Nonextensive Statistical Mechanics

30 2 Learning with Boltzmann–Gibbs Statistical MechanicsLet us mention also that, for discrete probabilities, definition (2.36) leads toI 1 (p, p (0) ) ≡W∑i=1[ pi]p i lnp (0) =−iW∑i=1[ p(0) ]ip i ln . (2.38)p iVarious other interesting related properties can be found in [95, 96].2.3 Constraints and Entropy OptimizationThe most simple entropic optimization cases are those worked out in what follows.2.3.1 Imposing the Mean Value of the VariableIn addition to∫ ∞we might know the mean value of the variable, i.e.,〈x〉 ≡0∫ ∞0dx p(x) = 1 , (2.39)dx xp(x) = X (1) . (2.40)By using the Lagrange method to find the optimizing distribution, we define[p] ≡−∫ ∞0dx p(x)lnp(x) − α∫ ∞0∫ ∞dx p(x) − β (1) dx xp(x) , (2.41)and then impose δ[p]/δp(x) = 0. We straightforwardly obtain 1 + ln p opt + α +β (1) x = 0(opt stands for optimal), hencep opt =e −β(1) x∫ ∞0dx e −β(1) x = β(1) e −β(1)x , (2.42)where we have used condition (2.39) to eliminate the Lagrange parameter α. Byusing condition (2.40), we obtain the following relation for the Lagrange parameterβ (1) :hence, replacing in (2.42),β (1) = 1 , (2.43)X(1)0