Nonextensive Statistical Mechanics

Appendix BEscort Distributions and q-Expectation ValuesB.1 First ExampleIn order to illustrate the practical utility and peculiar properties of escort distributionsand their associated q-expectation values, we introduce and analyze here apedagogical example [884]. 1Let us assume that we have a set of empirical distributions { f n (x)} (n = 1, 2, 3, ...)defined as follows:f n (x) =A n(1 + λx) α (λ>0; α ≥ 0) , (B.1)if 0 ≤ x ≤ n, and zero otherwise. Normalization of f n (x) immediately yieldsA n =λ(α − 1). (B.2)1 − (1 + λ n)1−αIn order to have finite values for A n , ∀n, including n →∞(i.e., 0 < A ∞ < ∞),α>1 is needed. ConsequentlyA ∞ = λ(α − 1) .(B.3)By identifyingα = 1q − 1 ,λ = β(q − 1) ,(B.4)(B.5)1 The present illustration has greatly benefited from lengthy discussions with S. Abe, who launched[885] interesting questions regarding q-expectation values, and with E.M.F. Curado.335