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PROFESSOR: We looked at
the macro state, where

9
00:00:26,620 --> 00:00:31,915
the volume of the box, the
temperature were given,

10
00:00:31,915 --> 00:00:35,620
as well as the
chemical potential.

11
00:00:35,620 --> 00:00:38,305
So rather than looking at a
fixed number of particles,

12
00:00:38,305 --> 00:00:41,800
we were looking at the
grand canonical prescription

13
00:00:41,800 --> 00:00:45,900
where the chemical
potential was specified.

14
00:00:45,900 --> 00:00:48,030
We said that when
we have a system

15
00:00:48,030 --> 00:00:53,218
of non-interacting
particles, we can construct

16
00:00:53,218 --> 00:01:00,900
a many bodied state based on
occupation of single particle

17
00:01:00,900 --> 00:01:07,984
states such that the action
of the Hamiltonian on this

18
00:01:07,984 --> 00:01:11,450
will give me a sum over
all single particle

19
00:01:11,450 --> 00:01:16,110
states, the occupation number
of that single particle

20
00:01:16,110 --> 00:01:25,330
state times the energy of that
single particle state, and then

21
00:01:25,330 --> 00:01:26,085
the state back.

22
00:01:31,160 --> 00:01:34,560
We said that in
quantum mechanics,

23
00:01:34,560 --> 00:01:40,210
we have to distinguish
between bosons and fermions,

24
00:01:40,210 --> 00:01:44,675
and in the occupation
number prescription.

25
00:01:44,675 --> 00:01:52,260
For the case of fermions,
corresponding theta v minus 1

26
00:01:52,260 --> 00:01:55,590
and k was 01.

27
00:01:55,590 --> 00:01:59,158
For the case of bosons,
corresponding theta

28
00:01:59,158 --> 00:02:02,116
plus 1 and k equals [INAUDIBLE].

29
00:02:09,511 --> 00:02:14,450
We said that within this
planned canonical prescription,

30
00:02:14,450 --> 00:02:23,945
these occupation numbers
were independently given.

31
00:02:23,945 --> 00:02:26,420
The probabilities
were independently

32
00:02:26,420 --> 00:02:31,283
distributed according
to exponential rules

33
00:02:31,283 --> 00:02:36,684
such that when we
calculated the average, nk.

34
00:02:36,684 --> 00:02:39,570
We commented on
the statistics what

35
00:02:39,570 --> 00:02:43,221
are the very simple forms,
which was 1 over [INAUDIBLE]

36
00:02:43,221 --> 00:02:49,820
into the beta
epsilon k minus eta,

37
00:02:49,820 --> 00:02:52,837
where z was constructed
from the temperature

38
00:02:52,837 --> 00:02:57,827
and the temporal potential as
[INAUDIBLE] to the beta mu.

39
00:03:00,821 --> 00:03:04,680
Now, for the case that
we are interested,

40
00:03:04,680 --> 00:03:10,491
for this gas that is
in a box of volume v,

41
00:03:10,491 --> 00:03:12,745
the one particle states
were characterized

42
00:03:12,745 --> 00:03:16,435
by [INAUDIBLE] waves
and their energies

43
00:03:16,435 --> 00:03:19,800
were f h bar squared k squared
over 2m, basically just

44
00:03:19,800 --> 00:03:22,140
the kinetic energy.

45
00:03:22,140 --> 00:03:25,750
And then at various
stages, we have

46
00:03:25,750 --> 00:03:31,360
to perform sums over k's,
and for a box of volume v,

47
00:03:31,360 --> 00:03:35,289
the sum over k was
replaced in the limit

48
00:03:35,289 --> 00:03:39,160
of a large box with an
integral over k times

49
00:03:39,160 --> 00:03:43,851
the density of states, which
was v divided by 2 pi cubed.

50
00:03:46,169 --> 00:03:47,710
And we said that
again, we recognized

51
00:03:47,710 --> 00:03:52,580
that quantum particles have
another characteristic, which

52
00:03:52,580 --> 00:03:57,330
is their spin, and there is a
degeneracy factor associated

53
00:03:57,330 --> 00:04:01,300
with that that we can
just stick over here

54
00:04:01,300 --> 00:04:03,920
as if we had [INAUDIBLE]
copies corresponding

55
00:04:03,920 --> 00:04:07,130
to the different
components of the spin.

56
00:04:10,000 --> 00:04:18,640
Now, once we put this form
of epsilon k over here,

57
00:04:18,640 --> 00:04:21,670
then we can calculate what
the mean number of particles

58
00:04:21,670 --> 00:04:24,310
is in the grand
canonical ensemble as sum

59
00:04:24,310 --> 00:04:30,920
over k, the average nk, given
the appropriate statistics.

60
00:04:30,920 --> 00:04:34,280
Of course, this is a
quantity that is extensive.

61
00:04:34,280 --> 00:04:37,450
We then construct a
density, which is intensive,

62
00:04:37,450 --> 00:04:41,870
simply by dividing by v, which
gets rid of that factor of v

63
00:04:41,870 --> 00:04:46,148
over here, and we found
that the answer could

64
00:04:46,148 --> 00:04:52,700
be written as g over
lambda cubed, again,

65
00:04:52,700 --> 00:04:57,380
coming from this g, lambda cubed
from the change of variables,

66
00:04:57,380 --> 00:05:02,354
where we defined lambda
and h over root 2 pi mkp.

67
00:05:06,620 --> 00:05:16,370
And then a function that
we indicated by s 3/2,

68
00:05:16,370 --> 00:05:18,441
depending on the
statistics of z.

69
00:05:21,600 --> 00:05:25,570
And this was a member of
our class of functions

70
00:05:25,570 --> 00:05:31,050
we defined in
general, f m h of z,

71
00:05:31,050 --> 00:05:36,535
to be 1 over m minus 1
factorial, integral 0

72
00:05:36,535 --> 00:05:41,880
to infinity, ex, x to
the m minus 1, z inverse,

73
00:05:41,880 --> 00:05:46,110
e to the x minus eta.

74
00:05:46,110 --> 00:05:50,755
So for each m, we
have a function of z.

75
00:05:50,755 --> 00:05:52,550
In fact, we have two
functions, depending

76
00:05:52,550 --> 00:05:55,510
on whether we are dealing
with the fermionic

77
00:05:55,510 --> 00:05:58,590
or the bosonic variety.

78
00:05:58,590 --> 00:06:01,830
We saw that these
functions we could expand

79
00:06:01,830 --> 00:06:08,953
for small z as z plus eta
z squared over 2 to the m.

80
00:06:08,953 --> 00:06:11,460
It's an alternating
series for fermions.

81
00:06:11,460 --> 00:06:16,802
All terms are positive for
bosons, zq, 3 to the m,

82
00:06:16,802 --> 00:06:21,194
eta z to the fourth, 4
to the m, and so forth.

83
00:06:25,100 --> 00:06:27,245
And we also noted
that these functions

84
00:06:27,245 --> 00:06:31,530
have a nice property
in that if you

85
00:06:31,530 --> 00:06:35,190
were to take a derivative with
respect to z and multiply by z,

86
00:06:35,190 --> 00:06:38,006
you get the function
with one lower index.

87
00:06:38,006 --> 00:06:40,247
So this is, in
fact, the same thing

88
00:06:40,247 --> 00:06:44,978
as z divided by zz of
f m plus 1 [INAUDIBLE].

89
00:07:29,465 --> 00:07:35,584
This just made the connection
between the number of particles

90
00:07:35,584 --> 00:07:37,000
and the chemical
potential that we

91
00:07:37,000 --> 00:07:41,660
need to use when we go to the
grand canonical prescription,

92
00:07:41,660 --> 00:07:44,840
but we are interested in
calculating various properties

93
00:07:44,840 --> 00:07:47,822
of this gas, such
as the pressure.

94
00:07:47,822 --> 00:07:50,880
We found that the
formula for the pressure

95
00:07:50,880 --> 00:07:55,465
was simply g over lambda
cubed, very similar to what

96
00:07:55,465 --> 00:08:00,420
we had over here, except that,
rather than having f 3/2,

97
00:08:00,420 --> 00:08:07,220
we had f 5/2 of z to
deal with, and if we

98
00:08:07,220 --> 00:08:13,050
were interested in energy,
we could simply use 3/2.

99
00:08:15,742 --> 00:08:20,795
Indeed, this was correct for
both fermions and bosons,

100
00:08:20,795 --> 00:08:25,770
irrespective of the
difficulties or the variations

101
00:08:25,770 --> 00:08:29,270
in pressure that you
would have, depending

102
00:08:29,270 --> 00:08:33,400
on your bosons or fermions
due to [INAUDIBLE] gases.

103
00:08:36,240 --> 00:08:39,285
So what is the task
here, if you are

104
00:08:39,285 --> 00:08:42,216
interested in getting
a formula for pressure

105
00:08:42,216 --> 00:08:45,370
or the energy or
the heat capacity?

106
00:08:45,370 --> 00:08:49,380
What should we do if we are
interested in a gas that

107
00:08:49,380 --> 00:08:52,970
has a fixed number of
particles or a fixed density?

108
00:08:52,970 --> 00:08:55,912
Clearly, the first
stage is to calculate

109
00:08:55,912 --> 00:08:59,800
z as a function of density.

110
00:08:59,800 --> 00:09:06,160
What we can do is to solve the
first equation graphically.

111
00:09:06,160 --> 00:09:14,510
We have to solve the
equation f 3/2 8 of z

112
00:09:14,510 --> 00:09:20,030
equals the combination
m lambda cubed over g,

113
00:09:20,030 --> 00:09:23,340
which we call the
degeneracy factor.

114
00:09:23,340 --> 00:09:25,270
So if I tell you what
the temperature is,

115
00:09:25,270 --> 00:09:26,700
you know lambda.

116
00:09:26,700 --> 00:09:29,135
If I tell you what
the density is,

117
00:09:29,135 --> 00:09:32,910
you know the combination
on the right hand side.

118
00:09:32,910 --> 00:09:34,110
There is this function.

119
00:09:34,110 --> 00:09:36,110
We have to plot this
function and find

120
00:09:36,110 --> 00:09:38,800
the value of the
argument of the function.

121
00:09:38,800 --> 00:09:42,850
That gives the value of
m lambda cubed over g

122
00:09:42,850 --> 00:09:46,200
as the value of the function.

123
00:09:46,200 --> 00:09:48,560
So graphically,
what we have to do

124
00:09:48,560 --> 00:09:55,290
is to plot as a function of
z these functions, f 3/2.

125
00:09:57,900 --> 00:10:03,592
I will generically
plot f m 8 of z,

126
00:10:03,592 --> 00:10:06,930
but for the case that
we are interested,

127
00:10:06,930 --> 00:10:09,890
we really need to
look at m equals 3/2.

128
00:10:12,630 --> 00:10:15,000
As we will see in
problem set, if you

129
00:10:15,000 --> 00:10:18,810
were to solve the
gas in d dimensions,

130
00:10:18,810 --> 00:10:22,780
this m rather than the
3/2 would be g over 2.

131
00:10:22,780 --> 00:10:25,800
So pictorially, the
same thing would

132
00:10:25,800 --> 00:10:28,510
work with slightly
different forms

133
00:10:28,510 --> 00:10:33,180
of these functions in
different dimensions.

134
00:10:33,180 --> 00:10:36,270
How do these
functions look like?

135
00:10:36,270 --> 00:10:43,090
Well, we see that initially,
they start to be linear,

136
00:10:43,090 --> 00:10:49,120
but then, the case of the bosons
and fermions, the next order

137
00:10:49,120 --> 00:10:52,020
term goes in
different directions.

