Scaling structure of the growth-probability distribution in diffusion-limited aggregation processes.

Abstract

In nonequilibrium growth such as diffusion-limited aggregation (DLA), the growth-site probability distribution characterizes these growth processes. By solving the Laplace equation numerically, we calculate the growth probability ${P}_{g}(x)$ at the perimeter site $x$ of clusters for the DLA and its generalized version called the $\ensuremath{\eta}$ model, and obtain the generalized dimension $D(q)$ and the $f\ensuremath{-}\ensuremath{\alpha}$ spectrum proposed by Halsey et al. [Phys. Rev. A 33, 1141 (1986)]. It is found that $D(q)$ depends strongly on $q$ and that the $f\ensuremath{-}\ensuremath{\alpha}$ spectrum is continuous. Our results suggest that these growth processes cannot be described by a simple scaling theory with a few scaling exponents. This is in clear contrast to the Botet-Jullien model [Phys. Rev. Lett. 55, 1943 (1985)] which yields equilibrium patterns whose $D(q)$ is constant. It is also found that the information dimension $D(1)$ which represents the properties of the unscreened surface is in good agreement with our recent theory.