Statistical Physics and Dynamical Systems pp 253-284 | Cite as

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

## 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, S_{t} denotes the time flow and (*,μ,S_{t}) 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_{−} ^{−1}Ω_{+} defines a non-trivial scattering matrix for the system. Ω_{+} define isomorphisms between the dynamical system (*°,μ°,S°_{t}) and (*,μ,S_{t}) (*°,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,S_{t}), 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 S_{t}x and S_{t}y become (locally) exponentially close.

## Keywords

Gibbs Measure Light Particle Wave Operator Ergodic Property Incoming Particle## Preview

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