A N -uniform quantitative Tanaka's theorem for the conservative Kac's N -particle system with Maxwell molecules
aa r X i v : . [ m a t h . P R ] A ug A N -uniform quantitative Tanaka’s theorem for theconservative Kac’s N -particle system with Maxwell molecules Mathias Rousset ∗ CERMICSINRIA Paris - Rocquencourt, Ecole des Ponts Paristech, & Université Paris-Est.
October 9, 2018
Abstract
This paper considers the space homogenous Boltzmann equation with Maxwell moleculesand arbitrary angular distribution. Following Kac’s program, emphasis is laid on the theassociated conservative Kac’s stochastic N -particle system, a Markov process with bi-nary collisions conserving energy and total momentum. An explicit Markov coupling (aprobabilistic, Markovian coupling of two copies of the process) is constructed, using simul-taneous collisions, and parallel coupling of each binary random collision on the sphere ofcollisional directions. The euclidean distance between the two coupled systems is almostsurely decreasing with respect to time, and the associated quadratic coupling creation (thetime variation of the averaged squared coupling distance) is computed explicitly. Then, afamily (indexed by δ > ) of N -uniform “weak” coupling / coupling creation inequalitiesare proven, that leads to a N -uniform power law trend to equilibrium of order ∼ t → + ∞ t − δ ,with constants depending on moments of the velocity distributions strictly greater than δ ) . The case of order moment is treated explicitly, achieving Kac’s program withoutany chaos propagation analysis. Finally, two counter-examples are suggested indicatingthat the method: (i) requires the dependance on > -moments, and (ii) cannot providecontractivity in quadratic Wasserstein distance in any case. Foreword
This paper is the rewritten, submitted version of the preliminary version: M.Rousset,
Scalable and Quasi-Contractive Markov Coupling of Maxwell Collision also availableon arXiv. The latter preliminary version is not to be published.
Contents ∗ Supported in part by
ERC MSMath . AMS 2000 subject classifications:
Primary 60J27; secondary 65C40.
Keywords : trend to equilibrium, Markov process, Kac’s particle system, coupling. Simultaneous Parallel Coupling 19 moment control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.5 Proof of Proposition 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Introduction
Kac’s particle system
The Kac’s conservative N -particle system is a Markov stochastic process describing theevolution of the velocities in ( R d ) N of N particles subject to random binary elastic collisions.The latter satisfies: (i) independence with respect to possible positions of particles (spacehomogeneity); (ii) conservation of momentum and kinetic energy (elasticity). Assuming prop-agation of chaos (asymptotic independence of subsets of particles), the formal N → ∞ limitof the probability distribution of a single velocity satisfies the classical Boltzmann kineticnon-linear equation, in its space homogenous simplified form. When the collision rate is con-stant, in particular independent of the relative speed of particle pairs, we speak of Maxwellmolecules .A classical problem consists in quantifying the speed at which the probability distributionof the latter process converges towards its large time limit. The latter limiting, invariantprobability, is the uniform distribution on the sphere S Nd − d − defined by the conservation ofmomentum and energy. Context
Let us recall standard strategies for probability flows solutions of evolution (linear or non-linear) equations defined by a reversible Markovian mechanism.(i) The (relative) entropy dissipation method. Denote t π t a probability flow with statespace E , expected to converge to π ∞ . The entropy method computes the variation ofthe relative entropy dd t Z E d π t d π ∞ ln d π t d π ∞ d π ∞ | {z } E ( π t ) = − D ( π t ) ≤ , and try to obtain exponential convergence to equilibrium by obtaining a so-called modi-fied log-Sobolev inequality of the form E ( π ) ≤ c ls D ( π ) ∀ π ∈ P ( E ) , (0.1)2or some constant c ls > . When t π t is the distribution flow of a reversible diffusion ona Riemannian manifold, the famous curvature condition CD ( c cd , ∞ ) of Bakry and Emery(a mixture of strong convexity of the diffusion drift’s potential, and uniform positivecurvature of the metric, with lower bound c cd > , see [4] and references therein) yieldssuch an exponential convergence by proving the inequality < c cd ≤ c ls . The latter isobtained using the inequality dd t D ( π t ) ≤ − c cd D ( π t ) , and integrating through time. This topic has received considerable interest recently,due to the following gradient’s flow interpretation: the probability flow of a reversiblediffusion on a manifold is in fact a gradient flow of the relative entropy E ( π ) withrespect to the probability metric given by the quadratic Wasserstein distance W (see themonographs [38, 1]). This has led to the interpretation of c cd has a uniform displacementconvexity constant, and yielded a conceptual explanation for the inequality < c cd ≤ c ls .(ii) The weaker ( < c ls ≤ c sg ) spectral gap method which computes dd t Z E (cid:18) − d π t d π ∞ (cid:19) d π ∞ | {z } E ( π t ) = − D ( π t ) ≤ , and try to obtain exponential convergence to equilibrium by obtaining a so-called spectralgap inequality of the form E ( π ) ≤ c sg D ( π ) ∀ π ∈ P ( E ) . When t π t is the flow of a reversible Markov process, the latter is indeed the spectralgap of D (the so-called Dirichlet form) seen as a self-adjoint operator in L ( E, π ∞ ) .(iii) The Markov coupling method, which amount to construct an explicit Markov coupling,a probabilistic coupling of two copies of the Markov process of interest which is itselfagain Markov: t ( U t , V t ) ∈ E × E. If the latter coupling contracts with respect to some distance in an average L p sense: dd t E ( d ( U t , V t ) p ) /p ≤ − c p E ( d ( U t , V t ) p ) /p for any initial condition, then the method yields an upper bound on the contractiv-ity (with constant c w p ≥ c p > ) with respect to the related probability Wassersteindistance W p . Exponential trend to equilibrium follows. Here again, for reversible dif-fusion on a manifold, the curvature condition CD ( c cd , ∞ ) is typically required (see [40]and references therein) to construct such a contractive coupling using paralleltransport . Using again the gradient’s flow interpretation, the CD ( c cd , ∞ ) condition isessentially equivalent to contractivity in quadratic Wasserstein distance c cd = c w , (seefor instance [39, 1]). 3 emark 0.1. (i) Up to our knowledge, there is no such settled general theory for jump processes, yieldinginequalities analogous to c cd = c w ≤ c ls , (see however [31, 26] and related papers forrecent approaches). This is of importance in our context, since space homogenous Boltz-mann’s collisions are reversible jump processes, and an the type of angular distribution(“small jumps”) is known to influence the trend to equilibrium (see discussion below).(ii) In the context of N -particle systems, such methods may be used either on the particlesystem (and one may look for N -uniform constants), or directly on the non-linear mean-field (here, kinetic) equation, seen as the formal N = + ∞ case.(iii) In practice, some more or less weakened versions of the above inequalities, especially ofthe modified log-Sobolev (“entropy / entropy dissipation”) inequality are obtained. Theyare of the form: E ( π ) /δ ≤ c ls ,δ ( π ) D ( π ) ∀ π ∈ P ( E ) , (0.2)and yields algebraic or power law trends of order t − δ , for δ ∈ ]0 , + ∞ ] ( δ = + ∞ formally stands for the exponential case). c ls ,δ ( π ) is typically dependent on moments of π . A priori moment propagation estimates on the probability flow have to be obtainedseparately in order to get quantitative convergence to equilibrium. In particular, inkinetic theory, “Cercignani’s conjecture” refers to the case c ls ,δ =+ ∞ ( π ) > , where thetype of dependence with respect to π (moments, regularity) is known to be propagatedby the probability flow. Usually, probabilists speak of modified log-Sobolev inequalitieswhen c ls is unconditionally bounded below (independent of π ).The mathematical literature studying the convergence to equilibrium of the space ho-mogenous Boltzmann kinetic equation, and its related Kac’s conservative N -particle systemis extremely vast, and we refer to the classical reviews [14, 36]. In the present work, theso-called Boltzmann collision kernel (the Markov generator defining random collisions on thesphere defined by the conservation laws of a pair velocities) will be denoted with Carleman’srepresentation b ( v − v ∗ , d n ′ v ) ≡ unif θ ( n v , d n ′ v ) β ( | v − v ∗ | , d θ ) , (0.3)where in the above ( v, v ∗ ) ∈ R d × R d are the incoming (pre-collisional) velocities of a pair par-ticle, ( n v , n ′ v ) = (cid:16) v − v ∗ | v − v ∗ | , v ′ − v ′∗ | v ′ − v ′∗ | (cid:17) ∈ S d − × S d − are respectively the pre- and post-collisional di-rections, n v · n ′ v = cos θ defines the scattering angle θ ∈ [0 , π ] , and unif θ ( n v , d n ′ v ) is the uniformprobability distribution on the sphere defined by a prescribed scattering angle. In [15], a reviewis provided about the different types of collisions classifying the possible large time behaviors.Following their convention, one can introduce two parameters ( γ, ν ) ∈ [ − d, + ∞ [ × ] − ∞ , and assume that β ( | v − v ∗ | , d θ ) ∼ | v − v ∗ | γ θ − ν − d θ, in the limit where | v − v ∗ | → + ∞ , as well as θ → + . From physical scattering theory, the case γ < is often called ”soft potential”, while γ > is called ”hard potential” and γ = 0 is called“Maxwell molecules”. The case ν ≤ corresponds usually to bounded kernels and is called4angular cut-off”. The case ν ∈ ]0 , corresponds to Levy generators associated to fractionalpseudo-differential operators, while formally the case ν = 2 is the diffusive case, the Boltzmannoperator becoming proportional to the Laplace-Beltrami operator on the sphere of collisionaldirections, also called the “Landau operator”. Finally, we also mention the important Kac’s“caricature” case, where d = 1 and momentum is not conserved; the latter is usually consideredwith Maxwell molecules and angular cut-off: ν < , γ = 0 . Literature
Entropy method, N = + ∞ . First, the most studied method for trend to equilibrium inkinetic theory is by far the entropy method, in the case of the kinetic ( N = + ∞ ) equa-tion. Some famous counterexamples (see [6, 5, 37]) have shown that a weak entropy-entropydissipation inequality of the form (0.2) (called “Cercignani’s conjecture” in kinetic theory)cannot not hold for δ = + ∞ ( i.e. c ls ,δ =+ ∞ ( π ) = 0 ), even when restricting to reasonableconditions on π (moments, regularity,...). This counter-example contains several physicallyrealistic collisions (for instance ν < , γ < , which includes Maxwell molecules in the non-diffusive case, of interest here). In fact, it has been conjectured in [15] from rigorous proofsin meaningful particular examples (for instance the Landau case γ = 0 , ν = 2 in [16]), that anecessary and sufficient criteria such that a modified log-Sobolev inequality holds is the fol-lowing: c ls > ⇔ γ + ν + ≥ . The latter suggests a common contribution of the probabilityof high energy collisions ( γ large), and of small scattering angle ( ν large).Meanwhile, many studies have been developped in the cases where exponential entropyconvergence is known to fail, say c ls = 0 . Some weakened versions of the “entropy / entropydissipation” analysis of the form (0.2) (here is a sample: [13, 9, 5, 35, 37]) in order to obtainalgebraic or power law trends with some a priori estimates on π that has to be obtainedseperately.These facts motivates the power-law behavior and moment dependence in theMaxwell case with angular cut-off ( γ = 0 , ν < , which are obtained in the presentpaper . Spectral gap and Wild’s expansion, N = + ∞ . For the case of interest in the presentpaper (Maxwell molecules, jump kernels: ( γ = 0 , ν < ), an expansion method, known asWild’s expansion, enables to give precise estimates using some refined form of the central limittheorem. It has been shown in [12], that arbitrary high moments of a velocity distributionnecessarily lead to arbitrary slow decay to equilibrium (in L ). In [12, 18, 19, 20] a full theoryof convergence to equilibrium for Maxwell molecules is then developed using Wild’s method,showing that the convergence is essentially exponential with rate given by the spectral gap,but requires some moment and regularity condition on the initial condition, and a constantwhich is sub-optimal for short time.In [29], the case of hard potentials is treated with a spectral method that essentially proveexponential convergence with rate given by the spectral gap of the linearized near equilibriumequation, and rely on moment creation in the case of hard potentials.These facts motivates moment dependence found in the present paper, but suggests thatthe associated power law behavior is sub-optimal for very large times. Spectral gap method
