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)
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.