Classical ${\varphi }^4$ Lattice Field Theory in Strong Thermal Gradients

The dynamics of classical ${\varphi }^4$ theory under weak and strong thermal gradients is studied. We obtain the thermal conductivity of the theory including its temperature dependence. Under moderately strong thermal gradients, the temperature profiles become visibly non-linear, yet the phenomenon can be understood using the linear response theory. When we move further away from equilibrium, we find that the linear response theory eventually breaks down, and the concept of local equilibrium also fails.