1
00:00:03,590 --> 00:00:09,580
Hello, bonjour, merhaba, assalomu alaykum,

2
00:00:10,400 --> 00:00:15,990
welcome to the third week of Statistical Mechanics: Algorithms and Computations

3
00:00:15,990 --> 00:00:19,210
from the Physics Department of Ecole Normale Superieure.

4
00:00:20,200 --> 00:00:26,690
Last week we discussed how the statistical approach provided an exact description

5
00:00:27,210 --> 00:00:28,870
for physical systems.

6
00:00:30,090 --> 00:00:34,300
As you found out by yourself in Homework Session 2,

7
00:00:35,370 --> 00:00:41,620
this statistical approach captures the system even for finite number of particles.

8
00:00:42,480 --> 00:00:46,580
The equivalence between Newton and Boltzmann

9
00:00:46,580 --> 00:00:50,580
is one of the great miracles of the natural sciences.

10
00:00:52,850 --> 00:00:59,310
It is also a breakthrough in mathematics, because there are now mathematical theorems

11
00:00:59,310 --> 00:01:03,310
that back up our numerical findings,

12
00:01:03,310 --> 00:01:07,310
at least for hard spheres systems.

13
00:01:07,310 --> 00:01:11,930
This week we will study phenomena in the physics of liquids

14
00:01:12,930 --> 00:01:15,930
and this object will accompany us

15
00:01:16,260 --> 00:01:18,250
a giant clothes-pin

16
00:01:19,580 --> 00:01:23,400
that resembles the object that many people use to hang up their clothes.

17
00:01:24,720 --> 00:01:31,670
For us, believe it or not, it will be the key for deep insights into physics.

18
00:01:32,540 --> 00:01:37,740
We place a first one randomly on a washing line

19
00:01:37,740 --> 00:01:39,550
between two poles

20
00:01:40,100 --> 00:01:41,530
like this

21
00:01:42,150 --> 00:01:44,900
and a second one like this

22
00:01:45,410 --> 00:01:47,420
and a third one like this

23
00:01:48,000 --> 00:01:54,160
and so on, until N clothes-pins have been placed onto the line.

24
00:01:55,910 --> 00:02:01,560
Each arrangement of N clothes-pins is equally likely

25
00:02:02,160 --> 00:02:05,560
and we use the tabula rasa rule

26
00:02:06,530 --> 00:02:10,650
to wipe-out a configuration that presents an overlap.

27
00:02:12,140 --> 00:02:17,170
What is the probability for a pin to be at position x?

28
00:02:19,050 --> 00:02:26,410
This simple question will get us to the heart of liquid and soft-matter physics.

29
00:02:28,270 --> 00:02:35,300
We will discover that although apparently the pin positions are independent,

30
00:02:37,170 --> 00:02:41,170
pins in fact experience a strong force:

31
00:02:42,250 --> 00:02:48,440
a force that can be seen in neutron scattering experiments or in X-ray diffraction,

32
00:02:49,180 --> 00:02:50,710
and that can be measured.

33
00:02:51,690 --> 00:03:00,260
This famous interaction was discovered by Asakura and Oosawa in 1954,

34
00:03:01,060 --> 00:03:08,320
and it is so important that some people take it to be the fifth force in nature.

35
00:03:09,370 --> 00:03:12,760
In this week's lecture we will study this interaction.

36
00:03:13,760 --> 00:03:15,960
It comes out of nowhere

37
00:03:15,960 --> 00:03:20,230
and it exists even though there are no charges,

38
00:03:20,230 --> 00:03:23,600
no currents, no mechanical strains.

39
00:03:23,600 --> 00:03:27,190
This is why it is called an entropic interaction.

40
00:03:28,400 --> 00:03:30,340
It can be so strong

41
00:03:30,940 --> 00:03:37,480
that it introduces phase transitions from the liquid to the solid.

42
00:03:38,500 --> 00:03:43,710
The clothes-pins model corresponds to one-dimensional hard disks.

43
00:03:44,960 --> 00:03:48,270
In this week's tutorial - Tutorial 3 -

44
00:03:48,270 --> 00:03:52,270
we will study an ingenious algorithm

45
00:03:53,100 --> 00:04:01,650
that allows to sample assemblies of tens, hundreds, thousands even millions of clothes-pins on a line.

46
00:04:03,120 --> 00:04:10,420
Remember: last week we had trouble simulating four hard disks in a square..

