Nonextensive Statistical Mechanics

3.3 Correlations, Occupancy of Phase-Space, and Extensivity of S q 613.3.2 The q-ProductIn relation with the pseudo-additive property (3.44) of S q , it has been recently introduced(independently and virtually simultaneously) [182, 183] a generalization ofthe product, which is called q-product. It is defined as follows:x ⊗ q y ≡[] 1x 1−q + y 1−q 1−q− 1+(x ≥ 0, y ≥ 0), 7 (3.78)or, equivalently,x ⊗ q y ≡ e ln q x+ln q yq (3.79)Let us list some of its main properties:(i) It recovers the standard product as a particular instance, namely,(ii) It is commutative, i.e.,x ⊗ 1 y = xy; (3.80)x ⊗ q y = y ⊗ q x ; (3.81)(iii) It is additive under q-logarithm (hereafter referred to as extensive), i.e.,whereas we remind thatConsistentlywhereasln q (x ⊗ q y) = ln q x + ln q y , (3.82)ln q (xy) = ln q x + ln q y + (1 − q)(ln q x)(ln q y) . (3.83)(iv) It has a (2 − q)-duality/inverse property, i.e.,e x q ⊗ q e y q = ex+y q , (3.84)e x q ey q = ex+y+(1−q)xyq ; (3.85)7 It is in fact easy to get rid of the requirement of non-negativity of x and y through the following[] 1extended definition: x ⊗ q y ≡ sign(x)sign(y) |x| 1−q +|y| 1−q 1−q− 1 . The correct q = 1 limit is+obtained by using sign(x)|x| =x (and similarly for y).