Transport-limited aggregation.

Diffusion-limited aggregation (DLA) and its variants provide the simplest models of fractal patterns, such as colloidal clusters, electrodeposits, and lightning strikes. The original model involves random walkers sticking to a growing cluster, but recently DLA (in the plane) has been reformulated in terms of stochastic conformal maps. This fruitful new perspective provides the exact Laplacian concentration field (colors) and diffusive flux lines (purple) around a growing DLA cluster (Fig. 1) and other analytical insights. Here, we exploit a conformal-invariance principle for non-Laplacian fields to extend the method to more general, transport-limited growth models, on flat and curved surfaces, such as DLA on a sphere (Fig. 2) and advectiondiffusion-limited aggregation (Fig. 3 and video) in a fluid flow (yellow streamlines). ADLA clusters grow preferentially upstream (to the left), and yet their fractal dimension remains that of DLA, since diffusion dominates over advection at small scales. T. A. Witten and L. M. Sander, “Diffusion-limited aggregation: A kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400 (1981). M. B. Hastings and L. S. Levitov, “Laplacian growth as one-dimensional turbulence,” Physica D 116, 244 (1998). M. Z. Bazant, “Conformal mapping of some non-harmonic functions in transport theory,” Proc. R. Soc. London, Ser. A 460, 1433 (2004). M. Z. Bazant, J. Choi, and B. Davidovitch, “Dynamics of conformal maps for a class of non-Laplacian growth phenomena,” Phys. Rev. Lett. 91, 045503 (2003). FIG. 1. FIG. 2.