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PROFESSOR: We established that,
essentially, what we want to do

9
00:00:25,050 --> 00:00:29,500
is to describe the
properties of a system that

10
00:00:29,500 --> 00:00:30,253
is in equilibrium.

11
00:00:33,340 --> 00:00:39,060
And a system in equilibrium
is characterized

12
00:00:39,060 --> 00:00:41,920
by a certain number
of parameters.

13
00:00:41,920 --> 00:00:48,090
We discussed
displacement and forces

14
00:00:48,090 --> 00:00:52,200
that are used for
mechanical properties.

15
00:00:52,200 --> 00:00:56,570
We described how when systems
are in thermal equilibrium,

16
00:00:56,570 --> 00:01:01,800
the exchange of heat requires
that there is temperature

17
00:01:01,800 --> 00:01:04,590
that will be the
same between them.

18
00:01:04,590 --> 00:01:07,000
So that was where the
Zeroth Law came and told us

19
00:01:07,000 --> 00:01:11,410
that there is another
function of state.

20
00:01:11,410 --> 00:01:15,040
Then, we saw that,
from the First Law,

21
00:01:15,040 --> 00:01:18,020
there was energy, which is
another important function

22
00:01:18,020 --> 00:01:19,180
of state.

23
00:01:19,180 --> 00:01:26,445
And from the Second Law,
we arrived at entropy.

24
00:01:29,920 --> 00:01:33,000
And then by
manipulating these, we

25
00:01:33,000 --> 00:01:37,100
generated a whole set of
other functions, free energy,

26
00:01:37,100 --> 00:01:41,830
enthalpy, Gibbs free energy, the
grand potential, and the list

27
00:01:41,830 --> 00:01:43,850
goes on.

28
00:01:43,850 --> 00:01:46,670
And when the system
is in equilibrium,

29
00:01:46,670 --> 00:01:49,920
it has a well-defined
values of these quantities.

30
00:01:49,920 --> 00:01:52,670
You go from one equilibrium
to another equilibrium,

31
00:01:52,670 --> 00:01:54,740
and these quantities change.

32
00:01:54,740 --> 00:01:57,790
But of course, we saw that the
number of degrees of freedom

33
00:01:57,790 --> 00:02:00,700
that you have to
describe the system

34
00:02:00,700 --> 00:02:08,030
is indicated through looking
at the changes in energy, which

35
00:02:08,030 --> 00:02:10,740
if you were only
doing mechanical work,

36
00:02:10,740 --> 00:02:15,620
you would write as sum over all
possible ways of introducing

37
00:02:15,620 --> 00:02:17,755
mechanical work into the system.

38
00:02:20,296 --> 00:02:21,670
Then, we saw that
it was actually

39
00:02:21,670 --> 00:02:24,330
useful to separate
out the chemical work.

40
00:02:24,330 --> 00:02:26,410
So we could also
write this as a sum

41
00:02:26,410 --> 00:02:31,040
of an alpha chemical
potential number of particles.

42
00:02:31,040 --> 00:02:34,800
But there was also
ways of changing

43
00:02:34,800 --> 00:02:40,270
the energy of the system
through addition of heat.

44
00:02:40,270 --> 00:02:44,040
And so ultimately,
we saw that if there

45
00:02:44,040 --> 00:02:49,530
were n ways of doing
chemical and mechanical work,

46
00:02:49,530 --> 00:02:56,400
and one way of introducing heat
into the system, essentially

47
00:02:56,400 --> 00:03:00,440
n plus 1 variables are
sufficient to determine

48
00:03:00,440 --> 00:03:02,430
where you are in
this phase space.

49
00:03:02,430 --> 00:03:05,370
Once you have n
plus 1 of that list,

50
00:03:05,370 --> 00:03:08,990
you can input, in
principle, determine others

51
00:03:08,990 --> 00:03:11,380
as long as you have
not chosen things

52
00:03:11,380 --> 00:03:13,470
that are really
dependent on each other.

53
00:03:13,470 --> 00:03:16,060
So you have to choose
independent ones,

54
00:03:16,060 --> 00:03:19,650
and we had some discussion
of how that comes into play.

55
00:03:22,190 --> 00:03:26,080
So I said that today
we will briefly

56
00:03:26,080 --> 00:03:29,140
conclude with the
last, or the Third Law.

57
00:03:33,580 --> 00:03:35,790
This is the statement
about trying

58
00:03:35,790 --> 00:03:39,440
to calculate the
behavior of entropy

59
00:03:39,440 --> 00:03:41,880
as a function of temperature.

60
00:03:41,880 --> 00:03:46,650
And in principle,
you can imagine

61
00:03:46,650 --> 00:03:51,545
as a function of some coordinate
of your system-- capital X

62
00:03:51,545 --> 00:03:55,520
could indicate pressure,
volume, anything.

63
00:03:55,520 --> 00:03:58,590
You calculate that at
some particular value

64
00:03:58,590 --> 00:04:03,620
of temperature, T, T the
difference in entropy

65
00:04:03,620 --> 00:04:07,255
that you would have
between two points

66
00:04:07,255 --> 00:04:08,895
parametrized by X1 and X2.

67
00:04:11,590 --> 00:04:15,220
And in principle,
what you need to do

68
00:04:15,220 --> 00:04:21,149
is to find some kind of a
path for changing parameters

69
00:04:21,149 --> 00:04:28,340
from X1 to X2 and calculate,
in a reversible process, how

70
00:04:28,340 --> 00:04:31,370
much heat you have to
put into the system.

71
00:04:31,370 --> 00:04:35,955
Let's say at this fixed
temperature, T, divide by T. T

72
00:04:35,955 --> 00:04:43,350
is not changing along the
process from say X1 to X2.

73
00:04:43,350 --> 00:04:46,680
And this would be a
difference between the entropy

74
00:04:46,680 --> 00:04:51,490
that you would have
between these two

75
00:04:51,490 --> 00:04:56,540
quantities, between
these two points.

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00:04:56,540 --> 00:04:59,690
You could, in principle,
then repeat this process

77
00:04:59,690 --> 00:05:06,540
at some lower temperature and
keep going all the way down

78
00:05:06,540 --> 00:05:09,530
to 0 temperature.

79
00:05:09,530 --> 00:05:20,120
What Nernst observed was that as
he went through this procedure

80
00:05:20,120 --> 00:05:28,250
to lower and lower temperatures,
this difference-- Let's

81
00:05:28,250 --> 00:05:35,840
call it delta s of T going
from X1 to X2 goes to 0.

82
00:05:42,320 --> 00:05:48,580
So it looks like, certainly
at this temperature,

83
00:05:48,580 --> 00:05:52,270
there is a change in entropy
going from one to another.

84
00:05:52,270 --> 00:05:53,680
There's also a change.

85
00:05:53,680 --> 00:05:56,530
This change gets
smaller and smaller

86
00:05:56,530 --> 00:05:59,330
as if, when you get
to 0 temperature,

87
00:05:59,330 --> 00:06:03,240
the value of your entropy is
independent of X. Whatever

88
00:06:03,240 --> 00:06:06,600
X you choose, you'll have
the same value of entropy.

89
00:06:09,390 --> 00:06:16,030
Now, that led to, after a while,
to a more ambitious version

90
00:06:16,030 --> 00:06:19,520
statement of the Third Law
that I will write down,

91
00:06:19,520 --> 00:06:31,515
which is that the entropy of
all substances at the zero

92
00:06:31,515 --> 00:06:44,680
of thermodynamic temperature is
the same and can be set to 0.

93
00:06:44,680 --> 00:06:52,500
Same universal
constant, set to 0.

94
00:06:56,200 --> 00:06:59,810
It's, in principle,
through these integration

95
00:06:59,810 --> 00:07:01,940
from one point to another
point, the only thing

96
00:07:01,940 --> 00:07:07,520
that you can calculate is the
difference between entropies.

97
00:07:07,520 --> 00:07:10,920
And essentially, this
suggests that the difference

98
00:07:10,920 --> 00:07:14,610
between entropies goes to 0,
but let's be more ambitious

99
00:07:14,610 --> 00:07:18,000
and say that even if you
look at different substances

100
00:07:18,000 --> 00:07:26,240
and you go to 0 temperature,
all of them have a unique value.

101
00:07:26,240 --> 00:07:30,920
And so there's more
evidence for being

102
00:07:30,920 --> 00:07:35,830
able to do this for
different substances via what

103
00:07:35,830 --> 00:07:41,930
is called allotropic state.

104
00:07:44,830 --> 00:07:50,450
So for example, some materials
can exist potentially

105
00:07:50,450 --> 00:07:53,440
in two different
crystalline states that

106
00:07:53,440 --> 00:07:58,390
are called allotropes,
for example, sulfur

107
00:07:58,390 --> 00:08:01,200
as a function of temperature.

108
00:08:01,200 --> 00:08:10,130
If you lower it's
temperature very slowly,

109
00:08:10,130 --> 00:08:19,740
it stays in some foreign all
the way down to 0 temperature.

110
00:08:19,740 --> 00:08:24,480
So if you change its temperature
rapidly, it stays in one form

111
00:08:24,480 --> 00:08:29,760
all the way to 0 temperature
in crystalline structure

112
00:08:29,760 --> 00:08:32,559
that is called monoclinic.

113
00:08:32,559 --> 00:08:35,500
If you cool it
very, very slowly,

114
00:08:35,500 --> 00:08:41,110
there is a temperature
around 40 degrees Celsius

115
00:08:41,110 --> 00:08:45,860
at which it makes a transition
to a different crystal

116
00:08:45,860 --> 00:08:48,140
structure.

117
00:08:48,140 --> 00:08:49,390
That is rhombohedral.

118
00:08:53,300 --> 00:08:57,116
And the thing that
I am plotting here,

119
00:08:57,116 --> 00:08:59,240
as a function of temperature,
is the heat capacity.

120
00:09:06,460 --> 00:09:11,320
And so if you are, let's
say, around room temperature,

121
00:09:11,320 --> 00:09:16,910
in principle you can say there's
two different forms of sulfur.

122
00:09:16,910 --> 00:09:20,670
One of them is truly stable,
and the other is metastable.

123
00:09:20,670 --> 00:09:25,950
That is, in principle, if
you rate what sufficiently

124
00:09:25,950 --> 00:09:27,700
is of the order of
[? centuries ?],

125
00:09:27,700 --> 00:09:31,900
you can get the transition from
this form to the stable form.

126
00:09:31,900 --> 00:09:34,770
But for our purposes,
at room temperature,

127
00:09:34,770 --> 00:09:37,470
you would say that
at the scale of times

128
00:09:37,470 --> 00:09:40,260
that I'm observing things, there
are these 2 possible states

129
00:09:40,260 --> 00:09:45,560
that are both equilibrium
states of the same substance.

130
00:09:45,560 --> 00:09:47,650
Now using these two
equilibrium states,

131
00:09:47,650 --> 00:09:52,780
I can start to test this
Nernst theorem generalized

132
00:09:52,780 --> 00:09:54,910
to different substances.

133
00:09:54,910 --> 00:09:57,500
If you, again, regard
these two different things

134
00:09:57,500 --> 00:09:59,440
as different substances.

135
00:09:59,440 --> 00:10:03,580
You could say that if I want
to calculate the entropy just

136
00:10:03,580 --> 00:10:08,270
slightly above the transition,
I can come from two paths.

137
00:10:08,270 --> 00:10:12,610
I can either come
from path number one.