N < + ∞ . Direct studies of the trend of equilibrium of the Kac’s N -particle system have been undertaken [17, 10, 11, 30]. The main striking feature of thelatter list is the difficulty to achieve the so-called “Kac’s program” for large timebehavior (see [28] ): obtaining a scalable ( N -uniform) analysis of the trend to equilibrium5f the N -particle system. A famous result (see [10, 11]) exactly computes the spectral gap for Maxwell molecules ( γ = 0 ), and proves that the latter is N -uniform ( lim N → + ∞ c N, sg > ).However, the L (( R d ) N , π ∞ ) -norm used in the spectral gap case, is usually thought to bean unsatisfactory N -scalable measure of trend to equilibrium (it is rather associated to the linearized kinetic equation for N = + ∞ , see the last section of [37] for a longer discussion).By extensivity of entropy, the modified log-Sobolev constant c N, ls is believed to be a morereliable quantity. Entropy method,
N < + ∞ . According to [37], it is conjectured (and partially provenin the case of Kac’s caricature) that the modified log-Sobolev constant of Kac’s N -particlesystem, at least for ν < , is of order c N, ls ∼ N − γ/ . This is of special interest in the(unphysical) case γ = 2 , since it shows that a modified log-Sobolev inequality holds, similarlyto the kinetic equation ( N = + ∞ ). In general (and in particular for the diffusive, Landaucase with Maxwell molecules, or more generally for γ + ν + ≥ ), the large N behavior of themodified log-Sobolev constant of the Kac’s particle system is an open problem. Coupling method, N ≤ + ∞ . The use of explicit coupling methods to study the trendto equilibrium of Markov processes (or Markov chains) is now a classical topic on its own,especially for discrete models (see e.g. the classical textbook [25]). It is also a well-establishedtopic for continuous models, as well as for non-linear partial differential equations that havean interpretation in terms of a Markovian mechanism. For the granular media equation (dif-fusive particles interacting through a smooth pairwise potential), and its related N -particlesystem, Markov coupling can give exponential trend to equilibrium, by using a “strong cou-pling/coupling creation inequality” (see for instance [27, 7, 8], using CD ( c N, cd , ∞ ) -type con-vexity assumptions on potentials, with lim N → + ∞ c N, cd > ). For the Kac’s N -particle systemof kinetic theory, the only paper known to us quantitatively using a Markov coupling is in [30].In the latter, the (almost optimal, and not N -uniform) estimate ( c N, w2 ∼ / ( N ln N ) ) isobtained for Kac’s caricature , in accordance with the result cited in [37]: c N, ls ∼ /N . Quantitative propagation of chaos.
Finally in [28], the authors have reversed the point ofview of Kac’s program, and proved indirectly the trend to equilibrium of Kac’s N -particlesystem by pulling-back the long time stability of the kinetic (mean-field N = + ∞ limit)equation using uniform in time propagation of chaos. Motivation of the paper.
The goal of this paper is to develop on the Kac’s particle system with Maxwell moleculesa “weak approach” of the (quadratic) coupling method, uniformly in the numberof particles N . The latter results extend in spirit the classical paper by Tanaka [33], wherethe quadratic Wasserstein distance between the solution of the kinetic equation with Maxwellcollisions and the equlibrium Gaussian distribution (the Mawellian) is shown to be decreasingthrough time, with a similar coupling argument, but without quantitative analysis. In a sense,the analysis in the present paper makes Tanaka’s argument quantitative (with respect to time),and available for the Kac’s N -particle system.More precisely, we will obtain power law trends to equilibrium with respect to a permuta-tion symmetrized version of the quadratic Wasserstein distance, and upon estimates on highermoments of the velocity distribution. Up to our knowledge, this is the first time thistype of estimate is obtained directly on the Kac’s particle systems . Such resultsare similar in spirit to the classical results mentioned above ([13, 9, 5, 35, 37]) that are ob-tained with entropy methods using weakened “entropy/entropy creation” inequalities. Yet, the6oupling method has some noticeable specificities:(i)
Maxwell restriction.
Because it requires coupling of simultaneous collisions, the analysisis restricted to Maxwell molecules ( γ = 0 ).(ii) Angular condition
The analysis is independent of the scattering angular distribution ν ∈ ] − ∞ , .(iii) Particle system size
The analysis is independent of the particle system size N . It canwork similarly for the kinetic equation N = + ∞ .(iv) A priori estimates
The analysis depends only on higher > moments of velocity distri-butions, and not on regularity estimates . Moments are known to exactly propagatesthrough time (finite and infinite moments remain so, see the classical paper [24]) for the N = + ∞ kinetic equation with Maxwell molecules, but unfortunately, not so directlyfor the Kac’s N -particle system. We will use the easiest case of order moments.(v) Constants
Constants explicitable.We finally also suggest some negative results in the form of two counterexamples to strongerversions of “coupling/coupling creation inequalities”. Although similar in spirit, the latterhave a different interpretation as compared to the counterexamples to Cercignani’s conjecture(see [6, 37]) in the entropy context: they provide information on the limitation of the specificchoice of the coupling, but not directly on the trend to equilibrium of the model. Here arethe counterexamples:(i) Velocity distributions with sufficiently heavy tails can make the coupling creation vanish.This first counterexample shows that the obtained “coupling/coupling creation inequal-ity” must involve some higher order (say, > ) velocity distribution moments.(ii) There exists a continuous perturbation of the identity coupling at equilibrium for whichhowever the coupling creation is sub-linearly smaller than the coupling itself. This secondtype of counterexample shows that even with moment restrictions, a sub-exponentialtrend is unavoidable .As discussed above, the latter facts are consistent with known power law behaviors dependingon moment conditions (see again [12]) in the angular cut-off case ( ν < , γ = 0) . Howeverthe sub-optimality of the considered coupling may be conjectured in the diffusive Landau case ( ν = 2 , γ = 0) , where exponential convergence with a log-Sobolev gap ( c ls > ) is known tooccur (see again [16]). Summary of results
Here, and in the rest of the paper, the following notation is used h o ( u, v, u ∗ , v ∗ ) i N def = 1 N N X n ,n =1 o ( u ( n ) , u ( n ) , u ( n ) , v ( n ) ) , in order to account for averages over particles of a two-body observable o : (cid:0) R d × R d (cid:1) → R .7he Markov coupling of Kac’s conservative N -particle system is a Markov process denoted t ( U t , V t ) ≡ ( U t, (1) , V t, (1) , . . . , U t, ( N ) , V t, ( N ) ) ∈ (cid:16) R d × R d (cid:17) N .U t ∈ (cid:0) R d (cid:1) N and V t ∈ (cid:0) R d (cid:1) N both satisfying the same normalized conservation laws: h V t i N = 0 a . s . [ centered momenta ] D | V t | E N = 1 a . s .. [ normalized energy ] (0.4)Throughout the paper, we will denote the associated probability distributions of the particlesystem: π t def = Law ( V t ) ∈ P (cid:16) ( R d ) N (cid:17) , π c,t def = Law ( U t , V t ) ∈ P (cid:18)(cid:16) R d × R d (cid:17) N (cid:19) , and assume that t U t is always taken to be distirbuted according to the equilibrium station-ary distribution, given by the uniform distribution on the sphere defined by the conservationlaws: Law ( U t ) = π ∞ = unif S Nd − d − . We also assume that the Kac’s N -particle systems are constructed from a Boltzmann kernelof the type (0.3) with Maxwell molecules ( γ = 0 ), and we will use Levy’s normalizationon scattering angular distribution Z π sin θβ (d θ ) = 1 . (0.5)The Markov dynamics of the coupled particle system can be described without ambiguitywith the related master equation dd t π c,t = L ∗ c,N π c,t , where ∗ refers to duality between measures and test functions. In the above, the dynamicsgenerator is of the form L c,N def = 12 N X ≤ n = m ≤ N L c, ( n,m ) , (0.6)where the two-body coupled generator L c, ( n,m ) is a coupled collision operator acting on theparticle pair ( n, m ) for n = m , and defined when acting on test functions ψ ∈ C ∞ c ( (cid:0) R d × R d (cid:1) ) by: L c ( ψ )( u, v ) def = Z S d − × [0 ,π ] (cid:0) ψ ( u ′ , v ′ ) − ψ ( u, v ) (cid:1) unif c,θ ( n u , n v , d n ′ u d n ′ v ) β (d θ ) . (0.7)In the above, the parallel spherical coupling unif c,θ is precisely defined in point ( iii ) of Defi-nition 0.3 below. Remark 0.2. unif c,θ ( n u , n v ; d n ′ u d n ′ v ) (there is a singularity on theextremity set (cid:8) n u , n v ∈ S d − | n u = − n v (cid:9) ), there is a difficulty to define rigorously thecoupled system without angular cut-off: b = Z [0 ,π ] β (d θ ) < + ∞ . However, the analysis of the present paper is independent of the latter cut-off value b ,and the proofs will be carried out for arbitrary angular singularity, using a continuityargument.(ii) By construction, such contracting Markov couplings cannot satisfy detailed balance (time symmetry is broken to obtain a contractive map) on the product space L (( R d × R d ) N ) , so that there is no simple way (at least known to us) to write a dual (to themaster equation (0.7)) kinetic equation on the density of π c,t .The latter coupling can be defined without ambiguity by requiring simultaneous colli-sions, and parallel coupling of each collision . This is specified using the following setof rules. Definition 0.3 (Simultaneous parallel coupling) . The Simultaneous Parallel Coupling between t U t and t V t is obtained by the following set of rules:(i) Collision times and collisional particles are the same (simultaneous collisions), as impliedby (0.6) .(ii) For each collision, the scattering angles θ ∈ [0 , π ] of are the same, as implied by (0.7) .(iii) For each coupled collision, the post-collisional directions n ′ u ∈ S d − and n ′ v ∈ S d − arecoupled using the elementary rotation along the great circle (the geodesic) of S d − joining n u and n v . The resulting coupled probability is denoted unif c,θ (cid:0) n u , n v ; d n ′ u d n ′ v (cid:1) . Since the post-collisional directions ( n ′ u , n ′ v ) are obtained using parallel coupling on asphere , a strictly (under the crucial assumption that d ≥ ) positively curved manifold,the latter coupling is bound to be almost surely decreasing, in the sense that for any initialcondition and ≤ t ≤ t + h D | U t + h − V t + h | E N ≤ D | U t − V t | E N a . s .. We then compute the quadratic coupling creation defined by dd t E D | U t − V t | E N = − E ( C ( U t , V t )) ≤ , where the “two-body coupling creation” functional satisfies C ( u, v ) = d − d − h| u − u ∗ | | v − v ∗ | − ( u − u ∗ ) · ( v − v ∗ ) i N ≥ , (0.8)9nd can be interpreted as a degree of alignement between the velocity difference v − v ∗ ∈ R d ,and its coupled counterpart u − u ∗ ∈ R d .In order to relate the coupling and the coupling creation, we have introduced in the presentpaper an original general sharp inequaIity, proved using brute force calculation: for any vectors u ∈ ( R d ) N and v ∈ ( R d ) N both satisfying the normalized conservation laws (0.4), it holds − h u · v i N ≤ min (cid:16) κ h u ⊗ u i N , κ h v ⊗ v i N (cid:17) × D | u − u ∗ | | v − v ∗ | − (( u − u ∗ ) · ( v − v ∗ )) E N , ∀ u, v ∈ S Nd − d − . (0.9)In the above, the spectral quantity κ S def = (1 − λ max ( S )) − ∈ [ dd − , + ∞ ] (0.10)is defined with the spectral radius λ max ( S ) ≤ of a positive trace symmetric matrix. It isfinite if and only if S is of rank at least (non-alignement condition). Note that if h u · v i N ≥ ,then the coupling distance satisfies D | u − v | E N ≤ (cid:16) − h u · v i N (cid:17) . The equality case in (0.9)is achieved (sharpness) under some strong isotropy and co-linear coupling conditions, detailedin Section 3.1. Remark 0.4.
