The Cook Book

Recipe for the week of July 22 - 28

Level Sets of Digital Heat

One of the most dramatic varieties of self-organization is clustering - the evolution of spatial systems toward ever-increasing length scales. An exotic instance of this process, surprisingly amenable to exact quantitative analysis, is the Stepping Stone Model featured on our main Kitchen Shelf page and described in our Particle Postcards notes. More mathemtically illusive, but ubiquitous in phenomena from many areas of science which involve symmetric nonlinear interface dynamics, is the clustering by surface tension which drives the Ising Model without external field, various selectively neutral competition rules, and one of our favorite cellular automata: Majority Vote.

In trying to understand the Big Bang of Majority Vote, it is illuminating to consider Discrete Heat, an even simpler update scheme in which the value of each cell is replaced by the average of its neighbors, rounded if necessary for a finite state implementation with N types. Heat diffuses over a uniform medium according to this averaging mechanism, whence the name. Suppose we start from a uniformly random distribution of heat levels over the array (white noise). Long ago our third recipe, the Rug Rule, showed short-term self-organization that results after a slight perturbation (add 1 to the average, mod N). This week's soup is a 256-color discretization of the unperturbed Digital Heat rule at time 12 (with wrap-around boundary).

Of course the heat should rapidly equilibriate to a uniform average temperature. Our palette is chosen so that below average temperatures appear in shades of green, whereas those above average range from orange to red (hottest). The average temperature level set, rendered in blue, undergoes an evolution qualitatively similar to motion by mean curvature. Note occasional 'pinches' along the blue boundaries where large heat clusters are connected only by narrow isthmi. Our regular visitors may suspect a lurking connection with percolation theory. We are currently trying to analyzse the frequency of these bottlenecks over time in order to explain mathematically the clustering of Digital Heat and Majority Vote.

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