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PROFESSOR: OK, let's start.

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00:00:25,438 --> 00:00:31,740
So we have been looking at the
problem of phase transitions

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00:00:31,740 --> 00:00:35,995
from the perspective of
a simple system which

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00:00:35,995 --> 00:00:37,360
is a piece of magnet.

12
00:00:40,550 --> 00:00:48,313
And we find that if we change
the temperature of the system,

13
00:00:48,313 --> 00:00:53,010
there is a critical
temperature, Tc,

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00:00:53,010 --> 00:00:56,840
that separates
paramagnetic behavior

15
00:00:56,840 --> 00:00:59,700
on the high temperature side
and ferromagnetic behavior

16
00:00:59,700 --> 00:01:03,080
on the low temperature side.

17
00:01:03,080 --> 00:01:08,680
Clearly in the
vicinity of this point,

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00:01:08,680 --> 00:01:11,010
whether you're on one
side or the other,

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00:01:11,010 --> 00:01:14,132
the magnetization
is small and we

20
00:01:14,132 --> 00:01:18,700
are relying on that to
make us a good parameter

21
00:01:18,700 --> 00:01:21,260
to expand things in.

22
00:01:21,260 --> 00:01:24,150
The other thing is
that we anticipated

23
00:01:24,150 --> 00:01:26,520
that over here there
are long wavelength

24
00:01:26,520 --> 00:01:29,970
fluctuations of the
magnetization field

25
00:01:29,970 --> 00:01:35,780
and so we did averaging and
defined a statistical field,

26
00:01:35,780 --> 00:01:39,180
this magnetization as
a function of position.

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00:01:42,120 --> 00:01:49,610
Then we said, OK, if I'm
just changing temperature,

28
00:01:49,610 --> 00:01:54,040
what potential is the
behavior of the probability

29
00:01:54,040 --> 00:01:57,806
that I will see in my sample
some particular configuration

30
00:01:57,806 --> 00:01:59,510
of this magnetization field?

31
00:01:59,510 --> 00:02:03,360
So there is a functional
that governs that.

32
00:02:03,360 --> 00:02:06,290
And the statement
that we made was

33
00:02:06,290 --> 00:02:09,220
that whatever this
functional is,

34
00:02:09,220 --> 00:02:11,800
I can write it as the
exponential of something

35
00:02:11,800 --> 00:02:12,660
if I want.

36
00:02:12,660 --> 00:02:15,220
This probability is positive.

37
00:02:15,220 --> 00:02:21,030
I will assume that it is--
locally there is a probability

38
00:02:21,030 --> 00:02:24,200
density that I will
integrate across the system.

39
00:02:24,200 --> 00:02:26,660
Probability density
is a function

40
00:02:26,660 --> 00:02:31,040
of whatever magnetization
I have around point x.

41
00:02:31,040 --> 00:02:34,210
And so then when we expanded
this, what did we have?

42
00:02:34,210 --> 00:02:37,850
We said that the terms that
are consistent with rotational

43
00:02:37,850 --> 00:02:44,380
symmetry have to be things like
m squared, m to the fourth,

44
00:02:44,380 --> 00:02:46,440
and so forth.

45
00:02:46,440 --> 00:02:48,420
In principal, there
is a long series

46
00:02:48,420 --> 00:02:50,570
but hopefully
since m is small, I

47
00:02:50,570 --> 00:02:54,620
don't have to include that
many terms in the series.

48
00:02:54,620 --> 00:02:58,870
And additionally, I can have
an expansion in gradient

49
00:02:58,870 --> 00:03:01,570
and the lowest order
term in that series

50
00:03:01,570 --> 00:03:05,700
was the gradient of m squared
and potentially higher

51
00:03:05,700 --> 00:03:08,280
order terms.

52
00:03:08,280 --> 00:03:10,990
OK?

53
00:03:10,990 --> 00:03:14,930
We said that we
could, if we relied

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00:03:14,930 --> 00:03:19,460
on looking at the most probable
configuration of this weight,

55
00:03:19,460 --> 00:03:21,470
make a connection
between what is going

56
00:03:21,470 --> 00:03:25,000
on here and the
experimental observation.

57
00:03:25,000 --> 00:03:27,960
And essentially the only
thing that we needed to do

58
00:03:27,960 --> 00:03:31,090
was to basically
start T from here,

59
00:03:31,090 --> 00:03:36,620
so T was made to be
proportional to T minus Tc.

60
00:03:36,620 --> 00:03:40,190
And then we could explain
these kinds of phenomena

61
00:03:40,190 --> 00:03:41,892
by looking at the
behavior of the most

62
00:03:41,892 --> 00:03:42,850
probable magnetization.

63
00:03:45,950 --> 00:03:50,240
Now I kind of said
that we are going

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00:03:50,240 --> 00:03:53,680
to have long wavelength
fluctuations.

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00:03:53,680 --> 00:03:57,580
There was one case
where we actually

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00:03:57,580 --> 00:04:01,220
saw a video of those long
wavelength fluctuations

67
00:04:01,220 --> 00:04:03,950
and that was for the case
of critical opalescence

68
00:04:03,950 --> 00:04:09,580
taking place at the liquid gas
mixture at its critical point.

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00:04:09,580 --> 00:04:13,850
Can we try to quantify
that a little bit better?

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00:04:13,850 --> 00:04:18,634
The answer is yes, we can do so
through scattering experiments.

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00:04:23,197 --> 00:04:25,990
And looking at the
sample was an example

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00:04:25,990 --> 00:04:27,690
of a scattering
experiment, which

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00:04:27,690 --> 00:04:30,320
if you want to do
more quantitatively,

74
00:04:30,320 --> 00:04:33,600
we can do the following-
we can say that there

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00:04:33,600 --> 00:04:41,090
is some incoming field,
electromagnetic wave that

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00:04:41,090 --> 00:04:43,500
is impingent on the system.

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00:04:43,500 --> 00:04:48,130
It's a pass on
incoming wave vector k.

78
00:04:48,130 --> 00:04:51,410
It sort of goes
through the sample

79
00:04:51,410 --> 00:04:54,860
and then when it comes out,
it gets scattered and so what

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00:04:54,860 --> 00:04:59,190
I will see is some
k [INAUDIBLE] that

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00:04:59,190 --> 00:05:01,960
comes from the other
part of the system.

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00:05:01,960 --> 00:05:07,700
In principle, I guess I can
put a probe here and measure

83
00:05:07,700 --> 00:05:10,120
what is coming out.

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00:05:10,120 --> 00:05:15,580
And essentially it will
depend on the angle

85
00:05:15,580 --> 00:05:19,740
towards which this has rotated.

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00:05:19,740 --> 00:05:25,940
If I asked well, how
much has been scattered?

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00:05:25,940 --> 00:05:28,720
We'd say, well it's
a complicated problem

88
00:05:28,720 --> 00:05:30,040
in quantum mechanics.

89
00:05:30,040 --> 00:05:32,620
Let's say this is a quantum
mechanical procedure-

90
00:05:32,620 --> 00:05:35,960
you would say that there's
an amplitude that you have

91
00:05:35,960 --> 00:05:47,270
the scattering that is
proportional to some overlap

92
00:05:47,270 --> 00:05:50,890
between the kind of state
that you started with,

93
00:05:50,890 --> 00:05:56,720
what we started is an
incoming wave with k initial,

94
00:05:56,720 --> 00:06:02,650
presumably there is the
initial state of my sample

95
00:06:02,650 --> 00:06:06,780
before the wave hits
it, and then I end up

96
00:06:06,780 --> 00:06:12,170
with the final configuration
which is k f, whatever

97
00:06:12,170 --> 00:06:17,570
the final version
of my system is.

98
00:06:17,570 --> 00:06:21,550
Now between these two,
I have to put whatever

99
00:06:21,550 --> 00:06:26,540
is responsible for
scattering this wave so there

100
00:06:26,540 --> 00:06:30,450
is in some sense some
overall potential

101
00:06:30,450 --> 00:06:33,970
that I have to put over here.

102
00:06:33,970 --> 00:06:36,690
Now let's think about the
case of this thing being,

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00:06:36,690 --> 00:06:41,180
say, a mixture of
gas and liquid,

104
00:06:41,180 --> 00:06:43,460
well what is scattering light?

105
00:06:43,460 --> 00:06:47,410
Well, it is the individual
atoms that are scattering light

106
00:06:47,410 --> 00:06:49,290
and there are lots of them.

107
00:06:49,290 --> 00:06:55,240
So basically I have to sum over
all of the scattering elements

108
00:06:55,240 --> 00:06:57,420
that I have my system.

109
00:06:57,420 --> 00:07:02,420
Let's say I have a u for a
scattering element, i that

110
00:07:02,420 --> 00:07:07,130
is located at position--
maybe bad choice,

111
00:07:07,130 --> 00:07:10,070
let's call it sum over alpha.

112
00:07:10,070 --> 00:07:12,150
X alpha is the
position of let's say

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00:07:12,150 --> 00:07:15,000
the atom that is scattered here.

114
00:07:15,000 --> 00:07:16,760
OK?

115
00:07:16,760 --> 00:07:21,870
So now since I'm dealing
with, say, linear order,

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00:07:21,870 --> 00:07:25,330
not multiple scattering,
what I can do

117
00:07:25,330 --> 00:07:29,230
is I can basically
take this sum outside.

118
00:07:29,230 --> 00:07:32,070
So this thing is
related to a sum

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00:07:32,070 --> 00:07:35,840
over alpha of the
scattering I would have

120
00:07:35,840 --> 00:07:39,200
for individual elements
that are scattering.

121
00:07:39,200 --> 00:07:43,350
And then roughly each
individual element

122
00:07:43,350 --> 00:07:47,900
will scatter an amount
that I will call sigma q.

123
00:07:47,900 --> 00:07:51,610
If you have elastic
scattering what happens

124
00:07:51,610 --> 00:07:58,680
is that essentially your
initial k simply gets rotated

125
00:07:58,680 --> 00:08:01,300
without changing its magnitude.

126
00:08:01,300 --> 00:08:05,460
So what happens is that
essentially everything

127
00:08:05,460 --> 00:08:10,480
will end up being a function
of this momentum transfer which

128
00:08:10,480 --> 00:08:15,970
is proportional
to q k f minus k i

129
00:08:15,970 --> 00:08:20,750
whose magnitude would be
twice the magnitude of your k

130
00:08:20,750 --> 00:08:23,740
sine of the half of
the angle if you just

131
00:08:23,740 --> 00:08:26,410
do the simple
geometry over there.

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00:08:26,410 --> 00:08:28,940
So this is for
elastic scattering

133
00:08:28,940 --> 00:08:31,415
which is what we will
be thinking about.

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00:08:37,370 --> 00:08:42,549
Now the amount that each
individual element scatters

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00:08:42,549 --> 00:08:44,950
like each atom is
indeed a function

136
00:08:44,950 --> 00:08:52,310
of your momentum transferred
from the scattering probe.

137
00:08:52,310 --> 00:08:55,600
But the thing that
you're scattering from

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00:08:55,600 --> 00:08:59,110
is something that is
very small like an atom,

139
00:08:59,110 --> 00:09:05,270
so it turns out that the
resulting q will give

140
00:09:05,270 --> 00:09:09,610
significant-- well, the
resulting sigma will vary all

141
00:09:09,610 --> 00:09:12,580
the over scales
where q is related

142
00:09:12,580 --> 00:09:15,360
to the inverse of
whatever is scattering

143
00:09:15,360 --> 00:09:18,560
which is something
that is very large.

144
00:09:18,560 --> 00:09:23,310
So most of this stuff that
is happening at small q, most

145
00:09:23,310 --> 00:09:25,900
of the variation
that is observed,

146
00:09:25,900 --> 00:09:28,950
comes from summing
over the contributions

147
00:09:28,950 --> 00:09:31,170
of the different elements.

