WEBVTT
00:00:00.090 --> 00:00:01.780
The following
content is provided
00:00:01.780 --> 00:00:04.019
under a Creative
Commons license.
00:00:04.019 --> 00:00:06.870
Your support will help MIT
OpenCourseWare continue
00:00:06.870 --> 00:00:10.730
to offer high quality
educational resources for free.
00:00:10.730 --> 00:00:13.330
To make a donation or
view additional materials
00:00:13.330 --> 00:00:17.217
from hundreds of MIT courses,
visit MIT OpenCourseWare
00:00:17.217 --> 00:00:17.842
at ocw.mit.edu.
00:00:20.550 --> 00:00:23.340
PROFESSOR: OK, let's start.
00:00:23.340 --> 00:00:27.160
So we said that the task
of statistical mechanics
00:00:27.160 --> 00:00:31.810
is to assign probabilities to
different microstates given
00:00:31.810 --> 00:00:33.620
that we have knowledge
of the microstate.
00:00:36.320 --> 00:00:43.310
And the most kind of simple
logical place to start
00:00:43.310 --> 00:00:51.470
was in the
microcanonical ensemble
00:00:51.470 --> 00:00:54.850
where the microstate
that we specified
00:00:54.850 --> 00:00:58.760
was one in which there was
no exchange of work or heat
00:00:58.760 --> 00:01:02.660
with the surroundings so
that the energy was constant,
00:01:02.660 --> 00:01:05.580
and the parameters,
such as x and N,
00:01:05.580 --> 00:01:08.270
that account for chemical
and mechanical work
00:01:08.270 --> 00:01:11.000
were fixed also.
00:01:11.000 --> 00:01:14.820
Then this assignment
was, we said,
00:01:14.820 --> 00:01:17.870
like counting how many
faces the dice has,
00:01:17.870 --> 00:01:21.110
and saying all of them
are equally likely.
00:01:21.110 --> 00:01:24.030
So we would say that
the probability here
00:01:24.030 --> 00:01:31.340
of a microstate is of
the form 0 or 1 depending
00:01:31.340 --> 00:01:36.450
on whether the energy
of that microstate
00:01:36.450 --> 00:01:46.280
is or is not the right energy
that is listed over here.
00:01:46.280 --> 00:01:50.660
And as any probability,
it has to be normalized.
00:01:50.660 --> 00:01:53.170
So we had this 1 over omega.
00:01:59.860 --> 00:02:06.080
And we also had a rule for
converting probabilities
00:02:06.080 --> 00:02:12.370
to entropies, which in
dimensionless form-- that
00:02:12.370 --> 00:02:18.370
is, if we divide by kB-- was
simply minus the expectation
00:02:18.370 --> 00:02:22.090
value of the log of the
probability, this probability
00:02:22.090 --> 00:02:24.520
being uniformly 1 over omega.
00:02:24.520 --> 00:02:32.000
This simply gave us
log of omega E, x, N.
00:02:32.000 --> 00:02:36.990
Once we had the entropy
as a function of E, x, N,
00:02:36.990 --> 00:02:41.870
we also identified the
derivatives of this quantity
00:02:41.870 --> 00:02:43.180
in equilibrium.
00:02:43.180 --> 00:02:46.540
And partial derivative
with respect to energy
00:02:46.540 --> 00:02:50.010
was identified as
1 So systems that
00:02:50.010 --> 00:02:55.580
were in equilibrium with each
other, this derivative dS by dE
00:02:55.580 --> 00:02:58.190
had to be the same
for all of them.
00:02:58.190 --> 00:03:03.280
And because of
mechanical stability,
00:03:03.280 --> 00:03:07.750
we could identify the next
derivative with respect to x.
00:03:07.750 --> 00:03:11.460
That was minus J/T.
And with respect to N
00:03:11.460 --> 00:03:20.020
by identifying a corresponding
thing, we would get this.
00:03:20.020 --> 00:03:22.810
So from here, we can
proceed and calculate
00:03:22.810 --> 00:03:28.830
thermodynamic properties using
these microscopic rules, as
00:03:28.830 --> 00:03:37.620
well as probabilities in the
entire space of microstates
00:03:37.620 --> 00:03:39.770
that you have.
00:03:39.770 --> 00:03:42.210
Now, the next thing
that we did was
00:03:42.210 --> 00:03:45.320
to go and look at the
different ensemble,
00:03:45.320 --> 00:03:53.240
the canonical ensemble,
in which we said that,
00:03:53.240 --> 00:03:56.385
again, thermodynamically,
I choose
00:03:56.385 --> 00:03:58.410
a different set of variables.
00:03:58.410 --> 00:04:02.630
For example, I can replace
the energy with temperature.
00:04:02.630 --> 00:04:06.215
And indeed, the characteristic
of the canonical ensemble
00:04:06.215 --> 00:04:09.190
is that rather than
specifying the energy,
00:04:09.190 --> 00:04:11.680
you specify the temperature.
00:04:11.680 --> 00:04:15.120
But there's still no
chemical or mechanical work.
00:04:15.120 --> 00:04:19.610
So the other two parameters
are kept fixed also.
00:04:19.610 --> 00:04:23.640
And then the
statement that we had
00:04:23.640 --> 00:04:29.810
was that by putting a system in
contact with a huge reservoir,
00:04:29.810 --> 00:04:33.480
we could ensure that it is
maintained at some temperature.
00:04:33.480 --> 00:04:40.130
And since system and reservoir
were jointly microcanonical,
00:04:40.130 --> 00:04:44.460
we could use the probabilities
that we had microcanonically.
00:04:44.460 --> 00:04:47.650
We integrated over the degrees
of freedom of the reservoir
00:04:47.650 --> 00:04:49.220
that we didn't care about.
00:04:49.220 --> 00:04:52.630
And we ended up
with the probability
00:04:52.630 --> 00:04:58.750
of a microstate in the
canonical ensemble, which
00:04:58.750 --> 00:05:02.000
was related to
the energy of that
00:05:02.000 --> 00:05:10.430
by the form exponential of minus
beta h where we introduced beta
00:05:10.430 --> 00:05:13.940
to be 1 over kT.
00:05:13.940 --> 00:05:18.670
And this probably, again,
had to be normalized.
00:05:18.670 --> 00:05:23.280
So the normalization
we called Z. And Z
00:05:23.280 --> 00:05:27.770
is attained by summing over
all of the microstates E
00:05:27.770 --> 00:05:32.190
to the minus beta H of the
microstate where this could
00:05:32.190 --> 00:05:35.120
be an integration over
the entire phase space
00:05:35.120 --> 00:05:39.520
if you're dealing with
continuous variables.
00:05:39.520 --> 00:05:44.920
OK, now the question
was, thermodynamically,
00:05:44.920 --> 00:05:49.790
these quantities T
and E are completely
00:05:49.790 --> 00:05:54.540
exchangeable ways of
identifying the same equilibrium
00:05:54.540 --> 00:05:58.160
state, whereas what
I have done now
00:05:58.160 --> 00:06:01.135
is I have told you what
the temperature is.
00:06:01.135 --> 00:06:05.090
But the energy of the
system is a random variable.
00:06:05.090 --> 00:06:08.270
And so I can ask, what
is the probability
00:06:08.270 --> 00:06:14.510
that I look at my system and
I find a particular energy E?
00:06:14.510 --> 00:06:17.006
So how can I get that?
00:06:17.006 --> 00:06:20.390
Well, that probability,
first of all,
00:06:20.390 --> 00:06:24.370
has to come from a microstate
that has the right energy.
00:06:24.370 --> 00:06:26.730
And so I will get
e to the minus beta
00:06:26.730 --> 00:06:31.960
E divided by Z, which is
the probability of getting
00:06:31.960 --> 00:06:35.040
that microstate that
has the right energy.
00:06:35.040 --> 00:06:37.740
But I only said something
about the energy.
00:06:37.740 --> 00:06:41.310
And there are a huge number
of microstates, as we've seen,
00:06:41.310 --> 00:06:43.150
that have the same energy.
00:06:43.150 --> 00:06:46.390
So I could have picked from
any one of those microstates,
00:06:46.390 --> 00:06:49.910
their number being omega
of E, the omega that we
00:06:49.910 --> 00:06:51.210
had identified before.
00:06:53.770 --> 00:07:00.120
So since omega can be
written as exponential of s
00:07:00.120 --> 00:07:04.350
over kB, this you can
also think of as being
00:07:04.350 --> 00:07:09.650
proportional to exponential
of minus beta E minus T,
00:07:09.650 --> 00:07:12.790
the entropy that I would
get for that energy
00:07:12.790 --> 00:07:19.564
according to the formula
above, divided by Z.
00:07:19.564 --> 00:07:22.600
Now, we said that
the quantity that I'm
00:07:22.600 --> 00:07:27.370
looking at here in
the numerator has
00:07:27.370 --> 00:07:34.470
a dependence on the size of
the system that is extensive.
00:07:34.470 --> 00:07:36.700
Both the entropy
and energy we expect
00:07:36.700 --> 00:07:40.670
to be growing proportionately
to the size of the system.
00:07:40.670 --> 00:07:45.750
And hence, this exponent
also grows proportionately
00:07:45.750 --> 00:07:47.620
to the size of the system.
00:07:47.620 --> 00:07:51.520
So I expect that if I
plot this probability
00:07:51.520 --> 00:07:57.100
as a function of energy, this
probability to get energy E,
00:07:57.100 --> 00:08:00.470
it would be one of
those functions.
00:08:00.470 --> 00:08:02.260
It's certainly positive.
00:08:02.260 --> 00:08:03.540
It's a probability.
00:08:03.540 --> 00:08:05.550
It has an exponential
dependence.
00:08:05.550 --> 00:08:09.280
So there's a part that maybe
from the density of states
00:08:09.280 --> 00:08:10.840
grows exponentially.
00:08:10.840 --> 00:08:14.060
e to the minus beta E
will exponential kill it.
00:08:14.060 --> 00:08:17.500
And maybe there is, because
of this competition,
00:08:17.500 --> 00:08:20.440
one or potentially more maxima.
00:08:20.440 --> 00:08:24.360
But if there are more maxima
locally, I don't really care.
00:08:24.360 --> 00:08:26.760
Because one will be
exponentially larger
00:08:26.760 --> 00:08:27.870
than the other.
00:08:27.870 --> 00:08:32.590
So presumably, there is
some location corresponding
00:08:32.590 --> 00:08:37.810
to some E star where the
probability is maximized.
00:08:42.900 --> 00:08:45.920
Well, how should I characterize
the energy of the system?
00:08:45.920 --> 00:08:49.900
Should I pick E star, or
given this probability,
00:08:49.900 --> 00:08:58.040
should I look at maybe the
mean value, the average of H?
00:08:58.040 --> 00:09:00.600
How can I get the average of H?
00:09:00.600 --> 00:09:05.210
Well, what I need to do is to
sum over all the microstates H
00:09:05.210 --> 00:09:07.460
of the microstate
e to the minus beta
00:09:07.460 --> 00:09:11.626
H divided by sum over
microstates e to the minus beta
00:09:11.626 --> 00:09:12.876
H, which is the normalization.
00:09:16.150 --> 00:09:19.880
The denominator is of course
the partition function.
00:09:19.880 --> 00:09:23.900
The numerator can be obtained
by taking a derivative of this
00:09:23.900 --> 00:09:27.060
with respect to beta
of the minus sign.
00:09:27.060 --> 00:09:32.060
So what this is is minus
the log Z by d beta.
00:09:34.900 --> 00:09:42.020
So if I have calculated
H, I can potentially
00:09:42.020 --> 00:09:45.720
go through this procedure
and maybe calculate
00:09:45.720 --> 00:09:48.880
where the mean value is.
00:09:48.880 --> 00:09:51.810
And is that a better
representation
00:09:51.810 --> 00:09:55.280
of the energy of the system than
the most likely value, which
00:09:55.280 --> 00:09:56.605
was E star?
00:09:56.605 --> 00:10:00.590
Well, let's see how
much energy fluctuates.
00:10:00.590 --> 00:10:04.050
Basically, we saw that if
I repeat this procedure
00:10:04.050 --> 00:10:10.120
many times, I can see that
the n-th moment is minus 1
00:10:10.120 --> 00:10:15.340
to the n 1 over Z the
n-th derivative of Z
00:10:15.340 --> 00:10:18.620
with respect to beta.
00:10:18.620 --> 00:10:22.000
And hence the
partition function,
00:10:22.000 --> 00:10:24.850
by expanding it
in powers of beta,
00:10:24.850 --> 00:10:28.540
will generate for me
higher and higher moments.