138
00:10:12,610 --> 00:10:15,930
Along path number
one, I would say

139
00:10:15,930 --> 00:10:22,200
that the entropy at
this Tc plus is obtained

140
00:10:22,200 --> 00:10:27,520
by integrating
degree heat capacity,

141
00:10:27,520 --> 00:10:39,020
so integral dT Cx
of T divided by T.

142
00:10:39,020 --> 00:10:42,760
This combination is
none other than dQ.

143
00:10:42,760 --> 00:10:47,180
Basically, the combination
of heat capacity dT

144
00:10:47,180 --> 00:10:49,240
is the amount of
heat that you have

145
00:10:49,240 --> 00:10:52,040
to put the substance to
change its temperature.

146
00:10:52,040 --> 00:10:58,100
And you do this all the
way from 0 to Tc plus.

147
00:10:58,100 --> 00:11:00,130
Let's say we go
along this path that

148
00:11:00,130 --> 00:11:05,290
corresponds to this
monoclinic way.

149
00:11:05,290 --> 00:11:11,260
And I'm using this Cm that
corresponds to this as opposed

150
00:11:11,260 --> 00:11:15,400
to 0 that corresponds to this.

151
00:11:15,400 --> 00:11:19,850
Another thing that I can
do-- and I made a mistake

152
00:11:19,850 --> 00:11:24,050
because what I really need
to do is to, in principle,

153
00:11:24,050 --> 00:11:27,710
add to this some
entropy that I would

154
00:11:27,710 --> 00:11:32,140
assign to this green state at 0
because this is the difference.

155
00:11:32,140 --> 00:11:36,270
So this is the
entropy that I would

156
00:11:36,270 --> 00:11:41,080
assign to the monoclinic
state at T close to 0.

157
00:11:41,080 --> 00:11:44,740
Going along the
orange path, I would

158
00:11:44,740 --> 00:11:49,820
say that S evaluated
at Tc plus is

159
00:11:49,820 --> 00:11:54,300
obtained by integrating from 0.

160
00:11:54,300 --> 00:11:59,110
Let's say to Tc
minus dT, the heat

161
00:11:59,110 --> 00:12:01,730
capacity of this rhombic phase.

162
00:12:06,580 --> 00:12:10,160
But when I get to just
below the transition,

163
00:12:10,160 --> 00:12:13,650
I want to go to just
above the transition.

164
00:12:13,650 --> 00:12:18,080
I have to actually be put in
certain amount of latent heat.

165
00:12:18,080 --> 00:12:25,130
So here I have to add
latent heat L, always

166
00:12:25,130 --> 00:12:29,120
at the temperatures Tc, to
gradually make the substance

167
00:12:29,120 --> 00:12:31,650
transition from
one to the other.

168
00:12:31,650 --> 00:12:35,470
So I have to add here L of Tc.

169
00:12:35,470 --> 00:12:40,650
This would be the
integration of dQ,

170
00:12:40,650 --> 00:12:46,160
but then I would have to add
the entropy that I would assign

171
00:12:46,160 --> 00:12:49,910
to the orange state
at 0 temperature.

172
00:12:49,910 --> 00:12:53,040
So this is something that
you can do experimentally.

173
00:12:53,040 --> 00:12:55,930
You can evaluate at these
integrals, and what you'll find

174
00:12:55,930 --> 00:12:59,970
is that these two
things are the same.

175
00:12:59,970 --> 00:13:03,250
So this is yet
another justification

176
00:13:03,250 --> 00:13:09,210
of this entropy
being independent

177
00:13:09,210 --> 00:13:13,000
of where you start
at 0 temperature.

178
00:13:13,000 --> 00:13:16,030
Again at this
point, if you like,

179
00:13:16,030 --> 00:13:18,100
you can by [INAUDIBLE]
state that this

180
00:13:18,100 --> 00:13:21,215
is 0 for everything
will start with 0.

181
00:13:27,730 --> 00:13:35,130
So this is a supposed new
law of thermodynamics.

182
00:13:35,130 --> 00:13:36,000
Is it useful?

183
00:13:36,000 --> 00:13:38,770
What can we deduce from that?

184
00:13:38,770 --> 00:13:40,830
So let's look at
the consequences.

185
00:13:46,280 --> 00:13:48,740
First thing is so what
I have established

186
00:13:48,740 --> 00:13:54,890
is that the limit
as T goes to 0 of S,

187
00:13:54,890 --> 00:13:57,440
irrespective of whatever
set of parameters

188
00:13:57,440 --> 00:14:02,920
I have-- so I pick T as one
of my n plus one coordinates,

189
00:14:02,920 --> 00:14:06,490
and I put some other
bunch of coordinates here.

190
00:14:06,490 --> 00:14:08,820
I take the limit
of this going to 0.

191
00:14:08,820 --> 00:14:09,870
This becomes 0.

192
00:14:13,410 --> 00:14:17,790
So that means, almost
by construction,

193
00:14:17,790 --> 00:14:24,540
that if I take the derivative of
S with respect to any of these

194
00:14:24,540 --> 00:14:30,650
coordinates-- if I take then
the limit as T goes to 0,

195
00:14:30,650 --> 00:14:33,790
this would be
fixed T. This is 0.

196
00:14:39,690 --> 00:14:40,190
Fine.

197
00:14:40,190 --> 00:14:44,220
So basically, this
is another way

198
00:14:44,220 --> 00:14:49,290
of stating that entropy
differences go through 0.

199
00:14:49,290 --> 00:14:52,890
But it does have a consequence
because one thing that you will

200
00:14:52,890 --> 00:14:55,797
frequently measure
are quantities,

201
00:14:55,797 --> 00:14:56,630
such as extensivity.

202
00:15:01,160 --> 00:15:03,700
What do I mean by that?

203
00:15:03,700 --> 00:15:05,050
Let's pick a displacement.

204
00:15:05,050 --> 00:15:07,650
Could be the length of a wire.

205
00:15:07,650 --> 00:15:11,210
Could be the volume of a gas.

206
00:15:11,210 --> 00:15:17,860
And we can ask if I were
to change temperature,

207
00:15:17,860 --> 00:15:21,030
how does that quantity change?

208
00:15:21,030 --> 00:15:26,230
So these are quantities
typically called alpha.

209
00:15:26,230 --> 00:15:29,020
Actually, usually
you would also divide

210
00:15:29,020 --> 00:15:35,115
by x to make them intensive
because otherwise x

211
00:15:35,115 --> 00:15:36,910
being extensive,
the whole quantity

212
00:15:36,910 --> 00:15:39,960
would have been extensive.

213
00:15:39,960 --> 00:15:44,840
Let's say we do this at fixed
corresponding displacement.

214
00:15:44,840 --> 00:15:48,430
So something that
is very relevant

215
00:15:48,430 --> 00:15:51,550
is you take the volume of
gas who changes temperature

216
00:15:51,550 --> 00:15:54,350
at fixed pressure, and
the volume of the gas

217
00:15:54,350 --> 00:15:59,070
will shrink or expand
according to this extensive.

218
00:15:59,070 --> 00:16:04,970
Now, this can be related to this
through Maxwell relationship.

219
00:16:04,970 --> 00:16:07,620
So let's see what I have to do.

220
00:16:07,620 --> 00:16:13,520
I have that dE is
something like Jdx plus,

221
00:16:13,520 --> 00:16:17,990
according to what I
have over there, TdS.

222
00:16:17,990 --> 00:16:21,940
I want to be able to write
a Maxwell relation that

223
00:16:21,940 --> 00:16:25,420
relates a derivative of x.

224
00:16:25,420 --> 00:16:28,510
So I want to make x
into a first derivative.

225
00:16:28,510 --> 00:16:31,640
So I look at E minus Jx.

226
00:16:31,640 --> 00:16:36,240
And this Jdx becomes minus xdJ.

227
00:16:36,240 --> 00:16:40,070
But I want to take a derivative
of x with respect not

228
00:16:40,070 --> 00:16:44,590
s, but with respect
to T. So I'll do that.

229
00:16:44,590 --> 00:16:48,130
This becomes a minus SdT.

230
00:16:48,130 --> 00:16:52,070
So now, I immediately see
that I will have a Maxwell

231
00:16:52,070 --> 00:16:58,110
relation that says dx
by dT at constant J

232
00:16:58,110 --> 00:17:02,850
is the same thing as
dS by dJ at constant T.

233
00:17:02,850 --> 00:17:09,321
So this is the same thing by
the Maxwell relation as dS

234
00:17:09,321 --> 00:17:23,000
by dJ at constant T. All right?

235
00:17:23,000 --> 00:17:27,599
This is one of these quantities,
therefore, as T goes 0,

236
00:17:27,599 --> 00:17:28,590
this goes to 0.

237
00:17:28,590 --> 00:17:33,370
And therefore, the
expansivity should go to 0.

238
00:17:33,370 --> 00:17:38,700
So any quantity that measures
expansion, contraction,

239
00:17:38,700 --> 00:17:42,070
or some other change as a
function of temperature,

240
00:17:42,070 --> 00:17:45,230
according to this law, as
you go through 0 temperature,

241
00:17:45,230 --> 00:17:46,350
should go to 0.

242
00:17:49,780 --> 00:17:54,100
There's one other quantity
that also goes to 0,

243
00:17:54,100 --> 00:17:57,810
and that's the heat capacity.

244
00:17:57,810 --> 00:18:04,160
So if I want to calculate the
difference between entropy

245
00:18:04,160 --> 00:18:10,820
at some temperature T
and some temperature

246
00:18:10,820 --> 00:18:15,570
at 0 along some particular
path corresponding

247
00:18:15,570 --> 00:18:18,810
to some constant
x for example, you

248
00:18:18,810 --> 00:18:20,640
would say that what
I need to do is

249
00:18:20,640 --> 00:18:27,310
to integrate from 0 to
T the heat that I have

250
00:18:27,310 --> 00:18:31,370
to put into the
system at constant x.

251
00:18:31,370 --> 00:18:34,700
And so if I do
that slowly enough,

252
00:18:34,700 --> 00:18:38,720
this heat I can write as CxdT.

253
00:18:38,720 --> 00:18:41,370
Cx, potentially,
is a function of T.

254
00:18:41,370 --> 00:18:44,750
Actually, since I'm indicating
T as the other point

255
00:18:44,750 --> 00:18:48,710
of integration, let me call
the variable of integration T

256
00:18:48,710 --> 00:18:50,610
prime.

257
00:18:50,610 --> 00:18:56,620
So I take a path in which
I change temperature.

258
00:18:56,620 --> 00:19:00,320
I calculate the heat
capacity at constant x.

259
00:19:00,320 --> 00:19:02,560
Integrate it.

260
00:19:02,560 --> 00:19:06,434
Multiply by dT to convert
it to T, and get the result.

261
00:19:09,220 --> 00:19:13,380
So all of these results that
they have been formulating

262
00:19:13,380 --> 00:19:18,630
suggest that the result that you
would get as a function of T,

263
00:19:18,630 --> 00:19:27,600
for entropy, is something that
as T goes to 0, approaches 0.

264
00:19:27,600 --> 00:19:31,640
So it should be a perfectly
nice, well-defined value

265
00:19:31,640 --> 00:19:35,210
at any finite temperature.

266
00:19:35,210 --> 00:19:38,170
Now, if you integrate
a constant divided

267
00:19:38,170 --> 00:19:42,420
by T, divided by dT, then
essentially the constant

268
00:19:42,420 --> 00:19:44,020
would give you a logarithm.

269
00:19:44,020 --> 00:19:48,100
And the logarithm would blow
up as we go to 0 temperature.

270
00:19:48,100 --> 00:19:52,240
So the only way that
this integral does not

271
00:19:52,240 --> 00:20:01,590
blow up on you-- so this is
finite only if the limit as T

272
00:20:01,590 --> 00:20:08,390
goes to 0 of the heat
capacities should also go to 0.

