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 --> 00:00:17,215 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,215 --> 00:00:17,840 at ocw.mit.edu. 8 00:00:22,010 --> 00:00:25,420 PROFESSOR: OK, let's start, so last lecture, 9 00:00:25,420 --> 00:00:30,790 we were talking about elasticity as an example of a field 10 00:00:30,790 --> 00:00:33,600 theory-- statistical field theory. 11 00:00:33,600 --> 00:00:38,110 And I ended the class by not giving a very good answer 12 00:00:38,110 --> 00:00:39,690 for a question was asked. 13 00:00:39,690 --> 00:00:42,280 So let's go back and revisit it. 14 00:00:42,280 --> 00:00:44,710 So the idea that we had was that maybe we 15 00:00:44,710 --> 00:00:48,560 have some kind of a lattice. 16 00:00:48,560 --> 00:00:55,250 And we make a distortion through a set 17 00:00:55,250 --> 00:01:02,570 of vectors in however many dimensional space we are. 18 00:01:02,570 --> 00:01:06,740 And in the continuum format, we basically 19 00:01:06,740 --> 00:01:12,590 regarded this u as being an average over some neighborhood. 20 00:01:12,590 --> 00:01:17,610 But before going to do that, we said that the energy cost 21 00:01:17,610 --> 00:01:21,100 of the distortion at the quadratic level-- of course, 22 00:01:21,100 --> 00:01:23,540 we can expand to go to a higher level-- 23 00:01:23,540 --> 00:01:28,620 was the sum over all pairs of positions, 24 00:01:28,620 --> 00:01:32,180 all pairs of components, alpha and beta data 25 00:01:32,180 --> 00:01:34,065 running from one to a higher whatever 26 00:01:34,065 --> 00:01:39,710 the dimensionality of space is u alpha at r or u beta 27 00:01:39,710 --> 00:01:40,880 at r prime. 28 00:01:43,460 --> 00:01:47,270 The components alphabet beta of this vector u at these two 29 00:01:47,270 --> 00:01:51,690 different locations-- let's say r and r prime here. 30 00:01:51,690 --> 00:01:58,720 And then there was some kind of an object that 31 00:01:58,720 --> 00:02:02,800 correlated the changes here and here maybe obtained 32 00:02:02,800 --> 00:02:06,070 as a second derivative of a potential energy 33 00:02:06,070 --> 00:02:09,690 as a function of the entirety of the coordinates. 34 00:02:09,690 --> 00:02:15,440 And this in principle or a general function 35 00:02:15,440 --> 00:02:18,250 of all of these distortions depends 36 00:02:18,250 --> 00:02:22,510 on all pairs of variables. 37 00:02:22,510 --> 00:02:28,980 But then we said that because we are dealing with a lattice 38 00:02:28,980 --> 00:02:32,200 and one pair of coordinates in the lattice 39 00:02:32,200 --> 00:02:34,930 is the same as another pair of coordinates 40 00:02:34,930 --> 00:02:40,590 that have the same spatial orientations that this function 41 00:02:40,590 --> 00:02:45,240 merely is a function of the separation 42 00:02:45,240 --> 00:02:48,890 r minus r prime, which of course can 43 00:02:48,890 --> 00:02:51,650 be a vector in this lattice. 44 00:02:51,650 --> 00:02:55,550 And the fact that it has this form 45 00:02:55,550 --> 00:02:59,410 then allows us to simplify this quadratic form 46 00:02:59,410 --> 00:03:04,990 rather than be a sum over pairs and squared, if you like, 47 00:03:04,990 --> 00:03:09,480 into a sum over one coordinate which 48 00:03:09,480 --> 00:03:14,550 is the wave vector obtained by Fourier transform. 49 00:03:14,550 --> 00:03:18,810 So once we Fourier transform, that V 50 00:03:18,810 --> 00:03:25,450 can be written as a sum over just one set of K vectors, 51 00:03:25,450 --> 00:03:29,110 of course appropriately discretized depending 52 00:03:29,110 --> 00:03:31,580 on the overall size of the system 53 00:03:31,580 --> 00:03:34,950 and confined to a [INAUDIBLE] zone that is determined 54 00:03:34,950 --> 00:03:40,290 by the type of wavelengths that lattice structure allows. 55 00:03:40,290 --> 00:03:40,790 And 56 00:03:40,790 --> 00:03:43,660 Then this becomes your alpha tilde. 57 00:03:43,660 --> 00:03:46,070 The Fourier transform's evaluated 58 00:03:46,070 --> 00:03:51,060 at K and the other point, which is minus K that if I write it 59 00:03:51,060 --> 00:03:54,730 again as K is really this. 60 00:03:54,730 --> 00:03:58,910 And this entity, once we Fourier transform it, 61 00:03:58,910 --> 00:04:07,160 becomes a function of the wave number K. Now, 62 00:04:07,160 --> 00:04:11,390 there is one statement that is correct, 63 00:04:11,390 --> 00:04:16,279 which is that if I have to take the whole lattice and move it, 64 00:04:16,279 --> 00:04:20,180 then essentially there would be no cost. 65 00:04:20,180 --> 00:04:23,820 And moving the entire lattice only 66 00:04:23,820 --> 00:04:26,840 changes the Fourier component that corresponds to K 67 00:04:26,840 --> 00:04:29,890 equals to 0-- translation of everything. 68 00:04:29,890 --> 00:04:39,105 So you know for sure that this K alpha beta at K equals to 0 69 00:04:39,105 --> 00:04:40,325 has to be 0. 70 00:04:44,540 --> 00:04:47,840 Now, we did a little bit more than that, then. 71 00:04:47,840 --> 00:04:55,120 We said, let's look not only at K equals to 0, but for small k. 72 00:04:55,120 --> 00:04:57,850 And then for small k, we are allowed 73 00:04:57,850 --> 00:05:00,800 to make an expansion of this. 74 00:05:00,800 --> 00:05:02,790 And this is the expansion K squared, 75 00:05:02,790 --> 00:05:05,020 K to the fourth, et cetera, that we 76 00:05:05,020 --> 00:05:10,570 hope to terminate at low values of K. Now, 77 00:05:10,570 --> 00:05:13,850 that itself is an assumption. 78 00:05:13,850 --> 00:05:16,830 That's part of the locality that i 79 00:05:16,830 --> 00:05:22,340 stated that I can make an expansion as a function of K. 80 00:05:22,340 --> 00:05:24,340 After all, let's say K to the 1/2 81 00:05:24,340 --> 00:05:27,720 is a function that goes to 0 at K equals to 0. 82 00:05:27,720 --> 00:05:30,140 But it is non analytic. 83 00:05:30,140 --> 00:05:34,075 Turns out that in order to generate that kind of non 84 00:05:34,075 --> 00:05:37,070 analyticity in this, this function 85 00:05:37,070 --> 00:05:39,170 in real as a function of separation 86 00:05:39,170 --> 00:05:41,420 should decay very slow. 87 00:05:41,420 --> 00:05:44,050 Sometimes something like a Coulomb interaction 88 00:05:44,050 --> 00:05:46,220 that is very long range will give you 89 00:05:46,220 --> 00:05:48,360 that kind of singularity. 90 00:05:48,360 --> 00:05:51,220 But things that are really just interacting 91 00:05:51,220 --> 00:05:55,340 within some neighborhood or even if they are long range, 92 00:05:55,340 --> 00:05:58,180 if the action falls of sufficiently rapidly-- 93 00:05:58,180 --> 00:06:00,700 and that's one of the things that we will discuss later 94 00:06:00,700 --> 00:06:03,830 on-- precisely what kind of potential 95 00:06:03,830 --> 00:06:05,800 allow you to make an expansion that 96 00:06:05,800 --> 00:06:09,235 is analytical in powers of k and hence consistent 97 00:06:09,235 --> 00:06:12,680 with this idea of locality. 98 00:06:12,680 --> 00:06:15,100 Let's also ignore cases where you 99 00:06:15,100 --> 00:06:18,350 don't have inversion symmetry. 100 00:06:18,350 --> 00:06:22,100 So we don't need to start and worry about anything 101 00:06:22,100 --> 00:06:26,110 that is linear in K. So the first thing that I can appear 102 00:06:26,110 --> 00:06:29,272 is quadratic in K. 103 00:06:29,272 --> 00:06:33,030 Now, when we have things that are quadratic in K, 104 00:06:33,030 --> 00:06:35,040 I have to make an object that has 105 00:06:35,040 --> 00:06:37,940 two indices-- alpha and beta. 106 00:06:37,940 --> 00:06:40,900 And, well, how can I do that? 107 00:06:40,900 --> 00:06:43,435 Well, the kinds of things that I mentioned last time, we 108 00:06:43,435 --> 00:06:46,060 could a term that is K squared. 109 00:06:46,060 --> 00:06:49,980 And then we have a delta alpha beta. 110 00:06:49,980 --> 00:06:54,350 If I insert that over here, what do I get? 111 00:06:54,350 --> 00:06:59,150 I get K squared delta alpha beta will give me U tilde squared. 112 00:06:59,150 --> 00:07:04,170 So that's the term that is in an isotropic solid identified 113 00:07:04,170 --> 00:07:07,840 with the shear modulus. 114 00:07:07,840 --> 00:07:12,250 Again, if the system is isotropic, 115 00:07:12,250 --> 00:07:16,960 I don't know any distance apart from any other distance, 116 00:07:16,960 --> 00:07:18,790 any direction, any other direction. 