A Hybrid Algorithm for Systems of Non-interacting Particles with an External Potential

TL;DR

This work tackles simulating non-interacting Brownian particles under an external potential by leveraging the Dean-Kawasaki SPDE and its regularized variants. It introduces a hybrid SPDE-particle algorithm (AMAR-inspired) that dynamically switches between finite-volume SPDE discretization and particle-based dynamics in low-density regions, using higher-order statistics to trigger refinement. The method is extended to multiple spatial dimensions with adaptive patches and is demonstrated on cases without and with external potentials, highlighting improvements in positivity preservation and higher-order statistics fidelity. The approach holds promise for scalable, accurate simulations of particle systems in regimes where local densities become small and rare events are relevant, with potential extensions to interactions and complex geometries.

Abstract

Our focus is on simulating the dynamics of non-interacting particles including the effects of an external potential, which, under certain assumptions, can be formally described by the Dean-Kawasaki equation. The Dean-Kawasaki equation can be solved numerically using standard finite volume methods. However, the numerical approximation implicitly requires a sufficiently large number of particles to ensure the positivity of the solution and accurate approximation of the stochastic flux. To address this challenge, we extend hybrid algorithms for particle systems to scenarios where the density is low. The aim is to create a hybrid algorithm that switches from a finite volume discretization to a particle-based method when the particle density falls below a certain threshold. We develop criteria for determining this threshold by comparing higher-order statistics obtained from the finite volume method with particle simulations. We then demonstrate the use of the resulting criteria for dynamic adaptation in both two- and three-dimensional spatial settings in the absence of an external potential. Finally we consider the dynamics when an external potential is included.

Figures (11)

• Figure 1: Sketch of the finite volume mesh in two dimensions. Number density is specified at cell center indicated with a circle. Fluxes are defined at edges denoted with squares.
• Figure 2: Sketch of hybrid algorithm in two dimensions. The red-shaded region correspond to the particle region; the blue-shaded region indicates boundary cells needed to advance particle region. The dark blue triangles in the blue-shaded region are generated probabilistically to provide a particle configuration consistent with the local SPDE solution. Arrows indicate particles that cross the boundary of the particle region during the time step that are used to compute particle fluxes into and out of the particle region. Here, the dark blue triangles connected to light blue circles represent particles entering the particle region and the dark red squares connecting to light red diamonds represent particles leaving the particle region.
• Figure 3: PDFs of particle, finite volume and Gaussian numerical methods at low number densities. Left is 5 particles per cells; right is 1 particle per cell. With five points per cell 1.3% of the cells are negative for Gaussian algorithm and 0.2% are negative for the finite volume algorithm. With one point per cell 15.9% of the cells are negative for Gaussian algorithm and 12.3% are negative for the finite volume algorithm.
• Figure 4: Skewness and kurtosis as a function of particles per cell. We have also included skewness and kurtosis for the Poisson distribution.
• Figure 5: Ensemble average of dynamics of particles diffusing into a void region. (a) Mean, (b) Variance, (c) Skewness, and (d) Kurtosis.