273
00:20:08,390 --> 00:20:12,550
So any heat capacity should
also essentially vanish

274
00:20:12,550 --> 00:20:14,890
as you go to lower
and lower temperature.

275
00:20:14,890 --> 00:20:17,940
This is something that you
will see many, many times when

276
00:20:17,940 --> 00:20:20,880
you look at different
heat capacities

277
00:20:20,880 --> 00:20:22,490
in the rest of the course.

278
00:20:25,030 --> 00:20:27,150
There is one other
aspect of this

279
00:20:27,150 --> 00:20:30,900
that I will not really explain,
but you can go and look

280
00:20:30,900 --> 00:20:36,410
at the notes or elsewhere, which
is that another consequence is

281
00:20:36,410 --> 00:20:47,222
unattainability of T equals
to 0 by any finite set

282
00:20:47,222 --> 00:20:47,805
of operations.

283
00:20:55,900 --> 00:21:00,130
Essentially, if you want
to get to 0 temperature,

284
00:21:00,130 --> 00:21:03,840
you'll have to do something
that cools you step by step.

285
00:21:03,840 --> 00:21:05,880
And the steps become
smaller and smaller,

286
00:21:05,880 --> 00:21:08,680
and you have to repeat
that many times.

287
00:21:08,680 --> 00:21:13,290
But that is another consequence.

288
00:21:13,290 --> 00:21:15,055
We'll leave that
for the time being.

289
00:21:18,780 --> 00:21:25,430
I would like to, however, end
by discussing some distinctions

290
00:21:25,430 --> 00:21:28,350
that are between
these different laws.

291
00:21:31,390 --> 00:21:36,140
So if you think
about whatever could

292
00:21:36,140 --> 00:21:42,210
be the microscopic
origin, after all, I

293
00:21:42,210 --> 00:21:46,240
have emphasized
that thermodynamics

294
00:21:46,240 --> 00:21:49,960
is a set of rules that you look
at substances as black boxes

295
00:21:49,960 --> 00:21:53,550
and you try to deduce a
certain number of things

296
00:21:53,550 --> 00:21:58,210
based on observations, such
as what Nernst did over here.

297
00:21:58,210 --> 00:22:00,730
But you say, these
black boxes, I

298
00:22:00,730 --> 00:22:03,750
know what is inside
them in principle.

299
00:22:03,750 --> 00:22:07,820
It's composed of atoms,
molecules, light, quark,

300
00:22:07,820 --> 00:22:09,670
whatever the
microscope theory is

301
00:22:09,670 --> 00:22:14,380
that you want to assign to
the components of that box.

302
00:22:14,380 --> 00:22:16,530
And I know the
dynamics that governs

303
00:22:16,530 --> 00:22:18,770
these microscopic
degrees of freedom.

304
00:22:18,770 --> 00:22:22,740
I should be able to get
the laws of thermodynamics

305
00:22:22,740 --> 00:22:27,000
starting from the
microscopic laws.

306
00:22:27,000 --> 00:22:30,170
Eventually, we will do
that, and as we do that,

307
00:22:30,170 --> 00:22:34,560
we will find the origin
of these different laws.

308
00:22:34,560 --> 00:22:39,670
Now, you won't be surprised
that the First Law is intimately

309
00:22:39,670 --> 00:22:44,390
connected to the fact that
any microscopic set of rules

310
00:22:44,390 --> 00:22:48,120
that you write down embodies
the conservation of energy.

311
00:22:52,570 --> 00:22:56,780
And all you have to make
sure is to understand

312
00:22:56,780 --> 00:23:01,750
precisely what heat is
as a form of energy.

313
00:23:01,750 --> 00:23:07,060
And then if we regard heat
as another form of energy,

314
00:23:07,060 --> 00:23:10,250
another component, it's
really the conservation law

315
00:23:10,250 --> 00:23:10,870
that we have.

316
00:23:14,630 --> 00:23:17,190
Then, you have the Zeroth
Law and the Second Law.

317
00:23:21,650 --> 00:23:25,530
The Zeroth Law and Second Law
have to do with equilibrium

318
00:23:25,530 --> 00:23:28,980
and being able to go in
some particular direction.

319
00:23:28,980 --> 00:23:35,170
And that always runs a fall of
the microscopic laws of motion

320
00:23:35,170 --> 00:23:39,370
that are typically things
that are time reversible where

321
00:23:39,370 --> 00:23:42,360
as the Zeroth Law and
Second Law are not.

322
00:23:42,360 --> 00:23:46,390
And what we will see later on,
through statistical mechanics,

323
00:23:46,390 --> 00:23:49,630
is that the origin
of these laws is

324
00:23:49,630 --> 00:24:00,180
that we are dealing with large
numbers of degrees of freedom.

325
00:24:03,760 --> 00:24:09,580
And once we adapt the
proper perspective

326
00:24:09,580 --> 00:24:12,960
to looking at properties
of large numbers of degrees

327
00:24:12,960 --> 00:24:15,330
of freedom, which
will be a start

328
00:24:15,330 --> 00:24:19,000
to do the elements of that
[? prescription ?] today,

329
00:24:19,000 --> 00:24:20,870
the Zeroth Law and
Second Law emerge.

330
00:24:23,990 --> 00:24:32,790
Now the Third Law, you
all know that once we

331
00:24:32,790 --> 00:24:37,590
go through this process,
eventually for example,

332
00:24:37,590 --> 00:24:41,560
we get things for the
description of entropy, which

333
00:24:41,560 --> 00:24:44,970
is related to some
number of states

334
00:24:44,970 --> 00:24:48,740
that the system
has indicated by g.

335
00:24:48,740 --> 00:24:55,880
And if you then want to
have S going through 0,

336
00:24:55,880 --> 00:24:59,690
you would require that
g goes to something

337
00:24:59,690 --> 00:25:04,600
that is order of 1-- of 1 if
you like-- as T goes to 0.

338
00:25:08,120 --> 00:25:10,520
And typically, you
would say that systems

339
00:25:10,520 --> 00:25:16,890
adopt their ground state, lowest
energy state, at 0 temperature.

340
00:25:16,890 --> 00:25:19,340
And so this is
somewhat a statement

341
00:25:19,340 --> 00:25:23,470
about the uniqueness of the
state of all possible systems

342
00:25:23,470 --> 00:25:26,120
at low temperature.

343
00:25:26,120 --> 00:25:31,420
Now, if you think about
the gas in this room,

344
00:25:31,420 --> 00:25:36,520
and let's imagine that
the particles of this gas

345
00:25:36,520 --> 00:25:40,830
either don't interact, which is
maybe a little bit unrealistic,

346
00:25:40,830 --> 00:25:42,640
but maybe repel each other.

347
00:25:42,640 --> 00:25:44,810
So let's say you have
a bunch of particles

348
00:25:44,810 --> 00:25:46,910
that just repel each other.

349
00:25:46,910 --> 00:25:51,180
Then, there is
really no reason why,

350
00:25:51,180 --> 00:25:54,850
as I go to lower and
lower temperatures,

351
00:25:54,850 --> 00:25:59,930
the number of configurations of
the molecules should decrease.

352
00:25:59,930 --> 00:26:03,180
All configurations that I
draw that they don't overlap

353
00:26:03,180 --> 00:26:05,900
have roughly the same energy.

354
00:26:05,900 --> 00:26:11,250
And indeed, if I look at say
any one of these properties,

355
00:26:11,250 --> 00:26:15,330
like the expansivity of a
gas at constant pressure

356
00:26:15,330 --> 00:26:19,230
which is given in fact
with a minus sign.

357
00:26:19,230 --> 00:26:23,850
dV by dT at constant pressure
would be the analog of one

358
00:26:23,850 --> 00:26:25,820
of these extensivities.

359
00:26:25,820 --> 00:26:32,290
If I use the Ideal Gas
Law-- So for ideal gas,

360
00:26:32,290 --> 00:26:35,530
we've seen that PV is
proportional to let's

361
00:26:35,530 --> 00:26:38,480
say some temperature.

362
00:26:38,480 --> 00:26:42,290
Then, dV by dT at
constant pressure

363
00:26:42,290 --> 00:26:46,390
is none other than V over
T. So this would give me

364
00:26:46,390 --> 00:26:59,270
1 over V, V over T. Probably
don't need it on this.

365
00:26:59,270 --> 00:27:03,000
This is going to
give me 1 over T.

366
00:27:03,000 --> 00:27:07,410
So not only doesn't it
go to 0 at 0 temperature,

367
00:27:07,410 --> 00:27:10,960
if the Ideal Gas
Law was satisfied,

368
00:27:10,960 --> 00:27:13,140
the extensivity would
actually diverge

369
00:27:13,140 --> 00:27:17,590
at 0 temperature as
different as you want.

370
00:27:17,590 --> 00:27:21,760
So clearly the Ideal Gas
Law, if it was applicable

371
00:27:21,760 --> 00:27:24,210
all the way down
to 0 temperature,

372
00:27:24,210 --> 00:27:27,330
would violate the Third
Law of thermodynamics.

373
00:27:27,330 --> 00:27:30,560
Again, not surprising
given that I have told you

374
00:27:30,560 --> 00:27:33,770
that a gas of classical
particles with repulsion

375
00:27:33,770 --> 00:27:36,880
has many states.

376
00:27:36,880 --> 00:27:39,850
Now, we will see
later on in the course

377
00:27:39,850 --> 00:27:48,220
that once we include quantum
mechanics, then as you

378
00:27:48,220 --> 00:27:50,910
go to 0 temperature,
these particles

379
00:27:50,910 --> 00:27:54,472
will have a unique state.

380
00:27:54,472 --> 00:27:59,177
If they are bosons, they will be
together in one wave function.

381
00:27:59,177 --> 00:28:01,260
If they are fermions, they
will arrange themselves

382
00:28:01,260 --> 00:28:05,190
appropriately so that,
because of quantum mechanics,

383
00:28:05,190 --> 00:28:08,530
all of these laws would
certainly breakdown at T equals

384
00:28:08,530 --> 00:28:09,750
to 0.

385
00:28:09,750 --> 00:28:13,080
You will get 0 entropy, and
you would get consistency

386
00:28:13,080 --> 00:28:16,010
with all of these things.

387
00:28:16,010 --> 00:28:18,650
So somehow, the nature
of the Third Law

388
00:28:18,650 --> 00:28:22,470
is different from the other
laws because its validity

389
00:28:22,470 --> 00:28:28,365
rests on being able to
be living in a world

390
00:28:28,365 --> 00:28:31,010
where quantum mechanics applies.

391
00:28:31,010 --> 00:28:34,350
So in principle, you could have
imagined some other universe

392
00:28:34,350 --> 00:28:36,700
where h-bar equals
to 0, and then

393
00:28:36,700 --> 00:28:39,580
the Third Law of thermodynamics
would not hold there

394
00:28:39,580 --> 00:28:41,990
whereas the Zeroth Law
and Second Law would hold.

395
00:28:41,990 --> 00:28:42,839
Yes?

396
00:28:42,839 --> 00:28:45,616
AUDIENCE: Are there any known
exceptions to the Third Law?

397
00:28:45,616 --> 00:28:47,240
Are we going to
[? account for them? ?]

398
00:28:52,330 --> 00:28:55,160
PROFESSOR: For
equilibrium-- So this

399
00:28:55,160 --> 00:28:59,010
is actually an
interesting question.

400
00:28:59,010 --> 00:29:03,050
What do I know
about-- classically,

401
00:29:03,050 --> 00:29:07,550
I can certainly come up with
lots of examples that violate.

