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August 2016 Markov Chains as Models in Statistical Mechanics
Eugene Seneta
Statist. Sci. 31(3): 399-414 (August 2016). DOI: 10.1214/16-STS568


The Bernoulli [Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae 14 (1769) 3–25]/Laplace [Théorie Analytique des Probabilités (1812) V. Courcier] urn model and the Ehrenfest and Ehrenfest [Physikalische Zeitschrift 8 (1907) 311–314] urn model for mixing are instances of simple Markov chain models called random walks. Both can be used to suggest a probabilistic resolution to the coexistence of irreversibility and recurrence in Boltzmann’s H-Theorem. Marian von Smoluchowski [In Sitzungsberichte der Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse (1914) 2381–2405 Hölder] also modelled by a simple Markov chain, with analogous properties, have fluctuations over time in the number of particles contained in a small element of volume in a solution.This paper explores the themes of entropy, recurrence and reversibility within the framework of such Markov chains.

A branching process with immigration, in this respect like Smoluchowski’s model, is introduced to accentuate common features of the spectral theory of all models. This is related to their reversibility, a key issue.


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Eugene Seneta. "Markov Chains as Models in Statistical Mechanics." Statist. Sci. 31 (3) 399 - 414, August 2016.


Published: August 2016
First available in Project Euclid: 27 September 2016

zbMATH: 06946232
MathSciNet: MR3552741
Digital Object Identifier: 10.1214/16-STS568

Keywords: branching process with immigration , Ehrenfest , entropy and recurrence , Krawtchouk, Hahn, Charlier, Meixner polynomials , reversible Markov chain , Smoluchowski , stochastic matrix

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.31 • No. 3 • August 2016
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