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PROFESSOR: OK.
00:00:21.840 --> 00:00:22.463
Let's start.
00:00:26.420 --> 00:00:30.000
So last time, we started
with kinetic theory.
00:00:36.120 --> 00:00:39.205
And we will focus for
gas systems mostly.
00:00:43.890 --> 00:00:45.730
And the question
that we would like
00:00:45.730 --> 00:00:50.140
to think about and answer
somehow is the following.
00:00:50.140 --> 00:00:56.260
You start with a gas
that is initially
00:00:56.260 --> 00:00:57.765
confined to one chamber.
00:01:01.400 --> 00:01:06.180
And you can calculate all of
its thermodynamic properties.
00:01:06.180 --> 00:01:11.560
You open at time 0 a
hole, allowing the gas
00:01:11.560 --> 00:01:17.290
to escape into a second
initially empty chamber.
00:01:17.290 --> 00:01:19.400
And after some time,
the whole system
00:01:19.400 --> 00:01:22.320
will come to a new
equilibrium position.
00:01:22.320 --> 00:01:25.040
It's a pretty reversible thing.
00:01:25.040 --> 00:01:27.620
You can do this experiment
many, many times.
00:01:27.620 --> 00:01:31.690
And you will always get
roughly the same amount of time
00:01:31.690 --> 00:01:34.280
for the situation to
start from one equilibrium
00:01:34.280 --> 00:01:36.820
and reach another equilibrium.
00:01:36.820 --> 00:01:38.250
So how do we describe that?
00:01:38.250 --> 00:01:41.010
It's slightly beyond what
we did in thermodynamics,
00:01:41.010 --> 00:01:43.990
because we want to go
from one equilibrium state
00:01:43.990 --> 00:01:46.910
to another equilibrium state.
00:01:46.910 --> 00:01:50.260
Now, we said, OK, we know
the equations of motion
00:01:50.260 --> 00:01:55.200
that governs the particles
that are described by this.
00:01:55.200 --> 00:02:00.970
So here we can say that we
have, let's say, N particles.
00:02:00.970 --> 00:02:05.930
They have their own
momenta and coordinates.
00:02:05.930 --> 00:02:09.160
And we know that these
momenta and coordinates evolve
00:02:09.160 --> 00:02:12.230
in time, governed
by some Hamiltonian,
00:02:12.230 --> 00:02:17.290
which is a function of all of
these momenta and coordinates.
00:02:17.290 --> 00:02:17.880
OK?
00:02:17.880 --> 00:02:19.270
Fine.
00:02:19.270 --> 00:02:21.400
How do we go from
a situation which
00:02:21.400 --> 00:02:23.100
describes a whole
bunch of things
00:02:23.100 --> 00:02:25.580
and coordinates that
are changing with time
00:02:25.580 --> 00:02:33.400
to some microscopic description
of macroscopic variables going
00:02:33.400 --> 00:02:36.850
from one equilibrium
state to another?
00:02:36.850 --> 00:02:39.740
So our first attempt
in that direction
00:02:39.740 --> 00:02:46.390
was to say, well, I could start
with many, many, many examples
00:02:46.390 --> 00:02:49.260
of the same situation.
00:02:49.260 --> 00:02:52.190
Each one of them would
correspond to a different
00:02:52.190 --> 00:02:55.500
trajectory of p's and q's.
00:02:55.500 --> 00:02:59.300
And so what we can
do is to construct
00:02:59.300 --> 00:03:05.720
some kind of an ensemble
average, or ensemble density,
00:03:05.720 --> 00:03:08.800
first, which is what we did.
00:03:08.800 --> 00:03:15.255
We can say that the very, very
many examples of this situation
00:03:15.255 --> 00:03:20.300
that I can have will
correspond to different points
00:03:20.300 --> 00:03:23.890
at some particular
instant of time,
00:03:23.890 --> 00:03:28.085
occupying this
6N-dimensional phase space,
00:03:28.085 --> 00:03:32.330
out of which I can construct
some kind of a density in phase
00:03:32.330 --> 00:03:34.950
space.
00:03:34.950 --> 00:03:39.320
But then I realize that since
each one of these trajectories
00:03:39.320 --> 00:03:42.490
is evolving according
to this Hamiltonian,
00:03:42.490 --> 00:03:45.320
this density could potentially
be a function of time.
00:03:48.030 --> 00:03:52.710
And we described the
equation for the evolution
00:03:52.710 --> 00:03:54.850
of that density with time.
00:03:54.850 --> 00:03:59.980
And we could write it in
this form-- V rho by dt
00:03:59.980 --> 00:04:05.350
is the Poisson bracket of
the Hamiltonian with rho.
00:04:05.350 --> 00:04:10.120
And this Poisson
bracket was defined
00:04:10.120 --> 00:04:13.650
as a sum over all
of your coordinates.
00:04:16.339 --> 00:04:16.839
Oops.
00:04:22.490 --> 00:04:34.660
We had d rho by d vector qi,
dot product with dH by dpi minus
00:04:34.660 --> 00:04:36.116
the other [? way. ?]
00:04:42.470 --> 00:04:47.740
So there are essentially
three N terms-- well,
00:04:47.740 --> 00:04:52.920
actually, six N terms, but three
N pairs of terms in this sum.
00:04:52.920 --> 00:04:55.890
I can either use
some index alpha
00:04:55.890 --> 00:04:59.460
running from one to
three N, or indicate them
00:04:59.460 --> 00:05:05.100
as contributions of things that
come from individual particles
00:05:05.100 --> 00:05:08.210
and then use this notation
with three vectors.
00:05:08.210 --> 00:05:11.280
So this is essentially
a combination
00:05:11.280 --> 00:05:14.440
of sum of three terms.
00:05:14.440 --> 00:05:16.900
OK?
00:05:16.900 --> 00:05:22.910
So I hope that
somehow this equation
00:05:22.910 --> 00:05:28.420
can describe the evolution
that goes on over here.
00:05:28.420 --> 00:05:32.660
And ultimately, when I wait
sufficiently long time,
00:05:32.660 --> 00:05:36.910
I will reach a situation
where d rho by dt
00:05:36.910 --> 00:05:39.455
does not change anymore.
00:05:39.455 --> 00:05:43.830
And I will find some density
that is invariant on time.
00:05:43.830 --> 00:05:46.650
I'll call that rho equilibrium.
00:05:46.650 --> 00:05:51.810
And so we saw that we could have
our rho equilibrium, which then
00:05:51.810 --> 00:05:58.620
should have zero Poisson bracket
with H to be a function of H
00:05:58.620 --> 00:06:00.140
and any other
conserved quantity.
00:06:05.900 --> 00:06:08.780
OK?
00:06:08.780 --> 00:06:14.582
So in principle,
we sort of thought
00:06:14.582 --> 00:06:21.030
of a way of describing
how the system will evolve
00:06:21.030 --> 00:06:24.660
in equilibrium--
towards an equilibrium.
00:06:24.660 --> 00:06:30.290
And indeed, we will find that in
statistical mechanics later on,
00:06:30.290 --> 00:06:35.340
we are going to rely heavily
on these descriptions
00:06:35.340 --> 00:06:38.370
of equilibrium
densities, which are
00:06:38.370 --> 00:06:41.090
governed by only
the Hamiltonian.
00:06:41.090 --> 00:06:43.850
And if there are any other
conserved quantities,
00:06:43.850 --> 00:06:45.150
typically there are not.
00:06:45.150 --> 00:06:49.650
So this is the general
form that we'll ultimately
00:06:49.650 --> 00:06:52.630
be using a lot.
00:06:52.630 --> 00:06:56.730
But we started
with a description
00:06:56.730 --> 00:06:58.820
of evolution of
these coordinates
00:06:58.820 --> 00:07:01.420
in time, which is
time-reversible.
00:07:05.870 --> 00:07:08.835
And we hope to end up
with a situation that
00:07:08.835 --> 00:07:12.180
is analogous to what we
describe thermodynamically,
00:07:12.180 --> 00:07:18.410
we describe something that
goes to some particular state,
00:07:18.410 --> 00:07:20.230
and basically
stays there, as far
00:07:20.230 --> 00:07:23.620
as the macroscopic
description is concerned.
00:07:23.620 --> 00:07:29.550
So did we somehow manage to just
look at something else, which
00:07:29.550 --> 00:07:32.470
is this density, and
achieve this transition
00:07:32.470 --> 00:07:35.520
from reversibility
to irreversibility?
00:07:35.520 --> 00:07:36.760
And the answer is no.
00:07:36.760 --> 00:07:40.610
This equation-- also,
if you find, indeed,
00:07:40.610 --> 00:07:47.230
a rho that goes from here to
here, you can set t to minus t
00:07:47.230 --> 00:07:49.705
and get a rho that goes
from back here to here.
00:07:49.705 --> 00:07:52.500
It has the same
time-reversibility.
00:07:52.500 --> 00:07:56.400
So somehow, we have to
find another solution.
00:07:56.400 --> 00:08:00.180
And the solution that we
will gradually build upon
00:08:00.180 --> 00:08:03.150
relies more on
physics rather than
00:08:03.150 --> 00:08:06.850
the rigorous mathematical
nature of this thing.
00:08:06.850 --> 00:08:09.310
Physically, we know
that this happens.
00:08:09.310 --> 00:08:11.560
All of us have seen this.
00:08:11.560 --> 00:08:16.026
So somehow, we should be able
to use physical approximation
00:08:16.026 --> 00:08:18.860
and assumptions that
are compatible with what
00:08:18.860 --> 00:08:20.260
is observed.
00:08:20.260 --> 00:08:24.280
And if, during that,
we have to sort of make
00:08:24.280 --> 00:08:27.450
mathematical
approximations, so be it.
00:08:27.450 --> 00:08:30.330
In fact, we have to make
mathematical approximations,
00:08:30.330 --> 00:08:35.620
because otherwise, there is
this very strong reversibility
00:08:35.620 --> 00:08:37.000
condition.
00:08:37.000 --> 00:08:41.030
So let's see how we are about
to proceed, thinking more
00:08:41.030 --> 00:08:43.820
in terms of physics.
00:08:43.820 --> 00:08:47.840
Now, the information
that I have over here,
00:08:47.840 --> 00:08:50.335
either in this
description of ensemble
00:08:50.335 --> 00:08:53.220
or in this description
in terms of trajectories,
00:08:53.220 --> 00:08:55.960
is just enormous.
00:08:55.960 --> 00:09:01.090
I can't think of any
physical process which
00:09:01.090 --> 00:09:05.990
would need to keep track of the
joined positions of coordinates
00:09:05.990 --> 00:09:08.900
and momenta of 10
to the 23 particles,
00:09:08.900 --> 00:09:13.190
and know them with infinite
precision, et cetera.
00:09:13.190 --> 00:09:15.280
Things that I make
observations with
00:09:15.280 --> 00:09:18.790
and I say, we see this,
well, what do we see?
00:09:18.790 --> 00:09:24.090
We see that something has some
kind of a density over here.
00:09:24.090 --> 00:09:26.010
It has some kind of pressure.
00:09:26.010 --> 00:09:28.720
Maybe there is, in the
middle of this process,
00:09:28.720 --> 00:09:30.530
some flow of gas particles.
00:09:30.530 --> 00:09:33.420
We can talk about the
velocity of those things.
00:09:33.420 --> 00:09:35.130
And let's say even
when we are sort
00:09:35.130 --> 00:09:39.460
of being very non-equilibrium
and thinking about velocity
00:09:39.460 --> 00:09:43.550
of those particles going from
one side to the other side,
00:09:43.550 --> 00:09:48.060
we really don't care which
particle among the 10 to the 23
00:09:48.060 --> 00:09:50.520
is at some instant
of time contributing
00:09:50.520 --> 00:09:55.130
to the velocity of the
gas that is swishing past.
00:09:55.130 --> 00:09:59.720
So clearly, any physical
observable that we make
00:09:59.720 --> 00:10:03.780
is something that
does not require
00:10:03.780 --> 00:10:05.890
all of those degrees of freedom.
00:10:05.890 --> 00:10:08.060
So let's just construct
some of those things
00:10:08.060 --> 00:10:10.480
that we may find
useful and see how
00:10:10.480 --> 00:10:13.170
we would describe
evolutions of them.
00:10:13.170 --> 00:10:16.020
I mean, the most useful
thing, indeed, is the density.
00:10:16.020 --> 00:10:19.850
So what I could do
is I can actually
00:10:19.850 --> 00:10:23.210
construct a
one-particle density.
00:10:23.210 --> 00:10:26.455
So I want to look at
some position in space
00:10:26.455 --> 00:10:30.530
at some time t and
ask whether or not
00:10:30.530 --> 00:10:34.960
there are particles there.
