Bose-Einstein condensation without chemical potential

Large-volume, high-temperature Bose-Einstein condensation is illustrated for a relativistic O(2)-invariant scalar field with fixed charge using the canonical ensemble. The standard, grand canonical results are reproduced for the infinite-volume limit. Finite-volume corrections are calculated and the results are found to differ from the grand canonical approximation in a consistent qualitative way. PACS numbers: 03.75Hh, 11.10Wx. 1 Conserved charges When dealing with conserved charges in a system in equilibrium the usual procedure is to employ the grand canonical ensemble [1, 2, 3]. The physical picture from which the grand canonical ensemble is derived is that of a system in contact with an infinite reservoir of particles with an associated chemical potential for flow of particles into and out-of the system. Averaging in the grand canonical ensemble is performed over all possible charge states. The average charge is then fixed by a definite choice of chemical potential. By contrast, in the canonical ensemble, charge cannot move into and outof the system. There is no chemical potential. The charge is fixed and any averaging must be performed only over states with this definite fixed charge. This provides a more reasonable model for a system which is insulated with respect to charge and a better approximation for experiments involving BoseEinstein condensation (BEC) of trapped atoms [4]. In this article, BEC is outlined using the canonical ensemble. After mathematically formulating the model in the remainder of this section we go on to consider the high-temperature, large-volume limit in section 2 (c.f. Refs.[1, 2, 3]). We calculate the proportion of the total charge that occupies the lowest energy state in section 3, and as expected, the canonical and grand canonical ensembles are found to give the same results in the infinite-volume limit. In section 4 we consider finite-volume corrections where a difference between the two ensembles is observed. email: [email protected]