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