148
00:09:31,170 --> 00:09:36,880
So going to the continuum
limit, this becomes an integral

149
00:09:36,880 --> 00:09:42,880
across your system of whatever
the density of the thing that

150
00:09:42,880 --> 00:09:44,760
is scattering is.

151
00:09:44,760 --> 00:09:47,870
Indeed if I'm thinking about
the light scattering experiment

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00:09:47,870 --> 00:09:51,300
that we saw with critical
opalescence, what you would

153
00:09:51,300 --> 00:09:57,080
be looking at is this
density of liquid

154
00:09:57,080 --> 00:10:01,250
versus gas, which if I
want to convert to q,

155
00:10:01,250 --> 00:10:03,280
I have to do a Fourier
transform here.

156
00:10:05,910 --> 00:10:13,900
And so this is the amplitude of
scattering that I expect and we

157
00:10:13,900 --> 00:10:17,350
can see that it is
directly probing

158
00:10:17,350 --> 00:10:20,670
the fluctuations of
the system, Fourier

159
00:10:20,670 --> 00:10:25,750
transform [INAUDIBLE] number q.

160
00:10:25,750 --> 00:10:28,680
Eventually of course
this is the amplitude

161
00:10:28,680 --> 00:10:33,050
what you will be seeing is
the amount that is scattered,

162
00:10:33,050 --> 00:10:39,580
s of q will be proportional
to this amplitude squared.

163
00:10:39,580 --> 00:10:46,100
We'll have a part that at
small q is roughly constant,

164
00:10:46,100 --> 00:10:51,230
so basically at small q I can
regard this as a constant.

165
00:10:51,230 --> 00:10:54,120
So at small q all
of the variations

166
00:10:54,120 --> 00:10:56,800
is going to come from
this Roth q squared.

167
00:10:59,770 --> 00:11:05,540
Of course again thinking about
the case of the liquid gas

168
00:11:05,540 --> 00:11:11,250
system, where we were seeing the
picture, there were variations,

169
00:11:11,250 --> 00:11:15,530
so there's essentially
lots and lots

170
00:11:15,530 --> 00:11:18,820
of these Roth q's depending
on which instant of time

171
00:11:18,820 --> 00:11:21,090
you're looking at.

172
00:11:21,090 --> 00:11:25,980
And then it would be useful to
do some kind of a time average

173
00:11:25,980 --> 00:11:29,920
and hope that the
time average comes

174
00:11:29,920 --> 00:11:34,880
from the result of a probability
measure such as this.

175
00:11:34,880 --> 00:11:35,520
OK?

176
00:11:35,520 --> 00:11:38,330
So that's the procedure
that we'll follow.

177
00:11:38,330 --> 00:11:43,540
We're going to go slightly
beyond what we did before.

178
00:11:43,540 --> 00:11:46,610
What we did before was we
started with the probability

179
00:11:46,610 --> 00:11:48,340
distribution, such
as this that we

180
00:11:48,340 --> 00:11:54,100
posed on the basis of symmetry
and then calculated singular

181
00:11:54,100 --> 00:11:57,070
behavior of various
thermodynamic functions

182
00:11:57,070 --> 00:12:00,890
such as heat capacity,
susceptibility, magnetization,

183
00:12:00,890 --> 00:12:05,060
et cetera, all of them at
macroscopic quantities.

184
00:12:05,060 --> 00:12:07,880
But this is a
probability that also

185
00:12:07,880 --> 00:12:10,850
works at the level
of microscopics.

186
00:12:10,850 --> 00:12:14,770
It's really a probability as a
function of our configurations

187
00:12:14,770 --> 00:12:17,470
and the way that
that is probed is

188
00:12:17,470 --> 00:12:19,830
through scattering experiments.

189
00:12:19,830 --> 00:12:22,980
So scattering experiments
really probe the Fourier

190
00:12:22,980 --> 00:12:28,220
transform of this probability
that we have posed over here.

191
00:12:28,220 --> 00:12:30,590
OK?

192
00:12:30,590 --> 00:12:36,340
Now again the full probability
that I have written down there

193
00:12:36,340 --> 00:12:38,120
is rather difficult.

194
00:12:38,120 --> 00:12:41,490
Let's say this in the case
of the liquid gas system,

195
00:12:41,490 --> 00:12:43,980
this row would be explicitly
the magnetization.

196
00:12:43,980 --> 00:12:47,880
It would be the fluctuations
of the magnetization

197
00:12:47,880 --> 00:12:50,880
around the mean.

198
00:12:50,880 --> 00:12:53,370
I should note that in
the case of the magnet,

199
00:12:53,370 --> 00:12:55,920
you can say, well how
do you probe things?

200
00:12:55,920 --> 00:12:59,870
In that case you need
something, some probe

201
00:12:59,870 --> 00:13:03,700
that scatters from
magnetization at each point.

202
00:13:03,700 --> 00:13:08,400
And the appropriate probe for
magnetization is neutrons.

203
00:13:08,400 --> 00:13:12,320
So you basically hit the
system with a beam of neutrons

204
00:13:12,320 --> 00:13:15,480
that may be polarizing them
in this particular direction,

205
00:13:15,480 --> 00:13:18,850
their spins-- and they
hit the spins of whatever

206
00:13:18,850 --> 00:13:21,630
is in your sample and
they get scattered

207
00:13:21,630 --> 00:13:23,730
according to this mechanism.

208
00:13:23,730 --> 00:13:26,610
And what you will
be seeing at small q

209
00:13:26,610 --> 00:13:32,770
is related to fluctuations
of this magnetization field.

210
00:13:37,390 --> 00:13:41,755
Think he was looking for that.

211
00:13:41,755 --> 00:13:52,780
Now I realize-- I'm not
going to run after him.

212
00:13:52,780 --> 00:13:53,520
OK.

213
00:13:53,520 --> 00:13:59,350
So teaches them to
leave the room earlier.

214
00:13:59,350 --> 00:14:00,750
OK.

215
00:14:00,750 --> 00:14:04,870
So let's see what
we have to-- we can

216
00:14:04,870 --> 00:14:08,662
do for the case of
calculating this quantity.

217
00:14:08,662 --> 00:14:11,600
Now I'm not going to calculate
it for the nonlinear form,

218
00:14:11,600 --> 00:14:13,510
it's rather difficult.

219
00:14:13,510 --> 00:14:17,780
What I'm going to do is to
sort of expand on the trick

220
00:14:17,780 --> 00:14:21,210
that we were using
last time which

221
00:14:21,210 --> 00:14:23,500
led to this other
point which is to look

222
00:14:23,500 --> 00:14:25,095
at the most probable state.

223
00:14:32,300 --> 00:14:36,240
So basically we looked
at that function

224
00:14:36,240 --> 00:14:38,640
where we were calculating
the subtle point

225
00:14:38,640 --> 00:14:40,310
integration and the
first thing that we

226
00:14:40,310 --> 00:14:42,610
did was to find
the configuration

227
00:14:42,610 --> 00:14:47,160
of the magnetization field
that was the most likely.

228
00:14:47,160 --> 00:14:50,830
And the answer was that
because k is positive,

229
00:14:50,830 --> 00:14:55,170
your magnetization that
extremizes that probability

230
00:14:55,170 --> 00:15:00,460
is something that is uniform,
does not depend on x,

231
00:15:00,460 --> 00:15:03,060
and is pointing all
in one direction.

232
00:15:03,060 --> 00:15:06,010
Let's call it e hat 1.

233
00:15:06,010 --> 00:15:08,670
That is indicating
this one thing

234
00:15:08,670 --> 00:15:12,790
is symmetry breaking in
the zero field limit.

235
00:15:12,790 --> 00:15:16,230
Of course that spontaneous
symmetry breaking

236
00:15:16,230 --> 00:15:24,796
only occurs when t is negative
and for t positive m bar is 0.

237
00:15:24,796 --> 00:15:28,860
For t negative just
minimizing the expression

238
00:15:28,860 --> 00:15:31,570
t m squared plus
u m to the fourth

239
00:15:31,570 --> 00:15:36,060
gives you minus t over 4u.

240
00:15:39,092 --> 00:15:41,380
OK?

241
00:15:41,380 --> 00:15:45,300
So that's the most
probable configuration.

242
00:15:45,300 --> 00:15:48,720
What this thing is
probing is fluctuations so

243
00:15:48,720 --> 00:15:52,960
let's expand around the
most probable configuration.

244
00:15:52,960 --> 00:15:58,420
So let's say that we-- say that
I have thermally excited and m

245
00:15:58,420 --> 00:16:05,690
of x which is m bar plus
a little bit that varies

246
00:16:05,690 --> 00:16:09,120
from each location
to another location,

247
00:16:09,120 --> 00:16:12,360
like if I'm looking at
this critical opalescence,

248
00:16:12,360 --> 00:16:14,620
it's the variation
in density from one

249
00:16:14,620 --> 00:16:17,660
location to another location.

250
00:16:17,660 --> 00:16:22,120
But that is true as
long as I'm looking

251
00:16:22,120 --> 00:16:26,770
at the case of something
that has only one component.

252
00:16:26,770 --> 00:16:29,640
If I have multiple
components, I can also

253
00:16:29,640 --> 00:16:34,770
have fluctuations in the
remaining m minus 1 directions,

254
00:16:34,770 --> 00:16:45,620
and this is n of let's say alpha
go from 2 of phi t of e alpha.

255
00:16:45,620 --> 00:16:51,250
So I have broken the
fluctuations into two types.

256
00:16:51,250 --> 00:16:56,845
I've said that let's say
if you were n equals to 2

257
00:16:56,845 --> 00:17:02,440
your m bar would be pointing
in some particular direction.

258
00:17:02,440 --> 00:17:09,530
And so phi l's correspond
to increasing the length

259
00:17:09,530 --> 00:17:13,700
or decreasing the length
whereas phi t corresponds

260
00:17:13,700 --> 00:17:18,550
to going in the
orthogonal direction which

261
00:17:18,550 --> 00:17:23,069
in general would be n minus
1, different components.

262
00:17:23,069 --> 00:17:25,000
OK?

263
00:17:25,000 --> 00:17:28,250
So I want to ask,
what's the probability

264
00:17:28,250 --> 00:17:31,190
of this set of fluctuations?

265
00:17:31,190 --> 00:17:34,940
All I need to do and
again this is x-dependent

266
00:17:34,940 --> 00:17:39,940
is to substitute into my general
expression for the probability

267
00:17:39,940 --> 00:17:42,480
so for that I need a few things.

268
00:17:42,480 --> 00:17:48,110
One of the things that I need
is the gradient of m squared.

269
00:17:48,110 --> 00:17:52,640
The uniform part has no
gradient so I will either

270
00:17:52,640 --> 00:17:58,760
get the gradient from phi l
squared or I will get gradient

271
00:17:58,760 --> 00:18:04,410
of the n minus 1 component
vector field phi t

272
00:18:04,410 --> 00:18:09,480
that I will simply write as
gradient of phi t squared.

273
00:18:09,480 --> 00:18:14,560
So phi t is an n
minus 1 component.

274
00:18:14,560 --> 00:18:18,430
I can ask, what is m squared?

275
00:18:18,430 --> 00:18:22,085
Basically the first term
I need to put over there,

276
00:18:22,085 --> 00:18:25,120
but m squared, I have to
square this expression.

277
00:18:25,120 --> 00:18:32,520
I will get m bar squared to m
bar phi l plus phi l squared

278
00:18:32,520 --> 00:18:35,930
that comes from the
component that is along e 1.

