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PROFESSOR: One more
lecture from me.

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And so remember that on
Thursday we'll be here.

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And Chris Wolverton from Ford
will be lecturing, straight

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live direct from Ford.

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And so you better have
good questions for him

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because he only
lectures about an hour.

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And so we usually--

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in the past, we've had
quite some good discussion

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afterwards, but that's
going to depend on you.

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So what I'm going to do
today is a variety of topics,

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but they all have to do with the
same idea of bigger and faster

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and better--

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not necessarily better.

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So when you deal with
atoms, both because

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of the size and the
time, if you go away,

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especially if you go away
from periodic systems,

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you always run into this problem
that it's too small [INAUDIBLE]

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system, that the
simulation is too short.

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I always like to quote
Professor Marzari on this.

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When people come to
him with nano problems,

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he always says, nano is just
a little too big for us.

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And so I think that's
telling you something

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about the state of what we do.

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So you've actually seen
certain solutions already

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to the sort of size
and time problem.

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Maybe they were fairly obvious.

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But if you think of the time
problem, what I've already

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showed you is how to essentially
coarse grain that away, which

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is it's one thing to coarse
grain over time and space.

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It's another thing, and
it's actually easier

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to almost remove it by,
integrating in a way.

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And when you essentially
integrate time away completely,

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you end up with thermodynamics.

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I mean, thin about it.

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That's what integration over
the partition function is.

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You're essentially assuming
the system has had enough time

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to sample all its
excitations, so we essentially

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integrate over that ensemble.

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You integrate over
all the excitations,

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and you end up with the
thermodynamic functions.

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And the examples that I showed
you with the cluster expansion

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were essentially
examples like that.

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So that part, I would say,
for crystalline systems

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is a solved problem.

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It's not a trivial solution.

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It's still a lot of work,
but it's essentially

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a solved problem.

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I would say most of
the spatial problem

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is solved if you're
going to deal

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with static properties,
static properties and things

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such as elastic behavior.

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You essentially coarse
grain fairly easily

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and end up with
continuum theory.

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The real problem lies in
what sort of happens in

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between here.

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When you want to keep some level
of detail that's not continuum

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or in the time max
is not thermodynamic,

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but you can afford to
stay all the way down

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at the atomistic scale and
the scale of the electrons,

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that's where I think the
real challenge exists.

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And I'll tell you a
little more about that.

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That's when you
break your computer.

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So first, I'm going to
tell you there's obviously

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the brute force approaches,
which is just throw

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more computer power at it.

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And these days the
way you do that is not

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buying faster computers.

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You buy more of them.

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So you have to parallelize.

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Parallelization over space is
obviously the most obvious one.

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If you have a really big system,
you can divide it in chunks.

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OK, you know?

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You'd probably do it a
little more systematic,

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but you can divide
it in chunks and give

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each chunk of atoms to a CPU,
CPU1, CPU2, CPU3, et cetera.

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Of course, that implies that
the coupling between regions

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is not too severe.

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Because when you calculate--

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let's say you're
doing electrodynamics.

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When you calculate forces on
the atoms in the region of CPU1,

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the atoms that live
in the boundary layer

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obviously see forces coming
from atoms in the other regions.

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So how efficient this method
is will depend very much

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on how well you can divide
the sort of spatial origin

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of the forces and the energy.

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So you can already
see that this is

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going to be easier when you do
it with pair potentials, say.

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With pair potentials,
you can directly assign,

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when you have the
force on an atom,

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the component coming
from other atoms.

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You can say the force
on this atom comes--

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I have a piece coming
from the potential

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with that atom, the potential
with that atom, et cetera.

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You'll see that in the
standard approximation,

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for example, the
quantum mechanics,

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this is virtually impossible.

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When we work with
block states, those

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are, by definition, extended
over the whole system, OK?

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So you can't partition
them over pieces of space.

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So to do this sort of
divide and conquer approach

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with quantum mechanics,
you have to go

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to alternative approaches.

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People invest a lot of time
now in just real space grids.

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In theory, the
Schrodinger equation

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is a partial
differential equation.

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Like any other partial
differential equation,

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you should be able to solve
it on a real space grid.

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There's all kinds of
complications that arise,

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but people are doing a lot--
there's a lot of work actually

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being done on that.

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The other is to go to
localized basis sets.

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And you know, maximally
localized Wannier functions

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fall in that category.

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And Professor Marzari
is an expert on that,

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almost essentially invented--

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tight binding approaches.

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So you'll see a lot of work
being done on real space

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methods, which is, in
essence, directed largely

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towards this sort of divide
and conquer approach.

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If you can go a real space
in quantum mechanics,

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you can divide space up.

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So space is maybe, in
the end, not the biggest

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problem we have to deal with.

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Time is a much harder one.

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Because how do you
parallelize over time?

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Time is, by nature, sequential.

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So it's really hard
to do something

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that's in the future when you
don't know the present yet.

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But you'll see in a second--

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well, not in a second, a
little more than a second,

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but later in the lecture.

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There are some approaches
to paralyzing time,

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but they're all, I
would say, much more

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limited in scope
than the methods

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for parallelizing over space.

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So you see time will
be the big problem.

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Time is easy to deal with when
it's either short or very long.

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Because when it's very long, you
essentially integrate it away.

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But in between is where
it tends to get harder.

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So let me give you sort
of a couple of examples

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of ideas that have
been kicked around

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for coarse-graining space.

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And then I'll say a little
more about the time.

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And with time comes
temperature problem.

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So you could imagine, if you
do an impurity calculation--

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so you take a sort of crystal
of these bluish atoms,

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and you stick a big red atom in.

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You want to calculate
the energy [INAUDIBLE]

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with quantum mechanics.

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Well, those shells around the
red atom kind of relax a lot.

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And then as you
sort of go further,

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this is largely an
elastic problem.

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You've put a ball in a
hole that's too small,

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and sort of the system
relaxes around it.

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We tend to sort of brute
force that typically.

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We do quantum mechanics on
bigger and bigger units cells.

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But it's actually
remarkable if you ever

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plot the displacements.

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You will find that these systems
behave remarkably elastically

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in very close shells
to the defect already.

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Often, from the
second, third neighbor,

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if you were to apply
elasticity theory,

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you'd still get it right.

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It's amazing how applicable
elasticity theory

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is at the atomistic level.

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So that gave people the
idea that really when

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they do calculations with broken
symmetry, so around defects,

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maybe you really don't
need all the atoms that we

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tend to put in calculations.

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So the idea that's
been kicked around

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is that let's say that this
is a piece of your simulation

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that's far away from whatever
perturbation you're looking at.

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Maybe it's a grain boundary.

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Maybe it's a dislocation moving.

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Maybe you just
did nuclear fusion

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somewhere in the material.

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But far away, you believe
that the system acts almost

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in a continuum mode.

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And so the idea is
that, what if I kept

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track only of these red atoms?

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Let's say I'm looking at energy
of an elastic displacement

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or even not elastic.

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If I knew how these
four items moved,

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I could probably know something
about how the other atoms move,

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OK?

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So I could interpolate,
say, their displacement

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from the displacement
of the corner atoms.

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So one thing you
can do is say, well,

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if I know how
these atoms move, I

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assume that the energy
of the atoms in between

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is essentially the same as an
energy of a sort of macroscopic

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solid that I deform
with that displacement.

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You can do things in between.

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You can actually
sort of calculate

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the energy of that unit
cell as it's displaced.

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But the whole idea
that essentially you

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start keeping only track of
a limited number of atoms.

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And you can think
of these as nodes.

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So these are called nodal atoms.

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And what happens to all
the atoms you removed--

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you essentially get
by interpolation.

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And once you're happy with
using elasticity theory,

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then essentially
you know the energy

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of that cube, how that changes
as you change the four corners.

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Typically, people use triangles
because people will tend

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to use triangular measures.

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But I wanted to give the
example with a square

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since it's a little simpler.

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So this is essentially the idea
of the quasi continuum method.

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Let's say out here you
have some perturbation,

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something you want to study
at the atomistic level.

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As you go farther and farther
away, you want to coarse grain

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more and more.

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So essentially, you want to
keep less and less nodal atoms

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and get more and
more information

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simply by interpolation.

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So you could say this is coarse
graining at the level of one

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out of four when you're here.

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And then you have a
few regions of that.

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And then you coarse
grain at a higher level.

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So you keep, say, I should
have removed-- sorry,

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these should not be red.

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These should be blue, sorry.

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So you only keep
these four atoms,

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and you sort of
assume that that solid

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undergoes a homogeneous
deformation within that box.

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What you see is that
somewhere in this limit

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you're going to end up
with a finite element

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approaches, where essentially
you do continuum theory.

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And you essentially
say that, if I

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know some displacements of
elements, of the corner points,

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the vertices of elements, I know
how to that element deforms.

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So what you have is essentially
a sort of continuous transition

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here in coarse graining
from our atomistic behavior

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to a finite element approach.

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And people have
put that together.

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And this is called a
quasi-continuum approach.

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This is sort of a
pictorial version of it.

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Let's say this is some boundary
where you do something.

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You have very fine resolution.

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You could say the nodes of
your finite element niche

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are the atoms.

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And then as you go
farther and farther away,

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you coarse grain more and more.

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And you keep less
and less information.

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I'll give you a
reference at the end,

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but the people
who developed this

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were essentially a
fairly small group

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of people, Shenoi,
Tadmor, and Rob Phillips,

250
00:12:43,230 --> 00:12:45,000
and Michael Ortiz at Caltech.

251
00:12:45,000 --> 00:12:47,520
And there's some excellent
review papers about this

252
00:12:47,520 --> 00:12:49,740
if you'd like to
read more about this.

253
00:12:54,010 --> 00:12:56,920
It's now been fairly
well-implemented,

254
00:12:56,920 --> 00:13:02,795
these kind of approaches, mainly
with empirical energy methods.

255
00:13:02,795 --> 00:13:05,170
People have tried to implement
it with quantum mechanics.

256
00:13:05,170 --> 00:13:06,753
And some people have
actually done it,

257
00:13:06,753 --> 00:13:09,700
but it's not all that
effective really.

258
00:13:09,700 --> 00:13:13,780
It's extremely hard to
couple the atomistic region

259
00:13:13,780 --> 00:13:16,060
where you do quantum
mechanics to a sort

260
00:13:16,060 --> 00:13:19,450
of more coarse grained region.

261
00:13:19,450 --> 00:13:22,583
But with potentials and with
things like embedded atom

262
00:13:22,583 --> 00:13:24,250
method, people have
coupled these things

263
00:13:24,250 --> 00:13:25,990
fairly well together.

264
00:13:25,990 --> 00:13:28,690
This is, for example,
a grain boundary--

265
00:13:28,690 --> 00:13:31,900
sorry, a crack impinging
on a grain boundary.

266
00:13:31,900 --> 00:13:34,400
See, this is a symmetric
grain boundary.

267
00:13:34,400 --> 00:13:36,310
And you have the crack
in the middle there.

268
00:13:36,310 --> 00:13:39,610
It comes in, hits
the grain boundary,

269
00:13:39,610 --> 00:13:43,510
and essentially deflects
in the grain boundary.

270
00:13:43,510 --> 00:13:48,070
So that's a simulation
done with quasi-continuum.

271
00:13:48,070 --> 00:13:51,310
You know, if you want to
think about it, what it really

272
00:13:51,310 --> 00:13:54,010
does for you
quasi-continuum, is that it's

273
00:13:54,010 --> 00:13:57,550
a way of getting the
boundary conditions right

274
00:13:57,550 --> 00:14:00,310
on your atomistic simulation.

275
00:14:00,310 --> 00:14:03,310
By slowly integrating
it, essentially

276
00:14:03,310 --> 00:14:08,440
with a continuum theory, you,
first of all, have adaptive--

277
00:14:08,440 --> 00:14:12,910
your boundary conditions or
your atomistic simulation

278
00:14:12,910 --> 00:14:15,460
change during the simulation.

279
00:14:15,460 --> 00:14:18,010
And because you have a
better embedding theory,

280
00:14:18,010 --> 00:14:20,200
you have a better
continuum theory,

281
00:14:20,200 --> 00:14:21,580
they're sort of
more appropriate.

282
00:14:21,580 --> 00:14:23,800
And especially in
mechanical problems,

283
00:14:23,800 --> 00:14:27,670
that is very important to have
the right elastic boundary

284
00:14:27,670 --> 00:14:28,698
conditions.

285
00:14:35,540 --> 00:14:39,100
So let me sort of
position this for you

286
00:14:39,100 --> 00:14:41,230
to show you where
we have the solution

287
00:14:41,230 --> 00:14:44,763
and where we still have
significant problems.

288
00:14:44,763 --> 00:14:46,180
You know, I would
essentially say,

289
00:14:46,180 --> 00:14:48,970
when we work purely at
the microscopic scale,

290
00:14:48,970 --> 00:14:52,960
we have most things
under control.

291
00:14:52,960 --> 00:14:55,810
We know well how to deal
with the energetics.

292
00:14:55,810 --> 00:14:58,162
We fairly well how to
deal with the dynamics.

293
00:14:58,162 --> 00:14:59,620
Because for pretty
much everything,

294
00:14:59,620 --> 00:15:02,980
except things like hydrogen, you
can use Newtonian dynamics even

295
00:15:02,980 --> 00:15:04,720
on the atomistic scale.

296
00:15:04,720 --> 00:15:06,220
You could argue we
don't really know

297
00:15:06,220 --> 00:15:08,200
how to deal well with
electron dynamics,

298
00:15:08,200 --> 00:15:10,210
but that's another problem.

299
00:15:10,210 --> 00:15:13,510
If you're willing to go all
the way to the continuum scale,

300
00:15:13,510 --> 00:15:17,210
we also know the equations
that describe matter.

