Distribution of the coalescence times in a system of diffusion-aggregation of particle clusters in one dimension

We consider the stochastic dynamics of a system of diffusing clusters of particles on a finite periodic chain. A given cluster of particles can diffuse to the right or left as a whole and merge with other clusters; this process continues until all the clusters coalesce. We examine the distribution of the cluster numbers evolving in time, by means of a general time-dependent master equation based on a Smoluchowski equation for local coagulation and diffusion processes. Further, the limit distribution of the coalescence times is evaluated when only one cluster survives.