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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2016, Volume 26, Issue 1, Pages 58–67 (Mi vuu518)

MATHEMATICS

Normal forms of the equations of thermodynamics

A. S. Vaganyan

Department of Differential Equations, Saint Petersburg State University, Universitetskii pr., 28, St. Petersburg, Petrodvorets, 198504, Russia

Abstract: In this article we consider applications of the theory of normal forms to the questions of thermodynamics of non-ideal media described by thermal equations of state. On the basis of the fundamental Gibbs–Duhem equation the notion of contact equivalence of such equations is introduced. Basic results from formal theory of normal forms for contact systems with a polynomial quasi-homogeneous unperturbed Hamiltonian are given, the definition of normal form of a contact Hamiltonian and the normalization theorem are formulated. From the application point of view, models for a mixture of non-ideal gases and classical hydrogen plasma are considered. For the equation of state of a mixture of non-ideal gases given in the form of a virial expansion it is shown that this equation is contact-equivalent to the equation of state of a mixture of ideal gases. Furthermore, explicit formulae for one of the possible normalizing transformations are given. Non-triviality of the physical effects that take place due to the impact of resonant perturbations on a model of ideal medium is illustrated by the example of perturbed equation for the Debye–Hückel model of hydrogen plasma. For this model the lowest terms of the perturbation in normal form are determined and their physical meaning is explained.

Keywords: normal forms, equations of state, non-ideal media, virial expansion, Debye–Hückel plasma.

DOI: https://doi.org/10.20537/vm160105

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Bibliographic databases:

UDC: 517.9
MSC: 34C20

Citation: A. S. Vaganyan, “Normal forms of the equations of thermodynamics”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 26:1 (2016), 58–67

Citation in format AMSBIB
\Bibitem{Vag16} \by A.~S.~Vaganyan \paper Normal forms of the equations of thermodynamics \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2016 \vol 26 \issue 1 \pages 58--67 \mathnet{http://mi.mathnet.ru/vuu518} \crossref{https://doi.org/10.20537/vm160105} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3485573} \elib{https://elibrary.ru/item.asp?id=25681785}