Derivation of a Boltzmann equation with higher-order collisions from a generalized Kac model

TL;DR

This paper extends Kac's stochastic model to derive a Boltzmann equation incorporating higher-order collisions, proving convergence and propagation of chaos for the generalized model.

Contribution

It introduces a generalized Kac model that includes higher-order collisional terms and proves convergence to a Boltzmann equation with Maxwell-type collision kernel.

Findings

Proves convergence from finite hierarchy to infinite hierarchy.

Establishes propagation of chaos for the generalized model.

Derives a Boltzmann equation with higher-order interactions.

Abstract

In this work, we generalize M. Kac's original many-particle binary stochastic model to derive a space homogeneous Boltzmann equation that includes a linear combination of higher-order collisional terms. First, we prove an abstract theorem about convergence from a finite hierarchy to an infinite hierarchy of coupled equations. We apply this convergence theorem on hierarchies for marginals corresponding to the generalized Kac model mentioned above. As a corollary, we prove propagation of chaos for the marginals associated to the generalized Kac model. In particular, the first marginal converges towards the solution of a Boltzmann equation including interactions up to a finite order, and whose collision kernel is of Maxwell-type with cut-off.