Solutions to the mean-field equations of branchless diffusion-limited aggregation

A problem with the quasistatic mean-field equations for branchless diffusion-limited aggregation, which were given and solved by Cates [Phys. Rev. A 34, 5007 (1986)], is pointed out and resolved. In particular, it is shown that the exponent {γ} describing the time dependence of the growing aggregate (which was set infinite by Cates) equals 1/2 on the mean-field level. An approximate analytical solution to the full mean-field equations is presented and compared with its numerically exact counterpart. Properties of the aggregate boundary as described by the exact solution are derived analytically. It is shown that scaling of the aggregate density at all times implies that the random-walker density cannot satisfy simple constant (nonzero) density or flux boundary conditions at infinity.