WEBVTT
00:00:00.060 --> 00:00:02.500
The following content is
provided under a Creative
00:00:02.500 --> 00:00:04.019
Commons license.
00:00:04.019 --> 00:00:06.360
Your support will help
MIT OpenCourseWare
00:00:06.360 --> 00:00:10.730
continue to offer high quality
educational resources for free.
00:00:10.730 --> 00:00:13.340
To make a donation or
view additional materials
00:00:13.340 --> 00:00:17.217
from hundreds of MIT courses,
visit MIT OpenCourseWare
00:00:17.217 --> 00:00:17.842
at ocw.mit.edu.
00:00:22.350 --> 00:00:25.440
PROFESSOR: OK, let's start.
00:00:25.440 --> 00:00:34.410
So we were talking about
melting in two dimensions,
00:00:34.410 --> 00:00:38.750
and the picture that
you had was something
00:00:38.750 --> 00:00:48.570
like a triangular lattice,
which at zero temperature
00:00:48.570 --> 00:00:55.520
has particles sitting
at precise sites--
00:00:55.520 --> 00:00:59.190
let's say, on this
triangular lattice--
00:00:59.190 --> 00:01:01.135
but then at finite
temperature, the particles
00:01:01.135 --> 00:01:05.370
will to start to deform.
00:01:05.370 --> 00:01:12.600
And the deformations were
indicated by a vector u.
00:01:12.600 --> 00:01:19.630
And the idea was that this
is like an elastic material,
00:01:19.630 --> 00:01:23.310
as long as we're thinking
about these long wavelength
00:01:23.310 --> 00:01:24.090
deformations.
00:01:24.090 --> 00:01:28.140
u and the energy
costs can be written
00:01:28.140 --> 00:01:32.830
for an isotropic material
in two dimensions in terms
00:01:32.830 --> 00:01:34.955
of two invariants.
00:01:37.750 --> 00:01:40.460
And traditionally, it
is written in terms
00:01:40.460 --> 00:01:46.855
of the so called lame
coefficients, mu and lambda.
00:01:52.800 --> 00:02:03.010
Where this uij,
which is the strain,
00:02:03.010 --> 00:02:08.800
is obtained by taking
derivatives of the deformation,
00:02:08.800 --> 00:02:12.494
the iuj, and symmetrizing it.
00:02:12.494 --> 00:02:16.610
This symmetrization essentially
eliminates an energy
00:02:16.610 --> 00:02:18.475
at a cost for rotations.
00:02:20.990 --> 00:02:25.650
And then because of this
simple quadratic translation
00:02:25.650 --> 00:02:29.090
of invariant form,
we could also express
00:02:29.090 --> 00:02:32.000
this in terms of fullier mode.
00:02:32.000 --> 00:02:37.670
And I'm going to write the
fullier description slightly
00:02:37.670 --> 00:02:42.370
differently than last time.
00:02:42.370 --> 00:02:44.920
Basically, this
whole form can be
00:02:44.920 --> 00:02:55.625
written as u plus 2 lambda over
2 q dot u tilde of q squared.
00:02:59.160 --> 00:03:04.410
And the other term-- other than
previously I had written things
00:03:04.410 --> 00:03:06.630
in terms of q dot
u and q squared
00:03:06.630 --> 00:03:14.320
u squared-- we write it in terms
of q crossed with u tilde of q
00:03:14.320 --> 00:03:14.820
squared.
00:03:23.750 --> 00:03:27.790
Essentially, you can see that
this ratifies that they're
00:03:27.790 --> 00:03:32.560
going to have modes that
are in the direction of q,
00:03:32.560 --> 00:03:35.290
the longitudinal modes.
00:03:35.290 --> 00:03:38.160
Cost is nu plus 2
lambda, and those
00:03:38.160 --> 00:03:42.030
that are transfers or orthogonal
to the direction of q,
00:03:42.030 --> 00:03:45.050
whose cost is just mu.
00:03:45.050 --> 00:03:49.030
And clearly if I were
to go into real space,
00:03:49.030 --> 00:03:53.720
this is kind of related
to a divergence of u.
00:03:53.720 --> 00:03:56.750
And the divergence
of u corresponds
00:03:56.750 --> 00:04:02.780
to essentially squeezing or
expanding this deformation.
00:04:02.780 --> 00:04:04.670
So what these measures
is essentially
00:04:04.670 --> 00:04:08.410
the cost of changing
the density.
00:04:08.410 --> 00:04:12.090
And this combination is
related to the bark modulus.
00:04:12.090 --> 00:04:14.270
You have that even for a liquid.
00:04:14.270 --> 00:04:16.329
So if you have a liquid,
you try to squeeze it.
00:04:16.329 --> 00:04:19.130
There will be a
bulk energy cost.
00:04:19.130 --> 00:04:24.580
And this term, which in the
real space is kind of related
00:04:24.580 --> 00:04:29.360
to kern u, you would say
is corresponding to making
00:04:29.360 --> 00:04:31.630
the rotations.
00:04:31.630 --> 00:04:37.480
So if you try to rotate
this material locally,
00:04:37.480 --> 00:04:41.530
then the corresponding
sheer cost of the formation
00:04:41.530 --> 00:04:46.030
has a cost that is indicated
by mu, the sheer modulus.
00:04:46.030 --> 00:04:50.590
And basically what really
makes a solid is this term.
00:04:50.590 --> 00:04:55.050
Because as I said, a liquid
also has the bark modulus,
00:04:55.050 --> 00:04:58.740
but lacks the resistance
to try to sheer it,
00:04:58.740 --> 00:05:02.260
which is captured by this, that
is unique and characteristic
00:05:02.260 --> 00:05:03.090
of a solid.
00:05:06.640 --> 00:05:10.640
So this is the energy cost.
00:05:10.640 --> 00:05:14.470
The other part of
this whole story
00:05:14.470 --> 00:05:19.640
is that this
structure has order.
00:05:19.640 --> 00:05:24.190
And we can
characterize that order
00:05:24.190 --> 00:05:29.370
which makes it distinct from a
liquid or gas a number of ways.
00:05:29.370 --> 00:05:32.390
One was to do an
x-ray scattering,
00:05:32.390 --> 00:05:35.010
and then you would
see the back peaks.
00:05:35.010 --> 00:05:38.870
And really that type of
order is translational.
00:05:43.110 --> 00:05:45.961
And you characterize that
by an order parameter.
00:05:49.050 --> 00:05:51.280
It's kind of like
a spin that you
00:05:51.280 --> 00:05:56.130
have in the case of a
magnet being up or down.
00:05:56.130 --> 00:06:01.990
In this case, this
object was e to the i g
00:06:01.990 --> 00:06:05.597
dot u-- the deformation that
you have that's on location r.
00:06:10.570 --> 00:06:18.520
And then these g's are chosen to
be the inverse lattice vectors.
00:06:22.080 --> 00:06:24.410
It doesn't really matter
whether I write here
00:06:24.410 --> 00:06:27.500
u of r or the actual position.
00:06:27.500 --> 00:06:31.550
Because the actual positions
starts at zero temperature,
00:06:31.550 --> 00:06:36.530
we devalue r 0, such that
the dot product of that g
00:06:36.530 --> 00:06:38.470
is a multiple of 2pi.
00:06:38.470 --> 00:06:43.990
And so essentially,
that's what captures this.
00:06:43.990 --> 00:06:47.580
Clearly, if I start with
a zero temperature picture
00:06:47.580 --> 00:06:51.890
and just move this
around, the phase
00:06:51.890 --> 00:06:55.400
of this order parameter
over here will change,
00:06:55.400 --> 00:06:58.640
but it will be the
same across the system.
00:06:58.640 --> 00:07:02.200
And so this is long
range correlation that
00:07:02.200 --> 00:07:04.580
is present at zero
temperature, you
00:07:04.580 --> 00:07:08.350
can ask what happens to
it at finite temperature.
00:07:08.350 --> 00:07:13.650
So we can look at the row
g at some position, row g
00:07:13.650 --> 00:07:17.000
star at some other position.
00:07:17.000 --> 00:07:23.560
And so that was related to
exponential of minus g squared
00:07:23.560 --> 00:07:27.380
over 2-- something
like u squared
00:07:27.380 --> 00:07:31.475
x, or u of x minus 0 squared.
00:07:34.410 --> 00:07:39.260
And what we saw
was that this thing
00:07:39.260 --> 00:07:44.570
had a characteristic that it
was falling off with distance
00:07:44.570 --> 00:07:51.460
according to some
kind of power law.
00:07:51.460 --> 00:07:55.830
The exponent of this power
law, when calculated,
00:07:55.830 --> 00:07:58.500
clearly is related
to this g squared.
00:07:58.500 --> 00:08:02.316
Because this is the quantity
that goes logarithmically.
00:08:02.316 --> 00:08:07.660
And so the answer was
g squared over 4 pi.
00:08:07.660 --> 00:08:12.390
Heat was dependent on these
two modes being present.
00:08:12.390 --> 00:08:19.134
So you have nu 2 nu plus lambda,
and then 2 nu plus lambda.
00:08:19.134 --> 00:08:20.300
You had a form such as this.
00:08:24.520 --> 00:08:30.450
Now this result was
obtained as long
00:08:30.450 --> 00:08:36.150
as we were treating this
field, u, as just the continuum
00:08:36.150 --> 00:08:40.799
field that satisfies this.
00:08:40.799 --> 00:08:45.190
And this result is
really different, also,
00:08:45.190 --> 00:08:48.365
from the expectation that
at very high temperature
00:08:48.365 --> 00:08:52.370
the particle in a liquid
should not know anything
00:08:52.370 --> 00:08:54.870
about the particle
further out in the liquid,
00:08:54.870 --> 00:09:00.050
as long as they're beyond
some small correlation links.
00:09:00.050 --> 00:09:03.530
So we expect this to
actually decay exponentially
00:09:03.530 --> 00:09:05.540
at high temperatures.
00:09:05.540 --> 00:09:09.570
And we found that
we could account
00:09:09.570 --> 00:09:25.150
for that by addition
of these locations,
00:09:25.150 --> 00:09:31.900
can cause a transition
to a high temperature
00:09:31.900 --> 00:09:39.931
phase in which row g, row
g star, between x and 0,
00:09:39.931 --> 00:09:40.806
decays exponentially.
00:09:45.160 --> 00:09:48.580
As opposed to this
algebraic behavior,
00:09:48.580 --> 00:09:53.920
indicating that these
locations-- once you
00:09:53.920 --> 00:09:57.320
go to sufficiently
high temperature,
00:09:57.320 --> 00:10:00.700
such that the entropy of
creating and rearranging
00:10:00.700 --> 00:10:05.655
these dislocations overcomes
the large cost of creating them
00:10:05.655 --> 00:10:10.130
in the first place,
then you'll have
00:10:10.130 --> 00:10:15.300
this absence of
translational order,
00:10:15.300 --> 00:10:22.540
and some kind of exponential
decay of this order parameter.
00:10:22.540 --> 00:10:27.370
So at this stage, you may
feel comfortable enough
00:10:27.370 --> 00:10:30.760
to say that addition
of these dislocation
00:10:30.760 --> 00:10:36.480
causes our solid to melt
and become a liquid.
00:10:36.480 --> 00:10:43.970
Now, I indicated,
however, that the sun also
00:10:43.970 --> 00:10:45.670
has an orientational role.
