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PROFESSOR: OK.
00:00:22.380 --> 00:00:25.250
Let's start.
00:00:25.250 --> 00:00:30.090
So if we have been thinking
about critical points.
00:00:35.050 --> 00:00:40.140
And these arise in
many phased diagrams
00:00:40.140 --> 00:00:47.440
such as that we have for the
liquid gas system where there's
00:00:47.440 --> 00:00:52.846
a coexistence line, let's say,
between the gas and the liquid
00:00:52.846 --> 00:01:00.480
that terminate, or we looked
in the case of a magnet
00:01:00.480 --> 00:01:07.930
where as a function of
[INAUDIBLE] temperature
00:01:07.930 --> 00:01:10.920
there was in some
sense coexistence
00:01:10.920 --> 00:01:15.200
between magnetizations
in different directions
00:01:15.200 --> 00:01:16.655
terminating at the
critical point.
00:01:20.290 --> 00:01:25.180
So why is it interesting
to take it a whole phase
00:01:25.180 --> 00:01:27.540
diagram that we have over here?
00:01:27.540 --> 00:01:33.132
For example, for this system, we
can also have solid, et cetera.
00:01:33.132 --> 00:01:35.090
AUDIENCE: So isn't this
[INAUDIBLE] [INAUDIBLE]
00:01:35.090 --> 00:01:35.480
PROFESSOR: [INAUDIBLE].
00:01:35.480 --> 00:01:35.740
Yes.
00:01:35.740 --> 00:01:36.440
Thank you.
00:01:39.540 --> 00:01:43.154
And focus on just the one point.
00:01:43.154 --> 00:01:44.570
In the vicinity
of this one point.
00:01:47.380 --> 00:01:52.920
And the reason for that was
this idea of universality.
00:01:59.090 --> 00:02:02.430
There many things
that are happening
00:02:02.430 --> 00:02:04.660
in the vicinity of
this point as far
00:02:04.660 --> 00:02:07.435
as singularitities,
correlations, et cetera,
00:02:07.435 --> 00:02:10.840
are concerned that are
independent of whatever
00:02:10.840 --> 00:02:14.530
the consequence
of the system are.
00:02:14.530 --> 00:02:20.460
And these singularities, we try
to capture through some scaling
00:02:20.460 --> 00:02:23.730
laws for the singularities.
00:02:23.730 --> 00:02:28.210
And I've been kind
of constructing
00:02:28.210 --> 00:02:31.090
a table of your singularities.
00:02:31.090 --> 00:02:34.340
Let's do it one more time here.
00:02:34.340 --> 00:02:39.180
So we could look at system
such as the liquid gas--
00:02:39.180 --> 00:02:44.079
so let's have here
system-- and then we
00:02:44.079 --> 00:02:46.230
could look at the liquid gas.
00:02:49.100 --> 00:02:54.514
And for that, we can look
at a variety of exponents.
00:02:54.514 --> 00:03:02.380
We have alpha, beta,
gamma, delta, mu, theta.
00:03:05.900 --> 00:03:11.390
And for the liquid gas,
I write you some numbers.
00:03:11.390 --> 00:03:14.955
The heat capacity
diverges with an exponent
00:03:14.955 --> 00:03:18.905
that is 0.11--
slightly more accurate
00:03:18.905 --> 00:03:21.360
than I had given you before.
00:03:21.360 --> 00:03:29.675
The case for beta is 0.33 gamma.
00:03:29.675 --> 00:03:30.175
OK.
00:03:30.175 --> 00:03:34.020
I will give you a little
bit more digits just
00:03:34.020 --> 00:03:36.856
to indicate the
accuracy of experiments.
00:03:36.856 --> 00:03:47.390
This is 1.238 minus plus 0.012.
00:03:47.390 --> 00:03:52.250
So these exponents are obtained
by looking at the fluid system
00:03:52.250 --> 00:03:56.920
with light scattering-- doing
this critical opalescence
00:03:56.920 --> 00:03:59.615
that we were talking
about in more detail
00:03:59.615 --> 00:04:05.430
and accurately, Delta is 4.8.
00:04:05.430 --> 00:04:14.726
The mu is, again from light
scattering 0.629 minus plus
00:04:14.726 --> 00:04:18.822
0.003.
00:04:18.822 --> 00:04:33.970
Theta is 0.032 0
minus plus 0.013.
00:04:33.970 --> 00:04:39.350
And essentially these three are
[INAUDIBLE] light scattered.
00:04:48.780 --> 00:04:49.570
Sorry.
00:04:49.570 --> 00:04:55.010
Another case that I mentioned
is that of the super fluid.
00:05:00.300 --> 00:05:04.270
And in this general
construction of the lambda
00:05:04.270 --> 00:05:07.436
gives [INAUDIBLE]
theories that we had,
00:05:07.436 --> 00:05:10.525
liquid gas would
be in question one.
00:05:10.525 --> 00:05:14.980
Superfluid would
be in question two.
00:05:14.980 --> 00:05:21.690
And I just want to mention
that actually the most
00:05:21.690 --> 00:05:25.746
experimentally accurate exponent
that has been determined
00:05:25.746 --> 00:05:30.660
is the heat capacity for dry
superfluid helium transition.
00:05:30.660 --> 00:05:34.720
I had said that it kind of looks
like a logarithmic divergence.
00:05:34.720 --> 00:05:36.880
You look at it very closely.
00:05:36.880 --> 00:05:41.850
And it is in fact a cusp, and
does not diverge all the way
00:05:41.850 --> 00:05:45.740
to infinity, so it corresponds
to a slightly negative value
00:05:45.740 --> 00:05:58.346
of alpha, which is the
0.0127 minus plus 0.0003.
00:05:58.346 --> 00:06:01.950
And the way that this
has been data-mined
00:06:01.950 --> 00:06:06.604
is they took superfluid
helium to the space shuttle,
00:06:06.604 --> 00:06:09.508
and this experiments
were done away
00:06:09.508 --> 00:06:12.160
from the gravity of
the earth in order
00:06:12.160 --> 00:06:15.600
to not to have to worry
about the density difference
00:06:15.600 --> 00:06:19.470
that we would have
across the system.
00:06:19.470 --> 00:06:23.133
Other exponents that you
have for this system--
00:06:23.133 --> 00:06:27.882
let me write down--
beta is around 0.35.
00:06:27.882 --> 00:06:32.896
Gamma is 1.32.
00:06:32.896 --> 00:06:37.520
Delta is 4.79.
00:06:37.520 --> 00:06:40.920
Mu is is 0.67.
00:06:40.920 --> 00:06:46.990
Theta is 0.04.
00:06:46.990 --> 00:06:52.470
And we don't need
this for system.
00:06:56.250 --> 00:07:03.580
Any questions we could
do players kind of
00:07:03.580 --> 00:07:12.010
add the exponents here
I've booked usability
00:07:12.010 --> 00:07:18.820
even if it's minus
1 is a research data
00:07:18.820 --> 00:07:32.650
0.7 down all those are long this
is more to say about new ideas
00:07:32.650 --> 00:07:49.650
and so on I think is that
these numbers aren't you
00:07:49.650 --> 00:08:02.930
think that is simplest
way for us is net
00:08:02.930 --> 00:08:10.480
my position and the question
is why these numbers are all
00:08:10.480 --> 00:08:14.160
of the same as all the
systems is therefore profound.
00:08:14.160 --> 00:08:16.200
These are dimensionless numbers.
00:08:16.200 --> 00:08:20.240
So in some sense, it is a
little bit of mathematics.
00:08:20.240 --> 00:08:22.775
It's not like you calculate
the charge of the electron
00:08:22.775 --> 00:08:25.080
and you get a number.
00:08:25.080 --> 00:08:29.240
These don't depend on
a specific material.
00:08:29.240 --> 00:08:31.200
Therefore, what is
important about them
00:08:31.200 --> 00:08:34.640
is that they must
somehow be capturing
00:08:34.640 --> 00:08:37.440
some aspect of the
collective behavior of all
00:08:37.440 --> 00:08:40.880
of these degrees of freedom,
in which the details of what
00:08:40.880 --> 00:08:44.090
the degrees of freedom
are is not that important.
00:08:44.090 --> 00:08:47.810
Maybe the type of synergy
rating is important.
00:08:47.810 --> 00:08:52.860
So unless we understand
and derive these numbers,
00:08:52.860 --> 00:08:55.950
there is something important
about the collective behavior
00:08:55.950 --> 00:09:00.440
of many degrees of freedom
that we have not understood.
00:09:00.440 --> 00:09:03.711
And it is somehow a
different question
00:09:03.711 --> 00:09:08.160
if you are thinking
about phase transitions.
00:09:08.160 --> 00:09:11.660
So let's say you're thinking
about superconductors.
00:09:11.660 --> 00:09:14.220
There's a lot of interest
in making high temperature
00:09:14.220 --> 00:09:18.600
superconductor pushing TC
further and further up.
00:09:18.600 --> 00:09:21.240
So that's certainly
a material problem.
00:09:21.240 --> 00:09:23.050
We are asking a
different problem.
00:09:23.050 --> 00:09:26.320
Why is it, whether you have a
high temperature superconductor
00:09:26.320 --> 00:09:30.590
or any other type of system,
the collective behavior
00:09:30.590 --> 00:09:34.940
is captured by the
same set of exponents.
00:09:34.940 --> 00:09:39.690
So in an attempt to
try to answer that,
00:09:39.690 --> 00:09:49.450
we did this Landau-Ginzburg and
try to calculate its singular
00:09:49.450 --> 00:09:53.720
behavior using this other
point of approximation.
00:09:53.720 --> 00:09:56.150
And the numbers
that we got, alpha
00:09:56.150 --> 00:10:00.680
was 0, meaning that
there was discontinuity.
00:10:00.680 --> 00:10:09.483
Beta was 1/2, gamma was 1,
delta was 3, my was 1/2,
00:10:09.483 --> 00:10:15.170
theta was 0, which don't
quite match with these numbers
00:10:15.170 --> 00:10:18.070
that we have up there.
00:10:18.070 --> 00:10:22.660
So question is,
what should you do?
00:10:22.660 --> 00:10:28.000
We've made an attempt and that
attempt was not successful.
00:10:28.000 --> 00:10:33.480
So we are going to completely
for a while forget about that
00:10:33.480 --> 00:10:36.440
and try to approach the problem
from a different perspective
00:10:36.440 --> 00:10:38.770
and see how far we
can go, whether we
00:10:38.770 --> 00:10:43.800
can gain any new insights.
00:10:43.800 --> 00:10:49.805
So that new approach
I put on there
00:10:49.805 --> 00:10:51.847
the name of the
scaling hypothesis.
00:10:59.640 --> 00:11:06.090
And the reason for that will
become apparent shortly.
00:11:06.090 --> 00:11:12.550
So what we have in common
in both of these examples
00:11:12.550 --> 00:11:17.570
is that there is
a line where there
00:11:17.570 --> 00:11:21.700
are discontinuities
in calculating
00:11:21.700 --> 00:11:24.630
some thermodynamic
function that terminates
00:11:24.630 --> 00:11:26.730
at a particular point.
00:11:26.730 --> 00:11:30.030
And in the case of
the magnetic system,
00:11:30.030 --> 00:11:35.980
we can look at the singularities
approaching that point either
00:11:35.980 --> 00:11:38.580
along the direction
that corresponds
00:11:38.580 --> 00:11:44.040
to change in temperature and
parametrize that through heat,
00:11:44.040 --> 00:11:48.280
or we can change
the magnetic field
00:11:48.280 --> 00:11:52.030
and approach the problem
from this other direction.
00:11:52.030 --> 00:11:55.220
And we saw that there
were analogs for doing
00:11:55.220 --> 00:11:59.195
so in the liquid
gas system also.
00:11:59.195 --> 00:12:03.830
And in particular, let's say
we calculated a magnetization,
00:12:03.830 --> 00:12:07.520
we found that there was one form
of singularity coming this way,
00:12:07.520 --> 00:12:10.430
one form of singularity
coming that way.
00:12:10.430 --> 00:12:14.005
We look at the picture
for the liquid gas system
00:12:14.005 --> 00:12:18.400
that I have up there, and
it's not necessarily clear
00:12:18.400 --> 00:12:24.310
which direction would
correspond to this nice symmetry
00:12:24.310 --> 00:12:26.780
breaking or
non-symmetry breaking
00:12:26.780 --> 00:12:29.740
that you have for
the magnetic system.
00:12:29.740 --> 00:12:33.610
So you may well ask, suppose
I approach the critical point
00:12:33.610 --> 00:12:35.440
along some other direction.
00:12:35.440 --> 00:12:38.950
Maybe I come in along
the path such as this.
00:12:38.950 --> 00:12:41.120
I still go to the
critical point.
00:12:41.120 --> 00:12:44.170
We can imagine that for
the liquid gas system.
00:12:44.170 --> 00:12:46.955
And what's the structure
of the singularities?
00:12:46.955 --> 00:12:51.221
I know that there are different
singularities in the t and h
00:12:51.221 --> 00:12:51.720
direction.
00:12:51.720 --> 00:12:53.802
What is it if I
come and approach
00:12:53.802 --> 00:12:55.670
the system along a
different direction,
00:12:55.670 --> 00:13:00.400
which we may well do
for a liquid gas system?
00:13:00.400 --> 00:13:03.270
Well, we could actually
answer that if we go back
00:13:03.270 --> 00:13:07.680
to our graph saddlepoint
approximation.
00:13:07.680 --> 00:13:10.050
In the saddlepoint
approximation,
00:13:10.050 --> 00:13:15.134
we said that ultimately,
the singularities in terms
00:13:15.134 --> 00:13:19.090
of these two
parameters t and h--
00:13:19.090 --> 00:13:20.400
so this is in the saddlepoint.
00:13:26.050 --> 00:13:33.185
Part obtained by minimizing
this function that
00:13:33.185 --> 00:13:37.405
was appearing in the
expansion in the exponent.
00:13:37.405 --> 00:13:39.470
There was a t over 2m squared.
00:13:39.470 --> 00:13:42.378
There was a mu n to the 4th.
00:13:42.378 --> 00:13:43.770
And there is an hm.
00:13:46.554 --> 00:13:52.325
So we had to minimize
this with respect to m.
00:13:52.325 --> 00:13:56.840
And clearly, what
that gives us is m.
00:13:56.840 --> 00:14:00.280
If I really solve
the equation, that
00:14:00.280 --> 00:14:03.220
corresponds to
this minimization,
00:14:03.220 --> 00:14:05.820
which is a function of t and h.
00:14:05.820 --> 00:14:10.970
And in particular, approaching
two directions that's
00:14:10.970 --> 00:14:16.000
indicated, if I'm along the
direction where h equals 0,
00:14:16.000 --> 00:14:18.869
I essentially balance
these two terms.
00:14:18.869 --> 00:14:20.660
Let's just write this
as a proportionality.
00:14:20.660 --> 00:14:23.515
I don't really care
about the numbers.
00:14:26.580 --> 00:14:30.110
Along the direction
where h equals 0,
00:14:30.110 --> 00:14:36.120
I have to balance m to
the 4th and tm squared.
00:14:36.120 --> 00:14:41.150
So m squared will scale like e.
00:14:41.150 --> 00:14:47.048
m will scale like
square root of t.
00:14:47.048 --> 00:14:49.495
And more precisely,
we calculated
00:14:49.495 --> 00:14:53.480
this formula for t
negative and h equals to 0.
00:14:56.220 --> 00:14:59.300
If I, on the other hand,
come along the direction
00:14:59.300 --> 00:15:08.050
that corresponds to t equals
to 0, along that direction
00:15:08.050 --> 00:15:09.510
I don't have a first term.
00:15:09.510 --> 00:15:14.440
I have to balance
um the 4th and hm.
00:15:14.440 --> 00:15:20.143
So we immediately see that
m will scale like h over u.
00:15:20.143 --> 00:15:25.008
In fact, more correctly h over
4u to the power of one third.
00:15:28.950 --> 00:15:31.030
You substitute this
in the free energy
00:15:31.030 --> 00:15:34.464
and you find that the singular
part of the free energy
00:15:34.464 --> 00:15:41.590
as a function of t and h in
this saddlepoint approximation
00:15:41.590 --> 00:15:45.960
has the [INAUDIBLE] to the
form of proportionality.
00:15:45.960 --> 00:15:49.740
If I substitute this in
the formula for t negative,
00:15:49.740 --> 00:15:52.905
I will get something
like minus t squared
00:15:52.905 --> 00:15:58.400
over 4 we have-- forget about
the number t squared over u.
00:15:58.400 --> 00:16:02.430
If I go along the t
equals to 0 direction,
00:16:02.430 --> 00:16:08.500
substitute that over there,
I will get n to the 4th.
00:16:08.500 --> 00:16:15.570
I will get h to the 4 thirds
divided by mu to the one third.
00:16:15.570 --> 00:16:18.170
Even the mu dependence
I'm not interested.
00:16:18.170 --> 00:16:22.850
I'm really interested in the
behavior close to t and h
00:16:22.850 --> 00:16:24.930
as a function of t and h.
00:16:24.930 --> 00:16:27.820
Mu is basically some
non-universal number
00:16:27.820 --> 00:16:29.960
that doesn't go to 0.
00:16:29.960 --> 00:16:33.980
I could in some sense
capture these two expressions
00:16:33.980 --> 00:16:39.050
by a form that is
t squared and then
00:16:39.050 --> 00:16:41.430
some function--
let's call it g sub
00:16:41.430 --> 00:16:47.055
f which is a function
of-- let's see
00:16:47.055 --> 00:16:52.338
how I define the delta
h over t to the delta.
00:16:58.510 --> 00:17:03.610
So my claim is that I toyed
with the behavior coming
00:17:03.610 --> 00:17:07.190
across these two different
special direction.
00:17:07.190 --> 00:17:13.420
In general, anywhere else
where t and h are both nonzero,
00:17:13.420 --> 00:17:17.980
the answer for m will be some
solution of a cubic equation,
00:17:17.980 --> 00:17:21.099
but we can arrange it to
only be a function of h
00:17:21.099 --> 00:17:25.410
over [INAUDIBLE]
and have this form.
00:17:25.410 --> 00:17:30.880
Now I could maybe
rather than explicitly
00:17:30.880 --> 00:17:33.210
show you how that
arises, which is not
00:17:33.210 --> 00:17:36.580
difficult-- you can do that--
since there's something
00:17:36.580 --> 00:17:39.460
that we need to
do later on, I'll
00:17:39.460 --> 00:17:43.030
show it in the following manner.
00:17:43.030 --> 00:17:48.980
I have not specified what
this function g sub f is.
00:17:48.980 --> 00:17:53.710
But I know its behavior
along h equals to 0 here.
00:17:53.710 --> 00:17:57.238
And so if I put h equals to 0,
the argument of the function
00:17:57.238 --> 00:17:59.580
goes to 0.
00:17:59.580 --> 00:18:02.810
So if I say that the
argument of the function
00:18:02.810 --> 00:18:06.950
is a constant-- the constant
let's say is minus 1
00:18:06.950 --> 00:18:10.270
over u on one side,
0 on the other side,
00:18:10.270 --> 00:18:12.170
then everything's fine.
00:18:12.170 --> 00:18:20.060
So I have is that the limit
as its argument goes to 0
00:18:20.060 --> 00:18:21.330
should be some constant.
00:18:26.890 --> 00:18:29.020
Well, what about
the other direction?
00:18:29.020 --> 00:18:32.520
How can I reproduce
from a form such as this
00:18:32.520 --> 00:18:37.510
the behavior when t equals to 0?
00:18:37.510 --> 00:18:40.400
Because I see that
when t equals to 0,
00:18:40.400 --> 00:18:43.560
the answer of course
cannot depend on t itself,
00:18:43.560 --> 00:18:47.840
but as a power law
as a function of h.
00:18:47.840 --> 00:18:51.790
Is it consistent with this form?
00:18:51.790 --> 00:18:54.010
Well, as t goes
to 0 in this form,
00:18:54.010 --> 00:18:58.580
the numerator here goes to 0,
the argument of the function
00:18:58.580 --> 00:19:01.660
goes to infinity, I
need to know something
00:19:01.660 --> 00:19:05.480
about the behavior of
the function of infinity.
00:19:05.480 --> 00:19:09.800
So let's say that the limiting
behavior as the argument
00:19:09.800 --> 00:19:13.400
of the function goes
to infinity of gf
00:19:13.400 --> 00:19:21.438
of x is proportional to the
argument to some other peak.
00:19:21.438 --> 00:19:24.432
And I don't know
where that power is.
00:19:24.432 --> 00:19:28.230
Then if I look at this
function, the whole function
00:19:28.230 --> 00:19:32.775
in this limit where t
goes to 0 will behave.
00:19:32.775 --> 00:19:37.530
There's a t squared out
front the goes to 0,
00:19:37.530 --> 00:19:39.890
the argument of the
function goes to infinity.
00:19:39.890 --> 00:19:44.860
So the function will go like
the argument to some power.
00:19:44.860 --> 00:19:49.313
So I go like h t to the
delta to some other peak.
00:19:56.720 --> 00:20:00.190
So what do I know?
00:20:00.190 --> 00:20:02.320
I know that the
answer should really
00:20:02.320 --> 00:20:06.360
be proportional to h
to the four thirds.
00:20:09.210 --> 00:20:15.496
So I immediately know that
my t should be four thirds.
00:20:20.480 --> 00:20:21.810
But what about this delta?
00:20:21.810 --> 00:20:24.660
I never told you what delta was.
00:20:24.660 --> 00:20:28.100
Now I can figure out what delta
is, because the answer should
00:20:28.100 --> 00:20:31.860
not depend on t.
t has gone to 0.
00:20:31.860 --> 00:20:34.460
And so what power
of t do I have?
00:20:34.460 --> 00:20:42.630
I have 2 minus
delta p should be 0.
00:20:42.630 --> 00:20:50.950
So my delta should be 2
over p, 2 over four thirds,
00:20:50.950 --> 00:20:52.365
so it should be three halves.
00:21:01.610 --> 00:21:08.920
Why is this exponent relevant to
the question that I had before?
00:21:08.920 --> 00:21:13.840
You can see that the function
that describes the free energy
00:21:13.840 --> 00:21:16.580
as a function of
these two coordinates.
00:21:16.580 --> 00:21:21.600
If I look at the combination
where h and t are non-zero,
00:21:21.600 --> 00:21:28.625
is very much dependent on this
h divided by t to the delta,
00:21:28.625 --> 00:21:31.650
and that delta is three halves.
00:21:31.650 --> 00:21:34.420
So, for example, if
I were to draw here
00:21:34.420 --> 00:21:43.630
curves where h goes like
3 to the three halves--
00:21:43.630 --> 00:21:48.770
it's some coefficient, I don't
know what that coefficient is--
00:21:48.770 --> 00:21:55.320
then essentially, everything
that is on the side
00:21:55.320 --> 00:22:01.160
that hogs the vertical axis
behaves like the h singularity.
