Nonextensive Statistical Mechanics

154 5 Deterministic Dynamical Foundations of Nonextensive Statistical MechanicsFig. 5.2 Numerical corroboration (full circles)oftheq-generalized Pesin-like identity K (k)q= λ q(k)t),at the edge of chaos the logistic map. On the ordinate we plot the q-logarithm of ξ tk (equal to λ q(k)and in the abscissa S q (equal to K q(q) t), both for q = 0.2445 .... The dashed line is a linear fit.Inset: The full lines are from the analytic result (from [147]).For unimodal maps with inflection z, negative Schwarzian derivative in thebounded interval, and partition scale b we havewhere α F is the so-called Feigenbaum constant. Henceα max (z) =ln bln α F (z) ,ln bα min (z) =z ln α F (z) , (5.10)11 − q sen (z) = (z − 1)ln α F(z). (5.11)ln bFor the universality class of the z-logistic map, we have b = 2 hence11 − q sen (z) = (z − 1)ln α F(z). (5.12)ln 2