# Hyperbolicity and Möller-Morphism for a Model of Classical Statistical Mechanics

• E. Presutti
• Ya. G. Sinai
• M. R. Soloviechik
Chapter
Part of the Progress in Physics book series (PMP, volume 10)

## 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.

## Keywords

Gibbs Measure Light Particle Wave Operator Ergodic Property Incoming Particle
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
Aizenmann M., Goldstein S., Lebowitz J. L. Ergodic properties of infinite systems. Springer Lect. Notes in Physics 38, 112 (1975).
2. [2]
Aizenmann M., Goldstein S., Lebowitz J. L. Ergodic Properties of an infinite one dimensional hard rods system. Comm. Math. Phys.Google Scholar
3. [3]
Arnold V. I., Avez A. Problemes ergodiques de la mecanique classique. Paris, Gauthier-Villars, 1967.Google Scholar
4. [4]
Boldrighini C., De Masi A. Ergodic properties of a class of one dimensional systems of Statistical mechanics. In preparation.Google Scholar
5. [5]
Boldrighini C., pobrushijx R. L., Sukhov Yu. One dimensional hard rod caricature of hydrodynamics. J. Stat. Phys. 31, 577 (1983}.Google Scholar
6. [6]
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.Google Scholar
7. [7]
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.Google Scholar
8. [8]
Boldrighini C., Pellegrinotti A., Triolo L. Convergence to stationary states for infinite harmonic systems. J. Stat. Phys. 30 (1983).Google Scholar
9. [9]
Cornfeld I. P., Fomin S. V., Sinai Ya. G. Ergodic theory. Springer-Verlag, 1982.Google Scholar
10. [10]
De Pazzis O. Ergodic properties of a semi-infinite hard rods system. Commun. Math. Phys. 22, 121 (1971).Google Scholar
11. [11]
Dobrushin R. L., Pellegrinotti A., Sukhov Yu., Triolo L. In preparation.Google Scholar
12. [12]
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).
13. [13]
Farmer J., Goldstein S., Speer E. R., Invariant states of a thermally conducting barrier. Preprint, 1983.Google Scholar
14. [14]
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).
15. [15]
Goldstein S., Lebowitz J. L., Ravishankar K. Approach to equilibrium in models of a system in contact with a heat bath. Preprint.Google Scholar
16. [16]
Landau L. D., Lifschitz E. M. Statistical physics. Pergamon Press, London-Paris, 1959.Google Scholar
17. [17]
Botnic, Malishev Commun. Math. Phys. ~(1983–84).Google Scholar
18. [18]
Narnhofer T., Requardt M., Thirring W. Quasi particles at finite temperature. Commun. Math. Phys. 92, 24 7 (1983).Google Scholar
19. [19]
Nogueira A. Ergodic properties of a one dimensional open system of Statistical mechanics. Preprint, 1984.Google Scholar
20. [20]
Ornstein D. S. Ergodic theory, randomness and dynamical systems. Yale University Press, New Häven and London 1974.Google Scholar
21. [21]
Reed M., Simon B. Methods of modern mathematical physics: III scattering theory. Academic Press, 1979.Google Scholar
22. [22]
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).Google Scholar
23. [23]
Sinai Ya. G. Construction of dynamics in one dimensional systems of Statistical mechanics. Theor. Math. Phys. 11, 248 (1972J.Google Scholar
24. [24]
Sinai Ya. G. Introduction to ergodic theory. Princeton University Press, 1977.Google Scholar
25. [25]
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).
26. [26]
Bowen R. Equilibrium states of the ergodic theory of Anosov diffeomorphisms. Springer, Lect. Notes in Math. 470 (1975).Google Scholar
27. [27]
Ruelle D. Thermodynamics formalism. Addison Wesley, Boston, 1978. Encyclopedia of mathematics and its ap-plications.Google Scholar