Nonextensive Statistical Mechanics

2.3 Constraints and Entropy Optimization 31p opt =e−x/ X (1)X (1) . (2.44)2.3.2 Imposing the Mean Value of the Squared VariableAnother simple and quite frequent case is when we know that 〈x〉 =0. In such case,in addition to∫ ∞−∞we might know the mean value of the squared variable, i.e.,〈x 2 〉≡∫ ∞−∞dx p(x) = 1 , (2.45)dx x 2 p(x) = X (2) > 0 . (2.46)By using, as before, the Lagrange method to find the optimizing distribution, wedefine∫ ∞∫ ∞∫ ∞[p] ≡− dx p(x)lnp(x) − α dx p(x) − β (2) dx x 2 p(x) , (2.47)−∞−∞−∞and then impose δ[p]/δp(x) = 0. We straightforwardly obtain 1 + ln p opt + α +β (2) x 2 = 0, hencep opt =e −β(2) x 2∫ ∞−∞ dx e−β(2) x 2 = √β(2)π e−β(2) x 2 , (2.48)where we have used condition (2.45) to eliminate the Lagrange parameter α.By using condition (2.46), we obtain the following relation for the Lagrangeparameter β (2) :hence, replacing in (2.48),β (2) = 1 , (2.49)2X(2)p opt = e− x2 /(2X (2) )√2π X(2). (2.50)We thus see the very basic connection between Gaussian distributions and BGentropy.