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Journal of Statistical Physics

Grand Canonical Versus Canonical Ensemble: Universal Structure of Statistics and Thermodynamics in a Critical Region of Bose–Einstein Condensation of an Ideal Gas in Arbitrary Trap

Sergey V. Tarasov, Vl. V. Kocharovsky, Vitaly V. Kocharovsky

Published Sep 7, 2015• 13 citations
Publisher
DOI
10.1007/s10955-015-1361-3

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The analytical theory of Bose–Einstein condensation of an ideal gas in mesoscopic systems has been briefly reviewed in application to traps with arbitrary shapes and dimension. This theory describes the phases of the classical gas and the formed Bose–Einstein condensate, as well as the entire vicinity of the phase transition point. The statistics and thermodynamics of Bose–Einstein condensation have been studied in detail, including their self-similar structure in the critical region, transition to the thermodynamic limit, effect of boundary conditions on the properties of a system, and nonequivalence of the description of Bose–Einstein condensation in different statistical ensembles. The complete classification of universality classes of Bose–Einstein condensation has been given.
For an ideal Bose gas confined in a trap of arbitrary dimension and shape, the features of probabilistic distributions of the number of particles at the ground and all excited levels are studied within canonical and grand canonical ensembles. It is shown that in the critical range of system parameters the statistics is self-similar, and its characteristic scales (mathematical expectation, variance, etc.) are independent of an ensemble used in the description. At the same time, the selfsimilar distributions themselves differ for different ensembles.