About Kac's Program in Kinetic Theory
Mouhot, Clément; Mischler, Stéphane (2011), About Kac's Program in Kinetic Theory, Comptes rendus mathématique, 349, 23-24, p. 1245-1250. http://dx.doi.org/10.1016/j.crma.2011.11.012
Type
Article accepté pour publication ou publiéExternal document link
http://hal.archives-ouvertes.fr/hal-00641197/fr/Date
2011Journal name
Comptes rendus mathématiqueVolume
349Number
23-24Publisher
Elsevier
Pages
1245-1250
Publication identifier
Metadata
Show full item recordAbstract (EN)
In this Note we present the main results from the recent work arxiv:1107.3251, which answers several conjectures raised fifty years ago by Kac. There Kac introduced a many-particle stochastic process (now denoted as Kac's master equation) which, for chaotic data, converges to the spatially homogeneous Boltzmann equation. We answer the three following questions raised in \cite{kac}: (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.Subjects / Keywords
propagation of chaos; mean-field limit; Boltzmann equation; jump process; relaxation; KacRelated items
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