Brownian coagulation and a version of Smoluchowski's equation on the circle

TL;DR

This paper models particles diffusing and coagulating on a circle, deriving a spatially inhomogeneous version of Smoluchowski's equation and proving the uniqueness of solutions in this setting.

Contribution

It introduces a stochastic particle system on the circle and rigorously connects it to a novel inhomogeneous coagulation equation, establishing uniqueness of solutions.

Findings

Derivation of a spatially inhomogeneous mass flow equation

Proof of uniqueness for the derived coagulation equation

Convergence of the stochastic system to the solution of the equation

Abstract

We introduce a one-dimensional stochastic system where particles perform independent diffusions and interact through pairwise coagulation events, which occur at a nontrivial rate upon collision. Under appropriate conditions on the diffusion coefficients, the coagulation rates and the initial distribution of particles, we derive a spatially inhomogeneous version of the mass flow equation as the particle number tends to infinity. The mass flow equation is in one-to-one correspondence with Smoluchowski's coagulation equation. We prove uniqueness for this equation in a broad class of solutions, to which the weak limit of the stochastic system is shown to belong.