Three Mechanisms for Randomness

In nature one of the single most common things one sees is apparent randomness. And indeed, there are a great many different kinds of systems that all exhibit randomness. And it could be that in each case the cause of randomness is different. But from my investigations of simple programs I have come to the conclusion that one can in fact identify just three basic mechanisms for randomness, as illustrated in the pictures below.

In the first mechanism, randomness is explicitly introduced into the underlying rules for the system, so that a random color is chosen for every cell at each step.

This mechanism is the one most commonly considered in the traditional sciences. It corresponds essentially to assuming that there is a random external environment which continually affects the system one is looking at, and continually injects randomness into it.

In the second mechanism shown below, there is no such interaction with the environment. The initial conditions for the system are chosen randomly, but then the subsequent evolution of the system is assumed to follow definite rules that involve no randomness.

Three possible mechanisms that can be responsible for randomness. The diagonal arrows represent external input. In the first case, there is random input from the environment at every step. In the second case, there is random input only in the initial conditions. And in the third case, there is effectively no random input at all. Yet despite their different underlying structure, each of these mechanisms leads to randomness in the column shown at the left. The first mechanism corresponds to randomness produced by external noise, as captured in so-called stochastic models. The second mechanism is essentially the one suggested by chaos theory. The third mechanism is new, and is suggested by the results on the behavior of simple programs in this book. I will give evidence that this third mechanism is the most common one in nature.

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From Stephen Wolfram: A New Kind of Science [citation]