Constructive Approximation

, Volume 47, Issue 1, pp 163–210 | Cite as

Equidistribution of Jellium Energy for Coulomb and Riesz Interactions

  • Mircea PetracheEmail author
  • Simona Rota Nodari


For general dimension d, we prove the equidistribution of energy at the micro-scale in \(\mathbb {R}^{d}\) for the optimal point configurations appearing in Coulomb gases at zero temperature. More precisely, we show that, after blow-up at the scale corresponding to the interparticle distance, the value of the energy in any large enough set is completely determined by the macroscopic density of points. This uses the “jellium energy” which was previously shown to control the next-order term in the large particle number asymptotics of the minimum energy. As a corollary, we obtain sharp error bounds on the discrepancy between the number of points and its expected average of optimal point configurations for Coulomb gases, extending previous results valid only for 2-dimensional log-gases. For Riesz gases with interaction potentials \(g(x)=|x|^{-s}, s\in ]\min \{0,d-2\},d[\), we prove the same equidistribution result under an extra hypothesis on the decay of the localized energy, which we conjecture to hold for minimizing configurations. In this case, we use the Caffarelli–Silvestre description of the nonlocal fractional Laplacians in \(\mathbb {R}^{d}\) to render the problem local.


Coulomb gases Riesz gases Renormalized energy Equidistribution Crystallization 

Mathematics Subject Classification

82B05 82B21 82D05 


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Max-Planck Institute for MathematicsBonnGermany
  2. 2.Institut de Mathématiques de Bourgogne (IMB), CNRS, UMR 5584Université Bourgogne Franche-ComtéDijonFrance

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