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PROFESSOR: Let's start.
00:00:23.166 --> 00:00:24.255
Are there any questions?
00:00:29.290 --> 00:00:34.220
We would like to have a
perspective for this really
00:00:34.220 --> 00:00:41.350
common observation that if you
have a gas that is initially
00:00:41.350 --> 00:00:45.730
in one half of a box, and
the other half is empty,
00:00:45.730 --> 00:00:50.900
and some kind of a partition
is removed so that the gas can
00:00:50.900 --> 00:00:54.620
expand, and it can
flow, and eventually we
00:00:54.620 --> 00:00:57.400
will reach another
equilibrium state where
00:00:57.400 --> 00:01:00.950
the gas occupies more chambers.
00:01:00.950 --> 00:01:04.450
How do we describe
this observation?
00:01:04.450 --> 00:01:07.720
We can certainly characterize
it thermodynamically
00:01:07.720 --> 00:01:11.940
from the perspectives
of atoms and molecules.
00:01:11.940 --> 00:01:16.510
We said that if I
want to describe
00:01:16.510 --> 00:01:22.410
the configuration of the gas
before it starts, and also
00:01:22.410 --> 00:01:25.240
throughout the expansion,
I would basically
00:01:25.240 --> 00:01:30.200
have to look at all sets
of coordinates and momenta
00:01:30.200 --> 00:01:32.790
that make up this particle.
00:01:32.790 --> 00:01:34.710
There would be some
point in this [? six ?],
00:01:34.710 --> 00:01:36.910
and I mention our
phase space, that
00:01:36.910 --> 00:01:39.710
would correspond to where
this particle was originally.
00:01:42.690 --> 00:01:46.110
We can certainly follow
the dynamics of this point,
00:01:46.110 --> 00:01:48.480
but is that useful?
00:01:48.480 --> 00:01:50.730
Normally, I could
start with billions
00:01:50.730 --> 00:01:54.690
of different types of
boxes, or the same box
00:01:54.690 --> 00:01:56.780
in a different
instance of time, and I
00:01:56.780 --> 00:02:01.130
would have totally different
initial conditions.
00:02:01.130 --> 00:02:03.730
The initial
conditions presumably
00:02:03.730 --> 00:02:08.150
can be characterized to a
density in this phase space.
00:02:08.150 --> 00:02:11.435
You can look at some volume
and see how it changes,
00:02:11.435 --> 00:02:13.790
and how many points
you have there,
00:02:13.790 --> 00:02:16.610
and define this
phase space density
00:02:16.610 --> 00:02:21.330
row of all of the
Q's and P's, and it
00:02:21.330 --> 00:02:24.790
works as a function of time.
00:02:24.790 --> 00:02:28.400
One way of looking at how it
works as a function of time
00:02:28.400 --> 00:02:32.580
is to look at this box
and where this box will
00:02:32.580 --> 00:02:34.560
be in some other
instance of time.
00:02:38.800 --> 00:02:45.250
Essentially then, we are
following a kind of evolution
00:02:45.250 --> 00:02:48.540
that goes along this streamline.
00:02:48.540 --> 00:02:54.340
Basically, the derivative
that we are going to look at
00:02:54.340 --> 00:03:00.150
involves changes both
explicitly in the time variable,
00:03:00.150 --> 00:03:03.020
and also increasingly
to the changes
00:03:03.020 --> 00:03:05.640
of all of the
coordinates and momenta,
00:03:05.640 --> 00:03:09.360
according to the Hamiltonian
that governs the system.
00:03:09.360 --> 00:03:13.700
I have to do, essentially,
a sum over all coordinates.
00:03:13.700 --> 00:03:21.080
I would have the
change in coordinate i,
00:03:21.080 --> 00:03:27.420
Qi dot, dot, d row by dQi.
00:03:27.420 --> 00:03:35.500
Then I would have Pi, dot-- I
guess these are all vectors--
00:03:35.500 --> 00:03:39.150
d row by dPi.
00:03:39.150 --> 00:03:44.010
There are six end coordinates
that implicitly depend on time.
00:03:44.010 --> 00:03:46.500
In principle, if I am
following along the streamline,
00:03:46.500 --> 00:03:48.325
I have to look at
all of these things.
00:03:51.080 --> 00:03:56.240
The characteristic of evolution,
according to some Hamiltonian,
00:03:56.240 --> 00:04:01.540
was that this volume of
phase space does not change.
00:04:01.540 --> 00:04:05.370
Secondly, we could
characterize, once we
00:04:05.370 --> 00:04:20.740
wrote Qi dot, as dH by dP,
and the i dot as the H by dQ.
00:04:20.740 --> 00:04:24.560
This combination of derivatives
essentially could be captured,
00:04:24.560 --> 00:04:30.230
and be written as 0 by
dt is the Poisson bracket
00:04:30.230 --> 00:04:31.870
of H and [? P. ?]
00:04:35.270 --> 00:04:39.110
One of the things,
however, that we emphasize
00:04:39.110 --> 00:04:44.050
is that as far as evolution
according to a Hamiltonian
00:04:44.050 --> 00:04:50.740
and this set of
dynamics is concerned,
00:04:50.740 --> 00:04:54.480
the situation is completely
reversible in time
00:04:54.480 --> 00:04:57.020
so that some
intermediate process,
00:04:57.020 --> 00:05:01.170
if I were to reverse all of
the momenta, then the gas
00:05:01.170 --> 00:05:07.130
would basically come back
to the initial position.
00:05:07.130 --> 00:05:08.040
That's true.
00:05:08.040 --> 00:05:10.170
There is nothing to do about it.
00:05:10.170 --> 00:05:13.410
That kind of seems to
go against the intuition
00:05:13.410 --> 00:05:16.556
that we have from
thermodynamics.
00:05:19.210 --> 00:05:23.120
We said, well, in
practical situations,
00:05:23.120 --> 00:05:27.460
I really don't care about
all the six end pieces
00:05:27.460 --> 00:05:30.270
of information that
are embedded currently
00:05:30.270 --> 00:05:33.370
in this full phase
space density.
00:05:33.370 --> 00:05:36.340
If I'm really trying
to physically describe
00:05:36.340 --> 00:05:39.470
this gas expanding,
typically the things
00:05:39.470 --> 00:05:43.170
that I'm interested in are
that at some intermediate time,
00:05:43.170 --> 00:05:45.190
whether the particles
have reached
00:05:45.190 --> 00:05:47.820
this point or that
point, and what
00:05:47.820 --> 00:05:51.090
is this streamline velocity
that I'm seeing before the thing
00:05:51.090 --> 00:05:54.570
relaxes, presumably,
eventually into zero velocity?
00:05:54.570 --> 00:05:57.200
There's a lot of
things that I would
00:05:57.200 --> 00:06:00.980
need to characterize
this relaxation process,
00:06:00.980 --> 00:06:03.560
but that is still
much, much, much
00:06:03.560 --> 00:06:06.600
less than all of the information
that is currently encoded
00:06:06.600 --> 00:06:10.050
in all of these six end
coordinates and momenta.
00:06:10.050 --> 00:06:14.880
We said that for things that
I'm really interested in,
00:06:14.880 --> 00:06:17.610
what I could, for
example, look at,
00:06:17.610 --> 00:06:20.960
is a density that involves
only one particle.
00:06:25.650 --> 00:06:29.820
What I can do is to
then integrate over
00:06:29.820 --> 00:06:35.000
all of the positions and
coordinates of particles
00:06:35.000 --> 00:06:36.746
that I'm not interested in.
00:06:50.230 --> 00:06:55.020
I'm sort of repeating this to
introduce some notation so as
00:06:55.020 --> 00:06:58.120
to not to repeat all of
these integration variables,
00:06:58.120 --> 00:07:02.085
so I will call dVi the phase
place contribution of particle
00:07:02.085 --> 00:07:02.585
i.
00:07:06.090 --> 00:07:10.320
What I may be interested in
is that this is something
00:07:10.320 --> 00:07:12.790
that, if I integrate
over P1 and Q1,
00:07:12.790 --> 00:07:17.390
it is clearly
normalized to unity
00:07:17.390 --> 00:07:22.160
because my row, by definition,
was normalized to unity.
00:07:22.160 --> 00:07:24.690
Typically we may be
interested in something else
00:07:24.690 --> 00:07:32.180
that I call F1, P1
Q1 P, which is simply
00:07:32.180 --> 00:07:37.510
n times this-- n times
the integral product
00:07:37.510 --> 00:07:47.620
out i2 to n, dVi, the full row.
00:07:50.470 --> 00:07:52.740
Why we do that is
because typically you
00:07:52.740 --> 00:07:56.110
are interested or used
to calculating things
00:07:56.110 --> 00:07:58.330
in [? terms ?] of
a number density,
00:07:58.330 --> 00:08:02.190
like how many particles are
within some small volume here,
00:08:02.190 --> 00:08:06.660
defining the density so
that when I integrate over
00:08:06.660 --> 00:08:11.080
the entire volume
of f1, I would get
00:08:11.080 --> 00:08:13.990
the total number of
particles, for example.
00:08:13.990 --> 00:08:19.270
That's the kind of normalization
that people have used for f.
00:08:19.270 --> 00:08:22.340
More generally,
we also introduced
00:08:22.340 --> 00:08:31.600
fs, which depended on
coordinates representing
00:08:31.600 --> 00:08:35.299
s sets of points, or s
particles, if you like,
00:08:35.299 --> 00:08:37.059
that was normalized to be--
00:08:56.710 --> 00:09:00.710
We said, OK, what I'm really
interested in, in order
00:09:00.710 --> 00:09:05.220
to calculate the properties of
the gases it expands in terms
00:09:05.220 --> 00:09:08.530
of things that I'm
able to measure, is f1.
00:09:11.550 --> 00:09:14.860
Let's write down the
time evolution of f1.
00:09:14.860 --> 00:09:16.500
Actually, we said,
let's write down
00:09:16.500 --> 00:09:19.430
the time evolution
of fs, along with it.
00:09:19.430 --> 00:09:22.520
So there's the time
evolution of fs.
00:09:22.520 --> 00:09:26.740
If I were to go
along this stream,
00:09:26.740 --> 00:09:32.900
it would be the fs
by dt, and then I
00:09:32.900 --> 00:09:37.855
would have contributions
that would correspond
00:09:37.855 --> 00:09:43.760
to a the changes in
coordinates of these particles.
00:09:43.760 --> 00:09:47.930
In order to progress
along this direction,
00:09:47.930 --> 00:09:51.732
we said, let's define
the total Hamiltonian.
00:09:51.732 --> 00:09:55.995
We will have a simple form,
and certainly for the gas,
00:09:55.995 --> 00:09:59.050
it would be a good
representation.
00:09:59.050 --> 00:10:03.260
I have the kinetic energies
of all of the particles.
00:10:03.260 --> 00:10:07.840
I have the box that confines
the particles, or some other one
00:10:07.840 --> 00:10:12.450
particle potential, if you like,
but I will write in this much.
00:10:12.450 --> 00:10:16.720
Then you have the interactions
between all pairs of particles.
00:10:16.720 --> 00:10:22.669
Let's write it as sum over i,
less than j, V of Qi minus Qj.
00:10:31.520 --> 00:10:36.840
This depends on n set of
particles, coordinates,
00:10:36.840 --> 00:10:39.170
and momenta.
00:10:39.170 --> 00:10:42.480
Then we said that for purposes
of manipulations that you have
00:10:42.480 --> 00:10:45.950
to deal with, since
there are s coordinates
00:10:45.950 --> 00:10:48.880
that are appearing here
whose time derivatives I have
00:10:48.880 --> 00:10:52.600
to look at, I'm going
to simply rewrite
00:10:52.600 --> 00:10:54.590
this as the
contribution that comes
00:10:54.590 --> 00:10:58.310
from those s particles,
the contribution that comes
00:10:58.310 --> 00:11:01.310
from the remaining
n minus s particles,
00:11:01.310 --> 00:11:04.571
and some kind of [? term ?]
that covers the two
00:11:04.571 --> 00:11:05.320
sets of particles.
00:11:11.720 --> 00:11:15.430
This, actually, I didn't quite
need here until the next stage
00:11:15.430 --> 00:11:18.910
because what I write
here could, presumably,
00:11:18.910 --> 00:11:23.460
be sufficiently general,
like we have here some n
00:11:23.460 --> 00:11:26.670
running from 1 to s.
