Nonextensive Statistical Mechanics

3.2 Nonadditive Entropy S q 51with ∑ W Bj=1 (p ij/p Ai) = 1. Symmetrically, Eq. (3.40) can be also written asS q [A + B] = S q [B] + S q [A|B] + (1 − q)S q [B]S q [A|B] . (3.43)If A and B are independent, then p ij = p Aip B j(∀)i, j)), hence S q [A|B] = S q [A]and S q [B|A] = S q [B], therefore both Eqs. (3.40) and (3.43) yield the well-knownpseudo-additivity property of the nonadditive entropy S q , namelyS q [A + B] = S q [A] + S q [B] + (1 − q)S q [A]S q [B] . (3.44)We thus see that Eqs. (3.40) and (3.43) nicely compress into one property twoimportant properties of the entropic form S q , namely Eqs. (3.28) and (3.44). Someof the axiomatic implications of these relations have been discussed by Abe [115].3.2.2.9 Santos Uniqueness TheoremThe Santos theorem [117] generalizes that of Shannon (addressed in Section 2.1.2).Let 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 }; (3.45)(ii) S(p i = 1/W, ∀i) monotonically increases with the totalnumber of possibilities W; (3.46)S(A + B)(iii) = S(A) + S(B) + (1 − q) S(A) S(B)(3.47)k k kk kif p A+Bij= pi A p B j ∀(i, j), withk > 0;(iv) S({p i }) = S(p L , p M ) + p q L S({p i/p L }) + p q M S({p i/p M }) (3.48)with p L ≡∑p i , p L ≡∑p i ,LtermsMtermsL + M = W, and p L + p M = 1 .Then and only then [117]S({p i }) = k 1 − ∑ Wi=1 pq iq − 1. (3.49)3.2.2.10 Abe Uniqueness TheoremThe Abe theorem [115] generalizes that of Khinchin (addressed in Section 2.1.2).Let us assume that an entropic form S({p i }) satisfies the following properties: