Dynamic Scaling of Growing Interfaces

TLDR

The authors propose a model for the evolution of a growing interface and predict its behavior in higher dimensions. The model is analyzed deterministically and stochastically, using dynamic renormalization‑group methods and mappings to Burgers’s equation and a random directed‑polymer problem. Exact solutions reveal nontrivial relaxation and dynamic scaling, with the one‑dimensional scaling form agreeing with numerical simulations.

Abstract

A model is proposed for the evolution of the profile of a growing interface. The deterministic growth is solved exactly, and exhibits nontrivial relaxation patterns. The stochastic version is studied by dynamic renormalization-group techniques and by mappings to Burgers's equation and to a random directed-polymer problem. The exact dynamic scaling form obtained for a one-dimensional interface is in excellent agreement with previous numerical simulations. Predictions are made for more dimensions.