Nonextensive Statistical Mechanics

88 3 Generalizing What We Learnt1 ( [I q p(x1 , x 2 ,...,x d ), p (0) (x 1 , x 2 ,...,x d ) )2+ I q(p (0) (x 1 , x 2 ,...,x d ), p(x 1 , x 2 ,...,x d ) )] ≥ 0 (q ≥ 1/2) . (3.173)The equalities in (3.171) and (3.173) hold if and only if all the variables x 1 , x 2 ,...,x d are independent among them (almost everywhere).Before closing this Section, let us mention that the discrete version of definition(3.155) naturally isI q (p, p (0) ) ≡W∑i=1[p i /p (0)i] q−1 − 1p i . (3.174)q − 13.5 Constraints and Entropy OptimizationAs we did with the BG entropy, let us work out here the most simple entropicoptimization cases.3.5.1 Imposing the Mean Value of the VariableIn addition to∫ ∞0dx p(x) = 1 , (3.175)we might know the following mean value of the variable (referred to as the q-meanvalue):〈x〉 q ≡∫ ∞where the escort distribution P(x) is defined through [212]0dx x P(x) = X (1)q , (3.176)[p(x)] qP(x) ≡ ∫ ∞0dx ′ [p(x ′ )] . (3.177)qWe immediately verify that also P(x) is normalized, i.e.,∫ ∞0dx P(x) = 1 . (3.178)The reasons for which we use P(x) instead of p(x) to express the constraint(3.176) are somewhat subtle and will be discussed later on. At the present stage, we