Generalization and the Fractal Dimensionality of Diffusion-Limited Aggregation

Abstract

Diffusion-limited aggregation (DLA) model is generalized to incorporate the dielectric breakdown model proposed by Niemeyer et al. , and the new simulation method is proposed. While a growing cluster is still in the diffusion (Laplace) field, the local growth probability at a perimeter site P ps of the cluster is now given by p g ( P ps )∼ | ∇φ( P ps ) | n , where φ( P ) is the probability of finding at a point P a random walker launched far away from the cluster. Ordinary DLA corresponds to η=1. Based on the theory of DLA proposed by Honda et al. , the fractal dimension d f for this generalized DLA is derived as d f ={ d s 2 +η( d w -1)} / { d s +η( d w -1)}, where d s is the dimension of space in which aggregation processes take place and d w is the fractal dimension of random walker trajectory. Both d s and d w are allowed to take any number larger than or equal to one. This formula is also applicable to Eden model (η=0) correctly, which means that the generalized DLA model naturally bridges a gap betw...