Anisotropic growth of diffusion-limited aggregates

Abstract

Two problems in the analysis of diffusion-limited aggregation are studied here: the isotropy of the growing aggregate and the meaning of the large-$N$ limit as a practical matter. The isotropy of the aggregate is quantified by the ratio of the radii of gyration of the aggregate about its principal axes. The ratio varies from ∼0.70 for $N=200\mathrm{} \mathrm{to} \sim 0.90$ for $N=50\mathrm{}000$. The next particle capture radius is directionally dependent with respect to the principal axes and scales with the respective radii of gyration. A lower bound is obtained for the size $N$ necessary for the emergence of a single dominant length scale by taking this as the size for which the radii of gyration are equal. In $d=2\mathrm{}$ the lower bound is $N\simeq 1.5\times {10}^6$; in $d=3\mathrm{}$ it is $N\simeq 8\times {10}^5$.