Nonextensive Statistical Mechanics

42 3 Generalizing What We Learnt3q = –1q = 0S q2q = 11q = 201 2 3 4 w 5Fig. 3.5 The equiprobability entropy S q as a function of the number of states W (with k = 1),for typical values of q. Forq > 1, S q saturates at the value 1/(q − 1) if W →∞;forq ≤ 1, itdiverges. For q →∞(q →−∞), it coincides with the abscissa (ordinate).This is precisely the form postulated in [39] as a possible basis for generalizingBG statistical mechanics. See Table 3.1. One possible manner for checking thatS 1 ≡ lim q→1 S q = S BG is to directly replace into Eq. (3.18) the equivalence p q i=p i p q−1i= p i e (q−1) ln p i∼ p i [1 + (q − 1) ln p i ].It turned out that this generalized entropic form, first with a different and thenwith the same multiplying factor, had already appeared outside the literature ofphysics, namely in that of cybernetics and control theory [107]. It was rediscoveredindependently in [39], when it was for the first time proposed as a starting point togeneralize the standard statistical mechanics itself. This was done for the canonicalensemble, by optimizing S q in the presence of an additional constraint, namely thatrelated to the mean value of the energy. We shall focus on this calculation later on.Table 3.1 S BG and S q entropies (S 1 = S BG )Entropy Equal probabilities Generic probabilities(p i = 1/W , ∀i) (∀{p i })S BG k ln W −k ∑ Wi=1 p i ln 2−q p i = k ∑ Wi=1 p i ln(1/p i )S q k ln q W k 1−∑ Wi=1 pq iq−1= k ∑ Wi=1 p i ln q (1/p i )(q ∈ R) =−k ∑ Wi=1 pq iln q p i=−k ∑ Wi=1 p i ln 2−q p i