>>Dr. John Cahn: The first one is a group called...what we call
dendrites; it's how alloy metals solidify.
But it shows up in snowflakes, it shows up in frost
on the windshield, on patterns that occur
on new galvanized iron pails;
it's just a very common phenomenon.
The other one is segregation where like things come together.
And this occurred very importantly in metals;
it's what makes them strong and hard.
But it also occurred in the early universe.
After the big bang, once things began to cool down enough
that gravity began to be more important
than the random motion due to the high temperature,
the early universe began to clump up
and we still see this structure.
It is applied widely to things like population dynamics
where you get segregation in cities where people move
in the direction of where they're more comfortable;
where their neighbors are like them.
And it leads to very interesting patterns, geometric patterns,
and how they develop in time and when they develop.
So that's a whole thing.
Then another topic is really a study
of surfaces near a critical point
where the surface has become very broad and diffuse.
And this has led to an amazing insight into...
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...to the spreading of liquids; a thing that we call wetting
over surfaces where we can get a liquid to spread
over a surface without detergent.
And this is coming into use now in the use of carbon dioxide
at high pressure where the liquid carbon dioxide has a very
very low surface tension and spreads and can be used
for dry cleaning without the use of very poisonous chemicals
like carbon tetrachloride that had been used in the past.
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My father had been a young lawyer,
and he had taken cases...he had brought suit
against the Nazi's before they were in power
because they were really street gangs
that were doing outrageous things.
And as soon as they took power, he took the trolley car
from his home to his office and was met by a fellow lawyer
at the street car stop and said, "Don't go to your office,
the SS is waiting for you."
So we fled right away, and we were lucky because people
who fled in 1933 had a much better chance of surviving
than those who waited until later.
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Early in my career I had become aware of the fact
that the theory of diffusion was inadequate.
In diffusion, you have a mathematical equation
that describes how density of a particular material smoothes
out if you have concentrations where the matter is more dense
and places where it's less dense; that the matter diffuses
from the regions of high density to low density
and the end result is that everything
in time becomes quite uniform.
But there were situations in which matter was attracted
to the regions where the density
of that particular component was higher.
So that diffusion instead of going from regions
of high density to low density, went the other way.
And the mathematical equation for that, the diffusion equation
for that situation, had no solution.
And yet I felt it was a very common thing.
It was certainly something that we saw in metals.
And the discovery that I made was a modification
to the diffusion equation, which is now known
as the Cahn-Hilliard equation, which deals with this problem.
And it deals with the case where material diffuses spontaneously
from low density to nearby regions of high density.
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There had been a mathematics of diffusion and this is
for the problem of, let's say, put a drop of milk in water
and it gradually spreads out and in time,
it becomes quite uniform.
There were opposite phenomena where instead
of the spreading out, which is part of this phenomenon,
you have a clumping together like if the milk curdles,
if there was acid in there.
And the equations of diffusion
for this particular situation gave no solution.
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They are to a layman, they are indeed very simple,
but they hadn't been studied.
They are what the mathematicians call
"nonlinear fourth order partial differential equations" and each
of these words "nonlinear," "partial,"
and "fourth order" were part of their charm
for the mathematicians.
Here was an equation.
They had not studied many equations of this type,
and they knew very little
about what interesting things would happen with them.
And the equation that I wrote down is the simplest
of those kinds of equations that you could possibly write down,
and yet it has proved the subject of 50 theses
and PhD theses in mathematics alone.
It's a fascinating equation to mathematicians.
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Well, you know, having been the father of this equation,
it really is very similar to being a father
because it has a life of its own
and I'd long ago lost the ability to keep guiding it.
It's just going and I'm very proud
of it, but it's on its own.
This equation that I wrote down was amazingly simple.
I had difficulty solving it; I'm not a mathematician.
I knew that it would explain the phenomenon in metallurgy
that I was looking for, but it turned
out to be an extremely rich equation.
There were a large number of phenomena
that could be explained by it.
There were also a large number of different properties,
quite general properties that mathematicians could work on
and have been working for decades now,
and they tell me they still have several decades to go.
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Oh, I think forever.
I mean as long as there's mathematics,
there are standard equations that keep coming up
and this is going to be one of them.
It's a little bit more complicated, but you know,
let me give you an example: price times the number
of things you buy is equal to the cost.
The same equation is the rate
at which you travel times the number of...the time
that you travel is the distance covered.
Ohm's law is simple.
Mathematically they're all the same;
you multiply two things together to get something else.
And so this equation is, at its deepest level, is a very simple,
generic equation for phenomena that occur all the time,
and so I think this equation will join those equations
which kids will learn in their physics classes,
or in their economics classes, or in their sociology classes.
And as we learn more about its properties
and how they are solved, and, of course,
the computer will give you numerical solutions quite
easily, it will persist.
It's not something that will disappear.