A probabilistic model for the equilibration of an ideal gas

TL;DR

This paper introduces a probabilistic model simulating how a dilute gas approaches equilibrium, extending the Ehrenfest urn model to include different volumes, cell occupancy, and elastic collisions, illustrating entropy behavior.

Contribution

It generalizes the Ehrenfest urn model to better mimic an ideal gas with volume differences, velocity vectors, and collisions, providing a probabilistic basis for the maximum-entropy principle.

Findings

Particles reach an equilibrium distribution described by binomial, hypergeometric, or beta-like densities.

The model's entropy matches the thermodynamic entropy of an ideal gas.

Analytical and numerical results confirm the approach to equilibrium.

Abstract

I present a generalization of the Ehrenfest urn model that is aimed at simulating the approach to equilibrium in a dilute gas. The present model differs from the original one in two respects: 1) the two boxes have different volumes and are divided into identical cells with either multiple or single occupancy; 2) particles, which carry also a velocity vector, are subjected to random, but elastic, collisions, both mutual and against the container walls. I show, both analytically and numerically, that the number and energy of particles in a given urn evolve eventually to an equilibrium probability density $W$ which, depending on cell occupancy, is binomial or hypergeometric in the particle number and beta-like in the energy. Moreover, the Boltzmann entropy $\ln W$ takes precisely the same form as the thermodynamic entropy of an ideal gas. This exercise can be useful for pedagogical purposes in that it provides, although in an extremely simplified case, a probabilistic justification for the maximum-entropy principle.