Diffusion-Limited Aggregation

Dielectric Breakdown Mathematical Sketch

In an electric field, work must be done to move a charged particle from one point to another.

If W((x₁,y₁),(x₂,y₂)) is the work to move a particle of charge q₀ from a point (x₂,y₂) to a point (x₁,y₁), the electric potential relative to (x₁,y₁) is the function f which assigns to each point (x₂,y₂) the number f((x₂,y₂)) = W((x₁,y₁),(x₂,y₂))/q₀.

The potential is related to the strength of the electric field around (x₁,y₁).

Like simple DLA, the dielectric breakdown model is constructed on a grid of square cells c(i,j).

Select a large circle, S, on the grid.

Assign to the potential f the value 1 at each cell on this circle, and the value 0 to the cell c(0,0) in the center of the circle.

The boundary conditions of the problem are that the potential keeps these values at these locations.

Consistent with these conditions, the value of the potential is determined at each cell of the grid (by a discrete version of Laplace's equation - details are not necessary here) and then for each of the four cells c(-1,0), c(1,0), c(0,-1), and c(0,1) adjacent to c(0,0), the growth probability at c(a,b) is defined as

p(a,b) = f(c(a,b))^t(f(c(a-1,b))^t + f(c(a+1,b))^t + f(c(a,b-1))^t + f(c(a,b+1))^t)

One of these four points is selected at random, with probabilities given by the p(a,b).

Together with the point c(0,0), this point forms the beginning of the DBM cluster.

Compute the potential again with boundary conditions f = 1 on the same large circle S, and f = 0 on the cluster.

Compute the growth probability for each of the 6 cells adjacent to the cluster and add another cell randomly, according to these probabilities.

Continuing in this way grows a DBM cluster.

Return to DLA.