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MEHRAN KARDAR: You
decide at first look
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00:00:22,610 --> 00:00:28,830
at the our simple system,
which was the ideal gas.
10
00:00:33,930 --> 00:00:38,250
And imagine that we
have this gas contained
11
00:00:38,250 --> 00:00:44,630
in a box of volume V that
contains N particles.
12
00:00:44,630 --> 00:00:48,970
And it's completely isolated
from the rest of the universe.
13
00:00:48,970 --> 00:00:55,240
So you can know the amount
of energy that it has.
14
00:00:55,240 --> 00:00:58,770
So the macroscopic
description of the system
15
00:00:58,770 --> 00:01:02,015
consists of these three
numbers, E, V, and N.
16
00:01:02,015 --> 00:01:07,540
And this was the characteristics
of a microcanonical ensemble
17
00:01:07,540 --> 00:01:10,600
in which there was no
exchange of heat or work.
18
00:01:10,600 --> 00:01:13,540
And therefore,
energy was conserved.
19
00:01:13,540 --> 00:01:17,710
And our task was somehow to
characterize the probability
20
00:01:17,710 --> 00:01:21,370
to find the system
in some microstate.
21
00:01:21,370 --> 00:01:26,410
Now if you have N
particles in the system,
22
00:01:26,410 --> 00:01:30,180
there is at the microscopic
level, some microstate that
23
00:01:30,180 --> 00:01:36,520
consists of a description of
all of the positions of momenta.
24
00:01:36,520 --> 00:01:39,690
And there is
Hamiltonian that governs
25
00:01:39,690 --> 00:01:43,850
how that microstate evolves
as a function of time.
26
00:01:43,850 --> 00:01:49,660
And for the case of ideal gas,
the particles don't interact.
27
00:01:49,660 --> 00:01:55,070
So the Hamiltonian can be
written as the sum of n terms
28
00:01:55,070 --> 00:01:59,680
that describe essentially
the energy of the end
29
00:01:59,680 --> 00:02:04,770
particle composed of
its kinetic energy.
30
00:02:04,770 --> 00:02:08,490
And the term that is really
just confining the particle
31
00:02:08,490 --> 00:02:10,240
is in this box.
32
00:02:10,240 --> 00:02:12,990
And so the volume of
the box is contained.
33
00:02:12,990 --> 00:02:14,510
Let's say that in
this potential,
34
00:02:14,510 --> 00:02:19,050
that it's zero inside the
box and infinity outside.
35
00:02:19,050 --> 00:02:25,900
And we said, OK, so given that
I know what the energy, volume,
36
00:02:25,900 --> 00:02:29,910
number of particles
are, what's the chance
37
00:02:29,910 --> 00:02:35,350
that I will find the system
in some particular microstate?
38
00:02:35,350 --> 00:02:38,490
And the answer was
that, obviously, you
39
00:02:38,490 --> 00:02:45,450
will have to put zero if the
particles are outside box.
40
00:02:50,320 --> 00:02:55,520
Or if the energy, which is
really just the kinetic energy,
41
00:02:55,520 --> 00:03:00,560
does not much the energy that
we know is in the system,
42
00:03:00,560 --> 00:03:06,030
we sum over i of P
i squared over 2m
43
00:03:06,030 --> 00:03:10,071
is not equal to the
energy of the system.
44
00:03:14,790 --> 00:03:18,170
Otherwise, we say that if
the microstate corresponds
45
00:03:18,170 --> 00:03:22,010
to exactly the right
amount of energy, then
46
00:03:22,010 --> 00:03:24,710
I have no reason to exclude it.
47
00:03:24,710 --> 00:03:26,960
And just like saying
that the dice can
48
00:03:26,960 --> 00:03:29,400
have six possible
faces, you would
49
00:03:29,400 --> 00:03:33,650
assign all of those possible
phases equal probability.
50
00:03:33,650 --> 00:03:36,050
I will give all of
the microstates that
51
00:03:36,050 --> 00:03:37,720
don't conflict
with the conditions
52
00:03:37,720 --> 00:03:40,800
that I have set out
the same probability.
53
00:03:40,800 --> 00:03:43,400
I will call that
probability 1 over
54
00:03:43,400 --> 00:03:47,530
some overall constant omega.
55
00:03:47,530 --> 00:03:49,140
And so this is one otherwise.
56
00:03:56,780 --> 00:03:59,590
So then the next
statement is well
57
00:03:59,590 --> 00:04:04,980
what is this number omega
that you have put it?
58
00:04:04,980 --> 00:04:07,960
And how do we determine it?
59
00:04:07,960 --> 00:04:12,100
Well, we know that this
P is a probability so
60
00:04:12,100 --> 00:04:15,640
that if I were to integrate
over the entirety of the face
61
00:04:15,640 --> 00:04:25,480
space of this probability,
the answer should be 1.
62
00:04:29,580 --> 00:04:32,700
So that means this
omega, which is
63
00:04:32,700 --> 00:04:36,250
a function of these parameters
that I set out from the outside
64
00:04:36,250 --> 00:04:39,190
to describe the
microstate, should
65
00:04:39,190 --> 00:04:53,660
be obtained by integrating over
all q and p of this collection
66
00:04:53,660 --> 00:04:57,620
of 1s and 0s that
I have out here.
67
00:04:57,620 --> 00:05:02,330
So I think this box of
1s and 0s, put it here,
68
00:05:02,330 --> 00:05:03,060
and I integrate.
69
00:05:07,630 --> 00:05:09,940
So what do I get?
70
00:05:09,940 --> 00:05:13,850
Well, the integration
over the q's is easy.
71
00:05:13,850 --> 00:05:17,890
The places that I get 1 are
when the q's are inside the box.
72
00:05:17,890 --> 00:05:22,280
So each one of them will
give me a factor of V.
73
00:05:22,280 --> 00:05:24,290
And there are N of them.
74
00:05:24,290 --> 00:05:28,490
So I would get V to the N.
75
00:05:28,490 --> 00:05:32,090
The integrations over
momenta essentially
76
00:05:32,090 --> 00:05:36,960
have to do with seeing whether
or not the condition sum
77
00:05:36,960 --> 00:05:42,730
over i Pi squared over 2m
equals to E is satisfied or not.
78
00:05:42,730 --> 00:05:46,232
So this I can write
also as sum over i
79
00:05:46,232 --> 00:05:55,180
P i squared equals to 2mE,
which I can write as R squared.
80
00:05:55,180 --> 00:06:00,690
And essentially, in
this momentum space,
81
00:06:00,690 --> 00:06:05,200
I have to make sure that the
sum of the components of all
82
00:06:05,200 --> 00:06:09,320
of the momenta squared
add up to this R. squared,
83
00:06:09,320 --> 00:06:11,950
which as we discussed
last time, is
84
00:06:11,950 --> 00:06:29,620
the surface of hypershpere
in 3N dimensions of radius
85
00:06:29,620 --> 00:06:31,370
R, which is square root of 2mE.
86
00:06:35,850 --> 00:06:40,350
So I have to integrate
over all of these momenta.
87
00:06:40,350 --> 00:06:42,820
And most of the
time I will get 0,
88
00:06:42,820 --> 00:06:47,850
except when I heat the
surface of this sphere.
89
00:06:47,850 --> 00:06:50,500
There's kind of a little
bit of singularity
90
00:06:50,500 --> 00:06:53,010
here because you have a
probability that there's 0,
91
00:06:53,010 --> 00:06:57,960
except at the very sharp
interval, and then 0 again.
92
00:06:57,960 --> 00:07:02,660
So it's kind of like a delta
function, which is maybe
93
00:07:02,660 --> 00:07:05,930
a little bit hard to deal with.
94
00:07:05,930 --> 00:07:09,860
So sometimes we
will generalize this
95
00:07:09,860 --> 00:07:13,900
by adding a little
bit be delta E here.
96
00:07:13,900 --> 00:07:18,130
So let's say that the energy
does not have to be exactly E,
97
00:07:18,130 --> 00:07:20,970
but E minus plus
a little bit, so
98
00:07:20,970 --> 00:07:25,850
that when we look at
this surface in three
99
00:07:25,850 --> 00:07:28,520
n dimensional space--
let's say this was two
100
00:07:28,520 --> 00:07:31,220
dimensional space--
rather than having
101
00:07:31,220 --> 00:07:34,940
to deal with an
exact boundary, we
102
00:07:34,940 --> 00:07:42,720
have kind of smoothed that
out into an interval that
103
00:07:42,720 --> 00:07:47,340
has some kind of a thickness
R, that presumably is related
104
00:07:47,340 --> 00:07:49,690
to this delta E
that I put up there.
105
00:07:49,690 --> 00:07:51,860
Turns out it doesn't
really make any difference.
106
00:07:51,860 --> 00:07:53,970
The reason it doesn't
make any difference I
107
00:07:53,970 --> 00:07:56,960
will tell you shortly.
108
00:07:56,960 --> 00:07:59,750
But now when I'm integrating
over all of these
109
00:07:59,750 --> 00:08:03,360
P's-- so there's P. There's
another P. This could be P1.
110
00:08:03,360 --> 00:08:04,650
This could be P2.
111
00:08:04,650 --> 00:08:06,730
And there are
different components.
112
00:08:06,730 --> 00:08:13,530
I then get 0, except
when I hit this interval
113
00:08:13,530 --> 00:08:16,150
around the surface
of this hypersphere.
114
00:08:16,150 --> 00:08:22,280
So what do I get as a result
of the integration over this 3N
115
00:08:22,280 --> 00:08:24,110
dimensional space?
116
00:08:24,110 --> 00:08:29,250
I will get the volume
of this element, which
117
00:08:29,250 --> 00:08:32,950
is composed of the
surface area, which
118
00:08:32,950 --> 00:08:37,380
has some kind of a solid
angle in 3N dimensions.
119
00:08:37,380 --> 00:08:41,429
The radius raised to
the power of dimension
120
00:08:41,429 --> 00:08:45,330
minus 1, because it's a surface.
121
00:08:45,330 --> 00:08:49,620
And then if I want to
really include a delta R
122
00:08:49,620 --> 00:08:54,960
to make it into a volume, this
would be the appropriate volume
123
00:08:54,960 --> 00:08:58,640
of this interval
in momentum space.
124
00:08:58,640 --> 00:08:59,395
Yes.
125
00:08:59,395 --> 00:09:01,135
AUDIENCE: Just to
clarify, you're
126
00:09:01,135 --> 00:09:02,910
asserting that
there's no potential
127
00:09:02,910 --> 00:09:05,782
inside the [INAUDIBLE] that
comes from the hard walls.
128
00:09:05,782 --> 00:09:06,740
MEHRAN KARDAR: Correct.
129
00:09:06,740 --> 00:09:10,490
We can elaborate
on that later on.
130
00:09:10,490 --> 00:09:13,710
But for the description of the
ideal gas without potential,
131
00:09:13,710 --> 00:09:16,300
like in the box, I have
said that potential
132
00:09:16,300 --> 00:09:19,610
to be just 0 infinity.
133
00:09:19,610 --> 00:09:20,110
OK?
134
00:09:24,170 --> 00:09:26,350
OK, so fine.
135
00:09:26,350 --> 00:09:28,080
So this is the description.
136
00:09:28,080 --> 00:09:29,620
There was one
thing that I needed
137
00:09:29,620 --> 00:09:41,550
to tell you, which is the d,
dimension, of solid angle,
138
00:09:41,550 --> 00:09:47,020
which is 2pi to the
d over 2 divided
139
00:09:47,020 --> 00:09:51,100
by d over 2 minus 1
factorial So again,
140
00:09:51,100 --> 00:09:55,330
in two dimensions, such as the
picture that I drew over here,
141
00:09:55,330 --> 00:10:00,210
the circumference of a
circle would be 2 pi r.
142
00:10:00,210 --> 00:10:03,065
So this s sub 2--
right there'd be 2 pi--
143
00:10:03,065 --> 00:10:06,380
and you can show that
it is 2 pi divided
144
00:10:06,380 --> 00:10:09,490
by 0 factorial, which is 1.
145
00:10:09,490 --> 00:10:13,850
In three dimensions it should
give you 4 pi r squared.
146
00:10:13,850 --> 00:10:15,340
Kind of looks
strange because you
147
00:10:15,340 --> 00:10:20,090
get 2 pi to the 3/2 divided
by 1/2 half factorial.
148
00:10:20,090 --> 00:10:22,530
But the 1/2 factorial
is in fact root 2
149
00:10:22,530 --> 00:10:25,670
over pi-- root pi over 2.
150
00:10:25,670 --> 00:10:27,670
And so this will work out fine.
151
00:10:30,340 --> 00:10:33,840
Again, the definition of
this factorial in general
152
00:10:33,840 --> 00:10:36,280
is through the gamma
function and an integral
153
00:10:36,280 --> 00:10:38,270
that we saw already.
154
00:10:38,270 --> 00:10:43,910
And factorial is the integral
0 to infinity, dx, x to the n
155
00:10:43,910 --> 00:10:46,680
into the minus x.
