Nonextensive Statistical Mechanics

212 6 Generalizing Nonextensive Statistical Mechanicsconstraints must have an admissible connection with the concept of energy (another,trivial, constraint is of course normalization). These various steps are going to beillustrated in the next subsections.6.2.1 Spectral StatisticsEquation (6.4) can be naturally generalized into∫dydx =− dκ F(κ) y κ , (6.7)∫where the nonnegative q-spectral function F(κ) (QSF) must be integrable, i.e.,dκ F(κ) must be finite. This (positive) integral does not need to be unity, i.e.,F(κ) is generically unnormalized. The particular caseF(κ) = a r δ(κ − r) + (a q − a r ) δ(κ − q) , (6.8)δ(x) being Dirac’s delta distribution, recovers Eq. (6.4). Unless specified otherwise,for simplicity we shall from now on assume that F(κ) is normalized (see in [383]details about how an unnormalized F(κ) can be transformed into a normalized one).The possible solution of Eq. (6.7) will be noted exp {F} (x). In other words,d exp {F} (x)dxBy setting x = ln {F} y ,wehavehenceln {F} x =∫ ∞=− dκ F(κ)[exp {F} (x)] κ . (6.9)−∞∫dy+∞d [ ln {F} y ] =∫ x{∫ +∞1−∞−∞F(κ) y κ dκ, (6.10)F(κ) u κ dκ} −1du (∀x ∈ (0, ∞)) , (6.11)which is the generic expression of the inverse function of exp {F} (x).With this definition we can generalize the entropy S q as follows:S {F} =W∑i=1p i ln {F}1p i≡W∑s {F} . (6.12)i=1At equiprobability (i.e., p i = 1/W )wehave