00:10:28.540 --> 00:10:35.620
And we therefore concluded that
the cumulant would be obtained
00:10:35.620 --> 00:10:39.650
by the same procedure, except
that I will replace this
00:10:39.650 --> 00:10:42.320
by log Z. So this will
be the n-th derivative
00:10:42.320 --> 00:10:43.570
of log Z with respect to beta.
00:10:49.090 --> 00:10:57.910
So the variance H squared
c is a second derivative
00:10:57.910 --> 00:11:00.800
of log Z. The first
derivative gives me
00:11:00.800 --> 00:11:03.870
the first cumulant,
which was the mean.
00:11:03.870 --> 00:11:08.616
So this is going to
give me d by d beta
00:11:08.616 --> 00:11:15.140
with a minus sign of the
expectation value of H,
00:11:15.140 --> 00:11:19.980
which is the same
thing as kBT squared,
00:11:19.980 --> 00:11:23.600
because the derivative
of 1 over kT
00:11:23.600 --> 00:11:28.670
is 1 over T squared times
derivative with respect to T.
00:11:28.670 --> 00:11:31.740
And it goes to the other
side, to the numerator.
00:11:31.740 --> 00:11:34.380
And so then I have
the derivative
00:11:34.380 --> 00:11:38.450
of this object with
respect to temperature,
00:11:38.450 --> 00:11:40.550
which is something
like a heat capacity.
00:11:40.550 --> 00:11:43.820
But most importantly,
the statement
00:11:43.820 --> 00:11:46.600
is that all of
these quantities are
00:11:46.600 --> 00:11:51.310
things that are of the order
of the size of the system.
00:11:51.310 --> 00:11:54.710
And just like we did in
the central limit theorem,
00:11:54.710 --> 00:11:57.040
with the addition
of random variables,
00:11:57.040 --> 00:12:00.220
you have a situation that
is very much like that.
00:12:00.220 --> 00:12:02.690
The distribution,
in some sense, is
00:12:02.690 --> 00:12:07.830
going to converge more and more
towards the Gaussian dominated
00:12:07.830 --> 00:12:11.770
by the first and
second cumulant.
00:12:11.770 --> 00:12:15.130
And in fact, even the
second cumulant we see
00:12:15.130 --> 00:12:17.980
is of the order of
square root of n.
00:12:17.980 --> 00:12:21.370
And hence the fluctuations
between these two quantities,
00:12:21.370 --> 00:12:22.970
which are each one
of them order of n,
00:12:22.970 --> 00:12:25.700
is only of the order
of square root of n.
00:12:25.700 --> 00:12:27.700
And when the limit of
n goes to infinity,
00:12:27.700 --> 00:12:30.150
we can ignore any
such difference.
00:12:30.150 --> 00:12:33.690
We can essentially
identify either one of them
00:12:33.690 --> 00:12:37.430
with the energy of the
system thermodynamically.
00:12:37.430 --> 00:12:40.260
And then this
quantity, if I identify
00:12:40.260 --> 00:12:42.850
this with energy of
the system, is simply
00:12:42.850 --> 00:12:47.520
the usual heat
capacity of constant x.
00:12:47.520 --> 00:12:51.090
So the scale of the
fluctuations over here
00:12:51.090 --> 00:12:56.926
is set by square root of kBT
squared the heat capacity.
00:12:59.890 --> 00:13:02.800
By the way, which
is also the reason
00:13:02.800 --> 00:13:06.540
that heat capacities
must be positive.
00:13:06.540 --> 00:13:09.750
Because statistically,
the variances
00:13:09.750 --> 00:13:12.410
are certainly
positive quantities.
00:13:12.410 --> 00:13:16.060
So you have a
constraint that we had
00:13:16.060 --> 00:13:18.760
seen before on the
sign emerging in
00:13:18.760 --> 00:13:20.570
its statistical interpretation.
00:13:25.350 --> 00:13:33.030
What I did over here was to
identify an entropy associated
00:13:33.030 --> 00:13:36.120
with a particular form
of the probability.
00:13:36.120 --> 00:13:41.160
What happens if I look at an
entropy with this probability
00:13:41.160 --> 00:13:42.500
that I have over here?
00:13:42.500 --> 00:13:44.870
OK, so this is a
probability phase space.
00:13:44.870 --> 00:13:46.860
What is its entropy?
00:13:46.860 --> 00:13:55.450
So S over k is expectation
value of log p.
00:13:55.450 --> 00:13:57.645
And what is log of p?
00:13:57.645 --> 00:14:02.920
Well, there is log of this minus
beta H. So what I will have,
00:14:02.920 --> 00:14:08.880
because of the change in sign--
beta expectation value of H.
00:14:08.880 --> 00:14:13.050
And then I have
minus log Z here.
00:14:13.050 --> 00:14:21.930
The sign changes, and
I will get plus log Z.
00:14:21.930 --> 00:14:27.030
So yeah, that's correct.
00:14:31.150 --> 00:14:34.090
If I were to rearrange
this, what do I get?
00:14:34.090 --> 00:14:41.311
I can take this
minus-- let's see.
00:14:45.207 --> 00:14:50.900
Yeah, OK, so if I take
this to the other side,
00:14:50.900 --> 00:14:53.730
then I will get
that log Z equals--
00:14:53.730 --> 00:14:57.090
this goes to the other
side-- minus beta.
00:14:57.090 --> 00:15:01.740
Expectation value of
H we are calling E.
00:15:01.740 --> 00:15:07.585
And then I have
multiplying this by beta.
00:15:07.585 --> 00:15:10.790
The kB's disappear, and
I will get minus TS.
00:15:15.100 --> 00:15:21.620
So we see that I can identify
log Z with the combination E
00:15:21.620 --> 00:15:33.480
minus TS, which is the
Helmholtz free energy.
00:15:37.900 --> 00:15:46.880
So log Z, in the same way that
the normalization here that
00:15:46.880 --> 00:15:51.470
was omega, gave us the entropy.
00:15:51.470 --> 00:15:55.110
The normalization
of the probability
00:15:55.110 --> 00:15:57.940
in the canonical ensemble, which
is also called the partition
00:15:57.940 --> 00:16:02.390
function if I take its log,
will give me the free energy.
00:16:02.390 --> 00:16:05.600
And I can then go and
compute various thermodynamic
00:16:05.600 --> 00:16:08.460
quantities based on that.
00:16:08.460 --> 00:16:13.640
There's an alternative way of
getting the same result, which
00:16:13.640 --> 00:16:18.450
is to note that actually the
same quantity is appearing over
00:16:18.450 --> 00:16:24.032
here, right?
00:16:24.032 --> 00:16:28.940
And really, I should be
evaluating the probability here
00:16:28.940 --> 00:16:31.550
at the maximum.
00:16:31.550 --> 00:16:33.930
Maximum is the
energy of the system.
00:16:33.930 --> 00:16:39.730
So if you like, I can call this
variable over here epsilon.
00:16:39.730 --> 00:16:42.360
This is the
probability of epsilon.
00:16:42.360 --> 00:16:45.290
And it is only when
I'm at the maximum
00:16:45.290 --> 00:16:55.030
that I can replace this epsilon
with E. Then since with almost
00:16:55.030 --> 00:17:00.129
probability 1, I'm
going to see this state
00:17:00.129 --> 00:17:01.670
and none of the
other states, because
00:17:01.670 --> 00:17:05.250
of this exponential dependence.
00:17:05.250 --> 00:17:09.950
When this expression is
evaluated at this energy,
00:17:09.950 --> 00:17:12.680
it should give me probability 1.
00:17:12.680 --> 00:17:20.270
And so you can see again that
Z has to be-- so basically what
00:17:20.270 --> 00:17:22.880
I'm saying is that
this quantity has
00:17:22.880 --> 00:17:27.849
to be 1 so that you get
this relationship back
00:17:27.849 --> 00:17:29.250
from that perspective also.
00:17:36.437 --> 00:17:37.260
AUDIENCE: Question.
00:17:37.260 --> 00:17:38.960
PROFESSOR: Yes.
00:17:38.960 --> 00:17:42.560
AUDIENCE: So I agree
that if in order
00:17:42.560 --> 00:17:45.440
to plug in d, mean energy,
into that expression,
00:17:45.440 --> 00:17:50.140
you would get a probability
of 1 at that point.
00:17:50.140 --> 00:17:54.820
But because even though
the other energies are
00:17:54.820 --> 00:17:59.320
exponentially less probable,
they strictly speaking
00:17:59.320 --> 00:18:01.567
aren't 0 probability, are they?
00:18:01.567 --> 00:18:02.150
PROFESSOR: No.
00:18:02.150 --> 00:18:04.460
AUDIENCE: So how does
this get normalized?
00:18:04.460 --> 00:18:06.920
How does this probability
expression get normalized?
00:18:06.920 --> 00:18:09.336
PROFESSOR: OK, so
what are we doing?
00:18:09.336 --> 00:18:14.920
So Z, I can also write it as
the normalization of the energy.
00:18:14.920 --> 00:18:20.580
So rather than picking the one
energy that maximizes things,
00:18:20.580 --> 00:18:26.530
you say, you should
really do this, right?
00:18:26.530 --> 00:18:30.522
Now, I will evaluate this
by the saddle point method.
00:18:30.522 --> 00:18:35.090
The saddle point method says,
pick the maximum of this.
00:18:35.090 --> 00:18:40.680
So I have e to the minus beta
F evaluated at the maximum.
00:18:40.680 --> 00:18:43.850
And then I have
to integrate over
00:18:43.850 --> 00:18:46.840
variations around that maximum.
00:18:46.840 --> 00:18:49.320
Variations around
that maximum I have
00:18:49.320 --> 00:18:51.870
to expand this to second order.
00:18:51.870 --> 00:18:55.960
And if I really correctly
expand this to second order,
00:18:55.960 --> 00:19:03.220
I will get delta E squared
divided by this 2kTCx,
00:19:03.220 --> 00:19:08.440
because we already established
what this variance is.
00:19:08.440 --> 00:19:10.840
I can do this
Gaussian integration.
00:19:10.840 --> 00:19:15.170
I get e to the minus
beta F of e star.
00:19:15.170 --> 00:19:17.810
And then the variance
of this object
00:19:17.810 --> 00:19:26.565
is going to give me root of
2 pi kTC of x-- T-square.
00:19:31.590 --> 00:19:39.190
So what do I mean when I
say that this quantity is Z?
00:19:39.190 --> 00:19:42.810
Really, the thing that we are
calculating always in order
00:19:42.810 --> 00:19:47.070
to make computation is
something like a free energy,
00:19:47.070 --> 00:19:48.800
which is log Z.
00:19:48.800 --> 00:19:52.260
So when I take the
log, log of Z is
00:19:52.260 --> 00:19:58.870
going to be this minus beta
F star S. But you're right.
00:19:58.870 --> 00:19:59.870
There was a weight.
00:19:59.870 --> 00:20:01.720
All of those things
are contributing.
00:20:01.720 --> 00:20:03.640
How much are they contributing?
00:20:03.640 --> 00:20:12.300
1/2 log of 2 pi kBT squared C.
00:20:12.300 --> 00:20:15.340
Now, in the limit
of large N, this
00:20:15.340 --> 00:20:23.550
is order of N. This is order
of log N. And I [INAUDIBLE].
00:20:23.550 --> 00:20:25.510
So it's the same saddle point.
00:20:25.510 --> 00:20:29.040
The idea of saddle point
was that there is a weight.
00:20:29.040 --> 00:20:31.940
But you can ignore it.
00:20:31.940 --> 00:20:33.690
There maybe another
maximum here.
00:20:33.690 --> 00:20:34.795
You say, what about that?
00:20:34.795 --> 00:20:37.990
Well, that will be
exponentially small.
00:20:37.990 --> 00:20:43.050
So everything I keep emphasizing
only works because of this N
00:20:43.050 --> 00:20:44.240
goes to infinity limit.
00:20:46.990 --> 00:20:50.130
And it's magical, you see?
00:20:50.130 --> 00:20:53.440
You can replace this sum
with just one, the maximum.
00:20:53.440 --> 00:20:55.910
And everything is
fine and consistent.
00:20:55.910 --> 00:20:57.950
Yes.
00:20:57.950 --> 00:20:59.480
AUDIENCE: Not to
belabor this point,
00:20:59.480 --> 00:21:06.570
but if you have an expected
return for catastrophe where
00:21:06.570 --> 00:21:11.630
this outlier causes an event
that brings the system down,
00:21:11.630 --> 00:21:16.020
couldn't that chase this limit
in the sense that as that goes
00:21:16.020 --> 00:21:18.500
to 0, that still
goes to infinity,
00:21:18.500 --> 00:21:21.570
and thus you're-- you
understand what I'm saying.
