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... ) is f -Kac’s chaotic. The most unexpected and interesting implication is maybe (b)⇒ (c). It is a simple consequence of the HWI inequality H(f2|ρ) ≤ H(f1|ρ) + √ I(f0|ρ)W2(f0, f1) of Otto and Villani =-=[35]-=-, which is itself a kind of (dimensionless) variant of the wellkonwn (and mere consequence of the Cauchy-Schwarz inequality) interpolation inequality ‖g‖L2 ≤ ‖g‖1/2H1 ‖g‖1/2H−1 . Indeed, using twice t...

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...pic description of the system, providing a possible answer to Problem 3 by Kac. Our approach developed in [34, 33] gives an alternative method to the classical coupling method initiate by A. Sznitman =-=[38]-=-. For that later, we refer to the work [4] and the reference therein for recent development of coupling method for the McKean-Vlasov model, as well as to the work in collaboration with N. Fournier [17...

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...he quantitative and qualitative propagation of chaos for the Boltzmann-Kac system obtained in collaboration with C. Mouhot in [33] which gives a possible answer to some questions formulated by Kac in =-=[25]-=-. We also present some related recent results about Kac’s chaos and Kac’s program obtained in [34, 23, 13] by K. Carrapatoso, M. Hauray, C. Mouhot, B. Wennberg and myself. Manuscript version of a talk...

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...os estimate 9 5. Kac’s chaos and related problems 11 6. Conclusion and open problems 14 References 16 1. Introduction 1.1. 6th Hilbert Problem. The Boltzmann equation was introduced by Maxwell (1867, =-=[30]-=-) and Boltzmann (1872, [5]) in order to describe the evolution of a rarefied gas in which particles uniquely interact through binary collisions. That equation governs the time evolution of the statist...

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...h explain how to get the Boltzmann equation from such a microscopic description was identified by Grad (1958, [19]) (thanks to the BBGKY method) and mathematically rigorously proved by Lanford (1975, =-=[27]-=-) on a very small time interval (smaller that the necessary waiting time before half of all the particles collide once). Of course the “BoltzmannGrad limit” is very interesting and very difficult to j...

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...s and related problems 11 6. Conclusion and open problems 14 References 16 1. Introduction 1.1. 6th Hilbert Problem. The Boltzmann equation was introduced by Maxwell (1867, [30]) and Boltzmann (1872, =-=[5]-=-) in order to describe the evolution of a rarefied gas in which particles uniquely interact through binary collisions. That equation governs the time evolution of the statistical distribution f(t, x, ...

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... KAC’S CHAOS AND KAC’S PROGRAM 7 (b) The entropy fits better for a N →∞ asymptotic. However in that case, the “spectral” gap ∆′N := inf{ − 〈(log dG/dγN)/N,ΛNG〉/H(G|γN )}, satisfies ∆′N ≥ 1/N (Villani =-=[43]-=-) and lim sup∆′N = 0 (Carlen et al [9]), and that cannot answer Problem 2 either. (c) On the other hand, the exponential rate of convergence to the equilibrium of the solutions to the nonlinear Boltzm...

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...in the sense that (A5) W1(ft, gt) ≤ Θ(W1(f0, g0)) ∀ f0, g0 ∈ Pexp(E) for Θ(u) = u (M model) and Θ(u) = | log |u| ∧ 1|−1 (HS model). Such an estimate has been proved by Tanaka [39] and Toscani-Villani =-=[41]-=- in the case M and it is mainly a consequence of Fournier-Mouhot [18] in the case of HS. As a consequence, and similarly as for the empirical measures method, we find |T3| = ∣∣〈GN0 , Rϕ(SNLt µNV )〉− 〈...

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...t is Θ-Holder continuous in the sense that (A5) W1(ft, gt) ≤ Θ(W1(f0, g0)) ∀ f0, g0 ∈ Pexp(E) for Θ(u) = u (M model) and Θ(u) = | log |u| ∧ 1|−1 (HS model). Such an estimate has been proved by Tanaka =-=[39]-=- and Toscani-Villani [41] in the case M and it is mainly a consequence of Fournier-Mouhot [18] in the case of HS. As a consequence, and similarly as for the empirical measures method, we find |T3| = ∣...

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...mics of molecules governed by the Newton’s law of motions? The “Boltzmann-Grad limit” which explain how to get the Boltzmann equation from such a microscopic description was identified by Grad (1958, =-=[19]-=-) (thanks to the BBGKY method) and mathematically rigorously proved by Lanford (1975, [27]) on a very small time interval (smaller that the necessary waiting time before half of all the particles coll...

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...tem in large increasing dimension. This has motivated beautiful works on the “Kac spectral gap problem”, i.e. the study of this relaxation rate in a L2 setting, for the Kac’s N -particle system first =-=[25, 24, 29, 9, 7]-=- and next for the Boltzmann-Kac system [9, 11]. Theorem 2.3. For both M and HS models, there exists δ > 0 such that for any N ≥ 1 ∆N := inf{ − 〈h,ΛNh〉L2 , 〈h, 1〉L2 = 0, ‖h‖2L2 = 1} ≥ δ, where 〈·, ·〉L2...

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...nction associated to the initial datum f0. There is a so huge number of works on that topics that we cannot quote all of them. Let us just say that the story began with the work by T. Carleman (1933, =-=[6]-=-) and we refer to [28] and the references therein for the HS model and to [10] and the references therein for the M model. 3. A reverse answer to Kac’s program 3.1. Our contributions to Kac’s program....

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... on that topics that we cannot quote all of them. Let us just say that the story began with the work by T. Carleman (1933, [6]) and we refer to [28] and the references therein for the HS model and to =-=[10]-=- and the references therein for the M model. 3. A reverse answer to Kac’s program 3.1. Our contributions to Kac’s program. We give some possible answers to the three problems formulated by Kac. • In c...

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...ption (of the physical system). For theMG model, the propagation of chaos (without rate and next with rate) has been proved by Kac [25], McKean [32, 31], Grunbaum [21], Tanaka [40], Graham, Méléard =-=[20]-=- using (except in [21]) some tree arguments (Wild sum, stochastic tree). These kind of arguments are very specific to the MG model. For the HS model, the propagation of chaos result (without rate) has...

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...e arguments (Wild sum, stochastic tree). These kind of arguments are very specific to the MG model. For the HS model, the propagation of chaos result (without rate) has been proved by Sznitman (1984, =-=[37]-=-) using a nonlinear Martingale approach, some compactness of the system and uniqueness of the limit arguments. An alternative proof is suggested in Arkeryd et al (1991, [1]) following a “BBGKY hierarc...

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...rd spheres models (without quantitative estimate) as well as the many works (Kac [25], McKean [32], Grunbaum [21], Tanaka [40], Graham and Méléard [20], Fournier and Méléard [15, 16], Kolokoltsov =-=[26]-=-, Peyre [36]) which deal with the Maxwell molecules model with Grad cutoff. ⊲ Our estimate is uniform in time and that make possible to answer Problem 2 by Kac on the convergence to equilibrium with u...

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...omogeneous) Boltzmann equation from a microscopic description (of the physical system). For theMG model, the propagation of chaos (without rate and next with rate) has been proved by Kac [25], McKean =-=[32, 31]-=-, Grunbaum [21], Tanaka [40], Graham, Méléard [20] using (except in [21]) some tree arguments (Wild sum, stochastic tree). These kind of arguments are very specific to the MG model. For the HS model...

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...possible answer to Problem 3 by Kac. Our approach developed in [34, 33] gives an alternative method to the classical coupling method initiate by A. Sznitman [38]. For that later, we refer to the work =-=[4]-=- and the reference therein for recent development of coupling method for the McKean-Vlasov model, as well as to the work in collaboration with N. Fournier [17] and the references therein for the appli...

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...utiful works on the “Kac spectral gap problem”, i.e. the study of this relaxation rate in a L2 setting, for the Kac’s N -particle system first [25, 24, 29, 9, 7] and next for the Boltzmann-Kac system =-=[9, 11]-=-. Theorem 2.3. For both M and HS models, there exists δ > 0 such that for any N ≥ 1 ∆N := inf{ − 〈h,ΛNh〉L2 , 〈h, 1〉L2 = 0, ‖h‖2L2 = 1} ≥ δ, where 〈·, ·〉L2 and ‖ · ‖L2 stand for the scalar product and ...

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...for Θ(u) = u (M model) and Θ(u) = | log |u| ∧ 1|−1 (HS model). Such an estimate has been proved by Tanaka [39] and Toscani-Villani [41] in the case M and it is mainly a consequence of Fournier-Mouhot =-=[18]-=- in the case of HS. As a consequence, and similarly as for the empirical measures method, we find |T3| = ∣∣〈GN0 , Rϕ(SNLt µNV )〉− 〈(SNLt f0)⊗k, ϕ〉∣∣ = ∣∣〈GN0 , Rϕ(SNLt µNV )−Rϕ(SNLt f0)〉∣∣ ≤ [Rϕ]C0,1 ...

