Section

Mathematics

Title

Normal forms of the equations of thermodynamics

Author(s)

Vaganyan A.S.^{a}

Affiliations

Saint Petersburg State University^{a}

Abstract

In this article we consider applications of the theory of normal forms to the questions of thermodynamics of nonideal media described by thermal equations of state. On the basis of the fundamental GibbsDuhem 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 quasihomogeneous 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 nonideal gases and classical hydrogen plasma are considered. For the equation of state of a mixture of nonideal gases given in the form of a virial expansion it is shown that this equation is contactequivalent to the equation of state of a mixture of ideal gases. Furthermore, explicit formulae for one of the possible normalizing transformations are given. Nontriviality 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 DebyeHückel 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, nonideal media, virial expansion, DebyeHückel plasma

UDC

517.9

MSC

34C20

DOI

10.20537/vm160105

Received

29 February 2016

Language

Russian

Citation

Vaganyan A.S. Normal forms of the equations of thermodynamics, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 1, pp. 5867.

References

 Krichevskii I.R. Ponyatiya i osnovy termodinamiki (Concepts and fundamentals of thermodynamics), Moscow: Khimiya, 1970, 440 p.
 Basov V.V., Vaganyan A.S. Normal forms of Hamiltonian systems, Differential Equations and Control Processes, 2010, no. 4, pp. 86107. http://www.math.spbu.ru/diffjournal/pdf/basovve.pdf
 Belitskii G.R. Invariant normal forms of formal series, Functional Analysis and Applications, 1979, vol. 13, no. 1, pp. 4647.
 Landau L.D., Lifshitz E.M. Teoreticheskaya fizika. Tom V. Statisticheskaya fizika. Chast' 1 (Course of Theoretical Physics. Vol. 5. Statistical Physics. Part I), 5th ed., Moscow: Fizmatlit, 2002, 616 p.

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