Oscillations in aggregation-shattering processes

TL;DR

This paper investigates oscillatory behaviors in aggregation-shattering systems, revealing conditions under which persistent oscillations occur or decay, and provides analytical and numerical insights into these dynamics.

Contribution

It introduces a model with specific kernels and identifies parameter regimes where oscillations emerge or vanish, including conditions for never-ending oscillations.

Findings

Oscillations occur for 1/2<a<1 in the model.

Large shattering rates suppress oscillations.

Persistent oscillations can exist without external particle sources.

Abstract

We observe never-ending oscillations in systems undergoing aggregation and collision-controlled shattering. Specifically, we investigate aggregation-shattering processes with aggregation kernels K_{i,j} = (i/j)^a+(j/i)^a and shattering kernels F_{i,j}=\lambda K_{i,j}, where i and j are cluster sizes and parameter \lambda quantifies the strength of shattering. When 0<a<1/2, there are no oscillations and the system monotonically approaches to a steady state for all values of \lambda; in this region we obtain an analytical solution for the stationary cluster size distribution. Numerical solutions of the rate equations show that oscillations emerge in the 1/2<a<1 range. When the shattering rate is sufficiently large oscillations decay and eventually disappear, while for \lambda<\lambda_c(a) oscillations apparently persist forever. Thus never-ending oscillations can arise in closed aggregation-shattering processes without sinks and sources of particles.