Nonextensive Statistical Mechanics

68 3 Generalizing What We LearntThe situation is more complex for q ≠ 1, and here we focus on q < 1. Indeed,it appears (as we shall verify in the next Subsection) that, in such cases, a few ormany of the states of the entire system become forbidden (in the sense that theircorresponding probabilities vanish), either for finite N or in the limit N → ∞.This is precisely why W eff (N) < W (N) = W1 N . So, if we assume that all statesof each subsystem are equally probable (with probability 1/W 1 ), then the statesof the entire system are not. Reciprocally, if we assume that the allowed states ofthe entire system are equally probable (with probability 1/W eff (N) > 1/W (N) =1/W1 N ), then the states of each of the subsystems are not. We see here the seed ofnonergodicity, hence the failure of the BG statistical mechanical basic hypothesisfor systems of this sort.Let us first consider the possibility in which the states of each subsystem areequally probable. Then k ln q W 1 is the entropy S q (1) associated with one subsystem.In other words Eq. (3.94) impliesk ln q [W eff (N)] = NS q (1) . (3.118)Let us then consider the other possibility, namely that in which it is the allowedstates of the entire system that are equally probable. Then k ln q W eff (N) is the entropyS q (N) associated with the entire system. In other words Eq. (3.94) impliesS q (N) = Nkln q W 1 . (3.119)We may say that we are now very close to answer a crucial question: Can S q forq ≠ 1 generically be strictly or asymptotically proportional to N in the presenceof these strong correlations, i.e., can it be extensive? The examples that we presentnext exhibit that the answer is yes. Bygenerically we refer to the most commoncase, in which neither the states of each subsystem are equally probable, nor theallowed states of the entire system are equally probable. This is what we address inthe next Section.But before that, let us summarize the knowledge that we acquired in the presentSubsection. We assume, for simplicity, that the W eff (N) joint states of a system areequally probable. See [190](i) If W eff (N) ∼ Aμ N (N →∞) with A > 0 and μ>1, the entropy whichis extensive is S BG (N) = ln W eff (N), i.e., lim N→∞ S BG (N)/N = ln μ ∈ (0, ∞).The nonadditive entropy S q for q ≠ 1 is, in contrast, nonextensive. It is primarilysystems like this that are addressed within the BG scenario.(ii) If W eff (N) ∼ BN ρ (N →∞) with B > 0 and ρ>0, the entropy which isextensive is S q (N) = ln q W eff (N) ∝ N ρ(1−q) withq = 1 − 1 ρ , (3.120)i.e., lim N→∞ S 1−(1/ρ) (N)/N = B 1−q /(1 − q) isfinite. For any other value of q (includingq = 1!), S q is nonextensive (e.g., S BG ∼ ρ ln N). It is primarily systems