Canonical Bose-Einstein condensation in a parabolic well

Abstract

The Bose-Einstein condensation is investigated for N particles in a parabolic confining potential. The partition function in the grand-canonical as well as in the canonical ensemble is obtained analytically, and the thermodynamical quantities are studied for the one-dimensional (1D) and for the three-dimensional (3D) version of the model. In 1D, a dramatic rise in the ground state occupancy with decreasing temperature is predicted. Both the grand-canonical and the canonical ensemble yield the same behavior of the specific heat as a function of the temperature and the number of particles. There is no indication of a critical phenomenon for a finite number of particles, neither in the ground state occupancy nor in the specific heat. Also in 3D, both ensembles give an identical contribution for the specific heat, but showing thereby a phase transition with a critical temperature which scales with the cube root of the number of particles, as predicted before by semi-classical continuum models.