Scaling theory for systems with instantaneous gelation: partial results

We study a system of kinetic equations describing irreversible aggregation-the so-called Smoluchowski equations-for a peculiar case in which even starting from initial conditions for which only a monomer is present, at arbitrarily small times an infinite cluster appears and causes the total mass of the system to decrease. This phenomenon, known as instantaneous gelation, is known to arise in cases where the reaction rates between large clusters increase strongly as the cluster size goes to infinity. We resolve two puzzles linked to this behaviour: first, from general results it is well known that a power-law behaviour must form instantly at large masses for arbitrarily small times. We show that the small time behaviour can be analysed for these systems, but this must be done in an essentially different manner than in the regular case. The correct treatment provides a mechanism for the instantaneous appearance of a power-law tail at large masses in the aggregate size distribution. We additionally show that instantaneous gelation is connected to the presence of an essential singularity in the solution and also show how, from the numerical study of the formal power series describing the solution, the nature of the singularity can be guessed and the time dependence of the mass scale for the onset of the power-law behaviour conjectured.