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Physical Review A

Pseudovolume and equation of state for an ideal trapped Bose gas in n dimensions

Zijun Yan

Published May 16, 2000• 22 citations
Publisher

Abstract

The equation of state of a trapped atomic gas, which is an important and fundamental property for a thermodynamic system, has been elusive to date. We address here this long-standing problem by introducing the concept of pseudovolume which can play an important role as the volume. The equation of state of an ideal Bose gas trapped in an n-dimensional generic power-law potential is derived. It is not only a long-sought equation of state for the trapped Bose gas but also including that of a free Bose gas. Moreover, we find that the thermal wavelength for the particles in n-dimensional space with any kinematic characteristic must be redefined.

DOI
10.1103/physreva.61.063607

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The equation of state of a trapped atomic gas, which is an important and fundamental property for a thermodynamic system, has been elusive to date. We address here this long-standing problem by introducing the concept of pseudovolume which can play an important role as the volume. The equation of state of an ideal Bose gas trapped in an n-dimensional generic power-law potential is derived. It is not only a long-sought equation of state for the trapped Bose gas but also including that of a free Bose gas. Moreover, we find that the thermal wavelength for the particles in n-dimensional space with any kinematic characteristic must be redefined.
By introducing the concept of pseudovolume, an ideal Bose gas trapped in a harmonic potential is treated as a free one. This means the equation of state of the trapped system, which has been elusive to date, can be derived and all thermodynamic functions of the system can be expressed in a compact form similar to those of a free ideal Bose system. Thus, a unified description of the two classes of systems is achieved and the internal relation and difference between the thermodynamic properties of the trapped and free ideal Bose systems are further revealed. Moreover, some significant and novel conclusions are obtained.
The density of states and thermodynamic properties of an ideal gas system trapped in a generic power-law potential in an n -dimensional space are studied. A unified description for the Bose, Fermi and classical gases is given by using the grand potential of the system. Consequently, not only the results in current textbooks of statistical mechanics but also some new general important conclusions, such as the conditions for the occurence of Bose-Eisntein condensation in a trapped Bose system in any dimensional space, can be derived directly using the results of this paper.
The critical temperature for Bose-Einstein condensation, ground-state fraction, and heat capacity of an ideal gas of Bose particles that are confined by an n-dimensional generic power-law potential trap are derived. The conditions for Bose-Einstein condensation and the discontinuous conditions of the heat capacity at critical temperature are obtained. All these quantities and conditions are found to vary markedly with the dimensionality of space, characteristics of particles, and shape of the external potential. The results obtained here are universal.
Thermodynamic properties of ideal Bose gas trapped in an external generic power law potential are investigated systematically from the grand thermodynamic potential in ddd dimensional space. The most general conditions for Bose-Einstein condensate and the discontinuous conditions of heat capacity at the critical temperature in presence of generic power law potential are presented in this manuscript. The dependence of the physical quantities on external potential, particle characteristics and space dimensionality are discussed. The more general results obtained in this paper presents an unified illustration of Bose-Einstein condensation of ideal Bose systems as they reduces to the expressions and conclusions available in the literature with appropiate choice of power law exponent.
Abstract The changes in characteristics of Bose condensation of ideal Bose gas due to an external generic power law potential U = ∑ i = 1 d c i | x i / a i | n i are studied carefully. Detailed calculation of Kim et al. (J. Phys. Condens. Matter 11 (1999) 10269) yielded the hierarchy of condensation transitions with changing fractional dimensionality. In this manuscript, some theorems regarding specific heat at constant volume C V are presented. Careful examination of these theorems reveal the existence of hidden hierarchy of the condensation transition in trapped systems as well.