117 00:07:18,790 --> 00:07:23,540 I can still make another tensor between this 118 00:07:23,540 --> 00:07:28,240 is alpha and beta by multiplying k alpha and k beta. 119 00:07:28,240 --> 00:07:31,480 If I then multiply this with this, 120 00:07:31,480 --> 00:07:35,640 I will get K.U-- the dot product squared. 121 00:07:35,640 --> 00:07:39,010 So that's the other term that in an isotropic solid 122 00:07:39,010 --> 00:07:43,610 was identified with nu plus lambda over 2. 123 00:07:43,610 --> 00:07:49,380 Now, as long as any direction for K in this lattice 124 00:07:49,380 --> 00:07:53,080 is the same, those are really the only terms 125 00:07:53,080 --> 00:07:55,830 that you can write down. 126 00:07:55,830 --> 00:08:00,210 But suppose that your system was unisotropic, 127 00:08:00,210 --> 00:08:03,290 such as for example a rectangular lattice. 128 00:08:03,290 --> 00:08:07,360 Or let's imagine we put a whole bunch of coins 129 00:08:07,360 --> 00:08:10,250 here to emphasize that rather than something 130 00:08:10,250 --> 00:08:12,750 like a square lattice, we have something 131 00:08:12,750 --> 00:08:16,112 that is very rectangular. 132 00:08:16,112 --> 00:08:19,900 In this case, we can see that the x and y directions 133 00:08:19,900 --> 00:08:21,940 are not the same. 134 00:08:21,940 --> 00:08:26,760 And there is no reason why this K squared here could not 135 00:08:26,760 --> 00:08:30,440 have been separately kx squared plus some other number 136 00:08:30,440 --> 00:08:31,000 ky squared. 137 00:08:33,770 --> 00:08:37,150 And hence, the type of elasticity that you would have 138 00:08:37,150 --> 00:08:41,010 would no longer depend on just two numbers. 139 00:08:41,010 --> 00:08:45,470 But, say, over here with three, in fact there would be more. 140 00:08:45,470 --> 00:08:50,110 And precisely how many independent elastic constants 141 00:08:50,110 --> 00:08:57,110 are given depend on the point group symmetry-- the way 142 00:08:57,110 --> 00:08:59,470 that your lattice is structured over here. 143 00:08:59,470 --> 00:09:02,880 But that's another story that I don't want to get into. 144 00:09:02,880 --> 00:09:05,520 But last time, I was rather careless 145 00:09:05,520 --> 00:09:08,920 and I said that just rotational symmetry will give you 146 00:09:08,920 --> 00:09:11,360 this k squared, k alpha, k beta. 147 00:09:11,360 --> 00:09:14,860 I have to be more precise about that. 148 00:09:14,860 --> 00:09:17,790 So this is what's happened. 149 00:09:17,790 --> 00:09:25,290 Now, given that you have an isotropic material, 150 00:09:25,290 --> 00:09:27,940 then the conclusion is that the potential energy 151 00:09:27,940 --> 00:09:30,870 will have two types of terms. 152 00:09:30,870 --> 00:09:32,680 There is the type of term that goes 153 00:09:32,680 --> 00:09:39,080 with k squared u tilde squared. 154 00:09:39,080 --> 00:09:40,830 And then there's a type of term that 155 00:09:40,830 --> 00:09:48,600 goes with u plus lambda over 2k tilde k times u tilde squared. 156 00:09:51,497 --> 00:09:53,080 Now you can see that immediately there 157 00:09:53,080 --> 00:09:57,140 is a distinction between two types of distortion. 158 00:09:57,140 --> 00:10:02,170 Because if I select my wave vector K here, 159 00:10:02,170 --> 00:10:07,420 any distortion that is orthogonal to that 160 00:10:07,420 --> 00:10:10,630 then will not get the contribution from this. 161 00:10:10,630 --> 00:10:16,840 And so the cost of those so called transverse distortions 162 00:10:16,840 --> 00:10:20,160 only come from u. 163 00:10:20,160 --> 00:10:27,600 Whereas if I make a distortion that is along k, 164 00:10:27,600 --> 00:10:31,570 it will get contribution from both of those two terms. 165 00:10:31,570 --> 00:10:40,430 And hence, the cost of those longitudinal distortions 166 00:10:40,430 --> 00:10:43,450 would be different. 167 00:10:43,450 --> 00:10:48,280 So when we go back to our story of how 168 00:10:48,280 --> 00:10:56,620 does the frequency depend on a whole bunch of ks, 169 00:10:56,620 --> 00:10:59,400 you can see that when we go to the small k 170 00:10:59,400 --> 00:11:03,130 limit in an isotropic material, it 171 00:11:03,130 --> 00:11:07,420 you've established that there will be longitudinal mode. 172 00:11:07,420 --> 00:11:11,240 And we'll some kind of longitudinal sound velocity. 173 00:11:11,240 --> 00:11:13,790 And in three dimensions, there will 174 00:11:13,790 --> 00:11:17,200 be two branches of the transverse mode that 175 00:11:17,200 --> 00:11:20,170 fall on top of each-other. 176 00:11:20,170 --> 00:11:23,820 As I said, once you go to further larger K, 177 00:11:23,820 --> 00:11:26,640 then you can include all kinds of other things. 178 00:11:26,640 --> 00:11:29,680 In fact, the types of things that you can include 179 00:11:29,680 --> 00:11:32,040 at fourth order, sixth order, et cetera 180 00:11:32,040 --> 00:11:36,510 again are constraints somewhat by the symmetry. 181 00:11:36,510 --> 00:11:38,850 But their number proliferates. 182 00:11:38,850 --> 00:11:42,160 And eventually you get all kinds of things 183 00:11:42,160 --> 00:11:47,290 that could potentially describe how these modes depend. 184 00:11:47,290 --> 00:11:51,050 And those things are dependent on your potential 185 00:11:51,050 --> 00:11:54,570 I don't know too much about. 186 00:11:54,570 --> 00:11:59,270 And why was this relevant to what we were discussing? 187 00:11:59,270 --> 00:12:05,560 We said that basically, once you go to some temperature other 188 00:12:05,560 --> 00:12:10,630 than zero, you start putting energy into the modes. 189 00:12:10,630 --> 00:12:14,940 And how far in frequency you can go given your temperature, 190 00:12:14,940 --> 00:12:18,460 typically, the maximum frequency at the particular temperature 191 00:12:18,460 --> 00:12:20,990 is of the order of kt over hr. 192 00:12:24,360 --> 00:12:26,950 So that if you are at very high temperature, 193 00:12:26,950 --> 00:12:31,110 all of the r1 equals [INAUDIBLE] and all of the vibrations 194 00:12:31,110 --> 00:12:32,680 are relevant. 195 00:12:32,680 --> 00:12:35,100 But if you go to low temperatures, 196 00:12:35,100 --> 00:12:38,590 eventually you hit the regime where 197 00:12:38,590 --> 00:12:43,840 only these long wavelength, low frequency modes are excited. 198 00:12:43,840 --> 00:12:46,020 And your heat capacity was really 199 00:12:46,020 --> 00:12:50,080 proportional to how many modes are excited. 200 00:12:50,080 --> 00:12:52,720 And from here, you can see since omega 201 00:12:52,720 --> 00:12:56,610 goes like T, either the case, for longitudinal 202 00:12:56,610 --> 00:12:59,740 or the k for transverse has to be 203 00:12:59,740 --> 00:13:02,750 proportional to T the maximum 1. 204 00:13:02,750 --> 00:13:04,940 Larger values of k would correspond 205 00:13:04,940 --> 00:13:07,840 to frequencies that are not excited. 206 00:13:07,840 --> 00:13:12,140 So all of the nodes that are down here 207 00:13:12,140 --> 00:13:14,370 are going to be excited. 208 00:13:14,370 --> 00:13:16,150 How many of them are there? 209 00:13:16,150 --> 00:13:20,215 Up to from 0 to this k max that is 210 00:13:20,215 --> 00:13:25,120 of the order of kt over h bar and some sound velocity. 211 00:13:25,120 --> 00:13:27,490 So you would say that the heat capacity which 212 00:13:27,490 --> 00:13:32,920 is proportional to the number of oscillators that you can have, 213 00:13:32,920 --> 00:13:37,960 you kt-- kb per oscillator. 214 00:13:37,960 --> 00:13:41,650 The number of oscillators was roughly volume 215 00:13:41,650 --> 00:13:50,770 times kt h bar v, cubed if you are in three dimensions. 216 00:13:50,770 --> 00:13:53,220 You saw that it was one in one dimension. 217 00:13:53,220 --> 00:13:57,010 So in general, it will be something like d. 218 00:13:57,010 --> 00:14:02,270 Of course, here, I was kind of not very precise because I 219 00:14:02,270 --> 00:14:05,300 really have to separate out the contribution 220 00:14:05,300 --> 00:14:09,340 of longitudinal modes and transverse modes, et cetera. 221 00:14:09,340 --> 00:14:13,320 So ultimately, the amplitude that I have 222 00:14:13,320 --> 00:14:15,692 will depend on a whole bunch of things. 223 00:14:18,350 --> 00:14:25,500 But the functional form is this universal t to the d. 224 00:14:25,500 --> 00:14:30,090 So the lesson that we would like to get from this simple, well 225 00:14:30,090 --> 00:14:34,870 known example is that sometimes you 226 00:14:34,870 --> 00:14:39,320 get results that are independent in form 227 00:14:39,320 --> 00:14:43,280 for very different materials. 