Ergodic properties of classical dissipative systems I

Abstract

We consider a class of models in which a Hamiltonian system A, with a finite number of degrees of freedom, is brought into contact with an infinite heat reservoir B. We develop the formalism required to describe these models near thermal equilibrium. Using a combination of abstract spectral techniques and harmonic analysis we investigate the singular spectrum of the Liouvillean L of the coupled system A+B. We provide a natural set of conditions which ensure that the spectrum of L is purely absolutely continuous except for a simple eigenvalue at zero. It then follows from the spectral theory of dynamical systems (Koopmanism) that the system A+B is strongly mixing. From a probabilistic point of view, we study a new class of random processes on finite dimensional manifolds: non-Markovian Ornstein-Uhlenbeck processes. The paths of such a process are solutions of a random integro-differential equation with Gaussian noise which is a natural generalization of the well known Langevin equation. In this context we establish that, under appropriate conditions, the OU process is strongly mixing even when the Langevin equation has memory and is driven by non-white noise.