S ep 2 01 6 Approach to the steady state in kinetic models with thermal reservoirs at different temperatures

We investigate kinetic models of a system in contact with several spatially homogeneous thermal reservoirs at different temperatures. We explicitly find all spatially homogeneous non-equilibrium stationary states (NESS). We then consider the question of whether there are also non spatially uniform NESS. This remains partly open in general, except for a class of models to which we can apply probabilistic methods. For these models we show not only that all NESS are spatially homogeneous, but that solutions of these kinetic equation relax exponentially fast, at an explictely computable rate, to the spatially homogeneous NESS also for general spatially inhomogeneous initial data.