# A Kac model for kinetic annihilation

## TL;DR

This paper introduces a Kac-like particle system model with annihilation and elastic collisions, establishing well-posedness, deriving kinetic equations in the thermodynamic limit, and confirming propagation of chaos for bounded kernels.

## Contribution

It provides a rigorous analysis of a stochastic particle system with annihilation, deriving the associated kinetic equations and proving propagation of chaos for bounded collision kernels.

## Key findings

- Well-posedness of the particle system established
- Kinetic hierarchy derived in the thermodynamic limit
- Propagation of chaos proven for bounded collision kernels

## 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 $\alpha \in (0,1)$ or collide elastically with probability $1-\alpha$. 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.