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Progress of Theoretical Physics

On the Bose-Einstein Condensation

Shigetoshi Katsura

Published Dec 1, 1956• 11 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.16.589

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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.
ABSTRACT This paper contains a proof that the description of the phenomenon of Bose-Einstein condensation is the same whether (1) an open system is contemplated and treated on the basis of the grand canonical ensemble, or (2) a closed system is contemplated and treated on the basis of the canonical ensemble without recourse to the method of steepest descents, or (3) a closed system is contemplated and treated on the basis of the canonical ensemble using the method of steepest descents. Contrary to what is usually believed, it is shown that the crucial factor governing the incidence of the condensation phenomenon of a system (open or closed) having an infinity of energy levels is the density of statesN(E) ∝Enfor high quantum numbers, a condition for condensation beingn> 0. These results are obtained on the basis of the following assumptions: (i) For large volumesV(a) all energy levels behave likeV−θ, and (b) there exists a finite integerMsuch that it is justifiable to put for thejth energy levelEj=c V−θand to use the continuous spectrum approximation, wheneverj ≥ Mc θ τ are positive constants, (ii) All results are evaluated in the limit in which the volume of the gas is allowed to tend to infinity, keeping the volume density of particles a finite and non-zero constant. The present paper also serves to coordinate much of previously published work, and corrects a current misconception regarding the method of steepest descents.
Abstract This chapter describes the mechanism of Bose–Einstein condensation in the simplest ideal Bose gas case. An explicit expression for the critical temperature is derived in the analytically soluble case of a gas confined in a box. Various thermodynamic quantities are calculated above and below the critical temperature. Emphasis is given to the isothermal compressibility which diverges below the critical temperature. The behaviour of the off-diagonal one-body density matrix is discussed. Results for the fluctuations of the single-particle occupation numbers are derived in both the canonical and grand canonical ensembles.
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.
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.
It is shown by a rigorous development from first principles that a relatively simple form for the partition function of an imperfect Bose-Einstein gas can be obtained. This form is that of a 3N3N3N-dimensional integral (NNN being the total number of particles) the integrand of which can be completely determined for any given interparticle interaction provided certain, in general small, many-body quantum mechanical effects are ignored. The final evaluation of the partition function in closed form requires that this integrand be approximated by functions sufficiently tractable that the 3N3N3N-dimensional integral can be performed.
The self-consistent random-phase approximation is used to derive an integral equation for the quasiparticle energy of a charged Bose gas near a critical point. An approximate analytical solution, valid at long wavelengths, is used to compute the critical exponents which are those of an ideal Bose gas with the single-particle spectrum $\ensuremath{\epsilon}(p)=A{p}^{2\ensuremath{-}\ensuremath{\eta}},\ensuremath{\eta}\ensuremath{\simeq}0.2$. The accuracy of the analytical solution is checked by an exact numerical solution which is also used in a calculation of the transition curve. A necessary condition for the correctness of the theory is found to be $\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\gg}0.01$.