Relaxation to steady states and dynamical exponents in deposition models

Considering some deposition models with limited mobility, we show that the typical decay of the interface width to its saturation value is exponential, which defines the crossover or saturation time τ. We present a method to calculate a characteristic time τ0 proportional to τ and estimate the dynamical exponent z. In one-dimensional substrates of lengths L⩽2048, the method is applied to the Family model, the restricted solid-on-solid (RSOS) model and the ballistic deposition. Effective exponents zL converge to asymptotic values consistent with the corresponding continuum theories. For the two-dimensional Family model, the expected dynamic scaling hypothesis suggests a particular definition of τ0 that leads to z=2, improving previous calculations based on data collapse methods. For the two-dimensional RSOS model, we obtain z≈1.6 and α<0.4, in agreement with recent large-scale simulations.