The inequality (0.9) can be interpreted as a way to bounde from above theeuclidean distance D | u − v | E N with a quadratic average of alignement between the velocitydifference v − v ∗ ∈ R d and u − u ∗ ∈ R d .It is then of interest to compare that the alignement functional in the right hand sideof (0.9), and the coupling creation functional (0.8). They differ by a weight of the form | u − u ∗ | | v − v ∗ | which implies that the strong “coupling/coupling creation” constant c ,N def = inf u,v ∈ S Nd − d − C ( u, v )2 D | u − v | E N is degenerated when the number of particles becom large: lim N → + ∞ c ,N = 0 (see the coun-terexamples of Section 3.2 for more details). However, a direct Hölder inequality yields someweaker power law versions (see details in Section 1.3) for any δ > , of the form c ( δ, u, v ) ≤ C ( u, v )2 D | u − v | E / δN , (0.11)where the constant c ( δ, u, v ) can be lower bounded by ( N -averaged) moments of the velocitydistributions, of any order strictly greater then δ ) . Additional control on the posi-tive correlation condtion h u · v i N ≥ , and on the isotropy of h u ⊗ u i N are required by theinequality (0.9).Upon a priori control of such moments, this leads to a power law trend to equilibriumof the Kac’s system distribution , of order ∼ t → + ∞ t − δ . The trend to equilibrium is obtained10ith respect to a permutation symmetrized quadratic Wasserstein distance defined by thequotient distance on ( R d ) N / Sym N : d sym ( u, v ) = def inf σ ∈ Sym N (cid:16)D(cid:12)(cid:12) u − v σ ( . ) (cid:12)(cid:12) E N (cid:17) / . The lat-ter is natural for exchangeable distributions, and necessary to handle the positive correlationassumption h u · v i N ≥ .The case of order -moments is finally treated explicitly, and sub-linear trends are esti-mated. For any < δ < , we prove that: d sym ,W ( π t , π ∞ ) ≤ (cid:16) d sym ,W ( π , π ∞ ) − /δ + c δ ( t − t ∗ ) + (cid:17) − δ , where the cut-off time satisfies depends logarithmically on the initial order : t ∗ = 2 (cid:18) ln (cid:18) dd + 2 E D | V | E N − (cid:19)(cid:19) + . and c δ > is explicitly computable. For instance, we find that lim δ → lim d → + ∞ lim N → + ∞ c δ,N ≥ − , which although sub-optimal, is not unreasonably small. Contents
In Section 1, we recall some notation and basic concepts related to kinetic theory andprobabilistic couplings for Markov particle systems. We then detail the results of the presentwork.In Section 2, the parallel, spherical coupling of interest is detailed, together with the precisecalculation of the associated quadratic coupling creation.In Section 3, the special inequality between coupling distance and colinearity of coupledpairs is proven.In Section 4, some details of proofs are given.
As usual, the velocities of a pair of collisional particles are denoted ( v, v ∗ ) ∈ R d × R d , and the post-collisional quantities are denoted by adding the superscrpit ′ . All particles areassumed to have the same mass so that the conservation of momentum imposes v ′ + v ′∗ = v + v ∗ , and conservation of energy imposes (cid:12)(cid:12) v ′ (cid:12)(cid:12) + (cid:12)(cid:12) v ′∗ (cid:12)(cid:12) = | v | + | v ∗ | .
11s a consequence, the relative speed is also conserved (cid:12)(cid:12) v ′ − v ′∗ (cid:12)(cid:12) = | v − v ∗ | . The post-collisional velocities of a particle pair are thus given by the standard collision mapping ( v ′ = ( v + v ∗ ) + | v − v ∗ | n ′ v ,v ′∗ = ( v + v ∗ ) − | v − v ∗ | n ′ v , (1.1)where ( n v , n ′ v ) = (cid:18) v − v ∗ | v − v ∗ | , v ′ − v ′∗ | v ′ − v ′∗ | (cid:19) ∈ S d − × S d − denote the pre-collisional/post-collisional directions. The scattering or deviation angle θ ∈ [0 , π ] of the collision is then uniquely defined as the half-line angle between the pre-collisionaland the post-collisional directions: cos θ def = n ′ v · n v . The binary collisions are then specified by the following operator (a Markov generator) actingon test functions ϕ ∈ C ∞ c (( R d ) ) : L ( ϕ )( v, v ∗ ) def = Z S d − × [0 ,π ] (cid:0) ϕ ( v ′ , v ′∗ ) − ϕ ( v, v ∗ ) (cid:1) b ( v − v ∗ , d n ′ v ) , (1.2)where in the above the Boltzmann collision kernel b can be decomposed as b ( v − v ∗ , d n ′ v ) def = Z θ ∈ [0 ,π ] unif θ ( n v , d n ′ v ) β ( | v − v ∗ | , d θ ) , with (i) β ( | v − v ∗ | , d θ ) the angular collisional kernel , a positive measure on [0 , π ] satisfyingthe Levy normalization condition (0.5), and (ii) unif θ is the uniform probability distributionon the sphere of collisional directions S d − with prescribed scattering (or deviation) angle θ .More formally: unif θ ( n v , d n ′ v ) def = unif { n ′ v ∈ S d − | n v · n ′ v =cos θ } (cid:0) d n ′ v (cid:1) , (1.3)where unif S denotes the uniform probability distribution on a sphere S in euclidean space.The introduction of unif θ will be convenient to describe the coupled collision unif c,θ .By construction, unif θ satisfies the detailed balance condition (micro-reversibility) withinvariant probability the uniform distribution on the sphere S d − . Formally: d n v unif θ ( n v , d n ′ v ) = d n ′ v unif θ ( n ′ v , d n v ) ∈ P (cid:16) S d − × S d − (cid:17) , where we implicitly define d n v = unif S d − (d n v ) . It is convenient to keep in mind that (1.1)extends by measure decomposition to the following version of detailed balance in the euclideanambient space d v d v ∗ unif θ ( n v , d n ′ v ) = d v ′ d v ′∗ unif θ ( n ′ v , d n v ) as (unbounded) positive measures in R d × R d ; by a tensorization argument, the latter yields reversibility (see below) of the conservative Kac’s particle system.12he conservative Kac’s N -particle system is then defined as a Markov process t V t ≡ ( V t, (1) , . . . , V t, ( N ) ) ∈ (cid:16) R d (cid:17) N , (1.4)whose probability distribution is described without ambiguity with the related master equation dd t π t = L ∗ N π t , (1.5)holding on the probability distribution flow: π t ≡ π Nt (cid:0) d v (1) . . . d v ( N ) (cid:1) def = Law( V t ≡ (cid:0) V t, ( N ) , . . . , V t, ( N ) (cid:1) ) ∈ P sym (( R d ) N ) t ≥ , (1.6)where P sym (( R d ) N ) denotes permutation symmetric probability distributions (assuming theinitial condition π is already permutation symmetric). Each of the V t, ( n ) , ≤ n ≤ N representthe velocity of a physical particle, the whole system of particles being subject to the randombinary elastic collisions.For arbitrary N ≥ , the Markov generator in ( R d ) N have the following structure: L N def = 12 N X ≤ n = m ≤ N L ( n,m ) , (1.7)where L ( n,m ) is the Markov generator (1.2) with state space (cid:0) R d × R d (cid:1) , the subscript ( n, m ) denoting the action on the corresponding pair of particles. The N scaling in (1.7) can bephysically understood by stating that each individual particle is subject to a collision mecha-nism with O (1) rate and a uniformly picked other particle. For elastic collisions, we have that L ( ϕ )( v, v ∗ = v ) = 0 , so that if we consider test functions ψ ∈ C ∞ c (( R d ) N ) of average type: ψ ( v ) = h ϕ ( v ) i N , we get, thanks to the factor / N in (1.7): L ψ ( v ) = 12 h L (( v, v ∗ ) ϕ ( v ) + ϕ ( v ∗ )) i N . = h L ( ϕ ⊗ ll) ( v, v ∗ ) i N . By construction, the process (1.4) satisfies the physical conservation laws of momentumand energy, that will be taken centered and normalized according to (0.4) throughout thepaper. Moreover, the fundamental detailed balance condition (1.1) implies detailed balanceat the level of the particle system. More precisely:(i) The unique stationary probability distributions is given by the sphere of conservationlaws: π ∞ = unif S d ( N − − (d v (1) . . . d v ( N ) ) ∈ P (( R d ) N ); where we have implicitly define the unit sphere with normalization condition (0.4) S d ( N − − = n v ∈ ( R d ) N | h v i N = 0 , D | v | E N = 1 o . π ∞ is in fact an equilibrium , in the sense that the process in stationary distribution is time reversible Law ( V ) = π ∞ ⇒ Law ( V t , ≤ t ≤ T ) = Law ( V T − t , ≤ t ≤ T ) ∀ T > . (iii) Equivalently to ( ii ) , on has L ∗ N = L N in the sense of self-adjointness in the Hilbert space L (cid:0) ( R d ) N , d v (1) . . . d v ( N ) (cid:1) , or alternatively in L (cid:0) S d ( N − − (cid:1) .Let us also recall that the latter process can be constructed explicitly in the case of Maxwellmolecules ( β ( | v − v ∗ | , d θ ) ≡ β (d θ ) ), and angular cut-off ( R [0 ,π ] β (d θ ) < + ∞ ):(i) Each particle perform a collision with a fixed rate b := R [0 ,π ] β (d θ ) , and with a uniformlyrandomly chosen other particle.(ii) The scattering angle of each collision is independently sampled according to the proba-bility defined by β (d θ ) /b .