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

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

2002-03-16

arxiv.org/abs/cond-mat/0203342v2

Phys. Rev. Lett. 89, 254103 (2002)

Physics

Condensed Matter

Statistical Mechanics

Final version with new Title and small modifications. REVTeX, 8 pages and 4 eps figures

Scientific paper

10.1103/PhysRevLett.89.254103

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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