Heat Conduction and Entropy Production in Anharmonic Crystals with Self-Consistent Stochastic Reservoirs

We investigate a class of anharmonic crystals in d dimensions, d >= 1, coupled to both external and internal heat baths of the Ornstein-Uhlenbeck type. The external heat baths, applied at the boundaries in the 1-direction, are at specified, unequal, temperatures T-L and T-R. The temperatures of the internal baths are determined in a self-consistent way by the requirement that there be no net energy exchange with the system in the non-equilibrium stationary state (NESS). We prove the existence of such a stationary self-consistent profile of temperatures for a finite system and show that it minimizes the entropy production to leading order in (T-L-T-R). In the NESS the heat conductivity. is defined as the heat flux per unit area divided by the length of the system and (T-L-T-R). In the limit when the temperatures of the external reservoirs go to the same temperature T, k(T) is given by the Green-Kubo formula, evaluated in an equilibrium system coupled to reservoirs all having the temperature T. This k(T) remains bounded as the size of the system goes to infinity. We also show that the