Nonequilibrium probabilistic dynamics of the logistic map at the edge of chaos -: art. no. 254103

We consider nonequilibrium probabilistic dynamics in logisticlike maps x(t+1)=1-aparallel tox(t)parallel to(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-Sigma(i=1)(W)p(i)(q))/(q-1) linearly increases with time. We then verify that S-qsen(t)-S-qsen(infinity) vanishes like t(rel)(-1/[q)(W)-1] [q(rel)(W)>1]. We finally exhibit a new finite-size scaling, q(rel)(infinity)-q(rel)(W)proportional toW(sen)(-\q)\. 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.