00:11:26.670 --> 00:11:30.260
Let me be consistent
with my S's.
00:11:30.260 --> 00:11:48.795
Then I have Qn, dot, dFs by
dQn, plus Pn, dot, dFs by dPn.
00:11:58.240 --> 00:12:05.780
If I just look at the
coordinates that appear here,
00:12:05.780 --> 00:12:10.470
and say, following this
as they move in time,
00:12:10.470 --> 00:12:13.380
there is the explicit
time dependence
00:12:13.380 --> 00:12:15.490
on all of the implicit
time dependence,
00:12:15.490 --> 00:12:17.790
this would be the
total derivative moving
00:12:17.790 --> 00:12:20.600
along the streamline.
00:12:20.600 --> 00:12:26.410
Qn dot I know is
simply the momentum.
00:12:26.410 --> 00:12:29.192
It is the H by dPn.
00:12:29.192 --> 00:12:33.720
The H by dPn I have from
this formula over here.
00:12:33.720 --> 00:12:37.470
It is simply Pn divided by m.
00:12:37.470 --> 00:12:40.300
It's the velocity--
momentum divided by mass.
00:12:40.300 --> 00:12:43.010
This is the velocity
of the particle.
00:12:43.010 --> 00:12:45.460
Pn dot, the rate of
change of momentum
00:12:45.460 --> 00:12:47.350
is the force that is
acting on the particle.
00:12:51.020 --> 00:12:54.720
What I need to do is to take
the derivatives of various terms
00:12:54.720 --> 00:12:55.680
here.
00:12:55.680 --> 00:12:59.550
So I have minus dU by dPn.
00:13:03.120 --> 00:13:03.782
What is this?
00:13:03.782 --> 00:13:05.740
This is essentially the
force that the particle
00:13:05.740 --> 00:13:07.990
feels from the
external potential.
00:13:07.990 --> 00:13:09.620
If you are in the
box in this room,
00:13:09.620 --> 00:13:13.820
It is zero until you
hit the edge of the box.
00:13:13.820 --> 00:13:20.230
I will call this Fn to represent
external potential that
00:13:20.230 --> 00:13:22.570
is acting on the system.
00:13:22.570 --> 00:13:24.100
What else is there?
00:13:24.100 --> 00:13:28.270
I have the force that will
come from the interaction
00:13:28.270 --> 00:13:35.150
with all other [? guys. ?] I
will write here a sum over m,
00:13:35.150 --> 00:13:44.675
dV of Qm minus Qn,
by dQn-- dU by dQm.
00:13:44.675 --> 00:13:45.375
I'm sorry.
00:13:49.920 --> 00:13:50.795
What is this?
00:13:50.795 --> 00:13:58.460
This Is basically the
sum of the forces that
00:13:58.460 --> 00:14:03.900
is exerted by the n
particle on the m particle.
00:14:03.900 --> 00:14:05.655
Define it in this fashion.
00:14:10.790 --> 00:14:15.060
If this was the entire
story, what I would have had
00:14:15.060 --> 00:14:21.610
here is a group of
s particles that
00:14:21.610 --> 00:14:26.070
are dominated by
their own dynamics.
00:14:26.070 --> 00:14:29.450
If there is no other
particle involved,
00:14:29.450 --> 00:14:33.760
they basically have to
satisfy the Liouville equation
00:14:33.760 --> 00:14:39.070
that I have written, now
appropriate to s particles.
00:14:39.070 --> 00:14:42.810
Of course, we know that
that's not the entire story
00:14:42.810 --> 00:14:46.430
because there are
all these other terms
00:14:46.430 --> 00:14:50.700
involving the interactions
with particles that I have not
00:14:50.700 --> 00:14:51.800
included.
00:14:51.800 --> 00:14:54.360
That's the whole
essence of the story.
00:14:54.360 --> 00:14:57.610
Let's say I want to think
about one or two particles.
00:14:57.610 --> 00:15:00.510
There is the interaction
between the two particles,
00:15:00.510 --> 00:15:03.670
and they would be evolving
according to some trajectories.
00:15:03.670 --> 00:15:05.670
But there are all of
these other particles
00:15:05.670 --> 00:15:09.830
in the gas in this room
that will collide with them.
00:15:09.830 --> 00:15:13.390
So those conditions
are not something
00:15:13.390 --> 00:15:17.980
that we had in the
Liouville equation,
00:15:17.980 --> 00:15:19.600
with everything considered.
00:15:19.600 --> 00:15:21.400
Here, I have to
include the effect
00:15:21.400 --> 00:15:23.600
of all of those other particles.
00:15:23.600 --> 00:15:25.210
We saw that the
way that it appears
00:15:25.210 --> 00:15:32.270
is that I have to imagine that
there's another particle whose
00:15:32.270 --> 00:15:36.390
coordinates and
momenta are captured
00:15:36.390 --> 00:15:41.340
through some volume for
the s plus 1 particle.
00:15:41.340 --> 00:15:45.350
This s plus 1
particle can interact
00:15:45.350 --> 00:15:47.870
with any of the particles
that are in the set
00:15:47.870 --> 00:15:50.710
that I have on the other side.
00:15:50.710 --> 00:15:55.870
There is an index
that runs from 1 to s.
00:15:55.870 --> 00:16:03.470
What I would have here is the
force that will come from this
00:16:03.470 --> 00:16:10.180
s plus 1 particle, acting on
particle n the same way that
00:16:10.180 --> 00:16:13.650
this force was deriving
the change of the momentum,
00:16:13.650 --> 00:16:19.390
this force will derive the
change of the momentum of-- I
00:16:19.390 --> 00:16:20.730
guess I put an m here--
00:16:23.410 --> 00:16:26.230
The thing that I
have to put here
00:16:26.230 --> 00:16:32.960
is now a density that also
keeps track of the probability
00:16:32.960 --> 00:16:38.282
to find the s plus 1 particle
in the location in phase place
00:16:38.282 --> 00:16:39.740
that I need to
integrate with both.
00:16:39.740 --> 00:16:44.030
I have to integrate
over all positions.
00:16:44.030 --> 00:16:48.170
One particle is moving along
a straight line by itself,
00:16:48.170 --> 00:16:50.000
let's say.
00:16:50.000 --> 00:16:53.670
Then there are all of the
other particles in the system.
00:16:53.670 --> 00:16:56.280
I have to ask, what is
the possibility that there
00:16:56.280 --> 00:16:59.690
is a second particle with
some particular momentum
00:16:59.690 --> 00:17:02.310
and coordinate that I
will be interacting with.
00:17:07.180 --> 00:17:14.989
This is the general set up
of these D-B-G-K-Y hierarchy
00:17:14.989 --> 00:17:15.530
of equations.
00:17:19.829 --> 00:17:23.130
At this stage, we
really have just
00:17:23.130 --> 00:17:28.640
rewritten what we had for
the Liouville equation.
00:17:28.640 --> 00:17:34.750
We said, I'm really, really
interested only one particle
00:17:34.750 --> 00:17:36.730
[? thing, ?] row one and F1.
00:17:36.730 --> 00:17:37.860
Let's focus on that.
00:17:37.860 --> 00:17:41.210
Let's write those
equations in more detail
00:17:41.210 --> 00:17:46.450
In the first equation, I
have that the explicit time
00:17:46.450 --> 00:17:52.920
dependence, plus the time
dependence of the position
00:17:52.920 --> 00:17:59.510
coordinate, plus the time
dependence of the momentum
00:17:59.510 --> 00:18:05.210
coordinate, which is driven
by the external force,
00:18:05.210 --> 00:18:07.850
acting on this one
particle density, which
00:18:07.850 --> 00:18:10.900
is dependent on
p1, q1 at time t.
00:18:14.420 --> 00:18:17.590
On the right hand
side of the equation.
00:18:17.590 --> 00:18:24.890
I need to worry about a second
particle with momenta P2
00:18:24.890 --> 00:18:28.130
at position Q2 that
will, therefore,
00:18:28.130 --> 00:18:29.910
be able to exert a force.
00:18:29.910 --> 00:18:33.820
Once I know the position,
I can calculate the force
00:18:33.820 --> 00:18:36.580
that particle exerts.
00:18:36.580 --> 00:18:38.160
What was my notation?
00:18:38.160 --> 00:18:45.785
The order was 2 and
1, dotted by d by dP1.
00:18:45.785 --> 00:18:54.220
I need now f2, p1, q2 at time t.
00:18:54.220 --> 00:18:58.390
We say, well, this
is unfortunate.
00:18:58.390 --> 00:19:03.300
I have to worry about
dependence on F2,
00:19:03.300 --> 00:19:05.710
but maybe I can get
away with things
00:19:05.710 --> 00:19:08.260
by estimating
order of magnitudes
00:19:08.260 --> 00:19:10.450
of the various terms.
00:19:10.450 --> 00:19:13.620
What is the left hand
side set of operations?
00:19:13.620 --> 00:19:15.550
The left hand side
set of operations
00:19:15.550 --> 00:19:22.310
describes essentially one
particle moving by itself.
00:19:22.310 --> 00:19:27.690
If that particle has to cross
a distance of this order of L,
00:19:27.690 --> 00:19:32.710
and I tell you that the typical
velocity of these particles
00:19:32.710 --> 00:19:36.980
is off the order of V, then
that time scale is going
00:19:36.980 --> 00:19:41.550
to be of the order of L
over V. The operations
00:19:41.550 --> 00:19:53.190
here will give me
a V over L, which
00:19:53.190 --> 00:19:56.660
is what we call the
inverse of Tau u.
00:19:59.910 --> 00:20:04.750
This is a reasonably
long macroscopic time.
00:20:04.750 --> 00:20:06.850
OK, that's fine.
00:20:06.850 --> 00:20:09.500
How big is the right hand side?
00:20:09.500 --> 00:20:12.050
We said that the
right hand side has
00:20:12.050 --> 00:20:15.430
something to do with collisions.
00:20:15.430 --> 00:20:19.030
I have a particle in my system.
00:20:19.030 --> 00:20:24.080
Let's say that particle has
some characteristic dimension
00:20:24.080 --> 00:20:25.530
that we call d.
00:20:28.720 --> 00:20:33.520
This particle is moving with
velocity V. Alternatively,
00:20:33.520 --> 00:20:36.820
you can think of this
particle as being stationary,
00:20:36.820 --> 00:20:39.895
and all the other
particles are coming at it
00:20:39.895 --> 00:20:42.970
with some velocity V.
00:20:42.970 --> 00:20:47.340
If I say that the density
of these particles is n,
00:20:47.340 --> 00:20:52.150
then the typical time for which,
as I shoot these particles,
00:20:52.150 --> 00:20:56.130
they will hit this
target is related
00:20:56.130 --> 00:21:00.800
to V squared and V, the
volume of particles.
00:21:03.330 --> 00:21:09.570
Over time t, I have to
consider this times V tau x.
00:21:09.570 --> 00:21:15.640
V tau xn V squared should
be of the order of one.
00:21:15.640 --> 00:21:21.240
This gave us a
formula for tau x.
00:21:21.240 --> 00:21:23.480
The inverse of tau
x that controls
00:21:23.480 --> 00:21:35.100
what's happening on this
side is n V squared V.
00:21:35.100 --> 00:21:39.450
Is the term on the right
hand side more important,
00:21:39.450 --> 00:21:41.920
or the term on the
left hand side?
00:21:41.920 --> 00:21:44.340
The term on the
right hand side has
00:21:44.340 --> 00:21:47.050
to do with the two body term.
00:21:47.050 --> 00:21:49.050
There's a particle
that is moving,
00:21:49.050 --> 00:21:50.710
and then there's
another particle
00:21:50.710 --> 00:21:56.190
with a slightly different
velocity that it is behind it.
00:21:56.190 --> 00:21:58.670
In the absence of
collisions, these particles
00:21:58.670 --> 00:22:01.410
would just go along
a straight line.
00:22:01.410 --> 00:22:04.040
They would bounce off the
walls, but the magnitude
00:22:04.040 --> 00:22:05.930
of their energy,
and hence, velocity,
00:22:05.930 --> 00:22:09.590
would not change from
these elastic collisions.
