Nonextensive Statistical Mechanics

4.7 Central Limit Theorems 141Fig. 4.20 The index q 2,n vs. q for typical values of n. The countable infinite family merges on asingle point only for q = 1. This reflects the fact that the structure of BG statistical mechanics isconsiderably simpler than that of the nonextensive one.q α,n = 2 − 1q α,n+2(n = 0, ±1, ±2,...) , (4.92)which, quite intriguingly, coincides with Eq. (4.38).Let us finally introduce one more definition. Two random variables X (with distributionf X (x)) and Y (with distribution f Y (y)) having zero q-mean values are saidq-independent ifF q [X + Y ](ξ) = F q [X](ξ) ⊗ 1+q3−qF q [Y ](ξ) , (4.93)i.e., if∫[ ∫ ] [ ∫ ]dzeq i ξ z f X+Y (z) = dx eqi ξ x f X (x) ⊗ 1+q dyeqi ξ y f Y (y) , (4.94)3−qwith∫ ∫∫∫f X+Y (z) = dx dy h(x, y) δ(x +y−z) = dx h(x, z−x) = dy h(z−y, y) ,(4.95)where h(x, y) is the joint distribution. Therefore, q-independence means independencefor q = 1 (i.e., h(x, y) = f X (x) f Y (y)), and it means strong correlation (of acertain class) for q ≠ 1 (i.e., h(x, y) ≠ f X (x) f Y (y)).