1
00:00:00,060 --> 00:00:01,780
The following
content is provided

2
00:00:01,780 --> 00:00:04,019
under a Creative
Commons license.

3
00:00:04,019 --> 00:00:06,870
Your support will help MIT
OpenCourseWare continue

4
00:00:06,870 --> 00:00:10,730
to offer high quality
educational resources for free.

5
00:00:10,730 --> 00:00:13,340
To make a donation or
view additional materials

6
00:00:13,340 --> 00:00:17,217
from hundreds of MIT courses
visit MIT OpenCourseWare

7
00:00:17,217 --> 00:00:17,842
at ocw.mit.edu.

8
00:00:21,680 --> 00:00:22,380
PROFESSOR: OK.

9
00:00:22,380 --> 00:00:25,250
Let's start.

10
00:00:25,250 --> 00:00:30,090
So if we have been thinking
about critical points.

11
00:00:35,050 --> 00:00:40,140
And these arise in
many phased diagrams

12
00:00:40,140 --> 00:00:47,440
such as that we have for the
liquid gas system where there's

13
00:00:47,440 --> 00:00:52,846
a coexistence line, let's say,
between the gas and the liquid

14
00:00:52,846 --> 00:01:00,480
that terminate, or we looked
in the case of a magnet

15
00:01:00,480 --> 00:01:07,930
where as a function of
[INAUDIBLE] temperature

16
00:01:07,930 --> 00:01:10,920
there was in some
sense coexistence

17
00:01:10,920 --> 00:01:15,200
between magnetizations
in different directions

18
00:01:15,200 --> 00:01:16,655
terminating at the
critical point.

19
00:01:20,290 --> 00:01:25,180
So why is it interesting
to take it a whole phase

20
00:01:25,180 --> 00:01:27,540
diagram that we have over here?

21
00:01:27,540 --> 00:01:33,132
For example, for this system, we
can also have solid, et cetera.

22
00:01:33,132 --> 00:01:35,090
AUDIENCE: So isn't this
[INAUDIBLE] [INAUDIBLE]

23
00:01:35,090 --> 00:01:35,480
PROFESSOR: [INAUDIBLE].

24
00:01:35,480 --> 00:01:35,740
Yes.

25
00:01:35,740 --> 00:01:36,440
Thank you.

26
00:01:39,540 --> 00:01:43,154
And focus on just the one point.

27
00:01:43,154 --> 00:01:44,570
In the vicinity
of this one point.

28
00:01:47,380 --> 00:01:52,920
And the reason for that was
this idea of universality.

29
00:01:59,090 --> 00:02:02,430
There many things
that are happening

30
00:02:02,430 --> 00:02:04,660
in the vicinity of
this point as far

31
00:02:04,660 --> 00:02:07,435
as singularitities,
correlations, et cetera,

32
00:02:07,435 --> 00:02:10,840
are concerned that are
independent of whatever

33
00:02:10,840 --> 00:02:14,530
the consequence
of the system are.

34
00:02:14,530 --> 00:02:20,460
And these singularities, we try
to capture through some scaling

35
00:02:20,460 --> 00:02:23,730
laws for the singularities.

36
00:02:23,730 --> 00:02:28,210
And I've been kind
of constructing

37
00:02:28,210 --> 00:02:31,090
a table of your singularities.

38
00:02:31,090 --> 00:02:34,340
Let's do it one more time here.

39
00:02:34,340 --> 00:02:39,180
So we could look at system
such as the liquid gas--

40
00:02:39,180 --> 00:02:44,079
so let's have here
system-- and then we

41
00:02:44,079 --> 00:02:46,230
could look at the liquid gas.

42
00:02:49,100 --> 00:02:54,514
And for that, we can look
at a variety of exponents.

43
00:02:54,514 --> 00:03:02,380
We have alpha, beta,
gamma, delta, mu, theta.

44
00:03:05,900 --> 00:03:11,390
And for the liquid gas,
I write you some numbers.

45
00:03:11,390 --> 00:03:14,955
The heat capacity
diverges with an exponent

46
00:03:14,955 --> 00:03:18,905
that is 0.11--
slightly more accurate

47
00:03:18,905 --> 00:03:21,360
than I had given you before.

48
00:03:21,360 --> 00:03:29,675
The case for beta is 0.33 gamma.

49
00:03:29,675 --> 00:03:30,175
OK.

50
00:03:30,175 --> 00:03:34,020
I will give you a little
bit more digits just

51
00:03:34,020 --> 00:03:36,856
to indicate the
accuracy of experiments.

52
00:03:36,856 --> 00:03:47,390
This is 1.238 minus plus 0.012.

53
00:03:47,390 --> 00:03:52,250
So these exponents are obtained
by looking at the fluid system

54
00:03:52,250 --> 00:03:56,920
with light scattering-- doing
this critical opalescence

55
00:03:56,920 --> 00:03:59,615
that we were talking
about in more detail

56
00:03:59,615 --> 00:04:05,430
and accurately, Delta is 4.8.

57
00:04:05,430 --> 00:04:14,726
The mu is, again from light
scattering 0.629 minus plus

58
00:04:14,726 --> 00:04:18,822
0.003.

59
00:04:18,822 --> 00:04:33,970
Theta is 0.032 0
minus plus 0.013.

60
00:04:33,970 --> 00:04:39,350
And essentially these three are
[INAUDIBLE] light scattered.

61
00:04:48,780 --> 00:04:49,570
Sorry.

62
00:04:49,570 --> 00:04:55,010
Another case that I mentioned
is that of the super fluid.

63
00:05:00,300 --> 00:05:04,270
And in this general
construction of the lambda

64
00:05:04,270 --> 00:05:07,436
gives [INAUDIBLE]
theories that we had,

65
00:05:07,436 --> 00:05:10,525
liquid gas would
be in question one.

66
00:05:10,525 --> 00:05:14,980
Superfluid would
be in question two.

67
00:05:14,980 --> 00:05:21,690
And I just want to mention
that actually the most

68
00:05:21,690 --> 00:05:25,746
experimentally accurate exponent
that has been determined

69
00:05:25,746 --> 00:05:30,660
is the heat capacity for dry
superfluid helium transition.

70
00:05:30,660 --> 00:05:34,720
I had said that it kind of looks
like a logarithmic divergence.

71
00:05:34,720 --> 00:05:36,880
You look at it very closely.

72
00:05:36,880 --> 00:05:41,850
And it is in fact a cusp, and
does not diverge all the way

73
00:05:41,850 --> 00:05:45,740
to infinity, so it corresponds
to a slightly negative value

74
00:05:45,740 --> 00:05:58,346
of alpha, which is the
0.0127 minus plus 0.0003.

75
00:05:58,346 --> 00:06:01,950
And the way that this
has been data-mined

76
00:06:01,950 --> 00:06:06,604
is they took superfluid
helium to the space shuttle,

77
00:06:06,604 --> 00:06:09,508
and this experiments
were done away

78
00:06:09,508 --> 00:06:12,160
from the gravity of
the earth in order

79
00:06:12,160 --> 00:06:15,600
to not to have to worry
about the density difference

80
00:06:15,600 --> 00:06:19,470
that we would have
across the system.

81
00:06:19,470 --> 00:06:23,133
Other exponents that you
have for this system--

82
00:06:23,133 --> 00:06:27,882
let me write down--
beta is around 0.35.

83
00:06:27,882 --> 00:06:32,896
Gamma is 1.32.

84
00:06:32,896 --> 00:06:37,520
Delta is 4.79.

85
00:06:37,520 --> 00:06:40,920
Mu is is 0.67.

86
00:06:40,920 --> 00:06:46,990
Theta is 0.04.

87
00:06:46,990 --> 00:06:52,470
And we don't need
this for system.

88
00:06:56,250 --> 00:07:03,580
Any questions we could
do players kind of

89
00:07:03,580 --> 00:07:12,010
add the exponents here
I've booked usability

90
00:07:12,010 --> 00:07:18,820
even if it's minus
1 is a research data

91
00:07:18,820 --> 00:07:32,650
0.7 down all those are long this
is more to say about new ideas

92
00:07:32,650 --> 00:07:49,650
and so on I think is that
these numbers aren't you

93
00:07:49,650 --> 00:08:02,930
think that is simplest
way for us is net

94
00:08:02,930 --> 00:08:10,480
my position and the question
is why these numbers are all

95
00:08:10,480 --> 00:08:14,160
of the same as all the
systems is therefore profound.

96
00:08:14,160 --> 00:08:16,200
These are dimensionless numbers.

97
00:08:16,200 --> 00:08:20,240
So in some sense, it is a
little bit of mathematics.

98
00:08:20,240 --> 00:08:22,775
It's not like you calculate
the charge of the electron

99
00:08:22,775 --> 00:08:25,080
and you get a number.

100
00:08:25,080 --> 00:08:29,240
These don't depend on
a specific material.

101
00:08:29,240 --> 00:08:31,200
Therefore, what is
important about them

102
00:08:31,200 --> 00:08:34,640
is that they must
somehow be capturing

103
00:08:34,640 --> 00:08:37,440
some aspect of the
collective behavior of all

104
00:08:37,440 --> 00:08:40,880
of these degrees of freedom,
in which the details of what

105
00:08:40,880 --> 00:08:44,090
the degrees of freedom
are is not that important.

106
00:08:44,090 --> 00:08:47,810
Maybe the type of synergy
rating is important.

107
00:08:47,810 --> 00:08:52,860
So unless we understand
and derive these numbers,

108
00:08:52,860 --> 00:08:55,950
there is something important
about the collective behavior

109
00:08:55,950 --> 00:09:00,440
of many degrees of freedom
that we have not understood.

110
00:09:00,440 --> 00:09:03,711
And it is somehow a
different question

111
00:09:03,711 --> 00:09:08,160
if you are thinking
about phase transitions.

112
00:09:08,160 --> 00:09:11,660
So let's say you're thinking
about superconductors.

113
00:09:11,660 --> 00:09:14,220
There's a lot of interest
in making high temperature

114
00:09:14,220 --> 00:09:18,600
superconductor pushing TC
further and further up.

115
00:09:18,600 --> 00:09:21,240
So that's certainly
a material problem.

116
00:09:21,240 --> 00:09:23,050
We are asking a
different problem.

117
00:09:23,050 --> 00:09:26,320
Why is it, whether you have a
high temperature superconductor

118
00:09:26,320 --> 00:09:30,590
or any other type of system,
the collective behavior

119
00:09:30,590 --> 00:09:34,940
is captured by the
same set of exponents.

120
00:09:34,940 --> 00:09:39,690
So in an attempt to
try to answer that,

121
00:09:39,690 --> 00:09:49,450
we did this Landau-Ginzburg and
try to calculate its singular

122
00:09:49,450 --> 00:09:53,720
behavior using this other
point of approximation.

123
00:09:53,720 --> 00:09:56,150
And the numbers
that we got, alpha

124
00:09:56,150 --> 00:10:00,680
was 0, meaning that
there was discontinuity.

125
00:10:00,680 --> 00:10:09,483
Beta was 1/2, gamma was 1,
delta was 3, my was 1/2,

126
00:10:09,483 --> 00:10:15,170
theta was 0, which don't
quite match with these numbers

127
00:10:15,170 --> 00:10:18,070
that we have up there.

128
00:10:18,070 --> 00:10:22,660
So question is,
what should you do?

129
00:10:22,660 --> 00:10:28,000
We've made an attempt and that
attempt was not successful.

130
00:10:28,000 --> 00:10:33,480
So we are going to completely
for a while forget about that

131
00:10:33,480 --> 00:10:36,440
and try to approach the problem
from a different perspective

132
00:10:36,440 --> 00:10:38,770
and see how far we
can go, whether we

133
00:10:38,770 --> 00:10:43,800
can gain any new insights.

134
00:10:43,800 --> 00:10:49,805
So that new approach
I put on there

135
00:10:49,805 --> 00:10:51,847
the name of the
scaling hypothesis.

136
00:10:59,640 --> 00:11:06,090
And the reason for that will
become apparent shortly.

137
00:11:06,090 --> 00:11:12,550
So what we have in common
in both of these examples

138
00:11:12,550 --> 00:11:17,570
is that there is
a line where there

139
00:11:17,570 --> 00:11:21,700
are discontinuities
in calculating

140
00:11:21,700 --> 00:11:24,630
some thermodynamic
function that terminates

141
00:11:24,630 --> 00:11:26,730
at a particular point.

