Comptes Rendus
Partial Differential Equations/Probability Theory
About Kacʼs program in kinetic theory
Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1245-1250.

In this Note we present the main results from the recent work of Mischler and Mouhot (2011) [15], which answers several conjectures raised fifty years ago by Kac (1956) [10]. There Kac introduced a many-particle stochastic process (now denoted as Kacʼs master equation) which, for chaotic data, converges to the spatially homogeneous Boltzmann equation. We answer the three following questions raised in Kac (1956) [10]: (1) prove the propagation of chaos for realistic microscopic interactions (i.e. in our results: hard spheres and true Maxwell molecules); (2) relate the time scales of relaxation of the stochastic process and of the limit equation by obtaining rates independent of the number of particles; (3) prove the convergence of the many-particle entropy towards the Boltzmann entropy of the solution to the limit equation (microscopic justification of the H-theorem of Boltzmann in this context). These results crucially rely on a new theory of quantitative uniform in time estimates of propagation of chaos.

Dans cette Note, nous présentons les résultats principaux du travail récent Mischler and Mouhot (2011) [15], qui répond à plusieurs conjectures proposées il y a une cinquantaine dʼannées par Kac (1956) [10]. Dans ce travail Kac introduit un processus stochastique à grand nombre de particules (aujourdʼhui appelé équation maîtresse de Kac) qui converge, pour des données chaotiques, vers lʼéquation de Boltzmann spatialement homogène. Nous répondons aux trois questions suivantes soulevées dans cet article : (1) prouver la propagation du chaos pour des processus de collision réalistes (dans notre cas : sphères dures et « vraies » molécules maxwelliennes), (2) connecter les vitesses de relaxation du processus stochastique et de lʼéquation limite en obtenant des taux indépendants du nombre de particules, (3) prouver la convergence de lʼentropie en grand nombre de particules vers lʼentropie de Boltzmann pour la solution de lʼéquation limite (justification microscopique du théorème H dans ce contexte). Tous ces résultats font appel de manière cruciale à une nouvelle théorie dʼestimations quantitatives et uniformes en temps de propagation du chaos.

Published online:
DOI: 10.1016/j.crma.2011.11.012

Stéphane Mischler 1; Clément Mouhot 2

1 Ceremade (UMR CNRS no. 7534), Université Paris-Dauphine, Place de-Lattre-de-Tassigny, 75775 Paris cedex 16, France
2 DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB2 0WA, UK
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Stéphane Mischler; Clément Mouhot. About Kacʼs program in kinetic theory. Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1245-1250. doi : 10.1016/j.crma.2011.11.012.

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