Nonextensive Statistical Mechanics

2.1 Boltzmann–Gibbs Entropy 25where S(A + B) ≡ S({p A+Bij}), S(A) ≡ S({pi A })(piA ≡and S(B) ≡ S({p B j })(pB j ≡∑W Ai=1p A+Bij);∑W Bj=1p A+Bij) ,(iv) S({p i }) = S(p L , p M ) + p L S({p i /p L }) + p M S({p i /p M }) (2.21)with p L ≡ ∑p i , p M ≡∑p i ,LtermsL + M = W , and p L + p M = 1 .Then and only then [25]S({p i }) =−kMtermsW∑p i ln p i (k > 0) . (2.22)i=12.1.2.8 Khinchin Uniqueness TheoremLet us assume that an entropic form S({p i }) satisfies the following properties:(i) S({p i }) is a continuous f unction of {p i }; (2.23)(ii) S(p i = 1/W, ∀i) monotonically increases with the totalnumber of possibilities W; (2.24)(iii) S(p 1 , p 2 ,...,p W , 0) = S(p 1 , p 2 ,...,p W ); (2.25)(iv) S(A + B) = S(A) + S(B|A), (2.26)where S(A + B) ≡ S({p A+Bij}), S(A) ≡ S({pi A })(piA ≡and the conditional entropy S(B|A) ≡Then and only then [81]2.1.2.9 ComposabilityS({p i }) =−k∑W Api A S({p A+Biji=1∑W Bj=1p A+Bij) ,/p Ai }) .W∑p i ln p i (k > 0) . (2.27)i=1A dimensionless entropic form S({p i }) (i.e., whenever expressed in appropriate conventionalunits, e.g., in units of k) issaidcomposable if the entropy S(A + B)corresponding to a system composed of two independent subsystems A and B canbe expressed in the form