Nonextensive Statistical Mechanics

64 3 Generalizing What We LearntFig. 3.13 Representation of the q-product, x ⊗ q x for q =−∞, −5, 0, 1, 2, ∞. Excluding q = 1,there is a special value x ∗ = 2 1/(q−1) ,forwhichq < 1 represents the lower bound [in figurex ∗ (q =−5) = 2 −1/6 ≃ 0.89089 and x ∗ (q = 0) = 1/2], and for q > 1 the upper bound [in figurex ∗ (q = 2) = 2]. For q =±∞, x ⊗ q x lies on the diagonal of bisection, but following the lowerand upper limits mentioned above.x ⊗y q≡ [yx 1−q − (y − 1)] 1/(1−q) , (3.93)where both x and y can be real numbers (with y(x 1−q − 1) ≥−1). From this, aninteresting, extensive-like property follows, namelyln q (x ⊗y q) = y ln q x . (3.94)It will gradually become clear that the peculiar mathematical structure associatedwith the q-product appears to be at the “heart” of the nonadditive entropy S q(which is nevertheless extensive for a special class of correlations) and its associatedstatistical mechanics (see also [185]).3.3.3 The q-SumAnalogously to the q-product we can define the q-sumx ⊕ q y ≡ x + y + (1 − q)xy. (3.95)It has the following main properties:(i) It recovers the standard sum as a particular instance, i.e.,x ⊕ 1 y = x + y ; (3.96)