Nonextensive Statistical Mechanics

156 5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanicssystems, we expect the straightforwardly q-generalized Kolmogorov–Sinai entropyrate to coincide with K qent .The properties that have been exhibited here for the sensitivity to the initialconditions and the entropy production have also been checked [143, 144] for otherentropies directly related to S q . The scheme remains the same, excepting for theslope K qent , which does depend on the particular entropy. The slope for S q turns outto be the maximal one among those that have been analyzed. For all these q ≠ 1examples, the Renyi entropy SqR fails in providing a linear time dependence: it providesinstead a logarithmic time dependence.5.1.1.4 RelaxationIn the previous paragraphs, we were dealing with the value of q, q sen , associatedwith the sensitivity to the initial conditions, and also with multifractality and theentropy production. We address now a different property, namely relaxation. As weshall see, a new value of q, denoted q rel (where rel stands for relaxation), emerges.Typically q rel ≥ 1, the equality holding for strongly chaotic systems (i.e., whenq sen = 1). Relaxation was systematically studied for the z-logistic map in [148]. Theprocedure consists in starting, at the edge of chaos, with a distribution of M >> 1initial conditions which is uniform in phase-space (x 0 ∈ [−1, 1] for the z-logisticmap), and let evolve the ensemble towards the multifractal attractor. A partitionof the phase-space is established with W (0) >> 1 little equal cells, and then thecovering is followed along time by only counting those cells which have at least onepoint at time t. This determines W (t), which gradually decreases since the measureof the multifractal attractor is zero. In the M →∞and W (0) →∞limits, anddisregarding small oscillations, it is verifiedW (t)W (0) ≃ e−t/τ q relq rel, (5.16)with q rel (z) ≥ 1 and τ qrel (z) > 0. If it is taken into account the fact that, for thez-logistic map, also the Hausdorff dimension depends on z, it can be numericallyverified the following quite intriguing, and yet unexplained, relation:1q rel (z) − 1 ≃ a [1 − d H (z)] 2 (z ∈ [1.1, 5.0]) , (5.17)with a = 3.3 ± 0.3. See Fig. 5.3. Higher precision calculations are available forz = 2, namely 1/[q rel (2) − 1] = 0.800138194 ..., hence q rel (2) = 2.249784109 ...[149, 152]. 22 These two references concern the approach to the multifractal attractor as a function of time.However, [149] contains a general criticism concerning also the time evolution within the attractor.This is rebutted in [150] (see also [151]).