279
00:18:35,930 --> 00:18:39,505
All the other components
here will add up

280
00:18:39,505 --> 00:18:46,330
to give me the magnitude of
this transverse field that

281
00:18:46,330 --> 00:18:51,390
exists in the other
m minus 1 directions.

282
00:18:51,390 --> 00:18:55,450
The other term that I
need is m to the fourth

283
00:18:55,450 --> 00:18:59,650
and in particular we saw that
it is absolutely necessary

284
00:18:59,650 --> 00:19:04,590
if t is negative to include
the m to the fourth term

285
00:19:04,590 --> 00:19:06,920
because otherwise the
probability that we were

286
00:19:06,920 --> 00:19:09,740
writing just didn't
make sense and we need

287
00:19:09,740 --> 00:19:12,590
to write expressions for
probability that are physically

288
00:19:12,590 --> 00:19:14,510
sensible.

289
00:19:14,510 --> 00:19:18,700
So I just take the line
above and square it.

290
00:19:18,700 --> 00:19:23,520
But I want to only keep
terms to quadratic order.

291
00:19:23,520 --> 00:19:28,670
So to zero order I have m bar
to the fourth to third order

292
00:19:28,670 --> 00:19:36,680
I have-- to first order I
have 4 m bar cubed phi l

293
00:19:36,680 --> 00:19:41,840
then there are a bunch of terms
that are order of phi squared.

294
00:19:41,840 --> 00:19:45,400
Squaring this will give
me 4 m bar squared phi l

295
00:19:45,400 --> 00:19:49,250
squared, but the dot
product of these two terms

296
00:19:49,250 --> 00:19:52,940
will also gives me two
m bar squared phi l

297
00:19:52,940 --> 00:20:01,230
squared for a total of six
m bar squared phi l squared.

298
00:20:01,230 --> 00:20:05,210
And then the phi t
squared comes simply

299
00:20:05,210 --> 00:20:08,065
from twice m bar
squared phi t squared.

300
00:20:12,587 --> 00:20:19,470
And then there's higher order
terms cubic and fourth order m

301
00:20:19,470 --> 00:20:22,960
phi t and phi l
that I don't write

302
00:20:22,960 --> 00:20:27,610
assuming that the fluctuations
that I'm looking at our small

303
00:20:27,610 --> 00:20:30,220
around the most probable state.

304
00:20:30,220 --> 00:20:32,150
OK?

305
00:20:32,150 --> 00:20:35,610
So if I stick with
this quadratics

306
00:20:35,610 --> 00:20:39,370
then the probability
of fluctuations

307
00:20:39,370 --> 00:20:45,650
across my system characterized
by phi l of x and phi t

308
00:20:45,650 --> 00:20:52,700
of x is proportional
to exponential

309
00:20:52,700 --> 00:20:59,975
of minus integral dd x.

310
00:21:02,850 --> 00:21:08,790
I have an overall factor-- well
I have a factor of K over 2

311
00:21:08,790 --> 00:21:09,910
for the first term.

312
00:21:09,910 --> 00:21:15,830
I have a gradient
of phi l squared

313
00:21:15,830 --> 00:21:24,230
and then you can see that
I have a bunch of them.

314
00:21:24,230 --> 00:21:30,200
Let's put the K over 2,
let's put the one-half here.

315
00:21:30,200 --> 00:21:36,380
I have K phi l gradient
of phi l squared.

316
00:21:36,380 --> 00:21:39,870
I'm going to put everything
that has phi l squared in it.

317
00:21:39,870 --> 00:21:46,330
I have here t over 2 phi l
squared from t over 2m squared.

318
00:21:46,330 --> 00:21:50,250
So I have phi l squared.

319
00:21:50,250 --> 00:21:53,800
I have t over 2.

320
00:21:53,800 --> 00:21:57,855
Actually I have taken--
I'm going to make mistakes

321
00:21:57,855 --> 00:22:03,010
unless I put the
one-half over here, too.

322
00:22:03,010 --> 00:22:09,830
I have another phi l
squared from over here.

323
00:22:09,830 --> 00:22:14,320
That gets multiplied
by u so I will get here

324
00:22:14,320 --> 00:22:20,230
plus 12 u m bar squared,
12 rather than 6

325
00:22:20,230 --> 00:22:21,648
because I divided by 2.

326
00:22:24,580 --> 00:22:28,660
And then I have a term that is
K over 2 gradient of the vector

327
00:22:28,660 --> 00:22:29,872
phi t squared.

328
00:22:32,550 --> 00:22:39,010
And then I have t
over 2 phi t squared

329
00:22:39,010 --> 00:22:42,980
and then 2 m bar
squared multiplied by u

330
00:22:42,980 --> 00:22:47,250
becomes 4u m bar squared.

331
00:22:47,250 --> 00:22:51,246
And there are higher order terms
that I will not write down.

332
00:22:56,066 --> 00:22:57,012
Yes?

333
00:22:57,012 --> 00:22:57,512
Did I--

334
00:22:57,512 --> 00:23:01,380
AUDIENCE: You said
phi l [INAUDIBLE].

335
00:23:01,380 --> 00:23:02,070
PROFESSOR: Good.

336
00:23:02,070 --> 00:23:05,780
Yes, so the question
is I immediately

337
00:23:05,780 --> 00:23:11,330
jumped to second order, so what
happened to the linear term?

338
00:23:11,330 --> 00:23:14,536
There's a linear term here and
there's a linear term here.

339
00:23:14,536 --> 00:23:16,400
AUDIENCE: [INAUDIBLE]

340
00:23:16,400 --> 00:23:18,420
PROFESSOR: Let's
write them down.

341
00:23:18,420 --> 00:23:21,010
So the coefficient
of phi l would

342
00:23:21,010 --> 00:23:34,280
be t over 2 m bar plus 4 u
m bar cubed and minimizing

343
00:23:34,280 --> 00:23:39,080
that expression is setting
this first derivative to zero

344
00:23:39,080 --> 00:23:43,050
so if you are expanding around
an extreme on the most probable

345
00:23:43,050 --> 00:23:46,570
state then by
construction you're

346
00:23:46,570 --> 00:23:50,260
not going to get any terms
that are linear either m phi l

347
00:23:50,260 --> 00:23:52,822
or m phi t.

348
00:23:52,822 --> 00:23:55,808
OK?

349
00:23:55,808 --> 00:23:56,308
Yes?

350
00:23:56,308 --> 00:23:58,798
AUDIENCE: Is there-- what's
the reason for not including

351
00:23:58,798 --> 00:24:03,280
a term in our
general probability,

352
00:24:03,280 --> 00:24:06,525
a term like [INAUDIBLE]
of m squared?

353
00:24:06,525 --> 00:24:08,750
PROFESSOR: We could.

354
00:24:08,750 --> 00:24:13,110
He said that
essentially that amount

355
00:24:13,110 --> 00:24:18,500
over here to an expansion in
powers of the gradient, which

356
00:24:18,500 --> 00:24:22,270
if I go to Fourier space
this would become q

357
00:24:22,270 --> 00:24:27,240
squared plus n squared would
be q to the fourth, et cetera.

358
00:24:27,240 --> 00:24:31,570
If you are looking at more q
or large wavelengths and so

359
00:24:31,570 --> 00:24:35,410
we are going to focus
on the first few terms.

360
00:24:35,410 --> 00:24:37,640
But they exist just
as they existed

361
00:24:37,640 --> 00:24:39,200
for the phonon spectrum.

362
00:24:39,200 --> 00:24:42,170
We looked at the linear
portion and then realized

363
00:24:42,170 --> 00:24:44,810
that going away
from q equals to 0

364
00:24:44,810 --> 00:24:47,000
generates all kinds
of other steps.

365
00:24:50,740 --> 00:24:51,240
OK?

366
00:24:51,240 --> 00:24:56,930
I mean, these are very important
things to ask again and again

367
00:24:56,930 --> 00:25:00,640
to ultimately convince
yourself, because in reality

368
00:25:00,640 --> 00:25:03,180
that expansion
that I have written

369
00:25:03,180 --> 00:25:06,380
has an infinity of terms in it.

370
00:25:06,380 --> 00:25:09,520
You have to always
convince yourself

371
00:25:09,520 --> 00:25:13,230
that close enough to
the critical point all

372
00:25:13,230 --> 00:25:16,185
of those other terms
that I don't write down

373
00:25:16,185 --> 00:25:20,670
are not going to
make any difference.

374
00:25:20,670 --> 00:25:23,390
OK?

375
00:25:23,390 --> 00:25:25,000
All right.

376
00:25:25,000 --> 00:25:27,490
So that's the weight.

377
00:25:27,490 --> 00:25:31,460
What I'm going to do is
the same thing that we did

378
00:25:31,460 --> 00:25:33,950
last time for the case
of Goldstone modes,

379
00:25:33,950 --> 00:25:38,930
et cetera, which is to go
to a Fourier presentation

380
00:25:38,930 --> 00:25:44,230
so any one of the components, be
the longitudinal or transverse,

381
00:25:44,230 --> 00:25:52,110
I will write as a sum over q e
to the i q dot, x some Fourier

382
00:25:52,110 --> 00:25:59,790
component and then
get a square root of v

383
00:25:59,790 --> 00:26:04,430
just so that the normalization
would look simple.

384
00:26:04,430 --> 00:26:11,910
And if I substitute for phi
of x in terms of phi of q,

385
00:26:11,910 --> 00:26:16,525
just as we saw last
time the probability

386
00:26:16,525 --> 00:26:21,550
will decompose into independent
contribution for each q

387
00:26:21,550 --> 00:26:24,380
because once you
substitute it here,

388
00:26:24,380 --> 00:26:30,220
every quadratic term will have
both a sum over x and-- sorry,

389
00:26:30,220 --> 00:26:32,650
and integral over x
and sums over q and q

390
00:26:32,650 --> 00:26:37,930
prime, e to the i q plus q prime
x the integration over x forces

391
00:26:37,930 --> 00:26:42,730
q and q prime to be
the same up to a sign.

392
00:26:42,730 --> 00:26:46,811
So then we find that the
probability distribution

393
00:26:46,811 --> 00:26:50,340
as a function of these
Fourier amplitudes

394
00:26:50,340 --> 00:26:59,140
phi alpha and phi t
decomposes into a product.

395
00:26:59,140 --> 00:27:03,020
Basically each q mode
is acting independently

396
00:27:03,020 --> 00:27:05,330
of all the others.

397
00:27:05,330 --> 00:27:07,570
And also at the
quadratic level we

398
00:27:07,570 --> 00:27:11,080
see that there is no
crosstalk between transverse

399
00:27:11,080 --> 00:27:16,690
and longitudinal so we will have
one weight for the transverse,

400
00:27:16,690 --> 00:27:19,370
one weight for the longitudinal.

401
00:27:19,370 --> 00:27:22,230
And what's it actually
going to look like?

402
00:27:22,230 --> 00:27:24,315
Essentially it's going
to look like something

403
00:27:24,315 --> 00:27:31,776
that is proportional,
let's say, for phi l,

404
00:27:31,776 --> 00:27:36,190
it is proportional to
phi l of q squared.

405
00:27:36,190 --> 00:27:42,440
I have here something and
then I have k q squared over 2

406
00:27:42,440 --> 00:27:46,386
so then a Fourier transform that
I will get k q squared over 2.

407
00:27:46,386 --> 00:27:52,710
So let's write it in this
fashion, q over 2 q squared

408
00:27:52,710 --> 00:27:59,400
and then that's something
that's not q dependent.

409
00:27:59,400 --> 00:28:05,346
And by convention I will
write it as x e l minus 2.

410
00:28:05,346 --> 00:28:10,070
It has dimensions of
inverse length scale

411
00:28:10,070 --> 00:28:13,630
because q has dimensions
of inverse length scale

412
00:28:13,630 --> 00:28:19,840
so I will shortly define
a length so that these two

413
00:28:19,840 --> 00:28:23,340
terms would have the
same dimensional,

414
00:28:23,340 --> 00:28:26,080
so c l is defined
in that fashion.

