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Limit distribution of the energy of a quantum ideal gas from the viewpoint of the theory of partitions of natural numbers. (English. Russian original) Zbl 0927.60089

Russ. Math. Surv. 52, No. 2, 379-386 (1997); translation from Usp. Mat. Nauk 52, No. 2, 139-146 (1997).
Many problems of mathematics and statistical physics have their natural counterparts in the theory of multiplicative statistics of partitions of natural numbers and vectors. Such an equivalent representation is quite useful; it reduces the study of the limiting behaviour of certain characteristics of physical and mathematical systems to the investigation of typical partitions with respect to the corresponding statistics, which often can be performed by using the formalism of generating functions.
To illustrate this point of view, the author studies the limiting behaviour of the energy distribution of a \(d\)-dimensional quantum ideal gas (with statistics of Bose-Einstein and Fermi-Dirac type). The corresponding multiplicative statistics on partitions can be viewed as distributions on Young diagrams, and the question about the typical structure of the former is naturally posed in terms of the limiting shape of the normalised Young diagrams. The main result of the paper asserts that under the appropriate scaling the microcanonical and the grand canonical measures converge, in the limit of large \(n\), to a degenerate distribution concentrated on a special curve. All possible limits form a one-parametric family \(\Gamma_\alpha\) with the parameter \(\alpha\) expressed through the region of analyticity of the Dirichlet series related to the statistics under consideration. As a direct consequence of the main theorem, one obtains the principle of equivalence of the microcanonical and grand canonical ensembles. It is interesting to notice that the logarithmic singularity of the limiting curve is closely related to the classical result by Erdős about the \(c\sqrt{n}\log n\) asymptotics of the number of components in a typical partition of a natural number \(n\).
In a restricted situation, when the number of components of partitions grows like \(N=vn^{\alpha/(1+\alpha)}\), the limiting curve \(\Gamma_{\alpha,v}\) can also be found. The limiting distribution, however, has an additional atom if \(\alpha>1\). Such a phenomenon is known in physics as the “Bose-Einstein condensation”. The limiting functions \(\Gamma_\alpha\) and \(\Gamma_{\alpha,v}\) contain the information about the leading term of the asymptotics of any functional of a typical partition.
The paper is dedicated to the memory of R. L. Dobrushin whose scientific interests were closely related to the discussed topic.

MSC:

60K40 Other physical applications of random processes
82B10 Quantum equilibrium statistical mechanics (general)
11P82 Analytic theory of partitions
11Z05 Miscellaneous applications of number theory
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