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PROFESSOR: And to
this purpose, we
00:00:23.670 --> 00:00:28.470
need to calculate
some thermodynamics.
00:00:28.470 --> 00:00:31.160
And we usually do that
in statistical mechanics
00:00:31.160 --> 00:00:35.450
by calculating for some kind
of a partition function.
00:00:35.450 --> 00:00:37.320
And we saw last
time that it would
00:00:37.320 --> 00:00:42.180
be useful to calculate this
grand partition function, which
00:00:42.180 --> 00:00:44.400
is the ensemble
where you specify
00:00:44.400 --> 00:00:48.920
the temperature, the chemical
potential, and the volume
00:00:48.920 --> 00:00:51.250
of the gas.
00:00:51.250 --> 00:00:55.120
And in this ensemble,
your task is
00:00:55.120 --> 00:01:00.060
to look at within
this box of volume V
00:01:00.060 --> 00:01:04.690
the possibility of there
being any number of particles.
00:01:04.690 --> 00:01:09.400
So you have to sum over all
possible N particle states.
00:01:09.400 --> 00:01:12.640
The contribution of
each N particle states
00:01:12.640 --> 00:01:17.710
is exponentially related
to the number of particles
00:01:17.710 --> 00:01:19.980
through the chemical potential.
00:01:19.980 --> 00:01:24.730
And then given that you are in
a segment that has N particles,
00:01:24.730 --> 00:01:27.690
you have to look at all
possible configurations
00:01:27.690 --> 00:01:30.890
of those particles
and integrate over
00:01:30.890 --> 00:01:32.790
all of those possibilities.
00:01:32.790 --> 00:01:36.350
And that amounts to calculating
the partition function.
00:01:36.350 --> 00:01:40.540
So for the partition function
of an N particle system,
00:01:40.540 --> 00:01:43.880
you have to integrate
over all of the momenta.
00:01:43.880 --> 00:01:47.020
Integration over each
component of the momentum
00:01:47.020 --> 00:01:52.340
gives you a factor of 1 over
lambda, where lambda again
00:01:52.340 --> 00:01:58.880
was related to the mass of
the particle and temperature
00:01:58.880 --> 00:02:02.440
through this formula
with h what we
00:02:02.440 --> 00:02:07.640
use to make these integrations
over pq combinations
00:02:07.640 --> 00:02:09.120
dimensionless.
00:02:09.120 --> 00:02:11.860
There are 3N such integrations.
00:02:11.860 --> 00:02:16.680
So that's a contribution
from the momenta.
00:02:16.680 --> 00:02:19.320
And then we have to
do the integration
00:02:19.320 --> 00:02:21.470
over all of the coordinates.
00:02:21.470 --> 00:02:25.970
And as long as these
particles are identical,
00:02:25.970 --> 00:02:30.140
we decided to divide by
the number of permutations.
00:02:30.140 --> 00:02:33.150
Because we cannot
tear them apart.
00:02:33.150 --> 00:02:37.710
So having done that, I need to
now integrate over all of the N
00:02:37.710 --> 00:02:43.700
particles spanning
a box of size V
00:02:43.700 --> 00:02:47.180
so the integration
is within the box.
00:02:47.180 --> 00:02:51.080
And when I have
interactions, then I
00:02:51.080 --> 00:02:54.200
have to worry about the
Boltzmann weight that
00:02:54.200 --> 00:02:55.550
comes from the interaction.
00:02:55.550 --> 00:02:58.630
So here I should really
put some kind of e
00:02:58.630 --> 00:03:02.920
to the minus beta times
the interaction U.
00:03:02.920 --> 00:03:05.480
And what we did was we
said that let's assume
00:03:05.480 --> 00:03:09.290
that this interaction comes
from pairs of particles.
00:03:09.290 --> 00:03:13.640
And so this U is the sum
over all possible pairs
00:03:13.640 --> 00:03:16.750
of particles, which
when exponentiated
00:03:16.750 --> 00:03:24.790
will then give you a product
over all possible pairs j, k
00:03:24.790 --> 00:03:29.450
and a factor that is related
to the potential interaction
00:03:29.450 --> 00:03:30.980
between these.
00:03:30.980 --> 00:03:35.350
And we found it useful
to write that factor as 1
00:03:35.350 --> 00:03:45.840
plus fjk, where this fjk
stood for e to the minus
00:03:45.840 --> 00:03:53.150
beta of V qj minus qk minus 1.
00:03:53.150 --> 00:03:55.950
So basically if I
add the 1 to that,
00:03:55.950 --> 00:03:59.190
I just get the exponentiated
interaction potential.
00:03:59.190 --> 00:04:03.150
And then I have a
forum such as this.
00:04:03.150 --> 00:04:05.780
So basically, this
is the quantity
00:04:05.780 --> 00:04:07.980
that we wanted to compute.
00:04:07.980 --> 00:04:11.390
And so then we said,
well, let's imagine
00:04:11.390 --> 00:04:17.250
expanding these factors of
1 plus f1, 2, 1 plus f1, 3,
00:04:17.250 --> 00:04:19.910
1 plus f2, 3, all
of these factors,
00:04:19.910 --> 00:04:24.880
and organize them according to
the powers of f that they have.
00:04:24.880 --> 00:04:27.990
So the leading term would
be taking 1 from everybody.
00:04:27.990 --> 00:04:29.970
So that would give
me V to the N, which
00:04:29.970 --> 00:04:32.730
would be the 0-th
order partition
00:04:32.730 --> 00:04:35.000
function, the
non-interacting system.
00:04:35.000 --> 00:04:37.590
And then I would start
to get corrections
00:04:37.590 --> 00:04:41.080
where there's order of f
integrated, order of f squared
00:04:41.080 --> 00:04:44.070
integrated, all kinds of things.
00:04:44.070 --> 00:04:49.330
And that being a somewhat
difficult object to look at,
00:04:49.330 --> 00:04:53.860
we said, let's imagine
graphically what we would get.
00:04:53.860 --> 00:04:58.800
And the typical contribution
that we would get to this
00:04:58.800 --> 00:05:04.640
would involve having to iterate
over all of these N particles.
00:05:04.640 --> 00:05:10.560
So we have to somehow imagine
that we have particles 1 to N.
00:05:10.560 --> 00:05:16.370
And then for a particular term,
we either pick 1's-- and there
00:05:16.370 --> 00:05:19.770
will be some points that are
not connected to any f that is
00:05:19.770 --> 00:05:21.410
integrated.
00:05:21.410 --> 00:05:24.590
There will be a bunch of
things that will be connected
00:05:24.590 --> 00:05:28.260
to things where there
are pair of f's.
00:05:28.260 --> 00:05:30.100
There will be things
later on maybe
00:05:30.100 --> 00:05:35.450
where there are triplets of
f, and so forth and so on.
00:05:35.450 --> 00:05:40.500
And then we said that when
I do the integrations that
00:05:40.500 --> 00:05:44.180
correspond to this,
what do I get?
00:05:44.180 --> 00:05:48.140
I will get the contribution
that comes from one particle
00:05:48.140 --> 00:05:50.650
by itself integrated.
00:05:50.650 --> 00:05:56.940
Let's call that b1
to the power of n1.
00:05:56.940 --> 00:06:02.620
I will get the contribution
from this pair integrated.
00:06:02.620 --> 00:06:07.290
I will get b2 to
the power of n2.
00:06:07.290 --> 00:06:10.820
I will get the contributions
from these entities.
00:06:10.820 --> 00:06:14.310
And in general, I said,
well, OK, somewhere in this,
00:06:14.310 --> 00:06:16.730
I will get bl to
the power of nl.
00:06:19.418 --> 00:06:22.650
Now of course, I have
a big constraint here
00:06:22.650 --> 00:06:30.080
that is that the
sum over l of lnl
00:06:30.080 --> 00:06:34.750
has to be the total
number of points 1 to n.
00:06:34.750 --> 00:06:40.330
So however I partition this,
I will have for each graph
00:06:40.330 --> 00:06:43.090
that particular
constraint acting.
00:06:46.180 --> 00:06:53.320
We said that clearly there's a
lot of graphs and combinations
00:06:53.320 --> 00:06:56.710
that give you precisely
this same factor.
00:06:56.710 --> 00:07:00.010
But all I had to do was
to sort of rearrange
00:07:00.010 --> 00:07:02.120
the numbers and
ordering, et cetera,
00:07:02.120 --> 00:07:04.750
and I would get all of this.
00:07:04.750 --> 00:07:07.520
So it would be very nice
if I could figure out
00:07:07.520 --> 00:07:09.320
what the overall
factor is out here.
00:07:12.450 --> 00:07:18.680
So we said that the factor is
something like N factorial.
00:07:18.680 --> 00:07:22.500
Because what I can do is I can
permute all of these numbers,
00:07:22.500 --> 00:07:25.950
and I would get
exactly the same thing.
00:07:25.950 --> 00:07:29.600
But then I have to make sure
that I don't over count.
00:07:29.600 --> 00:07:35.350
And not over counting
required me to divide
00:07:35.350 --> 00:07:37.350
by the number of
permutations that I
00:07:37.350 --> 00:07:40.050
have within each subgroup.
00:07:40.050 --> 00:07:43.380
So I have bl to the power of n.
00:07:43.380 --> 00:07:51.360
I have l factorial
to the power of nl.
00:07:51.360 --> 00:07:55.340
And then I have the change.
00:07:55.340 --> 00:07:57.370
Let's say this is
1, 2, this is 3, 4.
00:07:57.370 --> 00:08:01.170
I could have called one of
them 3, 4, the other one 1, 2.
00:08:01.170 --> 00:08:06.925
So basically I will have nl
factorial from the permutations
00:08:06.925 --> 00:08:07.835
within each.
00:08:12.390 --> 00:08:16.950
But I would have gotten exactly
the same numerical factor
00:08:16.950 --> 00:08:23.510
out front if I had
the same configuration
00:08:23.510 --> 00:08:25.070
but I had this diagram.
00:08:27.770 --> 00:08:29.860
I would have gotten
exactly the same factor.
00:08:29.860 --> 00:08:33.080
I would call the
contribution to this b3.
00:08:33.080 --> 00:08:36.559
I didn't say
currently what b3 is.
00:08:36.559 --> 00:08:39.640
I would have gotten
exactly the same factor.
00:08:39.640 --> 00:08:44.080
So maybe then what I
did was to sort of group
00:08:44.080 --> 00:08:46.080
all of those things
that would come
00:08:46.080 --> 00:08:50.100
with the same numerical factor
into this [INAUDIBLE] sum
00:08:50.100 --> 00:08:51.020
that I call bl.
00:08:51.020 --> 00:09:00.770
So I call bl to be the sum
over all l-particle clusters.
00:09:09.250 --> 00:09:12.730
And then, of course
here, I have to sum over
00:09:12.730 --> 00:09:15.610
all configurations
of nl that are
00:09:15.610 --> 00:09:19.122
consistent with this constraint
that I have to put up there.
00:09:24.430 --> 00:09:28.260
So the rest of it
then was algebra.
00:09:28.260 --> 00:09:35.080
We said that if I
constrain the total number,
00:09:35.080 --> 00:09:36.800
it's difficult for me to do.
00:09:36.800 --> 00:09:41.310
That's why I didn't go and
calculate the partition
00:09:41.310 --> 00:09:44.880
function and switch to the
grand partition function.
00:09:44.880 --> 00:09:48.550
Because in the grand
partition, I can essentially
00:09:48.550 --> 00:09:54.910
make this N that constrains
the values of these nl's
00:09:54.910 --> 00:09:57.260
to be all over the place.
00:09:57.260 --> 00:10:02.870
And therefore summing over
things with nl unconstrained
00:10:02.870 --> 00:10:07.480
is equivalent to summing over
terms with nl constrained,
00:10:07.480 --> 00:10:12.010
and then summing over whatever
the final constraint is.
