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PROFESSOR: Let's
say that I tell you
00:00:31.640 --> 00:00:37.870
that I'm interested in a gas
that has some temperature.
00:00:37.870 --> 00:00:39.910
I specify, let's
say, temperature.
00:00:39.910 --> 00:00:42.560
Pressure is room
temperature, room pressure.
00:00:42.560 --> 00:00:45.560
And I tell you how
many particles I have.
00:00:45.560 --> 00:00:50.870
So that's [INAUDIBLE] for
you what the macro state is.
00:00:50.870 --> 00:00:55.690
And I want to then see what the
corresponding micro state is.
00:00:55.690 --> 00:01:01.010
So I take one of these
boxes, whereas this is a box
00:01:01.010 --> 00:01:05.290
that I draw in three dimensions,
I can make a correspondence
00:01:05.290 --> 00:01:09.060
and draw this is 6N-dimensional
coordinate space, which
00:01:09.060 --> 00:01:10.960
would be hard for me to draw.
00:01:10.960 --> 00:01:13.850
But basically, a space
of six n dimensions.
00:01:13.850 --> 00:01:17.030
I figure out where the
position, and the particles,
00:01:17.030 --> 00:01:18.590
and the momenta are.
00:01:18.590 --> 00:01:22.270
And I sort of find that there
is a corresponding micro state
00:01:22.270 --> 00:01:25.270
that corresponds to
this macro state.
00:01:25.270 --> 00:01:26.180
OK, that's fine.
00:01:26.180 --> 00:01:28.500
I made the correspondence.
00:01:28.500 --> 00:01:37.410
But the thing is that I can
imagine lots and lots of boxes
00:01:37.410 --> 00:01:40.910
that have exactly the same
macroscopic properties.
00:01:40.910 --> 00:01:45.660
That is, I can imagine
putting here side by side
00:01:45.660 --> 00:01:48.460
a huge number of these boxes.
00:01:48.460 --> 00:01:53.230
All of them are described
by exactly the same volume,
00:01:53.230 --> 00:01:55.600
pressure, temperature,
for example.
00:01:55.600 --> 00:01:57.760
The same macro state.
00:01:57.760 --> 00:02:02.260
But for each one of them, when
I go and find the macro state,
00:02:02.260 --> 00:02:04.090
I find that it is
something else.
00:02:06.880 --> 00:02:12.210
So I will be having
different micro states.
00:02:12.210 --> 00:02:15.390
So this correspondence
is certainly
00:02:15.390 --> 00:02:17.980
something where
there should be many,
00:02:17.980 --> 00:02:25.160
many points here that correspond
to the same thermodynamic
00:02:25.160 --> 00:02:27.510
representation.
00:02:27.510 --> 00:02:31.330
So faced with that,
maybe it makes sense
00:02:31.330 --> 00:02:35.610
to follow Gibbs and
define an ensemble.
00:02:40.250 --> 00:02:42.950
So what we say is,
we are interested
00:02:42.950 --> 00:02:46.140
in some particular macro state.
00:02:46.140 --> 00:02:49.170
We know that they correspond
to many, many, many
00:02:49.170 --> 00:02:52.070
different potential
micro states.
00:02:52.070 --> 00:02:57.380
Let's try to make a map of
as many micro states that
00:02:57.380 --> 00:03:00.190
correspond to the
same macro state.
00:03:00.190 --> 00:03:13.620
So consider n copies of
the same macro state.
00:03:13.620 --> 00:03:17.370
And this would correspond
to n different points
00:03:17.370 --> 00:03:21.820
that I put in this
6N-dimensional phase space.
00:03:21.820 --> 00:03:24.940
And what I can do is I can
define an ensemble density.
00:03:33.440 --> 00:03:40.160
I go to a particular
point in this space.
00:03:40.160 --> 00:03:44.230
So let's say I pick some
point that corresponds
00:03:44.230 --> 00:03:46.510
to some set of p's and q's here.
00:03:52.370 --> 00:03:55.007
And what I do is
I draw a box that
00:03:55.007 --> 00:04:01.890
is 6N-dimensional
around this point.
00:04:01.890 --> 00:04:06.100
And I define a density
in the vicinity
00:04:06.100 --> 00:04:08.455
of that point, as follows.
00:04:12.520 --> 00:04:16.880
Actually, yeah.
00:04:16.880 --> 00:04:23.280
What I will do is I will
count how many of these points
00:04:23.280 --> 00:04:33.200
that correspond to micro
states fall within this box.
00:04:33.200 --> 00:04:45.165
So at the end is the number
of mu points in this box.
00:04:49.010 --> 00:04:56.900
And what I do is I divide
by the total number.
00:04:56.900 --> 00:05:00.220
I expect that the result
will be proportional
00:05:00.220 --> 00:05:02.220
to the volume of the box.
00:05:02.220 --> 00:05:05.650
So if I make the box
bigger, I will have more.
00:05:05.650 --> 00:05:09.950
So I divide by the
volume of the box.
00:05:09.950 --> 00:05:14.280
So this is, let's call d
gamma is the volume of box.
00:05:19.280 --> 00:05:22.190
Of course, I have
to do this in order
00:05:22.190 --> 00:05:27.430
to get a nice result by
taking the limit where
00:05:27.430 --> 00:05:32.330
the number of members of the
ensemble becomes quite large.
00:05:32.330 --> 00:05:37.830
And then presumably, this will
give me a well-behaved density.
00:05:37.830 --> 00:05:41.870
In this limit, I guess I want
to also have the size of the box
00:05:41.870 --> 00:05:44.650
go to 0.
00:05:44.650 --> 00:05:47.600
OK?
00:05:47.600 --> 00:05:50.370
Now, clearly, with
the definitions
00:05:50.370 --> 00:05:54.090
that I have made, if
I were to integrate
00:05:54.090 --> 00:06:05.290
this quantity against the
volume d gamma, what I would get
00:06:05.290 --> 00:06:09.990
is the integral dN over N.
N is, of course, a constant.
00:06:09.990 --> 00:06:12.850
And the integral of dN
is the total number.
00:06:12.850 --> 00:06:15.490
So this is 1.
00:06:15.490 --> 00:06:19.410
So we find that this quantity
rho that I have constructed
00:06:19.410 --> 00:06:21.180
satisfies two properties.
00:06:21.180 --> 00:06:23.020
Certainly, it is
positive, because I'm
00:06:23.020 --> 00:06:24.720
counting the number of points.
00:06:24.720 --> 00:06:27.030
Secondly, it's normalized to 1.
00:06:27.030 --> 00:06:29.970
So this is a nice
probability density.
00:06:29.970 --> 00:06:37.250
So this ensemble density is a
probability density function
00:06:37.250 --> 00:06:41.040
in this phase space
that I have defined.
00:06:41.040 --> 00:06:43.490
OK?
00:06:43.490 --> 00:06:44.620
All right.
00:06:44.620 --> 00:06:48.850
So once I have a
probability, then I
00:06:48.850 --> 00:06:52.230
can calculate various
things according
00:06:52.230 --> 00:06:55.260
to the rules of probability
that we defined before.
00:06:55.260 --> 00:06:59.185
So for example, I can
define an ensemble average.
00:07:03.000 --> 00:07:05.560
Maybe I'm interested
in the kinetic energy
00:07:05.560 --> 00:07:08.320
of the particles of the gas.
00:07:08.320 --> 00:07:12.090
So there is a function
O that depends
00:07:12.090 --> 00:07:15.260
on the sum of all
of the p squareds.
00:07:15.260 --> 00:07:20.570
In general, I have some function
of O that depends on p and q.
00:07:20.570 --> 00:07:25.350
And what I defined the ensemble
average would be the average
00:07:25.350 --> 00:07:28.010
that I would calculate
with this probability.
00:07:28.010 --> 00:07:31.690
Because I go over all of
the points in phase space.
00:07:31.690 --> 00:07:34.900
And let me again emphasize
that what I call d gamma
00:07:34.900 --> 00:07:41.230
then really is the
product over all points.
00:07:41.230 --> 00:07:46.290
For each point, I have to
make a volume in both momentum
00:07:46.290 --> 00:07:48.090
and in coordinate.
00:07:48.090 --> 00:07:51.650
It's a 6N-dimensional
volume element.
00:07:51.650 --> 00:07:55.380
I have to multiply
the probability, which
00:07:55.380 --> 00:07:59.980
is a function of p and
q against this O, which
00:07:59.980 --> 00:08:04.310
is another function of p and q.
00:08:04.310 --> 00:08:05.241
Yes?
00:08:05.241 --> 00:08:07.696
AUDIENCE: Is the
division by M necessary
00:08:07.696 --> 00:08:09.958
to make it into a
probability density?
00:08:09.958 --> 00:08:10.583
PROFESSOR: Yes.
00:08:10.583 --> 00:08:13.544
AUDIENCE: Otherwise, you
would still have a density.
00:08:13.544 --> 00:08:14.210
PROFESSOR: Yeah.
00:08:14.210 --> 00:08:18.210
When I would integrate then,
I would get the total number.
00:08:18.210 --> 00:08:20.380
But the total
number is up to me,
00:08:20.380 --> 00:08:22.650
how many members of
the ensemble I took,
00:08:22.650 --> 00:08:25.250
it's not a very
well-defined quantity.
00:08:25.250 --> 00:08:26.710
It's an arbitrary quantity.
00:08:26.710 --> 00:08:29.940
If I set it to become very
large and divide by it,
00:08:29.940 --> 00:08:33.100
then I will get something
that is nicely a probability.
00:08:33.100 --> 00:08:35.590
And we've developed
all of these tools
00:08:35.590 --> 00:08:37.299
for dealing with probabilities.
00:08:37.299 --> 00:08:40.982
So that would go to waste
if I don't divide by.
00:08:40.982 --> 00:08:41.944
Yes?
00:08:41.944 --> 00:08:42.906
AUDIENCE: Question.
00:08:42.906 --> 00:08:45.792
When you say that
you have set numbers,
00:08:45.792 --> 00:08:49.250
do you assume that you have
any more informations than just
00:08:49.250 --> 00:08:52.070
the microscopic
variables GP and--
00:08:52.070 --> 00:08:53.390
PROFESSOR: No.
00:08:53.390 --> 00:08:57.735
AUDIENCE: So how can we put a
micro state in correspondence
00:08:57.735 --> 00:09:00.310
with a macro state if
there is-- on the-- like,
00:09:00.310 --> 00:09:01.920
with a few variables?
00:09:01.920 --> 00:09:04.930
And do you need
to-- from-- there's
00:09:04.930 --> 00:09:07.057
like five variables,
defined down
00:09:07.057 --> 00:09:09.655
to 22 variables for
all the particles?
00:09:09.655 --> 00:09:11.280
PROFESSOR: So that's
what I was saying.
00:09:11.280 --> 00:09:13.792
It is not a one-to-one
correspondence.
00:09:13.792 --> 00:09:19.100
That is, once I specify
temperature, pressure,
00:09:19.100 --> 00:09:21.080
and the number of particles.
00:09:21.080 --> 00:09:21.580
OK?
00:09:24.075 --> 00:09:24.575
Yes?
00:09:24.575 --> 00:09:27.535
AUDIENCE: My question is, if
you generate identical macro
00:09:27.535 --> 00:09:33.004
states, and create--
which macro states--
00:09:33.004 --> 00:09:34.828
PROFESSOR: Yes.
00:09:34.828 --> 00:09:37.820
AUDIENCE: Depending on
some kind of a rule on how
00:09:37.820 --> 00:09:39.261
you make this
correspondence, you
00:09:39.261 --> 00:09:42.870
can get different
ensemble densities, right?
00:09:42.870 --> 00:09:44.270
PROFESSOR: No.
00:09:44.270 --> 00:09:48.551
That is, if I, in principle
and theoretically,
00:09:48.551 --> 00:09:54.180
go over the entirety of all
possible macroscopic boxes
00:09:54.180 --> 00:09:57.125
that have these
properties, I will
00:09:57.125 --> 00:10:00.200
be putting infinite
number of points in this.
