Nonextensive Statistical Mechanics

146 4 Stochastic Dynamical Foundations of Nonextensive Statistical MechanicsFig. 4.26 Top: Probability distribution P (Y N ) vs. Y N ,withY N ≡ ∑ Ni=1 X i , X i being ( )q = 5 3 -independent random variables associated with a G 3 (X) distribution with β = 1(left), and the2-squaredrespective ( q = 3 )2 -Fourier Transform, ˜P (k), vs. k (right). Middle: Same as above, in ln 32-squared scale (right). The straight lines indicate that P (Y N ) and ˜P (k) arescale (left), and ln 53q-Gaussians with q = 3 2 and q = 5 3and β ′ q ∗(N) for right panel curves. Bottom: β −1, respectively. Their slopes are β−1q ∗=3/2(N) for left panel curvesq (N) vs. N 2 ∗=3/2, which is a straight line with slope 1(left); βq ′ 3−q∗=3/2(N) vs. N which is also a straight line but with slope(right) (from [253]).∗8 Cq∗2(q∗−1)∣∣q ∗=3/2= 0.088844 ...This historical equation has been generalized in very many ways. Some ofthem yield exact solutions which are q-Gaussians. Two such examples are describedin [304] (simultaneous presence of uncorrelated additive and multiplicativenoises) and in [306] (dichotomous colored noise). Because of its particularly simplenature, we shall present here the first example in detail. Let us consider the followinggeneralization of Eq. (4.96):ẋ = f (x) + g(x) ξ(t) + η(t) , (4.98)where g(x) is an arbitrary function satisfying g(0) = 0, and ξ(t) is a Gaussiandistributedzero-mean white noise satisfying