Nonextensive Statistical Mechanics

122 4 Stochastic Dynamical Foundations of Nonextensive Statistical Mechanics0.4400.43–0.1β(N)0.42ln –4/3 [p(x)/p(0)]–0.2–0.3N = 50N = 80N = 100N = 150N = 200N = 300N = 400N = 500N = 10000.410 0.02 0.04 0.06 0.081/N–0.40 0.2 0.4 0.6 0.8 1Fig. 4.5 ln −4/3p(x)p(0)vs x 2 for (q, p) = (3/10, 1/2) and various system sizes N. INSET: N-dependence of the (negative) slopes of the ln qe vs x 2 straight lines. We find that, for p = 1/2and N >> 1, 〈(n −〈n〉) 2 〉∼N 2 /β(N) ∼ a(q)N + b(q)N 2 .Forq = 1wefinda(1) = 1andb(1) = 0, consistent with normal diffusion as expected. For q < 1wefinda(q) > 0andb(q) > 0,thus yielding ballistic diffusion. The linear correlation factor of the q − log vs. x 2 curves rangefrom 0.999968 up to near 0.999971 when N increases from 50 to 1000. The very slight lack oflinearity that is observed could be expected to vanish in the limit N →∞(from [239]).x 21q e0–1–2–3–4–5–6–7q e=2–1/qq e10.80.60.40.2–80–90.5 0.6 0.7 0.8 0.9 1q–100 0.2 0.4 0.6 0.8 1qFig. 4.6 Relation between the index q from the q-product definition, and the index q e resultingfrom the numerically calculated probability distribution. The agreement with the analytical conjectureq e = 2 − 1 q is remarkable. INSET: Detail for the range 0 < q e < 1 (from [239]).