00:10:34.960 --> 00:10:37.220
I don't care which
one of the particles.
00:10:37.220 --> 00:10:40.300
Actually, let's put in a
little bit more information;
00:10:40.300 --> 00:10:44.090
also, keep track of whether,
at some instant of time,
00:10:44.090 --> 00:10:47.350
I see at this
location particles.
00:10:47.350 --> 00:10:49.370
And these particles,
I will also ask
00:10:49.370 --> 00:10:51.020
which direction they are moving.
00:10:51.020 --> 00:10:53.650
Maybe I'm also going
to think about the case
00:10:53.650 --> 00:10:57.640
where, in the intermediate,
I am flowing from one side
00:10:57.640 --> 00:10:58.270
to another.
00:10:58.270 --> 00:11:01.840
And keeping track of both
of these is important.
00:11:01.840 --> 00:11:06.510
I said I don't care which
one of these particles
00:11:06.510 --> 00:11:09.340
is contributing.
00:11:09.340 --> 00:11:18.180
So that means that I have
to sum over all particles
00:11:18.180 --> 00:11:23.345
and ask whether
or not, at time t,
00:11:23.345 --> 00:11:29.818
there is a particle at
location q with momentum p.
00:11:32.890 --> 00:11:37.490
So these delta functions are
supposed to enforce that.
00:11:37.490 --> 00:11:41.860
And again, I'm not thinking
about an individual trajectory,
00:11:41.860 --> 00:11:44.750
because I think, more or
less, all of the trajectories
00:11:44.750 --> 00:11:46.970
are going to behave
the same way.
00:11:46.970 --> 00:11:50.940
And so what I will do
is I take an ensemble
00:11:50.940 --> 00:11:53.161
average of this quantity.
00:11:53.161 --> 00:11:53.660
OK?
00:11:53.660 --> 00:11:55.300
So what does that mean?
00:11:55.300 --> 00:11:57.650
To do an ensemble
average, we said
00:11:57.650 --> 00:12:04.200
I have to integrate the
density function, rho,
00:12:04.200 --> 00:12:06.970
which depends on all
of the coordinates.
00:12:06.970 --> 00:12:09.910
So I have coordinate
for particle one,
00:12:09.910 --> 00:12:14.780
coordinate-- particle and
momentum, particle and momentum
00:12:14.780 --> 00:12:17.690
for particle two,
for particle three,
00:12:17.690 --> 00:12:22.750
all the way to
particle number N.
00:12:22.750 --> 00:12:24.970
And this is something
that depends on time.
00:12:24.970 --> 00:12:26.920
That's where the time
dependence comes from.
00:12:26.920 --> 00:12:29.030
And again, let's think
about the time dependence
00:12:29.030 --> 00:12:31.620
where 0 is you
lift the partition
00:12:31.620 --> 00:12:34.750
and you allow things to
go from one to the other.
00:12:34.750 --> 00:12:38.950
So there is a non-equilibrium
process that you are following.
00:12:38.950 --> 00:12:44.120
What I need to do is I have
to multiply this density
00:12:44.120 --> 00:12:50.130
with the function that I
have to consider the average.
00:12:50.130 --> 00:12:53.790
Well, the function being a
delta function, it's very nice.
00:12:53.790 --> 00:12:59.980
The first term in the sum,
we'll set simply q1 equals to q.
00:12:59.980 --> 00:13:04.440
And it will set p1
equals to p when
00:13:04.440 --> 00:13:08.490
I do the integration
over p1 and q1.
00:13:08.490 --> 00:13:12.940
But then I'm left to have to do
the integration over particle
00:13:12.940 --> 00:13:20.155
two, particle three,
particle four, and so forth.
00:13:20.155 --> 00:13:21.580
OK?
00:13:21.580 --> 00:13:24.930
But this is only the
first term in the sum.
00:13:24.930 --> 00:13:27.480
Then I have to write the
sum for particle number two,
00:13:27.480 --> 00:13:30.220
particle number
three, et cetera.
00:13:30.220 --> 00:13:33.070
But all of the particles,
as far as I'm concerned,
00:13:33.070 --> 00:13:34.720
are behaving the same way.
00:13:34.720 --> 00:13:44.510
So I just need to multiply
this by a factor of N.
00:13:44.510 --> 00:13:49.700
Now, this is something that
actually we encountered before.
00:13:49.700 --> 00:13:54.500
Recall that this rho is a
joint probability distribution.
00:13:54.500 --> 00:13:56.600
It's a joint
probability distribution
00:13:56.600 --> 00:14:01.010
of all N particles, their
locations, and momentum.
00:14:01.010 --> 00:14:10.510
And we gave a name to taking
a joint probability density
00:14:10.510 --> 00:14:13.180
function, which this
is, and integrating
00:14:13.180 --> 00:14:16.266
over some subset of variables.
00:14:16.266 --> 00:14:21.150
The answer is an unconditional
probability distribution.
00:14:21.150 --> 00:14:24.380
So essentially, the
end of this story
00:14:24.380 --> 00:14:27.630
is if I integrate
over coordinates
00:14:27.630 --> 00:14:32.060
of two all the way to N, I will
get an unconditional result
00:14:32.060 --> 00:14:34.840
pertaining to the first
set of coordinate.
00:14:34.840 --> 00:14:38.470
So this is the same thing
up to a factor of N--
00:14:38.470 --> 00:14:43.110
the unconditional probability
that I will call rho one.
00:14:43.110 --> 00:14:44.885
Actually, I should
have called this f1.
00:14:47.830 --> 00:14:50.570
So this is something
that is defined
00:14:50.570 --> 00:14:52.470
to be the one particle density.
00:14:58.540 --> 00:15:04.430
I don't know why this name stuck
to it, because this one, that
00:15:04.430 --> 00:15:07.220
is up to a factor of
N different from it,
00:15:07.220 --> 00:15:13.340
is our usual unconditional
probability for one particle.
00:15:13.340 --> 00:15:28.944
So this is rho of p1 being
p, q1 being q at time.
00:15:28.944 --> 00:15:30.850
OK?
00:15:30.850 --> 00:15:35.930
So essentially, what I've said
is I have a joint probability.
00:15:35.930 --> 00:15:37.990
I'm really interested
in one of the particles,
00:15:37.990 --> 00:15:41.890
so I just integrate
over all the others.
00:15:41.890 --> 00:15:47.190
And this is kind of
not very elegant,
00:15:47.190 --> 00:15:49.100
but that's somehow
the way that it
00:15:49.100 --> 00:15:51.090
has appeared in the literature.
00:15:51.090 --> 00:15:53.790
The entity that I
have on the right
00:15:53.790 --> 00:15:59.730
is the properly
normalized probability.
00:15:59.730 --> 00:16:05.960
Once I multiply by N, it's a
quantity this is called f1.
00:16:05.960 --> 00:16:10.130
And it's called a density
and used very much
00:16:10.130 --> 00:16:12.170
in the literature.
00:16:12.170 --> 00:16:13.236
OK?
00:16:13.236 --> 00:16:16.240
And that's the thing that
I think will be useful.
00:16:16.240 --> 00:16:20.420
Essentially, all of
the things that I
00:16:20.420 --> 00:16:23.020
know about observations
of a system,
00:16:23.020 --> 00:16:26.870
such as its density,
its velocity, et cetera,
00:16:26.870 --> 00:16:30.080
I should be able
to get from this.
00:16:30.080 --> 00:16:33.800
Sometimes we need a little
bit more information,
00:16:33.800 --> 00:16:36.610
so I allow for the
possibility that I may also
00:16:36.610 --> 00:16:41.570
need to focus on a
two-particle density, where
00:16:41.570 --> 00:16:48.710
I keep information pertaining
to two particles at time t
00:16:48.710 --> 00:16:52.570
and I integrate over everybody
else that I'm not interested.
00:16:52.570 --> 00:16:58.180
So I introduce an integration
over coordinates number
00:16:58.180 --> 00:17:03.720
two-- sorry, number
three all the way to N
00:17:03.720 --> 00:17:09.280
to not have to repeat this
combination all the time.
00:17:09.280 --> 00:17:12.520
The 6N-dimensional
phase space for particle
00:17:12.520 --> 00:17:15.839
i I will indicate by dVi.
00:17:15.839 --> 00:17:24.329
So I take the full N particle
density, if you like,
00:17:24.329 --> 00:17:28.160
integrate over
everybody except 2.
00:17:28.160 --> 00:17:32.950
And up to a normalization
of NN minus 1,
00:17:32.950 --> 00:17:35.940
this is called a
two-particle density.
00:17:35.940 --> 00:17:41.160
And again, absent
this normalization,
00:17:41.160 --> 00:17:44.750
this is simply the
unconditional probability
00:17:44.750 --> 00:17:47.592
that I would construct out
of this joint probability
00:17:47.592 --> 00:17:48.092
[INAUDIBLE].
00:17:55.970 --> 00:18:00.340
And just for some mathematical
purposes, the generalization
00:18:00.340 --> 00:18:05.370
of this to S particles,
we will call fs.
00:18:05.370 --> 00:18:11.510
So this depends on p1
through qs at time t.
00:18:11.510 --> 00:18:14.740
And this is going to
be N factorial divided
00:18:14.740 --> 00:18:18.910
by N minus S
factorial in general.
00:18:18.910 --> 00:18:24.110
And they join the unconditional
probability that depends on S
00:18:24.110 --> 00:18:29.466
coordinates p1
through qs at time t.
00:18:34.312 --> 00:18:34.812
OK?
00:18:37.740 --> 00:18:40.860
You'll see why I need
these higher ones
00:18:40.860 --> 00:18:47.620
although, in reality, all my
interest is on this first one,
00:18:47.620 --> 00:18:50.120
because practically
everything that I need
00:18:50.120 --> 00:18:54.190
to know about this initial
experiment that I set up
00:18:54.190 --> 00:18:57.550
and how a gas expands I
should be able to extract
00:18:57.550 --> 00:19:00.004
from this one-particle density.
00:19:00.004 --> 00:19:02.220
OK?
00:19:02.220 --> 00:19:05.350
So how do I calculate this?
00:19:05.350 --> 00:19:08.770
Well, what I would
need to do is to look
00:19:08.770 --> 00:19:15.150
at the time variation
of fs with t
00:19:15.150 --> 00:19:20.290
to calculate the time dependence
of any one of these quantities.
00:19:20.290 --> 00:19:24.040
Ultimately, I want to have
an equation for df1 by dt.
00:19:24.040 --> 00:19:26.330
But let's write the general one.
00:19:26.330 --> 00:19:33.850
So the general one is up
to, again, this N factorial,
00:19:33.850 --> 00:19:44.510
N minus s factorial
and integral over
00:19:44.510 --> 00:19:50.740
coordinates that I'm not
interested, of d rho by dt.
00:19:55.830 --> 00:19:56.380
OK?
00:19:56.380 --> 00:20:01.080
So I just take the time
derivative inside the integral,
00:20:01.080 --> 00:20:04.580
because I know how
d rho by dt evolves.
00:20:04.580 --> 00:20:11.900
And so this is going to be
simply the Poisson bracket of H
00:20:11.900 --> 00:20:13.190
and rho.
00:20:13.190 --> 00:20:14.540
And I would proceed from there.
00:20:17.980 --> 00:20:20.150
Actually, to proceed
further and in order
00:20:20.150 --> 00:20:25.870
to be able to say things
that are related to physics,
00:20:25.870 --> 00:20:28.410
I need to say something
about the Hamiltonian.
00:20:28.410 --> 00:20:32.940
I can't do so in the
most general case.
00:20:32.940 --> 00:20:35.030
So let's write the
kind of Hamiltonian
00:20:35.030 --> 00:20:39.930
that we are interested
and describes the gas-- so
00:20:39.930 --> 00:20:44.530
the Hamiltonian for
the gas in this room.
00:20:44.530 --> 00:20:46.640
One term is simply
the kinetic energy.
00:20:52.360 --> 00:20:55.420
Another term is
that gas particles
00:20:55.420 --> 00:20:57.520
are confined by some potential.
00:20:57.520 --> 00:21:01.740
The potential could be as easy
as the walls of this room.
00:21:01.740 --> 00:21:05.370
Or it could be some
more general potential
00:21:05.370 --> 00:21:09.660
that could include gravity,
whatever else you want.
00:21:09.660 --> 00:21:14.850
So let's include
that possibility.
00:21:14.850 --> 00:21:17.680
So these are so-called
one-body terms,
00:21:17.680 --> 00:21:21.670
because they pertain to the
coordinates of one particle.
00:21:21.670 --> 00:21:24.225
And then there are
two-body terms.
00:21:24.225 --> 00:21:27.840
So for example, I could look
at all pairs of particles.
00:21:27.840 --> 00:21:33.480
Let's say i not equal to j
to avoid self-interaction.
