  We study how local equilibrium, and linear response predictions of transport
coefficients are violated as systems move far from equilibrium. This is done by
studying heat flow in classical lattice models with and without bulk transport
behavior, in 1--3 dimensions. We see that linear response and local equilibrium
assumptions break down at the same rate. The equation of state is also found to
develop non-local corrections in the steady state. We quantify the breakdown
through the analysis of both microscopic and macroscopic observables, which are
found to display non-trivial size dependence.
