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arXiv:cond-mat/0111186

Probability Distributions and Particle Number Fluctuations of Trapped Bose-Einstein Condensates in Different Dimensions

Hongwei Xiong, Shujuan Liu, Guoxiang Huang, Fengmin Wu, Zhijun Xu

Published Nov 10, 2001• 0 citations
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Abstract

The analytical probability distribution of finite systems obeying Bose-Einstein statistics in one, two, and three dimensions are investigated by using a canonical ensemble approach. Starting from the canonical partition function of the system, a unified approach is provided to study the probability distribution of a condensate for various confinements and in different dimensions. Based on the probability distribution function, it is straightforward to obtain the mean ground state occupation number and particle number fluctuations of the condensate. It is found that the particle number fluctuations in a trapped Bose gas are strongly dependent on the type of confining potential and on the dimensionality of the system.

DOI
10.48550/arxiv.cond-mat/0111186

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The mean ground-state occupation number and condensate fluctuations of interacting and noninteracting Bose gases confined in a harmonic trap are considered by using a canonical ensemble approach. To obtain the mean ground-state occupation number and the condensate fluctuations, an analytical description for the probability distribution function of the condensate is provided directly starting from the analysis of the partition function of the system. For the ideal Bose gas, the probability distribution function is found to be a Gaussian one for the case of the harmonic trap. For the interacting Bose gas, using a unified approach the condensate fluctuations are calculated based on the lowest-order perturbation method and on Bogoliubov theory. It is found that the condensate fluctuations based on the lowest-order perturbation theory follow the law $〈{\ensuremath{\delta}}^{2}{N}_{0}〉\ensuremath{\sim}N,$ while the fluctuations based on Bogoliubov theory behave as N4/3.{N}^{4/3}.N4/3.
We question the validity of the grand canonical ensemble for the description of Bose-Einstein condensation of small ideal Bose gas samples in isolated harmonic traps. While the ground state fraction and the specific heat capacity can be well approximated with the help of the conventional grand canonical arguments, the calculation of the fluctuation of the number of particles contained in the condensate requires a microcanonical approach. Resorting to the theory of restricted partitions of integer numbers, we present analytical and numerical results for such fluctuations in one- and three-dimensional traps, and show that their magnitude is essentially independent of the total particle number.
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Recursive approaches determining the canonical ideal Bose gas partition function are reviewed that enable the Bose–Einstein condensate occupation probability to be calculated for a finite number of particles ensemble, where the thermodynamic limit approximation fails. In addition to the earlier known method recursive with respect to the number of particles, an iteration procedure with respect to the number of quantum states is proposed. The efficiency of both methods is demonstrated for an ideal Bose gas in a three-dimensional isotropic harmonic trap.