A simple particle model for a system of coupled equations with absorbing collision term

TL;DR

This paper introduces a particle model for a coupled PDE system involving moving light particles and fixed absorbing obstacles, proving convergence of the particle densities to PDE solutions under specific asymptotic conditions.

Contribution

It presents a novel particle system model for coupled transport-reaction equations with absorbing collision terms and establishes convergence results in the asymptotic regime.

Findings

Particle densities converge to PDE solutions almost surely.

Convergence holds under specific asymptotic conditions on obstacle range and particle number.

The model captures the dynamics of light particles interacting with absorbing obstacles.

Abstract

We study a particle model for a simple system of partial differential equations describing, in dimension $d\geq 2$, a two component mixture where light particles move in a medium of absorbing, fixed obstacles; the system consists in a transport and a reaction equation coupled through pure absorption collision terms. We consider a particle system where the obstacles, of radius $\var$, become inactive at a rate related to the number of light particles travelling in their range of influence at a given time and the light particles are instantaneously absorbed at the first time they meet the physical boundary of an obstacle; elements belonging to the same species do not interact among themselves. We prove the convergence (a.s. w.r.t. the product measure associated to the initial datum for the light particle component) of the densities describing the particle system to the solution of the system of partial differential equations in the asymptotics $ a_n^d n^{-\kappa}\to 0$ and $a_n^d \var^{\zeta}\to 0$, for $\kappa\in(0,\frac 12)$ and $\zeta\in (0,\frac12 - \frac 1{2d})$, where $a_n^{-1}$ is the effective range of the obstacles and $n$ is the total number of light particles.