Kang-Redner Anomaly in Cluster-Cluster Aggregation

TL;DR

This paper investigates the asymptotic behavior of the average mass distribution in diffusing aggregating particles, revealing a novel anomaly in cluster-cluster aggregation in dimensions 1 and 2 through analytical and numerical methods.

Contribution

It introduces a renormalization group approach and direct re-summation to analyze the large-time, small-mass behavior, uncovering the Kang-Redner anomaly in cluster-cluster aggregation.

Findings

Derived the asymptotic form of the mass distribution in 1D and 2D.

Identified the Kang-Redner anomaly with specific scaling behavior.

Supported analytical results with numerical simulations in two dimensions.

Abstract

The large time, small mass, asymptotic behavior of the average mass distribution $\pb$ is studied in a $d$-dimensional system of diffusing aggregating particles for $1\leq d \leq 2$. By means of both a renormalization group computation as well as a direct re-summation of leading terms in the small reaction-rate expansion of the average mass distribution, it is shown that $\pb \sim \frac{1}{t^d} (\frac{m^{1/d}}{\sqrt{t}})^{e_{KR}}$ for $m \ll t^{d/2}$, where $e_{KR}=\epsilon +O(\epsilon ^2)$ and $\epsilon =2-d$. In two dimensions, it is shown that $\pb \sim \frac{\ln(m) \ln(t)}{t^2}$ for $ m \ll t/ \ln(t)$. Numerical simulations in two dimensions supporting the analytical results are also presented.