402
00:29:07,550 --> 00:29:11,320
So your question then amounts if
I say that quantum mechanics is

403
00:29:11,320 --> 00:29:15,130
necessary, do I
know that the ground

404
00:29:15,130 --> 00:29:18,790
state of a quantum
mechanical system is unique.

405
00:29:18,790 --> 00:29:22,830
And I don't know of a proof of
that for interacting system.

406
00:29:22,830 --> 00:29:27,380
I don't know of a case that's
violated, but as far as I know,

407
00:29:27,380 --> 00:29:32,240
there is no proof that I give
you an interacting Hamiltonian

408
00:29:32,240 --> 00:29:36,550
for a quantum system, and
there's a unique ground state.

409
00:29:36,550 --> 00:29:39,110
And I should say
that there'd be no--

410
00:29:39,110 --> 00:29:42,350
and I'm sure you know of cases
where the ground state is not

411
00:29:42,350 --> 00:29:44,880
unique like a ferromagnet.

412
00:29:44,880 --> 00:29:50,040
But the point is not that
g should be exactly one,

413
00:29:50,040 --> 00:29:54,460
but that the limit
of log g divided

414
00:29:54,460 --> 00:29:56,460
by the number of
degrees of freedom

415
00:29:56,460 --> 00:30:02,280
that you have should go to
0 as n goes to infinity.

416
00:30:02,280 --> 00:30:07,620
So something like a ferromagnet
may have many ground states,

417
00:30:07,620 --> 00:30:09,590
but the number of
ground states is not

418
00:30:09,590 --> 00:30:13,440
proportional to the number of
sites, the number of spins,

419
00:30:13,440 --> 00:30:15,770
and this entity will go to 0.

420
00:30:15,770 --> 00:30:19,330
So all the cases that we
know, the ground state

421
00:30:19,330 --> 00:30:24,100
is either unique
or is order of one.

422
00:30:24,100 --> 00:30:27,300
But I don't know a theorem that
says that should be the case.

423
00:30:34,600 --> 00:30:36,550
So this is the last
thing that I wanted

424
00:30:36,550 --> 00:30:39,100
to say about thermodynamics.

425
00:30:39,100 --> 00:30:40,910
Are there any
questions in general?

426
00:30:45,850 --> 00:30:50,170
So I laid out the
necessity of having

427
00:30:50,170 --> 00:30:56,240
some kind of a description of
microscopic degrees of freedom

428
00:30:56,240 --> 00:31:00,680
that ultimately will
allow us to prove

429
00:31:00,680 --> 00:31:03,140
the laws of thermodynamics.

430
00:31:03,140 --> 00:31:08,570
And that will come through
statistical mechanics, which

431
00:31:08,570 --> 00:31:12,410
as the name implies, has
to have certain amount

432
00:31:12,410 --> 00:31:16,010
of statistic characters to it.

433
00:31:16,010 --> 00:31:18,230
What does that mean?

434
00:31:18,230 --> 00:31:22,240
It means that you have to
abandon a description of motion

435
00:31:22,240 --> 00:31:26,360
that is fully
deterministic for one

436
00:31:26,360 --> 00:31:27,725
that is based on probability.

437
00:31:30,860 --> 00:31:34,760
Now, I could have told you
first the degrees of freedom

438
00:31:34,760 --> 00:31:37,090
and what is the
description that we

439
00:31:37,090 --> 00:31:40,610
need for them to
be probabilistic,

440
00:31:40,610 --> 00:31:43,740
but I find it more
useful to first lay out

441
00:31:43,740 --> 00:31:45,720
what the language
of probability is

442
00:31:45,720 --> 00:31:48,915
that we will be
using and then bring

443
00:31:48,915 --> 00:31:52,200
in the description of the
microscopic degrees of freedom

444
00:31:52,200 --> 00:31:55,690
within this language.

445
00:31:55,690 --> 00:32:06,590
So if we go first
with definitions--

446
00:32:06,590 --> 00:32:12,610
and you could, for example, go
to the branch of mathematics

447
00:32:12,610 --> 00:32:16,290
that deals with probability,
and you will encounter something

448
00:32:16,290 --> 00:32:21,970
like this that what probability
describes is a random variable.

449
00:32:27,050 --> 00:32:36,470
Let's call it X, which has a
number of possible outcomes,

450
00:32:36,470 --> 00:32:49,740
which we put together
into a set of outcomes, S.

451
00:32:49,740 --> 00:33:00,410
And this set can be discrete as
would be the case if you were

452
00:33:00,410 --> 00:33:05,030
tossing a coin, and
the outcomes would

453
00:33:05,030 --> 00:33:11,680
be either a head or a tail,
or we were throwing a dice,

454
00:33:11,680 --> 00:33:15,215
and the outcomes would
be the faces 1 through 6.

455
00:33:19,050 --> 00:33:23,180
And we will encounter
mostly actually cases

456
00:33:23,180 --> 00:33:25,150
where S is continuous.

457
00:33:28,610 --> 00:33:34,300
Like for example, if I want to
describe the velocity of a gas

458
00:33:34,300 --> 00:33:39,950
particle in this room, I need
to specify the three components

459
00:33:39,950 --> 00:33:44,530
of velocity that can
be anywhere, let's say,

460
00:33:44,530 --> 00:33:46,210
in the range of real numbers.

461
00:33:51,130 --> 00:34:10,929
And again, mathematicians
would say that to each event,

462
00:34:10,929 --> 00:34:20,080
which is a subset of
possible outcomes,

463
00:34:20,080 --> 00:34:30,480
is assigned a value which we
must satisfy the following

464
00:34:30,480 --> 00:34:30,980
properties.

465
00:34:35,360 --> 00:34:41,210
First thing is the
probability of anything

466
00:34:41,210 --> 00:34:44,440
is a positive number.

467
00:34:44,440 --> 00:34:46,763
And so this is positivity.

468
00:34:55,100 --> 00:34:57,550
The second thing is additivity.

469
00:35:01,360 --> 00:35:08,180
That is the probability
of two events, A or B,

470
00:35:08,180 --> 00:35:13,360
is the sum total of
the probabilities

471
00:35:13,360 --> 00:35:17,395
if A and B are
disjoint or distinct.

472
00:35:23,389 --> 00:35:24,930
And finally, there's
a normalization.

473
00:35:31,110 --> 00:35:33,980
That if you're event is
that something should happen

474
00:35:33,980 --> 00:35:38,888
the entire set, the probability
that you assign to that is 1.

475
00:35:42,310 --> 00:35:45,100
So these are formal statements.

476
00:35:45,100 --> 00:35:49,200
And if you are a mathematician,
you start from there,

477
00:35:49,200 --> 00:35:52,020
and you prove theorems.

478
00:35:52,020 --> 00:35:59,900
But from our perspective,
the first question to ask

479
00:35:59,900 --> 00:36:08,960
is how to determine this
quantity probability

480
00:36:08,960 --> 00:36:12,070
that something should happen.

481
00:36:12,070 --> 00:36:18,670
If it is useful and I want to do
something real world about it,

482
00:36:18,670 --> 00:36:23,340
I should be able to measure
it or assign values to it.

483
00:36:23,340 --> 00:36:28,290
And very roughly
again, in theory,

484
00:36:28,290 --> 00:36:34,240
we can assign probabilities
two different ways.

485
00:36:34,240 --> 00:36:36,940
One way is called objective.

486
00:36:41,010 --> 00:36:45,300
And from the perspective
of us as physicists

487
00:36:45,300 --> 00:36:48,390
corresponds to what would be
an experimental procedure.

488
00:36:51,700 --> 00:37:06,020
And if it is assigning p of e
as the frequency of outcomes

489
00:37:06,020 --> 00:37:16,400
in large number of
trials, i.e. you

490
00:37:16,400 --> 00:37:22,770
would say that the probability
that event A is obtained

491
00:37:22,770 --> 00:37:27,940
is the number of times you
would get outcome A divided

492
00:37:27,940 --> 00:37:34,040
by the total number of
trials as n goes to infinity.

493
00:37:34,040 --> 00:37:40,790
So for example, if you want to
assign a probability that when

494
00:37:40,790 --> 00:37:48,120
you throw a dice that face 1
comes up, what you could do

495
00:37:48,120 --> 00:37:52,590
is you could make a table
of the number of times

496
00:37:52,590 --> 00:37:57,690
1 shows up divided by the number
of times you throw the dice.

497
00:37:57,690 --> 00:38:04,050
Maybe you throw it 100
times, and you get 15.

498
00:38:04,050 --> 00:38:08,767
You throw it 200
times, and you get--

499
00:38:08,767 --> 00:38:09,850
that is probably too much.

500
00:38:09,850 --> 00:38:14,340
Let's say 15-- you get 35.

501
00:38:14,340 --> 00:38:19,765
And you do it 300 times, and
you get something close to 48.

502
00:38:19,765 --> 00:38:24,640
The ratio of these
things, as the number

503
00:38:24,640 --> 00:38:27,330
gets larger and
larger, hopefully

504
00:38:27,330 --> 00:38:30,130
will converge to something that
you would call the probability.

505
00:38:32,770 --> 00:38:38,380
Now, it turns out that
in statistical physics,

506
00:38:38,380 --> 00:38:42,660
we will assign things through
a totally different procedure

507
00:38:42,660 --> 00:38:46,330
which is subjective.

508
00:38:46,330 --> 00:38:51,140
If you like, it's
more theoretical,

509
00:38:51,140 --> 00:39:07,020
which is based on uncertainty
among all outcomes.

510
00:39:11,340 --> 00:39:16,224
Because if I were to
subjectively assign

511
00:39:16,224 --> 00:39:21,740
to throwing the dice and
coming up with value of 1,

512
00:39:21,740 --> 00:39:28,010
I would say, well, there's six
possible faces for the dice.

513
00:39:28,010 --> 00:39:30,900
I don't know anything about
this dice being loaded,

514
00:39:30,900 --> 00:39:34,750
so I will say they
are all equally alike.

515
00:39:34,750 --> 00:39:37,570
Now, that may or may not
be a correct assumption.

516
00:39:37,570 --> 00:39:38,520
You could test it.

517
00:39:38,520 --> 00:39:40,180
You could maybe
throw it many times.

518
00:39:40,180 --> 00:39:43,250
You will find that either the
dice is loaded or not loaded

519
00:39:43,250 --> 00:39:45,050
and this is correct or not.

520
00:39:45,050 --> 00:39:48,830
But you begin by
making this assumption.

521
00:39:48,830 --> 00:39:52,620
And this is actually, we
will see later on, exactly

522
00:39:52,620 --> 00:39:54,890
the type of assumption
that you would

523
00:39:54,890 --> 00:39:57,210
be making in
statistical physics.

524
00:40:05,979 --> 00:40:07,520
Any question about
these definitions?

525
00:40:10,980 --> 00:40:17,020
So let's again
proceed slowly to get

526
00:40:17,020 --> 00:40:25,430
some definitions established by
looking at one random variable.

527
00:40:25,430 --> 00:40:31,890
So this is the next section
on one random variable.

528
00:40:34,770 --> 00:40:38,770
And I will assume that
I'll look at the case

529
00:40:38,770 --> 00:40:40,670
of the continuous
random variable.

530
00:40:40,670 --> 00:40:49,270
So x can be any real number
minus infinity to infinity.

531
00:40:49,270 --> 00:40:52,170
Now, a number of definitions.