00:22:09.590 --> 00:22:12.450
But if the particles
can catch up
00:22:12.450 --> 00:22:19.040
and interact, which is governed
by V2, V on the other side,
00:22:19.040 --> 00:22:22.700
then what happens is that the
particles, when they interact,
00:22:22.700 --> 00:22:25.140
would collide and
go different ways.
00:22:25.140 --> 00:22:29.020
Quickly, their velocities,
and momenta, and everything
00:22:29.020 --> 00:22:31.760
would get mixed up.
00:22:31.760 --> 00:22:36.130
How rapidly that happens depends
on this collision distance,
00:22:36.130 --> 00:22:41.430
which is much less than
the size of the system,
00:22:41.430 --> 00:22:44.670
and, therefore, the term that
you have on the right hand
00:22:44.670 --> 00:22:48.770
side in magnitude is
much larger than what
00:22:48.770 --> 00:22:50.910
is happening on
the left hand side.
00:22:50.910 --> 00:22:55.230
There is no way in
order to describe
00:22:55.230 --> 00:22:59.590
the relaxation of the gas
that I can neglect collisions
00:22:59.590 --> 00:23:01.010
between gas particles.
00:23:01.010 --> 00:23:04.060
If I neglect collisions
between gas particles,
00:23:04.060 --> 00:23:06.530
there is no reason why
the kinetic energies
00:23:06.530 --> 00:23:08.380
of individual particles
should change.
00:23:08.380 --> 00:23:11.690
They would stay
the same forever.
00:23:11.690 --> 00:23:13.330
I have to keep this.
00:23:13.330 --> 00:23:16.305
Let's go and look at the second
equation in the hierarchy.
00:23:16.305 --> 00:23:17.140
What do you have?
00:23:17.140 --> 00:23:30.200
You have d by dT, P1 over m d
by d Q1, P2 over m, P d by d Q2.
00:23:30.200 --> 00:23:36.650
Then we have F1 d
by d Q1, plus F2, d
00:23:36.650 --> 00:23:42.180
by d Q2 coming from
the external potential.
00:23:42.180 --> 00:23:44.770
Then we have the force
that the involves
00:23:44.770 --> 00:23:47.320
the collision between
particles one and two.
00:23:50.600 --> 00:23:55.750
When I write down the
Hamiltonian for two particles,
00:23:55.750 --> 00:23:58.310
there is going to be
already for two particles
00:23:58.310 --> 00:24:00.640
and interactions between them.
00:24:00.640 --> 00:24:03.940
That's where the
F1 2 comes from.
00:24:03.940 --> 00:24:09.850
F1 2 changes d by the
momentum of particle one.
00:24:09.850 --> 00:24:11.910
I should write, it's
2 1 that changes
00:24:11.910 --> 00:24:13.960
momentum of particle two.
00:24:13.960 --> 00:24:19.500
But as 2 1 is simply minus F1
2, I can put the two of them
00:24:19.500 --> 00:24:22.220
together in this fashion.
00:24:22.220 --> 00:24:28.900
This acting on F2 is
then equal to something
00:24:28.900 --> 00:24:43.370
like integral over V3, F3 1,
d by dP1, plus F3 2, d by dP2.
00:24:43.370 --> 00:24:49.694
[INAUDIBLE] on F3 P1
and Q3 [INAUDIBLE].
00:24:53.890 --> 00:24:56.510
Are we going to do this forever?
00:24:56.510 --> 00:25:00.120
Well, we said, let's
take another look
00:25:00.120 --> 00:25:04.010
at the magnitude of
the various terms.
00:25:04.010 --> 00:25:06.770
This term on the
right hand side still
00:25:06.770 --> 00:25:10.660
involves a collision that
involves a third particle.
00:25:10.660 --> 00:25:13.450
I have to find that
third particle,
00:25:13.450 --> 00:25:17.120
so I need to have,
essentially, a third particle
00:25:17.120 --> 00:25:19.150
within some
characteristic volume,
00:25:19.150 --> 00:25:22.740
so I have something
that is of that order.
00:25:22.740 --> 00:25:26.790
Whereas on the
left hand side now,
00:25:26.790 --> 00:25:30.590
I have a term that
from all perspectives,
00:25:30.590 --> 00:25:34.580
looks like the kinds of terms
that I had before except
00:25:34.580 --> 00:25:39.730
that it involves the collision
between two particles.
00:25:39.730 --> 00:25:43.770
What it describes is the
duration that collision.
00:25:43.770 --> 00:25:45.890
We said this is of the
order of 1 over tau
00:25:45.890 --> 00:25:50.100
c, which replaces
the n over there
00:25:50.100 --> 00:25:52.940
with some characteristic
dimension.
00:25:52.940 --> 00:25:56.040
Suddenly, this term is very big.
00:25:58.570 --> 00:25:59.916
We should be able to use that.
00:25:59.916 --> 00:26:00.790
There was a question.
00:26:00.790 --> 00:26:04.513
AUDIENCE: On the left hand
side of both of your equations,
00:26:04.513 --> 00:26:07.892
for F1 and F2, shouldn't
all the derivatives that
00:26:07.892 --> 00:26:11.328
are multiplied by your
forces be derivatives
00:26:11.328 --> 00:26:13.958
of the effects of momentum?
[INAUDIBLE] the coordinates?
00:26:13.958 --> 00:26:15.986
[INAUDIBLE] reasons?
00:26:15.986 --> 00:26:20.170
PROFESSOR: Let's go back here.
00:26:20.170 --> 00:26:27.640
I have a function that
depends on P, Q, and t.
00:26:27.640 --> 00:26:31.750
Then there's the explicit
time derivative, d by dt.
00:26:31.750 --> 00:26:38.370
Then there is the Q dot here,
which will go by d by dQ.
00:26:38.370 --> 00:26:42.610
Then there's the P dot term
that will go by d by dP.
00:26:42.610 --> 00:26:45.560
All of things have to be there.
00:26:45.560 --> 00:26:48.270
I should have derivatives
in respect to momenta,
00:26:48.270 --> 00:26:50.990
and derivatives with
respect to coordinate.
00:26:50.990 --> 00:26:53.530
Dimensions are, of
course, important.
00:26:53.530 --> 00:26:57.070
Somewhat, what I write
for this and for this
00:26:57.070 --> 00:26:59.080
should make up for that.
00:26:59.080 --> 00:27:02.630
As I have written it now,
it's obvious, of course.
00:27:02.630 --> 00:27:06.080
This has dimensions of Q
over T. The Q's cancel.
00:27:06.080 --> 00:27:09.170
I would have one over
T. D over Dps cancel.
00:27:09.170 --> 00:27:12.750
I have 1 over P. Here,
dimensionality is correct.
00:27:12.750 --> 00:27:15.610
I have to just make sure
I haven't made a mistake.
00:27:15.610 --> 00:27:17.830
Q dot is a velocity.
00:27:17.830 --> 00:27:21.200
Velocity is momentum
divided by mass.
00:27:21.200 --> 00:27:23.730
So that should
dimensionally work out.
00:27:23.730 --> 00:27:26.500
P dot is a force.
00:27:26.500 --> 00:27:27.665
Everything here is force.
00:27:30.579 --> 00:27:31.745
In a reasonable coordinate--
00:27:31.745 --> 00:27:33.266
AUDIENCE: [INAUDIBLE]
00:27:33.266 --> 00:27:35.101
PROFESSOR: What did I do here?
00:27:35.101 --> 00:27:36.100
I made mistakes?
00:27:36.100 --> 00:27:37.100
AUDIENCE: [INAUDIBLE]
00:27:49.119 --> 00:27:50.910
PROFESSOR: Why didn't
you say that in a way
00:27:50.910 --> 00:27:57.800
that-- If I don't
understand the question,
00:27:57.800 --> 00:28:01.730
please correct me before I
spend another five minutes.
00:28:05.450 --> 00:28:10.950
Hopefully, this is now
free of these deficiencies.
00:28:10.950 --> 00:28:12.223
This there is very big.
00:28:15.150 --> 00:28:17.660
Now, compared to the
right hand side in fact,
00:28:17.660 --> 00:28:20.140
we said that the
right hand side is
00:28:20.140 --> 00:28:23.650
smaller by a factor that
measures how many particles
00:28:23.650 --> 00:28:26.060
are within an
interaction volume.
00:28:26.060 --> 00:28:27.950
And for a typical
gas, this would
00:28:27.950 --> 00:28:31.810
be a number that's of the
order of 10 to the minus 4.
00:28:31.810 --> 00:28:34.560
Using 10 to the minus
4 being this small,
00:28:34.560 --> 00:28:38.270
we are going to set the
right hand side to zero.
00:28:38.270 --> 00:28:40.410
Now, I don't have to
write the equation for F2.
00:28:45.770 --> 00:28:51.550
I'll answer a question here that
may arise, which is ultimately,
00:28:51.550 --> 00:28:54.700
we will do sufficient
manipulations
00:28:54.700 --> 00:28:58.650
so that we end up with a
particular equation, known
00:28:58.650 --> 00:29:00.685
as the Boltzmann
Equation, that we
00:29:00.685 --> 00:29:06.660
will show does not obey
the time reversibility
00:29:06.660 --> 00:29:08.890
that we wrote over here.
00:29:08.890 --> 00:29:14.520
Clearly, that is built in to the
various approximations I make.
00:29:14.520 --> 00:29:17.140
The first question
is, the approximation
00:29:17.140 --> 00:29:21.950
that I've made here,
did I destroy this time
00:29:21.950 --> 00:29:23.310
reversibility?
00:29:23.310 --> 00:29:25.110
The answer is no.
00:29:25.110 --> 00:29:28.600
You can look at this
set of equations,
00:29:28.600 --> 00:29:31.390
and do the manipulations
necessary to see
00:29:31.390 --> 00:29:35.110
what happens if P goes to minus
P. You will find that you will
00:29:35.110 --> 00:29:38.520
be able to reverse your
trajectory without any problem.
00:29:38.520 --> 00:29:39.020
Yes?
00:29:41.750 --> 00:29:44.144
AUDIENCE: Given that it
is only an interaction
00:29:44.144 --> 00:29:46.096
from our left side
that's very big,
00:29:46.096 --> 00:29:48.980
that's the reason why we can
ignore the stuff on the right.
00:29:48.980 --> 00:29:50.810
Why is it that we
are then keeping
00:29:50.810 --> 00:29:55.820
all of the other terms that
were even smaller before?
00:29:55.820 --> 00:29:57.300
PROFESSOR: I will ignore them.
00:29:57.300 --> 00:29:57.800
Sure.
00:29:57.800 --> 00:30:00.290
AUDIENCE: [LAUGHTER]
00:30:10.285 --> 00:30:14.570
PROFESSOR: There was the
question of time reversibility.
00:30:14.570 --> 00:30:19.050
This term here has to do
with three particles coming
00:30:19.050 --> 00:30:23.330
together, and how
that would modify what
00:30:23.330 --> 00:30:26.590
we have for just
two-body collisions.
00:30:26.590 --> 00:30:29.140
In principle, there
is some probability
00:30:29.140 --> 00:30:31.380
to have three particles
coming together
00:30:31.380 --> 00:30:34.400
and some combined interactions.
00:30:34.400 --> 00:30:37.050
You can imagine some
fictitious model,
00:30:37.050 --> 00:30:40.810
which in addition to these
two-body interactions,
00:30:40.810 --> 00:30:43.810
you cook up some body
interaction so that it
00:30:43.810 --> 00:30:47.190
precisely cancels what
would have happened
00:30:47.190 --> 00:30:48.910
when three particles
come together.
00:30:48.910 --> 00:30:51.110
We can write a computer
program in which
00:30:51.110 --> 00:30:52.890
we have two body conditions.
00:30:52.890 --> 00:30:56.360
But if three bodies come
close enough to each other,
00:30:56.360 --> 00:30:59.440
they essentially become ghosts
and pass through each other.
00:30:59.440 --> 00:31:03.140
That computer program
would be fully reversible.
00:31:03.140 --> 00:31:05.550
That's why sort of
dropping this there
00:31:05.550 --> 00:31:07.817
is not causing any
problems at this point.
00:31:12.990 --> 00:31:17.640
What is it that you
have included so far?
