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Mat. Zametki, 2008, Volume 83, Issue 2, Pages 232–263 (Mi mz4418)  

This article is cited in 36 scientific papers (total in 36 papers)

On the Distribution of Integer Random Variables Related by a Certain Linear Inequality. 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: We consider the mathematical problem of the allocation of indistinguishable particles to integer energy levels under the condition that the number of particles can be arbitrary and the total energy of the system is bounded above. Systems of integer as well as fractional dimension are considered. The occupation numbers can either be arbitrary nonnegative integers (the case of “Bose particles”) or lie in a finite set $\{0,1,dots,R\}$ (the case of so-called parastatistics; for example, $R=1$ corresponds to the Fermi–Dirac statistics). Assuming that all allocations satisfying the given constraints are equiprobable, we study the phenomenon whereby, for large energies, most of the allocations tend to concentrate near the limit distribution corresponding to the given parastatistics.

Keywords: Bose-Einstein statistics, parastatistics, system of fractional dimension, cumulative distribution, convergence, limit distribution

DOI: https://doi.org/10.4213/mzm4418

Full text: PDF file (720 kB)
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English version:
Mathematical Notes, 2008, 83:2, 211–237

Bibliographic databases:

UDC: 519.2+531.19
Received: 11.01.2008

Citation: V. P. Maslov, V. E. Nazaikinskii, “On the Distribution of Integer Random Variables Related by a Certain Linear Inequality. I”, Mat. Zametki, 83:2 (2008), 232–263; Math. Notes, 83:2 (2008), 211–237

