# Dynamical thermalization in time-dependent Billiards

## TL;DR

This paper investigates how particles in a time-dependent billiard reach thermal equilibrium through inelastic collisions, revealing three distinct statistical regimes and providing analytical descriptions that match numerical results.

## Contribution

It introduces a comprehensive analysis of dynamical thermalization in time-dependent billiards, identifying regimes and deriving analytical expressions for speed distribution evolution.

## Key findings

- Identification of three statistical regimes: diffusion plateau, exponential growth/decay, stagnation.
- Analytical derivation of speed distribution transition from Gauss-like to Boltzmann-like.
- Matching of analytical results with numerical experiments confirming the thermalization mechanism.

## Abstract

Numerical experiments of the statistical evolution of an ensemble of non-interacting particles in a time-dependent billiard with inelastic collisions, reveals the existence of three statistical regimes for the evolution of the speeds ensemble, namely, diffusion plateau, normal growth/exponential decay and stagnation. These regimes are linked numerically to the transition from Gauss-like to Boltzmann-like speed distributions. Further, the different evolution regimes are obtained analytically through velocity-space diffusion analysis. From these calculations the asymptotic root mean square of speed, initial plateau, and the growth/decay rates for intermediate number of collisions are determined in terms of the system parameters. The analytical calculations match the numerical experiments and point to a dynamical mechanism for thermalization, where inelastic collisions and a high-dimensional phase space lead to a bounded diffusion in the velocity space towards a stationary distribution function with a kind of reservoir temperature determined by the boundary oscillation amplitude and the restitution coefficient.