Reaction Diffusion Simulations
Lecture, Dr. Lawlor
An enormous number of complicated systems are classified as
"reaction-diffusion" systems. The "reaction" part could be a
simple chemical reaction like oxidation during respiration, or a
more complex interaction like "plants uptake nitrogen from soil"
or "predators feed on prey". The "diffusion" part
corresponds to the local neighborhood averaging that happens in a
variety of systems.
A huge number of real things follow a
In each case, the physical process is
driven by the interactions of immediate neighbors, but the effects
spread out across the entire domain. Do keep in mind
diffusion does not need to work equally in all places or
directions--real obstacles like glaciers or mountains simply slow
the diffusion rate, possibly to zero.
- Heat conduction in solids follows a diffusion law, where the
diffusion rate corresponds to the thermal conductivity.
This is due to the random kinetic interactions of the molecules
of the solid with their immediate neighbors. The net
result is heat
uniform materials, with a small sharp hot region blurring out
into a large fuzzy warm zone.
translucent materials like wax or milk, whenever the scattering
rate is high enough that the "mean free path" of a photon is
- Between the cells in living tissue, most of the
millimeter-scale transport of nutrients, waste, or oxygen beyond
the capillary level happens via diffusion. Essentially all
transport inside a cell is via random brownian diffusion.
signals diffuse through
growing tissue, directing the growth of that same tissue.
For example, the Gray
Scott equations represent
- Assuming animals move in random directions, the time average
of animal concentration will diffuse outward.
- Assuming random wind directions, the time average of pollen
transport follows a diffusion law, spreading from the source
plant out in all directions.
- Assuming you talk with friends about random topics, ideas
diffuse between people via a diffusion-type process: if most of
your friends have heard of BitCoin, eventually you probably will
The basic idea in diffusion, or
blurring, is to get rid of the high-spatial-frequency detail in an
image; a "low-pass" filter. See the fourier transform discussion
what I mean by high and low frequencies (small and large details
in an image).
The way blurring gets rid of small
details is by averaging them with nearby pixels--because nearby
pixels will have the same overall low frequency colors, but
different high-frequency details, the details get averaged away
but the low frequencies remain.
You can implement blurring in a 2D
texture in many different ways:
One curious result of the central limit theorem: (almost) any blurring technique
results in a Gaussian blur when applied repeatedly (same image
blurred over and over again). This is good, because it means
we can just repeatedly apply a cheap lumpy blur (with few taps) to
get results similar to a high-quality blur (with exponentially
- Change the mipmap LOD bias: texture2D(tex,coords,bias);.
This will make the mipmapping hardware include additional
blurring by artificially shifting mipmap levels, essentially
sampling a 2^bias pixel region. Clearly, this only works
if you've built mipmaps for your texture, but THREE.js does this
- Draw a few copies of the image slightly shifted from one
another, carefully adjusting the alpha each time so the copies
end up equally weighted onscreen--surprisingly, the alpha values
you need are 1.0, then 1.0/2, then 1.0/3, then 1.0/4, etc.
- Read a few nearby pixels ("filter taps") in a GLSL pixel
shader, and average them together in a single shader pass.
This is a little faster than the multipass rendering method, and
it's easier to write and expand to do other processing, so it's
what I usually do. (Of course, that doesn't mean it's the
- Read a whole bunch of nearby pixels, and weight them by a
Gaussian curve to get Gaussian
blur. This is the default blur performed by most
image editing programs, although it's not totally clear why
folks choose this.
Given both original and blurred images,
you can subtract the two to find just the image details (the high
frequencies alone). This is useful for several interesting
tasks, including HDR Tone Mapping. You can even mix high and low
frequencies from different images for a hideous effect.
A variety of processes can result in a
nonlinear interaction after the diffusion step:
Generally speaking, you can get
interesting behavior whenever you combine blurring (which brings
neighborhoods together) with almost *any* nonlinear reaction
(which drives neighbors farther apart).
- For ice solidification,
the interesting physics is in the interaction of the diffusing
heat and freezing ice fields. Both have thresholds that
drive the growth of solid dendrites.
- Predator-prey interactions can, depending on the parameters,
reach either a steady state, oscillate, or diverge wildly.
The number of animals, even time averaged, cannot ever go
negative. This limits the population swings, but can lead
to localized extinction until new animals diffuse in from
outside. For example, cicada
emergence is almost always delayed by a prime number of years,
such as 13 or 17, to avoid amplifying predator oscillations of
any shorter period.
- Interactions between plant species, such as the ongoing local
battle between spruce and the aspen/birch/cottonwood alliance,
occur mostly at the level of soil chemistry (spruce acidifies
the soil; deciduous trees smother their competitors with leaves)
and wildfires (the secret weapon of the spruce).
- Reactions between chemical signals inside living tissue result
in the formation of differentiated tissue like bones and
muscles, the layered neurons in your cerebral cortex, the spots
on a tiger or the stripes on a zebra (Turing patterns).