Nonextensive Statistical Mechanics

336 Appendix B Escort Distributions and q-Expectation ValuesFig. B.1 The distributions f n (x) forn = 1, 2, 3, ∞ (from top to bottom) for (λ, α) = (2, 3/2)(from [884]).Equation (B.1) can be rewritten asf n (x) = A n e −β xq (β >0; q ≥ 1) . (B.6)The variable x ≥ 0 could be a physical quantity, say earthquake intensity, measuredalong small intervals, say 10 −6 , so small that sums can be replaced by integralswithin an excellent approximation. The empiric distribution f n (x) could correspondto different seismic regions, say region 1 (for n = 1), region 2 (for n = 2), and soon. See Fig. B.1. Suppose we want to characterize the distribution f n (x) through itsmean value. A straightforward calculation yields〈x〉 (n) ≡∫ n0dx xf n (x) = 1 − (1 + λn)α + λn[α + (α − 1)λn](α − 2)λ(1 + λn)[1 − (1 + λn) α−1 ] . (B.7)This quantity is finite for all n (including n →∞)forα>2, but 〈x〉 (∞) divergesfor 1 2, butwe cannot for 1