Hello and welcome to unit 9. The topic
of unit 9 is pattern formation. So far in this course the
main results have centered around the phenomenon
of chaos. That's where a deterministic dynamical system
produces bounded orbits that are aperiodic and have
sensitive dependence on initial conditions -
the butterfly effect. But there's a lot more to
dynamical systems than just chaos and unpredictability.
Dynamical systems can produce not only disordered
behavior/unpredictable behavior, but also structures and
patterns and order. For that matter, there's a lot more in
the world than just chaos and unpredictability -
we look all around us and see all sorts of phenomena
that are patterned, have structure, order, or some sort
of organization to them. And so we'll see that just
as simple dynamical systems can produce apparently
random behavior, simple dynamical systems can
produce patterns as well. The structure of this unit
will be a little different than the previous units. I
won't be able to go into as much of the math, because
the math for the example that I want to cover is
a good bit more complex and I don't know how to
do it without a lot of calculus and differential equations.
Nevertheless, I think I can describe the general
phenomena and the basic mathematical ideas at play
without using the full machinery of differential equations.
We'll give it a try. The main phenomena that we'll study
is something known as reaction-diffusion systems, and
these are a class of dynamical systems somewhat
different than we have studied so far that are capable
of producing a wide range of different patterns. These
aren't the only pattern-forming systems at all, there are
many many different ones and there are a lot of
techniques used for studying them. But I think
studying these reaction-diffusion systems is a
nice example to use to illustrate some of what goes
on in pattern-formation and some of the other things
that dynamical systems can do. So we'll get started with
reaction-diffusion in a moment, first, just a side-note,
you may be wondering what happened here. Last night,
there was a territorial dispute between me and one
of my cats, Ancho, he's very gentle but a little bit
clumsy and I think he was trying to get under the covers
climbing over me and he slipped and grabbed ahold of
my face as he was falling. Anyway, it's just a flesh
wound, it will heal quickly and I'm sorry if it's distracting
or looks a little silly. Anyway, let's get started with
reaction-diffusion. Before we talk about reaction-
diffusion equations, I want to talk about plain-
old diffusion. Diffusion is a tendency for particles to
disperse - to become uniform in distribution or
concentration. Let me start with a demonstration
of diffusion. This is a container with just plain, clear,
regular water and this here has some purple food-
coloring in it. I'm going to put a few drops in the
water and what we should observe it that the food-
coloring tends to spread-out. It diffuses and
eventually we will reach a uniform concentration.
So let's give this a try. Here's a drop.
It's falling. But as it's falling, it's spreading out.
It's actually an amazing pattern.
Okay, so the falling motion is actually not diffusion.
But those strands, they should sort of become
thinner and fainter. And you know, they're
actually not quite doing that. Alright, so maybe
this wasn't the world's best demonstration of
diffusion, but it's a really cool pattern and this is
a unit on pattern formation, so we'll
just go with it. I'm going to put this down for a moment
and maybe it will diffuse a little bit. Maybe it needs
a little more time, we'll see. Alright, so in diffusion,
if we had a region, say, where the purple or whatever it
is...we'll just call it purple. Where the purple is highly
concentrated, it will tend to spread-out - to diffuse -
things want to be at a uniform concentration. One way
to think about why diffusion is the way it is - why
diffusion occurs - is that it's just a consequence of
the random motion of the individual particles. If we
imagine the individual purple particles are moving
around in the liquid, just like everything is randomly
moving around in a fluid - a gas or a liquid -
the consequence of all of those random motions is
to tend to smooth-out any non-uniformities. So if
we imagine a situation where we have 100 people
in a room and they're all walking around at random,
we would find it very unlikely if everybody suddenly met
in the middle or in a corner or something. If they were
walking around at random, we would expect that they
would spend roughly equal time everywhere. Similarly
if we started with a whole bunch of people randomly
walking around, maybe 100 drunk people in a room,
we would put them all very close in one location
and we let them go, they're just going
to stagger around at random and we would
expect that they would more or less fill
the room uniformly. So that's how I think about
diffusion. It's just a logical consequence - an
inevitable consequence, really - of the random
motion of a bunch of particles. Another way
to think of it at a somewhat less microscopic level
is simply that chemicals tend to move from regions
of high concentration to lower concentration.
If the concentration is the same, on average,
a chemical won't move. But if there's higher
concentration here and lower concentration there,
there'll be motion - diffusive motion - in this way.
So the main thing is that diffusion is a transport process
which tends to even things out - which tends to smooth-
out all sorts of differences. And I'll describe that
mathematically in a moment. Okay, a little
more about this. So, about 10 minutes have past since
I did that first experiment. And here you can take a
look at it - and I promise I didn't shake this up.
You can still see there's a little bit of some strands
in there, but as I claimed, it just took longer
than I expected, the purple tended to make
itself uniform. And if I waited another 10 minutes
or 20 minutes, it would get more uniform still.
So diffusion is a process - it just happens all by itself -
that tends to smooth-out differences in concentration
of something. And the last thing I guess to say
about this example is that it illustrates that
things diffuse at different rates. I hadn't done this
experiment with this type of food-coloring before.
I just got it at the store a couple of hours ago. Whatever
is in this food coloring, it doesn't diffuse very quickly.
So it took a lot longer than I expected, but it's
still diffusing. We could also imagine something
that diffuses much more quickly - maybe something
that's a little less viscous - the molecules were
lighter. But the main point is that diffusion tends
to even-out differences in concentration and
different quantities, different chemicals, different
things might diffuse at different rates.