Can Smoluchowski equation account for gelation transition?

We revisit the scaling theory of the Smoluchowski equation with special emphasis on the dimensional analysis to derive the scaling ansatz and to give an insightful foundation to it. It has long been argued that the homogeneity exponent λ of the aggregation kernel divides the aggregation process into two regimes (i) λ ≤ 1 nongelling and (ii) λ > 1 gelling. However, our findings contradict with this result. In particular, we find that the Smoluchowski equation is valid if and only if λ < 1. We show that beyond this limit i.e. at λ ≥ 1, it breaks down and hence it fails to describe a gelation transition. This also happens to be accompanied by violation of scaling. PACS number(s): 05.20.Dd,02.50.-r,05.40-y The kinetics of irreversible and sequential aggregation of particles occurs in a variety of physical processes and it is of wide interest in physics, chemistry, biology and in many other disciplines of science and technology [1, 2, 3]. Due to its cross-disciplinary importance, statistical physics has offered a number of models: diffusion limited aggregation (DLA) [4], percolation [5], diffusion limited cluster cluster aggregation (DLCA), ballistic aggregation (BA) [6] etc., to name just a few, and these are systems far from equilibrium. There hardly exists any systematic standard theoretical framework for describing the systems out of equilibrium, which is indeed in sharp contrast to its equilibrium counterpart. However, stochastic processes seem to some extent to rescue this shortcoming and appear to capture a wide class of non-equilibrium phenomena [7]. The dynamics of these processes often evolves in time following some conservation rules and can be expressed in the form of a master equation that constitutes a relation between the spatial and its temporal variables. As far as kinetics of cluster-cluster aggregation is concerned, much of the theoretical understanding is provided by the rate equation approach proposed by von Smoluchowski [8] almost a century ago which reads as ∂tψ(x, t) = −a(x, t)ψ(x, t) + 1