Dynamics of a rate equation describing cluster-size evolution (Q2568186)

The evolution of a cluster-size distribution (aggregation process), widely studied in physics, can be described as follows. On a two-dimensional surface monomers (being considered as clusters of size one) are deposited and move randomly. If a monomer meets a cluster of size \(s\) it may join to the cluster, the latter becomes a cluster of size \(s+1\). Inversely, a monomer may leave a cluster of size \(s\), so the latter becomes a cluster of size \(s-1\). It is assumed that \(n\) is the maximal cluster size. If \(N_s(t)\) is the number of the clusters of size \(s\) at time \(t\), and if a dissociation is allowed, then the dynamics of the process is described by the following Smoluchowski-type rate equation (system of equations): \[ \begin{aligned} N_1^\prime &=F+2D_2N_2-2K_1N_1^2+ \sum_{s=3}^n D_s N_s -N_1\sum_{s=2}^{n-1}K_sN_s, \\ N_s^\prime &=K_{s-1}N_1 N_{s-1}+D_{s+1}N_{s+1}-D_sN_s-K_sN_1N_s, \quad 2\leq s\leq n-1, \\ N_n^\prime &=K_{n-1}N_1 N_{n-1}- D_n N_n. \end{aligned} \] Here, \(F>0\) (the rate of deposition of monomers), \(D_s\geq 0\) (the rate at which monomers leave clusters of size \(s\)), \(K_s>0\) (the rate at which monomers join clusters of size \(s\)), and \(N_s^\prime=dN_s(t)/dt\). The positiveness of the solutions of the considered system is proved. For the case without dissociation, and when the monomer deposition stops, the uniqueness and the stability of the equilibrium are proved. Also, for the case with dissociation, but without monomer deposition, and maximum cluster size three, the equilibrium stability is proved. Numerical simulations are described, showing that such a result can be expected also for arbitrary maximum cluster size.