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Nuclear Science and Engineering

Kinetic Theory Study of Pressure and Equation of State in a Strongly Degenerate Fermi Gas

Vincenzo G. Molinari, D. Mostacci, Francesco Teodori

Published Apr 19, 2023• 0 citations
Publisher

Abstract

Starting from the Fermi-Dirac distribution fuunction, Kinetic Theory prescriptions are applied to express number density and pressure as integrals, which are then solved for nonvanishing temperature. The results are then combined to arrive at an equation of state, given in closed form, yielding pressure as a function of number density.
DOI
10.1080/00295639.2023.2194199

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distribution in an external potential directly probes the equation of state under the local density approximation. The equation of state is obtained in two ways: from the local density of the gas, reconstructed using the inverse Abel transformation; and from the local pressure, directly deduced from the 2D column density via an exact relation including the Gaussian potential. Regions of low density allow us to extract the chemical potential and the temperature using the virial expansion of the equation of state. We validate our method using the non-interacting Fermi gas. The experimental results are compared to recent Monte-Carlo calculations. 1 AFOSR-MURI, DARPA-OLE, Alfred P. Sloan Foundation, NSF
The singlet and pair distribution functions of a Fermi gas are evaluated for low temperatures. It is assumed that the particles interact with a hard-sphere potential and have an arbitrary spin, and the evaluation is made to first order in the hard-sphere diameter. The binary-kernel method developed by Siegert, Lee, and Yang is used and the cluster-expansion theory given recently by Isihara and Yee for the distribution functions is applied. The pair distribution function is determined by a dimensionless variable $\frac{r}{\ensuremath{\lambda}}$, with $\ensuremath{\lambda}$ being the thermal de Broglie wavelength. Thus, changing the distance corresponds to varying the temperature. Moreover, the pair distribution function depends on the density. Therefore, its asymptotic expression for $T\ensuremath{\rightarrow}0$ depends on the ways to approach the limit. If we keep the density finite and reduce the temperature, the gas will be degenerate when $\ensuremath{\lambda}$ is of order of ${n}^{\ensuremath{-}\frac{1}{3}}$, where nnn is the number density. The pair distribution function for this case is an oscillating function of the Fermi momentum. On the other hand, if we reduce the density first and then decrease the temperature, the pair distribution function decays as ${r}^{\ensuremath{-}4}$, and the free-particle diagrams give a contribution of the same decay.
The momentum distribution in a low-temperature Fermi gas is investigated using the methods of quantum statistics developed by Lee and Yang together with the linked-pair expansions of the previous paper. It is shown that in order to determine the momentum distribution at very low temperatures two coupled integral equations must be considered, one in momentum variables and due to Lee and Yang, and the other in temperature variables. It is also shown that the dominant low-temperature behavior of the momentum distribution can be extracted in terms of a certain function ${\ensuremath{\nu}}^{\ensuremath{'}}(\mathrm{k})$. For a low-density Fermi gas with strong, short-range, two-body interactions, it is shown to third order in the scattering parameters of the interaction that at T=0T=0T=0 the function ${\ensuremath{\nu}}^{\ensuremath{'}}(\mathrm{k})$ is equal to the free particle momentum distribution. Also, the energy and other thermodynamic quantities are expressed in terms of ${\ensuremath{\nu}}^{\ensuremath{'}}(\mathrm{k})$, so that the theory permits a generalization of perturbation theoretic results to nonzero temperatures. The ground-state energy, momentum distribution, and thermodynamic potential are calculated to third order in the scattering parameters.