Bose–Einstein condensation temperature of finite systems. (10th May 2018)

• Record Type: Journal Article
• Authors: Xie, Mi
• Is Part Of: Journal of statistical mechanics. (2018:May)
• Publication Date: 2018-05-10
• Subjects: 3 -- 1; Statistical mechanics -- Periodicals; Mechanics -- Statistical methods -- Periodicals; 530.1305
• DOI: 10.1088/1742-5468/aabbbd
• Languages: English
• ISSNs: 1742-5468

Abstract

Abstract: In studies of the Bose–Einstein condensation of ideal gases in finite systems, the divergence problem usually arises in the equation of state. In this paper, we present a technique based on the heat kernel expansion and zeta function regularization to solve the divergence problem, and obtain the analytical expression of the Bose–Einstein condensation temperature for general finite systems. The result is represented by the heat kernel coefficients, where the asymptotic energy spectrum of the system is used. Besides the general case, for systems with exact spectra, e.g. ideal gases in an infinite slab or in a three-sphere, the sums of the spectra can be obtained exactly and the calculation of corrections to the critical temperatures is more direct. For a system confined in a bounded potential, the form of the heat kernel is different from the usual heat kernel expansion. We show that as long as the asymptotic form of the global heat kernel can be found, our method works. For Bose gases confined in three- and two-dimensional isotropic harmonic potentials, we obtain the higher-order corrections to the usual results of the critical temperatures. Our method can also be applied to the problem of generalized condensation, and we give the correction of the boundary on the second critical temperature in a highly anisotropic slab.