Nonextensive Statistical Mechanics

6.2 Further Generalizing 217If f (β ′ ) is the Gamma-distribution, i.e.,f (β ′ n( nβ′ ) n/2−1 }) =2β ( )nexp{− nβ′2β2β2(n = 1, 2, 3, ...) , (6.38)we obtainP(E) = e −β Eq , (6.39)and q BC = q, withq = n + 2n≥ 1 . (6.40)Several other examples of f (β ′ ) are discussed in [384], and it is eventuallyestablished the following important result: all narrowly peaked distributions f (β ′ )behave, in the first nontrivial leading order, as q-statistics with q = q BC . Further detailsand various applications to real systems are now available[21, 385–395] of this theory (which, unless f (β ′ ) is deduced from first principles,remains phenomenological).As we mentioned previously, the above discussion concerns the statistics. Morethan that is needed to have a statistical mechanical theory, namely it is necessaryto introduce an associated generalized entropic functional, as well as an appropriateconstraint related to the energy. This program has in fact completely been carried outfor superstatistics, and details can be seen in [263, 264, 396]. An interesting point isworthy mentioning: of all admissible f (β ′ ), only Eq. (6.38) yields a stationary-statedistribution optimizing the associated entropy within which the Lagrange parameter(usually noted α) corresponding to the normalization constraint factorizes from theterm containing the β Lagrange parameter. In other words, of all superstatistics,only q-statistics admits a partition function on the usual grounds, i.e., depending onβ but not on α.Let us conclude this subsection by focusing on the connection of spectral statisticswith the Beck–Cohen superstatistics. Quite recently, an entropic functionalhas been derived that corresponds to superstatistics. This functional is of the formS = ∑ i s(p i), withwheres(y) =K (y) =∫ x0a + K −1 (y), (6.41)1 − K −1 (y)E ∗P(y)∫ +∞0P(u)du . (6.42)