228 00:14:43,280 --> 00:14:46,480 I will use the term universality. 229 00:14:46,480 --> 00:14:49,480 And the origin of that universality 230 00:14:49,480 --> 00:14:52,960 is because we are dealing with phenomena 231 00:14:52,960 --> 00:14:55,450 that involve collective behavior. 232 00:14:55,450 --> 00:14:59,260 As k becomes small, you're dealing with large wavelengths 233 00:14:59,260 --> 00:15:02,550 encompassing the collective motion of lots 234 00:15:02,550 --> 00:15:05,045 of things that are vibrating together. 235 00:15:05,045 --> 00:15:08,480 And so this kind of statistical averaging 236 00:15:08,480 --> 00:15:13,330 over lots of things that are collectively working together 237 00:15:13,330 --> 00:15:18,600 allows a lot of the details to in some sense be washed out 238 00:15:18,600 --> 00:15:21,990 and gives you a universal form just 239 00:15:21,990 --> 00:15:25,110 like adding random variables will give you a Gaussian if you 240 00:15:25,110 --> 00:15:27,920 add sufficient many of them together. 241 00:15:27,920 --> 00:15:30,420 It's that kind of phenomenon. 242 00:15:30,420 --> 00:15:32,880 Typically, we will be dealing with these kinds 243 00:15:32,880 --> 00:15:34,940 of relationships between quantities 244 00:15:34,940 --> 00:15:36,330 that have dimensions. 245 00:15:36,330 --> 00:15:38,420 Heat capacity has dimensions. 246 00:15:38,420 --> 00:15:40,730 Temperature has dimensions. 247 00:15:40,730 --> 00:15:44,410 So the amplitude will have to have 248 00:15:44,410 --> 00:15:47,620 things that have dimensions that come from the material 249 00:15:47,620 --> 00:15:49,640 that you are dealing with. 250 00:15:49,640 --> 00:15:53,325 So the thing that is universal is typically 251 00:15:53,325 --> 00:15:58,200 the exponents that appear in these functional forms. 252 00:15:58,200 --> 00:16:01,080 So those are the kinds of things that we 253 00:16:01,080 --> 00:16:03,350 would like to capture in different contexts. 254 00:16:08,640 --> 00:16:10,270 Now, as I said, this was supposed 255 00:16:10,270 --> 00:16:14,546 to be just an illustration of the typical approach. 256 00:16:14,546 --> 00:16:16,950 The phenomenon that we will be grappling 257 00:16:16,950 --> 00:16:20,405 with for a large part of this course 258 00:16:20,405 --> 00:16:22,080 has to do with face transactions. 259 00:16:28,240 --> 00:16:32,410 And again, this is a manifest result 260 00:16:32,410 --> 00:16:37,010 of having interaction among degrees of freedom that 261 00:16:37,010 --> 00:16:41,550 causes them to collectively behave separately differently 262 00:16:41,550 --> 00:16:45,810 from how a single individual would behave. 263 00:16:45,810 --> 00:16:52,930 And the example that we discussed 264 00:16:52,930 --> 00:16:55,370 in the previous course and is familiar 265 00:16:55,370 --> 00:17:06,540 is of course that of ice, water, steam. 266 00:17:06,540 --> 00:17:10,660 So here, we can look at things from two 267 00:17:10,660 --> 00:17:12,720 different perspectives. 268 00:17:12,720 --> 00:17:17,890 One perspective, let's say we start by-- actually, 269 00:17:17,890 --> 00:17:23,460 let's start with the perspective in which I show you 270 00:17:23,460 --> 00:17:26,500 pressure and temperature. 271 00:17:26,500 --> 00:17:30,330 And as a function of pressure and temperature, 272 00:17:30,330 --> 00:17:33,410 low temperatures and high pressures 273 00:17:33,410 --> 00:17:41,690 would respond to having some kind of a-- actually, 274 00:17:41,690 --> 00:17:45,830 let me not incur the wrath of people 275 00:17:45,830 --> 00:17:50,400 who know the slopes of these curves precisely. 276 00:17:50,400 --> 00:17:55,730 So there is at low temperatures, you have ice. 277 00:17:55,730 --> 00:17:59,030 And then the more interesting part 278 00:17:59,030 --> 00:18:02,960 is that you have at high temperatures and low pressures 279 00:18:02,960 --> 00:18:04,350 gas. 280 00:18:04,350 --> 00:18:06,860 And then at immediate values, you 281 00:18:06,860 --> 00:18:08,410 have, of course, the liquid. 282 00:18:11,170 --> 00:18:16,630 And the other perspective to look at this 283 00:18:16,630 --> 00:18:23,800 is to look at isoterms of the system for pressure 284 00:18:23,800 --> 00:18:25,290 versus velocity. 285 00:18:25,290 --> 00:18:30,930 So basically, what I'm doing is I'm staking at some temperature 286 00:18:30,930 --> 00:18:35,300 and calculating the behavior along that isoterm of how 287 00:18:35,300 --> 00:18:38,950 pressure and volume of the system are related. 288 00:18:38,950 --> 00:18:42,120 So then you are far out to the right, 289 00:18:42,120 --> 00:18:45,540 you'll have behavior that is releasing of an ideal gas. 290 00:18:45,540 --> 00:18:52,100 You have PV be proportional to the number of particles 291 00:18:52,100 --> 00:18:53,530 in temperature. 292 00:18:53,530 --> 00:18:58,830 As you lower the temperatures and you 293 00:18:58,830 --> 00:19:08,330 hit this critical isoterm at TC, the shape of the curve 294 00:19:08,330 --> 00:19:11,710 gets modified to something like this. 295 00:19:11,710 --> 00:19:15,260 And if you're looking at things at isoterms 296 00:19:15,260 --> 00:19:21,220 at an even lower temperature, you encounter a discontinuity. 297 00:19:21,220 --> 00:19:24,490 And that discontinuity is manifested 298 00:19:24,490 --> 00:19:29,700 in a coexistent interval between the liquid and gas. 299 00:19:29,700 --> 00:19:35,600 So isoterms for T and less than TC 300 00:19:35,600 --> 00:19:37,070 has this characteristic form. 301 00:19:46,560 --> 00:19:49,280 So here, I have not bothered to go along the way down 302 00:19:49,280 --> 00:19:51,040 to the case of the solid. 303 00:19:51,040 --> 00:19:54,850 Because our focus is going to be mostly as to what 304 00:19:54,850 --> 00:20:01,150 happens at this critical point-- TC and TC, 305 00:20:01,150 --> 00:20:08,310 where the distinction between liquid and gas disappears. 306 00:20:08,310 --> 00:20:17,670 So there is here the first case of a transition 307 00:20:17,670 --> 00:20:21,050 between different types of material 308 00:20:21,050 --> 00:20:26,130 that is manifested here, the solid line indicating 309 00:20:26,130 --> 00:20:31,250 a discontinuity in various thermodynamic properties. 310 00:20:31,250 --> 00:20:34,910 Discontinuity here being, say, the density 311 00:20:34,910 --> 00:20:36,820 as a function of pressure. 312 00:20:36,820 --> 00:20:39,790 Rather than having a nice curve, you have a discontinuity. 313 00:20:39,790 --> 00:20:44,720 In these curves, isoterms are suddenly discontinuous. 314 00:20:44,720 --> 00:20:49,320 And the question that we posed last time last semester 315 00:20:49,320 --> 00:20:52,580 was that essentially, all the properties 316 00:20:52,580 --> 00:20:55,960 of the system, the thermodynamic properties, I 317 00:20:55,960 --> 00:20:59,940 should be able to obtain through the partition function, 318 00:20:59,940 --> 00:21:03,490 log of the partition function, which involves an integral, 319 00:21:03,490 --> 00:21:05,610 let's say, over all of the coordinates 320 00:21:05,610 --> 00:21:13,180 and momenta of some kind of energy. 321 00:21:16,930 --> 00:21:20,050 And this energy in the part about momenta 322 00:21:20,050 --> 00:21:22,851 is not particularly important. 323 00:21:22,851 --> 00:21:25,260 Let's just get rid of that. 324 00:21:25,260 --> 00:21:28,470 And the part about the coordinates 325 00:21:28,470 --> 00:21:32,305 involves some kind of potential interaction 326 00:21:32,305 --> 00:21:35,220 between pairs of particles. 327 00:21:35,220 --> 00:21:37,110 That is not that difficult. 328 00:21:37,110 --> 00:21:39,980 Maybe particles are slightly attracted to each other 329 00:21:39,980 --> 00:21:41,370 when they're close enough. 330 00:21:41,370 --> 00:21:43,170 And they have a hard core. 331 00:21:43,170 --> 00:21:48,960 But somehow, after I do this calculation bunch of integrals, 332 00:21:48,960 --> 00:21:51,540 all of them are perfectly well behaved. 333 00:21:51,540 --> 00:21:52,630 There is no divergence. 334 00:21:52,630 --> 00:21:53,130 As 335 00:21:53,130 --> 00:21:55,590 The range of integration goes to zero 336 00:21:55,590 --> 00:21:58,820 and infinity, I get this discontinuity. 