Entropic and gradient flow formulations for nonlinear diffusion

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.

References in corpus (3)

• Large deviations and gradient flows
• Lyapunov functionals for boundary-driven nonlinear drift-diffusions
• Geometrical interpretation of fluctuating hydrodynamics in diffusive systems

Cited by in corpus (5)

• Jump processes as Generalized Gradient Flows
• Computing diffusivities from particle models out of equilibrium
• Nonlinear diffusion equations with nonlinear gradient noise
• Ill-posedness of the pure-noise Dean-Kawasaki equation
• Well-Posedness of the generalised Dean-Kawasaki Equation with correlated noise on bounded domains