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