Entropic and gradient flow formulations for nonlinear diffusion

TL;DR

This paper explores entropic and gradient flow formulations for nonlinear diffusion equations, linking thermodynamics, large deviations, and hydrodynamic limits to provide a unified analytical framework.

Contribution

It introduces a thermodynamic metric derived from entropy considerations for nonlinear diffusion, connecting microscopic zero range processes to macroscopic PDEs.

Findings

Thermodynamic entropy can be associated with certain nonlinear diffusion equations.

A new thermodynamic metric is derived from entropy considerations.

Connections between large deviations, fluctuating hydrodynamics, and gradient flows are established.

Abstract

Nonlinear diffusion $\partial_t \rho = \Delta(\Phi(\rho))$ is considered for a class of nonlinearities $\Phi$. It is shown that for suitable choices of $\Phi$, an associated Lyapunov functional can be interpreted as thermodynamics entropy. This information is used to derive an associated metric, here called thermodynamic metric. The analysis is confined to nonlinear diffusion obtainable as hydrodynamic limit of a zero range process. The thermodynamic setting is linked to a large deviation principle for the underlying zero range process and the corresponding equation of fluctuating hydrodynamics. For the latter connections, the thermodynamic metric plays a central role.