Mean field theory for skewed height profiles in nonlinear growth processes

We propose a mean field theory for interfaces growing according to the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions. The mean field equations are formulated in terms of densities at different heights, taking surface tension and the influence of the nonlinear term in the KPZ equation into account. Although spatial correlations are neglected, the mean field equations still reflect the spatial dimension-ality of the system. In the special case of Edwards-Wilkinson growth, our mean field theory correctly reproduces all features. In the presence of a nonlinear term one observes a crossover to a KPZ-like behavior with the correct dynamical exponent z = 3/2. In particular we are interested in the skewed interface profile during roughening, which is a universal function in KPZ growth processes. In addition, we study the influence of a co-moving reflecting wall, which has been discussed recently in the context of nonequilibrium wetting and synchronization transitions. Also here the mean field approximation reproduces all qualitative features of the full KPZ equation, although with different values of the surface exponents.