  On networks representing probability currents between states of a system, we
generalize Schnakenberg's theory of nonequilibrium observables to nonsteady
states, with the introduction of a new set of macroscopic observables that, for
planar graphs, are related by a duality. We apply this duality to the linear
regime, obtaining a dual proposition for the minimum entropy production
principle, and to discrete electromagnetism, finding that it exchanges fields
with sources. We interpret duality as reversing the role of system and
environment, and discuss generalization to nonplanar graphs. The results are
based on two theorems regarding the representation of bilinear and quadratic
forms over the edge vector space of an oriented graph in terms of observables
associated to cycles and cocycles.
