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 TMF, 1972, Volume 11, Number 2, Pages 248–258 (Mi tmf2857)

Construction of dynamics in one-dimensional systems of statistical mechanics

Ya. G. Sinai

Abstract: It is well known that in one-dimensional systems the microcanonical, small canonical, and grand canonical distributions have the same thermodynamic limit. This limit can be regarded as a measure on the phase space of an infinite system of particles. Under the assumption that the binary interaction potential has compaet support, it is shown that one can find a one- parametric group of transformations in the phase space that preserve this measure and are related in a natural manner to the infinite system of Hamiltonian equations that describe the motion of the particles. This result has been previously proved by Lanford under the assumption that the potential has bounded modulus and finite range.

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English version:
Theoretical and Mathematical Physics, 1972, 11:2, 487–494

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Citation: Ya. G. Sinai, “Construction of dynamics in one-dimensional systems of statistical mechanics”, TMF, 11:2 (1972), 248–258; Theoret. and Math. Phys., 11:2 (1972), 487–494

Citation in format AMSBIB
\Bibitem{Sin72} \by Ya.~G.~Sinai \paper Construction of dynamics in one-dimensional systems of statistical mechanics \jour TMF \yr 1972 \vol 11 \issue 2 \pages 248--258 \mathnet{http://mi.mathnet.ru/tmf2857} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=475512} \transl \jour Theoret. and Math. Phys. \yr 1972 \vol 11 \issue 2 \pages 487--494 \crossref{https://doi.org/10.1007/BF01028564} 

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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. A. N. Zemlyakov, “Postroenie dinamiki v odnomernykh sistemakh statisticheskoi fiziki v sluchae nefinitnykh potentsialov”, UMN, 28:1(169) (1973), 239–240
2. B. M. Gurevich, Ya. G. Sinai, Yu. M. Sukhov, “On invariant measures of dynamical systems of one-dimensional statistical mechanics”, Russian Math. Surveys, 28:5 (1973), 49–86
3. Ya. G. Sinai, Yu. M. Sukhov, “Existence theorem for solutions of the Bogolyubov equations”, Theoret. and Math. Phys., 19:3 (1974), 560–573
4. D. Ya. Petrina, “Mathematical description of the evolution of infinite systems of classical statistical physics. I. Locally perturbed one-dimensional systems”, Theoret. and Math. Phys., 38:2 (1979), 153–166
5. P. V. Malyshev, “Mathematical description of the evolution of an infinite classical system”, Theoret. and Math. Phys., 44:1 (1980), 603–611
6. B. I. Shubov, “On the unique solvability of the Cauchy problem for the equations of motion of discrete analogs of multidimensional chiral fields taking values on compact symmetric spaces”, Theoret. and Math. Phys., 49:2 (1981), 966–974
7. D. Ya. Petrina, V. I. Gerasimenko, “A mathematical description of the evolution of the state of infinite systems of classical statistical mechanics”, Russian Math. Surveys, 38:5 (1983), 1–61
8. V. A. Chulaevskii, “Stationary measures of integrable systems in statistical physics”, Russian Math. Surveys, 38:6 (1983), 115–116
9. V. A. Chulaevskii, “Inverse scattering method in statistical physics”, Funct. Anal. Appl., 17:1 (1983), 40–47
10. M. R. Soloveichik, “Ergodic properties of systems with an external potential in classical statistical mechanics”, Math. USSR-Izv., 34:1 (1990), 181–200
11. S. P. Novikov, L. A. Bunimovich, A. M. Vershik, B. M. Gurevich, E. I. Dinaburg, G. A. Margulis, V. I. Oseledets, S. A. Pirogov, K. M. Khanin, N. N. Chentsova, “Yakov Grigor'evich Sinai (on his sixtieth birthday)”, Russian Math. Surveys, 51:4 (1996), 765–778
12. B. M. Gurevich, “Dynamical aspects of statistical physics in Dobrushin's works”, Russian Math. Surveys, 52:2 (1997), 257–264
13. V. A. Malyshev, “Analytic Dynamics of a One-Dimensional System of Particles with Strong Interaction”, Math. Notes, 92:2 (2012), 237–248
14. A. I. Bufetov, B. M. Gurevich, K. M. Khanin, F. Cellarosi, “The Abel Prize award to Ya. G. Sinai”, Russian Math. Surveys, 69:5 (2014), 931–956
15. M. V. Tantsiura, “On strong solutions to countable systems of SDEs with interaction and non-Lipschitz drift”, Theory Stoch. Process., 21(37):1 (2016), 91–101
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