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PROFESSOR: OK, let's start.

00:00:24.530 --> 00:00:29.980
So last time, having
dealt with the ideal gas

00:00:29.980 --> 00:00:33.225
as much as possible, we started
with interacting systems.

00:00:39.420 --> 00:00:42.660
Let's say interacting gas.

00:00:42.660 --> 00:00:46.760
And the first topic
that we did was

00:00:46.760 --> 00:00:51.860
what I called an approach
towards interaction

00:00:51.860 --> 00:00:54.451
through the Cumulant expansion.

00:00:57.400 --> 00:01:03.720
The idea is that we certainly
solved the problem where

00:01:03.720 --> 00:01:14.550
we had particles in a box and
it was just a trivial system.

00:01:14.550 --> 00:01:20.690
Basically, the particles were
independent of each other

00:01:20.690 --> 00:01:23.520
in the canonical ensemble.

00:01:23.520 --> 00:01:27.540
And things become interesting
if you put interactions

00:01:27.540 --> 00:01:30.710
among all of the particles.

00:01:30.710 --> 00:01:33.480
Very soon, we will
have a specific form

00:01:33.480 --> 00:01:35.520
of this interaction.

00:01:35.520 --> 00:01:37.270
But for the beginning,
let's maintain

00:01:37.270 --> 00:01:40.190
it as general as possible.

00:01:40.190 --> 00:01:42.240
And the idea was
that we are trying

00:01:42.240 --> 00:01:46.120
to calculate the partition
function for a system that

00:01:46.120 --> 00:01:49.400
has a given temperature.

00:01:49.400 --> 00:01:54.410
The box has some volume V and
there are N particles in it.

00:01:54.410 --> 00:01:58.250
And to do so, we have
to integrate over

00:01:58.250 --> 00:02:02.140
the entirety of the phase
space of N particles.

00:02:02.140 --> 00:02:09.100
So we have to integrate over all
of the momenta and coordinates

00:02:09.100 --> 00:02:11.710
of e to the minus beta h.

00:02:11.710 --> 00:02:15.580
But we said that we are
going to first of all,

00:02:15.580 --> 00:02:18.330
note that for
identical particles,

00:02:18.330 --> 00:02:21.720
we can't tell apart
the phase space

00:02:21.720 --> 00:02:25.340
if you were to make these in
[? factorial ?] permutations.

00:02:25.340 --> 00:02:27.650
And we also made
it dimensionless

00:02:27.650 --> 00:02:30.620
by dividing by h to the 3n.

00:02:30.620 --> 00:02:33.250
And then we have e
to the minus beta

00:02:33.250 --> 00:02:39.340
h, which has a part that is
e to the minus sum over i pi

00:02:39.340 --> 00:02:43.240
squared over 2m, that
depends on the momenta.

00:02:43.240 --> 00:02:48.660
And a part that depends
on the interaction U.

00:02:48.660 --> 00:02:52.390
The integrals over the
momenta, we can easily perform.

00:02:52.390 --> 00:02:55.770
They give us the
result of Gaussians

00:02:55.770 --> 00:03:03.240
that we can express as
lambda to the power of 3n

00:03:03.240 --> 00:03:09.322
where we introduced this lambda
to be h over root 2 pi m k t.

00:03:12.770 --> 00:03:18.170
And actually, sort
of keeping in mind

00:03:18.170 --> 00:03:22.620
the result that we had for
the case of the ideal gas,

00:03:22.620 --> 00:03:28.700
let's rewrite this as V over
lambda cubed to the power of N.

00:03:28.700 --> 00:03:34.770
So I essentially divided
and multiplied by V to the N

00:03:34.770 --> 00:03:37.200
so that this part
becomes the partition

00:03:37.200 --> 00:03:42.010
function than I would
have had if U was absent.

00:03:42.010 --> 00:03:47.650
And so then the remainder is the
integral over all of the q's,

00:03:47.650 --> 00:03:50.580
but I divided by
V. So basically,

00:03:50.580 --> 00:03:53.580
with each integration,
I'm uniformly

00:03:53.580 --> 00:03:57.890
sampling all of the points
inside the box with probability

00:03:57.890 --> 00:04:02.540
density 1/V e to
the minus beta U.

00:04:02.540 --> 00:04:05.760
And we said that
this quantity we

00:04:05.760 --> 00:04:10.470
can regard as sampling
e to the minus beta U

00:04:10.470 --> 00:04:12.540
through 0 to all
other probability

00:04:12.540 --> 00:04:17.149
distribution in which the
particles ideal-like are

00:04:17.149 --> 00:04:19.160
uniformly distributed.

00:04:19.160 --> 00:04:23.870
And we call this e
to the minus beta U.

00:04:23.870 --> 00:04:27.810
And the index 0
again to indicate

00:04:27.810 --> 00:04:31.265
that we are dealing
with a uniform 0

00:04:31.265 --> 00:04:33.750
to order distribution
with respect

00:04:33.750 --> 00:04:35.584
to which this average is taken.

00:04:38.190 --> 00:04:44.060
And then, this quantity
we can write as Z0.

00:04:44.060 --> 00:04:46.760
And this exponential
we can formally

00:04:46.760 --> 00:04:51.160
express as a sum
over all, let's say,

00:04:51.160 --> 00:04:54.750
l running from 0 to
infinity minus beta

00:04:54.750 --> 00:04:58.080
to the l divided by l factorial.

00:04:58.080 --> 00:05:02.290
Then, expanding this, a quantity
that depends on the coordinates

00:05:02.290 --> 00:05:06.930
is U. So I could have U raised
to the l power, this kind

00:05:06.930 --> 00:05:10.360
of average taken
with respect to that.

00:05:10.360 --> 00:05:13.720
And immediately, my
writing something like this

00:05:13.720 --> 00:05:17.110
suggests that what
I'm after eventually

00:05:17.110 --> 00:05:20.120
is some kind of a
perturbation theory.

00:05:20.120 --> 00:05:24.460
Because I know how to solve
the problem where U is absent.

00:05:24.460 --> 00:05:26.370
That's the ideal gas problem.

00:05:26.370 --> 00:05:29.900
Maybe we can start to move
away from the ideal gas

00:05:29.900 --> 00:05:33.160
and calculate things
perturbatively

00:05:33.160 --> 00:05:34.850
in some quantity.

00:05:34.850 --> 00:05:36.910
And exactly what
that quantity is

00:05:36.910 --> 00:05:39.090
will become apparent shortly.

00:05:39.090 --> 00:05:43.260
At this time, it looks like
it's an expansion in beta U

00:05:43.260 --> 00:05:46.850
having to be a small quantity.

00:05:46.850 --> 00:05:51.050
Now, of course, for calculating
thermodynamic functions

00:05:51.050 --> 00:05:55.520
and behaviors, we don't
really rely usually on Z,

00:05:55.520 --> 00:05:59.120
but log Z, which can be
related to the free energy,

00:05:59.120 --> 00:06:01.110
for example.

00:06:01.110 --> 00:06:04.230
So taking the log
of that expression,

00:06:04.230 --> 00:06:07.490
I have a term that is log of Z0.

00:06:07.490 --> 00:06:11.190
And a term that is
the log of that sum.

00:06:11.190 --> 00:06:14.530
But we recognize that
that sum, in some sense,

00:06:14.530 --> 00:06:20.240
is a generator of moments
of this quantity U.

00:06:20.240 --> 00:06:22.020
And we have
experienced that when

00:06:22.020 --> 00:06:25.565
we take the logarithm of a
generator of moments, what

00:06:25.565 --> 00:06:29.350
we will get is a
generator of Cumulants,

00:06:29.350 --> 00:06:33.460
now starting with l
equals to 1 minus beta

00:06:33.460 --> 00:06:38.180
to the power of l
divided by l factorial U

00:06:38.180 --> 00:06:42.910
to the l0 with a subscript c.

00:06:42.910 --> 00:06:48.880
Of course, this subscript c sort
of captures all of the various

00:06:48.880 --> 00:06:51.600
subtractions that you
have to make in order

00:06:51.600 --> 00:06:54.120
to go from a moment
to a Cumulant.

00:06:54.120 --> 00:06:56.070
But that's quite
general and something

00:06:56.070 --> 00:07:00.720
that we know essentially is
relating log of the expansion

00:07:00.720 --> 00:07:02.120
to the expansion itself.

00:07:02.120 --> 00:07:03.170
But that's well-known.

00:07:05.790 --> 00:07:10.230
So presumably, I can
perturbatively calculate these.

00:07:10.230 --> 00:07:13.410
And I will have,
therefore, progressively

00:07:13.410 --> 00:07:17.130
better approximations
to log Z in the presence

00:07:17.130 --> 00:07:19.810
of this interaction.

00:07:19.810 --> 00:07:22.270
Now at this point,
we have to start

00:07:22.270 --> 00:07:29.150
thinking about a particular
expression for U.

00:07:29.150 --> 00:07:34.850
So let's imagine that our U is a
sum over pairwise interactions.

00:07:34.850 --> 00:07:37.290
So if I think about
the particles,

00:07:37.290 --> 00:07:41.030
there are molecules that
are in the gas in this room.

00:07:41.030 --> 00:07:43.880
Basically, the most important
thing is when the two of them

00:07:43.880 --> 00:07:47.960
approach each other and I have
to put a pairwise interaction

00:07:47.960 --> 00:07:50.710
between them of this form.

00:07:50.710 --> 00:07:53.280
In principle, I could add
three-point and higher-order

00:07:53.280 --> 00:07:54.530
interactions.

00:07:54.530 --> 00:07:57.290
But for all intents
and purposes,

00:07:57.290 --> 00:07:59.140
this should be enough.

00:07:59.140 --> 00:08:04.190
And then again, what I
have here to evaluate,

00:08:04.190 --> 00:08:07.735
the first term
would be minus beta,

00:08:07.735 --> 00:08:14.730
the average of the first
power, and the next one

00:08:14.730 --> 00:08:17.810
will be beta squared over 2.

00:08:17.810 --> 00:08:20.090
For the first Cumulant,
it is the same thing

00:08:20.090 --> 00:08:21.870
as the first moment.

00:08:21.870 --> 00:08:24.550
The second Cumulant is the
thing that is the variance.

00:08:24.550 --> 00:08:28.520
So this would have been
U squared 0 minus average

00:08:28.520 --> 00:08:32.633
of U squared, and then
there will be higher orders.

00:08:38.760 --> 00:08:43.669
So let's calculate these
first two moments explicitly

00:08:43.669 --> 00:08:46.250
for this potential.

00:08:46.250 --> 00:08:54.110
So the first term U in this 0
to order average, what is it?

00:08:54.110 --> 00:08:59.270
Well, that's my U. So I
have to do a sum over i

00:08:59.270 --> 00:09:06.570
and j of V of qi minus qj.

00:09:06.570 --> 00:09:08.380
What does the averaging mean?

00:09:08.380 --> 00:09:14.670
It means I have to integrate
this over all qi divided

00:09:14.670 --> 00:09:19.030
by V. Let's call this qk divided
V and this is [INAUDIBLE].

00:09:22.958 --> 00:09:28.400
OK, so let's say we look at the
first term in this series which

00:09:28.400 --> 00:09:33.390
involves V of q1 minus q2.

00:09:33.390 --> 00:09:36.800
Let's say we pick
that pair, q1 and q2.

00:09:36.800 --> 00:09:45.940
And explicitly, what I have
here is integrals over q1, q2,

00:09:45.940 --> 00:09:54.520
qn, each one of them
coming with a factor of V.

00:09:54.520 --> 00:09:58.100
Now, of course, it doesn't
matter which pair I pick.

00:09:58.100 --> 00:09:59.580
Everything is symmetric.

00:09:59.580 --> 00:10:03.090
The weight that I start
with uniformly is symmetric.

00:10:03.090 --> 00:10:06.510
So clearly, my
choice of this pair

00:10:06.510 --> 00:10:08.500
was arbitrary and
the answer when

00:10:08.500 --> 00:10:12.950
I do the sum is the answer for
1 times the number of pairs,

00:10:12.950 --> 00:10:14.680
which is N N minus 1 over 2.

00:10:19.250 --> 00:10:23.510
Now, over here I have
integrals over q3 or qn,

00:10:23.510 --> 00:10:26.260
et cetera, that don't
appear in the function

00:10:26.260 --> 00:10:27.840
that I'm integrating.

