An analytical formulation for roughness based on celular automata

TL;DR

This paper develops an analytical model for surface roughness based on cellular automata, deriving expressions for roughness evolution and addressing discrepancies with experimental data through probability adjustments.

Contribution

It introduces a novel analytical approach to compute roughness from cellular automata dynamics, improving agreement with experimental results.

Findings

Derived roughness as a function of time for cellular automata surfaces.

Identified divergence from KPZ predictions due to equiprobability assumption.

Adjusted roughness definition to match experimental observations.

Abstract

We present a method to derive the analytical expression of the roughness of a fractal surface whose dynamics is ruled by cellular automata. Starting from the automata, we write down the the time derivative of the height's average and variance. By assuming the equiprobability of the surface configurations and taking the limit of large substrates we find the roughness as a function of time. As expected, the function behaves as $t^\beta$ when $t\ll t_\times$ and saturate at $w_s$ when $t\gg t_\times$. We apply the methodology to describe the etching model \citep{Bernardo}, however, the value of $\beta$ we obtained are not the one predicted by the KPZ equation and observed in numerical experiments. That divergence may be due to the equiprobability assumption. We redefine the roughness with an exponent that compensate the nonuniform probability generated by the celular automata, resulting in an expression that perfectly matches the experimental results.