Nonextensive Statistical Mechanics

4.6 Probabilistic Models with Correlations – Numerical and Analytical Approaches 129Fig. 4.11 TMNT model: The ρ-dependence of the fitting parameter q ∞ for N = 1000. Theseresults provide numerical support to the heuristic relation (4.57) (continuous curve, where φ =5/6).this would be just perfect, but – there is a but! –, as we shall show in the next subsection,the N →∞distributions of the MTG and TMNT probabilistic modelsare not exactly q-Gaussians, even if numerically extremely close to them. 44.6.3 Analytical Approach of the MTG and TMNT ModelsHere we follow [241], where the MTG and the α = 0 TMNT models are analyticallydiscussed. As we shall see, the N →∞limiting distributions are notq-Gaussians, but distributions instead which numerically are amazingly close toq-Gaussians, although distinctively differing from them. It is of course trivial, –and ubiquitous in experimental, observational, and computational sciences –, thefact that a finite number of finite-precision values can never guarantee analyticalresults. History of science is full of such illustrations. Nevertheless, the present twoexamples are particularly instructive. Indeed, the numbers are strongly consistentwith q-Gaussians. Nevertheless, the exact distributions conspire in such a way asto be numerically extremely close to q-Gaussians, and still differing from them!4 Anticipating the notion of q-independence that will soon be introduced in the context of theq-generalization of the Central Limit Theorem, this means that the N random variables introducedin the present two models are not exactly, butonly approximately q-independent. If they wereexactly q-independent, the attractors ought to exactly be q-Gaussians.