138
00:10:52,020 --> 00:10:55,010
And in particular, for
the case of fermions,

139
00:10:55,010 --> 00:10:59,710
the quadratic term
starts to bring you down,

140
00:10:59,710 --> 00:11:03,820
whereas for the case of
bosons, the quadratic term

141
00:11:03,820 --> 00:11:09,640
is the opposite direction and
will tend to make you large.

142
00:11:09,640 --> 00:11:13,160
This is what these
functions look.

143
00:11:13,160 --> 00:11:17,280
For the case of fermions, I have
additional terms in the series.

144
00:11:17,280 --> 00:11:19,680
It's an alternating series.

145
00:11:19,680 --> 00:11:21,600
And then the question
is, what happens

146
00:11:21,600 --> 00:11:24,560
to this at large values of z?

147
00:11:24,560 --> 00:11:30,760
And what we found was
that at large values,

148
00:11:30,760 --> 00:11:34,670
this function satisfies
the asymptotic form

149
00:11:34,670 --> 00:11:41,690
that is provided by
Sommerfeld formula, which

150
00:11:41,690 --> 00:11:47,150
states that f m of z, for the
case of fermions, so this is

151
00:11:47,150 --> 00:11:51,790
eta equals minus 1,
is log z to the power

152
00:11:51,790 --> 00:11:54,895
of m divided by m factorial.

153
00:11:54,895 --> 00:11:58,890
Of course, we are
interested in m goes to 3/2.

154
00:11:58,890 --> 00:12:07,490
1 plus pi squared
over 6 m, m minus 1,

155
00:12:07,490 --> 00:12:14,980
log z squared, and higher order
inverse powers of log z, which

156
00:12:14,980 --> 00:12:18,390
one can compute, but
for our purposes,

157
00:12:18,390 --> 00:12:22,060
these two terms are sufficient.

158
00:12:22,060 --> 00:12:29,960
And if I'm interested in doing
the inversion for fermions,

159
00:12:29,960 --> 00:12:36,420
what I need to do is to
plot a line that corresponds

160
00:12:36,420 --> 00:12:40,680
to n lambda cubed
over g and find

161
00:12:40,680 --> 00:12:43,740
the intersection of
that line, and that

162
00:12:43,740 --> 00:12:46,870
would give me the value of z.

163
00:12:46,870 --> 00:12:49,415
When I'm very close
to the origin,

164
00:12:49,415 --> 00:12:52,310
I start with the
linear behavior,

165
00:12:52,310 --> 00:12:55,700
and then I can systematically
calculate the corrections

166
00:12:55,700 --> 00:12:59,850
to the linear behavior, which
would have been z equals n

167
00:12:59,850 --> 00:13:03,420
lambda cubed over g in higher
powers of n lambda cubed

168
00:13:03,420 --> 00:13:04,420
over g.

169
00:13:04,420 --> 00:13:07,620
And we saw that by doing
that, I can gradually

170
00:13:07,620 --> 00:13:10,030
construct a visual
expansion that

171
00:13:10,030 --> 00:13:14,670
is appropriate for these gases.

172
00:13:14,670 --> 00:13:17,790
I can do the same thing, of
course, for the case of bosons

173
00:13:17,790 --> 00:13:23,060
also, but now I'm interested
in the limiting form where

174
00:13:23,060 --> 00:13:26,510
the density goes higher and
higher or the temperature

175
00:13:26,510 --> 00:13:30,030
goes lower and lower so that
this horizontal line that I

176
00:13:30,030 --> 00:13:34,520
have drawn will go to
larger and larger values,

177
00:13:34,520 --> 00:13:37,300
and I'm interested in the
intersections that I have

178
00:13:37,300 --> 00:13:41,460
when z is large and this
asymptotic formula is

179
00:13:41,460 --> 00:13:43,760
being satisfied.

180
00:13:43,760 --> 00:13:50,410
So what do I need to do for the
case of degenerate fermions?

181
00:13:57,960 --> 00:14:03,850
I have this quantity, n lambda
cubed over g, which I claim

182
00:14:03,850 --> 00:14:09,130
is much larger than
1, so I have to look

183
00:14:09,130 --> 00:14:16,950
for values of the function, f
3/2 minus of z that are large,

184
00:14:16,950 --> 00:14:21,520
and those values are achieved
when the argument is large,

185
00:14:21,520 --> 00:14:27,520
and then it takes the form
log z to the 3/2 divided

186
00:14:27,520 --> 00:14:35,370
by 3/2 factorial, 1
plus pi squared over 6.

187
00:14:35,370 --> 00:14:40,160
m in this case is 3/2,
so I have 3/2 times m

188
00:14:40,160 --> 00:14:47,660
minus 1, which is 1/2,
divided by log z over 2.

189
00:14:47,660 --> 00:14:50,560
And then I will have
potentially higher order terms.

190
00:15:01,420 --> 00:15:10,890
So I can invert this
formula to find log

191
00:15:10,890 --> 00:15:15,730
z as a function of density.

192
00:15:15,730 --> 00:15:19,420
And to lowest order,
what I have is

193
00:15:19,420 --> 00:15:29,710
3/2 factorial n lambda cubed
over g raised to the 2/3 power.

194
00:15:29,710 --> 00:15:32,450
If I were to ignore
everything here,

195
00:15:32,450 --> 00:15:34,430
just arrange things
this way, that's

196
00:15:34,430 --> 00:15:38,000
the formula that I would get.

197
00:15:38,000 --> 00:15:42,980
But then I have this
correction, so I divide by this.

198
00:15:42,980 --> 00:15:47,770
I have 1 plus-- this combination
is equivalent to pi squared

199
00:15:47,770 --> 00:15:52,630
over 8, 1 over log z squared.

200
00:15:52,630 --> 00:15:56,370
I have taken it to the other
side, so I put the minus sign

201
00:15:56,370 --> 00:15:58,615
and raise again
to the 2/3 power.

202
00:16:06,380 --> 00:16:12,980
Now, the way that I have defined
z up there, clearly log z

203
00:16:12,980 --> 00:16:14,700
is proportional to beta nu.

204
00:16:18,644 --> 00:16:21,360
We can see that
lambda is proportional

205
00:16:21,360 --> 00:16:25,600
to beta to the 1/2,
so lambda cubed

206
00:16:25,600 --> 00:16:30,350
is beta to the 3/2, which,
when raised to the 2/3,

207
00:16:30,350 --> 00:16:33,110
gives me a factor of beta.

208
00:16:33,110 --> 00:16:37,840
So basically, the quantity
that we have over here

209
00:16:37,840 --> 00:16:42,610
is of the order of
beta, and then we

210
00:16:42,610 --> 00:16:47,940
saw that the remainder,
which depends on properties

211
00:16:47,940 --> 00:16:51,210
such as the mass
of the gas, gh, et

212
00:16:51,210 --> 00:16:56,380
cetera, which is independent
of temperature, is a constant.

213
00:16:56,380 --> 00:16:59,440
That is the Fermi
energy that we can

214
00:16:59,440 --> 00:17:03,720
compute by usual route
of filling up a Fermi c.

215
00:17:03,720 --> 00:17:09,819
And the value of
epsilon f is h bar

216
00:17:09,819 --> 00:17:15,400
squared over 2m
times kf squared,

217
00:17:15,400 --> 00:17:21,430
and the kf was 6 pi squared
n over g to the 1/3 power,

218
00:17:21,430 --> 00:17:26,510
so kf squared would be
something like this.

219
00:17:26,510 --> 00:17:31,230
So once we know the density
of the fermion, its spin

220
00:17:31,230 --> 00:17:34,580
and its mass, we can figure
out what this quantity is.

221
00:17:41,140 --> 00:17:44,520
The zeroth order solution
for the chemical potential

222
00:17:44,520 --> 00:17:46,520
is clearly this epsilon f.

223
00:17:53,780 --> 00:17:56,400
What do I have to do
if I want to calculate

224
00:17:56,400 --> 00:17:58,372
the next correction?

225
00:17:58,372 --> 00:18:02,420
Well, what I can do is I
can put the zeroth order

226
00:18:02,420 --> 00:18:05,800
solution for log z
in this expression,

227
00:18:05,800 --> 00:18:08,910
and then that will give me
the first order correction.

228
00:18:08,910 --> 00:18:12,940
I have to raise this whole
thing to the minus 2/3 power,

229
00:18:12,940 --> 00:18:18,470
so pi squared over 8
becomes 1 minus 2/3

230
00:18:18,470 --> 00:18:22,340
of pi squared over h, which
is minus pi squared over 12.

231
00:18:25,820 --> 00:18:32,595
1 over log of z squared--
log z is again mu over kt,

232
00:18:32,595 --> 00:18:38,540
so its inverse is kt
over the zeroth order

233
00:18:38,540 --> 00:18:45,850
value of mu, which is epsilon
f raised to second power,

234
00:18:45,850 --> 00:18:48,753
and we expect that there
will be higher order terms.

235
00:18:56,330 --> 00:18:57,710
So what have I stated?

236
00:18:57,710 --> 00:19:01,680
If I really take
this curve and try

237
00:19:01,680 --> 00:19:14,290
to solve for mu as a function
of temperature for a given

238
00:19:14,290 --> 00:19:24,030
density, I find that at
very low temperatures,

239
00:19:24,030 --> 00:19:31,460
the value that I
get is epsilon f,

240
00:19:31,460 --> 00:19:34,070
and then the value
starts to get reduced

241
00:19:34,070 --> 00:19:38,060
as I go to higher temperatures
and the initial fall

242
00:19:38,060 --> 00:19:39,980
is quadratic.

243
00:19:39,980 --> 00:19:43,100
Of course, the function will
have higher and higher order

244
00:19:43,100 --> 00:19:43,760
corrections.

245
00:19:43,760 --> 00:19:46,310
It will not stay quadratic.

246
00:19:46,310 --> 00:19:51,130
And I also know that
at very large values,

247
00:19:51,130 --> 00:19:57,320
it essentially converges
to a form that is minus

248
00:19:57,320 --> 00:20:04,270
kt log of n lambda cubed,
because down here, I

249
00:20:04,270 --> 00:20:09,550
know that this solution is z is
simply n lambda cubed over g.

250
00:20:09,550 --> 00:20:12,910
So at low densities and high
temperatures, basically,

251
00:20:12,910 --> 00:20:14,970
I will have something
that is almost

252
00:20:14,970 --> 00:20:18,800
linear with logarithmic
corrections and negative,

253
00:20:18,800 --> 00:20:24,160
and so presumably, the function
looks something like this.

254
00:20:31,760 --> 00:20:35,350
Of course, I can always convert.

255
00:20:35,350 --> 00:20:37,590
We can say that
what I see here is

256
00:20:37,590 --> 00:20:40,710
the combination of
kt over epsilon f.

257
00:20:40,710 --> 00:20:43,835
I can define epsilon f
over kv to be something

258
00:20:43,835 --> 00:20:47,820
that I'll call tf,
and this correction

259
00:20:47,820 --> 00:20:51,340
I can write as t
over tf squared.

260
00:20:51,340 --> 00:20:54,456
So basically, I have
defined epsilon f

261
00:20:54,456 --> 00:20:57,580
to be some kv times
a Fermi temperature.

262
00:21:03,140 --> 00:21:07,730
Just on the basis of
dimensional argument,

263
00:21:07,730 --> 00:21:12,500
you would expect that the place
where this zero is occurring

264
00:21:12,500 --> 00:21:15,090
is of the order of this tf.

265
00:21:15,090 --> 00:21:17,820
I don't claim that
it is exactly tf.

266
00:21:17,820 --> 00:21:20,420
It will have some
multiplicative factor,

267
00:21:20,420 --> 00:21:23,080
but it will be of
the order of tf,

268
00:21:23,080 --> 00:21:25,890
which can be related
to the density, mass,

269
00:21:25,890 --> 00:21:29,420
and other properties of
the gas by this formula.