156
00:10:46,680 --> 00:10:49,990
Now, the thing is that
this is a quantity that
157
00:10:49,990 --> 00:10:55,990
for large values of dimension
grows exponentially v E d.
158
00:10:55,990 --> 00:11:00,750
So what I claim is that if I
take the log of this surface
159
00:11:00,750 --> 00:11:06,610
area and take the limit that
d is much larger than 1,
160
00:11:06,610 --> 00:11:09,280
the quantity that I
will get-- well, let's
161
00:11:09,280 --> 00:11:10,340
take the log of this.
162
00:11:10,340 --> 00:11:13,390
I will get log of 2.
163
00:11:13,390 --> 00:11:23,840
I will get d over 2 log
pi and minus the log
164
00:11:23,840 --> 00:11:26,680
of this large factorial.
165
00:11:26,680 --> 00:11:30,650
And the log of the factorial
I will use Sterling's formula.
166
00:11:30,650 --> 00:11:32,870
I will ignore in
that large limit
167
00:11:32,870 --> 00:11:36,920
the difference between d
over 2 and d over 2 minus 1.
168
00:11:36,920 --> 00:11:38,530
Or actually I guess
at the beginning
169
00:11:38,530 --> 00:11:46,380
I may even write it d over 2
minus 1 log of d over 2 minus 1
170
00:11:46,380 --> 00:11:49,337
plus d over 2 minus 1.
171
00:11:53,000 --> 00:11:56,340
Now if I'm in this
limit of large d again,
172
00:11:56,340 --> 00:12:00,210
I can ignore the 1s.
173
00:12:00,210 --> 00:12:06,020
And I can ignore the log 2
with respect to this d over 2.
174
00:12:06,020 --> 00:12:10,070
And so the answer in this
limit is in fact proportional
175
00:12:10,070 --> 00:12:11,950
to d over 2.
176
00:12:11,950 --> 00:12:14,390
And I have the log.
177
00:12:14,390 --> 00:12:16,550
I have pi.
178
00:12:16,550 --> 00:12:21,290
I have this d over 2 that
will carry in the denominator.
179
00:12:21,290 --> 00:12:26,460
And then this d over 2 times 1
I can write as d over 2 log e.
180
00:12:26,460 --> 00:12:30,520
And so this is
the answer we get.
181
00:12:30,520 --> 00:12:38,590
So you can see that the
answer is exponentially large
182
00:12:38,590 --> 00:12:41,530
if I were to again write s of d.
183
00:12:41,530 --> 00:12:46,500
S of d grows like
an exponential in d.
184
00:12:46,500 --> 00:12:49,930
OK, so what do I
conclude from that?
185
00:12:49,930 --> 00:12:59,740
I conclude that s over Kd--
and we said that the entropy,
186
00:12:59,740 --> 00:13:05,080
we can regard as the entropy of
this probability distribution.
187
00:13:05,080 --> 00:13:08,065
So that's going to give
me the log of this omega.
188
00:13:12,050 --> 00:13:16,460
And log off this omega,
we'll get a factor
189
00:13:16,460 --> 00:13:18,970
from this v to the n.
190
00:13:18,970 --> 00:13:24,190
So I will get N
log V. I will get
191
00:13:24,190 --> 00:13:30,130
a factor from log of S of 3N.
192
00:13:30,130 --> 00:13:35,260
I figured out what that
log was in the limit
193
00:13:35,260 --> 00:13:37,110
of large dimensions.
194
00:13:37,110 --> 00:13:46,340
So I essentially have 3N over 2
because my d is now roughly 3N.
195
00:13:46,340 --> 00:13:48,980
It's in fact exactly 3N, sorry.
196
00:13:48,980 --> 00:13:53,645
I have the log of 2 pi e.
197
00:13:53,645 --> 00:13:56,900
For d I have 3N.
198
00:13:56,900 --> 00:14:00,940
And then I actually
have also from here
199
00:14:00,940 --> 00:14:09,600
a 3N log R, which I can write
as 3N over 2 log of R squared.
200
00:14:09,600 --> 00:14:13,506
And my R squared is 2mE.
201
00:14:13,506 --> 00:14:14,464
The figure I have here.
202
00:14:17,430 --> 00:14:24,950
And then you say, OK,
we added this delta R.
203
00:14:24,950 --> 00:14:27,250
But now you can
see that I can also
204
00:14:27,250 --> 00:14:33,230
ignore this delta R, because
everything else that I have
205
00:14:33,230 --> 00:14:37,240
in this expression is
something that grows radially
206
00:14:37,240 --> 00:14:43,400
with N. What's the worse
that I can do for delta R?
207
00:14:43,400 --> 00:14:49,910
I could make delta R even as big
as the entirety of this volume.
208
00:14:49,910 --> 00:14:52,680
And then the
typical volume would
209
00:14:52,680 --> 00:14:57,530
be of the order of the energy--
sorry, the typical value of R
210
00:14:57,530 --> 00:15:00,220
would be like the
square root of energy.
211
00:15:00,220 --> 00:15:02,310
So here I would
have to put this log
212
00:15:02,310 --> 00:15:04,680
of the square root
of the energy.
213
00:15:04,680 --> 00:15:07,495
And log of a square roots
of an extensive quantity
214
00:15:07,495 --> 00:15:10,400
is much less than the
extensive quantity.
215
00:15:10,400 --> 00:15:12,470
I can ignore it.
216
00:15:12,470 --> 00:15:17,870
And actually this reminds me,
that some 35 years ago when
217
00:15:17,870 --> 00:15:22,680
I was taking this course,
from Professor Felix Villiers,
218
00:15:22,680 --> 00:15:25,370
he said that he
had gone to lunch.
219
00:15:25,370 --> 00:15:30,515
And he had gotten to this
very beautiful, large orange.
220
00:15:30,515 --> 00:15:32,780
And he was excited.
221
00:15:32,780 --> 00:15:36,700
And he opened up the
orange, and it was all skin.
222
00:15:36,700 --> 00:15:39,500
And there was just a
little bit in the middle.
223
00:15:39,500 --> 00:15:41,560
He was saying it is like this.
224
00:15:41,560 --> 00:15:43,370
It's all in the surface.
225
00:15:43,370 --> 00:15:48,560
So if Professor Villiers had
an orange in 3N dimension,
226
00:15:48,560 --> 00:15:52,040
he would have exponentially
hard time extracting an orange.
227
00:15:55,070 --> 00:16:00,880
So this is our formula for
the entropy of this gas.
228
00:16:00,880 --> 00:16:04,110
Essentially the
extensive parts, n log v
229
00:16:04,110 --> 00:16:07,590
and something that
depends on n log E.
230
00:16:07,590 --> 00:16:10,100
And that's really all
we need to figure out
231
00:16:10,100 --> 00:16:12,950
all of the thermodynamic
properties,
232
00:16:12,950 --> 00:16:18,540
because we said that
we can construct--
233
00:16:18,540 --> 00:16:25,710
that's in thermodynamics--
dE is TdS minus PdV plus YdN
234
00:16:25,710 --> 00:16:29,440
in the case of a gas.
235
00:16:29,440 --> 00:16:32,950
And so we can
rearrange that to dS
236
00:16:32,950 --> 00:16:43,410
be dE over T plus P over
T dV minus Y over T dN.
237
00:16:43,410 --> 00:16:46,630
And the first thing
that we see is
238
00:16:46,630 --> 00:16:50,090
by taking the derivative of S
with respect to the quantities
239
00:16:50,090 --> 00:16:53,010
that we have
established, E, V, and N,
240
00:16:53,010 --> 00:16:57,780
we should be able to read
off appropriate quantities.
241
00:16:57,780 --> 00:17:01,040
And in particular,
let's say 1 over T
242
00:17:01,040 --> 00:17:06,819
would be dS by dE
of constant v and n.
243
00:17:06,819 --> 00:17:10,480
S will be proportional to kB.
244
00:17:10,480 --> 00:17:14,829
And then they dependents
of this object on E
245
00:17:14,829 --> 00:17:19,599
only appears on
this log E. Except
246
00:17:19,599 --> 00:17:23,210
that there's a factor
of 3N over 2 out front.
247
00:17:23,210 --> 00:17:30,110
And the derivative of log E
with respect to E is 1 over E.
248
00:17:30,110 --> 00:17:32,820
So I can certainly
immediately rearrange
249
00:17:32,820 --> 00:17:37,720
this and to get that
the energy is 3/2 N k
250
00:17:37,720 --> 00:17:46,160
T in this system of ideal point
particles in three dimensions.
251
00:17:46,160 --> 00:17:51,067
And then the
pressure, P over T, is
252
00:17:51,067 --> 00:17:54,380
the S by dV at constant e and n.
253
00:17:54,380 --> 00:17:56,520
And it's again, kB.
254
00:17:56,520 --> 00:18:00,850
The only dependence on V
is through this N log V.
255
00:18:00,850 --> 00:18:06,365
So I will get a factor of N over
V, which I can rearrange to PV
256
00:18:06,365 --> 00:18:12,150
is N kB T by the ideal gas law.
257
00:18:12,150 --> 00:18:15,530
And in principle,
the next step would
258
00:18:15,530 --> 00:18:17,670
be to calculate the
chemical potential.
259
00:18:17,670 --> 00:18:21,800
But we will leave that
for the time being
260
00:18:21,800 --> 00:18:23,830
for reasons that
will become apparent.
261
00:18:34,260 --> 00:18:46,480
Now, one thing to note is that
what you have postulated here,
262
00:18:46,480 --> 00:18:51,640
right at the beginning, is
much, much, more information
263
00:18:51,640 --> 00:18:56,390
than what we extracted here
about thermodynamic properties.
264
00:18:56,390 --> 00:19:00,760
It's a statement about a
joint probability distribution
265
00:19:00,760 --> 00:19:04,160
in this six N
dimensional face space.
266
00:19:04,160 --> 00:19:08,210
So it has huge amount
of information.
267
00:19:08,210 --> 00:19:11,290
Just to show you part
of it, let's take
268
00:19:11,290 --> 00:19:14,670
a note at the following.
269
00:19:14,670 --> 00:19:19,820
What it is a probability
as a function of all
270
00:19:19,820 --> 00:19:23,540
coordinates and momenta
across your system.
271
00:19:23,540 --> 00:19:26,470
But let me ask a
specific question.
272
00:19:26,470 --> 00:19:31,420
I can ask what's the probability
that some particular particle--
273
00:19:31,420 --> 00:19:36,160
say particle number
one-- has a momentum P1.
274
00:19:36,160 --> 00:19:37,990
It's the only
question that I care
275
00:19:37,990 --> 00:19:42,460
to ask about this huge
amount of degrees of freedom
276
00:19:42,460 --> 00:19:45,090
that are encoded in P of mu.
277
00:19:45,090 --> 00:19:48,070
And so what do I do if I
don't really care about all
278
00:19:48,070 --> 00:19:51,690
of the other degrees of freedom
is I will integrate them.
279
00:19:51,690 --> 00:19:57,810
So I don't really care where
particle number one is located.
280
00:19:57,810 --> 00:20:00,470
I didn't ask where
it is in the box.
281
00:20:00,470 --> 00:20:05,750
I don't really care where part
because numbers two through N
282
00:20:05,750 --> 00:20:08,680
are located or which
momenta they have.
283
00:20:08,680 --> 00:20:12,410
So I integrate over
all of those things
284
00:20:12,410 --> 00:20:17,200
of the full joint
probability, which
285
00:20:17,200 --> 00:20:20,161
depends on the entirety
of the face space.
286
00:20:23,460 --> 00:20:25,140
Fine, you say, OK.
287
00:20:25,140 --> 00:20:29,450
This joint probability actually
has a very simple form.
288
00:20:29,450 --> 00:20:35,670
It is 1 over this
omega E, V, and N,
289
00:20:35,670 --> 00:20:38,990
multiplying either 1 or 0.
290
00:20:38,990 --> 00:20:45,210
So I have to integrate
over all of these q1's, all
291
00:20:45,210 --> 00:20:49,020
of these qi P i.
292
00:20:51,630 --> 00:20:56,420
Of 1 over omega or 0
over omega, this delta
293
00:20:56,420 --> 00:20:58,890
like function that
we put in a box
294
00:20:58,890 --> 00:21:02,070
up there-- so this is
this delta function that
295
00:21:02,070 --> 00:21:05,850
says that the particular
should be inside the box.
296
00:21:05,850 --> 00:21:07,690
And the sum of
the momenta should
297
00:21:07,690 --> 00:21:09,900
be on the surface
of this hypershpere.
298
00:21:14,760 --> 00:21:18,390
Now, let's do
these integrations.
299
00:21:18,390 --> 00:21:19,555
Let's do it here.
300
00:21:19,555 --> 00:21:22,020
I may need space.
301
00:21:22,020 --> 00:21:24,830
The integration over
Q1 is very simple.
