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About Kac's Program in Kinetic Theory

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Date
2011
Link to item file
http://hal.archives-ouvertes.fr/hal-00641197/fr/
Dewey
Probabilités et mathématiques appliquées
Sujet
propagation of chaos; mean-field limit; Boltzmann equation; jump process; relaxation; Kac
Journal issue
Comptes rendus mathématique
Volume
349
Number
23-24
Publication date
2011
Article pages
1245-1250
Publisher
Elsevier
DOI
http://dx.doi.org/10.1016/j.crma.2011.11.012
URI
https://basepub.dauphine.fr/handle/123456789/7531
Collections
  • CEREMADE : Publications
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Author
Mouhot, Clément
Mischler, Stéphane
Type
Article accepté pour publication ou publié
Abstract (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.

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