301
00:15:17,210 --> 00:15:20,818
It's essentially
thermodynamics and elasticity.

302
00:15:20,818 --> 00:15:22,360
And you could think
of thermodynamics

303
00:15:22,360 --> 00:15:25,240
as the integrator of those.

304
00:15:25,240 --> 00:15:28,390
It's when we live at this
inhomogeneous coarse grain

305
00:15:28,390 --> 00:15:30,560
scale that we're
really in trouble.

306
00:15:30,560 --> 00:15:32,720
And I'll give you an example.

307
00:15:32,720 --> 00:15:35,140
Think of temperature.

308
00:15:35,140 --> 00:15:39,370
In continuum theory,
temperature is a field.

309
00:15:39,370 --> 00:15:42,160
In a microscopic
simulation, it's

310
00:15:42,160 --> 00:15:45,220
kinetic energy of the atoms.

311
00:15:45,220 --> 00:15:47,110
You can already see
a problem appear.

312
00:15:47,110 --> 00:15:50,800
Essentially, as you
inhomogeneously coarse grain

313
00:15:50,800 --> 00:15:53,590
around that atomistic
region, that temperature

314
00:15:53,590 --> 00:15:57,760
has to evolve from being
the kinetic energy of motion

315
00:15:57,760 --> 00:15:59,290
to a field.

316
00:15:59,290 --> 00:16:03,148
And it's [INAUDIBLE] tricky to
know what you do in between.

317
00:16:03,148 --> 00:16:05,440
I mean, I know how to deal
with temperature as a field.

318
00:16:05,440 --> 00:16:07,273
I know how to deal with
this kinetic energy.

319
00:16:07,273 --> 00:16:08,380
But what is it in between?

320
00:16:08,380 --> 00:16:11,290
What is it when you say a
factor of 10 coarse grain,

321
00:16:11,290 --> 00:16:14,220
so you've removed
every 10th atom?

322
00:16:14,220 --> 00:16:16,720
And I'm going to go a little
deeper in some of these things,

323
00:16:16,720 --> 00:16:18,460
but those are some
of the problems.

324
00:16:18,460 --> 00:16:23,380
We also don't know what the
dynamics here is at this scale.

325
00:16:23,380 --> 00:16:26,760
Because think of how do
you dissipate energy.

326
00:16:26,760 --> 00:16:30,870
How do you dissipate energy
in a microscopic system?

327
00:16:30,870 --> 00:16:35,790
You dissipate it, essentially,
by non-harmonic vibrations.

328
00:16:35,790 --> 00:16:37,770
That's how you dissipate energy.

329
00:16:37,770 --> 00:16:40,350
How do you dissipate it here?

330
00:16:40,350 --> 00:16:44,260
Actually, you don't unless
you put it inexplicitly.

331
00:16:44,260 --> 00:16:47,650
And in perfectly
elasticity theory,

332
00:16:47,650 --> 00:16:50,380
you don't have dissipation.

333
00:16:50,380 --> 00:16:54,220
So again, energy
dissipation is a big deal

334
00:16:54,220 --> 00:16:57,370
in inhomogeneously
coarse-grained systems.

335
00:16:57,370 --> 00:17:00,940
Because the mechanism by which
it happens at the atomistic

336
00:17:00,940 --> 00:17:03,894
and at the continuum
scale is different.

337
00:17:08,349 --> 00:17:13,510
So quasi-continuum approaches
have been extremely successful

338
00:17:13,510 --> 00:17:16,720
for elastic static
properties and even

339
00:17:16,720 --> 00:17:20,440
nonlinear elastic problems.

340
00:17:20,440 --> 00:17:22,540
It's when people try
to put temperature

341
00:17:22,540 --> 00:17:26,000
and dynamics in that life
gets a lot more complicated.

342
00:17:26,000 --> 00:17:27,880
So there are essentially
two approaches that

343
00:17:27,880 --> 00:17:29,890
have been suggested and tried.

344
00:17:29,890 --> 00:17:32,680
And both sort of
seem reasonable,

345
00:17:32,680 --> 00:17:35,110
but have significant failures.

346
00:17:35,110 --> 00:17:36,610
One is to simply--

347
00:17:36,610 --> 00:17:39,700
let's say, how would you
do molecular dynamics

348
00:17:39,700 --> 00:17:41,650
on a coarse grain
system like this?

349
00:17:41,650 --> 00:17:45,640
Well, remember, you've
removed these atoms.

350
00:17:45,640 --> 00:17:49,020
You could lump their
mass into the nodes.

351
00:17:49,020 --> 00:17:52,280
So essentially, the nodes of
your mesh now become heavier.

352
00:17:52,280 --> 00:17:56,118
And then you do MD on those.

353
00:17:56,118 --> 00:17:57,660
The problem with
that is that, as you

354
00:17:57,660 --> 00:17:58,910
coarse grain more
and more, you get

355
00:17:58,910 --> 00:18:00,160
the heavier and heavier nodes.

356
00:18:00,160 --> 00:18:02,150
And after a while, they
don't move anymore.

357
00:18:02,150 --> 00:18:05,020
Because if they have
the same temperature,

358
00:18:05,020 --> 00:18:07,990
mv squared average over
2 is the temperature.

359
00:18:07,990 --> 00:18:13,300
So at the same temperature,
the v becomes very small.

360
00:18:13,300 --> 00:18:16,270
The other one is a sort
of often used approach

361
00:18:16,270 --> 00:18:20,320
to use static optimizations.

362
00:18:20,320 --> 00:18:24,580
But rather than say get all
the positions of these atoms--

363
00:18:24,580 --> 00:18:26,620
rather than get them
from the energy,

364
00:18:26,620 --> 00:18:30,130
get them from minimizing
the free energy.

365
00:18:30,130 --> 00:18:34,210
Say, the force would
be now the gradient

366
00:18:34,210 --> 00:18:36,940
of something like
the free energy

367
00:18:36,940 --> 00:18:39,250
rather than the
gradient of the energy.

368
00:18:39,250 --> 00:18:42,560
So you have some amount of
temperature effects in there

369
00:18:42,560 --> 00:18:45,580
then, but you don't
truly have dynamics.

370
00:18:50,290 --> 00:18:52,960
So one thing this would do
for you, for example, is you'd

371
00:18:52,960 --> 00:18:55,660
get the right bond length
and lattice parameters.

372
00:19:06,030 --> 00:19:07,140
OK.

373
00:19:07,140 --> 00:19:11,290
I want to show you some other
ideas that have been out there.

374
00:19:11,290 --> 00:19:14,580
This is one that I've
worked on for a short while.

375
00:19:14,580 --> 00:19:17,782
But the generic idea has
really being out there

376
00:19:17,782 --> 00:19:18,990
in the community a long time.

377
00:19:18,990 --> 00:19:21,120
And it's very similar
to what we've done

378
00:19:21,120 --> 00:19:23,950
in the coarse graining of time.

379
00:19:23,950 --> 00:19:27,120
The idea is essentially think
of three atoms on a chain,

380
00:19:27,120 --> 00:19:29,196
1, 2, 3 there.

381
00:19:29,196 --> 00:19:33,210
What if you want to remove
atom 2, and so coarse grain

382
00:19:33,210 --> 00:19:35,670
and only keep 1
and 3 as the nodes?

383
00:19:35,670 --> 00:19:39,607
The question really is,
what should your potential--

384
00:19:39,607 --> 00:19:41,190
let's say you do
this with potentials.

385
00:19:41,190 --> 00:19:42,898
Don't even worry about
quantum mechanics.

386
00:19:42,898 --> 00:19:44,780
What should your
potential between 1 and 3

387
00:19:44,780 --> 00:19:50,310
be, so that they behave the
same way as if 2 were there?

388
00:19:50,310 --> 00:19:52,450
So you're going to
take 2 out, but you

389
00:19:52,450 --> 00:19:55,420
want to set up a
potential between 1 and 3

390
00:19:55,420 --> 00:19:57,890
that acts like if 2 were there.

391
00:19:57,890 --> 00:20:00,910
Well, you can get the solution
again from thermodynamics,

392
00:20:00,910 --> 00:20:04,660
essentially requiring
that the partition

393
00:20:04,660 --> 00:20:07,570
function for the
motion of 1 and 3

394
00:20:07,570 --> 00:20:10,570
is the same with
or without 2 there.

395
00:20:10,570 --> 00:20:13,220
And that's essentially
what's written out here.

396
00:20:13,220 --> 00:20:16,180
If you think of q's and
p's for the coordinates,

397
00:20:16,180 --> 00:20:19,790
remember q's are spatial
coordinates. p's are

398
00:20:19,790 --> 00:20:22,750
velocities,
essentially the moment.

399
00:20:22,750 --> 00:20:24,250
Essentially, what
you're saying is

400
00:20:24,250 --> 00:20:27,670
that the free energy
of the Hamiltonian

401
00:20:27,670 --> 00:20:29,630
with only q1 and q3--

402
00:20:29,630 --> 00:20:32,830
so now, you keep only the
positions of 1 and 3--

403
00:20:32,830 --> 00:20:37,390
is obtained by integrating
away the coordinates of 2.

404
00:20:37,390 --> 00:20:39,190
So you integrate
over the position

405
00:20:39,190 --> 00:20:41,440
and over the momentum of 2.

406
00:20:41,440 --> 00:20:43,660
And if the motions
are classical,

407
00:20:43,660 --> 00:20:46,300
then the integration over
the momentum is trivial.

408
00:20:46,300 --> 00:20:48,560
You always get the same
factors from momentum.

409
00:20:48,560 --> 00:20:52,910
So it's really the integration
over the coordinates.

410
00:20:52,910 --> 00:20:56,710
And so if you integrate
this away, then

411
00:20:56,710 --> 00:21:01,840
formally this integral has no
q2 dependence by definition.

412
00:21:01,840 --> 00:21:06,100
You've essentially integrated
over all possible displacements

413
00:21:06,100 --> 00:21:11,050
of 2 to get the interaction
between 1 and 3.

414
00:21:11,050 --> 00:21:16,848
And so that then defines a
potential just between 1 and 3.

415
00:21:16,848 --> 00:21:18,640
Now, there's all kinds
of practical issues.

416
00:21:18,640 --> 00:21:20,980
You can really only
do this effectively

417
00:21:20,980 --> 00:21:25,932
if you have interactions
that are spatially limited.

418
00:21:25,932 --> 00:21:27,640
Because, otherwise,
you really don't know

419
00:21:27,640 --> 00:21:29,300
what you all have to integrate.

420
00:21:29,300 --> 00:21:32,890
So this works great with
pair potentials then.

421
00:21:32,890 --> 00:21:36,320
So you know, what does
this physically mean?

422
00:21:36,320 --> 00:21:41,080
It really means that, if
you, say, displace atom three

423
00:21:41,080 --> 00:21:44,920
and you look at the force
that it produces on atom 1,

424
00:21:44,920 --> 00:21:47,560
that you're looking at
the kind of screening

425
00:21:47,560 --> 00:21:49,810
by atom 2 is included.

426
00:21:49,810 --> 00:21:51,760
In the real system, atom
2 would sort of maybe

427
00:21:51,760 --> 00:21:53,980
screen that displacement
away somewhat.

428
00:21:53,980 --> 00:21:55,210
And that's not included.

429
00:21:55,210 --> 00:21:56,920
And people have
done this before,

430
00:21:56,920 --> 00:21:59,560
looking at interactions of--
when you look at interactions

431
00:21:59,560 --> 00:22:02,912
of ions and fluids,
the way to look

432
00:22:02,912 --> 00:22:04,370
at the interaction
between two ions

433
00:22:04,370 --> 00:22:07,397
is by integrating over the
possible displacement of all

434
00:22:07,397 --> 00:22:08,480
the other ones in between.

435
00:22:08,480 --> 00:22:10,820
And that's how you
end up with screening.

436
00:22:10,820 --> 00:22:13,700
So this is sort of the
screening equivalent

437
00:22:13,700 --> 00:22:15,065
in displacement fields.

438
00:22:21,300 --> 00:22:25,080
And what you get at is very
much what you'd expected.

439
00:22:25,080 --> 00:22:27,150
This is the potentials
for different levels

440
00:22:27,150 --> 00:22:29,990
of coarse graining in
normalized distances.

441
00:22:29,990 --> 00:22:35,370
So if the length of the bond
length is in units of 1,

442
00:22:35,370 --> 00:22:37,410
so this would be the
normal potential.

443
00:22:37,410 --> 00:22:40,530
If you coarse grain one level,
so you remove every other atom,

444
00:22:40,530 --> 00:22:42,030
you actually end
up with a potential

445
00:22:42,030 --> 00:22:44,280
that's twice the bond length.

446
00:22:44,280 --> 00:22:48,450
Because, remember, now
you have atom 1, 2, 3.

447
00:22:48,450 --> 00:22:50,670
You've removed 2, so
the equilibrium distance

448
00:22:50,670 --> 00:22:53,962
between 1 and 3 is about
2 times the bond length.

449
00:22:53,962 --> 00:22:55,920
And you keep on coarse
graining, and you end up

450
00:22:55,920 --> 00:22:58,560
with a potential that's 4
times the bond length, 8

451
00:22:58,560 --> 00:23:00,470
times the bond length.

452
00:23:00,470 --> 00:23:00,970
OK.

453
00:23:05,500 --> 00:23:12,430
It turns out that special
coarse graining always

454
00:23:12,430 --> 00:23:15,190
tends to cause some
kind of time quartering.

455
00:23:15,190 --> 00:23:16,900
That's a much more
difficult issue.

456
00:23:16,900 --> 00:23:19,510
I don't want to say much.