00:10:12.010 --> 00:10:16.310
So once I did that, I was
liberated from this constraint.
00:10:16.310 --> 00:10:21.360
I could do the sum for each
value of nl separately.
00:10:21.360 --> 00:10:24.380
And the thing nicely
broke up into pieces.
00:10:24.380 --> 00:10:27.970
And so then what I
could do is I could
00:10:27.970 --> 00:10:32.360
show that each term
in the sum, there
00:10:32.360 --> 00:10:36.880
would be a product over
different contributions l.
00:10:36.880 --> 00:10:43.290
For each l, I could sum over
nl running from 0 to infinity.
00:10:43.290 --> 00:10:49.840
I had the 1 over nl
factorial from out here.
00:10:49.840 --> 00:10:53.320
I had an e to the
beta mu divided
00:10:53.320 --> 00:10:58.730
by lambda cubed from the
combination of these things
00:10:58.730 --> 00:11:02.060
raised to the
power of lnl, which
00:11:02.060 --> 00:11:05.700
is how big N would
have been composed.
00:11:05.700 --> 00:11:08.550
So I would have here lnl.
00:11:08.550 --> 00:11:17.980
And I also had an l factorial
raised to the power of nl.
00:11:17.980 --> 00:11:19.952
So I can write things
in this fashion.
00:11:22.840 --> 00:11:25.740
So this thing then
became the same thing
00:11:25.740 --> 00:11:31.795
as an exponential of a sum over
l running from 1 to infinity.
00:11:35.160 --> 00:11:37.380
1 over l factorial,
of if you like
00:11:37.380 --> 00:11:43.220
e to the beta mu over lambda
cubed to the power of l 1
00:11:43.220 --> 00:11:44.920
over l factorial.
00:11:44.920 --> 00:11:46.610
And then I had bl.
00:11:52.690 --> 00:11:58.610
So it was this very nice
result somehow summing
00:11:58.610 --> 00:12:03.820
over all kinds of things, and
then taking the logarithm.
00:12:03.820 --> 00:12:06.732
The logarithm
really depends only
00:12:06.732 --> 00:12:10.480
on the contributions
of single clusters.
00:12:10.480 --> 00:12:12.250
And again, the
reason it had to be
00:12:12.250 --> 00:12:15.090
that way is because
the ultimate thing
00:12:15.090 --> 00:12:18.170
that I calculated
in this ensemble
00:12:18.170 --> 00:12:22.960
is that the answer
should be e to the beta V
00:12:22.960 --> 00:12:23.940
times the pressure.
00:12:26.470 --> 00:12:30.950
And so the expression that
we have over here better
00:12:30.950 --> 00:12:36.890
have terms which are all
proportional to volume.
00:12:36.890 --> 00:12:40.460
Sorry, they're all made
extensive by proportionality
00:12:40.460 --> 00:12:41.690
to volume.
00:12:41.690 --> 00:12:45.380
And indeed, when I do the
integrations over a single any
00:12:45.380 --> 00:12:48.620
cluster, there is one degree
of freedom, if you like,
00:12:48.620 --> 00:12:52.080
associated with the center
of mass of the cluster that
00:12:52.080 --> 00:12:54.790
can go and explore
the entire volume.
00:12:54.790 --> 00:12:59.700
And so all of these things are
in fact in the large end limit
00:12:59.700 --> 00:13:05.739
proportional to V and
something that I call bl bar.
00:13:17.340 --> 00:13:24.610
So once I divide by this
volume, the final outcome
00:13:24.610 --> 00:13:27.940
of my calculation is
that I can calculate
00:13:27.940 --> 00:13:32.340
the pressure of
an interacting gas
00:13:32.340 --> 00:13:41.860
by summing over a series whose
terms are this e to the beta mu
00:13:41.860 --> 00:13:48.100
over lambda cubed raised to
the power of l bl bar divided
00:13:48.100 --> 00:13:50.280
by l factorial.
00:13:59.480 --> 00:14:05.710
OK, this is correct, but not
particularly illuminating.
00:14:05.710 --> 00:14:11.370
Because the thing that we said
we have some intuition for
00:14:11.370 --> 00:14:15.640
is that the pressure
of a gas-- let's
00:14:15.640 --> 00:14:19.630
say if I look at it
in terms of beta P,
00:14:19.630 --> 00:14:26.900
actually P is the density
times kt, which is 1 over beta,
00:14:26.900 --> 00:14:28.210
if you like, or nkT.
00:14:28.210 --> 00:14:30.930
It doesn't matter.
00:14:30.930 --> 00:14:32.880
And then there will
be corrections.
00:14:32.880 --> 00:14:35.270
There will be a term that
is order of n squared.
00:14:37.980 --> 00:14:42.640
There will be a term
that is order of n cubed.
00:14:42.640 --> 00:14:44.660
And there are
these coefficients,
00:14:44.660 --> 00:14:49.100
which are functions
of temperature,
00:14:49.100 --> 00:14:52.460
that are called the
virial coefficient.
00:14:52.460 --> 00:14:55.220
And this is a virial expansion.
00:14:55.220 --> 00:15:00.370
Essentially what
it is is a fitting
00:15:00.370 --> 00:15:02.240
of the form of the
pressure of the gas
00:15:02.240 --> 00:15:04.190
as a function of density.
00:15:04.190 --> 00:15:08.545
In the very low density limit
you get ideal gas result.
00:15:08.545 --> 00:15:11.110
And presumably because
of these interactions,
00:15:11.110 --> 00:15:14.650
you will get corrections
that presumably also
00:15:14.650 --> 00:15:17.740
know about the
potential that went
00:15:17.740 --> 00:15:23.100
into the construction of all
of these b bars, et cetera.
00:15:23.100 --> 00:15:28.990
So how do we relate these things
that are, say, experimentally
00:15:28.990 --> 00:15:35.040
accessible to this expansion
that we have over here?
00:15:35.040 --> 00:15:38.970
And the reason it is
not obvious immediately
00:15:38.970 --> 00:15:46.010
is because this is an expansion
in chemical potential,
00:15:46.010 --> 00:15:49.720
whereas over here,
I have density.
00:15:49.720 --> 00:15:51.730
So what do I do?
00:15:51.730 --> 00:15:56.040
Well, I realize that the density
can be obtained as follows.
00:15:56.040 --> 00:16:02.150
In the grand canonical ensemble,
the number of particles
00:16:02.150 --> 00:16:06.810
is in principal a
random variable.
00:16:06.810 --> 00:16:10.650
But that random variable
is governed by this e
00:16:10.650 --> 00:16:14.350
to the beta mu N that
controls how many you have.
00:16:14.350 --> 00:16:21.330
And if I take the log of
Q with respect to beta mu,
00:16:21.330 --> 00:16:23.840
I will generate
the average number,
00:16:23.840 --> 00:16:26.050
which in the
thermodynamic limit we
00:16:26.050 --> 00:16:28.940
expect to be the same thing
as what we thermodynamically
00:16:28.940 --> 00:16:30.750
would call the number.
00:16:30.750 --> 00:16:36.080
And this log Q we just
established is beta PV.
00:16:36.080 --> 00:16:40.600
So what we have here is
the derivative of beta VP
00:16:40.600 --> 00:16:43.270
with respect to beta mu.
00:16:43.270 --> 00:16:45.450
And if I do this at
constant temperature,
00:16:45.450 --> 00:16:47.240
the betas disappear.
00:16:47.240 --> 00:16:50.870
This is the same
thing as V dP by d
00:16:50.870 --> 00:16:54.190
mu at constant temperature.
00:16:54.190 --> 00:17:00.970
But also I can take the
derivative right here.
00:17:00.970 --> 00:17:02.800
So what happens?
00:17:02.800 --> 00:17:10.660
I will find that the
density which is N over V,
00:17:10.660 --> 00:17:14.660
is the derivative of this
expression with respect
00:17:14.660 --> 00:17:16.609
to beta mu.
00:17:16.609 --> 00:17:20.520
I go and I find that
there's a beta mu.
00:17:20.520 --> 00:17:24.060
If I take a derivative of
this with respect to beta mu,
00:17:24.060 --> 00:17:27.300
really it's exponential
of beta mu l.
00:17:27.300 --> 00:17:31.420
The derivative of it will
give me l e to the beta mu l.
00:17:31.420 --> 00:17:37.400
So what I get is a sum
over l e to the beta mu
00:17:37.400 --> 00:17:39.460
over lambda cubed.
00:17:39.460 --> 00:17:42.300
l gets repeated.
00:17:42.300 --> 00:17:45.590
And then I will
have l times this.
00:17:45.590 --> 00:17:50.300
So I will have vl bar divided
by l minus 1 factorial.
00:17:53.790 --> 00:17:56.370
So the series for
density is very
00:17:56.370 --> 00:17:58.680
much like the
series for pressure,
00:17:58.680 --> 00:18:03.980
except that you replace 1
over l factorial with 1 over l
00:18:03.980 --> 00:18:05.009
minus 1 factorial.
00:18:09.410 --> 00:18:12.690
So now my task is clear.
00:18:12.690 --> 00:18:18.350
What I should do
is I should solve
00:18:18.350 --> 00:18:24.090
given a particular
density for what mu is.
00:18:24.090 --> 00:18:27.100
Once I have mu as a
function of density,
00:18:27.100 --> 00:18:28.900
I can substitute it back here.
00:18:28.900 --> 00:18:32.860
And I will have pressure
as a function of density.
00:18:32.860 --> 00:18:36.090
Of course it's clear that
the right variable to look at
00:18:36.090 --> 00:18:43.710
is not mu, but x, which is e to
the beta mu over lambda cubed.
00:18:43.710 --> 00:18:45.150
This actually has a name.
00:18:45.150 --> 00:18:46.960
It's sometimes called fugacity.
00:18:54.230 --> 00:18:58.130
So the second
equation is telling me
00:18:58.130 --> 00:19:05.700
that the density I can write in
terms of the fugacity as a sum
00:19:05.700 --> 00:19:13.140
over l x to the l bl bar divided
by l minus 1 factorial, which
00:19:13.140 --> 00:19:16.860
if I were to sort of
write in its full details,
00:19:16.860 --> 00:19:24.430
it starts with b1 bar x
plus b2 bar x squared.
00:19:24.430 --> 00:19:28.890
Because there I have 0 factorial
or 1 factorial respectively.
00:19:28.890 --> 00:19:33.940
The next term will be
b3 bar over 2 x cubed.
00:19:33.940 --> 00:19:36.110
And this will go on and on.
00:19:41.880 --> 00:19:45.780
Also, let me remind
you what b1 bar is.
00:19:45.780 --> 00:19:51.740
b1 bar, for every
one of these b1 bars,
00:19:51.740 --> 00:19:55.140
I have to divide by 1
over V because of this V.
00:19:55.140 --> 00:19:57.980
And then I have to do the
integration that corresponds
00:19:57.980 --> 00:20:02.050
to the one cluster, which is
essentially one cluster going
00:20:02.050 --> 00:20:05.250
over the entirety of space,
which will give me V.
00:20:05.250 --> 00:20:09.230
So this is in fact 1.
00:20:09.230 --> 00:20:14.290
So basically this
coefficient here is 1.
00:20:14.290 --> 00:20:16.100
And then I will have
the corrections.
00:20:20.400 --> 00:20:23.600
So you say, well, I have
n as a function of x.
00:20:23.600 --> 00:20:26.890
I want x as a function of n.
00:20:26.890 --> 00:20:29.360
And I say, OK, that's
not that difficult.
00:20:29.360 --> 00:20:34.340
I will write that x
equals n minus b2 bar
00:20:34.340 --> 00:20:40.690
x squared minus b3 bar x
cubed over 2, and so forth.