00:10:00.200 --> 00:10:02.950
And I will get some
kind of a density.
00:10:02.950 --> 00:10:06.540
AUDIENCE: What if you, say,
generate infinite number
00:10:06.540 --> 00:10:10.560
of points, but all in
the case when, like,
00:10:10.560 --> 00:10:13.960
all molecules of gas are
in right half of the box?
00:10:13.960 --> 00:10:15.282
PROFESSOR: OK.
00:10:15.282 --> 00:10:17.602
Is that a thermodynamically
equilibrium state?
00:10:17.602 --> 00:10:19.352
AUDIENCE: Did you
mention it needed to be?
00:10:19.352 --> 00:10:20.990
PROFESSOR: Yes.
00:10:20.990 --> 00:10:23.500
I said that-- I'm
talking about things
00:10:23.500 --> 00:10:27.186
that can be described
macroscopically.
00:10:27.186 --> 00:10:28.560
Now, the thing
that you mentioned
00:10:28.560 --> 00:10:32.300
is actually something that
I would like to work with,
00:10:32.300 --> 00:10:36.450
because ultimately,
my goal is not only
00:10:36.450 --> 00:10:40.540
to describe equilibrium, but
how to reach equilibrium.
00:10:40.540 --> 00:10:45.520
That is, I would like precisely
to answer the question, what
00:10:45.520 --> 00:10:49.670
happens if you start in a
situation where all of the gas
00:10:49.670 --> 00:10:53.650
is initially in one
half of the room?
00:10:53.650 --> 00:10:55.920
And as long as there
is a partition,
00:10:55.920 --> 00:10:59.170
that's a well-defined
macroscopic state.
00:10:59.170 --> 00:11:01.930
And then I remove the partition.
00:11:01.930 --> 00:11:05.050
And suddenly, it is a
non-equilibrium state.
00:11:05.050 --> 00:11:10.620
And presumably, over time,
this gas will occupy that.
00:11:10.620 --> 00:11:16.200
So there is a physical process
that we know happens in nature.
00:11:16.200 --> 00:11:19.030
And what I would
like eventually to do
00:11:19.030 --> 00:11:22.660
is to also describe
that physical process.
00:11:22.660 --> 00:11:24.690
So what I will do
is I will start
00:11:24.690 --> 00:11:28.300
with the initial
configuration with everybody
00:11:28.300 --> 00:11:30.400
in the half space.
00:11:30.400 --> 00:11:33.360
And I will calculate
the ensemble
00:11:33.360 --> 00:11:35.400
that corresponds to that.
00:11:35.400 --> 00:11:37.270
And that's unique.
00:11:37.270 --> 00:11:39.670
Then I remove the partition.
00:11:39.670 --> 00:11:44.690
Then each member of the ensemble
will follow some trajectory
00:11:44.690 --> 00:11:48.010
as it occupies eventually
the entire box.
00:11:48.010 --> 00:11:52.020
And we would like to follow
how that evolution takes place
00:11:52.020 --> 00:11:55.720
and hopefully show that
you will always have,
00:11:55.720 --> 00:11:59.060
eventually, at the end of
the day, the gas occupying
00:11:59.060 --> 00:12:02.080
the system uniform.
00:12:02.080 --> 00:12:04.080
AUDIENCE: Yeah,
but just would it
00:12:04.080 --> 00:12:08.195
be more correct to generate
many, many different micro
00:12:08.195 --> 00:12:12.908
states as the macro states
which correspond to them?
00:12:12.908 --> 00:12:15.350
And how many different--
00:12:15.350 --> 00:12:17.702
PROFESSOR: What rule do
you use for generating
00:12:17.702 --> 00:12:21.040
many, many micro states?
00:12:21.040 --> 00:12:26.050
AUDIENCE: Like, all uniformly
arbitrary perturbations
00:12:26.050 --> 00:12:30.610
of particles always to
put them in phase space.
00:12:30.610 --> 00:12:39.020
And look to-- like, how
many different micro states
00:12:39.020 --> 00:12:40.562
give rise to the
same macro state?
00:12:40.562 --> 00:12:42.520
PROFESSOR: Oh, but you
are already then talking
00:12:42.520 --> 00:12:45.018
about the macro state?
00:12:45.018 --> 00:12:48.120
AUDIENCE: A portion--
which description
00:12:48.120 --> 00:12:51.720
do you use as the first one
to generate the second one?
00:12:51.720 --> 00:12:55.967
So in your point of view,
let's do-- [INAUDIBLE]
00:12:55.967 --> 00:12:59.320
to macro states, and
go to micro state.
00:12:59.320 --> 00:13:00.980
But can you reverse?
00:13:03.765 --> 00:13:04.650
PROFESSOR: OK.
00:13:04.650 --> 00:13:09.360
I know that the way that
I'm presenting things
00:13:09.360 --> 00:13:14.980
will lead ultimately to a useful
description of this procedure.
00:13:14.980 --> 00:13:16.720
You are welcome
to try to come up
00:13:16.720 --> 00:13:18.710
with a different prescription.
00:13:18.710 --> 00:13:22.430
But the thing that I want
to ensure that you agree
00:13:22.430 --> 00:13:26.710
is that the procedure
that I'm describing here
00:13:26.710 --> 00:13:30.120
has no logical inconsistencies.
00:13:30.120 --> 00:13:32.450
I want to convince you of that.
00:13:32.450 --> 00:13:35.420
I am not saying that this
is necessarily the only one.
00:13:35.420 --> 00:13:37.350
As far as I know,
this is the only one
00:13:37.350 --> 00:13:38.640
that people have worked with.
00:13:38.640 --> 00:13:43.770
But maybe somebody can come up
with a different prescription.
00:13:43.770 --> 00:13:45.710
So maybe there is another one.
00:13:45.710 --> 00:13:47.430
Maybe you can work on it.
00:13:47.430 --> 00:13:49.050
But I want you
to, at this point,
00:13:49.050 --> 00:13:51.440
be convinced that this is
a well-defined procedure.
00:13:56.180 --> 00:13:58.550
OK?
00:13:58.550 --> 00:14:01.646
AUDIENCE: But because it's
a well-defined procedure,
00:14:01.646 --> 00:14:03.986
if you did exist
on another planet
00:14:03.986 --> 00:14:07.024
or in some universe where
the physics were different,
00:14:07.024 --> 00:14:08.619
the point is, you can use this.
00:14:08.619 --> 00:14:10.660
But can't you use this
for information in general
00:14:10.660 --> 00:14:14.220
when you want to-- like, if
you have-- the only requirement
00:14:14.220 --> 00:14:18.370
is that at a fine
scale, you have
00:14:18.370 --> 00:14:21.640
a consistent way of describing
things; and at a large scale,
00:14:21.640 --> 00:14:25.092
you have a way of making
sense of generalizing.
00:14:25.092 --> 00:14:25.800
PROFESSOR: Right.
00:14:25.800 --> 00:14:28.070
AUDIENCE: So it's sort
of like a compression
00:14:28.070 --> 00:14:31.420
of data, or I use [INAUDIBLE].
00:14:31.420 --> 00:14:33.200
PROFESSOR: Yeah.
00:14:33.200 --> 00:14:38.040
Except that part of this was
starting with some physics
00:14:38.040 --> 00:14:39.550
that we know.
00:14:39.550 --> 00:14:42.980
So indeed, if you were
in a different universe--
00:14:42.980 --> 00:14:46.013
and later on in the course, we
will be in a different universe
00:14:46.013 --> 00:14:47.512
where the rules are
not classical or
00:14:47.512 --> 00:14:48.303
quantum-mechanical.
00:14:50.289 --> 00:14:52.080
And you have to throw
away this description
00:14:52.080 --> 00:14:53.850
of what a micro state is.
00:14:53.850 --> 00:14:57.150
And you can still go through
the entire procedure.
00:14:57.150 --> 00:15:00.250
But I want to do is to
follow these set of equations
00:15:00.250 --> 00:15:02.610
of motion and this
description of micro state,
00:15:02.610 --> 00:15:04.600
and see where it leads us.
00:15:04.600 --> 00:15:08.220
And for the gas in this room, it
is a perfectly good description
00:15:08.220 --> 00:15:10.170
of what's happened.
00:15:10.170 --> 00:15:10.670
Yes?
00:15:10.670 --> 00:15:12.542
AUDIENCE: Maybe a
simpler question.
00:15:12.542 --> 00:15:15.023
Is big rho defined
only in spaces
00:15:15.023 --> 00:15:17.327
where there are micro states?
00:15:17.327 --> 00:15:19.660
Like, is there anywhere where
there isn't a micro state?
00:15:19.660 --> 00:15:20.890
PROFESSOR: Yes, of course.
00:15:20.890 --> 00:15:23.420
So if I have--
thinking about a box,
00:15:23.420 --> 00:15:28.100
and if I ask what
is rho out here,
00:15:28.100 --> 00:15:30.180
I would say the
answer is rho is 0.
00:15:30.180 --> 00:15:33.270
But if you like, you can
say rho is defined only
00:15:33.270 --> 00:15:35.170
within this space of the box.
00:15:35.170 --> 00:15:38.090
So the description
of the macro state
00:15:38.090 --> 00:15:39.980
which has something
to do with the box,
00:15:39.980 --> 00:15:42.010
over which I am
considering, will also
00:15:42.010 --> 00:15:45.280
limit what I can describe.
00:15:45.280 --> 00:15:47.290
Yes.
00:15:47.290 --> 00:15:54.320
And certainly, as far as if I
were to change p with velocity,
00:15:54.320 --> 00:15:56.800
let's say, then you
would say, a space
00:15:56.800 --> 00:16:00.580
where V is greater than speed
of light is not possible.
00:16:00.580 --> 00:16:02.150
That's the point.
00:16:02.150 --> 00:16:06.580
So your rules of physics
will also define implicitly
00:16:06.580 --> 00:16:10.270
the domain over
which this is there.
00:16:10.270 --> 00:16:12.590
But that's all
part of mechanics.
00:16:12.590 --> 00:16:14.980
So I'm going to assume that
the mechanics part of it
00:16:14.980 --> 00:16:16.110
is you are comfortable.
00:16:16.110 --> 00:16:17.452
Yes?
00:16:17.452 --> 00:16:20.060
AUDIENCE: In your definition
of the ensemble average,
00:16:20.060 --> 00:16:23.790
are you integrating over all
6N dimensions of phase space?
00:16:23.790 --> 00:16:24.530
PROFESSOR: Yes.
00:16:24.530 --> 00:16:29.530
AUDIENCE: So why would your
average depend on p and q?
00:16:32.480 --> 00:16:33.300
If you integrate?
00:16:33.300 --> 00:16:36.600
PROFESSOR: The average is
of a function of p and q.
00:16:36.600 --> 00:16:38.650
So in the same sense
that, let's say,
00:16:38.650 --> 00:16:41.520
I have a particle of
gas that is moving on,
00:16:41.520 --> 00:16:45.070
and I can write the
symbol, p squared over 2m,
00:16:45.070 --> 00:16:47.100
what is this average?
00:16:47.100 --> 00:16:49.890
The answer will be KT
over 2, for example.
00:16:49.890 --> 00:16:51.380
It will not depend on p.
00:16:51.380 --> 00:16:56.530
But the quantity that I'm
averaging inside the triangle
00:16:56.530 --> 00:16:58.580
is a function of p and q.
00:17:01.144 --> 00:17:01.644
Yes?
00:17:01.644 --> 00:17:03.685
AUDIENCE: So if it's an
integration, or basically
00:17:03.685 --> 00:17:04.579
the--?
00:17:04.579 --> 00:17:06.970
PROFESSOR: The physical
limit of the problem.
00:17:06.970 --> 00:17:10.281
AUDIENCE: Given a macro state?
00:17:10.281 --> 00:17:13.130
PROFESSOR: Yes.
00:17:13.130 --> 00:17:16.430
So typically, we
will be integrating q
00:17:16.430 --> 00:17:20.619
over the volume of a box,
and p from minus infinity
00:17:20.619 --> 00:17:22.740
to infinity,
because classically,
00:17:22.740 --> 00:17:26.814
without relativity,
this is a lot.