337 00:21:58,820 --> 00:22:02,410 And the question of how that appears 338 00:22:02,410 --> 00:22:06,210 is something that clearly is a consequence of interactions. 339 00:22:06,210 --> 00:22:08,050 If we didn't have interactions, we 340 00:22:08,050 --> 00:22:10,560 would have ideal gas behavior. 341 00:22:10,560 --> 00:22:18,710 And maybe this place is really a better place 342 00:22:18,710 --> 00:22:21,080 to go and try to figure out what's 343 00:22:21,080 --> 00:22:26,350 going on than any other place in the phase diagram. 344 00:22:26,350 --> 00:22:30,100 The reason for that is in the vicinity of this point, 345 00:22:30,100 --> 00:22:34,400 we can see that the difference between liquid and gas 346 00:22:34,400 --> 00:22:37,650 is gradually disappearing. 347 00:22:37,650 --> 00:22:40,975 So in some sense, we have a small parameter. 348 00:22:40,975 --> 00:22:44,590 There's some small difference that appears here. 349 00:22:44,590 --> 00:22:49,130 And so maybe the idea that we can start with some phase 350 00:22:49,130 --> 00:22:51,600 that we understand fully and then perturb it 351 00:22:51,600 --> 00:22:56,520 and see how the singularity appears is a good idea. 352 00:22:56,520 --> 00:22:59,720 I will justify for you why that's the case 353 00:22:59,720 --> 00:23:04,060 and maybe why we can even construct a statistical field 354 00:23:04,060 --> 00:23:06,220 theory here. 355 00:23:06,220 --> 00:23:12,030 But the reason it is also interesting 356 00:23:12,030 --> 00:23:16,580 is that there is a lot of experimental evidence as to why 357 00:23:16,580 --> 00:23:18,640 you should be doing this. 358 00:23:18,640 --> 00:23:20,770 We discussed the phase diagrams. 359 00:23:20,770 --> 00:23:23,430 And we noticed that a interesting feature of these 360 00:23:23,430 --> 00:23:32,150 phase diagrams is that below TC when the liquid and gas first 361 00:23:32,150 --> 00:23:36,070 manifest themselves as different phases , 362 00:23:36,070 --> 00:23:40,200 there is a coexistence interval. 363 00:23:40,200 --> 00:23:44,530 And this coexistence interval is bounded on two sides 364 00:23:44,530 --> 00:23:50,630 by the gas and liquid volumes or alternatively densities. 365 00:23:50,630 --> 00:23:53,420 Now, the interesting experimental fact 366 00:23:53,420 --> 00:24:00,770 is that when you go and observe the shape of this curve, 367 00:24:00,770 --> 00:24:04,410 for the whole bunch of different gases-- here, 368 00:24:04,410 --> 00:24:09,600 you have neon, argon, krypton, xenon, oxygen, carbon dioxide, 369 00:24:09,600 --> 00:24:12,330 methane, things that are very different-- 370 00:24:12,330 --> 00:24:17,550 and scale them appropriately so that all of the vertical axes 371 00:24:17,550 --> 00:24:18,440 comes to one. 372 00:24:18,440 --> 00:24:20,580 So you divide P by PC. 373 00:24:20,580 --> 00:24:23,890 You appropriately normalize the horizontal axis 374 00:24:23,890 --> 00:24:26,380 so the maximum is at 1. 375 00:24:26,380 --> 00:24:30,240 You see that all of the curves you get from very, very 376 00:24:30,240 --> 00:24:34,430 different systems after you just do this simple scaling 377 00:24:34,430 --> 00:24:37,960 fall right on top of each-other. 378 00:24:37,960 --> 00:24:41,800 Now, clearly something like carbon dioxide and neon, 379 00:24:41,800 --> 00:24:45,320 they have very different inter atomic potentials 380 00:24:45,320 --> 00:24:49,010 that I have to put in this calculation. 381 00:24:49,010 --> 00:24:53,560 Yet despite that, there has emerged 382 00:24:53,560 --> 00:24:57,380 some kind of a universal law. 383 00:24:57,380 --> 00:25:00,780 And so I should be able to describe that. 384 00:25:00,780 --> 00:25:04,250 Why is that happening? 385 00:25:04,250 --> 00:25:08,490 Now I'll try to convince you that what I need to do 386 00:25:08,490 --> 00:25:10,710 is similar to what I did there. 387 00:25:10,710 --> 00:25:14,300 I need to construct a statistical field theory. 388 00:25:14,300 --> 00:25:16,550 But we said that statistical field theories 389 00:25:16,550 --> 00:25:20,050 rely on averaging over many things. 390 00:25:20,050 --> 00:25:23,450 So why is that justified in this context? 391 00:25:23,450 --> 00:25:27,920 So there is a phenomenon called critical opalescence. 392 00:25:27,920 --> 00:25:29,320 Let's look at it here. 393 00:25:29,320 --> 00:25:32,710 Also, this serves to show something that people sometimes 394 00:25:32,710 --> 00:25:36,950 don't believe, which is that since our experience comes 395 00:25:36,950 --> 00:25:42,330 from no pressures, where we cool the gas and it goes to a liquid 396 00:25:42,330 --> 00:25:44,790 or heat it and liquid goes to a gas 397 00:25:44,790 --> 00:25:47,960 and know that these are different things, 398 00:25:47,960 --> 00:25:50,290 it's kind of-- they find it hard to believe 399 00:25:50,290 --> 00:25:52,770 that if I repeat the same thing at high pressure, 400 00:25:52,770 --> 00:25:55,190 there is no difference between liquid and gas. 401 00:25:55,190 --> 00:25:57,330 They are really the same thing. 402 00:25:57,330 --> 00:26:00,260 And so this is an experiment that 403 00:26:00,260 --> 00:26:03,670 is done going more or less around here where 404 00:26:03,670 --> 00:26:05,060 the critical point is. 405 00:26:05,060 --> 00:26:09,970 And you start initially at the low temperatures side 406 00:26:09,970 --> 00:26:12,580 where you can see that at the bottom of your wire, 407 00:26:12,580 --> 00:26:14,190 you have a liquid. 408 00:26:14,190 --> 00:26:19,540 And a meniscus separates it nicely from a gas. 409 00:26:19,540 --> 00:26:23,170 As you heat it up, you approach the critical temperature. 410 00:26:23,170 --> 00:26:29,250 Well, the liquid expands and keeps going up. 411 00:26:29,250 --> 00:26:33,040 And once you hit TC and beyond, that well, one is the liquid 412 00:26:33,040 --> 00:26:35,660 and which one's the gas? 413 00:26:35,660 --> 00:26:38,040 You can't tell the difference anymore, right? 414 00:26:38,040 --> 00:26:39,960 There is a variation in density. 415 00:26:39,960 --> 00:26:44,800 Because of gravity, there's more density down here up there. 416 00:26:44,800 --> 00:26:47,330 OK, now what happens when we start to cool this? 417 00:26:47,330 --> 00:26:49,220 Ah, what happened? 418 00:26:49,220 --> 00:26:51,270 That's called critical opalescence. 419 00:26:51,270 --> 00:26:54,880 You can't see through that. 420 00:26:54,880 --> 00:26:55,640 And why? 421 00:26:55,640 --> 00:26:58,480 Because there are all of these fluctuations 422 00:26:58,480 --> 00:27:01,190 that you can now see. 423 00:27:01,190 --> 00:27:04,860 Right at TC when the thing became black, 424 00:27:04,860 --> 00:27:08,490 there were so many fluctuations covering so different length 425 00:27:08,490 --> 00:27:11,970 scales that light could not get through. 426 00:27:11,970 --> 00:27:15,660 But now gradually, the size of the fluctuations 427 00:27:15,660 --> 00:27:18,670 which exists both in the liquid and in the gas 428 00:27:18,670 --> 00:27:23,450 are becoming small but still quite visible. 429 00:27:23,450 --> 00:27:27,830 So clearly, despite the fact that whatever atoms 430 00:27:27,830 --> 00:27:31,660 and molecules you have over here have very short range 431 00:27:31,660 --> 00:27:36,220 interactions, somehow in the vicinity of this point, 432 00:27:36,220 --> 00:27:39,286 they decided to move together and have 433 00:27:39,286 --> 00:27:40,536 these collective fluctuations. 434 00:27:43,080 --> 00:27:47,750 So that's why we should be able to do a statistical theory. 435 00:27:47,750 --> 00:27:52,510 And that's why there is a hope that once we have done that, 436 00:27:52,510 --> 00:27:55,710 because the important fluctuations are covering 437 00:27:55,710 --> 00:27:59,540 so many large numbers of atoms or molecules, 438 00:27:59,540 --> 00:28:04,230 we should be able to explain what's going on over here. 439 00:28:04,230 --> 00:28:07,170 So that's the task that we set ourselves. 440 00:28:07,170 --> 00:28:11,750 Anything about this before I close the video? 441 00:28:14,360 --> 00:28:16,930 OK. 442 00:28:16,930 --> 00:28:17,500 Yes? 443 00:28:17,500 --> 00:28:22,179 AUDIENCE: There are Gaussians that when rescaled, 444 00:28:22,179 --> 00:28:27,100 their plots still comply to that? 445 00:28:27,100 --> 00:28:27,960 PROFESSOR: No. 446 00:28:27,960 --> 00:28:28,820 No. 447 00:28:28,820 --> 00:28:32,870 In the vacinity-- so maybe the range of the things 448 00:28:32,870 --> 00:28:36,530 that collapse on top of each-other is small. 