WEBVTT
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NICOLA MARZARI: To calculate
ensemble averages--
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that is to calculate
thermodynamical properties.
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And we have seen
also a few movies
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on how we can calculate
reaction barriers
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and how we can follow in
real time chemical kinetics.
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The last application
of molecular dynamics
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is likely more different.
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In particular, one can
use a final temperature,
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classic in molecular dynamics,
as an optimization scheme.
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In general, suppose that you
have a complex potential energy
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surface.
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This is obviously
in one dimension,
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but you could
think this as being
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in a multi-dimensional space.
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And actually, your
potential energy function
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could represent a
complex cost function.
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Again, it could be,
say, the problem
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of optimizing the
usage of your planes
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on several routes for
one airline company.
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And obviously, depending on
where you put your planes,
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you have a certain cost.
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And so this cost function
tends to be very complex
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depending on your coordinates,
how you arrange your planes.
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And finding either the
global minimum or sort of one
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of the lowest minima can
be a very complex affair.
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And one of the
powerful techniques
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that has been introduced
in the early '80s
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is something called a
simulated annealing.
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That is basically a
technique that basis itself
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on a thermodynamical analogy.
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That is suppose that you
want to find this minimum.
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And if you were to use
a deterministic recipe,
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it would be very
difficult. You would
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start from a certain
point and then
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go down according
to the gradient
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until you end up in what
would be a local minima.
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And so what, in particular,
Kilpatrick, Gelatt, and Vecchi
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introduced was a thermodynamical
technique in which you really
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populate your phase space
with what we call walkers.
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So these are sort of
dynamical systems.
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So those are the
points that represent
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those dynamical systems.
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You give them some coordinates.
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That is they are going to
have some potential energy.
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And then you give
them some temperature.
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So they are going to
have some kinetic energy.
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You can think this
as a swarm of skiers
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sort of going around
your phase space.
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And if you have a lot
of them and if these
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have a lot of temperature, they
will move all over phase space.
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But by the time you start
sort of very slowly cooling
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this dynamical systems down,
you start removing temperature
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for them, they are going
to condensate really
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in whichever local
minima they find.
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So some of them may
end up, say, here.
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And you cool them
down, and slowly they
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find themselves here.
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But if you cool them
sufficiently slow--
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and really in order to make
this into an exact schema,
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you have to pull them
almost infinitely slowly.
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So it's never going to be
an exact approach, but just
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a stochastic approach.
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But, you know, with some luck,
you'll find that some of them
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start condensing in
interesting minima.
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And then you choose basically
your lowest minima possible.
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And this is actually a very
sort of practical and useful
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approach.
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And so it's sort
of fairly widely
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used in optimization problems.
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OK, so this sort of
concludes the sort
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of set of applications
of molecular dynamics.
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Now, I wanted to
sort of hint at some
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of the advanced statistical
mechanics techniques
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that you'll actually see later
in the class that, generally
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speaking, go under the umbrella
name of Green-Kubo techniques.
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And sort of I'm taking
one of the simplest
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examples in which, actually,
Albert Einstein was involved.
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So it's an interesting approach.
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Well, this has to do on starting
a macroscopic property that
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is really diffusion in a solid.
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So you could think
of a silicon crystal.
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And we are doping that silicon
with gallium or phosphorus,
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and so you are
interested in studying
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how the impurities in
this system move around.
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And so if you
want, what you have
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is that you have a perturbation.
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You put a sort of gallium
on the surface of silicon,
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and then this diffuses
into the solid.
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And sort of the general
Green-Kubo approach
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is a formalism that relates
some of these macroscopic
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properties, like a
diffusion coefficient,
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a transpose property, to
microscopic properties.
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And you'll see that as sort of
fluctuations of the equilibrium
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distribution.
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OK.
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So suppose that we are looking
at the diffusion of, again,
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something like
gallium and silicon.
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What you would define as your
sort of fundamental variable
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is the concentration
of impurities.
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So that's a sort of
four-dimensional function is
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a concentration function
of the [INAUDIBLE] space
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you are interested
and function of time.
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And then, obviously,
there is sort
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of the diffusion
law for this that
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says that basically
the current is
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really proportional to the
gradient of the concentration.
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And then you put this together
with a sort of continuity
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equation that tells you that the
derivative of the concentration
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with respect to time plus the
divergence of your current--
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that is your flux--
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needs to be constant.
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That equals-- let's say
it's going to be equal to 0.
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I mean, this very
simply says that,
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if an infinitesimal volume
the concentration changes,
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it's that because there
is a current going out
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of that infinitesimal volume.
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And so the divergence of
the current measures this.
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And with fixed diffusion law,
we can put the 2 and 2 together
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and basically obtain this
relation, this differential
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equation, for the
concentration profile in time.
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So if you want,
these are starting
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macroscopic relationship.
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And then what
becomes interesting
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is the derivation that
Einstein did in order
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to recover the connection
with the microscopic dynamics.
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And this is just a little
bit of algebra, which
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I will go over fairly quickly.
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But, basically, what
we are doing here,
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we are actually
multiplying the left term
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and the right-hand
terms by r square
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and integrating
it over all space.
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And once we do
that, well, what we
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have is that on the
right-hand side,
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this requires some calculus.
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So I won't go really
into the derivation.
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But by sort of
integration by parts,
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you can actually show that
this integral provided,
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say, the concentration
is normalized to 1
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when integrated in all
space, what we really obtain
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is just 2 times the
dimensionality of your problem.
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That is, if you are in one
dimension, sort of small d
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is going to be equal to 1.
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If you are in two
dimensions, it's
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going to be equal to 2, to
3, and so on and so forth.
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So this integral
really just gives us
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the dimension of the space.
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And again, we won't go
into how we prove this.
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What we have on the left-hand
side, on the contrary,
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is really, you see, the
average value of r square.
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What we mean by this
bracket that you see
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is the integral
over all space of r
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square with the probability.
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If you want the
concentration, it's
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identical to the
probability of having
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a certain particle in a certain
position in a certain time.
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So what this integral
means is really
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the average value over
all space of r square,
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how much your
particles has moved.
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And we have the
derivative and the time.
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And so you see we
start seeing somehow
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a connection between a
diffusion coefficient
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that's the macroscopic property
and the microscopic properties,
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the dynamic piece, how much
the particles are moving
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once you put them in somewhere.
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Let me actually
give you an example
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of what would this
quantity look like
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and how we would calculate it in
a molecular dynamic simulation.
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Again, you can think
of your gallium atoms
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being at the surface.
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At the beginning, you
put everything in motion.
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And they are going to move
with a sort of random walk
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towards the inside.
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And if you want to calculate
this sort of average value
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of what is called the
mean square displacement,
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the average value
of r square, well,
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you need really to do the
average over all your particle
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at a certain time
of delta r square.
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Delta r square is really how far
all these particles have moved.
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And you know, this
would look at something
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like this in a
typical, say, solid.
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Actually, the example
that I have here
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is slightly more exotic, but
it was a sort of good example
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to show you the difference
between a liquid and a solid.
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And it's the case
of silver iodide
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that is, in particular,
an ionic crystal
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in which the blue iodine
atoms sit on a BCC lattice.
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And for each iodine,
there is a silver atom.
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But above, say, 420
Kelvin, this system
00:10:26.400 --> 00:10:28.680
is in a very peculiar
state of matter, what
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is called a super ionic state
in which this iodine atom
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oscillates around an
equilibrium position exactly
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as atoms do in a solid, while
the silver atoms sort of move
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around exactly like a liquid.
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So it's a system
that is really solid.
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It's crystalline.
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But one sublattice,
the silver sublattice,
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is sort of moving
around as a liquid.
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And so if you calculate what
are the average mean square
00:10:58.710 --> 00:11:01.590
displacements-- that
is if you average, say,
00:11:01.590 --> 00:11:07.170
all the iodine atoms, what is
their average delta r squared,
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how much they move around--
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well, you'll discover
that really, in a sense,
00:11:12.240 --> 00:11:15.360
that they are a
crystalline solid.
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They are just going to oscillate
around their equilibrium
00:11:18.390 --> 00:11:19.480
position.
00:11:19.480 --> 00:11:21.900
And so you calculate
that average.
00:11:21.900 --> 00:11:24.660
During a molecular
dynamic simulation,
00:11:24.660 --> 00:11:28.710
you see there is a sort
of first very short period
00:11:28.710 --> 00:11:29.940
of thermalization.
00:11:29.940 --> 00:11:32.280
The atoms start
moving, but then they
00:11:32.280 --> 00:11:33.840
are just oscillating around.
00:11:33.840 --> 00:11:36.310
They are not going anywhere.
00:11:36.310 --> 00:11:39.180
So their instantaneous
mean squared displacement
00:11:39.180 --> 00:11:42.010
as a value that is
just a constant.
00:11:42.010 --> 00:11:45.480
So this is sort of typical
of a crystalline solid.
00:11:45.480 --> 00:11:48.630
If you do the same
thing for the silvers,
00:11:48.630 --> 00:11:52.110
you see that, as a function
of increase in temperature,
00:11:52.110 --> 00:11:57.330
you see, at 250 Kelvin below
the superionic transition, also
00:11:57.330 --> 00:12:00.877
the silvers sort of look
more or less flatter.
00:12:00.877 --> 00:12:02.460
But when you increase
the temperature,
00:12:02.460 --> 00:12:05.310
their mean square
displacement starts becoming
00:12:05.310 --> 00:12:07.270
a linear function of time.
00:12:07.270 --> 00:12:09.810
So the silvers are
really moving around.
00:12:09.810 --> 00:12:12.330
They are diffusing
away as a liquid.
00:12:12.330 --> 00:12:15.780
And so in the mean
square displacement,
00:12:15.780 --> 00:12:19.080
you can see the
signature of a liquid
00:12:19.080 --> 00:12:23.010
when they increase
linearly with time.
00:12:23.010 --> 00:12:25.530
Or you can see the
signature of a solid
00:12:25.530 --> 00:12:27.500
where they just are
not going anywhere.
00:12:27.500 --> 00:12:31.065
They just vibrate around
the equilibrium position.
00:12:36.920 --> 00:12:39.160
And so this is the
quantity that we
00:12:39.160 --> 00:12:42.070
could calculate in a
molecular dynamic simulation.
00:12:42.070 --> 00:12:45.280
The average of the mean
square displacement,
00:12:45.280 --> 00:12:46.600
I have written it here.
00:12:46.600 --> 00:12:50.170
Remember, this is going to be
equal to 2 times the diffusion
00:12:50.170 --> 00:12:54.190
coefficient times the
dimensionality of your system.
00:12:54.190 --> 00:12:58.240
And now, we sort of work
a little bit with this.
00:12:58.240 --> 00:13:02.620
And in particular, we
introduce this sort
00:13:02.620 --> 00:13:06.590
of alternative definition of
the displacement of an atom.
00:13:06.590 --> 00:13:08.440
This is actually very useful.
00:13:08.440 --> 00:13:12.490
If you are sort of working in
periodic boundary conditions,
00:13:12.490 --> 00:13:14.020
you need to be careful.
00:13:14.020 --> 00:13:18.790
Often, your molecular dynamics
codes, for simplicity,
00:13:18.790 --> 00:13:23.200
always show you the
coordinates of the ions
00:13:23.200 --> 00:13:25.720
as they were sitting in
your first unit cells.
00:13:25.720 --> 00:13:28.330
If an atom is
diffusing as a liquid
00:13:28.330 --> 00:13:31.300
and it sort of moves
around and from one cell
00:13:31.300 --> 00:13:33.970
moves in the second unit
cell, your code usually
00:13:33.970 --> 00:13:36.040
will actually sort
of bring it back
00:13:36.040 --> 00:13:38.890
by a lattice translation
in the first unit cell.
00:13:38.890 --> 00:13:43.480
So a universal way to
sort of take into account
00:13:43.480 --> 00:13:46.360
what the actual
position of an atom is
00:13:46.360 --> 00:13:50.890
is actually write the
position that is usually
00:13:50.890 --> 00:13:53.680
a vector that satisfies this
periodic boundary condition
00:13:53.680 --> 00:13:55.760
as an integral of the velocity.
00:13:55.760 --> 00:13:59.890
So in that sense, the
position of your particle
00:13:59.890 --> 00:14:04.150
is always taking correctly
into account all the distance
00:14:04.150 --> 00:14:05.650
that has been traversed.
00:14:05.650 --> 00:14:08.140
So if you have a
unit cell really
00:14:08.140 --> 00:14:11.590
and your particle
is moving, you'll
00:14:11.590 --> 00:14:14.680
sort of integrate
correctly your trajectory
00:14:14.680 --> 00:14:17.830
by doing the integral of
the velocity, while sort
00:14:17.830 --> 00:14:23.200
of your algorithm would actually
bring back the particle once it
00:14:23.200 --> 00:14:24.220
crosses a boundary.
00:14:24.220 --> 00:14:24.790
OK.
00:14:24.790 --> 00:14:26.320
Nothing deep here,
we are just sort
00:14:26.320 --> 00:14:30.670
of writing the position as
the integral of the velocity.
00:14:30.670 --> 00:14:33.580
But with this definition,
we can actually
00:14:33.580 --> 00:14:37.210
go back and look at
what is the expression
00:14:37.210 --> 00:14:41.220
for the average of the
mean square displacement.
00:14:41.220 --> 00:14:42.760
That is the delta x square.
00:14:42.760 --> 00:14:46.090
And now, we write it
actually as the square
00:14:46.090 --> 00:14:49.910
of the integral of the
velocity at the instant of t.
00:14:49.910 --> 00:14:53.830
And so I'm just sort of writing
this integral out explicitly
00:14:53.830 --> 00:14:56.890
when I sort of
expand the square.
00:14:56.890 --> 00:14:58.810
I have an integral
in t and t prime.
00:14:58.810 --> 00:15:03.460
And again, the bracket means
just the ensemble average
00:15:03.460 --> 00:15:06.370
on all your particles, so it
commutes with the integral.
00:15:06.370 --> 00:15:09.130
And so what we are
calculating here
00:15:09.130 --> 00:15:16.210
is really the sum over all the
particles and averaging that.
00:15:16.210 --> 00:15:20.260
That is dividing by the
total number of particles
00:15:20.260 --> 00:15:24.970
of the velocity at certain
instant t prime times
00:15:24.970 --> 00:15:30.570
the velocity of a
certain instant t2 prime.
00:15:30.570 --> 00:15:34.870
And we can actually, for
computational convenience,
00:15:34.870 --> 00:15:37.200
this is really an
integral on a square,
00:15:37.200 --> 00:15:40.920
integral from 0 to t in one
dimension and the integral
00:15:40.920 --> 00:15:42.780
from 0 to t in the
other dimension.
