Nonextensive Statistical Mechanics

46 3 Generalizing What We LearntFig. 3.7 The p-dependence of the W = 2 entropies S q , Sq R, S q E,andS qN [110], where the Renyientropy Sq R({p i }) ≡ ln ∑ W({ })i=1 pq i ln[1+(1−q)Sq ({pi })]= ,theescort entropy S E 1−q 1−q q ({p pi }) ≡ S q i q =1−[ ∑ Wi=1 p1/q i ] −qq−1∑ Wj=1 pq j,andtheLandsberg–Vedral–Rajagopal–Abe entropy, orjustnormalized entropySq N ({p Sq ({pi }) Si }) ≡ ∑ W =q ({p i })i=1 pq 1+(1−q)Siq ({p i }) . We verify that, among these four entropic forms, only S qis concave for all q > 0.D q f (x) ≡f (qx) − f (x)qx − x. (3.24)We immediately verify that D 1 f (x) = df(x)/dx.Forq ≠ 1, this operator replacesthe usual (infinitesimal) translation operation on the abscissa x of the functionf (x) byadilatation operation.Abe noticed a remarkable property [112]. In the same way that we can easilyverify thatS BG =− ddxW∑i=1p xi | x=1 , (3.25)