(iii) The random post-collisional directions n ′ v is uniformly sampled with unif θ and scatteringangle prescribed by ( ii ) .The general angular collisions can then be obtained (rigorously, see [21]) as the limit of thelatter.If we denote the probability π t as a (generalized) probability density function with referencemeasure d v (1) . . . d v ( N ) π t ≡ f t ( v (1) . . . v ( N ) )d v (1) . . . d v ( N ) , then the detailed balance conditions yields the usual explicit dual kinetic equation , for any v ∈ ( R d ) N : dd t f t ( v ) = 12 N N X n,m =1 Z S d − × [0 ,π ] (cid:16) f t ( v ′ ) − f t ( v ) (cid:17) b (cid:16) n v ( n,m ) , d n ′ v ( n,m ) (cid:17) = L N f t ( v ) , where in the above, the subscript v ( n,m ) = ( v ( n ) , v ( m ) ) ∈ R d × R d refers to the correspondingpair of particles.Finally, one says that weak propagation of chaos holds, if for any time t ≥ , the marginaldistribution of k given particles of the above particle system ( k being fixed) is converging (inprobability distribution) to a product measure when N → + ∞ (independence). Under thisassumption, the large N limit of the one body marginal distribution π t ∈ P ( R d ) of the particlesystem satisfies an evolution equation in closed form with a quadratic non-linearity given by: dd t Z R d ϕ d π t = Z R d × R d L ( ϕ ⊗ ll) d π t ⊗ d π t , (1.8)where in the above ϕ is a test function of R d . The latter can be easily derived from the masterequation (1.5), by choosing tests functions in ( R d ) N of the form h ϕ ( v ) i N with ϕ ∈ C ∞ c ( R d ) .The dual kinetic equation of the non-linear equation (1.8) is the famous Boltzmann equation in R d with Maxwell collision kernel b . The usual expression on the one particle velocity density,denoted f t ( v )d v ≡ π t (d v ) , is then: dd t f t ( v ) = Z R d × S d − (cid:0) f t ( v ′ ) f t ( v ′∗ ) − f t ( v ) f t ( v ∗ ) (cid:1) d v ∗ b ( n v , d n ′ v ) . (1.9)14 .2 Coupling Let ( E, d ) denote a Polish state space ( E := (cid:0) R d (cid:1) N euclidean with d ( u, v ) := D | u − v | E N inthe present paper). We say that a time-homogenous Markov process in the product space t ( U t , V t ) ∈ E × E, is a Markov coupling , if the marginal probability distribution of the two processes t U t ∈ E and t V t ∈ E are two instances of the same Markov dynamics, with possibly different initialdistributions. We will be interested in weakly contracting couplings, where the couplingdistance is almost surely decreasing: d ( U t + h , V t + h ) ≤ d ( U t , V t ) a . s ., ∀ t, h ≥ , (1.10)and will especially consider quadratic coupling creation defined by: C ( u, v ) def = − dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 E ( U ,V )=( u,v ) (cid:0) d ( U t , V t ) (cid:1) ≥ . From an analytic point of view, if L c denotes the Markov generator of the coupled process t ( U t , V t ) , and L the Markov generator of the marginal process t U t (or t V t ) it isuseful to keep in mind that:(i) Coupling amounts to consider the compatibility condition: for any ( u, v ) ∈ E × E , and ϕ a test function: L c ( ϕ ⊗ ll)( u, v ) = L ( ϕ )( u ) , L c (ll ⊗ ϕ )( u, v ) = L ( ϕ )( v ) . (ii) Couplings generators L c invariant by permutation of the role of the two variables ( u, v ) ∈ E are called “symmetric” (if ϕ ( u, v ) = ϕ ( v, u ) , then L c ( ϕ )( u, v ) = L c ( ϕ )( v, u ) ). We willonly use symmetric couplings, although this fact is unimportant in the analysis.(iii) The quadratic coupling creation functional can be defined as C ( u, v ) def = −L c (cid:0) d (cid:1) ( u, v ) . (1.11)In the most favorable situation, one can expect a contractive coupling / coupling creationinequality with constant < c < + ∞ : d ( u, v ) ≤ c C ( u, v ) , ∀ ( u, v ) ∈ E . (1.12)The latter leads to contractivity with respect to the quadratic Wasserstein distance ( c ≤ c w with the notation of the introduction): d W ( π ,t , π ,t ) ≤ d W ( π , , π , ) e − c t , (1.13)where in the above t ( π ,t , π ,t ) are two probability flows solution of the master equation dd t π t = L ∗ π t , d W ( π , π ) def = inf π ∈ Π( π ,π ) (cid:18)Z E d ( u, v ) π (d u, d v ) (cid:19) / , (1.14) Π( π , π ) denoting the set of all possible couplings with marginal distributions π and π .Finally, if π ∞ ∈ P ( E ) denotes a stationary probability distribution for L , then (1.13) yieldsexponential convergence of the flow t π t towards π ∞ with respect to Wasserstein distance.In the context of the present paper, contractivity estimates as (1.13) are too strong toohold. We will seek for a power law trend to equilibrium in the form d + d t d W ( π t , π ∞ ) ≤ − c δ ( π t ) d W ( π t , π ∞ ) /δ , or equivalently d W ( π t , π ∞ ) ≤ (cid:18) d W ( π , π ∞ ) − /δ + 1 δ Z t c δ ( π s ) d s (cid:19) − δ . where we denote d + d t x def = lim inf h → + x t + h − x t h . Remark 1.1. (i) The limit δ → + ∞ gives back the exponential trend (1.13).(ii) The present paper will compute precise estimates of c δ ( π ) in terms of moments of π oforder q (1 + δ ) , for any q > .Consider now the case of an exchangeable (particle permutation symmetric) N -particlesystem as a random vector U ∈ ( R d ) N , where R d is euclidean. Strictly speaking, the statespace is obtained by quotienting out the symmetric group Sym N : E := ( R d ) N / Sym N , or equivalently considering the subset of empirical distributions: E := P N ( R d ) = ( π ∈ P ( R d ) | ∃ u ∈ ( R d ) N , π = 1 N N X n =1 δ u ( n ) ) . The former can be endowed with the associated orbifold distance d sym ( u, v ) def = inf σ ∈ Sym N (cid:16)D(cid:12)(cid:12) u − v σ ( . ) (cid:12)(cid:12) E N (cid:17) / , which is by definition equivalently the quadratic Wasserstein distance induced by P ( R d ) : d sym ( u, v ) = d W , P N N N X n =1 δ u ( n ) , N N X n =1 δ v ( n ) ! . This leads to the following definition. 16 efinition 1.2.
Let P sym (( R d ) N ) ≃ P (( R d ) N / Sym N ) the set of symmetric (exchangeable)probabilities of ( R d ) N . The “two-step” or “symmetric” quadratic Wasserstein distance on P sym (( R d ) N ) denoted d W , sym is defined as the usual quadratic Wasserstein distance (1.14) on the quotient space ( R d ) N / Sym N endowed with the distance d sym . However, in the present paper, the Markov couplings of two particle systems in ( R d ) N won’t be constructed on the product space (cid:16) R d (cid:17) N / Sym N × (cid:16) R d (cid:17) N / Sym N , but on the non-quotiented space (cid:0) R d × R d (cid:1) N . The generator L c,N of the coupled systemconserve permutation invariance only globally , and the exchangeability of particle will bebroken at the initial coupling (see the proof in Section (4.3)). This corresponds to the intuitivepicture of pairing particles of two exchangeable sets once and for all.More precisely, we will use the symmetrized Wasserstein distance by picking an initialcondition as follows Lemma 1.3.
Let π , π ∈ P sym (cid:0) ( R d ) N (cid:1) be two exchangeable probabilities. Then there existsa random variable representation ( U , V , Σ) ∈ ( R d × R d ) N × Sym N with Law( U ) = π , Law( V ) = π such that:(i) d W , sym ( π , π ) = E D(cid:12)(cid:12) U − V , Σ( . ) (cid:12)(cid:12) E N .(ii) If V is almost surely centered ( h V i N = 0 a . s . ), then (cid:10) U · V , Σ( . ) (cid:11) N ≥ . s . .Proof. First, ( i ) . Since Sym N is finite, (cid:16)(cid:0) R d (cid:1) N / Sym N , d sym (cid:17) is a Polish quotient metric space,so that existence of an optimal coupling is known to hold (see [38]). Then an exchangeablerepresentative V of the quotient can be picked uniformly at random, and then Σ can bedefined such that: D(cid:12)(cid:12) U − V Σ( . ) (cid:12)(cid:12) E N = inf σ ∈ Sym N D(cid:12)(cid:12) U − V ,σ ( . ) (cid:12)(cid:12) E N a . s .. Second, ( ii ) . By the centering assumption on V : N ! X σ ∈ Sym N (cid:10) U · V ,σ ( . ) (cid:11) N = 0 , and the result follows by definition of quotient metric d sym . We can now detail the results of the present paper.We first give the special inequality that will enable to derive coupling / coupling creationinequalities.
Theorem 1.4.
Denote κ S = def (1 − λ max ( S )) − ∈ [ d/ ( d − , + ∞ ] where λ max ( S ) is the max-imal eigenvalue of a trace symmetric positive matrix S . Let ( U, V ) ∈ R d × R d be a couple of entered and normalized (with E | U | = E | V | = 1 ) random variables in euclidean space. Let ( U ∗ , V ∗ ) ∈ R d × R d be an i.i.d. copy. Then the following inequality holds: − E ( U · V ) ≤ min (cid:0) κ E ( U ⊗ U ) , κ E ( V ⊗ V ) (cid:1) E (cid:16) | U − U ∗ | | V − V ∗ | − (( U − U ∗ ) · ( V − V ∗ )) (cid:17) . (1.15) Note that min (cid:0) κ E ( U ⊗ U ) , κ E ( V ⊗ V ) (cid:1) < + ∞ if and only if either E ( U ⊗ U ) or E ( V ⊗ V ) haverank at least (i.e. are not degenerate on a line).Moreover, a sufficient condition for the equality case in (1.15) is given by the followingisotropy and co-linear coupling conditions(i) U | U | = V | V | a . s .. (ii) Either E ( U ⊗ V ) = E ( U ⊗ U ) = d Id or E ( U ⊗ V ) = E ( V ⊗ V ) = d Id . Remark 1.5. (i) If E ( U · V ) ≥ (positive correlation condition), then E (cid:16) | U − V | (cid:17) ≤ − E ( U · V ) . (ii) Inequality (1.15) controls the averaged square coupling distance | U − V | with the average parallelogram area spanned by the pair ( U − U ∗ , V − V ∗ ) .(iii) The key point to obtain Theorem 1.6 below is to apply inequality (1.15) using theprobability space (Ω , P ) ≡ ([1 , N ] , h . i N ). This yields an inequality of the form (0.11).We thus obtain the main theorem on the power law trend to equilibrium with respect tothe quadratic Wassertsein distance: Theorem 1.6.
