1 00:00:00,060 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high quality educational resources for free. 5 00:00:10,730 --> 00:00:13,330 To make a donation, or view additional materials 6 00:00:13,330 --> 00:00:17,236 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,236 --> 00:00:17,861 at ocw.mit.edu. 8 00:00:20,689 --> 00:00:21,480 PROFESSOR: Alright. 9 00:00:21,480 --> 00:00:27,790 So what we are doing is covering 8.334, 10 00:00:27,790 --> 00:00:30,225 which is statistical physics. 11 00:00:35,740 --> 00:00:41,160 ANd let me remind you that basically statistical physics 12 00:00:41,160 --> 00:00:52,630 is a bridge from microscopic to macroscopic perspectives. 13 00:00:52,630 --> 00:00:58,330 And I'm going to emphasize a lot on changing perspectives. 14 00:00:58,330 --> 00:01:03,210 So at the level of the micro, you 15 00:01:03,210 --> 00:01:08,760 have the microstate that is characterized, maybe, 16 00:01:08,760 --> 00:01:14,070 by a collection of momenta and coordinates of particles, 17 00:01:14,070 --> 00:01:17,390 such as particles of gas in this room. 18 00:01:17,390 --> 00:01:20,830 Maybe those particles have spins. 19 00:01:20,830 --> 00:01:24,870 If you are absorbing things on a surface, 20 00:01:24,870 --> 00:01:27,840 you may have variables that denote the occupation-- 21 00:01:27,840 --> 00:01:31,820 whether a particle or site is occupied or not. 22 00:01:31,820 --> 00:01:36,600 And when you're looking at the microscopic world, 23 00:01:36,600 --> 00:01:39,340 typically you are interested in how 24 00:01:39,340 --> 00:01:41,800 things change as a function of time. 25 00:01:41,800 --> 00:01:45,170 You have a kind of dynamics that is governed 26 00:01:45,170 --> 00:01:48,340 by some kind of a [INAUDIBLE] which 27 00:01:48,340 --> 00:01:53,110 is dependent on the microstate that you are looking at, 28 00:01:53,110 --> 00:01:57,071 and tells you about the evolution of the microstate. 29 00:01:57,071 --> 00:02:00,460 At the other extreme, you look at the world 30 00:02:00,460 --> 00:02:05,780 around you and the macro world. 31 00:02:05,780 --> 00:02:09,919 You're dealing with things that are described 32 00:02:09,919 --> 00:02:11,910 by completely different things. 33 00:02:11,910 --> 00:02:14,320 For example, if you're thinking about the gas, 34 00:02:14,320 --> 00:02:15,340 you have the pressure. 35 00:02:15,340 --> 00:02:17,310 You have the volume. 36 00:02:17,310 --> 00:02:20,450 So somehow, the coordinates managed 37 00:02:20,450 --> 00:02:26,900 to define and macrostate, which is characterized 38 00:02:26,900 --> 00:02:29,860 by a few parameters in equilibrium. 39 00:02:29,860 --> 00:02:33,680 If you have something like a collection of spins, 40 00:02:33,680 --> 00:02:37,030 maybe you will have a magnetization. 41 00:02:37,030 --> 00:02:38,870 And again, another important thing 42 00:02:38,870 --> 00:02:41,280 that characterizes the equilibrium we 43 00:02:41,280 --> 00:02:44,140 describe as temperature. 44 00:02:44,140 --> 00:02:45,980 And when you're looking at things 45 00:02:45,980 --> 00:02:50,870 from the perspective of macro systems in equilibrium, 46 00:02:50,870 --> 00:02:56,170 then you have the laws of thermodynamics 47 00:02:56,170 --> 00:03:01,810 that govern constraints that are placed on these variables. 48 00:03:01,810 --> 00:03:05,290 So totally different perspectives. 49 00:03:05,290 --> 00:03:10,580 And what we have is that we need to build a bridge from one 50 00:03:10,580 --> 00:03:13,710 to the other and this bridge is provided 51 00:03:13,710 --> 00:03:16,490 by statistical mechanics. 52 00:03:16,490 --> 00:03:21,650 And the way that we described it in the previous course 53 00:03:21,650 --> 00:03:25,710 is that to what you need is a probabilistic prescription. 54 00:03:25,710 --> 00:03:28,950 So rather than following the time evolution 55 00:03:28,950 --> 00:03:31,180 of all of these degrees of freedom, 56 00:03:31,180 --> 00:03:33,280 you'll have a probability assigned 57 00:03:33,280 --> 00:03:36,700 to the different microstates that 58 00:03:36,700 --> 00:03:39,640 are dependent on the macro state. 59 00:03:39,640 --> 00:03:41,470 And for example, if you are dealing 60 00:03:41,470 --> 00:03:44,240 in the canonical ensemble, then you 61 00:03:44,240 --> 00:03:46,660 know you're at the particular temperature 62 00:03:46,660 --> 00:03:51,040 this form has E to the minus beta, this Hamiltonian 63 00:03:51,040 --> 00:03:54,930 that we you have here governing the dynamics. 64 00:03:54,930 --> 00:03:57,320 Beta was one over KT. 65 00:03:57,320 --> 00:04:01,070 So I could write it in this fashion. 66 00:04:01,070 --> 00:04:05,880 And the thing that enabled us to make 67 00:04:05,880 --> 00:04:11,030 a connection between this deterministic perspective 68 00:04:11,030 --> 00:04:15,780 and this equilibrium description wire probabilities abilities 69 00:04:15,780 --> 00:04:21,910 was relying on the limit where the number of degrees 70 00:04:21,910 --> 00:04:25,070 of freedom was very large. 71 00:04:25,070 --> 00:04:28,350 And this very large number of degrees of freedom 72 00:04:28,350 --> 00:04:32,840 enabled us to, although we had something that was probably 73 00:04:32,840 --> 00:04:36,810 realistic, to really make very precise statements about what 74 00:04:36,810 --> 00:04:39,080 was happening. 75 00:04:39,080 --> 00:04:45,036 Now, when we were doing this program in 8-333, 76 00:04:45,036 --> 00:04:50,490 we looked at very simple systems that were essentially non 77 00:04:50,490 --> 00:04:54,730 interacting, like the ideal gas. 78 00:04:54,730 --> 00:04:58,470 Or we put a little bit of weak perturbation 79 00:04:58,470 --> 00:05:04,430 and by some manipulations, we got things like liquids. 80 00:05:04,430 --> 00:05:07,690 But the important thing is that where is this program 81 00:05:07,690 --> 00:05:12,380 we could carry out precisely when we had no interactions. 82 00:05:12,380 --> 00:05:18,320 In the presence of interactions we 83 00:05:18,320 --> 00:05:22,350 encountered, even in our simplified perspective, 84 00:05:22,350 --> 00:05:23,140 new things. 85 00:05:23,140 --> 00:05:25,910 We went from the gas state to a liquid state. 86 00:05:25,910 --> 00:05:28,430 We didn't discuss it, but we certainly 87 00:05:28,430 --> 00:05:30,850 said a few things about solids. 88 00:05:30,850 --> 00:05:36,700 And clearly there are much more interesting things 89 00:05:36,700 --> 00:05:39,430 that can happen when you have interactions. 90 00:05:39,430 --> 00:05:43,060 You could have other phases of matter that we didn't discuss, 91 00:05:43,060 --> 00:05:50,000 such as liquid crystals, superconductors, 92 00:05:50,000 --> 00:05:55,050 and many, many other things. 93 00:05:55,050 --> 00:06:02,570 So the key idea is that you can solve things that are not 94 00:06:02,570 --> 00:06:04,950 interacting going their own ways, 95 00:06:04,950 --> 00:06:08,260 but when you put interactions, you 96 00:06:08,260 --> 00:06:10,990 get interesting collective behaviors. 97 00:06:16,900 --> 00:06:22,780 And what we want to do in 8-334, as opposed 98 00:06:22,780 --> 00:06:26,990 to just building the machinery in 8-333, 99 00:06:26,990 --> 00:06:30,150 is to think about all of the different types 100 00:06:30,150 --> 00:06:33,060 of collective behavior that are possible 101 00:06:33,060 --> 00:06:39,020 and how to describe them in the realm of classical systems. 102 00:06:39,020 --> 00:06:42,610 So we won't go much into the realm of quantum systems, which 103 00:06:42,610 --> 00:06:46,420 is also quite interesting as far as its own collective behaviors 104 00:06:46,420 --> 00:06:48,310 is concerned. 105 00:06:48,310 --> 00:06:51,390 So that's the program. 106 00:06:51,390 --> 00:06:54,630 Now, what I would like to do is to go 107 00:06:54,630 --> 00:06:58,800 to the description of the organization of the class, 108 00:06:58,800 --> 00:07:03,770 and hopefully this web page will come back online. 109 00:07:13,930 --> 00:07:20,700 So repeating what I was saying before, the idea is 110 00:07:20,700 --> 00:07:24,310 that interactions give rise to a variety 111 00:07:24,310 --> 00:07:27,950 of interesting collective behaviors, 112 00:07:27,950 --> 00:07:32,440 starting from pretty much simple degrees of freedom 113 00:07:32,440 --> 00:07:35,130 such as atoms and molecules and adding a little bit 114 00:07:35,130 --> 00:07:37,960 of interactions and complexity to them, 115 00:07:37,960 --> 00:07:43,570 you could get a variety of, for example, liquid crystal phases. 116 00:07:43,570 --> 00:07:49,150 But one of things to think about is that you think about all 117 00:07:49,150 --> 00:07:52,260 of the different atoms and molecules that you 118 00:07:52,260 --> 00:07:55,670 can put together, and the different interactions that you 119 00:07:55,670 --> 00:07:59,270 could put among them-- you could even imagine things 120 00:07:59,270 --> 00:08:02,970 that you construct in the lab that didn't exist in nature-- 121 00:08:02,970 --> 00:08:08,570 and you put them together, but the types of new behavior 122 00:08:08,570 --> 00:08:11,390 that you encounter is not that much. 123 00:08:11,390 --> 00:08:15,380 At some level, you learn about the three phases of matter, 124 00:08:15,380 --> 00:08:20,150 gas, liquids, solids, of course, as I said, there are many more, 125 00:08:20,150 --> 00:08:22,540 but there are not hundreds of them. 126 00:08:22,540 --> 00:08:24,270 There are just a few. 127 00:08:24,270 --> 00:08:28,790 So the question is why, given the complexity of interactions 128 00:08:28,790 --> 00:08:31,530 that you can have at the microscopic scale, 129 00:08:31,530 --> 00:08:33,919 you put things together, and you really 130 00:08:33,919 --> 00:08:37,370 don't get that many variety of things 131 00:08:37,370 --> 00:08:39,330 at the macroscopic level. 132 00:08:39,330 --> 00:08:44,190 And the answer has to do with mathematical consistency, 133 00:08:44,190 --> 00:08:47,370 and has at its heart something that you already 134 00:08:47,370 --> 00:08:51,570 saw in 8-333, which is the central limit theorem. 135 00:08:51,570 --> 00:08:56,880 You take many random variables of whatever ilk 136 00:08:56,880 --> 00:08:59,820 and you add them together, and the sum 137 00:08:59,820 --> 00:09:03,010 has a nice simple Gaussian distribution. 138 00:09:03,010 --> 00:09:08,830 So somehow mathematics forces things to become simplified 139 00:09:08,830 --> 00:09:11,740 when you put many of them together. 140 00:09:11,740 --> 00:09:13,680 So the same thing is happening when 141 00:09:13,680 --> 00:09:16,950 you put lots of interacting pieces together. 