Large systems with Coulomb interactions: Variational study and statistical mechanics

Sylvia Serfaty

Research output: Contribution to journalArticlepeer-review

Abstract

Systems with Coulomb and logarithmic interactions arise in various settings: an instance is the classical Coulomb gas which in some cases happens to be a random matrix ensemble, another is vortices in the Ginzburg-Landau model of superconductivity, where one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattice patterns named Abrikosov lattices, a third is the study of Fekete points which arise in approximation theory. In this review, we describe tools to study such systems and derive a next order (beyond mean field limit) ''renormalized energy'' that governs microscopic patterns of points. We present the derivation of the limiting problem and the question of its minimization and its link with the Abrikosov lattice and crystallization questions. We also discuss generalizations to Riesz interaction energies and the statistical mechanics of such systems. This is based on joint works with Etienne Sandier, Nicolas Rougerie, Simona Rota Nodari, Mircea Petrache, and Thomas Leblé.

Original languageEnglish (US)
Pages (from-to)247-278
Number of pages32
JournalPortugaliae Mathematica
Volume73
Issue number4
DOIs
StatePublished - 2016

Keywords

  • Abrikosov lattices
  • Coulomb gases
  • Coulomb systems
  • Fekete points
  • Large Deviations Principle
  • Log gases
  • Random matrices

ASJC Scopus subject areas

  • Mathematics(all)

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