# ERGODIC PROPERTIES OF A PARTICLE IN CONTACT WITH A HEAT BATH

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## Abstract

We study the ergodic properties of a system of point particles on the semi-infinite line. The particles evolve under a dynamics where the only interaction is between the first particle and each of the other particles through elastic collision. The first particle is constrained to stay within the spatial region {lcub}0,1{rcub} by a reflective boundary condition. We choose the Grand Canonical Gibbs measure (mu) appropriately normalized as our invariant measure.;The motivation for this work comes from physics. This system can be considered as a model for a finite system (first particle) in equilibrium with a reservoir (the rest of the particles). We show that the system, starting from an initial state (measure) concentrated on any point in its phase-space with the reservoir being in the equilibrium state, evolves under the time evolution described above to the equilibrium state (defined appropriately w.r.t. the reservoir state).;In order to study the ergodic properties of the finite system under the stochastic time evolution defined by the reservoir, we study the stochastic process (Q(w,t,),V(w,t)) of the position and velocity of the first particle. This is a non-Markovian process. By appropriately enlarging the state space (giving more information!) we obtain a Markov process. We prove that this process is an ergodic, Harris process with no cyclic classes. This we do by essentially showing that the transition probability grows an absolutely continuous component. This enables us to invoke the powerful theorems from erodic theory of Harris processes, which yield strong convergence of measures under time evolution. We then use this convergence result on the Markov process to show that.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;in absolute variations norm for all (q,v) (ELEM) {lcub}0,1{rcub} X (//R). This result implies in particular that the whole semi-infinite system is a Bernoulli flow.;Most of the results obtained are valid in the case where the mass of the first particle is either equal or different from the mass of the other particles.