Chaos and randomness

Abstract

It is well-known that a rigorous operational definition of randomness is very hard to formulate in terms of classical mathematical primitives. This difficulty is reflected in the difficulty of deciding whether a given (pseudo-)random number sequence is “sufficiently random”. Intuitively, we want the sequence to possess all the properties that a truly random sequence would have, where these properties are well-defined but uncountably infinite in number. This kind of reasoning invariably leads to an infinite number of conditions which must be satisfied, and which in addition are not independent. A more appealing way to approach the problem is through the concepts of chaos and fractals. Certainly a sequence of random numbers is the ultimate self-similar set, since it is (statistically) self-similar at all scales and in all permutations. The idea of applying chaos theory to randomness is not new, but as far as I know, it has only recently given rise to demonstrably “good” random number generators of practical usefulness in massive Monte Carlo calculations. The best of these is probably the algorithm of Martin Lüscher which will be described in some detail.