142 00:09:16,950 --> 00:09:20,290 New collective behaviors emerge, but because 143 00:09:20,290 --> 00:09:22,580 of the simplification, in the same sense 144 00:09:22,580 --> 00:09:25,390 the type of simplification that we see with the central limit 145 00:09:25,390 --> 00:09:28,900 theorem, there aren't that many consistent mathematical 146 00:09:28,900 --> 00:09:31,750 descriptions that can emerge. 147 00:09:31,750 --> 00:09:36,480 Of course, there are nice new ones, 148 00:09:36,480 --> 00:09:40,050 and the question is how to describe them. 149 00:09:40,050 --> 00:09:46,500 So this issue of what are possible consistent 150 00:09:46,500 --> 00:09:49,800 mathematical forms is what we will 151 00:09:49,800 --> 00:09:53,190 address through constructing these statistical field 152 00:09:53,190 --> 00:09:54,120 theories. 153 00:09:54,120 --> 00:09:57,830 So one of the important things that I 154 00:09:57,830 --> 00:10:05,090 will try to impose upon you is to change your perspective. 155 00:10:05,090 --> 00:10:08,640 In the same way that there is a big change of perspective 156 00:10:08,640 --> 00:10:12,970 thinking about the microscopic and macroscopic board, 157 00:10:12,970 --> 00:10:16,770 there is also a change of perspective involved 158 00:10:16,770 --> 00:10:22,240 in the idea of starting from interacting degrees of freedom, 159 00:10:22,240 --> 00:10:24,730 and changing perspective and constructing 160 00:10:24,730 --> 00:10:26,770 a statistical future. 161 00:10:26,770 --> 00:10:29,550 And I'll give you one example of that today 162 00:10:29,550 --> 00:10:32,370 that you're hopefully all familiar with, 163 00:10:32,370 --> 00:10:36,710 but it shows very much how this change of perspective works, 164 00:10:36,710 --> 00:10:38,720 and that's the kind of metallurgy 165 00:10:38,720 --> 00:10:41,800 that we will apply in this course. 166 00:10:41,800 --> 00:10:47,780 Basically the syllabus is as follows. 167 00:10:47,780 --> 00:10:56,650 So initially I will try to emphasize to you how, 168 00:10:56,650 --> 00:11:00,360 by looking at things, by averaging over 169 00:11:00,360 --> 00:11:06,660 many degrees of freedom, at long lens scales and time scales, 170 00:11:06,660 --> 00:11:11,660 you get simplified statistical field theory descriptions. 171 00:11:11,660 --> 00:11:14,070 The simplest one of these that appears 172 00:11:14,070 --> 00:11:18,120 in many different contexts is this Landau-Ginzburg model 173 00:11:18,120 --> 00:11:22,820 that will occupy us for quite a few lectures. 174 00:11:22,820 --> 00:11:25,955 Now once you construct a description of one 175 00:11:25,955 --> 00:11:27,680 of these statistical field theories, 176 00:11:27,680 --> 00:11:30,510 the question is how do you solve it. 177 00:11:30,510 --> 00:11:32,850 And there are a number of approaches 178 00:11:32,850 --> 00:11:37,450 that we will follow such as mean field theory, et cetera. 179 00:11:37,450 --> 00:11:40,830 And what we'll find is that those descriptions 180 00:11:40,830 --> 00:11:43,430 fail in certain dimensions. 181 00:11:43,430 --> 00:11:46,070 And then to do better than that, you 182 00:11:46,070 --> 00:11:50,160 have to rely on things such as perturbation theory. 183 00:11:50,160 --> 00:11:52,490 So this is kind of perturbation theory 184 00:11:52,490 --> 00:11:55,350 that builds upon the types of perturbation theory 185 00:11:55,350 --> 00:11:58,130 that we did in the previous semester, what 186 00:11:58,130 --> 00:12:00,860 is closer to the kind of perturbation theory 187 00:12:00,860 --> 00:12:05,330 that you would be doing in quantum field theories. 188 00:12:05,330 --> 00:12:07,885 Alongside these continuum theories, 189 00:12:07,885 --> 00:12:13,480 I will also develop some lattice models, that in certain limits, 190 00:12:13,480 --> 00:12:19,040 are simpler and admit either numerical approaches 191 00:12:19,040 --> 00:12:22,380 or exact solutions. 192 00:12:22,380 --> 00:12:26,940 The key idea that we will try to learn about 193 00:12:26,940 --> 00:12:31,110 is that of Kadanoff's perspective of renormalization 194 00:12:31,110 --> 00:12:35,480 group and how you can, in some sense, 195 00:12:35,480 --> 00:12:38,600 change your perspective continuously 196 00:12:38,600 --> 00:12:42,160 by looking at how a system would look over 197 00:12:42,160 --> 00:12:44,730 larger and larger landscape, and how 198 00:12:44,730 --> 00:12:48,150 the mathematical description that you have for the system 199 00:12:48,150 --> 00:12:51,250 changes as a function of the scale of observation, 200 00:12:51,250 --> 00:12:57,250 and hopefully becomes simple at sufficiently large scales. 201 00:12:57,250 --> 00:13:03,060 So we will conclude by looking at a variety of applications 202 00:13:03,060 --> 00:13:05,260 of the methodologies that we have developed. 203 00:13:05,260 --> 00:13:08,640 For example, in the context of two dimensional films, 204 00:13:08,640 --> 00:13:15,180 and potentially if we have time, a number of other systems. 205 00:13:15,180 --> 00:13:20,930 The first thing that I'm going-- the rest of the lecture today 206 00:13:20,930 --> 00:13:24,900 has essentially nothing in common with the material 207 00:13:24,900 --> 00:13:29,000 that we will start to cover as of the second lecture. 208 00:13:29,000 --> 00:13:32,380 But it illustrates this change in perspective 209 00:13:32,380 --> 00:13:35,880 that is essential to the way of thinking 210 00:13:35,880 --> 00:13:38,860 about the material in this course. 211 00:13:38,860 --> 00:13:45,490 And the context that I will use to introduce this perspective 212 00:13:45,490 --> 00:13:47,040 is phonons and elasticity. 213 00:13:55,810 --> 00:14:00,680 So you look around you, there's a whole bunch 214 00:14:00,680 --> 00:14:03,360 of different solids. 215 00:14:03,360 --> 00:14:07,090 There's metals, there's wood, et cetera. 216 00:14:07,090 --> 00:14:09,380 And for each one of them, you can 217 00:14:09,380 --> 00:14:13,330 ask things about their thermodynamic properties, heat 218 00:14:13,330 --> 00:14:17,230 content, for example, heat capacity. 219 00:14:17,230 --> 00:14:21,560 They are constructed of various very different materials. 220 00:14:21,560 --> 00:14:24,900 And so let's try to think, if you're 221 00:14:24,900 --> 00:14:29,130 going to start from this left side 222 00:14:29,130 --> 00:14:31,470 from the microscopic perspective, 223 00:14:31,470 --> 00:14:35,000 how we would approach the problem of the heat 224 00:14:35,000 --> 00:14:38,200 content of these solids. 225 00:14:38,200 --> 00:14:47,570 So you would say that solids, if I were to go to 0 temperature 226 00:14:47,570 --> 00:14:52,073 before I put any heat into them, they would be perfect crystals. 227 00:14:56,060 --> 00:14:57,570 What does that mean? 228 00:14:57,570 --> 00:15:06,000 It means that if I look at the positions of these atoms 229 00:15:06,000 --> 00:15:09,300 or particles that are making up the crystal-- I guess 230 00:15:09,300 --> 00:15:15,180 I have to be more specific since a metal is composed of nuclei 231 00:15:15,180 --> 00:15:16,940 and electrons-- let's imagine that we 232 00:15:16,940 --> 00:15:19,860 look at the positions of the ions. 233 00:15:19,860 --> 00:15:23,320 Then in the perfect position they 234 00:15:23,320 --> 00:15:27,450 will form a lattice where I will pick three integers, L M 235 00:15:27,450 --> 00:15:35,110 N and three unit vectors, and I can 236 00:15:35,110 --> 00:15:40,960 list the positions of all of these ions in the crystal. 237 00:15:40,960 --> 00:15:47,180 So this combination L M N that indicates 238 00:15:47,180 --> 00:15:52,620 the location of some particular ion in the perfect solid 239 00:15:52,620 --> 00:15:56,820 that's indicated by, let's say, the vector r. 240 00:15:56,820 --> 00:16:03,580 So this is the position that I would have ideally for the ion 241 00:16:03,580 --> 00:16:07,530 at zero temperature that is labelled by r. 242 00:16:07,530 --> 00:16:10,460 Now, of course when we go to finite temperatures, 243 00:16:10,460 --> 00:16:16,815 the particles, rather than forming this nice lattice, 244 00:16:16,815 --> 00:16:20,910 let's imagine a square lattice in two dimension, 245 00:16:20,910 --> 00:16:24,040 starts to move around. 246 00:16:24,040 --> 00:16:29,980 And they're no longer going to be perfect positions. 247 00:16:29,980 --> 00:16:37,800 And these distortions we can indicate by some vector u. 248 00:16:37,800 --> 00:16:43,980 So when we have perfect crystals across deformations, 249 00:16:43,980 --> 00:16:48,930 this Q star changes to a new position, 250 00:16:48,930 --> 00:16:57,110 Q of r, which is it's ideal position plus a distortion 251 00:16:57,110 --> 00:17:02,350 field u, at each location. 252 00:17:02,350 --> 00:17:04,260 OK? 253 00:17:04,260 --> 00:17:07,940 Now associated with these change of positions, 254 00:17:07,940 --> 00:17:12,619 you have moved away from the lowest energy configuration. 255 00:17:12,619 --> 00:17:15,030 You have put energy in the system. 256 00:17:15,030 --> 00:17:19,680 And you would say that the energy of the system 257 00:17:19,680 --> 00:17:22,470 is going to be composed of the following parts. 258 00:17:22,470 --> 00:17:26,240 There's always going to be some kind of a kinetic energy. 259 00:17:26,240 --> 00:17:32,650 That's sum over r p r squared over 2 m. 260 00:17:32,650 --> 00:17:36,780 Let's imagine particles all have the same mass m. 261 00:17:36,780 --> 00:17:43,760 And then the reason that these particles 262 00:17:43,760 --> 00:17:46,830 form this perfect crystal was presumably 263 00:17:46,830 --> 00:17:49,620 because of the overlap of electronic wave 264 00:17:49,620 --> 00:17:50,950 functions, et cetera. 265 00:17:50,950 --> 00:17:54,980 Eventually, if you're looking at these coordinates, 266 00:17:54,980 --> 00:17:58,070 there's some kind of a many body potential 267 00:17:58,070 --> 00:18:01,546 as a function of the positions of all of these particles. 268 00:18:07,720 --> 00:18:13,970 Now, what we are doing is we are looking 269 00:18:13,970 --> 00:18:17,970 at distortions that are small. 270 00:18:17,970 --> 00:18:19,690 So let's imagine that you haven't 271 00:18:19,690 --> 00:18:24,180 gone to such high temperatures where this crystal completely 272 00:18:24,180 --> 00:18:27,300 melts and disappears and gives us something else. 273 00:18:27,300 --> 00:18:32,490 But we have small changes from what 274 00:18:32,490 --> 00:18:35,740 we had at zero temperature. 275 00:18:35,740 --> 00:18:39,690 So presumably the crystal corresponds 276 00:18:39,690 --> 00:18:44,140 to the configuration that gives us the minimum energy. 277 00:18:44,140 --> 00:18:50,680 And then if I make a distortion and expand this potential 278 00:18:50,680 --> 00:18:54,880 in the distortion field, the lowest order term proportional 279 00:18:54,880 --> 00:18:58,010 to various u's will disappear because I'm 280 00:18:58,010 --> 00:19:00,130 expanding around the minimum. 