00:22:01.160 --> 00:22:06.510
Everything that is over here
depends like a t singularity.
00:22:06.510 --> 00:22:12.230
So a path that, for example,
comes along a straight line,
00:22:12.230 --> 00:22:16.960
if I, let's say, call
the distance that I have
00:22:16.960 --> 00:22:21.500
to the critical point
s, then t is something
00:22:21.500 --> 00:22:27.220
like s cosine of theta. h is
something like s sine theta.
00:22:27.220 --> 00:22:29.630
You can see however
that the information
00:22:29.630 --> 00:22:36.840
h over t to the delta as
s goes to 0 will diverge,
00:22:36.840 --> 00:22:39.745
because I have other
three halves down here
00:22:39.745 --> 00:22:44.550
for s that will overcome the
linear cover I have over there.
00:22:44.550 --> 00:22:49.570
So for any linear path that
goes through the critical point,
00:22:49.570 --> 00:22:53.760
eventually for small s I will
see the type of singularity
00:22:53.760 --> 00:22:57.770
that is characteristic
of the magnetic field
00:22:57.770 --> 00:23:03.050
if the exponents are
according to this other point.
00:23:03.050 --> 00:23:05.230
We have this
assumption, of course.
00:23:05.230 --> 00:23:10.040
But if I therefore
knew the correct delta
00:23:10.040 --> 00:23:14.070
for all of those systems, I
would be also able to answer,
00:23:14.070 --> 00:23:17.530
let's say for the
liquid gas, whether if I
00:23:17.530 --> 00:23:20.260
take a linear path that goes
through the critical point
00:23:20.260 --> 00:23:22.920
I would see one set of
singularities or deltas
00:23:22.920 --> 00:23:25.980
that have singularities.
00:23:25.980 --> 00:23:34.000
So this delta which is
called a gap exponent,
00:23:34.000 --> 00:23:35.290
gives you the answer to that.
00:23:38.420 --> 00:23:43.530
But of course I don't
know the other exponents.
00:23:43.530 --> 00:23:46.995
There is no reason for me
to trust the gap exponent
00:23:46.995 --> 00:23:51.010
that I obtained in this fashion.
00:23:51.010 --> 00:24:02.740
So what I say is let's assume
that for any critical point,
00:24:02.740 --> 00:24:06.620
the singular part
of the free energy
00:24:06.620 --> 00:24:10.080
on approaching the
critical point which
00:24:10.080 --> 00:24:14.710
depends on this
pair of coordinates
00:24:14.710 --> 00:24:18.730
has a singular behavior
that is similar to what
00:24:18.730 --> 00:24:22.460
we had over here, except that
I don't know the exponent.
00:24:22.460 --> 00:24:26.620
So rather than
putting 2 t squared,
00:24:26.620 --> 00:24:30.070
I write t to the 2
minus alpha for reason
00:24:30.070 --> 00:24:32.736
that will become
apparent shortly,
00:24:32.736 --> 00:24:39.820
and some function of h t
to the delta and for some
00:24:39.820 --> 00:24:41.227
alpha and delta.
00:24:47.580 --> 00:24:49.700
So this is certainly
already an assumption.
00:24:52.680 --> 00:24:57.730
This mathematically
corresponds to having
00:24:57.730 --> 00:24:59.015
homogeneous functions.
00:25:06.190 --> 00:25:09.490
Because if I have a
function of x and y,
00:25:09.490 --> 00:25:14.930
I can certainly write lots of
functions such as x squared
00:25:14.930 --> 00:25:19.110
plus y squared plus a constant
plus x cubed y cubed that I
00:25:19.110 --> 00:25:22.800
cannot rearrange into this form.
00:25:22.800 --> 00:25:25.560
But there are certain
functions of x and y
00:25:25.560 --> 00:25:29.150
that I can rearrange
so that I can pull out
00:25:29.150 --> 00:25:32.370
some factor of let's
say x squared out front,
00:25:32.370 --> 00:25:34.480
and everything that
is then in a series
00:25:34.480 --> 00:25:37.870
is a function of let's
say y over x cubed.
00:25:37.870 --> 00:25:39.910
Something like that.
00:25:39.910 --> 00:25:42.440
So there's some
class of functions
00:25:42.440 --> 00:25:46.074
of two arguments that
have this homogeneity.
00:25:46.074 --> 00:25:50.400
So we are going to assume that
the singular behavior close
00:25:50.400 --> 00:25:55.732
to the critical point is
described by such a function.
00:25:55.732 --> 00:25:57.181
That's an assumption.
00:26:00.080 --> 00:26:05.190
But having made that assumption,
let's follow its consequence
00:26:05.190 --> 00:26:07.120
and let's see if we
learned something
00:26:07.120 --> 00:26:10.520
about that table of exponents.
00:26:10.520 --> 00:26:13.570
Now the first thing
to note is clearly
00:26:13.570 --> 00:26:16.455
I chose this alpha
over here so that when
00:26:16.455 --> 00:26:23.820
I take two derivatives
with respect to t,
00:26:23.820 --> 00:26:27.050
I would get something
like a heat capacity,
00:26:27.050 --> 00:26:32.550
for which I know what
the divergence is.
00:26:32.550 --> 00:26:36.580
That's a divergence
called alpha.
00:26:36.580 --> 00:26:37.980
But there's one
thing that I have
00:26:37.980 --> 00:26:42.100
to show you is that when
I take a derivative of one
00:26:42.100 --> 00:26:48.032
of these homogeneous
functions, with respect
00:26:48.032 --> 00:26:52.270
to one of its arguments,
I will generate
00:26:52.270 --> 00:26:53.850
another homogeneous function.
00:26:53.850 --> 00:26:57.080
If I take one derivative
with respect to t,
00:26:57.080 --> 00:27:03.770
that derivative can
either act on this,
00:27:03.770 --> 00:27:08.370
leaving the function
unchanged, or it
00:27:08.370 --> 00:27:11.600
can act on the argument
of the function
00:27:11.600 --> 00:27:16.740
and give me d to
the 2 minus alpha.
00:27:16.740 --> 00:27:21.950
I will have minus h t to
the power of delta plus 1.
00:27:21.950 --> 00:27:24.366
There will be a factor
of delta and then
00:27:24.366 --> 00:27:28.713
I will have the derivative
function ht to the delta.
00:27:32.530 --> 00:27:34.630
So I just took derivatives.
00:27:34.630 --> 00:27:37.687
I can certainly pull
out a factor of t
00:27:37.687 --> 00:27:40.750
to the 1 minus alpha.
00:27:40.750 --> 00:27:46.420
Then the first term is
just 2 minus alpha times
00:27:46.420 --> 00:27:47.365
the original function.
00:27:50.820 --> 00:27:59.240
The second term is minus delta
h divided by t to the delta.
00:27:59.240 --> 00:28:01.500
Because I pulled out
the 1 minus alpha,
00:28:01.500 --> 00:28:05.310
this t gets rid of
the factor of 1 there.
00:28:05.310 --> 00:28:06.400
And I have the derivative.
00:28:12.000 --> 00:28:13.990
So this is completely
different function.
00:28:13.990 --> 00:28:16.825
It's not the derivative
of the original function.
00:28:16.825 --> 00:28:20.015
But whatever it is it
is still only a function
00:28:20.015 --> 00:28:23.984
of the combination h
over t to the delta.
00:28:23.984 --> 00:28:26.650
So the derivative of
a homogeneous function
00:28:26.650 --> 00:28:28.973
is some other
homogeneous function.
00:28:28.973 --> 00:28:30.452
Let's call it g2.
00:28:30.452 --> 00:28:31.438
It doesn't matter.
00:28:31.438 --> 00:28:36.370
Let's call it g1
ht to the delta.
00:28:36.370 --> 00:28:39.560
And this will happen if I
take a second derivative.
00:28:39.560 --> 00:28:41.920
So I know that if I
take two derivatives,
00:28:41.920 --> 00:28:44.960
I will get t to the minus alpha.
00:28:44.960 --> 00:28:49.100
I will basically drop
two factors over there.
00:28:49.100 --> 00:28:56.730
And then some other
function, ht to the delta.
00:28:56.730 --> 00:29:00.150
Clearly again, if I say
that I'm looking at the line
00:29:00.150 --> 00:29:05.020
where h equals to
zero for a magnet,
00:29:05.020 --> 00:29:08.300
then the argument of
the function goes to 0.
00:29:08.300 --> 00:29:12.030
If I say that the function
of the argument goes to 0
00:29:12.030 --> 00:29:15.790
is a constant, like
we had over here,
00:29:15.790 --> 00:29:19.840
then I will have the singularity
t to the minus alpha.
00:29:19.840 --> 00:29:24.160
So I've clearly engineered
whatever the value of alpha
00:29:24.160 --> 00:29:28.410
is in this table,
I can put over here
00:29:28.410 --> 00:29:33.630
and I have the right singularity
for the heat capacity.
00:29:33.630 --> 00:29:36.660
Essentially I've put
it there by hand.
00:29:36.660 --> 00:29:40.810
Let me comment on one
other thing, which
00:29:40.810 --> 00:29:47.869
is when we are looking
at just the temperature,
00:29:47.869 --> 00:29:49.410
let's say we are
looking at something
00:29:49.410 --> 00:29:51.360
like a superfluid,
the only parameter
00:29:51.360 --> 00:29:58.060
that we have at our disposal
is temperature and tens of ITC.
00:29:58.060 --> 00:30:03.050
Let's say we plug
the heat capacity
00:30:03.050 --> 00:30:06.631
and then we see divergence of
the heat capacity on the two
00:30:06.631 --> 00:30:07.130
sides.
00:30:09.910 --> 00:30:15.430
Who said that I should have
the same exponent on this side
00:30:15.430 --> 00:30:16.865
and on this side?
00:30:19.380 --> 00:30:22.230
So we said that
generally, in principle,
00:30:22.230 --> 00:30:26.370
I could say I would do that.
00:30:26.370 --> 00:30:31.090
And in principle, there
is no problem with that.
00:30:31.090 --> 00:30:33.980
If there is function that
has one behavior here,
00:30:33.980 --> 00:30:38.820
another behavior there,
who says that two exponents
00:30:38.820 --> 00:30:41.710
have to be the same?
00:30:41.710 --> 00:30:44.550
But I have said something more.
00:30:44.550 --> 00:30:49.310
I have said that in all of
the cases that I'm looking at,
00:30:49.310 --> 00:30:54.440
I know that there
is some other axis.
00:30:57.820 --> 00:31:02.850
And for example, if I am
in the liquid gas system,
00:31:02.850 --> 00:31:06.760
I can start from down
here, go all the way around
00:31:06.760 --> 00:31:10.940
back here without
encountering a singularity.
00:31:10.940 --> 00:31:15.120
I can go from the liquid
all the way to gas
00:31:15.120 --> 00:31:16.650
without encountering
a singularity.
00:31:19.450 --> 00:31:24.630
So that says that the system
is different from a system
00:31:24.630 --> 00:31:29.120
that, let's say, has a
line of singularities.
