  The mathematical physics of mechanical systems in thermal equilibrium is a
well studied, and relatively easy, subject, because the Gibbs distribution is
in general an adequate guess for the equilibrium state. On the other hand, the
mathematical physics of {\em non-equilibrium} systems, such as that of a chain
of masses connected with springs to two (infinite) heat reservoirs is more
difficult, precisely because no such {\em a priori} guess exists. Recent work
has, however, revealed that under quite general conditions, such states can not
only be shown to exist, but are {\em unique}, using the H\"ormander conditions
and controllability. Furthermore, interesting properties, such as energy flux,
exponentially fast convergence to the unique state, and fluctuations of that
state have been successfully studied. Finally, the ideas used in these studies
can be extended to certain stochastic PDE's using Malliavin calculus to prove
regularity of the process.
