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TMF, 2008, Volume 157, Number 1, Pages 141–148 (Mi tmf6269)  

This article is cited in 6 scientific papers (total in 6 papers)

Thermodynamics from the differential geometry standpoint

V. P. Pavlova, V. M. Sergeevb

a Steklov Mathematical Institute, Russian Academy of Sciences
b Moscow State Institute of International Relations (University) of the Ministry for Foreign Affairs of Russia

Abstract: We study the differential-geometric structure of the space of thermodynamic states in equilibrium thermodynamics. We demonstrate that this space is a foliation of codimension two and find variables in which the foliation fibers are flat. We show that we can introduce a symplectic structure on this space: the external derivative of the $1$-form of the heat source, which has the form of the skew-symmetric product $dT\wedge dS$ in the found variables. The entropy $S$ then plays the role of the Lagrange function (or Hamiltonian) in mechanics, completely determining the thermodynamic system.

Keywords: symplectic structure, space of states, dynamical principle

DOI: https://doi.org/10.4213/tmf6269

Full text: PDF file (328 kB)
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English version:
Theoretical and Mathematical Physics, 2008, 157:1, 1484–1490

Bibliographic databases:

Document Type: Article
Received: 19.11.2007

Citation: V. P. Pavlov, V. M. Sergeev, “Thermodynamics from the differential geometry standpoint”, TMF, 157:1 (2008), 141–148; Theoret. and Math. Phys., 157:1 (2008), 1484–1490

Citation in format AMSBIB
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  • https://doi.org/10.4213/tmf6269
  • http://mi.mathnet.ru/eng/tmf/v157/i1/p141

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Arteaga B J.R., Malakhaltsev M.A., “Symmetries of sub-Riemannian surfaces”, J Geom Phys, 61:1 (2011), 290–308  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    2. Pinheiro M.J., “A Variational Method in Out-of-Equilibrium Physical Systems”, Sci Rep, 3 (2013), 3454  crossref  isi  scopus  scopus
    3. Barbaresco F., “Koszul Information Geometry and Souriau Geometric Temperature/Capacity of Lie Group Thermodynamics”, Entropy, 16:8 (2014), 4521–4565  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    4. Barbaresco F., “Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families”, Entropy, 18:11 (2016), 386  crossref  isi  elib  scopus
    5. Fronsdal Ch., “Relativistic thermodynamics, a Lagrangian field theory for general flows including rotation”, Int. J. Geom. Methods Mod. Phys., 14:2 (2017), 1750017  crossref  mathscinet  zmath  isi  scopus
    6. D. A. Leites, “Two problems in the theory of differential equations”, Theoret. and Math. Phys., 198:2 (2019), 271–283  mathnet  crossref  crossref  elib
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