∆u + f in R N × (0, ∞), u(x, 0) = ϕ(x) in R N ,
where ϕ ∈ L 1 k . Then there holds the following:
(i) For any i ∈ {0, · · · , [k]}, the function v = [P i (t)u(t)](x) satisfies ∂ t v = ∆v + P i (t)f (t) in R N × (0, ∞); (ii) For any multi-index α with |α| ≤ [k], M α (u(t), t) − M α (u(s), s) = t s M α (f (τ ), τ )dτ for all t > s ≥ 0. In particular, if f ≡ 0, M α (u(t), t) = M α (ϕ, 0), |α| ≤ [k], t > 0.
Next we give one lemma on the volume potential. Let T > 0 and H ∈ L ∞ (0, T : L ∞ (R N )). Let w be the the volume potential of H defined by
(2.8) w(x, t) := t 0 R N G(x − ξ, t − τ )H(ξ, τ )dξdτ, t ∈ (0, T ).
Then we have:
Lemma 2.4 Let T > 0 and H ∈ L ∞ (0, T : L ∞ (R N ))
. Then w and ∇ x w are continuous functions in R N × (0, T ) and
(2.9) (∇ x w)(x, t) = t 0 R N (∇ x G)(x − ξ, t − τ )H(ξ, τ )dξdτ
holds for all (x, t) ∈ R N × (0, T ). Furthermore there exists a constant C 1 such that
(2.10) sup 0<t<T w(t) ∞ + sup 0<t<T (∇ x w)(t) ∞ ≤ C 1 H L ∞ (0,T :L ∞ (R N )) .
In addition, for any ν ∈ (0, 1) and |α| ≤ 1, there exists a constant C 2 such that
(2.11) |∂ α x w(x, t) − ∂ α x w(y, s)| |x − y| ν + |t − s| ν/2 ≤ C 2 H L ∞ (0,T :L ∞ (R N )) for all (x, t), (y, s) ∈ R N × (0, T ) with (x, t) = (y, s).
Lemma 2.4 is proved by the same argument as in [9,Chapter 1]. We give the proof in Section 7 for completeness of this paper.
At the end of this section we recall one lemma on an integral inequality. See [18,Lemma 2.4].
Lemma 2.5 Let ζ be a nonnegative function in (0, ∞) such that sup 0<t<1 ζ(t) < ∞. Let A > 1 and σ > 0. If, for any δ > 0, there holds
ζ(2t) ≤ (1 + δ)ζ(t) + C 1 2t t s −A ζ(s)ds + C 1 t σ , t ≥ 1/2,
for some constant C 1 , then there exists a constant C 2 such that ζ(t) ≤ C 2 t σ for all t ≥ 1.
Large time behavior of solutions
Consider the Cauchy problem
(3.1) ∂ t u = ∆u + f (x, t, u, ∇u) in R N × (0, ∞), u(x, 0) = ϕ(x) in R N ,
where f ∈ C(R N × (0, ∞) × R × R N ) and ϕ ∈ L 1 K for some K ≥ 0. In this section we assume that there exist constants C > 0 and A > 1 such that
(3.2) |f (x, t, p, q)| ≤ C(1 + t) −A (|p| + (1 + t) 1/2 |q|)
for all (x, t, p, q) ∈ R N × (0, ∞) × R × R N , and prove the following theorem, which ensures the well-definedness of P [K] (t)u(t) and P [K] (t)F (·, t, u(t), ∇u(t)) for the solution u of (1.1) in Section 4.
Theorem 3.1 Assume ϕ ∈ L 1 K for some K ≥ 0 and condition (3.2). Then there exists a solution u of (3.1) with the following properties:
(i) u, ∇u ∈ C(R N × (0, ∞)); (ii) For any q ∈ [1, ∞] and l ∈ [0, K], there hold sup 0<t<∞ t N 2 (1− 1 q ) u(t) q + t 1 2 (∇ x u)(t) q < ∞, (3.3) sup 0<t<∞ (1 + t) − l 2 |||u(t)||| l + t 1 2 |||(∇ x u)(t)||| l < ∞; (3.4) (iii) There exists a limit M := lim t→∞ R N u(x, t)dx = R N ϕ(x)dx + ∞ 0 R N f (x, t, u, ∇u)dxdt such that (3.5) lim t→∞ t N 2 (1− 1 q )+ j 2 ∇ j [u(t) − M g(t)
] q = 0 for any q ∈ [1, ∞] and j = 0, 1.
In order to prove Theorem 3.1, we first construct approximate solutions of (3.1), and prove the following lemma.
sup 0<t≤T t N 2 (1− 1 q ) u(t) q + t 1 2 (∇ x u)(t) q < ∞, (3.6) sup 0<t≤T |||u(t)||| l + t 1 2 |||(∇ x u)(t)||| l < ∞, (3.7)
for any T > 0, q ∈ [1, ∞], and l ∈ [0, K].
Proof. Let q ∈ [1, ∞] and ϕ ∈ L 1 (R N ). Put
(3.8) u 1 (x, t) := (e t∆ ϕ)(x), u n+1 (x, t) := (e t∆ ϕ)(x) + t 0 e (t−s)∆ f n (s)ds,
for (x, t) ∈ R N × (0, ∞), where n = 1, 2, . . . and f n (y, s) := f (y, s, u n (y, s), (∇u n )(y, s)). Let c 0 and c 1 be the constants given in (G1) and put C :
= c 0 + c 1 + 2 N+1 2 c 0 c 1 . By (G1) we have sup 0<t<∞ t N 2 (1− 1 q ) [ u 1 (t) q + t 1 2 ∇u 1 (t) q ] (3.9) = sup 0<t<∞ t N 2 (1− 1 q ) [ e t∆ ϕ q + t 1 2 ∇e t∆ ϕ q ] ≤ (c 0 + c 1 ) ϕ 1 ≤ C ϕ 1 .
This together with (3.2) implies that
(3.10) sup 0<t≤T t N 2 (1− 1 q )+ 1 2 f 1 (t) q ≤ CC 1 (1 + T ) 1 2 ϕ 1 , T > 0,
for some constant C 1 . By (G1) and (3.10) we have
t 0 e (t−s)∆ f 1 (s)ds q ≤ t/2 0 e (t−s)∆ f 1 (s) q ds + t t/2 e (t−s)∆ f 1 (s) q ds (3.11) ≤ c 0 t/2 0 (t − s) − N 2 (1− 1 q ) f 1 (s) 1 ds + t t/2 f 1 (s) q ds ≤ CC 2 (1 + T ) 1 2 t − N 2 (1− 1 q )+ 1 2 ϕ 1
for all t ∈ (0, T ) and T > 0, where C 2 is a constant. Then, by (G1), (3.8), and (3.11) we have
(3.12) sup 0<t≤T t N 2 (1− 1 q ) u 2 (t) q ≤ c 0 ϕ 1 + CC 2 (1 + T ) 1 2 T 1 2 ϕ 1 , T > 0. Furthermore, since (3.13) u 2 (x, t) = [e (t/2)∆ u 2 (t/2)](x) + t t/2 e (t−s)∆ f 1 (s)ds, (x, t) ∈ R N × (0, ∞),
applying (3.10) and (3.12) to (3.13), by (G1) we obtain
∇u 2 (t) q ≤ ∇e (t/2)∆ u 2 (t/2) q + t t/2 ∇e (t−s)∆ f 1 (s) q ds (3.14) ≤ c 1 (t/2) − 1 2 u 2 (t/2) q + c 1 t t/2 (t − s) − 1 2 f 1 (s) q ds ≤ c 0 c 1 (t/2) − N 2 (1− 1 q )− 1 2 ϕ 1 + CC 3 (1 + T ) 1 2 T 1 2 t − N 2 (1− 1 q )− 1 2 ϕ 1
for all t ∈ (0, T ) and T > 0, where C 3 is a constant. Therefore, by (3.12) and (3.14) we have
sup 0<t≤T t N 2 (1− 1 q ) [ u 2 (t) q + t 1 2 ∇u 2 (t) q ] (3.15) ≤ C ϕ 1 + C(C 2 + C 3 )(1 + T ) 1 2 T 1 2 ϕ 1 ≤ C ϕ 1 + CC T ϕ 1 , T > 0, where C T := (C 2 + C 3 )T 1 2 (1 + T ) 1 2 .
Furthermore we apply the same argument as in (3.15) to obtain
sup 0<t≤T t N 2 (1− 1 q ) [ u 3 (t) q + t 1 2 ∇u 3 (t) q ] ≤ C ϕ 1 + CC T (1 + C T ) ϕ 1 ≤ C(1 + C T + C 2 T ) ϕ 1 , T > 0.
Repeating the argument above, for any n = 1, 2, . . . , we have
(3.16) sup 0<t≤T t N 2 (1− 1 q ) [ u n (t) q + t 1 2 (∇ x u n )(t) q ] ≤ C(1 + C T + · · · + C n−1 T ) ϕ 1 and (3.17) u n+1 (x, t) = [e (t−T )∆ u n+1 (T )](x) + t T e (t−s)∆ f n (s)ds for all (x, t) ∈ R N × (T, ∞) and all T > 0. Let T 1 be a positive constant such that C T 1 ≤ 2 −1 . By (3.16) we have (3.18) sup 0<t≤T 1 t N 2 (1− 1 q ) [ u n (t) q + t 1 2 (∇ x u n )(t) q ] ≤ 2C ϕ 1 .
Applying the same argument as in the proof of (3.18) to (3.17) with T = T 1 /2, we have sup
T 1 /2<t≤3T 1 /2 (t − T 1 /2) N 2 (1− 1 q ) [ u n (t) ∞ + (t − T 1 /2) 1 2 (∇ x u n )(t) ∞ ] ≤ 2C u n (T 1 /2) 1 for n = 1, 2, . . . . This together with (3.18) implies that sup 0<t≤3T 1 /2 t N 2 (1− 1 q ) [ u n (t) q + t 1 2 (∇ x u n )(t) q ] ≤ C 4 ϕ 1
for some constant C 4 . Repeating this argument, for any T > 0, we can find a constant C 5 satisfying
(3.19) sup 0<t≤T t N 2 (1− 1 q ) [ u n (t) q + t 1 2 (∇ x u n )(t) q ] ≤ C 5 ϕ 1 , n = 1, 2, . . . .
This together with (3.2) implies that
(3.20) sup 0<t≤T t N 2 (1− 1 q )+ 1 2 f n (t) q ≤ C 6 , n = 1, 2, . . . .
for some constant C 6 . Next, by (3.20) we apply Lemma 2.4 and (G1) to (3.17), and we see that, for any ν ∈ (0, 1) and T > 0, there exists a constant C 7 , independent of n, such that
(3.21) |u n+1 (x, t) − u n+1 (y, s)| |x − y| ν + |t − s| ν/2 + |(∇ x u n+1 )(x, t) − (∇ x u n+1 )(y, s)| |x − y| ν + |t − s| ν/2 ≤ C 7
for all (x, t), (y, s) ∈ R N × (T /2, T ) with (x, t) = (y, s). Then, by (3.19) and (3.21), applying the Ascoli-Arzelà theorem and the diagonal argument to {u n } and taking a subsequence if necessary, we see that there exists a function u
∈ C ν,ν/2 (R N × (0, ∞)) such that ∇ x u ∈ C ν,ν/2 (R N × (0, ∞)) and (3.22) lim n→∞ u n (x, t) = u(x, t), lim n→∞ (∇u n )(x, t) = (∇ x u)(x, t)
uniformly on any compact set in R N × (0, ∞). Furthermore, by (3.2), (3.19), and (3.20) we have
(3.23) sup 0<t≤T t N 2 (1− 1 q ) [ u(t) q + t 1 2 (∇ x u)(t) q ] < ∞, sup 0<t≤T t N 2 (1− 1 q )+ 1 2 f (t) q < ∞, for any T > 0, where f (x, t) = f (x, t, u, ∇u).
In addition, we have
(3.24) u(x, t) = [e (t−T )∆ u(T )](x) + t T e (t−s)∆ f (s)ds for all (x, t) ∈ R N × (T, ∞)
and T > 0. This together with (3.23) implies that u is a solution of (3.1). It remains to prove (3.7). Put
w n (t) = |||u n (t)||| K + t 1 2 |||∇u n (t)||| K .
Then, applying (G2) and (G3) to (3.8), we have
(3.25) sup 0<t<1 w 1 (t) ≤ C ′ 1 w 1 (0) = C ′ 1 |||ϕ||| K < ∞ for some constant C ′ 1 . Furthermore, by (3.8) we have w 2 (t) ≤ R N (1 + |x|) K (|e t∆ ϕ| + t 1 2 |∇e t∆ ϕ|)dx (3.26) + t 0 R N (1 + |x|) K e (t−s)∆ f 1 (s) dx ds +t 1 2 t 0 R N (1 + |x|) K ∇e (t−s)∆ f 1 (s) dx ds =: I 1 (t) + I 2 (t) + I 3 (t)
for all t > 0. Let T 2 be a sufficiently small constant to be chosen later such that 0 < T 2 < 1.
Then, since I 1 (t) = w 1 (t), by (3.25) we have (3.27) sup 0<t≤T 2 I 1 (t) ≤ C ′ 1 |||ϕ||| K .
On the other hand, by (G2), (3.2), and (3.27) we have
I 2 (t) ≤ C ′ 2 t 0 (1 + (t − s) K 2 ) [|||f 1 (s)||| K + f 1 (s) 1 ] ds (3.28) ≤ C ′ 3 t 0 s − 1 2 w 1 (s)ds ≤ C ′ 1 C ′ 4 T 1/2 |||ϕ||| K for all 0 < t ≤ T < 1, where C ′ 2 , C ′ 3 ,I 3 (t) ≤ C ′ 5 T 1/2 t 0 (t − s) − 1 2 (1 + (t − s) K 2 ) [|||f 1 (s)||| K + f 1 (s) 1 ] ds (3.29) ≤ C ′ 6 T 1 2 t 0 (t − s) − 1 2 s − 1 2 w 1 (s)ds ≤ C ′ 1 C ′ 7 T 1 2 |||ϕ||| K for all 0 < t ≤ T < 1, where C ′ 5 , C ′ 6 , and C ′ 7 are constants. By (3.26)-(3.29), taking a sufficiently small T 2 > 0 so that (C ′ 4 + C ′ 7 )T 1/2 2 ≤ 2 −1 , we have sup 0<t≤T 2 w 2 (t) ≤ C ′ 1 [1 + (C ′ 4 + C ′ 7 )T 1 2 2 ]|||ϕ||| K ≤ C ′ 1 (1 + 2 −1 )|||ϕ||| K .
Repeating the argument above, we have
(3.30) sup 0<t≤T 2 w n (t) ≤ C ′ 1 (1 + 2 −1 + · · · + 2 −(n−1) )|||ϕ||| K ≤ 2C ′ 1 |||ϕ||| K , n = 1, 2, . . . .
Furthermore, applying the same argument to (3.24)
with T = T 2 /2, by (3.30) we have sup T 2 /2<t≤3T 2 /2 [|||u n (t)||| K + (t − T 2 /2) 1/2 |||∇u n (t)||| K ] ≤ 2C ′ 1 |||u n (T 2 /2)||| K ≤ (2C ′ 1 ) 2 |||ϕ||| K for n = 1, 2, . . . . This together with (3.30) yields sup 0<t≤3T 2 /2 w n (t) ≤ sup 0<t≤T 2 w n (t) + sup T 2 <t≤3T 2 /2 w n (t) ≤ C ′ 8 |||ϕ||| K < ∞, n = 1, 2, . . . ,
for some constant C 8 . Repeating this argument, for any T > 0, we have
sup n≥1 sup 0<t≤T w n (t) < ∞. This together with (3.22) implies sup 0<t≤T |||u(t)||| K + t 1 2 |||(∇u)(t)||| K < ∞ for any T > 0.
Thus we obtain (3.7), and the proof of Lemma 3.1 is complete. ✷ Next we prove the following lemma.
(3.31) sup t>T ( u(t) q + t 1 2 ∇u(t) q ) < +∞
for any T > 0 and q ∈ [1, ∞].
Proof. We use the same notation as in the proof of Lemma 3.1. Let q ∈ [1, ∞]. By (3.2) we have
(3.32) f (t) q ≤ C 1 t −A ( u(t) q + t 1 2 ∇u(t) q )
for all t ≥ 1, where C 1 is a constant. Let T 1 be a constant to be chosen later such that T 1 > 1. By (G1), (3.24), and (3.32) we have
u(t) q ≤ u(T 1 ) q + t T 1 f (s) q ds ≤ u(T 1 ) q + C 1 t T 1 s −A u(s) q + s 1 2 ∇u(s) q ds, t ≥ T 1 .
This inequality together with A > 1 implies that
(3.33) u(t) q ≤ u(T 1 ) q + C 2 T −A+1 1 sup T 1 ≤s≤t u(s) q + s 1 2 ∇u(s) q for all t ≥ T 1 , where C 2 is a constant. On the other hand, since t 1 2 t T 1 (t − s) − 1 2 s −A ds = t 1 2 t/2 T 1 (t − s) − 1 2 s −A ds + t t/2 (t − s) − 1 2 s −A ds ≤ t 1 2 t 2 − 1 2 t/2 T 1 s −A ds + t 2 −A t t/2 (t − s) − 1 2 ds T −A+1 1
for all t ≥ 2T 1 , by (G1), (3.24), and (3.32) we have
t 1 2 ∇u(t) q ≤ c 1 t 1 2 (t − T 1 ) − 1 2 u(T 1 ) q + c 1 t 1 2 t T 1 (t − s) − 1 2 f (s) q ds (3.34) ≤ C 3 u(T 1 ) q + C 1 c 1 t 1 2 t T 1 (t − s) − 1 2 s −A u(s) q + s for all t ≥ 2T 1 . Then, taking a sufficiently large T 1 so that 2C 2 T −A+1 1 ≤ 1/2 if necessary, we can find a constant C 5 satisfying sup 2T 1 ≤s<∞ u(s) q ≤ C 5 u(T 1 ) q + C 5 sup T 1 ≤s≤2T 1 u(s) q + s 1 2 ∇u(s) q < ∞.
This inequality together with (3.6) implies that Proof of Theorem 3.1. Let ϕ ∈ L 1 K with K ≥ 0. Let u be a solution of (3.1) given in Lemma 3.1. We first prove (3.3). Let q ∈ [1, ∞] and assume
(3.37) sup t>1 t γ u(t) q + t 1 2 ∇u(t) q < ∞ for some γ ≥ 0. Applying (G1), (3.31), (3.32), and (3.37) to inequality (3.24) with T = t/2, we obtain u(t) q ≤ e (t/2)∆ u(t/2) q + t t/2 f (s) q ds (3.38) t − N 2 (1− 1 q ) u(t/2) 1 + t t/2 s −A ( u(s) q + s 1 2 ∇u(s) q )ds t − N 2 (1− 1 q ) + t −γ−A+1 for all t ≥ 2. Similarly we have t 1 2 ∇u(t) q ≤ t 1 2 ∇e (t/2)∆ u(t/2) q + t 1 2 t t/2 ∇e (t−s)∆ f (s) q ds (3.39) t − N 2 (1− 1 q ) u(t/2) 1 + t 1 2 t t/2 (t − s) − 1 2 s −A ( u(s) q + s 1 2 ∇u(s) q )ds t − N 2 (1− 1 q ) + t −γ−A+1
for all t ≥ 2. Then, under assumption (3.37), by (3.31), (3.38), and (3.39) we have
sup t>1 t κ u(t) q + t 1 2 ∇u(t) q < ∞, where κ = min γ + A − 1, N 2 1 − 1 q .
Since (3.37) holds with γ = 0 by Lemma 3.2, applying the argument above several times, we obtain (3.37) with γ = (N/2)(1 − 1/q). This together with (3.6) implies (3.3).
Next we prove (3.4). For any l ∈ [0, K], we put
U l (t) := R N |x| l |u(x, t)| + t 1 2 |(∇ x u)(x, t)| dx.
Let T be a sufficiently large constant to be chosen later such that T ≥ 1. By (3.24) we have
U l (t) ≤ R N |x| l (|e (t−T )∆ u(T )| + t 1 2 |∇e (t−T )∆ u(T )|)dx (3.40) + t T R N |x| l e (t−s)∆ f (s) + t 1 2 ∇e (t−s)∆ f (s) dx ds =: I 1 (t) + I 2 (t)
for all t > T . By (G2), (G3), and Lemma 3.1 we have
I 1 (t) R N |x| l |u(x, T )|dx + (t − T ) l 2 u(T ) 1 (3.41) +t 1 2 (t − T ) − 1 2 R N |x| l |u(x, T )|dx + (t − T ) l−1 2 u(T ) 1 t l 2
for all t > 2T . Similarly, by (G2), (G3), (3.7), (3.32), and Lemma 3.2 we obtain
I 2 (t) t T R N (|y| l + (t − s) l 2 )|f (y, s)|dyds (3.42) +t 1 2 t T R N (|y| l (t − s) − 1 2 + (t − s) l−1 2 )|f (y, s)|dyds t T R N (|y| l + (t − s) l 2 )s −A (|u(y, s)| + s 1 2 |∇u(y, s)|)dyds +t 1 2 t T R N (|y| l (t − s) − 1 2 + (t − s) l−1 2 )s −A (|u(y, s)| + s 1 2 |∇u(y, s)|)dyds sup T <s<t s − l 2 U l (s) t T s −A+ l 2 ds + t T s −A (t − s) l 2 ds +t 1 2 sup T <s<t s − l 2 U l (s) t T s −A+ l 2 (t − s) − 1 2 ds + t 1 2 t T s −A (t − s) l−1 2 ds T −A+1 t l 2 sup T <s<t s − l 2 U l (s) + t l 2
for all t > 2T . By (3.40)-(3.42) we see that there exists a constant C 1 such that
sup 2T <s<t s − l 2 U l (s) ≤ C 1 T −A+1 sup T <s<t s − l 2 U l (s) + C 1
for all t > 2T ≥ 2. Then, taking a sufficiently large T so that
C 1 T −A+1 ≤ 1/2 if necessary, we have sup 2T <s<∞ s − l 2 U l (s) ≤ 2 sup T <s≤2T s − l 2 U l (s) + 2C 1 .
This together with (3.7) implies (3.4). It remains to prove (3.5). Let j = 0, 1. For any q ∈ [1, ∞], by (3.2) and (3.3) we have
(3.43) sup t>0 (1 + t) A− 1 2 t N 2 (1− 1 q )+ 1 2 f (t) q < ∞.
Then, by (2.3) and (3.43) we apply Lemma 2.2 (i) and Lemma 2.3 (ii) to obtain
|M 0 (u(t), t) − M 0 (u(t 0 ), t 0 )| = t t 0 M 0 (f (s), s)ds t t 0 (1 + s) −A+ 1 2 s − 1 2 ds
for all t ≥ t 0 ≥ 0. This together with A > 1 implies that there exists a constant M such that
(3.44) |M 0 (u(t), t) − M | = O(t −(A−1) )
as t → ∞. Then, by (2.7) and (3.44) we obtain
(3.45) lim t→∞ t N 2 (1− 1 q )+ j 2 ∇ j [M 0 (u(t), t)g(t) − M g(t)] q = 0 for any q ∈ [1, ∞]. Let (3.46) R(x, t) := u(x, t) − M 0 (u(t), t)g(t) = u(x, t)− R N u(x, t)dx g(x, t).
By Lemma 2.3 we see that
∂ t R = ∆R +f in R N × (0, ∞), where (3.47)f (x, t) := [P 0 (t)f (t)](x) = f (x, t)− R N f (x, t)dx g(x, t).
This implies that
∇ j R(t) = ∇ j e t∆ R(0) + ∇ j t 0 e (t−s)∆f (s)ds = ∇ j e t∆ R(0)+ t t/2 + t/2 L + L 0 ∇ j e (t−s)∆f (s)ds =: ∇ j e t∆ R(0) + J 1 (t) + J 2 (t) + J 3 (t) for t ≥ 2L, where L > 0. Since R N R(x, 0)dx = 0,(1 + t) A− 1 2 t N 2 (1− 1 q )+ 1 2 f (t) q < ∞, by (G1) we have t N 2 (1− 1 q )+ j 2 J 1 (t) q t N 2 (1− 1 q )+ j 2 t t/2 (t − s) − j 2 f (s) q ds (3.50) t j 2 −A t t/2 (t − s) − j 2 ds t −A+1 = o(1)
as t → ∞. Furthermore, by (G1) and (3.49) we have
t N 2 (1− 1 q )+ j 2 J 2 (t) q (3.51) ≤ t N 2 (1− 1 q )+ j 2 t/2 L ∇ j e (t−s)∆f (s) q ds t/2 L f (s) 1 ds t/2 L s −A ds L −A+1
for all sufficiently large t. Similarly, by (G3) we have
t N 2 (1− 1 q )+ j 2 J 3 (t) q (3.52) ≤ t N 2 (1− 1 q )+ j 2 L 0 ∇ j e (t−s) 2 ∆ e (t−1 = 0, (3.53) e (t−s) 2 ∆f (s) 1 ≤ f (s) 1 < ∞, t ≥ 2L,(1− 1 q )+ j 2 ∇ j R(t) q ≤ C 2 L −A+1
for some constant C 2 . Therefore, since L is arbitrary, by A > 1 we have
lim t→∞ t N 2 (1− 1 q )+ j 2 ∇ j R(t) q = 0.
This together with (3.45) and (3.46) yields (3.5), and Theorem 3.1 follows. ✷ By an argument similar to the proof of Theorem 3.1 and with the aid of (1.6) we can obtain the following theorem.
∂ t u = ∆u + ∇ · F (x, t, u) in R N × (0, ∞), u(x, 0) = ϕ(x) in R N ,
where F ∈ C(R N × (0, ∞) × R : R N ) and ϕ ∈ L 1 K for some K ≥ 0. Assume that there exist constants C > 0 and A > 1 such that
|F (x, t, p)| ≤ C(1 + t) −A+1/2 |p|, (x, t, p) ∈ R N × (0, ∞) × R.
Then there exists a function u ∈ C(R N × (0, ∞)) with the following properties:
(i) For any q ∈ [1, ∞] and l ∈ [0, K],
sup 0<t<∞ t N 2 (1− 1 q ) u(t) q + sup 0<t<∞ (1 + t) − l 2 |||u(t)||| l < ∞; (ii) u satisfies u(x, t) = e t∆ ϕ(x) + t 0 ∇ · e (t−s)∆ F (·, s, u(·, s))ds for almost all (x, t) ∈ R N × (0, ∞); (iii) There holds lim t→∞ t N 2 (1− 1 q ) u(t) − M g(t) q = 0, q ∈ [1, ∞], where M = R N ϕ(x)dx. Remark 3.1 Assume ϕ ∈ L ∞ (R N ) ∩ L 1 K for some K ≥ 0.
Let u be the solution of (3.1), given in Theorem 3.1. Then, by an argument similar to the proof of Lemma 3.1 we have
sup 0<t≤T u(t) ∞ + t 1 2 (∇ x u)(t) ∞ < ∞
for any T > 0. This together with assertion (i) of Theorem 3.1 implies that
sup 0<t<∞ (1 + t) N 2 (1− 1 q ) u(t) q + t 1 2 (∇ x u)(t) q < ∞
for any q ∈ [1, ∞]. This also holds for the solution of (3.56), given in Theorem 3.2.
Main Theorems
In this section we state the main results of this paper, and give the higher order asymptotic expansions of the solution u of Cauchy problem (1.1).
Let u be a solution of Cauchy problem (1.1) with ϕ ∈ L 1 K for some K ≥ 0. Assume that the solution u satisfies (3.3), (3.4) and condition (C A ) for some A > 1. Put
F (x, t) := F (x, t, u(x, t), ∇u(x, t))
for simplicity. Then, by (3.4), for any multi-index α with |α| ≤ [K], we can define M α (u(t), t) for all t ≥ 0 (see (2.3)). Furthermore, by (C A ), (3.3), and (3.4) we have
F (t) q (1 + t) −A u(t) q + (1 + t) 1 2 ∇ x u(t) q (1 + t) −A+ 1 2 t − N 2 (1− 1 q )− 1 2 , (4.1) |||F (t)||| l (1 + t) −A |||u(t)||| l + (1 + t) 1 2 |||∇ x u(t)||| l (1 + t) l+1 2 −A t − 1 2 , (4.2)
for all t > 0, where q ∈ [1, ∞] and l ∈ [0, K]. Therefore, applying Lemma 2.2 (i) and Lemma 2.3 (ii), we obtain
(4.3) |M α (u(t), t) − M α (u(t 0 ), t 0 )| = t t 0 M α (F (s), s)ds t t 0 (1 + s) −A+ |α|+1 2 s − 1 2 ds
for all t ≥ t 0 ≥ 0. This implies the following: (i) For any multi-index α with |α| ≤ [K], if A > 1 + |α|/2, there exists a constant M α such that
(4.4) |M α (u(t), t) − M α | (1 + t) −(A−1)+|α|/2 for all t > 0; (ii) For any multi-index α with |α| ≤ [K], if 1 < A ≤ 1 + |α|/2, then (4.5) M α (u(t), t) = O(t −(A−1)+|α|/2 ) if A < 1 + |α|/2, O(log t) if A = 1 + |α|/2,
as t → ∞. Now, following [18], we introduce the function U n = U n (x, t) defined inductively by t)). In particular, since e (t−s)∆ g α (s) = g α (t) for t > s ≥ 0, by (2.2) and (4.6) we have
(4.6) U 0 (x, t) := |α|≤[K] M α (u(t), t)g α (x, t), U n (x, t) := U 0 (x, t) + t 0 e (t−s)∆ P [K] (s)F n−1 (s)ds, n = 1, 2, . . . , where F n−1 (x, t) = F (x, t, U n−1 (x, t), (∇ x U n−1 )(x,U n (x, t) = |α|≤[K] M α (u(t), t)g α (x, t) + t 0 e (t−s)∆ F n−1 (s) − |α|≤[K] M α (F n−1 (s), s)g α (s) ds = |α|≤[K] M α (u(t), t) − t 0 M α (F n−1 (s), s)ds g α (x, t) + t 0 e (t−s)∆ F n−1 (s)ds.
Now we are ready to state the main theorems of this paper. (i) The function U n defined by (4.6) satisfies (ii) For any q ∈ [1, ∞] and j = 0, 1,
sup t>0 t N 2 (1− 1 q ) U n (t) q + t 1 2 ∇ x U n (t) q < ∞, (4.7) sup t>0 (1 + t) − l 2 |||U n (t)||| l + t 1 2 |||∇ x U n (t)||| l < ∞,(4.9) t N 2 (1− 1 q )+ j 2 ∇ j u(t) − U n (t) q (1 + t) − K 2 + (1 + t) −(n+1)(A−1) if 2(n + 1)(A − 1) = K, (1 + t) − K 2 log(2 + t) if 2(n + 1)(A − 1) = K,
for all t > 0; (iii) If 2(n + 1)(A − 1) > K, then, for any q ∈ [1, ∞] and j = 0, 1,
(4.10) t N 2 (1− 1 q )+ j 2 ∇ j u(t) − U n (t) q = o(t − K 2 ) if K = [K], O(t − K 2 ) if K > [K],
as t → ∞;
(iv) For any l ∈ [0, K], σ > 0, and j = 0, 1,
(4.11) t j 2 (1 + t) − l 2 ∇ j u(t) − U n (t) l (1 + t) − K 2 +σ + (1 + t) −(n+1)(A−1)
for all t > 0.
We remark that:
• U n (n = 1, 2, . . . ) gives the ([K] + 2)-th order asymptotic expansion of the solution u and is determined systematically by the function U 0 ;
• If 2(n + 1)(A − 1) > K, then the decay estimate of u(t) − U n (t) q as t → ∞ in (4.10) is the same as in (2.5);
• U 0 is represented as a linear combination of
U J (x, t) := 0≤|α|<J A M α g α (x, t) if J ≥ 1, M g(x, t)
if J = 0, and we write F (U J (x, t)) = F (x, t, U J (x, t), ∇U J (x, t)) for simplicity.
M α (u(t), t)g α (x, t) + t 0 e (t−s)∆ P [K] (s)F (U J (s))ds.
Then, for any q ∈ [1, ∞] and j = 0, 1,
(4.13) t N 2 (1− 1 q )+ j 2 ∇ j [u(t) −ũ(t)] q = O(t −2(A−1) ) if K > 4(A − 1), O(t − K 2 log t) if K = 4(A − 1), O(t − K 2 ) if K < 4(A − 1), K = [K], o(t − K 2 ) if K < 4(A − 1), K = [K],
as t → ∞.
Furthermore, as a corollary of Theorem 4.2, we have:
M α (u(t), t)g α (x, t) + t 0 e (t−s)∆ P [K] (s)F M (s)ds (4.14) = M − ∞ 0 R N F M (x, t)dxdt g(x, t) + |α|≤[K] c α (t)g α (x, t)c 0 (t) := ∞ t R N [F M (x, s) − F (x, s)]dxds, c α (t) := M α (u(t), t) − t 0 M α (F M (s), s)ds if 1 ≤ |α| ≤ [K].
Then (4.13) holds withũ replaced byû.
Proof of Main Theorems
In this section we prove Theorems 4.1, 4.2, and Corollary 4.1. We first prove assertions (i), (ii), and (iv) of Theorem 4.1.
Proof of assertions (i), (ii), and (iv). By (3.4) we apply Lemma 2.2 (i) with β = γ = 0 to the function U 0 (see (4.6)), and obtain
|∇ j U 0 (x, t)| ≤ |α|≤[K]
|M α (u(t), t)||∇ j g α (x, t)|
|α|≤[K]
(1 + t) |α| 2 |∇ j g α (x, t)| for all (x, t) ∈ R N × (0, ∞) and j = 0, 1. This inequality together with (2.7) implies (4.7) and (4.8) for the case n = 0, and assertion (i) follows for the case n = 0. Let n = −1, 0, 1, 2, . . . and j = 0, 1. We assume, without loss of generality, that σ ∈ (0, A − 1). Put
σ n = σ if 2n(A − 1) ≥ K, (K/2) − n(A − 1) if 2n(A − 1) < K, γ n = A + K 2 − σ n .
Let U −1 ≡ 0 and F −1 ≡ 0 in R N × (0, ∞). Then (4.6) holds for n = 0, 1, 2, . . . . Furthermore, since the solution u satisfies (3.3)-(3.5), assertions (i), (ii), and (iv) hold with n = −1 and σ = σ 0 . We prove assertions (i), (ii), and (iv) under condition (F A ). Assume that there exists a number n * ∈ {−1, 0, 1, 2, · · · } such that assertions (i), (ii), and (iv) hold with n = n * and σ = σ n * +1 . We first prove assertion (i) for n = n * + 1. Since U n * ∈ S and 0 ∈ S, by (F A ) we have
|F n * (x, t)| = |F (x, t, U n * , ∇U n * ) − F (x, t, 0, 0)| (1 + t) −A (|U n * (x, t)| + (1 + t) 1/2 |∇U n * (x, t)|)
for all (x, t) ∈ R N × (0, ∞). Then, since assertion (i) holds with n = n * , we obtain
sup t>0 (1 + t) A− 1 2 t 1 2 t N 2 (1− 1 q ) F n * (t) q + (1 + t) − l 2 |||F n * (t)||| l < ∞
for any q ∈ [1, ∞] and l ∈ [0, K]. This together with Lemma 2.2 (i) implies that
(5.1) sup t>0 (1 + t) A− 1 2 t 1 2 t N 2 (1− 1 q ) P [K] (t)F n * (t) q + (1 + t) − l 2 |||P [K] (t)F n * (t)||| l < ∞
for any q ∈ [1, ∞] and l ∈ [0, K]. Therefore, since A > 1, by (G1), (4.6), (4.7) with n = 0, and (5.1) we have
∇ j U n * +1 (t) q ≤ ∇ j U 0 (t) q + ∇ j t 0 e (t−s)∆ P [K] (s)F n * (s)ds q t − N 2 (1− 1 q )− j 2 + t/2 0 (t − s) − N 2 (1− 1 q )− j 2 P [K] (s)F n * (s) 1 ds + t t/2 (t − s) − j 2 P [K] (s)F n * (s) q ds t − N 2 (1− 1 q )− j 2 + t − N 2 (1− 1 q )− j 2 t/2 0 (1 + s) −A+ 1 2 s − 1 2 ds +t − N 2 (1− 1 q )− 1 2 (1 + t) −A+ 1 2 t t/2 (t − s) − j 2 ds t − N 2 (1− 1 q )− j 2
for all t > 0. Furthermore, by (G2), (G3), (4.6), (4.8) with n = 0, and (5.1) we have
|||∇ j U n * +1 (t)||| l ≤ |||∇ j U 0 (t)||| l + t 0 ∇ j e (t−s)∆ P [K] (s)F n * (s)ds l (5.2) t − j 2 (1 + t) l 2 + t 0 (t − s) j 2 |||P [K] (s)F n * (s)||| l ds + t 0 (t − s) − j 2 (1 + (t − s) l 2 ) P [K] (s)F n * (s) 1 ds t − j 2 (1 + t) l 2 + t/2 0 + t t/2 (t − s) − j 2 (1 + s) −A+ l+1 2 s − 1 2 ds + t/2 0 + t t/2 (t − s) − j 2 (1 + (t − s) l 2 )(1 + s) −A+ 1 2 s − 1 2 ds t − j 2 (1 + t) l 2
for all t > 0. These imply that assertion (i) holds with n = n * + 1. On the other hand, due to u ∈ S, by (F A ) we have
|F n * (x, t) − F (x, t)| (5.3) (1 + t) −A (|u(x, t) − U n * (x, t)| + (1 + t) 1/2 |∇u(x, t) − ∇U n * (x, t)|)
for all (x, t) ∈ R N × (0, ∞). Then, since assertions (ii) and (iv) hold with n = n * and σ = σ n * +1 , by (5.3) we obtain
sup t>0 t N 2 (1− 1 q )+(γ n * +1 − 1 2 )+ 1 2 F (t) − F n * (t) q (5.4) + sup t>0 (1 + t) − l 2 +(γ n * +1 − 1 2 ) t 1 2 |||F (t) − F n * (t)||| l < ∞
for any q ∈ [1, ∞] and l ∈ [0, K]. This together with Lemma 2.2 (i) implies that
sup t>0 t N 2 (1− 1 q )+(γ n * +1 − 1 2 )+ 1 2 P [K] (t)[F (t) − F n * (t)] q (5.5) + sup t>0 (1 + t) − l 2 +(γ n * +1 − 1 2 ) t 1 2 |||P [K] (t)[F (t) − F n * (t)]||| l < ∞ for any q ∈ [1, ∞] and l ∈ [0, K].
Next we prove that assertions (ii) and (iv) hold with n = n * + 1 and σ = σ n * +2 . Recall that the solution u satisfies (3.3) and (3.4). Then, due to assertion (i) with n = n * + 1, it suffices to prove that (4.9) and (4.11) hold with n = n * + 1 and σ = σ n * +2 for all sufficiently large t. Put z(t) := u(t) − U n * +1 (t). Then, by (2.2) and (4.6) we have
(5.6) z(x, t) = P [K] (t)u(t) − t 0 e (t−s)∆ P [K] (s)F n * (s)ds.
Then, by Lemma 2.3 (i) we obtain
∂ t z = ∆z + P [K] (t)[F (t) − F n * (t)] in R N × (0, ∞).
This implies that
(5.7) z(t) = e (t−t 0 )∆ z(t 0 ) + t t 0 e (t−s)∆ P [K] (s)[F (s) − F n * (s)]ds, t ≥ t 0 ≥ 0. Let q ∈ [1, ∞]. By (G1) we have (5.8) t N 2 (1− 1 q )+ j 2 ∇ j e t∆ z(0) q = t N 2 (1− 1 q )+ j 2 ∇ j e (t/2)∆ e (t/2)∆ z(0) q e (t/2)∆ z(0) 1 for all t > 0. Furthermore, it follows from (2.4) that R N x α z(x, 0)dx = R N x α P [K] (0)u(0)dx = 0, |α| ≤ K,
hence, we apply (5.8) and Lemma 2.1 (ii) to obtain
(5.9) t N 2 (1− 1 q )+ j 2 ∇ j e t∆ z(0) q t − K 2
for all t > 0. On the other hand, applying Lemma 2.2 (ii) with γ ′ = γ n * +1 − 1/2 and β ′ = 1/2 with the aid of (5.4), we obtain
t N 2 (1− 1 q )+ j 2 ∇ j t 0 e (t−s)∆ P [K] (s)[F (s) − F n * (s)]ds q (5.10) t − K 2 t 0 (1 + s) K 2 −γ n * +1 + 1 2 s − 1 2 ds = t − K 2 t 0 (1 + s) −A+σ n * +1 + 1 2 s − 1 2 ds t − K 2 + t − K 2 t 1 s −A+σ n * +1 ds = t − K 2 if 2(n * + 2)(A − 1) > K, t − K 2 log t if 2(n * + 2)(A − 1) = K, t −(n * +2)(A−1) if 2(n * + 2)(A − 1) < K,
for all sufficiently large t. Therefore we apply (5.9) and (5.10) to (5.7) with t 0 = 0, and obtain inequality (4.9) with n = n * + 1 for any sufficiently large t. Thus assertion (ii) holds with n = n * + 1.
On the other hand, for any l ∈ [0, K], we have
(1 + t) − l 2 |||∇ j z(t)||| l = R N 1 + |x| (1 + t) 1/2 l |∇ j z(t)|dx R N 1+ 1 + |x| (1 + t) 1/2 K |∇ j z(t)|dx = ∇ j z(t) 1 + (1 + t) − K 2 |||∇ j z(t)||| K
for all t > 0. Then, by (4.9) with q = 1 and n = n * + 1 we see that, if there holds (4.11) with l = K, then we have (4.11) for l ∈ [0, K]. Thus it suffices to prove (4.11) with l = K, n = n * + 1, and σ = σ n * +2 . Put Z j (t) = |||∇ j z(t)||| K . By (5.7) we have
(5.11) Z j (2t) ≤ |||∇ j e t∆ z(t)||| K + 2t t |||∇ j e (2t−s)∆ P [K] (s)[F (s) − F n * (s)]||| K ds
for all t > 0. Let δ > 0. Then, by (G2), (G3), and (4.9) with n = n * + 1 we have
|||e t∆ z(t)||| K ≤ (1 + δ)|||z(t)||| K + C 2 (1 + t K 2 ) z(t) 1 ≤ (1 + δ)Z 0 (t) + C 3 t σ n * +2 , (5.12) t 1 2 |||∇e t∆ z(t)||| K |||z(t)||| K + (1 + t K 2 ) z(t) 1 Z 0 (t) + t σ n * +2 , (5.13)
for all t ≥ 1/2, where C 2 and C 3 constants. Furthermore, by (G2), (G3), and (5.5) we have
2t t |||∇ j e (2t−s)∆ P [K] (s)[F (s) − F n * (s)]||| K ds (5.14) 2t t (2t − s) − j 2 |||P [K] (s)[F (s) − F n * (s)]||| K ds + 2t t (2t − s) − j 2 1 + (2t − s) K 2 P [K] (s)[F (s) − F n * (s)] 1 ds 2t t (2t − s) − j 2 (1 + s) K 2 −γ n * +1 ds + 2t t (2t − s) − j 2 1 + (2t − s) K 2 (1 + s) −γ n * +1 ds t − j 2 + K 2 −γ n * +1 +1 = t − j 2 −(A−1)+σ n * +1 t − j 2 +σ n * +2
for all t ≥ 1/2. Therefore, by (5.11), (5.12), and (5.14) we can find a constant C 4 satisfying (5.15) Z 0 (2t) ≤ (1 + δ)Z 0 (t) + C 4 t σ n * +2 , t ≥ 1/2.
Furthermore, since it follows from (3.4) and (4.8) with n = n * + 1 that sup 0<t<1 Z 0 (t) < ∞, we apply Lemma 2.5 to inequality (5.15), and obtain
(5.16) Z 0 (t) t σ n * +2
for all t ≥ 1. This together with (5.11), (5.13), and (5.14) implies that
(5.17) t 1 2 Z 1 (t) Z 0 (t) + t σ n * +2 t σ n * +2
for all t ≥ 1. By (5.16) and (5.17) we have inequality (4.11) with n = n * + 1, σ = σ n * +2 for any sufficiently large t. Therefore assertions (ii) and (iv) hold with n = n * + 1 for all t > 0. Thus, by induction we see that (4.8), (4.9) and (4.11) hold with σ = σ n+1 for all n = 0, 1, 2, . . . , and assertions (i), (ii), and (iv) of Theorem 4.1 follow under condition (F A ). Furthermore, for the case n = 0, since F −1 ≡ 0, the proof of (4.8), (4.9) and (4.11) with σ = σ 1 remains true without condition (F A ). Therefore we obtain assertions (i), (ii), and (iv) for the case n = 0 without condition (F A ), and the proof of assertions (i), (ii), and (iv) is complete. ✷
We complete the proof of Theorem 4.1. Then we can take a positive constant σ so that
(5.19) K 2 − n(A − 1) < σ < A − 1,
and put ǫ := A − 1 − σ > 0. By (4.7) we see U n ∈ S for n ∈ {−1, 0, 1, . . . }. By condition (F A ) and (5.19) we apply Theorem 4.1 (ii) and (iv) to obtain
t N 2 (1− 1 q ) F (t) − F n−1 (t) q + (1 + t) − l 2 |||F (t) − F n−1 (t)||| l (1 + t) −A j=0,1 (1 + t) j 2 t N 2 (1− 1 q ) ∇ j u(t) − U n−1 (t) q +(1 + t) − l 2 ∇ j u(t) − U n−1 (t) l t − 1 2 (1 + t) −A+ 1 2 [(1 + t) − K 2 +σ + (1 + t) −n(A−1) ] t − 1 2 (1 + t) −A+ 1 2 − K 2 +σ t − 1 2 (1 + t) − K 2 − 1 2 −ǫ for all t > 0, where q ∈ [1, ∞] and l ∈ [0, K]. Then, puttingF n−1 (t) = P K (t)[F (t) − F n−1 (t)], by Lemma 2.2 (i) we have (5.20) t N 2 (1− 1 q ) F n−1 (t) q + (1 + t) − l 2 |||F n−1 (t)||| l t − 1 2 (1 + t) − K 2 − 1 2 −ǫ
for all t > 0. Let j = 0, 1 and put z n (t) = u(t) − U n (t). By (5.7), for any L > 0, we have On the other hand, by (G1) and (5.20) we have
t N 2 (1− 1 q )+ j 2 I 1 (t) q ≤ t N 2 (1− 1 q )+ j 2 t t/2 (t − s) − j 2 F n−1 (s) q ds (5.23) t j 2 − K 2 −ǫ−1 t t/2 (t − s) − j 2 ds t − K 2 −ǫ = o(t − K 2 )
as t → ∞. Furthermore, by Lemma 2.1 (ii), (G1), (2.4), and (5.20) we have
t N 2 (1− 1 q )+ j 2 I 2 (t) q ≤ t N 2 (1− 1 q )+ j 2 t/2 L ∇ j e (t−s) 2 ∆ e (t−s) 2 ∆F n−1 (s) q ds (5.24) t/2 L e (t−s) 2 ∆F n−1 (s) 1 ds t/2 L (t − s) − K 2 |||F n−1 (s)||| K ds t − K 2 t/2 L s −1−ǫ ds t − K 2 L −ǫ
for all sufficiently large t. Similarly, by (G1) we have
t N 2 (1− 1 q )+ j 2 I 3 (t) q (5.25) ≤ t N 2 (1− 1 q )+ j 2 L 0 ∇ j e (t−s) 2 ∆ e (t−= lim t→∞ (t − s) K 2 e (t−s) 2 ∆F n−1 (s) 1 = 0, (5.26) e (t−s) 2 ∆F n−1 (s) 1 (t − s) − K 2 |||F n−1 (s)||| K t − K 2 s − 1 2 , t ≥ 2L, (5.27)
for all s ∈ (0, L). By (5.26) and (5.27) we apply the Lebesgue dominated convergence theorem to (5.25), and obtain
(5.28) t N 2 (1− 1 q )+ j 2 I 3 (t) q = o(t − K 2 )
as t → ∞. Therefore, by (5.21)-(5.24) and (5.28) we see that there exists a constant C 3 such that lim sup
t→∞ t N 2 (1− 1 q )+ K+j 2 ∇ j z n (t) q ≤ C 3 L −ǫ .
Then, since L is arbitrary, we have
lim t→∞ t N 2 (1− 1 q )+ K+j 2 ∇ j z n (t) q = 0.
Thus we have (4.10) for the case K = [K] under condition (F A ). Furthermore, similarly as in the proof of assertions (i), (ii), (iv), for the case n = 0, we have F −1 ≡ 0, and the proof of (4.10) with K = [K] remains true without condition (F A ). Therefore we have Proof of Theorem 4.2. Let K ≥ 0. By (2.7) and (4.4), for any q ∈ [1, ∞], l ∈ [0, K], and j = 0, 1, we have
sup t>0 t N 2 (1− 1 q )+γ+ j 2 ∇ j [U 0 (t) − U J (t)] q (5.29) +(1 + t) − l 2 +γ t j 2 |||∇ j [U 0 (t) − U J (t)]||| l < ∞, where γ = A − 1. Since |F 0 (t) − F (U J (t))| (1 + t) −A |U 0 (x, t) − U J (x, t)| + (1 + t) 1 2 |∇ [U 0 (x, t) − U J (x, t)] | in R N × (0, ∞), by (5.29) we have sup t>0 t N 2 (1− 1 q )+(A+γ− 1 2 )+ 1 2 F 0 (t) − F (U J (t)) q (5.30) + sup t>0 (1 + t) − l 2 +(A+γ− 1 2 ) t 1 2 |||F 0 (t) − F (U J (t))||| l < ∞
for any q ∈ [1, ∞] and l ∈ [0, K]. Then, by (5.30), applying Lemma 2.2 (ii) with γ ′ = A + γ − 1/2 and β ′ = 1/2, we obtain
t N 2 (1− 1 q )+ j 2 ∇ j t 0 e (t−s)∆ P [K] [F 0 (s) − F (U J (t))]ds q (5.31) t − K 2 t 0 (1 + s) K 2 −A−γ+ 1 2 s − 1 2 ds = O(t − K 2 ) + O(t −2(A−1) ) if K = 4(A − 1), O(t − K 2 log t) if K = 4(A − 1),
for all sufficiently large t. Furthermore, if K < 4(A − 1) and K = [K], then, by the same argument as in the proof of Theorem 4.1 (iii) with the aid of (5.30) we have
(5.32) t N 2 (1− 1 q )+ j 2 ∇ j t 0 e (t−s)∆ P [K] [F 0 (s) − F (U J (s))]ds q = o(t − K 2 )
for all sufficiently large t. Therefore, since
u(t) −ũ(t) = [u(t) − U 1 (t)] + t 0 e (t−s)∆ P [K] [F 0 (s) − F (U J (t))]ds,
by Theorem 4.1, (5.31), and (5.32) we have
t N 2 (1− 1 q )+ j 2 ∇ j [u(t) −ũ(t)] q = O(t −2(A−1) ) if K > 4(A − 1), O(t − K 2 log t) if K = 4(A − 1), O(t − K 2 ) if K < 4(A − 1), K = [K], o(t − K 2 ) if K < 4(A − 1), K = [K]
for all sufficiently large t. Thus we obtain (4.13), and Theorem 4.2 follows. ✷ Proof of Corollary 4.1 We apply Theorem 4.2 with J = 0. Then, sincẽ
u(x, t) = M − ∞ t R N F (s)dxds g(x, t) + 1≤|α|≤[K] M α (u(t), t)g α (x, t) + t 0 e (t−s)∆ F M (s)ds − g(x, t) t 0 R N F M (s)dxds − 1≤|α|≤[K] g α (x, t) t 0 M α (F M (s), s)ds = M − ∞ 0 R N F M (t)dxdt g(x, t) + t 0 e (t−s)∆ F M (s)ds + 1≤|α|≤[K] M α (u(t), t) − t 0 M α (F M (s), s)ds g α (x, t) − ∞ t R N F (s)dxds − ∞ t R N F M (s)dxds g(x, t) =û(x, t),
we see that (4.13) holds withũ replaced byû, and Corollary 4.1 follows. ✷
Applications to nonlinear parabolic equations
In this section we apply the main results of this paper, which are given in Section 4, to some selected nonlinear parabolic equations.
Convection-diffusion equation
Consider the Cauchy problem for the convection-diffusion equation
(6.1) ∂ t u = ∆u + a · ∇(|u| p−1 u) in R N × (0, ∞), u(x, 0) = ϕ(x) in R N ,
where N ≥ 1, a ∈ R N , p > 1, and ϕ ∈ L ∞ (R N ) ∩ L 1 K for some K ≥ 0. Then there exists a unique bounded solution u of (6.1), and the large time behavior of the solution u has been studied in several papers (see for example [1], [4], [5], [7], [8], [19], [35], and references therein). In particular, it is known that, if p > 1 + 1/N , then the solution u behaves like the Gauss kernel and (1.3) holds.
Let p > 1 + 1/N . Then we can easily see that conditions (C A ) and (F A ) hold with
A = A * := N 2 (p − 1) + 1 2 > 1.
Furthermore, by Theorem 3.1 and Remark 3.1 we see that the unique bounded solution u of (6.1) satisfies (3.3) and (3.4). These mean that all of the assertions in Section 4 hold for the solution u with A = A * . In particular, noticing that
M = R N ϕ(x)dx = R N u(x, t)dx for t > 0,
we have:
Theorem 6.1 Assume p > 1 + 1/N and ϕ ∈ L ∞ (R N ) ∩ L 1 K for some K ≥ 0. Let u be a bounded solution of (6.1) and A = A * . Then there holds (4.13) withũ replaced by
M g(x, t) + |M | p−1 M t 0 a · ∇e (t−s)∆ g(s) p ds + 1≤|α|≤[K] c α (t)g α (x, t).
Theorem 6.1 is a direct consequence of Corollary 4.1. We remark that, for the case K = 1, a result similar to Theorem 6.1 has been already obtained by Duro and Carpio in [4] (see also [35]). However, as far as we know, for the case K ∈ {0, 1}, there are no results corresponding to Theorem 6.1 for the convection-diffusion equation (6.1). We emphasize that the asymptotic expansion given in Theorem 6.1 is a simple modification of the function U 1 , and Theorem 4.1 can give the other higher order asymptotic expansions by the use of U n (n = 2, 3, . . . ).
Remark 6.1 Let 1 < p ≤ 1 + 1/N and M = 0. Then, since 0 < A * ≤ 1, we can not apply the arguments in this paper to problem (6.1). On the other hand, in this case, it is known that the solution of (6.1) does not behave like the Gauss kernel as t → ∞ (see for example [7], [8], and [19]), and we can not expect that the assertions of Theorem 6.1 hold.
The decay estimate between the solution and its asymptotic expansion can give the following theorem on the classification of the decay rate of L q -norm of the solution u.
∇ j u(t) q ≍ t − N 2 (1− 1 q )− d 2 − j 2
as t → ∞; or (ii) for any q ∈ [1, ∞] and j = 0, 1,
lim t→∞ t N 2 (1− 1 q )+ [K] 2 + j 2 ∇ j u(t) q = 0.
Keller-Segel System
Consider the Keller-Segel system of parabolic-parabolic type 4) where N ≥ 1 and
∂ t u = ∆u − ∇ · (u∇v) in R N × (0, ∞), (6.2) ∂ t v = ∆v − v + u in R N × (0, ∞), (6.3) u(x, 0) = ϕ(x), v(x, 0) = ψ(x) in R N ,(6.(6.5) ϕ, ψ, ∂ x ψ ∈ L 1 (R N ) ∩ B(R N ).
Here B(R N ) is the Banach space of all bounded and uniformly continuous functions on R N . Cauchy Problem (6.2)-(6.4) is a mathematical model describing the motion of some species due to chemotaxis (see [26]), and the asymptotics of solution (u, v) of (6.2)-(6.4) has been studied intensively in many papers, see for example [21], [22], [27], [28], [32], [33], and references therein. In particular, it is known that, for any L > 0, there exists a positive constant δ such that, if
ϕ ∞ ≤ L, ϕ 1 ≤ δ, ∇ψ 1 ≤ δ, ∇ψ ∞ ≤ δ,
then Cauchy problem (6.2)-(6.4) has a unique classical solution (u, v) satisfying
(6.6) sup t>0 ( u(t) p + v(t) p ) < ∞ for p ∈ {1, ∞}.
(See [27, Theorem 1.2].) Let (u, v) be a classical solution of (6.2)-(6.4) satisfying (6.6). Assume ϕ ∈ L 1 K for some K ≥ 0. Then we show that higher order asymptotic expansions of the solution of (6.2)-(6.4) are given as a corollary of our results. By [28,Proposition 4
.1] we have sup t>0 (1 + t) N 2 (1− 1 q ) u(t) q + sup t≥1 t N 2 (1− 1 q )+ 1 2 ∇u(t) q (6.7) + sup t>0 (1 + t) N 2 (1− 1 q )+ 1 2 ∇v(t) q < ∞ for any q ∈ [1, ∞].
Furthermore, applying arguments similar to the proof [28, Proposition 4.1], we can easily obtain
(6.8) sup t≥1 t N 2 (1− 1 q )+1 ∇ 2 u(t) q < ∞ for any q ∈ [1, ∞].
In addition, by (6.7) we can apply Theorem 3.2 to (6.2), and see that the solution u satisfies all of the assertions of Theorem 3.2. On the other hand, since it follows from (6.3) that (6.9) v(t) = e −t e t∆ ψ + t 0 e −t+s e (t−s)∆ u(s)ds, t > 0, by (G1), (6.7), and (6.8) we have
(6.10) sup t≥1 t N 2 (1− 1 q )+1 ∇ 2 v(t) q < ∞ for any q ∈ [1, ∞].
Therefore, putting (6.11) F (x, t, u, ∇u) := −∇ · (u∇v) = −∇v · ∇u − (∆v)u, by (6.7) and (6.10) we see that, in (6.2), there hold conditions (C A ) and
(F A ) in R N × (1, ∞) with A = N 2 + 1 ≥ 3 2 .
Furthermore, by Theorem 3.2 (i) we have u(1) ∈ L 1 K . Therefore, taking the function u(1) as the initial function of parabolic equation (6.2), we see that all of the assertions in Section 4 hold with A = N/2 + 1 for the solution u. In particular, we have Lemma 6.1 Let (u, v) be a global in time solution of (6.2)-(6.4) satisfying (6.6). Assume ϕ ∈ L 1 K for some K ≥ 0. Let c α (t) be the functions given in Corollary 4.1. Then there holds the following:
c α (t) = c α + O(t − N 2 + |α| 2 ) as t → ∞; (c) If |α| ≤ [K] and 1 ≤ |α| = N , then c α (t) = O(log t) as t → ∞; (d) t N 2 (1− 1 q )+ j 2 ∇ j t 0 e (t−s)∆ F M (s)ds q = O(t − N 2 ) as t → ∞.
Proof. Assertion (a) follows from (6.11) and the definition of c 0 (t). Furthermore, since
sup t>0 |M α (f, t)| |||f ||| |α| for f ∈ L 1 (R N , (1 + |x|) |α| dx),
by (2.7), (6.7), (6.10), and (6.11) we have
|M α (F M (t), t)| ∇v(t) ∞ |||∇g(t)||| |α| + ∆v(t) ∞ |||g(t)||| |α| t − N 2 −1+ |α| 2
for all sufficiently large t. Then, by using (4.4) and (4.5) with A = N/2 + 1 we have assertions (b) and (c). In addition, by (G1), (6.7), (6.10), and (6.11) we have
t N 2 (1− 1 q )+ j 2 ∇ j t 0 e (t−s)∆ F M (s)ds q t/2 0 F M (s) 1 ds + t N 2 (1− 1 q ) t t/2 (t − s) − j 2 F M (s) q ds t/2 0 (1 + s) − N 2 −1 ds + t N 2 (1− 1 q )+ j 2 t t/2 (t − s) − j 2 s − N 2 −1− N 2 (1− 1 q ) ds t − N 2
for all sufficiently large t. This gives assertion (d), and Lemma 6.1 follows. ✷
Then, since M ≡ R N ϕ(x)dx = R N u(x, t)dx for t > 0,
by Lemma 6.1 we apply Corollary 4.1 with N ≥ K to obtain the following theorem. Theorem 6.3 Let (u, v) be a global in time solution of (6.2)-(6.4), satisfying (6.6). Let N ≥ K and assume ϕ ∈ L 1 K . Then, for any j = 0, 1, there holds the following: (i) If N > K, then
(6.12) t N 2 (1− 1 q )+ j 2 ∇ j u(t) − M g(t) − 1≤|α|≤[K] c α g α (t) q = o(t − K 2 ) if K = [K], O(t − K 2 ) if K > [K], as t → ∞; (ii) if N = K, then t N 2 (1− 1 q )+ j 2 ∇ j u(t) − M g(t) − 1≤|α|≤K−1 c α g α (t) − |α|=K c α (t)g α (t) q = o(t − K 2 ) and (6.13) t N 2 (1− 1 q )+ j 2 ∇ j u(t) − M g(t) − 1≤|α|≤K−1 c α g α (t)] q = O(t − K 2 log t), as t → ∞; (iii) if N = K = 1, then (6.14) t 1 2 (1− 1 q )+ j 2 ∇ j [u(t) − M g(t)] q = O(t − 1 2 ) as t → ∞;
(iv) The same assertions as in (6.12)-(6.14) hold for v.
Proof of Theorem 6.3. Assertions (i) and (ii) follow from Corollary 4.1 and Lemma 6.1. Furthermore, by (6.9) we see that (6.12) and (6.13) hold with u replaced by v.
We prove assertion (iii). For this aim, by (2.7) and assertion (ii) we have only to prove
|M α (F (t), t)| = R x(u(x, t)v x (x, t)) x dx = R u(x, t)v x (x, t)dx = R u(x, t)(v(x, t) − M g(x, t)) x dx + R (M g(x, t)) x (u(x, t) − M g(x, t))dx ≤ u(t) ∞ (v(t) − M g(t)) x 1 + (M g(t)) x ∞ u(t) − M g(t) 1 = o(t − 3 2 log t)
as t → ∞. Similarly we have
|M α (F M (t), t)| = R x(M g(t)v x (x, t)) x dx = R M g(x, t)v x (x, t)dx = o(t − 3 2 log t)
as t → ∞. These together with Lemma 2.3 (ii) implies (6.15), and assertion (iii) follows. Then, by (6.9) we see that (6.14) holds with u replaced by v, and Theorem 6.3 follows. ✷ Remark 6.2 (i) Under assumption (6.6), Kato in [22] and Yamada in [32] and [33] recently studied the asymptotic expansions of the solution of (6.2)-(6.4) in detail, and obtained some asymptotic expansions given in Theorem 6.3. We emphasize that Theorem 6.3 is easily obtained by Corollary 4.1 with the aid of some global bounds of the solution and that Theorems 4.1 and 4.2 can systematically give the other higher order asymptotic expansions of the solution and the decay estimates between the solution and its asymptotic expansions.
(ii) Due to the decay estimates in Theorem 6.3, we can obtain the result similar to Theorem 6.2, and by using Theorems 4.1 and 4.2 we can also give the higher order asymptotic expansions of the solutions decaying faster than the Gauss kernel.
System of semilinear parabolic equations
Our arguments in this paper are also applicable to systems of parabolic equations under suitable assumptions. In this subsection we focus on the Cauchy problem for a system for semilinear parabolic equations,
(6.16) ∂ t u = ∆u + F (u) in R N × (0, ∞), u(x, 0) = Φ(x) in R N ,
where m = 1, 2, . . . , u = (u 1 , · · · , u m ), F = (F 1 (u), · · · F m (u)), and Φ = (ϕ 1 , · · · , ϕ m ) ∈ (L 1 K ∩ L ∞ (R N )) m for some K ≥ 0, and we study the asymptotics of the solution u. Throughout this subsection we assume F ∈ C(R N : R m ) and that there exist constants C > 0 and a > 1 + 2/N such that (6.17) |F (v)| ≤ C|v| a , v ∈ R m .
Let u be a unique global in time solution of (6.16) such that (6.18) u(t) ∞ (1 + t) − N 2 , t > 0.
Then, by (6.17) and (6.18) we have (6.19) |F (u(x, t))| (1 + t) − N(a−1) 2 |u(x, t)| for all (x, t) ∈ R N × (0, ∞). Therefore, similarly to Section 6.1, we can apply the same arguments as in the previous sections to the solution u with A = N (a − 1)/2 > 1. This means that all of the assertions in Section 4 hold with A = N (a − 1)/2 > 1. In particular, we apply Corollary 4.1 with K ∈ (0, 1] to obtain the following result. This is an extension of [18, Theorem 5.1], which treats the case m = 1.
Appendix
For convenience we present the proof of Lemma 2.4 by the same arguments as in Chapter 1 in [9]. We first prove (2.9) and (2.10).
Proof of (2.9) and (2.10). The C 1 -regularity of w and the representation (2.9) are easily obtained by a argument similar to Chapter 1 of [9]. Put C H = H L ∞ (0,T :L ∞ (R N )) . Then, by (2.6), (2.8), and (2.9) we see that there exist constants C 1 , C 2 , and C 3 , independent of C H and T , such that
|w(x, t)| ≤ t 0 R N G(x − ξ, τ )dξ H(τ ) ∞ dτ ≤ t 0 H(τ ) ∞ dτ ≤ C 1 C H T, |(∇ x w)(x, t)| ≤ t 0 R N |(∇ x G)(x − ξ, τ )|dξ H(τ ) ∞ dτ ≤ C 2 t 0 (t − τ ) − 1 2 H(τ ) ∞ dτ ≤ C 3 C H T 1/2
for all (x, t) ∈ R N × (0, T ), and we obtain (2.10). ✷ Next we prove (2.11). For this aim, we prove the following lemmas. Put G α (x, t) = (∂ α x G)(x, t).
Lemma 7.1 Let 0 < ν < 1 and |α| ≤ 1. Then there exists a constant C such that (7.1) Π 1 (x, y : t) := |G α (x, t) − G α (y, t)| |x − y| ν ≤ C{h(x, t) + h(y, t)}
for all x, y ∈ R N with x = y and all t > 0, where
(7.2) h(x, t) = t − N 2 − |α|+ν 2 1 + (t − 1 2 |x|) −ν + (t − 1 2 |x|) |α|+2 e − |x| 2 16t .
Proof. Let x, y ∈ R N with x = y and t > 0. If |x − y| ≥ t 1/2 , then, by (2.6) we have
Π 1 (x, y : t) ≤ t − ν 2 {|G α (x, t)| + |G α (y, t)|} ≤ C 1 [h(x, t) + h(y, t)]
for some constant C 1 , and obtain inequality (7.1). So it suffices to prove inequality (7.1) for the case |x − y| < t 1/2 . In this case, if y ∈ B(x, |x|/2), the mean value theorem implies the existence of the point x * ∈ B(x, |x|/2) such that
Π 1 (x, y : t) ≤ |(∇ x G α )(x * , t)||x − y| 1−ν ≤ t 1−ν 2 |(∇ x G α )(x * , t)|.
Then, since |x|/2 ≤ |x * | ≤ 3|x|/2, by (2.6) we have
Π 1 (x, y : t) ≤ C 2 t − N 2 − |α|+ν 2 1 + (t − 1 2 |x * |) |α|+1 e − |x * | 2 4t (7.3) ≤ C 3 t − N 2 − |α|+ν 2 1 + (t − 1 2 |x|) |α|+1 e − |x| 2 16t ≤ C 4 h(x, t) if y ∈ B(x, |x|/2),
where C 2 , C 3 , and C 4 are constants independent of x, y and t. Similarly we have (7.4) Π 1 (x, y : t) ≤ C 4 h(y, t) if x ∈ B(y, |y|/2).
On the other hand, if y ∈ B(x, |x|/2) and x ∈ B(y, |y|/2), then we have |x − y| ≥ (1/2) min{|x|, |y|}, and obtain
Π 1 (x, y : t) ≤ t − ν 2 (t − 1 2 |x|) −ν |G α (x, t)| + (t − 1 2 |y|) −ν |G α (y, t)| .
This together with (2.6) implies that
(7.5) Π 1 (x, y : t) ≤ C 5 [h(x, t) + h(y, t)],
where C 5 is a constant independent of x, y and t. Therefore, by (7.3)-(7.5) we have inequality (7.1) for the case |x − y| ≤ t 1/2 . Thus Lemma 7.1 follows. ✷ Lemma 7.2 Let 0 < ν < 1 and |α| ≤ 1. Then there exists a constant C such that (7.6) Π 2 (t, s : x) := |G α (x, t) − G α (x, s)| |t − s| ν/2 ≤ C{h(x, t) + h(x, s)}
for all x ∈ R N and all 0 < s < t.
Lemma 3. 1
1Assume the same conditions as in Theorem 3.1. Then there exists a solution of (3.1) such that
Lemma 3. 2
2Assume the same conditions as in Theorem 3.1. Let u be a solution of (3.1) given in Lemma 3.1. Then there holds
(s) q < ∞ for any T > 0, and obtain inequality (3.31). Thus Lemma 3.2 follows. ✷ Now we are ready to prove Theorem 3.1.
all t > 0. On the other hand, by Lemma 2.1 (ii), (G1), (2.4), and (3.
s ∈ (0, L). By (3.53) and (3.54) we apply the Lebesgue dominated convergence theorem to(3.52)
Theorem 3. 2
2Consider the Cauchy problem
Theorem 4. 1
1Let u be a solution of Cauchy problem (1.1) with ϕ ∈ L 1 K for some K ≥ 0. Assume that the solution u satisfies (3.3), (3.4), and condition (C A ) for some A > 1. Let n = 0, 1, 2, . . . and assume condition (F A ) if n ≥ 1. Then there holds the following:
(4.8) for any q ∈ [1, ∞] and l ∈ [0, K];
{g α (x, t)} |α|≤[K] , and plays a role of projection of the solution onto the space spanned by {g α (x, t)} |α|≤[K] . Furthermore we remark that the condition A > 1 in Theorem 4.1 is crucial. Indeed, even if conditions (C A ) and (F A ) hold for some A ∈ (0, 1], the solution of (1.1) does not necessarily behave like the Gauss kernel as t → ∞, that is, the conclusions of Theorem 4.1 does not necessarily hold. See Remark 6.1 and [18, Remark 1.1]. Theorem 4.1 is an extension of [18, Theorem 3.1], and is a result for general parabolic equations. Next, by Theorem 4.1 we give other higher order asymptotic expansions of the solution of (1.1), which are simple modifications of the function U 1 . Let J ∈ {0, . . . , [K]} and put J A = min{J, 2(A − 1)}. Then, by (4.4) we can define the function
Theorem 4. 2
2Let u be a solution of Cauchy problem (1.1) with ϕ ∈ L 1 K for some K ≥ 0. Assume that the solution u satisfies (3.3), (3.4), and conditions (C A ) and (F A ) for some A > 1. Let J ∈ {0, . . . , [K]} and put
Corollary 4. 1
1Assume the same conditions as in Theorem 4.1 and K ≥ 0. Put u(x, t) := |α|≤[K]
e
(t−s)∆ F M (s)ds where M = M 0 , F M (x, t) := F (x, t, M g(x, t), M ∇g(x, t)), and
Proof of Theorem 4 . 1 .
41It suffices to prove assertion (iii) of Theorem 4.1. Since there holds (4.10) for the case K > [K] by Theorem 4.1 (ii), it suffices to prove (4.10) for the case K = [K]. Let K = [K] and assume (F A ). Let n ∈ {0, 1, 2, . . . } be such that (5.18) 2(n + 1)(A − 1) > K.
∇
j z n (t) = ∇ j e t∆ z n (0) ∇ j e t∆ z n (0) + I 1 (t) + I 2 (t) + I 3 (t) for t ≥ 2L. Since z n (0) = P[K]
( 4 .
410) for the case K = [K], and the proof of Theorem 4.1 is complete. ✷ Next, by arguments similar to the proof of [18, Theorem 5.1] and Theorem 4.1 (iii) we prove Theorem 4.2.
Theorem 6. 2
2Assume the same conditions as in Theorem 6.1. Then the solution u satisfies either (i) there exists an integer d ∈ {0, . . . , [K]} such that, for any q ∈ [1, ∞] and j = 0, 1,
Theorem 6. 2
2is proved by the same argument as in the proof of [16, Corollary 1.2] with Theorem 6.1, and we leave the details of the proof to the reader. We remark that, if u(t) ∞ = O(t −(N +d)/2 ) as t → ∞ for some j ∈ {1, 2, . . . }, then conditions (C A ) and (F A ) hold with A = A d := (N + d)(p − 1)/2 + 1/2 > 1 and all of assertions of Theorems 4.1 and 4.2 hold with A = A d .
If |α| ≤ [K] and 1 ≤ |α| < N , then there exists a constant c α such that
( 6 .
615) c α (t) = O(1) as t → ∞ for the case K = N = |α| = 1. Since R gg x dx = 0 and (6.13) hold for u and v, by (2.3) and (6.7) we have
Theorem 6. 4 eq
4Let m ∈ {1, 2, . . . } and K ≥ 0. Assume (6.17) and Φ = (ϕ 1 , · · · , ϕ m ) ∈ (L 1 K ∩ L ∞ (R N )) m . Let u be a global in time solution of Cauchy problem (6.16), satisfying (6.18). Then there exists the limitM := lim t→∞ R N u(x, t)dx such that lim t→∞ t N 2 (1− 1 q ) u(t) − M g(t) q = 0for any q ∈ [1, ∞]. Furthermore there holds the following:(i) If K ∈ + O(t −(A−1) ) if 2(A − 1) = K, O(t − K 2 log t) if 2(A − 1) = K,as t → ∞, for any q ∈ [1, ∞];(ii) If K ∈ () if 4(A − 1) = K,as t → ∞, for any q ∈ [1, ∞] and σ > 0, whereu 1 (x, t) (t−s)∆ F (M g(x, s))ds; (iii) Assume that R N xF (M g(t))dx = 0 for all t > 0. Let K > 1. ) u(t) − M g(t) q = O(t − 1 2 ) + O(t −(A−1) ) as t → ∞ for any q ∈ [1, ∞].Proof of Theorem 6.4. This theorem is proved by Corollary 4.1 with minor modifications. We leave the details of the proof to the reader. (See also the proof of [18, Proposition 5.1].) ✷
2 ∇u(s) q ds ≤ C 3 u(T 1 ) q + C 4 T −A+1 1 sup T 1 ≤s≤t u(s) q + s 1 2 ∇u(s) qfor all t ≥ 2T 1 , where C 3 and C 4 are constants independent of T 1 . Let T 1 be a sufficiently large constant such that C 4 T −A+1 1 ≤ 1/2. Then inequality (3.34) together with (3.6) yields(3.35) sup2T 1 ≤s≤t s 1 2 ∇u(s) q ≤ 2C 3 u(T 1 ) q + sup T 1 ≤s≤t u(s) q + sup T 1 ≤s≤2T 1 s 1 2 ∇u(s) q < ∞ for all t ≥ 2T 1 . Furthermore, combining (3.33) with (3.35), we have sup 2T 1 ≤s≤t u(s) q ≤ u(T 1 ) q + C 2 T −A+1 1 sup T 1 ≤s≤t u(s) q + s 1 2 ∇u(s) q ≤ u(T 1 ) q + C 2 T −A+1 1 sup T 1 ≤s≤2T 1 u(s) q + s 1 2 ∇u(s) q +C 2 T −A+1 1 2C 3 u(T 1 ) q + sup T 1 ≤s≤2T 1 u(s) q + sup T 1 ≤s≤2T 1 s 1 2 ∇u(s) q +2C 2 T −A+1 1 sup 2T 1 ≤s≤t u(s) q
T 1− |α|+ν 2for all (x, y, t, s) ∈ E(T ). Therefore, by (7.8)-(7.10) we have inequality(2.11), and the proof of Lemma 2.4 is complete. ✷
Proof. If 0 < s ≤ t/2, then t/(t − s) ≤ 2 and s/(t − s) ≤ 1, and we obtain Π 2 (t, s :This together with (2.6) yields inequality (7.6) for the case 0 < s ≤ t/2. On the other hand, if t/2 < s < t, then, by the mean value theorem there exists a constant t * ∈ (t/2, t) such thatThis together with (2.6) implies thatfor some constants C 1 and C 2 , and we obtain inequality (7.6) for the case t/2 < s < t. Thus Lemma 7.2 follows. ✷We are ready to complete the proof of Lemma 2.4.Proof of Lemma 2.4. It suffices to prove (2.11). We can assume, without loss of generality, that C H = 1. Let |α| ≤ 1 andBy Lemmas 7.1 and 7.2 we haveh(x, t) + h(y, t) + h(x, s) + h(y, s) for all (x, y, t, s) ∈ E(T ). On the other hand, by (2.9) we havefor all (x, y, t, s) ∈ E(T ). Then, by (7.2) and (7.7) we havefor all (x, y, t, s) ∈ E(T ). Furthermore, by (2.6) we have (7.10) I 2 t s (t − τ ) −|α|/2 (t − s) ν/2 dτ (t − s) 1− |α|+ν
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"Integrable pseudopotentials related to elliptic curves",
"Integrable pseudopotentials related to elliptic curves"
] | [
"A V Odesskii [email protected] \nBrock University\nCanada\n\nLandau Institute for Theoretical Physics\nRussia\n\nInstitute for Theoretical Physics of Russian Academy of Sciences\nKosygina 2119334MoscowRussia\n",
"V V Sokolov [email protected] \nLandau Institute for Theoretical Physics\nRussia\n\nInstitute for Theoretical Physics of Russian Academy of Sciences\nKosygina 2119334MoscowRussia\n",
"A V Odesskii [email protected] \nBrock University\nCanada\n\nLandau Institute for Theoretical Physics\nRussia\n\nInstitute for Theoretical Physics of Russian Academy of Sciences\nKosygina 2119334MoscowRussia\n",
"V V Sokolov [email protected] \nLandau Institute for Theoretical Physics\nRussia\n\nInstitute for Theoretical Physics of Russian Academy of Sciences\nKosygina 2119334MoscowRussia\n"
] | [
"Brock University\nCanada",
"Landau Institute for Theoretical Physics\nRussia",
"Institute for Theoretical Physics of Russian Academy of Sciences\nKosygina 2119334MoscowRussia",
"Landau Institute for Theoretical Physics\nRussia",
"Institute for Theoretical Physics of Russian Academy of Sciences\nKosygina 2119334MoscowRussia",
"Brock University\nCanada",
"Landau Institute for Theoretical Physics\nRussia",
"Institute for Theoretical Physics of Russian Academy of Sciences\nKosygina 2119334MoscowRussia",
"Landau Institute for Theoretical Physics\nRussia",
"Institute for Theoretical Physics of Russian Academy of Sciences\nKosygina 2119334MoscowRussia"
] | [
"MSC numbers: 17B80, 17B63",
"MSC numbers: 17B80, 17B63"
] | We construct integrable pseudopotentials with an arbitrary number of fields in terms of elliptic generalization of hypergeometric functions in several variables. These pseudopotentials yield some integrable (2+1)-dimensional hydrodynamic type systems. An interesting class of integrable (1+1)dimensional hydrodynamic type systems is also generated by our pseudopotentials. | null | [
"https://arxiv.org/pdf/0810.3879v1.pdf"
] | 16,523,932 | 0810.3879 | 151dfe0281858d7a941413b22fb007fb610e1483 |
Integrable pseudopotentials related to elliptic curves
21 Oct 2008
A V Odesskii [email protected]
Brock University
Canada
Landau Institute for Theoretical Physics
Russia
Institute for Theoretical Physics of Russian Academy of Sciences
Kosygina 2119334MoscowRussia
V V Sokolov [email protected]
Landau Institute for Theoretical Physics
Russia
Institute for Theoretical Physics of Russian Academy of Sciences
Kosygina 2119334MoscowRussia
Integrable pseudopotentials related to elliptic curves
MSC numbers: 17B80, 17B63
328121 Oct 2008Address: L.D. Landau 1
We construct integrable pseudopotentials with an arbitrary number of fields in terms of elliptic generalization of hypergeometric functions in several variables. These pseudopotentials yield some integrable (2+1)-dimensional hydrodynamic type systems. An interesting class of integrable (1+1)dimensional hydrodynamic type systems is also generated by our pseudopotentials.
Introduction
In [1] a wide class of 3-dimensional integrable PDEs of the form m j=1 a ij (u) u j,t 1 + where u = (u 1 , . . . , u m ) was constructed. The coefficients of these PDEs were written in terms of generalized hypergeometric functions [2]. By the integrability of (1.1) we mean the existence of a pseudopotential representation 1
ψ t 2 = A(p, u), ψ t 3 = B(p, u), where p = ψ t 1 . (1.2)
Such a pseudopotential representation is a dispersionless version [3,4] of the zero curvature representation, which is a basic notion in the integrability theory of solitonic equations (see [5]). One of the interesting and attractive features of the theory of integrable systems (1.1) is that the dependence of the pseudopotentials on p can be much more complicated then in the solitonic case. In [6,7] some important examples of pseudopotentials A, B related to the Whitham averaging procedure for integrable dispersion PDEs and to the Frobenious manifolds were found. These examples are related to the universal algebraic curve of genus g with M punctures for arbitrary g, M. More precisely, the point Appp
A 2
pp , A p runs over a curve of genus g while p runs over C and u are some coordinates on the moduli space M g,M of curves of genus g with M punctures.
The pseudopotentials from [1] (see also [8]) were written in the following parametric form:
A = F 1 (ξ, u), p = F 2 (ξ, u),
where the ξ-dependence of the functions F i is defined by the ODE
F i,ξ = φ i (ξ, u) · ξ −s 1 (ξ − 1) −s 2 (ξ − u 1 ) −s 3 ...(ξ − u m ) −s m+2 . (1.3)
Here s 1 , ..., s m+2 are arbitrary constants and φ i are polynomials in ξ of degree m − k. These pseudopotentials are related to rational algebraic curves. If s 1 = ... = s m+2 = 0 and k = 0, then they coincide with pseudopotentials from [6] related to M 0,m+3 .
In this paper we construct integrable systems (1.1) and pseudopotentials related to elliptic curve. For these systems u = (u 1 , . . . , u n , τ ), where τ is the parameter of the elliptic curve. Note that τ is also an unknown function in our systems (1.1). The coefficients of the systems are expressed in terms of some elliptic generalization of hypergeometric functions in several variables. These elliptic hypergeometric functions can be defined as solutions of the following compatible linear overdetermined system of PDEs: g uαu β = s β ρ(u β − u α ) + ρ(u α + η) − ρ(u β ) − ρ(η) g uα + s α ρ(u α − u β ) + ρ(u β + η) − ρ(u α ) − ρ(η) g u β , g uαuα = s α β =α ρ(u α ) + ρ(η) − ρ(u α − u β ) − ρ(u β + η) g u β + β =α s β ρ(u α − u β ) + (s α + 1)ρ(u α + η) + s α ρ(−η) + (s 0 − s α − 1)ρ(u α ) + 2πir g uα − s 0 s α (ρ ′ (u α ) − ρ ′ (η))g,
g τ = 1 2πi β ρ(u β + η) − ρ(η) g u β − s 0 2πi ρ ′ (η)g (1.4)
for a single function g(u 1 , . . . , u n , τ ). Here and in the sequel η = s 1 u 1 + ... + s n u n + rτ + η 0 , s 0 = −s 1 − ... − s n , where s 1 , ..., s n , r, η 0 are arbitrary constants, and
θ(z) = α∈Z (−1) α e 2πi(αz+ α(α−1) 2 τ ) , ρ(z) = θ ′ (z) θ(z) . (1.5)
In the above formulas and in the sequel we omit the second argument τ of the functions θ, ρ and use the notation
ρ ′ (z) = ∂ρ(z) ∂z , ρ τ (z) = ∂ρ(z) ∂τ , θ ′ (z) = ∂θ(z) ∂z , θ τ (z) = ∂θ(z) ∂τ .
It turns out that the dimension of the space of solutions for (1.4) equals n + 1.
The paper is organized as follows.
In Section 2 we describe some properties of elliptic hypergeometric functions needed for our purposes. In particular, we present an integral representation similar to the representation for the generalized hypergeometric function (see, for example [1]).
In Section 3 for any n we construct pseudopotentials (1.2) with k = 0 related to the elliptic hypergeometric functions. The pseudopotential A n (p, u 1 , ..., u n , τ ) is defined in a parametric form by
A n = P n (g 1 , ξ), p = P n (g 0 , ξ),(1.6)
where g 1 , g 0 be linearly independent solutions of (1.4),
P n (g, ξ) = ξ 0 S n (g, ξ)e 2πir(τ −ξ) θ ′ (0) −s 1 −...−sn θ(u 1 ) s 1 ...θ(u n ) sn θ(ξ) −s 1 −...−sn θ(ξ − u 1 ) s 1 ...θ(ξ − u n ) sn dξ,(1.7)
and
S n (g, ξ) = 1≤α≤n θ(u α )θ(ξ − u α − η) θ(u α + η)θ(ξ − u α ) g uα − (s 1 + ... + s n ) θ ′ (0)θ(ξ − η) θ(η)θ(ξ) g. (1.8)
We call them elliptic pseudopotential of defect 0. Such pseudopotentials define integrable systems of the form (1.1) with m = l = n + 1. In the case s 1 = ... = s n = r = 0, η 0 → 0 our pseudopotentials coincide with elliptic pseudopotentials constructed in [6].
In Section 4 for k < n we construct pseudopotentials of defect k. These pseudopotentials define systems (1.1) with m = n + 1, l = n + k + 1.
A special class of solutions for integrable systems (1.1) depending on several arbitrary functions of one variable can be constructed by the method of hydrodynamic reductions [9,10]. The hydrodynamic reductions are defined by pairs of integrable compatible (1+1)-dimensional hydrodynamic type systems of the form
r i t = v i (r 1 , ..., r N )r i x , i = 1, 2, ..., N. (1.9)
These integrable systems also are of interest themselves. A general theory of such type integrable systems was developed in [11,12].
Section 5 is devoted to hydrodynamic reductions of systems (1.1) constructed in Sections 3,4. In the case k = 0 the corresponding systems (1.9) are defined by
r i t = S n (g 1 (u), ξ i ) S n (g 2 (u), ξ i ) r i x ,(1.10)
where g 1 , g 2 are linearly independent solutions of (1.4). For the pseudopotentials of defect k > 0 the corresponding formula is similar. The functions τ (r 1 , ..., r N ), ξ i (r 1 , ..., r N ), u i (r 1 , ..., r N ) are defined by the following universal overdetermined compatible system of PDEs of the Gibbons-Tsarev type [9,13]:
∂ α ξ β = 1 2πi ρ(ξ α − ξ β ) − ρ(ξ α ) ∂ α τ, ∂ α = ∂ ∂r α , (1.11) ∂ α ∂ β τ = − 1 πi ρ ′ (ξ α − ξ β )∂ α τ ∂ β τ,(1.12)
and
∂ α u β = 1 2πi ρ(ξ α − u β ) − ρ(ξ α ) ∂ α τ, α = 1, ..., N, β = 1, ..., n. (1.13)
Recall that here τ is the second argument of the function ρ. It would be interesting to compare formulas (1.11), (1.12) with formulas (3.23)-(3.26) from [14].
It is easy to verify that the system (1.11)-(1.13) is consistent. Therefore our (1+1)dimensional systems (1.10) admit a local parameterization by 2N arbitrary functions of one variable.
For some very special values of parameters s α in (1.4) our systems (1.10) are related to the Whitham hierarchies [6], to the Frobenious manifolds [7,15], and to the associativity equation [7,15].
Elliptic hypergeometric functions
Define a function θ in two variables z, τ by (1.5). We assume that Imτ > 0. The function θ is called theta-function of order one in one variable. Recall the following useful formulas:
θ(z + 1) = θ(z) θ(z + τ ) = −e −2πiz θ(z), θ(−z) = −e −2πiz θ(z), θ τ (z) = 1 4πi θ ′′ (z) − 1 2 θ ′ (z).
The following statements can be verified straightforwardly. Remark 1. Some coefficients of (1.4) can be written in a factorized form using the following identity
ρ(u β − u α ) + ρ(u α + η) − ρ(u β ) − ρ(η) = − θ ′ (0)θ(u α − u β + η)θ(u α )θ(u β + η) θ(u α − u β )θ(u α + η)θ(u β )θ(η) .
We call elements of H elliptic hypergeometric functions.
Proposition 2. Define a function F (u 1 , ..., u n , τ ) by the following integral representation F (u 1 , ..., u n , τ ) = 1 0 θ(u 1 − t) s 1 ...θ(u n − t) sn θ ′ (0) s 1 +...+sn+1 θ(t + η) θ(u 1 ) s 1 ...θ(u n ) sn θ(t) s 1 +...+sn+1 θ(η) e 2πi(s 1 +...+sn+r)t dt.
Then the function F satisfies system (1.4).
Proposition 3.
Let H = H s 1 ,...,sn,r,η 0 and H = H s 1 ,...,sn,0,r,η 0 . Then H is spanned by H and by the function
Z(u 1 , ..., u n , u n+1 , τ ) = u n+1 0 θ(u 1 − t) s 1 ...θ(u n − t) sn θ ′ (0) s 1 +...+sn+1 θ(t + η)
θ(u 1 ) s 1 ...θ(u n ) sn θ(t) s 1 +...+sn+1 θ(η) e 2πi(s 1 +...+sn+r)t dt.
(2.14) Moreover, the space H s 1 ,...,sn,0,...,0,r,η 0 (m zeros) is spanned by H and by Z(u 1 , ..., u n , u n+1 , τ ), Z(u 1 , ..., u n , u n+2 , τ ), ..., Z(u 1 , ..., u n , u n+m , τ ).
In the simplest case n = 1 system (1.4) for a function g(u 1 , τ ) has the following form
g u 1 u 1 = (s 1 + 1) θ ′ (u 1 + η) θ(u 1 + η) + s 1 θ ′ (−η) θ(−η) − (2s 1 + 1) θ ′ (u 1 ) θ(u 1 ) + 2πir g u 1 − s 2 1 θ ′ (0) 2 θ(u 1 − η)θ(u 1 + η) θ(u 1 ) 2 θ(−η)θ(η) g, g τ = 1 2πi θ ′ (u 1 + η) θ(u 1 + η) − θ ′ (η) θ(η) g u 1 + s 1 2πi (ln θ(η)) ′′ g.
If s 1 = 0, then the functions g = 1 and
g = Z(u 1 , τ ) = u 1 0 θ ′ (0)θ(t + η) θ(t)θ(η) e 2πirt dt
span the space of solutions. Moreover, Proposition 3 implies that the functions 1, Z(u 1 , τ ), ..., Z(u n , τ ) span the space of solutions of (1.4) in the case s 1 = · · · = s n = 0.
Elliptic pseudopotentials of defect 0
For any elliptic hypergeometric function g ∈ H we put
S n (g, ξ) = 1≤α≤n θ(u α )θ(ξ − u α − η) θ(u α + η)θ(ξ − u α ) g uα − (s 1 + ... + s n ) θ ′ (0)θ(ξ − η) θ(η)θ(ξ) g. (3.15)
Define P n (g, ξ) by the formula (1.7) if Re (s 1 + ... + s n ) > 1 and as the analytic continuation of (1.7) otherwise.
Proposition 4. The following relations hold
(P n (g, ξ)) uα = − g uα θ(u α )θ(ξ − u α − η) θ(u α + η)θ(ξ − u α ) e 2πir(τ −ξ) θ ′ (0) −s 1 −...−sn θ(u 1 ) s 1 ...θ(u n ) sn θ(ξ) −s 1 −...−sn θ(ξ − u 1 ) s 1 ...θ(ξ − u n ) sn , (3.16) (P n (g, ξ)) τ = 1 2πi ( 1≤α≤n θ(u α )θ ′ (ξ − u α − η) θ(u α + η)θ(ξ − u α ) g uα − (s 1 + ... + s n ) θ ′ (0)θ ′ (ξ − η) θ(η)θ(ξ) g)− θ ′ (−η) 2πiθ(−η) ( 1≤α≤n θ(u α )θ(ξ − u α − η) θ(u α + η)θ(ξ − u α ) g uα − (s 1 + ... + s n ) θ ′ (0)θ(ξ − η) θ(η)θ(ξ) g) × e 2πir(τ −ξ) θ ′ (0) −s 1 −...−sn θ(u 1 ) s 1 ...θ(u n ) sn θ(ξ) −s 1 −...−sn θ(ξ − u 1 ) s 1 ...θ(ξ − u n ) sn . (3.17)
Proof. Taking the derivatives of (3.16), (3.17) with respect to ξ, one arrives at thetafunctional identities, which can be proved straightforwardly. Moreover, the values of the left and the right hand sides of (3.16) and (3.17) are equal to zero at ξ = 0.
Let g 1 , g 0 be linearly independent elements of H. A pseudopotential A n (p, u 1 , ..., u n , τ ) defined in a parametric form by (1.6) is called elliptic pseudopotential of defect 0. Relations (1.6) mean that to find A n (p, u 1 , ..., u n , τ ), one has to express ξ from the second equation and substitute the result into the first equation.
Let g 0 , g 1 , ..., g n ∈ H be a basis in H. Define pseudopotentials B α (p, u 1 , ..., u n , τ ) of defect 0, where α = 1, ..., n, by B α = P n (g α , ξ), p = P n (g 0 , ξ), α = 1, ..., n.
(3.18)
Suppose that u 1 , ..., u n , τ are functions of t 0 = x, t 1 , ..., t n .
Theorem 1. The compatibility conditions ψ tαt β = ψ t β tα for the system ψ tα = B α (ψ x , u 1 , ..., u n , τ ), α = 1, ..., n, (3.19) are equivalent to the following system of PDEs for u 1 , ..., u n , τ :
1≤β≤n (g q g r,u β − g r g q,u β )(u β,ts + 1 2πi ( θ ′ (u β + η) θ(u β + η) − θ ′ (η) θ(η) )τ ts )+ 1≤β≤n (g r g s,u β − g s g r,u β )(u β,tq + 1 2πi ( θ ′ (u β + η) θ(u β + η) − θ ′ (η) θ(η) )τ tq ) + (3.20) 1≤β≤n (g s g q,u β − g q g s,u β )(u β,tr + 1 2πi ( θ ′ (u β + η) θ(u β + η) − θ ′ (η) θ(η) )τ tr ) = 0, 1≤β≤n,β =α θ(u β )θ(u α − u β − η) θ(u β + η)θ(u α − u β ) (g r,uα g q,u β −g q,uα g r,u β )(u α,ts −u β,ts + 1 2πi ( θ ′ (u α − u β − η) θ(u α − u β − η) − θ ′ (−η) θ(−η) )τ ts )+ 1≤β≤n,β =α θ(u β )θ(u α − u β − η) θ(u β + η)θ(u α − u β ) (g s,uα g r,u β −g r,uα g s,u β )(u α,tq −u β,tq + 1 2πi ( θ ′ (u α − u β − η) θ(u α − u β − η) − θ ′ (−η) θ(−η) )τ tq )+ 1≤β≤n,β =α θ(u β )θ(u α − u β − η) θ(u β + η)θ(u α − u β ) (g q,uα g s,u β −g s,uα g q,u β )(u α,tr −u β,tr + 1 2πi ( θ ′ (u α − u β − η) θ(u α − u β − η) − θ ′ (−η) θ(−η) )τ tr )− (s 1 + ... + s n ) θ ′ (0)θ(u α − η) θ(η)θ(u α ) (g q g r,uα − g r g q,uα )(u α,ts + 1 2πi ( θ ′ (u α − η) θ(u α − η) − θ ′ (−η) θ(−η) )τ ts )− (s 1 + ... + s n ) θ ′ (0)θ(u α − η) θ(η)θ(u α ) (g r g s,uα − g s g r,uα )(u α,tq + 1 2πi ( θ ′ (u α − η) θ(u α − η) − θ ′ (−η) θ(−η) )τ tq ) − (3.21) (s 1 + ... + s n ) θ ′ (0)θ(u α − η) θ(η)θ(u α ) (g s g q,uα − g q g s,uα )(u α,tr + 1 2πi ( θ ′ (u α − η) θ(u α − η) − θ ′ (−η) θ(−η) )τ tr ) = 0,
where α = 1, ..., n. Here q, r, s run from 0 to n.
Proof. Taking into account (3.18), we find that the compatibility conditions for (3.19) are equivalent to n α=1 ((P n (g q , ξ)) ξ (P n (g r , ξ)) uα − (P n (g r , ξ)) ξ (P n (g q , ξ)) uα )u α,ts + ((P n (g r , ξ)) ξ (P n (g s , ξ)) uα − (P n (g s , ξ)) ξ (P n (g r , ξ)) uα )u α,tq + ((P n (g s , ξ)) ξ (P n (g q , ξ)) uα − (P n (g q , ξ)) ξ (P n (g s , ξ)) uα )u α,tr + (3.22) ((P n (g q , ξ)) ξ (P n (g r , ξ)) τ − (P n (g r , ξ)) ξ (P n (g q , ξ)) τ )τ ts + ((P n (g r , ξ)) ξ (P n (g s , ξ)) τ − (P n (g s , ξ)) ξ (P n (g r , ξ)) τ )τ tq + ((P n (g s , ξ)) ξ (P n (g q , ξ)) τ − (P n (g q , ξ)) ξ (P n (g s , ξ)) τ )τ tr = 0.
Using (1.7), (3.17), we rewrite (3.22) as follows:
1≤β≤n θ(u β )θ(ξ − u β − η) θ(u β + η)θ(ξ − u β ) (S n (g q , ξ)g r,u β − S n (g r , ξ)g q,u β )u β,ts − 1 2πi 1≤β≤n θ(u β )θ ′ (ξ − u β − η) θ(u β + η)θ(ξ − u β ) (S n (g q , ξ)g r,u β − S n (g r , ξ)g q,u β )τ ts + (3.23) 1 2πi (s 1 + ... + s n ) θ ′ (0)θ ′ (ξ − η) θ(η)θ(ξ) (S n (g q , ξ)g r − S n (g r , ξ)g q )τ ts + (q, r, s) = 0,
where (q, r, s) means the cyclic permutation of q, r, s. Denote the left hand side of (3.23) by Λ(ξ). One can check that
Λ(ξ + 1) = Λ(ξ), Λ(ξ + τ ) = e 4πiη Λ(ξ),
and the only singularities of Λ(ξ) are poles of order one at the points ξ = 0, u 1 , ..., u n modulo 1, τ . This implies that Λ(ξ) = 0 iff the residues at these points are equal to zero. Calculating the residue at ξ = 0, we get (3.20). The calculation of the residue at ξ = u α leads to (3.21).
Remark 2. Given t 1 , t 2 , t 3 , Theorem 1 yields a 3-dimensional system of the form (1.1) with l = m = n + 1 possessing a pseudopotential representation. where c 0 , ..., c n are constants. Therefore,
S n (g, ξ) = 1≤α≤n c α e 2πiruα θ ′ (0)θ(ξ − u α − η 0 ) θ(η 0 )θ(ξ − u α ) .
If we assume r = 0 and c 1 + ... + c n = 0, then in the limit η 0 → 0 we obtain
S n (g, ξ) = 1≤α≤n c α ρ(ξ − u α ).
A system of PDEs equivalent to compatibility conditions for equations of the form (3.22), was called in [6] a Whitham hierarchy. In this paper I.M. Krichever constructed some Whitham hierarchies related to algebraic curves of arbitrary genus g. The hierarchy corresponding to g = 1 is equivalent to one described by Theorem 1 if r = s 1 = . . . = s n = 0, c 1 + .. + c n = 0, and η 0 → 0 as described above.
Elliptic pseudopotentials of defect k > 0
In this section we construct elliptic pseudopotentials of defect k. Fix k linearly independent elliptic hypergeometric functions h 1 , ..., h k ∈ H. For any g ∈ H define P n,k (g, ξ) by the formula
P n,k (g, ξ) = 1 ∆ det P n (g, ξ) P n (h 1 , ξ) ... P n (h k , ξ) g u n−k+1 h 1,u n−k+1 ... h k,u n−i → c i1 h 1 + ... + c ik h k , g → g + d 1 h 1 + ... + d k h k with constant coefficients c ij , d i do not change P n,k (g, ξ)
.
