About Kac's Program in Kinetic Theory

Stéphane Mischler (1) , Clément Mouhot (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 DPMMS - Department of Pure Mathematics and Mathematical Statistics

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.

Mots clés: jump process; Boltzmann equation; mean-field limit; propagation of chaos; Kac; relaxation

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]; Physique mathématique [math-ph]

Stéphane Mischler, Clément Mouhot. About Kac's Program in Kinetic Theory. Comptes rendus de l'Académie des sciences. Série I, Mathématique, 2011, 349 (23-24), pp.1245-1250. ⟨10.1016/j.crma.2011.11.012⟩. ⟨hal-00641197⟩