415
00:28:26,080 --> 00:28:28,956
And similarly I
have an exponential

416
00:28:28,956 --> 00:28:37,210
that governs k over 2 q
squared plus c t to the minus 2

417
00:28:37,210 --> 00:28:41,510
phi tilde of t of q squared
and this is a vector

418
00:28:41,510 --> 00:28:45,152
so there are n minus
1 components there.

419
00:28:45,152 --> 00:28:46,500
OK?

420
00:28:46,500 --> 00:28:51,310
And you can see that basically
potentially these two terms,

421
00:28:51,310 --> 00:28:55,010
c l and c t are different.

422
00:28:55,010 --> 00:28:59,010
In fact, let's just
write down what they are.

423
00:28:59,010 --> 00:29:06,340
So this coefficient
K over c l squared

424
00:29:06,340 --> 00:29:14,645
is defined to be t plus
12 u m bar squared.

425
00:29:14,645 --> 00:29:15,144
OK?

426
00:29:18,856 --> 00:29:20,192
Question?

427
00:29:20,192 --> 00:29:21,500
AUDIENCE: [INAUDIBLE]

428
00:29:21,500 --> 00:29:23,210
PROFESSOR: OK.

429
00:29:23,210 --> 00:29:25,170
Now this depends
on whether you are

430
00:29:25,170 --> 00:29:29,620
for t positive or t
negative-- better be positive

431
00:29:29,620 --> 00:29:34,140
since I have written it as one
over some positive quantity--

432
00:29:34,140 --> 00:29:40,850
for t positive, m bar
is 0 so this is just t.

433
00:29:40,850 --> 00:29:50,490
For t negative, then m bar
squared is minus t over 4 u

434
00:29:50,490 --> 00:29:56,450
so this becomes minus 3t,
so this becomes minus 2t.

435
00:29:59,250 --> 00:29:59,750
OK?

436
00:30:02,910 --> 00:30:12,960
And k over c t squared is
t plus 4u m bar squared.

437
00:30:12,960 --> 00:30:16,940
It is t for t positive.

438
00:30:16,940 --> 00:30:20,550
For t negative,
substitute it for m bar

439
00:30:20,550 --> 00:30:22,304
squared, it will give me 0.

440
00:30:26,980 --> 00:30:27,480
OK?

441
00:30:30,130 --> 00:30:33,910
Actually the top one
hopefully you recognize

442
00:30:33,910 --> 00:30:35,820
or you remember from last time.

443
00:30:35,820 --> 00:30:39,450
We had exactly this
expression t and minus

444
00:30:39,450 --> 00:30:44,460
2t when we calculated
the susceptibility.

445
00:30:44,460 --> 00:30:49,010
This was the inverse
susceptibility.

446
00:30:49,010 --> 00:30:51,460
In fact, I can be
now more precise

447
00:30:51,460 --> 00:30:56,570
and call that the inverse of
the longitudinal susceptibility.

448
00:30:56,570 --> 00:31:01,070
And what we have
here is the inverse

449
00:31:01,070 --> 00:31:04,780
of the transverse
susceptibility.

450
00:31:04,780 --> 00:31:06,820
What does that mean?

451
00:31:06,820 --> 00:31:10,230
Let me remind you what
susceptibility is.

452
00:31:10,230 --> 00:31:12,560
Susceptibility is
you have a system

453
00:31:12,560 --> 00:31:15,210
and you put on a
little bit of the field

454
00:31:15,210 --> 00:31:19,390
and then see how the
magnetization responds.

455
00:31:19,390 --> 00:31:24,500
If you are in the ordered
phase so that your system is

456
00:31:24,500 --> 00:31:27,635
spontaneously pointing
in one direction,

457
00:31:27,635 --> 00:31:30,990
then if you put the
field in this direction,

458
00:31:30,990 --> 00:31:34,900
you have to climb this
Mexican hat potential

459
00:31:34,900 --> 00:31:38,680
and you have to pay
a cost to do so.

460
00:31:38,680 --> 00:31:44,010
Whereas if I put it the field
perpendicular, all that happens

461
00:31:44,010 --> 00:31:47,090
is that the
magnetization rotates,

462
00:31:47,090 --> 00:31:51,970
so it can respond without any
cost and that's what this is.

463
00:31:51,970 --> 00:31:54,280
These are really
the Goldstone modes

464
00:31:54,280 --> 00:31:58,720
that we were discussing are the
transverse fluctuations that I

465
00:31:58,720 --> 00:32:00,345
have written before.

466
00:32:00,345 --> 00:32:03,290
So again, we discussed
last time you

467
00:32:03,290 --> 00:32:07,060
break a continual symmetry
you will have Goldstone modes

468
00:32:07,060 --> 00:32:09,830
and these Goldstone
modes are the ones that

469
00:32:09,830 --> 00:32:15,578
are perpendicular to the average
magnetization, if you like.

470
00:32:15,578 --> 00:32:16,078
OK?

471
00:32:22,870 --> 00:32:27,090
So now we have a prediction.

472
00:32:27,090 --> 00:32:34,370
We say that if I look at
these phi phi fluctuations,

473
00:32:34,370 --> 00:32:37,616
I can pick a particular
q, let's say,

474
00:32:37,616 --> 00:32:40,870
and here I have to
put q star in order

475
00:32:40,870 --> 00:32:45,810
to get something
that is non-zero.

476
00:32:45,810 --> 00:32:51,240
Actually that's put here q prime
and then pick two different

477
00:32:51,240 --> 00:32:54,556
and this is alpha and beta.

478
00:32:54,556 --> 00:32:58,910
Well, if I look at this
average since the weight is

479
00:32:58,910 --> 00:33:03,020
the product of contributions
from different q's the answer

480
00:33:03,020 --> 00:33:08,270
will be 0 unless q and
q prime add up to 0.

481
00:33:11,280 --> 00:33:14,420
And if I'm looking
at the same q,

482
00:33:14,420 --> 00:33:17,990
I better make sure that I'm
looking at the same component

483
00:33:17,990 --> 00:33:21,680
because the longitudinal and
transverse component or any

484
00:33:21,680 --> 00:33:25,410
of the n minus 1 transverse
components among each other

485
00:33:25,410 --> 00:33:30,090
have completely
independent Gaussian rates.

486
00:33:30,090 --> 00:33:34,790
If I'm now looking at the same
Gaussian, for the same Gaussian

487
00:33:34,790 --> 00:33:38,410
I can just immediately
read off it's variance

488
00:33:38,410 --> 00:33:46,600
which is K q squared plus
whatever the appropriate c is

489
00:33:46,600 --> 00:33:49,060
for that direction
whether it's c l

490
00:33:49,060 --> 00:33:53,660
or c t potentially
would make a difference.

491
00:33:53,660 --> 00:33:54,160
OK?

492
00:33:57,040 --> 00:33:57,730
Right.

493
00:33:57,730 --> 00:34:04,360
So now we have a prediction
for our experimentals.

494
00:34:04,360 --> 00:34:09,280
I said that these guys
can go and measure

495
00:34:09,280 --> 00:34:13,090
the scattering as a function
of angle at small angle

496
00:34:13,090 --> 00:34:16,750
and they can fit how
much is scattered

497
00:34:16,750 --> 00:34:23,320
as you go as a function of angle
and fit it as a function of q.

498
00:34:23,320 --> 00:34:28,600
So we predict that if
they look at something

499
00:34:28,600 --> 00:34:33,150
that's say like phi l
squared, if you're thinking

500
00:34:33,150 --> 00:34:35,870
about the liquid gas
system that's really

501
00:34:35,870 --> 00:34:39,269
the only thing that
you have because there

502
00:34:39,269 --> 00:34:43,530
is no transverse component if
you have a Scalar variable.

503
00:34:43,530 --> 00:34:51,320
We claim that if you go and look
at those critical opalescent

504
00:34:51,320 --> 00:34:57,270
pictures that we saw,
and do it more precisely

505
00:34:57,270 --> 00:35:02,145
and see what happens as a
function of the scattered wave

506
00:35:02,145 --> 00:35:05,860
number q, that you
will get a shape that

507
00:35:05,860 --> 00:35:10,920
is 1 over q squared
plus c to the minus 2.

508
00:35:10,920 --> 00:35:19,040
This kind of shape that
is called a Lorentzian

509
00:35:19,040 --> 00:35:23,210
is indeed what you commonly
see for all kinds of scattering

510
00:35:23,210 --> 00:35:25,420
line shapes.

511
00:35:25,420 --> 00:35:26,280
OK?

512
00:35:26,280 --> 00:35:30,230
So we have a prediction.

513
00:35:30,230 --> 00:35:32,440
Of course, the
reason that it works

514
00:35:32,440 --> 00:35:36,160
is because in principle we
know that this series will have

515
00:35:36,160 --> 00:35:37,825
higher order terms
as we discussed

516
00:35:37,825 --> 00:35:41,150
like q to the fourth, q
to the sixth, et cetera,

517
00:35:41,150 --> 00:35:44,960
but they fall way down
here where you're not

518
00:35:44,960 --> 00:35:48,910
going to be seeing
all that much anyway.

519
00:35:48,910 --> 00:35:54,440
Now the place where
this curve turns around

520
00:35:54,440 --> 00:36:01,860
from being something that
is dominated by 1 over K c

521
00:36:01,860 --> 00:36:09,410
to the minus 2 to something
that falls off as 1

522
00:36:09,410 --> 00:36:14,010
over K q squared or
maybe even faster,

523
00:36:14,010 --> 00:36:20,160
the borderline is this inverse
length scale that we indicated,

524
00:36:20,160 --> 00:36:21,464
c l minus 1.

525
00:36:25,550 --> 00:36:31,480
So now what happens if I go
closer to the phase transition

526
00:36:31,480 --> 00:36:33,160
point?

527
00:36:33,160 --> 00:36:36,930
As I go closer to the
phase transition point,

528
00:36:36,930 --> 00:36:44,850
t goes to zero, this c
inverse goes towards zero.

529
00:36:44,850 --> 00:36:50,110
So if this is for some
temperature above Tc

530
00:36:50,110 --> 00:36:54,040
and I go to some lower
temperature, then what happens

531
00:36:54,040 --> 00:37:05,460
is that the curve will start
higher and then [INAUDIBLE].

532
00:37:08,998 --> 00:37:11,960
Yeah?

533
00:37:11,960 --> 00:37:14,900
Actually it doesn't
cross the other curve

534
00:37:14,900 --> 00:37:17,210
which is what I wrote down.

535
00:37:17,210 --> 00:37:24,550
Just because it starts
higher it can bend and go

536
00:37:24,550 --> 00:37:29,090
and join this curve
at a further point.

537
00:37:29,090 --> 00:37:34,970
Eventually when you go through
exactly the critical point then

538
00:37:34,970 --> 00:37:40,010
you get the union of all of
these curves, which is a 1

539
00:37:40,010 --> 00:37:43,650
over q squared type of curve.

540
00:37:43,650 --> 00:37:49,660
So right at the point
where t equals to 0

541
00:37:49,660 --> 00:37:54,270
the prediction is that
the Lorentzian shape,

542
00:37:54,270 --> 00:37:58,870
the coefficient that appears
in front of q squared

543
00:37:58,870 --> 00:38:02,640
vanishes you will see
one over q squared.

544
00:38:02,640 --> 00:38:04,666
OK?

545
00:38:04,666 --> 00:38:08,060
Now the results of experiments
in reality that are very

546
00:38:08,060 --> 00:38:12,780
happily fitted, the
Lorentzian, when you're away

547
00:38:12,780 --> 00:38:16,000
from the critical
point they claim

548
00:38:16,000 --> 00:38:18,570
that when you are exactly
at the critical point,

549
00:38:18,570 --> 00:38:21,320
it's not quite 1 over q squared.