00:31:29.120 --> 00:31:33.550
So if I now take the
functions that in principle
00:31:33.550 --> 00:31:37.014
have two different
singularities,
00:31:37.014 --> 00:31:43.740
t to the minus alpha minus t to
the minus alpha plus on the h
00:31:43.740 --> 00:31:48.950
equals to 0 axis and
try to elevate them
00:31:48.950 --> 00:31:54.930
into the entire space by putting
this homogeneous functions
00:31:54.930 --> 00:32:00.430
in front of them, there
is one and only one way
00:32:00.430 --> 00:32:05.405
in which the two functions
can match exactly on this t
00:32:05.405 --> 00:32:09.680
equals to 0 line, and that's if
the two exponents are the same
00:32:09.680 --> 00:32:12.518
and you are dealing
with the same function.
00:32:12.518 --> 00:32:15.830
So that we put in
a bit of physics.
00:32:15.830 --> 00:32:21.266
So in principle, mathematically
if you don't have the h axis
00:32:21.266 --> 00:32:25.839
and you look at the one line
and there's a singularity,
00:32:25.839 --> 00:32:27.630
there's no reason why
the two singularities
00:32:27.630 --> 00:32:29.700
should be the same.
00:32:29.700 --> 00:32:31.920
But we know that we are
looking at the class
00:32:31.920 --> 00:32:35.430
of physical systems where
there is the possibility
00:32:35.430 --> 00:32:39.200
to analytically go from
one side to the other side.
00:32:39.200 --> 00:32:42.270
And that immediately
imposes this constraint
00:32:42.270 --> 00:32:47.590
that alpha plus should be
alpha minus, and one alpha
00:32:47.590 --> 00:32:49.205
is in fact sufficient.
00:32:49.205 --> 00:32:54.850
And I gave you the correct
answer for why that is.
00:32:54.850 --> 00:32:57.930
If you want to see the
precise mathematical details
00:32:57.930 --> 00:33:00.130
step by step, then
that's in the notes.
00:33:04.330 --> 00:33:06.020
So fine.
00:33:06.020 --> 00:33:08.360
So far we haven't learned much.
00:33:08.360 --> 00:33:10.790
We've justified why
the two alphas should
00:33:10.790 --> 00:33:13.480
be the same above
and below, but we
00:33:13.480 --> 00:33:17.770
put the alpha, the one
alpha, then by hand.
00:33:17.770 --> 00:33:19.945
And then we have this
unknown delta also.
00:33:19.945 --> 00:33:23.046
But let's proceed.
00:33:23.046 --> 00:33:26.170
Let's see what other consequence
emerge, because now we
00:33:26.170 --> 00:33:27.970
have a function
of two variables.
00:33:27.970 --> 00:33:30.490
I took derivatives
in respect to t.
00:33:30.490 --> 00:33:33.546
I can take derivatives
with respect to m.
00:33:33.546 --> 00:33:39.910
And in particular,
the magnetization
00:33:39.910 --> 00:33:46.340
m as a function of
t and h is obtained
00:33:46.340 --> 00:33:52.800
from a derivative of the free
energy with respect to h.
00:33:52.800 --> 00:33:55.230
There's potential.
00:33:55.230 --> 00:33:59.140
It's the response to
adding a field could
00:33:59.140 --> 00:34:01.250
be some factor of
beta c or whatever.
00:34:01.250 --> 00:34:03.440
It's not important.
00:34:03.440 --> 00:34:06.460
The singular part
will come from this.
00:34:06.460 --> 00:34:10.690
And so taking a derivative
of this function
00:34:10.690 --> 00:34:14.510
I will get t this to
the 2 minus alpha.
00:34:14.510 --> 00:34:17.159
The derivative of a
can be respect to h,
00:34:17.159 --> 00:34:19.849
but h comes in the
combination h over t
00:34:19.849 --> 00:34:24.610
to the delta will bring down a
factor of minus delta up front.
00:34:24.610 --> 00:34:29.058
Then the derivative function--
let's call it gf1, for example.
00:34:36.210 --> 00:34:41.906
So now I can look at this
function in the limit
00:34:41.906 --> 00:34:49.400
where h goes to 0, climb
along the coexistence line,
00:34:49.400 --> 00:34:51.280
h2 goes to 0.
00:34:51.280 --> 00:34:55.550
The argument of the
function has gone to 0.
00:34:55.550 --> 00:34:58.150
Makes sense that the
function should be constant
00:34:58.150 --> 00:35:00.140
when its argument goes to 0.
00:35:00.140 --> 00:35:03.192
So the answer is going
to be proportional to t
00:35:03.192 --> 00:35:06.108
to the 2 minus
alpha minus delta.
00:35:09.940 --> 00:35:13.720
But that's how beta was defined.
00:35:13.720 --> 00:35:21.400
So if I know my beta
and alpha, then I
00:35:21.400 --> 00:35:25.988
can calculate my delta from
this exponent identity.
00:35:28.550 --> 00:35:31.760
Again, so far you
haven't done much.
00:35:31.760 --> 00:35:37.075
You have translated two unknown
exponents, this singular form,
00:35:37.075 --> 00:35:40.540
this gap exponent
that we don't know.
00:35:40.540 --> 00:35:44.630
I can also look
at the other limit
00:35:44.630 --> 00:35:48.280
where t goes to 0
that is calculating
00:35:48.280 --> 00:35:51.550
the magnetization along
the critical isotherm.
00:35:55.280 --> 00:36:01.140
So then the argument of the
function has gone to infinity.
00:36:01.140 --> 00:36:03.990
And whatever the answer
is should not depend on t,
00:36:03.990 --> 00:36:06.470
because I have said t goes to 0.
00:36:06.470 --> 00:36:09.970
So I apply the same trick
that I did over here.
00:36:09.970 --> 00:36:13.420
I say that when the
argument goes to infinity,
00:36:13.420 --> 00:36:19.330
the function goes like
some power of its argument.
00:36:24.191 --> 00:36:29.430
And clearly I have to choose
that power such that the t
00:36:29.430 --> 00:36:34.130
dependence, since t is going
to 0, I have to get rid of it.
00:36:34.130 --> 00:36:40.060
The only way that I can do
that is if p is 2 minus alpha
00:36:40.060 --> 00:36:45.720
minus delta divided by that.
00:36:52.710 --> 00:36:56.980
So having done that, the
whole thing will then
00:36:56.980 --> 00:36:59.866
be a function of h to the p.
00:37:02.610 --> 00:37:05.146
But the shape of
the magnetization
00:37:05.146 --> 00:37:08.520
along the critical
isotherm, which was also
00:37:08.520 --> 00:37:13.590
the shape of the isotherm
of the liquid gas system,
00:37:13.590 --> 00:37:16.200
we were characterizing
by an exponent
00:37:16.200 --> 00:37:18.242
that we were calling
1 over delta.
00:37:22.100 --> 00:37:25.930
So we have now a
formula that says
00:37:25.930 --> 00:37:31.540
my delta shouldn't in
fact be the inverse of p.
00:37:31.540 --> 00:37:39.170
It should be the delta 2
minus alpha minus delta.
00:37:39.170 --> 00:37:39.670
Yes?
00:37:39.670 --> 00:37:42.309
AUDIENCE: Why isn't
the exponent t minus 1
00:37:42.309 --> 00:37:43.975
after you've
differentiated [INAUDIBLE]?
00:37:47.090 --> 00:37:51.115
Because g originally was
defined as [INAUDIBLE].
00:37:51.115 --> 00:37:54.070
PROFESSOR: Let's call it pr.
00:37:58.010 --> 00:38:01.440
Because actually, you're right.
00:38:01.440 --> 00:38:05.300
If this is the same g and this
has particular singularity
00:38:05.300 --> 00:38:07.220
[INAUDIBLE].
00:38:07.220 --> 00:38:09.446
But at the end of the
day, it doesn't matter.
00:38:15.550 --> 00:38:18.770
So now I have gained something
that I didn't have before.
00:38:18.770 --> 00:38:21.060
That is, in
principle I hit alpha
00:38:21.060 --> 00:38:25.320
and beta, my two exponents,
I'm able to figure out
00:38:25.320 --> 00:38:27.190
what delta is.
00:38:27.190 --> 00:38:30.320
And actually I can also
figure out what gamma is ,
00:38:30.320 --> 00:38:34.714
because gamma describes
the divergence
00:38:34.714 --> 00:38:35.630
of the susceptibility.
00:38:43.560 --> 00:38:48.215
[INAUDIBLE] which is the
derivative of magnetization
00:38:48.215 --> 00:38:51.980
with respect to
field, I have to take
00:38:51.980 --> 00:38:55.680
another derivative
of this function.
00:38:55.680 --> 00:38:58.270
Taking another derivative
with respect to h
00:38:58.270 --> 00:39:00.774
will bring down another
factor of delta.
00:39:00.774 --> 00:39:04.390
So this becomes minus 2 delta.
00:39:04.390 --> 00:39:10.850
Some other double derivative
function h 2 to the delta.
00:39:10.850 --> 00:39:12.660
And susceptibilities,
we are typically
00:39:12.660 --> 00:39:18.110
interested in the limit
where the field goes to 0.
00:39:18.110 --> 00:39:23.420
And we define them to
diverge with exponent gamma.
00:39:23.420 --> 00:39:32.640
So we have identified gamma to
be 2 delta plus alpha minus 2.
00:39:40.990 --> 00:39:42.860
So we have learned something.
00:39:42.860 --> 00:39:45.126
Let's summarize it.
00:39:45.126 --> 00:39:55.400
So the consequences--
one is we established
00:39:55.400 --> 00:39:59.854
that same critical
exponents above and below.
00:40:12.400 --> 00:40:15.800
Now since various
quantities of interest
00:40:15.800 --> 00:40:18.730
are obtained by
taking derivatives
00:40:18.730 --> 00:40:21.216
of our homogeneous
function and they
00:40:21.216 --> 00:40:25.530
turn into homogeneous
functions, we
00:40:25.530 --> 00:40:40.500
conclude that all quantities
are homogeneous functions
00:40:40.500 --> 00:40:45.270
of the same combination
ht to the delta.
00:40:45.270 --> 00:40:46.687
Same delta governs it.
00:40:53.160 --> 00:40:58.020
And thirdly, once we make
this answer our assumption
00:40:58.020 --> 00:41:02.600
for the free energy, we can
calculate the other exponents
00:41:02.600 --> 00:41:04.170
on the table.
00:41:04.170 --> 00:41:22.920
So all, of almost all other
exponents related to 2,
00:41:22.920 --> 00:41:27.070
in this case alpha and delta.
00:41:27.070 --> 00:41:30.490
Which means that if you have a
number of different exponents
00:41:30.490 --> 00:41:34.440
that all depend
on 2, there should
00:41:34.440 --> 00:41:38.680
be some identities,
exponent identities.
00:41:47.670 --> 00:41:51.620
It's these numbers in the
table, we predict if all of this
00:41:51.620 --> 00:41:56.390
is varied have some
relationships with t.
00:41:56.390 --> 00:42:00.700
So let's show a couple
of these relationships.
00:42:00.700 --> 00:42:07.330
So let's look at the combination
alpha plus 2 beta plus gamma.
00:42:07.330 --> 00:42:10.930
Measurement of heat capacity,
magnetization, susceptibility.
00:42:10.930 --> 00:42:13.140
Three different things.
00:42:13.140 --> 00:42:16.550
So alpha is alpha 2.
00:42:16.550 --> 00:42:21.530
My beta up there is 2
minus alpha minus delta.
00:42:21.530 --> 00:42:28.140
My gamma is 2 delta
plus alpha minus 2.
00:42:28.140 --> 00:42:30.350
We got algebra.
00:42:30.350 --> 00:42:33.370
There's one alpha minus
2 alpha plus alpha.
00:42:33.370 --> 00:42:36.120
Alpha is cancelled.