One can verify that
(P n,k (g, ξ)) ξ = S n,k (g, ξ)e 2πir(τ −ξ) θ ′ (0) −s 1 −...−sn θ(u 1 ) s 1 ...θ(u n ) sn θ(ξ) −s 1 −...−sn θ(ξ − u 1 ) s 1 ...θ(ξ − u n ) sn , (4.25) where S n,k (g, ξ) = 1 ∆ ( 1≤α≤n−k θ(u α )θ(ξ − u α − η) θ(u α + η)θ(ξ − u α ) ∆ α (g) − (s 1 + ... + s n ) θ ′ (0)θ(ξ − η) θ(η)θ(ξ) ∆ 0 (g)) (4.26) and ∆ α (g) = det g uα h 1,uα ... h k,uα g u n−k+1 h 1,u n−k+1 ... h k,u n− , ∆ 0 (g) = det g h 1 ... h k g u n−k+1 h 1,u n−k+1 ... h k,u n−(P n,k (g, ξ)) uα = − ∆ α (g)θ(u α ) ∆θ(u α + η) n−k+1≤β≤n θ(u β − u α − η)(P n,k (g, ξ)) u β θ(u β − u α )S n,k (g, u β ) − (4.27) ∆ α (g)θ(u α )θ(ξ − u α − η) ∆θ(u α + η)θ(ξ − u α ) e 2πir(τ −ξ) θ ′ (0) −s 1 −...−sn θ(u 1 ) s 1 ...θ(u n ) sn θ(ξ) −s 1 −...−sn θ(ξ − u 1 ) s 1 ...θ(ξ − u n ) sn , where 1 ≤ α ≤ n − k, and (P n,k (g, ξ)) τ = 1 2πi n−k+1≤β≤n (P n,k (g, ξ)) u β S n,k (g, u β ) (S ′ n,k (g, u β ) − θ ′ (−η) θ(−η) S n,k (g, u β )) + (4.28) 1 2πi S ′ n,k (g, ξ) − θ ′ (−η) θ(−η) S n,k (g, ξ) e 2πir(τ −ξ) θ ′ (0) −s 1 −...−sn θ(u 1 ) s 1 ...θ(u n ) sn θ(ξ) −s 1 −...−sn θ(ξ − u 1 ) s 1 ...θ(ξ − u n ) sn , where S ′ n,k (g, ξ) = 1 ∆ ( 1≤α≤n−k θ(u α )θ ′ (ξ − u α − η) θ(u α + η)θ(ξ − u α ) ∆ α (g) − (s 1 + ... + s n ) θ ′ (0)θ ′ (ξ − η) θ(η)θ(ξ) ∆ 0 (g)).
Moreover, (P n,k (g, ξ)) u β S n,k (g, u β ) does not depend on g if n − k + 1 ≤ β ≤ n.
Proof. Taking the derivatives of (4.27), (4.28) with respect to ξ, one arrives at thetafunctional identities, which can be proved straightforwardly. Moreover, the values of the left and the right hand sides of (4.27) and (4.28) are equal to zero at ξ = 0.
Let g 1 , g 2 ∈ H. Assume that g 1 , g 2 , h 1 , ..., h k are linearly independent. Define pseudopotential A n,k (p, u 1 , ..., u n , τ ) in the parametric form by A n,k = P n,k (g 1 , ξ), p = P n,k (g 2 , ξ). To construct A n,k (p, u 1 , ..., u n , τ ), one has to find ξ from the second equation and substitute into the first one. The pseudopotential A n,k (p, u 1 , ..., u n , τ ) is called elliptic pseudopotential of defect k.
Theorem 2. Let g 0 , g 1 , ..., g n−k , h 1 , ..., h k ∈ H be a basis in H and pseudopotentials B α , α = 1, ..., n − k are defined by B α = P n,k (g α , ξ), p = P n,k (g 0 , ξ), α = 1, ..., n − k.
Then the compatibility conditions for (3.19) are equivalent to the following system of PDEs for u 1 , ..., u n , τ :
1≤β≤n−k (∆ 0 (g q )∆ β (g r ) − ∆ 0 (g r )∆ β (g q ))(u β,ts + 1 2πi ( θ ′ (u β + η) θ(u β + η) − θ ′ (η) θ(η) )τ ts )+ 1≤β≤n−k (∆ 0 (g r )∆ β (g s ) − ∆ 0 (g s )∆ β (g r ))(u β,tq + 1 2πi ( θ ′ (u β + η) θ(u β + η) − θ ′ (η) θ(η) )τ tq ) + (4.30) 1≤β≤n−k (∆ 0 (g s )∆ β (g q ) − ∆ 0 (g q )∆ β (g s ))(u β,tr + 1 2πi ( θ ′ (u β + η) θ(u β + η) − θ ′ (η) θ(η) )τ tr ) = 0, 1≤β≤n−k,β =α θ(u β )θ(u α − u β − η) θ(u β + η)θ(u α − u β ) (∆ α (g r )∆ β (g q ) − ∆ α (g q )∆ β (g r ))× (u α,ts − u β,ts + 1 2πi ( θ ′ (u α − u β − η) θ(u α − u β − η) − θ ′ (−η) θ(−η) )τ ts )+ 1≤β≤n−k,β =α θ(u β )θ(u α − u β − η) θ(u β + η)θ(u α − u β ) (∆ α (g s )∆ β (g r ) − ∆ α (g r )∆ β (g s ))× (u α,tq − u β,tq + 1 2πi ( θ ′ (u α − u β − η) θ(u α − u β − η) − θ ′ (−η) θ(−η) )τ tq )+ 1≤β≤n−k,β =α θ(u β )θ(u α − u β − η) θ(u β + η)θ(u α − u β ) (∆ α (g q )∆ β (g s ) − ∆ α (g s )∆ β (g q ))× (u α,tr − u β,tr + 1 2πi ( θ ′ (u α − u β − η) θ(u α − u β − η) − θ ′ (−η) θ(−η) )τ tr )− (s 1 +...+s n ) θ ′ (0)θ(u α − η) θ(η)θ(u α ) (∆ 0 (g q )∆ α (g r )−∆ 0 (g r )∆ α (g q ))(u α,ts + 1 2πi ( θ ′ (u α − η) θ(u α − η) − θ ′ (−η) θ(−η) )τ ts )− (s 1 +...+s n ) θ ′ (0)θ(u α − η) θ(η)θ(u α ) (∆ 0 (g r )∆ α (g s )−∆ 0 (g s )∆ α (g r ))(u α,tq + 1 2πi ( θ ′ (u α − η) θ(u α − η) − θ ′ (−η) θ(−η) )τ tq )− (4.31) (s 1 +...+s n ) θ ′ (0)θ(u α − η) θ(η)θ(u α ) (∆ 0 (g s )∆ α (g q )−∆ 0 (g q )∆ α (g s ))(u α,tr + 1 2πi ( θ ′ (u α − η) θ(u α − η) − θ ′ (−η) θ(−η) )τ tr ) = 0,
where α = 1, ..., n − k and
n−k α=1 ∆ α (g r )θ(u α )θ(u β − u α − η) ∆θ(u α + η)θ(u β − u α ) u α,ts −S n,k (g r , u β )u β,ts − 1 2πi (S ′ n,k (g r , u β )− θ ′ (−η) θ(−η) S n,k (g r , u β ))τ ts = (4.32) n−k α=1 ∆ α (g s )θ(u α )θ(u β − u α − η) ∆θ(u α + η)θ(u β − u α ) u α,tr −S n,k (g s , u β )u β,tr − 1 2πi (S ′ n,k (g s , u β )− θ ′ (−η) θ(−η) S n,k (g s , u β ))τ tr ,
where β = n − k + 1, ..., n. Here q, r, s run from 0 to n and t 0 = x.
Proof is similar to the proof of Theorem 1.
Remark 4. Given t 1 , t 2 , t 3 , Theorem 2 yields a 3-dimensional system of the form (1.1) with m = n + 1, l = n + k + 1 possessing a pseudopotential representation. Indeed, formula (4.32) gives 3k linearly independent equations if q, r, s = 1, 2, 3. Formulas (4.30), (4.31) give n − k + 1 equations. On the other hand, one can construct exactly k linear combinations of equations (4.32) with q, r, s = 1, 2, 3 such that derivatives of u i , i = n − k + 1, ..., n cancel out. Moreover, these linear combinations belong to the span of equations (4.30), (4.31). Therefore there exist (n − k + 1) + 3k − k = n + k + 1 linearly independent equations.
Remark 5. We have to assume n ≥ k + 2 in (4.30), (4.31), (4.32). Indeed, for n = k + 1 we cannot construct more then one pseudopotential and therefore there is no any system of the form (1.1) associated with this case. However, the corresponding pseudopotential generates interesting integrable (1+1)-dimensional systems of hydrodynamic type (see Section 5). Probably these pseudopotentials for k = 0, 1, ... are also related to some infinite integrable chains of the Benney type [16,17]. System (4.30)-(4.32) possesses many conservation laws of the hydrodynamic type. In particular, the following statement can be verified by a straightforward calculation. Proposition 6. For any r = s = 0, 1, ..., n, system (4.30)-(4.32) has k conservation laws of the form:
∆(g r , h 1 , ...î...h k ) ∆(h 1 , ..., h k ) ts = ∆(g s , h 1 , ...î...h k ) ∆(h 1 , ..., h k ) tr , (4.33)
where i = 1, ..., k.
Here for all i = 1, ..., k and r = 0, 1, ..., n.
∆(f 1 , ..., f k ) = det f 1,u n−k+1 ... f k,u n−
Suppose n + 1 ≥ 3k; then the system of the form (1.1) obtained from (4.30)-(4.32) with q, r, s = 1, 2, 3 consists of 3k equations (4.32) (they are equivalent to (4.33)) and n + 1 − 2k equations of the form (4.30), (4.31). Indeed, only n + 1 − 2k equations (4.30), (4.31) are linearly independent from (4.32). Expressing τ, u 1 , ..., u 3k−1 in terms of z i,t 1 , z i,t 2 , z i,t 3 , i = 1, ..., k from (4.34) and substituting into n + 1 − 2k equations of the form (4.30), (4.31), we obtain a 3dimensional system of n + 1 − 2k equations for n + 1 − 2k unknowns z 1 , ..., z k , u 3k , ...u n . This is a quasi-linear system of the second order with respect to z i and of the first order with respect to u j , whose coefficients depend on z i,t 1 , z i,t 2 , z i,t 3 , i = 1, ..., k, and u 3k , ...u n . It is clear that the general solution of the system can be locally parameterized by n + 1 − k functions in two variables.
In the case 2k ≤ n + 1 < 3k the functions z i,t 1 , z i,t 2 , z i,t 3 , i = 1, ..., k are functionally dependent. We have 3k − n − 1 equations of the form
R i (z 1,t 1 , z 1,t 2 , z 1,t 3 , ..., z k,t 1 , z k,t 2 , z k,t 3 ) = 0, i = 1, ..., 3k − n − 1
and n+1−2k second order quasi-linear equations. Totally we have (3k−n−1)+(n+1−2k) = k equations for k unknowns z 1 , ..., z k . It is clear that the general solution of this system can be locally parameterized by n + 1 − k functions in two variables.
Suppose n + 1 < 2k; then we have n + 1 + k < 3k, which means that 3k equations of the form (4.32) are linearly dependent. Probably in this case the general solution of the system can also be locally parameterized by n + 1 − k functions in two variables.
One of the most interesting cases is n + 1 = 3k, when we have a system of k quasi-linear second order equations for the functions z 1 , ..., z k . The simplest case is k = 2.
Integrable (1+1)-dimensional hydrodynamic-type systems and hydrodynamic reductions
In this section we present integrable (1+1)-dimensional hydrodynamic type systems (1.9) constructed in terms of elliptic hypergeometric functions. These systems appear as the so-called hydrodynamic reductions of our elliptic pseudopotentials A n,k . Results and formulas of this section look similar to the rational case (see [1]). By integrability of (1.9) we mean the existence of infinite number of hydrodynamic commuting flows and conservation laws. It is known [12] that this is equivalent to the following relations for the velocities v i (r 1 , ..., r N ):
∂ j ∂ i v k v i − v k = ∂ i ∂ j v k v j − v k , i = j = k. (5.35)
Here ∂ α = ∂ ∂r i , α = 1, . . . , N. The system (1.9) is called semi-Hamiltonian if conditions (5.35) hold.
The main geometrical object related to any semi-Hamiltonian system (1.9) is a diagonal metric g kk , k = 1, . . . , N, where
1 2 ∂ i log g kk = ∂ i v k v i − v k , i = k. (5.36)
In view of (5.35), the overdetermined system (5.36) is compatible and the function g kk is defined up to an arbitrary factor η k (r k ). The metric g kk is called the metric associated with (1.9). It is known that two hydrodynamic type systems are compatible iff they possess a common associated metric [12].
A diagonal metric g kk is called a metric of Egorov type if for any i, j
∂ i g jj = ∂ j g ii . (5.37)
Note that if an Egorov-type metric associated with a hydrodynamic-type system of the form (1.9) exists, then it is unique. For any Egorov's metric there exists a potential G such that g ii = ∂ i G. Semi-Hamiltonian systems possessing associated metrics of Egorov type play important role in the theory of WDVV associativity equations and in the theory of Frobenious manifolds [7,15,18].
Let τ (r 1 , ..., r N ), ξ 1 (r 1 , ..., r N ), ..., ξ N (r 1 , ..., r N ) be a solution of the system (1.11), (1.12). It can be easily verified that this system is in involution and therefore its solution admits a local parameterization by 2N functions of one variable. Let u 1 (r 1 , ..., r N ), ..., u n (r 1 , ..., r N ) be a solution of the system (1.13). It is easy to verify that this system is in involution for each fixed β and therefore has an one-parameter family of solutions for fixed ξ i , τ . Consider the following system
r i t = S n,k (g 1 , ξ i ) S n,k (g 2 , ξ i ) r i x , (5.38)
where g 1 , g 2 are linearly independent solutions of (1.4), the polynomials S n,k , k > 0 are defined by (4.26), and S n,0 = S n (see (3.15)).
Theorem 3. The system (5.38) is semi-Hamiltonian. The associated metric is given by Proposition 7. Suppose that a solution ξ 1 , ..., ξ N , τ, u 1 , ..., u n of (1.11)-(1.13) is fixed. Then the hydrodynamic type systems
g ii = S n,k (g, ξ)e 2πir(τ −ξ) θ ′ (0) −s 1 −...−sn θ(u 1 ) s 1 ...θ(u n ) sn θ(ξ) −s 1 −...−sn θ(ξ − u 1 ) s 1 ...θ(ξ − u n ) sn 2 ∂ i τ.r i t 1 = S n,k (g 1 , ξ i ) S n,k (g 3 , ξ i ) r i x , r i t 2 = S n,k (g 2 , ξ i ) S n,k (g 3 , ξ i ) r i x (5.39)
are compatible for all g 1 , g 2 .
Proof. Indeed, the metric associated with (5.38) does not depend on g 2 . Therefore the systems (5.39) has a common metric depending on g 3 and on a solution of (1.11)-(1.13).
Remark 7. One can also construct some compatible systems of the form (5.39) using Proposition 3. Set g 2 = Z(u 1 , ..., u n , u n+1 , τ ) in (5.39). Here u n+1 is an arbitrary solution of (1.13) (with n replaced by n + 1) distinct from u 1 , ..., u n . It is clear that the flows (5.39) are compatible for such g 2 and any g 1 ∈ H. Moreover, Proposition 3 implies that the flows (5.39) are compatible if we set g 1 = Z(u 1 , ..., u n , u n+1 , τ ), g 2 = Z(u 1 , ..., u n , u n+2 , τ ) for two arbitrary solutions u n+1 , u n+2 of (1.13).
All members of the hierarchy constructed in Proposition 7 possess a dispersionless Lax representation of the form
L t = {L, A},(5.40)
where {L, A} = A p L x −A x L p , with common L = L(p, r 1 , ..., r N ). Define a function L(ξ, r 1 , ..., r N ) by the following system
∂ α L = − 1 2πi ρ(ξ α − ξ) − ρ(ξ α ) L ξ ∂ α τ, α = 1, ..., N. (5.41)
Note that the system (5.41) is in involution and therefore the function L(ξ, r 1 , ..., r N ) is uniquely defined up to inessential transformations L → λ(L). To find the function L(p, r 1 , ..., r N ) one has to express ξ in terms of p by (1.6) for k = 0 or by (4.29) for k > 0.
Proposition 8. Let u 1 , . . . , u n be arbitrary solution of (1.13). Then system (5.38) admits the dispersionless Lax representation (5.40), where A = A n,k is defined by (1.6) for k = 0 and by (4.29) for k > 0.
Proof. Define A = A n,k by (1.6) for k = 0 and by (4.29) for k > 0. Substituting A into (5.40) and calculating L t by virtue of (5.38), we obtain that (5.40) is equivalent to
∂ i L = ∂ i P n,k (g 2 , ξ) · S n,k (g 1 , ξ i ) − ∂ i P n,k (g 1 , ξ) · S n,k (g 2 , ξ i ) P n,k (g 2 , ξ) ξ · S n,k (g 1 , ξ i ) − P n,k (g 1 , ξ) ξ · S n,k (g 2 , ξ i ) L ξ .
Taking into account (4.25),(5.41) and writing down P n,k (g i , ξ) u 1 , ..., P n,k (g i , ξ) u n−k and P n,k (g i , ξ) τ in terms of P n,k (g 1 , ξ) u n−k+1 , ..., P n,k (g 1 , ξ) un by (4.27), (4.28), one can readily verify this equality.
Let us show that integrable (1+1)-dimensional systems (5.38) define hydrodynamic reductions for pseudopotentials and 3-dimensional systems from Sections 3 and 4.
In [10,13,19] a definition of integrability for equations (5.40), (1.2) and (1.1) is given in terms of hydrodynamic reductions.
Suppose there exists a pair of compatible semi-Hamiltonian hydrodynamic-type systems of the form
r i t 1 = v i 1 (r 1 , ..., r N )r i x , r i t 2 = v i 2 (r 1 , ..., r N )r i x (5.42)
and functions u i = u i (r 1 , ..., r N ) such that these functions satisfy (1.1) for any solution of (5.42). Then (5.42) is called a hydrodynamic reduction for (1.1).
Definition 1 [10]. A system of the form (1.1) is called integrable if equation (1.1) possesses sufficiently many hydrodynamic reductions for each N ∈ N. "Sufficiently many" means that the set of hydrodynamic reductions can be locally parameterized by 2N functions of one variable. Note that due to gauge transformations r i → λ i (r i ) we have only N essential functional parameters for hydrodynamic reductions.
Suppose there exists a semi-Hamiltonian hydrodynamic-type system (1.9) and functions u i = u i (r 1 , ..., r N ), L = L(p, r 1 , ..., r N ) such that these functions satisfy dispersionless Lax equation (5.40) for any solution r 1 (x, t), ..., r N (x, t) of the system (1.9). Then (1.9) is called a hydrodynamic reduction for (5.40).
Definition 2 [19]. A dispersionless Lax equation (5.40) is called integrable if equation (5.40) possesses sufficiently many hydrodynamic reductions for each N ∈ N.
We also call the corresponding pseudopotential A(p, u 1 , ..., u n ) integrable. If A 1 , A 2 are compatible, then A = c 1 A 1 + c 2 A 2 is integrable for any constants c 1 , c 2 . Indeed, the system r i t = (c 1 v i 1 (r) + c 2 v i 2 (r)) r i x is a hydrodynamic reduction of (5.40).
Definition 4. By 3-dimensional system associated with compatible functions A 1 , A 2 we mean the system of the form (1.1) equivalent to the compatibility conditions for the system ψ t 2 = A 1 (ψ t 1 , u 1 , ..., u n ), ψ t 3 = A 2 (ψ t 1 , u 1 , ..., u n ). (5.43) It is clear that any system associated with a pair of compatible functions possesses sufficiently many hydrodynamic reductions and therefore it is integrable in the sense of Definition 1.
The following statement is a reformulation of Proposition 8.
Theorem 4. The system (5.38) is a hydrodynamic reduction of the pseudopotential A n,k defined by (1.6) if k = 0 and by (4.29) if k > 0. Recall that we use the notation S n ≡ S n,0 , A n ≡ A n,0 , P n ≡ P n,0 . Proposition 9. Suppose g 1 , g 2 , g 3 , h 1 , ..., h k ∈ H are linearly independent. Define pseudopotentials A 1 , A 2 by A 1 = P n,k (g 1 , ξ), A 2 = P n,k (g 2 , ξ), p = P n,k (g 3 , ξ).
Then A 1 and A 2 are compatible.
Proof. Note that the system (1.11)-(1.13), (5.41) does not depend on g 1 , g 2 , g 3 and therefore we have a family of functions L, ξ i , u i , τ giving hydrodynamic reductions of the form (5.38) for both A 1 and A 2 . Moreover, according to Proposition 7 the systems r i t 1 = S n,k (g 1 , ξ i ) S n,k (g 3 , ξ i ) r i x , r i t 2 = S n,k (g 2 , ξ i ) S n,k (g 3 , ξ i ) r i x are compatible.
Remark 8. This result implies that 3-dimensional hydrodynamic type systems constructed in Sections 3, 4 possess sufficiently many hydrodynamic reductions and therefore are integrable in the sence of Definition 1.
Remark 9. Using Proposition 3, one can construct compatible pseudopotentials A 1 and A 2 depending on different number of variables u i . Indeed, let g 1 , g 3 , h 1 , ..., h k ∈ H and g 2 = Z(u 1 , ..., u n , u n+1 , τ ). Then A 2 depends on u 1 , ..., u n , u n+1 , τ and A 1 depends on u 1 , ..., u n , τ only.
Proposition 1 .
1The system of linear equations (1.4) is compatible for any constants s 1 , . . . , s n , r, η 0 . The dimension of the linear space H of solutions for system (1.4) is equal to n + 1.
Remark 3 .
3Consider the case s 1 = ... = s n = 0. We have g = c 0 + c 1 Z(u 1 , τ ) + ... + c n Z(u n , τ ),
Proof. Substituting the expression for the metric into (5.36), where v i are specified by (5.38), one obtains the identity by virtue of (1.4) and (1.11)-(1.13). Remark 6. The system (5.38) does not possess the associated metric of the Egorov type in general. However, for very special values of the parameters s i in (1.4) there exists g 2 ∈ H such that the metric is of the Egorov type for all solutions of the system (1.11)-(1.13).
Definition 3
3[1]. Two integrable pseudopotentials A 1 , A 2 are called compatible if the systemL t 1 = {L, A 1 }, L t 2 = {L, A 2 }possesses sufficiently many hydrodynamic reductions (5.42) for each N ∈ N.
k+1 ............
... .........
g un
h 1,un
...
h k,un
This means that (1.1) is equivalent to the compatibility conditions for (1.2).
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| [] |
[
"Associated production of Higgs boson with a photon at electron-positron colliders",
"Associated production of Higgs boson with a photon at electron-positron colliders"
] | [
"Mehmet Demirci \nDepartment of Physics\nKaradeniz Technical University\nTR61080TrabzonTurkey\n"
] | [
"Department of Physics\nKaradeniz Technical University\nTR61080TrabzonTurkey"
] | [] | A complete one-loop prediction for the single production of the neutral Higgs bosons in association with a photon in electron-positron collisions is presented in the framework MSSM, paying special attention to the individual contribution from each type of diagram. This process has no amplitude at tree level and is hence directly sensitive to one-loop impacts and the underlying dynamics of Higgs. In order to investigate the effect of the new physics, four different scenarios, which include a Higgs boson with mass and couplings consistent with those of the discovered Higgs boson and a considerable part of parameter space allowed by the bounds from the researches for additional Higgs bosons and sparticles, are chosen in the MSSM. The dependence of the cross section in both SM and MSSM on the center-of-mass energy is examined by considering the polarizations of the initial electron and positron beams. The effect of individual contributions from each type of oneloop diagrams on the total cross section is also investigated in detail. Furthermore, the total cross section of e − e + → γh 0 as well as e − e + → γA 0 are scanned over the plane mA − tan β for each scenario. The full one-loop contributions are crucial for the analysis of BSM physics at a future electron-positron collider. 12.60.Jv, 13.66.Fg, 14.80.Da * The diagrams were drawn by using JaxoDraw[32,33]. | 10.1103/physrevd.100.075006 | [
"https://arxiv.org/pdf/1905.09363v1.pdf"
] | 162,184,341 | 1905.09363 | a5c91126beae18cfa193750e68f75ef4c7d08c35 |
Associated production of Higgs boson with a photon at electron-positron colliders
22 May 2019
Mehmet Demirci
Department of Physics
Karadeniz Technical University
TR61080TrabzonTurkey
Associated production of Higgs boson with a photon at electron-positron colliders
22 May 2019(Dated: May 24, 2019)numbers: 1215-y1260Jv1366Fg1480Da
A complete one-loop prediction for the single production of the neutral Higgs bosons in association with a photon in electron-positron collisions is presented in the framework MSSM, paying special attention to the individual contribution from each type of diagram. This process has no amplitude at tree level and is hence directly sensitive to one-loop impacts and the underlying dynamics of Higgs. In order to investigate the effect of the new physics, four different scenarios, which include a Higgs boson with mass and couplings consistent with those of the discovered Higgs boson and a considerable part of parameter space allowed by the bounds from the researches for additional Higgs bosons and sparticles, are chosen in the MSSM. The dependence of the cross section in both SM and MSSM on the center-of-mass energy is examined by considering the polarizations of the initial electron and positron beams. The effect of individual contributions from each type of oneloop diagrams on the total cross section is also investigated in detail. Furthermore, the total cross section of e − e + → γh 0 as well as e − e + → γA 0 are scanned over the plane mA − tan β for each scenario. The full one-loop contributions are crucial for the analysis of BSM physics at a future electron-positron collider. 12.60.Jv, 13.66.Fg, 14.80.Da * The diagrams were drawn by using JaxoDraw[32,33].
I. INTRODUCTION
Despite its many success, the Standard Model (SM) leaves us with a lot of questions to be answered, such as the hierarchy problem, the origin of flavor, etc. Since the discovery of the Higgs boson at the LHC [1,2] and until the date it has not yet been found any evidence of new physics beyond the SM (BSM). However, the observations of neutrino oscillations, matter-antimatter asymmetry, relic density of dark matter (DM), and so on, open the door to new physics BSM. Additionally, there are strong motivations to extend the scalar sector of the SM by introducing more than one Higgs doublet. Therefore, it seems compulsory the development of new attempts concentrated on the research of data which provide a hint about new physical degrees of freedom. This is the main goal of proposals at future e + e − colliders such as the International Linear Collider (ILC) [3][4][5], Compact Linear Collider (CLIC) [5,6], Circular Electron-Positron Collider (CEPC) [7] and Future Circular Collider (FCC) [8]. On the other hand, they are mainly designed to provide a high precision and complete picture of the Higgs boson and its couplings. The e + e − colliders compared to the hadron colliders have a cleaner background, and and hence the new physics signals are easily separated from the background. The ILC is one of the most developed linear collider planned to be a Higgs factory in the centre of mass energies of √ s = 250 − 500 GeV (extendable up to a 1 TeV). The CLIC is a TeV-scale high luminosity linear collider planned to be operated at centre-ofmass energies of √ s = 380 GeV, 1.5 TeV and 3 TeV. The CEPC collider with a circumference of 100 km is designed to operate at √ s = 240 GeV. The potential for * [email protected] CEPC to probe a suite of loop-level corrections to Higgs and electroweak observables in supersymmetric models is comprehensively studied in [9].
Even at √ s = 250 GeV with a total integrated luminosity of 2 ab −1 for the electron-positron collider, there are some suggests to accurately determine the couplings of Higgs boson to gauge bosons, leptons and quarks [10,11] with an accuracy of order one percent compared the 0.2% accuracy based on the SM predicted couplings in terms of m h . That amplified precision may allow detecting the small deviations for BSM scenarios. A very precise prediction of Higgs boson production involving additional interactions which come from BSM scenarios can provide significant hints about new physics. There are many important motivations to choose the Minimal Supersymmetric Standard Model (MSSM) as BSMscenario that could identify these new interactions.
The MSSM [12][13][14][15], one of the most attractive and widely considered extensions of SM, keeps the number of new fields and couplings to a minimum. It provides a solution for the hierarchy problem of the SM, offers a candidate for the DM postulated to explain astrophysical observations, and a prediction for the mass of the scalar resonance observed at the LHC. The MSSM has two Higgs doublets, which leads to a physical spectrum that include a couple of charged Higgs bosons H ± , a CPodd Higgs boson A 0 , and the light/heavy CP -even Higgs bosons h 0 /H 0 in the CP conserving case. In Higgs sector of MSSM, all couplings and masses at tree-level can be described by only two parameters: the mass of pseudoscalar Higgs boson m A 0 , and the ratio of the vacuum expectation values of the two doublets, tan β. The discovered Higgs with mass of around 125 GeV could be interpreted naturally as one of the two neutral CP -even Higgs bosons in the MSSM [16][17][18]. Moreover, many new particles in the MSSM are predicted such as scalar lep-tons l, scalar quarks q k , neutralinos χ 0 i and charginos χ ± j . In R-parity conserving models [19], supersymmetric particles (or sparticles, for short) are pair-produced and their decay chains end in the stable, lightest sparticle (LSP). The lightest neutralino χ 0 1 is considered as LSP in many models. As a neutral, weakly interacting and stable particle, χ 0 1 is consistent with the properties required of a DM candidate [20].
The associated Higgs production with a photon, e − e + → γh 0 is well suited to study the Higgs to neutral gauge boson couplings such as the hγγ and hZγ couplings. Future e + e − colliders are optimal for studying e − e + → hγ, where the cross section in the SM has a peak about √ s = 250 GeV [21,22]. Because the treelevel contribution of the process is highly suppressed by the electron mass and the process is protected by the electromagnetic gauge symmetry, it occurs at the oneloop level for the first time. Therefore, the visible size of cross section is an order of magnitude of 10 −1 fb at √ s = 250 GeV, which is rather small. However, since the signal is very clean, it can be observed at the future e + e − colliders with the design luminosity. Furthermore, new physics contributions can considerably amplify the rate of production relative to the SM case; namely, the process is potentially very sensitive to new physics. There are few studies dedicated to the investigations of new physics effects on the process in the framework of an effective field theory or anomalous Higgs-boson couplings [23,24] as well as the extended Higgs models (inert doublet/triplet model, and two Higgs doublet model (THDM)) [25,26] and the MSSM [22,[27][28][29]. However, owing to the most recent constraints on the parameter space of the MSSM from Run 2 of the LHC, the size of the MSSM contributions to e − e + → hγ should be reevaluated in the allowedparameter space.
In this work, the single production of the neutral Higgs bosons in association with a photon in electronpositron collisions is reinvestigated in the framework MSSM, paying a special attention to the individual contribution from each type of diagram. In this aim, it is examined that how and how much the individual contributions from each type of diagrams could amplify or lessen the h 0 γ signal at a future e − e + collider. For this aim, four different benchmark scenarios, which have a Higgs boson with mass and couplings consistent with those of the discovered Higgs boson and a considerable part of their parameter space is allowed by the bounds from the researches for additional Higgs bosons and supersymmetric particles, are chosen. These scenarios are named as m 125 h , m 125 h (light τ ), m 125 h (light χ) and m 125 h (alignment) [30,31]. Distributions for the total cross sections are computed as a function of the center-ofmass energy and the polarization of the incoming beams. Furthermore, the total cross section of both e − e + → γh 0 and e − e + → γA 0 are scanned over the plane m A − tan β for each scenario, and regions in which the cross section is large enough to be detectable at a future e − e + collider, are determined in this paper.
The contents of the present work are the following: Section II provides the corresponding Feynman diagrams and the analytical expressions related to the process e − e + → hγ. This section then gives information on how the numerical evaluation is done. Section III provides details of the considered benchmark scenarios. In Sec. IV, numerical results are presented and the some parameter dependencies of the cross section are discussed in detail. Finally, Section V presents the conclusions of the present study.
II. THEORETICAL FRAMEWORK
The associated production of single Higgs boson with a photon in an electron-positron collision is indicated by
e + (p 1 )e − (p 2 ) → h(k 1 )γ(k 2 ), (2.1)
where after each particle, as usual, its 4-momenta is written in parentheses. The Mandelstam variables can be written as
s = (p 1 + p 2 ) 2 , t = (p 1 − k 1 ) 2 , u = (p 1 − k 2 ) 2 . (2.2)
At tree level, the process occurs via t-channel electronexchange diagram suppressed by the mass of electron. So the tree-level amplitude of the process is neglected, i.e., for the first time, the process is mediated by one-loop diagrams, and hence it is sensitive to all virtual particles inside the loop.
The total amplitude of one-loop level can be written as a linear sum of box, triangle, and bubble one-loop integrals. According to the type of loop-correction, the oneloop diagrams contributing to the process e + e − → hγ can be classified into four different types: the boxtype, self-energy, quartic coupling-type, triangle-type diagrams. A complete set of one-loop Feynman diagrams and the corresponding amplitudes in the SM and MSSM are generated by the FeynArts [34]. For MSSM, the diagrams * are shown in Figure 1 to Figure 3. Since selfenergy diagrams consist of loop corrections to the electron, and hence are highly suppressed by the electron mass, they are not explicitly shown here. Moreover, it is also possible another set of diagrams where particles in each loop are running counterclockwise. The bracket [. . .] represents that all possible combinations of the particles in the bracket can be written. In the Feynman diagrams, the label f m ( f x m ) refers to fermions (scalar fermions) e m , u m , d m , ( e x m , u x m , d x m ) and the label S 0 represents all neutral Higgs/Goldstone bosons h 0 , H 0 , A 0 , G 0 . The indexes m and x represent the generation of (scalar-)quark and the scalar-quark mass eigenstates, respectively. In loop diagrams, scalar particles such as neutral Figure 2 shows all quartic coupling-type diagrams which consist of bubbles (q 1−7 ) attached to the initial state via an intermediate γ or Z or neutral Higgs bosons (h 0 , H 0 , A 0 , G 0 ), and triangle loop (q 8 ) of the neutrino ν e , charged Higgs/Goldstone boson, W -boson directly attached to the final state. Finally, Figure 3 shows all triangle-type contributions which consist of triangle-vertices (t 1−7 ) attached to the initial state via an intermediate γ or Z or neutral Higgs bosons, and also include triangle-corrections to the top/bottom vertex. Feynman diagrams q 1−7 and t 1−6 are s-channel diagrams. Owing to the intermediated neutral Higgs bosons, the resonant effects could be observed at the triangle-type (t 1−6 ) and bubbles-type (q 1−7 ) diagrams for some specific center of mass energy.
G ± /W ± /W ± γ h 0 ν e [H ± , G ± ] W ± /G ± /W ± e e (b 7 ) G ± /W ± /W ± h 0 γ ν e [H ± , G ± ] W ± /G ± e e /W ± (b 1 ) e/ χ ±H ± /[G ± , W ± ] h 0 γ ν e W ± H ± / e e [G ± , W ± ] (b 9 )h 0 W (q 1 ) S 0 e H ± , G ± e γ h 0 H ± /G ± (q 4 ) h 0 , H 0 e H ± /G ± e γ / f w m / f w m h 0 (q 3 ) γ , Z e W e γ W h 0 H ± /G ± (q 2 ) γ , Z e H ± / G ± / f w m e γ / f w m h 0 W (q 5 ) h 0 e W e γ h 0 G ± (q 6 ) γ, Z e W e γ h 0 H ± /G ± (q 7 ) S 0 / e W e γ γ, Z h 0 H ± , G ± / W (q 8 )
In this study, the process e + e − → A 0 γ is also examined. Due to the CP nature of the pseudoscalar Higgs boson A 0 , the process e + e − → A 0 γ has no W and Zcontributions in the box-type diagrams, and no contribution from Z-boson, W-boson, f x m and H ± in the triangletype and bubble-type diagrams, compared to the process e + e − → h 0 γ. Therefore, the process e + e − → A 0 γ occurs only via s-channel triangle-type diagrams which involve loops of fermions and charginos, as well as t-channel triangle-type and box-type diagrams which involve loops of sneutrino/chargino and selectron/neutralino.
S 0 e e f m / χ ± m f m / χ ± m fm / χ ± m h 0 γ (t 3 ) S 0 e e [ H ± , G ± , W ± ] W ± ] [ H ± , G ± , W ± ] [H ± , G ± , h 0 γ (t 8 ) e e e e / χ ± m e/ χ ± m γ, Z, S 0 / ν e h 0 γ (t 6 ) γ, Z e e f w m / H ± / u ± f x m / H ± / u ± f w m /H ± /u ± h 0 γ (t 7 ) γ, Z e e [ G ± , W ± ] [ G ± , W ± ] [G ± , W ± ] h 0 γ (t 9 )S 0 / ν e h 0 γ (t 4 ) h 0 , H 0 , e e u ± u ± u ± G 0 , γ, Z h 0 γ (t 5 ) γ, Z e e f m / χ ± m f m / χ ± m fm / χ ± m FIG. 3.
Triangle-type diagrams contributing to the process e + e − → hγ at one-loop level.
The Higgs sector of MSSM at the tree level is described by two parameters, tan β and a mixing angle α in the CP-even Higgs sector. Furthermore, the angle α could be also given in terms of m A and tan β as follows:
tan(2α) = tan(2β) m 2 A + m 2 Z m 2 A − m 2 Z , − π 2 < α < 0. (2.3)
Once m A and tan β are given and the leading radiative correction is involved in α, all the couplings of the Higgs bosons to gauge bosons, fermions, and Higgs bosons are fixed. Table I lists the couplings of the neutral Higgs boson to gauge bosons and Higgs bosons which are included in each type of diagrams for the process e + e − → h 0 γ. The Feynman diagrams are dominated by triple couplings
λ h 0 G + H − , λ h 0 H + W − , λ h 0 G + W − and λ h 0 G + G −
which are proportional to mixing angles cos(β − α) and sin(β − α). Some couplings of Higgs boson are given by †
λ MSSM h 0 h 0 h 0 = − 3igm W 2c 2 W c 2α s β+α , (2.4) λ MSSM h 0 h 0 H 0 = igm W 2c 2 W c 2α c α+β − 2s 2α s α+β , (2.5) λ MSSM h 0 H − H + = − igm W 2 c 2β s β+α c 2 W + 2s β−α ,(2.
6) † The short-hand notation sx and cx are used for sin(x) and cos(x), respectively. For example, s α+β = sin(α + β) for x = α + β.
λ MSSM h 0 G + H − = − igm W 2 s 2β s α+β c 2 W − c β−α , (2.7) λ MSSM h 0 G − G + = igm W 2c 2 W c 2β s α+β , (2.8) λ MSSM h 0 A 0 A 0 = − igm W 2c 2 W c 2β s α+β , (2.9) λ MSSM h 0 H − W + = − ig 2 c β−α , (2.10) λ MSSM h 0 G − W + = − ig 2 s β−α , (2.11) λ MSSM h 0 W − W + = igm W s β−α , (2.12) λ MSSM h 0 h 0 H − H + = − ig 2 4 1 + c 2α c 2β s 2 W c 2 W − s 2α s 2β , (2.13) λ MSSM h 0 H − γW + = ig 2 s W 2 c β−α . (2.14)
where the gauge coupling constant g = e/s W and m W is the mass of W boson. All these couplings have a strong dependence on the mixing angles α and β. In this study,
λ h 0 [h 0 ,H 0 ][h 0 ,H 0 ] (b10) (t11,16) λ h 0 [A 0 ,G 0 ][A 0 ,G 0 ] (b10) (t11,16) λ h 0 H + H − (b2,6) (t3,6,15) (q2) λ h 0 G + H − (b3,6,7) (t3,11,15) λ h 0 G + G − (b3,6,7) (t3,6,7,11,15) (q2) λ H 0 H − G + (t3) λ H 0 G − G + (t3) λ A 0 H − G + (t3) VSS λ h 0 H + W − (b3,4,7,8) (t3,13) (q1) λ h 0 G + W − (b3,4,7,8) (t3,7,13) (q1) λ H 0 [G + ,H + ]W − (t3) λ A 0 H + W − (t3) λ G 0 G + W − (t3) λ h 0 [A 0 ,G 0 ]Z (b11) (t12) λ [γ,Z]H + H − (t6,9,18) (q4) λ [γ,Z]G + G − (t7,9,18) (q4) VVS λ h 0 ZZ (b11) (t12) λ h 0 W W (b3,4,8) (t3,7,13,15) (q3) λ H 0 W W (t3) λ [γ,Z]G + W (t7,9,18) (q6) SSSS/VVSS λ [h 0 ,H 0 ]h 0 H + H − (q4) λ [h 0 ,H 0 ]h 0 G + G − (q4) λ h 0 h 0 W W (q5) λ [γ,Z]H + H − γ (q2) λ [γ,Z]G + G − γ (q2) λ h 0 [H + ,G + ]W γ (q1,6,7) (q8) λ [H 0 ,A 0 ,G 0 ][H + ,G + ]W γ (q1) λ H + H − Zγ (q2) λ G + G − Zγ (q2) λ Zh 0 W G + (q6)
in particular, triple Higgs couplings and couplings of the scalar to gauge bosons are interested.
Since the tree-level amplitudes of process e + e − → h 0 γ suppressed by the electron mass are neglected and hence the process has only one-loop contributions as the lowest order, its one-loop amplitude can be easily obtained by summing all unrenormalized reducible and irreducible contributions. Consequently, the finite and gauge invariant results are obtained. Furthermore, the total amplitude is ultraviolet finite and this has been checked both numerically and analytically.
The corresponding total amplitude can be given in the form
M = M △ + M + M (2.15)
as a sum over all contributions from triangle-, box-and quartic-type diagrams. The differential cross section of the process, summing over the polarization of the photon, can be calculated by
dσ d cos θ (e + e − → h 0 γ) = s − m 2 h 32πs 2 pol |M| 2 ,(2.16)
where √ s are the center-of-mass energy of e + e − collisions and θ is the scattering angle between the photon and the electron in the centre-of-mass frame. The integrated cross section over all θ angles is given by
σ(e + e − → h 0 γ) = +1 −1 d cos θ dσ d cos θ . (2.17)
At an e + e − collider, the photon in association with the Higgs is produced as monochromatic with an energy of
E γ = s − m 2 h 2 √ s . (2.18)
At √ s = 250 GeV, this gives a "spectral-line" at E γ = 93.75 GeV. The signal is easy to separate from the backgrounds.
With the help of FeynArts [34] and FormCalc [35] packages ‡ , the numerical calculation is carried out in the 't Hooft-Feynman gauge using dimensional regularization. The corresponding amplitudes are generated by FeynArts. The analytical results of the squared amplitude are provided by FormCalc. The scalar integrals in ‡ Using the same tools, we have previously done a few more recent studies [38][39][40] and achieved significant results. loop amplitudes are evaluated by LoopTools [36]. The integration over phase space of 2 → 2 is numerically evaluated by using CUBA library. The properties of MSSM Higgs bosons are obtained by using FeynHiggs [37]. The polarisation effects are significant at electronpositron colliders and can be used to confer important advantages. In this study, the effect of beam polarisations on cross section is also analyzed. Since the electron and positron have only 2 spin-states, the cross section for general beam polarisations is defined by [11] σ P
e − P e + = 1 4 (1 − P e − )(1 − P e + )σ LL + (1 + P e − )(1 + P e + )σ RR + (1 − P e − )(1 + P e + )σ LR + (1 + P e − )(1 − P e + )σ RL ,(2.19)
where P e − /P e + indicates to the longitudinal polarisations of the initial electron/positron beam, equals to −1 (+1) for completely polarised left(right)-handed beam. σ LL , σ RR , σ LR and σ RL indicate the cross sections with completely polarised beams of the 4 possible cases. At schannel e − e + annihilation processes, only σ LR and σ RL are nonzero under condition of helicity conservation. The intrinsic left-right asymmetry of the cross section can be calculated with
A LR = σ LR − σ RL σ LR + σ RL (2.20)
for a given process. For the purpose of discussing the expected accuracy of future measurements, the centre of mass energies, integrated luminosities L and beam polarisations, proposed at future electron-positron colliders are presented in Table II. [30,31], are used to illustrate the effect of the new physics based on MSSM. Note that all of the considered scenarios include a CP-even Higgs boson with mass about 125 GeV and couplings consistent with those of the discovered Higgs boson and a considerable part of their parameter space is allowed by the bounds from the researches for additional Higgs bosons and supersymmetric particles. In particular, they are consistent with the most recent experimental results from the LHC-Run 2. The lighter CP-even Higgs h 0 of MSSM is SM-like in all scenario. In each scenario, two free parameters are left: tan β and m A . Hence, for each scenario the cross section can be presented in plane of m A − tan β. The parameters tan β and m A are varying in the range of 100 GeV ≤ m A ≤ 2 TeV and 0.5 ≤ tan β ≤ 50 for the first three scenarios, and 200 GeV ≤ m A ≤ 2 TeV and 1 ≤ tan β ≤ 20 for the last scenario.
Considering the bounds placed on SUSY parameters from current experimental results, especially the direct SUSY research results from ATLAS [41,42] and CMS experiments [43][44][45], a common soft SUSY-breaking mass parameter is fixed as M f = 2 TeV for the 1st and 2ndgenerations in the slepton and squark sector. The remaining parameters, which are the gaugino mass parameters M 1 , M 2 and M 3 ; the Higgsino mass parameter µ, the 3rd generation slepton mass parameters M L3 and M E3 ; the 3rd-generation squark mass parameters M Q3 , M U3 and M D3 ; the 3rd-generation the trilinear couplings A t,b,τ , are separately determined for each scenario. However, in the first three scenarios, the parameter X t = A t − µ cot β is set as input parameter instead of the parameter A t . In this study, for CP-conserving MSSM, all SUSY input parameters are chosen to be real and positive. We give a list of the input parameters for each scenario in Table III. In the m 125 h scenario, all sparticles are relatively heavy, hence they have only mildly effect on productions and decays of the Higgs bosons. Therefore, the phenomenology of this scenario is similar to that of a type-2 of THDM with MSSM-inspired couplings of Higgs. The masses of the gluino and 3rd-generation squarks are allowed by the available bounds from direct researches at the LHC. In the m 125 h (light τ ) and m 125 h (light χ) scenarios, the colorless sparticles (staus, and in one case, neutralinos and charginos) are relatively light, and the LSP is the lightest neutralino. The masses of gluino, sbottom and stop are the same as in the m 125 h scenario, but trilinear interaction term for the staus and the stop mixing parameter X t are reduced in the m 125 h (light τ ) and m 125 h (light χ) scenarios, respectively. These two scenarios can be a considerably effective on the Higgs phenomenology (see e.g. Refs. [30,46]) via loop contributions to the couplings of Higgs boson to particles of SM, as well as via direct decays of the Higgs bosons into sparticles if kinematically possible [31]. At low values of tan β, the m 125 h (align) scenario is defined by alignment without decoupling. In order to obtain an acceptable prediction for m h as well as to achieve alignment without decoupling, in the m 125 h (align) scenario the parameters determining the stop masses are remarkably larger values than in the other scenarios. For each scenario, taking both theory and experimental uncertainties into account, the impact on the parameter space of the available constraints from Higgs researches at LHC, Tevatron and LEP have been investigated in Ref. [31]. In this study, it is chosen four benchmark points (BPs) which are compatible with the most recent results of the LHC for the bounds on masses and couplings of new particles, and the Higgs-boson properties. These are checked by the HiggsBounds [47] and HiggsSignals [20]. For all scenarios, the masses of Higgs bosons are computed to two-loop accuracy with help of the FeynHiggs. The theoretical uncertainty in the prediction of FeynHiggs is estimated as ∆m theory h = ±3 GeV for the Higgs masses [49,50]. The dependence of properties of the Higgs boson on other lepton and quark masses is not very pronounced, and the default values of FeynHiggs are considered. The values of the input flags of FeynHiggs are set such that the evaluation includes full next-to-leading logarithms (NLL) and partial next-to-NLL resummation of the logarithmic corrections.
IV. NUMERICAL RESULTS AND DISCUSSION
In this section, the numerical predictions of the single production of the neutral Higgs bosons in association with a photon in electron-positron collisions are discussed in detail for both SM and MSSM, focusing on individual contributions from each type of one-loop diagrams, and polarizations of initial beams. For each benchmark scenario given in Table III, the numerical evaluation of the total cross sections and individual contributions of one-loop amplitudes for the process e + e − → hγ have been carried out as a function of the center-of-mass energy. Furthermore, the total cross section of e − e + → γh 0 as well as e − e + → γA 0 are scanned over the plane m A − tan β.
A. The process e − e + → γh 0 in the SM In Fig. 4, the SM cross section of e + e − → hγ is given as a function centre-of-mass energy in the range from 100 to 1500 GeV, focusing on individual contributions coming from different types of diagrams, and polarizations of initial beams. Note that the total cross section in- creases quickly with the opening of the phase-space and then decreases near √ s ∼ 2 × m t (2 times mass of top quark) close to the t + t − threshold with increments of √ s. It is seen that the cross section is very sensitive to the magnitudes of each amplitude and the relativephases between them. At high center-of-mass energy, where the triangle-and the bubble-type contributions are suppressed, σ SM (e + e − → hγ) is dominated by the boxtype contributions. However, the bubble-type contribution is larger than triangle-type contribution, and their interference (bub+tri) make a much smaller contribution from each of them in the region of √ s ≤ 700 GeV, since they nearly destroy each other. The (bub+tri) contribution is enhanced by the threshold effect when √ s is close to 2 × m t , since the t + t − threshold is seen at this energy. The top quark and W-boson contributions make destructive interference and the top contribution is maximal near the t + t − threshold. After passing the threshold of t + t − , the cross section scales like 1/s and hence drop steeply. Combining all the contributions, the total cross section ultimately has the first peak near √ s = 200 GeV, and the second one near √ s = 500 GeV. Note that the total cross section with completely polarised left-handed electron e − L and right-handed positron e + R , σ SM (e − L e + R → hγ), can be enhanced by about a factor between 2 and 4, compared with the unpolarized case. The longitudinal polarization of both the electron and positron beams is therefore significant to enhance the cross section. At √ s = 250 GeV, σ SM (e − L e + R → hγ) reaches to value of 0.88 fb. However, as expected, the cross sections for polarization cases of e − L e + L and e − R e + R are very small (see, the insert figure in Fig. 4(b)). The left-right asymmetry A LR has a peak at the region of √ s ≤ 340 GeV. After passing the t + t − threshold, its value remains almost constant which is equal to 0.98 with increments of √ s.
In Fig. 5(a)-(b), the cross section of process e − e + → γh 0 in the SM is also presented in plane of P e − and P e + by varying from −1 to +1. Especially, the cross section reaches its sizable values in the region of 0 < P e + ≤ +1 and −1 ≤ P e − < 0. The polarized cross section is maximum at point (P e − , P e + ) = (−1, +1), namely completely polarised left-handed electron and completely right-handed positron. The enhancement is raised up to a factor of 3. Fig. 6, total cross section of process e − e + → γh 0 in the MSSM is presented as a function of √ s from 150 GeV to 1.5 TeV for each benchmarks point scenario given in Table III. Additionally, to see deviations from the predictions of SM, the ratio of the total cross sections
∆R = (σ MSSM − σ SM ) σ SM × 100 (4.1)
is evaluated for each scenario, and this is presented in the lower panel of Fig. 6 The left-right asymmetry A LR has a peak at the region of √ s ≤ 340 GeV. After passing the t + t − threshold, its value remains almost constant which is equal to 0.98 with increments of √ s. Furthermore, the cross sections are sorted according to various polarizations of initial beams as follows: σ(LR)> σ(-0.8,+0.3)> σ(-0.8,0.0)> σ(UU)> σ(RL) for each benchmark point. It is seen that the basic size of the total cross sections enhance by a factor of 4, depending on the polarizations of the initial e − and e + beams. Figure 8 shows the effect of the individual contributions coming from different types of diagrams on total cross section of process e − e + → γh 0 as a function of the center-of-mass energy ranging from 100 GeV to 3 TeV for each benchmarks point scenario. Here, the abbreviations "box", "bub","qua","s triang" ,"t triang" and "all" indicate to the contributions of box-type (diagrams b 1−12 in Fig. 1), bubble-type (diagrams q 1−7 in Fig. 2), quartictype (diagram q 8 in Fig. 2), s-channel triangle-type (diagrams t 1−7 in Fig. 3) and t-and u-channel triangle-type (diagrams t 8−18 in Fig. 3) diagrams, and all of Feynman diagrams, respectively. The "t channels" denotes total contribution from all t-and u-channels diagrams. The "box W" represent to contributions from the box-type diagrams with one or more W-bosons (diagrams b 3 , b 4 , b 7 , and b 8 in Fig. 1). Also, the "bub+triang" is denoted to the contribution from interference between bubble-type and triangle-type diagrams.
It is clearly seen that the cross section is very sensitive to the magnitudes of each amplitude and the relative phases between them in all of the considered scenarios. The σ MSSM (e + e − → hγ) is dominated by the s triangleand the bubble-type contributions at low center-of-mass energy. On the other hand, at high energies, √ s ≥ 1600 GeV, these contributions are suppressed as 1/s and hence the box-type contributions become greater than them. However, the triangle-and bubble-type contributions are almost equal, and the interference among them (bub+tria) makes a much smaller contribution than any of them (by 2 orders of magnitude) because they make a destructive interference. On the other hand, the box-type contribution is larger by 1 order of magnitude than that of this interference. The t channels contribution is consist of box-type and t triang contributions, however the contribution of box-type diagrams is reduced by t triang diagrams. The t channels contribution is smaller than schannels contributions (buble-type and s triangle-type).
Note that the s triangle-type Feynman diagrams are dominated by triple couplings λ h 0 G + W − , λ h 0 H + W − λ h 0 G + H − , and λ h 0 G + G − . The bubble-type contributions are mainly determined by triple couplings λ h 0 G + W − and λ h 0 H + W − , quatric couplings λ h 0 G + W − γ and λ h 0 H + W − γ as shown in Table I. They are proportional to mixing angles cos(β − α) and sin(β − α).
The box W contribution ends up dominating over all the other ones because the masses of the sfermion and charged Higgs boson are fixed at the TeV scale. About 70% of the total contribution of box-type diagrams comes from box-type diagrams b 3 , b 4 , b 7 , and b 8 with one or more W-bosons in the loop. Therefore, it is also possible to assess the contribution of box-type diagrams in terms of a single coupling (the higgs-W-W coupling) given in Eq. (2.12) which is proportional to the term sin(β − α). Table IV shows numerical results over the scenarios for center-of-mass energies of the planned CEPC (at √ s = 240 GeV), FCC ee (at √ s = 350 GeV) and ILC (at √ s = 500 GeV) projects. Overall, the s-channel diagrams make a dominant contribution to the total cross section in all scenario. Therefore, the s triangle-and the bubble-type diagrams have a remarkable impact on production rate. Particularly, the total cross section of the production of e − e + → γh 0 reaches a value of 0.325 fb at √ s = 240 GeV and is more observable compared to others at the CEPC or ILC-250 for BP3. Additionally, the cross sections are sorted according to the BPs as follows σ(BP3)> σ(BP1)> σ(BP2)> σ(BP4) at low energies. Note that the basic size of the total cross-sections is not very sensitive according to the BPs. It is well known that the total cross section of e − e + → γh 0 depend on couplings of the Higgs to other particles and masses of corresponding particles. All the couplings of the Higgs bosons to gauge bosons, fermions, and Higgs bosons are determined by the parameters m A and tan β. The regions of the parameter space where the enhancement of cross section is large enough to be detectable at a future collider can be found by the behavior with these parameters. In this context, total cross section of e − e + → γh 0 is scanned over the plane of m A − tan β at √ s = 250 GeV as depicted in Fig. 9. The parameters m A and tan β are varying in the range of In the scenario m 125 h , the predictions for the mass of light CP-even Higgs boson m h are always below 126 GeV all over the plane of m A − tan β, and at tan β < 6 remain outside the window 125.09 ± 3 GeV [as shown in Fig. 9(a)]. At low m A , the decay and production rates of h have been obtained to be incompatible with the LHC results [31] Table III. m A ≥ 200, and this value decreases with increasing value of tan β.
In the scenario m 125 h (alignment), the predictions m h which are compatible with the measured Higgs mass are placed in the region of 4 < tan β < 13 and any values of m A [as shown in Fig. 9(d)]. In the allowed parameter region, the cross section reaches about 0.27 fb.
Overall, for production of the CP-even Higgs boson h 0 in association with a photon, the total cross section could reach a level of 10 −1 fb, depending on the model parameters. This renders the loop-induced process e − e + → h 0 γ in principle observable at a future electronpositron collider. Particularly, FCC ee produces high luminosity for Higgs, W, Z and top-quark researches, has multiple detectors, and could reach energies up to the top-pair threshold and beyond. As comparison with other linear e − e + colliders such as ILC and CLIC, the expected-luminosity at FCC ee is a factor of between 3 and 5 orders of magnitude larger than that proposed for a linear collider (as shown in Table II), at all energies from the Z-pole to the top-pair threshold, where precision measurements are to be made, hence where the collected statistics will be a key feature. For the expected high luminosity L = 10 4 fb −1 , the FCC ee can provide around two thousand events at √ s = 240 GeV. Therefore, the FCC ee can be expected to have a promising potential to detect the process e − e + → γh 0 . However, the basic size of the total cross section of e − e + → γh 0 can be enhanced by a factor of 4, depending on the polarizations of the initial e − and e + beams. Therefore, the ILC and CLIC which will be operated with beam polarization, also have an essential role to detect the process e − e + → γh 0 . The size of the total cross section is at a visible level of 10 −2 to 10 −1 fb. In particular, the total cross section reaches its largest values at low m A and tan β into the scan region. In the scenario m 125 h , the total cross section of the production of e − e + → γA 0 reaches about 1.22×10 −4 fb for the region of tan β ≥ 6 and value of m A = 400 GeV. For the scenario m 125 h ( τ ), the total cross section reaches about 5.9×10 −4 fb in the region of tan β ≥ 5 and m A = 200 GeV. In the m 125 h ( χ) scenario, the total cross section is about 7.7×10 −4 fb for tan β = 5 and m A = 200 GeV, and this value decreases with increasing value of tan β. In the m 125 h (alignment) scenario, the total cross section is about 7.9×10 −4 fb for tan β = 8 and m A = 400 GeV, and this value decreases with increasing value of tan β.
At the allowed parameter space of the considered scenarios, the size of total cross section of e − e + → γA 0 for √ s = 1 TeV is at a visible level of 10 −4 fb, and rather small. This renders the process e − e + → γA 0 at the border of observability. Consequently, the single production of the neutral Higgs bosons in association with a photon in electron-positron collisions appears to be observable for γh 0 , but it is very challenging for γA 0 at a e − e + collider.
V. CONCLUSION
In this study, the single production of the neutral Higgs bosons in association with a photon in electron-positron collisions has been analyzed in detail for both SM and MSSM, focusing on individual contributions from each type of one-loop diagrams, and polarizations of initial beams. This process has no amplitude at tree level and is hence directly sensitive to one-loop effects and the underlying Higgs dynamics. The four different benchmark scenarios m 125 h , m 125 h (light τ ), m 125 h (light χ) and m 125 h (alignment), have been used to illustrate the effect of the new physics in the framework of the MSSM. Note that all of the considered scenarios include a CP-even Higgs boson with mass about 125 GeV and couplings consistent with those of the discovered Higgs boson and a considerable part of their parameter space is allowed by the bounds from the researches for additional Higgs bosons and supersymmetric particles.
A remarkable deviations from the predictions of SM is seen at m 125 h (light χ) scenario where the production rate could be significantly enhanced. This scenario induces an enhancement in the production rate up to 25% of that predicted in the SM. In both SM and MSSM, the cross section is increased up to about 4 times by the longitudinal polarizations of the initial beams, compared with the unpolarized case. It is clear that the cross section is significantly dependent on the magnitudes of each oneloop amplitude and the relative phases between them in all of the considered scenarios. The σ MSSM (e + e − → hγ) is dominated by the s-channel triangle-and the bubbletype contributions at low center of mass energy. Note that the s triangle-type amplitude is mainly determined by triple couplings λ h 0 G + W − , λ h 0 H + W − λ h 0 G + H − , and λ h 0 G + G − . The bubble-type amplitude is mostly determined by triple couplings λ h 0 G + W − and λ h 0 H + W − , quatric couplings λ h 0 G + W − γ and λ h 0 H + W − γ .
The total cross section of e − e + → γh 0 as well as e − e + → γA 0 were scanned over the plane m A − tan β for all scenarios. The regions of the parameter space where the enhancement of the cross section is large enough to be observable at a future collider have been presented.
It should be emphasized that precise and modelindependent measurements for the single production of the neutral Higgs bosons in association with a photon would be possible at the future e − e + colliders ILC, CLIC, CEPC and FCC, and therefore the results of this study will be useful in detecting new physics signals based on MSSM, and in providing more precise limits on the corresponding couplings and masses.
FIG. 1 .
1Box-type diagrams contributing to the process e + e − → hγ at one-loop level.
FIG. 2 .
2Quartic interaction diagrams contributing to the process e + e − → hγ at one-loop level. and charged Higgs/Goldstone bosons, sfermions, are denoted by dashed-lines, and γ, Z, W bosons are denoted by wavy-lines.
Figure 1
1shows all possible box-type contributions, which have the loops of neutrino, electron, selectron e 1,2 , charginos χ ± 1,2 , neutralinos χ 0 1,2,3,4 , neutral Higgs bosons (h 0 , H 0 , A 0 , G 0 ), Z-boson, W -boson and charged Higgs/Goldstone bosons (H ± , G ± ).
[48] with results of 86 analyses.Moreover, the input parameters for SM are fixed as m h = 125.09 GeV, m W = 80.385 GeV, m Z = 91.1876 GeV, m pole t = 172.5 GeV, m b (m b ) = 4.18 GeV, α −1 = 137.036, and α(m 2 Z ) −1 = 127.934
FIG. 4 .
4(color online). The cross sections of process e − e + → γh 0 in the SM as a function of center of mass energy for a) individual amplitude from each type of one-loop diagrams, and b) various polarizations cases of initial beams. Also, the left-right asymmetry ALR is shown at the lower panel of figure (b).
FIG. 5 .
5(color online). The cross section of process e − e + → γh 0 in the SM as a 2D function of P e − and P e + at a) √ s = 250 GeV and b) √ s = 500 GeV, where the colour heat map corresponds to the total cross section (in fb) in the scan region. The red stars denote the unpolarized cross section at point (P e − , P e + ) = (0, 0).
4 at the left top corner, compared with the unpolarized case. At √ s = 250 GeV, the polarized cross section σ SM (P e − , P e + ) reaches up to σ SM (−0.8, +0.3) = 0.52 fb and σ SM (−0.8, +0.6) = 0.64 fb. At √ s = 500 GeV, the polarized cross section σ SM (P e − , P e + ) reaches up to σ SM (−0.8, +0.3) = 0.14 fb and σ SM (−0.8, +0.6) = 0.17 fb. B. The process e − e + → γh 0 in the MSSM In
FIG. 6 .FIG. 7 .
67(color online). Total cross section of process e − e + → γh 0 in the MSSM as a function of center of mass energy for each benchmark point defined in the scenarios of m 125 h , m 125 h (alignment), m 125 h (light τ ) and m 125 h (light χ). The vertical solid-lines indicate to the proposed energy of each future lepton colliders. The lower panel shows the ∆R ratio, the deviations from the predictions of SM. runs from 250 to 500 GeV, total cross section decreases from 0.27 to 0.064 fb in the m 125 h , m 125 h (light τ ) and m 125 h (alignment) scenarios, while 0.33 to 0.036 fb in the m 125 h (light χ) scenario. The remarkable deviations from the predictions of SM are seen at m 125 h (light χ) scenario such that at √ s = 250 GeV the production rate is enhanced by 25 percent whereas at √ s = 500 GeV reduced by 44 percent. In other scenarios, i.e. m 125 h , m 125 h (light τ ) and m 125 h (alignment), however, there is a deviation of about 4 percent from the predictions of SM. For the scenarios where all sparticles are too heavy to be produced directly at the selected center of mass energy, the MSSM contributions are small and will, therefore, be difficult to be detected. For m 125 h (light χ) scenario, the unpolarized cross section is around 0.32 fb, 0.09 fb, 0.07 fb and 0.036 fb for the planned CEPC (at √ s = 240 GeV), FCC ee (at √ s = 350 GeV), CLIC (at √ s = 380 GeV) and ILC (at √ s = 500 GeV) projects, respectively. For other scenarios, the unpolarized cross section reaches to values of 0.29 fb, 0.087 fb, 0.079 fb and 0.063 fb for the planned CEPC, FCC ee , CLIC-380 and ILC-500 projects, respectively. The energy-dependent structure of the cross section is appeared at value of √ s which is close to the 2 times mass of some particles, i.e., threshold effects. In Fig. 7, the polarized cross sections of e + e − → hγ are given as a function centre-of-mass energy in the range Note that for the other two scenarios, distributions of the polarized cross section are not shown here because they are similar to that of the m 125 h scenario. The total cross section with completely polarised left-handed electron and right-handed positron, σ MSSM (e − L e + R → hγ) for each benchmark points, can be enhanced by about a factor between 2.5 and 4, compared with the unpolarized case. The longitudinal polarization of both the positron and electron beams is, hence, significant to enhance the cross section. At √ s = 250 GeV, σ MSSM (e − L e + R → hγ) reaches to values of 0.93 fb and 1.16 fb for the benchmark points scenarios m 125 h and m 125 h (light χ), re-(color online). The polarized cross section of process e − e + → γh 0 for various polarizations of the initial beams as a function of center of mass energy for two benchmark points defined in the scenarios of a) m 125 h and b) m 125 h (light χ). Also, the left-right asymmetry ALR is shown at the lower panel of each figure.
. On the other hand, for BP3(m 125 h ( χ)), the polarized cross section σ MSSM (P e − , P e + ) reaches up to σ MSSM (
FIG. 8 .
8(color online). The individual contributions from each type of diagram to total cross section of process e − e + → γh 0 as a function of center of mass energy for benchmark points defined in the scenarios of a) m 125 h , b) m 125 h (light τ ), c) m 125 h (light χ) and d) m 125 h (alignment). 100 GeV ≤ m A ≤ 2 TeV and 0.5 ≤ tan β ≤ 50 for the m 125 h , m 125 h (light τ ) and m 125 h (light χ) scenarios, and 200 GeV ≤ m A ≤ 2 TeV and 1 ≤ tan β ≤ 20 for the m 125 h (alignment) scenario. Additionally, the predictions for the mass of light CP-even Higgs boson h 0 , m h = 122 GeV, 124 GeV, 125 GeV, 126 GeV and 127 GeV, are presented by the contour lines. The corresponding benchmark points are also marked by the red stars. It is clear that the total cross section decreases when both m A and tan β increase for the all cases. In particular, the cross section reaches its maximum values at small values of tan β into the scan region.
FIG. 9 .
9. The total cross section of the production of e − e + → γh 0 reaches about 0.27 fb for the region of tan β ≥ 6 and any values of m A . In the scenario m 125 h (light τ ), the m h predictions are smaller than 126 GeV all over the plane of m A − tan β, and the smallest value of tan β allowed by the uncertainty ∆m theory h = ±3 GeV is around 5 [as shown in Fig. 9(b)]. The total cross section of the production of e − e + → γh 0 reaches about 0.28 fb for the region of tan β ≥ 5 and m A ≥ 200. Additionally, at large tan β and low m A , total cross section reaches its biggest values. In the m 125 h (light χ) scenario, in spite of the decrement in X t , the predictions of m h display a mild increment with respect to the m 125 h scenario. However, these are m h < 127 GeV, except for upper-left corner in the plane m A − tan β. The smallest value of tan β allowed by ∆m theory h = ±3 GeV is around 5 [as shown in Fig. 9(c)]. The total cross section of the production of e − e + → γh 0 is about 0.34 fb for the region of 5 ≤ tan β ≤ 8 and (color online). Total cross section of process e − e + → h 0 γ as a 2D function of mA and tan β at √ s = 250 GeV for a) m 125 h scenario, b) m 125 h (light τ ) scenario, c) m 125 h (light χ) scenario and d) m 125 h (alignment) scenario. The colour heat map corresponds to the total cross section (in fb) in the scan region. The contour lines indicate predictions for the mass of the light CP-even higgs boson h 0 . The red stars denote the corresponding benchmark points from the last two rows in
FIG. 10. (color online). Total cross section of process e − e + → A 0 γ as a 2D function of mA and tan β at √ s = 1 TeV for each scenario. The colour heat map corresponds to the total cross section (in fb) in the scan region. The contour lines indicate predictions for the mass of the light CP-even higgs boson h 0 . C. The process e − e + → γA 0 in the MSSM In this study, the single production of the pseudoscalar Higgs boson in association with a photon in electronpositron collisions is also examined in the framework MSSM, considering full one-loop diagrams. Total cross section of e − e + → γA 0 is scanned over the plane of m A − tan β at √ s = 1 TeV as depicted in Fig. 10. The parameters m A and tan β are varying in the range of 100 GeV ≤ m A ≤ 900 GeV and 0.5 ≤ tan β ≤ 50 for the m 125 h , m 125 h (light τ ) and m 125 h (light χ) scenarios, and 200 GeV ≤ m A ≤ 900 GeV and 1 ≤ tan β ≤ 20 for the m 125 h (alignment) scenario. The total cross section decreases with increments of m A in all scenarios.
TABLE I .
ITriple and quartic Higgs couplings and couplings of the Higgs bosons to gauge-bosons which are included in each
type of diagrams.
Couplings
Box-type Triangle-type Bubble-type Quartic-type
SSS
TABLE II .
IIThe centre of mass energies √ s, integrated luminosities L and beam polarizations P , proposed at the future e + e − colliders ILC[3], CLIC[6], CEPC[7] and FCCee[8].III. DEFINITION OF THE BENCHMARK SCENARIOS IN MSSMThis section provides details of the benchmark scenarios considered in the present study. The four different benchmark scenarios, which are called as m 125 h ,Collider
√
s [GeV]
L [ab −1 ] (P e − ,P e + ) [%]
ILC250
250
2.0
(±80%, ∓30%)
ILC500
250 + 500
6.0
CLIC380
380
1.0
(±80%, 0)
CLIC1500
380 + 1500
2.5
CLIC3000 380 + 1500 + 3000
5.0
CEPC
240
5.6
(0, 0)
FCCee 240
240
5.0
(0, 0)
FCCee 365
240+365
6.5
m 125
h (light τ ), m 125
h (light χ) and m 125
h (alignment) pro-
posed in Refs.
TABLE III .
IIIThe input parameters for each scenario, where all masses are given in TeV. The symbol "*" means that At is taken as At = Xt + µ cot β. In the last two rows, the values of tan β and mA are given for benchmark points corresponding to each scenario.m 125
h
m 125
h (light τ ) m 125
h (light χ) m 125
h (align)
M Q 3 , D 3 , U 3 1.5
1.5
1.5
2.5
M L 3 , E 3
2.0
0.350
2.0
2.0
µ
1.0
1.0
0.180
7.5
M1
1.0
0.180
0.160
0.5
M2
1.0
0.300
0.180
1.0
M3
2.5
2.5
2.5
2.5
Xt
2.8
2.8
2.5
· · ·
At
*
*
*
6.25
A b
At
At
At
At
Aτ
At
0.800
At
At
BP1
BP2
BP3
BP4
tan β
10
10
10
10
mA
1.5
1.5
1.5
0.5
TABLE IV .
IVThe individual contributions from each type of diagram to total cross section of process e − e + → γh 0 at the different center of mass energies for all benchmark points corresponding to scenarios m 125h , m 125
h (light τ ), m 125
h (light χ)
and m 125
h (alignment).
BPs
√
s (GeV) box s triang bubble t channels All
BP1
240
0.227 23.93
23.42
0.108
0.270
350
0.185
6.07
6.98
0.093
0.087
500
0.116
1.70
1.79
0.061
0.063
BP2
240
0.228 23.54
23.03
0.107
0.269
350
0.185
5.98
6.87
0.093
0.087
500
0.117
1.67
1.76
0.061
0.064
BP3
240
0.227 23.56
22.52
0.107
0.325
350
0.186
5.78
6.73
0.093
0.090
500
0.118
1.46
1.73
0.061
0.036
BP4
240
0.227 26.40
25.89
0.107
0.268
350
0.186
6.79
7.75
0.093
0.087
500
0.118
1.89
1.99
0.061
0.063
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| [] |
[
"A Variance Reduction Method for Non-Convex Optimization with Improved Convergence under Large Condition Number",
"A Variance Reduction Method for Non-Convex Optimization with Improved Convergence under Large Condition Number",
"A Variance Reduction Method for Non-Convex Optimization with Improved Convergence under Large Condition Number",
"A Variance Reduction Method for Non-Convex Optimization with Improved Convergence under Large Condition Number"
] | [
"Zaiyi Chen \nDepartment of Computer Science\nUniversity of Science and Technology of China\nThe University of Iowa\n\n",
"Tianbao Yang [email protected] \nDepartment of Computer Science\nUniversity of Science and Technology of China\nThe University of Iowa\n\n",
"Zaiyi Chen \nDepartment of Computer Science\nUniversity of Science and Technology of China\nThe University of Iowa\n\n",
"Tianbao Yang [email protected] \nDepartment of Computer Science\nUniversity of Science and Technology of China\nThe University of Iowa\n\n"
] | [
"Department of Computer Science\nUniversity of Science and Technology of China\nThe University of Iowa\n",
"Department of Computer Science\nUniversity of Science and Technology of China\nThe University of Iowa\n",
"Department of Computer Science\nUniversity of Science and Technology of China\nThe University of Iowa\n",
"Department of Computer Science\nUniversity of Science and Technology of China\nThe University of Iowa\n"
] | [
"Workshop and Conference Proceedings",
"Workshop and Conference Proceedings"
] | In this paper, we propose a new SVRG-style accelerated stochastic algorithm for solving a family of non-convex optimization problems whose objective consists of a sum of n smooth functions and a non-smooth convex function. Our major goal is to improve the convergence of SVRG-style stochastic algorithms to stationary points under a setting with a large condition number c -the ratio between the smoothness constant and the negative curvature constant. The proposed algorithm achieves the best known gradient complexity when c ≥ Ω(n), which was achieved previously by a SAGA-style accelerated stochastic algorithm. Compared with the SAGA-style accelerated stochastic algorithm, the proposed algorithm is more practical due to its low memory cost that is inherited from previous SVRG-style algorithms. Compared with previous studies on SVRG-style stochastic algorithms, our theory provides much stronger results in terms of (i) reduced gradient complexity under a large condition number; and (ii) that the convergence is proved for a sampled stagewise averaged solution that is selected from all stagewise averaged solutions with increasing sampling probabilities instead of for a uniformly sampled solutions across all iterations. | null | [
"https://arxiv.org/pdf/1809.06754v3.pdf"
] | 119,735,969 | 1809.06754 | 34c2892fafc99badaaf59dada20912b8053dff56 |
A Variance Reduction Method for Non-Convex Optimization with Improved Convergence under Large Condition Number
14 Oct 2018
Zaiyi Chen
Department of Computer Science
University of Science and Technology of China
The University of Iowa
Tianbao Yang [email protected]
Department of Computer Science
University of Science and Technology of China
The University of Iowa
A Variance Reduction Method for Non-Convex Optimization with Improved Convergence under Large Condition Number
Workshop and Conference Proceedings
14 Oct 2018arXiv:1809.06754v2 [math.OC]
In this paper, we propose a new SVRG-style accelerated stochastic algorithm for solving a family of non-convex optimization problems whose objective consists of a sum of n smooth functions and a non-smooth convex function. Our major goal is to improve the convergence of SVRG-style stochastic algorithms to stationary points under a setting with a large condition number c -the ratio between the smoothness constant and the negative curvature constant. The proposed algorithm achieves the best known gradient complexity when c ≥ Ω(n), which was achieved previously by a SAGA-style accelerated stochastic algorithm. Compared with the SAGA-style accelerated stochastic algorithm, the proposed algorithm is more practical due to its low memory cost that is inherited from previous SVRG-style algorithms. Compared with previous studies on SVRG-style stochastic algorithms, our theory provides much stronger results in terms of (i) reduced gradient complexity under a large condition number; and (ii) that the convergence is proved for a sampled stagewise averaged solution that is selected from all stagewise averaged solutions with increasing sampling probabilities instead of for a uniformly sampled solutions across all iterations.
Introduction
The target of interest in this paper is to solve the following non-convex optimization problem:
min x∈R d φ(x) := 1 n n i=1 f i (x) + ψ(x)(1)
where each f i is a L-smooth and µ-weakly convex function, and ψ(x) is a "simple" closed convex function. The above problem covers constrained smooth optimization as a special case when ψ(x) is the indicator function of a convex set. A function f (·) is said to be µweakly convex if f (x)+ µ 2 x 2 is a convex function for µ > 0, where · denotes the Euclidean norm. By "simple", we mean that the proximal mapping for ψ(x) is easy to compute. This problem has broad applications in machine learning, and has been studied by a number of papers (Reddi et al., 2016b,a;Lan and Yang, 2018;Allen-Zhu, 2018;Allen-Zhu and Hazan, 2016). Several stochastic algorithms were proposed by utilizing the finite-sum structure of the problem to derive faster convergence than stochastic gradient methods. These algorithms are based on two well-known variance-reduction techniques, namely the SVRG-style variance reduction (Johnson and Zhang, 2013) and the SAGA-style variance reduction (Defazio et al., Table 1: Comparison of gradient complexities of variance reduction based algorithms for finding ǫ-stationary point of (1). The best complexity result for each setting is marked in red color. The top two algorithms, namely SAGA and RapGrad use the SAGA-style variance reduction technique, while others use the SVRG-style variance reduction technique. O(·) hides some logarithmic factor.
Algorithms L/µ ≥ Ω(n) L/µ ≤ O(n) SAGA (Reddi et al., 2016b) O(n 2/3 L/ǫ 2 ) O(n 2/3 L/ǫ 2 ) RapGrad (Lan and Yang, 2018) O
( √ nLµ/ǫ 2 ) O((µn + √ nLµ)/ǫ 2 ) SVRG (Reddi et al., 2016a) O(n 2/3 L/ǫ 2 ) O(n 2/3 L/ǫ 2 ) Natasha1 (Allen-Zhu, 2017a) NA O(n log(L/(ǫµ)) + n 2/3 L 1/3 µ 2/3 /ǫ 2 ) RepeatSVRG (Allen-Zhu, 2017a) O(n 3/4 √ Lµ/ǫ 2 ) O((µn + n 3/4 √ Lµ)/ǫ 2 ) Stagewise-Katyusha (this work) O( √ nLµ/ǫ 2 ) O((µn + L)/ǫ 2 )
2014; Roux et al., 2012). The key difference between these two variance reduction techniques it that SVRG uses a full gradient that is computed periodically and SAGA uses a full gradient that is computed from its maintained historical gradients for each component f i . Due to this difference, SAGA might require much higher memory than SVRG for many problems (e.g., learning deep neural networks), which renders algorithms of SVRG-style more favorable than algorithms of SAGA-style. This paper focuses on the case when the condition number L/µ is very large (e.g., L/µ > n), and aims to provide faster convergence for a SVRG-style stochastic algorithm. We summarize below the key features of the proposed algorithm and its convergence guarantee in comparison with previous results.
The proposed algorithm is following a stagewise framework, which has been commonly adopted for solving problems with weakly convex functions. At each stage, an accelerated SVRG-style algorithm is called to minimize φ(x) + µ 2 x 2 to a certain degree. We propose a variant based on Katyusha for convex problems (Allen-Zhu, 2017b). We refer to the proposed algorithm as Stagewise-Katyusha. The novelty of the proposed algorithm lies on a tailored setting of the involved parameters in Katyusha to adapt to the µ-weakly convexity of the problem. By a refined analysis of the gradient complexity of the proposed variant of Katyusha, we establish that for finding an ǫ-level stationary point (i.e., the magnitude of the proximal gradient is less than ǫ in expectation) the proposed algorithm has a gradient complexity of O( √ nµL log(L/µ)/ǫ 2 ) when L/µ ≥ Ω(n), and a gradient complexity of
O((nµ + L) log(L/µ)/ǫ 2 ) when L/µ ≤ Ω(n).
We compare our results with previous works on SVRG-style and SAGA-style stochastic algorithms for (1) in Table 1 1 . It is notable that for the setting of large condition number, the complexity of Stagewise-Katyusha matches that of RapGrad (Lan and Yang, 2018). However, RapGrad is a SAGA-style stochastic algorithm, hence requiring a large memory cost. Moreover, the complexity of Stagewise-Katyusha is better than SVRG and SAGA when L/µ ≥ log(L/µ)n 1/3 . Natasha1 has a better complexity for a small condition number, but it is not applicable for problems with a large condition number. Finally, we note that 1. After our first manuscript was posted online, it was brought to our attention that the updated arXiv manuscript (Allen-Zhu, 2018, V5) reported a new result for RepeatSVRG for our considered problem, which is in the same order as the result achieved in this work.
our algorithm is more practical than Natasha1/SVRG/SAGA/RepeatSVRG/RapGrad in that we do not use randomly selected intermediate solutions to proceed as in Natasha1, do not return a uniformly sampled solution across all iterations as in SVRG/SAGA, do not require setting ǫ aprior as in RepeatSVRG, and do not require a large memory cost as in SAGA/RapGrad. For our algorithm, the returned solution is a stagewise average solution that is sampled from all stagewise average solutions with increasing sampling probabilities, which also provides a better justification of using the last averaged solution in practice.
Stagewise-Katyusha
In this section we present the Stagewise-Katyusha algorithm and its analysis. We first present some notations, which is almost duplicate of that from (Chen et al., 2018).
For problem (1), a point x ∈ dom(ψ) is a first-order stationary point if 0 ∈ ∂φ(x), where ∂φ denotes the partial gradient of φ. Moreover, a point x is said to be ǫ-stationary if dist(0, ∂φ(x)) ≤ ǫ.(2)
where dist denotes the Euclidean distance from a point to a set. For any function f and λ > 0, the following function is called a Moreau envelope of f
f λ (x) = min z f (z) + 1 2λ z − x 2 .
Further, the optimal solution to the above problem denoted by
prox λf (x) = arg min z f (z) + 1 2λ z − x 2
is called a proximal mapping of f . A small norm of ∇f λ (x) has an interpretation that x is close to a point that is nearly stationary. In particular for any x ∈ R d , let x = prox λf (x), then we have
f ( x) ≤ f (x), x − x = λ ∇f λ (x) , dist(0, ∂f ( x)) ≤ ∇f λ (x) .(3)
This means that a point x satisfying ∇f λ (x) ≤ ǫ is close to a point in distance of O(ǫ) that is ǫ-stationary. Below, we will prove the convergence in terms of ∇φ γ (x) for some γ > 0 and discuss its implication for solving smooth problems with ψ = 0. To connect with the convergence measures in (Lan and Yang, 2018), we will also measure the convergence in terms of x − x .
Algorithm
The Stagewise-Katyusha algorithm is presented in Algorithm 1, which falls into the same framework presented in (Chen et al., 2018) with a modified Katyusha employed at each stage for solving the regularized subproblem f s (x), which is assumed to be σ-strongly convex and haveL-Lipschitz continuous gradients for the smooth components. The modified Katyusha is presented in Algorithm 2. Given the way that f s is constructed, we can write it as
f s (x) = 1 n n i=1 (f i (x) + µ 2 x − x s−1 2 f i (x) ) + γ −1 − µ 2 x − x s−1 2 + ψ(x) ψ (x) Algorithm 1 Stagewise-Katyusha 1: Initialize: non-decreasing positive weights {w s }, x 0 ∈ dom(ψ), γ = (2µ) −1 2: for s = 1, . . . , S + 1 do 3: Let f s (·) = φ(·) + 1 2γ · −x s−1 2 4: x s = Katyusha(f s , x s−1 , K s , µ, L + µ) 5: end for 6: Return: x τ +1 , τ is randomly chosen from {0, . . . , S} according to probabilities p τ = w τ +1 S k=0 w k+1 , τ = 0, . . . , S. Algorithm 2 Katyusha(f, x 0 , K, σ,L) 1: Initialize: τ 2 = 1 2 , τ 1 = min{ nσ 3L , 1 2 }, η = 1 3τ 1 L , θ = 1 + ησ, m = ⌈ log(2τ 1 +2/θ−1) log θ ⌉ + 1 2: y 0 = ζ 0 = x 0 ← x 0 3: for k = 0, . . . , K − 1 do 4: u k = ∇f ( x k ) 5: for t = 0, . . . , m − 1 do 6: j = km + t 7: x j = τ 1 ζ j + τ 2 x k + (1 − τ 1 − τ 2 )y j 8: ∇ j+1 = u k + ∇f i (x j+1 ) − ∇f i ( x k ) 9: ζ j+1 = arg min ζ 1 2η ζ − ζ j 2 + ∇ j+1 , ζ + ψ(ζ) 10: y j+1 = arg min y 3L 2 y − x j+1 2 + ∇ j+1 ,compute x k+1 = m−1 t=0 θ t y sm+t+1 m−1 j=0 θ t 13: end for 14: Output x K It is easy to see thatf i (x) is convex andL = (L + µ)-smooth, andψ(x) is σ = (γ −1 − µ)
strongly convex, which satisfy the conditions made in (Allen-Zhu, 2017b). In each call of the modified Katyusha,f i is considered as the smooth component, andψ is considered as the non-smooth regularizer. The key difference between our modified Katyusha and the original Katyusha algorithm for solving smooth and strongly convex problems in (Allen-Zhu, 2017b) lies at the setting of τ 1 , m and K. For example in (Allen-Zhu, 2017b), the value of τ 1 is set to τ 1 = min( mσ/3L, 1/2). However, in our modified Katyusha the value of τ 1 is independent of m. The value of m is also different from that suggested in (Allen-Zhu, 2017b), which is suggested to 2n. The value of K (the number of epochs) in the original Katyusha is chosen such that the objective gap is less than ǫ. In our modified Katyusha, it is set to make sure that the objective function f s (x) is decreased by a sufficient amount. Actually, we do not solve min x f s (x) to an ǫ-accuracy level in terms of the objective value. Below, we present the gradient complexity of Stagewise-Katyusha (i.e., the order of number of evaluations of ∇φ i (x)).
Assumption 1 For problem (1), we assume that (i) f i (·) is L-smooth and µ-weakly convex, (ii) ψ is a non-smooth convex function, and (iii) there exists
∆ φ > 0 such that φ(x 0 ) − min x φ(x) ≤ ∆ φ .
Theorem 1 Suppose Assumption 1 holds. Let w s = s α , α > 0, γ = 1 2µ ,L = L + µ, σ = µ, and in each call of Katyusha let τ 1 = min{ nσ 3L , 1 2 }, step size η = 1 3τ 1L
, τ 2 = 1/2, θ = 1 + ησ,
and K s = log(D s ) m log(θ) , m = log(2τ 1 + 2/θ − 1) log θ + 1,
where D s = max{12L/µ,L 3 /µ 3 , 4L 2 s/µ 2 }. Then we have that
max{E[ ∇φ γ (x τ +1 ) 2 ], E[L 2 x τ +1 − z τ +1 2 ]} ≤ 34µ∆ φ (α + 1) S + 1 + 49µ∆ φ (α + 1) (S + 1)α I α<1 ,
where z = prox γφ (x), τ is randomly chosen from {0, . . . , S} according to probabilities p τ = w τ +1 S k=0 w k+1 , τ = 0, . . . , S. Furthermore, the total gradient complexity for finding
x τ +1 such that max(E[ ∇φ γ (x τ +1 ) 2 ], L 2 E[ x τ +1 − z τ +1 2 ]) ≤ ǫ 2 is N (ǫ) = O (µn + nµL) log L µǫ 1 ǫ 2 , n ≥ 3L 4µ , O nLµ log L µǫ 1 ǫ 2 , n ≤ 3L 4µ .
Indeed, when ψ = 0 we can derive a slightly stronger result stated in the following theorem.
Theorem 2 Suppose Assumption 1 holds and ψ = 0. With the same parameter values as in Theorem 1 except that K = log(D) m log(θ) , where D = max(8L/µ,L 3 /µ 3 ). The total gradient complexity for finding
x τ +1 such that E[ ∇φ(x τ +1 ) 2 ] ≤ ǫ 2 is N (ǫ) = O (µn + nµL) log L µ 1 ǫ 2 , n ≥ 3L 4µ , O nLµ log L µ 1 ǫ 2 , n ≤ 3L 4µ .
Remark: Our results in the above two theorems match that in (Lan and Yang, 2018). Indeed, our result in Theorem 1 is slightly more general than that in (Lan and Yang, 2018), which only considers the constrained smooth optimization with ψ being the indicator function of a convex set.
Analysis
We need the following lemma for our analysis.
Lemma 3 (Allen-Zhu, 2017b) Regarding the modified-Katyusha algorithm, suppose that
τ 1 ≤ 1 3ηL , τ 2 = 1/2. Defining D t := f (y t ) − f (x), D k := f ( x k ) − f (x)
for any x, conditioned on iterations {0, . . . , t − 1} in k-th epoch and all iterations before k-th epoch, we have that
0 ≤ (1 − τ 1 − τ 2 ) τ 1 D t − 1 τ 1 D t+1 + τ 2 τ 1 E[ D k ] + 1 2η ζ t − x 2 − 1 + ησ 2η E[ ζ t+1 − x 2 ](4)
Proof [of Theorem 1] First, we verify K has a valid value. Overall, we need
θ −mK ≤ min µ 4L , µ 3 L 3 , µ 2 L 2 s
Define D s = max{4L/µ,L 3 /µ 3 , L 2 s/µ 2 } ≥ 16. We can set K = ⌈ log(Dmax) m log θ ⌉. Then,
K ≥ log(D s ) m log(1 + ησ) ≥ log(4) log(2τ 1 + 2/θ − 1) + log θ ≥ 1
where the last inequality follows that 2/θ ≥ 2τ 1 + 2/θ − 1 ≥ 1 always according to the setting of τ 1 = min{ nµ
3L
, 1 2 } and η = 1 3τ 1L
.
One call of Katyusha Define θ = 1 + ησ and multiply (4) by θ t on both side. By summing up the inequalities in (4) in the k-th epoch, we have that
0 ≤E k 1 − τ 1 − τ 2 τ 1 m−1 t=0 D km+t θ t − 1 τ 1 m−1 t=0 D km+t+1 θ t + τ 2 τ 1 E k+1 [ D k ] m−1 t=1 θ t + 1 2η ζ km − x * 2 − θ m 2η E k+1 [ ζ (k+1)m − x * 2 ]
where x * = arg min x f (x), E k [·] denotes expectation in k-th epoch conditional on 0, . . . , k −1 epochs. Using the convexity of f (·), we have that
τ 1 + τ 2 − 1 + 1/θ τ 1 θE k [ D k+1 ] m−1 t=0 θ t + 1 − τ 1 − τ 2 τ 1 θ m D (k+1)m + θ m 2η E k [ ζ (k+1)m − x * 2 ] ≤ τ 2 τ 1 D k m−1 t=0 θ t + 1 − τ 1 − τ 2 τ 1 D km + 1 2η ζ km − x * 2(5)
Substituting τ 2 = 1/2 and m ≤ log(2τ 1 +2/θ−1) log θ + 1, we have that
θ m 1 2τ 1 E k [ D k+1 ] m−1 t=0 θ t + 1/2 − τ 1 τ 1 θ m D (k+1)m + θ m 2η E k [ ζ (k+1)m − x * 2 ] ≤ 1 2τ 1 D k m−1 t=0 θ t + 1/2 − τ 1 τ 1 D km + 1 2η ζ km − x * 2
Telescoping above inequality over all epochs k = 0, . . . , K − 1 we have that
E k [ D K ] ≤ 2τ 1 θ −mK 1 2τ 1 D 0 + 1/2 − τ 1 τ 1 m−1 t=0 θ t D 0 + 1 2η m−1 t=0 θ t ζ 0 − x * 2 Since m−1 t=0 θ t ≥ 1, and τ 1 ≤ 1 2 we have E k [ D K ] ≤ 2τ 1 θ −mK ( 1 − τ 1 τ 1 D 0 + 1 2η ζ 0 − x * 2 )(6)
We can use the same analysis by plugging
x = ζ 0 in (4) to prove that E[f ( x K ) − f ( x 0 )] ≤ 0
-an objective value decreasing property that will be used later.
Convergence of ∇φ γ (·) : Let z s = arg min x f s (x) and x * denote the global minimum of min x φ(x). It is notable that x s−1 − z s /γ = ∇φ γ (x s−1 ). Below, we will use K to denote K s . Applying the above analysis to the s-th call of Katyusha, we have
E s [f s (x s ) − f s (z s )] ≤ 2θ −mK (f s (x s−1 ) − f s (z s )) + θ −mK 2η x s−1 − z s 2(7)
It is easy to see that
f s (x s−1 ) − f s (z s )) = φ(x s−1 ) − φ(z s ) − 1 2γ x s−1 − z s 2 ≤ φ(x s−1 ) − φ(x * ) − 1 2γ x s−1 − z s 2(8)
Thus, we have
E s [f s (x s ) − f s (z s )] ≤ 2θ −mK (φ(x s−1 ) − φ(x * )) + θ −mK 1 2η x s−1 − z s 2 ≤ 2θ −mK (φ(x s−1 ) − φ(x * )) + θ −mKL x s−1 − z s 2 Es(9)
By the strong convexity of f s , we have
E[ x s − z s 2 ] ≤ 2 σ E s(10)
Similar to the analysis in (Chen et al., 2018), we have
(1 − α s ) 2γ E s x s−1 − z s 2 ≤ E s [∆ s ] + (α −1 s − 1) 2γ E s [ x s − z s 2 ] + E s ≤ E s [∆ s ] + (α −1 s − 1) + γσ γσ E s ≤ E s [∆ s ] + (α −1 s − 1) + γσ γσ 2θ −mK (φ(x s−1 ) − φ(x * )) + θ −mKL x s−1 − z s where ∆ s = φ(x s−1 ) − φ(x s )
. Substituting α s = 1/2, γ = 1/(2µ), and σ = µ,L ≤ 2L and θ −mK ≤ µ/(12L), we have that
1 8γ x s−1 − z s 2 ≤ E s [∆ s ] + 6θ −mK (φ(x s−1 ) − φ(x * )),(11)
i.e.,
∇φ γ (x s−1 ) 2 ≤ E s [8∆ s /γ] + 48θ −mK (φ(x s−1 ) − φ(x * ))/γ,(12)
Multiplying both sides by w s , we have that
w s E s [ ∇φ γ (x s−1 ) 2 ] ≤ E s 8w s ∆ s /γ + 48θ −mK w s (φ(x s−1 ) − φ(x * ))/γ
By summing over s = 1, . . . , S + 1, we have
E[ S+1 s=1 w s ∇φ γ (x s−1 ) 2 ] ≤ E 8 γ S+1 s=1 w s ∆ s + 48 γ S+1 s=1 w s θ −mK (φ(x s−1 ) − φ(x * ))
Taking the expectation w.r.t. τ ∈ {0, . . . , S}, we have that
E[ ∇φ γ (x τ ) 2 ]] ≤ E 8 S+1 s=1 w s ∆ s γ S+1 s=1 w s + 48 S+1 s=1 w s θ −mK (φ(x s−1 ) − φ(x * )) γ S+1
s=1 w s Next, we bound the numerators of the two terms in the above bound. For the first term in the above bound, we have
S+1 s=1 w s ∆ s = S+1 s=1 w s (φ(x s−1 ) − φ(x s )) = S+1 s=1 (w s−1 φ(x s−1 ) − w s φ(x s )) + S+1 s=1 (w s − w s−1 )φ(x s−1 ) = w 0 φ(x 0 ) − w S+1 φ(x S+1 ) + S+1 s=1 (w s − w s−1 )φ(x s−1 ) = S+1 s=1 (w s − w s−1 )(φ(x s−1 ) − φ(x S+1 ))
Taking expectation on both sides, we have
E S+1 s=1 w s ∆ s = S+1 s=1 (w s − w s−1 )E[(φ(x s−1 ) − φ(x S+1 ))] ≤ S+1 s=1 (w s − w s−1 )[φ(x 0 ) − φ(x * )] ≤ ∆ φ w S+1 where we use the fact that E[f s (x s ) − f s (x s−1 )] ≤ 0 (this is the objective value decreasing property of Katyusha) implying E[φ(x s ) − φ(x s−1 )] ≤ 0 and hence E[φ(x s )] ≤ φ(x 0 ) for s ≥ 0.
For the second term, we can do similar analysis having
E S+1 s=1 w s θ −mK (φ(x s−1 ) − φ(x * )) ≤ S+1 s=1 w s θ −mK E[φ(x s−1 ) − φ(x * )] ≤ ∆ φ S+1 s=1 w s θ −mK
As a result,
E[ ∇φ γ (x τ ) 2 ]] ≤ 8∆ φ w S+1 γ S+1 s=1 w s + 48∆ φ S+1 s=1 w s θ −mK γ S+1 s=1 w s ≤ 8∆ φ w S+1 γ S+1 s=1 w s + 12∆ φ S+1 s=1 w s s −1 γ S+1 s=1 w s
where we use the fact θ −mK ≤ 1/(4s). Then by simple algebra (cf. (Chen et al., 2018)), we have
E[ ∇φ γ (x τ ) 2 ] ≤ 16µ∆ φ (α + 1) S + 1 + 24µ∆ φ (α + 1) (S + 1)α I α<1
Due to the objective decreasing property, we have
E[φ(x s ) + 1 2γ x s − x s−1 2 − φ(x s−1 )] ≤ 0,
which implies by a similar analysis
1 2γ E[ x τ +1 − x τ 2 ] ≤ ∆ φ (α + 1) S + 1
Since φ γ (x) has (γ −1 −µ)-Lipschitz continuous gradient (cf. Lemma 2.1 in (Drusvyatskiy and Paquette, 2018)), then we have
E[ ∇φ γ (x τ +1 ) 2 ] ≤ 2E[ ∇φ γ (x τ ) 2 ] + 2(γ −1 − µ) 2 E[ x τ +1 − x τ 2 ] ≤ 2E[ ∇φ γ (x τ ) 2 ] + 2µ∆ φ (α + 1) S + 1 ≤ 34µ∆ φ (α + 1) S + 1 + 48µ∆ φ (α + 1) (S + 1)α I α<1
To proceed, from (10) we have
L 2 x s − z s 2 ≤ 2L 2 σ (2θ −mK (φ(x s−1 ) − φ(x * )) + θ −mK 4L x s−1 − z s 2 ) ≤ 4L 2 θ −mK σ (φ(x s−1 ) − φ(x * )) + 2L 3 θ −mK µ 3 ∇φ γ (x s−1 ) 2 ≤ 4L 2 θ −mK σ (φ(x s−1 ) − φ(x * )) + 2 ∇φ γ (x s−1 )| 2
where we use the fact x s−1 − z s /γ = ∇φ γ (x s−1 ) and θ −mK ≤ µ 3 /L 3 . Then following the same analysis as above,
E[L 2 x τ +1 − z τ +1 2 ] ≤ 4L 2 ∆ φ S+1 s=1 w s θ −mK σ S+1 s=1 w s + 2E[ ∇φ γ (x τ )| 2 ]
Since θ −mK ≤ µ 2 /(4L 2 s), then
E[L 2 x τ +1 − z τ +1 2 ] ≤ 16µ∆ φ (α + 1) S + 1 + 49µ∆ φ (α + 1) (S + 1)α I α<1
When ψ(·) = 0 and considering α as a constant, we have
E[ ∇φ(x τ +1 ) 2 ] ≤E[ ∇φ(x τ +1 ) − ∇φ(z τ +1 ) + ∇φ(z τ +1 ) 2 ] ≤ E[2L 2 x τ +1 − z τ +1 2 + 2 ∇φ γ (x τ ) 2 ] ≤O µ∆ φ S + 1
Indeed, for ψ(·) = 0, we can do slightly better by bounding f s (x s−1 )−f s (z s ) ≤L 2 x s−1 −z s 2 .
Then E s becomes 2θ −mKL x s−1 −z s 2 and θ −mK (φ(x s−1 )−φ(z s )) in the proceeding analysis is gone, which removes the requirement θ −mK ≤ µ 2 /(4L 2 s). As a result, we can set K = ⌈log(D)/(m log θ)⌉, where D = max(24L/µ,L 3 /µ 3 ).
Gradient Complexity:
Finally, we analyze the gradient complexity. Let us consider the gradient complexity at the s-th stage, which is (n + m)K ≤ 2 log(D s ) log(2τ 1 + 2τ 1 ηµ + 1 − ηµ) n + 2 log(D s ) log(1 + ηµ)
Let τ 1 = c ηµ , where 0 ≤ c = µ 3L ≤ 1 3 . We have that (n + m)K = nK + mK = log(D s ) log(2τ 1 + 2τ 1 ηµ + 1 − ηµ) n + log(D s ) log(1 + ηµ) ≤ log(D s ) log(2τ 1 + 2c + 1 − c τ 1 ) n + log(D s ) log(1 + c τ 1 )
We analyze two cases. Case 1: If n ≥ 3L 4µ , then τ 1 = 1 2 , we have that (n + m)K ≤ O log(D s )n + log D s log(1 + 2c)
Since c ≤ 1/3 so log(1 + 2c) ≥ c, then (n + m)K ≤ O (n +L µ ) log D s .
Then the total gradient complexity for finding E ∇φ γ (x τ ) 2 ≤ ǫ 2 is O((µn+L) log(L/(µǫ)).
