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Chinese Physics B

Thermodynamic stability of a weakly interacting Fermi gas trapped in a harmonic potential

Men Fu-Dian, Liu Hui, Zhu Hou-Yu

Published Sep 1, 2008• 12 citations
Publisher

Abstract

Based on the theoretical results derived from pseudopotential method and local approximation, this paper studies the thermodynamic stability of a weakly interacting Fermi gas trapped in a harmonic potential by using analytical method of thermodynamics. The effects of the interparticle interactions as well as external potential on the thermodynamic stability of the system are discussed. It is shown that the system is stable as for the complete average, but as for local parts, the system is unstable anywhere. This instability shows that the stability conditions of mechanics cannot be satisfied anywhere, and the stability conditions of thermostatics cannot be satisfied somewhere. In addition, the interactions and external potential have direct effects on the local stability of the system.

DOI
10.1088/1674-1056/17/9/015

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Based on the theoretical results derived from pseudopotential method and local approximation, this paper studies the thermodynamic stability of a weakly interacting Fermi gas trapped in a harmonic potential by using analytical method of thermodynamics. The effects of the interparticle interactions as well as external potential on the thermodynamic stability of the system are discussed. It is shown that the system is stable as for the complete average, but as for local parts, the system is unstable anywhere. This instability shows that the stability conditions of mechanics cannot be satisfied anywhere, and the stability conditions of thermostatics cannot be satisfied somewhere. In addition, the interactions and external potential have direct effects on the local stability of the system.
The energy density of an interacting Fermi gas in a harmonic potential is derived within localdensity approximation framework,in which the gas is assumed at each point to be locally homogenous. Based on the derived expressions,the thermodynamic stability conditions of trapped gas at zero temperature are studied,and the analytical expressions of stability conditions for d=l,2,3 dimensions are given. Furthermore,the stability phase diagrams are plotted for different dimension.It is shown that it is always stable when the system has repulsive interactions or has no interactions in condition of certain density of particle number,no mater what the dimension is.However,the system with attractive interactions is stable only under the condition of weak interaction strength.For d=1 and d = 3 cases,the critical values of interaction strength have close relationship with particle number density,but as for d= 2 case,the corresponding critical value does not depend on the particle number density explicitly.
On the basis of the results derived from pseudopotential method and ensemble theory, thermal stability of a weakly interacting Fermi gas in a weak magnetic field is studied by using analytical method of thermodynamics. The exact analytical expressions of stability conditions at different temperatures are given, and the effects of interactions as well as magnetic field on the stability of the system are discussed. It is shown that there is an upper-limit magnetic field for the stability of the system at low temperatures, and there is an attractive dividing value at high temperatures. If attractive interaction is lower than the critical value, the stability of the system has no request for magnetic field, but if attractive interaction is higher than the dividing value, a lower-limit magnetic field exists for the stability of the system.
The analytical expressions of internal energy, chemical potential and heat capacity of the little parameter r for a weakly interacting Fermi gas in weak magnetic field are derived by using “pseudopotential" method and ensemble theory. Based on the derived expressions, the thermodynamic properties of a weakly interacting Fermi gas in weak magnetic field at both high and low temperatures are given. The effects of magnetic field and interparticle interactions on the thermodynamic properties are discussed. The difference in the effects of magnetic trap and three-dimensional harmonic trap on the properties of the system and their reasons are analyzed.
Based on the pseudopotential method and the local-density approximation, the thermodynamic properties of a weakly interacting Fermi gas in a strong magnetic filed are studied, the integrated analytical expressions of thermodynamic quantities of the system are derived, and the effects of magnetic field as well as interparticle interactions on the thermodynamic properties of the system are analyzed. It is shown that at both high and low temperatures, magnetic field may adjust the effects of interacting. At low temperatures, magnetic field can lower the chemical potential, total energy and heat capacity of the system compared with the situation of Fermi gas in the absence of the magnetic field. The repulsive interactions may increase the chemical potential, but reduce the total energy and heat capacity of the system compared with the situation of non-interacting Fermi gas. At high temperatures, magnetic field as well as repulsive interactions can reduce the total energy and increase heat capacity of the system, moreover, strong magnetic field may change the effects of interaction on the total energy and the heat capacity of the system.