Collective Oscillations in Irreversible Coagulation Driven by Monomer Inputs and Large-Cluster Outputs

We describe collective oscillatory behavior in the kinetics of irreversible coagulation with a constant input of monomers and removal of large clusters. For a broad class of collision rates, this system reaches a nonequilibrium stationary state at large times and the cluster size distribution tends to a universal form characterized by a constant flux of mass through the space of cluster sizes. Universality, in this context, means that the stationary state becomes independent of the cutoff as the cutoff grows. This universality is lost, however, if the aggregation rate between large and small clusters increases sufficiently steeply as a function of cluster sizes. We identify a transition to a regime in which the stationary state vanishes as the cutoff grows. This nonuniversal stationary state becomes unstable as the cutoff is increased. It undergoes a Hopf bifurcation after which the stationary state is replaced by persistent and periodic collective oscillations. These oscillations, which bear some similarities to relaxation oscillations in excitable media, carry pulses of mass through the space of cluster sizes such that the average mass flux through any cluster size remains constant. Universality is partially restored in the sense that the scaling of the period and amplitude of oscillation is inherited from the dynamical scaling exponents of the universal regime.

Figure 1

Comparison of asymptotic approximation, Eq. (5) (solid lines), to the true stationary state of Eq. (1) obtained using our minimization algorithm (symbols). The kernels are given by Eq. (2) with values of $\nu$ and $\mu$ chosen in the nonlocal regime. The cutoff is $\Lambda ={10}^4$.

Figure 2

Main panel: Total mass $M_1(t)$ versus time for different values of $\Lambda$ with $\nu =-\mu =\frac{3}{4}$. Inset: Collapse obtained by rescaling the data according to Eq. (8).

Figure 3

Numerical evolution of a perturbation of the stationary state for $\nu =1$, $\mu =-\frac{7}{8}$, and $\Lambda =300$. Main panel: Amplitude of successive maxima of the perturbation (circles). The solid line is the prediction of linear stability analysis. Inset: The total output mass flux oscillates about its mean value of $J=1$.

Figure 4

Main panel: $\mathrm{Re}[\zeta {]}_{\max }$ for kernels $\mu =-\nu$ plotted as a function of $\nu$ for different values of the cutoff $\Lambda$. Inset: $\mathrm{Re}[\zeta {]}_{\max }$ as a function of $\Lambda$ for different values of $\nu$.

Figure 5

Total mass versus time in a Monte Carlo simulation of the Markus-Lushnikov model with source and sink. Kernel is Eq. (2) with $\mu =-\nu =-0.95$ and $\Lambda =300$.