Transparent diffusion-limited aggregation in one dimension

Abstract

We introduce a transparent version of diffusion-limited aggregation (DLA) wherein the walkers stick to the aggregate with probability μ less than unity and are allowed to penetrate the aggregate. We study this model in one spatial dimension. We show that the ensemble average of this process is in the steady state only when one takes the average in the instuntaneous frame of the lead particle. We calculate this average exactly for μ near one. For small sticking probability, we introduce a mean-field treatment. As opposed to the standard mean-field treatments of DLA, our version, based on our novel ensemble average, shows no singular behaviour and no need for ad-hoc cut-off procedures. This mean-field treatment in quantitatively accurate as long as μ is not too small and qualitatively correct for all μ. We discuss the quantitative breakdown of the mean-field treatment for very small μ and directions towards improvement. The implications of these findings and future directions for research are also discussed.