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

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


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.


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.


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© Springer Science+Business Media New York 1985

Authors and Affiliations

  • E. Presutti
  • Ya. G. Sinai
  • M. R. Soloviechik

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