270
00:21:29,420 --> 00:21:34,540
And if you look at a
metal, such as copper,

271
00:21:34,540 --> 00:21:37,820
where the electrons are
approximately described

272
00:21:37,820 --> 00:21:43,120
by this Fermi gas, this tf
is of the order of, say,

273
00:21:43,120 --> 00:21:44,660
10 to the 4 degrees Kelvin.

274
00:21:48,620 --> 00:21:50,110
Yes?

275
00:21:50,110 --> 00:21:53,750
AUDIENCE: So you were saying
that at a high temperature,

276
00:21:53,750 --> 00:21:57,380
the chemical potential
should go as minus kvt

277
00:21:57,380 --> 00:22:00,430
times ln of n
lambda cubed, yeah?

278
00:22:00,430 --> 00:22:01,490
PROFESSOR: Yes.

279
00:22:01,490 --> 00:22:05,460
AUDIENCE: Now, the lambda itself
has temperature dependence,

280
00:22:05,460 --> 00:22:10,130
so how can you claim that
it is behaving linearly?

281
00:22:10,130 --> 00:22:12,530
PROFESSOR: I said
almost linearly.

282
00:22:12,530 --> 00:22:16,760
As you say, the correct behavior
is something like minus 3/2 t

283
00:22:16,760 --> 00:22:22,120
log t, so log t
changes the exponent

284
00:22:22,120 --> 00:22:24,660
to be slightly different from 1.

285
00:22:24,660 --> 00:22:26,490
This is not entirely linear.

286
00:22:26,490 --> 00:22:28,920
It has some curvature
due to this logarithm.

287
00:22:31,960 --> 00:22:35,800
In fact, something that people
sometimes get confused at

288
00:22:35,800 --> 00:22:41,620
is when you go to high
temperatures, beta goes to 0.

289
00:22:41,620 --> 00:22:45,060
So why doesn't z go to 1?

290
00:22:45,060 --> 00:22:48,280
Of course, beta goes
to 0 but mu goes

291
00:22:48,280 --> 00:22:52,600
to infinity such that the
product of 0 and infinity

292
00:22:52,600 --> 00:22:55,520
is actually still pretty
large, partly because

293
00:22:55,520 --> 00:22:56,402
of this logarithm.

294
00:23:05,709 --> 00:23:08,044
AUDIENCE: Could you just
explain again why it's linear?

295
00:23:11,720 --> 00:23:13,730
PROFESSOR: Down
here it is linear,

296
00:23:13,730 --> 00:23:20,000
so I know that z is n
lambda cubed over g.

297
00:23:20,000 --> 00:23:23,480
z is e to the mu over kt.

298
00:23:26,580 --> 00:23:35,490
So mu is kt log of n
lambda cubed over g.

299
00:23:35,490 --> 00:23:38,420
Remember lambda
depends on temperature,

300
00:23:38,420 --> 00:23:43,430
so it is almost linear, except
that really what is inside here

301
00:23:43,430 --> 00:23:45,570
has a temperature dependence.

302
00:23:45,570 --> 00:23:50,485
So it's really
more like t log t.

303
00:23:50,485 --> 00:23:52,370
AUDIENCE: When z is small?

304
00:23:52,370 --> 00:23:55,160
PROFESSOR: When z is small
and only when z is small,

305
00:23:55,160 --> 00:23:59,860
and z small corresponds
to being down here.

306
00:23:59,860 --> 00:24:01,800
Again, that's what I was saying.

307
00:24:01,800 --> 00:24:06,330
Down here, beta
goes to zero but mu

308
00:24:06,330 --> 00:24:11,160
goes to minus infinity such
that the product of 0 and minus

309
00:24:11,160 --> 00:24:17,110
infinity is still something that
is large and negative because

310
00:24:17,110 --> 00:24:18,413
of this.

311
00:24:18,413 --> 00:24:22,200
AUDIENCE: If the product is
large, then isn't z large?

312
00:24:22,200 --> 00:24:25,300
PROFESSOR: If the product
is large and negative,

313
00:24:25,300 --> 00:24:27,130
then z is exponentially small.

314
00:24:38,330 --> 00:24:44,120
In fact, if I were to
plot this function just

315
00:24:44,120 --> 00:24:46,770
without the higher
order corrections,

316
00:24:46,770 --> 00:24:51,600
if I plot the function, kt
log of n lambda cubed over g,

317
00:24:51,600 --> 00:24:58,890
you can see that it goes
to 0 when the combination n

318
00:24:58,890 --> 00:25:04,770
lambda cubed over g is
1 because log of 1 is 0.

319
00:25:04,770 --> 00:25:08,830
So that function by
itself, asymptotically this

320
00:25:08,830 --> 00:25:14,730
is what I have, it will come
and do something like this,

321
00:25:14,730 --> 00:25:21,100
but it really is not something
that we are dealing with.

322
00:25:21,100 --> 00:25:24,400
It's in some sense
the classical version.

323
00:25:24,400 --> 00:25:28,620
So classically, your mu
is given by this formula,

324
00:25:28,620 --> 00:25:30,190
except that
classically, you don't

325
00:25:30,190 --> 00:25:33,530
know what the volume factor
is that you have to put there

326
00:25:33,530 --> 00:25:35,464
because you don't
know what h is.

327
00:25:47,860 --> 00:25:50,020
So that's the
chemical potential,

328
00:25:50,020 --> 00:25:53,550
but the chemical
potential is really,

329
00:25:53,550 --> 00:25:56,380
for our intents and
purposes, a device

330
00:25:56,380 --> 00:25:59,740
that I need to put here
in order to calculate

331
00:25:59,740 --> 00:26:03,260
the pressure, to calculate
the energy, et cetera.

332
00:26:03,260 --> 00:26:05,940
So let's go to the next step.

333
00:26:05,940 --> 00:26:07,860
We are again in this limit.

334
00:26:07,860 --> 00:26:16,020
Beta p is g over lambda
cubed f 5/2 plus z.

335
00:26:19,180 --> 00:26:25,100
That's going to give me g over
lambda cubed in the limit where

336
00:26:25,100 --> 00:26:30,770
z is large-- we established that
z is large-- log z to the 5/2

337
00:26:30,770 --> 00:26:34,940
divided by 5/2 factorial
using the Sommerfeld

338
00:26:34,940 --> 00:26:41,910
expansion, 1 plus pi over 6.

339
00:26:41,910 --> 00:26:44,400
Now my m is 5/2.

340
00:26:44,400 --> 00:26:50,280
m minus 1 would be 3/2
divided by log z squared

341
00:26:50,280 --> 00:26:51,720
in higher order terms.

342
00:26:59,769 --> 00:27:00,560
AUDIENCE: Question.

343
00:27:00,560 --> 00:27:01,620
PROFESSOR: Yes?

344
00:27:01,620 --> 00:27:04,960
AUDIENCE: Why is that plus?

345
00:27:04,960 --> 00:27:06,912
PROFESSOR: Because
I made a mistake.

346
00:27:17,330 --> 00:27:21,390
Just to make my life and
manipulations easier,

347
00:27:21,390 --> 00:27:22,470
I'll do the following.

348
00:27:22,470 --> 00:27:26,070
I write under this
the formula for n

349
00:27:26,070 --> 00:27:32,280
being g over lambda cubed,
f 3/2 minus z, which

350
00:27:32,280 --> 00:27:38,050
is g over lambda
cubed log z to the 3/2

351
00:27:38,050 --> 00:27:43,170
divided by 3/2 factorial,
1 plus-- I calculated

352
00:27:43,170 --> 00:27:50,670
the correction-- pi
squared over 6 divided

353
00:27:50,670 --> 00:27:53,950
by various combination,
which is pi squared over 8,

354
00:27:53,950 --> 00:27:55,800
1 over log z squared.

355
00:28:02,050 --> 00:28:07,480
Now what I will do is to
divide these two equations.

356
00:28:07,480 --> 00:28:12,340
If I were to divide the
numerator by the denominator,

357
00:28:12,340 --> 00:28:14,080
what do I get?

358
00:28:14,080 --> 00:28:20,300
The left hand side
becomes beta p over n.

359
00:28:20,300 --> 00:28:24,200
The right hand side, I can
get rid of the ratio of log

360
00:28:24,200 --> 00:28:27,500
z 5/2 to log z.

361
00:28:27,500 --> 00:28:32,310
3/2 is just one factor of log z.

362
00:28:32,310 --> 00:28:37,350
What do I have when I divide
5/2 factorial by 3/2 factorial?

363
00:28:37,350 --> 00:28:39,960
I have 5/2.

364
00:28:39,960 --> 00:28:45,510
This division allows me to get
rid of some unwanted terms.

365
00:28:45,510 --> 00:28:47,840
And then what do I have here?

366
00:28:53,360 --> 00:29:00,270
This combination here is 5
pi squared over 8 minus 1 pi

367
00:29:00,270 --> 00:29:01,290
squared over 8.

368
00:29:01,290 --> 00:29:04,710
Because of the division, I
will put a minus sign here,

369
00:29:04,710 --> 00:29:07,240
so that becomes 4
pi squared over 8,

370
00:29:07,240 --> 00:29:09,480
which is pi squared over 2.

371
00:29:09,480 --> 00:29:13,660
So I will get 1 plus
pi squared over 2,

372
00:29:13,660 --> 00:29:16,290
and at the order
that we are dealing,

373
00:29:16,290 --> 00:29:20,650
we can replace 1
over log z squared

374
00:29:20,650 --> 00:29:22,690
with kt over epsilon f squared.

375
00:29:35,240 --> 00:29:44,800
Now, for log z over here,
I can write beta mu,

376
00:29:44,800 --> 00:29:48,940
and for mu, I can write the
formula that I have up here.

377
00:29:53,630 --> 00:29:58,970
So you can see that once I
do that, the betas cancel

378
00:29:58,970 --> 00:30:04,080
from the two sides of the
equation, and what I get

379
00:30:04,080 --> 00:30:11,270
is that p is 2/5,
the inverse of 5/2.

380
00:30:11,270 --> 00:30:15,990
The n I will take to the
other side of the equation.

381
00:30:15,990 --> 00:30:18,720
To the lowest
order mu is epsilon

382
00:30:18,720 --> 00:30:25,515
f, so I will put epsilon f, but
mu is not exactly epsilon f.

383
00:30:25,515 --> 00:30:36,220
It is 1 minus pi squared over
12, kt over epsilon f squared.

384
00:30:36,220 --> 00:30:39,476
And actually, there was
this additional factor of 1

385
00:30:39,476 --> 00:30:45,250
plus pi squared over 2,
kt over epsilon f squared.

386
00:30:51,140 --> 00:30:53,770
So what do we find?

387
00:30:53,770 --> 00:30:58,220
We find that to
the zeroth order,

388
00:30:58,220 --> 00:31:03,760
pressure is related
to the density

389
00:31:03,760 --> 00:31:11,420
but multiplied by epsilon f, so
that even at zero temperature,

390
00:31:11,420 --> 00:31:13,580
there is a finite
value of pressure left.

391
00:31:13,580 --> 00:31:16,590
So we are used to ideal
gases that are classical,

392
00:31:16,590 --> 00:31:18,880
and when we go to
zero temperature,

393
00:31:18,880 --> 00:31:21,806
they start to stop
moving around.

394
00:31:21,806 --> 00:31:23,180
Since they are
not moving around,

395
00:31:23,180 --> 00:31:28,430
there is no pressure that is
exerted, but for the Fermi gas,

396
00:31:28,430 --> 00:31:32,390
you cannot say that all of the
particles are not moving around

397
00:31:32,390 --> 00:31:37,805
because that violates this
exclusion of n being zero

398
00:31:37,805 --> 00:31:38,670
or one.

