Relaxation of ideal classical particles in a one-dimensional box

TL;DR

This paper investigates how classical particles in a one-dimensional box relax to equilibrium, analyzing the effects of wall interactions and identifying conditions for thermal velocity distributions through numerical simulations.

Contribution

It introduces a model with dynamical walls as independent scatterers and explores conditions under which the velocity distribution becomes Maxwellian.

Findings

Walls induce uniform spatial distribution at large times.

Stationary velocity distribution can be Maxwellian under certain conditions.

Chaotic dynamics observed in the Fermi-Ulam model for specific parameters.

Abstract

We study the deterministic dynamics of non-interacting classical gas particles confined to a one-dimensional box as a pedagogical toy model for the relaxation of the Boltzmann distribution towards equilibrium. Hard container walls alone induce a uniform distribution of the gas particles at large times. For the relaxation of the velocity distribution we model the dynamical walls by independent scatterers. The Markov property guarantees a stationary but not necessarily thermal velocity distribution for the gas particles at large times. We identify the conditions for physical walls where the stationary velocity distribution is the Maxwell distribution. For our numerical simulation we represent the wall particles by independent harmonic oscillators. The corresponding dynamical map for oscillators with a fixed phase (Fermi--Ulam accelerator) is chaotic for mesoscopic box dimensions.