Nonextensive Statistical Mechanics

4.7 Central Limit Theorems 139Fig. 4.19 The function K (q). At q = 1 we recover the well-known transformation, throughstandard Fourier transform, of widths of Gaussians, mathematically involved in the Heisenberguncertainty principle.〈(...)〉 n ≡∫ ∞−∞dx (...) P (n) (x) =we can rewrite the set of Eq. (4.81) as follows:∫ ∞−∞dx (...)[ f (x)]1+n(q−1)∫ ∞−∞ dx [ f (n = 0, 1, 2,...) ,(x)]1+n(q−1)(4.83)1 d n ∣ { n−1F q [ f ](ξ) ∣∣∣ξ=0 ∏}= (i) n [1 + m(q − 1)] 〈x n 〉ν qn dξ nn (n = 1, 2, 3,...) , (4.84)m=0whereν qn≡∫ ∞−∞dx [ f (x)] q n(n = 0, 1, 2,...) , (4.85)withNotice thatq n = 1 + (q − 1)n (n = 0, 1, 2,...) . (4.86)(i) For q = 1, we recover the well-known relations involving the generating functionin theory of probabilities;(ii) For n = 1, we obtain q 1 = q, hence the usual escort distribution (used to definethe energy-related constraint under which S q is to be extremized) emergesnaturally;(iii) All q-expectation values in nonextensive statistical mechanics are well defined(i.e., finite) up to one and the same value of q (more precisely, for q < 2foradiscrete energy spectrum);