47
00:04:12,080 --> 00:04:16,560
This week we can study so large systems

48
00:04:16,560 --> 00:04:20,560
that we can even approach the infinite-system limit:

49
00:04:20,860 --> 00:04:23,090
the thermodynamic limit.

50
00:04:23,090 --> 00:04:27,090
As you will see several times in this course,

51
00:04:27,090 --> 00:04:29,930
a great algorithmic success

52
00:04:29,930 --> 00:04:36,800
(the fact that we can directly sample thousands or millions of particles without rejections)

53
00:04:37,480 --> 00:04:41,770
is mirrored by the fact that we can solve this model analytically.

54
00:04:42,600 --> 00:04:46,160
This is what you will do in the tutorial of this week.

55
00:04:46,920 --> 00:04:52,250
The analytic solution allows us to drive home essential messages

56
00:04:52,250 --> 00:04:57,680
about liquids and solids, and about the phase transition between them.

57
00:04:58,650 --> 00:05:02,900
The providential Monte Carlo algorithm that we alluded to

58
00:05:02,900 --> 00:05:05,400
works only in one dimension.

59
00:05:07,010 --> 00:05:12,640
In two dimensions and higher (for hard disks or hard spheres)

60
00:05:12,640 --> 00:05:16,640
we must resort to the tools of last week:

61
00:05:17,470 --> 00:05:22,520
Markov chain Monte Carlo algorithms or Molecular Dynamics.

62
00:05:22,780 --> 00:05:30,850
At small densities, everybody understands that the system of hard disks or spheres

63
00:05:30,850 --> 00:05:32,520
is a liquid.

64
00:05:33,420 --> 00:05:37,110
But will it solidify, at higher density?

65
00:05:38,380 --> 00:05:44,070
This is a question that has motivated some of the best work

66
00:05:44,070 --> 00:05:46,500
in computational physics

67
00:05:48,180 --> 00:05:53,040
and that has also brought about the revolution in numerical algorithms:

68
00:05:53,040 --> 00:05:56,270
in Monte Carlo and in Molecular dynamics.

69
00:05:56,810 --> 00:05:59,590
In this week's homework session

70
00:05:59,590 --> 00:06:02,640
you will retrace these classic papers.

71
00:06:03,200 --> 00:06:07,840
Our algorithms and our very simple Python implementations

72
00:06:07,840 --> 00:06:10,010
are just good enough

73
00:06:10,010 --> 00:06:13,630
to get a flavor of what is going on in the system

74
00:06:13,630 --> 00:06:17,090
and to observe our first phase transition.

75
00:06:17,980 --> 00:06:24,680
So let's get started with week 3 of Statistical Mechanics: Algorithms and Computations.

76
00:06:30,070 --> 00:06:32,800
We take up the random clothes-pin model

77
00:06:33,460 --> 00:06:39,930
and place pins on positions x_0, x_1, x_2 and so on

78
00:06:39,930 --> 00:06:41,950
as shown in this program

79
00:06:43,580 --> 00:06:47,410
Each time, we check for the distances of the new pin

80
00:06:47,410 --> 00:06:50,420
with all the pins already present

81
00:06:51,530 --> 00:06:55,230
and if one of these distances is below (2 sigma)

82
00:06:56,040 --> 00:07:01,340
we have an overlap and we reject the configuration through the tabula rasa rule.

83
00:07:03,100 --> 00:07:08,980
This is what allows us to sample configurations x_0, .., x_(N-1)

84
00:07:08,980 --> 00:07:16,990
with a probability pi=constant if the configuration is legal, and pi=0 if it is illegal.

85
00:07:18,320 --> 00:07:22,870
As we are on a line (that is, one dimension)

86
00:07:22,870 --> 00:07:30,460
it is more efficient to sample the position x_0, x_1, x_2, x_3 and so on

87
00:07:30,920 --> 00:07:38,490
then to sort them such that x_0 < x_1 < x_2 < ..

88
00:07:39,820 --> 00:07:45,100
and then check only the (N-1) overlaps between neighboring pins.

89
00:07:45,850 --> 00:07:53,690
So we check for the distance between x_1 and x_0, x_2 and x_1, x_3 and x_2, and so on.

90
00:07:55,010 --> 00:08:00,500
This is written in the program direct_pins_improved.py.