00:17:26.814 --> 00:17:27.314
Yes?
00:17:27.314 --> 00:17:28.103
AUDIENCE: Sorry.
00:17:28.103 --> 00:17:36.007
So why is the [INAUDIBLE] from
one end for every particle,
00:17:36.007 --> 00:17:37.983
instead of just
scattering space,
00:17:37.983 --> 00:17:40.453
would you have a [INAUDIBLE]?
00:17:40.453 --> 00:17:42.923
Or is that the same thing?
00:17:42.923 --> 00:17:45.630
PROFESSOR: I am not sure
I understand the question.
00:17:45.630 --> 00:17:50.560
So if I want to, let's say,
find out just one particle that
00:17:50.560 --> 00:17:55.600
is somewhere in this box, so
there is a probability that it
00:17:55.600 --> 00:17:58.180
is here, there is a
probability that it is there,
00:17:58.180 --> 00:18:00.930
there is a probability
that it is there.
00:18:00.930 --> 00:18:06.070
The integral of that probability
over the volume of the room
00:18:06.070 --> 00:18:06.850
is one.
00:18:06.850 --> 00:18:08.150
So how do I do that?
00:18:08.150 --> 00:18:12.640
I have to do an integral
over dx, dy, dz, probability
00:18:12.640 --> 00:18:14.690
as a function of x, y, and z.
00:18:14.690 --> 00:18:16.590
Now I just repeat that 6N times.
00:18:20.890 --> 00:18:21.390
OK?
00:18:25.390 --> 00:18:26.020
All right.
00:18:28.990 --> 00:18:31.440
So that's the description.
00:18:31.440 --> 00:18:35.660
But the first question
to sort of consider
00:18:35.660 --> 00:18:44.940
is, what is equilibrium
in this perspective?
00:18:55.730 --> 00:19:00.100
Now, we can even be
generous, although it's
00:19:00.100 --> 00:19:04.370
a very questionable thing,
to say that, really,
00:19:04.370 --> 00:19:08.320
when I sort of talk about the
kinetic energy of the gas,
00:19:08.320 --> 00:19:14.230
maybe I can replace that
by this ensemble average.
00:19:14.230 --> 00:19:17.240
Now, if I'm in
equilibrium, the results
00:19:17.240 --> 00:19:20.640
should not depend as
a function of time.
00:19:20.640 --> 00:19:24.020
So I expect that if
I'm calculating things
00:19:24.020 --> 00:19:28.530
in equilibrium, the result
of equations such as this
00:19:28.530 --> 00:19:33.200
should not depend on time,
which is actually a problem.
00:19:33.200 --> 00:19:39.740
Because we know that if I take
a picture of all of these things
00:19:39.740 --> 00:19:42.440
that I am constructing
my ensemble with
00:19:42.440 --> 00:19:48.670
and this picture is at
time t, at time t plus dt,
00:19:48.670 --> 00:19:51.420
all of the particles
have moved around.
00:19:51.420 --> 00:19:56.150
And so the point that was
here, the next instant of time
00:19:56.150 --> 00:19:58.020
is going to be somewhere else.
00:19:58.020 --> 00:19:59.670
This is going to
be somewhere else.
00:19:59.670 --> 00:20:04.392
Each one of these is flowing
around as a function of time.
00:20:04.392 --> 00:20:05.770
OK?
00:20:05.770 --> 00:20:08.175
So the picture that I
would like you to imagine
00:20:08.175 --> 00:20:12.260
is you have a box, and there's
a huge number of bees or flies
00:20:12.260 --> 00:20:14.615
or whatever your
preferred insect
00:20:14.615 --> 00:20:17.010
is, are just moving around.
00:20:17.010 --> 00:20:18.860
OK?
00:20:18.860 --> 00:20:22.610
Now, you can sort of then
take pictures of this cluster.
00:20:22.610 --> 00:20:25.535
And it's changing potentially
as a function of time.
00:20:25.535 --> 00:20:28.490
And therefore, this
density should potentially
00:20:28.490 --> 00:20:30.090
change as a function of time.
00:20:33.510 --> 00:20:38.590
And then this answer could
potentially depend on time.
00:20:38.590 --> 00:20:42.540
So let's figure out how
is this density changing
00:20:42.540 --> 00:20:44.762
as a function of time.
00:20:44.762 --> 00:20:49.100
And hope that ultimately,
we can construct a solution
00:20:49.100 --> 00:20:52.970
for the equation that
governs the change in density
00:20:52.970 --> 00:20:58.700
as a function of time that
is in fact invariant in time.
00:20:58.700 --> 00:21:02.040
It is going back to my
flies or bees or whatever,
00:21:02.040 --> 00:21:04.330
you can imagine a
circumstance in which
00:21:04.330 --> 00:21:07.710
the bees are constantly
moving around.
00:21:07.710 --> 00:21:12.020
Each individual bee is now
here, then somewhere else.
00:21:12.020 --> 00:21:15.810
But all of your pictures have
the same density of bees,
00:21:15.810 --> 00:21:18.160
because for every bee
that left the box,
00:21:18.160 --> 00:21:21.220
there was another bee
that came in its place.
00:21:21.220 --> 00:21:24.290
So one can imagine
a kind of situation
00:21:24.290 --> 00:21:26.300
where all of these
points are moving around,
00:21:26.300 --> 00:21:29.730
yet the density
is left invariant.
00:21:29.730 --> 00:21:33.030
And in order to find whether
such a density is possible,
00:21:33.030 --> 00:21:36.010
we have to first know
what is the equation that
00:21:36.010 --> 00:21:38.494
governs the evolution
of that density.
00:21:38.494 --> 00:21:40.410
And that is given by the
Liouville's equation.
00:21:50.180 --> 00:22:01.420
So this governs evolution
of rho with time.
00:22:01.420 --> 00:22:01.920
OK.
00:22:07.430 --> 00:22:12.060
So let's kind of blow off the
picture that we had over there.
00:22:12.060 --> 00:22:15.310
Previously, they're all
of these coordinates.
00:22:15.310 --> 00:22:20.260
There is some point
in coordinate space
00:22:20.260 --> 00:22:22.140
that I am looking at.
00:22:22.140 --> 00:22:26.560
Let's say that the point
that I am looking at is here.
00:22:26.560 --> 00:22:30.480
And I have constructed
a box around it
00:22:30.480 --> 00:22:35.510
like this in the
6N-dimensional space.
00:22:35.510 --> 00:22:39.540
But just to be precise, I will
be looking at some particular
00:22:39.540 --> 00:22:45.280
coordinate q alpha and the
conjugate momentum p alpha.
00:22:45.280 --> 00:22:48.820
So this is my original
point corresponds to, say,
00:22:48.820 --> 00:22:53.620
some specific version
of q alpha p alpha.
00:22:53.620 --> 00:22:59.260
And that I have, in
this pair of dimensions,
00:22:59.260 --> 00:23:02.570
created a box that
in this direction
00:23:02.570 --> 00:23:09.110
has size dq alpha, in this
direction has size dp alpha.
00:23:09.110 --> 00:23:11.420
OK?
00:23:11.420 --> 00:23:17.602
And this is the picture
that I have at some time t.
00:23:17.602 --> 00:23:18.640
OK?
00:23:18.640 --> 00:23:24.030
Then I look at an instant of
time that is slightly later.
00:23:24.030 --> 00:23:29.360
So I go to a time
that is t plus dt.
00:23:29.360 --> 00:23:33.430
I find that the point that
I had initially over here
00:23:33.430 --> 00:23:37.350
as the center of this
box has moved around
00:23:37.350 --> 00:23:39.890
to some other
location that I will
00:23:39.890 --> 00:23:42.610
call q alpha prime,
p alpha prime.
00:23:45.560 --> 00:23:48.700
If you ask me what is q alpha
prime and p alpha prime,
00:23:48.700 --> 00:23:50.330
I say, OK, I know
that, because I
00:23:50.330 --> 00:23:51.990
know with equations of motion.
00:23:51.990 --> 00:23:54.220
If I was running
this on a computer,
00:23:54.220 --> 00:24:03.090
I would say that q alpha prime
is q alpha plus the velocity
00:24:03.090 --> 00:24:09.730
q alpha dot dt, plus
order of dt squared,
00:24:09.730 --> 00:24:12.480
which hopefully, I will choose
a sufficient and small time
00:24:12.480 --> 00:24:14.876
interval I can ignore.
00:24:14.876 --> 00:24:19.580
And similarly, p
alpha prime would
00:24:19.580 --> 00:24:26.460
be p alpha plus p alpha
dot dt order of dt squared.
00:24:29.446 --> 00:24:29.946
OK?
00:24:35.440 --> 00:24:39.885
So any point that was in
this box will also move.
00:24:39.885 --> 00:24:42.015
And presumably,
close-by points will
00:24:42.015 --> 00:24:45.140
be moving to close-by points.
00:24:45.140 --> 00:24:47.200
And overall, anything
that was originally
00:24:47.200 --> 00:24:52.210
in this square actually
projected from a larger
00:24:52.210 --> 00:24:57.830
dimensional cube will be part
of a slightly distorted entity
00:24:57.830 --> 00:24:58.330
here.
00:24:58.330 --> 00:25:03.421
So everything that was
here is now somewhere here.
00:25:03.421 --> 00:25:06.410
OK?
00:25:06.410 --> 00:25:14.850
I can ask, well, how wide is
this new distance that I have?
00:25:14.850 --> 00:25:19.120
So originally, the two
endpoints of the square
00:25:19.120 --> 00:25:22.020
were a distance dq alpha apart.
00:25:22.020 --> 00:25:26.430
Now they're going to be a
distance dq alpha prime apart.
00:25:26.430 --> 00:25:28.740
What is dq alpha prime?
00:25:28.740 --> 00:25:40.220
I claim that dq alpha prime
is whatever I had originally,
00:25:40.220 --> 00:25:43.930
but then the two N's
are moving at slightly
00:25:43.930 --> 00:25:48.080
different velocities,
because the velocity depends
00:25:48.080 --> 00:25:50.700
on where you are in phase space.
00:25:50.700 --> 00:25:53.590
And so the difference in
velocity between these two
00:25:53.590 --> 00:26:01.900
points is really the derivative
of the velocity with respect
00:26:01.900 --> 00:26:06.880
to the separation that I have
between those two points, which
00:26:06.880 --> 00:26:10.715
is dq times dq alpha.
00:26:13.950 --> 00:26:16.490
And this is how
much I would have
00:26:16.490 --> 00:26:18.910
expanded plus higher order.
00:26:21.694 --> 00:26:25.010
And I can apply the same thing
in the momentum direction.
00:26:28.330 --> 00:26:40.860
The new vertical
separation dp alpha prime
00:26:40.860 --> 00:26:44.230
is different from what
it was originally,
00:26:44.230 --> 00:26:47.710
because the two
endpoints got stretched.
00:26:47.710 --> 00:26:49.330
The reason they
got stretched was
00:26:49.330 --> 00:26:53.210
because their velocities
were different.
00:26:53.210 --> 00:26:55.270
And their difference
is just the derivative
00:26:55.270 --> 00:26:57.640
of velocity with
respect to separation
00:26:57.640 --> 00:26:59.800
times their small separation.
00:26:59.800 --> 00:27:02.650
And if I make
everything small, I
00:27:02.650 --> 00:27:05.092
can, in principle, write
higher order terms.
00:27:05.092 --> 00:27:06.633
But I don't have to
worry about that.
00:27:13.378 --> 00:27:13.878
OK?
00:27:16.790 --> 00:27:24.940
So I can ask, what is the area
of this slightly distorted
00:27:24.940 --> 00:27:26.440
square?
00:27:26.440 --> 00:27:30.680
As long as the dt is
sufficiently small,
00:27:30.680 --> 00:27:34.010
all of the distortions, et
cetera, will be small enough.
00:27:34.010 --> 00:27:37.930
And you can convince
yourself of this.
