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European Journal of Physics

Treating an ideal trapped Bose gas as a free one

Zijun Yan

Published Apr 26, 2001• 4 citations
Publisher

Abstract

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.

DOI
10.1088/0143-0807/22/3/306

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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.
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 have investigated the thermodynamic behaviour of ideal Bose gases with an arbitrary number of particles confined in a harmonic potential. By taking into account the conservation of the total number N of particles and using a saddle-point approximation, we derive analytically the simple explicit expression of mean occupation number in any state of the finite system. The temperature dependence of the chemical potential, specific heat, and condensate fraction for the trapped finite-size Bose system is obtained numerically. We compare our results with the usual treatment which is based on the grand canonical ensemble. It is shown that there exists a considerable difference between them at sufficiently low temperatures, especially for the relative small numbers of Bose atoms. The finite-size scaling at the transition temperature for the harmonically trapped systems is also discussed. We find that the scaled condensate fractions for various system sizes and temperatures collapse onto a single scaled form.
Abstract From the unified statistical thermodynamics of quantum gases, the virial coefficients of ideal Bose and Fermi gases, trapped under generic power law potential are derived systematically. From the general result of virial coefficients, one can produce the known results in d = 3 and d = 2. But more importantly we found that, the virial coefficients of Bose and Fermi gases become identical (except the second virial coefficient, where the sign is different) when the gases are trapped under harmonic potential in d = 1. This result suggests the equivalence between Bose and Fermi gases established in d = 1 (J. Stat. Phys. DOI 10.1007/s10955-015-1344-4). Also, it is found that the virial coefficients of two-dimensional free Bose (Fermi) gas are equal to the virial coefficients of one-dimensional harmonically trapped Bose (Fermi) gas.
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 thermodynamic properties of an ideal q-boson gas trapped in an n-dimensional harmonic potential are studied, based on the distribution function of q-bosons. The critical temperature Tc,q of Bose-Einstein condensation (BEC) and the heat capacity C of the system are derived analytically. It is shown that for the q-boson gas trapped in a harmonic potential, BEC may occur in any dimension when q ≠ 1, the critical temperature is always higher than that of an ordinary Bose gas (q = 1), and the heat capacity is continuous at Tc,q for a one-dimensional system but discontinuous at Tc,q for a two- or multi-dimensional system.