281 00:19:00,130 --> 00:19:02,050 And the first thing that I will get 282 00:19:02,050 --> 00:19:04,340 is from the second order term. 283 00:19:04,340 --> 00:19:09,685 So I have one half sum over the different positions. 284 00:19:09,685 --> 00:19:13,290 I have to do the second derivative of the potential 285 00:19:13,290 --> 00:19:14,756 with respect to these deformations. 286 00:19:19,060 --> 00:19:22,250 Of course, if I am in three dimensions, 287 00:19:22,250 --> 00:19:27,080 I also have three spatial indices, x, y, and z, 288 00:19:27,080 --> 00:19:31,170 so I would have to take derivatives with respect 289 00:19:31,170 --> 00:19:35,200 to the different coordinates, alpha and beta, 290 00:19:35,200 --> 00:19:37,130 and summed over them. 291 00:19:37,130 --> 00:19:42,560 And then I have u alpha of r, u beta of r prime. 292 00:19:42,560 --> 00:19:45,060 And then, of course, I will have higher order terms 293 00:19:45,060 --> 00:19:46,600 in this expansion. 294 00:19:46,600 --> 00:19:50,270 This is a general potential, so then in higher order terms, 295 00:19:50,270 --> 00:19:54,040 it would be order of u cubed and higher. 296 00:19:54,040 --> 00:19:56,730 OK? 297 00:19:56,730 --> 00:19:58,550 Fine? 298 00:19:58,550 --> 00:20:02,860 So I have a system of this form. 299 00:20:02,860 --> 00:20:08,900 Now, typically, the next stage if I 300 00:20:08,900 --> 00:20:11,520 stop at the quadratic level-- this 301 00:20:11,520 --> 00:20:14,940 I would do for a molecule also, not only for a solid-- 302 00:20:14,940 --> 00:20:19,060 is to try to find the normal modes of the system. 303 00:20:19,060 --> 00:20:23,260 Normal modes I have to obtain by diagonalizing 304 00:20:23,260 --> 00:20:27,150 this matrix of second derivatives. 305 00:20:27,150 --> 00:20:31,300 Now, there are a few things that I know that help me, 306 00:20:31,300 --> 00:20:33,650 and one of the things that I know 307 00:20:33,650 --> 00:20:39,090 is that because the original structure was a perfect solid, 308 00:20:39,090 --> 00:20:46,120 let's say, then there will be a matrix-- sorry, 309 00:20:46,120 --> 00:20:49,180 there will be an element of the second derivative that 310 00:20:49,180 --> 00:20:52,670 corresponds to that r and this r prime. 311 00:20:52,670 --> 00:20:55,720 That's going to be the same as the second derivative that 312 00:20:55,720 --> 00:20:59,120 connects these two points, because this pair of points 313 00:20:59,120 --> 00:21:01,050 is obtained by the first pair of points 314 00:21:01,050 --> 00:21:03,400 by simple translation among the lattice. 315 00:21:03,400 --> 00:21:06,540 The environment for this is exactly the same 316 00:21:06,540 --> 00:21:08,960 as the environment for this. 317 00:21:08,960 --> 00:21:15,250 So essentially, what I'm stating is that this function does not 318 00:21:15,250 --> 00:21:20,170 depend on r and r prime separately, but only 319 00:21:20,170 --> 00:21:23,216 on the difference between r and r prime. 320 00:21:26,920 --> 00:21:32,180 So, I know a lot-- so this is not 321 00:21:32,180 --> 00:21:36,420 if I had n atoms in my system, this 322 00:21:36,420 --> 00:21:40,560 is not something like n squared over 2 independent things, 323 00:21:40,560 --> 00:21:43,970 it is a much lower number. 324 00:21:43,970 --> 00:21:47,450 The fact that I have such a lower number 325 00:21:47,450 --> 00:22:00,720 allows me to calculate the normal modes of the system 326 00:22:00,720 --> 00:22:01,625 by Fourier transform. 327 00:22:12,950 --> 00:22:16,530 I won't be very precise about how 328 00:22:16,530 --> 00:22:18,800 we perform Fourier transforms. 329 00:22:18,800 --> 00:22:22,380 Basically I start with this k alpha beta, 330 00:22:22,380 --> 00:22:25,750 which is a function of separation. 331 00:22:25,750 --> 00:22:32,465 I can do a sum over all of these separations r of e 332 00:22:32,465 --> 00:22:39,150 to the i K dot r, for appropriately chosen k. 333 00:22:39,150 --> 00:22:43,660 And summing over all pairs of differences, so the argument 334 00:22:43,660 --> 00:22:49,360 r here now, is what was previously r minus r prime. 335 00:22:49,360 --> 00:22:53,340 So basically, what I can do is I can pick one point 336 00:22:53,340 --> 00:23:00,560 and go and look at all of the separations from that point. 337 00:23:00,560 --> 00:23:06,520 Construct this object, this will give me the Fourier transformed 338 00:23:06,520 --> 00:23:10,422 object that depends on the vague number k. 339 00:23:14,040 --> 00:23:16,600 So if I look at the potential energy 340 00:23:16,600 --> 00:23:24,940 of this system minus its value at zero temperature, which 341 00:23:24,940 --> 00:23:29,890 from one perspective was one half sum over r r 342 00:23:29,890 --> 00:23:37,075 prime alpha beta, is k alpha beta, r minus r prime, u alpha 343 00:23:37,075 --> 00:23:46,380 of r, u beta of r prime, in the quadratic approximation. 344 00:23:46,380 --> 00:23:55,700 If I do Fourier transforms what happens because it is only 345 00:23:55,700 --> 00:23:58,910 a function of r minus r prime, and not r and r prime 346 00:23:58,910 --> 00:24:01,935 separately, is that in Fourier space, 347 00:24:01,935 --> 00:24:08,630 it separates out into a sum that depends only 348 00:24:08,630 --> 00:24:11,110 on individual K modes. 349 00:24:11,110 --> 00:24:14,450 There's no coupling between k and k prime. 350 00:24:14,450 --> 00:24:19,300 So here we have r and r prime, but by the time we get here, 351 00:24:19,300 --> 00:24:22,520 we just have one k, alpha and beta. 352 00:24:22,520 --> 00:24:26,545 We'll do one example of that in more detail later on. 353 00:24:26,545 --> 00:24:32,130 I have the Fourier transformed object, 354 00:24:32,130 --> 00:24:36,950 and then I have u alpha k Fourier transform. 355 00:24:36,950 --> 00:24:40,550 So in the same manner that I Fourier transformed 356 00:24:40,550 --> 00:24:44,565 this kernel, k alpha beta, I can put here a u 357 00:24:44,565 --> 00:24:54,690 and end up here with a U tilde and u tilde beta of k star. 358 00:25:01,270 --> 00:25:05,910 So we start over here, if you like. 359 00:25:05,910 --> 00:25:11,790 If I have n particles with a matrix that is n by n-- 360 00:25:11,790 --> 00:25:17,630 actually 3n by 3n if I account for the three 361 00:25:17,630 --> 00:25:23,300 different orientations-- and by going to Fourier transforms, 362 00:25:23,300 --> 00:25:29,500 we have separated it out for each of n potential Fourier 363 00:25:29,500 --> 00:25:34,020 components, we just have a three by three matrix. 364 00:25:34,020 --> 00:25:38,230 And so then we can potentially diagonalize this three 365 00:25:38,230 --> 00:25:51,620 by three matrix to get two eigenvalues, lambda alpha of K. 366 00:25:51,620 --> 00:25:55,280 Once I have the eigenvalues of the system, 367 00:25:55,280 --> 00:25:57,430 then I can find frequencies or eigenfrequencies. 368 00:26:05,660 --> 00:26:09,380 Omega alpha of K, which would be related 369 00:26:09,380 --> 00:26:12,586 to this lambda alpha of K divided by m. 370 00:26:18,360 --> 00:26:22,340 So you go through this entire process. 371 00:26:22,340 --> 00:26:25,562 The idea was you start with different solids. 372 00:26:25,562 --> 00:26:29,220 You want to know what the heat content of the solid is. 373 00:26:29,220 --> 00:26:31,890 You have to make various approximations 374 00:26:31,890 --> 00:26:35,040 to even think about the normal modes. 375 00:26:35,040 --> 00:26:36,828 You can see that you have to figure out 376 00:26:36,828 --> 00:26:38,286 what this kernel of the interaction 377 00:26:38,286 --> 00:26:42,830 is-- Fourier transform it, diagonalize it, et cetera 378 00:26:42,830 --> 00:26:45,730 And ultimately thing that you're after is that there are 379 00:26:45,730 --> 00:26:50,550 these frequencies as a function of wave number. 380 00:26:50,550 --> 00:26:52,290 Actually, it's really a wave vector, 381 00:26:52,290 --> 00:26:56,660 because there will be three different-- Kx, Ky and Kz. 382 00:26:56,660 --> 00:26:59,360 And at each one of these K values, 383 00:26:59,360 --> 00:27:03,350 you will have three eigenfrequencies. 384 00:27:03,350 --> 00:27:07,610 And presumably as you span K, you 385 00:27:07,610 --> 00:27:12,400 will have a whole bunch of lines that 386 00:27:12,400 --> 00:27:16,260 will correspond to the variations of these frequencies 387 00:27:16,260 --> 00:27:18,910 as a function of K. 388 00:27:18,910 --> 00:27:21,030 Why is that useful? 389 00:27:21,030 --> 00:27:23,180 Well, the reason that it's useful 390 00:27:23,180 --> 00:27:26,840 is that as you go to high temperature, 391 00:27:26,840 --> 00:27:31,770 you put the energies into these normal modes and frequencies. 392 00:27:31,770 --> 00:27:35,380 That's why this whole lattice is vibrating. 393 00:27:35,380 --> 00:27:38,150 And the amount of energy that you 394 00:27:38,150 --> 00:27:45,110 have put at temperature T on top of this V0 395 00:27:45,110 --> 00:27:48,470 that you had at zero temperature, 396 00:27:48,470 --> 00:27:51,290 up to constants of proportionality 397 00:27:51,290 --> 00:27:54,520 that I don't want to bother with is 398 00:27:54,520 --> 00:27:59,410 a sum over all of these normal modes that are characterized 399 00:27:59,410 --> 00:28:06,480 by K and alpha, the polarization and the wave vector. 400 00:28:06,480 --> 00:28:08,370 And the amount of energy, then, that you 401 00:28:08,370 --> 00:28:14,030 would put in one harmonic oscillator of frequency omega. 402 00:28:14,030 --> 00:28:16,010 And that is something that's we know 403 00:28:16,010 --> 00:28:23,235 to be h bar omega alpha of K, divided by e to the beta h 404 00:28:23,235 --> 00:28:26,960 bar omega alpha of K minus 1. 405 00:28:30,970 --> 00:28:33,650 So the temperature dependence then 406 00:28:33,650 --> 00:28:37,480 appears in this factor of beta over here. 407 00:28:37,480 --> 00:28:40,550 So the energy content went there, 408 00:28:40,550 --> 00:28:43,350 and if you want to, for example, ultimately 409 00:28:43,350 --> 00:28:46,640 calculate heat capacity, I have to calculate 410 00:28:46,640 --> 00:28:49,020 this whole quantity as a function of temperature. 411 00:28:49,020 --> 00:28:52,150 And then take the derivative. 412 00:28:52,150 --> 00:28:57,380 So it seems like, OK, I have to do this for every single solid, 413 00:28:57,380 --> 00:29:01,970 whether it's copper, aluminium, wood, or whatever. 414 00:29:01,970 --> 00:29:05,630 I have to figure out what these frequencies are, 415 00:29:05,630 --> 00:29:08,320 what's the energy content in each frequency. 