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Complex behavior in the rule 110 cellular automaton starting from a random initial condition.

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Other examples of cellular automata that never settle down to stable states when started from random initial conditions.

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Patterns produced after 500 steps in the evolution of a sequence of two-dimensional cellular automata starting from random initial conditions.

Note (c) for Three Mechanisms for Randomness…History [of randomness]
In antiquity, it was often assumed that all events must be governed by deterministic fate—with any apparent randomness being the result of arbitrariness on the part of the gods. … But the significance of this for randomness in nature was never recognized. … And in fact the widespread assumption emerged that between randomness in the environment, quantum randomness and chaos theory almost any observed randomness in nature could be accounted for.

Note (a) for Defining the Notion of Randomness…As mentioned in the main text, any data generated by a simple program can by definition never be algorithmically random. And so even though algorithmic randomness is often considered in theoretical discussions (see note below ) it cannot be directly relevant to the kind of randomness we see in so many systems in this book—or, I believe, in nature.
… But even though one knows that almost all long sequences must be algorithmically random, it turns out to be undecidable in general whether any particular sequence is algorithmically random.

Note (a) for Randomness from the Environment…Power spectra [of random processes]
Many random processes in nature show power spectra Abs[Fourier[data]] 2 with fairly simple forms. Most common are white noise uniform in frequency and 1/f 2 noise associated with random walks. … Mechanisms that generally seem able to give α ≃ 1 include random walks with exponential waiting times, power-law distributions of step sizes (Lévy flights), or white noise variations of parameters, as well as random processes with exponentially distributed relaxation times (as from Boltzmann factors for uniformly distributed barrier heights), fractional integration of white noise, intermittency at transitions to chaos, and random substitution systems.

Note (f) for The Intrinsic Generation of Randomness…Cellular automata [as randomness generators]
From the discussion here it should not be thought that in general there is necessarily anything better about generating randomness with cellular automata than with systems based on numbers. But the point is that the specific method used for making practical linear congruential generators does not yield particularly good randomness and has led to some incorrect intuition about the generation of randomness. … In addition, one should recognize that while the complete evolution of the cellular automaton may effectively generate perfect randomness, there may be deviations from randomness introduced when one constructs a practical random number generator with a limited number of cells.

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Another example of a cellular automaton that produces a nested pattern even from random initial conditions.

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Examples of the evolution of two-dimensional cellular automata with various totalistic rules starting from random initial conditions.

Note (c) for Randomness from the Environment…Randomness in biology
Thermal fluctuations in chemical reactions lead to many kinds of microscopic randomness in biological systems, sometimes amplified when organisms grow. … Random changes in single DNA molecules can have global effects on the development of an organism. … Another reason is that egg and sperm cells get half the genetic material of an organism, somewhat at random.