457
00:23:19,510 --> 00:23:23,770
In 1D, it's easy to
remove atoms and add up.

458
00:23:23,770 --> 00:23:26,410
So when you remove atom
2 that's between 1 and 3,

459
00:23:26,410 --> 00:23:28,990
essentially that defines a
potential between 1 and 3.

460
00:23:28,990 --> 00:23:32,140
In 2D, it's a little harder
how you partition bonds,

461
00:23:32,140 --> 00:23:35,710
but you can do all kinds of
bond moving approximations.

462
00:23:35,710 --> 00:23:37,900
This is a simple one
in a triangular lattice

463
00:23:37,900 --> 00:23:44,250
where, if you have a
triangle, you essentially

464
00:23:44,250 --> 00:23:48,260
want to remove these guys here.

465
00:23:48,260 --> 00:23:50,040
And so you can
define some scheme

466
00:23:50,040 --> 00:23:59,230
by which you sort of collapse
the bonds onto these sides.

467
00:23:59,230 --> 00:24:01,300
You can also define
other schemes.

468
00:24:01,300 --> 00:24:04,840
So here's an example of an
inhomogeneously coarse-grained

469
00:24:04,840 --> 00:24:08,840
system in 2D where, in this
case, it's fine at the edges

470
00:24:08,840 --> 00:24:10,090
and it's coarse in the middle.

471
00:24:18,000 --> 00:24:19,850
OK.

472
00:24:19,850 --> 00:24:22,500
There are essentially two
major problems with almost

473
00:24:22,500 --> 00:24:24,090
any coarse-graining scheme.

474
00:24:24,090 --> 00:24:30,640
And the first one is the
worst, is that whenever

475
00:24:30,640 --> 00:24:32,050
you go from a--

476
00:24:32,050 --> 00:24:34,810
let's say you're
atomistic out here.

477
00:24:34,810 --> 00:24:39,070
Here, you've coarse-grained
level 4, coarse-grained

478
00:24:39,070 --> 00:24:42,250
another factor of 4.

479
00:24:42,250 --> 00:24:45,700
You cannot sustain
short wavelength phonons

480
00:24:45,700 --> 00:24:47,410
in the coarse regions.

481
00:24:47,410 --> 00:24:49,750
Let's say you have a
phonon come in here

482
00:24:49,750 --> 00:24:55,090
that's, say, has a
wavelength like this.

483
00:25:02,120 --> 00:25:07,200
Since here you only
have these nodes,

484
00:25:07,200 --> 00:25:11,100
this is essentially the minimum
wavelength you can have.

485
00:25:11,100 --> 00:25:12,420
It's the minimum bond distance.

486
00:25:12,420 --> 00:25:18,710
So at every interface
between a fine region

487
00:25:18,710 --> 00:25:20,930
and of coarser
region, what happens

488
00:25:20,930 --> 00:25:22,520
is that there's a
series of phonons.

489
00:25:22,520 --> 00:25:26,540
The ones that have a wavelength
that's too short get reflected

490
00:25:26,540 --> 00:25:32,405
back because they cannot
go into the coarser region.

491
00:25:32,405 --> 00:25:34,280
And ultimately, you hit
the continuum region,

492
00:25:34,280 --> 00:25:36,750
and you can't have any phonons.

493
00:25:36,750 --> 00:25:40,410
Because there they should all
be dissipated as temperature.

494
00:25:40,410 --> 00:25:43,340
So what does that cause?

495
00:25:43,340 --> 00:25:47,900
Well, if you look at
dynamical processes,

496
00:25:47,900 --> 00:25:50,300
let's say you have a fine
region in which you have

497
00:25:50,300 --> 00:25:55,380
some reaction that dissipates
energy or that creates energy,

498
00:25:55,380 --> 00:25:57,980
the way that that
energy has to get out

499
00:25:57,980 --> 00:26:00,770
is by phonon transmission.

500
00:26:00,770 --> 00:26:02,270
You have to send
lattice vibrations

501
00:26:02,270 --> 00:26:04,200
out which, through
their n harmonicity,

502
00:26:04,200 --> 00:26:07,040
have to get
dissipated somewhere.

503
00:26:07,040 --> 00:26:10,610
And so any time
you block phonons,

504
00:26:10,610 --> 00:26:14,030
you're essentially reducing
the thermal conductivity

505
00:26:14,030 --> 00:26:16,258
of the system.

506
00:26:16,258 --> 00:26:18,800
So this coarse graining gives
you lower thermal conductivity.

507
00:26:18,800 --> 00:26:21,332
And I'll show you some
examples in a second.

508
00:26:21,332 --> 00:26:22,790
But you know, it
does worse things.

509
00:26:22,790 --> 00:26:25,910
It's essentially
elastically confining, also,

510
00:26:25,910 --> 00:26:29,670
your system somewhat, but
only in a dynamical sense.

511
00:26:29,670 --> 00:26:33,230
So it's like you're
having, for certain,

512
00:26:33,230 --> 00:26:35,880
dynamical nodes,
a small unit cell.

513
00:26:35,880 --> 00:26:38,360
And so that means
that they reflect back

514
00:26:38,360 --> 00:26:41,390
of the coarse
graining interface.

515
00:26:41,390 --> 00:26:44,390
And they hit your
region of action

516
00:26:44,390 --> 00:26:46,550
fairly quickly as soon as
they're reflected back.

517
00:26:50,180 --> 00:26:55,640
So it's exactly the same
as putting your dynamics

518
00:26:55,640 --> 00:26:59,120
in a very small box except
that the size of the box

519
00:26:59,120 --> 00:27:00,830
that the system
fields is different

520
00:27:00,830 --> 00:27:03,140
for different wavelengths.

521
00:27:03,140 --> 00:27:06,080
So a certain amount of
elastic behavior gets

522
00:27:06,080 --> 00:27:09,630
reflected back fairly quickly.

523
00:27:09,630 --> 00:27:12,430
So the phonon transmission
problem is a serious one.

524
00:27:18,750 --> 00:27:21,420
And the phonon
transmission one is usually

525
00:27:21,420 --> 00:27:23,280
recognized very well.

526
00:27:23,280 --> 00:27:25,380
There's one that's not
as well recognized,

527
00:27:25,380 --> 00:27:28,020
but is probably in
the long-term just as

528
00:27:28,020 --> 00:27:31,590
severe, is any time
you remove degrees

529
00:27:31,590 --> 00:27:34,283
of freedom you remove entropy.

530
00:27:34,283 --> 00:27:35,700
Because the entropy
is essentially

531
00:27:35,700 --> 00:27:38,670
a count of your
degrees of freedom.

532
00:27:38,670 --> 00:27:40,290
You may not worry about that.

533
00:27:40,290 --> 00:27:43,470
But the problem is that,
if you do dynamics,

534
00:27:43,470 --> 00:27:47,460
your system will equilibrate
along certain derivatives

535
00:27:47,460 --> 00:27:48,570
of the entropy.

536
00:27:48,570 --> 00:27:49,740
And you'll get them wrong.

537
00:27:49,740 --> 00:27:51,240
If you get the
entropy wrong, you'll

538
00:27:51,240 --> 00:27:52,530
get its derivatives wrong.

539
00:27:52,530 --> 00:27:55,950
And some of the ones
that you may worry about

540
00:27:55,950 --> 00:27:58,830
are heat capacity,
which is the temperature

541
00:27:58,830 --> 00:28:02,208
derivative of the entropy,
but also this one, which

542
00:28:02,208 --> 00:28:03,750
you may worry a lot
more about if you

543
00:28:03,750 --> 00:28:07,080
do mechanical behavior, thermal
expansion, which is the volume

544
00:28:07,080 --> 00:28:09,540
derivative of the entropy.

545
00:28:09,540 --> 00:28:11,040
Remember your Maxwell relations?

546
00:28:11,040 --> 00:28:15,760
The sdv is essentially
dv dt up to some factors.

547
00:28:15,760 --> 00:28:17,820
So how the entropy
changed with volume

548
00:28:17,820 --> 00:28:20,580
determines the
thermal expansion.

549
00:28:20,580 --> 00:28:23,670
And that's why, if you
actually look back,

550
00:28:23,670 --> 00:28:27,430
the slide I showed
you, when we define

551
00:28:27,430 --> 00:28:31,340
the effective potential
between atoms 1 and 3,

552
00:28:31,340 --> 00:28:33,420
there's actually a potential.

553
00:28:33,420 --> 00:28:37,800
And then there's a part
that doesn't contain

554
00:28:37,800 --> 00:28:40,440
the coordinates of 1 and 3.

555
00:28:40,440 --> 00:28:45,450
And it's essentially the entropy
you've lost by removing node 2.

556
00:28:45,450 --> 00:28:50,960
You have to actually keep
that entropy in the system

557
00:28:50,960 --> 00:28:55,030
if you want to get the total
thermodynamic qualities right.

558
00:28:55,030 --> 00:28:59,410
It's sort of like you
see that this is obvious

559
00:28:59,410 --> 00:29:01,900
if you take the complete
limit of removing all nodes

560
00:29:01,900 --> 00:29:02,830
or going to 1 node.

561
00:29:02,830 --> 00:29:06,510
Then you end up with
continuum behavior.

562
00:29:06,510 --> 00:29:09,750
And the degrees of freedom
of continuing behavior

563
00:29:09,750 --> 00:29:11,340
are not the entropy.

564
00:29:11,340 --> 00:29:14,850
You've actually removed
all the dynamic motion

565
00:29:14,850 --> 00:29:16,570
when you go through
continuum theory.

566
00:29:16,570 --> 00:29:19,230
So unless you've kept track
of the degrees of freedom

567
00:29:19,230 --> 00:29:21,780
and the entropy
that they include,

568
00:29:21,780 --> 00:29:24,390
you've lost all your
information about that.

569
00:29:28,970 --> 00:29:32,590
So when you do something
like this here,

570
00:29:32,590 --> 00:29:35,890
these will actually, if
you define the potentials

571
00:29:35,890 --> 00:29:39,370
at a given temperature, because
the renormalization potential

572
00:29:39,370 --> 00:29:42,560
depends on temperature--

573
00:29:42,560 --> 00:29:44,510
if you define this at
a given temperature

574
00:29:44,510 --> 00:29:46,970
and you run this at
any other temperature,

575
00:29:46,970 --> 00:29:50,240
these regions will have
different thermal expansions.

576
00:29:50,240 --> 00:29:52,768
And so you'll build up
internal strain in your system

577
00:29:52,768 --> 00:29:53,560
without knowing it.

578
00:29:58,590 --> 00:30:01,000
I get a new color scheme.

579
00:30:01,000 --> 00:30:02,650
There we go.

580
00:30:02,650 --> 00:30:07,230
So if you take a simple
pair potential model,

581
00:30:07,230 --> 00:30:09,900
these coarse graining
schemes work pretty well.

582
00:30:09,900 --> 00:30:11,610
Here's essentially
a static property

583
00:30:11,610 --> 00:30:13,152
even though it has
temperature in it,

584
00:30:13,152 --> 00:30:18,780
but this is the strain in that
2D system versus the stress.

585
00:30:18,780 --> 00:30:22,960
And the black curve
underlying this

586
00:30:22,960 --> 00:30:26,770
is the full system,
so no coarse graining.

587
00:30:26,770 --> 00:30:29,950
And the red curve is that coarse
grain system that I showed you.

588
00:30:29,950 --> 00:30:32,950
And you'll see even in the
non-linear part does this

589
00:30:32,950 --> 00:30:33,790
very well.

590
00:30:33,790 --> 00:30:35,125
And it makes sense, you know.

591
00:30:35,125 --> 00:30:39,580
This kind of mechanical
behavior, strain versus stress,

592
00:30:39,580 --> 00:30:42,070
is largely dominated
by the potentials

593
00:30:42,070 --> 00:30:43,600
in this kind of solid system.

594
00:30:43,600 --> 00:30:46,850
There's very little dynamical
factors that play a role in it.

595
00:30:46,850 --> 00:30:50,230
So as long as you normalize
the potentials right,

596
00:30:50,230 --> 00:30:54,400
you will get the stress-strain
relation right even

597
00:30:54,400 --> 00:30:56,170
in the non-linear regime.

598
00:30:56,170 --> 00:30:57,520
So that's not a big surprise.

599
00:31:00,530 --> 00:31:02,960
If you keep track of
the entropy right,

600
00:31:02,960 --> 00:31:06,830
you'll get things like
the heat capacities right.

601
00:31:06,830 --> 00:31:10,850
This is the ratio
between the heat capacity

602
00:31:10,850 --> 00:31:14,240
in the non-homogeneous system,
which is the coarse grain

603
00:31:14,240 --> 00:31:17,150
system, divided by the
exact result, you could say,

604
00:31:17,150 --> 00:31:18,860
the heat capacity
in the homogeneous.

605
00:31:18,860 --> 00:31:21,500
It essentially
oscillates around one

606
00:31:21,500 --> 00:31:23,190
as a function of temperature.

607
00:31:23,190 --> 00:31:25,790
So by construction you
pretty much have that right.

608
00:31:30,090 --> 00:31:33,030
So here gets more interesting.

609
00:31:33,030 --> 00:31:36,180
If you look at heat
conduction through

610
00:31:36,180 --> 00:31:38,650
a coarse grained interface--

611
00:31:38,650 --> 00:31:40,950
so let's say you're
fine at this side.

612
00:31:40,950 --> 00:31:42,480
Let's say these are atoms.

613
00:31:42,480 --> 00:31:47,820
And here we've removed a bunch
of nodes with coarse grains.

614
00:31:47,820 --> 00:31:50,940
And so you look at the heat
conduction through this.

