Symmetry-based determination of space-time functions in nonequilibrium growth processes

We study the space-time correlation and response functions in nonequilibrium growth processes described by linear stochastic Langevin equations. Exploiting exclusively the existence of space- and time-dependent symmetries of the noiseless part of these equations, we derive expressions for the universal scaling functions of two-time quantities which are found to agree with the exact expressions obtained from the stochastic equations of motion. The usefulness of the space-time functions is illustrated through the investigation of two atomistic growth models, the Family model and the restricted Family model, which are shown to belong to a unique universality class in $1+1$ and $2+1$ space dimensions. This corrects earlier studies which claimed that in $2+1$ dimensions the two models belong to different universality classes.

Figure 1

(Color online) Dynamical scaling of the space-time correlation function $C(r^2∕s,t∕s)$ for the Family model in $1+1$ dimensions with different values of the waiting time $s$: (a) $C$ vs $r^2∕s$ for some fixed values of $t∕s$, (b) $C$ vs $t∕s$ for some fixed values of $r^2∕s$. The green curves are obtained from the exact result (19) derived from the continuum EW equations with uncorrelated Gaussian white noise. Numerical error bars are smaller than the sizes of the symbols.

Figure 2

(Color online) The same as in Fig. 1, but now for the Family model in $2+1$ dimensions. Numerical error bars are comparable to the sizes of the symbols.

Figure 3

(Color online) The same as in Fig. 2, but now for the restricted Family model in $2+1$ dimensions. The inset in (a) shows the correlation function in the (10) and (11) directions for the case $s=25$ and $t∕s=1.04$. The change of the diffusion rule has a strong impact on the autocorrelation with $r=0$ and on the nearest-neighbor correlations. Numerical error bars are comparable to the sizes of the symbols.