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This article is cited in 9 scientific papers (total in 9 papers)
Large Systems
Theorems on Concentration for the Entropy of Free Energy
V. V. V'yugina, V. P. Maslovb
a Institute for Information Transmission Problems, Russian Academy of Sciences
b M. V. Lomonosov Moscow State University, Faculty of Physics
Abstract:
Jaynes's entropy concentration theorem states that, for most words $\omega_1…\omega_N$
of length $N$ such that
$\sum\limits_{i=1}^Nf(\omega_i)\approx vN$,
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 fluctuations of energy levels at equilibrium points.
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English version:
Problems of Information Transmission, 2005, 41:2, 134–149
Bibliographic databases:
  
UDC:
621.391.1:519.2
Received: 14.09.2004
Revised: 01.03.2005
Citation:
V. V. V'yugin, V. P. Maslov, “Theorems on Concentration for the Entropy of Free Energy”, Probl. Peredachi Inf., 41:2 (2005), 72–88; Problems Inform. Transmission, 41:2 (2005), 134–149
Citation in format AMSBIB
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\jour Problems Inform. Transmission
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\issue 2
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\crossref{https://doi.org/10.1007/s11122-005-0019-1}
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V. P. Maslov, “Nonlinear Averages in Economics”, Math. Notes, 78:3 (2005), 347–363
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Maslov V.P., “On the principle of increasing complexity in portfolio formation on the stock exchange”, Dokl. Math., 72:2 (2005), 718–722
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Maslov V.P., V'yugin V.V., “A sufficient condition for a riskless distribution of investments”, Dokl. Math., 75:2 (2007), 299–303
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Maslov V.P., “Quantum economics”, Russ. J. Math. Phys., 12:2 (2005), 219–231
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V. V. V'yugin, V. P. Maslov, “Distribution of Investments in the Stock Market, Information
Types, and Algorithmic Complexity”, Problems Inform. Transmission, 42:3 (2006), 251–261
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V. P. Maslov, V. E. Nazaikinskii, “On the Distribution of Integer Random Variables Related by a Certain Linear Inequality. I”, Math. Notes, 83:2 (2008), 211–237
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Maslov V.P., “Theory of chaos and its application to the crisis of debts and the origin of inflation”, Russ. J. Math. Phys., 16:1 (2009), 103–120
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Maslov V., “Dequantization, Statistical Mechanics and Econophysics”, Tropical and Idempotent Mathematics, Contemporary Mathematics, 495, 2009, 239–279
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Maslov V.P., Maslova T.V., “Probability Theory for Random Variables with Unboundedly Growing Values and its Applications”, Russ. J. Math. Phys., 19:3 (2012), 324–339
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