Section 4
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Section 6
5. Diffusion-limited aggregation
In 1983 Witten and Sander [Wit1983] modified Eden's model by
assuming that the particles originate far away from the developing
structure, and perform a random walk in the surrounding space. Once a
particle encounters the structure, it sticks to it. This model has been
termed diffusion-limited aggregation (DLA for short),
since the random walk of the particles can be viewed as a simulation of
diffusion at the molecular level. Although the difference between the
operation of Eden's model and diffusion-limited aggregation may seem rather
minor, the difference in the resulting patterns is striking. In contrast
to Eden's approximately circular patterns, the DLA model produces fractal
branching structures as shown in Plate
9 (see caption). Mandelbrot
and Evertsz [Man1990] proposed
a visual explanation of this form, reproduced in Animation 1 (see caption). They depicted the average
concentration of particles in the space surrounding the growing structure
by solving Laplace's equation in this space. The boundary conditions
imposed high concentrations of particles at the outer boundary of the space
and zero on the boundary of the fractal. The green background color
indicates high concentration of particles in space, and black indicates low
concentrations. Some lines of constant concentration have been highlighted
to facilitate the interpretation of the image. It can be clearly seen that
the concentration of particles between the branches is always very low,
which is why the branches do not grow there; a bay remains a bay. On the
other hand, the tips of the structure are very close to the areas of high
concentration of particles, which is why the branches grow the fastest
there. Plate 10 (see caption) presents the result of a
diffusion-limited aggregation process, in which the initial structure is a
horizontal line, and particles come from above. A ``forest'' of branching
structures is produced in this case.
The traveling particles can be viewed as nutrients, used by a simple
growing structure (such as a bacterial colony) to form its body. The
distinction between diffusion-limited aggregation (where particles are
simply attached to the growing cluster) and diffusion-limited
growth (where particles are transformed into the organism's body) was
first characterized by Meakin [Mea1986]. In this context, Eden's
model can be viewed as describing the growth of a bacterial colony in an
abundance of nutrients. The structure develops everywhere it has space to
expand. In contrast, diffusion-limited growth occurs where nutrients are a
scarce resource. The structure uses all nutrients in its proximity, and
grows most vigorously in the directions exposed to the incoming nutrients.
Matsushita and Fujikawa [Mat1990, Fuj1991] showed experimentally
that, depending on the availability of nutrients, a bacterial colony may
adopt either an approximately circular or a branching structure.
Section 4
Table of Contents
Section 6