532
00:40:52,170 --> 00:40:59,260
I will use the term
Cumulative-- make

533
00:40:59,260 --> 00:41:25,420
sure I'll use the-- Cumulative
Probability Function, CPF,

534
00:41:25,420 --> 00:41:28,690
that for this one
random variable,

535
00:41:28,690 --> 00:41:32,460
I will indicate
by capital P of x.

536
00:41:35,070 --> 00:41:37,710
And the meaning of
this is that capital P

537
00:41:37,710 --> 00:41:46,700
of x is the probability
of outcome less than x.

538
00:41:54,130 --> 00:42:00,390
So generically,
we say that x can

539
00:42:00,390 --> 00:42:04,020
take all values
along the real line.

540
00:42:04,020 --> 00:42:05,820
And there is this
function that I

541
00:42:05,820 --> 00:42:12,710
want to plot that I will
call big P of x Now big P

542
00:42:12,710 --> 00:42:16,290
of x is a probability,
therefore,

543
00:42:16,290 --> 00:42:20,030
it has to be positive according
to the first item that we

544
00:42:20,030 --> 00:42:21,840
have over there.

545
00:42:21,840 --> 00:42:27,020
And it will be less than 1
because the net probability

546
00:42:27,020 --> 00:42:30,440
for everything toward
here is equal to 1.

547
00:42:30,440 --> 00:42:33,320
So asymptotically,
where I go all the way

548
00:42:33,320 --> 00:42:35,450
to infinity, the
probability that I

549
00:42:35,450 --> 00:42:41,590
will get some number along the
line-- I have to get something,

550
00:42:41,590 --> 00:42:45,530
so it should automatically
go to 1 here.

551
00:42:45,530 --> 00:42:51,450
And every element of
probability is positive,

552
00:42:51,450 --> 00:42:55,590
so it's a function that
should gradually go down.

553
00:42:55,590 --> 00:42:58,454
And presumably, it
will behave something

554
00:42:58,454 --> 00:42:59,370
like this generically.

555
00:43:06,170 --> 00:43:10,110
Once we have the Cumulative
Probability Function,

556
00:43:10,110 --> 00:43:24,530
we can immediately construct the
Probability Density Function,

557
00:43:24,530 --> 00:43:33,350
PDF, which is the
derivative of the above.

558
00:43:33,350 --> 00:43:42,290
P of x is the derivative of big
P of x with respect to the x.

559
00:43:42,290 --> 00:43:48,730
And so if I just take here
the curve that I have above

560
00:43:48,730 --> 00:43:51,850
and take its derivative,
the derivative

561
00:43:51,850 --> 00:43:54,360
will look something like this.

562
00:44:01,470 --> 00:44:07,340
Essentially, clearly by the
definition of the derivative,

563
00:44:07,340 --> 00:44:13,260
this quantity is
therefore ability

564
00:44:13,260 --> 00:44:23,120
of outcome in the
interval x to x plus dx

565
00:44:23,120 --> 00:44:25,274
divided by the size
of the interval dx.

566
00:44:29,050 --> 00:44:34,680
couple of things to
remind you of, one of them

567
00:44:34,680 --> 00:44:39,690
is that the Cumulative
Probability is a probability.

568
00:44:39,690 --> 00:44:43,540
It's a dimensionless
number between 0 and 1.

569
00:44:43,540 --> 00:44:48,440
Probability Density is obtained
by taking a derivative,

570
00:44:48,440 --> 00:44:54,570
so it has dimensions that are
inverse of whatever this x is.

571
00:44:54,570 --> 00:44:59,270
So if I change my variable
from meters to centimeters,

572
00:44:59,270 --> 00:45:01,570
let's say, the value
of this function

573
00:45:01,570 --> 00:45:04,530
would change by a factor of 100.

574
00:45:04,530 --> 00:45:09,490
And secondly, while
the Probability Density

575
00:45:09,490 --> 00:45:14,680
is positive, its
value is not bounded.

576
00:45:14,680 --> 00:45:16,648
It can be anywhere
that you like.

577
00:45:23,490 --> 00:45:27,050
One other, again,
minor definition

578
00:45:27,050 --> 00:45:28,600
is expectation value.

579
00:45:34,810 --> 00:45:39,990
So I can pick some
function of x.

580
00:45:39,990 --> 00:45:42,000
This could be x itself.

581
00:45:42,000 --> 00:45:43,970
It could be x squared.

582
00:45:43,970 --> 00:45:49,810
It could be sine x, x
cubed minus x squared.

583
00:45:49,810 --> 00:45:52,880
The expectation value
of this is defined

584
00:45:52,880 --> 00:45:58,310
by integrating the
Probability Density

585
00:45:58,310 --> 00:46:00,906
against the value
of the function.

586
00:46:06,620 --> 00:46:15,260
So essentially,
what that says is

587
00:46:15,260 --> 00:46:30,440
that if I pick some
function of x-- function

588
00:46:30,440 --> 00:46:33,160
can be positive,
negative, et cetera.

589
00:46:33,160 --> 00:46:42,640
So maybe I have a function
such as this-- then the value

590
00:46:42,640 --> 00:46:44,430
of x is random.

591
00:46:44,430 --> 00:46:47,460
If x is in this
interval, this would

592
00:46:47,460 --> 00:46:50,870
be the corresponding
contribution to f of x.

593
00:46:50,870 --> 00:46:55,605
And I have to look at
all possible values of x.

594
00:47:00,100 --> 00:47:02,710
Question?

595
00:47:02,710 --> 00:47:06,935
Now, very associated to this
is a change of variables.

596
00:47:14,660 --> 00:47:23,040
You would say that if x is
random, then f of x is random.

597
00:47:23,040 --> 00:47:26,420
So if I ask you what is
the value of x squared,

598
00:47:26,420 --> 00:47:29,780
and for one random
variable, I get this.

599
00:47:29,780 --> 00:47:32,220
The value of x
squared would be this.

600
00:47:32,220 --> 00:47:34,660
If I get this, the
value of x squared

601
00:47:34,660 --> 00:47:37,160
would be something else.

602
00:47:37,160 --> 00:47:41,035
So if x is random, f of x
is itself a random variable.

603
00:47:45,210 --> 00:47:52,510
So f of x is a random
variable, and you

604
00:47:52,510 --> 00:47:55,220
can ask what is the
probability, let's

605
00:47:55,220 --> 00:48:00,000
say, the Probability Density
Function that I would associate

606
00:48:00,000 --> 00:48:02,040
with the value of this.

607
00:48:02,040 --> 00:48:05,320
Let's say what's the
probability that I will find it

608
00:48:05,320 --> 00:48:10,710
in the interval between
small f and small f plus df.

609
00:48:10,710 --> 00:48:12,916
This will be Pf f of f.

610
00:48:17,052 --> 00:48:21,950
You would say that the
probability that I would find

611
00:48:21,950 --> 00:48:27,560
the value of the function
that is in this interval

612
00:48:27,560 --> 00:48:33,190
corresponds to finding a value
of x that is in this interval.

613
00:48:33,190 --> 00:48:36,020
So what I can do,
the probability

614
00:48:36,020 --> 00:48:38,990
that I find the value
of f in this interval,

615
00:48:38,990 --> 00:48:43,260
according to what I have here,
is the Probability Density

616
00:48:43,260 --> 00:48:44,912
multiplied by df.

617
00:48:44,912 --> 00:48:45,745
Is there a question?

618
00:48:48,250 --> 00:48:50,790
No.

619
00:48:50,790 --> 00:48:53,380
So the probability that
I'm in this interval

620
00:48:53,380 --> 00:48:56,520
translates to the probability
that I'm in this interval.

621
00:48:56,520 --> 00:49:01,965
So that's probability p of x dx.

622
00:49:04,650 --> 00:49:07,500
But that's boring.

623
00:49:07,500 --> 00:49:09,920
I want to look at the
situation maybe where

624
00:49:09,920 --> 00:49:11,565
the function is
something like this.

625
00:49:15,350 --> 00:49:19,730
Then, you say that f is
in this interval provided

626
00:49:19,730 --> 00:49:27,220
that x is either
here or here or here.

627
00:49:27,220 --> 00:49:33,964
So what I really need
to do is to solve

628
00:49:33,964 --> 00:49:40,200
f of x equals to f for x.

629
00:49:40,200 --> 00:49:43,010
And maybe there
will be solutions

630
00:49:43,010 --> 00:49:48,030
that will be x1,
x2, x3, et cetera.

631
00:49:48,030 --> 00:49:50,960
And what I need to
do is to sum over

632
00:49:50,960 --> 00:49:53,850
the contributions of
all of those solutions.

633
00:49:57,420 --> 00:49:58,830
So here, it's three solutions.

634
00:50:01,560 --> 00:50:08,010
Then, you would say
the Probability Density

635
00:50:08,010 --> 00:50:15,440
is the sum over i P
of xi, the xi by df,

636
00:50:15,440 --> 00:50:17,460
which is really the slopes.

637
00:50:17,460 --> 00:50:21,050
The slopes translate the
size of this interval

638
00:50:21,050 --> 00:50:23,010
to the size of that interval.

639
00:50:23,010 --> 00:50:25,770
You can see that here,
the slope is very sharp.

640
00:50:25,770 --> 00:50:28,190
The size of this
interval is small.

641
00:50:28,190 --> 00:50:31,450
It could be wider
accordingly, so I

642
00:50:31,450 --> 00:50:34,937
need to multiply by dxi by df.

643
00:50:39,160 --> 00:50:43,940
So I have to multiply by
dx by df evaluated at xi.

644
00:50:43,940 --> 00:50:47,165
That's essentially the value
of the derivative of f.

645
00:50:50,430 --> 00:50:57,070
Now, sometimes, it is easy
to forget these things

646
00:50:57,070 --> 00:50:59,300
that I write over here.

647
00:50:59,300 --> 00:51:01,640
And you would say,
well obviously,

648
00:51:01,640 --> 00:51:06,000
the probability of
something that is positive.

649
00:51:06,000 --> 00:51:09,890
But without being careful,
it is easy to violate

650
00:51:09,890 --> 00:51:11,620
such basic condition.

651
00:51:11,620 --> 00:51:13,020
And I violated it here.

652
00:51:16,740 --> 00:51:19,160
Anybody see where I violated it.

653
00:51:22,120 --> 00:51:24,710
Yeah, the slope
here is positive.

654
00:51:24,710 --> 00:51:26,060
The slope here is positive.

655
00:51:26,060 --> 00:51:28,240
The slope here is negative.

656
00:51:28,240 --> 00:51:33,130
So I am subtracting
a probability here.

657
00:51:33,130 --> 00:51:34,920
So what I really
should do-- it really

658
00:51:34,920 --> 00:51:37,490
doesn't matter whether the
slope is this way or that way.

659
00:51:37,490 --> 00:51:39,590
I will pick up
the same interval,

660
00:51:39,590 --> 00:51:44,144
so make sure you don't forget
the absolute values that

661
00:51:44,144 --> 00:51:45,596
go accordingly.

662
00:51:45,596 --> 00:51:48,210
So this is the
standard way that you

663
00:51:48,210 --> 00:51:50,480
would make change of variables.

664
00:51:50,480 --> 00:51:52,162
Yes?

665
00:51:52,162 --> 00:51:52,828
AUDIENCE: Sorry.

666
00:51:52,828 --> 00:51:56,701
In the center of that board,
on the second line, it says Pf.

667
00:51:56,701 --> 00:51:58,084
Is that an x or a times?

668
00:52:00,715 --> 00:52:02,340
PROFESSOR: In the
center of this board?

669
00:52:02,340 --> 00:52:03,678
This one?

670
00:52:03,678 --> 00:52:06,140
AUDIENCE: Yeah.

