[À propos du programme de Kac en théorie cinétique]
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.
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.
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@article{CRMATH_2011__349_23-24_1245_0,
author = {Mischler, St\'ephane and Mouhot, Cl\'ement},
title = {About {Kac's} program in kinetic theory},
journal = {Comptes Rendus. Math\'ematique},
pages = {1245--1250},
publisher = {Elsevier},
volume = {349},
number = {23-24},
year = {2011},
doi = {10.1016/j.crma.2011.11.012},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.11.012/}
}
TY - JOUR AU - Mischler, Stéphane AU - Mouhot, Clément TI - About Kacʼs program in kinetic theory JO - Comptes Rendus. Mathématique PY - 2011 SP - 1245 EP - 1250 VL - 349 IS - 23-24 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.11.012/ DO - 10.1016/j.crma.2011.11.012 LA - en ID - CRMATH_2011__349_23-24_1245_0 ER -
%0 Journal Article %A Mischler, Stéphane %A Mouhot, Clément %T About Kacʼs program in kinetic theory %J Comptes Rendus. Mathématique %D 2011 %P 1245-1250 %V 349 %N 23-24 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.11.012/ %R 10.1016/j.crma.2011.11.012 %G en %F CRMATH_2011__349_23-24_1245_0
Mischler, Stéphane; Mouhot, Clément. About Kacʼs program in kinetic theory. Comptes Rendus. Mathématique, Tome 349 (2011) no. 23-24, pp. 1245-1250. doi : 10.1016/j.crma.2011.11.012. http://www.numdam.org/articles/10.1016/j.crma.2011.11.012/
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