Geometrical interpretation of fluctuating hydrodynamics in diffusive systems

TL;DR

This paper explores a geometric framework for understanding fluctuating hydrodynamics in diffusive systems, linking the evolution of density profiles to steepest descent in Wasserstein space and connecting it to mesoscopic fluctuations.

Contribution

It introduces a geometric formulation of hydrodynamic limits using Wasserstein geometry, relating deterministic evolution to free energy minimization and mesoscopic fluctuations.

Findings

Hydrodynamic evolution corresponds to steepest descent in Wasserstein space.

The geometric approach connects deterministic hydrodynamics with stochastic fluctuations.

A saddle point argument links fluctuating hydrodynamics to the geometric formulation.

Abstract

We discuss geometric formulations of hydrodynamic limits in diffusive systems. Specifically, we describe a geometrical construction in the space of density profiles --- the Wasserstein geometry --- which allows the deterministic hydrodynamic evolution of the systems to be related to steepest descent of the free energy, and show how this formulation can be related to most probable paths of mesoscopic dissipative systems. The geometric viewpoint is also linked to fluctuating hydrodynamics of these systems via a saddle point argument.