An asymptotic expansions for the grand partition function of ideal Bose gas in the canonical ensemble with arbitrary number of particles is obtained. It is shown that the expressions found are valid in the whole temperature region, the critical temperature included. A comparison between the asymptotic formulas for Bose-condensate fluctuations and the exact ones is carried out and their quantitative agreement is established.
An asymptotic expansions for the grand partition function of ideal Bose gas in the canonical ensemble with arbitrary number of particles is obtained. It is shown that the expressions found are valid in the whole temperature region, the critical temperature included. A comparison between the asymptotic formulas for Bose-condensate fluctuations and the exact ones is carried out and their quantitative agreement is established.
The ideal Bose-gas with finite number N of particles is investigated. The exact expressions for the partition functions and occupation numbers in the grand canonical, canonical and microcanonical ensembles are found. The asymp\-totic expressions (in the case N≫1) for the partition functions and occupation numbers in the canonical and microcanonical ensembles are evaluated. It is shown that the chemical potential μ of the ideal Bose-gas can lie in the range $-\infty
The paper is devoted to the theory of an ideal Bose-gas with a finite number N of particles. The exact expressions for the partition functions and occupation numbers of the model in the grand canonical, canonical, and microcanonical ensembles are found. From the calculations, it is followed that, oppositely to the accepted opinion that the chemical potential μ of an ideal Bose-gas is only negative, it can take values in the range −∞ < μ < ∞. The asymptotic expressions (in the case N ≫ 1) for the partition functions and occupation numbers for all above-mentioned thermodynamic ensembles are also evaluated.
Based on counting statistics and Bogoliubov theory, we present a recurrence relation for the microcanonical partition function for a weakly interacting Bose gas with a finite number of particles in a cubic box. According to this microcanonical partition function, we calculate numerically the distribution function, condensate fraction, and condensate fluctuations for a finite and isolated Bose-Einstein condensate. For ideal and weakly interacting Bose gases, we compare the condensate fluctuations with those in the canonical ensemble. The present approach yields an accurate account of the condensate fluctuations for temperatures close to the critical region. We emphasize that the interactions between excited atoms turn out to be important for moderate temperatures.
In this work we present a large-deviations approach to the calculation of the canonical partition function for free bosons. Three-dimensional Bose-Einstein condensation is studied in the fixed-density ensemble as a function of the dimensionless density ϱ=ρλT3, with ρ=N/L3 the standard particle density, λT the thermal wavelength, L the linear size of the box and N the total number of particles. A large-deviations approach in terms of the dimensionless parameter ℓ=L/λT allows us to provide exact asymptotic estimates of the canonical partition function both above and below the critical density ϱc for Bose-Einstein condensation. We show how this approach allows to explicitly account for finite-size effects and how it fully captures the first-order aspects of the transition, allowing us to explicitate its driving mechanism in terms of the competing probabilities of normal and condensed phases. The proposed large-deviations approach allows then to obtain in all regimes explicit and simple analytical expressions, at the leading order in the large parameter ℓ, for both the average fraction of particles in the ground state, the condensate fraction ⟨n0(ϱ)⟩=⟨N0(ϱ)⟩/N, and for its fluctuations, σ0(ϱ)=⟨N02(ϱ)⟩−⟨N0(ϱ)⟩2/N, retrieving for instance the anomalous scaling σ0(ϱ)∼1/V1/3 in the condensed regime, ϱ>ϱc. Our large-deviations asymptotic estimate, by analytically clarifying the mixed-order nature of Bose-Einstein condensation, allows then to reveal the similarity between this transition and other mixed-order transitions, as for instance the localization transition in the Discrete Non-Linear Schrödinger Equation.
The fluctuations of a number of particles in the Bose-Einstein condensate are studied in the grand canonical ensemble with an effective single-mode Hamiltonian, which is derived from an assumption that the mode corresponding to the Bose-Einstein condensate does not asymptotically correlate with other modes. The fluctuations are evaluated in the dilute limit with a proposed simple method, which is beyond the mean-field approximation. The accuracy of the latter is estimated; it is shown that the mean-field scheme does not work for the single-mode Hamiltonian, while for the Hartree Hamiltonian it allows us to estimate the condensate fluctuations up to a numerical factor. As a hypothesis, a formula is proposed that relates the fluctuations in the canonical ensemble with that of the grand canonical one.
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.