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Physical Review

Statistical Distributions for the Condensate of a Degenerate Boson Fluid in Thermodynamic Equilibrium

A. E. Glassgold, H. Sauermann

Published Dec 5, 1969• 3 citations
Publisher

Abstract

Using a previously developed theory, various distribution functions are obtained for the fluctuations in the condensate of a degenerate boson fluid in homogeneous thermodynamic equilibrium. The distribution for the occupation number of the condensate is shown to be Gaussian. The Wigner distribution and the characteristic function for the order parameter are also derived. The nonexistence of Glauber's $P$ representation for this problem is demonstrated for a finite range of temperatures starting at absolute zero.

DOI
10.1103/physrev.188.515

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arXiv:cond-mat/0001074•Physical Review A•Jul 20, 2000•9 citations
Using a previously developed theory, various distribution functions are obtained for the fluctuations in the condensate of a degenerate boson fluid in homogeneous thermodynamic equilibrium. The distribution for the occupation number of the condensate is shown to be Gaussian. The Wigner distribution and the characteristic function for the order parameter are also derived. The nonexistence of Glauber's PPP representation for this problem is demonstrated for a finite range of temperatures starting at absolute zero.
A general theory is formulated for the fluctuations in the condensate of a degenerate boson fluid in homogeneous thermodynamic equilibrium. The point of view adopted is based on Bogoliubov's description of the condensate in terms of quasiaverages and on his "Principle of Diminishing Correlation." An essential aspect of the theory is the consistent treatment of the fluctuation operators ccc and ${c}^{\ifmmode\dagger\else\textdagger\fi{}}$, where $c={a}_{0}\ensuremath{-}\ensuremath{\alpha}$. a0{a}_{0}a0​ is the annihilation operator for the zero momentum state, and the order parameter $\ensuremath{\alpha}$ is the quasiaverage of a0{a}_{0}a0​. After the Hamiltonian for the fluctuation operators is obtained, a stability condition is derived, which we then demonstrate is satisfied under very general conditions. The energy spectrum of the condensate fluctuations is shown to be discrete, and explicit expressions are derived for the quantum states of the condensate. The ground state of the condensate is not a coherent state of the type studied by Glauber. A detailed analysis is given of the fluctuations in the order parameter and in the number of particles in the condensate. The physical properties of the fluctuations are determined by two parameters, which are the averages of the second derivatives of the Hamiltonian with respect to the order parameter. These quantities are shown to be related to the one-particle Green's functions introduced by Beliaev. The implications of this general theory are also worked out for Bogoliubov's low-density, weak-coupling model. The limit of the ideal Bose gas is then shown to possess pathological features, namely an unnormalizable ground state and infinite fluctuations, in contrast to the interacting fluid where all properties are well-behaved.
The equilibrium and quasi-equilibrium properties of a system of interacting bosons are studied from a microscopic point of view. For equilibrium, the model of Bogolyubov is generalized to finite temperature by using the grand partition function. The thermodynamic properties and the pair-correlation function are calculated. The statistical mechanics for moving systems is then developed and applied to the problem of a rotating fluid. For quasi-equilibrium, general transport equations are derived from first principles, independent of statistics and model. For the Bogolyubov model, the familiar two-fluid hydrodynamics is then derived, leading to the phenomena of first and second sound.
We obtain analytical formulas for the statistics, in particular, for the characteristic function and all cumulants, of the Bose-Einstein condensate in dilute weakly interacting and ideal equilibrium gases in the canonical ensemble via the particle-number-conserving operator formalism of Girardeau and Arnowitt. We prove that the ground-state occupation statistics is not Gaussian even in the thermodynamic limit. We calculate the effect of Bogoliubov coupling on suppression of ground-state occupation fluctuations and show that they are governed by a pair-correlation, squeezing mechanism.
The distribution function w 0 ( n 0 ) of the number n 0 of particles is found for the condensate of an ideal gas of free bosons with a fixed total number N of particles. It is shown that above the critical temperature ( T > T c ) this function has the usual form w 0 ( n 0 ) = (1 — e μ )e μ n 0 , where μ is the chemical potential in temperature units. In a narrow vicinity of the critical temperature | T/T c — 1| ≤ N -1/3 , this distribution changes and at T < T c acquires the form of a resonance. The width of the resonance depends on the shape of the volume occupied by the gas and it has exponential (but not the Gaussian) wings. As the temperature is lowered, the resonance maximum shifts to larger values of n 0 and its width tends to zero, which corresponds to the suppression of fluctuations. For N → ∞, this change occurs abruptly. The distribution function of the number of particles in excited states for the systems with a fixed and a variable number of particles (when only a mean number of particles is fixed) prove to be identical and have the usual form.
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.
A grand canonical system of hard-core bosons, subject to thermal fluctuations, is studied on a lattice. Starting from the slave-boson representation with fields for occupied and unoccupied sites, an effective field theory is derived in which a complex field corresponds with the order parameter of the condensate and a real field with the total density of bosons. Near the boundary between the normal and the superfluid phase, we obtain the Ginzburg-Landau functional for the superfluid order parameter. A mean-field calculation shows that the critical temperature Tc{T}_{c}Tc​ increases with increasing density up to a maximum and decreases with further increasing density.