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

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 $c_{\text{e}}(x,y)=\mathrm{\ell }\left(\frac{\mathrm{\mid }x-y\mathrm{\mid }}{\text{e}}\right)$ where $\mathrm{\ell }:{\mathbb{R}}_{+}\to [0,\mathrm{\infty }]$ is decreasing to zero at infinity and parameter $\text{e}>0$ scales the interaction distance. In this paper we identify the mean-field energy of such a model in the short-range regime $\text{e}\ll 1$ under the sole assumption that $\mathrm{\exists }{r}_0>0:{\int }_{r_0}^{\mathrm{\infty }}\mathrm{\ell }(r)r^{d-1}dr<+\mathrm{\infty }$. This extends recent results \cite{hardin2021, HardSerfLebl, Lewin} obtained in the homogeneous case $\mathrm{\ell }(r)=r^{-s}$ where $s>d$.