449 00:28:36,530 --> 00:28:38,860 Maybe it is so small that you have 450 00:28:38,860 --> 00:28:42,940 to do things at low pressure differences 451 00:28:42,940 --> 00:28:44,020 in order to see that. 452 00:28:44,020 --> 00:28:46,300 But as far as we know, if you have 453 00:28:46,300 --> 00:28:49,160 sufficient high resolution, everything 454 00:28:49,160 --> 00:28:52,011 will collapse on top of that [INAUDIBLE]. 455 00:28:52,011 --> 00:28:56,220 And what is more interesting than that-- that is not 456 00:28:56,220 --> 00:28:57,355 confined to gases. 457 00:29:00,290 --> 00:29:03,650 If I had done an experiment that involved 458 00:29:03,650 --> 00:29:10,815 mixing some protein that has a dense phase and a dilute phase 459 00:29:10,815 --> 00:29:13,020 and I went to the point where there 460 00:29:13,020 --> 00:29:17,150 is a separation between dense and dilute and I plotted that, 461 00:29:17,150 --> 00:29:20,700 that would also fall exactly on top of this [INAUDIBLE]. 462 00:29:20,700 --> 00:29:22,575 So it's not even gases. 463 00:29:22,575 --> 00:29:24,380 It's everything. 464 00:29:24,380 --> 00:29:35,937 And so-- so yes? 465 00:29:35,937 --> 00:29:38,811 AUDIENCE: I heard you built a system 466 00:29:38,811 --> 00:29:43,595 where some [INAUDIBLE] damages? 467 00:29:43,595 --> 00:29:44,220 PROFESSOR: Yes. 468 00:29:44,220 --> 00:29:46,080 And we will discuss those too. 469 00:29:46,080 --> 00:29:48,230 AUDIENCE: And the exponents will change? 470 00:29:48,230 --> 00:29:48,990 PROFESSOR: Yes. 471 00:29:48,990 --> 00:29:52,460 AUDIENCE: So it's [INAUDIBLE]. 472 00:29:52,460 --> 00:29:54,540 PROFESSOR: You may anticipate from that T 473 00:29:54,540 --> 00:29:56,390 to the D over there, the dimension is 474 00:29:56,390 --> 00:30:00,332 an important point, yes. 475 00:30:00,332 --> 00:30:00,832 OK? 476 00:30:03,700 --> 00:30:04,735 Anything else? 477 00:30:07,540 --> 00:30:09,300 All right. 478 00:30:09,300 --> 00:30:12,245 I'm going to rather than construct 479 00:30:12,245 --> 00:30:17,320 this theory, this statistical filter 480 00:30:17,320 --> 00:30:21,080 for the case of this liquid gas system, 481 00:30:21,080 --> 00:30:24,540 I'm going to do it for the case of the ferromagnet 482 00:30:24,540 --> 00:30:30,590 just to emphasize this fact that this result, this set 483 00:30:30,590 --> 00:30:36,090 of results spans much more than simple liquid gas 484 00:30:36,090 --> 00:30:39,960 phenomena-- a lot of different things-- and secondly, 485 00:30:39,960 --> 00:30:43,640 because writing it for the case of ferromagnet 486 00:30:43,640 --> 00:30:46,980 is much simpler because of the inherent symmetries 487 00:30:46,980 --> 00:30:49,550 that I will describe to you shortly. 488 00:30:49,550 --> 00:30:52,540 So what is the phenomenon that I would like to describe? 489 00:30:52,540 --> 00:30:54,330 What's the phase transition in the case 490 00:30:54,330 --> 00:30:58,300 of the ferromagnet like a piece of iron? 491 00:30:58,300 --> 00:31:04,480 Where one axis that is always of interest to us is temperature. 492 00:31:04,480 --> 00:31:11,600 And if I have a piece of iron or nickel or some other material, 493 00:31:11,600 --> 00:31:14,150 out here at high temperatures, it is a paramagnet. 494 00:31:17,910 --> 00:31:21,700 And there is a critical temperature TC 495 00:31:21,700 --> 00:31:31,330 below which it becomes a ferromagnet, which 496 00:31:31,330 --> 00:31:35,240 means that it has some permanent magnetization. 497 00:31:35,240 --> 00:31:40,040 And actually, the reason I drew the second axis is 498 00:31:40,040 --> 00:31:46,990 that a very nice way to examine the discontinuities 499 00:31:46,990 --> 00:31:52,480 is to put an external magnetic field and see what happens. 500 00:31:52,480 --> 00:31:56,280 Because if you put an external magnetic field 501 00:31:56,280 --> 00:32:08,730 like you put your piece of magnet enclosed in some parent 502 00:32:08,730 --> 00:32:11,435 [INAUDIBLE] loop so that you have field 503 00:32:11,435 --> 00:32:14,240 in one direction or the other direction, 504 00:32:14,240 --> 00:32:17,130 then as you change this [INAUDIBLE] 505 00:32:17,130 --> 00:32:21,060 of the field on the high temperature side, 506 00:32:21,060 --> 00:32:27,190 what you'll find is that if I plot where the magnetization is 507 00:32:27,190 --> 00:32:31,380 pointing, well, if you put the magnetic field on top 508 00:32:31,380 --> 00:32:34,160 of a paramagnet, it also magnetizes. 509 00:32:34,160 --> 00:32:38,440 It does pull in the direction of the field. 510 00:32:38,440 --> 00:32:40,510 The amount of magnetization that you 511 00:32:40,510 --> 00:32:45,680 have in the system when you are on the high temperatures side 512 00:32:45,680 --> 00:32:47,200 looks something like this. 513 00:32:47,200 --> 00:32:50,600 At very high fields, all of your spins 514 00:32:50,600 --> 00:32:52,990 are aligned with the field. 515 00:32:52,990 --> 00:32:55,950 As you low the field because of entropy and thermal 516 00:32:55,950 --> 00:32:58,660 fluctuations, they start to move around. 517 00:32:58,660 --> 00:33:00,710 They have too much fluctuation when 518 00:33:00,710 --> 00:33:06,090 you're in the phase that is a paramagment. 519 00:33:06,090 --> 00:33:11,350 And it then when the field goes to 0, 520 00:33:11,350 --> 00:33:13,846 the magnetization reverses itself. 521 00:33:13,846 --> 00:33:17,100 And you will get a structure such as this. 522 00:33:17,100 --> 00:33:20,390 So this is the behavioral of magnetization 523 00:33:20,390 --> 00:33:23,920 as a function of the field if you go along 524 00:33:23,920 --> 00:33:27,685 the path such as this which corresponds to the paramagnet. 525 00:33:32,240 --> 00:33:35,260 Now, what happens if I do the same thing 526 00:33:35,260 --> 00:33:40,110 but I do it along a looped route such as here? 527 00:33:40,110 --> 00:33:43,730 So again, when you are out here at high magnetic field, 528 00:33:43,730 --> 00:33:44,510 not much happens. 529 00:33:47,170 --> 00:33:57,080 But when you hit 0, then you have a piece of magnet. 530 00:33:57,080 --> 00:33:58,830 It has some magnetization. 531 00:33:58,830 --> 00:34:04,480 So basically, it goes to some particular value. 532 00:34:04,480 --> 00:34:05,690 There is hysteresis. 533 00:34:05,690 --> 00:34:10,110 So let's imagine that you start with a system down here 534 00:34:10,110 --> 00:34:12,130 and then reduce the magnetic field. 535 00:34:12,130 --> 00:34:16,940 And you would be getting a curve such as this. 536 00:34:16,940 --> 00:34:22,280 So in some sense, right at H equals to 0 537 00:34:22,280 --> 00:34:27,000 when you are 4T less than TC, there is a discontinuity. 538 00:34:27,000 --> 00:34:30,247 You don't know whether you are one side or the other side. 539 00:34:32,929 --> 00:34:39,550 And in fact, these curves are really the same as the isoterms 540 00:34:39,550 --> 00:34:41,380 that we had for the liquid gas system 541 00:34:41,380 --> 00:34:45,949 if I were to turn them around by 90 degrees. 542 00:34:45,949 --> 00:34:51,420 The paramagnetic curve looks very much topologically 543 00:34:51,420 --> 00:34:57,240 like what we have for their isoterm at high temperatures 544 00:34:57,240 --> 00:35:00,230 where the discontinuity that we have 545 00:35:00,230 --> 00:35:03,440 for the magnetization of the ferromagnet 546 00:35:03,440 --> 00:35:08,130 looks like the isoterm that we would have at low temperatures. 547 00:35:08,130 --> 00:35:11,180 And indeed, separating these two, 548 00:35:11,180 --> 00:35:15,810 there will be some kind of a critical isoterm. 549 00:35:15,810 --> 00:35:20,120 So if I were to go exactly down here, 550 00:35:20,120 --> 00:35:25,630 you see what I would get is a curve that 551 00:35:25,630 --> 00:35:31,780 is kind of like this-- comes and hogs the vertical axis 552 00:35:31,780 --> 00:35:36,650 and then does that, which is in the sense 553 00:35:36,650 --> 00:35:41,100 that we were discussing before identical to the curve 554 00:35:41,100 --> 00:35:47,530 that you have for the isoterm of the liquid gas coexistence. 555 00:35:47,530 --> 00:35:48,740 What do I mean? 556 00:35:48,740 --> 00:35:53,040 I could do the same type of collapse of data 557 00:35:53,040 --> 00:35:54,710 that I showed you before that I was 558 00:35:54,710 --> 00:35:57,030 doing for the coexistence curve. 