Einstein's fluctuation relation and Gibbs states far from equilibrium

We examine a class of one-dimensional lattice-gases characterised by a gradient condition which guarantees the existence of Gibbs-type homogeneous stationary states. We show how, defining appropriate boundary conditions, this leads to a special symmetry of the system under time and space reversal which, rephrased in terms of the large deviations function of stationary currents of conserved quantities, yields a novel fluctuation relation under reservoir exchange, unrelated to the Gallavotti-Cohen symmetry. We then show that this relation can be interpreted as a nonequilibrium and nonlinear generalisation Einstein's relation, leading to Onsager reciprocity relations in the limit of a small reservoir imbalance. Finally, we illustrate our results with two examples.