00:31:17.640 --> 00:31:23.680
What we have is a situation
where the change in F1
00:31:23.680 --> 00:31:26.330
is governed by a
process in which I
00:31:26.330 --> 00:31:29.760
have a particle that I
describe on the left hand
00:31:29.760 --> 00:31:34.930
side with momentum one, and
it collides with some particle
00:31:34.930 --> 00:31:39.710
that I'm integrating over, but
in some particular instance
00:31:39.710 --> 00:31:43.000
of integration, has momentum P2.
00:31:43.000 --> 00:31:46.690
Presumably they come
close enough to each other
00:31:46.690 --> 00:31:50.990
so that afterwards, the
momenta have changed over so
00:31:50.990 --> 00:31:55.165
that I have some P1 prime,
and I have some P2 prime.
00:32:04.410 --> 00:32:10.270
We want to make sure that we
characterize these correctly.
00:32:10.270 --> 00:32:15.330
There was a question about
while this term is big,
00:32:15.330 --> 00:32:18.770
these kinds of terms are small.
00:32:18.770 --> 00:32:22.580
Why should I basically
bother to keep them?
00:32:22.580 --> 00:32:25.040
It is reasonable.
00:32:25.040 --> 00:32:29.040
What we are following here are
particles in my picture that
00:32:29.040 --> 00:32:31.550
were ejected by the
first box, and they
00:32:31.550 --> 00:32:33.250
collide into each
other, or they were
00:32:33.250 --> 00:32:35.320
colliding in the first box.
00:32:35.320 --> 00:32:41.725
As long as you are away from
the [? vols ?] of the container,
00:32:41.725 --> 00:32:43.980
you really don't care
about these terms.
00:32:43.980 --> 00:32:47.450
They don't really
moved very rapidly.
00:32:47.450 --> 00:32:51.250
This is the process of
collision of two particles,
00:32:51.250 --> 00:32:55.550
and it's also the same process
that is described over here.
00:32:55.550 --> 00:32:59.150
Somehow, I should be able
to simplify the collision
00:32:59.150 --> 00:33:02.720
process that is going on
here with the knowledge
00:33:02.720 --> 00:33:05.650
that the evolution
of two particles
00:33:05.650 --> 00:33:09.190
is now completely deterministic.
00:33:09.190 --> 00:33:12.680
This equation by itself
says, take two particles
00:33:12.680 --> 00:33:14.800
as if they are the only
thing in the universe,
00:33:14.800 --> 00:33:17.570
and they would follow some
completely deterministic
00:33:17.570 --> 00:33:20.690
trajectory, that if you
put lots of them together,
00:33:20.690 --> 00:33:24.420
is captured through
this density.
00:33:24.420 --> 00:33:28.360
Let's see whether we can
massage this equation
00:33:28.360 --> 00:33:30.440
to look like this equation.
00:33:30.440 --> 00:33:33.050
Well, the force term, we
have, except that here we
00:33:33.050 --> 00:33:34.510
have dP by P1 here.
00:33:34.510 --> 00:33:39.240
We have d by dP
1 minus d by dP2.
00:33:39.240 --> 00:33:41.720
So let's do this.
00:33:41.720 --> 00:33:50.030
Minus d by dP2, acting on F2.
00:33:50.030 --> 00:33:52.050
Did I do something wrong?
00:33:52.050 --> 00:33:57.250
The answer is no, because I
added the complete derivative
00:33:57.250 --> 00:33:59.070
over something that
I'm integrating over.
00:34:01.610 --> 00:34:03.590
This is perfectly
legitimate mathematics.
00:34:06.420 --> 00:34:08.880
This part now looks like this.
00:34:08.880 --> 00:34:12.500
I have to find what is the
most important term that
00:34:12.500 --> 00:34:14.580
matches this.
00:34:14.580 --> 00:34:16.280
Again, let's think
about this procedure.
00:34:19.610 --> 00:34:24.020
What I have to make
sure of is what
00:34:24.020 --> 00:34:26.699
is the extent of the
collision, and how important is
00:34:26.699 --> 00:34:27.650
the collision?
00:34:27.650 --> 00:34:29.900
If I have one
particle moving here,
00:34:29.900 --> 00:34:32.960
and another particle off there,
they will pass each other.
00:34:32.960 --> 00:34:35.469
Nothing interesting
could happen.
00:34:35.469 --> 00:34:38.199
The important thing is how
close they come together.
00:34:38.199 --> 00:34:41.260
It Is kind of
important that I keep
00:34:41.260 --> 00:34:45.100
track of the relative
coordinate, Q,
00:34:45.100 --> 00:34:49.570
which is Q2 minus Q1,
as opposed to the center
00:34:49.570 --> 00:34:55.030
of mass coordinate, which
is just Q1 plus Q2 over 2.
00:34:59.320 --> 00:35:01.860
That kind of also
indicates maybe it's
00:35:01.860 --> 00:35:05.320
a good thing for me to
look at this entire process
00:35:05.320 --> 00:35:06.930
in the center of mass frame.
00:35:06.930 --> 00:35:08.440
So this is the lab frame.
00:35:11.730 --> 00:35:15.770
If I were to look at this
same picture in the center
00:35:15.770 --> 00:35:20.370
of mass frame,
what would I have?
00:35:20.370 --> 00:35:25.880
In the center of mass frame, I
would have the initial particle
00:35:25.880 --> 00:35:32.480
coming with P1 prime, P1
minus P center of mass.
00:35:35.340 --> 00:35:39.180
The other particle that
you are interacting with
00:35:39.180 --> 00:35:47.520
comes with P2 minus
P center of mass.
00:35:47.520 --> 00:35:53.110
I actually drew these
vectors that are hopefully
00:35:53.110 --> 00:35:55.312
equal and opposite,
because you know
00:35:55.312 --> 00:35:57.610
that in the center of
mass, one of them, in fact,
00:35:57.610 --> 00:35:59.690
would be P1 minus P2 over 2.
00:35:59.690 --> 00:36:01.850
The other would be
P2 minus P1 over 2.
00:36:01.850 --> 00:36:03.620
They would, indeed,
in the center of mass
00:36:03.620 --> 00:36:05.610
be equal and opposite momenta.
00:36:08.310 --> 00:36:15.250
Along the direction
of these objects,
00:36:15.250 --> 00:36:20.850
I can look at how close
they come together.
00:36:20.850 --> 00:36:23.680
I can look at some
coordinate that I
00:36:23.680 --> 00:36:32.860
will call A, which measures
the separation between them
00:36:32.860 --> 00:36:35.000
at some instant of time.
00:36:35.000 --> 00:36:38.280
Then there's another
pair of coordinates
00:36:38.280 --> 00:36:43.110
that I could put into a vector
that tells me how head to head
00:36:43.110 --> 00:36:44.590
they are.
00:36:44.590 --> 00:36:48.120
If I think about they're
being on the center of mass,
00:36:48.120 --> 00:36:50.690
two things that are
approaching each other,
00:36:50.690 --> 00:36:52.950
they can either
approach head on-- that
00:36:52.950 --> 00:36:55.200
would correspond
to be equal to 0--
00:36:55.200 --> 00:37:02.590
or they could be slightly
off a head-on collision.
00:37:02.590 --> 00:37:11.940
There is a so-called
impact parameter
00:37:11.940 --> 00:37:17.420
B, which is a measure
of this addition fact.
00:37:20.030 --> 00:37:23.830
Why is that going to
be relevant to us?
00:37:23.830 --> 00:37:29.100
Again, we said that there
are parts of this expression
00:37:29.100 --> 00:37:31.900
that all of the
order of this term,
00:37:31.900 --> 00:37:35.030
they're kind of
not that important.
00:37:35.030 --> 00:37:39.530
If I think about the collision,
and what the collision does,
00:37:39.530 --> 00:37:45.250
I will have forces that
are significant when
00:37:45.250 --> 00:37:49.780
I am within this
range of interactions,
00:37:49.780 --> 00:37:54.730
D. I really have to look at
what happens when the two
00:37:54.730 --> 00:37:56.400
things come close to each other.
00:37:56.400 --> 00:38:01.736
It Is only when this relative
parameter A has approached D
00:38:01.736 --> 00:38:05.110
that these particles
will start to deviate
00:38:05.110 --> 00:38:07.840
from their straight
line trajectory,
00:38:07.840 --> 00:38:12.775
and presumably go, to say in
this case, P2 prime minus P
00:38:12.775 --> 00:38:14.580
center of mass.
00:38:14.580 --> 00:38:16.865
This one occurs
[? and ?] will go,
00:38:16.865 --> 00:38:22.610
and eventually P1 prime
minus P center of mass.
00:38:22.610 --> 00:38:25.545
These deviations will
occur over a distance
00:38:25.545 --> 00:38:31.610
that is of the order of
this collision and D.
00:38:31.610 --> 00:38:37.640
The important changes that
occur in various densities,
00:38:37.640 --> 00:38:40.530
in various
potentials, et cetera,
00:38:40.530 --> 00:38:46.210
are all taking place when this
relative coordinate is small.
00:38:46.210 --> 00:38:50.570
Things become big when the
relative coordinate is small.
00:38:50.570 --> 00:38:54.590
They are big as a function
of the relative coordinate.
00:38:54.590 --> 00:38:59.140
In order to get big
things, what I need to do
00:38:59.140 --> 00:39:03.130
is to replace these
d by dQ's with
00:39:03.130 --> 00:39:06.120
the corresponding
derivatives with respect
00:39:06.120 --> 00:39:07.990
to the center of mass.
00:39:07.990 --> 00:39:10.670
One of them would come
be the minus sign.
00:39:10.670 --> 00:39:12.990
The other would come
be the plus sign.
00:39:12.990 --> 00:39:15.195
It doesn't matter
which is which.
00:39:15.195 --> 00:39:17.270
It depends on the
definition, whether I
00:39:17.270 --> 00:39:19.980
make Q2 minus Q1,
or Q1 minus Q2.
00:39:23.490 --> 00:39:28.190
We see that the big
terms are the force that
00:39:28.190 --> 00:39:31.840
changes the momenta
and the variations
00:39:31.840 --> 00:39:35.760
that you have over these
relative coordinates.
00:39:35.760 --> 00:39:40.260
What I can do now
is to replace this
00:39:40.260 --> 00:39:44.610
by equating the two big
terms that I have over here.
00:39:44.610 --> 00:40:03.280
The two big terms
are P2 minus P1
00:40:03.280 --> 00:40:08.240
over m, dotted by d by dQ of F2.
00:40:17.950 --> 00:40:23.750
There is some other
approximation that I did.
00:40:23.750 --> 00:40:28.540
As was told to me before,
this is the biggest term,
00:40:28.540 --> 00:40:30.155
and there is the
part of this that
00:40:30.155 --> 00:40:32.380
is big and compensates for that.
00:40:32.380 --> 00:40:34.750
But there are all these
other bunches of terms.
00:40:34.750 --> 00:40:38.260
There's also this d by dt.
00:40:38.260 --> 00:40:43.900
What I have done
over here is to look
00:40:43.900 --> 00:40:49.250
at this slightly coarser
perspective on time.
00:40:49.250 --> 00:40:53.130
Increasing all the equations
that I have over there
00:40:53.130 --> 00:40:55.830
tells me everything
about particles
00:40:55.830 --> 00:40:59.260
approaching each
other and going away.
00:40:59.260 --> 00:41:03.240
I can follow through
the mechanics
00:41:03.240 --> 00:41:06.190
precisely everything
that is happening, even
00:41:06.190 --> 00:41:08.930
in the vicinity
of this collision.
00:41:08.930 --> 00:41:13.480
If I have two squishy balls,
and I run my hand through them
00:41:13.480 --> 00:41:15.450
properly, I can
see how the things
00:41:15.450 --> 00:41:17.670
get squished then released.
00:41:17.670 --> 00:41:20.350
There's a lot of
information, but again, a lot
00:41:20.350 --> 00:41:22.140
of information
that I don't really
00:41:22.140 --> 00:41:25.610
care to know as far as
the properties of this gas
00:41:25.610 --> 00:41:29.650
expansion process is concerned.
00:41:29.650 --> 00:41:35.386
What you have done is to forget
about the detailed variations
00:41:35.386 --> 00:41:39.130
in time and space that
are taking place here.
00:41:39.130 --> 00:41:42.530
We're going to shortly make
that even more explicit
00:41:42.530 --> 00:41:45.110
by noting the following.