Case 2: If n ≤ 3L 4µ , then τ 1 = nµ 3L ∈ (0, 1 2 ]. We have that following inequalities hold
τ 1 c = 3nL µ ≤ 3L 2µ
, log( τ 1 + c c ) ≤ log 3L µ , log(1 + c/τ 1 ) ≥ c/(2τ 1 ) and due to if 2τ 1 + 2c − c τ 1 ≤ 1/2, then log(2τ 1 + 2c + 1 − c τ 1 ) ≥ τ 1 + c − c/(2τ 1 )
if 2τ 1 + 2c − c τ 1 ≥ 1/2, then log(2τ 1 + 2c + 1 − c τ 1 ) ≥ log(1.5)
we have 1 log(2τ 1 + 2c + 1 − c τ 1 ) ≤ max 1 log(1.5)
, 1 nµ 3L
+ µ 3L − µ 3nL ≤ O L nµ .
Thus we have (n + m)K ≤ O nL µ logL µ , and the total gradient complexity for finding E ∇φ γ (x τ ) 2 ≤ ǫ 2 is O( √ µnL log(L/(µǫ)).
Conclusion
In this paper, we have developed a SVRG-style accelerated stochastic algorithm for solving a family of non-convex optimization problems whose objective consists of a finite-sum of smooth functions and a non-smooth convex function. We proved that the gradient complexity can be improved when the condition number is very large compared to the number of smooth components, which matches an existing result based SAGA-style stochastic algorithm.
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| [] |
[
"ON THE RADIUS OF SPATIAL ANALYTICITY FOR SOLUTIONS OF THE DIRAC-KLEIN-GORDON EQUATIONS IN TWO SPACE DIMENSIONS",
"ON THE RADIUS OF SPATIAL ANALYTICITY FOR SOLUTIONS OF THE DIRAC-KLEIN-GORDON EQUATIONS IN TWO SPACE DIMENSIONS"
] | [
"Sigmund Selberg "
] | [] | [] | We consider the initial value problem for the Dirac-Klein-Gordon equations in two space dimensions. Global regularity for C ∞ data was proved by Grünrock and Pecher. Here we consider analytic data, proving that if the initial radius of analyticity is σ 0 > 0, then for later times t > 0 the radius of analyticity obeys a lower bound σ(t) ≥ σ 0 exp(−At). This provides information about the possible dynamics of the complex singularities of the holomorphic extension of the solution at time t. The proof relies on an analytic version of Bourgain's Fourier restriction norm method, multilinear space-time estimates of null form type and an approximate conservation of charge. | 10.1016/j.anihpc.2018.12.002 | [
"https://arxiv.org/pdf/1901.08416v1.pdf"
] | 119,317,245 | 1901.08416 | 15ad41c89f99ae3487a1b4b472bf1265b3776766 |
ON THE RADIUS OF SPATIAL ANALYTICITY FOR SOLUTIONS OF THE DIRAC-KLEIN-GORDON EQUATIONS IN TWO SPACE DIMENSIONS
24 Jan 2019
Sigmund Selberg
ON THE RADIUS OF SPATIAL ANALYTICITY FOR SOLUTIONS OF THE DIRAC-KLEIN-GORDON EQUATIONS IN TWO SPACE DIMENSIONS
24 Jan 2019
We consider the initial value problem for the Dirac-Klein-Gordon equations in two space dimensions. Global regularity for C ∞ data was proved by Grünrock and Pecher. Here we consider analytic data, proving that if the initial radius of analyticity is σ 0 > 0, then for later times t > 0 the radius of analyticity obeys a lower bound σ(t) ≥ σ 0 exp(−At). This provides information about the possible dynamics of the complex singularities of the holomorphic extension of the solution at time t. The proof relies on an analytic version of Bourgain's Fourier restriction norm method, multilinear space-time estimates of null form type and an approximate conservation of charge.
Introduction
We consider the Cauchy problem for the Dirac-Klein-Gordon (DKG) equations in two space dimensions, (1) (−i∂ t − iα · ∇ + M β)ψ = φβψ, ψ(0, x) = ψ 0 (x),
(∂ 2 t − ∆ + m 2 )φ = βψ, ψ , (φ, ∂ t φ)(0, x) = (φ 0 , φ 1 )(x),
where the unknowns ψ (the Dirac spinor) and φ (the meson field) are functions of (t, x) ∈ R × R 2 and take values in C 2 and R, respectively, and ψ = (ψ 1 , ψ 2 ) t is considered as a column vector upon which the Dirac matrices (in fact, the Pauli matrices) α 1 = 0 1 1 0 , α 2 = 0 −i i 0 and β = 1 0 0 −1 may act. The standard inner product on C 2 is denoted ·, · . We write x = (x 1 , x 2 ), ∂ j = ∂ ∂x j , ∇ = (∂ 1 , ∂ 2 ), ∆ = ∂ 2 1 + ∂ 2 2 and α · ∇ = α 1 ∂ 1 + α 2 ∂ 2 . The masses M and m are given real constants. We shall assume m > 0.
In particle physics, DKG arises as a model for forces between nucleons, mediated by mesons; see [3]. The well-posedness of the Cauchy problem in space dimensions d ≤ 3 with data in the family of Sobolev spaces H s (R d ) = W s,2 (R d ) has been extensively studied; see [9,12,11,16,21,28,2,7] and the references therein.
Our aim in this article is to add to the large-data global regularity theory in space dimension d = 2. Global regularity for C ∞ (R 2 ) data was proved by Grünrock and Pecher [16]. Our focus here is on spatial analyticity, with a uniform radius of analyticity σ(t) > 0 for each time t. By this we mean that the solution at time t has a holomorphic extension to the complex strip S σ = x + iy ∈ C 2 : x, y ∈ R 2 and |y 1 |, |y 2 | < σ with σ = σ(t).
Heuristically, the picture one should have in mind is that σ(t) is the distance from R 2
x to the nearest complex singularity of the holomorphic extension of the solution at time t. We will prove a lower bound
σ(t) ≥ σ 0 exp(−At)
as t → ∞, providing some information about the possible dynamics of the complex singularities.
The proof of global C ∞ regularity in [16] makes use of Bourgain's Fourier restriction norm method, and a key motivation behind the present paper was to investigate to which extent the analytic version of this method-introduced by Bourgain in [6, Section 8]-can yield refined information about the regularity of the solution for analytic data. A further motivation was a recent result of Cappiello, D'Ancona and Nicola [8] (see also [1]) on persistence of spatial analyticity for C ∞ solutions of semilinear symmetric hyperbolic systems, which in the special case of DKG 1 yields a lower bound
σ(t) ≥ σ 0 exp −A t 0 (1 + ψ(s) L ∞ + φ(s) L ∞ + ∂φ(s) L ∞ ) ds .
This is weaker than our lower bound σ(t) ≥ σ 0 exp(−At), since the best estimate known on the L ∞ norm of the solution of (1) appears to be O(exp(Ct)), which can be obtained from the global existence proof in [16], hence one would get a double exponential decay rate σ(t) ≥ σ 0 exp(−A exp(Ct)).
The investigation of spatially uniform lower bounds on the radius of analyticity for nonlinear evolutionary PDE was initiated by Kato and Masuda [19], and by now there is an extensive catalog of results along these lines for various nonlinear PDE, including the Kadomtsev-Petviashvili equation [6], the (generalized) Korteweg-de Vries equation [18,4,24], the Euler equations [20], the cubic Szegő equation [15] and the nonlinear Schrödinger equation [17,5,27].
Since the radius of analyticity can be related to the asymptotic decay of the Fourier transform, it is natural to use Fourier methods to study the type of problem outlined above. We take data in the analytic Gevrey class G σ,s = G σ,s (R 2 ), defined for σ > 0 and s ∈ R by
G σ,s (R 2 ) = f ∈ L 2 (R 2 ) : f G σ,s < ∞ , f G σ,s = e σ ξ ξ s f (ξ) L 2 ξ .
Here we denote, for ξ = (ξ 1 , ξ 2 ) ∈ R 2 ,
ξ = |ξ 1 | + |ξ 2 |, |ξ| = ξ 2 1 + ξ 2 2 1/2 , ξ = (1 + |ξ| 2 ) 1/2 , and f (ξ) = F f (ξ) = R 2 e −ix·ξ f (x) dx
is the Fourier transform. Note that G σ,s = F −1 (e −σ · · −s L 2 ) is isometrically isomorphic to L 2 and hence is a Banach space. We recall the fact that every f ∈ G σ,s has a uniform radius of analyticity σ, that is, f has a holomorphic extension to S σ (for a proof see, e.g., [23]). Our main result is the following.
1 DKG can be written as a semilinear symmetric hyperbolic system with unknown u = (ψ 1 , ψ 2 , φ, ∂ 1 φ, ∂ 2 φ, ∂tφ) ⊺ .
Theorem 1. Consider (1) with m > 0. Let σ 0 > 0. Given initial data (2) (ψ 0 , φ 0 , φ 1 ) ∈ G σ0,0 (R 2 ; C 2 ) × G σ0,1/2 (R 2 ; R) × G σ0,−1/2 (R 2 ; R),
let (ψ, φ) be the unique global C ∞ solution of (1), as obtained in [16]. Then for any T > 0 we have
(ψ, φ, ∂ t φ) ∈ C([−T, T ]; G σ(T ),0 × G σ(T ),1/2 × G σ(T ),−1/2 ), where σ(T ) = σ 0 e −AT
for some constant A > 0 depending on σ 0 and the norm of the data. Thus, for any time t ∈ R, the solution has a uniform radius of analyticity at least σ(|t|).
We have no reason to expect that this bound is optimal, but it does appear to be the best possible with the technique used in the proof, which is based on an analytic version of Bourgain's Fourier restriction norm method, multilinear spacetime estimates of null form type and an approximate version of the conservation of charge
(3) R 2 |ψ(t, x)| 2 dx = const.
We now describe in more detail the method of proof. The point of departure is the observation that the norm on G σ,s is obtained from the Sobolev norm
f H s = ξ s f (ξ) L 2 ξ by the substitution f −→ e σ D f = F −1 e σ · F f .
Applying the same substitution in the setting of Bourgain's Fourier restriction norm method, the space X s,b then yields the analytic space X σ,s,b . This idea was used by Bourgain [6,Theorem 8.12] to study spatial analyticity for the Kadomtsev-Petviashvili equation, but the argument applies to a class of dispersive PDE in general, as discussed in [23]. In brief summary, the consequences that can be abstracted from Bourgain's argument are the following.
(B1) If local well-posedness of some nonlinear dispersive PDE can be proved for H s initial data by a contraction argument in X s,b , then one also has local well-posedness for data in G σ0,s for any σ 0 > 0. (B2) If, moreover, the solution extends globally in time (so the H s norm does not blow up in finite time), then the solution remains spatially analytic for all time, but no lower bound is obtained on σ(t) > 0 as t → ∞. An additional observation, proved in [23], is that:
(B3) If the H s norm is conserved, then σ(t) ≥ σ 0 exp(−At) is obtained.
We emphasize that (B3) does not apply to DKG, since there is no conservation law for the field φ. Thus a more involved argument is needed to prove our main result. The first and easiest step of the proof is to use the idea behind (B1) to obtain a local well-posedness result for data (2), analogous to the local result from [16] with H s data. To reach any time T > 0, we then iterate the local result, and to control the growth of the data norms in each step we rely on an approximate conservation law for ψ(t, ·) in G σ,0 , involving the parameter σ > 0 and reducing to the conservation law (3) in the limit σ → 0. Superficially, this parallels the approach used by Tesfahun and the author in [25] for the 1d DKG problem, for which an algebraic lower bound was obtained, but the function spaces and estimates are much more involved in the 2d case. See Remark 2 below for an explanation of why we only get an exponential lower bound instead of an algebraic one in 2d. In 3d, on the other hand, global C ∞ regularity for large data remains an open problem.
We now turn to the proof of Theorem 1. Since m > 0, we may assume m = 1 by a rescaling.
The paper is organized as follows. In the next section we reformulate the system in a way which makes it easy to see the null structure. In Section 3.1 we state the analytic local existence theorem and the approximate conservation law and prove that they imply the main result, Theorem 1. In Section 4 we introduce the function spaces that we use. In Section 5 we prove some multilinear space-time estimates of null-form type, which are then used to prove the local existence in Section 6 and the approximate conservation law in Section 7.
Reformulation of the system
Set D = ∇/i. For a given function ξ → h(ξ) on R 2 we denote by h(D) the Fourier multiplier defined by
h(D)f = F −1 h(ξ)F f (ξ) .
Using the Dirac projections
Π ± = Π(±D), Π(ξ) = 1 2 I + ξ |ξ| · α we now write ψ = ψ + + ψ − , where ψ ± = Π ± ψ.
Further we set
φ = 1 2 (φ + + φ − ) , where φ ± = φ ± i D −1 ∂ t φ,
and note that φ = Re φ + , since φ is real valued (hence so is D −1 ∂ t φ). Since
|D|Π + − |D|Π − = −iα · ∇ + β,
one then obtains the following formulation of (1) (with m = 1):
(4) (−i∂ t + |D|)ψ + = Π + −M βψ + (Re φ + )βψ , ψ + (0, x) = f + (x), (−i∂ t − |D|)ψ − = Π − −M βψ + (Re φ + )βψ , ψ − (0, x) = f − (x), (−i∂ t + D )φ + = D −1 βψ, ψ , φ + (0, x) = g + (x),
where f ± = Π ± ψ 0 and g + = φ 0 + i D −1 φ 1 . As shown in [12], each bilinear term in (4) has a spinorial null structure encoded in the estimate
(5) Π(−s 2 η)Π(s 1 ξ) = O(∠(s 1 ξ, s 2 η)),
where ξ, η ∈ R 2 , s 1 , s 2 ∈ {+, −} and −s 1 denotes the reverse sign of s 1 . This estimate will be used in tandem with the sign-reversing identity (6) Π(ξ)β = βΠ(−ξ).
Proof of the main theorem
In this section we first state the analytic local existence theorem and the approximate conservation law, and then we show that they imply the main result, Theorem 1.
We start with the local existence result (the proof is given in Section 6).
Theorem 2.
There exists a constant c 0 > 0 such that for any σ 0 ≥ 0 and any data
(7) (f + , f − , g + ) ∈ X 0 := G σ0,0 (R 2 ; C) × G σ0,0 (R 2 ; C) × G σ0,1/2 (R 2 ; C),
the Cauchy problem (4) has a unique solution
(ψ + , ψ − , φ + ) ∈ C([−δ, δ]; X 0 ) on (−δ, δ) × R 2 , where δ = c 0 1 + f + 2 G σ 0 ,0 + f − 2 G σ 0 ,0 + g + 2 G σ 0 ,1/2 . Remark 1.
The uniqueness is immediate since the solution is certainly C ∞ .
Remark 2. If the dependence of the local existence time in Theorem 2 could be improved to δ = c 0
1 + f + 2 G σ 0 ,0 + f − 2 G σ 0 ,0 + g + ρ G σ 0 ,1/2
for some ρ < 2, then the argument in subsection 3.1 below would give an algebraic lower bound on σ(t) instead of an exponential one. But in order to get the improved existence time we would need to improve the estimate (34) used in the proof of the local existence theorem, more precisely the factor δ 1/2 in that estimate would have to be replaced by δ 1/ρ , and in view of (38) this does not seem possible using the (sharp) estimates in Theorem 4.
The conservation of charge
|ψ(t, x)| 2 dx = |ψ + (t, x)| 2 + |ψ − (t, x)| 2 dx = const.
does not hold for Ψ = e σ D ψ with σ > 0, but we can nevertheless obtain an approximate conservation law. Indeed, we have the following (proved in Section 7).
Theorem 3. Let σ 0 > 0. Consider the local solution from Theorem 2, with time of existence δ > 0, and define
M σ (t) = ψ + (t, ·) 2 G σ,0 + ψ − (t, ·) 2 G σ,0 , N σ (t) = φ + (t, ·) G σ,1/2 , for t ∈ [0, δ] and σ ∈ [0, σ 0 ]. Assume a ∈ (1/4, 1/2] and set (8) p = min a, 3(a − 1/4) . Then for all σ ∈ [0, σ 0 ] we have sup t∈[0,δ] M σ (t) ≤ M σ (0) + cδ p σ 1/2−a M σ (0) M σ (0) 1/2 + N σ (0) ,(9)sup t∈[0,δ] N σ (t) ≤ N σ (0) + cδ 1/2 ψ(0, ·) 2 L 2 + cδ p σ 1/2−a M σ (0),(10)
where the constant c > 0 depends only on a and M .
We now have all the tools needed to prove the main result, Theorem 1.
3.1.
Proof of Theorem 1. Without loss of generality we restrict attention to t ≥ 0. We must prove the lower bound σ(t) ≥ σ 0 e −At for all t ≥ 0, for some constant A > 0 depending on σ 0 and the norm of the data. But by Theorem 2 there exists t 0 > 0 such that σ(t) ≥ σ 0 for all t ∈ [0, t 0 ], hence it suffices to show a lower bound σ(t) ≥ ce −Bt for some constants c, B > 0 depending on σ 0 and the data norm. We split the proof into two steps.
Fix a ∈ (1/4, 1/2), define p by (8) and set q = 1/2 − a and r = (3/2 − p)/q. Let c 0 and c be the constants from Theorems 2 and 3. We will denote by
K = ψ 0 2 L 2 the conserved charge (3). We fix R 0 ≥ 1 so large that σ q 0 R 3/2−p 0 ≥ 1, cc 1/2 0 K ≤ R 0 and 11 3/2 cc p 0 ≤ R 0 . 3.1.1. Step 1. Let R ≥ R 0 be so large that M σ0 (0) + N σ0 (0) 2 ≤ R, and set δ = c 0 12R ,
where c 0 is as in the local existence theorem, Theorem 2. Iterating that result, with σ 0 replaced by a parameter σ ∈ (0, σ 0 ], we cover successive time intervals
[0, δ], [δ, 2δ] etc. In fact, we choose σ so that σ q R 3/2−p = 1, that is, σ = R −r = R −(3/2−p)/q .
Proceeding inductively, let us assume that for some n ∈ N we have
sup t∈[0,(n−1)δ] M σ (t) + N σ (t) 2 ≤ 11R.
Then by Theorem 2 (with σ 0 replaced by σ) we can extend the solution to [0, nδ], and by Theorem 3,
sup t∈[0,nδ] M σ (t) ≤ M σ (0) + ncδ p σ q (11R) 3/2 , sup t∈[0,nδ] N σ (t) ≤ N σ (0) + ncKδ 1/2 + ncδ p σ q 11R. Since M σ (0) ≤ M σ0 (0) ≤ R and N σ (0) ≤ N σ0 (0) ≤ R 1/2 , we then get sup t∈[0,nδ] M σ (t) ≤ R + ncδ p σ q (11R) 3/2 , sup t∈[0,nδ] N σ (t) ≤ R 1/2 + ncKδ 1/2 + ncδ p σ q 11R.
Thus, if
(11) ncδ p σ q 11 3/2 R 1/2 ≤ 1 and ncKδ 1/2 ≤ R 1/2 , it follows that sup t∈[0,nδ] M σ (t) + N σ (t) 2 ≤ 11R.
Note that (11) certainly holds for n = 1, by the choice σ = R −r , R ≥ R 0 and the assumptions on R 0 . Setting T = nδ and using δ = c 0 /12R and σ q R 3/2−p = 1, we rewrite (11) as
T · max 11 3/2 c c 0 12 p−1 , c c 0 12 −1/2 K ≤ 1.
The induction stops at time T = nδ, where n is the largest natural number such that (11) holds. It follows that
T ≥ T 0 := 1 2µ , where µ = max 11 3/2 c c 0 12 p−1 , c c 0 12 −1/2 K > 0.
Indeed, since (11) fails when n is replaced by n + 1, we have 1 < (T + δ)µ ≤ 2T µ.
To summarize, what we have proved in Step 1 is that there exists T 0 > 0, depending only on a, µ and the conserved charge, such that for any R ≥ R 0 and for any data at t = 0 satisfying M σ0 (0) + N σ0 (0) 2 ≤ R, the solution has radius of analyticity at least σ = R −r for all t ∈ [0, T 0 ], and we have the final-time bound
M σ (T 0 ) + N σ (T 0 ) 2 ≤ 11R. 3.1.2.
Step 2. We iterate the result of Step 1. Proceeding inductively we cover intervals [(n − 1)T 0 , nT 0 ] for n = 1, 2, . . . , on each of which the radius of analyticity is at least σ = σ n = (11 n−1 R) −r and we have the final-time bound
M σn (nT 0 ) + N σn (nT 0 ) 2 ≤ 11 n R.
Thus
σ(t) ≥ R −r e −(ln 11/T0)t
for t ≥ 0, as desired. This concludes the proof of Theorem 1.
Function spaces
We impose the convention that the letters N and L (and indexed versions of these) denote elements of the set of dyadic numbers 2 N0 = {2 n : n ∈ N 0 }, and that sums, unions and supremums over N or L are tacitly understood to be restricted to this set. Define disjoint dyadic sets S N by
S 1 = (−1, 1), S 2 n = (−2 n , −2 n−1 ] ∪ [2 n−1 , 2 n ) for n = 1, 2, . . . , so that R = ∪ N S N .
Note that each equation in (4) is of the form (−i∂ t + h(D))u = F , with h(ξ) = ±|ξ| or ± ξ . In general, given a continuous h : R 2 → R of polynomial growth we consider the family of norms, for σ ≥ 0, s, b ∈ R and 1 ≤ p < ∞,
u X σ,s,b;p h(ξ) = L L bp e σ ξ ξ s χ SL (τ + h(ξ)) u(τ, ξ) p L 2 τ,ξ 1/p , and for p = ∞, u X σ,s,b;∞ h(ξ) = sup L L b e σ ξ ξ s χ SL (τ + h(ξ)) u(τ, ξ) L 2 τ,ξ .
Here χ SL denotes the characteristic function of S L and
u(τ, ξ) = F u(τ, ξ) = R×R 2 e −i(tτ +x·ξ) u(t, x) dt dx (τ ∈ R, ξ ∈ R 2 )
is the space-time Fourier transform. The above norms are the analytic counterparts of the norms used in [16], the only difference being that we insert the exponential factor e σ ξ .
Definition 1. Let σ ≥ 0, s, b ∈ R and 1 ≤ p ≤ ∞. The space X σ,s,b;p h(ξ) is the set of u ∈ S ′ (R 1+2 ) such that u ∈ L 1 loc (R 1+2 ) and u X σ,s,b;p h(ξ) < ∞.
In the case σ = 0 we simplify the notation to X s,b;p h(ξ) = X 0,s,b;p h(ξ) . We define multipliers P N and
Q L = Q h(ξ) L by P N u(τ, ξ) = χ SN (|ξ|) u(τ, ξ), Q L u(τ, ξ) = χ SL (τ + h(ξ)) u(τ, ξ).
We also write Q ≤L0 = L≤L0 Q L and Q >L0 = Id − Q ≤L0 , and similarly for P . For convenience we shall use the shorthand u N = P N u, u L = Q L u and u N,
L = P N Q L u.
It is easy to see that the norms corresponding to h(ξ) = ±|ξ| and h(ξ) = ± ξ are comparable. The spaces X σ,s,b;p ±|ξ| and X σ,s,b;p ± ξ therefore coincide and have equivalent norms, so for our purposes they can be used interchangeably and we will denote either of them by X σ,s,b;p ± . We will also write Q ± L = Q ±|ξ| L . We now discuss the main properties of the above spaces. For this purpose it is just as well to work in the general setting of a given continuous h : R 2 → R of polynomial growth, and for the remainder of this section we fix such a function.
Lemma 1. X σ,s,b;p h(ξ) is a Banach space. Proof. It suffices to exhibit an isometric isomorphism u → g = (g L ) L∈2 N 0 from X σ,s,b;p h(ξ) onto a closed subspace M of l p (2 N0 ; Y ), where Y = L 2 (e 2σ ξ ξ 2s dτ dξ). The map is given by g L (τ, ξ) = L b u L (τ, ξ), and M is the subspace of all g = (g L ) ∈ l p (2 N0 ; Y ) such that each g L is supported in A L = {(τ, ξ) : τ + h(ξ) ∈ S L }.
To prove that the map is onto M, let g ∈ M and define U :
R 1+2 → C by U (τ, ξ) = L −b g L (τ, ξ) for (τ, ξ) ∈ A L .
By the assumption that h(ξ) has polynomial growth, it is easy to see that U is a tempered function, hence u = F −1 U is well defined and belongs to X σ,s,b;p h(ξ) .
Lemma 2. X σ,s,1/2;1 h(ξ)
embeds continuously into C(R; G σ,s ).
Proof. This follows from
u(t) G σ,s ≤ L e σ ξ ξ s | u L (τ, ξ)| dτ L 2 ξ ≤ L L 1/2 e σ ξ ξ s u L (τ, ξ) L 2 τ,ξ .
Similarly, we bound u(t + h) − u(t) G σ,s by the right-hand side with the factor e iτ h − 1 inserted in the L 2 τ,ξ norm, and the resulting expression converges to zero as h → 0, by the dominated convergence theorem. , and these terms are arbitrarily small for L 0 and N 0 large enough, by the dominated convergence theorem. If p = ∞, the convergence in S ′ follows from dominated convergence on the Fourier side of the Plancherel identity (see (12) below) when u is tested on any v ∈ S.
We remark that the Schwartz class
S(R 1+2 ) is contained in X σ,s,b;p h(ξ) if σ = 0, but not if σ > 0.
Recall that we simplify the notation to X 0,s,b;p h(ξ) = X s,b;p h(ξ) when σ = 0. We now prove some density and duality results for this case.
Lemma 4. S is dense in X s,b;p h(ξ) if 1 ≤ p < ∞, but not if p = ∞.
Proof. If 1 ≤ p < ∞ and u ∈ X s,b;p h(ξ) , then by Lemma 3, v = L≤L0 u L can be made arbitrarily close to u in X s,b;p h(ξ) by choosing L 0 large enough. But the index set of L now being finite, v belongs to L 2 ( ξ 2s dτ dξ), in which S is dense. Moreover, approximating v from S in L 2 ( ξ 2s dτ dξ), one approximates also in X s,b;p h(ξ) .
If p = ∞, set u(τ, ξ) = τ −b−1/2 χ S1 (|ξ|). Then L b ξ s u L L 2 ∼ 1 for large L, so u ∈ X s,b;∞ h(ξ) . Moreover, for any v ∈ S we have L b ξ s ( u L − v L ) L 2 1 for large L, so approximation from S is impossible in X s,b;∞ h(ξ) .
A duality pairing between X s,b;p h(ξ) and
X −s,−b;p ′ h(ξ)
can be defined in a natural way as an extension of the pairing sending (u, v) ∈ S ′ × S to
(12) uv dt dx = (2π) −3 u(τ, ξ) v(τ, ξ) dτ dξ,
where equality holds by Plancherel's theorem. But the right side is well defined as an absolutely convergent integral for any (u, v) ∈ X s,b;p h(ξ) × X −s,−b;p ′ h(ξ)
, since by Cauchy-Schwarz and Hölder we can bound in absolute value by
(13) L ξ s L b u L (τ, ξ) L 2 ξ −s L −b v L (τ, ξ) L 2 ≤ u X s,b;p h(ξ) v X −s,−b;p ′ h(ξ) . For (u, v) ∈ X s,b;p h(ξ) × X −s,−b;p ′ h(ξ)
we can therefore consistently define uv dt dx by (12). This bilinear pairing is bounded, and hence continuous, in view of (13). With this definition, we have the following.
Lemma 5. Let 1 ≤ p ≤ ∞. For any u ∈ X s,b;p h(ξ) we have (14) u X s,b;p h(ξ) = sup (2π) 3 uv dt dx : v ∈ X −s,−b;p ′ h(ξ) , v X −s,−b;p ′ h(ξ) = 1 ,
where 1 ≤ p ′ ≤ ∞ is the Hölder conjugate of p, defined by 1 p + 1 p ′ = 1. Moreover, the set over which the supremum is taken can be further restricted as follows:
(i) if p > 1, we can restrict to v ∈ S; (ii) if p < ∞, we can restrict to v such that v ∈ L 2 with compact support.
Proof. By (12) and (13), LHS(14) ≥ RHS (14). Conversely, if 1 ≤ p < ∞, then defining v by
(15) v L (τ, ξ) = ξ 2s L 2b u L (τ, ξ) · s L b u L 2−p L 2 u p/p ′ X s,b;p h(ξ)
for all L for which · s L b u L L 2 > 0, and v L (τ, ξ) = 0 for all other L, we have v X −s,−b;p ′ h(ξ) = 1 (we assume of course that LHS (14) is not zero) and equality holds in (14). If p = ∞, then fixing L and defining v by
v(τ, ξ) = ξ 2s L 2b u L (τ, ξ) · s L b u L L 2 we have v X −s,−b;1 h(ξ) = 1 and (2π) 3 uv dt dx = · s L b u L L 2 . It follows that RHS(14) ≥ · s L b u L L 2 for all L, hence RHS(14) ≥ u X s,b;∞ h(ξ)
. This concludes the proof of (14). The claim (i) follows since S is dense in
X −s,−b;p ′ h(ξ)
for p ′ < ∞. Finally, to prove (ii) we assume p < ∞ and note that by Lemma 3 we can reduce to the case where u has compact support, hence v given by (15) also has compact support. Moreover, v ∈ L 2 .
The restriction of X σ,s,b;p h(ξ) to a time interval (−δ, δ) is denoted X σ,s,b;p h(ξ) (δ). It can be viewed as the quotient space X σ,s,b;p h(ξ) /M, where M is the closed subspace consisting of those u ∈ X σ,s,b;p h(ξ) which vanish on (−δ, δ) × R 2 . The norm (16) u Proof. In view of the definition (16) of the restriction norm, it suffices to prove
X σ,s,b;p h(ξ) (δ) = inf v X σ,s,b;p h(ξ) : v ∈ X σ,s,b;p h(ξ) , u = v on (−δ, δ) × R 2 makes X σ,s,b;p h(ξ) (δ) a Banach space. As before, we write X 0,s,b;p h(ξ) (δ) = X s,b;p h(ξ) (δ). Lemma 6. Let σ ≥ 0, s ∈ R, 0 < b ≤ 1/2 and 0 < δ ≤ 1. Then u X σ,s,b;1 h(ξ) (δ) ≤ cδ 1/2−b u X σ,χ I u X s,b;p h(ξ) ≤ c u X s,b;p h(ξ) .
We adapt an argument from [10, Lemma 3.2]. Since p < ∞, S is dense in X s,b;p h(ξ) by Lemma 4, so it is enough to prove the estimate for u ∈ S. Replacing u by D s u, we may assume s = 0. Writing χ I (t) in terms of signum functions and applying Lemma 5, we then reduce to proving
(18) sgn(t)u(t, x)v(t, x) dt dx ≤ c u X 0,b;p h(ξ) v X 0,−b;p ′ h(ξ)
for u ∈ S and v ∈ X 0,−b;p ′ h(ξ) ∩ L 2 . We bound the left side by sgn(t)u L1 (t, x)v L2 (t, x) dt dx and separate the cases L 1 ∼ L 2 , L 1 ≪ L 2 and L 2 ≪ L 1 . For L 1 ∼ L 2 we bound by
L1∼L2 u L1 L 2 v L2 L 2 ∼ L1∼L2 L b 1 u L1 L 2 L −b 2 v L2 L 2 u X 0,b;p h(ξ) v X 0,−b;p ′ h(ξ) , while for L 1 ≪ L 2 we write sgn(t)u L1 (t, x)v L2 (t, x) dt dx = lim n→∞ φ t n u L1 (t, x)v L2 (t, x) dt dx = c lim n→∞ n φ(n[τ − λ]) u L1 (λ, ξ) v L2 (τ, ξ) dλ dτ dξ, where φ(t) = sgn(t)χ [−1,1] (t) has Fourier transform φ(τ ) = O(|τ | −1 ) and |τ − λ| = |(τ + h(ξ)) − (λ + h(ξ))| ∼ L 2 ,
hence we dominate in this case by
L1≪L2 L −1 2 | u L1 (λ, ξ)|| v L2 (τ, ξ)| dλ dτ dξ ≤ c L1≪L2 L 1 L 2 1/2 u L1 (λ, ξ) L 2 λ v L2 (τ, ξ) L 2 τ dξ ≤ c L1≪L2 L 1 L 2 1/2−b L b 1 u L1 L 2 L −b 2 v L2 L 2 ≤ c ∞ l=0 ∞ j=0 2 −lε α j β j+l ≤ c α l p β l p ′ = c u X 0,b;p h(ξ) v X 0,−b;p ′ h(ξ) , where ε = 1/2 − b > 0, L 1 = 2 j , L 2 = 2 j+l , α j = L b 1 u L1 L 2 and β j+l = L −b 2 v L2 L 2 .
Here we used b < 1/2. The remaining case L 2 ≪ L 1 works out similarly, but relies on −b < 1/2.
In terms of the free propagator U (t) = e −ith(D) the solution of (19) (
−i∂ t + h(D)) u = F, u(0, x) = f (x),
is given, for sufficiently regular F (t, x) and f (x), by Duhamel's formula (20)
u(t) = U (t)f + i t 0 U (t − t ′ )F (t ′ ) dt ′ ,
and satisfies the following estimate.
Lemma 8. Let σ ≥ 0, s ∈ R, −1/2 < b < 1/2 and 0 < δ ≤ 1. For any f ∈ G σ,s and F ∈ X σ,s,b;∞ h(ξ) (δ) there is a unique u ∈ X σ,s,1/2;1 h(ξ) (δ) satisfying the initial value problem (19) on (−δ, δ) × R 2 . Moreover, (21) u X σ,s,1/2;1 h(ξ) (δ) ≤ c f G σ,s + δ 1/2+b F X σ,s,b;∞ h(ξ) (δ) ,
where c depends only on b.
Proof. By the substitution u → e σ D D s u we reduce to the case σ = s = 0. The proof now follows more or less along the lines of the proof of the analogous result for the standard X s,b = X s,b;2 spaces, but some care must be taken since S is not dense in X 0,b;∞ h(ξ) . Assuming for the moment F ∈ S, then (20) can be rewritten, via the Fourier transform, as
u(t) = U (t)f + (T F )(t), where (T F )(t) = ∞ n=1 t n n! U (t)f n + U (t)g + F −1 F {Q L>δ −1 F }(τ, ξ) τ + h(ξ) , f n (ξ) = c (τ + h(ξ)) n−1 F {Q L≤δ −1 F }(τ, ξ) dτ, g(ξ) = c (τ + h(ξ)) −1 F {Q L>δ −1 F }(τ, ξ) dτ.
Now one observes that T F is well-defined for any F ∈ X 0,b;∞ h(ξ) and that (21) holds; see [13,Section 13.2]. However, it is not obvious that T F then satisfies (19) with
f = 0. But choosing b ′ ∈ (−1/2, b) we have F ∈ X 0,b;∞ h(ξ) ⊂ X 0,b ′ ;2 h(ξ) .
In the latter space, S is dense, and by a well-known result the linear operator T is bounded from
X 0,b ′ ;2 h(ξ) (δ) into X 0,b ′ +1;2 h(ξ) (δ) and T F satisfies (19) on (−δ, δ) × R 2 with f = 0.
Corollary 1. Under the assumptions of Lemma 8 we have
sup t∈[−δ,δ] u(t) G σ,s ≤ f G σ,s + cδ 1/2+b F X σ,s,b;∞ h(ξ) (δ) .
Proof. For the first term in (20) we use U (t)f G σ,s ≤ f G σ,s , and for the second term we use Lemma 2 and Lemma 8.
Multilinear space-time estimates
Estimating the solution of (4) via duality (Lemma 5), the need arises for the following trilinear space-time estimates, which we shall prove by combining dyadic bilinear L 2 space-times estimates (stated in Lemma 9 below) with the null form estimate (5). The special case σ = 0, a = 1/2 and b 0 = b 1 = b 2 = 1/3 of the following theorem was proved in [16].
Theorem 4. Assume that
• a ∈ (1/4, 3/4],
• b 0 , b 1 , b 2 ≥ max(1/4, 3/4 − a) • b 0 + b 1 + b 2 ≥ 3/2 − a.
Then there exists a constant c > 0 such that the following estimates hold for all signs s 0 , s 1 , s 2 ∈ {+, −} and for all σ ≥ 0:
R 1+2 e σ D φ βΠ s1 ψ 1 , Π s2 ψ 2 dt dx ≤ c φ X 0,a,b 0 ;1 s 0 ψ 1 X σ,0,b 1 ;1 s 1 ψ 2 X σ,0,b 2 ;1 s 2 ,(22)R 1+2 φ βΠ s1 ψ 1 , Π s2 e σ D ψ 2 dt dx ≤ c φ X σ,a,b 0 ;1 s 0 ψ 1 X σ,0,b 1 ;1 s 1 ψ 2 X 0,0,b 2 ;1 s 2 .(23)
The proof is given at the end of this section. Before proceeding we record the following consequence of Theorem 4.
βΠ s1 ψ 1 , Π s2 ψ 2 X σ,−a,−b 0 ;∞ s 0 (δ) ≤ cδ 1−b1−b2 ψ 1 X σ,0,1/2;1 s 1 (δ) ψ 2 X σ,0,1/2;1 s 2 (δ) , Π s2 (φβΠ s1 ψ 1 ) X σ,0,−b 2 ;∞ s 2 (δ) ≤ cδ 1−b0−b1 φ X σ,a,1/2;1 s 0 (δ) ψ 1 X σ,0,1/2;1 s 1 (δ) .
Proof. We only give the details for the first estimate. By Lemma 6 we reduce to
βΠ s1 ψ 1 , Π s2 ψ 2 X σ,−a,−b 0 ;∞ s 0 (δ) ≤ c ψ 1 X σ,0,b 1 ;1 s 1 (δ) ψ 2 X σ,0,b 2 ;1 s 2 (δ) .
Working with extensions, we note that it suffices to prove the estimate without the restriction to the time interval (−δ, δ). Thus we need to prove
βΠ s1 ψ 1 , Π s2 ψ 2 X σ,−a,−b 0 ;∞ s 0 ≤ c ψ 1 X σ,0,b 1 ;1 s 1 ψ 2 X σ,0,b 2 ;1 s 2 ,
but this follows from Theorem 4 via Lemma 5.
There is no L 4 space-time estimate for free solutions of the wave equation in two space dimensions, and hence no L 2 product estimate. As observed in [22], one can nevertheless prove Fourier restriction estimates on truncated thickened null cones in space-time, such as the ones in the following lemma, which will be used to prove Theorem 4.
Some notation: Given dyadic numbers N 0 , N 1 , N 2 , L 0 , L 1 , L 2 ≥ 1, we denote by L min , L med and L max the minimum, median and maximum of L 0 , L 1 and L 2 , and similarly for the N 's. Moreover, for j, k ∈ {0, 1, 2}, j < k, we denote by L jk min (resp. L jk max ) the minimum (resp. the maximum) of L j and L k , and similarly for the N 's. We also write N = (N 0 , N 1 , N 2 ) and L = (L 0 , L 1 , L 2 ). We will use the notation N N ′ , N ≪ N ′ and N ∼ N ′ as shorthand for, respectively, N ≤ cN ′ , N ≤ c −1 N ′ and c −1 N ′ ≤ N ≤ cN ′ , where c is a sufficiently large absolute constant. From now on we use the notation Q ± L for the modulation operator Q ±|ξ| L defined in the previous subsection (note that we could also have used Q ± ξ L ).
Lemma 9. There exists c > 0 such that for all dyadic numbers N j , L j ≥ 1, j, k ∈ {0, 1, 2}, and all signs s 0 , s 1 , s 2 ∈ {+, −} we have the bilinear L 2 space-time estimate 65)] via the transfer principle (by observing that the multiplier D − is of size λ, in the notation of that paper).
P N0 Q s0 L0 P N1 Q s1 L1 u 1 · P N2 Q s2 L2 u 2 L 2 (R 1+2 ) ≤ C(N, L) u 1 L 2 (R 1+2 ) u 2 L 2 (R 1+2 ) ,
Remark 3. By Plancherel's theorem, the estimate in Lemma 9 is equivalent to
I(τ, ξ) L 2 τ,ξ ≤ C(N, L) u 1 L 2 (R 1+2 ) u 2 L 2 (R 1+2 ) , where I(τ, ξ) = χ SN 0 (|ξ|)χ SL 0 (τ + s 0 |ξ 0 |) × P N1 Q s1 L1 u 1 (τ − λ, ξ − η) P N2 Q s2 L2 u 2 (λ, η) dλ dη,
and it is in this form that we will now apply the estimate.
We are now in a position to prove the trilinear estimates.
5.1. Proof of Theorem 4. Using Plancherel's theorem, the self-adjointness of Π(ξ), the sign-reversing identity (6) and the null estimate (5), we bound the left side of (22) by
e σ ξ φ(τ, ξ) βΠ (s 1 (η − ξ)) ψ 1 (λ − τ, η − ξ), Π (s 2 η) ψ 2 (λ, η) dλ dτ dη dξ = e σ ξ φ(τ, ξ) Π (s 2 η) βΠ (s 1 (η − ξ)) ψ 1 (λ − τ, η − ξ), ψ 2 (λ, η) dλ dτ dη dξ = e σ ξ φ(τ, ξ) βΠ (−s 2 η) Π (s 1 (η − ξ)) ψ 1 (λ − τ, η − ξ), ψ 2 (λ, η) dλ dτ dη dξ ≤ c θ 12 φ(τ, ξ) e σ η−ξ ψ 1 (λ − τ, η − ξ) e σ η ψ 2 (λ, η) dλ dτ dη dξ, where (24) θ 12 = ∠ (s 1 (η − ξ), s 2 η)
and we used the triangle inequality to write e σ ξ ≤ e σ η−ξ e σ η .
Similarly, the left side of (23) can be bounded by c θ 12 e σ ξ φ(τ, ξ) e σ η−ξ ψ 1 (λ − τ, η − ξ) ψ 2 (λ, η) dλ dτ dη dξ.
Thus both (22) and (23) reduce to the estimate (without σ)
(25) θ 12 φ(τ, ξ) ψ 1 (λ − τ, η − ξ) ψ 2 (λ, η) dλ dτ dη dξ ≤ c φ X a,b 0 ;1 s 0 ψ 1 X 0,b 1 ;1 s 1 ψ 2 X 0,b 2 ;1 s 2 ,
which we now prove. By dyadic decomposition we bound the left side by a constant times
N,L θ 12 P N0 Q s0 L0 φ(τ, ξ) P N1 Q s1 L1 ψ 1 (λ − τ, η − ξ) P N2 Q s2 L2 ψ 2 (λ, η) dλ dτ dη dξ,
where the sum is over dyadic N j , L j ≥ 1, j = 0, 1, 2. The integral vanishes unless the two largest N 's are comparable, so we reduce to the cases (i)
N 0 ≪ N 1 ∼ N 2 , (ii) N 1 N 0 ∼ N 2 or (iii) N 2 N 0 ∼ N 1 .
By symmetry, it suffices to consider cases (i) and (ii).
To estimate the integral we will apply Cauchy-Schwarz with respect to (τ, ξ) followed by Lemma 9 with u 1 (τ − λ, ξ − η) = | ψ 1 (λ − τ, η − ξ)| and u 2 (λ, η) = | ψ 2 (λ, η)|, cp. Remark 3. It should be kept in mind that due to the sign change in the argument of u 1 , the sign s 1 is reversed when we apply Lemma 9.
By [
C(N, L) N a 0 L b0 0 L b1 1 L b2 2 α N0,L0 β N1,L1 γ N2,L2 , where α N0,L0 = N a 0 L b0 0 P N0 Q s0 L0 φ L 2 (R 1+2 ) , β N1,L1 = L b1 1 P N1 Q s1 L1 ψ 1 L 2 (R 1+2 ) , γ N2,L2 = L b2 2 P N2 Q s2 L2 ψ 2 L 2 (R 1+2 )
, and C(N, L) is as in Lemma 9. It remains to prove that (27) S
≤ c L N0 α 2 N0,L0 1/2 N1 β 2 N1,L1 1/2 N2 γ 2 N2,L2 1/2 . 5.1.1. Case (ii), N 1 N 0 ∼ N 2 . Then C(N, L) ≤ cN 1/2 1 N 1/4 0 L 1/2 min L 1/4
med , hence we bound the corresponding part of the sum S by a constant times
N,L 1 N1 N0∼N2 L max N 1 µ N 1/2 1 N 1/4 0 L 1/2 min L 1/4 med N a 0 L b0 0 L b1 1 L b2 2 α N0,L0 β N1,L1 γ N2,L2
for any µ ∈ [0, 1/2]. Clearly
(28) L 1/2 min L 1/4 med L µ max L b0 0 L b1 1 L b2 2 ≤ 1 provided that b 0 + b 1 + b 2 ≥ 3 4 + µ, (29) b 0 , b 1 , b 2 ≥ max 1 4 , µ .(30)
Then we are left with
N,L 1 N1 N0∼N2 N 1/2−µ 1 N a−1/4 0 α N0,L0 β N1,L1 γ N2,L2 .
Assuming (31) 0 ≤ µ < 1 2 we sum N 1 and bound by
L sup N1 β N1,L1 N0,N2 1 N0∼N2 N 1/2−µ 0 N a−1/4 0 α N0,L0 γ N2,L2 ,
so if a+ µ− 3/4 ≥ 0, we can sum N 1 ∼ N 2 by Cauchy-Schwarz to obtain the desired estimate (27). We therefore choose µ = 3/4 − a. Then the conditions (29), (30) and (31) correspond exactly to the assumptions of the lemma. This concludes the proof in case (ii).
5.1.2. Case (i), N 0 ≪ N 1 ∼ N 2 . First, if L max = L 1 or L max = L 2 ,N,L 1 N0≪N1∼N2 L max N 1 µ N 3/4 0 L 1/2 min L 1/4 med N a 0 L b0 0 L b1 1 L b2 2 α N0,L0 β N1,L1 γ N2,L2 .
Taking µ = 3/4 − a as above, we apply (28) and reduce to
N,L 1 N0≪N1∼N2 N 0 N 1 3/4−a α N0,L0 β N1,L1 γ N2,L2 ,
so if a < 3/4, we can sum N 0 and then sum N 1 ∼ N 2 by Cauchy-Schwarz to get (27). If a = 3/4, we use instead C(N, L) ≤ cN 0 L 1/2 min and take µ = 1/4, yielding
(33) N,L 1 N0≪N1∼N2 L max N 1 1/4 N 0 L 1/2 min N 3/4 0 L b0 0 L b1 1 L b2 2 α N0,L0 β N1,L1 γ N2,L2 .
Now we use the fact that L
1/2 min L 1/4 max L b0 0 L b1 1 L b2 2 ≤ 1 if b 0 + b 1 + b 2 ≥ 3/4 and b 0 , b 1 , b 2 ≥ 1/4
, which are consistent with the assumptions of the lemma when a = 3/4, so we reduce to
N,L 1 N0≪N1∼N2 N 0 N 1 1/4 α N0,L0 β N1,L1 γ N2,L2 ,
and again obtain the desired bound (27). It remains to consider the subcase L max = L 0 of case (i). The argument used for a = 3/4 above still applies and yields (33), so it remains to consider a < 3/4. If s 1 = s 2 , then by Lemma 9 (with signs −s 1 and s 2 , so equal signs) we have the estimate C(N, L) ≤ cN Taking µ = 3/4 − a and applying (28) we reduce to
N,L 1 N0≪N1∼N2 N 0 N 1 a α N0,L0 β N1,L1 γ N2,L2 ,
so we only need a > 0 to sum N 0 , and then we sum N 1 ∼ N 2 by Cauchy-Schwarz. This concludes the proof of case (i) and of Theorem 4.
Local existence
In this section we prove the following local existence result, which is an extended version of Theorem 2.
It remains to prove the claimed estimates. By Lemma 8,
A n+1 ≤ ca 0 + ± cδ 1/2 Π ± M βψ (n) X σ,0,0;∞ ± (δ) + ± cδ 1/2−b2 Π ± Re φ (n) + βψ (n) X σ,0,−b 2 ;∞ ± (δ) .
Using X σ,0,0;∞ ± (δ) = X σ,0,0;∞ ∓ (δ), the identity (6) and Lemma 6, we bound the second term on the right by
± cδ 1/2 M βψ (n) ∓ X σ,0,0;∞ ∓ (δ) ≤ c ε δ 1−ε M A n
for any ε > 0. The third term we bound by, applying Corollary 2 with a = 1/2,
s1,s2
cδ 1/2−b2 Π s2 Re φ (n) + βΠ s1 ψ (n) X σ,0,−b 2 ;∞ ± (δ) ≤ cδ 3/2−b0−b1−b2 B n A n , which requires b 0 , b 1 , b 2 ≥ 1/4 and b 0 + b 1 + b 2 ≥ 1. We choose b 0 = b 1 = b 2 = 1/3.
Finally, the estimate (35) similarly reduces to
βΠ s1 ψ (n) , Π s2 ψ (n) X σ,−1/2,−1/3;∞ + (δ) ≤ cδ 1/3 A 2 n ,
which also follows from Corollary 2. Finally, the estimates (36) and (37) follow from the same considerations by linearity.
Approximate conservation of charge
In this section we prove Theorem 3. We need the following key estimate. Proof. By Plancherel's theorem we bound the left side by
Λ(ξ, η) φ(τ, ξ) βΠ (s 1 (η − ξ)) ψ 1 (λ − τ, η − ξ), Π (s 2 η) ψ 2 (λ, η) dλ dτ dη dξ where Λ(ξ, η) = e σ η − e σ η−ξ = e σ η−ξ e σ( η − η−ξ ) − 1 .
As in the proof of Theorem 4 we then bound by c |Λ(ξ, η)| θ 12 φ(τ, ξ) ψ 1 (λ − τ, η − ξ) ψ 2 (λ, η) dλ dτ dη dξ
Applying the inequality |e x − 1| ≤ |x| θ e |x| (x ∈ R, θ ∈ [0, 1]), and the triangle inequality η − η − ξ ≤ ξ , we finally bound by cσ θ θ 12 ξ θ e σ ξ φ(τ, ξ) e σ η−ξ ψ 1 (λ − τ, η − ξ) ψ 2 (λ, η) dλ dτ dη dξ and the desired estimate then follows from (25).
We now have all the tools needed to prove the approximate conservation law.
7.1. Proof of Theorem 3. By Theorem 5 (applied with σ 0 replaced by σ ∈ [0, σ 0 ]) there exist constants c, c 0 > 0 such that for all σ ∈ [0, σ 0 ] we have the bounds ψ + X σ,0,1/2;1 + (δ(σ)) + ψ − X σ,0,1/2;1 − (δ(σ)) ≤ cM σ (0) 1/2 ,
φ + X σ,1/2,1/2;1
+ (δ(σ)) ≤ c M σ (0) 1/2 + N σ (0) ,(40)
where (41) δ(σ) = c 0 1 + M σ (0) + N σ (0) 2 .
But clearly, δ(σ) ≥ δ := δ(σ 0 ) for σ ∈ [0, σ 0 ], so we may replace δ(σ) by δ in (39) and (40). 7.1.1. Proof of (9). Set Ψ ± = e σ D ψ ± and Ψ = Ψ + + Ψ − . Then (4) where we wrote 2 Re φ + = φ + + φ + and used φ − = φ + . Taking θ = 1/2 − a and invoking Lemma 7 followed by Lemma 6, we bound the summands by where b 0 ∈ [0, 1/2] remains to be chosen. Separating low frequencies, ξ ≤ σ −1 , and high frequencies, ξ > σ −1 , we estimate the last term by Taking a = 1/2 − θ and choosing the b's as in (42), we similarly bound δ 1/2−b0 σ θ D θ−1 βψ, ψ X σ,1/2,−b 0 ;∞ + (δ) ≤ cδ p σ 1/2−a M σ (0), concluding the proof of (10) and of Theorem 3.
Lemma 3 .
3Assume 1 ≤ p < ∞ and let u ∈ X σ,s,b;p h(ξ) . Then u = L u L and u = N,L u N,L hold in X σ,s,b;p h(ξ) . If p = ∞, the convergence holds in S ′ . Proof. The Fourier transforms of u − L≤L0 Q L u and u − N ≤N0 L≤L0 P N Q L u equal u multiplied by the characteristic functions of the regions, respectively, (i) |τ + h(ξ)| ≥ L 0 and (ii) |τ + h(ξ)| ≥ L 0 or |ξ| ≥ N 0 . If p < ∞, the X σ,s,
L bp e σ ξ ξ sL
bp e σ ξ ξ s χ |ξ|≥N0 u
where c depends only on b.
Proof.
Replacing u by e σ D D s u we reduce to the case σ = s = 0, which is proved in [16, Proposition 2.1(iii)].Lemma 7. Let s ∈ R, −1/2 < b < 1/2, 1 ≤ p < ∞ and 0 < δ ≤ 1. Then for any time interval I ⊂ [−δ, δ] we have the estimate(17) χ I u X s,b;p h(ξ) ≤ c u X s,b;p h(ξ) (δ) , where χ I (t) is the characteristic function of I, and c depends only on b.
Corollary 2 .
2Under the assumptions of Theorem 4 there exists c > 0 such that for all σ ≥ 0, δ ∈ (0, 1] and signs s 0 , s 1 , s 2 ∈ {+, −} we have the estimates
where C(N, L) = c min (N min ) Moreover, in the case s 1 = s 2 and N 0 ≪ N 1 ∼ N 2 , the above estimate holds also with C(N, L) = c(N 0 L 1 L 2 ) 1/2 . Proof. The estimate is proved in [22, Theorem 2.1], except for the last statement about the special case s 1 = s 2 and N 0 ≪ N 1 ∼ N 2 , which is included in [26, Proposition 9.1, Eq. (66)] or alternatively can be deduced from the free-wave estimate in [14, Theorem 12.1, Eq. (
α
hence we get again (32). This leaves us with s 1 = s 2 in case (i) with L max = L 0 . From (24) we have θ 12 ≤ cN 0 /N 1 , since |ξ| N 0 ≪ |η| ∼ N 1 . Interpolating this with (26N0,L0 β N1,L1 γ N2,L2 .
Lemma 10 .
10Assume that a, b 0 , b 1 , b 2 satisfy the assumptions of Theorem 4. Then there exists a constant c > 0 such that for all signs s 0 , s 1 , s 2 ∈ {+, −}, all σ ≥ 0 and all θ ∈ [0, 1] we have the estimatee σ D φβΠ s1 ψ 1 − φ βΠ s1 e σ D ψ 1 , Π s2 ψ 2 dt dx ≤ cσ θ φ X σ,a+θ,b
gives (−i∂ t + |D|)Ψ + = Π + −M βΨ + (Re φ + )βΨ + Π + F,(−i∂ t − |D|)Ψ − = Π − −M βΨ + (Re φ + )βΨ + Π − F, where F = e σ D (Re φ + )βψ − (Re φ + )βΨ. Now we calculate d dt M σ (t) = d dt Ψ + (t, x), Ψ + (t, x) + Ψ − (t, x), Ψ − (t, x) dx = 2 Im i∂ t Ψ + , Ψ + + i∂ t Ψ − , Ψ − dx = 2 Im (i∂ t − |D|)Ψ + , Ψ + + (i∂ t + |D|)Ψ − , Ψ − dx + 2 Im |D|Ψ + , Ψ + + −|D|Ψ − , Ψ − dx = 2 Im (M − Re φ + ) βΨ, Ψ − F, Ψ dx = −2 Im F, Ψ dx,where we used Plancherel to see that |D|Ψ ± , Ψ ± dx = 0 and the self-adjointness of β to see that βΨ, Ψ is real valued. Integrating over the time interval [0, T ] for any T ∈ [0, δ] we then getM σ (T ) ≤ M σ (0) + 2 χ [0,T ] (t) F, Ψ (t,the integral term by c s0,s1,s2∈{+,−} σ θ χ [0,T ] φ s0 X σ,a+θ,b 0 ;1 s 0 χ [0,T ] ψ s1 X σ,0,b 1 ;1 s 1 χ [0,T ] ψ s2 X σ,0,b 2 ;1 s 2 ,
≤ 4
4cT p σ 1/2−a M σ (0) M σ (0) 1/2 + N σ (0) ,where we applied the bounds (39) and (40) and used the fact that φ − X σ,) , on account of φ − = φ + . This concludes the proof of (9). 7.1.2. Proof of(10). Applying Corollary 1 to the last equation in (
θ ∈ [0, 1] remains to be chosen. We are going to estimate both terms using Corollary 2 and the bound (39). First, taking a = 1/2 and b 1 = b 2 = (1 − b 0 )/2 for any b 0 ∈ [0, 1/2], and setting σ = 0, we get δ 1/2−b0 D −1 βψ, ψ X
Theorem 5. There exist c, c 0 > 0 such that for any σ 0 ≥ 0 and any data(7), the Cauchy problem (4) has a unique local solution (ψ + , ψ − , φ + ) ∈ C([−δ, δ]; X 0 ), whereProof. To simplify the notation we write σ = σ 0 . Define the Picard iterates (ψwhere ψ (n) := ψandThen by induction one obtains A n ≤ 2ca 0 and B n ≤ 2c(a 0 + b 0 ) for all n, and further ∆A n+1 + ∆B n+1 ≤ (1/2)(∆A n + ∆B n ), with δ as in the statement of the theorem, for a sufficiently small c 0 > 0 depending on c and M . The sequence of iterates therefore converges and the conclusion of the theorem follows.
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| [] |
[
"Finite key effects in satellite quantum key distribution",
"Finite key effects in satellite quantum key distribution"
] | [
"Jasminder S Sidhu \nSUPA Department of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom\n",
"Thomas Brougham \nSUPA Department of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom\n",
"Duncan Mcarthur \nSUPA Department of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom\n",
"Roberto G Pousa \nSUPA Department of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom\n",
"Daniel K L Oi \nSUPA Department of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom\n"
] | [
"SUPA Department of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom",
"SUPA Department of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom",
"SUPA Department of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom",
"SUPA Department of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom",
"SUPA Department of Physics\nUniversity of Strathclyde\nG4 0NGGlasgowUnited Kingdom"
] | [] | Quantum key distribution (QKD) can provide secure means of communication that are robust to general quantum computing attacks. Satellite QKD (SatQKD) presents the means to overcome range limitations in fibre optic-based systems and achieve global coverage but raises a different set of challenges. For low-Earth orbit SatQKD, a major limitation is the restricted time window for quantum signal transmission and highly variable channel loss during a satellite overpass of an optical ground station. Here, we provide a systematic analysis of the finite block size effects on secret key length generation for low latency operation using BB84 weak coherent pulse decoy state protocols. In particular, we look at how the achievable single pass secret key length depends on various system parameters for different overpass geometries and calculate the total long-term-average key length. We find that optimisation of basis bias, pulse probabilities and intensities, and data selection, is crucial for extending the range of satellite trajectories and link efficiencies for which finite-block size keys can be extracted. The results also serve as a guide for system sizing of future SatQKD systems and the performance levels required for sources and detectors. * These two authors contributed equally † [email protected] | 10.1038/s41534-022-00525-3 | [
"https://arxiv.org/pdf/2012.07829v1.pdf"
] | 229,153,423 | 2012.07829 | f066a9cf7e4e7a6e3325aaa6dd42e56061f4b515 |
Finite key effects in satellite quantum key distribution
Jasminder S Sidhu
SUPA Department of Physics
University of Strathclyde
G4 0NGGlasgowUnited Kingdom
Thomas Brougham
SUPA Department of Physics
University of Strathclyde
G4 0NGGlasgowUnited Kingdom
Duncan Mcarthur
SUPA Department of Physics
University of Strathclyde
G4 0NGGlasgowUnited Kingdom
Roberto G Pousa
SUPA Department of Physics
University of Strathclyde
G4 0NGGlasgowUnited Kingdom
Daniel K L Oi
SUPA Department of Physics
University of Strathclyde
G4 0NGGlasgowUnited Kingdom
Finite key effects in satellite quantum key distribution
(Dated: 15th December 2020)
Quantum key distribution (QKD) can provide secure means of communication that are robust to general quantum computing attacks. Satellite QKD (SatQKD) presents the means to overcome range limitations in fibre optic-based systems and achieve global coverage but raises a different set of challenges. For low-Earth orbit SatQKD, a major limitation is the restricted time window for quantum signal transmission and highly variable channel loss during a satellite overpass of an optical ground station. Here, we provide a systematic analysis of the finite block size effects on secret key length generation for low latency operation using BB84 weak coherent pulse decoy state protocols. In particular, we look at how the achievable single pass secret key length depends on various system parameters for different overpass geometries and calculate the total long-term-average key length. We find that optimisation of basis bias, pulse probabilities and intensities, and data selection, is crucial for extending the range of satellite trajectories and link efficiencies for which finite-block size keys can be extracted. The results also serve as a guide for system sizing of future SatQKD systems and the performance levels required for sources and detectors. * These two authors contributed equally † [email protected]
I. INTRODUCTION
The second quantum revolution promises to deliver new technologies operating at the fundamental limits of Physics that offer enhanced capabilities over conventional classical devices [1]. These include quantum sensors [2][3][4], secure quantum communications [5,6], and quantum computers that open up new computational paradigms and facilitate a speed-up of particular problems [7,8]. A quantum communication infrastructure could network quantum devices, creating the quantum internet [9] while also enabling distributed quantum sensors [10][11][12][13], clocks [14,15], and navigation systems [16]. However, creating the necessary long distance links is a considerable challenge, since exponential absorption in optical fibres places severe limits for direct transmission of quantum photonic signals to under 1000 km with the most recent records still falling significantly short of that limit [17][18][19][20].
Quantum repeaters have been proposed as a method of overcoming the direct transmission limit but are still in the early stages of development with long-distance deployment likely to be over a decade away [21]. Hence, alternative methods of spanning intercontinental and global distances are required. Satellite-based quantum communication systems [22][23][24][25][26][27] is such a method that has received considerable recent interest with the pioneering proof-of-concept in-orbit demonstrations by Micius particularly noteworthy [28].
Space-ground quantum communication introduces particular challenges. Low-Earth orbit satellites have a limited time window in which they can contact an optical ground station in order to establish and maintain a quantum channel. For satellite quantum key distribution (SatQKD), this is particularly acute as protocols typically assume large amounts of data transfer from which a secret key is extracted. This asymptotic resource assumption, where statistical uncertainties of the channel parameters are assumed negligible, is impractical. In contrast, if the quantum transmission data is limited, then the security of the final secret key rests on a thorough treatment of statistical fluctuations in the estimation of key quantities, such as the phase error rate and number of received single photon signals [29][30][31].
With limited data, there is also a trade-off between the amount of received signal sacrificed for parameter estimation and the remaining raw key for further processing. This exacerbates the problem of secret key distillation. Further post-processing operations, such as error correction, reduce the amount of extractable secret key, with small block lengths leading to additional inefficiencies over the asymptotic limit [32,33].
To date, there has been little systematic analysis of achievable secret key lengths that take into account finite block size effects in SatQKD. Initial SatQKD studies employed the standard deviation of measured quantities to estimate statistical uncertainty [34,35]. These were used to derive upper and lower bounds on the gain and phase error rate of the received single photons in order to derive correction terms to the secret key rate. More rigorous methods were developed to account for statistical fluctuations based in smooth entropies [29] leading to better finite-key bounds [36] and their application to free-space quantum communication [37]. Recently, tight bounds [33] and small block length analyses [38] for QKD have been developed, which further improves the achievable rates with restricted amounts of signal.
Here, we apply recent results to estimate the achievable secret key length from minimal number of satellite overpasses, ideally a single pass. Our study accounts for different satellite trajectories, realistic descriptions for the sources and detectors, and quantum link efficiencies comparable with experimental demonstrations from the Micius satellite. We include practical finite key statistics to determine the optimised key rates per pass for WCP-source QKD protocols, comparing the performance of symmetric and asymmetric variants. We also investigate the influence of different system characteristics on the secret key length.
For simplicity, the majority of the presented results are for a zenith overpass, though our analysis can consider any pass geometry and optimising protocol parameters such as biases and arXiv:2012.07829v1 [quant-ph] 14 Dec 2020 Figure 1. Geometry of satellite pass. We consider a satellite in a circular sun synchronous orbit at an altitude h. The ground track of the satellite passes the OGS with a minimum distance of min and reaches a maximum elevation of max , the corresponding minimum zenith angle is min = 90 • − max . We define the smallest max for which a key can be generated as − max , this defines the footprint width of the satellite within which an OGS passing through can establish a finite key. A comprehensive model of satellite-OGS orbital geometry can be found in [39].
intensities to maximise the secret length that can be generated. We compare the annual average secret key length for different system link efficiencies and quantify the improvement of normalised (per pass) secret key we can generate from combining data from multiple satellite passes, extending the range of acceptable system link efficiencies and increasing the key length per pass. Our work serves as the basis for preliminary SatQKD system sizing and performance analysis and when combined with a size vs cost models, helps determine the optimum OGS size for a fixed space segment.
II. SYSTEM MODEL
We consider a satellite in a circular low Earth Sun-synchronous orbit of altitude ℎ = 500 km, similar to the Micius satellite [28]. We perform downlink QKD operations during night overpasses of the optical ground station to reduce errors due to background light. Our simulation generates time, elevation, and range, as a function of different angles between the orbital plane and that of the optical ground station (OGS) that represents different ground track off-sets and maximum elevations for the overpass (Fig. 1). We simulate the time from when the satellite is first visible above 0 • elevation until is passes below 0 • and calculate the instantaneous link efficiency depending on the elevation and range in order to generate expected detector count statistics. We simplify our model by combining all losses into a single link efficiency value link = −10 log 10 d (dB). This characterises the probability d that a single photon transmitted by the satellite is detected. A lower dB value of link represents better intrinsic system electro-optical efficiency. This is determined by the transmit and receive aperture diameters, pointing accuracy, atmospheric absorption and turbulence, receiver internal losses, and detector efficiencies. We do not consider transmit telescope internal optical efficiency since WCP source intensities can be adjusted to maintain the desired average photon number at the exit aperture [34]. For each elevation, we also do not consider explicitly time-varying transmittance, since changes in channel losses are only due to changes in elevation with time. For discrete variable (DV) protocols, such as BB84, link efficiency fluctuations do not directly impact on the secret key rate, unlike in continuous variable (CV) QKD where this can appear as excess noise leading to a reduction in key [41,42]. We have taken a simplified approach to the modelling of losses due to pointing and atmospheric effects. We base our link efficiency vs elevation values on concrete experimental measurements (see Extended data Fig. 3a in [43] and 3b for elevations 20 • , 26 • , 36 • , and 55 • ) and construct a representative link efficiency with time curve extrapolating from 0 • to 90 • elevations or over the entire ±350 horizon to horizon passage time for a zenith overpass (Fig. 3). Link efficiency vs elevation curves are highly dependent on system performance and OGS site conditions, the Micius data should be regarded as a best-case scenario since the OGSs are situated in dark sky conditions at high altitudes of ∼3000 m where the effects of atmospheric turbulence and attenuation are minimised. We also note that the link efficiency presented in [43] is for the entangled pair distribution experiments, not the prepare-measure QKD downlink system, but are still representative of the spaceground quantum channel link efficiencies that can be achieved with practical and current technologies. Figure 2. Illustration of a zenith pass. The simplest trajectory corresponds to when the ground track passes directly over the OGS position, hence the maximum elevation is 90 • . This represents the ideal overpass with the maximum transmission time and lowest average channel loss. The elevation changes non-uniformly with time, with the greatest angular velocity at zenith of 0.87 • −1 for ℎ = 500 km and the satellite-OGS range ( ) varies from 2600 km at the horizon to 500 km at zenith (14.32 dB change in far-field diffraction loss). The ground track distance from the OGS is denoted ( ) and we take as = 0 as the time of closet approach and maximum elevation max .
To allow formal comparison of our simulated results with different performing SatQKD systems, we introduce a system link efficiency sys link . This parameterises the overall performance of the SatQKD system, which is specified as the link efficiency at zenith (Fig. 2). The nominal value for sys link considered is 27 dB, which corresponds to the improved Micius system using a 1.2 m diameter OGS receiver at Delingha. If smaller OGS receivers are used, or else there are greater constant losses, the per pass secret key length can be determined using worse (i.e. higher) plotted sys link values. We estimate the scaling of sys link with OGS aperture size straightforwardly. Under the assumption that the long term average ground spot size at minimum range is much larger than the OGS diameter , we model the link efficiency change as 20 log 10 ( / 0 ) (dB) where 0 is the reference OGS diameter taken to be 1.2 m (Delingha OGS receiver diameter).
For changes to the transmitter aperture diameter, the timeaveraged ground spot diameter may not be solely dominated by the far-field diffraction beam width. Specifically, contributions from the pointing performance and turbulence can also lead to significant additional beam broadening [44,45]. The Micius mission reported sub-rad pointing performance and 10 rad beam widths from transmit apertures of 180 mm and 300 mm [43]. It is unclear the contribution of turbulence to the report link efficiencies as a function of range/elevation. However it may be expected that the presence of non-diffractive beam spreading effects results in a smaller dependence of sys link on transmitter diameter , i.e. using a smaller should result in a smaller increase in sys link than given by purely diffractive beam broadening. As the focus of this paper is not a detailed analysis of atmospheric turbulence [46] or extinction [47], we refer the reader to alternative recent works, where the effects on satellite quantum communication are considered [39,44,48].
It should be noted that we can estimate the effect of increasing the source repetition rate by incorporating a correction factor to sys link . As the secret key length is only a function of the total number of detection events per intensity level and Figure 3. Loss vs elevation. Adapted from data in [43]. The link efficiency varies throughout the overpass due to the effects of changing range that leads to increased diffraction, and changes in elevation that alters the optical depth of the atmosphere and exposing the beam to varying turbulence effects. The peak link = 27 dB occurs at zenith, which we call sys link to denote a characteristic system link performance level. We investigate different sized systems by adding a constant dB offset to the link efficiency curve to take into account different elevation independent losses such as a smaller or larger OGS receiver collection aperture or the use of detectors with a different photon detection efficiency. the observed error rates, this only depends on the integrated product of the source rate and the link efficiency, keeping all other system parameters the same. Thus a 100 MHz source (as assumed in all simulations presented here) with a given sys link would provide the same amount of key as 1 GHz source with a 10 dB worse system link efficiency, e.g. with a 3 times smaller OGS receiver diameter.
We consider the quantum bit error rate (QBER) arising from dark counts, background light, source quality, basis misalignment, and receiver measurement fidelity. Again, we simplify our model by combining several of these together. The dark and background light count rates are assumed to be constant and independent of elevation (in practice background light will depend not just on elevation but also azimuth and the local environment). We combine all other error terms into an intrinsic quantum bit error rate QBER I that does not vary with channel loss or elevation.
III. SATQKD OPERATIONS
A standard model of SatQKD has been the use of large, fixed, and long-term OGSs with which satellites regularly establish QKD links [26]. More demanding is where the OGS may only be able to communicate sporadically with a particular satellite and may not be able to accumulate sufficient data to process as a single large block. For example, smaller, mobile OGS terminals may be required to generate a key from a limited number of passes, possibly only one, due to operational constraints .
We contrast this to fibre-based QKD, where the quantum channel is assumed to be stable (or can be made stable) and can be operated continuously until It is common for the instantaneous (but asymptotic) key rate ∞ ( ) to be calculated as a function of time during an overpass and integrated to arrive at the secret key length (SKL) [25,26],
SKL Cont. ∞ = ∫ end start ∞ ( ) ,(1)
where start,end define the start and end times of the quantum transmission period of the overpass. For a meaningful SKL, segments of data with the same statistical features from many passes are combined into an asymptotically large block for post-processing. In practice, small blocks from different passes are combined to give the following SKL,
SKL Block ∞ = ∑︁ ∞ ,(2)
where ∞ is the asymptotic key rate for a small segment , and its length. This restricts the operational flexibility and leads to considerable latency between the first contact between a satellite and an OGS and the generation of the first set of secret keys. A less restrictive mode of operation is to combine all data from several passes (without segmenting into similar sections of similar asymptotic rates) and process using asymptotically determined or assumed parameters, however the security promise of such a procedure would need to be closely examined.
Here, we consider a more flexible mode of operation whereby data from an overpass is processed as a single block taking into account finite length key statistics and uncertainties to maintain high levels of composable security (Fig. 4),
SKL finite = SKL , ,(3)
where { , } are agglomerated count data without partitioning into sub-segments. This operational mode is considerably more practical for large constellations of satellites [25,49] and number of OGSs where it would be resource intensive to keep track of and store a combinatorially large number of individual segmented link data until each had attained a sufficiently large block size for asymptotic key extraction. For these reasons, it is important to analyse the performance of SatQKD systems without assuming asymptotic key rates or infeasibly long operational periods to establish post-processing blocks.
IV. FINITE KEY LENGTH ANALYSIS
In this section, we determine the achievable finite key for different satellite passes using the optimised, decoy-state BB84 protocol with weak coherent pulses. Tight finite-key security bounds for this protocol have been derived for one [31] and two [30] decoy states. The difference between the use of one a sufficiently large block size is collected. The minimum range, hence loss, can be made arbitrarily small thus high count rates and large block sizes, e.g. 10 12 , are feasible. and two-decoy state is small, however the use of two decoy states allows better estimation of the vacuum yield, which is important in the high loss regime. We optimise the two-decoy state protocol parameters and the amount of pass data to use in a block, accounting for practical constraints. This determines the average achievable key lengths that can be expected under different network operations. We also consider the sensitivity of the generated key length on different experimental parameters and satellite trajectories in order to derive a long term average amount of finite key.
The efficient BB84 protocol [50] encodes signals in X and Z bases with probabilities X and 1 − X respectively. It is a matter of convention which basis is used for key generation and the other for parameter estimation. In this work, the key is generated from X basis signals and the error rate in the Z basis is used to bound an eavesdroppers information on the raw key. The use of biased basis choice improves the sifting ratio without compromising the security. In the asymptotic regime, the sifting ratio tends to 1, compared with 0.5 for symmetric basis choice as in the original formulation of BB84. As we demonstrate later in section IV E, this sifting ratio advantage persists in the finite key regime, leading to longer raw sifted key for a given number of transmitted signals. This reduces parameter estimation uncertainties as well as provides more distillable raw key in the first place. The key is generated only from events corresponding to cases where the two authorised parties choose the X basis, and its phase error rate is determined from joint Z basis choices.
Practical BB84 implementations generally use phaserandomised laser-generated weak coherent pulses (WCPs) instead of true single photon sources as the latter devices are currently unavailable for field deployment [51]. Phase randomised WCPs are a probabilistic mixture of photon number (Fock) states, even weak pulses will have a non-trivial fraction of multi-photon emissions. Pulses that contain two or more photons compromise the security of basic BB84 due to the possibility of photon number splitting (PNS) attacks unless the channel loss is kept relatively low [52]. So-called decoy state methods were proposed as a method to detect PNS attacks and have rapidly become the de facto method of implementing prepare and measure QKD using WCP sources [6].
For the three-intensity (or two-decoy state) BB84 protocol, the sender chooses to send one of three intensities for ∈ {1, 2, 3} with probabilities . For the purposes of the security proof, we assume that 3 = 0 is the vacuum state, and the other intensities satisfy 1 > 2 and 2 > 0. The finite key attainable with two decoy states is given by [30],
ℓ = X,0 + X,1 (1 − ℎ( X )) − EC − 6 log 2 21 s − log 2 2 c ,(4)
where X,0 , X,1 and X , are the vacuum yield, single-photon yield, and the phase error rate associated with the singlephoton events respectively. For a satellite, unlike in the fibre case considered in [30], one cannot fix the size of the sifted X-basis data block. Instead, we have a fixed number of pulses, , sent per pass. This number is determined by the laser rep-
Algorithm 1 {η sys link (j), [j], QBER} = Optimised skl (N ,M ,L) 1: C 1 ← 0 < p X < 1 2: C 2,3,4 ← 0 < p j < 1 for j = 1, 2, 3 3: C 5,6 ← 0 < µ j < 1 for j = 1, 2 4: C 7 ← µ 1 − µ 3 > µ 2 5: C 8 ← µ 2 > µ 3 6
: N ← number of pulses per second 7: M ← number of satellite passes 8: L t ← elevation dependent loss at time t 9: t 0 ← time at max elevation 10: τ n ← probability of n-photon transmission 11: v ← 0 12: for all j = 0 to 15 dB do 13: for all ∆t = 0 to t max do 14: η sys link (j) ← L t0 + l sj 15:
[D(j)] kl ← detection probability of pulse k at time l 16: [E(j)] kl ← error probability of pulse k at time l -Generate data (error) block sizes for pulse k-17: Figure 4. Pseudocode to determine the optimised finite key length for a satellite pass. The optimisation is performed over all protocol parameters and the transmission time window Δ over which raw key is acquired. The correction terms ± X(Z) , for the bases X(Z) and the intensities come from the multiplicative Chernoff bound that accounts for finite statistics [33]. etition rate and the fraction of the orbit from which we collect data. The sizes of the X and Z basis data blocks are calculated from , the probability of detecting a pulse (which is function of time), and the sifting ratios. Next, we look into the security conditions of this secret key.
[n X(Z) ] k ← t0+∆t l=t0−∆t N p k p 2 X(Z) [D(j)] kl 18: n ± X(Z),k ← [n x(z) ] k ± δ ± X(Z),k 19: n X(Z) ← k [n x(z) ] k 20: s x,0 ← τ 0 µ2n − X,3 −µ3n + X,µ 2 µ2−µ3 21: s x,1 ← τ1µ1 n − X,2 −n + X,3 − µ 2 2 −µ 2 3 µ 2 1 n + X,1 − s X,0 τ 0 µ1(µ2−µ3)−µ 2 2 +µ 2 3 22: φ X ← phase error rate 23: [j, ∆t] ← 1 M s X,0 + s X,1 (1 − h(φ X )) − λ EC − 6
For error correction, error syndrome information is exchanged over a public channel to correct key strings. The number of bits leaked during error correction is denoted EC , and is accounted for during privacy amplification. In the finite key regime, this information leakage has a fundamental upper bound EC < log |M |, where M characterises the set of syndromes in the information reconciliation [32]. We use an estimate of EC that varies with block size [32] (details in Appendix A).
The security of the key rate for the decoy state efficient BB84 protocol is defined via the following statement: For small security parameters, c , s > 0, the protocol is = c + s -secure if it is c -correct and s -secret [30,53]. For numerical optimisation, we take c = 10 −15 and s = 10 −9 .
Conditioned on passing the checks in the error-estimation and error-verification steps, an s -secret key of length ℓ can be attained.
We implement a numerical optimiser to determine the achievable secret key length for a satellite pass, illustrated by pseudocode in Fig. 4, using improved finite key analysis to that in Ref. [30]. The main difference, is in estimating the terms ± X(Z) , , which are corrections due to the finite statistics. Instead of using the Hoeffding bound, we employ the tighter, multiplicative Chernoff bound [33], to estimate ± X(Z) , (see Appendix A). In the following, we provide a detailed analysis of how different factors, such as system parameters and pass geometry, influence the achievable finite key length.
A. Transmission time window optimisation
After the quantum transmission phase of an overpass (or several), we group together the sifted raw key to process in a single block. In addition to optimising the protocol parameters, we can also optimise the over elevations over which we include data in the processing block, i.e. trimming data from the start and end of the overpass to produce a truncated block. For small elevation angles (around the horizon), the high losses mean that the signal count rate is lower and a relatively higher proportion of extraneous counts when compared to larger elevation angles (where the satellite passes nearer to zenith). The QBER is thus greater at lower elevation angles than at higher angles. Excluding data from the start and end of the overpass results in a lower average QBER for the raw key. The reduction in QBER, however, comes at a cost of reducing the length of the raw key. This unveils a trade-off between length of raw key and QBER. This trade-off necessitates an optimisation of the protocol parameters (biased probabilities and intensities) and the length of each transmission window over which the finite key is assimilated.
For any satellite pass, we define = 0 as the time where the satellite reaches maximum elevation, and ± max as the time at which the satellite is at local horizon, i.e. where the elevation angle is zero. We consider a transmission window, Δ (0 < Δ ≤ max ) over which data is collected into a raw key block. We assume that the transmission window extends from −Δ to +Δ , i.e. the region surrounding maximum elevation.
For intermediate values of Δ , a partial satellite trajectory path is used to construct a finite key. Fig. 5 shows for different sys link the achievable secret key length (SKL) for varying values of Δ , hence data used, for a zenith pass.
We observe that for systems with good sys link , the QBER at low elevations does not rise greatly above QBER I . As such, it is generally better to keep more data by keeping Δ close to . For poor sys link , however, restricting the elevations over which data is included to near zenith can lead to a longer SKL (see Fig. [28], sys link = 35 dB). This is due to the improvement in error rate counteracting the smaller raw key length and slightly larger uncertainties due to a smaller sample size.
Our values of QBER are calculated for a whole pass, and thus represent an average of the instantaneous QBER. Nevertheless, the values presented in Fig. 5 are consistent with the instantaneous QBERs found in the Micius satellite [28]. For each Δ , the protocol parameters were optimised for maximum SKL using data in the interval −Δ to +Δ . Increasing Δ to large values only leads to minor increases in the SKL. For example restricting a sys link = 27 dB system to −150 ≤ ≤ +150 instead of the full ±350 leads to only a ∼10% drop in SKL and using only −50 ≤ ≤ +50 achieves 50% of the maximum SKL for a single pass. For worse sys link , inclusion of data from low elevations is detrimental since they have high error rates due to low signal to noise (SNR). This contaminates the data from around zenith that have higher raw count rates, higher SNR and thus much lower QBER. Satellite passes around the horizon yield large error rates, this is seen in the rise of the truncated block QBER, especially for sys link = 35 dB where increasing the transmission window into the low elevation region degrades the total SKL. Although our QBER is for the whole pass, and thus represents an average, the values are nevertheless consistent with the QBER found in the Micius satellite [28]. Numerical instabilities in the QBER for sys link = 35 dB appear since it is not the objective function of the optimisation program. p ec = 5 × 10 −7 p ec = 1 × 10 −6 p ec = 5 × 10 −6 p ec = 1 × 10 −5 Figure 6. SKL with link efficiency for different extraneous count rates. Solid lines represent the finite key attainable with a single satellite pass over zenith, while dashed lines represent the finite key, per pass, for two passes. For each curve, QBER I is 0.5% and the after pulse probability 0.1%. We consider 20% additional extraneous count rates over the baseline considered by Miscius, which could represent operation near a full Moon or severe light pollution.
B. SKL dependence on system parameters
In this section, we determine the dependence of the optimised finite key length to different system parameters: extraneous counts (detector dark count and background light); and intrinsic quantum bit errors QBER I . For each analysis, we optimise over all protocol parameters in addition to the satellite transmission window. The achievable SKL is then determined for different sys link representing differently performing SatQKD systems, e.g. using smaller OGS diameters, that introduce additional losses (independent of elevation) over the reference system with sys link = 27 dB. We denote changes to sys link by adding excess such that excess = 10 (− /10) , where is the corresponding difference in sys link in dB. We consider excess up to 15 dB (corresponding to sys link = 42 dB) equivalent to using a = 21.3 cm diameter OGS instead of the reference = 1.2 m, keeping all other system parameters unchanged unchanged. Notably we only consider worse total link efficiencies than the concrete nominal system loss demonstrated by the Micius satellite-OGS link. Negative excess , corresponding to the use of more efficient detectors or larger OGSs, would result in an increase in the attainable key.
We first consider the effects of different extraneous count rates ec on the achievable key lengths (Fig. 6). For OGS receivers, the constraints on detector size, weight, and power (SWaP) are not as severe as for space-based detectors, however it is desirable to reduce the cost and complexity of such systems. Although superconducting nanowire single photon detectors exhibit excellent photon detection efficiency with improved quantum efficiencies across a wide range of wavelengths with very low jitter and dark count rates [54], the need for cryogenic coolers and single mode coupling (necessitating the use of adaptive optics) greatly restricts widespread deployment. Silicon SPADs are the preferred option for Vis/NIR photons with moderately high PDE and sufficiently low dark count rate with conventional thermoelectric cooling of the order of 10s of counts per second, leading to a few error counts per second taking into account temporal filtering [55].
In typical OGS environments (i.e. not under dark sky conditions), we can assume that background light will be a significant contribution to the extraneous count rate. In our analysis, we account for stray light from celestial bodies and human activity adding to the detector dark count which is lumped together into a single extraneous count rate. Notice, that the extraneous count rate has a significant impact on the SKL. While it can increase the vacuum yield this effect is more than compensated by the increase in X and EC . The net effect is to decrease the SKL. Specifically, a factor of 10 increase in the ec reduces the achievable SKL by 40% at 27 dB. Further, the effects of extraneous counts are compounded by increased loss. In particular, in the high loss regime, an increase in extraneous count can result in no finite key being generated. This indicates that extraneous counts have a strong influence on the QBER, when sys link becomes worse. We now look at the effect of intrinsic errors, e.g. due to a less than unit fidelity of the source states, misalignment of the satellite and receiver reference frames, or non-ideal projective measurements by the OGS (Fig. 7). We lump all these effects into a quantity QBER I that characterises the intrinsic error of the system that is independent of the count rate. We observe that the finite key length is not as susceptible to changes in the QBER I as compared with ec .
Other factors affecting the SKL include signal intensities , their probabilities , and basis bias X . The optimised values of these system parameters generally depend on the excess loss considered. Fig. 8 shows how the optimised protocol parameters change with the total link efficiency. We see that as sys link increases (worsens), X decreases meaning that Alice needs to send more pulses in the Z basis. This comes from the fact that as the loss increases, the OGS detects fewer photons leading to worse parameter estimation of X,0 , X,1 and X from fewer statistics. To compensate, we need to collect more Z basis events by increasing Z i.e. by decreasing X . The reduced number of key generation events (X basis detections) is outweighed by better bounds on the key length parameters. This implies that the uncertainties in the parameter estimation from finite statistics dominates the SKL compared with raw key length when sys link becomes poor.
C. Non-zenith passes
The typical pass will not pass over zenith, hence it is necessary to analysis the SKL achievable from non-ideal overpasses (Fig. 1). Here, we explore the effect of sys link on the lowest elevation pass − max that still results in non-zero finite key and how this affects the long-term average SKL. In Fig. 9 we illustrate the achievable SKL depending on the minimum zenith angle min = 90 • − max attained during the pass. As expected larger minimum zenith angles min leads to smaller SKL due to shorter transmission times as well as lower count rates due to worse average link at lower elevations and longer ranges. The SKL vanishes when the satellite pass is above a critical zenith angle + min . This is a crucial point for any QKD satellite mission design as we can only extract a finite key for satellite passes that yield sufficient data.
In order to estimate the average amount of secret key that can be generate with single pass blocks, we first integrate the area under the SKL vs min curve,
SKL int = 2 ∫ + min 0 SKL min min ,(5)
where + min is the maximum OGS ground track offset that results in finite key (see Fig. 9). We can then estimate the annual average key (in the absence of weather effects and variation of ec , e.g. due to time of year, phase of Moon) by,
SKL year = year orbits SKL int lat ,(6)
where year orbits is the number of orbits of the satellite per year, and is the longitudinal circumference at the latitude of the OGS. This assumes that, on average, min is evenly distributed (unless in an Earth synchronous orbit [25]) and that the OGS is at low to moderate latitudes as the approximation becomes worse at the poles due to the orbital inclination (∼97 • ) of an SSO orbit. For ℎ = 500 km, year orbits ∼5500 and the circumference along a line of longitude at the position of Glasgow (55.9 • N) is 55 • ∼ 2.25 × 10 7 m resulting in SKL year = 2.44 × 10 −4 SKL int .
D. Multiple satellite passes
We now explore how combining separate satellite passes can improved secret key lengths for situations where some degree of key generation latency can be tolerated and the single-pass finite key assumption can be relaxed. Fig. 7 illustrates the finite and asymptotic secret key-lengths for a single zenith satellite pass. The asymptotic key rate corresponds to many satellite passes where the block sizes used to determine the key rate tend to infinity. Note that we do not use the process given by Eq. 2, instead we the put together all the data from the passes into a single block for processing and do not use Figure 9. SKL vs ground track offset. We plot the finite key length attainable with varying minimum ground track distances between the satellite and OGS for different sys link . The system parameters were ec = 5 × 10 −7 and QBER I = 0.5%. The key generation footprint is given by the maximum min that still results in non-zero finite key. The maximum footprint is limited by visibility above the horizon, for ℎ = 500 km the maximum total width is 4887 km. Here, we have assumed that quantum transmission can occur horizon to horizon. In practice, quantum transmission may not be possible immediately after the satellite rises above the horizon due to the need for initiation and stabilisation of tracking, handshake protocols, and local skyline obstructions. Similarly, as the satellite sets the quantum transmission may need to terminate before reaches 0 • to allow for post-processing and finalisation of all reconciliation steps. If we assume a nonzero minimum elevation below which quantum transmission does not occur, this would decrease + min and move all the curves down. It may be reasonable to set + max = 80 • for advanced SatQKD systems leading to a maximum key generation footprint of 3200 km total width ( + min = 1600 km). Fig. 9 and is in units of bit-metres (bm). We have not included weather effects or considered a minimum elevation for quantum transmission so these values should be considered an optimistic upper bound.
System loss
a segmented or instantaneous asymptotic key rate; we only assume that the aggregated block parameters are known with negligible uncertainty. The process of taking the block size to infinity requires care for satellites, as each orbit yields a fixed amount of data. A detailed discussion of the asymptotic limit is given in Appendix C. As expected, the key length increases significantly if the block asymptotic regime can be reached. Generating key from a single pass constrains the achievable secret key length. One way to approach the asymptotic key rate is to accumulate more signals through multiple satellite passes. This approach assumes that latency in key generation and the storage of raw key can be tolerated. Fig. 10 illustrates the improvement from multiple zenith passes where non-zero finite key cannot be generated in a single zenith pass for different combinations of system parameters. The relative improvement depends on the system link efficiency. The origin of this improvement is purely due to smaller parameter estimation uncertainties owing to increased statistics since averaging over several pass does not improve the actual QBER or phase error ( X ) rates, though is a small improvement due to better error correction efficiency and reduced EC . This clearly illustrates the principle difficulty in satellite QKD; fluctuations in finite statistics severely limit the average attainable SKL.
Combining the data from multiple passes can improve the key length per pass. In particular, we can obtain non-zero key when sys link is too poor for key generation from a single pass. However, this must be balanced against greater key generation latency and potential security issues from needing to store large amounts of raw data for a longer period of time between passes. Ensuring the security of the OGS for several passes is more difficult than ensuring its integrity over a single pass. We assume that key is consumed soon after generation so the attack horizon is dominated by the period in which secret data is at rest.
E. Protocol Selection
To illustrate the effect of statistic uncertainties in constraining the key length under finite block size assumptions, we compare the performance of two variants of BB84: efficient BB84 (as considered thus far) [50] and standard BB84 [56]. The main difference is in the choice of bases, standard BB84 chooses both X and Z bases with equal (symmetric) probability whilst efficient BB84 is allowed to favour one basis over the other (asymmetric basis choice). standard BB84 also uses both bases to generate key and perform parameter estimation, whilst efficient BB84 uses only one basis for key generation whilst the other is only used solely for parameter estimation. Due to these differences, we will refer to the two protocols as symmetric and asymmetric BB84 respectively. Fig. 11 illustrates the attainable key length for a single satellite pass over zenith with varying levels of system link efficiency in the link efficiency. Notice that the asymmetric protocol achieves both a better key length for all values of system link efficiency. Fig. 12 shows the SKL as a function of maximum zenith angle for non-zenith passes and shows that the asymmetric protocol generates more keys for all satellite trajectories. This amounts to an increased average key generation per year of 86% and 180% for sys link = 37 dB and sys link = 40 dB respectively under the most generous assumptions (no weather of minimum elevation effects).
The advantages of the asymmetric protocol stem form several factors. Firstly, the asymmetric protocol retains more raw bits at the sifting stage. This obviously leads to a larger final secret key. However, retaining more of the raw key bits also allows us to perform better parameter estimation. This is a crucial advantage for finite-key QKD. As we have seen, finite key analysis is extremely sensitive to how well we estimate the parameters. In particular, a small increase in loss can result in the key length dropping to zero.
The next advantage of the asymmetric protocol is more subtle. In asymmetric BB84, we use all the Z-basis events to estimate the vacuum and single photon yields. In contrast, for symmetric BB84, we reveal a random sub-sample of results for each basis, where each amount is optimised. The vacuum and single photon yields for the X or Z bases are found using only results corresponding to that basis. This means that only half of the revealed results are used to estimate the yields for each basis. In contrast, in the asymmetric protocol, all of the revealed results are used to estimate the yields. If minimum elevation limits for quantum transmission is considered, the relative advantage can only increase.
One final advantage is that asymmetric BB84 requires less classical communication. In particular, we do not need to communicate the choice of which bits to reveal to estimate the QBER and other parameters. Instead, this task is achieved by the choice of basis. One slight disadvantage of asymmetric BB84 is in the biased basis choices. A quantum random number generator needs to generate several unbiased bits to produce a single biased bit (at worst 2 unbiased bits per biased bit on average using a non-deterministic algorithm). However, after this stage, the storage requirements for these basis bits is identical in both symmetric and asymmetric BB84.
This analysis shows that in the finite key regime, asymmetric BB84 is both more efficient and practical than symmetric BB84. It is able to deliver improved key rates for a wider range of system parameters and system link efficiencies. Additionally, it can provide a non-zero key for more orbits than symmetric BB84.
V. CONCLUSIONS AND DISCUSSIONS
We have analysed how the attainable secret key length varies with a single or multiple overpasses under the finite key block size assumption. This type of analysis is important for establishing the security of realistic operational models for practical SatQKD. Our study explores the influence of different parameters, including: the system link efficiency performance level characterised by sys link , intrinsic source and measurement quality characterised by QBER I , extraneous count rate including detector dark counts and background light with combined probability ec per pulse, pass geometry, and raw data block length. We use as the basis for sys link the most recent data from the Micius satellite as representative of the current state of the art performance.
We show that extraneous count rates are more detrimental than worse intrinsic QBER to the finite size secret key length due to the the effect of worse link efficiencies at low elevations affecting the raw key error rates. We find that only a few passes are required to raise zero finite SKL to positive values quickly approaching asymptotic rates. Small efficiency gains can lead to large average SKL amounts due to the expanded key generation footprint enabled. This can be seen in the SKL year averages that shows a large dependence on sys link . For example, for the nominal system parameters, a change from sys link = 37 dB to sys link = 40 dB (factor of 2) results in a 7.7× reduction in the yearly average rate since not only is the SKL for a given max reduced, the key generation footprint, compounding the reduction.
Further work could include more general loss vs elevation models so that more diverse systems and OGS locations can be studied. Operational constraints on the transmission time may impact on achievable key rates for poor sys link at low elevations and requires further study. For example if post-processing and all reconciliations steps need to be completed before the end of the overpass, this will introduce a minimum gap between the end of the quantum transmission and the time the satellite passes below the horizon that will impact upon SKL and − max . The effect of source rate is implicit in the analysis via a change of sys link axis origin as previously discussed. Our model can be combined with higher fidelity orbital and constellation modelling for more precise long term average SKL prediction, taking into account local atmospheric and weather conditions [25,49]. We can also extend our finite key analysis to other protocols, such as CV-QKD with appropriate modification of the SKL and optimisation procedures. The Bennett-Brassard 1984 (BB84) quantum key distribution (QKD) protocol has become widely implemented owing to its simplicity, overall performance and provable security [56]. Practical implementations, however, depart from the idealised single-photon sources required in the original theoretical proposal. Instead, weak pulsed laser sources are used due to availability and ease of implementation compared with true single photon sources. While this delivers improved repetition rates over current single photon sources, the unmodified BB84 protocol is vulnerable to photon-number-splitting (PNS) attacks that exploit multi-photon pulse fraction present in each weak coherent pulse signal [57].
To overcome PNS attacks and extend the range of WCP sources, the so-called decoy-state protocols were developed. These employ multiple Poisson photon number distributions associated with different phase randomised coherent state intensities that allow more accurate characterisation of the photon number distribution of the transmitted pulses associated with detection events [58]. In this way, the secure fraction of the sifted raw key (vacuum and single photon yields) can be reliably estimated, which makes the decoy-state BB84 protocol a secure and practical implementation of QKD .
The security of decoy-state QKD was initially developed assuming the asymptotic-key regime [59,60]. Such an analysis neglects fluctuations due to finite statistics. To compensate, security analyses for finite-length keys have been developed [61][62][63]. However, early approaches used a Gaussian assumption to bound the difference between the asymptotic and finite results. As such, their validity is restricted to collective attacks, rather general coherent attacks. Security analyses for general attacks have also been developed [64]. The multiplicative Chernoff bound [36,65] and Hoeffding inequality [30] can be used to bound the fluctuations between the observed values and the true expectation value. A more complete finite-key analysis for decoy-state based BB84, with composable security, has recently been presented in [33]. This uses the multiplicative Chernoff bound, and presents simple analytic expressions, which are still reasonably tight.
The approach we adopt is a modification of [30]. We use the same structure but replace the Hoeffding bound with the inverse multiplicative Chernoff bound, as proposed in [33,65], specifically, using equations (9) and (10) of [33] to obtain new expressions for the finite statistical corrections terms, ± X(Z) , , that appear in equations (2)-(4) of [30].
Though the term "decoy state" has been used to describe such protocols, it is a slight misnomer when applied to current protocols. The original idea was to use decoy states of differing intensities to identify the presence of an eavesdropper performing PNS attacks, the results of the decoy states were discarded and only the detection event corresponding to "signal'; states were used for the final key. However, it is not only possible but also more efficient to utilise all intensity pulses to derive the final key, hence they can all be considered as "signal". In light of this, it may be more accurate to describe such protocols as "multi-intensity" WCP QKD and relegate the term "decoy" as a historical footnote. But to maintain consistency with historical usage, we will continue to refer to these protocols as "decoy-state". An important step in any QKD protocol is error correction. This necessitates classical communication, of EC bits, which are known to Eve. This must be taken account of in the privacy amplification stage.
In an actual run of the protocol, one would know EC , however, for optimisation, we must estimate it. If this estimate is too pessimistic, then we could conclude that no key can be extracted in a region where one is still viable. Conversely, an overly optimistic value for EC can lead to spurious results.
It is standard to model the channel as a bit-flip channel. This leads to EC = EC X ℎ 2 ( ), where is the QBER and EC is the reconciliation factor. The value of EC is typically slightly above 1, e.g. 1.16. The use of a constant reconciliation factor is a simple way of accounting for inefficiency in the error correction protocol. This approach is normally sufficient when determining the optimal secret key length. However, for satellite QKD, one operates with high losses that are at the limit of where one can extract a key. As such, it is beneficial to use a more refined estimate of EC .
A better estimate of EC is given by [32]. In this approach, the correction to X ℎ( ) depends on the data block size. In this work, we utilise the approach of [32] to estimate the information leaked during error correction. In particular,
EC = X ℎ( ) + X (1 − ) log (1 − ) −[ −1 ( ; X , 1 − , ) − 1] log (1 − ) − 1 2 log( X ) − log(1/ ),(A1)
where X is the data block size, is the QBER and −1 is the inverse of the cumulative distribution function of the binomial distribution.
In this section we give an operational definition for the asymptotic key length per pass and briefly sketch its derivation. One can combine data from multiple satellite passes, and extract a secret key from the combined results. The optimisation over system parameters would then be performed for the combined results, over several passes. In principle, we can combine results from different satellite orbits. However, for simplicity, here we only consider the case were we combine data from passes with the same orbit and under similar conditions. Let ℓ 1 be the length of key we extract from a single pass. Naively, one might expect that the length of key for two passes would equal 2ℓ 1 . This intuition, however, is false. This is illustrated in Fig. 10, where for a single pass we obtain no key, but a non-zero key is recovered for two or more passes. In this figure, we plotted the key length per pass, which allows us to clearly see the the advantage of combining data. Another thing to note about Fig. 10 is that while there is a gain in combining three or more passes, this gain seems to decrease.
A natural question to ask is: what is largest possible SKL, per pass, that one can obtain by combining arbitrarily many passes? This is given by the quantity ℓ ∞ = lim →∞ ℓ / , where is the number of passes and ℓ is the SKL one can extract by combining passes. The key length, ℓ , is found using equation (4), together with equations (2)-(5) of [30]. The quantity ℓ ∞ is found by examining the asymptotic scaling of ℓ / . We will not give the complete derivation, but will illustrate the method by looking at the first term: X,0 / .
