Scaling properties of driven interfaces: Symmetries, conservation laws, and the role of constraints

One-dimensional models of surface dynamics are studied analytically and by numerical simulation. In all cases the number of particles that form the deposit is conserved. We find that an initially flat surface, in general, roughens as a function of time and can be characterized by a width ξ which obeys the scaling form ξ(L,t)=${\mathit{L}}^{\mathrm{\chi }}$f(t/${\mathit{L}}^{\mathit{z}}$) for a deposit on a substrate of linear dimension L. Restricted solid-on-solid (RSOS) -type models in which the microscopic dynamics obeys detailed balance are shown to be in the ‘‘free-field’’ universality class with exponents z=4 and χ=1/2. On the other hand, if detailed balance is broken, several universality classes exist. As the ‘‘maximum-height-difference’’ constraint in a RSOS model is varied, one can observe a phase transition between a flat phase (z=2, χ=0) and a grooved phase characterized by a steady-state exponent χ=1 but with no scaling in the relaxation process. At the transition point, a nontrivially rough surface emerges with exponents (z≊3.67, χ≊0.33) close to those of the conserved Kardar-Parisi-Zhang equation. We propose a phenomenological model that may account for these observations.