00:21:21.570 --> 00:21:24.770
If an outlier causes this
simulation-- that's my word--
00:21:24.770 --> 00:21:30.307
causes this system to crumble,
then-- so is there a paradox
00:21:30.307 --> 00:21:30.806
there?
00:21:30.806 --> 00:21:33.170
PROFESSOR: No, there
is here the possibility
00:21:33.170 --> 00:21:36.345
of a catastrophe in the
sense that all the oxygen
00:21:36.345 --> 00:21:38.420
in this room could
go over there,
00:21:38.420 --> 00:21:42.870
and you and I will suffocate
after a few minutes.
00:21:42.870 --> 00:21:44.940
It's possible, that's true.
00:21:44.940 --> 00:21:47.040
You just have to
wait many, many ages
00:21:47.040 --> 00:21:49.040
of the universe
for that to happen.
00:21:52.760 --> 00:21:53.643
Yes.
00:21:53.643 --> 00:21:56.058
AUDIENCE: So when
you're integrating,
00:21:56.058 --> 00:21:58.956
that introduces units
into the problem.
00:21:58.956 --> 00:22:02.938
So we have to divide by
something to [INAUDIBLE].
00:22:08.850 --> 00:22:10.650
PROFESSOR: Yes, and
when we are doing
00:22:10.650 --> 00:22:12.440
this for the case
of the ideal gas,
00:22:12.440 --> 00:22:15.020
I will be very
careful to do that.
00:22:15.020 --> 00:22:17.980
But it turns out
that those issues,
00:22:17.980 --> 00:22:20.435
as far as various
derivatives are concerned,
00:22:20.435 --> 00:22:23.220
will not make too
much difference.
00:22:23.220 --> 00:22:26.050
But when we are looking
at the specific example,
00:22:26.050 --> 00:22:29.120
such as the ideal gas, I
will be careful about that.
00:22:37.130 --> 00:22:41.070
So we saw how this
transition occurs.
00:22:41.070 --> 00:22:44.420
But when we were
looking at the case
00:22:44.420 --> 00:22:46.750
of thermodynamical
descriptions, we
00:22:46.750 --> 00:22:51.410
looked at a couple
of other microstates.
00:22:51.410 --> 00:22:53.340
One of them was the
Gibbs canonical.
00:22:58.830 --> 00:23:00.690
And what we did
was we said, well,
00:23:00.690 --> 00:23:03.900
let's allow now some
work to take place
00:23:03.900 --> 00:23:06.450
on the system, mechanical work.
00:23:06.450 --> 00:23:11.170
And rather than saying
that, say, the displacement
00:23:11.170 --> 00:23:14.500
x is fixed, I allow it to vary.
00:23:14.500 --> 00:23:20.100
But I will say what the
corresponding force is.
00:23:20.100 --> 00:23:22.435
But let's keep the number
of particles fixed.
00:23:25.430 --> 00:23:28.040
So essentially, the
picture that we have
00:23:28.040 --> 00:23:33.060
is that somehow my system is
parametrized by some quantity
00:23:33.060 --> 00:23:34.370
x.
00:23:34.370 --> 00:23:46.380
And I'm maintaining the system
at fixed value of some force J.
00:23:46.380 --> 00:23:51.300
So x is allowed potentially
to find the value that
00:23:51.300 --> 00:23:54.380
is consistent with
this particular J
00:23:54.380 --> 00:23:56.320
that I impose on the system.
00:23:56.320 --> 00:23:59.900
And again, I have put the
whole system in contact
00:23:59.900 --> 00:24:03.630
with the reservoir
temperature T so
00:24:03.630 --> 00:24:08.630
that if I now say
that I really maintain
00:24:08.630 --> 00:24:16.550
the system at
variable x, but fix J,
00:24:16.550 --> 00:24:20.800
I have to put this
spring over there.
00:24:20.800 --> 00:24:25.630
And then this spring
plus the system
00:24:25.630 --> 00:24:32.320
is jointly in a
canonical perspective.
00:24:32.320 --> 00:24:34.310
And what that really
means is that you
00:24:34.310 --> 00:24:41.950
have to keep track of the
energy of the microstate,
00:24:41.950 --> 00:24:49.830
as well as the energy that
you extract from the spring,
00:24:49.830 --> 00:24:54.150
which is something like J dot x.
00:24:54.150 --> 00:24:59.170
And since the joint system
is in the canonical state,
00:24:59.170 --> 00:25:00.940
it would say that
the probability
00:25:00.940 --> 00:25:07.370
for the joint system, which
has T and J and N specified,
00:25:07.370 --> 00:25:09.240
but I don't know the
microstate, and I
00:25:09.240 --> 00:25:13.090
don't know the actual
displacement x,
00:25:13.090 --> 00:25:18.270
this joint probability
is canonical.
00:25:18.270 --> 00:25:22.790
And so it is proportional
to e to the minus beta
00:25:22.790 --> 00:25:27.680
H of the microstate plus
this contribution Jx
00:25:27.680 --> 00:25:31.530
that I'm getting from the
other degree of freedom, which
00:25:31.530 --> 00:25:35.490
is the spring, which keeps
the whole thing at fixed J.
00:25:35.490 --> 00:25:39.560
And I have to divide by
some normalization that
00:25:39.560 --> 00:25:42.990
in addition includes
integration over x.
00:25:42.990 --> 00:25:50.610
And this will depend on T, J, N.
00:25:50.610 --> 00:25:56.320
So in this system, both x
and the energy of the system
00:25:56.320 --> 00:25:57.230
are variables.
00:25:57.230 --> 00:26:00.880
They can potentially change.
00:26:00.880 --> 00:26:06.980
What I can now do is to either
characterize like I did over
00:26:06.980 --> 00:26:11.310
here what the probability of x
is, or go by this other route
00:26:11.310 --> 00:26:14.120
and calculate what the
entropy of this probability
00:26:14.120 --> 00:26:19.250
is, which would be minus
the log of the corresponding
00:26:19.250 --> 00:26:21.080
probability.
00:26:21.080 --> 00:26:23.840
And what is the
log in this case?
00:26:23.840 --> 00:26:30.870
I will get beta
expectation value of H.
00:26:30.870 --> 00:26:38.330
And I will get minus beta
J expectation value of x.
00:26:38.330 --> 00:26:44.274
And then I will get plus
log of this Z tilde.
00:26:48.630 --> 00:26:55.360
So rearranging things, what I
get is that log of this Z tilde
00:26:55.360 --> 00:27:00.200
is minus beta.
00:27:00.200 --> 00:27:04.580
I will call this E
like I did over there.
00:27:04.580 --> 00:27:09.920
Minus J-- I would call this
the actual thermodynamic
00:27:09.920 --> 00:27:12.530
displacement x.
00:27:12.530 --> 00:27:15.400
And from down here,
the other side,
00:27:15.400 --> 00:27:16.775
I will get a factor of TS.
00:27:22.370 --> 00:27:25.000
So this should
remind you that we
00:27:25.000 --> 00:27:30.270
had called a Gibbs free
energy the combination
00:27:30.270 --> 00:27:37.990
of E minus TS minus Jx.
00:27:37.990 --> 00:27:40.770
And the natural
variables for this G
00:27:40.770 --> 00:27:43.800
were indeed, once we
looked at the variation
00:27:43.800 --> 00:27:54.180
DE, T. They were J, and N that
we did not do anything with.
00:27:54.180 --> 00:28:03.870
And to the question of, what
is the displacement, since it's
00:28:03.870 --> 00:28:06.900
a random variable
now, I can again
00:28:06.900 --> 00:28:11.480
try to go to the same
procedure as I did over here.
00:28:11.480 --> 00:28:15.110
I can calculate the
expectation value
00:28:15.110 --> 00:28:18.390
of x by noting that
this exponent here
00:28:18.390 --> 00:28:21.160
has a factor of beta Jx.
00:28:21.160 --> 00:28:25.895
So if I take a derivative
with respect to beta J,
00:28:25.895 --> 00:28:29.840
I will bring down a factor of x.
00:28:29.840 --> 00:28:35.182
So if I take the derivative
of log of this Z tilde
00:28:35.182 --> 00:28:39.680
with respect to beta
J, I would generate
00:28:39.680 --> 00:28:42.370
what the mean value is.
00:28:42.370 --> 00:28:49.490
And actually, maybe
here let me show you
00:28:49.490 --> 00:28:52.440
that dG was indeed
what I expected.
00:28:52.440 --> 00:28:54.990
So dG would be dE.
00:28:54.990 --> 00:28:56.860
dE has a TdS.
00:28:56.860 --> 00:29:02.160
This will make that
into a minus SdT.
00:29:02.160 --> 00:29:04.110
dE has a Jdx.
00:29:04.110 --> 00:29:08.110
This will make it minus xdJ.
00:29:08.110 --> 00:29:12.320
And then I have mu dN.
00:29:12.320 --> 00:29:14.050
So thermodynamically,
you would have
00:29:14.050 --> 00:29:17.630
said that x is obtained
as a derivative of G
00:29:17.630 --> 00:29:23.040
with respect to J.
00:29:23.040 --> 00:29:26.670
Now, log Z is something
that is like beta
00:29:26.670 --> 00:29:33.410
G. At fixed temperature,
I can remove these betas.
00:29:33.410 --> 00:29:34.150
What do I get?
00:29:34.150 --> 00:29:39.380
This is the same
thing as dG by dJ.
00:29:39.380 --> 00:29:41.740
And I seem to have
lost a sign somewhere.
00:29:45.310 --> 00:29:51.250
OK, log Z was minus G.
So there's a minus here.
00:29:51.250 --> 00:29:52.320
Everything is consistent.
00:29:56.310 --> 00:29:59.780
So you can play around with
things in multiple ways,
00:29:59.780 --> 00:30:05.040
convince yourself, again,
that in this ensemble,
00:30:05.040 --> 00:30:08.810
I have fixed the
force, x's variable.
00:30:08.810 --> 00:30:12.850
But just like the energy
here was well defined up
00:30:12.850 --> 00:30:15.300
to something that
was square root of N,
00:30:15.300 --> 00:30:17.670
this x is well defined
up to something
00:30:17.670 --> 00:30:19.750
that is of the order
square root of N.
00:30:19.750 --> 00:30:25.280
Because again, you can
second look at the variance.
00:30:25.280 --> 00:30:30.150
It will be related to two
derivatives of log G, which
00:30:30.150 --> 00:30:33.970
would be related to
one derivative of x.
00:30:33.970 --> 00:30:35.960
And ultimately,
that will give you
00:30:35.960 --> 00:30:43.561
something like, again,
kBT squared d of x by dJ.
00:30:43.561 --> 00:30:44.060
Yes.
00:30:44.060 --> 00:30:46.483
AUDIENCE: Can you mention
again, regarding the probability
00:30:46.483 --> 00:30:47.024
distribution?
00:30:47.024 --> 00:30:48.510
What was the idea?
00:30:48.510 --> 00:30:50.360
PROFESSOR: How did I
get this probability?
00:30:50.360 --> 00:30:51.562
AUDIENCE: Yes.
00:30:51.562 --> 00:30:57.660
PROFESSOR: OK, so I
said that canonically, I
00:30:57.660 --> 00:31:02.260
was looking at a system
where x was fixed.
00:31:02.260 --> 00:31:04.950
But now I have
told you that J-- I
00:31:04.950 --> 00:31:07.540
know what J is,
what the force is.
00:31:07.540 --> 00:31:12.600
And so how can I make
sure that the system
00:31:12.600 --> 00:31:15.020
is maintained at a fixed J?
00:31:15.020 --> 00:31:18.860
I go back and say,
how did I ensure
00:31:18.860 --> 00:31:23.120
that my microcanonical system
was at a fixed temperature?
00:31:23.120 --> 00:31:25.740
I put it in contact
with a huge bath
00:31:25.740 --> 00:31:27.750
that had the right temperature.
00:31:27.750 --> 00:31:30.000
So here, what I
will do is I will
00:31:30.000 --> 00:31:35.090
connect the wall of my
system to a huge spring that
00:31:35.090 --> 00:31:37.690
will maintain a particle
[? effect ?] force
00:31:37.690 --> 00:31:39.570
J on the system.
00:31:39.570 --> 00:31:41.710
When we do it for
the case of the gas,
00:31:41.710 --> 00:31:44.510
I will imagine
that I have a box.
00:31:44.510 --> 00:31:47.980
I have a piston on top
that can move up and down.
