CHAOS AND RANDOMNESS

It is well-known that a rigorous operational definition of randomness is very hard to formulate in terms of classical mathematical primitive s. This difficulty is reflected in the difficulty of deciding whether a given (pseudo-)random number sequence is ''sufficiently random''. In tuitively, we want the sequence to possess all the properties that a t ruly random sequence would have, where these properties are well-defin ed but uncountably infinite in number. This kind of reasoning invariab ly leads to an infinite number of conditions which must be satisfied, and which in addition are not independent. A more appealing way to app roach the problem is through the concepts of chaos and fractals. Certa inly a sequence of random numbers is the ultimate self-similar set, si nce it is (statistically) self-similar at all scales and in all permut ations. The idea of applying chaos theory to randomness is not new, bu t as far as I know,, it has only recently given rise to demonstrably ' 'good'' random number generators of practical usefulness in massive Mo nte Carlo calculations. The best of these is probably the algorithm of Martin Luscher which will be described in some detail.