559 00:35:57,030 --> 00:36:01,570 I can do that for these inverted coexistence curves. 560 00:36:01,570 --> 00:36:05,900 I can do the same collapse for this critical isoterm 561 00:36:05,900 --> 00:36:07,660 of the ferromagnet. 562 00:36:07,660 --> 00:36:11,490 And it would fall on top of the critical isoterm 563 00:36:11,490 --> 00:36:16,700 that I would have for neon, argon, krypton, anything else. 564 00:36:16,700 --> 00:36:19,970 So that is that kind of overall universality. 565 00:36:23,670 --> 00:36:28,098 Now-- yes? 566 00:36:28,098 --> 00:36:30,222 AUDIENCE: In this [INAUDIBLE], why didn't you 567 00:36:30,222 --> 00:36:33,300 draw the full loop of hysteresis? 568 00:36:33,300 --> 00:36:35,780 PROFESSOR: Because I'm interested at this point 569 00:36:35,780 --> 00:36:37,990 with equilibrium phenomena. 570 00:36:37,990 --> 00:36:41,050 Hysteresis is a non equilibrium thing. 571 00:36:41,050 --> 00:36:45,670 So depending on how rapid you cool things or not cool things, 572 00:36:45,670 --> 00:36:49,070 you will get larger curves-- larger hysteresis curves. 573 00:36:49,070 --> 00:36:51,236 AUDIENCE: So what would it have been 574 00:36:51,236 --> 00:36:54,364 if we had high-- wasn't [INAUDIBLE] fields 575 00:36:54,364 --> 00:36:58,220 that were slowly reduce it to 0? 576 00:36:58,220 --> 00:37:00,450 And then go a little bit beyond and then 577 00:37:00,450 --> 00:37:04,050 just stay in that state for a long time? 578 00:37:04,050 --> 00:37:04,650 Will the-- 579 00:37:04,650 --> 00:37:05,274 PROFESSOR: Yes. 580 00:37:05,274 --> 00:37:08,636 If you wait sufficiently long time, which 581 00:37:08,636 --> 00:37:10,010 is the definition of equilibrium, 582 00:37:10,010 --> 00:37:13,450 you would follow the curve that I have drawn. 583 00:37:13,450 --> 00:37:16,060 There would be-- so the size of the hysteresis loop 584 00:37:16,060 --> 00:37:21,180 will go down as the amount of time that you wait. 585 00:37:21,180 --> 00:37:23,080 Except that in order to see this, 586 00:37:23,080 --> 00:37:25,789 you may have to wait longer than the age of the universe. 587 00:37:25,789 --> 00:37:26,330 I don't know. 588 00:37:26,330 --> 00:37:30,644 But in principle, that's what's going to happen. 589 00:37:30,644 --> 00:37:33,380 AUDIENCE: I have a question here. 590 00:37:33,380 --> 00:37:37,040 PROFESSOR: Yes 591 00:37:37,040 --> 00:37:41,200 AUDIENCE: When you create the flat line for temperature 592 00:37:41,200 --> 00:37:42,627 below the critical one-- 593 00:37:42,627 --> 00:37:43,210 PROFESSOR: Yes 594 00:37:43,210 --> 00:37:45,390 AUDIENCE: --are you essentially doing 595 00:37:45,390 --> 00:37:47,180 sort of a Maxwell construction again? 596 00:37:47,180 --> 00:37:51,840 Or is that too much even for this? 597 00:37:51,840 --> 00:37:52,480 PROFESSOR: No. 598 00:37:52,480 --> 00:37:54,720 I mean, I haven't done any theoretical work. 599 00:37:54,720 --> 00:37:57,920 At this point, I'm just giving you observation. 600 00:37:57,920 --> 00:38:00,420 Once we start to give theory, there 601 00:38:00,420 --> 00:38:03,370 is a type of theory for the magnet that 602 00:38:03,370 --> 00:38:06,140 is the analog of Maxwell's construction 603 00:38:06,140 --> 00:38:10,210 that you would do for the case of the liquid gas. 604 00:38:10,210 --> 00:38:14,730 Indeed, you will have that as the first problem set. 605 00:38:14,730 --> 00:38:16,435 So you can figure it out for yourself. 606 00:38:19,000 --> 00:38:22,430 Anything else? 607 00:38:22,430 --> 00:38:23,980 OK. 608 00:38:23,980 --> 00:38:29,830 So I kind of emphasized that functional forms 609 00:38:29,830 --> 00:38:32,720 are the things that are universal. 610 00:38:32,720 --> 00:38:35,465 And so in the context of the magnet, 611 00:38:35,465 --> 00:38:40,970 it is more clear how to characterize 612 00:38:40,970 --> 00:38:43,090 these functional forms. 613 00:38:43,090 --> 00:38:47,010 So one of the things that we have over here 614 00:38:47,010 --> 00:38:52,850 is that if I plot the true equilibrium magnetization 615 00:38:52,850 --> 00:38:57,230 as a function of temperature for fields that are 0-- 616 00:38:57,230 --> 00:38:58,660 so H equals to 0. 617 00:38:58,660 --> 00:39:03,450 If I start along the 0 field line, 618 00:39:03,450 --> 00:39:09,230 then all the way up to TC, magnetization 619 00:39:09,230 --> 00:39:12,180 at high temperatures is of course 0. 620 00:39:12,180 --> 00:39:13,830 You're dealing with a paramagnet. 621 00:39:13,830 --> 00:39:16,090 By definition, it has no magnetization. 622 00:39:16,090 --> 00:39:18,950 When you go below TC, you'll have 623 00:39:18,950 --> 00:39:22,210 a system that is spontaneously magnetized. 624 00:39:22,210 --> 00:39:25,500 Again, exactly what that means is 625 00:39:25,500 --> 00:39:28,260 it needs a little bit of clarification. 626 00:39:28,260 --> 00:39:31,810 Because it does depend on the direction of your field. 627 00:39:31,810 --> 00:39:35,150 The magnitude of it is well defined. 628 00:39:35,150 --> 00:39:38,230 If I go from H to minus H, the sine of it 629 00:39:38,230 --> 00:39:39,870 could potentially change. 630 00:39:39,870 --> 00:39:43,786 But the magnitude has a form such as this. 631 00:39:43,786 --> 00:39:48,030 It is again very similar, if you like, 632 00:39:48,030 --> 00:39:51,020 to half of that co existence curve 633 00:39:51,020 --> 00:39:53,560 that we had for the liquid gas. 634 00:39:53,560 --> 00:40:00,180 And we look at the behavior that you have in this vicinity. 635 00:40:00,180 --> 00:40:04,630 And we find that is well-described by a power law. 636 00:40:04,630 --> 00:40:13,880 So we say that M as T goes to TC for H equals to 0 637 00:40:13,880 --> 00:40:20,630 is characterized or is proportional to TC minus T 638 00:40:20,630 --> 00:40:25,345 to an exponent that is given the symbol beta. 639 00:40:29,370 --> 00:40:32,880 Now, sometimes you can't write this as TC minus T 640 00:40:32,880 --> 00:40:35,220 to the-- make it dimension. 641 00:40:35,220 --> 00:40:36,070 It doesn't matter. 642 00:40:36,070 --> 00:40:37,730 It's the same exponent. 643 00:40:37,730 --> 00:40:40,950 Sometimes in order not to have to write all of this thing 644 00:40:40,950 --> 00:40:44,440 again and again, you call this small T. 645 00:40:44,440 --> 00:40:48,160 It's just that reduced temperature 646 00:40:48,160 --> 00:40:50,620 from the critical point. 647 00:40:50,620 --> 00:40:55,440 And so basically, what we have is that T to the beta 648 00:40:55,440 --> 00:40:59,860 characterizes the singularity of the coexistence line, 649 00:40:59,860 --> 00:41:04,030 the magnetization [INAUDIBLE]. 650 00:41:04,030 --> 00:41:11,750 Now, rather then sitting at H equals to 0, 651 00:41:11,750 --> 00:41:16,060 I could have sat exactly at T equals to TC 652 00:41:16,060 --> 00:41:22,310 and merely H. So essentially, this is the curve. 653 00:41:22,310 --> 00:41:23,850 I say that T equals to TC. 654 00:41:23,850 --> 00:41:26,340 This is the blue curve. 655 00:41:26,340 --> 00:41:31,300 We see that new blue has this characteristic form that it 656 00:41:31,300 --> 00:41:35,380 also comes to 0 not linearly. 657 00:41:35,380 --> 00:41:39,902 So there is an exponent that characterizes that. 658 00:41:39,902 --> 00:41:45,200 that again is typically written as 1 over delta. 659 00:41:45,200 --> 00:41:50,260 So you can see that if I go back to the critical isoterm 660 00:41:50,260 --> 00:41:52,760 of the liquid gas system. 661 00:41:52,760 --> 00:41:57,080 For the liquid gas system, I would conclude that, say, 662 00:41:57,080 --> 00:42:02,630 delta v goes like delta P to the power of delta or one 663 00:42:02,630 --> 00:42:04,870 goes with the other to the power of 1 if it. 664 00:42:04,870 --> 00:42:10,680 So essentially, the shapes of these two things 665 00:42:10,680 --> 00:42:12,757 are very much related, characterized 666 00:42:12,757 --> 00:42:13,715 by these two exponents. 667 00:42:18,470 --> 00:42:19,405 What else? 668 00:42:23,890 --> 00:42:28,159 Another thing that we can certainly measure for a magnet 669 00:42:28,159 --> 00:42:29,075 is the susceptibility. 670 00:42:36,020 --> 00:42:41,130 So chi-- let's measure it as a function of temperature 671 00:42:41,130 --> 00:42:44,960 but for field equal to 0. 672 00:42:44,960 --> 00:42:51,370 So basically, I sit eventually at the equals to TC. 673 00:42:51,370 --> 00:42:53,660 But I put on a small magnetic field 674 00:42:53,660 --> 00:42:56,970 and see how the magnetization changes. 