Chapter 7: Mechanisms in Programs and Nature

Section 5: The Intrinsic Generation of Randomness

Noisy cellular automata

In correspondence with electronics, the continuous cellular automata used here can be thought of as analog models for digital cellular automata. The specific form of the continuous generalization of the modulo 2 function used is

λ[x_] := Exp[-10 (x - 1)2] + Exp[-10 (x - 3)2]

Each cell in the system is then updated according to λ[a + c] for rule 90, and λ[a + b + c + b c] for rule 30. Perturbations of size δ are then added using v + Sign[v - 1/2] Random[] δ.

Note that the basic approach used here can be extended to allow discrete cellular automata to be approximated by partial differential equations where not only color but also space and time are continuous. (Compare page 464.)

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From Stephen Wolfram: A New Kind of Science [citation]