Nonextensive Statistical Mechanics

32 2 Learning with Boltzmann–Gibbs Statistical Mechanics2.3.3 Imposing the Mean Values of both the Variableand Its SquareLet us unify here the two preceding subsections. We impose∫ ∞−∞and, in addition to this, we know thatdx p(x) = 1 (2.51)〈x〉 ≡and〈(x −〈x〉) 2 〉≡∫ ∞−∞∫ ∞−∞dx xp(x) = X (1) , (2.52)dx (x −〈x〉) 2 p(x) = X (2) − (X (1) ) 2 ≡ M (2) > 0 . (2.53)By using once again the Lagrange method, we define∫ ∞∫ ∞[p] ≡− dx p(x)lnp(x) − α dx p(x)−∞−∞∫ ∞∫ ∞−β (1) dx x p(x) − β (2) dx (x −〈x〉) 2 p(x) , (2.54)−∞and then impose δ[p]/δp(x) = 0. We straightforwardly obtain 1 + ln p opt + α +β (1) x + β (2) (x −〈x〉) 2 = 0, hence−∞p opt =√e −β(1) x−β (2) (x−〈x〉) 2β(2)(x−〈x〉)∫ ∞−∞ dx =2e−β(1) x−β (2) (x−〈x〉) 2 π e−β(2) , (2.55)where we have used condition (2.51) to eliminate the Lagrange parameter α.Byusingconditions (2.52) and (2.53), we obtain the following relations for the Lagrangeparameters β (1) and β (2) :andβ (2) =β (1) = 1 , (2.56)X(1)12[X (2) − (X (1) ) 2 ] . (2.57)