  Onsager's 1931 `reciprocity relations' result connects microscopic
time-reversibility with a symmetry property of corresponding macroscopic
evolution equations. Among the many consequences is a variational
characterization of the macroscopic evolution equation as a gradient-flow,
steepest-ascent, or maximal-entropy-production equation. Onsager's original
theorem is limited to close-to-equilibrium situations, with a Gaussian
invariant measure and a linear macroscopic evolution. In this paper we
generalize this result beyond these limitations, and show how the microscopic
time-reversibility leads to natural generalized symmetry conditions, which take
the form of generalized gradient flows.
