Fluctuations of the Bose-Einstein condensate

Abstract

This article gives a rigorous analysis of the fluctuations of the Bose-Einstein condensate for a system of non-interacting bosons in an arbitrary potential, assuming that the system is governed by the canonical ensemble. As a result of the analysis, we are able to tell the order of fluctuations of the condensate fraction as well as its limiting distribution upon proper centering and scaling. This yields interesting results. For example, for a system of $n$ bosons in a 3D harmonic trap near the transition temperature, the order of fluctuations of the condensate fraction is $n^{-1/2}$ and the limiting distribution is normal, whereas for the 3D uniform Bose gas, the order of fluctuations is $n^{-1/3}$ and the limiting distribution is an explicit non-normal distribution. For a 2D harmonic trap, the order of fluctuations is $n^{-1/2}(\log n{)}^{1/2}$, which is larger than $n^{-1/2}$ but the limiting distribution is still normal. All of these results come as easy consequences of a general theorem.

26 pages. Minor changes in new version

References in corpus (1)

• Self-similar analytical solution of the critical fluctuations problem for the Bose-Einstein condensation in an ideal gas

Cited by in corpus (13)

• Randomization and the Gross-Pitaevskii hierarchy
• Bose-Einstein Condensation in a Dilute, Trapped Gas at Positive Temperature
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• Random partitions in statistical mechanics
• Upper bound for the grand canonical free energy of the Bose gas in the Gross-Pitaevskii limit
• Ensemble nonequivalence and Bose-Einstein condensation in weighted networks
• The shape of the emerging condensate in effective models of condensation
• Upper bound for the grand canonical free energy of the Bose gas in the Gross-Pitaevskii limit for general interaction potentials
• Limit shapes for Gibbs ensembles of partitions
• Law of large numbers unifying Maxwell-Boltzmann, Bose-Einstein and Zipf-Mandelbort distributions, and related fluctuations
• Limit shapes for Gibbs partitions of sets
• The role of the density of states in Bose-Einstein condensation
• Bose-Einstein statistics for a finite number of particles