00:10:56.130 --> 00:11:03.970
What I could do is-- at
each location in the solid,
00:11:03.970 --> 00:11:11.470
I can ask how much has
the angle been deformed,
00:11:11.470 --> 00:11:13.320
and look at the bond angle.
00:11:13.320 --> 00:11:15.300
So maybe this
particle moved here,
00:11:15.300 --> 00:11:17.270
and this particle moved here.
00:11:17.270 --> 00:11:19.700
Somewhere else, the
particles may have
00:11:19.700 --> 00:11:21.900
moved in a different fashion.
00:11:21.900 --> 00:11:24.600
And the angle that
was originally,
00:11:24.600 --> 00:11:28.630
say, along the x direction,
had rotated somewhere else.
00:11:28.630 --> 00:11:32.630
And clearly, again,
at zero temperature,
00:11:32.630 --> 00:11:39.040
I can look at the correlations
of this angular order,
00:11:39.040 --> 00:11:42.660
and they would be the
same across the system.
00:11:42.660 --> 00:11:46.510
I can ask what happens when
I include these deformations
00:11:46.510 --> 00:11:49.330
and then the dislocations.
00:11:49.330 --> 00:11:53.480
So in the same way that we
defined the translational order
00:11:53.480 --> 00:11:57.750
parameter, I can define an
orientational order parameter.
00:12:07.310 --> 00:12:14.080
Let's call it sci
at some location, r,
00:12:14.080 --> 00:12:16.965
which is e to the i.
00:12:16.965 --> 00:12:20.750
Theta at that location r--
00:12:20.750 --> 00:12:24.780
Except that when I look
at the triangular lattice,
00:12:24.780 --> 00:12:28.440
it may be that the triangles
have actually rotated
00:12:28.440 --> 00:12:31.860
by 60 degrees or 120 degrees.
00:12:31.860 --> 00:12:36.250
And I can't really tell whether
I clicked once, zero times,
00:12:36.250 --> 00:12:38.310
twice, et cetera.
00:12:38.310 --> 00:12:44.420
So because of this symmetry of
the original lattice on their
00:12:44.420 --> 00:12:50.320
on their theta going to
theta plus 2 pi over 6,
00:12:50.320 --> 00:12:53.570
I have to use something
like this that will not
00:12:53.570 --> 00:12:57.480
be modified if I make
this transformation,
00:12:57.480 --> 00:12:58.930
even at zero temperature.
00:12:58.930 --> 00:13:03.918
If I miscount some angle by 60
degrees, this will become fine.
00:13:07.670 --> 00:13:13.500
Now I want to calculate the
correlations of this theta
00:13:13.500 --> 00:13:17.710
from one part of this system
to another part of the system.
00:13:17.710 --> 00:13:20.440
So for that, what
I need to do is
00:13:20.440 --> 00:13:27.330
to look at the
relationship between theta
00:13:27.330 --> 00:13:32.470
and the distortion field,
u, that I told you before.
00:13:32.470 --> 00:13:37.490
Now you can see that right
on the top right corner
00:13:37.490 --> 00:13:41.790
I took the distortion field,
and I took it's derivative,
00:13:41.790 --> 00:13:44.150
and then symmetrized
the result in pencil.
00:13:44.150 --> 00:13:48.750
And that symmetrization
actually removes any rotation
00:13:48.750 --> 00:13:50.430
that I would have.
00:13:50.430 --> 00:13:53.000
So in order to bring
back the notation,
00:13:53.000 --> 00:13:55.390
I just have to put a minus sign.
00:13:55.390 --> 00:14:05.150
And indeed, one can show that
the distortion or displacement
00:14:05.150 --> 00:14:09.450
u or r across my
system-- let's call it
00:14:09.450 --> 00:14:16.210
u of x-- leads to a
corresponding angular
00:14:16.210 --> 00:14:24.903
distortion, theta, at x, which
is minus one half-- let's
00:14:24.903 --> 00:14:31.180
call it z hat dotted
with curve of u.
00:14:31.180 --> 00:14:35.340
So if, rather than doing
the i u j plus d j u i,
00:14:35.340 --> 00:14:38.210
if I put a minus sign,
you can see that I
00:14:38.210 --> 00:14:40.600
have the structure of a curve.
00:14:40.600 --> 00:14:43.280
In two dimensions,
actually curve
00:14:43.280 --> 00:14:45.640
would be something that
would be pointing only
00:14:45.640 --> 00:14:48.710
along the z direction.
00:14:48.710 --> 00:14:51.830
And so I just make
a scale on my dot,
00:14:51.830 --> 00:14:54.990
without taking that
in the z direction.
00:14:54.990 --> 00:14:59.120
And so you can do
some distortion,
00:14:59.120 --> 00:15:02.440
and convince yourself
that for each distortion
00:15:02.440 --> 00:15:04.282
you will get an
angle that is this.
00:15:04.282 --> 00:15:06.282
AUDIENCE: Do we need some
kind of thermalization
00:15:06.282 --> 00:15:08.218
to fix the dimensions of this?
00:15:08.218 --> 00:15:12.100
Because that can go u has
dimensions of fields, and u--
00:15:12.100 --> 00:15:16.630
PROFESSOR: I'm only talking
about two dimensions.
00:15:16.630 --> 00:15:20.860
And in any case,
you can see that u
00:15:20.860 --> 00:15:23.530
is a distortion--
is a displacement--
00:15:23.530 --> 00:15:26.730
the gradient is reduced
by the displacement,
00:15:26.730 --> 00:15:29.830
so this thing is
dimensionless as long as you
00:15:29.830 --> 00:15:31.924
have these dimensions.
00:15:31.924 --> 00:15:32.590
AUDIENCE: Sorry.
00:15:32.590 --> 00:15:33.464
PROFESSOR: Yes?
00:15:33.464 --> 00:15:34.630
AUDIENCE: That's a 2, right?
00:15:34.630 --> 00:15:35.244
Not a c?
00:15:35.244 --> 00:15:36.160
PROFESSOR: That's a 2.
00:15:38.760 --> 00:15:42.810
It's the same 2 that I have for
the definition of the strain.
00:15:42.810 --> 00:15:44.575
Rather than a plus,
you put a minus.
00:15:47.214 --> 00:15:48.630
AUDIENCE: So can
we think of these
00:15:48.630 --> 00:15:51.405
as two sets of Goldstone
modes, or is that not a way
00:15:51.405 --> 00:15:52.190
to interpret it?
00:15:52.190 --> 00:15:54.672
Is it like two order parameters?
00:15:54.672 --> 00:15:58.690
I mean, you have a think
that has u dependence, but--
00:15:58.690 --> 00:16:02.210
PROFESSOR: OK, so let's look
at this picture over here.
00:16:02.210 --> 00:16:05.000
You do have two sets of
Goldstone modes corresponding
00:16:05.000 --> 00:16:07.720
to longitudinal transfers.
00:16:07.720 --> 00:16:10.160
You can see that this
curve is the thing
00:16:10.160 --> 00:16:12.540
is that I call the angle.
00:16:12.540 --> 00:16:16.360
So if you like, you can
put the angle over here.
00:16:16.360 --> 00:16:20.170
But the difference between
putting an angle here,
00:16:20.170 --> 00:16:23.420
and this term, is that
in terms of the angle,
00:16:23.420 --> 00:16:26.000
there is no q dependence.
00:16:26.000 --> 00:16:28.200
So it is not a ghost.
00:16:28.200 --> 00:16:35.390
Because the cost of making a
distortion of wave number q
00:16:35.390 --> 00:16:37.190
does not vanish as
q squared works.
00:16:44.010 --> 00:16:48.850
All right, so then I can look
at the correlation between, say,
00:16:48.850 --> 00:16:54.200
sci of x, sci star of zero.
00:16:54.200 --> 00:16:59.660
And what I will be calculating
is expectation value of e
00:16:59.660 --> 00:17:03.350
to the i 6.
00:17:03.350 --> 00:17:05.680
And then I will have
this factor of--
00:17:13.040 --> 00:17:15.460
So let me write it
in this fashion.
00:17:15.460 --> 00:17:18.990
Theta of x minus theta of 0.
00:17:23.750 --> 00:17:27.960
Since u is Gaussian
distributed, theta
00:17:27.960 --> 00:17:30.680
in the Gaussian distributor.
00:17:30.680 --> 00:17:34.130
So for any Gaussian
distributed entity,
00:17:34.130 --> 00:17:37.550
we can write the
exponential of e
00:17:37.550 --> 00:17:45.300
to the something as its average
as exponential of minus 1/2
00:17:45.300 --> 00:17:49.040
the average of whatever
is in the exponent.
00:17:49.040 --> 00:17:54.380
So I will get 36 divided by 2.
00:17:54.380 --> 00:18:00.480
I do have the expectation
value of delta theta squared.
00:18:00.480 --> 00:18:07.800
But delta theta is related
up to this factor of 1/4
00:18:07.800 --> 00:18:10.900
to some expectation
value of kern u.
00:18:10.900 --> 00:18:18.370
So I would need to calculate
kern mu at x minus kern u
00:18:18.370 --> 00:18:23.870
at 0, the whole thing squared
with the Gaussian average.
00:18:27.940 --> 00:18:35.580
Now, this entity--
clearly what I can do
00:18:35.580 --> 00:18:38.570
is to go back and
look at these things
00:18:38.570 --> 00:18:45.820
in terms of Fourier space,
rather than position space.
00:18:45.820 --> 00:18:54.830
So this becomes an integral
d 2 q 2 pi to the d.
00:18:54.830 --> 00:19:01.280
I will get e to the
i q dot x minus 1.
00:19:01.280 --> 00:19:07.090
And then I have something
like q cross u tilde of q.
00:19:07.090 --> 00:19:09.620
And I have to do that twice.
00:19:09.620 --> 00:19:13.480
When I do that twice, I find
that the different q's are
00:19:13.480 --> 00:19:14.530
uncorrelated.
00:19:14.530 --> 00:19:18.100
So I will get, rather than
two of these integrals, one
00:19:18.100 --> 00:19:19.555
of these integrals.
00:19:19.555 --> 00:19:23.960
And because the q and q prime
are said to be the same,
00:19:23.960 --> 00:19:28.430
the product of those two
factors will be the integral 2
00:19:28.430 --> 00:19:33.010
minus 2 cosine of q dot x
term that we are used to.
00:19:33.010 --> 00:19:36.440
And so that's where the
x dependence appears.
00:19:36.440 --> 00:19:43.870
And then I need the
average of q cross mu of q.
00:19:43.870 --> 00:19:47.752
And that I can read off
the beta [INAUDIBLE],
00:19:47.752 --> 00:19:51.010
root the energy over here.
00:19:51.010 --> 00:19:53.360
You can see that there
is a Gaussian cost
00:19:53.360 --> 00:19:56.770
for q plus u of q
squared, which is simply
00:19:56.770 --> 00:19:59.240
1 of a lingering variance.
00:19:59.240 --> 00:20:03.141
So basically, this term
you'll the sum of 1 over u.
00:20:05.787 --> 00:20:09.230
Now the difference between
all of the calculations
00:20:09.230 --> 00:20:13.010
that we were doing previously,
as was asked regarding
00:20:13.010 --> 00:20:17.220
Goldstone modes-- if I was just
looking at u squared, which
00:20:17.220 --> 00:20:19.720
is what I was doing
up here, I would
00:20:19.720 --> 00:20:23.670
need to put another factor of
1 over q squared [INAUDIBLE].
00:20:23.670 --> 00:20:26.180
And then I would have
the coulomb integral that
00:20:26.180 --> 00:20:27.960
would grow logarithmically.
