Higher Dimensional Coulomb Gases and Renormalized Energy Functionals

TL;DR

This paper develops a new renormalized energy functional to analyze fluctuations and energy in high-dimensional Coulomb gases, extending previous two-dimensional results to higher dimensions.

Contribution

It introduces a novel renormalized energy functional for Coulomb gases in dimensions greater than two, enabling detailed analysis of fluctuations and energy beyond mean-field approximation.

Findings

Derived the next-to-leading order term in ground state energy.

Provided a method to compute Coulomb energy of jellium in any dimension.

Extended fluctuation and energy estimates to higher dimensions.

Abstract

We consider a classical system of n charged particles in an external confining potential, in any dimension d larger than 2. The particles interact via pairwise repulsive Coulomb forces and the coupling parameter scales like the inverse of n (mean-field scaling). By a suitable splitting of the Hamiltonian, we extract the next to leading order term in the ground state energy, beyond the mean-field limit. We show that this next order term, which characterizes the fluctuations of the system, is governed by a new "renormalized energy" functional providing a way to compute the total Coulomb energy of a jellium (i.e. an infinite set of point charges screened by a uniform neutralizing background), in any dimension. The renormalization that cuts out the infinite part of the energy is achieved by smearing out the point charges at a small scale, as in Onsager's lemma. We obtain consequences for the statistical mechanics of the Coulomb gas: next to leading order asymptotic expansion of the free energy or partition function, characterizations of the Gibbs measures, estimates on the local charge fluctuations and factorization estimates for reduced densities. This extends results of Sandier and Serfaty to dimension higher than two by an alternative approach.