Mean-Field Theory for Diffusion-Limited Cluster Formation

A mean-field theory for the diffusion-controlled cluster formation is presented by considering the competition among the different portions of a growing cluster for the incoming diffusive particles. This competition is shown to introduce a screening length which depends inversely on the density of the cluster. The Hausdorff dimensionality $D$ of these clusters is shown to be $\frac{(d^2+1)}{\mathrm{}(d+1\mathrm{})\mathrm{}}$ where $d$ is the Euclidean dimensionality. This result is in excellent agreement with that of the computer simulations of Witten and Sander and of Meakin.