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28 March 2024
 
  » arxiv » 1208.5063

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An analytical formulation for roughness based on celular automata
Ismael V. L. Costa ; Henrique A. Fernandes ; Bernardo A. Mello ; Fernando A. Oliveira ;
Date 24 Aug 2012
AbstractWe 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 $t^eta$ when $tll t_ imes$ and saturate at $w_s$ when $tgg t_ imes$. We apply the methodology to describe the etching model citep{Bernardo}, however, the value of $eta$ 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.
Source arXiv, 1208.5063
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