00:15:42.780 --> 00:15:44.580
But this expression
is symmetrical.
00:15:44.580 --> 00:15:48.540
So we can actually integrate
this only on half the square.
00:15:48.540 --> 00:15:51.420
We are integrating it
only on the triangle.
00:15:51.420 --> 00:15:58.680
If this is were t and t prime
and our integration interval
00:15:58.680 --> 00:16:01.740
in this, we can just
sort, for simplicity,
00:16:01.740 --> 00:16:04.960
integrate into half of this.
00:16:04.960 --> 00:16:06.150
OK.
00:16:06.150 --> 00:16:09.240
So what we have
achieved here is we
00:16:09.240 --> 00:16:12.450
have written this
average mean squared
00:16:12.450 --> 00:16:18.540
displacement as an integral
of the average product
00:16:18.540 --> 00:16:22.260
between the velocity
at a certain instant
00:16:22.260 --> 00:16:25.290
and the velocity at
a different instant.
00:16:25.290 --> 00:16:29.230
And this is sort of where our
connection with the equilibrium
00:16:29.230 --> 00:16:31.050
properties is
starting to emerge.
00:16:31.050 --> 00:16:32.520
And you'll see this in a moment.
00:16:32.520 --> 00:16:37.140
This is called velocity
autocorrelation function.
00:16:37.140 --> 00:16:40.740
And I'm writing it
more explicitly.
00:16:40.740 --> 00:16:44.160
All of this has been done, again
to keep the algebra simple,
00:16:44.160 --> 00:16:45.730
in two dimensions.
00:16:45.730 --> 00:16:48.930
So remember, the
Einstein relation.
00:16:48.930 --> 00:16:50.890
Sorry, we are in one dimension.
00:16:50.890 --> 00:16:54.000
So remember, the
Einstein relation.
00:16:54.000 --> 00:16:57.090
The small d, the dimensionality
of your system, is 1.
00:16:57.090 --> 00:17:01.620
So we have the 2 times
the diffusion coefficient
00:17:01.620 --> 00:17:05.880
is equal to the derivative of
the mean square displacement.
00:17:05.880 --> 00:17:09.630
And mean square
displacement in itself
00:17:09.630 --> 00:17:12.420
has been written as
the double integral.
00:17:12.420 --> 00:17:14.700
But when you take the
derivative with respect to t,
00:17:14.700 --> 00:17:17.849
one of the integral cancels out.
00:17:17.849 --> 00:17:20.920
So if you want, this
is our final relation.
00:17:20.920 --> 00:17:25.020
Let me actually sort
of write it over here
00:17:25.020 --> 00:17:27.780
and remove the factor of 2.
00:17:27.780 --> 00:17:31.860
We have the diffusion
coefficient as
00:17:31.860 --> 00:17:33.390
written as this integral.
00:17:33.390 --> 00:17:36.930
We can sort of exploit the
translational invariance
00:17:36.930 --> 00:17:37.860
in time.
00:17:37.860 --> 00:17:40.170
Really, if we are
looking at what
00:17:40.170 --> 00:17:44.730
is the average value of
the product of the velocity
00:17:44.730 --> 00:17:47.490
at a certain instant, d
prime, times the velocity
00:17:47.490 --> 00:17:53.160
of another instant d2 prime,
well, that average product
00:17:53.160 --> 00:17:57.330
is not going to be different
if we translate it in time.
00:17:57.330 --> 00:18:01.540
So we can refer it to
an arbitrary origin.
00:18:01.540 --> 00:18:04.550
And so we just do a
translation in which we shift
00:18:04.550 --> 00:18:07.930
d2 prime into 0 and so on.
00:18:07.930 --> 00:18:10.770
So this is sort of
our final expression.
00:18:10.770 --> 00:18:14.340
And this is what is
called a velocity-velocity
00:18:14.340 --> 00:18:16.330
autocorrelation function.
00:18:16.330 --> 00:18:21.030
So you see, what we have is
that the macroscopic property,
00:18:21.030 --> 00:18:23.880
the diffusion coefficient,
that is really
00:18:23.880 --> 00:18:27.990
telling you how a system
responded to a perturbation.
00:18:27.990 --> 00:18:31.470
You put some gallium on
your surface of silicon,
00:18:31.470 --> 00:18:36.390
and then there is this
macroscopic diffusion transfer
00:18:36.390 --> 00:18:37.440
process.
00:18:37.440 --> 00:18:40.500
That is there is a perturbation,
and the system evolves.
00:18:40.500 --> 00:18:44.190
It can actually be
related to an equilibrium
00:18:44.190 --> 00:18:46.170
property of the system.
00:18:46.170 --> 00:18:47.970
That is really,
ultimately, what are
00:18:47.970 --> 00:18:52.560
the mean square displacements,
how atom microscopically move
00:18:52.560 --> 00:18:53.400
around.
00:18:53.400 --> 00:18:57.810
And in particular, the
quantity that is playing here
00:18:57.810 --> 00:19:00.510
is this velocity-velocity
autocorrelation.
00:19:00.510 --> 00:19:05.370
You see, what this is
looking at is suppose
00:19:05.370 --> 00:19:09.360
that you have a velocity
at a certain instant.
00:19:09.360 --> 00:19:11.870
And then you look
at that instant
00:19:11.870 --> 00:19:18.150
that is sort of tau away in time
from sort of your instant 0.
00:19:18.150 --> 00:19:22.350
And you look at the product
of these two velocities.
00:19:22.350 --> 00:19:27.270
Now, if tau is very
small, the velocity
00:19:27.270 --> 00:19:29.990
is not going to have
changed very much.
00:19:29.990 --> 00:19:34.920
So in the limit of small
tau, your velocity at 0
00:19:34.920 --> 00:19:38.223
and your velocity at tau are
going to be very similar.
00:19:38.223 --> 00:19:40.140
So if you want, there
is a lot of correlation.
00:19:40.140 --> 00:19:42.690
When you take this
product, it's going
00:19:42.690 --> 00:19:46.440
to look a lot like v0 squared.
00:19:46.440 --> 00:19:48.690
And then all these things
are normalized properly.
00:19:48.690 --> 00:19:51.510
So it could look like 1.
00:19:51.510 --> 00:19:54.270
But as you sort of
move away in time,
00:19:54.270 --> 00:19:57.540
you go towards longer
and longer times,
00:19:57.540 --> 00:19:59.610
there is going to
be no correlation.
00:19:59.610 --> 00:20:02.580
Your velocity at 0
and your velocity
00:20:02.580 --> 00:20:05.580
at an instant that is
very far away in time
00:20:05.580 --> 00:20:07.320
is not going to be correlated.
00:20:07.320 --> 00:20:12.960
And so the product of this can
be, if you want, any number.
00:20:12.960 --> 00:20:14.910
And when you take
the average, you
00:20:14.910 --> 00:20:17.730
see the sort of ensemble
average, the bracket.
00:20:17.730 --> 00:20:21.510
When you average this
quantity on all your particles
00:20:21.510 --> 00:20:23.640
in the system,
this is just going
00:20:23.640 --> 00:20:26.910
to be 0 because
this thing can have
00:20:26.910 --> 00:20:29.010
any positive or negative value.
00:20:29.010 --> 00:20:31.820
Because there is no correlation
between the velocity
00:20:31.820 --> 00:20:33.050
at the large tau.
00:20:33.050 --> 00:20:35.990
And so this thing
is going to be 0.
00:20:35.990 --> 00:20:36.530
OK.
00:20:36.530 --> 00:20:39.860
So the limit of this
velocity-velocity
00:20:39.860 --> 00:20:44.690
autocorrelation for very large
tau is going to be equal to 0.
00:20:44.690 --> 00:20:49.910
The limit for sort of very small
tau is going to be equal to 1
00:20:49.910 --> 00:20:52.670
or whatever your
normalization factor is.
00:20:52.670 --> 00:20:55.610
And then there should be
some interesting structure
00:20:55.610 --> 00:20:57.680
at a certain time.
00:20:57.680 --> 00:21:00.110
And we'll see it
in a moment that
00:21:00.110 --> 00:21:03.500
suppose your system is
actually sort of oscillating.
00:21:03.500 --> 00:21:05.660
You are looking not
at the liquid part
00:21:05.660 --> 00:21:08.210
of your silver iodide,
but you are looking just
00:21:08.210 --> 00:21:11.090
at the sort of
crystalline iodine
00:21:11.090 --> 00:21:13.130
sort of oscillating around.
00:21:13.130 --> 00:21:17.420
Well, suppose that your iodine
atom has a certain period
00:21:17.420 --> 00:21:18.390
of oscillation.
00:21:18.390 --> 00:21:20.900
So it sort of keeps
going back and forth.
00:21:20.900 --> 00:21:23.570
Well, then there will
be a lot of correlation
00:21:23.570 --> 00:21:26.840
if you look at your
velocity at a time 0
00:21:26.840 --> 00:21:29.930
and your velocity
at an instant that
00:21:29.930 --> 00:21:32.360
is roughly equal to a
period of oscillation.
00:21:32.360 --> 00:21:35.660
Because if this thing
were to oscillate exactly
00:21:35.660 --> 00:21:39.150
around this equilibrium,
every sort of period
00:21:39.150 --> 00:21:41.700
you would have that the
velocity has become the same.
00:21:41.700 --> 00:21:42.320
OK.
00:21:42.320 --> 00:21:45.710
So you will see a very
definite structure
00:21:45.710 --> 00:21:48.080
in this correlation function.
00:21:48.080 --> 00:21:50.630
If your system is not a
crystal, if your system is
00:21:50.630 --> 00:21:55.430
sort of diffusing away as a
liquid, as your time increases,
00:21:55.430 --> 00:21:57.800
the correlation becomes 0.
00:21:57.800 --> 00:22:01.010
And even if it's a crystal,
but it's not really
00:22:01.010 --> 00:22:05.750
oscillating perfectly,
like a harmonic
00:22:05.750 --> 00:22:08.300
oscillator around the
period, when you really
00:22:08.300 --> 00:22:10.730
got to very, very
long time away,
00:22:10.730 --> 00:22:14.930
you start losing this
correlation that sort of takes
00:22:14.930 --> 00:22:16.580
place at every beta.
00:22:16.580 --> 00:22:20.510
And that average quantity
starts in itself to be 0.
00:22:20.510 --> 00:22:23.940
So how does our
velocity-velocity
00:22:23.940 --> 00:22:26.110
autocorrelation
function looks like?
00:22:29.020 --> 00:22:31.120
Well, it looks like
something like this.
00:22:31.120 --> 00:22:33.950
I need to sort of graph it here.
00:22:33.950 --> 00:22:38.410
So again, when you are really
sort of far away in time--
00:22:38.410 --> 00:22:40.270
so, you know, what
we are plotting
00:22:40.270 --> 00:22:47.260
is the ensemble average
of vdt times vf0.
00:22:47.260 --> 00:22:51.340
So very, very far away in
time, it's going to be 0.
00:22:51.340 --> 00:22:55.930
There is no correlation between
the velocity of the atoms.
00:22:55.930 --> 00:23:02.680
At very, very small times, there
is really maximum correlation.
00:23:02.680 --> 00:23:05.800
And then what is
in between really
00:23:05.800 --> 00:23:08.500
depends on the
dynamics of the system.
00:23:08.500 --> 00:23:14.110
And it can look very different
in different systems.
00:23:14.110 --> 00:23:18.250
But Einstein relation and
the Green-Kubo formula
00:23:18.250 --> 00:23:20.170
that we have seen
before actually
00:23:20.170 --> 00:23:24.520
relates, first of all,
the integral from 0
00:23:24.520 --> 00:23:29.120
to infinity of the expansion
to the diffusion coefficient.
00:23:29.120 --> 00:23:31.720
So again, these functions
really represent
00:23:31.720 --> 00:23:34.510
a [INAUDIBLE]
microscopic fluctuation
00:23:34.510 --> 00:23:38.290
at equilibrium, sort of how the
velocities are correlated, so
00:23:38.290 --> 00:23:40.870
how your vibrate around.
00:23:40.870 --> 00:23:43.780
But all the sort of
algebra we have seem before
00:23:43.780 --> 00:23:47.350
shows also that the
integral of this quantity
00:23:47.350 --> 00:23:50.780
gives you the diffusion
coefficient [INAUDIBLE]..
00:23:50.780 --> 00:23:53.270
And that's one way of
calculating the diffusion
00:23:53.270 --> 00:23:54.410
coefficient.
00:23:54.410 --> 00:23:57.500
The other way, if you
go back to your slides,
00:23:57.500 --> 00:24:01.350
would be just calculating
the derivative with respect
00:24:01.350 --> 00:24:03.740
to time of the mean
squared displacement.
00:24:03.740 --> 00:24:05.750
And so in a liquid,
you have seen
00:24:05.750 --> 00:24:09.080
that your mean square
displacements sort of become
00:24:09.080 --> 00:24:12.350
linear if you sort of weighed
enough time with respect
00:24:12.350 --> 00:24:13.080
to time.
00:24:13.080 --> 00:24:17.150
And so the diffusion coefficient
is also equivalently given
00:24:17.150 --> 00:24:20.000
by the slope of this system.
00:24:20.000 --> 00:24:22.850
One of the interesting
things that you
00:24:22.850 --> 00:24:26.510
obtain from the sort
of velocity-velocity
00:24:26.510 --> 00:24:30.020
autocorrelation function
that you don't obtain
00:24:30.020 --> 00:24:33.380
from the slope of your
mean square displacement
00:24:33.380 --> 00:24:36.980
is that you can obtain what
we call the power spectrum.
00:24:36.980 --> 00:24:40.670
That is you can look at
a system like a liquid
00:24:40.670 --> 00:24:44.870
and figure out what are
the typical vibrations
00:24:44.870 --> 00:24:46.130
in your system.
00:24:46.130 --> 00:24:49.400
Again, this is because, you
know, what we have said.
00:24:49.400 --> 00:24:54.150
If a system, if an atom,
is oscillating around, say,
00:24:54.150 --> 00:24:56.300
an equilibrium
position, there is going
00:24:56.300 --> 00:24:58.460
to be a lot of correlation.
00:24:58.460 --> 00:25:01.860
That is, if we look at time 0
and if you look after a period
00:25:01.860 --> 00:25:05.900
t, the velocities are, again,
going to be very similar.
00:25:05.900 --> 00:25:08.550
If you look again at
time 2t, they are, again,
00:25:08.550 --> 00:25:09.560
going to be similar.
00:25:09.560 --> 00:25:11.400
If you look at the time
3t, they are, again,
00:25:11.400 --> 00:25:12.690
going to be similar.
