### abstract

- We review the phenomenon of equilibrium fluctuations in the number of condensed atoms n0 in a trap containing N atoms total. We start with a history of the Bose-Einstein distribution, a similar grand canonical problem with an indefinite total number of particles, the Einstein-Uhlenbeck debate concerning the rounding of the mean number of condensed atoms over(n, )0 near a critical temperature Tc, and a discussion of the relations between statistics of BEC fluctuations in the grand canonical, canonical, and microcanonical ensembles. First, we study BEC fluctuations in the ideal Bose gas in a trap and explain why the grand canonical description goes very wrong for all moments (n0 - over(n, )0)m , except of the mean value. We discuss different approaches capable of providing approximate analytical results and physical insight into this very complicated problem. In particular, we describe at length the master equation and canonical-ensemble quasiparticle approaches which give the most accurate and physically transparent picture of the BEC fluctuations. The master equation approach, that perfectly describes even the mesoscopic effects due to the finite number N of the atoms in the trap, is quite similar to the quantum theory of the laser. That is, we calculate a steady-state probability distribution of the number of condensed atoms pn0 (t = ) from a dynamical master equation and thus get the moments of fluctuations. We present analytical formulas for the moments of the ground-state occupation fluctuations in the ideal Bose gas in the harmonic trap and arbitrary power-law traps. In the last part of the review, we include particle interaction via a generalized Bogoliubov formalism and describe condensate fluctuations in the interacting Bose gas. In particular, we show that the canonical-ensemble quasiparticle approach works very well for the interacting gases and find analytical formulas for the characteristic function and all cumulants, i.e., all moments, of the condensate fluctuations. The surprising conclusion is that in most cases the ground-state occupation fluctuations are anomalously large and are not Gaussian even in the thermodynamic limit. We also resolve the Giorgini, Pitaevskii and Stringari (GPS) vs. Idziaszek et al. debate on the variance of the condensate fluctuations in the interacting gas in the thermodynamic limit in favor of GPS. Furthermore, we clarify a crossover between the ideal-gas and weakly-interacting-gas statistics which is governed by a pair-correlation, squeezing mechanism and show how, with an increase of the interaction strength, the fluctuations can now be understood as being essentially 1/2 that of an ideal Bose gas. We also explain the crucial fact that the condensate fluctuations are governed by a singular contribution of the lowest energy quasiparticles. This is a sort of infrared anomaly which is universal for constrained systems below the critical temperature of a second-order phase transition. 2006 Elsevier Inc. All rights reserved.