Persistence in nonequilibrium surface growth

TL;DR

This study investigates the persistence probabilities of surface height in nonequilibrium growth models within the MBE universality class using simulations, revealing differences in positive and negative persistence exponents and their relation to growth dynamics.

Contribution

The paper provides the first detailed numerical analysis of persistence exponents in MBE class surface growth models, including their relation to dynamic exponents and finite-time effects.

Findings

Positive and negative persistence exponents differ in steady state.

Persistence exponents relate to the dynamic growth exponent in nonlinear models.

Short-time simulations can accurately estimate steady-state persistence exponents.

Abstract

Persistence probabilities of the interface height in (1+1)- and (2+1)-dimensional atomistic, solid-on-solid, stochastic models of surface growth are studied using kinetic Monte Carlo simulations, with emphasis on models that belong to the molecular beam epitaxy (MBE) universality class. Both the initial transient and the long-time steady-state regimes are investigated. We show that for growth models in the MBE universality class, the nonlinearity of the underlying dynamical equation is clearly reflected in the difference between the measured values of the positive and negative persistence exponents in both transient and steady-state regimes. For the MBE universality class, the positive and negative persistence exponents in the steady-state are found to be $\theta^S_{+} = 0.66 \pm 0.02$ and $\theta^S_{-} = 0.78 \pm 0.02$, respectively, in (1+1) dimensions, and $\theta^S_{+} = 0.76 \pm 0.02$ and $\theta^S_{-} =0.85 \pm 0.02$, respectively, in (2+1) dimensions. The noise reduction technique is applied on some of the (1+1)-dimensional models in order to obtain accurate values of the persistence exponents. We show analytically that a relation between the steady-state persistence exponent and the dynamic growth exponent, found earlier to be valid for linear models, should be satisfied by the smaller of the two steady-state persistence exponents in the nonlinear models. Our numerical results for the persistence exponents are consistent with this prediction. We also find that the steady-state persistence exponents can be obtained from simulations over times that are much shorter than that required for the interface to reach the steady state. The dependence of the persistence probability on the system size and the sampling time is shown to be described by a simple scaling form.