00:41:45.110 --> 00:41:49.340
This integration over
here is an integration
00:41:49.340 --> 00:41:52.750
over phase space of
the second particle.
00:41:52.750 --> 00:41:57.440
I had written before d
cubed, P2, d cubed, Q2,
00:41:57.440 --> 00:41:59.530
but I can change
coordinates and look
00:41:59.530 --> 00:42:05.020
at the relative
coordinate, Q, over here.
00:42:09.070 --> 00:42:12.300
What I'm asking is,
I have one particle
00:42:12.300 --> 00:42:14.090
moving through the gas.
00:42:14.090 --> 00:42:17.230
What is the chance that
the second particle comes
00:42:17.230 --> 00:42:23.510
with momentum P2, and the
appropriate relative distance
00:42:23.510 --> 00:42:28.740
Q, and I integrate over both the
P and the relative distance Q?
00:42:28.740 --> 00:42:31.400
This is the quantity
that I have to integrate.
00:42:38.190 --> 00:42:41.280
Let's do one more
calculation, and then we
00:42:41.280 --> 00:42:46.430
will try to give a
physical perspective.
00:42:46.430 --> 00:42:50.670
In this picture of the center
of mass, what did I do?
00:42:50.670 --> 00:43:02.010
I do replaced the coordinate, Q,
with a part that was the impact
00:43:02.010 --> 00:43:05.320
parameter, which
had two components,
00:43:05.320 --> 00:43:07.505
and a part that was
the relative distance.
00:43:10.820 --> 00:43:14.100
What was this relative distance?
00:43:14.100 --> 00:43:17.360
The relative
distance was measured
00:43:17.360 --> 00:43:23.640
along this line that was
giving me the closest approach.
00:43:23.640 --> 00:43:26.270
What is the direction
of this line?
00:43:26.270 --> 00:43:30.330
The direction of this
line is P1 minus P2.
00:43:30.330 --> 00:43:32.150
This is P1 minus P2 over 2.
00:43:32.150 --> 00:43:32.900
It doesn't matter.
00:43:32.900 --> 00:43:36.380
The direction is P1 minus P2.
00:43:36.380 --> 00:43:43.860
What I'm doing here is I am
taking the derivative precisely
00:43:43.860 --> 00:43:46.920
along this line of
constant approach.
00:43:50.260 --> 00:43:54.410
I'm taking a derivative, and
I'm integrating along that.
00:43:57.790 --> 00:44:03.310
If I were to rewrite the
whole thing, what do I have?
00:44:03.310 --> 00:44:14.180
I have d by dt, plus P1 over m,
d by dQ1, plus F1, d by dP1--
00:44:14.180 --> 00:44:22.870
don't make a mistake--
acting on F1, P1, Q1, t.
00:44:22.870 --> 00:44:25.580
What do I have to write
on the right hand side?
00:44:25.580 --> 00:44:29.500
I have an integral
over the momentum
00:44:29.500 --> 00:44:33.620
of this particle with which
I'm going to make a collision.
00:44:33.620 --> 00:44:37.220
I have an integral over
the impact parameter
00:44:37.220 --> 00:44:41.910
that tells me the distance
of closest approach.
00:44:41.910 --> 00:44:47.110
I have to do the
magnitude of P2 minus P1
00:44:47.110 --> 00:44:50.180
over n, which is
really the magnitude
00:44:50.180 --> 00:44:53.170
of the relative velocity
of the two particles.
00:44:53.170 --> 00:44:57.710
I can write it as P2
minus P1, or P1 minus P2.
00:44:57.710 --> 00:44:59.070
These are, of course, vectors.
00:44:59.070 --> 00:45:02.070
and I look at the modulus.
00:45:02.070 --> 00:45:04.813
I have the integral
of the derivative.
00:45:07.610 --> 00:45:11.700
Very simply, I will
write the answer
00:45:11.700 --> 00:45:19.130
as F2 that is evaluated
at some large distance,
00:45:19.130 --> 00:45:23.150
plus infinity minus F2
evaluated at minus infinity.
00:45:23.150 --> 00:45:24.240
I have infinity.
00:45:24.240 --> 00:45:26.730
In principle, I have
to integrate over F2
00:45:26.730 --> 00:45:29.560
from minus infinity
to plus infinity.
00:45:29.560 --> 00:45:34.250
But once I am beyond
the range of where
00:45:34.250 --> 00:45:37.090
the interaction changes,
then the two particles
00:45:37.090 --> 00:45:39.000
just move away forever.
00:45:39.000 --> 00:45:41.030
They will never see each other.
00:45:41.030 --> 00:45:47.180
Really, what I should write here
is F2 of-- after the collision,
00:45:47.180 --> 00:45:54.990
I have P1 prime, P2
prime, at some Q plus,
00:45:54.990 --> 00:46:04.280
minus F2, P1, P2, at
some position minus.
00:46:04.280 --> 00:46:08.210
What I need to do is to
do the integration when
00:46:08.210 --> 00:46:12.420
I'm far away from the
collision, or wait
00:46:12.420 --> 00:46:16.070
until I am far
after the collision.
00:46:16.070 --> 00:46:21.630
Really, I have to just
integrate slightly below, after,
00:46:21.630 --> 00:46:25.270
and before the collision occurs.
00:46:25.270 --> 00:46:28.430
In principle, if I
just go a few d's
00:46:28.430 --> 00:46:31.080
in one direction or
the other direction,
00:46:31.080 --> 00:46:32.050
this should be enough.
00:46:34.560 --> 00:46:38.670
Let's see physically
what this describes.
00:46:38.670 --> 00:46:41.390
There is a connection
between this
00:46:41.390 --> 00:46:43.930
and this thing that I
had over here, in fact.
00:46:46.680 --> 00:46:51.420
This equation on the left
hand side, if it was zero,
00:46:51.420 --> 00:46:53.600
it would describe
one particle that
00:46:53.600 --> 00:46:57.840
is just moving by itself until
it hits the wall, at which
00:46:57.840 --> 00:47:01.160
point it basically
reverses its trajectory,
00:47:01.160 --> 00:47:04.100
and otherwise goes forward.
00:47:04.100 --> 00:47:06.080
But what you have on
the right hand side
00:47:06.080 --> 00:47:07.980
says that suddenly
there could be
00:47:07.980 --> 00:47:10.830
another particle with
which I interact.
00:47:10.830 --> 00:47:13.350
Then I change my direction.
00:47:13.350 --> 00:47:16.830
I need to know the
probability, given
00:47:16.830 --> 00:47:21.200
that I'm moving
with velocity P1,
00:47:21.200 --> 00:47:24.400
that there is a second
particle with P2 that
00:47:24.400 --> 00:47:25.500
comes close enough.
00:47:30.010 --> 00:47:32.540
There is this additional factor.
00:47:32.540 --> 00:47:35.320
From what does this
additional factor come?
00:47:35.320 --> 00:47:38.330
It's the same factor
that we have over here.
00:47:38.330 --> 00:47:42.140
It is, if you have a
target of size d squared,
00:47:42.140 --> 00:47:48.780
and we have a set of
bullets with a density of n,
00:47:48.780 --> 00:47:52.650
the number of collisions that
I get depends both on density
00:47:52.650 --> 00:47:55.720
and how fast these things go.
00:47:55.720 --> 00:48:01.020
The time between collisions, if
you like, is proportional to n,
00:48:01.020 --> 00:48:07.320
and it is also related to
V. That's what this is.
00:48:07.320 --> 00:48:10.760
I need some kind of a time
between the collisions
00:48:10.760 --> 00:48:12.750
that I make.
00:48:12.750 --> 00:48:15.120
I have already
specified that I'm
00:48:15.120 --> 00:48:17.580
only interested in the
set of particles that
00:48:17.580 --> 00:48:20.450
have momentum P2 for this
particular [? point in ?]
00:48:20.450 --> 00:48:26.270
integration, and that they
have this kind of area or cross
00:48:26.270 --> 00:48:27.500
section.
00:48:27.500 --> 00:48:31.930
So I replace this
V squared and V
00:48:31.930 --> 00:48:33.570
with the relative coordinates.
00:48:33.570 --> 00:48:37.790
This is the corresponding
thing to V squared,
00:48:37.790 --> 00:48:41.150
and this is really a
two particle density.
00:48:41.150 --> 00:48:43.850
This is a subtraction.
00:48:43.850 --> 00:48:46.480
The addition is
because it is true
00:48:46.480 --> 00:48:51.370
that I'm going with velocity P1,
and practically, any collisions
00:48:51.370 --> 00:48:54.830
that are significant
will move me off kilter.
00:48:54.830 --> 00:48:58.480
So there has to be a
subtraction for the channel that
00:48:58.480 --> 00:49:02.630
was described by P1
because of this collision.
00:49:02.630 --> 00:49:05.230
This then, is the
addition, because it
00:49:05.230 --> 00:49:09.240
says that it could be that
there is no particle going
00:49:09.240 --> 00:49:10.620
in the horizontal direction.
00:49:10.620 --> 00:49:15.140
I was actually coming along
the vertical direction.
00:49:15.140 --> 00:49:17.650
Because of the
collision, I suddenly
00:49:17.650 --> 00:49:21.150
was shifted to move
along this direction.
00:49:21.150 --> 00:49:28.560
The addition comes from having
particles that would correspond
00:49:28.560 --> 00:49:33.550
to momenta that somehow,
if I were in some sense
00:49:33.550 --> 00:49:35.850
to reverse this, and
then put a minus sign,
00:49:35.850 --> 00:49:38.990
a reverse collision
would create something
00:49:38.990 --> 00:49:40.860
that was along the
direction of P1.
00:49:46.110 --> 00:49:49.440
Here I also made
several approximations.
00:49:49.440 --> 00:49:56.570
I said, what is chief among
them is that basically I
00:49:56.570 --> 00:50:01.470
ignored the details of the
process that is taking place
00:50:01.470 --> 00:50:05.520
at scale the order of
d, so I have thrown away
00:50:05.520 --> 00:50:08.490
some amount of detail
and information.
00:50:08.490 --> 00:50:12.540
It is, again, legitimate
to say, is this
00:50:12.540 --> 00:50:15.970
the stage at which you
made an approximation
00:50:15.970 --> 00:50:19.790
so that the time
reversibility was lost?
00:50:19.790 --> 00:50:21.920
The answer is still no.
00:50:21.920 --> 00:50:26.890
If you are careful enough with
making precise definitions
00:50:26.890 --> 00:50:30.520
of what these Q's are before
and after the collision,
00:50:30.520 --> 00:50:35.220
and follow what happens if you
were to reverse everything,
00:50:35.220 --> 00:50:38.640
you'll find that the
equations is fully reversible.
00:50:38.640 --> 00:50:43.890
Even at this stage, I have
not made any transition.
00:50:43.890 --> 00:50:47.560
I have made approximations,
but I haven't made something
00:50:47.560 --> 00:50:49.920
to be time irreversible.
00:50:49.920 --> 00:50:52.490
That comes at the
next stage where
00:50:52.490 --> 00:50:55.660
we make the so-called
assumption of molecular chaos.
00:51:08.390 --> 00:51:12.990
The assumption is
that what's the chance
00:51:12.990 --> 00:51:16.512
that I have a particle
here and a particle there?
00:51:16.512 --> 00:51:17.970
You would say, it's
a chance that I
00:51:17.970 --> 00:51:20.985
have one here and one there.
00:51:20.985 --> 00:51:34.070
You say that if two of
any P1, P2, Q1, Q2, t
00:51:34.070 --> 00:51:45.760
is the same thing as the product
of F1, P1, Q1, t, F1, P2, Q2,
00:51:45.760 --> 00:51:46.260
t.
00:51:50.981 --> 00:51:55.940
Of course, this assumption
is generally varied.
00:51:55.940 --> 00:51:58.700
If I were to look
at the probability
00:51:58.700 --> 00:52:02.750
that I have two particles
as a function of, let's say,
00:52:02.750 --> 00:52:08.140
the relative
separation, I certainly
00:52:08.140 --> 00:52:12.040
expect that if
they are far away,
00:52:12.040 --> 00:52:16.290
the density should be the
product of the one particle
00:52:16.290 --> 00:52:16.790
densities.
