Nonextensive Statistical Mechanics

3.2 Nonadditive Entropy S q 453.2.2.5 Concavity and ConvexityWe refer to the concepts introduced in Eqs. (2.11), (2.12), and (2.13), which naturallyextend for arbitrary q. The second derivative of the (continuous) functionx(1 − x q−1 )/(q − 1) is negative (positive) forq > 0(q < 0). Consequently, forq > 0, we havepi ′′ [1 − (p′′i )q−1 ]q − 1>λ pi[1 − p q−1i]+ (1 − λ) p′ i [1 − (p′ i )q−1 ]q − 1q−(∀i;0 0) . (3.23)These inequalities are obviously reversed for q < 0. It is therefore proved thatS q is concave (convex)forq > 0(q < 0). An immediate corollary is, as announcedpreviously, that the case of equal probabilities corresponds to a maximum for q > 0,whereas it corresponds to a minimum for q < 0. See in Fig. 3.6 an illustration ofthis property. See also Fig. 3.7.2S⎯ qk1.510.5–1–0.5–0.100.1131000 0.2 0.4 0.6 0.8 1pFig. 3.6 The p-dependence of the W = 2entropyS q = [1 − p q − (1 − p) q ]/(q − 1) for typicalvalues of q (with S 1 =−p ln p − (1 − p)ln(1− p)).3.2.2.6 Connection with Jackson DerivativeOne century ago, the mathematician Jackson generalized [111] the concept ofderivative of a generic function f (x). He introduced his differential operator D qas follows: