Nonextensive Statistical Mechanics

3.6 Nonextensive Statistical Mechanics and Thermodynamics 93precisely coincides with that constant. For similar reasons, if we consider E A+Bij=Ei A + E B jwith p A+Bij= pi A p B j ,18 we do not generically obtain U q (2) (A + B) =U q (2)(2)(A) + U q (B). These features led finally to a new formulation of the energyconstraint.The third form for the constraint was introduced in 1998 [60]. It is written asfollows:〈E i 〉 q ≡W∑i=1P i E i = U (3)q , (3.195)where we have used the escort distributionP i ≡p q i∑ Wj=1 pq j. (3.196)The extremization of S q with constraints (3.190) and (3.195) yieldsp (3)i[1 − (1 − q)β(3) q= (E i − U q(3)¯Z q)]1/(1−q)= e−β(3)q (E i −U q (3) )q¯Z q, (3.197)withβ (3)q≡β (3)∑ Wj=1 [p(3) j] q , (3.198)and¯Z q ≡W∑ie −β(3) q (E i −U q (3) )q , (3.199)β (3) being the Lagrange parameter associated with constraint (3.195). This formulationsimultaneously solves all the difficulties mentioned above, namely (i) the αLagrange parameter factorizes, hence we can define a partition function dependingonly on β q(3) , hence we can make a simple junction with thermodynamics;(ii) the average of a constant coincides with that constant; (iii) if we considerE A+Bij= Ei A + E B jwith p A+Bij= pi A p B (3)j, we generically obtain U q (A + B) =U q (3)(3)(A) + U q (B); and (iv) since the difference E i − U q(3) does not depend on thechoice of zero for energies, the probability p (3)iis invariant, for fixed β q(3),withregard to the changement of that zero.18 Notice however that E A+BijEq. (3.194), unless q = 1.= Ei A + E B jand p A+Bij= pi A p B jare, of course, inconsistent with