PHYSICAL REVIEW A 83, 033609 (2011) Condensate statistics and thermodynamics of weakly interacting Bose gas: Recursion relation approach K. E. Dorfman, M. Kim, and A. A. Svidzinsky Department of Physics, Texas A&M University, College Station, Texas 77843-4242, USA (Received 15 September 2010; published 14 March 2011) We study condensate statistics and thermodynamics of weakly interacting Bose gas with a fixed total number N of particles in a cubic box. We find the exact recursion relation for the canonical ensemble partition function. Using this relation, we calculate the distribution function of condensate particles for N = 200. We also calculate the distribution function based on multinomial expansion of the characteristic function. Similar to the ideal gas, both approaches give exact statistical moments for all temperatures in the framework of Bogoliubov model. We compare them with the results of unconstraint canonical ensemble quasiparticle formalism and the hybrid master equation approach. The present recursion relation can be used for any external potential and boundary conditions. We investigate the temperature dependence of the first few statistical moments of condensate fluctuations as well as thermodynamic potentials and heat capacity analytically and numerically in the whole temperature range. DOI: 10.1103/PhysRevA.83.033609 PACS number(s): 03.75.Hh, 64.60.an, 05.70.Fh, 05.70.Ln I. INTRODUCTION Statistics of weakly interacting Bose gas attracted much at- tention within recent years due to the experimental realization of Bose-Einstein condensation (BEC) of trapped gases [1–4]. The experiments on trapped BEC are dealing with a finite number of particles, which makes the system mesoscopic. The grand-canonical description is not appropriate for the analysis of BEC fluctuations, even in mesoscopic ideal Bose gas (IG). For example, the widely used grand-canonical approach yields the grand-canonical catastrophe [5]. The restricted ensemble approach [6–9], which fixes the amplitude and the phase of the condensate wave function, is unable to analyze the fluctuation problem at all. To study fluctuations, one should fix only external macroscopical and global topolog- ical parameters of the system, such as temperature, pressure, number of particles, superfluid flow pattern, boundary condi- tions, etc. A physically motivated master equation approach based on the analogy between BEC and the laser threshold was studied in the series of papers [10–12]. In particular, the hybrid approach of Ref. [11] gives very accurate quantitative results in the whole temperature range; however, it is phenomenological and thus not very insightful. In contrast, the microscopic ap- proach of Ref. [12] for interacting gas is physically appealing but less accurate for the third central moment of condensate fluctuations. BEC statistics in canonical ensemble was investigated extensively [10–28] for weakly interacting gas (WIG), mostly in the framework of the Bogoliubov approximation [6,8,13, 14,29–31]. For the IG, the exact canonical recursion relation for a partition function has been known for a long time and is an effective tool in studying the condensate statistics [32–35]. Numerous papers [27,28,36] offered the canonical recursion relation for WIG, but unfortunately, none of those relations is as clear and transparent as for the IG. In some approaches the recursion is made not over the number of particles but rather over the number of states [36]. This technique gives relatively good result for a very large number of particles but is unable to describe the critical temperature region and the occurrence of phase transition. In another series of papers [27,28] the recursion is made over the excitations. In this case the summation involves noninteger limits and is not very transparent compared to the particle number recursion. In this paper we obtain a canonical ensemble recursion relation for the partition function of WIG utilizing the canonical ensemble quasiparticle formalism and constraint nonlinearity responsible for the phase transition by restricting the multiparticle Hilbert space. Using this relation, we cal- culate statistics of mesoscopic WIG BEC. Alternatively, the distribution function of the condensate particle number can be found from the multinomial expansion of the derivatives of the characteristic function and the particle number constraint. Our paper is organized as follows. In Sec. II we give a brief scope of the constrained cutoff formalism for WIG in a box in Bogoliubov approximation. Section III is devoted to the direct calculation of the distribution function in terms of the multinomial coefficients and spectral function. Then we obtain the canonical recursion relation for the N -particle partition function and use it to find the condensate statistics valid for all temperatures, particle numbers, and volumes (Sec. IV). Both approaches are physically insightful, and the crossover between an IG and WIG along with squeezed mode effects [11,15,37] is transparent. In Sec. V we compare our results with those of the hybrid master equation approach [11] and the unconstrained canonical ensemble quasiparticle formalism [15]. In Sec. VI we calculate thermodynamical properties such as the Gibbs free energy, mean energy, entropy, and heat capacity and compare them with the IG limit. We estimate the contribution coming from interaction and a finite number of particles. II. CONSTRAINED CUTOFF DISTRIBUTION IN WEAKLY INTERACTING BOSE GAS One way to handle the problem of a mesoscopic system with an exactly conserved number of particles is to consider a constrained many-body Hilbert space. 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