Chaos Theory and Randomness from Initial Conditions…But if we could watch the solar system for a few million years, then there should be significant randomness that could be attributed to sensitive dependence on the digit sequences of initial conditions—and whose presence in the past may explain some observed present-day features of our solar system.

In a class 2 system with random initial conditions, a similar thing happens: since different parts of the system do not communicate with each other, they all behave like separate patterns of limited size.

[Properties of] random networks The pictures below show networks in which each of a set of n nodes has as its successor a node that is chosen at random from the set. … Note that processes such as cellular automaton evolution do not yield networks whose properties are particularly close to those of purely random ones.

Chaos Theory and Randomness from Initial Conditions…If balls are assumed to fall randomly on each side of each peg then with a large number of balls the final positions will approximate a binomial distribution.

Note (d) for The Intrinsic Generation of Randomness…Algorithmic randomness The idea of there being no simple procedure that can generate a particular sequence can be stated more precisely by saying that there is no program shorter than the sequence itself which can be used to generate the sequence, as discussed in more detail on page 1067 .

Both rules start on step 1 from random initial conditions in which all sequences of black and white cells are allowed.

Note (f) for Randomness from the Environment…Even though an electric current may have a definite overall direction, the individual charge carriers within it will exhibit random motions. … (At very low temperatures, quantum mechanical fluctuations still yield random motion in most materials.)

Properties of [initially random cellular automaton] patterns For a random initial condition, the average density of black cells is exactly 1/2. … Even after many steps, individual lines in the patterns produced by rules 30 and 150 remain in general completely random.

Note (b) for Chaos Theory and Randomness from Initial Conditions…Randomness in the solar system Most motion observed in the solar system on human timescales is highly regular—though sometimes intricate, as in the sequence of numbers of days between successive new moons shown below. … But there are so far no observational signs of randomness resulting from this, and indeed the planets—at least now—mostly just seem to have orbits that are within a few percent of circles. If a planet moved in too random a way then it would tend to collide or escape from the solar system.

Note (b) for Chaos Theory and Randomness from Initial Conditions…To give evidence that this is not merely a reflection of continual injection of randomness from the environment what is normally done is to show that at least some aspect of the behavior of the system can be fit by a definite simple iterated map or differential equation. But inevitably the fit will only be approximate, so there will always be room for effects from randomness in the environment. And in general this kind of approach can never establish that sensitive dependence on initial conditions is actually the dominant source of randomness in a given system—say as opposed to intrinsic randomness generation.