302
00:21:24,830 --> 00:21:28,490
It will give me a factor of V.
303
00:21:28,490 --> 00:21:31,505
I have this omega E, V,
N, in the denominator.
304
00:21:35,320 --> 00:21:41,340
And I claim that the numerator
is simply the following.
305
00:21:41,340 --> 00:21:50,550
Omega E minus P1 squared
over 2m V N minus 1.
306
00:21:50,550 --> 00:21:52,250
Why?
307
00:21:52,250 --> 00:21:59,230
Because what I need to do over
here in terms of integrations
308
00:21:59,230 --> 00:22:03,320
is pretty much what I would
have to integrate over here
309
00:22:03,320 --> 00:22:08,040
that gave rise to that surface
and all of those factors
310
00:22:08,040 --> 00:22:09,900
with one exception.
311
00:22:09,900 --> 00:22:14,540
First of all, I integrated
over one particle already,
312
00:22:14,540 --> 00:22:18,520
so the coordinate momenta here
that I'm integrating pertains
313
00:22:18,520 --> 00:22:21,280
to the remaining N minus 1.
314
00:22:21,280 --> 00:22:25,470
Hence, the omega
pertains to N minus 1.
315
00:22:25,470 --> 00:22:27,900
It's still in the
same box of volume V.
316
00:22:27,900 --> 00:22:31,250
So V, the other
argument, is the same.
317
00:22:31,250 --> 00:22:33,530
But the energy is changed.
318
00:22:33,530 --> 00:22:34,030
Why?
319
00:22:34,030 --> 00:22:37,430
Because I told you
how much momentum
320
00:22:37,430 --> 00:22:40,460
I want the first
particle to carry.
321
00:22:40,460 --> 00:22:42,520
So given the knowledge
that I'm looking
322
00:22:42,520 --> 00:22:47,030
at the probability of the first
particle having momentum P1,
323
00:22:47,030 --> 00:22:50,810
then I know that the
remainder of the energy
324
00:22:50,810 --> 00:22:54,570
should be shared among the
momenta of all the remaining
325
00:22:54,570 --> 00:22:57,650
N minus 1 particles.
326
00:22:57,650 --> 00:23:03,580
So I have already calculated
these omegas up here.
327
00:23:03,580 --> 00:23:06,710
All I need to do is to
substitute them over here.
328
00:23:06,710 --> 00:23:09,500
And I will get this probability.
329
00:23:09,500 --> 00:23:16,000
So first of all, let's check
that the volume part cancels.
330
00:23:16,000 --> 00:23:19,090
I have one factor a volume here.
331
00:23:19,090 --> 00:23:23,540
Each of my omegas is in fact
proportional to V to the N.
332
00:23:23,540 --> 00:23:26,080
So the denominator
has V to the N.
333
00:23:26,080 --> 00:23:29,060
The numerator has a
V to the N minus 1.
334
00:23:29,060 --> 00:23:32,930
And all of the V's
would cancel out.
335
00:23:32,930 --> 00:23:39,900
So the interesting thing really
comes from these solid angle
336
00:23:39,900 --> 00:23:42,720
and radius parts.
337
00:23:42,720 --> 00:23:47,240
The solid angle is a ratio of--
let's write the denominator.
338
00:23:47,240 --> 00:23:48,300
It's easier.
339
00:23:48,300 --> 00:23:54,130
It is 2 pi to the 3N over
2 divided by 3N over 2
340
00:23:54,130 --> 00:23:56,200
minus one factorial.
341
00:23:56,200 --> 00:24:00,670
The numerator would be 2
pi to the 3 N and minus 1
342
00:24:00,670 --> 00:24:09,745
over 2 divided by 3 N minus
1 over 2 minus 1 factorial.
343
00:24:13,200 --> 00:24:18,620
And then I have these
ratio of the radii.
344
00:24:18,620 --> 00:24:34,980
In the denominator I have 2mE to
the power of 3 N minus 1 over 2
345
00:24:34,980 --> 00:24:36,130
minus 1.
346
00:24:36,130 --> 00:24:48,770
So minus 3N-- it is
3N minus 1 over 2.
347
00:24:48,770 --> 00:24:52,010
Same thing that we have
been calculating so far.
348
00:24:52,010 --> 00:24:59,490
And in the numerator it is 2m
E minus P1 squared over 2m.
349
00:24:59,490 --> 00:25:05,350
So I will factor out the E.
I have 1 minus P1 squared
350
00:25:05,350 --> 00:25:09,380
over 2m E. The whole
thing raised to something
351
00:25:09,380 --> 00:25:14,338
that is 3 N minus
1 minus 1 over 2.
352
00:25:25,954 --> 00:25:31,300
Now, the most
important part of this
353
00:25:31,300 --> 00:25:39,360
is the fact that the dependence
on P1 appears as follows.
354
00:25:39,360 --> 00:25:45,320
I have this factor of 1
minus P1 squared over 2m E.
355
00:25:45,320 --> 00:25:48,400
That's the one place
that P1, the momentum
356
00:25:48,400 --> 00:25:51,760
of the particle that
I'm interested appears.
357
00:25:51,760 --> 00:25:54,280
And it raised to
the huge problem,
358
00:25:54,280 --> 00:25:57,080
which is of the
order of 3N over 2.
359
00:25:57,080 --> 00:25:59,160
It is likely less.
360
00:25:59,160 --> 00:26:01,290
But it really doesn't
make any difference
361
00:26:01,290 --> 00:26:06,050
whether I write 3N over 2, 3N
over minus 1 over 2, et cetera.
362
00:26:06,050 --> 00:26:08,720
Really, ultimately,
what I will have
363
00:26:08,720 --> 00:26:13,260
is 1 minus the very small
number, because presumably
364
00:26:13,260 --> 00:26:15,920
the energy of one
part they can is
365
00:26:15,920 --> 00:26:19,030
less than the energy of the
entirety of the particle.
366
00:26:19,030 --> 00:26:22,670
So this is something that is
order of 1 out of N raised
367
00:26:22,670 --> 00:26:25,420
to something that is
order of N. So that's
368
00:26:25,420 --> 00:26:28,520
where an exponentiation
will come into play.
369
00:26:28,520 --> 00:26:31,680
And then there's a whole
bunch of other factors
370
00:26:31,680 --> 00:26:36,320
that if I don't make any
mistake I can try to write down.
371
00:26:36,320 --> 00:26:39,300
There is the 2s
certainly cancel when
372
00:26:39,300 --> 00:26:42,480
I look at the factors of pi.
373
00:26:42,480 --> 00:26:45,670
The denominator with
respect to the numerator
374
00:26:45,670 --> 00:26:49,780
has an additional
factor of pi to the 3/2.
375
00:26:49,780 --> 00:26:51,430
In fact, I will
have a whole bunch
376
00:26:51,430 --> 00:26:54,840
of things that are raised
to the powers of 3/2.
377
00:26:54,840 --> 00:26:57,980
I also have this
2mE that compared
378
00:26:57,980 --> 00:27:01,420
to the 2mE that
comes out front has
379
00:27:01,420 --> 00:27:03,940
an additional factor of 3/2.
380
00:27:03,940 --> 00:27:06,220
So let's put all
of them together.
381
00:27:06,220 --> 00:27:09,760
2 pi mE raised to
the power of 3/2.
382
00:27:13,260 --> 00:27:18,230
And then I have the ratio
of these factorials.
383
00:27:18,230 --> 00:27:22,350
And again, the factorial that
I have in the denominator
384
00:27:22,350 --> 00:27:27,940
has one and a half times
more or 3/2 times more
385
00:27:27,940 --> 00:27:31,420
than what is in the numerator.
386
00:27:31,420 --> 00:27:34,480
Roughly it is something
like the ratio
387
00:27:34,480 --> 00:27:39,390
of 3 N over 2
factorial divided by 3
388
00:27:39,390 --> 00:27:42,065
N minus 1 over 2 factorial.
389
00:27:46,870 --> 00:27:50,390
And I claim that, say,
N factorial compared
390
00:27:50,390 --> 00:27:55,430
to N minus 1 factorial is
larger by a factor of N.
391
00:27:55,430 --> 00:27:59,990
If I go between N factorial
N minus 2 factorial is
392
00:27:59,990 --> 00:28:02,960
a factor that is
roughly N squared.
393
00:28:02,960 --> 00:28:08,810
Now this does not shift either
by 1 or by 2, but by 1 and 1/2.
394
00:28:08,810 --> 00:28:12,250
And if you go through
Sterling formula, et cetera,
395
00:28:12,250 --> 00:28:16,650
you can convince yourself that
this is roughly 3 N over 2
396
00:28:16,650 --> 00:28:19,335
to the power of 1 and 1/2-- 3/2.
397
00:28:21,930 --> 00:28:28,180
And so once you do all of your
arrangements, what do you get?
398
00:28:28,180 --> 00:28:32,210
1 minus a small quantity
raised to a huge power,
399
00:28:32,210 --> 00:28:34,580
that's the definition
of the exponential.
400
00:28:34,580 --> 00:28:41,650
So I get exponential of
minus P1 squared over 2m.
401
00:28:41,650 --> 00:28:46,060
And the factor that
multiplies it is E.
402
00:28:46,060 --> 00:28:50,260
And then I have 3N over 2.
403
00:28:55,660 --> 00:29:02,160
And again, if I have
not made any mistake
404
00:29:02,160 --> 00:29:05,890
and I'm careful with all of
the other factors that remain,
405
00:29:05,890 --> 00:29:12,340
I have here 2 pi m
E. And this E also
406
00:29:12,340 --> 00:29:17,040
gets multiplied by the
inverse of 3N over 2.
407
00:29:17,040 --> 00:29:20,890
So I will have this
replaced by 2E over 3N.
408
00:29:29,140 --> 00:29:34,650
So statement number
one, this assignment
409
00:29:34,650 --> 00:29:37,500
of probabilities according
to just throwing the dice
410
00:29:37,500 --> 00:29:41,970
and saying that everything
that has the same right energy
411
00:29:41,970 --> 00:29:45,080
is equally likely is
equivalent to looking
412
00:29:45,080 --> 00:29:47,680
at one of the
particles and stating
413
00:29:47,680 --> 00:29:50,455
that the momentum of that part
again is Gaussian distributed.
414
00:29:53,660 --> 00:30:00,200
Secondly, you can check that
this combination, 2E divided
415
00:30:00,200 --> 00:30:03,330
by 3N is the same thing as kT.
416
00:30:03,330 --> 00:30:05,360
So essentially
this you could also
417
00:30:05,360 --> 00:30:09,030
if you want to
replace 1 over kT.
418
00:30:09,030 --> 00:30:13,120
And you would get the
more familiar kind
419
00:30:13,120 --> 00:30:16,370
of Maxwell type of
distribution for the momentum
420
00:30:16,370 --> 00:30:20,320
of a single particle
in an ideal gas.
421
00:30:20,320 --> 00:30:23,540
And again, since
everything that we did
422
00:30:23,540 --> 00:30:26,760
was consistent with the
laws of probability,
423
00:30:26,760 --> 00:30:30,200
if we did not mix up the
orders of N, et cetera,
424
00:30:30,200 --> 00:30:32,860
the answer should be
properly normalized.
425
00:30:32,860 --> 00:30:38,520
And indeed, you can check that
this is the three dimensional
426
00:30:38,520 --> 00:30:43,490
normalization that you
would require for this gas.
427
00:30:43,490 --> 00:30:49,790
So the statement of saying
that everything is allowed
428
00:30:49,790 --> 00:30:56,220
is equally likely is a
huge statement in space
429
00:30:56,220 --> 00:30:58,390
of possible configurations.
430
00:30:58,390 --> 00:31:01,980
On the one hand, it gives
you macroscopic information.
431
00:31:01,980 --> 00:31:04,580
On the other hand, it
retains a huge amount
432
00:31:04,580 --> 00:31:06,640
of microscopic information.
433
00:31:06,640 --> 00:31:08,340
The parts of it
that are relevant,
434
00:31:08,340 --> 00:31:10,257
you can try to
extract this here.
435
00:31:13,361 --> 00:31:13,860
OK?
436
00:31:22,680 --> 00:31:25,820
So those were the successes.
437
00:31:25,820 --> 00:31:29,470
Question is why didn't
I calculate for you
438
00:31:29,470 --> 00:31:31,840
this u over T?
439
00:31:31,840 --> 00:31:35,300
It is because this
expression as we wrote down
440
00:31:35,300 --> 00:31:39,450
has a glaring problem
with it, which in order
441
00:31:39,450 --> 00:31:43,870
to make it explicit, we will
look at mixing entropies.
442
00:31:52,490 --> 00:31:54,580
So the idea is
this is as follows.
443
00:31:54,580 --> 00:31:59,585
Let's imagine that we
start with two gases.
444
00:32:03,780 --> 00:32:11,080
Initially, I have N1 particles
of one type in volume 1.
445
00:32:11,080 --> 00:32:16,640
And I have N2 particles of
another type in volume 2.
