Nonextensive Statistical Mechanics

50 3 Generalizing What We LearntLet us further elaborate on Eq. (3.34). It can be also rewritten in a more symmetricform, namely as1 + (1 − q)S q ({p i }) = [1 + (1 − q)S q ({π k }][1 + (1 − q)〈S q ({ p i /π k })〉 q ] . (3.37)Since the Renyi entropy (associated with the probabilities {p i }) is defined as Sq R({p i})≡ (ln ∑ Wi=1 pq i)/(1−q), we can conveniently define the (monotonically increasing)function R q [x] ≡ ln[1 + (1 − q)x]/[1 − q] = ln{[1 + (1 − q)x] [1/(1−q)] } (withR 1 [x] = x), hence, for any distribution of probabilities, we have Sq R = R q [S q ].Equation (3.37) can now be rewritten asor equivalently,R q [S q ({p i })] = R q [S q ({π k })] + R q [〈S q ({ p i /π k })〉 q ] , (3.38)S R q ({p i}) = S R q ({π k}) + R q [〈R −1q [S R q ({ p i/π k })]〉 q ] , (3.39)where the inverse function is defined as Rq−1[y]≡ [(ey ) (1−q) − 1]/[1 − q] (withR −11 [y] = y). Notice that, in general, R q[〈...〉 q ] ≠〈R q [...]〉 q .Everything we have said in this Section is valid for arbitrary partitions (in Knonintersecting subsets) of the ensemble of W possibilities. Let us from now onaddress the particular case where the W possibilities correspond to the joint possibilitiesof two subsystems A and B, having respectively W A and W B possibilities(hence W = W A W B ). Let us denote by {p ij } the probabilities associated with thetotal system A + B, with i = 1, 2, ...,W A , and j = 1, 2, ...,W B .Themarginalprobabilities {pi A } associated with subsystem A are given by piA = ∑ W Bj=1 p ij, andthose associated with subsystem B are given by p B j= ∑ W Ai=1 p ij. A and B are saidto be independent if and only if p ij = pi A p B j(∀(i, j)). We can now identify the Ksubsets which we were previously analyzing with the W A possibilities of subsystemA, hence the probabilities {π k } correspond to {pi A }. Consistently, Eq. (3.34) impliesnowS q [A + B] = S q [A] + S q [B|A] + (1 − q)S q [A]S q [B|A] , (3.40)where S q [A + B] ≡ S q ({p ij }), S q [A] ≡ S q ({p Ai}) and the conditional entropywhereS q [B|A] ≡∑ WAi=1 (p Ai) q S q [B|A i ]∑ WAi=1 (p ≡〈Si Aq [B|A i 〉 q , (3.41)) qS q [B|A i ] ≡ 1 − ∑ W Bj=1 (p ij/pi A ) qq − 1(i = 1, 2,...,W A ) , (3.42)