416 00:29:08,320 --> 00:29:13,415 And it seems like a complicated engineering problem, 417 00:29:13,415 --> 00:29:14,900 if you like. 418 00:29:14,900 --> 00:29:19,150 Is that it about this that transcends 419 00:29:19,150 --> 00:29:24,990 having to look at all of these details and come to this? 420 00:29:24,990 --> 00:29:27,760 And of course, you know the answer already, 421 00:29:27,760 --> 00:29:31,920 which is that if I go to sufficiently low temperature, 422 00:29:31,920 --> 00:29:37,930 I know that the heat capacity due to phonons for all solids 423 00:29:37,930 --> 00:29:40,540 goes like t cubed. 424 00:29:40,540 --> 00:29:44,100 So somehow, all of this complexity, 425 00:29:44,100 --> 00:29:48,010 if I go to low enough temperature, disappears. 426 00:29:48,010 --> 00:29:53,610 And some universal law emerges that is completely independent 427 00:29:53,610 --> 00:29:56,390 of all of these details-- microscopics, 428 00:29:56,390 --> 00:29:58,890 interactions, et cetera. 429 00:29:58,890 --> 00:30:03,400 So our task-- this is the change of perspective-- 430 00:30:03,400 --> 00:30:08,660 is to find a way to circumvent all of these things 431 00:30:08,660 --> 00:30:13,110 and get immediately to the heart of the matter, the part that 432 00:30:13,110 --> 00:30:16,510 is independent of the details. 433 00:30:16,510 --> 00:30:18,320 Not that the details are irrelevant. 434 00:30:18,320 --> 00:30:20,900 Because after all, if you want to give some material 435 00:30:20,900 --> 00:30:24,030 that functions at some particular set of temperatures, 436 00:30:24,030 --> 00:30:28,320 you would need to know much more than this t cubed law 437 00:30:28,320 --> 00:30:30,260 that I'm telling you about. 438 00:30:30,260 --> 00:30:33,460 But maybe from the perspective of what 439 00:30:33,460 --> 00:30:37,610 I was saying before-- how many independent forms there 440 00:30:37,610 --> 00:30:41,470 are, in the same sense that adding up random variables 441 00:30:41,470 --> 00:30:42,896 always gives you a Gaussian. 442 00:30:42,896 --> 00:30:45,270 Of course, you don't know where the mean and the variance 443 00:30:45,270 --> 00:30:48,590 of the Gaussian is, but you are sure that it's a Gaussian form. 444 00:30:48,590 --> 00:30:52,420 So similarly, there is some universality in the knowledge 445 00:30:52,420 --> 00:30:55,130 that, no matter how complicated the material is, 446 00:30:55,130 --> 00:30:58,230 its low temperature heat capacity is t cubed. 447 00:30:58,230 --> 00:31:04,750 Can we get that by an approach that circumvents the details? 448 00:31:04,750 --> 00:31:07,330 So I'm going to do that. 449 00:31:07,330 --> 00:31:10,580 But before, since I did a little bit of hand-waving, 450 00:31:10,580 --> 00:31:18,580 to be more precise, let's do the one-dimensional example 451 00:31:18,580 --> 00:31:21,080 in a little bit more detail. 452 00:31:21,080 --> 00:31:24,970 So my one-dimensional solid is going 453 00:31:24,970 --> 00:31:29,790 to be a bunch of ions or molecules 454 00:31:29,790 --> 00:31:32,890 or whatever, whose zero temperature 455 00:31:32,890 --> 00:31:42,060 positions is uniformly separated by some lattice spacing, 456 00:31:42,060 --> 00:31:44,470 a, around one dimension. 457 00:31:44,470 --> 00:31:49,260 And if I look at the formations, I'm 458 00:31:49,260 --> 00:31:54,720 going to indicate them by the one-dimensional distortion 459 00:31:54,720 --> 00:31:59,370 Un of the nth one along this chain. 460 00:31:59,370 --> 00:32:04,550 So then I would say, OK, the potential energy of this system 461 00:32:04,550 --> 00:32:08,730 minus whatever it is at zero, just because of the distortion, 462 00:32:08,730 --> 00:32:14,400 I will write as follows-- it is a sum over n. 463 00:32:14,400 --> 00:32:20,090 And one thing that I can do is to say 464 00:32:20,090 --> 00:32:25,010 that if I look at two of these things that are originally 465 00:32:25,010 --> 00:32:30,180 at distance a, and then they go and the deform by Un and Un 466 00:32:30,180 --> 00:32:34,210 plus 1, the additional deformation from a 467 00:32:34,210 --> 00:32:37,520 is actually Un plus 1 minus Un. 468 00:32:37,520 --> 00:32:39,710 So I can put some kind of Hookean 469 00:32:39,710 --> 00:32:44,950 elasticity and write it in this fashion. 470 00:32:44,950 --> 00:32:49,450 Now of course, there could be an interaction that goes to second 471 00:32:49,450 --> 00:32:50,750 neighbours. 472 00:32:50,750 --> 00:32:55,240 So I can write that as K2 over 2, Un plus 2, 473 00:32:55,240 --> 00:32:59,710 minus Un squared and third neighbors, and so forth. 474 00:32:59,710 --> 00:33:03,990 I can add as many of these as I like 475 00:33:03,990 --> 00:33:07,930 to make it as general as possible. 476 00:33:07,930 --> 00:33:13,060 So in some sense, this is a kind of rewriting of the form 477 00:33:13,060 --> 00:33:19,030 that I had written over here, where these things that 478 00:33:19,030 --> 00:33:22,010 where a function of the separation-- 479 00:33:22,010 --> 00:33:27,220 these K alpha beta of separation or these K1, K2, K3, et 480 00:33:27,220 --> 00:33:30,970 cetera-- in this series that you would write down. 481 00:33:30,970 --> 00:33:36,770 Now if you go to Fourier space, what 482 00:33:36,770 --> 00:33:45,210 you can do is each Un of Un, is the distortion 483 00:33:45,210 --> 00:33:47,940 in the original perspective, you can Fourier transform. 484 00:33:54,170 --> 00:34:02,890 And write it as a sum over K e to the ik position the nth 485 00:34:02,890 --> 00:34:09,184 particle is na, times u tilde of k. 486 00:34:13,150 --> 00:34:17,770 And once you make this Fourier transform 487 00:34:17,770 --> 00:34:21,570 in the expression over here, you get an expression 488 00:34:21,570 --> 00:34:25,639 for V minus V0 in terms of the Fourier modes. 489 00:34:25,639 --> 00:34:28,420 So rather than having an expression 490 00:34:28,420 --> 00:34:33,270 in terms of the amplitudes u sub n, after Fourier transform, 491 00:34:33,270 --> 00:34:37,150 I will have an expiration in terms of u tilde of k. 492 00:34:37,150 --> 00:34:39,280 So let's see what that is. 493 00:34:39,280 --> 00:34:41,860 Forget about various proportionality. 494 00:34:41,860 --> 00:34:44,590 I have the sum over n. 495 00:34:44,590 --> 00:34:48,550 Each one of the Un's I can write in this fashion in terms 496 00:34:48,550 --> 00:34:52,920 of U tilde of K. Since this is a quadratic form, 497 00:34:52,920 --> 00:34:55,380 I need to have two of these. 498 00:34:55,380 --> 00:34:57,815 So I will have the sum of k and k prime. 499 00:35:00,470 --> 00:35:05,980 I have the factor of 1/2. 500 00:35:05,980 --> 00:35:12,480 Each Un goes with a factor of e to the i nak. 501 00:35:15,280 --> 00:35:17,260 But then I had 2 Un. 502 00:35:17,260 --> 00:35:21,100 There's a term here, if I do the expansion, which is Un squared. 503 00:35:21,100 --> 00:35:24,670 So I have one from k and one from k prime. 504 00:35:29,910 --> 00:35:33,431 However, if I have Un plus 1 minus Un, 505 00:35:33,431 --> 00:35:40,000 what I have is e to the ika minus 1. 506 00:35:40,000 --> 00:35:43,010 I already took the contribution that was e to the ink. 507 00:35:46,060 --> 00:35:48,326 From the second factor, I will get e 508 00:35:48,326 --> 00:35:51,570 to the ik prime a minus 1. 509 00:35:51,570 --> 00:35:56,770 This multiplies K1 over 2. 510 00:35:56,770 --> 00:36:01,260 And then I will have something that's K2 over 2, 511 00:36:01,260 --> 00:36:10,560 e to the 2ika minus 1, e to 2i k prime a minus 1, and so forth. 512 00:36:10,560 --> 00:36:14,730 Multiplying at the end of the day U tilde of k, 513 00:36:14,730 --> 00:36:17,156 U tilde of k prime. 514 00:36:25,060 --> 00:36:29,680 Now when I do the sum over n, and the only n dependence 515 00:36:29,680 --> 00:36:35,670 appears over here, then this is the thing 516 00:36:35,670 --> 00:36:38,920 that forces k and k prime to add up to 0, 517 00:36:38,920 --> 00:36:40,680 because if they don't add up to 0, 518 00:36:40,680 --> 00:36:43,360 then I'm adding lots of random phases together. 519 00:36:43,360 --> 00:36:45,220 And the answer will be 0. 520 00:36:45,220 --> 00:36:47,700 So essentially, this sum will give me 521 00:36:47,700 --> 00:36:53,370 a delta function that forces k plus k prime to be 0. 522 00:36:53,370 --> 00:36:58,800 And so then the additional potential energy that you have 523 00:36:58,800 --> 00:37:04,930 because of the distortion ends up being proportional to 1/2, 524 00:37:04,930 --> 00:37:09,040 sum over the different k's. 525 00:37:09,040 --> 00:37:11,800 Only one k will remain, because k prime 526 00:37:11,800 --> 00:37:14,300 is forced to be minus k. 527 00:37:14,300 --> 00:37:20,240 And so I have U of k, U of minus k, which 528 00:37:20,240 --> 00:37:23,920 is the same thing as U of k complex conjugate. 529 00:37:23,920 --> 00:37:26,320 So I will get that. 530 00:37:26,320 --> 00:37:32,200 And then from here, I will get K1. 531 00:37:32,200 --> 00:37:34,810 Now then, k prime is set to minus k. 532 00:37:34,810 --> 00:37:37,370 And when I multiply these two factors, 533 00:37:37,370 --> 00:37:41,940 I will get 1 plus 1 minus e to the ika minus 534 00:37:41,940 --> 00:37:43,480 e to the minus ika. 535 00:37:43,480 --> 00:37:46,880 So I will get 2 minus 2 cosine of ka. 536 00:37:49,740 --> 00:37:56,888 And then I will have K2, 2 minus cosine of 2ka, and so forth. 537 00:38:02,791 --> 00:38:03,874 Why are the lights not on? 538 00:38:37,806 --> 00:38:42,408 OK, still visible. 539 00:38:42,408 --> 00:38:45,750 So yes? 540 00:38:45,750 --> 00:38:48,690 AUDIENCE: So in your last slide, you have an absolute value Uk, 541 00:38:48,690 --> 00:38:53,860 but wouldn't it be-- right above it, 542 00:38:53,860 --> 00:38:59,280 is it U of k times U star of k prime, or how does that work? 543 00:38:59,280 --> 00:39:01,140 PROFESSOR: OK, so the way that I have 544 00:39:01,140 --> 00:39:04,820 written is each Un I have written 545 00:39:04,820 --> 00:39:08,380 in terms of U tilde of k. 546 00:39:08,380 --> 00:39:15,020 And at this first stage, the two factors of Un that I have here, 547 00:39:15,020 --> 00:39:18,860 I treat them completely equivalently with indices k 548 00:39:18,860 --> 00:39:20,030 and k prime. 549 00:39:20,030 --> 00:39:23,860 So there is no complex conjugation involved here. 550 00:39:23,860 --> 00:39:27,920 But when k prime is set to be minus k, 551 00:39:27,920 --> 00:39:33,180 then I additionally realize that if I Fourier transform here, 552 00:39:33,180 --> 00:39:37,245 I will find that U tilde of minus k 553 00:39:37,245 --> 00:39:41,590 is the same thing as U tilde of k star. 554 00:39:41,590 --> 00:39:46,180 Because essentially, the complex conjugation appears over here. 555 00:39:52,144 --> 00:39:54,640 It's not that important a point. 