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 selforganize 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 selforganization 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 halfsize 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 quartersize 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 selforganizing
systems.
