Nonextensive Statistical Mechanics

4.6 Probabilistic Models with Correlations – Numerical and Analytical Approaches 133Table 4.1 Merging of the Pascal triangle (the set of all left members) with the ν = 2 triangle (theset of all right members) associated with N equal subsystems(N = 0) (1, 1)(N = 1) (1, 1 2 ) (1, 1 2 )(N = 2) (1, 310 ) (2, 1 5 ) (1, 310 )(N = 3) (1, 1 5 ) (3, 110 ) (3, 110 ) (1, 1 5 )(N = 4) (1, 1 7 ) (4, 235 ) (6, 370 ) (4, 2 35 ) (1, 1 7 )q = ν − 2ν − 1 = 1 − 1ν − 1 . (4.67)Also, if we associate σ 1 =±1(i = 1, 2,...,N) with the N random variables,we can easily obtain (in addition to 〈σ i 〉=0 , ∀i) the following interesting result:〈σ i σ j 〉= 12ν + 1(∀ i ≠ j; ∀N) . (4.68)As expected, for the case of independence, i.e., when ν →∞, the correlationvanishes.This model, such as the MTG and TMNT ones, is strictly scale-invariant. But, invariance with those two, it asymptotically approaches a q-Gaussian. 54.6.5 The RST2 Model and Its Numerical ApproachWe shall now define a model by discretizing (symmetrically) a q-Gaussian into (N +1) values (identified by n = 0, 1, 2,...,N) [244]. These values can be interpretedas the probabilities corresponding to N equal and distinguishable binary randomvariables. This model, referred to as the RST2 one, will approach by construction theq-Gaussian that has been discretized (in fact, two slightly different discretizationshave been used). The interest of such a model is of course not its limit (since this isimposed), but how the limit is approached for increasingly large values of N. Therelation (3.124) corresponds to strict scale-invariance. We can numerically (and insome cases analytically) follow the ratioQ N,n ≡r N,nr N+1,n + r N+1,n+1. (4.69)We verify that Q N,n tends to 1 (or equivalently (Q N,n − 1) → 0) as N increases,i.e., the model is asymptotically scale-invariant. Note that Q 0,0 = Q 1,0 = Q 1,1 = 1for arbitrary values of r 0,0 , r 1,n , and r 2,n . See Figs. 4.15, 4.16, and 4.17.5 Using the Laplace-de Finetti representation, the present RST1 model has been recently extendedto real values of q, both above and below unity [R. Hanel, S. Thurner and C. Tsallis, Scale-invariantcorrelated probabilistic model yields q-Gaussians in the thermodynamic limit (2008), preprint].