00:21:33.480 --> 00:21:36.730
V of qi minus qj.
00:21:36.730 --> 00:21:40.910
So certainly, two particles
in this gas in this room,
00:21:40.910 --> 00:21:44.760
when they get close
enough, they certainly
00:21:44.760 --> 00:21:46.290
can pass through each other.
00:21:46.290 --> 00:21:47.840
They each have a size.
00:21:47.840 --> 00:21:51.970
But even when they are
a few times their sizes,
00:21:51.970 --> 00:21:54.120
they start to feel
some interaction, which
00:21:54.120 --> 00:21:55.270
causes them to collide.
00:21:55.270 --> 00:21:56.269
Yes.
00:21:56.269 --> 00:21:58.145
AUDIENCE: The whole
series of arguments
00:21:58.145 --> 00:22:00.021
that we're developing,
do these only hold
00:22:00.021 --> 00:22:01.897
for time-independent potentials?
00:22:01.897 --> 00:22:04.620
PROFESSOR: Yes.
00:22:04.620 --> 00:22:07.820
Although, it is not that
difficult to sort of
00:22:07.820 --> 00:22:10.200
go through all of
these arguments
00:22:10.200 --> 00:22:13.940
and see where the corresponding
modifications are going to be.
00:22:13.940 --> 00:22:15.860
But certainly, the
very first thing
00:22:15.860 --> 00:22:21.030
that we are using, which is
this one, we kind of implicitly
00:22:21.030 --> 00:22:23.810
assume the time-independent
Hamiltonian.
00:22:23.810 --> 00:22:27.370
So you have to start changing
things from that point.
00:22:30.615 --> 00:22:31.115
OK?
00:22:34.090 --> 00:22:37.071
So in principle, you could
have three-body and higher body
00:22:37.071 --> 00:22:37.570
terms.
00:22:37.570 --> 00:22:40.990
But essentially, most
of the relevant physics
00:22:40.990 --> 00:22:42.330
is captured by this.
00:22:42.330 --> 00:22:44.970
Even for things
like plasmas, this
00:22:44.970 --> 00:22:48.400
would be a reasonably
good approximation,
00:22:48.400 --> 00:22:51.630
with the Coulomb
interaction appearing here.
00:22:51.630 --> 00:22:53.860
OK?
00:22:53.860 --> 00:23:00.500
Now, what I note is that what
I have to calculate over here
00:23:00.500 --> 00:23:04.710
is a Poisson bracket.
00:23:04.710 --> 00:23:08.760
And then I have to integrate
that Poisson bracket.
00:23:08.760 --> 00:23:10.860
Poisson bracket
involves a whole bunch
00:23:10.860 --> 00:23:14.620
of derivatives over
all set of quantities.
00:23:14.620 --> 00:23:20.200
And I realize that when I'm
integrating over derivatives,
00:23:20.200 --> 00:23:22.280
there are
simplifications that can
00:23:22.280 --> 00:23:25.410
take place, such as
integration by parts.
00:23:25.410 --> 00:23:30.260
But that only will take
place when the derivative
00:23:30.260 --> 00:23:34.590
is one of the variables
that is being integrated.
00:23:34.590 --> 00:23:41.370
Now, this whole process has
separated out arguments of rho,
00:23:41.370 --> 00:23:44.100
as far as this
expression is concerned,
00:23:44.100 --> 00:23:48.980
into two sets-- one set, or the
set that is appearing out here,
00:23:48.980 --> 00:23:53.820
the first s ones that don't
undergo the integration,
00:23:53.820 --> 00:23:58.250
and then the remainder that
do undergo the integration.
00:23:58.250 --> 00:24:02.020
So this is going to be relevant
when we do our manipulations.
00:24:02.020 --> 00:24:06.400
And therefore, it is useful
to rewrite this Hamiltonian
00:24:06.400 --> 00:24:07.725
in terms of three entities.
00:24:12.560 --> 00:24:15.660
The first one that I call
H sub s-- so if you like,
00:24:15.660 --> 00:24:18.550
this is an end
particle Hamiltonian.
00:24:18.550 --> 00:24:22.720
I can write an H sub s, which
is just exactly the same thing,
00:24:22.720 --> 00:24:25.560
except that it
applies to coordinates
00:24:25.560 --> 00:24:27.920
that I'm not integrating over.
00:24:27.920 --> 00:24:30.390
And to sort of
make a distinction,
00:24:30.390 --> 00:24:35.870
I will label them by N. And
it includes the interaction
00:24:35.870 --> 00:24:37.770
among those particles.
00:24:45.900 --> 00:24:48.930
And I can similarly
write something
00:24:48.930 --> 00:24:55.780
that pertains to coordinates
that I am integrating over.
00:24:55.780 --> 00:24:58.930
I will label them by j and k.
00:24:58.930 --> 00:25:10.010
So this is s plus 1 to N, pj
squared over 2m plus u of qj
00:25:10.010 --> 00:25:17.820
plus 1/2 sum over j and
k, V of qj minus qk.
00:25:20.858 --> 00:25:23.790
OK?
00:25:23.790 --> 00:25:27.420
So everything that involves one
set of coordinates, everything
00:25:27.420 --> 00:25:29.310
that involves the other
set of coordinates.
00:25:29.310 --> 00:25:35.090
So what is left are terms
that [? copy ?] one set of
00:25:35.090 --> 00:25:37.090
coordinates to another.
00:25:37.090 --> 00:25:41.680
So N running from 1 to s,
j running from s plus 1
00:25:41.680 --> 00:25:48.340
to N, V between qm and qj.
00:25:48.340 --> 00:25:48.840
OK?
00:25:57.970 --> 00:26:01.980
So let me rewrite this
equation rather than in terms
00:26:01.980 --> 00:26:09.450
of-- f in terms of the
probabilities' rhos.
00:26:09.450 --> 00:26:13.890
So the difference
only is that I don't
00:26:13.890 --> 00:26:17.210
have to include this
factor out front.
00:26:17.210 --> 00:26:19.610
So I have dVi.
00:26:19.610 --> 00:26:27.150
I have the d rho by dt, which
is the commutator of H with rho,
00:26:27.150 --> 00:26:33.636
which I have been writing
as Hs plus HN minus s.
00:26:33.636 --> 00:26:35.920
I'm changing the way I write s.
00:26:35.920 --> 00:26:37.080
Sorry.
00:26:37.080 --> 00:26:41.376
Plus H prime and rho.
00:26:41.376 --> 00:26:44.250
OK?
00:26:44.250 --> 00:26:46.650
So there is a bit
of mathematics to be
00:26:46.650 --> 00:26:50.160
performed to analyze this.
00:26:50.160 --> 00:26:53.400
There are three terms
that I will label a, b,
00:26:53.400 --> 00:26:59.160
and c, which are
the three Poisson
00:26:59.160 --> 00:27:02.450
brackets that I
have to evaluate.
00:27:02.450 --> 00:27:06.310
So the contribution
that I call a
00:27:06.310 --> 00:27:15.430
is the integral
over coordinates s
00:27:15.430 --> 00:27:24.020
plus 1 to N, of the Poisson
bracket of Hs with rho.
00:27:24.020 --> 00:27:29.130
Now, the Poisson bracket
is given up here.
00:27:29.130 --> 00:27:35.650
It is a sum that involves N
terms over all N particles.
00:27:35.650 --> 00:27:42.540
But since, if I'm
evaluating it for Hs and Hs
00:27:42.540 --> 00:27:46.270
will only give nonzero
derivatives with respect
00:27:46.270 --> 00:27:49.400
to the coordinates
that are present in it,
00:27:49.400 --> 00:27:51.755
this sum of N terms
actually becomes simply
00:27:51.755 --> 00:27:54.850
a sum of small N terms.
00:27:54.850 --> 00:28:01.860
So I will get a sum over
N running from 1 to s.
00:28:01.860 --> 00:28:03.570
These are the only terms.
00:28:03.570 --> 00:28:11.715
And I will get the things that
I have for d rho by dpN-- sorry,
00:28:11.715 --> 00:28:25.496
rho by dqN, dHs by dpN minus
d rho by dpN, dHs by dqN.
00:28:33.207 --> 00:28:33.707
OK?
00:28:39.040 --> 00:28:40.634
Did I make a mistake?
00:28:40.634 --> 00:28:46.418
AUDIENCE: Isn't it the Poisson
bracket of rho H, not H rho?
00:28:46.418 --> 00:28:47.382
PROFESSOR: Rho.
00:28:51.250 --> 00:28:51.970
Oh, yes.
00:28:51.970 --> 00:28:54.720
So I have to put
a minus sign here.
00:28:54.720 --> 00:28:55.220
Right.
00:28:55.220 --> 00:28:56.200
Good.
00:28:56.200 --> 00:28:57.180
Thank you.
00:29:01.590 --> 00:29:02.580
OK.
00:29:02.580 --> 00:29:08.740
Now, note that these
derivatives and operations that
00:29:08.740 --> 00:29:13.800
involve Hs involve
coordinates that are not
00:29:13.800 --> 00:29:18.460
appearing in the
integration process.
00:29:18.460 --> 00:29:20.770
So I can take
these entities that
00:29:20.770 --> 00:29:23.190
do not depend on
variables that are part
00:29:23.190 --> 00:29:30.080
of the integration outside the
integration, which means that I
00:29:30.080 --> 00:29:35.590
can then write the
result as being
00:29:35.590 --> 00:29:44.480
an exchange of the order
of the Poisson bracket
00:29:44.480 --> 00:29:47.850
and the integration.
00:29:47.850 --> 00:30:04.750
So I would essentially have the
integration only appear here,
00:30:04.750 --> 00:30:17.940
which is the same
thing as Hs and rho s.
00:30:17.940 --> 00:30:19.350
This should be rho s.
00:30:24.800 --> 00:30:28.200
This is the definition of rho s,
which is the same thing as rho.
00:30:31.492 --> 00:30:31.992
OK.
00:30:34.807 --> 00:30:36.015
What does it mean physically?
00:30:36.015 --> 00:30:40.430
So it's actually much
easier to tell you
00:30:40.430 --> 00:30:43.810
what it means physically
than to do the math.
00:30:43.810 --> 00:30:48.500
So what we saw was
happening was that if I
00:30:48.500 --> 00:30:52.960
have a Hamiltonian that
describes N particles, then
00:30:52.960 --> 00:30:58.370
for that Hamiltonian,
the corresponding density
00:30:58.370 --> 00:30:59.980
satisfies this
Liouville equation.
00:30:59.980 --> 00:31:02.810
D rho by dt is HN rho.
00:31:02.810 --> 00:31:07.620
And this was a consequence of
this divergence-less character
00:31:07.620 --> 00:31:09.810
of the flow that we
have in this space,
00:31:09.810 --> 00:31:13.620
that the equations that
we write down over here
00:31:13.620 --> 00:31:17.220
for p dot and q
dot in terms of H
00:31:17.220 --> 00:31:21.820
had this character that
the divergence was 0.
00:31:21.820 --> 00:31:23.410
OK?
00:31:23.410 --> 00:31:28.230
Now, this is true if I have
any number of particles.
00:31:28.230 --> 00:31:34.410
So if I focus simply
on s of the particles,
00:31:34.410 --> 00:31:38.416
and they are governed
by this Hamiltonian,
00:31:38.416 --> 00:31:41.640
and I don't have anything
else in the universe,
00:31:41.640 --> 00:31:44.400
as far as this
Hamiltonian is concerned,
00:31:44.400 --> 00:31:48.740
I should have the analog
of a Liouville equation.
00:31:48.740 --> 00:31:52.720
So the term that I have obtained
over there from this first term
00:31:52.720 --> 00:31:56.430
is simply stating
that d rho s by dt,
00:31:56.430 --> 00:31:59.620
if there was no other
interaction with anybody else,
00:31:59.620 --> 00:32:02.960
would simply satisfy the
corresponding Liouville
00:32:02.960 --> 00:32:06.170
equation for s particles.
00:32:06.170 --> 00:32:09.620
And because of that, we
also expect and anticipate--
00:32:09.620 --> 00:32:11.990
and I'll show that
mathematically--
00:32:11.990 --> 00:32:13.890
that the next term
in the series,
00:32:13.890 --> 00:32:18.800
that is the Poisson bracket
of N minus s and rho,
00:32:18.800 --> 00:32:23.760
should be 0, because as
far as this s particles
00:32:23.760 --> 00:32:28.780
that I'm focusing on
and how they evolve,
00:32:28.780 --> 00:32:33.290
they really don't care about
what all the other particles
00:32:33.290 --> 00:32:35.970
are doing if they
are not [INAUDIBLE].
00:32:35.970 --> 00:32:37.685
So anything interesting
should ultimately
00:32:37.685 --> 00:32:39.940
come from this third term.
