Markov Chains as Models in Statistical Mechanics

Autores: Eugene Seneta
Localización: Statistical science, ISSN 0883-4237, Vol. 31, Nº. 3, 2016, págs. 399-414
Idioma: inglés
DOI: 10.1214/16-sts568
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Resumen
- 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.