Citation in format AMSBIB
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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 Related by a Certain Linear Inequality. II”, Math. Notes, 83:3 (2008), 345–363  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. V. P. Maslov, V. E. Nazaikinskii, “On the Distribution of Integer Random Variables Related by Two Linear Inequalities: I”, Math. Notes, 83:4 (2008), 512–529  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. V. P. Maslov, “Solution of the Gibbs Paradox in the Framework of Classical Mechanics (Statistical Physics) and Chrystalizations of the Gas $C_{60}$”, Math. Notes, 83:5 (2008), 716–722  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. V. P. Maslov, “Taking into Account the Interaction between Particles in the New Nucleation Theory, Quasiparticles, Quantization of Vortices, and the Two-Particle Distribution Function”, Math. Notes, 83:6 (2008), 790–803  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. V. P. Maslov, V. E. Nazaikinskii, “On the Distribution of Integer Random Variables Related by a Certain Linear Inequality. III”, Math. Notes, 83:6 (2008), 804–820  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. V. P. Maslov, V. E. Nazaikinskii, “On the Distribution of Integer Random Variables Satisfying Two Linear Relations”, Math. Notes, 84:1 (2008), 73–99  mathnet  crossref  crossref  mathscinet  isi  elib
    7. V. P. Maslov, “New Distributions in Classical Physics”, Math. Notes, 84:2 (2008), 290–296  mathnet  crossref  crossref  mathscinet  isi
    8. V. P. Maslov, “New concept of the nucleation process”, Theoret. and Math. Phys., 156:1 (2008), 1101–1102  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    9. V. P. Maslov, “Clustering in an ideal gas in nanostructures as a Bose-condensation-type phenomenon in an asymptotically probabilistically quantized space”, Theoret. and Math. Phys., 157:3 (2008), 1760–1761  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    10. Maslov V.P., “On the superfluidity of classical liquid in nanotubes. IV”, Russ. J. Math. Phys., 15:2 (2008), 280–290  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    11. Maslov V.P., “On a new universal constant in ideal gas theory in a nanoporous medium”, Math. Notes, 84:3 (2008), 439–441  crossref  mathscinet  zmath  isi  elib  scopus
    12. Maslov V.P., “New theory of nucleation”, Russ. J. Math. Phys., 15:3 (2008), 332–342  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    13. V. P. Maslov, “Liquefaction of a gas governed by the microcanonical distribution”, Theoret. and Math. Phys., 160:2 (2009), 1144–1145  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    14. Maslov V., “Dequantization, statistical mechanics and econophysics”, Tropical and idempotent mathematics, Contemp. Math., 495, ed. Litvinov G. Sergeev S., Amer. Math. Soc., Providence, RI, 2009, 239–279  crossref  mathscinet  zmath  isi
    15. V. P. Maslov, “A new approach to phase transitions, thermodynamics, and hydrodynamics”, Theoret. and Math. Phys., 165:3 (2010), 1699–1720  mathnet  crossref  crossref  isi
    16. Molotkov I.A., “Maslov distribution and formulas for the entropy”, Russ. J. Math. Phys., 17:4 (2010), 476–485  crossref  mathscinet  zmath  isi  elib  scopus
    17. Maslov V.P., “New global distributions in number theory and their applications”, J. Fixed Point Theory Appl., 8:1 (2010), 81–111  crossref  mathscinet  zmath  isi  scopus
    18. Molotkov I.A., “Maslov parastatistic distribution and entropy formula in dimensions less than three”, Dokl. Math., 81:3 (2010), 481–484  crossref  mathscinet  zmath  isi  elib  elib  scopus
    19. V. P. Maslov, “Phase transitions in real gases and ideal Bose gases”, Theoret. and Math. Phys., 167:2 (2011), 654–667  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    20. Maslov V.P., “Mathematical conception of “phenomenological” equilibrium thermodynamics”, Russ. J. Math. Phys., 18:4 (2011), 440–464  crossref  mathscinet  zmath  isi  elib  scopus
    21. Maslov V.P. Maslov A.V. Maslova T.V., “New paradigm in thermodynamics and its connection with economics and linguistics”, Russ. J. Math. Phys., 18:3 (2011), 329–337  crossref  mathscinet  zmath  isi  elib  scopus
    22. Maslov V.P., “Number-theoretic internal energy for a gas mixture”, Russ. J. Math. Phys., 18:2 (2011), 163–175  crossref  mathscinet  zmath  isi  elib  scopus
    23. Maslov V.P., “Mixture of new ideal gases and the solution of the Gibbs and Einstein paradoxes”, Russ. J. Math. Phys., 18:1 (2011), 83–101  crossref  mathscinet  zmath  isi  elib  scopus
    24. V. P. Maslov, T. V. Maslova, “Unbounded probability theory and its applications”, Theory Probab. Appl., 57:3 (2013), 444–467  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    25. Maslov V.P., “Ideal gas/liquid transition as a generalization of the problem of “partitio numerorum””, Russ. J. Math. Phys., 19:4 (2012), 484–498  crossref  mathscinet  zmath  isi  elib  scopus
    26. Maslov V.P., “New probability theory compatible with the new conception of modern thermodynamics. Economics and crisis of debts”, Russ. J. Math. Phys., 19:1 (2012), 63–100  crossref  mathscinet  zmath  isi  elib  scopus
    27. Maslov V.P., “Old Mathematical Errors in Statistical Physics”, Russ. J. Math. Phys., 20:2 (2013), 214–229  crossref  mathscinet  zmath  isi  elib  scopus
    28. 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  crossref  mathscinet  zmath  isi  elib  scopus
    29. Molotkov I.A., “The Union of Two Systems of Identical Maslov Elements With Different Dimensions”, Dokl. Math., 90:1 (2014), 518–520  crossref  mathscinet  zmath  isi  scopus
    30. V. P. Maslov, “UD-Statistics in the subcritical region”, Theoret. and Math. Phys., 182:2 (2015), 308–310  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    31. Maslov V.P., “Bubbles in the Negative Pressure Domain”, Russ. J. Math. Phys., 22:4 (2015), 483–490  crossref  mathscinet  zmath  isi  scopus
    32. D. S. Minenkov, V. E. Nazaikinskii, “Remark on the Inverse Abstract Prime Number Theorem”, Math. Notes, 100:4 (2016), 633–635  mathnet  crossref  crossref  mathscinet  isi  elib
    33. V. P. Maslov, S. Yu. Dobrokhotov, V. E. Nazaikinskii, “Volume and Entropy in Abstract Analytic Number Theory and Thermodynamics”, Math. Notes, 100:6 (2016), 828–834  mathnet  crossref  crossref  mathscinet  isi  elib
    34. D. S. Minenkov, V. E. Nazaikinskii, V. L. Chernyshev, “On the Asymptotics of the Element Counting Function in an Additive Arithmetic Semigroup with Exponential Counting Function of Prime Generators”, Funct. Anal. Appl., 50:4 (2016), 291–307  mathnet  crossref  crossref  mathscinet  isi  elib
    35. Maslov V.P., “Negative energy, debts, and disinformation from the viewpoint of analytic number theory”, Russ. J. Math. Phys., 23:3 (2016), 355–368  crossref  mathscinet  zmath  isi  elib  scopus
    36. Minenkov D.S., Nazaikinskii V.E., Chernyshev V.L., “On the Limit Shape of Elements of An Arithmetic Semigroup With An Exponentially Growing Counting Function of Basis Elements”, Dokl. Math., 95:3 (2017), 226–229  crossref  mathscinet  zmath  isi  scopus
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