Nonextensive Statistical Mechanics

8.2 Frequently Asked Questions 321Let us illustrate with the q = 2 q-Gaussian (i.e., the Cauchy–Lorentz distribution)p 2 (x) ∝ 1/(1+β x 2 ). Its width is characterized by 1/ √ β. However, its secondmoment diverges. At variance, its q = 2 q-expectation value is finite and given by〈x 2 〉 q ∝ 1/β. This is therefore a natural constraint to be used for extremizing theentropy S q .Further arguments yielding consistently the escort distributions as the appropriateones for expressing the constraints under which the entropy S q is to be extremizedcan be found in [259, 803], and in Appendix B.(m) Is it q just a fitting parameter? Does it characterize universality classes?From a first-principle standpoint, the basic universal constants of contemporaryphysics, namely c, h, G, and k B , are fitting parameters, but q is not. The indices qare in principle determined a priori from the microscopic or mesoscopic dynamicsof the system. Very many examples illustrate this fact. However, when the microormeso-scopic dynamics are unknown (which is virtually always the case in real,empirical systems), or when, even if known, the problem turns out to be mathematicallyuntractable (also this case is quite frequent), then and only then q is to behandled, faute de mieux, as a fitting parameter.To make this point clear cut, let us remind here a nonexhaustive list of examplesin which q is analytically known in terms of microscopic or mesoscopic quantities,or similar indices:Standard critical phenomena at finite critical temperature: q = 1+δ (see2Eq. (5.58));√Zero temperature critical phenomena of quantum entangled systems: q = 9+c 2 −3c(see Eq. (3.145));Lattice Lotka–Volterra models: q = 1 − 1 (see Eq. (7.22));DBoltzmann lattice models: q = 1 − 2 (see Eq. (7.4));DProbabilistic correlated models with cutoff: q = 1 − 1 (see Eq. (3.137));dProbabilistic correlated models without cutoff: q = ν−2 (see Eq. (4.67));ν−11Unimodal maps:1−q = 1α min− 1α max(see Eq. (5.9));1The particular case of the z-logistic family of maps:1−q(z) = (z − 1) ln α F (z)(seeln bEq. (5.11));The z = 2 particular case of the z-logistic maps: q = 0.244487701341282066198.... (see Eq. (5.13));Scale-free networks: q = 2m(2−r)+1−p−r (see Eq. (7.36));m(3−2r)+1−p−rNonlinear Fokker–Planck equation: q = 2 − ν (see Eq. (4.10)), and q = 3 − 2 μ(see Eq. (4.16));Langevin equation including multiplicative noise: q = τ+3M (see Eq. (4.107));τ+MLangevin equation including colored symmetric dichotomous noise: q = 1−2γ/λ1−γ/λ(see Eq. (4.109));Ginzburg–Landau discussion of point kinetics for n = d ferromagnets: q = d+4d+2(see Eq. (4.111));The q-generalized central limit theorems: q α,n = (2+α)q α,n+2−22q α,n+2(see Eq. (4.91)).+α−2