Nonextensive Statistical Mechanics

66 3 Generalizing What We LearntLet us make, at this point, a mathematical digression. If, to the relation ln(xy) =ln x +ln y, we add relations (3.103) and (3.104), we feel tempted to find out whethera further generalized logarithmic function exists which would elegantly unify all ofthem in the formln q,q ′(x ⊗ q y) = ln q,q ′ x ⊕ q ′ ln q,q ′ y . (3.107)It turns out that it does exist, and is given by [187]ln q,q ′ x ≡ ln q ′ e ln q x = 1 [ ( 1 − q′ (exp x 1−q − 1 )) ]− 1 . (3.108)1 − q ′ 1 − qThe relations ln q,1 x = ln 1,q x = ln q x are easily recovered by evaluatingEq. (3.108) in the limits q → 1 and q ′ → 1. From Eq. (3.108), the inverse functioneq,q x ′ can be easily obtained as well.Finally, let us end by mentioning some related open problems. Does a generalizedsum x ⊕ (q) y exist such as a q-generalized distributivity like the followingholds?x ⊗ q (y ⊕ (q) z) = (x ⊗ q y) ⊕ (q) (x ⊗ q z) . (3.109)Could it be ⊕ (q) =⊕ f (q) , f being some specific function?Analogously, does a generalized product x ⊗ (q) y exist such as a q-generalizeddistributivity like the following holds?x ⊗ (q) (y ⊕ q z) = (x ⊗ (q) y) ⊕ q (x ⊗ (q) z) . (3.110)Could it be ⊗ (q) =⊗ g(q) , g being some specific function?These questions are presently open. However, preliminary results suggest thatno equality (3.109) can generically exist with a generalized sum that would beassociative.3.3.4 Extensivity of S q – Effective Number of StatesSuppose we are composing the discrete states of two subsystems A and B, whosetotal numbers of states are, respectively, W A ≥ 1 and W B ≥ 1. To be more specific,W A (W B ) is the total number of states of A (B) whose associated probability is notzero. The total number of states of the system A + B is thenW A+B = W A W B . (3.111)Let us now denote by W effA+Bthe effective number of states of the system A + B,where by effective we mean the number of states whose joint probability is not zero.It will in general be