  Purely dissipative evolution equations are often cast as gradient flow
structures, $\dot{\mathbf{z}}=K(\mathbf{z})DS(\mathbf{z})$, where the variable
$\mathbf{z}$ of interest evolves towards the maximum of a functional $S$
according to a metric defined by an operator $K$. While the functional often
follows immediately from physical considerations (e.g., the thermodynamic
entropy), the operator $K$ and the associated geometry does not necessarily so
(e.g., Wasserstein geometry for diffusion). In this paper, we present a
variational statement in the sense of maximum entropy production that directly
delivers a relationship between the operator $K$ and the constraints of the
system. In particular, the Wasserstein metric naturally arises here from the
conservation of mass or energy, and depends on the Onsager resistivity tensor,
which, itself, may be understood as another metric, as in the Steepest Entropy
Ascent formalism. This new variational principle is exemplified here for the
simultaneous evolution of conserved and non-conserved quantities in open
systems. It thus extends the classical Onsager flux-force relationships and the
associated variational statement to variables that do not have a flux
associated to them. We further show that the metric structure $K$ is intimately
linked to the celebrated Freidlin-Wentzell theory of stochastically perturbed
gradient flows, and that the proposed variational principle encloses an
infinite-dimensional fluctuation-dissipation statement.
