Violations of local equilibrium and linear response in classical lattice theories

We study the dynamics of $\phi^4$ theory and the FPU $\beta$ model under thermal gradients, from first principles. We analyze quantitatively how local equilibrium and linear response are violated, paying special care to how we find observables that unambiguously display these violations. Relations between these quantities to equations of state are also examined. Further, we discuss how we can approach similar dynamical problems in continuum quantum field theory. We analyze how close we are to obtaining the continuum results.