Hyperbolicity and Möller-Morphism for a Model of Classical Statistical Mechanics
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.
KeywordsGibbs Measure Light Particle Wave Operator Ergodic Property Incoming Particle
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