142
00:11:26,730 --> 00:11:30,030
And in the case of
the magnetic system,

143
00:11:30,030 --> 00:11:35,980
we can look at the singularities
approaching that point either

144
00:11:35,980 --> 00:11:38,580
along the direction
that corresponds

145
00:11:38,580 --> 00:11:44,040
to change in temperature and
parametrize that through heat,

146
00:11:44,040 --> 00:11:48,280
or we can change
the magnetic field

147
00:11:48,280 --> 00:11:52,030
and approach the problem
from this other direction.

148
00:11:52,030 --> 00:11:55,220
And we saw that there
were analogs for doing

149
00:11:55,220 --> 00:11:59,195
so in the liquid
gas system also.

150
00:11:59,195 --> 00:12:03,830
And in particular, let's say
we calculated a magnetization,

151
00:12:03,830 --> 00:12:07,520
we found that there was one form
of singularity coming this way,

152
00:12:07,520 --> 00:12:10,430
one form of singularity
coming that way.

153
00:12:10,430 --> 00:12:14,005
We look at the picture
for the liquid gas system

154
00:12:14,005 --> 00:12:18,400
that I have up there, and
it's not necessarily clear

155
00:12:18,400 --> 00:12:24,310
which direction would
correspond to this nice symmetry

156
00:12:24,310 --> 00:12:26,780
breaking or
non-symmetry breaking

157
00:12:26,780 --> 00:12:29,740
that you have for
the magnetic system.

158
00:12:29,740 --> 00:12:33,610
So you may well ask, suppose
I approach the critical point

159
00:12:33,610 --> 00:12:35,440
along some other direction.

160
00:12:35,440 --> 00:12:38,950
Maybe I come in along
the path such as this.

161
00:12:38,950 --> 00:12:41,120
I still go to the
critical point.

162
00:12:41,120 --> 00:12:44,170
We can imagine that for
the liquid gas system.

163
00:12:44,170 --> 00:12:46,955
And what's the structure
of the singularities?

164
00:12:46,955 --> 00:12:51,221
I know that there are different
singularities in the t and h

165
00:12:51,221 --> 00:12:51,720
direction.

166
00:12:51,720 --> 00:12:53,802
What is it if I
come and approach

167
00:12:53,802 --> 00:12:55,670
the system along a
different direction,

168
00:12:55,670 --> 00:13:00,400
which we may well do
for a liquid gas system?

169
00:13:00,400 --> 00:13:03,270
Well, we could actually
answer that if we go back

170
00:13:03,270 --> 00:13:07,680
to our graph saddlepoint
approximation.

171
00:13:07,680 --> 00:13:10,050
In the saddlepoint
approximation,

172
00:13:10,050 --> 00:13:15,134
we said that ultimately,
the singularities in terms

173
00:13:15,134 --> 00:13:19,090
of these two
parameters t and h--

174
00:13:19,090 --> 00:13:20,400
so this is in the saddlepoint.

175
00:13:26,050 --> 00:13:33,185
Part obtained by minimizing
this function that

176
00:13:33,185 --> 00:13:37,405
was appearing in the
expansion in the exponent.

177
00:13:37,405 --> 00:13:39,470
There was a t over 2m squared.

178
00:13:39,470 --> 00:13:42,378
There was a mu n to the 4th.

179
00:13:42,378 --> 00:13:43,770
And there is an hm.

180
00:13:46,554 --> 00:13:52,325
So we had to minimize
this with respect to m.

181
00:13:52,325 --> 00:13:56,840
And clearly, what
that gives us is m.

182
00:13:56,840 --> 00:14:00,280
If I really solve
the equation, that

183
00:14:00,280 --> 00:14:03,220
corresponds to
this minimization,

184
00:14:03,220 --> 00:14:05,820
which is a function of t and h.

185
00:14:05,820 --> 00:14:10,970
And in particular, approaching
two directions that's

186
00:14:10,970 --> 00:14:16,000
indicated, if I'm along the
direction where h equals 0,

187
00:14:16,000 --> 00:14:18,869
I essentially balance
these two terms.

188
00:14:18,869 --> 00:14:20,660
Let's just write this
as a proportionality.

189
00:14:20,660 --> 00:14:23,515
I don't really care
about the numbers.

190
00:14:26,580 --> 00:14:30,110
Along the direction
where h equals 0,

191
00:14:30,110 --> 00:14:36,120
I have to balance m to
the 4th and tm squared.

192
00:14:36,120 --> 00:14:41,150
So m squared will scale like e.

193
00:14:41,150 --> 00:14:47,048
m will scale like
square root of t.

194
00:14:47,048 --> 00:14:49,495
And more precisely,
we calculated

195
00:14:49,495 --> 00:14:53,480
this formula for t
negative and h equals to 0.

196
00:14:56,220 --> 00:14:59,300
If I, on the other hand,
come along the direction

197
00:14:59,300 --> 00:15:08,050
that corresponds to t equals
to 0, along that direction

198
00:15:08,050 --> 00:15:09,510
I don't have a first term.

199
00:15:09,510 --> 00:15:14,440
I have to balance
um the 4th and hm.

200
00:15:14,440 --> 00:15:20,143
So we immediately see that
m will scale like h over u.

201
00:15:20,143 --> 00:15:25,008
In fact, more correctly h over
4u to the power of one third.

202
00:15:28,950 --> 00:15:31,030
You substitute this
in the free energy

203
00:15:31,030 --> 00:15:34,464
and you find that the singular
part of the free energy

204
00:15:34,464 --> 00:15:41,590
as a function of t and h in
this saddlepoint approximation

205
00:15:41,590 --> 00:15:45,960
has the [INAUDIBLE] to the
form of proportionality.

206
00:15:45,960 --> 00:15:49,740
If I substitute this in
the formula for t negative,

207
00:15:49,740 --> 00:15:52,905
I will get something
like minus t squared

208
00:15:52,905 --> 00:15:58,400
over 4 we have-- forget about
the number t squared over u.

209
00:15:58,400 --> 00:16:02,430
If I go along the t
equals to 0 direction,

210
00:16:02,430 --> 00:16:08,500
substitute that over there,
I will get n to the 4th.

211
00:16:08,500 --> 00:16:15,570
I will get h to the 4 thirds
divided by mu to the one third.

212
00:16:15,570 --> 00:16:18,170
Even the mu dependence
I'm not interested.

213
00:16:18,170 --> 00:16:22,850
I'm really interested in the
behavior close to t and h

214
00:16:22,850 --> 00:16:24,930
as a function of t and h.

215
00:16:24,930 --> 00:16:27,820
Mu is basically some
non-universal number

216
00:16:27,820 --> 00:16:29,960
that doesn't go to 0.

217
00:16:29,960 --> 00:16:33,980
I could in some sense
capture these two expressions

218
00:16:33,980 --> 00:16:39,050
by a form that is
t squared and then

219
00:16:39,050 --> 00:16:41,430
some function--
let's call it g sub

220
00:16:41,430 --> 00:16:47,055
f which is a function
of-- let's see

221
00:16:47,055 --> 00:16:52,338
how I define the delta
h over t to the delta.

222
00:16:58,510 --> 00:17:03,610
So my claim is that I toyed
with the behavior coming

223
00:17:03,610 --> 00:17:07,190
across these two different
special direction.

224
00:17:07,190 --> 00:17:13,420
In general, anywhere else
where t and h are both nonzero,

225
00:17:13,420 --> 00:17:17,980
the answer for m will be some
solution of a cubic equation,

226
00:17:17,980 --> 00:17:21,099
but we can arrange it to
only be a function of h

227
00:17:21,099 --> 00:17:25,410
over [INAUDIBLE]
and have this form.

228
00:17:25,410 --> 00:17:30,880
Now I could maybe
rather than explicitly

229
00:17:30,880 --> 00:17:33,210
show you how that
arises, which is not

230
00:17:33,210 --> 00:17:36,580
difficult-- you can do that--
since there's something

231
00:17:36,580 --> 00:17:39,460
that we need to
do later on, I'll

232
00:17:39,460 --> 00:17:43,030
show it in the following manner.

233
00:17:43,030 --> 00:17:48,980
I have not specified what
this function g sub f is.

234
00:17:48,980 --> 00:17:53,710
But I know its behavior
along h equals to 0 here.

235
00:17:53,710 --> 00:17:57,238
And so if I put h equals to 0,
the argument of the function

236
00:17:57,238 --> 00:17:59,580
goes to 0.

237
00:17:59,580 --> 00:18:02,810
So if I say that the
argument of the function

238
00:18:02,810 --> 00:18:06,950
is a constant-- the constant
let's say is minus 1

239
00:18:06,950 --> 00:18:10,270
over u on one side,
0 on the other side,

240
00:18:10,270 --> 00:18:12,170
then everything's fine.

241
00:18:12,170 --> 00:18:20,060
So I have is that the limit
as its argument goes to 0

242
00:18:20,060 --> 00:18:21,330
should be some constant.

243
00:18:26,890 --> 00:18:29,020
Well, what about
the other direction?

244
00:18:29,020 --> 00:18:32,520
How can I reproduce
from a form such as this

245
00:18:32,520 --> 00:18:37,510
the behavior when t equals to 0?

246
00:18:37,510 --> 00:18:40,400
Because I see that
when t equals to 0,

247
00:18:40,400 --> 00:18:43,560
the answer of course
cannot depend on t itself,

248
00:18:43,560 --> 00:18:47,840
but as a power law
as a function of h.

249
00:18:47,840 --> 00:18:51,790
Is it consistent with this form?

250
00:18:51,790 --> 00:18:54,010
Well, as t goes
to 0 in this form,

251
00:18:54,010 --> 00:18:58,580
the numerator here goes to 0,
the argument of the function

252
00:18:58,580 --> 00:19:01,660
goes to infinity, I
need to know something

253
00:19:01,660 --> 00:19:05,480
about the behavior of
the function of infinity.

254
00:19:05,480 --> 00:19:09,800
So let's say that the limiting
behavior as the argument

255
00:19:09,800 --> 00:19:13,400
of the function goes
to infinity of gf

256
00:19:13,400 --> 00:19:21,438
of x is proportional to the
argument to some other peak.

257
00:19:21,438 --> 00:19:24,432
And I don't know
where that power is.

258
00:19:24,432 --> 00:19:28,230
Then if I look at this
function, the whole function

259
00:19:28,230 --> 00:19:32,775
in this limit where t
goes to 0 will behave.

260
00:19:32,775 --> 00:19:37,530
There's a t squared out
front the goes to 0,

261
00:19:37,530 --> 00:19:39,890
the argument of the
function goes to infinity.

262
00:19:39,890 --> 00:19:44,860
So the function will go like
the argument to some power.

263
00:19:44,860 --> 00:19:49,313
So I go like h t to the
delta to some other peak.

264
00:19:56,720 --> 00:20:00,190
So what do I know?

265
00:20:00,190 --> 00:20:02,320
I know that the
answer should really

266
00:20:02,320 --> 00:20:06,360
be proportional to h
to the four thirds.

267
00:20:09,210 --> 00:20:15,496
So I immediately know that
my t should be four thirds.

268
00:20:20,480 --> 00:20:21,810
But what about this delta?

269
00:20:21,810 --> 00:20:24,660
I never told you what delta was.

270
00:20:24,660 --> 00:20:28,100
Now I can figure out what delta
is, because the answer should

271
00:20:28,100 --> 00:20:31,860
not depend on t.
t has gone to 0.

272
00:20:31,860 --> 00:20:34,460
And so what power
of t do I have?

273
00:20:34,460 --> 00:20:42,630
I have 2 minus
delta p should be 0.

274
00:20:42,630 --> 00:20:50,950
So my delta should be 2
over p, 2 over four thirds,

275
00:20:50,950 --> 00:20:52,365
so it should be three halves.

276
00:21:01,610 --> 00:21:08,920
Why is this exponent relevant to
the question that I had before?

277
00:21:08,920 --> 00:21:13,840
You can see that the function
that describes the free energy

278
00:21:13,840 --> 00:21:16,580
as a function of
these two coordinates.

279
00:21:16,580 --> 00:21:21,600
If I look at the combination
where h and t are non-zero,

280
00:21:21,600 --> 00:21:28,625
is very much dependent on this
h divided by t to the delta,

281
00:21:28,625 --> 00:21:31,650
and that delta is three halves.

