Non-Gaussian statistics of the Bose gas in canonical and microcanonical ensembles

statistics of a Bose gas has been studied for decades; however, the significance of the choice of statistical ensemble for its description is often underappreciated. We develop a method, which we call the segmented contour method, allowing for a precise description of statistics in the canonical and microcanonical ensembles. We first present results for the ideal Bose gas, demonstrating differences in condensate fluctuations and correlation functions between ensembles. In the case of a 3D harmonic trap, the asymptotic limit of the ratio of the maximal fluctuations in the microcanonical and canonical ensembles is reached very slowly: even for atom numbers of order 10¹⁰, it is only close to the limiting value of 0.39. The difference between the canonical and grand canonical ensembles is already apparent in the g₂(x) correlation function, whereas the microcanonical and canonical ensembles differ significantly in g₃(x), even in the thermodynamic limit. We also show results for the local second-order correlation function g₂(0) in the canonical ensemble in the Lieb–Liniger model in the thermodynamic limit, described by the Yang–Yang integral equations. At fixed temperature, the canonical result for g₂(0) approaches the grand-canonical one at larger interaction strengths, in the interaction regime where the chemical potential becomes positive.