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Because of its important phenomena in nature, a good deal of research into fractal growth has been carried out. Some of these works were carried out by computer simulations. In 1981, Witten and Sander [*] introduced a computer model called the diffusion limited aggregation (DLA)
to produce fractal objects.
Basically, a DLA is a model of irreversible growth process whereby individual particles stick to one another to form clusters or aggregates. It was found to be very useful for a wide range of growth processes in which diffusion (some kind of random motions) is the important limiting step and rearrangement of the material within a cluster is not allowed.
Diagram below shows a typical Witten-Sander (WS) cluster model grown on two dimensional square lattices (grids) which consists of 10 000 particles (or pixels as are generated on the computer screen).
The simulation procedure for the DLA is quite straightforward: Imagine a computer screen is the 'ground' for the fractal cluster to grow and the screen is made of tiny grids called pixels. Only one particle can occupy a pixel. Starting with a single-seed, fixed particle at the center of the screen, a second moving particle is 'created' randomly at some distance from the
origin and move randomly on the pixel grids until it reaches a grid adjacent to the seed and becomes part of growing cluster. A third moving particle is then generated like the second particle and allows to wander randomly. The particle will stick if it finds itself adjacent to any sticked particles. The procedure is repeated many times and the end result, instead a lump as one may expect, is (surprise) a fractal.
In fact, a number of experiments have been carried out which give WS-like aggregates. Examples are metal electrodeposition and dielectric break down experiment (shown below). Notice the similarity of the image to that of computer-generated WS cluster. In addition, such fractal shapes are also frequently found as a result of mineral depositions in rocks.
[*] T.A. Witten and L.M. Sander, Physical Review Letter 47, 1400 (1981)
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