Open Classical Systems

Open systems are usually understood as a small Hamiltonian system (i.e. with a finite number of degrees of freedom) in contact with one or several large reservoirs. There are several ways to model reservoirs and we will take the point of view that the reservoirs are also Hamiltonian systems themselves. It is a convenient physical and mathematical idealization to separate scales and assume that the reservoirs have infinitely many degrees of freedom. We will also assume that, to start with, the reservoirs are in equilibrium, i.e., the initial states of the reservoirs are distributed according to Gibbs distribution with given temperatures. It is also mathematically convenient to assume that the Gibbs measures of the reservoir have very good ergodic properties. This is, in general, a mathematically difficult problem and we will circumvent it by assuming that our reservoirs have a linear dynamics (i.e the Gibbs measures are Gaussian measures). Our model of a reservoir will be the classical field theory given by a linear wave equation in R ∂ t φt(x) = ∆φt(x) . (1.1)