Let t V t ∈ (cid:0) R d (cid:1) N any Kac’s conservative particle system with Maxwellmolecules and normalization conditions (0.4) - (0.5) . Denote π t = def Law( V t ) . For any δ > , q > , the following trend to equilibrium holds: d + d t d W , sym ( π t , π ∞ ) ≤ − c δ,q,N ( π t ) d W , sym ( π t , π ∞ ) /δ , where in the above c δ,q,N ( π t ) = k δ,q,N E (cid:16)D | V t | q (1+ δ ) E(cid:17) − / qδ > , with k δ,q,N a numerical constant (independent of the initial condition and of the angular kernel)satisfying lim inf N → + ∞ k δ,q,N > , and explicitly bounded below (NB: k δ,q,N → polynomiallywhen q → ). The moment can be explicitly estimated, uniformly in N , in the case of order moments.18 roposition 1.7. Consider the case < δ < , q (1 + δ ) = 4 , in Theorem 1.6. We have thelower bound estimate: d W ( π t , π ∞ ) ≤ (cid:16) d W ( π , π ∞ ) − /δ + c N,δ ( t − t ∗ ) + (cid:17) − δ . where the cut-off time depends logarithmically on the initial radial order moment and isdefined by: t ∗ = 2 (cid:18) ln (cid:18) dd + 2 E D | V | E N − (cid:19)(cid:19) + . and c δ,N is a numerical constant (independent of the initial condition and of the angular kernel)satisfying lim inf N → + ∞ c N,δ > and explicitly bounded below . Remark 1.8.
Finally, note that similar results as Theorem 1.6 and Proposition 1.7, with thesame constants, can be obtained directly on the associated non-linear kinetic equation. Thesketch of proof is the following.(i) Construct a coupled non-linear equation in P ( R d × R d ) with kernel defined from unif c,θ ,and under angular cut-off (using, say, total variation distance).(ii) Prove the analogue of Theorem 1.6 using the usual Wasserstein d W in R d .(iii) Prove the continuity (in Wasserstein distance) of the solution non-coupled kinetic equa-tion with respect to the angular cut-off parameter (this is done in Section of [34], andtypically requires an appropriate uniqueness theory).(iv) Prove uniform (with respect to the angular cut-off parameter) control on higher moments.For Maxwell molecules, explicit computations of the latter can be carried out (see theclassical paper [24]).The details are left for future work. A coupled collision can then be described by expressing the post-collisional velocities ( u ′ , u ′∗ , v ′ , v ′∗ ) ∈ R d × R d using coupled collision parameters. It is sufficient in order to obtain the above cou-pling to express, using the same collision random parameters, the collision and post-collisionaldirections ( n u , n ′ u , n v , n ′ v ) ∈ (cid:0) S d − (cid:1) × (cid:0) S d − (cid:1) . This is done using parallel coupling. We statewithout proof (the reader may resort to a simple drawing here) two equivalent elementarydescriptions of the parallel coupling on the sphere. The latter parallel spherical coupling . Definition 2.1.
Let ( n u , n v ) ∈ (cid:0) S d − (cid:1) satisfying n u = − n v . There is a unique rotation of R d denoted n ′ u n ′ v = coupl n u ,n v ( n ′ u ) ∈ S d − , called spherical parallel coupling , satisfying n ′ u = n ′ v if n u = n v , and equivalently defined asfollows for n u = n v . NB: for instance, we found lim δ → lim d → + ∞ lim N → + ∞ c δ,N > − . i) n ′ v is obtained from n ′ u by performing the elementary rotation in Span( n u , n v ) bringing n u to n v .(ii) Denote by t u a tangent vector of S d − at base point n u of a geodesic of length θ bringing n u to n ′ u . Generate t v from t u by using parallel transport in Span( n u , n v ) from base point n u to base point n v . Generate n ′ v as the endpoint of the geodesic of length θ and tangentto t v at base point n v .Moreover, it satisfies by construction the symmetry condition coupl n v ,n u = coupl − n u ,n v . (2.1) If n u = − n v and σ ∈ S d − , then we will denote by coupl σn u ,n v the unique rotation of R d satisying ( i ) − ( ii ) above, but with an elementary rotation, or a geodesic taken in the plane Span( n u , σ ) = Span( n v , σ ) . It is necessary to keep in mind that the full mapping ( n u , n v ) coupl n u ,n v is smooth,except at a singularity on the extremity set (cid:8) n u , n v ∈ S d − | n u = − n v (cid:9) . This fact has alreadybeen pointed out ([33, 23, 22]) in slightly different contexts, and causes difficulty in order todefine uniquely regular Levy generators and associated kinetic non-linear equations. However,we will avoid such technical issues by considering angular cut-off, and we will consider coupledLevy or diffusive generators only at the formal level.Anyway, it is possible to define a coupled probability transition by randomly generatingthe coupling geodesic when n u = − n v . This yields: Definition 2.2.
The spherical parallel coupling of a random collision with deviation angle θ ∈ [0 , π ] is defined by the following probability transition on S d − × S d − : unif c,θ ( n u , n v , d n ′ u d n ′ v ) def = (cid:16) ll n u = − n v δ coupl nu,nv ( n ′ u ) (d n ′ v ) + ll n u = − n v δ coupl σnu,nv ( n ′ u ) (d n ′ v )unif(d σ ) (cid:17) × unif θ (cid:0) n u , d n ′ u (cid:1) , (2.2) Lemma 2.3.
The probability transition (2.2) verifies the symmetry condition unif c,θ ( n u , n v , d n ′ u d n ′ v ) = unif c,θ ( n v , n u , d n ′ v d n ′ u ) . (2.3) It is thus a symmetric Markov coupling of the uniform probability transition unif θ ( n, d n ′ ) .Proof. By construction, coupl n u ,n v and coupl n u ,σ are isometries. On the other hand, byisotropy, for any isometry R and vector n v ∈ S d − we have R − unif θ ( Rn v , . ) = unif θ ( n v , . ) .Finally, the symmetry condition (2.1) yields (2.3). We give a special description of the isotropic probability transition with scattering angle θ . Lemma 2.4.
Let θ ∈ [0 , π ] be given, as well as ( n v , m v ) two orthonormal vectors in S d − .Consider the spherical change of variable n ′ v = cos θ n v + sin θ cos ϕ m v + sin θ sin ϕ l ∈ S d − (2.4)20 here ϕ ∈ [0 , π ] is an azimuthal angle and l ∈ S d − is such that ( n v , m v , l ) is an orthonormaltriplet. Then the image by the transformation (2.4) of the probability distribution sin d − ϕ d ϕw d − Unif ( n v ,m v ) ⊥ ∩ S d − (d l ) , (2.5) is the isotropic probability transition unif θ ( n v , d n ′ v ) with initial state n v and scattering angle θ ( w d − denotes the Wallis integral normalization). In particular, the latter does not dependon the choice of m v .Proof. unif θ ( n v , d n ′ v ) is defined as the uniform distribution induced by the euclidean structureon the submanifold of S d − defined by n ′ v · n v = cos θ . Moreover the expression of volumeelements in (hyper)spherical coordinates implies that for any m v ∈ S d − , the vector cos ϕ m v +sin ϕ l ∈ S d − is distributed (under (2.5)) uniformly in the d − -dimensional sphere n ⊥ v ∩ S d − .The result follows.Of course in the above, only the scattering angle θ has an intrinsic physical meaning, theazimuthal angle ϕ being dependent of the arbitrary choice of the pair ( m v , l ) . This leads tothe core analysis of a spherical coupling. Lemma 2.5.
Let ( n u , n v ) ∈ S d − × S d − be given. A pair ( n ′ u , n ′ v ) ∈ S d − × S d − is sphericallycoupled (the spherical coupling mapping is defined in Definition 2.1), in the sense that n ′ v =coupl n u ,n v ( n ′ u ) if n u = n v and n ′ v = coupl n u ,σ ( n ′ u ) for some σ ∈ S d − otherwise, if and only if ( n ′ u = cos θ n u + sin θ cos ϕ m u + sin θ sin ϕ l,n ′ v = cos θ n v + sin θ cos ϕ m v + sin θ sin ϕ l, (2.6) where in the above ( n u , m u , l ) and ( n v , m v , l ) are both orthonormal sets of vectors such that ( n u , m u ) and ( n v , m v ) belong to the same plane have the same orientation with respect to l .Note that if n u = n v , the pair ( m u , m v ) and the angle ϕ are defined uniquely up to a commoninvolution (a change of sign of the vectors and the reflexion ϕ → π − ϕ ).Proof. Assume n u = n v . Denote by R θ the unique elementary rotation bringing n u to n v . Byconstruction R θ m u = m v , and R θ l = l This immediately implies that the coupled probability transition unif c,θ ( n u , n v , d n ′ u d n ′ v ) isthe image using the mapping (2.6) above of the uniform probability described in ( ϕ, l ) -variablesand given by (2.5). Lemma 2.6.
Let ( n u , n v ) ∈ S d − × S d − be given. If n u = − n v pick ( m u , m v ) in the plane Span( n u , n v , σ ) for some σ ∈ S d − . Then the image under the mapping (2.6) of the probabilitydistribution sin d − ϕ d ϕw d − unif ( n v ,m v ) ⊥ ∩ S d − (d l )unif(d σ ) , is given by unif c,θ ( n u , n v , d n ′ u d n ′ v ) . .3 Contractivity of spherical couplings We can first calculate the quadratic contractivity (”coupling creation”) equation satisfied byparallel spherical couplings.
Lemma 2.7.
Consider coupled collisional and post-collisional velocities ( u, u ∗ , v, v ∗ ) ∈ R d × R d , and a parallel spherical coupling using the coordinate expression of Lemma 2.5. Then wehave: (cid:12)(cid:12) u ′ − v ′ (cid:12)(cid:12) + (cid:12)(cid:12) u ′∗ − v ′∗ (cid:12)(cid:12) − | u − v | − | u ∗ − v ∗ | = − sin θ sin ϕ ( | u − u ∗ | | v − v ∗ | − ( u − u ∗ ) · ( v − v ∗ )) ≤ . (2.7) Proof.
We use the following change of variable: s v def = 12 ( v + v ∗ ) d v def = 12 ( v − v ∗ ) ⇔ ( v = s v + d v v ∗ = s v − d v . First remark that | u − v | + | u ∗ − v ∗ | = | s u − s v + d u − d v | + | s u − s v − d u + d v | = 2 | s u − s v | + 2 | d u − d v | (2.8)Developing the left hand side of (2.7), and using the conservation laws ( s ′ = s and | d ′ | = | d | ),we obtain (cid:12)(cid:12) u ′ − v ′ (cid:12)(cid:12) + (cid:12)(cid:12) u ′∗ − v ′∗ (cid:12)(cid:12) − | u − v | − | u ∗ − v ∗ | = 2 (cid:12)(cid:12) d ′ u − d ′ v (cid:12)(cid:12) − | d u − d v | = − ( u ′ − u ′∗ ) · ( v ′ − v ′∗ ) + ( u − u ∗ ) · ( v − v ∗ )= − | u − u ∗ | | v − v ∗ | (cid:0) n ′ u · n ′ v − n u · n v (cid:1) Next, we expand n ′ u .n ′ v using (2.6) and obtain: n ′ u .n ′ v = (cos θ n u + sin θ cos ϕ m u ) . (cos θ n v + sin θ cos ϕ m v ) + sin θ sin ϕ. Next by construction, ( m u , m v ) is obtained from a π -rotation of ( n u , n v ) , so that n u .n v = m u .m v and n u .m v = − m u .n v and n ′ u .n ′ v = (cid:0) cos θ + sin θ cos ϕ (cid:1) n u .n v + sin θ sin ϕ. Using θ + sin θ cos ϕ + sin θ sin ϕ we obtain n ′ u .n ′ v − n u .n v = − sin θ sin ϕ ( n u .n v − , and the result follows.We can finally compute the coupling creation functional for the coupled Kac’s particlesystem. 22 emma 2.8. Consider the coupled Kac’s particle particle system as defined by (0.6) - (0.7) .Then the coupling creation functional C is given by (0.8) .Proof. By definition C ( u, v ) = 12 D L c (cid:16) ( u, v, u ∗ , v ∗ )
7→ | u − v | + | u ∗ − v ∗ | (cid:17)E N = − Z [0 ,π ] sin θ sin ϕ β (d θ ) sin d − ϕ d ϕw d − h| u − u ∗ | | v − v ∗ | − ( u − u ∗ ) · ( v − v ∗ ) i N and the result follows from the well known formula for Wallis integrals w d − /w d − = ( d − / ( d − . If ( Z , Z ) are two centered random vectors, we will use the notation C Z ,Z def = E ( Z ⊗ Z ) . Let ( U, V ) be two centered random vectors in a Euclidean space, and ( U ∗ , V ∗ ) an i.i.d.copy. The goal is to bound from above the quadratic coupling distance E (cid:16) | U − V | (cid:17) with thefollowing alignement average E (cid:16) | U − U ∗ | | V − V ∗ | − (( U − U ∗ ) · ( V − V ∗ )) (cid:17) . The latter can be interpreted as the average parallelogram squared area spanned by the twovector differences U − U ∗ and V − V ∗ .The main computation is based on a brute force expansion and a general trace inequalityapplied to the full covariance C ( U,V ) , ( U,V ) = (cid:18) C U,U C U,V C V,U C V,V (cid:19) . The trace inequality is detailed in the following lemma.
Lemma 3.1.
Let U and V be two centered random vectors in R d . Then we have Tr ( C U,U C V,V ) − Tr ( C U,V C V,U ) ≤ min (cid:18) λ max ( C U,U )Tr ( C U,U ) , λ max ( C V,V )Tr ( C V,V ) (cid:19) (cid:16) Tr ( C U,U ) Tr ( C V,V ) − Tr ( C U,V ) (cid:17) , where λ max ( . ) denotes the spectral radius (maximal eigenvalue) of a symmetric non-negativeoperator. Moreover, the equality case holds if (sufficient condition) either C U,U and C U,V or C V,V and C U,V are co-linear to the identity matrix.Proof.
First assume that C U,U has only strictly positive eigenvalues. In an orthonormal basiswhere C U,U is diagonal, we have the expression, for i, j, k ∈ [[1 , d ]] : Tr ( C U,U C V,V − C U,V C V,U ) = X i C i,iU,U C i,iV,V − X j,k C j,kU,V C k,jV,U , = X i (cid:18) C i,iU,U C i,iV,V − (cid:16) C i,iU,V (cid:17) (cid:19) − X j = k (cid:16) C j,kU,V (cid:17) | {z } ≤ . C U,U : X i C i,iU,U C i,iV,V − (cid:16) C i,iU,V (cid:17) ≤ λ max ( C U,U ) Tr ( C V,V ) − X i (cid:16) C i,iU,V (cid:17) C i,iU,U , so that by Cauchy-Schwarz inequality X i C i,iU,V ! ≤ X i (cid:16) C i,iU,V (cid:17) C i,iU,U × X i C i,iU,U ! , and we eventually get Tr ( C U,U C V,V − C U,V C V,U ) ≤ λ max ( C U,U )Tr ( C U,U ) (cid:16) Tr ( C V,V ) Tr ( C U,U ) − Tr ( C U,V ) (cid:17) . The general case of degenerate eigenvalues is obtained by density.The expansion of the average square paralellogram area is detailed in the next lemma.
Lemma 3.2.
Let U and V be two centered random vector in R d . Let ( U ∗ , V ∗ ) be a i.i.d. copy.Then we have the following decomposition: E (cid:16) | U − U ∗ | | V − V ∗ | − (( U − U ∗ ) · ( V − V ∗ )) (cid:17) = E (cid:16) | U | | V | − ( U · V ) (cid:17)| {z } ≥ + Tr (( C U,V − C V,U ) ( C V,U − C U,V )) | {z } ≥ + 2 (cid:16) Tr ( C U,U ) Tr ( C V,V ) − Tr ( C U,V ) − Tr ( C U,U C V,V ) + Tr ( C U,V C V,U ) (cid:17)| {z } ( Lemma 3.1 ) ≥ (cid:18) − min (cid:18) λ max ( C U,U )Tr ( C U,U ) , λ max ( C V,V )Tr ( C V,V ) (cid:19)(cid:19) (cid:16) Tr ( C U,U ) Tr ( C V,V ) − Tr ( C U,V ) (cid:17) . (3.1) Proof.
Before computing terms, recall that if
M, N are two square matrices, then
Tr (
M N ) = Tr (
N M ) = Tr (cid:0) M T N T (cid:1) = Tr (cid:0) N T M T (cid:1) . Let us expand the alignement functional (the left hand side of (3.1)). We have first, E (cid:16) | U − U ∗ | | V − V ∗ | (cid:17) = 2 E (cid:16) | U | | V | (cid:17) + 2 E (cid:16) | U | (cid:17) E (cid:16) | V | (cid:17) + 4 E ( U · U ∗ V · V ∗ ) + 8 ×
0= 2 E (cid:16) | U | | V | (cid:17) + 2Tr ( C U,U ) Tr ( C V,V ) + 4Tr ( C U,V C V,U ) , and second, E (cid:16) (( U − U ∗ ) · ( V − V ∗ )) (cid:17) =2 E (cid:0) U · V (cid:1) + 2 E ( U · V ) + 2 E (cid:0) U · V ∗ (cid:1) + 2 E ( U ∗ · V U · V ∗ ) + 8 × E (cid:0) U · V (cid:1) + 2Tr ( C U,V ) + 2Tr ( C U,U C V,V ) + 2Tr (cid:0) C U,V (cid:1) .
24n the other hand,
Tr (( C U,V − C V,U ) ( C V,U − C U,V )) = − (cid:0) C U,V (cid:1) + 2Tr ( C U,V C V,U ) , and the result then follows.Two remarks. Remark 3.3.
For normalized ( E ( | U | ) = E ( | V | ) = 1 ) random vectors, Tr ( C U,U ) = Tr ( C V,V ) =1 , and Tr ( C U,U ) Tr ( C V,V ) − Tr ( C U,V ) = 1 − E ( U · V ) . Then Lemma 3.1 and 3.2 immediately yield Theorem 1.4.
Remark 3.4.
Assuming that
Tr ( C V,V ) = 1 and C U,U = 1 d Id (isotropy), then (3.1) becomes − E ( U · V ) ≤ dd − E (cid:16) | U − U ∗ | | V − V ∗ | − (( U − U ∗ ) · ( V − V ∗ )) (cid:17) (3.2)Moreover, a sufficient condition for equality in (3.2) is given by strongly istropic distributionswith co-linear coupling, defined by the fact that the lengths ( | U | , | V | ) ∈ R are independantof the identically coupled and uniformly distributed direction U | U | (= V | V | a . s . ) . Let us finally present two negative results, that demonstrates, in coupling / coupling creationinequalities, the necessity of sub-exponential estimates on the one hand, and the necessity to resort on higher moments of velocity distributions on the other hand.We give the counter-examples in the form lemmas, with proofs. In both cases, we consider acoupled distribution in the form of random variables ( U, V ) ∈ R d × R d , U ∼ N (0 , d Id) , E V = 0 , E | V | = 1 . (3.3) ( U ∗ , V ∗ ) is an i.i.d. copy. Lemma 3.5 (The necessity of higher order moments) . Let (3.3) holds. Denote the momentof order < q < : m q def = E ( | V | q ) /q , Then we have ( lim m q → E (cid:16) | U − V | (cid:17) = 2lim m q → E ( | U − U ∗ | | V − V ∗ | − ( U − U ∗ ) · ( V − V ∗ )) = 0 . Proof.
Hölder inequality implies that lim m q → E (cid:16) | V − U | (cid:17) = 2 = 0 . Moreover, we have,with again Hölder inequality, E ( | U − U ∗ | | V − V ∗ | − ( U − U ∗ ) · ( V − V ∗ )) ≤ E ( | U − U ∗ | | V − V ∗ | ) ≤ E /p ( | U − U ∗ | p ) E /q ( | V − V ∗ | q ) −−−−−−→ m q → + ∞ . emma 3.6 (The necessity of sub-exponential rates) . Let (3.3) holds. Consider the co-linearcoupling V | V | def = U | U | a . s ., with moreover the following radial coupling perturbation on some interval < r − < r + < + ∞ : | V | def = | U | ll | U | / ∈ [ r − ,r + ] + E / (cid:16) | U | | | U | ∈ [ r − , r + ] (cid:17) ll | U |∈ [ r − ,r + ] . Then (i) the moments of
Law( V ) are uniformly bounded in r − , r + ; (ii) we have the followingdegeneracy of the coupling - coupling creation estimate lim r − → + ∞ lim r + → r − E ( | U − U ∗ | | V − V ∗ | − ( U − U ∗ ) · ( V − V ∗ )) E (cid:16) | V − U | (cid:17) = 0 . Proof.
First, for such isotropic ( U is normally distributed) and co-linear couplings, the keyinequality (1.15) is in fact an equality. Denoting: A def = ( U − U ∗ ) · ( V − V ∗ ) | U − U ∗ | | V − V ∗ | , we obtain R ( r − , r + ) def = E ( | U − U ∗ | | V − V ∗ | − ( U − U ∗ ) · ( V − V ∗ )) E (cid:16) | V − U | (cid:17) ≤ d − d E ( | U − U ∗ | | V − V ∗ | (1 − A )) E (cid:16) | U − U ∗ | | V − V ∗ | (1 − A ) (cid:17) . Since | A | ≤ and A = 1 when both | U | / ∈ [ r − , r + ] and | U ∗ | / ∈ [ r − , r + ] , we have (1 − A ) ≤ (1 − A ) (cid:0) ll | U |∈ [ r − ,r + ] + ll | U ∗ |∈ [ r − ,r + ] (cid:1) a . s ., − A ) ≥ (1 − A ) (cid:0) ll | U |∈ [ r − ,r + ] + ll | U ∗ |∈ [ r − ,r + ] (cid:1) a . s ., and the smoothness of Gaussian density yields lim r + → r − R ( r − , r + ) ≤ d − d E ( | R U − U ∗ | | R U − U ∗ | ) E (cid:16) | R U − U ∗ | | R U − U ∗ | (cid:17) , where R U is distributed uniformly on the sphere with radius r − and is independant of U ∗ . Inthe limit r − → + ∞ , dominated convergence implies lim r + → r − R ( r − , r + ) = O ( r − − ) , hence theresult. In this section, we prepare the trend to equilibrium analysis. For this puprose, we will applyHölder’s inequality two times to the special inequality (1.15); a first time with respect toparticle averaging h . i N , and second time with respect to the expectation E ( . ) .26 emma 4.1. Let ( u, v ) ∈ ( R d × R d ) N satisfying the centering and normalization condi-tion (0.4) ( h u i N = h v i N = 0 and D | u | E N = D | v | E N = 1 ); as well as the positive correlationassumption h u · v i N ≥ . For any δ > , and p, q > with /p + 1 /q = 1 , we have theinequality: c δ,p ( u, v ) ≤ C ( u, v ) D | u − v | E / δN , with c δ,p ( u, v ) = k ,δ min (cid:16) κ h u ⊗ u i N , κ h v ⊗ v i N (cid:17) − − / δ D | u − u ∗ | p (1+ δ ) E − / pδN D | v − v ∗ | q (1+ δ ) E − / qδN , and constant k ,δ = 2 − − / δ d − d − δ ) /δ (1 + 2 δ ) / δ . Note that c δ,p ( u, v ) = 0 if and only if h u ⊗ u i N and h v ⊗ v i N are of (minimal) rank .Proof. We first apply (1.15) on the pair ( u, v ) with respect to the probability space gen-erated by the particle averaging operator h i N . We obtain using the positive correlationcondition h u · v i N ≥ : min (cid:16) κ h u ⊗ u ∗ i N , κ h v ⊗ v ∗ i N (cid:17) − D | u − v | E N ≤ D | u − u ∗ | | v − v ∗ | − (( u − u ∗ ) · ( v − v ∗ )) E N def = I. Next, let us denote for ε ∈ { +1 , − }A ε = | u − u ∗ | | v − v ∗ | + ε ( u − u ∗ ) · ( v − v ∗ ) , and introduce b = 1 + 2 δ , a = 1 + 1 / δ so that /a + 1 /b = 1 . Using Hölder inequality yields I = hA + A − i N = D A + A /b − A /a − E N ≤ D A b + A − E /bN hA − i /aN . The elementary (sharp) inequality (1 + θ )(1 − θ ) /b ≤ b (cid:18) b + 1 (cid:19) /b +1 ∀ θ ∈ [ − , , used for θ = ( u − u ∗ ) · ( v − v ∗ ) / | u − u ∗ | | v − v ∗ | then yields I ≤ b (cid:18) b + 1 (cid:19) /b +1 D | u − u ∗ | b | v − v ∗ | b E /bN hA − i /aN . Finally, remarking that b (cid:16) b + 1 (cid:17) /b +1 = 1 + 2 δ (1 + δ ) δ δ , and applying again Hölder inequalitywith /p + 1 /q = 1 yields I b =1+2 δ ≤ (1 + 2 δ ) δ (1 + δ ) δ D | u − u ∗ | p (1+ δ ) E /pN D | v − v ∗ | q (1+ δ ) E /pN hA − i b/a =2 δN . The result follows. 27he latter lemma implies the following corollary on random vectors with positive correla-tions.
