Aggregation of Particles with Deterministic Trajectories
The diffusion-limited aggregation (DLA) is currently of major interest in the field of physics of pattern formation, because it is a simple model of several physical phenomena: metal leaves, viscous fingerings, and dielectric breakdown, etc. In all of the phenomena, growth probabilities depend on the gradient of a solution of the Laplace equation. DLA and these physical phenomena are known to show fractal patterns. Hence much effort has been made to obtain the fractal dimension d f of DLA. Remarkably, in some dimension analysis treatments [2,3], though the Laplace equation is not taken into account at all, the result (d f = 5/3) agrees with that of numerical simulations (d f = 1.7) . The fact they make use of is that only trajectories of particles have fractal dimension d w of two. Here we notice a possibility that other trajectories with d w of two can also generate fractal patterns with the same d f as that of DLA. To clear up this point, we consider trajectories which are not diffusive but have d w of 2.