Collective behaviors of an electron gas in the mean-field regime

TL;DR

This paper analyzes the momentum distribution of an electron gas in a three-dimensional torus using mean-field approximations, extending previous formulas to high and metallic densities and applying to general singular potentials.

Contribution

It provides a mean-field analogue of momentum distribution formulas for electron gases, generalizing previous results and applying to a broad class of potentials.

Findings

Derived mean-field momentum distribution formulas for high and metallic densities.

Extended analysis to general singular potentials beyond Coulomb.

Connected results to recent independent studies in the field.

Abstract

In this paper, we study the momentum distribution of an electron gas in a $3$-dimensional torus. The goal is to compute the occupation number of Fourier modes for some trial state obtained through random phase approximation. We obtain the mean-field analogue of momentum distribution formulas for electron gas in [Daniel and Voskov, Phys. Rev. \textbf{120}, (1960)] in high density limit and [Lam, Phys. Rev. \textbf{3}, (1971)] at metallic density. The analysis in the present paper is majorly based on the work [Christiansen, Hainzl, Nam, Comm. Math. Phys. \textbf{401}, (2023)]. Our findings are related to recent results obtained independently by Benedikter, Lill and Naidu, and the analysis applies to a general class of singular potentials rather than just the Coulomb case.