00:27:37.930 --> 00:27:41.530
And what you will find
is that the product
00:27:41.530 --> 00:27:46.250
of dq alpha prime,
dp alpha prime,
00:27:46.250 --> 00:27:48.580
if I were to multiply
these things,
00:27:48.580 --> 00:27:51.340
you can see that dq
alpha and dp alpha
00:27:51.340 --> 00:27:53.125
is common to the two of them.
00:27:53.125 --> 00:27:57.200
So I have dq alpha, dp alpha.
00:27:57.200 --> 00:28:01.620
From multiplying these
two terms, I will get one.
00:28:01.620 --> 00:28:06.610
And then I will get two
terms that are order of dt,
00:28:06.610 --> 00:28:15.682
that I will get from dq alpha
dot with respect to dq alpha,
00:28:15.682 --> 00:28:19.750
plus dp alpha dot with
respect to dp alpha.
00:28:19.750 --> 00:28:23.900
And then there will be terms
that are order of dt squared
00:28:23.900 --> 00:28:25.562
and higher order
types of things.
00:28:28.640 --> 00:28:29.140
OK?
00:28:32.090 --> 00:28:38.680
So the distortion in the
area of this particle,
00:28:38.680 --> 00:28:44.410
or cross section, is
governed by something
00:28:44.410 --> 00:28:46.680
that is proportional to dt.
00:28:46.680 --> 00:28:53.170
And dq alpha dot over dq alpha
plus dp alpha dot dp alpha.
00:28:53.170 --> 00:29:02.060
But I have formally here for
what P i dot and qi dot are.
00:29:02.060 --> 00:29:06.460
So this is the dot notation
for the time derivative
00:29:06.460 --> 00:29:08.580
other than the one that
I was using before.
00:29:08.580 --> 00:29:11.435
So q alpha dot, what
do I have for it?
00:29:11.435 --> 00:29:14.490
It is dh by dp alpha.
00:29:14.490 --> 00:29:20.205
So this is d by dq alpha
of the H by dp alpha.
00:29:22.880 --> 00:29:28.620
Whereas p alpha dot, from which
I have to evaluate d by dp
00:29:28.620 --> 00:29:35.780
alpha, p alpha dot is
minus dH by dq alpha.
00:29:35.780 --> 00:29:37.620
So what do I have?
00:29:37.620 --> 00:29:43.360
I have two second derivatives
that appear with opposite sign
00:29:43.360 --> 00:29:45.285
and hence cancel each other out.
00:29:49.085 --> 00:29:50.520
OK?
00:29:50.520 --> 00:29:57.190
So essentially, what we find
is that the volume element
00:29:57.190 --> 00:30:01.590
is preserved under this process.
00:30:01.590 --> 00:30:05.970
And I can apply this to
all of my directions.
00:30:11.240 --> 00:30:18.050
And hence, conclude that
the initial volume that
00:30:18.050 --> 00:30:24.039
was surrounding my point
is going to be preserved.
00:30:24.039 --> 00:30:24.538
OK?
00:30:27.820 --> 00:30:34.200
So what that means is that these
classical equations of motion,
00:30:34.200 --> 00:30:40.030
the Hamiltonian equations, for
this description of micro state
00:30:40.030 --> 00:30:43.410
that involves the
coordinates and momenta
00:30:43.410 --> 00:30:47.330
have this nice property
that they preserve volume
00:30:47.330 --> 00:30:50.071
of phase space as
they move around.
00:30:50.071 --> 00:30:50.570
Yes?
00:30:50.570 --> 00:30:53.741
AUDIENCE: If the Hamiltonian has
expontential time dependence,
00:30:53.741 --> 00:30:55.100
that doesn't work anymore.
00:30:55.100 --> 00:30:55.710
PROFESSOR: No.
00:30:55.710 --> 00:30:58.459
So that's why I did not
put that over there.
00:30:58.459 --> 00:30:58.959
Yes.
00:31:06.070 --> 00:31:10.840
And actually, this is
sometimes referred to
00:31:10.840 --> 00:31:14.980
as being something like
an incompressible fluid.
00:31:24.770 --> 00:31:31.680
Because if you kind of deliver
hydrodynamics for something
00:31:31.680 --> 00:31:34.910
like water if you regard
it as incompressible,
00:31:34.910 --> 00:31:37.810
the velocity field
has the condition
00:31:37.810 --> 00:31:42.050
that the divergence
of the velocity is 0.
00:31:42.050 --> 00:31:46.440
Here, we are looking at a
6N-dimensional mention velocity
00:31:46.440 --> 00:31:51.790
field that is composed of q
alpha dot and p alpha dot.
00:31:51.790 --> 00:31:55.170
And this being 0 is
really the same thing
00:31:55.170 --> 00:32:00.190
as the divergence in this
6N-dimensional space is 0.
00:32:00.190 --> 00:32:04.230
And that's a property of
the Hamiltonian dynamics
00:32:04.230 --> 00:32:04.960
that one has.
00:32:07.540 --> 00:32:08.040
Yes?
00:32:08.040 --> 00:32:09.960
AUDIENCE: Could
you briefly go over
00:32:09.960 --> 00:32:15.240
why you have to divide
by the separation
00:32:15.240 --> 00:32:20.126
when you expand the times
between the displacement?
00:32:20.126 --> 00:32:23.620
PROFESSOR: Why do I have to
multiply by the separation?
00:32:23.620 --> 00:32:24.991
AUDIENCE: Divide by.
00:32:24.991 --> 00:32:28.280
PROFESSOR: Where do I divide?
00:32:28.280 --> 00:32:30.420
AUDIENCE: dq alpha by--
00:32:30.420 --> 00:32:32.180
PROFESSOR: Oh, this.
00:32:32.180 --> 00:32:35.804
Why do I have to
take a derivative.
00:32:35.804 --> 00:32:39.740
So I have two points here.
00:32:39.740 --> 00:32:43.060
All of my points
are moving in time.
00:32:43.060 --> 00:32:47.200
So if these things were
moving with uniform velocity,
00:32:47.200 --> 00:32:50.360
one second later, this
would have moved here,
00:32:50.360 --> 00:32:52.620
this would have moved
the same distance,
00:32:52.620 --> 00:32:54.470
so that the separation
between them
00:32:54.470 --> 00:32:56.640
would have been
maintained if they
00:32:56.640 --> 00:32:59.070
were moving with
the same velocity.
00:32:59.070 --> 00:33:00.935
So if you are following
somebody and you
00:33:00.935 --> 00:33:03.650
are moving with the
same velocity as them,
00:33:03.650 --> 00:33:06.210
thus, your separation
does not change.
00:33:06.210 --> 00:33:11.500
But if one of you is going
faster than the other one,
00:33:11.500 --> 00:33:14.750
then the difference
in velocity will
00:33:14.750 --> 00:33:17.400
determine how you separate.
00:33:17.400 --> 00:33:20.390
And what is the
difference in velocity?
00:33:20.390 --> 00:33:23.200
The difference in velocity
depends, in this case,
00:33:23.200 --> 00:33:26.210
to how far apart the points are.
00:33:26.210 --> 00:33:29.300
So the difference between
velocity here and velocity
00:33:29.300 --> 00:33:32.530
here is the
derivative of velocity
00:33:32.530 --> 00:33:34.900
as a function of
this coordinate.
00:33:34.900 --> 00:33:38.560
Derivative of velocity as a
function of that coordinate
00:33:38.560 --> 00:33:40.145
multiplied by the separation.
00:33:50.520 --> 00:33:52.530
OK?
00:33:52.530 --> 00:33:54.960
So what does this
incompressibility condition
00:33:54.960 --> 00:33:56.250
mean?
00:33:56.250 --> 00:34:01.310
It means that however many
points I had over here,
00:34:01.310 --> 00:34:06.990
they end up in a box that has
exactly the same volume, which
00:34:06.990 --> 00:34:13.040
means that the density is going
to be the same around here
00:34:13.040 --> 00:34:16.050
and around this new point.
00:34:16.050 --> 00:34:25.420
So essentially, what we have is
that the rho at the new point,
00:34:25.420 --> 00:34:32.630
p prime, q prime, and
time, t, plus dt, is
00:34:32.630 --> 00:34:41.670
the same thing as the rho at
the old point, p, q, at time, t.
00:34:41.670 --> 00:34:46.310
Again, this is the
incompressibility condition.
00:34:46.310 --> 00:34:47.234
Now we do mathematics.
00:34:51.690 --> 00:34:53.510
So let's write it again.
00:34:53.510 --> 00:34:59.380
So I've said, in other
words, that rho p, q, t
00:34:59.380 --> 00:35:02.510
is the same as the
rho at the new point.
00:35:02.510 --> 00:35:05.270
What's the momentum
at the new point?
00:35:05.270 --> 00:35:10.180
It is p plus.
00:35:10.180 --> 00:35:16.330
For each component, it is
p alpha plus p alpha dot.
00:35:16.330 --> 00:35:19.810
Let's put it this
way. p plus p dot
00:35:19.810 --> 00:35:30.118
dt, q plus q dot
dt, and t plus dt.
00:35:30.118 --> 00:35:35.440
That is, if I look at
the new location compared
00:35:35.440 --> 00:35:40.040
to the old location, the time
changed, the position changed,
00:35:40.040 --> 00:35:42.870
the momentum changed.
00:35:42.870 --> 00:35:45.810
They all changed--
in each arguments
00:35:45.810 --> 00:35:47.880
changed infinitesimally
by an amount that
00:35:47.880 --> 00:35:50.660
is proportional to dt.
00:35:50.660 --> 00:35:54.320
And so what I can do is I
can expand this function
00:35:54.320 --> 00:35:56.320
to order of dt.
00:35:56.320 --> 00:36:01.390
So I have rho at
the original point.
00:36:01.390 --> 00:36:03.590
So this is all mathematics.
00:36:03.590 --> 00:36:06.310
I just look at
variation with respect
00:36:06.310 --> 00:36:07.930
to each one of these arguments.
00:36:07.930 --> 00:36:12.470
So I have a sum
over alpha, p alpha
00:36:12.470 --> 00:36:20.910
dot d rho by dp
alpha plus q alpha
00:36:20.910 --> 00:36:28.720
dot d rho by dq alpha plus
the explicit derivative,
00:36:28.720 --> 00:36:30.400
d rho by dt.
00:36:30.400 --> 00:36:35.050
This entirety is going
to be multiplied by dt.
00:36:35.050 --> 00:36:37.760
And then, in principle,
the expansion
00:36:37.760 --> 00:36:39.574
would have higher order terms.
00:36:43.350 --> 00:36:45.260
OK?
00:36:45.260 --> 00:36:48.242
Now, of course, the
first term vanishes.
00:36:48.242 --> 00:36:51.050
It is the same on both times.
00:36:51.050 --> 00:36:53.780
So the thing that I
will have to set to 0
00:36:53.780 --> 00:36:55.510
is this entity over here.
00:36:58.270 --> 00:37:05.785
Now, quite generally, if
you have a function of p,
00:37:05.785 --> 00:37:11.710
q, and t, you evaluate
it at the old point
00:37:11.710 --> 00:37:14.860
and then evaluate
at the new point.
00:37:14.860 --> 00:37:21.110
One can define what is
called a total derivative.
00:37:21.110 --> 00:37:24.640
And just like here,
the total derivative
00:37:24.640 --> 00:37:28.000
will come from variations
of all of these arguments.
00:37:28.000 --> 00:37:30.470
We'll have a partial
derivative with respect
00:37:30.470 --> 00:37:33.970
to time and partial
derivatives with respect
00:37:33.970 --> 00:37:35.395
to any of the other arguments.
00:37:46.940 --> 00:37:50.320
So I wrote this to sort
of make a distinction
00:37:50.320 --> 00:37:53.950
between the symbol
that is commonly used,
00:37:53.950 --> 00:37:58.360
sometimes d by dt, which is
straight, sometimes Df by Dt.
00:38:02.620 --> 00:38:09.120
And this is either
a total derivative
00:38:09.120 --> 00:38:13.470
or a streamline derivative.
