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 $F_{i,j}=\lambda K_{i,j}$, where $i$ and $j$ are cluster sizes, and parameter $\lambda$ quantifies the strength of shattering. When $0\le a<1/2$, there are no oscillations, and the system monotonically approaches a steady state for all values of $\lambda$; 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 $1/2<a\le 1$ range. When $\lambda$ is sufficiently large, oscillations decay and eventually disappear, while for $\lambda <{\lambda }_c(a)$, oscillations apparently persist forever. Thus, never-ending oscillations can arise in closed aggregation-shattering processes without sinks and sources of particles.

Figure 1

Aggregation (a) and shattering (b) of clusters.

Figure 2

Comparison of the steady-state numerical solution of Eq. (3) for the kernel (4) with $a=0.05$, $\lambda =0.003$ (top) and $a=0.1$, $\lambda =0.02$ (bottom) with the analytical steady-state solution Eq. (8).

Figure 3

Time dependence of the cluster density $N(t)$ for $a=0.7$ (top) and $a=0.75$ (bottom) and different $\lambda$. For all these systems, damped oscillations are found that tend to a steady state. The oscillations become more pronounced with increasing $a$ and decreasing $\lambda$.

Figure 4

Time dependence of the cluster density $N(t)$ for $a=0.9$ and different $\lambda$. For small $\lambda <{\lambda }_c(a)$, stable oscillations emerge.

Figure 5

Stationary oscillations of the aggregate concentrations for $a=0.95$ (top) and $a=1$ (bottom) and $\lambda <{\lambda }_c(a)$. The shape of the oscillations depends on the aggregates’ size (see the Supplemental Material for details). Inset: The amplitude of the oscillations decreases with the size $k$; the dashed line is $k^{-7/4}$.

Figure 6

Limit cycle for the steady-state oscillations in terms of $n_1(t)$ and $n_2(t)$ for $a=0.95$ and $\lambda =0.005$. For the total mass density $M=5.5$, the initial conditions are monodisperse $n_k=M{\delta }_{1,k}$ and stepwise (5). For $M=3$, the initial and current values of $n_1(t)$ and $n_2(t)$ are rescaled accordingly. The relaxation to the unique limit cycle is clearly visible.