Dynamical generation of fractals: Diffusion Limited Aggregation

Some random fractals, such as the clusters describing a bacterial colony, can be generated by a physically motivated model called 'diffusion limited aggregation' (DLA). Consider for simplicity the formation of such a cluster in the plane, with the initial (seed) particle located at the origin. Other particles are then released far from the origin, at random locations, and allowed to diffuse: Mathematically this is done using an algorithm such as a 'random walk' to simulate the diffusion process. When the diffusing particle encounters the seed particle it is made to stick to it. The process is repeated with other diffusing particles, leading to the formation of a cluster. As the cluster forms, there is a greater probability for particles to stick to the ends than to penetrate the interior. Hence this leads to the formation of a branch-like structure emanating from the origin.



Some other examples of DLA are in the growth of crystals (e.g. snowflakes) and coral reefs. (Note: A 'random walk', colloquially referred to as the 'drunken man's walk' is a path generated by a random process. Consider a two dimensional random walk starting at the origin. The location of the the next step is generated, for example, by two random numbers which give respectively the direction (angle) and length of the walk. The random process is repeated at the following time intervals. See the Exercises.) Another dynamical 'explanation' of the ubiquitous occurrence of fractals and power laws in nature is the idea of 'self-organised criticality' that we will discuss later in the course.