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

2002/03/31 by Ernesto P. Borges, Constantino Tsallis, Garin F. J. Ananos +3
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Fractional Differential Equations Solutions #Statistical Mechanics and Entropy #cond-mat.stat-mech

paper · pdf · doi:10.1103/physrevlett.89.254103

published as Phys. Rev. Lett. 89, 254103 (2002) · Final version with new Title and small modifications. REVTeX, 8 pages and 4 eps figures

openalex publication_date 2002/12/05 · arxiv created 2002/12/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider nonequilibrium probabilistic dynamics in logisticlike maps x_t+1=1\ensuremath-a|x_t|^z, (z>1) at their chaos threshold: We first introduce many initial conditions within one among W\ensuremath≫1 intervals partitioning the phase space and focus on the unique value q_sen<1 for which the entropic form S_q\ensuremath≡(1\ensuremath-\ensuremath∑_i=1^Wp_i^q)/(q\ensuremath-1) linearly increases with time. We then verify that S_q_sen(t)\ensuremath-S_q_sen(\ensuremath∞) vanishes like t^\ensuremath-1/[q_rel(W)\ensuremath-1] [q_rel(W)>1]. We finally exhibit a new finite-size scaling, q_rel(\ensuremath∞)\ensuremath-q_rel(W)\ensuremath∝W^\ensuremath-|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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