Particle approximation of nonlocal interaction energies

TL;DR

This paper proves that discretized particle systems approximate nonlocal Riesz energies through $ ext{Gamma}$-convergence and establishes the existence of minimal particle configurations.

Contribution

It introduces a general framework for particle discretizations of nonlocal energies and proves their convergence and existence of minimizers.

Findings

Discretized energies $ ext{Gamma}$-converge to the continuous Riesz energies.

Existence of minimal particle configurations is established.

The results apply to general interaction kernels and settings.

Abstract

We consider Riesz-type nonlocal energies with general interaction kernels and their discretizations related to particle systems. We prove that the discretized energies $\Gamma$-converge in the weak-$*$ topology to the Riesz functional defined over the space of probability measures. We also address the minimization problem for the discretized energies, and prove the existence of minimal configurations of particles in a very general and natural setting.