00:31:47.980 --> 00:31:49.610
I put a weight on top of it.
00:31:49.610 --> 00:31:51.820
And that weight will
ensure that the pressure
00:31:51.820 --> 00:31:54.850
is at some particular
value inside the gas.
00:31:54.850 --> 00:31:58.270
But then the piston
can slide up and down.
00:31:58.270 --> 00:32:01.040
So a particular
state of the system
00:32:01.040 --> 00:32:05.850
I have to specify where the
piston is, how big x is,
00:32:05.850 --> 00:32:08.820
and what is the
microstate of the system.
00:32:08.820 --> 00:32:15.020
So the variables that I'm
not sure of are mu and x.
00:32:15.020 --> 00:32:18.640
Now, once I have said
what mu and x is, and I've
00:32:18.640 --> 00:32:21.570
put the system in contact
with a bath at temperature T,
00:32:21.570 --> 00:32:26.370
I can say, OK, the whole
thing is canonical,
00:32:26.370 --> 00:32:31.110
the energy of the entire system
composed of the energy of this,
00:32:31.110 --> 00:32:36.850
which is H of mu, and the energy
of the spring, which is Jx.
00:32:36.850 --> 00:32:41.880
And so the canonical
probability for the joint system
00:32:41.880 --> 00:32:44.630
is composed of the
net energy of the two.
00:32:44.630 --> 00:32:47.910
And I have to normalize it.
00:32:47.910 --> 00:32:50.180
Yes.
00:32:50.180 --> 00:32:52.080
AUDIENCE: Are you
taking x squared
00:32:52.080 --> 00:32:58.535
to be a scalar, so
that way the dx over dJ
00:32:58.535 --> 00:32:59.410
is like a divergence?
00:32:59.410 --> 00:33:03.483
Or are you taking it to
be a covariance tensor?
00:33:03.483 --> 00:33:06.290
PROFESSOR: Here, I have
assumed that it is a scalar.
00:33:06.290 --> 00:33:12.840
But we can certainly do-- if
you want to do it with a vector,
00:33:12.840 --> 00:33:16.660
then I can say something
about positivity.
00:33:16.660 --> 00:33:20.460
So I really wanted this to
be a variance and positive.
00:33:20.460 --> 00:33:23.650
And so if you like, then it
would be the diagonal terms
00:33:23.650 --> 00:33:28.900
of whatever
compressibility you have.
00:33:28.900 --> 00:33:34.030
So you certainly can generalize
this expression to be a vector.
00:33:34.030 --> 00:33:37.830
You can have xI xJ, and you
would have d of xI with respect
00:33:37.830 --> 00:33:40.120
to JJ or something like this.
00:33:40.120 --> 00:33:42.900
But here, I really
wanted the scalar case.
00:33:42.900 --> 00:33:45.410
Everything I did was
scalar manipulation.
00:33:45.410 --> 00:33:48.696
And this is now the variance
of something and is positive.
00:33:58.420 --> 00:34:01.260
OK, there was one
other ensemble,
00:34:01.260 --> 00:34:02.510
which was the grand canonical.
00:34:09.139 --> 00:34:16.739
So here, I went from the
energy to temperature.
00:34:16.739 --> 00:34:18.800
I kept N fixed.
00:34:18.800 --> 00:34:21.940
I didn't make it into
a chemical potential.
00:34:21.940 --> 00:34:23.760
But I can do that.
00:34:23.760 --> 00:34:25.480
Rather than having
fixed number, I
00:34:25.480 --> 00:34:27.610
can have fixed
chemical potential.
00:34:27.610 --> 00:34:33.280
But then I can't allow
the other variable
00:34:33.280 --> 00:34:35.250
to be J. It has to be x.
00:34:35.250 --> 00:34:37.788
Because as we have discussed
many times, at least one
00:34:37.788 --> 00:34:38.954
of them has to be extensive.
00:34:41.780 --> 00:34:45.679
And then you can follow this
procedure that you had here.
00:34:45.679 --> 00:34:50.020
Chemical work is just an
analog of mechanical work
00:34:50.020 --> 00:34:51.510
mathematically.
00:34:51.510 --> 00:34:53.560
So now I would have
the probability
00:34:53.560 --> 00:34:58.550
that I have specified
this set of variables.
00:34:58.550 --> 00:35:01.640
But now I don't know
what my microstate is.
00:35:01.640 --> 00:35:03.550
Actually, let me put mu s.
00:35:03.550 --> 00:35:07.020
Because I introduced a
chemical potential mu.
00:35:07.020 --> 00:35:10.460
So mu sub s is the
microstate of the system.
00:35:10.460 --> 00:35:15.510
I don't know how many particles
are in that microstate.
00:35:15.510 --> 00:35:18.670
The definition of mu s
would potentially cover N.
00:35:18.670 --> 00:35:21.110
Or I can write it explicitly.
00:35:21.110 --> 00:35:29.920
And then, what I will have is
e to the beta mu N minus beta
00:35:29.920 --> 00:35:34.260
H of the microstate's
energy divided
00:35:34.260 --> 00:35:38.870
by the normalization, which is
the grand partition function
00:35:38.870 --> 00:35:42.440
Q. So this was the Gibbs
partition function.
00:35:42.440 --> 00:35:48.930
This is a function of T, x
at the chemical potential.
00:35:48.930 --> 00:35:55.240
And the analog of all of these
expressions here would exist.
00:35:55.240 --> 00:35:58.370
So the average
number in the system,
00:35:58.370 --> 00:36:01.160
which now we can use as
the thermodynamic number,
00:36:01.160 --> 00:36:09.530
as we've seen, would be d of log
of Q with respect to d beta mu.
00:36:09.530 --> 00:36:12.430
And you can look at the
variances, et cetera.
00:36:12.430 --> 00:36:14.450
Yes.
00:36:14.450 --> 00:36:16.450
AUDIENCE: So do
you define, then,
00:36:16.450 --> 00:36:19.180
what the microstate
of the system
00:36:19.180 --> 00:36:24.010
really is if the particle
number is [INAUDIBLE]?
00:36:24.010 --> 00:36:27.760
PROFESSOR: OK, so let's say
I have five particles or six
00:36:27.760 --> 00:36:29.530
particles in the system.
00:36:29.530 --> 00:36:33.070
It would be, in the
first case, the momentum
00:36:33.070 --> 00:36:35.380
coordinates of five particles.
00:36:35.380 --> 00:36:37.095
In the next case,
in the momentum,
00:36:37.095 --> 00:36:39.750
I have coordinates
of six particles.
00:36:39.750 --> 00:36:43.780
So these are spaces,
as it has changed,
00:36:43.780 --> 00:36:46.410
where the dimensionality
of the phase space
00:36:46.410 --> 00:36:49.990
is also being modified.
00:36:49.990 --> 00:36:53.316
And again, for the ideal gas, we
will explicitly calculate that.
00:36:57.880 --> 00:37:04.940
And again, you can identify what
the log of this grand partition
00:37:04.940 --> 00:37:06.220
is going to be.
00:37:06.220 --> 00:37:12.590
It is going to be
minus beta E minus TS.
00:37:12.590 --> 00:37:16.660
Rather than minus Jx,
it will be minus mu N.
00:37:16.660 --> 00:37:24.400
And this combination is what
we call the grand potential g,
00:37:24.400 --> 00:37:34.850
which is E minus TS minus mu
N. You can do thermodynamics.
00:37:34.850 --> 00:37:36.580
You can do probability.
00:37:36.580 --> 00:37:41.520
Essentially, in the
large end limit, and only
00:37:41.520 --> 00:37:46.670
in the large end limit, there's
a consistent identification
00:37:46.670 --> 00:37:52.320
of most likely states according
to the statistical description
00:37:52.320 --> 00:37:53.835
and thermodynamic parameters.
00:38:05.631 --> 00:38:06.130
Yes.
00:38:06.130 --> 00:38:11.210
AUDIENCE: Can one build a more
general ensemble or description
00:38:11.210 --> 00:38:14.640
where, say, J is
not a fixed number,
00:38:14.640 --> 00:38:17.430
but I know how it varies?
00:38:17.430 --> 00:38:21.340
I know that it's subject to--
say the spring is not exactly
00:38:21.340 --> 00:38:22.753
a Hookean spring.
00:38:22.753 --> 00:38:23.695
It's not linear.
00:38:23.695 --> 00:38:25.810
The proportion is not linear.
00:38:25.810 --> 00:38:28.310
PROFESSOR: I think you are then
describing a [? composing ?]
00:38:28.310 --> 00:38:30.040
system, right?
00:38:30.040 --> 00:38:33.820
Because it may be
that your system
00:38:33.820 --> 00:38:37.470
itself has some
nonlinear properties
00:38:37.470 --> 00:38:40.090
for its force as a
function of displacement.
00:38:40.090 --> 00:38:42.130
And most generally, it will.
00:38:42.130 --> 00:38:43.980
Very soon, we will
look at the pressure
00:38:43.980 --> 00:38:47.380
of a gas, which will be
some complicated function
00:38:47.380 --> 00:38:50.700
of its density-- does not
necessarily have to be linear.
00:38:50.700 --> 00:38:53.240
So the system itself
could have all kinds
00:38:53.240 --> 00:38:55.720
of nonlinear dependencies.
00:38:55.720 --> 00:38:59.190
But you calculate the
corresponding force
00:38:59.190 --> 00:39:00.390
through this procedure.
00:39:00.390 --> 00:39:05.270
And it will vary in some
nonlinear function of density.
00:39:05.270 --> 00:39:08.260
If you're asking,
can I do something
00:39:08.260 --> 00:39:11.950
in which I couple this
to another system, which
00:39:11.950 --> 00:39:14.720
has some nonlinear
properties, then I
00:39:14.720 --> 00:39:16.960
would say that really
what you're doing
00:39:16.960 --> 00:39:18.980
is you're putting
two systems together.
00:39:18.980 --> 00:39:23.290
And you should be doing separate
thermodynamical calculations
00:39:23.290 --> 00:39:25.730
for each system, and
then put the constraint
00:39:25.730 --> 00:39:27.430
that the J of this
system should equal
00:39:27.430 --> 00:39:31.280
the J of the other system.
00:39:31.280 --> 00:39:35.186
AUDIENCE: I don't think that's
quite what I'm getting at.
00:39:35.186 --> 00:39:41.260
I'm thinking here,
the J is not given
00:39:41.260 --> 00:39:42.721
by another
thermodynamical system.
00:39:42.721 --> 00:39:45.230
We're just applying
it-- like we're not
00:39:45.230 --> 00:39:49.072
applying thermodynamics to
whatever is asserting to J.
00:39:49.072 --> 00:39:54.000
PROFESSOR: No, I'm trying
to mimic thermodynamics.
00:39:54.000 --> 00:39:56.805
So in thermodynamics,
I have a way
00:39:56.805 --> 00:39:59.070
of describing
equilibrium systems
00:39:59.070 --> 00:40:02.080
in terms of a certain
set of variables.
00:40:02.080 --> 00:40:06.420
And given that set of variables,
there are conjugate variables.
00:40:06.420 --> 00:40:08.420
So I'm constructing
something that
00:40:08.420 --> 00:40:11.230
is analogous to thermodynamics.
00:40:11.230 --> 00:40:14.270
It may be that you want
to do something else.
00:40:14.270 --> 00:40:17.210
And my perspective
is that the best way
00:40:17.210 --> 00:40:20.560
to do something else
is to sort of imagine
00:40:20.560 --> 00:40:23.380
different systems that are
in contact with each other.
00:40:23.380 --> 00:40:25.360
Each one of them you
do thermodynamics,
00:40:25.360 --> 00:40:28.130
and then you put
equilibrium between them.
00:40:28.130 --> 00:40:30.570
If I want to solve
some complicated system
00:40:30.570 --> 00:40:36.000
mechanically, then I sort of
break it down into the forces
00:40:36.000 --> 00:40:38.780
that are acting on one, forces
that are acting on the other,
00:40:38.780 --> 00:40:40.790
and then how this
one is responding,
00:40:40.790 --> 00:40:42.750
how that one is responding.
00:40:42.750 --> 00:40:45.600
I don't see any
advantage to having
00:40:45.600 --> 00:40:48.850
a more complicated
mechanical description
00:40:48.850 --> 00:40:52.780
of an individual system.
00:40:52.780 --> 00:40:54.990
AUDIENCE: It's not
like in reality.
00:40:54.990 --> 00:40:57.245
It's hard to obtain
situations where
00:40:57.245 --> 00:40:58.720
the external force is fixed.
00:40:58.720 --> 00:41:00.470
Really, if we're doing
an experiment here,
00:41:00.470 --> 00:41:04.466
the pressure in the
atmosphere is a fixed number.
