Relaxation after a change in the interface growth dynamics

Abstract

The global effects of sudden changes in the interface growth dynamics are studied using models of the Edwards-Wilkinson (EW) and Kardar-Parisi-Zhang (KPZ) classes during their growth regimes in dimensions $d=1$ and $d=2$. Scaling arguments and simulation results are combined to predict the relaxation of the difference in the roughness of the perturbed and the unperturbed interfaces, $\Delta W^2 \sim s^c t^{-\gamma}$, where $s$ is the time of the change and $t>s$ is the observation time after that event. The previous analytical solution for the EW-EW changes is reviewed and numerically discussed in the context of lattice models, with possible decays with $\gamma=3/2$ and $\gamma=1/2$. Assuming the dominant contribution to $\Delta W^2$ to be predicted from a time shift in the final growth dynamics, the scaling of KPZ-KPZ changes with $\gamma = 1-2\beta$ and $c=2\beta$ is predicted, where $\beta$ is the growth exponent. Good agreement with simulation results in $d=1$ and $d=2$ is observed. A relation with the relaxation of a local autoresponse function in $d=1$ cannot be discarded, but very different exponents are shown in $d=2$. We also consider changes between different dynamics, with the KPZ-EW as a special case in which a faster growth, with dynamical exponent $z_i$, changes to a slower one, with exponent $z$. A scaling approach predicts a crossover time $t_c\sim s^{z/z_i}\gg s$ and $\Delta W^2 \sim s^c F\left( t/t_c\right)$, with the decay exponent $\gamma=1/2$ of the EW class. This rules out the simplified time shift hypothesis in $d=2$ dimensions. These results help to understand the remarkable differences in EW smoothing of correlated and uncorrelated surfaces, and the approach may be extended to sudden changes between other growth dynamics.