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TMF, 2003, Volume 135, Number 1, Pages 3–54 (Mi tmf178)  

This article is cited in 7 scientific papers (total in 7 papers)

Generalized Informational Entropy and Noncanonical Distribution in Equilibrium Statistical Mechanics

Yu. G. Rudoi

Peoples Friendship University of Russia

Abstract: Based on the Jaynes principle of maximum for informational entropy, we find a generalized probability distribution and construct a generalized equilibrium statistical mechanics (ESM) for a wide class of objects to which the usual (canonical) ESM cannot be applied. We consistently consider the case of a continuous, not discrete, random variable characterizing the state of the object. For large values of the argument, the resulting distribution is characterized by a power-law, not exponential, asymptotic behavior, and the corresponding power asymptotic expression agrees with the empirical laws established for these objects. The $\varepsilon$-deformed Boltzmann–Gibbs–Shannon functional satisfying the requirements of the entropy axiomatics and leading to the canonical ESM for $\varepsilon=0$ is used as the original entropy functional. We also consider nonlinear transformations of this functional. We show that depending on how the averages of the dynamical characteristics of the object are defined, the different (Tsallis, Renyi, and Hardy–Littlewood–Pólya) versions of the generalized ESM can be used, and we give their comparative analysis. We find conditions under which the Gibbs–Helmholtz thermodynamic relations hold and the Legendre transformation can be applied to the generalized entropy and the Massieu–Planck function. We consider the Tsallis and Renyi ESM versions in detail for the case of a one-dimensional probabilistic object with a single dynamical characteristic whose role is played by a generalized positive “energy” with a monotonic power growth. We obtain constraints on the Renyi index under which the equilibrium distribution relates to a definite class of stable Gaussian or Levy–Khinchin distributions.

Keywords: Shannon entropy, Renyi entropy, Tsallis entropy, Levy–Khinchin distribution, Jaynes maximum entropy principle, equilibrium statistical mechanics

DOI: https://doi.org/10.4213/tmf178

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English version:
Theoretical and Mathematical Physics, 2003, 135:1, 451–496

Bibliographic databases:

Received: 21.11.2001
Revised: 05.07.2002

Citation: Yu. G. Rudoi, “Generalized Informational Entropy and Noncanonical Distribution in Equilibrium Statistical Mechanics”, TMF, 135:1 (2003), 3–54; Theoret. and Math. Phys., 135:1 (2003), 451–496

Citation in format AMSBIB
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\transl
\jour Theoret. and Math. Phys.
\yr 2003
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\issue 1
\pages 451--496
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    This publication is cited in the following articles:
    1. Pykh YA, “Construction of entropy characteristics based on replicator equations with nonsymmetric interaction matrices”, Doklady Mathematics, 72:2 (2005), 780–782  mathscinet  zmath  isi  elib
    2. Figueiredo, A, “On the statistical interpretation of generalized entropies”, Physica A-Statistical Mechanics and Its Applications, 367 (2006), 191  crossref  mathscinet  adsnasa  isi  scopus  scopus
    3. A. I. Olemskoi, O. V. Yushchenko, A. Yu. Badalyan, “Statistical field theory of a nonadditive system”, Theoret. and Math. Phys., 174:3 (2013), 386–405  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. V. P. Koverda, V. N. Skokov, A. V. Vinogradov, “Stability of low-frequency pulsations in a transient heat transfer regime upon phase transitions”, High Temperature, 51:3 (2013), 421–425  mathnet  crossref  isi  elib  elib
    5. Tawfik A., “Dynamical Fluctuations in Baryon-Meson Ratios”, J. Phys. G-Nucl. Part. Phys., 40:5 (2013), 055109  crossref  adsnasa  isi  elib  scopus  scopus
    6. Magomedov R.A. Meilanov R.P. Akhmedov E.N. Aliverdiev A.A., “Calculation of Multicomponent Compound Properties Using Generalization of Thermodynamics in Derivatives of Fractional Order”, Xxxi International Conference on Equations of State For Matter (Elbrus 2016), Journal of Physics Conference Series, 774, IOP Publishing Ltd, 2016, UNSP 012025  crossref  isi  scopus
    7. A. V. Kolesnichenko, “K razrabotke statisticheskoi termodinamiki i tekhniki fraktalnogo analiza dlya neekstensivnykh sistem na osnove entropii i razlichayuschei informatsii Reni”, Preprinty IPM im. M. V. Keldysha, 2018, 060, 44 pp.  mathnet  crossref
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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