00:20:27.960 --> 00:20:32.720
But here you can see that
the whole thing-- the cosine
00:20:32.720 --> 00:20:34.620
integrated against
the constant--
00:20:34.620 --> 00:20:36.280
will average out to 0.
00:20:36.280 --> 00:20:39.250
So I will think you
have 2 over u times
00:20:39.250 --> 00:20:41.440
this integral is a constant.
00:20:41.440 --> 00:20:43.750
So the whole thing,
at the end of the day,
00:20:43.750 --> 00:20:51.050
is exponential of-- that
becomes a 9 divided by 2.
00:20:56.190 --> 00:21:00.830
There's a factor
of 1 over the mu,
00:21:00.830 --> 00:21:06.260
and then I have twice the
integral of d 2 q over 2 pi
00:21:06.260 --> 00:21:06.960
squared.
00:21:06.960 --> 00:21:08.730
Which is-- you can
convince yourself
00:21:08.730 --> 00:21:13.695
simply the density of the
system a number of times.
00:21:16.940 --> 00:21:25.490
So as opposed to the
translational order, which
00:21:25.490 --> 00:21:29.420
was decaying as above
our lot, then we
00:21:29.420 --> 00:21:31.460
include the phonon modes.
00:21:31.460 --> 00:21:34.450
When we include
these phonon modes,
00:21:34.450 --> 00:21:40.940
we find that the orientational
order decays much more weakly.
00:21:40.940 --> 00:21:43.980
So that was falling off as
I went further and further.
00:21:43.980 --> 00:21:46.360
This, as I go
further and further,
00:21:46.360 --> 00:21:49.310
eventually reaches a
constant that is less than 1,
00:21:49.310 --> 00:21:51.020
but it is something.
00:21:51.020 --> 00:21:53.690
Using conversely
proportional to temperature--
00:21:53.690 --> 00:21:59.500
so as I go to 0
temperature, these go to 1.
00:21:59.500 --> 00:22:04.280
And basically because
this order parameter,
00:22:04.280 --> 00:22:06.745
with respect to--
well, this measure
00:22:06.745 --> 00:22:10.080
of distortion with respect
to that measure of distortion
00:22:10.080 --> 00:22:12.940
has an additional
factor of gradient.
00:22:12.940 --> 00:22:16.810
I will get an additional
factor of q squared, and then
00:22:16.810 --> 00:22:19.200
everything changes accordingly.
00:22:19.200 --> 00:22:26.650
So orientational order
is much more robust.
00:22:26.650 --> 00:22:29.010
This phase that we were
calling the analogue of a two
00:22:29.010 --> 00:22:33.570
dimensional solid had only
quasi long range order.
00:22:33.570 --> 00:22:36.890
The long range order was
decaying as a power law.
00:22:39.510 --> 00:22:40.980
Yes?
00:22:40.980 --> 00:22:44.620
AUDIENCE: Is n dependent
on the position, or--
00:22:44.620 --> 00:22:46.340
PROFESSOR: No.
00:22:46.340 --> 00:22:50.370
So basically, if you were to
remember the number of points
00:22:50.370 --> 00:22:53.650
it should be the same as the
number of allowed fullier
00:22:53.650 --> 00:22:55.150
modes.
00:22:55.150 --> 00:22:57.790
And this goes to
an integral-- 2 q
00:22:57.790 --> 00:23:04.610
over 2 pi squared-- when I put
the area in two dimensions.
00:23:04.610 --> 00:23:06.540
So the integral over
whatever [INAUDIBLE]
00:23:06.540 --> 00:23:09.460
zone you have over
the fullier modes
00:23:09.460 --> 00:23:12.800
is the same thing as
the number of points
00:23:12.800 --> 00:23:17.050
that you have in the original
lattice, divided by area,
00:23:17.050 --> 00:23:23.125
or 1 over the size of one
of those triangles squared.
00:23:23.125 --> 00:23:23.625
Yes?
00:23:23.625 --> 00:23:26.940
AUDIENCE: Where is the x
dependent in that expression?
00:23:26.940 --> 00:23:28.810
PROFESSOR: OK, the x
dependence basically
00:23:28.810 --> 00:23:33.940
disappears because you integrate
over the cosine of q x.
00:23:33.940 --> 00:23:38.297
And if x is sufficiently large,
those fluctuations disappear.
00:23:38.297 --> 00:23:40.672
AUDIENCE: Oh, so we're really
looking at the [INAUDIBLE].
00:23:40.672 --> 00:23:41.660
PROFESSOR: Yes, that's right.
00:23:41.660 --> 00:23:43.256
So at short distances,
there are going
00:23:43.256 --> 00:23:45.610
to be some oscillations
or whatever.
00:23:45.610 --> 00:23:48.530
But it gradually-- we are
interested in the long distance
00:23:48.530 --> 00:23:50.300
behavior.
00:23:50.300 --> 00:23:52.330
At very short
distances, I can't even
00:23:52.330 --> 00:23:54.930
use the continuum
description for things
00:23:54.930 --> 00:23:57.451
that are three or
lattice spacings apart.
00:24:05.310 --> 00:24:10.790
So maybe I should explicitly
say that this is usually
00:24:10.790 --> 00:24:19.430
called quasi long range order,
versus this dependence, which
00:24:19.430 --> 00:24:20.640
is two long ranges.
00:24:32.120 --> 00:24:40.050
So given that this is more
robust than these forum-like
00:24:40.050 --> 00:24:43.370
fluctuations, the
next question is, well
00:24:43.370 --> 00:24:49.314
does it completely disappear
when I include these locations.
00:24:49.314 --> 00:24:54.930
So again, this calculation,
based on Gaussian's, relies
00:24:54.930 --> 00:25:01.130
on just the fullier modes of
that line that I have up there.
00:25:01.130 --> 00:25:05.090
It does not include the
dislocations, which, in order
00:25:05.090 --> 00:25:07.530
to properly account,
you saw that we
00:25:07.530 --> 00:25:13.760
need to look at a collections
of these locations appearing
00:25:13.760 --> 00:25:16.120
at different positions
on the lattice.
00:25:16.120 --> 00:25:19.850
And they had these vectorial
nature of the fullier
00:25:19.850 --> 00:25:22.410
interactions among them.
00:25:22.410 --> 00:25:30.566
So presumably, when I
go into the base where
00:25:30.566 --> 00:25:45.150
these locations unbind--
and by unbinding--
00:25:45.150 --> 00:25:47.342
as I said, in the
low temperature
00:25:47.342 --> 00:25:48.716
picture of the
dislocations, they
00:25:48.716 --> 00:25:51.950
should appear very
close to each other
00:25:51.950 --> 00:25:55.920
because it is costly to separate
them by an amount that grows
00:25:55.920 --> 00:25:57.870
invariably in the separation.
00:25:57.870 --> 00:26:02.220
In the unbound phase, you
have essentially a gas
00:26:02.220 --> 00:26:06.160
of dislocations that
can be anywhere.
00:26:06.160 --> 00:26:11.860
So the picture here
now is that indeed this
00:26:11.860 --> 00:26:15.680
is a phase, that if I just
focus on the dislocations,
00:26:15.680 --> 00:26:17.230
there is a whole bunch of them.
00:26:17.230 --> 00:26:19.910
In a triangular
lattice, they could
00:26:19.910 --> 00:26:26.240
be pointing in any one of three
directions, plus or minus.
00:26:26.240 --> 00:26:34.190
And then there is certainly
an additional contribution
00:26:34.190 --> 00:26:41.550
to the angle that comes from the
presence of these dislocations.
00:26:41.550 --> 00:26:46.830
So you calculate-- if you
have a dislocation that
00:26:46.830 --> 00:26:54.323
has inverse b, let's say at the
origin, what kind of angular
00:26:54.323 --> 00:26:56.630
distortion does it cause.
00:26:56.630 --> 00:27:01.530
And you find that it goes
like v dot x, divided
00:27:01.530 --> 00:27:03.650
by the absolute value of x.
00:27:07.700 --> 00:27:10.120
This is for one dislocation.
00:27:10.120 --> 00:27:15.690
This is the theta that you
would get for that dislocation
00:27:15.690 --> 00:27:17.580
at location x.
00:27:23.400 --> 00:27:27.270
Essentially, you
can see that if I
00:27:27.270 --> 00:27:34.320
were to replace the u that
I have here with the u that
00:27:34.320 --> 00:27:39.060
was caused by dislocation,
you would get something
00:27:39.060 --> 00:27:40.880
like this formula.
00:27:40.880 --> 00:27:46.050
Because remember the u that
was caused by dislocation
00:27:46.050 --> 00:27:50.410
was something like the
gradient of the log potential.
00:28:07.120 --> 00:28:10.070
It's kind of hard to
work, but maybe I'll
00:28:10.070 --> 00:28:12.070
make an attempt to write it.
00:28:15.540 --> 00:28:19.230
So let's take a
gradient of theta.
00:28:21.840 --> 00:28:25.360
Gradient of theta, if
I use that formula,
00:28:25.360 --> 00:28:33.640
you would say, OK, I
have minus 1/2 z hat dot
00:28:33.640 --> 00:28:38.520
kern of something.
00:28:38.520 --> 00:28:44.410
And if I take a gradient of the
kern, the answer should be 0.
00:28:44.410 --> 00:28:49.020
But that's as long as this
u is a well defined object.
00:28:49.020 --> 00:28:54.100
And our task was to say
that this u, then you
00:28:54.100 --> 00:28:57.530
have these dislocations,
is not a well defined
00:28:57.530 --> 00:28:59.890
object, in the sense
that you take the kern,
00:28:59.890 --> 00:29:02.350
and the gradient
then you would get 0.
00:29:02.350 --> 00:29:05.260
So essentially, I will
transport the gradient
00:29:05.260 --> 00:29:08.210
all the way over here,
and the part of u
00:29:08.210 --> 00:29:12.880
that will survive that is
the one that is characterized
00:29:12.880 --> 00:29:14.220
by this dislocation feat.
00:29:18.060 --> 00:29:24.540
Now, you can see
that this object
00:29:24.540 --> 00:29:30.510
kind of looks like a
Laplacian of this distortion.
00:29:30.510 --> 00:29:34.090
It's two the derivatives
of this distortion field
00:29:34.090 --> 00:29:36.720
that had this logarithm in it.
00:29:36.720 --> 00:29:39.480
And when you take two
derivatives of a logarithm,
00:29:39.480 --> 00:29:41.530
you get the delta function.
00:29:41.530 --> 00:29:43.250
So if you do things
correctly, you
00:29:43.250 --> 00:29:48.940
will find that this answer
here becomes a sum over i
00:29:48.940 --> 00:29:57.870
v i delta function
of x minus xi.
00:29:57.870 --> 00:30:05.010
So basically, each dislocation
at location x i-- again,
00:30:05.010 --> 00:30:07.630
depending on its v being
in each direction--
00:30:07.630 --> 00:30:11.950
gives a contribution to
the gradient of theta.
00:30:11.950 --> 00:30:17.540
And if I were to take the
gradient of the expression
00:30:17.540 --> 00:30:23.350
that I have over here, the
gradient of this object
00:30:23.350 --> 00:30:25.450
is also-- this is like
the field that you
00:30:25.450 --> 00:30:27.790
have for the
logarithmic potential--
00:30:27.790 --> 00:30:29.490
will give you the data function.
00:30:29.490 --> 00:30:33.380
So that's where the
similarity comes.
