The modified Sutherland-Einstein relation for diffusive non-equilibria

There remains a useful relation between diffusion and mobility for a Langevin particle in a periodic medium subject to non-conservative forces. The usual fluctuation-dissipation relation can be easily modified and the mobility matrix is no longer proportional to the diffusion matrix, with a correction term depending explicitly on the (non-equilibrium) forces. We discuss this correction by considering various simple examples and we visualize the various dependencies on the applied forcing and on the time by means of simulations. For example, in all cases the diffusion depends on the external forcing more strongly than does the mobility. We also give an explicit decomposition of the symmetrized mobility matrix as the difference between two positive matrices, one involving the diffusion matrix and the other involving force-force correlations.