Dynamics of fractal dimension during phase ordering of a geometrical multifractal

TL;DR

This paper introduces a model explaining how the fractal dimension evolves during phase ordering in multifractal systems, highlighting early-time stability and late-time dynamics following the multifractal spectrum.

Contribution

It presents a simple multifractal coarsening model that captures the dynamical behavior of fractal dimensions in various coarsening fractal systems.

Findings

Fractal dimension remains constant at early times.

At late times, the dimension follows the $f(\u03b1)$-curve.

Model is demonstrated with a two-scale Cantor dust.

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 $f(\alpha)$-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 $f(\alpha)$ 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.