Ensemble-dependent condensate statistics and correlations in an ideal Bose gas
The choice of statistical ensemble has a measurable impact on the statistical description of Bose-Einstein condensates beyond the well-known grand canonical anomaly. We develop a segmented-contour approach for the ideal Bose gas, which refines the saddle-point approximation by including non-Gaussian corrections in the canonical and microcanonical ensembles. The method applies to a broad class of noninteracting spectra, and provides efficient access to condensate fluctuations, higher moments, and correlation functions. For a three-dimensional harmonic trap with fixed anisotropy, we find that the ratio of the maxima of the microcanonical and canonical condensate-number variances approaches the theoretically predicted asymptotic value $1-3ζ(3)^2/[4ζ(2)ζ(4)]\approx 0.391$ only extremely slowly with increasing particle number. We also show that ensemble effects are visible in the second-order correlation function $g^{(2)}(x)$ near the condensation crossover. After ballistic release from a three-dimensional harmonic trap, the time evolution of $g^{(2)}(x)$ preserves the differences between statistical ensembles.