부산대학교 도서관

The canonical steady-state (s-s) particle coalescence model (PCM) in one dimension is solved for the cluster mass distribution. In this model single particles are fed into the system and no clusters are removed. The increase in the total cluster concentration is balanced by the continual coalescence of clusters. The concentration of clusters of any fixed mass reaches a steady state but the large-mass tail of the cluster mass distribution evolves forever. More precisely, the point clusters hop randomly from bond to bond on a linear lattice. Two clusters aggregate when they hop onto the same bond. A $(k+p)$-mer, i.e.\ a cluster of $(k+p)$ particles, is formed if a $k$-mer occupying one bond aggregates with a $p$-mer. The hopping rate of the $k$-mers is $2\alpha$ hops per unit of time, independently of $k$. Monomers are added to the lattice at random at the rate $\beta$ per unit time per bond. Let the probability $P^N_{ij}=$Prob\{exactly $N$ particles occupy the bonds $(ij)$\}. The aggregation of point clusters is described by the equations $(d/dt)P^0_k=2\alpha (P^0_{k-1}-2P^0_k+P^0_{k+1})-\beta kP^0_k$, and $(d/dt)P^N_k=2\alpha(P^N_{k-1}-2P^N_k+P^N_{k+1})+\beta k P^{N-1}_k-\beta kP^N_k$. In the continuous limit, i.e.\ when the lattice spacing $\Delta\to 0$ and the diffusion constant for individual $k$-mers is $D=\alpha\Delta^2$, the steady-state continuous system equations are $0=2D(d^2/dx^2)P^0(x)-\beta^*xP^0(x)$, and $0=2D(d^2/dx^2) P^N(x)+\beta^*xP^{N-1}(x)-\beta^*xP^N(x)$, where the feed rate per unit distance, $\beta^*=\beta/\Delta$, remains constant. The aggregating system reaches a steady state in which $P^N(x)$ is constant.