  We study the structure of stationary non equilibrium states for interacting
particle systems from a microscopic viewpoint. In particular we discuss two
different discrete geometric constructions. We apply both of them to determine
non reversible transition rates corresponding to a fixed invariant measure. The
first one uses the equivalence of this problem with the construction of
divergence free flows on the transition graph. Since divergence free flows are
characterized by cyclic decompositions we can generate families of models from
elementary cycles on the configuration space. The second construction is a
functional discrete Hodge decomposition for translational covariant discrete
vector fields. According to this, for example, the instantaneous current of any
interacting particle system on a finite torus can be canonically decomposed in
a gradient part, a circulation term and an harmonic component. All the three
components are associated to functions on the configuration space. This
decomposition is unique and constructive. The stationary condition can be
interpreted as an orthogonality condition with respect to an harmonic discrete
vector field and we use this decomposition to construct models having a fixed
invariant measure.