671
00:52:06,140 --> 00:52:10,460
PROFESSOR: So the
value of the function

672
00:52:10,460 --> 00:52:12,930
is a random variable, right?

673
00:52:12,930 --> 00:52:14,540
It can come up to be here.

674
00:52:14,540 --> 00:52:16,390
It can come up to be here.

675
00:52:16,390 --> 00:52:21,720
And so there is, as any other
one parameter random variable,

676
00:52:21,720 --> 00:52:25,210
a Probability Density
associated with that.

677
00:52:25,210 --> 00:52:30,350
That Probability Density
I have called P of f

678
00:52:30,350 --> 00:52:32,440
to indicate that
it is the variable

679
00:52:32,440 --> 00:52:34,910
f that I'm
considering as opposed

680
00:52:34,910 --> 00:52:37,370
to what I wrote
originally that was

681
00:52:37,370 --> 00:52:39,810
associated with the value of x.

682
00:52:39,810 --> 00:52:42,282
AUDIENCE: But what you have
written on the left-hand side,

683
00:52:42,282 --> 00:52:44,750
it looks like your
x [? is random. ?]

684
00:52:44,750 --> 00:52:47,160
PROFESSOR: Oh, this was
supposed to be a multiplication

685
00:52:47,160 --> 00:52:48,110
sign, so sorry.

686
00:52:48,110 --> 00:52:49,635
AUDIENCE: Thank you.

687
00:52:49,635 --> 00:52:50,510
PROFESSOR: Thank you.

688
00:52:56,445 --> 00:52:56,945
Yes?

689
00:52:56,945 --> 00:53:01,900
AUDIENCE: CP-- that function,
is this [INAUDIBLE]?

690
00:53:01,900 --> 00:53:04,020
PROFESSOR: Yes.

691
00:53:04,020 --> 00:53:07,670
So you're asking whether this--
so I constructed something,

692
00:53:07,670 --> 00:53:11,660
and my statement is that the
integral from minus infinity

693
00:53:11,660 --> 00:53:21,100
to infinity df Pf of f better be
one which is the normalization.

694
00:53:21,100 --> 00:53:24,420
So if you're asking
about this, essentially,

695
00:53:24,420 --> 00:53:30,020
you would say the
integral dx p of x

696
00:53:30,020 --> 00:53:34,200
is the integral dx
dP by dx, right?

697
00:53:34,200 --> 00:53:36,710
That was the definition p of x.

698
00:53:36,710 --> 00:53:38,930
And the integral
of the derivative

699
00:53:38,930 --> 00:53:43,830
is the value of the function
evaluated at its two extremes.

700
00:53:43,830 --> 00:53:47,360
And this is one minus 0.

701
00:53:47,360 --> 00:53:51,845
So by construction, it
is, of course, normalized

702
00:53:51,845 --> 00:53:54,100
in this fashion.

703
00:53:54,100 --> 00:53:55,310
Is that what you were asking?

704
00:53:55,310 --> 00:53:58,226
AUDIENCE: I was asking
about the first possibility

705
00:53:58,226 --> 00:54:00,970
of cumulative
probability function.

706
00:54:00,970 --> 00:54:04,220
PROFESSOR: So the
cumulative probability,

707
00:54:04,220 --> 00:54:09,540
its constraint is that
the limit as its variable

708
00:54:09,540 --> 00:54:14,030
goes to infinity,
it should go to 1.

709
00:54:14,030 --> 00:54:15,650
That's the normalization.

710
00:54:15,650 --> 00:54:19,660
The normalization here
is that the probability

711
00:54:19,660 --> 00:54:23,300
of the entire set is 1.

712
00:54:23,300 --> 00:54:26,000
Cumulative adds
the probabilities

713
00:54:26,000 --> 00:54:29,270
to be anywhere up to point x.

714
00:54:29,270 --> 00:54:32,160
So I have achieved being
anywhere on the line

715
00:54:32,160 --> 00:54:35,700
by going through this point.

716
00:54:35,700 --> 00:54:41,477
But certainly, the integral
of P of xdx is not equal to 1

717
00:54:41,477 --> 00:54:42,685
if that's what you're asking.

718
00:54:45,370 --> 00:54:48,494
The integral of
small p of x is 1.

719
00:54:51,973 --> 00:54:53,464
Yes?

720
00:54:53,464 --> 00:54:56,446
AUDIENCE: Are we assuming
the function is invertible?

721
00:55:00,430 --> 00:55:03,380
PROFESSOR: Well, rigorously
speaking, this function

722
00:55:03,380 --> 00:55:08,090
is not invertible
because for a value of f,

723
00:55:08,090 --> 00:55:10,810
there are three
possible values of x.

724
00:55:10,810 --> 00:55:14,150
So it's not a function,
but you can certainly

725
00:55:14,150 --> 00:55:18,070
solve for f of x equals to
f to find particular values.

726
00:55:27,630 --> 00:55:31,520
So again, maybe it
is useful to work

727
00:55:31,520 --> 00:55:33,230
through one example of this.

728
00:55:37,320 --> 00:55:44,220
So let's say that you
have a probability that

729
00:55:44,220 --> 00:55:50,890
is of the form e to the minus
lambda absolute value of x.

730
00:55:50,890 --> 00:55:58,550
So as a function of x,
the Probability Density

731
00:55:58,550 --> 00:56:04,430
falls off exponentially
on both sides.

732
00:56:04,430 --> 00:56:08,340
And again, I have to ensure that
when I integrate this from 0

733
00:56:08,340 --> 00:56:11,280
to infinity, I will get one.

734
00:56:11,280 --> 00:56:14,090
The integral from
0 to infinity is

735
00:56:14,090 --> 00:56:16,580
1 over lambda,
from minus infinity

736
00:56:16,580 --> 00:56:19,420
to zero by symmetry
is 1 over lambda.

737
00:56:19,420 --> 00:56:26,940
So it's really I have to divide
by 2 lambda-- to lambda over 2.

738
00:56:26,940 --> 00:56:27,440
Sorry.

739
00:56:34,680 --> 00:56:39,650
Now, suppose I change variables
to F, which is x squared.

740
00:56:42,350 --> 00:56:46,840
So I want to know what
the probability is

741
00:56:46,840 --> 00:56:53,070
for a particular value of x
squared that I will call f.

742
00:56:53,070 --> 00:56:57,720
So then what I have to
do is to solve this.

743
00:56:57,720 --> 00:57:03,920
And this will give me x is minus
plus square root of small f.

744
00:57:03,920 --> 00:57:10,440
If I ask for what f of-- for
what value of x, x squared

745
00:57:10,440 --> 00:57:15,500
equals to f, then I have
these two solutions.

746
00:57:15,500 --> 00:57:18,670
So according to the
formula that I have,

747
00:57:18,670 --> 00:57:22,480
I have to, first of
all, evaluate this

748
00:57:22,480 --> 00:57:26,540
at these two possible routes.

749
00:57:26,540 --> 00:57:33,160
In both cases, I will get
minus lambda square root of f.

750
00:57:33,160 --> 00:57:35,610
Because of the absolute
value, both of them

751
00:57:35,610 --> 00:57:38,290
will give you the same thing.

752
00:57:38,290 --> 00:57:43,350
And then I have to look
at this derivative.

753
00:57:43,350 --> 00:57:52,394
So if I look at this, I can
see that df by dx equals to 2x.

754
00:57:52,394 --> 00:57:57,180
The locations that I have to
evaluate are at plus minus

755
00:57:57,180 --> 00:57:58,800
square root of f.

756
00:57:58,800 --> 00:58:04,470
So the value of the slope is
minus plus to square root of f.

757
00:58:04,470 --> 00:58:07,840
And according to that
formula, what I have to do

758
00:58:07,840 --> 00:58:09,600
is to put the inverse of that.

759
00:58:09,600 --> 00:58:13,680
So I have to put for one
solution, 1 over 2 square root

760
00:58:13,680 --> 00:58:15,460
of f.

761
00:58:15,460 --> 00:58:19,920
For the other one, I have to
put 1 over minus 2 square root

762
00:58:19,920 --> 00:58:24,560
of f, which would be a disaster
if I didn't convert this

763
00:58:24,560 --> 00:58:27,650
to an absolute value.

764
00:58:27,650 --> 00:58:30,900
And if I did convert that
to an absolute value,

765
00:58:30,900 --> 00:58:36,322
what I would get is lambda
over 2 square root of f

766
00:58:36,322 --> 00:58:41,000
e to the minus lambda root f.

767
00:58:41,000 --> 00:58:46,180
It is important to note
that this solution will only

768
00:58:46,180 --> 00:58:51,340
exist only if f is positive.

769
00:58:51,340 --> 00:58:55,040
And there's no solution
if f is negative,

770
00:58:55,040 --> 00:59:00,200
which means that if I
wanted to plot a Probability

771
00:59:00,200 --> 00:59:03,770
Density for this
function f, which

772
00:59:03,770 --> 00:59:08,420
is x squared as a function
of f, it will only

773
00:59:08,420 --> 00:59:12,510
have values for positive
values of x squared.

774
00:59:12,510 --> 00:59:16,300
There's nothing for
negative values.

775
00:59:16,300 --> 00:59:19,300
For positive values,
I have this function

776
00:59:19,300 --> 00:59:22,200
that's exponentially decays.

777
00:59:22,200 --> 00:59:27,380
Yet at f equals to 0 diverges.

778
00:59:27,380 --> 00:59:30,770
One reason I chose
that example is

779
00:59:30,770 --> 00:59:34,170
to emphasize that these
Probability Density

780
00:59:34,170 --> 00:59:39,206
functions can even go
all the way infinity.

781
00:59:39,206 --> 00:59:42,880
The requirement,
however, is that you

782
00:59:42,880 --> 00:59:47,790
should be able to integrate
across the infinity because

783
00:59:47,790 --> 00:59:51,840
integrating across the infinity
should give you a finite number

784
00:59:51,840 --> 00:59:53,660
less than 1.

785
00:59:53,660 --> 00:59:58,490
And so the type of divergence
that you could have is limited.

786
00:59:58,490 --> 01:00:00,500
1 over square root of f is fine.

787
01:00:00,500 --> 01:00:02,000
1/f is not accepted.

788
01:00:06,380 --> 01:00:07,555
Yes?

789
01:00:07,555 --> 01:00:09,680
AUDIENCE: I have a doubt
about [? the postulate. ?]

790
01:00:09,680 --> 01:00:16,850
It says that if you raise
the value of f slowly,

791
01:00:16,850 --> 01:00:20,260
you will eventually get to--
yeah, that point right there.

792
01:00:20,260 --> 01:00:22,982
So if the prescription that
we have of summing over

793
01:00:22,982 --> 01:00:27,025
the different roots, at
some point, the roots,

794
01:00:27,025 --> 01:00:27,715
they converge.

795
01:00:27,715 --> 01:00:28,340
PROFESSOR: Yes.

796
01:00:28,340 --> 01:00:30,552
AUDIENCE: So at some point,
we stop summing over 2

797
01:00:30,552 --> 01:00:31,956
and we start summing over 1.

798
01:00:31,956 --> 01:00:35,240
It just seems a
little bit strange.

799
01:00:35,240 --> 01:00:36,070
PROFESSOR: Yeah.

800
01:00:36,070 --> 01:00:40,280
If you are up here, you have
only one term in the sum.

801
01:00:40,280 --> 01:00:43,200
If you are down here,
you have three terms.

802
01:00:43,200 --> 01:00:46,120
And that's really just
the property of the curve

803
01:00:46,120 --> 01:00:47,870
that I have drawn.

