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

We consider nonequilibrium probabilistic dynamics in logisticlike 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≡(1-∑_i=1^Wp_i^q)/(q-1) linearly increases with time. We then verify that S_qsen(t)-S_qsen(∞) vanishes like t^{-1/[q_rel(W)-1]} [q_rel(W)>1]. We finally exhibit a new finite-size scaling, q_rel(∞)-q_rel(W)∝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.