00:25:12.690 --> 00:25:16.250
So in the velocity-velocity
autocorrelation function,
00:25:16.250 --> 00:25:19.820
actually you will see not
only a peak at 0, but a peak
00:25:19.820 --> 00:25:24.620
at t, at 2t, and 3t and so on,
and then sort of slowly decay.
00:25:24.620 --> 00:25:26.900
But so that means that
the velocity-velocity
00:25:26.900 --> 00:25:32.000
autocorrelation function
will sort of show somehow
00:25:32.000 --> 00:25:34.940
some periodic features
that are related
00:25:34.940 --> 00:25:38.970
to what is the typical
dynamics of your particles.
00:25:38.970 --> 00:25:41.900
And so if you do the Fourier
transform of that, the Fourier
00:25:41.900 --> 00:25:44.340
transform of a
function, it picks up
00:25:44.340 --> 00:25:47.640
what are the relevant
frequencies in that function.
00:25:47.640 --> 00:25:49.580
And so if you do
that, you actually
00:25:49.580 --> 00:25:52.760
find out that what is
the vibrational density
00:25:52.760 --> 00:25:55.700
of states, what are the typical
frequencies of your system.
00:25:55.700 --> 00:25:58.640
And this is sort
of an example that
00:25:58.640 --> 00:26:02.580
comes from a molecular
dynamics simulation of water.
00:26:02.580 --> 00:26:05.570
So what you have is you
have your liquid water.
00:26:05.570 --> 00:26:09.060
Remember, that in a
system like liquid water,
00:26:09.060 --> 00:26:13.760
you actually need only 30, 40,
50 molecules in a unit cell
00:26:13.760 --> 00:26:17.150
to basically simulate the
infinite system as long as you
00:26:17.150 --> 00:26:21.020
are sort of enough far away
from the melting or freezing
00:26:21.020 --> 00:26:22.100
transition.
00:26:22.100 --> 00:26:25.040
So you take the system,
you let it evolve.
00:26:25.040 --> 00:26:29.210
And then you calculate at
every instant in time t
00:26:29.210 --> 00:26:31.220
the product of the velocity.
00:26:31.220 --> 00:26:33.080
You average that on
all the molecules,
00:26:33.080 --> 00:26:34.950
and you have the
velocity-velocity
00:26:34.950 --> 00:26:36.590
autocorrelation function.
00:26:36.590 --> 00:26:40.310
And that, as a
function of time, will
00:26:40.310 --> 00:26:44.960
show typical beatings that
have to do with the frequencies
00:26:44.960 --> 00:26:46.160
in your system.
00:26:46.160 --> 00:26:48.770
You do the Fourier
transform of this,
00:26:48.770 --> 00:26:50.930
and you find a power spectrum.
00:26:50.930 --> 00:26:55.430
You find a vibrational spectrum
of your liquid in which you see
00:26:55.430 --> 00:26:58.465
you can identify
a very large peak,
00:26:58.465 --> 00:27:02.990
a very sort of typical set of
vibrations that have really
00:27:02.990 --> 00:27:07.340
to do-- this is actually heavy
water, so it's got deuterium--
00:27:07.340 --> 00:27:12.710
with the stretching mode of
the deuterium oxygen distance
00:27:12.710 --> 00:27:16.430
in the water molecule, so the
modes in which the two atoms
00:27:16.430 --> 00:27:18.720
vibrate one against the other.
00:27:18.720 --> 00:27:22.400
So this would be the optical
intermolecular modes.
00:27:22.400 --> 00:27:24.530
And then you see
another peak that
00:27:24.530 --> 00:27:27.020
has to do with the
vibration modes, the fact
00:27:27.020 --> 00:27:29.990
that the water molecules
act as a scissors.
00:27:29.990 --> 00:27:34.230
And then you find a lot
of lower energy modes.
00:27:34.230 --> 00:27:36.260
But again, just the
Fourier transform
00:27:36.260 --> 00:27:38.840
of this velocity-velocity
autocorrelation function
00:27:38.840 --> 00:27:43.040
gives you right away what is the
vibrational density of states
00:27:43.040 --> 00:27:46.490
and, again, sort of
a snapshot of what
00:27:46.490 --> 00:27:48.740
are the important
vibrations in the system.
00:27:48.740 --> 00:27:51.710
And a lot of
spectroscopic properties
00:27:51.710 --> 00:27:55.070
would be correlated with this
vibrational density of states.
00:27:55.070 --> 00:27:57.740
Because suppose that
some of these states
00:27:57.740 --> 00:28:00.590
are, say, what we
call infrared active.
00:28:00.590 --> 00:28:02.810
That is, when the
atoms move around,
00:28:02.810 --> 00:28:05.420
they create a
little polarization.
00:28:05.420 --> 00:28:08.730
They create a little
local electric field.
00:28:08.730 --> 00:28:12.290
Well, then those
modes would interact
00:28:12.290 --> 00:28:14.720
and couple very
strongly with sort
00:28:14.720 --> 00:28:17.190
of electromagnetic radiation.
00:28:17.190 --> 00:28:18.890
And so depending
on your frequency
00:28:18.890 --> 00:28:21.150
of your electromagnetic
radiation,
00:28:21.150 --> 00:28:24.170
you would couple
very strongly with
00:28:24.170 --> 00:28:27.570
the appropriate frequencies
of your liquid system.
00:28:27.570 --> 00:28:30.650
And so this is, again, one
of the sort of very important
00:28:30.650 --> 00:28:33.320
quantities that you
might want to extract
00:28:33.320 --> 00:28:35.910
from a molecular
dynamics simulation.
00:28:35.910 --> 00:28:39.680
And if, instead of having a
liquid, you had a solid, again,
00:28:39.680 --> 00:28:42.590
that would be the sort
of vibrational density
00:28:42.590 --> 00:28:45.930
of states of your phonon modes.
00:28:45.930 --> 00:28:46.430
OK.
00:28:46.430 --> 00:28:48.950
This sort of concludes
this parenthesis
00:28:48.950 --> 00:28:51.450
on Green-Kubo approach.
00:28:51.450 --> 00:28:55.660
You'll see more of this in
one of the later classes.
00:28:55.660 --> 00:28:58.140
But again, if there is one
thing that you need to remember,
00:28:58.140 --> 00:29:03.090
it's this, that this general
sort of set of relation
00:29:03.090 --> 00:29:06.810
make a connection between
a macroscopic property
00:29:06.810 --> 00:29:10.320
and, in particular, a response
property of the system.
00:29:10.320 --> 00:29:13.200
The diffusion coefficient
is a response property
00:29:13.200 --> 00:29:17.730
of the system to a
concentration inhomogeneity.
00:29:17.730 --> 00:29:21.600
And these macroscopic property,
these response properties,
00:29:21.600 --> 00:29:25.950
can be connected to equilibrium
fluctuations or something
00:29:25.950 --> 00:29:29.700
like the velocity-velocity
autocorrelation function.
00:29:29.700 --> 00:29:32.310
And there are a lot of
interesting quantities
00:29:32.310 --> 00:29:34.530
that you can obtain from
Green-Kubo relations
00:29:34.530 --> 00:29:37.300
besides the diffusion
coefficient.
00:29:37.300 --> 00:29:39.780
So in particular,
you could find out
00:29:39.780 --> 00:29:41.670
the viscosity of your system.
00:29:41.670 --> 00:29:44.370
Suppose that you want to study
liquid iron because you want
00:29:44.370 --> 00:29:47.190
to understand how
seismic waves propagate
00:29:47.190 --> 00:29:50.210
in this sort of liquid
inner core of Earth.
00:29:50.210 --> 00:29:53.430
Well, you are very interested
in the viscosity of the system
00:29:53.430 --> 00:29:56.130
because you want to know how
sort of shear propagation
00:29:56.130 --> 00:29:57.210
takes place.
00:29:57.210 --> 00:30:02.460
And just from the fluctuations
in the stress tensor,
00:30:02.460 --> 00:30:05.910
if you have your unit cell
with all the iron liquid atoms
00:30:05.910 --> 00:30:09.210
moving around, the
fluctuations instantaneous
00:30:09.210 --> 00:30:11.985
due to the thermal motion
in the stress tensor
00:30:11.985 --> 00:30:14.820
for that system give
you the sheer viscosity.
00:30:14.820 --> 00:30:17.970
Or say you want to
study how, say, one
00:30:17.970 --> 00:30:20.640
of the systems like
water would couple
00:30:20.640 --> 00:30:22.920
and what would be its
infrared absorption.
00:30:22.920 --> 00:30:27.690
Well, you can actually look at
the instantaneous fluctuations
00:30:27.690 --> 00:30:30.150
in the total polarization
in the system.
00:30:30.150 --> 00:30:33.960
Because what is the microscopic
local electric field there
00:30:33.960 --> 00:30:35.330
and how it couples?
00:30:35.330 --> 00:30:39.060
So again, you can sort of find
out macroscopic properties
00:30:39.060 --> 00:30:41.550
from microscopic
fluctuation or sort
00:30:41.550 --> 00:30:44.790
of electrical thermal
conductivities.
00:30:44.790 --> 00:30:47.610
Again, sort of macroscopic
transfer properties
00:30:47.610 --> 00:30:50.700
can be found out
from fluctuations
00:30:50.700 --> 00:30:54.150
in the autocorrelation
functions for the sort
00:30:54.150 --> 00:31:01.320
of electrical charge or a
thermal carriers in a system.
00:31:01.320 --> 00:31:02.460
OK.
00:31:02.460 --> 00:31:07.480
This basically concludes the
classical molecular dynamic
00:31:07.480 --> 00:31:08.490
part.
00:31:08.490 --> 00:31:11.220
And what I wanted
to show you next
00:31:11.220 --> 00:31:15.640
is how we actually do first
principle molecular dynamics.
00:31:15.640 --> 00:31:19.260
That is how we sort of
evolve atoms in time
00:31:19.260 --> 00:31:22.800
not using a classic
field, but using
00:31:22.800 --> 00:31:25.080
our favorite electronic
structure methods.
00:31:25.080 --> 00:31:26.940
That, actually for
most of this class,
00:31:26.940 --> 00:31:28.620
has been density
functional theory,
00:31:28.620 --> 00:31:31.230
but doesn't really have to
be density functional theory.
00:31:31.230 --> 00:31:35.102
Any of the electronic
structure methods would work.
00:31:35.102 --> 00:31:37.560
Density functional theory tends
to be the simplest and most
00:31:37.560 --> 00:31:40.110
efficient to implement.
00:31:40.110 --> 00:31:42.360
Sadly, in order
to do this, I need
00:31:42.360 --> 00:31:49.740
to give you some other reminders
of formal classical mechanics.
00:31:49.740 --> 00:31:53.310
Because especially in first
principle molecular dynamics,
00:31:53.310 --> 00:31:57.750
we use a lot of the concept
of extended Lagrangian and
00:31:57.750 --> 00:31:58.980
extended Hamiltonians.
00:31:58.980 --> 00:32:02.400
That is we'll derive
the equation of motion
00:32:02.400 --> 00:32:06.540
from an appropriate functional
that includes sometimes very
00:32:06.540 --> 00:32:08.530
exotic degrees of freedom.
00:32:08.530 --> 00:32:10.890
So let me remind
you how you have
00:32:10.890 --> 00:32:14.280
seen this in some of your
physics or mechanics class.
00:32:14.280 --> 00:32:17.730
But let me show you
how, in general, one
00:32:17.730 --> 00:32:21.810
can think at evolution
in phase space
00:32:21.810 --> 00:32:23.700
and sort of find
out the equations
00:32:23.700 --> 00:32:25.860
that integrated the trajectory.
00:32:25.860 --> 00:32:28.530
And up to now, we
have really just seen
00:32:28.530 --> 00:32:30.840
Newton equation
of motion, forces
00:32:30.840 --> 00:32:33.270
equal to the mass
times acceleration.
00:32:33.270 --> 00:32:36.240
But there is a sort of more
complex and, if you want,
00:32:36.240 --> 00:32:41.190
more elegant formalism to
derive the equation of motion
00:32:41.190 --> 00:32:42.240
for a system.
00:32:42.240 --> 00:32:44.160
And in particular,
what I'm showing here
00:32:44.160 --> 00:32:47.640
is sort of what is called
Lagrangian dynamics.
00:32:47.640 --> 00:32:50.230
In fact, this is actually
very, very simple.
00:32:50.230 --> 00:32:54.490
And the way Lagrangian
dynamics works is this.
00:32:54.490 --> 00:32:58.920
First of all, you have to
construct your Lagrangian.
00:32:58.920 --> 00:33:02.640
That is the
functional that drives
00:33:02.640 --> 00:33:05.580
the evolution of your systems.
00:33:05.580 --> 00:33:10.560
And there are various ways
in which one of constructs
00:33:10.560 --> 00:33:12.090
these functionals.
00:33:12.090 --> 00:33:14.160
And there are even sort
of equivalent ways.
00:33:14.160 --> 00:33:16.950
One can construct
different Lagrangians
00:33:16.950 --> 00:33:19.380
that give you the same
equation of motion
00:33:19.380 --> 00:33:20.910
of the same trajectories.
00:33:20.910 --> 00:33:24.780
But, see, what is important
here is the standard way.
00:33:24.780 --> 00:33:27.340
That is the way we
construct this functional,
00:33:27.340 --> 00:33:29.400
it's not very different
from thermodynamics.
00:33:29.400 --> 00:33:32.040
We just take the kinetic
energy of the system,
00:33:32.040 --> 00:33:36.070
T. We subtract the potential
energy of the system.
00:33:36.070 --> 00:33:39.930
And this is sort of our
Lagrangian, T minus V.
00:33:39.930 --> 00:33:45.090
In general, the potential
energy, as you have seen it,
00:33:45.090 --> 00:33:49.300
tends to be a function
of position only.
00:33:49.300 --> 00:33:51.970
So usually, we
written as a function
00:33:51.970 --> 00:33:55.720
of, if we have n
particles, r1 to rn.
00:33:55.720 --> 00:33:59.230
So you know, this is what is
called a conservative field.
00:33:59.230 --> 00:34:01.270
If you are sort of
a particle living
00:34:01.270 --> 00:34:03.430
in a gravitational
field, well, you
00:34:03.430 --> 00:34:05.230
are going to be in
a certain position.
00:34:05.230 --> 00:34:07.300
You are going to feel
a certain potential,
00:34:07.300 --> 00:34:09.070
or you are going to
feel a certain force.
00:34:09.070 --> 00:34:11.020
That's the gradient
of that potential.
00:34:11.020 --> 00:34:13.420
You go somewhere else, you'll
feel a different potential.
