Abstract
We consider a gas of point particles in IR+. The first particle has mass M, the others m and M>m. The particles interact by elastic collisions (among themselves and with the wall at the origin). Let * be the phase space and μ a Gibbs measure for the system, St denotes the time flow and (*,μ,St) is a dynamical system.requires a Solution of the Milne problem.
We identify the m -particles during their evolution so that they keep the same velocity until they collide with the M -particle. Hence the motion is free, asymptotically far from the origin: free particles come from, +∞ interact with the M -particle and then move back free to +∞. We prove that the Möller wave operators exist, asymptotic Ω± completeness holds and that Ω− −1Ω+ defines a non-trivial scattering matrix for the system. Ω+ define isomorphisms between the dynamical system (*°,μ°,S°t) and (*,μ,St) (*°,μ°,S°t) refers to the case when all the particles have mass m and μ° has the same thermodynamical parameters as μ.
An independent generating partition is explicitely known for the system (*°,μ°,S°t) and Ω± transform it in an independent generating partition for (*,μ,St), thereby proving that this is a Bernoulli flow.
The proof of the existence of the wave operator is based on the (almost everywhere) existence of contractive manifolds. Namely we prove that for almost all configurations x∈* the following holds. Fix any finite subset I of particles in x and consider all the configurations y obtained by changing the coordinates of the particles in I while leaving all the others fixed. Then if the change is small enough Stx and Sty become (locally) exponentially close.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Aizenmann M., Goldstein S., Lebowitz J. L. Ergodic properties of infinite systems. Springer Lect. Notes in Physics 38, 112 (1975).
Aizenmann M., Goldstein S., Lebowitz J. L. Ergodic Properties of an infinite one dimensional hard rods system. Comm. Math. Phys.
Arnold V. I., Avez A. Problemes ergodiques de la mecanique classique. Paris, Gauthier-Villars, 1967.
Boldrighini C., De Masi A. Ergodic properties of a class of one dimensional systems of Statistical mechanics. In preparation.
Boldrighini C., pobrushijx R. L., Sukhov Yu. One dimensional hard rod caricature of hydrodynamics. J. Stat. Phys. 31, 577 (1983}.
Boldrighini C., De Masi A., Nogueira A., Presutti E. The dynamics of a particle interacting with a semiinfinite ideal gas is a Bernoulli flow. Preprint, 1984.
Boldrighini C., Pellegrinotti A., Presutti E., Sinai Ya. G., Solovietchic M. R. Ergodic properties of a one dimensional semi-infinite system of Statistical mechanics. Preprint, 1984.
Boldrighini C., Pellegrinotti A., Triolo L. Convergence to stationary states for infinite harmonic systems. J. Stat. Phys. 30 (1983).
Cornfeld I. P., Fomin S. V., Sinai Ya. G. Ergodic theory. Springer-Verlag, 1982.
De Pazzis O. Ergodic properties of a semi-infinite hard rods system. Commun. Math. Phys. 22, 121 (1971).
Dobrushin R. L., Pellegrinotti A., Sukhov Yu., Triolo L. In preparation.
Dobrushin R. L., Sukhov Yu. The asymptotics for some degenerate models of evolution of systems with an infinite number of particles. J. Soviet Math. 16, 1277 (1981).
Farmer J., Goldstein S., Speer E. R., Invariant states of a thermally conducting barrier. Preprint, 1983.
Goldstein S., Lebowitz J. L., Ravishankar K. Ergodic properties of a system in contact with a heat baths a one dimensional model. Comm. Math. Phys. 85, 419 (1982).
Goldstein S., Lebowitz J. L., Ravishankar K. Approach to equilibrium in models of a system in contact with a heat bath. Preprint.
Landau L. D., Lifschitz E. M. Statistical physics. Pergamon Press, London-Paris, 1959.
Botnic, Malishev Commun. Math. Phys. ~(1983–84).
Narnhofer T., Requardt M., Thirring W. Quasi particles at finite temperature. Commun. Math. Phys. 92, 24 7 (1983).
Nogueira A. Ergodic properties of a one dimensional open system of Statistical mechanics. Preprint, 1984.
Ornstein D. S. Ergodic theory, randomness and dynamical systems. Yale University Press, New Häven and London 1974.
Reed M., Simon B. Methods of modern mathematical physics: III scattering theory. Academic Press, 1979.
Sinai Ya. G. Ergodic properties of a gas of one dimensional hard rods with an infinite number of degrees of freedom. Funct. Anal. Appl. 6, 35 (1972).
Sinai Ya. G. Construction of dynamics in one dimensional systems of Statistical mechanics. Theor. Math. Phys. 11, 248 (1972J.
Sinai Ya. G. Introduction to ergodic theory. Princeton University Press, 1977.
Sinai Ya. G., Volkovysski K. Ergodic Properties of an ideal gas with an infinite number of degrees of free-dom. Funct. Anal. Appl. 5, 19 (1971).
Bowen R. Equilibrium states of the ergodic theory of Anosov diffeomorphisms. Springer, Lect. Notes in Math. 470 (1975).
Ruelle D. Thermodynamics formalism. Addison Wesley, Boston, 1978. Encyclopedia of mathematics and its ap-plications.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1985 Springer Science+Business Media New York
About this chapter
Cite this chapter
Presutti, E., Sinai, Y.G., Soloviechik, M.R. (1985). Hyperbolicity and Möller-Morphism for a Model of Classical Statistical Mechanics . In: Fritz, J., Jaffe, A., Szász, D. (eds) Statistical Physics and Dynamical Systems. Progress in Physics, vol 10. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-6653-7_15
Download citation
DOI: https://doi.org/10.1007/978-1-4899-6653-7_15
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4899-6655-1
Online ISBN: 978-1-4899-6653-7
eBook Packages: Springer Book Archive