Diffusion-Limited Aggregation

To test if the similarity of DAL and phisical clusters is more than appearance, we compare the mass dimensions of DLA clusters and of these objects.

At scales smaller than the size of the diffusing particles, a DLA cluster does not reveal any additional structure and so the dimension cannot be computed from covers by smaller and smaller boxes.

So instead of the box-counting dimension, for DLA clusters the mass dimension d_m is used.

If N(r) denotes the number of particles in a circle (or sphere) of radius r, then for large r we expect

N(r) = k*r^d

for some constant k and for d = d_m.

There is a range of r values, the scaling range, over which this relation is valid.

Early computer simulations gave

d_m = 1.71 for clusters in the plane, and

d_m = 2.5 for clusters in space.

In electrodeposition and dielectric breakdown, we compare these calculations with physical measurements.

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