Surface Dean--Kawasaki equations
This work provides a theoretical framework for stochastic particle systems on curved surfaces, which is relevant for modeling biological and physical phenomena on membranes or interfaces.
The paper derives surface Dean-Kawasaki equations for stochastic particle dynamics on hypersurfaces, incorporating geometry through induced metrics, and establishes weak uniqueness for non-interacting cases. Numerical experiments demonstrate equilibrium and dynamical properties influenced by surface geometry.
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.