Nonextensive Statistical Mechanics

52 3 Generalizing What We Learnt(i) S({p i })is a continuous f unction of {p i }; (3.50)(ii) S(p i = 1/W, ∀i) monotonically increases with the totalnumber of possibilities W; (3.51)(iii) S(p 1 , p 2 ,...,p W , 0) = S(p 1 , p 2 ,...,p W ); (3.52)S(A + B)(iv) = S(A) + S(B|A) + (1 − q) S(A) S(B|A)k k kk k(3.53)∑W Bwhere S(A + B) ≡ S({p A+Bij}), S(A) ≡ S({ p A+Bij}), and theconditional entropy S(B|A) ≡∑ WAi=1 (p Aj=1i) q S({p A+Bij/pi A })∑ WAi=1 (p (k > 0)i A ) qThen and only then [115] 2 S({p i }) = k 1 − ∑ Wi=1 pq iq − 1. (3.54)Notice that, interestingly enough, what enters in the definition of the conditionalentropy is the escort distribution, and not the original one.3.2.2.11 ComposabilityThe entropy S q is, like the BG one, composable (see also [116]). Indeed, it satisfiesEq. (3.21). In other words, we have F(x, y; q) = x + y + (1 − q)x y.The Renyi entropy SqR is composable since it is additive. In other words, in thatcase we have F(x, y; q) = x + y.As examples of the various noncomposable entropic forms that exist in the literature,we may cite the Curado entropy S C [120] and the Anteneodo–Plastino entropyS AP [121]. Since these two forms have some quite interesting mathematical properties,it would be thermodynamically valuable in principle to construct entropiesfollowing along the lines of these ones, but which would be composable instead.3.2.2.12 Sensitivity to the Initial Conditions, Entropy Production Per UnitTime, and the q-generalized Pesin-Like IdentityLet us focus on a one-dimensional nonlinear dynamical system (characterized by thevariable x) whose Lyapunov exponent λ 1 vanishes (e.g., the edge of chaos for typicalunimodal maps such as the logistic one). The sensitivity to the initial conditions ξdefined in Eq. (2.29) is conjectured to satisfy the equation2 The possibility of existence of such a theorem through the appropriate generalization ofKhinchin’ s fourth axiom had already been considered by Plastino and Plastino [118, 119]. Abeestablished [115] the precise form of this generalized fourth axiom, and proved the theorem.