### Citations

241 | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality
- Otto, Villani
(Show Context)
Citation Context ... ) 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... |

177 |
Topics in propagation of chaos, École d’Été de Probabilités de Saint-Flour
- Sznitman
- 1991
(Show Context)
Citation Context ...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... |

120 |
Foundations of kinetic theory
- Kac
- 1956
(Show Context)
Citation Context ...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... |

116 |
On the dynamical theory of gases
- Maxwell
- 1867
(Show Context)
Citation Context ...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... |

111 |
Time evolution of large classical systems
- Lanford
- 1975
(Show Context)
Citation Context ...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... |

90 |
Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Sitzungsberichte der keiserlichen Akademie der Wissenschaften 1872, 66, 275–370. Translation: Further studies on the thermal equilibrium of gas molecules
- Boltzmann
- 2003
(Show Context)
Citation Context ...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, ... |

59 |
Cercignani’s conjecture is sometimes true and always almost true
- Villani
- 2003
(Show Context)
Citation Context ... 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... |

55 |
Probability metrics and uniqueness of the solution of the Boltzmann equation for a Maxwell gas
- Toscani, Villani
- 1999
(Show Context)
Citation Context ...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 )〉− 〈... |

51 |
Probabilistic treatment of the Boltzmann equation of Maxwellian molecules
- Tanaka
- 1978
(Show Context)
Citation Context ...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| = ∣... |

50 |
Principles of the Kinetic Theory of Gases,” Handbuch der Physik, edited by
- Grad
- 1958
(Show Context)
Citation Context ...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... |

48 | Determination of the spectral gap for Kac’s master equation and related stochastic evolution
- Carlen, Carvalho, et al.
- 2003
(Show Context)
Citation Context ...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... |

45 |
Sur la théorie de l’equation intégrodifferentielle de Boltzmann
- Carleman
- 1932
(Show Context)
Citation Context ...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.... |

43 |
Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas
- Carlen, Gabetta, et al.
- 1999
(Show Context)
Citation Context ... 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... |

33 |
S.: Stochastic particle approximations for generalized Boltzmann models and convergence
- Graham, M'el'eard
- 1997
(Show Context)
Citation Context ...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... |

29 | Spectral gap for Kac’s model of Boltzmann equation - Janvresse |

29 |
Équations de type de Boltzmann, spatialement homogènes
- Sznitman
- 1984
(Show Context)
Citation Context ...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... |

25 | Nonlinear Markov processes and kinetic equations
- Kolokoltsov
- 2010
(Show Context)
Citation Context ...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... |

25 |
The central limit theorem for Carleman’s equation
- McKean
- 1975
(Show Context)
Citation Context ...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... |

24 | Florent Malrieu. Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation
- Bolley, Guillin
(Show Context)
Citation Context ...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... |

20 | Determination of the spectral gap in the Kac model for physical momentum and energy-conserving collisions
- Carlen, Geronimo, et al.
- 2008
(Show Context)
Citation Context ...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 ... |

18 | On the well-posedness of the spatially homogeneous boltzmann equation with a moderate angular singularity
- Fournier, Mouhot
(Show Context)
Citation Context ...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 ... |

17 |
The eigenvalues of Kac’s master equation
- Maslen
(Show Context)
Citation Context ...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... |

15 | Entropy and chaos in the Kac model
- Carlen, Carvalho, et al.
(Show Context)
Citation Context ...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... |

14 |
Propagation of chaos for the Boltzmann equation
- Grünbaum
- 1971
(Show Context)
Citation Context ...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... |

14 | Kac’s program in kinetic theory
- Mischler, Mouhot
- 2013
(Show Context)
Citation Context ...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, ... |

11 | The Brownian motion as the limit of a deterministic system of hard-spheres. arXiv:1305.3397 [math.AP
- Bodineau, Gallagher, et al.
- 2013
(Show Context)
Citation Context ...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... |

11 | Propagation of chaos for the 2d viscous vortex model - Fournier, Hauray, et al. - 2012 |

10 | S.: A stochastic particle numerical method for 3D Boltzmann equations without cutoff
- Fournier, M'el'eard
- 2001
(Show Context)
Citation Context ...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 ... |

10 |
On Kac’s chaos and related problems
- Hauray, Mischler
(Show Context)
Citation Context ...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... |

10 |
An exponential formula for solving Boltmann’s equation for a Maxwellian gas
- McKean
- 1967
(Show Context)
Citation Context ...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... |

9 | Rate of convergence of the Nanbu particle system for hard potentials
- Fournier, Mischler
- 2013
(Show Context)
Citation Context ...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 ... |

8 | On the Mean Speed of Convergence of Empirical and Occupation Measures in Wasserstein Distance. Available at arXiv:1105.5263v1
- Boissard, Gouic
- 2011
(Show Context)
Citation Context ...), 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... |

8 |
Some probabilistic problems in the spatially homogeneous Boltzmann equation. In Theory and application of random fields
- Tanaka
- 1982
(Show Context)
Citation Context ...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... |

8 | Fisher information estimates for Boltzmann’s collision operator
- Villani
- 1998
(Show Context)
Citation Context ... << 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... |

7 |
A new approach to quantitative chaos propagation for drift, diffusion and jump processes
- Mischler, Mouhot, et al.
- 2011
(Show Context)
Citation Context ...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... |

6 |
The homogeneous Boltzmann hierarchy and statistical solutions to the homogeneous Boltzmann equation
- Arkeryd, Caprino, et al.
- 1991
(Show Context)
Citation Context ... 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... |

6 | Propagation of chaos for the spatially homogenous Landau equation for Maxwellian molecules
- Carrapatoso
- 2014
(Show Context)
Citation Context ...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... |

4 | Quantitative and qualitative Kac’s chaos on the Boltzmann’s sphere
- Carrapatoso
(Show Context)
Citation Context ...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... |

4 |
Monte Carlo approximations and fluctuations for 2d Boltzmann equations without cutoff. Markov Process. Related Fields 7
- Fournier, Méléard
- 2001
(Show Context)
Citation Context ...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 ... |

4 | Some ideas about quantitative convergence of collision models to their mean field limit
- Peyre
(Show Context)
Citation Context ...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 ... |

2 |
Spectral gap for the Kac model with hard collisions. arXiv:1304.5124
- Carlen, Carvalho, et al.
(Show Context)
Citation Context ...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... |

2 | On measure solutions of the Boltzmann equation, Part II: Rate of convergence to equilibrium. arXiv:1306.0764
- Lu, Mouhot
(Show Context)
Citation Context ...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... |

1 |
Fisher information decay for the boltzman-kac system associtaed to maxwell molecules. Personnal communication
- Hauray
(Show Context)
Citation Context ...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... |