399
00:31:38,670 --> 00:31:42,520
You have to give particles
more and more momentum

400
00:31:42,520 --> 00:31:46,670
so that they don't violate the
condition that they should have

401
00:31:46,670 --> 00:31:50,030
different values of the
occupation numbers, the Pauli

402
00:31:50,030 --> 00:31:51,160
exclusion.

403
00:31:51,160 --> 00:31:53,910
And therefore, even
at zero temperature,

404
00:31:53,910 --> 00:31:57,610
you will have particles
that in the ground state

405
00:31:57,610 --> 00:32:00,880
are zipping around, and
because they're moving around,

406
00:32:00,880 --> 00:32:03,540
they can hit the wall
and exert pressure on it,

407
00:32:03,540 --> 00:32:06,910
and this is the value
of the pressure.

408
00:32:06,910 --> 00:32:10,990
As you go to higher temperature,
that pressure gets modified.

409
00:32:10,990 --> 00:32:14,190
We can see that balancing
these two terms,

410
00:32:14,190 --> 00:32:16,920
there is an increase
in pressure, which

411
00:32:16,920 --> 00:32:27,688
is 1 plus 5 pi squared over
6, kt over epsilon f squared.

412
00:32:37,570 --> 00:32:42,100
So you expect the
pressure, if I plot it

413
00:32:42,100 --> 00:32:48,195
as a function of temperature,
at zero temperature,

414
00:32:48,195 --> 00:32:53,400
it will have this
constant value, pf.

415
00:32:53,400 --> 00:32:57,420
As I put a higher
temperature, there

416
00:32:57,420 --> 00:33:00,680
will be even more energy
and more kinetic energy,

417
00:33:00,680 --> 00:33:03,380
and the pressure will rise.

418
00:33:03,380 --> 00:33:06,190
Eventually, at very
high temperatures,

419
00:33:06,190 --> 00:33:09,270
I will regain the
classical, rather

420
00:33:09,270 --> 00:33:11,100
pressure is proportional
to temperature.

421
00:33:19,090 --> 00:33:24,230
That's the pressure of
this ideal Fermi gas.

422
00:33:24,230 --> 00:33:25,860
The energy.

423
00:33:25,860 --> 00:33:29,750
Well, the energy
is simply what we

424
00:33:29,750 --> 00:33:33,940
said over here, always 3/2 pv.

425
00:33:33,940 --> 00:33:38,980
So it is 3/2, the pressure
over there multiplied

426
00:33:38,980 --> 00:33:45,950
by v. When I multiply
the density by v,

427
00:33:45,950 --> 00:33:48,650
I will get the
number of particles.

428
00:33:48,650 --> 00:33:56,480
3/2 times 5/2 will give me
3/5, so I will 3/5 and epsilon

429
00:33:56,480 --> 00:33:59,930
f at the lowest order.

430
00:33:59,930 --> 00:34:03,790
Again, you've probably
seen this already

431
00:34:03,790 --> 00:34:10,880
where you draw diagrams
for filling up a Fermi c.

432
00:34:10,880 --> 00:34:14,900
Up to some particular
kf, you would

433
00:34:14,900 --> 00:34:16,600
say that all states
are occupied.

434
00:34:19,489 --> 00:34:23,810
When epsilon f is the
energy of this state that

435
00:34:23,810 --> 00:34:28,110
is sitting right at the
edge of the Fermi c,

436
00:34:28,110 --> 00:34:30,889
and clearly, the
energies of the particles

437
00:34:30,889 --> 00:34:36,389
vary all the way from
zero up to epsilon f,

438
00:34:36,389 --> 00:34:41,060
so the average energy is going
to be less than epsilon f.

439
00:34:41,060 --> 00:34:44,560
Dimensionally, it works
out to be 3/5 of that.

440
00:34:44,560 --> 00:34:45,750
That's what that says.

441
00:34:48,380 --> 00:34:50,540
But the more
important part is how

442
00:34:50,540 --> 00:34:52,949
it changes as a
function of temperature.

443
00:34:52,949 --> 00:35:00,360
What you find is that it goes
to 1 plus 5 pi squared over 6,

444
00:35:00,360 --> 00:35:05,140
kt over epsilon f squared
in high order terms.

445
00:35:07,795 --> 00:35:08,586
AUDIENCE: Question.

446
00:35:08,586 --> 00:35:11,630
Should the 6 be 12?

447
00:35:11,630 --> 00:35:14,240
PROFESSOR: 6 should be 12?

448
00:35:14,240 --> 00:35:15,570
Let's see.

449
00:35:15,570 --> 00:35:23,560
This is 6/12 minus
1/12 is indeed 5/12.

450
00:35:23,560 --> 00:35:24,532
Thank you.

451
00:35:37,530 --> 00:35:42,630
So if I ask what's the heat
capacity at constant volume,

452
00:35:42,630 --> 00:35:49,060
this is dE by dT, so just
taking the derivative.

453
00:35:49,060 --> 00:35:52,110
The zeroth order term,
of course, is irrelevant.

454
00:35:52,110 --> 00:35:54,400
It doesn't vary the temperature.

455
00:35:54,400 --> 00:35:58,770
It's this t squared that
will give you the variations.

456
00:35:58,770 --> 00:36:00,190
So what does it give me?

457
00:36:00,190 --> 00:36:04,680
I will have n epsilon f.

458
00:36:04,680 --> 00:36:11,810
I have 3/5 times 5
pi squared over 12.

459
00:36:11,810 --> 00:36:14,070
The derivative of
this combination

460
00:36:14,070 --> 00:36:22,500
will give me 2kb squared t
divided by epsilon f squared.

461
00:36:22,500 --> 00:36:26,340
There will be, of
course, higher order,

462
00:36:26,340 --> 00:36:30,130
but this combination you can
see I can write as follows.

463
00:36:30,130 --> 00:36:31,510
It is extensive.

464
00:36:31,510 --> 00:36:35,710
It is proportional to n.

465
00:36:35,710 --> 00:36:39,720
There is one kb that I can
take out, which is nice,

466
00:36:39,720 --> 00:36:42,850
because natural units
of heat capacity,

467
00:36:42,850 --> 00:36:44,610
as we have emphasized, are kb.

468
00:36:47,190 --> 00:36:51,590
This combination of numbers,
I believe the 5 cancels.

469
00:36:51,590 --> 00:36:56,030
3 times 2 divided
by 12 gives me 1/2,

470
00:36:56,030 --> 00:36:58,710
so I have pi squared over 2.

471
00:36:58,710 --> 00:37:02,350
And then I have the
combination, kt over epsilon

472
00:37:02,350 --> 00:37:04,730
f, which I can also
write as t over tf.

473
00:37:11,290 --> 00:37:15,080
So the heat capacity
of the Fermi gas

474
00:37:15,080 --> 00:37:19,140
goes to 0 as I go
to zero temperature,

475
00:37:19,140 --> 00:37:22,320
in accord with the third
law of thermodynamics, which

476
00:37:22,320 --> 00:37:25,810
we said is a consequence
of quantum mechanics.

477
00:37:25,810 --> 00:37:32,050
So we have this result,
that the proportionality

478
00:37:32,050 --> 00:37:36,440
is given by the inverse of tf.

479
00:37:36,440 --> 00:37:42,700
If I were to plot the heat
capacity in units of nkb

480
00:37:42,700 --> 00:37:48,400
as a function of t, at
very high temperatures,

481
00:37:48,400 --> 00:37:53,460
I will get the classical
result, which is 3/2,

482
00:37:53,460 --> 00:37:56,770
and then I will start to
get corrections to that.

483
00:37:56,770 --> 00:38:00,430
Those corrections will
gradually reduce this,

484
00:38:00,430 --> 00:38:04,580
so eventually, the function
that I will get starts linearly

485
00:38:04,580 --> 00:38:07,630
and then gets matched to that.

486
00:38:07,630 --> 00:38:12,650
So the linear behavior is
of the order of t over tf,

487
00:38:12,650 --> 00:38:15,860
and presumably at
some temperature

488
00:38:15,860 --> 00:38:19,160
that is of the order
of tf, you will

489
00:38:19,160 --> 00:38:22,390
switch to some
classical behavior.

490
00:38:22,390 --> 00:38:29,810
And as I said, if you look at
metals, this tf is very large.

491
00:38:29,810 --> 00:38:33,410
So when you look at the
heat capacity of something

492
00:38:33,410 --> 00:38:37,580
like copper or some
other metal, you

493
00:38:37,580 --> 00:38:40,240
find that there is a
contribution from the electrons

494
00:38:40,240 --> 00:38:41,815
to the heat capacity
that is linear.

495
00:38:48,420 --> 00:38:50,750
Now, the reason for
this linear behavior

496
00:38:50,750 --> 00:38:53,730
is also good to understand.

497
00:38:53,730 --> 00:38:56,890
This calculation that
I did is necessary

498
00:38:56,890 --> 00:39:01,620
in order to establish what this
precise factor of pi squared

499
00:39:01,620 --> 00:39:03,775
over 2 or appropriate
coefficient

500
00:39:03,775 --> 00:39:08,560
is, but the physical reason
for the linearity you

501
00:39:08,560 --> 00:39:10,010
should be able to know.

502
00:39:10,010 --> 00:39:15,780
Basically, we saw that
the occupation numbers

503
00:39:15,780 --> 00:39:20,310
for the case of fermions
have this form that

504
00:39:20,310 --> 00:39:27,520
is 1 over z inverse, e
to the epsilon plus 1.

505
00:39:31,060 --> 00:39:38,980
If I plot the occupation
numbers as a function of energy,

506
00:39:38,980 --> 00:39:42,050
for a case that is
at zero temperature,

507
00:39:42,050 --> 00:39:45,635
you can see that the
occupation number is 1 or 0,

508
00:39:45,635 --> 00:39:49,810
so you have a picture
that is like this.

509
00:39:49,810 --> 00:39:54,130
You switch from 1
to 0 at epsilon f.

510
00:39:58,700 --> 00:40:03,890
Oops, there was a beta here
that I forgot, beta epsilon.

511
00:40:03,890 --> 00:40:08,360
When I go to finite
temperature, this becomes fuzzy

512
00:40:08,360 --> 00:40:11,170
because this expectation
value for the number,

513
00:40:11,170 --> 00:40:14,630
rather than being a
one zero step function,

514
00:40:14,630 --> 00:40:19,040
becomes smoothed out,
becomes a function

515
00:40:19,040 --> 00:40:27,040
such as this, and the width
over which this smoothing takes

516
00:40:27,040 --> 00:40:29,910
place is of the order of kt.

517
00:40:34,810 --> 00:40:42,380
So what happens is that rather
than having this short Fermi c,

518
00:40:42,380 --> 00:40:47,090
you will start having
less particles here

519
00:40:47,090 --> 00:40:52,210
and more occupied
particles over here.

520
00:40:52,210 --> 00:40:55,300
To do that, you have
created energies

521
00:40:55,300 --> 00:40:58,060
going from here to here
that you have stored

522
00:40:58,060 --> 00:41:00,410
that is of the order of this kt.

523
00:41:03,870 --> 00:41:07,030
What fraction of
the total number

524
00:41:07,030 --> 00:41:10,380
have you given this energy?

525
00:41:10,380 --> 00:41:14,565
Well, the fraction
is given from here.

526
00:41:14,565 --> 00:41:19,230
It is the ratio of
this to the entirety,

527
00:41:19,230 --> 00:41:23,540
and that ratio is
t, or kt, divided

528
00:41:23,540 --> 00:41:29,590
by epsilon f is
the fraction that

529
00:41:29,590 --> 00:41:31,980
has been given energy this.

530
00:41:31,980 --> 00:41:35,290
So the excitation
energy that you have

531
00:41:35,290 --> 00:41:39,840
is this number, the
total times this fraction

532
00:41:39,840 --> 00:41:43,850
times the energy that
you have, which is kt.

