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: a linear chain of planar spins with nearest neighbor couplings and a Lorentz gas. We look at the steady state of the system when the two ends are connected to heat baths at temperatures T-1 and T-2. If T-1 = T-2, the system reaches thermal equilibrium. If T-1 not equal T-2, there is a heat current through the system, but there is no local thermal equilibrium, 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. [S0031-9007(98)08254-4].