00:20:40.690 --> 00:20:41.860
And I have now x.
00:20:46.910 --> 00:20:50.560
Maybe a few of
you are skeptical.
00:20:50.560 --> 00:20:52.330
Some of you don't
seem to be bothered.
00:20:55.440 --> 00:20:58.810
OK, so my claim is
that this is indeed
00:20:58.810 --> 00:21:04.860
a systematic way of solving
a series in which when
00:21:04.860 --> 00:21:10.475
the density goes to 0, you
expect x to also go to 0.
00:21:10.475 --> 00:21:14.500
And to lowest order, x will
be of the order of density.
00:21:14.500 --> 00:21:18.450
And these will be higher
order powers in density.
00:21:18.450 --> 00:21:24.620
So to get a systematic series
in density, all you need to do
00:21:24.620 --> 00:21:29.150
is to sort of work with a
series such as this keeping
00:21:29.150 --> 00:21:33.030
in mind what order you
have solved things to.
00:21:33.030 --> 00:21:35.900
So I claim that to
lowest order, this really
00:21:35.900 --> 00:21:37.960
just says that x is n.
00:21:37.960 --> 00:21:42.780
And then there are corrections
that are order of n squared.
00:21:42.780 --> 00:21:46.770
And to get the next order
term, all I need to do
00:21:46.770 --> 00:21:50.310
is to substitute the lower
order in this equation.
00:21:50.310 --> 00:21:55.300
So basically if I substitute
x equals n in this equation,
00:21:55.300 --> 00:22:00.900
I will get n minus b2 n square.
00:22:00.900 --> 00:22:04.990
If I substitute x over n here,
it will be a higher order term.
00:22:04.990 --> 00:22:06.970
And I claim that this
is the correct result
00:22:06.970 --> 00:22:08.380
to order of n cubed.
00:22:11.940 --> 00:22:15.250
And then to get the
result to order of n
00:22:15.250 --> 00:22:20.420
cubed, I substitute this back
into the original equation.
00:22:20.420 --> 00:22:26.050
So I start with n minus
b2 bar, the square
00:22:26.050 --> 00:22:29.750
of the previous solution
at the right order.
00:22:29.750 --> 00:22:34.630
So squaring this, I will
get n squared minus twice
00:22:34.630 --> 00:22:38.030
b2 bar n cubed.
00:22:38.030 --> 00:22:40.660
And the next term that would
be the square of this, which
00:22:40.660 --> 00:22:44.210
is order of n to the
fourth, I don't write down.
00:22:44.210 --> 00:22:49.200
And then I have
minus b3 bar over 2,
00:22:49.200 --> 00:22:51.670
the cube of the
previous solution.
00:22:51.670 --> 00:22:53.600
And at the right
order that I have,
00:22:53.600 --> 00:22:58.610
that's the same thing
as b3 bar over 2 n cubed
00:22:58.610 --> 00:23:02.700
and order of n to the fourth.
00:23:02.700 --> 00:23:05.750
And all I really need
to do at this order
00:23:05.750 --> 00:23:08.870
is to recognize that
I have two terms that
00:23:08.870 --> 00:23:10.500
are order of n cubed.
00:23:10.500 --> 00:23:16.590
Putting them together, I have
n minus b2 bar n squared.
00:23:16.590 --> 00:23:22.370
And then I have
plus 2b2 bar squared
00:23:22.370 --> 00:23:31.180
minus b3 bar over 2 n cubed
and order of n to the fourth.
00:23:41.000 --> 00:23:42.180
So now we erase this.
00:23:53.420 --> 00:23:58.700
OK, so now I have solved for
x in a power series in n.
00:23:58.700 --> 00:24:00.910
All I need to do
is to substitute
00:24:00.910 --> 00:24:03.160
in the power series for p.
00:24:03.160 --> 00:24:04.840
So let's write that down.
00:24:04.840 --> 00:24:11.720
The power series for
beta p is b1 bar times
00:24:11.720 --> 00:24:16.810
the density b2 bar
squared over 2 density
00:24:16.810 --> 00:24:23.670
square, b3 bar over
3 factorial, which
00:24:23.670 --> 00:24:27.480
is 6 n cubed, and so forth.
00:24:27.480 --> 00:24:30.190
But I only calculated
things to order of n cubed.
00:24:32.930 --> 00:24:36.740
Also, b1 bar is the
same thing as 1.
00:24:36.740 --> 00:24:40.740
So basically that's what
I should be working with.
00:24:40.740 --> 00:24:44.900
And so all I need to do
is to substitute-- oops,
00:24:44.900 --> 00:24:46.420
except that these are all x's.
00:24:54.170 --> 00:24:56.890
The series here
for beta p that I
00:24:56.890 --> 00:25:01.600
have is in powers
of this quantity x.
00:25:01.600 --> 00:25:07.450
And it is x that I had
calculated as a function of n.
00:25:07.450 --> 00:25:15.430
To the order that I calculated,
it is n minus b2n squared.
00:25:15.430 --> 00:25:26.360
And then I have 2b2
squared minus b3 over b3.
00:25:26.360 --> 00:25:30.850
Somehow this sounds
incorrect to me.
00:25:30.850 --> 00:25:33.222
Nope, that's fine.
00:25:33.222 --> 00:25:35.010
Because here I have 2.
00:25:35.010 --> 00:25:37.020
So that's correct.
00:25:37.020 --> 00:25:43.970
b3 over 2 n cubed--
so essentially,
00:25:43.970 --> 00:25:47.910
I just substituted for x here.
00:25:47.910 --> 00:25:51.630
The next order
term is going to be
00:25:51.630 --> 00:25:56.280
b2 over 2 times the square of x.
00:25:56.280 --> 00:26:00.550
The square of x will give
me a term that is n squared,
00:26:00.550 --> 00:26:07.110
a term that is from
here 2b2n cubed.
00:26:07.110 --> 00:26:11.800
Order of n to the fourth I
haven't calculated correctly.
00:26:11.800 --> 00:26:17.440
Order of n cubed
I have b3 over 6.
00:26:17.440 --> 00:26:21.850
And just x cubed is the same
thing as n cubed to this order.
00:26:21.850 --> 00:26:26.562
And I have not calculated
anything at the next order.
00:26:29.830 --> 00:26:33.544
So let's see what we have.
00:26:33.544 --> 00:26:36.120
We have n.
00:26:36.120 --> 00:26:40.550
At the next order, there are
two terms that are n squared.
00:26:40.550 --> 00:26:43.570
There is this term,
and there is that term.
00:26:43.570 --> 00:26:49.040
Putting them together, I will
get minus b2 over 2 n squared.
00:26:52.030 --> 00:26:55.140
At the next order, at
the order of n cubed,
00:26:55.140 --> 00:26:57.540
I have a bunch of terms.
00:26:57.540 --> 00:27:03.470
First of all, there
is this 2b2 squared.
00:27:03.470 --> 00:27:09.810
But then multiplying this with
this will subtract one b2.
00:27:09.810 --> 00:27:14.500
So I'm going to be
left with b2 squared.
00:27:14.500 --> 00:27:20.030
And then I have minus 1/2 b3.
00:27:20.030 --> 00:27:22.320
So that's one term.
00:27:22.320 --> 00:27:26.140
And then I have plus 1/6 b3.
00:27:26.140 --> 00:27:30.560
Minus 1/2 plus 1/6 is minus 1/3.
00:27:30.560 --> 00:27:35.210
So this is minus
b3 over 3 n cubed.
00:27:35.210 --> 00:27:39.325
And I haven't calculated
order of n to the fourth.
00:27:42.170 --> 00:27:44.095
And this is the formula for BP.
00:27:51.680 --> 00:27:55.865
So lowest order, I have
the ideal gas result.
00:27:59.170 --> 00:28:01.570
Actually, let me,
for simplicity,
00:28:01.570 --> 00:28:05.445
define the virial
coefficients in this fashion.
00:28:08.030 --> 00:28:13.410
The second order, I will
get a correction B2,
00:28:13.410 --> 00:28:16.210
which is minus 1/2 of b2 bar.
00:28:18.755 --> 00:28:24.990
So this is minus 1/2
of the diagram that
00:28:24.990 --> 00:28:30.640
corresponds to essentially
one of these lines
00:28:30.640 --> 00:28:32.950
that I have up here.
00:28:32.950 --> 00:28:34.770
And so what is it?
00:28:34.770 --> 00:28:42.140
It is minus 1/2 the integral
over the relative coordinate
00:28:42.140 --> 00:28:45.900
of e to the minus
beta v as a function
00:28:45.900 --> 00:28:47.940
of the relative
coordinate minus 1.
00:28:52.340 --> 00:28:57.580
Now earlier, we had in fact
calculated this result directly
00:28:57.580 --> 00:28:59.510
through the partition function.
00:28:59.510 --> 00:29:04.300
I did a calculation in which
I calculated the first term
00:29:04.300 --> 00:29:06.740
in the other way of
looking at things,
00:29:06.740 --> 00:29:12.060
in the cumulant expansion,
as a function of expansions
00:29:12.060 --> 00:29:13.280
in the potential.
00:29:13.280 --> 00:29:15.110
We saw that there
was a term that all
00:29:15.110 --> 00:29:16.840
was order of density squared.
00:29:16.840 --> 00:29:20.140
We summed all of the
terms to get this factor.
00:29:20.140 --> 00:29:23.270
And there was precisely
a factor of minus 1/2
00:29:23.270 --> 00:29:25.710
as a correction to
pressure once we
00:29:25.710 --> 00:29:27.772
took the derivative of
the partition function
00:29:27.772 --> 00:29:28.730
with respect to volume.
00:29:31.340 --> 00:29:38.600
OK, so the thing that is new is
really the next order term, B3,
00:29:38.600 --> 00:29:44.360
which is b2 squared
minus b3 bar over 3.
00:29:44.360 --> 00:29:48.230
OK, so what is that?
00:29:48.230 --> 00:29:52.340
Diagrammatically,
B2 was this factor
00:29:52.340 --> 00:29:58.480
that we calculated above and
is the square of this pair.
00:29:58.480 --> 00:30:03.360
And then I have to subtract
from that 1/3 of whatever
00:30:03.360 --> 00:30:06.290
goes into B3.
00:30:06.290 --> 00:30:10.770
Now remember, we said that B3,
I have to pick three points
00:30:10.770 --> 00:30:14.740
and make sure I link them
in all possible ways.
00:30:14.740 --> 00:30:19.910
So the diagrams that go into
B3, one of them is this.
00:30:19.910 --> 00:30:22.160
But then there are
three other ways
00:30:22.160 --> 00:30:26.743
of making a linked object,
which is these things.
00:30:31.950 --> 00:30:35.150
We also saw that if
I were to calculate
00:30:35.150 --> 00:30:39.010
the contribution of any
one of these objects,
00:30:39.010 --> 00:30:43.480
I can very easily choose
to measure my coordinates
00:30:43.480 --> 00:30:47.690
with respect to, say, the
point that is at their apex.
00:30:47.690 --> 00:30:49.730
And then I would
have one variable
00:30:49.730 --> 00:30:51.710
which is this distance,
one variable which
00:30:51.710 --> 00:30:53.000
is that distance.
00:30:53.000 --> 00:30:55.150
And independently,
each one of them
00:30:55.150 --> 00:31:00.030
would give me the factor
that I calculated before.
00:31:00.030 --> 00:31:06.050
In essence, all of these one
particle reducible graphs
00:31:06.050 --> 00:31:09.362
give a contribution,
which is the product
00:31:09.362 --> 00:31:13.690
of these single line graphs.
00:31:13.690 --> 00:31:14.770
There are three of them.