Lemma 4.2.
Let ( U, V ) ∈ ( R d ) N × ( R d ) N satisfying the centering and normalization condition(conservation laws) (0.4) , as well as the positive correlation assumption h U · V i N ≥ . s . .Denote Law( U ) = π , Law( V ) = π . Let δ > and q > be given. We have: c δ,p ( π , π ) ≤ E C ( U, V ) E (cid:16)D | U − V | E N (cid:17) / δ , (4.1) where in the above: c δ,p ( π , π ) = k ,δ E κ p (1+2 δ ) h U ⊗ U i N *(cid:12)(cid:12)(cid:12)(cid:12) U − U ∗ √ (cid:12)(cid:12)(cid:12)(cid:12) p (1+ δ ) + N ! − / pδ E (cid:16)D | V | q (1+ δ ) E N (cid:17) − / qδ , with constant k ,δ = 2 − / − /δ d − d − δ ) /δ (1 + 2 δ ) / δ . Proof.
Hölder’s inequality with /p + 1 /q = 1 implies E (cid:0) d ( U, V ) (cid:1) ≤ E (cid:16) d ( U, V ) q C ( U, V ) − q/p (cid:17) /q E ( C ( U, V )) /p . Then the result follows by taking p = 1 + 1 / δ and q = 1 + 2 δ , together with the standardinequality for n ≥ : | v − v ∗ | n ≤ n − ( | v | n + | v ∗ | n ) . In this section, we detail an estimate of the moments of the random condition number κ ( h U ⊗ U i N ) = (1 − λ max ( h U ⊗ U i N )) − , when U is distributed uniformly on the sphere S dN − d − defined by the centering and normalization condition used in the present paper: h U ⊗ U i N = 0 , and Tr ( h U ⊗ U i N ) = 1 .We start with a well-known result from random matrix theory: Lemma 4.3.
Let U ∼ unif S dN − d − , the uniform probability distribution on the sphere S dN − d − defined by h U ⊗ U i N = 0 , and Tr ( h U ⊗ U i N ) = 1 . The order statistics ≤ L N ≤ . . . ≤ L Nd ≤ of the eigenvalues of h U ⊗ U i N are distributed according to Z ( N − , d ) d Y i =1 l ( N − − d ) / i Y ≤ i 2) Γ (( N − − i ) / . roof. Step (i). The point is to rewrite the distribution of h U ⊗ U i N as a rescaled Wishartdistribution, using a sample co-variance matrix associated to normal idependent random vari-ables. For this purpose, denote: ( G (1) , · · · , G ( N ) ) ∼ N ( d, N ) , a normal random matrix of size d × N with centered and normalized i.i.d. entries. ThenCochran’s theorem ensures that ( G (1) − h G i N , · · · , G ( N ) − h G i N ) , is a normal vector with identity co-variance in the sub-vector space defined by h g i N = 0 , sothat D | G − h G i N | E N ( G (1) − h G i N , · · · , G ( N ) − h G i N ) ∈ S dN − d − , is uniformly distributed.On the other hand, h ( G − h G i N ) ⊗ ( G − h G i N ) i (called a sample co-variance of a mul-tivariate normal distribution), and is well-known to be distributed according to a Wishartdistribution of dimension d with N − degrees of freedom, denoted W d ( N − see for in-stance [3].As a consequence, the spectrum of h U ⊗ U i N , ≤ L N ≤ . . . ≤ L Nd ≤ , is distributedaccording to ≤ M N P i M Ni ≤ . . . ≤ M Nd P i M Ni ≤ , where ≤ M N ≤ . . . ≤ M Nd ≤ is the spectrum of a Wishart distribution W d ( N − .Step (ii). The spectrum of Wishart distributions, and related quantities, can be computedexplicitly. It is a classical topic of random matrix theory (see e.g. [3, 2, 32].This leads to the following property Lemma 4.4. For any p ≥ , and L Nd as in Lemma 4.3. Then, d − d ≥ E (cid:0) (1 − L Nd ) − p (cid:1) − /p ∼ N → + ∞ d − d . Proof. By Jensen’s inequality (1 − L Nd ) − p = ( L N + · · · + L Nd ) − p ≤ d − p d Y i =2 ( L Ni ) − p/ ( d − | {z } =( L Nd ) p/ ( d − Q di =1 ( L Ni ) − p/ ( d − , so that, using the explicit formula (4.2), E (cid:0) (1 − L Nd ) − p (cid:1) ≤ Z ( N eff , d ) Z ( N, d ) E (cid:16) ( L N eff d ) p/ ( d − (cid:17) , where N eff = N − p/ ( d − is an effective sample size. Then by dominated convergence E (cid:16) ( L N eff d ) p/ ( d − (cid:17) → N → + ∞ d − p/ ( d − ; Γ (( N − d/ N eff − d/ ∼ N → + ∞ (( N − d/ ( N − N eff ) d/ , as well as Γ (( N − / − i/ N eff − / − i/ ∼ N → + ∞ (( N − / ( N − N eff ) / , so that Z ( N eff − , d ) Z ( N − , d ) → N → + ∞ d pd/ ( d − . The result follows. Step 1: Initial condition. Let π t denotes the distribution of the Kac’s particle system attime t ≥ . We fix t ≥ , and following Lemma 1.3, we take as a new initial condition (cid:16) e U t, (1) , e V t, (1) , . . . , e U t, ( N ) , e V t, ( N ) (cid:17) = (cid:0) U t, (Σ(1)) , V t, (Σ(1)) , . . . , U t, (Σ( N )) , V t, (Σ( N )) (cid:1) ∈ ( R d × R d ) N , where ( U t , V t , Σ) ∈ ( R d × R d ) N × Sym N is a representative of the d sym ,W -optimal couplingbetween π ∞ = Law( U t ) and π t = Law( V t ) . We denote e π t = Law( e V t ) and remark thatalthough π t = ˜ π t , the distributions e π t and π t have the same permutation invariant momentsor observables. Finally, Lemma 1.3 implies the positive correlation assumption D e U t · e V t E N ≥ . s . . Step 2: Propagation. Let b = R β (d θ ) < + ∞ a given angular cut-off. Consider the so-lution h ( e U t + h , e V t + h ) of the coupled Kac’s particle system with the initial condition ( e U t , e V t ) and angular cut-off b . We get: d sym ,W ( π t , π ∞ ) = E (cid:28)(cid:12)(cid:12)(cid:12) e U t − e V t (cid:12)(cid:12)(cid:12) (cid:29) N = E (cid:28)(cid:12)(cid:12)(cid:12) e U t + h − e V t + h (cid:12)(cid:12)(cid:12) (cid:29) N + Z h E (cid:16) C (cid:16) e U t + h ′ , e V t + h ′ (cid:17)(cid:17) d h ′ . Next, by definition of the Wasserstein distance: E (cid:28)(cid:12)(cid:12)(cid:12) e U t + h − e V t + h (cid:12)(cid:12)(cid:12) (cid:29) N ≥ d W ( e π t + h , π ∞ ) ; Now, since the coupling distance is almost surely decreasing (and energy is conserved), thepositive correlation condition propagtes so that D e U t + h ′ · e V t + h ′ E N ≥ . s . ∀ h ′ ≥ , and we can apply Lemma 4.2 to get E (cid:16) C (cid:16) e U t + h ′ , e V t + h ′ (cid:17)(cid:17) ≥ c δ,q,N ( e π t + h ′ , π ∞ ) (cid:18) E (cid:28)(cid:12)(cid:12)(cid:12) e U t + h ′ − e V t + h ′ (cid:12)(cid:12)(cid:12) (cid:29) N (cid:19) / δ ≥ c δ,q,N ( e π t + h ′ , π ∞ ) d W ( e π t + h , π ∞ ) /δ . b → + ∞ of vanishing cut-off in the above inequality.Indeed, it is known (see for instance [21]) that any well-defined Makov process (includingdiffusion and Levy processes) on a manifold can be approximated by a bounded jump process,in the sense of convergence of distribution on trajectory (Skorokhod) space. As a consequence,we can assume that if e π t is a solution without angular cut-off, it is possible to construct asequence e π b ,n t with angular cut-off such that e π b ,n t → e π t in distribution. Since the Wassersteindistance d W metrizes weak convergence, the state space S Nd − N − is compact, and momentsare continuous observables, it is possible to remove the angular cut-off in the above inequality.Finally, using the inequality d sym ,W ( π t + h , π ∞ ) = d sym ,W ( e π t + h , π ∞ ) ≤ d W ( e π t + h , π ∞ ) , we obtain the result: dd t d ,W ( π t , π ∞ ) ≤ − c δ,q,N ( π t , π ∞ ) d ,W ( π t , π ∞ ) . Step 3: N -uniform control of the constant. Let us introduce the notation c δ,q,N ( π t , π ∞ ) = k δ,q,N E (cid:16)D | V t | q (1+ δ ) E N (cid:17) − / qδ , with k δ,q,N = 2 − / − δ d − d − δ ) /δ (1 + 2 δ ) / δ × E κ p (1+2 δ ) h U ⊗ U i N *(cid:12)(cid:12)(cid:12)(cid:12) U − U ∗ √ (cid:12)(cid:12)(cid:12)(cid:12) p (1+ δ ) + N ! − / pδ , (4.3)where U is distributed uniformly on sphere S Nd − N − . Using the integrability property inLemma 4.4, and the well-known convergence with large dimension of the uniform distributionon spheres towards Gaussian distributions, we obtain: lim N → + ∞ E κ p (1+2 δ ) h U ⊗ U i N *(cid:12)(cid:12)(cid:12)(cid:12) U − U ∗ √ (cid:12)(cid:12)(cid:12)(cid:12) p (1+ δ ) + N ! − / pδ = (cid:18) d − d (cid:19) / δ E ( | G d | p (1+ δ ) ) − / pδ . where G d a d -dimensional centered normal distribution with E ( | G d | ) = 1 . The result follows. moment control In this section, the evolution of the radial order moment of the Kac’s particle system iscalculated. ( v, v ∗ , v ′ , v ′∗ ) ∈ ( R d ) denotes a solution of the collision mapping (1.1), and we will use again the following quantities: n v def = ( v − v ∗ ) / | v − v ∗ | , s v def = v + v ∗ , d v def = v − v ∗ . 