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

Abstract

In models of $N$ interacting particles in ${\mathbb{R}}^d$ 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 $\ell :{\mathbb{R}}_{+}\to [0,\infty ]$ 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 $s>d$.