In this paper we discuss mesoscopic models describing pattern formation mechanisms for a prototypical model of surface processes that involves multiple microscopic mechanisms. We focus on a mean field partial differential equation, which contains qualitatively microscopic information on particle-particle interactions and multiple particle dynamics, and we rigorously derive the macroscopic cluster evolution laws and transport structure. We show that the motion by mean curvature is given by V = μ σ κ, where k is the mean curvature, σ is the surface tension and μ is an effective mobility that depends on the presence of the multiple mechanisms and speeds up the cluster evolution. This is in contrast with the Allen-Cahn equation where V = κ. © 2007 Elsevier Inc. All rights reserved.
Karali, G., & Katsoulakis, M. A. (2007). The role of multiple microscopic mechanisms in cluster interface evolution. Journal of Differential Equations, 235(2), 418–438. https://doi.org/10.1016/j.jde.2006.12.021