675 00:42:56,970 --> 00:43:00,310 You can see that as long as I am above TC 676 00:43:00,310 --> 00:43:05,430 in the paramagnetic phase, there is a linear relationship here. 677 00:43:05,430 --> 00:43:11,060 Chi is proportional to H. But the proportionality here 678 00:43:11,060 --> 00:43:12,990 is the susceptibility. 679 00:43:12,990 --> 00:43:17,820 As I get closer to C, that susceptibility diverges. 680 00:43:17,820 --> 00:43:23,820 OK, so chi is proportional to T diverges 681 00:43:23,820 --> 00:43:27,230 with an exponent that is indicated by gamma. 682 00:43:27,230 --> 00:43:30,180 Actually, let me be a little bit more precise. 683 00:43:30,180 --> 00:43:35,980 So what I have said is that if I plot the susceptibility 684 00:43:35,980 --> 00:43:41,800 as a function of temperature at TC, it diverges. 685 00:43:45,230 --> 00:43:50,550 I could also calculate something similar to that below TC. 686 00:43:50,550 --> 00:43:52,290 Below TC, it is true that I already 687 00:43:52,290 --> 00:43:54,626 have some magnetization. 688 00:43:54,626 --> 00:43:57,480 But if I put on a magnetic field, 689 00:43:57,480 --> 00:44:00,580 the magnetization will go up. 690 00:44:00,580 --> 00:44:03,510 And I can define the slope of that 691 00:44:03,510 --> 00:44:07,920 as being the susceptibility below TC. 692 00:44:07,920 --> 00:44:11,302 And again, as I approach TC from below, 693 00:44:11,302 --> 00:44:14,600 that susceptibility also diverges. 694 00:44:14,600 --> 00:44:19,520 So there is a susceptibility that comes like this. 695 00:44:19,520 --> 00:44:21,770 So you would say, OK. 696 00:44:21,770 --> 00:44:24,870 There's a susceptibility above. 697 00:44:24,870 --> 00:44:27,340 And there is a susceptibility below. 698 00:44:27,340 --> 00:44:30,975 And maybe they diverge with different exponents. 699 00:44:30,975 --> 00:44:34,100 at this point, we don't know. 700 00:44:34,100 --> 00:44:35,620 Yes or no. 701 00:44:35,620 --> 00:44:40,970 We will show shortly that indeed the two gammas are really 702 00:44:40,970 --> 00:44:45,476 the same and using one exponent is sufficient for that story. 703 00:44:49,050 --> 00:44:53,650 And analog of susceptibility in the case of the liquid gas 704 00:44:53,650 --> 00:44:55,860 would be the compressibility. 705 00:44:55,860 --> 00:44:58,740 And the compressibility is related somehow 706 00:44:58,740 --> 00:45:03,260 to the inverse of these PV curve slopes. 707 00:45:03,260 --> 00:45:07,590 We know that the sign of it has to be negative for stability. 708 00:45:07,590 --> 00:45:09,980 But right on the critical isoterm, 709 00:45:09,980 --> 00:45:14,180 you see that this slope goes to 0 or its inverse diverges. 710 00:45:14,180 --> 00:45:19,550 So there is, again, the same exponent gamma for some magnets 711 00:45:19,550 --> 00:45:20,897 also describes that divergence. 712 00:45:24,560 --> 00:45:28,340 Susceptibility is an example of a response function. 713 00:45:28,340 --> 00:45:31,360 You perturb the system and see how it responds. 714 00:45:31,360 --> 00:45:33,250 Another response function that we've 715 00:45:33,250 --> 00:45:39,780 seen is the heat capacity where you put heat into this system 716 00:45:39,780 --> 00:45:41,560 or you try to change the temperature 717 00:45:41,560 --> 00:45:44,350 and see how the heat energy is modified. 718 00:45:47,330 --> 00:45:53,400 And again, we experimentally observe. 719 00:45:53,400 --> 00:45:55,590 We already saw something like this 720 00:45:55,590 --> 00:45:57,470 when we were discussing the super fluid 721 00:45:57,470 --> 00:46:00,640 transition, the lambda transition-- that the heat 722 00:46:00,640 --> 00:46:06,700 capacity as a function of temperature at some transition 723 00:46:06,700 --> 00:46:09,785 temperatures happens to diverge. 724 00:46:12,560 --> 00:46:16,450 And in principle, one can again look 725 00:46:16,450 --> 00:46:19,250 at the two sides of the transition 726 00:46:19,250 --> 00:46:24,460 and characterize them by divergences that are typically 727 00:46:24,460 --> 00:46:27,300 indicated by the exponent alpha. 728 00:46:27,300 --> 00:46:30,990 Again, you will see that there are reasons 729 00:46:30,990 --> 00:46:34,980 why there is only one exponent [INAUDIBLE]. 730 00:46:34,980 --> 00:46:39,310 So this is essentially the only part 731 00:46:39,310 --> 00:46:42,640 where there is some zoology involved. 732 00:46:42,640 --> 00:46:45,330 You have to remember and learn where 733 00:46:45,330 --> 00:46:47,990 the different exponents come from. 734 00:46:47,990 --> 00:46:51,390 The rest of it, I hope, is very logical-- but which 735 00:46:51,390 --> 00:46:54,139 one is alpha, which one's beta, which one's gamma. 736 00:46:54,139 --> 00:46:56,472 There's four things, five things you should [INAUDIBLE]. 737 00:46:59,310 --> 00:47:01,900 OK? 738 00:47:01,900 --> 00:47:06,320 Now, the next thing that I want to do 739 00:47:06,320 --> 00:47:11,000 is to show you that the two things that we looked 740 00:47:11,000 --> 00:47:13,480 at about the liquid gas transition-- 741 00:47:13,480 --> 00:47:16,870 in fact, one of them implies the other. 742 00:47:16,870 --> 00:47:20,440 So we said that essentially by continuity, 743 00:47:20,440 --> 00:47:24,040 when I look at the shape of this critical isoterm, 744 00:47:24,040 --> 00:47:27,260 it come down in this zero slope. 745 00:47:27,260 --> 00:47:30,530 When, again, to continuously join the type of curve 746 00:47:30,530 --> 00:47:33,465 that I have for T greater than TC of magnetization 747 00:47:33,465 --> 00:47:36,230 and T less than TC below, I should 748 00:47:36,230 --> 00:47:39,750 have a curve that comes and hogs the axis 749 00:47:39,750 --> 00:47:41,510 or has infinite susceptibility. 750 00:47:44,300 --> 00:47:48,220 I will show you that the infinite susceptibility does, 751 00:47:48,220 --> 00:47:52,510 in fact imply, that you must have collective behavior. 752 00:47:52,510 --> 00:47:56,130 That this critical opalescence that we saw 753 00:47:56,130 --> 00:47:59,870 is inseparable from the fact that you 754 00:47:59,870 --> 00:48:03,130 have diverging susceptibility. 755 00:48:03,130 --> 00:48:04,940 I'll show that in the case of the magnet. 756 00:48:04,940 --> 00:48:09,110 But it also applies, of course, to the critical opalescence. 757 00:48:09,110 --> 00:48:12,290 So let's do a little bit of thermodynamics. 758 00:48:12,290 --> 00:48:18,770 So I can imagine if I have some kind of Hamiltonia that 759 00:48:18,770 --> 00:48:23,810 describes my interactions among the spins in my argon 760 00:48:23,810 --> 00:48:26,540 or any other magnet that I have. 761 00:48:26,540 --> 00:48:31,060 And if I was to calculate a partition function, what 762 00:48:31,060 --> 00:48:34,170 I need to do would be to trace over 763 00:48:34,170 --> 00:48:38,310 all degrees of freedom of this system. 764 00:48:38,310 --> 00:48:41,600 Now, for the case of the magnet, it is more convenient 765 00:48:41,600 --> 00:48:46,030 actually to look at the ensemble that is corresponding to this. 766 00:48:46,030 --> 00:48:49,810 That is, fix the magnetic field and allow the magnetization 767 00:48:49,810 --> 00:48:51,960 to decide where it wants to be. 768 00:48:51,960 --> 00:48:54,110 So I really want to evaluate things 769 00:48:54,110 --> 00:48:58,560 in this ensemble, which really to be precise, 770 00:48:58,560 --> 00:49:03,280 is not the [INAUDIBLE] ensemble, but the Gibbs ensemble. 771 00:49:03,280 --> 00:49:05,180 So this should be a Gibbs free energy. 772 00:49:05,180 --> 00:49:07,290 This should be a Gibbs partition function. 773 00:49:07,290 --> 00:49:11,100 But traditionally, most texts including my notes 774 00:49:11,100 --> 00:49:13,750 ignore that difference and log of this. 775 00:49:13,750 --> 00:49:16,380 Rather than calling it G, we will call F. 776 00:49:16,380 --> 00:49:20,600 But that doesn't make much difference. 777 00:49:20,600 --> 00:49:25,760 This clearly is a function of temperature and H. 778 00:49:25,760 --> 00:49:32,040 Now, if I wanted-- let's imagine that always we 779 00:49:32,040 --> 00:49:33,810 put the magnetic field in one direction. 780 00:49:33,810 --> 00:49:35,932 So I don't have to worry for the time being 781 00:49:35,932 --> 00:49:37,510 about vectorial aspect. 