what probability a given deviation from the average should occur for a sequence that is genuinely chosen at random. … So even though individual block frequencies seem to suggest that sequences (d) and (e) are random, the lack of any spread in these frequencies provides evidence that in fact they are not. So are sequences (g) and (h) in the end truly random?

There are many seemingly random fluctuations in this path, though in general it tends to grow to the right.

Some quickly randomize, as the Second Law of Thermodynamics would suggest.

Statistical physics cover The pictures show disks representing idealized molecules bouncing around in a box, and the book claims that as time goes on there is almost inevitably increasing randomization. … From the point of view of this book the randomization seen in these pictures is in large part just a reflection of the fact that a random sequence of digits were used in the initial conditions. But what the discoveries in this book show is that such randomness can also be generated without any such random input—finally clarifying some very basic issues in statistical physics.

Despite the simple definition of π as the ratio of the circumference to the diameter of a circle, its digit sequence is sufficiently complicated as to seem for practical purposes random.

[No text on this page] Examples of random walks on square and hexagonal lattices.

But as I will discuss at length in Chapter 7 one must realize that on its own this cannot explain why randomness—or complexity—should occur in any particular case. And indeed, for the shift map what we have seen is that randomness will occur only when the initial conditions that are given happen to be a number whose digit sequence is random.

One might imagine, however, that if such systems were just to try patterns at random, then even though incredibly few of these patterns would satisfy any given constraint exactly, a reasonable number might at least still come close. … And as an example, the pictures below show what fraction of patterns chosen at random have a given percentage of squares that violate the constraints described on page 211 . … And what this means is that a process based on picking patterns at random will be incredibly unlikely to yield results that are even close to satisfying the constraints.

And as soon as the underlying rule for the cellular automaton is such that information will eventually propagate from one entity to all others—in effect a minimal version of an efficient market hypothesis—it is essentially inevitable that running totals of numbers of cells will exhibit significant randomness. … But although this will make it more difficult to recognize definite rules even if one looks at the complete behavior of every element in the system, it does not affect the basic point that there is randomness that can intrinsically be generated by the evolution of the system. … And although there are patches of predictability that can be seen in the complete behavior of the system the bottom-right plot looks in many respects random.

For even though the detailed placement of black and white cells in the first two pictures does not seem simple to describe, at an overall level these pictures still admit a quite simple description: in essence they just involve a kind of uniform randomness in which every region looks more or less the same as every other. … And if one is interested in descriptions in which no information is lost—as in the discussion of randomness in the previous section —then as I Examples of pictures that at an everyday level one might typically describe either as being "quite random" or as being "quite complex".