Anomalous Convergence of the Time Average in Semi-Markovian Chaotic Dynamics

Abstract

Two state semi-Markovian process and basic properties General theory of the semiMarkovian chaotic dynamics was developed in the previous paper,I) where the origin of long time tails was clearly explained_ In this article we are going to elucidate the large deviation for a dichotomous semi-Markovian process, which is the simplest model of intermittent chaos. We consider a random process a(t) which can take two sates A and B. The residence time r in each state is the statistically independent random variable. Denote the residence time distributions by a( r) and b( r) for A and B states respectively, and let a(t) be the indicator defined by a(t)=O in state A and a(t)=l in state B. We define a random variable AN = (l/N)JoNa(t)dt, which means the time average of aU) during the interval [0, N] and O::::::AN::::::L Denote the probability of the start at A (or B) state by PA (or PB) (h + PB=l), and the first pausing time distribution in each state by al(r) and bl(r) respectively, then the distribution density defined by p(N; il) =prob {il<AN::::::il+dil}/dil becomes