Asymptotic of Grazing Collisions and Particle Approximation for the Kac Equation without Cutoff

Abstract

The subject of this article is the Kac equation without cutoff. We first show that in the asymptotic of grazing collisions, the Kac equation can be approximated by a Fokker-Planck equation. The convergence is uniform in time and we give an explicit rate of convergence. Next, we replace the small collisions by a small diffusion term in order to approximate the solution of the Kac equation and study the resulting error. We finally build a system of stochastic particles undergoing collisions and diffusion, that we can easily simulate, which approximates the solution of the Kac equation without cutoff. We give some estimates on the rate of convergence. (2000): 82C40, 60K35. Mathematics Subject Classification

Quantitative propagation of chaos for generalized Kac particle systems

“…A similar result as the one of Fournier and Godinho (2012) can be found in Fournier and Mischler (2014) where, motivated by the numerical approximation of the Boltzmann equation for hard spheres, hard potentials and Maxwellian gases, a pathwise coupling argument was developed for Nanbu particle systems, which extends a coupling construction based on optimal transport developed in Fontbona, Guérin and Méléard (2009). That pathwise approach, however, does not readily extend to the particle systems of Bird type we are interested in, which in turn provide a physically more transparent description of the relevant interaction phenomena.…”

Asymptotic of grazing collisions for the spatially homogeneous Boltzmann equation for soft and Coulomb potentials

“…In all cases (soft or Coulomb potentials), we expect to get a bound for W 2 (f ǫ t , g t ) of order π 0 θ 4 β ǫ (θ)dθ as for the Kac equation (see [19]). For soft potentials, the rate of convergence that we get is ǫ 1/2− (if f 0 is nice) instead of ǫ.…”

“…In this article, we will also use a result of Zaitsev [32] in order to obtain a bound for the Wasserstein distance between a compensated Poisson integral and a Gaussian random variable. Such an idea comes from the paper of Fournier [18] about the approximation of Lévy-driven stochastic differential equations in one dimension, see also [19]. Since we work here in dimension 3, such a result is much more difficult to obtain.…”

A new approach to quantitative propagation of chaos for drift, diffusion and jump processes

“…It is classical to prove that V N t is a Feller process and we refer to the textbooks [13,34] where the theory is set up with full details (one can also refer to [37,30,15] where similar processes are considered).…”

“…A conceivable alternative approach would be to use the general nonlinear martingale approach, but that would most likely not provide any quantitative rate of propagation of chaos. The techniques developed recently in [15] for the elastic Kac equation without cut-off is yet another alternative technique that could be tried on this model, but we have not made any attempts in this direction, and, would it work, it is not clear as to what kind of convergence rate one could hope to achieve. 7.3.…”