00:31:14.770 --> 00:31:18.630
Minus 1/3 precisely
cancels that.
00:31:18.630 --> 00:31:24.450
And so the calculation that
goes into the third virial
00:31:24.450 --> 00:31:31.670
coefficient ultimately will
only depend on this one graph.
00:31:31.670 --> 00:31:34.720
And we had anticipated
this before.
00:31:34.720 --> 00:31:36.640
When we were doing
things previously
00:31:36.640 --> 00:31:39.880
using the expansion of
the partition function,
00:31:39.880 --> 00:31:43.270
we saw that in this
cumulant expansion,
00:31:43.270 --> 00:31:46.210
only the cumulants
were appearing.
00:31:46.210 --> 00:31:48.360
And for calculation
of the cumulants,
00:31:48.360 --> 00:31:50.860
I had to do lots
of subtractions.
00:31:50.860 --> 00:31:53.280
And those subtractions
were genetically
00:31:53.280 --> 00:32:00.880
removing these one
particle reducible graphs.
00:32:00.880 --> 00:32:04.230
And this continues
to all orders.
00:32:04.230 --> 00:32:10.280
And the general formula is
that the l-th contribution here
00:32:10.280 --> 00:32:17.060
will be a factor of l minus 1
over l factorial, which is not
00:32:17.060 --> 00:32:19.750
so difficult to get by
following the procedures
00:32:19.750 --> 00:32:24.940
that I have described over here,
and then the part of bl that
00:32:24.940 --> 00:32:28.150
is one particle irreducible.
00:32:33.360 --> 00:32:38.600
So this is the
eventual result for how
00:32:38.600 --> 00:32:44.760
you would be able, given some
particular form of the two body
00:32:44.760 --> 00:32:51.120
interaction, to calculate
an expansion for pressure
00:32:51.120 --> 00:32:55.960
in powers of density, how the
coefficients of that expansion
00:32:55.960 --> 00:32:59.261
are related to properties
of this potential
00:32:59.261 --> 00:33:00.760
through this
diagrammatic expansion.
00:33:10.160 --> 00:33:13.100
So this is formally correct.
00:33:13.100 --> 00:33:16.440
And then the next question
is, is it practical?
00:33:16.440 --> 00:33:18.380
Is it useful?
00:33:18.380 --> 00:33:23.310
So we need to start computing
things to see something
00:33:23.310 --> 00:33:27.524
about the usefulness of this
series for some particular type
00:33:27.524 --> 00:33:28.065
of potential.
00:33:31.060 --> 00:33:34.790
So we are going to look
at the kind of potential
00:33:34.790 --> 00:33:37.370
that I already
described for you.
00:33:37.370 --> 00:33:43.800
That is, if I look at two
particles in a gas that
00:33:43.800 --> 00:33:47.720
are separated by
an amount r-- let's
00:33:47.720 --> 00:33:50.630
imagine the potential
only depends
00:33:50.630 --> 00:33:54.580
on the relative distance, not
orientation or anything else.
00:33:54.580 --> 00:33:57.610
We said that basically,
at large distances,
00:33:57.610 --> 00:34:02.080
the potential is attractive
because of van der Waals.
00:34:02.080 --> 00:34:05.890
At short distances, the
potential is repulsive.
00:34:05.890 --> 00:34:09.350
And so you have a general
form such as this.
00:34:12.900 --> 00:34:15.610
Now, if I want to
do calculations,
00:34:15.610 --> 00:34:18.530
it would be useful for
me to have something
00:34:18.530 --> 00:34:22.070
that I can do
calculations exactly
00:34:22.070 --> 00:34:24.909
with and get an estimate.
00:34:24.909 --> 00:34:27.659
So what I'm going to do is
to replace that potential,
00:34:27.659 --> 00:34:31.130
essentially, with a hard wall.
00:34:31.130 --> 00:34:38.469
So my approximation to the
potential is that my v of r
00:34:38.469 --> 00:34:42.135
is infinite for
distances, separations
00:34:42.135 --> 00:34:45.060
that are less than some r0.
00:34:45.060 --> 00:34:48.480
So basically, I define
some kind of r0,
00:34:48.480 --> 00:34:52.389
which is the-- if you think
of them as billiard balls,
00:34:52.389 --> 00:34:56.380
it's related to the closest
distance of approach.
00:34:56.380 --> 00:34:59.740
This potential at
large distances
00:34:59.740 --> 00:35:06.660
has typical van der
Waals form, by which
00:35:06.660 --> 00:35:10.580
I mean it falls off as
a function of separation
00:35:10.580 --> 00:35:13.180
with the sixth power.
00:35:13.180 --> 00:35:14.140
It is attractive.
00:35:14.140 --> 00:35:16.710
So I put a minus sign here.
00:35:16.710 --> 00:35:20.440
In order to make sure that I
get eventually the dimensions
00:35:20.440 --> 00:35:25.820
right, the coefficient that goes
here I write as r0 with a u0
00:35:25.820 --> 00:35:33.350
up here such that if I assume
that the potential is precisely
00:35:33.350 --> 00:35:42.870
this for all r that are
greater than r0, then
00:35:42.870 --> 00:35:47.410
I'm replacing the
actual potential
00:35:47.410 --> 00:35:50.710
by something like this.
00:35:50.710 --> 00:35:54.550
And the minimum
depth of potential
00:35:54.550 --> 00:35:58.010
will occur at this cusp
over here at minus u0.
00:36:08.190 --> 00:36:13.090
So why did I do that?
00:36:13.090 --> 00:36:16.410
Because now I can
calculate with that
00:36:16.410 --> 00:36:18.850
what the second
virial coefficient is.
00:36:18.850 --> 00:36:21.870
So what is B2?
00:36:21.870 --> 00:36:31.060
B2 is minus 1/2 the integral
of relative potential.
00:36:35.100 --> 00:36:40.620
Now this integral
will take two parts
00:36:40.620 --> 00:36:44.800
from these two
different contributions.
00:36:44.800 --> 00:36:50.050
One part is when I'm in the
regime where the particles are
00:36:50.050 --> 00:36:52.570
excluded, the
potential is infinity.
00:36:52.570 --> 00:36:57.430
And there, f is simply minus 1.
00:36:57.430 --> 00:37:02.180
So that minus 1 then gets
integrated from 0 to r,
00:37:02.180 --> 00:37:04.700
giving me the volume
that is excluded.
00:37:04.700 --> 00:37:07.750
So what I will get
here is a minus 1
00:37:07.750 --> 00:37:11.400
times the excluded volume.
00:37:11.400 --> 00:37:16.680
That's the first part, where
the volume is, of course,
00:37:16.680 --> 00:37:21.628
4 pi over 3 r0 cubed for
the volume of the sphere.
00:37:24.700 --> 00:37:27.180
And then I have to
add the part where
00:37:27.180 --> 00:37:32.000
I go from f0 all
the way to infinity.
00:37:32.000 --> 00:37:35.110
The potential is
vertically symmetric.
00:37:35.110 --> 00:37:41.930
So d cubed r becomes
4 pi r squared dr.
00:37:41.930 --> 00:37:49.390
And I have to integrate
e to the minus
00:37:49.390 --> 00:37:52.700
beta times the attractive
part of the potential.
00:37:52.700 --> 00:37:57.900
So it's beta u0 r0 over
r to the sixth minus 1.
00:38:03.390 --> 00:38:06.780
So what I will do,
I will additionally
00:38:06.780 --> 00:38:11.280
assume that I'm in the
range of parameters
00:38:11.280 --> 00:38:16.200
where this beta u0
is much less than 1,
00:38:16.200 --> 00:38:19.840
that is, at temperatures
that are higher
00:38:19.840 --> 00:38:22.880
compared to the depth of
this potential converted
00:38:22.880 --> 00:38:24.910
to units of kT.
00:38:24.910 --> 00:38:29.010
And if that is the case,
then I can expand this.
00:38:29.010 --> 00:38:31.700
And the expansion
to the lowest order
00:38:31.700 --> 00:38:37.380
will simply give me beta u0
r0 over r to the sixth power.
00:38:37.380 --> 00:38:40.485
And I will ignore
higher order terms.
00:38:40.485 --> 00:38:44.820
So this, if you like, is
order of beta u0 squared.
00:38:49.570 --> 00:38:54.530
Now, having done that, then the
second integral becomes simple.
00:38:54.530 --> 00:38:56.700
Because I have to
integrate r squared
00:38:56.700 --> 00:39:01.330
divided by r to the fourth,
which is r to the minus 4.
00:39:01.330 --> 00:39:09.320
Integral of r to the minus
4 gives me r to the minus 3.
00:39:09.320 --> 00:39:13.020
And then there's a factor of
minus 1/3 from the integration.
00:39:13.020 --> 00:39:18.280
It has to be evaluated
between 0 and infinity.
00:39:18.280 --> 00:39:26.040
And so the final answer,
then, for B2 is minus 1/2.
00:39:26.040 --> 00:39:32.211
I have minus 4 pi over 3 r0
cubed from the excluded volume
00:39:32.211 --> 00:39:32.710
part.
00:39:35.850 --> 00:39:49.180
From here, I will get a 4
pi beta u0 r0 to the sixth.
00:39:52.700 --> 00:39:57.130
And then I will get this
factor of r the minus 3
00:39:57.130 --> 00:39:59.500
over 3 evaluated
between infinity,
00:39:59.500 --> 00:40:05.740
which gives me 0 and r0.
00:40:13.720 --> 00:40:16.530
So you can see that with
this potential, actually
00:40:16.530 --> 00:40:22.900
both terms are proportional
to 4 pi r0 cubed over 3.
00:40:22.900 --> 00:40:29.200
And so I can write the
answer as minus omega over 2
00:40:29.200 --> 00:40:33.760
where I have defined omega
to be this 4 pi r0 cubed
00:40:33.760 --> 00:40:36.505
over 3, which is
the excluded volume.
00:40:39.240 --> 00:40:46.404
Basically, it's the volume that
is excluded from exploration
00:40:46.404 --> 00:40:47.570
when you have two particles.
00:40:47.570 --> 00:40:52.900
Because their center of mass
cannot come as close as r0.
00:40:52.900 --> 00:40:58.330
So that 1/2 appears here as 1.
00:40:58.330 --> 00:41:01.810
Because I already
took care of that.
00:41:01.810 --> 00:41:05.400
This rest of the coefficients
are proportional to this
00:41:05.400 --> 00:41:08.690
except with a factor
of minus beta u0.
00:41:08.690 --> 00:41:12.280
So I get minus beta u0.
00:41:12.280 --> 00:41:19.351
So this is your second
virial coefficient for this.
00:41:19.351 --> 00:41:21.100
AUDIENCE: It should
be positive out front.
00:41:21.100 --> 00:41:21.849
PROFESSOR: Pardon?
00:41:21.849 --> 00:41:26.544
AUDIENCE: It should be positive
out front, [INAUDIBLE]?
00:41:26.544 --> 00:41:28.516
PROFESSOR: Yes, because
there was a minus,
00:41:28.516 --> 00:41:29.502
and there's a minus.
00:41:29.502 --> 00:41:30.488
There's a plus.
00:41:50.701 --> 00:41:51.687
Let's keep that.
00:42:00.620 --> 00:42:05.620
So what we have
so far, we've said
00:42:05.620 --> 00:42:11.940
that beta times the pressure
starts with ideal gas behavior
00:42:11.940 --> 00:42:13.644
n.
00:42:13.644 --> 00:42:17.440
And then I have the
next correction,
00:42:17.440 --> 00:42:22.860
which is this B2
multiplying n squared.
00:42:22.860 --> 00:42:27.970
So I will have n squared
times this coefficient
00:42:27.970 --> 00:42:31.070
that I have calculated for
this type of gas, which
00:42:31.070 --> 00:42:34.710
is omega over 2 1 minus beta u0.
