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APS

Fourth Statistical Ensemble for the Bose-Einstein Condensate

Patrick Navez, Dmitri Bitouk, Mariusz Gajda, Zbigniew Idziaszek, Kazimierz Rza̧żewski, Christoph Weiß, Martin Wilkens

Published May 1, 1998• 0 citations
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We study the Bose-Einstein condensate (BEC) with a fixed number of particles. On the basis of conventional statistical mechanics we introduce the, so called, Maxwell's demon ensemble, where only particle transfer (without energy exchange) is allowed. We show that this new ensemble can be used for the microcanonical description of the system. We apply our formalism to the case of BEC in a harmonic trap and give the analytic expressions for the ground state fluctuations. We compare these expressions with the exact numerical results obtained for relatively small condensates.
We show various aspects of finite size effects on Bose-Einstein condensation(BEC). In the first section we introduce very briefly the BEC of harmonically trapped ideal Bose gas. In the second section we theoretically argued that Bose-Einstein(B-E) statistics needs a correction for finite system at ultralow temperatures. As a corrected statistics we introduced a Tsallis type of generalized B-E statistics. The condensate fraction calculated with this generalized B-E statistics, is satisfied well with the experimental result. In the third section we show how to apply the scaling theory in an inhomogeneous system like harmonically trapped Bose condensate at finite temperatures. We calculate the temperature dependence of the critical number of particles by a scaling theory within the Hartree-Fock approximation and find that there is a dramatic increase in the critical number of particles as the condensation point is approached. Our results support the experimental result which was obtained well below the condensation temperature. In the fourth section we concentrate on the thermodynamic Casimir force on the Bose-Einstein condensate. We explored the temperature dependence of the Casimir force.
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.