Cluster persistence in one-dimensional diffusion--limited cluster--cluster aggregation

TL;DR

This paper investigates how the probability of a cluster remaining unaggregated evolves over time in one-dimensional diffusion-limited cluster-cluster aggregation, revealing different behaviors depending on the size dependence of diffusion.

Contribution

It introduces a mean-field approach to analyze cluster persistence with size-dependent diffusion, highlighting the role of spatial fluctuations for negative gamma values.

Findings

Persistence probability decays differently for gamma ≥ 0 and gamma < 0.

For 0 < gamma < 2, the cluster size distribution is flat and independent of gamma.

Mean-field theory accurately describes the system for gamma ≥ 0.

Abstract

The persistence probability, $P_C(t)$, of a cluster to remain unaggregated is studied in cluster-cluster aggregation, when the diffusion coefficient of a cluster depends on its size $s$ as $D(s) \sim s^\gamma$. In the mean-field the problem maps to the survival of three annihilating random walkers with time-dependent noise correlations. For $\gamma \ge 0$ the motion of persistent clusters becomes asymptotically irrelevant and the mean-field theory provides a correct description. For $\gamma < 0$ the spatial fluctuations remain relevant and the persistence probability is overestimated by the random walk theory. The decay of persistence determines the small size tail of the cluster size distribution. For $0 < \gamma < 2$ the distribution is flat and, surprisingly, independent of $\gamma$.