Nonextensive Statistical Mechanics

3.7 About the Escort Distribution and the q-Expectation Values 101using the standard expectation value. If we do that, the energy mean value divergesfor a value of q different (smaller in fact) than that at which the norm diverges.(iv) An interesting analysis was recently done [259,260] which exhibited thatthe relative entropy I q (p, p (0) ) that we introduced in Section 3.4 is directly associatedwith differences of free energies calculated with the q-expectation values (i.e.,ordinary expectation values but using {P i } instead of {p i }), whereas some differentspecific relative entropy is directly associated with differences of free energiescalculated with the ordinary expectation values (i.e., just using {p i }). Then theyshow that I q (p, p (0) satisfies three important properties that the other relative entropyviolates. The first of these properties is to be jointly convex with regard toeither p or p (0) . The second of these properties is to be composable. And the thirdof these properties is to satisfy the Shore–Johnson axioms [261] for the principleof minimal relative entropy to be consistent as a rule of statistical inference. Itisthen concluded in [259] that these arguments select the q-expectation values, andexclude the ordinary expectation values whenever we wish to use the entropy S q .These arguments clearly are quite strong. Some further clarification would howeverbe welcome. Indeed, stated in this strong sense, there would be contradiction withthe arguments presented in [60, 323], which lead to the conclusion that the variousexisting formulations of the optimization problem using S q are mathematicallyequivalent, in the sense that they can be transformed one into the other (as long asall the involved quantities are finite, of course).(v) It has been shown in various systems that the theory based on q-expectationvalues exhibits thermodynamic stability (see, for instance, [318, 319, 321, 496]).(vi) The Beck–Cohen superstatistics [384] (see Chapter 6) is a theory whichgeneralizes nonextensive statistics, in the sense that its stationary state distributioncontains the q-exponential one as a particular case. In order to go one step furtheralong the same line, i.e., for this approach to become a statistical mechanics with apossible connection to thermodynamics, it also needs to have a corresponding entropy.This step was accomplished in [263,264,396] by generalizing the entropy S q .But it became clear in this extension that generalizing the entropy was not enough:the mathematical form of the energy constraint had to be generalized as well. To bemore precise, in order to make some contact with the macroscopic level, the onlysolution that was found was to simultaneously generalize the entropy and the formof the constraint. This fact suggests of course that, from an information-theoreticalstandpoint, it is kind of natural to generalize not only the entropic functional butalso the expression of the constraints.(vii) Let us anticipate that, in the context of the q-generalization of the centrallimit theorem that we present later on, a natural generalization emerges for theFourier transform. This is given, for q ≥ 1, byF q [p](ξ) ≡∫ ∞−∞dx eixξ [p(x)]q−1q p(x) , (3.238)