Generalization and the Fractal Dimensionality of Diffusion-Limited Aggregation

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)|^η, 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 [formula omitted], 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 between ordinary DLA and Eden models.