Nonextensive Statistical Mechanics

100 3 Generalizing What We Learntq-expectation values, i.e., through the escort distributions. So, if we have an observableO whose possible values are {O i }, the associated constraint is to be writtenas〈O〉 q ≡W∑P i O i = O q , (3.235)i=1where O q is a known finite quantity.Regretfully, it is not yet totally transparent what is the geometrical/probabilisticreason which makes it convenient to express the constraints as q-expectation values.We do know, however, a set of properties that surely are directly related to thiselusive reason. Let us next list some of them that are particularly suggestive.(i) The derivative of eq x is not the same function (unless q = 1), but (ex q )q .Thissimple property makes naturally appear P i instead of p i in the steepest descentmethod developed in [227].(ii) The conditional entropy (3.41) naturally appears as a q-expectation value,without involving any optimizing operation.(iii) The norm constraint involves the quantity ∑ Wi=1 p i with p i ∝ 1/[1 + (q − 1)¯β E i ] 1/(q−1) . A case, which frequently appears, concerns W →∞, with E i increasinglylarge with increasing i. In such a case, we have that p i ∝ 1/E 1/(q−1)ifor highvalues of E i . Therefore, q must be such that ∑ ∞i=i 0E −1/(q−1)iis finite, where i 0 issome value of the index i. Equivalently, in the continuous limit, q must be such that∫ ∞constantdE g(E) E −1/(q−1) < ∞ , (3.236)where g(E) is the density of states. A typical case is g(E) ∝ E δ in the E →∞limit. In such a case, the theory is mathematically well posed if 1/(q − 1) − δ>1,i.e., ifq < 2 + δ1 + δ . (3.237)For the simplest case, namely for δ = 0, this implies q < 2.Let us make the same analysis for the constraint U q = [ ∑ Wi=1 pq i E i]/[ ∑ Wj=1 pq j ].Under the same circumstances analyzed just above, we must have the finitenessof ∫ ∞constant dE g(E) EE−q/(q−1) . But this equals ∫ ∞constant dE g(E) E −1/(q−1) . Consequently,remarkably enough, we arrive to the same condition (3.236)! In otherwords, the entire theory is valid up to an unique value of q, namely that which guaranteescondition (3.236). This nice property disappears if we impose the constrainteither in the eigenvectors (Schroedinger representation) or in the operators (Dirac representation),or even partially in both (Heisenberg representation).