Quantitative uniform in time chaos propagation for Boltzmann collision processes

Stéphane Mischler (1) , Clément Mouhot (2, 3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 DAMTP - Department of Applied Mathematics and Theoretical Physics

This paper is devoted to the study of mean-field limit for systems of indistinguables particles undergoing collision processes. As formulated by [Kac, 1956] this limit is based on the chaos propagation, and we (1) prove and quantify this property for Boltzmann collision processes with unbounded collision rates (hard spheres or long-range interactions), (2) prove and quantify this property \emph{uniformly in time}. This yields the first chaos propagation result for the spatially homogeneous Boltzmann equation for true (without cut-off) Maxwell molecules whose "Master equation" shares similarities with the one of a Lévy process and the first quantitative chaos propagation result for the spatially homogeneous Boltzmann equation for hard spheres (improvement of the convergence result of [Sznitman, 1984]. Moreover our chaos propagation results are the first uniform in time ones for Boltzmann collision processes (to our knowledge), which partly answers the important question raised by Kac of relating the long-time behavior of a particle system with the one of its mean-field limit, and we provide as a surprising application a new proof of the well-known result of gaussian limit of rescaled marginals of uniform measure on the N-dimensional sphere as N goes to infinity (more applications will be provided in a forthcoming work). Our results are based on a new method which reduces the question of chaos propagation to the one of proving a purely functional estimate on some generator operators (consistency estimate) together with fine stability estimates on the flow of the limiting non-linear equation (stability estimates).

Mots clés: hard spheres; mean-field limit; quantitative; uniform in time; jump process; collision process; Boltzmann equation; Maxwell molecules; non cutoff; hard spheres.

Domaines: Equations aux dérivées partielles [math.AP]

Stéphane Mischler, Clément Mouhot. Quantitative uniform in time chaos propagation for Boltzmann collision processes. 2010. ⟨hal-00447988⟩