Nonextensive Statistical Mechanics

4.9 Time-Dependent Ginzburg–Landau d-Dimensional O(n) Ferromagnet with n = d 149with 10andq = τ + 3Mτ + M , (4.107)β ≡ 1kT = τ + M2A . (4.108)(T can be generically seen as the amplitude of an effective noise). For the typicalcase τ > 0, we have that q ≥ 1ifM ≥ 0, the value q = 1 corresponding tovanishing multiplicative noise. If |g(u)| grows, for |u| →∞, faster than |u| 1+τ/M ,P(u, ∞) decreases faster than 1/|u|, and is therefore normalizable. The condition(4.105) is in fact not necessary for having solutions of the q-exponential form. Theinterested reader can see the details in [304].Let us mention that we have discussed here a case in which q-Gaussian distributionsemerge from a linear Fokker–Planck equation (Eq. (4.104)). It is clear thatthis mechanism differs from the one focused on in Eq. (4.9), which is a nonlinearFokker–Planck equation. For the Langevin discussion, i.e., from mesoscopic firstprinciples, of this nonlinear Fokker–Planck equation, see [305] (with the notationchange q → 2 − q). It turns out that if we consider a mechanism involving stronglynon-Markovian processes, i.e., long memory effects, the nonlinear Fokker–Planckequation (Eq. (4.9)) naturally comes out.Finally, as mentioned above, another Langevin process has been studied [306]which includes a colored symmetric dichotomous noise. Although not identified inthis manner by the author, the stationary state has a q-Gaussian distribution withwhere γ and λ are mesoscopic parameters of the model.q = 1 − 2γ/λ1 − γ/λ , (4.109)4.9 Time-Dependent Ginzburg–Landau d-Dimensional O(n)Ferromagnet with n = dThe standard Langevin and Fokker–Planck equations are by no means equivalentto BG statistical mechanics, but they surely are consistent with it. One expectssomething similar to occur in the case of nonextensive statistical mechanics. It isour purpose here to exhibit one such example, even if the authors did not make theconnection in their original papers [350, 351].An interesting short-range-interacting d-dimensional ferromagnetic system isthat whose symmetry is dictated by rotations in n dimensions, i.e., the so-called10 Incidentally, if we had used the Itô convention, we would have obtained q = τ+4Mτ+2M .