Growing interfaces: A brief review on the tilt method

M. F. Torres, R. C. Buceta

Published 2017-11-27Version 1

The tilt method applied to models of growing interfaces is a useful tool to characterize the nonlinearities of their associated equation. Growing interfaces with average slope $m$, in models and equations belonging to Kardar-Parisi-Zhang (KPZ) universality class, have average saturation velocity $\mathcal{V}_\mathrm{sat}=\Upsilon+\frac{1}{2}\Lambda\,m^2$ when $|m|\ll 1$. This property is sufficient to ensure that there is a nonlinearity type square height-gradient. Usually, the constant $\Lambda$ is considered equal to the nonlinear coefficient $\lambda$ of the KPZ equation. In this paper, we show that the mean square height-gradient $\langle |\nabla h|^2\rangle=a+b \,m^2$, where $b=1$ for the continuous KPZ equation and $b\neq 1$ otherwise, e.g. ballistic deposition (BD) and restricted-solid-on-solid (RSOS) models. In order to find the nonlinear coefficient $\lambda$ associated to each system, we establish the relationship $\Lambda=b\,\lambda$ and we test it through the discrete integration of the KPZ equation. We conclude that height-gradient fluctuations as function of $m^2$ are constant for continuous KPZ equation and increasing or decreasing in other systems, such as BD or RSOS models, respectively.