00:38:13.470 --> 00:38:16.580
That is, you are
taking derivatives
00:38:16.580 --> 00:38:20.560
as you are moving
along with the flow.
00:38:20.560 --> 00:38:25.860
And that is to be distinguished
from these partial derivatives,
00:38:25.860 --> 00:38:29.650
which is really sitting
at some point in space
00:38:29.650 --> 00:38:33.360
and following how, from one
time instant to another time
00:38:33.360 --> 00:38:36.930
instant, let's say
the density changes.
00:38:36.930 --> 00:38:40.190
So Df by Dt with
a partial really
00:38:40.190 --> 00:38:43.240
means sit at the same point.
00:38:43.240 --> 00:38:48.060
Whereas this big Df by Dt
means, go along with the flow
00:38:48.060 --> 00:38:50.880
and look at the changes.
00:38:50.880 --> 00:38:56.955
Now, what we have established
here is that for the density,
00:38:56.955 --> 00:38:59.990
the density has some
special character because
00:38:59.990 --> 00:39:03.125
of this Liouville's theorem,
that the streamlined derivative
00:39:03.125 --> 00:39:04.500
is 0.
00:39:04.500 --> 00:39:09.330
So what we have is
that d rho by dt is 0.
00:39:11.990 --> 00:39:16.530
And this d rho by dt
I can also write down
00:39:16.530 --> 00:39:25.445
as d rho by dt plus sum
over all 6N directions, d
00:39:25.445 --> 00:39:28.550
rho by dp alpha.
00:39:28.550 --> 00:39:34.870
But then I substitute for
p alpha dot from here.
00:39:34.870 --> 00:39:36.880
p dot is minus dH by dq.
00:39:44.010 --> 00:39:49.460
And then I add d
rho by dq alpha.
00:39:49.460 --> 00:39:52.740
q alpha dot is dH by dp r.
00:39:56.556 --> 00:39:57.056
OK?
00:40:01.120 --> 00:40:07.110
So then I can take this
combination with the minus sign
00:40:07.110 --> 00:40:14.520
to the other side and
write it as d rho by dt
00:40:14.520 --> 00:40:20.780
is something that I will
call the Poisson bracket of H
00:40:20.780 --> 00:40:27.680
and rho, where,
quite generally, if I
00:40:27.680 --> 00:40:31.480
have two functions
in phase space
00:40:31.480 --> 00:40:36.530
that is defending on B and
q, this scalary derivative
00:40:36.530 --> 00:40:42.700
the Poisson bracket is
defined as the sum over all 6N
00:40:42.700 --> 00:40:44.710
possible variation.
00:40:44.710 --> 00:40:49.740
The first one with respect to
q, the second one with respect
00:40:49.740 --> 00:40:51.100
to p.
00:40:51.100 --> 00:40:55.190
And then the whole thing
with the opposite sign.
00:40:55.190 --> 00:40:58.950
dA, dP, dB, dq.
00:41:03.270 --> 00:41:04.895
So this is the Poisson bracket.
00:41:10.450 --> 00:41:12.110
And again, from
the definition, you
00:41:12.110 --> 00:41:14.540
should be able to
see immediately
00:41:14.540 --> 00:41:16.870
that Poisson bracket
of A and B is
00:41:16.870 --> 00:41:24.279
minus the Poisson
bracket of B and A. OK?
00:42:01.440 --> 00:42:06.290
Again, we ask the question
that in general, I
00:42:06.290 --> 00:42:14.410
can construct in principle
a rho of p, q, let's say,
00:42:14.410 --> 00:42:16.280
for an equilibrium ensemble.
00:42:16.280 --> 00:42:20.780
But then I did something,
like I removed a partition
00:42:20.780 --> 00:42:23.940
in the middle of the gas,
and the gas is expanding.
00:42:23.940 --> 00:42:28.980
And then presumably, this
becomes a function of time.
00:42:28.980 --> 00:42:33.810
And since I know exactly how
each one of the particles,
00:42:33.810 --> 00:42:35.680
and hence each one
of the micro states
00:42:35.680 --> 00:42:37.740
is evolving as a
function of time,
00:42:37.740 --> 00:42:41.200
I should be able to tell how
this density in phase space
00:42:41.200 --> 00:42:42.430
is changing.
00:42:42.430 --> 00:42:46.970
So this perspective is,
again, this perspective
00:42:46.970 --> 00:42:50.100
of looking at all of these
bees that are buzzing around
00:42:50.100 --> 00:42:54.190
in this 6N-dimensional space,
and asking the question, if I
00:42:54.190 --> 00:42:58.350
look at the particular point in
this 6N-dimensional space, what
00:42:58.350 --> 00:43:00.600
is the density of bees?
00:43:00.600 --> 00:43:03.890
And the answer is
that it is given
00:43:03.890 --> 00:43:08.820
by the Poisson bracket
of the Hamiltonian
00:43:08.820 --> 00:43:11.709
that governs the evolution
of each micro state
00:43:11.709 --> 00:43:12.750
and the density function.
00:43:20.940 --> 00:43:21.470
All right.
00:43:21.470 --> 00:43:23.880
So what does it mean?
00:43:23.880 --> 00:43:25.320
What can we do with this?
00:43:29.520 --> 00:43:32.170
Let's play around with it and
look at some consequences.
00:43:37.870 --> 00:43:43.402
But before that, does
anybody have any questions?
00:43:43.402 --> 00:43:44.840
OK.
00:43:44.840 --> 00:43:45.340
All right.
00:43:49.350 --> 00:43:51.300
We had something
that I just erased.
00:43:51.300 --> 00:43:55.730
That is, if I have a
function of p and q,
00:43:55.730 --> 00:43:58.210
let's say it's not
a function of time.
00:43:58.210 --> 00:44:01.360
Let's say it's the kinetic
energy for this system where,
00:44:01.360 --> 00:44:04.290
at t equals to 0, I
remove the partition,
00:44:04.290 --> 00:44:06.280
and the particles are expanding.
00:44:06.280 --> 00:44:08.462
And let's say the other
place you have a potential,
00:44:08.462 --> 00:44:11.600
so your kinetic energy on
average is going to change.
00:44:11.600 --> 00:44:14.380
You want to know what's
happening to that.
00:44:14.380 --> 00:44:17.980
So you calculate at
each instant of time
00:44:17.980 --> 00:44:21.380
an ensemble average
of the kinetic energy
00:44:21.380 --> 00:44:24.340
or any other quantity that
is interesting to you.
00:44:24.340 --> 00:44:29.460
And your prescription for
calculating an ensemble average
00:44:29.460 --> 00:44:36.440
is that you integrate against
the density the function
00:44:36.440 --> 00:44:42.260
that you are
proposing to look at.
00:44:42.260 --> 00:44:45.580
Now, in principle,
we said that there
00:44:45.580 --> 00:44:48.620
could be situations where
this is dependent on time,
00:44:48.620 --> 00:44:53.040
in which case, your average
will also depend on time.
00:44:53.040 --> 00:44:57.130
And maybe you want to know how
this time dependence occurs.
00:44:57.130 --> 00:44:59.720
How does the kinetic
energy of a gas that
00:44:59.720 --> 00:45:04.100
is expanding into some
potential change on average?
00:45:04.100 --> 00:45:04.600
OK.
00:45:04.600 --> 00:45:05.810
So let's take a look.
00:45:05.810 --> 00:45:07.990
This is a function of
time, because, as we said,
00:45:07.990 --> 00:45:09.220
these go inside the average.
00:45:09.220 --> 00:45:12.710
So really, the only explicit
variable that we have here
00:45:12.710 --> 00:45:13.680
is time.
00:45:13.680 --> 00:45:15.430
And you can ask,
what is the time
00:45:15.430 --> 00:45:16.765
dependence of this quantity?
00:45:22.000 --> 00:45:24.480
OK?
00:45:24.480 --> 00:45:35.380
So the time dependence is
obtained by doing this,
00:45:35.380 --> 00:45:39.950
because essentially, you
would be adding things
00:45:39.950 --> 00:45:43.210
at different points in p, q.
00:45:43.210 --> 00:45:45.830
And at each point, there
is a time dependence.
00:45:45.830 --> 00:45:49.290
And you take the derivative
in time with respect
00:45:49.290 --> 00:45:51.860
to the contribution
of that point.
00:45:51.860 --> 00:45:54.940
So we get something like this.
00:45:54.940 --> 00:45:58.420
Now you say, OK, I know
what d rho by dt is.
00:45:58.420 --> 00:46:05.080
So this is my integration
over all of the phase space.
00:46:05.080 --> 00:46:09.400
d rho by dt is this Poisson
bracket of H and rho.
00:46:09.400 --> 00:46:14.400
And then I have O. OK?
00:46:14.400 --> 00:46:16.930
Let's write this explicitly.
00:46:16.930 --> 00:46:19.340
This is an integral
over a whole bunch
00:46:19.340 --> 00:46:20.530
of coordinates and momenta.
00:46:26.340 --> 00:46:30.725
There is, for the
Poisson bracket,
00:46:30.725 --> 00:46:32.690
a sum over derivatives.
00:46:32.690 --> 00:46:39.390
So I have a sum over
alpha-- dH by dq alpha,
00:46:39.390 --> 00:46:48.030
d rho by dp alpha minus dH by
dp alpha, d rho by dq alpha.
00:46:48.030 --> 00:46:50.970
And that Poisson
bracket in its entirety
00:46:50.970 --> 00:46:57.210
then gets multiplied by this
function of phase space.
00:46:57.210 --> 00:47:00.080
OK.
00:47:00.080 --> 00:47:04.475
Now, one of the
mathematical manipulations
00:47:04.475 --> 00:47:07.760
that we will do a
lot in this class.
00:47:07.760 --> 00:47:11.160
And that's why I do this
particular step, although it's
00:47:11.160 --> 00:47:13.795
not really necessary to
the logical progression
00:47:13.795 --> 00:47:17.470
that I'm following,
is to remind you
00:47:17.470 --> 00:47:21.260
how you would do an integration
by parts when you're
00:47:21.260 --> 00:47:23.290
faced with something like this.
00:47:23.290 --> 00:47:26.700
An integration by
parts is applicable
00:47:26.700 --> 00:47:31.660
when you have variables that you
are integrating that are also
00:47:31.660 --> 00:47:35.340
appearing as derivatives.
00:47:35.340 --> 00:47:39.740
And whenever you are
integrating Poisson brackets,
00:47:39.740 --> 00:47:43.130
you will have derivatives
for the Poisson bracket.
00:47:43.130 --> 00:47:45.360
And the integration
would allow you
00:47:45.360 --> 00:47:48.530
to use integration by parts.
00:47:48.530 --> 00:47:52.130
And in particular,
what I would like to do
00:47:52.130 --> 00:47:55.510
is to remove the derivative
that acts on the densities.
00:47:58.060 --> 00:48:05.730
So I'm going to
essentially rewrite that as
00:48:05.730 --> 00:48:11.090
minus an integral
that involves-- again.
00:48:11.090 --> 00:48:14.190
I don't want to keep
rewriting that thing.
00:48:14.190 --> 00:48:20.340
I want to basically take
the density out and then
00:48:20.340 --> 00:48:24.070
have the derivative, which is
this d by dp in the first term
00:48:24.070 --> 00:48:28.790
and d by dq in the second
term, act on everything else.
00:48:28.790 --> 00:48:36.890
So in the first case, d
by dp alpha will act on O
00:48:36.890 --> 00:48:38.540
and dH by dq alpha.
00:48:41.150 --> 00:48:48.260
And in the second case, d
by dq alpha will act on O
00:48:48.260 --> 00:48:49.691
and dH by dp alpha.
00:48:53.459 --> 00:48:57.000
Again, there is a sum over
alpha that is implicit.
00:48:59.916 --> 00:49:01.380
OK?
00:49:01.380 --> 00:49:03.080
Again, there is a minus sign.
00:49:03.080 --> 00:49:07.230
So every time you do this
procedure, there is this.
