Universality in surface growth: Scaling functions and amplitude ratios

Abstract

A scaling analysis of a variety of nonlinear equations for surface growth is presented. It predicts the existence of universal scaling functions and amplitude ratios for the surface width w(L,t) on length scale L at time t and for the height-difference correlation function G(x,t). This analysis is applied to the Kardar-Parisi-Zhang (KPZ) equation for driven interface growth in d=2, in order to derive explicit scaling forms for the amplitudes associated with the scaling of the surface width, correlation function, and saturation velocity as a function of the hydrodynamical parameters in the KPZ equation. A mode-coupling calculation that estimates the values of the various universal amplitude ratios, as well as the associated universal scaling functions is also presented. Our predictions are confirmed by simulations of three different surface-growth models in d=2 from which the amplitude ratios as well as the universal scaling function for the surface width w(L,t) (for the case of periodic boundary conditions) are numerically determined. These results are also supported by numerical integration of the KPZ equation in d=2. The universality of the height-fluctuation distribution function is also discussed. Our scaling analysis is expected to be useful in the analysis of experiments and in the study of a variety of models of surface growth as well as in establishing a more detailed connection between continuum surface-growth equations and microscopic models.