Absence of local thermal equilibrium in two models of heat conduction

A crucial assumption in the conventional description of thermal conduction is the existence of local thermal equilibrium. We test this assumption in two simple models of heat conduction. Our first model is a linear chain of planar spins with nearest neighbour couplings, and the second model is that of a Lorentz gas. We look at the steady state of the system when the two ends are connected to heat baths at temperatures T1 and T2. If T1 = T2, the system reaches thermal equilibrium. If T1 6= T2, there is a heat current through the system, but there is no local thermal equilibrium. This is true even in the limit of large system size, when the heat current goes to zero. We argue that this is due to the existence of an infinity of local conservation laws in their dynamics.