The estimate for the vacuum counts per pass is [30],
X,0 = 0 2 − 3 2 Γ 3 X,3 − − X,3 − 3 Γ 2 X,2 + + X,2 , (C1)
where X, is the number of sifted counts in the X-basis, from pulses of intensity , 0 is averaged probability that a vacuum state is transmitted by the laser, Γ = exp( )/ and ± X, are a correction terms resulting from the finite statistics. In [30], ± X, comes from Hoeffding's inequality. In our case, we use the multiplicative Chernoff bound (see equations (9) and (10) of [33]). However, for both the Hoeffding and Chernoff bound, the asymptotic scaling of ± X, goes O ( √ X ). This means that the scaling with the number of satellite passes is O ( √ ). Hence, ± X, / scales O (1/ √ ) and thus tends to zero as → ∞. The finite statistics correction terms thus go to zero. By assumption, each satellite pass is for the same orbit and under similar conditions. We should expect that X, equals times the number of counts for a single pass, (1) X, . From this observation, we obtain
lim →∞ X,0 = 0 2 − 3 2 Γ 3 (1) X,3 − 3 Γ 2 (1) X,2 = * X,0 , (C2)
where * X,0 is the asymptotic estimate of the vacuum counts for a single pass. By following a similar process for each term in ℓ / , we obtain the result
ℓ ∞ = * X,0 + * X,1 1 − ℎ( * X ) − ∞ EC ,(C3)
where * X = * Z,1 / * Z,1 is the phase error rate, * X,1 , * Z,1 and * Z,1 are respectively, the single pass asymptotic estimates for: the single photon counts in the X basis, the single photon counts in the Z basis and the single photon errors in the Z basis. The expressions for * X,1 , * Z,1 and * Z,1 are identical to the equations for the equivalent asymptotic quantities given in Appendix A of [30]. The term ∞ EC is the limit of EC / . This depends on how we estimate the information leaked during error correction.
A refined estimate of EC is presented in [32]. Equation (5) of [32] gives a upper bound on the asymptotic behaviour of EC . Using this expression, we find that ∞ EC = (1) X ℎ( ), where is the QBER for a single pass. When running the numerical code for the asymptotic key length per pass, we set ∞ EC = 1.16 (1) ℎ( ). This was to take account of the fact that even in the asymptotic limit, our error correction protocol might not be perfectly efficient. However, in all other runs, we used the expression for EC given in equation (6) of [32].
Rather than looking at the key length per pass, it is also common to consider the key rate, i.e. the number of secret key bits per transmitted pulse. Let be the total number of pulses transmitted by the satellite during a single pass. The key rate for passes is just SKR = ℓ /( ). In the limit of infinitely many passes, the asymptotic key rate is given by
SKR ∞ = ℓ ∞ / .
We conclude by noting that the asymptotic secret key rate, SKR ∞ , we present is rather different from the one often discussed in the literature. It is more common to calculate an asymptotic key rate for each angle of elevation and then combine the asymptotic key rates in a manner similar to that shown in equation (1). While such an approach is sensible if we extract keys for each angle of elevation, it is not appropriate for the current analysis, where we group all the data for a pass and then extract a key from the combined data.
Figure 5 .
5Secret key length and QBER vs zenith pass segment size. Solid lines are for different sys link and dashed lines represent corresponding truncated block QBERs. System parameters are QBER I = 0.5% and ec = 10 −6 .
Figure 7 .
7SKL with link efficiency for different values of QBER I . Solid lines represent the finite key attainable with a single satellite pass over zenith, while dashed lines represent the normalised (per pass) finite key, using combined data from two passes. The dotted lines represent the normalised block asymptotic SKL (Sect. IV D). The extraneous count rate is ec = 5 × 10 −7 per pulse and the after pulse probability 0.1%.
Figure 8 .
8Optimisation parameters vs sys link for zenith pass. System parameters are ec = 5 × 10 −7 and QBER I of 0.5%.
Figure 10 .
10Plot of normalised (per pass) SKL vs number of passes. The plot shows the SKL achievable by combining data from multiple zenith passes before processing as a single block. The different curves correspond to different sets of system parameters with different { sys link , ec , QBER I }: A={45.7 dB, 10 −7 , 0.5%}, B={44.8 dB, 10 −7 , 1%}, and C={40.5 dB, 5 × 10 −7 , 1%}.
Figure 11 .
11SKL vs sys link for asymmetric and symmetric BB84 protocols. System parameters are ec = 5 × 10 −7 , and QBER I = 0.5%. The vertical lines at sys link = 37 dB and 40 dB indicates the system configurations for which we explore the elevation dependence of both protocols inFig. 12.
Figure 12 .
12Plot of key length (in bits) attainable using the asymmetric and symmetric BB84 protocols with varying max altitude satellite elevation angles. Solid lines correspond to a system link efficiency of sys link = 40 dB, and dashed lines with sys link = 37 dB, both with ec = 5 × 10 −7 , QBER I = 0.5%. As in Sec. IV C, we can calculate int and SKL 55 • N year to arrive at {1.76 × 10 11 bm, 42.9 Mb} and {9.43 × 10 10 bm, 23.0 Mb} for sys link = 37 dB, and {2.28×10 10 bm, 5.58 Mb} and {8.14 × 10 9 bm, 1.99 Mb} for sys link = 40 dB for the asymmetric and asymmetric protocols respectively. The inefficiency of symmetric BB84 is apparent, asymmetric BB4 is able to generate 1.86× and 2.80× more key on average for sys link = 37 dB and sys link = 40 dB respectively. The advantage is seen to grow as sys link become worse.
ACKNOWLEDGMENTS
We acknowledge support from the UK NQTP and the Quantum Technology Hub in Quantum Communications (EP-SRC Grant Ref: EP/T001011/1), the UK Space Agency (NSTP3-FT-063, NSTP3-FT2-065, NSIP ROKS Payload Flight Model), the Innovate UK project ReFQ (Project number: 78161), and QTSPACE (COST CA15220). DO is an EPSRC Researchers in Residence at the Satellite Applications Catapult (EPSRC Grant Ref: EP/T517288/1). DO and TB acknowledge support from the Innovate UK project AirQKD (Application number: 45364). DO and DM acknowledge support from the Innovate UK project ViSatQT (Project number: 43037). RP aknowledge support from the EPSRC Research Excellence Award (REA) Studentship. The authors thank J. Rarity, D. Lowndes, S. K. Joshi, E. Hastings, P. Zhang, and L. Mazzarella. for insightful discussions. DO also acknowledges discussion with S. Mohapatra, Craft Prospect Ltd., and support from the EPSRC Impact Acceleration Account. Appendix A: Finite key analysis for decoy-state BB84
Maximize [j, ∆t] over constraints {C 1 , . . . , C 8 }log 2
21
s − 6 log 2
2
c
24:
25:
v j ← {η sys
link (j), sup ∆t [j, ∆t], QBER}
26: Return v
1
Appendix B: Protocol optimisationWe detail the numerical optimisation that we implement for the finite key length. The finite key rate is a complex function of these five protocol parameters: { X , 1 , 2 , 1 , 2 }, where Z = 1 − X and 3 = 1 − 1 − 2 are fixed by the optimised quantities, and we set 3 = 0. We generate an optimised key length by optimising over the parameter space of these five variables. We do so by implementing a constrained numerical optimisation program, which ensures the generated finite key length satisfies the following criteria[30]:The security of the key rate derived from the asymmetric, decoy state BB84 protocol has a specific definition. Notably, for small protocol errors, c , s > 0, the protocol is c + s -secure if it is c -correct and s -secret. For the numerical optimisation, we take c = 10 −15 and s = 10 −9 .In addition, we take into account some fixed parameters for the numerical optimisation as shown inTable Ithat define a baseline for system performance. The numerical optimisation is performed separately for different values of Δ , we then chose the value of Δ that yields the largest secret key length, for the particular value of system link efficiency. The full optimisation over { X , 1 , 2 , 1 , 2 } and Δ , occurs for each value of system efficiency. For example, infigure 6, each point of the graph corresponds to different optimised parameters.Appendix C: Asymptotic key length per pass
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| [] |
[
"Athermal jamming of soft frictionless Platonic solids",
"Athermal jamming of soft frictionless Platonic solids"
] | [
"Kyle C Smith \nSchool of Mechanical Engineering\nMaurice Zucrow Laboratories\nBirck Nanotechnology Center\nPurdue University\n47907West LafayetteINUSA\n",
"Meheboob Alam \nEngineering Mechanics Unit\nJawaharlal Nehru Centre for Advanced Scientific Research\nJakkur Campus560064BangaloreIndia\n",
"Timothy S Fisher \nSchool of Mechanical Engineering\nMaurice Zucrow Laboratories\nBirck Nanotechnology Center\nPurdue University\n47907West LafayetteINUSA\n"
] | [
"School of Mechanical Engineering\nMaurice Zucrow Laboratories\nBirck Nanotechnology Center\nPurdue University\n47907West LafayetteINUSA",
"Engineering Mechanics Unit\nJawaharlal Nehru Centre for Advanced Scientific Research\nJakkur Campus560064BangaloreIndia",
"School of Mechanical Engineering\nMaurice Zucrow Laboratories\nBirck Nanotechnology Center\nPurdue University\n47907West LafayetteINUSA"
] | [] | A mechanically-based structural optimization method is utilized to explore the phenomena of jamming for assemblies of frictionless Platonic solids. Systems of these regular convex polyhedra exhibit mechanically stable phases with density substantially less than optimal for a given shape, revealing that thermal motion is necessary to access high density phases. We confirm that the large system jamming threshold of 0.623 ± 0.003 for tetrahedra is consistent with experiments on tetrahedral dice. Also, the extremely short-ranged translational correlations of packed tetrahedra observed in experiments are confirmed here, in contrast with those of thermally simulated glasses. Though highly ordered phases are observed to form for small numbers of cubes and dodecahedra, the short correlation length scale suppresses ordering in large systems, resulting in packings that are mechanically consistent with 'orientationally disordered' contacts (point-face and edge-edge contacts). Mild nematic ordering is observed for large systems of cubes, whereas angular correlations for the remaining shapes 2 are ultra short-ranged. In particular the angular correlation function of tetrahedra agrees with that recently observed experimentally for tetrahedral dice. Power-law scaling exponents for energy with respect to distance from the jamming threshold exhibit a clear dependence on the 'highest order' percolating contact topology. These nominal exponents are 6, 4, and 2 for configurations having percolating point-face (or edge-edge), edge-face, and face-face contacts, respectively. Jamming contact number is approximated for small systems of tetrahedra, icosahedra, dodecahedra, and octahedra with order and packing representative of larger systems. These Platonic solids exhibit hypostatic behavior, with average jamming contact number between the isostatic value for spheres and that of asymmetric particles. These shapes violate the isostatic conjecture, displaying contact number that decreases monotonically with sphericity. The common symmetry of dual polyhedra results in local translational structural similarity. Systems of highly spherical particles possessing icosahedral symmetry, such as icosahedra or dodecahedra, exhibit structural behavior similar to spheres, including jamming contact number and radial distribution function. These results suggest that though continuous rotational symmetry is broken by icosahedra and dodecahedra, the structural features of disordered packings of these particles are well replicated by spheres. Octahedra and cubes, which possess octahedral symmetry, exhibit similar local translational ordering, despite exhibiting strong differences in nematic ordering. In general, the structural features of systems with tetrahedra, octahedra, and cubes differ significantly from those of sphere packings.(e.g., system energy, pressure, contact number) denoted as X vary with density φ relative to the jammed density φ J as:Scaling exponents for mechanical properties β have not been shown to be universal. For instance, bulk and shear modulus scaling exponents depend on the contact force model employed in simulations. O'Hern et al. [3] utilized various potential interaction models between | 10.1103/physreve.82.051304 | [
"https://arxiv.org/pdf/1008.2475v4.pdf"
] | 2,227,825 | 1008.2475 | e0948af62540ee5204ef23b09b7b56246a793b8d |
Athermal jamming of soft frictionless Platonic solids
Kyle C Smith
School of Mechanical Engineering
Maurice Zucrow Laboratories
Birck Nanotechnology Center
Purdue University
47907West LafayetteINUSA
Meheboob Alam
Engineering Mechanics Unit
Jawaharlal Nehru Centre for Advanced Scientific Research
Jakkur Campus560064BangaloreIndia
Timothy S Fisher
School of Mechanical Engineering
Maurice Zucrow Laboratories
Birck Nanotechnology Center
Purdue University
47907West LafayetteINUSA
Athermal jamming of soft frictionless Platonic solids
1
A mechanically-based structural optimization method is utilized to explore the phenomena of jamming for assemblies of frictionless Platonic solids. Systems of these regular convex polyhedra exhibit mechanically stable phases with density substantially less than optimal for a given shape, revealing that thermal motion is necessary to access high density phases. We confirm that the large system jamming threshold of 0.623 ± 0.003 for tetrahedra is consistent with experiments on tetrahedral dice. Also, the extremely short-ranged translational correlations of packed tetrahedra observed in experiments are confirmed here, in contrast with those of thermally simulated glasses. Though highly ordered phases are observed to form for small numbers of cubes and dodecahedra, the short correlation length scale suppresses ordering in large systems, resulting in packings that are mechanically consistent with 'orientationally disordered' contacts (point-face and edge-edge contacts). Mild nematic ordering is observed for large systems of cubes, whereas angular correlations for the remaining shapes 2 are ultra short-ranged. In particular the angular correlation function of tetrahedra agrees with that recently observed experimentally for tetrahedral dice. Power-law scaling exponents for energy with respect to distance from the jamming threshold exhibit a clear dependence on the 'highest order' percolating contact topology. These nominal exponents are 6, 4, and 2 for configurations having percolating point-face (or edge-edge), edge-face, and face-face contacts, respectively. Jamming contact number is approximated for small systems of tetrahedra, icosahedra, dodecahedra, and octahedra with order and packing representative of larger systems. These Platonic solids exhibit hypostatic behavior, with average jamming contact number between the isostatic value for spheres and that of asymmetric particles. These shapes violate the isostatic conjecture, displaying contact number that decreases monotonically with sphericity. The common symmetry of dual polyhedra results in local translational structural similarity. Systems of highly spherical particles possessing icosahedral symmetry, such as icosahedra or dodecahedra, exhibit structural behavior similar to spheres, including jamming contact number and radial distribution function. These results suggest that though continuous rotational symmetry is broken by icosahedra and dodecahedra, the structural features of disordered packings of these particles are well replicated by spheres. Octahedra and cubes, which possess octahedral symmetry, exhibit similar local translational ordering, despite exhibiting strong differences in nematic ordering. In general, the structural features of systems with tetrahedra, octahedra, and cubes differ significantly from those of sphere packings.(e.g., system energy, pressure, contact number) denoted as X vary with density φ relative to the jammed density φ J as:Scaling exponents for mechanical properties β have not been shown to be universal. For instance, bulk and shear modulus scaling exponents depend on the contact force model employed in simulations. O'Hern et al. [3] utilized various potential interaction models between
I. Introduction
Granular materials are observed in nature and widely utilized in modern industrial processes. Aside from materials possessing solid granular particles, other discrete network systems such as foams, emulsions, and glasses display similar mechanical and structural behavior [1]. The mechanics of granular materials can exhibit gas, liquid, and solid-like behavior [2]. Density, defined as the fraction of volume occupied by grains, and state of stress are critical quantities of interest for such systems of particles. For periodic systems, density φ is defined as p cell NV V , where N, V p , and V cell are the number of particles in the primary cell, volume of each particle, and volume of the primary cell, respectively. In particular, for systems of particles with purely repulsive interactions the jamming threshold density φ J is of interest. The jamming threshold marks the thermodynamic transition between flowing and static states of granular materials [3]. In practice the onset of non-zero potential energy has been used to identify the jamming threshold density (e.g., in [3]).
In addition to the jamming threshold, which could potentially depend on consolidation path, the optimal density for a given particle shape is also an important quantitative characteristic of granular assemblies. Optimal density differs from jamming threshold in that optimal density is the maximum possible density for an assembly of particles irrespective of its accessibility via mechanical or thermal means. A proof of optimal density for spheres was not accomplished until recently [4]. Recent non-mechanically-based approaches have been employed to determine the optimal density of Platonic and Archimedean solids [5][6], tetrahedra [5][6][7][8][9][10][11][12][13], ellipsoids [14], and superballs [15]. Both geometric [7,[9][10][11][12][13] and thermodynamic Monte
Carlo [5,6,8] methods have been utilized to probe high density phases of rigid particles, but incremental random motions that particles undergo in such simulations are not induced by mechanical interaction. As a result, such methods enable systems to bypass low density jammed configurations. Accordingly, these methods are capable of achieving densities and phases not obtainable through purely athermal mechanical interactions. In particular, several groups have focused intensely on demonstrating increases in the predicted maximum density of tetrahedra packings, employing Monte Carlo and geometric methods. A Monte Carlo method was employed to find a densest packing of φ = 0.823 [5], exceeding previous record densities obtained through strictly geometric approaches (see [10][11][12]). Haji-Akbari et al. [8] reported the spontaneous formation of a quasi-crystal for a fluid of tetrahedra at φ = 0.8230, and by compressing an approximant of that structure they obtained a structure with φ = 0.8503. Kallus et al. [13] utilized a 'divide and conquer' search of non-overlapping positions and orientations of four tetrahedra, and thereby discovered a one-parameter family of dense double-dimer packings with φ = 0.8547. Chen et al. [9] extended the double-dimer system to produce the highest density structure of tetrahedra with φ = 0.856347, which was subsequently shown to be the optimal packing for a six-parameter family of double-dimers [7]. These densities are much higher than the experimentally reported value of φ = 0.76 ± 0.02 for tetrahedral dice [16].
Further, Jiao and Torquato [5,6] showed that optimal packings of icosahedra, dodecahedra, and octahedra are crystalline using Monte Carlo optimization, and they can form structures with φ > 0.8.
Though Monte Carlo and geometric approaches are useful for finding optimal packings of particles, many granular systems cannot access the highest density phases. For example, although spheres possess an optimal closest packed density of φ = 0.74, the maximally random jammed state of spheres is known to produce a density of φ = 0.64 [17]. Thus, to obtain granular configurations that are commonly encountered in nature and industry the methods utilized to consolidate granular materials should be replicated. Athermal, mechanically-based studies of spheres [3] and ellipses [18] have been performed recently through energy based approaches and have yielded sphere packing densities consistent with the precisely-defined maximally random jammed density (see [17]). Though these methods do not incorporate friction, they do replicate the quasi-static nature of motion during consolidation.
Properties other than jamming threshold are of interest at the jamming point, as well.
For instance, the average number of contacts per particle at the jamming point exhibits values that depend on particle shape. The isostatic conjecture attempts to describe structure at the jamming point through kinematic constraint counting. The conjecture suggests that the average number of contacts between particles is equal to twice the average number of particle degrees of freedom. Ellipses [18,19], ellipsoids [19], and tetrahedra [16] for instance display hypostatic behavior, i.e. having average jamming contact numbers less than their respective isostatic values. The departure of ellipsoids from isostatic behavior has been attributed to the presence of floppy modes, which provide vanishing restoring force [18], whereas for tetrahedra it has been attributed to the varying degrees of rotational constraint by discrete contact topologies (e.g., vertex-face, edge-edge, edge-face, face-face contacts) [16].
Mechanical properties of granular systems as they are perturbed to densities above the jamming threshold have exhibited power law scaling [3]. Expressed mathematically, properties bi-disperse disks in which potential energy scaled with inter-particle separation via power-law exponents of 5/2 (Hertzian) and 2 (harmonic), respectively. For the different potential models utilized by O'Hern et al., bulk modulus displayed scaling exponents of 1/2 and zero, while shear modulus displayed scaling of 1 and 1/2, respectively. In contrast to scaling of mechanical properties the excess contact number, Z-Z J , exhibited 1/2 power-law scaling for a variety of contact interaction models [3].
Ellenbroek et al. [20] explained this dependence by relating the probability of developing a new contact with the scaling of non-affine displacements away from the jamming point. Universality of this scaling across particle morphologies is not guaranteed though.
In this work, we explore the jamming of assemblies of soft Platonic solids ( Fig. 1) through mechanically-based structural optimization. Particular emphasis in Section II is given to the methods of structural optimization and the calculation of forces on particles. In Section III results are presented for the distribution of jamming threshold for small configurations of particles, scaling of mechanical properties upon perturbation from the jamming point, system ordering, and isostaticity.
II. Methodology
At the onset of jamming during consolidation, total energy is infinitesimal and any motion of a collection of particles necessarily increases total energy. Approaching this point is quite difficult because the finite range of potentials employed in soft particle simulations requires finite energy to impart forces between particles to induce motion. Consequently small perturbations from the jamming point, having finite, non-zero total energy are required for numerical simulation. To obtain such configurations, a sequence of compressive and expansive volumetric strain steps is applied to an initially dilute (φ = 0.05) three-dimensional configuration of N particles in the primary cubic-shaped cell. Periodic boundary conditions are often employed to neglect wall effects in jamming simulations, as reflected in the recent review on jamming of hard particles by Stillinger and Torquato [21]. Accordingly, we employ periodic boundary conditions for all configurations studied in this work. Following each strain step, conjugate gradient energy minimization is utilized to relax the system of particles toward static equilibrium. The conjugate gradient method admits simultaneous motion of particles, resulting in equilibrium configurations that are collectively jammed upon convergence, i.e. energy of these configurations cannot be reduced by individual or concerted particle motions [23].
Here the initial dilute configuration is chosen with pseudo-random particle positions and orientations. In contrast to the work of O'Hern et al. [3] to find inherent structures of particle systems, our approach attempts to replicate the stepwise consolidation of random particle assemblies to dense, jammed states. Configurations are compressed past the jamming point near a target energy E t and subsequently expanded toward the jamming point. The initial configuration is first compressed in an affine manner by an isotropic volumetric strain increment ε V . Conjugate gradient minimization then proceeds until (1) average energy E falls below a threshold value of 0.25E t or (2) equilibrium is achieved. Equilibrium convergence criterion is described in sub-section B. When criterion (1) is satisfied (E < 0.25E t ) the configuration is further compressed by ε V to approach a mechanically stable state near the consolidation target energy. Above the jamming point, criterion (2) will be satisfied, and if energy is less than the target energy (Ε < E t ), the jammed configuration possesses acceptable energy. In contrast, if the equilibrium energy is greater than the target energy (Ε > E t ) the configuration is expanded by -0.75ε V . Following expansion, ε V is recursively assigned a value of 0.25ε V in order to asymptotically approach the target energy with further strain-relaxation sequences. This procedure is similar to that of Mailman et al. [18] and Gao et al. [22] in which volumetric strain is applied and subsequently relaxed via conjugate gradient iterations.
Following consolidation, the assembly is sequentially expanded to approach the jamming point (Ε → 0 + ). To approach the jamming point asymptotically, the strain increment is chosen to depend on the current configuration's energy E and the contact model exponent m:
( ) 1 3m V E ε γ = (2)
Here, γ is a constant chosen for a particular system of particles to achieve sufficient system energy variation during the expansion process. As will be shown later, this strain is generally insufficient to reduce energy directly to zero for disordered packings, and will result in an asymptotic approach toward the jamming point.
A. Conjugate Gradient Method
The conjugate gradient method is employed to search for a local minimizer of the positions and orientations of particles in the configuration. The conjugate gradient method is concatenation of those for each particle i g . The conjugate gradient minimization procedure starts with a trajectory in the steepest descent direction [27]:
( ) ( ) 1 1 = − r g(5)
The elements of ( ) k r are actually displacement coordinates in real-space and must be converted to angular-space to perform rotation operations on particles. The properly scaled search trajectory ( ) k s is given by:
( ) ( ) 1/2 k k − = s S r (6)
where S is a block-diagonal matrix incorporating S i sub-matrices intended to scale rotational coordinates relative to translation coordinates:
1 0 0 0 0 0 0 N = ⋅ S S S(7)
Translation and rotation components for each particle are decomposed from ( ) k s and are then applied to particles in order to compute energy at the corresponding position. Details of the quaternion-based rotation operation utilized are described in Appendix A.
A one-dimensional line search for the minimum along this direction is performed. Details of this line search procedure employed are described in the Appendix B. The trajectory for the next conjugate gradient search is determined with the following update formula
( ) ( ) ( ) ( ) 1 1 k k k k β + + = − + r g r(8)
where ( ) k β is determined by the formula of Polak and Ribiere [28]:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 T T k k k k k k β + + = − g g g g g (9)
The formula of Fletcher and Reeves [29] was also tested and found to substantially underperform. Though both formulas produce quadratic rates of convergence when applied to a quadratic objective functional, the Fletcher-Reeves formula can slow to linear convergence for non-quadratic objective functionals if not sequentially restarted (see [30]). The Polak-Ribiere update formula tends to restart automatically toward the steepest descent direction
( ( ) ( ) 1 1 k k + + → − r g when ( ) ( ) 1 k k + → g g ),
and as a result it is superior for the minimization of energy functionals employed in this work.
Convergence of conjugate gradient iterations is assessed by considering the relative change in energy D E (k) at iteration k:
( ) ( ) ( ) ( ) 1 1 k k k E k E E D E − − − =(10)
This convergence criterion has been utilized by Mailman et al. [18] and Gao et al. [22] to obtain mechanically stable packings of ellipsoids and disks, respectively, and we confirm that when set to an adequately small value, residual forces also become small relative to contact forces, satisfying static equilibrium.
B. Contact energy model
Determining an expression for energy and force between contacting polyhedral particles in terms of inter-particle separation, as is done for spheres (cf. [3]) and ellipsoids (cf. [18]), is difficult due to the non-smooth surfaces of faceted particles. We utilize a simplified model proposed by Feng and Owen [31] whose energy-based contact model assumes that the elastic energy resulting from contact between two particles depends only on the volume of intersection V. Variations of this model have been presented in two dimensions for discrete contact interaction cases by Pöschel and Schwager [32] and Feng and Owen [33]. Similar models have been applied in the literature wherein contact forces and moments are formulated heuristically in terms of contact geometry [34], but they lack energy conservation [32]. In contrast, the derivation of forces and moments from an energy basis results in energy conservation, though particle interpenetration may lead to unphysical behavior [32]. For the athermal, quasi-static simulations employed here, interpenetrating contacts are not expected because our implementation explicitly enforces that energy descent occurs during conjugate gradient iterations, preventing configurations from settling into local maxima during interpenetration.
The conjugate gradient method requires evaluation of contact forces and moments to determine search directions. Expressions for the forces and moments between a pair of particles, A and B, are hereafter presented. The force of particle A on particle B AB F is the gradient of energy E AB with respect to the translational displacement of particle A A x .
A AB AB dE V dV = ∇ x F (11)
where V is the intersection volume between particles A and B. As indicated above in Eq. 11, the dependence of energy on volume (i.e., ( ) AB E V ) influences the magnitude of force, but not its direction. The force's direction is determined entirely by the gradient of intersection volume.
Intersection volume gradient is the sum of the product of area A i and inward pointing unit normal vectorˆi n for each face belonging to particle A on the intersection volume [31]:
 x i i i V A n ∇ = ∑ (12)
Similarly, the moment about the centroid of particle B can be expressed as the sum from each face belonging to particle A on the intersection volume [31]:
 B AB i i i i dE A n dV = × ∑ M r(13)
where i r is the distance from the centroid of particle A to the centroid of face i.
The form of the energy functional is important in representing the true dynamical character of the contacting particles, as well as in achieving stable numerical solutions. We assume that the energy functional takes the general form of a power law with exponent m, elastic modulus Y, and volumes of the particles in contact, V A and V B :
( ) 1 / m m AB A B E YV V V m − = +(14)
The value of m = 2 utilized here ensures consistency with Hooke's law for mechanical interaction exhibited in uniaxial compression of aligned bars. Numerical details of the computation of intersection volume between contacting polyhedra are described in the Appendix C.
III. Results and discussion
A sample curve displaying the variation of energy with density during the process described above is shown in Fig. 2. During the consolidation phase, system energy is maintained just below the target energy by performing conjugate gradient relaxation. At the final consolidation point, the system becomes mechanically stable near the target energy and is subsequently expanded through stable states toward the jamming point. We extrapolate the average contact depth with respect to density in order to approximate jamming threshold, as described in Appendix D. For each conjugate gradient iteration D E (k) = 10 -12 is used to assess convergence of 25 particle systems.
Choice of the target energy can influence the jamming threshold for a given initial configuration. Consolidated configurations of each shape were generated with E t = 3.2 . 10 -5 , 10 -6 , and 3.2 . 10 -8 . The volumetric strain step ε V was chosen to be -0.036, -0.020, and -0.011 for the respective target energies. Following consolidation selected configurations were expanded to estimate jamming thresholds. Figure 3 displays the resulting dependence of jamming threshold on target energy with the same initial positions for each value of E t . Increased target energy enables configurations to access states with high jamming threshold. Hereafter E t = 3.2 . 10 -5 is utilized to obtain consolidated configurations having high jamming thresholds.
A. Configurational distributions
To study the distribution of jamming thresholds for small assemblies of Platonic solids, 11 random initial configurations were consolidated for each shape. Selected configurations were then expanded in order to find the jamming threshold; the jamming thresholds of remaining configurations were determined by extrapolation of energy with the scaling behavior of other configurations in the ensemble. Phases form with distinct scaling of energy with respect to density; evidence and discussion of which will follow in the next section. The resulting jamming threshold distributions are shown in Fig. 4 grouped according to these phases. Tetrahedra, icosahedra, and octahedra exhibit uni-modal distributions, while dodecahedra and cubes display bi-modal and tri-modal distributions, respectively. In particular dodecahedra exhibited a 60 % probability of crystallizing at φ J = 0.838. Though cubes did not display crystallization, highly ordered layer structures formed for 50 % of the configurations at φ J = 0.926. The densities of these ordered structures are lower than the expected optimal packing densities due to the insufficient number of particles in the assembly. Such layered structures are similar to the irregular square tessellations proposed by Torquato and Jiao [5] but with vacancy defects due to finite system size. At intermediate density, cubes form another phase which is primarily distinguished by its mechanical properties and slight orientational disorder resulting from edgeface contacts.
Statistical properties of small system phases are listed in Table I. In addition, the particle sphericity ψ listed is defined as the ratio of surface area between a sphere and a particle of the same volume (see [35]). With the exception of cubes, average jamming thresholds increase with sphericity for these small systems. As presented in the next section, this trend does not persist as system size increases. The jammed densities of disordered packings are substantially lower than the optimal densities recently reported in the literature. In contrast the average density for tetrahedra is reasonable when compared to the experimental density obtained for tetrahedral dice. Considering that tetrahedral dice fill 16 % more volume than inscribed tetrahedra, the inscribed tetrahedra of the experimental packing of Jaoshvili et al. [16] would have a density of 0.64 ± 0.02, which is very close to the average here. With such a substantial difference between optimal and jammed densities it is clear that optimal packing is improbable for athermal grains. Table II. Phases lacking order generally exhibited a nominal scaling exponent of 6 with respect to deviation from the jamming threshold that we denote as excess density (∆φ = φ φ J ), i.e. E ~ (∆φ) 6 . Such scaling is considered very soft when compared to the scaling of sphere packings. Among small ensembles, crystallization of dodecahedra and layering of cubes were observed, and, as a result, energy scaling displayed a nominal scaling exponent of 2 (E ~ (∆φ) 2 ) with respect to excess density, distinct from that of disordered phases. The 'cubes edge' structure previously referred to exhibits a differing scaling exponent than layered cubes. This phase possessed sufficient density of edge-face contacts for mechanical percolation of the edge-face contact network to occur. As a result, these configurations exhibit an energy scaling exponent of 4 (E ~ (∆φ) 4 ). As indicated in Fig. 5c, scaling of the cubes edge structure energy becomes softer as the jamming point is approached. We attribute this softening to decreased numerical accuracy at low ∆φ. As will be shown later, these highly ordered phases are suppressed as system size increases.
B. Energy scaling
Shown in
These scaling exponents can be understood in terms of the energy scaling of discrete contact topologies. A faceted particle possesses discrete geometric features (vertices, edges, and faces) that intersect the features of a contacting particle. Distinct combinations of these intersecting features we refer to as 'contact topologies'. The contact topology hierarchy possesses three levels with increasing degree of restraint between contacting particles - (1) vertex-face and edge-edge contacts, (2) edge-face contacts, and (3) and electrical conductivity, of these systems will also display topologically dependent percolation physics.
C. System size dependence and ordering
The number of particles was varied to determine limiting structures for each particle system with increasing degrees of freedom. For systems of 100 and 400 particles convergence tolerances of D E (k) = 2.5 . 10 -13 and 6.25 . 10 -14 were respectively employed for consolidation.
Coarser convergence tolerances during expansion of 10 -8 were utilized for large systems and jamming threshold was approximated by extrapolation of average contact depth. Shown in Fig. 6, jamming threshold stabilizes as the number of particles increases. Also no crystallization was observed for configurations with 100 or 400 particles. Thus, the high probability of crystallization for dodecahedra in 25 particle configurations was enabled by the small system size. Therefore, it is desirable to understand the structural mechanisms behind order frustration, as well as a precise description of the disordered sub-structures present in these packings.
As discussed in the previous sections, disordered configurations of Platonic solids produce drastically lower jamming threshold than their ordered counterparts. Also, the scaling of energy with excess density changes dramatically between ordered and disordered phases.
One expects the underlying structure of disordered and ordered configurations to be significantly different. Figure 7 compares such configurations at densities just above the jamming point ( 0 0.02 φ < ∆ < ). Crystallization is visually apparent for ordered dodecahedra, while its disordered counterpart possesses significant orientational and translational disorder. In contrast, only a few particles with orientational disorder disturb the translational order of cube systems. Specifically, only slight orientational disorder in the percolating edge-face cube system is necessary to induce structures that exhibit drastically different mechanical scaling.
Differences in these structural phases can be understood in terms of nematic ordering. We therefore calculate the nematic order parameter S for configurations containing cubes, as described in Appendix E. This parameter approaches unity for highly oriented nematic phases.
Configurations of layered cubes with scaling exponents of [36], superballs [37], and superellipsoids [38], but this is the first such study to confirm such a phase for athermal cubes. In contrast to small ordered systems, large systems of cubes exhibit substantially lower nematic order parameters of 0.93 and 0.87 for 100 and 400 cubes, respectively. Though the octahedron and cube form a dual pair and both possess octahedral symmetry, we find that octahedra tend to exhibit nematic order parameters of ~0.6, indicating little nematic order. Such behavior is in contrast with the nematic phase formed prior to crystallization of octahedra via Monte Carlo simulation (see [5]). This finding reveals that the triaxial symmetry of cube faces, rather than the triaxial symmetry of octahedron vertices, results
in nematic order at the jamming point. Fig. 9 were calculated by averaging over all particles and normalizing by the number of particles in an ideal gas volume element of the same density φ, as is conventional practice (see [39]). As a result of homogeneous correlation, each RDF approaches unity at large radii. All RDFs are presented in terms of radius r normalized by the nearest distance between contacting polyhedra R min , and were calculated for configurations with energy near E t = 3.2 . 10 -5 . At this minimum radius, all shapes display strong peaks as a result of face-face contacts. The RDFs for each particle shape vary dramatically, and are therefore grouped with RDFs displaying similar features.
Radial distribution functions (RDFs) of large periodic systems presented in
Firstly, the RDFs of dodecahedra and icosahedra, displayed in Fig. 9a, exhibit short-range translational order with features resembling those of the RDF for randomly packed spheres.
The RDFs of these highly spherical faceted particles exhibit correlation peaks at positions similar to those of spheres. In particular, these dual polyhedra share a distinguishing feature with sphere RDFs differing from those of other shapes presented -closely-grouped secondary and tertiary peaks. Finally, peak intensities are reduced to 20 % above the mean value at a radius of 3.5R min consistent with spheres. Subsequent peaks are offset from multiples of R min .
Packings of the self-dual tetrahedron exhibit a RDF, shown in Fig. 9b, with ultra-low translational order dissimilar to the other Platonic solids. Such behavior is consistent with that observed experimentally for tetrahedral dice (see [16]), as displayed. RDFs of both shapes exhibit a peak between R min and 2R min decaying to values less than 20% above the mean value at 2R min . Strong decay of the RDF of this disordered system contrasts with that of the higherdensity thermal glasses generated by Haji-Akbari et al. [8] via thermodynamic Monte Carlo simulation. Finally, the dual pair of cubes and octahedra exhibit similar RDFs (Fig. 9c) with peaks at integer multiples of R min . Peaks of the octahedra packing are broadened relative to those of cubes due to the lack of nematic order. Cubes and octahedra exhibit peaks with intensity greater than 20% above the mean up to 4R min and 3R min , respectively. In addition to the translational extent of ordering, the extent of orientational ordering is also interesting to assess. In particular we calculate the face-face angular correlation function F(r) displayed in Fig. 11, according to the procedure described in [16], by averaging over all particles in large periodic systems. Calculation details are described in Appendix E. F(r) measures the average anti-alignment of face normal vectors on particle surfaces. Aligned faceface contacts exhibit F(r) = -1, and F(r) increases with decreasing alignment of faces. In the vicinity of r ~ R min the angular correlation function for each shape exhibits high face-face antialignment. Octahedra, dodecahedra, and icosahedra exhibit step-like F(r) immediately following R min . Thus, these particle systems exhibit nearly homogeneous orientational correlation, confirming orientational disorder. The nematic order of cubes is also reflected in their angular correlation function exhibiting multiple peaks. Tetrahedra exhibit orientational correlation diminishing to a constant value only after 3R min in agreement with recent experimental results.
D. Isostaticity and jamming contact number
The variation of contact number with respect to excess density for sphere packings is known to exhibit square-root scaling for sphere systems independent of contact interaction model due to the tremendous degree of non-affine motion near the jamming point [3,20]. We seek to investigate this dependence for packings of non-spherical Platonic solids. To perform this study, the average contact number during expansion from the high-energy consolidated state was calculated, as shown in Fig. 12. A single sample for each shape was chosen from the 25 particle ensemble with jamming threshold and radial distribution function similar to that of large systems for the same shape. Due to the high nematic order in small systems of cubes relative to large systems, they have been omitted from this analysis. As the jamming point is approached, these systems become ill-conditioned, and the number of iterations required to relax structures diverges. Therefore, average contact number could only be studied over a limited range of excess density. Over this limited range square-root scaling does appear to be consistent with the variation of average contact number. Statistical fluctuations relative to the curve fit are present that would likely be suppressed for larger systems.
Finally, square-root curve fits are used to extrapolate contact number to the jamming point. In Fig. 13 the jamming contact number Z J is plotted as a function of shape sphericity.
The isostatic conjecture purports that the jamming contact number of three-dimensional particles Z J,iso depends on the number of particles N and the number of degrees of freedom per particle n:
, 2 6 J iso nN Z N − = (15)
For spheres with continuous rotational symmetries, n = 3 and Z J,iso = 5.76 for a system of 25 particles. In contrast, for the Platonic solids studied here, continuous rotational symmetry is broken, and as a result n = 6 and Z J,iso = 11.76. The actual calculated values of Z J are much lower than the value expected from the broken rotational symmetry of these shapes. Instead, values between the isostatic conjecture predictions are found for the shapes. Shapes possessing high sphericity (icosahedra and dodecahedra) display Z J just above the isostatic value for spheres with average contact number decreasing monotonically with sphericity ψ. The value of 8.6 ± 0.1 for tetrahedra is larger than the value of 6.3 ± 0.5 measured for a large system of tetrahedral dice [16]. The difference in average contact number is likely due to the high asphericity of tetrahedra relative to tetrahedral dice.
IV. Conclusions
An energy-based approach to modeling the mechanical behavior of non-smooth particles has been implemented and utilized to study jamming of frictionless Platonic solids.
The method does not incorporate thermal fluctuations as in the thermodynamic Monte Carlo approaches utilized to find optimal densities of non-smooth particles. For small particle systems, average jamming thresholds were obtained through configurational ensembles, and the resulting densities were substantially less than the previously reported optimal densities for each shape. In particular, our simulations produce tetrahedra with a jamming threshold consistent with experimental results. For small particle systems, dodecahedra can crystallize and cubes can order into layered structures with finite probability. These ordered structures are similar to the optimal structures previously predicted, but their formation is suppressed with increased system size. No prior reports have indicated such behavior, which is critical to the understanding of granular materials with non-smooth particle surfaces.
Radial distribution functions of the jammed structures were examined to quantify the extent of ordering in large configurations. The common symmetry of dual polyhedra results in similarity of radial distribution functions. The length scale of ordering for icosahedra, dodecahedra, and octahedra is approximately 3.5R min ; also, icosahedra and dodecahedra display similar local structure to spheres. Tetrahedra exhibit order only over a length scale of 2R min , with a radial distribution function similar to that observed in recent experiments. The short-range orientational correlation of tetrahedra is consistent with experimentally measured correlation. In contrast, cubes exhibit much longer range order up to a length scale of 4R min .
Large systems of cubes exhibit nematic ordering along a particular axis, despite lack of longrange translational order. Local ordering or the lack of it is crucial to description of these systems, and correlating these structures to macroscopic properties will be a subject of future investigation.
Aside from structural evidence based on the radial distribution function, ordered phases displayed power law scaling exponents for energy of 2 and 4 versus a scaling exponent of 6 for disordered phases. These effects are all linked to percolation of orientational order of faceted particles possessing discrete rotational symmetry. As the dramatic differences in mechanical properties resultant from topological contact networks affect mechanical properties, topological contact networks are expected to affect thermal and electrical transport properties within these microstructures. This topologically-dependent scaling phenomena is absent from assemblies of smooth particles, and this is the first such observance of this behavior.
Though all the Platonic solids possess broken continuous rotational symmetry, all possess contact numbers less than the isostatic value for asymmetric particles and are, hence, hypostatic. As evidenced by the close agreement of contact number for icosahedra and dodecahedra with the sphere isostatic value, the influence of rotational degrees of freedom is substantially less for these particles. Conversely, the remaining shapes studied display strong departure from the sphere isostatic value, and the average contact number at jamming correlates monotonically with shape sphericity. Prior work has revealed the hypostatic nature of smooth non-spherical particles [19] and tetrahedra [16], but ours is the first to correlate the degree of hypostaticity with sphericity of non-smooth particles.
The methods utilized herein provide a platform for further mechanical analysis of jamming of non-smooth particles. The findings also help to provide a more comprehensive picture of jamming phenomena across particle morphologies. Further study of the structural and local ordering phenomena in packings of faceted particles will aid development of granular materials possessing microstructures tailored for applications.
Appendices
A. Cumulative quaternion-based rotations
Particles are numerically rotated from initial positions after each compression step to prevent accumulation of round-off error due to sequential rotation operator application. To facilitate rotation, the cumulative rotation is stored as a quaternion vector q 0 , where
[ ] 0, sin , cos 2 2 s θ θ θ = = q v(16)
Here s and v are the scalar and vector components of the quaternion, respectively.
= + + × − ⋅ q q v v v v v v(18)
The rotation operator is then applied to the coordinates of the particle for computation of intersection with other particles. This approach helps to prevent numerical distortion of particles and to maintain stability.
B. Line search methods and termination criteria
A line search method that explicitly minimizes energy was utilized along each conjugate gradient search direction; the method employs combinations of golden searches and quadratic interpolations (see [42,43]). The first Goldstein condition and a two-sided slope test (see [27])
were incorporated to assess convergence:
( ) ( ) ( ) 0 ' 0 f f f α αρ − ≤ (19) ( ) ( ) ' ' 0 f f α σ ≤ −(20)
Here α and ( ) ' f α are the position along the search direction and the projection of the gradient of energy along the search direction. Eq. 20 tends to be the more restrictive condition and is essential to finding an accurate minimum. Eq. 19 essentially ensures that α is not a maximum.
In practice, a value ρ = 0 was required to achieve adequate computational efficiency, and since σ = 0.033 resulted in satisfactory convergence rates, this value was used in all simulations.
Upon each line search we initially guess that the minimum is contained in the region of α satisfying:
( ) ( ) ( ) 1 1 0 2 ' 0 k k E E D f α − − < ≤(21)
Here D E (k-1) is the relative energy after the previous conjugate gradient iteration. If energy calculated at the upper bound of the bracket does not satisfy Eq. 19, then the search interval is restricted in multiples of ten until it is satisfied. After this restriction, it is possible that the minimum is not contained within the bracket. Therefore, the search interval is then expanded in multiples of two until
C. Intersection volume computation
Computation of intersections among polyhedra is essential to the structural optimization methods described. Overlap volume can be calculated with ease analytically for simple contact types such as vertex-face and edge-edge contacts, but for more complex types of contacts,
analytical determination is very difficult to generalize. For instance, the drastic differences in contact topology require that generalized methods for determining intersection for arbitrary contacts are utilized. Methods from computational geometry are employed to generalize and stabilize such calculations.
In our implementation, candidate intersections between polyhedral particles are screened by performing bounding sphere contact detection (see [44]). Vertices on intersection volumes are found by determining intersections of the edges of both particles with the opposing particle via the ray intersection algorithm of Haines [45]. Convexity of the particles ensures that the intersection of the two particles is in fact convex. We therefore compute the Delaunay triangulation of the intersection points and calculate its volume through the resulting triangulated representation of the intersection geometry. Faces belonging originally to only one particle participating in the contact are identified as well. Triangulated representations of those faces are then utilized to calculate areas and centroids required for force and moment computations.
Because volume and area of the intersection geometry are calculated with floating point arithmetic, volume and area of each simplex are sorted in ascending order prior to summation in order to minimize round-off error. This general approach to determining contact geometry without contact planes does not require explicit declaration of contact topology.
D. Jamming threshold extrapolation
After
E. Nematic order and angular correlation computations
Nematic order parameters were calculated in order to assess uniaxial and biaxial ordering of cube and octahedron systems as in [36][37][38]. To identify the dominant nematic director vector for the system, orientational direction vectors for each particle must be grouped into sets having similar direction. For cubes, these direction vectors correspond to face unit normals, while for octahedra they correspond to vertex unit position vectors relative to particle centroids. To do this, we choose one particle's axes as a reference and match axes of the other particles to those reference axes. From this procedure three sets of aligned axes are obtained Face-face angular correlation functions were calculated according to the approach described by Jaoshvili et al. [16] to quantify orientational alignment. The face-face angular correlation between particles q and l is given as:
-{ } i u , { } i v ,( )m in ql qi lj ij F n n = ⋅(23)
where i and j represent the set of all faces of particles q and l, respectively. n represents the unit normal vector of a given face. ij represents the set of all possible combinations of i and j.
The face-face angular correlation function is determined as:
( ) ( ) ql q l F r F r δ = − − r r(24)
where δ(r) is the Dirac delta function. To numerically evaluate this function we compute F ql for all particle combinations of q and l. The set of all F ql are then binned with respect to q l − r r , and the average value for the bin at radius r is assigned to F(r). The inset shows how average contact depth for the expansion process was linearly extrapolated to approximate the jamming threshold. Energy is normalized by Energy is normalized by For N=100 and 400, φ J for only one configuration is presented with uncertainty due to the standard error of contact depth extrapolation. average contact number, Z J sphericity, ψ tetrahedra octahedra dodecahedra icosahedra asymmetric particle limit (n = 6)
Tables
1/3 p V Y , where V p is particle volume.1/3 p V Y , where V p is particle volume.
Fig. 5a is the variation in energy with density for selected samples from the configurational distributions presented. Figures 5b and 5c reveal the scaling of the different phases observed. Samples in each phase group possess scaling exponents ranging over the values listed in
.
face-face contacts. Within the hierarchy contact intersection volume varies with contact depth d as ~n V d ; levels 1, 2, and 3 of the hierarchy possess volume scaling exponents of n = 3, 2, and 1, respectively. Within the contact mechanics model employed, energy scales as ~m E V and therefore ~n m E d Thus by utilizing m = 2 in all simulations here, the nominal scaling exponents of 6, 4, and 2 are observed for the configurations exhibiting mechanical percolation of contacts with level 1, 2, and 3, respectively. This percolation phenomenon is unique to systems of soft, faceted particles and is essential to the scaling of mechanical properties in these systems. Scalar transport properties, such as effective thermal
all phases of cubes with 1.0000 ≥ S > 0.9999. Cubes edge structures with scaling exponents of ~4 β exhibit lesser nematic order with 0.999 > S > 0.99. Finally, disordered structures with scaling exponents of ~6 β exhibit S < 0.99 with an average value of S = 0.96, displaying lowest nematic order. Thus only slight orientational disorder induces drastic contrast in the scaling of mechanical properties.
Figure 8
8contains representations of the largest stable systems of particles investigated near the target energy E t = 3.2 . 10 -5 . Disordered structures are apparent for large systems of tetrahedra, icosahedra, dodecahedra, and octahedra. Such visual evidence is consistent with translational and orientational correlation functions presented subsequently in Figs. 9, 10, and 11. In contrast, cubes exhibit mild nematic ordering along the dominant nematic director vector shown in the figure. Similar phases have been observed for thermal cuboids
Through visual observation of particle packing and orientational order parameter analysis, we have shown that large cube systems possess moderate nematic ordering. The influence of this nematic ordering is not clear from the calculation of RDF, because it exhibits diminishing correlation with radius. For nematic systems it is more appropriate to analyze the density correlation function with respect to coordinates longitudinal and transverse to the dominant nematic axis. These measures of translational correlation anisotropy have been utilized to distinguish between nematic and columnar phases of cut spheres[41] and are plotted inFig. 10. The values shown were biased and normalized relative the mean values for the respective function. The fluctuations in both functions are very small relative to the mean values; intensity of the longitudinal function does not diminish with radius. In contrast, the transverse function exhibits strong correlation peaks which diminish with radius. Thus, translational order is present along the direction possessing nematic order.
Incremental rotational trajectories determined during a conjugate gradient iteration are then converted to a quaternion vector q 1 . The angle of rotation θ about the unit vector axis θ is determined from the rotational component of the conjugate gradient trajectory( )
α > is satisfied. It is possible though that the initial interval satisfies Eq. 19 but does not in fact bracket the minimum. When( ) ' 0 f α < this situation occurs and the search interval is correspondingly expanded in multiples of two until satisfied. These bracketing procedures are essential to convergence of the line search procedure. With these parameters, 2 bracketing function evaluations with 4 line search function evaluations are often required per conjugate gradient iteration.
obtaining jamming configurations near the target energy, configurations are expanded toward the jamming point. As a result of floating-point precision employed in our simulations, equilibrium configurations can only be simulated to finite values of ∆φ. This is a result of higher order contact topologies (face-face and edge-face contacts) being in equilibrium with lower order contact topologies (vertex-face and edge-face contacts).Therefore, to accurately estimate the jamming threshold extrapolation techniques must be employed. As a result of affine motion during expansion toward the jamming point, we expect average contact depth to scale linearly with excess density, i.e. d φ ∆ . Numerically, we calculate depth for each contact as the ratio of contact volume to normal-projected surface area. We find the jamming threshold for a particular configuration by calculating the leastsquares linear intercept for ∆φ as a function of d.
FIG. 1
1-(color online) Platonic solids from left to right: tetrahedron, icosahedron, dodecahedron, octahedron, and cube.FIG. 2 -(color online) Average energy as a function of density for a jammed assembly of 25 cubes with E t ε V = -0.036 during consolidation and subsequent expansion toward the jamming threshold.
FIG. 3
3-(color online) Jamming threshold as a function of target energy for configurations of 25 Platonic solids. 95 % confidence intervals based on extrapolation of contact depth are represented by error bars.
FIG. 4
4-(color online) Configurational spectrum of jamming threshold for assemblies of 25 Platonic solids produced via 11 random initial configurations.
FIG. 5
5-(color online) Average energy for jammed assemblies of 25 Platonic solids. (a) average energy as a function of density. Average energy as a function of excess density for (b) disordered phases and (c) ordered phases. Energy is normalized by 1/3 p V Y , where V p is particle volume.
FIG. 6
6-(color online) Scaling of jamming threshold as a function of number of particles for jammed assemblies of 25 Platonic solids. 60 % confidence intervals are represented by error bars. Statistics for N=25 configurations were determined from the sample set of φ J obtained for different initial configurations.
FIG. 7
7-(color online) Depictions of the ordered and disordered configurations of dodecahedra and cubes in the primary cubic cell with boundaries indicated in black. Highly ordered (a) crystallized dodecahedra, (c) layered cubes, and (d) semi-ordered cubes with percolating edge-face contacts. Highly disordered (b) dodecahedra and (e) cubes. Arrows indicate the dominant uniaxial nematic director vector for each phase of cubes.
FIG. 8
8-(color online) Jammed assemblies of 400 (a) tetrahedra, (b) icosahedra, (c) dodecahedra, (d) octahedra, and (e) cubes. (f) cubes are oriented clearly displaying nematic order in the direction of the arrow. Boundaries of primary cubic cells are indicated in black.
FIG. 9
9-(color online) Radial distribution functions for jammed assemblies of 400 (a) icosahedra, dodecahedra, and spheres, (b) tetrahedra, tetrahedral dice, and tetrahedra thermal glass, and (c) octahedra and cubes. The RDF of the tetrahedra thermal glass has been scaled for viewing purposes.
FIG. 10 -
10Anisotropic radial distribution functions for a jammed assembly of 400 cubes. The longitudinal distribution function is calculated along the dominant direction of nematic order, while the transverse distribution function is calculated in the plane normal to that direction. The peak at r/R min < 1 is a consequence of projecting transverse (longitudinal) neighbors onto the longitudinal (transverse) direction.
FIG. 11 -
11(color online) Face-face angular correlation functions for jammed assemblies of 400 Platonic solids.
FIG. 12 -
12(color online) Scaling of average contact number as a function of excess density for jammed assemblies of 25 Platonic solids. Curves represent square-root scaling fits. FIG. 13 -(color online) Jammed contact number as a function of particle sphericity for assemblies of 25 Platonic solids. Error bars were calculated based on the standard error for square-root extrapolation.
Figure 1
1Figure 1 (two-column format)
Figure 3
3Figure 3
Figure 4
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Figure 12
Figure 13
We calculate Q vv and Q ww for axes sets { } i v and { } i w as well. For each nematic tensor we determine the dominant eigenvalues and eigenvectors and assign the maximal eigenvalue among those three sets as the uniaxial nematic order parameter S. The nematic director vector is then assigned as the eigenvector corresponding to the maximal eigenvalue among the three axes sets.and { }
i
w . For axes set { }
i
u we calculate the nematic tensor Q uu as a sum over all
particles [36]:
1
1
3
1
2
2
N
uu
i
i
i
N
u u
αβ
α β
αβ
δ
−
=
=
−
∑
Q
(22)
Table I -
IAverage jamming threshold for jammed assemblies of 25 Platonic solids determined via 11 random initial configurations. Uncertainties shown represent the 95 % confidence intervals for each spectrum.ψ
φ J
φ max
tetrahedra
0.671
0.611 ± 0.037
0.856 (a)
octahedra
0.846
0.677 ± 0.011
0.947 (b)
dodecahedra
0.910
0.684 ± 0.009
0.904 (b)
icosahedra
0.939
0.727 ± 0.029
0.836 (b)
cubes
0.806
0.773 ± 0.057
1.000
cubes edge
0.806
0.796 ± 0.051
1.000
dodecahedra crystal 0.910
0.838
0.904 (b)
layered cubes
0.806
0.926
1.000
(a) ref. 9
(b) ref. 5
Table II -
IIAverage energy scaling exponents for jammed assemblies of 25 Platonic solids. Uncertainties shown represent the maximal deviation of the sample exponents from the average.β
disordered phases
tetrahedra
6.34 ± 0.92
dodecahedra
5.44 ± 0.08
octahedra
5.67 ± 0.24
icosahedra
5.47 ± 0.58
cubes
5.49 ± 0.91
ordered phases
cubes edge
4.61 ± 0.29
dodecahedra crystal 2.22 ± 0.22
layered cubes
1.99 ± 0.14
Figure captions
Graduate School for financial support via the Charles C. Chappelle fellowship. The authors thank Damian Sheehy of The Mathworks for insightful correspondence regarding computational geometry, Adam Carlyle of the Rosen Center for Advanced Computing at Purdue for computing aid, Jayathi Murthy for utilization of computing resources, Andriy Kyrylyuk and Gary Delaney for valuable discussion regarding jamming phenomena, Mitch Mailman and Carl Shreck for insightful discussion about numerical structural optimization of granular materials, and Salvatore Torquato and Yang Jiao for helpful, encouraging comments.
Figure 7(two-column format)
Figure 8(two-column format)
AcknowledgementsThe authors acknowledge the Indo-US Science and Technology Forum for supporting Purdue-JNCASR exchanges through the Joint Networked Centre on Nanomaterials for Energy
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| [] |
[
"Mixture Hidden Markov Models for Sequence Data: The seqHMM Package in R",
"Mixture Hidden Markov Models for Sequence Data: The seqHMM Package in R"
] | [
"Satu Helske \nUniversity of Oxford\nUK\n",
"Jouni Helske \nUniversity of Jyväskylä\nFinland\n"
] | [
"University of Oxford\nUK",
"University of Jyväskylä\nFinland"
] | [] | Sequence analysis is being more and more widely used for the analysis of social sequences and other multivariate categorical time series data. However, it is often complex to describe, visualize, and compare large sequence data, especially when there are multiple parallel sequences per subject. Hidden (latent) Markov models (HMMs) are able to detect underlying latent structures and they can be used in various longitudinal settings: to account for measurement error, to detect unobservable states, or to compress information across several types of observations. Extending to mixture hidden Markov models (MHMMs) allows clustering data into homogeneous subsets, with or without external covariates.The seqHMM package in R is designed for the efficient modeling of sequences and other categorical time series data containing one or multiple subjects with one or multiple interdependent sequences using HMMs and MHMMs. Also other restricted variants of the MHMM can be fitted, e.g., latent class models, Markov models, mixture Markov models, or even ordinary multinomial regression models with suitable parameterization of the HMM.Good graphical presentations of data and models are useful during the whole analysis process from the first glimpse at the data to model fitting and presentation of results. The package provides easy options for plotting parallel sequence data, and proposes visualizing HMMs as directed graphs. | 10.18637/jss.v088.i03 | [
"https://arxiv.org/pdf/1704.00543v2.pdf"
] | 14,192,465 | 1704.00543 | 832b1d7eb6977901bb571e44c46c0dd7ed4fffc4 |
Mixture Hidden Markov Models for Sequence Data: The seqHMM Package in R
Satu Helske
University of Oxford
UK
Jouni Helske
University of Jyväskylä
Finland
Mixture Hidden Markov Models for Sequence Data: The seqHMM Package in R
Submitted to Journal of Statistical Softwaremultichannel sequencescategorical time seriesvisualizing sequence datavisual- izing modelslatent Markov modelslatent class modelsR
Sequence analysis is being more and more widely used for the analysis of social sequences and other multivariate categorical time series data. However, it is often complex to describe, visualize, and compare large sequence data, especially when there are multiple parallel sequences per subject. Hidden (latent) Markov models (HMMs) are able to detect underlying latent structures and they can be used in various longitudinal settings: to account for measurement error, to detect unobservable states, or to compress information across several types of observations. Extending to mixture hidden Markov models (MHMMs) allows clustering data into homogeneous subsets, with or without external covariates.The seqHMM package in R is designed for the efficient modeling of sequences and other categorical time series data containing one or multiple subjects with one or multiple interdependent sequences using HMMs and MHMMs. Also other restricted variants of the MHMM can be fitted, e.g., latent class models, Markov models, mixture Markov models, or even ordinary multinomial regression models with suitable parameterization of the HMM.Good graphical presentations of data and models are useful during the whole analysis process from the first glimpse at the data to model fitting and presentation of results. The package provides easy options for plotting parallel sequence data, and proposes visualizing HMMs as directed graphs.
Introduction
Social sequence analysis is being more and more widely used for the analysis of longitudinal data consisting of multiple independent subjects with one or multiple interdependent sequences (channels). Sequence analysis is used for computing the (dis)similarities of sequences, and often the goal is to find patterns in data using cluster analysis. However, describing, visualizing, and comparing large sequence data is often complex, especially in the case of multiple channels. Hidden (latent) Markov models (HMMs) can be used to compress and visualize information in such data. These models are able to detect underlying latent structures. Extending to mixture hidden Markov models (MHMMs) allows clustering via latent classes, possibly with additional covariate information. One of the major benefits of using hidden Markov modeling is that all stages of analysis are performed, evaluated, and compared in a probabilistic framework.
The seqHMM package for R (R Core Team 2015) is designed for modeling sequence data and other categorical time series with one or multiple subjects and one or multiple channels using HMMs and MHMMs. The package provides functions for the estimation and inference of models, as well as functions for the easy visualization of multichannel sequences and HMMs. Even though the package was originally developed for researchers familiar with social sequence analysis and the examples are related to life course, knowledge on sequence analysis or social sciences is not necessary for the usage of seqHMM. The package is available on Comprehensive R Archive Repository (CRAN) and easily installed via install.packages("seqHMM"). Development versions can be obtained from GitHub 1 .
There are also other R packages in CRAN for HMM analysis of categorical data. The HMM package (Himmelmann 2010) is a compact package designed for fitting an HMM for a single observation sequence. The hmm.discnp package (Turner and Liu 2014) can handle multiple observation sequences with possibly varying lengths. For modeling continuous-time processes as hidden Markov models, the msm package (Jackson 2011) is available. Both hmm.discnp and msm support only single-channel observations. The depmixS4 package (Visser and Speekenbrink 2010) is able to fit HMMs for multiple interdependent time series (with continuous or categorical values), but for one subject only. In the msm and depmixS4 packages, covariates can be added for initial and transition probabilities. The mhsmm package (O'Connell and Højsgaard 2011) allows modeling of multiple sequences using hidden Markov and semi-Markov models. There are no ready-made options for modeling categorical data, but users can write their own extensions for arbitrary distributions. The LMest package (Bartolucci and Pandolfi 2015) is aimed to panel data with a large number of subjects and a small number of time points. It can be used for hidden Markov modeling of multivariate and multichannel categorical data, using covariates in emission and transition processes. LMest also supports mixed latent Markov models, where the latent process is allowed to vary in different latent subpopulations. This differs from mixture hidden Markov models used in seqHMM, where also the emission probabilities vary between groups. The seqHMM package also supports covariates in explaining group memberships. A drawback in the LMest package is that the user cannot define initial values or zero constraints for model parameters, and thus important special cases such as left-to-right models cannot be used.
We start with describing data and methods: a short introduction to sequence data and sequence analysis, then the theory of hidden Markov models for such data, an expansion to mixture hidden Markov models and a glance at some special cases, and then some propositions on visualizing multichannel sequence data and hidden Markov models. After the theoretic part we take a look at features of the seqHMM package and at the end show an example on using the package for the analysis of life course data. The appendix shows the list of notations.
Methods
Sequences and sequence analysis
By the term sequence we refer to an ordered set of categorical states. It can be a time series, such as a career trajectory or residential history, or any other series with ordered categorical observations, e.g., a DNA sequence or a structure of a story. Typically, sequence data consist of multiple independent subjects (multivariate data). Sometimes there are also multiple interdependent sequences per subject, often referred to as multichannel or multidimensional sequence data.
As an example we use the biofam data available in the TraMineR package (Gabadinho, Ritschard, Müller, and Studer 2011). It is a sample of 2000 individuals born in 1909-1972, constructed from the Swiss Household Panel survey in 2002 (Müller, Studer, and Ritschard 2007). The data set contains sequences of annual family life statuses from age 15 to 30. Eight observed states are defined from the combination of five basic states: living with parents, left home, married, having children, and divorced. To show a more complex example, we split the original data into three separate channels representing different life domains: marriage, parenthood, and residence. The data for each individual now includes three parallel sequences constituting of two or three states each: single/married/divorced, childless/parent, and living with parents / having left home.
Sequence analysis (SA), as defined in the social science framework, is a model-free data-driven approach to the analysis of successions of states. The approach has roots in bioinformatics and computer science (see e.g. Durbin, Eddy, Krogh, and Mitchison 1998), but during the past few decades SA has also become more common in other disciplines for the analysis of longitudinal data. In social sciences SA has been used increasingly often and is now "central to the life-course perspective" (Blanchard, Bühlmann, and Gauthier 2014).
SA is used for computing (dis)similarities of sequences. The most well-known method is optimal matching (McVicar and Anyadike-Danes 2002), but several alternatives exist (see e.g. Aisenbrey and Fasang 2010;Elzinga and Studer 2014;Gauthier, Widmer, Bucher, and Notredame 2009;Halpin 2010;Hollister 2009;Lesnard 2010). Also a method for analyzing multichannel data has been developed (Gauthier, Widmer, Bucher, and Notredame 2010). Often the goal in SA is to find typical and atypical patterns in trajectories using cluster analysis, but any approach suitable for compressing information on the dissimilarities can be used. The data are usually presented also graphically in some way. So far the TraMineR package has been the most extensive and frequently used software for social sequence analysis.
Hidden Markov models
In the context of hidden Markov models, sequence data consists of observed states, which are regarded as probabilistic functions of hidden states. Hidden states cannot be observed directly, but only through the sequence(s) of observations, since they emit the observations on varying probabilities. A discrete first order hidden Markov model for a single sequence is characterized by the following:
• Observed state sequence y = (y 1 , y 2 , . . . , y T ) with observed states m ∈ {1, . . . , M }.
• Hidden state sequence z = (z 1 , z 2 , . . . , z T ) with hidden states s ∈ {1, . . . , S}.
• Initial probability vector π = {π s } of length S, where π s is the probability of starting from the hidden state s: π s = P (z 1 = s); s ∈ {1, . . . , S}.
• Transition matrix A = {a sr } of size S × S, where a sr is the probability of moving from the hidden state s at time t − 1 to the hidden state r at time t: a sr = P (z t = r|z t−1 = s); s, r ∈ {1, . . . , S}.
We only consider homogeneous HMMs, where the transition probabilities a sr are constant over time. The (first order) Markov assumption states that the hidden state transition probability at time t only depends on the hidden state at the previous time point t − 1:
• Emission matrix B = {b s (m)} of size S × M , where b s (m)P (z t |z t−1 , . . . , z 1 ) = P (z t |z t−1 ).(1)
Also, the observation at time t is only dependent on the current hidden state, not on previous hidden states or observations:
P (y t |y t−1 , . . . , y 1 , z t , . . . , z 1 ) = P (y t |z t ).
(2)
For a more detailed description of hidden Markov models, see e.g., Rabiner (1989), MacDonald and Zucchini (1997), and Durbin et al. (1998).
HMM for multiple sequences
We can also fit the same HMM for multiple subjects; instead of one observed sequence y we have N sequences as Y = (y 1 , . . . , y N ) , where the observations y i = (y i1 , . . . , y iT ) of each subject i take values in the observed state space. Observed sequences are assumed to be mutually independent given the hidden states. The observations are assumed to be generated by the same model, but each subject has its own hidden state sequence.
HMM for multichannel sequences
In the case of multichannel sequence data, such as the example described in Section 2.1, for each subject i there are C parallel sequences. Observations are now of the form y itc , i = 1, . . . , N ; t = 1 . . . , T ; c = 1 . . . , C, so that our complete data is Y = {Y 1 , . . . , Y C }. In seqHMM, multichannel data are handled as a list of C data frames of size N × T . We also define Y i as all the observations corresponding to subject i.
We apply the same latent structure for all channels. In such a case the model has one transition matrix A but several emission matrices B 1 , . . . , B C , one for each channel. We assume that the observed states in different channels at a given time point t are independent of each other given the hidden state at t, i.e., P (y it |z it ) = P (y it1 |z it ) · · · P (y itC |z it ).
Sometimes the independence assumption does not seem theoretically plausible. For example, even conditioning on a hidden state representing a general life stage, are marital status and parenthood truly independent? On the other hand, given a person's religious views, could their opinions on abortion and gay marriage be though as independent?
If the goal is to use hidden Markov models for prediction or simulating new sequence data, the analyst should carefully check the validity of independence assumptions. However, if the goal is merely to describe structures and compress information, it can be useful to accept the independence assumption even though it is not completely reasonable in a theoretical sense. When using multichannel sequences, the number of observed states is smaller, which leads to a more parsimonious representation of the model and easier inference of the phenomenon. Also due to the decreased number of observed states, the number of parameters of the model is decreased leading to the improved computational efficiency of model estimation.
The multichannel approach is particularly useful if some of the channels are only partially observed; combining missing and non-missing information into one observation is usually problematic. One would have to decide whether such observations are coded completely missing, which is simple but loses information, or whether all possible combinations of missing and non-missing states are included, which grows the state space larger and makes the interpretation of the model more difficult. In the multichannel approach the data can be used as it is.
Missing data
Missing observations are handled straightforwardly in the context of HMMs. When observation y itc is missing, we gain no additional information regarding hidden states. In such a case, we set the emission probability b s (y itc ) = 1 for all s ∈ 1, . . . , S. Sequences with varying lengths are handled by setting missing values before and/or after the observed states.
Log-likelihood and parameter estimation
The unknown transition, emission and initial probabilities are commonly estimated via maximum likelihood. The log-likelihood of the parameters M = {π, A, B 1 , . . . , B C } for the HMM is written as
log L = N i=1 log P (Y i |M) ,(3)
where Y i are the observed sequences in channels c = 1, . . . , C for subject i. The probability of the observation sequence of subject i given the model parameters is
P (Y i |M) = all z P (Y i |z, M) P (z|M) = all z P (z 1 |M)P (y i1 |z 1 , M) T t=2 P (z t |z t−1 , M)P (y it |z t , M) = all z π z 1 b z 1 (y i11 ) · · · b z 1 (y i1C ) T t=2 a z t−1 zt b zt (y it1 ) · · · b zt (y itC ) ,(4)
where the hidden state sequences z = (z 1 , . . . , z T ) take all possible combinations of values in the hidden state space {1, . . . , S} and where y it are the observations of subject i at t in channels 1, . . . , C; π z 1 is the initial probability of the hidden state at time t = 1 in sequence z; a z t−1 zt is the transition probability from the hidden state at time t − 1 to the hidden state at t; and b zt (y itc ) is the probability that the hidden state of subject i at time t emits the observed state at t in channel c.
For direct numerical maximization (DNM) of the log-likelihood, any general-purpose optimization routines such as BFGS or Nelder-Mead can be used (with suitable reparameterizations). Another common estimation method is the expectation-maximization (EM) algorithm, also known as the Baum-Welch algorithm in the HMM context. The EM algorithm rapidly converges close to a local optimum, but compared to DNM, the converge speed is often slow near the optimum.
The probability (4) is efficiently calculated using the forward part of the forward-backward algorithm (Baum and Petrie 1966;Rabiner 1989). The backward part of the algorithm is needed for the EM algorithm, as well as for the computation of analytic gradients for derivative based optimization routines. For more information on the algorithms, see a supplementary vignette on CRAN (Helske 2017a).
The estimation process starts by giving initial values to the estimates. Good starting values are needed for finding the optimal solution in a reasonable time. In order to reduce the risk of being trapped in a poor local maximum, a large number of initial values should be tested.
Inference on hidden states
Given our model and observed sequences, we can make several interesting inferences regarding the hidden states. Forward probabilities α it (s) (Rabiner 1989) are defined as the joint probability of hidden state s at time t and the observation sequences y i1 , . . . , y it given the model M, whereas backward probabilities β it (s) are defined as the joint probability of hidden state s at time t and the observation sequences y i(t+1) , . . . , y iT given the model M.
From forward and backward probabilities we can compute the posterior probabilities of states, which give the probability of being in each hidden state at each time point, given the observed sequences of subject i. These are defined as
P (z it = s|Y i , M) = α it β it P (Y i |M) .(5)
Posterior probabilities can be used to find the locally most probable hidden state at each time point, but the resulting sequence is not necessarily globally optimal. To find the single best hidden state sequenceẑ i (Y i ) =ẑ i1 ,ẑ i2 , . . . ,ẑ iT for subject i, we maximize P (z|Y i , M) or, equivalently, P (z, Y i |M). A dynamic programming method, the Viterbi algorithm (Rabiner 1989), is used for solving the problem.
Model comparison
Models with the same number of parameters can be compared with the log-likelihood. For choosing between models with a different number of hidden states, we need to take account of the number of parameters. We define the Bayesian information criterion (BIC) as
BIC = −2 log(L d ) + p log N i=1 T t=1 1 C C c=1 I(y itc observed) ,(6)
where L d is computed using Equation 3, p is the number of estimated parameters, I is the indicator function, and the summation in the logarithm is the size of the data. If data are completely observed, the summation is simplified to N × T . Missing observations in multichannel data may lead to non-integer data size.
Clustering by mixture hidden Markov models
There are many approaches for finding and describing clusters or latent classes when working with HMMs. A simple option is to group sequences beforehand (e.g., using sequence analysis and a clustering method), after which one HMM is fitted for each cluster. This approach is simple in terms of HMMs. Models with a different number of hidden states and initial values are explored and compared one cluster at a time. HMMs are used for compressing information and comparing different clustering solutions, e.g., finding the best number of clusters. The problem with this solution is that it is, of course, very sensitive to the original clustering and the estimated HMMs might not be well suited for borderline cases.
Instead of fixing sequences into clusters, it is possible to fit one model for the whole data and determine clustering during modeling. Now sequences are not in fixed clusters but get assigned to clusters with certain probabilities during the modeling process. In this section we expand the idea of HMMs to mixture hidden Markov models (MHMMs). This approach was formulated by van de Pol and Langeheine (1990)
Mixture hidden Markov model
Assume that we have a set of HMMs
M = {M 1 , . . . , M K }, where M k = {π k , A k , B k 1 , .
. . , B k C } for submodels k = 1, . . . , K. For each subject Y i , denote P (M k ) = w k as the prior probability that the observation sequences of a subject follow the submodel M k . Now the log-likelihood of the parameters of the MHMM is extended from Equation 3 as
log L = N i=1 log P (Y i |M) = N i=1 log K k=1 P (M k ) all z P Y i |z, M k P z|M k = N i=1 log K k=1 w k all z π k z 1 b k z 1 (y i11 ) · · · b k z 1 (y i1C ) T t=2 a k z t−1 zt b k zt (y it1 ) · · · b k zt (y itC ) .(7)
Compared to the usual hidden Markov model, there is an additional summation over the clusters in Equation 7, which seems to make the computations less straightforward than in the non-mixture case. Fortunately, by redefining MHMM as a special type HMM allows us to use standard HMM algorithms without major modifications. We combine the K submodels into one large hidden Markov model consisting of K k=1 S k states, where the initial state vector contains elements of the form w k π k . Now the transition matrix is block diagonal
A = A 1 0 · · · 0 0 A 2 · · · 0 . . . . . . . . . . . . 0 0 · · · A K ,(8)
where the diagonal blocks A k , k = 1, . . . , K, are square matrices containing the transition probabilities of one cluster. The off-diagonal blocks are zero matrices, so transitions between clusters are not allowed. Similarly, the emission matrices for each channel contain stacked emission matrices B k .
Covariates and cluster probabilities
Covariates can be added to MHMM to explain cluster memberships as in latent class analysis. The prior cluster probabilities now depend on the subject's covariate values x i and are defined as multinomial distribution:
P (M k |x i ) = w ik = e x i γ k 1 + K j=2 e x i γ j .(9)
The first submodel is set as the reference by fixing γ 1 = (0, . . . , 0) .
As in MHMM without covariates, we can still use standard HMM algorithms with a slight modification; we now allow initial state probabilities π to vary between subjects, i.e., for subject i we have π i = (w i1 π 1 , . . . , w iK π K ) . Of course, we also need to estimate the coefficients γ. For direct numerical maximization the modifications are straightforward. In the EM algorithm, regarding the M-step for γ, seqHMM uses iterative Newton's method with analytic gradients and Hessian which are straightforward to compute given all other model parameters. This Hessian can also be used for computing the conditional standard errors of coefficients. For unconditional standard errors, which take account of possible correlation between the estimates of γ and other model parameters, the Hessian is computed using finite difference approximation of the Jacobian of the analytic gradients.
The posterior cluster probabilities P (M k |Y i , x i ) are obtained as
P (M k |Y i , x i ) = P (Y i |M k , x i )P (M k |x i ) P (Y i |x i ) = P (Y i |M k , x i )P (M k |x i ) K j=1 P (Y i |M j , x i )P (M j |x i ) = L i k L i ,(10)
where L i is the likelihood of the complete MHMM for subject i, and L i k is the likelihood of cluster k for subject i. These are straightforwardly computed from forward probabilities. Posterior cluster probabilities are used e.g., for computing classification tables.
Important special cases
The hidden Markov model is not the only important special case of the mixture hidden Markov model. Here we cover some of the most important special cases that are included in the seqHMM package.
Markov model
The Markov model (MM) is a special case of the HMM, where there is no hidden structure. It can be regarded as an HMM where the hidden states correspond to the observed states perfectly. Now the number of hidden states matches the number of the observed states. The emission probability P (y it ) = 1 if z t = y it and 0 otherwise, i.e., the emission matrices are identity matrices. Note that for building Markov models the data must be in a single-channel format.
Mixture Markov model
Like MM, the mixture Markov model (MMM) is a special case of the MHMM, where there is no hidden structure. The likelihood of the model is now of the form
log L = N i=1 log P (y i |x i , M k ) = N i=1 log K k=1 P (M k |x i )P (y i |x i , M k ) = N i=1 log K k=1 P (M k |x i )P (y i1 |x i , M k ) T t=2 P (y it |y i(t−1) , x i , M k ).(11)
Again, the data must be in a single-channel format.
Latent class model
Latent class models (LCM) are another class of models that are often used for longitudinal research. Such models have been called, e.g., (latent) growth models, latent trajectory models, or longitudinal latent class models (Vermunt et al. 2008;Collins and Wugalter 1992). These models assume that dependencies between observations can be captured by a latent class, i.e., a time-constant variable which we call cluster in this paper.
The seqHMM includes a function for fitting an LCM as a special case of MHMM where there is only one hidden state for each cluster. The transition matrix of each cluster is now reduced to a scalar 1 and the likelihood is of the form
log L = N i=1 log P (Y i |x i , M k ) = N i=1 log K k=1 P (M k |x i )P (Y i |x i , M k ) = N i=1 log K k=1 P (M k |x i ) T t=1 P (y it |x i , M k ).(12)
For LCMs, the data can consist of multiple channels, i.e., the data for each subject consists of multiple parallel sequences. It is also possible to use seqHMM for estimating LCMs for non-longitudinal data with only one time point, e.g., to study multiple questions in a survey.
Package features
The purpose of the seqHMM package is to offer tools for the whole HMM analysis process from sequence data manipulation and description to model building, evaluation, and visualization. Naturally, seqHMM builds on other packages, especially the TraMineR package designed for sequence analysis. For constructing, summarizing, and visualizing sequence data, TraMineR provides many useful features. First of all, we use the TraMineR's stslist class as the sequence data structure of seqHMM. These state sequence objects have attributes such as color palette and alphabet, and they have specific methods for plotting, summarizing, and printing. Many other TraMineR's features for plotting or data manipulation are also used in seqHMM.
Usage
Functions/methods On the other hand, seqHMM extends the functionalities of TraMineR, e.g., by providing easy-to-use plotting functions for multichannel data and a simple function for converting such data into a single-channel representation.
Other significant packages used by seqHMM include the igraph package (Csardi and Nepusz 2006), which is used for drawing graphs of HMMs, and the nloptr package (Ypma, Borchers, and Eddelbuettel 2014;Johnson 2014), which is used in direct numerical optimization of model parameters. The computationally intensive parts of the package are written in C++ with the help of the Rcpp (Eddelbuettel and François 2011;Eddelbuettel 2013) and RcppArmadillo (Eddelbuettel and Sanderson 2014) packages. In addition to using C++ for major algorithms, seqHMM also supports parallel computation via the OpenMP interface (Dagum and Enon 1998) by dividing computations for subjects between threads. Table 1 shows the functions and methods available in the seqHMM package. The package includes functions for estimating and evaluating HMMs and MHMMs as well as visualizing data and models. There are some functions for manipulating data and models, and for simulating model parameters or sequence data given a model. In the next sections we discuss the usage of these functions more thoroughly.
As the straightforward implementation of the forward-backward algorithm poses a great risk of under-and overflow, typically forward probabilities are scaled so that there should be no underflow. seqHMM uses the scaling as in Rabiner (1989), which is typically sufficient for numerical stability. In case of MHMM though, we have sometimes observed numerical issues in the forward algorithm even with proper scaling. Fortunately this usually means that the backward algorithm fails completely, giving a clear signal that something is wrong. This is especially true in the case of global optimization algorithms which can search unfeasible areas of the parameter space, or when using bad initial values often with large number of zero-constraints. Thus, seqHMM also supports computation on the logarithmic scale in most of the algorithms, which further reduces the numerical instabilities. On the other hand, as there is a need to back-transform to the natural scale during the algorithms, the log-space approach is somewhat slower than the scaling approach. Therefore, the default option is to use the scaling approach, which can be changed to the log-space approach by setting the log_space argument to TRUE in, e.g., fit_model.
Building and fitting models
A model is first constructed using an appropriate build function. As Table 1 illustrates, several such functions are available: build_hmm for hidden Markov models, build_mhmm for mixture hidden Markov models, build_mm for Markov models, build_mmm for mixture Markov models, and build_lcm for latent class models.
The user may give their own starting values for model parameters, which is typically advisable for improved efficiency, or use random starting values. Build functions check that the data and parameter matrices (when given) are of the right form and create an object of class hmm (for HMMs and MMs) or mhmm (for MHMMs, MMMs, and LCMs). For ordinary Markov models, the build_mm function automatically estimates the initial probabilities and the transition matrix based on the observations. For this type of model, starting values or further estimation are not needed. For mixture models, covariates can be omitted or added with the usual formula argument using symbolic formulas familiar from, e.g., the lm function. Even though missing observations are allowed in sequence data, covariates must be completely observed.
After a model is constructed, model parameters may be estimated with the fit_model function. MMs, MMMs, and LCMs are handled internally as their more general counterparts, except in the case of print methods, where some redundant parts of the model are not printed.
In all models, initial zero probabilities are regarded as structural zeroes and only positive probabilities are estimated. Thus it is easy to construct, e.g., a left-to-right model by defining the transition probability matrix as an upper triangular matrix.
The fit_model function provides three estimation steps: 1) EM algorithm, 2) global DNM, and 3) local DNM. The user can call for one method or any combination of these steps, but should note that they are performed in the above-mentioned order. At the first step, starting values are based on the model object given to fit_model. Results from a former step are then used as starting values in the latter. Exceptions to this rule include some global optimization algorithms, which do not use initial values (because of this, performing just the local DNM step can lead to a better solution than global DNM with a small number of iterations).
We have used our own implementation of the EM algorithm for MHMMs whereas the DNM steps (2 and 3) rely on the optimization routines provided by the nloptr package. The EM algorithm and computation of gradients were written in C++ with an option for parallel computation between subjects. The user can choose the number of parallel threads (typically, the number of cores) with the threads argument.
In order to reduce the risk of being trapped in a poor local optimum, a large number of initial values should be tested. The seqHMM package strives to automatize this. One option is to run the EM algorithm multiple times with more or less random starting values for transition or emission probabilities or both. These are called for in the control_em argument. Although not done by default, this method seems to perform very well as the EM algorithm is relatively fast compared to DNM.
Another option is to use a global DNM approach such as the multilevel single-linkage method (MLSL) (Rinnooy Kan and Timmer 1987a,b). It draws multiple random starting values and performs local optimization from each starting point. The LDS modification uses low-discrepancy sequences instead of random numbers as starting points and should improve the convergence rate (Kucherenko and Sytsko 2005).
By default, the fit_model function uses the EM algorithm with a maximum of 1000 iterations and skips the local and global DNM steps. For the local step, the L-BFGS algorithm (Nocedal 1980;Liu and Nocedal 1989) is used by default. Setting global_step = TRUE, the function performs MSLS-LDS with the L-BFGS as the local optimizer. In order to reduce the computation time spent on non-global optima, the convergence tolerance of the local optimizer is set relatively large, so again local optimization should be performed at the final step.
Unfortunately, there is no universally best optimization method. For unconstrained problems, the computation time for a single EM or DNM rapidly increases as the model size increases and at the same time the risk of getting trapped in a local optimum or a saddle point also increases. As seqHMM provides functions for analytic gradients, the optimization routines of nloptr which make use of this information are likely preferable. In practice we have had most success with randomized EM, but it is advisable to try a couple of different settings; e.g.,
State and model inference
In seqHMM, forward and backward probabilities are computed using the forward_backward function, either on the logarithmic scale or in the form of scaled probabilities, depending on the argument log_space. Posterior probabilities are obtained from the posterior_probs function. In seqHMM, the most probable paths are computed with the hidden_paths function. For details of the Viterbi and the forward-backward algorithm, see e.g., Rabiner (1989).
The seqHMM package provides the logLik method for computing the log-likelihood of a model. The method returns an object of class logLik which is compatible with the generic information criterion functions AIC and BIC of R. When constructing the hmm and mhmm objects via model building functions, the number of observations and the number of parameters of the model are stored as attributes nobs and df which are extracted by the logLik method for the computation of information criteria. The number of model parameters defined from the initial model by taking account of the parameter redundancy constraints (stemming from sum-to-one constraints of transition, emission, and initial state probabilities) and by defining all zero probabilities as structural, fixed values.
The summary method automatically computes some features for the MHMM, MMM, and the latent class model, e.g., standard errors for covariates and prior and posterior cluster probabilities for subjects. A print method for this summary shows an output of the summaries: estimates and standard errors for covariates, log-likelihood and BIC, and information on most probable clusters and prior probabilities.
Visualizing sequence data
Good graphical presentations of data and models are useful during the whole analysis process from the first glimpse into the data to the model fitting and presentation of results. The TraMineR package provides nice plotting options and summaries for simple sequence data, but at the moment there is no easy way of plotting multichannel data. We propose to use a so-called stacked sequence plot (ssp), where the channels are plotted on top of each other so that the same row in each figure matches the same subject. Figure 1 illustrates an example of a stacked sequence plot with the ten first sequences of the biofam data set. The code for creating the figure is shown in Section 4.1.
The ssplot function is the simplest way of plotting multichannel sequence data in seqHMM. It can be used to illustrate state distributions or sequence index plots. The former is the default option, since index plots can take a lot of time and memory if data are large. Figure 2 illustrates a default plot which the user can modify in many ways (see the code in Section 4.1). More examples are shown in the documentation pages of the ssplot function.
Another option is to define function arguments with the ssp function and then use previously saved arguments for plotting with a simple plot method. It is also possible to combine several ssp figures into one plot with the gridplot function. Figure 3 illustrates an example of such a plot showing sequence index plots for women and men (see the code in Section 4.1). Sequences are ordered in a more meaningful order using multidimensional scaling scores of observations (computed from sequence dissimilarities). After defining the plot for one group, a similar plot for other groups is easily defined using the update function.
The gridplot function is useful for showing different features for the same subjects or the same features for different groups. The user has a lot of control over the layout, e.g., dimen- We also provide a function mc_to_sc_data for the easy conversion of multichannel sequence data into a single channel representation. Plotting combined data is often useful in addition to (or instead of) showing separate channels.
Visualizing hidden Markov models
For the easy visualization of the model structure and parameters, we propose plotting HMMs as directed graphs. Such graphs are easily called with the plot method, with an object of class hmm as an argument. Figure 4 illustrates a five-state HMM. The code for producing the plot is shown in Section 4.4.
Hidden states are presented with pie charts as vertices (or nodes), and transition probabilities are shown as edges (arrows, arcs). By default, the higher the transition probability, the thicker the stroke of the edge. Emitted observed states are shown as slices in the pies. For gaining a simpler view, observations with small emission probabilities (less than 0.05 by default) can be combined into one category. Initial state probabilities are given below or next to the respective vertices. In the case of multichannel sequences, the data and the model are converted into a single-channel representation with the mc_to_sc function.
A simple default plot is easy to call, but the user has a lot of control over the layout. Figure 5 illustrates another possible visualization of the same model. The code is shown in Section 4.4.
The ssplot function (see Section 3.2) also accepts an object of class hmm. The plot method works for mhmm objects as well. The user can choose between an interactive mode, where the model for each (chosen) cluster is plotted separately, and a combined plot with all models in one plot. The equivalent to the ssplot function for MHMMs is mssplot. It plots stacked sequence plots separately for each cluster. If the user asks to plot more than one cluster, the function is interactive by default.
Examples with life course data
In this section we show examples of using the seqHMM package. We start by constructing and visualizing sequence data, then show how HMMs are built and fitted for single-channel and multichannel data, then move on to clustering with MHMMs, and finally illustrate how to plot HMMs.
Throughout the examples we use the same biofam data described in Section 2.1. We use both the original single-channel data and a three-channel modification named biofam3c, which is included in the seqHMM package. For more information on the conversion, see the documentation of the biofam3c data.
Sequence data
Before getting to the estimation, it is good to get to know the data. We start by loading the original biofam data as well as the three-channel version of the same data, biofam3c. We Observed and hidden state sequences, n = 2000 Figure 6: Using the ssplot function for an hmm object makes it easy to plot the observed sequences together with the most probable paths of hidden states given the model. convert the data into the stslist form with the seqdef function. We set the starting age at 15 and set the order of the states with the alphabet argument (for plotting). Colors of the states can be modified and stored as an attribute in the stslist object -this way the user only needs to define them once.
R> library("seqHMM") R> R> data("biofam", package = "TraMineR") R> biofam_seq <-seqdef(biofam[, 10:25], start = 15, labels = c("parent", + "left", "married", "left+marr", "child", "left+child", "left+marr+ch", + "divorced")) R> R> data("biofam3c") R> marr_seq <-seqdef(biofam3c$married, start = 15, alphabet = c("single", + "married", "divorced")) R> child_seq <-seqdef(biofam3c$children, start = 15, + alphabet = c("childless", "children")) R> left_seq <-seqdef(biofam3c$left, start = 15, alphabet = c("with parents", + "left home")) R> R> attr(marr_seq, "cpal") <-c("violetred2", "darkgoldenrod2", "darkmagenta") R> attr(child_seq, "cpal") <-c("darkseagreen1", "coral3") R> attr(left_seq, "cpal") <-c("lightblue", "red3")
Here we show codes for creating Figures 2, 1, and 3. Such plots give a good glimpse into multichannel data.
Figure 2: Plotting state distributions
We start by showing how to call the simple default plot of Figure 2 in Section 3.3. By default the function plots state distributions (type = "d"). Multichannel data are given as a list where each component is an stslist corresponding to one channel. If names are given, those will be used as labels in plotting.
R> ssplot(list("Marriage" = marr_seq, "Parenthood" = child_seq, + "Residence" = left_seq)) Figure 1: Plotting sequences Figure 1 with the whole sequences requires modifying more arguments. We call for sequence index plots (type = "I") and sort sequences according to the first channel (the original sequences), starting from the beginning. We give labels to y and x axes and modify the positions of y labels. We give a title to the plot but omit the number of subjects, which by default is printed. We set the proportion of the plot given to legends and the number of columns in each legend. R> ssplot(seq_data, type = "I", sortv = "from.start", sort.channel = 1, + ylab = c("Original", "Marriage", "Parenthood", "Residence"), + xtlab = 15:30, xlab = "Age", ylab.pos = c(1, 1.5), title.n = FALSE, + title = "Ten first sequences", legend.prop = 0.63, + ncol.legend = c(3, 1, 1, 1))
Figure 3: Plotting sequence data in a grid
For using the gridplot function, we first need to specify the ssp objects of the separate plots. Here we start by defining the first plot for women with the ssp function. It stores the features of the plot, but does not draw anything. We want to sort sequences according to multidimensional scaling scores. These are computed from optimal matching dissimilarities for observed sequences. Any dissimilarity method available in TraMineR can be used instead of the default (see the documentation of the seqdef function for more information). We want to use the same legends for the both plots, so we remove legends from the ssp objects.
Since we are going to plot to two similar figures, one for women and one for men, we can pass the first ssp object to the update function. This way we only need to define the changes and omit everything that is similar.
These two ssp objects are then passed on to the gridplot function. Here we make a 2 × 2 grid, of which the bottom row is for the legends, but the function can also automatically determine the number of rows and columns and the positions of the legends.
R> ssp_f <-ssp(list(marr_seq[biofam3c$covariates$sex == "woman",], + child_seq[biofam3c$covariates$sex == "woman",], + left_seq[biofam3c$covariates$sex == "woman",]), + type = "I", sortv = "mds.obs", withlegend = FALSE, title = "Women", + ylab.pos = c(1, 2, 1), xtlab = 15:30, ylab = c("Married", "Children", + "Residence")) R> R> ssp_m <-update(ssp_f, title = "Men", + x = list(marr_seq[biofam3c$covariates$sex == "man",], + child_seq[biofam3c$covariates$sex == "man",], + left_seq[biofam3c$covariates$sex == "man",])) R> R> gridplot(list(ssp_f, ssp_m), ncol = 2, nrow = 2, byrow = TRUE, + legend.pos = "bottom", legend.pos2 = "top", row.prop = c(0.65, 0.35))
For more examples on visualization, see a supplementary vignette on CRAN (Helske 2017c).
Hidden Markov models
We start by showing how to fit an HMM for single-channel biofam data. The model is initialized with the build_hmm function which creates an object of class hmm. The simplest way is to use automatic starting values by giving the number of hidden states.
R> sc_initmod_random <-build_hmm(observations = biofam_seq, n_states = 5)
It is, however, often advisable to set starting values for initial, transition, and emission probabilities manually. Here the hidden states are regarded as more general life stages, during which individuals are more likely to meet certain observable life events. We expect that the life stages are somehow related to age, so constructing starting values from the observed state frequencies by age group seems like an option worth a try (these are easily computed using the seqstatf function in TraMineR). We construct a model with four hidden states using age groups 15-18, 19-21, 22-24, 25-27 and 28-30. The fit_model function uses the probabilities given by the initial model as starting values when estimating the parameters. Only positive probabilities are estimated; zero values are fixed to zero. Thus, the amount of 0.1 is added to each value in case of zero-frequencies in some categories (at this point we do not want to fix any parameters to zero). Each row is divided by its sum, so that the row sums equal to 1. Now, the build_hmm checks that the data and matrices are of the right form.
R> sc_initmod <-build_hmm(observations = biofam_seq, initial_probs = sc_init, + transition_probs = sc_trans, emission_probs = sc_emiss)
We then use the fit_model function for parameter estimation. Here we estimate the model using the default options of the EM step.
R> sc_fit <-fit_model(sc_initmod)
The fitting function returns the estimated model, its log-likelihood, and information on the optimization steps.
R> sc_fit$logLik Emission probabilities : symbol_names state_names 0 1 2 3 4 5 6 7 State 1 1 0 0.00000 0.000 0.00000 0.0000 0.000 0.0000 State 2 1 0 0.00000 0.000 0.00000 0.0000 0.000 0.0000 State 3 0 1 0.00000 0.000 0.00000 0.0000 0.000 0.0000 State 4 0 0 0.00195 0.992 0.00581 0.0000 0.000 0.0000 State 5 0 0 0.21508 0.000 0.00000 0.0246 0.713 0.0474
As a multichannel example we fit a 5-state model for the 3-channel data. Emission probabilities are now given as a list of three emission matrices, one for each channel. The alphabet function from the TraMineR package can be used to check the order of the observed statesthe same order is used in the build functions. Here we construct a left-to-right model where transitions to earlier states are not allowed, so the transition matrix is upper-triangular. This seems like a valid option from a life-course perspective. Also, in the previous single-channel model of the same data the transition matrix was estimated almost upper triangular. We also give names for channels -these are used when printing and plotting the model.
We estimate model parameters using the local step with the default L-BFGS algorithm using parallel computation with 4 threads.
R> mc_init <-c(0.9, 0.05, 0.02, 0.02, 0.01) R> R> mc_trans <-matrix(c(0.80, 0.10, 0.05, 0.03, 0.02, 0, 0.90, 0.05, 0.03, + 0.02, 0, 0, 0.90, 0.07, 0.03, 0, 0, 0, 0.90, 0.10, 0, 0, 0, 0, 1), + nrow = 5, ncol = 5, byrow = TRUE) R> R> mc_emiss_marr <-matrix(c(0.90, 0.05, 0.05, 0.90, 0.05, 0.05, 0.05, 0.90, + 0.05, 0.05, 0.90, 0.05, 0.30, 0.30, 0.40), nrow = 5, ncol = 3, + byrow = TRUE) R> R> mc_emiss_child <-matrix(c(0.9, 0.1, 0.9, 0.1, 0.1, 0.9, 0.1, 0.9, 0.5, + 0.5), nrow = 5, ncol = 2, byrow = TRUE) R> R> mc_emiss_left <-matrix(c(0.9, 0.1, 0.1, 0.9, 0.1, 0.9, 0.1, 0.9, 0.5, + 0.5), nrow = 5, ncol = 2, byrow = TRUE) R> R> mc_obs <-list(marr_seq, child_seq, left_seq) R> R> mc_emiss <-list(mc_emiss_marr, mc_emiss_child, mc_emiss_left) R> R> mc_initmod <-build_hmm(observations = mc_obs, initial_probs = mc_init, + transition_probs = mc_trans, emission_probs = mc_emiss, + channel_names = c("Marriage", "Parenthood", "Residence")) R> R> mc_initmod We fit a model using 100 random restarts of the EM algorithm followed by the local L-BFGS method. Again we use parallel computation.
R> mc_init2 <-c(0.9, 0.05, 0.03, 0.02) R> R> mc_trans2 <-matrix(c(0.85, 0.05, 0.05, 0.05, 0, 0.90, 0.05, 0.05, 0, 0, + 0.95, 0.05, 0, 0, 0, 1), nrow = 4, ncol = 4, byrow = TRUE) R> R> mc_emiss_marr2 <-matrix(c(0.90, 0.05, 0.05, 0.90, 0.05, 0.05, 0.05, + 0.85, 0.10, 0.05, 0.80, 0.15), nrow = 4, ncol = 3, byrow = TRUE) R> R> mc_emiss_child2 <-matrix(c(0.9, 0.1, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5), + nrow = 4, ncol = 2, byrow = TRUE) R> R> mc_emiss_left2 <-matrix(c(0.9, 0.1, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5), + nrow = 4, ncol = 2, byrow = TRUE) R> R> mhmm_init <-list(mc_init, mc_init2) R> R> mhmm_trans <-list(mc_trans, mc_trans2) R> R> mhmm_emiss <-list(list(mc_emiss_marr, mc_emiss_child, mc_emiss_left), + list(mc_emiss_marr2, mc_emiss_child2, mc_emiss_left2)) R> R> biofam3c$covariates$cohort <-cut(biofam3c$covariates$birthyr, + c (1908,1935,1945,1957)) R> biofam3c$covariates$cohort <-factor(biofam3c$covariates$cohort, + labels=c ("1909-1935", "1936-1945", "1946-1957")) R> R> init_mhmm <-build_mhmm(observations = mc_obs, initial_probs = mhmm_init, + transition_probs = mhmm_trans, emission_probs = mhmm_emiss, + formula =~sex + cohort, data = biofam3c$covariates, + channel_names = c("Marriage", "Parenthood", "Residence"), + cluster_names = c("Cluster 1", "Cluster 2")) R> R> set.seed(1011) R> mhmm_fit <-fit_model(init_mhmm, local_step = TRUE, threads = 4, + control_em = list(restart = list(times = 100))) R> mhmm <-mhmm_fit$model
The summary method automatically computes some features for an MHMM, e.g., standard errors for covariates and prior and posterior cluster probabilities for subjects. A print method shows some summaries of these: estimates and standard errors for covariates (see Section 2.3), log-likelihood and BIC, and information on most probable clusters and prior probabilities. Parameter estimates for transitions, emissions, and initial probabilities are omitted by default. The classification table shows mean probabilities of belonging to each cluster by the most probable cluster (defined from posterior cluster probabilities). A good model should have values close to 1 on the diagonal.
R> summary(mhmm, conditional_se = FALSE)
Covariate effects : Cluster 1 is the reference.
Visualizing hidden Markov models
The figures in Section 3.3 illustrate the five-state multichannel HMM fitted in Section 4.2.
A basic HMM graph is easily called with the plot method. Figure 7 illustrates the default plot.
R> plot(hmm_biofam)
A simple default plot is a convenient way of visualizing the models during the analysis process, but for publishing it is often better to modify the plot to get an output that best illustrates Figure 7: A default plot of a hidden Markov model.
the structure of the model in hand. Figure 4 and Figure 5 show two variants of the same model.
Figure 4: HMM plot with modifications
In Figure 4 we draw larger vertices, control the distances of initial probabilities (vertex labels), set the curvatures of the edges, give a more descriptive label for the combined slices and give less space for the legend.
R> plot(hmm_biofam, vertex.size = 50, vertex.label.dist = 1.5, + edge.curved = c(0, 0.6, -0.8, 0.6, 0, 0.6, 0), legend.prop = 0.3, + combined.slice.label = "States with prob. < 0.05")
Figure 5: HMM plot with a different layout
Here we position the vertices using given coordinates. Coordinates are given in a two-column matrix, with x coordinates in the first column and y coordinates in the second. Arguments xlim and ylim set the lengths of the axes, and rescale = FALSE prevents rescaling the coordinates to the [−1, 1] × [−1, 1] interval (the default). We modify the positions of initial probabilities, fix edge widths to 1, reduce the size of the arrows in edges, position legend on top of the figure, and print labels in two columns in the legend. Parameter values are shown with one significant digit. All emission probabilities are shown regardless of their value (combine.slices = 0).
