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

Borges, Ernesto P.; Tsallis, Constantino; Ananos, Garin F. J.; de Oliveira, Paulo Murilo C.

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

Authors

Borges, Ernesto P.

Tsallis, Constantino

Ananos, Garin F. J.

de Oliveira, Paulo Murilo C.

Description

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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