Kac’s program in kinetic theory

Abstract

Abstract. 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 Grünbaum [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 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 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 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.

Kinetic Theory beyond the Stosszahlansatz

“…To this end, we need to define collisional invariants that are appropriate to the scheme developed here. The form assumed by the collision term in Equation (21) makes it necessary (see Section 7) to introduce bilocal invariants χ that are quantities conserved in a bilocal collision of particle pairs (1,2) and (3,4), which occur in x 1 and x 2 for the (1,3) and (2,4) collisions, respectively. A collisional invariant has to be defined in this case such that (in this context, spatial arguments are not relevant and can be dropped out).…”

“…Still, however, although exact results have been obtained regarding its range of validity [2,3], from a purist's perspective, the Stosszahlansatz is little more than an ad hoc assumption. Unfortunately, the way it could be complemented or generalized is anything but obvious, so it might seem that the ansatz is here to stay.…”

“…We first substitute p 1 ↔ p 3 in the first integral of the collision term and p 2 ↔ p 4 in the second (which is allowed as both are dummy variables) so as to have two equivalent forms of the collision term, of which we take the average. In this averaged collision term, we now relabel p 1 ↔ p 1 , p 2 ↔ p 2 , p 3 ↔ p 3 and p 4 ↔ p 4 . Using the fact that dp 1 dp 2 dp 3 = dp 1 dp 2 dp 3 and dp 1 dp 2 dp 4 = dp 1 dp 2 dp 4 , and that |p 3 − p 1 | = |p 3 − p 1 | and |p 4 − p 2 | = |p 4 − p 2 |, both terms can be averaged again and we finally obtain the collision term as 4 dp 1 dp 2 dp…”

A few ways to destroy entropic chaoticity on Kac’s Sphere

“…In a remarkable recent paper, [14], Mischler and Mouhot introduced a new abstract method that allowed them to tackle many unsolved questions in the subject, including the velocity dependent cases mentioned above. They managed to show quantitative and uniform in time propagation of chaos in weak measure distance, propagation of entropic chaos (soon to be defined), and quantitative estimation of relaxation rates that are independent of the number of particles.…”

“…More information about the topic and the related spectral gap problem and entropy-entropy production ratio can be found in [2,3,4,5] and the excellent [16,14].…”