Diffusion-Limited Aggregation: A Model for Pattern Formation

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Abstract
Ature confronts us at every turn with pat- terns—whether the stately spiral shapes of galaxies and hurricanes or the beautiful symmetries of snowflakes and silicon. A host of processes can play a role in forming natural patterns, though they usually involve an interaction between the transport and the thermody- namic properties of the matter and radiation involved. Typically, convection dominates the transport, in both terrestrial and astrophysical contexts. A classical example is Rayleigh-Bénard convection. The instabilities and pat- terns generated in a fluid that is convectively transport- ing heat have implications in contexts as far-flung as lab- oratory fluid dynamics and solar physics. In many natural settings, however, convection simply cannot occur. In those cases, diffusion usually dominates the transport. Consider the formation of river networks, frost on glass, or veins of minerals in geologic formations. Similarly, convection plays no role in many patterns in laboratory settings—for example, during ion deposition, electrodeposition, or other solidification processes. The patterns occurring in this type of system have some general features, which are captured by a number of simple models. The most famous of these models is diffu- sion-limited aggregation. 1

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