Entropy production and the geometry of dissipative evolution equations

Abstract

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.

References in corpus (6)

• The large deviation approach to statistical mechanics
• Large deviations for infinite dimensional stochastic dynamical systems
• On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion
• Steepest Entropy Ascent Model for Far-Non-Equilibrium Thermodynamics. Unified Implementation of the Maximum Entropy Production Principle
• On the general equation of motion of quantum thermodynamics and the distinction between quantal and nonquantal uncertainties
• Essential equivalence of the GENERIC and Steepest Entropy Ascent models of dissipation for non-equilibrium thermodynamics

Cited by in corpus (7)

• A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility
• The fourth law of thermodynamics: steepest entropy ascent
• Deriving GENERIC from a generalized fluctuation symmetry
• Instability, rupture and fluctuations in thin liquid films: Theory and computations
• Computing diffusivities from particle models out of equilibrium
• Cosh gradient systems and tilting
• Heat distribution in open quantum systems with maximum entropy production