Hydrodynamic limit and stochastic PDEs related to interface motion

Abstract

The hydrodynamic limit gives a link between microscopic and macroscopic systems via a space–time scaling. Its notable feature is the averaging effect due to the local ergodicity under the local equilibria. In this article, as the microscopic system, we consider several types of interacting particle systems, in which particles perform random walks with interaction. We derive, under the hydrodynamic limit or its nonlinear fluctuation limit, three different objects: the motion by mean curvature, Stefan free boundary problem, and coupled KPZ equation. These are all related to the interface motion. The Boltzmann–Gibbs principle plays a fundamental role. We discuss the coupled KPZ equation from the aspect of singular SPDEs and renormalizations. Ginzburg–Landau $\nabla \varphi$-interface model, stochastic motion by mean curvature, and stochastic eight-vertex model are also briefly discussed.