| To test if the similarity of DLA and physical 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 dm is used. |
| Recall that 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⋅rd |
| for some constant k and for d = dm. |
| For physical fractals, there is a range of r values,
the scaling range,
over which this relation is valid. |
| Early computer simulations give |
| dm ≈ 1.71 for clusters in the plane, and |
| dm ≈ 2.5 for clusters in space. |
|
| In the next sections, electrodeposition and
dielectric breakdown, we shall compare these calculations with
physical measurements. |