00:42:34.710 --> 00:42:37.540
And presumably there are
higher order terms in this.
00:42:42.860 --> 00:42:47.840
So whenever you see
something like this,
00:42:47.840 --> 00:42:51.760
then you have to
start thinking about,
00:42:51.760 --> 00:42:55.680
is this a good expansion?
00:42:55.680 --> 00:42:59.690
So let's think about
how this expansion could
00:42:59.690 --> 00:43:00.790
have become problematic.
00:43:03.690 --> 00:43:08.850
Actually, there is already one
thing that I should have noted
00:43:08.850 --> 00:43:17.790
and I didn't, which is short
versus long range potential.
00:43:21.890 --> 00:43:25.680
Now, it was nice
for this potential
00:43:25.680 --> 00:43:29.460
that I got an answer
that was in the form
00:43:29.460 --> 00:43:33.440
that I could write
down, factor out omega.
00:43:33.440 --> 00:43:35.570
You can say, well,
would you have always
00:43:35.570 --> 00:43:38.630
been able to do something
similar to that?
00:43:38.630 --> 00:43:40.630
Because you see,
ultimately what this
00:43:40.630 --> 00:43:47.950
says is that in order for all
of these terms in this expansion
00:43:47.950 --> 00:43:54.560
to be dimensionally correct,
the next term is a density,
00:43:54.560 --> 00:43:57.210
has dimensions of
inverse volume,
00:43:57.210 --> 00:43:58.720
density squared
compared to this.
00:43:58.720 --> 00:44:02.780
So it should be compensated
by some factor of volume.
00:44:02.780 --> 00:44:05.530
And we can see that
this factor of volume
00:44:05.530 --> 00:44:08.250
came from something
that was of the order
00:44:08.250 --> 00:44:12.986
of this size of the
molecule, this r0.
00:44:12.986 --> 00:44:16.860
OK, so it's interesting.
00:44:16.860 --> 00:44:20.830
It says that it's really
the short-range part
00:44:20.830 --> 00:44:26.770
of the potential that seems
to be setting the correction.
00:44:26.770 --> 00:44:29.500
Now, where did that come from?
00:44:29.500 --> 00:44:31.670
Well, there was one
place that I had
00:44:31.670 --> 00:44:34.920
to do an integration
over the potential.
00:44:34.920 --> 00:44:36.850
And I found that
the integration was
00:44:36.850 --> 00:44:40.876
dominated by the lower range.
00:44:40.876 --> 00:44:43.710
That was over here.
00:44:43.710 --> 00:44:45.395
Where would this
become difficult?
00:44:49.454 --> 00:44:52.300
AUDIENCE: In a small box?
00:44:52.300 --> 00:44:53.876
PROFESSOR: In a
small box, but we
00:44:53.876 --> 00:44:56.050
are always taking the
thermodynamic limit
00:44:56.050 --> 00:44:57.721
where V goes to infinity.
00:44:57.721 --> 00:44:59.685
AUDIENCE: Or inverse square log?
00:44:59.685 --> 00:45:01.790
PROFESSOR: Inverse square
log, yes, that's right.
00:45:01.790 --> 00:45:05.090
So who says that this
integral should converge?
00:45:05.090 --> 00:45:10.760
I'm doing an integral d cubed
r, something like v of r.
00:45:10.760 --> 00:45:14.240
And convergence was the
reason why this integral
00:45:14.240 --> 00:45:16.695
was dominated by short distance.
00:45:16.695 --> 00:45:22.490
If this potential goes
like 1 over r cubed,
00:45:22.490 --> 00:45:26.050
then it will logarithmically
depend on the size of the box.
00:45:26.050 --> 00:45:29.320
For Coulomb interaction,
1/r potential
00:45:29.320 --> 00:45:30.940
you can't even think about it.
00:45:30.940 --> 00:45:34.030
It's just too divergent.
00:45:34.030 --> 00:45:38.640
So this expansion will
fail for potentials
00:45:38.640 --> 00:45:43.300
that have tails that
are decaying as 1
00:45:43.300 --> 00:45:46.680
over r cubed or even slower.
00:45:46.680 --> 00:45:50.670
Fortunately, that's not
the case for van der Waals'
00:45:50.670 --> 00:45:53.320
potential and
typical potentials.
00:45:53.320 --> 00:45:56.100
But if you have a plasma,
you have to worry about this.
00:45:56.100 --> 00:45:58.610
And that's why I was
also saying last time
00:45:58.610 --> 00:46:01.597
that the typical expansions
that you have to do for plasmas
00:46:01.597 --> 00:46:02.180
are different.
00:46:04.930 --> 00:46:10.050
So given that we are
dominated by the short range,
00:46:10.050 --> 00:46:14.380
what is the correction
that I've obtained
00:46:14.380 --> 00:46:16.510
compared to the first term?
00:46:16.510 --> 00:46:19.915
So essentially, I have
calculated a second order term
00:46:19.915 --> 00:46:23.450
that is of the order of this
divided by the first order
00:46:23.450 --> 00:46:25.940
term and pressure
that was density.
00:46:25.940 --> 00:46:31.020
And we find that this is
of the order of n omega.
00:46:31.020 --> 00:46:36.810
How many particles are within
the range of interaction?
00:46:36.810 --> 00:46:44.720
And this would be
kind of like the ratio
00:46:44.720 --> 00:46:53.390
of the density of liquid
to the density of gas.
00:46:53.390 --> 00:46:59.648
Because the density of
liquid would be related-- oh,
00:46:59.648 --> 00:47:00.731
it's the other way around.
00:47:04.030 --> 00:47:09.260
The density of liquid
would be 1 over the volume
00:47:09.260 --> 00:47:10.860
that one particle occupies.
00:47:10.860 --> 00:47:15.410
So the density of liquid
would go in the denominator.
00:47:15.410 --> 00:47:24.020
The density of gas is this
n that I have over here.
00:47:24.020 --> 00:47:26.420
And again, for the
gas in this room,
00:47:26.420 --> 00:47:29.980
this ratio is of the order
of 10 to the minus 3.
00:47:29.980 --> 00:47:31.780
And we are safe.
00:47:31.780 --> 00:47:35.300
But if I were to
start compressing this
00:47:35.300 --> 00:47:38.500
so that I go to higher
and higher densities,
00:47:38.500 --> 00:47:41.690
ultimately I say that this
second order term becomes
00:47:41.690 --> 00:47:43.300
of the order of the
first order term.
00:47:43.300 --> 00:47:45.730
And then perturbation
doesn't make sense.
00:47:45.730 --> 00:47:49.430
Naturally, the reason for
that is I haven't calculated.
00:47:49.430 --> 00:47:51.530
But typically what you
find is that if you
00:47:51.530 --> 00:47:53.360
go to higher and
higher order terms
00:47:53.360 --> 00:47:57.400
in the series, in most cases,
but clearly not in all cases,
00:47:57.400 --> 00:48:01.220
the ratio of successive
terms is more or less
00:48:01.220 --> 00:48:03.990
set by the ratio
of the first terms.
00:48:03.990 --> 00:48:06.770
It has to be
dimensionally correct.
00:48:06.770 --> 00:48:09.170
And we've established that
the typical dimension that
00:48:09.170 --> 00:48:15.450
is controlling everything is
the volume of the particle.
00:48:15.450 --> 00:48:20.590
So this expansion will fail
for long range potentials,
00:48:20.590 --> 00:48:23.980
for going to
liquid-like densities.
00:48:23.980 --> 00:48:27.570
But also I made
another thing, which
00:48:27.570 --> 00:48:30.540
is that I assumed that that
could expand this exponential.
00:48:33.140 --> 00:48:35.820
And then in principle,
there are other difficulties
00:48:35.820 --> 00:48:40.070
that could arise if this
condition that I wrote here
00:48:40.070 --> 00:48:41.080
is violated.
00:48:41.080 --> 00:48:46.080
If beta u0 is greater than 1,
then this coefficient by itself
00:48:46.080 --> 00:48:49.180
becomes large exponentially.
00:48:49.180 --> 00:48:53.160
And then you would expect that
higher order terms will also
00:48:53.160 --> 00:48:56.350
get more factors of this
exponential and potential.
00:48:56.350 --> 00:48:58.130
And things will blow up on you.
00:48:58.130 --> 00:49:01.390
So it will also
have difficulties
00:49:01.390 --> 00:49:06.550
at low temperatures for
attractive potentials.
00:49:10.400 --> 00:49:13.010
Again, the reason
for that is obvious.
00:49:13.010 --> 00:49:15.320
If you have an
attractive potential,
00:49:15.320 --> 00:49:17.380
you go to low
enough temperature,
00:49:17.380 --> 00:49:20.330
and the ground state is
everybody sticking together--
00:49:20.330 --> 00:49:25.560
looks nothing like a gas, looks
like a solid or something.
00:49:25.560 --> 00:49:28.170
So these are the
kind of limitations
00:49:28.170 --> 00:49:29.667
that one has in this series.
00:49:54.060 --> 00:50:01.650
OK, let's be brave
and do some more
00:50:01.650 --> 00:50:03.500
rearrangements of this equation.
00:50:03.500 --> 00:50:06.110
So I have that to
this order, beta P
00:50:06.110 --> 00:50:10.930
is n plus n squared
this excluded
00:50:10.930 --> 00:50:14.680
volume over 2 1 minus beta u0.
00:50:17.870 --> 00:50:22.430
And then there's higher
order terms of course.
00:50:22.430 --> 00:50:25.800
And then I notice that there are
two terms on this equation that
00:50:25.800 --> 00:50:27.860
are proportional to beta.
00:50:27.860 --> 00:50:30.680
And I say, why not put
both of them together?
00:50:30.680 --> 00:50:32.650
So I will have beta.
00:50:32.650 --> 00:50:34.910
Bring that term to
this side, and it
00:50:34.910 --> 00:50:42.800
becomes P plus n
squared omega over 2 u0.
00:50:45.420 --> 00:50:48.510
And then what is left
on the other side?
00:50:48.510 --> 00:50:49.950
Let's factor out the n.
00:50:49.950 --> 00:50:53.650
I expect this to be a series in
higher and higher powers of n.
00:50:53.650 --> 00:50:56.420
And the first
correction to 1 comes
00:50:56.420 --> 00:50:59.130
from here, which
is n omega over 2.
00:50:59.130 --> 00:51:01.550
And I expect there to
be higher order terms.
00:51:13.830 --> 00:51:18.660
Now again, to order of n squared
that I have calculated things
00:51:18.660 --> 00:51:22.310
correctly, this
expression is no different
00:51:22.310 --> 00:51:27.870
from the following expression--
1 minus n omega over 2.
00:51:27.870 --> 00:51:31.910
Again, there will be
higher order terms in n.
00:51:31.910 --> 00:51:35.100
But the order of n squared,
both of these equations,
00:51:35.100 --> 00:51:36.325
expressions, are equivalent.
00:51:41.220 --> 00:51:46.740
So if I now ignore
higher order terms,
00:51:46.740 --> 00:51:54.700
this whole thing is equivalent
to P plus n squared u--
00:51:54.700 --> 00:51:57.790
well, let's write this in
the following fashion-- p
00:51:57.790 --> 00:52:05.220
plus u0 omega over 2 N over
V, which is density squared.
00:52:07.790 --> 00:52:13.350
And actually this side, if
I were to multiply by V,
00:52:13.350 --> 00:52:14.860
what do I get?
00:52:14.860 --> 00:52:20.055
This becomes numerator
V, denominator V minus N
00:52:20.055 --> 00:52:21.710
omega over 2.
00:52:21.710 --> 00:52:27.840
Let's multiply by V
minus N omega over 2.