00:42:36.120 --> 00:42:40.410
Minus 2 deltas plus 2
deltas then it does cancel.
00:42:40.410 --> 00:42:45.342
I have 2 times 2
minus 2, so that 2.
00:42:45.342 --> 00:42:51.250
So the prediction is that you
take some line on the table,
00:42:51.250 --> 00:42:55.000
add alpha, beta, 2
beta plus gamma, they
00:42:55.000 --> 00:42:56.390
should add up to one.
00:42:56.390 --> 00:42:59.820
So let's pick something.
00:42:59.820 --> 00:43:04.500
Let's pick a first--
actually, let's
00:43:04.500 --> 00:43:08.070
pick the last line that
has a negative alpha.
00:43:08.070 --> 00:43:11.270
So let's do n equals to 3.
00:43:11.270 --> 00:43:17.160
For n equals to 3 I have
alpha which is minus .12.
00:43:17.160 --> 00:43:25.800
I have twice beta, that is
.37, so that becomes 74.
00:43:25.800 --> 00:43:33.860
And then I have
gamma, which is 1.39.
00:43:33.860 --> 00:43:34.856
So this is 74.
00:43:37.880 --> 00:43:43.990
I have 9 plus 413
minus 2, which is 1.
00:43:43.990 --> 00:43:52.880
I have 3 plus 7, which is
10, minus 1, which is 9.
00:43:52.880 --> 00:43:56.625
But then I had a 1 that was
carried over, so I will have 0.
00:43:56.625 --> 00:44:00.160
So then I have 1, so 201.
00:44:00.160 --> 00:44:02.650
Not bad.
00:44:02.650 --> 00:44:07.705
Now this goes by the name
of the Rushbrooke identity.
00:44:16.010 --> 00:44:19.830
The Rushbrooke made
a simple manipulation
00:44:19.830 --> 00:44:24.960
based on thermodynamics and you
have a relationship with these.
00:44:24.960 --> 00:44:27.980
Let's do another one.
00:44:27.980 --> 00:44:33.450
Let's do delta and
subtract 1 from it.
00:44:33.450 --> 00:44:34.820
What is my delta?
00:44:34.820 --> 00:44:39.360
I have delta to the delta
2 plus alpha minus delta.
00:44:41.864 --> 00:44:45.550
This is small delta
versus big delta.
00:44:45.550 --> 00:44:47.358
And then I have minus 1.
00:44:50.350 --> 00:44:55.184
Taking that into the numerator
with the common denominator
00:44:55.184 --> 00:44:59.880
of 2 plus alpha minus
delta, this minus delta
00:44:59.880 --> 00:45:04.300
becomes plus delta, which
this becomes 2 delta minus
00:45:04.300 --> 00:45:07.730
alpha minus 2.
00:45:07.730 --> 00:45:15.362
2 delta
00:45:15.362 --> 00:45:17.860
AUDIENCE: Should that be a
minus alpha in the denominator?
00:45:17.860 --> 00:45:19.696
PROFESSOR: It better be.
00:45:19.696 --> 00:45:20.195
Yes.
00:45:25.430 --> 00:45:28.060
2 delta plus alpha minus 2.
00:45:28.060 --> 00:45:30.952
Then we can read off the gamma.
00:45:30.952 --> 00:45:33.097
So this is gamma over beta.
00:45:35.720 --> 00:45:39.740
And let's check this, let's
say for m equals to 2.
00:45:42.770 --> 00:45:46.140
No, let's check it for m
plus 21, for the following
00:45:46.140 --> 00:45:53.870
reason, that for n equals to
1, what we have for delta is
00:45:53.870 --> 00:46:01.500
4.8 minus 1, which would be 3.8.
00:46:01.500 --> 00:46:06.170
And on the other side,
we have gamma over beta.
00:46:06.170 --> 00:46:12.150
Gamma is 1.24, roughly,
divided by beta .33,
00:46:12.150 --> 00:46:14.870
which is roughly one third.
00:46:14.870 --> 00:46:17.520
So I multiply this by 3.
00:46:17.520 --> 00:46:27.420
And that becomes 3.72.
00:46:27.420 --> 00:46:34.438
This one is known after another
famous physicist, Ben Widom,
00:46:34.438 --> 00:46:35.935
as the Widom identity.
00:46:39.930 --> 00:46:41.690
So that's nice.
00:46:41.690 --> 00:46:48.620
We can start learning that
although we don't know anything
00:46:48.620 --> 00:46:52.910
about this table, these are
not independent numbers.
00:46:52.910 --> 00:46:55.060
There's relationship
between them.
00:46:55.060 --> 00:46:58.975
And they're named after
famous physicists.
00:46:58.975 --> 00:46:59.475
Yes?
00:46:59.475 --> 00:47:03.075
AUDIENCE: Can we briefly go
over again what extra assumption
00:47:03.075 --> 00:47:06.024
we had put in to get
these in and these out?
00:47:08.690 --> 00:47:11.642
Is it just that we have
this homogeneous function
00:47:11.642 --> 00:47:12.415
[INAUDIBLE]?
00:47:12.415 --> 00:47:13.980
PROFESSOR: That's right.
00:47:13.980 --> 00:47:18.230
So you assume that
the singularity
00:47:18.230 --> 00:47:22.600
in the vicinity of the
critical point as a function
00:47:22.600 --> 00:47:26.300
of deviations from
that critical point
00:47:26.300 --> 00:47:29.302
can be expressed as a
homogeneous function.
00:47:29.302 --> 00:47:33.900
The homogeneous function, you
can rearrange any way you like.
00:47:33.900 --> 00:47:37.610
One nice way to rearrange
it is in this fashion.
00:47:37.610 --> 00:47:41.940
It will depend, the homogeneous
function on two exponents.
00:47:41.940 --> 00:47:44.540
I chose to write
it as 2 minus alpha
00:47:44.540 --> 00:47:48.290
so that one of the exponents
would immediately be alpha.
00:47:48.290 --> 00:47:50.530
The other one I
couldn't immediately
00:47:50.530 --> 00:47:52.770
write in terms of beta or gamma.
00:47:52.770 --> 00:47:55.740
I had to do these
manipulations to find out
00:47:55.740 --> 00:47:59.290
what the relationship
[INAUDIBLE].
00:47:59.290 --> 00:48:03.700
But the physics of it is simple.
00:48:03.700 --> 00:48:09.000
That is, once you know the
singularity of a free energy,
00:48:09.000 --> 00:48:11.110
various other
quantities you obtain
00:48:11.110 --> 00:48:13.090
by taking derivatives
of the free energy.
00:48:13.090 --> 00:48:17.460
That's [INAUDIBLE]
And so then you
00:48:17.460 --> 00:48:19.862
would have the singular
behavior of [INAUDIBLE].
00:48:27.110 --> 00:48:34.530
So I started by saying
that all other exponents,
00:48:34.530 --> 00:48:39.640
but then I realized we have
nothing so far that tells us
00:48:39.640 --> 00:48:43.840
anything about mu and eta.
00:48:43.840 --> 00:48:48.490
Because mu and eta
relate to correlations.
00:48:48.490 --> 00:48:51.190
They are in
microscopic quantities.
00:48:51.190 --> 00:48:55.740
Alpha, beta, gamma depend
on macroscopic thermodynamic
00:48:55.740 --> 00:48:59.140
quantities, magnetization
susceptibility.
00:48:59.140 --> 00:49:03.440
So there's no way that I will
be able to get information,
00:49:03.440 --> 00:49:04.400
almost.
00:49:04.400 --> 00:49:07.290
No easy way or no direct
way to get information
00:49:07.290 --> 00:49:09.080
about mu and eta.
00:49:11.680 --> 00:49:19.630
So I will go to assumption 2.0.
00:49:19.630 --> 00:49:25.720
Go to the next version of the
homogeneity assumption, which
00:49:25.720 --> 00:49:29.580
is to emphasize
that we certainly
00:49:29.580 --> 00:49:32.930
know, again from physics
and the relationship
00:49:32.930 --> 00:49:35.330
between susceptibility
and correlations,
00:49:35.330 --> 00:49:37.810
that the reason
for the divergence
00:49:37.810 --> 00:49:42.200
of the susceptibility is that
the correlations become large.
00:49:42.200 --> 00:49:45.740
So we'll emphasize that.
00:49:45.740 --> 00:49:51.170
So let's write our ansatz
not about the free energy,
00:49:51.170 --> 00:49:54.730
but about the
correlation length.
00:49:54.730 --> 00:50:02.044
So let's replace that
ansatz with homogeneity
00:50:02.044 --> 00:50:04.474
of correlation length.
00:50:13.240 --> 00:50:17.680
So once more, we
have a structure
00:50:17.680 --> 00:50:21.784
where is a line
that is terminate
00:50:21.784 --> 00:50:25.780
when two parameters,
t and h go to 0.
00:50:25.780 --> 00:50:30.650
And we know that on
approaching this point,
00:50:30.650 --> 00:50:33.260
the system will become cloudy.
00:50:33.260 --> 00:50:36.880
There's a correlation
length that
00:50:36.880 --> 00:50:41.430
diverges on approaching that
point a function of these two
00:50:41.430 --> 00:50:42.640
arguments.
00:50:42.640 --> 00:50:45.280
I'm going to make the same
homogeneity assumption
00:50:45.280 --> 00:50:46.520
for the correlation length.
00:50:46.520 --> 00:50:48.460
And again, this
is an assumption.
00:50:48.460 --> 00:50:52.805
I say that this is
a to the minus mu.
00:50:52.805 --> 00:50:56.770
The exponent mu was a divergence
of the correlation length.
00:50:56.770 --> 00:51:00.830
Some other function, it's not
that first g that we wrote.
00:51:00.830 --> 00:51:06.610
Let's call it g psi
of ht to the delta.
00:51:12.380 --> 00:51:17.650
So we never discussed it,
but this function immediately
00:51:17.650 --> 00:51:22.170
also tells me if you
approach the critical point
00:51:22.170 --> 00:51:25.820
along the criticalizer term,
how does the correlation length
00:51:25.820 --> 00:51:31.540
diverge through the various
tricks that we have discussed?
00:51:31.540 --> 00:51:35.976
But this is going to be
telling me something more
00:51:35.976 --> 00:51:44.670
if from here, I can reproduce
my scaling assumption 1.0.
00:51:44.670 --> 00:51:49.860
So there is one other
step that I can make.
00:51:49.860 --> 00:52:02.670
Assume divergence of
c is responsible--
00:52:02.670 --> 00:52:11.275
let's call it even solely
responsible-- for singular
00:52:11.275 --> 00:52:11.775
behavior.
00:52:17.475 --> 00:52:21.670
And you say, what
does all of this mean?
00:52:21.670 --> 00:52:27.150
So let's say that I have a
system could be my magnet,
00:52:27.150 --> 00:52:33.014
could be my liquid gas that
has size l on each search.
00:52:36.170 --> 00:52:44.830
And I calculate the
partition function log z.
00:52:44.830 --> 00:52:49.935
Log z will certainly have
the part that is regular.
00:52:49.935 --> 00:52:55.280
Well-- log z will have a part
that is certainly-- let's
00:52:55.280 --> 00:52:57.620
say the contribution
phonons, all kinds
00:52:57.620 --> 00:53:00.880
of other regular things
that don't have anything
00:53:00.880 --> 00:53:03.570
to do with singularity
of the system.
00:53:03.570 --> 00:53:09.550
Those things will give
you some regular function.
00:53:09.550 --> 00:53:11.580
But one thing that
I know for sure
00:53:11.580 --> 00:53:14.120
is that the answer is
going to be extensive.
