Nonextensive Statistical Mechanics

44 3 Generalizing What We Learntas soon as we establish that S q ({p i })isconcave (convex) forq > 0(q < 0), whichwill be done below. The q = 0 case is marginal: the entropy is a constant. In thatcase we have thatS 0 = k(W − 1) (∀{p i }) (3.19)3.2.2.3 ExpansibilityIt is straightforwardly verified that S q is expansible, ∀q, sinceS q (p 1 , p 2 ,...,p W , 0) = S q (p 1 , p 2 ,...,p W ) . (3.20)This property trivially follows from the definition (3.18) if q > 0. For q < 0, itfollows from the fact that the sum in (3.18) runs only for states whose probability ispositive.3.2.2.4 NonadditivityIt is straightforwardly verified that, if A and B are independent, i.e., if the jointprobability satisfies p A+Bij= pi A p B j(∀(ij)), thenS q (A + B)k= S q(A)k+ S q(B)k+ (1 − q) S q(A)kS q (B)k. (3.21)It is due to this property that, for q ≠ 1, S q is said to be nonadditive. 1 However,drastic modifications occur when the subsystems A and B are correlated in a specialmanner. We shall see that in this case, a value of q might exist such that, eitherstrictly or asymptotically (N →∞), S q (A + B) = S q (A) + S q (B). In other words,the nonadditive entropy S q can be extensive for q ≠ 1! This is a nontrivial issue thatwill be addressed in detail in Section 3.3.Still, given the nonnegativity of S q ,itfollowsthat,forindependent subsystems,S q (A + B) ≥ S q (A) + S q (B) ifq < 1, and S q (A + B) ≤ S q (A) + S q (B) ifq > 1.Consistently, the q < 1 and q > 1 cases are occasionally referred in the literatureas the superadditive and subadditive ones, respectively.1 During many years, this property has been referred in the literature as nonextensivity. Thisis,in some sense, unfortunate. Indeed, it will become clear that, for a vast class of systems, a specialvalue of q exists for which the nonadditive entropy S q is extensive. The name “nonextensivestatistical mechanics” itself had historically been coined from this property. At the level of statisticalmechanics, this name is in fact not inadequate, since the Hamiltonian systems for which thistheory is expected to apply are those with long-range interactions, whose total energy is preciselynonextensive in the thermodynamical sense.