Nonextensive Statistical Mechanics

164 5 Deterministic Dynamical Foundations of Nonextensive Statistical MechanicsN = 2x10 6 ; n ini =2x10 6N = 100; n ini =10x10 6eq.(9); N = 100eq.(9); N = 2x10 610 010 –1a = 2ρ y ( y)10 –210 –310 –4–4 –2 0 2 4yFig. 5.12 Probability density of rescaled sums of iterates of the logistic map with a = 2; N =2 · 10 6 and N = 100. The number of initial values contributing to the histogram is n ini = 2 ·10 6 , respectively n ini = 10 7 .Thesolid lines correspond to the analytical expressions for finite N(from [370]).more transparent later on) [370]. See Figs. 5.15 and 5.16. In these figures we canappreciate relatively well the tails of the distributions. The central part can be seenin Fig. 5.17. 4Let us summarize the case of simple one-dimensional dissipative maps at theedge of chaos by reminding that we have established the existence of a basicq-triplet. In particular, for the z = 2logisticmapwehave(q sen , q rel , q stat ) ≃(0.24, 2.2, 1.7). Later on, we turn back onto q-triplets (as well as other values ofq; see, for instance, [150, 155]).5.1.2 Two-Dimensional Dissipative MapsAlthough not with the same detail as for the one-dimensional ones, some twodimensionaldissipative maps have been studied as well [156–158]. More specifically,the Henon and the Lozi maps. Let us illustrate with the Henon map. It isdefined as follows:4 An illustration on a different map can be seen in G. Ruiz and C. Tsallis, Nonextensivity at theedge fo chaos of a new universality class of one-dimensional maps, Eur. Phys. J. B. (2009), inpress, 0901.4292 [cond-mat.stat-mech].