00:52:19.990 --> 00:52:24.500
But you would say that if the
two particles come to distances
00:52:24.500 --> 00:52:29.510
that are closer than
their separation d,
00:52:29.510 --> 00:52:32.800
then the probability and
the range of interaction d--
00:52:32.800 --> 00:52:34.490
and let's say the
interaction is highly
00:52:34.490 --> 00:52:40.660
repulsive like hardcore-- then
the probability should go to 0.
00:52:40.660 --> 00:52:44.360
Clearly, you can
make this assumption,
00:52:44.360 --> 00:52:47.560
but up to some degree.
00:52:47.560 --> 00:52:51.220
Part of the reason we
went through this process
00:52:51.220 --> 00:52:57.520
was to indeed make sure that
we are integrating things
00:52:57.520 --> 00:53:00.500
at the locations where
the particles are
00:53:00.500 --> 00:53:03.000
far away from each other.
00:53:03.000 --> 00:53:06.630
I said that the range of
that integration over A
00:53:06.630 --> 00:53:08.640
would be someplace
where they are
00:53:08.640 --> 00:53:11.720
far apart after the
collision, and far apart
00:53:11.720 --> 00:53:12.680
before the collision.
00:53:15.290 --> 00:53:19.300
You have an
assumption like that,
00:53:19.300 --> 00:53:21.700
which is, in
principle, something
00:53:21.700 --> 00:53:23.670
that I can insert into that.
00:53:27.140 --> 00:53:30.940
Having to make a distinction
between the arguments that
00:53:30.940 --> 00:53:37.570
are appearing in this equation
is kind of not so pleasant.
00:53:37.570 --> 00:53:41.070
What you are going to do is
to make another assumption.
00:53:41.070 --> 00:53:45.296
Make sure that everything is
evaluated at the same point.
00:53:53.610 --> 00:53:59.850
What we will eventually now have
is the equation that d by dt,
00:53:59.850 --> 00:54:11.110
plus P1 over n, d by dQ1,
plus F1, dot, d by dP1,
00:54:11.110 --> 00:54:19.010
acting on F1, on
the left hand side,
00:54:19.010 --> 00:54:24.380
is, on the right hand side,
equal to all collisions
00:54:24.380 --> 00:54:29.245
in the particle of
momentum P2, approaching
00:54:29.245 --> 00:54:34.350
at all possible cross
sections, calculating
00:54:34.350 --> 00:54:36.870
the flux of the
incoming particle
00:54:36.870 --> 00:54:38.880
that corresponds to
that channel, which
00:54:38.880 --> 00:54:42.550
is proportional to V2 minus V1.
00:54:42.550 --> 00:54:49.460
Then here, we subtract the
collision of the two particles.
00:54:49.460 --> 00:54:55.050
We write that as F1 of
P1 at this location,
00:54:55.050 --> 00:55:06.390
Q1, t, F1 of t2 at the
same location Q1, t.
00:55:06.390 --> 00:55:21.610
Then add F1 prime, P1 prime, Q1
t, F1 prime, P2 prime, Q2, t.
00:55:24.730 --> 00:55:27.850
In order to make the
equation eventually
00:55:27.850 --> 00:55:38.070
manageable, what you
did is to evaluate all
00:55:38.070 --> 00:55:41.200
off the coordinates that
we have on the right hand
00:55:41.200 --> 00:55:46.680
side at the same location, which
is the same Q1 that you specify
00:55:46.680 --> 00:55:48.950
on the left hand side.
00:55:48.950 --> 00:55:53.260
That immediately means
that what you have done
00:55:53.260 --> 00:55:56.280
is you have changed the
resolution with which you
00:55:56.280 --> 00:55:58.060
are looking at space.
00:55:58.060 --> 00:56:00.250
You have kind of washed
out the difference
00:56:00.250 --> 00:56:03.080
between here and here.
00:56:03.080 --> 00:56:08.100
Your resolution has to
put this whole area that
00:56:08.100 --> 00:56:12.740
is of the order of d squared
or d cubed in three dimensions
00:56:12.740 --> 00:56:14.320
into one pixel.
00:56:14.320 --> 00:56:17.630
You have changed the
resolution that you have.
00:56:17.630 --> 00:56:20.340
You are not looking at things
at this [? fine ?] [? state. ?]
00:56:23.330 --> 00:56:27.370
You are losing additional
information here
00:56:27.370 --> 00:56:31.670
through this change of
the resolution in space.
00:56:31.670 --> 00:56:34.980
You have also lost
some information
00:56:34.980 --> 00:56:39.960
in making the assumption that
the two [? point ?] densities
00:56:39.960 --> 00:56:44.090
are completely within always
as the product one particle
00:56:44.090 --> 00:56:45.530
densities.
00:56:45.530 --> 00:56:49.680
Both of those things
correspond to taking something
00:56:49.680 --> 00:56:52.730
that is very precise
and deterministic,
00:56:52.730 --> 00:56:57.380
and making it kind of vague
and a little undefined.
00:56:57.380 --> 00:57:01.930
It's not surprising then,
that if you have in some sense
00:57:01.930 --> 00:57:05.450
changed the precision of
your computer-- let's say,
00:57:05.450 --> 00:57:08.450
that is running the particles
forward-- at some point,
00:57:08.450 --> 00:57:10.920
you've changed the resolution.
00:57:10.920 --> 00:57:13.320
Then you can't
really run backward.
00:57:13.320 --> 00:57:17.180
In fact, to sort of precisely
be able to run the equations
00:57:17.180 --> 00:57:19.430
forward and backward,
you would need
00:57:19.430 --> 00:57:22.460
to keep resolution
at all levels.
00:57:22.460 --> 00:57:26.270
Here, we have sort of removed
some amount of resolution.
00:57:26.270 --> 00:57:29.000
We have a very good guess
that the equation that you
00:57:29.000 --> 00:57:33.510
have over here no longer
respects time reversal
00:57:33.510 --> 00:57:37.680
inversions that you
had originally posed.
00:57:37.680 --> 00:57:42.840
Our next task is to prove
that you need this equation.
00:57:42.840 --> 00:57:46.000
It goes in one particular
direction in time,
00:57:46.000 --> 00:57:49.730
and cannot be drawn
backward, as opposed to all
00:57:49.730 --> 00:57:55.340
of the predecessors that I
had written up to this point.
00:57:55.340 --> 00:57:56.480
Are there any questions?
00:57:59.714 --> 00:58:00.682
AUDIENCE: [INAUDIBLE]
00:58:08.105 --> 00:58:13.532
PROFESSOR: Yes, Q prime
and Q1, not Q1 prime.
00:58:13.532 --> 00:58:14.240
There is no dash.
00:58:14.240 --> 00:58:15.170
AUDIENCE: Oh, I see.
00:58:15.170 --> 00:58:16.425
It is Q1.
00:58:16.425 --> 00:58:18.750
PROFESSOR: Yes, it is.
00:58:18.750 --> 00:58:20.060
Look at this equation.
00:58:20.060 --> 00:58:22.460
On the left hand side,
what are the arguments?
00:58:22.460 --> 00:58:28.110
The arguments are P1 and Q1.
00:58:28.110 --> 00:58:30.730
What is it that I have
on the other side?
00:58:30.730 --> 00:58:32.800
I still have P1 and Q1.
00:58:32.800 --> 00:58:36.350
I have introduced
P1 and b, which
00:58:36.350 --> 00:58:40.490
is simply an impact parameter.
00:58:40.490 --> 00:58:42.590
What I will do is
I will evaluate
00:58:42.590 --> 00:58:48.110
all of these things, always
at the same location, Q1.
00:58:48.110 --> 00:58:50.650
Then I have P1 and P2.
00:58:50.650 --> 00:58:56.460
That's part of my story of
the change in resolution.
00:58:56.460 --> 00:58:59.910
When I write here Q1,
and you say Q1 prime,
00:58:59.910 --> 00:59:01.330
but what is Q1 prime?
00:59:01.330 --> 00:59:02.745
Is it Q1 plus b?
00:59:02.745 --> 00:59:03.860
Is it Q1 minus b?
00:59:03.860 --> 00:59:06.020
Something like this
I'm going to ignore.
00:59:09.800 --> 00:59:11.520
It's also legitimate,
and you should
00:59:11.520 --> 00:59:16.027
ask, what is P1
prime and Q2 prime?
00:59:16.027 --> 00:59:16.610
What are they?
00:59:20.120 --> 00:59:26.780
What I have to do, is I
have to run on the computer
00:59:26.780 --> 00:59:29.300
or otherwise, the
equations for what
00:59:29.300 --> 00:59:35.430
happens if I have P1
and P2 come together
00:59:35.430 --> 00:59:38.604
at an impact parameter
that is set by me.
00:59:38.604 --> 00:59:41.430
I then integrate
the equations, and I
00:59:41.430 --> 00:59:44.760
find that deterministically,
that collision will
00:59:44.760 --> 00:59:48.230
lead to some P1
prime and P2 prime.
00:59:48.230 --> 00:59:59.590
P1 prime and P2 prime are
some complicated functions
00:59:59.590 --> 01:00:02.760
of P1, P2, and b.
01:00:05.560 --> 01:00:10.040
Given that you know two
particles are approaching
01:00:10.040 --> 01:00:16.940
each other at distance d with
momenta P1 P2, in principle,
01:00:16.940 --> 01:00:20.300
you can integrate
Newton's equations,
01:00:20.300 --> 01:00:22.790
and figure out with what
momenta they end up.
01:00:25.600 --> 01:00:30.700
This equation, in fact, hides a
very, very complicated function
01:00:30.700 --> 01:00:33.790
here, which describes
P1 prime and P2 prime
01:00:33.790 --> 01:00:36.760
as a function of P1 and P2.
01:00:36.760 --> 01:00:40.815
If you really needed all of
the details of that function,
01:00:40.815 --> 01:00:44.360
you would surely be in trouble.
01:00:44.360 --> 01:00:45.620
Fortunately, we don't.
01:00:45.620 --> 01:00:50.700
As we shall see shortly, you
can kind of get a lot of mileage
01:00:50.700 --> 01:00:51.630
without knowing that.
01:00:51.630 --> 01:00:52.755
Yes, what is your question?
01:00:52.755 --> 01:00:54.622
AUDIENCE: There
was an assumption
01:00:54.622 --> 01:00:57.210
that all the interactions
between different molecules
01:00:57.210 --> 01:00:59.770
are central potentials
[INAUDIBLE].
01:00:59.770 --> 01:01:02.850
Does the force of the
direction between two particles
01:01:02.850 --> 01:01:04.990
lie along the [INAUDIBLE]?
01:01:04.990 --> 01:01:07.760
PROFESSOR: For the things that
I have written, yes it does.
01:01:07.760 --> 01:01:10.160
I should have been more precise.
01:01:10.160 --> 01:01:14.950
I should have put
absolute value here.
01:01:19.054 --> 01:01:20.740
AUDIENCE: You have
particles moving
01:01:20.740 --> 01:01:23.368
along one line
towards each other,
01:01:23.368 --> 01:01:25.326
and b is some arbitrary vector.
01:01:25.326 --> 01:01:27.990
You have two directions,
so you define a plane.
01:01:27.990 --> 01:01:31.460
Opposite direction particles
stay at the same plane.
01:01:31.460 --> 01:01:34.040
Have you reduced--
01:01:34.040 --> 01:01:36.426
PROFESSOR: Particles
stay in the same plane?
01:01:36.426 --> 01:01:42.280
AUDIENCE: If the two particles
were moving towards each other,
01:01:42.280 --> 01:01:45.080
and also you have
in the integral
01:01:45.080 --> 01:01:50.040
your input parameter,
which one is [INAUDIBLE].
01:01:50.040 --> 01:01:54.000
There's two directions.
01:01:54.000 --> 01:01:55.880
All particles align,
and all b's align.
01:01:55.880 --> 01:01:57.039
They form a plane.
01:01:57.039 --> 01:01:59.330
[? Opposite ?] direction
particles [? stand ?] in the--
01:01:59.330 --> 01:02:02.690
PROFESSOR: Yes, they
stand in the same plane.
01:02:02.690 --> 01:02:04.362
AUDIENCE: My
question is, what is
01:02:04.362 --> 01:02:06.070
[INAUDIBLE] use the
integral on the right
01:02:06.070 --> 01:02:08.930
from a two-dimensional
integral [? in v ?]
