  Classical $\phi^4$ theory in weak and strong thermal gradients is studied on
the lattice in (1+1) dimensions. Classical $\phi^4$ theory in weak and strong
thermal gradients is studied on the lattice in (1+1) dimensions. The steady
state physics of the theory is investigated from first principles and
classified into dynamical regimes. We derive the bulk properties associated
with thermal transport, and explore in detail the non-equilibrium statistical
mechanics of the theory as well as connections to equilibrium and irreversible
thermodynamics. Linear response predictions are found to be valid for systems
quite far from equilibrium and are seen to eventually break down simultaneously
with local equilibrium.
