- 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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