Can We Distinguish Between the Grand Canonical and the Canonical Ensemble in a BEC Experiment?

Abstract

It is well known that at the thermodynamic limit there are no observable differences in the results obtained by grand canonical and canonical descriptions of a many-body system. In the present paper, we test the validity of this statement for finite systems using as an example an ensemble of bosons trapped in a 1D 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 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 3D "cigar-shape" trap in the range of parameters corresponding to current BEC experiments.