533
00:41:43,850 --> 00:41:46,230
And if I take the
derivative of this,

534
00:41:46,230 --> 00:41:48,490
I will get precisely
this formula over

535
00:41:48,490 --> 00:41:51,670
here up to this factor
of pi squared over 2,

536
00:41:51,670 --> 00:41:55,050
which I was very cavalier,
but the scaling everything

537
00:41:55,050 --> 00:41:57,380
comes from this.

538
00:41:57,380 --> 00:42:00,120
And to all intents
and purposes, this

539
00:42:00,120 --> 00:42:04,670
is the characteristic
of all Fermi systems,

540
00:42:04,670 --> 00:42:08,990
that as you go to
low temperatures,

541
00:42:08,990 --> 00:42:13,320
there is a small fraction
of electrons, if you like,

542
00:42:13,320 --> 00:42:15,180
that can be excited.

543
00:42:15,180 --> 00:42:17,780
That fraction goes to
0 at low temperatures

544
00:42:17,780 --> 00:42:22,290
as kt over epsilon f, and the
typical energies that they have

545
00:42:22,290 --> 00:42:24,060
is of the order of kt.

546
00:42:24,060 --> 00:42:29,590
So essentially, many of the
results for the degenerate

547
00:42:29,590 --> 00:42:35,230
Fermi gas you can obtain
by taking classical results

548
00:42:35,230 --> 00:42:38,390
and substituting for
the number of particles

549
00:42:38,390 --> 00:42:41,710
nkt over epsilon f.

550
00:42:41,710 --> 00:42:45,330
You will see in the problem
set that this kind of argument

551
00:42:45,330 --> 00:42:51,470
tells you something also
about the magnetic response

552
00:42:51,470 --> 00:42:57,310
of a system of electrons,
what is the susceptibility

553
00:42:57,310 --> 00:43:01,480
and why does the susceptibility
saturate at low temperatures.

554
00:43:01,480 --> 00:43:05,430
It's just substituting for
the number of particles

555
00:43:05,430 --> 00:43:08,050
this fraction, nkt
over epsilon f,

556
00:43:08,050 --> 00:43:10,240
will give you lots of things.

557
00:43:10,240 --> 00:43:14,990
This picture is also
valid in all dimensions,

558
00:43:14,990 --> 00:43:19,230
whereas next, we'll be
discussing the case of bosons,

559
00:43:19,230 --> 00:43:22,530
where the dependence
is on temperature.

560
00:43:22,530 --> 00:43:25,370
There is an exponent
here that determines

561
00:43:25,370 --> 00:43:28,380
which dimension you are
or depends on dimensions.

562
00:43:28,380 --> 00:43:32,105
For the case of fermions, this
linearity exists independently.

563
00:43:38,630 --> 00:43:41,110
So now let's think
about bosons and what's

564
00:43:41,110 --> 00:43:44,180
going to happen for bosons.

565
00:43:44,180 --> 00:43:46,980
For bosons, we saw that
the series, rather than

566
00:43:46,980 --> 00:43:49,990
being alternating,
all of the terms

567
00:43:49,990 --> 00:43:54,480
are adding up together so
that the parabolic correction,

568
00:43:54,480 --> 00:43:57,110
rather than reducing
the function

569
00:43:57,110 --> 00:44:01,130
for the case of bosons,
will increase it.

570
00:44:01,130 --> 00:44:05,120
So for a particular
value of density,

571
00:44:05,120 --> 00:44:11,640
we find that the z of fermions
is larger than the z of bosons.

572
00:44:11,640 --> 00:44:14,400
So for a given
density, what we will

573
00:44:14,400 --> 00:44:17,850
find is that the chemical
potential, whereas for the case

574
00:44:17,850 --> 00:44:24,300
of fermions was deviating from
the classical form by going up,

575
00:44:24,300 --> 00:44:26,540
for the case of
bosons, we'll start

576
00:44:26,540 --> 00:44:29,735
to deviate from the
classical form and go down.

577
00:44:34,850 --> 00:44:37,930
Now, there is
something, however,

578
00:44:37,930 --> 00:44:41,450
that the chemical potential
cannot do for the case

579
00:44:41,450 --> 00:44:46,160
of bosons which it did it
for the case of fermions,

580
00:44:46,160 --> 00:44:49,690
and that was to change sign.

581
00:44:49,690 --> 00:44:51,260
Why is that?

582
00:44:51,260 --> 00:44:55,315
Well, we have that over
there, the occupation number

583
00:44:55,315 --> 00:45:01,480
for a boson is 1
over z inverse, which

584
00:45:01,480 --> 00:45:06,500
is e to the beta
epsilon k minus mu.

585
00:45:06,500 --> 00:45:10,450
z inverse is e to the
minus meta mu minus 1.

586
00:45:14,130 --> 00:45:17,480
Now, this result was
obtained from summing

587
00:45:17,480 --> 00:45:20,970
a geometric series, and
the condition certainly

588
00:45:20,970 --> 00:45:25,660
is that this object, that
is, the inverse of what

589
00:45:25,660 --> 00:45:29,770
was multiplying different
terms in the series,

590
00:45:29,770 --> 00:45:32,090
has to be larger than 1.

591
00:45:32,090 --> 00:45:34,410
Of course, it has
to be larger than 1

592
00:45:34,410 --> 00:45:37,390
so that I get a positive
occupation number.

593
00:45:37,390 --> 00:45:39,850
If it was less than 1,
the geometric series

594
00:45:39,850 --> 00:45:41,820
was never convergent.

595
00:45:41,820 --> 00:45:46,050
Clearly, the negativity
of the occupation number

596
00:45:46,050 --> 00:45:48,520
has no meaning.

597
00:45:48,520 --> 00:45:55,600
So this being positive
strictly immediately implies

598
00:45:55,600 --> 00:46:02,557
that mu has to be less
than epsilon k for all k.

599
00:46:05,700 --> 00:46:10,540
So it has to be certainly less
than the minimum of epsilon k

600
00:46:10,540 --> 00:46:15,000
with respect to all k.

601
00:46:15,000 --> 00:46:17,700
And for the case that I'm
looking at where my epsilon

602
00:46:17,700 --> 00:46:24,000
k's are h bar squared
k squared over 2m,

603
00:46:24,000 --> 00:46:27,350
the lowest one
corresponds to k equals 0,

604
00:46:27,350 --> 00:46:32,930
so mu has to be less than 0.

605
00:46:32,930 --> 00:46:37,580
It can never go to the
other side of this.

606
00:46:37,580 --> 00:46:42,420
Or alternatively, z
has to be less than

607
00:46:42,420 --> 00:46:44,300
or approaches maybe 1.

608
00:46:48,780 --> 00:46:51,470
So there is certainly
a barrier here

609
00:46:51,470 --> 00:46:56,870
that we are going to encounter
for the case of bosons at z

610
00:46:56,870 --> 00:46:58,940
equals 1.

611
00:46:58,940 --> 00:47:01,200
We should not go
beyond that point.

612
00:47:04,700 --> 00:47:19,196
So let's see what's
happening for z equals 1

613
00:47:19,196 --> 00:47:20,185
to this function.

614
00:47:30,830 --> 00:47:34,520
You see, my task
remains the same.

615
00:47:34,520 --> 00:47:38,700
In order to find z, I
have to solve graphically

616
00:47:38,700 --> 00:47:41,350
for the intersection of
the curve that corresponds

617
00:47:41,350 --> 00:47:46,870
to f plus 3/2 and the curve that
corresponds to the density n

618
00:47:46,870 --> 00:47:49,630
lambda cubed over g.

619
00:47:49,630 --> 00:47:52,730
So what happens as I go to
higher and higher values

620
00:47:52,730 --> 00:47:55,420
of n lambda cubed over g?

621
00:47:55,420 --> 00:47:58,150
You can see one possibility
is that this curve just

622
00:47:58,150 --> 00:48:02,020
diverges as z approaches 1.

623
00:48:02,020 --> 00:48:04,850
Then for every value of
n lambda cubed over g,

624
00:48:04,850 --> 00:48:07,460
you will find some value
of z that will gradually

625
00:48:07,460 --> 00:48:10,790
become closer and closer to 1.

626
00:48:10,790 --> 00:48:13,390
But is that the scenario?

627
00:48:13,390 --> 00:48:16,580
For that, we need to know what
the value of this f function

628
00:48:16,580 --> 00:48:27,840
is at z equals 1, so the limit
f m plus of z as z goes to 1.

629
00:48:27,840 --> 00:48:29,890
How do you obtain that?

630
00:48:29,890 --> 00:48:35,810
Well, that is 1 over m minus
1 factorial, integral 0

631
00:48:35,810 --> 00:48:44,770
to infinity, dx, x to the m
minus 1, z inverse, which is 1,

632
00:48:44,770 --> 00:48:46,450
e to the x minus 1.

633
00:48:49,570 --> 00:48:54,040
I have to integrate
this function.

634
00:48:54,040 --> 00:48:58,050
The integrand, how
does it look like?

635
00:48:58,050 --> 00:49:02,090
x to the m minus 1,
e to the x minus 1.

636
00:49:02,090 --> 00:49:04,770
Well, at large x,
there is no problem.

637
00:49:04,770 --> 00:49:06,365
It goes to 0 exponentially.

638
00:49:08,945 --> 00:49:13,710
At small x, it goes to
0 as x to the m minus 2.

639
00:49:16,890 --> 00:49:19,800
So basically, it's
a curve such as this

640
00:49:19,800 --> 00:49:23,350
that I have to integrate.

641
00:49:23,350 --> 00:49:27,710
And if the curve is like I have
drawn it, there is no problem.

642
00:49:27,710 --> 00:49:31,380
I can find the
integral underneath it

643
00:49:31,380 --> 00:49:35,870
and there actually is a
finite value that has a name.

644
00:49:35,870 --> 00:49:39,920
It's called a zeta function, so
this is some tabulated function

645
00:49:39,920 --> 00:49:42,560
that you can look at.

646
00:49:42,560 --> 00:49:47,280
But you can see that
if m, let's say, is 0,

647
00:49:47,280 --> 00:49:49,490
then it is dx over x squared.

648
00:49:49,490 --> 00:49:52,390
So the other
possibility is that this

649
00:49:52,390 --> 00:49:55,710
is a function that diverges
at the origin, which

650
00:49:55,710 --> 00:49:58,470
then may or may
not be integrable.

651
00:49:58,470 --> 00:50:07,800
And we can see that this
is finite and exists only

652
00:50:07,800 --> 00:50:13,150
for m that is larger then 1.

653
00:50:21,070 --> 00:50:24,930
So in particular, we are
interested in the case

654
00:50:24,930 --> 00:50:28,170
of m equals 3/2.

655
00:50:28,170 --> 00:50:32,160
So then at z equals 1,
I have a finite value.

656
00:50:32,160 --> 00:50:35,340
It is zeta of 3/2.

657
00:50:35,340 --> 00:50:44,010
So basically, the function will
come up to a finite value at z

658
00:50:44,010 --> 00:50:48,650
equals 1, which is
this zeta of 3/2,

659
00:50:48,650 --> 00:50:50,180
which you can look up in tables.

660
00:50:50,180 --> 00:50:59,610
It's 2.612.

661
00:50:59,610 --> 00:51:06,010
Now, I tried to draw this
curve as if it comes and hugs

662
00:51:06,010 --> 00:51:11,810
the vertical line tangentially,
that is, with infinite slope,

663
00:51:11,810 --> 00:51:13,140
and that is the case.

664
00:51:13,140 --> 00:51:15,460
Why do I know that?

665
00:51:15,460 --> 00:51:19,050
Because the derivative
of the function

666
00:51:19,050 --> 00:51:23,750
will be related to the
function at one lower index.

