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arXiv:2011.04625

On the theory of ideal Bose-gas at a finite particle number

A. I. Bugrij, В. М. Локтев

Published Nov 9, 2020• 0 citations
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Abstract

The ideal Bose-gas with finite number $N$ of particles is investigated. The exact expressions for the partition functions and occupation numbers in the grand canonical, canonical and microcanonical ensembles are found. The asymp\-totic expressions (in the case $N\gg1$) for the partition functions and occupation numbers in the canonical and microcanonical ensembles are evaluated. It is shown that the chemical potential $μ$ of the ideal Bose-gas can lie in the range $-\infty

DOI
10.48550/arxiv.2011.04625

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The ideal Bose-gas with finite number NNN of particles is investigated. The exact expressions for the partition functions and occupation numbers in the grand canonical, canonical and microcanonical ensembles are found. The asymp-totic expressions (in the case N≫1N\gg1N≫1) for the partition functions and occupation numbers in the canonical and microcanonical ensembles are evaluated. It is shown that the chemical potential μμμ of the ideal Bose-gas can lie in the range $-\infty
The paper is devoted to the theory of an ideal Bose-gas with a finite number N of particles. The exact expressions for the partition functions and occupation numbers of the model in the grand canonical, canonical, and microcanonical ensembles are found. From the calculations, it is followed that, oppositely to the accepted opinion that the chemical potential μ of an ideal Bose-gas is only negative, it can take values in the range −∞ < μ < ∞. The asymptotic expressions (in the case N ≫ 1) for the partition functions and occupation numbers for all above-mentioned thermodynamic ensembles are also evaluated.
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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.
Employing asymptotic formulas from the partition theory of numbers, we derive the microcanonical probability distribution of the ground state occupation number for a one-dimensional ideal Bose gas confined at low temperatures by a harmonic potential. We compare the grand canonical analysis to the microcanonical one, and show how the fluctuation catastrophe characteristic for the grand canonical ensemble is avoided by the proper microcanonical approach. \textcopyright{} 1996 The American Physical Society.
A new method is proposed for a treatment of ideal quantum gases in the grand microcanonical ensemble near the thermodynamic limit. The method allows rigorous asymptotic calculations of the average number of particles and particle number fluctuations in the system with exactly fixed total energy. It also gives the finite-volume corrections for the total average number of particles and for higher moments of the particle number distribution in a grand microcanonical system approaching the thermodynamic limit. The present consideration confirms our previous findings that the scaled variance for particle number fluctuations in the grand microcanonical ensemble is different from that in the grand canonical ensemble even in the thermodynamic limit.