Nonequilibrium probabilistic dynamics of the logistic map at the edge of chaos

Abstract

We consider nonequilibrium probabilistic dynamics in logisticlike maps xt+1=1-a|xt|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 qsen<1 for which the entropic form Sq≡(1 ∑i=1Wpiq)/(q-1) linearly increases with time. We then verify that Sqsen(t)-Sqsen(∞) vanishes like t-1/[qrel(W)-1] [qrel(W)>1]. We finally exhibit a new finite-size scaling, qrel(∞)-qrel(W)∝W-|qsen|. 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.