Recipe for the week of October 9

The Cook Book

Recipe for the week of October 9 - 15

Fossils Again: a lesson in length scale

Let's continue a thread begun with the recipes of November 28, 1994 and August 28, 1995. Recall that the Cyclic dynamics under discussion self-organize to a complex mix of broad waves and intricate 'fossils.' The first recipe described the basic phenomenology, while the second announced the formation of remarkably regular and stable triangle wave parades on 512 by 512 arrays after tens of thousands of updates. Our previous installment concluded thus:

What have we learned? Mainly that our array isn't large enough, and that some quite interesting self-organization should occur if the dynamics have more room to move. We'll carry out the same CAM8 experiment on a larger array, say 2K by 2K, and report back later with our findings...

This week's soup shows a 1K by 1K CAM8 experiment (reduced to half-size in each dimension) after nearly one hundred thousand updates. Again, we see alternating bands of triangular waves and fossilized debris that wrap around the screen. On the basis of this larger experiment one might suspect that the dynamics on an infinite array would eventually cluster to arbitrarily large length scale in some exotic fashion. However, when we ran a 2K by 2K CAM8 experiment for more than a half a million time steps, no such clustering occurred. Rather, the length scale seems to stabilize, and the dynamics appear to achieve a complex turbulent equilibrium. So what about the triangles? Check out The Big Picture for a typical scene (reduced to quarter-size in each dimension) from the steady state evolution that kept our mixmaster busy for more than a week. Right in the middle you'll see a little ensemble of triangles. At other times the triangles and fossils achieved somewhat greater concentrations, but we are convinced that they do not aggregate indefinitely. Rather, this Cyclic Cellular Automaton rule gives a particularly dramatic illustration of the role of system size in determining the phenomenology of self-organizing systems.