Abstract
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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References
L. Boltzmann, “Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen,” Akad. Sitzungsber Wien 66, 275–370 (1872).
L. Boltzmann, “Über die Beziehung zwischen dem zweiten Hauptsatze der Mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung, respektive den Satzen über das Wärmegleichgewicht,” Akad. Sitzungsber Wien 76, 373–435 (1878)
H. Poincaré, “Remarks on the Kinetic Theory of Gases,” in Selected Works (Nauka, Moscow, 1974), Vol. 3 [in Russian].
V. V. Kozlov, Heat Equilibrium in the Sense of Gibbs and Poincare (Inst. Komp’yuternykh Issledovanii, Moscow, 2002) [in Russian].
V. V. Kozlov and D. V. Treshchev, “Weak Convergence of Solutions of the Liouville Equation for Nonlinear Hamiltonian Systems,” Theor. Math. Phys. 134, 339–350 (2003).
F. Riesz and B. Sz.-Nagy, Functional Analysis (Mir, Moscow, 1979; Dover, New York, 1991).
V. V. Vedenyapin, “Time Averages and Boltzmann Extremals,” Dokl. Math. 78, 686–688 (2008).
T. Morimoto, “Markov Processes and the H-Theorem,” J. Phys. Soc. Jpn. 18, 328–331 (1963).
I. Csiszár, “Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizitat von Markoffschen Ketten,” Magyar. Tud. Akad. Mat. Kutató Int. Kózl. 8, 85–108 (1963).
E. R. Lorch, “Means of Iterated Transformations in Reflexive Vector Spaces,” Bull. Am. Math. Soc. 45, 945–947 (1939).
M. Kac, Some Stochastic Problems in Physics and Mathematics (Magnolia Petrolium Co., Dallas, Texas, 1956; Nauka, Moscow, 1967).
V. V. Vedenyapin, Kinetic Theory According to Maxwell, Boltzmann, and Vlasov: Notes of Lectures (MGOU, Moscow, 2005) [in Russian].
S. Kullback and R. A. Leibler, “On Information and Sufficiency,” Ann. Math. Stat. 22(1), 79–86 (1951).
N. N. Sanov, “On Probabilities of Large Deviations of Random Variables,” Mat. Sb. 42(1), 11–44 (1957).
N. N. Chentsov, “Asymmetric Distance between Probability Distributions, Entropy, and the Pythagorean Theorem,” Mat. Zametki 4, 323–332 (1968).
S. Z. Adzhiev, S. A. Amosov, and V. V. Vedenyapin, “One-Dimensional Discrete Models of Kinetic Equations for Mixtures,” Comput. Math. Math. Phys. 44, 523–528 (2004).
A. V. Bobylev and M. C. Vinerean, “Construction of Discrete Kinetic Models with Given Invariants,” J. Stat. Phys. 132, 153–170 (2008).
H. Cornille and C. Cercignani, “A Class of Planar Discrete Velocity Models for Gas Mixtures,” J. Stat. Phys. 99, 967–991 (2000).
Ya. G. Batishcheva and V. V. Vedenyapin, “The Second Law of Thermodynamics for Chemical Kinetics,” Mat. Model. 17(8), 106–110 (2005).
V. A. Malyshev and S. A. Pirogov, “Reversibility and Irreversibility in Stochastic Chemical Kinetics,” Usp. Mat. Nauk 63(1), 3–36 (2008).
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Original Russian Text © S.Z. Adzhiev, V.V. Vedenyapin, 2011, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2011, Vol. 51, No. 11, pp. 2063–2074.
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Adzhiev, S.Z., Vedenyapin, V.V. Time averages and Boltzmann extremals for Markov chains, discrete Liouville equations, and the Kac circular model. Comput. Math. and Math. Phys. 51, 1942–1952 (2011). https://doi.org/10.1134/S0965542511110029
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DOI: https://doi.org/10.1134/S0965542511110029