Numerical study of roughness distributions in nonlinear models of interface growth

We analyze the shapes of roughness distributions of discrete models in the Kardar, Parisi, and Zhang (KPZ) and in the Villain, Lai, and Das Sarma (VLDS) classes of interface growth, in one and two dimensions. Three KPZ models in $d=2$ confirm the expected scaling of the distribution and show a stretched exponential tail approximately as $\exp (-x^{0.8})$, with a significant asymmetry near the maximum. Conserved restricted solid-on-solid models belonging to the VLDS class were simulated in $d=1$ and $d=2$. The tail in $d=1$ has the form $\exp (-x^2)$ and, in $d=2$, has a simple exponential decay, but is quantitatively different from the distribution of the linear fourth-order (Mullins-Herring) theory. It is not possible to fit any of the above distributions to those of $1∕f^{\alpha }$ noise interfaces, in contrast with recently studied models with depinning transitions.

Figure 1

(Color online) Scaled roughness distributions [according to Eq. (4)] at the steady states of the models: RSOS in $L=256$ (solid line), etching in $L=256$ (squares), and RSOS2 in $L=128$ (triangles).

Figure 2

(Color online) (a) Skewness and (b) kurtosis of the interface width distributions at the steady states of the RSOS (squares) and etching (triangle) models as a function of inverse lattice length.

Figure 3

(Color online) Effective exponents $\gamma \left(x\right)$ of the exponential decay of the scaling functions versus $1∕x^2$, for three values of the exponent $\beta$ and the models RSOS (squares) and etching (crosses). The dotted (dashed) curve is a linear fit of the data of the RSOS (etching) model.

Figure 4

(Color online) (a) Scaled roughness distributions [according to Eq. (3)] at the steady states of the CRSOS1 (squares) and CRSOS2 (crosses) models in $d=1$, with $L=256$. The solid curve is the scaled distribution of the MH theory in $d=1$. (b) The same distributions for CRSOS1 and CRSOS2 models as a function of $y^2$ and a linear fit (dashed line) of the data for the CRSOS1 model. (c) Distributions of the CRSOS1 model in $d=2$, with $L=64$, and a linear fit of the data (dashed line) in the range $2<y<4$. The solid curve is the distribution of the MH theory in $d=2$.