Origin of the spontaneous oscillations in a simplified coagulation-fragmentation system driven by a source

TL;DR

This paper investigates how a simplified coagulation-fragmentation system with a constant monomer source exhibits spontaneous oscillations driven by a fluctuating damping coefficient, linked to the Liénard oscillator mechanism.

Contribution

It introduces a reduced nonlinear differential equation model that explains the emergence of self-oscillations in coagulation-fragmentation systems with mass-dependent rates.

Findings

System exhibits self-oscillations within specific parameter ranges.

Oscillations are due to a fluctuating damping coefficient.

The mechanism is related to the Liénard oscillator dynamics.

Abstract

We consider a system of aggregated clusters of particles, subjected to coagulation and fragmentation processes with mass dependent rates. Each monomer particle can aggregate with larger clusters, and each cluster can fragment into individual monomers with a rate directly proportional to the aggregation rate. The dynamics of the cluster densities is governed by a set of Smoluchowski equations, and we consider the addition of a source of monomers at constant rate. The whole dynamics can be reduced to solving a unique non-linear differential equation which displays self-oscillations in a specific range of parameters, and for a number of distinct clusters in the system large enough. This collective phenomenon is due to the presence of a fluctuating damping coefficient and is closely related to the Li\'enard self-oscillation mechanism observed in a more general class of physical systems such as the van der Pol oscillator.