Abstract. 
In this talk I will present our new results in nonequilibrium
steady-states for a class of 1-D heat conduction models. I will start with
the randomization of 1-D mechanical particle system coupled to unequal heat
baths. Next, I will justify that the randomization preserves the main
features of the mechanical chain model. Then I'll introduce our results
about the existence and uniqueness of nonequilibrium steady states, tail
bounds of nonequilibrium steady states and the exponential convergence to
steady states. In the end I will present the main technique of the proof.
If time permits, I will give a brief review of technical difficulties in
this problem. This is a joint work with Lai-Sang Young.