Nonextensive Statistical Mechanics

4.8 Generalizing the Langevin Equation 145Fig. 4.24 Both panels represent probability density function P (Y ) vs. Y (properly scaled) in twodifferent log–log scales, where Y represents the sum of N independent variables X each of themhaving a q-Gaussian distribution with q = 9/5 (> 5/3). Since the variables are independent andthevariancediverges,P (Y ) converges to a Lévy distribution as it is visible (from [252]).−b |ξ|αFig. 4.25 Outline of (q,α)-stable distributions (inverse q-Fourier transforms of aeq)forthe case in which the correlation is given by q 1 = 2. As α approaches 2, the (q,α)-stable distributions become closer and closer to a q-Gaussian with = 5/3, with an exponent[2 (q − 1) + α (3 − q)] / [2 (q − 1)]. However, since α ≠ 2, for some value X ∗ , a crossover occursthrough which the distribution changes from the intermediate regime towards the distant regimewith a tail exponent (α + 1) / (1 + α q − α). The inequalities 2/(q − 1) ≥ [2(q − 1) + α(3 −q)]/[2(q − 1)] > (1 + α)/[1 + α(q − 1)] are satisfied (from [253]).