Entropy production and the geometry of dissipative evolution equations

Purely dissipative evolution equations are often cast as gradient flow structures, $\dot{\mathbf{z}}=K\left(\mathbf{z}\right)DS\left(\mathbf{z}\right)$, 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 do 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 variational principle is exemplified here for the simultaneous evolution of conserved and nonconserved 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.

Figure 1

Summary of the connections established in this work between the maximum entropy production principle, gradient flow structure, large deviation principle, and fluctuation dissipation relation $K\propto \sigma {\sigma }^{*}$, with $\sigma$ defined by the stochastic gradient flow.

Figure 2

Evolution of a closed system at $z\left(t\right)$ (a) without and (b) with the presence of a constraint. The equilibrium configuration coincides with the maximum of the entropy functional $S\left(z\right)$.