Nonextensive Statistical Mechanics

96 3 Generalizing What We Learntwhereln q Z q = ln q ¯Z q − β U q . (3.212)This relation takes into account the trivial fact that, in contrast with what is usuallydone in BG statistics, the energies {E i } are here referred to U q in Eq. (3.195).From Eqs. (3.211) and (3.212), we immediately obtain the anticipated relation(3.206). It can also be provedas well as relations such asU q =− β ln q Z q , (3.213)C q ≡ T S qT = U qT=−T 2 F qT 2 . (3.214)In fact the entire Legendre transformation structure of thermodynamics is q-invariant, which no doubt is remarkable and welcome.Let us stress an important fact. The temperatures T ≡ 1/(kβ) and T q ≡ 1/(kβ q )do not depend on the choice of the zero of energies, and are therefore susceptibleof physical interpretation (even if they do not necessarily coincide). Not so thetemperature T q ′ ≡ 1/(kβ′ q ).In addition to the Legendre structure, various other important theorems and propertiesare q-invariant. Let us briefly mention some of them.(i) H-theorem (macroscopic time irreversibility). Under a variety of irreversibleequations such as the master equation, Fokker–Planck equation, and others, it hasbeen proved (see, for instance, [213–215]) thatq dS qdt≥ 0 (∀q), (3.215)the equality corresponding to (meta)equilibrium. In other words, the arrow timeinvolved in the second principle of thermodynamics basically holds in the usualway. It is appropriate to remind at this point that, for q > 0(q < 0), the entropytends to attain its maximum (minimum) since it is a concave (convex) functional, asalready shown.(ii) The Clausius relation is verified ∀q, and the second principle of thermodynamicsremains the same [337].(iii) Ehrenfest theorem (correspondence principle between quantum and classicalmechanics). It can be shown [216] thatd〈Ô〉 qdt= i 〈[Ĥ, Ô]〉 q (∀q), (3.216)where Ô is any observable of the system.