550
00:38:21,320 --> 00:38:23,572
Seems to be slightly different.

551
00:38:26,940 --> 00:38:30,670
At Tc, the scattering
appears to be

552
00:38:30,670 --> 00:38:37,760
more similar to 1 over q
to 2 minus a small amount.

553
00:38:37,760 --> 00:38:41,740
That's where another
critical exponent theta

554
00:38:41,740 --> 00:38:46,240
is introduced so that's another
thing that ultimately you

555
00:38:46,240 --> 00:38:49,766
have to try to figure
out and understand.

556
00:38:49,766 --> 00:38:52,580
OK?

557
00:38:52,580 --> 00:38:59,410
Of course I drew the curves
for the longitudinal component.

558
00:38:59,410 --> 00:39:02,410
If I look at the curves for
the transverse components,

559
00:39:02,410 --> 00:39:08,000
and again, by appropriate choice
of spin polarized neutrons

560
00:39:08,000 --> 00:39:11,190
you can decompose
different components

561
00:39:11,190 --> 00:39:14,180
of scattering from
the magnetization

562
00:39:14,180 --> 00:39:17,160
field of a piece of
iron, for example.

563
00:39:17,160 --> 00:39:19,350
If you are above
Tc, there is really

564
00:39:19,350 --> 00:39:23,410
no difference between
longitudinal and transverse

565
00:39:23,410 --> 00:39:26,500
because there is no
direction that is selected.

566
00:39:26,500 --> 00:39:30,450
And you can see that the forms
that you will get above Tc

567
00:39:30,450 --> 00:39:32,990
would be exactly the same.

568
00:39:32,990 --> 00:39:37,020
When you go below Tc, that's
where the difference appears

569
00:39:37,020 --> 00:39:40,670
because the length
scale that would appear

570
00:39:40,670 --> 00:39:44,380
for the longitudinal
parameters would be finite

571
00:39:44,380 --> 00:39:48,100
and it corresponds
to having to push

572
00:39:48,100 --> 00:39:53,840
the magnetization above this
bottom of the Mexican hat

573
00:39:53,840 --> 00:39:59,060
potential whereas there is no
cost in the other direction.

574
00:39:59,060 --> 00:40:03,510
So if you can probe in
the fluctuations that

575
00:40:03,510 --> 00:40:08,040
would correspond to
these Goldstone modes,

576
00:40:08,040 --> 00:40:13,282
you would see the 1 over q
squared type of behavior.

577
00:40:13,282 --> 00:40:15,580
OK?

578
00:40:15,580 --> 00:40:21,410
So the story that we were
talking about last time around

579
00:40:21,410 --> 00:40:23,920
about the Goldstone
modes and they're

580
00:40:23,920 --> 00:40:29,620
fluctuating a lot because of
their low cost also certainly

581
00:40:29,620 --> 00:40:31,600
remains in this case.

582
00:40:31,600 --> 00:40:34,630
We have now explicitly
separated out

583
00:40:34,630 --> 00:40:37,620
the longitudinal
fluctuations that

584
00:40:37,620 --> 00:40:41,150
are finite because
they are controlled

585
00:40:41,150 --> 00:40:46,300
by this stiffness of going up
the bottom of the potential

586
00:40:46,300 --> 00:40:49,380
whereas there is no
stiffness associated

587
00:40:49,380 --> 00:40:55,430
with the transverse ones.

588
00:40:55,430 --> 00:40:57,900
OK?

589
00:40:57,900 --> 00:40:59,420
All right, so good.

590
00:40:59,420 --> 00:41:03,120
So we've talked about
some of the things that

591
00:41:03,120 --> 00:41:07,806
are experimentally observed.

592
00:41:07,806 --> 00:41:08,610
Any questions?

593
00:41:14,210 --> 00:41:21,530
Now we looked at things here in
the Fourier space corresponding

594
00:41:21,530 --> 00:41:25,350
to this momentum transfer in
the scattering experiment,

595
00:41:25,350 --> 00:41:30,030
but we can also ask about what
is happening in physical space.

596
00:41:30,030 --> 00:41:33,270
That is if I have a
fluctuation at one point,

597
00:41:33,270 --> 00:41:36,690
how much does the influence
of that fluctuation

598
00:41:36,690 --> 00:41:38,960
propagate in space?

599
00:41:38,960 --> 00:41:46,070
So for that I need to calculate
things like phi-- let's say,

600
00:41:46,070 --> 00:41:49,270
do we need to put an
invert, why not?--

601
00:41:49,270 --> 00:41:54,580
phi l of x phi l of x prime.

602
00:41:54,580 --> 00:41:59,293
Let's say we want to
calculate this quantity.

603
00:41:59,293 --> 00:41:59,792
OK?

604
00:42:02,580 --> 00:42:05,400
Now I can certainly
decompose phi l

605
00:42:05,400 --> 00:42:09,970
of x in terms of these
Fourier components.

606
00:42:09,970 --> 00:42:11,510
And so what do I get?

607
00:42:11,510 --> 00:42:18,490
I will get a sum over q-- maybe
I should write it in one case

608
00:42:18,490 --> 00:42:25,350
explicitly, q and q
prime e to i cube of x e

609
00:42:25,350 --> 00:42:30,130
to the i q prime the x prime.

610
00:42:30,130 --> 00:42:33,770
Two factors of root
3 giving me the V

611
00:42:33,770 --> 00:42:42,300
and then I have phi l
of q phi l of q prime

612
00:42:42,300 --> 00:42:48,484
and the expectation
value will go over here.

613
00:42:48,484 --> 00:42:53,110
Now we said that the
different q's and q primes

614
00:42:53,110 --> 00:42:55,230
are uncorrelated.

615
00:42:55,230 --> 00:43:00,130
So here I immediately
will have a delta function

616
00:43:00,130 --> 00:43:06,530
mq plus q prime but if I'm
looking at the same q and q

617
00:43:06,530 --> 00:43:14,150
prime, I have this factor
of 1 over K q squared

618
00:43:14,150 --> 00:43:20,940
plus c to the minus
2-- c l to the minus 2.

619
00:43:20,940 --> 00:43:22,830
OK?

620
00:43:22,830 --> 00:43:31,580
So then the whole thing becomes
due to the delta function

621
00:43:31,580 --> 00:43:46,410
the sum over 1 q of 1
over V each with i q

622
00:43:46,410 --> 00:43:52,800
dot x minus x prime because
q prime was 2 minus q.

623
00:43:52,800 --> 00:44:00,598
And then I have K 2 squared
plus c l to the minus.

624
00:44:00,598 --> 00:44:03,560
OK?

625
00:44:03,560 --> 00:44:07,350
So then I go to the continuum
limit of a large size,

626
00:44:07,350 --> 00:44:10,400
the sum over q gets
replaced with integral

627
00:44:10,400 --> 00:44:16,790
over q d times the
density of state.

628
00:44:16,790 --> 00:44:19,620
The V's disappear,
I will have a factor

629
00:44:19,620 --> 00:44:30,228
of 2 pi to the d and what I have
is the Fourier transform of k q

630
00:44:30,228 --> 00:44:34,220
squared plus c to the
minus-- l to the minus 2.

631
00:44:38,220 --> 00:44:48,870
And I will write this
as minus 1 over K

632
00:44:48,870 --> 00:44:54,470
a function I that
depends on d dimension

633
00:44:54,470 --> 00:44:58,410
and clearly depends
on the separation

634
00:44:58,410 --> 00:45:04,960
x minus x prime at the
correlation length c l.

635
00:45:04,960 --> 00:45:10,490
Why I said correlation length
shortly becomes apparent.

636
00:45:10,490 --> 00:45:16,825
So I introduce a function I
d which depends on x and c

637
00:45:16,825 --> 00:45:27,900
to be minus integral over q 2 pi
to the d Fourier transform of q

638
00:45:27,900 --> 00:45:30,730
squared plus c to the minus 2.

639
00:45:33,310 --> 00:45:37,270
If that c was not there
that's the integral that we

640
00:45:37,270 --> 00:45:39,890
did last time and
it was the Coulomb

641
00:45:39,890 --> 00:45:42,790
potential in d dimension.

642
00:45:42,790 --> 00:45:46,970
So this presumably
is related to that.

643
00:45:46,970 --> 00:45:51,720
And we can use the same
trick that we employed

644
00:45:51,720 --> 00:45:56,230
to make it explicit
last time around.

645
00:45:56,230 --> 00:46:03,040
We can take the Laplacian of
this potential i d and what

646
00:46:03,040 --> 00:46:06,910
happens is that I will
bring two factors of i

647
00:46:06,910 --> 00:46:10,370
q squared so the
minus goes away.

648
00:46:10,370 --> 00:46:15,210
I will have the integral
d d cubed 2 pi to the d.

649
00:46:15,210 --> 00:46:19,310
I will have a q
squared, denominator

650
00:46:19,310 --> 00:46:22,660
is q squared plus
c to the minus 2.

651
00:46:22,660 --> 00:46:28,420
I add and subtract the c to
the minus 2 to the numerator

652
00:46:28,420 --> 00:46:30,652
and I have to Fourier transform.

653
00:46:34,620 --> 00:46:37,600
OK?

654
00:46:37,600 --> 00:46:44,370
The first part if I divide by
the denominator is simply 1.

655
00:46:44,370 --> 00:46:46,370
Integral of Fourier
transform of 1

656
00:46:46,370 --> 00:46:48,038
will give me a delta function.

657
00:46:52,920 --> 00:46:58,470
And then what I have is minus
c squared, the same integral

658
00:46:58,470 --> 00:47:03,576
that I used to define i d.

659
00:47:03,576 --> 00:47:12,960
So this becomes plus i d
of x divided by c squared.

660
00:47:12,960 --> 00:47:15,430
OK?

661
00:47:15,430 --> 00:47:21,010
So whereas in the
absence of c you have

662
00:47:21,010 --> 00:47:27,934
the potential do to a
charge, the presence of c

663
00:47:27,934 --> 00:47:32,080
adds this additional
term that corresponds

664
00:47:32,080 --> 00:47:34,390
to some kind of a damping.

665
00:47:34,390 --> 00:47:37,560
So this equation
you probably have

666
00:47:37,560 --> 00:47:42,120
seen in the context of
screened Coulomb interaction

667
00:47:42,120 --> 00:47:44,700
and giving rise to the
[INAUDIBLE] potential

668
00:47:44,700 --> 00:47:46,780
in three dimensions.

669
00:47:46,780 --> 00:47:50,240
We would like to look
at it in d dimension

670
00:47:50,240 --> 00:47:54,180
so that we know what the
behavior is in general.

671
00:47:54,180 --> 00:47:54,680
OK?

672
00:48:01,400 --> 00:48:11,660
So again what I'm looking
at is the potential

673
00:48:11,660 --> 00:48:16,420
that is due to a
charge at the origin

674
00:48:16,420 --> 00:48:20,620
so this idea of x
in principle only

675
00:48:20,620 --> 00:48:26,850
depends on the magnitude of x
and not on the direction of it.

676
00:48:26,850 --> 00:48:30,870
It is something that has
general spherical symmetry

677
00:48:30,870 --> 00:48:31,825
in d dimensions.

678
00:48:34,580 --> 00:48:44,660
So I use that fact
of spherical symmetry

679
00:48:44,660 --> 00:48:48,600
to write down what the
expression for the Laplacian

680
00:48:48,600 --> 00:48:49,980
is.

681
00:48:49,980 --> 00:48:52,360
OK?

