Approximating particle-based clustering dynamics by stochastic PDEs

TL;DR

This paper demonstrates that stochastic partial differential equations can effectively approximate complex particle-based clustering dynamics, capturing both initial formation and long-term merging of clusters more efficiently than traditional models.

Contribution

It introduces a novel SPDE-based approach to replicate particle clustering effects, providing a computationally efficient alternative to detailed particle simulations.

Findings

SPDEs successfully reproduce clustering dynamics over time.

SPDE approach is computationally efficient for large-scale simulations.

Mean-field models fail to capture long-term cluster merging.

Abstract

This work proposes stochastic partial differential equations (SPDEs) as a practical tool to replicate clustering effects of more detailed particle-based dynamics. Inspired by membrane-mediated receptor dynamics on cell surfaces, we formulate a stochastic particle-based model for diffusion and pairwise interaction of particles, leading to intriguing clustering phenomena. Employing numerical simulation and cluster detection methods, we explore the approximation of the particle-based clustering dynamics through mean-field approaches. We find that SPDEs successfully reproduce spatiotemporal clustering dynamics, not only in the initial cluster formation period, but also on longer time scales where the successive merging of clusters cannot be tracked by deterministic mean-field models. The computational efficiency of the SPDE approach allows us to generate extensive statistical data for parameter estimation in a simpler model that uses a Markov jump process to capture the temporal evolution of the cluster number.