00:10:27.840 --> 00:10:32.100
So all of those integrals
will give me factors of 1.

00:10:32.100 --> 00:10:34.660
And really, the only thing
that I am going to be left with

00:10:34.660 --> 00:10:39.020
is these two integrals
over q1 and q2.

00:10:39.020 --> 00:10:42.310
But even then, the function
that I'm integrating

00:10:42.310 --> 00:10:45.230
is only a function of
the relative coordinate

00:10:45.230 --> 00:10:47.270
that I can call q.

00:10:47.270 --> 00:10:54.470
And then I can, if I want,
integrate over q1 and q.

00:10:54.470 --> 00:10:56.970
The integral over q1
would also give me

00:10:56.970 --> 00:10:59.520
1 when I divide by the volume.

00:10:59.520 --> 00:11:04.280
So the answer is going
to be N N minus 1 over 2,

00:11:04.280 --> 00:11:06.600
which is the number of pairs.

00:11:06.600 --> 00:11:11.420
The integral over the
relative coordinate times

00:11:11.420 --> 00:11:14.330
the potential-- there
is a factor of 1 over V

00:11:14.330 --> 00:11:17.780
that I can put out here.

00:11:17.780 --> 00:11:21.790
So you tell me what the
first pair potential is

00:11:21.790 --> 00:11:25.790
and I tell you that this is the
first correction that you will

00:11:25.790 --> 00:11:29.780
get once you multiply
by minus beta

00:11:29.780 --> 00:11:32.890
to log Z from which you
can calculate the partition

00:11:32.890 --> 00:11:35.450
functions.

00:11:35.450 --> 00:11:36.642
Yes.

00:11:36.642 --> 00:11:39.222
AUDIENCE: How did you turn the
product of two differentials

00:11:39.222 --> 00:11:44.670
d q1 times d q2 into
just one differential?

00:11:44.670 --> 00:11:49.590
PROFESSOR: OK, I have
the integral dq q1 dq q2,

00:11:49.590 --> 00:11:53.000
some function of q1 minus q2.

00:11:53.000 --> 00:11:58.160
I change variables keeping
q1 as one of my variables

00:11:58.160 --> 00:12:03.800
and q [INAUDIBLE] replace
q2 with q minus q2.

00:12:03.800 --> 00:12:06.950
So then I have the
integration over q1.

00:12:06.950 --> 00:12:10.200
I have the integration over q2.

00:12:10.200 --> 00:12:14.160
Not q2, but q f of q.

00:12:14.160 --> 00:12:15.520
And then everything's fine.

00:12:23.280 --> 00:12:25.950
All right, so that's
the first term.

00:12:25.950 --> 00:12:29.140
Let's calculate the next term.

00:12:29.140 --> 00:12:33.540
Next term becomes
somewhat interesting.

00:12:33.540 --> 00:12:39.520
I have to calculate this
quantity as a Cumulant.

00:12:39.520 --> 00:12:41.785
It's the variance.

00:12:45.900 --> 00:12:49.480
V itself is a sum over pairs.

00:12:49.480 --> 00:12:53.200
So to get V squared
to U squared,

00:12:53.200 --> 00:12:56.160
I have to essentially
square this expression.

00:12:56.160 --> 00:12:59.780
So rather than having
sum over one pair ij,

00:12:59.780 --> 00:13:02.690
I have the sum over
two pairs ij and kl.

00:13:05.890 --> 00:13:09.120
And then what I
need to calculate

00:13:09.120 --> 00:13:12.910
is an average that if
I were to square this,

00:13:12.910 --> 00:13:22.230
I will get v of qi minus
qj, v of qk minus ql 0.

00:13:22.230 --> 00:13:25.720
This will be the
average of U squared.

00:13:25.720 --> 00:13:28.900
But when I calculate
the variance,

00:13:28.900 --> 00:13:32.380
I have to subtract from it
the average of U squared.

00:13:32.380 --> 00:13:40.502
So I have to subtract V of
qi minus qj V of qk minus ql.

00:13:49.810 --> 00:13:52.510
Fine.

00:13:52.510 --> 00:13:57.190
So this is just rewriting
of that expression

00:13:57.190 --> 00:14:00.000
in terms of the sum.

00:14:00.000 --> 00:14:06.590
Now, each one of these is
N N minus 1 over two terms.

00:14:06.590 --> 00:14:10.016
And so this, when I sum
over all possibility,

00:14:10.016 --> 00:14:15.640
is the square of N N minus 1
over two possible terms that

00:14:15.640 --> 00:14:20.220
will occur here in the
series that I've written.

00:14:20.220 --> 00:14:26.190
Now, let's group those terms
that can occur in this as

00:14:26.190 --> 00:14:27.620
follows.

00:14:27.620 --> 00:14:33.680
One class of terms
will be when the pair

00:14:33.680 --> 00:14:45.260
ij that I pick for the first sum
is distinct from the pair kl.

00:14:45.260 --> 00:14:49.140
So huge number of terms
as I go through the pairs

00:14:49.140 --> 00:14:52.720
will have this possibility.

00:14:52.720 --> 00:14:55.400
OK, then what happens?

00:14:55.400 --> 00:15:02.640
Then when I calculate the
average of qi minus qj qk minus

00:15:02.640 --> 00:15:09.580
ql, which is this part, what
I need to do is to do all

00:15:09.580 --> 00:15:19.420
of these integral of
this form over all q's.

00:15:19.420 --> 00:15:26.630
And then I have this V of qi
minus qj V of qk minus ql.

00:15:29.745 --> 00:15:32.980
Now, my statement is that
I have, let's say here,

00:15:32.980 --> 00:15:37.450
q1 and q2 and somewhere
else q7 and q8.

00:15:37.450 --> 00:15:39.740
Or, q7 and q974.

00:15:39.740 --> 00:15:41.710
It doesn't matter.

00:15:41.710 --> 00:15:43.880
The point is that
this integral that

00:15:43.880 --> 00:15:45.810
involves this pair
of variables has

00:15:45.810 --> 00:15:47.310
nothing to do with
the integral that

00:15:47.310 --> 00:15:49.650
involves the other
pair of variables.

00:15:49.650 --> 00:15:54.460
So this answer is the same
thing as the average of qi

00:15:54.460 --> 00:16:00.120
minus qj average of qk minus ql.

00:16:00.120 --> 00:16:04.000
Essentially, in the 0 order
probability that we are using,

00:16:04.000 --> 00:16:06.620
the particles are
completely independently

00:16:06.620 --> 00:16:11.040
exploring this space and these
averages will independently

00:16:11.040 --> 00:16:13.160
rely on one pair
and independently

00:16:13.160 --> 00:16:15.380
rely on another pair.

00:16:15.380 --> 00:16:21.060
And once this kind of
factorization occurs,

00:16:21.060 --> 00:16:24.640
it's clear that this
subtraction that I

00:16:24.640 --> 00:16:28.430
need to do for the variance
will get rid of this term.

00:16:28.430 --> 00:16:33.090
So these pairs do
not contribute.

00:16:33.090 --> 00:16:33.765
No contribution.

00:16:38.450 --> 00:16:41.510
If you like, this is this thing.

00:16:45.320 --> 00:16:53.678
All right, so let's do the
next possible term, ij and ik,

00:16:53.678 --> 00:16:57.710
with k not equal to this k.

00:16:57.710 --> 00:17:02.720
So basically, I make the pairs
have one point in common.

00:17:02.720 --> 00:17:05.460
So previously, I had a
pair here, a pair here.

00:17:05.460 --> 00:17:08.390
Now I joined one
of their points.

00:17:08.390 --> 00:17:11.760
So what do I get here?

00:17:11.760 --> 00:17:20.520
I will get for this average
qi minus qj qi minus ql.

00:17:20.520 --> 00:17:24.470
You can see that essentially,
the three integrals that I need

00:17:24.470 --> 00:17:27.060
to worry about are
the integrals that

00:17:27.060 --> 00:17:28.510
involve these three indices.

00:17:28.510 --> 00:17:30.810
All the others will give me 0.

00:17:30.810 --> 00:17:40.370
So this is going to be the
integral qi qj ql divided by V.

00:17:40.370 --> 00:17:47.170
I have V of qi minus
qj V of qi minus ql.

00:17:50.026 --> 00:17:52.237
AUDIENCE: You
switched [INAUDIBLE].

00:17:52.237 --> 00:17:52.820
PROFESSOR: OK.

00:17:58.720 --> 00:18:00.050
I think that's fine.

00:18:04.440 --> 00:18:06.640
All right.

00:18:06.640 --> 00:18:11.150
Now, I can do the same
thing that I did over here.

00:18:11.150 --> 00:18:13.760
I have three variables.

00:18:13.760 --> 00:18:17.370
I will pick qi to stay
as one of my variables.

00:18:17.370 --> 00:18:23.920
I replace qj with this
difference, q j i.

00:18:23.920 --> 00:18:29.440
I replace ql with
its distance to i.

00:18:32.120 --> 00:18:34.610
And again, I can
independently integrate

00:18:34.610 --> 00:18:37.810
over these three variables.

00:18:37.810 --> 00:18:41.520
So this again,
becomes the same thing

00:18:41.520 --> 00:18:50.670
as V of qi minus qj average
V of qi minus ql average.

00:18:50.670 --> 00:18:52.430
And again, there
is no contribution.

00:19:02.430 --> 00:19:06.180
Now, the first class it was
obvious because this pair

00:19:06.180 --> 00:19:09.800
and this pair were
completely distinct.

00:19:09.800 --> 00:19:12.830
This class is a
little bit more subtle

00:19:12.830 --> 00:19:17.300
because I joined one
of the points together,

00:19:17.300 --> 00:19:23.090
and then I used this and
this as well as this point

00:19:23.090 --> 00:19:25.280
as independent variables.

00:19:25.280 --> 00:19:29.450
And I saw that the sum
breaks into pieces.

00:19:29.450 --> 00:19:32.680
And you won't be
surprised if no matter

00:19:32.680 --> 00:19:37.870
how complicated I make various
more interactions over here,

00:19:37.870 --> 00:19:40.780
I can measure all of the
coordinates with respect

00:19:40.780 --> 00:19:44.150
to the single point and
same thing would happen.

00:19:44.150 --> 00:19:52.548
This class of diagrams are
called 1-particle reducible.

00:19:52.548 --> 00:19:56.160
And the first part had to
do with distinct graphs.

00:19:56.160 --> 00:19:58.090
You didn't have to
worry about them.

00:19:58.090 --> 00:20:01.170
But if I were to sort of
convert this expression--

00:20:01.170 --> 00:20:04.150
and we will do so
shortly-- into graphs,

00:20:04.150 --> 00:20:06.575
it corresponds to
graphs where there

00:20:06.575 --> 00:20:10.620
is a point from which
everything else is hanging.

00:20:10.620 --> 00:20:12.760
And measuring
coordinates with respect

00:20:12.760 --> 00:20:18.490
to that point will allow the
average to break into pieces,

00:20:18.490 --> 00:20:23.520
and then to be removed
through this subtraction.

00:20:23.520 --> 00:20:28.190
So this is simple example of
something that is more general.

00:20:28.190 --> 00:20:31.560
So at the level of this
second-order theorem,

00:20:31.560 --> 00:20:34.540
the only type of thing
that will survive

00:20:34.540 --> 00:20:40.030
is if the pairs are
identical, ij and other term

00:20:40.030 --> 00:20:41.780
is also the same pair ij.

00:20:44.380 --> 00:20:46.250
What do I get then?

00:20:46.250 --> 00:20:58.613
I get V of qi minus
qj squared minus V

00:20:58.613 --> 00:21:04.740
of qi minus qj 0 order squared.

00:21:04.740 --> 00:21:09.260
So it's the variance of a
single one of these bond

00:21:09.260 --> 00:21:11.060
contributions.

00:21:11.060 --> 00:21:14.390
If I write that in
terms of integrals,

00:21:14.390 --> 00:21:19.200
this becomes the
integral d cubed q1,

00:21:19.200 --> 00:21:26.580
d cubed q2-- well, qi qj.

00:21:26.580 --> 00:21:30.210
I will have factors
of V. Then, I

00:21:30.210 --> 00:21:39.250
would have V squared qi
minus qj for the first term

00:21:39.250 --> 00:21:51.360
and then the square of something
like this for the second term.