00:34:13.420 --> 00:34:14.920
You see a different force.
00:34:14.920 --> 00:34:18.159
And the work that you sort of
make in going from one place
00:34:18.159 --> 00:34:20.889
to the other, it's just
the integral of that force.
00:34:20.889 --> 00:34:23.300
And it's independent
of the trajectory.
00:34:23.300 --> 00:34:25.630
So this is sort of, you
know, a very general sort
00:34:25.630 --> 00:34:27.820
of potential function
that you have seen.
00:34:27.820 --> 00:34:30.710
And you know, again, the
kinetic energy, you have seen it
00:34:30.710 --> 00:34:37.245
and tends to be a function of
the square of the velocities.
00:34:37.245 --> 00:34:37.745
OK.
00:34:42.040 --> 00:34:45.280
So again, if you have
only one particle,
00:34:45.280 --> 00:34:48.250
its kinetic energy is
going to be 1/2 times
00:34:48.250 --> 00:34:52.389
the mass times the
square velocity.
00:34:52.389 --> 00:34:55.600
We usually, in the
Lagrangian formulation,
00:34:55.600 --> 00:35:01.420
don't use the sort of notation
for the positions r1, rn.
00:35:01.420 --> 00:35:02.860
But instead, say
in particular, we
00:35:02.860 --> 00:35:06.460
use the other notation in
which the coordinates are
00:35:06.460 --> 00:35:09.460
given by q1, q2, qn.
00:35:09.460 --> 00:35:13.780
And then the velocities we
just indicate them as q dot.
00:35:13.780 --> 00:35:17.950
And the reason we call them
q is that what you sometimes
00:35:17.950 --> 00:35:22.990
want to do is not use
your regular coordinates
00:35:22.990 --> 00:35:26.350
as the description
of your position
00:35:26.350 --> 00:35:28.810
for your dynamical
system, but you might want
00:35:28.810 --> 00:35:31.420
to use generalized coordinates.
00:35:31.420 --> 00:35:34.870
Say you study water molecules.
00:35:34.870 --> 00:35:38.920
And all of a sudden you want to
describe this liquid of water
00:35:38.920 --> 00:35:41.720
molecules as rigid molecules.
00:35:41.720 --> 00:35:45.130
So you want to
say that the angle
00:35:45.130 --> 00:35:48.700
between the hydrogen,
oxygen, and the hydrogen
00:35:48.700 --> 00:35:49.670
doesn't change.
00:35:49.670 --> 00:35:53.140
And if you want to say that the
distance between the hydrogen
00:35:53.140 --> 00:35:55.270
and oxygen doesn't
change, well, then you
00:35:55.270 --> 00:35:59.440
want to sort of develop
a dynamic in which what
00:35:59.440 --> 00:36:03.200
you really move around are
not the position of the atoms,
00:36:03.200 --> 00:36:05.740
but you move around
the center of mass
00:36:05.740 --> 00:36:07.360
of your water molecules.
00:36:07.360 --> 00:36:10.197
And you move around
the orientation.
00:36:10.197 --> 00:36:11.530
This is actually very important.
00:36:11.530 --> 00:36:14.080
If you remember in
sort of last class,
00:36:14.080 --> 00:36:17.560
I've told you that, when
we study water, actually
00:36:17.560 --> 00:36:20.770
because water at
regular temperature
00:36:20.770 --> 00:36:22.780
is still a quantum
system, is still
00:36:22.780 --> 00:36:27.220
a system that has most
of its vibrational states
00:36:27.220 --> 00:36:31.150
frozen in the sort of 0
point motion quantum state,
00:36:31.150 --> 00:36:34.870
you actually tend to
describe better liquid water
00:36:34.870 --> 00:36:39.940
if you describe it as a set of
rigid molecules moving around.
00:36:39.940 --> 00:36:41.990
This is, again,
an approximation.
00:36:41.990 --> 00:36:43.960
But it's actually a
better approximation
00:36:43.960 --> 00:36:47.210
of the true dynamics of the
system than an approximation
00:36:47.210 --> 00:36:50.270
which you let also the internal
degrees of freedom change.
00:36:50.270 --> 00:36:52.030
So suppose you want
to sort of simulate
00:36:52.030 --> 00:36:53.350
the rigid water around.
00:36:53.350 --> 00:36:56.500
You need to find out what
are the equation of motion
00:36:56.500 --> 00:36:59.620
for this generalized
set of coordinates
00:36:59.620 --> 00:37:02.680
in which what you really move
around when you move water
00:37:02.680 --> 00:37:05.990
is the center of mass
and their orientation.
00:37:05.990 --> 00:37:08.050
And you know, it would
be very difficult to do
00:37:08.050 --> 00:37:11.860
sort of using Newton's equation
of motion with a constraint.
00:37:11.860 --> 00:37:15.550
And so what you do, you use
your Lagrangian formulation
00:37:15.550 --> 00:37:19.450
in a generalized formalism
of generalized coordinates q
00:37:19.450 --> 00:37:22.270
and generalized velocity q dot.
00:37:22.270 --> 00:37:24.580
So what Lagrangian
dynamics tell us
00:37:24.580 --> 00:37:27.790
is that we construct our
Lagrangian function T
00:37:27.790 --> 00:37:32.410
minus V. And then the equation
of motion are given by these.
00:37:32.410 --> 00:37:33.880
These are the
Lagrangian equation.
00:37:33.880 --> 00:37:37.810
We want to derive them, but
this is how they are written.
00:37:37.810 --> 00:37:40.975
The derivative with respect to
time of the partial derivatives
00:37:40.975 --> 00:37:44.540
of the Lagrangian with respect
to the generalized velocity q
00:37:44.540 --> 00:37:47.530
dot minus the partial
derivatives of the Lagrangian
00:37:47.530 --> 00:37:49.480
with respect to the
generalized coordinates
00:37:49.480 --> 00:37:51.550
needs to be equal to 0.
00:37:51.550 --> 00:37:54.490
And you know, this
also focuses us
00:37:54.490 --> 00:37:58.210
on just constructing
the two scalar
00:37:58.210 --> 00:38:04.038
function, kinetic energy and
potential energy, for a system.
00:38:04.038 --> 00:38:05.830
And that is just, you
know, straightforward
00:38:05.830 --> 00:38:08.440
algebra to derive this equation.
00:38:08.440 --> 00:38:10.660
And I've actually
done the derivation
00:38:10.660 --> 00:38:15.550
for the sort of simple
case of Newtonian dynamics.
00:38:15.550 --> 00:38:20.680
You actually see how trivially
the Lagrange equation
00:38:20.680 --> 00:38:25.330
that is written here turns into
Newton's equation of motion
00:38:25.330 --> 00:38:29.950
when you plug in for your
Lagrangian 1/2 nV squared
00:38:29.950 --> 00:38:32.480
minus your potential energy.
00:38:32.480 --> 00:38:34.810
And you see, when you take
the partial derivatives
00:38:34.810 --> 00:38:38.410
of the Lagrangian with respect
to the generalized velocity,
00:38:38.410 --> 00:38:40.510
since it is a
partial derivative,
00:38:40.510 --> 00:38:42.790
you only need to
take the derivative
00:38:42.790 --> 00:38:46.030
of the kinetic energy with
respect to the velocity,
00:38:46.030 --> 00:38:47.710
with respect to x dot.
00:38:47.710 --> 00:38:50.890
And then you need to
take partial derivatives
00:38:50.890 --> 00:38:52.330
with respect to the position.
00:38:52.330 --> 00:38:55.190
And there is no position in
the kinetic energy functional.
00:38:55.190 --> 00:38:58.760
So there is only the position
in that potential energy.
00:38:58.760 --> 00:39:02.155
So what you see is that,
from the left-hand term,
00:39:02.155 --> 00:39:03.760
you have the
derivative with respect
00:39:03.760 --> 00:39:07.850
to time of mass times velocity.
00:39:07.850 --> 00:39:11.260
And so that's nothing else
than mass times acceleration.
00:39:11.260 --> 00:39:14.710
And then on the right,
the right-hand term,
00:39:14.710 --> 00:39:20.140
you have minus the gradient
of your conservative potential
00:39:20.140 --> 00:39:20.800
field.
00:39:20.800 --> 00:39:22.970
And so this is nothing
else than the force.
00:39:22.970 --> 00:39:27.430
So we have force equal to
mass times acceleration.
00:39:27.430 --> 00:39:29.380
And we have recovered.
00:39:29.380 --> 00:39:33.910
Just by applying
Lagrange equations
00:39:33.910 --> 00:39:38.110
to the Lagrangian of
classical dynamics, kinetics
00:39:38.110 --> 00:39:42.710
minus potential, we have derived
Newton's equation of motion.
00:39:42.710 --> 00:39:48.950
So this is one way of deriving
equation of motion, but, again,
00:39:48.950 --> 00:39:52.060
are going to be
second-order differential
00:39:52.060 --> 00:39:54.190
equation with respect to time.
00:39:54.190 --> 00:39:56.560
Because, if you
think, you are taking
00:39:56.560 --> 00:40:00.430
the derivative of a kinetic
energy with respect to q dot.
00:40:00.430 --> 00:40:03.310
There is a second formulation
of classical mechanics--
00:40:03.310 --> 00:40:05.920
and I also need to sort
of present this here--
00:40:05.920 --> 00:40:08.920
that is called the
Hamiltonian formulation.
00:40:08.920 --> 00:40:12.370
And it's sort of,
again, very easy
00:40:12.370 --> 00:40:16.630
to see this if you think
of a thermodynamic analogy.
00:40:16.630 --> 00:40:20.320
Say, when you're sort of
looking at thermodynamics,
00:40:20.320 --> 00:40:23.230
suppose that you are in the
microcanonical ensemble.
00:40:23.230 --> 00:40:26.090
Your thermodynamical
functional is the energy.
00:40:26.090 --> 00:40:30.850
And so when you sort of look at
that thermodynamical ensemble,
00:40:30.850 --> 00:40:33.100
it means that, say, you are
looking at the system that
00:40:33.100 --> 00:40:35.980
has a constant energy,
constant number of particle,
00:40:35.980 --> 00:40:37.600
and constant volume.
00:40:37.600 --> 00:40:41.500
But many times it becomes
appropriate to look
00:40:41.500 --> 00:40:43.690
at, actually, sort
of a system that
00:40:43.690 --> 00:40:47.170
lives at constant pressure
or constant temperature.
00:40:47.170 --> 00:40:50.020
And so you transform your
thermodynamical functional
00:40:50.020 --> 00:40:53.020
from the energy to one, say,
of the Helmholtz or Gibbs
00:40:53.020 --> 00:40:53.890
free energies.
00:40:53.890 --> 00:40:58.030
You do e minus ds to obtain a
thermodynamic functional that
00:40:58.030 --> 00:41:01.090
depends on temperature instead
of depending on entropy.
00:41:01.090 --> 00:41:04.720
Or you do e plus pv to
obtain the enthalpy that
00:41:04.720 --> 00:41:07.900
sort of depends on
pressure and not on volume.
00:41:07.900 --> 00:41:12.860
And this is this general concept
of Legendre transformation.
00:41:12.860 --> 00:41:15.010
If you have a
function, let's say,
00:41:15.010 --> 00:41:20.200
for the Lagrangian was a
functional of q and q dot,
00:41:20.200 --> 00:41:27.580
you can construct a new one that
doesn't depend, say, on q dot,
00:41:27.580 --> 00:41:31.060
but depends only
on a new variable
00:41:31.060 --> 00:41:35.050
that we call it the
conjugate variable 2q dot.
00:41:35.050 --> 00:41:39.760
So pressure and volume,
temperature and entropy,
00:41:39.760 --> 00:41:42.670
chemical potential and
number of particles
00:41:42.670 --> 00:41:44.740
are all conjugate variables.
00:41:44.740 --> 00:41:49.750
And so, say, if you take the
Lagrangian and you derive it
00:41:49.750 --> 00:41:51.470
with respect to
q dot-- remember,
00:41:51.470 --> 00:41:53.680
the Lagrangian is a
function of q dot--
00:41:53.680 --> 00:41:56.710
what you obtain is
at conjugate variable
00:41:56.710 --> 00:42:00.010
that we call a
conjugate momentum.
00:42:00.010 --> 00:42:03.220
And then if you do this
operation, this Legendre
00:42:03.220 --> 00:42:06.070
transform in which you take--
00:42:06.070 --> 00:42:08.450
the sign doesn't really matter.
00:42:08.450 --> 00:42:10.330
In this case, you
take a minus sign.
00:42:10.330 --> 00:42:14.710
And you sum the product
of the conjugate variable
00:42:14.710 --> 00:42:16.690
times your original variable.
00:42:16.690 --> 00:42:20.650
You get a new function
that doesn't depend
00:42:20.650 --> 00:42:24.580
on your original q dot
variable, but depends only
00:42:24.580 --> 00:42:26.530
on its conjugate variable p.
00:42:26.530 --> 00:42:29.620
That's how you remove the
dependence, say, on the volume
00:42:29.620 --> 00:42:32.080
and you put in the
dependence on the pressure,
00:42:32.080 --> 00:42:33.700
making the Legendre
transformation
00:42:33.700 --> 00:42:36.370
which you have pv, or that's
how you sort of move on
00:42:36.370 --> 00:42:40.720
from entropy to temperature
in the Helmholtz free energy.
00:42:40.720 --> 00:42:42.760
And this is just
very simple to do
00:42:42.760 --> 00:42:47.440
when you take the
differential of this quantity.
00:42:47.440 --> 00:42:49.600
Because you are going
to see that really,
00:42:49.600 --> 00:42:54.220
because of this relation,
you remove all the dependency
00:42:54.220 --> 00:42:54.910
in q dot.
00:42:54.910 --> 00:42:59.020
And you put into your
system an independence on p.
00:42:59.020 --> 00:43:00.800
Well, why do we do this?
00:43:00.800 --> 00:43:03.350
Well, because, again
with some algebra,
00:43:03.350 --> 00:43:07.750
we can find that, for this
new function that is now
00:43:07.750 --> 00:43:11.500
called the Hamiltonian,
another alternative
00:43:11.500 --> 00:43:13.780
sets of equation of motions.
00:43:13.780 --> 00:43:16.120
Remember, from
the Lagrangian, we
00:43:16.120 --> 00:43:20.080
had obtained equation of motion
basically for q dot and q.
00:43:20.080 --> 00:43:22.930
And from the
Hamiltonian formulation,
00:43:22.930 --> 00:43:26.860
if we work this out, we
obtain equation of motion
00:43:26.860 --> 00:43:29.470
for q and for p.
