Nonequilibrium Probabilistic Dynamics of the Logistic Map at the Edge of Chaos

Abstract

We consider nonequilibrium probabilistic dynamics in logistic-like maps $x_{t+1}=1-a|x_t{|}^z$, $(z>1)$ at their chaos threshold: We first introduce many initial conditions within one among $W>>1$ intervals partitioning the phase space and focus on the unique value $q_{sen}<1$ for which the entropic form $S_q\equiv \frac{1-{\sum }_{i=1}^W{p}_i^q}{q-1}$ {\it linearly} increases with time. We then verify that $S_{q_{sen}}(t)-S_{q_{sen}}(\infty )$ vanishes like $t^{-1/[q_{rel}(W)-1]}$ [$q_{rel}(W)>1$]. We finally exhibit a new finite-size scaling, $q_{rel}(\infty )-q_{rel}(W)\propto W^{-|q_{sen}|}$. This establishes quantitatively, for the first time, a long pursued relation between sensitivity to the initial conditions and relaxation, concepts which play central roles in nonextensive statistical mechanics.

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