Large Deviations for the Bose–Einstein Condensate of the Ideal Gas in the Canonical Ensemble

This paper investigates the statistical behavior of the Bose-Einstein condensate (BEC) in an ideal, non-interacting Bose gas within the canonical ensemble. Specifically, the authors seek to determine the large-deviation behavior of the number of particles in the condensate, $N_0$. While previous work has characterized the central limit behavior of $N_0$, this study focuses on the exponential decay rates of the probability that $N_0$ deviates significantly from its mean as the total particle number $N$ tends to infinity.

To analyze the canonical ensemble, the authors utilize a mathematical framework where the system is defined by a one-particle Hamiltonian with a discrete spectrum. They employ large-deviation theory to quantify the asymmetry between the left-tail (probabilities of $N_0$ being significantly smaller than the mean) and the right-tail (probabilities of $N_0$ being significantly larger than the mean). The proof strategy involves relating the occupation numbers in the canonical ensemble to an independent family of geometric random variables, conditioned on the total particle number, and then applying asymptotic analysis to the resulting rate functions.

The study reveals that the left- and right-tail probabilities of the condensate particle number decay at different exponential rates. This asymmetry is a notable departure from the symmetric Gaussian fluctuations observed in some regimes. The authors provide explicit formulas for these decay rates, which depend on the underlying energy level structure of the Hamiltonian (characterized by the parameter $\alpha$). The results confirm that the canonical ensemble, despite not being exactly solvable, exhibits a well-defined large-deviation structure that can be rigorously determined.

Understanding the fluctuations of the BEC is crucial for interpreting experimental data from cold alkali gases, where precise measurements of condensate-number fluctuations have sometimes diverged from theoretical predictions. By providing a rigorous mathematical foundation for these large deviations, this paper helps clarify the statistical properties of quantum gases and offers a framework that could be tested in future experiments involving ultracold atoms.