Aggregation of particles which move on deterministic trajectories with fractal dimension two. I. A simple and new model for DLA

The author simulates a new type of aggregation process: particles follow deterministic trajectories. The trajectories have a fractal dimension, d w , of two. d w is defined as S(r) approximately r dw , where S(r) is the total number of sites a particle visits while travelling a distance r. When the growth process is started with a single stationary particle on a square lattice, a large aggregate looks like diffusion-limited aggregation (DLA). The fractal dimension, d f , of the aggregate is nearly equal to 1.7; that of DLA is 1.67+or-0.05. Moreover, the aggregate has axial anisotropy, although it includes less than one thousand particles. Therefore the aggregation belongs to the same universality class of DLA and is supposed to be the highly noise-reduced model for DLA. Because the Laplace equation has no relationship to the aggregation process, the Laplace equation does not seem to be a necessary condition for producing aggregates like DLA.