Serfaty Sylvia. Coulomb Gases and Ginzburg-Landau Vortices

European Mathematical Society, 2015. — 167 p. — (Zurich Lectures in Advanced Mathematics 21). — ISBN 978-3-03719-152-1.The topic of this book is systems of points in Coulomb interaction, in particular, the classical Coulomb gas, and vortices in the Ginzburg–Landau model of superconductivity. The classical Coulomb and Log gases are classical statistical mechanics models, which have seen important developments in the mathematical literature due to their connection with random matrices and approximation theory. At low temperature, these systems are expected to “cristallize” to so-called Fekete sets, which exhibit microscopically a lattice structure.
The Ginzburg–Landau model, on the other hand, describes superconductors. In superconducting materials subjected to an external magnetic field, densely packed point vortices emerge, forming perfect triangular lattice patterns, so-called Abrikosov lattices.The leading order behavior of the Coulomb gas
Splitting the Hamiltonian
Definition(s) and properties of renormalized energy
DerivingW as the large n limit: lower bound via a general abstract method
DerivingW as the large n limit: screening, upper bound, and consequences
The Ginzburg–Landau functional: presentation and heuristics
Main mathematical tools for Ginzburg–Landau
The leading order behavior for Ginzburg–Landau
The splitting and the next order behavior for Ginzburg–Landau