Influence of Boundary Conditions on Statistical Properties of Ideal Bose
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PHYSICAL REVIEW E, VOLUME 65, 036129 Influence of boundary conditions on statistical properties of ideal Bose-Einstein condensates Martin Holthaus* Fachbereich Physik, Carl von Ossietzky Universita¨t, D-26111 Oldenburg, Germany Kishore T. Kapale and Marlan O. Scully Max-Planck-Institut fu¨r Quantenoptik, D-85748 Garching, Germany and Department of Physics and Institute for Quantum Studies, Texas A&M University, College Station, Texas 77843 ͑Received 23 October 2001; published 27 February 2002͒ We investigate the probability distribution that governs the number of ground-state particles in a partially condensed ideal Bose gas confined to a cubic volume within the canonical ensemble. Imposing either periodic or Dirichlet boundary conditions, we derive asymptotic expressions for all its cumulants. Whereas the conden- sation temperature becomes independent of the boundary conditions in the large-system limit, as implied by Weyl’s theorem, the fluctuation of the number of condensate particles and all higher cumulants remain sensi- tive to the boundary conditions even in that limit. The implications of these findings for weakly interacting Bose gases are briefly discussed. DOI: 10.1103/PhysRevE.65.036129 PACS number͑s͒: 05.30.Ch, 05.30.Jp, 03.75.Fi ϭ When London, in 1938, wrote his now-famous papers on with 1/(kBT), where kB denotes Boltzmann’s constant. Bose-Einstein condensation of an ideal gas ͓1,2͔, he simply Note that the product runs over the excited states only, ex- considered a free system of N noninteracting Bose particles cluding the ground state ϭ0; the frequencies of the indi- without an external trapping potential, imposing periodic vidual oscillators being given by the excited-states energies boundary conditions on a cubic volume V. The rationale for ͓ ͔ relative to the ground-state level 0. Note further that the so doing lies in Weyl’s theorem 3 . In the thermodynamic equivalence of a partially condensed ideal Bose gas and a limit the particular boundary conditions are not supposed to harmonic-oscillator system holds regardless of the precise play any role. Yet, for any box size the half-sine single- form of the single-particle spectrum, so that it does not mat- particle ground state in a box with hard walls is quite differ- ter whether or not the trapping potential itself is harmonic. ent from the plane-wave ground state in a box with periodic The only approximation entering into the derivation of the boundary conditions, so that, intuitively, one expects some ͑ ͒ influence of the boundary conditions on the physics of Bose- canonical partition function 1 is a replacement of the ac- Einstein condensation even in the large-system limit ͑that is, tual, finite number of condensed particles by a condensate ͓ ͔ for N!ϱ and V!ϱ, such that the density N/V remains containing infinitely many particles 6 . For temperatures constant͒. In this paper we will explain that, indeed, all such that a sizeable fraction of the particles is condensed, ‘‘higher’’ statistical properties of an ideal, free Bose-Einstein this enlargement of the condensate does not appreciably condensate, such as the fluctuation ͓4,5͔ of the number of change the thermal properties of the system. On the other condensed particles, remain sensitive to the boundary condi- hand, assuming an inifinite supply of condensed particles tions even in the large-system limit, even though the conden- means excluding the onset of condensation from the analy- sation temperature does not. We speculate, however, that this sis; this onset can be treated by a different approach based on unusual behavior constitutes a pathology of the ideal gas that a master equation ͓4,5͔. In short, the validity of Eq. ͑1͒ is will be cured by interparticle interactions, although so far no restricted to the condensate regime. The same approximation detailed proof for this surmise exists. had already been used by Fierz in 1956 ͓7͔ for computing the Let us consider an ideal, N-particle Bose gas stored in fluctuation of an ideal Bose-Einstein condensate; it also un- some trap with single-particle energies (ϭ0,1,2,...;in derlies the ‘‘Maxwell’s demon ensemble’’ suggested by general, abbreviates a multi-index͒. We assume further that Navez et al. ͓8͔. In a different guise, it has been utilized in the setup conforms to the canonical ensemble, which means the canonical quasiparticle approach formulated by Ko- that the trapped gas is kept in thermal contact with an exter- charovsky, Kocharovsky, and Scully ͓9,10͔. nal