Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Sichuan University

The stability conditions of interacting Fermions trapped in a harmonic potential

Yu Wang

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

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.

Related papers

The instability conditions of a weakly interacting Fermi gas trapped in weak magnetic field

Men Fu-Dian, Liu Hui

In this paper the analytical expression of free energy expressed by small parameter r of a weakly interacting Fermi gas trapped in weak magnetic field is derived by using `the maximum approximation' method and the ensemble theory. Based on the derived expression, the exact instability conditions of a weakly interacting Fermi gas trapped in weak magnetic field at both high and low temperatures are given. From the instability conditions we get the following two results. (1) At the whole low-temperature extent, whether the interactions are repulsive or attractive with (α n +4ε F /3) ( n and ε F denote the particle-number density and the Fermi energy respectively, α = 4π a ℏ 2 / m , and a is s-wave scattering length) positive, there is a lower-limit magnetic field of instability; in addition, there is an upper-limit magnetic field for the system of attractive interactions with (α n +4ε F /3) negative. (2) At the whole high-temperature extent, the system with repulsive interactions is always stable, but for the system with attractive interactions, the greater the scattering length of attractive interactions | a | is, the stronger the magnetic field is and the larger the particle-number density is, the bigger the possibility of instability in the system will be.

Chinese Physics•Nov 2, 2006•5 citations

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

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

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.

Chinese Physics B•Sep 1, 2008•12 citations

Thermodynamic Properties of A Weakly Interacting Fermi Gas Trapped in Homogenous GravitationaI Field

Men Fu-dian

The properties of a weakly interactlng Fermi gas in homogenous gravitational field are studied when the temperature is higher than Fermi temperature,based on the local-density approximation(LDA).The general analytical expressions of several important thermodynamic quantities are derived and the rules of distribution of particle-number density and energy density of the system are analyzed.Both of systems in homogenous gravitational field with and without internal interactlons are compared.The effects of gravitational field and inter-particle interactions on properties of Fermi gas are discussed.

Jiangxi kexue•Jan 1, 2007•0 citations

Thermodynamic properties of a weakly interacting Fermi gas

Guozhen Su, Lixuan Chen, 厦门大学物理系，厦门 361005

The analytical expressions of chemical potential, total energy and heat capacity at a constant volume for a weakly interacting Fermi gas in the absence of external potential are derived by using “pseudopotential” method. Based on the derived expressions, the thermodynamic properties of a weakly interacting Fermi gas trapped in a harmonic potential are studied under the local-density approximation. The effects of interparticle interactions on the properties of the systems are discussed.

Acta Physica Sinica•Jan 1, 2004•13 citations

Thermodynamic Properties of a Weakly Interacting Bose Gas Trapped in a Harmonic Potential

Lixuan Chen

The thermodynamic properties of a weakly interacting Bose gas trapped in a harmonic potential are studied based on the local-density approximation (LDA). The spatial distributions of particle number and energy and the expressions of some important thermodynamic quantities are derived analytically. The effects of the interatomic interaction on the thermodynamic properties of the system are discussed.

Journal of Xiamen University•Jan 1, 2003•1 citation

Simple variational approach for an interacting Fermi trapped gas

Rocío Jáuregui, R. Paredes, G. Toledo Sánchez

Quantum states of a two-component Fermi trapped gas are described by introducing an effective trap frequency, determined via variational techniques. Closed expressions for the contribution of a contact interaction potential to the total energy and the pairing interaction are derived. They are valid for both few and large number of particles, given the discrete nature of the formulation, and therefore richer than the continuous expressions, which are perfectly matched. Pairing energies within a shell are explicitly evaluated and its allowed values at a given energy level delimited. We show the importance of the interaction over the trap energy as the number of particles N grows and the temperature decreases. At zero temperature we find a polynomial dependence of the interaction energy on the Fermi energy, whose dominant term at large N corresponds to the mean-field approximation result. In addition, the role of the strength of an attractive potential on the total energy is exhibited.

arXiv:cond-mat/0405045•Physical Review A•Jan 1, 2004•2 citations
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.
In this paper the analytical expression of free energy expressed by small parameter r of a weakly interacting Fermi gas trapped in weak magnetic field is derived by using `the maximum approximation' method and the ensemble theory. Based on the derived expression, the exact instability conditions of a weakly interacting Fermi gas trapped in weak magnetic field at both high and low temperatures are given. From the instability conditions we get the following two results. (1) At the whole low-temperature extent, whether the interactions are repulsive or attractive with (α n +4ε F /3) ( n and ε F denote the particle-number density and the Fermi energy respectively, α = 4π a ℏ 2 / m , and a is s-wave scattering length) positive, there is a lower-limit magnetic field of instability; in addition, there is an upper-limit magnetic field for the system of attractive interactions with (α n +4ε F /3) negative. (2) At the whole high-temperature extent, the system with repulsive interactions is always stable, but for the system with attractive interactions, the greater the scattering length of attractive interactions | a | is, the stronger the magnetic field is and the larger the particle-number density is, the bigger the possibility of instability in the system will be.
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 properties of a weakly interactlng Fermi gas in homogenous gravitational field are studied when the temperature is higher than Fermi temperature,based on the local-density approximation(LDA).The general analytical expressions of several important thermodynamic quantities are derived and the rules of distribution of particle-number density and energy density of the system are analyzed.Both of systems in homogenous gravitational field with and without internal interactlons are compared.The effects of gravitational field and inter-particle interactions on properties of Fermi gas are discussed.
Quantum states of a two-component Fermi trapped gas are described by introducing an effective trap frequency, determined via variational techniques. Closed expressions for the contribution of a contact interaction potential to the total energy and the pairing interaction are derived. They are valid for both few and large number of particles, given the discrete nature of the formulation, and therefore richer than the continuous expressions, which are perfectly matched. Pairing energies within a shell are explicitly evaluated and its allowed values at a given energy level delimited. We show the importance of the interaction over the trap energy as the number of particles N grows and the temperature decreases. At zero temperature we find a polynomial dependence of the interaction energy on the Fermi energy, whose dominant term at large N corresponds to the mean-field approximation result. In addition, the role of the strength of an attractive potential on the total energy is exhibited.