615
00:31:50,940 --> 00:31:52,540
So you set up a
temperature gradient

616
00:31:52,540 --> 00:31:55,600
and essentially look at the
heat flux when you do dynamics.

617
00:31:55,600 --> 00:32:03,450
So what this is showing
here is the heat conduction

618
00:32:03,450 --> 00:32:07,260
in the non-homogeneous system
divided by the heat conduction

619
00:32:07,260 --> 00:32:08,850
in the homogeneous system.

620
00:32:08,850 --> 00:32:11,490
And this is pretty much
always lower than 1.

621
00:32:11,490 --> 00:32:14,670
So the non-homogeneous system
has less heat conduction.

622
00:32:14,670 --> 00:32:18,950
And the reason is you scatter
phonons on this interface.

623
00:32:18,950 --> 00:32:20,710
So here there is
a certain amount

624
00:32:20,710 --> 00:32:22,910
of phonons that
get reflected back.

625
00:32:22,910 --> 00:32:25,090
But it's interesting.

626
00:32:25,090 --> 00:32:26,860
If you show this
here as a function

627
00:32:26,860 --> 00:32:30,920
of the length of the
homogeneous system,

628
00:32:30,920 --> 00:32:35,030
you do better and better
as this system gets bigger.

629
00:32:35,030 --> 00:32:37,550
And that makes sense because
then the interface becomes

630
00:32:37,550 --> 00:32:39,230
less and less important.

631
00:32:39,230 --> 00:32:42,560
But, you know, look
at the size here.

632
00:32:42,560 --> 00:32:44,630
To get up to sort
of about 1, you

633
00:32:44,630 --> 00:32:51,640
need about 1,500 atoms in
the homogeneous region.

634
00:32:51,640 --> 00:32:56,110
So this is a major problem
with full phonons in general.

635
00:32:56,110 --> 00:33:01,180
I think of any atomistic level
phenomena phonons are probably

636
00:33:01,180 --> 00:33:05,032
the ones that have the
largest length scale.

637
00:33:05,032 --> 00:33:07,240
The phonon mean free
[INAUDIBLE] at a low temperature

638
00:33:07,240 --> 00:33:09,520
is enormous.

639
00:33:09,520 --> 00:33:11,500
It's thousands of atoms far.

640
00:33:11,500 --> 00:33:15,040
That means that phonons
are usually the first thing

641
00:33:15,040 --> 00:33:18,015
to see something far away.

642
00:33:18,015 --> 00:33:20,140
They're the first thing to
see finite size effects.

643
00:33:20,140 --> 00:33:22,480
They're the first to see
scattering of interfaces.

644
00:33:22,480 --> 00:33:25,570
Because the reason is
that a lot of systems

645
00:33:25,570 --> 00:33:28,830
behave at low temperature
fairly harmonic.

646
00:33:28,830 --> 00:33:30,420
That means that
these phonons just

647
00:33:30,420 --> 00:33:31,860
travel and travel and travel.

648
00:33:31,860 --> 00:33:33,870
And their probability
of being scattered

649
00:33:33,870 --> 00:33:36,340
is pretty low because it's
only as they start becoming

650
00:33:36,340 --> 00:33:39,180
enharmonic that they scatter.

651
00:33:39,180 --> 00:33:41,077
Of course, they could
scatter off disorder.

652
00:33:41,077 --> 00:33:42,660
So if you have the
disordered systems,

653
00:33:42,660 --> 00:33:44,340
the mean free pattern
is a lot lower.

654
00:33:44,340 --> 00:33:48,030
But phonons you often
see hundreds of angstrom

655
00:33:48,030 --> 00:33:49,887
far, essentially.

656
00:34:00,160 --> 00:34:04,790
OK, here's the same result
as a function of temperature.

657
00:34:04,790 --> 00:34:10,030
So the heat capacity versus the
normalized temperature, this

658
00:34:10,030 --> 00:34:12,429
is the real system.

659
00:34:12,429 --> 00:34:15,520
So this is a fully
atomistic level system.

660
00:34:15,520 --> 00:34:19,570
And this, in red here,
is the partially coarse

661
00:34:19,570 --> 00:34:21,800
grained system, in this
case with two interfaces.

662
00:34:21,800 --> 00:34:24,175
Now, what you see is the coarse
grained system always has

663
00:34:24,175 --> 00:34:27,100
lower heat conductance.

664
00:34:27,100 --> 00:34:29,560
They tend to come a
little closer together

665
00:34:29,560 --> 00:34:30,995
at higher temperature.

666
00:34:30,995 --> 00:34:32,620
And the reason is at
higher temperature

667
00:34:32,620 --> 00:34:34,510
you have more enharmonicity.

668
00:34:34,510 --> 00:34:38,210
So the phonons have a
shorter mean free path.

669
00:34:38,210 --> 00:34:42,130
So they tend to
homogenizer easier

670
00:34:42,130 --> 00:34:45,040
and don't see the
interfaces as much.

671
00:34:51,540 --> 00:34:53,199
OK, let me skip this.

672
00:34:53,199 --> 00:34:55,080
OK.

673
00:34:55,080 --> 00:34:59,850
So that's some of the efforts
going on in coarse graining

674
00:34:59,850 --> 00:35:00,675
over space.

675
00:35:05,260 --> 00:35:08,950
I think if there's
an area in which you

676
00:35:08,950 --> 00:35:12,490
want to work and have high
impact, this is probably one.

677
00:35:12,490 --> 00:35:14,140
I haven't seen a
single good idea

678
00:35:14,140 --> 00:35:16,257
in this field in
the last 10 years.

679
00:35:16,257 --> 00:35:17,840
I probably shouldn't
say this on tape,

680
00:35:17,840 --> 00:35:20,920
but, you know, this
is a field where

681
00:35:20,920 --> 00:35:22,510
progress is desperately needed.

682
00:35:22,510 --> 00:35:25,630
And it's a really,
really hard problem.

683
00:35:25,630 --> 00:35:28,010
Because you're essentially
asking the question,

684
00:35:28,010 --> 00:35:31,120
what is the dynamics of
partially coarse-grained

685
00:35:31,120 --> 00:35:32,920
systems?

686
00:35:32,920 --> 00:35:36,310
And the reason it's
so hard is that it's

687
00:35:36,310 --> 00:35:38,740
a mixture of sort of
Newtonian mechanics, which

688
00:35:38,740 --> 00:35:43,900
is energy conserving,
and dissipated dynamics.

689
00:35:43,900 --> 00:35:45,520
And you have to
sort of-- and that

690
00:35:45,520 --> 00:35:47,500
how you mix those
essentially depends

691
00:35:47,500 --> 00:35:50,550
on the frequency of your motion.

692
00:35:50,550 --> 00:35:52,500
And that's why this is
such a hard problem.

693
00:35:52,500 --> 00:35:55,530
People have tried
things with boundaries

694
00:35:55,530 --> 00:35:59,130
that have complex
impedances, where

695
00:35:59,130 --> 00:36:01,755
essentially your transmission
through the boundary

696
00:36:01,755 --> 00:36:04,120
is frequency dependent.

697
00:36:04,120 --> 00:36:05,590
And you can see,
with that one, you

698
00:36:05,590 --> 00:36:10,130
can definitely sort of tailor
better what the phonons do.

699
00:36:10,130 --> 00:36:12,800
Because, now, you have a
much more complex interface

700
00:36:12,800 --> 00:36:15,052
literally to control.

701
00:36:15,052 --> 00:36:17,510
The problem is that some of
these things you can make work,

702
00:36:17,510 --> 00:36:20,690
but they get so complicated
to implement that they're not

703
00:36:20,690 --> 00:36:22,160
necessarily very practical.

704
00:36:22,160 --> 00:36:25,760
Other people do things with
large overlapping regions.

705
00:36:25,760 --> 00:36:28,430
You could say one way
to solve this problem is

706
00:36:28,430 --> 00:36:31,070
to not have these
sharp boundaries

707
00:36:31,070 --> 00:36:34,100
between different levels
of coarse graining,

708
00:36:34,100 --> 00:36:38,030
but overlap the two regions
and slowly force them

709
00:36:38,030 --> 00:36:39,000
in the overlap region.

710
00:36:39,000 --> 00:36:40,940
The solution, say, to
the displacements fields

711
00:36:40,940 --> 00:36:41,510
are the same.

712
00:36:44,520 --> 00:36:45,610
OK.

713
00:36:45,610 --> 00:36:50,740
The next thing I want to talk
about is accelerating time.

714
00:36:50,740 --> 00:36:53,010
I said without Einstein,
but I think Einstein only

715
00:36:53,010 --> 00:36:54,060
could slow down time.

716
00:36:54,060 --> 00:36:54,930
Is isn't that right?

717
00:36:54,930 --> 00:36:56,510
I'm not sure we
can accelerate, no?

718
00:36:56,510 --> 00:36:59,310
I don't know.

719
00:36:59,310 --> 00:37:03,180
I don't know my theory of
relativity too well anymore.

720
00:37:03,180 --> 00:37:06,390
I think I don't have to
convince you-- you've done MD--

721
00:37:06,390 --> 00:37:10,135
why you need to speed
up time sometimes.

722
00:37:10,135 --> 00:37:11,760
Most of what I'm
going to say comes out

723
00:37:11,760 --> 00:37:14,670
of this review
article by Art Voter,

724
00:37:14,670 --> 00:37:19,440
who's probably been the guy to
work on accelerated MD methods.

725
00:37:19,440 --> 00:37:21,540
And I'll give you the
references again at the end.

726
00:37:21,540 --> 00:37:22,957
This is a great
review if you want

727
00:37:22,957 --> 00:37:27,700
to read about this
because there's

728
00:37:27,700 --> 00:37:30,310
some really clever stuff
in here that I think,

729
00:37:30,310 --> 00:37:32,860
even if you don't want
to do accelerated MD,

730
00:37:32,860 --> 00:37:37,580
that you could use
for other things.

731
00:37:37,580 --> 00:37:40,780
So first of all,
what's the problem?

732
00:37:40,780 --> 00:37:44,140
The problem is obviously
that, in most systems that

733
00:37:44,140 --> 00:37:48,850
are fairly dense, you have
well-defined minimum phase

734
00:37:48,850 --> 00:37:49,630
space.

735
00:37:49,630 --> 00:37:52,790
And often the barriers
between them are pretty high.

736
00:37:52,790 --> 00:37:55,690
So you sample the
transitions between them

737
00:37:55,690 --> 00:37:57,760
at a very low rate.

738
00:37:57,760 --> 00:38:00,590
The rate is essentially
proportional to the exponential

739
00:38:00,590 --> 00:38:02,590
of the activation barrier
between the minimum.

740
00:38:02,590 --> 00:38:04,548
And like I said, in a
lot of condensed systems,

741
00:38:04,548 --> 00:38:07,720
those activation
barriers are quite high

742
00:38:07,720 --> 00:38:11,365
and lead to timescales that
you can't quite sample.

743
00:38:16,430 --> 00:38:19,870
So I'm going to talk briefly
about three different methods

744
00:38:19,870 --> 00:38:23,050
to speed up time
parallel replica

745
00:38:23,050 --> 00:38:26,450
dynamics, hyperdynamics,
and temperature accelerated

746
00:38:26,450 --> 00:38:26,950
dynamics.

747
00:38:30,180 --> 00:38:31,110
OK.

748
00:38:31,110 --> 00:38:34,110
Let me sort of quickly
review what you would do

749
00:38:34,110 --> 00:38:36,190
with transition state theory.

750
00:38:36,190 --> 00:38:39,750
So if you knew the transitions
that a system had to make, like

751
00:38:39,750 --> 00:38:43,170
in, say, a simple
molecular reaction,

752
00:38:43,170 --> 00:38:45,300
then you really don't
need to do a simulation.

753
00:38:45,300 --> 00:38:50,220
Because if the activation
barrier, this quantity and you

754
00:38:50,220 --> 00:38:53,910
have some idea of the frequency
with which the system tries

755
00:38:53,910 --> 00:38:58,110
to cross that barrier, you
can do transition state theory

756
00:38:58,110 --> 00:39:02,100
and essentially say that
the crossing rate is going

757
00:39:02,100 --> 00:39:05,160
to be something like
the attempt rate,

758
00:39:05,160 --> 00:39:09,150
nu, times the success rate,
which is this Boltzmann factor.

759
00:39:09,150 --> 00:39:12,047
And the chemists among
you and chemical engineers

760
00:39:12,047 --> 00:39:14,130
know there's all kinds of
approximations in there.

761
00:39:14,130 --> 00:39:16,530
But none of them are
particularly severe for most

762
00:39:16,530 --> 00:39:17,370
things to study.

763
00:39:17,370 --> 00:39:21,390
Like, one approximation just
assumes you don't cross back.

764
00:39:21,390 --> 00:39:23,610
Often, when you sit at
the transition state,

765
00:39:23,610 --> 00:39:25,110
you're essentially
assuming that you

766
00:39:25,110 --> 00:39:27,240
fall over and don't cross back.

767
00:39:27,240 --> 00:39:29,460
The other approximation
is that this

768
00:39:29,460 --> 00:39:32,580
assumes that the
transitions are infrequent

769
00:39:32,580 --> 00:39:34,950
compared to the
vibrations in the minimum.

770
00:39:34,950 --> 00:39:38,580
So you assume that when the
system goes from here to here

771
00:39:38,580 --> 00:39:41,340
it's sort of equilibrates
here, vibrates

772
00:39:41,340 --> 00:39:45,240
around until it makes
the next transition.

773
00:39:45,240 --> 00:39:47,040
And the reason you
need to do that

774
00:39:47,040 --> 00:39:51,130
is that you need to equilibrate
first in every potential well

775
00:39:51,130 --> 00:39:52,380
before you go to the next one.