446
00:32:16,640 --> 00:32:20,600
And for simplicity I will
assume that both of them
447
00:32:20,600 --> 00:32:23,430
are of the same temperature.
448
00:32:23,430 --> 00:32:27,590
So this is my initial state.
449
00:32:27,590 --> 00:32:30,720
And then I remove the partition.
450
00:32:30,720 --> 00:32:33,770
And I come up with
this situation
451
00:32:33,770 --> 00:32:38,950
where the particles are mixed.
452
00:32:38,950 --> 00:32:44,070
So the particles of type
1 could be either way.
453
00:32:44,070 --> 00:32:48,100
Particles of type 2
could be in either place.
454
00:32:48,100 --> 00:32:55,715
And let's say I have a box
of some toxic gas here.
455
00:32:55,715 --> 00:32:57,510
And I remove the lid.
456
00:32:57,510 --> 00:33:00,080
And it will get
mixed in the room.
457
00:33:00,080 --> 00:33:03,740
It's certainly an
irreversible situation
458
00:33:03,740 --> 00:33:07,075
where is an increase of
entropy i associated with that.
459
00:33:07,075 --> 00:33:10,310
And we can calculate
that increase of entropy,
460
00:33:10,310 --> 00:33:14,400
because we know what the
expression for entropy is.
461
00:33:14,400 --> 00:33:18,060
So what we have to do is to
compare the entropy initially.
462
00:33:20,930 --> 00:33:23,315
So this is the initial entropy.
463
00:33:23,315 --> 00:33:26,430
And I calculate
everything in units of kB
464
00:33:26,430 --> 00:33:30,610
so I don't have to write
kB all over the place.
465
00:33:30,610 --> 00:33:33,940
For particle number
one, what do I have?
466
00:33:33,940 --> 00:33:36,030
I have N1 log V1.
467
00:33:40,240 --> 00:33:50,100
And then I have a contribution,
which is 3 N 1 over 2.
468
00:33:50,100 --> 00:33:53,970
But I notice that whatever
appears here is really
469
00:33:53,970 --> 00:33:59,690
only a function of
E over N. E over N
470
00:33:59,690 --> 00:34:02,970
is really only a
function of temperature.
471
00:34:02,970 --> 00:34:11,840
So this is something that I can
call a sigma of T over here.
472
00:34:11,840 --> 00:34:21,139
And the contribution of box 2
is N2 log V plus 3N 2 over 2.
473
00:34:21,139 --> 00:34:28,179
This-- huh-- let's
say that they are--
474
00:34:28,179 --> 00:34:30,000
we ignore the
difference in masses.
475
00:34:30,000 --> 00:34:33,840
You could potentially have
here sigma 1, sigma 2.
476
00:34:33,840 --> 00:34:37,330
It really doesn't
make any difference.
477
00:34:37,330 --> 00:34:45,780
The final state,
what do we have?
478
00:34:45,780 --> 00:34:50,179
Essentially, the one
thing that changed
479
00:34:50,179 --> 00:34:53,770
is that the N1 particles now
are occupying the box of volume
480
00:34:53,770 --> 00:35:00,420
V. So if call V to the
V1 plus V2, what we have
481
00:35:00,420 --> 00:35:10,620
is that we have N1 log
of V plus N2 log of V.
482
00:35:10,620 --> 00:35:15,240
My claim is that all
of these other factors
483
00:35:15,240 --> 00:35:16,500
really stay the same.
484
00:35:22,920 --> 00:35:28,150
Because essentially what is
happening in these expressions
485
00:35:28,150 --> 00:35:35,900
are various ratios of E over N.
And by stating that initially I
486
00:35:35,900 --> 00:35:38,480
had the things at
the same temperature,
487
00:35:38,480 --> 00:35:44,850
what I had effectively
stated was that E1 over N1
488
00:35:44,850 --> 00:35:48,430
is the same thing as E2 over N2.
489
00:35:48,430 --> 00:35:51,640
I guess in the ideal
gas case this E over N
490
00:35:51,640 --> 00:35:54,350
is the same thing as 3/2 kT.
491
00:36:01,530 --> 00:36:07,180
But if I have a
ratio such as this,
492
00:36:07,180 --> 00:36:14,330
that is also the same as E1
plus E2 divided by N1 plus N2.
493
00:36:14,330 --> 00:36:19,090
This is a simple manipulation
of fractions that I can make.
494
00:36:19,090 --> 00:36:26,510
And E1 plus E2 over N1 plus N2,
by the same kinds of arguments,
495
00:36:26,510 --> 00:36:29,220
would give me the
final temperature.
496
00:36:29,220 --> 00:36:35,060
So what I have to compute is
that the final temperature
497
00:36:35,060 --> 00:36:37,210
is the same thing as
the initial temperature.
498
00:36:37,210 --> 00:36:40,320
Essentially, in this
mixing of the ideal gases,
499
00:36:40,320 --> 00:36:42,880
temperature does not change.
500
00:36:42,880 --> 00:36:48,870
So basically, these
factors of sigma
501
00:36:48,870 --> 00:36:51,630
are the same before and after.
502
00:36:51,630 --> 00:36:56,630
And so when we calculate
the increase in entropy, Sf
503
00:36:56,630 --> 00:37:00,690
minus Si, really
the contribution
504
00:37:00,690 --> 00:37:05,720
that you care about comes
from these volume factors.
505
00:37:05,720 --> 00:37:10,620
And really the statement is
that in one particle currently
506
00:37:10,620 --> 00:37:13,190
are occupying a
volume of size V,
507
00:37:13,190 --> 00:37:15,990
whereas previously
they were in V1.
508
00:37:15,990 --> 00:37:17,760
And similarly for
the N2 particles.
509
00:37:22,670 --> 00:37:26,550
And if you have more of these
particles, more of these boxes,
510
00:37:26,550 --> 00:37:28,980
you could see how the
general expression
511
00:37:28,980 --> 00:37:32,550
for the mixing entropy goes.
512
00:37:32,550 --> 00:37:34,930
And so that's fine.
513
00:37:34,930 --> 00:37:38,110
V is certainly
greater than V1 or V2.
514
00:37:38,110 --> 00:37:40,940
Each of these logs gives
you a positive contribution.
515
00:37:40,940 --> 00:37:43,045
There's an increase in
entropy as we expect.
516
00:37:48,080 --> 00:37:52,430
Now, there is the following
difficulty however.
517
00:37:57,830 --> 00:38:04,050
What if the gases are
identical-- are the same?
518
00:38:11,180 --> 00:38:14,930
We definitely have to do this
if I take a box of methane here
519
00:38:14,930 --> 00:38:18,615
and I open it, we all know
that something has happened.
520
00:38:18,615 --> 00:38:21,920
There is an irreversible
process that has occured.
521
00:38:21,920 --> 00:38:25,020
But if the box--
I have essentially
522
00:38:25,020 --> 00:38:27,310
taken the air in this
room, put it in this box,
523
00:38:27,310 --> 00:38:30,632
whether I open the lid
or not open the lid,
524
00:38:30,632 --> 00:38:32,710
it doesn't make any difference.
525
00:38:32,710 --> 00:38:35,250
There is no
additional work that I
526
00:38:35,250 --> 00:38:38,300
have to do in order to
close or open the lid.
527
00:38:38,300 --> 00:38:42,430
Is no there no increase of
entropy one way or the other.
528
00:38:42,430 --> 00:38:45,450
Whereas if I look
at this expression,
529
00:38:45,450 --> 00:38:48,530
this expression only
depends on the final volume
530
00:38:48,530 --> 00:38:50,860
and the initial
volumes, and says
531
00:38:50,860 --> 00:38:53,690
that there should an
increase in entropy
532
00:38:53,690 --> 00:38:57,650
when we know that
there shouldn't be.
533
00:38:57,650 --> 00:39:02,500
And of course, the
resolution for that
534
00:39:02,500 --> 00:39:03,780
is something like this.
535
00:39:06,390 --> 00:39:15,280
That if I look at my two boxes--
and I said maybe one of them
536
00:39:15,280 --> 00:39:19,420
is a box that contains methane.
537
00:39:19,420 --> 00:39:22,000
Let's call it A.
And the other is
538
00:39:22,000 --> 00:39:27,820
the room that contains the air.
539
00:39:27,820 --> 00:39:31,480
Now this situation
where all of the methane
540
00:39:31,480 --> 00:39:35,750
is in the box and the oxygen
freely floating in the room
541
00:39:35,750 --> 00:39:39,820
is certainly different
from a configuration
542
00:39:39,820 --> 00:39:44,140
where I exchange these two
and the methane is here
543
00:39:44,140 --> 00:39:48,148
and the oxygen
went into the box.
544
00:39:48,148 --> 00:39:50,340
They're different
configurations.
545
00:39:50,340 --> 00:39:54,320
You can definitely
tell them apart.
546
00:39:54,320 --> 00:40:01,570
Whereas if I do the same
thing, but the box and outside
547
00:40:01,570 --> 00:40:11,710
contain the same entity, and
the same entity is, let's say,
548
00:40:11,710 --> 00:40:16,520
oxygen, then how can you tell
apart these two configurations?
549
00:40:16,520 --> 00:40:19,740
And so the meaning of-- yes.
550
00:40:19,740 --> 00:40:21,192
AUDIENCE: Are you
thinking quantum
551
00:40:21,192 --> 00:40:22,644
mechanically or classically.
552
00:40:22,644 --> 00:40:25,064
Classically we can
tell them apart, right?
553
00:40:27,980 --> 00:40:29,820
MEHRAN KARDAR:
This is currently I
554
00:40:29,820 --> 00:40:32,810
am making a
macroscopic statement.
555
00:40:32,810 --> 00:40:36,410
Now when I get to the
distinction of microstates
556
00:40:36,410 --> 00:40:40,590
we have to-- so I was
very careful in saying
557
00:40:40,590 --> 00:40:43,210
whether or not you
could tell apart
558
00:40:43,210 --> 00:40:45,970
whether it is methane or oxygen.
559
00:40:45,970 --> 00:40:48,960
So this was a very
macroscopic statement as to
560
00:40:48,960 --> 00:40:50,570
whether or not you
can distinguish
561
00:40:50,570 --> 00:40:53,660
this circumstance versus
that circumstance.
562
00:40:53,660 --> 00:40:57,090
So as far as our senses of
this macroscopic process
563
00:40:57,090 --> 00:41:02,940
is concerned, these two cases
have to be treated differently.
564
00:41:02,940 --> 00:41:08,560
Now, what we have calculated
here for these factors
565
00:41:08,560 --> 00:41:12,130
are some volume of phase space.
566
00:41:12,130 --> 00:41:15,190
And where in the
evening you might
567
00:41:15,190 --> 00:41:18,790
say that following
this procedure
568
00:41:18,790 --> 00:41:23,690
you counted these as
two distinct cases.
569
00:41:23,690 --> 00:41:26,730
In this case, these
were two distinct cases.
570
00:41:26,730 --> 00:41:30,050
But here, you can't
really tell them apart.
571
00:41:30,050 --> 00:41:31,990
So if you can't
tell them apart, you
572
00:41:31,990 --> 00:41:34,580
shouldn't call them
two distinct cases.
573
00:41:34,580 --> 00:41:39,950
You have over counted phase
space by a factor of two here.
574
00:41:39,950 --> 00:41:43,620
And here, I just looked
at two particles.
575
00:41:43,620 --> 00:41:47,080
If I have N
particles, I have over
576
00:41:47,080 --> 00:41:50,980
counted the phase space
of identical particles
577
00:41:50,980 --> 00:41:56,110
by all possible permutations of
n objects, it is n factorial.
578
00:41:56,110 --> 00:42:03,680
So there is an over
counting of phase space
579
00:42:03,680 --> 00:42:19,020
or configurations of
N identical particles
580
00:42:19,020 --> 00:42:24,300
by a factor of N factorial.
581
00:42:31,210 --> 00:42:38,390
I.e., when we said that particle
number one can be anywhere
582
00:42:38,390 --> 00:42:42,060
in the box, particle
number two can be anywhere
583
00:42:42,060 --> 00:42:46,250
in the box, all the way
to particle number n,
584
00:42:46,250 --> 00:42:50,370
well, in fact, I can't
tell which each is which.
585
00:42:50,370 --> 00:42:53,060
If I can't tell which
particle is which,
586
00:42:53,060 --> 00:42:57,195
I have to divide by the number
of permutations and factors.
587
00:42:59,950 --> 00:43:02,105
Now, as somebody was
asking the question,
588
00:43:02,105 --> 00:43:06,950
as you were asking the
question, classically,
589
00:43:06,950 --> 00:43:11,240
if I write a
computer program that
590
00:43:11,240 --> 00:43:14,620
looks at the trajectories
of N particles
591
00:43:14,620 --> 00:43:18,780
in the gas in this room,
classically, your computer
592
00:43:18,780 --> 00:43:22,670
would always know the particle
that started over here
593
00:43:22,670 --> 00:43:24,430
after many collisions
or whatever
594
00:43:24,430 --> 00:43:28,960
is the particle that
ended up somewhere else.
