Exact asymptotic solution of an aggregation model with a bell-shaped distribution

TL;DR

This paper derives an exact asymptotic solution for an aggregation model with a specific reaction rate, providing detailed insights into its scaling behavior and differences from earlier approximations.

Contribution

It presents the exact solution of the scaling function for a particular aggregation model, enhancing understanding of its asymptotic behavior and large-aggregate dynamics.

Findings

Exact scaling function expressed as a differential equation solution

Numerical solutions confirm earlier scaling theories

Refined analysis of large-aggregate behavior at fixed times

Abstract

We present in a detailed manner the scaling theory of irreversible aggregation characterized by the set of reaction rates $K(k,l)=1/k+1/l$, as well as a minor generalisation thereof. In this case, it is possible to evaluate the scaling function exactly. By this we mean that it is expressed as the unique solution of an ordinary differential equation with given boundary conditions. This can be solved numerically to high accuracy, making a highly detailed analysis of the scaling behaviour possible. The results confirm the far more general results of earlier work concerning a general scaling theory for so-called reaction rates of Type III. On the other hand, the behaviour of large aggregates at fixed time, that is, not in the scaling limit, which had been up to now analysed in an approximation valid in the limit of small times, can be determined more precisely in this case, and is shown to display subtle differences from the small-time approximation.