00:49:07.230 --> 00:49:11.000
But every time, you also have
to worry about surface terms.
00:49:11.000 --> 00:49:14.050
So on the surface,
you would potentially
00:49:14.050 --> 00:49:19.850
have to evaluate things that
involve rho, O, and these d
00:49:19.850 --> 00:49:21.305
by d derivatives.
00:49:25.020 --> 00:49:28.290
But let's say we are
integrating over momentum
00:49:28.290 --> 00:49:30.650
from minus infinity to infinity.
00:49:30.650 --> 00:49:35.240
Then the density evaluated at
infinity momenta would be 0.
00:49:35.240 --> 00:49:38.760
So practicality, in all
cases that I can think of,
00:49:38.760 --> 00:49:43.280
you don't have to worry
about the boundary terms.
00:49:43.280 --> 00:49:47.200
So then when you look
at these kinds of terms,
00:49:47.200 --> 00:49:51.380
this d by dp alpha
can either act on O.
00:49:51.380 --> 00:49:58.000
So I will get dO by dp
alpha, dH by dq alpha.
00:49:58.000 --> 00:50:03.070
Or it can act on dH by d alpha.
00:50:03.070 --> 00:50:08.530
So I will get plus O d2
H, dp alpha dq alpha.
00:50:08.530 --> 00:50:11.950
And similarly, in
this term, either I
00:50:11.950 --> 00:50:20.720
will have dO by dq alpha, dH by
dp alpha, or O d2 H, dq alpha
00:50:20.720 --> 00:50:23.740
dp alpha.
00:50:23.740 --> 00:50:25.770
Once more, the second
derivative terms
00:50:25.770 --> 00:50:31.620
of the Hamiltonian, the
order is not important.
00:50:31.620 --> 00:50:34.600
And what is left
here is this set
00:50:34.600 --> 00:50:39.590
of objects, which is none
other than the Poisson bracket.
00:50:39.590 --> 00:50:44.440
So I can rewrite the whole thing
as d by dt of the expectation
00:50:44.440 --> 00:50:47.630
value of this quantity,
which potentially
00:50:47.630 --> 00:50:49.860
is a function of time
because of the time
00:50:49.860 --> 00:50:53.410
dependence of my density
is the same thing
00:50:53.410 --> 00:51:00.090
as minus an integration over
the entire phase space of rho
00:51:00.090 --> 00:51:03.170
against this entity,
which is none
00:51:03.170 --> 00:51:09.200
other than the Poisson
bracket of H with O.
00:51:09.200 --> 00:51:13.830
And this integration over
density of this entire space
00:51:13.830 --> 00:51:17.710
is just our definition
of the expectation value.
00:51:17.710 --> 00:51:27.650
So we get that the time
derivative of any quantity
00:51:27.650 --> 00:51:32.370
is related to the average
of its Poisson bracket
00:51:32.370 --> 00:51:36.110
with the Hamiltonian, which
is the quantity that is really
00:51:36.110 --> 00:51:37.437
governing time dependences.
00:51:40.913 --> 00:51:41.413
Yes?
00:51:41.413 --> 00:51:43.350
AUDIENCE: Could
you explain again
00:51:43.350 --> 00:51:46.420
why the time derivative
when N is the integral,
00:51:46.420 --> 00:51:48.688
it's rho as a
partial derivative?
00:51:48.688 --> 00:51:50.140
PROFESSOR: OK.
00:51:50.140 --> 00:51:53.190
So suppose I'm doing a
two-dimensional integral
00:51:53.190 --> 00:51:56.060
over p and q.
00:51:56.060 --> 00:52:01.010
So I have some contribution
from each point in this p and q.
00:52:01.010 --> 00:52:05.220
And so my integral
is an integral dpdq,
00:52:05.220 --> 00:52:08.770
something evaluated
at each point in p,
00:52:08.770 --> 00:52:12.420
q that could potentially
depend on time.
00:52:12.420 --> 00:52:14.600
Imagine that I discretize this.
00:52:14.600 --> 00:52:17.230
So I really-- if you
are more comfortable,
00:52:17.230 --> 00:52:20.310
you can think of this
as a limit of a sum.
00:52:20.310 --> 00:52:23.170
So this is my integral.
00:52:23.170 --> 00:52:26.800
If I'm interested in the time
dependence of this quantity--
00:52:26.800 --> 00:52:29.130
and I really depends
only on time,
00:52:29.130 --> 00:52:32.290
because I integrated
over p and q.
00:52:32.290 --> 00:52:34.970
So if I'm interested
in something
00:52:34.970 --> 00:52:38.990
that is a sum of
various terms, each term
00:52:38.990 --> 00:52:41.760
is a function of time.
00:52:41.760 --> 00:52:43.880
Where do I put the
time dependence?
00:52:43.880 --> 00:52:49.990
For each term in this sum, I
look at how it depends on time.
00:52:49.990 --> 00:52:53.805
I don't care on its points
to the left and to the right.
00:52:59.030 --> 00:52:59.990
OK?
00:52:59.990 --> 00:53:04.420
Because the big D by Dt involves
moving with the streamline.
00:53:04.420 --> 00:53:07.050
I'm not doing any moving
with the streamline.
00:53:07.050 --> 00:53:12.350
I'm looking at each point in
this two-dimensional space.
00:53:12.350 --> 00:53:15.730
Each point gives a
contribution at that point
00:53:15.730 --> 00:53:17.380
that is time-dependent.
00:53:17.380 --> 00:53:20.580
And I take the derivative with
respect to time at that point.
00:53:23.200 --> 00:53:23.720
Yes?
00:53:23.720 --> 00:53:27.672
AUDIENCE: Couldn't you say
that you have function O
00:53:27.672 --> 00:53:30.636
as just some
function of p and q,
00:53:30.636 --> 00:53:34.457
and its time derivative
would be Poisson bracket?
00:53:34.457 --> 00:53:35.082
PROFESSOR: Yes.
00:53:35.082 --> 00:53:37.291
AUDIENCE: And does the
average of the time derivative
00:53:37.291 --> 00:53:38.957
would be the average
of Poisson bracket,
00:53:38.957 --> 00:53:40.826
and you don't have to
go through all the--
00:53:40.826 --> 00:53:41.570
PROFESSOR: No.
00:53:41.570 --> 00:53:43.280
But you can see the
sign doesn't work.
00:53:45.942 --> 00:53:47.304
AUDIENCE: How come?
00:53:47.304 --> 00:53:48.720
PROFESSOR: [LAUGHS]
Because of all
00:53:48.720 --> 00:53:50.390
of these manipulations,
et cetera.
00:53:50.390 --> 00:53:54.030
So the statement that you
made is manifestly incorrect.
00:53:54.030 --> 00:53:58.190
You can't say that the time
dependence of this thing
00:53:58.190 --> 00:54:03.366
is the-- whatever you
were saying. [LAUGHS]
00:54:03.866 --> 00:54:06.794
AUDIENCE: [INAUDIBLE].
00:54:06.794 --> 00:54:07.510
PROFESSOR: OK.
00:54:07.510 --> 00:54:08.760
Let's see what you are saying.
00:54:08.760 --> 00:54:11.090
AUDIENCE: So
Poisson bracket only
00:54:11.090 --> 00:54:14.402
counts for in place
for averages, right?
00:54:14.402 --> 00:54:15.360
PROFESSOR: OK.
00:54:15.360 --> 00:54:17.560
So what do we have here?
00:54:17.560 --> 00:54:25.630
We have that dp by dt is the
Poisson bracket of H and rho.
00:54:25.630 --> 00:54:26.630
OK?
00:54:26.630 --> 00:54:33.730
And we have that O is an
integral of rho O. Now,
00:54:33.730 --> 00:54:37.460
from where do you conclude
from this set of results
00:54:37.460 --> 00:54:44.900
that d O by dt is the
average of a Poisson bracket
00:54:44.900 --> 00:54:47.390
that involves O
and H, irrespective
00:54:47.390 --> 00:54:48.670
of what we do with the sign?
00:54:48.670 --> 00:54:53.230
AUDIENCE: Or if you look not
at the average failure of O
00:54:53.230 --> 00:54:57.745
but at the value of O
and point, and take-- I
00:54:57.745 --> 00:55:01.050
guess it would be
streamline derivative of it.
00:55:01.050 --> 00:55:07.214
So that's assuming that you're
just like assigning value of O
00:55:07.214 --> 00:55:10.570
to each point, and making
power changes with time
00:55:10.570 --> 00:55:13.076
as this point moves
across the phase space.
00:55:13.076 --> 00:55:13.810
PROFESSOR: OK.
00:55:13.810 --> 00:55:20.230
But you still have to do some
bit of derivatives, et cetera,
00:55:20.230 --> 00:55:21.652
because--
00:55:21.652 --> 00:55:24.520
AUDIENCE: But if you know
that the volume of the
00:55:24.520 --> 00:55:28.756
in phase space is
conserved, then we basically
00:55:28.756 --> 00:55:30.980
don't care much
that the function
00:55:30.980 --> 00:55:34.340
O is any much different
from probability density.
00:55:34.340 --> 00:55:35.080
PROFESSOR: OK.
00:55:35.080 --> 00:55:38.070
If I understand correctly,
this is what you are saying.
00:55:38.070 --> 00:55:41.150
Is that for each
representative point,
00:55:41.150 --> 00:55:46.190
I have an O alpha, which
is a function of time.
00:55:46.190 --> 00:55:51.880
And then you want to say
that the average of O
00:55:51.880 --> 00:55:57.370
is the same thing as the sum
over alpha of O alpha of t's
00:55:57.370 --> 00:55:59.690
divided by N,
something like this.
00:55:59.690 --> 00:56:02.150
AUDIENCE: Eh.
00:56:02.150 --> 00:56:08.020
Uh, I want to first calculate
what does time derivative of O?
00:56:08.020 --> 00:56:12.406
O remains in a function
of time and q and p.
00:56:12.406 --> 00:56:13.390
So I can calculate--
00:56:13.390 --> 00:56:14.380
PROFESSOR: Yes.
00:56:14.380 --> 00:56:20.834
So this O alpha is a function of
00:56:20.834 --> 00:56:22.176
AUDIENCE: So if I said--
00:56:22.176 --> 00:56:22.800
PROFESSOR: Yes.
00:56:22.800 --> 00:56:23.140
OK.
00:56:23.140 --> 00:56:23.640
Fine.
00:56:23.640 --> 00:56:27.292
AUDIENCE: O is a function
of q and p and t,
00:56:27.292 --> 00:56:29.890
and I take a streamline
derivative of it.
00:56:29.890 --> 00:56:31.616
So filter it with respect to t.
00:56:31.616 --> 00:56:37.500
And then I average that
thing over phase space.
00:56:37.500 --> 00:56:39.994
And then I should get
the same version--
00:56:39.994 --> 00:56:40.910
PROFESSOR: You should.
00:56:40.910 --> 00:56:42.185
AUDIENCE: --perfectly.
00:56:42.185 --> 00:56:43.035
But--
00:56:43.035 --> 00:56:46.890
PROFESSOR: Very quickly,
I don't think so.
00:56:46.890 --> 00:56:49.150
Because you are already
explaining things a bit
00:56:49.150 --> 00:56:51.455
longer than I think I went
through my derivation.
00:56:51.455 --> 00:56:54.096
But that's fine. [LAUGHS]
00:56:55.004 --> 00:56:57.290
AUDIENCE: Is there any special--
00:56:57.290 --> 00:56:59.600
PROFESSOR: But I
agree in spirit, yes.
00:56:59.600 --> 00:57:02.920
That each one of these will
go along its streamline,
00:57:02.920 --> 00:57:06.280
and you can calculate the
change for each one of them.
00:57:06.280 --> 00:57:08.810
And then you have to do an
average of this variety.
00:57:08.810 --> 00:57:09.705
Yes.
00:57:09.705 --> 00:57:15.254
AUDIENCE: [INAUDIBLE] when
you talk about time derivative
00:57:15.254 --> 00:57:19.300
of probability density, it's
Poisson bracket of H and rho.