New colors are set from the ready-defined colorpalette data. The seqHMM package uses these palettes when determining colors automatically, e.g., in the mc_to_sc function. Since here there are 10 combined states, the default color palette is number 10. To get different colors, we choose the ten first colors from palette number 14.
R> vertex_layout <-matrix(c(1, 2, 2, 3, 1, 0, 0.5, -0.5, 0, -1), + ncol = 2) R> R> plot(hmm_biofam, layout = vertex_layout, xlim = c(0.5, 3.5), + ylim = c(-1.5, 1), rescale = FALSE, vertex.size = 50, + vertex.label.pos = c("left", "top", "bottom", "right", "left"), + edge.curved = FALSE, edge.width = 1, edge.arrow.size = 1, + withlegend = "left", legend.prop = 0.4, label.signif = 1, + combine.slices = 0, cpal = colorpalette [[30]][c(14:5)]) Figure 6: ssplot for an HMM object
Plotting observed and hidden state sequences is easy with the ssplot function: the function accepts an hmm object instead of (a list of) stslists. If hidden state paths are not provided, the function automatically computes them when needed.
R> ssplot(hmm_biofam, plots = "both", type = "I", sortv = "mds.hidden", + title = "Observed and hidden state sequences", xtlab = 15:30, + xlab = "Age")
Visualizing mixture hidden Markov models
Objects of class mhmm have similar plotting methods to hmm objects. The default way of visualizing a model is to plot in an interactive mode, where the model for each cluster is plotted separately. Another option is a combined plot with all models in one plot, although it can be difficult to fit several graphs and legends in one figure. Figure 8 illustrates the MHMM fitted in Section 4.3. By setting interactive = FALSE and nrow = 2 we plot graphs in a grid with two rows. The rest of the arguments are similar to basic HMM plotting and apply for all the graphs.
R> plot(mhmm, interactive = FALSE, nrow = 2, legend.prop = 0.45, + vertex.size = 50, vertex.label.cex = 1.3, cex.legend = 1.3, + edge.curved = 0.65, edge.label.cex = 1.3, edge.arrow.size = 0.8)
The equivalent of the ssplot function for mhmm objects is mssplot. It shows data and/or hidden paths one cluster at a time. The function is interactive if more than one cluster is plotted (thus omitted here). Subjects are allocated to clusters according to the most probable hidden state paths.
R> mssplot(mhmm, ask = TRUE) Figure 8: Plotting submodels of an MHMM with the plot method.
If the user wants more control than the default mhmm plotting functions offer, they can use the separate_mhmm function to convert a mhmm object into a list of separate hmm objects. These can then be plotted as any hmm objects, e.g., use ssp and gridplot for plotting sequences and hidden paths of each cluster into the same figure.
Conclusion
Hidden Markov models are useful in various longitudinal settings with categorical observations. They can be used for accounting measurement error in the observations (e.g., drug use as in Vermunt et al. 2008), for detecting true unobservable states (e.g., different periods of the bipolar disorder as in Lopez 2008), and for compressing information across several types of observations (e.g., finding general life stages as in Helske, Helske, and Eerola 2016).
The seqHMM package is designed for analyzing categorical sequences with hidden Markov models and mixture hidden Markov models, as well as their restricted variants Markov models, mixture Markov models, and latent class models. It can handle many types of data from a single sequence to multiple multichannel sequences. Covariates can be included in MHMMs to explain cluster membership. The package also offers versatile plotting options for sequence data and HMMs, and can easily convert multichannel sequence data and models into singlechannel representations.
Parameter estimation in (M)HMMs is often very sensitive to starting values. To deal with that, seqHMM offers several fitting options with global and local optimization using direct numerical estimation and the EM algorithm. Almost all intensive computations are done in C++. The package also supports parallel computation.
Especially combined with the TraMineR package, seqHMM is designed to offer tools for the whole analysis process from data preparation and description to model fitting, evaluation, and visualization. In future we plan to develop MHMMs to deal with time-varying covariates in transition and emission matrices (Bartolucci, Farcomeni, and Pennoni 2012), and add an option to incorporate sampling weights for model estimation. Also, the computational efficiency of the restricted variants of (M)HMMs, such as latent class models, could be improved by taking account of the restricted structure of those models in EM and log-likelihood computations.
A. Notations
Symbol Meaning Y i Observation sequences of subject i, i = 1 . . . , N y it Observations of subject i at time t, t = 1, . . . , T y itc Observation of subject i at time t in channel c, c = 1, . . . , C m c ∈ {1, . . . , M c } Observed state space for channel c z it Hidden state at time t for subject i s ∈ {1, . . . , S} Hidden state space A = {a sr } Transition matrix of size S × S a sr = P (z t = r|z t−1 = s)
Transition probability between hidden states s and r B c = {b s (m c )} Emission matrix of size S × M c for channel c b s (m c ) = P (y itc = m c |z it = s) Emission probability of observed state m c in channel c given hidden state s b s (y it ) = b s (y it1 ) · · · b s (y itC ) Joint emission probability of observations at time t in channels 1, . . . , C given hidden state s π = (π 1 , . . . , π S ) Vector of initial probabilities π s = P (z 1 = s) Initial probability of hidden state ŝ z i (Y i )
The most probable hidden state sequence for subject i x i
Covariates of subject i M k , k = 1, . . . , K Submodel for cluster k (latent class/cluster) w ik Probability of cluster k for subject i γ k Regression coefficients for cluster k {π k , A k , B k 1 , . . . , B k C , γ k } Model parameters for cluster k
is the probability of the hidden state s emitting the observed state m: b s (m) = P (y t = m|z t = s); s ∈ {1, . . . , S}, m ∈ {1, . . . , M }.
randomized EM, EM followed by global DNM, and only global DNM, perhaps with different optimization routines. Documentation of the fit_model function gives examples of different optimization strategies and how they can lead to different solutions. For examples on model estimation and starting values, see a supplementary vignette on CRAN (Helske 2017b).
Figure 1 :
1Stacked sequence plot of the first ten individuals in the biofam data plotted with the ssplot function. The top plot shows the original sequences, and the three bottom plots show the sequences in the separate channels for the same individuals. The sequences are in the same order in each plot, i.e., the same row always matches the same individual.
Figure 2 :
2Stacked sequence plot of annual state distributions in the three-channel biofam data. This is the default output of the ssplot function. The labels for the channels are taken from the named list of state sequence objects, and the labels for the x axis ticks are taken from the column names of the first object.
Figure 3 :Figure 4 :
34Showing state distribution plots for women and men in the biofam data. Two figures were defined with the ssp function and then combined into one figure with the gridplot function. Illustrating a hidden Markov model as a directed graph. Pies represent five hidden states, with slices showing emission probabilities of combinations of observed states. States with emission probability less than 0.05 are combined into one slice. Edges show the transtion probabilities. Initial probabilities of hidden states are given below the pies. sions of the grid, widths and heights of the cells, and positions of the legends.
Figure 5 :
5Another version of the hidden Markov model ofFigure 4with a different layout and modified labels, legends, and colors. All observed states are shown.automatically computes hidden paths if the user does not provide them.
Figure 6
6shows observed sequences with the most probable paths of hidden states given the model. Sequences are sorted according to multidimensional scaling scores computed from hidden paths. The code for creating the plot is shown in Section 4.4.
R> seq_data <-list(biofam_seq[1:10,], marr_seq[1:10,], child_seq[1:10,], + left_seq[1:10,])
Time-constant covariates deal with unobserved heterogeneity and they are used for predicting cluster memberships of subjects.as a mixed Markov latent class model and
later generalized to include time-constant and time-varying covariates by Vermunt, Tran, and
Magidson (2008) (who named the resulting model as mixture latent Markov model, MLMM).
The MHMM presented here is a variant of MLMM where only time-constant covariates are
allowed.
Table 1 :
1Functions and methods in the seqHMM package.
The user can easily choose to plot observations, most probable paths of hidden states, or both. The function0.05
0.03
0.01
0.01
0.08
0.03
0.2
1
0.01
0
0
0
divorced/childless/left home
divorced/childless/with parents
divorced/children/left home
married/childless/left home
married/childless/with parents
married/children/left home
single/childless/left home
single/childless/with parents
single/children/left home
single/children/with parents
seqHMM :
seqHMMMixture Hidden Markov Models for Sequence Data single/childless/with parents single/childless/left home married/childless/left home married/children/left home divorced/childless/left home divorced/children/left home others single/childless/with parents single/childless/left home single/children/left home single/children/with parents married/childless/with parents divorced/childless/with parents othersCluster 1
0.065
0.041
0.014
0.084
0.024
0.00018
0.2
0.013
0.006
0.98
0.016
0
0
0
Cluster 2
0.0092
0.077
0.0098
1
0
0
0
https://github.com/helske/seqHMM
AcknowledgementsSatu Helske is grateful for support for this research from the John Fell Oxford University Press (OUP) Research Fund and the Department of Mathematics and Statistics at the University of Jyväskylä, Finland, and Jouni Helske for the Emil Aaltonen Foundation and the Academy of Finland (research grant 284513).We also wish to thank Mervi Eerola and Jukka Nyblom as well as the editor and two anonymous referees for their helpful comments and suggestions. Comments, suggestions, and bug reports from various users of seqHMM have also been highly appreciated.
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nloptr: R Interface to NLopt. J Ypma, H W Borchers, D Eddelbuettel, R Package Version 1.0.4Ypma J, Borchers HW, Eddelbuettel D (2014). nloptr: R Interface to NLopt. R Package Version 1.0.4, URL http://CRAN.R-project.org/package=nloptr.
| [
"https://github.com/helske/seqHMM"
] |
[
"Fermi-Dirac-Fokker-Planck Equation: Well-posedness & Long-time Asymptotics",
"Fermi-Dirac-Fokker-Planck Equation: Well-posedness & Long-time Asymptotics"
] | [
"José A Carrillo ",
"Philippe Laurençot ",
"Jesús Rosado "
] | [] | [] | A Fokker-Planck type equation for interacting particles with exclusion principle is analysed. The nonlinear drift gives rise to mathematical difficulties in controlling moments of the distribution function. Assuming enough initial moments are finite, we can show the global existence of weak solutions for this problem. The natural associated entropy of the equation is the main tool to derive uniform in time a priori estimates for the kinetic energy and entropy. As a consequence, long-time asymptotics in L 1 are characterized by the Fermi-Dirac equilibrium with the same initial mass. This result is achieved without rate for any constructed global solution and with exponential rate due to entropy/entropy-dissipation arguments for initial data controlled by Fermi-Dirac distributions. Finally, initial data below radial solutions with suitable decay at infinity lead to solutions for which the relative entropy towards the Fermi-Dirac equilibrium is shown to converge to zero without decay rate. | 10.1016/j.jde.2009.07.018 | [
"https://arxiv.org/pdf/0801.2888v1.pdf"
] | 14,685,238 | 0801.2888 | 75869d57942dc20599d1f5f575d4c8517398e23b |
Fermi-Dirac-Fokker-Planck Equation: Well-posedness & Long-time Asymptotics
18 Jan 2008 February 2, 2008
José A Carrillo
Philippe Laurençot
Jesús Rosado
Fermi-Dirac-Fokker-Planck Equation: Well-posedness & Long-time Asymptotics
18 Jan 2008 February 2, 2008
A Fokker-Planck type equation for interacting particles with exclusion principle is analysed. The nonlinear drift gives rise to mathematical difficulties in controlling moments of the distribution function. Assuming enough initial moments are finite, we can show the global existence of weak solutions for this problem. The natural associated entropy of the equation is the main tool to derive uniform in time a priori estimates for the kinetic energy and entropy. As a consequence, long-time asymptotics in L 1 are characterized by the Fermi-Dirac equilibrium with the same initial mass. This result is achieved without rate for any constructed global solution and with exponential rate due to entropy/entropy-dissipation arguments for initial data controlled by Fermi-Dirac distributions. Finally, initial data below radial solutions with suitable decay at infinity lead to solutions for which the relative entropy towards the Fermi-Dirac equilibrium is shown to converge to zero without decay rate.
Introduction
Kinetic equations for interacting particles with exclusion principle, such as fermions, have been introduced in the physics literature in [9,12,13,15,14,23] and the review [10].
Spatially inhomogeneous equations appear from formal derivations of generalized Boltzmann equations and Uehling-Uhlenbeck kinetic equations both for fermionic and bosonic particles. The most relevant questions related to these problems concern their long-time asymptotics and the rate of convergence towards global equilibrium if any.
The spatially inhomogeneous situation has been recently studied in [22], where the long time asympotics of these models in the torus is shown to be given by spatially homogeneous equilibrium given by Fermi-Dirac distributions when the initial data is not far from equilibrium in a suitable Sobolev space. This nice result is based on techniques developed in previous works [20,21]. Other related mathematical results for Boltzmanntype models have appeared in [7,19].
In this work, we focus on the global existence of solutions and the convergence of solutions towards global equilibrium in the spatially homogeneous case without any smallness assumption on the initial data. Preliminary results in the one-dimensional setting were reported in [5]. More precisely, we analyse in detail the following Fokker-Planck equation for fermions, see for instance [10],
∂f ∂t = ∆ v f + div v [vf (1 − f )], v ∈ R N , t > 0,(1.1)
with initial condition f (0, v) = f 0 (v) ∈ L 1 (R N ), 0 ≤ f 0 ≤ 1 and suitable moment conditions to be specified below. Here, f = f (t, v) is the density of particles with velocity v at time t ≥ 0. This equation has been proposed in order to describe the dynamics of classical interacting particles, obeying the exclusion-inclusion principle in [12]. In fact, equation (1.1) is formally equivalent to
∂f ∂t = div v f (1 − f )∇ v log f 1 − f + |v| 2 2
from which it is easily seen that Fermi-Dirac distributions defined by Another striking property of this equation is the existence of a formal Liapunov functional, related to the standard entropy functional for linear and nonlinear Fokker-Planck models [4,2], given by
H(f ) := 1 2 R N |v| 2 f (v) dv + R N [(1 − f ) log(1 − f ) + f log(f )] dv.
We will show that this functional plays the same role as the H-functional for the spatially homogeneous Boltzmann equation, see for instance [24]. In particular it will be crucial to characterize long-time asymptotics of (1.1). In fact, the entropy method will be the basis of the main results in this work; more precisely by taking the formal time derivative of H(f ), we conclude that
d dt H(f ) = − R N f (1 − f ) v + ∇ v log f 1 − f 2 dv ≤ 0.
Therefore, to show the global equilibration of solutions to (1.1) we need to find the right functional setting to show the entropy dissipation. Furthermore, if we succeed in relating functionally the entropy and the entropy dissipation, we will be able to give decay rates towards equilibrium. These are the main objectives of this work. Let us finally mention that these equations are of interest as typical examples of gradient flows with respect to euclidean Wasserstein distance of entropy functionals with nonlinear mobility, see [1,3] for other examples and related problems.
In section 2, we will show the global existence of solutions for equation (1.1) based on fixed point arguments, estimates involving moment bounds and the conservation of certain properties of the solutions. The suitable functional setting is reminiscent of the one used in equations sharing a similar structure and technical difficulties as those treated in [8,11]. The main technical obstacle for the Fermi-Dirac-Fokker-Planck equation (1.1) lies in the control of moments. Next, in section 3, we show that the constructed solutions verify that the entropy is decreasing, and from that, we prove the convergence towards global equilibrium without rate. Again, here the uniform-in-time control of the second moment is crucial. Finally, we obtain an exponential rate of convergence towards equilibrium if the initial data is controlled by Fermi-Dirac distributions and the convergence to zero of the relative entropy when controlled by radial solutions.
Global Existence of Solutions
In this section, we will show the global existence of solutions to the Cauchy problem to (1.1). We start by proving local existence of solutions together with a characterization of the time-span of these solutions. Later, we show further regularity properties of these solutions with the help of estimates on derivatives. Based on these estimates we can derive further properties of the solutions: conservation of mass, positivity, L ∞ bounds, comparison principle, moment estimates and entropy estimates. All of these uniform estimates allow us to show that solutions can be extended and thus exist for all times.
Local Existence
We will prove the local existence and uniqueness of solutions using contraction-principle arguments as in [1,8,11] for instance. As a first step, let us note that we can write (1.1) as
∂f ∂t = ∆ v f + div v (vf ) − div v (vf 2 ) (2.1)
and, due to Duhamel's formula, we are led to consider the corresponding integral equation
f (t, v) = R N F (t, v, w)f 0 (w)dw − t 0 R N F (t − s, v, w)(div w (wf (s, w) 2 )) dw ds (2.2)
where F (t, v, w) is the fundamental solution for the homogeneous Fokker-Planck equation:
∂f ∂t = div v (vf + ∇ v f ) given by F (t, v, w) := a(t) − N 2 M ν(t) (a(t) − 1 2 v − w) with a(t) := e −2t , ν(t) := e 2t − 1 and M l (ξ) := (2πl) − N 2 e − |ξ| 2 2l
for any λ > 0. Let us define the operator F (t, v)[g] acting on functions g as:
F (t, v)[g(w)] = R N F (t, v, w)g(w) dw. (2.3)
Note that by integration by parts, the expression F (t, v)[div w (wf 2 (w))] is equivalent to:
R N e N t (2π (e 2t − 1)) N 2 e − |e t v−w| 2 2(e 2t −1) div w (wf (w) 2 ) dw = − R N ∇ w e N t (2π (e 2t − 1)) N 2 e − |e t v−w| 2 2(e 2t −1) · w f (w) 2 dw = − R N e −t (∇ v F (t, v, w) · w) f (w) 2 dw =: −e −t ∇ v F (t, v)[wf (w) 2 ] so that (2.2) becomes f (t, v) = F (t, v)[f 0 (w)] + t 0 e −(t−s) ∇ v F (t − s, v)[wf (s, w) 2 ] ds. (2.4)
We will now define a space in which the functional induced by (2.4)
T [f ](t, v) := F (t, v)[f 0 (w)] + t 0 e −(t−s) ∇ v F (t − s, v)[wf (s, w) 2 ] ds (2.5)
has a fixed point. To this end, we define the spaces Υ :
= L ∞ (R N ) ∩ L 1 1 (R N ) ∩ L p m (R N ) and Υ T := C([0, T ]; Υ) with norms f (t) Υ := max{ f (t) ∞ , f (t) L 1 1 , f (t) L p m } and f Υ T := max 0≤t≤T f (t) Υ
for any T > 0, where we omit the N-dimensional euclidean space R N for notational convenience and
f L p m := (1 + |v| m )f p and f p := R N |f | p dv 1 p .
In the following, we will see that for p > N, p ≥ 2, and m ≥ 1 we can choose q and r satisfying Np
N + p < p 2 ≤ r ≤ mp m + 1 < p and p 2 ≤ q ≤ p (2.6) such that T [f ] Υ T is bounded by f Υ T .
Let us fix such parameters p, m, r, q and 0 ≤ t ≤ T . Due to Proposition A.1 and q ≤ p ≤ 2q, we can compute
T [f ](t) ∞ ≤ Ce N t f 0 ∞ + t 0 C e N (t−s) ν(t − s) N 2q + 1 2 |w|f 2 (s) q ds ≤ Ce N t f 0 ∞ + t 0 C e N (t−s) ν(t − s) N 2q + 1 2 f (s) 2− p q ∞ f (s) p q L p m ds ≤ Ce N t f 0 ∞ + t 0 C e N (t−s) ν(t − s) N 2q + 1 2 ds f 2 Υ T ≤ Ce N t f 0 ∞ + C I 1 (t) f 2 Υ T , where I 1 (t) := 1 e −2t χ − 1 2 (N− N q −1)−1 (1 − χ) − 1 2 ( N q +1) dχ < ∞
by the choice (2.6) of q. In the same way, since r satisfies (m + 1)r ≤ mp and 2r ≥ p, we
get
T [f ](t) L p m ≤ Ce N p ′ t f 0 L p m + t 0 C e N p ′ (t−s) ν(t − s) N 2 ( 1 r − 1 p )+ 1 2 |w|f 2 (s) L r m ds ≤ Ce N p ′ t f 0 L p m + t 0 C e N p ′ (t−s) ν(t − s) N 2 ( 1 r − 1 p )+ 1 2 f (s) 2− p r ∞ f (s) p r L p m ds ≤ Ce N p ′ t f 0 L p m + t 0 C e N p ′ (t−s) ν(t − s) N 2 ( 1 r − 1 p )+ 1 2 ds f 2 Υ T ≤ Ce N p ′ t f 0 L p m + C I 2 (t) f 2 Υ T , where I 2 (t) := 1 e −2t χ − 1 2 h N p ′ −(N ( 1 r − 1 p )+1) i −1 (1 − χ) − N 2 ( 1 r − 1 p )− 1 2 dχ < ∞ by the choice (2.6) of r.
Finally we can estimate
T [f ](t) L 1 1 ≤ C f 0 L 1 1 + t 0 C ν(t − s) 1 2 |w|f 2 (s) L 1 1 ds
where by interpolation, we get as p ≥ 2 and m ≥ 1
|w|f 2 L 1 1 = R N (1 + |w|)|w|f 2 dw ≤ R N (1 + |w|) 2 f 2 dw ≤ R N (1 + |w|)f dw p−2 p−1 R N (1 + |w|) p f p dw 1 p−1 ≤ f p−2 p−1 L 1 1 f p p−1 L p m .
(2.7)
Consequently
T [f ](t) L 1 1 ≤ C f 0 L 1 1 + C 1 e −2t χ − 3 2 (1 − χ) − 1 2 dχ f 2 Υ T .
We next check the existence of a fixed point of (2.5) in Υ T . To this end, we define a sequence (f n ) n≥1 by f n+1 = T [f n ] for n ≥ 0. Collecting all the above estimates, we can write
f n+1 (t) Υ ≤ C 1 (N, t) f 0 Υ + C 2 (N, p, q, r, t) f n 2 Υ T
for any 0 ≤ t ≤ T and any T > 0, with
C 1 (N, t) := Ce N t C 2 (N, p, q, r, t) := C max I 1 (t), I 2 (t), 1 e −2t χ − 3 2 (1 − χ) − 1 2 dχ
which are clearly increasing with t and C 2 (t) tends to 0 as t does. Thus, for any T > 0
f n+1 Υ T ≤ C 1 (T ) f 0 Υ + C 2 (T ) f n 2 Υ T
with C 1 (T ) = C 1 (N, T ) and C 2 (T ) = C 2 (N, p, q, r, T ), both being increasing functions of T . We may also assume that C 1 (T ) ≥ 1 without loss of generality. From now on, we will follow the arguments in [18]. We will first show that if T is small enough, the functional T is bounded in Υ T , which will in turn imply the convergence. Let us take T > 0 and δ > 0 which verify
f 0 Υ < δ and 0 < δ < 1 4C 1 (T )C 2 (T ) .
Then, let us prove by induction that f n Υ T < 2C 1 (T )δ for all n. By the choice of T and δ we have
f 0 Υ < C 1 (T )δ < 2C 1 (T )δ. If we suppose that f n Υ T < 2C 1 (T )δ, we have f n+1 Υ T < C 1 (T )δ + 4C 2 1 (T )C 2 (T )δ 2 < 2C 1 (T )δ,
hence the claim. Now, computing the difference between two consecutive iterations of the functional and proceeding with the same estimates as above, we can see for any 0 ≤ t ≤ T that
f n+1 − f n Υ T = t 0 e −(t−s) ∇ v F (t − s, v) w f 2 n − f 2 n−1 ds Υ T ≤ C 2 (T ) sup [0,T ] f n + f n−1 ∞ f n − f n−1 Υ T ≤ C 2 (T ) f n Υ T + f n−1 Υ T f n − f n−1 Υ T ≤ 4C 1 (T )C 2 (T )δ f n − f n−1 Υ T ≤ (4C 1 (T )C 2 (T )δ) n f 1 − f 0 Υ T .
Since 4C 1 (T )C 2 (T )δ < 1 we can conclude that there exists a function f * in Υ T which is a fixed point for T , and hence a solution to the integral equation (2.2). It is not difficult to check that the solution f ∈ Υ T to the integral equation is a solution of (1.1) in the sense of distributions defining our concept of solution. We summarize the results of this subsection in the following result.
Remark 2.2 The previous theorem is also valid for
f 0 ∈ (L ∞ ∩ L p m ∩ L 1 )(R N ), with a solution defined in C([0, T ]; (L ∞ ∩ L p m ∩ L 1 )(R N )
) but we will need to have the first moment of the solution bounded in order to be able to extend it to a global in time solution. We thus include here this additional condition.
Remark 2.3 With the same arguments used to prove Theorem 2.1 we can prove an equivalent result for the Bose-Einstein-Fokker-Planck equation
∂f ∂t = ∆ v f + div v [vf (1 + f )], v ∈ R N , t > 0.
Estimates on Derivatives
Let us now work on estimates on the derivatives. By taking the gradient in the integral equation, we obtain
∇ v f (t, v) = ∇ v F (t, v)[f (w)] − t 0 ∇ v F (t − s, v)[div w (wf 2 (s, w))] ds. (2.8) where ∇ v F (t, v)[g]
is defined as the vector:
∇ v F (t, v)[g] := R N ∇ v F (t, v, w)g(w)dw
for the real-valued function g. Here, we will consider a space X T with suitable weighted norms for the derivatives
f X T = max f Υ T , sup 0<t<T ν(t) 1 2 ∇ v f (t) L p m , sup 0<t<T ν(t) 1 2 ∇ v f (t) L 1 1
where for notational simplicity we refer to |∇ v f | L p m as ∇ v f L p m . Let us estimate the L p mand L 1 -norms of ∇ v f using again the results in Proposition A.1 as follows: for r ∈ [1, p) satisfying (2.6)
∇ v f (t) L p m ≤ C e " N p ′ +1 " t ν(t) 1 2 f 0 L p m + t 0 ∇ v F [2f (w · ∇ w f )] + Nf 2 L p m ds ≤ C e " N p ′ +1 " t ν(t) 1 2 f 0 L p m + C t 0 e " N p ′ +1 " (t−s) ν(t − s) 1 2 f (s) L p m f (s) ∞ ds +C t 0 e " N p ′ +1 " (t−s) ν(t − s) N 2 ( 1 r − 1 p )+ 1 2 f (w · ∇ w f ) L r m ds ≤ C e " N p ′ +1 " t ν(t) 1 2 f 0 L p m + C f 2 Υ T 1 e −2t χ − N+2p ′ 2p ′ (1 − χ) − 1 2 ds +C sup 0<s<T ν(s) 1/2 f (s)(w · ∇ w f (s)) L r m I(t)
where
I(t) ≤ e −t 2 1 e −2t e t " N+2r ′ r ′ " (1 − χ) −( N 2 ( 1 r − 1 p )+ 1 2 ) (χ − e −2t ) − 1 2 dχ ≤ 1 2 e t( N+r ′ r ′ ) 1+e −2t 2 e −2t 1 − e −2t 2 − N 2 ( 1 r − 1 p )− 1 2 (χ − e −2t ) − 1 2 dχ + 1 1+e −2t 2 (χ − e −2t ) −( N 2 ( 1 r − 1 p )− 1 2 1 − e −2t 2 − 1 2 dχ ≤ Ce t N+r ′ r ′ (1 − e −2t ) − N 2 ( 1 r − 1 p ) ≤ Ce t(N − N r +1+ N r − N p ) ν(t) − 1 2 ν(t) 1 2 − N 2 ( 1 r − 1 p ) ≤ Ch(t)ν(t) − 1 2 with h(t) := e t( N+p ′ p ′ ) ν(t) 1 2 − N 2 ( 1 r − 1 p ) which is an increasing function of time with h(0) = 0 since p > r > Np/(N + p). It remains to estimate f (w · ∇ w f ) L r m : f (w · ∇ w f ) L r m ≤ C R N f r |∇ w f | r dw + R N |w| (m+1)r f r |∇ w f | r dw 1 r
Now, we can bound these integrals by using Hölder's inequality to obtain
R N f r |∇ w f | r dw ≤ R N f pr p−r dw p−r p R N |∇ w f | p dw r p and R N |w| (m+1)r f r |∇ w f | r dw ≤ R N |w| pr p−r f pr p−r dw p−r p R N |w| mp |∇ w f | p dw r p .
Since p < pr/(p − r) ≤ mp or equivalently (m + 1)r/m ≤ p < 2r by (2.6), we have for
any 0 < t ≤ T R N f r |∇ w f | r dw ≤ f 2r−p ∞ f p−r p ∇ w f r p ≤ f 2r X T ν(t) r 2 and R N |w| (m+1)r f r |∇ w f | r dw ≤ f 2r−p ∞ f p−r L p m ∇ w f r L p m ≤ f 2r X T ν(t) r 2 .
Putting together the above estimates we have shown that,
ν(t) 1/2 f (t)(w · ∇ w f (t)) L r m ≤ C f 2 X T and ν(t) 1 2 ∇ v f (t) L p m ≤ C 1 1 (T, N, p) f 0 L p m + C 1 2 (T, N, p, r) f 2 X T (2.9)
with C 1 1 and C 1 2 increasing functions of T and for any 0 < t ≤ T . Analogously, we reckon
∇ v f (t) L 1 1 ≤ C e t ν(t) 1 2 f 0 L 1 1 + C t 0 e t−s ν(t − s) 1 2 f (s) ∞ f (s) L 1 1 ds + C t 0 e (t−s) ν(t − s) 1 2 f (w · ∇ w f )(s) L 1 1 ds
where by taking p ≥ 2 and by interpolation as in (2.7), we have
f (w · ∇ w f ) L 1 1 ≤ |w| 1 2 f 2 |w| 1 2 |∇ w f | 2 ≤ f p−2 2(p−1) L 1 1 f p 2(p−1) L p m ∇ w f p−2 2(p−1) L 1 1 ∇ w f p 2(p−1) L p m ≤ f 2 X T ν(t) 1/2 .
Putting together the last estimates, we deduce
ν(t) 1 2 ∇ v f (t) L 1 1 ≤ C 3 1 (T, N, p) f 0 L 1 1 + C 3 2 (T, N, p, r) f 2 X T (2.10)
with C 3 1 and C 3 2 increasing functions of T , for any 0 < t ≤ T . From (2.9) and (2.10) and all the estimates of the previous section, we finally get
f X T ≤ C 1 (T, N, p) f 0 Υ + C 2 (T, N, p, r) f 2 X T
for any T > 0. From these estimates and proceeding as at the end of the previous section, it is easy to show that we have uniform estimates in X T of the iteration sequence and the convergence of the iteration sequence in the space X T . From the uniqueness obtained in the previous section, we conclude that the solution obtained in this new procedure is the same as before and lies in X T . Summarizing, we have shown:
Theorem 2.4 Let m ≥ 1, p > N, p ≥ 2, and f 0 ∈ Υ. Then there exists T > 0 depending only on the norm of the initial condition f 0 in Υ such that (1.1) has a unique solution in C([0, T ]; Υ) with f (0) = f 0 and velocity gradients verifying that t → ν(t) 1 2 |∇ v f (t)| ∈ BC((0, T ), (L p m ∩ L 1 )(R N )).
Properties of the solutions
As (1.1) belongs to the general class of convection-diffusion equation, it enjoys several classical properties which we gather in this section. The proof of these results uses classical approximation arguments, see [8,25] for instance. Since these arguments are somehow standard we will only give the detailed proof of the L 1 -contraction property below.
Lemma 2.5 (Positivity and Boundedness) Let f ∈ X T be the solution of the Cauchy
problem (1.1) with initial condition f 0 ∈ Υ. If 0 ≤ f 0 ≤ 1 in R N , then 0 ≤ f (t) ≤ 1 for any 0 < t ≤ T .
Lemma 2.6 (L 1 -Contraction and Comparison Principle) Let f ∈ X T and g ∈ X T be the solutions of the Cauchy problem (1.1) with respective initial data f 0 ∈ Υ and g 0 ∈ Υ.
Then
f (t) − g(t) 1 ≤ f 0 − g 0 1 (2.11) for all 0 < t ≤ T . Furthermore, if f 0 ≤ g 0 then f (t, v) ≤ g(t, v) for all 0 < t ≤ T and v ∈ R N .
Proof.-Since f and g solve (1.1),
d dt (f − g) = ∆ v (f − g) + ∇ v (v(f − g)) − ∇ v (v(f 2 − g 2 )) (2.12)
holds. We will obtain this result from the time evolution of |f − g| ε where | · | ε is the primitive vanishing at zero of sign ε (s), the latter being an increasing smooth approximation of the sign function defined by sign (s) = 1 if s > 0, sign (0) = 0 and sign (s) = −1 if s < 0. Multiplying both sides of equation (2.12)
by ζ n (v) sign ε (f − g) and integrating over R N , where ζ n ∈ C ∞ 0 (R N ) is a cut-off function satisfying 0 ≤ ζ n ≤ 1, ζ n (v) = 1 if |v| ≤ n, ζ n (v) = 0 if |v| ≥ 2n, and |∇ v ζ n | ≤ 1 n , we obtain d dt R N ζ n (v)|f − g| ε dv ≤ − R N ζ n (v) sign ′ ε (f − g)(v · ∇ v (f − g))(f − g) dv + R N ζ n (v) sign ′ ε (f − g)(v · ∇ v (f − g))(f 2 − g 2 ) dv − R N ∇ v ζ n sign ε (f − g)(∇ v (f − g) + v(f − g − (f 2 − g 2 ))) dv = − R N ζ n (v)(v · ∇ v ((f − g) sign ε (f − g) − |f − g| ε )) dv + R N ζ n (v)(f + g)(v · ∇ v ((f − g) sign ε (f − g) − |f − g| ε )) dv − R N ∇ v ζ n sign ε (f − g)(∇ v (f − g) + v(f − g − (f 2 − g 2 ))) dv.
Integrating by parts, we finally get
d dt R N ζ n (v)|f − g| ε dv ≤ R N div v (vζ n (v))((f − g) sign ε (f − g) − |f − g| ε ) dv − R N div v (ζ n (v)v(f + g))((f − g) sign ε (f − g) − |f − g| ε ) dv + 1 n R N |∇ v (f − g) + v(f − g − (f 2 − g 2 ))| dv.
For every n, the first two integrals become zero as ε → 0, since f and g are in
X T whence f (t), g(t) ∈ L 1 1 ∩ L ∞ (R N ) and ∇ v f (t), ∇ v g(t) ∈ L 1 1 (R N ) for any 0 < t ≤ T , allowing for a Lebesgue dominated convergence argument. We have that ∇ v f + vf (1 − f ) ∈ L 1 (R N ) and ∇ v g + vg(1 − g) ∈ L 1 (R N )
for any 0 < t ≤ T , and thus the third integral vanishes as n → ∞, getting finally
d dt R N |f − g| dv ≤ 0 (2.13)
which concludes the proof of the first assertion of the lemma. Similar arguments show the conservation of mass. Finally, we establish time dependent bounds on moments of the solution to (1.1). More precisely, we will show that moments increase at most as a polynomial on t. First, let us note that given a, b ≥ 1 and f ∈ L 1
ab (R N ) ∩ L ∞ (R N ) then f L b a ≤ C f 1 b L 1 ab f 1− 1 b ∞ .
(2.14)
Indeed,
f L b a = R N (1 + |v| a ) b f b dv 1 b ≤ C R N (1 + |v| ab )f b dv 1 b ≤ C f b−1 ∞ R N (1 + |v| ab )f dv 1 b = C f 1 b L 1 ab f 1− 1 b ∞ .
In particular, (L 1 mp ∩ L ∞ )(R N ) ⊂ Υ. We next define ⌈γ⌉ to be the smallest integer larger or equal than γ. Proof.-We will prove it by induction on γ. First, we will see that the second moment is bounded, and afterward that we can bound every moment of order smaller than pm in terms of a γ th * moment with 0 < γ * ≤ 2, which can in turn be bounded in terms of the second moment.
Let (ζ n ) n≥1 be a sequence of smooth cut-off functions satisfying 0 ≤ ζ n ≤ 1, ζ n (v) = 1 if |v| ≤ n, ζ n (v) = 0 if |v| ≥ 2n, |∇ v ζ n | ≤ 1/n and |∆ v ζ n | ≤ 1/n 2 . We multiply (1.1) by |v| 2 ζ n (v) and integrate over R N to get
d dt R N ζ n (v)|v| 2 f dv = R N ζ n (v)|v| 2 ∆ v f dv + R N ζ n (v)|v| 2 div v (vf (1 − f ))dv ≤ R N ∆ v ζ n |v| 2 + 4∇ v ζ n v + 2Nζ n f dv + R N |∇ v ζ n ||v| 3 f (1 − f )dv − 2 R N ζ n |v| 2 f dv + 2 R N ζ n |v| 2 f 2 dv ≤ 5 n<|v|<2n f dv + 2N R N ζ n f dv + n<|v|<2n |v| 2 f dv.
Now, letting n → ∞ and noticing that f ½ {n<|v|<2n} and |v| 2 f ½ {n<|v|<2n} converge pointwise to zero and are bounded by f and |v| 2 f respectively with f ∈ X T , we infer from the Lebesgue dominated convergence theorem that the first and the last integrals converge to zero. Finally, integrating in time, we get
R N |v| 2 f (t, v)dv ≤ R N |v| 2 f 0 (v)dv + 2NMt (2.15)
for all 0 ≤ t ≤ T . Now, for the moment 2γ we can see in the same way
d dt R N ζ n (v)|v| 2γ f dv = R N ζ n (v)|v| 2γ ∆ v f dv + R N ζ n (v)|v| 2γ div v (vf (1 − f ))dv ≤ R N ∆ v ζ n |v| 2γ + 4γ∇ v ζ n |v| 2(γ−1) v + 2γ(2(γ − 1) + N)|v| 2(γ−1) ζ n f dv + R N |∇ v ζ n ||v| 2γ+1 f (1 − f )dv − 2γ R N ζ n |v| 2γ f dv + 2γ R N ζ n |v| 2γ f 2 dv ≤ C n<|v|<2n |v| 2(γ−1) f dv + 2γ(2(γ − 1) + N) R N ζ n |v| 2(γ−1) f dv + n<|v|<2n
|v| 2γ f dv and we again let n go to infinity. If 2γ ≤ mp, the previous argument ensures that only the second integral remains, and integrating in time, we conclude
R N |v| 2γ f (t, v)dv ≤ R N |v| 2γ f 0 (v)dv + 2γ(2(γ − 1) + N) t 0 R N |v| 2(γ−1) f (s, v)dv ds (2.16) for all 0 ≤ t ≤ T .
Whence, if we assume by induction that the hypothesis of the lemma holds true for the 2(γ − 1)-moment,
R N |v| 2γ f (v, t)dv ≤ R N |v| 2γ f 0 (v)dv + 2γ(2(γ − 1) + N) t 0 P ⌈γ−1⌉ (s) ds (2.17)
for all 0 ≤ t ≤ T , defining by induction the polynomial P ⌈γ⌉ .
Global existence
Given an initial condition f 0 ∈ L 1 mp (R N ), p > N, p ≥ 2, m ≥ 1 such that 0 ≤ f 0 ≤ 1, we have f 0 ∈ Υ and we have shown in the previous subsections that there exists a unique local solution of (1.1) on an interval [0, T ). In fact, we can extend this solution to be global in time. If there exists T max < ∞ such that the solution does not exist out of (0, T max ), then the Υ-norm of it shall go to infinity as t goes to T max ; as we will see, that situation cannot happen.
Due to Lemma 2.5, we have that 0 ≤ f (t, v) ≤ 1 for any 0 ≤ t < T and any v ∈ R N , and thus a bound for the L ∞ -norm of f (t). Also, the conservation of the mass in Lemma 2.7 together with the positivity in Lemma 2.5 provide us with a bound for the L 1 -norm. Finally, due to (2.14) and Lemma 2.8 the L p m -norm is also bounded on any finite time interval.
Theorem 2.9 (Global Existence) Let f 0 ∈ L 1 mp (R N ), p > N, p ≥ 2, m ≥ 1 be such that 0 ≤ f 0 ≤ 1. Then the Cauchy problem (1.1) with initial condition f 0 has a unique solution defined in [0, ∞) belonging to X T for all T > 0. Also, we have 0 ≤ f (t, v) ≤ 1, for all t ≥ 0 and v ∈ R N and f (t) 1 = f 0 1 = M for all t ≥ 0.
Remark 2.10 Note that for any K > 0 we can consider (1.1) restricted to the cylinder C K := [0, ∞) × {|v| ≤ K}. Then, due to the fact that the solutions to (1.1) we have constructed are in L ∞ , we can show that the solution is indeed C ∞ (C K ) by applying regularity results in [16] for quasilinear parabolic equations.
Corollary 2.11 If f 0 ∈ L 1 mp (R N ) ∩ L ∞ (R N )
is a radially symmetric and non-increasing function (that is, f 0 (v) = ϕ 0 (|v|) for some non-increasing function ϕ 0 ), then so is f (t) for all t ≥ 0, that is, f (t, v) = ϕ(t, |v|) and r → ϕ(t, r) is non-increasing for all t ≥ 0. In addition, ϕ solves
∂ϕ ∂t = 1 r N −1 ∂ ∂r r N −1 ∂ϕ ∂r + r N ϕ(1 − ϕ) with ∂ϕ ∂r (t, 0) = 0 (2.18)
and ϕ(0, r) = ϕ 0 (r).
Proof.-The uniqueness part of Theorem 2.9 and the rotational invariance of (1.1) imply that f (t) is radially symmetric for all t ≥ 0. The other properties are proved by classical arguments, the monotonicity of r → ϕ(t, r) being a consequence of the comparison principle applied to the equation solved by ∂ϕ/∂r.
Asymptotic Behaviour
Now that we have shown that under the appropriate assumptions equation (1.1) have a unique solution which is global in time, we are interested in how does this solution behave when the time is large. For that we will define an appropriate entropy functional for the solution and study its properties.
Associated Entropy Functional
In this section, we will show that the solutions constructed above satisfy an additional dissipation property, the entropy decay. For g ∈ Υ such that 0 ≤ g ≤ 1, we define the functional H(g) := S(g) + E(g) (3.1) with the entropy given by
S(g) := R N s(g(v)) dv (3.2) where s(r) := (1 − r) log(1 − r) + r log(r) ≤ 0, r ∈ [0, 1],(3.3)
and the kinetic energy given by
E(g) := 1 2 R N |v| 2 g(v) dv. (3.4)
We first check that H(g) is indeed well defined and establish a control of the entropy in terms of the kinetic energy. for every g ∈ L 1 2 (R N ) such that 0 ≤ g ≤ 1.
Proof.-For ε ∈ (0, 1) and v ∈ R N , we put z ε (v) := 1/(1 + e ε|v| 2 /2 ). The convexity of s ensures that
s(g(v)) − s(z ε (v)) ≥ s ′ (z ε (v))(g(v) − z ε (v)) −s(z ε (v)) + s(g(v)) ≥ log z ε (v) 1 − z ε (v) (g(v) − z ε (v)) for v ∈ R N . Since z ε (v)/(1 − z ε (v)) = e −ε|v| 2 /2 , we end up with −s(g(v)) ≤ ε|v| 2 2 g(v) − s(z ε (v)) − ε|v| 2 2 z ε (v) = ε|v| 2 2 g(v) + (1 − z ε (v)) log 1 + e −ε|v| 2 /2 + z ε (v) log 1 + e −ε|v| 2 /2 ≤ ε|v| 2 2 g(v) + e −ε|v| 2 /2 (3.6)
for v ∈ R N , where we used log(1 + a) ≤ a for a ≥ 0 and 0 ≤ z ε ≤ 1. Integrating the previous inequality yields (3.5).
We next recall that F M is the unique Fermi-Dirac equilibrium state satisfying F M 1 = M := f 0 1 ; then we can introduce the next property for H.
H(f 0 ) ≥ H(f (t)) ≥ H(F M ) with M := f 0 1 . (3.7)
Proof.-We first give a formal proof of the time monotonicity of H(f ) and supply additional details at the end of the proof. First of all, we observe that we can formulate (1.1) as
∂f ∂t = div v f (1 − f )∇ v s ′ (f ) + |v| 2 2 .
We multiply the previous equation by s ′ (f ) + |v| 2 /2 and integrate over R N to obtain that
d dt H(f ) = − R N f (1 − f )|v + ∇ v s ′ (f )| 2 dv ≤ 0. (3.8)
Consequently, the function t −→ H(f (t)) is a non-increasing function of time, whence the first inequality in (3.7). To prove the second inequality, we observe that the convexity of s entails that
s(f (t, v)) − s(F M (v)) ≥ s ′ (F M (v))(f (t, v) − F M (v)) s(F M (v)) − s(f (t, v)) ≤ log β(M) + |v| 2 2 (f (t, v) − F M (v)) for (t, v) ∈ [0, ∞) × R N .
The second inequality in (3.7) now follows from the integration of the previous inequality over R N since F M 1 = f (t) 1 by Lemma 2.7.
We shall point out that, in order to justify the previous computations leading to the time monotonicity of the entropy, one should first start with an initial condition f ε 0 , ε ∈ (0, 1), given by
f ε 0 (v) = max min f 0 (v), 1 1 + εe |v| 2 /2 , ε ε + e |v| 2 /2 ∈ ε ε + e |v| 2 /2 , 1 1 + εe |v| 2 /2 , v ∈ R N .
Owing to the comparison principle (Lemma 2.6), the corresponding solution f ε to (1.1) satisfies
0 < ε ε + e |v| 2 /2 ≤ f ε (t, v) ≤ 1 1 + εe |v| 2 /2 < 1 , (t, v) ∈ (0, ∞) × R N ,
for which the previous computations can be performed since the solutions are immediately smooth and fast decaying at infinity for all t > 0, and thus H(f ε (t)) ≤ H(f ε 0 ) for all t ≥ 0. Since f ε 0 → f 0 in Υ and in L 1 mp (R N ) as ε → 0, it is not difficult to see that redoing all estimates in subsections 2.1 and 2.2, we have continuous dependence of solutions with respect to the initial data, and thus, f ε converges towards f in X T for any T > 0. Moreover, we have uniform bounds with respect to ε of the moments in finite time intervals using Lemma 2.8. Direct estimates easily show that
H(f ε 0 ) → H(f 0 ) as ε → 0. Let us now prove that H(f ε (t)) → H(f (t)) as ε → 0 for t > 0. Let us fix R > 0. Since f ε (t) → f (t) in L 1 (R N )
and we have uniform estimates in ε of moments of order mp > 2 then
R N |v| 2 (f ε (t) − f (t)) dv ≤ |v|≥R |v| 2 |f ε (t) − f (t)| dv + |v|<R |v| 2 (f ε (t) − f (t)) dv ≤ 1 R mp−2 |v|≥R |v| mp (f ε (t) + f (t)) dv + R 2 f ε (t) − f (t) 1 ≤ C(t) R mp−2 + R 2 f ε (t) − f (t) 1 .
Since the above inequality is valid for all R > 0, we conclude that E(f ε (t)) → E(f (t)) as ε → 0. Now, taking into account that
(1 + |v| 2 )f ε (t) → (1 + |v| 2 )f (t) in L 1 (R N ), we deduce that there exists h ∈ L 1 (R N ) such that ||v| 2 f ε (t)| ≤ h and f ε (t) → f (t)
a.e. in R N , for a subsequence that we denote with the same index. Using inequality (3.6), we deduce that
0 ≤ −s(f ε (t, v)) ≤ 1 4 h(v) + e −|v| 2 /4 ∈ L 1 (R N )
and that −s(f ε (t, v)) → −s (f (t, v)) a.e. in R N . Thus, by the Lebesgue dominated convergence theorem, we finally deduce that S(f ε (t)) → S(f (t)) as ε → 0. The convergence as ε → 0 of S(f ε (t)) to S(f (t)) is actually true for the whole family (and not only for a subsequence) thanks to the uniqueness of the limit. As a consequence, we showed H(f ε (t)) → H(f (t)) as ε → 0 and passing to the limit ε → 0 in the inequality H(f ε (t)) ≤ H(f ε 0 ), we get the desired result. Now, it is easy to see the existence of a uniform in time bound for the kinetic energy E(f (t)), or equivalently, of the solutions in L 1 2 (R N ). If we take equations (3.1), (3.5) (with ε = 1/2) and (3.7) we get that
E(f (t)) = H(f (t)) − S(f (t)) ≤ 1 2 E(f (t)) + C 1/2 + H(f 0 ) for t ≥ 0 whence E(f (t)) ≤ 2 C 1/2 + H(f 0 ) . (3.9)
Convergence to the Steady State
1) with initial condition f 0 in L 1 mp (R N ), p > max(N, 2), m ≥ 1 satisfying 0 ≤ f 0 ≤ 1. Then {f (t)} t≥0 converges strongly in L 1 (R N ) towards F M as t → ∞ with M := f 0 1 .
For the proof, we first need a technical lemma.
Lemma 3.4 Let f be the solution to the Cauchy problem
(1.1) with initial condition f 0 in L 1 mp (R N ), p > max(N, 2), m ≥ 1 satisfying 0 ≤ f 0 ≤ 1. If A is a measurable subset of R N , we have ∞ 0 A |vf (1 − f ) + ∇ v f | dv 2 dt ≤ H(F M ) sup t≥0 A f (t, v)dv (3.10)
Proof.-Owing to the second inequality in (3.7) and the finiteness of H(f 0 ), we also infer from (3.8)
that (t, v) −→ f (1 − f ) |v + ∇ v s ′ (f )| 2 belongs to L 1 ((0, ∞) × R N ).
Working again with the regularized solutions f ε , it then follows from Lemma 2.7 and the Cauchy-Schwarz inequality that, if A is a measurable subset of R N , we can compute
∞ 0 A |vf ε (1 − f ε )+∇ v f ε |dv 2 dt = ∞ 0 A |vf ε (1 − f ε ) + ∇ v f ε | (f ε (1 − f ε )) 1/2 (f ε (1 − f ε )) 1/2 dv 2 dt ≤ ∞ 0 A |vf ε (1 − f ε ) + ∇ v f ε | 2 f ε (1 − f ε ) dv A f ε (1 − f ε )dv dt,
and thus,
∞ 0 A |vf ε (1 − f ε )+∇ v f ε |dv 2 dt ≤ sup t≥0 A f ε (t, v)dv ∞ 0 A f ε (1 − f ε ) [v + ∇ v s ′ (f ε )] 2 dvdt ≤ H(F M ε ) sup t≥0 A f ε (t, v)dv .
Here, M ε := f ε 0 1 so that F M ε is the Fermi-Dirac distribution with the mass of the regularized initial condition f ε 0 . It is easy to check that H(F M ε ) → H(F M ) as ε → 0 since M ε → M as ε → 0. Passing to the limit as ε → 0, f ε → f in X T for any T > 0, and thus we get the conclusion.
Proof of Theorem 3.3.-We first establish that
{f (t)} t≥0 is bounded in L 1 2 (R N ) ∩ L ∞ (R N ) . (3.11)
From (3.9) and Theorem 2.9, it is straightforward that E(f (t)) is bounded in [0, ∞).
Recalling the mass conservation, the boundedness of {f (t)} t≥0 in L 1 2 (R N )∩L ∞ (R N ) follows. We next turn to the strong compactness of {f (t)} t≥0 in L 1 (R N ). For that purpose, we put f (t, v)) for (t, v) ∈ (0, ∞) × R N and deduce from Theorem 2.9 and (3.11) that sup t≥0
R(t, v) := vf (t, v)(1 −R(t) 1 + R(t) 2 2 ≤ 2 sup t≥0 R N (1 + |v| 2 )f (t, v)dv < ∞ . (3.12)
Denoting the linear heat semigroup on R N by (e t∆ ) t≥0 , it follows from (1.1) that f is given by the Duhamel formula
f (t) = e t∆ f 0 + t 0 ∇ v e (t−s)∆ R(s)ds , t ≥ 0 . (3.13)
It is straightforward to check by direct Fourier transform techniques that
e t∆ g Ḣα ≤ C(α) min t −α/2 g 2 , t −(2α+N )/4 g 1 for t ∈ (0, ∞), g ∈ L 1 (R N ) ∩ L 2 (R N ) and α ∈ [0, 2] with g Ḣα := R N |ξ| 2α | g(ξ)| 2 dξ 1/2
and g being the Fourier transform of g. Thus, we deduce from (3.13) that, if t ≥ 1 and α ∈ ((1 − (N/2)) + , 1), we have
f (t) Ḣα ≤ C(α)t −(2α+N )/4 f 0 1 + C(α + 1) t−1 0 (t − s) −(2+2α+N )/4 R(s) 1 ds + C(α + 1) t t−1 (t − s) −(1+α)/2 R(s) 2 ds ≤ C 1 + t 1 s −(2+2α+N )/4 ds + 1 0 s −(1+α)/2 ds ≤ C ,
thanks to the choice of α. Consequently, {f (t)} t≥1 is also bounded inḢ α for α ∈ ((1 − (N/2)) + , 1). Owing to the compactness of the embedding of (Ḣ α ∩ L 1 2 )(R N ) in L 1 (R N ), we finally conclude that
{f (t)} t≥0 is relatively compact in L 1 (R N ) .
(3.14)
Consider now a sequence {t n } n∈N of positive real numbers such that t n → ∞ as n → ∞. Owing to (3.14), there are a subsequence of {t n } (not relabelled) and g ∞ ∈ L 1 (R N ) such that {f (t n )} n∈N converges towards g ∞ in L 1 (R N ) as n → ∞. Putting f n (t) = f (t n + t), t ∈ [0, 1] and denoting by g the unique solution to (1.1) with initial datum g ∞ , we infer from the contraction property (2.11) that
lim n→∞ sup t∈[0,1] f n (t) − g(t) 1 = 0 . (3.15)
Next, on one hand, we deduce from the proof of Lemma 3.
4 with A = R N that (t, v) −→ vf (t, v)(1 − f (t, v)) + ∇ v f (t, v) belongs to L 2 ((0, ∞); L 1 (R N )). Since 1 0 R N |vf n (1 − f n ) + ∇ v f n | dv 2 dt = tn+1 tn R N |vf (1 − f ) + ∇ v f | dv 2 dt ,
we end up with
lim n→∞ 1 0 R N |vf n (1 − f n ) + ∇ v f n | dv 2 dt = 0 . (3.16)
On the other hand, it follows from the mass conservation and (3.10) that, if A is a measurable subset of R N with finite measure |A|, we have
1 0 A |vf n (1 − f n ) + ∇ v f n | dv 2 dt ≤ H(F M )|A| ,
which implies that {vf n (1−f n )+∇ v f n } n∈N is weakly relatively compact in L 1 ((0, 1)×R N ) by the Dunford-Pettis theorem. Since {vf n (1 −f n )} n∈N converges strongly towards vg(1 − g) in L 1 ((0, 1)×R N ) by (3.11) and (3.15), we conclude that {∇ v f n } n≥0 is weakly relatively compact in L 1 ((0, 1) × R N ). Upon extracting a further subsequence, we may thus assume that {∇ v f n } n≥0 converges weakly towards ∇ v g in L 1 ((0, 1) × R N ). Consequently, By now, we have seen that the solution of (1.1) with initial condition f 0 converges to the Fermi-Dirac distribution F M with the same mass as f 0 as t → ∞, but we are also interested in how fast this happens. We will answer that question with the next result, which was already proved in [5] in the one dimensional case, and easily extends to any dimension based on the existence and entropy decay results established above.
1 0 R N |vg(1 − g) + ∇ v g| dv dt ≤ lim inf n→∞ 1 0 R N |vf n (1 − f n ) + ∇ v f n | dv
Theorem 3.5 (Entropy Decay Rate) Let f be the solution to the Cauchy problem
(1.1) with initial condition f 0 in L 1 mp (R N ), p > max(N, 2), m ≥ 1 satisfying 0 ≤ f 0 ≤ F M * ≤ 1 for some M * .
Then (3.17) and Proof.
H(f (t)) − H(F M ) ≤ (H(f 0 ) − H(F M ))e −2Ctf (t) − F M 1 ≤ C 2 (H(f 0 ) − H(F M )) 1/2 e −Ct
-Since 0 ≤ f 0 ≤ F M * , then the initial condition satisfies all the hypotheses of Theorems 2.9 and 3.3. In order to show the exponential convergence, we use the same arguments as in [5]. We first remark that the entropy functional H coincides with the one introduced in [2] for the nonlinear diffusion equation
∂g ∂t = div x [g∇ x (x + h(g))] (3.19) for the function 0 ≤ g(t, x) ≤ 1, x ∈ R, t > 0, where h(g) = s ′ (g) = log g − log(1 − g).
Let us point out that the relation between the entropy dissipation for the solutions of the nonlinear diffusion equation (3.19), given by
−D 0 (g) = d dt H(g) = − R N g x + ∂ ∂x h(g)
and the entropy dissipation for the solutions of (1.1), given by (3.8), is the basic idea of the proof. Indeed, one can check that, once restricted to the range f ∈ (0, 1), h(f ) verifies the hypotheses of the Generalized Logarithmic Sobolev Inequality [2,Theorem 17]. The Generalized Logarithmic Sobolev Inequality then asserts that
H(g) − H(F M ) ≤ 1 2 D 0 (g) (3.20)
for all integrable positive g with mass M for which the right-hand side is well-defined and finite. We can now, by the same regularization argument as before, compare the entropy dissipation D(f ) = − d dt H(f ) of equation (1.1) and the one D 0 (f ) of equation (3.19). Thanks to Lemma 2.6 we have f (t, v) ≤ F M * (v) ≤ (β(M * ) + 1) −1 a.e. in R N , and thus
D(f ) = R N f (1 − f ) |v + ∇ v h(f )| 2 dv ≥ C R N f |v + ∇ v h(f )| 2 dv (3.21)
where C = 1 − (β(M * ) + 1) −1 . Applying the Generalized Logarithmic Sobolev Inequality (3.20) to the solution f and taking into account the previous estimates, we conclude
H(f (t)) − H(F M ) ≤ (2C) −1 D(f (t)). (3.22)
Finally, coming back to the entropy evolution:
Propagation of Moments and Consequences
There is a large gap between Theorem 3.3 which only provides the L 1 -convergence to the equilibrium and Theorem 3.5 which warrants an exponential decay to zero of the relative entropy for a restrictive class of initial data. This last section is devoted to an intermediate result where we prove the convergence to zero of the relative entropy but without a rate for a larger class of initial data than in Theorem 3.5.
Lemma 3.6 (Time independent bound for Moments) Let g 0 ∈ L 1 mp (R N ) with m ≥ 1, p > max (p, 2) such that 0 ≤ g 0 ≤ 1, and assume further that g 0 is a radially symmetric and non-increasing function, i.e., there is a non-increasing function ϕ 0 such that g 0 (v) = ϕ 0 (|v|) for v ∈ R N . Then, for the unique solution g of the Cauchy problem (1.1) with initial condition g 0 , the control of moments propagates in time, i.e., there exists C > 0 depending on N and g 0 , but not on time, such that
lim R→∞ sup t≥0 {|v|≥R} |v| mp g(t, v)dv = 0. (3.23)
Proof.-We have already seen in Corollary 2.11 the existence and uniqueness of g and that g(t, v) = ϕ(t, |v|) for t ≥ 0 and v ∈ R N for some function ϕ such that r → ϕ(t, r) is non-increasing. Furthermore, we have that its moments are given by
M := R N g(t, v) dv = Nω N ∞ 0 r N −1 ϕ(t, r) dr (3.24) and R N |v| mp g(t, v) dv = Nω N ∞ 0 r N +mp−1 ϕ(t, r) dr (3.25)
for t ≥ 0, where ω N denotes the volume of the unit ball of R N . Next, since |v| mp g 0 ∈ L 1 (R N ), the map v → |v| mp belongs to L 1 (R N ; g 0 (v) dv) and a refined version of de la Vallée-Poussin theorem [6,17] ensures that there is a nondecreasing, non-negative and convex function ψ ∈ C ∞ ([0, ∞)) such that ψ(0) = 0, ψ ′ is concave,
lim r→∞ ψ(r) r = ∞ and R N ψ(|v| mp )g 0 (v) dv < ∞. (3.26)
Observe that, since ψ(0) = 0 and ψ ′ (0) ≥ 0, the convexity of ψ and the concavity of ψ ′ ensure that for r ≥ 0 rψ ′′ (r) ≤ ψ ′ (r) and ψ(r) ≤ rψ ′ (r). (3.27) Then, after integration by parts, it follows from (2.18) that
1 mp d dt ∞ 0 ψ(r mp )r N −1 ϕ dr = − ∞ 0 r mp−1 ψ ′ (r mp ) r N −1 ∂ϕ ∂r + r N ϕ(1 − ϕ) dr = I 1 + I 2 ,(3.28)
where
I 1 = ∞ 0 ϕ (mp + N − 2)r mp+N −3 ψ ′ (r mp ) + mpr 2mp+N −3 ψ ′′ (r mp ) dr I 2 = − ∞ 0 r N +mp−1 ψ ′ (r mp )ϕ(1 − ϕ) dr.
We now fix R > 0 such that ω N R N ≥ 4M and R 2 ≥ 4(2mp + N − 2), and note that due to the monotonicity of ϕ with respect to r and (3.24)-(3.25) the inequality
M ≥ Nω N R 0 r N −1 ϕdr ≥ ω N R N ϕ(R) (3.29)
holds. Therefore, we first use the monotonicity of ψ ′ and ϕ together with (3.29) to obtain
I 2 ≤ − ∞ R r N +mp−1 ψ ′ (r mp )ϕ(1 − ϕ) dr ≤ (ϕ(R) − 1) ∞ R r N +mp−1 ψ ′ (r mp )ϕ dr ≤ M ω N R N − 1 ∞ R r N +mp−1 ψ ′ (r mp )ϕ dr ≤ − 3 4 ∞ R r N +mp−1 ψ ′ (r mp )ϕ dr ≤ 3 4 R 0 r N +mp−1 ψ ′ (r mp )ϕ dr − 3 4 ∞ 0 r N +mp−1 ψ ′ (r mp )ϕ dr ≤ 3MR mp ψ ′ (R mp ) 4Nω N − 3 4 ∞ 0 r N +mp−1 ψ ′ (r mp )ϕ dr.
On the other hand, from (3.24),(3.25), (3.27), (3.29) and the monotonicity of ψ ′
I 1 ≤ (N + 2m − 2) ∞ 0 r N +mp−3 ψ ′ (r mp )ϕ dr ≤ (N + 2mp − 2)ψ ′ (R mp )R mp−2 R 0 r N −1 ϕ dr + N + 2mp − 2 R 2 ∞ R r N +mp−1 ψ ′ (r mp )ϕ dr ≤ (N + 2mp − 2)ψ ′ (R mp )R mp−2 M Nω N + 1 4 ∞ R r N +mp−1 ψ ′ (r mp )ϕ dr.
Inserting these bounds for I 1 and I 2 in (3.28) and using (3.27) we end up with
1 mp d dt ∞ 0 ψ(r mp )r N −1 ϕ dr ≤ ψ ′ (R mp )MR mp−2 Nω N 3R 2 4 + N + 2mp − 2 − 1 2 ∞ 0 r N +mp−1 ψ ′ (r mp )ϕ dr ≤ ψ ′ (R mp )MR mp−2 Nω N 3R 2 4 + N + 2mp − 2 − 1 2 ∞ 0 r N −1 ψ(r mp )ϕ dr.
We then use the Gronwall lemma to conclude that there exists C > 0 depending on N, M, m, p, g 0 and ψ such that sup t≥0 ψ(|v| mp )g(t, v)dv ≤ C from which (3.23) readily follows by (3.26). Proof.-Due to [19,Theorem 3] we know that Here we follow similar arguments as in [11] to show some bounds for ∂ α F f (t) L p m which were useful in the fixed point argument in Section 2.1. We recall the well-known Young inequality: Let g 1 ∈ L r (R N ), g 2 ∈ L q (R N ) with 1 ≤ p, r, q ≤ ∞ and 1 p + 1 = 1 r + 1 q , then g 1 * g 2 ∈ L p (R N ) and g 1 * g 2 p ≤ g 1 r g 2 q .
|H(f (t)) − H(F M )| ≤ C R N |v| 2 |f (t, v) − F (v)|dw ≤ R 2 f (t) − F 1 + sup t≥0 |v|≥R |v| 2 |f (t) − F |dv
Proposition A.1 Let 1 ≤ q ≤ p ≤ ∞, m ≥ 0 and α ∈ N N . Then for t > 0, f (e t w) dw = e t(2N +|α|)
∂ α F (t)[f ] L p m ≤ Ce " N p ′ +|α| " t ν(t)ν(t) N+|α| 2 R N φ α v − w e −t ν(t) 1/2 f (e t w) dw (A.2)
where φ α (χ) = ∂ α χ (φ 0 ) (χ) = P |α| (χ)φ 0 (χ), being P |α| (χ) a polynomial of degree |α| which we can recursively reckon by P 0 (χ) = 1, P |α| (χ) = P ′ |α|−1 (χ) − χP |α|−1 (χ) and φ 0 (χ) = (2π) − N 2 e − |χ| 2 2 .
Since 1 + |v| m ≤ C(1 + |v − w| m )(1 + |w| m ), we deduce 3) as before, since 1 ≤ q ≤ p with r given by 1 p + 1 = 1 r + 1 q to get the desired bound.
(1+|v| m )|(∂ α F * f )(t)| ≤ ≤ C e t(2N +|α|) ν(t) N+|α| 2 R N (1 + |v − w| m ) φ α v − w e −
β ≥ 0 are stationary solutions. Moreover, for each value of M ≥ 0, there exists a unique β = β(M) ≥ 0 such that F β(M ) has mass M, that is, F β(M ) 1 = M. Throughout the paper we shall denote F β(M ) by F M .
Theorem 2. 1 (
1Local Existence) Let m ≥ 1, p > N, p ≥ 2, and f 0 ∈ Υ. Then there exists T > 0 depending only on the norm of the initial condition f 0 in Υ, such that (1.1) has a unique solution f in C([0, T ]; Υ) with f (0) = f 0 .
Lemma 2. 7 (
7Mass Conservation) Let f ∈ X T be the solution of the Cauchy problem (1.1) with non-negative initial condition f 0 ∈ Υ, then the L 1 -norm of f is conserved, i.e. f (t) 1 = f 0 1 for all t ∈ [0, T ].
Lemma 2. 8 (
8Moments Bound) Let f ∈ X T be the solution of the Cauchy problem(1.1) with initial condition f 0 ∈ L 1 mp (R N ) for some m ≥ 1, p > N, p ≥ 2,and satisfying 0 ≤ f 0 ≤ 1. Then, for 0 ≤ t ≤ T and 1 ≤ γ ≤ mp/2 the 2γ-moment of f (t) is bounded by a polynomial P ⌈γ⌉ (t) of degree ⌈γ⌉, which depends only on the moments of f 0 .
Lemma 3. 1 (
1Entropy Control) For ε ∈ (0, 1), there exists a positive constant C ε such that 0 ≤ −S(g) ≤ εE(g) + C ε (3.5)
Lemma 3. 2 (
2Entropy Monotonicity) Assume that f is the solution to the Cauchy problem (1.1) with initial condition f 0 in L 1 mp (R N ) for some p > max(N, 2), m ≥ 1 and satisfying 0 ≤ f 0 ≤ 1. Then, the function H is a non-increasing function of time satisfying for all t > 0 that
Theorem 3. 3 (
3Convergence) Let f be the solution to the Cauchy problem (1.
), from which we readily deduce that vg(1 − g) + ∇ v g = 0 a.e. in (0, 1) × R N . Since g(t) 1 = M for each t ∈ [0, 1] by Lemma 2.7 and (3.15), standard arguments allow us to conclude that g(t) = F M for each t ∈ [0, 1]. We have thus proved that F M is the only possible cluster point in L 1 (R N ) of {f (t)} t≥0 as t → ∞, which, together with the relative compactness of {f (t)} t≥0 in L 1 (R N ), implies the assertion of Theorem 3.3.
( 3 .
318) for all t ≥ 0, where C depends on M * and M := f 0 1 .
d dt [H(f (t)) − H(F M )] = −D(f (t)) ≤ −2C [H(f (t)) − H(F M )] , and the result follows from Gronwall's lemma. The convergence in L 1 is obtained by a Csiszár-Kullback type inequality proven in [5, Corollary 4.3], its proof being valid for any space dimension. It is actually a consequence of a direct application of the Taylor theorem to the relative entropy H(f ) − H(F M ) giving: f − F M 2 1 ≤ 2M(H(f ) − H(F M )).
Let f be the solution of the Cauchy problem (1.1) with initial condition f 0 ∈ L 1 mp (R N ) such that there exists a radially symmetric and non-increasing functiong 0 ∈ L 1 mp (R N ) with 0 ≤ f 0 ≤ g 0 ≤ 1. Then H(f ) → H(F M ) as t → ∞ where M = f 0 1 .
that H(f (t)) → H(F M ) as t → ∞.A L p m -bounds for the Fokker-Planck Operator
Proof.-For all α ∈ N N , we have ∂ α F (t, v)[f ] = ∂ α
|w| m ) p f (e tN t ( 1 + |e −t χ| m ) p f (
1t ν(t) 1/2 (1 + |w| m ) f (e t w) dw. |v − w| m ) r φ α v − we −t ν(t) with (A.5), we can use Young's inequality in (A.
dx,
Acknowledgements.-JAC and JR acknowledge partial support from DGI-MEC (Spain) project MTM2005-08024 and 2005SGR00611 from Generalitat de Catalunya. We thank the Centre de Recerca Matemàtica (Barcelona) for partial funding and for providing an excellent atmosphere for research.
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| [] |
[
"Lexical Based Semantic Orientation of Online Customer Reviews and Blogs",
"Lexical Based Semantic Orientation of Online Customer Reviews and Blogs"
] | [
"Aurangzeb Khan \nInstitute of Engineering and Computing Sciences\nUniversity Of Science and Technology Bannu\nPakistan\n",
"Khairullah Khan ",
"Shakeel Ahmad \nInstitute of Engineering and Computing Sciences\nUniversity Of Science and Technology Bannu\nPakistan\n\nInstitute of Computing and Information Technology\nGomal University\n\n",
"Fazal Masood Kundi \nInstitute of Computing and Information Technology\nGomal University\n\n",
"Irum Tareen \nInstitute of Computing and Information Technology\nGomal University\n\n",
"Muhammad Zubair Asghar \nInstitute of Computing and Information Technology\nGomal University\n\n",
"Pakistan.D I Khan "
] | [
"Institute of Engineering and Computing Sciences\nUniversity Of Science and Technology Bannu\nPakistan",
"Institute of Engineering and Computing Sciences\nUniversity Of Science and Technology Bannu\nPakistan",
"Institute of Computing and Information Technology\nGomal University\n",
"Institute of Computing and Information Technology\nGomal University\n",
"Institute of Computing and Information Technology\nGomal University\n",
"Institute of Computing and Information Technology\nGomal University\n"
] | [] | Rapid increase in internet users along with growing power of online review sites and social media has given birth to sentiment analysis or opinion mining, which aims at determining what other people think and comment. Sentiments or Opinions contain public generated content about products, services, policies and politics. People are usually interested to seek positive and negative opinions containing likes and dislikes, shared by users for features of particular product or service. This paper proposed sentence-level lexical based domain independent sentiment classification method for different types of data such as reviews and blogs. The proposed method is based on general lexicons i.e. WordNet, SentiWordNet and user defined lexical dictionaries for semantic orientation. The relations and glosses of these dictionaries provide solution to the domain portability problem. The method performs better than word and text level corpus based machine learning methods for semantic orientation. The results show the proposed method performs better as it showsprecision of 87% and83% at document and sentence levels respectively for online comments. | null | [
"https://arxiv.org/pdf/1607.02355v1.pdf"
] | 15,684,960 | 1607.02355 | a394cc2a158cd89d31ccb80750794e73e76dbc0c |
Lexical Based Semantic Orientation of Online Customer Reviews and Blogs
Aurangzeb Khan
Institute of Engineering and Computing Sciences
University Of Science and Technology Bannu
Pakistan
Khairullah Khan
Shakeel Ahmad
Institute of Engineering and Computing Sciences
University Of Science and Technology Bannu
Pakistan
Institute of Computing and Information Technology
Gomal University
Fazal Masood Kundi
Institute of Computing and Information Technology
Gomal University
Irum Tareen
Institute of Computing and Information Technology
Gomal University
Muhammad Zubair Asghar
Institute of Computing and Information Technology
Gomal University
Pakistan.D I Khan
Lexical Based Semantic Orientation of Online Customer Reviews and Blogs
sentiment analysisopinion miningclassificationsemantic orientation
Rapid increase in internet users along with growing power of online review sites and social media has given birth to sentiment analysis or opinion mining, which aims at determining what other people think and comment. Sentiments or Opinions contain public generated content about products, services, policies and politics. People are usually interested to seek positive and negative opinions containing likes and dislikes, shared by users for features of particular product or service. This paper proposed sentence-level lexical based domain independent sentiment classification method for different types of data such as reviews and blogs. The proposed method is based on general lexicons i.e. WordNet, SentiWordNet and user defined lexical dictionaries for semantic orientation. The relations and glosses of these dictionaries provide solution to the domain portability problem. The method performs better than word and text level corpus based machine learning methods for semantic orientation. The results show the proposed method performs better as it showsprecision of 87% and83% at document and sentence levels respectively for online comments.
Introduction
Sentiment analysis or opinion mining is a sub-discipline within text mining, to identify subjectivity, sentiments, affects and other states of emotions within the text found in the other online resources. Opinion mining is in reference to computational techniques utilized to extract, assess, understand and classify the numerous opinions that are expressed in a variety of online social media comments, news sources and other content created by the user (Chen and Zimbra, 2010). Sentiment is a view, feeling, opinion or assessment of a person for some product, event or service. Sentiment Analysis or Opinion Mining is a challenging Text Mining and Natural Language Processing problem for automatic extraction, classification and summarization of sentiments and emotions expressed in online text (Pang, Lee and Vaithyanathan, 2002) )(Asghar, Khan, Ahmad and Kundi, 2013). Sentiment analysis is replacing traditional and web based surveys conducted by companies for finding public opinion about entities like products and services. Sentiment Analysis also assists individuals and organizations interested in knowing what other people comment about a particular product, service topic, issue and event to find an optimal choice for which they are looking for. By the end of 2011, over 181 million blogs were tracked with 6.5 million personal blogs and 12 million blogs written on social networks with majority of users seeking opinions on products and services(Balahur, Steinberger, Kabadjov, Zavarella, Van Der Goot, Halkia and Belyaeva, 2013) (Andreevskaia, and Bergler, 2008).Sentiment analysis is of great value for business intelligence applications, where business analysts can analyze public sentiments about products, services, and policies (Funk, A., Li, Saggion, Bontcheva, and Leibold, 2008). Sentiment Analysis in the context of Government Intelligence aims at extracting public views on government policies and decisions to infer possible public reaction on implementation of certain policies(Stylios, Christodoulakis, Besharat, Vonitsanou, Kotrotsos, Koumpouri and Stamou, 2010). Feature based sentiment analysis include feature extraction, sentiment prediction, sentiment classification and optional summarization modules . Feature extraction identifies those product aspects which are being commented by customers, sentiment prediction identifies the text containing sentiment or opinion by deciding sentiment polarity as positive, negative or neutral and finally summarization module aggregates the results obtained from previous two steps. Feature extraction process takes text as input and generates the extracted features in any of the forms like Lexico-Syntactic or Stylistic, Syntactic and Discourse based (Ohana, B., 2009).Sentiment analysis allows for a better understanding of customers' feelings regarding various companies, their products and services or the way they handle customer services, as well as the behavior of their individual agents. It can be used to help in customer relationship management, employees training, identifying and resolving difficult problems as they appear (Asghar, Khan, Kundi, Qasim, Khan, RahmanUllah, Nawaz, 2014) (Pang, and Lee, 2008). This work, presents an online customer reviews classification method extracting sentiments from blogs comments. The results describes, that the method performs better, as compared to other techniques.
Background And Related Work
Sentiment analysis deals with online customer reviews, blog comments, and other related online social network contents (views and news). People recognizing the usefulness, in the immense expansion of the Web, are being drawn more and more towards online services like, shopping, e-banking, e-commerce etc. as well as to the feedback given in the form of reviews and comments about various products and services. Online reviews and comments added on a daily basis to various online sites, like epinion.com, cnet.com, amazon.com, facebook.com, and twitter.com are quite helpful for consumers in making decisions and for companies planning market strategies (Zhu and Zhang, 2010). This has attracted a lot of attention of research communities from industries as well as academia. Consequently, the steady flow of interest towards online resources in recent times has resulted in a tremendous amount of research activity in the field of sentiment analysis and opinion mining (Liu, 2010a). This has led to the appearance of Web 2.0 which, combined with the vast social media content, has caused quite a bit of excitement as it provide ample opportunities to get a better understanding of what the general public, especially consumers, think about company strategies, product preferences and marketing campaigns as well as social events and political movements. Analysis of the thousands, possibly millions, of reviews, comments and other feedbacks expressed in various forums (Yahoo Forums), blogs (blogosphere), social network and social media sites (including You Tube, Flikr, and Facebook, Twitter etc) and virtual worlds (like Second Life) can potentially answer the numerous new and interesting research questions regarding social, economical, cultural, geo political and business issues (Chen, and Zimbra, 2010).
The early work of sentiment analysis began with subjectivity detection, dating back to the late 1990's. Afterward, it shifted its focus towards the interpretation of metaphors, point of views, narrations, affects, evidentiality in text and other related areas. With the increase in internet usage, the Web became an important source of text repositories. Consequently, a switch was slowly made away from the use of subjectivity analysis and towards the use of sentiment analysis of the Web content. The early work in subjective and objective analysis and classification, the separating of subjective, objective and natural sentences, was a very hard task (Pang, and Lee, 2008) (Turney, 2002) (Wiebe, 1994) (Wiebe, 1990).
In sentence level sentiment analysis, the text document or reviews are split into sentences and each sentence is checked for its semantic orientation by using lexical or statistical techniques. It can be associated with two tasks. The first of these two tasks is to identify whether the sentence is subjective or objective. In the second task, subjective sentences are classified into positive, negative or neutral polarity. Sentence level semantic orientation is important because it takes each sentence individually for semantic orientation. NLP methods are useful for such types of semantic orientations. Sentence level analysis decides what the primary or comprehensive semantic orientation of a sentence is while the primary or comprehensive semantic orientation of the entire document is handled by the document level analysis (Pang, and Lee, 2008) .In addition to sufficient work being performed in text analytics, feature extraction in sentiment analysis is now becoming an active area of research. A review paper presented by (Asghar, Khan, Ahmad and Kundi, 2014) discusses existing techniques and approaches for feature extraction in sentiment analysis and opinion mining. In this review, the main focus is on state-of-art paradigms used for feature extraction in sentiment analysis. Further evaluation of existing techniques is done and challenges to be solved in this area are addressed.
Many approaches have been adopted for performing sentiment analysis on social media sites. Knowledge based approaches classify the sentiments through dictionaries defining the sentiment polarity of words and linguistic patterns (Asghar, Qasim, Ahmad, Ahmad, Khan, Khan, 2013).However, the text documents or reviews are broken down into sentences for sentiment analysis at the sentence level. These sentences are then evaluated by utilizing lexical or statistical methods in order to determine their semantic orientation. This process involves two functions; first is to determine the subjectivity or objectivity of a sentence and the next function is of taking the subjective sentences for an opinion orientation. Some existing work involves analysis at different levels. Particularly, the level of semantic orientation involving words regarding opinion as well as the phrase level. Semantic orientation can be accumulated from the words and phrases to find out the overall Semantic Orientation of a particular sentence or review (Leung, (Liu, 2010b). A rule based subjectivity classifier, capable of mining user tweets shared on twitter during some key political event, was designed to isolate subjective and objective sentences (Asghar, RahmanUllah, Ahmad, Khan, Ahmad and Nawaz, 2014). The framework for subjectivity and objectivity classification is compatible with both annotated and un-annotated dataset.
However sentence level semantic orientation along with extraction of sentence sense using WSD is not being considered so far. This work, presents a technique for sentence level rule based method for sentiments classification considering semantics strength of sentence from blogs and online reviews. The strength of each sentence in a review is obtained considering all parts of speech.
Materials and Methods
In this section, rule based sentiment classification method of online reviews and blogs comments arepresented. The following four steps describe the overall process for semantic orientation for different genre and domains using sentence level lexical dictionaries. 1) Collecting data (text), processing and removal of noise form text data. 2) Developing and using knowledge base which is the collection of lexical dictionaries.
3) Processing of text data at sentence level using WSD for extraction of sentence sense. 4) Checking the polarity of each sentence according to sentence structure and deciding about its opinion orientation (positive, negative or neutral).
This work creates a combination of dictionaries called knowledge base which conations SentiWordNet, WordNet and predefined intensifier dictionaries for rule based polarity classification of positive, negative and neutral opinions. It combines and interlinks the lexical dictionaries (WordNet, SentiWordNet, intensifiers etc.) to make a knowledge base and extract the sense of terms, and semantic score, as described in the next section (Esuli and Sebastiani, 2006). The dictionary database information is described as follows. The pair (POS, offset) uniquely identifies a WordNet synset. Numeric ID called offset associated with POS uniquely identified a synset in a database. The values PosScore and NegScore are the positive and negative scores assigned by SentiWordNet to the synset The objectivity score can be calculated as :
Last column of the dictionary database includes synsets (separated by spaces) and gloss information associated with the term.
(Where NegScore= negative Score, PosScore=Positive Score, Pos(s) = Positive score of synset s., Neg(s) = Negative score of synset s., Obj(s) = Objectiveness score of synset s.)
Results and Discussions
For evaluation of the proposed method, public feedbacks in term of comments are extracted from cricinfo 1 ; the 2011 blog for cricket worldcup. Table-1 shows the blog comment dataset information. The reviews are spilt into sentences and after separation of sentences, thelexical terms for semantic orientation were extracted.Furthermore, the Part of Speech (POS) tagger was applied to classify the sentences into subjective and objective sentences. The subjective sentences are considered for further processing to find the semantic orientation at the individual sentence level. To evaluate the proposed method157 Cricket blog feedbacks are taken from www.cricinfo.com as a dataset. The dataset is split into 592 sentences which are manually evaluated for positive, negative and neutral sentiments. Out of these manuallyevaluated sentences, 266 are labelled as positive, 206 as negative and 120 as neutral sentences as shown in Table-2
.
When the proposed method is evaluated on this dataset for sentiment orientation, an accuracy of 83% is achieved at sentence level.
The blog comments of the above dataset are manually evaluated for performance checking at the feedback level. Among 157 feedbacks, 86 comments as whole feedback are judged as positive, 53 as negative and 18 as neutral feedbacks as described in Table-3.The objective of this work is to evaluate the capability of the proposed method to correctly classify the semantic orientationof sentences and also to access the positive, negative or neutral sentiments from the dataset. The proposed method achieved 87% results at the feedback level from the sports blog. It is observed that number of sentences in blogs can affect the accuracy at feedback level. If feedback or blog contains more sentences, its accuracy could be higher compared to those having less number of sentences. The results of proposed method were compared with other machine learning methods presented by (Go, Bhayani and Huang, 2009) and (Andreevskaia, Bergler, 2008)who achieved the accuracy of 80% for classifying positive and negative sentiments. The proposed method describes that pre-processing is more important to remove noisy text in the case of short messages and comments to achieve high accuracy. (Shamma, D. A., Kennedy and Churchill, 2009, October)investigated the twitter blogs comments for the 2008 American Presidential Electoral debates. They illustrated that the analysis of twitter usage is important and closely yield the semantic structure and contents of the media objects. The twitter can be a predictor of the change in any media event. So mining blogs comments play an important role that can be leveraged to evaluate and analyse any activity.The method is compared withthe methods proposed by (Andreevskaia, and Bergler, 2008, February)(Go, Bhayani and Huang, 2009); the proposed method achieved better results than this approach as shown in Table-4 Figure 1: Comparison of proposed method with other techniques for blogs comments
Conclusion and Future Work
In this work, a sentence level rule based classification method of online reviews and blogs comments is introduced. In this paper the process of developing and using knowledge base which is the collection of lexical dictionaries is presented for processing of text data at sentence level using WSD for extraction of sentence sense. The polarity of each sentence is checked according to sentence structure and deciding about opinion orientation (positive, negative or neutral).
From the results the proposed method performs better as compared to other methods as it is clear that the proposed method achieves an accuracy of 83% and 87% at the sentence and document level respectively for blogs short comments. In future, extraction of the acute sense of sentence and remove noisy text for an efficient semantic orientation. Furthermore, the knowledgebase need to improve for the semantic scores of all parts of speech.
and Fig-1.
and Chan, 2008.)(Westerski, 2007)(Hu, and Liu, 2004)(Kundi, Ahmad, Khan and Asghar, 2014)(Andreevskaia, and Bergler, 2008)
Table1 :
Table1Sum of Opinion SentencesDatasets
Comment
Sentence
Subjective
Objective
Percent
Cricket World Cup
2011
500
1630
1238
392
76/24
Airlines Reviews
1000
7730
5405
2325
70/30
Table 3 :
3Sentiment Orientation of Cricket Blog Comments at Feedback LevelActual Orientation
Positive
Negative
Neutral
Total
System
Positive
80
7
3
90
Assigned
Negative
5
44
2
51
Neutral
1
2
13
16
Total
86
53
18
157
Overall Accuracy : 0.87
Table 2 :
2Sentiment Orientation of Cricket Blog Comments at Sentence LevelActual Orientation
Positive
Negative
Neutral
Total
System
Positive
222
26
12
260
Assigned
Negative
30
170
8
208
Neutral
14
10
100
124
Total
266
206
120
592
Overall Accuracy
0.83
Table 4 :
4Compression with Other Related Works on Blog DatasetsAndreevskaia
Bergler,(2008)
Go, Bhayani,
Huang,(2009)
Proposed Method
http://www.cricinfo.com
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"Vortex and half-vortex stability in coherently driven spinor polariton fluid",
"Vortex and half-vortex stability in coherently driven spinor polariton fluid"
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"Francesca M Marchetti \nDepartamento de Física Teórica de la Materia Condensada\nUAM\n28049MadridSpain\n",
"Bruno Piccirillo \nDipartimento di Fisica\nUniversità Federico II di Napoli\n80126NapoliItaly\n",
"Lorenzo Marrucci \nDipartimento di Fisica\nUniversità Federico II di Napoli\n80126NapoliItaly\n",
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"Jonathan M Fellows \nDepartment of Physics\nUniversity of Warwick\nCV47ALCoventryUK\n",
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"Francesca M Marchetti \nDepartamento de Física Teórica de la Materia Condensada\nUAM\n28049MadridSpain\n",
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] | [] | Spinor or multi-component Bose-Einstein condensates are recently attracting noticeable interest because of the possibility of accessing a rich variety of spin textures. An exclusive feature of spinor condensates is that their fundamental unit of circulation can be fractional, thus allowing to generate topological excitations such as fractional quantum vortices. Matter-light quantum fluids such as microcavity polaritons offer a unique test bed for realising and studying two-component spinor condensates under strongly interacting out-of-equilibrium conditions. This motivates the search for fractional vorticity with half integer windings of phase and polarization. Particularly interesting is the quest of whether half vortices-rather than full integer vortices-are the elementary topological excitations of a spinor polariton condensate by the study of their dynamical stability. Here, by making use of a versatile and tunable injection scheme, we are able to generate either full or half vortex states and observe their stability by following their coherent evolution. By using defect-free microcavity samples, we can ultimately determine the conditions upon which a full and a half integer vortex do or do not demonstrate stability in a condensate of polariton gas. * Electronic address: [email protected] † Electronic address: [email protected] 1 arXiv:1403.0487v1 [cond-mat.quant-gas] | 10.1126/sciadv.1500807 | [
"https://arxiv.org/pdf/1403.0487v2.pdf"
] | 118,397,855 | 1403.0487 | 02487155b1d1b50e9041e63212a09b6804ad0f27 |
Vortex and half-vortex stability in coherently driven spinor polariton fluid
3 Mar 2014
Lorenzo Dominici
Istituto Italiano di Tecnologia
IIT-Lecce
Via Barsanti73010LecceItaly
NNL
Istituto Nanoscienze-CNR
Via Arnesano73100LecceItaly
Jonathan M Fellows
Department of Physics
University of Warwick
CV47ALCoventryUK
Stefano Donati
Istituto Italiano di Tecnologia
IIT-Lecce
Via Barsanti73010LecceItaly
NNL
Istituto Nanoscienze-CNR
Via Arnesano73100LecceItaly
Dario Ballarini
Istituto Italiano di Tecnologia
IIT-Lecce
Via Barsanti73010LecceItaly
NNL
Istituto Nanoscienze-CNR
Via Arnesano73100LecceItaly
Milena De Giorgi
Istituto Italiano di Tecnologia
IIT-Lecce
Via Barsanti73010LecceItaly
NNL
Istituto Nanoscienze-CNR
Via Arnesano73100LecceItaly
Francesca M Marchetti
Departamento de Física Teórica de la Materia Condensada
UAM
28049MadridSpain
Bruno Piccirillo
Dipartimento di Fisica
Università Federico II di Napoli
80126NapoliItaly
Lorenzo Marrucci
Dipartimento di Fisica
Università Federico II di Napoli
80126NapoliItaly
Alberto Bramati
Laboratoire Kastler Brossel
UPMC-Paris 6
ENS et CNRS
75005ParisFrance
Giuseppe Gigli
Istituto Italiano di Tecnologia
IIT-Lecce
Via Barsanti73010LecceItaly
NNL
Istituto Nanoscienze-CNR
Via Arnesano73100LecceItaly
Marzena H Szymańska
Department of Physics and Astronomy
UCL
WC1E6BTLondonUK
Daniele Sanvitto
Istituto Italiano di Tecnologia
IIT-Lecce
Via Barsanti73010LecceItaly
NNL
Istituto Nanoscienze-CNR
Via Arnesano73100LecceItaly
Vortex and half-vortex stability in coherently driven spinor polariton fluid
3 Mar 2014
Spinor or multi-component Bose-Einstein condensates are recently attracting noticeable interest because of the possibility of accessing a rich variety of spin textures. An exclusive feature of spinor condensates is that their fundamental unit of circulation can be fractional, thus allowing to generate topological excitations such as fractional quantum vortices. Matter-light quantum fluids such as microcavity polaritons offer a unique test bed for realising and studying two-component spinor condensates under strongly interacting out-of-equilibrium conditions. This motivates the search for fractional vorticity with half integer windings of phase and polarization. Particularly interesting is the quest of whether half vortices-rather than full integer vortices-are the elementary topological excitations of a spinor polariton condensate by the study of their dynamical stability. Here, by making use of a versatile and tunable injection scheme, we are able to generate either full or half vortex states and observe their stability by following their coherent evolution. By using defect-free microcavity samples, we can ultimately determine the conditions upon which a full and a half integer vortex do or do not demonstrate stability in a condensate of polariton gas. * Electronic address: [email protected] † Electronic address: [email protected] 1 arXiv:1403.0487v1 [cond-mat.quant-gas]
Quantum vortices play a crucial role in our understanding of quantum fluids from hydrodynamics to phase transitions. The discovery of Bose-Einstein condensation of exciton polaritons in semiconductor microcavities [1] has paved the way for a prolific series of studies into quantum hydrodynamics in two-dimensional systems [2][3][4][5][6]. Polaritons are hybrid light-matter particles consisting of strongly coupled excitons and photons. The ±1 spin components of the excitons couple to different polarisation states of light making the Bosedegenerate polariton gas a spinor condensate.
For the equilibrium spinor polariton fluid, in which the drive and decay processes are ignored, the lowest energy topological excitations have been predicted to be "half vortices" (HV) [7]. These carry a phase singularity in only one circular polarisation, such that in the linear polarisation basis they have a half-integer winding number for both the phase and field direction. Such an excitation is complementary to a "full vortex" (FV), which instead has a singularity in each circular polarisation. Even in this simplified equilibrium scenario the question of whether HVs or FVs are dynamically stable has led to some debate [8][9][10] due to the presence of an inherent TE-TM splitting, which often arises in semiconductor microcavities and couples the opposite HVs [8]. The issue is even more complicated in a real polariton system, which is always subject to drive and dissipation, and is intrinsically out of equilibrium [11]. Indeed, in the case of an incoherently pumped polariton superfluid, in contrast to the equilibrium predictions, it has been theoretically demonstrated [12] that both FV and HV are dynamically stable in the absence of a symmetry breaking between the linear polarisation states, while in its presence only full vortex states are seen to be stable. On the experimental side, the recent work of Manni et al [13] shows the splitting of a spontaneously formed linear polarised vortex state (FV) into two circularly polarised vortices (HVs) under non-resonant pulsed excitation. However, in this case, formation and motion/pinning of these vortices are caused by strong inhomogeneities and disorder in specific locations of the sample rather than by any fundamental fluid-intrinsic phenomena.
In general, the fundamental issue of the stability or otherwise of the vortex states in polariton condensates remains an open question. At the same time, this is an issue of fundamental importance given that the nature of the elementary excitations is likely to affect the macroscopic properties of the system such as, for example, the conditions for the Berezinsky-Kosterlitz-Thouless (BKT) transitions to the superfluid state.
In this work for the first time we have been able to study the stability and dynamics of half and full vortices injected into a polariton condensate in a variety of initial conditions and in a controlled manner, taking advantage of the versatility of resonant pumping. We take care to imprint the vortex in a specific position on the sample with sufficiently weak disorder that the biasing effects of sample inhomogeneities can be neglected.
We used a q-plate ( Fig. 1 a), recently developed to study laser windings and optical vorticity [14][15][16], to generate coherent vortex states. The q-plate allows us, through appropriate optical and electrical tuning, to directly inject either a full or half vortex into the polariton condensate, as shown in Fig. 1 b and Fig. 1 c. Using a homodyne interferometric detection with numeric Fourier transform and digital off-axis filtering [17,18], we measure both the instantaneous local modulus and phase of the polariton condensate [19] in all its polarisation components so as to observe the evolution of the resonantly created vortices after the initial pulse has gone but before the population has decayed away.
In our microcavity sample [20], the polariton lifetime is 10 ps, while the laser source is a pulse with 4 ps time length, 0.5 nm bandwidth, centered at 836 nm resonant on the lower polariton branch. In Fig. 2 we show the generation of a vortex with winding number m = 1 in each circular polarisation -i.e., a FV -that can then be detected separately. The 2D maps of panels (a-c) represent the overlapped intensity of the two populations (in red and yellow colours), with superposition of the trajectories of the opposite singularities, for three increasing pulse powers. The evolution of the primary vortices has been shown also using 3D plots, i.e., (x(t), y(t), t) curves, in the panels (d-f) corresponding to (a-c), respectively. At low powers, panels (a) and (d), in the linear regime at which the polariton density is low, we observe that the two polarisation-resolved vortices evolve jointly for the first few picoseconds once the pulse has gone. As the density starts to drop, the vortex cores show a significant separation in space (panel (g), orange), adopting independent trajectories. This suggests that the FV state is unstable at the low polariton densities for which spin-dependent disorder is expected to play an important role in triggering its split.
In contrast, at larger polariton densities, panels (b) and (e), where the disorder is expected to be screened out, the twin singularities of the full vortex appear to move together, never increasing in separation to more than twice the healing length along a 30 ps evolution (panel (g), violet), i.e., 3 times the lifetime, hence indicating the stability of the full vortex state.
Increasing the polariton density further, panels (c) and (f), leads to an ever growing proliferation of vortex-antivortex pairs in both polarisations. In particular, a second generation of vortex-antivortex (V-AV) pairs nucleate in the low density regions of circular ripples which appear due to nonlinear radial currents (i.e., radial simmetry breaking). Any potential instability of a full vortex, and the consequent tendency to split into two half vortices, is not observed in this regime (panel (g), cyan), this is in contrast to what was observed in [13], where the splitting is possibly due to marked sample inhomogeneities. On the contrary our results show that at high densities, for which the superfluid currents should prevail, there is a strong inclination for the system to stay in the full vortex state, despite the successive proliferation of vortex pairs in both polarisations (panel (h)), which could additionally disrupt the original vortex stability. Moreover, this stability also holds for the newly created full V-AV pairs. As an example, the time evolution of the case at P = 1.8 mW is shown in Fig. 3, with the second generation of two additional V-AV couples, created and moving together between the two σ states at least until t = 20 ps. The panels (a-d) represent the evolution of the joint density and vortices at different time frames. At the current density, the system finds a transient stability with the 5 FVs placed in a 4 fold symmetry inside the dark ripple and the excess FV out of the quadrangle. This can be seen also in the phase map of panel (f) simultaneous to panel (d). The primary FV undergoes a displacement a moment before the creation of the first V-AV couple, as represented in the branch structure of Fig. 3(e), where the vortices trajectories are reported in a xy map versus time. We found that at different densities, localized transient structures with 3, 4 or 6 fold symmetry may arise (see also [21]). 8 mW). The frames (a-d) are taken at t = 7.5, 12, 18 and 24 ps, respectively, and the instantaneous vortices shown as symbols. In (f) the phase map relative to t=24 ps and σ + polarisation is reported as well. The evolution of the density shows no polarisation splitting. While at lower power (see Fig. 2 and movie M1) the condensate is in a linear regime and qualitatively preserves its shape throughout the dynamics, here the initial condensate (a, orange due to overlap of red and yellow for the two σ) is subject to a spontaneous radial symmetry-breaking associated with the formation of contractive concentric ripples (b-d) (see also movie M3). In the valley of these ripples we see spontaneous full V-AV formation. The time evolution of the vortices is shown as (x(t), y(t), t) branches with time step of 0.5 ps in (e). Each HV component stays close to its counterpart in the opposite polarisation until quite late into the dynamics.
On the other hand, the injection of a HV -equivalent to a single vortex in one circular polarisation state and a gaussian profile in the other-, shown in Fig. 4 is not seen to be followed by any drag towards the vortex core of an opposite spin singularity, even for those densities for which many vortex pairs are spontaneously generated. In Fig. 4 (a) the trajectory of the injected vortex appears overlapped to the map of the opposite polarisation density (background frame taken at t=30 ps). At this power no secondary vortex is generated. However also in this set of measurements at higher powers V-AV pairs start to form. In panel (b) of Fig. 4 we show both σ densities plus the singularity states at t=20 ps.
The initially vortex-free gaussian component now is in the intensity regime as to develop FIG. 4: Evolution of singularities upon HV injection. The gaussian σ is shown 30ps into the evolution as a density map (a), together with the core trajectory in the opposite σ, at the power of 0.77 mW. In (b) both the densities are overlapped and the instantaneous vortices shown at a larger power (1.8 mW) and t=20 ps. The (x(t), y(t), t) trajectories for (a) and (b) are shown in (c) and (d), respectively, with the bigger blue spheres representing the gaussian center of mass. In both cases, the primary HV singularity is seen to travel around the maximum of the gaussian, mantaining a near-constant distance (e) (red and blue dots refer to cases (a,c) and (b,d), respectively, and lines are guides for the eyes). This may be indicative of interactions between the opposite populations. Note again the development of concentric ripples in the initially gaussian density (radial symmetry breaking), and the formation of two spontaneous V-AV couples arranging in a quadrilater for the case (b,d) (azimuthal breaking). See movies M4 and M5 for the whole sequences.
concentric density ripples and secondary spontaneous vortices are born disposing in a 4-fold symmetry. For both powers, the singularity of the primary HV remains not bound to any opposite spin counterpart, moving along a circular trajectory around the density maximum of the opposite polarisation gaussian state, keeping itself orbiting at the boundary of the very central lobe. Hence the previous case together with this one suggest the lack of any observable intrinsic preference of the system for either HV or FV states. Instead, we observe an unexpected attraction of the vortex core toward the center of the vortex-free polarisation component. Such curves are better depicted in Fig. 4 (c) and (d), which are the (x(t), y(t), t) trajectories relative to cases (a) and (b), respectively, and in panel (e) reporting the distance between the primary HV core and the gaussian center of mass (see also movie M4 and M5).
