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Journal of Physics A Mathematical and Theoretical

A new approach to treat the problems of trapped ideal gases

Guozhen Su, Yanhua Cai, Jincan Chen

Published Mar 2, 2009• 2 citations
Publisher

Abstract

By comparing the thermodynamic properties of ideal atomic gases in a power-law potential and in a rigid box, it is found that the power-law potential is equivalent to the rigid box as far as the macroscopic behaviors of the system are concerned. The dimensionality and volume of the equivalent box are dependent on the parameters charactering the power-law potential. This equivalent relation enables us to treat a trapped ideal gas as a free one, and consequently, several useful conclusions of the trapped-gas system can be easily derived from the corresponding results of the free-gas system.

DOI
10.1088/1751-8113/42/12/125003

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By comparing the thermodynamic properties of ideal atomic gases in a power-law potential and in a rigid box, it is found that the power-law potential is equivalent to the rigid box as far as the macroscopic behaviors of the system are concerned. The dimensionality and volume of the equivalent box are dependent on the parameters charactering the power-law potential. This equivalent relation enables us to treat a trapped ideal gas as a free one, and consequently, several useful conclusions of the trapped-gas system can be easily derived from the corresponding results of the free-gas system.
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.
We investigate ideal quantum gases in D-dimensional space and confined in a generic external potential by using the semiclassical approximation. In particular, we derive density of states, density profiles and critical temperatures for Fermions and Bosons trapped in isotropic power-law potentials. From such results, one can easily obtain those of quantum gases in a rigid box and in a harmonic trap. Finally, we show that the Bose–Einstein condensation can set up in a confining power-law potential if and only if D/2+D/n>1, where D is the space dimension and n is the power-law exponent.
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 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.
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.