00:57:19.300 --> 00:57:23.156
But when you talk about
time derivative of average,
00:57:23.156 --> 00:57:27.330
you have to add the minus sign.
00:57:27.330 --> 00:57:29.840
PROFESSOR: And if you
do this correctly here,
00:57:29.840 --> 00:57:31.134
you should get the same result.
00:57:33.858 --> 00:57:34.766
AUDIENCE: Oh, OK.
00:57:34.766 --> 00:57:35.945
PROFESSOR: Yes.
00:57:35.945 --> 00:57:39.005
AUDIENCE: Well, along
that line, though,
00:57:39.005 --> 00:57:43.455
are you using the fact
that phase space volume is
00:57:43.455 --> 00:57:47.074
incompressible to then
argue that the total time
00:57:47.074 --> 00:57:49.900
derivative of the
ensemble average
00:57:49.900 --> 00:57:54.045
is the same as the ensemble
average of the total time
00:57:54.045 --> 00:57:56.520
derivative of O, or not?
00:58:08.326 --> 00:58:09.700
PROFESSOR: Could
you repeat that?
00:58:09.700 --> 00:58:11.810
Mathematically,
you want me to show
00:58:11.810 --> 00:58:16.875
that the time derivative
of what quantity?
00:58:16.875 --> 00:58:19.515
AUDIENCE: Of the average of O.
00:58:19.515 --> 00:58:21.970
PROFESSOR: Of the
average of O. Yes.
00:58:21.970 --> 00:58:24.370
AUDIENCE: Is it in any
way related to the average
00:58:24.370 --> 00:58:27.065
of dO over dt?
00:58:27.065 --> 00:58:29.520
PROFESSOR: No, it's not.
00:58:29.520 --> 00:58:34.660
Because O-- I mean, so
what do you mean by that?
00:58:34.660 --> 00:58:37.850
You have to be careful, because
the way that I'm defining this
00:58:37.850 --> 00:58:41.640
here, O is a
function of p and q.
00:58:41.640 --> 00:58:46.250
And what you want to do
is to write something
00:58:46.250 --> 00:58:56.990
that is a sum over all
representative points, divided
00:58:56.990 --> 00:59:00.780
by the total number, some
kind of an average like this.
00:59:00.780 --> 00:59:04.114
And then I can define
a time derivative here.
00:59:04.114 --> 00:59:05.030
Is that what you are--
00:59:05.030 --> 00:59:06.710
AUDIENCE: Well, I
mean, I was thinking
00:59:06.710 --> 00:59:10.230
that even if you start out with
your observable being defined
00:59:10.230 --> 00:59:14.080
for every point in phase
space, then if you were to,
00:59:14.080 --> 00:59:16.150
before doing any
ensemble averaging,
00:59:16.150 --> 00:59:19.970
if you were to take the total
time derivative of that,
00:59:19.970 --> 00:59:25.030
then you would be accounted for
p dot and q dot as well, right?
00:59:25.030 --> 00:59:26.960
And then if you were
to take the ensemble
00:59:26.960 --> 00:59:29.710
average of that
quantity, could you
00:59:29.710 --> 00:59:31.420
arrive at the same
result for that?
00:59:31.420 --> 00:59:34.570
PROFESSOR: I'm pretty sure that
if you do things consistently,
00:59:34.570 --> 00:59:35.320
yes.
00:59:35.320 --> 00:59:37.230
That is, what we have
done is essentially
00:59:37.230 --> 00:59:41.830
we started with a collection
of trajectories in phase space
00:59:41.830 --> 00:59:45.235
and recast the result in
terms of density and variables
00:59:45.235 --> 00:59:49.950
that are defined only as
positions in phase space.
00:59:49.950 --> 00:59:53.200
The two descriptions
completely are equivalent.
00:59:53.200 --> 00:59:55.810
And as long as one
doesn't make a mistake,
00:59:55.810 --> 00:59:58.022
one can get one or the other.
00:59:58.022 --> 01:00:01.635
This is actually a kind
of a well-known thing
01:00:01.635 --> 01:00:05.490
in hydrodynamics, because
typically, you write down,
01:00:05.490 --> 01:00:09.070
in hydrodynamics, equations
for density and velocity
01:00:09.070 --> 01:00:11.850
at each point in phase space.
01:00:11.850 --> 01:00:14.220
But there is an
alternative description
01:00:14.220 --> 01:00:18.400
which we can say that there's
essentially particles that are
01:00:18.400 --> 01:00:19.700
flowing.
01:00:19.700 --> 01:00:22.800
And particle that was
here at this location
01:00:22.800 --> 01:00:25.860
is now somewhere else
at some later time.
01:00:25.860 --> 01:00:28.820
And people have tried hard.
01:00:28.820 --> 01:00:30.570
And there is a
consistent definition
01:00:30.570 --> 01:00:34.620
of hydrodynamics that follows
the second perspective.
01:00:34.620 --> 01:00:37.700
But I haven't seen it
as being practical.
01:00:37.700 --> 01:00:41.440
So I'm sure that everything
that you guys say is correct.
01:00:41.440 --> 01:00:45.250
But from the experience of
what I know in hydrodynamics,
01:00:45.250 --> 01:00:47.450
I think this is the more
practical description
01:00:47.450 --> 01:00:49.090
that people have been using.
01:00:53.170 --> 01:00:56.060
OK?
01:00:56.060 --> 01:00:58.810
So where were we?
01:00:58.810 --> 01:00:59.310
OK.
01:01:01.920 --> 01:01:05.810
So back to our buzzing bees.
01:01:09.960 --> 01:01:13.240
We now have a way
of looking at how
01:01:13.240 --> 01:01:17.070
densities and various quantities
that you can calculate,
01:01:17.070 --> 01:01:22.670
like ensemble averages, are
changing as a function of time.
01:01:22.670 --> 01:01:24.145
But the question
that I had before
01:01:24.145 --> 01:01:25.270
is, what about equilibrium?
01:01:31.700 --> 01:01:35.170
Because the thermodynamic
definition of equilibrium,
01:01:35.170 --> 01:01:38.550
and my whole ensemble idea,
was that essentially, I
01:01:38.550 --> 01:01:41.920
have all of these
boxes and pressure,
01:01:41.920 --> 01:01:44.120
volume, everything
that I can think of,
01:01:44.120 --> 01:01:46.510
as long as I'm not
doing something that's
01:01:46.510 --> 01:01:52.230
like opening the box, is
perfectly independent of time.
01:01:52.230 --> 01:01:56.260
So how can I ensure that
various things that I calculate
01:01:56.260 --> 01:01:58.170
are independent of time?
01:01:58.170 --> 01:02:03.640
Clearly, I can do that by having
this density not really depend
01:02:03.640 --> 01:02:05.950
on time.
01:02:05.950 --> 01:02:07.830
OK?
01:02:07.830 --> 01:02:10.180
Now, of course, each
representative point
01:02:10.180 --> 01:02:11.370
is moving around.
01:02:11.370 --> 01:02:13.210
Each micro state
is moving around.
01:02:13.210 --> 01:02:14.710
Each bee is moving around.
01:02:14.710 --> 01:02:21.100
But I want the density that is
characteristic of equilibrium
01:02:21.100 --> 01:02:23.340
be something that does
not change in time.
01:02:26.030 --> 01:02:26.560
It's 0.
01:02:29.220 --> 01:02:34.830
And so if I posit that this
particular function of p and q
01:02:34.830 --> 01:02:38.150
is something that is not
changing as a function of time,
01:02:38.150 --> 01:02:40.900
I have to require that
the Poisson bracket
01:02:40.900 --> 01:02:44.310
of that function of p and
q with the Hamiltonian,
01:02:44.310 --> 01:02:48.435
which is another function
of p and q, is 0.
01:02:48.435 --> 01:02:49.330
OK?
01:02:49.330 --> 01:02:52.290
So in principle,
I have to go back
01:02:52.290 --> 01:02:57.000
to this equation over here,
which is a partial differential
01:02:57.000 --> 01:03:03.190
equation in 6N-dimensional
space and solve with equal to 0.
01:03:03.190 --> 01:03:06.640
Of course, rather than doing
that, we will guess the answer.
01:03:06.640 --> 01:03:12.390
And the guess is clear from
here, because all I need to do
01:03:12.390 --> 01:03:18.500
is to make rho equilibrium
depend on coordinates in phase
01:03:18.500 --> 01:03:24.210
space through some functional
dependence on Hamiltonian,
01:03:24.210 --> 01:03:28.354
which depends on the
coordinates in phase space.
01:03:28.354 --> 01:03:29.820
OK?
01:03:29.820 --> 01:03:31.730
Why does this work?
01:03:31.730 --> 01:03:36.580
Because then when I take the
Poisson bracket of, let's
01:03:36.580 --> 01:03:45.640
say, this function of H with
H, what do I have to do?
01:03:45.640 --> 01:03:50.590
I have to do a sum over
alpha d rho with respect
01:03:50.590 --> 01:03:55.510
to, let's say-- actually,
let's write it in this way.
01:03:55.510 --> 01:04:02.850
I will have the dH by
dp alpha from here.
01:04:02.850 --> 01:04:06.770
I have to multiply
by d rho by dq alpha.
01:04:06.770 --> 01:04:10.510
But rho is a function
of only H. So I
01:04:10.510 --> 01:04:14.500
have to take a derivative of rho
with respect to its argument H.
01:04:14.500 --> 01:04:16.270
And I'll call that rho prime.
01:04:16.270 --> 01:04:18.610
And then the derivative
of the argument,
01:04:18.610 --> 01:04:22.090
which is H, with
respect to q alpha.
01:04:22.090 --> 01:04:24.580
That would be the first term.
01:04:24.580 --> 01:04:29.755
The next term would be, with
the minus sign, from the H here,
01:04:29.755 --> 01:04:37.520
dH by dq alpha, the derivative
of rho with respect to p alpha.
01:04:37.520 --> 01:04:42.460
But rho only depends on H, so I
will get the derivative of rho
01:04:42.460 --> 01:04:44.650
with respect to
its one argument.
01:04:44.650 --> 01:04:49.610
The derivative of that argument
with respect to p alpha.
01:04:49.610 --> 01:04:51.410
OK?
01:04:51.410 --> 01:04:54.470
So you can see that up to
the order of the terms that
01:04:54.470 --> 01:04:59.020
are multiplying here, this is 0.
01:04:59.020 --> 01:05:02.020
OK?
01:05:02.020 --> 01:05:10.650
So any function I choose of H,
in principle, satisfies this.
01:05:10.650 --> 01:05:14.710
And this is what we
will use consistently
01:05:14.710 --> 01:05:17.680
all the time in
statistical mechanics,
01:05:17.680 --> 01:05:20.560
in depending on the
ensemble that we have.
01:05:20.560 --> 01:05:22.275
Like you probably
know that when we
01:05:22.275 --> 01:05:25.000
are in the micro
canonical ensemble,
01:05:25.000 --> 01:05:28.110
we look at-- in the
micro canonical ensemble,
01:05:28.110 --> 01:05:31.630
we'd say what the energy is.
01:05:31.630 --> 01:05:35.620
And then we say
that the density is
01:05:35.620 --> 01:05:41.100
a delta function--
essentially zero.
01:05:41.100 --> 01:05:44.820
Except places, it's
the surface that
01:05:44.820 --> 01:05:47.410
corresponds to having
the right energy.
01:05:47.410 --> 01:05:50.160
So you sort of
construct in phase space
01:05:50.160 --> 01:05:53.100
the surface that has
the right energy.
01:05:53.100 --> 01:05:56.889
So it's a function
of these four.
01:05:56.889 --> 01:05:58.180
So this is the micro canonical.
01:06:00.790 --> 01:06:02.750
When you are in the
canonical, I use
01:06:02.750 --> 01:06:05.822
a rho this is proportional
to e to the minus beta H
01:06:05.822 --> 01:06:07.610
and other functional features.
01:06:07.610 --> 01:06:10.175
So it's, again, the same idea.
01:06:13.140 --> 01:06:13.640
OK?
