Oscillations in Aggregation-Shattering Processes

We observe never-ending oscillations in systems undergoing collision-controlled aggregation and 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}=λ K_{i ,j}, where i and j are cluster sizes, and parameter λ quantifies the strength of shattering. When 0 ≤a <1 /2 , there are no oscillations, and the system monotonically approaches 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 λ 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.