595
00:43:28,960 --> 00:43:33,010
So if you ask the
computer, the computer
596
00:43:33,010 --> 00:43:37,600
can certainly distinguish
these classical trajectories.
597
00:43:37,600 --> 00:43:42,180
And then it is kind of
strange to say that, well, I
598
00:43:42,180 --> 00:43:44,860
have to divide by N factorial
because all of these
599
00:43:44,860 --> 00:43:45,760
are identical.
600
00:43:45,760 --> 00:43:48,950
Again, classically
these particles
601
00:43:48,950 --> 00:43:51,070
are following
specific trajectories.
602
00:43:51,070 --> 00:43:54,170
And you know where in
phase space they are.
603
00:43:54,170 --> 00:43:58,530
Whereas quantum mechanically,
you can't tell that apart.
604
00:43:58,530 --> 00:44:02,670
So quantum mechanically, as
we will describe later, rather
605
00:44:02,670 --> 00:44:04,820
than classical statistical
mechanics-- when
606
00:44:04,820 --> 00:44:07,880
we do quantum statistical
mechanics-- if you have
607
00:44:07,880 --> 00:44:10,580
identical particles,
you have to write down
608
00:44:10,580 --> 00:44:13,820
of a wave function that
is either symmetric or
609
00:44:13,820 --> 00:44:16,960
anti-symmetric under the
exchange of particles.
610
00:44:16,960 --> 00:44:19,500
And when we do eventually
the calculations
611
00:44:19,500 --> 00:44:23,070
for these factors of
1 over N factorial
612
00:44:23,070 --> 00:44:26,070
will emerge very naturally.
613
00:44:26,070 --> 00:44:31,070
So I think different people
have different perspectives.
614
00:44:31,070 --> 00:44:34,540
My own perspective is
that this factor really
615
00:44:34,540 --> 00:44:39,130
is due to the quantum
origin of identity.
616
00:44:39,130 --> 00:44:42,290
And classically, you have to
sort of fudge it and put it
617
00:44:42,290 --> 00:44:43,670
over there.
618
00:44:43,670 --> 00:44:46,710
But some people say that really
it's a matter of measurements.
619
00:44:46,710 --> 00:44:50,090
And if you can't really tell
A and B sufficiently apart,
620
00:44:50,090 --> 00:44:51,800
then you don't know.
621
00:44:51,800 --> 00:44:53,890
I always go back
to the computer.
622
00:44:53,890 --> 00:44:56,260
And say, well, the
computer can tell.
623
00:44:56,260 --> 00:44:58,990
But it's kind of
immaterial at this stage.
624
00:44:58,990 --> 00:45:03,390
It's obvious that for all
practical purposes for things
625
00:45:03,390 --> 00:45:08,040
that are identical you have
to divide by this factor.
626
00:45:08,040 --> 00:45:12,210
So what happens if you
divide by that factor?
627
00:45:12,210 --> 00:45:17,180
So I have changed all
of my calculations now.
628
00:45:17,180 --> 00:45:24,040
So when I do the log of--
previously I had V to the N.
629
00:45:24,040 --> 00:45:29,300
And it gave me N log
V. Now, I have log of V
630
00:45:29,300 --> 00:45:32,550
to the N divided by N factorial.
631
00:45:32,550 --> 00:45:36,780
So I will get my Sterling's
approximation additional factor
632
00:45:36,780 --> 00:45:42,300
of minus N log N plus N, which
I can sort of absorb here
633
00:45:42,300 --> 00:45:43,110
in this fashion.
634
00:45:48,825 --> 00:45:53,450
Now you say, well, having
done that, you have to first
635
00:45:53,450 --> 00:45:57,610
of all show me that you
fixed the case of this change
636
00:45:57,610 --> 00:46:00,960
in entropy for
identical particles,
637
00:46:00,960 --> 00:46:04,380
but also you should show me
that the previous case where
638
00:46:04,380 --> 00:46:07,570
we know there has to be an
increase in entropy just
639
00:46:07,570 --> 00:46:12,640
because of the gas being
different that that is not
640
00:46:12,640 --> 00:46:15,530
changed because of this
modification that you make.
641
00:46:15,530 --> 00:46:17,280
So let's check that.
642
00:46:17,280 --> 00:46:25,490
So for distinct gases, what
would be the generalization
643
00:46:25,490 --> 00:46:29,648
of this form Sf minus
Si divided by kV?
644
00:46:35,310 --> 00:46:38,820
Well, what happens here?
645
00:46:38,820 --> 00:46:51,000
In the case of the final object,
I have to divide N1 log of V.
646
00:46:51,000 --> 00:46:58,640
But that V really becomes
V divided by N1, because
647
00:46:58,640 --> 00:47:02,920
in the volume of size
V, I have N1 oxygen
648
00:47:02,920 --> 00:47:04,630
that I can't tell apart.
649
00:47:04,630 --> 00:47:08,630
So I divide by the N1
factorial for the oxygens.
650
00:47:08,630 --> 00:47:13,020
And then I have N2 methanes
that I can't tell apart
651
00:47:13,020 --> 00:47:18,560
in that volume, so I divide by
essentially N2 factorial that
652
00:47:18,560 --> 00:47:20,840
goes over there.
653
00:47:20,840 --> 00:47:30,450
The initial change is over
here I would have N1 log of V1
654
00:47:30,450 --> 00:47:32,440
over N1.
655
00:47:32,440 --> 00:47:41,980
And here I would have
had N2 log of V2 over N2.
656
00:47:41,980 --> 00:47:46,100
So every one of these
expressions that was previously
657
00:47:46,100 --> 00:47:51,870
log V, and I had four
of them, gets changed.
658
00:47:51,870 --> 00:47:57,150
But they get change precisely in
a manner that this N1 log of N1
659
00:47:57,150 --> 00:48:00,530
here cancels this N1
log of and N1 here.
660
00:48:00,530 --> 00:48:05,140
This N2 log of N2 here cancels
this N2 log of N2 here.
661
00:48:05,140 --> 00:48:08,320
So the delta S that I get
is precisely the same thing
662
00:48:08,320 --> 00:48:09,960
as I had before.
663
00:48:09,960 --> 00:48:18,230
I will get N1 log of V over
V1 plus N2 log of V over V2.
664
00:48:21,230 --> 00:48:27,240
So this division,
because the oxygens were
665
00:48:27,240 --> 00:48:29,370
identical to
themselves and methanes
666
00:48:29,370 --> 00:48:31,670
were identical to
themselves, does not
667
00:48:31,670 --> 00:48:37,110
change the mixing entropy
of oxygen and nitrogen.
668
00:48:37,110 --> 00:48:40,520
But let's say that both
gases are the same.
669
00:48:40,520 --> 00:48:42,380
They're both oxygen.
670
00:48:42,380 --> 00:48:43,130
Then what happens?
671
00:48:50,890 --> 00:48:55,500
Now, in the final
state, I have a box.
672
00:48:55,500 --> 00:48:59,110
It has a N1 plus N2 particles
that are all oxygen.
673
00:48:59,110 --> 00:49:01,370
I can't tell them apart.
674
00:49:01,370 --> 00:49:07,140
So the contribution
from the phase space
675
00:49:07,140 --> 00:49:14,450
would be N1 plus N2 log of the
volume divided by N1 plus N2
676
00:49:14,450 --> 00:49:15,240
factorial.
677
00:49:15,240 --> 00:49:18,460
That ultimately will give me
a factor of N1 plus N2 here.
678
00:49:22,550 --> 00:49:24,800
The initial entropy
is exactly the one
679
00:49:24,800 --> 00:49:26,706
that I calculated before.
680
00:49:26,706 --> 00:49:32,040
For the line above,
I have N1 log of V1
681
00:49:32,040 --> 00:49:37,157
over N1 minus N2
log of V2 over N2.
682
00:49:45,440 --> 00:49:48,840
Now certainly, I still expect
to see some mixing entropy
683
00:49:48,840 --> 00:49:53,870
if I have a box of oxygen
that is at very low pressure
684
00:49:53,870 --> 00:49:56,340
and is very dilute,
and I open it
685
00:49:56,340 --> 00:49:59,185
into this room, which is
at much higher pressure
686
00:49:59,185 --> 00:50:01,600
and is much more dense.
687
00:50:01,600 --> 00:50:03,520
So really, the
case where I don't
688
00:50:03,520 --> 00:50:06,210
expect to see any
change in entropy
689
00:50:06,210 --> 00:50:09,505
is when the two boxes
have the same density.
690
00:50:12,680 --> 00:50:14,860
And hence, when I mix
them, I would also
691
00:50:14,860 --> 00:50:16,255
have exactly the same density.
692
00:50:21,170 --> 00:50:26,090
And you can see that, therefore,
all of these factors that
693
00:50:26,090 --> 00:50:30,370
are in the log are of the
inverse of the same density.
694
00:50:30,370 --> 00:50:33,320
And there's N1 plus N2
of them that's positive.
695
00:50:33,320 --> 00:50:36,100
And N1 plus N2 of
them that is negative.
696
00:50:36,100 --> 00:50:38,830
So the answer in
this case, as long
697
00:50:38,830 --> 00:50:44,020
as I try to mix identical
particles of the same density,
698
00:50:44,020 --> 00:50:46,340
if I include this
correction to the phase
699
00:50:46,340 --> 00:50:48,586
space of identical
particles, the answer
700
00:50:48,586 --> 00:50:54,438
will be [? 0. ?] Yes?
701
00:50:54,438 --> 00:50:58,422
AUDIENCE: Question,
[INAUDIBLE] in terms
702
00:50:58,422 --> 00:51:04,398
of the revolution
of the [INAUDIBLE]
703
00:51:04,398 --> 00:51:09,400
there is no
transition [INAUDIBLE]
704
00:51:09,400 --> 00:51:13,388
so that your temporary, and say
like, oxygen and nitrogen can
705
00:51:13,388 --> 00:51:15,778
catch a molecule, put it
in a [? aspertometer. ?]
706
00:51:15,778 --> 00:51:18,168
and have different isotopes.
707
00:51:18,168 --> 00:51:20,668
You can take like closed
isotopes of oxygen
708
00:51:20,668 --> 00:51:22,604
and still tell them apart.
709
00:51:22,604 --> 00:51:25,992
But this is like
their continuous way
710
00:51:25,992 --> 00:51:28,684
of choosing a pair
of gases which
711
00:51:28,684 --> 00:51:31,980
would be arbitrarily
closed in atomic mass.
712
00:51:31,980 --> 00:51:34,500
MEHRAN KARDAR: So,
as I said, there
713
00:51:34,500 --> 00:51:38,030
are alternative explanations
that I've heard.
714
00:51:38,030 --> 00:51:40,530
And that's precisely
one of them.
715
00:51:40,530 --> 00:51:44,620
And my counter is that
what we are putting here
716
00:51:44,620 --> 00:51:47,670
is the volume of phase space.
717
00:51:47,670 --> 00:51:52,180
And to me that has a
very specific meaning.
718
00:51:52,180 --> 00:51:55,120
That is there's a set of
coordinates and momenta
719
00:51:55,120 --> 00:51:58,960
that are moving according
to Hamiltonian trajectories.
720
00:51:58,960 --> 00:52:03,690
And in principle, there
is a computer nature
721
00:52:03,690 --> 00:52:06,060
that is following
these trajectories,
722
00:52:06,060 --> 00:52:08,800
or I can actually put
them on the computer.
723
00:52:08,800 --> 00:52:11,920
And then no matter
how long I run
724
00:52:11,920 --> 00:52:14,980
and they're identical
oxygen molecules,
725
00:52:14,980 --> 00:52:18,010
I start with number one
here, numbers two here.
726
00:52:18,010 --> 00:52:21,420
The computer will say that this
is the trajectory of number one
727
00:52:21,420 --> 00:52:23,760
and this is the
trajectory of numbers two.
728
00:52:23,760 --> 00:52:28,030
So unless I change my
definition of phase space
729
00:52:28,030 --> 00:52:32,710
and how I am calculating
things, I run into this paradox.
730
00:52:32,710 --> 00:52:36,050
So what you're saying
is forget about that.
731
00:52:36,050 --> 00:52:39,720
It's just can tell isotopes
apart or something like that.
732
00:52:39,720 --> 00:52:41,140
And I'm saying that that's fine.
733
00:52:41,140 --> 00:52:43,480
That's perspective,
but it has nothing
734
00:52:43,480 --> 00:52:45,091
to do with phase space counting.
735
00:52:51,300 --> 00:52:51,800
OK?
736
00:52:54,560 --> 00:52:57,100
Fine, now, why didn't
I calculate this?
737
00:52:57,100 --> 00:53:02,220
It was also for the
same reason, because we
738
00:53:02,220 --> 00:53:07,460
expect to have quantities
that are extensive
739
00:53:07,460 --> 00:53:11,362
and quantities
that are intensive.
