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International Journal of Progressive Research in Science and Engineering

Green’s Function Theory on A Fluctuation of Condensed Particle Number in Bose-Einstein Condensed Systems

Sung-Gyu Pak, Gi-Song Kim, Tae-Hwi Ko, Ryong Kim, Ung-Chol Song, Hyon-Ui Pak

Published Mar 20, 2026• 0 citations
Publisher

Abstract

By using the Green’s function theory, we studied the fluctuation of condensed particle number for Bose-Einstein condensed systems in the canonical ensemble. The expressions for the fluctuations can be expressed by the Green’s functions. Our results are consistent with previous theories in Bogoliubov approximation. In addition, a single-particle and a pair condensation equally contribute to the mean-square fluctuation of the condensates.

DOI
10.65138/ijprse.2026.v7i03.1250

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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 discuss the exact particle number counting statistics of degenerate ideal Bose gases in the microcanonical, canonical, and grand-canonical ensemble, respectively, for various trapping potentials. We then invoke the Maxwell's Demon ensemble [Navez et el., Phys. Rev. Lett. (1997)] and show that for large total number of particles the root-mean-square fluctuation of the condensate occupation scales n0 / [T=Tc] r N s with scaling exponents r = 3=2, s = 1=2 for the3D harmonic oscillator trapping potential, and r = 1,s= 2=3 for the 3D box. We derive an explicit expression for r and s in terms of spatial dimension D and spectral index sigma of the single-particle energy spectrum. Our predictions also apply to systems where Bose-Einstein condensation does not occur. We point out that the condensate fluctuations in the microcanonical and canonical ensemble respect the principle of thermodynamic equivalence.