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Journal of the Physical Society of Japan

Coherence Theory of the Bose-Einstein Condensation. II

Yutaka Kano

Published Aug 1, 1974• 2 citations
Publisher

Abstract

The properties of the coherent state appearing in the Bose-Einstein condensation of an ideal Bose gas are discussed. The thermodynamic functions in the presence of such a coherent state are obtained. The density correlation function is derived. It is shown that the density correlation function reduces to the London-Placzek formula as temperature rises above the critical temperature of the Bose-Einstein condensation.

DOI
10.1143/jpsj.37.310

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The properties of the coherent state appearing in the Bose-Einstein condensation of an ideal Bose gas are discussed. The thermodynamic functions in the presence of such a coherent state are obtained. The density correlation function is derived. It is shown that the density correlation function reduces to the London-Placzek formula as temperature rises above the critical temperature of the Bose-Einstein condensation.
The coherent-state representation of the density operator for an ideal Bose gas in a thermal equilibrium is introduced. It is shown that a pure coherent state can be an eigenstate of the Hamiltonian of the non-interacting boson system when the symmetry breaking term is added to it. However, at the limit of the vanishing “classical field” only zero-energy states can be a coherent state. It is also proved that the pure coherent state is the lowest energy state among all the zero-momentum states. The temperature dependence of the coherent state is then discussed. The diminution of the coherent state can clearly be seen as the temperature rises from zero to the critical temperature.
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.
The thermodynamic properties of the noninteracting Bose gas in the neighborhood of its transition are examined in detail. The order parameter is a complex extensive variable, but the thermodynamic properties depend only on its amplitude under simple boundary conditions. As the dimensionality or the single-particle energy spectrum is varied, the critical singularity displays a variety of forms. The equation of state has a simple structure, different from the homogeneous form often discussed for critical systems but asymptotically reducing to the latter except when logarithmic singularities are involved. The correlation function in the critical region is a homogeneous function of the distance and a correlation length. Only for a quadratic energy spectrum is the Ornstein-Zernike theory result valid at the critical temperature. A precise correspondence is noted between the asymptotic properties of the ideal Bose gas transition and those of the spherical model of ferromagnetism.
The calculation of the thermodynamic properties in the Bose-Einstein distribution is presented.The behaviour of the Bose-gas in the Bose-Einstein distribution is analysed based on high and low temperature dependence.At > all particles are in the excited state and the ground state is essentially occupied.But at B T T < particles gradually falls to the ground state and eventually contains all the particles in the system as .0 → TThe analytical expressions for some important parameters such as Bose-Einstein condensation of an ideal Bose gas, the population density of the excited and ground state particles are also presented.Heart capacity of an ideal Bose gas and total internal energy are also calculated and the results presented graphically.The results obtained are consistent with those found in the literatures.
An exact expression for the single-particle reduced density matrix of a translationally invariant system of bosons is obtained, as an average of a functional of the density and current operators. An approximation scheme, based on small fluctuations in density and current, is used in its lowest order, to approximate the single-particle reduced density matrix and the condensate density by a functional of the structure factor, which is a measurable quantity. This is done for the ground state and for finite temperatures. The resulting expressions for the condensate density are discussed. For the ideal gas the result is correct below the condensation temperature. For the case of an interacting liquid, Bose-Einstein condensation exists, in the ground states for two- and three-dimensional systems, and for finite temperatures it does exist only in three dimensions. The present experimental knowledge of the structure factor enables only a crude estimate of the condensed fraction which is found to be 0.1-0.2.
The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∊/ k , where Δ∊ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest-energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermodynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced. From this we obtain the density profile and the velocity distribution of particles, and use these to determine the shape of an anisotropic cloud after free expansion. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. Finally, Sec. 2.5 treats corrections to the transition temperature due to a finite particle number.