The Average Shape of Transport-Limited Aggregates

TL;DR

This paper derives a nonlinear equation describing the average shape of transport-limited aggregates, bridging stochastic and continuous models, and explains discrepancies observed in different growth patterns.

Contribution

It introduces a nonlinear integro-differential equation for the average shape of stochastic aggregates, connecting mean-field approximations with actual growth models.

Findings

The derived equation accurately models advection-diffusion-limited aggregation.

Average ADLA clusters differ from exact mean-field solutions due to nonlinear effects.

Results explain discrepancies between DLA clusters and viscous fingers.

Abstract

We study the relation between stochastic and continuous transport-limited growth models, which generalize conformal-mapping formulations of diffusion-limited aggregation (DLA) and viscous fingering, respectively. We derive a nonlinear integro-differential equation for the asymptotic shape (average conformal map) of stochastic aggregates, whose mean-field approximation is the corresponding continuous equation, where the interface moves at its local expected velocity. Our equation accurately describes advection-diffusion-limited aggregation (ADLA), and, due to nonlinear averaging over fluctuations, the average ADLA cluster is similar, but not identical, to an exact solution of the mean-field dynamics. Similar results should apply to all models in our class, thus explaining the known discrepancies between average DLA clusters and viscous fingers in a channel geometry.