682
00:48:52,360 --> 00:48:54,740
We can again use Gauss's
law if you've forgotten

683
00:48:54,740 --> 00:48:58,580
or whatever but in the
presence of spherical symmetry

684
00:48:58,580 --> 00:49:08,000
the general expression for
Laplacian in d dimension

685
00:49:08,000 --> 00:49:11,444
is this.

686
00:49:11,444 --> 00:49:12,430
OK?

687
00:49:12,430 --> 00:49:16,330
So if this d was
equal to 1 it would

688
00:49:16,330 --> 00:49:18,820
be a simple second derivative.

689
00:49:18,820 --> 00:49:23,350
And in higher dimensions you
would have additional factors

690
00:49:23,350 --> 00:49:28,300
if you basically apply Gauss's
law to shares around here.

691
00:49:28,300 --> 00:49:31,340
You can very easily
convince yourself of that.

692
00:49:31,340 --> 00:49:33,730
This is some kind
of an aerial term

693
00:49:33,730 --> 00:49:37,390
that comes in d
minus 1 dimension.

694
00:49:37,390 --> 00:49:43,980
And then I can write this as
either the second derivative

695
00:49:43,980 --> 00:49:47,845
if d by the x acts on
this, the x to the minus 1

696
00:49:47,845 --> 00:49:51,830
disappears or if it
acts on this one,

697
00:49:51,830 --> 00:49:54,730
it will gives me d
minus 1 x to the d

698
00:49:54,730 --> 00:50:02,130
minus 2, x to the d minus 1
gives me an x, the i by d x.

699
00:50:02,130 --> 00:50:05,460
So the equation
that I have to solve

700
00:50:05,460 --> 00:50:13,129
is this object equals to i over
c squared plus a delta function

701
00:50:13,129 --> 00:50:13,629
[INAUDIBLE].

702
00:50:18,398 --> 00:50:18,898
OK?

703
00:50:24,180 --> 00:50:29,010
Now if you vary one
dimension you wouldn't

704
00:50:29,010 --> 00:50:33,310
have this term at all and you
would have-- except that x

705
00:50:33,310 --> 00:50:36,880
equals to 0, the second
derivative proportional

706
00:50:36,880 --> 00:50:40,040
to the function
divided by c squared.

707
00:50:40,040 --> 00:50:41,690
So you will
immediately write a way

708
00:50:41,690 --> 00:50:48,060
x equals to 0 that the answer
is e to the minus x over c.

709
00:50:48,060 --> 00:50:50,290
OK?

710
00:50:50,290 --> 00:50:51,880
Actually proportional
because you

711
00:50:51,880 --> 00:50:54,380
have to fit out with the
amplitude, et cetera.

712
00:50:57,630 --> 00:51:03,650
Now in higher
dimensions what happens

713
00:51:03,650 --> 00:51:06,590
is that this solution
gets modified,

714
00:51:06,590 --> 00:51:11,430
falls off with some
additional x to the p

715
00:51:11,430 --> 00:51:13,920
but we have to be somewhat
careful with this.

716
00:51:13,920 --> 00:51:19,210
So let's look at this a
little bit more closely.

717
00:51:19,210 --> 00:51:24,810
If I were to substitute this
ansatz into this expression,

718
00:51:24,810 --> 00:51:26,980
what would happen?

719
00:51:26,980 --> 00:51:30,360
What I need to do is to take
the first and the second

720
00:51:30,360 --> 00:51:30,860
derivative.

721
00:51:35,800 --> 00:51:39,880
Now if I take the
first derivative,

722
00:51:39,880 --> 00:51:42,940
the derivative either
acts on this factor,

723
00:51:42,940 --> 00:51:46,490
gives me a factor of
minus 1 over c and then

724
00:51:46,490 --> 00:51:50,720
the exponential back,
so I can get the i back.

725
00:51:50,720 --> 00:51:53,100
If I had an exponential
I take a derivative,

726
00:51:53,100 --> 00:51:56,870
I will get just minus
1 over psi exponential.

727
00:51:56,870 --> 00:52:02,690
If I act on x to the
minus b I will get minus p

728
00:52:02,690 --> 00:52:06,870
x to the p minus 1, which is
different from the original

729
00:52:06,870 --> 00:52:10,565
solution by a
factor of p over x.

730
00:52:10,565 --> 00:52:13,547
OK?

731
00:52:13,547 --> 00:52:22,600
If I now take two derivatives I
can take the second derivative

732
00:52:22,600 --> 00:52:26,380
on I itself and then d
I by d x will give me I

733
00:52:26,380 --> 00:52:29,420
back with this factor.

734
00:52:29,420 --> 00:52:38,390
So I will get 1 over c squared
plus 2 P c x plus P squared

735
00:52:38,390 --> 00:52:44,115
over x squared with
I but that's not

736
00:52:44,115 --> 00:52:48,125
the whole story because the
derivative can also leave I

737
00:52:48,125 --> 00:52:52,420
aside and act on P over
x, which if it does so,

738
00:52:52,420 --> 00:52:55,500
it will get P over
x squared so that

739
00:52:55,500 --> 00:52:58,960
will be an additional term here.

740
00:52:58,960 --> 00:53:02,340
So that's the second derivative.

741
00:53:02,340 --> 00:53:09,120
So now what I have done
is I have evaluated

742
00:53:09,120 --> 00:53:12,870
with this ansatz the
terms that should appear

743
00:53:12,870 --> 00:53:16,390
in that equation of a from
x equals to 0, so let's

744
00:53:16,390 --> 00:53:17,740
substitute it.

745
00:53:17,740 --> 00:53:22,010
Everything now I have is
proportional to I so I just

746
00:53:22,010 --> 00:53:23,690
forget about the I.

747
00:53:23,690 --> 00:53:26,450
I have the second
derivative 1 over c

748
00:53:26,450 --> 00:53:31,960
squared plus 2 P
divided by x c plus p p

749
00:53:31,960 --> 00:53:37,130
plus 1 divided by x squared.

750
00:53:37,130 --> 00:53:40,300
And then I have d minus
1, the first derivative,

751
00:53:40,300 --> 00:53:50,720
so I have minus d minus 1 over
c minus d minus 1 p over x.

752
00:53:50,720 --> 00:53:53,160
Both of these terms get
an additional factor

753
00:53:53,160 --> 00:53:58,650
of x because of here so I
will get x c and x squared

754
00:53:58,650 --> 00:54:03,050
and what I have on the right
hand side from the origin

755
00:54:03,050 --> 00:54:04,710
is I over c squared.

756
00:54:04,710 --> 00:54:07,575
Divide by the I, I
have 1 over c squared.

757
00:54:12,430 --> 00:54:16,930
Now if I'm moving away
from x I can organize

758
00:54:16,930 --> 00:54:20,080
things in powers of 1 over x.

759
00:54:20,080 --> 00:54:23,110
The most important
term is the constant

760
00:54:23,110 --> 00:54:27,740
and clearly you can see
that I chose the decay

761
00:54:27,740 --> 00:54:31,920
constant of the
exponential correctly as

762
00:54:31,920 --> 00:54:36,590
evidenced by the
absence or removal of 1

763
00:54:36,590 --> 00:54:41,210
over the constant
term on the two sides.

764
00:54:41,210 --> 00:54:45,040
But now I have two terms,
two types of terms.

765
00:54:45,040 --> 00:54:50,490
Terms that are proportional
to x squared and terms

766
00:54:50,490 --> 00:54:54,650
that are proportional
to x psi and there's

767
00:54:54,650 --> 00:55:01,860
no way that I can simultaneously
satisfy both of these.

768
00:55:01,860 --> 00:55:07,170
So the assumption that the
solution of this equation

769
00:55:07,170 --> 00:55:10,606
is a single exponential
divided by a power law

770
00:55:10,606 --> 00:55:13,950
is in fact not correct.

771
00:55:13,950 --> 00:55:17,910
But it can be correct
in two regions.

772
00:55:17,910 --> 00:55:26,980
So for x that is much
less than psi then

773
00:55:26,980 --> 00:55:35,130
the more important term is
the 1 over x squared part.

774
00:55:35,130 --> 00:55:38,540
For x but going towards
0 the 1 over x squared

775
00:55:38,540 --> 00:55:41,560
is more important than 1 over x.

776
00:55:41,560 --> 00:55:44,560
OK so then what I do is
I will match these two

777
00:55:44,560 --> 00:55:49,300
terms and those two terms that
are 1 over x squared tell me

778
00:55:49,300 --> 00:55:59,435
that P p plus 1
should be P d minus 1.

779
00:55:59,435 --> 00:56:01,770
OK?

780
00:56:01,770 --> 00:56:06,340
And that immediately getting
rid of the p's tells me

781
00:56:06,340 --> 00:56:13,485
that the P in this
regime is d minus 2.

782
00:56:13,485 --> 00:56:14,762
OK?

783
00:56:14,762 --> 00:56:18,320
Now the d minus 2
you recall is what

784
00:56:18,320 --> 00:56:21,723
we had for the
Coulomb potential.

785
00:56:21,723 --> 00:56:22,670
Right?

786
00:56:22,670 --> 00:56:27,080
So basically at
short distances you

787
00:56:27,080 --> 00:56:31,560
are still not screened
by this additional term.

788
00:56:31,560 --> 00:56:36,490
You don't see its effect
and you get essentially

789
00:56:36,490 --> 00:56:45,930
the standard Coulomb potential
whereas if you are away

790
00:56:45,930 --> 00:56:50,520
what you get is that you
have to match the terms that

791
00:56:50,520 --> 00:56:53,900
are proportional to x c because
they're more important than 1

792
00:56:53,900 --> 00:56:55,300
over x squared.

793
00:56:55,300 --> 00:56:59,482
And there you get that
2 P should be d minus 1

794
00:56:59,482 --> 00:57:02,990
or P should be d minus 1 over 2.

795
00:57:06,448 --> 00:57:09,420
OK?

796
00:57:09,420 --> 00:57:15,800
So let's just plot that
function over here.

797
00:57:15,800 --> 00:57:23,260
So if I plot this function as
a function of the separation x

798
00:57:23,260 --> 00:57:26,630
and it only depends
on the magnitude,

799
00:57:26,630 --> 00:57:29,790
in fact, what I
should plot is minus i

800
00:57:29,790 --> 00:57:32,170
d because it's
the minus i d that

801
00:57:32,170 --> 00:57:37,900
depends on the
fluctuations once I divide

802
00:57:37,900 --> 00:57:42,250
by K. I find that
it has two regimes.

803
00:57:42,250 --> 00:57:47,530
Let's say above two dimensions
you have one regime that

804
00:57:47,530 --> 00:57:51,040
is a simple Coulomb
type of potential

805
00:57:51,040 --> 00:57:52,950
and the Coulomb
potential last time

806
00:57:52,950 --> 00:57:55,260
actually we normalized properly.

807
00:57:55,260 --> 00:58:03,600
We saw that it is x to the
2 minus d S d d minus 2.

808
00:58:08,550 --> 00:58:11,160
The e to x into the
minus x over c I

809
00:58:11,160 --> 00:58:12,910
can in fact ignore
in this regime

810
00:58:12,910 --> 00:58:15,870
because I'm at distance x
that is much less than c

811
00:58:15,870 --> 00:58:20,010
so the exponential term
has not kicked in yet

812
00:58:20,010 --> 00:58:25,950
whereas I go at large distances
and the exponential term does

813
00:58:25,950 --> 00:58:27,360
kick in.

814
00:58:27,360 --> 00:58:32,230
So the overall behavior is
e to the minus x over c.

815
00:58:32,230 --> 00:58:36,330
That's the most dominant
behavior that you have.

816
00:58:36,330 --> 00:58:38,570
On top of that, we
have a power log

817
00:58:38,570 --> 00:58:42,857
which is x to the power
of d minus 1 over 2.

