We show with a direct numerical analysis that a dilute Bose gas in an external potential—which is chosen for simplicity as a radial parabolic well—undergoes at a certain temperature Tc a phase transition to a state supporting a macroscopic fraction of particles at the origin of the phase space (r=0, p=0). Quantization of particle motion in a well wipes out the sharp transition but supports a distribution of a radial particle density ρ(r) peaked at r=0 (a real-space condensate) as well as a phase-space Wigner distribution density W(r,p) peaked at r=0 and p=0 below a crossover temperature Tc* of order of Tc. A fixed-particle-number canonical ensemble, which is a combination of the fixed-N condensate part and the fixed-μ excitation part, is suggested to resolve the difficulty of large fluctuation of the particle number (δN∼N) in the Bose-Einstein condensation problem treated within the orthodox grand canonical ensemble formalism.
We show with a direct numerical analysis that a dilute Bose gas in an external potential—which is chosen for simplicity as a radial parabolic well—undergoes at a certain temperature Tc a phase transition to a state supporting a macroscopic fraction of particles at the origin of the phase space (r=0, p=0). Quantization of particle motion in a well wipes out the sharp transition but supports a distribution of a radial particle density ρ(r) peaked at r=0 (a real-space condensate) as well as a phase-space Wigner distribution density W(r,p) peaked at r=0 and p=0 below a crossover temperature Tc* of order of Tc. A fixed-particle-number canonical ensemble, which is a combination of the fixed-N condensate part and the fixed-μ excitation part, is suggested to resolve the difficulty of large fluctuation of the particle number (δN∼N) in the Bose-Einstein condensation problem treated within the orthodox grand canonical ensemble formalism.
We consider Bose-Einstein condensation of noninteracting homogeneous three-dimensional gas in canonical ensemble when both particle number N and total momentum P of all particles are fixed. Using the saddle-point method, we derive the large-N analytical approximations for partition function, free energy, and statistical distributions of occupation numbers of different single-particle energy levels. At temperatures below the critical point of phase transition, we predict, in some ranges of P, fragmentation of the condensate, when more than one single-particle level is macroscopically occupied. The occupation number distributions have approximately Gaussian shapes for the levels hosting the condensate, and exponential shapes for other, noncondensate levels. Our analysis demonstrates breaking of Galilean invariance of moving finite-temperature many-particle system in the presence of Bose-Einstein condensation and extends the theory of moving and rotating quantum systems to the finite-temperature large-N limit.
We study the Bose-Einstein condensation phase transition in a weakly interacting gas through a perturbative analysis of finite systems. In both the grand canonical and the canonical ensembles, perturbation theory suffers from infrared divergences and cannot directly determine the transition temperature in the thermodynamic limit. However, in conjunction with finite-size scaling, perturbation theory provides a powerful calculation tool. Here we use this combination of perturbation theory and scaling to estimate a shift in the transition temperature in the canonical ensemble consistent with grand canonical calculations.
Bose-Einstein condensation in the grand canonical ensemble admits a formulation in terms of a phase-density decomposition of the condensate mode operator ψ^0. In the presence of macroscopic condensate number fluctuations this representation presents nontrivial implications. In particular, we show that, for the ideal gas, under the assumption of a well-defined phase and a fluctuating condensate density, the square modulus of the anomalous average ⟨ψ^0⟩ can provide only a fraction of the whole condensate density ρ0 and for the grand canonical statistics of the ideal Bose gas one obtains the value ∣⟨ψ^0⟩∣2=(π/4)ρ0. The remaining part is supplemented by the (macroscopic) fluctuations of ψ^0, which become a distinctive feature of the BEC in this setting. This provides a transparent physical picture of a condensate of photons with a well-defined phase but large number fluctuations, as observed in dye-filled microcavity photon experiments. We also propose a way to access the square modulus of the anomalous average to test theoretical predictions.
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.
Bose-Einstein condensation (BEC) of a noninteracting Bose gas of N particles in a two-dimensional box with Dirichlet boundary conditions is studied. Confirming previous work, we find that BEC occurs at finite N at low temperatures T without the occurrence of a phase transition. The conventionally-defined transition temperature T(E) for an infinite three-dimensional (3D) system is shown to correspond in a 2D system with finite N to a crossover temperature between a slow and rapid increase in the fractional boson occupation N(0)/N of the ground state with decreasing T. We further show that T(E)∼1/logN at fixed area per boson, so in the thermodynamic limit there is no significant BEC in 2D at finite T. Thus, paradoxically, BEC only occurs in 2D at finite N with no phase transition associated with it. Calculations of thermodynamic properties versus T and area A are presented, including Helmholtz free energy, entropy S, pressure p, ratio of p to the energy density U/A, heat capacity at constant volume (area) C(V) and at constant pressure C(p), isothermal compressibility κ(T) and thermal expansion coefficient α(p), obtained using both the grand-canonical ensemble (GCE) and canonical ensemble (CE) formalisms. The GCE formalism gives acceptable predictions for S, p, p/(U/A), κ(T) and α(p) at large N, T and A but fails for smaller values of these three parameters for which BEC becomes significant, whereas the CE formalism gives accurate results for all thermodynamic properties of finite systems even at low T and/or A where BEC occurs.
Condensation of the ideal Bose gas in a closed volume having the shape of a rectangular parallel-epiped of length L with a square base of side length l ( L ≫ l ) is theoretically studied within the framework of the Bose-Einstein statistics (grand canonical ensemble) and within the statistics of a canonical ensemble of bosons. Under the condition N ( l/L ) 4 ≪ l, where N is the total number of gas particles, dependence of the average number of particles in the condensate on the temperature T in both statistics is expressed as a function of the ratio t = T/T 1 , where T 1 is a certain characteristic temperature depending only on the longitudinal size L . Therefore, the condensation process exhibits a one-dimensional (1D) character. In the 1D regime, the average numbers of particles in condensates of the grand canonical and canonical ensembles coincide only in the limiting cases of t → 0 and t → ∞. The distribution function of the number of particles in the condensate of a canonical ensemble of bosons at t ≤1 has a resonance shape and qualitatively differs from the Bose-Einstein distribution. The former distribution begins to change in the region of t ∼ 1 and acquires the shape of the Bose-Einstein distribution for t ≫ 1. This transformation proceeds gradually that is, the 1D condensation process exhibits no features characteristic of the phase transition in a 3D system. For N ( l/L ) 4 ≫ 1, the process acquires a 3D character with respect to the average number of particles in the condensate, but the 1D character of the distribution function of the number of particles in the condensate of a canonical ensemble of bosons is retained at all N values.