Nonextensive Statistical Mechanics

4.6 Probabilistic Models with Correlations – Numerical and Analytical Approaches 123Transformations (4.39) and (4.40) enable the construction of an interesting algebra.3 Indeed, the following properties can be easily established:andμ 2 = 1 , (4.41)ν 2 = 1 , (4.42)where 1 represents the identity, i.e., 1(q) = q , ∀q. These properties justify the nameduality.We immediately verify that(μν) n (νμ) n = (νμ) n (μν) n = 1 (n = 0, 1, 2,...) . (4.43)Consistently, we may define (μν) −n ≡ (νμ) n and (νμ) −n ≡ (μν) n .We verify also that, for z = 0, ±1, ±2,..., and ∀q,(μν) z (q) =ν(μν) z (q) =(μν) z μ(q) =z − (z − 1) qz + 1 − zq , (4.44)z + 2 − (z + 1) q,z + 1 − zq(4.45)−z + 1 + zq−z + (z + 1) q . (4.46)These three expressions have the formq ∗ = A + BqC + Dq , (4.47)q = q ∗ = 1 being a fixed point, hence A + B = D + C. The constants A, B, C, andD generically (but not necessarily) do not vanish. In such a case, these expressionscan be rewritten in the formwithQ ∗ = f λ (Q) , (4.48)Q ∗ ≡ C A q∗ , (4.49)3 Private discussions with M. Gell-Mann in the context of a possible understanding of the numericalvalues determined (for the solar wind) in [361] for the q-triplet that will be discussed inSection 5.4.4.