00:52:27.840 --> 00:52:32.900
And the right-hand
side will be--
00:52:32.900 --> 00:52:36.810
AUDIENCE: Is that where
it should be there?
00:52:36.810 --> 00:52:37.866
PROFESSOR: What happened?
00:52:37.866 --> 00:52:40.346
AUDIENCE: This one, [INAUDIBLE].
00:52:45.802 --> 00:52:52.860
PROFESSOR: Yes, so if I multiply
this, which is N over V, by V,
00:52:52.860 --> 00:52:56.920
I will get N,
good, which removes
00:52:56.920 --> 00:52:58.490
the difficulty that I had.
00:52:58.490 --> 00:53:02.790
Because now I multiply by kT.
00:53:02.790 --> 00:53:05.460
And the left hand
side disappears.
00:53:05.460 --> 00:53:07.650
And on the right hand
side, I will get NkT.
00:53:13.940 --> 00:53:18.830
So we'll spend some
time on this equation
00:53:18.830 --> 00:53:22.105
that you will likely recognize
as the van der Waals equation.
00:53:28.790 --> 00:53:34.810
I kind of justify it by
rearranging this series.
00:53:34.810 --> 00:53:37.390
But van der Waals
himself, of course,
00:53:37.390 --> 00:53:40.890
had a different way
of justifying it,
00:53:40.890 --> 00:53:45.080
which is that basically if
we think about the ideal gas,
00:53:45.080 --> 00:53:51.260
and you have particles that are
moving within some volume V,
00:53:51.260 --> 00:53:54.600
if you have excluded
volume interactions,
00:53:54.600 --> 00:53:59.420
then some of this volume
is no longer available.
00:53:59.420 --> 00:54:01.690
And so maybe what
you should do is
00:54:01.690 --> 00:54:05.300
you should reduce the
volume by an amount that
00:54:05.300 --> 00:54:07.470
is proportional to the
number of particles.
00:54:10.320 --> 00:54:14.410
This factor of 1/2 is
actually very interesting.
00:54:14.410 --> 00:54:16.510
Because it is correct.
00:54:16.510 --> 00:54:20.610
And I see that a
number of people
00:54:20.610 --> 00:54:24.020
in well known
journals, et cetera,
00:54:24.020 --> 00:54:27.950
write that the excluded
volume should be essentially
00:54:27.950 --> 00:54:31.630
n times the volume that is
excluded around each particle.
00:54:31.630 --> 00:54:32.560
It's not that.
00:54:32.560 --> 00:54:34.480
It is 1/2 of that.
00:54:34.480 --> 00:54:36.600
And I'll leave you
to mull on that.
00:54:36.600 --> 00:54:39.790
Because we will
justify it later on.
00:54:39.790 --> 00:54:41.990
But in the meantime,
you can think
00:54:41.990 --> 00:54:46.860
about why the factor
of 1/2 is there.
00:54:46.860 --> 00:54:50.450
The other issue
is that-- so there
00:54:50.450 --> 00:54:54.100
has to be a correction
to the volume.
00:54:54.100 --> 00:54:57.070
And the kind of
hand-waving statement
00:54:57.070 --> 00:54:59.960
that you make about the
correction to pressure
00:54:59.960 --> 00:55:05.730
is that if you think about the
particle that is in the middle,
00:55:05.730 --> 00:55:10.190
it is being attracted by
everybody, whereas when
00:55:10.190 --> 00:55:13.750
it comes to the surface, it
is really being attracted
00:55:13.750 --> 00:55:17.760
by things that are
half of the space.
00:55:17.760 --> 00:55:19.810
So there is an
effective potential
00:55:19.810 --> 00:55:23.850
that the particles feel
from the collective action
00:55:23.850 --> 00:55:27.790
of all the others, which
is slightly less steep when
00:55:27.790 --> 00:55:31.140
you approach the boundaries.
00:55:31.140 --> 00:55:36.030
And therefore, you can either
think that because of this,
00:55:36.030 --> 00:55:38.590
there's less density that
you have at the boundary.
00:55:38.590 --> 00:55:41.050
Less density will give
you less pressure.
00:55:41.050 --> 00:55:42.770
Or if you have a
particle that is
00:55:42.770 --> 00:55:45.780
kind of moving
towards the wall, it
00:55:45.780 --> 00:55:47.840
is being pulled
back so it doesn't
00:55:47.840 --> 00:55:51.110
hit the wall as strongly
as you would expect,
00:55:51.110 --> 00:55:53.440
that would give the
pressure of the ideal gas.
00:55:53.440 --> 00:55:57.730
So there is a pressure that
has to be reduced related
00:55:57.730 --> 00:56:00.910
to the strength of the
potential and something that
00:56:00.910 --> 00:56:03.160
has to do with all of
the other particles.
00:56:03.160 --> 00:56:06.470
And there's density
squared will appear there.
00:56:06.470 --> 00:56:10.470
We will have a more full
justification of this equation
00:56:10.470 --> 00:56:11.690
later on.
00:56:11.690 --> 00:56:16.140
But for the time being, let's
sort of sit with this equation
00:56:16.140 --> 00:56:18.510
and think about its
consequences for awhile.
00:56:25.200 --> 00:56:27.560
Because the thing that
we would like to do
00:56:27.560 --> 00:56:35.060
is we have come from a
perspective of looking
00:56:35.060 --> 00:56:39.710
at the ideal gas and how the
pressure of the ideal gas
00:56:39.710 --> 00:56:43.610
starts to get corrected
because of the interactions.
00:56:43.610 --> 00:56:46.340
Of course, things
become interesting
00:56:46.340 --> 00:56:48.470
when you go to the dense limit.
00:56:48.470 --> 00:56:51.040
And then the gas becomes
something like a liquid.
00:56:51.040 --> 00:56:53.830
And you have transitions
and things like that.
00:56:53.830 --> 00:56:58.570
So really, it's the other limit,
the dense, highly interacting
00:56:58.570 --> 00:57:00.740
limited that is interesting.
00:57:00.740 --> 00:57:05.640
And to get that, we
have few choices.
00:57:05.640 --> 00:57:10.770
Either I have to somehow sum
many, many terms in the series,
00:57:10.770 --> 00:57:14.700
which will be very difficult,
and we can't do that,
00:57:14.700 --> 00:57:18.155
or you can make some kind of
approximation, rearrangement,
00:57:18.155 --> 00:57:19.490
and a guess.
00:57:19.490 --> 00:57:24.100
And this is what the van der
Waals equation is based on.
00:57:24.100 --> 00:57:27.950
I made the guess here
by somewhat rearranging
00:57:27.950 --> 00:57:30.490
and re-summing the
terms in this series.
00:57:30.490 --> 00:57:34.390
But I will give you shortly
a different justification
00:57:34.390 --> 00:57:37.920
that is more transparent
and tells you immediately
00:57:37.920 --> 00:57:39.740
what the limitations are.
00:57:39.740 --> 00:57:42.550
But basically that's
why we are going
00:57:42.550 --> 00:57:45.110
to spend some time
with this equation.
00:57:45.110 --> 00:57:48.640
Because ultimately, we
are hoping to transition
00:57:48.640 --> 00:57:51.502
from the weakly interacting
case to the strongly interacting
00:57:51.502 --> 00:57:52.002
case.
00:57:58.360 --> 00:57:59.277
Yes.
00:57:59.277 --> 00:58:00.152
AUDIENCE: [INAUDIBLE]
00:58:09.050 --> 00:58:12.190
PROFESSOR: No, no,
no, no, omega was
00:58:12.190 --> 00:58:16.030
defined as the volume
excluded around one particle.
00:58:16.030 --> 00:58:19.070
So if you're thinking
about billiard balls,
00:58:19.070 --> 00:58:21.490
r0 is the diameter.
00:58:21.490 --> 00:58:23.320
It's not the radius.
00:58:23.320 --> 00:58:28.780
And 4 pi over 3 r0 cubed
is 8 times the diameter
00:58:28.780 --> 00:58:30.980
of a single billiard ball.
00:58:30.980 --> 00:58:34.030
So the correction that
we get, if you like,
00:58:34.030 --> 00:58:38.860
is 4 times the volume of
a billiard ball multiplied
00:58:38.860 --> 00:58:40.472
by the number of billiard balls.
00:58:51.350 --> 00:58:52.249
Where was I?
00:59:10.840 --> 00:59:17.060
OK, so what this equation
gives you, the van der Waals,
00:59:17.060 --> 00:59:20.350
is an expression for
how the pressure behaves
00:59:20.350 --> 00:59:21.765
as a function of volume.
00:59:26.390 --> 00:59:31.970
Actually, it would be nicer
if we were to sort of replace
00:59:31.970 --> 00:59:35.150
this by volume per
particle, which
00:59:35.150 --> 00:59:36.670
would be the inverse density.
00:59:36.670 --> 00:59:39.980
But you can use
one or the other.
00:59:39.980 --> 00:59:42.320
It doesn't matter.
00:59:42.320 --> 00:59:46.410
Now, what you find is
that there is, first
00:59:46.410 --> 00:59:51.510
of all, a limitation
to the volume.
00:59:51.510 --> 00:59:54.880
So basically, none
of your cares are
00:59:54.880 --> 00:59:59.520
going to go to volumes that
are lower than n omega over 2.
00:59:59.520 --> 01:00:04.040
So basically there's a barrier
here that occurs over here.
01:00:04.040 --> 01:00:06.190
But if you go to
the other limit,
01:00:06.190 --> 01:00:10.720
where you go to large volumes,
you can ignore terms like this.
01:00:10.720 --> 01:00:15.610
And then you get back the
kind of ideal gas behavior.
01:00:15.610 --> 01:00:19.060
So basically, in
one limit, where
01:00:19.060 --> 01:00:22.720
you are either at
high temperatures--
01:00:22.720 --> 01:00:27.340
and at high temperatures,
essentially, the correction
01:00:27.340 --> 01:00:30.660
here will also be negligible.
01:00:30.660 --> 01:00:34.090
Or you are at high
values of the volume.
01:00:34.090 --> 01:00:38.160
You get isotherms that are
very much like the isotherms
01:00:38.160 --> 01:00:41.180
that you have for the
ideal gas, except that
01:00:41.180 --> 01:00:45.760
rather than asymptote
to 0, PV going like NkT,
01:00:45.760 --> 01:00:50.910
you asymptote to
this excluded volume.
01:00:50.910 --> 01:00:54.440
So this is for high T, T large.
01:00:57.170 --> 01:00:58.680
T larger than what?
01:00:58.680 --> 01:01:01.910
Well essentially,
what happens is
01:01:01.910 --> 01:01:07.020
that if I look at the pressure,
I can write it in this fashion
01:01:07.020 --> 01:01:07.520
also.
01:01:07.520 --> 01:01:15.060
It is NkT divided by V
minus N omega over 2.
01:01:15.060 --> 01:01:18.180
So that's the term that
dominates at high temperature.
01:01:18.180 --> 01:01:20.640
It's proportional to kT.
01:01:20.640 --> 01:01:26.850
But then there's the subtraction
u0 omega over 2 N/V squared.
01:01:26.850 --> 01:01:30.270
And again, this term is not
so important at large volume.
01:01:30.270 --> 01:01:32.300
Because at large
volume, this 1/V
01:01:32.300 --> 01:01:35.350
is more dominant than
1 over V squared.
01:01:35.350 --> 01:01:38.880
But as you go to
lower temperatures
01:01:38.880 --> 01:01:42.980
and intermediate volumes,
then essentially you
01:01:42.980 --> 01:01:47.140
have potentially a correction
that falls off as minus 1
01:01:47.140 --> 01:01:49.110
over V squared.
