Nonextensive Statistical Mechanics

258 7 Thermodynamical and Nonthermodynamical Applicationsensembles. This has been done in [572] and elsewhere [573–577], and interestinggeneralizations of the semi-circle law for the eigenvalue density, and of Wigner’ssurmise for the level-spacing distribution are obtained. The index q determines thedegree of confinement, in such a way that q ≤ 1 corresponds to strong localizationand q > 1 corresponds to weak localization.7.2 Chemistry7.2.1 Generalized Arrhenius Law and Anomalous DiffusionThe Arrhenius law plays a fundamental role in chemistry. It has been interestinglygeneralized in [578].Let us consider the following nonlinear Fokker–Planck equationρ(x, t)t= [ U(x)]ρ(x, t) + D 2 [ρ(x, t)] 2−qx xx 2= [ U(x)]ρ(x, t) + (2 − q) D {1−qρ(x, t)[ρ(x, t)]x xxx[q ∈ R; (2− q)D > 0; t ≥ 0] ,}(7.19)with∫dx ρ(x, t) = 1 , ∀t , (7.20)U(x) being a potential whose global minimal value is U 0 . The stationary solution isgiven byV (x)e−β qρ s (x) ≡ lim ρ(x, t) = , (7.21)t→∞ ZZ 1−qwith V (x) ≡ U(x) − U 0 , β = , Z being a positive normalization constant.(2−q)DSee Fig. 7.49. By using the associated Ito–Langevin equation one can consider alarge amount of stochastic trajectories each of them starting at x L . We note T (x) ≡T (x L → x) the average time for the first passage to a value larger than x, withx ≥ x L : see Fig. 7.50. In particular, we can focus on the escape time T ≡ T (x R ):see Fig. 7.51.