Nonextensive Statistical Mechanics

160 5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanics10.8x t+1= 1− a c(z) | x t|zq0.60.40.2x t+1= 1− a c(z) e −1/ | x t |z0−0.2−0.40 0.5 1 1.5 2 2.5Fig. 5.6 z-dependence of q avsen (empty circles and squares: present work) and q sen (filled circles:from [128, 129]; filled squares: from [146]). Dotted lines are guides to the eye.belong to the same universality class in the sense that q sen (z) is one and the samefor all of them. But it is not so for qsen av (z). The situation is depicted in Figs. 5.7, 5.8,and 5.9. Also, we numerically verify an intriguing relation between qsen av (cycle n; z)and q rel (cycle n; z), namely (see Fig. 5.10).1/Zq rel (cycle n; z) − 1 ≃ A n [1 − q avsen (cycle n; z)]α n(A n > 0; α n > 0; n = 2, 3, 5,...) . (5.23)The limit q rel (cycle n; z) = qsen av (cycle n; z) = 1 corresponds to the BG case.Finally, we verify (see Fig. 5.11) thatqsen av (cycle 3; z) ≃ 2.5 qav sen (cycle 2; z) − 0.03 , (5.24)◭Fig. 5.5 (continued) Time dependence of 〈ln q ξ〉 and 〈S q 〉: z = 2 logistic map for strong [(a)a = 2] and weak [(c) a = 1.401155189] chaos, and z = 0.5 exponential map for strong [(b)a = 4] and weak [(d) a = 3.32169594] chaos. Sensitivity function 〈ln q ξ〉(t): averages over10 5 (10 7 ) runs for (a) and(b) ((c) and(d)); we use x(0) = 10 −12 as the initial discrepancyunless otherwise indicated; in the insets, we show the linear tendency of the sensitivity functionfor qsen av with various values of x(0); at the edge of chaos ((c)and(d)) we exhibit the q = 1curvenonlinearity. Entropy 〈S q 〉(t): (a,b) 3000 runs with N = 10W with W = 10 5 and W = 3.10 5(empty and filled symbols, respectively); 50,000 runs with N = 10W with W = 10 5 (for (c)) andW = 5.10 4 and W = 10 5 (for (d)). (c) inset: determination of qsen av (see text). (d) inset: we exhibitthe q = 1curvenonlinearity.