The concavity of entropy and extremum principles in thermodynamics

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 deri-vation aims at providing a coherent formal framework for such principles which may be also pedagogically useful as it fully exploits and highlights the equiva-lence between different schemes. We also elucidate the consequences of the extremum principles for the general shape of thermodynamic potentials in rela-tion to first-order phase transitions. KEY WORDS: Equilibrium thermodynamics; entropy concavity; maximum entropy principle; thermodynamic representation theory