Nonextensive Statistical Mechanics

4.6 Probabilistic Models with Correlations – Numerical and Analytical Approaches 131Fig. 4.13 Exact distribution (dots) forρ = 7/10 and its best q-Gaussian approximant with q =−5/9 (continuous curve) (from [241]).Fig. 4.14 MTG model. Left: The ρ-dependence of the index q of the best q-Gaussian approximant(dots), compared to Eq. (4.63). Right: Exact limiting distribution for ρ = 7/10 (hence q correlation =3/10 (continuous curve), and its best q-Gaussian approximant with q =−4/3(dots) (from [241]).Let us start with the α = 0 TMNT model. The N →∞distribution is givenby [241]( 2 − ρP(U) =ρ) 1/2exp(−2(1 − ρ)[er f −1 (2U)] 2) (− 1 ρ2 ≤ U ≤ 1 2 ) . (4.60)Clearly, this distribution is not a q-Gaussian, even if numerically it is amazinglyclose to it: see Fig. 4.13. If we approximate it by the best q-Gaussian (by imposingthe matching of the second and fourth moments), we obtain for q precisely theconjectural Eq. (4.57)!Let us address now the MTG model. The N →∞distribution is given by [241]R(y) = A −1ρ (1 − y)a ρ[− ln(1 − y)] (1−ρ)/ρ (0 ≤ y ≤ 1) , (4.61)a ρ ≡ 2 − 2ρ2 ρ − 1 , (4.62)A ρ being a normalizing constant. Once again, this distribution is not a q-Gaussian,even if numerically it is very close to it: see Fig. 4.14. If we approximate it by thebest q-Gaussian (by imposing the matching of the second and fourth moments), weobtain