00:32:39.940 --> 00:32:44.530
But let's actually go and do the
calculation for the second term
00:32:44.530 --> 00:32:49.510
to show that this anticipation
that the answer should be 0
00:32:49.510 --> 00:32:51.190
does hold up and why.
00:32:55.100 --> 00:33:02.410
So for the second term, I need
to calculate a similar Poisson
00:33:02.410 --> 00:33:05.690
bracket, except that
this second Poisson
00:33:05.690 --> 00:33:09.290
bracket involves H of N minus s.
00:33:09.290 --> 00:33:14.790
And H of N minus s, when
I put in the full sum,
00:33:14.790 --> 00:33:19.530
will only get contribution from
terms that start from s plus 1.
00:33:19.530 --> 00:33:22.100
So the same way that
that started from N,
00:33:22.100 --> 00:33:27.840
this contribution starts
from s plus 1 to N.
00:33:27.840 --> 00:33:31.360
And actually, I can just write
the whole thing as above, d
00:33:31.360 --> 00:33:50.570
rho by d qj dotted by d HN minus
s by dpj, plus d rho by-- no,
00:33:50.570 --> 00:34:01.990
this is the rho. d rho by d pj
dotted by dHN minus s by dqj.
00:34:11.340 --> 00:34:16.010
So now I have a totally
different situation
00:34:16.010 --> 00:34:19.010
from the previous case,
because the previous case,
00:34:19.010 --> 00:34:22.630
the derivatives were over
things I was not integrating.
00:34:22.630 --> 00:34:26.500
I could take outside
the integral.
00:34:26.500 --> 00:34:29.010
Now all of the
derivatives involve things
00:34:29.010 --> 00:34:31.252
that I'm integrating over.
00:34:31.252 --> 00:34:36.090
Now, when that happens, then
you do integration by parts.
00:34:36.090 --> 00:34:42.620
So what you do is
you take rho outside
00:34:42.620 --> 00:34:47.400
and let the derivative
act on everything else.
00:34:47.400 --> 00:34:49.080
OK?
00:34:49.080 --> 00:34:54.219
So what do we end up with if
we do integration by parts?
00:35:02.640 --> 00:35:06.140
I will get surface terms.
00:35:06.140 --> 00:35:09.130
Surface terms are essentially
rho-evaluated when
00:35:09.130 --> 00:35:11.660
the coordinates are at
infinity or at the edge
00:35:11.660 --> 00:35:14.360
of your space, where rho is 0.
00:35:14.360 --> 00:35:17.080
So there is no surface term.
00:35:17.080 --> 00:35:19.140
There is an overall
change in sign,
00:35:19.140 --> 00:35:23.540
so I will get a product
i running from s plus 1
00:35:23.540 --> 00:35:27.420
to N, dVi.
00:35:27.420 --> 00:35:30.640
Now the rho comes outside.
00:35:30.640 --> 00:35:34.840
And the derivative acts on
everything that is left.
00:35:34.840 --> 00:35:37.470
So the first term will
give me a second derivative
00:35:37.470 --> 00:35:45.930
of HN minus s, with respect
to p, with respect to q.
00:35:45.930 --> 00:35:49.560
And the second term will be
essentially the opposite way
00:35:49.560 --> 00:35:50.560
of doing the derivative.
00:35:57.130 --> 00:35:58.787
And these two are,
of course, the same.
00:35:58.787 --> 00:35:59.620
And the answer is 0.
00:36:03.106 --> 00:36:04.600
OK?
00:36:04.600 --> 00:36:10.845
So we do expect that
the evolution of all
00:36:10.845 --> 00:36:13.650
the other particles
should not affect
00:36:13.650 --> 00:36:15.920
the subset that
we are looking at.
00:36:15.920 --> 00:36:18.550
And that's worn out also.
00:36:18.550 --> 00:36:20.710
So the only thing
that potentially
00:36:20.710 --> 00:36:26.470
will be relevant and exciting
is the last term, number c.
00:36:26.470 --> 00:36:29.030
So let's take a look at that.
00:36:29.030 --> 00:36:34.170
So here, I have to do an
integration over variables
00:36:34.170 --> 00:36:38.580
that I am not interested.
00:36:38.580 --> 00:36:49.470
And then I need now,
however, to do a full Poisson
00:36:49.470 --> 00:36:53.080
bracket of a whole
bunch of terms,
00:36:53.080 --> 00:36:55.900
because now the terms
that I'm looking at
00:36:55.900 --> 00:36:58.510
have coordinates from both sets.
00:36:58.510 --> 00:37:01.200
So I have to be a
little bit careful.
00:37:01.200 --> 00:37:10.160
So let me just make sure that
I follow the notes that I
00:37:10.160 --> 00:37:11.720
have here and don't
make mistakes.
00:37:15.620 --> 00:37:17.140
OK.
00:37:17.140 --> 00:37:19.935
So this H prime
involves two sums.
00:37:23.260 --> 00:37:31.730
So I will write the first
sum, N running from 1 to s.
00:37:31.730 --> 00:37:37.480
And then I have the second
sum, j running from s
00:37:37.480 --> 00:37:42.890
plus 1 to N. What do I need?
00:37:42.890 --> 00:38:01.774
I need the-- OK.
00:38:04.740 --> 00:38:07.490
Let's do it the following way.
00:38:07.490 --> 00:38:12.910
So what I have to do
for the Poisson bracket
00:38:12.910 --> 00:38:16.690
is a sum that involves
all coordinates.
00:38:16.690 --> 00:38:20.070
So let's just write
this whole expression.
00:38:20.070 --> 00:38:23.700
But first, for
coordinates 1 through s.
00:38:23.700 --> 00:38:27.500
So I have a sum N
running from 1 to s.
00:38:27.500 --> 00:38:29.360
And then I will
write the term that
00:38:29.360 --> 00:38:32.456
corresponds to
coordinates s plus 1 to m.
00:38:32.456 --> 00:38:37.400
For the first set of
coordinates, what do I have?
00:38:37.400 --> 00:38:43.520
I have d rho by dqn.
00:38:43.520 --> 00:38:47.510
And then I have
d H prime by dpn.
00:38:47.510 --> 00:38:50.410
So I didn't write
H prime explicitly.
00:38:50.410 --> 00:38:54.150
I'm just breaking
the sum over here.
00:38:54.150 --> 00:39:04.590
And then I have sum j
running from s plus 1 to N,
00:39:04.590 --> 00:39:14.381
d rho by dqj times
d H prime by dpj.
00:39:14.381 --> 00:39:20.690
And again, my H prime
is this entity over here
00:39:20.690 --> 00:39:25.100
that [? copies ?]
coordinates from both sets.
00:39:25.100 --> 00:39:26.780
OK.
00:39:26.780 --> 00:39:32.970
First thing is I claim that one
of these two sets of sums is 0.
00:39:32.970 --> 00:39:34.926
You tell me which.
00:39:34.926 --> 00:39:36.300
AUDIENCE: The first.
00:39:36.300 --> 00:39:37.830
PROFESSOR: Why first?
00:39:37.830 --> 00:39:42.764
AUDIENCE: Because H prime is
independent of p [? dot. ?]
00:39:48.416 --> 00:39:50.200
PROFESSOR: That's true.
00:39:50.200 --> 00:39:50.700
OK.
00:39:50.700 --> 00:39:52.170
That's very good.
00:39:52.170 --> 00:39:55.580
And then it sort of brings
up a very important question,
00:39:55.580 --> 00:39:57.710
which is, I forgot to
write two more terms.
00:39:57.710 --> 00:40:01.787
[LAUGHTER]
00:40:01.787 --> 00:40:19.630
Running to s of d rho by dpn,
d H prime by dqn minus sum j
00:40:19.630 --> 00:40:32.320
s plus 1 to N of d rho by
dpj dot dH prime by dqj.
00:40:32.320 --> 00:40:35.760
So indeed, both answers
now were correct.
00:40:35.760 --> 00:40:37.620
Somebody said that
the first term
00:40:37.620 --> 00:40:41.770
is a 0, because H prime
does not depend on pn.
00:40:41.770 --> 00:40:45.310
And somebody over here
said that this term is 0.
00:40:45.310 --> 00:40:48.844
And maybe they can explain why.
00:40:48.844 --> 00:40:51.149
AUDIENCE: [INAUDIBLE].
00:40:51.149 --> 00:40:53.250
PROFESSOR: Same
reason as up here.
00:40:53.250 --> 00:40:55.460
That is, I can do
integration by parts.
00:40:55.460 --> 00:40:57.196
AUDIENCE: [INAUDIBLE].
00:40:57.196 --> 00:41:03.890
PROFESSOR: To get rid of this
term plus this term together.
00:41:03.890 --> 00:41:08.310
So it's actually
by itself is not 0.
00:41:08.310 --> 00:41:11.710
But if I do
integration by parts,
00:41:11.710 --> 00:41:14.550
I will have-- actually,
even by itself, it is 0,
00:41:14.550 --> 00:41:19.183
because I would have d
by dqj, d by pj, H prime.
00:41:19.183 --> 00:41:22.900
And H prime, you cannot
have a double derivative pj.
00:41:22.900 --> 00:41:25.740
So each one of them,
actually, by itself is 0.
00:41:25.740 --> 00:41:28.480
But in general, they would
also cancel each other
00:41:28.480 --> 00:41:30.621
through their single process.
00:41:30.621 --> 00:41:31.120
Yes.
00:41:31.120 --> 00:41:34.100
AUDIENCE: Do you have the sign
of [? dH ?] of [? H prime? ?]
00:41:34.100 --> 00:41:35.820
PROFESSOR: Did I have
the sign incorrect?
00:41:35.820 --> 00:41:36.570
Yes.
00:41:36.570 --> 00:41:39.120
Because for some
reason or other,
00:41:39.120 --> 00:41:46.048
I keep reading from here, which
is rho and H. So let's do this.
00:41:49.476 --> 00:41:49.976
OK?
00:41:49.976 --> 00:41:50.824
AUDIENCE: Excuse me.
00:41:50.824 --> 00:41:51.449
PROFESSOR: Yes?
00:41:51.449 --> 00:41:54.886
AUDIENCE: [INAUDIBLE].
00:41:54.886 --> 00:41:55.920
PROFESSOR: OK.
00:41:55.920 --> 00:41:57.170
Yes, it is different.
00:41:57.170 --> 00:41:57.670
Yes.
00:41:57.670 --> 00:42:03.430
So what I said, if I had
a more general Hamiltonian
00:42:03.430 --> 00:42:07.200
that also depended
on momentum, then
00:42:07.200 --> 00:42:11.020
this term would, by itself
not 0, but would cancel,
00:42:11.020 --> 00:42:13.370
be the corresponding
term from here.
00:42:13.370 --> 00:42:15.290
But the way that I
have for H prime,
00:42:15.290 --> 00:42:19.010
indeed, each term by
itself would be 0.
00:42:19.010 --> 00:42:19.510
OK.
00:42:19.510 --> 00:42:24.330
So hopefully-- then
what do we have?
00:42:27.150 --> 00:42:34.240
So actually, let's keep the
sign correct and do this,
00:42:34.240 --> 00:42:37.290
because I need this right
sign for the one term
00:42:37.290 --> 00:42:38.040
that is preserved.
00:42:38.040 --> 00:42:39.330
So what does that say?
00:42:39.330 --> 00:42:47.870
It is a sum, n
running from 1 to s.
00:42:51.030 --> 00:42:52.330
OK?
00:42:52.330 --> 00:42:56.940
I have d H prime by dqn.
00:42:59.500 --> 00:43:01.100
And I have this integration.
00:43:01.100 --> 00:43:10.150
I have the integration i running
from s plus 1 to N dV of i.
00:43:10.150 --> 00:43:16.480
I have d rho by
dpn dot producted
00:43:16.480 --> 00:43:19.410
with d H prime by dqN.
00:43:19.410 --> 00:43:23.760
d H prime by dqn I can
calculate from here,
00:43:23.760 --> 00:43:31.940
is a sum over terms j
running from s plus 1
00:43:31.940 --> 00:43:42.280
to N of V of qn minus qj.
00:43:52.440 --> 00:43:53.010
All right.
00:43:56.433 --> 00:43:57.411
AUDIENCE: Question.
00:43:57.411 --> 00:43:58.878
PROFESSOR: Yes.
00:43:58.878 --> 00:44:02.045
AUDIENCE: Why aren't you
differentiating [? me ?]
00:44:02.045 --> 00:44:04.095
if you're
differentiating H prime?
00:44:07.884 --> 00:44:10.200
PROFESSOR: d by dqj.
00:44:13.400 --> 00:44:14.442
All right?
