Nonextensive Statistical Mechanics

3.2 Nonadditive Entropy S q 49S BG ({p i }) = S BG ({π k }) +K∑π k S BG ({p i /π k }) . (3.29)This property constitutes in fact the fourth axiom of the Shannon theorem.The nonnegative entropies S q ({p i }), S q ({π k }), and S q ({p i /π k }) depend, respectively,on W , K , and W k probabilities. Equation (3.28) can be rewritten ask=1S q ({p i }) = S q ({π k }) +〈S q ({p i /π k })〉 (u)q , (3.30)where the unnormalized q-expectation value (u stands for unnormalized) of the conditionalentropy is defined as〈S q ({p i /π k })〉 (u)qK∑≡ π q k S q({p i /π k }) , (3.31)k=1Also, since the definition of S q ({π k }) implies1 + (1 − q)S q ({π k })∑ Kk ′ =1 π q k ′ = 1 , (3.32)Equation 3.28 can be rewritten as follows:S q ({p i }) = S q ({π k }) +ConsequentlyK∑k=1π q k1 + (1 − q)S q ({π k })∑ Kk ′ =1 π q k ′ S q ({p i /π k }) . (3.33)S q ({p i }) = S q ({π k }) +〈S q ({ p i /π k })〉 q + (1 − q) S q ({π k }) 〈S q ({p i /π k })〉 q , (3.34)where the normalized q-expectation value of the conditional entropy is defined as〈S q ({p i /π k })〉 q ≡with the escort probabilities [212]K∑ k S q ({p i /π k }) , (3.35)k=1 k ≡π q k∑ Kk ′ =1 π q k ′ (k = 1, 2,...,K ) . (3.36)Property (3.34) is, as we shall see later on, a very useful one, and it exhibitsa most important fact, namely that the definition of the nonextensive entropic form(3.18) naturally leads to normalized q-expectation values and to escort distributions.