# Entropy Production in Random Billiards

## TL;DR

This paper introduces random billiards as a model for nonequilibrium systems, deriving a general entropy production formula that links mathematical and thermodynamic entropy, with explicit examples involving boundary thermostats.

## Contribution

It develops a general entropy production rate formula for random billiards and connects it to thermodynamic entropy, including explicit models with boundary thermostats.

## Key findings

- Derived a universal entropy production rate formula for random billiards.
- Established a relation between entropy production and thermodynamic entropy.
- Proved ergodicity and computed stationary distributions for specific models.

## Abstract

We introduce a class of random mechanical systems called random billiards to study the problem of quantifying the irreversibility of nonequilibrium macroscopic systems. In a random billiard model, a point particle evolves by free motion through the interior of a spatial domain, and reflects according to a reflection operator, specified in the model by a Markov transition kernel, upon collision with the boundary of the domain. We derive a formula for entropy production rate that applies to a general class of random billiard systems. This formula establishes a relation between the purely mathematical concept of entropy production rate and textbook thermodynamic entropy, recovering in particular Clausius' formulation of the second law of thermodynamics. We also study an explicit class of examples whose reflection operator, referred to as the Maxwell-Smolukowski thermostat, models systems with boundary thermostats kept at possibly different temperatures. We prove that, under certain mild regularity conditions, the class of models are uniformly ergodic Markov chains and derive formulas for the stationary distribution and entropy production rate in terms of geometric and thermodynamic parameters.