776
00:39:52,380 --> 00:39:54,660
Because, otherwise, you
have dynamical memory left

777
00:39:54,660 --> 00:39:57,450
from the previous transition,
which would essentially

778
00:39:57,450 --> 00:40:00,870
change your statistic, would
make your statistical mechanics

779
00:40:00,870 --> 00:40:03,540
approach invalid.

780
00:40:03,540 --> 00:40:08,545
But usually those are
easy to deal with.

781
00:40:08,545 --> 00:40:09,420
It's sort of obvious.

782
00:40:09,420 --> 00:40:12,180
Because if you think about
it, if the system crosses very

783
00:40:12,180 --> 00:40:15,090
rapidly compared to
the time it vibrates,

784
00:40:15,090 --> 00:40:16,638
then you just do
molecular dynamics

785
00:40:16,638 --> 00:40:18,430
because then you have
very fast transition.

786
00:40:18,430 --> 00:40:20,470
So who cares about
transition state theory?

787
00:40:26,710 --> 00:40:27,820
OK.

788
00:40:27,820 --> 00:40:30,490
So the parallel replica
method is really

789
00:40:30,490 --> 00:40:35,440
kind of elegant in
its triviality almost.

790
00:40:35,440 --> 00:40:40,120
If you think of what is time,
time is waiting for events

791
00:40:40,120 --> 00:40:41,930
to happen.

792
00:40:41,930 --> 00:40:44,877
And if you think of a
really trivial example,

793
00:40:44,877 --> 00:40:46,960
let's say the system wants
to move from this state

794
00:40:46,960 --> 00:40:48,220
to that state.

795
00:40:48,220 --> 00:40:50,230
You're just sitting there.

796
00:40:50,230 --> 00:40:51,640
The atoms are vibrating around.

797
00:40:51,640 --> 00:40:54,580
And you're waiting for
a system to transition.

798
00:40:54,580 --> 00:40:57,920
Why not wait on many
processors at a time?

799
00:40:57,920 --> 00:41:00,310
So the idea is you
run this simulation

800
00:41:00,310 --> 00:41:02,680
on a bunch of processors.

801
00:41:02,680 --> 00:41:05,080
And if you run on,
say, 100, you could

802
00:41:05,080 --> 00:41:07,930
say that's kind of like
waiting 100 times the time you

803
00:41:07,930 --> 00:41:09,790
wait on one processor.

804
00:41:09,790 --> 00:41:15,100
And then let's say it
happens on one processor.

805
00:41:15,100 --> 00:41:16,840
The system transitions.

806
00:41:16,840 --> 00:41:19,930
Then you essentially add up
all the time all the systems

807
00:41:19,930 --> 00:41:22,660
have waited, and that's
your total waiting time.

808
00:41:22,660 --> 00:41:25,240
And in a statistical sense,
that's actually right.

809
00:41:25,240 --> 00:41:27,550
If you do queuing theory
and things like that,

810
00:41:27,550 --> 00:41:29,520
you'll find that
this is the solution.

811
00:41:29,520 --> 00:41:31,270
It's actually easy to
see if you turn this

812
00:41:31,270 --> 00:41:33,460
all into probabilities
for transfer.

813
00:41:33,460 --> 00:41:37,750
Essentially, you could say,
when you do it 100 times,

814
00:41:37,750 --> 00:41:40,360
you've 100 times
attempted or given

815
00:41:40,360 --> 00:41:44,290
the system chances in a given
time to make a transition.

816
00:41:44,290 --> 00:41:48,790
So you add up all the time,
and then you just restart.

817
00:41:48,790 --> 00:41:52,270
So you put the system
in the next state.

818
00:41:52,270 --> 00:41:54,453
And you again run
all those replicas.

819
00:41:54,453 --> 00:41:56,620
You have to introduce a
little bit of randomization,

820
00:41:56,620 --> 00:41:57,250
of course.

821
00:41:57,250 --> 00:42:00,130
You don't want these to follow
exactly the same trajectory

822
00:42:00,130 --> 00:42:01,420
through phase space.

823
00:42:01,420 --> 00:42:05,530
So there's a certain amount
of randomization that goes on.

824
00:42:05,530 --> 00:42:10,510
So essentially, this
is linear time scaling.

825
00:42:10,510 --> 00:42:13,810
Overall, you pretty much--
if you have n processors,

826
00:42:13,810 --> 00:42:16,840
time is accelerated by a
factor of n you could say.

827
00:42:16,840 --> 00:42:19,390
It's a little less
because there's overhead.

828
00:42:19,390 --> 00:42:23,050
You have to, first of all,
do the non-trivial thing,

829
00:42:23,050 --> 00:42:25,300
which is detect the transition.

830
00:42:25,300 --> 00:42:28,570
When you think about it, this
is kind of not that trivial.

831
00:42:28,570 --> 00:42:30,160
You're doing your MD simulation.

832
00:42:30,160 --> 00:42:33,970
And you've got to sort
of ask your system, oh,

833
00:42:33,970 --> 00:42:35,800
has something happened?

834
00:42:35,800 --> 00:42:39,040
And maybe it's, like, an
atom diffusing, whatever.

835
00:42:39,040 --> 00:42:41,590
So you've got to do quite a
bit of work to sort of check

836
00:42:41,590 --> 00:42:43,007
whether a transition
that happens.

837
00:42:43,007 --> 00:42:44,500
So you've got overhead there.

838
00:42:44,500 --> 00:42:49,600
And then you have overhead when
you sort of copy the transition

839
00:42:49,600 --> 00:42:52,180
state into all the other ones.

840
00:42:52,180 --> 00:42:54,500
You have to do a little
bit of randomization here.

841
00:42:54,500 --> 00:42:59,170
So typically you may restart
with an actual Boltzmann

842
00:42:59,170 --> 00:43:00,760
distribution of
velocities and so.

843
00:43:00,760 --> 00:43:02,488
But you have to initialize that.

844
00:43:02,488 --> 00:43:04,280
You have to run that
for a very short time.

845
00:43:04,280 --> 00:43:06,100
And so you have a bit
of overhead there.

846
00:43:06,100 --> 00:43:10,690
So you'll get a little
less than end scaling.

847
00:43:10,690 --> 00:43:12,640
But it's not bad.

848
00:43:12,640 --> 00:43:14,050
It's not great either.

849
00:43:14,050 --> 00:43:16,150
But like I said, with
1,000 processors,

850
00:43:16,150 --> 00:43:19,360
you can go from nanoseconds
to microseconds.

851
00:43:19,360 --> 00:43:22,270
So yeah, that's not bad.

852
00:43:22,270 --> 00:43:23,830
I mean, you can see
that you're never

853
00:43:23,830 --> 00:43:27,730
going to make it to seconds
because of the linear scaling.

854
00:43:27,730 --> 00:43:33,650
But it is a sort of intermediate
timescale that can be useful.

855
00:43:33,650 --> 00:43:34,910
Here's an example.

856
00:43:34,910 --> 00:43:39,700
This is planarization
of silver on silver.

857
00:43:39,700 --> 00:43:43,930
And this comes straight
out of Art Voter's article.

858
00:43:43,930 --> 00:43:46,780
So essentially, this is a
1, 1, 1 layer of silver.

859
00:43:46,780 --> 00:43:49,060
That's the white atoms.

860
00:43:49,060 --> 00:43:51,800
And then on top of
that is a first layer,

861
00:43:51,800 --> 00:43:53,110
which is the blue atoms.

862
00:43:53,110 --> 00:43:55,430
And then the yellow atoms
is another layer on top.

863
00:43:55,430 --> 00:43:57,430
OK, so you're
seeing three layers.

864
00:43:57,430 --> 00:44:03,340
And so this study tracks
how the system planarizes

865
00:44:03,340 --> 00:44:05,030
to lower its surface energy.

866
00:44:05,030 --> 00:44:07,510
So the top atoms should
sort of come down

867
00:44:07,510 --> 00:44:10,250
and form all one island.

868
00:44:10,250 --> 00:44:13,910
And so this is perfect
for accelerated MD.

869
00:44:13,910 --> 00:44:16,670
I mean, researchers know
why they pick problems

870
00:44:16,670 --> 00:44:18,950
because they can solve them.

871
00:44:18,950 --> 00:44:22,510
Because this is a set
of discrete events.

872
00:44:22,510 --> 00:44:25,580
You know, atoms diffuse and
maybe even collectively.

873
00:44:25,580 --> 00:44:27,770
You see, the nice thing
about MD is that you're not

874
00:44:27,770 --> 00:44:30,590
imposing the kinetic mechanism.

875
00:44:30,590 --> 00:44:33,410
But it's still a set
of discrete events.

876
00:44:33,410 --> 00:44:35,810
So atoms may help a
little-- you can see it

877
00:44:35,810 --> 00:44:37,970
in the first few slides
there-- and ultimately

878
00:44:37,970 --> 00:44:39,020
sort of planarize.

879
00:44:39,020 --> 00:44:45,050
People got all the way
up to 1 microsecond.

880
00:44:45,050 --> 00:44:47,600
Although it was done
with empirical potential.

881
00:44:47,600 --> 00:44:51,260
And according to Art Voter,
this took only about five days

882
00:44:51,260 --> 00:44:52,580
on 32 Pentiums.

883
00:44:52,580 --> 00:44:53,390
And these are old.

884
00:44:53,390 --> 00:44:55,015
I mean, these are 1
gigahertz Pentiums,

885
00:44:55,015 --> 00:44:58,030
so probably do it in
maybe one to two days now.

886
00:45:02,120 --> 00:45:06,260
I think you can already
see, if I go back,

887
00:45:06,260 --> 00:45:07,790
when this will not work.

888
00:45:10,700 --> 00:45:14,720
This will not work if the
overhead becomes excessive.

889
00:45:14,720 --> 00:45:17,030
And how is it going
to become excessive--

890
00:45:17,030 --> 00:45:22,840
if you start getting
a lot of transitions.

891
00:45:22,840 --> 00:45:25,180
If only after a little
bit of simulation

892
00:45:25,180 --> 00:45:27,190
you get a transition,
then your overhead

893
00:45:27,190 --> 00:45:29,035
starts to weigh heavily.

894
00:45:29,035 --> 00:45:30,410
And when is that
going to happen?

895
00:45:30,410 --> 00:45:32,368
Well, it's going to
happen, of course, when you

896
00:45:32,368 --> 00:45:33,580
have low activation barriers.

897
00:45:33,580 --> 00:45:35,050
But there's another
problem which

898
00:45:35,050 --> 00:45:36,910
is a little more
severe, if you make

899
00:45:36,910 --> 00:45:39,570
your system bigger and bigger.

900
00:45:39,570 --> 00:45:42,590
See the probability-- let's
say you have a system that's

901
00:45:42,590 --> 00:45:43,790
sort of locally identical.

902
00:45:43,790 --> 00:45:46,540
If you make it
bigger and bigger,

903
00:45:46,540 --> 00:45:49,100
the probability that
a transition occurs--

904
00:45:49,100 --> 00:45:50,600
let's say an atom diffuses--

905
00:45:50,600 --> 00:45:53,150
is proportional to
the system size.

906
00:45:53,150 --> 00:45:56,010
The probability that
an event happens

907
00:45:56,010 --> 00:45:57,900
is proportional to
the system size.

908
00:45:57,900 --> 00:45:59,870
So as you make the
system bigger and bigger,

909
00:45:59,870 --> 00:46:03,080
this essentially becomes
less and less efficient

910
00:46:03,080 --> 00:46:05,790
because you get more
and more transitions.

911
00:46:05,790 --> 00:46:07,850
And that's something you'll
see in other versions

912
00:46:07,850 --> 00:46:10,180
of accelerated MD as well.

913
00:46:12,820 --> 00:46:13,960
OK.

914
00:46:13,960 --> 00:46:16,990
Hyperdynamics, I like
the word very much,

915
00:46:16,990 --> 00:46:19,300
but I'm not going to
say much about it.

916
00:46:19,300 --> 00:46:22,150
Hyperdynamics is really
a class of methods

917
00:46:22,150 --> 00:46:25,900
that is getting very popular
not just in accelerating MD now.

918
00:46:25,900 --> 00:46:29,770
But the whole idea of
modifying the potential surface

919
00:46:29,770 --> 00:46:31,960
is getting sort of
very popular also

920
00:46:31,960 --> 00:46:34,840
for pure optimization methods.

921
00:46:34,840 --> 00:46:42,000
If you think that this is the
surface of the original system,

922
00:46:42,000 --> 00:46:45,210
this is the energy
surface, your problem

923
00:46:45,210 --> 00:46:46,650
is that these
wells are too deep.

924
00:46:46,650 --> 00:46:49,170
Well, solve the problem
by lifting them up.

925
00:46:49,170 --> 00:46:51,780
So if you can define
a potential that

926
00:46:51,780 --> 00:46:58,170
tracks the original
potential in most of space,

927
00:46:58,170 --> 00:47:02,480
especially the activated
pieces, but then lifts up

928
00:47:02,480 --> 00:47:08,813
the minima of the well,
if your system moves

929
00:47:08,813 --> 00:47:10,730
on the red potential,
it's going to transition

930
00:47:10,730 --> 00:47:15,170
a lot faster because the
activation barrier goes down

931
00:47:15,170 --> 00:47:17,000
a lot.

932
00:47:17,000 --> 00:47:21,050
And you can actually run
the system on the red curve

933
00:47:21,050 --> 00:47:23,720
and correct for the
transition rate.

934
00:47:23,720 --> 00:47:27,340
Because, see, the cool
thing is that once

935
00:47:27,340 --> 00:47:31,600
the MD finds the transitions--

936
00:47:31,600 --> 00:47:34,650
so MD is going back
and forth here.

