Nonextensive Statistical Mechanics

5.1 Low-Dimensional Dissipative Maps 157Fig. 5.3 The exponentμ ≡ 1/(q rel − 1) for thez-logistic map, as a functionof z (top), and of the fractaldimension d H ≡ d f (bottom.From [148]).For the z-circular map, it is numerically found [148] q rel (z) →∞and d H (z) = 1,∀z, which also is consistent with a relation such as Eq. (5.17) (q rel →∞suggests alogarithmic behavior instead of the asymptotic power-law in Eq. (5.16)). 3An alternative way for studying q rel has been proposed in [140]. If we considerS 1 (t) for a map which is strongly chaotic (or S qent (t) for a map which is weaklychaotic) for a given number W (0) of little cells within which the phase-spacehas been partitioned, we typically observe the following behavior. For small valuesof t there is a transient; for intermediate values of t there is a linear regime(which enables the calculation of the entropy production per unit time, and becomeslonger and longer with increasing W (0)); finally, for larger values of t theentropy approaches (typically from above!) its saturation value S q (∞). Therefore,S q (t) − S q (∞) vanishes with diverging t, and it does so as follows:S qent (t) − S qent (∞) ∝ e −t/τ q relq rel, (5.18)which enables the determination of q rel , as well as that of τ qrel . See Fig. 5.4.3 The d-dimensional generalization of Eq. (5.17) might well be 1/(q rel −1) ∝ (d−d H ) 2 . Therefore,all the so-called fat-fractal dynamical attractors (i.e., d H = d) would yield q rel →∞.