The orbital-like trajectories suggests the presence of some interactions between the vortex state and the oppositely polarised population. Such dynamical configuration resembles that of a metastable rotating vortex state supported by a harmonic trap [22], which could be rep-
ih ∂φ ± ∂t = −h 2 2m φ ∇ 2 − ih τ φ + g|φ ± | 2 φ ± +h Ω R 2 ψ ± + α|φ ∓ | 2 φ ± (1) ih ∂ψ ± ∂t = −h 2 2m ψ ∇ 2 − ih τ ψ ψ ± +h Ω R 2 φ ± +β ∂ ∂x ± i ∂ ∂y 2 ψ ∓ + F ±
where φ ± and ψ ± represent the field of the ± polarisation component of the excitons and photons respectively. Since the effective mass of the excitons, m φ , is an order of magnitude greater than that of the microcavity photons, m ψ , we may safely neglect the kinetic energy of the excitons. We take the Rabi splitting to be Ω R = 6.6meV and exciton-exciton interaction strength g = 2.5µeV·µm 2 [23]. The exciton and photon lifetimes relevant for this experiment are τ φ = 500ps and τ ψ = 10ps respectively. We take the strength of the inter-spin exciton interaction to be an order of magnitude weaker than the intra-spin interaction, so that 2m ψ . The initial laser pulse is modelled as a pulsed Gauss-Laguerre probe F ± :
F ± (r) = f ± r |n ± | e − 1 2 r 2 σ 2 r e in ± θ e − 1 2 (t−t 0 ) 2 σ 2 t e i(kp·r−ωpt)
so that the winding number of the vortex component in the ± polarisation is n ± . The strength, f , has been selected so as to replicate the observed total photon output. The width and duration (FWHM) of the probe were chosen to be σ r = 40µm and σ t = 500fs in line with the experimental reality. The probe is instantiated some time into the simulation, reaching its maximum at t 0 = 1.7ps and cut out completely after 5σ t so as to negate any unintented phase-locking. The equations of motion (1) were solved numerically on a finite grid starting from a weakly noisy initial condition so as to explicilty break the symmetry between the two polarisations and better replicate the inherent noisiness of the experimental system.
We have performed simulations for the half and full vortex configurations at k p = 0, probing on the lower polariton spectrum (ω p = −1). Results are qualitatively similar over a realisitic range of pump powers, representative examples are shown in Fig. 5. We have also increased both α and β up to an order of magnitude with no stark divergence from the results shown.
Our simulations show that both the half and full vortex configurations are dynamically stable, i.e. immune to any internal splitting, a conclusion which is in agreement with the present experiment in the regime of high polariton densities. This conclusion holds in our simulations even with artificially enlarged α and β suggesting that any splitting that might occur is an external rather than an intrinsic effect. The splitting of the full vortex, observed in the experiment towards the end of the dynamics, occurs only once the density is very low. In such conditions the sample disorder is expected to play a pivotal role, and is likely to be different for the two polarisations. Furthermore also in the simulations we see the emergence of density ripples as observed in the experiment. It is in the very bottom of these ripples, where the density is almost zero, that spontaneous V-AV pairs form, leading us to conclude that this effect is not connected with an inherent instability of the condensate to pair formation.
To conclude, we have investigated the stability and dynamics of half and full vortices in a polariton condensate. As a main observation, we may state that surprisingly both the vortices configurations are stable for polariton condensates. This holds true when no external factor is acting in breaking the fluid symmetry, as potential (depending on sample)
or density dishomogeneities (depending on power and non-linearities). At low powers (long times) we observe splitting between the two half vortex components of a full vortex, indicating a sample-driven instability when the disorder landscape is prevalent; at intermediate and high powers (short times) we see no indication of FV splitting or HV joining, such that both half and full vortices are stable, a conclusion supported by numerical simulations. Additionally in such regime, we observe the nonlinear drive of circular ripples (radial symmetry breaking), with the proliferation of vortex-antivortex pairs in the regions of vanishing density (azimuthal breaking). We see no indication of an attraction or repulsion between the primary or secondary vortices in different polarisations, however, we do see some evidence of an attraction of a vortex in one polarisation to the density of the opposing polarisation such that the core becomes localised in an orbit around the maximum. With both types of
FIG. 1 :
1Experimental setup for the creation of FVs and HVs. (a) Use of a q-plate for the generation of a full-(b) and half-vortex (c) state via a q-plate. In red (yellow) the σ + (σ − ) polarisation. Bottom row shows in colour scale the spatial evolution of the phase distribution of each polarisation component.
FIG. 2 :
2Trajectory of phase singularities upon resonant injection of FV states at different power regimes. Density maps (a-c) are the overlap of σ + (red) and σ − (yellow) components taken at t = 30 ps. The position and type of the phase singularities are given by symbols (circle for V, star for AV, taken every 2 ps) and their trajectories superimposed to the maps (see also movies M1-M3 for the whole dynamics). We report the trajectories of the primary vortices only (d-f) as3D curves (x(t), y(t), t) with time step δt = 0.5 ps, and the time evolution of their mutual distance in (g) (orange, violet and cyan for (d), (e) and (f), respectively). The final panel (h) shows the proliferation of secondary pairs, detected at an intermediate time (t = 30 ps) upon increasing pump power, P 1−5 = 0.17, 0.77, 1.8, 3.1 & 4.4 mW, being P 1 , P 2 and P 3 the values relevant for (a,d), (b,e) and (c,f), respectively.
FIG. 3 :
3Dynamics of a polariton condensate containing a FV injected at an intermediate power regime (1.
resented in our case by the opposite population, though the effective total potential could be somehow complicated by the intra-spin forces −at this stage is just speculation− yet the two keep closer ((b,d) and (e) blue line) within the largest of the two shown densities than in ((a,c) and (e) red line). To treat this system theoretically, we consider a coupled Gross-Pitaevskii equation for the individual spin components of the excitons and the microcavity photons
α = − 0 . 1g .
01gFollowing Hivet et al[24], we treat the mass of the transverse electric component of the cavity mode, m TE ψ as being around 95% that of the transverse magnetic component m TM ψ (≈ 5 × 10 −5 m e ), so that the strength of the TE-TM splitting is taken to be β =
FIG. 5 :
5Representative example of the simulated time evolution of a HV. The three columns are different time frames (t=4 ps, t=32 ps and t=60 ps) of a half vortex with a phase singularity in the − polarisation (upper row) and a gaussian profile in the + polarisation (lower row). The density data is rescaled such that the maximum value of the density is 1 and maps to white, the relative scale with respect to the initial images (a,d) is 1 · 10 −2 and 2 · 10 −4 for (b,e) and (c,f), respectively. The case of a full vortex injection displays the same maps as those in the first row for both the polarisation components. The simulations show both the HV and FV configurations to be stable and demonstrate the formation of circular ripples.
topological excitation seemingly stable, an interesting question left to be addressed is which excitation is relevant in a Kosterlitz-Thouless-type transition. topological excitation seemingly stable, an interesting question left to be addressed is which excitation is relevant in a Kosterlitz-Thouless-type transition.
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| [] |
[
"Non-unitarity effects in the minimal inverse seesaw model",
"Non-unitarity effects in the minimal inverse seesaw model"
] | [
"Michal Malinský \nDepartment of Theoretical Physics\nTheoretical Particle Physics Group\nRoyal Technical Institute (KTH)\nRoslagstullsbacken 21SE-106 91StockholmSweden\n"
] | [
"Department of Theoretical Physics\nTheoretical Particle Physics Group\nRoyal Technical Institute (KTH)\nRoslagstullsbacken 21SE-106 91StockholmSweden"
] | [] | A minimal version of the inverse seesaw model featuring only two pairs of TeV-scale singlet neutrinos is discussed from the perspective of non-standard neutrino interactions.A particular attention is paid to the non-standard patterns of flavour and CP violation emerging due to the possibly enhanced non-decoupling effects of the heavy sector and the associated non-unitarity of the effective lepton mixing matrix. | null | [
"https://arxiv.org/pdf/0909.1953v1.pdf"
] | 16,984,806 | 0909.1953 | f829fbadda84f338fe82353fd2b8112f54e9112b |
Non-unitarity effects in the minimal inverse seesaw model
10 Sep 2009
Michal Malinský
Department of Theoretical Physics
Theoretical Particle Physics Group
Royal Technical Institute (KTH)
Roslagstullsbacken 21SE-106 91StockholmSweden
Non-unitarity effects in the minimal inverse seesaw model
10 Sep 2009(Dated: September 10, 2009)
A minimal version of the inverse seesaw model featuring only two pairs of TeV-scale singlet neutrinos is discussed from the perspective of non-standard neutrino interactions.A particular attention is paid to the non-standard patterns of flavour and CP violation emerging due to the possibly enhanced non-decoupling effects of the heavy sector and the associated non-unitarity of the effective lepton mixing matrix.
I. INTRODUCTION
With the upcoming generation of dedicated precision neutrino experiments and a continuing data-taking from the ongoing facilities the next decade offers very good prospects of measuring some of the observables underpinning the flavour violationg in the lepton sector to an unprecedented accuracy level. Improving sufficiently our current knowledge about the neutrino oscillation parameters one can in principle even attempt to ask to what extent the current picture of three flavours oscillating into each other during the neutrino propagation from the source to the detector is complete or if there could be any substential room for physics beyond the standard framework.
Indeed, the seesaw mechanism furnishing perhaps the most compelling understanding of the very tiny light neutrino mass scale relies on new dynamics coming to play at a certain scale, which could exhibit itself as various non-standard effects in the neutrino behavior. Although the standard lore suggests such a new physics scale should be presumably very high (typically at around 10 13 GeV complying nicely with the unification paradigm) there are theoretical scenarios that admit bringing it down into the TeV domain. This would not only provide a very attractive option to probe the underlying dynamics directly at the LHC but also give us a chance to observe various non-decoupling effects of this kind of a new physics relatively close to the electroweak scale. * Electronic address: [email protected];
In collaboration with Tommy Ohlsson and Zhang He (KTH Stockholm) and Zhi-zhong Xing (IHEP Beijing). One of the most profound implications of such scenarios is the possibility that the mixing matrix entering the leptonic charged-current interactions need not be unitary, i.e. the transformation matrix V αi connecting the flavour (α) and mass (i) bases |ν α = V αi |ν i , when contracted only to the dynamically accessible light degrees of freedom, does not obey the unitarity relation (V V † ) αβ = 1.
In such case the traditional survival/transition probability formula receives the following form [1]:
P α→β ≈ i,j F i αβ F j * αβ − 4 i>j Re(F i αβ F j * αβ ) sin 2 ∆m 2 ij L 4E + 2 i>j Im(F i αβ F j * αβ ) sin ∆m 2 ij L 2E (1) where F i αβ ≡ γρ (R * ) αγ (R * ) −1 ρβ U * γi U ρi with R αβ ≡ (1 − η) αβ /[(1 − η)(1 − η † )
] αα and η denoting the "departure" of V from unitarity: V = (1 − η)U . The first two terms in (1) correspond to the so called zero-distance effects and the "standard" oscillation term respectively (giving rise to the traditional simple oscillation formula for a unitary V in the η → 0 limit) while the last term leads to extra CP violating effects for η = 0. The entries of the η matrix are constrained from various experimental inputs like e.g. rare leptonic decays, invisible Z-boson width, neutrino oscillations etc.
For illustration let us quote the 90% C.L. limits given in [2]:
|η ee | < ∼ 2.0 × 10 −3 , |η µµ | < ∼ 8.0 × 10 −4 , |η τ τ | < ∼ 2.7 × 10 −3 , |η eµ | < ∼ 3.5 × 10 −5 , |η eτ | < ∼ 8 × 10 −3 , |η µτ | < ∼ 5.1 × 10 −3 .
III. NON-UNITARITY EFFECTS IN SEESAW MODELS
Let us now inspect in brief the prospects of accommodating non-negligible non-unitarity effects within the three "canonical" realizations of the seesaw idea:
Type-I: In type-I seesaw, the structure F ≡ M D M −1 R (with M D and M R denoting the Dirac and Majorana masses respectively) determining the overall scale of the η-matrix obeying η = 1 2 F F † governs also the scale of the light neutrino masses m ν = F M −1 R F T . Thus, non-unitarity effects can be sizeable only if M D is relatively close to M R . This, however, implies a need for a severe structural fine-tuning in order to bring the eigenvalues of m ν down to the observed level. A detailed discussion of this option can be found for instance in [3]. However, if M R is at around 10 13 GeV the heavy sector essentially decouples and the non-unitary effects are extremely tiny.
Type-II: The "canonical" type-II realization of the seesaw mechanism does not bring in any extra sector that could mix with the SM leptons; the unitarity of the mixing matrix remains unaffected.
Type-III: The situation in the type-III seesaw is a close analogue of the type-I case with the only difference that SU (2) L -triplet fermions are employed in type-III rather than singlets in type-I.
Thus, in order to accommodate naturally the sub-eV light neutrino masses together with potentially sizeable non-unitary effects in the lepton sector one has to go beyond the simple "canonical" realizations of the seesaw idea. In particular, there is a need to disentangle the scale providing the suppression for the neutrino masses (like M R in the type-I & III seesaw) from the scale entering F .
To our best knowledge, the simplest model of this kind can be constructed along the lines of the inverse seesaw [4] approach in which a third scale (usually denoted by µ besides M D and M R ) associated to an extra SM-singlet sector enters the full neutrino mass-matrix:
M ν = 0 M D 0 . 0 M T R . . µ .(2)
where the undisplayed entries follow from symmetries of M ν . One of the virtues of this strucure consists in the fact that µ is the only lepton-number-violating mass term around and as such can be made arbitrarily small with no implications for naturalness (in the t'Hooft sense, c.f. [5]). For reads again V ≃ 1 − 1 2 F F † U , see e.g. [6], and the basic "type-I" formula η = 1 2 F F † remains intact. Thus, while the non-unitarity effects are driven (as in the type-I case) by F = M D M −1 R , one has an option here to pull M R down to lower scales, perhaps even into the TeV region, because in such case the neutrinos can be kept naturally light if µ falls into about the keV domain.
The non-unitarity effects emerging in the original inverse seesaw with three pseudo-Dirac heavy neutrinos have been discussed at length in a recent paper [7] so here we will confine ourselves only to the salient points of the analysis. In full generality, one can parametrize the F matrix (à la Cassas-Ibarra [8]) in terms of the measurable light neutrino masses grouped into a diagonal matrix d ν , a complex orthogonal matrix O and the observed lepton mixing matrix U (including possible Majorana phases) so that η = 1
2 F F † = 1 2 U √ d ν Oµ −1 O † √ d ν U † .µ −1 0 U √ d ν exp(2 iA) √ d ν U † with
A being a real antisymmetric matrix [9] parametrized by three real numbers only. This degeneracy admits to cast more stringent bounds on |η eτ | < 2.3 × 10 −3 and |η µτ | < 1.5 × 10 −3 while saturating the 90% C.L. limits for the others.
B. Minimal inverse seesaw & non-unitarity effects
The situation becomes even more interesting in the minimal inverse seesaw (MISS) framework [10] with only two pairs of heavy pseudo-Dirac neutrinos around the TeV scale. This, indeed, is enough to generate two non-zero light neutrino mass-squared-differences while one of the light neutrinos remains massless. Due to the 2 × 2 shape of the µ and M R matrices in this case η is obtained by replacing the original (3 × 3 complex orthogonal) matrix O above by a rectangular cos 2 θ 23 ± 1 2 sin 2θ 23 .
(3×2 complex) matrix R instead: η = 1 2 U √ d ν Rµ −1 R † √ d ν U † ,
.
sin 2 θ 23 + O(ε 2−x ).(3)
Here x ∈ {1, 0} where the former value (the upper sign above) corresponds to the normal hierarchy of the neutrino spectrum while the latter (the lower sign in (3)) to the inverted hierarchy case.
The structure (3) has two immediate implications: 1) Due to the observed approximate maximality of the atmospheric mixing one has |η µµ | ≃ |η µτ | ≃ |η τ τ | and |η eµ | ≃ |η eτ | which is a genuine testable prediction of the MISS model. The correlation is stronger in the inverse hierarchy case because the sub-leading term in (3) for that would be a neutrino factory with an OPERA-like near detector within up to few tens of kilometers from the source. In such case an aparatus with 5kt fiducial volume exposed by neutrinos from 5+5 years of muon+antimuon runs (corresponding to around 10 21 useful muon decays) could probe |η µτ | down to the level of about 5×10 −4 . There are also reasonable prospects of observing extra CP features in such situation provided the phase of η µτ is around δ µτ = ±π/4 (otherwise the third (CP) term in formula (1) is further suppressed by a small product |η µτ | sin δ µτ ). For more details an interested reader is deferred to the original work [7]. 2 Let us remark that for very small values of |η αβ | the uncertainities in determination of the neutrino mass-squaredifferences and the lepton mixing angles dominate over the effects subsumed by formula (3) which leads to a typical smearing of the patterns obtained from a full numerical simulation towards the |η αβ | → 0 limit, c.f. Figure 1.
The 'traditional' seesaw models of neutrino masses often resort on a directly inaccessible dynamics in the vicinity of the unification scale. However, in many extended scenarios the new physics scale can be as low as few TeV and the enhanced non-decoupling effects can be testable in the upcoming experimental facilities. From this perspective, the minimal inverse seesaw provides an appealing low-energy framework accommodating naturally the observed neutrino masses and mixings. Moreover, the distinct pattern of non-standard flavour and CP effects due to the non-unitarity of the lepton mixing matrix can be observable in the near future neutrino experiments.
µ
≪ M D < M R the neutrino spectrum consists of three light Majorana neutrinos with a mass matrix m ν ≃ M D M −1 R µ(M T R ) −1 M T D = F µF T , and (usually) three pairs of heavy (∼ M R ) almost degenrate Majorana neutrinos with oposite CP (and mass splitting proportional to µ) which can be viewed as (three) heavy pseudo-Dirac neutrinos. It can be shown that the heavy states are admixed into the light flavours with an effective "strength" proportional to F and the 3×3 sub-matrix of the full (usually 9×9) neutrino sector diagonalization matrix (in the basis of diagonal charged leptons)
Interesting non-trivial correlations between the entries of η emerge in special cases in which further assumptions are made about the shape of O and/or µ matrices (reflecting various physically interesting situations at the level of an underlying theory). For example, for a flavour-blind µ = µ 0 × 1 and arbitrary O, the relevant formula reads η = 1 2
which can be parametrized by means of a single (complex) angle 1 . Expanding in powers of ε ≡ (∆m 2 21 /|∆m 2 31 |) 1/4 ≃ 0.42 one obtains the following approximate formula for the η matrix: |x|) −x cos θ 23 x sin θ 23 .
is smaller. 2 )
2Only the |η µτ | off-diagonal entry can be sizeable, which yields a rather specific pattern of non-standard flavour and CP violation effects potentially emmerging in the current scenario. Moreover, in the inverse hierarchy case the phase of η µτ is nearly maximal (corresponding to the minus sign of the 23 entry of the leading term in(3)perturbed only at the O(ε 2 ) level while for the normal hierarchy it is preferably small (zero at the leading level perturbed by the subleading O(ε) terms). One can observe this kind of behavior in FIG. 1: Correlations among various parameters governing the non-unitarity effects in the MISS. Red points (simulated) and regions (extrapolated) correspond to the normal neutrino mass hierarchy, while the blue ones stand for the inverted mass hierarchy. Generic experimental constraints are indicated by the green dashed lines. For sake of simplicity, the correlations in their determination have been neglected).
Figure 1
1depicting the results of the numerical analysis 2 . Concerning the prospects of measuring the value of |η µτ | at the near future experimental facilities, one can conclude that an ideal tool
Talk given at The 2009 Europhysics Conference on High Energy Physics, July 16 -22 2009 Krakow, Poland
The shape of R, however, differs for normal and inverted hierarchy of the light neutrino spectrum; an interested reader is deferred to the original work[7] adopting the parametrization from[11] for further details.
. S Antusch, hep-ph/0607020JHEP. 1084S. Antusch et al., JHEP 10, 084 (2006), hep-ph/0607020.
. S Antusch, J P Baumann, E Fernandez-Martinez, Nucl. Phys. 810S. Antusch, J. P. Baumann, and E. Fernandez-Martinez, Nucl. Phys. B810, 369 (2009), 0807.1003.
. J Kersten, A Y Smirnov, hep-ph/0705.3221Phys. Rev. 7673005J. Kersten and A. Y. Smirnov, Phys. Rev. D76, 073005 (2007), hep-ph/0705.3221.
. R N Mohapatra, J W F Valle, Phys. Rev. 341642R. N. Mohapatra and J. W. F. Valle, Phys. Rev. D34, 1642 (1986).
. G Hooft, Lecture given at Cargese Summer Inst. G. 't Hooft et al., Lecture given at Cargese Summer Inst., Cargese, France, Aug 26 -Sep 8, 1979.
. J Schechter, J W F Valle, Phys. Rev. 25774J. Schechter and J. W. F. Valle, Phys. Rev. D25, 774 (1982).
. M Malinsky, T Ohlsson, H Zhang, arXiv:0903.1961Phys. Rev. 7973009M. Malinsky, T. Ohlsson, and H. Zhang, Phys. Rev. D79, 073009 (2009), arXiv:0903.1961.
. J A Casas, A Ibarra, hep-ph/0103065Nucl. Phys. 618J. A. Casas and A. Ibarra, Nucl. Phys. B618, 171 (2001), hep-ph/0103065.
. S Pascoli, S T Petcov, C E Yaguna, hep-ph/0301095Phys. Lett. 564241S. Pascoli, S. T. Petcov, and C. E. Yaguna, Phys. Lett. B564, 241 (2003), hep-ph/0301095.
. M Malinsky, T Ohlsson, Z Xing, H Zhang, arXiv:0905.2889Phys. Lett. 679242M. Malinsky, T. Ohlsson, Z.-z. Xing, and H. Zhang, Phys. Lett. B679, 242 (2009), arXiv:0905.2889.
. A Ibarra, G G Ross, hep-ph/0312138Phys. Lett. 591285A. Ibarra and G. G. Ross, Phys. Lett. B591, 285 (2004), hep-ph/0312138.
| [] |
[
"An electrically injected photon-pair source at room temperature",
"An electrically injected photon-pair source at room temperature"
] | [
"Fabien Boitier \nLaboratoire Matériaux et Phénomènes Quantiques\nUMR 7162\nUniversité Paris Diderot\nSorbonne Paris Cité\nCNRS\nCase courrier7021, 75205Paris Cedex 13France\n",
"Adeline Orieux \nLaboratoire Matériaux et Phénomènes Quantiques\nUMR 7162\nUniversité Paris Diderot\nSorbonne Paris Cité\nCNRS\nCase courrier7021, 75205Paris Cedex 13France\n",
"Claire Autebert \nLaboratoire Matériaux et Phénomènes Quantiques\nUMR 7162\nUniversité Paris Diderot\nSorbonne Paris Cité\nCNRS\nCase courrier7021, 75205Paris Cedex 13France\n",
"Aristide Lemaître \nLaboratoire de Photonique et Nanostructures\nCNRS-UPR20\nRoute de Nozay91460MarcoussisFrance\n",
"Elisabeth Galopin \nLaboratoire de Photonique et Nanostructures\nCNRS-UPR20\nRoute de Nozay91460MarcoussisFrance\n",
"Christophe Manquest \nLaboratoire Matériaux et Phénomènes Quantiques\nUMR 7162\nUniversité Paris Diderot\nSorbonne Paris Cité\nCNRS\nCase courrier7021, 75205Paris Cedex 13France\n",
"Carlo Sirtori \nLaboratoire Matériaux et Phénomènes Quantiques\nUMR 7162\nUniversité Paris Diderot\nSorbonne Paris Cité\nCNRS\nCase courrier7021, 75205Paris Cedex 13France\n",
"Ivan Favero \nLaboratoire Matériaux et Phénomènes Quantiques\nUMR 7162\nUniversité Paris Diderot\nSorbonne Paris Cité\nCNRS\nCase courrier7021, 75205Paris Cedex 13France\n",
"Giuseppe Leo \nLaboratoire Matériaux et Phénomènes Quantiques\nUMR 7162\nUniversité Paris Diderot\nSorbonne Paris Cité\nCNRS\nCase courrier7021, 75205Paris Cedex 13France\n",
"Sara Ducci \nLaboratoire Matériaux et Phénomènes Quantiques\nUMR 7162\nUniversité Paris Diderot\nSorbonne Paris Cité\nCNRS\nCase courrier7021, 75205Paris Cedex 13France\n"
] | [
"Laboratoire Matériaux et Phénomènes Quantiques\nUMR 7162\nUniversité Paris Diderot\nSorbonne Paris Cité\nCNRS\nCase courrier7021, 75205Paris Cedex 13France",
"Laboratoire Matériaux et Phénomènes Quantiques\nUMR 7162\nUniversité Paris Diderot\nSorbonne Paris Cité\nCNRS\nCase courrier7021, 75205Paris Cedex 13France",
"Laboratoire Matériaux et Phénomènes Quantiques\nUMR 7162\nUniversité Paris Diderot\nSorbonne Paris Cité\nCNRS\nCase courrier7021, 75205Paris Cedex 13France",
"Laboratoire de Photonique et Nanostructures\nCNRS-UPR20\nRoute de Nozay91460MarcoussisFrance",
"Laboratoire de Photonique et Nanostructures\nCNRS-UPR20\nRoute de Nozay91460MarcoussisFrance",
"Laboratoire Matériaux et Phénomènes Quantiques\nUMR 7162\nUniversité Paris Diderot\nSorbonne Paris Cité\nCNRS\nCase courrier7021, 75205Paris Cedex 13France",
"Laboratoire Matériaux et Phénomènes Quantiques\nUMR 7162\nUniversité Paris Diderot\nSorbonne Paris Cité\nCNRS\nCase courrier7021, 75205Paris Cedex 13France",
"Laboratoire Matériaux et Phénomènes Quantiques\nUMR 7162\nUniversité Paris Diderot\nSorbonne Paris Cité\nCNRS\nCase courrier7021, 75205Paris Cedex 13France",
"Laboratoire Matériaux et Phénomènes Quantiques\nUMR 7162\nUniversité Paris Diderot\nSorbonne Paris Cité\nCNRS\nCase courrier7021, 75205Paris Cedex 13France",
"Laboratoire Matériaux et Phénomènes Quantiques\nUMR 7162\nUniversité Paris Diderot\nSorbonne Paris Cité\nCNRS\nCase courrier7021, 75205Paris Cedex 13France"
] | [] | One of the main challenges for future quantum information technologies is miniaturization and integration of high performance components in a single chip. In this context, electrically driven sources of non-classical states of light have a clear advantage over optically driven ones. Here we demonstrate the first electrically driven semiconductor source of photon pairs working at room temperature and telecom wavelength. The device is based on type-II intracavity Spontaneous Parametric Down-Conversion in an AlGaAs laser diode and generates pairs at 1.57 µm. Time-correlation measurements of the emitted pairs give an internal generation efficiency of 7 × 10 −11 pairs/injected electron. The capability of our platform to support generation, manipulation and detection of photons opens the way to the demonstration of massively parallel systems for complex quantum operations.PACS numbers: 42.65. Lm, 03.67.Bg, 42.55.Px, Photons have a peculiar advantage in the development of quantum information technologies [1-3], since they behave naturally as flying qubits presenting a high speed transmission over long distances and being almost immune to decoherence[4,5]. The intrinsic scalability and reliability of integrated photonic circuits has recently given rise to a new generation of devices for quantum communication, computation and metrology[6]. Nevertheless even if great progress have been made in the manipulation[7,8]and detection[9]of nonclassical state of light on chip, a complete integration of the light source in the photonic circuitry stays one of the main challenges on the way towards large scale applications; such devices would have a clear advantage over optically driven ones in terms of portability, energy consumption and integration. Semiconductor materials are ideal to achieve extremely compact and massively parallel devices: concerning photon-pair sources, the bi-exciton cascade of a quantum dot has been used to demonstrate an entangledlight-emitting diode at a wavelength of 890 nm [10]. However, even if the use of a single emitter guarantees a deterministic emission, these devices operate at cryogenic temperature, greatly limiting their potential for applications.Optical parametric conversion offers an alternative approach. Despite its non-deterministic nature, this process is the most widely used to produce photon pairs for quantum information and communications protocols. Up to now, entangled photon pairs have been generated by optical pumping in passive semiconductor waveguides by exploiting four-wave mixing in Silicon[11]or SPDC in Aluminium Gallium Arsenide (AlGaAs)[12,13]. Thanks to its direct band gap, the latter platform presents an evident interest for the electrical injection. In order to deal with the isotropic structure of this crystal, several solutions have been proposed to achieve nonlinear optical conversion in AlGaAs waveguides[14][15][16][17][18]; among these, modal phase matching, in which the phase velocity mismatch is compensated by multimode waveguide dispersion, is one of the most promising to monolithically integrate the laser source and the nonlinear medium into a single device[19,20]. In this scheme, the interacting modes can either be confined by homogeneous claddings[21]or by photonic band gap[22], this latter option avoiding aging problems via the reduction of the total aluminum content.In this letter we present an electrically injected Al-GaAs device that emits photons pairs at telecom wavelength and operates at room temperature. Our device, shown inFig. 1(a), has been engineered for simultaneous lasing around 785 nm and efficient type-II internal SPDC with photon pairs around 1.57 µm. Two Bragg mirrors provide both a photonic band gap vertical confinement for the laser mode -a Transverse Electric Bragg (TEB) mode -and total internal reflection claddings for the photon-pairs modes (one TE 00 and one TM 00 ). The nonlinear process is possible thanks to the interaction of the TEB pump mode and the two twin photon modes verifying the equations of energy conservation and type-II phase matching:hω TEB =hω TE00 +hω TM00 n TEB (ω TEB ) ω TEB = n TE00 (ω TE00 ) ω TE00 + n TM00 (ω TM00 ) ω TM00where ω i and n i (with i = TEB, TE 00 , TM 00 ) are the angular optical frequency and the effective index of the i-th mode. The simulated tuning curves based on Ref.[23,24], solutions of the above system, are shown inFig. 1(b). Due to the strong dispersion of the TEB mode arising from the proximity to the energy band gap of the waveguide core, small shifts of the laser wavelength arXiv:1312.6137v2 [quant-ph] | 10.1103/physrevlett.112.183901 | [
"https://arxiv.org/pdf/1312.6137v2.pdf"
] | 39,188,474 | 1312.6137 | 1dab3f4626219ce2e99f680a531dfe82405320f6 |
An electrically injected photon-pair source at room temperature
(Dated: May 13, 2014) 25 Dec 2013
Fabien Boitier
Laboratoire Matériaux et Phénomènes Quantiques
UMR 7162
Université Paris Diderot
Sorbonne Paris Cité
CNRS
Case courrier7021, 75205Paris Cedex 13France
Adeline Orieux
Laboratoire Matériaux et Phénomènes Quantiques
UMR 7162
Université Paris Diderot
Sorbonne Paris Cité
CNRS
Case courrier7021, 75205Paris Cedex 13France
Claire Autebert
Laboratoire Matériaux et Phénomènes Quantiques
UMR 7162
Université Paris Diderot
Sorbonne Paris Cité
CNRS
Case courrier7021, 75205Paris Cedex 13France
Aristide Lemaître
Laboratoire de Photonique et Nanostructures
CNRS-UPR20
Route de Nozay91460MarcoussisFrance
Elisabeth Galopin
Laboratoire de Photonique et Nanostructures
CNRS-UPR20
Route de Nozay91460MarcoussisFrance
Christophe Manquest
Laboratoire Matériaux et Phénomènes Quantiques
UMR 7162
Université Paris Diderot
Sorbonne Paris Cité
CNRS
Case courrier7021, 75205Paris Cedex 13France
Carlo Sirtori
Laboratoire Matériaux et Phénomènes Quantiques
UMR 7162
Université Paris Diderot
Sorbonne Paris Cité
CNRS
Case courrier7021, 75205Paris Cedex 13France
Ivan Favero
Laboratoire Matériaux et Phénomènes Quantiques
UMR 7162
Université Paris Diderot
Sorbonne Paris Cité
CNRS
Case courrier7021, 75205Paris Cedex 13France
Giuseppe Leo
Laboratoire Matériaux et Phénomènes Quantiques
UMR 7162
Université Paris Diderot
Sorbonne Paris Cité
CNRS
Case courrier7021, 75205Paris Cedex 13France
Sara Ducci
Laboratoire Matériaux et Phénomènes Quantiques
UMR 7162
Université Paris Diderot
Sorbonne Paris Cité
CNRS
Case courrier7021, 75205Paris Cedex 13France
An electrically injected photon-pair source at room temperature
(Dated: May 13, 2014) 25 Dec 2013
One of the main challenges for future quantum information technologies is miniaturization and integration of high performance components in a single chip. In this context, electrically driven sources of non-classical states of light have a clear advantage over optically driven ones. Here we demonstrate the first electrically driven semiconductor source of photon pairs working at room temperature and telecom wavelength. The device is based on type-II intracavity Spontaneous Parametric Down-Conversion in an AlGaAs laser diode and generates pairs at 1.57 µm. Time-correlation measurements of the emitted pairs give an internal generation efficiency of 7 × 10 −11 pairs/injected electron. The capability of our platform to support generation, manipulation and detection of photons opens the way to the demonstration of massively parallel systems for complex quantum operations.PACS numbers: 42.65. Lm, 03.67.Bg, 42.55.Px, Photons have a peculiar advantage in the development of quantum information technologies [1-3], since they behave naturally as flying qubits presenting a high speed transmission over long distances and being almost immune to decoherence[4,5]. The intrinsic scalability and reliability of integrated photonic circuits has recently given rise to a new generation of devices for quantum communication, computation and metrology[6]. Nevertheless even if great progress have been made in the manipulation[7,8]and detection[9]of nonclassical state of light on chip, a complete integration of the light source in the photonic circuitry stays one of the main challenges on the way towards large scale applications; such devices would have a clear advantage over optically driven ones in terms of portability, energy consumption and integration. Semiconductor materials are ideal to achieve extremely compact and massively parallel devices: concerning photon-pair sources, the bi-exciton cascade of a quantum dot has been used to demonstrate an entangledlight-emitting diode at a wavelength of 890 nm [10]. However, even if the use of a single emitter guarantees a deterministic emission, these devices operate at cryogenic temperature, greatly limiting their potential for applications.Optical parametric conversion offers an alternative approach. Despite its non-deterministic nature, this process is the most widely used to produce photon pairs for quantum information and communications protocols. Up to now, entangled photon pairs have been generated by optical pumping in passive semiconductor waveguides by exploiting four-wave mixing in Silicon[11]or SPDC in Aluminium Gallium Arsenide (AlGaAs)[12,13]. Thanks to its direct band gap, the latter platform presents an evident interest for the electrical injection. In order to deal with the isotropic structure of this crystal, several solutions have been proposed to achieve nonlinear optical conversion in AlGaAs waveguides[14][15][16][17][18]; among these, modal phase matching, in which the phase velocity mismatch is compensated by multimode waveguide dispersion, is one of the most promising to monolithically integrate the laser source and the nonlinear medium into a single device[19,20]. In this scheme, the interacting modes can either be confined by homogeneous claddings[21]or by photonic band gap[22], this latter option avoiding aging problems via the reduction of the total aluminum content.In this letter we present an electrically injected Al-GaAs device that emits photons pairs at telecom wavelength and operates at room temperature. Our device, shown inFig. 1(a), has been engineered for simultaneous lasing around 785 nm and efficient type-II internal SPDC with photon pairs around 1.57 µm. Two Bragg mirrors provide both a photonic band gap vertical confinement for the laser mode -a Transverse Electric Bragg (TEB) mode -and total internal reflection claddings for the photon-pairs modes (one TE 00 and one TM 00 ). The nonlinear process is possible thanks to the interaction of the TEB pump mode and the two twin photon modes verifying the equations of energy conservation and type-II phase matching:hω TEB =hω TE00 +hω TM00 n TEB (ω TEB ) ω TEB = n TE00 (ω TE00 ) ω TE00 + n TM00 (ω TM00 ) ω TM00where ω i and n i (with i = TEB, TE 00 , TM 00 ) are the angular optical frequency and the effective index of the i-th mode. The simulated tuning curves based on Ref.[23,24], solutions of the above system, are shown inFig. 1(b). Due to the strong dispersion of the TEB mode arising from the proximity to the energy band gap of the waveguide core, small shifts of the laser wavelength arXiv:1312.6137v2 [quant-ph]
One of the main challenges for future quantum information technologies is miniaturization and integration of high performance components in a single chip. In this context, electrically driven sources of non-classical states of light have a clear advantage over optically driven ones. Here we demonstrate the first electrically driven semiconductor source of photon pairs working at room temperature and telecom wavelength. The device is based on type-II intracavity Spontaneous Parametric Down-Conversion in an AlGaAs laser diode and generates pairs at 1.57 µm. Time-correlation measurements of the emitted pairs give an internal generation efficiency of 7 × 10 −11 pairs/injected electron. The capability of our platform to support generation, manipulation and detection of photons opens the way to the demonstration of massively parallel systems for complex quantum operations. Photons have a peculiar advantage in the development of quantum information technologies [1][2][3], since they behave naturally as flying qubits presenting a high speed transmission over long distances and being almost immune to decoherence [4,5]. The intrinsic scalability and reliability of integrated photonic circuits has recently given rise to a new generation of devices for quantum communication, computation and metrology [6]. Nevertheless even if great progress have been made in the manipulation [7,8] and detection [9] of nonclassical state of light on chip, a complete integration of the light source in the photonic circuitry stays one of the main challenges on the way towards large scale applications; such devices would have a clear advantage over optically driven ones in terms of portability, energy consumption and integration. Semiconductor materials are ideal to achieve extremely compact and massively parallel devices: concerning photon-pair sources, the bi-exciton cascade of a quantum dot has been used to demonstrate an entangledlight-emitting diode at a wavelength of 890 nm [10]. However, even if the use of a single emitter guarantees a deterministic emission, these devices operate at cryogenic temperature, greatly limiting their potential for applications.
Optical parametric conversion offers an alternative approach. Despite its non-deterministic nature, this process is the most widely used to produce photon pairs for quantum information and communications protocols. Up to now, entangled photon pairs have been generated by optical pumping in passive semiconductor waveguides by exploiting four-wave mixing in Silicon [11] or SPDC in Aluminium Gallium Arsenide (AlGaAs) [12,13]. Thanks to its direct band gap, the latter platform presents an evident interest for the electrical injection. In order to deal with the isotropic structure of this crystal, several solutions have been proposed to achieve nonlinear optical conversion in AlGaAs waveguides [14][15][16][17][18]; among these, modal phase matching, in which the phase velocity mismatch is compensated by multimode waveguide dispersion, is one of the most promising to monolithically integrate the laser source and the nonlinear medium into a single device [19,20]. In this scheme, the interacting modes can either be confined by homogeneous claddings [21] or by photonic band gap [22], this latter option avoiding aging problems via the reduction of the total aluminum content.
In this letter we present an electrically injected Al-GaAs device that emits photons pairs at telecom wavelength and operates at room temperature. Our device, shown in Fig. 1(a), has been engineered for simultaneous lasing around 785 nm and efficient type-II internal SPDC with photon pairs around 1.57 µm. Two Bragg mirrors provide both a photonic band gap vertical confinement for the laser mode -a Transverse Electric Bragg (TEB) mode -and total internal reflection claddings for the photon-pairs modes (one TE 00 and one TM 00 ). The nonlinear process is possible thanks to the interaction of the TEB pump mode and the two twin photon modes verifying the equations of energy conservation and type-II phase matching:
hω TEB =hω TE00 +hω TM00 n TEB (ω TEB ) ω TEB = n TE00 (ω TE00 ) ω TE00 + n TM00 (ω TM00 ) ω TM00
where ω i and n i (with i = TEB, TE 00 , TM 00 ) are the angular optical frequency and the effective index of the i-th mode. The simulated tuning curves based on Ref. [23,24], solutions of the above system, are shown in Fig. 1 from degeneracy produce a large wavelength separation between the generated photons. For this reason, taking into account the sensitivity range of our single-photon avalanche photodiodes, our spectral window to detect the two photons of each pair is limited to the region of frequency degeneracy.
The sample was grown by molecular beam epitaxy on a (100) n-doped GaAs substrate. It consists of a n-doped 6-period Al 0.80 Ga 0.20 As/Al 0.25 Ga 0.75 As Bragg reflector (lower mirror), a 298 nm Al 0.45 Ga 0.55 As core with a 8.5 nm Al 0.11 Ga 0.89 As quantum well (QW) in the middle, and a p-doped 6-period Al 0.25 Ga 0.75 As/Al 0.80 Ga 0.20 As Bragg reflector (upper mirror). The Bragg reflectors are gradually doped from 1 × 10 −17 cm −3 to 2 × 10 −18 cm −3 . A 230 nm GaAs cap layer (2 × 10 −19 cm −3 p-doped) protects the structure and facilitates the upper contact. Waveguides are fabricated using wet chemical etching to define 5.5-6 µm wide and 2 µm deep ridges along the (011) crystalline axis, in order to exploit the maximum nonzero nonlinear coefficient and a natural cleavage plane. Processing is completed by sample thinning and contact metallization with Au alloys. Samples are cleaved into 2 mm long stripes. age characteristics of the device as a function of the injected current. The device is mounted epi-side-up on a copper heat-sink; the temperature can be tuned between 15 and 40 • C with a standard Peltier module. In order to avoid unwanted thermal drifts, we employ current pulses of duration 120 ns and the repetition rate is set to 10 kHz. The laser internal peak power is evaluated by taking into account the modal reflectivity of the TEB mode (79%), numerically simulated by 2D FDTD. We observe that the turn-on voltage is ∼1.6 V, which is very close to the QW bandgap (∼1.58 eV), thus meaning that no current-blocking effects occur at the hetero-interfaces. The threshold current is around 420 mA, corresponding to a threshold currentdensity of 3.3 kA/cm 2 . This value is higher than state-of-the-art laser diodes in this spectral range [25] probably because of the crudely optimized doping of the Bragg mirrors. The spatial intensity distribution of the laser beam is studied by imaging the output facet; the recorded near-field distribution is reported in Fig. 2(b) together with the corresponding numerical simulation, showing a clear evidence of emission on the TEB mode. Figure 2(c) displays the laser emission intensity spectra as a function of heat-sink temperature, for an injected current of 650 mA. Apart from the longitudinal mode hopping -typical of laser diode -, the general trend corresponds to the theoretical temperature dependence of the QW bandgap (0.23 nm/ • C).
Optical propagation losses in the waveguide, a key issue for photon sources intended for quantum information, are measured via a standard Fabry-Perot technique [26]: the values obtained for the TE 00 and TM 00 modes in the telecom range are around 2 cm −1 . Similar measurements on an undoped waveguide giving a value of 0.1 cm −1 , the losses on the active device are mainly attributed to doping. The nonlinear optical properties of the sample are first explored through a Second Harmonic (SH) generation measurement performed without electrical injection. An input beam at the fundamental wavelength is polarized at 45 • and is injected in the waveguide in order to couple TE and TM modes simultaneously. Figure 3(a) shows a clear growth of the SH power for an input beam wavelength around 1.57 µm at T = 19 • C; the inset shows the expected quadratic dependence of the SH power with the fundamental power. The observed modulation as a function of the input wavelength is due to Fabry-Perot interferences between the waveguide facets. The solid curve results from a fit taking into account propagation losses and modal reflectivities of the three interacting modes [27]. The inferred internal SH generation efficiency is ∼ 35 %W −1 cm −2 and the FWHM of the phasematching bandwidth is ∼ 0.6 nm. Figure 3(b) reports the variation of the SH peak wavelength with temperature. The comparison between these data and those of Fig. 2(c) shows that the tunability curves of the laser emission and of the SH signal intersect in the explored temperature range.
In order to confirm the existence of a working region of the device and to demonstrate the emission of photon pairs around 1.57 µm, time-correlation measurements are performed under electrical injection (see Fig. 4(a)). The detected SPDC signal is optimized by tuning the temperature. Figure 4(b) shows a histogram of the detection time delays between TE and TM polarized photons at T = 25 • C. The sharp peak emerging from the background is a clear evidence of pairs production. From these data, taking into account the overall transmission along the optical path, we can estimate that the internal generation efficiency of the device is ∼ 7×10 −11 pairs per injected electron above the threshold. This value corresponds to a SPDC efficiency ∼ 10 −9 pairs/pump photon: these results are in agreement with our SH generation efficiency, letting expect ∼ 6 × 10 −9 pairs/pump photon, and consistent with our numerical simulation on an undoped structure giving ∼ 1.8 × 10 −8 pairs/pump photon for a 2 mm-long waveguide. Note here that such efficiency compare well with those obtained in a completely passive device based on the same kind of phase matching [17].
The Signal to Noise Ratio (SNR) is evaluated by taking the number of true coincidences within the FWHM of the peak over the background signal on the same time window; data presented in Fig. 4(b) give a SNR of 13.5, mainly limited by the luminescence noise of the device. In this respect, an optimization work leading to smaller laser threshold will be beneficial to reduce spurious luminescence and, thus, to increase the SNR. Our result enables to estimate the fidelity F to the Bell state |ψ + that can be produced with our device. Assuming that the source emits a Werner state [28,29]-which is reasonable since the noise is not polarized -, the associated density matrix isρ W = P |ψ + ψ + | + (1 − P )/4 × 1 1 with P = SN R/(2 + SN R). This leads to a maximal fidelity estimation F to |ψ + = (1 + 3P )/4 ∼ 90 %, which is compliant with future experimental violation of Bell's The emitted signal, collected through a 63X microscope objective, is focussed into a fibred 1.2 nm-FWHM interferential filter centered at 1.57 µm then sent in a fibred polarizing beam-splitter. The emerging TE and TM photons are detected with two InGaAs single-photon avalanche photodiodes having 20% detection efficiency and 50 ns gate, synchronized with the current pulses. A time-to-digital converter is used to analyze the time correlations between detected photons. (b) Time-correlation histogram of TE/TM photons around 1.57 µm at T= 25 • C. The sample is electrically injected with current pulses having an intensity of 700 mA, a duration of 60 ns and a repetition rate of 10 kHz. The data were accumulated during 1200 s with a sampling resolution of 162 ps. The inset shows a zoom on the sharp central peak.
inequality.
These results open the way towards large scale photonic-circuit-based quantum computation. Indeed one application of this source could be the controlled onchip electrical injection of an arbitrary number of heralded single photons or photon pairs on an arbitrary number of input modes of an integrated photonic circuit. This could be achieved by fabricating a monolithic device consisting of equally spaced laser diodes independently injected through a control electronics, which is allowed by the mature III-V technology. Interfacing it with multiport reconfigurable circuits [1] would allow practical medium size reconfigurable on-chip quantum photonics computation, such as boson sampling [30][31][32][33] and multiple photon quantum walks allowing medium size optical simulations [34].
This work was partly supported by Région Ile-de-France in the framework of C'Nano IdF with the TWI-LIGHT project. F.B. acknowledges the Labex SEAM 'Science and Engineering for Advanced Materials and devices' for financial support. A.O. et C.A. acknowledge the Délégation Générale de l'Armement (DGA) for financial support. We acknowledge G. Boucher and A. Eckstein for help with the experimental setup and A. Andronico for discussions on numerical simulations. S.D. and C.S. are members of Institut Universitaire de France.
PACS numbers: 42.65.Lm, 03.67.Bg, 42.55.Px, 42.82.-m
. 1. (Color online) Working principle of the device. (a) Schematic view of the source. The laser light emitted by the quantum well is converted into telecom photon pairs by intracavity spontaneous parametric down-conversion. (b) Simulated tuning curves of the type-II phase matching at T = 20 • C. Energy conservation imposes pair generation either on the thick or the thin branches of the curves.
Figure 2 (FIG. 2 .
22a) shows the internal peak power and volt-(Color online) Laser operation. (a) Voltage (cross) and internal optical power (circle) versus current. Measurements are performed with a current pulse duration of 120 ns and a repetition rate of 10 kHz for a heat-sink temperature of 19 • C. The solid line is a linear fit for current values above laser threshold. The laser diode has an electrical resistivity of 3.1 Ω, a turn-on voltage of 1.6 V, a laser threshold of 0.420 A and an efficiency of 267 mW/A. (b) Measured (top) and simulated (bottom) near-field emission of the laser mode. (c) Normalized laser emission intensity as a function of wavelength and heat-sink temperature measured with a fibred optical spectrum analyser. The dashed line shows the expected temperature variation of the QW bandgap.
online) Second harmonic generation. (a) SH spectrum as a function of the fundamental wavelength at T = 19 • C. The curve is a fit taking into account propagation losses and modal facet reflectivities. The inset shows the peak SH power as a function of the fundamental beam power. The solid line shows the expected squared power law function. (b) SH peak wavelength versus temperature. The solid line is a linear fit of the experimental data whereas the shaded area indicates the FWHM of the phase-matching bandwidth. The experimental slope of 0.09 nm/K is consistent with the theoretical slope of our numerical modeling (0.07 nm/K). The dashed line reports the expected variation of the QW bandgap presented in Fig. 2(c).
online) Coincidence measurement. (a) Experimental setup.
(b). Due to the strong dispersion of the TEB mode arising from the proximity to the energy band gap of the waveguide core, small shifts of the laser wavelength arXiv:1312.6137v2 [quant-ph] 25 Dec 2013Quantum well
Bragg mirror
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| [] |
[
"Multiaxial Fatigue Behaviour of A356-T6",
"Multiaxial Fatigue Behaviour of A356-T6"
] | [
"M Roy \nDept. of Materials Engineering\nThe University of British Columbia\nV6T 1Z4VancouverBCCanada\n",
"Y Nadot [email protected] \nInstitut PPRIME\nFUTUROSCOPE CHASSENEUIL Cedex\nCNRS\nUniversité de Poitiers -ENSMA -UPR\n3346 -Départment Mécanique des Matériaux -Téléport 2 -1 Avenue Clément Ader -BP 4019 -86961France\n",
"D M Maijer [email protected] \nDept. of Materials Engineering\nThe University of British Columbia\nV6T 1Z4VancouverBCCanada\n",
"G Benoit \nInstitut PPRIME\nFUTUROSCOPE CHASSENEUIL Cedex\nCNRS\nUniversité de Poitiers -ENSMA -UPR\n3346 -Départment Mécanique des Matériaux -Téléport 2 -1 Avenue Clément Ader -BP 4019 -86961France\n"
] | [
"Dept. of Materials Engineering\nThe University of British Columbia\nV6T 1Z4VancouverBCCanada",
"Institut PPRIME\nFUTUROSCOPE CHASSENEUIL Cedex\nCNRS\nUniversité de Poitiers -ENSMA -UPR\n3346 -Départment Mécanique des Matériaux -Téléport 2 -1 Avenue Clément Ader -BP 4019 -86961France",
"Dept. of Materials Engineering\nThe University of British Columbia\nV6T 1Z4VancouverBCCanada",
"Institut PPRIME\nFUTUROSCOPE CHASSENEUIL Cedex\nCNRS\nUniversité de Poitiers -ENSMA -UPR\n3346 -Départment Mécanique des Matériaux -Téléport 2 -1 Avenue Clément Ader -BP 4019 -86961France"
] | [] | Aluminum alloy A356-T6 was subjected to fully reversed cyclic loading under tension, torsion and combined loading. Results indicate that endurance limits are governed by maximum principal stress. Fractography demonstrates long shear mode III propagation with multiple initiation sites under torsion. Under other loadings, fracture surfaces show unique initiation sites coincidental to defects and mode I crack propagation. Using the replica technique, it has been shown that the initiation life is negligible for fatigue lives close to 10 6 cycles for combined loading. The natural crack growth rate has also been shown to be comparable to long cracks in similar materials. Nomenclature a Crack length A Average fatigue limit in torsion √ area Defect size parameter defined as the square root of a defect cross-sectional area C Average tensile fatigue limit J 1max Maximum hydrostatic stress J 2,a Amplitude of the second invariant of the deviatoric stress K Stress intensity factor ∆K Positive load cycle stress intensity range ∆K eff Effective stress range for crack growth K op Stress intensity required to open a crack N f , N Number of cycles to failure, number of cycles R Load ratio where R = σ min /σ max | 10.1111/j.1460-2695.2012.01702.x | [
"https://arxiv.org/pdf/1406.1204v1.pdf"
] | 118,546,090 | 1406.1204 | 42461ea63c273003435d6ef80370f5a34b9f5a6d |
Multiaxial Fatigue Behaviour of A356-T6
4 Jun 2014
M Roy
Dept. of Materials Engineering
The University of British Columbia
V6T 1Z4VancouverBCCanada
Y Nadot [email protected]
Institut PPRIME
FUTUROSCOPE CHASSENEUIL Cedex
CNRS
Université de Poitiers -ENSMA -UPR
3346 -Départment Mécanique des Matériaux -Téléport 2 -1 Avenue Clément Ader -BP 4019 -86961France
D M Maijer [email protected]
Dept. of Materials Engineering
The University of British Columbia
V6T 1Z4VancouverBCCanada
G Benoit
Institut PPRIME
FUTUROSCOPE CHASSENEUIL Cedex
CNRS
Université de Poitiers -ENSMA -UPR
3346 -Départment Mécanique des Matériaux -Téléport 2 -1 Avenue Clément Ader -BP 4019 -86961France
Multiaxial Fatigue Behaviour of A356-T6
4 Jun 2014* Corresponding author,A356-T6multiaxial fatiguecasting defectshigh cycle fatigue
Aluminum alloy A356-T6 was subjected to fully reversed cyclic loading under tension, torsion and combined loading. Results indicate that endurance limits are governed by maximum principal stress. Fractography demonstrates long shear mode III propagation with multiple initiation sites under torsion. Under other loadings, fracture surfaces show unique initiation sites coincidental to defects and mode I crack propagation. Using the replica technique, it has been shown that the initiation life is negligible for fatigue lives close to 10 6 cycles for combined loading. The natural crack growth rate has also been shown to be comparable to long cracks in similar materials. Nomenclature a Crack length A Average fatigue limit in torsion √ area Defect size parameter defined as the square root of a defect cross-sectional area C Average tensile fatigue limit J 1max Maximum hydrostatic stress J 2,a Amplitude of the second invariant of the deviatoric stress K Stress intensity factor ∆K Positive load cycle stress intensity range ∆K eff Effective stress range for crack growth K op Stress intensity required to open a crack N f , N Number of cycles to failure, number of cycles R Load ratio where R = σ min /σ max
Introduction
Cast aluminum alloys have been adopted over cast iron in many different applications to realize the benefits of weight reduction and part consolidation. Their relative low fatigue resistance can be an obstacle for structural components. A direct link between microstructure and fatigue resistance has been demonstrated by many authors [1,2,3,4,5,6]. Coarse microstructures, as characterized by large secondary dendrite arm spacing (λ 2 ), lead to diminished fatigue resistance. In defect-free material, the first cracks are known to initiate either inside the primary α-Al phase [7], at Fe-rich intermetallic particles [1], or in the secondary phase [4]. In material containing defects, fatigue behaviour is affected by gas pores, shrinkage porosity, and/or oxides. The studies on the high-cycle fatigue behaviour of cast aluminum primarily report that defects are of primary importance for a fatigue life assessment.
While it is clear that defects are observed at the initiation point on fracture surfaces, very few studies have identified the critical defect size that diminishes the fatigue limit. Brochu [7] suggested that the critical defect size is 150 µm for A357 and McDowell [4] found a critical defect size of 200 µm for A356-T6. With few exceptions, all of the preceding studies considered the fatigue behaviour under tensile loading. The multiaxial fatigue behaviour of A356-T6 was studied by De-Feng et al. [8] with thin-walled tubular specimens but under loading conditions leading to very low cycle fatigue; as such these results are not directly applicable to high cycle fatigue (HCF). Fan et al. [4] analyzed fatigue data under tension, torsion and combined loading and found that the equivalent von Mises deformation was able to correlate the results. However, these tests were conducted using deformation control leading to the possibility of a different material response.
The aim of the current study is address the scarcity of multiaxial HCF data for A356-T6. Tension, tension-torsion and torsion fatigue tests were performed for fully reversed loading and fatigue mechanisms were analyzed through fracture surface observation and replica studies. Basic fatigue criteria are also compared to discuss the multiaxial behaviour of the material.
Material and Experimental Methodology
The material employed in this study was strontium modified A356 (Al-7Si-0.3Mg) in the T6 condition that was taken from the melt supply of a North American aluminum alloy wheel manufacturer.
The typical chemical composition of this material is given in Table 1. While all specimens came from castings made with permanent steel dies, the majority of specimens came from a wedge-shaped casting produced for this investigation, and a lesser number were cut directly from an automotive wheel removed from the production stream. The wheel casting was actively cooled during solidification while the wedge casting was passively left to cool. Thus, these two castings represent a wide range of solidification conditions that lead to a range of defects and microstructure in the specimens cut from them. Considering the range of conditions, the fatigue behaviour characterized in this work is directly applicable to commercial castings.
Material preparation
The motivation for using a wedge-shaped casting (Fig 1a) was to create a cooling rate gradient during solidification that varied with height in the casting. A more refined microstructure was expected at the base of the casting where the cooling rate was the highest and a coarser microstructure occurred at the top where the cooling rate was the lowest. One half of the wedge was sectioned to produce tension-torsion fatigue specimens, while the other half was used to produce a mixture of solid tension, torsion and tension-torsion fatigue specimens (Fig 1b). Solid specimens were employed as opposed to tubular specimens in order to retain sampling within specific microstructural regions of the wedge.
Metallographic specimens were taken such that there was at least two samples per height where fatigue specimens were cut in the wedge (Fig 1a). Additional tension-torsion fatigue and metallographic specimens were sectioned from the spokes of a Low Pressure Die Cast (LPDC) automotive wheel. All specimens were subjected to a T6 heat treatment: solutionized at 538 • C for 3 hours, quenched in
Microstructure
Aluminum alloy A356 has an as-cast microstructure consisting of a primary dendritic structure, α-Al, filled with Al-Si eutectic phases. As the cooling rates decrease and solidification time increases, the dendritic structure coarsens resulting in larger secondary dendrite arm spacings. The T6 heat treatment serves to modify and refine the Al-Si eutectic structure. Depending on the casting conditions, the as-cast material may contain a variety of defects include gas and shrinkage porosity, and/or oxides which remain after heat treatment. Metallographic analysis did not present any evidence of Fe-rich intermetallic phases known to affect fatigue properties.
The secondary dendrite arm spacing (λ 2 ) and porosity were measured on each of the metallographic specimens from the wedge and wheel with an optical microscope and the Clemex Vision PE software.
Each measurement was made on a series of images representing a composite area of over 250 mm 2 .
The measured microstructure data is presented in Table 2 with data arranged according to height from the bottom of the wedge. The specimens were also grouped as four families of specimens: one for the wheel, and one each for the bottom, middle and top regions within the wedge. Due to the structure being equiaxed, λ 2 was taken to be the average distance between the centers of secondary dendrite arms based on a minimum of 60 measurements per sample. Equivalent pore diameter was calculated by equating the area of measured pores to equivalent circular geometry.
Fatigue test conditions
All fatigue tests were performed with fully reversed (R = −1) loading conditions under load control with a sinusoidal signal. The tension-torsion specimens were loaded in phase on an Instron servohydraulic test platform at 11 Hz. The pure torsion and pure tension specimens were tested on a Amsler-Vibraphore machine at 45 Hz. Testing was conducted using the step technique to target the fatigue limit (σ f , τ f ) at 10 6 cycles. For this technique, each specimen undergoes cyclic loading at an initial load level estimated based on previous testing experience. Samples that did not fail after 10 6 cycles were then cycled again at a higher stress amplitude. The details of the loading conditions and the number of loading steps experience by each sample are listed in Table 3. For specimens that failed before 10 6 cycles without a preceding loading step, the fatigue limit was found using a Basquin coefficient of 0.17 based on experimental results found in the literature [1,6,7,9,10]. For specimens that failed before 10 6 cycles after a minimum of one loading step, the fatigue limit was calculated as the average stress amplitude of the failure and prior step.
Fatigue Test Results & Criteria Comparison
The testing conditions and results are summarized in Table 3. In the majority of the fatigue tests, failure occurred after at least one loading step. However, four specimens failed during the first loading step (Specimens W2, W3, M7, and T6). There were two instances of specimens arriving at the same fatigue limit albeit with different numbers of steps: i) Specimens T1 and T2, both from the top of the wedge, failed at σ a , τ a = 65 MPa after 5 and 2 loading steps, respectively. Specimen T3 employed in the crack growth study (Section 6.2) was conducted as a run-out test and failed after 907 000 cycles at σ a , τ a = 65 MPa. Furthermore, samples W4 and M2 failed at σ a = 90 and τ a = 52 MPa after 5 and 2 loading steps, respectively. The non-ferrous nature of this material coupled with these limited observations provide speculative insight that the coaxing effect [11,12] is not prominent. While the coaxing effect may be present under other loading scenarios, it is asserted that the step testing method was able to ascertain the fatigue limit for 10 6 cycles to within 5 MPa. decreasing λ 2 from top to bottom of the wedge, the general trend is that specimens with the largest λ 2 exhibit the lowest fatigue limit. The exception was specimens extracted from the wheel, which showed a lower fatigue limit than the wedge material at a tension/torsion ratio slightly below pure torsion.
However, this difference is less than the range of results shown by the pure torsion testing of the wheel specimens. Fig 4 depicts σ a versus τ a for all of the loading scenarios (Table 3) origin for all ratios of tension to torsion.
Under pure torsion loading, the straight-gauged tension-torsion type specimens demonstrating a higher fatigue limit as compared to torsion-type specimens. The defect population at different levels of the wedge ( Table 2) demonstrate that the largest range of defect sizes was observed in the middle of the casting. This is where ∼80 % of the samples loaded under pure torsion were extracted.
During the torsion tests, it was also observed that multiple shear cracks were active at the same time, with the dominant crack not appearing until late in the test. The fractographic observations for this loading condition (Section 4) combined with the porosity measurements preclude specimen configuration being responsible for sample-type variations in fatigue limit under pure torsion.
In order to gauge the existing multiaxial fatigue criteria, the maximum fatigue limit from all data sets (i.e. each specimen family) at each tension/torsion ratio was determined to characterize the behaviour for A356-T6. The extracted fatigue limit data was compared to the Crossland and Maximum Principal Stress (MPS) fatigue criteria. This was done to compare the respective criteria to the entire breadth of experimental results. This is more representative of the microstructural differences in a cast component due to inherent variations in cooling rate. Furthermore, it has been shown that defects play a larger role in determining fatigue resistance as opposed to microstructure [4,13].
Crossland [14] proposed that the second invariant of the deviatoric stress and the maximum hydrostatic stress are the main parameters determining fatigue resilience:
J 2,a + ρJ 1max A(1)
where J 2,a is the amplitude of the second invariant of the deviatoric stress and J 1max is the maximum hydrostatic stress. The constant ρ is a function of the fatigue limit in pure tension and torsion and
A is the fatigue limit in pure torsion. Using the average σ f = 89 MPa and τ f = 67 MPa found in this study, ρ and A were determined to be 0.54 and 67 MPa, respectively.
The MPS criterion asserts that the maximum principal stress must be below a critical threshold such that:
σ 1max ≤ C (2)
where C is taken to be the average tensile fatigue limit, σ f = 89 MPa.
These two criteria are plotted versus σ a and τ a in Fig 4. Both the Crossland and the MPS criteria underestimate the measured fatigue limit for combined loading. It should be noted that the Crossland criteria approaches the MPS criteria under tension. As a result, the variance in the pure torsion results may show that the Crossland criteria applied to these results is overly conservative.
Since the Crossland criterion is reliant on the determination of an accurate fatigue limit under pure torsion, the MPS criteria is the most conservative criteria to describe these results [1,15].
Fracture surfaces
A representative summary of the fracture surfaces formed for each type of loading is presented in Fig 5. Under pure tension (Fig 5a), the fracture plane was found to be always normal to the direction of applied stress and thus, coincident with the maximum principal stress (σ 1max ) plane. Specimens from both the wheel and wedge casting exhibited this behaviour indicating that this observation is The fractographic observations indicate that the fracture morphology is independent of the sample family. Therefore, the microstructural features and the defect characteristics, such as λ 2 and average pore diameter (
Initiation sites
There are various multi-scale microstructural features that can cause fatigue initiation in A356-T6 The analysis performed on each sample followed a systematic methodology to reproducibly identify initiation sites and crack surface features. The methodology employed was as follows:
• Use optical microscopy to observe the fracture surface and identify the fatigue and fast-fracture zones. The fast-fracture zone refers to that portion of the crack surface which developed in the last fatigue cycles.
• Observe the fatigue zone using an SEM to determine the initiation site (1 mm 2 ) where river marks on the fracture surface converge.
• If a clear defect is identified, measured the size of the defect using the parameter √ area on the fracture surface. This is performed regardless of the position of the defect relative to the gage surface.
In a number of samples, the initiation site could not be accurately identified or characterized. This was true for pure torsion samples with multiple initiation sites, but also for multiaxial tests where model III shear cracking lead to contact and damage of fracture surfaces.
Due to the readily available tensile fatigue data for A356-T6, only two specimens were tested under these conditions. Each of these specimens exhibited a gas or shrinkage pore as the initiation site for the fatal crack. The first, M3 (Table 3) Under cyclic torsional loading, it is apparent that fatigue mechanisms are related to small, diffuse damage. This suggests that samples tested under pure shear conditions are more susceptible to distributed porosity as compared to the other loading scenarios. The lack of hydrostatic stress in this loading scenario may play a role in shear susceptibility. For steel, it has been shown that pure torsion, or a lack of hydrostatic stress, leads to small distributed shear cracks on the surface of the sample while the other loading states cause more localized fatigue damage [17,18]. However, when a crack in steel initiates on a shear plane, it bifurcates into mode I propagation after a relatively short length (100 -300 µm). In the present study, the pre-bifurcation crack length was much larger (on the order of millimeters as shown in Fig 5e).
Crack propagation
The propagation of cracks intersecting the surfaces of samples under different loading conditions was studied using the replica technique. In the replica technique, an elastomer is applied to the surface of a specimen at different cycle counts to make a negative imprint of the specimen surface [21]. The imprints, which show the various stages of crack evolution, are sputter-coated and analyzed via an SEM. In this study, replicas were applied to a tensile specimen that had an artificial defect introduced on its surface. This specimen is independent from those specimens listed in Table 3. The replica technique was also applied to a combined tension-torsion specimen (T3 in Table 3) to study natural crack propagation. When translating the replica measurements to the fracture surface, the specimen curvature was taken into account. Crack measurements where the crack length was greater than 75% of the replica perimeter have been omitted due to replica curvature complications.
Crack propagation under pure tension
The artificial defect specimen was drawn from the bottom of the wedge and had a 416 µm defect introduced to the middle of the gage section via Electro-Discharge Machining (EDM). This technique of generating artificial defects has been qualified in other crack propagation investigations [22,23].
The specimen was then cycled and cycled under pure tension at 90 MPa with the fatal crack localized at the site of the defect. This stress level is comparable to the highest fatigue limit found for the two other samples tested under pure tension (M3 and T5).
The resulting fatigue life of this sample, 673 000 cycles, indicates that the defect had an influence It is necessary to discard the first stages of crack propagation to study the evolution of an isolated crack far form the influence of the defect. The crack emanating from a spherical defect has a stress intensity factor that is unaffected by its presence at distances greater than 25% the radius of the defect [24,25]. Therefore, when the crack is deeper than 520 µm, the stress intensity factor can be calculated using Linear Elastic Fracture Mechanics (LEFM). As the crack front geometry is semi-circular, the appropriate shape factor is 2/π. The crack observed on Fig 7 indicates that crack path is perpendicular to the loading direction and locally influenced by the microstructure. Analysis including propagation rates are discussed further in Section 6.3. Damage accumulation (iii) was found to occur at the initiation zone and a 300 µm defect (iv) was found on the fracture surface.
Natural crack propagation under tension-torsion
presented in Fig 8. The crack path is regular and the propagation plane normal of both cracks is oriented at 31 • from the central axis of the specimen. This is consistent with other tests of the same type (Section 4). Fracture surface analysis revealed a gas pore with an equivalent diameter of 300 µm just below the gage surface. Since this specimen had the same fatigue limit (68 MPa) as a specimen where a defect could not be identified, it is assumed that this pore had minimal effect on the fatigue limit. The secondary crack observed on this sample initiated at a gas pore, visible on the surface of the specimen, near the beginning of loading and grew in the same orientation as the main crack.
The natural crack growth rate measurements were disregarded until the crack depth was 25% larger than the radius of the defect, and crack length measurements were only tabulated when the crack depth was greater than 625 µm (i.e. the depth of the defect was 500 µm from the surface). A similar practice was followed for the secondary crack. Due to the loading type, it was not possible to identify markings on the fracture surface and the exact shape of the crack in the bulk is unknown.
However, since the fracture plane is governed by the principal stress, it is assumed that the crack front is semi-circular and identical to the one observed under tension.
It has been observed by others that the first stage of crack propagation links the silicon particles in the brittle eutectic [1,26,4,10]. This is contrary to the findings of other studies [7,27] that have identified the first crack as appearing in the more ductile α-Al primary phase. The phase where crack nucleation occurred is not evident as the initiation zone of the replica at high magnification ( Fig 8) shows several cracks that are smaller than λ 2 . However, the very first stages of crack initiation represent a small fraction of the overall fatigue life for HCF and may be therefore discounted since the fatigue life is clearly governed by crack propagation rather than crack initiation. surements is ±5 µm. While the crack length data in pure tension is for a test with σ 1 = 90 MPa, the multiaxial test data is for σ 1 = 105 MPa. Assuming that σ 1 is the governing parameter for crack growth, these two tests are not strictly comparable as the stress state for each is different. However, this difference is relatively minor and the natural crack growth behaviour exhibits a very small difference in crack growth versus cycle count despite the different stress states. Therefore, the crack growth data in Fig 9 demonstrates the intrinsic short crack growth behaviour of this alloy. Consequently, the aggregate crack growth data has been used to fit an exponential relationship according to: of A356-T6 has demonstrated that fatigue initiation can be neglected for pure tension [4]. For fatigue lives close to 1 million cycles, the present findings assert that fatigue initiation may be considered negligible regardless of loading condition.
Comparison of natural and long crack propagation
a = α exp(βN )(3)
The crack growth rates are plotted against the amplitude of the stress intensity factor in Fig. 10. The crack growth rates for each stress state were determined from the fitted data (Eq. 3). The stress intensity factor was computed assuming a semi-circular crack front, where K = φσ 1 √ πa with φ = 0.643 [22,33] difference between ∆ K and ∆K eff for A356-T6. The tensile and multiaxial crack growth rates from the current study are approximately the same as that of ∆K eff Skallerud et al., which reinforces the short crack categorization microstructure, as it has been found appropriate to employ a closure-free K value to describe short crack behaviour [34,35].
The long crack growth data for A319 shows the same effective long crack threshold for A356-T6
as the reported effective long crack threshold is equal to 1.5 MPa √ m in A356-T6 [5,30,27]. This indicates that the natural crack growth rate is similar to the effective long crack growth rate. This is contrary to the short crack effect observed in many metallic materials [31,32], where effective long crack growth requires a completely positive load cycle (R > 0). The natural cracks in this study were cycled with fully reversed loading (R = −1) conditions.
With the exception of the pure torsion results, all of the preceding results and observations have indicated that the principal stress is the governing mechanical parameter for crack propagation even under multiaxial loading. Therefore, the Paris law may be used to describe natural crack growth in A356-T6 under multi-axial loading as:
da dN = 2 × 10 −10 ∆K 2(4)
where da/dN is the crack growth rate in meters per cycle and ∆K is the positive component of the load cycle in MPa √ m. As da/dN versus ∆K for both natural cracks analyzed via the replica technique (Fig 10) show good agreement with aggregate long crack data for similar alloys, the MPS assertion is further validated.
Summary and Conclusions
Employing the step technique, A356-T6 specimens with a wide range of microstructure were submitted to multiaxial HCF loading. The application of various fully reversed tension-torsion ratios generated fatigue limits that were compared to classical fatigue criteria. Fatigue cracks were found to initiate either on casting defects or inside the microstructure. Both scales are in competition for the localization of cyclic plastic deformation that induces the initiation of the crack that leads to failure.
When the crack initiates on a defect, it is typically of two different types: oxides or pores (gas or shrinkage). The distance to the free surface as well as the morphology are important parameters.
This study has provided the following conclusions: • For one million cycles, the fatigue life is governed by crack propagation rather than initiation life regardless of the loading condition.
• The natural crack growth rate has been found to be the same as for long cracks for similar materials.
Figure 1 :
1Wedge casting dimensions and specimen locations (a) and (b) the different types of fatigue specimens employed: tension-torsion (i), torsion (ii) and tension (iii) type.
Figure 2 :
2Typical A356-T6 microstructure showing α-Al, Al-Si eutectic phases and porosity. 60 • C water, and artificially aged at 150 • C for 3 hours.
Fig 2 shows optical micrographs of the typical microstructure.
Grouped according to family, Fig 3 depicts σ f versus τ f for each specimen. Following the trend of
Figure 3 :
3independent of family type. Considering the maximum fatigue limit for the entire experimental data set independent of specimen type and microstructure, σ f and τ f for R = −1 are approximately equidistant from the Fatigue limit grouped by material family type.
Figure 4 :Figure 5 :
45Fatigue test points compared to Crossland and Maximum Principal Stress criteria. independent of microstructure. Furthermore, SEM observations on gage section material far from the initiation site did not reveal any other cracks. Thus, multiple initiation sites did not manifest under pure tension, regardless of microstructure. Under combined tension-torsion loading, the fracture surface features are similar to pure tension including regular fracture planes with no bifurcation (Fig 5b) regardless of the sample location. SEM observation of the gage section far from the fracture surface reveals only small secondary cracks less than 50 µm long. The orientation of the fracture surface in Fig 5d is shown by the fracture surface normal. In this combined loading case where σ a = τ a , the fracture surface normal is oriented 26 • from the axis of the specimen. With the maximum principal stress acting at 31 • from the specimen axis, the fracture surface normal is close to being parallel. This difference of 5 • is the largest discrepancy observed over the 18 multiaxial specimens tested. Thus, the orientation of the macroscopic fracture plane under combined loading conditions correlates to the loading condition and more specifically, to the direction of the maximum principal stress (σ 1max ). Therefore, the tension component of the loading must play a major role in deciding the path of crack propagation. Under pure torsion, the failures surfaces observed were very different and showed no similarities to Failure types: (a) pure tension (b) combined tension-torsion (σ a = τ a ) (c) pure torsion (d) Macroscopic crack plane orientation combined tension-torsion (σ a = τ a ) (e) shear crack on the gauge section of the torsion sample far from the fracture surface. that of pure tension and combined tension-torsion loading. Fig 5c is an example of the fracture surfaces observed. The tortuous fracture surface has two major crack planes activated: one aligned with the axis of the specimen and the other one perpendicular to this axis. This highlights the difficulty in finding a unique initiation site on the fracture surface. Macroscopic observations confirmed by SEM analysis indicate that there are many different initiation sites over the periphery of the gage section and that different crack planes link together to form the final fracture surface. Remarkably, there is no evidence of macroscopic cracks growing in the direction normal to the σ 1max in the pure torsion scenario. For this loading condition, the crack path is governed by shear as opposed to principal stress. A clear demonstration of the dominance of shear is shown in Fig 5e where cracks propagate in shear mode from early in the fatigue life to the final failure. The very long crack observed in Fig 5e was observed to propagate throughout the test under shear mode III and showed no evidence of bifurcation under mode I. When bifurcation did occur on this sample, a new crack plane extended from the original mode III shear plane and linked with another mode III shear crack in the opposing activated shear plane. The sample shown in Fig 5e exhibited more than 10 other cracks similar to the one depicted, and additional smaller cracks observed in both shear mode III planes.
[1, 4 ,
416]. Defects such as gas pores, shrinkage pores, oxides and inter-metallic particles can initiate fatigue cracks. At smaller length scales, the fatigue properties are dominated by the α-Al and eutectic characteristics. SEM observations were performed on each sample to identify fatigue crack initiation sites and to examine the crack propagation surfaces.
Figure 6 :
6Fracture surfaces: (a) specimen T5, σ f = 85 MPa (b) specimen B3, σ f = τ f = 68 MPa (c) specimen W5, σ f = τ f = 68 MPa (d) specimen M4, σ f = τ f = 63 MPa (e) specimen T4, σ f = τ f = 51 MPa (f) specimen T7, τ f = 45 MPa.
, had a fatigue limit of 90 MPa and contained a gas pore with an equivalent diameter of 88 µm at the initiation site. The other specimen, T5, had a shrinkage pore with an equivalent diameter of 370 µm (Fig 6a) at the initiation site and exhibited a fatigue limit of 85 MPa. Similar to pure tension, the combined tension-torsion samples exhibited fracture surfaces that were easily characterized. Figs 6b-e are examples of typical defects that initiated the fatal crack. Fig 6b shows the fracture surface of sample B3 that had a σ f of 68 MPa. The initiation area on this specimen was readily identified, but a root initiating defect could not be found. Fig 6c is the fracture surface for sample W5, which had the same fatigue limit as B3. The fracture surface is less clear but it was possible to identify the initiation area where no clear defects were observed. The fracture surface instead shows friction-generated oxide associated with crack propagation. Fig 6d reveals a 500 µm subsurface pore just below the surface of specimen M4. Remarkably, the fatigue limit for M5 was 63 MPa, which is close to the highest value observed. This may be caused by the root defect location relative to the surface of the sample. When a propagating crack does not intersect the sample surface, it is not under ambient environmental conditions, but under vacuum instead. This effect has been investigated in a cast Al-Si-Cu alloy [19] where the fatigue life under vacuum when a surface defect is present is the same as when failure initiates from an internal defect of the same size. These observations have also been confirmed with nodular cast iron [20]. Therefore, a fatigue assessment of cast A356 should account for the position of the defect with respect to the free surface as there is different damage accumulation dependant on whether the defect lies on the surface or within the bulk. Fig 6e shows a typical shrinkage pore at the surface of specimen T4. In this case, a 500 µm pore decreased the σ f to 51 MPa. This result suggests that the surface shrinkage pore is more detrimental to the fatigue life than the previous subsurface pore of the same size. However, the environmental effect is not the only factor in this case because the morphology of the defects are different.The fracture surfaces of the samples tested in pure torsion were much more difficult to analyze than the pure tension samples. As shown inFig 5c,the macroscopic topology of the surface is very complex with two perpendicular mode III cracks activated and multiple initiation sites. Initiation is spatially distributed and as a result, the fracture surface is a combination of different cracks that have coalesced to cause final failure of the sample. Thus, it was difficult to identify a single initiating feature. However, when the fracture surface was less tortuous, there were features that could be analyzed(Fig 6f). For these samples, it was non-trivial to separate the fatigue zone from final failure zone.Fig 6f showstwo features that are presumed fatigue initiation sites. The feature closest to the center of the specimen is clearly shrinkage porosity with an equivalent diameter of 265 µm, located well away from the surface. The second feature is possibly an oxide film, but the features have been destroyed by damage/oxidation of crack surfaces under mode III propagation.
on the fatigue limit. The artificial defect is easily identifiable as the origin of failure on the fracture surface shown in Fig 7. The replica results in Fig 7 show the crack along the surface of the sample at 150 000, 224 000, and 300 000 cycles. Indicators identifying the extent of the crack have been added to the replica images. Through the superposition of the fracture surface and the replicas, it is possible to correlate the crack extent indicators with the fracture surface. The exact configuration of the crack with both the surface and inner shape of the crack front is tractable. As shown by the markers on Fig 7, cracking occurs on either side of the defect and the crack front passes through the defect early in crack propagation.
Figure 7 :
7T3 was cycled at σ a = τ a =65 MPa until failure at N f =907 000 cycles. The entire gauge surface of the sample was captured via replicas throughout the test. The replicas revealed that two cracks were present on the gage section at failure. It was also noted that this sample lacked the multiple initiations observed optically on samples following pure torsion loading. The critical crack was located in the middle of the gage section and the secondary crack was at the edge of the gage section. The replica results, showing the evolution of both the primary and secondary cracks, are Tensile specimen with a 416 µm artificial defect, σ a = 90 MPa and N f = 6.73 × 10 5 cycles. The top image is the fracture surface with crack front marking compared to surface replica at (i) N = 3 × 10 5 , (ii) N = 2.25 × 10 5 and (iii) N = 1.50 × 10 5 cycles. Replicas are oriented normal to the loading direction.
Figure 8 :
8Multiaxial specimen T3, σ a , τ a = 65 MPa and N f = 9.07 × 10 5 cycles with the secondary crack (i) at N = 2.25 × 10 5 , N = 6.75 × 10 5 and N = 8.75 × 10 5 . The primary crack (ii) is shown at N = 2.25 × 10 5 , N = 3.75 × 10 5 , N = 6.75 × 10 5 and N = 8.75 × 10 5 .
Fig 9
9shows the evolution of the crack length a versus the number of loading cycles, where a is half of the crack length measured from the replica. The estimated accuracy of these crack length mea-
Figure 9 :
9Crack growth law under tension and tension-torsion. where 2a is the surface crack length, N the number of cycles with α and β fitting coefficients. As the short crack growth behaviour shows inherent variations, this exponential fit characterizes the average short crack growth curve with α = 470.4 and β = 2.32 × 10 −6 (Fig 9). The crack evolution described by Eq. 3 lacks the non-linear behaviour associated with initiation. However, based on the degree this equation represents the crack growth data, it is apparent that propagation dominates the fatigue life for A356-T6. The investigation of precise correlations for the initiation stage of the fatigue life
Figure 10 :
10Comparison between effective long crack growth behaviour and natural crack growth behaviour.
•
Standard criteria like Maximum Principal Stress (MPS) and Crossland provide mostly conservative estimates of the experimental fatigue limits. However, MPS provides the closest fit to the mechanical results. • Cracking mechanisms are very different depending on the loading type. Under pure torsion the material shows multiple initiation sites and long mode III/I shear crack propagation. The bifurcation to mode I is not observed at the end of the fatigue life. Under tension or combined loading the initiation is shorter and more localized such that crack propagation starts directly in mode I.
Table 1 :
1A356 composition in wt-%Element
Si
Mg
Na
Sr
Range (wt-%) 6.5-7.5 0.25-0.4 ∼0.002 ∼0.005
Table 2 :
2Secondary dendrite arm spacing and porosity measurements for material tested.Family
Height (mm) λ 2 (µm)
Mean Porosity
Area (%) Max. Dia. (µm)
Wheel (W)
N/A
36.7 ±8
0.0603
58.6
Wedge Bottom (B)
28.75
39.5 ±7.6
0.1237
36
58.75
39.7 ±9.1
61
88.75
47.6 ±14.0
124
Wedge Middle (M)
118.75
57.2 ±17.9
0.1244
105
144.73
58.5 ±21.0
45
174.73
59.7 ±21.2
93
Wedge Top (T)
204.73
62.6 ±21.6 0.1232
48
234.73
72.2 ±28.2
126
The measured λ 2 increases from the bottom (39.5 µm) to the top (72.2 µm) of the wedge, coinciding
with the variation in cooling rate imposed by the casting practice. The λ 2 measured in the wheel
specimen (36.7 µm) was similar to that measured at the bottom location in the wedge. The area
percent porosity throughout the wedge was uniform ( 0.12 %) and approximate double that measured
in the wheel (0.06 %). The reduced porosity content is consistent with the in-line degassing practices
employed by the wheel manufacturer which substantially reduce the hydrogen content. A range of
mean maximum pore diameters was measured in the wedge with the largest diameters observed in
the upper portion of the bottom region. The increase in pore diameter at this location was a result of
the double ladle pouring procedure used to fill the casting where the casting began to solidify during
the delay between the end of pouring the first ladle and the start of pouring the second.
Table 3 :
3Test history for all fatigue specimens TT: tension-torsion, To: torsion, Te: tension. Tension-torsion specimens W1, W2, B5 and B6 were tested in pure torsion; M3 was tested in pure tension.Specimen
Loading (MPa)
Steps
N f (×10 5 )
Name Type † σ a
τ a
Number MPa/step
W1
TT
0
90
3
5
7.22
W2
TT
0
85
1 ‡
N/A
3.00
W3
TT
45
78
1 ‡
N/A
8.35
W4
TT
90
52
5
5
1.45
W5
TT
70
70
2
5
1.05
B1
TT
50
87
2
5
1.18
B2
TT
90
52
2
5
2.04
B3
TT
70
70
2
5
0.76
B4
To
0
70
2
5
3.98
B5
TT
0
100
3
10
1.51
B6
TT
0
110
2
10
8.83
M1
TT
50
87
2
5
2.31
M2
TT
90
52
2
5
4.01
M3
TT
95
0
3
5
0.79
M4
TT
65
65
2
5
2.46
M5
TT
70
70
3
5
5.26
M6
To
0
60
2
10
3.25
M7
To
0
55
1 ‡
N/A
2.27
M8
To
0
60
4
10
2.61
T1
TT
65
65
5
5
4.24
T2
TT
65
65
2
5
1.29
T3
TT
65
65
1 ‡
N/A
9.08
T4
TT
60
60
1 ‡
N/A
4.05
T5
Te
90
0
5
10
6.63
T6
To
0
50
1 ‡
N/A
7.33
T7
To
0
50
2
10
4.84
† ‡ Failure before 10 6 cycles
Table 2 )
2do not dictate crack paths under multiaxial loading. The overriding observation from the following analysis is that the macroscopic crack path is governed by σ 1max under multiaxial loading except for pure torsion where shear mode III dominates. This observation conflicts with the general mechanical analysis performed in Section 3 which showed that the MPS criteria provided the closest description of the multiaxial fatigue results even under pure torsion.
. The results for the two stress states are compared against studies performed on A319 [28] by Dabayeh et al., B319-T6 and unmodified A356-T6[29] by Chan et al., and peak-aged A356-T6 [2] by Skallerud et al. [2]. These four data sets for similar materials with different load ratios are plotted to provide references for short, long, and unstable crack growth behaviours. Dabayeh et al. and Skallerud et al. found the crack opening stress intensity K op needed to attain similar ∆K eff for A319 and A356 respectively. The data from Skallerud et al. clearly shows the
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| [] |
[
"QCD topology and axion's properties from Wilson twisted mass lattice simulations",
"QCD topology and axion's properties from Wilson twisted mass lattice simulations"
] | [
"A Yu Kotov [email protected] ",
"M P Lombardo [email protected] ",
"A Trunin ",
"\nJülich Supercomputing Centre\nLaboratory of Theoretical Physics\nForschungszentrum Jülich\nD-52428JülichGermany Bogoliubov\n",
"\nRussia INFN\nJoint Institute for Nuclear Research\n141980Dubna\n",
"\nSezione di Firenze\n50019Sesto Fiorentino (FI)Italy\n",
"\nSamara National Research University\n443086SamaraRussia\n"
] | [
"Jülich Supercomputing Centre\nLaboratory of Theoretical Physics\nForschungszentrum Jülich\nD-52428JülichGermany Bogoliubov",
"Russia INFN\nJoint Institute for Nuclear Research\n141980Dubna",
"Sezione di Firenze\n50019Sesto Fiorentino (FI)Italy",
"Samara National Research University\n443086SamaraRussia"
] | [
"The 38th International Symposium on Lattice Field Theory"
] | We present the results on topological susceptibility and chiral observables in = 2 + 1 + 1 QCD for temperature range 120 < < 600 MeV. The lattice simulations are performed with Wilson twisted mass fermions at physical pion, strange and charm masses. In high-region 300 MeV the chiral observables are shown to follow leading order Griffith analyticity, and the topological susceptibility follows a power-law decay as in the instanton dilute gas models. The measured topological susceptibility is used to estimate the mass of QCD axion. The resulting axion mass constraints are in agreement with our previous studies at higher pion masses. | 10.22323/1.396.0032 | [
"https://arxiv.org/pdf/2111.15421v1.pdf"
] | 244,729,803 | 2111.15421 | 52ff8b35d8d9142c7877c32202da42a59e37f24e |
QCD topology and axion's properties from Wilson twisted mass lattice simulations
LATTICE2021 26th-30th July, 2021
A Yu Kotov [email protected]
M P Lombardo [email protected]
A Trunin
Jülich Supercomputing Centre
Laboratory of Theoretical Physics
Forschungszentrum Jülich
D-52428JülichGermany Bogoliubov
Russia INFN
Joint Institute for Nuclear Research
141980Dubna
Sezione di Firenze
50019Sesto Fiorentino (FI)Italy
Samara National Research University
443086SamaraRussia
QCD topology and axion's properties from Wilson twisted mass lattice simulations
The 38th International Symposium on Lattice Field Theory
LATTICE2021 26th-30th July, 2021Zoom/Gather@Massachusetts Institute of Technology * Speaker
We present the results on topological susceptibility and chiral observables in = 2 + 1 + 1 QCD for temperature range 120 < < 600 MeV. The lattice simulations are performed with Wilson twisted mass fermions at physical pion, strange and charm masses. In high-region 300 MeV the chiral observables are shown to follow leading order Griffith analyticity, and the topological susceptibility follows a power-law decay as in the instanton dilute gas models. The measured topological susceptibility is used to estimate the mass of QCD axion. The resulting axion mass constraints are in agreement with our previous studies at higher pion masses.
Introduction
The topological aspects of QCD play a pivotal role in many theoretical problems. Prominent examples include the explanation of the meson mass [1,2] and (possible) solution to the strong CP problem leading to the prediction of a new particle, the QCD axion [3][4][5]. This new particle is also considered as a promising candidate for Dark Matter constituent. Another wide topic of interest is the interplay between topology and various mechanisms of chiral and axial symmetry breaking/restoration in hot QCD [6][7][8].
Lattice simulations were first applied to the problem of axion properties in Ref. [9]. In particular, the axion mass can be extracted from lattice data on high-temperature topological susceptibility under certain assumptions about axion cosmological evolution (post-inflationary scenario). First results were obtained in [9] in quenched approximation, followed by numerous works with dynamical quarks [10][11][12][13][14][15].
In this Proceeding we report the preliminary results of our ongoing project on simulation of finite-QCD with Wilson twisted mass fermions at the physical point. We extend our previous study on axions performed at higher than physical pion masses in [14,15]. We calculate the temperature dependence of several chiral observables, including chiral condensate and susceptibility, and relate them to high-temperature topological susceptibility via QCD symmetry relations. Then, using the observed value of Dark Matter density as an input, we obtain the lower limit on (post-inflationary) axion mass.
Lattice setup
We perform simulations with = 2 + 1 + 1 Wilson twisted mass fermions tuned at maximal twist [16,17]. The summary of our lattice ensembles are given in Table 1. Strange and charm quark masses are set to the physical values, and four different pion masses are available including the physical point. For lattice spacing and other parameters we rely on ETMC = 0 results [18,19]. We employ fixed-scale approach for finite-simulations: for each ensemble the lattice spacing is fixed, and the temperature is varied by lattice size in temporal direction . Thus we cover the temperature range approximately 120 600 MeV. Additional details on our lattice simulations can be found in [15,20].
Observables
We consider the following chiral observables:
• Chiral condensate ¯ = ¯ + ¯ = = 1 3 Tr −1 .
• Chiral susceptiblity = ¯ = disc + conn consisting from connected and disconnected parts.
• By combining chiral condensate ¯ and its susceptibility we introduce the new observ-
able ¯ 3 = ¯ − ,(1)
which is free from linear additive renormalization as well as from linear correction to scaling. For additional details on ¯ 3 and its properties we refer to [20,21].
In order to measure the topological susceptibility top we employ its relation to the disconnected chiral susceptibility disc via the QCD symmetry arguments [22][23][24]. In particular, the following continuum relation is valid:
top = 2 = 2 5,disc ,(2)
where is topological charge, and 5,disc is disconnected pseudo-scalar susceptibility. The direct measurement of 5,disc on the lattice is difficult due to large fluctuations. Instead, we note that after the chiral transition 5,disc becomes equal to disc . Then, Eq. (2) reads as
top ( ) = 2 disc = 2 (¯) 2 − ¯ 2(3)
defining the topological susceptibility in high-region. Finally, we note that Eqs. (2)-(3) are exact only in the continuum limit, meaning that fine lattices should be used in order to avoid large artifacts.
Results
We present the results on topological susceptibility measured according to Eq. (3) at the physical pion mass in Fig. 1. We compare it with the results obtained in other lattice approaches [11,12,25,26] and also with our previous study at higher pion masses [15]. In order to set the common scale for comparison, the results from non-physical pion masses are rescaled according to top ∝ 4 . Such behavior is predicted by dilute instanton gas model (DIGA) and can also be obtained from more general considerations based on the analyticity of chiral condensate in light quark mass [15]. Fig. 1 shows that different studies lead to similar results following the same trend, but still lacking complete numerical agreement.
In order to obtain a simple analytical expression for topological susceptibility, we fit it in Fig. 2 with DIGA-inspired high-temperature behavior top − . Table 1 corresponding to the same value of pion mass are treated equally. The data are well described by the power-law decay (4) in the region 300 MeV. For higher than physical pion masses the fits are performed over the combined data from all available ensembles (see Table 1). The data show no apparent lattice spacing dependence suggesting that artifacts are small, as was also confirmed in [15] by more detailed analysis.
The temperature dependence of the ¯ 3 (1) is shown in Fig. 3. First, we rescale with the leading order Griffith analyticity prediction ¯ 3 ∝ 6 . Then, we fit with the universal scaling behavior ¯ 3 ∝ ( − 0 ) − −2 , where 0 is fixed to the critical temperature 0 = 138 MeV in the chiral limit [20,21]. The critical exponents , and are fixed to represent 3D (4) universality class. As expected, the universal behavior sets in near the transition and remains up to 300 MeV. After that, rescaled data from different pion masses merge to the single curve, indicating simple Griffith analyticity behavior. It is intriguing that this change of trend in ¯ 3 ( ) coincides with the onset of DIGA-like behavior for the topological susceptibility top ( ) mentioned above, both occurring at approximately the same temperature 300 MeV. Once we determined the temperature dependence of topological susceptibility (4) in highregion, we can use it to estimate the axion mass [9,27]:
( ) = √︁ top ( ) .(5)
Since the exact value of the axion decay constant is unknown, we take relation (5) at two moments of time (or, equivalently, temperatures) corresponding to the evolution of axions in the early Universe and to present day. The two time moments are connected by the axion equation of motion, allowing to obtain today's axion density Ω as a function of its mass. Then, assuming that axions are responsible for the observed Dark Matter density Ω DM , the axion mass can finally be extracted. For detailed derivation we refer to the original works [9,27] (see also the review [28] and references therein). In particular, we use the result of Ref. [15] Ω = ( , , . . .)
− 3.053+ /2 2.027+ /2 ,(6)
where is a function of topological susceptibility parameters (4) (amplitude and power decay constant ) and of relevant cosmological constants. We plot the result (6), using the parameters extracted from our fits, in Fig. 4. For physical pion ensemble we also explore the limiting cases by increasing or decreasing the amplitude by factor 10 4 and setting the decay constant = 8 and = 4 corresponding to pure DIGA prediction and to very slow decay of topological susceptibility, respectively. The actual fraction of axion density in Ω DM is unknown, so the ratio Ω /Ω DM plays the role of a free parameter. By setting Ω = Ω DM we can obtain the lower limit on the axion mass. The curves in Fig. 4 corresponding to different pion masses lead to virtually the same value of axion mass. Indeed, as was shown earlier in Fig. 1, the results for topological susceptibility from different ensembles lie close to each other. So, in this preliminary analysis we retain the result of Ref. [15] for the lower bound on axion mass = 20(5) eV.
Summary
We measured chiral observables and topological susceptibility in the region 120 600 MeV. The temperature dependence of ¯ 3 (1) shows clear threshold at 300 MeV, above which a trend consistent with 3D (4) scaling gives way to a simple leading order Griffith analytic behavior. Around the same point 300 MeV the topological susceptibility starts to follow DIGA-like power-law decay. The high-topological results from different studies are in the same ballpark, but lacking complete quantitative agreement. Still, the final prediction for axion mass is rather insensitive to these differences. The same holds for its dependence on pion mass, once the appropriate scaling is applied.
Figure 1 :
1Topological susceptibility vs temperature obtained in this work and in Refs.[11,12,15,25,26].The results for non-physical pion masses are rescaled as top ∝ 4 .
Figure 2 :
2Fits of the topological susceptibility with power-law decay (4). All ensembles from
Figure 3 :
3¯ 3 vs temperature, also fitted with the 3D (4) scaling behavior ¯ 3 ∝ ( − 0 ) − −2 . For higher than physical pion masses the data are rescaled as ¯ 3 ∝ 6 .
Figure 4 :
4The axion fraction in Dark Matter vs the axion mass. For physical pion ensemble the parameters and are varied as indicated in the legend.
Table 1 :
1Parameters of = 2 + 1 + 1 lattice ensembles used for the analysis[18,19].
AcknowledgmentsThis work is partially supported by STRONG-2020 under grant agreement No. 824093, RFBR grant 18-02-40126, and by the "BASIS" foundation. Numerical simulations have been carried out on computational resources of CINECA (INFN-CINECA agreement project INF21_sim and ISCRA project IsB20), the supercomputer of Joint Institute for Nuclear Research "Govorun", and the computing resources of the federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC "Kurchatov Institute", http://ckp.nrcki.ru/.
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| [] |
[
"Supersymmetric Higgs-portal and X-ray lines",
"Supersymmetric Higgs-portal and X-ray lines"
] | [
"Hyun Min Lee \nDepartment of Physics\nChung-Ang University\n156-756SeoulKorea\n",
"Chan Beom Park \nSchool of Physics\nInstitute for Advanced Study\n130-722SeoulKorea, Korea\n\nAsia Pacific Center for Theoretical Physics\n77 Cheongam-Ro, Nam-Gu790-784PohangKorea\n\nDepartment of Physics\n790-784PostechPohangKorea\n\nKavli IPMU (WPI)\nThe University of Tokyo\n277-8583KashiwaChibaJapan\n",
"Myeonghun Park "
] | [
"Department of Physics\nChung-Ang University\n156-756SeoulKorea",
"School of Physics\nInstitute for Advanced Study\n130-722SeoulKorea, Korea",
"Asia Pacific Center for Theoretical Physics\n77 Cheongam-Ro, Nam-Gu790-784PohangKorea",
"Department of Physics\n790-784PostechPohangKorea",
"Kavli IPMU (WPI)\nThe University of Tokyo\n277-8583KashiwaChibaJapan"
] | [] | We consider a Dirac singlet fermion as thermal dark matter for explaining the Xray line in the context of a supersymmetric Higgs-portal model or a generalized Dirac NMSSM. The Dirac singlet fermion gets a mass splitting due to their Yukawa couplings to two Higgs doublets and their superpartners, Higgsinos, after electroweak symmetry breaking. We show that a correct relic density can be obtained from thermal freeze-out, due to the co-annihilation with Higgsinos for the same Yukawa couplings. We discuss the phenomenology of the Higgsinos in this model such as displaced vertices at the LHC. * | 10.1016/j.physletb.2015.03.046 | [
"https://arxiv.org/pdf/1501.05479v2.pdf"
] | 119,258,320 | 1501.05479 | 57bc3f02518ad618ae030dbe5b60e3466a16c112 |
Supersymmetric Higgs-portal and X-ray lines
29 Jan 2015
Hyun Min Lee
Department of Physics
Chung-Ang University
156-756SeoulKorea
Chan Beom Park
School of Physics
Institute for Advanced Study
130-722SeoulKorea, Korea
Asia Pacific Center for Theoretical Physics
77 Cheongam-Ro, Nam-Gu790-784PohangKorea
Department of Physics
790-784PostechPohangKorea
Kavli IPMU (WPI)
The University of Tokyo
277-8583KashiwaChibaJapan
Myeonghun Park
Supersymmetric Higgs-portal and X-ray lines
29 Jan 2015
We consider a Dirac singlet fermion as thermal dark matter for explaining the Xray line in the context of a supersymmetric Higgs-portal model or a generalized Dirac NMSSM. The Dirac singlet fermion gets a mass splitting due to their Yukawa couplings to two Higgs doublets and their superpartners, Higgsinos, after electroweak symmetry breaking. We show that a correct relic density can be obtained from thermal freeze-out, due to the co-annihilation with Higgsinos for the same Yukawa couplings. We discuss the phenomenology of the Higgsinos in this model such as displaced vertices at the LHC. *
Introduction
Dark matter (DM) is a main component of matter in the Universe, confirmed by various observations such as galaxy rotation curves, gravitational lensing. Moreover, it is supported by the measurement of Cosmic Microwave Background Radiation and Large Scale Structure, and so on. However, we have no clue as to the DM mass and interactions other than gravity. Therefore, direct detection on earth, indirect detection in the sky, and direct production at particle colliders have been thought to be complementary for identifying the nature of DM. In particular, indirect detections look for the remnants of annihilations or decays of DM through cosmic rays coming from galaxies and galaxy clusters.
There has recently been a lot of interest in light DM models, after new detection of X-ray line coming from galaxies and galaxy clusters mainly by the XMM-Newton observatory [1]. There are on-going debates on the possibility of explaining the X-ray line excess with thermal atomic transition [3], but it is worthwhile to take it to be a signal for DM and study the consequences of decaying or annihilating DM models [4][5][6][7].
