The idea of aggregation structures is that a seed shape then grows by accumulating a dissolved resource from the medium that it is in. An example is a sponge or other coral, growing from the accumulation of calcium in the sea water.
More geometrically, we can think of an initial set in 2D or 3D space, which is a sink, and a surrounding distant source continues to replenish the space within it, modelling the effect of a large sea. Point sources or other shaped sources are possible, each producing the resource at various rates.
The resource is usually modelled as diffusing in the medium, the simplest diffusion is Laplacian, where the curvature of resource concentration is everywhere zero. This form of aggregation is therefore generally termed Laplacian growth (or Laplacian instability).
Now we have three parameters describing the process. This is best seen by thinking of the resource as millions of individual grains. Each grain is emitted from the source, and follows a random walk until it touches the aggregation structure, where it can stick and form part of the new structure. I'll look at each parameter in turn:
1. Determinism n
This describes how stochastic or deterministic the system is, with two extremes:- Stochastic: The particles can be released one at a time, and so there is always a larger structure for each subsequent particle. This is the Diffusion Limited Aggregation method: A similar method is called the Dialectric Breakdown Model, and treat the problem as an equivalent electrical one. It constrains the charge on the aggregation structure to be 0 (and positive on the source structures), then solves the Laplacian field for the (conductive) space in between. The probability of the particle attaching to any point on the structure is proportional to the gradient in the Laplacian at that point.
The results of DBM look very similar to DLA and so they should, as far as I can see this is the same algorithm. The Laplacian field is exactly the probability density of particles in the medium due to random walks.
- Deterministic: The particles are all laid down at once, layer by layer. This means that in any one layer, the aggregation for each particle is independent of the other particles. We can implement it like the DBM: instead of using the gradient of the Laplacian on the border as a probability distribution, you use it as the layer's thickness. This second extreme gives smooth results. The mathematics apparently gets unsolvable in finite time, but this can be fixed by adding a tiny amount of surface tension (next).
Figure: Laplacian growth: source is the two red dots, the start seed shape is the green half-plane.
These extremes also represent the difference between sequential and concurrent aggregation. We can find any process in between these two extreme options by choosing the number of points n released for each layer. With 1 release being option 1 and infinitely many being option 2.
Here is an image of option 2 iterated to high precision, with a disk as the seed and large ring as the source. You might expect the deterministic approach to add a uniform skin to the seed each iteration, and this is true, but any small inconsistencies escalate in what is called Laplacian instability, leading to a pattern like below:
Here is an image of option 2 iterated to high precision, with a disk as the seed and large ring as the source. You might expect the deterministic approach to add a uniform skin to the seed each iteration, and this is true, but any small inconsistencies escalate in what is called Laplacian instability, leading to a pattern like below:
Figure: showing the 'Laplacian instability' inherent in Laplacian growth.
Now this is starting to look a bit more like the stochastic option, and indeed I suspect that they tend to the same thing. As each is computed at higher and higher precision (and with smaller and smaller particles in the stochastic case), it seems they should converge to the same structure. Indeed this paper suggests that they are, here is a higher resolution comparison:
Notice that the high resolution DLA (50 million particles) starts to mimic the radially outward branching of the Laplacian growth.
Given that these two (probably) tend to the same structure, we could replace the parameter n with a particle size parameter. When it is 0, the result is equal to the deterministic case (with no surface tension).
2. Surface tension p (scale-dependent)
For the deterministic case we can specify a surface tension by setting the charge equal to p times the curvature of the shape boundary. Larger values of p show smoother, fatter fingers in what is now called viscous fingering.
Figure: Viscous fingering by squeezing a viscous fluid between two plates.
We can mimic the same surface tension by taking the charge some small skin width away from the surface, as the probability density in the stochastic case and as the growth speed in the deterministic case.
Figure: increasing surface tension
On a stochastic (grid-based) aggregation, this is mimicked by making the probability of particle attachment proportional to p^(3-N) where N is the number of neighbour cells that are part of the fixed structure. (The 3 changes depending on your neighbourhood size).
Figure: 3D DLA, left: low surface tension (sticks when hits one), right: higher surface tension (images from syntopia's blog).
The surface tension parameter p has units of distance, and fixes a feature scale on the structure, giving non-scale symmetry. However, for larger and larger structures it (should) tend towards the zero surface tension appearance.
3. Directedness d
Directedness appears in the DBM model by raising the distribution of the Laplacian gradient along the border to some power d. For n=1, the effect of increasing d above 1 is that the dendritic aggregation structure has longer lines between branches, and looks more like lightning, and ultimately cracks and then just a straight line. It takes the fractal dimension down from about 1.72 (for normal DBM/DLA) down to 1 as d approaches infinity. On the other extreme, when d is lower than one, the DBM gets denser, ultimately leading to an Eden cluster when d=0.
Figure: left d=0, middle d=1, right d=2
For the n=infinity (deterministic) case, large values of d tend to pointier and less branching fingers in the structure. As p tends upwards, the result seems to tend towards a greedy shortest path from the seed to the nearest point in the source, then onwards to the next nearest point. Giving a sort of spanning tree.
Figure: left d=1, right d=2 gives thinner branches
Figure: d=2, when the right source point is slightly higher, it grows to the nearest point first.
This d parameter has some sort of meaning under the dialectric breakdown model, but it also has meaning in terms of resource particles. It means that aggregation only occurs when p particles hit the surface point at the same time. This is reminiscent of the Reaction Diffusion Systems (particularly the Gray-Scott model), where powers of the density at each pixel correspond to number of simultaneous particles in the reaction. Here it means that you need d particles and one border point in the same pixel to create a new piece of the structure.
