Nonextensive Statistical Mechanics

118 4 Stochastic Dynamical Foundations of Nonextensive Statistical MechanicsThis equation has four classes of solutions (see Fig. 4.3) which provide interestinghints.First of all, the Gaussian class, corresponding to q = 1 and γ = 2. Its basicsolution is, as already shown, a Gaussian. This corresponds to the standard CentralLimit Theorem (G − CLT). This theorem essentially states that, if we add (or arithmeticallyaverage) N random variables, that are probabilistically independent andhave finite variance, then the distribution of the sum approaches, after appropriatecentering and rescaling, a Gaussian when N →∞.Second, the Lévy class, orα-class (with α ≡ γ ), corresponding to q = 1 and0 0, hence 5/3 < q < 3 (see [338,339,341–343], based on [344,345]).For instance, the dashed line which joins the (q,γ) points (1, 1) and (2, 2) schematically indicatesthose solutions of Eq. (4) which asymptotically decay as 1/x 2 , and the dashed line joining (1, 1/2)and (7/3, 2) indicates those solutions which decay as 1/|x| 3/2 . The dot slightly to the right of thepoint (5/3, 2) is joint to the point slightly below (1, 2).