TY - JOUR

T1 - Higher-Dimensional Coulomb Gases and Renormalized Energy Functionals

AU - Rougerie, Nicolas

AU - Serfaty, Sylvia

N1 - Publisher Copyright:
© 2016 Wiley Periodicals, Inc.

PY - 2016/3/1

Y1 - 2016/3/1

N2 - 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.

AB - 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.

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U2 - 10.1002/cpa.21570

DO - 10.1002/cpa.21570

M3 - Article

AN - SCOPUS:84955331619

VL - 69

SP - 519

EP - 605

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 3

ER -