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

TL;DR

This paper introduces a hybrid simulation algorithm that dynamically switches between finite volume and particle-based methods to efficiently model non-interacting particles under external potentials, ensuring accuracy and positivity.

Contribution

It develops criteria for adaptive switching in hybrid algorithms for particle systems, extending their applicability to low-density regimes with external potentials.

Findings

Effective criteria for switching between methods

Successful dynamic adaptation in 2D and 3D

Accurate simulation of external potential effects

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.