From number theory to statistical mechanics: Bose-Einstein condensation inisolated traps

We question the validity of the grand canonical ensemble for the descriptio n of Bose-Einstein condensation of small ideal Bose gas samples in isolated harmonic traps. While the ground state fraction and the specific heat capa city can be well approximated with the help of the conventional grand canon ical arguments, the calculation of the fluctuation of the number of particl es contained in the condensate requires a microcanonical approach. Resortin g to the theory of restricted partitions of integer numbers, we present ana lytical and numerical results for such fluctuations in one- and three-dimen sional traps, and show that their magnitude is essentially independent of t he total particle number. (C) 1999 Elsevier Science Ltd. All rights reserve d.