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

TL;DR

This paper investigates the stability of equilibrium states in microcanonical systems described by Sharma-Taneja-Mittal entropy, revealing that entropy concavity alone does not guarantee thermodynamic stability due to composability effects.

Contribution

It demonstrates that the traditional link between entropy concavity and stability is modified for Sharma-Taneja-Mittal entropy, depending on the system's additivity properties.

Findings

Concavity of entropy does not always imply stability for Sharma-Taneja-Mittal systems.

Super-additive systems maintain the stability-concavity relation.

Sub-additive systems can be stable even if entropy is not concave.

Abstract

It is generally assumed that the thermodynamic stability of equilibrium state is reflected by the concavity of entropy. We inquire, in the microcanonical picture, on the validity of this statement for systems described by the bi-parametric entropy $S_{_{\kappa, r}}$ of Sharma-Taneja-Mittal. We analyze the ``composability'' rule for two statistically independent systems, A and B, described by the entropy $S_{_{\kappa, r}}$ with the same set of the deformed 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 deformed parameters, when the relation $S_{_{\kappa, r}}({\rm A}\cup{\rm B})> S_{_{\kappa, r}}({\rm A})+S_{_{\kappa, r}}({\rm B})$ holds (super-additive systems), the concavity conditions does imply the thermodynamics stability. Otherwise, when the relation $S_{_{\kappa, r}}({\rm A}\cup{\rm B})<S_{_{\kappa, r}}({\rm A})+S_{_{\kappa, r}}({\rm B})$ holds (sub-additive systems), the concavity conditions does not imply the thermodynamical stability of the equilibrium state.