Fluctuation relations in simple examples of non-equilibrium steady states

We discuss fluctuation relations in simple cases of non-equilibrium Langevin dynamics. In particular, we show that, close to non-equilibrium steady states with non-vanishing probability currents, some of these relations reduce to a modified version of the fluctuation-dissipation theorem. The latter may be interpreted as the equilibrium-like relation in the reference frame moving with the mean local velocity determined by the probability current.