00:43:29.470 --> 00:43:31.570
And sort of the
slight difference
00:43:31.570 --> 00:43:37.300
is that instead of having
basically a second order
00:43:37.300 --> 00:43:40.990
differential equation from the
Lagrangian formulation, now
00:43:40.990 --> 00:43:45.430
we have a double
set of differential
00:43:45.430 --> 00:43:47.710
that are only first-order.
00:43:47.710 --> 00:43:49.840
So depending on
the your problem,
00:43:49.840 --> 00:43:52.180
they can actually
be easier to solve.
00:43:52.180 --> 00:43:54.340
They can actually be
different, but they could
00:43:54.340 --> 00:43:56.450
lead to the same trajectories.
00:43:56.450 --> 00:44:00.430
So again, all these formalisms
of Lagrangian and Hamiltonian
00:44:00.430 --> 00:44:04.870
is that's the very general
way to construct functions,
00:44:04.870 --> 00:44:09.370
either the Lagrangian T
minus V or the Hamiltonian
00:44:09.370 --> 00:44:12.670
via the Legendre
transform that give us
00:44:12.670 --> 00:44:15.640
equation of motion
in q and q dot
00:44:15.640 --> 00:44:19.390
for the case of the
Lagrangian and in q and p
00:44:19.390 --> 00:44:21.880
for the case of the Hamiltonian.
00:44:21.880 --> 00:44:23.530
And if you actually
work this out
00:44:23.530 --> 00:44:27.460
for the standard case
of Newtonian dynamics,
00:44:27.460 --> 00:44:30.850
you find out that your
Hamiltonian is, again,
00:44:30.850 --> 00:44:32.950
something very trivial.
00:44:32.950 --> 00:44:37.010
It's just the kinetic energy
plus the potential energy.
00:44:37.010 --> 00:44:40.480
So at the end of all of
this, for all the cases
00:44:40.480 --> 00:44:43.260
of interest to you, you see
that either you construct
00:44:43.260 --> 00:44:46.550
a Lagrangian T
minus V and you have
00:44:46.550 --> 00:44:48.610
the Lagrangian
equation of motion,
00:44:48.610 --> 00:44:52.750
or your construct your
Hamiltonian, p plus v.
00:44:52.750 --> 00:44:56.680
And you have the sort of
Hamiltonian equation of motion
00:44:56.680 --> 00:44:59.380
that I have written here.
00:44:59.380 --> 00:45:06.380
And one of the reasons
that we do this--
00:45:06.380 --> 00:45:15.070
so this is-- is that,
often, we want actually
00:45:15.070 --> 00:45:18.620
to simulate the
microscopic dynamics
00:45:18.620 --> 00:45:21.200
not in the
microcanonical ensemble,
00:45:21.200 --> 00:45:23.930
but in different
thermodynamical ensembles,
00:45:23.930 --> 00:45:26.510
say, in which we
control the temperature.
00:45:26.510 --> 00:45:28.940
Or we control the
pressure, or maybe we
00:45:28.940 --> 00:45:31.220
control the number of particles.
00:45:31.220 --> 00:45:34.110
Or maybe we control
the chemical potential.
00:45:34.110 --> 00:45:38.690
And so we need to have some
of this sort of formalism
00:45:38.690 --> 00:45:40.430
to do this effectively.
00:45:40.430 --> 00:45:44.000
In particular, remember when
we have discussed the sort
00:45:44.000 --> 00:45:45.710
of control of temperature.
00:45:45.710 --> 00:45:47.240
I've told you that
there are sort
00:45:47.240 --> 00:45:49.520
of three different
approaches in which you
00:45:49.520 --> 00:45:52.580
can do a canonical simulation,
which you can control
00:45:52.580 --> 00:45:54.050
the temperature in your system.
00:45:54.050 --> 00:45:56.300
And you could have
Langevin dynamics.
00:45:56.300 --> 00:45:59.510
You could have a stochastic
approach in which you randomly
00:45:59.510 --> 00:46:02.480
pick atoms in order
to accelerate them
00:46:02.480 --> 00:46:04.040
or to slow them down.
00:46:04.040 --> 00:46:06.710
So that in analogy
with a thermal bath,
00:46:06.710 --> 00:46:10.790
they sort of, on average,
have the right kinetic target,
00:46:10.790 --> 00:46:13.280
kinetic energy for your problem.
00:46:13.280 --> 00:46:16.940
Or you could do something that
is probably even more brutal,
00:46:16.940 --> 00:46:20.870
although it tends to be very
efficient in thermalizing,
00:46:20.870 --> 00:46:22.370
effectively, your system.
00:46:22.370 --> 00:46:25.490
You could actually do
a dynamics in which,
00:46:25.490 --> 00:46:28.250
every time you have
got new position,
00:46:28.250 --> 00:46:32.150
you renormalize those new
position by renormalizing
00:46:32.150 --> 00:46:35.540
the velocity of the
particle, so that the sum
00:46:35.540 --> 00:46:38.670
of the kinetic energy
is a actually constant.
00:46:38.670 --> 00:46:41.390
So a constraint
method would actually
00:46:41.390 --> 00:46:45.410
keep, strictly speaking, the
temperature of your system
00:46:45.410 --> 00:46:47.990
sort of constant to
your target value.
00:46:47.990 --> 00:46:51.290
That can be actually
very effective
00:46:51.290 --> 00:46:53.700
to thermalize your
system, to bring
00:46:53.700 --> 00:46:58.520
it really very close to your
equilibrium distribution.
00:46:58.520 --> 00:47:02.660
But it does have
some counter-effects.
00:47:02.660 --> 00:47:05.030
That is, if you
actually look at what
00:47:05.030 --> 00:47:09.080
is going to be your equilibrium
distribution in positions,
00:47:09.080 --> 00:47:12.410
it's really going to sort of
be a canonical distribution
00:47:12.410 --> 00:47:14.840
according to the Boltzmann
canonical ensemble.
00:47:14.840 --> 00:47:17.690
But if you look at your
distribution of velocity,
00:47:17.690 --> 00:47:19.710
it's only pseudo-canonical.
00:47:19.710 --> 00:47:21.470
So often, people,
especially those
00:47:21.470 --> 00:47:25.460
that do very complex and
long molecular dynamics,
00:47:25.460 --> 00:47:29.120
use the sort of most elegant
and most accurate approach.
00:47:29.120 --> 00:47:32.480
That is really
coupling your system
00:47:32.480 --> 00:47:35.180
to an additional
dynamical variable
00:47:35.180 --> 00:47:39.380
using an extended Lagrangian
or an extended Hamiltonian.
00:47:39.380 --> 00:47:43.160
So we have written our
Lagrangian or Hamiltonian
00:47:43.160 --> 00:47:45.470
in terms of generalized
coordinates.
00:47:45.470 --> 00:47:49.510
All of a sudden, you can add
one more generalized coordinate.
00:47:49.510 --> 00:47:52.490
So you can add, if you
want, a pseudo-particle
00:47:52.490 --> 00:47:56.510
in your system with its
own sort of kinetic energy
00:47:56.510 --> 00:47:58.940
and with its own
potential energy.
00:47:58.940 --> 00:48:01.370
And you can construct
the kinetic energy.
00:48:01.370 --> 00:48:04.230
And in particular, you can
construct the potential energy,
00:48:04.230 --> 00:48:08.390
so that this additional
dynamical system,
00:48:08.390 --> 00:48:11.540
this additional dynamical
variable, interacts
00:48:11.540 --> 00:48:13.820
with the other
dynamical variables,
00:48:13.820 --> 00:48:17.400
basically exchanging
temperature with it
00:48:17.400 --> 00:48:20.690
to bring the sort of
average temperature
00:48:20.690 --> 00:48:24.770
of the real classical
particles to the equilibrium
00:48:24.770 --> 00:48:25.868
distribution.
00:48:25.868 --> 00:48:27.410
And so this is
actually how you would
00:48:27.410 --> 00:48:31.460
write the extended
Lagrangian for the case
00:48:31.460 --> 00:48:34.130
of a canonical
simulation, something
00:48:34.130 --> 00:48:37.460
in which we want to keep
the temperature constant.
00:48:37.460 --> 00:48:40.640
And so you see that's
where the power of all
00:48:40.640 --> 00:48:43.680
this generalized
system comes into play.
00:48:43.680 --> 00:48:48.230
And again, you sort of write
out your kinetic energy.
00:48:48.230 --> 00:48:50.020
And note that there
is this sort of s
00:48:50.020 --> 00:48:54.440
squared term that couples
the kinetic energy
00:48:54.440 --> 00:48:58.670
of your real particles
with the kinetic energy
00:48:58.670 --> 00:49:02.160
of this new thermostat,
as we call it.
00:49:02.160 --> 00:49:05.730
So we have kinetic energy
minus potential energy.
00:49:05.730 --> 00:49:07.700
And then this is
the new particle.
00:49:07.700 --> 00:49:10.610
This is the new
extended coordinate
00:49:10.610 --> 00:49:12.860
that is described by
a generalized position
00:49:12.860 --> 00:49:17.720
s and the generalized
velocity s dot.
00:49:17.720 --> 00:49:20.600
And so, if you want,
its kinetic energy
00:49:20.600 --> 00:49:24.755
is written in a trivial way,
1/2 our generalized mass times
00:49:24.755 --> 00:49:26.060
the square velocity.
00:49:26.060 --> 00:49:30.500
And this is actually how Nosé
figured out the potential
00:49:30.500 --> 00:49:35.840
energy should look like for a
system of n classical particles
00:49:35.840 --> 00:49:38.890
that you want to keep at an
inverse temperature beta.
00:49:38.890 --> 00:49:42.260
Remember, beta is just 1 over
the Boltzmann constant times
00:49:42.260 --> 00:49:43.250
the temperature.
00:49:43.250 --> 00:49:45.710
And so it's a very
exotic potential energy.
00:49:45.710 --> 00:49:50.840
And it's sort of the logarithm
of this generalized coordinate.
00:49:50.840 --> 00:49:53.450
And so if you use
these Lagrangian,
00:49:53.450 --> 00:49:57.210
you can construct the
Lagrangian equation of motion.
00:49:57.210 --> 00:50:01.790
So you will have equation of
motion for your 3N particle
00:50:01.790 --> 00:50:04.580
that will be very similar
to your standard equation
00:50:04.580 --> 00:50:08.030
of motion, but we'll
have in there also terms
00:50:08.030 --> 00:50:09.140
that contain s.
00:50:09.140 --> 00:50:11.870
And then you will have a
sort of an equation of motion
00:50:11.870 --> 00:50:16.430
for your Nosé or Nosé-Hoover
variable that is equation
00:50:16.430 --> 00:50:18.020
of motion for s.
00:50:18.020 --> 00:50:20.810
And you let all
this system evolve.
00:50:20.810 --> 00:50:26.360
And what will happen is
that the kinetic energies
00:50:26.360 --> 00:50:32.240
of your classical particles
will start to sort of distribute
00:50:32.240 --> 00:50:36.020
themselves according to a
Maxwell-Boltzmann distribution.
00:50:36.020 --> 00:50:39.020
That is according to
the canonical ensemble.
00:50:39.020 --> 00:50:42.950
And people have worked out
all the statistical mechanics
00:50:42.950 --> 00:50:45.560
of your problem
and so worked out
00:50:45.560 --> 00:50:47.120
the appropriate coordination.
00:50:47.120 --> 00:50:50.720
You would need to have a sort of
Maxwell-Boltzmann distribution.
00:50:50.720 --> 00:50:52.610
This is in the moduli
of your system.
00:50:52.610 --> 00:50:54.200
And really, if
you do all of this
00:50:54.200 --> 00:50:58.070
and you sort of take an average
from your molecular dynamic
00:50:58.070 --> 00:51:01.880
simulation, you really
see that both the position
00:51:01.880 --> 00:51:05.420
and the velocity of
your classical particles
00:51:05.420 --> 00:51:08.460
are distributed in
the appropriate way.
00:51:08.460 --> 00:51:12.710
So it's very sort of useful
to calculate, say, response
00:51:12.710 --> 00:51:14.900
suppose property of the system.
00:51:14.900 --> 00:51:17.000
Say, again, if you want
to calculate the diffusion
00:51:17.000 --> 00:51:19.310
coefficient from the
velocity-velocity
00:51:19.310 --> 00:51:21.200
autocorrelation
function, you really
00:51:21.200 --> 00:51:23.870
need to have your
velocity distributed
00:51:23.870 --> 00:51:27.530
according to the appropriate
thermodynamical ensemble.
00:51:27.530 --> 00:51:32.000
So it's sort of the, if you
want, most accurate and careful
00:51:32.000 --> 00:51:35.660
way of doing the dynamics.
00:51:35.660 --> 00:51:39.170
Nosé-Hoover, so it's
probably the most common way
00:51:39.170 --> 00:51:41.480
of thermostarting your problem.
00:51:41.480 --> 00:51:44.465
It's not the most robust.
00:51:44.465 --> 00:51:50.030
It's the approach that
gives you the long time
00:51:50.030 --> 00:51:52.610
thermodynamical
properties correctly.
00:51:52.610 --> 00:51:57.350
But it tends to be
very poorly, say,
00:51:57.350 --> 00:52:01.040
for a system that
is very harmonic.
00:52:01.040 --> 00:52:01.670
OK.
00:52:01.670 --> 00:52:05.960
A system is very harmonic when
its potential energy is really
00:52:05.960 --> 00:52:09.420
a quadratic function
of its coordinate.
00:52:09.420 --> 00:52:11.930
So if you have, say,
a single particle
00:52:11.930 --> 00:52:14.090
that sits in a
parabolic well, what
00:52:14.090 --> 00:52:16.480
we call a harmonic
oscillator also
00:52:16.480 --> 00:52:21.120
from a standard pendulum, that
is a perfect harmonic system.
00:52:21.120 --> 00:52:24.290
A solid at very,
very low temperature
00:52:24.290 --> 00:52:25.790
is also very harmonic.
00:52:25.790 --> 00:52:29.000
If you want the potential
energy of each particle,
00:52:29.000 --> 00:52:32.600
it's just a quadratic function
of the [INAUDIBLE] displacement
00:52:32.600 --> 00:52:34.590
from its equilibrium position.
00:52:34.590 --> 00:52:39.230
And so what are the trajectories
in a harmonic oscillator?
00:52:39.230 --> 00:52:41.660
Well, they look
something like this.
00:52:41.660 --> 00:52:46.600
If you look now, if you go back
to a one-dimensional position
00:52:46.600 --> 00:52:50.440
and momentum
representation, well,
00:52:50.440 --> 00:52:54.960
you know, a pendulum does
really this over and over again.
00:52:58.570 --> 00:53:02.410
So this would be the
trajectory moving in time.