556 00:39:54,640 --> 00:39:57,470 The important point is that we now 557 00:39:57,470 --> 00:40:00,660 have an expression for our frequencies, 558 00:40:00,660 --> 00:40:04,920 or omega alpha of k-- actually, there's no polarization. 559 00:40:04,920 --> 00:40:09,060 It's just omega of k-- are related 560 00:40:09,060 --> 00:40:12,760 to square root of something like a mass down here. 561 00:40:12,760 --> 00:40:14,620 Again, that's not particularly important, 562 00:40:14,620 --> 00:40:20,868 but something like k1, 2 minus 2 cosine of ka, k2, 2 563 00:40:20,868 --> 00:40:26,269 minus 2 cosine of 2ka, and so forth. 564 00:40:32,190 --> 00:40:37,244 So I can plot these frequencies omega as a function of k. 565 00:40:40,190 --> 00:40:43,180 One thing to note is first of all, 566 00:40:43,180 --> 00:40:45,740 the expression is clearly symmetric under k 567 00:40:45,740 --> 00:40:49,570 goes to minus k, so it only depends on cosine of k. 568 00:40:49,570 --> 00:40:52,570 So it is sufficient to draw one side. 569 00:40:52,570 --> 00:40:55,920 The other side for negative k would be the opposite. 570 00:40:55,920 --> 00:40:59,050 The other thing to note is that again, 571 00:40:59,050 --> 00:41:00,852 if I do this Fourier transformation, 572 00:41:00,852 --> 00:41:04,240 and I have things that are spaced by a, 573 00:41:04,240 --> 00:41:06,620 it effectively means that the shortest wavelength 574 00:41:06,620 --> 00:41:10,370 that I have to deal with are of the order of k, which 575 00:41:10,370 --> 00:41:17,910 means that the wave numbers are also kind of limited 576 00:41:17,910 --> 00:41:20,900 by something that I can't go beyond. 577 00:41:20,900 --> 00:41:25,130 So there's something that in the generalized over here, 578 00:41:25,130 --> 00:41:29,170 you recall that your k vectors are within the Brillouin zone. 579 00:41:29,170 --> 00:41:33,375 In one dimension, the Brillouin zone is simply between minus 5 580 00:41:33,375 --> 00:41:36,390 over a and 5 over a. 581 00:41:36,390 --> 00:41:43,920 Now the interesting thing to note is that as k goes to 0, 582 00:41:43,920 --> 00:41:45,130 omega goes to 0. 583 00:41:45,130 --> 00:41:48,350 Because all of these factors you can see, as k goes to 0, 584 00:41:48,350 --> 00:41:49,670 vanish. 585 00:41:49,670 --> 00:41:52,630 In particular, if I start expanding around k 586 00:41:52,630 --> 00:41:56,037 close to zero, what I find is that all of these things 587 00:41:56,037 --> 00:41:56,620 are quadratic. 588 00:41:56,620 --> 00:41:59,110 They go like k squared. 589 00:41:59,110 --> 00:42:00,855 So when I take the square root, they 590 00:42:00,855 --> 00:42:03,950 will have an absolute value of k. 591 00:42:03,950 --> 00:42:09,955 So I know for sure that these omegas start with that. 592 00:42:12,570 --> 00:42:16,380 What I don't know, since I have no idea what k1, 593 00:42:16,380 --> 00:42:21,147 k2, et cetera are, is what they do out here. 594 00:42:21,147 --> 00:42:23,230 So there could be some kind of a strange spaghetti 595 00:42:23,230 --> 00:42:27,210 or going on over here, I have no idea. 596 00:42:27,210 --> 00:42:29,190 There's all kinds of complexity. 597 00:42:29,190 --> 00:42:32,410 But they are away from the k close to 0 part. 598 00:42:34,970 --> 00:42:38,530 Again, why does it go 0? 599 00:42:38,530 --> 00:42:41,120 Of course, k equals to 0 corresponds 600 00:42:41,120 --> 00:42:44,950 to taking the entire chain and translating it. 601 00:42:44,950 --> 00:42:48,020 And clearly, I constructed this such 602 00:42:48,020 --> 00:42:50,410 that all of the U's are the same-- I 603 00:42:50,410 --> 00:42:54,660 take everything and translate it-- there's no energy cost. 604 00:42:54,660 --> 00:42:58,930 So there's no energy costs for k equals to 0. 605 00:42:58,930 --> 00:43:03,706 The energy costs for small k have to be small. 606 00:43:03,706 --> 00:43:06,470 By symmetry, they have to be quadratic in case, 607 00:43:06,470 --> 00:43:09,680 so I take the square root and we will get a linear. 608 00:43:09,680 --> 00:43:12,670 Of course, you know that this linear part-- 609 00:43:12,670 --> 00:43:19,140 we can say that omega is something like a sum velocity. 610 00:43:19,140 --> 00:43:22,670 So all of these chains, when I go 611 00:43:22,670 --> 00:43:29,900 to low enough case or low enough frequencies, 612 00:43:29,900 --> 00:43:31,790 admit these sound-like waves. 613 00:43:36,210 --> 00:43:38,390 Now heat content-- what am I supposed to do? 614 00:43:38,390 --> 00:43:41,620 I'm supposed to take these frequencies, 615 00:43:41,620 --> 00:43:46,080 put them in the expression that I have over here, 616 00:43:46,080 --> 00:43:49,840 and calculate what's going on. 617 00:43:49,840 --> 00:43:53,252 So again, if I want to look at the entirety of everything that 618 00:43:53,252 --> 00:43:56,800 is going on here, I would need to know 619 00:43:56,800 --> 00:44:00,040 the details of k2, k3, k4, et cetera. 620 00:44:00,040 --> 00:44:01,830 And I don't know all of that. 621 00:44:01,830 --> 00:44:03,830 So you would say I haven't really 622 00:44:03,830 --> 00:44:06,560 found anything universal yet. 623 00:44:06,560 --> 00:44:11,470 But if I look at one of these functions, 624 00:44:11,470 --> 00:44:17,621 and plot it as a function of the frequency, what do I see? 625 00:44:17,621 --> 00:44:19,820 Well, omega goes to 0. 626 00:44:19,820 --> 00:44:21,780 I can expand this. 627 00:44:21,780 --> 00:44:25,070 What I get is kt. 628 00:44:25,070 --> 00:44:29,670 Essentially, it's a statement that low frequencies 629 00:44:29,670 --> 00:44:32,030 behave like classical oscillators. 630 00:44:32,030 --> 00:44:35,670 A classic oscillator has an energy kt. 631 00:44:35,670 --> 00:44:37,840 Once I get to a frequency that is 632 00:44:37,840 --> 00:44:42,690 of the order of kt over h bar, then 633 00:44:42,690 --> 00:44:50,250 because of the exponential, I kind of drop down to 0. 634 00:44:50,250 --> 00:44:54,170 So very approximately, I can imagine 635 00:44:54,170 --> 00:44:58,100 that this is a function that is kind of like a step. 636 00:44:58,100 --> 00:44:59,950 It is either 1 or 0. 637 00:45:02,920 --> 00:45:06,330 And the change from 1 to 0 occurs 638 00:45:06,330 --> 00:45:10,980 at the frequency that is related to temperatures by k2 over a. 639 00:45:10,980 --> 00:45:18,300 So if I'm at some high temperature, up here, 640 00:45:18,300 --> 00:45:27,216 and I want to-- so this omega is k v t sum i over h bar-- 641 00:45:27,216 --> 00:45:29,820 that's the corresponding high frequency-- 642 00:45:29,820 --> 00:45:32,870 I need to know all of these frequencies 643 00:45:32,870 --> 00:45:35,890 to know what's going on for the energy content. 644 00:45:35,890 --> 00:45:43,489 But if I go to lower and lower temperatures, 645 00:45:43,489 --> 00:45:46,740 eventually I will get to low enough temperatures 646 00:45:46,740 --> 00:45:52,980 where the only thing that I will see is this linear portion. 647 00:45:52,980 --> 00:45:55,910 And I'm guaranteed that I will see that. 648 00:45:55,910 --> 00:45:58,470 I know that I will see that eventually. 649 00:45:58,470 --> 00:46:00,520 And therefore, I know that eventually, 650 00:46:00,520 --> 00:46:04,145 if I go to low enough temperatures, 651 00:46:04,145 --> 00:46:10,580 the excitation energy becomes low enough, 652 00:46:10,580 --> 00:46:16,040 it's simply proportional to this integral from 0. 653 00:46:16,040 --> 00:46:20,600 I can change the upper part of the infinity if I like. 654 00:46:20,600 --> 00:46:31,520 dk h bar 1k divided by e to the beta h bar dk minus 1, 655 00:46:31,520 --> 00:46:36,220 And again, dimensionally, I have two factors of k here. 656 00:46:36,220 --> 00:46:39,280 Each k scales with kt. 657 00:46:39,280 --> 00:46:44,640 So I know that whole thing is proportional to kt squared. 658 00:46:44,640 --> 00:46:47,155 In fact, there's some proportionality constants 659 00:46:47,155 --> 00:46:50,390 that depend on h bar, t, et cetera. 660 00:46:50,390 --> 00:46:51,240 It doesn't matter. 661 00:46:51,240 --> 00:46:55,720 The point is this t squared. 662 00:46:55,720 --> 00:46:59,560 So I know immediately that my heat capacity 663 00:46:59,560 --> 00:47:04,560 is proportional to-- derivative of this 664 00:47:04,560 --> 00:47:06,723 is going to be proportional to t. 665 00:47:06,723 --> 00:47:10,680 The heat capacity of a linear chain independent 666 00:47:10,680 --> 00:47:11,900 of what you do. 667 00:47:11,900 --> 00:47:19,148 So no matter what the state of interactions 668 00:47:19,148 --> 00:47:24,380 is, if I start with a situation such as this at 0 temperature, 669 00:47:24,380 --> 00:47:28,600 I know if I put energy into it at low enough temperature, 670 00:47:28,600 --> 00:47:32,130 I would get this heat capacity that is linear. 671 00:47:32,130 --> 00:47:35,360 I don't know how low I have to do. 672 00:47:35,360 --> 00:47:39,060 Because how I have to go to depends 673 00:47:39,060 --> 00:47:42,520 on what this velocity is, what the other complication is, 674 00:47:42,520 --> 00:47:43,860 et cetera. 675 00:47:43,860 --> 00:47:46,850 So that's the part that I don't know. 676 00:47:46,850 --> 00:47:50,120 I know for sure of the functional form, 677 00:47:50,120 --> 00:47:53,870 but I don't know the amplitude of that functional form. 678 00:47:57,230 --> 00:48:02,540 So the question is, can we somehow get this answer 679 00:48:02,540 --> 00:48:05,260 in a slightly different way, without going 680 00:48:05,260 --> 00:48:07,240 through all of these things? 681 00:48:07,240 --> 00:48:12,410 And the idea is to do a coarse grain. 682 00:48:20,630 --> 00:48:24,030 So what's going on here? 683 00:48:24,030 --> 00:48:27,750 Why is it that I got is form? 684 00:48:27,750 --> 00:48:29,950 Well, the reason I got this form was I 685 00:48:29,950 --> 00:48:32,130 got to low enough temperature. 686 00:48:32,130 --> 00:48:36,660 At low enough temperature, I have only the possibility 687 00:48:36,660 --> 00:48:41,260 of exciting emote, whose frequencies were small. 688 00:48:41,260 --> 00:48:45,755 I find that frequencies small correspond to wave numbers 689 00:48:45,755 --> 00:48:48,520 k that are small, or they correspond 690 00:48:48,520 --> 00:48:52,830 to wavelengths that are very large. 691 00:48:52,830 --> 00:48:56,200 So essentially, if you have you solid, 692 00:48:56,200 --> 00:48:58,710 you go to low enough temperature, 693 00:48:58,710 --> 00:49:04,330 you will be exciting modes of some characteristic wavelength 694 00:49:04,330 --> 00:49:07,570 that are inversely proportional to temperature, 695 00:49:07,570 --> 00:49:10,850 and become larger and larger. 