Condensate fluctuations in trapped bose gases: Canonical vs. microcanonical ensemble

We study the fluctuation of the number of particles in ideal Bose-Einstein condensates, both within the canonical and the microcanonical ensemble. Emp loying the Mellin-Barnes transformation, we derive simple expressions that link the canonical number of condensate particles, its fluctuation, and the difference between canonical and microcanonical fluctuations to the poles of a Zeta function that is determined by the excited single-particle levels of the trapping potential. For the particular examples of one- and three-d imensional harmonic traps we explore the microcanonical statistics in detai l, with the help of the saddle-point method. Emphasizing the close connecti on between the partition theory of integer numbers and the statistical mech anics of ideal Bosons in one-dimensional harmonic traps, and utilizing ther modynamical arguments, we also derive an accurate formula for the fluctuati on of the number of summands that occur when a large integer is partitioned . (C) 1998 Academic Press.