00:53:14.120 --> 00:53:19.050
If I have any nice
thermodynamic system
00:53:19.050 --> 00:53:24.810
and I am in v
dimensions, then it
00:53:24.810 --> 00:53:28.686
will be proportional to
the volume of that system
00:53:28.686 --> 00:53:29.618
that I have.
00:53:32.420 --> 00:53:38.120
Now the way that I have written
it is not entirely nice,
00:53:38.120 --> 00:53:44.630
because log z is-- a log is
a dimensionless quantity.
00:53:44.630 --> 00:53:47.940
Maybe I measured my length
in meters or centimeters
00:53:47.940 --> 00:53:51.262
or whatever, so I
have dimensions here.
00:53:51.262 --> 00:53:56.900
So it makes sense to pick some
landscape to dimensionalize it
00:53:56.900 --> 00:54:00.470
before multiplying it by some
kind of irregular function
00:54:00.470 --> 00:54:04.525
of whatever I have,
t and h, for example.
00:54:10.370 --> 00:54:15.580
But what about
the singular part?
00:54:15.580 --> 00:54:18.770
For the singular
part, the statement
00:54:18.770 --> 00:54:21.270
was that somehow it was
a connective behavior.
00:54:21.270 --> 00:54:23.350
It involved many, many
degrees of freedom.
00:54:23.350 --> 00:54:27.665
We saw for the heat capacity of
the solid at low temperatures,
00:54:27.665 --> 00:54:32.040
it came from long wavelength
degrees of freedom.
00:54:32.040 --> 00:54:36.120
So no lattice parameter
is going to be important.
00:54:36.120 --> 00:54:41.272
So one thing that I could
do, maintaining extensivity,
00:54:41.272 --> 00:54:48.605
is to divide by l over
c times something.
00:54:52.750 --> 00:54:55.030
So that's the only
thing that I did
00:54:55.030 --> 00:55:00.100
to ensure that extensivity
is maintained when
00:55:00.100 --> 00:55:05.025
I have kind of benign landscape,
but in addition a landscape
00:55:05.025 --> 00:55:08.940
that is divergent.
00:55:08.940 --> 00:55:12.095
Now you can see that
immediately that says that log
00:55:12.095 --> 00:55:18.330
z singular as a
function of t and h,
00:55:18.330 --> 00:55:23.726
will be proportional
to c to the minus t.
00:55:23.726 --> 00:55:26.000
And using that
formula, it will be
00:55:26.000 --> 00:55:31.150
proportional to t to the du,
some other scaling function.
00:55:31.150 --> 00:55:34.718
And it's go back to
gf ht to the delta.
00:55:42.040 --> 00:55:46.650
Physically, what it's
saying is that when
00:55:46.650 --> 00:55:52.550
I am very close, but not
quite at the critical point,
00:55:52.550 --> 00:55:56.580
I have a long correlation
length, much larger
00:55:56.580 --> 00:56:00.280
than microscopic length
scale of my system.
00:56:00.280 --> 00:56:06.010
So what I can say is that
within a correlation length,
00:56:06.010 --> 00:56:11.375
my degrees of freedom for
magentization or whatever it is
00:56:11.375 --> 00:56:15.120
are very much coupled
to each other.
00:56:15.120 --> 00:56:17.940
So maybe what I can
do is I can regard
00:56:17.940 --> 00:56:21.480
this as an independent lock.
00:56:21.480 --> 00:56:25.130
And how many independent
locks do I have?
00:56:25.130 --> 00:56:28.060
It is l over c to the d.
00:56:28.060 --> 00:56:30.710
So the statement
roughly is a part
00:56:30.710 --> 00:56:34.390
of the assumption is that this
correlation length that is
00:56:34.390 --> 00:56:36.340
getting bigger and bigger.
00:56:36.340 --> 00:56:38.700
Because things are
correlated, the number
00:56:38.700 --> 00:56:40.670
of independent degrees
of freedom that you
00:56:40.670 --> 00:56:43.980
are having gets
smaller and smaller.
00:56:43.980 --> 00:56:47.550
And that's changing the
number of degrees of freedom
00:56:47.550 --> 00:56:51.400
is responsible for the singular
behavior of the free energy.
00:56:51.400 --> 00:56:56.520
If I make this assumption about
this correlation then diverges,
00:56:56.520 --> 00:56:57.784
then I will get this form.
00:57:01.090 --> 00:57:05.920
So now my ansatz 2.0
matches my ansatz 1.0
00:57:05.920 --> 00:57:09.940
provided du is 2 minus alpha.
00:57:09.940 --> 00:57:16.950
So I have du2 plus
2 minus alpha which
00:57:16.950 --> 00:57:20.734
is known after Brian
Josephson, so this
00:57:20.734 --> 00:57:26.662
is the Josephson relation.
00:57:26.662 --> 00:57:33.600
And it is different from the
other exponent identities
00:57:33.600 --> 00:57:37.100
that we have because
it explicitly
00:57:37.100 --> 00:57:39.790
depends on the
dimensionality of space.
00:57:39.790 --> 00:57:42.060
d appears in the problem.
00:57:42.060 --> 00:57:45.947
It's called hyperscale
for that reason.
00:57:50.420 --> 00:57:51.200
Yes?
00:57:51.200 --> 00:57:54.230
AUDIENCE: So does the assumption
that the divergence in c
00:57:54.230 --> 00:57:56.479
is solely responsible for
the singular behavior, what
00:57:56.479 --> 00:57:58.020
are we excluding
when we assume that?
00:57:58.020 --> 00:58:02.066
What else could happen that
would make that not true?
00:58:02.066 --> 00:58:05.440
PROFESSOR: Well, what
is appearing here maybe
00:58:05.440 --> 00:58:10.401
will have some singular
function of t and h.
00:58:10.401 --> 00:58:12.275
AUDIENCE: So this
similar to what
00:58:12.275 --> 00:58:15.581
we were assuming before when we
said that our free energy could
00:58:15.581 --> 00:58:17.890
have some regular
part that depends
00:58:17.890 --> 00:58:21.874
on [INAUDIBLE] the
part that [INAUDIBLE].
00:58:21.874 --> 00:58:23.297
PROFESSOR: Yes, exactly.
00:58:26.430 --> 00:58:30.620
But once again, the truth
is really whether or not
00:58:30.620 --> 00:58:33.120
this matches up
with experiments.
00:58:33.120 --> 00:58:39.300
So let's, for example,
pick anything in that
00:58:39.300 --> 00:58:41.040
table, v equals to t.
00:58:41.040 --> 00:58:45.660
Let's pick n goes to 2,
which we haven't done so far.
00:58:45.660 --> 00:58:54.610
And so what the formula
would say is 3 times mu.
00:58:54.610 --> 00:59:03.670
Mu for the superfluid
is 67 is 2 minus-- well,
00:59:03.670 --> 00:59:07.990
alpha is almost 0 but
slightly negative.
00:59:07.990 --> 00:59:16.240
So it is 0.01.
00:59:16.240 --> 00:59:17.270
And what do we have?
00:59:17.270 --> 00:59:26.440
3 times 67 is 2.01.
00:59:26.440 --> 00:59:27.750
So it matches.
00:59:27.750 --> 00:59:31.830
Actually, we say, well,
why do you emphasize
00:59:31.830 --> 00:59:34.300
that it's the
function of dimension?
00:59:34.300 --> 00:59:39.040
Well, a little bit
later on in the course,
00:59:39.040 --> 00:59:44.990
we will do an exact solution of
the so-called 2D Ising model.
00:59:47.670 --> 00:59:51.820
So this is a system that
first wants to be close to 2,
00:59:51.820 --> 00:59:53.120
n equals to 1.
00:59:53.120 --> 00:59:56.780
And it was an important thing
that people could actually
00:59:56.780 --> 01:00:00.860
solve an interacting problem,
not in three dimensions
01:00:00.860 --> 01:00:01.760
but in two.
01:00:01.760 --> 01:00:05.780
And the exponents
for that, alpha is 0,
01:00:05.780 --> 01:00:08.860
but it really is a
logarithmic divergence.
01:00:08.860 --> 01:00:11.700
Beta is 1/8.
01:00:11.700 --> 01:00:20.060
Gamma is 7/4, delta is 15,
mu is 1, and eta is 1/4.
01:00:20.060 --> 01:00:28.190
And we can check now
for this v equals to 2 n
01:00:28.190 --> 01:00:32.980
equals to 1 that
we have two times
01:00:32.980 --> 01:00:35.970
our mu, which exactly
is known to be
01:00:35.970 --> 01:00:43.410
1 is 2 minus logarithmic
divergence corresponding to 0.
01:00:43.410 --> 01:00:46.036
So again, there's
something that works.
01:00:50.600 --> 01:00:56.150
One thing that you may
want to see and look at
01:00:56.150 --> 01:01:01.500
is that the ansatz
that we made first also
01:01:01.500 --> 01:01:05.820
works for the result
of saddlepoint,
01:01:05.820 --> 01:01:09.930
not surprisingly because
again in the saddlepoint
01:01:09.930 --> 01:01:14.020
we start with a singular free
energy and go through all this.
01:01:14.020 --> 01:01:18.000
But it does not work for
this type of scaling,
01:01:18.000 --> 01:01:23.690
because 2 minus alpha
would be 0 is not
01:01:23.690 --> 01:01:29.925
equal to d times one half,
except in the case of four
01:01:29.925 --> 01:01:30.425
dimensions.
01:01:33.060 --> 01:01:37.390
So somehow, this
ansatz and this picture
01:01:37.390 --> 01:01:42.680
breaks down within the
saddlepoint approximation.
01:01:42.680 --> 01:01:47.180
If you remember what we did
when we calculated fluctuation
01:01:47.180 --> 01:01:49.815
corrections for the
saddlepoint, you
01:01:49.815 --> 01:01:54.800
got actually an exponent alpha
that was 2 minus mu over 2.
01:01:54.800 --> 01:02:00.400
So the fluctuating part that
we get around the saddlepoint
01:02:00.400 --> 01:02:02.420
does satisfy this.
01:02:02.420 --> 01:02:04.995
But on top of that
there's another part that
01:02:04.995 --> 01:02:07.730
is doe to the
saddlepoint value itself
01:02:07.730 --> 01:02:10.720
that violates this
hyperscaling solution.
01:02:10.720 --> 01:02:11.220
Yes?
01:02:11.220 --> 01:02:15.280
AUDIENCE: Empirically, how well
can we probe the dependence
01:02:15.280 --> 01:02:19.192
on dimensionality that we're
finding in these expressions?
01:02:19.192 --> 01:02:21.050
PROFESSOR:
Experimentally, we can
01:02:21.050 --> 01:02:23.730
do d equals to 2 d equals to 3.
01:02:23.730 --> 01:02:27.948
And computer simulations we
can also do d equals to 2 d
01:02:27.948 --> 01:02:28.840
equals to 3.
01:02:28.840 --> 01:02:31.695
Very soon, we will do
analytical expressions
01:02:31.695 --> 01:02:35.310
where we will be
in 3.99 dimensions.
01:02:35.310 --> 01:02:38.960
So we will be coming down
conservatively around 4.
01:02:38.960 --> 01:02:42.260
So mathematically, we can
play tricks such as that.
01:02:42.260 --> 01:02:45.983
But certainly empirically, in
the sense of experimentally
01:02:45.983 --> 01:02:48.456
we are at a disadvantage
in those languages.
01:02:51.490 --> 01:02:51.990
OK?
01:03:00.220 --> 01:03:03.190
So we are making progress.
