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

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We consider nonequilibrium probabilistic dynamics in logistic-like 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₌₁ʷ pᵢqq-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.