Nonextensive Statistical Mechanics

132 4 Stochastic Dynamical Foundations of Nonextensive Statistical Mechanicsq ≃ 1 − 2ρ1 − ρ . (4.63)Through the identification ρ ≡ 1 − q correlation , this relation becomesq ≃ 2 −1q correlation, (4.64)which, with the notation change (q correlation , q) → (q, q e ), recovers the conjecturalEq. (4.38)!Further understanding is obviously needed. Why these two strictly scale-invariantmodels (MTG and TMNT) areso close to q-Gaussians?, and why they do not preciselycoincide with them? Work is presently under progress in order to solve thisopen problem.4.6.4 The RST1 Model and Its Analytical ApproachIn Table 3.7, we have the celebrated Leibnitz triangle (merged in fact with the Pascaltriangle). It satisfies the recursive relation (3.124). Consequently, it is completelydetermined by the marginal coefficientr (1)N,0 = 1 (N = 1, 2, 3,...) . (4.65)N + 1Let us now generalize this triangle by still imposing relation (3.124), and neverthelessgeneralizing Eq. (4.65) as follows [244]:r (1)N,0 = 1N + 1 ,r (2)N,0 = 2 · 3(N + 2)(N + 3) ,r (3)N,0 = 3 · 4 · 5(N + 3)(N + 4)(N + 5) ,r (ν)N,0 = ν ···(2ν − 1) (2ν − 1)!(N + ν − 1)!=(N + ν) ···(N + 2ν − 1) (ν − 1)!(N + 2ν − 1)! . (4.66)We verify that, ∀ν, lim N→0 r (ν)(ν)N,0= 1, and that rN,0 ∼ (2ν−1)! (N →∞). Also,(ν−1)! N νlim ν→∞ r (ν)N,0 = 1 . As an example, the ν = 2 triangle (merged with the Pascal2 Ntriangle) is indicated in Table 4.1.It has been analytically shown [244] that, after appropriate centering and scaling,the N →∞limit of these distributions is exactly a q-Gaussian with