Diffusion, fragmentation and merging: Rate equations, distributions and critical points

The fundamental processes of diffusion, fragmentation and merging are very common in many physical systems. We study situations where either two or all three of these processes are present in the dynamical evolution of the system. Specifically, we formulate rate equations in terms of the distribution N(x,t) of fragments of linear size x at time t which include a combination of diffusive growth, size fragmentation and fragment coagulation. Our goal is to obtain analytical solutions for N(x,t) in varying situations and in specific limits of x and t.