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 TMF, 2003, Volume 136, Number 3, Pages 496–506 (Mi tmf1914)

Evolution of Measures in the Phase Space of Nonlinear Hamiltonian Systems

V. V. Kozlov, D. V. Treschev

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We establish the existence of weak limits of solutions (in the class $L_p$, $p\ge1$) of the Liouville equation for nondegenerate quasihomogeneous Hamilton equations. We find the limit probability distributions in the configuration space. We give conditions for a uniform distribution of Gibbs ensembles for geodesic flows on compact manifolds.

Keywords: quasihomogeneous Hamiltonian system, geodesic flow, weak limit, Gibbs ensemble, uniform distribution

DOI: https://doi.org/10.4213/tmf1914

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English version:
Theoretical and Mathematical Physics, 2003, 136:3, 1325–1335

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Revised: 21.04.2003

Citation: V. V. Kozlov, D. V. Treschev, “Evolution of Measures in the Phase Space of Nonlinear Hamiltonian Systems”, TMF, 136:3 (2003), 496–506; Theoret. and Math. Phys., 136:3 (2003), 1325–1335

Citation in format AMSBIB
\Bibitem{KozTre03} \by V.~V.~Kozlov, D.~V.~Treschev \paper Evolution of Measures in the Phase Space of Nonlinear Hamiltonian Systems \jour TMF \yr 2003 \vol 136 \issue 3 \pages 496--506 \mathnet{http://mi.mathnet.ru/tmf1914} \crossref{https://doi.org/10.4213/tmf1914} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2025369} \zmath{https://zbmath.org/?q=an:1178.37049} \elib{http://elibrary.ru/item.asp?id=13435491} \transl \jour Theoret. and Math. Phys. \yr 2003 \vol 136 \issue 3 \pages 1325--1335 \crossref{https://doi.org/10.1023/A:1025607517444} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000185966500010} 

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• http://mi.mathnet.ru/eng/tmf/v136/i3/p496

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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. Kozlov VV, “Billiards, invariant measures, and equilibrium thermodynamics - II”, Regular & Chaotic Dynamics, 9:2 (2004), 91–100
2. V. V. Kozlov, D. V. Treschev, “Fine-grained and coarse-grained entropy in problems of statistical mechanics”, Theoret. and Math. Phys., 151:1 (2007), 539–555
3. Bogachev, VI, “On the ergodic theorem in the Kozlov-Treshchev form”, Doklady Mathematics, 75:1 (2007), 47
4. Kozlov, VV, “Gibbs ensembles, equidistribution of the energy of sympathetic oscillators and statistical models of thermostat”, Regular & Chaotic Dynamics, 13:3 (2008), 141
5. Kozlov V.V., “Vorticity equation of 2D-hydrodynamics, Vlasov steady-state kinetic equation and developed turbulence”, Iutam Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, Iutam Bookseries, 6, 2008, 27–37
6. A. V. Korolev, “On the Convergence of Nonuniform Ergodic Means”, Math. Notes, 87:6 (2010), 912–915
7. Korolev A.V., “On the Ergodic Theorem in the Kozlov-Treshchev Form for An Operator Semigroup”, Ukrainian Math J, 62:5 (2010), 809–815
8. Bogachev V.I., Korolev A.V., Pilipenko A.Yu., “Non Uniform Averagings in the Ergodic Theorem for Stochastic Flows”, Doklady Mathematics, 81:3 (2010), 422–425
9. Trushechkin A., “Microscopic and Soliton-Like Solutions of the Boltzmann Enskog and Generalized Enskog Equations For Elastic and Inelastic Hard Spheres”, Kinet. Relat. Mod., 7:4 (2014), 755–778
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