Power law tail in the radial growth probability distribution for DLA

TL;DR

This study combines analytic and numerical approaches to analyze the radial growth probability distribution in DLA clusters, revealing a power law tail and conditions under which the distribution's width scales with the cluster size.

Contribution

It demonstrates that the radial growth probability distribution in DLA has a power law tail and clarifies the conditions for the distribution's width to scale with the cluster's radius of gyration.

Findings

Distribution has a power law tail in the interior.

Exponent α is approximately 2, smaller than previous estimates.

Width of distribution can scale as R_G if α > D_f - 1.

Abstract

Using both analytic and numerical methods, we study the radial growth probability distribution $P(r,M)$ for large scale off lattice diffusion limited aggregation (DLA) clusters. If the form of $P(r,M)$ is a Gaussian, we show analytically that the width $\xi(M)$ of the distribution {\it can not} scale as the radius of gyration $R_G$ of the cluster. We generate about $1750$ clusters of masses $M$ up to $500,000$ particles, and calculate the distribution by sending $10^6$ further random walkers for each cluster. We give strong support that the calculated distribution has a power law tail in the interior ($r\sim 0$) of the cluster, and can be described by a scaling Ansatz $P(r,M) \propto {r^\alpha\over\xi}\cdot g\left( {r-r_0}\over \xi \right)$, where $g(x)$ denotes some scaling function which is centered around zero and has a width of order unity. The exponent $\alpha$ is determined to be $\approx 2$, which is now substantially smaller than values measured earlier. We show, by including the power-law tail, that the width {\it can} scale as $R_G$, if $\alpha > D_f-1$.