00:44:14.442 --> 00:44:17.025
AUDIENCE: Where is the qn?
00:44:17.025 --> 00:44:19.161
PROFESSOR: d by dqn.
00:44:19.161 --> 00:44:20.604
Thank you.
00:44:20.604 --> 00:44:22.050
Right.
00:44:22.050 --> 00:44:25.390
Because always, pn of
qn would go together.
00:44:25.390 --> 00:44:26.764
Thank you.
00:44:26.764 --> 00:44:27.264
OK.
00:44:31.121 --> 00:44:31.620
All right.
00:44:31.620 --> 00:44:34.870
So we have to slog
through these derivations.
00:44:34.870 --> 00:44:38.980
And then I'll give you
the physical meaning.
00:44:38.980 --> 00:44:43.075
So I can rearrange this.
00:44:46.750 --> 00:44:49.910
Let's see what's happening here.
00:44:49.910 --> 00:44:55.180
I have here a sum
over particles that
00:44:55.180 --> 00:44:58.980
are not listed on
the left-hand side.
00:44:58.980 --> 00:45:02.660
So when I wrote
this d rho by dt,
00:45:02.660 --> 00:45:07.090
I had listed coordinates
going from p1 through qs that
00:45:07.090 --> 00:45:09.940
were s coordinates
that were listed.
00:45:09.940 --> 00:45:12.520
If you like, you can think
of them as s particles.
00:45:17.030 --> 00:45:21.460
Now, this sum involves
the remaining particles.
00:45:21.460 --> 00:45:22.540
What is this?
00:45:22.540 --> 00:45:23.690
Up to a sign.
00:45:23.690 --> 00:45:28.000
This is the force that
is exerted by particle j
00:45:28.000 --> 00:45:30.150
from the list of
particles that I'm not
00:45:30.150 --> 00:45:33.410
interested on one of the
particles on the list
00:45:33.410 --> 00:45:35.262
that I am interested.
00:45:35.262 --> 00:45:38.090
OK?
00:45:38.090 --> 00:45:42.140
Now, I expect that at the end
of the day, all of the particles
00:45:42.140 --> 00:45:46.950
that I am not interested
I can treat equivalently,
00:45:46.950 --> 00:45:49.460
like everything
that we had before,
00:45:49.460 --> 00:45:55.140
like how I got this factor
of N or N minus 1 over there.
00:45:55.140 --> 00:46:01.700
I expect that all of these will
give me the same result, which
00:46:01.700 --> 00:46:06.410
is proportional to the number
of these particles, which
00:46:06.410 --> 00:46:08.462
is N minus s.
00:46:08.462 --> 00:46:11.110
OK?
00:46:11.110 --> 00:46:14.950
And then I can focus on just
one of the terms in this sum.
00:46:14.950 --> 00:46:17.495
Let's say the term
that corresponds to j,
00:46:17.495 --> 00:46:18.450
being s plus 1.
00:46:23.480 --> 00:46:28.870
Now, having done that,
I have to be careful.
00:46:28.870 --> 00:46:32.210
I can do separately
the integration
00:46:32.210 --> 00:46:37.660
over the volume of this one
coordinate that I'm keeping,
00:46:37.660 --> 00:46:39.930
V of s plus 1.
00:46:39.930 --> 00:46:42.270
And what do I have here?
00:46:42.270 --> 00:46:49.180
I have the force that
exerted on particle number N
00:46:49.180 --> 00:46:52.520
by the particle that
is labelled s plus 1.
00:46:57.880 --> 00:47:03.540
And this force is dot
producted with a gradient
00:47:03.540 --> 00:47:12.390
along the momentum in direction
of particle N of its density.
00:47:12.390 --> 00:47:15.820
Actually, this is the
density of all particles.
00:47:15.820 --> 00:47:20.510
This is the rho that
corresponds to the joint.
00:47:20.510 --> 00:47:24.950
But I had here s
plus 1 integrations.
00:47:24.950 --> 00:47:27.860
One of them I wrote
down explicitly.
00:47:27.860 --> 00:47:30.028
All the others I do over here.
00:47:36.860 --> 00:47:38.540
Of the density.
00:47:38.540 --> 00:47:43.770
So basically, I change the
order of the derivative
00:47:43.770 --> 00:47:46.890
and the integrations over
the variables not involved
00:47:46.890 --> 00:47:48.670
in the remainder.
00:47:48.670 --> 00:47:51.450
And the reason I
did that, of course,
00:47:51.450 --> 00:47:54.875
is that then this
is my rho s plus 1.
00:47:58.590 --> 00:48:00.930
OK?
00:48:00.930 --> 00:48:05.160
So what we have at
the end of the day
00:48:05.160 --> 00:48:13.130
is that if I take the time
variation of an s particle
00:48:13.130 --> 00:48:21.380
density, I will get one
term that I expected,
00:48:21.380 --> 00:48:26.060
which is if those s particles
were interacting only
00:48:26.060 --> 00:48:30.620
with themselves, I would
write the Liouville equation
00:48:30.620 --> 00:48:34.460
that would be
appropriate to them.
00:48:34.460 --> 00:48:36.510
But because of the
collisions that I
00:48:36.510 --> 00:48:41.930
can have with particles
that are not over here,
00:48:41.930 --> 00:48:46.010
suddenly, the momenta that
I'm looking at could change.
00:48:46.010 --> 00:48:50.400
And because of that, I
have a correction term here
00:48:50.400 --> 00:48:53.170
that really describes
the collisions.
00:48:53.170 --> 00:48:57.980
It says here that
these s particles
00:48:57.980 --> 00:49:00.870
were following the
trajectory that
00:49:00.870 --> 00:49:03.400
was governed by the
Hamiltonian that
00:49:03.400 --> 00:49:06.330
was peculiar to the s particles.
00:49:06.330 --> 00:49:11.210
But suddenly, one of them had
a collision with somebody else.
00:49:11.210 --> 00:49:12.800
So which one of them?
00:49:12.800 --> 00:49:14.200
Well, any one of them.
00:49:14.200 --> 00:49:20.160
So I could get a contribution
from any one of the s particles
00:49:20.160 --> 00:49:24.970
that is listed over here, having
a collision with somebody else.
00:49:24.970 --> 00:49:27.970
How do I describe
the effect of that?
00:49:27.970 --> 00:49:37.020
I have to do an integration over
where this new particle that I
00:49:37.020 --> 00:49:39.160
am colliding with could be.
00:49:39.160 --> 00:49:43.090
I have to specify both where the
particle is that I am colliding
00:49:43.090 --> 00:49:47.750
with, as well as its momentum.
00:49:47.750 --> 00:49:50.100
So that's this.
00:49:50.100 --> 00:49:55.190
Then I need to know the
force that this particle is
00:49:55.190 --> 00:49:56.420
exerting on me.
00:49:56.420 --> 00:50:05.680
So that's the V of qs plus
1 minus qN divided by dqN.
00:50:05.680 --> 00:50:10.240
This is the force that is
exerted by this particle
00:50:10.240 --> 00:50:13.980
that I don't see on myself.
00:50:13.980 --> 00:50:18.990
Then I have to multiply this,
or a dot product of this,
00:50:18.990 --> 00:50:23.840
with d by dpN, because what
happens in the process,
00:50:23.840 --> 00:50:28.150
because of this force, the
momentum of the N particle
00:50:28.150 --> 00:50:29.880
is changing.
00:50:29.880 --> 00:50:34.300
The variation of
that is captured
00:50:34.300 --> 00:50:38.080
through looking at
the density that
00:50:38.080 --> 00:50:44.370
has all of these particles in
addition to this new particle
00:50:44.370 --> 00:50:46.250
that I am colliding with.
00:50:46.250 --> 00:50:50.270
But, of course, I am not really
interested in the coordinate
00:50:50.270 --> 00:50:54.970
of this new particle,
so I integrate over it.
00:50:54.970 --> 00:50:57.670
There are N minus
s such particles.
00:50:57.670 --> 00:51:00.280
So I really have to put
a factor of N minus s
00:51:00.280 --> 00:51:04.640
here for all of
potential collisions.
00:51:04.640 --> 00:51:05.775
And so that's the equation.
00:51:09.010 --> 00:51:13.900
Again, it is more common,
rather than to write
00:51:13.900 --> 00:51:20.150
the equation for rho, to
write the equation for f.
00:51:20.150 --> 00:51:22.730
And the f's and the
rhos where simply
00:51:22.730 --> 00:51:27.270
related by these factors of
N factorial over N minus 1
00:51:27.270 --> 00:51:28.880
s factorial.
00:51:28.880 --> 00:51:36.550
And the outcome of that
is that the equation for f
00:51:36.550 --> 00:51:41.160
simply does not have this
additional factor of N minus s,
00:51:41.160 --> 00:51:44.190
because that disappears
in the ratio of rho
00:51:44.190 --> 00:51:46.230
of s plus 1 and rho s.
00:51:46.230 --> 00:51:53.690
And it becomes a sum over
N running from 1 to s.
00:51:53.690 --> 00:52:01.790
Integral over coordinates and
momenta of a particle s plus 1.
00:52:04.620 --> 00:52:08.660
The force exerted
by particle s plus 1
00:52:08.660 --> 00:52:19.540
on particle N used to vary the
momentum of the N particle.
00:52:19.540 --> 00:52:25.040
And the whole thing would depend
on the density that includes,
00:52:25.040 --> 00:52:30.850
in addition to the s
particles that I had before,
00:52:30.850 --> 00:52:33.810
the new particle that
I am colliding with.
00:52:36.642 --> 00:52:39.010
OK?
00:52:39.010 --> 00:52:42.190
So there is a set
of equations that
00:52:42.190 --> 00:52:48.730
relates the different densities
and how they evolve in time.
00:52:48.730 --> 00:52:53.660
The evolution of f1, which is
the thing that I am interested,
00:52:53.660 --> 00:52:55.420
will have, on the
right-hand side,
00:52:55.420 --> 00:52:57.850
something that involves f2.
00:52:57.850 --> 00:53:01.100
The evolution of
f2 will involve f3.
00:53:01.100 --> 00:53:05.910
And this whole thing is
called a BBGKY hierarchy,
00:53:05.910 --> 00:53:09.760
after people whose names
I have in the notes.
00:53:09.760 --> 00:53:11.500
[SOFT LAUGHTER]
00:53:13.620 --> 00:53:19.450
But again, what have
we learned beyond what
00:53:19.450 --> 00:53:21.315
we had in the original case?
00:53:21.315 --> 00:53:23.720
And originally,
we had an equation
00:53:23.720 --> 00:53:29.370
that was governing a function
in 6N-dimensional space, which
00:53:29.370 --> 00:53:30.770
we really don't need.
00:53:30.770 --> 00:53:33.500
So we tried our
best to avoid that.
00:53:33.500 --> 00:53:36.700
We said that all of the physics
is in one particle, maybe
00:53:36.700 --> 00:53:38.560
two particle densities.
00:53:38.560 --> 00:53:41.770
Let's calculate the evolution
of one-particle and two-particle
00:53:41.770 --> 00:53:42.800
densities.
00:53:42.800 --> 00:53:45.980
Maybe they will tell us about
this non-equilibrium situation
00:53:45.980 --> 00:53:47.340
that we set up.
00:53:47.340 --> 00:53:50.860
But we see that the time
evolution of the first particle
00:53:50.860 --> 00:53:53.410
density requires
two-particle density.
00:53:53.410 --> 00:53:56.700
Two-particle density requires
three-particle densities.
00:53:56.700 --> 00:54:01.250
So we sort of made this
ladder, which ultimately will
00:54:01.250 --> 00:54:03.740
[? terminate ?] at the
Nth particle densities.
00:54:03.740 --> 00:54:08.040
And so we have not
really gained much.
00:54:08.040 --> 00:54:13.050
So we have to now look at these
equations a little bit more
00:54:13.050 --> 00:54:17.150
and try to inject more physics.
00:54:17.150 --> 00:54:20.755
So let's write down the
first two terms explicitly.
00:54:23.340 --> 00:54:28.580
So what I will do is I will
take this Poisson bracket of H
00:54:28.580 --> 00:54:32.850
and f, to the left-hand
side, and use the Hamiltonian
00:54:32.850 --> 00:54:37.760
that we have over here
to write the terms.
00:54:37.760 --> 00:54:41.680
So the equation that we
have for f1-- and I'm
00:54:41.680 --> 00:54:44.360
going to write it as a
whole bunch of derivatives
00:54:44.360 --> 00:54:46.980
acting on f1.
00:54:46.980 --> 00:54:52.930
f1 is a function of p1 q1 t.
00:54:52.930 --> 00:54:57.100
And essentially, what
Liouville's theorem
00:54:57.100 --> 00:55:01.100
says is that as you move
along the trajectory,
00:55:01.100 --> 00:55:06.290
the total derivative is
0, because the expansion
00:55:06.290 --> 00:55:08.720
of the flows is incompressible.
