As part of my MPhys degree I undertook a project to simulate three-dimensional dielectric breakdown. The work is not related to my PhD however it was very enjoyable and resulted in some nice images.
An example of dielectric breakdown that we are all familiar with is lightning. Although the example of lightning has other complications, in essence, charge wants to get from A to B whilst avoiding itself. The result is a fractal pattern. The mathematics that describes dielectric breakdown are the same as those describing diffusion-limited aggregation (DLA). We can therefore model dielectric breakdown by using DLA algorithms. Viewing videos in high definition is recommended.
Diffusion limited aggregation
The basic mechanism of the model is very simple. We start with a particle in some space. We then allow another particle to go on a random walk in the vacinity of the first. If the walker collides with the first it sticks. This process is repeated to form a cluster of particles.
The following video is of a cluster of 5 million particles. The colour denotes the age of the particles; red particles are the oldest and magenta particles are the youngest. As you can see the oldest particles are in the centre and the youngest particles on the tips. The cluster is rotated to show three-dimensionalilty.
Dielectric breakdown by DLA
If we make the growth of a cluster proportional to its age, the model turns into the dielectric breakdown model. If the proportionalilty constant, eta, is 1 we have DLA; if eta < 1 the clusters become more dense; if eta > 1 they are less dense. The following video is of a cluster grown with eta = 3.