Mean curvature interface limit from Glauber+Zero-range interacting particles

TL;DR

This paper derives a mean-curvature flow as a hydrodynamic limit of Glauber+Zero-range particle systems, linking microscopic particle interactions to macroscopic interface evolution with homogenized parameters.

Contribution

It introduces a novel derivation of mean-curvature flow from combined Glauber and Zero-range particle dynamics, incorporating multiple time scales and homogenized surface tension.

Findings

Mean-curvature interface flow emerges from microscopic particle systems.

The system can be approximated by a discretized Allen-Cahn PDE.

Interface properties are characterized through the PDE analysis.

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.