Nonextensive Statistical Mechanics

4.5 Stable Solutions of Fokker–Planck-Like Equations 117L γ (|x|) ∝|x| d−1 /|x| d+γ = 1/|x| 1+γ ) and p q (x) ∝ 1/|x| 2/(q−1) , where x is a d-dimensional variable. By identifying the exponents we obtainγ ={2 ifq ≤4+d2+d ,24+d− d ifq−1 2+d < q < 2+dd , (4.30)where we have taken into account that p q (x) is normalizable only if q < 2+dd, andthat its variance is finite only if q < 4+d . The particular instance γ = 1 corresponds2+dto the distribution of the radial component |x| of the d-dimensional Cauchy–Lorentzdistribution (proportional to 1/(a 2 +|x| 2 ), a being a constant). The value q = (3 +d)/(1 + d) precisely leads to γ = 1. Similarly, when q approaches (2 + d)/dfrom below, γ approaches zero from above. The similarities and differences betweenLévy distributions and q-Gaussians are illustrated in Fig. 4.2.4.4.1 Further Generalizing the Fokker–Planck EquationEquation (4.1) can of course be generalized even more, as follows: β p(x, t)|t| β= D β,γ,q γ [p(x, t)] 2−q|x| γ (0