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This basic mechanism may well be the main one at work in many biological systems: each cell or each organism becomes separated from others, and while it survives, it can exhibit organized behavior.

But looking at the pictures of rule 37R on page 454 one may ask whether perhaps the effects we see are just transients, and that if we waited long enough something different would happen.

It is an inevitable feature of having a closed system of limited size that in the end the behavior one gets must repeat itself. And in rules like 0R and 90R shown on page 452 the period of repetition is always very short. But for rule 37R it usually turns out to be rather long. Indeed, for the specific example shown on page 454, the period is 293,216,266.

In general, however, the maximum possible period for a system containing a certain number of cells can be achieved only if the evolution of the system from any initial condition eventually visits all the possible states of the system, as discussed on page 258. And if this in fact happens, then at least eventually the system will inevitably spend most of its time in states that seem quite random.

But in rule 37R there is no such ergodicity. And instead, starting from any particular initial condition, the system will only ever visit a tiny fraction of all possible states. Yet since the total number of states is astronomically large—about 10^60 for size 100—the number of states visited by rule 37R, and therefore the repetition period, can still be extremely long.

There are various subtleties involved in making a formal study of the limiting behavior of rule 37R after a very long time. But irrespective of these subtleties, the basic fact remains that so far as I can tell, rule 37R simply does not follow the predictions of the Second Law.

And indeed I strongly suspect that there are many systems in nature which behave in more or less the same way. The Second Law is an important and quite general principle—but it is not universally valid. And by thinking in terms of simple programs we have thus been able in this section not only to understand why the Second Law is often true, but also to see some of its limitations.

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