Kinetic Aggregation and Real-Space Renormalization Group

This dissertation addresses the formation of large clusters from small particles by irreversible aggregation. Special emphasis is placed on studying diffusion limited aggregation and dielectric breakdown models within the framework of the kinetic real-space renormalization group theory. Based on the combined length and time scale invariance of the clusters, a new kinetic real-space renormalization group (KRG) theory and an improved version (IKRG) for kinetic aggregation are presented. Its application to the diffusion -limited aggregation (DLA) on various lattices and to the dielectric breakdown model on a square lattice are presented. The probabilities relating different local configurations under rescaling are determined from the growth process itself. Recursion relations between the masses of the cluster and its interface on consecutive scales are obtained. The common fractal dimension (D) of the cluster and its interface is calculated. From the subleading eigenvalue a new correction to scaling exponent (D_{i}) is derived for the first time. The values of D derived for lattices in two and three dimensions are in excellent agreement with the most extensive numerical results. The approach becomes exact on a regular fractal lattice with finite ramification. It is also applied to investigate the dependence of the fractal dimension on the non-linearity parameter eta in a model for dielectric breakdown. Moreover the whole multifractal spectrums D(q) of the q-moments of the local growth probability distribution are calculated in all these cases. From them f-alpha curves for the fractal measure of sites with singularities strength alpha, are derived. In addition, a new simple kinetic aggregation called the shortest path aggregation (SPA) is proposed. In this process, particles are released sequentially from randomly chosen sites; each particle attaches to the cluster at the closest site(s) along the luster's perimeter. This model generates random fractals which have dendritic structures. Simulations on a two dimensional square lattice with line seed and particle (point) seed have been studied. We find D = 1.21 +/- 0.01 for the fractal dimension of the clusters in two-dimensional Euclidean space. Initial steps in two analytic approaches to this model are presented.