heat bath of temperature T. The starting point of our Having stepped from the original trapped Bose gas to a analysis is the recognition that for temperatures below the harmonic-oscillator system, we now exploit the fact that onset of Bose-Einstein condensation the canonical partition there exist powerful mathematical tools for evaluating function Z() reduces, to excellent approximation ͓6͔,tothe harmonic-oscillator sums ͓11͔ in order to quantify the statis- partition function of a system of infinitely many independent tics of the ground-state occupation number of the partially ͑i.e., Boltzmannian͒ harmonic oscillators, condensed Bose gas described by Eq. ͑1͒. Specifically, we ϱ study the canonical probability distribution p (N ;) for 1 N ex Z͑͒ϭ ͑ ͒ finding Nex of the N particles in an excited state at a given ͟ Ϫ Ϫ Ϫ ͒ , 1 ϭ1 1 exp͓ ͑ 0 ͔ inverse temperature , so that the number of condensate par- ϭ Ϫ ticles is n0 N Nex . This distribution is characterized by its  ͓ ͔  cumulants k( ) 9,10 . The first cumulant 1( ) is the *Electronic address: [email protected] canonical-ensemble expectation value ͗Nex͘ of the number 1063-651X/2002/65͑3͒/036129͑5͒/$20.0065 036129-1 ©2002 The American Physical Society HOLTHAUS, KAPALE, AND SCULLY PHYSICAL REVIEW E 65 036129  ͓ ͔ of excited particles, the second cumulant 2( ) corresponds Analytically continuing this function S(t) 13,14 , one finds ͑ to the mean-square fluctuation of Nex thus equalling the that it possesses merely one simple pole, located at ͒  fluctuation of the number of ground-state particles , 3( )   ϭ ͑ ͒ equals the third central moment 3( ); similarly, 4( ) t 3/2 with residue 2 . 7 ϭ  Ϫ  2 ͓ ͔ 4( ) 3 2( ) 12 . The reason for focusing on the cu-  For kϭ1 this pole of Z(,t) lies to the right of the pole of mulants k( ), rather than on the more familiar central mo-  (t)attϭ1 ͓16͔, and, thus, dominates the temperature de- ments k( ), lies in the independence of the Boltzmannian oscillators: The kth order cumulant of a sum of independent pendence of the number of excited particles. Defining the ϵ ប⍀ stochastic variables is given by the sum of the individual scaled temperature kBT/( ), the canonical-ensemble ϭ  cumulants. expectation value ͗Nex͘ 1( ) of the number of excited Without any further approximation, one can then derive a particles then takes, for ӷ1, the form compact integral representation for (). Introducing the k ͗ ͘ϳ3/2͑ ͒3/2ϩ ͑ ͒ ͑ ͒ generalized zeta function Nex 3/2 S 1 , 8 ϱ 1 with S(1)ϷϪ8.914. This asymptotic expression, valid as Z͑,t͒ϵ ͚ , ͑2͒ long as ͗N ͘рN ͑that is, in the condensate regime͒, yields ϭ ͓͑ Ϫ ͔͒t ex 1 0 excellent agreement with exact numerical data even for merely moderately large particle numbers ͓6͔. we find ͓6͔ In contrast, for all higher cumulants, kу2, it is no longer ϩ ϱ ͑ ͒   1 t0 i the pole 7 of Z( ,t) that dominates k( ) in the ͑͒ϭ ͵ ⌫͑ ͒ ͑ ͒͑ ϩ Ϫ ͒ ͑ ͒ k dt t Z ,t t 1 k . 3 asymptotic regime ӷ1, but rather the pole of (tϩ1Ϫk)at 2i Ϫ ϱ t0 i ϭ  t k. Hence, to leading order k( ) becomes proportional to k Here, ⌫(t) and (t) denote the gamma function and the Rie- : mann zeta function, respectively; the real number t has to 0 ͑͒ϳ͑ Ϫ ͒ ͑ ͒kϩ3/2͑ Ϫ ͒3/2 ͑ ͒ be chosen such that the path of integration parallel to the k k 1 !S k 5/2 k . 9 imaginary axis of the complex t plane lies to the right of all In the next step we repeat this analysis for the case of poles of the integrand. Note that the properties of the particu- Dirichlet boundary conditions, that is, for an ideal N-particle lar trap ͑that is, the single-particle spectrum͒ enter into Bose gas stored in a cubic volume VϭL3 with hard, impen- () only through Z(,t), whereas the cumulant order k is k etrable walls. This implies a nonzero ground-state energy determined only by the argument of the Riemann zeta func- ϭ ប⍀ ͑ ͒ 0 3 /4, so that the corresponding generalized zeta func- tion. The usefulness of this cumulant formula 3 stems from ͑ ͒ the fact that there exist well-established techniques ͓13͔ for tion 2 becomes continuing the ‘‘spectral’’ zeta functions ͑2͒ analytically to ͑ ͒ϭ͑ប⍀ ͒Ϫt˜͑ ͒ ͑ ͒ the complex t plane, so that simply collecting the residues at Z ,t /4 S t , 10 the poles of the product ⌫(t)Z(,t)(tϩ1Ϫk), taken from ͓ ͔ right to left, results in a systematic asymptotic expansion of with a modified, inhomogeneous Epstein function 14,15  the desired canonical cumulants k( ). The large-system ϱ limit is governed by the rightmost pole alone ͓6͔. 1 ˜S͑t͒ϵ ͚Ј . ͑11͒ We now apply these general findings to an ideal Bose gas ϭ ͑ 2ϩ 2ϩ 2Ϫ ͒t n1 ,n2 ,n3 1 n1 n2 n3 3 of N particles with mass m in a cubic volume VϭL3.