00:30:33.380 --> 00:30:36.470
So the full answer
comes out to be--
00:30:36.470 --> 00:30:40.270
if you have a sum
over the dislocations,
00:30:40.270 --> 00:30:44.700
the sum over the distortion
fields that each one of them
00:30:44.700 --> 00:30:56.910
is causing-- and you will
have a form such as this.
00:30:56.910 --> 00:30:57.410
Yes?
00:30:57.410 --> 00:30:59.243
AUDIENCE: Should the
denominator be squared?
00:31:08.750 --> 00:31:12.990
PROFESSOR: Yes, that's right.
00:31:12.990 --> 00:31:16.430
The potential goes
logarithmically.
00:31:16.430 --> 00:31:19.090
The field, which is the
gradient of the potential,
00:31:19.090 --> 00:31:21.910
falls off as 1 over separation.
00:31:21.910 --> 00:31:23.930
So since I put the
separation out there,
00:31:23.930 --> 00:31:25.625
I have to put the
separation squared.
00:31:34.300 --> 00:31:39.520
So you can see that the
singular part, the part that
00:31:39.520 --> 00:31:44.200
arises from dislocations-- if
I have a soup of dislocations,
00:31:44.200 --> 00:31:45.770
I can figure out what theta is.
00:31:49.720 --> 00:31:55.220
Now what I did look
for-- actually, I
00:31:55.220 --> 00:32:02.680
was kind of hinting at that-- if
I take the gradient of theta--
00:32:02.680 --> 00:32:05.800
and I forgot to put
the factor of 1/2pi
00:32:05.800 --> 00:32:15.770
here-- does the 4 vertices
that had charged to pi--
00:32:15.770 --> 00:32:16.900
I had the potential.
00:32:16.900 --> 00:32:18.220
That was 1/r.
00:32:18.220 --> 00:32:21.440
So for dislocations
it becomes d/2pi.
00:32:21.440 --> 00:32:25.970
If I take the gradient,
then the gradient translates
00:32:25.970 --> 00:32:34.630
to sum over pi, the i
data function of x minus x
00:32:34.630 --> 00:32:39.750
i-- the expression that I
have written over there.
00:32:39.750 --> 00:32:44.810
And if I do the
fullier transform,
00:32:44.810 --> 00:32:47.445
you see what I did over
here was essentially
00:32:47.445 --> 00:32:49.980
to look at theta
in fullier space.
00:32:49.980 --> 00:32:53.559
So let's do something
similar here.
00:32:53.559 --> 00:32:55.350
So when I do the fullier
transform of this,
00:32:55.350 --> 00:33:04.110
I will get pi q-- the fullier
transform of this angular feat.
00:33:04.110 --> 00:33:09.195
And on the right hand
side, what I would get
00:33:09.195 --> 00:33:12.315
is essentially the
fullier transform
00:33:12.315 --> 00:33:15.580
of the field of dislocations.
00:33:15.580 --> 00:33:28.080
So I have defined my v of q to
be sum over i into the i q dot
00:33:28.080 --> 00:33:34.470
position of the i
dislocation, the vector that
00:33:34.470 --> 00:33:37.776
characterizes the dislocation.
00:33:37.776 --> 00:33:41.240
And it would make
sense to also tap
00:33:41.240 --> 00:33:43.480
into the normalization
that gives 1
00:33:43.480 --> 00:33:45.640
over the square root of area.
00:33:45.640 --> 00:33:48.520
If you don't do that,
then at some other point
00:33:48.520 --> 00:33:50.290
you have to worry about
the normalization.
00:33:58.090 --> 00:34:03.720
So if I just multiply
both sides by q--
00:34:03.720 --> 00:34:08.050
and I think I forgot the
minus sign throughout,
00:34:08.050 --> 00:34:17.260
which is not that important--
but theta tilde of q
00:34:17.260 --> 00:34:24.340
becomes i q dot b of
q, divided-- maybe
00:34:24.340 --> 00:34:31.022
I should've been calling this
b tilde-- divided by q squared.
00:34:37.288 --> 00:34:38.734
So this is important.
00:34:42.120 --> 00:34:49.580
Essentially, you take the
collection of dislocations
00:34:49.580 --> 00:34:52.929
in this picture
and you calculate
00:34:52.929 --> 00:34:56.790
what the fullier transform
is, call that the tilde of q.
00:34:56.790 --> 00:34:59.010
Essentially, you divide
by 1 factor of q,
00:34:59.010 --> 00:35:02.490
and you can get the
corresponding angle of feat.
00:35:07.300 --> 00:35:12.720
Now what I needed
to evaluate for here
00:35:12.720 --> 00:35:18.790
was the average of theta
tilde of q squared.
00:35:23.190 --> 00:35:28.050
And you can see that if I
write this explicitly, let's
00:35:28.050 --> 00:35:36.440
say, q i for be tilde i 4,
then the two of them I will
00:35:36.440 --> 00:35:40.152
get q beta b tilde of beta.
00:35:40.152 --> 00:35:42.562
And then I would
have a q to the side.
00:35:45.460 --> 00:35:50.840
And the average over
here becomes the average
00:35:50.840 --> 00:35:55.200
over all contributions
of these dislocations
00:35:55.200 --> 00:35:57.360
that I can put across my system.
00:36:00.990 --> 00:36:06.772
Now, explicitly I'm interested
in the limit where q goes to 0.
00:36:06.772 --> 00:36:10.710
So these things depend on q.
00:36:10.710 --> 00:36:14.520
What I'm interested
in is the limit
00:36:14.520 --> 00:36:20.270
as q goes to 0, especially
what happens to this average.
00:36:28.280 --> 00:36:36.040
It becomes-- multiplying two
of these things together--
00:36:36.040 --> 00:36:41.070
actually, in the limit
where q goes to 0,
00:36:41.070 --> 00:36:46.930
what I have is the sum
over all of the b's.
00:36:46.930 --> 00:36:49.030
So in the limit
where q goes to 0,
00:36:49.030 --> 00:36:53.740
this becomes an integral or sum.
00:36:53.740 --> 00:36:56.770
It doesn't matter which
one of them I write.
00:36:56.770 --> 00:36:59.820
q has gone to 0,
so I basically need
00:36:59.820 --> 00:37:06.620
to look at the average of
the alpha of x, the beta of x
00:37:06.620 --> 00:37:09.170
[INAUDIBLE], divided by area.
00:37:12.730 --> 00:37:17.390
So what is there
in the numerator?
00:37:21.160 --> 00:37:26.410
We can see that in the
numerator, sq goes to 0.
00:37:26.410 --> 00:37:32.780
What I'm looking at is the sum
of all of these dislocations
00:37:32.780 --> 00:37:35.940
that I have in the system.
00:37:35.940 --> 00:37:39.275
Now the average up the
sum is 0, because in all
00:37:39.275 --> 00:37:42.710
of our calculations, we've been
restricting the configurations
00:37:42.710 --> 00:37:44.550
that we moved from.
00:37:44.550 --> 00:37:46.465
Because if I go
beyond that strategy,
00:37:46.465 --> 00:37:48.115
it's going to cost too much.
00:37:50.920 --> 00:37:52.965
But what I'm looking
at is not the average
00:37:52.965 --> 00:37:57.460
of b, which is 0, but
the average of b squared,
00:37:57.460 --> 00:37:59.770
which is the variance.
00:37:59.770 --> 00:38:06.540
So essentially I have a system
that has a large area, a.
00:38:06.540 --> 00:38:09.540
It is on average neutral.
00:38:09.540 --> 00:38:12.530
And the question is, what is
the variance of the net charge.
00:38:15.040 --> 00:38:19.120
And my claim is that the
variance of the net charge
00:38:19.120 --> 00:38:25.870
is, by central limit theorem,
proportional to the area--
00:38:25.870 --> 00:38:28.710
actually, it is proportional
to the units that are
00:38:28.710 --> 00:38:30.950
independent from each other.
00:38:30.950 --> 00:38:39.450
So roughly I would expect that
in this high temperature phase,
00:38:39.450 --> 00:38:42.520
I have a correlation that is c.
00:38:42.520 --> 00:38:48.570
And within each portion of
side c, will be neutral.
00:38:48.570 --> 00:38:53.540
But when I go within things
that are more than c apart,
00:38:53.540 --> 00:38:57.120
there's no reason to
maintain the strategy.
00:38:57.120 --> 00:39:02.560
So overall I have something
like throwing coins,
00:39:02.560 --> 00:39:05.060
but at each one of
them, the average
00:39:05.060 --> 00:39:09.040
is 0, with probability
being up or down.
00:39:09.040 --> 00:39:12.550
But when I look at the
variance for the entire thing,
00:39:12.550 --> 00:39:16.301
the average will be
proportional to the area
00:39:16.301 --> 00:39:22.280
in units of these things that
are independent of each other.
00:39:22.280 --> 00:39:26.090
It was from the normalization
factor of 1 over area.
00:39:26.090 --> 00:39:29.120
And these, really, I should
write as a proportionality,
00:39:29.120 --> 00:39:32.030
because I don't
know precisely what
00:39:32.030 --> 00:39:35.550
the relationship between
these independent sides
00:39:35.550 --> 00:39:37.000
that correlation [INAUDIBLE].
00:39:37.000 --> 00:39:39.500
But they have to be
roughly proportional.
00:39:43.500 --> 00:39:45.318
So what do you compute?
00:39:45.318 --> 00:39:50.670
You compute that
the limit as q goes
00:39:50.670 --> 00:40:02.580
to 0, of the average of my
theta tilde of q squared
00:40:02.580 --> 00:40:04.350
is a structure such as this.
00:40:04.350 --> 00:40:07.650
I forgot to put one
more thing here.
00:40:07.650 --> 00:40:10.970
I don't expect to
be any correlations
00:40:10.970 --> 00:40:13.780
between the x
component and the y
00:40:13.780 --> 00:40:16.390
component of this
answer-- the variance,
00:40:16.390 --> 00:40:18.940
the covariance of the
dislocations in one
00:40:18.940 --> 00:40:20.990
direction and the
other direction--
00:40:20.990 --> 00:40:23.570
so I put the delta
function there.
00:40:23.570 --> 00:40:26.820
If I put this over here,
I would get the q squared
00:40:26.820 --> 00:40:28.700
divided by q to the 4th.
00:40:28.700 --> 00:40:34.160
So I will get a 1
over4 q squared.
00:40:34.160 --> 00:40:37.931
And I have the c squared and
then some unknown coefficient
00:40:37.931 --> 00:40:38.431
up here.
00:40:48.270 --> 00:40:55.740
So it's interesting,
because we started
00:40:55.740 --> 00:41:00.470
without thinking about
dislocations, just in terms
00:41:00.470 --> 00:41:01.983
of the distortion field.
00:41:01.983 --> 00:41:09.540
And we said that this object
is related to the angle.
00:41:09.540 --> 00:41:12.640
And indeed, we had
this distortion,
00:41:12.640 --> 00:41:16.270
that energy cost of distortions
is proportional to angle
00:41:16.270 --> 00:41:17.950
squared.
00:41:17.950 --> 00:41:21.940
And that angle, therefore,
is not the Goldstone mode
00:41:21.940 --> 00:41:24.050
because it doesn't
go like q squared.
00:41:27.050 --> 00:41:29.180
Now we go to this
other phase now,
00:41:29.180 --> 00:41:32.530
with dislocations
all over the place,
00:41:32.530 --> 00:41:38.210
and we calculate the expectation
value of theta squared.