00:21:58.680 --> 00:22:04.560
So putting everything
together up to this order,

00:22:04.560 --> 00:22:05.580
what do I get?

00:22:05.580 --> 00:22:09.530
I will get log Z,
which is log of Z0.

00:22:09.530 --> 00:22:13.560
Let me remind you,
log of Z0 was N log

00:22:13.560 --> 00:22:24.400
of V over lambda cubed N.
That's on form such as this.

00:22:24.400 --> 00:22:26.865
And then I have
these corrections.

00:22:29.490 --> 00:22:33.520
Note that both of the
terms that have survived

00:22:33.520 --> 00:22:37.240
correspond to
looking at one pair.

00:22:37.240 --> 00:22:42.620
So the corrections will be
of the order of N N minus 1

00:22:42.620 --> 00:22:43.850
over 2.

00:22:43.850 --> 00:22:48.000
Because in both cases, I
really look at one pair.

00:22:48.000 --> 00:22:55.230
The first contribution was minus
beta integral d cubed q v of q.

00:22:55.230 --> 00:22:57.600
There was a factor of
1 over V. Actually,

00:22:57.600 --> 00:23:01.270
I can take-- well, let's
put the factor of 1 over V

00:23:01.270 --> 00:23:03.970
here for the time being.

00:23:03.970 --> 00:23:10.290
The next term will be
beta squared over 2

00:23:10.290 --> 00:23:12.740
because I am looking at
the second-order term.

00:23:12.740 --> 00:23:14.701
And what do I have?

00:23:14.701 --> 00:23:19.610
I have the difference
of integral d cubed q

00:23:19.610 --> 00:23:21.240
over V V squared.

00:23:25.010 --> 00:23:32.210
And the square of the integral
of d cubed q over V V.

00:23:32.210 --> 00:23:34.910
And presumably, there
will be higher-order terms

00:23:34.910 --> 00:23:36.367
as we will discuss.

00:23:40.510 --> 00:23:45.760
Now, I'm interested in the
limit of thermodynamics

00:23:45.760 --> 00:23:47.800
where N and V are large.

00:23:51.480 --> 00:23:53.250
And in that limit,
what do I get?

00:23:53.250 --> 00:23:56.800
I will get log Z is log of Z0.

00:24:00.520 --> 00:24:10.340
And then I can see that these
terms I can write as beta.

00:24:10.340 --> 00:24:14.900
N N minus 1 I will
replace will N squared.

00:24:14.900 --> 00:24:18.250
The factors-- here I
have a 1 over V. Here,

00:24:18.250 --> 00:24:20.310
I have 1 over V squared.

00:24:20.310 --> 00:24:23.920
So this factor is
smaller by an amount that

00:24:23.920 --> 00:24:28.560
is order of V in
the large V-limit.

00:24:28.560 --> 00:24:32.770
I will bring the
factor of V outside,

00:24:32.770 --> 00:24:35.310
so I have something like this.

00:24:35.310 --> 00:24:38.090
And then, what do I have here?

00:24:38.090 --> 00:24:43.530
Actually, let's keep
this in this form,

00:24:43.530 --> 00:24:50.455
minus beta integral
d cubed q V plus beta

00:24:50.455 --> 00:24:57.460
squared over 2 integral d
cubed q V squared [INAUDIBLE].

00:25:04.590 --> 00:25:06.690
So the kinds of things
that we are interested

00:25:06.690 --> 00:25:09.980
and we can measure are
the energy of the system,

00:25:09.980 --> 00:25:11.585
but let's say we
focus on pressure.

00:25:15.700 --> 00:25:21.140
And if you look through various
things that we derived before,

00:25:21.140 --> 00:25:23.510
beta times the
pressure you can get

00:25:23.510 --> 00:25:35.220
from taking a derivative
of log Z with respect to V.

00:25:35.220 --> 00:25:39.160
We can express the log Z in
terms of the free energy,

00:25:39.160 --> 00:25:41.940
and then the derivative of free
energy with respect to volume

00:25:41.940 --> 00:25:44.830
will give you the pressure of
the various factors of beta

00:25:44.830 --> 00:25:47.660
[? and signs, ?] we
will come up with this.

00:25:47.660 --> 00:25:53.360
So from the first term, N long
V, we get our ideal gas result.

00:25:53.360 --> 00:25:57.920
Beta p is the same
thing as density.

00:25:57.920 --> 00:26:01.830
And we see that we get a
correction from here when

00:26:01.830 --> 00:26:05.420
I divide by-- when I take
a derivative with respect

00:26:05.420 --> 00:26:09.760
to V of 1 over V, I will
get minus 1 over V squared.

00:26:09.760 --> 00:26:13.040
So the next term would be
the square of the density.

00:26:16.260 --> 00:26:18.310
And then, there is
a series that we

00:26:18.310 --> 00:26:27.650
will encounter which depends
on the interaction potential.

00:26:32.960 --> 00:26:37.070
Now, it turns out that
so far I have calculated

00:26:37.070 --> 00:26:41.400
terms that relied on
only two points, ij.

00:26:41.400 --> 00:26:43.740
And hence, they
become at the end

00:26:43.740 --> 00:26:46.750
proportional to density squared.

00:26:46.750 --> 00:26:48.930
If I go further
in my expansion, I

00:26:48.930 --> 00:26:52.770
will encounter things that I
will need triangular points.

00:26:52.770 --> 00:26:56.470
For example, i j k
forming a triangle.

00:26:56.470 --> 00:26:58.490
And then, that would
give me something

00:26:58.490 --> 00:27:00.750
that would be of the order
of the density cubed.

00:27:03.300 --> 00:27:11.030
So somehow, I can also see
that ultimately this series

00:27:11.030 --> 00:27:14.080
in perturbation theory,
as we shall see,

00:27:14.080 --> 00:27:18.980
also more precisely can be
organized in powers of density.

00:27:18.980 --> 00:27:21.210
Why is that important?

00:27:21.210 --> 00:27:23.930
Because typically,
as we discussed right

00:27:23.930 --> 00:27:27.580
at the beginning, when you
look at the pressure of a gas

00:27:27.580 --> 00:27:31.050
when it is dilute and
over V is very small,

00:27:31.050 --> 00:27:34.140
it is always ideal gas-like.

00:27:34.140 --> 00:27:36.620
As you make it more
dense, you start

00:27:36.620 --> 00:27:41.300
to get corrections that you can
express in powers of density.

00:27:41.300 --> 00:27:45.020
And the coefficients of that
are called Virial coefficients.

00:27:45.020 --> 00:27:46.810
So in some sense,
we have already

00:27:46.810 --> 00:27:50.850
started in calculating the
second Virial coefficient,

00:27:50.850 --> 00:27:53.620
and there are higher-order
Virial coefficients,

00:27:53.620 --> 00:27:56.960
which together give you the
equation of states relating

00:27:56.960 --> 00:28:01.059
pressure to density
by a power series.

00:28:01.059 --> 00:28:01.850
AUDIENCE: Question.

00:28:01.850 --> 00:28:02.632
PROFESSOR: Yes.

00:28:02.632 --> 00:28:04.048
AUDIENCE: The
Virial coefficients,

00:28:04.048 --> 00:28:06.670
if they have integral
over the whole volume,

00:28:06.670 --> 00:28:09.956
then they will be
functions of volume, right?

00:28:09.956 --> 00:28:10.580
PROFESSOR: Yes.

00:28:10.580 --> 00:28:14.170
But imagine that I'm thinking
about the gas in this room

00:28:14.170 --> 00:28:16.590
and the interaction of
two oxygen molecules.

00:28:16.590 --> 00:28:20.290
So what I am saying is you
pick an oxygen molecule.

00:28:20.290 --> 00:28:22.940
There is interaction and
I have to integrate over

00:28:22.940 --> 00:28:25.530
where the other
oxygen molecule is.

00:28:25.530 --> 00:28:28.250
And by the time it
is tiny, tiny bit

00:28:28.250 --> 00:28:30.800
[? away, ?] the
interaction is 0.

00:28:30.800 --> 00:28:33.886
It doesn't really matter.

00:28:33.886 --> 00:28:36.353
AUDIENCE: If the characteristic
volume of interaction

00:28:36.353 --> 00:28:37.561
is much smaller than volume--

00:28:37.561 --> 00:28:38.741
PROFESSOR: Of the space.

00:28:38.741 --> 00:28:41.032
AUDIENCE: [INAUDIBLE], then
the integrals just converge

00:28:41.032 --> 00:28:41.660
to [? constant. ?]

00:28:41.660 --> 00:28:42.660
PROFESSOR: That's right.

00:28:42.660 --> 00:28:45.170
So if you want to be
even more precise,

00:28:45.170 --> 00:28:48.610
you will get corrections
when your particles are

00:28:48.610 --> 00:28:50.840
close to the wall and
uniformity, et cetera,

00:28:50.840 --> 00:28:52.790
is violated.

00:28:52.790 --> 00:28:55.740
So there will, actually,
be corrections to, say, log

00:28:55.740 --> 00:29:00.800
Z that are not only proportional
ultimately to number volume,

00:29:00.800 --> 00:29:03.620
but the area of the enclosure
and other [? subleading ?]

00:29:03.620 --> 00:29:04.270
factors.

00:29:04.270 --> 00:29:07.740
But in the thermodynamic
limit, we ignore all of that.

00:29:07.740 --> 00:29:11.860
So a lot of that is resolved
by this statement here.

00:29:23.240 --> 00:29:28.510
Again, you won't be surprised
that if I were to go ahead,

00:29:28.510 --> 00:29:30.740
then there will be a
term in this series that

00:29:30.740 --> 00:29:34.280
would be beta cubed over 3
factorial integral over q

00:29:34.280 --> 00:29:36.740
of vq, et cetera.

00:29:36.740 --> 00:29:43.290
And that's actually very good
because as I have written

00:29:43.290 --> 00:29:48.850
for you currently, this
expression is totally useless.

00:29:48.850 --> 00:29:50.580
Why is it useless?

00:29:50.580 --> 00:29:55.820
Because let's think back
about these two oxygen

00:29:55.820 --> 00:29:58.130
molecules in the
gas in this room

00:29:58.130 --> 00:30:01.700
and what the potential of
interaction between them

00:30:01.700 --> 00:30:04.960
would look like.

00:30:04.960 --> 00:30:08.340
Presumably, it is a function
of the relative separation.

00:30:08.340 --> 00:30:12.650
We can call it r or
q, it doesn't matter.

00:30:12.650 --> 00:30:19.260
And if you bring them
closer than the typical size

00:30:19.260 --> 00:30:24.610
of these molecules, their
potential will go to infinity.

00:30:24.610 --> 00:30:29.440
So basically, they don't want
to come close to each other.

00:30:29.440 --> 00:30:31.820
If you go very far
away, typically you

00:30:31.820 --> 00:30:33.660
have the van der
Waals attraction

00:30:33.660 --> 00:30:35.440
between the particles.

00:30:35.440 --> 00:30:37.280
So out here, it is attractive.

00:30:37.280 --> 00:30:39.820
The potential is
negative and falls off,

00:30:39.820 --> 00:30:42.980
typically, as 1 over
r to the 6 related

00:30:42.980 --> 00:30:46.360
to the polarizabilities
of the particles.

00:30:46.360 --> 00:30:51.570
And these, when you come
very short distances,

00:30:51.570 --> 00:30:54.830
the electronic
clouds will overlap

00:30:54.830 --> 00:30:57.910
and the potential
goes to infinity.

00:30:57.910 --> 00:31:02.310
And if you want to sort of think
about numbers, typical scales

00:31:02.310 --> 00:31:05.120
that we have here--
let's say here,

00:31:05.120 --> 00:31:08.730
are presumably of the
order of angstroms.

00:31:08.730 --> 00:31:11.900
And the typical depths
of these potentials

00:31:11.900 --> 00:31:16.720
in units that make sense
to statistical physics

00:31:16.720 --> 00:31:20.370
are of the order of
100 degrees Kelvin.

00:31:20.370 --> 00:31:26.370
So it's why typical
gases liquidy

00:31:26.370 --> 00:31:29.020
at range of temperatures
that is of the order of 100

00:31:29.020 --> 00:31:31.790
degrees Kelvin.

00:31:31.790 --> 00:31:37.020
Now, if I want to take this
potential and calculate this,

00:31:37.020 --> 00:31:38.920
I have trouble.

00:31:38.920 --> 00:31:44.000
Because the integral over
here will give me infinity.

00:31:44.000 --> 00:31:47.620
So this expression
is, in general, fine.

00:31:47.620 --> 00:31:49.780
But whenever you have
perturbation theory,

00:31:49.780 --> 00:31:51.670
you start to evaluate
things and you

00:31:51.670 --> 00:31:54.430
have to see whether
the correction indeed

00:31:54.430 --> 00:31:56.100
is sufficiently small.

00:31:56.100 --> 00:31:58.500
And it seems we can't make
it sufficiently small.

00:31:58.500 --> 00:31:59.384
Yes.

00:31:59.384 --> 00:32:05.312
AUDIENCE: If you're modeling
interatomic or intermolecular

00:32:05.312 --> 00:32:07.782
potential as a
Lennard-Jones potential,

00:32:07.782 --> 00:32:11.960
isn't the [INAUDIBLE]
term at the--

00:32:11.960 --> 00:32:17.150
or near the origin, a
phenomenological choice?

00:32:17.150 --> 00:32:20.210
PROFESSOR: If you are
using as a formula

00:32:20.210 --> 00:32:22.390
for this the
Lennard-Jones potential,

00:32:22.390 --> 00:32:24.820
people phenomenologically
write something

00:32:24.820 --> 00:32:27.780
like 1 over r to the
12th power, et cetera.

00:32:27.780 --> 00:32:30.153
But I don't have to assume that.

00:32:30.153 --> 00:32:30.980
AUDIENCE: Right.

00:32:30.980 --> 00:32:33.842
Because I've seen forms of
the Lennard-Jones potential

00:32:33.842 --> 00:32:37.500
which is exponentially
decaying at the origin.

00:32:37.500 --> 00:32:41.890
So it reaches a finite,
albeit probably high value

00:32:41.890 --> 00:32:42.840
at the origin.

00:32:42.840 --> 00:32:46.620
Wouldn't that take
care of the issue?

00:32:46.620 --> 00:32:49.740
PROFESSOR: If you believe in
that kind of potential, yes.

00:32:49.740 --> 00:32:52.650
So what you are
saying is that if I

00:32:52.650 --> 00:32:55.000
make this come to
a finite value,

00:32:55.000 --> 00:32:58.250
I will be able to find
the temperature that

00:32:58.250 --> 00:33:01.170
is sufficiently high
compared to this,

00:33:01.170 --> 00:33:02.610
and then things would be fine.

00:33:06.820 --> 00:33:09.750
If we had that, then we would
have fusion right here going

00:33:09.750 --> 00:33:10.930
on, right?

00:33:10.930 --> 00:33:14.950
So clearly, there
is some better truth

00:33:14.950 --> 00:33:18.520
to a potential that is really
very, very high compared

00:33:18.520 --> 00:33:21.940
to temperatures that
we are looking at.

00:33:21.940 --> 00:33:25.790
So we don't have to
worry about that.

00:33:25.790 --> 00:33:28.256
Well, we have to worry
about this issue.

00:33:28.256 --> 00:33:30.570
OK?

00:33:30.570 --> 00:33:31.930
All right.

00:33:31.930 --> 00:33:36.880
So you say, well, how about
the following-- let's say

00:33:36.880 --> 00:33:39.940
that I am something like
the gas in this room

00:33:39.940 --> 00:33:42.700
and I am sufficiently dilute.

00:33:42.700 --> 00:33:46.010
I'm going to forget
about these terms.

00:33:46.010 --> 00:33:50.880
But since this potential
becomes essentially large,

00:33:50.880 --> 00:33:55.550
I can never ignore a
particular term in this series.

00:33:55.550 --> 00:33:59.710
And I keep going adding more
and more terms in the series.

00:33:59.710 --> 00:34:02.660
If I sort of think
back about the picture

00:34:02.660 --> 00:34:07.930
that I was generating before
diagrammatically-- which again,

00:34:07.930 --> 00:34:11.780
we will clarify
later on-- this term

00:34:11.780 --> 00:34:14.800
corresponds to taking a
pair of points and a V

00:34:14.800 --> 00:34:17.449
that connects them.

00:34:17.449 --> 00:34:21.080
And this term
corresponds to two V's

00:34:21.080 --> 00:34:24.090
going between the
same pair of points.

00:34:24.090 --> 00:34:26.020
And you say, well,
there will be terms

00:34:26.020 --> 00:34:30.000
that will involve three
V's, four V's, and they

00:34:30.000 --> 00:34:35.949
will contribute in the series in
a manner that I can recognize.

00:34:35.949 --> 00:34:40.350
So why don't I add all
of those terms together?

00:34:40.350 --> 00:34:46.699
And if you like, we can call
the resulting object something

00:34:46.699 --> 00:34:49.340
else.

00:34:49.340 --> 00:34:51.219
What is that something else?

00:34:51.219 --> 00:34:55.585
Well, what I have done
is I have minus beta V.

00:34:55.585 --> 00:34:58.430
Well, all of them--
so what is this?

00:34:58.430 --> 00:35:03.120
It is the integral d cubed q.

00:35:03.120 --> 00:35:11.740
I have minus beta V
plus 1/2 beta V squared.

00:35:11.740 --> 00:35:13.840
The next term is
going to be minus 1

00:35:13.840 --> 00:35:18.240
over 3 factorial beta V
cubed because it will come

00:35:18.240 --> 00:35:21.050
from the third-order
term in the expansion.

00:35:21.050 --> 00:35:23.740
And I can go on.

00:35:23.740 --> 00:35:27.850
And you say, well,
obviously this whole thing

00:35:27.850 --> 00:35:31.820
came from the expansion
of e to the minus beta V

00:35:31.820 --> 00:35:35.960
except that I don't have
the 0 order term, 1.

00:35:35.960 --> 00:35:44.070
So this quantity that I will
call f of q rather than V of q

00:35:44.070 --> 00:35:49.740
is obtained as e to the
minus beta V of q minus 1.

00:35:56.220 --> 00:36:00.420
So if I were to, in fact,
add up an infinite number

00:36:00.420 --> 00:36:03.500
of terms in the series
rather than having

00:36:03.500 --> 00:36:07.770
to integrate V V squared
V cubed, each one of them

00:36:07.770 --> 00:36:09.390
is divergence.

00:36:09.390 --> 00:36:12.600
I have to integrate this f of q.

00:36:12.600 --> 00:36:17.980
So let's see what this f of
q-- or correspondingly, f of r

00:36:17.980 --> 00:36:19.670
looks like as a function of r.

00:36:22.910 --> 00:36:28.970
So in the range that
I have my hard core

00:36:28.970 --> 00:36:35.650
and V is large and positive,
this is going to give me 0.

00:36:35.650 --> 00:36:41.898
So the function is
minus 1 down here.

00:36:41.898 --> 00:36:49.480
At exactly whatever point it
is that the potential is 0,

00:36:49.480 --> 00:36:51.200
then f is also 0.

00:36:51.200 --> 00:36:54.505
So basically, I will
come to the same 0.

00:36:57.180 --> 00:37:03.570
When the potential is negative,
this will be larger than 1.

00:37:03.570 --> 00:37:05.530
My f will be positive.

00:37:05.530 --> 00:37:07.340
So presumably, it
will have some kind

00:37:07.340 --> 00:37:11.400
of a peak around
where this peak is.

00:37:11.400 --> 00:37:15.690
And then at large distances,
it goes back towards 0.

00:37:15.690 --> 00:37:21.065
So this is going to go back
towards 1 minus 1, which is 0.

00:37:21.065 --> 00:37:23.100
So actually, this
end of the potential

00:37:23.100 --> 00:37:25.243
is the same thing as minus beta.

00:37:30.800 --> 00:37:36.330
And whereas I couldn't integrate
any one term in this series,

00:37:36.330 --> 00:37:40.460
I can certainly integrate
the sum over all positions

00:37:40.460 --> 00:37:43.320
and it will give
me a number, which

00:37:43.320 --> 00:37:45.500
will depend on
temperature, the properties

00:37:45.500 --> 00:37:47.790
of the potential, et cetera.

00:37:47.790 --> 00:37:51.080
And it is that number
that will tell me

00:37:51.080 --> 00:37:54.660
what the density squared term in
the expression for the pressure

00:37:54.660 --> 00:37:59.735
is and what the second
Virial coefficient is.

00:37:59.735 --> 00:38:00.235
Yes.

00:38:00.235 --> 00:38:03.894
AUDIENCE: Is this basically
an excluded volume of sorts?

00:38:03.894 --> 00:38:04.560
PROFESSOR: Yeah.

00:38:04.560 --> 00:38:06.870
So this part if I
sent it to infinity,

00:38:06.870 --> 00:38:09.040
corresponds to an
excluded volume.

00:38:09.040 --> 00:38:12.670
So if you want really excluded
volume, this would be minus 1

00:38:12.670 --> 00:38:14.309
up to some point.

00:38:14.309 --> 00:38:15.225
AUDIENCE: [INAUDIBLE].

00:38:15.225 --> 00:38:15.891
PROFESSOR: Yeah.

00:38:15.891 --> 00:38:18.800
But I'm talking
about very general,

00:38:18.800 --> 00:38:21.730
so I would say that
excluded volume can

00:38:21.730 --> 00:38:24.095
be very easily captured
within this [INAUDIBLE].

00:38:31.720 --> 00:38:34.400
All right.

00:38:34.400 --> 00:38:40.340
So we want to sort of
follow-up from here.

00:38:40.340 --> 00:38:42.370
But since you will
have a problem set,

00:38:42.370 --> 00:38:46.150
I will mention one other thing.

00:38:46.150 --> 00:38:50.640
That this reorganization of
the series that I made here

00:38:50.640 --> 00:38:54.220
is appropriate to the limit
of low densities, where

00:38:54.220 --> 00:38:58.950
I would have a nice expansion
in powers of density.

00:38:58.950 --> 00:39:01.100
The problem that
you will deal with

00:39:01.100 --> 00:39:04.370
has to do with plasmas,
where the interaction

00:39:04.370 --> 00:39:06.350
range is very large.

00:39:06.350 --> 00:39:09.710
And you already saw
something along those lines

00:39:09.710 --> 00:39:12.350
when we had the Vlasov
equation as opposed

00:39:12.350 --> 00:39:13.900
to Boltzmann equation.

00:39:13.900 --> 00:39:16.450
There was a regime where
you had to reorganize

00:39:16.450 --> 00:39:17.970
the series different ways.

00:39:17.970 --> 00:39:21.720
In that case, it was
the BBGKY hierarchy

00:39:21.720 --> 00:39:25.205
whether you were looking at the
dense limit or a dilute limit.

00:39:25.205 --> 00:39:28.890
So this is the analog of where
the Boltzmann equation would

00:39:28.890 --> 00:39:30.640
have been inappropriate.

00:39:30.640 --> 00:39:32.550
The analog of the
regime where you

00:39:32.550 --> 00:39:43.020
are dense and something
like the Vlasov equation

00:39:43.020 --> 00:39:44.550
would be appropriate.

00:39:44.550 --> 00:39:47.150
So there is some kind
of interaction range,

00:39:47.150 --> 00:39:50.770
n d cubed if it is
much larger than 1.

00:39:50.770 --> 00:39:55.100
Then it turns out that rather
than looking at diagrams that

00:39:55.100 --> 00:39:58.630
have the fewest number of
points-- in this case, 2--

00:39:58.630 --> 00:40:00.420
you have to look
at diagrams that

00:40:00.420 --> 00:40:02.570
have the largest
number of points.

00:40:02.570 --> 00:40:05.200
Because each additional
point will give you

00:40:05.200 --> 00:40:07.960
an additional
factor of N over V.

00:40:07.960 --> 00:40:13.140
You can see that these
factors of N N minus 1 over 2

00:40:13.140 --> 00:40:15.810
came from the number of
points that I had selected.

00:40:15.810 --> 00:40:17.850
So if I had to
select three points,

00:40:17.850 --> 00:40:23.100
I would have N cubed and the
corresponding V. So the more

00:40:23.100 --> 00:40:25.010
points that I have
in my series, I

00:40:25.010 --> 00:40:27.540
will have more powers
of the density.

00:40:27.540 --> 00:40:29.870
So in that case, it
turns out that rather

00:40:29.870 --> 00:40:33.910
than looking at two points
and all lines between them,

00:40:33.910 --> 00:40:36.040
you organize things
in terms of what

00:40:36.040 --> 00:40:40.910
are called ring
diagrams, which are

00:40:40.910 --> 00:40:46.330
things such as this, this, this.

00:40:46.330 --> 00:40:54.090
So basically, for a
given number of lines,

00:40:54.090 --> 00:40:59.270
the most number of points is
obtained by creating your ring.

00:40:59.270 --> 00:41:03.980
And so one of the
problems that you have

00:41:03.980 --> 00:41:07.510
is to sort of sum these ring
diagrams and see what happens.

00:41:13.900 --> 00:41:17.150
But it seems like what
I'm telling you here

00:41:17.150 --> 00:41:20.100
is while we calculated
order of density squared,

00:41:20.100 --> 00:41:23.370
but maybe I want to calculate
order of density cubed.

00:41:23.370 --> 00:41:27.180
And it makes much more
sense rather than when

00:41:27.180 --> 00:41:30.910
faced with potentials
of this form arranging

00:41:30.910 --> 00:41:34.310
the series in powers of
the potential to arrange it

00:41:34.310 --> 00:41:40.490
in powers of this quantity e
to the minus beta V minus 1.

00:41:40.490 --> 00:41:45.640
So let's go through a
route in which I directly

00:41:45.640 --> 00:41:49.850
expand everything in
terms of this quantity.

00:41:49.850 --> 00:41:55.440
And that's the second
type of expansion

00:41:55.440 --> 00:41:58.596
that I will call the
cluster expansion.

00:42:05.750 --> 00:42:09.470
So once more, what
we want to calculate

00:42:09.470 --> 00:42:11.110
is the partition function.

00:42:11.110 --> 00:42:14.030
It depends on temperature,
volume, number

00:42:14.030 --> 00:42:19.200
of particles obtained
by integrating over

00:42:19.200 --> 00:42:21.580
all degrees of freedom.

00:42:21.580 --> 00:42:25.190
The integration over momenta
we saw is very simple.

00:42:25.190 --> 00:42:30.710
Ultimately, will give us this
N factorial lambda to the 3N.

00:42:30.710 --> 00:42:33.530
This time, I won't pull
out the factor of V

00:42:33.530 --> 00:42:36.380
to the N at this point.

00:42:36.380 --> 00:42:43.130
And then I have the integration
over all of these q's.

00:42:43.130 --> 00:42:46.180
I did not divide by
V because I did not

00:42:46.180 --> 00:42:51.030
multiply by V to the N. OK.

00:42:51.030 --> 00:42:54.170
And then I have e
to the minus beta U,

00:42:54.170 --> 00:43:00.190
but this is my U. U is
the sum of various terms.

00:43:00.190 --> 00:43:06.630
So e to the minus beta U
will be a product over all,

00:43:06.630 --> 00:43:15.068
let's say-- let's
call this k pairs ij.

00:43:15.068 --> 00:43:20.480
e to the minus beta
V of qi minus qj.

00:43:23.380 --> 00:43:27.020
So basically, the
only thing that I did

00:43:27.020 --> 00:43:29.860
was e to the minus
beta-- this quantity

00:43:29.860 --> 00:43:32.220
I wrote as a product
of contributions

00:43:32.220 --> 00:43:33.400
from the different pairs.

00:43:38.490 --> 00:43:42.810
Now, this quantity I have
it in the line above.

00:43:42.810 --> 00:43:45.560
This is also a
relative position.

00:43:45.560 --> 00:43:50.650
Clearly, I can
think of this as 1

00:43:50.650 --> 00:43:55.420
plus the f that would correspond
to the distance between qi

00:43:55.420 --> 00:43:57.310
and qj.

00:43:57.310 --> 00:44:02.510
And again, I can either write
it as Vf of qi minus qj,

00:44:02.510 --> 00:44:07.630
or simplify of i-- my notation--
and write it as f i j.

00:44:07.630 --> 00:44:13.900
So f i j means-- so let
me maybe write it here.

00:44:13.900 --> 00:44:18.020
f i j I have defined to
be e to the minus beta

00:44:18.020 --> 00:44:21.518
V of qi minus qj minus 1.

00:44:28.190 --> 00:44:31.210
So what do I expect?

00:44:31.210 --> 00:44:33.760
I saw that the first
correction to something

00:44:33.760 --> 00:44:39.150
that was interesting to me
had one power of f in it

00:44:39.150 --> 00:44:41.480
that I had to
ultimately integrate.

00:44:41.480 --> 00:44:46.410
So maybe what I should do is I
should start organizing things

00:44:46.410 --> 00:44:49.520
in terms of how many f's I have.

00:44:49.520 --> 00:44:52.970
So an expansion in powers of f.

00:44:52.970 --> 00:44:54.610
So what's going to happen here?

00:44:54.610 --> 00:44:59.100
I would have 1 over N
factorial lambda to the 3N.

00:44:59.100 --> 00:45:05.460
I have a product of all
of the integrations.

00:45:05.460 --> 00:45:10.010
And I have all of these
factors of 1 plus f1 2 times 1.

00:45:10.010 --> 00:45:15.240
So basically, this is maybe--
it's really a product of 1

00:45:15.240 --> 00:45:20.460
plus f1 2, 1 plus f1 3, 1 plus
all of one of these things.

00:45:20.460 --> 00:45:23.690
So the thing that has
least number of f's is

00:45:23.690 --> 00:45:26.570
when I pick the 1 from
all of these brackets

00:45:26.570 --> 00:45:29.240
that are multiplying each other.

00:45:29.240 --> 00:45:37.690
The next term is to
pick 1f from one pair

00:45:37.690 --> 00:45:40.580
and all the others would be 1.

00:45:40.580 --> 00:45:49.840
The next term would be sum
over ij kl f i j f k l,

00:45:49.840 --> 00:45:53.270
and then there will be diagrams
that would progressively

00:45:53.270 --> 00:45:55.790
have more and more factors of f.

00:46:05.670 --> 00:46:09.350
Now, what I will do is to
represent the various terms

00:46:09.350 --> 00:46:11.546
that I generate in this
series diagrammatically.

00:46:25.440 --> 00:46:30.790
So first of all, I have to
integrate over N points.

00:46:30.790 --> 00:46:43.925
So I put points 1 through
N. So I have 1, 2, 3, 4.

00:46:43.925 --> 00:46:48.980
It doesn't matter
how I put them-- N.

00:46:48.980 --> 00:46:53.840
And then, this 1
would correspond

00:46:53.840 --> 00:46:56.970
to just this diagram.

00:46:56.970 --> 00:47:03.330
The next thing is I
put a line for f i j.

00:47:03.330 --> 00:47:05.700
So let's say that I
picked in this series

00:47:05.700 --> 00:47:07.400
the term that was f 2 3.

00:47:07.400 --> 00:47:13.180
I will represent that by a
line that goes between 2 and 3.

00:47:13.180 --> 00:47:16.930
Later on in the
picture, maybe I will

00:47:16.930 --> 00:47:21.580
pick a term that is f 2 3 f 4 5.

00:47:21.580 --> 00:47:26.130
So some second-order term
could be something like this.

00:47:26.130 --> 00:47:28.510
Maybe later on in
the series, I will

00:47:28.510 --> 00:47:31.120
pick a term that is
connecting f3 and f4.

00:47:31.120 --> 00:47:34.760
So this would be a third-order
term in the series.

00:47:34.760 --> 00:47:40.540
Maybe later on the series, I
have some other pair over here.

00:47:40.540 --> 00:47:44.870
So any one of the
huge number of terms

00:47:44.870 --> 00:47:48.460
that I have generated
here has one diagram

00:47:48.460 --> 00:47:50.020
that is appearing over here.

00:47:53.120 --> 00:47:56.630
Now, the next step is, of course
I have to integrate over all

00:47:56.630 --> 00:47:57.440
of the q's.

00:47:57.440 --> 00:48:00.520
I mean, these diagrams, what
they represent is essentially

00:48:00.520 --> 00:48:04.130
some f of q1 minus q3,
some f of whatever.

00:48:04.130 --> 00:48:07.570
And I have to integrate
over all of these q's.

00:48:07.570 --> 00:48:11.290
So the contribution to
the partition function

00:48:11.290 --> 00:48:14.770
that I will get would
be some value associated

00:48:14.770 --> 00:48:17.140
with one of these
graphs obtained by doing

00:48:17.140 --> 00:48:18.420
the integrations over q.

00:48:21.560 --> 00:48:24.870
Now, the thing that
I want you to notice

00:48:24.870 --> 00:48:38.760
is that contribution of one
of these graphs or diagrams

00:48:38.760 --> 00:48:50.640
is the product of contributions
of its linked clusters.

00:48:59.190 --> 00:49:02.910
So here I have one
particular diagram

00:49:02.910 --> 00:49:04.350
that is represented here.

00:49:04.350 --> 00:49:10.390
Well, let's say 2, 3,
4, 5 are linked together

00:49:10.390 --> 00:49:12.460
and separate from
whatever this is.

00:49:12.460 --> 00:49:13.950
Let's say this is 7, 9.

00:49:16.640 --> 00:49:21.520
So when I do the integrations
over q2, q3, q4, q5,

00:49:21.520 --> 00:49:25.530
I don't really rely on
any of the other things.

00:49:25.530 --> 00:49:28.120
So my integration
here, we break off

00:49:28.120 --> 00:49:29.760
into the integration
that involves

00:49:29.760 --> 00:49:32.690
this, the integration that
involves this, as well

00:49:32.690 --> 00:49:35.060
as integrations over
all of these points

00:49:35.060 --> 00:49:36.650
that are not
connected to anybody.

00:49:36.650 --> 00:49:41.520
All of those will simply
give me a factor of v.

00:49:41.520 --> 00:49:45.110
So is everybody happy
with this simple statement

00:49:45.110 --> 00:49:49.070
that the contribution-- if I
think of these linked clusters

00:49:49.070 --> 00:49:52.940
as some collection of points
that are linked together,

00:49:52.940 --> 00:49:55.100
the value that I would
get for this diagram

00:49:55.100 --> 00:49:58.365
would be the contribution--
product of the contributions

00:49:58.365 --> 00:49:59.490
that I would have for this.

00:50:04.060 --> 00:50:05.590
So yes?

00:50:05.590 --> 00:50:07.417
AUDIENCE: Can you
explain [INAUDIBLE]?

00:50:07.417 --> 00:50:08.000
PROFESSOR: OK.

00:50:08.000 --> 00:50:13.060
So let's pick a
particular thing.

00:50:13.060 --> 00:50:16.510
Let's say we do
something like this,

00:50:16.510 --> 00:50:25.795
where this is number
7, 245, 6, 5, 4.

00:50:25.795 --> 00:50:26.920
This is a particular thing.

00:50:26.920 --> 00:50:28.230
So what do I have to do?

00:50:28.230 --> 00:50:33.020
There's also a whole
bunch of points-- 1, 2, 3.

00:50:33.020 --> 00:50:36.980
What I am instructed to
do is to integrate over

00:50:36.980 --> 00:50:44.490
q1, q2, q3, et cetera.

00:50:44.490 --> 00:50:49.240
Now, the integral over q1
is just the integral of q1.

00:50:49.240 --> 00:50:52.110
So this integral by
itself will give me

00:50:52.110 --> 00:50:57.190
V. Any number of points
that are by themselves

00:50:57.190 --> 00:51:02.950
will also give me
V. So the first term

00:51:02.950 --> 00:51:04.870
that becomes
nontrivial is when I

00:51:04.870 --> 00:51:23.020
have to do the integration over
q4, q5, q6 of f 4 5, f 5 6,

00:51:23.020 --> 00:51:24.010
f 6 4.

00:51:26.990 --> 00:51:31.350
I don't know what
this is, but there

00:51:31.350 --> 00:51:33.960
is something that
will come from here.

00:51:33.960 --> 00:51:37.570
Then, later on I have
to integrate over 7

00:51:37.570 --> 00:51:39.310
and something else.

