Classical lattice field theories are studied in non-equilibrium steady states from first principles by placing them in strong thermal gradients. The breakdown of local equilibrium as well as general observables is quantified. We find that linear response predictions are violated at the same rate as the breakdown of local equilibrium, and that there is no threshold for this behavior. Further, the equation of state develops non-local contributions even in theories with bulk behavior in the non-equilibrium steady state. ©2001 American Institute of Physics.