Aggregation Driven by a Localized Source

TL;DR

This paper investigates how monomers aggregate around a localized source, revealing algebraic decay of densities and unique decay exponents in different dimensions, along with the dynamics of cluster growth over time.

Contribution

It introduces a model with mass-independent reaction rates and diffusion coefficients, deriving explicit algebraic decay exponents and analyzing cluster dynamics with a localized source.

Findings

Monomer density decays as a power law with irrational exponents in 2D and 3D.

Total cluster density and number of clusters exhibit specific asymptotic behaviors.

Cluster number grows logarithmically over time.

Abstract

We study aggregation driven by a localized source of monomers. The densities become stationary and have algebraic tails far away from the source. We show that in a model with mass-independent reaction rates and diffusion coefficients, the density of monomers decays as $r^{-\beta(d)}$ in $d$ dimensions. The decay exponent has irrational values in physically relevant dimensions: $\beta(3)=(\sqrt{17}+1)/2$ and $\beta(2)=\sqrt{8}$. We also study Brownian coagulation with a localized source and establish the behavior of the total cluster density and the total number of of clusters in the system. The latter quantity exhibits a logarithmic growth with time.