00:51:39.310 --> 00:51:44.652
So then I have the integral
over q7, q-- let's call this 8.

00:51:49.380 --> 00:51:50.760
f of 7 8.

00:51:50.760 --> 00:51:53.910
So this is something else.

00:51:53.910 --> 00:51:59.460
So the overall value of this
term in the perturbation theory

00:51:59.460 --> 00:52:02.800
is the product of contributions
from a huge number of one

00:52:02.800 --> 00:52:04.690
clusters.

00:52:04.690 --> 00:52:06.070
Here, I have a two cluster.

00:52:06.070 --> 00:52:07.740
Here, I have a three cluster.

00:52:07.740 --> 00:52:09.360
Maybe I will have more things.

00:52:13.410 --> 00:52:20.140
Also, notice that if I have
more of these two clusters,

00:52:20.140 --> 00:52:24.350
the result for them would
be exactly the same.

00:52:24.350 --> 00:52:27.550
So if I have lots of these
pairs, the same way that I

00:52:27.550 --> 00:52:30.080
had lots of single
points-- and it became V

00:52:30.080 --> 00:52:32.020
to the number of
single points-- it

00:52:32.020 --> 00:52:34.820
will become whatever that
y is to the number of pair

00:52:34.820 --> 00:52:38.530
points, x, whatever that is,
to the number of triplets

00:52:38.530 --> 00:52:40.680
that I have in triangles.

00:52:40.680 --> 00:52:44.970
So this is how this
object is built up.

00:52:44.970 --> 00:52:45.680
Yes?

00:52:45.680 --> 00:52:48.580
AUDIENCE: How are we
deciding how many triplets

00:52:48.580 --> 00:52:51.226
and how many pairs
we actually have?

00:52:51.226 --> 00:52:53.100
PROFESSOR: At this point,
I haven't told you.

00:52:53.100 --> 00:52:55.940
So there is a multiplicity
that we have to calculate.

00:52:55.940 --> 00:52:57.500
Yes.

00:52:57.500 --> 00:53:00.080
So at this point, all
I'm saying is given

00:53:00.080 --> 00:53:03.099
this, the answer is the
product of contribution.

00:53:03.099 --> 00:53:05.015
There is a multiplicity
factor-- you're right.

00:53:05.015 --> 00:53:07.185
We have to calculate that.

00:53:09.980 --> 00:53:11.780
OK?

00:53:11.780 --> 00:53:14.715
Anything else?

00:53:14.715 --> 00:53:16.040
All right.

00:53:16.040 --> 00:53:21.300
So I mean, think of
this as taking a number

00:53:21.300 --> 00:53:24.850
and writing it as the
product of its prime factors,

00:53:24.850 --> 00:53:27.500
factorizing into primes.

00:53:27.500 --> 00:53:31.770
So you sort of immediately know
that somehow prime factors,

00:53:31.770 --> 00:53:34.730
the prime numbers,
are more important.

00:53:34.730 --> 00:53:37.920
Because everything else you
can write as prime numbers.

00:53:37.920 --> 00:53:40.530
So clearly, what is
also buried in heart

00:53:40.530 --> 00:53:46.610
of calculating this partition
function is these clusters.

00:53:46.610 --> 00:53:56.950
So let me define the analog
of prime numbers as follows.

00:53:56.950 --> 00:54:08.870
I define bl to be the
sum over contributions

00:54:08.870 --> 00:54:17.430
of all linked
clusters of l-points.

00:54:39.350 --> 00:54:42.050
So let's go term by term.

00:54:42.050 --> 00:54:46.480
b1 corresponds essentially
to the one point

00:54:46.480 --> 00:54:50.460
by itself in the diagram
that I was drawing down here.

00:54:50.460 --> 00:54:53.970
And corresponds to
integrating over

00:54:53.970 --> 00:54:56.890
the coordinate that
goes all over the space.

00:54:56.890 --> 00:55:03.330
And hence, the volume is the
same thing as the volume V.

00:55:03.330 --> 00:55:08.810
b2 is a cluster of two points.

00:55:08.810 --> 00:55:11.160
So it is this.

00:55:11.160 --> 00:55:16.300
And so it is the
integral over q1 and q2,

00:55:16.300 --> 00:55:24.910
which are two endpoints of
this, of f of q1 minus q2.

00:55:24.910 --> 00:55:27.430
And this we have seen
already many times.

00:55:27.430 --> 00:55:33.550
It's the same thing as volume,
the integral over q f of q.

00:55:33.550 --> 00:55:36.270
It is the thing that was
related to our second Virial

00:55:36.270 --> 00:55:36.770
coefficient.

00:55:40.040 --> 00:55:43.600
Now, this is very
important for b3.

00:55:46.640 --> 00:55:52.040
I note that I
underlined here "all."

00:55:52.040 --> 00:55:54.580
And this is for
later convenience.

00:55:54.580 --> 00:55:57.610
It seems that I'm essentially
pushing complexity

00:55:57.610 --> 00:56:00.080
from one place to another.

00:56:00.080 --> 00:56:03.090
So there are a
number of diagrams

00:56:03.090 --> 00:56:08.130
that have three things
linked together.

00:56:08.130 --> 00:56:09.440
This is one of them.

00:56:09.440 --> 00:56:14.380
Let's say, think of 1, 2,
3 connected to each other.

00:56:14.380 --> 00:56:16.920
But then, I have diagrams.

00:56:16.920 --> 00:56:19.380
At this stage, I
don't really care

00:56:19.380 --> 00:56:22.430
that this is one
particle irreducible.

00:56:22.430 --> 00:56:26.800
Here, I have no constraints on
one particle irreducibility.

00:56:26.800 --> 00:56:29.710
And in fact, this comes
in three varieties

00:56:29.710 --> 00:56:33.640
because the bond that is
missing can be one of three.

00:56:33.640 --> 00:56:38.040
So once I pick the triplets
of three points, sum

00:56:38.040 --> 00:56:42.360
over all clusters that involve
these three points is this,

00:56:42.360 --> 00:56:45.560
in particular with this
one, appearing three times.

00:56:45.560 --> 00:56:46.283
Yes.

00:56:46.283 --> 00:56:48.116
AUDIENCE: Do we still
count all the diagrams

00:56:48.116 --> 00:56:49.990
when particles are identical?

00:56:49.990 --> 00:56:51.220
PROFESSOR: Yes.

00:56:51.220 --> 00:56:56.230
Because here, at
this stage, there

00:56:56.230 --> 00:56:57.830
is an N factorial out here.

00:56:57.830 --> 00:57:00.410
All I'm really doing
is transcribing

00:57:00.410 --> 00:57:02.700
a mathematical formula.

00:57:02.700 --> 00:57:04.600
The mathematical
formula says you

00:57:04.600 --> 00:57:06.840
have to do a bunch
of integrations.

00:57:06.840 --> 00:57:09.580
And I am just
following translating

00:57:09.580 --> 00:57:11.730
that mathematical
formula to diagrams.

00:57:11.730 --> 00:57:13.600
At this stage, we forget.

00:57:13.600 --> 00:57:17.510
This has nothing to do with
this N factorial or identity.

00:57:17.510 --> 00:57:24.990
It's just a transcription
of the mathematics

00:57:24.990 --> 00:57:31.710
So at this stage, don't try
to correct my conceptual parts

00:57:31.710 --> 00:57:37.090
but try to make sure that my
mathematical steps are correct.

00:57:37.090 --> 00:57:39.115
OK?

00:57:39.115 --> 00:57:39.615
Yes.

00:57:39.615 --> 00:57:41.070
AUDIENCE: I didn't
understand what

00:57:41.070 --> 00:57:43.690
that missing links represents.

00:57:43.690 --> 00:57:45.040
PROFESSOR: OK.

00:57:45.040 --> 00:57:53.340
So at the end of this story,
what I want to write down

00:57:53.340 --> 00:57:57.390
is that the value of some
particular term in this series

00:57:57.390 --> 00:58:00.610
is related to product
of contributions.

00:58:00.610 --> 00:58:03.360
I described to you how
productive contributions

00:58:03.360 --> 00:58:05.450
comes along.

00:58:05.450 --> 00:58:08.880
And each contribution
is a bunch of points

00:58:08.880 --> 00:58:12.400
that are connected together.

00:58:12.400 --> 00:58:16.360
So what I want to do, in order
to make my algebra easier

00:58:16.360 --> 00:58:19.360
later on, is to say
that-- let's say

00:58:19.360 --> 00:58:22.760
these three points
are part of a cluster.

00:58:22.760 --> 00:58:25.320
They are connected
together somehow.

00:58:25.320 --> 00:58:28.950
So they will be
giving me some factor.

00:58:28.950 --> 00:58:32.030
Now, they are connected, if
they are connected like this,

00:58:32.030 --> 00:58:35.360
like this, like this, like that.

00:58:35.360 --> 00:58:38.080
So basically, any
one of these is

00:58:38.080 --> 00:58:41.690
a way of connecting these
three points so that they

00:58:41.690 --> 00:58:43.250
will make a
contribution together.

00:59:00.960 --> 00:59:01.510
Yes.

00:59:01.510 --> 00:59:03.764
AUDIENCE: So the one
with all three bonds

00:59:03.764 --> 00:59:05.252
comes with a
different [INAUDIBLE]

00:59:05.252 --> 00:59:08.200
in the series than the ones
with only two bonds, right?

00:59:08.200 --> 00:59:09.030
PROFESSOR: Yes.

00:59:09.030 --> 00:59:11.020
Now, there is
reason, ultimately,

00:59:11.020 --> 00:59:15.300
why I want to group
all of them together.

00:59:15.300 --> 00:59:17.760
Because there was a
question before about

00:59:17.760 --> 00:59:19.980
multiplicity factors.

00:59:19.980 --> 00:59:23.190
I can write down a closed
form, nice expression

00:59:23.190 --> 00:59:27.320
for the multiplicity factor
if I group them together.

00:59:27.320 --> 00:59:30.660
Otherwise, I have to also worry
that there are three of these

00:59:30.660 --> 00:59:32.650
and there is one
of them, et cetera.

00:59:32.650 --> 00:59:36.220
There is an additional
layer of multiplicity.

00:59:36.220 --> 00:59:40.140
And I want to separate those
two layers of multiplicity.

00:59:40.140 --> 00:59:46.140
So here, the thing that I have
is I have these endpoints.

00:59:46.140 --> 00:59:49.870
I want to say that the
eventual result for endpoints

00:59:49.870 --> 00:59:55.630
comes from clusters that
involves 1's, clusters that

00:59:55.630 --> 01:00:01.350
involve pairs, and cluster
that involve triplets.

01:00:01.350 --> 01:00:05.010
Now, I realize suddenly that if
I made the multiplicity count

01:00:05.010 --> 01:00:10.140
here, then for this triplet,
I could have put all of them

01:00:10.140 --> 01:00:12.440
together like this or like that.

01:00:12.440 --> 01:00:14.190
And the next order
of business when

01:00:14.190 --> 01:00:17.330
I am thinking
about four objects,

01:00:17.330 --> 01:00:21.440
I can connect them together
into a cluster in multiple ways.

01:00:21.440 --> 01:00:25.350
And there is a separate
way of calculating

01:00:25.350 --> 01:00:27.370
the relative
multiplicity of what

01:00:27.370 --> 01:00:31.650
comes outside and the
multiplicity of this quartic

01:00:31.650 --> 01:00:33.040
with respect to entirety.

01:00:36.690 --> 01:00:39.350
So you have to think
about this a little bit.

01:00:39.350 --> 01:00:44.240
So I'll proceed with this.

01:00:44.240 --> 01:00:53.640
So this mathematically would
be the integral q1, q2, q3, f

01:00:53.640 --> 01:01:05.310
1 2, f 2 3, f 3 1 plus, say,
f 1 2, f 2 3, plus f 1 3 f 3 2

01:01:05.310 --> 01:01:11.250
plus f 2 3, f 3 1,
something like this.

01:01:14.080 --> 01:01:17.230
Well, 1, 2.

01:01:17.230 --> 01:01:17.945
2 is repeated.

01:01:17.945 --> 01:01:18.750
3 is repeated.

01:01:18.750 --> 01:01:20.026
I should repeat 1.

01:01:24.130 --> 01:01:24.780
OK.