667
00:51:23,750 --> 00:51:29,990
So the derivative of f
3/2 is really an f 1/2.

668
00:51:29,990 --> 00:51:32,280
f 1/2 does not exist.

669
00:51:32,280 --> 00:51:34,490
It's a function that
goes to infinity.

670
00:51:34,490 --> 00:51:40,394
So essentially, this curve
comes with an infinite slope.

671
00:51:40,394 --> 00:51:44,120
Now, it will turn out
that this is the scenario

672
00:51:44,120 --> 00:51:46,710
that we have in
three dimensions.

673
00:51:46,710 --> 00:51:49,710
If you are in two dimensions,
then what you need to do

674
00:51:49,710 --> 00:51:55,080
is look at f that corresponds
to m equals 2 over 2 or 1,

675
00:51:55,080 --> 00:51:59,390
and in that case, the
function diverges.

676
00:51:59,390 --> 00:52:05,050
So then, you have no problem in
finding some intersection point

677
00:52:05,050 --> 00:52:08,850
for any combination of
n lambda cubed over g.

678
00:52:08,850 --> 00:52:10,770
But currently in
three dimensions,

679
00:52:10,770 --> 00:52:13,950
we have a problem because
for n lambda cubed

680
00:52:13,950 --> 00:52:19,310
over g that falls
higher than zeta of 3/2,

681
00:52:19,310 --> 00:52:24,940
it doesn't hit the
curve at any point,

682
00:52:24,940 --> 00:52:29,110
and we have to interpret
what that means.

683
00:52:29,110 --> 00:52:44,100
So for d equals 3,
encounter singularity

684
00:52:44,100 --> 00:52:52,780
when n lambda cubed
over g is greater than

685
00:52:52,780 --> 00:52:55,760
or equal to zeta of 3/2.

686
00:53:00,710 --> 00:53:07,210
This corresponds at a fixed
density to temperatures

687
00:53:07,210 --> 00:53:13,830
that are less than some critical
temperature that depends on n,

688
00:53:13,830 --> 00:53:19,305
and that I can read
off as being 1 over kb.

689
00:53:22,340 --> 00:53:25,550
This is going to give
me a combination.

690
00:53:25,550 --> 00:53:32,940
Lambda cubed is proportional
inversely to 3/2,

691
00:53:32,940 --> 00:53:46,530
so this will give me nz
3/2 over g to the 2/3,

692
00:53:46,530 --> 00:53:51,740
and then I have
h squared 2 pi m.

693
00:53:51,740 --> 00:53:53,695
I put the kb over here.

694
00:53:58,930 --> 00:54:05,030
So the more dense
your system is,

695
00:54:05,030 --> 00:54:13,050
the lower-- I guess that's why
I have it the opposite way.

696
00:54:13,050 --> 00:54:19,370
It should be the more dense it
is, the higher the temperature.

697
00:54:19,370 --> 00:54:20,223
That's fine.

698
00:54:29,850 --> 00:54:30,920
But what does that mean?

699
00:54:33,530 --> 00:54:38,210
The point is that if we go
and look at the structure

700
00:54:38,210 --> 00:54:43,670
that we have developed,
for any temperature that

701
00:54:43,670 --> 00:54:49,980
is high enough, or any density
that is low enough, so that I

702
00:54:49,980 --> 00:54:53,270
hit the curve on
its continuous part,

703
00:54:53,270 --> 00:54:57,930
it means that I can find the
value of z that is strictly

704
00:54:57,930 --> 00:55:01,670
less than 1, and
for that value of z,

705
00:55:01,670 --> 00:55:08,040
I can occupy states according
to that probability,

706
00:55:08,040 --> 00:55:11,830
and when I calculate
the net mean occupation,

707
00:55:11,830 --> 00:55:15,260
I will get the actual density
that I'm interested in.

708
00:55:17,860 --> 00:55:22,540
As I go to temperatures
that are lower,

709
00:55:22,540 --> 00:55:25,010
this combination goes up and up.

710
00:55:25,010 --> 00:55:34,010
The value of z that I have to
get gets pushed more towards 1.

711
00:55:34,010 --> 00:55:39,830
As it gets pushed more towards
1, I see that mu going to 0,

712
00:55:39,830 --> 00:55:46,718
the state that corresponds to
the lowest energy, k equals 0,

713
00:55:46,718 --> 00:55:47,884
gets more and more occupied.

714
00:55:50,030 --> 00:55:55,460
But still, nothing special about
that occupation except what

715
00:55:55,460 --> 00:55:59,080
happens when I am
at higher values.

716
00:55:59,080 --> 00:56:02,870
The most natural thing is that
when I am at higher values,

717
00:56:02,870 --> 00:56:08,810
I should pick z equals
1 minus a little bit.

718
00:56:11,460 --> 00:56:24,650
And if I choose z to be 1 minus
a very small quantity, then

719
00:56:24,650 --> 00:56:28,470
when I do the
integration over here,

720
00:56:28,470 --> 00:56:33,500
I didn't get a value
for the density, which

721
00:56:33,500 --> 00:56:37,470
is g over lambda cubed,
the limiting value

722
00:56:37,470 --> 00:56:41,130
of this f function,
which is zeta of 3/2.

723
00:56:45,750 --> 00:56:50,050
I will call this n star
because this is strictly

724
00:56:50,050 --> 00:56:54,240
less than the total
density that I have.

725
00:56:54,240 --> 00:56:55,640
That was the problem.

726
00:56:55,640 --> 00:56:59,550
If I could make up the total
density with z equals 1,

727
00:56:59,550 --> 00:57:00,780
I would be satisfied.

728
00:57:00,780 --> 00:57:03,160
I would be making
it here, but I'm not

729
00:57:03,160 --> 00:57:10,770
making it up with the spectrum
that I have written over here.

730
00:57:10,770 --> 00:57:16,000
But on the other hand, if
epsilon is incredibly small,

731
00:57:16,000 --> 00:57:20,410
what I find is that the
occupation number of the k

732
00:57:20,410 --> 00:57:25,840
equals to 0 state-- let's
write it n of k equals 0.

733
00:57:25,840 --> 00:57:26,670
What is that?

734
00:57:26,670 --> 00:57:33,150
It is 1 over z inverse, which
is the inverse of this quantity,

735
00:57:33,150 --> 00:57:36,280
which is 1 plus epsilon,
which is approximately

736
00:57:36,280 --> 00:57:42,725
e to the epsilon minus 1.

737
00:57:46,110 --> 00:57:51,510
Actually, maybe what I should
do is to write it in the form e

738
00:57:51,510 --> 00:57:57,450
to the beta mu minus 1 and
realize that this beta mu is

739
00:57:57,450 --> 00:57:59,270
a quantity that is very small.

740
00:57:59,270 --> 00:58:03,370
It is the same thing that I
was calling epsilon before.

741
00:58:03,370 --> 00:58:08,770
And if it is very small, I can
make it to be 1 over beta mu.

742
00:58:14,010 --> 00:58:20,195
So by making mu arbitrarily
close to the origin or epsilon

743
00:58:20,195 --> 00:58:24,340
mu arbitrarily close to
1, I could in principle

744
00:58:24,340 --> 00:58:30,370
pump a lot of particles
in the k equals 0 state,

745
00:58:30,370 --> 00:58:32,660
and that does not
violate anything.

746
00:58:32,660 --> 00:58:36,470
Bosons, you can put as many
particles as you like in the k

747
00:58:36,470 --> 00:58:39,160
equals 0 state.

748
00:58:39,160 --> 00:58:46,090
So essentially, what I can do
is I can make this, but you say,

749
00:58:46,090 --> 00:58:50,590
well, isn't this already
covered by this curve

750
00:58:50,590 --> 00:58:51,760
that you have over here?

751
00:58:51,760 --> 00:58:54,060
Are you doing
something different?

752
00:58:54,060 --> 00:58:58,410
Well, let's follow this line
of thought a little bit more.

753
00:58:58,410 --> 00:59:01,580
How much do I have to put here?

754
00:59:01,580 --> 00:59:10,670
This n is the total number of
particles divided by volume,

755
00:59:10,670 --> 00:59:19,030
and so this is going to be
this quantity, n star, which

756
00:59:19,030 --> 00:59:27,270
is gv over lambda cubed
divided by volume.

757
00:59:27,270 --> 00:59:35,140
Let's write it in this fashion,
z over lambda cubed zeta of 3/2

758
00:59:35,140 --> 00:59:39,670
and then whatever is left
over, and the leftover

759
00:59:39,670 --> 00:59:41,900
I can write as n0.

760
00:59:47,280 --> 00:59:54,680
So what we know is if
we take z equals 1,

761
00:59:54,680 --> 00:59:59,140
this function will
tell me how many things

762
00:59:59,140 --> 01:00:04,100
I have put in everything
except k equals 0

763
01:00:04,100 --> 01:00:07,060
that I will
calculate separately.

764
01:00:07,060 --> 01:00:09,380
That amount is here.

765
01:00:09,380 --> 01:00:13,100
And I will put some
more in k equals 0,

766
01:00:13,100 --> 01:00:15,340
and you can see
that the amount I

767
01:00:15,340 --> 01:00:22,930
have to put there is going
to be n minus g over lambda

768
01:00:22,930 --> 01:00:26,490
cubed zeta of 3/2.

769
01:00:26,490 --> 01:00:28,310
Actually, I will have
to put a volume here.

770
01:00:32,370 --> 01:00:34,880
Why is that?

771
01:00:34,880 --> 01:00:43,350
The reason I have to put the
volume is because the volume

772
01:00:43,350 --> 01:00:46,680
that I had here,
that ultimately I

773
01:00:46,680 --> 01:00:51,660
divided the number of particles
by volume to get the density,

774
01:00:51,660 --> 01:00:56,440
came from replacing the
sum with an integration.

775
01:00:56,440 --> 01:01:01,290
So what I have envisioned now
is that there are all of these

776
01:01:01,290 --> 01:01:04,930
points that correspond
to different k's.

777
01:01:04,930 --> 01:01:08,320
I'll replace the sum
with a integration,

778
01:01:08,320 --> 01:01:12,266
but then there was
one point, k equals 0,

779
01:01:12,266 --> 01:01:13,515
that I am treating separately.

780
01:01:16,210 --> 01:01:18,650
When I'm treating
that separately, it

781
01:01:18,650 --> 01:01:24,320
means that is really is the
same as the total number

782
01:01:24,320 --> 01:01:29,210
expectation value without having
to divide by the volume, which

783
01:01:29,210 --> 01:01:31,640
comes from this
density of state,

784
01:01:31,640 --> 01:01:34,645
and I have something like this.

785
01:01:34,645 --> 01:01:38,860
Now, the problem is that you
can see that suddenly, you

786
01:01:38,860 --> 01:01:43,410
have to pick a value
of mu that is inversely

787
01:01:43,410 --> 01:01:45,500
proportional to the
volume of the system.

788
01:01:52,800 --> 01:01:58,410
This is a problem in which
the thermodynamic limit

789
01:01:58,410 --> 01:02:03,210
is taken in this strange sense.

790
01:02:03,210 --> 01:02:08,800
You have all of these potential
values of the energy, epsilon

791
01:02:08,800 --> 01:02:13,642
of k, that correspond to h bar
squared, k squared over 2m.

792
01:02:13,642 --> 01:02:18,970
There is one that is at 0, and
then choosing k equals 2 to pi

793
01:02:18,970 --> 01:02:21,720
over l, you will have one
other state, another state,

794
01:02:21,720 --> 01:02:23,680
all of these states.

795
01:02:23,680 --> 01:02:28,000
All of these states are
actually very finely spaced.