282
00:21:31,650 --> 00:21:34,420
So, for example, if
I were to draw here

283
00:21:34,420 --> 00:21:43,630
curves where h goes like
3 to the three halves--

284
00:21:43,630 --> 00:21:48,770
it's some coefficient, I don't
know what that coefficient is--

285
00:21:48,770 --> 00:21:55,320
then essentially, everything
that is on the side

286
00:21:55,320 --> 00:22:01,160
that hogs the vertical axis
behaves like the h singularity.

287
00:22:01,160 --> 00:22:06,510
Everything that is over here
depends like a t singularity.

288
00:22:06,510 --> 00:22:12,230
So a path that, for example,
comes along a straight line,

289
00:22:12,230 --> 00:22:16,960
if I, let's say, call
the distance that I have

290
00:22:16,960 --> 00:22:21,500
to the critical point
s, then t is something

291
00:22:21,500 --> 00:22:27,220
like s cosine of theta. h is
something like s sine theta.

292
00:22:27,220 --> 00:22:29,630
You can see however
that the information

293
00:22:29,630 --> 00:22:36,840
h over t to the delta as
s goes to 0 will diverge,

294
00:22:36,840 --> 00:22:39,745
because I have other
three halves down here

295
00:22:39,745 --> 00:22:44,550
for s that will overcome the
linear cover I have over there.

296
00:22:44,550 --> 00:22:49,570
So for any linear path that
goes through the critical point,

297
00:22:49,570 --> 00:22:53,760
eventually for small s I will
see the type of singularity

298
00:22:53,760 --> 00:22:57,770
that is characteristic
of the magnetic field

299
00:22:57,770 --> 00:23:03,050
if the exponents are
according to this other point.

300
00:23:03,050 --> 00:23:05,230
We have this
assumption, of course.

301
00:23:05,230 --> 00:23:10,040
But if I therefore
knew the correct delta

302
00:23:10,040 --> 00:23:14,070
for all of those systems, I
would be also able to answer,

303
00:23:14,070 --> 00:23:17,530
let's say for the
liquid gas, whether if I

304
00:23:17,530 --> 00:23:20,260
take a linear path that goes
through the critical point

305
00:23:20,260 --> 00:23:22,920
I would see one set of
singularities or deltas

306
00:23:22,920 --> 00:23:25,980
that have singularities.

307
00:23:25,980 --> 00:23:34,000
So this delta which is
called a gap exponent,

308
00:23:34,000 --> 00:23:35,290
gives you the answer to that.

309
00:23:38,420 --> 00:23:43,530
But of course I don't
know the other exponents.

310
00:23:43,530 --> 00:23:46,995
There is no reason for me
to trust the gap exponent

311
00:23:46,995 --> 00:23:51,010
that I obtained in this fashion.

312
00:23:51,010 --> 00:24:02,740
So what I say is let's assume
that for any critical point,

313
00:24:02,740 --> 00:24:06,620
the singular part
of the free energy

314
00:24:06,620 --> 00:24:10,080
on approaching the
critical point which

315
00:24:10,080 --> 00:24:14,710
depends on this
pair of coordinates

316
00:24:14,710 --> 00:24:18,730
has a singular behavior
that is similar to what

317
00:24:18,730 --> 00:24:22,460
we had over here, except that
I don't know the exponent.

318
00:24:22,460 --> 00:24:26,620
So rather than
putting 2 t squared,

319
00:24:26,620 --> 00:24:30,070
I write t to the 2
minus alpha for reason

320
00:24:30,070 --> 00:24:32,736
that will become
apparent shortly,

321
00:24:32,736 --> 00:24:39,820
and some function of h t
to the delta and for some

322
00:24:39,820 --> 00:24:41,227
alpha and delta.

323
00:24:47,580 --> 00:24:49,700
So this is certainly
already an assumption.

324
00:24:52,680 --> 00:24:57,730
This mathematically
corresponds to having

325
00:24:57,730 --> 00:24:59,015
homogeneous functions.

326
00:25:06,190 --> 00:25:09,490
Because if I have a
function of x and y,

327
00:25:09,490 --> 00:25:14,930
I can certainly write lots of
functions such as x squared

328
00:25:14,930 --> 00:25:19,110
plus y squared plus a constant
plus x cubed y cubed that I

329
00:25:19,110 --> 00:25:22,800
cannot rearrange into this form.

330
00:25:22,800 --> 00:25:25,560
But there are certain
functions of x and y

331
00:25:25,560 --> 00:25:29,150
that I can rearrange
so that I can pull out

332
00:25:29,150 --> 00:25:32,370
some factor of let's
say x squared out front,

333
00:25:32,370 --> 00:25:34,480
and everything that
is then in a series

334
00:25:34,480 --> 00:25:37,870
is a function of let's
say y over x cubed.

335
00:25:37,870 --> 00:25:39,910
Something like that.

336
00:25:39,910 --> 00:25:42,440
So there's some
class of functions

337
00:25:42,440 --> 00:25:46,074
of two arguments that
have this homogeneity.

338
00:25:46,074 --> 00:25:50,400
So we are going to assume that
the singular behavior close

339
00:25:50,400 --> 00:25:55,732
to the critical point is
described by such a function.

340
00:25:55,732 --> 00:25:57,181
That's an assumption.

341
00:26:00,080 --> 00:26:05,190
But having made that assumption,
let's follow its consequence

342
00:26:05,190 --> 00:26:07,120
and let's see if we
learned something

343
00:26:07,120 --> 00:26:10,520
about that table of exponents.

344
00:26:10,520 --> 00:26:13,570
Now the first thing
to note is clearly

345
00:26:13,570 --> 00:26:16,455
I chose this alpha
over here so that when

346
00:26:16,455 --> 00:26:23,820
I take two derivatives
with respect to t,

347
00:26:23,820 --> 00:26:27,050
I would get something
like a heat capacity,

348
00:26:27,050 --> 00:26:32,550
for which I know what
the divergence is.

349
00:26:32,550 --> 00:26:36,580
That's a divergence
called alpha.

350
00:26:36,580 --> 00:26:37,980
But there's one
thing that I have

351
00:26:37,980 --> 00:26:42,100
to show you is that when
I take a derivative of one

352
00:26:42,100 --> 00:26:48,032
of these homogeneous
functions, with respect

353
00:26:48,032 --> 00:26:52,270
to one of its arguments,
I will generate

354
00:26:52,270 --> 00:26:53,850
another homogeneous function.

355
00:26:53,850 --> 00:26:57,080
If I take one derivative
with respect to t,

356
00:26:57,080 --> 00:27:03,770
that derivative can
either act on this,

357
00:27:03,770 --> 00:27:08,370
leaving the function
unchanged, or it

358
00:27:08,370 --> 00:27:11,600
can act on the argument
of the function

359
00:27:11,600 --> 00:27:16,740
and give me d to
the 2 minus alpha.

360
00:27:16,740 --> 00:27:21,950
I will have minus h t to
the power of delta plus 1.

361
00:27:21,950 --> 00:27:24,366
There will be a factor
of delta and then

362
00:27:24,366 --> 00:27:28,713
I will have the derivative
function ht to the delta.

363
00:27:32,530 --> 00:27:34,630
So I just took derivatives.

364
00:27:34,630 --> 00:27:37,687
I can certainly pull
out a factor of t

365
00:27:37,687 --> 00:27:40,750
to the 1 minus alpha.

366
00:27:40,750 --> 00:27:46,420
Then the first term is
just 2 minus alpha times

367
00:27:46,420 --> 00:27:47,365
the original function.

368
00:27:50,820 --> 00:27:59,240
The second term is minus delta
h divided by t to the delta.

369
00:27:59,240 --> 00:28:01,500
Because I pulled out
the 1 minus alpha,

370
00:28:01,500 --> 00:28:05,310
this t gets rid of
the factor of 1 there.

371
00:28:05,310 --> 00:28:06,400
And I have the derivative.

372
00:28:12,000 --> 00:28:13,990
So this is completely
different function.

373
00:28:13,990 --> 00:28:16,825
It's not the derivative
of the original function.

374
00:28:16,825 --> 00:28:20,015
But whatever it is it
is still only a function

375
00:28:20,015 --> 00:28:23,984
of the combination h
over t to the delta.

376
00:28:23,984 --> 00:28:26,650
So the derivative of
a homogeneous function

377
00:28:26,650 --> 00:28:28,973
is some other
homogeneous function.

378
00:28:28,973 --> 00:28:30,452
Let's call it g2.

379
00:28:30,452 --> 00:28:31,438
It doesn't matter.

380
00:28:31,438 --> 00:28:36,370
Let's call it g1
ht to the delta.

381
00:28:36,370 --> 00:28:39,560
And this will happen if I
take a second derivative.

382
00:28:39,560 --> 00:28:41,920
So I know that if I
take two derivatives,

383
00:28:41,920 --> 00:28:44,960
I will get t to the minus alpha.

384
00:28:44,960 --> 00:28:49,100
I will basically drop
two factors over there.

385
00:28:49,100 --> 00:28:56,730
And then some other
function, ht to the delta.

386
00:28:56,730 --> 00:29:00,150
Clearly again, if I say
that I'm looking at the line

387
00:29:00,150 --> 00:29:05,020
where h equals to
zero for a magnet,

388
00:29:05,020 --> 00:29:08,300
then the argument of
the function goes to 0.

389
00:29:08,300 --> 00:29:12,030
If I say that the function
of the argument goes to 0

390
00:29:12,030 --> 00:29:15,790
is a constant, like
we had over here,

391
00:29:15,790 --> 00:29:19,840
then I will have the singularity
t to the minus alpha.

392
00:29:19,840 --> 00:29:24,160
So I've clearly engineered
whatever the value of alpha

393
00:29:24,160 --> 00:29:28,410
is in this table,
I can put over here

394
00:29:28,410 --> 00:29:33,630
and I have the right singularity
for the heat capacity.

395
00:29:33,630 --> 00:29:36,660
Essentially I've put
it there by hand.

396
00:29:36,660 --> 00:29:40,810
Let me comment on one
other thing, which

397
00:29:40,810 --> 00:29:47,869
is when we are looking
at just the temperature,

398
00:29:47,869 --> 00:29:49,410
let's say we are
looking at something

399
00:29:49,410 --> 00:29:51,360
like a superfluid,
the only parameter

400
00:29:51,360 --> 00:29:58,060
that we have at our disposal
is temperature and tens of ITC.

401
00:29:58,060 --> 00:30:03,050
Let's say we plug
the heat capacity

402
00:30:03,050 --> 00:30:06,631
and then we see divergence of
the heat capacity on the two

403
00:30:06,631 --> 00:30:07,130
sides.

404
00:30:09,910 --> 00:30:15,430
Who said that I should have
the same exponent on this side

405
00:30:15,430 --> 00:30:16,865
and on this side?

406
00:30:19,380 --> 00:30:22,230
So we said that
generally, in principle,

407
00:30:22,230 --> 00:30:26,370
I could say I would do that.

408
00:30:26,370 --> 00:30:31,090
And in principle, there
is no problem with that.

409
00:30:31,090 --> 00:30:33,980
If there is function that
has one behavior here,

410
00:30:33,980 --> 00:30:38,820
another behavior there,
who says that two exponents

411
00:30:38,820 --> 00:30:41,710
have to be the same?

412
00:30:41,710 --> 00:30:44,550
But I have said something more.

413
00:30:44,550 --> 00:30:49,310
I have said that in all of
the cases that I'm looking at,

414
00:30:49,310 --> 00:30:54,440
I know that there
is some other axis.

415
00:30:57,820 --> 00:31:02,850
And for example, if I am
in the liquid gas system,

416
00:31:02,850 --> 00:31:06,760
I can start from down
here, go all the way around

417
00:31:06,760 --> 00:31:10,940
back here without
encountering a singularity.

418
00:31:10,940 --> 00:31:15,120
I can go from the liquid
all the way to gas

419
00:31:15,120 --> 00:31:16,650
without encountering
a singularity.

420
00:31:19,450 --> 00:31:24,630
So that says that the system
is different from a system

421
00:31:24,630 --> 00:31:29,120
that, let's say, has a
line of singularities.

