Surface Dean--Kawasaki equations

TL;DR

This paper derives and analyzes the surface Dean-Kawasaki equation for stochastic particle systems on hypersurfaces, incorporating geometry, interactions, and evolving surfaces, with theoretical and numerical insights.

Contribution

It introduces a surface DK equation framework that accounts for hypersurface geometry and particle interactions, extending existing models to evolving surfaces and providing discretization methods.

Findings

Derived the surface DK equation from Langevin dynamics.

Established weak uniqueness in the non-interacting case.

Developed a finite-volume discretization preserving fluctuation-dissipation.

Abstract

We consider stochastic particle dynamics on hypersurfaces represented in Monge gauge parametrization. Starting from the underlying Langevin system, we derive the surface Dean-Kawasaki (DK) equation and formulate it in the martingale sense. The resulting SPDE explicitly reflects the geometry of the hypersurface through the induced metric and its differential operators. Our framework accommodates both pairwise interactions and environmental potentials, and we extend the analysis to evolving hypersurfaces driven by an SDE that interacts with the particles, yielding the corresponding surface DK equation for the coupled surface-particle system. We establish a weak uniqueness result in the non-interacting case, and we develop a finite-volume discretization preserving the fluctuation-dissipation relation. Numerical experiments illustrate equilibrium properties and dynamical behavior influenced by surface geometry and external potentials.