Large deviations for Kac-like walks

TL;DR

This paper studies a Kac-like particle system with momentum conservation but energy dissipation, deriving a large deviation principle and providing a gradient flow formulation of the associated Boltzmann-Kac equation.

Contribution

It introduces a novel Kac-like walk with partial conservation laws and explicitly characterizes its large deviations, linking it to gradient flow structures.

Findings

Explicit large deviation rate function derived.

Gradient flow formulation of Boltzmann-Kac equation established.

Describes dynamics of particle systems with non-energy conserving collisions.

Abstract

We introduce a Kac's type walk whose rate of binary collisions preserves the total momentum but not the kinetic energy. In the limit of large number of particles we describe the dynamics in terms of empirical measure and flow, proving the corresponding large deviation principle. The associated rate function has an explicit expression. As a byproduct of this analysis, we provide a gradient flow formulation of the Boltzmann-Kac equation.