The Cook Book

Recipe for the week of February 12 - 18

Digital Boiling

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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