Branch order and ramification analysis of large diffusion-limited-aggregation clusters

Using a hierarchical ordering scheme, many off-lattice diffusion-limited-aggregation (DLA) clusters containing ${10}^6$ particles are separated into branches of different orders. For each order we measure the number, mass, length, and width of the branches. All of these branch properties depend, with an exponential law, on the branch order. This means that for all of them the ratios between properties of subsequent branch orders are constant. By relating length, width, and mass to each other we find that the length and width of the whole branches both depend, with the same exponents ${\nu }_{\mathrm{?}}$=${\nu }_{\mathrm{\perp }}$=0.60≃β=1/D, on the mass of the branch. By measuring the ramification matrix ${\mathit{R}}_{\mathit{i}\mathit{k}}$ of the clusters we find that the branches and stems are distributed in a self-similar way. Furthermore, we measure the angles enclosed by two branches. We find that this angle saturates to a finite value around 38°. All of these results indicate a statistical self-similarity between branches of different orders. This result is supported by a direct comparison of off-lattice DLA clusters of ${10}^5$, ${10}^6$, and ${10}^7$ particles.