91
00:08:01,580 --> 00:08:05,660
Output of this program for 5 pins

92
00:08:05,660 --> 00:08:12,200
of radius sigma=0.075 on a line of length 1,

93
00:08:12,820 --> 00:08:20,320
(that is with 75% of the line occupied by clothes-pins) is shown here.

94
00:08:21,580 --> 00:08:26,050
Each line shows an accepted configuration

95
00:08:26,940 --> 00:08:33,870
with the 5 positions x of the centers of the clothes-pins.

96
00:08:35,530 --> 00:08:39,690
It has taken about 500 trials

97
00:08:40,520 --> 00:08:44,440
each time to get one accepted configuration.

98
00:08:45,150 --> 00:08:52,500
As we discussed, for N=5, we'll have one acceptance out of about 500.

99
00:08:55,210 --> 00:09:02,810
For N=10, it will be really interesting to see that rejection rate will be so high

100
00:09:02,810 --> 00:09:08,190
that you accept only one out of 500000 samples.

101
00:09:09,520 --> 00:09:14,200
Output will be really interesting for N=15,

102
00:09:16,160 --> 00:09:19,760
but don't try to obtain it with this algorithm

103
00:09:21,860 --> 00:09:26,730
you will accept only one legal configuration

104
00:09:26,730 --> 00:09:30,190
out of 500000000 trials.

105
00:09:31,690 --> 00:09:41,300
Wait until Michael - in this week's tutorial - shows you how to sample this configuration

106
00:09:41,430 --> 00:09:50,800
for any N (as you like: hundred, thousand and millions) or any density, without any effort.

107
00:09:53,110 --> 00:10:02,950
Output of this program (direct_pins_noreject.py) is shown here for N=15

108
00:10:02,950 --> 00:10:13,260
Again, you see the 15 x positions in ordered way, for all the legal configurations that we obtained.

109
00:10:15,060 --> 00:10:19,040
Now, let us analyze these data

110
00:10:20,090 --> 00:10:27,770
and the way to do it is to do a histogram of all the x-positions in this table.

111
00:10:27,770 --> 00:10:31,770
This is shown here.

112
00:10:33,430 --> 00:10:36,430
Look in what a paradoxical situation we are

113
00:10:37,020 --> 00:10:43,530
We place our pins uniformly on an interval, yet the distribution is oscillatory

114
00:10:44,740 --> 00:10:52,870
Many people.. everybody finds this outcome counterintuitive

115
00:10:52,870 --> 00:10:54,900
and difficult to understand.

116
00:10:56,190 --> 00:11:04,170
How can it be that close to the poles we have 4 to 5 times more particles than in the center?

117
00:11:05,930 --> 00:11:09,880
The poles actually attract the particles.

118
00:11:11,310 --> 00:11:16,890
Furthermore, what you see here is not a mere boundary effect.

119
00:11:18,330 --> 00:11:24,250
Particles also attract each other, even though this is more difficult to see.

120
00:11:26,110 --> 00:11:33,450
What we see here (attraction of particles to a wall or the attraction of particles with the others)

121
00:11:33,450 --> 00:11:39,600
is what Asakura and Oosawa found in 1954 in their famous paper

122
00:11:40,260 --> 00:11:43,600
and this is what we will study in a moment.

123
00:11:43,600 --> 00:11:52,690
Before doing so, please take a moment to download, run and modify the two programs we discussed in this section.

124
00:11:53,810 --> 00:11:59,130
On the Coursera website, you'll find the program direct_pins.py

125
00:11:59,870 --> 00:12:06,680
that generates random configurations of N clothes-pins using the tabula rasa rule.

126
00:12:08,270 --> 00:12:13,860
Please check out also the program direct_pins_improved.py

127
00:12:13,860 --> 00:12:16,730
that uses the sorting trick.

128
00:12:17,660 --> 00:12:23,690
And please be patient, for the direct sampling algorithm without rejections

129
00:12:23,690 --> 00:12:29,990
(direct_pins_noreject.py) that we will discuss in this week's tutorial.

130
00:12:35,580 --> 00:12:39,100
When drawing configurations of pins

131
00:12:39,100 --> 00:12:44,660
we must realize that there are two types of excluded regions.

132
00:12:45,940 --> 00:12:53,170
The first type is the core, the space occupied by the pin itself,

133
00:12:54,040 --> 00:12:57,900
no other particles can penetrate into this core.

134
00:12:59,380 --> 00:13:06,190
But we must realize that the position of the pin is given by its center

135
00:13:07,490 --> 00:13:13,520
and this creates a second type of excluded region that we call the halo.