696 00:49:10,850 --> 00:49:14,240 So eventually, these long wavelength modes 697 00:49:14,240 --> 00:49:21,620 will encompass whole bunches of your atoms. 698 00:49:21,620 --> 00:49:26,190 So this lambda becomes much larger 699 00:49:26,190 --> 00:49:28,423 than the spacing of the particles in the chain 700 00:49:28,423 --> 00:49:30,640 that you were looking at. 701 00:49:30,640 --> 00:49:34,540 And what you're looking at. low temperature, 702 00:49:34,540 --> 00:49:40,180 is a collective behavior that encompasses lots of particles 703 00:49:40,180 --> 00:49:43,470 moving collectively and together. 704 00:49:43,470 --> 00:49:47,605 And again, because of some kind of averaging 705 00:49:47,605 --> 00:49:51,880 that is going on over here, you don't really 706 00:49:51,880 --> 00:49:55,700 care about the interactions among small particles. 707 00:49:55,700 --> 00:49:57,350 So it's the same idea. 708 00:49:57,350 --> 00:50:00,780 It's the same large n limit appearing 709 00:50:00,780 --> 00:50:02,500 in a different context. 710 00:50:02,500 --> 00:50:05,910 It's not the n that becomes very large, 711 00:50:05,910 --> 00:50:10,000 but n that becomes of the order of, let's say, 100 lattice 712 00:50:10,000 --> 00:50:13,395 spacings-- already much larger than an individual atom doing 713 00:50:13,395 --> 00:50:15,270 something, because it's a collection of atoms 714 00:50:15,270 --> 00:50:18,680 that are moving together. 715 00:50:18,680 --> 00:50:21,635 So what I drew here was an example of a mode. 716 00:50:21,635 --> 00:50:24,780 But I can imagine that I have some kind 717 00:50:24,780 --> 00:50:28,030 of a distortion in my system. 718 00:50:30,680 --> 00:50:34,250 Now, I started with the distortions Un 719 00:50:34,250 --> 00:50:37,430 that were defined at the level of each individual atom, 720 00:50:37,430 --> 00:50:41,950 or molecule, or variable that I have over here. 721 00:50:41,950 --> 00:50:45,170 But I know that things that are next to each other 722 00:50:45,170 --> 00:50:47,930 are more or less moving together. 723 00:50:47,930 --> 00:50:52,180 So what I can do is I can average. 724 00:50:52,180 --> 00:50:54,300 I can sort of pick aa distance-- let's 725 00:50:54,300 --> 00:50:59,050 call it the x-- and average of all of those Un's that 726 00:50:59,050 --> 00:51:03,470 are within that distance and find how that averages. 727 00:51:03,470 --> 00:51:07,100 And as I move my interval that I'm averaging, 728 00:51:07,100 --> 00:51:11,620 I'm constructing this core function U of x. 729 00:51:15,080 --> 00:51:20,740 So there is a moving window along the chain constructed 730 00:51:20,740 --> 00:51:24,470 with the Vx which is much larger than a, 731 00:51:24,470 --> 00:51:28,340 but is much less than this characteristic frequency. 732 00:51:28,340 --> 00:51:34,340 And using that, I can construct a distortion field. 733 00:51:34,340 --> 00:51:38,390 I started discrete variables. 734 00:51:38,390 --> 00:51:41,690 And I ended up with a continuous function. 735 00:51:41,690 --> 00:51:45,042 So this is an example of a statistical field. 736 00:51:45,042 --> 00:51:50,550 So this distortion appears to be defined continuously. 737 00:51:50,550 --> 00:51:53,920 But in fact, it has much less degrees of freedom, 738 00:51:53,920 --> 00:51:57,680 if you like, compared to all of the discrete variables that I 739 00:51:57,680 --> 00:51:58,920 started with. 740 00:51:58,920 --> 00:52:02,360 Because this continuous function certainly 741 00:52:02,360 --> 00:52:05,380 does not have, when I fully transform it, 742 00:52:05,380 --> 00:52:08,660 variations at short length scales. 743 00:52:08,660 --> 00:52:11,370 So we are going to be constructing 744 00:52:11,370 --> 00:52:18,350 a lot of these coarse grained statistical fields. 745 00:52:18,350 --> 00:52:20,630 If you think about the temperature in this room, 746 00:52:20,630 --> 00:52:24,010 bearing from one location to another location pressure, 747 00:52:24,010 --> 00:52:26,970 so that we don't strike sound waves, et cetera. 748 00:52:26,970 --> 00:52:31,240 All of these things are examples of a continuous field. 749 00:52:31,240 --> 00:52:33,130 But clearly, that continuous field 750 00:52:33,130 --> 00:52:38,450 comes from averaging things that exist at the microscopic level. 751 00:52:38,450 --> 00:52:41,650 So it's kind of counter-intuitive 752 00:52:41,650 --> 00:52:43,930 that I start with discrete variables, 753 00:52:43,930 --> 00:52:47,250 and I can replace them with some continuous function. 754 00:52:47,250 --> 00:52:50,350 But again, the emphasis is that this continuous function 755 00:52:50,350 --> 00:52:53,486 has a limited set of available and surveyed numbers 756 00:52:53,486 --> 00:52:56,680 over which it is defined. 757 00:52:56,680 --> 00:52:57,600 OK. 758 00:52:57,600 --> 00:53:02,920 So we are going to describe the system in terms of this. 759 00:53:02,920 --> 00:53:08,560 So that analog of this potential that we have over here 760 00:53:08,560 --> 00:53:13,730 is some b function of this U of x. 761 00:53:16,310 --> 00:53:20,730 And I want to construct that function. 762 00:53:20,730 --> 00:53:28,280 And so the next step after you decided 763 00:53:28,280 --> 00:53:32,290 what your statistical field is to construct 764 00:53:32,290 --> 00:53:36,110 some relevant thing, such as an potential energy that 765 00:53:36,110 --> 00:53:41,250 is appropriate with that statistical field, 766 00:53:41,250 --> 00:53:46,970 putting as limited amount of information as possible 767 00:53:46,970 --> 00:53:48,920 in construction of that. 768 00:53:48,920 --> 00:53:50,480 So what are the things that we are 769 00:53:50,480 --> 00:53:55,200 going to put in constructing this function at. 770 00:53:55,200 --> 00:53:56,750 The first thing that I will do is 771 00:53:56,750 --> 00:54:01,694 I will assume that there is a kind of locality. 772 00:54:01,694 --> 00:54:04,390 By which, I mean the following. 773 00:54:04,390 --> 00:54:08,882 While this is in principle the function 774 00:54:08,882 --> 00:54:13,070 of the entire function, locality means 775 00:54:13,070 --> 00:54:22,570 that I will write it as an integral of some density, where 776 00:54:22,570 --> 00:54:26,190 the density at location x that I'm integrating 777 00:54:26,190 --> 00:54:30,790 depends on you at that location. 778 00:54:30,790 --> 00:54:34,703 But not just you, also including derivatives of you. 779 00:54:43,100 --> 00:54:45,720 And you can see that this is really 780 00:54:45,720 --> 00:54:50,380 a continuum version of what I have written here, this. 781 00:54:50,380 --> 00:54:54,510 If I go to the continuum, this goes like a derivative. 782 00:54:54,510 --> 00:54:57,225 And if I look at further and further distances, 783 00:54:57,225 --> 00:55:01,250 I can construct higher and higher derivatives. 784 00:55:01,250 --> 00:55:07,390 So in the sense that this is a quite general description, 785 00:55:07,390 --> 00:55:10,680 I can construct any kind of potential in here 786 00:55:10,680 --> 00:55:13,330 by choosing interactions. 787 00:55:13,330 --> 00:55:18,140 K1, K2, K3, K100 that go further and further apart, 788 00:55:18,140 --> 00:55:21,750 you would say that if I include sufficiently high derivatives 789 00:55:21,750 --> 00:55:25,300 here, I can also include interactions 790 00:55:25,300 --> 00:55:29,980 that are extending over far away distances. 791 00:55:29,980 --> 00:55:34,760 The idea of locality is that while you make this expansion, 792 00:55:34,760 --> 00:55:36,720 our hope is that at the end of the day, 793 00:55:36,720 --> 00:55:40,080 we can terminate this expansion without needing 794 00:55:40,080 --> 00:55:43,130 to go to many, many higher orders. 795 00:55:43,130 --> 00:55:45,940 So locality is two parts. 796 00:55:45,940 --> 00:55:48,420 One, that you will write it in this form. 797 00:55:48,420 --> 00:55:51,315 And secondly, that this function will not 798 00:55:51,315 --> 00:55:56,260 depend on many, many high derivatives. 799 00:55:56,260 --> 00:56:00,710 The second part of it is symmetries. 800 00:56:07,890 --> 00:56:11,860 Now, one of the things that I constructed in here 801 00:56:11,860 --> 00:56:16,120 and ultimately was very relevant to the result that I had 802 00:56:16,120 --> 00:56:18,230 was that if I take a distortion U 803 00:56:18,230 --> 00:56:27,310 of x and I add a constant to everybody, 804 00:56:27,310 --> 00:56:32,132 so if I replace all of my Uns to plus Un plus 5, for example, 805 00:56:32,132 --> 00:56:34,320 the energy does not change. 806 00:56:34,320 --> 00:56:40,690 So V of this is the same thing as V of U of x. 807 00:56:40,690 --> 00:56:42,140 So that's a symmetry. 808 00:56:42,140 --> 00:56:45,220 Essentially, it's this translation of symmetry 809 00:56:45,220 --> 00:56:49,200 that I was saying right here at the beginning, 810 00:56:49,200 --> 00:56:52,540 that this only depends on the separation of two points. 811 00:56:52,540 --> 00:56:54,500 It's the same thing. 812 00:56:54,500 --> 00:56:59,740 But what that means is that when you write your density 813 00:56:59,740 --> 00:57:04,540 function, then the density cannot depend on U of x. 814 00:57:04,540 --> 00:57:05,950 Because that would violate this. 815 00:57:05,950 --> 00:57:08,870 So you can only be start with things 816 00:57:08,870 --> 00:57:15,162 that depend on the U of V IBX this second, et cetera. 817 00:57:15,162 --> 00:57:16,220 So this is 1. 818 00:57:16,220 --> 00:57:18,210 This is 2. 819 00:57:18,210 --> 00:57:20,800 Another thing is what I call stability. 820 00:57:24,680 --> 00:57:30,790 You are looking at distortions around a state that corresponds 821 00:57:30,790 --> 00:57:36,130 to being a stable configuration of your system. 822 00:57:36,130 --> 00:57:40,750 What that means is that you cannot have any pairs in this 823 00:57:40,750 --> 00:57:42,810 expansion that are linear. 824 00:57:42,810 --> 00:57:45,760 So again, this was implicit in everything 825 00:57:45,760 --> 00:57:47,620 that we did over here. 826 00:57:47,620 --> 00:57:49,370 We went to second order. 827 00:57:49,370 --> 00:57:52,920 But there's third order, et cetera, are not ruled out. 828 00:57:52,920 --> 00:57:54,610 It is more than that. 829 00:57:54,610 --> 00:57:58,590 Because you require the second order terms to have the right 830 00:57:58,590 --> 00:58:01,090 sign, so that your system corresponds 831 00:58:01,090 --> 00:58:04,116 to being at the bottom of a quadratic potential 832 00:58:04,116 --> 00:58:06,200 rather than the top of it. 833 00:58:06,200 --> 00:58:10,040 So there is a bit more than the absence of linear terms. 