Clustering of branching Brownian motions in confined geometries

TL;DR

This paper analyzes how particles undergoing Brownian motion, reproduction, and death form clusters in confined spaces, using a backward Green's function approach to derive key observables like concentration and correlations.

Contribution

It introduces a backward formalism based on Green's functions to explicitly analyze clustering phenomena in branching Brownian motions within confined geometries.

Findings

Explicit expressions for particle concentration and pair correlation functions.

Demonstration of clustering behavior due to fluctuations in confined systems.

Applicability to physical and biological systems like nuclear reactors and bacterial colonies.

Abstract

We study the evolution of a collection of individuals subject to Brownian diffusion, reproduction and disappearance. In particular, we focus on the case where the individuals are initially prepared at equilibrium within a confined geometry. Such systems are widespread in physics and biology and apply for instance to the study of neutron populations in nuclear reactors and the dynamics of bacterial colonies, only to name a few. The fluctuations affecting the number of individuals in space and time may lead to a strong patchiness, with particles clustered together. We show that the analysis of this peculiar behaviour can be rather easily carried out by resorting to a backward formalism based on the Green's function, which allows the key physical observables, namely, the particle concentration and the pair correlation function, to be explicitly derived.