标题： 平均场理论用于一类一般的短程相互作用泛函 显示英文标题

标题： Mean field theory for a general class of short-range interaction functionals

摘要： 在$N$个相互作用粒子的模型中，如密度泛函理论或人群运动中的$\R^d$，排斥成本通常由一个两点函数$c_\e(x,y) =\ell\Big(\frac{|x-y|}{\e}\Big)$描述，其中$\ell: \R_+ \to [0,\infty]$在无穷远处趋于零，参数$\e>0$调整相互作用距离。 在本文中，我们在仅假设$\exists r_0>0 \ : \ \int_{r_0}^\infty \ell(r) r^{d-1}\, dr <+\infty$的情况下，确定了此类模型在短程区域$\e\ll 1$的平均场能量。 这扩展了最近在均匀情况下获得的结果\cite{hardin2021, HardSerfLebl, Lewin}，其中$\ell(r) = r^{-s}$为$s>d$。 显示英文摘要

摘要： In models of $N$ interacting particles in $\R^d$ as in Density Functional Theory or crowd motion, the repulsive cost is usually described by a two-point function $c_\e(x,y) =\ell\Big(\frac{|x-y|}{\e}\Big)$ where $\ell: \R_+ \to [0,\infty]$ is decreasing to zero at infinity and parameter $\e>0$ scales the interaction distance. In this paper we identify the mean-field energy of such a model in the short-range regime $\e\ll 1$ under the sole assumption that $\exists r_0>0 \ : \ \int_{r_0}^\infty \ell(r) r^{d-1}\, dr <+\infty$. This extends recent results \cite{hardin2021, HardSerfLebl, Lewin} obtained in the homogeneous case $\ell(r) = r^{-s}$ where $s>d$.