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arXiv:math-ph/0405043 · Journal of Mathematical Physics

The canonical perfect Bose gas in Casimir boxes

Joseph V. Pulé, Valentin Anatol'evich Zagrebnov

Published Aug 6, 2004• 28 citations
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Abstract

We study the problem of Bose–Einstein condensation in the perfect Bose gas in the canonical ensemble, in anisotropically dilated rectangular parallelepipeds (Casimir boxes). We prove that in the canonical ensemble for these anisotropic boxes there is the same type of generalized Bose–Einstein condensation as in the grand-canonical ensemble for the equivalent geometry. However the amount of condensate in the individual states is different in some cases and so are the fluctuations.

DOI
10.1063/1.1777402

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We study the problem of Bose–Einstein condensation in the perfect Bose gas in the canonical ensemble, in anisotropically dilated rectangular parallelepipeds (Casimir boxes). We prove that in the canonical ensemble for these anisotropic boxes there is the same type of generalized Bose–Einstein condensation as in the grand-canonical ensemble for the equivalent geometry. However the amount of condensate in the individual states is different in some cases and so are the fluctuations.
Abstract The theory of quantum-mechanical grand canonical ensembles is used to derive for the case of a perfect Bose-Einstein gas the average number of particles in the different energy levels, the fluctuations in these numbers and the equation of state. The Einstein condensation phenomenon is then discussed, and it is shown that in a p-v diagram (v being the specific volume) the isotherm consists of two analytically different parts in the limit where the number of particles in the system, JV, goes to infinity. It is also shown that for finite N at the critical volume ∂np/∂vn is of the order N1/3(n-2) in accordance with a result obtained by Wergeland & Hove-Storhoug.
Grand-canonical fluctuations of Bose-Einstein condensates of light are accessible to state-of-the-art experiments [J. Schmitt et al., Phys. Rev. Lett. 112, 030401 (2014).PRLTAO0031-900710.1103/PhysRevLett.112.030401]. We phenomenologically describe these fluctuations by using the grand-canonical ensemble for a weakly interacting Bose gas at thermal equilibrium. For a two-dimensional harmonic trap, we use two models for which the canonical partition functions of the weakly interacting Bose gas are given by exact recurrence relations. We find that the grand-canonical condensate fluctuations for weakly interacting Bose gases vanish at zero temperature, thus behaving qualitatively similarly to an ideal gas in the canonical ensemble (or microcanonical ensemble) rather than the grand-canonical ensemble. For low but finite temperatures, the fluctuations remain considerably higher than for the canonical ensemble, as predicted by the ideal gas in the grand-canonical ensemble, thus clearly showing that we are not in a regime in which the ensembles are equivalent.
Condensation of the ideal Bose gas in a closed volume having the shape of a rectangular parallel-epiped of length L with a square base of side length l ( L ≫ l ) is theoretically studied within the framework of the Bose-Einstein statistics (grand canonical ensemble) and within the statistics of a canonical ensemble of bosons. Under the condition N ( l/L ) 4 ≪ l, where N is the total number of gas particles, dependence of the average number of particles in the condensate on the temperature T in both statistics is expressed as a function of the ratio t = T/T 1 , where T 1 is a certain characteristic temperature depending only on the longitudinal size L . Therefore, the condensation process exhibits a one-dimensional (1D) character. In the 1D regime, the average numbers of particles in condensates of the grand canonical and canonical ensembles coincide only in the limiting cases of t → 0 and t → ∞. The distribution function of the number of particles in the condensate of a canonical ensemble of bosons at t ≤1 has a resonance shape and qualitatively differs from the Bose-Einstein distribution. The former distribution begins to change in the region of t ∼ 1 and acquires the shape of the Bose-Einstein distribution for t ≫ 1. This transformation proceeds gradually that is, the 1D condensation process exhibits no features characteristic of the phase transition in a 3D system. For N ( l/L ) 4 ≫ 1, the process acquires a 3D character with respect to the average number of particles in the condensate, but the 1D character of the distribution function of the number of particles in the condensate of a canonical ensemble of bosons is retained at all N values.
We analyze the concept of generalized Bose–Einstein condensation (g-BEC), known since 1982 for the perfect Bose gas (PBG) in Casimir (or anisotropic) boxes. Our aim is to establish a relation between this phenomenon and two concepts: the concept of long cycles and the off-diagonal-long-range-order (ODLRO), which are usually considered as an adequate way to describe standard BEC on the ground state for cubic boxes. First, we show that these three criteria are equivalent in this latter case. Then, based on a scaling approach, we revise the formulation of these concepts to prove that the classification of g-BEC into three types I, II, III, implies a hierarchy of long cycles (depending on their size scale) as well as a hierarchy of ODLRO which depends on the coherence length of the condensate.
Abstract Particle fluctuations in systems, exhibiting Bose–Einstein condensation, are reviewed in order to clarify the basic points that attract high interest and often confront misunderstanding. It is explained that the so-called ‘grand canonical catastrophe’, claiming the occurrence of catastrophic particle fluctuations in the condensed phase, treated by grand canonical ensemble, does not exist. What exists is the incorrect use of the grand canonical ensemble, where gauge symmetry is not broken, while the correct description of the condensed phase necessarily requires gauge symmetry breaking. The ideal Bose gas has no catastrophic condensate fluctuations, and moreover there are no condensate fluctuations at all, as soon as gauge symmetry is broken. However it does have anomalous fluctuations of uncondensed particles, which implies its instability. For interacting particles, there are no condensate fluctuations, as soon as gauge symmetry is broken, and anomalous fluctuations of uncondensed particles, when correctly calculated, do not appear. Particle fluctuations in the systems of trapped atoms are discussed. Canonical ensemble and grand canonical ensemble with broken gauge symmetry are equivalent with respect to the number of particle scaling.