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Fluctuations of the Bose-Einstein condensate

Sourav Chatterjee, Persi W. Diaconis

Published Jan 1, 2014• 29 citations
OpenAlex

Abstract

Abstract. This article 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 conden-sate 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(logn)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. 1.

Related papers

Fluctuations of the Bose–Einstein condensate

Sourav Chatterjee, Persi W. Diaconis

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.

arXiv:1306.3625•Journal of Physics A Mathematical and Theoretical•Feb 10, 2014•25 citations

Condensation ofNbosons. III. Analytical results for all higher moments of condensate fluctuations in interacting and ideal dilute Bose gases via the canonical ensemble quasiparticle formulation

Vitaly V. Kocharovsky, Vl. V. Kocharovsky, Marlan O. Scully

We consider equilibrium canonical-ensemble fluctuations of the Bose-Einstein condensate (BEC) in a trap using the particle-number conserving operator formalism of Girardeau and Arnowitt [Phys. Rev. 113, 755 (1959)] and the concept of the canonical-ensemble quasiparticles. We find explicit expressions for the characteristic function and all cumulants of the ground-state occupation statistics both for the dilute weakly interacting and ideal Bose gases. We present analytical formulas for the moments of the ground-state occupation fluctuations in the weakly interacting and ideal Bose gases in a box (``homogeneous gas'') as well as in the ideal Bose gas in an arbitrary power-law trap, in particular, in an arbitrary harmonic trap. We calculate the effect of Bogoliubov coupling between quasiparticles on suppression of the ground-state occupation fluctuations at moderate temperatures and their enhancement at very low temperatures. We clarify a crossover between ideal-gas and weakly interacting gas statistics, which is governed by a pair-correlation, squeezing mechanism. The important conclusion is that in most cases the ground-state occupation fluctuations are anomalously large and are not Gaussian even in the thermodynamic limit.

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

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BEC in Nonextensive Statistical Mechanics: Some Additional Results

Luca Salasnich

In a recent paper [Int. J. Mod. Phys. B 14, 405 (2000)] we discussed the Bose-Einstein condensation (BEC) in the framework of Tsallis’s nonextensive statistical mechanics. In particular, we studied an ideal gas of bosons in a confining harmonic potential. In this memoir we generalize our previous analysis by investigating an ideal Bose gas in a generic power-law external potential. We derive analytical formulas for the energy of the system, the BEC transition temperature and the condensed fraction. PACS numbers: 05.30-d; 03.75.Fi 1 Recently we analyzed the consequences of weak nonextensivity on Bose-Einstein Condensation for an ideal Bose gas 1 by using Tsallis’s nonextensive statistical mechanics (NSM). 2 From the nonextensive Bose-Einstein distribution 3−5 we derived the BEC transition temperature, the condensed fraction and the energy per particle in three different cases: the homogeneous gas, the gas in a harmonic trap and the relativistic homogenous gas. In this short contribution we generalize our previous results by considering the ideal Bose gas in a generic power-law potential. All the calculations are performed by assuming a D-dimensional space. The NSM predicts that, for an ideal quantum gas of identical bosons in the grand canonical ensemble, the weak nonextensivity formula of the average number of particles with energy ǫ is given by 〈n(ǫ)〉q = 1

•Jan 1, 2001•7 citations

Condensation of N Interacting Bosons: A Hybrid Approach to Condensate Fluctuations

Anatoly A. Svidzinsky, Marlan O. Scully

We present a new method of calculating the distribution function and fluctuations for a Bose-Einstein condensate (BEC) of N interacting atoms. The present formulation combines our previous master equation and canonical ensemble quasiparticle techniques. It is applicable both for ideal and interacting Bogoliubov BEC and yields remarkable accuracy at all temperatures. For the interacting gas of 200 bosons in a box we plot the temperature dependence of the first four central moments of the condensate particle number and compare the results with the ideal gas. For the interacting mesoscopic BEC, as with the ideal gas, we find a smooth transition for the condensate particle number as we pass through the critical temperature.