00:53:02.410 --> 00:53:05.770
So the position sort of
oscillates back and forth.
00:53:05.770 --> 00:53:09.250
And the momentum or the velocity
oscillate back and forth.
00:53:09.250 --> 00:53:10.720
And they are in a
positional phase.
00:53:10.720 --> 00:53:13.630
When the elongation is
maximum, your momentum
00:53:13.630 --> 00:53:15.130
is linear and so on.
00:53:15.130 --> 00:53:22.000
I actually sort of made,
I guess, incorrect axis.
00:53:22.000 --> 00:53:27.100
But thanks to the power
of the [INAUDIBLE] tablet,
00:53:27.100 --> 00:53:31.650
now this looks much better.
00:53:31.650 --> 00:53:32.320
OK.
00:53:32.320 --> 00:53:36.730
Now, suppose that
you sort of studying
00:53:36.730 --> 00:53:38.950
not a single
oscillator, but you are
00:53:38.950 --> 00:53:42.850
studying a solid at
very low temperature.
00:53:42.850 --> 00:53:47.380
What your Nosé-Hoover thermostat
would give you is something
00:53:47.380 --> 00:53:50.900
that is very different from
the equilibrium distribution.
00:53:50.900 --> 00:53:54.490
So the sort of equilibrium
canonical distribution
00:53:54.490 --> 00:53:57.870
is going to look
something in which there's
00:53:57.870 --> 00:54:03.520
of momenta and velocity of
the particle are distributed.
00:54:03.520 --> 00:54:06.430
Because, now,
there is obviously,
00:54:06.430 --> 00:54:10.280
if we have a real solid, a
small amount of interaction.
00:54:10.280 --> 00:54:13.240
So each atom, each
harmonic oscillator,
00:54:13.240 --> 00:54:16.880
is going to be talking
with its neighbors.
00:54:16.880 --> 00:54:20.980
And so they are not going to
be all in phase and perpetually
00:54:20.980 --> 00:54:24.100
in phase, but they are going
to sort of exchange energy
00:54:24.100 --> 00:54:26.060
between one and the other.
00:54:26.060 --> 00:54:30.940
And so a large harmonic
solid, but slightly
00:54:30.940 --> 00:54:33.160
sort of different
from 0 temperature,
00:54:33.160 --> 00:54:35.620
will have a distribution
of velocity and momentum
00:54:35.620 --> 00:54:37.960
that doesn't sit in
a perfect circle,
00:54:37.960 --> 00:54:40.330
but is sort of
distributed around.
00:54:40.330 --> 00:54:43.750
If you do this with a
Nosé-Hoover thermostat,
00:54:43.750 --> 00:54:47.830
sadly there will be
no exchange of energy.
00:54:47.830 --> 00:54:51.250
There will be no thermalization
between all different atoms.
00:54:51.250 --> 00:54:54.130
They are going to
be doing perpetually
00:54:54.130 --> 00:54:56.020
their own thing in synchrony.
00:54:56.020 --> 00:55:01.120
And so the Nosé-Hoover
thermostat works very poorly
00:55:01.120 --> 00:55:03.990
for systems.
00:55:03.990 --> 00:55:06.130
The more harmonic
your system is,
00:55:06.130 --> 00:55:08.950
the poorer the equilibration
that comes from the Nosé
00:55:08.950 --> 00:55:10.760
thermostat is.
00:55:10.760 --> 00:55:13.480
And so you have to
pay a lot of attention
00:55:13.480 --> 00:55:15.880
to system at low temperature.
00:55:15.880 --> 00:55:19.180
And sort of one of the usual
solutions that people mention
00:55:19.180 --> 00:55:23.650
is using Nosé-Hoover chains
in which you actually have
00:55:23.650 --> 00:55:25.250
your dynamical system.
00:55:25.250 --> 00:55:26.740
You have your thermostat.
00:55:26.740 --> 00:55:28.810
And then you have
another thermostat,
00:55:28.810 --> 00:55:30.735
thermostarting your
thermostart to thermostart
00:55:30.735 --> 00:55:31.870
for your particle.
00:55:31.870 --> 00:55:33.370
And then you have
a third thermostat
00:55:33.370 --> 00:55:35.170
that thermostarts the
second thermostat.
00:55:35.170 --> 00:55:36.730
And suddenly, you
have to do this.
00:55:39.460 --> 00:55:42.890
Harmonic solids are
actually very important.
00:55:42.890 --> 00:55:45.520
So there is a reason
why a lot of effort
00:55:45.520 --> 00:55:46.998
has been put into this.
00:55:46.998 --> 00:55:48.790
Because in particular--
and this is, again,
00:55:48.790 --> 00:55:51.490
something that we'll see
in one of the last classes.
00:55:51.490 --> 00:55:55.990
In order to calculate the
free energy of a system
00:55:55.990 --> 00:55:58.420
at finite temperature,
often you need
00:55:58.420 --> 00:56:01.720
to do an integration
from 0 temperature
00:56:01.720 --> 00:56:03.430
to the present temperature.
00:56:03.430 --> 00:56:07.180
And so you really need to
start from a harmonic solid
00:56:07.180 --> 00:56:09.130
that you need to
describe correctly.
00:56:09.130 --> 00:56:11.050
And if you don't
do that, so if you
00:56:11.050 --> 00:56:14.590
don't use your proper
thermostating techniques,
00:56:14.590 --> 00:56:16.510
this is not going to work out.
00:56:16.510 --> 00:56:18.940
And there is really a
lot of dark [INAUDIBLE]
00:56:18.940 --> 00:56:19.960
in all of this.
00:56:19.960 --> 00:56:23.920
You really need to be careful
because the temperature is
00:56:23.920 --> 00:56:25.600
a global quantity.
00:56:25.600 --> 00:56:29.920
But if you have, say, a
system with 100 particles,
00:56:29.920 --> 00:56:32.320
you can have a
temperature of 300 Kelvin
00:56:32.320 --> 00:56:35.410
by having sort of all
particles distributed
00:56:35.410 --> 00:56:37.100
along the same temperature.
00:56:37.100 --> 00:56:39.160
But you can have also
the same temperature
00:56:39.160 --> 00:56:42.700
if part of your system is very
cold and part of your system
00:56:42.700 --> 00:56:43.910
is very hot.
00:56:43.910 --> 00:56:45.880
And if you wanted,
the thermostat
00:56:45.880 --> 00:56:50.380
has this role of making sure
that the energy flow goes
00:56:50.380 --> 00:56:53.350
around, so that all
part of your system
00:56:53.350 --> 00:56:54.970
are really at the equilibrium.
00:56:54.970 --> 00:56:59.590
And you know, this can be
actually a trickier thing
00:56:59.590 --> 00:57:02.320
than sort of it actually seems.
00:57:02.320 --> 00:57:04.890
Obviously, all these
dots and [INAUDIBLE]..
00:57:08.700 --> 00:57:09.200
OK.
00:57:09.200 --> 00:57:12.490
These are sort of a
summary of all the books
00:57:12.490 --> 00:57:15.980
in which you can find a
description of these concepts.
00:57:15.980 --> 00:57:18.730
If you have to read
one for this class--
00:57:18.730 --> 00:57:20.780
the sort of free primer
that Furio Ercolessi
00:57:20.780 --> 00:57:24.320
has put on his website.
00:57:24.320 --> 00:57:27.190
I'm actually very proud of this
because Furio moved recently
00:57:27.190 --> 00:57:29.290
to this very small
town in northern Italy
00:57:29.290 --> 00:57:30.790
that is sort of my hometown.
00:57:30.790 --> 00:57:33.280
But anyhow, this is
available and free.
00:57:33.280 --> 00:57:34.780
It's 30 or 40 pages.
00:57:34.780 --> 00:57:37.910
I posted it also on
the Stella website.
00:57:37.910 --> 00:57:40.810
And it gives you a
very clean introduction
00:57:40.810 --> 00:57:42.490
to molecular dynamics.
00:57:42.490 --> 00:57:46.030
And then I think this is a
field in which there are,
00:57:46.030 --> 00:57:49.180
in particular, two
exemplary books.
00:57:49.180 --> 00:57:52.420
So if you really want
to do this for a living,
00:57:52.420 --> 00:57:55.660
there are a couple of books
that are very, very good--
00:57:55.660 --> 00:57:58.600
the Allen and Tildesley book
that, by now, it's very old.
00:57:58.600 --> 00:58:01.030
It's sort of really
written at the beginning
00:58:01.030 --> 00:58:05.020
of the '80s when you can imagine
computer capabilities were
00:58:05.020 --> 00:58:06.310
very different.
00:58:06.310 --> 00:58:09.370
And it's actually
remarkable somehow how
00:58:09.370 --> 00:58:13.300
the clarity that you develop
by dealing with systems that
00:58:13.300 --> 00:58:16.060
are very primitive from the
computational point of view
00:58:16.060 --> 00:58:19.100
actually develops your
intellectual skills.
00:58:19.100 --> 00:58:21.370
So this is really
very, very good.
00:58:21.370 --> 00:58:23.860
And equally good is a
much more recent book
00:58:23.860 --> 00:58:27.925
that is in its third edition
from last year, the Frenkel
00:58:27.925 --> 00:58:33.040
and Smit that also has
many more advanced topics.
00:58:33.040 --> 00:58:35.620
But either of these
are really sort
00:58:35.620 --> 00:58:37.870
of very, very good references.
00:58:37.870 --> 00:58:40.690
And now, in the last
20 minutes, I'll
00:58:40.690 --> 00:58:43.930
give you really the flavor
on how we do first principle
00:58:43.930 --> 00:58:45.460
molecular dynamics.
00:58:45.460 --> 00:58:49.720
That is really nothing else than
a classical molecular dynamics
00:58:49.720 --> 00:58:53.390
with a lot of
additional variables.
00:58:53.390 --> 00:58:57.640
So we use this concept
of extended Hamiltonians.
00:58:57.640 --> 00:59:01.070
And you know, somehow it's,
by now, a little bit old.
00:59:01.070 --> 00:59:04.210
I had sort of once
downloaded this.
00:59:04.210 --> 00:59:09.490
This is actually the citations
of first principle molecular
00:59:09.490 --> 00:59:13.060
dynamics, of initial molecular
dynamics or the citation
00:59:13.060 --> 00:59:15.820
of the first paper that
developed this concept that
00:59:15.820 --> 00:59:18.840
goes under the name of the
[INAUDIBLE] techniques that
00:59:18.840 --> 00:59:20.497
is from 1985.
00:59:20.497 --> 00:59:22.330
And you see we are sort
of very pleased with
00:59:22.330 --> 00:59:24.970
this exponential explosion.
00:59:24.970 --> 00:59:28.330
It means that basically there is
a lot of useless literature out
00:59:28.330 --> 00:59:31.060
there that you can look at.
00:59:31.060 --> 00:59:33.700
So in order to do first
principle molecular dynamics,
00:59:33.700 --> 00:59:37.900
I need to give you a reminder
of your favorite topic.
00:59:37.900 --> 00:59:41.590
That is the expansion in plane
waves of the total energy
00:59:41.590 --> 00:59:42.790
of a system.
00:59:42.790 --> 00:59:46.270
And hopefully, you still
remind something about this.
00:59:46.270 --> 00:59:48.400
But if you have a
crystal that's going
00:59:48.400 --> 00:59:53.410
to be defined by the three
primitive direct lattice
00:59:53.410 --> 00:59:54.030
vectors--
00:59:54.030 --> 00:59:59.740
so you could have an FCC crystal
and they point to 1/2, 1/2, 0,
00:59:59.740 --> 01:00:02.080
0, 0, and the third one.
01:00:02.080 --> 01:00:05.810
And then we can define
reciprocal space
01:00:05.810 --> 01:00:11.080
and the reciprocal lattice whose
three primitive vectors, this g
01:00:11.080 --> 01:00:16.460
vectors, are just defined as
the duals of your direct lattice
01:00:16.460 --> 01:00:16.960
vector.
01:00:16.960 --> 01:00:19.960
So the scalar total
between the reciprocals
01:00:19.960 --> 01:00:24.380
and the direct lattice vectors
needs to be equal to 0 or 2 pi.
01:00:24.380 --> 01:00:24.910
OK.
01:00:24.910 --> 01:00:26.470
So this is the definition.
01:00:26.470 --> 01:00:30.550
Once we have this
reciprocal lattice vectors
01:00:30.550 --> 01:00:33.160
that, for simplicity,
I'll draw here
01:00:33.160 --> 01:00:38.530
as being two-dimensional
and square, what we have is,
01:00:38.530 --> 01:00:46.750
again, sort of these points here
represents all possible integer
01:00:46.750 --> 01:00:52.030
combination of reciprocal
lattice vectors.
01:00:52.030 --> 01:00:55.750
That, for my problem
at hand here,
01:00:55.750 --> 01:00:57.730
say, this would be
the Brillouin zone.
01:00:57.730 --> 01:01:01.210
This would be the unit cell
of the reciprocal lattice.
01:01:01.210 --> 01:01:04.810
And so all these
g vectors in blue,
01:01:04.810 --> 01:01:10.510
remember, are such that
the complex function e
01:01:10.510 --> 01:01:15.640
to the iG times
r, where G is any
01:01:15.640 --> 01:01:19.380
of these vectors represented
by one of these point,
01:01:19.380 --> 01:01:23.170
this function in real
space is compatible
01:01:23.170 --> 01:01:25.480
with your periodic
boundary condition.
01:01:25.480 --> 01:01:28.930
It's going to have either one
oscillation inside your unit
01:01:28.930 --> 01:01:32.590
cell, 2 oscillation, 3
oscillation, 10 oscillations.
01:01:32.590 --> 01:01:36.320
And it's going to be, generally
speaking, in three dimensions.
01:01:36.320 --> 01:01:40.240
And so if you remember, we
expand our wave function
01:01:40.240 --> 01:01:42.910
into linear combination
of these plane waves
01:01:42.910 --> 01:01:46.780
because these plane waves
are all the possible--
01:01:46.780 --> 01:01:49.060
it's a complete set
of functions that
01:01:49.060 --> 01:01:52.360
describe orbitals that
have the periodicity
01:01:52.360 --> 01:01:53.390
of the direct lattice.
01:02:01.720 --> 01:02:02.560
OK.
01:02:02.560 --> 01:02:05.810
So what was our quantum
mechanical system?
01:02:05.810 --> 01:02:07.070
We keep it simple.
01:02:07.070 --> 01:02:09.070
We won't go into density
functional theory.
01:02:09.070 --> 01:02:12.190
We'll just think
again at the operator.
01:02:12.190 --> 01:02:15.680
You see the Hamiltonian
raises its head again.
