ORIGIN OF THE LONG TIME TAIL IN CHAOTIC DYNAMICS

Abstract

One of the most striking phenomena in chaotic systems is the appearance of long time tails or f^<-v> spectral fluctuations. The main purpose of this paper is to discuss the recent studies concerning the long time behaviors in non-stationary regime. In the first part, some ergodic-theoretical quantities and indecies are concretely determined by use of the modified Bernoulli shift. The origin of f^<-v> spectral fluctuations and the stationary/non-stationary phase transition are characterized in terms of the large derivation theory. The break-up process not only in the law of large number, but also in the law of small number will be precisely described. In the second part, it will be shown that the nearly integrable Hamiltonian systems universally reveals a typical long time tail, and that the universal behavior can be theoretically explained by the stagnant layer theory based on the Nekhoroshev theorem. The importance of two key concepts- non-stationarity and multi-ergodicity will be emphasized. Finally, the origin of the f^<-v> noises in quartz crystal experiments will be discussed from the view-point of the stagnant layer theory.