Nonextensive Statistical Mechanics

94 3 Generalizing What We LearntBecause of all these remarkable properties, the third form is the most commonlyused nowadays. Before enlarging its discussion and presenting its connection withthermodynamics, let us finish the brief review of this instructive evolution of ideas.A few years later, it was noticed [77] that the constraint (3.195) can be rewritten inthe following compact manner:W∑p q i(E i − U q ) = 0 . (3.200)i=1This approach led to the so-called “optimal Lagrange multipliers,” a twist whichhas some interesting properties. A question obviously arrives: Which one is the correctone, if any of these? The answer is quite simple: basically all of them!. Indeed,as it was first outlined in [60], and discussed in detail recently [323], they can betransformed one into the other through simple operations redefining the q s and theβ q s. Further comments can be found in [322, 324–326].To avoid confusion, and also because of its convenient properties, we shall stickonto the third form [60]. Consistently, we shall from now on use the simplified notation(p (3)i, U q (3),β(3),β q (3))≡ (p i, U q ,β,β q ). Let us rewrite Eqs. (3.197), (3.198),and (3.199) with this simplified notation:withp i = [1 − (1 − q)β q(E i − U q )] 1/(1−q)= e−β q (E i −U q )q, (3.201)¯Z q¯Z qβ q ≡β∑ Wj=1 pq j, (3.202)and¯Z q ≡W∑ie −β q (E i −U q )q . (3.203)Notice that, from the definition of S q ,W∑p q j= 1 + (1 − q)S q /k , (3.204)j=1and also thatW∑p q j= ( ¯Z q ) 1−q . (3.205)j=1