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...tem in large increasing dimension. This has motivated beautiful works on the “Kac spectral gap problem”, i.e. the study of this relaxation rate in a L2 setting, for the Kac’s N -particle system first =-=[25, 24, 29, 9, 7]-=- and next for the Boltzmann-Kac system [9, 11]. Theorem 2.3. For both M and HS models, there exists δ > 0 such that for any N ≥ 1 ∆N := inf{ − 〈h,ΛNh〉L2 , 〈h, 1〉L2 = 0, ‖h‖2L2 = 1} ≥ δ, where 〈·, ·〉L2...

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...build a sequence of initial data GN0 which satisfies both properties suppFN0 ⊂ KSSN (orBSSN) and FN0 is f0-chaotic. A first answer to that issue is the following. Theorem 2.2 (Kac [25]; Carlen et al. =-=[8]-=-). Consider f0 ∈ L14(E) ∩ Lp(E), p > 1, E = R. There exists FN0 ∈ P(EN ) such that (a) suppFN0 ⊂ KSSN ; (b) FN0 is f0-chaotic; (c) FN0 is f0-entropy chaotic. Here, we say that a sequence (GN ) of Psym...

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...nn equation from a microscopic description (of the physical system). For theMG model, the propagation of chaos (without rate and next with rate) has been proved by Kac [25], McKean [32, 31], Grunbaum =-=[21]-=-, Tanaka [40], Graham, Méléard [20] using (except in [21]) some tree arguments (Wild sum, stochastic tree). These kind of arguments are very specific to the MG model. For the HS model, the propagati...

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...GRAM S. MISCHLER Abstract. In this note I present the main results about the quantitative and qualitative propagation of chaos for the Boltzmann-Kac system obtained in collaboration with C. Mouhot in =-=[33]-=- which gives a possible answer to some questions formulated by Kac in [25]. We also present some related recent results about Kac’s chaos and Kac’s program obtained in [34, 23, 13] by K. Carrapatoso, ...

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...tand how to get an irreversible equation (the Boltzmann equation) from a reversible equation (the Newton’s law of motions), and very few results are known on that major problem up to now. We refer to =-=[2]-=- and the references therein for updated results on that direction. 1.2. Kac’s approach. In order to circumvent the above difficulties, M. Kac (1956, [25]) suggested to derive the space homogeneous Bol...

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...on of chaos for the hard spheres models (without quantitative estimate) as well as the many works (Kac [25], McKean [32], Grunbaum [21], Tanaka [40], Graham and Méléard [20], Fournier and Méléard =-=[15, 16]-=-, Kolokoltsov [26], Peyre [36]) which deal with the Maxwell molecules model with Grad cutoff. ⊲ Our estimate is uniform in time and that make possible to answer Problem 2 by Kac on the convergence to ...

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...aboration with C. Mouhot in [33] which gives a possible answer to some questions formulated by Kac in [25]. We also present some related recent results about Kac’s chaos and Kac’s program obtained in =-=[34, 23, 13]-=- by K. Carrapatoso, M. Hauray, C. Mouhot, B. Wennberg and myself. Manuscript version of a talk given in séminaire Laurent Schwartz, 2012-2013 Keywords: Kac’s program; Kac’s chaos; kinetic theory; mas...

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...tochastic independence (Kac’s chaos) and thus to clarify the notion of molecular chaos on which Boltzmann’s work is based. 1.3. N-particle system. The approach by Kac was generalized by McKean (1967, =-=[32]-=-), and then many other people. Generally and roughly speaking, the approach is as follows. Consider a system of N indistinguishable particles, each particle being identified by its state (position, ve...

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...38]. For that later, we refer to the work [4] and the reference therein for recent development of coupling method for the McKean-Vlasov model, as well as to the work in collaboration with N. Fournier =-=[17]-=- and the references therein for the application of the coupling method to a Kac-Boltzmann related (but simpler) collisional system. It is worth mentioning that 8 S. MISCHLER the estimates obtained in ...

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...), and a similar result replacing the “Kac’s spheres” by the “Boltzmann’s spheres”. Let us make some comments: ⊲ In the case (i) and f has compact support, the estimate (5.4) with α = αc is true, see =-=[3]-=-. ⊲ Estimate (5.5) is nothing but an accurate variant to the (sometimes called) “Poincaré’s Lemma” which is attributed to Mehler 1866 in [8], and has also been considered by many other authors, among...

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...rom a microscopic description (of the physical system). For theMG model, the propagation of chaos (without rate and next with rate) has been proved by Kac [25], McKean [32, 31], Grunbaum [21], Tanaka =-=[40]-=-, Graham, Méléard [20] using (except in [21]) some tree arguments (Wild sum, stochastic tree). These kind of arguments are very specific to the MG model. For the HS model, the propagation of chaos r...

