The evolution of the cluster size distribution in a coagulation system

The coagulation equation with kernelK_ij=A+B(i+j)+C_ij and arbitrary initial conditions is studied analytically and a simple expression for the solution is found. For monodisperse initial conditions, we recover the known size distribution expressed in terms of a degeneracy factorN_k, which is determined by a recursion relation. For polydisperse initial conditions, a similar solution form is found, which includes a degeneracy factorN_kl, also determined by a recursion relation. The physical meaning ofN_kl and the recursion relation is given. A method to get explicit expressions forN_k andN_kl is illustrated. Finally, the pre-gel solution is given explicitly and a general method to get the post-gel solution is proposed.