00:41:38.210 --> 00:41:40.790
And it looks like it
came from a theory that
00:41:40.790 --> 00:41:42.016
was like Goldstone modes.
00:41:45.490 --> 00:41:50.750
So you would say that once
I am in this phase, where
00:41:50.750 --> 00:41:55.970
the dislocations are unbound,
there is an effective energy
00:41:55.970 --> 00:42:03.880
cost for these
changes in angle that
00:42:03.880 --> 00:42:11.307
is proportional to the
radiant of the angle squared.
00:42:15.710 --> 00:42:20.680
So that means fullier space,
this would go to k a over 2,
00:42:20.680 --> 00:42:28.222
integral into q 2 pi squared,
q squared theta tilde of q
00:42:28.222 --> 00:42:28.721
squared.
00:42:31.500 --> 00:42:36.170
So that if you had this
theory, you would definitely
00:42:36.170 --> 00:42:41.130
say that the expectation value
of theta tilde of q squared
00:42:41.130 --> 00:42:45.245
is 1 over k a q squared.
00:42:49.160 --> 00:42:53.950
The variance is k
a q squared invers.
00:42:53.950 --> 00:43:00.290
You compare those two things
and you find that once
00:43:00.290 --> 00:43:03.960
the dislocations have
unbound, and there
00:43:03.960 --> 00:43:07.710
is a correlation lend
that essentially tells you
00:43:07.710 --> 00:43:11.280
how far the dislocations
are talking to each other
00:43:11.280 --> 00:43:15.060
and maintaining neutrality,
that there is exactly
00:43:15.060 --> 00:43:19.290
an effective stiffness,
like a Goldstone note,
00:43:19.290 --> 00:43:23.410
for angular distortions, that
is proportional to c squared.
00:43:29.760 --> 00:43:33.750
And hence, if I were to look
at the orientation of all
00:43:33.750 --> 00:43:45.530
their correlations,
I would essentially
00:43:45.530 --> 00:43:51.150
have something like expectation
value of theta q squared,
00:43:51.150 --> 00:43:52.590
which is 1 over q squared.
00:43:52.590 --> 00:43:55.385
If I fully transform
that, I get the log.
00:43:55.385 --> 00:43:57.890
And so I will get
something that falls off
00:43:57.890 --> 00:44:02.340
in the distance to some
other exponent, if I recall
00:44:02.340 --> 00:44:02.840
[INAUDIBLE].
00:44:07.720 --> 00:44:11.940
If I have a true
liquid-- in a liquid,
00:44:11.940 --> 00:44:14.700
again, maybe in a
neighborhood of seven
00:44:14.700 --> 00:44:17.240
or eight particles,
neighbors, et cetera,
00:44:17.240 --> 00:44:20.830
they talk to each other and the
orientations are correlated.
00:44:20.830 --> 00:44:22.830
But then I go from
one part of the liquid
00:44:22.830 --> 00:44:24.810
to another part of
the liquid, there
00:44:24.810 --> 00:44:27.630
is no correlation
between bond angles.
00:44:27.630 --> 00:44:32.070
I expect these things
to decay exponentially.
00:44:32.070 --> 00:44:39.750
So what we've established
is that neither the phonon
00:44:39.750 --> 00:44:45.160
nor the dislocations
are sufficient to give
00:44:45.160 --> 00:44:51.250
the exponential decay that
you expect for the bond.
00:44:51.250 --> 00:44:57.690
So this object has
quasi long range order,
00:44:57.690 --> 00:45:00.756
versus what I expect to
happen in the liquid, which
00:45:00.756 --> 00:45:03.468
is exponential of
minus x over psi.
00:45:08.900 --> 00:45:15.650
So the unbinding of
dislocations gives rise
00:45:15.650 --> 00:45:20.120
to the new phase of matter
that has this quasi long range
00:45:20.120 --> 00:45:22.810
order in the orientations.
00:45:22.810 --> 00:45:25.060
It has no positional order.
00:45:25.060 --> 00:45:28.260
It's a kind of a liquid crystal
that is called a hexatic.
00:45:43.790 --> 00:45:45.060
Yes?
00:45:45.060 --> 00:45:47.992
AUDIENCE: So your correlation
where you got 1 over k q
00:45:47.992 --> 00:45:51.980
squared, doesn't that assume
that you're allowing the angle
00:45:51.980 --> 00:45:55.480
to vary in minus [INAUDIBLE]
when you do your averaging?
00:45:55.480 --> 00:45:57.300
What about the restriction--
00:45:57.300 --> 00:46:04.030
PROFESSOR: OK, so what is the
variance of the angle here?
00:46:04.030 --> 00:46:07.630
There's a variance of the angle
that is controlled by this 1
00:46:07.630 --> 00:46:09.430
over k a.
00:46:09.430 --> 00:46:13.830
So if I go back and calculate
these in real space,
00:46:13.830 --> 00:46:19.013
I will find that if I look at
theta at location x minus theta
00:46:19.013 --> 00:46:22.570
at location 0, the
answer is going to go
00:46:22.570 --> 00:46:28.210
like 1 over k logarithm x.
00:46:28.210 --> 00:46:30.690
So what it says
is that if things
00:46:30.690 --> 00:46:35.270
are close enough to each other--
and this is in units of 1/a--
00:46:35.270 --> 00:46:38.170
up to some factor, let's
say log 5, et cetera.
00:46:38.170 --> 00:46:40.480
So I don't go all
the way to infinity.
00:46:40.480 --> 00:46:46.840
The fluctuations in angle are
inversely set by a parameter
00:46:46.840 --> 00:46:52.440
that we see as I approach
right after the transition is
00:46:52.440 --> 00:46:54.310
very large.
00:46:54.310 --> 00:46:57.120
So in the same sense
that previously
00:46:57.120 --> 00:47:00.210
for the positional
correlations I
00:47:00.210 --> 00:47:04.400
had the temperature being small
and inverse temperature being
00:47:04.400 --> 00:47:08.880
large, limiting the size of
the translational fluctuations,
00:47:08.880 --> 00:47:13.930
here the same thing happens for
the bond angle fluctuations.
00:47:13.930 --> 00:47:19.080
Close to the transitions,
they are actually small.
00:47:19.080 --> 00:47:21.190
So the question
that you asked, you
00:47:21.190 --> 00:47:24.820
could have certainly
also asked over here.
00:47:24.820 --> 00:47:29.500
That is, when I'm thinking
about the distortion field,
00:47:29.500 --> 00:47:33.125
the distortion field is
certainly going to be limited.
00:47:33.125 --> 00:47:38.120
If it becomes as big as this,
then it doesn't make sense.
00:47:38.120 --> 00:47:41.280
So given that,
what sense or what
00:47:41.280 --> 00:47:44.830
justification do I have
in making these Gaussian
00:47:44.830 --> 00:47:46.210
integrals?
00:47:46.210 --> 00:47:48.205
And the answer is
that while it is true
00:47:48.205 --> 00:47:53.140
that it is fluctuating, as
I go to low temperature,
00:47:53.140 --> 00:47:56.270
the degree of fluctuations
is very small.
00:47:56.270 --> 00:48:01.580
So effectively
what I have is that
00:48:01.580 --> 00:48:05.190
I have to integrate over
some finite interval
00:48:05.190 --> 00:48:09.510
a function that kind
of looks like this.
00:48:09.510 --> 00:48:14.120
And the fact that I replaced
that with an integration
00:48:14.120 --> 00:48:17.340
from minus infinity to infinity
rather than from minus a
00:48:17.340 --> 00:48:19.130
to a just doesn't matter.
00:48:30.460 --> 00:48:36.410
So we know that ultimately
we should get this,
00:48:36.410 --> 00:48:42.730
but so far we've only got
this, so what should we do?
00:48:42.730 --> 00:48:45.810
Well, we say OK, we
encountered this difficulty
00:48:45.810 --> 00:48:52.550
before in something that looked
like an angle in the xy model--
00:48:52.550 --> 00:48:55.740
that low temperature
had power-law decay,
00:48:55.740 --> 00:48:57.860
whereas we knew that
at high temperatures
00:48:57.860 --> 00:49:00.430
they would have to
have exponential decay.
00:49:00.430 --> 00:49:03.620
And what we said was that we
need these topological defects
00:49:03.620 --> 00:49:06.060
in angle.
00:49:06.060 --> 00:49:18.950
So what you need-- topological
defects-- or in our case,
00:49:18.950 --> 00:49:20.505
theta is a bond angle.
00:49:27.640 --> 00:49:32.101
And these topological defects
in the bond angle have a name.
00:49:32.101 --> 00:49:33.350
They're called disconnections.
00:49:43.180 --> 00:49:47.950
And very roughly they correspond
to something like this.
00:49:47.950 --> 00:49:53.180
Suppose this is the center
one of these discriminations,
00:49:53.180 --> 00:49:57.490
and then maybe
next to this, here
00:49:57.490 --> 00:50:03.820
I have locally at the distance
r-- if I look at a point,
00:50:03.820 --> 00:50:07.840
I would see that the bonds
that connect it to its neighbor
00:50:07.840 --> 00:50:11.875
have an orientation such as the
one that I have indicated over
00:50:11.875 --> 00:50:13.590
here.
00:50:13.590 --> 00:50:21.210
Now what I want to do is, as I
go around and make a circuit,
00:50:21.210 --> 00:50:26.320
that this angle theta
that I have here to be 0,
00:50:26.320 --> 00:50:31.700
rotates and comes
back up to 60 degrees.
00:50:31.700 --> 00:50:36.140
So essentially what I
do is I take this line
00:50:36.140 --> 00:50:43.190
and I gradually shift it
around so that by the time
00:50:43.190 --> 00:50:47.517
I come back, I have
rotated by 60 degrees.
00:50:47.517 --> 00:50:49.100
It's kind of hard
for me to draw that,
00:50:49.100 --> 00:50:52.980
but you can imagine
what I have to do.
00:50:52.980 --> 00:50:57.200
So what I need to do
is to have the integral
00:50:57.200 --> 00:51:02.860
over a circuit that encloses
this discrimination such
00:51:02.860 --> 00:51:07.805
that when I do a d s dotted
with the gradient of the bond
00:51:07.805 --> 00:51:13.000
orientational angle,
I come back to pi over
00:51:13.000 --> 00:51:16.440
6 times some integer.
00:51:16.440 --> 00:51:21.590
And again, I expect the
[INAUDIBLE] dislocations that
00:51:21.590 --> 00:51:28.600
correspond to minus plus 1.
00:51:28.600 --> 00:51:32.320
Then the cost of
these is obtained
00:51:32.320 --> 00:51:36.220
by taking this distortion
fee, gradient of theta,
00:51:36.220 --> 00:51:40.910
whose magnitude at a distance r
from the center of this object
00:51:40.910 --> 00:51:52.580
is going to be 1 over 2
pi r times pi over 6 times
00:51:52.580 --> 00:51:55.710
whatever this integer n is.
00:51:55.710 --> 00:52:00.570
And then if I substitute
this 1 over r behavior
00:52:00.570 --> 00:52:04.330
in this expression, which
is the effective energy
00:52:04.330 --> 00:52:08.230
of this entity, I will
get the logarithmic cost
00:52:08.230 --> 00:52:11.550
for making a single
disclination.
00:52:11.550 --> 00:52:13.220
Which means that
at low temperature,
00:52:13.220 --> 00:52:15.940
I have to create
disclination pairs.
