Abstract

This describes joint work with Sylvia Serfaty and Cyrill Muratov. We study the asymptotic behavior of the screened sharp interface Ohta-Kawasaki energy in dimension 2 using the framework of \Gamma-convergence. In that model, two phases appear, and they interact via a nonlocal Coulomb type energy. We focus on the regime where one of the phases has very small volume fraction, thus creating “droplets” of that phase in a sea of the other phase. We consider perturbations to the critical volume fraction where droplets first appear, show the number of droplets increases monotonically with respect to the perturbation factor, and describe their arrangement in all regimes, whether their number is bounded or unbounded. When their number is unbounded, the most interesting case we compute the \Gamma limit of the “zeroth” order energy and yield averaged information for almost minimizers, namely that the density of droplets should be uniform. We then go to the next order, and derive a next order \Gamma-limit energy, which is exactly the “Coulombian renormalized energy W” introduced in the work of Sandier/Serfaty, and obtained there as a limiting interaction energy for vortices in Ginzburg-Landau. The derivation is based on their abstract scheme, that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing at all to the Euler-Lagrange equation, we establish here for all configurations which have “almost minimal energy,” the asymptotic roundness and radius of the droplets as done by Muratov, and the fact that they asymptotically shrink to points whose arrangement should minimize the renormalized energy W, in some averaged sense. This leads to expecting to see hexagonal lattices of droplets.