Nonextensive Statistical Mechanics

B.1 First Example 33721.5< x > (n)< x > q(n)10.500 1 2 3 4 5 6 7 8 9 10n54(σ (n) ) 2(< σ > 2q-1(n))232100 1 2 3 4 5 6 7 8 9 10nFig. B.2 The n-dependences of relevant average quantities of the model (λ, α) = (2, 3/2); q =1+ 1 α . Top: Expectation value 〈x〉(n) ≡ ∫ n0 dx xf n(x) (lim n→∞ 〈x〉 (n) =∞), and q-expectation value∫ n〈x〉 q(n) 0 dx x [ fn(x)]q≡ ∫ n (lim0 dx [ n→∞ 〈x〉 (n) 1fn(x)]q q =λ (α−1) ). Bottom: Variance [σ (n) ] 2 ≡〈x 2 〉 (n) − [〈x〉 (n) ] 2(lim n→∞ [σ (n) ] 2 =∞), and (2q−1)-variance [σ (n)2q−1 ]2 ≡〈x 2 〉 (n)2q−1 −[〈x〉(n) 2q−1 ]2 (lim n→∞ [σ (n)2q−1 ]2 =1+α) (from [884]).λ 2 α 2 (α−1)instead of with the original distribution f n (x). It follows immediately that〈x〉 q (∞) 1=λ(α − 1) ,(B.10)which is finite for all values α > 1, i.e., as long as the norm itself is finite. Theproblem that we exhibited with the standard mean value reappears, and even worse,if we are interested in the second moment of f n (x). We have that[σ (n) ] 2 ≡〈x 2 〉 (n) − [〈x〉 (n) ] 2 (B.11)