422
00:31:29,120 --> 00:31:33,550
So if I now take the
functions that in principle

423
00:31:33,550 --> 00:31:37,014
have two different
singularities,

424
00:31:37,014 --> 00:31:43,740
t to the minus alpha minus t to
the minus alpha plus on the h

425
00:31:43,740 --> 00:31:48,950
equals to 0 axis and
try to elevate them

426
00:31:48,950 --> 00:31:54,930
into the entire space by putting
this homogeneous functions

427
00:31:54,930 --> 00:32:00,430
in front of them, there
is one and only one way

428
00:32:00,430 --> 00:32:05,405
in which the two functions
can match exactly on this t

429
00:32:05,405 --> 00:32:09,680
equals to 0 line, and that's if
the two exponents are the same

430
00:32:09,680 --> 00:32:12,518
and you are dealing
with the same function.

431
00:32:12,518 --> 00:32:15,830
So that we put in
a bit of physics.

432
00:32:15,830 --> 00:32:21,266
So in principle, mathematically
if you don't have the h axis

433
00:32:21,266 --> 00:32:25,839
and you look at the one line
and there's a singularity,

434
00:32:25,839 --> 00:32:27,630
there's no reason why
the two singularities

435
00:32:27,630 --> 00:32:29,700
should be the same.

436
00:32:29,700 --> 00:32:31,920
But we know that we are
looking at the class

437
00:32:31,920 --> 00:32:35,430
of physical systems where
there is the possibility

438
00:32:35,430 --> 00:32:39,200
to analytically go from
one side to the other side.

439
00:32:39,200 --> 00:32:42,270
And that immediately
imposes this constraint

440
00:32:42,270 --> 00:32:47,590
that alpha plus should be
alpha minus, and one alpha

441
00:32:47,590 --> 00:32:49,205
is in fact sufficient.

442
00:32:49,205 --> 00:32:54,850
And I gave you the correct
answer for why that is.

443
00:32:54,850 --> 00:32:57,930
If you want to see the
precise mathematical details

444
00:32:57,930 --> 00:33:00,130
step by step, then
that's in the notes.

445
00:33:04,330 --> 00:33:06,020
So fine.

446
00:33:06,020 --> 00:33:08,360
So far we haven't learned much.

447
00:33:08,360 --> 00:33:10,790
We've justified why
the two alphas should

448
00:33:10,790 --> 00:33:13,480
be the same above
and below, but we

449
00:33:13,480 --> 00:33:17,770
put the alpha, the one
alpha, then by hand.

450
00:33:17,770 --> 00:33:19,945
And then we have this
unknown delta also.

451
00:33:19,945 --> 00:33:23,046
But let's proceed.

452
00:33:23,046 --> 00:33:26,170
Let's see what other consequence
emerge, because now we

453
00:33:26,170 --> 00:33:27,970
have a function
of two variables.

454
00:33:27,970 --> 00:33:30,490
I took derivatives
in respect to t.

455
00:33:30,490 --> 00:33:33,546
I can take derivatives
with respect to m.

456
00:33:33,546 --> 00:33:39,910
And in particular,
the magnetization

457
00:33:39,910 --> 00:33:46,340
m as a function of
t and h is obtained

458
00:33:46,340 --> 00:33:52,800
from a derivative of the free
energy with respect to h.

459
00:33:52,800 --> 00:33:55,230
There's potential.

460
00:33:55,230 --> 00:33:59,140
It's the response to
adding a field could

461
00:33:59,140 --> 00:34:01,250
be some factor of
beta c or whatever.

462
00:34:01,250 --> 00:34:03,440
It's not important.

463
00:34:03,440 --> 00:34:06,460
The singular part
will come from this.

464
00:34:06,460 --> 00:34:10,690
And so taking a derivative
of this function

465
00:34:10,690 --> 00:34:14,510
I will get t this to
the 2 minus alpha.

466
00:34:14,510 --> 00:34:17,159
The derivative of a
can be respect to h,

467
00:34:17,159 --> 00:34:19,849
but h comes in the
combination h over t

468
00:34:19,849 --> 00:34:24,610
to the delta will bring down a
factor of minus delta up front.

469
00:34:24,610 --> 00:34:29,058
Then the derivative function--
let's call it gf1, for example.

470
00:34:36,210 --> 00:34:41,906
So now I can look at this
function in the limit

471
00:34:41,906 --> 00:34:49,400
where h goes to 0, climb
along the coexistence line,

472
00:34:49,400 --> 00:34:51,280
h2 goes to 0.

473
00:34:51,280 --> 00:34:55,550
The argument of the
function has gone to 0.

474
00:34:55,550 --> 00:34:58,150
Makes sense that the
function should be constant

475
00:34:58,150 --> 00:35:00,140
when its argument goes to 0.

476
00:35:00,140 --> 00:35:03,192
So the answer is going
to be proportional to t

477
00:35:03,192 --> 00:35:06,108
to the 2 minus
alpha minus delta.

478
00:35:09,940 --> 00:35:13,720
But that's how beta was defined.

479
00:35:13,720 --> 00:35:21,400
So if I know my beta
and alpha, then I

480
00:35:21,400 --> 00:35:25,988
can calculate my delta from
this exponent identity.

481
00:35:28,550 --> 00:35:31,760
Again, so far you
haven't done much.

482
00:35:31,760 --> 00:35:37,075
You have translated two unknown
exponents, this singular form,

483
00:35:37,075 --> 00:35:40,540
this gap exponent
that we don't know.

484
00:35:40,540 --> 00:35:44,630
I can also look
at the other limit

485
00:35:44,630 --> 00:35:48,280
where t goes to 0
that is calculating

486
00:35:48,280 --> 00:35:51,550
the magnetization along
the critical isotherm.

487
00:35:55,280 --> 00:36:01,140
So then the argument of the
function has gone to infinity.

488
00:36:01,140 --> 00:36:03,990
And whatever the answer
is should not depend on t,

489
00:36:03,990 --> 00:36:06,470
because I have said t goes to 0.

490
00:36:06,470 --> 00:36:09,970
So I apply the same trick
that I did over here.

491
00:36:09,970 --> 00:36:13,420
I say that when the
argument goes to infinity,

492
00:36:13,420 --> 00:36:19,330
the function goes like
some power of its argument.

493
00:36:24,191 --> 00:36:29,430
And clearly I have to choose
that power such that the t

494
00:36:29,430 --> 00:36:34,130
dependence, since t is going
to 0, I have to get rid of it.

495
00:36:34,130 --> 00:36:40,060
The only way that I can do
that is if p is 2 minus alpha

496
00:36:40,060 --> 00:36:45,720
minus delta divided by that.

497
00:36:52,710 --> 00:36:56,980
So having done that, the
whole thing will then

498
00:36:56,980 --> 00:36:59,866
be a function of h to the p.

499
00:37:02,610 --> 00:37:05,146
But the shape of
the magnetization

500
00:37:05,146 --> 00:37:08,520
along the critical
isotherm, which was also

501
00:37:08,520 --> 00:37:13,590
the shape of the isotherm
of the liquid gas system,

502
00:37:13,590 --> 00:37:16,200
we were characterizing
by an exponent

503
00:37:16,200 --> 00:37:18,242
that we were calling
1 over delta.

504
00:37:22,100 --> 00:37:25,930
So we have now a
formula that says

505
00:37:25,930 --> 00:37:31,540
my delta shouldn't in
fact be the inverse of p.

506
00:37:31,540 --> 00:37:39,170
It should be the delta 2
minus alpha minus delta.

507
00:37:39,170 --> 00:37:39,670
Yes?

508
00:37:39,670 --> 00:37:42,309
AUDIENCE: Why isn't
the exponent t minus 1

509
00:37:42,309 --> 00:37:43,975
after you've
differentiated [INAUDIBLE]?

510
00:37:47,090 --> 00:37:51,115
Because g originally was
defined as [INAUDIBLE].

511
00:37:51,115 --> 00:37:54,070
PROFESSOR: Let's call it pr.

512
00:37:58,010 --> 00:38:01,440
Because actually, you're right.

513
00:38:01,440 --> 00:38:05,300
If this is the same g and this
has particular singularity

514
00:38:05,300 --> 00:38:07,220
[INAUDIBLE].

515
00:38:07,220 --> 00:38:09,446
But at the end of the
day, it doesn't matter.

516
00:38:15,550 --> 00:38:18,770
So now I have gained something
that I didn't have before.

517
00:38:18,770 --> 00:38:21,060
That is, in
principle I hit alpha

518
00:38:21,060 --> 00:38:25,320
and beta, my two exponents,
I'm able to figure out

519
00:38:25,320 --> 00:38:27,190
what delta is.

520
00:38:27,190 --> 00:38:30,320
And actually I can also
figure out what gamma is ,

521
00:38:30,320 --> 00:38:34,714
because gamma describes
the divergence

522
00:38:34,714 --> 00:38:35,630
of the susceptibility.

523
00:38:43,560 --> 00:38:48,215
[INAUDIBLE] which is the
derivative of magnetization

524
00:38:48,215 --> 00:38:51,980
with respect to
field, I have to take

525
00:38:51,980 --> 00:38:55,680
another derivative
of this function.

526
00:38:55,680 --> 00:38:58,270
Taking another derivative
with respect to h

527
00:38:58,270 --> 00:39:00,774
will bring down another
factor of delta.

528
00:39:00,774 --> 00:39:04,390
So this becomes minus 2 delta.

529
00:39:04,390 --> 00:39:10,850
Some other double derivative
function h 2 to the delta.

530
00:39:10,850 --> 00:39:12,660
And susceptibilities,
we are typically

531
00:39:12,660 --> 00:39:18,110
interested in the limit
where the field goes to 0.

532
00:39:18,110 --> 00:39:23,420
And we define them to
diverge with exponent gamma.

533
00:39:23,420 --> 00:39:32,640
So we have identified gamma to
be 2 delta plus alpha minus 2.

534
00:39:40,990 --> 00:39:42,860
So we have learned something.

535
00:39:42,860 --> 00:39:45,126
Let's summarize it.

536
00:39:45,126 --> 00:39:55,400
So the consequences--
one is we established

537
00:39:55,400 --> 00:39:59,854
that same critical
exponents above and below.

538
00:40:12,400 --> 00:40:15,800
Now since various
quantities of interest

539
00:40:15,800 --> 00:40:18,730
are obtained by
taking derivatives

540
00:40:18,730 --> 00:40:21,216
of our homogeneous
function and they

541
00:40:21,216 --> 00:40:25,530
turn into homogeneous
functions, we

542
00:40:25,530 --> 00:40:40,500
conclude that all quantities
are homogeneous functions

543
00:40:40,500 --> 00:40:45,270
of the same combination
ht to the delta.

544
00:40:45,270 --> 00:40:46,687
Same delta governs it.

545
00:40:53,160 --> 00:40:58,020
And thirdly, once we make
this answer our assumption

546
00:40:58,020 --> 00:41:02,600
for the free energy, we can
calculate the other exponents

547
00:41:02,600 --> 00:41:04,170
on the table.

548
00:41:04,170 --> 00:41:22,920
So all, of almost all other
exponents related to 2,

549
00:41:22,920 --> 00:41:27,070
in this case alpha and delta.

550
00:41:27,070 --> 00:41:30,490
Which means that if you have a
number of different exponents

551
00:41:30,490 --> 00:41:34,440
that all depend
on 2, there should

552
00:41:34,440 --> 00:41:38,680
be some identities,
exponent identities.

553
00:41:47,670 --> 00:41:51,620
It's these numbers in the
table, we predict if all of this

554
00:41:51,620 --> 00:41:56,390
is varied have some
relationships with t.

555
00:41:56,390 --> 00:42:00,700
So let's show a couple
of these relationships.

556
00:42:00,700 --> 00:42:07,330
So let's look at the combination
alpha plus 2 beta plus gamma.

557
00:42:07,330 --> 00:42:10,930
Measurement of heat capacity,
magnetization, susceptibility.

558
00:42:10,930 --> 00:42:13,140
Three different things.

559
00:42:13,140 --> 00:42:16,550
So alpha is alpha 2.

560
00:42:16,550 --> 00:42:21,530
My beta up there is 2
minus alpha minus delta.

561
00:42:21,530 --> 00:42:28,140
My gamma is 2 delta
plus alpha minus 2.

562
00:42:28,140 --> 00:42:30,350
We got algebra.

