Nonextensive Statistical Mechanics

3.5 Constraints and Entropy Optimization 89just assume that, for whatever reason, what we know is the mean value of x withthe escort distribution. We wish now to optimize S q with the constraints (3.178) and(3.176), or, equivalently, with the constraints (3.175) and (3.176).In order to use the Lagrange method to find the optimizing distribution, we define[p] ≡ 1 − ∫ ∞0dx [p(x)] q− αq − 1∫ ∞0dx p(x) − β (1)q∫ ∞0dx x [p(x)] q∫ ∞0dx [p(x)] q , (3.179)where α and β q(1) are the Lagrange parameters. We then impose [p]/p = 0, andstraightforwardly obtainp opt (x) =e −β(1) q (x−X q (1) )q∫ ∞q (x ′ −X q (1) )q0dx ′ e −β(1), (3.180)where opt stands for optimal, and where we have used condition (3.175) to eliminatethe Lagrange parameter α. Notice that the fact that Lagrange parameter α can befactorized, and therefore eliminated, constitutes a quite remarkable mathematicalproperty.3.5.2 Imposing the Mean Value of the Squared VariableAnother simple and quite frequent case is when we know that 〈x〉 qcase, in addition to= 0. In such∫ ∞−∞dx p(x) = 1 , (3.181)we might know the q-mean value of the squared variable, i.e.,〈x 2 〉 q ≡∫ ∞−∞dx x 2 P(x) = X (2)q > 0 . (3.182)In order to use, as before, the Lagrange method to find the optimizing distribution,we define[p] ≡ 1 − ∫ ∞∫ ∫−∞dx [p(x)]q ∞∞− α dx p(x) − β (2) −∞ dx x2 [p(x)] qq∫q − 1∞−∞−∞ dx . (3.183)[p(x)]qWe then impose [p]/p = 0, and straightforwardly obtainp opt (x) =e −β(2) q (x 2 −X q (2) )q∫ ∞−∞ dx′ e −β(2) (x ′2 −X (2)q )q, (3.184)