Nonextensive Statistical Mechanics

176 5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanics(a)200S q(n)q = –0.2150(b)4S ′ q (n) q = –0.2310050q = 0q = +0.200 20 40 60 80 100n(c)250S 0(n) W = 4 × 10 22001501005000 20 40 60 80 100n21q = 0q = +0.200 25 50 75 100nW = 4 × 10 4W = 4 × 10 6W = 1.6 × 10 7Fig. 5.26 Time-evolution of the statistical entropy S q for different values of q. The phase-spacehas been divided into W = 4000 × 4000 equal cells of size l = 5 × 10 −4 and the initial ensembleis characterized by N = 10 3 points randomly distributed inside a partition-square. Curves are theresult of an average over 100 different initial squares randomly chosen in phase-space. The analysisof the derivative of S q in (b) shows that only for q = 0 a linear behavior is obtained. In fact, a linearregression provides S 0 (n) = 1.029 n+1.997 with a correlation coefficient R = 0.99993. (c)showsthat the linear growth for S 0 is reached from above, in the limit W →∞. (from [358]).Fig. 5.26 that q ent = 0. The linear time-dependence [8,9] of the sensitivity ξ impliesq sen = 0, which, as usual, coincides with q ent . Furthermore, we can verify that theq-generalized Pesin-like identity is once again satisfied.In conclusion, while positivity of Lyapunov exponents is sufficient for a meaningfulstatistical description (the BG statistical mechanics), it might be not necessary.Indeed, we have illustrated, for a conservative, mixing and ergodic nonlinear dynamicalsystem, that the use of the more general entropy S q (with the value q = 0forthis case) provides a satisfactory frame for handling nonlinear dynamical systemswhose maximal Lyapunov exponent vanishes. In particular, we have shown that(the upper bound of) the coefficient λ q of the sensitivity to the initial conditionscoincides with the entropy production per unit time, in total analogy with the Pesintheorem for standard chaotic systems. These results suggest that a thermostatisticalapproach of such systems is possible. Indeed, the structure that we have exhibitedhere for the time dependence of S q is totally analogous to the one that has beenrecently exhibited [199] for the N-dependence of S q , where N is the number ofelements of a many-body system. When the number of nonzero-probability statesof the system increases as a power of N (instead of exponentially with N as usually),