The Cook Book

Recipe for the week of June 24 - 30

A Two-phase Waffle with Stable Interface

I've just returned from a week in Israel, and it will take some time to get back up to speed, so this week's recipe is a quickie. Our rule is range 2 Box Larger than Life with birth interval [7,18] and survival interval [11,22], rendered in a 200-color cycle of level-sets. Thanks to Kellie Evans for this one. As opposed to the Unstable Waffles of last spring, in which a metastable crystalline phase is eventually displaced by a turbulent one, this week's soup shows a stable interface between the highly-ordered regime that grows from some small seeds and the distinctly chaotic phase that grows from others. Our initial state here consisted of two very small lattice circles separated by a few cells in the diagonal direction. Similar stable boundaries also emerge as this rule nucleates from small random densities of occupied sites, yielding waffles that are often one-quarter regular and three-quarters irregular, though of course not with the perfect symmetry of our engineered creation.

It is rare that two qualitatively distinct phases coexist indefinitely in a complex spatial evolution since one usually manifests a competitive advantage over the other. In theoretical ecology, where the dynamics should presumably be robust under small random perturbations, this principle was enuncinated by Gause about sixty years ago:

The steady state of a mixed population consisting of two species occupying an identical 'ecological niche' will be the pure population of one of them, of the better adapted for the particular set of conditions.

Coexistence is more common in deterministic dynamics, not subject to stochastic fluctuations that tend to destabilize highly structured configurations, but even for cellular automata such a delicate balance is unusual.

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