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Surface Dean--Kawasaki equations

John Bell, Ana Djurdjevac, Nicolas Perkowski

TL;DR

This work develops a geometric stochastic framework for particle systems confined to evolving hypersurfaces by deriving a surface Dean–Kawasaki equation from Langevin dynamics in Monge-gauge coordinates. The approach yields a martingale formulation that naturally incorporates the surface geometry through the induced metric $G$, and it identifies two noise representations corresponding to Lebesgue and surface measures while ensuring a fluctuation-dissipation balance. The paper establishes weak uniqueness in the non-interacting moving-surface setting, extends the model to interacting particles with Gibbs-type drifts, and provides a structure-preserving finite-volume discretization that remains faithful to the stochastic geometry. Numerical experiments validate equilibrium statistics, reveal geometry-driven diffusion, and illustrate the impact of external potentials on particle distribution, highlighting the practical relevance of the framework for curved-domain diffusion problems. Overall, the work delivers a tractable SPDE description for finite-size particle systems on curved, evolving domains with geometry-coupled fluctuations, enabling efficient computation and analysis of surface-bound processes.

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.

Surface Dean--Kawasaki equations

TL;DR

This work develops a geometric stochastic framework for particle systems confined to evolving hypersurfaces by deriving a surface Dean–Kawasaki equation from Langevin dynamics in Monge-gauge coordinates. The approach yields a martingale formulation that naturally incorporates the surface geometry through the induced metric , and it identifies two noise representations corresponding to Lebesgue and surface measures while ensuring a fluctuation-dissipation balance. The paper establishes weak uniqueness in the non-interacting moving-surface setting, extends the model to interacting particles with Gibbs-type drifts, and provides a structure-preserving finite-volume discretization that remains faithful to the stochastic geometry. Numerical experiments validate equilibrium statistics, reveal geometry-driven diffusion, and illustrate the impact of external potentials on particle distribution, highlighting the practical relevance of the framework for curved-domain diffusion problems. Overall, the work delivers a tractable SPDE description for finite-size particle systems on curved, evolving domains with geometry-coupled fluctuations, enabling efficient computation and analysis of surface-bound processes.

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.
Paper Structure (18 sections, 2 theorems, 125 equations, 8 figures)

This paper contains 18 sections, 2 theorems, 125 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometric setting
  4. Brownian motion on the hypersurface
  5. Derivation of surface Dean--Kawasaki equation
  6. Surface Dean--Kawasaki equation on a fixed surface: non-interacting, potential-free case
  7. Intrinsic representation
  8. Fluctuation dissipation relation
  9. Surface Dean--Kawasaki equation for interacting particle system on fixed hypersurface
  10. The surface Dean--Kawasaki equation on a moving, interacting surface.
  11. Discretization
  12. Numerical experiments
  13. Equilibrium test
  14. Transient simulations
  15. Effect of an external potential
  16. ...and 3 more sections

Key Result

Theorem 8

Let $U = 0$ and let $a$ and $\sigma$ be independent of $\mu$. Let $\mathcal{G}_t =\mathcal{F}_t \vee \mathcal{F}^B_{\infty}$, $t\ge 0$, where $\mathcal{F}^B$ is the canonical filtration generated by $B$. We assume that for all $\mathcal{G}_0$-measurable $f \in C^2 (D)$ and $t \geq 0$ there exists a and that strong existence and weak uniqueness hold for eq:eta. Then the law of the martingale solut

Figures (8)

  • Figure 1: Mean of $\rho$. The left panel is the finite volume scheme, the middle panel is a Brownian particles simulation and the right panel is the theoretical result. Note that we have set the range to correspond to the theoretical mean $\pm 10\%$.
  • Figure 2: Mean number of particles in each cell. The left panel is the finite volume scheme, the middle panel is a Brownian particles simulation and the right panel is the theoretical result.
  • Figure 3: Variance of $\rho$. The left panel is the finite volume scheme, the middle panel is a Brownian particles simulation and the right panel is the theoretical result.
  • Figure 4: The variance of the number of particles in each cell. The left panel is the finite volume scheme, the middle panel is a Brownian particles simulation and the right panel is the theoretical result.
  • Figure 5: Time evolution of $\rho$. The range of $\rho$ in the image is restricted to $[0,0.4]$ so the values are clipped particularly at early times. The peak values of $\rho$, left to right, are 0.166, 0.0786, 0.0542 and 0.0457. The height of the peaks on the surface have been scaled by 0.3 so that $\rho$ is not obscured.
  • ...and 3 more figures

Theorems & Definitions (11)

  • Definition 1
  • Remark 2
  • Remark 3: On the interpretation of $W^G$
  • Remark 4
  • Remark 5
  • Definition 6
  • Definition 7: Martingale problem
  • Theorem 8: Weak uniqueness
  • proof
  • Lemma 9
  • ...and 1 more