Kac’s program in kinetic theory

By St'ephane Mischler, Clément Mouhot. Inventiones Mathematicae, 2011

157 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a1593bc211a13fc7ebd5cfbdab30e071c9d718ba

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Citations in the corpus

The derivation of swarming models: Mean-field limit and Wasserstein distances (2013, background): “The question of the derivation of the Boltzmann equation from particles with jump processes was also raised and solved by [53], and further results are given in the recent important work by [65].”
Mean Field Limit for Stochastic Particle Systems (2017, background): “After some major contributions in the 90’s, see in particular [26, 37, 38], significant progress was again achieved recently in solving these conjectures, see [41, 63, 64], and the earlier [16].”
Mean Field Limit and Propagation of Chaos for Vlasov Systems with Bounded Forces (2015, methodology): “Such stronger notions of propagation of chaos have recently been more thoroughly investigated and some of the connections between them elucidated, we refer to [29, 38, 39] for example or to the survey [31].”
Global Well‐Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games (2020, methodology): “Therefore, such equations are very natural, and they were successfully used in the literature in the context of mean field limits of large particle system (see for instance in [34, 14]).”
On Kac's chaos and related problems (2012, methodology): “stigate the inﬁnite number of particles limit N→ ∞ for these systems as it has been illustrated by many works since the seminal article by Kac [41]. Using the above deﬁnition of chaos, it is shown in [41, 49, 50, 35, 55] that if f(t) evolves according to the nonlinear space homogeneous Boltzmann equ”