Nonextensive Statistical Mechanics

3.7 About the Escort Distribution and the q-Expectation Values 99P ≡ T q [p] , (3.228)with the notation p ≡ (p 1 , p 2 ,...,p W ) and P ≡ (P 1 , P 2 ,...,P W ). With thenotationWe can easily verify the following properties:(i) Unit. The unit is given by T 1 . Indeed(T q ∗ T q ′)[p] ≡ T q [T q ′[p]]. (3.229)(ii) Inverse. TheinverseofT q is given by T 1/q . Indeed,p ≡ T 1 [p] . (3.230)T 1/q ∗ T q = T q ∗ T 1/q = T 1 . (3.231)(iii) Commutativity. This transformation is commutative. IndeedT q ∗ T q ′ = T q ′ ∗ T q . (3.232)(iv) Associativity. This transformation is associative. Indeed(v) Cloture. IndeedT q ∗ (T q ′ ∗ T q ′′) = (T q ∗ T q ′) ∗ T q ′′ ≡ T q ∗ T q ′ ∗ T q ′′ . (3.233)T q ∗ T q ′ = T qq ′ . (3.234)In other words, the set of transformations {T q } constitutes an Abelian continuousgroup.Two more properties deserve to be stated.(vi) Certainty is a fixed point of the transformation. Indeed, if one of the possiblestates has probability p equal to unity, hence all the others have probability zero, thesame happens with P.(vii) Equal probabilities is a fixed point of the transformation. Indeed, if p i =1/W (∀i), then (and only then) P i = 1/W (∀i).We have seen in the previous Subsection that the most convenient manner 20for performing the optimization of the entropy is to express the constraints as20 We have said “the most convenient manner,” and not “the manner,” because, as we have alreadyseen in the previous Subsection, the calculation can be done through various equivalent paths. Forexample, optimizing S q with fixed 〈O〉 q is equivalent to optimizing S 2−q with fixed 〈O〉 1 ≡〈O〉[325]. Both optimizations yield one and the same result, in this case, p i ∝ e − ¯β O iq ,where ¯β isunivocally determined by using the constraint (3.235). This freedom is kind of reminiscent of thefreedom one has in quantum mechanics, where we can equivalently include the time-dependence