A note on relations between convexity and concavity of thermodynamic functions
Mária Lukáčová-Medvid'ová, Ferdinand Thein, Gerald Warnecke, Yuhuan Yuan
TL;DR
The paper addresses how convexity and concavity properties of thermodynamic functions transform under changes of state-variable sets, establishing equivalence relations through convexity-preserving transformations. It develops and extends involutions—Legendre, reciprocal, and exchange of a variable with a function—plus affine maps, detailing how Hessian definiteness is preserved or flipped and how generating potentials arise in Godunov's framework. By applying these results to energy/entropy and to ideal polytropic gas, van der Waals, and Tait EOS, it shows that entropy density can be strictly concave (negative definite Hessian) in Euler-conserved variables, while the total energy remains convex in appropriate variable sets, independently of the EOS. The findings provide a compact toolkit for proving stability properties and for constructing symmetric hyperbolic formulations, with direct implications for PDE analysis and numerics in compressible fluid flows via generating potentials and relative energy methods. Overall, the work offers a unified perspective on how thermodynamic convexity structures propagate through variable transformations and their role in stability and numerical analysis.
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.