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Bose–Einstein condensation in two-dimensional traps

Mi Xie

Abstract In two-dimensional traps, since the theoretical study of Bose–Einstein condensation will encounter the problem of divergence, the actual contribution of the divergent terms is often estimated in some indirect ways with the accuracy to the leading order. In this paper, by using an analytical continuation method to solve the divergence problem, we obtain the analytical expressions of critical temperature and condensate fraction for Bose gases in a 2D anisotropic box and harmonic trap, respectively. They are consistent with or better than previous studies. Then, we further consider the nonvanishing chemical potential, and obtain the expressions of chemical potential and more precise condensate fraction. These results agree with the numerical calculation well, especially for the case of harmonic traps. The comparison between the grand canonical and canonical ensembles shows that our calculation in the grand canonical ensemble is reliable.

arXiv:1905.00830•Journal of Statistical Mechanics Theory and Experiment•Apr 1, 2019•2 citations
Abstract. This article 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 conden-sate 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(logn)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. 1.
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.
We consider equilibrium canonical-ensemble fluctuations of the Bose-Einstein condensate (BEC) in a trap using the particle-number conserving operator formalism of Girardeau and Arnowitt [Phys. Rev. 113, 755 (1959)] and the concept of the canonical-ensemble quasiparticles. We find explicit expressions for the characteristic function and all cumulants of the ground-state occupation statistics both for the dilute weakly interacting and ideal Bose gases. We present analytical formulas for the moments of the ground-state occupation fluctuations in the weakly interacting and ideal Bose gases in a box (``homogeneous gas'') as well as in the ideal Bose gas in an arbitrary power-law trap, in particular, in an arbitrary harmonic trap. We calculate the effect of Bogoliubov coupling between quasiparticles on suppression of the ground-state occupation fluctuations at moderate temperatures and their enhancement at very low temperatures. We clarify a crossover between ideal-gas and weakly interacting gas statistics, which is governed by a pair-correlation, squeezing mechanism. The important conclusion is that in most cases the ground-state occupation fluctuations are anomalously large and are not Gaussian even in the thermodynamic limit.
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.
In a recent paper [Int. J. Mod. Phys. B 14, 405 (2000)] we discussed the Bose-Einstein condensation (BEC) in the framework of Tsallis’s nonextensive statistical mechanics. In particular, we studied an ideal gas of bosons in a confining harmonic potential. In this memoir we generalize our previous analysis by investigating an ideal Bose gas in a generic power-law external potential. We derive analytical formulas for the energy of the system, the BEC transition temperature and the condensed fraction. PACS numbers: 05.30-d; 03.75.Fi 1 Recently we analyzed the consequences of weak nonextensivity on Bose-Einstein Condensation for an ideal Bose gas 1 by using Tsallis’s nonextensive statistical mechanics (NSM). 2 From the nonextensive Bose-Einstein distribution 3−5 we derived the BEC transition temperature, the condensed fraction and the energy per particle in three different cases: the homogeneous gas, the gas in a harmonic trap and the relativistic homogenous gas. In this short contribution we generalize our previous results by considering the ideal Bose gas in a generic power-law potential. All the calculations are performed by assuming a D-dimensional space. The NSM predicts that, for an ideal quantum gas of identical bosons in the grand canonical ensemble, the weak nonextensivity formula of the average number of particles with energy ǫ is given by 〈n(ǫ)〉q = 1
We present a new method of calculating the distribution function and fluctuations for a Bose-Einstein condensate (BEC) of N interacting atoms. The present formulation combines our previous master equation and canonical ensemble quasiparticle techniques. It is applicable both for ideal and interacting Bogoliubov BEC and yields remarkable accuracy at all temperatures. For the interacting gas of 200 bosons in a box we plot the temperature dependence of the first four central moments of the condensate particle number and compare the results with the ideal gas. For the interacting mesoscopic BEC, as with the ideal gas, we find a smooth transition for the condensate particle number as we pass through the critical temperature.
Abstract In two-dimensional traps, since the theoretical study of Bose–Einstein condensation will encounter the problem of divergence, the actual contribution of the divergent terms is often estimated in some indirect ways with the accuracy to the leading order. In this paper, by using an analytical continuation method to solve the divergence problem, we obtain the analytical expressions of critical temperature and condensate fraction for Bose gases in a 2D anisotropic box and harmonic trap, respectively. They are consistent with or better than previous studies. Then, we further consider the nonvanishing chemical potential, and obtain the expressions of chemical potential and more precise condensate fraction. These results agree with the numerical calculation well, especially for the case of harmonic traps. The comparison between the grand canonical and canonical ensembles shows that our calculation in the grand canonical ensemble is reliable.