937
00:47:34,650 --> 00:47:36,490
But once it's found
the transition,

938
00:47:36,490 --> 00:47:38,740
you could calculate back
what the real barrier

939
00:47:38,740 --> 00:47:41,560
should have been.

940
00:47:41,560 --> 00:47:43,630
So you know at what
rate the system should

941
00:47:43,630 --> 00:47:45,950
have transitioned.

942
00:47:45,950 --> 00:47:49,010
So boosting the potential
surface with this boost

943
00:47:49,010 --> 00:47:50,780
potential-- that's
this difference here,

944
00:47:50,780 --> 00:47:52,280
boosting it up--

945
00:47:52,280 --> 00:47:57,860
is essentially accelerating
the escape from the minimum.

946
00:47:57,860 --> 00:48:01,520
And you can correct back by
this Boltzmann factor, which

947
00:48:01,520 --> 00:48:07,790
is essentially comes from
the ratio of the proper rate

948
00:48:07,790 --> 00:48:10,838
to the rate that you actually
had in your MD simulation.

949
00:48:10,838 --> 00:48:12,380
There's all kinds
of tricks involved.

950
00:48:12,380 --> 00:48:15,950
This is a simplified
version of it.

951
00:48:15,950 --> 00:48:17,450
There's a lot of
questions about how

952
00:48:17,450 --> 00:48:20,300
you define that boost surface.

953
00:48:20,300 --> 00:48:23,850
For example, do you want to
keep the same frequencies?

954
00:48:23,850 --> 00:48:27,320
So then you'd like to keep
the curvature the same.

955
00:48:27,320 --> 00:48:30,110
Because then you keep the
same attempt frequencies,

956
00:48:30,110 --> 00:48:32,450
but you're only
changing the success

957
00:48:32,450 --> 00:48:34,107
rate of crossing the barrier.

958
00:48:34,107 --> 00:48:35,690
That's a very elegant
way of doing it.

959
00:48:35,690 --> 00:48:38,930
And people have ways
that they can do this.

960
00:48:38,930 --> 00:48:45,180
So doing this in
practice often means

961
00:48:45,180 --> 00:48:49,290
calculating derivatives
of potential surfaces,

962
00:48:49,290 --> 00:48:53,640
in many cases even Hessians, so
matrices of second derivatives.

963
00:48:53,640 --> 00:48:56,460
And that's easy to do
when you have potentials.

964
00:48:56,460 --> 00:48:58,975
It's a lot harder to do when
you're doing quantum mechanics.

965
00:48:58,975 --> 00:49:00,600
And so it's, again,
one of these things

966
00:49:00,600 --> 00:49:04,770
that is usually only done
with empirical potentials.

967
00:49:04,770 --> 00:49:07,170
Actually, I don't know of any
implementation with quantum

968
00:49:07,170 --> 00:49:09,810
mechanics, but it may
be something I've just

969
00:49:09,810 --> 00:49:12,720
missed in the literature.

970
00:49:12,720 --> 00:49:15,270
This is used for sort of
other schemes as well.

971
00:49:15,270 --> 00:49:19,200
If you for a second don't
care about dynamics,

972
00:49:19,200 --> 00:49:22,800
there are now schemes out there
to use this for optimization.

973
00:49:22,800 --> 00:49:25,077
If you think of
global optimization

974
00:49:25,077 --> 00:49:26,910
of a system is a big
problem because there's

975
00:49:26,910 --> 00:49:29,490
all kinds of local wells,
well, what you can do

976
00:49:29,490 --> 00:49:33,860
is, if you fall in a
well, you essentially

977
00:49:33,860 --> 00:49:35,030
start filling up the well.

978
00:49:39,570 --> 00:49:41,370
It's sort of like water
filling up the well

979
00:49:41,370 --> 00:49:44,100
until you fall out
into the next well.

980
00:49:44,100 --> 00:49:44,850
OK.

981
00:49:44,850 --> 00:49:49,720
And then you start
filling up that well

982
00:49:49,720 --> 00:49:51,100
until you fall in next well.

983
00:49:51,100 --> 00:49:53,500
And it's sort of like
water cascading down

984
00:49:53,500 --> 00:49:54,850
an energy surface.

985
00:49:54,850 --> 00:49:57,200
And it's a way to do
global optimization

986
00:49:57,200 --> 00:49:59,610
in a landscape of many minima.

987
00:49:59,610 --> 00:50:01,120
I think this method
was developed

988
00:50:01,120 --> 00:50:03,596
by people at Princeton.

989
00:50:09,660 --> 00:50:12,060
So that's sort of in the
category of hyperdynamics.

990
00:50:14,727 --> 00:50:16,310
The one that's
actually very practical

991
00:50:16,310 --> 00:50:20,360
is the next one, which is
temperature accelerated

992
00:50:20,360 --> 00:50:26,760
dynamics, which is, again,
a sort of obvious idea.

993
00:50:26,760 --> 00:50:29,600
If stuff runs too slow
at low temperature,

994
00:50:29,600 --> 00:50:30,850
run it at higher temperatures.

995
00:50:30,850 --> 00:50:33,300
It'll go faster.

996
00:50:33,300 --> 00:50:36,180
That's actually something
people have often done in MD

997
00:50:36,180 --> 00:50:38,700
without much justification.

998
00:50:38,700 --> 00:50:41,490
What Art Voter's group
did in Los Alamos

999
00:50:41,490 --> 00:50:43,380
is essentially give
this a justification

1000
00:50:43,380 --> 00:50:46,770
and show how you can
correct for that temperature

1001
00:50:46,770 --> 00:50:49,630
difference between where
you would like the result

1002
00:50:49,630 --> 00:50:51,780
and where you're
actually simulating.

1003
00:50:51,780 --> 00:50:55,360
And so the idea of
TAD, as it's called,

1004
00:50:55,360 --> 00:51:00,570
is to use the high temperature
to find the transitions,

1005
00:51:00,570 --> 00:51:03,300
but then execute them
with the proper rate

1006
00:51:03,300 --> 00:51:05,947
of the low temperature.

1007
00:51:05,947 --> 00:51:07,780
So essentially, you run
at high temperature.

1008
00:51:07,780 --> 00:51:11,770
So you fairly quickly scan
the potential surface.

1009
00:51:11,770 --> 00:51:14,520
And that tells you what
the transitions are.

1010
00:51:14,520 --> 00:51:16,830
And then you go back
to the low temperature,

1011
00:51:16,830 --> 00:51:18,970
and you do a sort of
statistical mechanics,

1012
00:51:18,970 --> 00:51:20,940
a kind of kinetic
Monte Carlo scheme

1013
00:51:20,940 --> 00:51:25,650
or a transition state
theory approach and execute

1014
00:51:25,650 --> 00:51:26,950
the transitions.

1015
00:51:26,950 --> 00:51:29,907
So I'll show you an example.

1016
00:51:29,907 --> 00:51:31,740
The idea is that you
run at high temperature

1017
00:51:31,740 --> 00:51:33,660
until a transition occurs.

1018
00:51:33,660 --> 00:51:35,790
Once you find the
transition, it's

1019
00:51:35,790 --> 00:51:40,290
a trivial matter of finding
the activation barrier.

1020
00:51:40,290 --> 00:51:42,960
Then you reverse the
transition, and you run again

1021
00:51:42,960 --> 00:51:44,070
at high temperature.

1022
00:51:44,070 --> 00:51:47,430
And the idea is that you get
this catalog of transitions.

1023
00:51:47,430 --> 00:51:49,860
So you get this catalog
of possible transitions.

1024
00:51:49,860 --> 00:51:52,780
Now, why do you need
more than one, you say?

1025
00:51:52,780 --> 00:51:56,370
Well, the reason is that the
one you find at high temperature

1026
00:51:56,370 --> 00:51:57,930
may not be the one
that actually gets

1027
00:51:57,930 --> 00:51:59,850
executed at low temperature.

1028
00:51:59,850 --> 00:52:02,310
If they have different
activation barriers,

1029
00:52:02,310 --> 00:52:04,845
then their rates may
cross with temperature.

1030
00:52:09,300 --> 00:52:10,830
Here's an example.

1031
00:52:10,830 --> 00:52:17,130
If I show the crossing
rate versus 1 over t,

1032
00:52:17,130 --> 00:52:19,860
let's say you run
at high temperature.

1033
00:52:19,860 --> 00:52:23,770
If these behave Arrhenius,
then their crossing rate,

1034
00:52:23,770 --> 00:52:26,550
which is 1 over the
time you have to wait,

1035
00:52:26,550 --> 00:52:29,340
goes like 1 over t.

1036
00:52:29,340 --> 00:52:32,820
You may find one that
has a high crossing rate

1037
00:52:32,820 --> 00:52:38,420
at high temperature and this
one that has a lower crossing

1038
00:52:38,420 --> 00:52:40,340
rate at high temperature.

1039
00:52:40,340 --> 00:52:43,070
But if they have a different
activation barrier,

1040
00:52:43,070 --> 00:52:45,440
they'll extrapolate differently
to lower temperature.

1041
00:52:45,440 --> 00:52:49,820
But what you see
at low temperature,

1042
00:52:49,820 --> 00:52:52,310
the one with the high
rate at high temperature

1043
00:52:52,310 --> 00:52:55,690
has the lower rate
at low temperature.

1044
00:52:55,690 --> 00:52:58,270
So because transitions have
different activation barriers,

1045
00:52:58,270 --> 00:53:02,950
they may not be in the same
order, so the same frequency,

1046
00:53:02,950 --> 00:53:05,620
the same rate at
different temperatures.

1047
00:53:05,620 --> 00:53:07,930
And that's why you
need a catalog of them.

1048
00:53:07,930 --> 00:53:16,140
You essentially need
a window of rates,

1049
00:53:16,140 --> 00:53:19,230
of transmission rates, that
give you a certain certainty

1050
00:53:19,230 --> 00:53:24,090
that at the lower temperature
you found the lowest one.

1051
00:53:24,090 --> 00:53:27,060
And you can kind of do back
of the envelope calculations.

1052
00:53:27,060 --> 00:53:31,950
If you make some assumptions
about the range in which

1053
00:53:31,950 --> 00:53:34,800
your EAs, your activation
barrier, can vary,

1054
00:53:34,800 --> 00:53:37,140
then you have the range of
slopes of these Arrhenius

1055
00:53:37,140 --> 00:53:38,460
walls.

1056
00:53:38,460 --> 00:53:40,500
So that can tell you a
little bit about how long

1057
00:53:40,500 --> 00:53:44,520
you should be running
at the high temperature

1058
00:53:44,520 --> 00:53:47,700
to make sure you found
all the transitions

1059
00:53:47,700 --> 00:53:50,010
or are likely to have found
all the transitions that

1060
00:53:50,010 --> 00:53:52,590
get executed at the
low temperature.

1061
00:53:52,590 --> 00:53:54,510
So does everyone see the idea?

1062
00:53:54,510 --> 00:53:58,920
You're essentially
using molecular dynamics

1063
00:53:58,920 --> 00:54:02,350
as a way of finding transitions.

1064
00:54:02,350 --> 00:54:04,350
And that's the hard problem
in any kinetic from.

1065
00:54:04,350 --> 00:54:05,700
It's finding the transitions.

1066
00:54:05,700 --> 00:54:08,490
Once you find them, you can
calculate their activation

1067
00:54:08,490 --> 00:54:09,270
barrier.

1068
00:54:09,270 --> 00:54:13,710
And you can execute them with
standard rate constant theory.

1069
00:54:13,710 --> 00:54:14,700
But it's finding them.

1070
00:54:14,700 --> 00:54:18,780
And MD, because it's unbiased,
is great at doing that.

1071
00:54:26,270 --> 00:54:28,810
OK.

1072
00:54:28,810 --> 00:54:32,200
So the approximations of the
method are fairly obvious.

1073
00:54:32,200 --> 00:54:37,480
To do this extrapolation, you
assume harmonic transition

1074
00:54:37,480 --> 00:54:38,330
state theory.

1075
00:54:38,330 --> 00:54:40,780
So you essentially assume that
you have a simple Arrhenius

1076
00:54:40,780 --> 00:54:43,180
extrapolation
between temperature,

1077
00:54:43,180 --> 00:54:47,200
so that the exponential
factor is constant.

1078
00:54:47,200 --> 00:54:50,230
And like I said before,
you have to make sure

1079
00:54:50,230 --> 00:54:53,852
that you found all
the mechanisms,

1080
00:54:53,852 --> 00:54:56,060
that you found enough
mechanisms at high temperature,

1081
00:54:56,060 --> 00:55:00,370
so you definitely have the
fastest one at low temperature.

1082
00:55:04,510 --> 00:55:05,500
OK.

1083
00:55:05,500 --> 00:55:09,340
And here's another application
out of Art Voter's work.

1084
00:55:09,340 --> 00:55:12,970
This is copper on
copper deposition.

1085
00:55:12,970 --> 00:55:16,240
So they literally, I think,
add atoms to a surface

1086
00:55:16,240 --> 00:55:18,910
and then equilibrate them.

1087
00:55:18,910 --> 00:55:24,020
And they run-- so they do
direct MD for 2 picoseconds.

1088
00:55:24,020 --> 00:55:28,120
That's how the deposit
the atom, but then run

1089
00:55:28,120 --> 00:55:34,210
0.3 seconds of
intervening time with TAD.

1090
00:55:34,210 --> 00:55:36,370
Now, you may think, 0.3
seconds, that's a lot.

1091
00:55:36,370 --> 00:55:38,470
Well, it's not
because you're really

1092
00:55:38,470 --> 00:55:40,780
doing transition state theory.

