Exact thermodynamic theory of an ideal boson gas in a one-dimensional harmonic trap

Abstract

.We employ the q-digamma function to calculate the internal energy E, the entropy S, the Helmholtz free energy F, and the heat capacity at constant number of ideal Bose atoms in a one-dimensional harmonic trap. The first main finding in this paper is that in a one-dimensional harmonic trap and when the temperature approaches zero, the entropy of a finite number of ideal Bose atoms is nonzero but the entropy of an infinite number of ideal Bose atoms is zero. The second main finding is that the heat capacity at constant number has a classic limit as . In the thermodynamic limit, we find that at critical temperature the heat capacity at constant number is discontinuous. Hence, at there is a jump in the heat capacity and this jump is calculated. Consequently, the transition between normal and condensed states is a second-order phase transition.