Trapped Bose gas: The canonical versus grand canonical statistics

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Abstract
For ensemble of bosons trapped in a one-dimensional harmonic potential well we have found an analytical formula for the canonical partition function and shown that, for 100 trapped atoms, the discrepancy between the grand canonical and the canonical predictions for the condensate fraction reaches 10% in the vicinity of the Bose-Einstein threshold. This discrepancy decreases only logarithmically as the number of atoms increases. Furthermore, we investigate numerically the case of a three-dimensional ``cigar-shape'' trap in the range of parameters corresponding to current Bose-Einstein condensation experiments.