Non-Equilibrium Statistical Mechanics of Classical Lattice $\phi^4$   Field Theory
  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. 
Kenichiro Aoki and Dimitri Kusnezov