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First we discuss the definition of the instantaneous current in interactingparticle systems, in particular in mass-energy systems and we point out itsrole in the derivation of the hydrodynamics. Later we present some geometricalstructures of the instantaneous current when the rates satisfy a commonsymmetry. These structures give some new ideas in non-gradient models and shownew phenomenology in diffusive interacting particle systems. Specifically, weintroduce models with vorticity and present some new perspectives on the linkbetween the Green-Kubo's formula and the hydrodynamics of non-gradient models.

Geometrical structures of the instantaneous current and their macroscopic effects: vortices and perspectives in non-gradient models