Nonextensive Statistical Mechanics

92 3 Generalizing What We LearntIf we want to formulate instead the statistical mechanics of the canonical ensemble,i.e., of a system in longstanding contact with a large thermostat at fixedtemperature, we need to add one more constraint (or even more than one, in fact,for more complex systems), namely that associated with the energy. The expressionof this constraint is less trivial than it seems at first sight! Indeed, it has been writtenin different forms since the first proposal of the theory. Let us describe here thesesuccessive forms since the underlying epistemological process is undoubtedly quiteinstructive.The first form was that adopted in 1988 [39], namely the simplest possible one(Eq. (2.63)):W∑i=1p i E i = U (1)q . (3.191)The extremization of S q with constraints (3.190) and (3.191) yieldsp (1)i∝ [1 − (q − 1)β (1) E i ] 1/(q−1) = e −β(1) E i2−q. (3.192)This expression already exhibits all the important facts of nonextensive statistics,namely the possibility (when q < 1) for an asymptotic power-law behavior at highenergies, and the possibility (when q > 1) of a cutoff. However, it can be seen thatit does not allow for a satisfactory connection with thermodynamics, in the sensethat no partition function can be defined which would not depend on the Lagrange. Moreover, p(1)iis not invariant, forfixed β q(1) , with regard to a changement of zero of energies. Indeed, ea+b q ≠ eq a eb q(if q ≠ 1), and therefore, as it stands, it is not possible to factorize the new zero ofenergy so that it becomes cancelled between numerator and denominator.The second form for the constraint was first indicated in [39] and developed in1991 [59]. It is written as follows:parameter α, but only on the parameter β (1)qW∑i=1p q iE i = U (2)q . (3.193)The extremization of S q with constraints (3.190) and (3.193) yieldsp (2)i∝ [1 − (1 − q)β (2) E i ] 1/(1−q) = e −β(2) E iq . (3.194)It can be seen that this result allows for a simple factorization of the Lagrangeparameter α, hence a partition function emerges which, as in BG statistics, onlydepends on β q(2) . Consistently, a smooth connection with classical thermodynamicsbecomes possible. However, p (2)iis still not invariant, for fixed β q(2) , with regard toa changement of zero of energies. Even more disturbing, the type of average usedin Eq. (3.193) violates the (a priori reasonable) result that the average of a constant