Exact thermodynamic theory of ideal Bose atoms in a two-dimensional isotropic harmonic trap

Abstract

Listen

Translate article

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 CN 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 Tc. The second main finding is that the internal energy of a finite number of ideal Bose atoms has a classic limit as . 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 Tc. The fourth main finding is that the heat capacity of a finite number of ideal Bose atoms has a classic limit as . The fifth main finding is that in the thermodynamic limit, at critical temperature Tc 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.