We study the nonequilibrium statistical mechanics of classical lattice 4 theory in thermal gradients of arbitrary strength. The steady-state physics of the theory in (1 + 1) dimensions is investigated from first principles and classified into dynamical regimes. The bulk properties associated with thermal transport are derived. Starting with thermal equilibrium, we examine the relation to the kinetic theory description of the behavior of the system. Green-Kubo transport coefficients are computed and compared to measured transport near and far from equilibrium, defining the scope of linear response and including an analysis of long-time tails. Local equilibrium is investigated and seen to explicitly break down under sufficiently strong gradients. Additional measures of the nonequilibrium steady state are also discussed as well as the connections to irreversible thermodynamics.
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