Motivated by a toy model suggested by one of us [5], we consider a concrete model for explaining the X-ray line with the magnetic dipole moment of a weakly interacting massive particle (WIMP) in the context of a generalized next-to-minimal supersymmetric standard model (NMSSM) with an additional Dirac singlet superfield, dubbed as Dirac NMSSM [8,9]. Unlike the toy model where a discrete Z 2 symmetry for stabilizing DM is broken by a small amount at the cutoff scale [5], the corresponding discrete parity, i.e. R-parity, in the supersymmetric (SUSY) version is assumed to be exact. Then, a singlet Dirac fermion or two Majorana fermions called the singlinos, introduced in the Dirac NMSSM, is the DM candidate, and it gets a small mass splitting for the X-ray line energy at 3.55 keV due to its small Yukawa couplings to the MSSM Higgses and their superpartners. In this case, a tiny magnetic transition dipole moment for decaying DM generates the X-ray line by the small Yukawa couplings of the singlinos. We regard the model as a SUSY Higgs-portal in the limit that gauginos, squarks and sleptons are heavy enough. We also include the effects of non-decoupled gauginos on the mass splitting of Higgsinos or singlinos. The lightness of Higgsinos and singlinos can be ensured by a chiral symmetry such as Peccei-Quinn symmetry while gauginos could be relatively light due to R-symmetry.
The Dirac singlet fermion can keep in thermal equilibrium with the Standard Model (SM) particles at freeze-out, due to the co-annihilation with the Higgsino-like fermions. Consequently, we show that the correct relic density can be attained, being compatible with the X-ray line. In the limit of heavy gauginos, the mass splitting of Higgsino states is about keV scale as for the singlino fermions, so neutral or charged Higgsinos decay into a singlino +Z * /W * , leaving a displaced vertex due to small Yukawa couplings of singlinos. We discuss the possibility of discovering Higgsinos at the LHC in this new topology.
The paper is organized as follows. We begin with the model description of the SUSY Higgs-portal for the low-energy mass spectra of DM and Higgsinos. Then, we present the results of the magnetic transition dipole moment between two singlinos at one loop in our model and show the parameter space that is consistent with both the energy and flux for the X-ray line. In turn, we discuss the bound from the DM relic density and its compatibility with the X-ray line. Finally, conclusions are drawn.
Supersymmetric Higgs-portal
The dark sector couples to the SM particles only through the Higgs and its superpartners. As an example, we consider an extension of the Higgs sector in the MSSM with a Dirac singlet chiral superfield containing two additional singlet superfields, S andS. We assume that the gauginos as well as the superpartners of quarks and leptons are sufficiently heavy so that they are not relevant for our discussion. Meanwhile, we also discuss the effects of non-decoupled gauginos in this section.
The part of the superpotential containing only Higgs doublets, H u and H d , and the singlet chiral superfields are
W 0 = λ S SH u H d + λSSH u H d + M S SS + µ H H u H d + µ S S + µSS.(1)
In this model, the Dirac singlet chiral superfield communicates with the SM only through the Higgs and Higgsino interactions. As for the Dirac singlino, the model can also be called the Higgino portal. In a Peccei-Quinn (PQ) symmetric realization of the above superpotential, the cubic couplings for the singlet chiral superfields are forbidden, while the bare Higgsino and singlino mass terms and the singlet tadpole terms can be generated after a spontaneous breaking of the PQ symmetry by non-renormalizable interactions with PQ-breaking fields.
When there is a U (1) S global symmetry or a Z 2 symmetry distinguishing S andS, the operatorSH u H d is forbidden. This case corresponds to the Dirac NMSSM that was discussed in Ref. [8,9], where even after integrating out the singlet scalar masses with keeping their fermion partners, the resulting Higgs potential gets an additional quartic potential, |λ S H u H d | 2 , and increases the Higgs mass as compared to the MSSM. When the singlet symmetry is broken spontaneously or explicitly, we can write a small Yukawa coupling forS such that |λS| |λ S | = O(1). Then, the feature of the Dirac NMSSM for the Higgs mass can be maintained.
On the other hand, if |λ S | and |λS| are comparable, the PQ symmetry only does not distinguish between S andS. Thus, there is no obvious reason to forbid Majorana mass terms such as S 2 andS 2 in the superpotential. But, if we ignore those Majorana mass terms under the assumption that such a flavor structure in the dark sector is determined by a flavor symmetry for singlinos at a high energy scale, we can explain a small mass splitting and a small flux required for the X-ray line for |λ S |, |λS| 1, as will be discussed in the next section. The neutralino mass matrix containing the gauginos in MSSM is given in the basis
( B, W 0 , H 0 d , H 0 u , S, S ) by M χ 0 = M 1 0 − 1 2 g v d 1 2 g v u 0 0 0 M 2 1 2 gv d − 1 2 gv u 0 0 − 1 2 g v d 1 2 gv d 0 −µ eff − 1 √ 2 λ S v u − 1 √ 2 λSv u 1 2 g v u − 1 2 gv u −µ eff 0 − 1 √ 2 λ S v d − 1 √ 2 λSv d 0 0 − 1 √ 2 λ S v u − 1 √ 2 λ S v d 0 M S 0 0 − 1 √ 2 λSv u − 1 √ 2 λSv d M S 0 ,(2)
where
v 2 u + v 2 d = v 2 (246 GeV) 2 , tan β = v u /v d ,
and the effective µ parameter is given by
µ eff = µ H + λ S S + λS S .
In order to keep a small mass splitting between singlinos, we take the gauginos to be much heavier than Higgsinos and singlinos, namely, M 1,2 µ eff , M S . Then, we can consider only the 4 × 4 sub-matrix for Higgsinos and singlinos and a mass splitting of the Dirac singlinos is attributed to a small coupling between Higgsinos and singlinos. Then, keeping all the other superpartners of the SM heavy enough, we can call the model the SUSY Higgs-portal.
In the limit of M 1,2 µ eff , M S , the mass eigenvalues for Higgsino-like neutralinos are
m χ 0 1 = µ eff − 1 8 (v u + v d ) 2 g 2 (M 1 − 2µ eff ) (M 1 − µ eff ) 2 + g 2 (M 2 − 2µ eff ) (M 2 − µ eff ) 2 , m χ 0 2 = µ eff + 1 8 (v u − v d ) 2 g 2 (M 1 + 2µ eff ) (M 1 + µ eff ) 2 + g 2 (M 2 + 2µ eff ) (M 2 + µ eff ) 2 ,(3)
while those for singlino-like neutralinos are, for λ S , λS 1,
m χ 0 3 = M S + 1 8 (λ S − λS) 2 (v u − v d ) 2 µ eff + M S − (v u + v d ) 2 µ eff − M S + 1 16 (λ S − λS) 2 (v 2 u − v 2 d ) 2 µ 2 eff (µ 2 eff − M 2 S ) 2 g 2 (M 1 + 2M S ) (M 1 + M S ) 2 + g 2 (M 2 + 2M S ) (M 2 + M S ) 2 , m χ 0 4 = M S + 1 8 (λ S + λS) 2 (v u + v d ) 2 µ eff + M S − (v u − v d ) 2 µ eff − M S − 1 16 (λ S + λS) 2 (v 2 u − v 2 d ) 2 µ 2 eff (µ 2 eff − M 2 S ) 2 g 2 (M 1 − 2M S ) (M 1 − M S ) 2 + g 2 (M 2 − 2M S ) (M 2 − M S ) 2 .(4)
Consequently, the mass differences between the nearest neutralinos are
∆m 21 ≡ m χ 0 2 − m χ 0 1 ≈ 1 4 v 2 g 2 M 1 + g 2 M 2 ,(5)
and
∆m 34 ≡ m χ 0 3 − m χ 0 4 ≈ 1 2 v 2 µ 2 eff − M 2 S (λ 2 + − λ 2 − )M S − (λ 2 + + λ 2 − )µ eff sin(2β) + 1 8 (λ 2 + + λ 2 − ) v 4 cos 2 (2β)µ 2 eff (µ 2 eff − M 2 S ) 2 g 2 M 1 + g 2 M 2 ,(6)
where
λ ± ≡ 1 √ 2 (λ S ± λS).(7)
We note that as far as λ S and λS are comparable, ∆m 34 is positive so that χ 0 4 is the Lightest Supersymmetric Particle (LSP) and χ 0 3 is Next-LSP in our model. When the singlino mass splitting is about a few keV and µ eff M S ∼ 100 GeV, the Yukawa couplings, λ S and λS, should be of order 10 −5 and the gaugino masses should be greater than about 1 TeV, unless there is an accidental cancellation. 1 In Fig. 1, we have illustrated the masses of Higgsino-like neutralinos as a function of the gaugino mass. For gaugino masses being greater than 1 TeV and DM mass being 300 GeV, the Higgsino mass splitting ∆m 21 is less than 6 GeV. In Fig. 2, we show the parameter space for the Yukawa couplings and the mass parameters satisfying the mass splitting between singlino-like neutralinos, |∆m 34 | = 3.55 keV, in blue dashed line.
� �� �� �� �� × �� -� �� × �� -� �� × �� -� �� × �� -� �� × �� -� �� × �� -� μ ��� -� � (���) |λ � | � � =������� λ � _ =����λ � ��� ��� ��� ��� ��� ��� ��� ��� ���� ���� ���� ���� ���� ���� ���� � �� (���) �-λ � _ /λ � λ � _ ≃���×�� -� � μ ��� ≃� � +������� Figure 2: (Left) Parameter space of µ eff − M S vs |λ S |. (Right) Parameter space of µ eff − M S vs 1 − λS/λ S .
In both figures, the parameter space explaining the X-ray line is shown between two red solid lines and the X-ray line energy at 3.55 keV is obtained for the blue dashed line. We took m H ± = 1 TeV, M 1 = 0.5M 2 = 3 TeV and tan β = 10.
The mass eigenstates are found by H 0
d = i N i1 χ 0 i , H 0 u = i N i2 χ 0 i , S = i N i3 χ 0 i and S = i N i4 χ 0
i . For M 1,2 µ eff , M S , and λ S , λS 1, they read
H 0 d = 1 √ 2 χ 0 1 + 1 √ 2 iγ 5 χ 0 2 − √ 2 4 λ − v u − v d µ eff + M S − v u + v d µ eff − M S iγ 5 χ 0 3 + √ 2 4 λ + v u − v d µ eff − M S − v u + v d µ eff + M S χ 0 4 ,(8)H 0 u = − 1 √ 2 χ 0 1 − 1 √ 2 iγ 5 χ 0 2 + √ 2 4 λ − v u − v d µ eff + M S + v u + v d µ eff − M S iγ 5 χ 0 3 − √ 2 4 λ + v u − v d µ eff − M S + v u + v d µ eff + M S χ 0 4 ,(9)S = − √ 2 4 (v u − v d ) λ − µ eff + M S + λ + µ eff − M S χ 0 1 − √ 2 4 (v u + v d ) λ + µ eff + M S + λ − µ eff − M S iγ 5 χ 0 2 − 1 √ 2 iγ 5 χ 0 3 + 1 √ 2 χ 0 4 ,(10)S = √ 2 4 (v u − v d ) λ − µ eff + M S − λ + µ eff − M S χ 0 1 − √ 2 4 (v u + v d ) λ + µ eff + M S − λ − µ eff − M S iγ 5 χ 0 2 + 1 √ 2 iγ 5 χ 0 3 + 1 √ 2 χ 0 4 .(11)
The chargino mass matrix in the basis (
W + , H + u , W − , H − d ) is M χ ± = M 2 1 √ 2 gv u 1 √ 2 gv d µ eff .(12)
Then, for M 2 µ eff , the mass eigenvalues for charginos are
m χ ± 1 = µ eff − µ eff + M 2 sin(2β) M 2 2 − µ 2 eff · sgn(µ eff )m 2 W , m χ ± 2 = M 2 + M 2 + µ eff sin(2β) M 2 2 − µ 2 eff · m 2 W .(13)
The mass difference between the lighter Higgsino-like neutralino and the lighter chargino is
m χ ± 1 − m χ 0 1 = m 2 W 2g 2 1 + sin(2β) g 2 M 1 + 1 + (1 − 2sgn(µ eff )) sin(2β) g 2 M 2 .(14)
In Fig. 1, we have also shown the masses of Higgsino-like chargino as a function of the gaugino mass. In this example, the mass difference between the lighter Higgsino-like neutralino and the Higgsino-like chargino is less than 2.5 GeV for gauginos being heavier than 1 TeV.
Before closing the section, we remark on the scalar sector of the SUSY Higgs-portal. Due to the small Yukawa couplings of the singlinos, their superpartners, singlet scalars, have only a small mixing with the MSSM Higgs fields so the Higgs sector is MSSMlike. Moreover, it would be hard to produce singlet scalars at the current LHC at a detectable level. On the other hand, singlet scalars may induce the self-annihilation and co-annihilation of DM through the s-channels. The self-annihilation is suppressed due to small Yukawa couplings outside the resonance, while the co-annihilation is sizable enough to keep DM in thermal equilibrium, as will be discussed in Sec. 4.
Magnetic dipole moments and the X-ray line
The magnetic (transition) dipole moments can be obtained for either Majorana [5,10] or Dirac [11] singlet DM. 2 In the SUSY Higgs-portal model, the heavier singlino χ 0 3 is a Majorana fermion that has an almost degenerate mass with the lighter singlino χ 0 4 . As gauginos, leptons, and squarks are assumed to be decoupled in our model, only charged Higgs and W -boson loops contribute to the magnetic dipole moment. In order to compute the magnetic transition dipole moment for singlinos, we choose the non-linear R ξ gauge [12], in which γ − W ± − G ∓ interactions are absent for the charged unphysical Goldstone boson G ± . In this case, we have to deal with only the Yukawa interactions for Goldstone bosons, thus simplifying the calculation.
The singlino Yukawa interactions with charged Higgs (H ± ) and charged Goldstone (G ± ) are
−L S = sin β χ − 2 P L (λ S S + λS S )H − + cos β χ + 2 P L (λ S S + λS S )H + − cos β χ − 2 P L (λ S S + λS S )G − + sin β χ + 2 P L (λ S S + λS S )G + + h.c.(15)
Then, from Eqs. (10) and (11), we get
−L S = χ − 2 (f 3L P L + f 3R P R ) χ 0 3 H − + χ − 2 (f 4L P L + f 4R P R ) χ 0 4 H − + h.c. + (H − → G − , sin β → − cos β, cos β → sin β) + h.c. + · · · ,(16)
where
f 3L = iλ − sin β, f 3R = −iλ − cos β, f 4L = λ + sin β, f 4R = λ + cos β.(17)
On the other hand, the interactions between the W -boson and singlino-like neutralinos come from the mixing with Higgsinos, given as follows.
− L V = g √ 2 χ − 2 γ µ P L H 0 d W − µ + g √ 2 χ + 2 γ µ P L H 0 u W + µ + h.c.(18)
Then, from Eqs. (8) and (9), we get the singlino-like interactions to the W -boson as
− L V = χ − 2 γ µ (g 3L P L + g 3R P R ) χ 0 3 W − µ + χ − 2 γ µ (g 4L P L + g 4R P R ) χ 0 4 W − µ + h.c. + · · · ,(19)
where
g 3L = i 4 gλ − v u − v d µ eff + M S − v u + v d µ eff − M S , g 3R = − i 4 gλ − v u − v d µ eff + M S + v u + v d µ eff − M S , g 4L = 1 4 gλ + v u − v d µ eff − M S − v u + v d µ eff + M S , g 4R = 1 4 gλ + v u − v d µ eff − M S + v u + v d µ eff + M S .(20)
Therefore, the magnetic transition dipole moment, generated from charged Higgs, Goldstone, and W -boson loops, is given by
L mdm = ef χ 2m χ 0 3 χ 0 4 iσ µν χ 0 3 F µν ,(21)
where
f χ ≡ f H χ + f G χ + f W χ with f H χ = − λ + λ − 16π 2 cos 2β 1 0 dx m 2 χ 0 3 x(1 − x) m 2 χ 0 3 x 2 + (m 2 H ± − m 2 χ 0 3 )x + m 2 χ ± 2 (1 − x) , f G χ = + λ + λ − 16π 2 cos 2β 1 0 dx m 2 χ 0 3 x(1 − x) m 2 χ 0 3 x 2 + (m 2 W − m 2 χ 0 3 )x + m 2 χ ± 2 (1 − x) , f W χ = − λ + λ − 32π 2 cos 2β m 2 W (µ 2 eff + M 2 S ) (µ 2 eff − M 2 S ) 2 1 0 dx m 2 χ 0 3 x(x + 2) m 2 χ 0 3 x 2 + (m 2 W − m 2 χ 0 3 )x + m 2 χ ± 2 (1 − x)
. (22) We note that due to the interchange between cos β and sin β, the unphysical Goldstone contribution is of the same magnitude but the opposite sign as compared to the charged Higgs contribution. For µ eff M S and m H ± ∼ m χ ± 2 , the W -boson loops tend to be suppressed by m 2 W . But, for µ eff ∼ M S , which is necessary for the co-annihilation of DM as will be discussed in the next section, the W -boson loops give a dominant contribution to the magnetic dipole moment of DM.
We take two singlino-like neutralinos to be lighter than Higgsino-like neutralinos and almost degenerate in mass. Then, the heavier singlino χ 0 3 can decay into the lighter one χ 0 4 through the transition magnetic moment or the mixing with Higgsinos. The decay modes are χ 0 3 → χ 0 4 γ and χ 0 3 → χ 0 4 νν, where neutrinos in the latter channel is due to the off-shell Z-boson. The energy of the monochromatic photon coming from χ 0 3 → χ 0 4 γ is given by
E γ m χ 0 3 − m χ 0 4 for m χ 0 3,4 E γ . For |m χ 0 3 − m χ 0 4 | m χ 0 4
, the decay rates of the heavier singlino are
Γ( χ 0 3 → χ 0 4 γ) = e 2 f 2 χ m χ 0 3 π 1 − m χ 0 4 m χ 0 3 3 ,(23)
and
Γ( χ 0 3 → χ 0 4 νν) = |v 34 | 2 G 2 F m 5 χ 0 3 10π 3 1 − m χ 0 4 m χ 0 3 5 ,(24)
where
v 34 ≡ N 31 N 41 − N 32 N 42 ≈ − 1 2 v 2 cos 2β M 2 S λ + λ − .(25)
Due to an extra factor (∆m 34 ) 2 , the decay rate for χ 0 3 → χ 0 4 νν is suppressed as compared to the one for χ 0 3 → χ 0 4 γ. Thus, it is sufficient to consider only the decay mode χ 0 3 → χ 0 4 γ to determine the decay rate of DM.
Suppose that the heavier singlino constitutes a fraction of the total DM by r ≡ Ω χ 0 3 /Ω DM . Then, for the X-ray line at 3.55 keV, we need to take the necessary value of the lifetime of the heavier singlino to be τ χ 0 3 = 0.20-1.8 × 10 28 sec (7.1 keV/m χ 0 3 )r [1,5], which is equivalent to Γ χ 0 3 = 0.36 -3.3 × 10 −52 GeV (m χ 0 3 /7.1 keV)r −1 . For comparably small λ S and λS, and a small mass splitting between singlinos, two singlinos contribute to the relic density equally, that is, r = 1/2. In Fig. 2, we show the parameter space for the mass splitting µ eff − M S vs |λ S | or 1 − λS/λ S , satisfying the X-ray line flux (in the region between two black solid lines) and the X-ray line energy (in the blue dashed line). Therefore, the singlino Yukawa couplings of order 10 −5 required for the X-ray line energy is consistent with the X-ray line flux, as far as both Yukawa couplings are of similar size, that is, λS/λ S 0.97 − 0.99.
Dark matter relic density
Depending on the singlino Yukawa couplings to Higgsinos, λ S and λS, the singlino DM may be in thermal equilibrium with the SM particles due to self-annihilation and/or coannhiliation with charged and neutral Higgsinos [7]. The annihilation channels for singlinos are χ 0 3,4). In the case with small λ S and λS, the self-annihilation cross sections would be too small to make DM in thermal equilibrium with the SM particles. However, DM can keep in thermal equilibrium until freeze-out, through the scattering off of the SM particles or due to a sizable co-annihilation with neutral or charged Higgsino by crossing symmetry [13]. In this case, we can obtain a correct relic density for DM, after Higgsinos are decoupled from the SM bath and decay into DM. Therefore, we need λ S , λS 10 −5 for thermal DM [7]. The mass splitting between singlinos is 3.55 keV, so it can be ignored in computing the relic density.
i χ 0 j → ff , ZH 0 (h 0 ), W + H − , W + W − , χ 0 i χ 0 1,2 → ff and χ 0 i χ ± 2 → ff (i, j =
The relic abundance is given by
Ω DM = 8.8 × 10 −11 GeV −2 √ g * ∞ x f dx σ eff v x −2 ,(26)
where g * is the effective number of relativistic degrees of freedom at freeze-out and x ≡ m DM /T which read x f ≈ 20 at freeze-out temperature. The effective cross section is a weighted average of the annihilation cross sections for the co-annihilating particles and is given [7,13] by Figure 3: Parameter space of m DM vs µ eff −M S , satisfying the relic density within Planck 3σ [15].
σ eff v = i,j σ ij w i w j ( i w i ) 2 ,(27)��� ��� ��� ��� ��� ��� ��� ��� � � � � � �� �� � �� (���) μ ��� -� �� (��� ) Ω �� � � (������ �σ)
where
w i ≡ (1 + ∆ i ) 3/2 e −x∆ i , ∆ i ≡ m i − m DM m DM ,(28)
and σ ij v = σ ij x −n with n = 0 (1) for s-wave (p-wave) annihilation.
For small λ S and λS, the effective annihilation cross section is dominated by the annihilation of neutral or charged Higgsino so we find that
σ eff v = 1 4( i w i ) 2 (σ χ 0 1 χ 0 1 w 2 χ 0 1 + σ χ 0 2 χ 0 2 w 2 χ 0 2 + 2σ χ ± 1 χ ∓ 1 w 2 χ ± 1 + 2σ χ 0 1 χ ± 1 w χ 0 1 w χ ± 1 + 2σ χ 0 2 χ ± 1 w χ 0 2 w χ ± 1 ),(29)where i w i = 1 2 w χ 0 1 + 1 2 w χ 0 2 + w χ ± 1 + 1.(30)
In the limit of decoupled gauginos, w ≡ w χ 0 1 ≈ w χ 0 2 ≈ w χ ± 1 , so we obtain [7]
σ eff v = σ H H w 2 (w + 1 2 ) 2 ,(31)
where
σ H H = 81g 4 + 12g 2 g 2 + 43g 4 2048πµ 2 eff , w = µ eff M S 3/2 exp −x µ eff M S − 1 .(32)
In this case, the relic density can be determined from Eqs. (26) and (31) and the parameter space for the DM mass and the mass splitting between Higgsinos and DM is shown in Fig. 3.
Below the red region, the relic density is smaller than the lower end of the Planck 3σ values so the parameter space in that region is consistent with those obtained for explaining the X-ray line in Fig. 2.
If the Higgsino mass splitting is not ignorable, the contribution of the lighter neutral/charged Higgsino to the effective annihilation cross section gets larger, while the one of the heavier Higgsino gets smaller. But, overall, the effective annihilation cross section would increase due to the lighter charged Higgsino. Therefore, the difference between the averaged Higgsino mass and the singlino can be larger than µ eff − M S as shown in Fig. 3.
Collider searches
When the gauginos are heavy enough, the mass splitting between neutral Higgsinos is of about keV scale, being as small as the one between singlinos, and the charged Higgsino is almost degenerate in mass with the neutral Higgsinos. On the other hand, the difference between Higgsino and singlino masses should be less than about 10 GeV for the co-annihilation with singlino DM. Thus, neutral Higgsinos and charged Higgsino decay dominantly into singlinos with three-body modes such as χ 0 2 → χ 0 1 νν, χ 0 1,2 → χ 0 3,4 Z * and χ ± 2 → χ 0 3,4 W ± * , as well as the modes containing Higgs fields. Since the singlino Yukawa couplings are small for the X-ray line, the charged/neutral Higgsinos decay modes into singlino plus off-shell W/Z leave displaced vertex plus missing energy [7]. When the decay length of the charged Higgsino is between about 1 cm and 100 m, there is a bound on the mass of the charged Higgsino from the disappearing tracks at the LHC [16]. In our case, when the charged Higgsino decays mostly into singlino plus off-shell W , the decay length of the charged Higgsino is about O(mm-m) as in Ref. [7]. Thus, a certain parameter space of a small mass splitting can be constrained. However, the LEP, Tevatron and the LHC Run I [17] are not sensitive enough to rule out the neutral Higgsinos.
When Higgsinos have a sizable mass splitting due to the non-decoupling effect of gauginos, the heavier neutral Higgsino can decay into the lighter neutral or charged Higgsino with a sizable branching fraction, and the charged Higgsino can decay into neutral Higgsinos as well. In this case, since Higgsinos have gauge interactions, there is no displaced vertex. However, depending on the mass splitting of the Higgsinos, missing energy plus collimated leptons at the primary vertices can be a signature. In order for the gauginos not to give a large contribution to the singlino mass splitting, their contribution to the Higgsino mass splitting is less than 5 GeV. In this case, the situation would be better due to larger efficiency of the lepton momentum cuts, as compared to the Higgsinos with keV mass splitting. The detailed discussion on the search for almost degenerate Higgsinos is outside the scope of this work, so it is left for a future publication [14].
Conclusions
We have considered a Dirac singlet fermion or singlinos with small mass splitting as thermal DM in the SUSY Higgs-portal model. In order to explain the X-ray line excess observed from the sky, we introduced small singlino Yukawa couplings with Higgses and Higgsinos, the SUSY version of Higgs-portal couplings, and showed that the mass splitting of 3.55 keV is made and at the same time a tiny magnetic transition dipole moment between the Majorana components of the singlino is generated. The singlino mass splitting requires gaugino masses to be heavier than about 1 TeV, leading to almost degenerate Higgsinos. The thermal production of the singlino DM restricts the Higgsino masses to be not greater than about 10 GeV as compared to the singlino masses. New search strategies for almost degenerate Higgsinos at the LHC Run II and future colliders are needed to probe the SUSY Higgs-portal models.
Figure 1 :
1Masses of Higgsino-like states as a function of gaugino mass M 1 . We ignored the mixing effect with singlinos and took tan β = 10 and µ eff > 0.
In the case of cancellation, the Yukawa couplings, λ S and λS, can be of order one so they can be used to increase the Higgs mass[14].
Similar studies on magnetic dipole moments have been done in light of the X-ray line in Ref.[6,7].
AcknowledgmentsThe work of HML is supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2013R1A1A2007919). MP was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan and acknowledges support from the Korea Ministry of Science, ICT and Future Planning, Gyeongsangbuk-Do and Pohang City for Independent Junior Research Groups at the Asia Pacific Center for Theoretical Physics.
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| [] |
[
"Simulating multi-exit evacuation using deep reinforcement learning",
"Simulating multi-exit evacuation using deep reinforcement learning"
] | [
"Dong Xu \nSchool of Geographic Sciences\nKey Laboratory of Geographical Information Science (Ministry of Education)\nEast China Normal University\nShanghaiChina\n",
"Xiao Huang \nDepartment of Geography\nGeoinformation and Big Data Research Laboratory\nUniversity of South Carolina\nSCUSA\n",
"Joseph Mango \nSchool of Geographic Sciences\nKey Laboratory of Geographical Information Science (Ministry of Education)\nEast China Normal University\nShanghaiChina\n",
"Xiang Li \nSchool of Geographic Sciences\nKey Laboratory of Geographical Information Science (Ministry of Education)\nEast China Normal University\nShanghaiChina\n",
"Zhenlong Li \nDepartment of Geography\nGeoinformation and Big Data Research Laboratory\nUniversity of South Carolina\nSCUSA\n"
] | [
"School of Geographic Sciences\nKey Laboratory of Geographical Information Science (Ministry of Education)\nEast China Normal University\nShanghaiChina",
"Department of Geography\nGeoinformation and Big Data Research Laboratory\nUniversity of South Carolina\nSCUSA",
"School of Geographic Sciences\nKey Laboratory of Geographical Information Science (Ministry of Education)\nEast China Normal University\nShanghaiChina",
"School of Geographic Sciences\nKey Laboratory of Geographical Information Science (Ministry of Education)\nEast China Normal University\nShanghaiChina",
"Department of Geography\nGeoinformation and Big Data Research Laboratory\nUniversity of South Carolina\nSCUSA"
] | [] | Conventional simulations on multi-exit indoor evacuation focus primarily on how to determine a reasonable exit based on numerous factors in a changing environment. Results commonly include some congested and other under-utilized exits, especially with massive pedestrians. We propose a multi-exit evacuation simulation based on Deep Reinforcement Learning (DRL), referred to as the MultiExit-DRL, which involves in a Deep Neural Network (DNN) framework to facilitate state-to-action mapping. The DNN framework applies Rainbow Deep Q-Network (DQN), a DRL algorithm that integrates several advanced DQN methods, to improve data utilization and algorithm stability, and further divides the action space into eight isometric directions for possible pedestrian choices. We compare MultiExit-DRL with two conventional multi-exit evacuation simulation models in three separate scenarios: 1) varying pedestrian distribution ratios, 2) varying exit width ratios, and 3) varying open schedules for an exit. The results show that MultiExit-DRL presents great learning efficiency while reducing the total number of evacuation frames in all designed experiments. In addition, the integration of DRL allows pedestrians to explore other potential exits and helps determine optimal directions, leading to a high efficiency of exit utilization. | 10.1111/tgis.12738 | [
"https://arxiv.org/pdf/2007.05783v1.pdf"
] | 220,495,754 | 2007.05783 | a8384d716724fc9cea2ffe66aa64f22e2da082fb |
Simulating multi-exit evacuation using deep reinforcement learning
Dong Xu
School of Geographic Sciences
Key Laboratory of Geographical Information Science (Ministry of Education)
East China Normal University
ShanghaiChina
Xiao Huang
Department of Geography
Geoinformation and Big Data Research Laboratory
University of South Carolina
SCUSA
Joseph Mango
School of Geographic Sciences
Key Laboratory of Geographical Information Science (Ministry of Education)
East China Normal University
ShanghaiChina
Xiang Li
School of Geographic Sciences
Key Laboratory of Geographical Information Science (Ministry of Education)
East China Normal University
ShanghaiChina
Zhenlong Li
Department of Geography
Geoinformation and Big Data Research Laboratory
University of South Carolina
SCUSA
Simulating multi-exit evacuation using deep reinforcement learning
1 Preprint submitted to Transactions in GIS 2agent-based modelingdeep reinforcement learningpedestrian simulationartificial intelligencemulti-exit simulation
Conventional simulations on multi-exit indoor evacuation focus primarily on how to determine a reasonable exit based on numerous factors in a changing environment. Results commonly include some congested and other under-utilized exits, especially with massive pedestrians. We propose a multi-exit evacuation simulation based on Deep Reinforcement Learning (DRL), referred to as the MultiExit-DRL, which involves in a Deep Neural Network (DNN) framework to facilitate state-to-action mapping. The DNN framework applies Rainbow Deep Q-Network (DQN), a DRL algorithm that integrates several advanced DQN methods, to improve data utilization and algorithm stability, and further divides the action space into eight isometric directions for possible pedestrian choices. We compare MultiExit-DRL with two conventional multi-exit evacuation simulation models in three separate scenarios: 1) varying pedestrian distribution ratios, 2) varying exit width ratios, and 3) varying open schedules for an exit. The results show that MultiExit-DRL presents great learning efficiency while reducing the total number of evacuation frames in all designed experiments. In addition, the integration of DRL allows pedestrians to explore other potential exits and helps determine optimal directions, leading to a high efficiency of exit utilization.
Introduction
Indoor pedestrian simulation in evacuation studies is one of the most critical components that has been receiving great attention due to its potential of rescuing people in any case of emergency (Sun andLi 2011, Chen andFeng 2009). A reasonable simulation should be able to guide the pedestrian to leave fast and safely through exits of the indoor environments. In order to achieve this objective, pedestrians need to choose the best route to pass through, usually depending on the number of sub-exits present in a particular complex. This problem can be regarded as a navigation planning problem where the shortest path can be therefore chosen by using sampling-based algorithms (Kuffner andLaValle 2000, Karaman andFrazzoli 2011) or geometry-based algorithms (Geraerts 2010, Kallmann 2014. The pedestrians are expected to reach the final exits in the shortest time as long as they have followed the calculated path. However, these assumptions are only effective and efficient in the environment where the number of pedestrians and sub-exits is low. In actual sense, there many complex indoor environments with a large number of pedestrians. In such circumstances, the evacuation process becomes more complex due to overcrowding, starting from the nearest sub-exits to the final exits. This overcrowding problem is usually due to the narrow widths of the sub-exits, which pose challenges for pedestrians to leave in a rapid manner. Moreover, this problem becomes more complex if there is a variation of the sub-exit widths with uneven distribution of the pedestrian. Therefore, how to reasonably allocate pedestrians to different exits is still a valid and challenging question in pedestrian evacuation simulation studies.
The existing multi-exit selection research can be broken into two categories of studies. The first category is focused on the building design, aiming at proposing a more reasonable and more efficient multi-exit layouts (Seyfried et al. 2009, Choi et al. 2014). However, the models designed from these studies are unable to accommodate pedestrian behaviors that are proven to be significant during the evacuation process (Bode and Codling 2013). The second is focused on designing a multi-exit selection model using realistic experimental simulation (Haghani et al. 2015, Heliövaara et al. 2012, Wagoum et al. 2017. As contrary to the first category, the models designed using the second approach often require high preliminary costs. Diverging from these two main streams, other researchers started to utilize virtual simulation to describe the multi-exit simulation process of pedestrians by setting motion in models to guide the pedestrian navigation via movement strategies. Following this direction, Zia and Ferscha (2009) proposed three different strategies to analyze the evacuation efficiency of pedestrians in the room. In the same domain, Hao et al. (2014) also proposed a mix-strategy to combine distance-based and time-based strategy based on an improved dynamic parameter model.
In the past decade, deep learning as one branch of the machine learning methods emerged to provide remarkable modeling results such as in image classification (Krizhevsky et al. 2012), object tracking (Bertinetto et al. 2016), and natural language processing (Sutskever et al. 2014). As one type of deep learning, Deep Reinforcement Learning (DRL) has also made significant successes in defeating human players in many competitions such as in Go and Arita games (Mnih et al. 2013). Unlike supervised learning and unsupervised learning, DRL learns a mapping (i.e., from state to action) to maximize the long-term reward (Sutton and Barto 2018). Depending on its learning methods, DRL can be classified into two categories of Deep Q-Network (DQN) and Policy Gradient (PG). To date, there are several advanced DRL algorithms which include: Rainbow DQN (Hessel et al. 2018), Proximal Policy Optimization (PPO) (Schulman et al. 2017), Soft Actor Critic (SAC) (Haarnoja et al. 2018), and Deep Deterministic Policy Gradient (DDPG) (Lillicrap et al. 2015). All these DRL algorithms have been applied in different areas, such as in robot visual navigation. Typical examples in this area of application are found in some studies such as the study by Gupta et al. (2017) that proposed a cognitive mapper and planner based on a neural architecture to navigate the robot using the first-person views as the state space. Another study by Zhang et al. (2017) presented the application of DRL-based robot navigation in a mazelike environment, where the robot has shown a robust adaptive ability for new environments. Apart from the examples of robotic applications, DRL algorithms have been also used in local motion planning as well. In this case for example, Lee et al. (2018) proposed a deep reinforcement learning method based on the AC framework for crowd simulation. In the same field, Long et al. (2018) presented a decentralized collision avoidance policy by directly detecting the environment using raw sensors and feeding the measurements to a DRL network. Despite these achievements, the research on multi-exit navigation based on DRL needs to be explored further in different environments.
In this paper, we focus on exploring the potential of DRL in multi-exit navigation within certain designed room environments. Each pedestrian is defined as a disc, characterized by its position and velocity. The simulation process in our research can be regarded as a series of ordered frames. At a given frame, the pedestrians' positions are determined based on their velocities (described in detail in Section 3). A total of eight possible movement directions are provided for each pedestrian, and the desired direction is determined using the DRL algorithm. The Optimal Reciprocal Collision Avoidance (ORCA) is adopted to avoid collision during the pedestrians' movement. Unlike other studies, we do not explicitly assign a reasonable exit. In our method, each pedestrian is considered as an intelligent person with self-judgment, who takes desired actions to facilitate its evacuation after millions of iterations of interacting with the environment. To illustrate the advantages of our method, we compared our proposed method with two popular methods from (Zheng et al. 2015) and (Guo et al. 2012). The main contributions of this paper are threefold: 1) we designed a hierarchical model to handle multi-exit navigation simulation in micro-scale, a more realistic method in describing pedestrian's behaviors compared with other simulations in mesoscale; 2) we developed a new method which abstracts the grey image of the room as the state space by distinguishing the target pedestrian and other pedestrians using greyscale values. Compared with the traditional ray casting methods that consider external environments (Lee et al. 2018), our method can noticeably speed up the simulation process; 3) we integrated Rainbow DQN, an effective DRL method that combines several advanced DQN algorithms. This training design largely increases the stability of the neural network and the speed to reach the convergence point. Furthermore, the Rainbow DQN in our method can be updated in One-Step or N-Steps without collecting a complete training trajectory. Thus, it provides a privilege in data collection compared with the traditional PG methods.
This work is organized into seven sections, including this part of the introduction. Section 2 summarizes the existing literature of multi-exit navigation and deep reinforcement learning algorithms. Section 3 presents more information about ORCA and the Rainbow DQN algorithm in network architecture, state-space, action-space, and reward functions. Section 4 introduces three designed scenarios to compare our proposed MultiExit-DRL model with two traditional models based on mathematical strategy navigation. It also describes the settings of the relevant parameters and computer configurations used when developing our model. Section 5 presents, analyzes, and discusses the results of the proposed MutliExit-DRL model. Section 6 discusses the advantages and limitations of the proposed model before concluding our work in Section 7.
Related Work
Pedestrian simulation
Nowadays, Pedestrian simulation has received large attention in geographic information science (GIS) field and widely used in path planning (Wu et al. 2007), emergency decision making (Tashakkori et al. 2015), and human behavior analysis (Li et al. 2010). Pedestrian simulation can be divided into two groups depending on their simulation scales: these are, the meso-scale simulation and the micro-scale simulation. Cellular Automata (CA) model, as a classic meso-scale model, has been applied to simulate the movement of pedestrians and reached decent results (Chopard and Droz 1998). In general, a CA model divides the research region into a series of regular cells, in which each of them holds different statuses, including "available", "occupied", and "obstacle". A pedestrian can move to the adjacent cells in the next frame if there are available cells surrounding the current cell. In the past decades, the advancement of the computational power of rough modeling facilitated many CA related studies (Dijkstra et al. 2001, Burstedde et al. 2001, Pelechano and Malkawi 2008. However, the disadvantages of the CA model are also known due to the limitation of rough modeling. In particular, the CA model discretizes pedestrian movements, falling short of describing the dynamic behaviors of pedestrians (Cao et al. 2016).
In light of this issue, micro-scale pedestrian simulation has gradually become one of the research hotspots. Unlike the rough modeling, commonly used in meso-scale simulations, microscopic models accurately simulate the environment and pedestrians with finer details via geometric expressions, thus to allow pedestrians to move arbitrarily within the configuration space. The aforementioned advantage made micro-scale simulation more popular and has attracted many types of research in different directions. For instance, Helbing and Molnar (1995) proposed a classic social force model (SFM), which characterizes the interactions between pedestrians using a mathematical model with pedestrian behaviors. The SFM has received widespread attention due to its convenient and effective nature (Mehran et al. 2009, Hou et al. 2014, Karamouzas et al. 2017. Nevertheless, SFM and other force-based methods fail to guarantee complete collision avoidance (Curtis and Manocha 2014). In comparison, geometrical based methods are considered to be better methods to avoid collisions and have been widely applied in robotics and motion simulation (Curtis and Manocha, 2014). Velocity Obstacle (VO) method is one of the traditional geometry-based collision avoidance method which is able to forecast potential collisions by calculating the VO region (Fiorini and Shiller 1998). As an improvement of VO approach, a reciprocal n-body collision avoidance approach (ORCA) by Van Den Berg et al. (2011) introduced a low-dimensional linear program for collision-free movements. As reviewed in many other pieces of literature, the ORCA approach proved to simulate collision-free actions for thousands of agents in a few milliseconds. Due to its efficiency, the ORCA has been applied in many applications with different modifications. For instance, Alonso-Mora et al. (2013) guaranteed a smooth and collision-free motion for non-holonomic robots via distributed collision avoidance simulation under non-holonomic constraints. Golas et al. (2013) presented an effective algorithm to perform long-range collision avoidance in crowd simulation by adopting a novel metric to quantify the smoothness of trajectories. Bareiss and van den Berg (2015) proposed a generalized reciprocal collision avoidance algorithm, not only for presenting an extension to control obstacles but also for generating collision-free motions under a non-linear and non-homogeneous system. Given the superiorities of ORCA method and its flexibility in different applications, we adopted it in our study as the local pedestrian simulation method to avoid collision during the simulation of the multi-exit evacuation process.
Multi-exit selection
Evacuation simulation has always been a hot topic in the fields of safety and robotics. Compared with common pedestrian simulations, evacuation simulation focuses more on the pedestrian behaviors given in various internal or external factors during the evacuating process. For example, Parisi and Dorso (2005) applied SFM to explore the 'faster is slower' effect under different degrees of panic or fear. Frank and Dorso (2011) investigated the efficiency of evacuation by placing some obstacles near the exit. Ben et al. (2013) presented an agent-based modeling approach to describe individual behavior during the evacuation. Wang et al. (2015a) developed a novel multi-agent based congestion evacuation model to investigate individual panic behavior at the individual level and further analyze the evacuation efficiency if a virtual leader is added in the model.
As an important component in evacuation simulation, multi-exit selection aims to mimic or analyze pedestrians' behaviors under complex local environments. In some studies, the evacuation data are extracted from realistic scenarios, and the decision rules are specified accordingly (Guo et al. 2012, Haghani andSarvi 2016). Such approaches can model pedestrian behavior in an accurate manner, however, it is difficult to archive since the realization of the realistic scenarios is always elusive. Alternatively, other studies focus on investigating the evacuation process via computer simulations by controlling the designed environments (Bode and Codling 2013, Kinateder et al. 2014, Davidich et al. 2013, Han et al. 2017, Zhou et al. 2019. Nonetheless, the choice of exits in this direction also remains to be a great challenge, especially when dealing with complex environments (Zheng et al. 2017, Zia andFerscha 2009).
In most studies, the choices of exits are determined by several factors, including the distance from pedestrians to the exits, the width of exits, the degree of congestion, and the familiarity of pedestrians with the environment. Fu et al. (2018) proposed an exit selection method during evacuation processes based on the CA mode; it considers the impact of the building design on the pedestrians' behaviors. To investigate the multi-exit selection, Wagoum et al. (2017) conducted three empirical experiments with several extracted indicators that link temporal information with the choice of exit behaviors. Kinateder et al. (2018) developed an exit selection model in an ambulatory virtual environment, aiming to reveal the 'movement to the familiar' behavior in the controlled experiments. Lo et al. (2006) proposed a novel dynamic exit selection process based on a non-cooperative game theory that describes the equilibrium between the number of evacuees and the number of exits. To investigate the influence of the asymmetry of exits on pedestrian behaviors, Hao et al. (2014) proposed a mixed strategy that fused the distance-based and time-based strategies using a cognitive coefficient. (Cao et al. 2018) described the exit selection based on a random utility theory in a room of two-exits. In addition, other studies applied more complex numerical operations to the multi-exit selection and achieved remarkable results. For example, Guo et al. (2012) proposed a nested logit discrete model (NLDM) to investigate the choices of exits considering different factors such as visibility, intimacy, and physical conditions of the exits. (Zheng et al. 2015) proposed an improved adaptive multi-factor model (AMFM) by considering the factors of spatial distance, density, and the exit width.
Deep Reinforcement Learning (DRL) and its applications in pedestrian simulation
Reinforcement Learning (RL) has attracted wide attention, largely due to its powerful performance in playing games like Go and Arita games (Silver et al. 2017, Mnih et al. 2013. Generally, RL consists of three essential parts: environment, agent, and the reward (Sutton and Barto 2018). An agent takes action based on the current state in the environment, then the environment proceeds to the next state and renders a corresponding reward, serving as an evaluation of the action taken by the agent. Further, the agent takes the next action based on the next state and receives the next reward with a newly updated state. This iteration continues until the stopping rules are reached. The purpose of RL is to discover a function, i.e., a state-to-action mapping, in order to increase the total reward under the current state. Unlike other machine learning methods, RL focuses on long-term rewards. That's why it is popular in many fields, including complicated decision-making, robot simulation, and intellectual games.
In recent years, many studies have been done to combine deep neural networks (DNN) and RL to come up with the so-called Deep Reinforcement Learning (DRL) for solving more complicated problems. For instance, Mnih et al. (2013) proposed a novel DRL method, named Deep Q Network (DQN), by combining Q-Learning (an important branch in RL methods) with DNN. The results showed that it was able to achieve excellent results that surpassed the human-level performances in multiple Atari games. To reduce the problem of overestimation in DQN, Double DQN was developed by decoupling the target value (Van Hasselt et al. 2016). Instead of generating action value directly using a fully connected network, Dueling DQN (DDQN) first decouples the action value in DNN with the state and advantage values and then, it combines the two values as a total action value (Wang et al. 2015b). Benefitting from this network architecture, DRL can easily learn the importance of the states.
On the other hand, there are researches that explored the learning efficiency of the DRL based on the sampling method. For example, Schaul et al. (2015) proposed a Prioritized Experience Replay (PER) algorithm that orders data according to the loss value, and in the end, the algorithm proved to have higher sample utilization. Instead of using epsilon greedy to balance the tradeoff between exploration and exploitation, Noisy DQN is proposed to encourage exploration by generating random noises in the DNN (Fortunato et al. 2017). The result showed that Noisy DQN could yield better scores in Atari games as it leads to more efficient exploration. To make the learning process more stable, Bellemare et al. (2017) proposed a Categorical DQN algorithm by approximating value distribution using expectation value. The result showed that Categorical DQN facilitates chattering reducing, state aliasing, and well-behaved optimization. Recently, Hessel et al. (2018) proposed a Rainbow DQN approach that integrates the several independent improvements of the DQN algorithm. Compared with other algorithms, Rainbow DQN has noticeable improvements in the reward and convergence speed. Given its great capability, Rainbow DQN has been widely applied in open car simulation (Güçkıran and Bolat 2019), adaptive traffic signals (Nawar et al. 2019), and predictive panoramic video streaming (Xiao et al. 2019).
Nowadays, DRL is extensively used for robot navigation and pedestrian simulation. In robot navigation, the robots are able to detect the environment with the support of raw cameras or other types of sensors. In this case, the DRL algorithm is responsible for generating an optimal action to navigate the robot to the destination without collisions with other obstacles. For example, Zhang et al. (2017) applied successor-feature-based DRL in robot navigation with maze-like environments. The results suggested that this algorithm can easily adapt to other new environments. Kahn et al. (2018) proposed a self-supervised DRL with a generalized computation graph to autonomously navigate the robot in real-world environments. It finally proved to have high sample efficiency in learning complex policies. For the crowd simulation, virtual pedestrians are the agents in the environment (usually a micro-scene), aiming to reach their destinations. During the moving process, each pedestrian should avoid static and dynamic obstacles and ensure the trajectory without noticeable oscillations. Numerous attempts have been made to use DRL in crowd simulation. Godoy et al. (2016) proposed a novel Coordinated Navigation (C-Nav), a distributed approach, to address the multi-agent navigation problem in complex environments. The result showed great generalization, and the agents can reach their goals faster than other advanced collision-avoidance frameworks. Long et al. (2018) proposed a decentralized collision avoidance policy based on DRL for multi-robot systems where the inputs are directly detected from the raw sensors. To smoothen the global trajectory of agents, Xu et al. (2020) presented a local motion simulation method that integrates ORCA and DRL, named as ORCA-DRL. The model has shown a great capability of generalization, smoothening the global trajectory, and fastening learning, compared with other DRL methods. Nevertheless, the proximal policy optimization (PPO) algorithm applied in Xu et al. (2020) falls short to adapt a dynamic number of agents. In addition, the state inputs based on the traditional raycasting method largely limit its performance. Despite the aforementioned advancements, the exploration of the DRL's potential in multi-exit evacuation simulation by involving room environments is still rare.
In order to fill the gap, we proposed a novel multi-exit evacuation simulation, named as MultiExit-DRL. In this prototype, the ORCA is applied to avoid collision of pedestrians during the simulation, and Rainbow DQN architecture is used for guiding pedestrians to choose the best direction. Unlike in other studies of multi-exit evacuation simulation, this study does not explicitly define the best choice of exits for the pedestrian to evacuate at each frame. Instead, we defined eight directions in each frame for them to learn the best choice using Rainbow DQN. We further designed several indoor environments to illustrate the advantages of integrating DRL in multi-exit environments. The designed MultiExit-DRL model has demonstrated good results after comparing it with the other two traditional models, and therefore, it can be applied in many local indoor environments to investigate the behavior of the pedestrian flows. It can also be applied as the component to evaluate the evacuation efficiency under different building designs.
Methods
Methodology overview
We set the simulation environment as a two-dimensional space ℝ 2 , which consists of randomly distributed pedestrians and exits with varying properties. The goal of each pedestrian is to evacuate the room as quickly as possible through one of the available exits without colliding with obstacles or other pedestrians. Each pedestrian is represented by a disc of the radius with the maximum speed of . The obstacles are composed of line segments defined by a series of counter-clockwise points. The pedestrian in frame is characterized by the following parameters:
• position as • velocity as • speed as • collision-free velocity, i.e., optimal velocity as • direction as Each exit holds two properties. For exit , represents the position of exit and represents the width of exit . The pedestrians in the room are able to observe the positions and velocities of other pedestrians without explicit communication. At each frame within a cycle of a total of frames, pedestrians update their positions based on the restrained from the kinetics rules ( Figure 1). The simulation process continues until all pedestrians successfully evacuated the room, or the number of frames reaches the horizon . At each frame , pedestrians interact with the environment and the resulted interaction data, including state, action, reward, and terminal information, are stored in the container with a capacity of ( Figure 1a). We further define an integer value named, learning start ( ), to encourage the pedestrian to explore the environment sufficiently. At each training step, we sample interaction data from the container with a defined batch size. DNN parameters are further updated by Rainbow DQN, where a crossing-entropy loss is implemented to minimize the KL divergence between predicted state-action probabilities and target state-action probabilities. At the end of each training step, the priorities are updated based on the loss weight. The interacting process (Figure 1b) at each frame follows the procedure described in Xu et al. (2020). More details regarding the simulation implementation are presented in the following sections. We assume that pedestrians are fully aware of the environment. Similar to the state space in the Arita game, we directly abstract the images in the last three frames as the current state. The image is resized to 84 × 84. In our method, we use two coordinate systems, the environment coordinate system (ECS) and the screen coordinate system (SCS). The ECS aims to describe the interaction environment while the SCS aims to visualize and convert to the state space. It should be understood that in the practical environments, this problem is a typical GIS problem since it involves space and pedestrians as the moving objects in case of emergency (Zheni et al. 2009, Xu and Güting 2013, Tryfona et al. 2003. Due to the fact that the inputs are greyscale images (one channel with the value ranging from [0, 255]), the difference between the walls and other moving agents is not distinguished. We set the grayscale images as follows:
1) The background of the images is set to be a zero-value matrix with row ℎ and column , where ℎ and are also regarded as the height and width of the image. 2) The value for a pixel representing the current pedestrian is set to 255.
3) The value for a pixel representing static obstacles and other pedestrians is set to 100.
Action space
In our method, we discretize the action space with eight directions, indexed from 0 to 7. Each agent chooses a direction from the DNN and then translates the direction to a normalized vector . The best velocity is calculated as × .
Reward function
A total of four reward functions are included in this study, including the goal reward function , the collision reward function , the smooth reward function ℎ , and penalty reward function . The total reward R is the aggregation of all four rewards:
= 1 + 2 + 3 ℎ + ,(1)
where 1 , 2 , 3 and represent the weighting parameters. defines the reward of one-step movement, described as:
= max � �1 − ( , ) 4 � =0 − max � �1 − ( −1 , ) 4 � =0(2)
where ( , ) is the Euclidean distance between and , and 4 is a hyperparameter with the range (0, 1]. The above function consists of two major parts. The first part is the reward based on the current position for pedestrian . The second part is the reward based on the last position −1 . The represents the reward that the current step is relative to the previous one.
defines the reward of the optimal velocity with the desired velocity towards to exit:
= max � ( , ( − )) =0(3)
where (•) represents the normalization function, ( , ) represents the dot product value, defining the similarity between and .
ℎ is used to evaluate the smoothness of the pedestrian by comparing the current optimal velocity with the previous optimal velocity: ℎ = ( , −1 ).
Rainbow DQN
Reinforcement Learning (RL) generally includes three components, the environment, the agents, and the reward. At each frame t, an agent obtains the state from the environment and conducts an action based on . Then the simulation transit to the next state +1 and the agent receives a reward , an evaluation of . The interaction continues until the agent reaches the goal, or the iteration number reaches horizon . Theoretically, RL is a branch of the Markov Process Decision (MDP) < , , , , >, where is state space, is action space, is the transition relationship, is the reward function, and is a discount factor that defines the foresight of the agents. An agent's movement satisfies the Markov property, i.e., the current state is only related to the previous state, not to the historical trajectory. The objective of RL is to find a stateaction pair mapping that maximizes the total expected return, i.e., = ∑ + =0
. In general, a deep neural network (DNN) is required to map the state-to-action function, as the state space and action space tend to be multi-dimensional. In the following part, we briefly introduce the Rainbow DQN algorithm to optimize .
Rainbow DQN is an integration of a set of advanced DQN approaches. In our experiments, we utilized Double DQN, DDQN, PER, Multi-Step DQN, Categorical DQN, and Noisy DQN. In DQN, the ( , ; ) represents the predicted return in state with action following the policy under the parameter setting in the predicted DNN architecture. A bootstrap method (Mnih et al., 2013) is used to represent the target expected return, i.e.,
[ | = , = ] ≈ + max +1 ( +1 , +1 ; � )(5)
where � represents the target DNN parameters, defined to reduce the correlation between predicted expected return and target expected return. To optimize the , a Mean Square Error (MSE) loss is defined as
( ) = � + max +1 ( +1 , +1 ; � ) − ( , ; ) � 2(6)
where we only update and � will be automatically updated to after a constant number of frames ( ). Double DQN is regarded as a decoupling approach to reduce the overestimation bias, where the target expected return is defined as,
[ | = , = ] ≈ + ( +1 , argmax +1 � +1 , +1 ; � �; ,(7)
DDQN decomposes the last output value, i.e., ( , ; ) into two streams, the state value ( ) and the advantage value ( , ), and combines the two streams by a special aggregator to a new output value:
( , ; ) = ( ) + ( , ) − ∑ ( , ′ ) ′(8)
where represents a total number of actions. Multi-Step DQN is adopted here to facilitate faster learning. We rewrite the step reward as:
( ) = � ( ) −1 =0 + +1(9)
Thus, the target expected return based on Double DQN can also be rewritten as
[ | = , = ] ≈ ( ) + ( ) � +1 , argmax +1 � +1 , +1 ; � �; �(10)
To describe the expected value in a more accurate way, Categorical DQN applies the distribution of values, ( , ), instead of a single value, where presents the parameters of the related DNN. ( , ) denotes a discrete distribution, parameterized by . and respectively denote the minimum of maximum boundaries. For each atom in the distribution , defined as = + ∆ , where is the atom index with the range [0,
) and ∆ is the interval distance between two joint values, calculated as ∆ = − −1 . The predicted return expectation ( , ) = ∑ ( , ; ), where is the probability of atom in state-action. The probability at each atom is updated as
̂= � �1 − ��̂� − � ∆ � 0 1 −1 =0 ( +1, argmax +1 � +1, , +1 ; � �(11)
where [•] represents the bound of argument in the range [ , ] and ̂ represents the estimated value for atom , equaling to:
̂= +(12)
The KL divergence described as (|| ), is applied to evaluate the difference between ̂ and and the cross-entropy term is considered to be the loss function ( ) as ( ) = − ∑̂log ( , ; )
Finally, Noisy DQN is applied to balance the tradeoff between exploration and exploitation. A general linear layer with input and output can be represented by = +
where and denote the weights and bias respectively. In Noisy DQN, and will be re-defined by a Gaussian distribution. Therefore, the related noisy linear layer can be described as
= ( + ⨀ ) + + ⨀(15)
where ⨀ represents element-wise multiplication to increase the noises in Gaussian distribution. and are the parameters in Gaussian distribution parameters for wights . and are the parameters for the Gaussian distribution parameters for bias .
To speed up learning efficiency, we applied PER, a sampling-based optimization algorithm. For each data pair < , , , , +1 >, where is a Boolean value to judge whether the agent reaches the destination at time . The corresponding priority is positively related to ( ) as ∝ ( )
Network architecture
As mentioned above, a state space is composed of the last three greyscale images with a size of 84 × 84 and the action space is a discrete space that includes eight different directions, i.e., = 8. Based on this setting, the input and output spaces of the DNN are respectively set to be 3 × 84 × 84 and 51 × 8. As shown in Figure 2, three twodimensional convolutional layers are applied to the input . After each convolution operation, Batch Normalization (BN) is further applied to prevent overfitting (Ioffe and Szegedy 2015). The output from the convolutional layers is flattened and decomposed as two components. The first component consists of two fully connected (FC) layer with 512 rectifier units and × rectifier units, respectively. The second component consists of two fully connected layers with 512 rectifier units and rectifier units, respectively. Noisy terms are added to all the FC layers to encourage exploration at each interaction. Finally, we aggregate the outputs of the two components as the final output, including values for each action in the action space. The activation function in both convolutional and FC layers are the ReLU nonlinearities (Nair and Hinton 2010).
Experiment environment and scenarios
Coding environment
The algorithm of this model is implemented using the Python programming language. PyTorch packages are applied to build a deep neural network (DNN) for mapping the relationship between the state and action. In addition, OpenCV packages are used to collect and visualize data. The program runs on a computer with Ubuntu 18.04 in an environment that consists of i7 CPU, 64G RAM, and two NVIDIA GTX 1080 Ti. The simulation process runs on CPU while the DNN is trained on GPUs. The hyperparameters setting used in this study can be found in Table 1.
Scenarios
A virtual indoor environment with two exits named and respectively, is designed as the research environment. The room geometry is square with a side length of 100 and a wall width of 2.0. The radius of each pedestrian ( ) equals to 2.0. A total of three scenarios are presented to evaluate the performance of our method. Those scenarios include 1) varying exit width ratio ( ) with a uniform pedestrian distribution, 2) varying pedestrian distribution ratio ( ) with a uniform exit width, and 3) varying exit opening times with a uniform exit width and a uniform pedestrian distribution. In the first scenario, the distribution of the pedestrians and the width of ( ) hold the same (4.0 × ) while the width of ( ) is set to be 4.0 × , 6.0 × , and 8.0 × , respectively. Assuming represents the ratio of the two exits, i.e., , the three sub scenarios include: = 1: 1 , = 1: 1.5 , and = 1: 2 . This scenario aims to investigate the model's performance under different exit width ratio with uniform distribution of pedestrians. In the second scenario, both exits are with the same width, equaling to 4.0 × . A total of pedestrians are within the room but with different distributions. Assuming = , where represents the number of pedestrians whose closest exit is while represents the number of pedestrians whose closest exit is , three different distributions are designed, namely = 1: 1, = 1: 2 and = 1: 3. This scenario investigates the model's performance in handling uneven congestions at the exits during the evacuation. In the third scenario, the distribution of pedestrians and the width of the two exits hold the same. However, doesn't open at the initial frame while keeps open throughout the entire simulation process. In this scenario, the open time for is set at the 15 th frame, the 30 th frame, and the 45 th frame, respectively. This scenario creates a dynamic simulation environment as the pedestrians are unaware of when another exit is open. To compare the effectiveness and efficiency of our proposed method, different numbers of pedestrians (m) are tested, i.e., = 12, m = 24, and m = 36 in all three scenarios. We investigate the performance of methods from two perspectives: 1) the total frames for evacuation, and 2) the utilization efficiency of two exits ( ). Since the and represent the number of pedestrians to evacuate from and , then the can be calculated as:
= min ( , ) max ( , )(17)
With a range of [0,1], represents how efficient and are utilized in general. The higher the , the more the efficiency of those two exits for the pedestrians to pass during the evacuation. We only investigate in the first two scenarios due to the delay imposed to open the in the last scenario. The proposed MultiExit-DRL method is compared against AMFM and NLDM from Zheng et al. (2015) and Guo et al. (2012), respectively.
Results
To compare the performances of this model, we obtain screenshots of different frames during the evacuation process from all three scenarios (Figure 3, Figure 4, and Figure 5). For the first two scenarios, an interval of 10 frames is used, while an interval of 15 frames is used for the third scenario given its longer simulation process. The total frames for pedestrians to evacuate the room and the utilization efficiency of the two exits are analyzed for the three designed scenarios.
Model performances under different exit width ratio ( )
We first evaluate the model performance under different with uniform distribution of pedestrians. Given different exit width, pedestrians are expected to find appropriate exits to prevent unnecessary congestions, thus leading to high exit utilization efficiency and low total frames (total time for all the pedestrians to evacuate). As expected, the total frames increase along with the increasing number of pedestrians with the same ( Table 2). It is reasonable that, with the same indoor environment, it takes a longer time for more pedestrians to evacuate the room. Compared with the other two methods, our proposed MultiExit-DRL method achieves the best performance in all conditions ( Table 2). As the number of pedestrians increases (e.g., in the 36-pedestrian case), a larger performance gap is found between MultiExit-DRL and the other two methods, suggesting that MultiExit-DRL can better handle environments with massive agents. For example, in the 36-pedestrian case with = 1, the evacuation via AMFM and NLDM takes 115 and 106 frames, respectively, while the evacuation via MultiExit-DRL only takes 60 frames. Since we hold consistently while incrementing the , a faster evacuation is achieved by MultiExit-DRL since the wider exit allows more pedestrians to evacuate. This phenomenon proves that, after training with the DRL algorithm, pedestrians are able to recognize the wider exit (the in this case) and use it evacuate faster. Note. Best performances among the three methods are highlighted in bold. The efficiency of exit utilization of this scenario is presented in Table 3. All the methods achieve decent performances with = 1: 1 (i.e., = ), suggesting that all three models can handle the situation where the two exits have the same width, creating the same attraction for all the pedestrians. However, the superiority of MultiExit-DRL is clear when the two exits become unbalanced. For example, in the 36-pedestrian case with = 1:2 (i.e., = 2 × ), the values of in AMFM and NLDM are 0.06 and 0.18, respectively while the value of in MultiExit-DRL is 1.00 (Table 3), suggesting that the efficiency of exit utilization in MultiExit-DRL is significantly better with an unbalanced exit width ratio compared with the other two methods. The superiority of MultiExit-DRL is well documented from the screenshots of frames during the evacuation process ( Figure 3). When = 1: 2, pedestrians in AMFM and NLDM are clearly crowded at , evidenced by the screenshots of the 20 th , the 30 th , and the 40 th frame (Figure 3). Even though the width of is twice larger than the width of , lacks the ability to handle overwhelming pedestrians. This congestion is significantly due to the low efficiency of exit utilization with many frames of the AMFM and the NLDM method. In comparison, pedestrians in MultiExit-DRL are able to choose exits more appropriately, given the unbalanced , leading to the perfect efficiency of exit utilization with few total frames.
Model performances under different pedestrian distribution ratio ( )
This scenario investigates the model's performance in handling different initial distributions of pedestrians. Given the different distribution patterns (i.e., a different ), pedestrians are expected to adjust their strategies during the evacuation to exit the room as fast as possible with high efficiency of exit utilization. As shown in Table 4, our proposed MultiExit-DRL method achieves the best performance in all of the designed conditions. Similar to the comparison in the first scenario, MultiExit-DRL significantly outperforms the other two methods as the number of pedestrians increases regardless of the variations of the initial distributions, thus demonstrating its great capability in handling massive agents. In addition, the insensitivity to the different initial distributions in our method suggests that pedestrians have learned to adjust their strategies appropriately to prevent congestions at different initial locations. For instance, in a crowded 36-pedestrian case, pedestrians in MultiExit-DRL evacuate the room in 60 and 58 frames for = 1:1 and = 1:3, respectively (Table 4). Despite the fact that = 1:1 suggests an even distribution while = 1:3 suggests the number of pedestrians whose initial locations are closer to is three times as many as , both rooms are evacuated within a similar number of frames (Table 4). Note. Best performances among the three methods are highlighted in bold. In terms of exit utilization, higher values are found for MultiExit-DRL in the crowded environments (cases with 24 and 36 pedestrians), where all values are above 0.8, suggesting great efficiency of exit utilization (Table 5). In the 12-pedestrian case, however, MuiltiExit-DRL presents low with uneven initial distributions ( = 1:2 and = 1:3). It indicates that, in the current hyperparameter setting, pedestrians in MultiExit-DRL prioritize closer exit in the uncrowded environment. Despite the low , pedestrians in MultiExit-DRL can still evacuate the room faster than the other two methods, as evidenced by the low number of the total frames (Table 4). It is observed that the initial uneven distribution might cause congestion at the exit for certain methods, which is well documented by the screenshots of frames. When = 1:2 and = 1:3, pedestrians in AMFM clearly congest at , as is the closest exit to the initial locations of most pedestrians. However, this congestion unavoidably results in low efficiency of exit utilization and longer evacuation time. In comparison, pedestrians in MultiExit-DRL can clearly adjust their evacuation strategies, leading to a balanced exit assignment. The 30 th frame in all cases shows that, with our proposed MultiExit-DRL method, both exits are targeted with a balanced amount of pedestrians, which largely increase the efficiency of exit utilization and reduces the evacuation time.
Model performances under different opening times of
The different opening times of a certain exit creates a dynamic simulation environment where pedestrians are expected to recognize the newly opened exit and adjust their evacuation strategies accordingly. Similar to the previous two scenarios, MultiExit-DRL shows the least total frames for pedestrians to evacuate the room compared with AMFM and NLDM in all conditions (Table 6). As the number of pedestrians increases, the superiority of MultiExit-DRL becomes obvious. In addition, the earlier the opens, the faster the pedestrians in MultiExit-DRL evacuate, especially in a more crowded environment, e.g., the 24-and 36-pedestrian case. Since is the only available exit before opening , all pedestrians are moving in one direction towards ( Figure 5). The discrepancy of pedestrians' behaviors occurs when pedestrians start to be aware of the availability of . In the case when opens at the 15 th frame, about half of the pedestrians in MultiExit-DRL switched to the new exit, thus to greatly speed up the evacuation process. In comparison, pedestrians in AMFM and NLDM usually fail to respond timely, and if they respond, they cause congestion due to lack of the unbalanced loads (see AMFM pedestrians in the 45 th frame when opens at the 15 th frame). To sum up, three different scenarios are designed to compare the performance of our proposed method, MultiExit-DRL against AMFM and NLDM. The model's performance is revealed from the total number of frames and the efficiency of exit utilization, i.e., the . The results indicate remarkable superiority of MultiExit-DRL in terms of total frames, as pedestrians under the MultiExit-DRL method are able to evacuate the room with the least frames in all designed conditions. With the room becoming more crowded by adding additional pedestrians, the performance gap becomes more obvious, suggesting great generalization capability of the MultiExit-DRL method. As for the efficiency of exit utilization, MultiExit-DRL shows consistent high values in the first scenario with a varying , indicating that pedestrians under MultiExit-DRL can adjust their strategies according to different exit widths once the evacuation phase starts. MultiExit-DRL also exhibits high values in crowded environments in the second scenario with a varying . Although it presents low in the 12-pedestrian case, the evacuation is still completed within fewer frames compared with AMFM and NLDM. The underperformance of AMFM can be explained by its intrinsic design. As an improved adaptive multi-factor model, it utilizes an additional judgment to prevent pedestrians from frequently changing the targeted exit. Despite its great performance in uncrowded environments, pedestrians in AMFM fail to modify their strategies timely when more pedestrians are added to the environment. Therefore, congestion at one exit and idleness at the other is found in AMFM, especially in crowded environments. NLDM, as a nested logit discrete model, uses non-strict logical judgment, potentially leading to frequent changes of targeted exits of its pedestrians, consequently leading to more frames. Different from the two methods above, MultiExit-DRL provides a proper direction for each pedestrian under the current state via Rainbow DQN, allowing the pedestrians to choose the optimal directions of movement by given reward functions. It further allows pedestrians to rapidly explore other options instead of congesting at a certain exit. Note. Best performances among the three methods are highlighted in bold.
Figure 5
Model performance for 36 pedestrians under different opening times of .
Discussion
In this study, we presented a novel multi-exit evacuation simulation by integrating ORCA and DRL. In general, our method shows great performances both in the training process and the evacuating process. Specifically, it outperforms other approaches due to its 1) efficient learning speed, 2) stability, 3) ability to scale up and 4) great exit utilization.
Transcending popular ray-based state acquisition approaches that require intensive computational resources (Lee et al. 2018, Xu et al. 2020, we directly capture screenshots of the environment via OpenCV as the source of information, allowing the pedestrians to learn their surroundings quickly. Besides, by packing adjacent frames as input to the DNN, no external information (such as position and velocity) is required. Given the exponential growth of the complexity in the ray-based methods when the number of pedestrians increases, the complexity of our method stays with linear growth that makes it more efficient to perform simulations with massive pedestrians. Stability is another merit of our approach. In the multi-exit simulation, pedestrians often hesitate when facing multiple exits, given the limited designs of traditional mathematical models. Although researches suggested that this problem can be mitigated by a proper probability assignment (Zheng et al. 2017), still the existing models fail to present stable results. In our method, pedestrians are granted a global perspective by the application of DRL for selecting the optimal action in the current state. Given the long-term reward maximum nature of DRL, pedestrians keep interacting with the environment for numerous times to learn the best action for the total return. Once the best state-to-action mapping function has been learned by the pedestrians, they take actions without hesitation, allowing the simulation process to be pretty stable. Our method is tested with a different number of pedestrians, and the results have shown great performance compared with other traditional methods. This scaling-up ability is due to the fact that the effectiveness of our approach largely depends on the reward function rather than the complexity of the environment. As the number of pedestrians increases, the state space expands accordingly, which in turn leads to longer training time. However, the increase of pedestrians does not affect pedestrians taking optimal action to maximize the value of the reward function in the given state. Finally, the results proved a great exit utilization as pedestrians in our approach hardly crowd at a certain exit. For traditional multi-exit selection methods, once a pedestrian chooses an exit at a certain frame, it moves towards the target exit, regardless of any changes, thus causing congestion that eventually reduces the evacuation efficiency. In our approach, however, instead of choosing a specific exit for a pedestrian in a given frame, an optimal direction is calculated based on DRL. This optimal direction prevents further crowding at a certain exit by allowing pedestrians to explore other opportunities. This behavior can be achieved via the implementation of an appropriate reward function via DRL but is difficult to be expressed via traditional multiexit selection simulation that utilizes mathematical models.
Despite the merits explained above, it worth noting that our model is designed under the following constraints. Limited by the discrete nature of Rainbow DQN, we adopted a discrete space with a total of eight possible directions. It turns out that, in a few cases, pedestrians move along with Manhattan distances instead of Euclidean distances with better efficiency. The application of DRL algorithms with a continuous action space like DDPG, PPO, or SAC might further improve the model performance. The total reward function in our method is composed of four sub-reward functions, i.e., the goal reward function, the collision reward function, the smooth reward function, and the penalty reward function, where each sub-reward function needs a hyperparameter to specify its importance. However, the balance between the four hyperparameters requires trial-anderror experiments which are not sufficiently covered in our work.
Future studies should focus on designing more simple and reliable reward functions. It is acknowledged that the performances of DRL algorithms greatly rely on the number of training samples. Our approach uses a single-core data collection method that basically causes sub-optimal simulations under the same experimental configuration and reward function. This can be regarded as the result of incomplete exploration of agents. In the future, the potentially more advanced DRL methods, such as the APeX-DQN (Horgan et al. 2018), need to be explored. Finally, the pedestrians in our method have the same size and velocity. However, in practice, each pedestrian is not uniform. Our future research will extend this method to accommodate heterogeneous pedestrians.
Conclusion
Multi-exit evacuation simulation is one of the key areas that need more attention due to its potential in public safety. Traditional multi-exit evacuation simulation methods largely rely on discrete multi-exit selection methods that are characterized by different factors such as distance, density, and exit width. These methods are, however, coupled with low exit utilization and congestion at the exits. In this article, we proposed a novel multi-exit evacuation simulation that integrates ORCA and DRL, referred to as the MultiExit-DRL, where local collision avoidance detection is achieved via ORCA, and movement direction is achieved via DRL. We further designed a DNN framework to facilitate state-to-action mapping. In the designed framework, successive screenshots (greyscale images) are used as the raw state, and they have proven to be faster in data collection compared to ray-based state acquisition. The action space is further divided into eight isometric directions for pedestrians to vacate. Rainbow DQN, a DRL algorithm that integrates several advanced DQN methods, is applied to improve data utilization and algorithm stability. We compared our proposed MultiExit-DRL method with two traditional multi-exit evacuation simulation models, i.e., the AMFM and the NLDM in three individual scenarios: 1) varying pedestrian distributions with a uniform exit width; 2) varying exit widths with a uniform pedestrian distribution; 3) varying exit opening schedules with a uniform exit width and a uniform pedestrian distribution. The results have shown that the proposed MultiExit-DRL presents great learning efficiency in all of the designed experiments. It further shows great utilization of exits regardless of the number of pedestrians. Nevertheless, MultiExit-DRL has some limitations, such as the utilization of discrete action space and homogeneous pedestrian design. Further researches should focus on solving these problems and investigate the model's performance in large areas with more complicated scenarios.
Data availability statement
No third-party data was used in this study. The source code and videos for demonstration can be found on GitHub at https://github.com/XD1227/MultiExit-Rainbow
Figure 1
1Methodology overview. (a) General simulation structure; (b) Interacting process for all agents at frame = (modified from Xu et al. (2020)) 3.2 Deep reinforcement learning 3.2.1 State space, action space, and reward function 3.2.1.1 State space
Figure 2
2Network architecture.
Figure 3
3Model performance for 36 pedestrians under different exit width ratios.
Figure 4
4Model performance for 36 pedestrians under different pedestrian distribution ratios.
Table 1 .
1Rainbow DQN hyperparameters used in Multi-Exit evacuation simulationHyperparameter
Value
Learning rate ( )
1e-4
Discount ( )
0.99
Horizon ( )
200
Multi-step ( )
3
Batch size
128
1000
1e+5
51
10
Table 2 .
2Totalframes for evacuation under different exit width ratio (
) with 12,
24, and 36 pedestrians.
Methods
12 pedestrians
24 pedestrians
36 pedestrians
=
=
=
1:1 1:1.5 1:2
1:1 1:1.5 1:2
1:1
1:1.5
1:2
AMFM
51
48
46
69
65
69
115
78
62
NLDM
53
41
47
90
59
70
106
88
54
MultiExit-DRL
38
39
34
50
45
45
60
57
52
Table 3 .
3The Note. Best performances among the three methods are highlighted in bold.utilization efficiency (
) under different exit width ratio (
) with 12, 24,
and 36 pedestrians.
Methods
12 pedestrians
24 pedestrians
36 pedestrians
=
=
=
1:1
1:1.5
1:2
1:1
1:1.5
1:2
1:1
1:1.5
1:2
AMFM
0.71
0.25
0.18
0.71
0.90
0.29
0.89 0.36
0.06
NLDM
1.00
0.93
0.67
1.00
0.75
0.29
0.89 0.50
0.18
MultiExit-DRL 1.00
0.93
0.70
0.85
0.93
0.70
0.80 0.83
1.00
Table 4 .
4Total frames for evacuation under different pedestrian distribution ratio ( ) with 12, 24, and 36 pedestrians.Methods
12 pedestrians
24 pedestrians
36 pedestrians
=
=
=
1:1
1:2
1:3
1:1
1:2
1:3
1:1
1:2
1:3
AMFM
51
53
58
69
103
121
115
117
119
NLDM
53
66
65
90
83
104
106
149
153
MultiExit-DRL
38
51
44
50
56
53
60
61
58
Table 5 .
5Theutilization efficiency (
) under different pedestrian distribution ratio ( )
with 12, 24, and 36 pedestrians.
Table 6 .
6Total frames for evacuation under different open frames for with 12, 24, and 36 pedestrians.Methods
12 pedestrians
24 pedestrians
36 pedestrians
opens at frame
opens at frame
opens at frame
15 th
30 th
45 th
15 th
30 th
45 th
15 th
30 th
45 th
AMFM
66
74
68
100
106
136
144
159
149
NLDM
70
68
68
97
141
183
153
183
173
MultiExit-DRL 48
49
49
74
86
100
84
104
121
Note. Best performances among the three methods are highlighted in bold.
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| [
"https://github.com/XD1227/MultiExit-Rainbow"
] |
[
"Interstellar C 2 in the Perseus molecular complex: excitation temperature and density of a molecular cloud with anomalous microwave emission",
"Interstellar C 2 in the Perseus molecular complex: excitation temperature and density of a molecular cloud with anomalous microwave emission"
] | [
"Susana Iglesias-Groth \nInstituto de Astrofisíca de Canarias\n38200 La Laguna, Canary IslandsTenerifeSpain\n\nUniversidad de La Laguna\nE-38205La Laguna, TenerifeSpain\n"
] | [
"Instituto de Astrofisíca de Canarias\n38200 La Laguna, Canary IslandsTenerifeSpain",
"Universidad de La Laguna\nE-38205La Laguna, TenerifeSpain"
] | [
"Mon. Not. R. Astron. Soc"
] | Interstellar absorption lines up to J"=10 in the (2,0) band and up to J"=6 in the (3,0) band of the C 2 A 1 Π u -X 1 Σ + g system are detected toward star Cernis 52 (BD+31 o 640) in the Perseus molecular complex. The star lies in a redenned line of sight where various experiments have detected anomalous microwave emission spatially correlated with dust thermal emission. The inferred total C 2 column density of N(C 2 ) = (10.5± 0.2) x 10 13 cm −2 is well correlated with that of CH as expected from theoretical models and is among the highest reported on translucent clouds with similar extinction. The observed rotational C 2 lines constrain the gas-kinetic temperature T and the density n=n(H)+n(H 2 ) of the intervening cloud to T = 40±10 K and n = 250 ± 50 cm −3 , respectively. This is the first determination of gas-kinetic temperature and particle density of a cloud with known anomalous microwave emission. | 10.1111/j.1365-2966.2010.17807.x | [
"https://arxiv.org/pdf/1011.3796v1.pdf"
] | 118,372,711 | 1011.3796 | 382ad764713c285b6b906aabbc98606a7b04176c |
Interstellar C 2 in the Perseus molecular complex: excitation temperature and density of a molecular cloud with anomalous microwave emission
1-?? (2010
Susana Iglesias-Groth
Instituto de Astrofisíca de Canarias
38200 La Laguna, Canary IslandsTenerifeSpain
Universidad de La Laguna
E-38205La Laguna, TenerifeSpain
Interstellar C 2 in the Perseus molecular complex: excitation temperature and density of a molecular cloud with anomalous microwave emission
Mon. Not. R. Astron. Soc
0001-?? (2010Accepted Received In original form(MN L A T E X style file v2.2)ISM:molecules-ISM:lines and bands-ISM:abundances
Interstellar absorption lines up to J"=10 in the (2,0) band and up to J"=6 in the (3,0) band of the C 2 A 1 Π u -X 1 Σ + g system are detected toward star Cernis 52 (BD+31 o 640) in the Perseus molecular complex. The star lies in a redenned line of sight where various experiments have detected anomalous microwave emission spatially correlated with dust thermal emission. The inferred total C 2 column density of N(C 2 ) = (10.5± 0.2) x 10 13 cm −2 is well correlated with that of CH as expected from theoretical models and is among the highest reported on translucent clouds with similar extinction. The observed rotational C 2 lines constrain the gas-kinetic temperature T and the density n=n(H)+n(H 2 ) of the intervening cloud to T = 40±10 K and n = 250 ± 50 cm −3 , respectively. This is the first determination of gas-kinetic temperature and particle density of a cloud with known anomalous microwave emission.
INTRODUCTION
The moderately reddened [E(B-V)=0.9] early A-type star Cernis 52 (R.A 03 43 01; Dec +31 58 10; Cernis 1993 ) lies in the line of sight of the Perseus molecular complex which is located at a distance of about 240 pc. The star is a likely member of the very young IC 348 cluster, about 1 degree in projection away from its core (Gonzánlez- Hernández et al. 2009). This line of sight towards Perseus is remarkable because of the anomalous microwave emission detected by Watson et al (2005) in the frequency range 10-60 GHz. According to recent data, this anomalous emission reaches a maximum in the line of sight of Cernis 52 (Tibbs et al. 2010). This new microwave emission process cannot be explained by classical mechanisms as synchrotron, free-free or thermal emission from dust particles, however it is spatially correlated with the distribution of dust traced by IRAS images. Draine and Lazarian (1998) postulated that the anomalous microwave emission could be due to electric dipole radiation of rapidly spinning small interstellar carbon based molecules, as for example polycyclic aromatic hydrocarbons (PAHs) (see also Iglesias-Groth 2005 for the potential contribution of hydrogenated forms of fullerenes). The recent detection of optical bands in the spectrum of Cernis 52 which ⋆ E-mail: [email protected] are consistent with transitions of the naphthalene C10H8 + and anthracene C14H10 + cations (Iglesias-Groth et al 2008, 2010 add support to this PAH hypothesis and call for a more extensive study of the physical and chemical conditions of the intervening material. Interstellar CH, CH + have also been detected toward Cernis 52 with high column densities suggesting that the H2 column density in the intervening cloud is also high (Iglesias-Groth et al. 2010). CH is empirically correlated with C2 in diffuse and translucent molecular clouds (see e.g. van Dishoeck & Black 1989, Federman et al. 1994, Gredel 1999) thus a high column density of C2 in the intervening cloud may be expected. The basic chemical processes which lead to the formation of diatomic carbon bearing species were reviewed by Federman & Huntress (1989) and the dominant C2 formation path is C + + CH → C2 + + H.
We report here the detection of several interstellar absorption lines of the simplest multicarbon molecule C2 toward Cernis 52 and the study of physical parameters such as gas-kinetic temperatures and particle densities in the intervening cloud. We identify many weak absorption lines of C2 leading to the estimation of column densities for levels J " < 12. The theory of C2 excitation developed by Chaffee et al. (1980) and van Dishoeck & Black (1982) has been considered by a variety of authors in their analysis of interstellar C2 lines (see e.g. Hobbs 1981, Gredel 1999, Gredel et al 2001, Sonnentrucker et al. 2007, Kaźmierczak et al. 2010 to derive densities and temperatures in diffuse and translucent molecular clouds. We use the van Dishoeck & Black formalism to infer the C2 total abundance, the kinetic temperature and the density of the gas in the intervening cloud. We also report the detection of an absorption feature in the spectrum of Cernis 52 which can be abscribed to the 4051.6 A band of C3.
OBSERVATIONS AND DATA REDUCTION
Interstellar absorption lines which arise from the (2,0) and (3,0) bands of the C2 Phillips system around 8765 abd 7720 A, respectively, were searched toward star Cernis 52. The observations were carried out in November 2008 using the High Resolution Spectrograph (HRS) of the 9.2m Hobby-Eberly Telescope (HET) at McDonald Observatory (Texas, USA). Exposures of the star's spectrum were obtained on each of four nights (November 4,7,9 and 29) at a resolving power R=40000. In all, 15 exposures lasting 20 minutes each were obtained and were subsequently combined during data reduction. Fast rotating stars were also observed with the same instrument on the same nights to allow removal of telluric lines and correction of the instrumental response. The data were reduced using IRAF and wavelength calibrated using ThAr lamps. The dispersion of the re-binned linear data was 96 mÅ/pixel. The fiber used to feed HRS led to a spectral resolution of 0.25Å in the spectral regions of interest. The accuracy of the wavelength calibration of each order was better than 10 mÅ as shown by measurements of the wavelengths of telluric lines recorded in various echelle orders. After wavelength calibration, the Cernis 52 spectra were corrected for telluric line contamination and possible instrumental effects dividing each individual spectrum by the featureless spectrum of a much brighter, hot and fast rotating star in a nearby line of sight observed with the same instrument configuration. Individual spectra where then combined to improve S/N. The final spectrum achieved S/N 300 per pixel.
Cernis 52 is embedded in a cloud responsible for significant visible extinction and the presence in the spectrum of molecular absorption bands caused by the intervening interstellar material is therefore expected. Figures 1 and 2 show interstellar absorption lines which arise from the (2,0) and (3,0) bands of the C2 A 1 Πu -X 1 Σ + g Phillips System, around 8765Å and 7720Å, respectively. We identified and measured absorption lines (P,Q and R branches) in bands (2,0) 8750-8849Å and some of (3,0) 7714-7793Å. The equivalent widths in the final normalized spectra were measured by fitting a Gaussian profile to each absorption line using the IRAF task SPLOT. In the case of unresolved line blends, such as the (2,0) R(10)+R(2) blend, we used the deblending option offered by the same routine to determine the equivalent width of each component. The equivalent widths with errors for all the measured interstellar lines of C2 are listed in Table 1. The observational uncertainties in the equivalent widths are typically ±1 mÅ, except for those lines marked as blends which have errors of ∼ 2 mÅ. The table lists rest wavelengths in air obtained from the wavenumbers in vacuum of Chauville, Maillard and Mantz (1977) and converted to air wavelengths using Birch & Downs (1993).
RESULTS AND ANALYSIS
Molecular parameters
The measured equivalent widths W λ of the C2 lines were converted into column densities for individual rotational levels N(J") assuming that the absorption lines are optically thin (i.e. that the absorption lines are on the linear part of the curve of growth). This approximation is accurate to better than 10 per cent for W λ < 10 mÅ and Doppler widths b > 1 km s −1 (Stromgren 1948). We used the following equation with W λ in units ofÅ and N in units of cm −2
N (J " ) = 1.13x10 20 W λ /(f J ′ J " λ 2 )(1)
where λ is wavelength and f J ′ J " is the absorption oscillator strength.
Line oscillator strengths f J ′ J " were calculated from the (2,0) band oscillator strength f20 from the relation f J ′ J " =f20 (ν J ′ J " /ν band ) S J ′ J " /(2(2J " + 1)), and Hönl-London rotational line intensity factors S J ′ J " of (J " +2), (2J " +1), and (J " -1) for the R, Q, and P lines, respectively. The general formulae for the Hönl-London factors (Herzberg 1950) are simplified to the former values and normalized such that for each J" the sum Σ J ′ S J ′ J " / g (2 J " +1) = 1, where g=2 is the value of the ratio of electronic degeneracy factor for the C2 A-X system. Here ν band is the wavenumber of the band, equal to 11413.91 cm −1 for the Phillips (2-0) band. For the absorption oscillator strength of the (2-0) Phillips band we used the value f20 = 1.36 (±0.15) x 10 −3 measured by Erman and Iwamae (1995) which compares well with 1.44 x 10 −3 as obtained from an ab initio calculation by van Dishoeck (1983) and with the value f20 = 1.2 x 10 −3 suggested by Lambert et al. (1995). We infer the band oscillator strength for the (3,0) band from f20 using the theoretical ratio of f20/f30 = 2.2 (van Dishoeck & Black 1982).
Column densities and Rotational excitation of C2
As discussed by Gredel et al. (2001) a curve of growth analysis shows that for a typical value of b= 1 km s −1 the C2 lines suffer from saturation for W λ > 15 mÅ. All our C2 lines are weaker than this value and therefore we have ignored saturation corrections. The derived column densities for each rotational level N (J") are listed in Table 1 with their uncertainties. An independent check on the reliability of the results can be made by a comparison of the column densities derived from the P, Q and R lines which arise from the same lower level J " . Because of the smaller oscillator strength of P lines they proved very difficult to measure. Thus, the empirical uncertainties in N(J") of order 20 % are obtained by comparing for each J" the values obtained from R(J) and Q(J) lines.
In order to obtain average column densities < N (J") > in rotational level J", the column densities inferred from the individual measurements in the R and Q lines of the (2,0) and (3,0) bands, when available, were combined. Following van Dishoeck and Black (1982), we obtain the C2 excitation diagram for Cernis 52 shown in Fig. 3 where we represent the weighted relative column densities -ln [5 N (J " ) / ((2J " +1) N(2))] versus excitation energy E(J") of rotational level J " (k is the Boltzmann constant). Rotation excitation temperatures were obtained from a linear fit to the logarithm of the column densities of the two, three and four lowest rotational Table 1. Summary of C 2 lines toward Cenis 52. (a-blend line).
Molecule
Band (4) 7714.58+7714.95 5(2) a +3 (2) levels starting from J"=0 and result T02=34 K, T04=39 K and T06=49 K, respectively.
Line λ air (Å ) W λ (mÅ ) N(J " ) (10 13 cm −2 ) C2 A-X (3-0) R(0) 7719.33 3(1) 0.91 (0.3) R(2) 7716.53 5(2) 3.56 (1.4) R(6)+R
As described by van Dishoeck and Black (1982), in general the populations of all rotational levels cannot be characterized by a single rotational temperature, the detailed behaviour of the rotational excitation temperature depends on the gas kinetic temperature T kin , on the intensity of the interstellar radiation field I and on the collisional rate nσ0 (the number density of collision partners n= n(H) + n(H2) times the cross section, σ0, for collision induced transitions for level J" to rotational level (J" -2)). To interpret the excitation diagram of Fig. 3 we assume the same σ0=2 x 10 −16 cm 2 and the same intensity of the interstellar radiation field than van Dishoeck and Black (1982) and adjusted for the differences with respect their adopted f values. We infer from these models an excitation temperatute of 34±10 K and for the effective density of collision partners n=300 ± 80 cm −3 (see Fig. 3). We checked this result using the code made available by B. Mc Call at the web site http:/dib.uiuc.edu/c2/. We generated a grid of models for gas kinetic temperatures ranging from 10 to 1000 K, with a step of 1 K and a range of particle densities from n=50 to 2000 cm −3 with a step of 50 cm −3 . The results were scaled to the f-values adopted above (a factor 1.36 higher than in the code by Mc Call) for the (2,0) series and we took for comparison the weighted mean of the column densities N(J) inferred from the (2,0) series of lines. Then, we determined from the best figure of merit (obtained from the minimum difference between predicted and observed values) the most likely values for T and n. The best fit was found at T=40±10 K and n=250± 50 cm −3 in very good agreement with our previous independent analysis.
Total observed column densities were derived from the sum Σ J " N(J") over the observed rotational levels of each individual band, resulting N obs = (9.2±1.4) x 10 13 cm −2 and (9.8±1.5) x 10 13 cm −2 for the (2,0) and (3,0) bands, respectively. The combination leads to N obs =(9.5±1.2) x 10 13 cm −2 . The total C2 column density, defined as the sum of the mean column densities of the observed levels and of the contribution of the unobserved levels estimated from the theoretical model characterized by the best-fitting parameters results Ntot= (10.5±1.2) x 10 13 cm −2 .
Heliocentric velocities for individual interstellar absorption lines of C2 were determined from the final combined spectrum obtained each of the four days that Cernis 52 was observed. Typically, the three or four most intense unblended lines for each band were used. We found 14.9±1.0 and 12.7±1.2 km s −1 as average values of the heliocentric velocities of the lines measured for the (2,0) and (3,0) bands, respectively. The error was obtained as the rms of the measurements. We find consistent results among the two bands and finally combine them to obtain an average heliocentric velocity of v hel =13.8±0.9 km s −1 as best estimate from the detected interstellar C2 lines. It appears, at this spectral resolution, that C2 lines rise in a single velocity component. The C2 lines and the K I line (see below) display the same heliocentric velocity within the uncertainties of our measurements.
Interstellar Potassium
The K I 7664 and 7698Å absorption lines are also present in our spectra of Cernis 52. The K I 7698 line is well separated from a telluric O2 absorption line but the K I 7664 line is not. The equivalent width of the K I 7698 A line is 153±3 mÅ. We have obtained heliocentric velocities for this K I line from the available set of spectra resulting a final mean value of 13.7 ± 0.4 km s −1 where the error is obtained from the dispersion of the measurements obtained in different days. The heliocentric velocity agrees well with those of C2 above and CH, CH + measured by Iglesias-Groth et al. (2010). Using the atomic parameters of Morton (1991), we infer N(K) = 1.4 x 10 12 cm −2 towards Cernis 52.
DISCUSSION
Out of 37 lines of sight towards translucent molecular clouds investigated by van Dishoeck and Black (1989, see their table 4) 19 present E(B-V) higher than Cernis 52 but only 4 show C2 column densities higher than this star. Remarkably, the column density of Cernis 52 is about 2 times larger than the average value observed toward stars with similar E(B-V). Out of 12 lines of sight investigated by Gredel (1999) with E(B-V) higher than Cernis 52 only 2 showed C2 column densities comparable or higher than Cernis 52. The C2 column density derived for Cernis 52 is one of the largest so far observed in translucent molecular clouds. Although its value is not as large as the one toward the much more heavily reddened star Cygnus OB2 No. 12 with E(B-V)=3.31 reported to be 20 x 10 13 cm −2 (Gredel et al. 2001). It is however comparable to the highest values reported for the stars in Gredel (1999) where star HD 172028 displays one of the highest C2 column densities (N(C2)= (11±0.5) x 10 13 cm −2 ) comparable to our measurement for Cernis 52. HD 172028 displays moderate extinction (AV =2.3) as and Gredel determines a kinetic temperature of 30 K and density of 300±100 cm −3 for the intervening cloud, both values are very similar to those found for the cloud in the line of sight of Cernis 52.
The heliocentric velocities of C2 and CH in Cernis 52 coincide so it is very likely that both molecules co-exist to a large extent in the same region. The C2 and CH column densities of translucent molecular clouds are empirically well correlated and appear to follow the relationship log N(C2) = 0.85 log N(CH)+2.2 ( van Dishoeck & Black 1989). Theoretical reasons for this correlation are discussed by Federman et al. (1994). Using the previous relationship and the CH column density N(CH) = 20 (± 2) x 10 13 cm −2 of Iglesias-Groth et al. (2010) we expected a column density of C2 of ∼ 22 x 10 13 cm −2 . This is about a factor 2 higher than our measurement, however the column density of CH derived for Cernis 52 is significantly affected by saturation of the observed absorption line and the applied correction may have led to an overestimation of the abundance by some 50%. With this consideration in mind and using the relation N(H2) = 2.6 x 10 7 N(CH) of Somerville & Smith (1989) we estimate a column density of N(H2)= 5 x 10 21 cm −2 , but cannot discard a value a factor two lower because of the saturation correction just mentioned. The fractional abundance f(C2) = N(C2)/N(H), that is f(C2)∼ N(C2)/2N(H2) ∼ N(C2)/ 2 N(H) is ∼ 1 x 10 −8 which compares well with values 0.5-1.5 x 10 −8 obtained by Gredel (1999) toward stars in the assocation surrounding NGC 2439, in Vela OB 1 and in Cen OB1 in spite of generally higher gas kinetic temperatures and densities towards these lines of sight.
Observations of the A 1 Πu -X 1 Σ + g transition of C3 at 4051.6Å in translucent sight lines have been reported by Oka et al. (2003Oka et al. ( ) andÁdámkovics et al. (2003. The observed C3 column densities range from 10 12 to 10 13 cm −2 and are well correlated with the corresponding C2 columns with a ratio N(C2)/N(C3)∼40. The observed strong correlation suggests that C3 and C2 are involved in the same chain of chemical reactions. As discussed by Oka et al., C3 formation may result from C2 by photoionization, ion neutral reactions and dissociative recombinations. The spectral range of our HET spectra do not cover this transition, but fortunately, the spectra of Cernis 52 obtained with the blue arm of ISIS at the 4.2m William Herschel Telescope (described in Iglesias-Groth et al. 2010) do include the relevant spectral range to search for this transition. We report the detection of an absorption feature with an equivalent widht of W λ = 6±1.5 mÅ fully consistent in wavelength with the transition from C3. The feature presents a weaker blend on the blue wing whose origin can be understood looking at the simulated spectra by Haffner and Meyer (1995) for the 4051.6 band of C3 in clouds of T=40 K which fits well the conditions of our cloud, or simulations for a larger range of temperatures byÁdámkovics et al. (2003). The blue line of the blend appears to be caused by the pileup of robovibronic transitions of the first lines of the R series in the wavelength range 4050-4051Å. The main feature is due to the stronger lines of the Q series in the range 4051-4052Å. Our equivalent width would mostly correspond to the pileup of these Qbranch lines. The resolving power of the WHT spectra is not sufficient to measure individual rotational lines and therefore does not allow us to derive the temperature of the cloud from C3 which would have provided an interesting comparison with the results derived from C2. Our data provides however a preliminary estimate of the column density for C3. Adopting the simplest assumption of a one-temperature rotational distribution, following the arguments given by Oka et al. (2003) (we multiply by a factor 2 the right term of the W λ -N expression in Section 3.1) and using their oscillator strength of f=0.016 we derive N(C3)= (5.2±1.3) x 10 12 cm −2 and a ratio of N(C2) / N(C3)= 20 +10 −6 which is comparable to those derived in the above papers for other lines of sight.
Models of quiescent translucent molecular clouds are presented by van Dishoeck and Black (1988) for a range of total visual extinction AV (1-5 mag), densities (500-1000 cm −3 ) and temperatures (15-40 K). Computed column densities for several species (CH,C2,CN,CO,etc...) in these models can also be found in van Dishoeck & Black (1988) which may help to predict the abundance of other species in the intervening cloud and prepare observing programmes for an extensive characterization. The constraints on density (250±50 cm −3 ) and gas kinetic temperature (40±10 K) that we have obtained provide already valuable information for the modelling of the anomalous microwave emission in the Perseus complex. The models computed by Iglesias-Groth (2005) to explain the spectral energy distribution of electric dipole radiation from hydrogenated carbon particles in this light of sight should be revised accordingly. However, we recall here that to infer the density of the cloud a major assumption was made on the value of the intensity of the radiation field in the near infrared, and since the excitation temperature curves are degeneratedversus the ratio of the density to the intensity field, we would still require an independent determination of the radiation field instead of an assumption on its value, to determine proper abundances. For instance, the strength of the radiation field was inferred to be enhanced by a factor of 3-5 in ζ Oph and other diffuse clouds with similar kinetic temperatures (van Dishoeck & Black 1988). The incident radiation field in the translucent cloud may also be enhanced because of the presence of Cernis 52. For the moment, with the limited number of molecular species so far observed toward Cernis 52, we cannot constrain the parameter of the radiation field any further. However, plausible models for the Cernis 52 cloud predict significant column densities for a variety of atomic and molecular species which may be observed in the near future (H, H + 3 , C2H, C2H2, OH, H2O, NH, CH2, etc.) and could in principle provide much better constrains on the basic parameters of the model. Simple steady state chemical kinetics (Oka et al.) indicate that the neutral molecules C2, C2H, C2H2 and C3 are more abundant than the ionic species by at least 2 orders of magnitude.
Comparison with other lines of sight in Perseus
The diffuse clouds toward ζ Per and omi Per have been extensively studied since early work on interstellar C2 (Black, Hartquist and Dalgarno 1978, Hobbs 1979, Chaffee et al. 1980, van Dishoeck & Black 1989). These two stars are located in the Perseus molecular complex, but in lines of sight where anomalous microwave emission is not detected, if this emission exists it is definitely much weaker than in the line of sight of Cernis 52. According to van Dishoeck & Black (1982) the molecular core of the ζ Per cloud would have n= 150cm −3 , T= 45 K and I∼=0.7 and the column density of C2 would be N(C2)= 1.6 x 10 13 cm −2 which compares well with the predictions in the early model by Black et al. (1978). A recent summary of results on gas kinetic temperatures and densities for these and other diffuse and translucent clouds can be found in Sonnentrucker et al. (2007). The physical conditions in these two clouds are apparently not very different to those estimated for Cernis 52 but the extinciton is 3 times higher and the abundance of C2 and CH are ∼10 times higher than in ζ and omi Per. It is possible that reactions causing destruction (C2 + H2 → C2H + H) and production (C2H + hν → C2 + H) of C2 work at very different rates at the lower temperature of the Cernis 52 cloud. A search for lines of C2H in the optical or radio would provide valuable insight on the reasons for the much higher column density of C2 in Cernis 52 and will contribute to a better understanding of the physical conditions there and in particular of the processes involved in the formation of naphthalene and anthracene cations in the intervening cloud (Iglesias-Groth et al. 2008, 2010.
