  We consider a model of heat conduction which consists of a finite nonlinear
chain coupled to two heat reservoirs at different temperatures. We study the
low temperature asymptotic behavior of the invariant measure. We show that, in
this limit, the invariant measure is characterized by a variational principle.
We relate the heat flow to the variational principle. The main technical
ingredient is an extension of Freidlin-Wentzell theory to a class of degenerate
diffusions.
