A probabilistic model for the equilibration of an ideal gas

Santi Prestipino

Physica A: Statistical Mechanics and its Applications, 2004, vol. 340, issue 1, 373-379

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 lnW 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.

Keywords: Urn models; Boltzmann entropy; Maximum-entropy principle (search for similar items in EconPapers)
Date: 2004
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:340:y:2004:i:1:p:373-379

DOI: 10.1016/j.physa.2004.04.029

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