01:02:15.680 --> 01:02:18.220
It's just going to be the
quantum kinetic energy
01:02:18.220 --> 01:02:22.510
minus 1/2 the Laplacian
plus the potential energy
01:02:22.510 --> 01:02:25.870
and with the caveat
that we develop,
01:02:25.870 --> 01:02:29.800
we expand, the periodic
part of your wave function,
01:02:29.800 --> 01:02:34.300
what we call sort of, via the
Bloch theorem, the periodic.
01:02:34.300 --> 01:02:36.250
Sometimes I call it u.
01:02:36.250 --> 01:02:40.120
But what is the
periodic component?
01:02:40.120 --> 01:02:41.690
We expand it.
01:02:41.690 --> 01:02:46.420
We write it as a linear
combination of plane waves
01:02:46.420 --> 01:02:48.800
with appropriate coefficients.
01:02:48.800 --> 01:02:53.230
So if you want, your
quantum mechanical problem
01:02:53.230 --> 01:02:56.380
is, yet again, nothing
else than trying
01:02:56.380 --> 01:02:58.360
to find that these numbers.
01:02:58.360 --> 01:03:01.180
Once you have defined
your basis sector,
01:03:01.180 --> 01:03:03.220
then you have
written an expansion.
01:03:03.220 --> 01:03:06.760
And then all your algebra and
all your computational problem
01:03:06.760 --> 01:03:10.600
is finding out what this
coefficient should be.
01:03:10.600 --> 01:03:12.700
And this tends to be a
very good basis sector
01:03:12.700 --> 01:03:16.000
because it can be made more
and more accurate that by
01:03:16.000 --> 01:03:19.570
including G vectors of
larger and larger models
01:03:19.570 --> 01:03:22.420
that correspond to
plane waves of higher
01:03:22.420 --> 01:03:24.700
and higher resolution.
01:03:24.700 --> 01:03:29.380
Not only that, but it's actually
a very manageable basis sector
01:03:29.380 --> 01:03:30.760
to use.
01:03:30.760 --> 01:03:34.910
Because it's very easy to
take first derivatives,
01:03:34.910 --> 01:03:38.590
second derivatives of a wave
function of square moduli.
01:03:38.590 --> 01:03:43.090
Because, say, if we take a first
derivative of psi with respect
01:03:43.090 --> 01:03:47.170
to r, the only thing that we
have is that each term in G
01:03:47.170 --> 01:03:51.100
gets down from here
a factor i times G.
01:03:51.100 --> 01:03:52.690
And the second
derivative gives us
01:03:52.690 --> 01:03:55.960
a factor iG scalar product iG.
01:03:55.960 --> 01:03:57.880
That is minus G squared.
01:03:57.880 --> 01:03:58.690
OK.
01:03:58.690 --> 01:04:02.140
So our quantum
mechanical problem basis
01:04:02.140 --> 01:04:04.480
for our first principle
molecular dynamics
01:04:04.480 --> 01:04:08.590
has really nothing else
to do than finding what
01:04:08.590 --> 01:04:10.840
is the ground state energy.
01:04:10.840 --> 01:04:14.145
That, not really in a
density functional formalism,
01:04:14.145 --> 01:04:16.750
but in a sort of
simplified formalism,
01:04:16.750 --> 01:04:21.610
I write out just as a
sum of the eigenvalues
01:04:21.610 --> 01:04:23.890
of our single particle orbitals.
01:04:23.890 --> 01:04:28.000
That is just the sum of all the
electrons of this expectation
01:04:28.000 --> 01:04:28.663
value.
01:04:28.663 --> 01:04:30.580
If you remember, in
density functional theory,
01:04:30.580 --> 01:04:34.840
the energy is slightly different
because the Hamiltonian
01:04:34.840 --> 01:04:39.640
in that approach has actually
become self-consistent.
01:04:39.640 --> 01:04:42.980
The Hamiltonian depends on
the charge density itself.
01:04:42.980 --> 01:04:45.790
And so there are some
additional collective terms.
01:04:45.790 --> 01:04:49.430
But you know, from the point of
view of our sort of [INAUDIBLE]
01:04:49.430 --> 01:04:51.640
here, it doesn't
really matter that we
01:04:51.640 --> 01:04:53.830
include these additional terms.
01:04:53.830 --> 01:04:57.220
And so, you know, what is that
we have to deal with here?
01:04:57.220 --> 01:05:00.640
Well, we have, again, a
quantum kinetic energy term
01:05:00.640 --> 01:05:02.950
and a potential energy term.
01:05:02.950 --> 01:05:05.440
And so the quantum
kinetic energy
01:05:05.440 --> 01:05:08.860
is going to be the sum of the
kinetic energy of each orbital,
01:05:08.860 --> 01:05:10.360
as written here.
01:05:10.360 --> 01:05:12.040
And you know what I
was saying before.
01:05:12.040 --> 01:05:16.450
It's actually trivial to
calculate the kinetic energy
01:05:16.450 --> 01:05:20.800
if you have a wave function
that is expanded in plane waves
01:05:20.800 --> 01:05:22.180
because this is it.
01:05:22.180 --> 01:05:23.830
This is the expectation value.
01:05:23.830 --> 01:05:27.970
Remember, it's the integral
over all your Hilbert space.
01:05:27.970 --> 01:05:31.230
That is the integral
in space of this.
01:05:31.230 --> 01:05:32.230
This is the [INAUDIBLE].
01:05:32.230 --> 01:05:36.550
So this is the complex
conjugate of the plane waves
01:05:36.550 --> 01:05:37.850
e to the iGr.
01:05:37.850 --> 01:05:42.490
So it's e to the minus
iGr times the action
01:05:42.490 --> 01:05:47.090
of this operator to the plane
wave e to the iG prime r.
01:05:47.090 --> 01:05:50.280
And so, you know, this
second derivative,
01:05:50.280 --> 01:05:53.020
so with respect to
r, gives us an iG
01:05:53.020 --> 01:05:56.800
prime times iG prime
that is a minus G square.
01:05:56.800 --> 01:05:59.150
The minus sign cancels out.
01:05:59.150 --> 01:06:03.700
And so we have 1/2 g square
times the integral with respect
01:06:03.700 --> 01:06:09.670
to space of e to the minus
iGr times e to the iG prime r.
01:06:09.670 --> 01:06:12.580
And actually, if G
is equal to G prime,
01:06:12.580 --> 01:06:15.130
that is just going to be 1.
01:06:15.130 --> 01:06:16.810
And so the integral,
it's actually--
01:06:16.810 --> 01:06:18.730
I sort of skipped all
the normalization.
01:06:18.730 --> 01:06:20.447
So the integral
is going to be 1.
01:06:20.447 --> 01:06:21.655
You can work out the algebra.
01:06:21.655 --> 01:06:25.090
If G is different from G
prime, it's going to be 0.
01:06:25.090 --> 01:06:29.950
So this expectation value of
the kinetic energy operator
01:06:29.950 --> 01:06:32.290
between two plane
waves is just going
01:06:32.290 --> 01:06:36.790
to be 1/2 G square between
two identical plane waves.
01:06:36.790 --> 01:06:40.480
And it's going to be 0 if the
plane waves are different.
01:06:40.480 --> 01:06:43.960
This is what we say when we
say that the kinetic energy
01:06:43.960 --> 01:06:48.370
operator in a matrix
representation in which
01:06:48.370 --> 01:06:53.860
the rows and the columns are
the different plane waves
01:06:53.860 --> 01:06:55.690
is actually diagonal.
01:06:55.690 --> 01:07:01.260
That is, if you calculate this,
so you would have, say, G here
01:07:01.260 --> 01:07:06.210
and G prime here, all the
terms in this matrix function
01:07:06.210 --> 01:07:09.720
of G and G prime has 0
outside the diagonal.
01:07:09.720 --> 01:07:13.860
And the diagonal is
1/2 G square, so very
01:07:13.860 --> 01:07:17.520
trivial to do in plane waves.
01:07:17.520 --> 01:07:21.280
If we want to calculate the
second term in the energy,
01:07:21.280 --> 01:07:24.255
we need to calculate
the potential energy.
01:07:32.770 --> 01:07:35.470
And the potential
energy is going to be,
01:07:35.470 --> 01:07:40.205
again, sort of the sum over
all the single particle
01:07:40.205 --> 01:07:42.520
terms in the potential energy.
01:07:42.520 --> 01:07:44.570
And I'm doing the
same thing here.
01:07:44.570 --> 01:07:48.520
I'm writing this expectation
value between two plane waves.
01:07:48.520 --> 01:07:51.400
And so I have same as
before, the exponential
01:07:51.400 --> 01:07:55.090
of the complex conjugate
minus iGr times e of r
01:07:55.090 --> 01:07:57.590
times e to the iG prime r.
01:07:57.590 --> 01:08:01.090
And if you look at this,
this is nothing else
01:08:01.090 --> 01:08:03.940
than the definition
of the Fourier
01:08:03.940 --> 01:08:08.080
transform coefficient of your
potential energy with respect
01:08:08.080 --> 01:08:11.230
to the wave vector
G minus G prime.
01:08:11.230 --> 01:08:17.410
So now, this part, the potential
energy in a plane wave basis
01:08:17.410 --> 01:08:19.784
sector, is not anymore diagonal.
01:08:22.330 --> 01:08:25.359
It's going to have
off-diagonal terms.
01:08:25.359 --> 01:08:28.990
That is, if you look in
the previous sort of matrix
01:08:28.990 --> 01:08:34.390
representation in G and G prime,
what we have along the diagonal
01:08:34.390 --> 01:08:39.310
is just the kinetic energy
terms T that are 1/2 G squared.
01:08:39.310 --> 01:08:44.470
But we are going to have
a number of diagonal terms
01:08:44.470 --> 01:08:49.540
and also on-diagonal diagonal
terms from the potential.
01:08:49.540 --> 01:08:53.950
Obviously, the more you
go outside the diagonal,
01:08:53.950 --> 01:08:58.399
the more G minus G prime
is going to be large.
01:08:58.399 --> 01:09:01.450
So the more you're going
to look at high frequency
01:09:01.450 --> 01:09:03.580
components of your potential.
01:09:03.580 --> 01:09:06.220
And in general,
unless your potential
01:09:06.220 --> 01:09:09.819
is really ill-defined, the
high frequency components
01:09:09.819 --> 01:09:11.899
are going to be
smaller and smaller.
01:09:11.899 --> 01:09:15.430
So again, sort of the
overall Hamiltonian matrix
01:09:15.430 --> 01:09:18.490
kinetic plus potential
in a plane wave basis
01:09:18.490 --> 01:09:20.229
tends to be diagonally dominant.
01:09:20.229 --> 01:09:21.939
As we move farther
and farther away,
01:09:21.939 --> 01:09:24.560
it tends to be
smaller and smaller.
01:09:24.560 --> 01:09:26.859
So we have actually
written all the terms.
01:09:26.859 --> 01:09:29.710
And then we can take the sum
over all these expectation
01:09:29.710 --> 01:09:30.340
value.
01:09:30.340 --> 01:09:33.819
And so we have here
sort of a [INAUDIBLE],,
01:09:33.819 --> 01:09:39.700
if you want, the total energy as
a function of this coefficient
01:09:39.700 --> 01:09:41.109
of the plane waves.
01:09:41.109 --> 01:09:45.310
So we have the kinetic energy
term and the potential energy
01:09:45.310 --> 01:09:46.210
term.
01:09:46.210 --> 01:09:50.770
And the fundamental approach
that we take in first principle
01:09:50.770 --> 01:09:55.720
calculation is not trying to
diagonalize this Hamiltonian
01:09:55.720 --> 01:09:58.300
to find eigenvalues
and eigenvectors
01:09:58.300 --> 01:10:00.820
because that
operation would scale
01:10:00.820 --> 01:10:03.820
as the cube of the
number of plane waves.
01:10:03.820 --> 01:10:08.620
And it gives us not only the
10 or 20 lowest eigenvalues
01:10:08.620 --> 01:10:10.360
that have the occupied states.
01:10:10.360 --> 01:10:13.720
But, say, if we have a simple
system like the silicon atom--
01:10:13.720 --> 01:10:16.240
remember, you have seen
semiconductors in your lab
01:10:16.240 --> 01:10:18.790
two, where, at the
end, what you need
01:10:18.790 --> 01:10:23.830
is really the four lowest
eigenvalues and the four lowest
01:10:23.830 --> 01:10:25.000
eigenvectors.
01:10:25.000 --> 01:10:27.430
Well, if you diagonalize
this Hamiltonian
01:10:27.430 --> 01:10:31.090
and you have plane wave basis
set with 1,000 elements,
01:10:31.090 --> 01:10:34.900
you get all 1,000
eigenvalues and eigenvectors.
01:10:34.900 --> 01:10:35.710
We don't need that.
01:10:35.710 --> 01:10:36.710
It's too expensive.
01:10:36.710 --> 01:10:39.540
We need only 4 out of 1,000.
01:10:39.540 --> 01:10:40.090
OK.
01:10:40.090 --> 01:10:43.960
So the approach that most
or all electronic structure
01:10:43.960 --> 01:10:46.630
approaches take is
actually an approach
01:10:46.630 --> 01:10:51.580
in which we find what
are these coefficients
01:10:51.580 --> 01:10:55.180
of the relevant
occupied orbitals.
01:10:55.180 --> 01:10:57.358
You see, here, the
total energy, that's
01:10:57.358 --> 01:10:58.900
the function of the
occupied orbital.
01:10:58.900 --> 01:11:01.900
So for silicon,
we would sum only
01:11:01.900 --> 01:11:04.240
on the four lowest orbitals.
01:11:04.240 --> 01:11:08.110
And by minimizing
this quantity, we
01:11:08.110 --> 01:11:10.610
find what are the coefficients.
01:11:10.610 --> 01:11:13.540
So the electronic structure
problem, again, it's
01:11:13.540 --> 01:11:17.800
a minimum problem in which
you have an expression
01:11:17.800 --> 01:11:20.320
for the energy that, in
density functional theory,
01:11:20.320 --> 01:11:24.070
is slightly more complex,
but at the end depends only
01:11:24.070 --> 01:11:27.760
on the coefficients of
your plane wave expansion
01:11:27.760 --> 01:11:31.690
and on the matrix elements of
the potential between plane
01:11:31.690 --> 01:11:35.170
waves and of the kinetic
energy between plane waves.
01:11:35.170 --> 01:11:39.370
You calculate these ones for
all or, in density functional
01:11:39.370 --> 01:11:40.900
theory, at every step.
01:11:40.900 --> 01:11:43.750
Because your potential,
being self-consistent,
01:11:43.750 --> 01:11:45.670
changes up every time step.