740
00:53:11,362 --> 00:53:13,950
And therefore, if I
were to, for example,
741
00:53:13,950 --> 00:53:16,630
calculate this object,
that it should be something
742
00:53:16,630 --> 00:53:18,672
that is intensive.
743
00:53:18,672 --> 00:53:22,420
Now the problem is that if I
take a derivative with respect
744
00:53:22,420 --> 00:53:26,470
N, I have log V.
And log V is clearly
745
00:53:26,470 --> 00:53:30,590
something that does not
grow proportionately to size
746
00:53:30,590 --> 00:53:33,890
but grows proportionately
to size logarithmically.
747
00:53:33,890 --> 00:53:36,640
So if I make volume
twice as big,
748
00:53:36,640 --> 00:53:40,590
I will get an additional factor
of log 2 here contribution
749
00:53:40,590 --> 00:53:42,470
to the chemical potential.
750
00:53:42,470 --> 00:53:45,500
And that does not make sense.
751
00:53:45,500 --> 00:53:52,520
But when I do this identity,
then this V becomes V over N.
752
00:53:52,520 --> 00:53:56,270
And then everything
becomes nicely intensive.
753
00:53:56,270 --> 00:54:02,520
So if I allowed now to
replace this V over N,
754
00:54:02,520 --> 00:54:14,774
then I can calculate V over T
as dS by dN at constant E and V.
755
00:54:14,774 --> 00:54:17,470
And so then
essentially I will get
756
00:54:17,470 --> 00:54:22,880
to drop the factor of log N that
comes in front, so I will get
757
00:54:22,880 --> 00:54:28,840
kT log of V over
N. And then I would
758
00:54:28,840 --> 00:54:40,060
have 3/2 log of something, which
I can put together as 4 pi N
759
00:54:40,060 --> 00:54:45,030
E over 3N raised
to the 3/2 power.
760
00:54:49,730 --> 00:54:53,980
And you can see that
there were these E's
761
00:54:53,980 --> 00:54:57,500
from Sterling's
approximation up there that
762
00:54:57,500 --> 00:55:01,265
got dropped here, because
you can also take derivative
763
00:55:01,265 --> 00:55:04,260
with respect to the
N's that are inside.
764
00:55:04,260 --> 00:55:07,310
And you can check that the
function of derivatives
765
00:55:07,310 --> 00:55:09,320
with respects to the
N's that are inside
766
00:55:09,320 --> 00:55:11,415
is precisely to get
rid of those factors.
767
00:55:15,841 --> 00:55:16,340
OK?
768
00:55:23,850 --> 00:55:26,610
Now, there is still
one other thing
769
00:55:26,610 --> 00:55:30,950
that is not wrong, but
kind of like jarring
770
00:55:30,950 --> 00:55:32,430
about the expressions
that they've
771
00:55:32,430 --> 00:55:39,280
had so far in that right
from the beginning,
772
00:55:39,280 --> 00:55:43,510
I said that you can certainly
calculate entropies out
773
00:55:43,510 --> 00:55:49,710
of probabilities as minus
log of P average if you like.
774
00:55:49,710 --> 00:55:51,610
But it makes sense
only if you're
775
00:55:51,610 --> 00:55:55,170
dealing with discrete
variables, because when you're
776
00:55:55,170 --> 00:55:57,220
dealing with continual
variables and you
777
00:55:57,220 --> 00:55:59,870
have a probability density.
778
00:55:59,870 --> 00:56:01,590
And the probability
density depends
779
00:56:01,590 --> 00:56:03,490
on the units of measurement.
780
00:56:03,490 --> 00:56:06,240
And if you were to change
measurement from meters
781
00:56:06,240 --> 00:56:08,580
to centimeters or
something else,
782
00:56:08,580 --> 00:56:13,460
then there will be changes
in the probability densities,
783
00:56:13,460 --> 00:56:17,750
which would then modify the
various factors over here.
784
00:56:17,750 --> 00:56:22,080
And that's really also
is reflected ultimately
785
00:56:22,080 --> 00:56:24,720
in the fact that these
combinations of terms
786
00:56:24,720 --> 00:56:28,890
that I have written
here have dimensions.
787
00:56:28,890 --> 00:56:32,250
And it is kind
of, again, jarring
788
00:56:32,250 --> 00:56:36,450
to have expressions inside the
logarithm or in the exponential
789
00:56:36,450 --> 00:56:39,550
that our not dimensionless.
790
00:56:39,550 --> 00:56:43,480
So it would be good
if we had some way
791
00:56:43,480 --> 00:56:46,500
of making all of
these dimensionless.
792
00:56:46,500 --> 00:56:50,570
And you say, well,
really the origin of it
793
00:56:50,570 --> 00:56:53,430
is all the way back
here, when I was
794
00:56:53,430 --> 00:56:56,050
calculating volumes
in phase space.
795
00:56:56,050 --> 00:56:59,060
And volumes in phase
space have dimensions.
796
00:56:59,060 --> 00:57:03,560
And that dimensions of
pq raised to the 3N power
797
00:57:03,560 --> 00:57:07,240
really survives all
the way down here.
798
00:57:07,240 --> 00:57:12,150
So I can say, OK, I
choose some quantity
799
00:57:12,150 --> 00:57:15,775
as a reference that has the
right dimensions of the product
800
00:57:15,775 --> 00:57:18,690
of p and q, which is an action.
801
00:57:18,690 --> 00:57:22,790
And I divide all
of my measurements
802
00:57:22,790 --> 00:57:28,140
by that reference unit, so
that, for example, here I
803
00:57:28,140 --> 00:57:32,130
have 3N factors of this.
804
00:57:32,130 --> 00:57:34,120
Or let's say each
one of them is 3.
805
00:57:34,120 --> 00:57:39,720
I divide by some quantity
that has units of action .
806
00:57:39,720 --> 00:57:41,680
And then I will be set.
807
00:57:41,680 --> 00:57:47,008
So basically, the units of this
h is the product of p and q.
808
00:57:51,160 --> 00:57:56,050
Now, at this point
we have no way
809
00:57:56,050 --> 00:57:59,330
of choosing some h as
opposed to another h.
810
00:57:59,330 --> 00:58:04,070
And so by adding that factor,
we can make things look nicer.
811
00:58:04,070 --> 00:58:08,610
But then things are undefined
after this factor of h.
812
00:58:08,610 --> 00:58:11,150
When we do quantum
mechanics, another thing
813
00:58:11,150 --> 00:58:14,640
that quantum mechanics
does is to provide us
814
00:58:14,640 --> 00:58:18,670
with precisely [? age ?] of
Planck's constant as a measure
815
00:58:18,670 --> 00:58:21,900
of these kinds of integrations.
816
00:58:21,900 --> 00:58:25,390
So when we eventually
you go to calculate, say,
817
00:58:25,390 --> 00:58:28,900
the ideal gas or any
other mechanic system
818
00:58:28,900 --> 00:58:32,180
that involves p and q
in quantum mechanics,
819
00:58:32,180 --> 00:58:34,400
then the phase space
becomes discretized.
820
00:58:34,400 --> 00:58:39,030
You would have-- The
appropriate description
821
00:58:39,030 --> 00:58:42,730
would have energies that are
discretized corresponding
822
00:58:42,730 --> 00:58:46,650
to various other discretization
that are eventually
823
00:58:46,650 --> 00:58:52,670
the equivalent to dividing
by this Planck's constant.
824
00:58:52,670 --> 00:58:56,400
Ultimately, I will
have additionally
825
00:58:56,400 --> 00:59:01,600
a factor of h squared
appearing here.
826
00:59:01,600 --> 00:59:05,920
And it will make everything
nicely that much [? less. ?]
827
00:59:05,920 --> 00:59:16,700
None of these other quantities
that I mentioned calculated
828
00:59:16,700 --> 00:59:18,572
would be affected by this.
829
00:59:28,990 --> 00:59:31,250
So essentially,
what I'm saying is
830
00:59:31,250 --> 00:59:38,340
that you are going to
use a measure for phase
831
00:59:38,340 --> 00:59:43,960
space of identical particles.
832
00:59:51,310 --> 00:59:59,270
Previously we had a product,
d cubed Pi, d cubed Qi.
833
00:59:59,270 --> 01:00:02,430
This is what we were
integrating and requiring
834
01:00:02,430 --> 01:00:06,540
that this integration
will give us [INAUDIBLE].
835
01:00:06,540 --> 01:00:11,950
Now, we will change this to
divide by this N factorial,
836
01:00:11,950 --> 01:00:14,920
if the particles are identical.
837
01:00:14,920 --> 01:00:21,560
And we divide by h
to the 3N because
838
01:00:21,560 --> 01:00:25,386
of the number of pairs of
pq that appear in this.
839
01:00:30,850 --> 01:00:35,020
The justification to come when
we ultimately do quantum study.
840
01:00:40,597 --> 01:00:41,180
Any questions?
841
01:00:59,640 --> 01:01:04,780
So I said that this
prescription when
842
01:01:04,780 --> 01:01:08,480
we look at a system
at complete isolation,
843
01:01:08,480 --> 01:01:12,360
and therefore, specify
fully its energy
844
01:01:12,360 --> 01:01:16,300
is the microcanonical
ensemble, as opposed
845
01:01:16,300 --> 01:01:27,440
to the canonical
ensemble, whereas the set
846
01:01:27,440 --> 01:01:32,145
of microscopic parameters
that you identified
847
01:01:32,145 --> 01:01:39,400
with your system, you replace
the energy with temperature.
848
01:01:39,400 --> 01:01:42,630
So in general,
let's say there will
849
01:01:42,630 --> 01:01:48,590
be some bunch of
displacements, x, that give you
850
01:01:48,590 --> 01:01:51,490
the work content to the system.
851
01:01:51,490 --> 01:01:54,550
Just like we fixed over there
the volume and the number
852
01:01:54,550 --> 01:02:00,420
of particles, let's say that
all of the work parameters,
853
01:02:00,420 --> 01:02:03,130
such as x
microscopically, we will
854
01:02:03,130 --> 01:02:06,130
fix in this canonical ensemble.
855
01:02:06,130 --> 01:02:09,040
So however, the
ensemble is one in which
856
01:02:09,040 --> 01:02:12,640
the energy is not specified.
857
01:02:12,640 --> 01:02:15,550
And so how do I
imagine that I can
858
01:02:15,550 --> 01:02:18,120
maintain a system
at temperature T?
859
01:02:18,120 --> 01:02:21,740
Well, if this room is at
some particular temperature,
860
01:02:21,740 --> 01:02:25,150
I assume that smaller objects
that I put in this room
861
01:02:25,150 --> 01:02:27,480
will come to the
same temperature.
862
01:02:27,480 --> 01:02:31,260
So the general prescription
for beginning something other
863
01:02:31,260 --> 01:02:34,860
than temperature T is
to put it in contact
864
01:02:34,860 --> 01:02:37,550
with something that
is much bigger.
865
01:02:37,550 --> 01:02:39,390
So let's call this
to be a reservoir.
866
01:02:43,330 --> 01:02:48,420
And we put our system, which
we assume to be smaller,
867
01:02:48,420 --> 01:02:49,580
in contact with it.
868
01:02:49,580 --> 01:02:53,810
And we allow it to exchange
heat with the reservoir.
869
01:02:58,570 --> 01:03:04,600
Now I did this way of
managing the system
870
01:03:04,600 --> 01:03:07,750
to come to a
temperature T, which
871
01:03:07,750 --> 01:03:09,990
is the characteristic
of a big reservoir.
872
01:03:09,990 --> 01:03:11,980
Imagine that you have a lake.
873
01:03:11,980 --> 01:03:15,390
And you put your gas or
something else inside the lake.
874
01:03:15,390 --> 01:03:21,140
And it will equilibrate to
the temperature of the lake.
875
01:03:21,140 --> 01:03:23,900
I will assume that
the two of them,
876
01:03:23,900 --> 01:03:26,610
the system and the
reservoir, just
877
01:03:26,610 --> 01:03:30,450
for the purpose of my being
able to do some computation,
878
01:03:30,450 --> 01:03:33,230
are isolated from the
rest of the universe,
879
01:03:33,230 --> 01:03:38,750
so that the system plus
reservoir is microcanonical.
880
01:03:44,400 --> 01:03:52,620
And the sum total of their
energies is sum E total.
881
01:03:59,320 --> 01:04:05,120
So now this system still
is something like a gas.
882
01:04:05,120 --> 01:04:08,970
It's a has a huge number of
potential degrees of freedom.
883
01:04:08,970 --> 01:04:12,270
And these potential number
of degrees of freedom
884
01:04:12,270 --> 01:04:16,020
can be captured through the
microstate of the system,
885
01:04:16,020 --> 01:04:17,660
u sub s.
886
01:04:17,660 --> 01:04:21,390
And similarly, the water
particle in the lake
887
01:04:21,390 --> 01:04:22,600
have their own state.
