Geometry and Dynamics of DiffusionLimited Growth
Abstract
Diffusionlimited aggregation (DLA) is a simple discrete growth model, applicable to literally hundreds of natural phenomena. DLA growth takes place by addition of random walkers to the perimeter of the aggregate. The probability that a perimeter site is hit first by an incident walker is called the site's growth probability. Since these growth probabilities provide an almost complete description of the DLA growth dynamics, they have initiated considerable research activity. However, so far, our theoretical understanding of DLA is limited. In particular, the interplay between the geometrical properties of the pattern's shape and the dependence of the growth probabilities on the size of the cluster is poorly understood. I propose a geometric voidneck model for the structure of two dimensional (2D) DLA which pictures the aggregate as a sequence of voids of increasing sizes separated by narrow necks. The model predicts a breakdown of power law behavior with system size for the smallest growth probability p_{rm min} of 2D DLA. This prediction is confirmed in a simulation of 2D aggregates. Additionally, I study the growth probability distribution (GPD) and find a functional form for its tail in agreement with the predicted behavior of p_ {rm min}. In 3D, constraints of topological nature inherent in the 2D voidneck model no longer apply. I confirm by simulation that powerlaw behavior of p_{rm min} and, in addition, of all moments of the GPD is restored. I propose different functional forms for the GPD and the density profile of the growing DLA cluster which are consistent with my simulation data. I analytically explore the consequences of these propositions for very large clusters and find interesting consequences of the interplay between cluster density and GPD. A direct geometrical test of the voidneck model motivates the development of a novel "glove algorithm." I describe sequences of "perimeters" and "accessible perimeters" (gloves) of DLA and related fractals by new scaling laws. I relate the quantities governing these scaling laws to already established properties of the investigated models. Moreover, I define a hierarchy of "lagoons" as regions that are inaccessible to particles of different sizes. A study of the lagoon size distribution supports the voidneck model and the fractal nature of DLA. Finally, I use the "skeleton" algorithm by Havlin and Nossal to determine the number of main branches of DLA, which is an important morphological characteristic of the cluster. In 2D the cluster displays ~ 9 main branches, whereas in 3D and 4D their number is seemingly not limited.
 Publication:

Ph.D. Thesis
 Pub Date:
 1994
 Bibcode:
 1994PhDT........55S
 Keywords:

 GROWTH PROBABILITY;
 Physics: General