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Condensation of Ideal Bose Gas Confined In… Physical Review a 76, 063604 ͑2007͒ PHYSICAL REVIEW A 76, 063604 ͑2007͒ Condensation of ideal Bose gas confined in a box within a canonical ensemble Konstantin Glaum* and Hagen Kleinert† Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany Axel Pelster‡ Fachbereich Physik, Universität Duisburg-Essen, Campus Duisburg, Lotharstrasse 1, 47048 Duisburg, Germany ͑Received 18 July 2007; published 5 December 2007͒ We set up recursion relations for the partition function and the ground-state occupancy for a fixed number of noninteracting bosons confined in a square box potential and determine the temperature dependence of the specific heat and the particle number in the ground state. A proper semiclassical treatment is set up which yields the correct small-T behavior in contrast to an earlier theory in Feynman’s textbook on statistical me- chanics, in which the special role of the ground state was ignored. The results are compared with an exact quantum-mechanical treatment. Furthermore, we derive the finite-size effect of the system. DOI: 10.1103/PhysRevA.76.063604 PACS number͑s͒: 03.75.Hh I. INTRODUCTION fixing the full particle number N, the system does not ex- change its particles with the environment at all and this situ- The thermodynamic properties of a Bose gas are usually ation, therefore, corresponds to one in a canonical ensemble. studied in the grand-canonical formulation where energy and The only exception is that excited states are then still de- particle number are fixed on the average. Present day experi- scribed by the grand-canonical Bose-Einstein distribution. ments, on the other hand, are performed in a magnetic or The entire canonical description will be presented below. optical trap which contain a roughly fixed number of par- The canonical treatment is conveniently performed in a ticles. Thus, the experimental situation may be better ap- first-quantized path-integral description. All particles are rep- proximated by a canonical ensemble with fixed particle num- resented by cycles winding around a cylinder in Euclidean ber. In the thermodynamic limit one usually expects grand- space time whose circumference is the imaginary-time axis canonical and canonical treatments to yield the same results ␶ʦ͑0,ប/k T͓͒10,11͔. For sufficiently low temperatures, the for all thermodynamic quantities and the condensate fraction. B This has indeed been proved in ͓1͔ in general and for har- indistinguishability of particles leads to long closed paths ͓ ͔ winding many times around this cylinder, giving rise to cor- monic traps in 2–5 . Some statistical quantities, however, do ͓ ͔ ͓ ͔ not have the same thermodynamic limits in the two en- related subsets of particles 12 . In Ref. 13 , this represen- sembles. One fundamental example is the fluctuation width tation was utilized to claim that condensation amounts di- ⌬N ϵͱ͗͑⌬N ͒2͘ of the particle number N in the ground rectly to a proliferation of long cycles. This claim can, 0 0 0 however, not be upheld. The formation of correlated collec- state. In the grand-canonical ensemble, this quantity has the ͱ ͑ ͒ tives is the origin of superfluidity, not condensation, as it was size N0 N0 +1 and is at low temperatures T of the order of ͓ ͔ correctly implemented for an ideal gas in a harmonic trap the total particle number N 1,2,6–8 . In the canonical en- ͓14͔. The essential signal for the Bose-Einstein condensation semble, on the other hand, this quantity can be shown to is, instead, associated with the occupancy of the single- vanish at T=0 since N0 becomes equal to the fixed particle particle ground state. Thus, not the winding numbers of the number N. The grand-canonical result is a consequence of cycles are relevant to condensation, but the weight of the the assumed statistical independence of all open grand- ground state in them. For a harmonic trap, this was properly canonical subsystems. This assumption is only fulfilled if shown in Refs. ͓15,16͔. there is no long-range ordered condensate, otherwise it is If the harmonic trap is replaced by a square box, whose wrong. The problem can be properly removed by fixing the infinite-volume limit is the homogeneous system, the sys- whole particle number and ignoring its fluctuations as was ͓ ͔ tems confined in it show some unusual features. It is the shown, e.g., in 1,6 . The particle number in the ground state purpose of this paper to exhibit these. The discussion starts is then N0 =N−Nex, where Nex is the excited particle number. ⌬ in Sec. II with a brief summary of the imaginary-time evo- Since, furthermore, the fluctuation Nex of the thermal gas, lution amplitude for