888
01:04:22,600 --> 01:04:25,330
An there's some
microstate that describes
889
01:04:25,330 --> 01:04:28,880
the positions and the momenta
of all of the particles
890
01:04:28,880 --> 01:04:32,150
that are in the lake.
891
01:04:32,150 --> 01:04:32,940
Yes?
892
01:04:32,940 --> 01:04:35,400
AUDIENCE: When you're writing
the set of particles used
893
01:04:35,400 --> 01:04:37,750
to describe it, why
don't you write N?
894
01:04:37,750 --> 01:04:39,874
Since it said the number
of particles in the system
895
01:04:39,874 --> 01:04:40,560
is not fixed.
896
01:04:40,560 --> 01:04:43,800
MEHRAN KARDAR: Yes, so I
did want to [INAUDIBLE]
897
01:04:43,800 --> 01:04:48,660
but in principle I could add N.
I wanted to be kind of general.
898
01:04:48,660 --> 01:04:51,800
If you like X,
[? it ?] is allowed
899
01:04:51,800 --> 01:05:04,210
to include chemical
work type of an X.
900
01:05:04,210 --> 01:05:09,020
So what do I know?
901
01:05:09,020 --> 01:05:14,940
I know that there is also if
I want to describe microstates
902
01:05:14,940 --> 01:05:17,260
and their revolution,
I need to specify
903
01:05:17,260 --> 01:05:21,260
that there's a
Hamiltonian that governs
904
01:05:21,260 --> 01:05:23,780
the evolution of
these microstates.
905
01:05:23,780 --> 01:05:25,930
And presumably
there's a Hamiltonian
906
01:05:25,930 --> 01:05:31,610
that describes the evolution
of the reservoir microstate.
907
01:05:31,610 --> 01:05:35,910
And so presumably the
allowed microstates
908
01:05:35,910 --> 01:05:40,180
are ones in which
E total is made up
909
01:05:40,180 --> 01:05:44,090
of the energy of the system plus
the energy of the reservoir.
910
01:05:53,350 --> 01:06:04,300
So because the whole thing
is the microcanonical,
911
01:06:04,300 --> 01:06:07,810
I can assign a probability,
a joint probability,
912
01:06:07,810 --> 01:06:15,070
to finding some particular
mu s, mu r combination,
913
01:06:15,070 --> 01:06:18,040
just like we were
doing over there.
914
01:06:18,040 --> 01:06:21,180
You would say that
essentially this
915
01:06:21,180 --> 01:06:25,120
is a combination
of these 1s and 0s.
916
01:06:25,120 --> 01:06:30,940
So it is 0 if H of--
again, for simplicity I
917
01:06:30,940 --> 01:06:38,030
drop the s on the
system-- H of mu s plus H
918
01:06:38,030 --> 01:06:43,660
reservoir of mu reservoir
is not equal to E total.
919
01:06:43,660 --> 01:06:52,595
And it is 1 over some omega
of reservoir in the system
920
01:06:52,595 --> 01:06:53,095
otherwise.
921
01:06:58,980 --> 01:07:02,010
So this is just again,
throwing the dice,
922
01:07:02,010 --> 01:07:05,860
saying that it has so many
possible configurations, given
923
01:07:05,860 --> 01:07:08,210
that I know what
the total energy is.
924
01:07:08,210 --> 01:07:13,290
All the ones that are consistent
with that are allowed.
925
01:07:13,290 --> 01:07:17,150
Which is to say, I don't
really care about the lake.
926
01:07:17,150 --> 01:07:21,720
All I care is about to
the states of my gas,
927
01:07:21,720 --> 01:07:24,550
and say, OK, no problem, if
I have the joint probability
928
01:07:24,550 --> 01:07:27,950
distribution just
like I did over here,
929
01:07:27,950 --> 01:07:29,980
I get rid of all of
the degrees of freedom
930
01:07:29,980 --> 01:07:32,190
that I'm not interested in.
931
01:07:32,190 --> 01:07:37,370
So if I'm interested only
in the states of the system,
932
01:07:37,370 --> 01:07:41,680
I sum over or integrate
over-- so this would be a sum.
933
01:07:41,680 --> 01:07:44,200
This would be an
integration, whatever--
934
01:07:44,200 --> 01:07:46,085
of the joint probability
distribution.
935
01:07:50,230 --> 01:07:54,330
Now actually follow the
steps that I had over here
936
01:07:54,330 --> 01:07:58,040
when we were looking at the
momentum of a gas particle.
937
01:07:58,040 --> 01:08:00,940
I say that what
I have over here,
938
01:08:00,940 --> 01:08:04,165
this probability see
is this 1 over omega R,
939
01:08:04,165 --> 01:08:09,110
S. This is a function
that is either 1 or 0.
940
01:08:09,110 --> 01:08:13,660
And then I have so sum
over all configurations
941
01:08:13,660 --> 01:08:16,120
of the reservoir.
942
01:08:16,120 --> 01:08:23,050
But given that I said what
the microstate of the system
943
01:08:23,050 --> 01:08:30,735
is, then I know that the
reservoir has to take energy
944
01:08:30,735 --> 01:08:35,149
in total minus the amount
the microstates has taken.
945
01:08:35,149 --> 01:08:39,040
And I'm summing over all
of the microstates that
946
01:08:39,040 --> 01:08:41,830
are consistent with
the requirement
947
01:08:41,830 --> 01:08:44,569
that the energy in
the reservoir is
948
01:08:44,569 --> 01:08:48,050
E total minus H of microstate.
949
01:08:48,050 --> 01:08:50,910
So what that is that
the omega that I
950
01:08:50,910 --> 01:08:54,390
have for the reservoir--
and I don't know what it is,
951
01:08:54,390 --> 01:08:59,010
but whatever it is,
evaluated at the total energy
952
01:08:59,010 --> 01:09:03,250
minus the energy that is
taken outside the microstate.
953
01:09:03,250 --> 01:09:07,390
So again, exactly the reason
why this became E minus Pi
954
01:09:07,390 --> 01:09:08,670
squared over 2.
955
01:09:08,670 --> 01:09:14,000
This becomes E total
minus H of mu S. Except
956
01:09:14,000 --> 01:09:16,640
that I don't know either
what this is or what this is.
957
01:09:20,200 --> 01:09:22,790
Actually, I don't really
even care about this
958
01:09:22,790 --> 01:09:26,695
because all of the H dependents
on microstate dependents
959
01:09:26,695 --> 01:09:28,359
is in the numerator.
960
01:09:28,359 --> 01:09:31,890
So I write that as
proportional to exponential.
961
01:09:31,890 --> 01:09:36,060
And the log of omega
is the entropy.
962
01:09:36,060 --> 01:09:40,470
So I have the entropy of the
[INAUDIBLE] in units of kB,
963
01:09:40,470 --> 01:09:54,980
evaluated at the argument that
this E total minus H of mu S.
964
01:09:54,980 --> 01:09:57,370
So my statement is
that when I look
965
01:09:57,370 --> 01:10:02,030
at the entropy of the
reservoir as a function of E
966
01:10:02,030 --> 01:10:06,650
total minus the energy
that is taken out
967
01:10:06,650 --> 01:10:11,490
by the system, my
construction I assume
968
01:10:11,490 --> 01:10:16,440
that I'm putting a small
volume of gas in contact
969
01:10:16,440 --> 01:10:19,150
with a huge lake.
970
01:10:19,150 --> 01:10:24,520
So this total energy is
overwhelmingly larger
971
01:10:24,520 --> 01:10:28,230
than the amount of energy
that the system can occupy.
972
01:10:28,230 --> 01:10:32,530
So I can make a Taylor
expansion of this quantity
973
01:10:32,530 --> 01:10:39,920
and say that this is S R of E
total minus the derivative of S
974
01:10:39,920 --> 01:10:41,290
with respect to its energy.
975
01:10:41,290 --> 01:10:44,310
So the derivative
of the S reservoir
976
01:10:44,310 --> 01:10:47,990
with respect to the energy
of the reservoir times
977
01:10:47,990 --> 01:10:54,010
H of the microstate and
presumably higher order terms
978
01:10:54,010 --> 01:10:55,090
that are negligible.
979
01:10:58,340 --> 01:11:02,090
Now the next thing that is
important about the reservoir
980
01:11:02,090 --> 01:11:04,150
is you have this huge lake.
981
01:11:04,150 --> 01:11:07,850
Let's say it's exactly at some
temperature of 30 degrees.
982
01:11:07,850 --> 01:11:11,030
And you take some small
amount of energy from it
983
01:11:11,030 --> 01:11:12,610
to put in the system.
984
01:11:12,610 --> 01:11:14,910
The temperature of the
lake should not change.
985
01:11:14,910 --> 01:11:17,590
So that's the definition
of the reservoir.
986
01:11:17,590 --> 01:11:22,930
It's a system that is so big
that for the range of energies
987
01:11:22,930 --> 01:11:26,420
that we are considering,
this S by dE
988
01:11:26,420 --> 01:11:30,706
is 1 over the temperature that
characterizes the reservoir.
989
01:11:34,600 --> 01:11:38,070
So just like here,
but eventually
990
01:11:38,070 --> 01:11:41,520
the answer that we
got was something
991
01:11:41,520 --> 01:11:44,830
like the energy of the
particle divided by kT.
992
01:11:44,830 --> 01:11:48,940
Once I exponentiate, I
find that the probability
993
01:11:48,940 --> 01:11:54,370
to find the system in some
microstate is proportional to E
994
01:11:54,370 --> 01:12:01,250
to the minus of the energy of
that microstate divided by kT.
995
01:12:04,330 --> 01:12:06,570
And of course, there's
a bunch of other things
996
01:12:06,570 --> 01:12:11,206
that I have to eventually
put into a normalization that
997
01:12:11,206 --> 01:12:11,705
will cause.
998
01:12:15,390 --> 01:12:21,310
So in the canonical
prescription you sort of replace
999
01:12:21,310 --> 01:12:23,550
this throwing of
the dice and saying
1000
01:12:23,550 --> 01:12:26,630
that everything is equivalent
to saying that well,
1001
01:12:26,630 --> 01:12:30,750
each microstates can have
some particular energy.
1002
01:12:30,750 --> 01:12:34,800
And the probabilities
are partitioned according
1003
01:12:34,800 --> 01:12:39,480
to the Boltzmann weights
of these energies.
1004
01:12:39,480 --> 01:12:43,650
And clearly this quantity,
Z, the normalization
1005
01:12:43,650 --> 01:12:48,630
is obtained by integrating
over the entire space
1006
01:12:48,630 --> 01:12:50,870
of microstates, or
summing over them
1007
01:12:50,870 --> 01:12:54,865
if they are discrete of this
factor of E to the minus beta
1008
01:12:54,865 --> 01:13:01,370
H of mu S. And we'll use
this notation beta 1 over kT
1009
01:13:01,370 --> 01:13:02,837
sometimes for simplicity.
1010
01:13:28,200 --> 01:13:33,660
Now, the thing is that
thermodynamically, we
1011
01:13:33,660 --> 01:13:38,520
said that you can choose
any set of parameters,
1012
01:13:38,520 --> 01:13:42,070
as long as they are
independent, to describe
1013
01:13:42,070 --> 01:13:45,510
the macroscopic equilibrium
state of the system.
1014
01:13:45,510 --> 01:13:50,670
So what we did in the
microcanonical ensemble is
1015
01:13:50,670 --> 01:13:54,330
we specified a number of
things, such as energy.
1016
01:13:54,330 --> 01:13:56,805
And we derived the other
things, such as temperature.
1017
01:13:59,360 --> 01:14:03,750
So here, in the
canonical ensemble,
1018
01:14:03,750 --> 01:14:07,349
we have stated what the
temperature of the system is.
1019
01:14:07,349 --> 01:14:08,390
Well, then what happened?
1020
01:14:11,950 --> 01:14:16,060
On one hand, maybe we have
to worry because energy
1021
01:14:16,060 --> 01:14:19,580
is constantly being
exchanged with the reservoir.
1022
01:14:19,580 --> 01:14:25,770
And so the energy of the system
does not have a specific value.
1023
01:14:25,770 --> 01:14:28,420
There's a probability for it.
1024
01:14:28,420 --> 01:14:36,740
So probability of
system having energy
1025
01:14:36,740 --> 01:14:44,940
epsilon-- it doesn't
have a fixed energy.
1026
01:14:44,940 --> 01:14:47,770
There is a probability
that it should have energy.
1027
01:14:47,770 --> 01:14:49,850
And this probability,
let's say we indicate
1028
01:14:49,850 --> 01:14:55,610
with P epsilon given that
we know what temperature is.
1029
01:14:55,610 --> 01:14:58,920
Well, on one hand we
have this factor of E
1030
01:14:58,920 --> 01:15:01,980
to the minus epsilon over kT.
1031
01:15:01,980 --> 01:15:03,560
That comes from the [INAUDIBLE].