563
00:42:30,350 --> 00:42:33,370
There's one alpha minus
2 alpha plus alpha.

564
00:42:33,370 --> 00:42:36,120
Alpha is cancelled.

565
00:42:36,120 --> 00:42:40,410
Minus 2 deltas plus 2
deltas then it does cancel.

566
00:42:40,410 --> 00:42:45,342
I have 2 times 2
minus 2, so that 2.

567
00:42:45,342 --> 00:42:51,250
So the prediction is that you
take some line on the table,

568
00:42:51,250 --> 00:42:55,000
add alpha, beta, 2
beta plus gamma, they

569
00:42:55,000 --> 00:42:56,390
should add up to one.

570
00:42:56,390 --> 00:42:59,820
So let's pick something.

571
00:42:59,820 --> 00:43:04,500
Let's pick a first--
actually, let's

572
00:43:04,500 --> 00:43:08,070
pick the last line that
has a negative alpha.

573
00:43:08,070 --> 00:43:11,270
So let's do n equals to 3.

574
00:43:11,270 --> 00:43:17,160
For n equals to 3 I have
alpha which is minus .12.

575
00:43:17,160 --> 00:43:25,800
I have twice beta, that is
.37, so that becomes 74.

576
00:43:25,800 --> 00:43:33,860
And then I have
gamma, which is 1.39.

577
00:43:33,860 --> 00:43:34,856
So this is 74.

578
00:43:37,880 --> 00:43:43,990
I have 9 plus 413
minus 2, which is 1.

579
00:43:43,990 --> 00:43:52,880
I have 3 plus 7, which is
10, minus 1, which is 9.

580
00:43:52,880 --> 00:43:56,625
But then I had a 1 that was
carried over, so I will have 0.

581
00:43:56,625 --> 00:44:00,160
So then I have 1, so 201.

582
00:44:00,160 --> 00:44:02,650
Not bad.

583
00:44:02,650 --> 00:44:07,705
Now this goes by the name
of the Rushbrooke identity.

584
00:44:16,010 --> 00:44:19,830
The Rushbrooke made
a simple manipulation

585
00:44:19,830 --> 00:44:24,960
based on thermodynamics and you
have a relationship with these.

586
00:44:24,960 --> 00:44:27,980
Let's do another one.

587
00:44:27,980 --> 00:44:33,450
Let's do delta and
subtract 1 from it.

588
00:44:33,450 --> 00:44:34,820
What is my delta?

589
00:44:34,820 --> 00:44:39,360
I have delta to the delta
2 plus alpha minus delta.

590
00:44:41,864 --> 00:44:45,550
This is small delta
versus big delta.

591
00:44:45,550 --> 00:44:47,358
And then I have minus 1.

592
00:44:50,350 --> 00:44:55,184
Taking that into the numerator
with the common denominator

593
00:44:55,184 --> 00:44:59,880
of 2 plus alpha minus
delta, this minus delta

594
00:44:59,880 --> 00:45:04,300
becomes plus delta, which
this becomes 2 delta minus

595
00:45:04,300 --> 00:45:07,730
alpha minus 2.

596
00:45:07,730 --> 00:45:15,362
2 delta

597
00:45:15,362 --> 00:45:17,860
AUDIENCE: Should that be a
minus alpha in the denominator?

598
00:45:17,860 --> 00:45:19,696
PROFESSOR: It better be.

599
00:45:19,696 --> 00:45:20,195
Yes.

600
00:45:25,430 --> 00:45:28,060
2 delta plus alpha minus 2.

601
00:45:28,060 --> 00:45:30,952
Then we can read off the gamma.

602
00:45:30,952 --> 00:45:33,097
So this is gamma over beta.

603
00:45:35,720 --> 00:45:39,740
And let's check this, let's
say for m equals to 2.

604
00:45:42,770 --> 00:45:46,140
No, let's check it for m
plus 21, for the following

605
00:45:46,140 --> 00:45:53,870
reason, that for n equals to
1, what we have for delta is

606
00:45:53,870 --> 00:46:01,500
4.8 minus 1, which would be 3.8.

607
00:46:01,500 --> 00:46:06,170
And on the other side,
we have gamma over beta.

608
00:46:06,170 --> 00:46:12,150
Gamma is 1.24, roughly,
divided by beta .33,

609
00:46:12,150 --> 00:46:14,870
which is roughly one third.

610
00:46:14,870 --> 00:46:17,520
So I multiply this by 3.

611
00:46:17,520 --> 00:46:27,420
And that becomes 3.72.

612
00:46:27,420 --> 00:46:34,438
This one is known after another
famous physicist, Ben Widom,

613
00:46:34,438 --> 00:46:35,935
as the Widom identity.

614
00:46:39,930 --> 00:46:41,690
So that's nice.

615
00:46:41,690 --> 00:46:48,620
We can start learning that
although we don't know anything

616
00:46:48,620 --> 00:46:52,910
about this table, these are
not independent numbers.

617
00:46:52,910 --> 00:46:55,060
There's relationship
between them.

618
00:46:55,060 --> 00:46:58,975
And they're named after
famous physicists.

619
00:46:58,975 --> 00:46:59,475
Yes?

620
00:46:59,475 --> 00:47:03,075
AUDIENCE: Can we briefly go
over again what extra assumption

621
00:47:03,075 --> 00:47:06,024
we had put in to get
these in and these out?

622
00:47:08,690 --> 00:47:11,642
Is it just that we have
this homogeneous function

623
00:47:11,642 --> 00:47:12,415
[INAUDIBLE]?

624
00:47:12,415 --> 00:47:13,980
PROFESSOR: That's right.

625
00:47:13,980 --> 00:47:18,230
So you assume that
the singularity

626
00:47:18,230 --> 00:47:22,600
in the vicinity of the
critical point as a function

627
00:47:22,600 --> 00:47:26,300
of deviations from
that critical point

628
00:47:26,300 --> 00:47:29,302
can be expressed as a
homogeneous function.

629
00:47:29,302 --> 00:47:33,900
The homogeneous function, you
can rearrange any way you like.

630
00:47:33,900 --> 00:47:37,610
One nice way to rearrange
it is in this fashion.

631
00:47:37,610 --> 00:47:41,940
It will depend, the homogeneous
function on two exponents.

632
00:47:41,940 --> 00:47:44,540
I chose to write
it as 2 minus alpha

633
00:47:44,540 --> 00:47:48,290
so that one of the exponents
would immediately be alpha.

634
00:47:48,290 --> 00:47:50,530
The other one I
couldn't immediately

635
00:47:50,530 --> 00:47:52,770
write in terms of beta or gamma.

636
00:47:52,770 --> 00:47:55,740
I had to do these
manipulations to find out

637
00:47:55,740 --> 00:47:59,290
what the relationship
[INAUDIBLE].

638
00:47:59,290 --> 00:48:03,700
But the physics of it is simple.

639
00:48:03,700 --> 00:48:09,000
That is, once you know the
singularity of a free energy,

640
00:48:09,000 --> 00:48:11,110
various other
quantities you obtain

641
00:48:11,110 --> 00:48:13,090
by taking derivatives
of the free energy.

642
00:48:13,090 --> 00:48:17,460
That's [INAUDIBLE]
And so then you

643
00:48:17,460 --> 00:48:19,862
would have the singular
behavior of [INAUDIBLE].

644
00:48:27,110 --> 00:48:34,530
So I started by saying
that all other exponents,

645
00:48:34,530 --> 00:48:39,640
but then I realized we have
nothing so far that tells us

646
00:48:39,640 --> 00:48:43,840
anything about mu and eta.

647
00:48:43,840 --> 00:48:48,490
Because mu and eta
relate to correlations.

648
00:48:48,490 --> 00:48:51,190
They are in
microscopic quantities.

649
00:48:51,190 --> 00:48:55,740
Alpha, beta, gamma depend
on macroscopic thermodynamic

650
00:48:55,740 --> 00:48:59,140
quantities, magnetization
susceptibility.

651
00:48:59,140 --> 00:49:03,440
So there's no way that I will
be able to get information,

652
00:49:03,440 --> 00:49:04,400
almost.

653
00:49:04,400 --> 00:49:07,290
No easy way or no direct
way to get information

654
00:49:07,290 --> 00:49:09,080
about mu and eta.

655
00:49:11,680 --> 00:49:19,630
So I will go to assumption 2.0.

656
00:49:19,630 --> 00:49:25,720
Go to the next version of the
homogeneity assumption, which

657
00:49:25,720 --> 00:49:29,580
is to emphasize
that we certainly

658
00:49:29,580 --> 00:49:32,930
know, again from physics
and the relationship

659
00:49:32,930 --> 00:49:35,330
between susceptibility
and correlations,

660
00:49:35,330 --> 00:49:37,810
that the reason
for the divergence

661
00:49:37,810 --> 00:49:42,200
of the susceptibility is that
the correlations become large.

662
00:49:42,200 --> 00:49:45,740
So we'll emphasize that.

663
00:49:45,740 --> 00:49:51,170
So let's write our ansatz
not about the free energy,

664
00:49:51,170 --> 00:49:54,730
but about the
correlation length.

665
00:49:54,730 --> 00:50:02,044
So let's replace that
ansatz with homogeneity

666
00:50:02,044 --> 00:50:04,474
of correlation length.

667
00:50:13,240 --> 00:50:17,680
So once more, we
have a structure

668
00:50:17,680 --> 00:50:21,784
where is a line
that is terminate

669
00:50:21,784 --> 00:50:25,780
when two parameters,
t and h go to 0.

670
00:50:25,780 --> 00:50:30,650
And we know that on
approaching this point,

671
00:50:30,650 --> 00:50:33,260
the system will become cloudy.

672
00:50:33,260 --> 00:50:36,880
There's a correlation
length that

673
00:50:36,880 --> 00:50:41,430
diverges on approaching that
point a function of these two

674
00:50:41,430 --> 00:50:42,640
arguments.

675
00:50:42,640 --> 00:50:45,280
I'm going to make the same
homogeneity assumption

676
00:50:45,280 --> 00:50:46,520
for the correlation length.

677
00:50:46,520 --> 00:50:48,460
And again, this
is an assumption.

678
00:50:48,460 --> 00:50:52,805
I say that this is
a to the minus mu.

679
00:50:52,805 --> 00:50:56,770
The exponent mu was a divergence
of the correlation length.

680
00:50:56,770 --> 00:51:00,830
Some other function, it's not
that first g that we wrote.

681
00:51:00,830 --> 00:51:06,610
Let's call it g psi
of ht to the delta.

682
00:51:12,380 --> 00:51:17,650
So we never discussed it,
but this function immediately

683
00:51:17,650 --> 00:51:22,170
also tells me if you
approach the critical point

684
00:51:22,170 --> 00:51:25,820
along the criticalizer term,
how does the correlation length

685
00:51:25,820 --> 00:51:31,540
diverge through the various
tricks that we have discussed?

686
00:51:31,540 --> 00:51:35,976
But this is going to be
telling me something more

687
00:51:35,976 --> 00:51:44,670
if from here, I can reproduce
my scaling assumption 1.0.

688
00:51:44,670 --> 00:51:49,860
So there is one other
step that I can make.

689
00:51:49,860 --> 00:52:02,670
Assume divergence of
c is responsible--

690
00:52:02,670 --> 00:52:11,275
let's call it even solely
responsible-- for singular

691
00:52:11,275 --> 00:52:11,775
behavior.

692
00:52:17,475 --> 00:52:21,670
And you say, what
does all of this mean?

693
00:52:21,670 --> 00:52:27,150
So let's say that I have a
system could be my magnet,

694
00:52:27,150 --> 00:52:33,014
could be my liquid gas that
has size l on each search.

695
00:52:36,170 --> 00:52:44,830
And I calculate the
partition function log z.

696
00:52:44,830 --> 00:52:49,935
Log z will certainly have
the part that is regular.

697
00:52:49,935 --> 00:52:55,280
Well-- log z will have a part
that is certainly-- let's

698
00:52:55,280 --> 00:52:57,620
say the contribution
phonons, all kinds

699
00:52:57,620 --> 00:53:00,880
of other regular things
that don't have anything

700
00:53:00,880 --> 00:53:03,570
to do with singularity
of the system.

701
00:53:03,570 --> 00:53:09,550
Those things will give
you some regular function.

702
00:53:09,550 --> 00:53:11,580
But one thing that
I know for sure

703
00:53:11,580 --> 00:53:14,120
is that the answer is
going to be extensive.

