Exactly solvable model for cluster-size distribution in a closed system.

We obtain an exact solution for the cluster-size distributions in a closed system described by nonlinear rate equations for irreversible homogeneous growth with size-linear agglomeration rates of the form K_{s}=D(a+s-1) for all s≥1, where D is the diffusion coefficient, s is the size, and a is a positive constant. The size spectrum is given by the Pólya distribution times a factor that normalizes the first moment of the distribution to unity and zeroes out the monomer concentration at t→∞. We show that the a value sets a maximum mean size that equals e for large a and tends to infinity only when a→0. The size distributions are monotonically decreasing in the initial stage, converting to different monomodal shapes with a maximum at s=2 in the course of growth. The variance of the distribution is narrower than Poissonian at large a and broader than Poissonian at small a, with the threshold occurring at a≅1. In most cases, the sizes present in the distributions are small and hence can hardly be described by continuum equations.