Nonextensive Statistical Mechanics

10 –5 1 10 100 100010 –5 1 10 100 1000168 5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanicsa10 010 –1a = 1.401175a = 1.401159451/(1–q);q = 1.7;β = 6.2P (y) / P (0)10 –210 –3N = 2 2nn ini = 10 810 –41 + (q−1)β[ y P(0)] 2b10 010 –1a = 1.4011644a = 1.401157161/(1–q);q = 1.63;β = 6.2P (y) / P (0)10 –210 –3N = 2 2nn ini = 10 810 –41 + (q−1)β[ y P(0)] 2Fig. 5.16 Probability density of the quantity y/σ at the critical point a c for z = 1.75, 2, 3(from [371]).lim t→∞ lim W →∞ lim M→∞ S BG (t)/t = λ (1)1. An analytic proof would naturally bemost welcome.The standard map (or kicked rotor map) is defined as follows [85, 356]:p t+1 = p t + a2π sin(2πθ t) (mod 1)θ t+1 = θ t + p t+1 (mod 1) (a ≥ 0) . (5.32)This map is only partially chaotic (i.e., the size of the “chaotic sea” is smallerthan the unit square), and the percentage of chaos increases for increasing a. Insidethe chaotic sea, it has been numerically verified [85] that lim t→∞ lim W →∞ lim M→∞S BG (t)/t = λ (1)1equals 0.98, 1.62, and 2.30 for a = 5, 10, and 20, respectively.It is instructive to define [356] a dynamical “temperature” T as the variance of theangular momentum, i.e., T ≡〈(p−〈p〉) 2 〉=〈p 2 〉−〈p〉 2 , where 〈〉 denotes ensemble