A note on relations between convexity and concavity of thermodynamic functions
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917048049729536 |
|---|---|
| author | Lukáčová-Medvid'ová, Mária Thein, Ferdinand Warnecke, Gerald Yuan, Yuhuan |
| author_facet | Lukáčová-Medvid'ová, Mária Thein, Ferdinand Warnecke, Gerald Yuan, Yuhuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24440 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on relations between convexity and concavity of thermodynamic functions Lukáčová-Medvid'ová, Mária Thein, Ferdinand Warnecke, Gerald Yuan, Yuhuan Analysis of PDEs Mathematical Physics 26B25, 80A17, 76A02, 76N15, 35L65 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. |
| title | A note on relations between convexity and concavity of thermodynamic functions |
| topic | Analysis of PDEs Mathematical Physics 26B25, 80A17, 76A02, 76N15, 35L65 |
| url | https://arxiv.org/abs/2510.24440 |