01:03:03.190 --> 01:03:05.600
We have made our way
across this table.
01:03:05.600 --> 01:03:09.110
We have also an identity
that involves mu.
01:03:09.110 --> 01:03:11.320
But so far I haven't
said anything about eta.
01:03:15.420 --> 01:03:21.950
I can say something about
the eta reasonably simply,
01:03:21.950 --> 01:03:24.255
but then you try to
build something profound
01:03:24.255 --> 01:03:27.290
based on that.
01:03:27.290 --> 01:03:35.046
So let's look at exactly at
tc, at the critical point.
01:03:37.860 --> 01:03:43.262
So let's say you are sitting
at t and h equals to 0.
01:03:43.262 --> 01:03:45.980
You have to prepare your
system at that point.
01:03:45.980 --> 01:03:49.450
There's nothing physically
that says you can't.
01:03:49.450 --> 01:03:53.640
At that point, you can
look at correlations.
01:03:53.640 --> 01:03:57.420
And the exponent eta for
example is a characteristic
01:03:57.420 --> 01:03:59.150
of those correlations.
01:03:59.150 --> 01:04:05.490
And one of the things that we
have is that m of x m of 0,
01:04:05.490 --> 01:04:10.966
the connected parts-- well,
actually at the critical point
01:04:10.966 --> 01:04:13.340
we don't even have to
put the connected part
01:04:13.340 --> 01:04:16.840
because the average
of n is going to be 0.
01:04:16.840 --> 01:04:19.725
But this is a
quantity that behaves
01:04:19.725 --> 01:04:24.570
as 1 over the separation
that's actually
01:04:24.570 --> 01:04:28.560
include two possible
points, x minus y.
01:04:28.560 --> 01:04:34.760
When we did the case
of the fluctuations
01:04:34.760 --> 01:04:39.280
at the critical point within
the saddlepoint method,
01:04:39.280 --> 01:04:42.445
we found that the behavior
was like the Coulomb law.
01:04:42.445 --> 01:04:45.830
It was falling off as
1x to the d minus 2.
01:04:45.830 --> 01:04:48.320
But we said that
experiment indicated
01:04:48.320 --> 01:04:52.940
that there is a small
correction for this
01:04:52.940 --> 01:04:54.580
that we indicate
with exponent eta.
01:04:54.580 --> 01:04:59.850
So that was how the
exponent eta was defined.
01:04:59.850 --> 01:05:04.560
So can we have an identity
that involves the exponent eta?
01:05:04.560 --> 01:05:08.330
We actually have seen
how to do this already.
01:05:08.330 --> 01:05:12.050
Because we know that in
general, the susceptibilities
01:05:12.050 --> 01:05:16.400
are related to integrals of
the correlation functions.
01:05:20.860 --> 01:05:25.850
Now if I put this
power law over here,
01:05:25.850 --> 01:05:28.110
you can see that the
answer is like trying
01:05:28.110 --> 01:05:32.100
to be integrate x squared
all the way to infinity down.
01:05:32.100 --> 01:05:35.512
So it will be divergent
and that's no problem.
01:05:35.512 --> 01:05:37.720
At the critical point we
know that the susceptibility
01:05:37.720 --> 01:05:40.010
is divergent.
01:05:40.010 --> 01:05:48.500
But you say, OK, if I'm away
from the critical point,
01:05:48.500 --> 01:05:53.940
then I will use this
formula, but only
01:05:53.940 --> 01:05:57.060
up to the correlation length.
01:05:57.060 --> 01:06:00.250
And I say that beyond
the correlation length,
01:06:00.250 --> 01:06:03.010
then the correlations
will decay exponentially.
01:06:03.010 --> 01:06:07.810
That's too rapid a falloff,
and essentially the only part
01:06:07.810 --> 01:06:10.155
that's contributing
is because what
01:06:10.155 --> 01:06:13.110
was happening at
the critical point.
01:06:13.110 --> 01:06:20.345
Once I do that, I have to
integrate ddx over x to the d
01:06:20.345 --> 01:06:24.290
minus 2 plus eta up to
the correlation length.
01:06:24.290 --> 01:06:27.040
The answer will be proportional
to the correlation length
01:06:27.040 --> 01:06:30.400
to the power of 2 minus eta.
01:06:30.400 --> 01:06:36.230
And this will be proportional
to p to the power of c goes
01:06:36.230 --> 01:06:38.180
[INAUDIBLE] to the minus mu.
01:06:38.180 --> 01:06:39.698
2 minus eta times mu.
01:06:44.920 --> 01:06:46.696
But we know that
the susceptibilities
01:06:46.696 --> 01:06:51.210
diverge as t to the minus gamma.
01:06:51.210 --> 01:06:55.320
So we have established
an exponent identity
01:06:55.320 --> 01:07:04.210
that tells us that gamma
is 2 minus eta times mu.
01:07:04.210 --> 01:07:09.170
And this is known as the Fisher
identity, after Michael Fisher.
01:07:13.341 --> 01:07:16.500
Again, you can see that in
all of the cases in three
01:07:16.500 --> 01:07:18.730
dimensions that we
are dealing with,
01:07:18.730 --> 01:07:20.668
exponent eta is roughly 0.
01:07:20.668 --> 01:07:24.350
It's 0.04 And all
of our gammas are
01:07:24.350 --> 01:07:27.650
roughly twice what our
mus are in that table.
01:07:27.650 --> 01:07:30.630
It's time we get
that table checked.
01:07:30.630 --> 01:07:34.790
The one case that I have on
that table where eta is not 0
01:07:34.790 --> 01:07:41.162
is when I'm looking at v
positive 2 where eta is 1/4.
01:07:41.162 --> 01:07:44.870
So I take 2 minus
1/4, multiply it
01:07:44.870 --> 01:07:47.640
by the mu that is
one in two dimension,
01:07:47.640 --> 01:07:51.162
and the answer is
the 7/4, which we
01:07:51.162 --> 01:07:53.830
have for the exponent
gamma over there.
01:08:02.075 --> 01:08:05.459
So we have now the
identity that is
01:08:05.459 --> 01:08:09.810
applicable to the
last exponents.
01:08:09.810 --> 01:08:13.160
So all of this works.
01:08:13.160 --> 01:08:16.359
Let's now take the
conceptual leap
01:08:16.359 --> 01:08:19.185
that then allows
us to do what we
01:08:19.185 --> 01:08:22.890
will do later on to
get the exponents.
01:08:22.890 --> 01:08:25.890
Basically, you can see
that what we have imposed
01:08:25.890 --> 01:08:30.569
here conceptually
is the following.
01:08:30.569 --> 01:08:35.109
That when I'm away from
the critical point,
01:08:35.109 --> 01:08:39.000
I look at the correlations
of this important statistical
01:08:39.000 --> 01:08:40.562
field.
01:08:40.562 --> 01:08:43.640
And I find that they
fall off with separation,
01:08:43.640 --> 01:08:46.810
according to some power.
01:08:46.810 --> 01:08:52.580
And the reason is that
at the critical point,
01:08:52.580 --> 01:08:54.970
the correlation length
has gone to infinity.
01:08:54.970 --> 01:08:57.279
That's not the length scale
that you have to play with.
01:08:57.279 --> 01:09:02.890
You can divide x minus y divided
by c, which is what we do away
01:09:02.890 --> 01:09:06.765
from the critical point.
c has gone to infinity.
01:09:06.765 --> 01:09:09.410
The other length scale
that we are worried about
01:09:09.410 --> 01:09:12.689
are things that go
into the microscopics.
01:09:12.689 --> 01:09:18.260
but we are assuming that
microscopics is irrelevant.
01:09:18.260 --> 01:09:21.939
It has been washed out.
01:09:21.939 --> 01:09:24.810
So if we don't have
a large length scale,
01:09:24.810 --> 01:09:27.670
if we don't have a short
length scale, some function
01:09:27.670 --> 01:09:30.960
of distance, how can it decay?
01:09:30.960 --> 01:09:33.510
The only way it can
decay is [INAUDIBLE].
01:09:36.510 --> 01:09:45.010
So this statement is that when
we are at a critical point,
01:09:45.010 --> 01:09:47.479
I look at some correlation.
01:09:47.479 --> 01:09:50.170
And this was the
magnetization correlation.
01:09:50.170 --> 01:09:54.020
But I can look at
correlation of anything else
01:09:54.020 --> 01:09:56.069
as a function of separation.
01:10:01.820 --> 01:10:06.990
And this will only fall off
as some power of separation.
01:10:06.990 --> 01:10:09.400
Another way of writing
it is that if I
01:10:09.400 --> 01:10:13.500
were to multiply this
by some length scale,
01:10:13.500 --> 01:10:17.570
so rather than looking at things
that are some distance apart
01:10:17.570 --> 01:10:20.030
at twice that distance
apart or hundred times
01:10:20.030 --> 01:10:24.006
that distance apart, I will
reproduce the correlation
01:10:24.006 --> 01:10:32.970
that I have up to some
other of the scale factor.
01:10:32.970 --> 01:10:35.520
So the scale factor
here we can read off
01:10:35.520 --> 01:10:38.730
has to be related to
t minus 2 plus eta.
01:10:38.730 --> 01:10:41.450
But essentially, this
is a statement again
01:10:41.450 --> 01:10:45.680
about homogeneity of
correlation functions
01:10:45.680 --> 01:10:47.968
when you are at
a critical point.
01:10:50.510 --> 01:10:54.496
So this is a symmetry here.
01:10:54.496 --> 01:11:00.260
It says you take your
statistical correlations
01:11:00.260 --> 01:11:02.670
and you look at them
at the larger scale
01:11:02.670 --> 01:11:04.860
or at the shorter scale.
01:11:04.860 --> 01:11:08.400
And up to some
overall scale factor,
01:11:08.400 --> 01:11:10.280
you reproduce what
you had before.
01:11:14.020 --> 01:11:20.190
So this is something to do
with invariance on the scale.
01:11:28.290 --> 01:11:32.930
This scaling variance
is some property
01:11:32.930 --> 01:11:37.190
that was popular a while
ago as being associated
01:11:37.190 --> 01:11:38.772
with the kind of
geometrical objects
01:11:38.772 --> 01:11:39.980
that you would call fractals.
01:11:46.100 --> 01:11:54.000
So the statement is that
if I go across my system
01:11:54.000 --> 01:11:58.590
and there is some pattern of
magnetization fluctuations,
01:11:58.590 --> 01:12:00.416
let's say I look at it.
01:12:00.416 --> 01:12:02.465
I'm going along
this direction x.
01:12:04.980 --> 01:12:11.240
And I plot at some
particular configuration
01:12:11.240 --> 01:12:13.580
that is dominant
and is contributing
01:12:13.580 --> 01:12:17.092
to my free energy,
the magnetization,
01:12:17.092 --> 01:12:21.490
that it has a shape that
has this characteristic self
01:12:21.490 --> 01:12:25.620
similarity kind of maybe looking
like a mountain landscape.
01:12:28.420 --> 01:12:31.510
And the statement
is that if I were
01:12:31.510 --> 01:12:40.650
to take a part of that
landscape and then blow it up,
01:12:40.650 --> 01:12:45.290
I will generate a pattern
that is of course not the same
01:12:45.290 --> 01:12:46.210
as the first one.
01:12:46.210 --> 01:12:48.970
It is not exactly
scale invariant.
01:12:48.970 --> 01:12:53.150
But it has the same kind
of statistics as the one
01:12:53.150 --> 01:12:57.220
that I had originally after
I multiplied this axis
01:12:57.220 --> 01:12:59.088
by some factor lambda.
