Exact thermodynamic theory of ideal Bose atoms in a two-dimensional isotropic harmonic trap. (6th November 2017)

• Record Type: Journal Article
• Authors: Cheng, Ze
• Is Part Of: Journal of statistical mechanics. (2017:Nov.)
• Publication Date: 2017-11-06
• Subjects: 1; Statistical mechanics -- Periodicals; Mechanics -- Statistical methods -- Periodicals; 530.1305
• DOI: 10.1088/1742-5468/aa9341
• Languages: English
• ISSNs: 1742-5468

Abstract

Abstract: We have proposed an analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, two-dimensional, and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a series of q -digamma functions in mathematics. We employ the analytical solution to calculate the internal energy E, the entropy S, the Helmholtz free energy F, and the heat capacity at constant number C N of ideal Bose atoms in a two-dimensional isotropic harmonic trap. The first main finding in this paper is that the variation with temperature of the internal energy of a finite number of ideal Bose atoms has an inflection point, which occurs at the transition temperature T c . The second main finding is that the internal energy of a finite number of ideal Bose atoms has a classic limit E = 2 N k B T as T ≫ T c . The third main finding is that the variation with temperature of the heat capacity of a finite number of ideal Bose atoms has a maximal value, which occurs at T c . The fourth main finding is that the heat capacity of a finite number of ideal Bose atoms has a classic limit C N / k B = 2 N as T ≫ T c . The fifth main finding is that in the thermodynamic limit, at critical temperature T c the heat capacity at constant number shows a cusp singularity, which is analogous to the λ -transition of liquid helium. Consequently, the transition between normal and condensed states is a second-order phase transition.