Nonextensive Statistical Mechanics

3.3 Correlations, Occupancy of Phase-Space, and Extensivity of S q 65(ii) It is commutative, i.e.,(iii) It is multiplicative under q-exponential, i.e.,(iv) It is associative, i.e.,(v) It admits zero, i.e.,x ⊕ q y = y ⊕ q x ; (3.97)e x⊕ q yq = e x q ey q ; (3.98)x ⊕ q (y ⊕ q z) = (x ⊕ q y) ⊕ q z = x ⊕ q y ⊕ q z= x + y + z + (1 − q)(xy + yz + zx) + (1 − q) 2 xyz; (3.99)(vi) By q-summing n equal terms we obtain:x ⊕ q 0 = x ; (3.100)x ⊕n q[n−2∑≡ x ⊕q x ⊕ q ...⊗ q x = nx (1 − q) i x i] + (1 − q) n−1 x n (n = 2, 3,...);i=0(3.101)(vii) It satisfies the following generalization of the distributive property of standardsum and product, i.e., of a(x + y) = ax + ay:a(x ⊕ q y) = (ax) ⊕ q+a−1a(ay) . (3.102)Interesting cross properties emerge from the q-generalizations of the product andof the sum, for instanceand, consistently, 8ln q (xy) = ln q x ⊕ q ln q y , (3.103)ln q (x ⊗ q y) = ln q x + ln q y , (3.104)eq x+y = eq x ⊗ q eq y , (3.105)e x⊕ q yq = eq x ey q . (3.106)8 While both the q-sum and the q-product are mathematically interesting structures, they play aquite different role within the deep structure of the nonextensive theory. The q-product reflects anessential property, namely the extensivity of the entropy in the presence of special global correlations.The q-sum instead only reflects how the entropies would compose if the subsystems wereindependent, even if we know that in such a case we only actually need q = 1.