Instruction: Clusters of Diffusion Limited Aggregation (DLA) can be generated by pressing the "Grow" button. To see more details about the growth process, press the "Grow slowly" button. Particles added to the aggregate at different time are shaded using different colors.
The model: DLA is one of the most important models of fractal growth. It was invented by two physicists, T.A. Witten and L.M. Sander , in 1981. The growth rule is remarkably simple. We start with an immobile seed on the plane. A walker is then launched from a random position far away and is allowed to diffuse. If it touches the seed, it is immobilized instantly and becomes part of the aggregate. We then launch similar walkers one-by-one and each of them stops upon hitting the cluster. After launching a few hundred particles, a cluster with intricate branch structures results.
"Grow slowly": This button initiates an illustration of the algorithm. We launch walkers from a "launching circle" which inscribes the cluster. They are discarded if they wander too far and go beyond a "killing circle". During launching or killing, the corresponding circle is shown in red or blue respectively. The diffusion is simulated by successive displacements each of one-tenth of the particle diameter in an independent random direction. After every step, all particles on the cluster are checked to detect any overlapping with the walker which would form an attachment.
"Grow": This button initiates a much faster simulation. The launching circle is set to be the smallest circle inscribing the cluster while the killing circle is 5 times as large to approximate an infinite killing circle. Dramatic improvement in the efficiency is due to allowing every walker to jump over the longest possible distance which is still clear of any part of the cluster. Again, we check the position of every particle in the cluster to determine how far a walker can jump.
Please click to thumbnail to see two large DLA clusters. More about physicists' views of DLA can be found in the book "Fractal Growth Phenomena" by Vicsek.
Author: Chi-Hang Lam, Applied Physics, Hong Kong Polytechnic University (complete package, java source code)