01:11:45.670 --> 01:11:48.130
But if you have this
and you have this,
01:11:48.130 --> 01:11:50.650
it's nothing else than
a problem of finding
01:11:50.650 --> 01:11:53.380
the minimum with respect
to these variables.
01:11:53.380 --> 01:11:56.320
Sadly, you have sort
of thousands or tens
01:11:56.320 --> 01:11:59.060
of thousands or hundreds
of thousands of them.
01:11:59.060 --> 01:12:04.060
So it tends to be a
very expansive problem
01:12:04.060 --> 01:12:07.330
of minimization of a
non-linear function that
01:12:07.330 --> 01:12:11.570
depends on basically
zillions of variables.
01:12:11.570 --> 01:12:15.200
So how do we do this?
01:12:15.200 --> 01:12:16.000
Well, let's see.
01:12:21.920 --> 01:12:25.550
We use a molecular
dynamics idea.
01:12:25.550 --> 01:12:30.230
That is we have an energy,
a function of thousands
01:12:30.230 --> 01:12:31.560
of variables.
01:12:31.560 --> 01:12:36.050
So we have
multi-dimensional space
01:12:36.050 --> 01:12:39.560
with really thousands
of coordinates.
01:12:39.560 --> 01:12:43.920
And as a function of those
thousands of coordinates,
01:12:43.920 --> 01:12:51.770
these c and G variables, what we
have is a total energy surface.
01:12:51.770 --> 01:12:56.600
So our problem looks
nothing else like this.
01:12:56.600 --> 01:12:59.120
We have a total energy
that is this sort
01:12:59.120 --> 01:13:04.010
of hyper surface in black
that is non-linear functional
01:13:04.010 --> 01:13:06.110
of these things.
01:13:06.110 --> 01:13:08.605
In density functional
theory, for a standard LDA,
01:13:08.605 --> 01:13:13.220
GGA calculation, this system
tends to have only one minimum.
01:13:13.220 --> 01:13:15.650
If you are doing a spin
polarized calculation,
01:13:15.650 --> 01:13:17.330
you will have
different minima that
01:13:17.330 --> 01:13:22.610
corresponds to a
non-magnetic solution or all
01:13:22.610 --> 01:13:26.030
the magnetic solution with
different set of values
01:13:26.030 --> 01:13:27.410
for the magnetization.
01:13:27.410 --> 01:13:29.810
Or maybe have
anti-ferromagnetic solution
01:13:29.810 --> 01:13:32.600
with a total magnetization
of 0, but different
01:13:32.600 --> 01:13:34.380
spins and so on and so forth.
01:13:34.380 --> 01:13:36.170
So this potential
energy surface can
01:13:36.170 --> 01:13:39.478
start to develop some
of the complex minima
01:13:39.478 --> 01:13:41.270
that I was showing you
when we were talking
01:13:41.270 --> 01:13:42.710
about simulated annealing.
01:13:42.710 --> 01:13:44.600
And that's why actually
finding the ground
01:13:44.600 --> 01:13:49.040
state of magnetic system tend to
become, very quickly, much more
01:13:49.040 --> 01:13:49.700
complex.
01:13:49.700 --> 01:13:52.580
In LDA or GGA, it
has only one minimum.
01:13:52.580 --> 01:13:55.310
And we need to find that
minimum as a function
01:13:55.310 --> 01:13:58.190
of this very many coordinates.
01:13:58.190 --> 01:14:02.630
And so we use molecular
dynamics techniques
01:14:02.630 --> 01:14:04.940
and molecular dynamic analogies.
01:14:04.940 --> 01:14:08.390
And you know, I'll do
this in one dimension,
01:14:08.390 --> 01:14:15.170
but really what we want to do
is find an equation of motion,
01:14:15.170 --> 01:14:20.870
something that evolves
our coefficients, c, G,
01:14:20.870 --> 01:14:24.500
so that they move
towards the minimum
01:14:24.500 --> 01:14:27.020
of that potential
energy surface.
01:14:27.020 --> 01:14:30.170
And in order to do that,
we have nothing else
01:14:30.170 --> 01:14:34.070
to do than to calculate the
derivative of that Bloch
01:14:34.070 --> 01:14:36.770
potential energy
surface with respect
01:14:36.770 --> 01:14:40.610
to each and every variable, with
respect to each and every plane
01:14:40.610 --> 01:14:42.800
wave coefficient.
01:14:42.800 --> 01:14:46.520
Again-- sort of lots of algebra
that I'm hitting in there.
01:14:46.520 --> 01:14:49.070
But actually, and
very pleasantly,
01:14:49.070 --> 01:14:50.810
it's actually very simple.
01:14:50.810 --> 01:14:54.860
And we sort of write
this generalized force.
01:14:54.860 --> 01:14:58.130
That is the derivative
with the minus signs,
01:14:58.130 --> 01:15:00.710
the gradient of the
energy with respect
01:15:00.710 --> 01:15:02.900
to each and every coefficient.
01:15:02.900 --> 01:15:06.080
And it's, by chance
and nothing else than,
01:15:06.080 --> 01:15:09.410
the application of
the Hamiltonian itself
01:15:09.410 --> 01:15:13.100
to the wave function
coefficient by coefficient.
01:15:13.100 --> 01:15:16.830
And so it's actually sort
of very easy to calculate.
01:15:16.830 --> 01:15:21.110
And then once we
have that, well,
01:15:21.110 --> 01:15:25.220
so if we look at this
problem in one dimension,
01:15:25.220 --> 01:15:29.720
what we have is our Bloch
potential energy surface.
01:15:29.720 --> 01:15:32.730
And we need to find the minimum.
01:15:32.730 --> 01:15:35.660
So this is the
energy as a function
01:15:35.660 --> 01:15:40.610
of all this coefficient
of plane waves.
01:15:40.610 --> 01:15:46.700
And we can choose a random
value for this coefficient,
01:15:46.700 --> 01:15:48.690
calculate the energy.
01:15:48.690 --> 01:15:52.220
And what we will have is
a value for this energy.
01:15:52.220 --> 01:15:56.930
And now, what we want
is really to evolve,
01:15:56.930 --> 01:16:01.770
to move the value of each
and every coefficient.
01:16:01.770 --> 01:16:05.840
So that the total energy of
our system reaches the minimum.
01:16:05.840 --> 01:16:08.930
And in order to do that,
we need the gradient
01:16:08.930 --> 01:16:13.070
that we have written in the
previous expression that
01:16:13.070 --> 01:16:16.670
is nothing than minus the
Hamiltonian applied to the wave
01:16:16.670 --> 01:16:17.540
function.
01:16:17.540 --> 01:16:18.500
That is the force.
01:16:18.500 --> 01:16:22.400
And it's a perfect
dynamical analogy.
01:16:22.400 --> 01:16:24.620
If you are somewhere
and you know
01:16:24.620 --> 01:16:28.190
what is the force that
drives you downhill,
01:16:28.190 --> 01:16:31.310
well, then you have
molecular dynamics techniques
01:16:31.310 --> 01:16:35.390
to get to the bottom of your
potential energy surface.
01:16:35.390 --> 01:16:39.810
And this is particularly
easy if, as in LDA or GGA,
01:16:39.810 --> 01:16:43.340
your potential energy
surface has only one minimum.
01:16:43.340 --> 01:16:46.490
And there are actually
two different approaches
01:16:46.490 --> 01:16:48.380
that you could use.
01:16:48.380 --> 01:16:52.680
You could use standard
molecular dynamics.
01:16:52.680 --> 01:16:56.210
So if you are here, you
put yourself in motion.
01:16:56.210 --> 01:16:59.240
And what you start
doing is going down.
01:16:59.240 --> 01:17:02.570
But in a pure conservative
molecular dynamics,
01:17:02.570 --> 01:17:04.760
you are going to
overshoot your minimum.
01:17:04.760 --> 01:17:08.300
Go back and oscillate
perennially.
01:17:08.300 --> 01:17:12.260
So what you do, you
actually put some friction,
01:17:12.260 --> 01:17:16.430
so that you sort of
slowly lose energy.
01:17:16.430 --> 01:17:19.850
And you basically,
again, condense down
01:17:19.850 --> 01:17:23.810
to your sort of lowest
energy solution.
01:17:23.810 --> 01:17:28.630
And so a standard molecular
dynamics technique
01:17:28.630 --> 01:17:32.380
applied on the coefficient
of the plane waves
01:17:32.380 --> 01:17:37.270
with a friction term
added is one approach that
01:17:37.270 --> 01:17:41.110
brings all these additional
degrees of freedom of plane
01:17:41.110 --> 01:17:44.140
waves down to the ground state.
01:17:44.140 --> 01:17:46.210
This would be one approach.
01:17:46.210 --> 01:17:50.500
The other approach would be
actually evolving your system
01:17:50.500 --> 01:17:55.870
in a way in which the velocity
is proportional to the force.
01:17:55.870 --> 01:17:58.870
And so, again, you
can think that, when
01:17:58.870 --> 01:18:02.290
you are at the minimum,
your force is 0.
01:18:02.290 --> 01:18:05.140
And so your velocity is 0.
01:18:05.140 --> 01:18:08.710
And so if instead of
accelerating your system
01:18:08.710 --> 01:18:11.770
proportionally to the force
and putting a friction that
01:18:11.770 --> 01:18:15.940
is something that will oscillate
and dump down to the minimum,
01:18:15.940 --> 01:18:20.230
you move with a velocity that
is proportional to your force.
01:18:20.230 --> 01:18:23.440
It's something that, again,
it will sort of move you
01:18:23.440 --> 01:18:25.750
towards the ground state.
01:18:25.750 --> 01:18:30.580
Once you are here, your
driving force is going to be 0.
01:18:30.580 --> 01:18:32.620
So your velocity
is going to be 0.
01:18:32.620 --> 01:18:35.140
So somehow it's a
different approach
01:18:35.140 --> 01:18:37.150
that also brings you to the 0.
01:18:37.150 --> 01:18:48.610
And sort of the two approaches
have a slightly different
01:18:48.610 --> 01:18:51.640
performance depending
on the kind of system
01:18:51.640 --> 01:18:52.660
that you are doing.
01:18:52.660 --> 01:18:56.440
In this case, in the
sort of dynamical way,
01:18:56.440 --> 01:19:00.520
you need to make sure that
your friction is good enough.
01:19:00.520 --> 01:19:03.340
That is you don't use
too much friction,
01:19:03.340 --> 01:19:06.310
so you sort of flow down
and never reach the minimum.
01:19:06.310 --> 01:19:07.810
And you have to
be sure that there
01:19:07.810 --> 01:19:11.410
is sort of enough friction, so
you don't oscillate forever.
01:19:11.410 --> 01:19:14.050
And that's one way in
this sort of approach
01:19:14.050 --> 01:19:18.100
that you need to make sure
that your system, again,
01:19:18.100 --> 01:19:20.800
doesn't really
sort of stop short
01:19:20.800 --> 01:19:21.970
of getting to the minimum.
01:19:21.970 --> 01:19:24.400
Because the closer you
get to the minimum,
01:19:24.400 --> 01:19:27.340
the closer your
velocity is to 0.
01:19:27.340 --> 01:19:31.570
And so you tend to just get very
close, but not right to there.
01:19:31.570 --> 01:19:33.940
And there are
advanced techniques
01:19:33.940 --> 01:19:36.520
to deal with both
of these systems.
01:19:36.520 --> 01:19:46.360
And with this, I think
I'll actually conclude here
01:19:46.360 --> 01:19:51.280
because I wanted to sort of tell
you more about how we actually
01:19:51.280 --> 01:19:56.860
evolve in time this problem
when we start moving the atoms.
01:19:56.860 --> 01:20:00.370
Up to now, we were sort of
thinking at just how to get out
01:20:00.370 --> 01:20:02.080
at the ground state solution.
01:20:02.080 --> 01:20:04.690
But then we have the
additional problem
01:20:04.690 --> 01:20:09.130
of having the atoms
move, so our potential
01:20:09.130 --> 01:20:11.620
starts changing in real time.
01:20:11.620 --> 01:20:14.920
And we need to sort of
follow in real time what
01:20:14.920 --> 01:20:16.120
happens to the solution.
01:20:16.120 --> 01:20:19.450
And again, there are sort of
two general approaches that
01:20:19.450 --> 01:20:22.810
go under the name of sort of
Born-Oppenheimer molecular
01:20:22.810 --> 01:20:26.140
dynamics or sort of
Car-Parrinello molecular
01:20:26.140 --> 01:20:26.830
dynamics.
01:20:26.830 --> 01:20:30.280
And I think we'll leave
that at this stage for one
01:20:30.280 --> 01:20:31.540
of the next lectures.
01:20:31.540 --> 01:20:34.150
So let me just
sort of remind you
01:20:34.150 --> 01:20:36.820
that another of the
terms that you'll
01:20:36.820 --> 01:20:41.140
see for this kind of dynamics
in which the velocity drives
01:20:41.140 --> 01:20:45.430
the system towards the
minimum goes actually
01:20:45.430 --> 01:20:49.060
under the name of steepest
descent or conjugate gradient
01:20:49.060 --> 01:20:49.990
minimization.
01:20:49.990 --> 01:20:51.970
And that's, again,
a sort of technique
01:20:51.970 --> 01:20:54.760
that you see very often
in minimization problem
01:20:54.760 --> 01:20:58.960
where sort of this is really
a molecular dynamic kind
01:20:58.960 --> 01:21:00.880
of simulated [INAUDIBLE].
01:21:00.880 --> 01:21:02.470
With this, I conclude.
01:21:02.470 --> 01:21:06.610
We'll keep some of the sort
of ab initio molecular dynamic
01:21:06.610 --> 01:21:08.860
techniques for one
of the last lectures
01:21:08.860 --> 01:21:12.700
when we actually go back and
look at case studies of this.
01:21:12.700 --> 01:21:15.610
And in the last transparencies,
in the last few graph
01:21:15.610 --> 01:21:19.430
of your notes, you have,
again, some freely available
01:21:19.430 --> 01:21:24.790
bibliography of quantum
molecular dynamics
01:21:24.790 --> 01:21:27.730
primers, in particular,
the sort of primer
01:21:27.730 --> 01:21:30.250
of Dominik Marx on his website.
01:21:30.250 --> 01:21:34.150
The last note is something
very completely and very well
01:21:34.150 --> 01:21:34.810
written.
01:21:34.810 --> 01:21:36.220
With this, I conclude.
01:21:36.220 --> 01:21:39.220
Have a very happy weekend,
and we'll see you on Tuesday
01:21:39.220 --> 01:21:42.315
in the lab in 115.