Dynamics of fractal dimension during phase ordering of a geometrical multifractal

Abstract

A simple multifractal coarsening model is suggested that can explain the observed dynamical behavior of the fractal dimension in a wide range of coarsening fractal systems. It is assumed that the minority phase (an ensemble of droplets) at $t=0$ represents a non-uniform recursive fractal set, and that this set is a geometrical multifractal characterized by a $f(α)$-curve. It is assumed that the droplets shrink according to their size and preserving their ordering. It is shown that at early times the Hausdorff dimension does not change with time, whereas at late times its dynamics follow the $f(α)$ curve. This is illustrated by a special case of a two-scale Cantor dust. The results are then generalized to a wider range of coarsening mechanisms.