01:01:49.110 --> 01:01:53.620
And so this correction that
falls off as 1 over V squared
01:01:53.620 --> 01:01:57.880
can potentially
modify your curve,
01:01:57.880 --> 01:02:03.370
bring it down, and then give
it a structure such as this.
01:02:03.370 --> 01:02:06.350
So this is T less than.
01:02:06.350 --> 01:02:10.320
And clearly, between these
two types of behavior,
01:02:10.320 --> 01:02:14.100
where there is
monotonic behavior
01:02:14.100 --> 01:02:16.880
or non-monotonic
behavior, there has
01:02:16.880 --> 01:02:20.860
to be a limiting curve that,
let's say, does something
01:02:20.860 --> 01:02:27.190
like this, comes tangentially
to the horizontal axis,
01:02:27.190 --> 01:02:30.800
and then goes on like this.
01:02:30.800 --> 01:02:33.870
So this would be
for T equals Tc.
01:02:33.870 --> 01:02:37.543
This is T greater than
Tc, T less than Tc.
01:02:47.700 --> 01:02:50.140
OK, so that's fine,
except that now we
01:02:50.140 --> 01:02:53.080
have encountered the difficulty.
01:02:53.080 --> 01:02:57.040
Because one of the things
that we had established
01:02:57.040 --> 01:03:10.210
for thermodynamic stability
was that delta P delta
01:03:10.210 --> 01:03:11.870
V had to be negative.
01:03:15.870 --> 01:03:20.690
And the ideal gas
curve and portions
01:03:20.690 --> 01:03:26.480
of this curve which
have a negative slope
01:03:26.480 --> 01:03:29.280
are all consistent with this.
01:03:29.280 --> 01:03:39.720
But this portion over here
where dP by dV is positive,
01:03:39.720 --> 01:03:42.350
it kind of violates
the condition
01:03:42.350 --> 01:03:48.380
that the compressibility
kappa T, which is minus 1/V dV
01:03:48.380 --> 01:03:51.550
by dP at constant temperature,
better be positive.
01:03:55.360 --> 01:04:02.830
And so clearly, the expression
that one gets through this van
01:04:02.830 --> 01:04:07.850
der Waals equation
has a limitation.
01:04:07.850 --> 01:04:11.460
And the most
natural way about it
01:04:11.460 --> 01:04:15.320
is to say, well, the
question that you wrote down
01:04:15.320 --> 01:04:19.214
is clearly incorrect.
01:04:19.214 --> 01:04:20.130
That's certainly true.
01:04:20.130 --> 01:04:22.710
Also, this equation
is incorrect,
01:04:22.710 --> 01:04:26.856
except that this
kind of reminds us
01:04:26.856 --> 01:04:34.780
with what actually happens if I
look at the isotherms of a gas,
01:04:34.780 --> 01:04:38.660
such as gas in this
room, or something that
01:04:38.660 --> 01:04:41.010
is more familiar, such as water.
01:04:41.010 --> 01:04:46.330
What you find is that
at high temperatures,
01:04:46.330 --> 01:04:50.440
you indeed have curves
that look like this.
01:04:53.130 --> 01:04:56.550
But at low temperatures,
you liquidify.
01:04:56.550 --> 01:05:00.632
And what you have is
a zone of coexistence,
01:05:00.632 --> 01:05:03.580
let's say something
like this, by which I
01:05:03.580 --> 01:05:07.950
mean that the
isotherm that you draw
01:05:07.950 --> 01:05:13.910
has a portion that
lives in the gas phase
01:05:13.910 --> 01:05:18.630
and is a slightly modified
version of the ideal gas.
01:05:18.630 --> 01:05:21.200
But then it has a
portion that corresponds
01:05:21.200 --> 01:05:24.090
to the more or less
incompressible liquid.
01:05:24.090 --> 01:05:28.870
Although this has clearly still
some finite compressibility.
01:05:28.870 --> 01:05:33.100
And then in between, there is a
region where if you have a box,
01:05:33.100 --> 01:05:36.090
part of your box would be
liquid, and part of your box
01:05:36.090 --> 01:05:37.240
would be gas.
01:05:37.240 --> 01:05:42.100
And as you compress the box,
the proportion of liquid and gas
01:05:42.100 --> 01:05:44.120
will change.
01:05:44.120 --> 01:05:48.140
And this happens for
T less than some Tc.
01:05:48.140 --> 01:05:53.030
And once more, there
is a curve that kind of
01:05:53.030 --> 01:05:58.660
looks like this at some
intermediate temperature Tc.
01:06:01.550 --> 01:06:06.220
So you look at the
comparison, you say, well,
01:06:06.220 --> 01:06:09.340
as long as I stay
above Tc-- of course
01:06:09.340 --> 01:06:13.865
I don't expect this equation
to give the right numbers
01:06:13.865 --> 01:06:15.750
for what Tc or whatever is.
01:06:15.750 --> 01:06:19.940
But qualitatively, I get
topologies and behaviors
01:06:19.940 --> 01:06:22.970
at that T greater
than Tc all the way up
01:06:22.970 --> 01:06:26.300
to Tc are not very different.
01:06:26.300 --> 01:06:29.660
But at T less than
Tc, they also signal
01:06:29.660 --> 01:06:31.980
that something bad is happening.
01:06:31.980 --> 01:06:37.120
And somehow, the original
description has to be modified.
01:06:37.120 --> 01:06:42.080
And so the thing that Maxwell
and van der Waals and company
01:06:42.080 --> 01:06:50.520
did was to somehow convert
this incorrect set of equations
01:06:50.520 --> 01:06:54.260
to something that
resembles this.
01:06:54.260 --> 01:06:57.724
So let's see how they
managed to do this.
01:07:23.960 --> 01:07:28.180
Actually, this may not be a
bad thing to keep in mind.
01:07:30.750 --> 01:07:33.770
Let's have a
thermodynamic-- well,
01:07:33.770 --> 01:07:40.200
I kind of emphasize that there
is thermodynamic reason for why
01:07:40.200 --> 01:07:43.560
this is not valid.
01:07:43.560 --> 01:07:45.760
Always there's a
corresponding reason
01:07:45.760 --> 01:07:48.100
if you look at things
from the perspective
01:07:48.100 --> 01:07:49.920
of statistical mechanics.
01:07:49.920 --> 01:07:51.590
So it may be useful
to sort of think
01:07:51.590 --> 01:07:54.820
about what's happening
from that perspective.
01:07:54.820 --> 01:07:57.510
Let's imagine that we are in
this grand canonical ensemble.
01:07:57.510 --> 01:08:02.040
In the grand canonical ensemble,
the number of particles,
01:08:02.040 --> 01:08:05.760
as we discussed, is not fixed.
01:08:05.760 --> 01:08:10.800
But the mean number of particles
is given by the expressions
01:08:10.800 --> 01:08:12.850
that we saw over
here, is related
01:08:12.850 --> 01:08:17.700
to the pressure through dP by
d mu at constant temperature.
01:08:22.560 --> 01:08:25.790
But there are fluctuations.
01:08:25.790 --> 01:08:32.340
So you would say that the
fluctuations are related
01:08:32.340 --> 01:08:38.490
by taking a second
derivative of this object.
01:08:38.490 --> 01:08:43.910
Because clearly, q, if I were to
expand it in powers of beta mu,
01:08:43.910 --> 01:08:47.779
will generate
various moments of n.
01:08:47.779 --> 01:08:50.880
Log of q will
generate cumulants.
01:08:50.880 --> 01:08:54.399
So the variance would
be one more derivative.
01:08:54.399 --> 01:08:59.880
And so that will
amount to taking
01:08:59.880 --> 01:09:02.310
a derivative of the
first derivative, which
01:09:02.310 --> 01:09:04.740
was the number itself.
01:09:04.740 --> 01:09:11.970
So I will get this to be
dN by d mu at constant T,
01:09:11.970 --> 01:09:15.250
except that there's an
additional factor of kT.
01:09:15.250 --> 01:09:18.529
So maybe I will write
this carefully enough.
01:09:18.529 --> 01:09:28.020
I have d2 log Q with
respect to beta mu squared
01:09:28.020 --> 01:09:30.632
will give me N squared.
01:09:30.632 --> 01:09:37.370
Now, the first derivative with
respect to beta mu of log Q
01:09:37.370 --> 01:09:42.200
gave me the mean, which I'm
actually thinking of as N
01:09:42.200 --> 01:09:43.870
itself.
01:09:43.870 --> 01:09:45.800
And then there's
this factor of beta.
01:09:45.800 --> 01:09:50.955
So this is really kT dN by d mu.
01:09:50.955 --> 01:09:53.290
All of these are done
at constant temperature.
01:10:04.680 --> 01:10:10.110
So the statistical
analog of stability
01:10:10.110 --> 01:10:15.270
really comes down to
variances being positive.
01:10:15.270 --> 01:10:21.320
So my claim is that this
variance being positive
01:10:21.320 --> 01:10:26.630
is related to this
stability condition.
01:10:26.630 --> 01:10:29.010
Let me do that in the
following fashion.
01:10:29.010 --> 01:10:30.880
I divide these two expressions.
01:10:30.880 --> 01:10:33.470
I will get the average
of N squared divided
01:10:33.470 --> 01:10:40.130
by average of N, which is really
N, so the variance over N.
01:10:40.130 --> 01:10:41.290
I have kT.
01:10:44.350 --> 01:10:51.300
And then I have the ratio
of two of these derivatives.
01:10:51.300 --> 01:10:54.090
And the reason I
wanted to do that ratio
01:10:54.090 --> 01:10:56.690
was to get rid of the d mu.
01:10:56.690 --> 01:10:59.160
Because d mu is not so nice.
01:10:59.160 --> 01:11:01.645
If I take the ratio of
those two derivatives,
01:11:01.645 --> 01:11:09.830
I will get dN by
dP at constant T.
01:11:09.830 --> 01:11:13.410
But what I really wanted was
to say something about dV
01:11:13.410 --> 01:11:17.800
by dP at constant T
rather than dN by dP.
01:11:17.800 --> 01:11:24.750
So what I'm going to do is to
somehow convert this dN by dP
01:11:24.750 --> 01:11:30.180
into dN by dV,
and then dV by dP.
01:11:32.890 --> 01:11:36.250
And actually to do that, I
need to use the chain rule.
01:11:36.250 --> 01:11:40.440
So this will be T,
P. This will be T, N.
01:11:40.440 --> 01:11:44.590
And the chain rule will give
me an additional minus sign.
01:11:44.590 --> 01:11:48.830
So this is a reminder of
how your partial derivatives
01:11:48.830 --> 01:11:51.580
and the chain rule goes.
01:11:51.580 --> 01:11:58.460
Finally, dN by dV is none
other than the density N/V.
01:11:58.460 --> 01:12:02.050
At constant temperature, N
and V will be proportional.
01:12:02.050 --> 01:12:07.980
And what I have here
is 1/V dV by dP,
01:12:07.980 --> 01:12:09.560
which is the compressibility.
01:12:09.560 --> 01:12:17.940
So the whole thing here
is NkT times kappa of T.
01:12:17.940 --> 01:12:23.730
And so this being positive,
this positivity of the variance,
01:12:23.730 --> 01:12:28.035
is-- the thermodynamic
analog of it
01:12:28.035 --> 01:12:31.770
was from the stability,
and something such as this.
01:12:31.770 --> 01:12:35.080
The statistical analog of
it is that the probability
01:12:35.080 --> 01:12:39.710
distribution that I'm looking at
in the grand canonical ensemble
01:12:39.710 --> 01:12:43.850
as a function of N has to be
peaked around some region.
01:12:43.850 --> 01:12:45.720
The variance around
that peak better
01:12:45.720 --> 01:12:51.360
be positive so that I'm looking
in the vicinity of a maximum.
01:12:51.360 --> 01:12:53.250
If it was negative,
it means that I'm
01:12:53.250 --> 01:12:55.620
looking in the
vicinity of a minimum.
01:12:55.620 --> 01:12:58.580
And I'm looking at the
least likely configuration.
01:12:58.580 --> 01:13:00.270
So it cannot be allowed.
01:13:00.270 --> 01:13:03.380
So that's the
statistical reason.
01:13:03.380 --> 01:13:06.690
OK, now I want to use
that in connection
01:13:06.690 --> 01:13:11.260
with what we have
over here and see
01:13:11.260 --> 01:13:12.985
what I can learn about this.
01:13:25.090 --> 01:13:28.680
I said that I wrote two
expressions and divided them,
01:13:28.680 --> 01:13:32.810
because I didn't want to deal
with the chemical potential.
01:13:32.810 --> 01:13:37.940
But let me redefine that
and say the following,
01:13:37.940 --> 01:13:44.330
that N is V dP by
d mu at constant T.
01:13:44.330 --> 01:13:48.290
And maybe I will start to
gain some idea about chemical
01:13:48.290 --> 01:13:51.430
potential if I
rearrange this as d mu
01:13:51.430 --> 01:14:02.500
is V/N dP along this surface
of constant P at constant T.
01:14:02.500 --> 01:14:08.340
And then I can calculate
mu as a function of P
01:14:08.340 --> 01:14:12.430
at some particular
temperature minus mu
01:14:12.430 --> 01:14:18.810
at some reference point by
integrating from the reference
01:14:18.810 --> 01:14:22.590
point to the
pressure of interest
01:14:22.590 --> 01:14:30.020
the quantity V/N dP prime
where V of P prime I
01:14:30.020 --> 01:14:32.360
take from this curve.
01:14:32.360 --> 01:14:37.950
So this curve gives me
P as a function of V.
01:14:37.950 --> 01:14:43.420
But I can invert it and think
of V as a function of P,
01:14:43.420 --> 01:14:47.310
put it here, and see what is
happening with the chemical
01:14:47.310 --> 01:14:51.420
potential if I walk along
one of these trajectories.
01:14:51.420 --> 01:14:58.250
And in particular, let's
kind of draw one of these
01:14:58.250 --> 01:15:01.520
curves that I am
unhappy with that
01:15:01.520 --> 01:15:06.620
have this kind of
form in general
01:15:06.620 --> 01:15:11.230
and calculate what happens
to the chemical potential
01:15:11.230 --> 01:15:14.060
if I, let's say, pick
this reference point
01:15:14.060 --> 01:15:18.100
A, and the corresponding
pressure PA,
01:15:18.100 --> 01:15:22.610
and go along this curve, which
corresponds to some temperature
01:15:22.610 --> 01:15:27.350
T that is less than this
instability temperature,
01:15:27.350 --> 01:15:29.485
and track the shape of
the chemical potential.
01:15:32.460 --> 01:15:36.330
So this formula says that the
change in chemical potential we
01:15:36.330 --> 01:15:41.000
obtained by calculating
V-- divide by N,
01:15:41.000 --> 01:15:44.320
but our N is fixed--
as a function of P,
01:15:44.320 --> 01:15:47.560
and integrating as
you go up in pressure.
01:15:47.560 --> 01:15:52.070
So essentially, I'm calculating
the integral starting
01:15:52.070 --> 01:15:55.250
from point A as I
go under this curve.
01:15:58.700 --> 01:16:04.120
So let's plot as a result
of that integration how
01:16:04.120 --> 01:16:09.460
that chemical potential at
P minus PA is going to look
01:16:09.460 --> 01:16:15.340
like as I go in P beyond PA.
01:16:15.340 --> 01:16:17.990
So at PA, that's my reference.
01:16:17.990 --> 01:16:21.230
The chemical potential
is some value.
01:16:21.230 --> 01:16:26.960
As I go up and up, because
of this area of this curve,
01:16:26.960 --> 01:16:29.810
I keep adding to the
chemical potential
01:16:29.810 --> 01:16:34.420
until I reach the maximum
here at this point C.
01:16:34.420 --> 01:16:43.380
So there is some curve that goes
from this A to some point C.
01:16:43.380 --> 01:16:51.110
Now, the thing is that when I
continue going along this curve
01:16:51.110 --> 01:16:56.900
down this potential all the way
to the bottom of this, which
01:16:56.900 --> 01:17:02.840
I will call D, my
DP's are negative.
01:17:02.840 --> 01:17:08.650
So I start subtracting
from what I had before.
01:17:08.650 --> 01:17:12.351
And so then the curve
starts to go down.
01:17:12.351 --> 01:17:16.600
So all the way to
D, I'm proceeding
01:17:16.600 --> 01:17:19.070
in the opposite direction.
01:17:19.070 --> 01:17:24.370
Once I hit D, I start going all
the way to the end of the curve
01:17:24.370 --> 01:17:26.340
however far I want to go.
01:17:26.340 --> 01:17:29.180
And I'm adding
some positive area.
01:17:29.180 --> 01:17:37.415
So the next part I also have a
variation in chemical potential
01:17:37.415 --> 01:17:38.620
that goes like this.
01:17:44.140 --> 01:17:51.384
So you ask, well, if I give
you what the temperature is
01:17:51.384 --> 01:17:53.300
and what the pressure
is-- and I have told you
01:17:53.300 --> 01:17:55.360
what the temperature
and pressure are--
01:17:55.360 --> 01:17:57.850
I should be able to calculate
the chemical potential.
01:17:57.850 --> 01:17:59.800
It's an intensive
function of the other two
01:17:59.800 --> 01:18:01.700
intensive functions.
01:18:01.700 --> 01:18:04.650
And this curve tells me
that, except that there
01:18:04.650 --> 01:18:07.710
are regions where I don't
know which one to pick.
01:18:11.060 --> 01:18:13.050
There are regions where
it's obvious there's
01:18:13.050 --> 01:18:15.630
one value of the
chemical potential.
01:18:15.630 --> 01:18:18.760
But there is in between,
because of this instability,
01:18:18.760 --> 01:18:21.880
three possible values.
01:18:21.880 --> 01:18:27.570
And typically, if
you have systems--
01:18:27.570 --> 01:18:31.200
think about different chemicals
that can go between each other.
01:18:31.200 --> 01:18:34.860
And there are potentials for
chemical transformations.
01:18:34.860 --> 01:18:38.450
You say, go and pick the
lowest chemical potential.
01:18:38.450 --> 01:18:41.900
The system will evolve
onto the condition that
01:18:41.900 --> 01:18:44.410
has the lowest
chemical potential.
01:18:44.410 --> 01:18:48.290
So that says that when you
have a situation such as this,
01:18:48.290 --> 01:18:52.490
you really have to
pick the lowest one.
01:18:52.490 --> 01:19:00.190
And so you have to bail out on
your first curve at the point
01:19:00.190 --> 01:19:08.600
B, and on your second curve at
a point E, where on this curve
01:19:08.600 --> 01:19:15.030
B and E are such that when
you integrate this BDP all
01:19:15.030 --> 01:19:19.450
the way from B to E, you will
get 0, which means that you
01:19:19.450 --> 01:19:27.280
have to find points B and E
such that the integral here
01:19:27.280 --> 01:19:30.470
is the same as
the integral here.
01:19:30.470 --> 01:19:33.416
And this is the so-called
Maxwell construction.
01:19:42.200 --> 01:19:47.820
So what that means is that
originally I was telling you
01:19:47.820 --> 01:19:54.700
over here that there is some
portion of the curve that
01:19:54.700 --> 01:19:56.860
violates all kinds
of thermodynamics.
01:19:56.860 --> 01:19:59.420
And you should not access it.
01:19:59.420 --> 01:20:03.500
And this other argument that
we are pursuing here says that
01:20:03.500 --> 01:20:07.550
actually the regions that you
cannot access are beyond that.
01:20:07.550 --> 01:20:10.200
There's a portion that
extends in this direction,
01:20:10.200 --> 01:20:13.160
and a portion that
extends in that direction.
01:20:13.160 --> 01:20:16.890
And really there's a portion
that you can access up to here,
01:20:16.890 --> 01:20:19.960
and a portion that you
can access up to here.
01:20:19.960 --> 01:20:22.010
And maybe you
should sort of then
01:20:22.010 --> 01:20:25.210
join them by a straight
line and make an analog
01:20:25.210 --> 01:20:28.980
with what we have over there.
01:20:28.980 --> 01:20:36.310
Now, in another thing,
another way of looking at this
01:20:36.310 --> 01:20:41.370
is that the green portion
is certainly unstable,
01:20:41.370 --> 01:20:44.950
violates all kinds of
thermodynamic conditions.
01:20:44.950 --> 01:20:48.850
And people argue that
these other portions that
01:20:48.850 --> 01:20:53.960
have the right slope are
in some sense metastable.
01:20:53.960 --> 01:21:01.040
So this would be a
stable equilibrium.
01:21:01.040 --> 01:21:04.540
This would be an
unstable equilibrium.
01:21:04.540 --> 01:21:07.700
And a metastable equilibrium
is that you are at a minimum,
01:21:07.700 --> 01:21:10.880
but there's a deeper
minimum somewhere else.
01:21:10.880 --> 01:21:23.350
And the picture is that if
you manage to take your gas
01:21:23.350 --> 01:21:27.270
and put additional
pressure on it,
01:21:27.270 --> 01:21:33.190
there is a region where there
is a coexistence with the liquid
01:21:33.190 --> 01:21:37.180
which is better in terms
of free energy, et cetera.
01:21:37.180 --> 01:21:40.250
But there could be some
kind of a kinetic barrier,
01:21:40.250 --> 01:21:44.990
like nucleation or whatever,
that prevents you to go there.
01:21:44.990 --> 01:21:48.180
And this again is something
that is experimentally observed.
01:21:48.180 --> 01:21:51.890
If you try to rapidly
pressurize a gas,
01:21:51.890 --> 01:21:56.900
you could sometimes avoid
and not make your transition
01:21:56.900 --> 01:21:59.680
to the liquid state
at the right pressure.
01:21:59.680 --> 01:22:02.595
But if you were to do
things sufficiently slowly,
01:22:02.595 --> 01:22:07.020
you would ultimately
reach this point
01:22:07.020 --> 01:22:09.230
that corresponds to
the true equilibrium
01:22:09.230 --> 01:22:13.660
pressure of making
this transition.
01:22:13.660 --> 01:22:18.020
Now, the thing is that
somehow this whole story
01:22:18.020 --> 01:22:21.690
is not very satisfactory.
01:22:21.690 --> 01:22:23.940
I started with
some rearrangement
01:22:23.940 --> 01:22:27.840
of some perturbative to
arrive at this equation.
01:22:27.840 --> 01:22:31.200
This equation has
this unstable portion.
01:22:31.200 --> 01:22:34.900
And somehow I'm trying to relate
something that makes absolutely
01:22:34.900 --> 01:22:37.880
no thermodynamic
sense and try to gain
01:22:37.880 --> 01:22:43.545
from it some idea about what is
happening in a real liquid gas
01:22:43.545 --> 01:22:45.250
system.
01:22:45.250 --> 01:22:48.580
So it would be
good if all of this
01:22:48.580 --> 01:22:53.250
could be more formally
described within a framework
01:22:53.250 --> 01:22:57.530
where all of the approximations,
et cetera, are clear,
01:22:57.530 --> 01:22:59.410
and we know what's going on.
01:22:59.410 --> 01:23:02.575
And so next time,
we will do that.
01:23:02.575 --> 01:23:06.910
We will essentially developed a
much more systematic formalism
01:23:06.910 --> 01:23:11.550
for calculating the properties
of an interacting gas,
01:23:11.550 --> 01:23:16.619
and see that we get this, and
we also understand why it fails.