On the limiting Markov process of energy exchanges in a rarely interacting ball-piston gas

TL;DR

This paper investigates the energy exchange process in a simplified billiard model with rare interactions, demonstrating convergence to a Markov jump process and deriving its generator, which advances understanding of heat transport in hybrid materials.

Contribution

It introduces a Markov process approximation for energy exchanges in a rare-interaction billiard model, providing a new analytical framework for heat transport in complex materials.

Findings

Convergence of the billiard process to a Markov jump process in the rare interaction regime

Explicit expression for the generator of the limiting Markov process

Numerical evidence supporting the theoretical convergence

Abstract

We analyse the process of energy exchanges generated by the elastic collisions between a point-particle, confined to a two-dimensional cell with convex boundaries, and a `piston', i.e. a line-segment, which moves back and forth along a one-dimensional interval partially intersecting the cell. This model can be considered as the elementary building block of a spatially extended high-dimensional billiard modeling heat transport in a class of hybrid materials exhibiting the kinetics of gases and spatial structure of solids. Using heuristic arguments and numerical analysis, we argue that, in a regime of rare interactions, the billiard process converges to a Markov jump process for the energy exchanges and obtain the expression of its generator.