Nonextensive Statistical Mechanics

216 6 Generalizing Nonextensive Statistical Mechanicshence( )s(x) = x 1 − e x−1x . (6.33)It is trivial to see that Eq.(6.33) fulfills all the criteria set above, and we can thusfind a normalized QSF for it. Using Eq.(6.30) we get:F(κ) = 1 e∞∑n=0δ(κ − n). (6.34)n!Although different, entropy (6.32) has some resemblance with that introduced byCurado [120]. We claim no particular physical justification for the form (6.32). Inthe present context, it has been chosen uniquely with the purpose of illustrating themathematical procedure involved in the inverse QSF problem.6.2.2 Beck–Cohen SuperstatisticsWe may say that Beck–Cohen superstatistics originated essentially from a mathematicalremark and its physical interpretation [327, 328]. The basic remark is thatthere is a simple link, described hereafter, between the q-exponential function (withq ≥ 1) and the so-called Gamma distribution with n degrees of freedom. Beckand Cohen [384] start from the standard Boltzmann factor but with β being itself arandom variable (whence the name “superstatistics”) due to possible spatial and/ortemporal fluctuations. They defineP(E) =∫ ∞0dβ ′ f (β ′ ) e −β′ E , (6.35)where f (β ′ ) is a normalized distribution, such that P(E) also is normalizable underthe same conditions as the Boltzmann factor e −β′ E itself is. They also define∫q BC ≡ 〈(β′ ) 2 ∞〉0dβ ′ (β ′ ) 2 f (β ′ )=〈β ′ 〉 2 [∫ ∞0dβ ′ β ′ f (β ′ ) ] 2 , (6.36)where we have introduced BC standing for Beck–Cohen.If f (β ′ ) = δ(β ′ − β) we obtain Boltzmann weightand q BC = 1.P(E) = e −β E , (6.37)