796
01:02:28,000 --> 01:02:30,790
The difference between
the ground state

797
01:02:30,790 --> 01:02:34,990
and the first excited state
is h bar squared over 2m,

798
01:02:34,990 --> 01:02:36,730
2 pi over l squared.

799
01:02:36,730 --> 01:02:41,480
So this distance over here is of
the order of 1 over l squared.

800
01:02:44,780 --> 01:02:47,250
But I see that in
order to occupy

801
01:02:47,250 --> 01:02:50,610
this with the
appropriate number,

802
01:02:50,610 --> 01:02:59,750
I have to choose my chemical
potential to be as close as 1

803
01:02:59,750 --> 01:03:00,997
over the volume.

804
01:03:04,340 --> 01:03:07,740
So in the limit where I
take the size of the system

805
01:03:07,740 --> 01:03:10,960
go to infinity, this
approaches this.

806
01:03:10,960 --> 01:03:14,630
It never touches it, but the
distance that I have here

807
01:03:14,630 --> 01:03:16,650
is much, much less
than the distance

808
01:03:16,650 --> 01:03:19,850
that I have in the spacings
that I made this replacement.

809
01:03:23,290 --> 01:03:26,930
So I can indeed treat
these separately.

810
01:03:26,930 --> 01:03:30,520
I can give a particular
weight to this state

811
01:03:30,520 --> 01:03:35,510
where the mu has come this
close to it as 1 over v,

812
01:03:35,510 --> 01:03:38,410
treat all of the
other ones as part

813
01:03:38,410 --> 01:03:41,640
of this replacement of the
summations with the integral.

814
01:03:44,400 --> 01:03:47,310
You can also see that this
trick is going to be problematic

815
01:03:47,310 --> 01:03:50,970
if I were to perform it in
two dimensions, because rather

816
01:03:50,970 --> 01:03:54,680
than 1 over l to
the third power,

817
01:03:54,680 --> 01:03:56,380
that is, the volume
in three dimensions,

818
01:03:56,380 --> 01:03:59,420
I would have had
1 over l squared.

819
01:03:59,420 --> 01:04:01,330
And indeed, that's
another reason

820
01:04:01,330 --> 01:04:03,660
why two dimensions is special.

821
01:04:03,660 --> 01:04:07,270
And as I said, in
two dimensions,

822
01:04:07,270 --> 01:04:09,830
the curve actually does go
all the way to infinity.

823
01:04:09,830 --> 01:04:12,970
You won't have this problem.

824
01:04:12,970 --> 01:04:15,220
So this is essentially
what happens

825
01:04:15,220 --> 01:04:24,890
with this Bose-Einstein
condensation,

826
01:04:24,890 --> 01:04:33,930
that the occupation number of
the excited state for bosons

827
01:04:33,930 --> 01:04:37,730
is such that you encounter
a singularity in three

828
01:04:37,730 --> 01:04:42,930
dimensions beyond a particular
density or temperatures

829
01:04:42,930 --> 01:04:46,530
lower than a certain amount
or highly degenerate case,

830
01:04:46,530 --> 01:04:51,920
you have to separately treat
all the huge number of particles

831
01:04:51,920 --> 01:04:55,445
that is proportional to the
volume that have now piled up

832
01:04:55,445 --> 01:04:59,870
in the single state, k equals 0,
the ground state in this case,

833
01:04:59,870 --> 01:05:01,915
but whatever the
ground state may

834
01:05:01,915 --> 01:05:05,740
be for your appropriate
system, and all

835
01:05:05,740 --> 01:05:11,930
the other particles go in the
corresponding excited states.

836
01:05:11,930 --> 01:05:16,450
Now, as you go towards
zero temperature,

837
01:05:16,450 --> 01:05:18,120
you can see that
this combination

838
01:05:18,120 --> 01:05:21,570
is proportional to
temperature goes to zero.

839
01:05:21,570 --> 01:05:25,780
So at 0 temperature,
essentially all of the particles

840
01:05:25,780 --> 01:05:29,870
will need to be placed
in this k equals 0 state.

841
01:05:29,870 --> 01:05:50,930
So if you like, a rough picture
is that the net density.

842
01:05:58,090 --> 01:06:03,200
This is the density,
n, at high temperature

843
01:06:03,200 --> 01:06:07,860
is entirely made up as being
part of the excited state.

844
01:06:07,860 --> 01:06:12,170
Let's draw it this way.

845
01:06:12,170 --> 01:06:22,580
When you hit Tc, you find that
the excited state can no longer

846
01:06:22,580 --> 01:06:24,290
accommodate the entire density.

847
01:06:24,290 --> 01:06:26,580
This is a coordination
that, as we said,

848
01:06:26,580 --> 01:06:31,440
goes to zero at zero
temperature as T to the 3/2.

849
01:06:31,440 --> 01:06:35,080
So there's a curve that
goes like T to the 3/2,

850
01:06:35,080 --> 01:06:38,840
hits 1 at exactly Tc, which
is a function of the density

851
01:06:38,840 --> 01:06:40,500
that you choose.

852
01:06:40,500 --> 01:06:43,640
This is the fraction that
corresponds to the excited

853
01:06:43,640 --> 01:06:45,940
states.

854
01:06:45,940 --> 01:06:49,960
And on top of that, there
is a macroscopic fraction

855
01:06:49,960 --> 01:06:54,720
that is occupying the
ground state, which

856
01:06:54,720 --> 01:06:57,130
is the compliment to this curve.

857
01:06:57,130 --> 01:06:59,912
Basically, it behaves
something like this.

858
01:07:03,850 --> 01:07:08,110
One thing that I
should emphasize

859
01:07:08,110 --> 01:07:16,530
is that what I see here makes it
look like at high temperatures,

860
01:07:16,530 --> 01:07:19,860
there is no occupation
of k equals 0 state.

861
01:07:19,860 --> 01:07:22,750
That is certainly not
correct because if you

862
01:07:22,750 --> 01:07:26,250
look at this function
at any finite z

863
01:07:26,250 --> 01:07:28,030
as a function of
epsilon, you can

864
01:07:28,030 --> 01:07:30,240
see that the largest
value of this function

865
01:07:30,240 --> 01:07:33,020
is still at epsilon equals 0.

866
01:07:33,020 --> 01:07:35,200
It is much more
likely that there

867
01:07:35,200 --> 01:07:39,820
is occupation of the ground
state than any other state,

868
01:07:39,820 --> 01:07:44,230
except that the fraction
that is occupying here,

869
01:07:44,230 --> 01:07:47,770
when I divide by the total
number, goes to zero.

870
01:07:47,770 --> 01:07:50,980
It becomes macroscopic when
I'm below the BEC transition.

871
01:07:57,820 --> 01:08:01,990
Now, the properties of what
is happening in this system

872
01:08:01,990 --> 01:08:06,310
below Tc is actually very
simple because below Tc,

873
01:08:06,310 --> 01:08:09,240
the chemical potential
in the macroscopic sense

874
01:08:09,240 --> 01:08:11,840
is stuck at 1.

875
01:08:11,840 --> 01:08:15,050
Essentially, what I'm saying
is that in this system,

876
01:08:15,050 --> 01:08:17,950
the chemical
potential comes down.

877
01:08:17,950 --> 01:08:23,319
It hits Tc of n,
and then it is 0.

878
01:08:23,319 --> 01:08:26,340
Now of course, 0, as
we have discussed here,

879
01:08:26,340 --> 01:08:32,600
if I put a magnifier here
and multiply by a factor of n

880
01:08:32,600 --> 01:08:35,800
or a factor of
volume, then I see

881
01:08:35,800 --> 01:08:39,710
that there is a distinction
between that and 0.

882
01:08:39,710 --> 01:08:44,910
So it doesn't quite go
to 0, but the distance

883
01:08:44,910 --> 01:08:47,229
is of the order of
1 over volume or 1

884
01:08:47,229 --> 01:08:49,100
over the number of particles.

885
01:08:49,100 --> 01:08:51,564
Effectively, from the
thermodynamic perspective,

886
01:08:51,564 --> 01:08:52,789
it is 0.

887
01:08:56,140 --> 01:09:00,920
But again, for the purposes
of thermodynamics, pressure,

888
01:09:00,920 --> 01:09:02,560
everything comes
from the particles

889
01:09:02,560 --> 01:09:04,360
that are moving around.

890
01:09:04,360 --> 01:09:07,149
The particles in the ground
state are frozen out.

891
01:09:07,149 --> 01:09:09,060
They don't do anything.

892
01:09:09,060 --> 01:09:12,950
So the pressure is very simple.

893
01:09:12,950 --> 01:09:21,960
Beta p is g over lambda
cubed, f 5/2, eta of z

894
01:09:21,960 --> 01:09:25,770
equals 1, which is some number.

895
01:09:25,770 --> 01:09:29,770
It is g over lambda
cubed, the zeta

896
01:09:29,770 --> 01:09:34,899
function at 5/2, which again
has some value that you can read

897
01:09:34,899 --> 01:09:39,366
of in tables, 1.34,
something like that.

898
01:09:42,649 --> 01:09:47,250
So you can see that this
lambda cubed is inversely

899
01:09:47,250 --> 01:09:52,800
proportional to t to
the 3/2, so pressure

900
01:09:52,800 --> 01:09:54,890
is proportional to t to the 5/2.

901
01:09:58,270 --> 01:10:08,790
If I were to plot the pressure
as a function of temperature,

902
01:10:08,790 --> 01:10:12,376
what I find is that
at low temperatures,

903
01:10:12,376 --> 01:10:17,350
the pressure is simply
given as t to the 5/2.

904
01:10:25,850 --> 01:10:32,080
And you can see that this
pressure knows nothing

905
01:10:32,080 --> 01:10:36,780
about the overall
density of the system

906
01:10:36,780 --> 01:10:40,860
because this is a pure number.

907
01:10:40,860 --> 01:10:43,830
It is only a function of
temperature, mass, et cetera.

908
01:10:43,830 --> 01:10:47,860
There is no factor
of density here.

909
01:10:47,860 --> 01:10:49,660
So what is happening?

910
01:10:49,660 --> 01:10:52,870
Because we certainly
know that when

911
01:10:52,870 --> 01:10:57,290
I am at very high temperatures,
ideal gas behavior

912
01:10:57,290 --> 01:10:59,090
dictates that the
pressure should

913
01:10:59,090 --> 01:11:02,720
be proportional to
temperature times density.

914
01:11:02,720 --> 01:11:09,240
So if I have two different
gases where the density here

915
01:11:09,240 --> 01:11:13,176
is greater than
the density here,

916
01:11:13,176 --> 01:11:14,800
these will be the
form of the isotherms

917
01:11:14,800 --> 01:11:17,920
that I have at
high temperatures.

918
01:11:17,920 --> 01:11:26,170
So what happens is that you
will start with the ideal gas

919
01:11:26,170 --> 01:11:27,940
behavior.

920
01:11:27,940 --> 01:11:30,100
We saw that for
the case of bosons,

921
01:11:30,100 --> 01:11:33,090
there is some kind of an
effective attraction that

922
01:11:33,090 --> 01:11:38,090
reduces the pressure, so the
pressure starts to go down.

923
01:11:38,090 --> 01:11:45,260
Eventually, it will join
this universal curve

924
01:11:45,260 --> 01:11:48,450
at the value that
corresponds to Tc of n2.

925
01:11:52,500 --> 01:11:56,860
Whereas if I see what's
happening with the lower

926
01:11:56,860 --> 01:12:00,245
density, at high
temperatures, again, I

927
01:12:00,245 --> 01:12:04,620
will have the linear behavior
of the ideal gas with a lower

928
01:12:04,620 --> 01:12:07,280
slope because I have
a lower density.

929
01:12:07,280 --> 01:12:12,140
As I go to lower temperatures,
quantum corrections

930
01:12:12,140 --> 01:12:14,520
will reduce the pressure.

