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Journal of Physics A Mathematical and General

General one-particle fluctuations of the ideal Bose gas

N Angelescu, J.G. Brankov, André F. Verbeure

Published Jul 7, 1996• 10 citations
Publisher

Abstract

Fluctuations of general one-particle observables, such as density, momentum, angular momentum and position, etc, are studied and their distribution functions are explicitly obtained for all non-zero temperatures. The normalization exponent for the fluctuation operators is explicitly obtained for the three distinct phases of the ideal Bose gas: normal, critical and condensed.

DOI
10.1088/0305-4470/29/13/010

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Fluctuations of general one-particle observables, such as density, momentum, angular momentum and position, etc, are studied and their distribution functions are explicitly obtained for all non-zero temperatures. The normalization exponent for the fluctuation operators is explicitly obtained for the three distinct phases of the ideal Bose gas: normal, critical and condensed.
Abstract This chapter describes the mechanism of Bose–Einstein condensation in the simplest ideal Bose gas case. An explicit expression for the critical temperature is derived in the analytically soluble case of a gas confined in a box. Various thermodynamic quantities are calculated above and below the critical temperature. Emphasis is given to the isothermal compressibility which diverges below the critical temperature. The behaviour of the off-diagonal one-body density matrix is discussed. Results for the fluctuations of the single-particle occupation numbers are derived in both the canonical and grand canonical ensembles.
The field fluctuations are studied for the free and imperfect Bose gas. In the phase where there is condensation, these field fluctuations are abnormal, with critical exponents depending on the way the symmetry-breaking external field tends to zero in the thermodynamic limit. On the critical line there is also abnormal behavior, but the dependence on the way the field disappears, is slightly different. For densities below the critical density, the fluctuations are normal and independent of the way the external field goes to zero.
The distribution function ω 0 ( n 0 ) of the number n 0 of particles in the condensate of an ideal Bose gas confined by a trap is found. It is shown that at the temperature above the critical one ( T > T c ) this function has the usual form ω 0 ( n 0 ) =(1 — e μ ) e μ n o , where μ is the chemical potential in the temperature units. For T < T c , this distribution changes almost in a jump to a Gaussian distribution, which depends on the trap potential only parametrically. The centre of this function shifts to larger values of n 0 with decreasing temperature and its width tends to zero, which corresponds to the suppression of fluctuations.
This paper 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.