834 00:58:10,040 --> 00:58:13,490 So given that, you would say that your potential 835 00:58:13,490 --> 00:58:18,480 for this system as a function now of this distortion 836 00:58:18,480 --> 00:58:23,780 is something like an integral over x. 837 00:58:23,780 --> 00:58:26,550 And the first thing that is consistent with everything 838 00:58:26,550 --> 00:58:29,960 we have written so far is that if we 839 00:58:29,960 --> 00:58:34,740 be proportional to du by dx squared. 840 00:58:34,740 --> 00:58:36,860 So there's a coefficient that I can put here. 841 00:58:36,860 --> 00:58:39,790 Let's call it k over 2. 842 00:58:39,790 --> 00:58:41,590 It cannot depend on you. 843 00:58:41,590 --> 00:58:44,530 It has to be quadratic function of derivative. 844 00:58:44,530 --> 00:58:46,520 That's the first thing I can write down. 845 00:58:46,520 --> 00:58:50,380 I can certainly write down something like d2u 846 00:58:50,380 --> 00:58:53,820 by dx to the fourth power. 847 00:58:53,820 --> 00:58:58,480 And if I consider higher order terms, why not something 848 00:58:58,480 --> 00:59:04,039 like the second derivative squared, 849 00:59:04,039 --> 00:59:10,830 first derivative squared, a whole bunch of other things. 850 00:59:10,830 --> 00:59:15,090 So again, there's still many, many, many terms 851 00:59:15,090 --> 00:59:16,570 that I can't write down. 852 00:59:16,570 --> 00:59:17,825 Yes? 853 00:59:17,825 --> 00:59:20,300 Is that second term supposed to be 854 00:59:20,300 --> 00:59:22,775 a second derivative to the fourth power? 855 00:59:22,775 --> 00:59:24,755 PROFESSOR: Yes. 856 00:59:24,755 --> 00:59:27,740 Thank you. 857 00:59:27,740 --> 00:59:33,460 So that when I fully transform this, 858 00:59:33,460 --> 00:59:40,450 the quadratic part becomes sum over k, k over 2. 859 00:59:40,450 --> 00:59:44,340 This fully transformed, becomes k squared. 860 00:59:44,340 --> 00:59:45,840 This fully transformed, as you said, 861 00:59:45,840 --> 00:59:49,560 is second derivative squared. 862 00:59:49,560 --> 00:59:51,490 So it becomes k to the fourth. 863 00:59:51,490 --> 00:59:54,416 I have a whole bunch of terms. 864 00:59:54,416 --> 00:59:58,912 And then I have U of k squared. 865 00:59:58,912 --> 01:00:01,190 And then I will have higher order terms from Fourier 866 01:00:01,190 --> 01:00:02,550 fully transform of this. 867 01:00:02,550 --> 01:00:03,100 Yes? 868 01:00:03,100 --> 01:00:05,820 AUDIENCE: Does this actually forbid odd derivatives? 869 01:00:05,820 --> 01:00:08,840 Are you saying the third derivative and stuff don't-- 870 01:00:08,840 --> 01:00:11,790 PROFESSOR: I didn't go into that, because that depends 871 01:00:11,790 --> 01:00:14,530 on some additional considerations, 872 01:00:14,530 --> 01:00:17,500 whether or not you have a mirror symmetry. 873 01:00:17,500 --> 01:00:20,650 If you have a mirror symmetry, you cannot have terms of that 874 01:00:20,650 --> 01:00:22,320 are odd and x. 875 01:00:22,320 --> 01:00:27,040 Whether or not you have some condition on U and minus U may 876 01:00:27,040 --> 01:00:30,580 or may not forbid third order terms in [? U ?] [? by the ?] 877 01:00:30,580 --> 01:00:31,330 x. 878 01:00:31,330 --> 01:00:35,370 So once I go beyond the quadratic level, 879 01:00:35,370 --> 01:00:39,500 I need to rely on some additional symmetry statement 880 01:00:39,500 --> 01:00:43,750 as to which additional terms I am allowed to write down. 881 01:00:43,750 --> 01:00:44,280 Yes? 882 01:00:44,280 --> 01:00:48,030 AUDIENCE: Also the coefficients could depend on x, right? 883 01:00:48,030 --> 01:00:48,640 PROFESSOR: OK. 884 01:00:48,640 --> 01:00:50,550 So one of the things that I assumed 885 01:00:50,550 --> 01:00:55,730 was this symmetry, which means that every position 886 01:00:55,730 --> 01:00:58,940 in the crystal is the same as any other position. 887 01:00:58,940 --> 01:01:02,905 So here, if I break and make the coefficient 888 01:01:02,905 --> 01:01:06,800 to be different from here, different from there, 889 01:01:06,800 --> 01:01:12,020 it amounts to the same thing, that the starting point was not 890 01:01:12,020 --> 01:01:13,282 the crystal. 891 01:01:13,282 --> 01:01:14,740 AUDIENCE: Shouldn't that be written 892 01:01:14,740 --> 01:01:16,906 as-- in the place where you wrote down the symmetry, 893 01:01:16,906 --> 01:01:21,187 it should be U of x plus c inside the parenthesis? 894 01:01:21,187 --> 01:01:21,770 PROFESSOR: No. 895 01:01:21,770 --> 01:01:22,790 No. 896 01:01:22,790 --> 01:01:24,270 So look at this. 897 01:01:24,270 --> 01:01:31,340 So if I take Un and I replace Un to Un plus 5, 898 01:01:31,340 --> 01:01:33,600 essentially I take the entire lattice 899 01:01:33,600 --> 01:01:36,620 and move it by a distance. 900 01:01:36,620 --> 01:01:39,380 Actually, 5 was probably not good. 901 01:01:39,380 --> 01:01:40,600 5.14. 902 01:01:40,600 --> 01:01:43,299 It's not just anything I can put over here. 903 01:01:43,299 --> 01:01:44,257 Energy will not change. 904 01:01:49,127 --> 01:01:51,730 OK? 905 01:01:51,730 --> 01:01:53,840 AUDIENCE: That must be different from adding 906 01:01:53,840 --> 01:02:00,140 to all the ends a constant displacement. 907 01:02:00,140 --> 01:02:04,230 PROFESSOR: Ends are labels of your variables. 908 01:02:04,230 --> 01:02:06,170 So I don't know what mean by-- 909 01:02:06,170 --> 01:02:07,670 AUDIENCE: OK, you're right. 910 01:02:07,670 --> 01:02:10,130 But in that picture, where instead of N's, we have X? 911 01:02:10,130 --> 01:02:10,370 PROFESSOR: Yes. 912 01:02:10,370 --> 01:02:12,161 AUDIENCE: It seems like displacing in space 913 01:02:12,161 --> 01:02:13,695 would mean adding up to x. 914 01:02:13,695 --> 01:02:14,771 PROFESSOR: No. 915 01:02:14,771 --> 01:02:15,270 No. 916 01:02:15,270 --> 01:02:17,930 It is this displacement. 917 01:02:17,930 --> 01:02:21,100 I take U1, U2, U3, U4. 918 01:02:21,100 --> 01:02:23,760 U1 becomes U1 plus .3. 919 01:02:23,760 --> 01:02:26,162 U2 becomes U2 plus .3. 920 01:02:26,162 --> 01:02:27,272 Everybody moves in step. 921 01:02:27,272 --> 01:02:29,230 AUDIENCE: So the conclusion is the coefficients 922 01:02:29,230 --> 01:02:31,280 don't depend on x? 923 01:02:31,280 --> 01:02:36,030 PROFESSOR: If you have a system that is uniform-- 924 01:02:36,030 --> 01:02:39,780 so this statement here actually depends on uniformity. 925 01:02:39,780 --> 01:02:42,265 This is an additional thing, uniform. 926 01:02:42,265 --> 01:02:45,150 So one part of the material is the same. 927 01:02:45,150 --> 01:02:47,990 Now, you have non-uniform systems. 928 01:02:47,990 --> 01:02:50,730 So you take your crystal and you bombard it 929 01:02:50,730 --> 01:02:52,510 with neutrons or whatever. 930 01:02:52,510 --> 01:02:55,300 Then you have defects all over the place. 931 01:02:55,300 --> 01:02:57,220 Then one location will be different 932 01:02:57,220 --> 01:02:58,330 from another location. 933 01:02:58,330 --> 01:03:02,160 You are not able to write that anymore. 934 01:03:02,160 --> 01:03:05,820 So uniformity is another symmetry 935 01:03:05,820 --> 01:03:08,774 that I kind of implicitly used. 936 01:03:08,774 --> 01:03:10,526 Yes? 937 01:03:10,526 --> 01:03:12,570 AUDIENCE: When your uniform is a separate part, 938 01:03:12,570 --> 01:03:15,318 why isn't it implied by translational symmetry? 939 01:03:19,270 --> 01:03:22,780 PROFESSOR: If I take this material that I neutron 940 01:03:22,780 --> 01:03:27,300 bombarded, and I translate it in space, 941 01:03:27,300 --> 01:03:33,389 it's internal energy will still not change, right? 942 01:03:33,389 --> 01:03:35,384 AUDIENCE: OK. 943 01:03:35,384 --> 01:03:35,884 OK. 944 01:03:45,380 --> 01:03:52,160 PROFESSOR: So again, once I come to this stage, 945 01:03:52,160 --> 01:04:00,100 what it amounts to is that I have constructed 946 01:04:00,100 --> 01:04:13,710 a kind of energy as a function of a deformation field, which 947 01:04:13,710 --> 01:04:16,800 in the limit of very long wavelengths 948 01:04:16,800 --> 01:04:21,535 has this very simple form which is the integral du by dx 949 01:04:21,535 --> 01:04:22,035 squared. 950 01:04:24,620 --> 01:04:26,190 There are higher order terms. 951 01:04:26,190 --> 01:04:29,460 But hopefully in the limit of long wavelengths, 952 01:04:29,460 --> 01:04:32,525 the higher derivatives will disappear. 953 01:04:32,525 --> 01:04:35,050 In the limit of small deformation, 954 01:04:35,050 --> 01:04:38,280 the higher order terms will disappear. 955 01:04:38,280 --> 01:04:41,230 So the lowest order term at long wavelengths, 956 01:04:41,230 --> 01:04:45,670 et cetera, is parametrized by this 1k. 957 01:04:45,670 --> 01:04:51,110 If I fully transport it, I will just get k over 2k squared. 958 01:04:51,110 --> 01:04:53,770 When I take the frequency that corresponds to that, 959 01:04:53,770 --> 01:04:57,590 I will get that kind of behavior. 960 01:04:57,590 --> 01:05:02,440 So by kind of relying on these kinds of statements 961 01:05:02,440 --> 01:05:07,230 about symmetry, et cetera, I was able to guess that. 962 01:05:07,230 --> 01:05:09,650 Now, let's go and do with this for the case 963 01:05:09,650 --> 01:05:15,660 of material in three dimensions. 964 01:05:15,660 --> 01:05:18,800 Actually, in any dimensions, in higher dimensions. 965 01:05:26,800 --> 01:05:31,120 So I take a solid in three dimensions, 966 01:05:31,120 --> 01:05:34,190 or maybe a film in 2 dimensions. 967 01:05:34,190 --> 01:05:40,380 I can still describe it's deformations 968 01:05:40,380 --> 01:05:46,320 at low enough temperature in terms of long wavelength modes. 969 01:05:46,320 --> 01:05:47,940 I do the coarse graining. 970 01:05:47,940 --> 01:05:49,010 I have U of x. 971 01:05:49,010 --> 01:05:53,760 Actually, both U and x are now vectors. 972 01:05:53,760 --> 01:05:57,310 And what I want to do is to construct 973 01:05:57,310 --> 01:06:02,680 a potential that corresponds to this end. 974 01:06:02,680 --> 01:06:05,200 OK? 975 01:06:05,200 --> 01:06:09,860 So I will use the idea of locality. 976 01:06:09,860 --> 01:06:13,515 And I write it as an integral over however many dimensions 977 01:06:13,515 --> 01:06:14,340 I have. 978 01:06:14,340 --> 01:06:17,880 So d is the dimensionality of space 979 01:06:17,880 --> 01:06:22,620 of some kind of an energy density. 980 01:06:22,620 --> 01:06:27,480 And the energy density now will depend on U. Actually, 981 01:06:27,480 --> 01:06:32,250 U has many components, U alpha of x. 982 01:06:32,250 --> 01:06:38,000 Derivatives of U-- so I will have du alpha by dx beta. 983 01:06:38,000 --> 01:06:40,650 And you can see that higher order derivatives, 984 01:06:40,650 --> 01:06:43,240 they're really more and more indices. 985 01:06:43,240 --> 01:06:47,458 So the complication now is that I 986 01:06:47,458 --> 01:06:49,270 have additional indices involved. 