Geodesic
Time averages and Boltzmann extremals for Markov chains, discrete Liouville equations, and the Kac circular model
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 11, pp. 2063-2074 Cet article a éte moissonné depuis la source Math-Net.Ru

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Time averages are proved to coincide with Boltzmann extremals for Markov chains, discrete Liouville equations, and their generalizations. A variational principle is proposed for finding stationary solutions in these cases. 
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S. Z. Adzhiev; V. V. Vedenyapin. Time averages and Boltzmann extremals for Markov chains, discrete Liouville equations, and the Kac circular model. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 11, pp. 2063-2074. http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_11_a9/

[1] Boltzmann L., “Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen”, Wien Akad. Sitzungsber, 66 (1872), 275–370; Boltsman L., “Dalneishie issledovaniya teplovogo ravnovesiya mezhdu molekulami gaza”, Izbrannye trudy, Nauka, M., 1984, 125–189 | MR

[2] Boltzmann L., “Über die Beziehung zwischen dem zweiten Hauptsatze der Mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung, respektive den Satzen über das Wärmegleichgewicht”, Wien Akad. Sitzungsber, 76 (1878), 373–435; Boltsman L., “O svyazi mezhdu vtorym nachalom mekhanicheskoi teorii teploty i teoriei veroyatnostei v teoremakh o teplovom ravnovesii”, Izbrannye trudy, Nauka, M., 1984, 190–235 | MR

[3] Puankare A., Zamechaniya o kineticheskoi teorii gazov. Izbrannye trudy, v. 3, Nauka, M., 1974

[4] Kozlov V. V., Teplovoe ravnovesie po Gibbsu i Puankare, In-t kompyuternykh issl., M.–Izhevsk, 2002 | MR | Zbl

[5] Kozlov V. V., Treschev D. V., “Slabaya skhodimost reshenii uravneniya Liuvillya dlya nelineinykh gamiltonovykh sistem”, Teor. i matem. fiz., 134:3 (2003), 388–400 | MR | Zbl

[6] Riss F., Sekefalvi-Nad B., Lektsii po funktsionalnomu analizu, Mir, M., 1979 | MR

[7] Vedenyapin V. V., “Vremennye srednie i ekstremali po Boltsmanu”, Dokl. RAN, 422:2 (2008), 161–163 | MR | Zbl

[8] Morimoto T., “Markov processes and the $H$-theorem”, J. Phys. Soc. Japan, 18:3 (1963), 328–331 | DOI | MR | Zbl

[9] Csiszár I., “Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten”, Magyar. Tud. Akad. Mat. Kutato Int. Kozl., 8 (1963), 85–108 | MR | Zbl

[10] Lorch E. R., “Means of iterated transformations in reflexive vector spaces”, Bull. Amer. Math. Soc., 45 (1939), 945–947 | DOI | MR

[11] Kats M., Neskolko veroyatnostnykh zadach fiziki i matematiki, Nauka, M., 1967 | Zbl

[12] Vedenyapin V. V., Kineticheskaya teoriya po Maksvellu, Boltsmanu i Vlasovu, Konspekt lektsii, MGOU, M., 2005

[13] Kullback S., Leibler R. A., “On information and sufficiency”, Ann. Math. Statist., 22:1 (1951), 79–86 | DOI | MR | Zbl

[14] Sanov N. N., “O veroyatnostyakh bolshikh otklonenii sluchainykh velichin”, Matem. sb., 42:1 (1957), 11–44 | MR | Zbl

[15] Chentsov N. N., “Nesimmetrichnoe rasstoyanie mezhdu raspredeleniyami veroyatnostei, entropiya i teorema Pifagora”, Matem. zametki, 4:3 (1968), 323–332 | Zbl

[16] Adzhiev S. Z., Amosov S. A., Vedenyapin V. V., “Odnomernye diskretnye modeli kineticheskikh uravnenii dlya smesei”, Zh. vychisl. matem. i matem. fiz., 44:3 (2004), 553–558 | MR | Zbl

[17] Bobylev A. V., Vinerean M. C., “Construction of discrete kinetic models with given invariants”, J. Statist. Phys., 132:1 (2008), 153–170 | DOI | MR | Zbl

[18] Cornille H., Cercignani C., “A class of planar discrete velocity models for gas mixtures”, J. Statist. Phys., 99:3,4 (2000), 967–991 | DOI | MR | Zbl

[19] Batischeva Ya. G., Vedenyapin V. V., “II-i zakon termodinamiki dlya khimicheskoi kinetiki”, Matem. modelirovanie, 17:8 (2005), 106–110 | MR

[20] Malyshev V. A., Pirogov S. A., “Obratimost i neobratimost v stokhasticheskoi khimicheskoi kinetike”, Uspekhi matem. nauk, 63:1 (2008), 3–36 | MR | Zbl