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Uspekhi Mat. Nauk, 2014, Volume 69, Issue 6(420), Pages 45–80 (Mi umn9635)  

This article is cited in 15 scientific papers (total in 15 papers)

Entropy in the sense of Boltzmann and Poincaré

V. V. Vedenyapina, S. Z. Adzhievb

a Keldysh Institute of Applied Mathematics of Russian Academy of Sciences
b Moscow State University

Abstract: The $H$-theorem is proved for generalized equations of chemical kinetics, and important physical examples of such generalizations are considered: a discrete model of the quantum kinetic equations (the Uehling–Uhlenbeck equations) and a quantum Markov process (a quantum random walk). The time means are shown to coincide with the Boltzmann extremals for these equations and for the Liouville equation.
Bibliography: 41 titles.

Keywords: Boltzmann equation, $H$-theorem, entropy, conservation laws, discrete model, Boltzmann extremal, Liouville equation, time mean, Cesáro mean, Markov chains, variational principle.


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English version:
Russian Mathematical Surveys, 2014, 69:6, 995–1029

Bibliographic databases:

Document Type: Article
UDC: 517.958
MSC: Primary 60K30, 80A30; Secondary 60J27, 82C22, 82C40, 92E20
Received: 13.10.2013

Citation: V. V. Vedenyapin, S. Z. Adzhiev, “Entropy in the sense of Boltzmann and Poincaré”, Uspekhi Mat. Nauk, 69:6(420) (2014), 45–80; Russian Math. Surveys, 69:6 (2014), 995–1029

Citation in format AMSBIB
\by V.~V.~Vedenyapin, S.~Z.~Adzhiev
\paper Entropy in the sense of Boltzmann and Poincar\'e
\jour Uspekhi Mat. Nauk
\yr 2014
\vol 69
\issue 6(420)
\pages 45--80
\jour Russian Math. Surveys
\yr 2014
\vol 69
\issue 6
\pages 995--1029

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    This publication is cited in the following articles:
    1. S. E. Zhukovskii, D. V. Petrov, “Ob odnoi zadache vypuklogo programmirovaniya”, Vestnik Tambovskogo universiteta. Seriya: Estestvennye i tekhnicheskie nauki, 20 (2015), 1150–1153  elib
    2. D. R. Baimurzina, A. V. Gasnikov, E. V. Gasnikova, “Teoriya makrosistem s tochki zreniya stokhasticheskoi khimicheskoi kinetiki”, Trudy Moskovskogo fiziko-tekhnicheskogo instituta, 7 (2015), 95–103  elib
    3. A. V. Gasnikov, E. V. Gasnikova, Yu. E. Nesterov, A. V. Chernov, “Efficient numerical methods for entropy-linear programming problems”, Comput. Math. Math. Phys., 56:4 (2016), 514–524  mathnet  crossref  crossref  mathscinet  isi  elib
    4. 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
    5. S. Z. Adzhiev, I. V. Melikhov, V. V. Vedenyapin, “The $H$-theorem for the physico-chemical kinetic equations with explicit time discretization”, Phys. A, 481 (2017), 60–69  crossref  mathscinet  isi  scopus
    6. S. Z. Adzhiev, I. V. Melikhov, V. V. Vedenyapin, “The $H$-theorem for the physico-chemical kinetic equations with discrete time and for their generalizations”, Phys. A, 480 (2017), 39–50  crossref  mathscinet  isi  scopus
    7. S. Adzhiev, I. Melikhov, V. Vedenyapin, “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, 012001  crossref  isi  scopus
    8. S. Z. Adzhiev, V. V. Vedenyapin, Yu. A. Volkov, I. V. Melikhov, “Generalized Boltzmann-type equations for aggregation in gases”, Comput. Math. Math. Phys., 57:12 (2017), 2017–2029  mathnet  crossref  crossref  isi  elib
    9. Ch. R. Kwang-Hua, “Thermodynamic aspect in using modified Boltzmann model as an acoustic probe for $\mathrm{URu}_2\mathrm{Si}_2$”, Phys. B, 537 (2018), 355–359  crossref  isi  scopus
    10. Vedenyapin V.V., Kazakova T.S., Kisselevskaya-Babinina V.Ya., Chetverushkin B.N., “Schrodinger Equation as a Self-Consistent Field”, Dokl. Math., 97:3 (2018), 240–242  crossref  zmath  isi  scopus
    11. Vedenyapin V.V. Andreeva A.A. Vorobyeva V.V., “Euler and Navier–Stokes Equations as Self-Consistent Fields”, Dokl. Math., 97:3 (2018), 283–285  crossref  zmath  isi  scopus
    12. 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
    13. V. V. Vedenyapin, “Uravnenie Vlasova–Maksvella–Einshteina”, Preprinty IPM im. M. V. Keldysha, 2018, 188, 20 pp.  mathnet  crossref
    14. V. V. Vedenyapin, N. N. Fimin, V. M. Chechetkin, “Ob uravnenii Vlasova–Maksvella–Einshteina i ego nerelyativistskikh i slaborelyativistskikh analogakh”, Preprinty IPM im. M. V. Keldysha, 2018, 265, 30 pp.  mathnet  crossref
    15. Kwang-Hua Ch.R., “Modeling of Helium Flux in Cosmic Rays With Alpha Magnetic Spectrometer Measurements”, Vacuum, 160 (2019), 123–127  crossref  isi
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