Note (a) for Chaos Theory and Randomness from Initial Conditions

Note (a) for Chaos Theory and Randomness from Initial Conditions

Note (a) for Chaos Theory and Randomness from Initial Conditions

Code 10 Rule 30 is by many measures the simplest cellular automaton that generates randomness from a single black initial cell. … And indeed among the 64 such rules, 13 show randomness.

Random programs See page 1182 .

It is common for animals to move in apparently random ways when they are trying to avoid predators. Yet I suspect that the randomness they use is often generated by quite simple rules (see page 1011 )—so that in principle it could be predictable. So it is then notable that biological evolution has apparently never made predators able to catch their prey by predicting anything that looks to us particularly random; instead strategies tend to be based on tricks that do not require predicting more than at most repetition.

[Meaning in] molecular biology DNA sequences of organisms can be thought of as artifacts created by biological evolution, and current data suggests that they contain some long-range correlations not present in typical random sequences. … But probably such patterns would also occur in purely random proteins—at least if their folding happened in the same cellular apparatus. … Note that the antibodies of the immune system are much like short random proteins—whose range of shapes must be sufficient to match any antigen.

Self-avoiding [random] walks Any walk where the probabilities for a given step depend only on a fixed number of preceding steps gives the same kind of limiting Gaussian distribution. … If one adds individual steps at random then in 2D one typically gets stuck after perhaps a few tens of steps. … They look in many ways similar to ordinary random walks, but their limiting distribution is no longer strictly Gaussian, and their root mean square displacement after t steps varies like t 3/4 .

Usually, however, it is turbulent near the ground—producing, for example, random gusting in wind—but becomes laminar at higher altitudes. Turbulent convection nevertheless occurs in most clouds, leading to random billowing shapes.

But out of a million randomly chosen rules, there will typically be a few that show complex behavior. Page 186 shows one example where the behavior seems in many respects completely random.