Approach to a non-equilibrium steady state

We consider a non-interacting one-dimensional gas accelerated by a constant and uniform external field. The energy absorbed from the field is transferred via elastic collisions to a bath of scattering obstacles. At gas-obstacle encounters the particles of the gas acquire a fixed kinetic energy. The approach to the resulting stationary state is studied within the Boltzmann kinetic theory. It is shown that the long time behavior is governed by the hydrodynamic mode of diffusion superposed on a convective flow. The diffusion coefficient is analytically calculated for any value of the field showing a minimum at intermediate field intensities. It is checked that the properly generalized Green-Kubo formula applies in the non-equilibrium stationary state.