  We construct a non reversible exclusion process with Bernoulli product
invariant measure and having, in the diffusive hydrodynamic scaling, a non
symmetric diffusion matrix, that can be explicitly computed. The antisymmetric
part does not affect the evolution of the density but it is relevant for the
evolution of the current. Switching on a weak external field we obtain a
symmetric mobility matrix that is related just to the symmetric part of the
diffusion matrix by an Einstein relation. We argue that this fact is typical
within a class of generalized gradient models. We consider for simplicity the
model in dimension $d=2$, but a similar behavior can be also obtained in higher
dimensions.
