The Coagulating Brownian Particles and Smoluchowski's Equation

#### F. Rezakhanlou

2006, v.12, Issue 2, 425-445

ABSTRACT

We study a model of mass-bearing coagulating Brownian particles in $R^d$. Coagulation occurs when two particles are within a distance of order $\epsilon$. We assume that a particle of mass $n$ has a radius of size $r=n^{\chi}$. The initial number of particles $N$ is of order $|\log \epsilon|$ when the dimension is $d=2$ and $\epsilon^{2-d}$ when $d\ge 3$. Under suitable assumptions on the initial distribution of particles and the microscopic coagulation propensities, we show that the macroscopic particle densities satisfy a Smoluchowski-type equation. This was shown in [A.M. Hammond and F. Rezakhanlou, The kinetic limit of a system of coagulating Brownian particles. Preprint 2005, arXiv:math.PR/0408395. To appear in Arch. Ration. Mech. Anal.] and [A.M. Hammond and F. Rezakhanlou, Kinetic limit for a system of coagulating planar Brownian particles. To appear in J. Stat. Phys.,2006] when $\chi=0$. In this article we review the arguments of these papers and extend the result to the case $\chi\in [0,1/(d-2))$.

Keywords: interacting particle systems,Brownian motion,Smoluchowski'sequation,coagulation