00:55:08.720 --> 00:55:10.670
So what does that mean?
00:55:10.670 --> 00:55:17.870
It means that d by dt, which
is this argument, plus q1 dot
00:55:17.870 --> 00:55:25.126
times d by dq1 plus
p1 dot by d by dp1.
00:55:27.960 --> 00:55:31.775
So here, I should write
p1 dot and q1 dot.
00:55:34.670 --> 00:55:37.340
In the absence of
everything else is 0.
00:55:37.340 --> 00:55:44.790
Then, of course, for q1
dot, we use the momentum
00:55:44.790 --> 00:55:46.820
that we would get out of this.
00:55:46.820 --> 00:55:53.060
q1 dot is momentum
divided by mass.
00:55:53.060 --> 00:55:55.790
So that's the velocity.
00:55:55.790 --> 00:55:59.740
And p1 dot, changing
momentum, is
00:55:59.740 --> 00:56:02.905
the force, is minus dH by dq1.
00:56:02.905 --> 00:56:09.360
So this is minus d of this
one particle potential divided
00:56:09.360 --> 00:56:14.630
by dq1 dotted by [? h ?] p1.
00:56:14.630 --> 00:56:21.240
So if you were asked to think
about one particle in a box,
00:56:21.240 --> 00:56:24.400
then you know its
equation of motion.
00:56:24.400 --> 00:56:27.470
If you have many,
many realizations
00:56:27.470 --> 00:56:32.080
of that particle in a box,
you can construct a density.
00:56:32.080 --> 00:56:34.186
Each one of the elements
of the trajectory,
00:56:34.186 --> 00:56:36.250
you know how they
evolve according
00:56:36.250 --> 00:56:38.010
to [? Newton's ?] equation.
00:56:38.010 --> 00:56:43.120
And you can see how the
density would evolve.
00:56:43.120 --> 00:56:45.540
It would evolve
according to this.
00:56:45.540 --> 00:56:48.280
I would have said, equal to 0.
00:56:48.280 --> 00:56:53.420
But I can't set it to 0 if I'm
really thinking about a gas,
00:56:53.420 --> 00:56:56.360
because my particle
can come and collide
00:56:56.360 --> 00:57:00.050
with a second
particle in the gas.
00:57:00.050 --> 00:57:02.980
The second particle
can be anywhere.
00:57:02.980 --> 00:57:11.570
And what it will do is that
it will exert a force, which
00:57:11.570 --> 00:57:15.000
would be like this,
on particle one.
00:57:15.000 --> 00:57:17.300
And this force will
change the momentum.
00:57:17.300 --> 00:57:20.040
So my variation of
the momentum will not
00:57:20.040 --> 00:57:22.410
come only from the
external force,
00:57:22.410 --> 00:57:27.100
but also from the
force that is coming
00:57:27.100 --> 00:57:30.380
from some other
particle in the medium.
00:57:30.380 --> 00:57:33.640
So that's where
this d by dp really
00:57:33.640 --> 00:57:37.080
gets not only the
external force but also
00:57:37.080 --> 00:57:39.890
the force from somebody else.
00:57:39.890 --> 00:57:42.830
But then I need to know
where this other particle is,
00:57:42.830 --> 00:57:45.830
given that I know where
my first particle is.
00:57:45.830 --> 00:57:50.340
So I have to include here a
two-particle density which
00:57:50.340 --> 00:57:56.740
depends on p1 as
well as q2 at time t.
00:57:56.740 --> 00:57:58.230
OK.
00:57:58.230 --> 00:58:00.546
Fine.
00:58:00.546 --> 00:58:05.420
Now you say, OK, let's write
down-- I need to know f2.
00:58:05.420 --> 00:58:07.220
Let's write down
the equation for f2.
00:58:07.220 --> 00:58:12.110
So I will write it more rapidly.
00:58:12.110 --> 00:58:15.500
I have p1 over m, d by dq1.
00:58:15.500 --> 00:58:22.232
I have p2 over m, d by dq2.
00:58:22.232 --> 00:58:27.450
I will have dU by dq1, d by dp1.
00:58:27.450 --> 00:58:35.580
I will have dU by dq2, d by dp2.
00:58:35.580 --> 00:58:40.470
I will have also a term from
the collision between q1 and q2.
00:58:44.500 --> 00:58:49.680
And once it will change the
momentum of the first particle,
00:58:49.680 --> 00:58:52.490
but it will change the
momentum of the second particle
00:58:52.490 --> 00:58:53.735
in the opposite direction.
00:58:53.735 --> 00:58:57.110
So I will put the
two of them together.
00:58:57.110 --> 00:59:03.050
So this is all of the terms that
I would get from H2, Poisson
00:59:03.050 --> 00:59:06.600
bracket with density acting
on the two-particle density.
00:59:09.530 --> 00:59:13.220
And the answer would be
0 if the two particles
00:59:13.220 --> 00:59:15.490
were the only thing in the box.
00:59:15.490 --> 00:59:17.910
But there's also
other particles.
00:59:17.910 --> 00:59:20.770
So there can be
interactions and collisions
00:59:20.770 --> 00:59:22.700
with a third particle.
00:59:22.700 --> 00:59:25.920
And for that, I
would need to know,
00:59:25.920 --> 00:59:29.680
let's actually try
to simplify notation.
00:59:29.680 --> 00:59:35.470
This is the force that is
exerted from two to one.
00:59:35.470 --> 00:59:41.822
So I will have here the
force from three to one
00:59:41.822 --> 00:59:44.075
dotted by d by dp1.
00:59:48.380 --> 00:59:49.240
Right.
00:59:49.240 --> 00:59:52.600
And the force that is
exerted from three to two
00:59:52.600 --> 00:59:57.910
dotted by d by dp2 acting on
a three-particle density that
00:59:57.910 --> 01:00:02.870
involves everything up to three.
01:00:02.870 --> 01:00:04.880
And now let's write
the third one.
01:00:04.880 --> 01:00:07.380
[LAUGHTER]
01:00:08.380 --> 01:00:10.390
So that, I will leave
to next lecture.
01:00:10.390 --> 01:00:14.250
But anyway, so this
is the structure.
01:00:14.250 --> 01:00:15.840
Now, this is the
point at which we
01:00:15.840 --> 01:00:23.340
would like to inject some
physics into the problem.
01:00:23.340 --> 01:00:30.080
So what we are going to do is to
estimate the various terms that
01:00:30.080 --> 01:00:33.880
are appearing in this
equation to see whether there
01:00:33.880 --> 01:00:37.582
is some approximation
that we can make
01:00:37.582 --> 01:00:42.680
to make the equations more
treatable and handle-able.
01:00:42.680 --> 01:00:45.090
All right?
01:00:45.090 --> 01:00:54.900
So let's try to look
at the case of a gas--
01:00:54.900 --> 01:00:58.160
let's say the gas in this room.
01:00:58.160 --> 01:01:01.790
A typical thing
that is happening
01:01:01.790 --> 01:01:03.690
in the particles of
the gas in this room
01:01:03.690 --> 01:01:06.240
is that they are zipping around.
01:01:06.240 --> 01:01:09.760
Their velocity is of
the order-- again,
01:01:09.760 --> 01:01:14.991
just order of magnitude,
hundreds of meters per second.
01:01:14.991 --> 01:01:15.490
OK?
01:01:15.490 --> 01:01:20.899
So we are going to, again,
be very sort of limited
01:01:20.899 --> 01:01:22.315
in what we are
trying to describe.
01:01:22.315 --> 01:01:24.110
There is this experiment.
01:01:24.110 --> 01:01:26.585
Gas expands into a chamber.
01:01:26.585 --> 01:01:30.802
In room temperature, typical
velocities are of this order.
01:01:30.802 --> 01:01:34.620
Now we are going to
use that to estimate
01:01:34.620 --> 01:01:36.710
the magnitude of the
various terms that
01:01:36.710 --> 01:01:39.830
are appearing in this equation.
01:01:39.830 --> 01:01:42.200
Now, the whole thing
about this equation
01:01:42.200 --> 01:01:44.870
is variation with time.
01:01:44.870 --> 01:01:46.920
So the entity that
we are looking
01:01:46.920 --> 01:01:51.450
at in all of these
brackets is this d by dt,
01:01:51.450 --> 01:01:55.830
which means that the various
terms in this differential
01:01:55.830 --> 01:01:58.980
equation, apart
from d by dt, have
01:01:58.980 --> 01:02:03.320
to have dimensions
of inverse time.
01:02:03.320 --> 01:02:06.620
So we are going to
try to characterize
01:02:06.620 --> 01:02:08.220
what those inverse times are.
01:02:12.430 --> 01:02:21.355
So what are the typical
magnitudes of various terms?
01:02:25.640 --> 01:02:28.740
If I look at the
first equation, I
01:02:28.740 --> 01:02:31.640
said what that first
equation describes.
01:02:31.640 --> 01:02:35.180
That first equation describes
for you a particle in a box.
01:02:35.180 --> 01:02:38.390
We've forgotten about
everything else.
01:02:38.390 --> 01:02:41.710
So if I have a
particle in a box,
01:02:41.710 --> 01:02:43.590
what is the characteristic time?
01:02:43.590 --> 01:02:46.730
It has to be set by the
size of the box, given
01:02:46.730 --> 01:02:49.740
that I am moving
with that velocity.
01:02:49.740 --> 01:02:52.550
So there is a
timescale that I would
01:02:52.550 --> 01:02:57.215
call extrinsic in the
sense that it is not really
01:02:57.215 --> 01:02:58.640
a property of the gas.
01:02:58.640 --> 01:03:02.020
It will be different if
I make the box bigger.
01:03:02.020 --> 01:03:04.010
There's a timescale
over which I would
01:03:04.010 --> 01:03:07.430
go from one side of the box
to another side of the box.
01:03:07.430 --> 01:03:09.920
So this is kind of
a timescale that
01:03:09.920 --> 01:03:13.100
is related to the term
that knows something
01:03:13.100 --> 01:03:17.140
about the box, which
is dU by dq, d by dp.
01:03:17.140 --> 01:03:21.480
I would say that if I were to
assign some typical magnitude
01:03:21.480 --> 01:03:24.170
to this type of
term, I would say
01:03:24.170 --> 01:03:31.980
that it is related to having
to traverse a distance that
01:03:31.980 --> 01:03:39.630
is of the order of
the size of the box,
01:03:39.630 --> 01:03:42.640
given the velocity
that I have specified.
01:03:42.640 --> 01:03:44.465
This is an inverse timescale.
01:03:44.465 --> 01:03:45.950
Right?
01:03:45.950 --> 01:03:50.190
And let's sort of
imagine that I have an--
01:03:50.190 --> 01:03:55.620
and I will call this
timescale 1 over tau U,
01:03:55.620 --> 01:04:00.610
because it is sort of
determined by my external U.
01:04:00.610 --> 01:04:03.070
Let's say we have
a typical size that
01:04:03.070 --> 01:04:04.760
is of the order of millimeter.
01:04:04.760 --> 01:04:06.890
If I make it larger,
it will be larger.
01:04:06.890 --> 01:04:12.520
So actually, let's say we
have 10 to the minus 3 meters.
01:04:12.520 --> 01:04:14.340
Actually, let's make it bigger.
01:04:14.340 --> 01:04:18.740
Let's make it of the order
of 10 to the minus 1 meter.
01:04:18.740 --> 01:04:20.900
Kind of reasonable-sized box.
01:04:20.900 --> 01:04:24.270
Then you would say
that this 1 over tau c
01:04:24.270 --> 01:04:29.010
is of the order of 10 to the 2
divided by 10 to the minus 1,
01:04:29.010 --> 01:04:31.110
which is of the order of 1,000.
01:04:31.110 --> 01:04:35.390
Basically, it
takes a millisecond
01:04:35.390 --> 01:04:38.530
to traverse a box
that is a fraction
01:04:38.530 --> 01:04:42.220
of a meter with
these velocities.
01:04:42.220 --> 01:04:42.720
OK?
01:04:42.720 --> 01:04:44.450
You say fine.
01:04:44.450 --> 01:04:46.360
That is the kind
of timescale that I
01:04:46.360 --> 01:04:50.100
have in the first equation
that I have in my hierarchy.
01:04:50.100 --> 01:04:52.070
And that kind of
term is certainly
01:04:52.070 --> 01:04:56.950
also present in the second
equation for the hierarchy.
01:04:56.950 --> 01:04:59.860
If I have two particles,
maybe these two particles
01:04:59.860 --> 01:05:01.990
are orbiting each
other, et cetera.
