Steady oscillations in aggregation-fragmentation processes

Finite-size effects in addition and chipping processes

We investigate analytically and numerically a system of clusters evolving via collisions with clusters of minimal mass (monomers). Each collision either leads to the addition of the monomer to the cluster or the chipping of a monomer from the cluster, and emerging behaviors depend on which of the two processes is more probable. If addition prevails, monomers disappear in a time that scales as lnN with the total mass N≫1, and the system reaches a jammed state. When chipping prevails, the system remains in a quasistationary state for a time that scales exponentially with N, but eventually, a giant fluctuation leads to the disappearance of monomers. In the marginal case, monomers disappear in a time that scales linearly with N, and the final supercluster state is a peculiar jammed state; i.e., it is not extensive.

Hopf bifurcation in addition-shattering kinetics

In aggregation-fragmentation processes, a steady state is usually reached. This indicates the existence of an attractive fixed point in the underlying infinite system of coupled ordinary differential equations. The next simplest possibility is an asymptotically periodic motion. Never-ending oscillations have not been rigorously established so far, although oscillations have been recently numerically detected in a few systems. For a class of addition-shattering processes, we provide convincing numerical evidence for never-ending oscillations in a certain region U of the parameter space. The processes which we investigate admit a fixed point that becomes unstable when parameters belong to U and never-ending oscillations effectively emerge through a Hopf bifurcation.