Violations of local equilibrium and stochastic thermostats

We quantitatively investigate the violations of local equilibrium in the $\phi^4$ theory under thermal gradients, using stochastic thermostats. We find that the deviations from local equilibrium can be quite well described by a behavior $\sim(\nabla T)^2$. The dependence of the quantities on the thermostat type is analyzed and its physical implications are discussed.