Steady oscillations in aggregation-fragmentation processes

N. V. Brilliantov, W. Otieno, S. A. Matveev, A. P. Smirnov, E. E. Tyrtyshnikov, and P. L. Krapivsky
Phys. Rev. E 98, 012109 – Published 11 July 2018

Abstract

We report surprising steady oscillations in aggregation-fragmentation processes. Oscillating solutions are observed for the class of aggregation kernels Ki,j=iνjμ+jνiμ homogeneous in masses i and j of merging clusters and fragmentation kernels, Fij=λKij, with parameter λ 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 simulation results when θνμ<1. For θ=νμ>1, however, the densities exhibit an oscillatory behavior. While these oscillations decay for not very small λ, they become steady if θ is close to 2 and λ is very small. Simulation results lead to a conjecture that for θ<1 the system has a stable fixed point, corresponding to the steady-state density distribution, while for any θ>1 there exists a critical value λc, such that for λ<λc, 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.

  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
5 More
  • Received 2 March 2018
  • Revised 14 May 2018

DOI:https://doi.org/10.1103/PhysRevE.98.012109

©2018 American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & Thermodynamics

Authors & Affiliations

N. V. Brilliantov1,2,*, W. Otieno1, S. A. Matveev2, A. P. Smirnov3,4, E. E. Tyrtyshnikov3,4, and P. L. Krapivsky5

  • 1Department of Mathematics, University of Leicester, Leicester LE1 7RH, United Kingdom
  • 2Skolkovo Institute of Science and Technology, Moscow, Russia
  • 3Faculty of Computational Mathematics and Cybernetics, Lomonosov MSU, Moscow, Russia
  • 4Institute of Numerical Mathematics RAS, Moscow, Russia
  • 5Department of Physics, Boston University, Boston, Massachusetts 02215, USA

  • *Corresponding author: nb144@leicester.ac.uk

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand
Issue

Vol. 98, Iss. 1 — July 2018

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review E

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×