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