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 $W\gg 1$ intervals partitioning the phase space and focus on the unique value $q_{\mathrm{s}\mathrm{e}\mathrm{n}}<1$ for which the entropic form $S_q\equiv (1-\sum _{i=1}^W{p}_i^q)/(q-1)$ linearly increases with time. We then verify that $S_{q_{\mathrm{s}\mathrm{e}\mathrm{n}}}(t)-S_{q_{\mathrm{s}\mathrm{e}\mathrm{n}}}(\infty )$ vanishes like $t^{-1/[q_{\mathrm{r}\mathrm{e}\mathrm{l}}(W)-1]}$ [$q_{\mathrm{r}\mathrm{e}\mathrm{l}}(W)>1$]. We finally exhibit a new finite-size scaling, $q_{\mathrm{r}\mathrm{e}\mathrm{l}}(\infty )-q_{\mathrm{r}\mathrm{e}\mathrm{l}}(W)\propto W^{-|q_{\mathrm{s}\mathrm{e}\mathrm{n}}|}$. 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.

• Received 16 March 2002

DOI:https://doi.org/10.1103/PhysRevLett.89.254103