By intrinsically generating randomness such systems in a sense have a certain fundamental stability: for whatever is done to their initial conditions, they still give the same overall random behavior, with the same large-scale properties. … Special Initial Conditions We have seen that cellular automata such as rule 30 generate seemingly random behavior when they are started both from random initial conditions and from simple ones. … Small patches with the same structures as shown here can be seen embedded in typical random patterns produced by rule 30.

And indeed the picture on the next page shows one of many examples in which starting from random initial conditions there continues to be very complicated behavior forever. And indeed the behavior that is produced appears in many respects completely random. … Examples of cellular automata that evolve from random initial conditions to produce a definite set of simple structures.

Chaos Theory and Randomness from Initial Conditions…And then in the first case, the subsequent motion looks quite random. … But after a while they reach the edge of the material, and although in the first case they then show quite random behavior, in the second case they instead just show simple repetitive behavior. What differs between the two cases is the detailed digit sequences of the positions of the points: in the first case these digit sequences are quite random, while in the second case they have a simple repetitive form.

But in this section I want to ask the more general question of what arbitrary cellular automata do when started from random initial conditions. … But the next few pages [ 232 , 233 , 234 ] show various sequences of cellular automata, all starting from random initial conditions. … Examples of the four basic classes of behavior seen in the evolution of cellular automata from random initial conditions.

Chaos Theory and Randomness from Initial Conditions…And once again, at least for a while, any randomness in the motion of the ball can be attributed to randomness in this initial digit sequence. But after at most ten or so collisions, many other effects, mostly associated with continual interaction with the environment, will always in practice become important, so that any subsequent randomness cannot solely be attributed to initial conditions. And indeed in any system, the amount of time over which the details of initial conditions can ever be considered the dominant source of randomness will inevitably be limited by the level of separation that exists between the large-scale features that one observes and small-scale features that one cannot readily control.

And the picture below shows that rule 184—unlike any of the additive rules—still produces recognizably nested patterns even when the initial conditions that are used are random. … And in general it is possible to find quite a few cellular automata that yield nested patterns like rule 184 even from random initial conditions. … Rule 184 evolving from a random initial condition.

[No text on this page] A cellular automaton that never settles down to a stable state, but instead continues to show behavior that seems in many respects random.

[No text on this page] A typical example of the behavior of the rule 110 cellular automaton with random initial conditions.

Note (a) for Randomness from the Environment…Quantum randomness It is usually assumed that even if all else fails a quantum process such as radioactive decay will yield perfect randomness. … Acceptable randomness has however been obtained at rates of tens of bits per second. Recent attempts have also been made to produce quantum randomness at megahertz rates by detecting paths of single photons.

Each picture shows 1500 steps of evolution from random initial conditions.