931
01:12:14,520 --> 01:12:17,700
Eventually, I find
that this curve

932
01:12:17,700 --> 01:12:22,400
will join my universal curve at
the point that would correspond

933
01:12:22,400 --> 01:12:26,250
to Tc of n1, which in
this case would be lower.

934
01:12:29,320 --> 01:12:32,700
But beyond that, it will
forget what the density was,

935
01:12:32,700 --> 01:12:36,146
the pressure would be the
same for all of these curves.

936
01:12:39,110 --> 01:12:43,350
And I have kind of indicated,
or tried to indicate,

937
01:12:43,350 --> 01:12:48,970
that the curves join
here in a manner

938
01:12:48,970 --> 01:12:53,750
that there is no
discontinuity in slope.

939
01:12:53,750 --> 01:12:57,100
So an interesting
exercise to do is

940
01:12:57,100 --> 01:12:59,990
to calculate derivatives
of this pressure

941
01:12:59,990 --> 01:13:02,610
as a function of
temperature and see

942
01:13:02,610 --> 01:13:05,720
whether they match coming
from the two sides.

943
01:13:05,720 --> 01:13:08,840
And you will find that if
you do the algebra correctly,

944
01:13:08,840 --> 01:13:10,674
there is a matching of the two.

945
01:13:16,810 --> 01:13:28,750
Now, if I have the pressure
as a function of temperature,

946
01:13:28,750 --> 01:13:32,577
then I also have the energy as a
function of temperature, right?

947
01:13:43,570 --> 01:13:52,320
Because we know that
the energy is 3/2 PV.

948
01:13:52,320 --> 01:13:56,360
So if I know P,
I can immediately

949
01:13:56,360 --> 01:13:59,150
know what the energy is.

950
01:13:59,150 --> 01:14:02,840
I know, therefore,
that in the condensate,

951
01:14:02,840 --> 01:14:07,080
the energy is proportional
to T to the 5/2.

952
01:14:07,080 --> 01:14:09,935
Take a derivative, I know
that the heat capacity

953
01:14:09,935 --> 01:14:12,625
will be proportional
to T to the 3/2.

954
01:14:17,480 --> 01:14:22,980
That T to the 3/2 is a
signature of this condensate

955
01:14:22,980 --> 01:14:23,720
that we have.

956
01:14:23,720 --> 01:14:30,430
Again, heat capacity will go to
0 as T goes to 0, as we expect.

957
01:14:30,430 --> 01:14:34,310
And as opposed to the
case of the fermions,

958
01:14:34,310 --> 01:14:38,473
where the vanishing of the heat
capacity was always linear,

959
01:14:38,473 --> 01:14:41,720
the vanishing of the
heat capacity for bosons

960
01:14:41,720 --> 01:14:47,510
will depend on dimensions, and
this vanishing as T to the 3/2

961
01:14:47,510 --> 01:14:51,390
is in general T to
the d over 2, as can

962
01:14:51,390 --> 01:14:56,000
be shown very easily by
following this algebra.

963
01:14:56,000 --> 01:14:58,650
I wanted to do a
little bit of work

964
01:14:58,650 --> 01:15:02,630
on calculating the heat
capacity because its shape is

965
01:15:02,630 --> 01:15:03,830
interesting.

966
01:15:03,830 --> 01:15:10,230
Let's write the formula a
little bit more accurately.

967
01:15:10,230 --> 01:15:30,490
I have 3/2 v. Pressure is
kTg over lambda cubed, f

968
01:15:30,490 --> 01:15:32,850
5/2 plus of z.

969
01:15:35,480 --> 01:15:40,515
This formula is valid,
both at high temperatures

970
01:15:40,515 --> 01:15:42,270
and at low temperatures.

971
01:15:42,270 --> 01:15:44,430
At high temperatures,
z will be varying

972
01:15:44,430 --> 01:15:50,525
as a function of temperature.

973
01:16:05,870 --> 01:16:09,600
Let me also write the formula
for the number of particles.

974
01:16:09,600 --> 01:16:15,720
The number of particles
is V times g over lambda

975
01:16:15,720 --> 01:16:19,920
cubed f 3/2 plus of z.

976
01:16:23,840 --> 01:16:26,520
Once more, I can get rid
of a number of things

977
01:16:26,520 --> 01:16:28,680
by dividing these two.

978
01:16:28,680 --> 01:16:44,321
So the energy per
particle is 3/2 kT, f 5/2

979
01:16:44,321 --> 01:16:49,540
of z divided by f 3/2 of z.

980
01:16:49,540 --> 01:16:52,255
Let's make sure I
didn't make any mistake.

981
01:17:02,000 --> 01:17:07,570
Now, I want to calculate the
heat capacity, let's say,

982
01:17:07,570 --> 01:17:15,228
per particle, which is d by
dT of the energy per particle.

983
01:17:17,970 --> 01:17:24,780
Now, I have one factor out
here, which is easy to evaluate.

984
01:17:24,780 --> 01:17:27,380
There's an explicit
temperature dependence here,

985
01:17:27,380 --> 01:17:29,990
so I will get 3/2 kb.

986
01:17:29,990 --> 01:17:45,260
I will have f 5/2 plus of z
divided by f 3/2 plus of z.

987
01:17:45,260 --> 01:17:48,330
Something here
that I don't like.

988
01:18:03,630 --> 01:18:06,930
Everything that I have written
here is clearly valid only

989
01:18:06,930 --> 01:18:12,445
for T greater than Tc
because for T less than Tc,

990
01:18:12,445 --> 01:18:15,140
I cannot write n
in this fashion.

991
01:18:15,140 --> 01:18:18,950
So what I'm writing for you is
correct for T greater than Tc.

992
01:18:22,210 --> 01:18:26,500
Point is that there is
also an implicit dependence

993
01:18:26,500 --> 01:18:30,240
on temperature because z is
a function of temperature.

994
01:18:30,240 --> 01:18:38,010
So what I can do is
I can do 3/2 kbT,

995
01:18:38,010 --> 01:18:42,230
and this is a function of z, and
z is a function of temperature.

996
01:18:42,230 --> 01:18:49,860
So what I have it is dz by
dT done at constant volume

997
01:18:49,860 --> 01:18:54,580
or number of particles times
the derivative of this function

998
01:18:54,580 --> 01:18:57,155
with respect to z.

999
01:18:57,155 --> 01:18:59,930
Now, when I take the
derivative with respect to z,

1000
01:18:59,930 --> 01:19:01,770
I know that I
introduce a function

1001
01:19:01,770 --> 01:19:06,532
with one lower derivative
up to a factor of z.

1002
01:19:06,532 --> 01:19:09,930
So I divide by the
1 over z, and then

1003
01:19:09,930 --> 01:19:21,890
the derivative of the numerator
is f 3/2 plus divided by f 3/2

1004
01:19:21,890 --> 01:19:26,880
plus in the denominator
minus the derivative

1005
01:19:26,880 --> 01:19:31,550
of the denominator,
which is f 1/2 plus of z

1006
01:19:31,550 --> 01:19:45,186
times the numerator divided
by f 3/2 plus of z squared.

1007
01:19:53,090 --> 01:19:56,540
So what you would need
to evaluate in order

1008
01:19:56,540 --> 01:20:00,170
to get an expression
that is meaningful

1009
01:20:00,170 --> 01:20:05,420
and we can eliminate z's as much
as possible is what dz by dT

1010
01:20:05,420 --> 01:20:07,540
is.

1011
01:20:07,540 --> 01:20:11,910
Well, if this is our formula
for the number of particles

1012
01:20:11,910 --> 01:20:15,560
and the number of particles
or density is fixed,

1013
01:20:15,560 --> 01:20:18,025
then we take a derivative
with respect to temperature,

1014
01:20:18,025 --> 01:20:24,600
and I will get that dN
by dT, which is 0, is v.

1015
01:20:24,600 --> 01:20:26,390
And then we have to
take the temperature

1016
01:20:26,390 --> 01:20:29,910
derivative of this
combination, g over lambda

1017
01:20:29,910 --> 01:20:34,200
cubed, f 3/2 plus of z.

1018
01:20:34,200 --> 01:20:41,910
Now, lambda cubed scales
like 1 over t to the 3/2,

1019
01:20:41,910 --> 01:20:44,345
so when I take a
derivative of this,

1020
01:20:44,345 --> 01:20:50,040
I will get 3/2 T to the
1/2, which I can again

1021
01:20:50,040 --> 01:20:57,520
combine and write in
this fashion, 1 over T,

1022
01:20:57,520 --> 01:20:59,655
f 3/2 plus of z.

1023
01:21:02,160 --> 01:21:06,540
Or then I take the implicit
derivative that I have here,

1024
01:21:06,540 --> 01:21:12,950
so just like there, I will get
dz by dT at constant density

1025
01:21:12,950 --> 01:21:15,310
times 1 over z.

1026
01:21:15,310 --> 01:21:23,960
The derivative of f 3/2 will
give me f 1/2 plus of z.

1027
01:21:23,960 --> 01:21:27,160
Setting it to zero,
we can immediately

1028
01:21:27,160 --> 01:21:34,960
see that this combination,
T over z dz by dT

1029
01:21:34,960 --> 01:21:38,890
at constant density over
the number of particles

1030
01:21:38,890 --> 01:21:48,584
is minus 3/2, f 3/2 plus
divided by f 1/2 plus of z.

1031
01:21:53,230 --> 01:22:03,910
So what I need to do is to
substitute this over here.

1032
01:22:03,910 --> 01:22:07,930
And then when I want to
calculate the various limiting

1033
01:22:07,930 --> 01:22:11,800
behaviors as I start with the
high temperature and approach

1034
01:22:11,800 --> 01:22:17,290
Tc, all I need to know
is that z will go to 1.

1035
01:22:17,290 --> 01:22:24,140
f 1/2 of 1 is divergent, so
this factor will go to 0.

1036
01:22:24,140 --> 01:22:29,180
This factor will disappear,
and so then these things

1037
01:22:29,180 --> 01:22:33,580
will take nice, simple forms.

1038
01:22:33,580 --> 01:22:38,250
Once you do that-- we will do
this more correctly next time

1039
01:22:38,250 --> 01:22:40,310
around-- we'll
find that the heat

1040
01:22:40,310 --> 01:22:45,520
capacity of the
bosons in units of kb

1041
01:22:45,520 --> 01:22:52,120
as a function of temperature
has a discontinuity at Tc of n.

1042
01:22:52,120 --> 01:22:54,470
It approaches the
classical result,

1043
01:22:54,470 --> 01:22:58,960
which is 3/2 at
high temperatures.

1044
01:22:58,960 --> 01:23:02,140
At low temperatures, we said
it is simply proportional

1045
01:23:02,140 --> 01:23:08,030
to T to the 3/2, so I
have this kind of behavior

1046
01:23:08,030 --> 01:23:11,080
just a simple T
to the 3/2 curve.

1047
01:23:11,080 --> 01:23:13,820
And one can show
through arguments,

1048
01:23:13,820 --> 01:23:16,970
such as the one that
I showed you above,

1049
01:23:16,970 --> 01:23:19,460
that the heat
capacity is continuous

1050
01:23:19,460 --> 01:23:23,450
but its derivative is
discontinuous at Tc.

1051
01:23:23,450 --> 01:23:28,990
So the behavior overall of the
heat capacity of this Bose gas

1052
01:23:28,990 --> 01:23:30,650
is something like this.

1053
01:23:30,650 --> 01:23:34,770
So next time, we'll elucidate
this a little bit better

1054
01:23:34,770 --> 01:23:38,560
and go on and talk about
experimental realizations

1055
01:23:38,560 --> 01:23:40,940
of Bose-Einstein condensation.