818
00:58:45,779 --> 00:58:53,350
Now those of you who know
what the screened Coulomb

819
00:58:53,350 --> 00:58:58,090
potential is know that the
screened Coulomb potential

820
00:58:58,090 --> 00:59:02,320
in three dimensions is the 1
over r, the Coulomb potential,

821
00:59:02,320 --> 00:59:05,070
and you put an exponential
on top of that.

822
00:59:05,070 --> 00:59:07,030
There is no difference
in the powers

823
00:59:07,030 --> 00:59:09,270
that you have whether
or not you are

824
00:59:09,270 --> 00:59:12,575
smaller than this
correlation length or larger.

825
00:59:12,575 --> 00:59:16,570
You can check here, if
I put d equals to 3,

826
00:59:16,570 --> 00:59:21,350
this becomes a 1 over x and
this becomes a 1 over x.

827
00:59:21,350 --> 00:59:24,020
So it's just an accident
of three dimensions

828
00:59:24,020 --> 00:59:26,220
that the screened
Coulomb potential

829
00:59:26,220 --> 00:59:29,470
is the 1 over r with
an exponential on top.

830
00:59:29,470 --> 00:59:35,160
In general dimensions you
have different powers.

831
00:59:35,160 --> 00:59:37,270
But having different
powers also means

832
00:59:37,270 --> 00:59:41,230
that somehow the
amplitude that goes over

833
00:59:41,230 --> 00:59:47,130
here has to carry dimensions so
that it can be matched to what

834
00:59:47,130 --> 00:59:50,910
we have here at
this distance of C.

835
00:59:50,910 --> 00:59:54,570
And so if I try to match
those terms, roughly

836
00:59:54,570 --> 00:59:58,000
when you're at order of c,
what I would do is I would put

837
00:59:58,000 --> 01:00:04,576
s d d minus 2 and c to
the 3 minus d over 2.

838
01:00:04,576 --> 01:00:09,610
And now you can check
that the two expressions

839
01:00:09,610 --> 01:00:12,840
will have the right dimension
and will match roughly

840
01:00:12,840 --> 01:00:16,310
at order of c.

841
01:00:16,310 --> 01:00:18,340
OK?

842
01:00:18,340 --> 01:00:29,170
So essentially what it says is
that if I ask in my system what

843
01:00:29,170 --> 01:00:31,810
is the nature of
these fluctuations,

844
01:00:31,810 --> 01:00:35,180
how correlated they
are, they would

845
01:00:35,180 --> 01:00:40,830
know to be more or less the same
although falling off as if you

846
01:00:40,830 --> 01:00:42,700
were at the critical
point because we said

847
01:00:42,700 --> 01:00:46,160
that the critical point or
when you have Goldstone modes,

848
01:00:46,160 --> 01:00:49,920
you have just this term.

849
01:00:49,920 --> 01:00:52,380
But then they know that
you are not exactly

850
01:00:52,380 --> 01:00:56,640
sitting at the
critical point and then

851
01:00:56,640 --> 01:00:59,850
they are no longer correlated.

852
01:00:59,850 --> 01:01:03,400
So basically there
is this length scale

853
01:01:03,400 --> 01:01:05,060
that we also saw
when we were looking

854
01:01:05,060 --> 01:01:07,720
at these critical
opalescence and we

855
01:01:07,720 --> 01:01:11,380
were seeing things that
were moving together.

856
01:01:11,380 --> 01:01:14,780
That length scale where
things are moving together

857
01:01:14,780 --> 01:01:19,240
is this parameter c that
we have defined over here.

858
01:01:19,240 --> 01:01:27,420
So what we have is-- where
do we want to put it?

859
01:01:27,420 --> 01:01:28,330
Let's put it here.

860
01:01:45,610 --> 01:01:55,800
A correlation length
which measures

861
01:01:55,800 --> 01:02:01,330
the extent to which things
are fluctuating together,

862
01:02:01,330 --> 01:02:03,920
although when I'm saying
fluctuating together,

863
01:02:03,920 --> 01:02:07,990
they are still correlations
that are falling off

864
01:02:07,990 --> 01:02:10,630
but they're not falling
off exponentially.

865
01:02:10,630 --> 01:02:12,800
They start to fall
off exponentially

866
01:02:12,800 --> 01:02:16,830
when you are beyond
this length scale c.

867
01:02:16,830 --> 01:02:20,670
And we have the formula for c.

868
01:02:20,670 --> 01:02:23,210
So what we find
is that if I were

869
01:02:23,210 --> 01:02:29,450
to invert that, for
example, what I find for c l

870
01:02:29,450 --> 01:02:44,530
as a function of t is that it is
simply square root of k over t

871
01:02:44,530 --> 01:02:48,320
when I am on the
t positive side.

872
01:02:48,320 --> 01:02:50,890
When I go to the
t negative side,

873
01:02:50,890 --> 01:03:02,296
it just becomes square
root of K minus 2 t.

874
01:03:02,296 --> 01:03:02,796
OK?

875
01:03:05,670 --> 01:03:14,480
So this correlation length
I indicated we could state

876
01:03:14,480 --> 01:03:17,980
has behavior close
to a transition,

877
01:03:17,980 --> 01:03:20,310
there's a divergence.

878
01:03:20,310 --> 01:03:25,190
We can parametrize those
divergences through something

879
01:03:25,190 --> 01:03:32,510
like t minus Tc to exponent
u, potentially different

880
01:03:32,510 --> 01:03:35,530
on the two sides
of the transition.

881
01:03:35,530 --> 01:03:38,630
But this t is simply
proportional to the real t

882
01:03:38,630 --> 01:03:43,520
minus Tc so we
conclude that u plus is

883
01:03:43,520 --> 01:03:47,190
the same as u minus we've
just indicated by u,

884
01:03:47,190 --> 01:03:48,310
should be one-half.

885
01:03:52,130 --> 01:03:56,630
The amplitudes themselves
depend on all kinds of things.

886
01:03:56,630 --> 01:03:58,840
We don't know much about them.

887
01:03:58,840 --> 01:04:01,850
But we can see that
the amplitude ratio B

888
01:04:01,850 --> 01:04:09,220
plus over B minus, if I
were to divide those two

889
01:04:09,220 --> 01:04:13,130
the ratio of those
two is universal,

890
01:04:13,130 --> 01:04:15,940
it gives me a factor
of square root of 2.

891
01:04:21,590 --> 01:04:23,760
OK?

892
01:04:23,760 --> 01:04:29,810
If I were to plot
c t, for example,

893
01:04:29,810 --> 01:04:36,340
on the high temperature side c t
and c l are of course the same.

894
01:04:36,340 --> 01:04:40,040
On the low temperature side, we
said that the Goldstone modes

895
01:04:40,040 --> 01:04:42,170
have these long
range correlations.

896
01:04:42,170 --> 01:04:46,260
They fall off or grow according
to the Coulomb potential

897
01:04:46,260 --> 01:04:49,280
but there is no length
scale so in some sense

898
01:04:49,280 --> 01:04:52,119
the correlation length
for the transverse modes

899
01:04:52,119 --> 01:04:52,910
is always infinity.

900
01:04:58,530 --> 01:04:59,030
OK.

901
01:05:13,430 --> 01:05:17,550
Now actually in the
second lecture what I said

902
01:05:17,550 --> 01:05:21,800
was that the fact
that the response

903
01:05:21,800 --> 01:05:25,660
function such as
susceptibility diverges

904
01:05:25,660 --> 01:05:29,260
immediately tells you
that there have to be long

905
01:05:29,260 --> 01:05:31,470
range correlations,
so we had predicted

906
01:05:31,470 --> 01:05:35,320
before that c has to diverge.

907
01:05:35,320 --> 01:05:40,530
But we were not sufficiently
precise about the way

908
01:05:40,530 --> 01:05:44,471
that it does, so
let's try to do that.

909
01:05:44,471 --> 01:05:44,970
Let's see.

910
01:05:44,970 --> 01:05:50,850
A relationship more
precisely with susceptibility

911
01:05:50,850 --> 01:05:55,900
and these correlation
lengths, so what we said more

912
01:05:55,900 --> 01:06:00,560
generally was that the
susceptibilities up

913
01:06:00,560 --> 01:06:03,540
to various factors of data,
et cetera, that are not

914
01:06:03,540 --> 01:06:11,330
that important are related to
the integrated magnetization

915
01:06:11,330 --> 01:06:15,130
to magnetization
connected correlation.

916
01:06:15,130 --> 01:06:17,505
So basically, what
I have to do is

917
01:06:17,505 --> 01:06:22,102
to look at m minus
its average at x,

918
01:06:22,102 --> 01:06:25,400
m minus its average at
some other point, which

919
01:06:25,400 --> 01:06:27,370
means that what I'm
really looking at

920
01:06:27,370 --> 01:06:29,387
is the phi phi averages.

921
01:06:34,229 --> 01:06:34,728
OK?

922
01:06:37,870 --> 01:06:41,850
Now what we have
shown right now is

923
01:06:41,850 --> 01:06:47,236
that these averages
are significant.

924
01:06:50,380 --> 01:06:54,480
These phi phi correlations
are significant

925
01:06:54,480 --> 01:06:59,720
only over a distance that
is this correlation length

926
01:06:59,720 --> 01:07:02,010
and then they die off.

927
01:07:02,010 --> 01:07:06,450
So we could basically as
far as scaling and things

928
01:07:06,450 --> 01:07:11,520
like that is concerned
terminate this integration at c.

929
01:07:11,520 --> 01:07:18,350
And that when we are looking at
distances that are below that,

930
01:07:18,350 --> 01:07:20,560
you don't see the effect
of the exponential,

931
01:07:20,560 --> 01:07:24,170
you just see the
Coulomb power law

932
01:07:24,170 --> 01:07:28,060
so you would see
here fluctuations

933
01:07:28,060 --> 01:07:35,850
that decay as x
to the 2 minus d.

934
01:07:35,850 --> 01:07:37,850
Right?

935
01:07:37,850 --> 01:07:43,880
So essentially what you're doing
is integrating x to the 2 minus

936
01:07:43,880 --> 01:07:46,790
d in d dimension of space.

937
01:07:46,790 --> 01:07:51,590
So you can see that
immediately gets related

938
01:07:51,590 --> 01:07:55,050
to the square of the
correlation length.

939
01:07:55,050 --> 01:07:59,910
X to the minus d n d d
x, the d part vanishes,

940
01:07:59,910 --> 01:08:02,970
there's a 2 that remains
and gives you c squared.

941
01:08:02,970 --> 01:08:05,790
If you like you can write
it in spherical coordinates,

942
01:08:05,790 --> 01:08:08,620
et cetera, but dimensions
have to work out

943
01:08:08,620 --> 01:08:10,640
to be something like this.

944
01:08:10,640 --> 01:08:12,240
So now we can-- yes?

945
01:08:12,240 --> 01:08:15,350
AUDIENCE: Just to clarify, when
you sat phi of x and phi of 0,

946
01:08:15,350 --> 01:08:19,740
are those both longitudinal
or both transverse?

947
01:08:19,740 --> 01:08:25,920
PROFESSOR: I wasn't precise
so if I, thinking about chi l,

948
01:08:25,920 --> 01:08:28,399
these will be both longitudinal.

949
01:08:28,399 --> 01:08:29,016
OK?

950
01:08:29,016 --> 01:08:31,859
And then we have
this expression.

951
01:08:31,859 --> 01:08:37,649
If I'm talking about chi t
and I'm above the transition

952
01:08:37,649 --> 01:08:40,340
temperature, there's no problem.

