# Coalescing particle systems and applications to nonlinear Fokker-Planck equations

## TL;DR

This paper introduces a stochastic particle system with singular interactions, relating it to nonlinear Fokker-Planck equations, and develops an efficient numerical method to simulate collision dynamics and singularity formation in models like Keller-Segel.

## Contribution

It presents a novel particle system model with inelastic collisions and a collision-detection numerical method that avoids pairwise computations, linking microscopic dynamics to nonlinear PDEs.

## Key findings

- The numerical method effectively detects collisions without pairwise calculations.
- Hydrodynamic limit of the particle system solves nonlinear Fokker-Planck equations.
- Numerical simulations capture finite-time singularities and blow-up behavior in Keller-Segel models.

## Abstract

We study a stochastic particle system with a logarithmically-singular inter-particle interaction potential which allows for inelastic particle collisions. We relate the squared Bessel process to the evolution of localized clusters of particles, and develop a numerical method capable of detecting collisions of many point particles without the use of pairwise computations, or very refined adaptive timestepping. We show that when the system is in an appropriate parameter regime, the hydrodynamic limit of the empirical mass density of the system is a solution to a nonlinear Fokker-Planck equation, such as the Patlak-Keller-Segel (PKS) model, or its multispecies variant. We then show that the presented numerical method is well-suited for the simulation of the formation of finite-time singularities in the PKS, as well as PKS pre- and post-blow-up dynamics. Additionally, we present numerical evidence that blow-up with an increasing total second moment in the two species Keller-Segel system occurs with a linearly increasing second moment in one component, and a linearly decreasing second moment in the other component.