Nonextensive Statistical Mechanics

114 4 Stochastic Dynamical Foundations of Nonextensive Statistical Mechanicsviii. x scales with t 1/(3−q) . Consequently, if 〈x 2 〉 is finite (i.e., if q < 5/3), it mustbeBy using Eq. (4.6), we obtainAnalogously we have that〈x 2 〉∝t 23−q . (4.15)μ = 23 − q . (4.16)〈x 2 〉 q ∝ t 23−q(q < 3) . (4.17)Consequently, we have superdiffusion for 1 < q < 3 (ballistic for q = 2),and subdiffusion for q < 1 (localization for q →−∞).ix. We see that D q > 0(D q < 0) if q < 2(2< q < 3). The q → 2 limit deservesa special comment. Equation (4.11) can be re-written in the followingform [272, 273, 275]:p(x, t)t= (2 − q)D q 2[p(x, t)] 2−q − 1x 2 2 − q].(4.18)In the limit q → 2, this equation becomesp(x, t)t={} 2lim [(2 − q)D q] ln p(x, t) , (4.19)q→2 x2{}with lim q→2 [(2 − q)D q ] > 0. This equation is known to have as solutionthe Cauchy–Lorentz distribution (more precisely, p 2 (x, t)).In the presence of an external drift, Eq. (4.11) is extended as follows [348, 349]:p(x, t)=− t x [F(x)p(x, t)] + D 2 [p(x, t)] 2−qq, (4.20)x 2where F(x) =−dV/dx is an external force associated with the potential V (x). Weshall consider the simple case whereF(x) = k 1 − k 2 x (k 2 ≥ 0); (4.21)k 2 = 0 corresponds to the important case of external constant force, and k 1 = 0corresponds to the Uhlenbeck–Ornstein process. For this simple force, it is possibleto find the analytical solution of the equation. It is given by [349]