Nonextensive Statistical Mechanics

172 5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanics(a)T0.080.061/t cT BG≡1/12 a = 201×10 –3a = 1.6a = 1.20.04a = 1.1t 25×10 –40.02a = 1.051a = 0.70a = 0010 0 10 1 10 2 10 3 10 4 10 50 0.01 0.02 0.03t(a–a c ) 2.7(c)(b)p10.5t = 0 t = t 1 t = t 2110.50.500 0.5 1PDF0.60.300 0.5 100 0.5 10.020.01θ00 0.5 1p00 0.5 10.020.0100 0.5 1Fig. 5.21 (a) Time evolution of the dynamical temperature T of a standard map, for typical valuesof a. We start with “water bag” initial conditions (M = 2500 points in 0 ≤ θ ≤ 1, p = 0.5 ±510 −4 ). In order to eliminate cyclical fluctuations, the dots represent average of 10 iteration steps;moreover, each curve is the average of 50 realizations. (b) Inverse crossover time t c (inflectionpoint between the QSS and the BG regimes) vs. 1/(a − a c ) 2.7 . No inflection points subsist if t islinearly represented. (c) Time evolution of the ensemble in (a) fora = 1.1 (first row) andPDFof its angular momentum (second row). t = 0: “water bag” initial conditions; t = t 1 = 500: theensemble is mostly restricted by cantori; t = t 2 = 10 5 : the ensemble is confined inside KAM-tori(from [356]).5.2.2 Strongly Chaotic Four-Dimensional Conservative MapsIn the previous subsection we considered N = 1 particle. Let us consider hereN = 2, on the road to the thermodynamic limit N → ∞ [85, 356, 357]. Weshall focus on a simple symplectic system of two coupled standard maps, defined asfollows:θ 1 (t + 1) = p 1 (t + 1) + θ 1 (t) + bp 2 (t + 1),p 1 (t + 1) = p 1 (t) + a 12π sin[2πθ 1(t)], (5.33)θ 2 (t + 1) = p 2 (t + 1) + θ 2 (t) + bp 1 (t + 1),p 2 (t + 1) = p 2 (t) + a 22π sin[2πθ 2(t)],