00:52:15.940 --> 00:52:18.430
And then there will be
an effective interaction
00:52:18.430 --> 00:52:21.230
between disclination
pairs, that is
00:52:21.230 --> 00:52:24.510
[INAUDIBLE] in exactly the
same way that we calculated
00:52:24.510 --> 00:52:26.150
for the x y model.
00:52:26.150 --> 00:52:33.040
So up to just this minor change
that the charge of a refect
00:52:33.040 --> 00:52:36.850
is reduced by a factor
of six, this theory
00:52:36.850 --> 00:52:41.680
is identical the theory of
the unbinding of the x y
00:52:41.680 --> 00:52:43.020
model [INAUDIBLE] defects.
00:52:43.020 --> 00:52:43.950
Yes?
00:52:43.950 --> 00:52:46.079
AUDIENCE: Why is it pi
over 6 and not 2 pi over 6?
00:52:46.079 --> 00:52:47.120
PROFESSOR: You are right.
00:52:47.120 --> 00:52:49.646
It should be 2 pi over 6.
00:52:49.646 --> 00:52:50.602
Thank you.
00:52:55.870 --> 00:52:57.520
Yes?
00:52:57.520 --> 00:52:59.918
AUDIENCE: So when
you were saying that
00:52:59.918 --> 00:53:05.960
the-- so the distance of this
hexatic phase would require
00:53:05.960 --> 00:53:12.880
the dislocations to occur
to [INAUDIBLE] [INAUDIBLE]
00:53:12.880 --> 00:53:15.420
orientational defects.
00:53:15.420 --> 00:53:18.200
Is there an analogous
case where--
00:53:18.200 --> 00:53:23.589
I guess you can't have
dislocations in the orientation
00:53:23.589 --> 00:53:24.630
without the dislocation--
00:53:24.630 --> 00:53:28.480
PROFESSOR: So if you try
to make these objects
00:53:28.480 --> 00:53:32.050
in the original case, in
the origin of lattice,
00:53:32.050 --> 00:53:34.540
you will find that their
cost grows actually
00:53:34.540 --> 00:53:39.640
like l squared log l, as
opposed to dislocations,
00:53:39.640 --> 00:53:43.450
whose cost only grows as log l.
00:53:43.450 --> 00:53:48.830
So these entities are
extremely unlikely to occur
00:53:48.830 --> 00:53:50.970
in the original system.
00:53:50.970 --> 00:53:54.590
If you sort of go back
and ask what they actually
00:53:54.590 --> 00:53:59.640
correspond to, if you have a
picture that you have generated
00:53:59.640 --> 00:54:01.290
on the computer,
they're actually
00:54:01.290 --> 00:54:04.530
reasonably easy to identify.
00:54:04.530 --> 00:54:07.200
Because the centers
of these disclinations
00:54:07.200 --> 00:54:09.250
correspond to
having points, that
00:54:09.250 --> 00:54:14.780
have, rather than 6
neighbors, 5 or 7 neighbors.
00:54:14.780 --> 00:54:18.800
So you generate the picture,
and you find mostly you have
00:54:18.800 --> 00:54:23.690
neighborhoods with 6 neighbors,
and then there's a site where
00:54:23.690 --> 00:54:26.147
there's 5 neighbours,
and another site that's 7
00:54:26.147 --> 00:54:27.550
neighbors.
00:54:27.550 --> 00:54:30.635
5 and 7 come more
or less in pairs,
00:54:30.635 --> 00:54:34.916
and you can identify these
disclination pairs reasonably
00:54:34.916 --> 00:54:35.416
easy.
00:54:39.250 --> 00:54:43.730
So at the end of the day,
the picture that we have
00:54:43.730 --> 00:54:46.400
is something like this.
00:54:46.400 --> 00:54:51.670
We are starting with
the triangular lattice
00:54:51.670 --> 00:54:55.890
that I drew at the
beginning, and you're
00:54:55.890 --> 00:54:58.130
increasing temperature.
00:54:58.130 --> 00:55:00.970
We're asking what happens.
00:55:00.970 --> 00:55:03.940
So this is 0 temperature.
00:55:03.940 --> 00:55:07.160
Close to 0 temperature,
what we have
00:55:07.160 --> 00:55:14.930
is an entity that has
translational quasi
00:55:14.930 --> 00:55:16.366
long range order.
00:55:16.366 --> 00:55:24.140
So this quantity goes like 1
over x to this power a to g.
00:55:27.010 --> 00:55:37.166
Whereas the orientations
go to a constant.
00:55:42.100 --> 00:55:48.340
Now, this a to g is
there because there's
00:55:48.340 --> 00:55:50.590
a shear modulus.
00:55:50.590 --> 00:55:55.430
And so throughout this phase,
I have a shear modulus.
00:55:55.430 --> 00:55:58.070
The parameter that
I'm calling u,
00:55:58.070 --> 00:56:00.730
I had scaled inversely
with temperature.
00:56:00.730 --> 00:56:04.590
So I have this
shear modulus u that
00:56:04.590 --> 00:56:08.440
diverges once we scale
by temperature as 1
00:56:08.440 --> 00:56:09.910
over temperature.
00:56:09.910 --> 00:56:16.160
But then as I come down, the
reduction is more than one
00:56:16.160 --> 00:56:19.870
over temperature
because I will have
00:56:19.870 --> 00:56:23.350
this effect of dislocations
appearing in pairs,
00:56:23.350 --> 00:56:26.290
and the system becomes softer.
00:56:26.290 --> 00:56:28.010
And eventually you
will find that there's
00:56:28.010 --> 00:56:32.030
a transition temperature at
which the shear modulus drops
00:56:32.030 --> 00:56:34.770
down to 0.
00:56:34.770 --> 00:56:39.180
And we said that
near this transition,
00:56:39.180 --> 00:56:42.790
there is this behavior
that mu approaches mu
00:56:42.790 --> 00:56:48.500
c, whatever it is,
with something-- let's
00:56:48.500 --> 00:56:51.240
call this t 1.
00:56:51.240 --> 00:56:56.590
T 1 minus t to this
exponent mu bar
00:56:56.590 --> 00:56:58.831
which was planned to be 6963.
00:57:05.800 --> 00:57:12.630
Now, once we are beyond
this temperature t 1, then
00:57:12.630 --> 00:57:22.940
our positional correlations
decay exponentially
00:57:22.940 --> 00:57:24.890
at some correlation, like c.
00:57:27.900 --> 00:57:32.480
And this c is
something that diverges
00:57:32.480 --> 00:57:34.090
on approaching this transition.
00:57:34.090 --> 00:57:40.450
So basically I have a c that
goes up here to infinity.
00:57:40.450 --> 00:57:46.230
And the fact that if
we calculate the c,
00:57:46.230 --> 00:57:50.190
it diverges according to
this strange formula that
00:57:50.190 --> 00:57:54.705
was 1 over t minus t
1 to this exponent mu
00:57:54.705 --> 00:57:58.530
bar Very strange
type of divergence.
00:58:03.850 --> 00:58:08.860
But then, associated with
the presence of this c
00:58:08.860 --> 00:58:12.160
is the fact that when you
look at the orientational
00:58:12.160 --> 00:58:22.380
correlations, they don't
decay as an exponential
00:58:22.380 --> 00:58:25.750
but as a power-law 8 of c.
00:58:28.870 --> 00:58:37.490
And this 8 of c is related to
this k a, and falls off as 1
00:58:37.490 --> 00:58:41.180
over c squared.
00:58:41.180 --> 00:58:45.870
So here it diverges as you
approach this transition.
00:58:45.870 --> 00:58:49.890
Now, as we go further
and further on,
00:58:49.890 --> 00:58:53.518
the disclinations will
appear-- disclinations
00:58:53.518 --> 00:58:56.990
with [INAUDIBLE]
resolve of the angles
00:58:56.990 --> 00:58:59.140
to be parallel to each other.
00:58:59.140 --> 00:59:01.150
And there's another
transition that
00:59:01.150 --> 00:59:08.820
is [INAUDIBLE], at which this is
going to suddenly go down to 0.
00:59:08.820 --> 00:59:13.096
And close to here,
we have that a to c
00:59:13.096 --> 00:59:18.820
reaches the critical
value of 1/4 v
00:59:18.820 --> 00:59:24.210
to square root
of-- let's call it
00:59:24.210 --> 00:59:32.320
t 2-- v the square
root singularity.
00:59:35.887 --> 00:59:42.560
And then finally we have
the ordinary liquid phase,
00:59:42.560 --> 00:59:53.790
where additionally I will find
that psi 6 of x psi star 6 of 0
00:59:53.790 --> 00:59:57.260
decays exponentially.
00:59:57.260 --> 01:00:00.520
Let's call it psi 6.
01:00:00.520 --> 01:00:04.100
And this psi 6 is
something that will diverge
01:00:04.100 --> 01:00:12.638
of this transition as an
exponential of minus 1
01:00:12.638 --> 01:00:15.584
over square root of t minus t2.
01:00:19.530 --> 01:00:27.020
So this is the current
scenario of how
01:00:27.020 --> 01:00:35.010
melting could occur for a system
of particles in two dimensions.
01:00:35.010 --> 01:00:37.376
If it is a continuous
phase transition,
01:00:37.376 --> 01:00:40.770
it has to go through
these two transitions
01:00:40.770 --> 01:00:44.850
with the intermediate
exotic phase.
01:00:44.850 --> 01:00:48.840
Of course, it is also
possible-- and typically people
01:00:48.840 --> 01:00:53.530
were seeing numerically
when they were doing hearts,
01:00:53.530 --> 01:00:55.580
spheres, et cetera,
that there is
01:00:55.580 --> 01:01:02.360
a direct transition from here
to here, which is discontinuous,
01:01:02.360 --> 01:01:03.960
like you have in
three dimensions.
01:01:03.960 --> 01:01:07.260
So that's an area of a
discontinuous transition
01:01:07.260 --> 01:01:09.270
that is not ruled out.
01:01:09.270 --> 01:01:11.305
But if you have
continuous transitions,
01:01:11.305 --> 01:01:15.524
it has to have this intermediate
phase in [INAUDIBLE].
01:01:25.444 --> 01:01:26.436
Any questions?
01:01:31.888 --> 01:01:32.388
Yes?
01:01:32.388 --> 01:01:36.356
AUDIENCE: [INAUDIBLE]
so the red one is mu.
01:01:36.356 --> 01:01:40.910
The yellow one is theta psi, and
the purple one is [INAUDIBLE].
01:01:40.910 --> 01:01:44.600
PROFESSOR: The correlation,
then, that I would put here.
01:01:44.600 --> 01:01:46.196
So they are three
different entities.
01:02:27.470 --> 01:02:30.490
So throughout the
course, we have
01:02:30.490 --> 01:02:33.550
been thinking about
systems that are
01:02:33.550 --> 01:02:36.800
described by some kind of
an equilibrium probability
01:02:36.800 --> 01:02:39.050
distribution.
01:02:39.050 --> 01:02:43.260
So what we did not
discuss is how the system
01:02:43.260 --> 01:02:46.190
comes to that equilibrium.
01:02:46.190 --> 01:02:52.750
So we're going to now very
briefly talk about dynamics,
01:02:52.750 --> 01:02:56.010
and the specific
type of dynamics
01:02:56.010 --> 01:02:58.670
that is common to
condensed matter
01:02:58.670 --> 01:03:01.785
systems at finite
temperature, which I
01:03:01.785 --> 01:03:03.790
will call precipative dynamics.
01:03:06.510 --> 01:03:16.084
And the prototype of this
is a Brownian particle
01:03:16.084 --> 01:03:22.300
that I will briefly
review for you.
