Constant-speed interface flow from unbalanced Glauber-Kawasaki dynamics

TL;DR

This paper derives the hydrodynamic limit of Glauber-Kawasaki dynamics, showing how an interface between two favored particle densities moves at constant speed, differing from previous models due to the dynamics' density preferences.

Contribution

It introduces a new hydrodynamic limit for Glauber-Kawasaki dynamics with density preferences, revealing a constant-speed interface evolution distinct from prior mean curvature flow results.

Findings

Interface propagates at constant speed

Hydrodynamic limit differs from previous models

Glauber dynamics preferences influence interface behavior

Abstract

We derive the hydrodynamic limit of Glauber-Kawasaki dynamics. The Kawasaki part is simple and describes independent movement of the particles with hard core exclusive interactions. It is speeded up in a diffusive space-time scaling. The Glauber part describes the birth and death of particles. It is set to favor two levels of particle density with a preference for one of the two. It is also speeded up in time, but at a lesser rate than the Kawasaki part. Under this scaling, the limiting particle density instantly takes either of the two favored density values. The interface which separates these two values evolves with constant speed (Huygens' principle). Similar hydrodynamic limits have been derived in four recent papers. The crucial difference with these papers is that we consider Glauber dynamics which has a preferences for one of the two favored density values. As a result, we observe limiting dynamics on a shorter time scale, and the evolution is different from the mean curvature flow obtained in the four previous papers. While several steps in our proof can be adopted from these papers, the proof for the propagation of the interface is new.