704
00:53:14,120 --> 00:53:19,050
If I have any nice
thermodynamic system

705
00:53:19,050 --> 00:53:24,810
and I am in v
dimensions, then it

706
00:53:24,810 --> 00:53:28,686
will be proportional to
the volume of that system

707
00:53:28,686 --> 00:53:29,618
that I have.

708
00:53:32,420 --> 00:53:38,120
Now the way that I have written
it is not entirely nice,

709
00:53:38,120 --> 00:53:44,630
because log z is-- a log is
a dimensionless quantity.

710
00:53:44,630 --> 00:53:47,940
Maybe I measured my length
in meters or centimeters

711
00:53:47,940 --> 00:53:51,262
or whatever, so I
have dimensions here.

712
00:53:51,262 --> 00:53:56,900
So it makes sense to pick some
landscape to dimensionalize it

713
00:53:56,900 --> 00:54:00,470
before multiplying it by some
kind of irregular function

714
00:54:00,470 --> 00:54:04,525
of whatever I have,
t and h, for example.

715
00:54:10,370 --> 00:54:15,580
But what about
the singular part?

716
00:54:15,580 --> 00:54:18,770
For the singular
part, the statement

717
00:54:18,770 --> 00:54:21,270
was that somehow it was
a connective behavior.

718
00:54:21,270 --> 00:54:23,350
It involved many, many
degrees of freedom.

719
00:54:23,350 --> 00:54:27,665
We saw for the heat capacity of
the solid at low temperatures,

720
00:54:27,665 --> 00:54:32,040
it came from long wavelength
degrees of freedom.

721
00:54:32,040 --> 00:54:36,120
So no lattice parameter
is going to be important.

722
00:54:36,120 --> 00:54:41,272
So one thing that I could
do, maintaining extensivity,

723
00:54:41,272 --> 00:54:48,605
is to divide by l over
c times something.

724
00:54:52,750 --> 00:54:55,030
So that's the only
thing that I did

725
00:54:55,030 --> 00:55:00,100
to ensure that extensivity
is maintained when

726
00:55:00,100 --> 00:55:05,025
I have kind of benign landscape,
but in addition a landscape

727
00:55:05,025 --> 00:55:08,940
that is divergent.

728
00:55:08,940 --> 00:55:12,095
Now you can see that
immediately that says that log

729
00:55:12,095 --> 00:55:18,330
z singular as a
function of t and h,

730
00:55:18,330 --> 00:55:23,726
will be proportional
to c to the minus t.

731
00:55:23,726 --> 00:55:26,000
And using that
formula, it will be

732
00:55:26,000 --> 00:55:31,150
proportional to t to the du,
some other scaling function.

733
00:55:31,150 --> 00:55:34,718
And it's go back to
gf ht to the delta.

734
00:55:42,040 --> 00:55:46,650
Physically, what it's
saying is that when

735
00:55:46,650 --> 00:55:52,550
I am very close, but not
quite at the critical point,

736
00:55:52,550 --> 00:55:56,580
I have a long correlation
length, much larger

737
00:55:56,580 --> 00:56:00,280
than microscopic length
scale of my system.

738
00:56:00,280 --> 00:56:06,010
So what I can say is that
within a correlation length,

739
00:56:06,010 --> 00:56:11,375
my degrees of freedom for
magentization or whatever it is

740
00:56:11,375 --> 00:56:15,120
are very much coupled
to each other.

741
00:56:15,120 --> 00:56:17,940
So maybe what I can
do is I can regard

742
00:56:17,940 --> 00:56:21,480
this as an independent lock.

743
00:56:21,480 --> 00:56:25,130
And how many independent
locks do I have?

744
00:56:25,130 --> 00:56:28,060
It is l over c to the d.

745
00:56:28,060 --> 00:56:30,710
So the statement
roughly is a part

746
00:56:30,710 --> 00:56:34,390
of the assumption is that this
correlation length that is

747
00:56:34,390 --> 00:56:36,340
getting bigger and bigger.

748
00:56:36,340 --> 00:56:38,700
Because things are
correlated, the number

749
00:56:38,700 --> 00:56:40,670
of independent degrees
of freedom that you

750
00:56:40,670 --> 00:56:43,980
are having gets
smaller and smaller.

751
00:56:43,980 --> 00:56:47,550
And that's changing the
number of degrees of freedom

752
00:56:47,550 --> 00:56:51,400
is responsible for the singular
behavior of the free energy.

753
00:56:51,400 --> 00:56:56,520
If I make this assumption about
this correlation then diverges,

754
00:56:56,520 --> 00:56:57,784
then I will get this form.

755
00:57:01,090 --> 00:57:05,920
So now my ansatz 2.0
matches my ansatz 1.0

756
00:57:05,920 --> 00:57:09,940
provided du is 2 minus alpha.

757
00:57:09,940 --> 00:57:16,950
So I have du2 plus
2 minus alpha which

758
00:57:16,950 --> 00:57:20,734
is known after Brian
Josephson, so this

759
00:57:20,734 --> 00:57:26,662
is the Josephson relation.

760
00:57:26,662 --> 00:57:33,600
And it is different from the
other exponent identities

761
00:57:33,600 --> 00:57:37,100
that we have because
it explicitly

762
00:57:37,100 --> 00:57:39,790
depends on the
dimensionality of space.

763
00:57:39,790 --> 00:57:42,060
d appears in the problem.

764
00:57:42,060 --> 00:57:45,947
It's called hyperscale
for that reason.

765
00:57:50,420 --> 00:57:51,200
Yes?

766
00:57:51,200 --> 00:57:54,230
AUDIENCE: So does the assumption
that the divergence in c

767
00:57:54,230 --> 00:57:56,479
is solely responsible for
the singular behavior, what

768
00:57:56,479 --> 00:57:58,020
are we excluding
when we assume that?

769
00:57:58,020 --> 00:58:02,066
What else could happen that
would make that not true?

770
00:58:02,066 --> 00:58:05,440
PROFESSOR: Well, what
is appearing here maybe

771
00:58:05,440 --> 00:58:10,401
will have some singular
function of t and h.

772
00:58:10,401 --> 00:58:12,275
AUDIENCE: So this
similar to what

773
00:58:12,275 --> 00:58:15,581
we were assuming before when we
said that our free energy could

774
00:58:15,581 --> 00:58:17,890
have some regular
part that depends

775
00:58:17,890 --> 00:58:21,874
on [INAUDIBLE] the
part that [INAUDIBLE].

776
00:58:21,874 --> 00:58:23,297
PROFESSOR: Yes, exactly.

777
00:58:26,430 --> 00:58:30,620
But once again, the truth
is really whether or not

778
00:58:30,620 --> 00:58:33,120
this matches up
with experiments.

779
00:58:33,120 --> 00:58:39,300
So let's, for example,
pick anything in that

780
00:58:39,300 --> 00:58:41,040
table, v equals to t.

781
00:58:41,040 --> 00:58:45,660
Let's pick n goes to 2,
which we haven't done so far.

782
00:58:45,660 --> 00:58:54,610
And so what the formula
would say is 3 times mu.

783
00:58:54,610 --> 00:59:03,670
Mu for the superfluid
is 67 is 2 minus-- well,

784
00:59:03,670 --> 00:59:07,990
alpha is almost 0 but
slightly negative.

785
00:59:07,990 --> 00:59:16,240
So it is 0.01.

786
00:59:16,240 --> 00:59:17,270
And what do we have?

787
00:59:17,270 --> 00:59:26,440
3 times 67 is 2.01.

788
00:59:26,440 --> 00:59:27,750
So it matches.

789
00:59:27,750 --> 00:59:31,830
Actually, we say, well,
why do you emphasize

790
00:59:31,830 --> 00:59:34,300
that it's the
function of dimension?

791
00:59:34,300 --> 00:59:39,040
Well, a little bit
later on in the course,

792
00:59:39,040 --> 00:59:44,990
we will do an exact solution of
the so-called 2D Ising model.

793
00:59:47,670 --> 00:59:51,820
So this is a system that
first wants to be close to 2,

794
00:59:51,820 --> 00:59:53,120
n equals to 1.

795
00:59:53,120 --> 00:59:56,780
And it was an important thing
that people could actually

796
00:59:56,780 --> 01:00:00,860
solve an interacting problem,
not in three dimensions

797
01:00:00,860 --> 01:00:01,760
but in two.

798
01:00:01,760 --> 01:00:05,780
And the exponents
for that, alpha is 0,

799
01:00:05,780 --> 01:00:08,860
but it really is a
logarithmic divergence.

800
01:00:08,860 --> 01:00:11,700
Beta is 1/8.

801
01:00:11,700 --> 01:00:20,060
Gamma is 7/4, delta is 15,
mu is 1, and eta is 1/4.

802
01:00:20,060 --> 01:00:28,190
And we can check now
for this v equals to 2 n

803
01:00:28,190 --> 01:00:32,980
equals to 1 that
we have two times

804
01:00:32,980 --> 01:00:35,970
our mu, which exactly
is known to be

805
01:00:35,970 --> 01:00:43,410
1 is 2 minus logarithmic
divergence corresponding to 0.

806
01:00:43,410 --> 01:00:46,036
So again, there's
something that works.

807
01:00:50,600 --> 01:00:56,150
One thing that you may
want to see and look at

808
01:00:56,150 --> 01:01:01,500
is that the ansatz
that we made first also

809
01:01:01,500 --> 01:01:05,820
works for the result
of saddlepoint,

810
01:01:05,820 --> 01:01:09,930
not surprisingly because
again in the saddlepoint

811
01:01:09,930 --> 01:01:14,020
we start with a singular free
energy and go through all this.

812
01:01:14,020 --> 01:01:18,000
But it does not work for
this type of scaling,

813
01:01:18,000 --> 01:01:23,690
because 2 minus alpha
would be 0 is not

814
01:01:23,690 --> 01:01:29,925
equal to d times one half,
except in the case of four

815
01:01:29,925 --> 01:01:30,425
dimensions.

816
01:01:33,060 --> 01:01:37,390
So somehow, this
ansatz and this picture

817
01:01:37,390 --> 01:01:42,680
breaks down within the
saddlepoint approximation.

818
01:01:42,680 --> 01:01:47,180
If you remember what we did
when we calculated fluctuation

819
01:01:47,180 --> 01:01:49,815
corrections for the
saddlepoint, you

820
01:01:49,815 --> 01:01:54,800
got actually an exponent alpha
that was 2 minus mu over 2.

821
01:01:54,800 --> 01:02:00,400
So the fluctuating part that
we get around the saddlepoint

822
01:02:00,400 --> 01:02:02,420
does satisfy this.

823
01:02:02,420 --> 01:02:04,995
But on top of that
there's another part that

824
01:02:04,995 --> 01:02:07,730
is doe to the
saddlepoint value itself

825
01:02:07,730 --> 01:02:10,720
that violates this
hyperscaling solution.

826
01:02:10,720 --> 01:02:11,220
Yes?

827
01:02:11,220 --> 01:02:15,280
AUDIENCE: Empirically, how well
can we probe the dependence

828
01:02:15,280 --> 01:02:19,192
on dimensionality that we're
finding in these expressions?

829
01:02:19,192 --> 01:02:21,050
PROFESSOR:
Experimentally, we can

830
01:02:21,050 --> 01:02:23,730
do d equals to 2 d equals to 3.

831
01:02:23,730 --> 01:02:27,948
And computer simulations we
can also do d equals to 2 d

832
01:02:27,948 --> 01:02:28,840
equals to 3.

833
01:02:28,840 --> 01:02:31,695
Very soon, we will do
analytical expressions

834
01:02:31,695 --> 01:02:35,310
where we will be
in 3.99 dimensions.

835
01:02:35,310 --> 01:02:38,960
So we will be coming down
conservatively around 4.

836
01:02:38,960 --> 01:02:42,260
So mathematically, we can
play tricks such as that.

837
01:02:42,260 --> 01:02:45,983
But certainly empirically, in
the sense of experimentally

838
01:02:45,983 --> 01:02:48,456
we are at a disadvantage
in those languages.

839
01:02:51,490 --> 01:02:51,990
OK?

840
01:03:00,220 --> 01:03:03,190
So we are making progress.

841
01:03:03,190 --> 01:03:05,600
We have made our way
across this table.

