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

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 W1 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 S_qₛₑₙ(t)-S_qₛₑₙ(∞) 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.