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We study the statistical mechanics of a finite-dimensional non-linearHamiltonian system (a chain of anharmonic oscillators) coupled to two heatbaths (described by wave equations). Assuming that the initial conditions ofthe heat baths are distributed according to the Gibbs measures at two differenttemperatures we study the dynamics of the oscillators. Under suitableassumptions on the potential and on the coupling between the chain and the heatbaths, we prove the existence of an invariant measure for any temperaturedifference, i.e., we prove the existence of steady states. Furthermore, if thetemperature difference is sufficiently small, we prove that the invariantmeasure is unique and mixing. In particular, we develop new techniques forproving the existence of invariant measures for random processes on anon-compact phase space. These techniques are based on an extension of thecommutator method of H\"ormander used in the study of hypoelliptic differentialoperators.Comment: 43 page

Non-Equilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures