Systems with Coulomb interactions
Journées équations aux dérivées partielles (2014), article no. 9, 23 p.

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.

DOI: 10.5802/jedp.112
Serfaty, Sylvia 1

1 UPMC Université Paris 6 UMR 7598 Laboratoire Jacques-Louis Lions Paris, F-75005 France & Courant Institute New York University 251 Mercer street NY NY 10012, USA
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Serfaty, Sylvia. Systems with Coulomb interactions. Journées équations aux dérivées partielles (2014), article  no. 9, 23 p. doi : 10.5802/jedp.112.

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