842
01:03:05,600 --> 01:03:09,110
We have also an identity
that involves mu.

843
01:03:09,110 --> 01:03:11,320
But so far I haven't
said anything about eta.

844
01:03:15,420 --> 01:03:21,950
I can say something about
the eta reasonably simply,

845
01:03:21,950 --> 01:03:24,255
but then you try to
build something profound

846
01:03:24,255 --> 01:03:27,290
based on that.

847
01:03:27,290 --> 01:03:35,046
So let's look at exactly at
tc, at the critical point.

848
01:03:37,860 --> 01:03:43,262
So let's say you are sitting
at t and h equals to 0.

849
01:03:43,262 --> 01:03:45,980
You have to prepare your
system at that point.

850
01:03:45,980 --> 01:03:49,450
There's nothing physically
that says you can't.

851
01:03:49,450 --> 01:03:53,640
At that point, you can
look at correlations.

852
01:03:53,640 --> 01:03:57,420
And the exponent eta for
example is a characteristic

853
01:03:57,420 --> 01:03:59,150
of those correlations.

854
01:03:59,150 --> 01:04:05,490
And one of the things that we
have is that m of x m of 0,

855
01:04:05,490 --> 01:04:10,966
the connected parts-- well,
actually at the critical point

856
01:04:10,966 --> 01:04:13,340
we don't even have to
put the connected part

857
01:04:13,340 --> 01:04:16,840
because the average
of n is going to be 0.

858
01:04:16,840 --> 01:04:19,725
But this is a
quantity that behaves

859
01:04:19,725 --> 01:04:24,570
as 1 over the separation
that's actually

860
01:04:24,570 --> 01:04:28,560
include two possible
points, x minus y.

861
01:04:28,560 --> 01:04:34,760
When we did the case
of the fluctuations

862
01:04:34,760 --> 01:04:39,280
at the critical point within
the saddlepoint method,

863
01:04:39,280 --> 01:04:42,445
we found that the behavior
was like the Coulomb law.

864
01:04:42,445 --> 01:04:45,830
It was falling off as
1x to the d minus 2.

865
01:04:45,830 --> 01:04:48,320
But we said that
experiment indicated

866
01:04:48,320 --> 01:04:52,940
that there is a small
correction for this

867
01:04:52,940 --> 01:04:54,580
that we indicate
with exponent eta.

868
01:04:54,580 --> 01:04:59,850
So that was how the
exponent eta was defined.

869
01:04:59,850 --> 01:05:04,560
So can we have an identity
that involves the exponent eta?

870
01:05:04,560 --> 01:05:08,330
We actually have seen
how to do this already.

871
01:05:08,330 --> 01:05:12,050
Because we know that in
general, the susceptibilities

872
01:05:12,050 --> 01:05:16,400
are related to integrals of
the correlation functions.

873
01:05:20,860 --> 01:05:25,850
Now if I put this
power law over here,

874
01:05:25,850 --> 01:05:28,110
you can see that the
answer is like trying

875
01:05:28,110 --> 01:05:32,100
to be integrate x squared
all the way to infinity down.

876
01:05:32,100 --> 01:05:35,512
So it will be divergent
and that's no problem.

877
01:05:35,512 --> 01:05:37,720
At the critical point we
know that the susceptibility

878
01:05:37,720 --> 01:05:40,010
is divergent.

879
01:05:40,010 --> 01:05:48,500
But you say, OK, if I'm away
from the critical point,

880
01:05:48,500 --> 01:05:53,940
then I will use this
formula, but only

881
01:05:53,940 --> 01:05:57,060
up to the correlation length.

882
01:05:57,060 --> 01:06:00,250
And I say that beyond
the correlation length,

883
01:06:00,250 --> 01:06:03,010
then the correlations
will decay exponentially.

884
01:06:03,010 --> 01:06:07,810
That's too rapid a falloff,
and essentially the only part

885
01:06:07,810 --> 01:06:10,155
that's contributing
is because what

886
01:06:10,155 --> 01:06:13,110
was happening at
the critical point.

887
01:06:13,110 --> 01:06:20,345
Once I do that, I have to
integrate ddx over x to the d

888
01:06:20,345 --> 01:06:24,290
minus 2 plus eta up to
the correlation length.

889
01:06:24,290 --> 01:06:27,040
The answer will be proportional
to the correlation length

890
01:06:27,040 --> 01:06:30,400
to the power of 2 minus eta.

891
01:06:30,400 --> 01:06:36,230
And this will be proportional
to p to the power of c goes

892
01:06:36,230 --> 01:06:38,180
[INAUDIBLE] to the minus mu.

893
01:06:38,180 --> 01:06:39,698
2 minus eta times mu.

894
01:06:44,920 --> 01:06:46,696
But we know that
the susceptibilities

895
01:06:46,696 --> 01:06:51,210
diverge as t to the minus gamma.

896
01:06:51,210 --> 01:06:55,320
So we have established
an exponent identity

897
01:06:55,320 --> 01:07:04,210
that tells us that gamma
is 2 minus eta times mu.

898
01:07:04,210 --> 01:07:09,170
And this is known as the Fisher
identity, after Michael Fisher.

899
01:07:13,341 --> 01:07:16,500
Again, you can see that in
all of the cases in three

900
01:07:16,500 --> 01:07:18,730
dimensions that we
are dealing with,

901
01:07:18,730 --> 01:07:20,668
exponent eta is roughly 0.

902
01:07:20,668 --> 01:07:24,350
It's 0.04 And all
of our gammas are

903
01:07:24,350 --> 01:07:27,650
roughly twice what our
mus are in that table.

904
01:07:27,650 --> 01:07:30,630
It's time we get
that table checked.

905
01:07:30,630 --> 01:07:34,790
The one case that I have on
that table where eta is not 0

906
01:07:34,790 --> 01:07:41,162
is when I'm looking at v
positive 2 where eta is 1/4.

907
01:07:41,162 --> 01:07:44,870
So I take 2 minus
1/4, multiply it

908
01:07:44,870 --> 01:07:47,640
by the mu that is
one in two dimension,

909
01:07:47,640 --> 01:07:51,162
and the answer is
the 7/4, which we

910
01:07:51,162 --> 01:07:53,830
have for the exponent
gamma over there.

911
01:08:02,075 --> 01:08:05,459
So we have now the
identity that is

912
01:08:05,459 --> 01:08:09,810
applicable to the
last exponents.

913
01:08:09,810 --> 01:08:13,160
So all of this works.

914
01:08:13,160 --> 01:08:16,359
Let's now take the
conceptual leap

915
01:08:16,359 --> 01:08:19,185
that then allows
us to do what we

916
01:08:19,185 --> 01:08:22,890
will do later on to
get the exponents.

917
01:08:22,890 --> 01:08:25,890
Basically, you can see
that what we have imposed

918
01:08:25,890 --> 01:08:30,569
here conceptually
is the following.

919
01:08:30,569 --> 01:08:35,109
That when I'm away from
the critical point,

920
01:08:35,109 --> 01:08:39,000
I look at the correlations
of this important statistical

921
01:08:39,000 --> 01:08:40,562
field.

922
01:08:40,562 --> 01:08:43,640
And I find that they
fall off with separation,

923
01:08:43,640 --> 01:08:46,810
according to some power.

924
01:08:46,810 --> 01:08:52,580
And the reason is that
at the critical point,

925
01:08:52,580 --> 01:08:54,970
the correlation length
has gone to infinity.

926
01:08:54,970 --> 01:08:57,279
That's not the length scale
that you have to play with.

927
01:08:57,279 --> 01:09:02,890
You can divide x minus y divided
by c, which is what we do away

928
01:09:02,890 --> 01:09:06,765
from the critical point.
c has gone to infinity.

929
01:09:06,765 --> 01:09:09,410
The other length scale
that we are worried about

930
01:09:09,410 --> 01:09:12,689
are things that go
into the microscopics.

931
01:09:12,689 --> 01:09:18,260
but we are assuming that
microscopics is irrelevant.

932
01:09:18,260 --> 01:09:21,939
It has been washed out.

933
01:09:21,939 --> 01:09:24,810
So if we don't have
a large length scale,

934
01:09:24,810 --> 01:09:27,670
if we don't have a short
length scale, some function

935
01:09:27,670 --> 01:09:30,960
of distance, how can it decay?

936
01:09:30,960 --> 01:09:33,510
The only way it can
decay is [INAUDIBLE].

937
01:09:36,510 --> 01:09:45,010
So this statement is that when
we are at a critical point,

938
01:09:45,010 --> 01:09:47,479
I look at some correlation.

939
01:09:47,479 --> 01:09:50,170
And this was the
magnetization correlation.

940
01:09:50,170 --> 01:09:54,020
But I can look at
correlation of anything else

941
01:09:54,020 --> 01:09:56,069
as a function of separation.

942
01:10:01,820 --> 01:10:06,990
And this will only fall off
as some power of separation.

943
01:10:06,990 --> 01:10:09,400
Another way of writing
it is that if I

944
01:10:09,400 --> 01:10:13,500
were to multiply this
by some length scale,

945
01:10:13,500 --> 01:10:17,570
so rather than looking at things
that are some distance apart

946
01:10:17,570 --> 01:10:20,030
at twice that distance
apart or hundred times

947
01:10:20,030 --> 01:10:24,006
that distance apart, I will
reproduce the correlation

948
01:10:24,006 --> 01:10:32,970
that I have up to some
other of the scale factor.

949
01:10:32,970 --> 01:10:35,520
So the scale factor
here we can read off

950
01:10:35,520 --> 01:10:38,730
has to be related to
t minus 2 plus eta.

951
01:10:38,730 --> 01:10:41,450
But essentially, this
is a statement again

952
01:10:41,450 --> 01:10:45,680
about homogeneity of
correlation functions

953
01:10:45,680 --> 01:10:47,968
when you are at
a critical point.

954
01:10:50,510 --> 01:10:54,496
So this is a symmetry here.

955
01:10:54,496 --> 01:11:00,260
It says you take your
statistical correlations

956
01:11:00,260 --> 01:11:02,670
and you look at them
at the larger scale

957
01:11:02,670 --> 01:11:04,860
or at the shorter scale.

958
01:11:04,860 --> 01:11:08,400
And up to some
overall scale factor,

959
01:11:08,400 --> 01:11:10,280
you reproduce what
you had before.

960
01:11:14,020 --> 01:11:20,190
So this is something to do
with invariance on the scale.

961
01:11:28,290 --> 01:11:32,930
This scaling variance
is some property

962
01:11:32,930 --> 01:11:37,190
that was popular a while
ago as being associated

963
01:11:37,190 --> 01:11:38,772
with the kind of
geometrical objects

964
01:11:38,772 --> 01:11:39,980
that you would call fractals.

965
01:11:46,100 --> 01:11:54,000
So the statement is that
if I go across my system

966
01:11:54,000 --> 01:11:58,590
and there is some pattern of
magnetization fluctuations,

967
01:11:58,590 --> 01:12:00,416
let's say I look at it.

968
01:12:00,416 --> 01:12:02,465
I'm going along
this direction x.

969
01:12:04,980 --> 01:12:11,240
And I plot at some
particular configuration

970
01:12:11,240 --> 01:12:13,580
that is dominant
and is contributing

971
01:12:13,580 --> 01:12:17,092
to my free energy,
the magnetization,

972
01:12:17,092 --> 01:12:21,490
that it has a shape that
has this characteristic self

973
01:12:21,490 --> 01:12:25,620
similarity kind of maybe looking
like a mountain landscape.

974
01:12:28,420 --> 01:12:31,510
And the statement
is that if I were

975
01:12:31,510 --> 01:12:40,650
to take a part of that
landscape and then blow it up,

976
01:12:40,650 --> 01:12:45,290
I will generate a pattern
that is of course not the same

977
01:12:45,290 --> 01:12:46,210
as the first one.

978
01:12:46,210 --> 01:12:48,970
It is not exactly
scale invariant.

979
01:12:48,970 --> 01:12:53,150
But it has the same kind
of statistics as the one

980
01:12:53,150 --> 01:12:57,220
that I had originally after
I multiplied this axis

981
01:12:57,220 --> 01:12:59,088
by some factor lambda.

982
01:13:03,770 --> 01:13:05,345
Yes?