01:03:22.300 --> 01:03:31.230
So what you have is that
you have a particle that
01:03:31.230 --> 01:03:36.510
is within some
kind of a solvent,
01:03:36.510 --> 01:03:40.060
and this particle
is moving around.
01:03:40.060 --> 01:03:43.580
So you would say,
let's for simplicity
01:03:43.580 --> 01:03:47.730
actually focus on
the one direction, x.
01:03:47.730 --> 01:03:50.830
And you would say that
the mass of the particle
01:03:50.830 --> 01:03:56.450
times its acceleration
is equal to the forces
01:03:56.450 --> 01:03:58.515
that it's experiencing.
01:04:02.280 --> 01:04:10.880
The forces-- well, if you
are moving in a fluid,
01:04:10.880 --> 01:04:14.780
you are going to be
subject to some kind
01:04:14.780 --> 01:04:21.490
of a dissipative force
which is typically
01:04:21.490 --> 01:04:24.800
portional to your velocity.
01:04:24.800 --> 01:04:27.670
If you, for example,
solve for the hydrodynamic
01:04:27.670 --> 01:04:31.470
of a sphere in a
fluid, you find that mu
01:04:31.470 --> 01:04:33.945
is related to
viscosity inversely
01:04:33.945 --> 01:04:36.760
to the size of the
particle, et cetera.
01:04:36.760 --> 01:04:38.410
But that behavior is generic.
01:04:38.410 --> 01:04:42.730
You're not going be
thinking about that.
01:04:42.730 --> 01:04:46.210
Now suppose that
additionally I put
01:04:46.210 --> 01:04:50.470
some kind of an optical
trap, or something that tries
01:04:50.470 --> 01:04:54.060
to localize this potential.
01:04:54.060 --> 01:04:58.680
So then there would be
an additional force v
01:04:58.680 --> 01:05:04.230
2, the derivative of the
potential with respect to x.
01:05:07.086 --> 01:05:10.950
And then we are talking
about Brownian particles.
01:05:10.950 --> 01:05:13.730
Brownian particles are
constantly jiggling.
01:05:13.730 --> 01:05:18.804
So there is also a random force
that is a function of time.
01:05:26.932 --> 01:05:34.370
Now we are going to be
interested in the dynamics that
01:05:34.370 --> 01:05:38.780
is very much controlled
by the dissipation term.
01:05:38.780 --> 01:05:44.770
And acceleration we can forget.
01:05:44.770 --> 01:05:50.930
And if we are in that limit,
we can write the equation
01:05:50.930 --> 01:06:00.005
as mu-- I can sort of rearrange
it slightly as-- actually,
01:06:00.005 --> 01:06:02.910
let me change location to this.
01:06:06.230 --> 01:06:13.210
So that the eventual
velocity x dot
01:06:13.210 --> 01:06:19.026
is going to be proportional
to the external force.
01:06:21.840 --> 01:06:26.950
mu the coefficient
that is the mobility.
01:06:26.950 --> 01:06:34.580
So mu essentially relates
the force to the velocity.
01:06:34.580 --> 01:06:38.100
Of course, this is
the average force.
01:06:38.100 --> 01:06:43.300
And there is a fluctuating
part, so essentially, I
01:06:43.300 --> 01:06:47.535
call mu times this to be
the a times function of t.
01:06:51.720 --> 01:06:57.360
Now, if I didn't have
this external force,
01:06:57.360 --> 01:07:02.130
the fluctuations of the
particles would be diffusive.
01:07:02.130 --> 01:07:04.700
And you can convince
yourself that you
01:07:04.700 --> 01:07:13.090
can get the diffusive result
provided that you relate
01:07:13.090 --> 01:07:18.230
the correlations of this
force that fluctuating
01:07:18.230 --> 01:07:23.440
and have 0 average, the
diffusion constant d
01:07:23.440 --> 01:07:28.372
of the particle in the medium
through delta of t minus t.
01:07:32.660 --> 01:07:38.170
So if their track was not there,
you solve of this equation
01:07:38.170 --> 01:07:42.200
without the track and find that
the prohibitive distribution
01:07:42.200 --> 01:07:50.590
for x grows as a Gaussian whose
width grows with time, as d t.
01:07:50.590 --> 01:07:55.090
d therefore must be
the diffusion constant.
01:07:55.090 --> 01:07:59.700
Now, in the presence
of the potential,
01:07:59.700 --> 01:08:02.930
this particle will
start to fluctuate.
01:08:02.930 --> 01:08:05.600
Eventually if you
wait long enough,
01:08:05.600 --> 01:08:07.600
there is a probability
that it will be here,
01:08:07.600 --> 01:08:10.100
a probability to
be somewhere else.
01:08:10.100 --> 01:08:13.870
So at long enough times,
there's a probability p
01:08:13.870 --> 01:08:17.340
of x to find the particle.
01:08:17.340 --> 01:08:23.120
And you expect
that t of x will be
01:08:23.120 --> 01:08:28.430
proportional to exponential
of minus v of x divided
01:08:28.430 --> 01:08:30.149
by whatever the temperature is.
01:08:33.109 --> 01:08:38.439
And you can show that in
order to have this occur,
01:08:38.439 --> 01:08:44.863
you need to relate mu and d
through the so called Einstein
01:08:44.863 --> 01:08:45.362
relation.
01:08:48.609 --> 01:08:52.540
So this is a brief review
of Brownian particles.
01:08:52.540 --> 01:08:54.420
Yes?
01:08:54.420 --> 01:09:01.960
AUDIENCE: The average and
time correlation of eta
01:09:01.960 --> 01:09:06.044
can be found by saying
the potential is 0, right?
01:09:06.044 --> 01:09:08.160
PROFESSOR: Mm hmm.
01:09:08.160 --> 01:09:09.729
AUDIENCE: Those
will still be true
01:09:09.729 --> 01:09:12.880
even if the potential
is not 0, right?
01:09:12.880 --> 01:09:14.779
PROFESSOR: Yes.
01:09:14.779 --> 01:09:22.120
So I just wanted to have an
idea of where this d comes from.
01:09:22.120 --> 01:09:27.200
But more specifically, this
is the important thing.
01:09:27.200 --> 01:09:32.430
That if at very
long times you want
01:09:32.430 --> 01:09:35.760
to have a probability
distribution coming
01:09:35.760 --> 01:09:41.660
from this equation, that
has the Boltzmann form
01:09:41.660 --> 01:09:46.210
with k t, the coefficients
of mu and the noise,
01:09:46.210 --> 01:09:49.830
you have to relate through the
so called Einstein relation.
01:09:49.830 --> 01:09:53.109
And once you do
that, this result
01:09:53.109 --> 01:09:57.990
is true no matter how
complicated this v of x is.
01:10:00.920 --> 01:10:03.960
So in general, for a
complicated v of x,
01:10:03.960 --> 01:10:08.020
you won't be able to solve
this equation analytically.
01:10:08.020 --> 01:10:10.390
You can only do it numerically.
01:10:10.390 --> 01:10:14.650
Yet you are guaranteed that
this equation with this noise
01:10:14.650 --> 01:10:18.080
correlator will have
asymptotically a probability
01:10:18.080 --> 01:10:19.423
distribution of [INAUDIBLE].
01:10:42.780 --> 01:10:45.580
The problem that we
have been looking at all
01:10:45.580 --> 01:10:50.285
along is something different.
01:10:50.285 --> 01:10:53.960
Let's say you have, let's
say, a piece of magnet
01:10:53.960 --> 01:10:59.400
or some other system
that we characterize,
01:10:59.400 --> 01:11:04.020
let's say, by something m of x.
01:11:04.020 --> 01:11:07.640
Again, you can do it for vector,
but for simplicity, let's do it
01:11:07.640 --> 01:11:10.110
for the scalar case.
01:11:10.110 --> 01:11:16.420
So we know, or we have stated,
that subject to the symmetries
01:11:16.420 --> 01:11:20.250
of the system, I
know the probability.
01:11:20.250 --> 01:11:27.890
For some configuration of this
field is governed by a form,
01:11:27.890 --> 01:11:30.990
let's say, that has
Landau Ginzburg character.
01:11:52.070 --> 01:11:54.500
So that has been
our starting point.
01:11:54.500 --> 01:11:58.350
We have said that I have a
prohibitive distribution that
01:11:58.350 --> 01:12:01.120
is of this form.
01:12:01.120 --> 01:12:05.025
So that statement is kind
of like this statement.
01:12:08.080 --> 01:12:12.050
But the way that I
came to that statement
01:12:12.050 --> 01:12:15.640
was to say that there
was a degree of freedom
01:12:15.640 --> 01:12:20.430
x, the position of the particle,
that was fluctuating subject
01:12:20.430 --> 01:12:24.790
to forces and external
variables from the particles
01:12:24.790 --> 01:12:30.645
of the fluid, that was given
by this so called Langevin
01:12:30.645 --> 01:12:31.145
equation.
01:12:37.770 --> 01:12:41.350
So I had a time
dependent prescription
01:12:41.350 --> 01:12:46.170
that eventually went to the
Boltzmann way that I wanted.
01:12:46.170 --> 01:12:51.910
Here I have started with
the final Boltzmann weight.
01:12:51.910 --> 01:12:57.620
And the question is, can
I think about a dynamics
01:12:57.620 --> 01:13:05.130
for a field that will
eventually give this state.
01:13:05.130 --> 01:13:11.680
So there are lots and
lots of different dynamics
01:13:11.680 --> 01:13:14.300
that I can impose.
01:13:14.300 --> 01:13:17.060
But I want to look
at the dynamics that
01:13:17.060 --> 01:13:21.370
is closest to the Brownian
particle that I wrote,
01:13:21.370 --> 01:13:25.240
and that's where this
word dissipative comes.
01:13:25.240 --> 01:13:30.390
So among the universe of
all possible dynamics,
01:13:30.390 --> 01:13:37.840
I'm going to look at one
that has a linear time
01:13:37.840 --> 01:13:40.378
derivative for the field n.
01:13:44.940 --> 01:13:48.850
So this is the
analog of the x dot.
01:13:48.850 --> 01:13:54.340
And so I write that it is
equal to some coefficient,
01:13:54.340 --> 01:13:58.230
that with their minds,
the ease with which
01:13:58.230 --> 01:14:02.020
that particle-- well the
field of that location x
01:14:02.020 --> 01:14:07.040
changes as a function of the
forces that is exerted on it.
01:14:07.040 --> 01:14:10.790
I assume that mu is the
same across my system.
01:14:10.790 --> 01:14:13.910
So here I'm already assuming
there's no x dependence.
01:14:13.910 --> 01:14:15.900
This system is uniform.
01:14:15.900 --> 01:14:19.130
And then there was a d v by d x.
01:14:19.130 --> 01:14:22.460
So v was ultimately the
thing that was appearing
01:14:22.460 --> 01:14:24.200
in the Boltzmann weight.
01:14:24.200 --> 01:14:27.590
So clearly the analog
of the v that I have
01:14:27.590 --> 01:14:30.190
is this Landau Ginzburg.
01:14:30.190 --> 01:14:37.070
So I will do a function of the
derivative of this quantity
01:14:37.070 --> 01:14:44.425
that I will call beta h,
with respect to m of x.
01:14:48.520 --> 01:14:53.640
Again, over there, I
had one variable, x.
01:14:53.640 --> 01:14:56.220
You can imagine that I could
have had a system where
01:14:56.220 --> 01:15:02.120
two particles, x 1 and x 2, also
have an interaction among them.