a fixed number N of bosons, where the consisting of excited states, can be shown to vanish for zero ⌬ indistinguishability of particles is explicitly taken into ac- temperature, the same is true for N0. Its temperature behav- ⌬ ϳ͑ / ͒3/2 1/2 count. In Sec. III we calculate the partition function of such ior is found to be N0 T Tc N for an ideal gas in a a canonical system from the trace of the N-particle ͓ ͔ ⌬ ϳ͑ / ͒ 2/3 harmonic trap 2 and N0 T Tc N for a homogeneous imaginary-time evolution amplitude written in the cycle rep- ͓ ͔ system 9 for temperatures T below its critical value Tc.By resentation of paths. Since this is hard to evaluate numeri- cally for a large particle number, we have derived an efficient recursion relation along the lines of Refs. ͓5,17–20͔ in Sec. *[email protected] IV. In Sec. V this relation is made specific by a spectral †[email protected] decomposition of the canonical partition function. This al- ‡[email protected] lows us in Sec. VI to extract the ground-state contribution to 1050-2947/2007/76͑6͒/063604͑12͒ 063604-1 ©2007 The American Physical Society GLAUM, KLEINERT, AND PELSTER PHYSICAL REVIEW A 76, 063604 ͑2007͒ the partition function from the cycle representation, from guishable particles: bosons with a completely symmetric and which we read off the probability to find a particle in the fermions with a completely antisymmetric N-particle wave ground state. This is used to define the condensate fraction function. According to Pauli’s spin-statistic theorem they are / N0 N for a system with a fixed particle number N. In Sec. associated with integer and half-integer spins, respectively. VII we apply our results to the Bose gas confined in the The corresponding modifications of the path integral ͑1͒ are square box potential. There we show that the semiclassical straightforward. For bosons, we must sum over all permuted ͑␯͒ approximation requires a delicate treatment of the ground final positions xP͑␯͒b, where P denotes one of the N! pos- state for finite systems. This is necessary in order to obtain sible permutations of the index ␯. For fermions, there is an the correct low-temperature behavior of the thermodynami- extra minus sign for odd permutations. Restricting ourselves cal functions in contrast to an earlier theory in Feynman’s in this paper to the bosonic case, we must calculate the original textbook ͓10͔. The results are then compared with an imaginary-time evolution amplitude exact quantum-mechanical calculation. In Sec. VIII we in- ͑ ␶ ͉ ␶ ͉͒B vestigate the finite-size effect in a box potential. We identify x1b, ...,xNb, b x1a, ...,xNa, a a would-be transition temperature as the finite-N version of 1 the critical temperature of Bose-Einstein condensation in a ͑ ␶ ͉ ␶ ͉͒ ͑ ͒ = ͚ xP͑1͒b, ...,xP͑N͒b, b x1a, ...,xNa, a . 6 canonical ensemble. Subsequently, canonical results calcu- N! P lated for these crossover temperatures are compared with grand-canonical ones. We end with an analytic calculation of the finite-size effect upon the critical temperature in the III. CYCLE EXPANSION grand-canonical ensemble. The canonical partition function of N noninteracting II. QUANTUM STATISTICS OF IDENTICAL PARTICLES bosons is given by the trace We start by briefly reviewing the quantum statistics of a fixed number N of noninteracting identical particles in a first B͑␤͒ ͵ 3 ͵ 3 ͑ ប␤͉ ͉͒B ͑ ͒ quantized approach ͓10,11͔. At first, they are treated as dis- ZN = d x1 ¯ d xN x1, ...,xN, x1, ...,xN,0 , 7 tinguishable particles. Their orbits in imaginary time ␶ are denoted by x␯͑␶͒ with the particle indices ␯=1,2,...,N. ␤ϵ͑ ͒−1 where kBT represents the reciprocal temperature. The Global and local thermodynamic properties of the many- calculation of this basic statistic quantity is nontrivial due to particle system are determined by the imaginary-time evolu- the indistinguishability of the particles. At first we consider tion amplitude which is given by the path integral the Bose gas with small particle numbers N. For N=1, the ͑ ␶ ͉ ␶ ͉͒ partition function ͑7͒ reduces to the well-known expression x1b, ...,xNb, b x1a, ...,xNa, a N ͑␶ ͒ x␯ b =x␯b A͓ ͔/ប ͩ ͵ D ͪ − x1,...,xN ͑ ͒ ͑␤͒ 3 ͑ ប␤͉ ͉͒ ͑ ͒ = ͟ x␯ e . 1 ͵ Z1 = d x x, x,0 . 8 ␯ ͑␶ ͒ =1 x␯ a =x␯a The Euclidean action in the exponent has the generic form Here we have omitted the index B, as the distinguishable N ␶ particles are not an issue for a single particle. For N=2, we b M A͓ ͔ ͵ ␶ͩ 2͑␶͒ ͓ ͑␶͔͒ͪ ͑ ͒ obtain from ͑7͒ according to the indistinguishability of ͑6͒ x1, ...,xN = ͚ d x˙ ␯ + V x␯ 2 ␯ ␶ 2 =1 a and the factorization property ͑3͒ with the particle mass M and the common background po- 1 tential V͑x␯͒ for all particles.
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