136
00:13:14,460 --> 00:13:20,630
The center of another particle can penetrate neither into the halo

137
00:13:20,630 --> 00:13:22,630
nor into the core.

138
00:13:27,120 --> 00:13:32,120
The poles also have a halo, of size sigma.

139
00:13:32,920 --> 00:13:37,790
Because all pins must be at x > sigma

140
00:13:37,790 --> 00:13:41,790
and x < L - sigma.

141
00:13:42,380 --> 00:13:47,570
The total core area of our N pins is fixed:

142
00:13:47,570 --> 00:13:51,570
it is equal to (2 N sigma).

143
00:13:53,720 --> 00:13:58,070
The total area of the halo is not fixed

144
00:14:00,270 --> 00:14:09,510
If we put the pin into the center (far from the poles) the total halo area is equal to 4 sigma

145
00:14:10,340 --> 00:14:15,390
if we put the pin to one of the poles

146
00:14:16,340 --> 00:14:20,260
the total halo area is equal to 2 sigma

147
00:14:21,200 --> 00:14:24,950
it follows that the configuration on the right

148
00:14:25,730 --> 00:14:30,300
has more accessible space for other particles to be put,

149
00:14:30,980 --> 00:14:37,180
and a smaller chance to undergo a tabula rasa wipe-out.

150
00:14:38,480 --> 00:14:45,850
This means that the configuration to the right has a higher probability than the configuration to the left

151
00:14:47,030 --> 00:14:52,500
or that the particle is attracted to the boundaries.

152
00:14:54,120 --> 00:14:59,390
You can also say that there is a force between the boundary and the pin.

153
00:15:00,420 --> 00:15:05,760
Let us now see for ourselves whether the halo picture

154
00:15:05,760 --> 00:15:09,760
predicts interactions between particles.

155
00:15:10,210 --> 00:15:16,040
We will consider a line of length L, with a pole to the left and a pole to the right

156
00:15:16,680 --> 00:15:23,410
and two pins, with radius sigma, which means core area 2 sigma.

157
00:15:25,050 --> 00:15:26,660
Question 1

158
00:15:27,730 --> 00:15:31,760
what is the available space for other particles

159
00:15:32,640 --> 00:15:39,740
if we place our two pins far from each other and far from the poles?

160
00:15:43,080 --> 00:15:48,470
Answer: for two pins that are far from each other and are far from the poles

161
00:15:49,070 --> 00:15:57,280
the available space for other particles is equal to (L - 10 sigma), as you can see from this figure.

162
00:15:59,560 --> 00:16:01,280
Question 2

163
00:16:01,470 --> 00:16:05,650
what is the available space for other particles

164
00:16:05,650 --> 00:16:12,120
if we place the two pins close together but far from the poles?

165
00:16:14,700 --> 00:16:15,580
Answer:

166
00:16:15,910 --> 00:16:25,240
the available space for the other particles is equal to L - 8 sigma, as you see in this figure.

167
00:16:29,310 --> 00:16:30,480
Third question:

168
00:16:31,590 --> 00:16:36,360
what is the available space for the other particles

169
00:16:36,890 --> 00:16:44,300
if we put one pin close to the left pole and another pin close to the right pole?

170
00:16:48,110 --> 00:16:48,970
Answer:

171
00:16:49,690 --> 00:16:58,030
the available space for the other particles is equal to (L - 6 sigma), as you see here.

172
00:16:59,560 --> 00:17:01,090
Final question:

173
00:17:02,430 --> 00:17:10,150
the halo picture, can it fully describe the density profile that we obtained earlier?

174
00:17:12,790 --> 00:17:14,400
The answer is no,

175
00:17:15,270 --> 00:17:22,930
the halo picture nicely describes the increase of density close to the boundaries

176
00:17:23,950 --> 00:17:30,030
but it cannot account for the intricate oscillations of the profile.

177
00:17:31,270 --> 00:17:38,470
Please be patient for this week's tutorial, where we will derive an analytical formula

178
00:17:39,250 --> 00:17:45,680
which exactly describes this density profile that we obtained from numerical simulations.

179
00:17:47,440 --> 00:17:53,370
The halo picture, although it is approximate, applies in any dimension

180
00:17:54,930 --> 00:18:00,630
and it explains why in Homework Session 2 (last week's homework)

181
00:18:01,210 --> 00:18:07,970
you obtained an inhomogeneous density profile for the 4 hard disks in a box.

