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.