Qualms concerning Tsallis’s condition of Pseudo-Additivity as a Definition of Non-Extensivity

The pseudo-additive relation that the Tsallis entropy satisfies has nothing whatsoever to do with the super-and sub-additivity properties of the entropy. The latter properties, like concavity and convexity, are couched in geometric inequalities and cannot be reduced to equalities. Rather, the pseudo-additivity relation is a functional equation that determines the functional forms of the random entropies. The Arimoto entropy satisfies a similar pseudo-additive relation and yet it is a first-order homogeneous form. Hence, no conclusions can be drawn on the extensive nature of the system from either the Tsallis or the Arimoto entropy based on the pseudo-additive functional equation. Tsallis [1] has used the pseudo-additivity condition, Sα(A+B) = Sα(A) + Sα(B) + (1− α)Sα(A)Sα(B) (1) where we work in energy units in which k = 1, Sα(P ) = 1− ∑m i=1 p α i α− 1 (2) is the Tsallis entropy ∀ α > 0, and the probability distribution P = (p1, . . . , pm) is complete. Supposedly [1], A and B are two independent systems in the sense that the probabilities of A+B factorize into those of A and B (i.e., pij(A+B) = pi(A)pj(B)). We immediately see that, since in all cases Sα ≥ 0 (nonnegativity property), α < 1, α = 1 and α > 1 respectively correspond to superadditivity (superextensivity), additivity (extensivity) and subadditivity (subextensivity). 1 Yet, it appears odd that criteria of super-and sub-additivity can be obtained through a functional equation rather than as a geometric inequality as are the criteria for convexity and concavity. The subadditive property is the ‘triangle inequality’ [2], whereas the geometrical interpretation of a concave function is one that never rises above its tangent plane at any point. There are classes of functions which are defined by inequalities that are weaker than convexity (concavity) and stronger than superadditivity (subadditivity) [3]. To prove that (2) is always subadditive, consider the sum ∑m i=1 p α i . Companions to Minkowski’s inequalities are [4] m ∑ i=1 (pi + qi) α > m ∑