804
01:00:47,870 --> 01:00:51,530
And so over here, I
have only one root.

805
01:00:51,530 --> 01:00:53,040
Over here, I have three roots.

806
01:00:53,040 --> 01:00:54,960
And this is not surprising.

807
01:00:54,960 --> 01:00:58,190
There are many situations
in mathematics or physics

808
01:00:58,190 --> 01:01:01,140
where you encounter
situations where,

809
01:01:01,140 --> 01:01:04,670
as you change some parameters,
new solutions, new roots,

810
01:01:04,670 --> 01:01:05,650
appear.

811
01:01:05,650 --> 01:01:11,030
And so if this was really some
kind of a physical system,

812
01:01:11,030 --> 01:01:13,690
you would probably
encounter some kind

813
01:01:13,690 --> 01:01:16,712
of a singularity of phase
transitions at this point.

814
01:01:20,960 --> 01:01:21,825
Yes?

815
01:01:21,825 --> 01:01:24,675
AUDIENCE: But how does the
equation deal with that when

816
01:01:24,675 --> 01:01:27,060
[INAUDIBLE]?

817
01:01:27,060 --> 01:01:28,570
PROFESSOR: Let's see.

818
01:01:28,570 --> 01:01:34,390
So if I am approaching
that point, what I find

819
01:01:34,390 --> 01:01:40,540
is that the f by
the x goes to 0.

820
01:01:40,540 --> 01:01:44,500
So the x by df has some kind
of infinity or singularity,

821
01:01:44,500 --> 01:01:46,590
so we have to deal with that.

822
01:01:46,590 --> 01:01:49,670
If you want, we can
choose a particular form

823
01:01:49,670 --> 01:01:52,140
of that function and
see what happens.

824
01:01:52,140 --> 01:01:55,230
But actually, we have
that already over here

825
01:01:55,230 --> 01:01:58,440
because the function f
that I plotted for you

826
01:01:58,440 --> 01:02:05,740
as a function of x
has this behavior

827
01:02:05,740 --> 01:02:09,035
that, for some range of
f, you have two solutions.

828
01:02:14,280 --> 01:02:17,940
So for negative values
of f, I have no solution.

829
01:02:17,940 --> 01:02:21,000
So this curve, after
having rotated,

830
01:02:21,000 --> 01:02:24,390
is precisely an example
of what is happening here.

831
01:02:24,390 --> 01:02:26,950
And you see what the
consequence of that is.

832
01:02:26,950 --> 01:02:29,910
The consequence of that
is that as I approach here

833
01:02:29,910 --> 01:02:33,810
and the two solutions merge,
I have the singularity that

834
01:02:33,810 --> 01:02:36,679
is ultimately
manifested in here.

835
01:02:47,450 --> 01:02:49,060
So in principle, yes.

836
01:02:49,060 --> 01:02:51,230
When you make these
changes of variables

837
01:02:51,230 --> 01:02:55,630
and you have functions
that have multiple solution

838
01:02:55,630 --> 01:02:58,840
behavior like that, you
have to worry about this.

839
01:03:02,940 --> 01:03:04,065
Let me go down here.

840
01:03:07,690 --> 01:03:10,860
One other definition
that, again, you've

841
01:03:10,860 --> 01:03:13,350
probably seen, before
we go through something

842
01:03:13,350 --> 01:03:16,483
that I hope you
haven't seen, moment.

843
01:03:19,430 --> 01:03:22,110
A form of this expectation
value-- actually, here we

844
01:03:22,110 --> 01:03:24,950
did with x squared,
but in general, we

845
01:03:24,950 --> 01:03:29,480
can calculate the expectation
value of x to the m.

846
01:03:29,480 --> 01:03:37,120
And sometimes, that is called
mth moment is the integral 0

847
01:03:37,120 --> 01:03:41,130
to infinity dx x
to the m p of x.

848
01:03:56,760 --> 01:04:00,300
Now, I expect that
after this point,

849
01:04:00,300 --> 01:04:01,940
you would have seen everything.

850
01:04:01,940 --> 01:04:06,820
But next one maybe
half of you have seen.

851
01:04:06,820 --> 01:04:10,870
And the next item,
which we will use a lot,

852
01:04:10,870 --> 01:04:12,720
is the characteristic function.

853
01:04:24,860 --> 01:04:29,590
So given that I have some
probability distribution

854
01:04:29,590 --> 01:04:34,220
p of x, I can calculate
various expectation values.

855
01:04:34,220 --> 01:04:38,470
I calculate the expectation
value of e to the minus ikx.

856
01:04:43,200 --> 01:04:47,140
This is, by definition
that you have,

857
01:04:47,140 --> 01:04:50,750
I have to integrate
over the domain of x--

858
01:04:50,750 --> 01:04:55,140
let's say from minus infinity
to infinity-- p of x against e

859
01:04:55,140 --> 01:04:56,140
to the minus ikx.

860
01:05:01,550 --> 01:05:05,560
And you say, well, what's
special about that?

861
01:05:05,560 --> 01:05:11,470
I know that to be the
Fourier transform of p of x.

862
01:05:11,470 --> 01:05:13,220
And it is true.

863
01:05:13,220 --> 01:05:17,790
And you also know how to
invert the Fourier transform.

864
01:05:17,790 --> 01:05:20,460
That is if you know the
characteristic function, which

865
01:05:20,460 --> 01:05:23,170
is another name for the Fourier
transform of a probability

866
01:05:23,170 --> 01:05:26,770
distribution, you
would get the p of x

867
01:05:26,770 --> 01:05:33,490
back by the integral
over k divided by 2pi,

868
01:05:33,490 --> 01:05:39,460
the way that I chose the things,
into the ikx p tilde of k.

869
01:05:39,460 --> 01:05:42,600
Basically, this is the
standard relationship

870
01:05:42,600 --> 01:05:45,520
between these objects.

871
01:05:45,520 --> 01:05:50,090
So this is just a
Fourrier transform.

872
01:05:50,090 --> 01:05:56,320
Now, something that appears a
lot in statistical calculations

873
01:05:56,320 --> 01:05:58,530
and implicit in lots
of things that we

874
01:05:58,530 --> 01:06:03,185
do in statistical mechanics
is a generating function.

875
01:06:12,250 --> 01:06:17,880
I can take the characteristic
function p tilde of k.

876
01:06:17,880 --> 01:06:21,620
It's a function of this
Fourrier variable, k.

877
01:06:21,620 --> 01:06:24,360
And I can do an
expansion in that.

878
01:06:24,360 --> 01:06:29,000
I can do the expansion
inside the expectation value

879
01:06:29,000 --> 01:06:34,460
because e to the minus ikx I can
write as a sum over n running

880
01:06:34,460 --> 01:06:39,810
from 0 to infinity minus ik
to the power of m divided by n

881
01:06:39,810 --> 01:06:42,540
factorial x to the nth.

882
01:06:45,330 --> 01:06:48,880
This is the expansion
of the exponential.

883
01:06:48,880 --> 01:06:55,170
The variable here is x, so I can
take everything else outside.

884
01:06:55,170 --> 01:07:01,040
And what I see is that
if I make an expansion

885
01:07:01,040 --> 01:07:06,370
of the characteristic
function, the coefficient

886
01:07:06,370 --> 01:07:11,240
of k to the n up to some
trivial factor of n factorial

887
01:07:11,240 --> 01:07:14,130
will give me the nth moment.

888
01:07:14,130 --> 01:07:16,910
That is once you have
calculated the Fourrier

889
01:07:16,910 --> 01:07:20,610
transform, or the characteristic
function, you can expand it.

890
01:07:20,610 --> 01:07:23,430
And you can, from out
of that expansion, you

891
01:07:23,430 --> 01:07:26,690
can extract all the
moments essentially.

892
01:07:26,690 --> 01:07:30,550
So this is expansion
generates for you the moments,

893
01:07:30,550 --> 01:07:34,250
hence the generating function.

894
01:07:34,250 --> 01:07:37,220
You could even do
something like this.

895
01:07:37,220 --> 01:07:47,880
You could multiply e to the
ikx0 for some x0 p tilde of k.

896
01:07:47,880 --> 01:07:51,510
And that would be the
expectation value of e

897
01:07:51,510 --> 01:07:56,780
to the ikx minus x0.

898
01:07:56,780 --> 01:08:00,450
And you can expand
that, and you would

899
01:08:00,450 --> 01:08:06,600
generate all of the moments
not around the origin,

900
01:08:06,600 --> 01:08:08,350
but around the point x0.

901
01:08:13,960 --> 01:08:17,830
So simple manipulations of
the characteristic function

902
01:08:17,830 --> 01:08:23,840
can shift and give you
other set of moments

903
01:08:23,840 --> 01:08:25,000
around different points.

904
01:08:34,800 --> 01:08:38,560
So the Fourier transform,
or characteristic function,

905
01:08:38,560 --> 01:08:42,290
is the generator of moments.

906
01:08:42,290 --> 01:08:46,240
An even more
important property is

907
01:08:46,240 --> 01:08:48,986
possessed by the cumulant
generating function.

908
01:08:58,229 --> 01:09:07,260
So you have the
characteristic function,

909
01:09:07,260 --> 01:09:08,899
the Fourier transform.

910
01:09:08,899 --> 01:09:13,609
You take its log, so
another function of k.

911
01:09:13,609 --> 01:09:16,810
You start expanding this
function in covers of k.

912
01:09:21,189 --> 01:09:29,370
Add the coefficients of
that, you call cumulants.

913
01:09:33,420 --> 01:09:38,060
So I essentially repeated the
definition that I had up there.

914
01:09:38,060 --> 01:09:44,497
I took a log, and all I did
is I put this subscript c

915
01:09:44,497 --> 01:09:47,729
to go from moments to cumulants.

916
01:09:47,729 --> 01:09:53,890
And also, I have to start the
series from 1 as opposed to 0.

917
01:09:53,890 --> 01:10:00,960
And essentially, I can find the
relationship between cumulants

918
01:10:00,960 --> 01:10:04,370
and moments by
writing this as a log

919
01:10:04,370 --> 01:10:08,980
of the characteristic
function, which

920
01:10:08,980 --> 01:10:14,150
is 1 plus some n
plus 1 to infinity

921
01:10:14,150 --> 01:10:21,190
of minus ik to the n over n
factorial, the nth moments.

922
01:10:21,190 --> 01:10:26,120
So inside the log,
I have the moments.

923
01:10:26,120 --> 01:10:30,020
Outside the log, I
have the cumulants.

924
01:10:30,020 --> 01:10:35,910
And if I have a log
of 1 plus epsilon,

925
01:10:35,910 --> 01:10:41,180
I can use the expansion
of this as epsilon minus

926
01:10:41,180 --> 01:10:45,380
epsilon squared over 2 epsilon
cubed over 3 minus epsilon

927
01:10:45,380 --> 01:10:49,500
to the fourth over 4, et cetera.

928
01:10:49,500 --> 01:10:55,940
And this will enable me to
then match powers of minus ik

929
01:10:55,940 --> 01:11:00,710
on the left and powers
of minus ik on the right.

930
01:11:00,710 --> 01:11:03,530
You can see that the first
thing that I will find

931
01:11:03,530 --> 01:11:09,600
is that the expectation
value of x-- the first power,

932
01:11:09,600 --> 01:11:13,800
the first term that I have
here is minus ik to the mean.

933
01:11:13,800 --> 01:11:16,630
Take the log, I will get that.

934
01:11:16,630 --> 01:11:22,230
So essentially, what I get
is that the first cumulant

935
01:11:22,230 --> 01:11:26,760
on the left is the
first moment that I

936
01:11:26,760 --> 01:11:29,660
will get from the
expansion on the right.

937
01:11:29,660 --> 01:11:32,460
And this is, of course, called
the mean of the distribution.

938
01:11:34,970 --> 01:11:40,680
The second cumulant, I will
have two contributions,

939
01:11:40,680 --> 01:11:45,304
one from epsilon, the other from
minus epsilon squared over 2.

940
01:11:45,304 --> 01:11:48,220
And If you go through
that, you will

941
01:11:48,220 --> 01:11:52,680
get that it is expectation
value of x squared

942
01:11:52,680 --> 01:11:57,320
minus the average of x,
the mean squared, which

943
01:11:57,320 --> 01:12:01,450
is none other than the
expectation value of x

944
01:12:01,450 --> 01:12:06,910
around the mean squared, which
is clearly a positive quantity.

945
01:12:06,910 --> 01:12:08,410
And this is the variance.

946
01:12:14,420 --> 01:12:16,300
And you can keep going.

947
01:12:16,300 --> 01:12:26,360
The third cumulant is x cubed
minus 3 average of x squared

948
01:12:26,360 --> 01:12:32,402
average of x plus 2
average of x itself cubed.

949
01:12:32,402 --> 01:12:33,915
It is called the skewness.

950
01:12:36,900 --> 01:12:40,340
I don't write the
formula for the next one

951
01:12:40,340 --> 01:12:42,050
which is called
a [? cortosis ?].

952
01:12:42,050 --> 01:12:45,910
And you keep going and so forth.

953
01:12:53,390 --> 01:13:01,220
So it turns out that this
hierarchy of cumulants,

954
01:13:01,220 --> 01:13:04,710
essentially, is a hierarchy
of the most important things

955
01:13:04,710 --> 01:13:09,540
that you can know about
a random variable.

956
01:13:09,540 --> 01:13:17,140
So if I tell you that the
outcome of some experiment

957
01:13:17,140 --> 01:13:23,409
is some number x,
distribute it somehow-- I

958
01:13:23,409 --> 01:13:25,450
guess the first thing that
you would like to know

959
01:13:25,450 --> 01:13:28,375
is whether the typical
values that you get

960
01:13:28,375 --> 01:13:33,100
are of the order of 1, are of
the order of million, whatever.

961
01:13:33,100 --> 01:13:36,380
So somehow, the
mean is something

962
01:13:36,380 --> 01:13:40,510
that tells you something that is
most important is zeroth order

963
01:13:40,510 --> 01:13:45,030
thing that you want to
know about the variable.

964
01:13:45,030 --> 01:13:47,130
But the next thing that
you might want to know

965
01:13:47,130 --> 01:13:50,250
is, well, what's the spread?

966
01:13:50,250 --> 01:13:52,830
How far does this thing go?

967
01:13:52,830 --> 01:13:58,090
And then the variance will tell
you something about the spread.

968
01:13:58,090 --> 01:13:59,750
So the next thing
that you want to do

969
01:13:59,750 --> 01:14:02,770
is maybe if given
the spread, am I

970
01:14:02,770 --> 01:14:06,080
more likely to get things
that are on one side or things

971
01:14:06,080 --> 01:14:08,390
that are on the other side.

972
01:14:08,390 --> 01:14:12,780
So the measure of its
asymmetry, right versus left,

973
01:14:12,780 --> 01:14:16,329
is provided by the third
cumulant, which is the skewness

974
01:14:16,329 --> 01:14:16,870
and so forth.

975
01:14:20,180 --> 01:14:24,380
So typically, the
very first few members

976
01:14:24,380 --> 01:14:28,340
of this hierarchy of
cumulants tells you

977
01:14:28,340 --> 01:14:30,985
the most important
information that you

978
01:14:30,985 --> 01:14:32,110
need about the probability.

979
01:14:37,100 --> 01:14:38,860
Now, I will mention
to you, and I

980
01:14:38,860 --> 01:14:44,040
guess we probably will deal
with it more next time around,

981
01:14:44,040 --> 01:14:51,700
the result that is in some
sense the backbone or granddaddy

982
01:14:51,700 --> 01:14:57,080
of all graphical expansions
that are carrying [INAUDIBLE].

983
01:14:57,080 --> 01:15:00,800
And that's a relationship
between the moments

984
01:15:00,800 --> 01:15:04,770
and cumulants that I
will express graphically.

985
01:15:04,770 --> 01:15:14,970
So this is graphical
representation

986
01:15:14,970 --> 01:15:20,860
of moments in
terms of cumulants.

987
01:15:26,470 --> 01:15:29,440
Essentially, what I'm
saying is that you

988
01:15:29,440 --> 01:15:33,100
can go through the
procedure as I outlined.

989
01:15:33,100 --> 01:15:37,240
And if you want to calculate
minus ik to the fifth power

990
01:15:37,240 --> 01:15:41,740
so that you find the description
of the fifth cumulant in terms

991
01:15:41,740 --> 01:15:44,475
of the moment, you'll
have to do a lot of work

992
01:15:44,475 --> 01:15:49,460
in expanding the log and powers
of this object and making sure

993
01:15:49,460 --> 01:15:53,790
that you don't make any
mistakes in the coefficient.

994
01:15:53,790 --> 01:15:58,570
There is a way to
circumvent that graphically

995
01:15:58,570 --> 01:16:00,460
and get the relationship.

996
01:16:00,460 --> 01:16:03,710
So how do we do that?

997
01:16:03,710 --> 01:16:16,010
You'll represent nth cumulant
as let's say a bag of endpoints.

998
01:16:20,100 --> 01:16:28,640
So let's say this entity will
represent the third cumulant.

999
01:16:28,640 --> 01:16:31,700
It's a bag with three points.

1000
01:16:31,700 --> 01:16:37,520
This-- one, two, three,
four, five, six--

1001
01:16:37,520 --> 01:16:39,391
will represent the
sixth cumulant.

1002
01:16:42,340 --> 01:16:54,645
Then, the nth moment
is some of all ways

1003
01:16:54,645 --> 01:17:04,680
of distributing end
points amongst bags.

1004
01:17:11,360 --> 01:17:13,664
So what do I mean?

1005
01:17:13,664 --> 01:17:19,140
So I want to calculate
the first moment x.

1006
01:17:19,140 --> 01:17:22,370
That would correspond
to one point.

1007
01:17:22,370 --> 01:17:25,260
And really, there's
only one diagram

1008
01:17:25,260 --> 01:17:28,470
I can put the bag
around it or not

1009
01:17:28,470 --> 01:17:31,280
that would correspond to this.

1010
01:17:31,280 --> 01:17:35,410
And that corresponds
to the first cumulant,

1011
01:17:35,410 --> 01:17:39,120
basically rewriting
what I had before.

1012
01:17:39,120 --> 01:17:43,010
If I want to look at the second
moment, the second moment

1013
01:17:43,010 --> 01:17:44,650
I need two points.

1014
01:17:44,650 --> 01:17:48,770
The two points I can either
put in the same bag or I

1015
01:17:48,770 --> 01:17:52,180
can put into two separate bags.

1016
01:17:52,180 --> 01:17:56,510
And the first one
corresponds to calculating

1017
01:17:56,510 --> 01:17:59,650
the second cumulant.

1018
01:17:59,650 --> 01:18:02,390
The second term
corresponds to two ways

1019
01:18:02,390 --> 01:18:05,460
in which their first
cumulant has appeared,

1020
01:18:05,460 --> 01:18:07,090
so I have to squared x.

1021
01:18:10,010 --> 01:18:17,080
if I want to calculate the
third moment, I need three dots.

1022
01:18:17,080 --> 01:18:21,400
The three dots I can
either put in one bag

1023
01:18:21,400 --> 01:18:27,950
or I can take one of them out
and keep two of them in a bag.

1024
01:18:27,950 --> 01:18:30,000
And here I had the
choice of three things

1025
01:18:30,000 --> 01:18:32,770
that I could've pulled out.

1026
01:18:32,770 --> 01:18:38,370
Or, I could have all of them in
individual bags of their own.

1027
01:18:38,370 --> 01:18:43,290
And mathematically, the first
term corresponds to x cubed c.

1028
01:18:43,290 --> 01:18:46,680
The third term corresponds
to three versions

1029
01:18:46,680 --> 01:18:49,460
of the variance times the mean.

1030
01:18:49,460 --> 01:18:54,040
And the last term is
just the mean cubed.

1031
01:18:54,040 --> 01:18:57,740
And you can massage
this expression

1032
01:18:57,740 --> 01:19:03,200
to see that I get the expression
that I have for the skewness.

1033
01:19:03,200 --> 01:19:05,860
I didn't offhand
remember the relationship

1034
01:19:05,860 --> 01:19:09,790
that I have to write down
for the fourth cumulant.

1035
01:19:09,790 --> 01:19:12,210
But I can graphically,
immediately get

1036
01:19:12,210 --> 01:19:15,140
the relationship for
the fourth moment

1037
01:19:15,140 --> 01:19:20,020
in terms of the fourth
cumulant which is this entity.

1038
01:19:20,020 --> 01:19:23,860
Or, four ways that I
can take one of the back

1039
01:19:23,860 --> 01:19:29,440
and maintain three in the bag,
three ways in which I have

1040
01:19:29,440 --> 01:19:37,190
two bags of two, six ways in
which I can have a bag of two

1041
01:19:37,190 --> 01:19:42,165
and two things that are
individually apart, and one

1042
01:19:42,165 --> 01:19:44,970
way in which there
are four things that

1043
01:19:44,970 --> 01:19:47,350
are independent of each other.

1044
01:19:47,350 --> 01:19:53,700
And this becomes x to the fourth
cumulant, the fourth cumulant,

1045
01:19:53,700 --> 01:19:58,800
4 times the third
cumulant times the mean,

1046
01:19:58,800 --> 01:20:03,740
3 times the square
of the variance,

1047
01:20:03,740 --> 01:20:10,010
6 times the variance
multiplied by the mean squared,

1048
01:20:10,010 --> 01:20:15,630
and the mean raised
to the fourth power.

1049
01:20:15,630 --> 01:20:16,690
And you can keep going.

1050
01:20:24,642 --> 01:20:29,130
AUDIENCE: Is the variance not
squared in the third term?

1051
01:20:29,130 --> 01:20:31,030
PROFESSOR: Did I forget that?

1052
01:20:31,030 --> 01:20:32,030
Yes, thank you.

1053
01:20:42,340 --> 01:20:43,830
All right.

1054
01:20:43,830 --> 01:20:48,010
So the proof of this is really
just the two-line algebra

1055
01:20:48,010 --> 01:20:51,960
exponentiating these expressions
that we have over here.

1056
01:20:51,960 --> 01:20:55,710
But it's much nicer to
represent that graphically.

1057
01:20:55,710 --> 01:20:59,700
And so now you can go
between things very easily.

1058
01:20:59,700 --> 01:21:05,310
And what I will show next time
is how, using this machinery,

1059
01:21:05,310 --> 01:21:09,960
you can calculate any
moment of a Gaussian,

1060
01:21:09,960 --> 01:21:12,650
for example, in just a
matter of seconds as opposed

1061
01:21:12,650 --> 01:21:16,980
to having to do integrations
and things like that.

1062
01:21:16,980 --> 01:21:19,900
So that's what we
will do next time will

1063
01:21:19,900 --> 01:21:23,120
be to apply this machinery
to various probability

1064
01:21:23,120 --> 01:21:25,290
distribution, such
as a Gaussian,

1065
01:21:25,290 --> 01:21:28,590
that we are likely to
encounter again and again.