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... << t. We then optimize that last estimate together with (2.7) which is a good estimate for N >> t and we conclude to (3.2) and (3.3). Kac’s Problem 3. For the M model, one can adapt Villani’s result =-=[42]-=- for the Boltzmann equation to the Kac-Boltzmann equation and obtain ([22, 33]) the uniform estimate on the Fisher information (6.6) sup t≥0 I(GNt |γN) ≤ I(GN0 |γN). We then conclude thanks to Theorem...

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...S PROGRAM 9 4. Uniformly in time chaos estimate 4.1. Weak uniform in time quantitative chaos propagation. We give the cornerstone estimate of the quantitative propagation of chaos method developed in =-=[34, 33]-=- for which we present next a sketch of the proof. Theorem 4.1 (HS & M, [33]). Under the assumptions and notations of Theorems 3.1 & 3.2, there holds for any ε > 0 and for some suitable modulus of cont...

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... proved by Sznitman (1984, [37]) using a nonlinear Martingale approach, some compactness of the system and uniqueness of the limit arguments. An alternative proof is suggested in Arkeryd et al (1991, =-=[1]-=-) following a “BBGKY hierarchy” approach. For the HS model again, in order to be able to apply the first part of Theorem 2.1, we have to build a sequence of initial data GN0 which satisfies both prope...

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...ntial (with rate) or even for the true soft potential (without rate to begin with). A first generalization of our method was obtained for the Landau equation (for Maxwell molecules) by Carrapatoso in =-=[12]-=-. (3) Consider singular models and generalize the propagation of chaos for the vortex model obtained in [14]. 16 S. MISCHLER (4) Generalize the coupling technics used for the asymmetric variant of the...

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...aboration with C. Mouhot in [33] which gives a possible answer to some questions formulated by Kac in [25]. We also present some related recent results about Kac’s chaos and Kac’s program obtained in =-=[34, 23, 13]-=- by K. Carrapatoso, M. Hauray, C. Mouhot, B. Wennberg and myself. Manuscript version of a talk given in séminaire Laurent Schwartz, 2012-2013 Keywords: Kac’s program; Kac’s chaos; kinetic theory; mas...

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...on of chaos for the hard spheres models (without quantitative estimate) as well as the many works (Kac [25], McKean [32], Grunbaum [21], Tanaka [40], Graham and Méléard [20], Fournier and Méléard =-=[15, 16]-=-, Kolokoltsov [26], Peyre [36]) which deal with the Maxwell molecules model with Grad cutoff. ⊲ Our estimate is uniform in time and that make possible to answer Problem 2 by Kac on the convergence to ...

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...odels (without quantitative estimate) as well as the many works (Kac [25], McKean [32], Grunbaum [21], Tanaka [40], Graham and Méléard [20], Fournier and Méléard [15, 16], Kolokoltsov [26], Peyre =-=[36]-=-) which deal with the Maxwell molecules model with Grad cutoff. ⊲ Our estimate is uniform in time and that make possible to answer Problem 2 by Kac on the convergence to equilibrium with uniform rate ...

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...tem in large increasing dimension. This has motivated beautiful works on the “Kac spectral gap problem”, i.e. the study of this relaxation rate in a L2 setting, for the Kac’s N -particle system first =-=[25, 24, 29, 9, 7]-=- and next for the Boltzmann-Kac system [9, 11]. Theorem 2.3. For both M and HS models, there exists δ > 0 such that for any N ≥ 1 ∆N := inf{ − 〈h,ΛNh〉L2 , 〈h, 1〉L2 = 0, ‖h‖2L2 = 1} ≥ δ, where 〈·, ·〉L2...

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...the initial datum f0. There is a so huge number of works on that topics that we cannot quote all of them. Let us just say that the story began with the work by T. Carleman (1933, [6]) and we refer to =-=[28]-=- and the references therein for the HS model and to [10] and the references therein for the M model. 3. A reverse answer to Kac’s program 3.1. Our contributions to Kac’s program. We give some possible...

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...good estimate for N >> t and we conclude to (3.2) and (3.3). Kac’s Problem 3. For the M model, one can adapt Villani’s result [42] for the Boltzmann equation to the Kac-Boltzmann equation and obtain (=-=[22, 33]-=-) the uniform estimate on the Fisher information (6.6) sup t≥0 I(GNt |γN) ≤ I(GN0 |γN). We then conclude thanks to Theorem 5.5. For the HS model, the proof is somewhat simpler, and in the same time le...