So even though the total of the energy of all particles remains the same, the distribution of this energy becomes progressively more random, just as the usual Second Law implies. … In the bottom group, however, it becomes progressively more random with time.

And of those that withstand scrutiny, most in my experience turn out to be transformed versions of statements that some quantity or another can be approximated by perfect randomness. Gaussian distributions typically arise when measurements involve sums of random quantities; other common distributions are obtained from products or other simple combinations of random quantities, or from the results of simple processes based on random quantities.

But superimposed on this is all sorts of elaborate and seemingly quite random behavior. … For as the pictures on the facing page indicate—and as common experience suggests—almost any time a fluid is made to flow rapidly, it tends to form complex patterns that seem in many ways random. … But from my discovery that complex and seemingly random behavior is in a sense easy to get even with very simple programs, the phenomenon of fluid turbulence immediately begins to seem much less surprising.

But with rules (e), (f) and (g) the networks are more complicated, and begin to seem somewhat random. … Like in so many other systems that we have studied in this book, the randomness that we find in causal networks will inevitably tend to wash out details of how the networks are constructed. And thus, for example, even though the underlying rules for a mobile automaton always treat space and time very differently, the causal networks that emerge nevertheless often exhibit a kind of uniform randomness in which space and time somehow work in many respects the same.

For any process like the one in the picture above must occur on top of a background of apparently random small-scale rearrangements of the network. And in effect what this background does is to introduce a kind of random environment that can make many different detailed patterns of behavior occur with certain probabilities even with the same initial configuration of particles. The idea that even a vacuum without particles will have a complicated and in some ways random form also exists in standard quantum field theory in traditional physics.

Randomness [in tag systems] To get some idea of the randomness of the behavior, one can look at the sequence of first elements produced on successive steps.

But the crucial point that I will discuss more in Chapter 7 is that the presence of sensitive dependence on initial conditions in systems like (a) and (b) in no way implies that it is what is responsible for the randomness and complexity we see in these systems. And indeed, what looking at the shift map in terms of digit sequences shows us is that this phenomenon on its own can make no contribution at all to what we can reasonably consider the ultimate production of randomness.

But what about random initial conditions? … And the basic mechanism is typically some kind of progressive annihilation of elements that are initially distributed randomly.

The pictures below show the typical behavior of rule 73—first with completely random initial conditions, and then with initial conditions in which no run of an even number of black squares occurs. … The top example uses completely random initial conditions; the bottom example uses initial conditions in which no run of an even number of black squares ever occurs.

Indeed, in about five out of every million rules of this kind, one gets patterns with features that seem in many respects random, as in the pictures on the next two pages [ 80 , 81 ]. … Apparent randomness becomes slightly more common, but otherwise the results are essentially the same.