Nonextensive Statistical Mechanics

4.6 Probabilistic Models with Correlations – Numerical and Analytical Approaches 121ln –4/3 [p(x)/p(0)]0–0.1–0.2–0.3p(x)*N/20.60.50.40.30.20.10–1 –0.5 0 0.5 1x = (n – N/2)/N/2–0.40 0.2 0.4 0.6 0.8 1 1.2x 2Fig. 4.4 ln −4/3p(x)p(0) vs x 2 for (q, p) = (3/10, 1/2) and N = 1000. Two branches are observed dueto the asymmetry emerging from the fact that we have imposed the q-product on the left side of thetriangle; we could have done otherwise. The mean value of the two branches is indicated in dashedline. It is through this mean line that we have numerically calculated q e (q) as indicated in Fig. 4.6.In order to minimize the tiny asymmetry, we have represented a variable x slightly displaced withregard to n−(N/2)N/2so that the center x = 0 precisely coincides with the location of the maximum ofp(x). INSET: Linear–linear representation of p(x) (from [239]).of the index q in the q-product (which, together with Leibnitz rule, introduces thescale-invariant correlations into the probability sets). The numbers strongly suggest(see Fig. 4.6)q e = 2 − 1 q(0 ≤ q ≤ 1) . (4.38)The particular case q = q e = 1 recovers of course the celebrated de Moivre–Laplace theorem. This transformation is a simple combination of the multiplicativedualityμ(q) ≡ 1/q , (4.39)and the additive dualityν(q) ≡ 2 − q . (4.40)In other words, relation (4.38) can be rewritten as q e = νμ(q) ≡ ν(μ(q)). Thisrelation as well as the two basic dualities appear again and again in the literature ofnonextensive statistical mechanics, in very many contexts (see, for instance, [284,417, 419, 420, 869]).