01:02:08.930 --> 01:02:12.170
into employing central symmetry?
01:02:12.170 --> 01:02:14.170
PROFESSOR: Yes, you could.
01:02:14.170 --> 01:02:19.901
You could, in principle, write
this as b db, if you like,
01:02:19.901 --> 01:02:20.900
if that's what you want.
01:02:24.390 --> 01:02:27.414
AUDIENCE: [INAUDIBLE]
01:02:27.414 --> 01:02:29.080
PROFESSOR: Yes, you
could do that if you
01:02:29.080 --> 01:02:30.945
have simple enough potential.
01:02:39.050 --> 01:02:47.180
Let's show that this equation
leads to irreversibility.
01:02:47.180 --> 01:02:49.020
That you are going to do here.
01:03:09.440 --> 01:03:12.835
This, by the way, is called
the Boltzmann equation.
01:03:18.880 --> 01:03:25.064
There's an associated
Boltzmann H-Theorem,
01:03:25.064 --> 01:03:38.400
which restates the following--
If F of P1, Q1, and t
01:03:38.400 --> 01:03:53.200
satisfies the above
Boltzmann equation,
01:03:53.200 --> 01:04:08.540
then there is a quantity H that
always decreases in time, where
01:04:08.540 --> 01:04:20.350
H is the integral over P
and Q of F1, log of F1.
01:04:26.340 --> 01:04:28.330
The composition of
irreversibility,
01:04:28.330 --> 01:04:30.940
as we saw in thermal
dynamics, was
01:04:30.940 --> 01:04:33.490
that there was a
quantity entropy that
01:04:33.490 --> 01:04:34.980
was always increasing.
01:04:34.980 --> 01:04:37.560
If you have calculated
for this system,
01:04:37.560 --> 01:04:41.700
entropy before for the half
box, and entropy afterwards
01:04:41.700 --> 01:04:45.490
for the space both boxes
occupy, the second one
01:04:45.490 --> 01:04:47.066
would certainly be larger.
01:04:50.030 --> 01:04:52.900
This H is a quantity like
that, except that when
01:04:52.900 --> 01:04:55.720
it is defined this
way, it always
01:04:55.720 --> 01:04:58.500
decreases as a function of time.
01:04:58.500 --> 01:05:01.605
But it certainly is very
much related to entropy.
01:05:04.330 --> 01:05:09.080
You may have asked,
why did Boltzmann
01:05:09.080 --> 01:05:11.110
come across such
a function, which
01:05:11.110 --> 01:05:15.790
is F log F, except that
actually right now,
01:05:15.790 --> 01:05:19.860
you should know
why you write this.
01:05:19.860 --> 01:05:22.020
When we were dealing
with probabilities,
01:05:22.020 --> 01:05:26.060
we introduced the entropy of the
probability distribution, which
01:05:26.060 --> 01:05:31.460
was related to something
like sum over iPi, log of Pi,
01:05:31.460 --> 01:05:34.900
with a minus sign.
01:05:34.900 --> 01:05:38.170
Up to this factor
of normalization N,
01:05:38.170 --> 01:05:42.610
this F1 really is a
one-particle probability.
01:05:42.610 --> 01:05:45.600
After this normalization
N, you have
01:05:45.600 --> 01:05:49.600
a one-particle probability,
the probability
01:05:49.600 --> 01:05:54.150
that you have occupation
of one-particle free space.
01:05:54.150 --> 01:05:57.380
This occupation of
one-particle phase space
01:05:57.380 --> 01:06:01.550
is changing as a
function of time.
01:06:01.550 --> 01:06:06.990
What this statement says is
that if the one-particle density
01:06:06.990 --> 01:06:09.790
evolves in time according
to this equation,
01:06:09.790 --> 01:06:12.880
the corresponding
minus entropy decreases
01:06:12.880 --> 01:06:15.970
as a function of time.
01:06:15.970 --> 01:06:19.230
Let's see if that's the case.
01:06:19.230 --> 01:06:24.340
To prove that, let's do this.
01:06:24.340 --> 01:06:30.810
We have the formula for H, so
let's calculate the H by dt.
01:06:30.810 --> 01:06:34.500
I have an integral
over the phase space
01:06:34.500 --> 01:06:44.520
of particle one, the particle
that I just called one.
01:06:44.520 --> 01:06:47.570
I could have
labeled it anything.
01:06:47.570 --> 01:06:52.060
After integration, H is
only a function of time.
01:06:52.060 --> 01:06:54.050
I have to take the
time derivative.
01:06:54.050 --> 01:06:57.950
The time derivative
can act on F1.
01:06:57.950 --> 01:07:04.010
Then I will get the F1
by dt, times log F1.
01:07:04.010 --> 01:07:10.870
Or I will have F1 times
the derivative of log F1.
01:07:10.870 --> 01:07:15.870
The derivative of log F1
would be dF1 by dt, and then 1
01:07:15.870 --> 01:07:16.720
over F1.
01:07:16.720 --> 01:07:18.420
Then I multiply by F1.
01:07:21.260 --> 01:07:22.910
This term is simply 1.
01:07:26.815 --> 01:07:29.180
AUDIENCE: Don't you want to
write the full derivative,
01:07:29.180 --> 01:07:31.630
F1 with respect [INAUDIBLE]?
01:07:31.630 --> 01:07:35.060
PROFESSOR: I thought we
did that with this before.
01:07:35.060 --> 01:07:40.560
If you have something that
I am summing over lots
01:07:40.560 --> 01:07:46.420
of [? points, ?] and these
[? points ?] can be positioned,
01:07:46.420 --> 01:07:50.040
then I have S at location
one, S at location two,
01:07:50.040 --> 01:07:56.230
S at location three,
discretized versions of x.
01:07:56.230 --> 01:07:58.460
If I take the time
derivative, I take
01:07:58.460 --> 01:08:00.480
the time derivative
of this, plus this,
01:08:00.480 --> 01:08:03.285
plus this, which are
partial derivatives.
01:08:14.940 --> 01:08:18.069
If I actually take the
time derivative here,
01:08:18.069 --> 01:08:24.950
I get the integral d cubed P1,
d cubed Q1, the time derivative.
01:08:24.950 --> 01:08:28.310
This would be that
partial dF1 by dt
01:08:28.310 --> 01:08:31.430
is the time derivative
of n, which is 0.
01:08:31.430 --> 01:08:34.479
The number of particles
does not change.
01:08:34.479 --> 01:08:38.970
Indeed, I realize that 1
integrated against dF1 by dt
01:08:38.970 --> 01:08:41.640
is the same thing that's here.
01:08:41.640 --> 01:08:43.890
This term gives you 0.
01:08:43.890 --> 01:08:47.149
All I need to worry
about is integrating
01:08:47.149 --> 01:08:51.460
log F against the Fydt.
01:08:51.460 --> 01:09:01.520
I have an integral over P1 and
Q1 of log F against the Fydt.
01:09:01.520 --> 01:09:10.389
We have said that F1 satisfies
the Boltzmann equation.
01:09:14.844 --> 01:09:21.149
So the F1 by dt, if I
were to rearrange it,
01:09:21.149 --> 01:09:25.370
I have the F1 by dt.
01:09:25.370 --> 01:09:29.899
I take this part to the
other side of the equation.
01:09:29.899 --> 01:09:34.830
This part is also
the Poisson bracket
01:09:34.830 --> 01:09:40.300
of a one-particle H with F1.
01:09:40.300 --> 01:09:42.620
If I take it to
the other side, it
01:09:42.620 --> 01:09:47.399
will be the Poisson
bracket of H with F1.
01:09:47.399 --> 01:09:49.080
Then there is this
whole thing that
01:09:49.080 --> 01:09:52.050
involves the collision
of two particles.
01:09:52.050 --> 01:09:56.550
So I define whatever is
on the right hand side
01:09:56.550 --> 01:10:02.060
to be some collision
operator that acts on two
01:10:02.060 --> 01:10:04.160
[? powers ?] of F1.
01:10:04.160 --> 01:10:11.060
This is plus a collision
operator, F1, F1.
01:10:11.060 --> 01:10:16.130
What I do is I
replace this dF1 by dt
01:10:16.130 --> 01:10:24.850
with the Poisson bracket of H,
or H1, if you like, with F1.
01:10:24.850 --> 01:10:28.340
The collision operator I will
shortly write explicitly.
01:10:28.340 --> 01:10:32.502
But for the time being, let
me just write it as C of F1.
01:10:42.460 --> 01:10:44.450
There is a first
term in this sum--
01:10:44.450 --> 01:10:50.860
let's call it number one--
which I claim to be 0.
01:10:50.860 --> 01:10:53.590
Typically, when you
get these integrations
01:10:53.590 --> 01:10:56.070
with Poisson brackets,
you would get 0.
01:10:56.070 --> 01:10:58.190
Let's explicitly show that.
01:10:58.190 --> 01:11:07.240
I have an integral over
P1 and Q1 of log of F1,
01:11:07.240 --> 01:11:11.330
and this Poisson
bracket of H1 and F1,
01:11:11.330 --> 01:11:14.220
which is essentially
these terms.
01:11:14.220 --> 01:11:27.110
Alternatively, I could write
it as dH1 by dQ1, dF1 by dt1,
01:11:27.110 --> 01:11:34.310
minus the H1 by
dt1, dF1, by dQ1.
01:11:37.770 --> 01:11:43.090
I've explicitly written this
form for the one-particle
01:11:43.090 --> 01:11:44.380
in terms of the Hamiltonian.
01:11:47.230 --> 01:11:49.030
The advantage of
that is that now I
01:11:49.030 --> 01:11:51.803
can start doing
integrations by parts.
01:11:55.830 --> 01:12:01.550
I'm taking derivatives
with respect to P,
01:12:01.550 --> 01:12:05.680
but I have integrations
with respect to P here.
01:12:05.680 --> 01:12:08.930
I could take the F1 out.
01:12:08.930 --> 01:12:10.820
I will have a minus.
01:12:10.820 --> 01:12:15.530
I have an integral, P1, Q1.
01:12:15.530 --> 01:12:18.100
I took F1 out.
01:12:18.100 --> 01:12:22.940
Then this d by dP1 acts on
everything that came before it.
01:12:22.940 --> 01:12:25.610
It can act on the H1.
01:12:25.610 --> 01:12:33.500
I would get d2 H1 with
respect to dP1, dQ1.
01:12:33.500 --> 01:12:38.490
Or it could act on the
log of F1, in which case
01:12:38.490 --> 01:12:43.380
I will get set dH1 by dQ1.
01:12:47.890 --> 01:12:51.900
Then I would have d
by dP acting on log
01:12:51.900 --> 01:13:00.990
of F, which would
give me dF1 by dP1,
01:13:00.990 --> 01:13:04.018
then the derivative of the
log, which is 1 over F1.
01:13:07.810 --> 01:13:10.300
This is only the first term.
01:13:10.300 --> 01:13:16.822
I also have this term, with
which I will do the same thing.
01:13:16.822 --> 01:13:22.553
AUDIENCE: [INAUDIBLE] The
second derivative [INAUDIBLE]
01:13:22.553 --> 01:13:27.598
should be multiplied
by log of F.
01:13:27.598 --> 01:13:28.806
PROFESSOR: Yes, it should be.
01:13:33.505 --> 01:13:34.540
It is Log F1.
01:13:38.180 --> 01:13:38.920
Thank you.
01:13:41.780 --> 01:13:46.160
For the next term, I have F1.
01:13:46.160 --> 01:13:53.620
I have d2 H1, and the other
order of derivatives, dQ1, dP1.
01:13:53.620 --> 01:13:57.720
Now I'll make sure I
write down the log of F1.
01:13:57.720 --> 01:14:04.510
Then I have dH1
with respect to dQ1.
01:14:13.110 --> 01:14:18.100
Then I have a dot product with
the derivative of log F, which
01:14:18.100 --> 01:14:24.008
is the derivative of F1 with
respect to Q1 and 1 over F1.
01:14:31.660 --> 01:14:33.400
Here are the terms
that are proportional
01:14:33.400 --> 01:14:34.680
to the second derivative.
01:14:34.680 --> 01:14:37.310
The order of the
derivatives does not matter.
