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

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 \[Daniel and Voskov, Phys. Rev. 120, (1960)\] in high density limit and \[Lam, Phys. Rev. 3, (1971)\] at metallic density. The analysis in the present paper is majorly based on the work \[Christiansen, Hainzl, Nam, Comm. Math. Phys. 401, (2023)\]. 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.