CONCLUSIONS
We have presented observations of interstellar absorption lines of C2 toward Cernis 52. We have identified six rotational lines of the (2,0) and five rotational lines of the (3,0) Phillips system. The population distribution in the lowest rotational levels indicates a rotational excitation temperature of Trot = 34 ± 12 K. A detailed study of the level population leads to a best fit gas kinetic temperature T = 40±10 K and particle density n = 250 ±50 cm −3 for the molecular gas in the intervening cloud. We obtain a total C2 column density of (10.5±1.2) x 10 13 cm −2 . We also report a tentative detection of an interstellar absorption feature at the wavelength (4051.6Å) expected for an unresolved blend of the strongest Q-branch lines arising from the ground vibrational state of the C3 A 1 Πu -X 1 Σ + g electronic transition and derive from this a column density for C3 of (5.2±1.3)
x 10 12 cm −2 . The ratio C2/C3 compares well with those reported in the literature on translucent clouds (Oka et al. 2003). Higher resolution, higher S/N data will be required to confirm the C3 detection and to determine the weak rotational structure of the band which could be used as an independent diagnostic of the cloud's kinetic gas temperature. In addition, it is worth to search for longer multicarbon chains (C4, C5,... C18) in this rather carbon-rich sight line. The growth of carbon chains from C2 and C3 via C + association followed by H abstraction and recombination may be fast after C4 (Freed et al. 1982). This line of sight is well suited to carry out such studies which will provide insight on carbon chemistry.
In the intervening cloud toward Cernis 52, C2 forms dominantly in cool material at gas kinetic temperatures of 40 ± 10 K. The C2 and CH column densities are apparently both high and their relative value follow the expectations from theoretical models of translucent molecular clouds and the well established empirical relationship for this two species. The C2 molecules mainly probe the dense cold core of the cloud, measurements of column densities of H, H2 in various rotational levels and C in various fine structure levels would be worth to model in detail the intervening cloud and gain insight on the physical processes causing the anomalous microwave emission, it is important to obtain also information on a possible warmer less dense envelope. Observations of C2H may help to explain the high content of C2. This molecule may provide pathways to the formation of long-chain carbon molecules and is likely that a high content may have also favoured the formation of PAHs to a detectable level. Additional observations may establish the chemistry of the cloud and the role that a high content of C2 and CH could have on the formation of PAHs.
Figure 1 .Figure 2 .
12Fig. 1. Spectrum covering the (2,0) band of the C2 Phillips System towards Cenis 52. Detected rotational lines are identified. Spectrum covering the (3,0) band of the C2 Phillips System towards Cernis 52. Detected rotational lines are identified
Figure 3 .
3C 2 excitation diagram for Cernis 52, with observed relative rotational populations with respect to that of the J " =2 level as function of the excitation energy E(J " ). Filled diamonds correspond to individual line detections. The 1σ error bars are indicated. The theoretical populations (dotted and dashed lines) at a kinetic temperature T = 34 K are shown for comparison at several densities n (see text). The dot-dashed line indicates the thermal distribution at T=34 K.
ACKNOWLEDGEMENTS I thank D.L. Lambert and R. Rebolo for useful comments on this paper. This work was partially supported by grant AYA-2007-64748 from the Spanish Ministry of Science and Innovation.
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| [] |
[
"Phase Synchronization of non-Abelian Oscillators on Small-World Networks",
"Phase Synchronization of non-Abelian Oscillators on Small-World Networks"
] | [
"Zhi-Ming Gu \nCollege of Science\nNanjing University of Aeronautics and Astronautics\n210016NanjingPR China\n",
"Ming Zhao \nDepartment of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiPR China\n",
"Tao Zhou \nDepartment of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiPR China\n",
"Chen-Ping Zhu \nCollege of Science\nNanjing University of Aeronautics and Astronautics\n210016NanjingPR China\n",
"Bing-Hong Wang \nDepartment of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiPR China\n"
] | [
"College of Science\nNanjing University of Aeronautics and Astronautics\n210016NanjingPR China",
"Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiPR China",
"Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiPR China",
"College of Science\nNanjing University of Aeronautics and Astronautics\n210016NanjingPR China",
"Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiPR China"
] | [] | In this paper, by extending the concept of Kuramoto oscillator to the left-invariant flow on general Lie group, we investigate the generalized phase synchronization on networks. The analyses and simulations of some typical dynamical systems on Watts-Strogatz networks are given, including the n-dimensional torus, the identity component of 3-dimensional general linear group, the special unitary group, and the special orthogonal group. In all cases, the greater disorder of networks will predict better synchronizability, and the small-world effect ensures the global synchronization for sufficiently large coupling strength. The collective synchronized behaviors of many dynamical systems, such as the integrable systems, the two-state quantum systems and the top systems, can be described by the present phase synchronization frame. In addition, it is intuitive that the lowdimensional systems are more easily to synchronize, however, to our surprise, we found that the high-dimensional systems display obviously synchronized behaviors in regular networks, while these phenomena can not be observed in low-dimensional systems. | 10.1016/j.physleta.2006.10.010 | [
"https://arxiv.org/pdf/cond-mat/0607100v1.pdf"
] | 119,095,006 | cond-mat/0607100 | 7e164a3add371f0487ce355c7ed4a1d73380faed |
Phase Synchronization of non-Abelian Oscillators on Small-World Networks
5 Jul 2006 (Dated: October 1, 2018)
Zhi-Ming Gu
College of Science
Nanjing University of Aeronautics and Astronautics
210016NanjingPR China
Ming Zhao
Department of Modern Physics
University of Science and Technology of China
230026HefeiPR China
Tao Zhou
Department of Modern Physics
University of Science and Technology of China
230026HefeiPR China
Chen-Ping Zhu
College of Science
Nanjing University of Aeronautics and Astronautics
210016NanjingPR China
Bing-Hong Wang
Department of Modern Physics
University of Science and Technology of China
230026HefeiPR China
Phase Synchronization of non-Abelian Oscillators on Small-World Networks
5 Jul 2006 (Dated: October 1, 2018)PACS numbers: 8975Hc, 0545Xt
In this paper, by extending the concept of Kuramoto oscillator to the left-invariant flow on general Lie group, we investigate the generalized phase synchronization on networks. The analyses and simulations of some typical dynamical systems on Watts-Strogatz networks are given, including the n-dimensional torus, the identity component of 3-dimensional general linear group, the special unitary group, and the special orthogonal group. In all cases, the greater disorder of networks will predict better synchronizability, and the small-world effect ensures the global synchronization for sufficiently large coupling strength. The collective synchronized behaviors of many dynamical systems, such as the integrable systems, the two-state quantum systems and the top systems, can be described by the present phase synchronization frame. In addition, it is intuitive that the lowdimensional systems are more easily to synchronize, however, to our surprise, we found that the high-dimensional systems display obviously synchronized behaviors in regular networks, while these phenomena can not be observed in low-dimensional systems.
I. INTRODUCTION
Synchronization is observed in many natural, social, physical and biological systems, and has found applications in a variety of fields [1]. The large number of networks of coupled dynamical systems that exhibit synchronized states are subjects of great interest. As a theoretical paradigm, the Kuramoto model is usually used to investigate the phase synchronization among nonidentical oscillators [2,3,4]. However, some real physical systems that display synchronized phenomenon (e.g. the coupled double planar pendulums, the two-state quantum system, etc.) can not be properly described by the simply Kuramoto model. Actually, the Kuramoto oscillator is equal to the left-invariant flow on the simplest nontrivial Lie group U (1) = {e iζ |ζ ∈ R} (see the Refs. [5,6] for the fundamental knowledge of Lie group). Accordingly, in this paper, by extending the concept of Kuramoto oscillator to left-invariant flow on general Lie group, we investigate the generalized phase synchronization on networks. In particular, the left-invariant flows corresponding to non-Abelian Lie groups are named non-Abelian oscillators, which can be applied on some non-Abelian dynamical systems like quantum systems. This paper is organized as follow: In section 2, the concept of synchronization of non-Abelian oscillators is introduced. In section 3, the analyses and simulations on some typical examples are given, including the n-dimensional torus T n , the identity component of 3-dimensional general linear group GL(3, R) (i.e. GL 0 (3, R)), the special unitary group SU (2) and the special orthogonal group SO(3). Finally, we sum up this paper and discuss the relevance of phase synchronization of non-Abelian oscillators to the real world in section 4. Some mathematic remarks are given in the Appendix.
II. SYNCHRONIZATION OF NON-ABELIAN OSCILLATORS
First of all, we review the simple oscillator:
e i(ωt+θ) = Ce iωt , C = e iθ .(1)
It is obvious that x(t) = ωt + θ is the solution of the equation
dx dt = ω, x(0) = θ,(2)
which relates to the dynamical system
Φ(t, c) = ce iω t(3)
on the circle S 1 = U (1) = {e iζ |ζ ∈ R}. In terms of Lie groups, S 1 = U (1) is an abelian Lie group with its Lie algebra iR = {iω |ω ∈ R }, and the map
exp : iR → S 1 , exp(iω) = e iω ,(4)
is the exponential map of S 1 . The system (3) is also called the left-invariant flow, of which the tangent vector field is called left-invariant vector field on S 1 . This fact motivates the idea that the left-invariant flows on the general Lie groups can be considered as general oscillators. In particular, the left-invariant flows on the non-Abelian Lie groups can be considered as non-Abelian oscillators. Let G be a Lie group and Γ its Lie algebra. If v ∈ Γ, then the left-invariant flow determined by v on the G is Φ(t, g) = g · exp tv, t ∈ R, g ∈ G.
Different from those defined on the linear Euclidean spaces, the above dynamical system (5) is defined on the manifold G. Next we consider the synchronization of network, of which each node is located a general oscillator on the Lie group G with its Lie algebra Γ. The collective synchronization behavior of the coupled general oscillators starts by randomly choosing an element v α = 0 in Γ and an initial phase g 0 ∈ G for each node α. In the case of the general oscillator v α and g 0 correspond to the frequency ω and c = e iθ in Eq. (3), respectively.
It is worthwhile to note that, in general, the phase space of our general oscillator is both non-Abelian group in algebra, and non-linear space, which is a manifold, in geometry. Thus one may expect that there will be more interesting phenomena in this model. For a dynamical system on the manifold G, its equations of motion should be written in a local coordinate system because, generally, there does not exist a global coordinate system on a manifold. Let (x 1 , x 2 , · · · , x n ) be a local coordinate system about the identity e of Lie group G, n = dim G, and (0, · · · , 0) the coordinates of e. If (x 1 , x 2 , · · · , x n ), (y 1 , y 2 , · · · , y n ) and (z 1 , z 2 , · · · , z n ) are the coordinates of g 1 , g 2 ∈ G and g 1 g 2 ∈ G in this coordinate system respectively, then the multiplication function f = (f 1 , f 2 , · · · , f n ) can be defined as:
z 1 = f 1 (x 1 , · · · x n ; y 1 , · · · y n ) · · · z n = f n (x 1 , · · · x n ; y 1 , · · · y n ).(6)
Suppose that v ∈ Γ, then the equations of the leftinvariant flow determined by v are
dx j (t) dt = i a i l j i (x(t)), 1 ≤ j ≤ n,(7)
where (a 1 , a 2 , · · · a n ) is the coordinates of v with respect to the basis { ∂ ∂x1 , ∂ ∂x2 , · · · , ∂ ∂xn } x=0 , and l = (l j i ) is the matrix valued function:
l j i (x) = ∂f j (x, y) ∂y i | y=0 , 1 ≤ i, j ≤ n.(8)
The set of equations (7) is the intrinsic dynamic of a single general oscillator. Note that, the multiplication functions are the local representation of the multiplication in the Lie group, and the matrix valued function (8) is the map induced by multiplying by x at left. This map translates every vector in the Lie algebra to x. Eqs.
(7) mean that the left-invariant flow determined by v are made from the vector field whose value at x is just the vector into which translating v. Similar to the case of coupled simple oscillators [12], the coupled dynamics of general oscillators are:
dx α j (t) dt = i a α i l j i (x α (t)) − 1 kα β∈Λα q j x α (t) − x β (t) x α j (0) = x α j0 ,(9)
where q j is the coupling function assumed smooth, Λ α is the set of α's neighboring nodes, and x α j0 is the coordinates of initial phase of node α. According to the existence and uniqueness theorem of ordinary differential equations, Eq. (9) has an unique solution. Then, by the theory of Lie groups, this solution can be extended to an one-parameter subgroup, which corresponds to a left-invariant flow.
III. SIMULATION RESULTS FOR SOME TYPICAL EXAMPLES
In this section, we will show some analyses and simulations on several typical examples. All the simulations are obtained based on the Watts-Strogatz (WS) network [8], which can be constructed by starting with one-dimensional lattice and randomly moving one endpoint of each edge with probability p. The network size We firstly investigate the n-dimensional torus, T n = S 1 × S 1 × · · · × S 1 n , which is a connected compact Lie group. Denote G 1 = T n and its Lie algebra Γ 1 = iR × iR × · · · × iR n with [, ] ≡ 0 (i.e. Abelian). This is a direct extension of the simple Kuramoto oscillators, and will degenerate to Karumoto oscillators when n = 1. It follows that an element in Γ 1 , that is an intrinsic frequency, is v = (iω 1 , iω 2 , · · · , iω n ).
It is easy to see that the general oscillation determined by v is periodic if ω 1 , ω 2 , · · · ω n are integer-linearly dependent, and quasi-periodic if ω 1 , ω 2 , · · · ω n are integerlinearly independent. For the equations of motion we have
d dt x α 1 . . . x α n = ω α 1 . . . ω α n − K k α β∈Λα sin(x α 1 − x β 1 ) . . . sin(x α n − x β n ), (10)
with the initial conditions x α j (0) = x α j0 , where K is the coupling strength. To characterize the synchronized states of the jth dimension, we use the order parameter
m j = 1 N α e ix α j , j = 1, 2, · · · , n,(11)
where {·} signifies the time averaging. Fig. 1 displays the relations between the phase order parameter and the coupling strength for various values of the rewiring probability p, where n = 2. Since the dynamical behaviors in different dimensions are independent, and each one is equal to a simply Kuramoto oscillator, the synchronization behavior in each dimension is the same as that of a Kuramoto oscillator [12]. In a word, the small-world effect ensures the global synchronization for sufficiently large coupling strength, and the higher disorder (i.e. larger p) will enhance the network synchronizability.
This example can be used to describe the collective behavior of the coupled double planar pendulums in a network. Furthermore, according to the Liouville-Arnold theorem, the phase space of a integrable system, if it is compact, is a n-torus. Therefore, this example can be widely applied to describe the synchronization phenomenon of integrable systems. In this circumstances the equations of motion are (12) with the initial conditions x α ij (0) = x α ij0 , where (a α ij ) ∈ Γ 2 and K is the coupling strength.
dx α ij (t) dt = 3 p=1 x α ip (t)a α pj − K k α β∈Λα (x α ij (t) − x β ij (t)),
Since G 2 is a subset of the Euclidean space R 9 , the coupling term of arbitrary pair of nodes α and β can be directly written as x α ij − x β ij . Therefore, the order parameter is defined as:
m = 2 N (N − 1) α<β exp −c x α −x β 2 ,(13)wherex α = (x α ij ) = (x α ij ) M and M = max x β (t) 1≤β≤N .
In Fig. 2, we report the simulation results about the order parameter m as a function of K for different rewiring probabilities. Similar to the situation of T n , this system can approach to the global synchronized state for sufficiently large K when p is large enough. It is worthwhile to emphasize that the order parameter of different systems can not compare to each other directly, since their definitions are not the same.
C. SU (2), the special unitary group SU (2) = S 3 , the special unitary group, is a connected compact Lie group consisted of all anti-Hermitian traceless matrices. Denote G 3 = SU (2), and Γ 3 its Lie algebra having the following basis
J 1 = 1 2 i 0 0 −i , J 2 = 1 2 0 1 −1 0 , J 3 = 1 2 0 i i 0 .
Give an arbitrary element g ∈ SU (2), its local coordinates (x 1 , x 2 , x 3 ) can be found from:
g = w 1 w 2 −w 2w1 = (exp x 1 J 1 )(exp x 2 J 2 ) (exp x 3 J 3 ) ,(14)
and
w 1w1 + w 2w2 = 1.(15)
The corresponding equations of motion are
dx α j (t) dt = 3 i=1 a α i l j i (x α (t)) − K k α β∈Λα sin 1 2 (x α j (t) − x β j (t)),(16)
with the initial condition x α j (0) = x α j0 , j = 1, 2, 3, where K is the coupling strength. It is worthwhile to emphasize that the behaviors of this dynamical system is independent to the local coordinate systems. It follows that the local coordinates (14) in SU (2) can be replaced by
g = exp (x 1 J 1 + x 2 J 2 + x 3 J 3 ) ,(17)
which is convenient for the description of two-state quantum systems since the matrices J 1 ,J 2 , and J 3 are relative to Pauli matrices as:
−2iJ 1 = σ 3 , −2iJ 2 = σ 2 , −2iJ 3 = σ 1 .(18)
Note that, a one-parameter subgroup U (t) ⊂ SU (2) can be considered as the time-evolution in the quantum mechanics. Since the subgroup e iζ U (t), where ζ is an arbitrary real number, represents the same time-evolution as U (t) does, we call two different nodes α and β are synchronous if they are only differ in a phase factor e iζ , that is
w α 1 w α 2 −w α 2w α 1 w β 1 w β 2 −w β 2w β 1 −1 = e iζ 0 0 e −iζ , ζ ∈ R.
(19) Therefore, if the equation
w α 1 w β 2 = w α 2 w β 1(20)
is hold for every 1 ≤ α < β ≤ N , the completely global synchronization is achieved. Consequently, for each pair of nodes α and β, define the dynamical "distance" between them as
µ αβ = w α 1 w β 2 − w α 2 w β 1 ,(21)
where | · | signifies the modulus of the complex number. Accordingly, the order parameter is
m 3 = [ exp (− max {µ αβ |1 ≤ α < β ≤ N }) ] .(22)
In Fig. 3, we report the simulation results about the order parameter m as a function of K for different rewiring probabilities. Analogously, this system can approach to the global synchronized state for sufficiently large K when p is large enough. Note that, e −1 ≤ m ≤ 1 for 0 ≤ |µ αβ | ≤ 1. Thus in the uncoupling case (i.e. K = 0), the order parameter m approaches to 0.37.
In terms of the Hopf bundle S 1 → S 3 p −→ S 2 , the base space S 2 is the set of states of a two-state quantum system (see the ref. [9] and Appendix B for details). Therefore, let a two-state quantum system be located at each node of a small-world network, the whole system can achieve the synchronized state if the coupling strength is large enough. Note that, the special orthogonal group SO(3) has an isomorphic Lie algebra to Γ 3 for the existence of the double covering SU (2) → SO(3), which is a homomorphism. Therefore, the dynamical system SO(3) has the same synchronized behavior as that of SU (2). The case where each node is located with a top can be represented by the system SO(3) [10].
IV. CONCLUSION
In this paper, we extended the concept of Kuramoto oscillator to the left-invariant flow on general Lie group, and studied the generalized phase synchronization on small-world networks. The left-invariant flow on Lie group can be used to represent many significant physical systems, such as the integrable systems (e.g. the double planar pendulums), the two-state quantum systems (e.g. Ising model), the motions of top, and so on. In particular, the dynamics of two-state quantum systems on complex networks are extensively studied recently [11,12,13], the present work can provide us a theoretical frame to investigate their collective synchronized behaviors.
It is intuitive that the low-dimensional systems are more easily to synchronize. However, one may note that in the regular networks (i.e. p = 0), the system T 2 does not exhibit synchronized behavior while the obvious synchronization is observed in the higher dimensional systems GL 0 (3, R) and SU (2) when the coupling strength gets large. This is an interesting phenomenon that worths a further study in the future.
FIG. 1 :
1(Color online) Order parameters m1 (a) and m2 (b) vs coupling strength K for different values of the rewiring probability p. All the simulation results are the average over 100 independent runs corresponding to the case of n = 2.
FIG. 2 :
2(Color online) Order parameter m vs the coupling strength K. All the simulation results are obtained by averaging 100 independent runs. N = 1000 and average degree k = 6 are fixed. In our numerical simulation below, the coordinates a i are chosen randomly and uniformly in the range (−0.5, 0.5), and the initial phases x α j0 in (0, 2π). All the numerical results are obtained by integrating the dynamical equations using the Runge-Kutta method with step size 0.01. The order parameters (see below) are averaged over 2000 time steps, excluding the former 2000 time steps, to allow for relaxation to a steady state.A. T n , the n-dimensional torus
B. GL0(3, R), the identity component of 3-dimensional general linear group GL(3, R) GL 0 (3, R), the identity component of GL(3, R), is a connected Lie group of dimension 9 (Actually, GL 0 (3, R)is a open subset of R 9 ). Denote G 2 = GL 0 (3, R), and Γ 2 its Lie algebra containing all the 3 × 3-real matrices. The coordinates (g ij ) for all g = (g ij ) ∈ G 2 can be addressed on the basis E ij of Γ 2 (see the case (a) of Appendix A
FIG. 3 :
3(Color online) Order parameter m vs the coupling strength K. All the simulation results are obtained by averaging 100 independent runs.
AcknowledgmentsWe thank Dr. Hui-Jie Yang for helpful discussions. This work was partially supported by the National Nat-APPENDIX A: HOW TO CHOOSE THE COORDINATE SYSTEMIn order to investigate the collective synchronization behaviors of the coupled non-Abelian oscillators, we have to integrate Eq. (9) numerically. Because of the using of local coordinate system, a question rises: Does the Eq. (9) express the behavior of the system in the whole manifold G? The answer is as follows.Case (a) Manifold G is a connected open subset of Euclidean space R n . In this case, we can directly use the natural coordinate system of R n as the local coordinate system of G.Case (b) G is a connected compact Lie group. In this case, we have exp(Γ) = G, so we can choose the coordinate system of Γ as the local coordinate system of G. According to the theory of fiber bundles, the condition (20) is equivalent to that g α (t) and g β (t) is all in the same fiber of the Hopf bundle S 1 → S 3 p −→ S 2 . In order to interpret the map p we consider the 2-sphere S 2 as the complex projective line CP 1 in which a point is written as [w 1 , w 2 ], where w 1 and w 2 are two complex numbers and not all zero. Here, the point [w 1 , w 2 ] is identified with [λw 1 , λw 2 ] for any nonzero complex number λ. Then the map p can be written asIt is easy to show that the inverse image p −1 [w 1 , w 2 ] is the sphere S 1[9]. This fact tells us that, in the sense of our definition, the collective synchronization behaviors of the coupled non-Abelian oscillators SU (2) can represent the two-state quantum system.
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| [] |
[
"On asymptotic power corrections to differential fluxes and generalization of optical theorem for potential scattering",
"On asymptotic power corrections to differential fluxes and generalization of optical theorem for potential scattering"
] | [
"S E Korenblit ",
"A V Sinitskaya \nIrkutsk National Research Technical University\n83 Lermontov str664074IrkutskRussia\n",
"\nDepartment of Physics\nIrkutsk State University\n20 Gagarin blvd664003IrkutskRussia\n"
] | [
"Irkutsk National Research Technical University\n83 Lermontov str664074IrkutskRussia",
"Department of Physics\nIrkutsk State University\n20 Gagarin blvd664003IrkutskRussia"
] | [] | In a wide class of potentials the exact asymptotic dependence on finite distance R from scattering center is established for outgoing differential flux. It is shown how this dependence is eliminated by integration over solid angle for total flux, unitarity relation, and optical theorem. Thus, their applicability domain extends naturally to the finite R. | 10.1142/s0217732317500663 | [
"https://arxiv.org/pdf/1607.00625v2.pdf"
] | 118,476,552 | 1607.00625 | 62065274ab9ce9f519bfef6ffd7122da7549d8e8 |
On asymptotic power corrections to differential fluxes and generalization of optical theorem for potential scattering
1 Apr 2017 April 4, 2018
S E Korenblit
A V Sinitskaya
Irkutsk National Research Technical University
83 Lermontov str664074IrkutskRussia
Department of Physics
Irkutsk State University
20 Gagarin blvd664003IrkutskRussia
On asymptotic power corrections to differential fluxes and generalization of optical theorem for potential scattering
1 Apr 2017 April 4, 2018arXiv:1607.00625v2 [quant-ph] 3:2 WSPC/INSTRUCTION FILE 17˙mpla Modern Physics Letters A c World Scientific Publishing Companyasymptotic expansiondifferential fluxescross-sectionsunitarity relation PACS Nos: 0365-w, 0365Bz, 0365Nk, 1180Et, 1180Fv
In a wide class of potentials the exact asymptotic dependence on finite distance R from scattering center is established for outgoing differential flux. It is shown how this dependence is eliminated by integration over solid angle for total flux, unitarity relation, and optical theorem. Thus, their applicability domain extends naturally to the finite R.
Introduction
According to the common rules, 1-5 the differential cross-section dσ for the scattering on hermitian scalar spherically symmetric potential U (R) is uniquely defined by onshell scattering amplitudes f ± (q; k). These amplitudes are defined as coefficients at outgoing or incoming spherical waves being the first order terms of asymptotic expansion of the scattering wave functions Ψ ± k (R) for R = |R| → ∞, R = Rn, q = kn, k = kω, n 2 = ω 2 = 1:
Ψ ± k (R) −→ R→∞ e i(k·R) + f ± (q; k) e ±ikR R + O(R −2 ),(1)Ψ − k (R) = Ψ + −k (R) * , f − (q; k) = f + (q; −k) * ,(2)dσ = f + (q; k) 2 dΩ(n), where: σ = f + (kn; kω) 2 dΩ(n),(3)
is the respective total (elastic) cross-section, which also does not depend on R. Of course the terms of order O(R −2 ) in Eq. (1) are unimportant 1-5 for both definitions (3). However, R is finite for real experiments, and the recent investigations 6-9 of (anti-) neutrino processes at short distances from the source reveal a possible violation of inverse-square law for event rate corresponding 7,8 to (1) and (3). Since the macroscopic parameter of distance R has very peculiar meaning when it is considered in the framework of quantum field theory, 7-10 it seems natural and convenient to elucidate this problem at first for nonrelativistic quantum-mechanical scattering.
In the following sections the closed formula and recurrent relation for coefficients of asymptotic expansion of wave function Ψ ± k (R) in all orders of R −s are obtained in terms of the on-shell scattering amplitudes f ± (q; k) only. This expansion together with obtained exact asymptotic expression for interference fluxes reveals for finite R the necessity to replace the differential cross-section (3) by the normalized outgoing differential flux. Nevertheless, the second definition of Eq. (3) for total cross-section, which thus is replaced by total outgoing flux, remains unchanged together with the unitarity relation and the optical theorem, as all their asymptotic power corrections precisely disappear.
Asymptotic expansion of scattering wave function
To show the nature of asymptotic expansion we have to recall some properties 1-5 of wave functions and amplitudes (2). The function Ψ ± k (R) (1), being solution of Schrödinger equation for the energy E > 0, satisfies Lippman-Schwinger equation:
∇ 2 R + k 2 Ψ ± k (R) = V (R)Ψ ± k (R), for: k 2 = 2M 2 E, V (R) = 2M 2 U (R),(4)Ψ ± k (R) = e i(k·R) − d 3 x e ±ik|R−x| 4π|R − x| V (|x|)Ψ ± k (x) ≡ e i(k·R) + J ± k (R).(5)
Here the differential vector-operator and the operator of angular momentum square in the spherical basis n, η ϑ , η ϕ have the following properties for R = Rn, n = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ), η ϑ = ∂ ϑ n, η ϕ sin ϑ = ∂ ϕ n :
∇ R = n∂ R + 1 R ∂ n , (n · ∇ R ) = ∂ R , ∂ n = η ϑ ∂ ϑ + η ϕ sin ϑ ∂ ϕ ,(6)
(n · ∂ n ) = 0, (∂ n · n) = 2, (n × ∂ n ) 2 = ∂ 2 n , (n × ∂ n ) = iL n ,
−∂ 2 n = L 2 n = 2R(n · ∇ R ) + R 2 (n · ∇ R ) 2 − ∇ 2 R , whence,(8)
for cos ϑ = c :
L n ≡ L 2 n = − ∂ c (1 − c 2 )∂ c + (1 − c 2 ) −1 ∂ 2 ϕ ,(10)
and the well-known representation also is used for arriving from point x to point R spherical wave being free 3-dimensional Green function: 1-5
e ±ik|R−x| 4π|R − x| = d 3 q (2π) 3 e i(q·(R−x)) (q 2 − k 2 ∓ i0) .(11)
When x = 0 it satisfies the well-known inhomogeneous equation:
∇ 2 R + k 2 e ±ikR 4πR = − δ 3 (R).(12)
Then the power index ±ikR is defined in the sense of analytic continuation with a small real negative admixture: 4, 5 ±ik → −(−k 2 ∓ i0) 1/2 = ±ik − 0, which is almost nowhere written but is everywhere assumed. The following Lemma is in order.
Lemma 1. When R = Rn, x = rv, v = (sin β cos α, sin β sin α, cos β), |x| = r < R and operator L n = L 2 n (or n → v) is defined by Eqs. (7)-(10) with positively defined On asymptotic power corrections to differential fluxes and generalization of optical theorem for potential scattering 3 operator L n + 1 4 = (Λ n + 1 2 ) 2 such that Λ n + 1 2 = L n + 1 4 is also positively defined, then:
e ±ik|R−x| 4π|R − x| = χ Λn (∓ikR + 0) 4πR e ∓ik(n·x) ∼ (13) ∼ e ±ikR 4πR 1 + ∞ s=1 s µ=1 [L n − µ(µ − 1)] s!(∓2ikR) s e ∓ik(n·x) .(14)
Proof. The expression (13) for R > r is a formal operator rewriting of the usual multipole expansion of free Green function 3 (11) with the help of self-adjoint operator formally introduced instead of l: l → Λ n but never really arising and with the help of multipole expansion of plane wave, 3 that are listed also in Ref., 11 formulae (8.533), (8.534):
e ±ik|R−x| 4π|R − x| = 1 4πkRr ∞ l=0 i ∓l χ l (∓ikR + 0) ψ l 0 (kr)(2l + 1)P l ((n · v)) ,(15)
e ∓ik(n·x) = 1 kr ∞ l=0 i ∓l ψ l 0 (kr)(2l + 1)P l ((n · v)) ,
Here the spherical functions Y m l (n) = n|l m and Legendre polynomials P l (c) being eigenfunctions of self-adjoint operator (10) on the unit sphere for c = (n · v) or c = cos ϑ satisfy the well-known orthogonality, parity, completeness and other conditions 1-5, 13, 14 (A.1) -(A.7) with the delta-function δ Ω (n, v) on the unit sphere.
The solutions χ l (∓ikr), ψ l 0 (kr) of free radial Schrödinger equation:
r 2 1 r ∂ 2 r r + k 2 ψ l 0 (kr) r = l(l + 1) ψ l 0 (kr) r ,(17)
are defined by Macdonald K λ (z) and Bessel J λ (y) functions 11-14 (A.8) -(A.13) that for integer l, i.e. half integer λ = l + 1 2 are reduced to elementary functions:
χ l (bR) = 2bR π 1/2 K l+ 1 2 (bR), χ l (bR) ==⇒ l=int e −bR l s=0 (l + s)! s!(l − s)!(2bR) s . (18)
The function K λ (z) (A.8) is entire function 11, 12 of λ 2 . This is the reason the welldefined operator Λ n introduced in Eq. (13) does not appear explicitly.
The expansion (14) for large R is the known asymptotic expansion of function (13), being infinite asymptotic version 11, 12 of the sum (18) for arbitrary non-integer l, |arg(bR)| < 3π/2, is supplemented by observation 11,12 for the product:
(l + s)! (l − s)! = s µ=1 (l − µ + 1)(l + µ) = s µ=1 [l(l + 1) − µ(µ − 1)] .(19)
Due to (A.1) it may be factored out 9 from the sum over l (15) as operator product in the right hand side of Eq. (14), thus converting this sum into the expansion (14).
Remark. The operator L n in Eq. (14) may be replaced by operator in square brackets of the left hand side of Eq. (17) or by the similar operator with interchange of r ⇋ k with the same result.
Theorem 1. Let the potential V (r) have finite first absolute moment and decrease at r → ∞ faster than any power of 1/r. Then the integral J ± k (R) in Eq. (5) for sufficiently large R admits asymptotic power expansion whose coefficients are defined by the on-shell scattering amplitudes f ± (q; k) only. This expansion has asymptotic sense 12 even though the potential V (r) in Eq. (4) has a finite support:
J ± k (R) ∼ e ±ikR R f ± (kn; k) + ∞ s=1 h ± s (kn; k) (∓2ikR) s ,(20)
with
: h ± s (kn; k) = 1 s! s µ=1 [L n − µ(µ − 1)] f ± (kn; k),(21)or: h ± s (kn; k) = L n − s(s − 1) s h ± s−1 (kn; k), k = kω,(22)
and is equivalent to infinite reordering of its asymptotic multipole expansion: 3-5
J ± k (R) ≃ 1 R ∞ j=0 χ j (∓ikR)(2j + 1)η ± j (k)P j (±(n · ω)) ,(23)with: h ± s (kn; k) = 1 s! ∞ j=s (j + s)! (j − s)! (2j + 1)η ± j (k)P j (±(n · ω)) ,(24)
and:
f ± (kn; k) = h ± 0 (kn; k) = ∞ j=0 (2j + 1)η ± j (k)P j (±(n · ω)) ,(25)for: f ± (kn; k) = − 1 4π d 3 x e ∓ik(n·x) V (|x|)Ψ ± k (x), x = rv,(26)
as the usual on-shell scattering amplitude. [1][2][3][4][5] Proof. Suppose at first the finite support for V (r) at r a. Then for R > a we can directly substitute the expression (13) into representation (5) for J ± k (R) with the following result after interchange of the order of differentiation and integration for Fourier transformation (26), what is justified 12 also for asymptotic series (14):
J ± k (R) ≃ χ Λn (∓ikR) R f ± (kn; k) ∼ (27) ∼ e ±ikR R 1 + ∞ s=1 s µ=1 [L n − µ(µ − 1)] s!(∓2ikR) s f ± (kn; k).(28)
On asymptotic power corrections to differential fluxes and generalization of optical theorem for potential scattering 5
This is exactly the asymptotic expansion (20) with the coefficients h ± s (kn; k) defined by Eq. (21). However, that is not the case for the potential V (r) with infinite support. Estimating it for r > R as |V (r)| < C N /r N with arbitrary finite N ≫ 1, two pieces of correction that should be added may be easy estimated as:
∆ R J ± = − r>R d 3 x e ±ik|R−x| 4π|R − x| V (r)Ψ ± k (x),(29)∆ R f ± = χ Λn (∓ikR) 4πR r>R d 3 x e ∓ik(n·x) V (r)Ψ ± k (x),(30)|∆ R J ± | < ||Ψ||C N (N − 2)R N−2 , |∆ R f ± | < ||Ψ||C N (N − 3)R N−2 1 + O(R −1 ) ,(31)
with the finite norm:
4, 5 ||Ψ|| = sup x |Ψ(x)| of functions Ψ ± k (x).(32)
Due to the arbitrariness of N ≫ 1 for these corrections the asymptotic expansion conserves its form (20), (28) but acquires additional asymptotic sense 12 compared with expansion (14). Indeed, due to the partial wave decomposition (25) of scattering amplitude f ± (kn; k), expression (27) The assumed potential V (r) for the case of infinite support has only finite effective radius 4 and provides a slowdown fall 4, 5 of partial waves at j → ∞, e.g. like |η j (k)| ∼ e −τ j , τ > 0 for potential of Yukawa-type. This is enough for convergence of partial wave decompositions (24), (25) but can not provide convergence of the multipole expansion (23) which now also acquires the asymptotical sense. Its infinite reordering (23) → (20), (24) given here simply "displaces" this asymptotic sense from the summation over angular momentum j onto the always asymptotic expansion on integer powers R −s whose coefficients now are well-defined as derivatives (21) of scattering amplitude with respect to c = (n · ω), or as convergent partial wave decompositions (24). Thus, all these coefficients are observable.
Remark. From the estimations (29) -(32) it is also clear 3-5 that even standard asymptotic (1) requires for the potential N > 3 at least. More generally these estimations mean that for |V (r)| C N /r N with r → ∞ the asymptotic expansion (28) is applied until s [N − 3]. Thus, the further consideration is possible only for potentials V (r), specified with the conditions of Theorem 1.
Differential fluxes and unitarity relation
To make a careful analysis of different fluxes, the following Lemma 2 is useful.
Lemma 2.
The function e ikr[1−(n·v)] as a distribution on the space of infinitely smooth functions H(n) on the unit sphere n(cos ϑ, ϕ), parametrized by (6), has the following exact operator representation for c = (n · v). Let H(c) be defined as:
H(c) == 1 −1 dc δ(1 − c) − e 2ikr δ(1 + c) (−ikr + ∂ c ) −1 H(c).(34)
Proof. With dΩ(n) = sin ϑ dϑ dϕ = −dc dϕ the result is obtained by using integration over c by parts infinite number of times. The operator in Eq. (35) has a sense of a formal series over powers of differential operator ∂ c . The well-known standard asymptotic relation 1-4 of the first order on 1/r corresponds here to ∂ c → 0. Now let's consider the elementary flux of non-diagonal current J q,k (R) through a small element of spherical surface nR 2 dΩ(n), for R = Rn, q = kv, k = kω, and
↔ ∇ R = → ∇ R − ← ∇ R , ↔ ∂ R = → ∂ R − ← ∂ R = n · ↔
∇ R according to (7) -(9). Total flux through any closed surface is zero because the current is conserved 4 due to Eq. (4):
J q,k (R) = 1 2i Ψ + q (R) * ↔ ∇ R Ψ + k (R) , (∇ R · J q,k (R)) = 0,(36)R 2 dΩ(n) (n · J q,k (R)) = R 2 dΩ(n) 1 2i Ψ + q (R) * ↔ ∂ R Ψ + k (R) −→ (37) −→ R 2 dΩ(n) k 2 e ikR(n·(ω−v)) (n · (ω + v)) + (38) + dΩ(n) 2i * f + (kn; kv) χ← Λ n (ikR) ↔ ∂ R χ→ Λ n (−ikR) f + (kn; kω) − (39) − dΩ(n) 2i e ikR[1−(n·v)] e z z(n · v) + 1 − z ∂ ∂z χ→ Λn (z)f + (kn; kω) z = 0−ikR − (40) − v ⇋ ω * .(41)
Here, for sufficiently large R, the expressions (5) and (27) On asymptotic power corrections to differential fluxes and generalization of optical theorem for potential scattering 7 of action for the operators ← L n and → L n (9). Integration of separate terms over solid angle dΩ(n) with fixed R gives here the following interesting results. For the flux of incoming plane waves (38), since ((ω − v) · (ω + v)) = ω 2 − v 2 = 0, one has: 4
R 2 dΩ(n) k 2 e ikR(n·(ω−v)) (n · (ω + v)) = 0.(42)
For the flux (39) we can ignore the arrows of Λ n because operator L n (10) is selfadjoint on the unit sphere. So, the Wronskian (A.12) leads to the total flux:
dΩ(n) 2i * f + (kn; kv) χ← Λn (ikR) ↔ ∂ R χ→ Λn (−ikR) f + (kn; kω) = (43) = k dΩ(n) * f + (kn; kv)f + (kn; kω).
This is the total non-diagonal outgoing flux for finite R, The lines (40), (41) represent the non-diagonal interference (v = ω) between incoming and outgoing fluxes. According to Lemma 2 for the first exponential of (40), it takes place only in corresponding forward and backward directions. Note that any averaging over R due to rapidly oscillating exponent e 2ikR eliminates 3 the contribution of backward direction in Eq. (35). With this elimination and definition (33), the line (40) for (n · v) = c gives:
− 2π 0 dϕ 2i 1 −1 dc δ(1 − c) e z (z + ∂ c ) zc + 1 − z ∂ ∂z χ→ Λn (z)f + (kn; kω) z = 0−ikR .(44)
By moving the operator from denominator into the exponential for z = 0 − ikR:
e z (z + ∂ c ) = ∞ 0 dξe z(1−ξ)−ξ∂c ,(45)
after simple commutations, one obtains for Eq. (44)
= − z 2i 2π 0 dϕ ∞ 0 dξ e −ξ∂c χ→ Λn (z) ↔ ∂ z + 1 z e z(1−ξ) f + (kn; kω) c=1 z = 0−ikR ,(46)
with c = 1 where possible. For the arbitrary term of partial wave decomposition (25) the scattering amplitude here is effectively replaced by Legendre polynomial: f + (kn; kω) → P j ((n · ω)). This substitution immediately replaces χ→ Λ n (z) → χ j (z), thus permitting to make all remaining ϕ-integration and ∂ c -differentiations in the closed form by using the relations (A.6), (A.11), (A.12), wherefrom for:
2π 0 dϕP j ((n · ω)) = 2πP j ((v · ω)) P j (c), e −ξ∂c P j (c) c=1 = P j (1 − ξ),(47)
it follows:
− z 2i 2π 0 dϕ ∞ 0 dξ e −ξ∂c χ j (z) ↔ ∂ z + 1 z e z(1−ξ) P j ((n · ω)) c=1 z = 0−ikR = (48) = − 2π 2i z P j ((v · ω)) χ j (z) ↔ ∂ z + 1 z ∞ 0 dξ e z(1−ξ) e −ξ∂c P j (c) c=1 z = 0−ikR = (49) = − 2π 2i z P j ((v · ω)) χ j (z) ↔ ∂ z + 1 z χ j (−z) z = − 4π 2i P j ((v · ω)) ,(50)
or, equivalently:
(44) = (46) = − 4π 2i f + (kv; kω).(51)
Thus, the contribution of the lines (40), (41) into the full flux in accordance with the left hand side of unitarity condition 1-5 becomes equal to:
− 4π 2i f + (kv; kω) − * f + (kω; kv) = −4π Im f + (kv; kω),(52)
but now taking into account all the possible asymptotic power corrections:
4π k Im f + (kv; kω) = dΩ(n) * f + (kn; kv)f + (kn; kω).
The diagonal case v = ω in Eqs. (43), (53) represents the optical theorem 1-5 with total cross-section σ of Eq. (3) in the right hand side and is not changed by these power corrections also. Moreover, since the operator of angular momentum square (10) depends for this case only on one variable: L n → ∂ c (c 2 − 1)∂ c , the result (51) may be checked in first several orders of R −s directly from Eqs. (27), (28), (44) on operator level. However, for finite R the differential cross-section of Eq. (3) has to be replaced now by diagonal outgoing differential flux dσ(R) (39) normalized 1-5 to the density k of incoming flux (38) for v = ω. It still contains asymptotic power corrections defined by Eqs. (27), (28) of Theorem 1:
dσ(R) dΩ(n) = 1 2ik * f + (kn; kω) χ← Λn (ikR) ↔ ∂ R χ→ Λn (−ikR) f + (kn; kω) = (54) = |f + (kn; kω)| 2 − 1 kR Im * f + (kn; kω) → L n f + (kn; kω) + 1 4(kR) 2 · · → L n f + (kn; kω) 2 − Re * f + (kn; kω) → L 2 n f + (kn; kω) + O 1 R 3 .
On asymptotic power corrections to differential fluxes and generalization of optical theorem for potential scattering 9
In terms of partial wave decomposition (25) with corresponding phase shifts δ j (k), for c = (n · ω), η j (k) = η + j (k), kη j (k) = e iδj (k) sin δ j (k), ∆ jl = j(j + 1) − l(l + 1), with the help of (A.13) -(A.15), it reads:
dσ(R) dΩ(n) = ∞ l,j=0 (2l + 1)(2j + 1) * η l (k)η j (k)P l (c)P j (c) (χ l (ikR) ↔ ∂ R χ j (−ikR)) 2ik ,(55)
where:
(χ l (ikR) ↔ ∂ R χ j (−ikR)) 2ik = 1 − ∆ jl 2ikR − ∆ 2 jl 8(kR) 2 + O 1 R 3 .(56)
The power corrections arising in (54) or in this two-fold series for j = l in (56), may be observable for slowly moving particles with k → 0. They contain only real or imaginary parts of the products * η l (k)η j (k) and automatically disappear for j = l in the total outgoing flux σ being the total cross-section (3) now, or in the limit R → ∞ for outgoing differential flux:
σ = σ ≡ dΩ(n) dσ(R) dΩ(n) , dσ = lim R→∞ dσ(R).(57)
Since for real potential the Born approximation for amplitudes f + (kn; kω), η j (k) is real, 1-4 it is not enough to obtain the non zero first order correction R −1 from Eqs.
Identical particles with spin
In case of mutual scattering of identical Bose-or Fermi-particles 1-3 with spin S one faces symmetrical or antisymmetrical scattering wave functions, amplitudes and respective cross-sections. The proper generalizations are straightforward, and instead of (5), (27) one has:
Ψ + k(±) (R) ≃ e i(k·R) ± e −i(k·R) + χ→ Λn (−ikR) R F + (±) (kn; k),(58)
with: F + (±) (kn; k) = f + (kn; k) ± f + (−kn; k), and then: (59)
dσ (±) (R) dΩ(n) = 1 2ik * F + (±) (kn; kω) χ← Λ n (ikR) ↔ ∂ R χ→ Λ n (−ikR) F + (±) (kn; kω) . (60)
Of course, the partial wave decompositions like (25), (55) contain now only even j for F + (+) and only odd j for F + (−) . The normalized outgoing differential fluxes (60) again have to replace corresponding differential cross-sections. For the scattering of nonpolarized identical particles the outgoing differential fluxes are defined by the usual way 2 as:
dσ S (R) = w (+) (S) dσ (+) (R) + w (−) (S) dσ (−) (R),(61)
with the well known 1, 2 probabilities w (±) (S) for Bose-and Fermi-particles. Similarly (57) integration of the flux (61), (60) over solid angle again obviously leads to the independent of R total cross-section σ S for identical particles with spin S.
Conclusions
As it is well-known, for a point-like, even anisotropic, stationary source of classical particles, or rays of light, or incompressible fluid, the radial flux of outgoing particles in a given solid angle does not depend on distance R at all, due to the local conservation of classical current density.
Turning to the wave picture, such independence is true only for the flux of pure spherical outgoing or incoming wave in Eq. (1) (see the first term in the right hand side of Eq. (54)). That all results in well-known inverse-square law for event rate, which explicitly contains 1/R 2 (see, e.g. Refs. 7,8 ). A possible violation of this law is the subject of our interest. We show this violation is a pure wave effect, arising from nonsphericity of the exact scattering wave, i.e. from the next terms R −s (s > 1) of asymptotic expansion. The last is investigated here up to all orders, also using again the conservation of corresponding current.
To this end, using the operator-valued asymptotic expansion for free Green function of Helmholtz equation, the asymptotic expansion for the wave function of potential scattering on inverse integer power of distance R from scattering center is obtained. It is shown how these power corrections affect the definition of outgoing differential flux and interference flux.
Surprisingly, these power corrections precisely entirely disappear in total outgoing flux, unitarity relation, and optical theorem due to integration over solid angle at finite R. Thus, the applicability domain of these relations a naturally extends to finite R for fast ehough decrising potentials.
It is worth to note that all obtained corrections are defined by observable onshell amplitude or partial phase shifts. Nevertheless, the real observation of this dependence involves reevaluation of the phase shifts extracted earlier in fact from outgoing differential flux at finite R (54), (55), (61) without taking into account any corrections on the finite distance.
Although asymptotic expansion by its nature has no sense as infinite sum, the obtained asymptotic expansions of the wave function and outgoing differential flux have a sense up to any finite order s of R −s if potential U (R) has a finite support or decreases for R → ∞ faster than any power of 1/R. Otherwise the maximal order s of their validity is governed directly by the potential according to the Remark for Theorem 1. For example, the first two corrections, given by Eq. (54), i.e. Eqs. (55), (56), may be applied to potentials with N > 5. Their disappearence on integration over solid angle in Eq. (57) obviously takes place separatly in each order of R −s . So, following the authors of Ref., 10 we come to conclusion, that "... the physics under consideration seems to comply naturally with the mathematical requirements".
is the formal operator rewriting of asymptotic multipole expansion 3-5 (23) of J ± k (R). Unlike its exact expression given by Eq. (5) the expansion (23) according to Eqs. (17), (18) is a solution of free Schrödinger equation like Eq. (12) with R > 0. When V (r) = 0 at r > a, the Schrödinger operators in (4) and (12) coincide for R > a. Then both asymptotic relations (23), (27) become exact expressions due to convergence of the expansion (23) for R > a in the usual sense 3-5 similarly to expansions (15), (16). At the same time, the expansion (20), i.e. (28), conserves its asymptotic sense acquired according to Lemma 1.
for the wave function Ψ + k (R) were used. As well as in Eqs. (36), (37), the arrows point out the directions of action for the operators ← Λ n and → Λ n from Lemma 1, that in fact are
obtained from the line (39), now taking into account all possible asymptotic power corrections. Nevertheless, it looks exactly like right hand side of unitarity relation 1-5 independent of R. It is clear the same result may be obtained using the partial wave decomposition (25) with the help of Eqs. (A.5), (A.12) (clf. (53), (55), (56) below).
(54) -(56). The relations (27), (43), (54) -(57) and the observed disappearance of asymptotic corrections to Eq. (53) are the main results of this work.
a
Calculation similar to (44)-(50) reveals exact disappearance of the contribution of backward direction in Eq. (35) even without averaging over R because (χ j (z) ↔ ∂ z χ j (z)) ≡ 0.For the integral (A.13) with integer j, l the Eq.
S. E. Korenblit and A. V. Sinitskaya
AcknowledgmentsAuthors thank V. Naumov, D. Naumov, A. Rastegin and N. Ilyin for useful discussions and Reviewer for constructive comments and valuable advice.The following relations for spherical functions and Legendre polynomials are necessary[3][4][5][11][12][13][14]for n 2 = v 2 = ω 2 = 1, with n(cos ϑ, ϕ), v(cos β, α) parametrized by Eq. (6) and Lemma 1, and c may be equal to any of values cos ϑ, (n · v), (n · ω): Macdonald and Bessel functions[11][12][13]in definitions (17), (18) are defined by the relations, with |arg u − β 1,2 | < π/2, as:By making use of (18) and (A.7) for integer l and z = 0 − ikr one finds:The following well-known expressions for Wronskians 3-5 ∀ j, l are used:
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Scattering Theory. J R Taylor, J. Wiley & Sons Inc. NYJ. R. Taylor, Scattering Theory. (J. Wiley & Sons Inc. NY, 1972).
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| [] |
[
"Shifted Matroid Optimization",
"Shifted Matroid Optimization"
] | [
"Asaf Levin \nShmuel Onn †\n\n"
] | [
"Shmuel Onn †\n"
] | [] | We show that finding lexicographically minimal n bases in a matroid can be done in polynomial time in the oracle model. This follows from a more general result that the shifted problem over a matroid can be solved in polynomial time as well. | 10.1016/j.orl.2016.05.013 | [
"https://arxiv.org/pdf/1507.00447v2.pdf"
] | 119,127 | 1507.00447 | 7b88dd96b63b4157dd5f1c7bafc2caab64c0cca6 |
Shifted Matroid Optimization
2 Oct 2015
Asaf Levin
Shmuel Onn †
Shifted Matroid Optimization
2 Oct 2015integer programmingcombinatorial optimizationunimodularmatroidspanning treematching
We show that finding lexicographically minimal n bases in a matroid can be done in polynomial time in the oracle model. This follows from a more general result that the shifted problem over a matroid can be solved in polynomial time as well.
Call two matrices x, y ∈ R d×n equivalent and write x ∼ y if each row of x is a permutation of the corresponding row of y. The shift of matrix x ∈ R d×n is the unique matrix x ∈ R d×n which satisfies x ∼ x and x 1 ≥ · · · ≥ x n , that is, the unique matrix equivalent to x with each row nonincreasing. Let |x j | := d i=1 |x i,j | and |x| := n j=1 |x j | be the sums of absolute values of the components of x j and x, respectively. The vulnerability vector of x ∈ {0, 1} d×n is (|x 1 |, . . . , |x n |) .
We then have the following nonlinear combinatorial optimization problem.
Lexicographic Combinatorial Optimization. Given S ⊆ {0, 1} d and n, solve lexmin{(|x 1 |, . . . , |x n |) : x ∈ S n } .
(1)
The complexity of this problem depends on the presentation of the set S. In [7] it was shown that it is polynomial time solvable for S = {z ∈ {0, 1} d : Az = b} for any totally unimodular A and any integer b. Here we solve the problem for matroids.
Theorem 1.1 The lexicographic combinatorial optimization problem (1) over the bases of any matroid given by an independence oracle and any n is polynomial time solvable.
Our spanning tree problem is the special case with S the set of indicators of spanning trees in a given connected graph with d edges, and hence is polynomial time solvable. We note that Theorem 1.1 provides a solution of a nonlinear optimization problem over matroids, adding to available solutions of other nonlinear optimization problems over matroids and independence systems in the literature, see e.g. [2,3,8,9] and the references therein.
We proceed as follows. In Section 2 we discuss the shifted combinatorial optimization problem and its relation to the lexicographic combinatorial optimization problem. In section 3 we solve the shifted problem over matroids in Theorem 3.4 and conclude Theorem 1.1. In Section 4 we discuss matroid intersections, partially solve the shifted problem over the intersection of two strongly base orderable matroids (which include gammoids) in Theorem 4.3, and leave open the complexity of the problem for the intersection of two arbitrary matroids. We conclude in Section 5 with some final remarks about the polynomial time solvability of the shifted and lexicographic problems over totally unimodular systems from [7], and show that these problems are NP-hard over matchings already for n = 2 and cubic graphs.
Shifted Combinatorial Optimization
Lexicographic combinatorial optimization can be reduced to the following problem.
Shifted Combinatorial Optimization. Given S ⊆ {0, 1} d and c ∈ Z d×n , solve max{c x : x ∈ S n } .(2)
The following lemma was shown in [7]. We include the proof for completeness.
Lemma 2.1 [7] The Lexicographic Combinatorial Optimization problem (1) can be reduced in polynomial time to the Shifted Combinatorial Optimization problem (2).
Proof. Define the following c ∈ Z d×n , and note that it satisfies c = c,
c i,j := −(d + 1) j−1 , i = 1, . . . , d , j = 1, . . . , n .
Consider any two vectors x, y ∈ S n , and suppose that the vulnerability vector (|x 1 |, . . . , |x n |) of x is lexicographically smaller than the vulnerability vector (|y 1 |, . . . , |y n |) of y. Let r be the largest index such that |x r | = |y r |. Then |y r | ≥ |x r | + 1. We then have
c x − c y = c x − c y = n j=1 (d + 1) j−1 |y j | − |x j | ≥ j<r (d + 1) j−1 |y j | − |x j | + (d + 1) r−1 ≥ (d + 1) r−1 − j<r d(d + 1) j−1 > 0 .
Thus, an optimal solution x for problem (2) is also optimal for problem (1).
We proceed to reduce the Shifted Combinatorial Optimization problem (2) in turn to two yet simpler auxiliary problems. For a set of matrices U ⊆ {0, 1} d×n let [U ] be the set of matrices which are equivalent to some matrix in U ,
[U ] := {x ∈ {0, 1} d×n : ∃ y ∈ U , x ∼ y} .
Consider the following two further algorithmic problems over a given S ⊆ {0, 1} d :
Shuffling. Given c ∈ Z d×n , solve max{cx : x ∈ [S n ]}.(3)
Fiber. Given x ∈ [S n ], find y ∈ S n such that x ∼ y. Proof. First solve the Shuffling problem (3) with profit matrix c and let x ∈ [S n ] be an optimal solution. Next solve the Fiber problem (4) for x and find y ∈ S n such that x ∼ y. We claim that y is optimal for the Shifted Combinatorial Optimization problem (2). To prove this, we consider any z which is feasible in (2), and prove that the following inequality holds,
c y = c x ≥ c x ≥ c z .
Indeed, the first equality follows since x ∼ y and therefore we have y = x. The middle inequality follows since c is nonincreasing. The last inequality follows since z ∈ S n implies that z ∈ [S n ] and hence z is feasible in (3). So y is indeed an optimal solution for problem (2).
Matroids
Define the n-union of a set S ⊆ {0, 1} E where E is any finite set to be the set Proposition 3.1 For any matroid S and any n we have that ∨ n S is also a matroid. Given an independence oracle for S, it is possible in polynomial time to realize an independence oracle for ∨ n S, and if x ∈ ∨ n S, to find x 1 , . . . , x n ∈ S with x = n k=1 x k .
∨ n S := {x ∈ {0, 1} E : ∃x 1 , . . . , x n ∈ S , x = n k=1 x k } . Call S ⊆ {0, 1} E a matroid if
Define the n-lift of a set of vectors S ⊆ {0, 1} d to be the following set of matrices,
↑ n S := {x ∈ {0, 1} d×n : n j=1 x j ∈ S} .
We need two lemmas. Proof. Consider x ∈ [S n ]. Then x ∼ y for some y ∈ S n , and thus y j ∈ S for j = 1, . . . , n. Since x ∼ y, each row of x is a permutation of the corresponding row of y. Assume that the i-th row of x is given by the permutation π i of the corresponding row of y. That is,
x i,j = y i,π i (j)
. For k = 1, . . . , n, we define a matrix z k ∈ {0, 1} d×n whose column sum satisfies n j=1 z j k = y k , by (z k ) i,j := x i,j if π i (j) = k, and otherwise (z k ) i,j := 0. Since y k ∈ S, we conclude that z k ∈↑ n S. Since the supports of the z k are pairwise disjoint, we have that n k=1 z k ∈ ∨ n ↑ n S. However, n k=1 z k = x by definition, and thus x ∈ ∨ n ↑ n S. In the other direction, assume that x ∈ ∨ n ↑ n S. Then there are x 1 , . . . , x n ∈↑ n S such that x = n k=1 x k . That is, there are x 1 , . . . , x n ∈ {0, 1} d×n such that for each k = 1, . . . , n, we have n j=1 x j k ∈ S, and x = n k=1 x k . Let y be a matrix whose k-th column is y k = n j=1 x j k . Then y k ∈ S, and therefore y ∈ S n . The matrices x and y are 0 − 1 matrices and n j=1 x j = n j=1 y j , and therefore x ∼ y. Thus, x ∈ [S n ], as required. 1. ↑ n S ⊆ {0, 1} d×n is also a matroid, for which an independence oracle can be realized.
2. The rank of the matroid ↑ n S equals to the rank of the matroid S.
3. The rank of the matroid [S n ] = ∨ n ↑ n S equals n times the rank of the matroid S.
Proof. We begin with the first claim. Let z be the 0 matrix (where z i,j = 0 for all i, j). Then n j=1 z j is the zero vector that belongs to S since S is a matroid, and therefore z ∈↑ n S. Let x ≥ y be a pair of 0 − 1 matrices such that x ∈↑ n S. Letx = n j=1 x j , andŷ = n j=1 y j . Thenx ≥ŷ. Since x ∈↑ n S,x ∈ S, and because S is a matroidŷ ∈ S, and thus y ∈↑ n S. Last, assume that x, y ∈↑ n S and |x| > |y|. Letx = n j=1 x j andŷ = n j=1 y j . Since |x| > |y| we conclude that |x| > |ŷ|. Since x, y ∈↑ n S,x,ŷ ∈ S and since S is a matroid there is i such thatx i = 1,ŷ i = 0 and changingŷ i to 1 results in an indicating vector of an independent set of S. Consider this value of i, and let j be such that x i,j = 1. Then, by changing y i,j to 1, we get a larger matrix whose column sum is the vector resulted fromŷ by changing its i-th component to 1, and thus the new matrix is in ↑ n S. Therefore, ↑ n S is a matroid. To present an independence oracle of ↑ n S, assume that we would like to test the independence of a 0 − 1 matrix x, then we compute n j=1 x j and test if the resulting vector is independent in S. Next we show the second claim. Let x be an indicating matrix of an independent set of the matroid ↑ n S, then its column sum is in S and its rank in S equals to the rank of x in ↑ n S. In the other direction, letx ∈ S. Define a matrix x whose first column isx and its all other columns are zero columns, then n j=1 x j =x ∈ S, and therefore x ∈↑ n S. The rank of x in ↑ n S equals the rank ofx in S, and thus the claim follows.
Last, consider the remaining claim. Given a matroid T it is always the case that the rank of the ground set according to ∨ n T is at most n times the rank of the ground set according to T . Thus, to show the claim using the second claim, it suffices to show that given a base z of S, there is an independent set of ∨ n ↑ n S whose rank (in ∨ n ↑ n S) is n times larger than the rank of z (in S). With a slight abuse of notation, assume that z is the indicator of the base z. Let x be a matrix such that for j = 1, . . . , n, x j = z. Then x ∈ S n , and therefore it is an indicator of an independent set of ∨ n ↑ n S whose rank is |x| = n|z| that equals n times the rank of the base z (according to the rank function of the matroid S).
Theorem 3.4 The shifted combinatorial optimization problem (2) over any matroid given by an independence oracle or over its set of bases and any n is polynomial time solvable.
Proof. We begin with the problem over (the independent sets of) a matroid S. Consider problem (3) over S. Since [S n ] = ∨ n ↑ n S by Lemma 3.2, it follows from Proposition 3.1 and Lemma 3.3 that [S n ] is a matroid for which we can realize an independence oracle. So the greedy algorithm over [S n ] solves problem (3) over S. Next consider problem (4) over S. Let x ∈ [S n ] be given. By Proposition 3.1 and Lemma 3.2 and Lemma 3.3 again, we can find x 1 , . . . , x n ∈↑ n S with x = n k=1 x k . Let y be the matrix with y k = n j=1 x j k for k = 1, . . . , n. Then This implies that x ∼ y. Now, since x k ∈↑ n S and y k = n j=1 x j k , we have that y k ∈ S for all k. So we can find y ∈ S n with x ∼ y and therefore solve problem (4) over S as well. It now follows from Lemma 2.2 that we can indeed solve problem (2) over S.
To solve problem (2) over the set of bases of S proceed as follows. Define a new profit matrix w by w i,j := c i,j + 2|c| + 1 for all i, j. Solve problem (3) over (the independent sets of) S with profit w by the greedy algorithm over [S n ]. Let r be the rank of S so that by Lemma 3.3 the rank of [S n ] is nr. Consider any x, y, z ∈ [S n ] with x, y bases and z not a basis. Then wx = cx + (2|c| + 1)|x| = cx + (2|c| + 1)nr ≥ −|c| + (2|c| + 1)nr , wz = cz + (2|c| + 1)|z| ≤ |c| + (2|c| + 1)(nr − 1) .
So wx − wz > 0 and hence the w-profit of any basis x is larger than that of any independent set z which is not a basis, so the algorithm will output a basis. Also, wx − wy = cx − cy and hence the w-profit of basis x is larger than that of basis y if and only if the c-profit of x is larger than that of y, so the algorithm will output a basis x of [S n ] with largest c-profit. Now solve problem (4) and find y ∈ S n with x ∼ y as before. Since S has rank r, each column y k ∈ S satisfies |y k | ≤ r. Since nr = |x| = |y| = n k=1 |y k | we must have in fact |y k | = r so that y k is a basis of S for all k. Therefore y is an optimal solution for problem (2) over the bases.
Matroid Intersections
We next consider the Shifted Combinatorial Optimization problem over matroid intersections. We can only provide a partial solution for the class of strongly base orderable matroids, We need the following elegant result of [5], see also [10,Theorem 42.13].
Proposition 4.1 For any two strongly base orderable matroids S 1 and S 2 we have
(∨ n S 1 ) ∩ (∨ n S 2 ) = ∨ n (S 1 ∩ S 2 ) .
We also need the following lemma.
Lemma 4.2 If
S is a strongly base orderable matroid, then ↑ n S is also a strongly base orderable matroid. Further, for any S 1 ,
S 2 ⊆ {0, 1} d we have that ↑ n S 1 ∩ ↑ n S 2 =↑ n (S 1 ∩ S 2 ).
Proof. We begin with the first part. Let S be a strongly base orderable matroid. Then by Lemma 3.3, ↑ n S is also a matroid. Let x, y ∈ {0, 1} d×n whose supports X, Y are two bases of ↑ n S. By Lemma 3.3, the supportsX andŶ ofx = n j=1 x j andŷ = n j=1 y j respectively are two bases of S. Since S is a strongly base orderable matroid there is a bijection π :X →Ŷ such that π(Î) ∪ (X \Î) is a base of S for everyÎ ⊆X. We define a bijection π ′ : X → Y as follows. For every (i, j) ∈ X, we let π ′ (i, j) = (π(i), j ′ ) where j ′ is the unique column containing 1 in the π(i) row of y (there is such a column as π(i) ∈Ŷ ). A support of a 0 − 1 matrix is a base in ↑ n S if and only if its column sum is a base in S. Thus, the required properties of π ′ follow from these properties of π.
Next we prove the second claim. Let x ∈↑ n S 1 ∩ ↑ n S 2 and denote its column sum bŷ x = n j=1 x j . Since x ∈↑ n S 1 , we havex ∈ S 1 , and similarly since x ∈↑ n S 2 , we havex ∈ S 2 . Thus,x ∈ S 1 ∩ S 2 , and thus x ∈↑ n (S 1 ∩ S 2 ). In the other direction, let x ∈↑ n (S 1 ∩ S 2 ) and denotex = n j=1 x j . Then,x ∈ S 1 ∩ S 2 . Thus,x ∈ S 1 , and therefore x ∈↑ n S 1 , and similarlŷ x ∈ S 2 , and therefore x ∈↑ n S 2 . Therefore, x ∈↑ n S 1 ∩ ↑ n S 2 .
Theorem 4.3
The optimal objective function value of the shifted combinatorial optimization problem (2) over the intersection of any two strongly base orderable matroids given by independence oracles and any n can be computed in polynomial time.
Proof. Let S 1 , S 2 ⊆ {0, 1} d be strongly base orderable matroids and let S := S 1 ∩ S 2 . By Lemma 4.2, ↑ n S 1 and ↑ n S 2 are strongly base orderable matroids. Applying Proposition 4.1 to ↑ n S 1 , ↑ n S 2 , and using Lemmas 3.2 and 4.2, we get
[S n 1 ] ∩ [S n 2 ] = (∨ n ↑ n S 1 ) ∩ (∨ n ↑ n S 2 ) = ∨ n (↑ n S 1 ∩ ↑ n S 2 ) = ∨ n ↑ n (S 1 ∩ S 2 ) = [S n ] .
By Proposition 3.1 applied to the matroids ↑ n S 1 and ↑ n S 2 , and using Lemma 3.2, we can realize in polynomial time independence oracles for [S n 1 ] and [S n 2 ]. So we can maximize a given profit matrix c over [S n ] by maximizing c over the intersection of the matroids [S n 1 ] and [S n 2 ]. This solves problem (3) over S and gives the optimal value of the shifted optimization problem (2) over S via the equality max{c x : x ∈ S n } = max{cx : x ∈ [S n ]}.
We raise the following natural questions. Is it possible to actually find an optimal solution (and not only the optimal value) for the shifted problem over the intersection of strongly base orderable matroids in polynomial time? What is the complexity of shifted combinatorial optimization over the intersection of two arbitrary matroids?
The answer to the first question is yes for transversal matroids since then the Fiber problem (4) over the intersection can also be solved. This implies in particular that the shifted problem over matchings in bipartite graphs is polynomial time solvable. However, if the Fiber problem cannot be solved efficiently, that is given x ∈ [S n ], it is hard to find y ∈ S n such that x ∼ y, then there are matrices c for which it is hard to solve the Shifted Combinatorial Optimization problem as well. To see this, let c be such that for all i, j we have c i,j = 1 if x i,j = 1 and c i,j = −1 if x i,j = 0. Then, the optimal profit will be |x|, and every optimal solution y to the Shifted Combinatorial Optimization problem must satisfy y ∈ S n and y ∼ x, and thus solves the Fiber problem as well. Thus, the Fiber problem is not harder than the Shifted Combinatorial Optimization problem.
Remarks
It was shown in [7] that problem (2) and hence also problem (1) are polynomial time solvable for S = {z ∈ {0, 1} d : Az = b} with A totally unimodular and b integer. We briefly give a new interpretation of this by showing how the new auxiliary problems (3) and (4) over such S can also be efficiently solved, so Lemma 2.2 can be applied.
[S n ] = {x ∈ {0, 1} d×n : A n j=1 x j = nb} .(5)
Moreover, problems (3) and (4) hence (2) and (1) over S are polynomial time solvable.
Proof. If x is in the left hand side of (5) then there is a y ∈ S n with x ∼ y. Then A n j=1 x j = A n j=1 y j = nb so x is also in the right hand side. If x is in the right hand side, then a decomposition theorem of [1] implies that there is a y ∈ S n with n j=1 y j = n j=1 x j . Then x ∼ y so x is also in the left hand side of (5). Moreover, by an algorithmic version of [7] of this decomposition theorem, such a y can be found in polynomial time, solving problem (4) over S. By the equality in (5), to solve problem (3) over S with a given profit c, we can maximize c over the right hand side of (5), and this can be done in polynomial time by linear programming since the defining matrix of this set is [A, ..., A], which is totally unimodular since A is. Lemmas 2.1 and 2.2 now imply that problems (2) and (1) over S can also be solved in polynomial time.
This in particular implies that lexicographic combinatorial optimization over s − t paths in digraphs or perfect matchings in bipartite graphs is polynomial time doable.
The shifted combinatorial optimization problem over S ⊆ {0, 1} d in the special case n = 1 is just the standard combinatorial optimization problem max{cx : x ∈ S} over S. So a solution of the former implies a solution of the latter. Is the converse true? The above results show that it is true for matroids, certain matroid intersections, and unimodular systems including s − t paths in digraphs and perfect matchings in bipartite graphs. Unfortunately, as we next show, the answer in general is negative.
Proposition 5.2 Let S ⊆ {0, 1} d be the set of perfect matchings in a cubic graph. Then the lexicographic problem (1) and shifted problem (2) over S are NP-hard already for fixed n = 2.
Proof. By Lemma 2.1 it is enough to show that solving lexmin{(|x 1 |, |x 2 |) : x ∈ S 2 } is NP-hard. We claim that deciding if there is an x ∈ S 2 with |x 2 | = 0 is NP-complete. Indeed, there is such an x if and only if there are two edge disjoint perfect matchings in the given graph. Now this holds if and only if there are three pairwise edge disjoint perfect matchings in the given graph, since it is cubic. This is equivalent to the existence of a 3 edge coloring, which is NP-complete to decide on cubic graphs [6].
We point out that the set of perfect matchings in a graph can be written in the form S = {z ∈ {0, 1} d : Az = b} with A the vertex-edge incidence matrix of the graph and b the all 1 vector in the vertex space. Thus, Proposition 5.2 shows, in contrast with Proposition 5.1, that if A is not totally unimodular, then the shifted problem is NP hard over such sets S, even if we can efficiently do linear optimization over S, as is the case for perfect matchings.
The shifted optimization problem max{c x : x ∈ S n } can be defined for any set S ⊂ Z d , and its complexity will depend on the structure and presentation of S. Consider sets of bounded cardinality. Note that even with |S| = 2, the number of feasible solutions is |S n | = 2 n , so exhaustive search is not polynomial. However, we do have the following simple statement.
Proposition 5.3
For any fixed positive integer s, the shifted optimization problem over any set S ⊂ Z d satisfying |S| ≤ s, any n, and any c ∈ Z d×n , can be solved in polynomial time.
Proof. Let S = {z 1 , . . . , z m } be the list of elements of S, with m ≤ s and z i ∈ Z d for all i. Given x ∈ S n , let n i := |{k : x k = z i }| for i = 1, . . . , m. Let y ∈ S n be the matrix with first n 1 columns equal to z 1 , next n 2 columns equal to z 2 , and so on, with last n m columns equal to z m . Then y ∼ x (in fact, y is obtained from x by applying the same permutation to each row), and therefore c x = c y. So the objective value of x can be computed from n 1 , . . . , n m . Now, any integers 0 ≤ n i ≤ n with m i=1 n i = n give a feasible y ∈ S n with these counts. So, it is enough to go over all such n 1 , . . . , n m , for each compute the corresponding y and the value c y, and pick the best. The number of such tuples is at most (n + 1) m−1 = O(n s−1 ), since n m = n − m−1 i=1 n i is determined by the others, which is polynomial in n for fixed s.
. 2
2The Shifted Combinatorial Optimization problem (2) can be reduced in polynomial time to the Shuffling and Fiber problems (3) and (4).
it is the set of indicators of independent sets of a matroid over E. We will have matroids over E = [d] := {1, . . . , d} and E = [d] × [n].The following facts about n-unions of matroids are well known, see e.g.[10].
Lemma 3. 2
2For any set S ⊆ {0, 1} d and any n we have that [S n ] = ∨ n ↑ n S in {0, 1} d×n .
Lemma 3. 3
3Let S ⊆ {0, 1} d be a matroid given by an independence oracle. Then we have:
Proof of Theorem 1.1. This follows at once from Lemma 2.1 and Theorem 3.4.
introduced in [4], which strictly includes the class of gammoids, see [10, Section 42.6c]. Let S ⊆ {0, 1} d be a matroid and let B be the set of subsets of [d] which are supports of bases of S. Then S is strongly base orderable if for every pair B 1 , B 2 ∈ B there is a bijection π : B 1 → B 2 such that for all I ⊆ B 1 we have π(I)∪(B 1 \I) ∈ B (where π(I) = {π(i) : i ∈ I}).
For S = {z ∈ {0, 1} d : Az = b} with A totally unimodular and b integer,
Integer rounding and polyhedral decomposition for totally unimodular systems. S Baum, L E Trotter, Jr, Lecture Notes in Economical and Mathematical Systems. 157Baum, S., Trotter, L.E., Jr.: Integer rounding and polyhedral decomposition for totally unimodular systems. Lecture Notes in Economical and Mathematical Systems 157:15-23 (1978)
Nonlinear matroid optimization and experimental design. Y Berstein, J Lee, H Maruri-Aguilar, S Onn, E Riccomagno, R Weismantel, H Wynn, SIAM Journal on Discrete Mathematics. 22Berstein, Y., Lee J., Maruri-Aguilar, H., Onn, S., Riccomagno, E., Weismantel, R., Wynn, H.: Nonlinear matroid optimization and experimental design. SIAM Journal on Discrete Mathematics 22:901-919 (2008)
Parametric nonlinear discrete optimization over well-described sets and matroid intersections. Y Berstein, J Lee, S Onn, R Weismantel, Mathematical Programming. 124Berstein, Y., Lee, J., Onn, S., Weismantel, R.: Parametric nonlinear discrete optimization over well-described sets and matroid intersections. Mathematical Programming 124:233- 253 (2010)
Common transversals and strong exchange systems. R A Brualdi, Journal of Combinatorial Theory. 8Brualdi, R.A.: Common transversals and strong exchange systems. Journal of Combina- torial Theory 8:307-329 (1970)
Disjoint common transversals and exchange structures. J Davies, C Mcdiarmid, The Journal of the London Mathematical Society. 14Davies, J., McDiarmid, C.: Disjoint common transversals and exchange structures. The Journal of the London Mathematical Society 14:55-62 (1976)
The NP-Completeness of Edge-Coloring. I Holyer, SIAM Journal on Computing. 10Holyer, I.: The NP-Completeness of Edge-Coloring. SIAM Journal on Computing 10:718- 720 (1981)
The unimodular intersection problem. V Kaibel, S Onn, P Sarrabezolles, Operations Research Letters. 43Kaibel, V., Onn, S., Sarrabezolles, P.: The unimodular intersection problem. Operations Research Letters 43:592-594 (2015)
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Nonlinear Discrete Optimization. S Onn, Zurich Lectures in Advanced Mathematics. European Mathematical SocietyOnn, S.: Nonlinear Discrete Optimization. Zurich Lectures in Advanced Mathematics, European Mathematical Society (2010), available online at: http://ie.technion.ac.il/∼onn/Book/NDO.pdf
. A Schrijver, SpringerSchrijver, A.: Combinatorial Optimization, Springer (2003)
| [] |
[
"Long Short-Term Memory Over Tree Structures",
"Long Short-Term Memory Over Tree Structures"
] | [
"Xiaodan Zhu \[email protected] National Research Council Canada\n1200 Montreal Road M-50K1A 0R6OttawaON\n",
"Hongyu Guo \nSchool of Electrical Engineering and Computer Science\[email protected] National Research Council Canada\nCANADA Parinaz Sobhani PSOBH090@UOTTAWA\nUniversity of Ottawa\n800 King Edward Avenue, 1200 Montreal Road M-50K1N 6N5, K1A 0R6Ottawa, OttawaON, ONCANADA, CANADA\n"
] | [
"[email protected] National Research Council Canada\n1200 Montreal Road M-50K1A 0R6OttawaON",
"School of Electrical Engineering and Computer Science\[email protected] National Research Council Canada\nCANADA Parinaz Sobhani PSOBH090@UOTTAWA\nUniversity of Ottawa\n800 King Edward Avenue, 1200 Montreal Road M-50K1N 6N5, K1A 0R6Ottawa, OttawaON, ONCANADA, CANADA"
] | [] | The chain-structured long short-term memory (LSTM) has showed to be effective in a wide range of problems such as speech recognition and machine translation. In this paper, we propose to extend it to tree structures, in which a memory cell can reflect the history memories of multiple child cells or multiple descendant cells in a recursive process. We call the model S-LSTM, which provides a principled way of considering long-distance interaction over hierarchies, e.g., language or image parse structures. We leverage the models for semantic composition to understand the meaning of text, a fundamental problem in natural language understanding, and show that it outperforms a state-of-theart recursive model by replacing its composition layers with the S-LSTM memory blocks. We also show that utilizing the given structures is helpful in achieving a performance better than that without considering the structures. | null | [
"https://arxiv.org/pdf/1503.04881v1.pdf"
] | 17,629,631 | 1503.04881 | 4b9af9b9ed1bfb6b0688019d9fa79875bbcd8f3f |
Long Short-Term Memory Over Tree Structures
16 Mar 2015
Xiaodan Zhu
[email protected] National Research Council Canada
1200 Montreal Road M-50K1A 0R6OttawaON
Hongyu Guo
School of Electrical Engineering and Computer Science
[email protected] National Research Council Canada
CANADA Parinaz Sobhani PSOBH090@UOTTAWA
University of Ottawa
800 King Edward Avenue, 1200 Montreal Road M-50K1N 6N5, K1A 0R6Ottawa, OttawaON, ONCANADA, CANADA
Long Short-Term Memory Over Tree Structures
16 Mar 2015
The chain-structured long short-term memory (LSTM) has showed to be effective in a wide range of problems such as speech recognition and machine translation. In this paper, we propose to extend it to tree structures, in which a memory cell can reflect the history memories of multiple child cells or multiple descendant cells in a recursive process. We call the model S-LSTM, which provides a principled way of considering long-distance interaction over hierarchies, e.g., language or image parse structures. We leverage the models for semantic composition to understand the meaning of text, a fundamental problem in natural language understanding, and show that it outperforms a state-of-theart recursive model by replacing its composition layers with the S-LSTM memory blocks. We also show that utilizing the given structures is helpful in achieving a performance better than that without considering the structures.
Introduction
Recent years have seen a revival of the long short-term memory (LSTM) (Hochreiter & Schmidhuber, 1997), with its effectiveness being demonstrated on a wide range of problems such as speech recognition (Graves et al., 2013), machine translation (Sutskever et al., 2014;Cho et al., 2014), and image-to-text conversion , On February 6th, 2015, this work was submitted to the International Conference on Machine Learning (ICML). among many others, in which history is summarized and coded in the memory cell in a full-order time sequence.
Recursion is a fundamental process associated with many problems-a recursive process and hierarchical structure so formed are common in different modalities. For example, semantics of sentences in human languages is believed to be carried by not merely a linear concatenation of words; instead, sentences have parse structures (Manning & Schütze, 1999). Image understanding, as another example, benefits from recursive modeling over structures, which yielded the state-of-the-art performance on tasks like scene segmentation (Socher et al., 2011).
In this paper, we extend LSTM to tree structures, in which we learn memory cells that can reflect the history memories of multiple child cells and hence multiple descendant cells. We call the model S-LSTM. Compared with previous recursive neural networks (Socher et al., 2013;2012), S-LSTM has the potentials of avoiding gradient vanishing and hence may model long-distance interaction over trees. This is a desirable characteristic as many of such structures are deep. S-LSTM can be considered as bringing the merits of a recursive neural network and a recurrent neural network together 1 . In short, S-LSTM wires memory blocks in a partial-order structures instead of in a full-order sequence as in a chain-structured LSTM.
Stanford Sentiment Tree Bank (Socher et al., 2013) to determine the sentiment for different granularities of phrases in a tree. The dataset has favorable properties: in addition to being a benchmark for much previous work, it provides with human annotations at all nodes of the trees, enabling us to comprehensively explore the properties of S-LSTM. We experimentally show that S-LSTM outperforms a stateof-the-art recursive model by simply replacing the original tensor-enhanced composition with the S-LSTM memory block we propose here. We showed that utilizing the given structures is helpful in achieving a better performance than that without considering the structures.
Related Work
Recursive neural networks Recursion is a fundamental process in different modalities. In recent years, recursive neural networks (RvNN) have been introduced and demonstrated to achieve state-of-the-art performances on different problems such as semantic analysis in natural language processing and image segmentation (Socher et al., 2013;2011). These networks are defined over recursive tree structures-a tree node is a vector computed from its children. In a recursive fashion, the information from the leaf nodes of a tree and its internal nodes are combined in a bottom-up manner through the tree. Derivatives of errors are computed with backpropagation over structures (Goller & Kchler, 1996).
In addition, the literature has also included many other efforts of applying feedforward-based neural network over structures, including (Goller & Kchler, 1996;Chater, 1992;Starzyk et al.;Hammer et al., 2004), amongst others. For instance, Legrand and Collobert leverage neural networks over greedy syntactic parsing (Pinheiro & Collobert, 2014). In (Irsoy & Cardie, 2014), a deep recursive neural network is proposed . Nevertheless, over the often deep structures, the networks are potentially subject to the vanishing gradient problem, resulting in difficulties in leveraging long-distance dependencies in the structures. In this paper, we propose the S-LSTM model that wires memory blocks in recursive structures. We compare our model with the RvNN models presented in (Socher et al., 2013), as we directly replaced the tensor-enhanced composition layer at each tree node with a S-LSTM memory block. We show the advantages of our proposed model in achieving significantly better results.
Recurrent neural networks and LSTM Unlike a feedforward network, a recurrent neural network (RNN) shares their hidden states across time. The sequential history is summarized in a hidden vector. RNN also suffers from the decaying of gradient, or less frequently, blowing-up of gradient problem. LSTM replaces the hidden vector of a recurrent neural network with memory blocks which are equipped with gates; it can in principle keep longterm memory by training proper gating weights (refer to (Graves, 2008) for intuitive illustrations and good discussions), and it has practically showed to be very useful, achieving the state of the art on a range of problems including speech recognition (Graves et al., 2013), digit handwriting recognition (Liwicki et al., 2007;Graves, 2012), and achieve interesting results on statistical machine translation (Sutskever et al., 2014;Cho et al., 2014) and music composition (Eck & Schmidhuber, 2002b;a). In (Graves et al., 2013), a deep LSTM network achieved the state-of-the-art results on the TIMIT phoneme recognition benchmark. In (Sutskever et al., 2014;Cho et al., 2014), a pair of LSTM networks are trained to encode and decode human language for automatic machine translation, which is in particular effective for the more challenging long sentence translation. In (Liwicki et al., 2007;Graves, 2012), LSTM networks are found to be very useful for digit writing recognition because of the network's capability of memorizing context information in a long sequence. In (Eck & Schmidhuber, 2002b;a), LSTM networks are trained to effectively capture global structures of the temporal data. With the memory cells, LSTM is able to keep track of temporally distant events that indicate global music structures. As a result, LSTM can be successfully trained to compose music, where other RNNs have failed to do so.
Although promising results have been observed by applying chain-structured LSTM, many other interesting problems are inherently associated with input structures that are more complicated than a sequence. For example, sentences in human languages are believed to be carried by not merely a linear sequence of words; instead, meaning is thought to interweave with structures. While a sequential application of LSTM may capture structural information implicitly, in practice it sometimes lacks the claimed power. For example, even simply reversing the input sequences may result in significant differences in modeling performances, in tasks such as machine translation and speech recognition. Unlike in previous work, we propose here to directly wire memory blocks in recursive structures. We show the proposed S-LSTM model does utilize the structures and achieve results better than those ignoring such priori structures.
The Model
Model brief In this paper, we extend LSTM to structures, in which a memory cell can reflect the history memories of multiple child cells and hence multiple descendant cells in a hierarchical structure. As intuitively showed in Figure 1, the root of the tree can in principle consider information from long-distance interactions over the tree-in this fig-ure, the gray and light-blue leaf. In the figure, the small circle ("•") or short line ("−") at each arrowhead indicates pass and block of information, respectively. Note that the figure shows a binary case, while in real models a soft version of gating is applied, where a gating signal is in the range of [0, 1], often enforced with a logistic sigmoid function. Through learning the gating signals, as detailed later in this section, S-LSTM provides a principled way of considering long-distance interplays over the input structures. The memory block Each node in Figure 1 is composed of a S-LSTM memory block. We present a specific wiring of such a block in Figure 2. Each memory block contains one input gate and one output gate. The number of forget gates depends on the structure, i.e., the number of children of a node. In this paper, we assume there are two children at each nodes, same as in (Socher et al., 2013) and therefore we use their data in our experiments. That is, we have two forget gates. Extension of the model to handle more children is rather straightforward.
As shown in the figure, the hidden vectors of the two children, denoted as h L t−1 for the left child and h R t−1 for the right, are taken in as input of the current block. The input gate i t consider four resources of information: the hidden vectors (h L t−1 and h R t−1 ) and cell vectors (c L t−1 and c R t−1 ) of its two children. These four sources of information are also used to form the gating signals for the left forget gate f L t−1 and right forget gate f R t−1 , where the weights used to combining them are specific to each of these gates, denoted as Figure 2. A S-LSTM memory block, consisting of an input gate, two forget gates, and an output gate. Hidden vectors h * t−1 and cell vectors c * t−1 from the left (red arrows) and right (blue arrows) children are deployed to compute ct and ht. ⊗ denotes a Hadamard product, and the "s" shaped sign is a squashing function (in this paper the tanh function). different W in the formulas below. Different from the process in a regular LSTM, the cell here considers the copies from both children's cell vectors (c L t−1 , c R t−1 ), gated with separated forget gates. The left and right forget gates can be controlled independently, allowing the pass-through of information from children's cell vectors. The output gate o t considers the hidden vectors from the children and the current cell vector. In turn, the hidden vector h t and the cell vector c t of the current block are passed to the parent and are used depending on if the current block is a left or right child of its parent. In this way, the memory cell, through merging the gated cell vectors of the children, can reflect multiple direct or indirect descendant cells. As a result, the long-distance interplays over the structures can be captured. More specifically, the forward computation of a S-LSTM memory block is specified in the following equations.
i t = σ(W L hi h L t−1 + W R hi h R t−1 + W L ci c L t−1 + W R ci c R t−1 + b i ) (1) f L t = σ(W L hf l h L t−1 + W R hf l h R t−1 + W L cf l c L t−1 + W R cf l c R t−1 + b f l ) (2) f R t = σ(W L hf r h L t−1 + W R hf r h R t−1 + W L cf r c L t−1 + W R cf r c R t−1 + b fr ) (3) x t = W L hx h L t−1 + W R hx h R t−1 + b x (4) c t = f L t ⊗ c L t−1 + f R t ⊗ c R t−1 + i t ⊗ tanh(x t ) (5) o t = σ(W L ho h L t−1 + W R ho h R t−1 + W co c t + b o ) (6) h t = o t ⊗ tanh(c t )(7)
where σ is the element-wise logistic function used to confine the gating signals to be in the range of [0, 1]; f L and f R are the left and right forget gate, respectively; b is bias and W is network weight matrices; the sign ⊗ is a Hadamard product, i.e., element-wise product. The subscripts of the weight matrices indicate what they are used for. For example, W ho is a matrix mapping a hidden vector to an output gate.
Backpropagation over structures During training, the gradient of the objective function with respect to each parameter can be calculated efficiently via backpropagation over structures (Goller & Kchler, 1996;Socher et al., 2013). The major difference from that of (Socher et al., 2013) is we use LSTM-like backpropagation, where unlike a regular LSTM, pass of error needs to discriminate between the left and right children, or in a topology with more than two children, needs to discriminate between children.
Obtaining the backprop formulas is tedious but we list them below to facilitate duplication of our work 2 . We will discuss the specific objective function later in experiments. For each memory block, assume that the error passed to the hidden vector is ǫ h t . The derivatives of the output gate δ o t , left forget gate δ f l t , right forget gate δ fr t , and input gate δ i t are computed as:
ǫ h t = ∂O ∂h t (8) δ o t = ǫ h t ⊗ tanh(c t ) ⊗ σ ′ (o t ) (9) δ f l t = ǫ c t ⊗ c L t−1 ⊗ σ ′ (f L t ) (10) δ fr t = ǫ c t ⊗ c R t−1 ⊗ σ ′ (f R t ) (11) δ i t = ǫ c t ⊗ tanh(x t ) ⊗ σ ′ (i t )(12)
2 The code will be published at www.icml-placeholderonly.com where σ ′ (x) is the element-wise derivative of the logistic function over vector x. Since it can be computed with the activation of x, we abuse the notation a bit to write it over the activated vectors in these equations. ǫ c t is the derivative over the cell vector. So if the current node is the left child of its parent, we use Equation (13) to calculate ǫ c t , otherwise Formula (14) is used:
ǫ c t =ǫ h t ⊗ o t ⊗ g ′ (c t ) + ǫ c t+1 ⊗ f L t+1 + (W L ci ) T δ i t+1 + (W L cf l ) T δ f l t+1 + (W L cfr ) T δ fr t+1 + (W co ) T δ o t (13) ǫ c t =ǫ h t ⊗ o t ⊗ g ′ (c t ) + ǫ c t+1 ⊗ f R t+1 + (W R ci ) T δ i t+1 + (W R cf l ) T δ f l t+1 + (W R cfr ) T δ fr t+1 + (W co ) T δ o t(14)
where g ′ (x) is the element-wise derivative of the tanh function. It can also be directly calculated from the tanh activation of x. The superscript T over the weight matrices means matrix transpose.
With derivatives at each gate computed, the derivatives of the weight matrices used in Formula (1)- (7) can be calculated accordingly, which is omitted here. We checked the correctness of the S-LSTM implementation with the standard approximated gradient approach.
Objective over trees The objective function defined over structures can be complicated, which could consider the output structures depending on the properties of problem. Following (Socher et al., 2013), the overall objective function we used to learn S-LSTM in this paper is simply minimizing the overall cross-entropy errors and a sum of that at all nodes.
Experiment Set-up
As discussed earlier, recursion is a basic process inherent to many problems. In this paper, we leverage the proposed model to solve semantic composition for the meanings of pieces of text, a fundamental problem in understanding human languages.
We specifically attempt to determine the sentiment of different granularities of phrases in a tree, within the Stanford Sentiment Tree Bank benchmark data (Socher et al., 2013). In obtaining the sentiment of a long piece of text, early work often factorized the problem to consider smaller pieces of component words or phrases with bag-of-words or bag-ofphrases models (Pang & Lee, 2008;Liu & Zhang, 2012).
More recent work has started to model composition (Moilanen & Pulman, 2007;Choi & Cardie, 2008;Socher et al., 2012;Kalchbrenner et al., 2014), a more principled approach to modeling the formation of semantics. In this paper, we put the proposed LSTM memory blocks at tree nodes-we replaced the tensorenhanced composition layer at each tree node presented in (Socher et al., 2013) with a S-LSTM memory block. We used the same dataset, the Stanford Sentiment Tree Bank, to evaluate the performances of the models. In addition to being a benchmark for much previous work, the data provide with human annotations at all nodes of the trees, facilitating a more comprehensive exploration of the properties of S-LSTM.
Data Set
The Stanford Sentiment Tree Bank (Socher et al., 2013) contains about 11,800 sentences from the movie reviews that were originally discussed in (Pang & Lee, 2005). The sentences were parsed with the Stanford parser (Klein & Manning, 2003). Phrases at all the tree nodes were manually annotated with sentiment values. We use the same split of the training and test data as in (Socher et al., 2013) to predict the sentiment categories of the roots (sentences) and all phrases (including sentences). For the root sentiment, the training, development, and test sentences are 8544, 1101, and 2210, respectively. The phrase sentiment task includes 318582, 41447, and 82600 phrases for the three sets. Following (Socher et al., 2013), we also use the classification accuracy to measure the performances.
Training Details
As mentioned before, we follow (Socher et al., 2013) to minimize the cross-entropy error for all nodes or for roots only, depending on specific experiment settings. For all phrases, the error is calculated as a regularized sum:
E(θ) = i j t i j logy seni j + λ θ 2 2(15)
where y seni ∈ R c×1 is predicted distribution and t i ∈ R c×1 the target distribution. c is the number of classes or categories, and j ∈ c denotes the j-th element of the multinomial target distribution; i iterates over nodes, θ are model parameters, and λ is a regularization parameter. We tuned our model against the development data set as split in (Socher et al., 2013).
Results
To understand the modeling advantages of S-LSTM over the structures, we conducted four sets of experiments.
Default setting
In the default setting, we conducted experiments as in (Socher et al., 2013). Table 1 shows the accuracies of different models on the test set of the Stanford Sentiment Tree Bank. We present the results on 5-category sentiment prediction at both the sentence level (i.e., the ROOTS column in the table) and for all phrases including roots (the PHRASES column) 3 . In Table 1, NB and SVM are naive Bayes and support vector machine classifiers, respectively; RvNN corresponds to RNN in (Socher et al., 2013). As described earlier, we refer to recursive neural networks to as RvNN to avoid confusion with recurrent neural networks. RNTN is different from RvNN in that when merging two nodes to obtain the hidden vector of their parent, tensor is used to obtain the second-degree polynomial interactions. Table 1 showed that S-LSTM achieved the best predictive performance, when compared to all the models reported in (Socher et al., 2013). The S-LSTM results reported here were obtained by setting the size of the hidden units to be 100, batch size to be 10, and learning rate to be 0.1.
In our experiments, we only tuned these hyper-parameters, and we feel that more finer tuning, such as discriminating the classification weights between the leaves (word embedding) and other nodes, using different numbers of hidden units for the memory blocks (e.g., for the hidden layers of words), or different initializations of word embedding, may further improve the performances reported here.
To evaluate the S-SLTM model's convergence behavior, Figure 3 depicts the converging time during training. More specifically, we show two sub-figures: one for roots (upper much less parameters than RNTN and the forward and backward propagation can be computed efficiently.
More real-life settings
We further compare S-LSTM with RNTN in two more experimental settings. In the first setting we only keep the training signals at the roots to train S-LSTM and RNTN, depicted as model (1) and (2) in Table 2. ROOT LBLS besides the model names stands for root labels; that is, only the gold labels of the sentence level are used to train the model. In most sentiment analysis circumstances, phrase level annotations are not available: most nodes in a tree are fragments that may not be that interesting; e.g., the fragment "of a good movie" 4 . Also, annotating all phrases is expensive. However, these should not be regarded as comments on the value of the Sentiment Tree Bank. Detailed annotations in the tree bank enable much interesting work to be possible, e.g., the study of the effect of negation in changing sentiment (Zhu et al., 2014).
The second setting, corresponding to model (3) and (4) in Table 2, is only slightly different, in which we keep annotation for the tree leafs as well, to simulate that a sentiment lexicon is available and it covers all leafs (words) (LEAF LBLS along the side of the model names stands for leaf labels), and so there is no out-of-vocabulary concern. Using real sentiment lexicons is expected to have a performance between the two settings here.
Results in the table show that in both settings, S-LSTM outperforms RNTN by a large margin. When only root labels are used to train the models, S-LSTM obtains an accuracy of 43.5, compared with 29.1 of RNTN. When the leaf labels are also used, S-LSTM achieves an accuracy of 44.1 and RNTN 34.9. All these improvements are statistically significant (p < 0.05). For the RNTN, without supervising signals from the internal nodes, the composition parameters may not be learned well, potentially because the tensor has much more parameters to learn. On the other hand, through controlling its gates, the S-LSTM shows a very good ability to learn from the trees. Table 2. Performances of models trained with only root labels (the first two rows) and models that use both root and leaf labels (the last two rows). Figure 4, we further depict the performances of models on different levels of nodes in the trees. In the Figure, the x-axis corresponds to different depths or lengths and y-axis is accuracy. The depth here is defined as the longest distance between the root of a phrase and their descendant leafs. The Length is simply the number of words of a node, where depth is not necessarily to be length-e.g., a balanced tree with 4 leafs has different depths than the unbalanced tree with the same number of leafs. The trends of the two figure are similar. In both figures, S-LSTM performs better at all depths, showing its advantages on nodes at depth. As the deeper levels of the tree tend to have more complicated syntax and semantics, S-LSTM can model such more complicated syntax and semantics better.
MODELS
Explicit structures vs. no structures Some efforts in the literature attempt to learn distributed representation by utilizing input structures when available, and others prefer to assume chain-structured recurrent neural networks can actually capture the structures implicitly though a linear coding process. In this paper, we attempt to give some empirical evidences in our experiment setting by comparing several different models. First, a special case for the S-LSTM model is considered, in which no sentential structures are given. Instead, words are read from left to right and combined in that order. We call it left recursive S-LSTM, or S- LSTM-LR in short. Similarly, we also experimented with a right recursive S-LSTM, S-LSTM-RR, in which words are read from right to left instead. Since for these models, phrase-level training signals are not available-the nodes here do not correspond to that in the original Standford Sentiment Tree Bank, but the roots and leafs annotations are still the same, so we run two versions of our experiments: one uses only training signals from roots and the other includes also leaf annotations.
It can be observed from Table 3 that the given parsing structure helps improve the predictive accuracy. In the case of using only root labels, the left recursive S-LSTM and right recursive S-LSTM have similar performance (40.2 and 40.3, respectively), both inferior to S-LSTM (43.5). When using gold leaf labels, the gaps are smaller, but still, using the parse structure are better. Note that in real applications, where there is no out-of-vocabulary issue (i.e., some leafs are not seen in the sentiment dictionaries), the difference between S-LSTM and the recursive version without using the structures are expected to be between the gaps we observed here.
Conclusions
We aim to extend the conventional chain-structured long short-term memory to explicitly consider structures. In this paper we particularly study tree structures, in which Table 3. Performances of models that do not use the given sentence structures. S-LSTM-LR is a degenerated version of S-LSTM that reads input words from left to right, and S-LSTM-RR reads words from right to left. the proposed S-LSTM memory cell can reflect the history memories of multiple descendants through gated copying of memory vectors. The model provides a principled way to consider long-distance interplays over the structures. We leveraged the model to learn distributed sentiment representations for texts, and showed that it outperforms a stateof-the-art recursive model by replacing its tensor-enhanced composition layers with the S-LSTM memory blocks. We showed that the structure information is useful in helping S-LSTM achieve the state-of-the-art performance.
MODELS
The research community seems to contain two lines of wisdom; one attempts to learn distributed representation by utilizing structures when available, and the other prefers to believe recurrent neural networks can actually capture the structures implicitly through a linear-chain coding process.
In this paper, we also attempt to give some empirical evidences toward answering the question. It is at least for the settings of our experiments that the explicit input structures are helpful in inferring the high-level (e.g., root) semantics.
Figure 1 .
1An example of S-LSTM, a long-short term memory network on tree structures. A tree node can consider information from multiple descendants. Information of the other nodes in white are blocked. The small circle ("•") or short line ("−") at each arrowhead indicates a pass or block of information, respectively, while in the real model the gating is a soft version of gating.
Figure 3 .
3Converging time during training for roots (the upper figure) and for all nodes (the lower figure).
S-LSTM (ROOT + LEAF LBLS) 44.1 † Performance over different levels of trees In
Figure 4 .
4Accuracies at different depths (the upper figure) in the trees, or by different lengths of the phrases (the lower figure).
Table 1 .
1Performances (accuracies) of different models on the test
set of Stanford Sentiment Tree Bank, at the sentence level (roots)
and the phrase level. † shows the performance are statistically
significantly better (p < 0.05) than the corresponding models.
MODELS
ROOTS
PHRASES
NB
41.0
67.2
SVM
40.7
64.3
RVNN
43.2
79.0
RNTN
45.7
80.7
S-LSTM
48.0 †
81.9 †
We leverage the S-LSTM model to solve a semantic composition problem that learns the meaning for a piece of texts-learning good representations for meaning of text is core to automatically understanding human languages. More specifically, we experiment with the models on the 1 As both of them can be shortened to be RNN, in the rest of this paper we refer to a Recurrent Neural Network as RNN and a Recursive Neural Network as RvNN.
sub-figure) and the other for all phrases (lower sub-figure). From these figures, we can observe that S-LSTM converge faster than the RNTN. For instance, for the phrase-level task, S-LSTM started to converge after about 20 minutes but the RNTN needed over 180 minutes. S-LSTM has3 The Stanford CoreNLP package (http://nlp.stanford.edu/sentiment/code.html) only gives approximate accuracies for 2-category sentiment, which are not included here in the table.
Phrase-level sentiment analysis is often defined over a very small subset of phrases of interest, such as in the phrase-level task defined in(Wilson et al., 2005;Mohammad et al., 2013).
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