Diffusion-Limited Aggregation (DLA) is a process whereby particles undergoing Brownian motion aggregate to form clusters of particles. “Diffusion” because the particles diffuse randomly before attaching themselves (“Aggregating”) to the structure. “Diffusion-limited” because the particles are considered to be in low concentrations so the structure grows one particle at a time.

DLA can be observed in many natural phenomenon, such as the formation of snowflakes and the formation of electrically conducting regions in a dielectric breakdown. These clusters are an example of a fractal, i.e. a pattern that replicates itself in any scale.

In ordinary geometry, the volume of an object scales up in a power law of the spatial dimension in which the object resides. \[V = k r^D\] for example, by making the side of a square twice as long we quadruple its volume and by making the side of a cube twice as long we multiply the volume by 8.

We can define the dimensionality of a fractal in the same way, interestingly, the dimensionality of a fractal differs from the space it resides in, and can take non-integer values. \[D = \frac{ln(V)}{ln(R)}\]

The primary means to study DLA is by computer simulations, where a random-walker diffuses in the lattice. When it diffuses into a “sticky” site it sticks to it and becomes “sticky” too. Other walkers that reaches it stick as well, creating clusters. The model is simple, but the patterns produced are rich and “organic” looking.

In the following section, we use a simple MATLAB script to demonstrate some properties of DLA and show the patterns that emerge.