1093
00:55:40,780 --> 00:55:46,900
If I have barriers of 1 electron
volt, then pretty much at room

1094
00:55:46,900 --> 00:55:49,450
temperature 1
electron volt, that

1095
00:55:49,450 --> 00:55:52,160
tends to give you rates
around 1 per second.

1096
00:55:52,160 --> 00:55:53,718
That's kind of the time scale.

1097
00:55:53,718 --> 00:55:55,510
So if you have barriers
of 1 electron volt,

1098
00:55:55,510 --> 00:55:57,280
then every time
you execute a step,

1099
00:55:57,280 --> 00:56:00,110
you're 1 second
further on average.

1100
00:56:00,110 --> 00:56:03,940
So the 0.3 seconds, that
is not that impressive.

1101
00:56:03,940 --> 00:56:06,250
You know, I can do 1,000
seconds with one step

1102
00:56:06,250 --> 00:56:09,310
just by having a system at a
very high activation barrier.

1103
00:56:13,790 --> 00:56:15,860
OK.

1104
00:56:15,860 --> 00:56:19,190
You can look at the different
events in the simulation.

1105
00:56:19,190 --> 00:56:23,090
And one of the nice things
is, because you're doing MD,

1106
00:56:23,090 --> 00:56:25,490
your system is unbiased.

1107
00:56:25,490 --> 00:56:28,130
And one thing we've
learned more and more

1108
00:56:28,130 --> 00:56:33,800
by doing MD on surfaces is that
the diffusive processes are not

1109
00:56:33,800 --> 00:56:36,380
at all the way people
thought they were.

1110
00:56:36,380 --> 00:56:39,080
People think of atoms
hopping by themselves.

1111
00:56:39,080 --> 00:56:41,720
But because surfaces
have so much open space,

1112
00:56:41,720 --> 00:56:43,310
have such a low
symmetry, you see

1113
00:56:43,310 --> 00:56:45,590
a lot of collective behavior.

1114
00:56:45,590 --> 00:56:48,110
For example, here's one.

1115
00:56:48,110 --> 00:56:54,470
This blue atom here
pushes in the surface,

1116
00:56:54,470 --> 00:57:01,070
pushing that orange atom or
kind of-- what is it-- brown.

1117
00:57:01,070 --> 00:57:03,470
And that pushes
then these two out.

1118
00:57:03,470 --> 00:57:06,650
And that's all one
collective event.

1119
00:57:06,650 --> 00:57:08,780
So these are not things
you would easily guess at.

1120
00:57:11,555 --> 00:57:13,180
Here's another
collective event of sort

1121
00:57:13,180 --> 00:57:18,820
of a kind of roll of atoms
three at a time at a step edge,

1122
00:57:18,820 --> 00:57:20,320
essentially kind of moving down.

1123
00:57:26,300 --> 00:57:28,190
OK.

1124
00:57:28,190 --> 00:57:37,250
So you typically need MD
methods whenever you don't know

1125
00:57:37,250 --> 00:57:39,770
or you're not absolutely
sure what the transmission

1126
00:57:39,770 --> 00:57:41,720
mechanisms are.

1127
00:57:41,720 --> 00:57:44,390
If you know what the
transmission mechanisms are,

1128
00:57:44,390 --> 00:57:48,530
then it tends to be much more
efficient to do Monte Carlo.

1129
00:57:48,530 --> 00:57:50,360
You can set up
kinetic Monte Carlo

1130
00:57:50,360 --> 00:57:56,750
schemes that execute transitions
with the right transfer rate.

1131
00:57:56,750 --> 00:57:59,480
It's just that you have to know
what those transitions are.

1132
00:57:59,480 --> 00:58:03,020
So you've already done Monte
Carlo on lattice models,

1133
00:58:03,020 --> 00:58:05,810
but we saw it as
a sampling method.

1134
00:58:05,810 --> 00:58:08,490
You could also, now, see
it as a kinetic method.

1135
00:58:08,490 --> 00:58:12,140
Let's say you have
a lattice of atoms.

1136
00:58:12,140 --> 00:58:14,430
Let's make it easier
even, atoms and vacancies.

1137
00:58:17,620 --> 00:58:20,950
So we've shown you how
to do the thermodynamics.

1138
00:58:20,950 --> 00:58:24,310
Well, let's say you
want to study diffusion.

1139
00:58:24,310 --> 00:58:27,910
If you know microscopically
how this atom would migrate

1140
00:58:27,910 --> 00:58:31,220
to there, through which
path it would do that,

1141
00:58:31,220 --> 00:58:34,340
then you just calculate
the energy along that path.

1142
00:58:34,340 --> 00:58:38,260
And you can now do
a Monte Carlo where

1143
00:58:38,260 --> 00:58:41,410
the way you travel
through phase space

1144
00:58:41,410 --> 00:58:43,840
is by a real kinetic mechanism.

1145
00:58:43,840 --> 00:58:44,740
OK.

1146
00:58:44,740 --> 00:58:48,100
So you could now take
your exchange rate,

1147
00:58:48,100 --> 00:58:50,582
dependent not only on
the energy of the initial

1148
00:58:50,582 --> 00:58:52,540
and the final state,
which is the way we did it

1149
00:58:52,540 --> 00:58:54,880
in a Metropolis
algorithm, but also

1150
00:58:54,880 --> 00:58:57,160
have a prefactor
now that's kinetic.

1151
00:58:57,160 --> 00:58:59,890
So that has a frequency
times an activation barrier.

1152
00:58:59,890 --> 00:59:02,560
And that's essentially
kinetic Monte Carlo.

1153
00:59:02,560 --> 00:59:03,430
OK.

1154
00:59:03,430 --> 00:59:05,200
But you see, kinetic
Monte Carlo implies

1155
00:59:05,200 --> 00:59:10,510
that you know what
your possible migration

1156
00:59:10,510 --> 00:59:12,610
mechanisms are between states.

1157
00:59:12,610 --> 00:59:16,990
And usually you know that
reasonably well in things

1158
00:59:16,990 --> 00:59:18,130
like crystalline solids.

1159
00:59:18,130 --> 00:59:20,890
So there kinetic Monte
Carlo is very applicable.

1160
00:59:20,890 --> 00:59:23,200
People also use it on
surfaces even though it

1161
00:59:23,200 --> 00:59:25,462
gets a little more dicey there.

1162
00:59:25,462 --> 00:59:26,920
But once, of course,
you go to sort

1163
00:59:26,920 --> 00:59:29,560
of very disordered
systems, it really

1164
00:59:29,560 --> 00:59:32,240
gets way out of hand what
the kinetic mechanisms are.

1165
00:59:32,240 --> 00:59:35,920
And this wouldn't work anymore.

1166
00:59:35,920 --> 00:59:38,550
So I was going to show you
one example to sort of end up.

1167
00:59:38,550 --> 00:59:40,838
And on the continuity,
I was going

1168
00:59:40,838 --> 00:59:43,380
to take the same example that
we did for the phase stability.

1169
00:59:43,380 --> 00:59:46,290
If you remember this when
we talked about the cluster

1170
00:59:46,290 --> 00:59:48,600
expansion, we looked
at the phase diagram

1171
00:59:48,600 --> 00:59:53,340
of lithium and lithium vacancies
and lithium cobalt oxide.

1172
00:59:53,340 --> 00:59:56,910
So remember the issue is
that, if you take lithium out,

1173
00:59:56,910 --> 00:59:58,500
you create vacancies there.

1174
00:59:58,500 --> 01:00:01,050
And the issue was how
did they organize.

1175
01:00:01,050 --> 01:00:03,630
And I showed you how to
use a cluster expansion

1176
01:00:03,630 --> 01:00:06,810
to do the phase diagram,
the thermodynamics of that.

1177
01:00:06,810 --> 01:00:09,860
Let's say you want to
worry about the kinetics.

1178
01:00:09,860 --> 01:00:12,972
So essentially, how fast can
you get that lithium in and out?

1179
01:00:12,972 --> 01:00:14,430
So you essentially
want a diffusion

1180
01:00:14,430 --> 01:00:17,160
constant in that material.

1181
01:00:17,160 --> 01:00:21,800
Well, this is fairly well-suited
for kinetic Monte Carlo rather

1182
01:00:21,800 --> 01:00:22,800
than molecular dynamics.

1183
01:00:22,800 --> 01:00:26,340
First of all, the
rates are fairly slow.

1184
01:00:26,340 --> 01:00:29,480
So doing anything with
transition state theory

1185
01:00:29,480 --> 01:00:32,020
is going to help you a lot.

1186
01:00:32,020 --> 01:00:37,380
And you have transitions
between well-defined positions.

1187
01:00:37,380 --> 01:00:39,510
The atoms go from
one well-defined site

1188
01:00:39,510 --> 01:00:40,380
to another site.

1189
01:00:44,920 --> 01:00:51,850
So if you had dilute
diffusion, you

1190
01:00:51,850 --> 01:00:54,110
could just use
random walk theory.

1191
01:00:54,110 --> 01:00:57,370
So in that case, diffusion
is sort of trivial.

1192
01:00:57,370 --> 01:01:00,730
All you need is an activation
barrier and a prefactor.

1193
01:01:00,730 --> 01:01:01,720
All the rest you know.

1194
01:01:01,720 --> 01:01:03,520
You have the lattice
constant, which

1195
01:01:03,520 --> 01:01:04,990
is actually the jump linked.

1196
01:01:04,990 --> 01:01:07,180
And you have the geometric
correlation factor.

1197
01:01:07,180 --> 01:01:08,920
These are all known.

1198
01:01:08,920 --> 01:01:12,010
The things in green you
would have to calculate.

1199
01:01:12,010 --> 01:01:14,080
But let's do calculation.

1200
01:01:14,080 --> 01:01:17,770
Random walks only apply when
you have dilute diffusion.

1201
01:01:17,770 --> 01:01:20,530
What I mean with
dilute diffusion--

1202
01:01:20,530 --> 01:01:23,950
it's when the carrier of
diffusion is very dilute,

1203
01:01:23,950 --> 01:01:28,070
so it doesn't
interact with itself.

1204
01:01:28,070 --> 01:01:30,190
So if you have a
substitutional diffusion

1205
01:01:30,190 --> 01:01:32,540
with very dilute
vacancy concentration,

1206
01:01:32,540 --> 01:01:35,283
then this would apply.

1207
01:01:35,283 --> 01:01:36,700
The problem, of
course, that we're

1208
01:01:36,700 --> 01:01:38,658
going to look at the
lithium cobalt [INAUDIBLE]

1209
01:01:38,658 --> 01:01:40,210
is by definition not dilute.

1210
01:01:40,210 --> 01:01:41,680
You essentially
can go all the way

1211
01:01:41,680 --> 01:01:46,150
from all lithium on the site
to all vacancies on the site.

1212
01:01:46,150 --> 01:01:49,162
So you go to very
non-dilute regimes.

1213
01:01:49,162 --> 01:01:51,370
If you want to do that, you
need to actually simulate

1214
01:01:51,370 --> 01:01:53,950
the diffusion-- and you may
have even done that in the MD

1215
01:01:53,950 --> 01:01:55,420
homework--

1216
01:01:55,420 --> 01:01:58,240
by doing some kinetic
model and then tracking

1217
01:01:58,240 --> 01:02:00,400
the root mean
square displacement.

1218
01:02:00,400 --> 01:02:03,947
This is sort of a simple
form for the self-diffusion.

1219
01:02:03,947 --> 01:02:05,530
The chemical diffusion
constant, which

1220
01:02:05,530 --> 01:02:08,110
is the one you would use
in macroscopic theories,

1221
01:02:08,110 --> 01:02:12,040
is the self-diffusion times
the thermodynamic factor.

1222
01:02:12,040 --> 01:02:15,575
But the thermodynamic factor
is essentially the derivative

1223
01:02:15,575 --> 01:02:17,450
of the chemical potential
of the composition,

1224
01:02:17,450 --> 01:02:21,108
so that you already have from
the thermodynamic calculation.

1225
01:02:25,500 --> 01:02:27,072
OK.

1226
01:02:27,072 --> 01:02:29,280
So I have a few more slides
here that I already used,

1227
01:02:29,280 --> 01:02:31,260
so I'm going to
[INAUDIBLE] through them.

1228
01:02:31,260 --> 01:02:33,180
You remember how we
did the thermodynamics?

1229
01:02:33,180 --> 01:02:35,370
We built a lattice model.

1230
01:02:35,370 --> 01:02:37,560
We calculate a lot of
lithium vacancy arrangement,

1231
01:02:37,560 --> 01:02:40,020
build a cluster expansion,
and do Monte Carlo on that.

1232
01:02:40,020 --> 01:02:43,820
And that gives you
the phase diagram.

1233
01:02:43,820 --> 01:02:45,620
Let me skip that since
we did that already.

1234
01:02:45,620 --> 01:02:46,820
Those are the interactions.

1235
01:02:46,820 --> 01:02:49,550
OK.

1236
01:02:49,550 --> 01:02:51,380
How would you do
kinetic Monte Carlo?

1237
01:02:51,380 --> 01:02:53,180
I sort of also mentioned that.

1238
01:02:53,180 --> 01:02:58,100
You would now execute all the
exchanges in the Monte Carlo

1239
01:02:58,100 --> 01:03:00,230
that look like
diffusive processes.

1240
01:03:00,230 --> 01:03:04,170
And you would execute
them with the proper rate.

1241
01:03:04,170 --> 01:03:07,650
So you would have an
activation barrier in there.

1242
01:03:10,640 --> 01:03:12,040
OK.

1243
01:03:12,040 --> 01:03:13,520
I think I said all this.

1244
01:03:13,520 --> 01:03:15,220
OK.

1245
01:03:15,220 --> 01:03:20,180
So here's an example
of how that works out.

