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Markov chains as models in statistical mechanics. (English) Zbl 1442.62764

Summary: The Bernoulli/Laplace urn model [D. Bernoulli, “Disquisitiones analyticae de novo problemate conjecturale”, Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae 14, 3–25 (1769); P. S. de Laplace, Théorie Analytique des Probabilités. Paris: V. Courcier (1812)] and the P. Ehrenfest and T. Ehrenfest urn model for mixing [Phys. Z. 8, 311–314 (1907; JFM 38.0931.01)] 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. In [“Studien über Molekularstatistik von Emulsionen und deren Zusammenhang mit der Brown’schen Bewegung”, in: Sitzungsberichte der Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse CXXIII. Band X, Abteilung IIA. Wien: Hölder. 2381–2405 (1914)], M. von Smoluchowski 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.

MSC:

62P35 Applications of statistics to physics
62M05 Markov processes: estimation; hidden Markov models
82-10 Mathematical modeling or simulation for problems pertaining to statistical mechanics

Citations:

JFM 38.0931.01
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References:

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