182
00:18:09,510 --> 00:18:11,710
For a disk of radius sigma,

183
00:18:12,400 --> 00:18:17,400
there is a ring of radius sigma that forms the halo

184
00:18:18,040 --> 00:18:26,750
No other center of disks can penetrate into the halo or the core area.

185
00:18:28,290 --> 00:18:35,190
But notice that the halo itself can penetrate into another halo

186
00:18:35,190 --> 00:18:41,190
or into another core area, isn't that curious?

187
00:18:41,800 --> 00:18:49,360
Here, on the screen, you see three configurations of two disks in a box.

188
00:18:51,870 --> 00:18:55,690
In configuration c, the two disks are in the corners

189
00:18:57,520 --> 00:19:04,860
and the available space for other particles is much larger than for configuration b,

190
00:19:07,560 --> 00:19:14,980
but configuration a has the least accessible space for other particles.

191
00:19:17,000 --> 00:19:22,770
If we have two particles already in configuration c, that means in the corners,

192
00:19:23,740 --> 00:19:26,420
we have much more available space for the other particles,

193
00:19:26,420 --> 00:19:31,820
and a lower chance to undergo a tabula rasa wipe-out.

194
00:19:33,210 --> 00:19:41,280
This means we expect a higher density in the corner of the box than in the center

195
00:19:41,950 --> 00:19:46,600
and this is exactly what you observed in last week's homework

196
00:19:47,590 --> 00:19:54,070
but you probably concluded that these inhomogeneities were a boundary effect.

197
00:19:55,360 --> 00:19:58,070
This misses the main point.

198
00:19:58,930 --> 00:20:04,600
Consider the configuration b: the two particles attract each other.

199
00:20:15,400 --> 00:20:22,950
In the one dimensional clothes-pin model, the oscillations that you see here decay exponentially

200
00:20:22,950 --> 00:20:26,950
on the scale of a few clothes-pins.

201
00:20:28,250 --> 00:20:36,570
This means that the boundary effects and the pair correlations decay on the scale of a few sigma

202
00:20:37,900 --> 00:20:42,960
and this implies that the Asakura-Oosawa interaction

203
00:20:42,960 --> 00:20:49,830
leads to local modulations of densities and pair correlations.

204
00:20:51,420 --> 00:20:57,050
On long length scales, the system is completely homogeneous

205
00:20:57,910 --> 00:21:02,080
and this is what defines the liquid state.

206
00:21:02,890 --> 00:21:07,480
The clothes-pin model is an example

207
00:21:07,480 --> 00:21:13,180
of a very general class of physical systems with short range interactions.

208
00:21:13,180 --> 00:21:23,280
Powerful mathematical theorems exclude the possibility of a phase transition in this systems in one spatial dimension.

209
00:21:25,290 --> 00:21:29,360
The question is now whether these interactions that we discussed

210
00:21:29,360 --> 00:21:36,300
can be strong enough in 2 and higher dimensions to be seen on arbitrary length scales

211
00:21:37,690 --> 00:21:40,520
and whether they can introduce phase transitions.

212
00:21:42,290 --> 00:21:50,020
This will be the subject of the next section and also of this week's homework session.

213
00:21:55,450 --> 00:22:02,240
In this week's tutorial, we will continue our detailed study of the random clothes-pin model,

214
00:22:02,900 --> 00:22:11,050
but here let me put what we have done and what we will do in the tutorial into a wider prospective.

215
00:22:12,460 --> 00:22:16,810
And I will start from a discussion of the two extremes

216
00:22:16,810 --> 00:22:20,810
the close-packed limit and the dilute system

217
00:22:20,810 --> 00:22:24,290
considering systems with periodic boundary conditions.

218
00:22:26,840 --> 00:22:34,660
In a one dimensional system, for random clothes-pin the close-packed limit eta=1 is special

219
00:22:35,630 --> 00:22:43,810
We have one pin at x_0, x_0 + 2 sigma, x_0 + 4 sigma, x_0 + 6 sigma, and so on..

220
00:22:45,080 --> 00:22:51,550
even though, because of the periodic boundary conditions, we do not know where is x_0

221
00:22:52,310 --> 00:22:55,920
we have position long-range order in the system

222
00:22:57,980 --> 00:23:02,340
but we also know (in fact we will prove, in Tutorial 3)

223
00:23:02,780 --> 00:23:07,950
that for all densities lower than eta=1

224
00:23:07,950 --> 00:23:11,950
the system is different, it is liquid-like

225
00:23:12,590 --> 00:23:16,220
like it is in a very low density limit.