1246
01:03:20,180 --> 01:03:22,670
If you calculate barriers
in lithium cobalt oxide

1247
01:03:22,670 --> 01:03:24,830
and you then do a kinetic
Monte Carlo simulation,

1248
01:03:24,830 --> 01:03:27,680
you keep track of the root
mean square displacement.

1249
01:03:27,680 --> 01:03:30,500
You get the macroscopic
diffusivity.

1250
01:03:30,500 --> 01:03:34,940
And as you see, it varies
by orders of magnitude.

1251
01:03:34,940 --> 01:03:38,940
It's 10 to the minus 7 here
centimeters squared per second.

1252
01:03:38,940 --> 01:03:42,173
And it's 10 to
the minus 13 here.

1253
01:03:42,173 --> 01:03:43,840
You can already see
that you would never

1254
01:03:43,840 --> 01:03:48,820
get to all these time scales
with a simple dynamics model.

1255
01:03:54,920 --> 01:03:56,690
OK.

1256
01:03:56,690 --> 01:04:00,830
How you get
activation barriers--

1257
01:04:00,830 --> 01:04:02,270
typically, we get
this with what's

1258
01:04:02,270 --> 01:04:06,590
called a nudge
elastic band model.

1259
01:04:06,590 --> 01:04:10,980
You know, what's the issue in
finding activation barrier?

1260
01:04:10,980 --> 01:04:12,800
Well, you know
the initial state,

1261
01:04:12,800 --> 01:04:14,600
and you know the final state.

1262
01:04:14,600 --> 01:04:16,760
This is not necessarily
the state of an atom.

1263
01:04:16,760 --> 01:04:20,330
You really should think of these
as two states in phase space.

1264
01:04:20,330 --> 01:04:23,030
Now, for simplicity,
you can think, well,

1265
01:04:23,030 --> 01:04:26,090
only one atom moves, is
different, between those two

1266
01:04:26,090 --> 01:04:26,810
states.

1267
01:04:26,810 --> 01:04:28,520
But in reality, more
stuff is different.

1268
01:04:28,520 --> 01:04:30,950
Because if the atom has
moved, the other atoms

1269
01:04:30,950 --> 01:04:32,930
may have relaxed around it.

1270
01:04:32,930 --> 01:04:36,300
So these are really
states in phase space.

1271
01:04:36,300 --> 01:04:40,220
So what you have to find
is the path between the two

1272
01:04:40,220 --> 01:04:44,240
with the lowest energy
maximum along that path.

1273
01:04:44,240 --> 01:04:46,820
That's essentially
the activated state.

1274
01:04:46,820 --> 01:04:48,950
And a very practical
way to find is

1275
01:04:48,950 --> 01:04:51,290
what's called a nudged
elastic band model, which

1276
01:04:51,290 --> 01:04:56,270
is essentially that you do
a simulation on many systems

1277
01:04:56,270 --> 01:04:57,890
at a time.

1278
01:04:57,890 --> 01:05:01,280
And all the systems live
at intermediate states

1279
01:05:01,280 --> 01:05:04,970
between the final state
and the initial state.

1280
01:05:04,970 --> 01:05:08,570
And the way you keep them
there is interesting.

1281
01:05:08,570 --> 01:05:13,310
Rather than optimizing the
energy of each system, what

1282
01:05:13,310 --> 01:05:17,060
you optimize is the
sum of their energies,

1283
01:05:17,060 --> 01:05:20,120
the sum of their
Hamiltonian values,

1284
01:05:20,120 --> 01:05:25,460
plus some spring constant,
basically plus some energy

1285
01:05:25,460 --> 01:05:29,660
penalty, which is associated
with the difference in

1286
01:05:29,660 --> 01:05:31,980
coordinates between
the two systems.

1287
01:05:31,980 --> 01:05:34,880
And what that tends
to do is actually,

1288
01:05:34,880 --> 01:05:38,090
since this is a quadratic in
the difference in corniness,

1289
01:05:38,090 --> 01:05:42,730
it tends to spread out
the systems along the path

1290
01:05:42,730 --> 01:05:44,027
between the two states.

1291
01:05:44,027 --> 01:05:44,860
I mean, think of it.

1292
01:05:44,860 --> 01:05:49,090
If I got rid of this,
so if I set k to 0,

1293
01:05:49,090 --> 01:05:52,030
then I'm going to minimize this.

1294
01:05:52,030 --> 01:05:54,968
So they're all going
to go down here.

1295
01:05:54,968 --> 01:05:57,010
But what do you actually
do is you hold one here.

1296
01:05:57,010 --> 01:05:59,170
You hold one there.

1297
01:05:59,170 --> 01:06:02,020
And then because of the
harmonic potential between them,

1298
01:06:02,020 --> 01:06:04,430
you tend to end up with
systems that live in between.

1299
01:06:04,430 --> 01:06:08,710
So it's a very elegant way
of finding minimum energy

1300
01:06:08,710 --> 01:06:11,050
paths between states.

1301
01:06:11,050 --> 01:06:12,640
It's called the
elastic band method

1302
01:06:12,640 --> 01:06:14,310
or the nudged
elastic band method.

1303
01:06:14,310 --> 01:06:18,730
And there's a whole bunch
of variants of it as well.

1304
01:06:18,730 --> 01:06:20,630
It's the perfect
parallelization.

1305
01:06:20,630 --> 01:06:21,505
You may have noticed.

1306
01:06:24,280 --> 01:06:26,270
If you want to do,
say, 10 replicas--

1307
01:06:26,270 --> 01:06:28,360
so you want to do 10
points along the path--

1308
01:06:28,360 --> 01:06:30,280
you run on 10 processors.

1309
01:06:30,280 --> 01:06:32,650
Because it's the same
system, they pretty much

1310
01:06:32,650 --> 01:06:35,590
run about the same
amount of time.

1311
01:06:35,590 --> 01:06:37,760
They never have to
talk to each other.

1312
01:06:37,760 --> 01:06:40,510
They have to talk to each other
on extremely rare occasion

1313
01:06:40,510 --> 01:06:42,030
when they exchange coordinates.

1314
01:06:42,030 --> 01:06:43,180
That's it.

1315
01:06:43,180 --> 01:06:45,360
So it's a perfect
parallelization tool.

1316
01:06:49,850 --> 01:06:54,110
If you do that in lithium cobalt
oxide, you have a complication.

1317
01:06:54,110 --> 01:06:56,600
Just to show you how the data
for that kinetic Monte Carlo

1318
01:06:56,600 --> 01:07:03,740
simulation was derived, the
yellow is the lithium here.

1319
01:07:03,740 --> 01:07:07,130
These are oxygens,
the big red balls.

1320
01:07:07,130 --> 01:07:09,680
The way you can diffuse
from this side to that side

1321
01:07:09,680 --> 01:07:11,330
is straight through
the oxygen bond.

1322
01:07:11,330 --> 01:07:14,780
And if you do that,
this is obtained

1323
01:07:14,780 --> 01:07:16,080
by elastic band method.

1324
01:07:16,080 --> 01:07:17,970
So these are the replicas.

1325
01:07:17,970 --> 01:07:19,640
So this is initial stage.

1326
01:07:19,640 --> 01:07:22,100
This is the final state.

1327
01:07:22,100 --> 01:07:29,530
And so you get an activation
barrier of about 0.85.

1328
01:07:29,530 --> 01:07:34,360
So it's very nice how you
get these activation curves.

1329
01:07:34,360 --> 01:07:36,980
But the problem is that there's
two paths in this material.

1330
01:07:36,980 --> 01:07:40,330
So already you're starting to
run into a particular problem

1331
01:07:40,330 --> 01:07:42,010
of kinetic Monte Carlo.

1332
01:07:42,010 --> 01:07:44,300
Let's say you thought
that was the path.

1333
01:07:44,300 --> 01:07:48,610
Well, you'd be very wrong
because there's another path.

1334
01:07:48,610 --> 01:07:50,680
And that's one when the
outcome actually, rather

1335
01:07:50,680 --> 01:07:52,840
than going straight
through this bond,

1336
01:07:52,840 --> 01:07:57,820
it takes a curve into this
tetrahedron and comes back out.

1337
01:07:57,820 --> 01:08:00,700
And that's actually a path,
if you do elastic band, that

1338
01:08:00,700 --> 01:08:02,360
is so much lower.

1339
01:08:02,360 --> 01:08:05,590
It's only about 200
millielectron volts.

1340
01:08:05,590 --> 01:08:07,780
And it sort of shows
the real problem

1341
01:08:07,780 --> 01:08:11,650
you can run into that, when
you do kinetic Monte Carlo,

1342
01:08:11,650 --> 01:08:16,149
you have to assume what your
paths are for transitions.

1343
01:08:16,149 --> 01:08:19,029
And if you don't get them right,
you can be seriously wrong.

1344
01:08:19,029 --> 01:08:20,740
And that's one of the
nice things of MD,

1345
01:08:20,740 --> 01:08:21,580
that it's unbiased.

1346
01:08:21,580 --> 01:08:24,710
So it would find
these things for you.

1347
01:08:24,710 --> 01:08:26,678
But these are actually
the only two paths.

1348
01:08:26,678 --> 01:08:28,720
And of course, because of
the activation barrier,

1349
01:08:28,720 --> 01:08:30,319
this is so much lower.

1350
01:08:30,319 --> 01:08:32,770
This is pretty much the one
that gets executed almost

1351
01:08:32,770 --> 01:08:34,100
all the time.

1352
01:08:34,100 --> 01:08:37,135
The difference between 0.2
electron volts and 0.8 electron

1353
01:08:37,135 --> 01:08:40,060
volts at room
temperature is amazing.

1354
01:08:40,060 --> 01:08:44,870
It's, I think, 5, 6 orders of
magnitude in rate constant.

1355
01:08:44,870 --> 01:08:48,220
You just have to calculate the
exponential of each of these.

1356
01:08:51,300 --> 01:08:53,790
One of the problems you have
whenever you have two rate

1357
01:08:53,790 --> 01:08:58,500
constants is that, when
you do your Monte Carlo,

1358
01:08:58,500 --> 01:09:01,819
you can only scale
the fastest one away.

1359
01:09:01,819 --> 01:09:03,800
So you have essentially
one scale factor

1360
01:09:03,800 --> 01:09:06,090
in the time of your
Monte Carlo simulation.

1361
01:09:06,090 --> 01:09:09,890
So when you do multiple
time scales, when

1362
01:09:09,890 --> 01:09:11,390
you have multiple
time scales, it

1363
01:09:11,390 --> 01:09:15,990
gets very inefficient to
do kinetic Monte Carlo.

1364
01:09:15,990 --> 01:09:16,770
But this works.

1365
01:09:16,770 --> 01:09:20,010
You can get diffusion constants.

1366
01:09:20,010 --> 01:09:24,340
This is with
elastic band method.

1367
01:09:24,340 --> 01:09:27,300
This is the activation barrier
for that low energy mechanism

1368
01:09:27,300 --> 01:09:29,319
as a function of
lithium concentration,

1369
01:09:29,319 --> 01:09:31,779
so very dependent
on concentration.

1370
01:09:31,779 --> 01:09:34,050
So you can do your
kinetic Monte Carlo.

1371
01:09:34,050 --> 01:09:36,569
And then you have these,
and you can go all the way

1372
01:09:36,569 --> 01:09:38,520
to macroscopic stimulation.

1373
01:09:38,520 --> 01:09:40,109
Then you can solve
a fixed equation

1374
01:09:40,109 --> 01:09:44,340
and actually look at diffusion
on the scale of microns.

1375
01:09:44,340 --> 01:09:47,399
So it's a fairly simplistic
way of course graining.

1376
01:09:50,399 --> 01:09:51,240
OK.

1377
01:09:51,240 --> 01:09:54,330
So I'm going to end just
giving you some reference.

1378
01:09:54,330 --> 01:09:57,150
If you want to read more about
the quasi-continuum method,

1379
01:09:57,150 --> 01:10:00,420
there's these great papers
by Miller and Tadmor.

1380
01:10:00,420 --> 01:10:02,010
These are actually
somewhat similar,

1381
01:10:02,010 --> 01:10:05,123
but the 202 version
is an update.

1382
01:10:05,123 --> 01:10:06,540
These are not the
original papers,

1383
01:10:06,540 --> 01:10:09,270
but they're actually
sometimes more pedagogical

1384
01:10:09,270 --> 01:10:10,855
than the original papers.

1385
01:10:10,855 --> 01:10:12,480
But you'll find the
references in there

1386
01:10:12,480 --> 01:10:14,550
to the original papers.

1387
01:10:14,550 --> 01:10:20,370
The problem of sort of dynamics
at that scale is my own work.

1388
01:10:20,370 --> 01:10:24,120
The accelerated MD, there's a
series of papers by Art Voter.

1389
01:10:24,120 --> 01:10:27,090
But this annual review
of materials research

1390
01:10:27,090 --> 01:10:30,540
is, I think, a great review
if you want to read about it.

1391
01:10:30,540 --> 01:10:34,800
And the lithium cobalt
application is in a few papers,

1392
01:10:34,800 --> 01:10:36,150
but it's quite elaborate.

1393
01:10:36,150 --> 01:10:37,860
The whole idea of
kinetic Monte Carlo

1394
01:10:37,860 --> 01:10:40,060
and parameterizing
activation barriers

1395
01:10:40,060 --> 01:10:43,390
is quite elaborately
discussed in this paper.

1396
01:10:43,390 --> 01:10:46,360
So I think we're done a
little early, but that's fine.

1397
01:10:46,360 --> 01:10:47,610
So I'm going to end here.

1398
01:10:47,610 --> 01:10:52,490
And I'll see you on Thursday
for the lecture from Ford.