Dimensions, Maximal Growth Sites and Optimization in the Dielectric Breakdown Model

TL;DR

This paper investigates the fractal growth patterns in the Dielectric Breakdown Model using conformal mappings, analyzing how growth parameters influence fractal dimensions and proposing a method to estimate model parameters from growth data.

Contribution

It provides new insights into the relationship between growth exponent and fractal dimension, and introduces an optimization method to estimate model parameters from experimental data.

Findings

No evidence of a phase transition from fractal to non-fractal growth at finite η.

The non-fractal growth limit corresponds to a Hölder exponent of 1/2.

An optimization recipe to estimate η from growth data is proposed.

Abstract

We study the growth of fractal clusters in the Dielectric Breakdown Model (DBM) by means of iterated conformal mappings. In particular we investigate the fractal dimension and the maximal growth site (measured by the Hoelder exponent $\alpha_{min}$) as a function of the growth exponent $\eta$ of the DBM model. We do not find evidence for a phase transition from fractal to non-fractal growth for a finite $\eta$-value. Simultaneously, we observe that the limit of non-fractal growth ($D\to 1$) is consistent with $\alpha_{min} \to 1/2$. Finally, using an optimization principle, we give a recipe on how to estimate the effective value of $\eta$ from temporal growth data of fractal aggregates.