Mean Curvature Interface Limit from Glauber+Zero-Range Interacting Particles

Abstract

We derive a continuum mean-curvature flow as a certain hydrodynamic scaling limit of a class of Glauber+Zero-range particle systems. The Zero-range part moves particles while preserving particle numbers, and the Glauber part governs the creation and annihilation of particles and is set to favor two levels of particle density. When the two parts are simultaneously seen in certain different time-scales, the Zero-range part being diffusively scaled while the Glauber part is speeded up at a lesser rate, a mean-curvature interface flow emerges, with a homogenized ‘surface tension-mobility’ parameter reflecting microscopic rates, between the two levels of particle density. We use relative entropy methods, along with a suitable ‘Boltzmann–Gibbs’ principle, to show that the random microscopic system may be approximated by a ‘discretized’ Allen–Cahn PDE with nonlinear diffusion. In turn, we show the behavior, especially generation and propagation of interface properties, of this ‘discretized’ PDE.