- This week's soup depicts one variety of ring dynamics that can
be represented by an excitable cellular automaton. In the 8-state form
shown here, dark blue is latent, orange is excited, and successively deeper
shades of red represent six refractory stages. Each time step, latent sites
become excited spontaneously with a small probability, p = .0001 in
this instance. In addition, a latent site always become excited if there is
at least one excited cell within its neighborhood, an octagon of 'radius' 3
here. With each update, excited cells become refractory and refractory cells
cycle until they again become latent. The resulting evolution consists of
expanding rings that nucleate at random locations and annihilate upon
collision, a kind of digital boiling. Our soup is a sized-down version
of the system on a 1024 by 768 array after 256 updates, by which time it has
achieved an apparent 'bubbling' equilibrium. The full-size graphic is
Bigboil.gif
- A range 1 Box digital boiling experiment is included among the demos for
our simulation software WinCA available from
The Kitchen Sink. To get a better feeling for these
intriguing ring dyynamics, run boil.xpt full screen in movie mode
(see the Release Notes in WinCA Help) on a fast machine and crank up the
accompanying Nirvana midi track. It really cooks! Then experiment with
other rings.rul parameter settings to discover an intriguing variety of
related behavior including the Poisson-Voronoi
crystallization mentioned last fall. Ball neighborhoods with large range
do a good job of approximating isotropic waves, but note that in annihilating
interactions the excitation threshold must be 1 to ensure that rings do not
break and create the characteristic wave ends of other exictable CA systems.
- My colleague Janko Gravner at UC-Davis is currently studying the
statistics of digital boiling when the spontaneous excitation parameter
p is small. He is able to show that such systems on infinite
arrays survive indefinitely, and that the origin is occupied about once every
p^(-1/3) time units on average. We conjecture that there is a unique
equilibrium for each p, but this assertion seems exceedingly difficult
to prove with currently available mathematical techniques.
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