  First we discuss the definition of the instantaneous current in interacting
particle systems, in particular in mass-energy systems and we point out its
role in the derivation of the hydrodynamics. Later we present some geometrical
structures of the instantaneous current when the rates satisfy a common
symmetry. These structures give some new ideas in non-gradient models and show
new phenomenology in diffusive interacting particle systems. Specifically, we
introduce models with vorticity and present some new perspectives on the link
between the Green-Kubo's formula and the hydrodynamics of non-gradient models.
