Abstract
Due to the first and second law of thermodynamics, the state of a thermodynamic system is described by a Legendrian manifold. This Legendrian manifold is locally determined by the information gain function. We describe the algebra of rational differential invariants of the information gain function under the action of two different Lie groups appearing naturally as a result of measuring random vectors, and we discuss our results in the context of ideal and van der Waals gases.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
P.V. Bibikov, V.V. Lychagin, Classification of Linear Actions of Algebraic Groups on Spaces of Homogeneous Forms, V.V. Dokl. Math. 85, 109–112 (2012), https://doi.org/10.1134/S1064562412010383.
I. Krasilshchik, V. Lychagin, A. Vinogradov, Geometry of jet spaces and nonlinear partial differential equations, Gordon and Breach (1986).
B. Kruglikov, V. Lychagin, Geometry of Differential equations, Handbook of Global Analysis, Ed. D.Krupka, D.Saunders, Elsevier, 725–772 (2008).
B. Kruglikov, V. Lychagin, Global Lie-Tresse theorem, Selecta Mathematica 22, 1357–1411 (2016).
V.V. Lychagin, Contact Geometry, Measurement and Thermodynamics, to appear in the same book as this paper.
P. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge (1995).
M. Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78, 401–443 (1956).
Acknowledgements
A part of this work was done during the summer school “Nonlinear PDEs, their geometry, and applications” in Wisła in August 2018. I would like to thank Valentin V. Lychagin for his guidance throughout this project.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Schneider, E. (2019). Differential Invariants in Thermodynamics. In: Kycia, R., Ułan, M., Schneider, E. (eds) Nonlinear PDEs, Their Geometry, and Applications. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-17031-8_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-17031-8_7
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-17030-1
Online ISBN: 978-3-030-17031-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)