Asymptotic Behavior of Thermal Non-Equilibrium Steady States for a   Driven Chain of Anharmonic Oscillators
  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. 
Luc Rey-Bellet and Lawrence E. Thomas