Thermodynamic equilibrium and its stability for microcanonical systems described by the Sharma-Taneja-Mittal entropy

It is generally assumed that the thermodynamic stability of equilibrium states is reflected by the concavity of entropy. We inquire, in the microcanonical picture, about the validity of this statement for systems described by the two-parametric entropy S,r of Sharma, Taneja, and Mittal. We analyze the "composability" rule for two statistically independent systems A and B, described by the entropy S,r with the same set of the deformation parameters. It is shown that, in spite of the concavity of the entropy, the "composability" rule modifies the thermodynamic stability conditions of the equilibrium state. Depending on the values assumed by the deformation parameters, when the relation Sk,r(AB)>Sk,r(A)+Sk,r(B) holds (superadditive systems), the concavity condition does imply thermodynamics stability. Otherwise, when the relation Sk,r(AB)<Sk,r(A) +Sk,r(B) holds (subadditive systems), the concavity condition does not imply thermodynamical stability of the equilibrium state.