Oscillations in aggregation-shattering processes

Abstract

We observe never-ending oscillations in systems undergoing aggregation and collision-controlled shattering. Specifically, we investigate aggregation-shattering processes with aggregation kernels K_{i,j} = (i/j)^a+(j/i)^a and shattering kernels F_{i,j}=λK_{i,j}, where i and j are cluster sizes and parameter λquantifies the strength of shattering. When 0<a<1/2, there are no oscillations and the system monotonically approaches to a steady state for all values of λ; in this region we obtain an analytical solution for the stationary cluster size distribution. Numerical solutions of the rate equations show that oscillations emerge in the 1/2<a<1 range. When the shattering rate is sufficiently large oscillations decay and eventually disappear, while for λ<λ_c(a) oscillations apparently persist forever. Thus never-ending oscillations can arise in closed aggregation-shattering processes without sinks and sources of particles.

5+5 pages, 6+1 figures

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• Finite Size Effects in Addition and Chipping Processes
• The role of energy in ballistic agglomeration
• Origin of the spontaneous oscillations in a simplified coagulation-fragmentation system driven by a source
• Stability condition of the steady oscillations in aggregation models with shattering process and self-fragmentation
• The role of zero-clusters in exchange-driven growth with and without input
• Explicit solutions to some fragmentation equations with growth or decay
• Gelation in input-driven aggregation
• Tensors in modelling multi-particle interactions