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Concentration Theorems for Entropy and Free Energy

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Abstract

Jaynes’s entropy concentration theorem states that, for most words ω1 ...ωN of length N such that \(\mathop \Sigma \limits_{i = 1}^{\rm N} \;f(\omega _i ) \approx vN\), empirical frequencies of values of a function f are close to the probabilities that maximize the Shannon entropy given a value v of the mathematical expectation of f. Using the notion of algorithmic entropy, we define the notions of entropy for the Bose and Fermi statistical models of unordered data. New variants of Jaynes’s concentration theorem for these models are proved. We also present some concentration properties for free energy in the case of a nonisolated isothermal system. Exact relations for the algorithmic entropy and free energy at extreme points are obtained. These relations are used to obtain tight bounds on uctuations of energy levels at equilibrium points.

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Authors and Affiliations

  1. Institute for Information Transmission Problems, RAS, Moscow

    V. V. V’yugin

  2. Physics Department, M.V. Lomonosov Moscow State University, Russia

    V. P. Maslov

Authors
  1. V. V. V’yugin
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  2. V. P. Maslov
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__________

Translated from Problemy Peredachi Informatsii, No. 2, 2005, pp. 72–88.

Original Russian Text Copyright © 2005 by V’yugin, Maslov.

Supported in part by Grants of the President of the Russian Federation for Leading Scientific Schools, nos. 358.2003.1 and 1678.2003.1, Russian Foundation for Basic Research, project no. 03-01-00475, and the joint Russian-French RFBR-CNRS grant, no. 02-02-22001.

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V’yugin, V.V., Maslov, V.P. Concentration Theorems for Entropy and Free Energy. Probl Inf Transm 41, 134–149 (2005). https://doi.org/10.1007/s11122-005-0019-1

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  • DOI: https://doi.org/10.1007/s11122-005-0019-1

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