Nonextensive Statistical Mechanics

120 4 Stochastic Dynamical Foundations of Nonextensive Statistical Mechanics4.6.1 The MTG Model and Its Numerical ApproachHere we follow [239]. The de Moivre–Laplace theorem is the simplest (and historicallythe earliest) form of the CLT. It consists in proving that the N →∞limit of the binomial distribution is, after centering and rescaling, a Gaussian. Moreprecisely, we consider N independent and distinguishable binary variables, each ofthem having two equally probable states. The joint probabilities are then given byr Nn = 12 N (n = 0, 1, 2,...,N; N = 1, 2,...). (4.33)This set of probabilities can be reobtained by assumingr N0 = 12 N (N = 1, 2,...) , (4.34)and the Leibnitz rule, i.e., Eq. (3.124). We remind that this rule guarantees scaleinvariance,as seen in Section 3.3.5. To avoid the Gaussian as the N →∞attractor,we need to introduce persistent correlations. We shall do this by generalizing Eq.(4.34) and re-written in the following form:1= 1r N0 1/2 × 11/2 × ...× 11/2(N factors) . (4.35)The generalization will consist in introducing the q-product as follows:1= 1r N0 1/2 ⊗ 1q1/2 ⊗ 1q ...⊗ q1/2 = [21−q N −(N −1)] 11−q(0 ≤ q ≤ 1) . (4.36)We have generalized the product between the inverse probabilities, and not theprobabilities themselves, in order to (conveniently) conform to the requirements ofthe q-product (see Eq. (3.78)). The Leibnitz rule is maintained, which enables us tocalculate the entire set {r Nn } by assuming Eq. (4.36).N!(N−n)! n! r Nn, and x ≡ n−(N/2)If we define p(x) ≡, we obtain the results exhibitedin Figs. 4.4 and 4.5. In other words, we verify that, in the limit N →∞,theN/2numerical results approachp(x) ={p(0) e −β + x 2q eif x ≥ 0,p(0) e −β − x 2q eif x ≤ 0 ,(4.37)where β + and β − are slightly different, i.e., the distribution is slightly asymmetric.This specific asymmetry is caused by the fact that we have imposed r N0 , insteadof say r NN , or something similar. By introducing β ≡ 1 2 (β + + β − ), we obtain thedashed line of Fig. 4.4, and the results of Fig 4.5. The index q e in the q e -Gaussian(apparent –butnot exact, as we shall see! – attractor for N →∞) is a function