31y Lemma (2.4), it is possible to: (i) pick ( n v , m v ) ∈ ( S d − ) , an orthonormal pair such that s v ∈ Span ( n v , m v ) ; (ii) consider spherical coordinates such that: n ′ v = cos θ n v + sin θ cos ϕ m v + sin θ sin ϕ l v ∈ S d − , where l v ∈ S d − ∩ Span ( n v , m v ) ⊥ ; (iii) and write the collision kernel in ( θ, ϕ, l v ) -coordinatesas: b ( n v , d n ′ v ) ≡ unif S d − ∩ Span( n v ,m v ) ⊥ (d l v ) sin d − ( ϕ ) d ϕw d − β (d θ ) . Lemma 4.5. Under the normalized Levy condition (0.5) , the the post-collisional order radialmoment satisfies: ∆ ( v, v ∗ ) def = 12 Z S d − (cid:12)(cid:12) v ′ (cid:12)(cid:12) + (cid:12)(cid:12) v ′∗ (cid:12)(cid:12) − | v | − | v ∗ | b ( n v , d n ′ v )= − (cid:16) | v | + | v ∗ | (cid:17) + d + 12( d − | v | | v ∗ | − d − v · v ∗ ) . (4.4) Proof. Straightforward calculation (that can be double-checked using the stationarity underthe Boltzmann kernel b of product Gaussian distributions of the form N (d v ) ⊗ N (d v ∗ ) ). Seealso the classical paper [24] for a general treatment of moments for Maxwell molecules.It then follows: Lemma 4.6. Let ( V t ) t ≥ ∈ (cid:0) R d (cid:1) N a centered and normalized conservative Kac’s particlesystem. Then for any t ≥ : E D | V t | E N ≤ e − t/ (cid:18) E D | V | E N − d + 2 d (cid:19) + d + 2 d . Moreover, denoting (convention: ln a = −∞ if a < ) : t ∗ def = 2 (cid:18) ln (cid:18) dd + 2 E D | V | E N − (cid:19)(cid:19) + then for any γ, t > : Z t (cid:16) E D | V s | E N (cid:17) − γ d s ≥ (cid:18) d + 4 d (cid:19) − γ ( t − t ∗ ) + . (4.5) Proof. The order formula (4.4) implies that dd t E D | V t | E N = h ∆ ( V t , V t, ∗ ) i N = − E D | V t | E N + d + 12( d − − d − E Tr (cid:16) h V t ⊗ V t i N (cid:17)| {z } Cauchy-Schwarz ≥ Tr(Id) = d ≤ − E D | V t | E N + d + 22 d , so that the first result follows by Gronwall’s argument.Now if t ∗ = 0 , then E D | V | E N ≤ d +4 d . Otherwise, since e − t ∗ / (cid:16) E D | V | E N − d +2 d (cid:17) = d +2 d we obtain E D | V t | E N ≤ d + 4 d . The result follows. 32 .5 Proof of Proposition 1.7 We can then apply Gronwall’s lemma to Theorem 1.6 with the estimate (4.5). Since δ (1+ p ) =4 , we take γ = 1 / pδ = 1 / / δ , and find: c δ,N = k δ, − δ ,N (cid:18) d + 4 d (cid:19) − / − / δ , where in the above k δ, − δ ,N is defined by (4.3). The result follows. Acknowledegement I thank C. Mouhot, N. Fournier, F. Bolley, and A. Guillin for their helpful comments. References [1] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré. Gradient flows: in metric spaces andin the space of probability measures . Springer, 2006.[2] Greg W Anderson, Alice Guionnet, and Ofer Zeitouni. An introduction to random ma-trices . Number 118. Cambridge University Press, 2010.[3] Theodore Wilbur Anderson. An introduction to multivariate statistical analysis , volume 2.Wiley New York, 1958.[4] Dominique Bakry, Ivan Gentil, Michel Ledoux, et al. Analysis and geometry of Markovdiffusion operators . Springer, 2014.[5] AlexanderV. Bobylev and Carlo Cercignani. On the rate of entropy production for theboltzmann equation. Journal of Statistical Physics , 94(3-4):603–618, 1999.[6] AV Bobylev. The theory of the nonlinear boltzmann equation for maxwell molecules. Mathematical physics reviews , 7:111, 1988.[7] François Bolley, Ivan Gentil, and Arnaud Guillin. Convergence to equilibrium in wasser-stein distance for fokker–planck equations. Journal of Functional Analysis , 2012.[8] François Bolley, Ivan Gentil, and Arnaud Guillin. Uniform convergence to equilibriumfor granular media. Archive for Rational Mechanics and Analysis , pages 1–17, 2012.[9] EA Carlen and MC Carvalho. Strict entropy production bounds and stability of the rateof convergence to equilibrium for the boltzmann equation. Journal of statistical physics ,67(3-4):575–608, 1992.[10] Eric A Carlen, Maria C Carvalho, and Michael Loss. Determination of the spectral gapfor kacś master equation and related stochastic evolution. Acta mathematica , 191(1):1–54,2003.[11] Eric A Carlen, Jeffrey S Geronimo, and Michael Loss. Determination of the spectralgap in the kac’s model for physical momentum and energy-conserving collisions. SIAMJournal on Mathematical Analysis , 40(1):327–364, 2008.3312] Eric A Carlen and Xuguang Lu. Fast and slow convergence to equilibrium for maxwellianmolecules via wild sums. Journal of statistical physics , 112(1-2):59–134, 2003.[13] C Cercignani. H-theorem and trend to equilibrium in the kinetic theory of gases. Archivof Mechanics, Archiwum Mechaniki Stosowanej , 34:231–241, 1982.[14] Carlo Cercignani. Mathematical methods in kinetic theory . Plenum Press New York, 1969.[15] Laurent Desvillettes, Clément Mouhot, Cédric Villani, et al. Celebrating cercignani’sconjecture for the boltzmann equation. Kinetic and related models , 4(1):277–294, 2011.[16] Laurent Desvillettes and Cédric Villani. On the spatially homogeneous landau equationfor hard potentials part ii: h-theorem and applications: H-theorem and applications. Communications in Partial Differential Equations , 25(1-2):261–298, 2000.[17] Persi Diaconis and Laurent Saloff-Coste. Bounds for kacś master equation. Communica-tions in Mathematical Physics , 209(3):729–755, 2000.[18] Emanuele Dolera, Ester Gabetta, and Eugenio Regazzini. Reaching the best possible rateof convergence to equilibrium for solutions of kac equation via central limit theorem. TheAnnals of Applied Probability , 19(1):186–209, 2009.[19] Emanuele Dolera and Eugenio Regazzini. The role of the central limit theorem in discov-ering sharp rates of convergence to equilibrium for the solution of the kac equation. TheAnnals of Applied Probability , 20(2):430–461, 2010.[20] Emanuele Dolera and Eugenio Regazzini. Proof of a mckean conjecture on the rateof convergence of boltzmann-equation solutions. Probability Theory and Related Fields ,pages 1–75, 2012.[21] S. N. Ethier and T. G. Kurtz. Markov Processes. Characterization and Convergence .Wiley Series in Probability and Mathematical Statistics, 1985.[22] Nicolas Fournier and Stéphane Mischler. Rate of convergence of the nanbu particle systemfor hard potentials. arXiv preprint arXiv:1302.5810 , 2013.[23] Nicolas Fournier and Clément Mouhot. On the well-posedness of the spatially homo-geneous boltzmann equation with a moderate angular singularity. Communications inMathematical Physics , 289(3):803–824, 2009.[24] E Ikenberry and C Truesdell. On the pressures and flux of energy in a gas according to { M } axwell \ ’s kinetic theory. J. Rat. Mech. Anal. , 5, 1956.[25] David Asher Levin, Yuval Peres, and Elizabeth Lee Wilmer. Markov chains and mixingtimes . AMS Bookstore, 2009.[26] Jan Maas. Gradient flows of the entropy for finite markov chains. Journal of FunctionalAnalysis , 261(8):2250–2292, 2011.[27] Florient Malrieu. Logarithmic sobolev inequalities for some nonlinear pde’s. Stochasticprocesses and their applications , 95(1):109–132, 2001.3428] Stéphane Mischler and Clément Mouhot. About kac’s program in kinetic theory. ComptesRendus Mathematique , 349(23):1245–1250, 2011.[29] Clément Mouhot. Rate of convergence to equilibrium for the spatially homogeneousboltzmann equation with hard potentials. Communications in mathematical physics ,261(3):629–672, 2006.[30] Roberto Imbuzeiro Oliveira. On the convergence to equilibrium of kacś random walk onmatrices. The Annals of Applied Probability , 19(3):1200–1231, 2009.[31] Yann Ollivier. Ricci curvature of markov chains on metric spaces. Journal of FunctionalAnalysis , 256(3):810–864, 2009.[32] FJ Schuurmann, PR Krishnaiah, and AK Chattopadhyay. On the distributions of theratios of the extreme roots to the trace of the wishart matrix. Journal of MultivariateAnalysis , 3(4):445–453, 1973.[33] Hiroshi Tanaka. Probabilistic treatment of the boltzmann equation of maxwellianmolecules. Probability Theory and Related Fields , 46(1):67–105, 1978.[34] G Toscani and C Villani. Probability metrics and uniqueness of the solution to theboltzmann equation for a maxwell gas. Journal of statistical physics , 94(3-4):619–637,1999.[35] G Toscani and C Villani. Sharp entropy dissipation bounds and explicit rate of trendto equilibrium for the spatially homogeneous boltzmann equation. Communications inmathematical physics , 203(3):667–706, 1999.[36] Cédric Villani. A review of mathematical topics in collisional kinetic theory. Handbookof mathematical fluid dynamics , 1:71–74, 2002.[37] Cédric Villani. Cercignani’s conjecture is sometimes true and always almost true. Com-munications in mathematical physics , 234(3):455–490, 2003.[38] Cédric Villani. Optimal transport: old and new , volume 338. Springer, 2008.[39] Max-K von Renesse and Karl-Theodor Sturm. Transport inequalities, gradient esti-mates, entropy and ricci curvature. Communications on pure and applied mathematics ,58(7):923–940, 2005.[40] Feng-Yu Wang. On estimation of the logarithmic sobolev constant and gradient estimatesof heat semigroups.