953
01:08:40,340 --> 01:08:42,460
If I'm below the
transition temperature,

954
01:08:42,460 --> 01:08:44,500
I can use the same
thing but have

955
01:08:44,500 --> 01:08:47,420
to set c to infinity
so I have to integrate

956
01:08:47,420 --> 01:08:49,047
all the way to infinity.

957
01:08:52,719 --> 01:08:54,076
OK?

958
01:08:54,076 --> 01:08:59,220
But now you can see that the
divergence of susceptibility

959
01:08:59,220 --> 01:09:03,760
is very much related to the
divergence of correlations,

960
01:09:03,760 --> 01:09:07,319
in some sense very
precisely in that

961
01:09:07,319 --> 01:09:13,180
if this goes like t
to the minus gamma

962
01:09:13,180 --> 01:09:19,630
and the correlation length
diverges as t to the minus u,

963
01:09:19,630 --> 01:09:22,076
then gamma should be 2 nu.

964
01:09:22,076 --> 01:09:24,859
And indeed, our ne is one-half.

965
01:09:24,859 --> 01:09:28,700
We had seen previously
that gamma was 2 nu.

966
01:09:28,700 --> 01:09:33,359
Secondly that the amplitude
ratio for susceptibility

967
01:09:33,359 --> 01:09:36,029
should be the square
of the amplitude ratio

968
01:09:36,029 --> 01:09:38,880
for the correlation
length and again this

969
01:09:38,880 --> 01:09:40,990
is something that
we have seen before.

970
01:09:40,990 --> 01:09:43,920
The amplitude ratio
for susceptibility

971
01:09:43,920 --> 01:09:46,288
was the square root of 2.

972
01:09:49,090 --> 01:09:50,840
Now it turns out
that all of this

973
01:09:50,840 --> 01:09:55,300
is a gain within this
[INAUDIBLE] point approximation

974
01:09:55,300 --> 01:09:58,720
looking at the most
probable state, et cetera.

975
01:09:58,720 --> 01:10:07,220
Because what we find in reality
is that at the critical point,

976
01:10:07,220 --> 01:10:11,390
the correlations don't decay
simply according to the Coulomb

977
01:10:11,390 --> 01:10:16,560
law but there is
this additional eta

978
01:10:16,560 --> 01:10:21,820
which is the same eta
that we had over here.

979
01:10:21,820 --> 01:10:23,100
OK?

980
01:10:23,100 --> 01:10:27,360
And that because of that
eta, here what you would have

981
01:10:27,360 --> 01:10:32,380
is 2 minus eta and you
would get an example

982
01:10:32,380 --> 01:10:36,440
of a number of things that
we will see a lot later on.

983
01:10:36,440 --> 01:10:40,530
That is there even if you don't
know what the exponents are,

984
01:10:40,530 --> 01:10:42,650
you know that there
are relationships

985
01:10:42,650 --> 01:10:44,020
among the exponents.

986
01:10:44,020 --> 01:10:47,585
This is an example of an
exponent identity called

987
01:10:47,585 --> 01:10:49,550
a Fisher exponent
and there are several

988
01:10:49,550 --> 01:10:51,236
of these exponents identities.

989
01:10:54,117 --> 01:10:54,617
OK?

990
01:10:58,490 --> 01:11:05,010
But that also brings
us to the following-

991
01:11:05,010 --> 01:11:11,550
that we did all of this
work and we came up

992
01:11:11,550 --> 01:11:23,990
with answers for the singular
behaviors at critical points

993
01:11:23,990 --> 01:11:27,650
and why they are universal.

994
01:11:27,650 --> 01:11:31,410
And actually as far as the
thermodynamic quantities

995
01:11:31,410 --> 01:11:34,850
were concerned, all
we ended up doing

996
01:11:34,850 --> 01:11:38,990
was to write some expression
that was analytical

997
01:11:38,990 --> 01:11:40,880
and then find its minimum.

998
01:11:40,880 --> 01:11:46,200
And we found that the minimum
of an analytical expression

999
01:11:46,200 --> 01:11:50,530
always has the same type
of singularities, which

1000
01:11:50,530 --> 01:11:53,530
we can characterize
by these exponents.

1001
01:11:53,530 --> 01:11:56,600
So maybe it's now a
good time to check

1002
01:11:56,600 --> 01:12:00,090
how these match
with the experiment.

1003
01:12:00,090 --> 01:12:11,500
So let's look at the various
types of phase transition,

1004
01:12:11,500 --> 01:12:16,540
an example of the material
that undergoes that phase

1005
01:12:16,540 --> 01:12:24,830
transition, and what the
exponents alpha, beta, gamma,

1006
01:12:24,830 --> 01:12:28,680
and u are that are
experimentally obtained.

1007
01:12:37,974 --> 01:12:40,920
AUDIENCE: What is this again?

1008
01:12:40,920 --> 01:12:43,660
PROFESSOR: The material
that undergoes a transition,

1009
01:12:43,660 --> 01:12:51,950
so for example when we are
talking about the ferromagnet

1010
01:12:51,950 --> 01:12:58,070
to paramagnet transition,
you could look at material

1011
01:12:58,070 --> 01:13:05,870
such as iron or
nickel and if we ask

1012
01:13:05,870 --> 01:13:09,620
in the context of
this systematics

1013
01:13:09,620 --> 01:13:12,620
that we were developing for
the Landau-Ginzburg what they

1014
01:13:12,620 --> 01:13:17,930
correspond to, they are things
that have three components

1015
01:13:17,930 --> 01:13:22,620
or fields so they
correspond to n equals to 3.

1016
01:13:22,620 --> 01:13:27,370
Of course everything that I will
be talking to in this column

1017
01:13:27,370 --> 01:13:29,800
will correspond to
3-dimensional systems.

1018
01:13:29,800 --> 01:13:32,930
Later we'll talk also about
2-dimensional and other

1019
01:13:32,930 --> 01:13:39,060
systems, but let's stick with
real 3-dimensional world.

1020
01:13:39,060 --> 01:13:41,360
So that would be one set.

1021
01:13:41,360 --> 01:13:43,543
We will look at super fluidity.

1022
01:13:47,170 --> 01:13:53,130
Let's say in helium, which
we discussed last semester,

1023
01:13:53,130 --> 01:13:55,500
that corresponds
to n equals to 2.

1024
01:13:58,030 --> 01:14:04,310
We will talk about various
examples of liquid gas

1025
01:14:04,310 --> 01:14:17,020
transition which correspond to
a scalar density difference.

1026
01:14:17,020 --> 01:14:23,415
And this could be anything
from say carbon dioxide, neon,

1027
01:14:23,415 --> 01:14:26,570
argon, whatever gas we like.

1028
01:14:26,570 --> 01:14:41,310
And also talk about
superconductors

1029
01:14:41,310 --> 01:14:43,840
which to all
intents and purposes

1030
01:14:43,840 --> 01:14:49,680
should have the same type of
symmetries as super fluids.

1031
01:14:49,680 --> 01:14:53,440
An example of a quantum system
should be n equals to 2,

1032
01:14:53,440 --> 01:14:57,692
gained lots of different cases
such as aluminum, copper,

1033
01:14:57,692 --> 01:14:58,192
whatever.

1034
01:15:01,110 --> 01:15:05,280
So what do we find
for the exponent?

1035
01:15:05,280 --> 01:15:07,270
Actually for
ferromagnetic system

1036
01:15:07,270 --> 01:15:09,510
the heat capacity
does not diverge.

1037
01:15:09,510 --> 01:15:14,300
It has a discontinuous
derivative at the transition

1038
01:15:14,300 --> 01:15:18,340
and kind of goes in a manner
that if you take its derivative

1039
01:15:18,340 --> 01:15:21,470
then the derivative appears
to be singular and corresponds

1040
01:15:21,470 --> 01:15:22,440
to an alpha.

1041
01:15:22,440 --> 01:15:28,935
If you try to fit it to
it's slightly negative.

1042
01:15:28,935 --> 01:15:34,170
The superfluid has this famous
lambda shape for its heat

1043
01:15:34,170 --> 01:15:39,012
capacity and a lambda
shape is very well fitted

1044
01:15:39,012 --> 01:15:42,446
to a logarithm type of function.

1045
01:15:42,446 --> 01:15:46,590
The logarithm is the
limit of a power law

1046
01:15:46,590 --> 01:15:50,400
as the exponent goes to 0 so we
can more or less indicate that

1047
01:15:50,400 --> 01:15:55,460
by an alpha of 0 or really
it's a divergent log.

1048
01:15:58,560 --> 01:16:02,060
These objects, the
liquid gas transition

1049
01:16:02,060 --> 01:16:06,210
does have weakly
divergent heat capacity

1050
01:16:06,210 --> 01:16:11,110
so the alpha is around 0.1.

1051
01:16:11,110 --> 01:16:14,970
The values of betas are
all less than one-half,

1052
01:16:14,970 --> 01:16:21,350
for ferromagnet system
is of the order of 0.4.

1053
01:16:21,350 --> 01:16:29,380
It is almost one-third, slightly
less for superfluid helium

1054
01:16:29,380 --> 01:16:34,610
and less for the
liquid gas system.

1055
01:16:34,610 --> 01:16:39,990
Gammas something like 1.4.

1056
01:16:39,990 --> 01:16:42,040
We don't have a
gamma for superfluid,

1057
01:16:42,040 --> 01:16:44,730
you can't put a magnetic
field on the superfluid.

1058
01:16:44,730 --> 01:16:48,300
There's nothing that is
conjugate to the quantum phase.

1059
01:16:48,300 --> 01:16:51,680
Here it is more like 1.3.

1060
01:16:51,680 --> 01:16:59,498
Mu is 0.-- it's
not-- [INAUDIBLE].

1061
01:17:06,758 --> 01:17:08,210
OK.

1062
01:17:08,210 --> 01:17:18,081
So what I have here it is more
like 1.3, 1.24, 0.7, 0.67,

1063
01:17:18,081 --> 01:17:20,962
0.63.

1064
01:17:20,962 --> 01:17:21,462
OK?

1065
01:17:24,360 --> 01:17:30,260
Now they are different from
the predictions that we had.

1066
01:17:30,260 --> 01:17:36,520
Predictions that we had where
alpha was 0 discontinuous.

1067
01:17:36,520 --> 01:17:38,970
Beta goes to one-half.

1068
01:17:38,970 --> 01:17:43,920
Gamma was 1, mu
equals to one-half.

1069
01:17:43,920 --> 01:17:49,420
And actually these
predictions that we just made

1070
01:17:49,420 --> 01:17:52,120
happen to match extremely
well with all kinds

1071
01:17:52,120 --> 01:17:55,164
of super conducting
systems that you look at.

1072
01:17:58,120 --> 01:18:00,870
So again it is
important to state

1073
01:18:00,870 --> 01:18:04,405
that within a particular
class like liquid gas

1074
01:18:04,405 --> 01:18:07,430
you can do a lot of
different systems.

1075
01:18:07,430 --> 01:18:10,090
We saw that curve in
the second lecture.

1076
01:18:10,090 --> 01:18:13,470
They all correspond to
this same set of exponents,

1077
01:18:13,470 --> 01:18:18,110
singularly for a different
magnet and so forth.

1078
01:18:18,110 --> 01:18:21,150
So there is something
that is universal

1079
01:18:21,150 --> 01:18:24,950
but our Landau-Ginzburg
approach with this looking

1080
01:18:24,950 --> 01:18:27,660
at the most probable state
and fluctuations around

1081
01:18:27,660 --> 01:18:31,120
it has not captured
it for most cases

1082
01:18:31,120 --> 01:18:33,070
but for some reason
has captured it

1083
01:18:33,070 --> 01:18:35,760
for the case of superconductors.

1084
01:18:35,760 --> 01:18:40,410
So we have that puzzle and
starting from next lecture

1085
01:18:40,410 --> 01:18:43,160
we'll start to unravel that.