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

TL;DR

This paper investigates the nonequilibrium probabilistic dynamics of the logistic map at the edge of chaos, revealing a quantitative relation between sensitivity to initial conditions and relaxation using nonextensive entropy.

Contribution

It introduces a finite-size scaling relation linking sensitivity and relaxation parameters in logistic maps at chaos threshold, advancing nonextensive statistical mechanics.

Findings

Identifies the unique entropic index q_{sen} for linear entropy growth.

Shows the decay of entropy difference as a power law with time.

Establishes a new finite-size scaling relation between q_{rel} and system size W.

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.