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 Regul. Chaotic Dyn., 1999, Volume 4, Issue 2, Pages 44–54 (Mi rcd901)

Canonical Gibbs distribution and thermodynamics of mechanical systems with a finite number of degrees of freedom

V. V. Kozlov

Faculty of Mechanics and Mathematics, Department of Theoretical Mechanics, Moscow State University, Vorob'ievy gory, 119899 Moscow, Russia

Abstract: Traditional derivation of Gibbs canonical distribution and the justification of thermodynamics are based on the assumption concerning an isoenergetic ergodicity of a system of n weakly interacting identical subsystems and passage to the limit $n \to \infty$. In the presented work we develop another approach to these problems assuming that n is fixed and $n \geqslant 2$. The ergodic hypothesis (which frequently is not valid due to known results of the KAM-theory) is substituted by a weaker assumption that the perturbed system does not have additional first integrals independent of the energy integral. The proof of nonintegrability of perturbed Hamiltonian systems is based on the Poincare method. Moreover, we use the natural Gibbs assumption concerning a thermodynamic equilibrium of bsystems at vanishing interaction. The general results are applied to the system of the weakly connected pendula. The averaging with respect to the Gibbs measure allows to pass from usual dynamics of mechanical systems to the classical thermodynamic model.

DOI: https://doi.org/10.1070/RD1999v004n02ABEH000106

Bibliographic databases:

MSC: 82C22, 70F07
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Citation: V. V. Kozlov, “Canonical Gibbs distribution and thermodynamics of mechanical systems with a finite number of degrees of freedom”, Regul. Chaotic Dyn., 4:2 (1999), 44–54

Citation in format AMSBIB
\Bibitem{Koz99} \by V. V. Kozlov \paper Canonical Gibbs distribution and thermodynamics of mechanical systems with a finite number of degrees of freedom \jour Regul. Chaotic Dyn. \yr 1999 \vol 4 \issue 2 \pages 44--54 \mathnet{http://mi.mathnet.ru/rcd901} \crossref{https://doi.org/10.1070/RD1999v004n02ABEH000106} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1781157} \zmath{https://zbmath.org/?q=an:1004.82002} 

• http://mi.mathnet.ru/eng/rcd901
• http://mi.mathnet.ru/eng/rcd/v4/i2/p44

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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. V. V. Kozlov, “Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas”, Russian Math. Surveys, 71:2 (2016), 253–290
2. Ivan A. Bizyaev, Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups”, Regul. Chaotic Dyn., 21:6 (2016), 759–774
3. I. A. Bizyaev, A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Integriruemost i neintegriruemost subrimanovykh geodezicheskikh potokov na gruppakh Karno”, Nelineinaya dinam., 13:1 (2017), 129–146