01:14:37.310 --> 01:14:38.760
One often is positive.
01:14:38.760 --> 01:14:43.010
One often is negative,
so they cancel out.
01:14:43.010 --> 01:14:44.910
Then I have these
additional terms.
01:14:44.910 --> 01:14:47.190
For the additional
terms, you'll note
01:14:47.190 --> 01:14:50.480
that the F1 and the
1 over F1 cancels.
01:14:54.770 --> 01:14:58.960
These are just a product
of two first derivatives.
01:14:58.960 --> 01:15:06.180
I will apply the five
parts process one more time
01:15:06.180 --> 01:15:11.760
to get rid of the derivative
that is acting on F1.
01:15:11.760 --> 01:15:17.150
The answer becomes plus
d cubed P1, d cubed Q1.
01:15:17.150 --> 01:15:29.160
Then I have F1, d2 H1, dP1,
dQ1, minus d2 H1, dQ1, dP1.
01:15:29.160 --> 01:15:34.202
These two cancel each other
out, and the answer is 0.
01:15:40.110 --> 01:15:43.560
So that first term vanishes.
01:15:43.560 --> 01:15:55.816
Now for the second term,
number two, what I have
01:15:55.816 --> 01:15:56.940
is the first term vanished.
01:15:56.940 --> 01:16:00.080
So I have the H by dt.
01:16:00.080 --> 01:16:06.260
It is the integral
over P1 and Q1.
01:16:09.490 --> 01:16:17.660
I have log of F1.
01:16:17.660 --> 01:16:22.060
F1 is a function of
P1, and Q1, and t.
01:16:22.060 --> 01:16:26.760
I will focus, and make sure I
write the argument of momentum,
01:16:26.760 --> 01:16:31.710
for reasons that will
become shortly apparent.
01:16:31.710 --> 01:16:35.180
I have to multiply with
the collision term.
01:16:35.180 --> 01:16:39.520
The collision term
involves integrations
01:16:39.520 --> 01:16:46.360
over a second particle,
over an impact parameter,
01:16:46.360 --> 01:16:51.450
a relative velocity, once I
have defined what P2 and P1 are.
01:16:51.450 --> 01:16:57.340
I have a subtraction
of F evaluated at P1,
01:16:57.340 --> 01:17:01.760
F evaluated at
P2, plus addition,
01:17:01.760 --> 01:17:06.705
F evaluated at P1 prime,
F evaluated at P2 prime.
01:17:14.430 --> 01:17:18.700
Eventually, this whole thing
is only a function of time.
01:17:18.700 --> 01:17:22.310
There are a whole bunch of
arguments appearing here,
01:17:22.310 --> 01:17:25.430
but all of those arguments
are being integrated over.
01:17:29.230 --> 01:17:35.500
In particular, I have arguments
that are indexed by P1 and P2.
01:17:35.500 --> 01:17:38.930
These are dummy
variables of integration.
01:17:38.930 --> 01:17:41.750
If I have a function
of x and y that I'm
01:17:41.750 --> 01:17:45.110
integrating over x and
y, I can call x "z."
01:17:45.110 --> 01:17:47.060
I can call y "t."
01:17:47.060 --> 01:17:48.990
I would integrate
over z and t, and I
01:17:48.990 --> 01:17:51.310
would have the same answer.
01:17:51.310 --> 01:17:54.280
I would have exactly
the same answer
01:17:54.280 --> 01:17:58.650
if I were to call all of the
dummy integration variable that
01:17:58.650 --> 01:18:01.700
is indexed 1, "2."
01:18:01.700 --> 01:18:04.420
Any dummy variable
that is indexed 2,
01:18:04.420 --> 01:18:10.130
if I rename it and call it 1,
the integral would not change.
01:18:10.130 --> 01:18:11.900
If I do that, what do I have?
01:18:11.900 --> 01:18:14.920
I have integral
over Q-- actually,
01:18:14.920 --> 01:18:19.030
let's get of the
integration number on Q.
01:18:19.030 --> 01:18:21.350
It really doesn't matter.
01:18:21.350 --> 01:18:24.840
I have the integrals
over P1 and P1.
01:18:24.840 --> 01:18:29.560
I have to integrate over
both sets of momenta.
01:18:29.560 --> 01:18:33.730
I have to integrate over
the cross section, which
01:18:33.730 --> 01:18:37.490
is relative between 1 and 2.
01:18:37.490 --> 01:18:43.840
I have V2 minus V1, rather
than V1 minus V2, rather than
01:18:43.840 --> 01:18:45.550
V2 minus V1.
01:18:45.550 --> 01:18:47.620
The absolute value
doesn't matter.
01:18:47.620 --> 01:18:52.970
If I were to replace these
indices with an absolute value,
01:18:52.970 --> 01:18:56.750
[? or do a ?] V2 minus V1
goes to minus V1 minus V2.
01:18:56.750 --> 01:18:59.400
The absolute value
does not change.
01:18:59.400 --> 01:19:01.250
Here, what do I have?
01:19:01.250 --> 01:19:04.720
I have minus F of P1.
01:19:04.720 --> 01:19:11.550
It becomes F of P2, F of
P1, plus F of P2 prime,
01:19:11.550 --> 01:19:13.640
f of P1 prime.
01:19:13.640 --> 01:19:14.530
They are a product.
01:19:14.530 --> 01:19:18.450
It doesn't really matter in
which order I write them.
01:19:18.450 --> 01:19:20.080
The only thing
that really matters
01:19:20.080 --> 01:19:26.930
is that the argument was
previously called F1 of P1
01:19:26.930 --> 01:19:31.870
for the log, and now it
will be called F1 of P2.
01:19:31.870 --> 01:19:32.965
Just its name changed.
01:19:37.150 --> 01:19:39.980
If I take this, and the
first way of writing things,
01:19:39.980 --> 01:19:43.350
which are really two ways of
writing the same integral,
01:19:43.350 --> 01:19:49.044
and just average them, I will
get 1/2 an integral d cubed Q,
01:19:49.044 --> 01:19:56.350
d cubed P1, d cubed P2,
d2 b, and V2 minus V1.
01:19:56.350 --> 01:20:09.100
I will have F1 of P1, F1 of
P2, plus F1 of P1 prime, F1
01:20:09.100 --> 01:20:11.630
of P2 prime.
01:20:11.630 --> 01:20:17.360
Then in one term, I
had log of F1 of P1,
01:20:17.360 --> 01:20:19.180
and I averaged it
with the other way
01:20:19.180 --> 01:20:24.630
of writing things, which was
log of F-- let's put the two
01:20:24.630 --> 01:20:28.320
logs together, multiplied by F1.
01:20:28.320 --> 01:20:30.400
So the sum of the
two logs I wrote,
01:20:30.400 --> 01:20:32.540
that's a log of the product.
01:20:32.540 --> 01:20:34.770
I just rewrote that equation.
01:20:34.770 --> 01:20:39.100
If you like, I symmetrized It
with respect to index 1 and 2.
01:20:39.100 --> 01:20:42.222
So the log of 1,
that previously had
01:20:42.222 --> 01:20:43.930
one argument through
this symmetrization,
01:20:43.930 --> 01:20:48.860
became one half
of the sum of it.
01:20:48.860 --> 01:20:55.790
The next thing one has to
think about, what I want to do,
01:20:55.790 --> 01:20:59.280
is to replace primed and
unprimed coordinates.
01:21:04.280 --> 01:21:07.570
What I would
eventually write down
01:21:07.570 --> 01:21:22.290
is d cubed P1 prime, d cubed P2
prime, d2 b, V2 prime minus V1
01:21:22.290 --> 01:21:33.760
prime, minus F1 of P1 prime,
F1 of P2 prime, plus F1 of P1,
01:21:33.760 --> 01:21:35.790
F1 of P2.
01:21:35.790 --> 01:21:42.308
Then log of F1 of P1
prime, F1 of P2 prime.
01:21:47.100 --> 01:21:52.050
I've symmetrized originally
the indices 1 and 2
01:21:52.050 --> 01:21:55.420
that were not quite
symmetric, and I end up
01:21:55.420 --> 01:22:00.210
with an expression that has
variables P1, P2, and functions
01:22:00.210 --> 01:22:04.170
P1 prime and P2 prime, which
are not quite symmetric again,
01:22:04.170 --> 01:22:11.360
because I have F's evaluated
for P's, but not for P primes.
01:22:11.360 --> 01:22:14.260
What does this mean?
01:22:14.260 --> 01:22:17.620
This mathematical expression
that I have written down here
01:22:17.620 --> 01:22:24.010
actually is not correct,
because what this amounts to,
01:22:24.010 --> 01:22:30.520
is to change variables
of integration.
01:22:30.520 --> 01:22:33.340
In the expression
that I have up here,
01:22:33.340 --> 01:22:37.910
P1 and P2 are variables
of integration.
01:22:37.910 --> 01:22:41.700
P1 prime and P2 prime are
some complicated functions
01:22:41.700 --> 01:22:43.770
of P1 and P2.
01:22:43.770 --> 01:22:47.730
P1 prime is some complicated
function that I don't know.
01:22:47.730 --> 01:22:52.360
P1, P2, and V, for which I
need to solve in principle,
01:22:52.360 --> 01:22:55.060
is Newton's equation.
01:22:55.060 --> 01:22:58.250
This is similarly for P2 prime.
01:22:58.250 --> 01:23:00.380
What I have done
is I have changed
01:23:00.380 --> 01:23:06.000
from my original variables
to these functions.
01:23:06.000 --> 01:23:11.620
When I write things over here,
now P1 prime and P2 prime
01:23:11.620 --> 01:23:14.365
are the integration variables.
01:23:14.365 --> 01:23:16.680
P1 and P2 are supposed
to be regarded
01:23:16.680 --> 01:23:20.810
as functions of P1
prime and P2 prime.
01:23:20.810 --> 01:23:22.770
You say, well, what
does that mean?
01:23:22.770 --> 01:23:26.790
You can't simply
take an integral dx,
01:23:26.790 --> 01:23:34.096
let's say F of some function of
x, and replace this function.
01:23:34.096 --> 01:23:35.470
You can't call it
a new variable,
01:23:35.470 --> 01:23:38.100
and do integral dx prime.
01:23:38.100 --> 01:23:43.020
You have to multiply with the
Jacobian of the transformation
01:23:43.020 --> 01:23:49.140
that takes you from the P
variables to the new variables.
01:23:55.680 --> 01:23:58.550
My claim is that this
Jacobian of the integration
01:23:58.550 --> 01:23:59.917
is, in fact, the unit.
01:24:03.110 --> 01:24:06.310
The reason is as follows.
01:24:06.310 --> 01:24:11.010
These equations that have
to be integrated to give me
01:24:11.010 --> 01:24:15.260
the correlation are
time reversible.
01:24:15.260 --> 01:24:20.020
If I give you two momenta, and
I know what the outcomes are,
01:24:20.020 --> 01:24:22.350
I can write the
equations backward,
01:24:22.350 --> 01:24:25.530
and I will have the
opposite momenta go back
01:24:25.530 --> 01:24:28.510
to minus the original momenta.
01:24:28.510 --> 01:24:33.400
Up to a factor of minus, you
can see that this equation has
01:24:33.400 --> 01:24:38.660
this character, that P1, P2
go to P1 prime, P2 prime, then
01:24:38.660 --> 01:24:44.850
minus P1 prime, minus P2
prime, go to P1, and P2.
01:24:44.850 --> 01:24:47.480
If you sort of
follow that, and say
01:24:47.480 --> 01:24:50.770
that you do the
transformation twice,
01:24:50.770 --> 01:24:53.840
you have to get back up
to where a sign actually
01:24:53.840 --> 01:24:55.800
disappears to where you want.
01:24:55.800 --> 01:24:58.820
You have to multiply
by two Jacobians,
01:24:58.820 --> 01:25:01.560
and you get the same unit.
01:25:01.560 --> 01:25:05.730
You can convince yourself that
this Jacobian has to be unit.
01:25:05.730 --> 01:25:08.340
Next time, I guess we'll
take it from there.
01:25:08.340 --> 01:25:11.180
I will explain this
stuff a little bit more,
01:25:11.180 --> 01:25:15.810
and show that this implies what
we had said about the Boltzmann
01:25:15.810 --> 01:25:17.084
equation.