01:15:02.120 --> 01:15:05.780
Then the equation that I
would have had over there
01:15:05.780 --> 01:15:09.740
would be the force that
is acting on particle 1,
01:15:09.740 --> 01:15:11.910
by taking the total
potential-- which
01:15:11.910 --> 01:15:17.350
is the external potential
plus the potential that
01:15:17.350 --> 01:15:19.440
comes from the inter
particle interaction.
01:15:19.440 --> 01:15:21.120
So I would have to
take a derivative
01:15:21.120 --> 01:15:26.630
of the net potential energy
v, divided with respect
01:15:26.630 --> 01:15:29.430
to either x 1 or x 2
to calculate the force
01:15:29.430 --> 01:15:31.470
on the first one
or the second one.
01:15:31.470 --> 01:15:34.856
So here for a
particular configuration
01:15:34.856 --> 01:15:38.120
m of the field
across the system,
01:15:38.120 --> 01:15:42.690
if I'm interested in the
dynamics of this position x,
01:15:42.690 --> 01:15:49.930
I have to take this total
internal potential energy,
01:15:49.930 --> 01:15:51.790
and take the
derivative with respect
01:15:51.790 --> 01:15:54.830
to the variable that is
sitting on that side.
01:15:54.830 --> 01:15:57.510
So that's why this
is a functional
01:15:57.510 --> 01:16:00.130
derivative of this end.
01:16:00.130 --> 01:16:06.950
And then I will have
to put a noise, eta.
01:16:06.950 --> 01:16:10.010
Well, again, if I had
multiple particles,
01:16:10.010 --> 01:16:14.520
I would subject each one of
them to an independent noise.
01:16:14.520 --> 01:16:20.710
So at each location, I
have an independent noise.
01:16:20.710 --> 01:16:24.490
So the noise is a
function of time,
01:16:24.490 --> 01:16:26.910
but it is also wearing
across my system.
01:16:30.140 --> 01:16:37.300
So if I take that form and do
the functional derivative--
01:16:37.300 --> 01:16:40.680
so if I take the derivative
with respect to m of x,
01:16:40.680 --> 01:16:43.055
I have to take the
derivative of these objects.
01:16:43.055 --> 01:16:49.550
So I will have minus derivative
of t m squared is t m.
01:16:49.550 --> 01:16:56.250
The u m to the 4th is
4 u m q, and so forth.
01:16:56.250 --> 01:17:00.500
Once I have gotten
rid of these terms,
01:17:00.500 --> 01:17:04.150
then I would have terms
that depend on the gradient.
01:17:04.150 --> 01:17:08.450
So I would have minus the
derivative of this object
01:17:08.450 --> 01:17:10.670
with respect to the gradient.
01:17:10.670 --> 01:17:14.300
So here I would get
k gradient of m.
01:17:14.300 --> 01:17:17.500
And then the next term would
be Laplacian derivative,
01:17:17.500 --> 01:17:18.660
with respect to Laplacian.
01:17:18.660 --> 01:17:23.480
So I would put l Laplacian
of m, and so forth
01:17:23.480 --> 01:17:28.822
with the methodologies of
taking function derivatives.
01:17:28.822 --> 01:17:31.690
And then I have the noise.
01:17:35.530 --> 01:17:41.110
So this leads to
an equation which
01:17:41.110 --> 01:17:48.589
is called a time
dependent, Landau Ginzberg.
01:17:55.491 --> 01:18:01.390
Because we started with
the Landau Ginzberg weight,
01:18:01.390 --> 01:18:04.200
and this equation,
as we see shortly,
01:18:04.200 --> 01:18:10.510
subject to similar restrictions
as we had before, will give us,
01:18:10.510 --> 01:18:12.503
eventually, this
probability distribution.
01:18:16.130 --> 01:18:19.410
This is a difficult
equation in the same sense
01:18:19.410 --> 01:18:21.605
that the original
Landau Ginzberg
01:18:21.605 --> 01:18:24.820
is difficult to look at
correlations, et cetera.
01:18:24.820 --> 01:18:28.100
This is a nonlinear equation,
causes various difficulties,
01:18:28.100 --> 01:18:30.655
and we need approaches
to be able to deal
01:18:30.655 --> 01:18:34.190
with the difficult,
non linealities.
01:18:34.190 --> 01:18:38.790
So what we did for
the Landau Ginzburg
01:18:38.790 --> 01:18:43.430
was to initially get insights
and simplify the system
01:18:43.430 --> 01:18:47.910
by focusing on the linearized
or Gaussian version.
01:18:47.910 --> 01:18:53.453
So let's look at the version
of this equation that
01:18:53.453 --> 01:18:54.439
is linearized.
01:18:58.390 --> 01:19:01.650
And when it is linearized, what
I have on the left hand side
01:19:01.650 --> 01:19:05.020
is d m by d t.
01:19:05.020 --> 01:19:09.930
What I have on the
right hand side is mu.
01:19:09.930 --> 01:19:12.060
I have t m.
01:19:12.060 --> 01:19:15.430
I got rid of the
non linear term,
01:19:15.430 --> 01:19:21.180
so the next term that I will
have will be k Laplacian of m,
01:19:21.180 --> 01:19:27.216
and then will be minus l 4th
derivative of m, and so forth.
01:19:27.216 --> 01:19:29.918
And then there will be
a noise [INAUDIBLE].
01:19:41.560 --> 01:19:46.040
One thing that I
can immediately do
01:19:46.040 --> 01:19:49.000
is to go to fullier transform.
01:19:49.000 --> 01:19:55.440
So m of x goes to m theta of q.
01:19:55.440 --> 01:20:00.530
And if I do that, but not
fullier transform in time,
01:20:00.530 --> 01:20:04.820
I will get that the time
derivative of m tilde of q
01:20:04.820 --> 01:20:11.020
is essentially what I have here.
01:20:11.020 --> 01:20:14.880
And I forgot the minus
that I have here.
01:20:14.880 --> 01:20:16.420
So this minus is important.
01:20:24.640 --> 01:20:30.970
And then this becomes negative,
this becomes positive.
01:20:30.970 --> 01:20:34.980
So that when I go fullier
transform, what I will get
01:20:34.980 --> 01:20:43.780
is minus t plus k q squared plus
l q to the 4th, and so forth.
01:20:43.780 --> 01:20:50.400
And tilde of q with
this mu out front.
01:20:50.400 --> 01:20:55.266
And then the fullier
transform of what my noise is.
01:21:00.150 --> 01:21:06.840
First thing to note is that
even in the absence of noise,
01:21:06.840 --> 01:21:11.180
there is a set of
relaxation times.
01:21:11.180 --> 01:21:14.360
That is, for eta it was to 0.
01:21:14.360 --> 01:21:18.660
Or in general, I would
have n tilde of q and p.
01:21:18.660 --> 01:21:24.220
I can solve this
equation kind of simply.
01:21:24.220 --> 01:21:27.910
It is the m by d t is
some constant times
01:21:27.910 --> 01:21:31.735
n-- let's call it
gama of q-- which
01:21:31.735 --> 01:21:33.880
has dimensions of 1 over time.
01:21:33.880 --> 01:21:38.160
So I can call that
1 over tau of q.
01:21:38.160 --> 01:21:40.020
If I didn't have
noise, if I started
01:21:40.020 --> 01:21:45.160
with some original
value at time 0,
01:21:45.160 --> 01:21:50.360
it is going to
decay exponentially
01:21:50.360 --> 01:21:54.270
with this characteristic time.
01:21:54.270 --> 01:21:56.770
And once I have
noise, it is actually
01:21:56.770 --> 01:21:59.940
easy to convince yourself
that the answer is
01:21:59.940 --> 01:22:07.590
0 2 t d t prime e to the minus
this gamma of q or inversify.
01:22:10.800 --> 01:22:17.799
Tau of q times eta
of q i t prime.
01:22:21.210 --> 01:22:30.970
So you see that you have a
hierarchy of relaxation times,
01:22:30.970 --> 01:22:40.900
ta of q, which are 1 over u t
plus k q squared, and so forth,
01:22:40.900 --> 01:22:46.800
which scale in two limits.
01:22:46.800 --> 01:22:52.890
Either the wavelength lambda,
which is the inverse of q
01:22:52.890 --> 01:22:57.630
is much larger than the
correlation length--
01:22:57.630 --> 01:23:00.260
and the correlation
length of this model
01:23:00.260 --> 01:23:04.980
you have seen to be the
square root of t over k,
01:23:04.980 --> 01:23:08.600
square root of k over
t-- or the other limit,
01:23:08.600 --> 01:23:13.470
where lambda is
much less than c.
01:23:13.470 --> 01:23:15.640
In this limit,
where we are looking
01:23:15.640 --> 01:23:20.640
at modes that are much shorter
than the correlation length,
01:23:20.640 --> 01:23:22.130
this term is dominant.
01:23:22.130 --> 01:23:25.996
This becomes 1 over
mu k q squared.
01:23:25.996 --> 01:23:32.340
In the other limit, it goes
to a constant 1 over u t.
01:23:32.340 --> 01:23:36.850
So this linear equation
has a whole bunch
01:23:36.850 --> 01:23:39.360
of modes that can
be characterized
01:23:39.360 --> 01:23:42.310
by their wavelength
or their wave number.
01:23:42.310 --> 01:23:45.680
You find that the
short wavelength modes
01:23:45.680 --> 01:23:50.060
have this characteristic, time,
that becomes longer and longer
01:23:50.060 --> 01:23:52.580
as the wavelength increases.
01:23:52.580 --> 01:23:57.280
So if you make the
wavelength twice as large,
01:23:57.280 --> 01:24:00.640
and you want to relax a
system that is linearly
01:24:00.640 --> 01:24:05.430
twice as large, this says that
it will take 4 times longer.
01:24:05.430 --> 01:24:09.840
Because the answer goes
like lambda squared.
01:24:09.840 --> 01:24:14.720
Whereas eventually you reach the
size of the correlation length.
01:24:14.720 --> 01:24:17.670
Once you are beyond the size
of the correlation length,
01:24:17.670 --> 01:24:18.680
it doesn't matter.
01:24:18.680 --> 01:24:21.030
It's the same time.
01:24:21.030 --> 01:24:24.280
But the interesting
thing, of course, to us
01:24:24.280 --> 01:24:26.750
is that there are
phase transitions that
01:24:26.750 --> 01:24:27.560
are continuous.
01:24:27.560 --> 01:24:29.330
And close to that
phase transition,
01:24:29.330 --> 01:24:32.250
the correlation length
goes to infinity.
01:24:32.250 --> 01:24:37.610
Which means that the relaxation
time also will go to infinity.
01:24:37.610 --> 01:24:39.520
So according to
this theory, there's
01:24:39.520 --> 01:24:44.080
a particular divergence
as 1 over t minus t c.
01:24:44.080 --> 01:24:49.560
But it will be modified, and
as I will discuss next time,
01:24:49.560 --> 01:24:53.770
this is only-- even within
to dissipative class--
01:24:53.770 --> 01:24:56.800
one type of dynamics
that you can have.
01:24:56.800 --> 01:24:59.310
And there are
additional dynamics,
01:24:59.310 --> 01:25:05.020
and this system
characterizes criticality
01:25:05.020 --> 01:25:08.630
as single universality
class in statics.
01:25:08.630 --> 01:25:13.160
There are many dynamic
universality classes that
01:25:13.160 --> 01:25:16.100
correspond to this same static.