TY - JOUR

T1 - Steady oscillations in aggregation-fragmentation processes

AU - Brilliantov, N. V.

AU - Otieno, W.

AU - Matveev, S. A.

AU - Smirnov, A. P.

AU - Tyrtyshnikov, E. E.

AU - Krapivsky, P. L.

PY - 2018/7/11

Y1 - 2018/7/11

N2 - 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 θ≡ν-μ<1. For θ=ν-μ>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.

AB - 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 θ≡ν-μ<1. For θ=ν-μ>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.

UR - http://www.scopus.com/inward/record.url?scp=85050157717&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.98.012109

DO - 10.1103/PhysRevE.98.012109

M3 - Article

C2 - 30110817

AN - SCOPUS:85050157717

VL - 98

JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

SN - 1539-3755

IS - 1

M1 - 012109

ER -