A nonlinear least squares method for the inverse droplet coagulation problem

TL;DR

This paper introduces a nonlinear least squares approach to infer coalescence rates in the Smoluchowski equation, focusing on homogeneous functions, with applications across physics and science.

Contribution

It develops a novel method to determine the coalescence kernel's scaling exponents and function from particle distribution data, simplifying the inverse problem.

Findings

Effective in polymer physics, cloud science, and astrophysics applications.

Handles stationary and self-similar particle distributions.

Demonstrates success in practical inverse problems.

Abstract

If the rates, $K(x,y)$, at which particles of size $x$ coalesce with particles of size $y$ is known, then the mean-field evolution of the particle-size distribution of an ensemble of irreversibly coalescing particles is described by the Smoluchowski equation. We study the corresponding inverse problem which aims to determine the coalescence rates, $K(x,y)$ from measurements of the particle size distribution. We assume that $K(x,y)$ is a homogeneous function of its arguments, a case which occurs commonly in practice. The problem of determining, $K(x,y)$, a function to two variables, then reduces to a simpler problem of determining a function of a single variable plus two exponents, $\mu$ and $\nu$, which characterise the scaling properties of $K(x,y)$. The price of this simplification is that the resulting least squares problem is nonlinear in the exponents $\mu$ and $\nu$. We demonstrate the effectiveness of the method on a selection of coalescence problems arising in polymer physics, cloud science and astrophysics. The applications include examples in which the particle size distribution is stationary owing to the presence of sources and sinks of particles and examples in which the particle size distribution is undergoing self-similar relaxation in time.