Nonextensive Statistical Mechanics

3.6 Nonextensive Statistical Mechanics and Thermodynamics 95Equation (3.205) can be established from Eq. (3.204) by usingS q = k ln q ¯Z q , (3.206)which is proved a few lines further on.The (meta)equilibrium or stationary state distribution (3.201) can be rewritten asfollows:p i = e−β′ q E iqZ q′, (3.207)withZ ′ q ≡ W ∑j=1e −β′ q E jq , (3.208)andβ ′ q ≡β q1 + (1 − q)β q U q. (3.209)This form is particularly convenient for many applications where comparisonwith experimental, observational, or computational data is involved. 19The connection to thermodynamics is established in what follows. It can beproved that1T = S qU q, (3.210)with T ≡ 1/(kβ), where, for clarity, k has been restored into the expressions. Alsowe prove, for the free energy,F q ≡ U q − TS q =− 1 β ln q Z q , (3.211)19 We may rewrite in fact the distribution (3.201) with regard to any referential energy that wewish, say E 0 .Itjustbecomesp i = [1 − (1 − q)β q (0)(Ei − E 0 )] 1/(1−q) /Z q (0) with β q(0) ≡ β q /[1 + (1 −q) β q (U q − E 0 )] and Z q(0) ≡ ∑ Wj=1[1 − (1 − q)β(0) q (E j − E 0 )] 1/(1−q) . If we choose E 0 = U q weget back Eq. (3.201); if we choose E 0 = 0 we recover Eq. (3.207). The preference of a particularreferential energy E 0 is dictated by convenience for specific applications. Notice also that theseexpressions, e.g., Eq. (3.201) are self-referential in the sense that Eq. (3.202) is itself expressedin terms of the set {p i }. This implies of course in a slight operational complication. There arehowever in the literature several procedures that conveniently overcome this difficulty. One ofthose procedures is indicated in [60].