Nonextensive Statistical Mechanics

72 3 Generalizing What We LearntTable 3.5 Two equal binary subsystems A and B. Joint probabilities r 20 , r 21 ,andr 22 ,withr 20 + 2 r 21 + r 22 = 1A\ B 1 21 r 20 ≡ p A+B11r 21 ≡ p A+B12= p A+B21r 20 + r 212 r 21 r 22 ≡ p A+B22r 21 + r 22r 20 + r 21 r 21 + r 22 1From now on we shall refer to this rule as “Leibnitz triangle rule”, or simply“Leibnitz rule”. 13 It should be clear that the Leibnitz triangle satisfies Leibnitz rule,but infinitely many different probabilistic triangles also satisfy it. As we shall see,this rule will turn out to play an important role in the discussion of the nature andapplicability of the entropy S q .Let us answer this crucial question: What is the probabilistic meaning of Leibnitzrule? If we compare the triangle representation (Table 3.4) with the hypercubicrepresentation (e.g., Tables 3.5 and 3.6), we immediately verify that the Leibnitzrule means that the marginal probabilities of the N-system coincide with thejoint probabilities of the (N − 1)-system. Generally speaking, if we calculate themarginal probabilities of the N-system where we have summed over the states of Msubsystems, we precisely obtain the joint probabilities of the (N − M)-subsystem.This is a remarkable property which implies in a specific form of scale-invariance.This invariance is in fact quite close to that emerging within analytical proceduressuch as the renormalization group, successfully applied in critical phenomena andelsewhere [208–211]. Equation 3.124 will be referred to as strict scale-invariance.It can and does happen that this relation is only asymptotically true for largeN, i.e.,r N,n + r N,n+1lim= 1 (∀n) . (3.125)N→∞ r N−1,nIn this case, we talk of asymptotic scale-invariance.Leibnitz rule is in fact stronger than it might look at first sight. If we give, forall N, the value of the probability r Nn for a single value of n, Leibnitz rule completelydetermines the entire set {r Nn }∀(N, n). A simple choice might be to giver N0 , ∀N.13 This rule should not be confused with Kolmogorov’ s consistency conditions characterizing astochastic process [297, 298]. Indeed, Kolmogorov conditions refer to the various marginal probabilitiesthat are associated with a given set of N random variables (e.g., observing the probabilitiesassociated with N ′ elements belonging to one and the same physical system with N elements,where N ′ < N), whereas the Leibnitz rule relates the marginal probabilities of a system withN variables with the joint probabilities of a different system with N ′ variables, where N ′ < N.Whereas Kolmogorov conditions are very generic, the Leibnitz rule is extremely restrictive.Another famous rule associated with Leibnitz is the so-called “Leibnitz chain rule” for derivationof a function of a function. These two rules are in principle unrelated. However, they both have arecurrent structure. Is this just a coincidence, or does it provide a hint on the manner through whichLeibnitz liked to think mathematics?