Generalized motion by mean curvature as a macroscopic limit of stochastic ising models with long range interactions and Glauber dynamics

Abstract

We study themacroscopic limit of an appropriately rescaledstochastic Ising model withlong range interactions evolving withGlauber dynamics as well as the correspondingmean field equation, which is nonlinear and nonlocal. In the limit we obtain an interface evolving with normal velocity ϑk, wherek isthe mean curvature and thetransport coefficient ϑ is identified by aneffective Green-Kubo type formula. The above assertions are valid for all positive times, the motion of the interface being interpreted in theviscosity sense after the onset of the geometric singularities.