Nonextensive Statistical Mechanics

106 3 Generalizing What We LearntTable 3.10 Comparative table of selected properties of selected entropies (with k = 1): S BG =− ∑ Wi=1 p i ln p i [1, 5, 25], S q is given by Eq. (3.18) [39], the Renyi entropy Sq R is given byLV RAEq. 3.246 [108], the Landsberg–Vedral–Rajagopal–Abe (or normalized) entropySqis givenby Eq. (3.251) [397, 398], and the escort entropy SqE is given by Eq. (3.252) [60]. A NO appearsto make the entropy unacceptable for thermodynamical purposes; not necessarily so a NO. Theq-exponential function (q-exp) has a cutoff for q < 1, and an asymptotic power-law for q > 1. By“special global correlations” we mean such that W ef f (N) ∝ N ρ (ρ>0). The additivity of S BGand SqR guarantees their extensivity for standard correlations, i.e., those which generically yieldW ef f (N) ∝ μ N (μ>1). The non-concavity of Sq R RA, SLV q ,andSq E is illustrated in Fig. 3.7 (forsee also [61]). By (–) we mean that it has not been addressed in detailS E qENTROPY S BG S q Sq R LV RASqSqEAdditive (∀q ≠ 1) YES NO YES NO NOq < 1 exists such that S is extensive for NO YES NO NO YESspecial global correlationsConcave (∀q > 0) YES YES NO NO NOLesche-stable (∀q > 0) YES YES NO NO –q < 1 exists such that entropy production per YES YES NO NO –unit time is finiteŜ exists, ∀q ≠ 1, such that Ŝ and S =〈Ŝ〉 YES YES NO NO –obey, for independent systems, the samecomposition lawŜ exists, ∀q ≠ 1, such that Ŝ(ˆρ −1 ) has the YES YES NO NO –same functional form as S(p i = 1/W )Same functional form for both Z q (β F q ) and YES YES NO NO –[Z q p(β E i )], ∀q ≠ 1Optimizing distribution, ∀q ≠ 1 exp q-exp q-exp q-exp q-expS K κW ≡− ∑p i lnκ K p i , (3.253)i=1withln K κ≡ x κ − x −κ2κ(ln K 0x = ln x) . (3.254)We straightforwardly verify thatlim S q = lim SqR q→1 q→1= limq→1 S N q= limq→1 S E q= lim S ηAP = lim Sκ K = S BG. (3.255)η→1 κ→0We can also verify that each of S q , Sq R, and S qN is a monotonic function of eachone of the others. Therefore, under the same constraints, they yield one and thesame extremizing probability distribution. Indeed, optimization is preserved throughmonotonicity. Not so, by the way, for concavity, convexity, and other properties. Forvarious comparisons, see Figs. 2.1, 3.6, 3.7, 3.8, and Table 3.10.Many other extensions of the classical BG entropy are available in the literaturethat follow along related lines: see for instance [186–189, 383, 396, 400, 401].