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Communications in Theoretical Physics

Effect of Spatial Dimension and External Potential on Joule–Thomson Coefficients of Ideal Bose Gases

Yuan Du-Qi, Canjun Wang

Published Apr 1, 2010• 2 citations

Abstract

Based on the form of the n-dimensional generic power-law potential, the state equation and the heat capacity, the analytical expressions of the Joule–Thomson coefficient (JTC) for an ideal Bose gas are derived in n-dimensional potential. The effect of the spatial dimension and the external potential on the JTC are discussed, respectively. These results show that: (i) For the free ideal Bose gas, when n/s ≤ 2 (n is the spatial dimension, s is the momentum index in the relation between the energy and the momentum), and T → T C (T C is the critical temperature), the JTC can obviously improve by means of changing the throttle valve's shape and decreasing the spatial dimension of gases. (ii) For the inhomogeneous external potential, the discriminant Δ = [1 – ∏[ n i = 1 (kT/ϖ i ) 1/t i Γ(1/t i + 1)] (k is the Boltzmann Constant, T is the thermodynamic temperature, ϖ i is the external field's energy), is obtained. The potential makes the JTC increase when Δ > 0, on the contrary, it makes the JTC decrease when Δ < 0. (iii) In the homogenous strong external potential, the JTC gets the maximum on the condition of kT/ϖ i < 1.

DOI: 10.1088/0253-6102/53/4/17

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Based on the form of the n-dimensional generic power-law potential, the state equation and the heat capacity, the analytical expressions of the Joule–Thomson coefficient (JTC) for an ideal Bose gas are derived in n-dimensional potential. The effect of the spatial dimension and the external potential on the JTC are discussed, respectively. These results show that: (i) For the free ideal Bose gas, when n/s ≤ 2 (n is the spatial dimension, s is the momentum index in the relation between the energy and the momentum), and T → T C (T C is the critical temperature), the JTC can obviously improve by means of changing the throttle valve's shape and decreasing the spatial dimension of gases. (ii) For the inhomogeneous external potential, the discriminant Δ = [1 – ∏[ n i = 1 (kT/ϖ i ) 1/t i Γ(1/t i + 1)] (k is the Boltzmann Constant, T is the thermodynamic temperature, ϖ i is the external field's energy), is obtained. The potential makes the JTC increase when Δ > 0, on the contrary, it makes the JTC decrease when Δ < 0. (iii) In the homogenous strong external potential, the JTC gets the maximum on the condition of kT/ϖ i < 1.

Thermodynamic properties of ideal Bose gas trapped in an external generic power law potential are investigated systematically from the grand thermodynamic potential in

d

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.

Considering the changes of internal energy caused by weak degeneracy and interaction between molecules,the internal energy of weak degenerate non-ideal Bose gas,partition function and equation of state were brought out.Using the results and relevant thermodynamic relations,Joule-Thomson coefficient was deduced.The results show that the Joule-Thomson coefficient of weak degenerate non-ideal Bose gas system is influenced by weak degeneracy and interaction between molecules.If it's a repulsive system,after throttling process,not only the change direction of the system temperature is influenced by weak degeneracy and interaction between molecular,but also the critical state of the temperature changing direction is influenced by them.If it's an attractive system,after throttling process,change of the system temperature has a limit,it's impossible to reach this limit temperature by throttling process.

The Joule–Thomson effect reflects the interaction among constituent particles of macroscopic system. For classical ideal gas, the corresponding Joule–Thomson coefficient is vanishing while it is non-zero for ideal quantum gas due to the quantum degeneracy. In recent years, much attention is paid to the unitary Fermi gas with infinite two-body scattering length. According to universal analysis, the thermodynamical law of unitary Fermi gas is similar to that of non-interacting ideal gas, which can be explored by the virial theorem P = 2 E /3 V . Based on previous works, we further study the unitary Fermi gas properties. The effective chemical potential is introduced to characterize the nonlinear levels crossing effects in a strongly interacting medium. The changing behavior of the rescaled Joule–Thomson coefficient according to temperature manifests a quite different behavior from that for ideal Fermi gas.

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.

The properties of an ideal Fermi gas in an external potential in any dimensional space are studied, based on the semiclassical (Thomas-Fermi) approximation. The general analytical expressions of the total particle number

N,

density of states $D(\ensuremath{\varepsilon}),$ Fermi energy

{E}_{F},

total energy

E,

and heat capacity

C

have been derived, where

N,

E,

and

C

are expressed by the Fermi integration. Moreover, the analytical expressions of the total energy

E,

chemical potential \ensuremath{\mu}, and heat capacity

C

in the high- and low-temperature approximations are given. From these results, how the characteristics of the Fermi gas depend on an external potential and the dimension of space is discussed.