Higher-Dimensional Coulomb Gases and Renormalized Energy Functionals

We consider a classical system of n charged particles in an external confining potential in any dimension d≥2. The particles interact via pairwise repulsive Coulomb forces and the coupling parameter is of order n^-1 (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 2 by an alternative approach.