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 Mat. Zametki, 2008, Volume 83, Issue 4, Pages 559–580 (Mi mz4576)

On the Distribution of Integer Random Variables Related by Two Linear Inequalities: I

V. P. Maslova, V. E. Nazaikinskiib

a M. V. Lomonosov Moscow State University, Faculty of Physics
b A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences

Abstract: The authors' two preceding papers deal with the problem on the allocation of indistinguishable particles to positive integer energy levels under the condition that the total energy of the system is bounded above by some constant $M$. The estimates proved there imply that, for large $M$, most of the allocations concentrate near a limit distribution (which is the Bose–Einstein distribution, provided that the particles obey the corresponding statistics). The present paper continues this trend of research by considering the case in which not only the total energy is constrained but also the overall number of particles is specified. We study both the Bose and the Gibbs distribution and analyze the phenomenon whereby the Bose distribution passes into the Gibbs distribution in the limit as the number of particles is relatively small.

Keywords: Bose–Einstein statistics, Boltzmann–Gibbs statistics, cumulative distribution, entropy, allocation of particles

DOI: https://doi.org/10.4213/mzm4576

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English version:
Mathematical Notes, 2008, 83:4, 512–529

Bibliographic databases:

UDC: 519.2+531.19

Citation: V. P. Maslov, V. E. Nazaikinskii, “On the Distribution of Integer Random Variables Related by Two Linear Inequalities: I”, Mat. Zametki, 83:4 (2008), 559–580; Math. Notes, 83:4 (2008), 512–529

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz4576
• https://doi.org/10.4213/mzm4576
• http://mi.mathnet.ru/eng/mz/v83/i4/p559

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. V. P. Maslov, V. E. Nazaikinskii, “On the Distribution of Integer Random Variables Satisfying Two Linear Relations”, Math. Notes, 84:1 (2008), 73–99
2. V. P. Maslov, “Transition to the Condensate State for Classical Gases and Clusterization”, Math. Notes, 84:6 (2008), 785–813
3. Ìàñëîâ Â.Ï., “Íîâàÿ êîíöåïöèÿ ïðîöåññà íóêëåàöèè”, ÒÌÔ, 156:1 (2008), 159–160
4. Maslov V., “Dequantization, Statistical Mechanics and Econophysics”, Tropical and Idempotent Mathematics, Contemporary Mathematics, 495, eds. Litvinov G., Sergeev S., Amer Mathematical Soc, 2009, 239–279
5. Maslov V.P., “The Relationship Between the Van-der-Waals Model and the Undistinguishing Statistics of Objectively Distinguishable Objects. the New Parastatistics”, Russ. J. Math. Phys., 21:1 (2014), 99–111
6. Wang X., Luo X., Zhang M., Guan X., “Distributed Detection and Isolation of False Data Injection Attacks in Smart Grids Via Nonlinear Unknown Input Observers”, Int. J. Electr. Power Energy Syst., 110 (2019), 208–222
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