Noise-Driven Interfaces and Their Macroscopic Representation

We study the macroscopic representation of noise-driven interfaces in stochastic interface growth models in (1+1) dimensions. The interface is characterized macroscopically by saturation, which represents the fluctuating sharp interface by a smoothly varying phase field with values between 0 and 1. We determine the one-point interface height statistics for the Edwards-Wilkinson (EW) and Kadar-Paris-Zhang (KPZ) models in order to determine explicit deterministic equations for the phase saturation for each of them. While we obtain exact results for the EW model, we develop a Gaussian closure approximation for the KPZ model. We identify an interface compression term, which is related to mass transfer perpendicular to the growth direction, and a diffusion term that tends to increase the interface width. The interface compression rate depends on the mesoscopic mass transfer process along the interface and in this sense provides a relation between meso- and macroscopic interface dynamics. These results shed light on the relation between mesoscale and macroscale interface models, and provide a systematic framework for the upscaling of stochastic interface dynamics.MD acknowledges the funding from the European Research Council through the project MHetScale (Grant agreement no. 617511). IN acknowledges funding from DFG (Ne824/6-2 under the research unit MUSIS FOR 1083). DMT was supported in part by Defense Advanced Research Projects Agency under the EQUiPS program, the Air Force Office of Scientific Research under grant FA9550-12-1-0185 and by the National Science Foundation under grant DMS-1522799.Peer reviewe