01:06:21.570 --> 01:06:25.910
That's almost but
not entirely true,
01:06:25.910 --> 01:06:29.894
because sometimes there are
also other conserved quantities.
01:06:39.640 --> 01:06:43.620
Let's say that, for example, we
have a collection of particles
01:06:43.620 --> 01:06:48.100
in space in a cavity that
has the shape of a sphere.
01:06:48.100 --> 01:06:50.740
Because of the symmetry
of the problem,
01:06:50.740 --> 01:06:56.300
angular momentum is going
to be a conserved quantity.
01:06:56.300 --> 01:06:58.570
Angular momentum
you can also write
01:06:58.570 --> 01:07:02.030
as some complicated
function of p and q.
01:07:02.030 --> 01:07:05.340
For example, p cross q summed
over all of the particles.
01:07:05.340 --> 01:07:10.130
But it could be some
other conserved quantity.
01:07:10.130 --> 01:07:13.680
So what does this
conservation law mean?
01:07:13.680 --> 01:07:20.410
It means that if you
evaluate L for some time,
01:07:20.410 --> 01:07:23.920
it is going to be the
same L for the coordinates
01:07:23.920 --> 01:07:26.010
and momenta at some other time.
01:07:26.010 --> 01:07:29.700
Or in other words,
dL by dt, which
01:07:29.700 --> 01:07:34.470
you obtained by summing
over all coordinates dL
01:07:34.470 --> 01:07:44.370
by dp alpha, p alpha dot, plus
dL by dq alpha q alpha dot.
01:07:44.370 --> 01:07:46.440
Essentially, taking
time derivatives
01:07:46.440 --> 01:07:48.150
of all of the arguments.
01:07:48.150 --> 01:07:51.780
And I did not put any
explicit time dependence here.
01:07:51.780 --> 01:07:57.440
And this is again sum
over alpha dL by dp alpha.
01:07:57.440 --> 01:08:04.670
p alpha dot is minus
dH by dq alpha.
01:08:04.670 --> 01:08:14.570
And dL by dq alpha, q alpha
dot is dH by dp alpha.
01:08:14.570 --> 01:08:17.834
So you are seeing that
this is the same thing
01:08:17.834 --> 01:08:24.540
as the Poisson
bracket of L and H.
01:08:24.540 --> 01:08:29.830
So essentially, conserved
quantities which
01:08:29.830 --> 01:08:33.990
are essentially functions
of coordinates and momenta
01:08:33.990 --> 01:08:37.680
that you calculate that don't
change as a function of time
01:08:37.680 --> 01:08:40.420
are also quantities
that have zero Poisson
01:08:40.420 --> 01:08:43.700
brackets with the Hamiltonian.
01:08:43.700 --> 01:08:47.569
So if I have a
conserved quantity,
01:08:47.569 --> 01:08:49.414
then I have a more
general solution.
01:08:55.890 --> 01:09:02.140
To my d rho by dt
equals to 0 requirement.
01:09:02.140 --> 01:09:05.410
I could make a rho
equilibrium which
01:09:05.410 --> 01:09:10.563
is a function of H of p and q,
as well as, say, L of p and q.
01:09:14.649 --> 01:09:20.979
And when you go through the
Poisson bracket process,
01:09:20.979 --> 01:09:34.120
you will either be
taking derivatives
01:09:34.120 --> 01:09:37.870
with respect to the
first argument here.
01:09:37.870 --> 01:09:43.319
So you would get rho prime with
respect to the first argument.
01:09:43.319 --> 01:09:47.859
And then you would get the
Poisson bracket of H and H.
01:09:47.859 --> 01:09:50.810
Or you would be getting
derivatives with respect
01:09:50.810 --> 01:09:53.819
to the second argument.
01:09:53.819 --> 01:09:59.280
And then you would be getting
the Poisson bracket of L and H.
01:09:59.280 --> 01:10:01.540
And both of them are 0.
01:10:01.540 --> 01:10:04.850
By definition, L is
a conserved quantity.
01:10:04.850 --> 01:10:10.440
So any solution that's a
function of Hamiltonian,
01:10:10.440 --> 01:10:12.340
the energy of the
system is conserved
01:10:12.340 --> 01:10:14.570
as a function of
time, as well as
01:10:14.570 --> 01:10:18.190
any other conserved quantities,
such as angular momentum,
01:10:18.190 --> 01:10:22.200
et cetera, is certainly
a valid thing.
01:10:22.200 --> 01:10:28.680
So indeed, when I drew here, in
the micro canonical ensemble,
01:10:28.680 --> 01:10:33.020
a surface that corresponds
to a constant energy,
01:10:33.020 --> 01:10:36.450
well, if I am in a
spherical cavity,
01:10:36.450 --> 01:10:39.270
only part of that
surface that corresponds
01:10:39.270 --> 01:10:43.490
to the right angular momentum
is going to be accessible.
01:10:43.490 --> 01:10:48.870
So essentially, what I know
is that if I have conserved
01:10:48.870 --> 01:10:55.050
quantities, my trajectories
will explore the subspace
01:10:55.050 --> 01:10:59.070
that is consistent with
those conservation laws.
01:10:59.070 --> 01:11:05.720
And this statement here is that
ultimately, those spaces that
01:11:05.720 --> 01:11:09.720
correspond to the
appropriate conservation law
01:11:09.720 --> 01:11:11.750
are equally populated.
01:11:11.750 --> 01:11:15.400
Rho is constant around.
01:11:15.400 --> 01:11:20.960
So in some sense, we started
with the definition of rho
01:11:20.960 --> 01:11:24.460
by putting these points
around and calculating
01:11:24.460 --> 01:11:28.410
probability that way, which
was my objective definition
01:11:28.410 --> 01:11:30.100
of probability.
01:11:30.100 --> 01:11:32.320
And through this
Liouville theorem,
01:11:32.320 --> 01:11:33.920
we have arrived
at something that
01:11:33.920 --> 01:11:38.770
is more consistent with
the subjective assignment
01:11:38.770 --> 01:11:40.040
of probability.
01:11:40.040 --> 01:11:42.790
That is, the only
thing that I know,
01:11:42.790 --> 01:11:44.360
forgetting about
the dynamics, is
01:11:44.360 --> 01:11:46.240
that there are some
conserved quantities,
01:11:46.240 --> 01:11:49.060
such as H, angular
momentum, et cetera.
01:11:49.060 --> 01:11:51.990
And I say that any point in
phase space that does not
01:11:51.990 --> 01:11:56.600
violate those constants in this,
say, micro canonical ensemble
01:11:56.600 --> 01:11:58.520
would be equally populated.
01:11:58.520 --> 01:12:00.860
There was a question somewhere.
01:12:00.860 --> 01:12:02.314
Yes?
01:12:02.314 --> 01:12:11.324
AUDIENCE: So I almost feel like
the statement that the rho has
01:12:11.324 --> 01:12:15.967
to not change in time is too
strong, because if you go over
01:12:15.967 --> 01:12:20.458
to the equation that
says the rate of change
01:12:20.458 --> 01:12:24.574
is observable is equal
to the integral that
01:12:24.574 --> 01:12:28.795
was with a Poisson
bracket of rho and H, then
01:12:28.795 --> 01:12:30.450
it means that for
any observable,
01:12:30.450 --> 01:12:33.138
it's constant in time, right?
01:12:33.138 --> 01:12:34.040
PROFESSOR: Yes.
01:12:34.040 --> 01:12:38.330
AUDIENCE: So rho of q means
any observable we can think of,
01:12:38.330 --> 01:12:40.232
its function of p
and q is constant?
01:12:40.232 --> 01:12:42.640
PROFESSOR: Yep.
01:12:42.640 --> 01:12:43.370
Yep.
01:12:43.370 --> 01:12:49.720
Because-- and that's
the best thing that we
01:12:49.720 --> 01:12:53.730
can think of in terms
of-- because if there
01:12:53.730 --> 01:12:56.660
is some observable that
is time-dependent--
01:12:56.660 --> 01:12:59.655
let's say 99 observables
are time-independent,
01:12:59.655 --> 01:13:03.830
but one is time-dependent,
and you can measure that,
01:13:03.830 --> 01:13:06.140
would you say your
system is in equilibrium?
01:13:06.140 --> 01:13:07.104
Probably not.
01:13:11.856 --> 01:13:12.356
OK?
01:13:12.356 --> 01:13:14.022
AUDIENCE: It seemed
like the same method
01:13:14.022 --> 01:13:17.155
that you did to show that
the density is a function of
01:13:17.155 --> 01:13:20.850
[INAUDIBLE], or
[INAUDIBLE] that it's
01:13:20.850 --> 01:13:27.860
the function of any observable
that's a function of q.
01:13:27.860 --> 01:13:28.360
Right?
01:13:28.360 --> 01:13:29.500
PROFESSOR: Observables?
01:13:29.500 --> 01:13:30.530
No.
01:13:30.530 --> 01:13:36.000
I mean, here, if
this answer is 0,
01:13:36.000 --> 01:13:39.580
it states something about
this quantity averaged.
01:13:39.580 --> 01:13:43.720
So if this quantity does not
change as a function of time,
01:13:43.720 --> 01:13:48.410
it is not a statement
that H and 0 is 0.
01:13:48.410 --> 01:13:52.700
A statement that H and O is 0
is different from its ensemble
01:13:52.700 --> 01:13:55.330
average being 0.
01:13:55.330 --> 01:13:59.450
What you can show-- and I think
I have a problem set for that--
01:13:59.450 --> 01:14:04.280
is that if this statement
is correct for every O
01:14:04.280 --> 01:14:08.870
that the average
is 0, then your rho
01:14:08.870 --> 01:14:11.940
has to satisfy
this theorem equals
01:14:11.940 --> 01:14:15.992
to-- Poisson bracket of
rho and H is equal to 0.
01:14:23.860 --> 01:14:24.420
OK.
01:14:24.420 --> 01:14:28.030
So now the big question
is the following.
01:14:28.030 --> 01:14:31.390
We arrived that the way of
thinking about equilibrium
01:14:31.390 --> 01:14:34.610
in a system of
particles and things
01:14:34.610 --> 01:14:38.000
that are this many-to-one
mapping, et cetera, in terms
01:14:38.000 --> 01:14:42.030
of the densities-- we arrived
that the definition of what
01:14:42.030 --> 01:14:44.990
the density is going
to be in equilibrium.
01:14:44.990 --> 01:14:49.730
But the thermodynamic statement
is much, much more severe.
01:14:49.730 --> 01:14:52.770
The statement, again,
is that if I have a box
01:14:52.770 --> 01:14:56.320
and I open the door
of the box, the gas
01:14:56.320 --> 01:15:01.300
expands to fill the empty space
or the other part of the box.
01:15:01.300 --> 01:15:04.850
And it will do so all the time.
01:15:04.850 --> 01:15:07.790
Yet the equations of motion
that we have over here
01:15:07.790 --> 01:15:10.090
are time reversal invariant.
01:15:10.090 --> 01:15:12.710
And we did not manage
to remove that.
01:15:12.710 --> 01:15:15.920
We can show that this
Liouville equation, et cetera,
01:15:15.920 --> 01:15:19.030
is also time reversal invariant.
01:15:19.030 --> 01:15:21.340
So for every case,
if you succeed
01:15:21.340 --> 01:15:24.340
to show that there is a density
that is in half of the box
01:15:24.340 --> 01:15:27.610
and it expands to
fill the entire box,
01:15:27.610 --> 01:15:30.940
there will be a density that
presumably goes the other way.
01:15:30.940 --> 01:15:34.760
Because that will be also a
solution of this equation.
01:15:34.760 --> 01:15:39.430
So when we sort of go back
from the statement of what
01:15:39.430 --> 01:15:42.670
is the equilibrium
solution and ask,
01:15:42.670 --> 01:15:46.720
do I know that I will eventually
reach this equilibrium
01:15:46.720 --> 01:15:50.210
solution as a function of
time, we have not shown that.
01:15:50.210 --> 01:15:55.040
And we will attempt
to do so next time.