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Section
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Mathematics
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Title
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Normal forms of the equations of thermodynamics
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Author(-s)
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Vaganyan A.S.a
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Affiliations
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Saint Petersburg State Universitya
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Abstract
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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.
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Keywords
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normal forms, equations of state, non-ideal media, virial expansion, Debye-Hückel plasma
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UDC
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517.9
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MSC
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34C20
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DOI
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10.20537/vm160105
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Received
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29 February 2016
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Language
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Russian
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Citation
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Vaganyan A.S. Normal forms of the equations of thermodynamics, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 1, pp. 58-67.
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References
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- 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. 86-107. 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. 46-47.
- 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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Full text
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