Statistical mechanics of ideal Bose atoms in a harmonic trap

Abstract

For ideal Bose atoms in an isotropic harmonic trap, we consider thermodynamic variables obtained from microcanonical, canonical, and grand canonical ensembles, each with certain variables specified and other variables fluctuating. For the first two of these ensembles, we derive recursion relations that link partition functions for different dimensions. We discuss fluctuations in general, and obtain expressions for variances of the atom number N, the chemical potential \ensuremath{\mu}, and the temperature T for small, Gaussian fluctuations in the grand canonical ensemble. Then from our recursion relations and others given elsewhere, we obtain probability distributions for N, for ground-state occupation number ${n}_{0}$, for \ensuremath{\mu}, and for T. Below the critical temperature, the shape of the distributions for N, ${n}_{0},$ and \ensuremath{\mu} are definitely not Gaussian for the grand canonical ensemble. For given temperature and small N, we find that the chemical potential values pertaining to the three ensembles differ. We compare the specific-heat function ${C}_{N}(T)$ for the three ensembles and propose to use the minimum of ${\mathrm{dC}}_{N}/dT$ to define the critical temperature to facilitate comparisons with similar configurations of interacting atoms.