Steady oscillations in aggregation-fragmentation processes

TL;DR

This paper discovers and analyzes surprising steady oscillations in aggregation-fragmentation processes, revealing conditions under which the system exhibits stable fixed points or limit cycles depending on kernel parameters and fragmentation intensity.

Contribution

It introduces the existence of steady oscillations in aggregation-fragmentation models with specific kernels, a phenomenon not previously documented in such closed systems.

Findings

Oscillations occur for kernel parameter difference > 1.

Steady oscillations depend on the fragmentation intensity λ.

System transitions from fixed point to limit cycle at a critical λ_c(θ).

Abstract

We report surprising steady oscillations in aggregation-fragmentation processes. Oscillating solutions are observed for the class of aggregation kernels $K_{i,j} = i^{\nu}j^{\mu} + j^{\nu}i^{\mu}$ homogeneous in masses $i$ and $j$ of merging clusters and fragmentation kernels, $F_{ij}=\lambda K_{ij}$, with parameter $\lambda$ quantifying the intensity of the disruptive impacts. We assume a complete decomposition (shattering) of colliding partners into monomers. We show that an assumption of a steady-state distribution of cluster sizes, compatible with governing equations, yields a power law with an exponential cutoff. This prediction agrees with simulations results when $ \theta \equiv \nu-\mu <1$. For $ \theta=\nu-\mu >1$, however, the densities exhibit an oscillatory behavior. While these oscillations decay for not very small $\lambda$, they become steady if $\theta$ is close to two and $\lambda$ is very small. Simulation results lead to a conjecture that for $ \theta <1$ the system has a stable fixed point, corresponding to the steady-state density distribution, while for any $\theta >1 $ there exists a critical value $\lambda_c(\theta )$, such that for $\lambda < \lambda_c(\theta)$, the system has an attracting limit cycle. This is rather striking for a closed system of Smoluchowski-like equations, lacking any sinks and sources of mass.