# DLA archive

This page collects unpublished simulation results on a stochastic field model of diffusion-limited aggregation. The published account of this work appears first, and then some unpublished pictures showing some peculiar attributes.
## Published paper

``Quasi-Continuum Variants of Diffusion-Limited Aggregation,"
Yacov Kantor, T.~A. Witten, and Robin C. Ball, {\sl Physical Review A},
\vol{33} 3341-3351 (1986) pdf 2.5megabytes.
Two diffusive-growth models with an indefinite range of local densities are studied, and their fractal scaling properties are compared with those of diffusion-limited aggregation (DLA). The first model, "penetrable DLA," differs from standard DLA in that each diffusing particle interacts arbitrarily weakly with an aggregated particle. We derive an analytic expression for the growth rates and find by simulation that the weak interaction leaves the DLA scaling properties unmodified. We explain the observed dependence of the density on the interaction strength. Our second model is a stochastic differential equation, without discrete particles. Simulations of this equation show fractal scaling properties consistent with those of DLA. The scaling of this model shows new subtleties; the average density and the spatial correlations are controlled by different exponents. We describe the reasons for this novel behavior.

Fig 5 from this paper shows four variants of the model. Pictures a) and b) show single quadrants with no stochastic noise. Picture a) implements the Laplacian operator on the lattice as a simple nearest neighbor difference, resulting in anisotropic growth. This representation has cubic anisotropy. Each band of symbols represents a (square root of e) range of density. Picture b) uses a corrected representation of the laplacian using nearest and next nearest neighbors, adjusted to have no cubic anisotropy. Figure c) shows a full-disk realization of the stochastic differential equation with maximal noise in the local growth rate. Picture d) represents the same simulation but with densities below a certain threshold represented by dots. Figures e and f show a larger growth implemented on a single quadrant.

## Weak noise

This picture was made with the same program as in the published paper above, but with the noise amplitude reduced by a factor 10.

## Detailed profiles

These pictures are variants of figures e and f above. Letters indicate logarithmic increments of aggregate density. The picture is elongated vertically because the characters are taller than they are wide. The blank line is a page boundary that should be removed. One notices periodic "puffs of density extending at equal intervals along the diagonal. This resembles recently identified droplet emission from other variants of laplacian growth. (need reference)

T. Witten, January 2007