Chapter 8: Implications for Everyday Systems

Section 2: The Growth of Crystals

Diffusion-limited aggregation (DLA)

DLA is a model for a variety of natural growth processes that was invented by Thomas Witten and Leonard Sander in 1981, and which at first seems quite different from a cellular automaton. The basic idea of DLA is to build up a cluster of black cells by starting with a single black cell and then successively introducing new black cells far away that undergo random walks and stick to the cluster as soon as they come into contact with it. The patterns that are obtained by this procedure turn out for reasons that are still not particularly clear to have a random but on average nested form. (Depending on precise details of the underlying model, very large clusters may sometimes not have nested forms, at least in 2D.) The basic reason that DLA patterns are not very dense is that once arms have formed on the outside of the cluster, they tend to catch new cells before these cells have had a chance to go inside. It turns out that at a mathematical level DLA can be reproduced by solving the Laplace equation at each step with a constant boundary condition on the cluster, and then using the result to give the probability for adding a new cell at each point on the cluster. To construct a cellular automaton analog of DLA one can introduce gray as well as black and white cells, and then have the gray cells represent pieces of solid that have not yet become permanently attached to the main cluster. Rapid rearrangement of gray cells on successive steps can then have a similar effect to the random walks that occur in the usual DLA model. Whether a pattern with all the properties expected in DLA is produced seems to depend in some detail on the rules for the gray cells. But so long as there is effective randomness in the successive positions of these cells, and so long as the total number of them is conserved, then it appears that DLA-like results are usually obtained. No doubt there are also simpler cellular automaton rules that yield similar results. (See also page 979.)

From Stephen Wolfram: A New Kind of Science [citation]