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Probl. Peredachi Inf., 2005, Volume 41, Issue 2, Pages 72–88 (Mi ppi98)  

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
\by V.~V.~V'yugin, V.~P.~Maslov
\paper Theorems on Concentration for the Entropy of Free Energy
\jour Probl. Peredachi Inf.
\yr 2005
\vol 41
\issue 2
\pages 72--88
\jour Problems Inform. Transmission
\yr 2005
\vol 41
\issue 2
\pages 134--149

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    This publication is cited in the following articles:
    1. V. P. Maslov, “Nonlinear Averages in Economics”, Math. Notes, 78:3 (2005), 347–363  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. Maslov V.P., “On the principle of increasing complexity in portfolio formation on the stock exchange”, Dokl. Math., 72:2 (2005), 718–722  mathnet  mathscinet  zmath  isi  elib  elib
    3. Maslov V.P., V'yugin V.V., “A sufficient condition for a riskless distribution of investments”, Dokl. Math., 75:2 (2007), 299–303  mathnet  crossref  mathscinet  zmath  isi  elib
    4. Maslov V.P., “Quantum economics”, Russ. J. Math. Phys., 12:2 (2005), 219–231  mathscinet  zmath  isi  elib
    5. 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  mathnet  crossref  mathscinet  elib  elib
    6. 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  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. 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  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. Maslov V., “Dequantization, Statistical Mechanics and Econophysics”, Tropical and Idempotent Mathematics, Contemporary Mathematics, 495, 2009, 239–279  crossref  mathscinet  zmath  isi
    9. 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  crossref  mathscinet  zmath  isi  elib
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