226
00:23:17,310 --> 00:23:21,810
In two dimensions, that is for systems of hard disks,

227
00:23:22,610 --> 00:23:29,700
the close-packed density is equal to pi / (2 sqrt(3))

228
00:23:30,350 --> 00:23:34,240
and this is about 0.907.

229
00:23:35,830 --> 00:23:43,270
And the close-packing configuration is hexagonal, as shown here in the picture.

230
00:23:45,930 --> 00:23:52,830
The mathematician Laszlo Fejes Toth proved in 1940

231
00:23:52,830 --> 00:23:58,670
that no other configuration than the hexagonal packing

232
00:23:58,670 --> 00:24:03,540
exists at this density pi / (2 sqrt(3)).

233
00:24:04,630 --> 00:24:12,470
So we are sure to have long range positional and orientational order

234
00:24:12,710 --> 00:24:15,950
in this system, at the close-packing density.

235
00:24:17,160 --> 00:24:20,550
There are no other mathematical results

236
00:24:21,070 --> 00:24:26,470
about this system, besides the fact that at very small density

237
00:24:26,470 --> 00:24:29,510
the system is liquid.

238
00:24:30,660 --> 00:24:36,150
To find out what is going on at intermediate densities

239
00:24:37,810 --> 00:24:41,490
we have to resort to numerical simulations:

240
00:24:43,200 --> 00:24:50,830
Markov chain Monte Carlo simulations and Molecular Dynamics simulations allow us to sample this.

241
00:24:51,710 --> 00:24:57,950
For example here you see a system of 256 disks

242
00:24:58,510 --> 00:25:06,900
at density 0.48, again with periodic boundary conditions in x and y.

243
00:25:08,160 --> 00:25:14,260
Here, the Asakura-Oosawa depletion interaction is at work,

244
00:25:15,540 --> 00:25:21,580
and it produces some unconnected local arrangements of disks,

245
00:25:21,580 --> 00:25:25,060
that resemble the close-packing limit

246
00:25:25,060 --> 00:25:32,570
at density 0.907. Look at the highlighted spots in the picture.

247
00:25:34,240 --> 00:25:40,820
At higher density, look here, the system all of a sudden changes.

248
00:25:43,000 --> 00:25:48,480
What you see here on the right is qualitatively different,

249
00:25:48,480 --> 00:25:51,630
from the configuration on the left.

250
00:25:52,680 --> 00:25:55,630
A phase transition has taken place.

251
00:25:57,450 --> 00:26:03,780
The discovery of this phase transition from the configuration on the left (the liquid one)

252
00:26:03,780 --> 00:26:08,390
to a denser phase is a computational fact,

253
00:26:08,870 --> 00:26:14,650
a computational achievement, and it also been observed in many experiments.

254
00:26:15,620 --> 00:26:19,420
But it is not backed by mathematical theorems.

255
00:26:19,910 --> 00:26:25,280
This transition at a finite density, below close-packing,

256
00:26:25,960 --> 00:26:32,250
takes place in two dimensions, but also in three dimensions for hard spheres.

257
00:26:34,170 --> 00:26:44,360
The existence of such a transition was first suggested in theoretical works by Kirkwood and Monroe in 1941.

258
00:26:48,930 --> 00:26:56,220
In conclusion, please note that in Lecture 3 of Statistical Mechanics: Algorithms and Computations

259
00:26:56,220 --> 00:27:03,210
we have remained within the tight conceptual framework of the equal probability principle.

260
00:27:03,830 --> 00:27:06,680
What a simple principle

261
00:27:07,080 --> 00:27:13,620
but what far reaching consequences and rich and surprising phenomena!

262
00:27:14,700 --> 00:27:20,490
And what a nice interplay between algorithms and theory!

263
00:27:21,840 --> 00:27:28,150
In coming weeks, we will continue to develop the conceptual framework,

264
00:27:28,150 --> 00:27:32,150
the consequences, and the algorithms.

265
00:27:33,420 --> 00:27:40,300
I hope to keep up your interest in Statistical Mechanics: Algorithms and Computations.

266
00:27:40,300 --> 00:27:44,300
In the meantime, let me thank you for listening.