983
01:13:05,345 --> 01:13:10,960
AUDIENCE: Under what length
scales are those subsimilarity

984
01:13:10,960 --> 01:13:15,032
properties evident and how
do they compare to the length

985
01:13:15,032 --> 01:13:16,740
scale over which you're
doing your course

986
01:13:16,740 --> 01:13:18,356
grading for this field?

987
01:13:18,356 --> 01:13:25,476
PROFESSOR: OK, so basically we
expect this to be applicable

988
01:13:25,476 --> 01:13:27,230
presumably at length
scales that are

989
01:13:27,230 --> 01:13:28,780
less than the size
of your system

990
01:13:28,780 --> 01:13:31,330
because once I get to
the size of the system

991
01:13:31,330 --> 01:13:33,950
I can't blow it up
further or whatever.

992
01:13:33,950 --> 01:13:37,560
It has to certainly be
larger than whatever

993
01:13:37,560 --> 01:13:41,870
the coarse-graining length is,
or the length scale at which

994
01:13:41,870 --> 01:13:45,825
I have confidence that I have
washed out the microscoping

995
01:13:45,825 --> 01:13:48,340
details.

996
01:13:48,340 --> 01:13:50,400
Now that depends on
the system in question,

997
01:13:50,400 --> 01:13:53,630
so I can't really give
you an answer for that.

998
01:13:53,630 --> 01:13:56,140
The answer will
depend on the system.

999
01:13:56,140 --> 01:13:58,040
But the point is
that I'm looking

1000
01:13:58,040 --> 01:14:01,720
in the vicinity of a point
where mathematically I'm

1001
01:14:01,720 --> 01:14:04,250
assured that there's a
correlation length that

1002
01:14:04,250 --> 01:14:05,580
goes to infinity.

1003
01:14:05,580 --> 01:14:09,580
So maybe there is some system
number 1 that average out

1004
01:14:09,580 --> 01:14:12,390
very easily, and
after a distance of 10

1005
01:14:12,390 --> 01:14:14,890
I can start applying this.

1006
01:14:14,890 --> 01:14:16,730
But maybe there's
some other system

1007
01:14:16,730 --> 01:14:19,520
where the microscopic degrees
of freedom are very problematic

1008
01:14:19,520 --> 01:14:22,950
and I have to go further and
further out before they average

1009
01:14:22,950 --> 01:14:23,730
out.

1010
01:14:23,730 --> 01:14:27,200
But in principle, since
my c has gone to infinity,

1011
01:14:27,200 --> 01:14:31,240
I can just pick a bigger and
bigger piece of my system

1012
01:14:31,240 --> 01:14:33,170
until that has happened.

1013
01:14:33,170 --> 01:14:36,280
So I can't tell you what
the short distance length

1014
01:14:36,280 --> 01:14:39,880
scale is in the same sense
that when [INAUDIBLE] says

1015
01:14:39,880 --> 01:14:44,230
that coast of Britain
is fractal, well,

1016
01:14:44,230 --> 01:14:47,040
I can't tell you whether
the short distance is

1017
01:14:47,040 --> 01:14:50,840
the size of a sand
particle, or is it

1018
01:14:50,840 --> 01:14:54,648
the size of, I don't know, a
tree or something like that.

1019
01:14:54,648 --> 01:14:55,624
I don't know.

1020
01:15:03,920 --> 01:15:13,740
So we started thinking
about our original problem.

1021
01:15:13,740 --> 01:15:19,360
And constructing
this Landau-Ginzburg,

1022
01:15:19,360 --> 01:15:25,270
[INAUDIBLE] that we worked
with on the basis of symmetries

1023
01:15:25,270 --> 01:15:30,010
such as invariance on
the rotation, et cetera.

1024
01:15:30,010 --> 01:15:32,895
Somehow we've discovered
that the point that we

1025
01:15:32,895 --> 01:15:37,700
are interested has an additional
symmetry that maybe we

1026
01:15:37,700 --> 01:15:41,706
didn't anticipate, which is
this self-similarity and scale

1027
01:15:41,706 --> 01:15:42,205
invariance.

1028
01:15:45,290 --> 01:15:49,240
So you say, OK, that's the
solution to the problem.

1029
01:15:49,240 --> 01:15:52,190
Let's go back to
our construction

1030
01:15:52,190 --> 01:15:55,930
of the Landau-Ginsburg
theory and add

1031
01:15:55,930 --> 01:15:58,010
to the list of
symmetries that have

1032
01:15:58,010 --> 01:16:03,250
to be obeyed, this additional
self-similarity of scaling.

1033
01:16:03,250 --> 01:16:07,990
And that will put us at t
equals to 0, h equals to 0.

1034
01:16:07,990 --> 01:16:10,880
And for example, we should
be able to calculate

1035
01:16:10,880 --> 01:16:13,530
this correlation.

1036
01:16:13,530 --> 01:16:15,987
Let me expand a little
bit on that because we

1037
01:16:15,987 --> 01:16:17,320
will need one other correlation.

1038
01:16:17,320 --> 01:16:19,550
Because we've said
that essentially,

1039
01:16:19,550 --> 01:16:22,530
all of the properties
of the system

1040
01:16:22,530 --> 01:16:25,760
I can get from two
independent exponents.

1041
01:16:25,760 --> 01:16:29,690
So suppose I constructed
this scale invariant theory

1042
01:16:29,690 --> 01:16:32,470
and I calculated this.

1043
01:16:32,470 --> 01:16:33,770
That would be on exponent.

1044
01:16:33,770 --> 01:16:36,420
I need another one.

1045
01:16:36,420 --> 01:16:39,560
Well, we had here a
statement about alpha.

1046
01:16:39,560 --> 01:16:43,830
We made the statement that
heat capacity diverges.

1047
01:16:43,830 --> 01:16:48,690
Now in the same sense that the
susceptibility is a response--

1048
01:16:48,690 --> 01:16:52,470
it came from two derivatives
of the free energy with respect

1049
01:16:52,470 --> 01:16:54,000
to the field.

1050
01:16:54,000 --> 01:16:56,540
The derivative of
magnetization with respect

1051
01:16:56,540 --> 01:16:59,400
to field magnetization is
one derivative [INAUDIBLE].

1052
01:16:59,400 --> 01:17:01,810
The heat capacity is
also two derivatives

1053
01:17:01,810 --> 01:17:05,400
of free energy with respect
to some other variable.

1054
01:17:08,400 --> 01:17:10,700
So in the same
sense that there is

1055
01:17:10,700 --> 01:17:14,011
a relationship between
the susceptibility

1056
01:17:14,011 --> 01:17:16,770
and an integrated
correlation function,

1057
01:17:16,770 --> 01:17:21,800
there is a relationship that
says that the heat capacity is

1058
01:17:21,800 --> 01:17:26,000
related to an integrated
correlation function.

1059
01:17:26,000 --> 01:17:31,860
So c as a function of say t and
h, let's say the singular part,

1060
01:17:31,860 --> 01:17:36,115
is going to be related to
an integral of something.

1061
01:17:40,080 --> 01:17:45,420
And again, we've
already seen this.

1062
01:17:45,420 --> 01:17:48,512
Essentially, you take one
derivative of the free energy

1063
01:17:48,512 --> 01:17:50,470
let's say with respect
the beta or temperature,

1064
01:17:50,470 --> 01:17:53,060
you get the energy.

1065
01:17:53,060 --> 01:17:55,860
And you take another
derivative of the energy

1066
01:17:55,860 --> 01:17:59,300
you will get the heat capacity.

1067
01:17:59,300 --> 01:18:02,120
And then that
derivative, if we write

1068
01:18:02,120 --> 01:18:05,320
in terms of the first derivative
of the partition function

1069
01:18:05,320 --> 01:18:09,780
becomes converted to
the variance in energy.

1070
01:18:09,780 --> 01:18:12,750
So in the same way that
the susceptibility was

1071
01:18:12,750 --> 01:18:16,220
the variance of the
net magnetization,

1072
01:18:16,220 --> 01:18:19,555
the heat capacity is
related to the variance

1073
01:18:19,555 --> 01:18:23,580
of the net energy of the
system at an even temperature.

1074
01:18:23,580 --> 01:18:26,090
The net energy of the
system we can write this

1075
01:18:26,090 --> 01:18:29,400
as an integral of
an energy density,

1076
01:18:29,400 --> 01:18:31,450
just as we wrote
the magnetization

1077
01:18:31,450 --> 01:18:34,050
as an integral of
magnetization density.

1078
01:18:34,050 --> 01:18:40,105
And then the heat capacity will
be related to the correlation

1079
01:18:40,105 --> 01:18:42,486
functions of the energy density.

1080
01:18:47,226 --> 01:18:53,380
Now once more, you say that
I'm at the critical point.

1081
01:18:53,380 --> 01:18:57,210
At the critical point
there is no length scale.

1082
01:18:57,210 --> 01:19:02,360
So any correlation function, not
only that of the magnetization,

1083
01:19:02,360 --> 01:19:08,580
should fall off as some
power of separation.

1084
01:19:08,580 --> 01:19:12,920
And you can call that
exponent whatever you like.

1085
01:19:12,920 --> 01:19:16,772
There is no definition
for it in the literature.

1086
01:19:16,772 --> 01:19:20,300
Let me write it in the
same way as magnetization

1087
01:19:20,300 --> 01:19:23,320
as d minus 2 plus eta prime.

1088
01:19:23,320 --> 01:19:28,320
So then when I go and say let's
terminate it at the correlation

1089
01:19:28,320 --> 01:19:30,800
length, the answer
is going to be

1090
01:19:30,800 --> 01:19:35,330
proportional to c to
the 2 minus eta prime,

1091
01:19:35,330 --> 01:19:38,792
which would be t
to the minus mu.

1092
01:19:38,792 --> 01:19:41,150
2 minus eta prime.

1093
01:19:41,150 --> 01:19:46,590
So then I would have alpha
being mu 2 minus eta.

1094
01:19:51,550 --> 01:19:56,100
So all I need to
do in principle is

1095
01:19:56,100 --> 01:20:00,160
to construct a theory,
which in addition

1096
01:20:00,160 --> 01:20:02,980
to rotational
invariance or there's

1097
01:20:02,980 --> 01:20:05,720
whatever is appropriate
to the system in question,

1098
01:20:05,720 --> 01:20:09,860
has this statistical
scale invariance.

1099
01:20:09,860 --> 01:20:13,710
Within that theory, calculate
the correlation functions

1100
01:20:13,710 --> 01:20:17,740
of two quantities, such as
magnetization and energy.

1101
01:20:17,740 --> 01:20:20,520
Extract two exponents.

1102
01:20:20,520 --> 01:20:22,936
Once we have two
exponents, then we

1103
01:20:22,936 --> 01:20:24,920
know why your
manipulations will be

1104
01:20:24,920 --> 01:20:26,597
able to calculate
all the exponents.

1105
01:20:29,520 --> 01:20:32,850
So why doesn't this
solve the problem?

1106
01:20:32,850 --> 01:20:36,270
The answer is that whereas I
can write immediately for you

1107
01:20:36,270 --> 01:20:40,800
a term such as m squared,
that is rotational invariant,

1108
01:20:40,800 --> 01:20:46,725
I don't know how to write down a
theory that is scale invariant.

1109
01:20:46,725 --> 01:20:50,180
The one case where people
have succeeded to do that

1110
01:20:50,180 --> 01:20:52,590
is actually two dimensions.

1111
01:20:52,590 --> 01:20:54,890
So in two dimensions,
one can show

1112
01:20:54,890 --> 01:20:56,835
that this kind of
scale invariance

1113
01:20:56,835 --> 01:21:00,615
is related to
conformal invariance

1114
01:21:00,615 --> 01:21:04,070
and that one can explicitly
write down conformal invariant

1115
01:21:04,070 --> 01:21:08,500
theories, extract exponents
et cetera out of those.

1116
01:21:08,500 --> 01:21:12,860
But say in three dimensions,
we don't know how to do that.

1117
01:21:12,860 --> 01:21:16,460
So we will still,
with that concept

1118
01:21:16,460 --> 01:21:19,020
in the back of our
mind, approach it

1119
01:21:19,020 --> 01:21:24,810
slightly differently by looking
at the effects of the scale

1120
01:21:24,810 --> 01:21:27,190
transformation on the system.

1121
01:21:27,190 --> 01:21:31,140
And that's the beginning of
this concept of normalization.