  Macroscopic equations arising out of stochastic particle systems in detailed
balance (called dissipative systems or gradient flows) have a natural
variational structure, which can be derived from the large-deviation rate
functional for the density of the particle system. While large deviations can
be studied in considerable generality, these variational structures are often
restricted to systems in detailed balance. Using insights from macroscopic
fluctuation theory, in this work we aim to generalise this variational
connection beyond dissipative systems by augmenting densities with fluxes,
which encode non-dissipative effects. Our main contribution is an abstract
framework, which for a given flux-density cost and a quasipotential, provides a
decomposition into dissipative and non-dissipative components and a generalised
orthogonality relation between them. We then apply this abstract theory to
various stochastic particle systems -- independent copies of jump processes,
zero-range processes, chemical-reaction networks in complex balance and
lattice-gas models.
