Beyond random walks, there are many other systems based on discrete components in which randomness at a microscopic level also leads to continuous behavior on a large scale. … Behavior of a simple aggregation model, in which a single new black cell is added at each step at a randomly chosen position adjacent to the existing cluster of black cells.

So what about randomness? The pictures on the next page show a few examples of images with various degrees of randomness. And just by looking at these images it is remarkably difficult to tell which of them is in fact the most random.

Note (d) for Chaos Theory and Randomness from Initial Conditions…Often this happens quickly, but sometimes all three bodies show complex and apparently random behavior for quite a while.

But if one finds that the value lies far from the average then one can take this as evidence that the sequence is not random. … So any deviation from such uniform gray potentially provides evidence for a deviation from randomness. … So from this it is fairly easy to conclude that the first three sequences are definitely not random, while the remaining sequences could still be random.

But in case (b), the force quickly randomizes, and there is no obvious way to obtain systematic mechanical work from it. … Case (b) corresponds to a box containing a fixed obstacle, and in this case rapid randomization is seen. … In case (b), however, it rapidly becomes for most practical purposes random.

And even though there is no explicit randomness inserted into the model in any way, the paths of the cracks that emerge nevertheless appear to be quite random. … But I nevertheless suspect that even when much more realistic models for specific materials are used, the fundamental mechanisms responsible for randomness will still be very much the same as in the extremely simple model shown here. … Even though no randomness is inserted from outside, the paths of the cracks that emerge from this model nevertheless appear to a large extent random.

Typical intuition from traditional science makes it difficult to understand how such randomness could possibly arise. … For there appears to be an irreversible increase in randomness as one goes down successive panels on the page. … Starting from an initial condition in which all black cells or particles lie at the center of a box, the distribution becomes progressively more random.

For our everyday experience is full of examples in which randomness increases much as in the second half of the picture below. But we essentially never see the kind of systematic decrease in randomness that occurs in the first half. … If one starts with this arrangement, then the randomness of the system will effectively increase whether one goes forwards or backwards in time from that point.

This simple rule produces randomness through the mechanism of intrinsic randomness generation, and this randomness in turn leads to a pattern of growth that takes on an increasingly smooth more-or-less circular form.

[No text on this page] Network systems in which the total number of nodes obtained on successive steps appears to vary in a largely random way forever. About one in 10,000 randomly chosen network systems seem to exhibit the kind of behavior shown here.