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

This paper is devoted to the study of propagation of chaos and mean-field limits for systems of indistinguable particles, undergoing collision processes. The prime examples we will consider are the many-particle jump processes of Kac and McKean [42, 53] giving rise to the Boltzmann equation. We solve the conjecture raised by Kac [42], 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 Gru ̈nbaum [35], 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 es- timates 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 in [10]. (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 phys- ical 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.

Domaines: Equations aux dérivées partielles [math.AP]

Stéphane Mischler, Clément Mouhot. Kac's program in kinetic theory. Inventiones Mathematicae, 2013, 193 (1), pp.1-147. ⟨hal-01067318⟩