Random initial conditions in other systems Whenever the initial conditions for a system can involve an infinite sequence of elements these elements can potentially be chosen at random. … Ordinary substitution systems can operate on infinite sequences of elements chosen at random. … Random networks (see pages 963 and 1038 ) can be used to provide random initial conditions for network systems.

Note (e) for Randomness from the Environment…Random walks See page 328 .

Note (c) for Randomness from the Environment…Stochastic models The mechanism for randomness discussed in this section is the basis for so-called stochastic models now widely used in traditional science. Typically the idea of these models is to approximate those elements of a system about which one does not know much by random variables. … But when electronic computers became available in the 1940s, the so-called Monte Carlo method became increasingly popular, in which instead explicit simulations are performed with different choices of random variables, and then statistical averages are found.

Continual injection of randomness [in cellular automata] In the main text we discuss what happens when one starts from random initial conditions and then evolves according to a definite cellular automaton rule. As an alternative one can consider starting with very simple initial conditions, such as all cells white, and then at each step randomly changing the color of the center cell. … The results are usually very similar to those obtained with random initial conditions.

Different runs [of initially random cellular automata] The qualitative behavior seen with a given cellular automaton rule will normally look exactly the same for essentially all different large random initial conditions—just as it does for different parts of a single initial condition. And as discussed on page 597 any obvious differences could in effect be thought of as revealing deviations from randomness in the initial conditions.

With most rules, including 90 and 150, such nested initial conditions typically yield results that are ultimately indistinguishable from those obtained with typical random initial conditions.

Note (b) for Randomness in Class 3 Systems

Note (c) for Randomness in Class 3 Systems

Note (b) for Randomness in Class 3 Systems

Note (c) for Randomness in Class 3 Systems