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arXiv:2403.08482 · Physical Review A

Bose-Einstein condensation in a canonical ensemble with fixed total momentum

Andrey S. Plyashechnik, Alexey A. Sokolik, Yu. E. Lozovik

Published Jul 2, 2024• 1 citation
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Abstract

We consider Bose-Einstein condensation of noninteracting homogeneous three-dimensional gas in canonical ensemble when both particle number $N$ and total momentum $\mathbf{P}$ 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 $\mathbf{P}$, 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.

DOI
10.1103/physreva.110.013301

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We consider Bose-Einstein condensation of noninteracting homogeneous three-dimensional gas in canonical ensemble when both particle number NNN and total momentum P\mathbf{P}P of all particles are fixed. Using the saddle-point method, we derive the large-NNN 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\mathbf{P}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-NNN limit.
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.
The choice of statistical ensemble has a measurable impact on the statistical description of Bose-Einstein condensates beyond the well-known grand canonical anomaly. We develop a segmented-contour approach for the ideal Bose gas, which refines the saddle-point approximation by including non-Gaussian corrections in the canonical and microcanonical ensembles. The method applies to a broad class of noninteracting spectra, and provides efficient access to condensate fluctuations, higher moments, and correlation functions. For a three-dimensional harmonic trap with fixed anisotropy, we find that the ratio of the maxima of the microcanonical and canonical condensate-number variances approaches the theoretically predicted asymptotic value 1−3ζ(3)2/[4ζ(2)ζ(4)]≈0.3911-3ζ(3)^2/[4ζ(2)ζ(4)]\approx 0.3911−3ζ(3)2/[4ζ(2)ζ(4)]≈0.391 only extremely slowly with increasing particle number. We also show that ensemble effects are visible in the second-order correlation function g(2)(x)g^{(2)}(x)g(2)(x) near the condensation crossover. After ballistic release from a three-dimensional harmonic trap, the time evolution of g(2)(x)g^{(2)}(x)g(2)(x) preserves the differences between statistical ensembles.
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\varrho = ρλ_T^3ϱ=ρλT3​, with ρ=N/L3ρ=N/L^3ρ=N/L3 the standard particle density, λTλ_TλT​ the thermal wavelength, LLL the linear size of the box and NNN the total number of particles. A large-deviations approach in terms of the dimensionless parameter ℓ=L/λT\ell=L/λ_Tℓ=L/λT​ allows us to provide exact asymptotic estimates of the canonical partition function both above and below the critical density ϱc\varrho_cϱc​ 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\ellℓ, for both the average fraction of particles in the ground state, the condensate fraction ⟨n0(ϱ)⟩=⟨N0(ϱ)⟩/N\langle n_0(\varrho) \rangle = \langle N_0(\varrho) \rangle/N⟨n0​(ϱ)⟩=⟨N0​(ϱ)⟩/N, and for its fluctuations, σ0(ϱ)=⟨N02(ϱ)⟩−⟨N0(ϱ)⟩2/Nσ_0(\varrho) = \sqrt{\langle N_0^2(\varrho)\rangle - \langle N_0(\varrho)\rangle^2}/Nσ0​(ϱ)=⟨N02​(ϱ)⟩−⟨N0​(ϱ)⟩2​/N, retrieving for instance the anomalous scaling σ0(ϱ)∼1/V1/3σ_0(\varrho)\sim 1/V^{1/3}σ0​(ϱ)∼1/V1/3 in the condensed regime, ϱ>ϱc\varrho > \varrho_cϱ>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.
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.
Recursive approaches determining the canonical ideal Bose gas partition function are reviewed that enable the Bose–Einstein condensate occupation probability to be calculated for a finite number of particles ensemble, where the thermodynamic limit approximation fails. In addition to the earlier known method recursive with respect to the number of particles, an iteration procedure with respect to the number of quantum states is proposed. The efficiency of both methods is demonstrated for an ideal Bose gas in a three-dimensional isotropic harmonic trap.
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.