The Kinetic Limit of a System of Coagulating Brownian Particles

Abstract

We consider a random model of diffusion and coagulation. A large number of small particles is randomly scattered in \(\mathbb{R}^d\) at an initial time. Each particle has some integer mass and moves as a Brownian motion whose rate of diffusion is determined by that mass. When any two particles are close, they are liable to combine into a single particle that bears the mass of each of them. The range of interaction between pairs of particles is chosen so that a typical particle is liable to interact with a unit order of other particles in a unit of time. We determine the macroscopic evolution of the system, in any dimension d ≧ 3. The density of particles evolves according to the Smoluchowski system of partial differential equations, indexed by the mass parameter, in which the interaction term is a sum of products of densities. Central to the proof is the task of establishing the so-called Stosszahlansatz, which asserts that, at any given time, the presence of particles of two given masses at any given point in macroscopic space is asymptotically independent, as the initial number of particles is taken to be high. Nonetheless, there is, in a microscopic region about each particle, a reduced probability of finding another particle. Determining this deficit precisely is necessary in computing the coefficients appearing in the interaction terms of the Smoluchowski partial differential equation.