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

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
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
\Bibitem{VyuMas05} \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 \mathnet{http://mi.mathnet.ru/ppi98} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2158687} \zmath{https://zbmath.org/?q=an:1090.94011} \elib{http://elibrary.ru/item.asp?id=9182293} \transl \jour Problems Inform. Transmission \yr 2005 \vol 41 \issue 2 \pages 134--149 \crossref{https://doi.org/10.1007/s11122-005-0019-1} \elib{http://elibrary.ru/item.asp?id=13479833} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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