  We study the statistical mechanics of a finite-dimensional non-linear
Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat
baths (described by wave equations). Assuming that the initial conditions of
the heat baths are distributed according to the Gibbs measures at two different
temperatures we study the dynamics of the oscillators. Under suitable
assumptions on the potential and on the coupling between the chain and the heat
baths, we prove the existence of an invariant measure for any temperature
difference, i.e., we prove the existence of steady states. Furthermore, if the
temperature difference is sufficiently small, we prove that the invariant
measure is unique and mixing. In particular, we develop new techniques for
proving the existence of invariant measures for random processes on a
non-compact phase space. These techniques are based on an extension of the
commutator method of H\"ormander used in the study of hypoelliptic differential
operators.
