A note on relations between convexity and concavity of thermodynamic functions

TL;DR

This paper establishes the fundamental equivalence of convexity and concavity properties of thermodynamic functions across different variables and transformations, aiding in the analysis of thermodynamic stability and mathematical properties.

Contribution

It provides general, equation-of-state independent equivalence relations for convexity and concavity of thermodynamic functions under variable transformations.

Findings

Proves equivalence of convexity and concavity properties for thermodynamic functions.

Shows how equations of state influence these convexity properties.

Demonstrates application to entropy density in Euler equations.

Abstract

The paper is concerned with proving the equivalence of convexity or concavity properties of thermodynamic functions, such as energy and entropy, depending on different sets of variables. These variables are the basic thermodynamic state variables, specific state variables or the densities of state variables that are used in continuum mechanics. We prove results for transformations of variables and functions in conjunction with convexity properties. We are concerned with convexity, strict convexity, positive definite Hessian matrices and the analogous forms of concavity. The main results are equivalence relations for these properties between functions. These equivalences are independent of the equations of state since they only use general properties of them. The results can be used for instance to easily prove that the entropy density function for the Euler equations in conservative variables in three space dimensions is strictly concave or even has a negative definite Hessian matrix. Further, we show how various equations of state imply these properties and how these properties are relevant to mathematical analysis.