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Using the theory of large deviations, macroscopic fluctuation theory providesa framework to understand the behaviour of non-equilibrium dynamics and steadystates in diffusive systems. We extend this framework to a minimal model ofnon-equilibrium non-diffusive system, specifically an open linear network on afinite graph. We explicitly calculate the dissipative bulk and boundary forcesthat drive the system towards the steady state, and non-dissipative bulk andboundary forces that drives the system in orbits around the steady state. Usingthe fact that these forces are orthogonal in a certain sense, we provide adecomposition of the large-deviation cost into dissipative and non-dissipativeterms. We establish that the purely non-dissipative force turns the dynamicsinto a Hamiltonian system. These theoretical findings are illustrated bynumerical examples.

Untangling Dissipative and Hamiltonian effects in bulk and boundary driven systems