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Quantum Electronics

The distribution function and fluctuations of the number of particles in an ideal Bose gas confined by a trap

В. А. Алексеев

Published Jan 31, 2001• 5 citations
Publisher

Abstract

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.

DOI
10.1070/qe2001v031n01abeh001884

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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.
The distribution function W0(n0) of the number n0 of particles in the condensate of an ideal Bose gas confined by a trap is found. It is shown that at the temperature below the critical temperature Tc this function has a Gaussian shape and depends on the trap potential via two parameters only. The center of this function shifts to larger values of n0 with decreasing temperature and its width tends to zero, which corresponds to the suppression of fluctuations. In the narrow vicinity of the critical temperature ∣T−Tc∣⩽Tc/N\left| {T - {T_c}} \right| \leqslant {T_c}/\sqrt N ∣T−Tc​∣⩽Tc​/N​ , where N is the number of particles in the trap, the distribution function changes and at the temperature above the critical one it takes the usual form W0(n0) = [1 − exp(μ)]exp(μn0), where μ is the chemical potential in temperature units. In the limit N→∞, this change occurs at a jump.
The distribution function w 0 ( n 0 ) of the number n 0 of particles is found for the condensate of an ideal gas of free bosons with a fixed total number N of particles. It is shown that above the critical temperature ( T > T c ) this function has the usual form w 0 ( n 0 ) = (1 — e μ )e μ n 0 , where μ is the chemical potential in temperature units. In a narrow vicinity of the critical temperature | T/T c — 1| ≤ N -1/3 , this distribution changes and at T < T c acquires the form of a resonance. The width of the resonance depends on the shape of the volume occupied by the gas and it has exponential (but not the Gaussian) wings. As the temperature is lowered, the resonance maximum shifts to larger values of n 0 and its width tends to zero, which corresponds to the suppression of fluctuations. For N → ∞, this change occurs abruptly. The distribution function of the number of particles in excited states for the systems with a fixed and a variable number of particles (when only a mean number of particles is fixed) prove to be identical and have the usual form.
We analyse occupation number fluctuations of an ideal Bose gas in a trap which is isolated from the environment with respect to particle exchange (canonical ensemble). We show that in contrast to the predictions of the grandcanonical ensemble, the counting statistics of particles in the trap ground state changes from monotonously decreasing above the condensation temperature to single-peaked below that temperature. For the exactly solvable case of a harmonic oscillator trapping potential in one spatial dimension we extract a Landau–Ginzburg functional which–despite the non-interacting nature of the system–displays the characteristic behaviour of a weakly interacting Bose gas. We also compare our findings with the usual treatment which is based on the grand-canonical ensemble. We show that for an ideal Bose gas neither the grand-canonical and canonical ensemble thermodynamically equivalent, nor the grand-canonical ensemble can be viewed as a small system in diffusive contact with a particle reservoir.
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.
We have investigated the thermodynamic behaviour of ideal Bose gases with an arbitrary number of particles confined in a harmonic potential. By taking into account the conservation of the total number N of particles and using a saddle-point approximation, we derive analytically the simple explicit expression of mean occupation number in any state of the finite system. The temperature dependence of the chemical potential, specific heat, and condensate fraction for the trapped finite-size Bose system is obtained numerically. We compare our results with the usual treatment which is based on the grand canonical ensemble. It is shown that there exists a considerable difference between them at sufficiently low temperatures, especially for the relative small numbers of Bose atoms. The finite-size scaling at the transition temperature for the harmonically trapped systems is also discussed. We find that the scaled condensate fractions for various system sizes and temperatures collapse onto a single scaled form.
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.