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TMF, 2003, Volume 134, Number 3, Pages 388–400 (Mi tmf164)  

This article is cited in 29 scientific papers (total in 29 papers)

Weak Convergence of Solutions of the Liouville Equation for Nonlinear Hamiltonian Systems

V. V. Kozlov, D. V. Treschev

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

Abstract: We suggest sufficient conditions for the existence of weak limits of solutions of the Liouville equation as time increases indefinitely. The presence of the weak limit of the probability distribution density leads to a new interpretation of the second law of thermodynamics for entropy increase.

Keywords: Hamiltonian system, Liouville equation, weak convergence, entropy

DOI: https://doi.org/10.4213/tmf164

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English version:
Theoretical and Mathematical Physics, 2003, 134:3, 339–350

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Received: 05.07.2002

Citation: V. V. Kozlov, D. V. Treschev, “Weak Convergence of Solutions of the Liouville Equation for Nonlinear Hamiltonian Systems”, TMF, 134:3 (2003), 388–400; Theoret. and Math. Phys., 134:3 (2003), 339–350

Citation in format AMSBIB
\by V.~V.~Kozlov, D.~V.~Treschev
\paper Weak Convergence of Solutions of the Liouville Equation for Nonlinear Hamiltonian Systems
\jour TMF
\yr 2003
\vol 134
\issue 3
\pages 388--400
\jour Theoret. and Math. Phys.
\yr 2003
\vol 134
\issue 3
\pages 339--350

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. V. Kozlov, D. V. Treschev, “Evolution of Measures in the Phase Space of Nonlinear Hamiltonian Systems”, Theoret. and Math. Phys., 136:3 (2003), 1325–1335  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. Kozlov VV, “Billiards, invariant measures, and equilibrium thermodynamics - II”, Regular & Chaotic Dynamics, 9:2 (2004), 91–100  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    3. Kozlov VV, “Gibbs and Poincaré statistical equilibria in systems with slowly varying parameters”, Doklady Mathematics, 69:2 (2004), 278–281  mathnet  mathnet  mathscinet  zmath  isi
    4. Kozlov VV, “To the piston problem”, Doklady Mathematics, 72:1 (2005), 634–637  mathnet  mathnet  mathscinet  zmath  isi  elib
    5. V. V. Vedenyapin, “Fotoforez i reaktivnye sily”, Matem. modelirovanie, 18:8 (2006), 77–85  mathnet  zmath
    6. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. Kozlov, VV, “Statistical properties of billiards in polytopes”, Doklady Mathematics, 76:2 (2007), 696  mathnet  mathnet  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    8. V. V. Kozlov, “The generalized Vlasov kinetic equation”, Russian Math. Surveys, 63:4 (2008), 691–726  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. Piftankin G, Treschev D, “Gibbs entropy and dynamics”, Chaos, 18:2 (2008), 023116  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    10. Kozlov, VV, “Gibbs ensembles, equidistribution of the energy of sympathetic oscillators and statistical models of thermostat”, Regular & Chaotic Dynamics, 13:3 (2008), 141  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    11. 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  crossref  mathscinet  zmath  isi
    12. Piftankin G., Treschev D., “Coarse-grained Entropy in Dynamical Systems”, Regular & Chaotic Dynamics, 15:4–5 (2010), 575–597  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    13. A. V. Kargovsky, L. S. Bulushova, O. A. Chichigina, “Theorem on energy distribution over degrees of freedom for a quasistable symmetric anharmonic oscillator”, Theoret. and Math. Phys., 167:2 (2011), 636–644  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    14. S. Z. Adzhiev, V. V. Vedenyapin, “Time averages and Boltzmann extremals for Markov chains, discrete Liouville equations, and the Kac circular model”, Comput. Math. Math. Phys., 51:11 (2011), 1942–1952  mathnet  crossref  mathscinet  isi
    15. I. V. Volovich, A. S. Trushechkin, “Asymptotic properties of quantum dynamics in bounded domains at various time scales”, Izv. Math., 76:1 (2012), 39–78  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    16. V. V. Vedenyapin, S. Z. Adzhiev, “Entropy in the sense of Boltzmann and Poincaré”, Russian Math. Surveys, 69:6 (2014), 995–1029  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    17. 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  crossref  mathscinet  zmath  isi  scopus  scopus
    18. Lykov A.A., Malyshev V.A., “a New Approach To Boltzmann'S Ergodic Hypothesis”, Dokl. Math., 92:2 (2015), 624–626  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    19. V. V. Kozlov, “Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas”, Russian Math. Surveys, 71:2 (2016), 253–290  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    20. V. V. Vedenyapin, M. A. Negmatov, N. N. Fimin, “Vlasov-type and Liouville-type equations, their microscopic, energetic and hydrodynamical consequences”, Izv. Math., 81:3 (2017), 505–541  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    21. Adzhiev S.Z., Melikhov I.V., Vedenyapin V.V., “The H-Theorem For the Physico-Chemical Kinetic Equations With Explicit Time Discretization”, Physica A, 481 (2017), 60–69  crossref  mathscinet  isi  scopus  scopus
    22. Adzhiev S.Z., Melikhov I.V., Vedenyapin V.V., “The H-Theorem For the Physico-Chemical Kinetic Equations With Discrete Time and For Their Generalizations”, Physica A, 480 (2017), 39–50  crossref  mathscinet  isi  scopus  scopus
    23. Adzhiev S., Melikhov I., Vedenyapin V., “The H-Theorem For the Chemical Kinetic Equations With Discrete Time and For Their Generalizations”, V International Conference on Problems of Mathematical and Theoretical Physics and Mathematical Modelling, Journal of Physics Conference Series, 788, IOP Publishing Ltd, 2017, UNSP 012001  crossref  isi  scopus  scopus
    24. V. V. Vedenyapin, S. Z. Adzhiev, V. V. Kazantseva, “Entropiya po Boltsmanu i Puankare, ekstremali Boltsmana i metod Gamiltona–Yakobi v negamiltonovoi situatsii”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 64, no. 1, Rossiiskii universitet druzhby narodov, M., 2018, 37–59  mathnet  crossref
    25. Andrea Carati, Luigi Galgani, Alberto Maiocchi, Fabrizio Gangemi, Roberto Gangemi, “The FPU Problem as a Statistical-mechanical Counterpart of the KAM Problem, and Its Relevance for the Foundations of Physics”, Regul. Chaotic Dyn., 23:6 (2018), 704–719  mathnet  crossref
    26. Qian H., Wang Sh., Yi Y., “Entropy Productions in Dissipative Systems”, Proc. Amer. Math. Soc., 147:12 (2019), 5209–5225  crossref  mathscinet  isi
    27. Carati A., Galgani L., Gangemi F., Gangemi R., “Relaxation Times and Ergodic Properties in a Realistic Ionic-Crystal Model, and the Modern Form of the Fpu Problem”, Physica A, 532 (2019), UNSP 121911  crossref  isi
    28. A. I. Komech, E. A. Kopylova, “Attractors of nonlinear Hamiltonian partial differential equations”, Russian Math. Surveys, 75:1 (2020), 1–87  mathnet  crossref  crossref  mathscinet  isi  elib
    29. Valery V. Kozlov, “Nonequilibrium Statistical Mechanics of Weakly Ergodic Systems”, Regul. Chaotic Dyn., 25:6 (2020), 674–688  mathnet  crossref  mathscinet
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