Time evolution of a two-dimensional model system. I. Invariant states and time correlation functions

Abstract

This paper is the first one of a series devoted to the study of a particularly simple two-dimensional system of classical particles. The model is presented and some general feautres of it are established. We prove that, among states without correlations between particles with different velocities, there is a unique time invariant state with given density and hydrodynamic velocity. This ``equilibrium state'' is studied in detail. In particular its ergodic and mixing properties are investigated. We propose an approximation in order to estimate the asymptotic part of the time correlation functions and show that the long time tail is ruled by the ``hydrodynamic'' behavior of the model, namely by the evolution of the long wavelength perturbations.