A Kac Model for Kinetic Annihilation

Abstract

In this paper we consider the stochastic dynamics of a finite system of particles in a finite volume (Kac-like particle system) which annihilate with probability α ∈ (0, 1) or collide elastically with probability 1 − α. We first establish the well-posedness of the particle system which exhibits no conserved quantities. We rigorously prove that, in some thermodynamic limit, a suitable hierarchy of kinetic equations is recovered for which tensorized solution to the homogenous Boltzmann with annihilation is a solution. For bounded collision kernels, this shows in particular that propagation of chaos holds true. Furthermore, we make conjectures about the limit behaviour of the particle system when hard-sphere interactions are taken into account.

Kinetic Description of a Rayleigh Gas with Annihilation

“…[2,4,5]). Recently, a first rigorous derivation of the spatially homogeneous nonlinear Boltzmann equation with annihilation from a Kac-like particle system has been obtained in [18], under suitable assumption on the collision kernel.…”