Effects of microscopic dynamics on Brownian coagulation

TL;DR

This paper investigates how microscopic particle dynamics influence Brownian coagulation, deriving collision estimates for different models and establishing the existence of unique solutions to the resulting equations.

Contribution

It introduces and analyzes two models for colloidal particle motion, deriving collision estimates and proving solution uniqueness for the associated coagulation-diffusion equations.

Findings

Collision estimates differ from classical Brownian case

Coagulation kernel and diffusivity are affected by microscopic dynamics

Unique solutions exist for the modified coagulation-diffusion equations

Abstract

We consider two different models for colloidal particles. In the first model, we consider their free motion to be diffusion while in the second model we take it to be integrated Ornstein-Uhlenbeck process. In both models, we derived collision estimates for pairs of particles. In particular, we found that these estimates would be different to the Brownian case even when the particles' free motion is Brownian at macroscopic scales. As a consequence, the coagulation kernel and diffusivity in the coagulation-diffusion equations would also be affected accordingly. We then proved that there exists a unique solution to the coagulation-diffusion equations in these cases under physically reasonable assumptions.