Qualms concerning Tsallis's condition of pseudo-additivity as a definition of non-extensivity

TL;DR

This paper critiques Tsallis's pseudo-additivity condition, arguing it does not define non-extensivity and is unrelated to key properties like additivity or concavity, thus questioning its use in characterizing entropy.

Contribution

It clarifies that Tsallis's pseudo-additivity is a functional equation for entropy forms and not indicative of system extensivity, contrasting it with other entropies like Arimoto's.

Findings

Pseudo-additivity is unrelated to additivity or concavity.

Arimoto entropy also satisfies a pseudo-additive relation.

Pseudo-additivity cannot determine system extensivity.

Abstract

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 Arimoto entropy based on the pseudo-additive equation.