00:41:04.466 --> 00:41:06.314
But maybe in other
circumstances--
00:41:06.314 --> 00:41:09.680
PROFESSOR: Yes, so if you
are in another circumstance,
00:41:09.680 --> 00:41:13.520
like you have one gas that
is expanding in a nonlinear
00:41:13.520 --> 00:41:16.520
medium, then what
I would do is I
00:41:16.520 --> 00:41:18.720
would calculate
the pressure of gas
00:41:18.720 --> 00:41:21.270
as a function of its
volume, et cetera.
00:41:21.270 --> 00:41:24.080
I would calculate the
response of that medium
00:41:24.080 --> 00:41:25.840
as a function of
stress, or whatever
00:41:25.840 --> 00:41:27.930
else I'm exerting on that.
00:41:27.930 --> 00:41:31.320
And I say that I am constrained
to move around the trajectory
00:41:31.320 --> 00:41:33.450
where this force
equals this force.
00:41:43.040 --> 00:41:46.970
So now let's carry out this
procedure for the ideal gas.
00:41:51.500 --> 00:41:58.410
And the microscopic
description of the ideal gas
00:41:58.410 --> 00:42:06.460
was that I have to sum over--
its Hamiltonian is composed
00:42:06.460 --> 00:42:09.750
of the kinetic energies
of the particles.
00:42:09.750 --> 00:42:12.450
So I have their momentum.
00:42:12.450 --> 00:42:17.050
Plus the potential that we
said I'm going to assume
00:42:17.050 --> 00:42:22.620
describes box of volume mu.
00:42:22.620 --> 00:42:27.290
And it is ideal,
because we don't
00:42:27.290 --> 00:42:29.010
have interactions
among particles.
00:42:29.010 --> 00:42:35.030
We will put that shortly
for the next version.
00:42:35.030 --> 00:42:37.720
Now, given this,
I can go through
00:42:37.720 --> 00:42:39.850
the various prescriptions.
00:42:39.850 --> 00:42:43.130
I can go
microcanonically and say
00:42:43.130 --> 00:42:47.850
that I know the energy volume
and the number of particles.
00:42:47.850 --> 00:42:52.870
And in this prescription,
we said that the probability
00:42:52.870 --> 00:42:59.320
is either 0 or 1
divided by omega.
00:42:59.320 --> 00:43:08.510
And this is if qi not
in box and sum over i Pi
00:43:08.510 --> 00:43:14.743
squared over 2m not equal
to E and one otherwise.
00:43:14.743 --> 00:43:16.118
AUDIENCE: Is there
any reason why
00:43:16.118 --> 00:43:19.494
we don't use a
direct delta there?
00:43:19.494 --> 00:43:20.910
PROFESSOR: I don't
know how to use
00:43:20.910 --> 00:43:24.100
the direct delta
for reading a box.
00:43:24.100 --> 00:43:27.210
For the energy,
the reason I don't
00:43:27.210 --> 00:43:30.960
like to do that is because
direct deltas have dimensions
00:43:30.960 --> 00:43:34.220
of 1 over whatever
thing is inside.
00:43:34.220 --> 00:43:35.950
Yeah, you could do that.
00:43:35.950 --> 00:43:41.010
It's just I prefer not to do it.
00:43:41.010 --> 00:43:42.640
But you certainly could do that.
00:43:42.640 --> 00:43:44.240
It is certainly
something that is
00:43:44.240 --> 00:43:48.505
0 or 1 over omega
on that surface.
00:43:53.950 --> 00:43:57.760
The S for this, what was it?
00:43:57.760 --> 00:44:00.350
It was obtained by
integrating over
00:44:00.350 --> 00:44:04.270
the entirety of the 6N
dimensional phase space.
00:44:04.270 --> 00:44:08.560
From the Q integrations,
we got V to the N.
00:44:08.560 --> 00:44:11.030
From the momentum
integrations, we
00:44:11.030 --> 00:44:15.580
got the surface area of
this hypersphere, which
00:44:15.580 --> 00:44:22.110
was 2 pi to the 3N over 2,
because it was 3N dimensional.
00:44:22.110 --> 00:44:25.410
And this was the solid angle.
00:44:25.410 --> 00:44:30.920
And then we got
the radius, which
00:44:30.920 --> 00:44:36.660
is 2mE square root raised
to the power of number
00:44:36.660 --> 00:44:37.840
of dimensions minus 1.
00:44:41.710 --> 00:44:45.730
Then we did something else.
00:44:45.730 --> 00:44:51.190
We said that the phase space
is more appropriately divided
00:44:51.190 --> 00:44:55.250
by N factorial for
identical particles.
00:44:55.250 --> 00:45:01.410
Here, I introduced
some factor of H
00:45:01.410 --> 00:45:06.130
to dimensionalize the
integrations over PQ space
00:45:06.130 --> 00:45:10.820
so that this quantity now
did not carry any dimensions.
00:45:10.820 --> 00:45:15.535
And oops, this was
not S. This was omega.
00:45:18.710 --> 00:45:27.900
And S/k, which
was log of omega--
00:45:27.900 --> 00:45:30.680
we took the log of
that expression.
00:45:30.680 --> 00:45:36.890
We got N log of V. From
the log of N factorial,
00:45:36.890 --> 00:45:40.500
Stirling's approximation
gave us a factor of N over e.
00:45:44.570 --> 00:45:48.400
All of the other factors were
proportional to 3N over 2.
00:45:48.400 --> 00:45:50.360
So the N is out front.
00:45:50.360 --> 00:45:53.850
I will have a factor
of 3 over 2 here.
00:45:53.850 --> 00:46:03.990
I have 2 pi mE
divided by 3N over 2.
00:46:03.990 --> 00:46:07.750
And then there's E from
Stirling's approximation.
00:46:07.750 --> 00:46:10.640
And that was the entropy
of the ideal gas.
00:46:10.640 --> 00:46:11.140
Yes.
00:46:11.140 --> 00:46:13.966
AUDIENCE: Does that
N factorial change
00:46:13.966 --> 00:46:17.952
when describing the system
with distinct particles?
00:46:17.952 --> 00:46:19.160
PROFESSOR: Yes, that's right.
00:46:19.160 --> 00:46:22.500
AUDIENCE: So that
definition changes?
00:46:22.500 --> 00:46:25.300
PROFESSOR: Yes, so
this factor of-- if I
00:46:25.300 --> 00:46:29.280
have a mixture of gas, and
one of them are one kind,
00:46:29.280 --> 00:46:31.760
and two of them
are another kind,
00:46:31.760 --> 00:46:35.760
I would be dividing by N1
factorial, N2 factorial,
00:46:35.760 --> 00:46:37.910
and not by N1 plus N2 factorial.
00:46:41.510 --> 00:46:46.780
AUDIENCE: So that is the correct
definition for all the cases?
00:46:46.780 --> 00:46:49.420
PROFESSOR: That's the
definition for phase space
00:46:49.420 --> 00:46:52.070
of identical particles.
00:46:52.070 --> 00:46:54.310
So all of them are
identical particle,
00:46:54.310 --> 00:46:55.910
is what I have written.
00:46:55.910 --> 00:46:57.370
All of them are distinct.
00:46:57.370 --> 00:46:59.330
There's no such factor.
00:46:59.330 --> 00:47:02.990
If half of them are of one type
and identical, half of them
00:47:02.990 --> 00:47:04.890
are another type
of identical, then
00:47:04.890 --> 00:47:10.678
I will have N over 2
factorial, N over 2 factorial.
00:47:10.678 --> 00:47:11.634
AUDIENCE: Question.
00:47:11.634 --> 00:47:12.518
PROFESSOR: Yeah.
00:47:12.518 --> 00:47:14.809
AUDIENCE: So when you're
writing the formula for number
00:47:14.809 --> 00:47:19.355
of microstates, just a
question of dimensions.
00:47:19.355 --> 00:47:22.334
You write V to the
N. It gives you
00:47:22.334 --> 00:47:25.610
something that mentions
coordinate to the power
00:47:25.610 --> 00:47:29.031
3N times something of
dimensions of [INAUDIBLE].
00:47:29.031 --> 00:47:33.385
The power is 3N minus
1 divided by H to 3N.
00:47:33.385 --> 00:47:37.110
So overall, this gives
you 1 over momentum.
00:47:37.110 --> 00:47:42.860
PROFESSOR: OK, let's put a delta
E over here-- actually, not
00:47:42.860 --> 00:47:49.810
a delta E, but a delta
R. OK, so now it's fine.
00:47:52.510 --> 00:47:56.080
It's really what I
want to make sure is
00:47:56.080 --> 00:47:59.305
that the extensive part is
correctly taken into account.
00:48:01.872 --> 00:48:03.455
I may be missing a
factor of dimension
00:48:03.455 --> 00:48:06.800
that is of the order of
1 every now and then.
00:48:06.800 --> 00:48:10.440
And then you can put some
[INAUDIBLE] to correct it.
00:48:10.440 --> 00:48:13.510
And it's, again, the
same story of the orange.
00:48:13.510 --> 00:48:19.690
It's all in the
skin, including--
00:48:19.690 --> 00:48:22.330
so the volume is the same
thing as the surface area,
00:48:22.330 --> 00:48:28.450
which is essentially
what I'm-- OK?
00:48:28.450 --> 00:48:32.170
So once you have this, you can
calculate various quantities.
00:48:32.170 --> 00:48:46.810
Again, the S is dE over T plus
PdV over T minus mu dN over T.
00:48:46.810 --> 00:48:51.920
So you can immediately
identify, for example,
00:48:51.920 --> 00:48:58.180
that 1 over T derivative
with respect to energy
00:48:58.180 --> 00:49:02.880
would give me a factor
of N/E. If I take
00:49:02.880 --> 00:49:08.060
a derivative with respect
to volume, P over T,
00:49:08.060 --> 00:49:10.310
derivative of this object
with respect to volume
00:49:10.310 --> 00:49:14.870
will give me N over V.
00:49:14.870 --> 00:49:20.430
And mu over T is--
oops, actually,
00:49:20.430 --> 00:49:23.620
in all of these cases,
I forgot the kB.
00:49:23.620 --> 00:49:25.040
So I have to restore it.
00:49:29.650 --> 00:49:35.750
And here I would get kB
log of the N out front.
00:49:35.750 --> 00:49:44.490
I can throw out V/N
4 pi m E over 3N.
00:49:44.490 --> 00:49:48.520
And I forgot the H's.
00:49:48.520 --> 00:49:51.490
So the H's would appear
as an H squared here.
00:49:51.490 --> 00:49:55.370
They will appear as an H squared
here raised to the 3/2 power.
00:50:05.440 --> 00:50:11.450
We also said that I can
look at the probability
00:50:11.450 --> 00:50:13.790
of a single particle
having momentum P1.
00:50:16.330 --> 00:50:18.520
And we showed that
essentially I have
00:50:18.520 --> 00:50:22.280
to integrate over everything
else if I'm not interested,
00:50:22.280 --> 00:50:27.110
such as the volume of
the particle number one.
00:50:27.110 --> 00:50:34.520
The normalization was omega
E, V, N. And integrating
00:50:34.520 --> 00:50:38.580
over particles numbers
two to N, where
00:50:38.580 --> 00:50:41.350
the energy that
is left to them is
00:50:41.350 --> 00:50:47.010
E minus the kinetic energy
of the one particle,
00:50:47.010 --> 00:50:49.920
gave me this expression.
00:50:49.920 --> 00:50:52.100
And essentially,
we found that this
00:50:52.100 --> 00:50:56.580
was proportional to one
side divided by E. 1
00:50:56.580 --> 00:51:01.690
minus P1 squared over 2m raised
to the power that was very
00:51:01.690 --> 00:51:04.875
close to 3N over 2
minus/plus something,
00:51:04.875 --> 00:51:08.170
and that this was
proportional, therefore,
00:51:08.170 --> 00:51:15.430
to this P1 squared
over 2m times-- P1
00:51:15.430 --> 00:51:20.135
squared 2mE-- 3N over 2E.
00:51:24.300 --> 00:51:32.242
And 3N over 2E from here we see
is the same thing as 1 over kT.
00:51:36.400 --> 00:51:38.870
So we can, with
this prescription,
00:51:38.870 --> 00:51:43.950
calculate all of the
properties of the ideal gas,
00:51:43.950 --> 00:51:50.830
including probability to see
one particle with some momentum.
