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

Abstract

We consider nonequilibrium probabilistic dynamics in logisticlike maps ${x}_{t+1}=1\ensuremath{-}a|{x}_{t}{|}^{z}$, $(z>1)$ at their chaos threshold: We first introduce many initial conditions within one among $W\ensuremath{\gg}1$ intervals partitioning the phase space and focus on the unique value ${q}_{\mathrm{s}\mathrm{e}\mathrm{n}}<1$ for which the entropic form ${S}_{q}\ensuremath{\equiv}(1\ensuremath{-}\ensuremath{\sum}_{i=1}^{W}{p}_{i}^{q})/(q\ensuremath{-}1)$ linearly increases with time. We then verify that ${S}_{{q}_{\mathrm{s}\mathrm{e}\mathrm{n}}}(t)\ensuremath{-}{S}_{{q}_{\mathrm{s}\mathrm{e}\mathrm{n}}}(\ensuremath{\infty})$ vanishes like ${t}^{\ensuremath{-}1/[{q}_{\mathrm{r}\mathrm{e}\mathrm{l}}(W)\ensuremath{-}1]}$ [${q}_{\mathrm{r}\mathrm{e}\mathrm{l}}(W)>1$]. We finally exhibit a new finite-size scaling, ${q}_{\mathrm{r}\mathrm{e}\mathrm{l}}(\ensuremath{\infty})\ensuremath{-}{q}_{\mathrm{r}\mathrm{e}\mathrm{l}}(W)\ensuremath{\propto}{W}^{\ensuremath{-}|{q}_{\mathrm{s}\mathrm{e}\mathrm{n}}|}$. 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.