01:13:03.770 --> 01:13:05.345
Yes?
01:13:05.345 --> 01:13:10.960
AUDIENCE: Under what length
scales are those subsimilarity
01:13:10.960 --> 01:13:15.032
properties evident and how
do they compare to the length
01:13:15.032 --> 01:13:16.740
scale over which you're
doing your course
01:13:16.740 --> 01:13:18.356
grading for this field?
01:13:18.356 --> 01:13:25.476
PROFESSOR: OK, so basically we
expect this to be applicable
01:13:25.476 --> 01:13:27.230
presumably at length
scales that are
01:13:27.230 --> 01:13:28.780
less than the size
of your system
01:13:28.780 --> 01:13:31.330
because once I get to
the size of the system
01:13:31.330 --> 01:13:33.950
I can't blow it up
further or whatever.
01:13:33.950 --> 01:13:37.560
It has to certainly be
larger than whatever
01:13:37.560 --> 01:13:41.870
the coarse-graining length is,
or the length scale at which
01:13:41.870 --> 01:13:45.825
I have confidence that I have
washed out the microscoping
01:13:45.825 --> 01:13:48.340
details.
01:13:48.340 --> 01:13:50.400
Now that depends on
the system in question,
01:13:50.400 --> 01:13:53.630
so I can't really give
you an answer for that.
01:13:53.630 --> 01:13:56.140
The answer will
depend on the system.
01:13:56.140 --> 01:13:58.040
But the point is
that I'm looking
01:13:58.040 --> 01:14:01.720
in the vicinity of a point
where mathematically I'm
01:14:01.720 --> 01:14:04.250
assured that there's a
correlation length that
01:14:04.250 --> 01:14:05.580
goes to infinity.
01:14:05.580 --> 01:14:09.580
So maybe there is some system
number 1 that average out
01:14:09.580 --> 01:14:12.390
very easily, and
after a distance of 10
01:14:12.390 --> 01:14:14.890
I can start applying this.
01:14:14.890 --> 01:14:16.730
But maybe there's
some other system
01:14:16.730 --> 01:14:19.520
where the microscopic degrees
of freedom are very problematic
01:14:19.520 --> 01:14:22.950
and I have to go further and
further out before they average
01:14:22.950 --> 01:14:23.730
out.
01:14:23.730 --> 01:14:27.200
But in principle, since
my c has gone to infinity,
01:14:27.200 --> 01:14:31.240
I can just pick a bigger and
bigger piece of my system
01:14:31.240 --> 01:14:33.170
until that has happened.
01:14:33.170 --> 01:14:36.280
So I can't tell you what
the short distance length
01:14:36.280 --> 01:14:39.880
scale is in the same sense
that when [INAUDIBLE] says
01:14:39.880 --> 01:14:44.230
that coast of Britain
is fractal, well,
01:14:44.230 --> 01:14:47.040
I can't tell you whether
the short distance is
01:14:47.040 --> 01:14:50.840
the size of a sand
particle, or is it
01:14:50.840 --> 01:14:54.648
the size of, I don't know, a
tree or something like that.
01:14:54.648 --> 01:14:55.624
I don't know.
01:15:03.920 --> 01:15:13.740
So we started thinking
about our original problem.
01:15:13.740 --> 01:15:19.360
And constructing
this Landau-Ginzburg,
01:15:19.360 --> 01:15:25.270
[INAUDIBLE] that we worked
with on the basis of symmetries
01:15:25.270 --> 01:15:30.010
such as invariance on
the rotation, et cetera.
01:15:30.010 --> 01:15:32.895
Somehow we've discovered
that the point that we
01:15:32.895 --> 01:15:37.700
are interested has an additional
symmetry that maybe we
01:15:37.700 --> 01:15:41.706
didn't anticipate, which is
this self-similarity and scale
01:15:41.706 --> 01:15:42.205
invariance.
01:15:45.290 --> 01:15:49.240
So you say, OK, that's the
solution to the problem.
01:15:49.240 --> 01:15:52.190
Let's go back to
our construction
01:15:52.190 --> 01:15:55.930
of the Landau-Ginsburg
theory and add
01:15:55.930 --> 01:15:58.010
to the list of
symmetries that have
01:15:58.010 --> 01:16:03.250
to be obeyed, this additional
self-similarity of scaling.
01:16:03.250 --> 01:16:07.990
And that will put us at t
equals to 0, h equals to 0.
01:16:07.990 --> 01:16:10.880
And for example, we should
be able to calculate
01:16:10.880 --> 01:16:13.530
this correlation.
01:16:13.530 --> 01:16:15.987
Let me expand a little
bit on that because we
01:16:15.987 --> 01:16:17.320
will need one other correlation.
01:16:17.320 --> 01:16:19.550
Because we've said
that essentially,
01:16:19.550 --> 01:16:22.530
all of the properties
of the system
01:16:22.530 --> 01:16:25.760
I can get from two
independent exponents.
01:16:25.760 --> 01:16:29.690
So suppose I constructed
this scale invariant theory
01:16:29.690 --> 01:16:32.470
and I calculated this.
01:16:32.470 --> 01:16:33.770
That would be on exponent.
01:16:33.770 --> 01:16:36.420
I need another one.
01:16:36.420 --> 01:16:39.560
Well, we had here a
statement about alpha.
01:16:39.560 --> 01:16:43.830
We made the statement that
heat capacity diverges.
01:16:43.830 --> 01:16:48.690
Now in the same sense that the
susceptibility is a response--
01:16:48.690 --> 01:16:52.470
it came from two derivatives
of the free energy with respect
01:16:52.470 --> 01:16:54.000
to the field.
01:16:54.000 --> 01:16:56.540
The derivative of
magnetization with respect
01:16:56.540 --> 01:16:59.400
to field magnetization is
one derivative [INAUDIBLE].
01:16:59.400 --> 01:17:01.810
The heat capacity is
also two derivatives
01:17:01.810 --> 01:17:05.400
of free energy with respect
to some other variable.
01:17:08.400 --> 01:17:10.700
So in the same
sense that there is
01:17:10.700 --> 01:17:14.011
a relationship between
the susceptibility
01:17:14.011 --> 01:17:16.770
and an integrated
correlation function,
01:17:16.770 --> 01:17:21.800
there is a relationship that
says that the heat capacity is
01:17:21.800 --> 01:17:26.000
related to an integrated
correlation function.
01:17:26.000 --> 01:17:31.860
So c as a function of say t and
h, let's say the singular part,
01:17:31.860 --> 01:17:36.115
is going to be related to
an integral of something.
01:17:40.080 --> 01:17:45.420
And again, we've
already seen this.
01:17:45.420 --> 01:17:48.512
Essentially, you take one
derivative of the free energy
01:17:48.512 --> 01:17:50.470
let's say with respect
the beta or temperature,
01:17:50.470 --> 01:17:53.060
you get the energy.
01:17:53.060 --> 01:17:55.860
And you take another
derivative of the energy
01:17:55.860 --> 01:17:59.300
you will get the heat capacity.
01:17:59.300 --> 01:18:02.120
And then that
derivative, if we write
01:18:02.120 --> 01:18:05.320
in terms of the first derivative
of the partition function
01:18:05.320 --> 01:18:09.780
becomes converted to
the variance in energy.
01:18:09.780 --> 01:18:12.750
So in the same way that
the susceptibility was
01:18:12.750 --> 01:18:16.220
the variance of the
net magnetization,
01:18:16.220 --> 01:18:19.555
the heat capacity is
related to the variance
01:18:19.555 --> 01:18:23.580
of the net energy of the
system at an even temperature.
01:18:23.580 --> 01:18:26.090
The net energy of the
system we can write this
01:18:26.090 --> 01:18:29.400
as an integral of
an energy density,
01:18:29.400 --> 01:18:31.450
just as we wrote
the magnetization
01:18:31.450 --> 01:18:34.050
as an integral of
magnetization density.
01:18:34.050 --> 01:18:40.105
And then the heat capacity will
be related to the correlation
01:18:40.105 --> 01:18:42.486
functions of the energy density.
01:18:47.226 --> 01:18:53.380
Now once more, you say that
I'm at the critical point.
01:18:53.380 --> 01:18:57.210
At the critical point
there is no length scale.
01:18:57.210 --> 01:19:02.360
So any correlation function, not
only that of the magnetization,
01:19:02.360 --> 01:19:08.580
should fall off as some
power of separation.
01:19:08.580 --> 01:19:12.920
And you can call that
exponent whatever you like.
01:19:12.920 --> 01:19:16.772
There is no definition
for it in the literature.
01:19:16.772 --> 01:19:20.300
Let me write it in the
same way as magnetization
01:19:20.300 --> 01:19:23.320
as d minus 2 plus eta prime.
01:19:23.320 --> 01:19:28.320
So then when I go and say let's
terminate it at the correlation
01:19:28.320 --> 01:19:30.800
length, the answer
is going to be
01:19:30.800 --> 01:19:35.330
proportional to c to
the 2 minus eta prime,
01:19:35.330 --> 01:19:38.792
which would be t
to the minus mu.
01:19:38.792 --> 01:19:41.150
2 minus eta prime.
01:19:41.150 --> 01:19:46.590
So then I would have alpha
being mu 2 minus eta.
01:19:51.550 --> 01:19:56.100
So all I need to
do in principle is
01:19:56.100 --> 01:20:00.160
to construct a theory,
which in addition
01:20:00.160 --> 01:20:02.980
to rotational
invariance or there's
01:20:02.980 --> 01:20:05.720
whatever is appropriate
to the system in question,
01:20:05.720 --> 01:20:09.860
has this statistical
scale invariance.
01:20:09.860 --> 01:20:13.710
Within that theory, calculate
the correlation functions
01:20:13.710 --> 01:20:17.740
of two quantities, such as
magnetization and energy.
01:20:17.740 --> 01:20:20.520
Extract two exponents.
01:20:20.520 --> 01:20:22.936
Once we have two
exponents, then we
01:20:22.936 --> 01:20:24.920
know why your
manipulations will be
01:20:24.920 --> 01:20:26.597
able to calculate
all the exponents.
01:20:29.520 --> 01:20:32.850
So why doesn't this
solve the problem?
01:20:32.850 --> 01:20:36.270
The answer is that whereas I
can write immediately for you
01:20:36.270 --> 01:20:40.800
a term such as m squared,
that is rotational invariant,
01:20:40.800 --> 01:20:46.725
I don't know how to write down a
theory that is scale invariant.
01:20:46.725 --> 01:20:50.180
The one case where people
have succeeded to do that
01:20:50.180 --> 01:20:52.590
is actually two dimensions.
01:20:52.590 --> 01:20:54.890
So in two dimensions,
one can show
01:20:54.890 --> 01:20:56.835
that this kind of
scale invariance
01:20:56.835 --> 01:21:00.615
is related to
conformal invariance
01:21:00.615 --> 01:21:04.070
and that one can explicitly
write down conformal invariant
01:21:04.070 --> 01:21:08.500
theories, extract exponents
et cetera out of those.
01:21:08.500 --> 01:21:12.860
But say in three dimensions,
we don't know how to do that.
01:21:12.860 --> 01:21:16.460
So we will still,
with that concept
01:21:16.460 --> 01:21:19.020
in the back of our
mind, approach it
01:21:19.020 --> 01:21:24.810
slightly differently by looking
at the effects of the scale
01:21:24.810 --> 01:21:27.190
transformation on the system.
01:21:27.190 --> 01:21:31.140
And that's the beginning of
this concept of normalization.