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Inventiones mathematicae

, Volume 193, Issue 1, pp 1–147 | Cite as

Kac’s program in kinetic theory

  • Stéphane Mischler
  • Clément Mouhot
Article

Abstract

This paper is devoted to the study of propagation of chaos and mean-field limits for systems of indistinguishable particles, undergoing collision processes. The prime examples we will consider are the many-particle jump processes of Kac and McKean (Kac in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, Vol. III, pp. 171–197, 1956; McKean in J. Comb. Theory 2:358–382, 1967) giving rise to the Boltzmann equation. We solve the conjecture raised by Kac (Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, Vol. III, pp. 171–197, 1956), motivating his program, on the rigorous connection between the long-time behavior of a collisional many-particle system and the one of its mean-field limit, for bounded as well as unbounded collision rates.

Motivated by the inspirative paper by Grünbaum (Arch. Ration. Mech. Anal. 42:323–345, 1971), we develop an abstract method that reduces the question of propagation of chaos to that of proving a purely functional estimate on generator operators (consistency estimates), along with differentiability estimates on the flow of the nonlinear limit equation (stability estimates). This allows us to exploit dissipativity at the level of the mean-field limit equation rather than the level of the particle system (as proposed by Kac).

Using this method we show: (1) Quantitative estimates, that are uniform in time, on the chaoticity of a family of states. (2) Propagation of entropic chaoticity, as defined by Carlen et al. (Kinet. Relat. Models 3:85–122, 2010). (3) Estimates on the time of relaxation to equilibrium, that are independent of the number of particles in the system. Our results cover the two main Boltzmann physical collision processes with unbounded collision rates: hard spheres and true Maxwell molecules interactions. The proof of the stability estimates for these models requires significant analytic efforts and new estimates.

Keywords

Kac’s program Kinetic theory Master equation Mean-field limit Quantitative Uniform in time Jump process Collision process Boltzmann equation Maxwell molecules Non-cutoff Hard spheres 

Mathematics Subject Classification

82C40 76P05 54C70 60J75 

Notes

Acknowledgements

We thank the Mathematics Department of Chalmers University for the invitation in November 2008, where the abstract method was devised and the related joint work [58] with Bernt Wennberg was initiated. We thank Ismaël Bailleul, Thierry Bodineau, François Bolley, Anne Boutet de Monvel, José Alfredo Cañizo, Eric Carlen, Nicolas Fournier, François Golse, Arnaud Guillin, Maxime Hauray, Joel Lebowitz, Pierre-Louis Lions, Richard Nickl, James Norris, Mario Pulvirenti, Judith Rousseau, Laure Saint-Raymond and Cédric Villani for fruitful comments and discussions, and Amit Einav for his careful proofreading of parts of the manuscript. We would also like to mention the inspiring courses by Pierre-Louis Lions at Collège de France on “Mean-Field Games” in 2007–2008 and 2008–2009, which triggered our interest in the functional analysis aspects of this topic. Finally we thank the anonymous referees for helpful suggestions on the presentation.

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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.CEREMADE, UMR CNRS 7534Université Paris-DauphineParis Cedex 16France
  2. 2.DPMMS, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK

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