01:01:24.780 --> 01:01:26.960
Of course, the last
three are the same thing,

01:01:26.960 --> 01:01:29.630
but this is the
expression for this.

01:01:29.630 --> 01:01:34.340
Now, before would be a
huge complex of things.

01:01:34.340 --> 01:01:37.060
It would have a diagram
that is of this form.

01:01:37.060 --> 01:01:40.720
It would have a number of
diagrams that are of this form.

01:01:40.720 --> 01:01:44.540
It would have diagrams that
are of this form, diagrams

01:01:44.540 --> 01:01:49.250
that are like
this, and so forth.

01:01:49.250 --> 01:01:53.600
Basically, even
within the choice

01:01:53.600 --> 01:01:55.470
of things that
have four clusters,

01:01:55.470 --> 01:01:57.760
there is a huge
number when you go

01:01:57.760 --> 01:01:59.975
to further and
further [? down. ?]

01:02:12.870 --> 01:02:20.340
OK, so maybe this statement
will now clarify things.

01:02:20.340 --> 01:02:26.510
So what I have is that
the partition function

01:02:26.510 --> 01:02:30.650
is 1 over N factorial
lambda to the 3N.

01:02:33.380 --> 01:02:42.440
And then it is all
of the terms that

01:02:42.440 --> 01:02:44.890
are obtained by
summing this series.

01:02:47.450 --> 01:02:54.170
And so I have to look
at all possible ways

01:02:54.170 --> 01:02:59.235
that I can break N-points
into these clusters.

01:03:01.880 --> 01:03:14.330
Suppose I created a situation
where I have clusters of size l

01:03:14.330 --> 01:03:16.100
and then I have nl of them.

01:03:19.120 --> 01:03:26.120
So I saw that the contribution
that I would get from the 1

01:03:26.120 --> 01:03:28.430
clusters was
essentially the product

01:03:28.430 --> 01:03:29.650
of the number of 1 clusters.

01:03:29.650 --> 01:03:31.830
It was V to the
number of points that

01:03:31.830 --> 01:03:34.250
are not connected together.

01:03:34.250 --> 01:03:38.290
So I have to take b1, raise
it to the power of N1,

01:03:38.290 --> 01:03:41.040
which is the number
of 1 cluster.

01:03:41.040 --> 01:03:47.240
Then, I have to do the same
thing for b2, b3, et cetera.

01:03:47.240 --> 01:03:50.160
And then I have to
make sure that I

01:03:50.160 --> 01:03:54.580
have looked at all
ways of partitioning

01:03:54.580 --> 01:03:56.860
N the numbers into
these clusters.

01:03:56.860 --> 01:04:02.470
So I have a sum over nl
l has to add up to n.

01:04:02.470 --> 01:04:06.130
And I have to sum
over all nl's that are

01:04:06.130 --> 01:04:09.060
consistent with this constraint.

01:04:11.650 --> 01:04:15.750
But then there is this
issue of the multiplicity.

01:04:15.750 --> 01:04:21.166
Given that I chose some
articular set of these,

01:04:21.166 --> 01:04:26.930
if I were to
reorganize the numbers,

01:04:26.930 --> 01:04:29.130
I would get the
same contribution.

01:04:29.130 --> 01:04:32.210
So let's say we pick
this diagram that

01:04:32.210 --> 01:04:34.320
has precisely a
contribution that

01:04:34.320 --> 01:04:39.580
is V, which is b1 to the
number of these one points.

01:04:39.580 --> 01:04:43.520
This to the number
of b2 to the power

01:04:43.520 --> 01:04:46.290
of the number of
these, this, et cetera.

01:04:46.290 --> 01:04:50.100
But then I can take these
labels-- 1, 2, 3, 4, 5,

01:04:50.100 --> 01:04:54.270
6-- that I assigned to these
things and permute them.

01:04:54.270 --> 01:04:58.510
If I permute them, I will get
exactly the same contribution.

01:04:58.510 --> 01:05:03.310
So there is a large
number of diagrams

01:05:03.310 --> 01:05:05.990
that have exactly this
same contribution because

01:05:05.990 --> 01:05:06.990
of this permutation.

01:05:06.990 --> 01:05:09.230
There was a question back there.

01:05:09.230 --> 01:05:11.086
AUDIENCE: You've
already answered it.

01:05:15.165 --> 01:05:16.040
PROFESSOR: All right.

01:05:19.120 --> 01:05:21.060
Actually, this is an
interesting thing--

01:05:21.060 --> 01:05:23.170
I don't know how many
of you recognize this.

01:05:23.170 --> 01:05:27.840
So essentially, I have
to take a huge number, N,

01:05:27.840 --> 01:05:31.210
and break it into a number
of 1 clusters, 2 clusters, et

01:05:31.210 --> 01:05:32.460
cetera.

01:05:32.460 --> 01:05:34.360
The number of ways
of doing that is

01:05:34.360 --> 01:05:36.700
called a partition
of an integer.

01:05:36.700 --> 01:05:41.520
And last century,
the [INAUDIBLE]

01:05:41.520 --> 01:05:43.370
calculated what that number is.

01:05:43.370 --> 01:05:47.160
So there is a [INAUDIBLE]
theorem associated with that.

01:05:47.160 --> 01:05:48.740
And actually,
later in the course

01:05:48.740 --> 01:05:50.930
I will give you a
problem to calculate

01:05:50.930 --> 01:05:53.445
the asymptotic version
of the [INAUDIBLE].

01:05:57.160 --> 01:05:58.390
But this is different story.

01:05:58.390 --> 01:06:01.830
So given that you had made
one of these partitions

01:06:01.830 --> 01:06:05.950
of this integer, what is
this degeneracy factor?

01:06:05.950 --> 01:06:09.090
So let me tell you what
this degeneracy factor is.

01:06:09.090 --> 01:06:13.960
So given this
choice of nl's, you

01:06:13.960 --> 01:06:18.550
say-- well, the first
thing is what I told you.

01:06:18.550 --> 01:06:21.690
For a particular
graph, the value

01:06:21.690 --> 01:06:26.320
is independent of how these
numbers were assigned.

01:06:26.320 --> 01:06:29.540
So I will permute those
numbers in all possible ways

01:06:29.540 --> 01:06:31.630
and I will get the same thing.

01:06:31.630 --> 01:06:33.272
So that will give
me N factorial.

01:06:33.272 --> 01:06:34.605
It's the number of permutations.

01:06:37.130 --> 01:06:41.860
But then, I have over-counted
things because within,

01:06:41.860 --> 01:06:45.850
let's say, a 2 cluster--
if I count 7, 8, 8,

01:06:45.850 --> 01:06:47.370
7 they are the same thing.

01:06:47.370 --> 01:06:50.520
And by multiplying
by this N factorial,

01:06:50.520 --> 01:06:53.320
I have over-counted that.

01:06:53.320 --> 01:06:58.990
So I have to divide by 2 for
every one of my 2 clusters.

01:06:58.990 --> 01:07:04.750
So I have to divide by
2 to the power of N2.

01:07:04.750 --> 01:07:07.700
For the 3 clusters, I have
the permutation of everything

01:07:07.700 --> 01:07:08.330
that is inside.

01:07:08.330 --> 01:07:10.930
So it is 3 factor here.

01:07:10.930 --> 01:07:14.050
So what I have here
is I have to divide

01:07:14.050 --> 01:07:21.020
by the over-counting, which
is l factorial-- labels

01:07:21.020 --> 01:07:24.740
within a cluster-- to
the number of clusters

01:07:24.740 --> 01:07:27.790
that I have that
are subject to this.

01:07:30.830 --> 01:07:40.195
There is another thing that this
pair is 100, 101 and this pair

01:07:40.195 --> 01:07:42.280
is 7, 8.

01:07:42.280 --> 01:07:46.310
Exchanging that pair of numbers
with that pair of numbers

01:07:46.310 --> 01:07:50.960
is also part of the symmetries
that I have now over-counted.

01:07:50.960 --> 01:07:54.210
So that has to be
taken into account

01:07:54.210 --> 01:07:59.630
because the number of
these 2 clusters here is 2.

01:07:59.630 --> 01:08:00.790
Here is actually 3.

01:08:00.790 --> 01:08:03.050
I have to divide by 3 factorial.

01:08:03.050 --> 01:08:06.406
In general, I have to
divide by nl factorial.

01:08:10.300 --> 01:08:11.490
OK.

01:08:11.490 --> 01:08:14.810
Again, this is one
of those things

01:08:14.810 --> 01:08:19.580
that the best advice that I
can give you is to draw five

01:08:19.580 --> 01:08:22.890
or six points so
your n is small.

01:08:22.890 --> 01:08:25.430
And draw some diagrams.

01:08:25.430 --> 01:08:27.640
And convince yourself
that what I told you

01:08:27.640 --> 01:08:30.510
here rapidly is correct.

01:08:30.510 --> 01:08:32.590
And it is correct.

01:08:35.192 --> 01:08:35.691
Yes.

01:08:35.691 --> 01:08:38.517
AUDIENCE: Can you just clarify
which is in the [INAUDIBLE]?

01:08:38.517 --> 01:08:39.100
PROFESSOR: OK.

01:08:39.100 --> 01:08:41.700
So the N factorial is outside.

01:08:41.700 --> 01:08:44.290
And then-- so maybe
write it better.

01:08:44.290 --> 01:08:49.819
For each l, I have to
multiply nl factorial and l

01:08:49.819 --> 01:08:52.422
factorial to the power of nl.

01:09:04.020 --> 01:09:07.954
So this is my partition function
in terms of these clusters.

01:09:10.760 --> 01:09:13.270
Even with all of
these definitions

01:09:13.270 --> 01:09:18.670
that I have up there, maybe
not so obvious an answer.

01:09:18.670 --> 01:09:22.590
And one of the reason it is
kind of an obscure answer

01:09:22.590 --> 01:09:28.240
is because here I have
to do a constrained sum.

01:09:28.240 --> 01:09:31.870
That is, I have all
of these variables.

01:09:31.870 --> 01:09:38.220
Let's call them n1,
n2, n3, et cetera.

01:09:38.220 --> 01:09:42.920
Each one of them can
go from 0, 1, 2, 3.

01:09:42.920 --> 01:09:44.890
But they are all
linked to each other

01:09:44.890 --> 01:09:49.399
because their sum is kind of
restricted by some total value

01:09:49.399 --> 01:09:52.500
l, by this constraint.

01:09:52.500 --> 01:09:57.530
And constrained
sums are hard to do.

01:09:57.530 --> 01:10:00.310
But in statistical
physics, we know

01:10:00.310 --> 01:10:03.170
how to gets rid of
constrained sums.

01:10:03.170 --> 01:10:06.290
The way that we do
that is we essentially

01:10:06.290 --> 01:10:09.560
allow this N to go all the way.

01:10:09.560 --> 01:10:13.420
So if I say, let's make--
it's very hard for me

01:10:13.420 --> 01:10:15.560
to do this n1 from
0 to something,

01:10:15.560 --> 01:10:19.760
n2 from 0 to something with
all of this constrained.

01:10:19.760 --> 01:10:22.800
My life would be very easier
if I could independently

01:10:22.800 --> 01:10:26.080
have n1 go take any
value, n2 take any value,

01:10:26.080 --> 01:10:28.450
n3 take any value.

01:10:28.450 --> 01:10:32.570
But if I do that essentially
for each choice of n1, n2, n3,

01:10:32.570 --> 01:10:38.010
I have shifted the
value of big N.

01:10:38.010 --> 01:10:39.970
But there is an
ensemble that I know

01:10:39.970 --> 01:10:42.650
which has possibly
any value of N,

01:10:42.650 --> 01:10:44.044
and that's the grand canonical.

01:10:50.820 --> 01:10:54.810
So rather than looking at
the partition function,

01:10:54.810 --> 01:10:57.840
I say I will look at the
grand partition function

01:10:57.840 --> 01:11:01.920
Q that is obtained
by summing over

01:11:01.920 --> 01:11:06.470
all N. Can take any
value from 0 to infinity.

01:11:06.470 --> 01:11:12.280
I have e to the beta mu N times
the partition function that

01:11:12.280 --> 01:11:13.395
is for N particles.

01:11:16.750 --> 01:11:21.130
So that's the definition of how
you would go from the canonical

01:11:21.130 --> 01:11:23.260
where you have fixed
N to grand canonical

01:11:23.260 --> 01:11:27.630
where you have fixed the
chemical potential mu.

