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Simulating Experiences: Excursions in Programming. Spreading Phenomena
Author

Richard J. Gaylord
Organization: University of Illinois at Urbana-Champaign
Department: Department of Material Science and Engineering
Journal / Anthology

Mathematica in Education
Year: 1992
Volume: 1
Issue: 4
Page range: 17-20
Description

Spreading phenomena are ubiquitous in nature and include epidemics, percolation, forest fires, gelation, tumor growth, rumor-mongering and fluid flow through porous media. While these phenomena are quite disparate in terms of the systems involved (both material and non-material)m it is in fact, possible to discern a common underlying mechanism. Using the subject-independent terminology of the lattice system, a “spreading” system consists of occupied “cluster” lattice sites and unoccupied “perimeter” sites adjacent to the cluster sites. In the spreading process, perimeter sites turn into cluster sites, giving rise to additional perimeter sites which in turn may become cluster sites, creating perimeter sites... and so forth. Spreading is therefore basically an inside -> outside process, in contrast to the mechanism of aggregation [Gaylord and Tyndall, 1992]. This common mechanism enables us to create a general model, known as the kinetic growth (KG) model, which can then be specialized for application to particular phenomena, based on the criteria used for the perimeter-cluster site transformation. In this column, we will first describe the general KG algorithm and then give an implementation which covers three cases: 1) Eden model (used for tumor growth, this was the first KG model and it is named after the biologist who proposed it). Perimeter sites are randomly selected and converted to cluster sites. 2) Percolation model: (used for gelation). This is a variation of the Eden model, in which perimeter sites are randomly selected, given associated values by random number generation and, based on that value, accepted into or rejected from the cluster. This is also known as the single percolation cluster' model. 3) Invasion percolation model: (used for fluid flow through porous media). This is a variation of the percolation model in which perimeter sites become cluster sites by virtue of being the perimeter site with the lowest associated value. This is said to “follow the path of least resistance”.
Subject

*Mathematics > Discrete Mathematics > Cellular Automata
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