Entropy production and the geometry of dissipative evolution equations

Area of Science:

• Thermodynamics
• Statistical Mechanics
• Non-equilibrium Systems

Background:

• Dissipative evolution equations often use gradient flow structures (ż=K(z)DS(z)) where S is a functional and K defines geometry.
• The operator K and its associated geometry (e.g., Wasserstein) are not always directly derived from physical principles.

Purpose of the Study:

• To present a variational principle based on maximum entropy production.
• To establish a direct relationship between the operator K and system constraints.
• To extend classical Onsager relations to non-flux variables in open systems.

Main Methods:

• Formulating a variational statement based on maximum entropy production.
• Demonstrating the emergence of Wasserstein metric from mass/energy conservation.
• Connecting the metric structure K to Freidlin-Wentzell theory.

Main Results:

• The Wasserstein metric arises naturally from conservation laws and depends on the Onsager resistivity tensor.
• The variational principle extends Onsager flux-force relationships to non-conserved quantities.
• The metric structure K is linked to stochastic perturbation theory of gradient flows.

Conclusions:

• The proposed variational principle provides a unified framework for understanding dissipative systems.
• It offers insights into the geometric structure of evolution equations and their relation to physical constraints.
• The principle encompasses an infinite-dimensional fluctuation-dissipation statement.