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arXiv:1409.8377

Study of Bose-Einstein condensation using generalized canonical partition function

R Sarath., P. C. Vinodkumar

Published Sep 30, 2014• 0 citations
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Abstract

We open a new discussion of generalized canonical partition function in standard statistical mechanics and apply it for the study of Bose-Einstein condensation. We discuss the possible cases for the generalized canonical partition function and arrives at a conclusion that the system of trapped bose gas will not be existing at absolute zero. We analyse the present study with an experimental result and point out the general difficulties in the analyses of experimental observations, which can possibly suppress the effect of generalized canonical partition function over standard canonical partition function. We mention that the experimental studies with ideal condensates at absolute zero with an unbiased approach towards the traditional Bose-Einstein condensation theory can bring out the effect of generalized canonical partition function.

DOI
10.48550/arxiv.1409.8377

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We open a new discussion of generalized canonical partition function in standard statistical mechanics and apply it for the study of Bose-Einstein condensation. We discuss the possible cases for the generalized canonical partition function and arrives at a conclusion that the system of trapped bose gas will not be existing at absolute zero. We analyse the present study with an experimental result and point out the general difficulties in the analyses of experimental observations, which can possibly suppress the effect of generalized canonical partition function over standard canonical partition function. We mention that the experimental studies with ideal condensates at absolute zero with an unbiased approach towards the traditional Bose-Einstein condensation theory can bring out the effect of generalized canonical partition function.
We calculate certain features of Bose–Einstein condensation in the ideal gas by using recurrence relations for the partition function. The grand canonical ensemble gives inaccurate results for certain properties of the condensate that are accurately provided by the canonical ensemble. Calculations in the latter can be made tractable for finite systems by means of the recurrence relations. The ideal one-dimensional harmonic Bose gas provides a particularly simple and pedagogically useful model for which detailed results are easily derived. An analysis of the Bose system via permutation cycles yields insight into the physical meaning of the recurrence relations.
This chapter focuses on the applications involving finite temperature theories. Technically, the grand canonical partition function is easiest to handle and in general it is the one used for the analysis of the thermodynamical properties. In fact, for properties like the energy and the specific heat of the gas this is justified because in the thermodynamical limit the different ensembles are known to give the same answer. The chapter uses the grand canonical ensemble to obtain properties like critical temperature, energy and specific heat. It summarizes the grand canonical description of ideal Bose gases trapped by magnetic fields and describes the formulation of Bose-Einstein condensation as a symmetry-breaking phenomenon.
For ensemble of bosons trapped in a one-dimensional harmonic potential well we have found an analytical formula for the canonical partition function and shown that, for 100 trapped atoms, the discrepancy between the grand canonical and the canonical predictions for the condensate fraction reaches 10% in the vicinity of the Bose-Einstein threshold. This discrepancy decreases only logarithmically as the number of atoms increases. Furthermore, we investigate numerically the case of a three-dimensional ``cigar-shape'' trap in the range of parameters corresponding to current Bose-Einstein condensation experiments.
We calculate the canonical partition function of the ideal Bose-Einstein gas by the modified method of steepest descent without the continuous spectrum approximation in the cases of periodic boundary condition and of perfectly reflecing well condition. Above the critical point the steepest curve is the ordinary one through the saddle point. At and below the lambda temperature the steepest curve is such that the singularity of the integrand lies on it as a cusp, and the ordinary saddle point method loses its validity. The evaluation of the residual term in the integration, however, justifies the usual expression of the extensive quantities per particle and intensive quantities as N and V sufficiently large. In deriving the above results, a sufficient condition for the justification of the substitution of bl0 for bl0(V) in canonical partition function of the imperfect gas is given and discussed.
The atom fluctuation statistics of an ideal, mesoscopic, Bose-Einstein condensate are investigated from several different perspectives. By generalizing the grand canonical analysis (applied to the canonical ensemble problem), we obtain a self-consistent equation for the mean condensate particle number that coincides with the microscopic result calculated from the laser master equation approach. For the case of a harmonic trap, we obtain an analytic expression for the condensate particle number that is very accurate at all temperatures, when compared with numerical canonical ensemble results. Applying a similar generalized grand canonical treatment to the variance, we obtain an accurate result only below the critical temperature. Analytic results are found for all higher moments of the fluctuation distribution by employing the stochastic path integral formalism, with excellent accuracy. We further discuss a hybrid treatment, which combines the master equation and stochastic path integral analysis with results obtained based on the canonical ensemble quasiparticle formalism [Kocharovsky et al., Phys. Rev. A 61, 053606 (2000)], producing essentially perfect agreement with numerical simulation at all temperatures.
Recursive approaches determining the canonical ideal Bose gas partition function are reviewed that enable the Bose–Einstein condensate occupation probability to be calculated for a finite number of particles ensemble, where the thermodynamic limit approximation fails. In addition to the earlier known method recursive with respect to the number of particles, an iteration procedure with respect to the number of quantum states is proposed. The efficiency of both methods is demonstrated for an ideal Bose gas in a three-dimensional isotropic harmonic trap.