The concavity of entropy and extremum principles in thermodynamics

TL;DR

This paper provides a formal proof of the concavity of thermodynamic entropy and explores extremum principles for thermodynamic potentials, clarifying their implications for phase transitions and offering a pedagogical framework.

Contribution

It offers a unified, rigorous proof of extremum principles in thermodynamics, emphasizing the equivalence of different thermodynamic schemes and their relation to phase transitions.

Findings

Proof of the concavity of thermodynamic entropy.

Formal derivation of minimum energy and other extremum principles.

Insights into the shape of thermodynamic potentials during phase transitions.

Abstract

We revisit the concavity property of the thermodynamic entropy in order to formulate a general proof of the minimum energy principle as well as of other equivalent extremum principles that are valid for thermodynamic potentials and corresponding Massieu functions under different constraints. The current derivation aims at providing a coherent formal framework for such principles which may be also pedagogically useful as it fully exploits and highlights the equivalence between different schemes. We also elucidate the consequences of the extremum principles for the general shape of thermodynamic potentials in relation to first-order phase transitions.