98-233 Eckmann J.-P., Pillet C.-A., Rey-Bellet L.

Non-Equilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures (419K, postscript) Mar 27, 98

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Abstract. We study the statistical mechanics of a finite-dimensional non-linear Hamiltonian system (a chain of {\it 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 {\it 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 {\it any} temperature difference, {\it i.e.}, we prove the existence of {\it steady states}. Furthermore, if the temperature difference is sufficiently small, we prove that the invariant measure is {\it unique} and {\it 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.

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