01:05:01.990 --> 01:05:04.880
Still, their center of
mass would move, typically,
01:05:04.880 --> 01:05:06.569
with this velocity.
01:05:06.569 --> 01:05:08.110
And it would take
this amount of time
01:05:08.110 --> 01:05:11.010
to go across the
size of the box.
01:05:11.010 --> 01:05:15.210
But there is another
timescale inside there
01:05:15.210 --> 01:05:25.815
that I would call intrinsic,
which involves dV/dq, d by dp.
01:05:31.380 --> 01:05:36.770
Now, if I was to see what
the characteristic magnitude
01:05:36.770 --> 01:05:42.470
of this term is, it would
have to be V divided
01:05:42.470 --> 01:05:47.730
by a lens scale that
characterizes the potential.
01:05:47.730 --> 01:05:50.040
And the potential, let's
say, is of the order
01:05:50.040 --> 01:05:52.780
of atomic size or
molecular size.
01:05:52.780 --> 01:05:54.230
Let's call it d.
01:05:54.230 --> 01:05:58.910
So this is an atomic
size-- or molecular size.
01:06:03.740 --> 01:06:07.240
More correctly, really, it's
the range of the interaction
01:06:07.240 --> 01:06:09.730
that you have between particles.
01:06:09.730 --> 01:06:13.280
And typical values of
these [? numbers ?]
01:06:13.280 --> 01:06:17.320
are of the order of 10
angstroms, or angstroms,
01:06:17.320 --> 01:06:18.080
or whatever.
01:06:18.080 --> 01:06:19.905
Let's say 10 to the
minus 10 meters.
01:06:22.920 --> 01:06:23.480
Sorry.
01:06:23.480 --> 01:06:27.250
The first one I would
like to call tau U.
01:06:27.250 --> 01:06:30.070
This second time,
that I will call
01:06:30.070 --> 01:06:36.414
1 over tau c for collisions,
is going to be the ratio of 10
01:06:36.414 --> 01:06:39.950
to the 2 to 10 to the minus 10.
01:06:39.950 --> 01:06:45.010
It's of the order of 10
to the 12 in both seconds.
01:06:45.010 --> 01:06:47.340
OK?
01:06:47.340 --> 01:06:54.700
So you can see that this term is
much, much larger in magnitude
01:06:54.700 --> 01:06:58.584
than the term that was
governing the first equation.
01:06:58.584 --> 01:06:59.400
OK?
01:06:59.400 --> 01:07:03.730
And roughly, what you
expect in a situation
01:07:03.730 --> 01:07:11.840
such as this-- let's imagine,
rather than shooting particles
01:07:11.840 --> 01:07:17.990
from here, you are
shooting bullets.
01:07:17.990 --> 01:07:20.370
And then the bullets
would come and basically
01:07:20.370 --> 01:07:23.440
have some kind of
trajectory, et cetera.
01:07:23.440 --> 01:07:27.360
The characteristic time
for a single one of them
01:07:27.360 --> 01:07:31.150
would be basically
something that
01:07:31.150 --> 01:07:33.230
is related to the
size of the box.
01:07:33.230 --> 01:07:38.170
How long does it take a bullet
to go over the size of the box?
01:07:38.170 --> 01:07:40.620
But if two of these
bullets happen
01:07:40.620 --> 01:07:45.130
to come together and
collide, then there's
01:07:45.130 --> 01:07:48.650
a very short period
of time over which
01:07:48.650 --> 01:07:51.100
they would go in
different directions.
01:07:51.100 --> 01:07:53.690
And the momenta
would get displaced
01:07:53.690 --> 01:07:55.600
from what they were before.
01:07:55.600 --> 01:07:59.600
And that timescale is
of the order of this.
01:08:02.910 --> 01:08:05.880
But in the situation
that I set up,
01:08:05.880 --> 01:08:09.510
this particular
time is too rapid.
01:08:09.510 --> 01:08:12.480
There is another
more important time,
01:08:12.480 --> 01:08:16.880
which is, how long
do I have to wait
01:08:16.880 --> 01:08:20.330
for two of these particles,
or two of these bullets,
01:08:20.330 --> 01:08:23.550
to come and hit each other?
01:08:23.550 --> 01:08:28.080
So it's not the duration of the
collision that is irrelevant,
01:08:28.080 --> 01:08:32.450
but how long it would be for
me to find another particle
01:08:32.450 --> 01:08:34.500
to collide with.
01:08:34.500 --> 01:08:38.779
And actually, that is what
is governed by the terms
01:08:38.779 --> 01:08:41.399
that I have on the other side.
01:08:41.399 --> 01:08:44.600
Because the terms on the
other side, what they say is I
01:08:44.600 --> 01:08:47.270
have to find another particle.
01:08:47.270 --> 01:08:55.470
So if I look at the terms that
I have on the right-hand side
01:08:55.470 --> 01:08:58.290
and try to construct
a characteristic time
01:08:58.290 --> 01:09:02.279
out of them, I have to compare
the probability that I will
01:09:02.279 --> 01:09:06.840
have, or the density that
I will have for s plus 1,
01:09:06.840 --> 01:09:16.080
integrated over some
volume, over which
01:09:16.080 --> 01:09:20.184
the force between these
particles is non-zero.
01:09:23.770 --> 01:09:27.580
And then in order to
construct a timescale for it,
01:09:27.580 --> 01:09:32.279
I know that the d by dt on
the left-hand side acts on fs.
01:09:32.279 --> 01:09:35.250
On the right-hand
side, I have fs plus 1.
01:09:35.250 --> 01:09:37.824
So again, just if I want
to construct dimensionally,
01:09:37.824 --> 01:09:44.529
it's a ratio that
involves s plus 1 to s.
01:09:44.529 --> 01:09:46.640
OK?
01:09:46.640 --> 01:09:52.569
So I have to do an
additional integration
01:09:52.569 --> 01:09:59.410
over a volume in phase space
over which two particles can
01:09:59.410 --> 01:10:01.820
have substantial interactions.
01:10:01.820 --> 01:10:03.710
Because that's where
this [? dv ?] by dq
01:10:03.710 --> 01:10:09.820
would be non-zero, provided
that there is a density for s
01:10:09.820 --> 01:10:13.300
plus 1 particle
compared to s particles.
01:10:13.300 --> 01:10:15.000
If you think about
it, that means
01:10:15.000 --> 01:10:18.390
that I have to look
at the typical density
01:10:18.390 --> 01:10:26.770
or particles times d cubed for
these additional operations
01:10:26.770 --> 01:10:32.480
multiplied by this collision
time that I had before.
01:10:32.480 --> 01:10:34.300
OK?
01:10:34.300 --> 01:10:39.945
And this whole thing I will
call 1 over tau collision.
01:10:42.620 --> 01:10:50.080
And another way of getting
the same result is as follows.
01:10:50.080 --> 01:10:56.190
This is typically how you get
collision times by pictorially.
01:10:56.190 --> 01:11:01.320
You say that I have something
that can interact over
01:11:01.320 --> 01:11:05.530
some characteristic size d.
01:11:05.530 --> 01:11:11.640
It moves in space
with velocity v
01:11:11.640 --> 01:11:17.050
so that if I wait a time
that I will call tau,
01:11:17.050 --> 01:11:19.840
within that time,
I'm essentially
01:11:19.840 --> 01:11:28.690
sweeping a volume of space that
has volume d squared v tau.
01:11:28.690 --> 01:11:33.000
So my cross section, if my
dimension is d, is d squared.
01:11:33.000 --> 01:11:36.680
I sweep in the other
direction by [? aman ?] d tau.
01:11:36.680 --> 01:11:40.290
And how many particles
will I encounter?
01:11:40.290 --> 01:11:42.130
Well, if I know
the density, which
01:11:42.130 --> 01:11:45.020
is the number of
particles per unit volume,
01:11:45.020 --> 01:11:48.450
I have to multiply this by n.
01:11:48.450 --> 01:11:51.930
So how far do I have
to go until I hit 1?
01:11:51.930 --> 01:11:54.340
I'll call that tau x.
01:11:54.340 --> 01:11:57.910
Then my formula
for tau x would be
01:11:57.910 --> 01:12:05.080
1 over nvd squared, which
is exactly what I have here.
01:12:05.080 --> 01:12:12.280
1 over tau x is
nd squared v. OK?
01:12:12.280 --> 01:12:19.020
So in order to compare
the terms that I
01:12:19.020 --> 01:12:22.710
have on the right-hand
side with the terms
01:12:22.710 --> 01:12:25.850
on the left-hand
side, I notice that I
01:12:25.850 --> 01:12:31.210
need to know something
about nd cubed.
01:12:31.210 --> 01:12:36.110
So nd cubed tells you
if I have a particle
01:12:36.110 --> 01:12:40.000
here and this particle has
a range of interactions
01:12:40.000 --> 01:12:44.950
that I call d, how many
other particles fall
01:12:44.950 --> 01:12:46.370
within that range
of interaction?
01:12:49.490 --> 01:12:53.750
Now, for the gas in this room,
the range of the interaction
01:12:53.750 --> 01:12:55.680
is of the order of the
size of the molecule.
01:12:55.680 --> 01:12:58.350
It is very small.
01:12:58.350 --> 01:13:01.180
And the distance between
molecules in this room
01:13:01.180 --> 01:13:03.340
is far apart.
01:13:03.340 --> 01:13:10.770
And indeed, you can estimate
that for gas, nd cubed
01:13:10.770 --> 01:13:14.480
has to be of the order
of 10 to the minus 4.
01:13:14.480 --> 01:13:16.260
And how do I know that?
01:13:16.260 --> 01:13:20.610
Because if I were to take all of
the gas particles in this room
01:13:20.610 --> 01:13:24.560
and put them together so
that they are touching,
01:13:24.560 --> 01:13:26.690
then I would have a liquid.
01:13:26.690 --> 01:13:29.600
And the density of,
say, typical liquid
01:13:29.600 --> 01:13:33.740
is of the order of 10,000 times
larger than the density of air.
01:13:33.740 --> 01:13:37.516
So basically, it has to
be a number of this order.
01:13:37.516 --> 01:13:39.960
OK?
01:13:39.960 --> 01:13:41.355
So fine.
01:13:44.180 --> 01:13:47.310
Let's look at our equations.
01:13:47.310 --> 01:13:53.170
So what I find is that
in this equation for f2,
01:13:53.170 --> 01:13:56.570
on the left-hand
side, I have a term
01:13:56.570 --> 01:14:00.850
that has magnitude that is
very large-- 1 over tau c.
01:14:04.020 --> 01:14:07.280
Whereas the term on
the right-hand side,
01:14:07.280 --> 01:14:12.210
in terms of magnitude,
is something like this.
01:14:15.420 --> 01:14:17.920
And in fact, this will be
true for every equation
01:14:17.920 --> 01:14:20.420
in the hierarchy.
01:14:20.420 --> 01:14:25.570
So maybe if I am in the
limit where nd cubed is much,
01:14:25.570 --> 01:14:29.950
much less than 1-- such as
the gas in this room, which
01:14:29.950 --> 01:14:34.490
is called the dilute limit-- I
can ignore the right-hand side.
01:14:34.490 --> 01:14:38.511
I can set the
right-hand side to 0.
01:14:38.511 --> 01:14:39.010
OK?
01:14:41.720 --> 01:14:45.380
Now, I can't do that
for the first equation,
01:14:45.380 --> 01:14:47.320
because the first
equation is really
01:14:47.320 --> 01:14:50.020
the only equation in the
hierarchy that does not
01:14:50.020 --> 01:14:53.484
have the collision term
on the left-hand side.
01:14:53.484 --> 01:14:54.950
Right?
01:14:54.950 --> 01:14:59.611
And so for the first equation,
I really need to keep that.
01:14:59.611 --> 01:15:03.650
And actually, it goes
back to all of the story
01:15:03.650 --> 01:15:07.340
that we've had over here.
01:15:07.340 --> 01:15:11.540
Remember that we said
this rho equilibrium has
01:15:11.540 --> 01:15:16.600
to be a function of H
and conserved quantities.
01:15:16.600 --> 01:15:20.170
Suppose I go to my
Hamiltonian and I
01:15:20.170 --> 01:15:24.290
ignore all of these
interactions, which
01:15:24.290 --> 01:15:26.710
is what I would
have done if I just
01:15:26.710 --> 01:15:31.370
look at the first term over
here and set the collision
01:15:31.370 --> 01:15:34.040
term on the
right-hand side to 0.
01:15:34.040 --> 01:15:36.380
What would happen then?
01:15:36.380 --> 01:15:40.180
Then clearly, for each
one of the particles,
01:15:40.180 --> 01:15:44.480
I have-- let's say it's energy
is going to be conserved.
01:15:44.480 --> 01:15:46.330
Maybe the magnitude
of its momentum
01:15:46.330 --> 01:15:49.450
is going to be conserved in
the appropriate geometry.
01:15:49.450 --> 01:15:52.390
And so there will
be a huge number
01:15:52.390 --> 01:15:55.870
of individual
conserved quantities
01:15:55.870 --> 01:15:58.300
that I would have
to put over there.
01:15:58.300 --> 01:16:00.560
Indeed, if I sort of
go back to the picture
01:16:00.560 --> 01:16:04.860
that I was drawing over
here, if I ignore collisions
01:16:04.860 --> 01:16:08.810
between the particles, then
the bullets that I send
01:16:08.810 --> 01:16:13.740
will always be following
a trajectory such as this
01:16:13.740 --> 01:16:17.200
forever, because momentum
will be conserved.
01:16:17.200 --> 01:16:19.990
You will always-- I mean,
except up to reflection.
01:16:19.990 --> 01:16:21.760
And say the
magnitude of velocity
01:16:21.760 --> 01:16:25.030
would be always
following the same thing.
01:16:25.030 --> 01:16:26.380
OK?
01:16:26.380 --> 01:16:30.050
So however, if there is
a collision between two
01:16:30.050 --> 01:16:34.090
of the particles-- so the
particles that come in here,
01:16:34.090 --> 01:16:37.290
they have different velocities,
they will hit each other.
01:16:37.290 --> 01:16:38.810
The moment they hit
each other, they
01:16:38.810 --> 01:16:41.080
go off different directions.
01:16:41.080 --> 01:16:43.400
And after a certain
number of hits,
01:16:43.400 --> 01:16:45.460
then I will lose all
of the regularity
01:16:45.460 --> 01:16:48.490
of what I had in the beginning.
01:16:48.490 --> 01:16:54.390
And so essentially,
this second term
01:16:54.390 --> 01:16:57.680
on the right-hand
with the collisions
01:16:57.680 --> 01:17:00.550
is the thing that
is necessary for me
01:17:00.550 --> 01:17:06.200
to ensure that my gas does come
to equilibrium in the sense
01:17:06.200 --> 01:17:09.870
that their momenta get
distributed and reversed.
01:17:09.870 --> 01:17:12.890
I really need to
keep track of that.
01:17:12.890 --> 01:17:16.350
And also, you can see that
the timescales for which
01:17:16.350 --> 01:17:18.800
this kind of
equilibration takes place
01:17:18.800 --> 01:17:22.770
has to do with this
collision time.
01:17:22.770 --> 01:17:28.030
But as far as this term
is concerned, for the gas
01:17:28.030 --> 01:17:31.570
or for that system of bullets,
it doesn't really matter.
01:17:31.570 --> 01:17:35.030
Because for this term
to have been important,
01:17:35.030 --> 01:17:37.510
it would have been
necessary for something
01:17:37.510 --> 01:17:41.470
interesting to physically occur
should three particles come
01:17:41.470 --> 01:17:44.520
together simultaneously.
01:17:44.520 --> 01:17:47.450
And if I complete
the-- say that never
01:17:47.450 --> 01:17:51.350
in the history of this
system three particles will
01:17:51.350 --> 01:17:55.900
come together, they do
come together in reality.
01:17:55.900 --> 01:17:58.800
It's not that big a difference.
01:17:58.800 --> 01:18:00.780
It's only a factor of
10 to the 4 difference
01:18:00.780 --> 01:18:03.290
between the right-hand side
and the left-hand side.
01:18:03.290 --> 01:18:05.290
But still, even if
they didn't, there
01:18:05.290 --> 01:18:07.990
was nothing about
equilibration of the gas that
01:18:07.990 --> 01:18:10.910
would be missed by this.
01:18:10.910 --> 01:18:15.710
So it's a perfectly reasonable
approximation and assumption,
01:18:15.710 --> 01:18:20.040
therefore, for us
to drop this term.
01:18:20.040 --> 01:18:23.020
And we'll see that although
that is physically motivated,
01:18:23.020 --> 01:18:26.980
it actually doesn't resolve
this question of irreversibility
01:18:26.980 --> 01:18:31.650
yet, because that's also
potentially a system
01:18:31.650 --> 01:18:33.130
that you could set up.
01:18:33.130 --> 01:18:35.880
You just eliminate all of
the three-body interactions
01:18:35.880 --> 01:18:37.030
from the problem.
01:18:37.030 --> 01:18:39.330
Still, you could have
a very reversible set
01:18:39.330 --> 01:18:42.570
of conditions and
deterministic process
01:18:42.570 --> 01:18:45.850
that you could reverse in time.
01:18:45.850 --> 01:18:48.180
But still, it's
sort of allows us
01:18:48.180 --> 01:18:51.250
to have something that
is more manageable,
01:18:51.250 --> 01:18:54.340
which is what we will
be looking at next.
01:18:54.340 --> 01:18:57.270
Before I go to what
is next, I also
01:18:57.270 --> 01:19:01.450
mentioned that there is
one other limit where
01:19:01.450 --> 01:19:04.420
one can do things,
which is when,
01:19:04.420 --> 01:19:07.580
within the range of
interaction of one particle,
01:19:07.580 --> 01:19:09.930
there are many other particles.
01:19:09.930 --> 01:19:15.180
So you are in the dense limit,
nd cubed greater than 1.
01:19:15.180 --> 01:19:18.000
This does not happen for a
liquid, because for a liquid,
01:19:18.000 --> 01:19:22.230
the range does not allow many
particles to come within it.
01:19:22.230 --> 01:19:24.100
But it happens
for a plasma where
01:19:24.100 --> 01:19:26.800
you have long-range
Coulomb interaction.
01:19:26.800 --> 01:19:29.780
And within the range
of Coulomb interaction,
01:19:29.780 --> 01:19:32.140
you could have many
other interactions.
01:19:32.140 --> 01:19:36.620
And so that limit you will
explore in the problem set
01:19:36.620 --> 01:19:41.290
leads to a different description
of approach to equilibrium.
01:19:41.290 --> 01:19:44.210
It's called the Vlasov equation.
01:19:44.210 --> 01:19:48.140
What we are going to proceed
with now will lead to something
01:19:48.140 --> 01:19:52.310
else, which is called--
in the dilute limit,
01:19:52.310 --> 01:19:54.306
it will get the
Boltzmann equation.
01:19:58.766 --> 01:19:59.266
OK?
01:20:03.250 --> 01:20:06.925
So let's see what we have.
01:20:19.200 --> 01:20:26.210
So currently, we
achieved something.
01:20:26.210 --> 01:20:34.140
We want to describe
properties of a few particles
01:20:34.140 --> 01:20:37.450
in the system-- densities
that describe only,
01:20:37.450 --> 01:20:41.990
say, one particle by itself
if one was not enough.
01:20:41.990 --> 01:20:44.040
But I can terminate
the equations.
01:20:44.040 --> 01:20:48.740
And with one-particle density
and two-particle density,
01:20:48.740 --> 01:20:51.240
I should have an
appropriate description
01:20:51.240 --> 01:20:54.730
of how the system evolves.
01:20:54.730 --> 01:20:57.310
Let's think about
it one more time.
01:20:57.310 --> 01:20:59.115
So what is happening here?
01:20:59.115 --> 01:21:02.450
There is the
one-particle description
01:21:02.450 --> 01:21:06.580
that tells you how the
density for one particle,
01:21:06.580 --> 01:21:10.430
or an ensemble, the probability
for one particle, its position
01:21:10.430 --> 01:21:12.740
and momentum evolves.
01:21:12.740 --> 01:21:15.300
But it requires
knowledge of what
01:21:15.300 --> 01:21:19.150
would happen with a
second particle present.
01:21:19.150 --> 01:21:23.220
But the equations that we
have for the density that
01:21:23.220 --> 01:21:28.020
involves two particles is
simply a description of things
01:21:28.020 --> 01:21:32.390
that you would do if you had
deterministic trajectories.
01:21:32.390 --> 01:21:35.130
There is nothing else
on the right-hand side.
01:21:35.130 --> 01:21:38.930
So basically, all you
need to do is in order
01:21:38.930 --> 01:21:43.240
to determine this, is to have
full knowledge of what happens
01:21:43.240 --> 01:21:46.810
if two particles come
together, collide together, go
01:21:46.810 --> 01:21:48.880
away, all kinds of things.
01:21:48.880 --> 01:21:52.410
So if you have those
trajectories for two particles,
01:21:52.410 --> 01:21:55.540
you can, in principle,
build this density.
01:21:55.540 --> 01:21:57.480
It's still not an easy task.
01:21:57.480 --> 01:22:00.050
But in principle,
one could do that.
01:22:00.050 --> 01:22:03.840
And so this is the
description of f2.
01:22:03.840 --> 01:22:09.050
And we expect f2 to
describe processes
01:22:09.050 --> 01:22:12.960
in which, over a very
rapid timescale, say,
01:22:12.960 --> 01:22:18.200
momenta gets shifted from one
direction to another direction.
01:22:18.200 --> 01:22:23.270
But then there is something
about the overall behavior that
01:22:23.270 --> 01:22:26.030
should follow, more or less, f1.
01:22:26.030 --> 01:22:27.920
Again, what do I mean?
01:22:27.920 --> 01:22:33.400
What I mean is the following,
that if I open this box,
01:22:33.400 --> 01:22:35.720
there is what you would observe.
01:22:35.720 --> 01:22:39.060
The density would kind
of gush through here.
01:22:39.060 --> 01:22:40.860
And so you can
have a description
01:22:40.860 --> 01:22:44.480
for how the density, let's
say in coordinate space,
01:22:44.480 --> 01:22:47.590
would be evolving as
a function of time.
01:22:47.590 --> 01:22:51.630
If I ask how does the
two-particle prescription
01:22:51.630 --> 01:22:55.780
evolve, well, the two-particle
prescription, part of it
01:22:55.780 --> 01:22:57.450
is what's the
probability that I have
01:22:57.450 --> 01:22:59.890
a particle here and
a particle there?
01:22:59.890 --> 01:23:04.120
And if the two particles, if
the separations are far apart,
01:23:04.120 --> 01:23:06.880
you would be justified to
say that that is roughly
01:23:06.880 --> 01:23:09.920
the product of the probabilities
that I have something here
01:23:09.920 --> 01:23:11.710
and something there.
01:23:11.710 --> 01:23:14.960
When you become very close
to each other, however,
01:23:14.960 --> 01:23:17.900
over the range of
interactions and collisions,
01:23:17.900 --> 01:23:21.510
that will have to be modified,
because at those descriptions
01:23:21.510 --> 01:23:24.760
from here, you
would have to worry
01:23:24.760 --> 01:23:28.830
about the collisions, and the
exchange of momenta, et cetera.
01:23:28.830 --> 01:23:35.610
So in that sense, part of f2
is simply following f1 slowly.
01:23:35.610 --> 01:23:42.340
And part of f1 captures all of
the collisions that you have.
01:23:42.340 --> 01:23:45.600
In fact, that part of f2 that
captures in the collisions,
01:23:45.600 --> 01:23:49.930
we would like to simplify
as much as possible.
01:23:49.930 --> 01:23:52.960
And that's the next
task that we do.
01:23:52.960 --> 01:24:02.340
So what I need to do
is to somehow express
01:24:02.340 --> 01:24:06.630
the f2 that appears
in the first equation
01:24:06.630 --> 01:24:09.690
while solving this equation
that is the second one.
01:24:12.870 --> 01:24:15.790
I will write the answer
that we will eventually
01:24:15.790 --> 01:24:18.740
deal with and
explain it next time.
01:24:18.740 --> 01:24:21.760
So the ultimate result would
be that the left-hand side,
01:24:21.760 --> 01:24:23.910
we will have the terms
that we have currently.
01:24:33.800 --> 01:24:36.280
f1.
01:24:36.280 --> 01:24:39.660
On the right-hand
side, what we find
01:24:39.660 --> 01:24:44.240
is that I need to
integrate over all momenta
01:24:44.240 --> 01:24:47.320
of a second particle
and something that
01:24:47.320 --> 01:24:53.940
is like a distance to the
target-- one term that
01:24:53.940 --> 01:24:56.700
is the flux of
incoming particles.
01:24:59.580 --> 01:25:11.930
And then we would have f2
after collision minus f2
01:25:11.930 --> 01:25:12.880
before collision.
01:25:18.580 --> 01:25:21.390
And this is really
the Boltzmann equation
01:25:21.390 --> 01:25:26.890
after one more approximation,
where we replace f2 with f1.
01:25:26.890 --> 01:25:28.420
f1.
01:25:28.420 --> 01:25:30.730
But what all of that
means symbolically
01:25:30.730 --> 01:25:34.780
and what it is we'll have
to explain next time.