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Probabilistic evolution matrix

Empirically it has been noted that automata go through three phases of evolution. If a ring is started out with an arbitrary initial configuration the randomness decreases fairly rapidly, followed by a longer period with a fairly constant density of states, and a final period which will be reached after a time depending on the length of the ring, in which the evolution is completely periodic.

Paying closer attention to the intermediate phase, it is seen that the density of states is not constant but that it fluctuates, and that there may be short range correlations between the states. Nevertheless, these densities, correlations, and even standard deviations can be calculated and they seem to agree fairly well with experience.

Harold V. McIntosh