987 01:06:52,120 --> 01:06:58,540 The symmetry that I will use is a slightly more complicated 988 01:06:58,540 --> 01:07:01,080 version of what I had before. 989 01:07:01,080 --> 01:07:05,870 I will take the crystal, U of x, and I 990 01:07:05,870 --> 01:07:10,895 can translate it just as I did before. 991 01:07:10,895 --> 01:07:15,020 And I just say that this translation of crystal 992 01:07:15,020 --> 01:07:17,894 does not change its energy. 993 01:07:17,894 --> 01:07:19,060 But you know something else? 994 01:07:19,060 --> 01:07:22,990 I can take my crystal, and I can rotate it. 995 01:07:22,990 --> 01:07:25,660 The internal energy should not change. 996 01:07:25,660 --> 01:07:32,730 So there is a rotation of this that also you 997 01:07:32,730 --> 01:07:34,590 can put the c inside of. 998 01:07:34,590 --> 01:07:37,450 Before or after rotation, it doesn't matter. 999 01:07:37,450 --> 01:07:43,050 The energy should not depend on that. 1000 01:07:43,050 --> 01:07:45,020 OK. 1001 01:07:45,020 --> 01:07:48,660 So let's see what we have to construct. 1002 01:07:48,660 --> 01:07:50,080 I can write down the answer. 1003 01:07:50,080 --> 01:07:55,720 So first of all, we know immediately that the energy 1004 01:07:55,720 --> 01:08:00,640 cannot depend on U itself for the same reason as before. 1005 01:08:00,640 --> 01:08:02,400 It can depend on derivatives. 1006 01:08:02,400 --> 01:08:05,100 But this rotation and derivatives 1007 01:08:05,100 --> 01:08:07,140 is a little bit strange. 1008 01:08:07,140 --> 01:08:11,200 So I'm going to use a trick. 1009 01:08:11,200 --> 01:08:19,210 If I do this in Fourier space, like I did over here, 1010 01:08:19,210 --> 01:08:24,230 I went from this to k over 2 integral dk 1011 01:08:24,230 --> 01:08:28,600 k squared U tilde of k squared. 1012 01:08:31,710 --> 01:08:35,260 If I stick with sufficiently low derivatives, 1013 01:08:35,260 --> 01:08:39,770 only at the level of the second order derivatives, 1014 01:08:39,770 --> 01:08:42,890 if I have a second order form that depends on something 1015 01:08:42,890 --> 01:08:47,160 like this, I can still go to Fourier space. 1016 01:08:47,160 --> 01:08:51,390 And the answer will be of the form integral d dk. 1017 01:08:51,390 --> 01:08:56,979 The different k modes will only get coupled from higher order 1018 01:08:56,979 --> 01:08:58,939 terms, third order terms, et cetera. 1019 01:08:58,939 --> 01:09:01,160 At the level of quadratic, I know 1020 01:09:01,160 --> 01:09:04,760 that the answer is proportional to d dk. 1021 01:09:04,760 --> 01:09:09,029 And for all of the reasons that we have been discussing so far, 1022 01:09:09,029 --> 01:09:13,640 the answer is going to be U of k squared 1023 01:09:13,640 --> 01:09:19,090 times some function of k, like k squared, k to the fourth. 1024 01:09:19,090 --> 01:09:24,300 Now, whatever I put over here has 1025 01:09:24,300 --> 01:09:28,300 to be invariant under rotations. 1026 01:09:28,300 --> 01:09:31,029 So, let's see. 1027 01:09:31,029 --> 01:09:34,590 I know that the answer that I write here 1028 01:09:34,590 --> 01:09:36,324 should be quadratic [INAUDIBLE] tilde. 1029 01:09:39,050 --> 01:09:41,920 It should be at least quadratic in k, 1030 01:09:41,920 --> 01:09:43,684 because I'm looking at derivatives. 1031 01:09:43,684 --> 01:09:47,510 In the same way that I had k here, 1032 01:09:47,510 --> 01:09:52,000 I should have factors of k here. 1033 01:09:52,000 --> 01:09:55,760 But k is a vector when I go to three dimensions. 1034 01:09:55,760 --> 01:09:59,590 U becomes a vector when I go to three dimensions. 1035 01:09:59,590 --> 01:10:03,120 So I want to construct something that 1036 01:10:03,120 --> 01:10:07,630 involves quadratic in vector k, quadratic in vector U, 1037 01:10:07,630 --> 01:10:09,830 and is also invariant under rotations. 1038 01:10:13,140 --> 01:10:16,340 One thing that I know is that if I 1039 01:10:16,340 --> 01:10:19,500 do a dot product of two vectors, that dot product 1040 01:10:19,500 --> 01:10:21,480 is invariant on the rotations. 1041 01:10:21,480 --> 01:10:23,050 So I have two vectors. 1042 01:10:23,050 --> 01:10:26,680 So I know, therefore, that k squared, 1043 01:10:26,680 --> 01:10:30,710 k dot k is a rotational invariant. 1044 01:10:30,710 --> 01:10:38,830 The tilde of k squared is rotationally invariant. 1045 01:10:38,830 --> 01:10:48,230 But also, k dot U tilde of k squared 1046 01:10:48,230 --> 01:10:51,230 is rotationally invariant. 1047 01:10:51,230 --> 01:10:52,630 OK? 1048 01:10:52,630 --> 01:10:57,290 So what I can do is I can say that the most general form 1049 01:10:57,290 --> 01:11:03,500 that I will write down will allow 2 terms. 1050 01:11:03,500 --> 01:11:07,830 The coefficients of that are traditionally called mu. 1051 01:11:07,830 --> 01:11:09,510 This is mu over 2. 1052 01:11:09,510 --> 01:11:13,540 This one is called mu plus lambda over 2. 1053 01:11:13,540 --> 01:11:15,992 Actually, I have to put an absolute value squared here. 1054 01:11:21,080 --> 01:11:26,270 So that's the most general theory 1055 01:11:26,270 --> 01:11:31,330 of elasticity in any number of dimensions 1056 01:11:31,330 --> 01:11:38,060 that is consistent with this symmetry that I have here. 1057 01:11:38,060 --> 01:11:39,990 And it turns out that this corresponds 1058 01:11:39,990 --> 01:11:43,140 to elasticity of materials that are isotropic. 1059 01:11:51,240 --> 01:11:55,280 And they are described by 2 elastic coefficients, 1060 01:11:55,280 --> 01:11:58,570 mu and lambda, that are called [INAUDIBLE] coefficient. 1061 01:11:58,570 --> 01:12:01,840 Mu is also related to shear modules. 1062 01:12:01,840 --> 01:12:07,000 Actually, mu and lambda combined are related to [INAUDIBLE] 1063 01:12:07,000 --> 01:12:10,380 And if I want, in fact, to fully transform 1064 01:12:10,380 --> 01:12:17,190 this back to the space, in real space the can be written 1065 01:12:17,190 --> 01:12:24,500 as mu over 2 mu alpha beta of x mu alpha 1066 01:12:24,500 --> 01:12:29,170 beta over x, where the sum over alpha and beta takes place. 1067 01:12:29,170 --> 01:12:32,060 Alpha and beta run from 1 to d. 1068 01:12:32,060 --> 01:12:34,710 And the other term, lambda over 2, 1069 01:12:34,710 --> 01:12:39,960 is U alpha alpha of x, U beta beta of x. 1070 01:12:39,960 --> 01:12:44,020 And this object, U alpha beta, is one half 1071 01:12:44,020 --> 01:12:51,395 of the symmeterized derivatives, du alpha by dx beta plus du 1072 01:12:51,395 --> 01:12:54,470 beta by dx alpha. 1073 01:12:54,470 --> 01:12:59,436 And it's called the strength 1074 01:12:59,436 --> 01:13:00,250 AUDIENCE: Question. 1075 01:13:00,250 --> 01:13:02,340 PROFESSOR: Yes? 1076 01:13:02,340 --> 01:13:05,360 AUDIENCE: So are you still looking 1077 01:13:05,360 --> 01:13:08,601 in the regime of low energy expectations? 1078 01:13:08,601 --> 01:13:09,226 PROFESSOR: Yes. 1079 01:13:09,226 --> 01:13:10,890 That's right. 1080 01:13:10,890 --> 01:13:13,130 AUDIENCE: So wouldn't the discreteness 1081 01:13:13,130 --> 01:13:17,470 of the allowed wave vectors become important? 1082 01:13:17,470 --> 01:13:19,280 And if so, why are you integrating 1083 01:13:19,280 --> 01:13:20,580 rather than discrete summon? 1084 01:13:23,340 --> 01:13:24,360 PROFESSOR: OK. 1085 01:13:24,360 --> 01:13:28,850 So let's go back to what we have over here. 1086 01:13:28,850 --> 01:13:32,140 The discreteness is present over here. 1087 01:13:32,140 --> 01:13:36,730 And what I am looking at for the discreteness, this spacing 1088 01:13:36,730 --> 01:13:41,700 that I have between these objects is 2pi over L. 1089 01:13:41,700 --> 01:13:44,840 The L is the size of the system. 1090 01:13:44,840 --> 01:13:48,183 So if you like, what we are looking at here 1091 01:13:48,183 --> 01:13:52,814 is the hierarchy of landscapes, where 1092 01:13:52,814 --> 01:13:57,710 L is much larger than the typical wavelengths 1093 01:13:57,710 --> 01:14:02,250 of these excitations that are set by the temperature, which 1094 01:14:02,250 --> 01:14:05,610 is turn much larger than the lattice place. 1095 01:14:05,610 --> 01:14:08,900 And so when we are talking about, say, 1096 01:14:08,900 --> 01:14:12,790 a solid at around 100 degrees temperature or so, 1097 01:14:12,790 --> 01:14:17,950 this-- then say, over here, it typically spans 1098 01:14:17,950 --> 01:14:22,440 10 to 100 atoms, where as the actual size of the system 1099 01:14:22,440 --> 01:14:24,970 spans billions of atoms or more. 1100 01:14:24,970 --> 01:14:28,902 And so the separations that are imposed 1101 01:14:28,902 --> 01:14:31,290 by the discreteness of k are irrelevant 1102 01:14:31,290 --> 01:14:34,134 to the considerations that we have. 1103 01:14:37,926 --> 01:14:38,880 Yes? 1104 01:14:38,880 --> 01:14:42,090 AUDIENCE: So before, with this adding a constant c, that 1105 01:14:42,090 --> 01:14:47,542 corresponds to translating almost crystal by some vector. 1106 01:14:47,542 --> 01:14:48,250 PROFESSOR: Right. 1107 01:14:48,250 --> 01:14:51,610 AUDIENCE: For the rotation, is this a rotation of the crystal 1108 01:14:51,610 --> 01:14:54,460 or is this a rotation of the displacement field? 1109 01:14:54,460 --> 01:14:56,790 PROFESSOR: It's the rotation of the entire crystal. 1110 01:14:56,790 --> 01:15:02,100 So you can see that essentially both x and U 1111 01:15:02,100 --> 01:15:03,730 have to be rotated together. 1112 01:15:03,730 --> 01:15:05,950 I didn't write it precisely enough. 1113 01:15:05,950 --> 01:15:10,550 But when I wrote the invariant as being k dot U, 1114 01:15:10,550 --> 01:15:13,190 the implicit thing was that the debate vector 1115 01:15:13,190 --> 01:15:16,100 and the distortion are rotated. 1116 01:15:16,100 --> 01:15:19,660 AUDIENCE: So does it require an isotropic crystal in that case? 1117 01:15:19,660 --> 01:15:21,101 PROFESSOR: Yes. 1118 01:15:21,101 --> 01:15:23,200 AUDIENCE: I would think if you're 1119 01:15:23,200 --> 01:15:24,725 rotating everything together, who 1120 01:15:24,725 --> 01:15:28,642 cares if one axis is different than another? 1121 01:15:28,642 --> 01:15:30,350 Because if I have a non-isotropic crystal 1122 01:15:30,350 --> 01:15:32,240 and I rotate it around, it shouldn't 1123 01:15:32,240 --> 01:15:33,470 change the internal energy. 1124 01:15:45,230 --> 01:15:46,410 PROFESSOR: OK. 1125 01:15:46,410 --> 01:15:50,880 Where it will make difference is at higher order terms. 1126 01:15:50,880 --> 01:15:54,970 And so then I have to think about the invariants that 1127 01:15:54,970 --> 01:15:57,560 are possible at the level of higher order terms. 1128 01:15:57,560 --> 01:15:58,720 But that's a good question. 1129 01:15:58,720 --> 01:16:02,920 Let me come back and try answer that more carefully next time 1130 01:16:02,920 --> 01:16:04,470 around.