01:11:27.630 --> 01:11:31.620
So let's apply this
sum over there.

01:11:31.620 --> 01:11:36.690
I have a sum over N, but
I said that if I allowed

01:11:36.690 --> 01:11:40.940
the nl's to vary
independently, it

01:11:40.940 --> 01:11:45.480
is equivalent to varying
that N, recognizing

01:11:45.480 --> 01:11:50.019
that this n is sum over l l nl.

01:11:50.019 --> 01:11:50.935
That's the constraint.

01:11:54.140 --> 01:11:55.800
So that's just the first term.

01:11:55.800 --> 01:12:01.000
I have rewritten this sum that
was constrained and this sum

01:12:01.000 --> 01:12:05.370
over the total number as
independent sums over the nl's.

01:12:05.370 --> 01:12:06.640
Got rid of the constraint.

01:12:09.570 --> 01:12:12.290
Now, I write W. Oh, OK.

01:12:12.290 --> 01:12:13.404
The partition function.

01:12:13.404 --> 01:12:14.820
Now, write the
partition function.

01:12:14.820 --> 01:12:18.680
I have 1 over N factorial.

01:12:18.680 --> 01:12:21.240
I have lambda cubed
raised to this power.

01:12:21.240 --> 01:12:25.980
So actually, let me put
this in this fashion.

01:12:25.980 --> 01:12:28.470
So both e to the
beta mu over lambda

01:12:28.470 --> 01:12:31.090
cubed that is raised
to that power.

01:12:31.090 --> 01:12:33.310
OK, so that's that part.

01:12:33.310 --> 01:12:36.140
The sum I have gotten rid of.

01:12:36.140 --> 01:12:40.350
I notice that my W
has an N factorial.

01:12:40.350 --> 01:12:43.160
So this is the N factorial
that came from here.

01:12:45.670 --> 01:12:49.790
But then there is an N factorial
that is up here from the W.

01:12:49.790 --> 01:12:54.440
So the two of them will cancel
each other, N factorial.

01:12:54.440 --> 01:13:00.770
And then I have a product
over all clusters.

01:13:00.770 --> 01:13:04.980
Part of it is this bl
to the power of nl,

01:13:04.980 --> 01:13:08.790
and then I have the contribution
that is nl factorial l

01:13:08.790 --> 01:13:10.236
factorial to the power of nl.

01:13:15.930 --> 01:13:17.150
So this part cancels.

01:13:19.840 --> 01:13:25.360
For each l, I can
now independently

01:13:25.360 --> 01:13:31.910
sum over the values of nl can
be anything, 0 to infinity.

01:13:31.910 --> 01:13:33.762
And what do I have?

01:13:33.762 --> 01:13:39.650
I have bl to the power of nl.

01:13:39.650 --> 01:13:43.790
I have division by nl factorial.

01:13:43.790 --> 01:13:48.630
I have l factorial
to the power of nl.

01:13:48.630 --> 01:13:55.920
And then I have e to the beta
mu over lambda cubed raised

01:13:55.920 --> 01:13:57.431
to the power of l nl.

01:14:08.500 --> 01:14:12.650
I recognize that each one
of these terms in the sum

01:14:12.650 --> 01:14:19.150
is 1 over nl factorial something
raised to the power of nl,

01:14:19.150 --> 01:14:22.240
which is the definition
of the exponential.

01:14:22.240 --> 01:14:27.880
So my q is a product
over all l's.

01:14:27.880 --> 01:14:35.690
Once I exponentiate, I
have e to the beta mu

01:14:35.690 --> 01:14:42.380
over lambda cubed raised
to the power of l.

01:14:42.380 --> 01:14:46.890
And then I have bl
divided by l factorial.

01:14:53.080 --> 01:14:59.770
Or, log of Q, which is the
quantity that I'm interested,

01:14:59.770 --> 01:15:04.240
is obtained by summing
over all clusters 1

01:15:04.240 --> 01:15:11.290
to infinity e to the beta
mu over lambda cubed raised

01:15:11.290 --> 01:15:16.366
to the power of l bl
divided by l factorial.

01:15:26.840 --> 01:15:30.330
So what does this
tell us, which is

01:15:30.330 --> 01:15:34.050
kind of nice and
fundamentally important?

01:15:34.050 --> 01:15:38.110
You see, we started at
the beginning over here

01:15:38.110 --> 01:15:40.510
with a huge number of graphs.

01:15:43.280 --> 01:15:49.240
These graphs could be organizing
all kinds of clusters.

01:15:49.240 --> 01:15:51.980
And they would give us
either the partition function

01:15:51.980 --> 01:15:56.700
or summing over N, the
grand partition function.

01:15:56.700 --> 01:16:05.230
But when we take the log, I get
only single connect objects.

01:16:05.230 --> 01:16:07.510
And this is something
that you had already

01:16:07.510 --> 01:16:10.440
seen as the connection
that we have

01:16:10.440 --> 01:16:13.750
between moments and Cumulants.

01:16:13.750 --> 01:16:17.010
So the way that we
got Cumulants was

01:16:17.010 --> 01:16:19.860
to look at the
expansion of the log.

01:16:19.860 --> 01:16:23.250
So the function itself was
the generator of all moments

01:16:23.250 --> 01:16:24.830
and we took the log.

01:16:24.830 --> 01:16:26.990
And graphically,
we presented that

01:16:26.990 --> 01:16:29.480
as getting the moments
by putting together

01:16:29.480 --> 01:16:34.920
all kinds of clusters of points
that corresponded to Cumulants.

01:16:34.920 --> 01:16:37.300
And this is again,
a representation

01:16:37.300 --> 01:16:38.660
of the same thing.

01:16:38.660 --> 01:16:41.490
That is, in the log
you have essentially

01:16:41.490 --> 01:16:44.290
individual contributions.

01:16:44.290 --> 01:16:49.790
Once you exponentiated, you
get multiple contributions.

01:16:49.790 --> 01:16:54.870
Now, the other thing is, that
if I am thinking about the gas,

01:16:54.870 --> 01:17:00.730
then log of Q is e
to the minus beta g.

01:17:00.730 --> 01:17:11.270
And g is E minus TS minus
mu N. Bur for a gas that

01:17:11.270 --> 01:17:16.840
has extensive properties,
E is TS mu N minus PV.

01:17:16.840 --> 01:17:19.210
So this is the same
thing as beta PV.

01:17:22.320 --> 01:17:25.730
So this quantity that
we have calculated here

01:17:25.730 --> 01:17:27.830
is, in fact, related
to the pressure

01:17:27.830 --> 01:17:30.001
directly through this formula.

01:17:32.810 --> 01:17:35.770
And the important part or
the important observation

01:17:35.770 --> 01:17:40.800
is that it says that it should
be proportional to volume.

01:17:40.800 --> 01:17:44.170
You can see that
each one of my b's

01:17:44.170 --> 01:17:48.500
that I calculated here will
have one free integral.

01:17:48.500 --> 01:17:50.010
If you like, its center of mass.

01:17:50.010 --> 01:17:52.310
You can go over
the entire volume.

01:17:52.310 --> 01:17:58.560
So all of my b's are indeed
proportionality to volume.

01:17:58.560 --> 01:18:00.440
And you can see what
disaster it would

01:18:00.440 --> 01:18:05.310
have been if there was a term
here that was not just a linked

01:18:05.310 --> 01:18:07.510
cluster but product of
two linked clusters.

01:18:07.510 --> 01:18:10.240
Then I would have something
that would go like V squared.

01:18:10.240 --> 01:18:12.010
It's not allowed.

01:18:12.010 --> 01:18:16.970
So essentially, this linked
cluster nature is also

01:18:16.970 --> 01:18:19.850
related to extensivity
in this sense.

01:18:23.350 --> 01:18:28.620
So in some sense, what I have
established here-- again,

01:18:28.620 --> 01:18:35.170
related to that-- is that
clearly all of my bl's

01:18:35.170 --> 01:18:38.650
are proportional to volume.

01:18:38.650 --> 01:18:40.510
And I can define
something that I

01:18:40.510 --> 01:18:48.540
will call bl bar, which is
divide the thing by the volume.

01:18:48.540 --> 01:18:54.770
And then what we
have established

01:18:54.770 --> 01:19:03.400
is that the pressure of this
interacting gas as an expansion

01:19:03.400 --> 01:19:09.142
beta p, which takes this
nice, simple form, sum over l

01:19:09.142 --> 01:19:14.770
e to the beta mu divided
by lambda cubed bl

01:19:14.770 --> 01:19:21.580
bar-- the intensive part of the
contribution of these cluster--

01:19:21.580 --> 01:19:23.300
divided by l factorial.

01:19:23.300 --> 01:19:26.540
This, of course, runs
from 1 to [INAUDIBLE].

01:19:29.510 --> 01:19:30.070
Yes.

01:19:30.070 --> 01:19:33.592
AUDIENCE: [INAUDIBLE]
is raised to the l?

01:19:33.592 --> 01:19:35.133
PROFESSOR: Who's
asking the question?

01:19:35.133 --> 01:19:37.999
AUDIENCE: The term in the
parentheses is raised to the l?

01:19:37.999 --> 01:19:38.790
PROFESSOR: Exactly.

01:19:38.790 --> 01:19:42.120
Thank you very much.

01:19:42.120 --> 01:19:43.534
Yes.

01:19:43.534 --> 01:19:44.450
AUDIENCE: [INAUDIBLE].

01:19:53.575 --> 01:19:55.930
But they don't, actually.

01:19:55.930 --> 01:19:58.870
Could that be true, for
example [INAUDIBLE]?

01:20:04.667 --> 01:20:05.250
PROFESSOR: OK.

01:20:05.250 --> 01:20:07.750
So the triangle, I have it here.

01:20:07.750 --> 01:20:12.320
Integral d q1, d q2, d
Q3, f 1 2, f 2 3, f 3.

01:20:12.320 --> 01:20:14.370
Let's write it explicitly.

01:20:14.370 --> 01:20:23.210
It is f of q1 minus q2, f of
q2 minus q3, f of q3 minus q1.

01:20:23.210 --> 01:20:25.725
I call this vector x.

01:20:25.725 --> 01:20:28.500
I call that vector y.

01:20:28.500 --> 01:20:30.700
This vector is x plus y.

01:20:35.310 --> 01:20:36.109
Yes.

01:20:36.109 --> 01:20:38.504
AUDIENCE: Does the
contribution from [INAUDIBLE]?

01:20:44.260 --> 01:20:45.290
PROFESSOR: OK.

01:20:45.290 --> 01:20:52.720
So you say that I have here
an expansion for the pressure.

01:20:52.720 --> 01:20:55.930
I had given you previously
an expansion for pressure

01:20:55.930 --> 01:20:59.240
that I said is sensible, which
is this Virial expansion, which

01:20:59.240 --> 01:21:01.870
is powers of density.

01:21:01.870 --> 01:21:06.160
This is not an expansion
in powers of density.

01:21:06.160 --> 01:21:10.320
So whether or not the
terms in this expansion

01:21:10.320 --> 01:21:13.230
becomes smaller or
larger will depend

01:21:13.230 --> 01:21:16.490
on density in some indirect way.

01:21:16.490 --> 01:21:21.100
So my next task, which
I will do next lecture,

01:21:21.100 --> 01:21:23.540
is within this ensemble
I have told you

01:21:23.540 --> 01:21:25.740
what the chemical potential is.

01:21:25.740 --> 01:21:28.160
Once I know what the
chemical potential is,

01:21:28.160 --> 01:21:30.770
I can calculate the
number of particles

01:21:30.770 --> 01:21:36.380
as d log Q by d beta mu.

01:21:36.380 --> 01:21:41.640
And hence, I can calculate
what the density is.

01:21:41.640 --> 01:21:46.390
And so the answer for this will
also be a series in powers of e

01:21:46.390 --> 01:21:47.980
to the beta mu.

01:21:47.980 --> 01:21:51.280
And what then I will do is
I will combine these two

01:21:51.280 --> 01:21:56.620
series to get an expansion for
pressure in powers of density,

01:21:56.620 --> 01:22:00.410
and then identify the
convergence of this series

01:22:00.410 --> 01:22:03.320
and all of that
via that procedure.