Steady oscillations in aggregation-fragmentation processes

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.

arXiv comment14 pages; 12 figs; extension of arXiv:1708.01604

References in corpus (14)

• Traffic and Related Self-Driven Many-Particle Systems
• Scaling Theory and Exactly Solved Models In the Kinetics of Irreversible Aggregation
• Size distribution of particles in Saturn's rings from aggregation and fragmentation
• Stochastic Coagulation and the Timescale for Runaway Growth
• Oscillations in aggregation-shattering processes
• A model of ballistic aggregation and fragmentation
• Instantaneous Gelation in Smoluchowski's Coagulation Equation Revisited
• Stationary mass distribution and nonlocality in models of coalescence and shattering
• Chiral and chemical oscillations in a simple dimerization model
• Universality of temperature distribution in granular gas mixtures with a steep particle size distribution
• Phase Transitions in Systems with Aggregation and Shattering
• Driven Brownian coagulation of polymers
• Universality properties of steady driven coagulation with collisional evaporation
• Analysis of Clumps in Saturn's F Ring from Voyager and Cassini

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• Dual-moon forced dynamics and nonlinear aggregation in Saturn's F ring: From quasi-periodicity to modulated oscillations