1032
01:15:06,710 --> 01:15:10,660
But there isn't a single
state that has that energy.
1033
01:15:10,660 --> 01:15:14,340
There's a whole
bunch of other states
1034
01:15:14,340 --> 01:15:18,770
of the system that
have that energy.
1035
01:15:18,770 --> 01:15:23,740
So as I scan the
microstates, there
1036
01:15:23,740 --> 01:15:28,800
will be a huge number of them,
omega of epsilon in number,
1037
01:15:28,800 --> 01:15:30,740
that have this right energy.
1038
01:15:30,740 --> 01:15:33,600
And so that's the
probability of the energy.
1039
01:15:33,600 --> 01:15:39,710
And I can write this
as E to the minus 1
1040
01:15:39,710 --> 01:15:42,610
over kT that I've called beta.
1041
01:15:42,610 --> 01:15:44,730
I have epsilon.
1042
01:15:44,730 --> 01:15:47,387
And then the log of
omega that I take
1043
01:15:47,387 --> 01:15:52,580
in the numerator
is S divided by Kb.
1044
01:15:52,580 --> 01:15:57,620
I can take that Kb here and
write this as T S of epsilon.
1045
01:15:57,620 --> 01:16:00,440
And so this kind of
should remind you
1046
01:16:00,440 --> 01:16:02,135
of something like a free energy.
1047
01:16:06,620 --> 01:16:14,980
But it tells you is
that this probability
1048
01:16:14,980 --> 01:16:22,310
to have some particular energy
is some kind of [? a form. ?]
1049
01:16:22,310 --> 01:16:25,300
Now note that again for
something like a gas
1050
01:16:25,300 --> 01:16:30,360
or whatever, we expect typical
values of both the energy
1051
01:16:30,360 --> 01:16:33,000
and entropy to be
quantities that
1052
01:16:33,000 --> 01:16:35,700
are proportional to
the size of the system.
1053
01:16:35,700 --> 01:16:39,470
As the size of the system
becomes exponentially large,
1054
01:16:39,470 --> 01:16:42,140
we would expect that
this probability would
1055
01:16:42,140 --> 01:16:45,800
be one of those things
that has portions that
1056
01:16:45,800 --> 01:16:49,190
let's say are exponentially
larger than any other portion.
1057
01:16:52,440 --> 01:16:55,720
There will be a factor of
E to the minus N something
1058
01:16:55,720 --> 01:16:58,870
that will really peak up,
let's say, the extremum
1059
01:16:58,870 --> 01:17:02,800
and make the extremum
overwhelmingly more likely
1060
01:17:02,800 --> 01:17:05,770
than other places.
1061
01:17:05,770 --> 01:17:08,450
Let's try to quantify
that a little bit better.
1062
01:17:10,990 --> 01:17:14,130
Once we have a
probability, we can also
1063
01:17:14,130 --> 01:17:16,920
start calculating averages.
1064
01:17:16,920 --> 01:17:22,566
So let's define what the
average energy of the system is.
1065
01:17:22,566 --> 01:17:25,590
The average energy
of the system is
1066
01:17:25,590 --> 01:17:30,380
obtained by summing
over all microstates.
1067
01:17:30,380 --> 01:17:34,020
The energy of that
microstate, the probability
1068
01:17:34,020 --> 01:17:40,580
of that microstate, which
is E to the minus beta H
1069
01:17:40,580 --> 01:17:44,900
microstate divided by the
partition function, which
1070
01:17:44,900 --> 01:17:48,400
is the sum-- OK,
the normalization,
1071
01:17:48,400 --> 01:17:51,760
which we will call the
partition function, which
1072
01:17:51,760 --> 01:17:54,074
is the sum over all
of these microstates.
1073
01:17:57,890 --> 01:18:03,290
Now this is something
that we've already seen.
1074
01:18:03,290 --> 01:18:08,180
If I look at this expression
in the denominator
1075
01:18:08,180 --> 01:18:13,610
that we call Z and has a
name, which is the partition
1076
01:18:13,610 --> 01:18:19,530
function, then it's
certainly a function of beta.
1077
01:18:19,530 --> 01:18:24,610
If I take a derivative of Z with
respect to beta, what happens
1078
01:18:24,610 --> 01:18:28,050
I'll bring down a
factor of H over here.
1079
01:18:28,050 --> 01:18:33,610
So the numerator up to
a sine is the derivative
1080
01:18:33,610 --> 01:18:36,330
of Z with respect to beta.
1081
01:18:36,330 --> 01:18:41,580
And the denominator is 1 over
Z. And so this is none other
1082
01:18:41,580 --> 01:18:47,090
than minus the log Z
with respect to beta.
1083
01:18:47,090 --> 01:18:52,900
So OK, fine, so the mean
value of this probability
1084
01:18:52,900 --> 01:18:57,890
is given by some
expression such as this.
1085
01:18:57,890 --> 01:19:07,850
Well, you can see that if I
were to repeat this process
1086
01:19:07,850 --> 01:19:11,980
and rather than
taking one derivative,
1087
01:19:11,980 --> 01:19:27,320
I will take n derivatives
and then divide by 1 over Z.
1088
01:19:27,320 --> 01:19:33,000
Each time I do that, I will
bring down a factor of H.
1089
01:19:33,000 --> 01:19:40,740
So this is going to give me
the average of H to the N.
1090
01:19:40,740 --> 01:19:43,490
The end moment of this
probability distribution
1091
01:19:43,490 --> 01:19:46,890
of energy is obtainable
by this procedure.
1092
01:19:50,920 --> 01:19:56,220
So now you recognize, oh, I've
seen things like such as this.
1093
01:19:56,220 --> 01:19:59,560
So clearly this
partition function
1094
01:19:59,560 --> 01:20:03,080
is something that
generates the moments
1095
01:20:03,080 --> 01:20:05,920
by taking subsequent
derivatives.
1096
01:20:05,920 --> 01:20:09,360
I can generate different
moments of this distribution.
1097
01:20:11,880 --> 01:20:13,570
But then there
was something else
1098
01:20:13,570 --> 01:20:16,040
that maybe this
should remind you,
1099
01:20:16,040 --> 01:20:19,450
which is that if there's
a quantity that generates
1100
01:20:19,450 --> 01:20:24,470
moments, then its log
generates cumulants.
1101
01:20:24,470 --> 01:20:31,590
So you would say,
OK, the nth cumulant
1102
01:20:31,590 --> 01:20:36,580
should be obtainable up to this
factor of minus 1 to the n,
1103
01:20:36,580 --> 01:20:41,630
as the nth derivative with
respect to the beta of logs.
1104
01:20:45,887 --> 01:20:48,560
And it's very easy
to check that indeed
1105
01:20:48,560 --> 01:20:50,970
if I were to take
two derivatives,
1106
01:20:50,970 --> 01:20:53,190
I will get the
expectation value of H
1107
01:20:53,190 --> 01:20:58,080
squared minus the average
of H squared, et cetera.
1108
01:20:58,080 --> 01:21:04,460
But the point is that
clearly this to log Z
1109
01:21:04,460 --> 01:21:08,730
is, again, something
that is extensive.
1110
01:21:08,730 --> 01:21:12,600
Another way of getting
the normalization-- I
1111
01:21:12,600 --> 01:21:15,600
guess I forgot to put
this 1 over Z here.
1112
01:21:21,990 --> 01:21:25,670
So now it is a perfectly
normalized object.
1113
01:21:25,670 --> 01:21:32,660
So another way to get
z would be to look
1114
01:21:32,660 --> 01:21:34,810
at the normalization
of the probability.
1115
01:21:34,810 --> 01:21:39,570
I could integrate over epsilon
this factor of E to the minus
1116
01:21:39,570 --> 01:21:45,580
beta epsilon minus
T S of epsilon.
1117
01:21:45,580 --> 01:21:51,710
And that would give
me Z. Now, again
1118
01:21:51,710 --> 01:21:54,750
the quantities that appear
in the exponent, energy--
1119
01:21:54,750 --> 01:21:57,010
entropy, their
difference, free energy--
1120
01:21:57,010 --> 01:21:59,180
are quantities
that are extensive.
1121
01:21:59,180 --> 01:22:02,850
So this Z is going
to be dominated again
1122
01:22:02,850 --> 01:22:04,950
by where this peak is.
1123
01:22:04,950 --> 01:22:08,370
And therefore, log of Z
will be proportional to log
1124
01:22:08,370 --> 01:22:10,170
of what we have over here.
1125
01:22:10,170 --> 01:22:12,660
And it be an extensive quantity.
1126
01:22:12,660 --> 01:22:16,720
So ultimately, my statement
is that this log of Z
1127
01:22:16,720 --> 01:22:23,510
is something that is order of N.
1128
01:22:23,510 --> 01:22:26,560
So we are, again,
kind of reminiscent
1129
01:22:26,560 --> 01:22:29,290
of the central limit theorem.
1130
01:22:29,290 --> 01:22:33,860
In a situation where we have
a probability distribution,
1131
01:22:33,860 --> 01:22:37,540
at large N, in which
all of the cumulants
1132
01:22:37,540 --> 01:22:42,150
are proportional to N. The
mean is proportional to N.
1133
01:22:42,150 --> 01:22:45,690
The variance is proportional
to N. All of the cumulants
1134
01:22:45,690 --> 01:22:48,010
are proportional
to N, which means
1135
01:22:48,010 --> 01:22:51,420
that essentially the extent of
the fluctuations that you have
1136
01:22:51,420 --> 01:23:00,380
over here are going to go the
order of the square root of N.
1137
01:23:00,380 --> 01:23:04,020
So the bridge, the thing
that again allows us,
1138
01:23:04,020 --> 01:23:09,130
while we have in principle in
the expression that we have
1139
01:23:09,130 --> 01:23:13,540
said, a variable
energy for the system.
1140
01:23:13,540 --> 01:23:18,530
In fact, in the limit of
things becoming extensive,
1141
01:23:18,530 --> 01:23:23,960
I know where that energy
is, up to fluctuations
1142
01:23:23,960 --> 01:23:25,680
or up to uncertainty
that is only
1143
01:23:25,680 --> 01:23:28,060
of the order of
square root of N.
1144
01:23:28,060 --> 01:23:32,580
And so the relative uncertainty
will vanish as the N goes
1145
01:23:32,580 --> 01:23:34,790
to infinity, limit
is approached.
1146
01:23:34,790 --> 01:23:36,570
So although again
we have something
1147
01:23:36,570 --> 01:23:39,410
that is in principle
probabilistic,
1148
01:23:39,410 --> 01:23:41,990
again, in the
thermodynamic sense
1149
01:23:41,990 --> 01:23:46,740
we can identify uniquely an
energy for our system as,
1150
01:23:46,740 --> 01:23:49,440
let's say, the mean value
or the most likely value.
1151
01:23:49,440 --> 01:23:53,250
They're all the same thing
of the order of 1 over N.
1152
01:23:53,250 --> 01:23:58,680
And again, to be more
precise, the variance
1153
01:23:58,680 --> 01:24:05,050
is clearly the second derivative
of log Z. 1 derivative of a log
1154
01:24:05,050 --> 01:24:07,560
Z is going to give
me the energy.
1155
01:24:07,560 --> 01:24:10,483
So this is going to
be d by d beta up
1156
01:24:10,483 --> 01:24:14,000
to a minus sign of the
energy or the expectation
1157
01:24:14,000 --> 01:24:17,820
value of the Hamiltonian,
which we identified
1158
01:24:17,820 --> 01:24:19,900
as the energy of the system.
1159
01:24:19,900 --> 01:24:21,615
The derivative with
respect to beta,
1160
01:24:21,615 --> 01:24:25,320
I can write as kB T squared.
1161
01:24:25,320 --> 01:24:28,530
The derivative of energy
with respect to T,
1162
01:24:28,530 --> 01:24:32,110
everything here we are doing
that conditions of no work.
1163
01:24:32,110 --> 01:24:37,030
So the variance is
in fact kB T squared,
1164
01:24:37,030 --> 01:24:40,880
the heat capacity of the system.
1165
01:24:40,880 --> 01:24:45,300
So the extent that these
fluctuations squared
1166
01:24:45,300 --> 01:24:50,680
is kB T squared times the
heat capacity the system.
1167
01:24:50,680 --> 01:24:53,790
OK, so next time
what we will do is
1168
01:24:53,790 --> 01:24:57,490
we will calculate the
results for the ideal gas.
1169
01:24:57,490 --> 01:24:59,790
First thing, the
canonical ensemble
1170
01:24:59,790 --> 01:25:02,990
to show that we get exactly
the same macroscopic
1171
01:25:02,990 --> 01:25:05,620
and microscopic descriptions.
1172
01:25:05,620 --> 01:25:09,460
And then we look
at other ensembles.
1173
01:25:09,460 --> 01:25:11,250
And that will
conclude the segment
1174
01:25:11,250 --> 01:25:13,700
that we have on
statistical mechanics
1175
01:25:13,700 --> 01:25:16,387
of non-interacting systems.