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

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 $s>d$.