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Quantum Electronics

Statistics of an ideal homogeneous Bose gas with a fixed number of particles

В. А. Алексеев

Published May 31, 2001• 2 citations
Publisher

Abstract

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.

DOI
10.1070/qe2001v031n05abeh001972

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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.
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 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.
Condensation of the ideal Bose gas in a closed volume having the shape of a rectangular parallel-epiped of length L with a square base of side length l ( L ≫ l ) is theoretically studied within the framework of the Bose-Einstein statistics (grand canonical ensemble) and within the statistics of a canonical ensemble of bosons. Under the condition N ( l/L ) 4 ≪ l, where N is the total number of gas particles, dependence of the average number of particles in the condensate on the temperature T in both statistics is expressed as a function of the ratio t = T/T 1 , where T 1 is a certain characteristic temperature depending only on the longitudinal size L . Therefore, the condensation process exhibits a one-dimensional (1D) character. In the 1D regime, the average numbers of particles in condensates of the grand canonical and canonical ensembles coincide only in the limiting cases of t → 0 and t → ∞. The distribution function of the number of particles in the condensate of a canonical ensemble of bosons at t ≤1 has a resonance shape and qualitatively differs from the Bose-Einstein distribution. The former distribution begins to change in the region of t ∼ 1 and acquires the shape of the Bose-Einstein distribution for t ≫ 1. This transformation proceeds gradually that is, the 1D condensation process exhibits no features characteristic of the phase transition in a 3D system. For N ( l/L ) 4 ≫ 1, the process acquires a 3D character with respect to the average number of particles in the condensate, but the 1D character of the distribution function of the number of particles in the condensate of a canonical ensemble of bosons is retained at all N values.
Based on the canonical ensemble, we suggested the simple scheme for taking into account Gaussian fluctuations in a finite system of ideal boson gas. Within framework of scheme we investigated the influence of fluctuations on the particle distribution in Bose -gas for two cases - with taking into account the number of particles in the ground state and without this assumption. The temperature and fluctuation parameter dependences of the modified Bose-Einstein distribution have been determined. Also the dependence of the condensation temperature on the fluctuation distribution parameter has been obtained.
By expanding the grand partition function, we obtain the expression for partition function of two dimensional free particles which is seen to be of the same form as that of one dimensional harmonic oscillators. For such systems it is shown that the difference in statistics does not appear in thermal behavior. When the energy levels of the particles are modified as ε n 1 , n 2 = Δ + b ( n 1 2 + n 2 2 ), Δ >0, the Bose condensation occurs and, if we set Δ =2 b , the condensation temperature qualitatively agrees with that of Osborne. The speciffic heat of free Bosons in a narrow box is calculated as the function of temperature, its width d being taken 7.1 Å and 14.2 Å. The results show that the maximum of the specific heat occurs at higher temperatures than that of the case d =∞, and its height becomes lower. The weak excitation of a Fermi system can be treated as excitons which obey Bose statistics. And we can show, by the method analogous to that of Ward and Wilks and Dingle, that the sound wave in a Fermi gas should be considered as the second sound, or the propagation of fluctuation in exciton distribution. Its propagation velocity is shown to become p0/3mp_{0}/\sqrt{3}mp0​/3​m, where p 0 is the momentum at the Fermi surface.
When the chemical potential μ limits to zero, the steady increase of particles in the ground state with e=0 under Bose-Einstein gas in equilbirium distribution. This is called Bose-Einstein condensation. Though Monte Carlo simulation in quantum ensenble, when the number of particles in the system in small we find: First, the fluctuation of the number of particles in the ground state is very quick and is analogous to energy level distribution. Second, when TT c , the system appears Bose-Einstein condensation, but whether T c is congruous with the theoretical value should be further verified.