00:51:50.830 --> 00:51:56.970
Now, if I do the same thing in
the canonical form, in which
00:51:56.970 --> 00:52:00.750
the microstate that I'm
looking at is temperature,
00:52:00.750 --> 00:52:04.950
volume, number of
particles, then
00:52:04.950 --> 00:52:08.410
the probability of
a microstate, given
00:52:08.410 --> 00:52:11.630
that I've specified
now the temperature,
00:52:11.630 --> 00:52:15.980
is proportional to the energy
of that microstate, which
00:52:15.980 --> 00:52:23.940
is e to the minus beta sum
over i Pi squared over 2m.
00:52:23.940 --> 00:52:27.480
Of course I would have to
have all of Qi's in box.
00:52:30.540 --> 00:52:33.720
Certainly they cannot
go outside the box.
00:52:33.720 --> 00:52:35.120
And this has to be normalized.
00:52:38.660 --> 00:52:41.480
Now we can see that in
this canonical ensemble,
00:52:41.480 --> 00:52:45.670
the result that we had to do
a couple of lines of algebra
00:52:45.670 --> 00:52:50.030
to get, which is that the
momentum of a particle
00:52:50.030 --> 00:52:54.510
is Gaussian distributed,
is automatically satisfied.
00:52:54.510 --> 00:52:58.260
And in this ensemble, each
one of the momenta you can see
00:52:58.260 --> 00:53:02.370
is independently distributed
according to this probability
00:53:02.370 --> 00:53:05.090
distribution.
00:53:05.090 --> 00:53:09.060
So somethings clearly
emerge much easier
00:53:09.060 --> 00:53:11.690
in this perspective.
00:53:11.690 --> 00:53:16.290
And if I were to look
at the normalization Z,
00:53:16.290 --> 00:53:21.875
the normalization Z I
obtained by integrating over
00:53:21.875 --> 00:53:23.830
the entirety of the phase space.
00:53:23.830 --> 00:53:31.720
So I have to do the integration
over d cubed Pi d cubed qi.
00:53:31.720 --> 00:53:35.240
Since this is the phase
space of identical particles,
00:53:35.240 --> 00:53:39.370
we said we have to
normalize it by N factorial.
00:53:39.370 --> 00:53:42.950
And I had this
factor of H to the 3N
00:53:42.950 --> 00:53:45.810
to make things dimensionless.
00:53:45.810 --> 00:53:52.400
I have to exponentiate this
energy sum over i Pi squared
00:53:52.400 --> 00:53:58.730
over 2m and just ensure that
the qi are inside the box.
00:53:58.730 --> 00:54:02.340
So if I integrate the
qi, what do I get?
00:54:02.340 --> 00:54:03.810
I will get V per particle.
00:54:03.810 --> 00:54:07.650
So I have V to the N
divided by N factorial.
00:54:10.320 --> 00:54:13.100
So that's the
volume contribution.
00:54:13.100 --> 00:54:15.000
And then what are
the P integrations?
00:54:15.000 --> 00:54:19.490
Each one of the P integrations
is an independent Gaussian.
00:54:19.490 --> 00:54:21.680
In fact, each component
is independent.
00:54:21.680 --> 00:54:24.780
So I have 3N Gaussian
integrations.
00:54:24.780 --> 00:54:27.310
And I can do Gaussian
integrations.
00:54:27.310 --> 00:54:31.420
I will get root 2 pi
m inverse of beta,
00:54:31.420 --> 00:54:36.030
which is kT per
Gaussian integration.
00:54:36.030 --> 00:54:37.820
And there are 3N of them.
00:54:42.950 --> 00:54:47.060
And, oh, I had the
factor of h to the 3N.
00:54:47.060 --> 00:54:47.870
I will put it here.
00:54:54.930 --> 00:55:00.630
So I chose these H's in order
to make this phase space
00:55:00.630 --> 00:55:02.060
dimensionless.
00:55:02.060 --> 00:55:06.680
So the Z that I have
now is dimensionless.
00:55:06.680 --> 00:55:10.100
So the dimensions
of V must be made up
00:55:10.100 --> 00:55:13.840
by dimensions of all of
these things that are left.
00:55:13.840 --> 00:55:17.550
So I'm going to make that
explicitly clear by writing it
00:55:17.550 --> 00:55:22.470
as 1 over N factorial V over
some characteristic volume
00:55:22.470 --> 00:55:25.070
raised to the N-th power.
00:55:25.070 --> 00:55:29.130
The characteristic volume comes
entirely from these factors.
00:55:29.130 --> 00:55:32.100
So I have introduced
lambda of T,
00:55:32.100 --> 00:55:41.740
which is h over root 2 pi mkT,
which is the thermal de Broglie
00:55:41.740 --> 00:55:42.240
wavelength.
00:55:45.960 --> 00:55:49.750
At this stage, this
h is just anything
00:55:49.750 --> 00:55:52.370
to make things
dimensionally work.
00:55:52.370 --> 00:55:54.530
When we do quantum
mechanics, we will
00:55:54.530 --> 00:55:57.920
see that this lens scale
has a very important
00:55:57.920 --> 00:55:59.260
physical meaning.
00:55:59.260 --> 00:56:03.310
As long as the separations
of particles on average
00:56:03.310 --> 00:56:06.380
is larger than this, you can
ignore quantum mechanics.
00:56:06.380 --> 00:56:08.000
When it becomes
less than this, you
00:56:08.000 --> 00:56:09.570
have to include quantum factors.
00:56:14.070 --> 00:56:17.540
So then what do we have?
00:56:17.540 --> 00:56:23.200
We have that the free
energy is minus kT log Z.
00:56:23.200 --> 00:56:31.440
So F is minus kT log of
this partition function.
00:56:31.440 --> 00:56:40.590
Log of that quantity will give
me N log V over lambda cubed.
00:56:40.590 --> 00:56:43.610
Stirling's formula,
log of N factorial,
00:56:43.610 --> 00:56:46.880
will give me N log N over e.
00:56:46.880 --> 00:56:48.105
So that's the free energy.
00:56:50.950 --> 00:56:55.520
Once I have the free energy,
I can calculate, let's say,
00:56:55.520 --> 00:56:57.090
the volume.
00:56:57.090 --> 00:57:02.040
Let's see, dF, which
is d of E minus TS,
00:57:02.040 --> 00:57:08.930
is minus SdT minus
PdV, because work
00:57:08.930 --> 00:57:11.800
of the gas we had
identified as minus PdV.
00:57:11.800 --> 00:57:14.820
I have mu dN.
00:57:14.820 --> 00:57:16.180
What do we have, therefore?
00:57:16.180 --> 00:57:27.780
We have that, for example, P
is minus dF by dV at constant T
00:57:27.780 --> 00:57:32.300
and N. So I have to go and
look at this expression
00:57:32.300 --> 00:57:35.880
where the V up here, it
just appears in log V.
00:57:35.880 --> 00:57:45.820
So the answer is going
to be NkT over V.
00:57:45.820 --> 00:57:49.740
I can calculate the
chemical potential.
00:57:49.740 --> 00:57:56.630
Mu will be dF by dN
at constant T and V,
00:57:56.630 --> 00:58:01.210
so just take the derivative with
respect to N. So what do I get?
00:58:01.210 --> 00:58:10.930
I will get minus kT log
of V over N lambda cubed.
00:58:10.930 --> 00:58:13.420
And this E will
disappear when I take
00:58:13.420 --> 00:58:16.920
the derivative with
respected to that log inside.
00:58:16.920 --> 00:58:22.340
So that's the formula for my mu.
00:58:22.340 --> 00:58:25.520
I won't calculate entropy.
00:58:25.520 --> 00:58:33.830
But I will calculate the energy,
noting that in this ensemble,
00:58:33.830 --> 00:58:39.350
a nice way of calculating energy
is minus d log Z by d beta.
00:58:39.350 --> 00:58:44.900
So this is minus
d log Z by d beta.
00:58:44.900 --> 00:58:59.170
And my Z, you can see, has
a bunch of things-- V, N, et
00:58:59.170 --> 00:58:59.690
cetera.
00:58:59.690 --> 00:59:04.140
But if I focus on temperature,
which is the inverse beta,
00:59:04.140 --> 00:59:07.610
you can see it appears
with a factor of 3N over 2.
00:59:07.610 --> 00:59:12.710
So this is beta to
the minus 3N over 2.
00:59:12.710 --> 00:59:18.050
So this is going to be 3N
over 2 derivative of log beta
00:59:18.050 --> 00:59:21.430
with respect to beta,
which is 1 over beta,
00:59:21.430 --> 00:59:24.250
which is the same
thing as 3 over 2 NkT.
00:59:58.080 --> 01:00:05.460
So what if I wanted to maintain
the system at some fixed
01:00:05.460 --> 01:00:07.820
temperature, but
rather than telling you
01:00:07.820 --> 01:00:10.200
what the volume of the
box is, I will tell you
01:00:10.200 --> 01:00:15.050
what its pressure is and
how many particles I have?
01:00:15.050 --> 01:00:18.690
How can I ensure that I
have a particular pressure?
01:00:18.690 --> 01:00:22.740
You can imagine that this is
the box that contains my gas.
01:00:22.740 --> 01:00:25.920
And I put a weight
on top of some kind
01:00:25.920 --> 01:00:32.300
of a piston that can
move over the gas.
01:00:32.300 --> 01:00:38.230
So then you would say
that the net energy that I
01:00:38.230 --> 01:00:42.820
have to look at is the
kinetic energy of the gas
01:00:42.820 --> 01:00:46.050
particles here plus
the potential energy
01:00:46.050 --> 01:00:48.910
of this weight that
is going up and down.
01:00:48.910 --> 01:00:50.680
If you like, that
potential energy
01:00:50.680 --> 01:00:58.310
is going to be mass times delta
H. Delta H times area gives you
01:00:58.310 --> 01:00:59.500
volume.
01:00:59.500 --> 01:01:04.320
Mass times G divided by
area will give you pressure.
01:01:04.320 --> 01:01:06.900
So the combination of
those two is the same thing
01:01:06.900 --> 01:01:08.830
as pressure times volume.
01:01:08.830 --> 01:01:10.690
So this is going to
be the same thing
01:01:10.690 --> 01:01:17.310
as minus sum over i
Pi squared over 2m.
01:01:17.310 --> 01:01:21.940
For all of the gas particles
for this additional weight that
01:01:21.940 --> 01:01:24.420
is going up and
down, it will give me
01:01:24.420 --> 01:01:27.910
a contribution that
is minus beta PV.
01:01:33.380 --> 01:01:40.090
And so this is the
probability in this state.
01:01:40.090 --> 01:01:43.660
It is going to be the
same as this provided
01:01:43.660 --> 01:01:49.240
that I divide by some
Gibbs partition function.
01:01:52.410 --> 01:01:55.840
So this is the probability
of the microstate.
01:01:55.840 --> 01:02:00.480
So what is the Gibbs
partition function?
01:02:00.480 --> 01:02:02.350
Well, what is the normalization?
01:02:02.350 --> 01:02:05.540
I now have one
additional variable,
01:02:05.540 --> 01:02:10.440
which is where this piston
is located in order to ensure
01:02:10.440 --> 01:02:12.950
that it is at the
right pressure.
01:02:12.950 --> 01:02:18.255
So I have to integrate also
over the additional volume.
01:02:21.680 --> 01:02:30.020
This additional factor
only depends on PV.
01:02:30.020 --> 01:02:33.280
And then I have
to integrate given
01:02:33.280 --> 01:02:35.640
that I have some
particular V that I then
01:02:35.640 --> 01:02:40.870
have to integrate over all
of the microstates that
01:02:40.870 --> 01:02:43.720
are confined within this volume.
01:02:43.720 --> 01:02:50.420
Their momenta and their
coordinates-- what is that?
01:02:50.420 --> 01:02:51.720
I just calculated that.
01:02:51.720 --> 01:02:57.280
That's the partition function
as a function of T, V, and N.
01:02:57.280 --> 01:03:01.700
So for a fixed V, I
already did the integration
01:03:01.700 --> 01:03:05.425
over all microscopic
degrees of freedom.
01:03:05.425 --> 01:03:08.850
I have one more integration
to do over the volume.
01:03:08.850 --> 01:03:10.240
And that's it.
01:03:10.240 --> 01:03:13.370
So if you like, this is like
doing a Laplace transform.
01:03:13.370 --> 01:03:20.750
To go from Z of T, V, and N,
to this Z tilde of T, P, and N
01:03:20.750 --> 01:03:23.700
is making some kind of
a Laplace transformation
01:03:23.700 --> 01:03:27.280
from one variable
to another variable.
01:03:27.280 --> 01:03:30.990
And now I know
actually what my answer
01:03:30.990 --> 01:03:34.330
was for the partition function.
01:03:34.330 --> 01:03:41.890
It was 1 over N factorial
V over lambda cubed raised
01:03:41.890 --> 01:03:50.900
to the power of N. So I have
1 over N factorial lambda
01:03:50.900 --> 01:03:52.450
to the power of 3N.
01:03:52.450 --> 01:03:55.430
And then I have to do
one of these integrals
01:03:55.430 --> 01:04:01.550
that we have seen many times,
the integral of V to the N
01:04:01.550 --> 01:04:04.540
against an exponential.
01:04:04.540 --> 01:04:06.050
That's something
that actually we
01:04:06.050 --> 01:04:12.180
used in order to define N
factorial, except that I have
01:04:12.180 --> 01:04:17.010
to dimensionalize this V. So
I will get a factor of beta P
01:04:17.010 --> 01:04:19.260
to the power of N plus 1.
01:04:22.040 --> 01:04:24.330
The N factorials cancel.
01:04:24.330 --> 01:04:28.250
And the answer is beta
P to the power of N
01:04:28.250 --> 01:04:44.410
plus 1 divided by lambda
cubed to the power of N.
01:04:44.410 --> 01:04:52.060
So my Gibbs free
energy, which is minus
01:04:52.060 --> 01:04:57.430
kT log of the Gibbs
partition function,
01:04:57.430 --> 01:05:10.100
is going to be minus
NkT log of this object.
01:05:10.100 --> 01:05:14.380
I have ignored the difference
between N and N plus 1.
01:05:14.380 --> 01:05:16.960
And what I will get here
is the combination beta
01:05:16.960 --> 01:05:20.490
P lambda cubed.
01:05:20.490 --> 01:05:21.760
Yes.
01:05:21.760 --> 01:05:24.170
AUDIENCE: Is your
beta P to the N plus 1
01:05:24.170 --> 01:05:26.150
in the numerator
or the denominator?
01:05:26.150 --> 01:05:31.341
PROFESSOR: Thank you, it should
be in the denominator, which
01:05:31.341 --> 01:05:41.220
means that-- OK,
this one is correct.
01:05:41.220 --> 01:05:43.750
I guess this one I was
going by dimensions.
01:05:43.750 --> 01:05:46.420
Because beta PV
is dimensionless.
01:05:49.120 --> 01:05:51.340
All right, yes.
01:05:51.340 --> 01:05:53.680
AUDIENCE: One other
thing, it seems
01:05:53.680 --> 01:05:57.386
like there's a dimensional
mismatch in your expression
01:05:57.386 --> 01:06:03.599
for Z. Because you have an
extra factor of beta P--
01:06:03.599 --> 01:06:09.480
PROFESSOR: Exactly right,
because this object
01:06:09.480 --> 01:06:15.110
is a probability density that
involves a factor of volume.
01:06:15.110 --> 01:06:18.220
And as a probability density,
the dimension of this
01:06:18.220 --> 01:06:20.060
will carry an extra
factor of volume.
01:06:22.780 --> 01:06:27.565
So if I really wanted to make
this quantity dimensionless
01:06:27.565 --> 01:06:31.900
also, I would need to
divide by something
01:06:31.900 --> 01:06:35.850
that has some
dimension of volume.
01:06:35.850 --> 01:06:41.100
But alternatively, I can
recognize that indeed this
01:06:41.100 --> 01:06:43.330
is a probability
density in volume.
01:06:43.330 --> 01:06:46.180
So it will have the
dimensions of volume.
01:06:46.180 --> 01:06:49.130
And again, as I said,
what I'm really always
01:06:49.130 --> 01:06:52.750
careful to make sure that is
dimensionless is the thing that
01:06:52.750 --> 01:06:57.280
is proportional to N. If there's
a log of a single dimension
01:06:57.280 --> 01:07:00.310
out here, typically we don't
have to worry about it.
01:07:00.310 --> 01:07:02.085
But if you think
about its origin,
01:07:02.085 --> 01:07:04.750
the origin is indeed
that this quantity
01:07:04.750 --> 01:07:06.034
is a probability density.
01:07:13.450 --> 01:07:17.110
But it will, believe
me, not change anything
01:07:17.110 --> 01:07:18.944
in your life to ignore that.
01:07:23.330 --> 01:07:27.960
All right, so once we have this
G, then we recognize-- again,
01:07:27.960 --> 01:07:29.920
hopefully I didn't
make any mistakes.
01:07:29.920 --> 01:07:35.210
G is E plus PV minus TS.
01:07:35.210 --> 01:07:46.952
So dG should be minus
SdT plus VdP plus mu dN.
01:07:52.710 --> 01:07:56.730
So that, for example, in this
ensemble I can ask-- well,
01:07:56.730 --> 01:07:58.170
I told you what the pressure is.
01:07:58.170 --> 01:07:59.820
What's the volume?
01:07:59.820 --> 01:08:04.356
Volume is going to be
obtained as dG by dP
01:08:04.356 --> 01:08:07.210
at constant
temperature and number.
01:08:07.210 --> 01:08:10.740
So these two are constant.
01:08:10.740 --> 01:08:15.420
Log P, its derivative is
going to give me NkT over P.
01:08:15.420 --> 01:08:18.439
So again, I get another
form of the ideal gas
01:08:18.439 --> 01:08:20.090
equation of state.
01:08:20.090 --> 01:08:22.700
I can ask, what's the
chemical potential?
01:08:22.700 --> 01:08:28.720
It is going to be dG by
dN at constant T and P.
01:08:28.720 --> 01:08:31.090
So I go and look at
the N dependence.
01:08:31.090 --> 01:08:34.609
And I notice that there's just
an N dependence out front.
01:08:34.609 --> 01:08:41.300
So what I will get is kT
log of beta P lambda cubed.
01:08:41.300 --> 01:08:44.020
And if you like, you
can check that, say,
01:08:44.020 --> 01:08:48.200
this expression for
the chemical potential
01:08:48.200 --> 01:08:54.399
and-- did we derive
it somewhere else?
01:08:54.399 --> 01:08:56.620
Yes, we derived it over here.
01:08:56.620 --> 01:08:59.790
This expression for
the chemical potential
01:08:59.790 --> 01:09:04.390
are identical once you take
advantage of the ideal gas
01:09:04.390 --> 01:09:06.590
equation of state
to convert the V
01:09:06.590 --> 01:09:16.069
over N in that
expression to beta P.
01:09:16.069 --> 01:09:23.420
And finally, we have
the grand canonical.
01:09:23.420 --> 01:09:24.697
Let's do that also.
01:09:38.140 --> 01:09:43.220
So now we are going to look at
an ensemble where I tell you
01:09:43.220 --> 01:09:46.470
what the temperature is
and the chemical potential.
01:09:46.470 --> 01:09:49.700
But I have to tell you
what the volume is.
01:09:49.700 --> 01:09:53.324
And then the statement
is that the probability
01:09:53.324 --> 01:09:57.640
of a particular microstate
that I will now indicate mu
01:09:57.640 --> 01:10:03.700
s force system-- not to be
confused with the chemical
01:10:03.700 --> 01:10:09.310
potential-- is proportional
to e to the beta mu
01:10:09.310 --> 01:10:16.890
N minus beta H of the
microstate energy.
01:10:16.890 --> 01:10:22.770
And the normalization is this Q,
which will be a grand partition
01:10:22.770 --> 01:10:28.160
function that is
function of T, V, and mu.
01:10:28.160 --> 01:10:31.330
How is this
probability normalized?
01:10:31.330 --> 01:10:38.330
Well, I'm spanning over
a space of microstates
01:10:38.330 --> 01:10:41.030
that have indefinite number.
01:10:41.030 --> 01:10:47.450
Their number runs, presumably,
all the way from 0 to infinity.
01:10:47.450 --> 01:10:53.280
And I have to multiply each
particular segment that
01:10:53.280 --> 01:10:58.090
has N of these present
with e to the beta mu N.
01:10:58.090 --> 01:11:02.930
Now, once I have that
segment, what I need to do
01:11:02.930 --> 01:11:06.820
is to sum over all
coordinates and momenta
01:11:06.820 --> 01:11:10.100
as appropriate to a
system of N particles.
01:11:10.100 --> 01:11:13.690
And that, once more, is
my partition function
01:11:13.690 --> 01:11:23.590
Z of T, V, and N. And so since
I know what that expression is,
01:11:23.590 --> 01:11:27.637
I can substitute it in
some e to the beta mu
01:11:27.637 --> 01:11:34.840
N. And Z is 1 over N factorial
V over lambda cubed raised
01:11:34.840 --> 01:11:41.240
to the power of N.
01:11:41.240 --> 01:11:45.530
Now fortunately, that's
a sum that I recognize.
01:11:45.530 --> 01:11:48.530
It is 1 over N factorial
something raised
01:11:48.530 --> 01:11:51.190
to the N-th power
summed over all N,
01:11:51.190 --> 01:11:53.900
which is the summation
for the exponential.
01:11:53.900 --> 01:11:58.770
So this is the exponential
of e to the beta mu
01:11:58.770 --> 01:11:59.850
V over lambda cubed.
01:12:08.910 --> 01:12:13.190
So once I have this,
I can construct
01:12:13.190 --> 01:12:24.150
my G, which is minus kT
log of Q, which is minus
01:12:24.150 --> 01:12:32.380
kT e to the beta mu V
divided by lambda cubed.
01:12:38.320 --> 01:12:42.615
Now, note that in all
of the other expressions
01:12:42.615 --> 01:12:47.030
that I had all of these logs of
something, they were extensive.
01:12:47.030 --> 01:12:50.680
And the extensivity was
ensured by having results
01:12:50.680 --> 01:12:52.570
that were ultimately
proportional
01:12:52.570 --> 01:12:55.205
to N for these logs.
01:12:55.205 --> 01:12:58.410
Let's say I have here,
I have an N here.
01:12:58.410 --> 01:13:00.380
For the s, I have an N there.
01:13:00.380 --> 01:13:04.770
Previously I had also an N here.
01:13:04.770 --> 01:13:06.850
Now, in this
ensemble, I don't have
01:13:06.850 --> 01:13:10.660
N. Extensivity is insured by
this thing being proportional
01:13:10.660 --> 01:13:22.770
to G. Now also remember that
G was E minus TS minus mu N.
01:13:22.770 --> 01:13:26.540
But we had another
result for extensivity,
01:13:26.540 --> 01:13:33.130
that for extensive systems,
E was TS plus mu N minus PV.
01:13:33.130 --> 01:13:38.930
So this is in fact
because of extensivity, we
01:13:38.930 --> 01:13:43.600
had expected it to be
proportional to the volume.
01:13:43.600 --> 01:13:47.270
And so this combination
should end up
01:13:47.270 --> 01:13:50.170
being in fact the pressure.
01:13:50.170 --> 01:13:54.070
I can see what the pressure
is in different ways.
01:13:54.070 --> 01:13:57.422
I can, for example,
look at what this dG is.
01:13:57.422 --> 01:13:59.780
It is minus SdT.
01:13:59.780 --> 01:14:04.243
It is minus PdV minus Nd mu.
01:14:07.530 --> 01:14:11.860
I could, for example,
identify the pressure
01:14:11.860 --> 01:14:15.390
by taking a derivative of
this with respect to volume.
01:14:15.390 --> 01:14:17.100
But it is proportional
to volume.
01:14:17.100 --> 01:14:21.090
So I again get that
this combination really
01:14:21.090 --> 01:14:23.760
should be pressure.
01:14:23.760 --> 01:14:26.530
You say, I don't recognize
that as a pressure.
01:14:26.530 --> 01:14:29.460
You say, well, it's because
the formula for pressure
01:14:29.460 --> 01:14:33.430
that we have been using is
always in terms of N. So let's
01:14:33.430 --> 01:14:34.030
check that.
01:14:34.030 --> 01:14:35.840
So what is N?
01:14:35.840 --> 01:14:42.560
I can get N from
minus dG by d mu.
01:14:42.560 --> 01:14:46.980
And what happens if I do that?
01:14:46.980 --> 01:14:49.030
When I take a derivative
with respect to mu,
01:14:49.030 --> 01:14:52.030
I will bring down
a factor of beta.
01:14:52.030 --> 01:14:53.480
Beta will kill the kT.
01:14:53.480 --> 01:15:00.640
I will get e to the beta mu
V over lambda cubed, which
01:15:00.640 --> 01:15:03.620
by the way is also
these expressions
01:15:03.620 --> 01:15:07.840
that we had previously for
the relationship between mu
01:15:07.840 --> 01:15:10.480
and N over V lambda
cubed if I just
01:15:10.480 --> 01:15:12.810
take the log of this expression.
01:15:12.810 --> 01:15:17.640
And then if I substitute e to
the beta mu V over lambda cubed
01:15:17.640 --> 01:15:20.860
to BN, you can see that I have
the thing that I was calling
01:15:20.860 --> 01:15:33.080
pressure is indeed NkT
over V. So everything
01:15:33.080 --> 01:15:35.070
is consistent with
the ideal gas law.
01:15:35.070 --> 01:15:39.370
The chemical potential comes
out consistently-- extensivity,
01:15:39.370 --> 01:15:44.300
everything is
correctly identified.
01:15:44.300 --> 01:15:47.390
Maybe one more thing
that I note here
01:15:47.390 --> 01:15:52.430
is that, for this ideal gas
and this particular form that I
01:15:52.430 --> 01:16:01.670
have for this object-- so
let's maybe do something here.
01:16:01.670 --> 01:16:06.960
Note that the N appears
in an exponential with e
01:16:06.960 --> 01:16:08.400
to the beta mu.
01:16:08.400 --> 01:16:11.790
So another way that I
could have gotten my N
01:16:11.790 --> 01:16:18.157
would have been d log Q
with respect to beta mu.
01:16:20.840 --> 01:16:29.030
And again, my log Q is
simply V over lambda cubed
01:16:29.030 --> 01:16:31.820
e to the beta mu.
01:16:31.820 --> 01:16:34.300
And you can check
that if I do that, I
01:16:34.300 --> 01:16:39.060
will get this formula
that I had for N. Well,
01:16:39.060 --> 01:16:44.380
the thing is that I can
get various cumulants
01:16:44.380 --> 01:16:49.200
of this object by continuing
to take derivatives.
01:16:49.200 --> 01:16:58.430
So I take m derivatives of log
of Q with respect to beta mu.
01:17:02.580 --> 01:17:05.245
So I have to keep taking
derivatives of the exponential.
01:17:08.080 --> 01:17:12.560
And as long as I keep taking the
derivative of the exponential,
01:17:12.560 --> 01:17:15.310
I will get the exponential back.
01:17:15.310 --> 01:17:17.820
So all of these things
are really the same thing.
01:17:20.380 --> 01:17:24.640
So all cumulants of the
number fluctuations of the gas
01:17:24.640 --> 01:17:27.550
are really the same
thing as a number.
01:17:27.550 --> 01:17:30.250
Can you remember what that says?
01:17:30.250 --> 01:17:33.140
What's the distribution?
01:17:33.140 --> 01:17:35.610
AUDIENCE: [INAUDIBLE]
01:17:35.610 --> 01:17:37.230
PROFESSOR: Poisson, very good.
01:17:37.230 --> 01:17:41.430
The distribution where all of
the cumulants were the same
01:17:41.430 --> 01:17:43.530
is Poisson distribution.
01:17:43.530 --> 01:17:46.550
Essentially it says
that if I take a box,
01:17:46.550 --> 01:17:52.070
or if I just look at that
imaginary volume in this room,
01:17:52.070 --> 01:17:54.150
and count the
number of particles,
01:17:54.150 --> 01:17:56.820
as long as it is
almost identical,
01:17:56.820 --> 01:18:00.060
the distribution of the number
of particles within the volume
01:18:00.060 --> 01:18:03.970
is Poisson.
01:18:03.970 --> 01:18:09.170
You know all of the
fluctuations, et cetera.
01:18:09.170 --> 01:18:09.875
Yes.
01:18:09.875 --> 01:18:11.250
AUDIENCE: So this
expression, you
01:18:11.250 --> 01:18:17.060
have N equals e to the beta mu V
divided by lambda to the third.
01:18:17.060 --> 01:18:19.550
Considering that
expression, could you then
01:18:19.550 --> 01:18:22.040
say that the exponential
quantity is proportional
01:18:22.040 --> 01:18:26.024
to the phase space
density [INAUDIBLE]?
01:18:26.024 --> 01:18:28.016
PROFESSOR: Let's rearrange this.
01:18:28.016 --> 01:18:35.150
Beta mu is log of N
over V lambda cubed.
01:18:35.150 --> 01:18:39.780
This is a single
particle density.
01:18:39.780 --> 01:18:44.780
So beta mu, or the chemical
potential up to a factor of kT,
01:18:44.780 --> 01:18:52.350
is the log of how many particles
fit within one de Broglie
01:18:52.350 --> 01:18:54.520
volume.
01:18:54.520 --> 01:18:58.240
And this expression is in
fact correct only in the limit
01:18:58.240 --> 01:18:59.880
where this is small.
01:18:59.880 --> 01:19:01.543
And as we shall
see later on, there
01:19:01.543 --> 01:19:03.560
will be quantum
mechanical corrections
01:19:03.560 --> 01:19:05.635
when this combination is large.
01:19:05.635 --> 01:19:10.490
So this combination is very
important in identifying
01:19:10.490 --> 01:19:12.362
when things become
quantum mechanic.
01:19:21.210 --> 01:19:25.220
All right, so let's
just give a preamble
01:19:25.220 --> 01:19:27.390
of what we will be
doing next time.
01:19:34.352 --> 01:19:38.570
I should have erased
the [INAUDIBLE].
01:19:38.570 --> 01:19:46.610
So we want to now do
interacting systems.
01:19:46.610 --> 01:19:51.160
So this one example
of the ideal gas I
01:19:51.160 --> 01:19:54.070
did for you to [INAUDIBLE]
in all possible ensembles.
01:19:54.070 --> 01:19:56.980
And I could do
that, because it was
01:19:56.980 --> 01:20:00.920
a collection of non-interacting
degrees of freedom.
01:20:00.920 --> 01:20:06.140
As soon as I have interactions
among my huge number of degrees
01:20:06.140 --> 01:20:08.860
of freedom, the story changes.
01:20:08.860 --> 01:20:13.270
So let's, for example, look
at a generalization of what
01:20:13.270 --> 01:20:24.680
I had written before-- a one
particle description which
01:20:24.680 --> 01:20:30.310
if I stop here gives me an
ideal system, and potentially
01:20:30.310 --> 01:20:35.640
some complicated interaction
among all of these coordinates.
01:20:35.640 --> 01:20:37.380
This could be a
pairwise interaction.
01:20:37.380 --> 01:20:39.880
It could have three particles.
01:20:39.880 --> 01:20:42.560
It could potentially--
at this stage
01:20:42.560 --> 01:20:44.710
I want to write down
the most general form.
01:20:49.120 --> 01:20:52.410
I want to see what I can
learn about the properties
01:20:52.410 --> 01:20:56.100
of this modified version,
or non-ideal gas,
01:20:56.100 --> 01:20:59.460
and the ensemble that I
will choose initially.
01:20:59.460 --> 01:21:01.910
Microcanonical is
typically difficult.
01:21:01.910 --> 01:21:05.700
I will go and do
things canonically,
01:21:05.700 --> 01:21:07.190
which is somewhat easier.
01:21:07.190 --> 01:21:11.310
And later on, maybe even
grand canonical is easier.
01:21:11.310 --> 01:21:14.860
So what do I have to do?
01:21:14.860 --> 01:21:19.580
I would say that the
partition function is obtained
01:21:19.580 --> 01:21:26.340
by integrating over the entirety
of this phase space-- product
01:21:26.340 --> 01:21:30.950
d cubed Pi d cubed qi.
01:21:30.950 --> 01:21:35.010
I will normalize
things by N factorial,
01:21:35.010 --> 01:21:38.830
dimensionalize them
by h to the 3N.
01:21:38.830 --> 01:21:42.840
And I have exponential
of minus beta
01:21:42.840 --> 01:21:46.360
sum over i Pi squared over 2m.
01:21:46.360 --> 01:21:56.120
And then I have the
exponential of minus U.
01:21:56.120 --> 01:22:00.300
Now, this time around, the
P integrals are [INAUDIBLE].
01:22:00.300 --> 01:22:02.590
I can do them immediately.
01:22:02.590 --> 01:22:07.200
Because typically the momenta
don't interact with each other.
01:22:07.200 --> 01:22:09.900
And practicality, no
matter how complicated
01:22:09.900 --> 01:22:12.260
a system of interactions
is, you will
01:22:12.260 --> 01:22:16.310
be able to integrate over the
momentum degrees of freedom.
01:22:16.310 --> 01:22:20.660
And what you get, you
will get this factor of 1
01:22:20.660 --> 01:22:25.512
over lambda to the power
of 3N from the 3N momentum
01:22:25.512 --> 01:22:26.053
integrations.
01:22:29.320 --> 01:22:32.610
The thing that is hard-- there
will be this factor of 1 over N
01:22:32.610 --> 01:22:38.280
factorial-- is the integration
over all of the coordinates,
01:22:38.280 --> 01:22:42.060
d cubed qi of this factor
of e to the minus beta
01:22:42.060 --> 01:22:45.680
U. I gave you the most general
possible U. There's no way
01:22:45.680 --> 01:22:48.930
that I can do this integration.
01:22:48.930 --> 01:22:53.300
What I will do is I will divide
each one of these integrations
01:22:53.300 --> 01:22:57.480
over coordinate of
particle by its volume.
01:22:57.480 --> 01:23:01.040
I will therefore
have V to the N here.
01:23:01.040 --> 01:23:05.400
And V to the N divided by
lambda to the 3N N factorial
01:23:05.400 --> 01:23:07.760
is none other than
the partition function
01:23:07.760 --> 01:23:11.490
that we had calculated
before for the ideal gas.
01:23:11.490 --> 01:23:15.830
And I will call Z0
for the ideal gas.
01:23:15.830 --> 01:23:21.410
And I claim that this
object I can interpret
01:23:21.410 --> 01:23:26.030
as a kind of average
of a function e
01:23:26.030 --> 01:23:30.470
to the beta U defined over
the phase space of all
01:23:30.470 --> 01:23:33.420
of these particles
where the probability
01:23:33.420 --> 01:23:38.360
to find each particle is
uniform in the space of the box,
01:23:38.360 --> 01:23:40.520
let's say.
01:23:40.520 --> 01:23:45.920
So what this says is
for the 0-th order case,
01:23:45.920 --> 01:23:49.490
for the ideal case that we
have discussed, once I have set
01:23:49.490 --> 01:23:53.970
the box, the particle can be
anywhere in the box uniformly.
01:23:53.970 --> 01:23:57.540
For that uniform
description probability,
01:23:57.540 --> 01:23:59.820
calculate what the average
of this quantity is.
01:24:02.630 --> 01:24:06.530
So what we have is that
Z is in fact Z0, that
01:24:06.530 --> 01:24:09.360
average that I can expand.
01:24:09.360 --> 01:24:11.920
And what we will
be doing henceforth
01:24:11.920 --> 01:24:15.420
is a perturbation
theory in powers of U.
01:24:15.420 --> 01:24:20.150
Because I know how to do things
for the case of U equals 0.
01:24:20.150 --> 01:24:26.530
And then I hope to calculate
things in various powers of U.
01:24:26.530 --> 01:24:30.460
So I will do that expansion.
01:24:30.460 --> 01:24:34.030
And then I say, no
really, what I'm
01:24:34.030 --> 01:24:37.860
interested in is something like
a free energy, which is log Z.
01:24:37.860 --> 01:24:43.540
And so for that, I will need log
of Z0 plus log of that series.
01:24:43.540 --> 01:24:45.540
But the log of these
kinds of series
01:24:45.540 --> 01:24:50.200
I know I can write as
minus beta to the l over l
01:24:50.200 --> 01:24:55.120
factorial, replacing, when
I go to the log, moments
01:24:55.120 --> 01:24:58.660
by corresponding cumulants.
01:24:58.660 --> 01:25:02.290
So this is called the
cumulant expansion,
01:25:02.290 --> 01:25:07.410
which we will carry
out next time around.
01:25:07.410 --> 01:25:07.980
Yes.
01:25:07.980 --> 01:25:09.916
AUDIENCE: In
general, [INAUDIBLE].
01:25:21.190 --> 01:25:24.210
PROFESSOR: For some
cases, you can.
01:25:24.210 --> 01:25:26.680
For many cases,
you find that that
01:25:26.680 --> 01:25:28.160
will give you wrong answers.
01:25:28.160 --> 01:25:31.900
Because the phase space
around which you're expanding
01:25:31.900 --> 01:25:33.570
is so broad.
01:25:33.570 --> 01:25:35.760
It is not like a
saddle point where
01:25:35.760 --> 01:25:38.880
you have one variable
you are expanding
01:25:38.880 --> 01:25:41.860
in a huge number of variables.