Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Experimental and Theoretical Physics

Distribution function of the number of particles in the condensate of an ideal Bose gas confined by a trap

В. А. Алексеев

Published Apr 1, 2001• 5 citations
Publisher

Abstract

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 $$\left| {T - {T_c}} \right| \leqslant {T_c}/\sqrt N $$ , 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.

DOI
10.1134/1.1371342

Related papers

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

В. А. Алексеев

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.

Quantum Electronics•Jan 31, 2001•5 citations

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

В. А. Алексеев

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.

Quantum Electronics•May 31, 2001•2 citations

Normal and generalized Bose condensation in traps: One dimensional examples

András Sütö

We prove the following results. (i) One-dimensional Bose gases which interact via unscaled integrable pair interactions and are confined in an external potential increasing faster than quadratically undergo a complete generalized Bose-Einstein condensation (BEC) at any temperature, in the sense that a macroscopic number of particles are distributed on a o(N) number of one-particle states. (ii) In a one dimensional harmonic trap the replacement of the oscillator frequency ω by ω ln N/N gives rise to a phase transition at a ≡ �ωβ = 1 in the noninteracting gas. For a < 1 the limit distribution of n0/N a is exponential and 〈n0〉/N a → 1. For a> 1 there is BEC with a condensate density 〈n0〉/N → 1 − a −1. For a ≥ 1, (lnN/N)(n0 − 〈n0〉) is asymptotically distributed following Gumbel’s law. For any a> 0 the free energy is −(π 2 /6βa)N / lnN + o(N/ln N), with no singularity at a = 1. (iii) In Model (ii) both above and below the critical temperature the gas undergoes a complete generalized BEC, thus providing a coexistence of ordinary and generalized condensates below the critical point. (iv) Adding an interaction 〈UN 〉 = o(N ln N) to Model (ii) we prove that a complete generalized BEC occurs for any β> 0.

•Jan 1, 2003•2 citations

Thermodynamic properties of a finite Bose gas in a harmonic trap

Wang Jian-Hui, Ma Yong-Li

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.

Chinese Physics B•May 1, 2010•5 citations

Probability Distributions and Particle Number Fluctuations of Trapped Bose-Einstein Condensates in Different Dimensions

Hongwei Xiong, Shujuan Liu, Guoxiang Huang +2 more

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.

arXiv:cond-mat/0111186•Nov 10, 2001•0 citations

Canonical statistics of trapped ideal and interacting Bose gases

Hongwei Xiong, Shujuan Liu, Guoxiang Huang +1 more

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 ${N}^{4/3}.$

arXiv:cond-mat/0111125•Physical Review A•Feb 8, 2002•57 citations
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.
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 prove the following results. (i) One-dimensional Bose gases which interact via unscaled integrable pair interactions and are confined in an external potential increasing faster than quadratically undergo a complete generalized Bose-Einstein condensation (BEC) at any temperature, in the sense that a macroscopic number of particles are distributed on a o(N) number of one-particle states. (ii) In a one dimensional harmonic trap the replacement of the oscillator frequency ω by ω ln N/N gives rise to a phase transition at a ≡ �ωβ = 1 in the noninteracting gas. For a < 1 the limit distribution of n0/N a is exponential and 〈n0〉/N a → 1. For a> 1 there is BEC with a condensate density 〈n0〉/N → 1 − a −1. For a ≥ 1, (lnN/N)(n0 − 〈n0〉) is asymptotically distributed following Gumbel’s law. For any a> 0 the free energy is −(π 2 /6βa)N / lnN + o(N/ln N), with no singularity at a = 1. (iii) In Model (ii) both above and below the critical temperature the gas undergoes a complete generalized BEC, thus providing a coexistence of ordinary and generalized condensates below the critical point. (iv) Adding an interaction 〈UN 〉 = o(N ln N) to Model (ii) we prove that a complete generalized BEC occurs for any β> 0.
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 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.
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.