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arXiv:2606.19301 · American Journal of Physics

An ideal Fermi gas under uniform gravity

Pattarapon Tanalikhit, Wittaya Kanchanapusakit

Published Apr 22, 2026• 0 citations
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Abstract

We consider an ideal Fermi gas in a container subject to a uniform gravitational field at absolute zero temperature. Under a semiclassical approximation, we examine the density profile of the particles and derive an expression for the chemical potential. A critical value of the chemical potential separates the weak- and strong-gravity regimes, and the kinetic and potential energies of the Fermi gas are determined in both regimes.

DOI
10.1119/5.0300463

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We consider an ideal Fermi gas in a container subject to a uniform gravitational field at absolute zero temperature. Under a semiclassical approximation, we examine the density profile of the particles and derive an expression for the chemical potential. A critical value of the chemical potential separates the weak- and strong-gravity regimes, and the kinetic and potential energies of the Fermi gas are determined in both regimes.
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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,N,N, density of states $D(\ensuremath{\varepsilon}),$ Fermi energy EF,{E}_{F},EF​, total energy E,E,E, and heat capacity CCC have been derived, where N,N,N, E,E,E, and CCC are expressed by the Fermi integration. Moreover, the analytical expressions of the total energy E,E,E, chemical potential \ensuremath{\mu}, and heat capacity CCC 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.
The study is on the thermodynamic properties of finite gravitational field weak degenerate ideal Fermi gas.Base on the Thomas-Fermi semiclassical approximation,after deriving the density of states of an ideal Fermi gas in finite gravitational field,the analytic formula of the chemical potential,internal energy and pressure under the condition of weak degeneration is obtained in this paper.The specific impact of the gravity field on the thermodynamic properties of weakly degenerate ideal Fermi gas is analyzed.The result showed that finite gravitational field has effect on the thermodynamic property of the system.
The low-temperature properties of an idea and a weakly interacting Fermi gas in homogenous gravitational field are studied,based on the semiclassical(Thomas-Fermi) approximation and local-density approximation.The general analytical expressions of several thermodynamic quantities have been derived.The free system,idea system in homogenous gravitational field,and interacting system in homogenous gravitational field are compared.The effects of gravitational field and interparticle interactions on properties of Fermi gas are discussed.
Abstract: We present an accurate analytical formulation to determine the chemical potential of an ultrarelativistic spherical degenerate Fermi gas interacting with nuclear matter by examining rigorously the three terms of the general mathematical expression for the chemical-potential energy namely, the Fermi energy, the optical-potential energy, and a third term which is negligible if the Fermi gas is dilute. In fact, we derive the above expression as a function of distance and absolute temperature. The Fermi velocity is also determined. In addition, essential aspects of the optical potential are discussed.