The pictures on the facing page show examples of various reversible cellular automata. And what we see immediately from these pictures is that while some systems exhibit exactly the kind of randomization implied by the Second Law, others do not.
The most obvious exceptions are cases like rule 0R and rule 90R, where the behavior that is produced has only a very simple fixed or repetitive form. And existing mathematical studies have indeed identified these simple exceptions to the Second Law. But they have somehow implicitly assumed that no other kinds of exceptions can exist.
The picture on the next page, however, shows the behavior of rule 37R over the course of many steps. And in looking at this picture, we see a remarkable phenomenon: there is neither a systematic trend towards increasing randomness, nor any form of simple predictable behavior. Indeed, it seems that the system just never settles down, but rather continues to fluctuate forever, sometimes becoming less orderly, and sometimes more so.
So how can such behavior be understood in the context of the Second Law? There is, I believe, no choice but to conclude that for practical purposes rule 37R simply does not obey the Second Law.
And as it turns out, what happens in rule 37R is not so different from what seems to happen in many systems in nature. If the Second Law was always obeyed, then one might expect that by now every part of our universe would have evolved to completely random equilibrium.
Yet it is quite obvious that this has not happened. And indeed there are many kinds of systems, notably biological ones, that seem to show, at least temporarily, a trend towards increasing order rather than increasing randomness.
How do such systems work? A common feature appears to be the presence of some kind of partitioning: the systems effectively break up into parts that evolve at least somewhat independently for long periods of time.
The picture on page 456 shows what happens if one starts rule 37R with a single small region of randomness. And for a while what one sees is that the randomness that has been inserted persists. But eventually the system instead seems to organize itself to yield just a small number of simple repetitive structures.