782 00:49:37,510 --> 00:49:40,670 When you go back to the vectorial aspect 783 00:49:40,670 --> 00:49:44,050 later on, clearly-- actually this 784 00:49:44,050 --> 00:49:48,420 is the net magnetization of the system. 785 00:49:48,420 --> 00:49:50,045 If I have a piece of iron, it would 786 00:49:50,045 --> 00:49:53,770 be the magnetization of the entire iron. 787 00:49:53,770 --> 00:49:57,190 And the average of that magnetization, 788 00:49:57,190 --> 00:50:00,040 given temperature and field, et cetera, 789 00:50:00,040 --> 00:50:07,890 I can obtain by taking the log z by D beta H. 790 00:50:07,890 --> 00:50:11,830 Because when I do that, I go back inside the trace, 791 00:50:11,830 --> 00:50:13,890 take a derivative in respect to beta H. 792 00:50:13,890 --> 00:50:23,410 And I have a trace of M into the minus beta H plus beta HM, 793 00:50:23,410 --> 00:50:29,720 which is how the different probabilities are rated once 794 00:50:29,720 --> 00:50:32,520 because of the log z. 795 00:50:32,520 --> 00:50:34,620 In a derivative, I have a 1 over z 796 00:50:34,620 --> 00:50:38,730 to make this more properly normalized probabilities. 797 00:50:38,730 --> 00:50:40,510 So that's the standard story. 798 00:50:43,780 --> 00:50:49,740 Now if I were to take another derivative, 799 00:50:49,740 --> 00:50:54,160 if I were to take a derivative of M 800 00:50:54,160 --> 00:50:59,320 with respect to H, which is what the susceptibility is, 801 00:50:59,320 --> 00:51:00,770 after all. 802 00:51:00,770 --> 00:51:02,790 So the sensitivity of the system-- 803 00:51:02,790 --> 00:51:04,660 there is the derivative of magnetization 804 00:51:04,660 --> 00:51:07,240 with respect to H. 805 00:51:07,240 --> 00:51:11,270 It is the same thing as beta derivative 806 00:51:11,270 --> 00:51:16,540 of M with respect to beta H, of course. 807 00:51:16,540 --> 00:51:19,690 So I have to go to this expression that I have on top 808 00:51:19,690 --> 00:51:23,260 and take another derivative with respect to beta H. 809 00:51:23,260 --> 00:51:29,220 The derivative can act on beta H that is in the numerator. 810 00:51:29,220 --> 00:51:32,850 And what that does is it essentially brings down 811 00:51:32,850 --> 00:51:42,750 another factor of M. And the Z is not touched. 812 00:51:42,750 --> 00:51:46,650 Or I leave the numerator as is and take 813 00:51:46,650 --> 00:51:49,105 a derivative of the denominator. 814 00:51:49,105 --> 00:51:53,090 And the derivative of z I've already taken. 815 00:51:53,090 --> 00:51:55,635 So essentially, because it's in the denominator, 816 00:51:55,635 --> 00:51:58,110 I will get the minus 1. 817 00:51:58,110 --> 00:52:01,810 The derivative of 1 over z will become 1 over z squared. 818 00:52:01,810 --> 00:52:04,020 It will also give me that fact factor 819 00:52:04,020 --> 00:52:07,350 that multiplies that factor itself. 820 00:52:07,350 --> 00:52:13,120 So I have e to the minus beta H plus beta HM. 821 00:52:13,120 --> 00:52:17,870 And this whole thing got squared. 822 00:52:17,870 --> 00:52:19,800 OK? 823 00:52:19,800 --> 00:52:23,590 So a very famous formula-- always true. 824 00:52:23,590 --> 00:52:27,530 Response functions such as susceptibilities 825 00:52:27,530 --> 00:52:31,880 are related to variances-- in this case, 826 00:52:31,880 --> 00:52:36,153 variance of the net magnetization 827 00:52:36,153 --> 00:52:41,910 of the system of course true for other responses. 828 00:52:41,910 --> 00:52:46,300 OK, this doesn't seem to tell us very much. 829 00:52:46,300 --> 00:52:49,782 But then I note the following that-- OK, so I have my magnet. 830 00:52:53,050 --> 00:52:58,560 What I have been asking is what is the net magnetization 831 00:52:58,560 --> 00:53:02,410 of piece of magnet and its response to adding 832 00:53:02,410 --> 00:53:05,150 a magnetic field. 833 00:53:05,150 --> 00:53:08,340 If I want to think more microscopically-- 834 00:53:08,340 --> 00:53:10,360 if I want to think, go back in terms 835 00:53:10,360 --> 00:53:12,200 of what we saw for the liquid gas 836 00:53:12,200 --> 00:53:14,640 system and the critical opalescence 837 00:53:14,640 --> 00:53:18,990 where there were fluctuations in density all over the place, 838 00:53:18,990 --> 00:53:23,610 I expect that the reality also at some particular instance, 839 00:53:23,610 --> 00:53:26,205 when I look at this, there will be fluctuations 840 00:53:26,205 --> 00:53:29,190 in magnetization from one location 841 00:53:29,190 --> 00:53:32,410 of the sample to another location of the sample. 842 00:53:32,410 --> 00:53:35,020 It's kind of building gradually in the direction 843 00:53:35,020 --> 00:53:36,720 of the statistical field. 844 00:53:36,720 --> 00:53:40,970 I kind of expect to have these long wavelength fluctuations. 845 00:53:40,970 --> 00:53:42,220 And that's where I want to go. 846 00:53:42,220 --> 00:53:45,530 But at this stage, I cannot even worry about that. 847 00:53:45,530 --> 00:53:48,150 I can say that I define in whatever 848 00:53:48,150 --> 00:53:51,440 way some kind of a local magnetization 849 00:53:51,440 --> 00:53:56,090 so that the net magnetization, let's say in V dimension, 850 00:53:56,090 --> 00:53:59,490 is the integrated version of the local magnetization. 851 00:53:59,490 --> 00:54:02,150 So magnetization density I integrate 852 00:54:02,150 --> 00:54:04,582 with the net magnetization. 853 00:54:04,582 --> 00:54:07,480 OK, I put that over here. 854 00:54:07,480 --> 00:54:10,920 I have two factors of big M. So I 855 00:54:10,920 --> 00:54:17,520 will get two factors of integration over R and R prime. 856 00:54:17,520 --> 00:54:20,380 Let's stick to three dimensions. 857 00:54:20,380 --> 00:54:21,170 Doesn't matter. 858 00:54:21,170 --> 00:54:24,120 We can generalize it to be dimensions. 859 00:54:24,120 --> 00:54:30,930 And here I have M of R, M of R prime. 860 00:54:30,930 --> 00:54:32,980 And average would give me the average 861 00:54:32,980 --> 00:54:38,340 of this M of R minus M of R prime. 862 00:54:38,340 --> 00:54:41,320 Whereas in the second part, there 863 00:54:41,320 --> 00:54:43,600 are the product of two averages-- M 864 00:54:43,600 --> 00:54:49,680 and R, average of M and R prime. 865 00:54:49,680 --> 00:54:50,726 So it is the covariance. 866 00:54:55,570 --> 00:54:58,662 So I basically have to look at the M 867 00:54:58,662 --> 00:55:02,680 at some point in my sample, the M at some other point R 868 00:55:02,680 --> 00:55:06,780 prime in the sample, calculate the covariance. 869 00:55:11,270 --> 00:55:13,900 Now, on average, this piece of iron 870 00:55:13,900 --> 00:55:19,070 doesn't know its left corner from its right corner. 871 00:55:19,070 --> 00:55:21,540 So just as in the case of the lattice, 872 00:55:21,540 --> 00:55:26,380 I expect once I do the averaging, on average, M of R, 873 00:55:26,380 --> 00:55:31,650 M of R prime should be only a function of R minus R prime. 874 00:55:31,650 --> 00:55:35,580 All right, so I expect this covariance that's 875 00:55:35,580 --> 00:55:43,840 indicated by MNC is really only a function of the separation 876 00:55:43,840 --> 00:55:50,160 between the two points, OK? 877 00:55:50,160 --> 00:55:52,340 Which means that one of these integrations 878 00:55:52,340 --> 00:55:53,565 I can indeed perform. 879 00:55:53,565 --> 00:55:55,320 There's the integral with respect 880 00:55:55,320 --> 00:55:58,650 to the relative coordinate and the center of mass coordinate. 881 00:55:58,650 --> 00:56:00,940 For getting boundary dependence, this 882 00:56:00,940 --> 00:56:04,350 will give you a factor of V. There was a factor of beta 883 00:56:04,350 --> 00:56:07,100 that I forgot. 884 00:56:07,100 --> 00:56:09,850 There is an overall beta. 885 00:56:09,850 --> 00:56:13,320 So I have chi beta. 886 00:56:13,320 --> 00:56:19,600 And then I have the integral over the relative coordinate 887 00:56:19,600 --> 00:56:26,150 of the correlations within two spins 888 00:56:26,150 --> 00:56:28,870 in the system as a function of separation. 889 00:56:59,490 --> 00:57:02,230 Now, of course, like any other quantity, 890 00:57:02,230 --> 00:57:05,450 susceptibility is proportional to how much material you have. 891 00:57:05,450 --> 00:57:07,490 It's an extensive quantity. 892 00:57:07,490 --> 00:57:10,520 So when I say that the susceptibility diverges, 893 00:57:10,520 --> 00:57:12,730 I really mean that the susceptibility 894 00:57:12,730 --> 00:57:15,050 per unit volume-- the intensive part 895 00:57:15,050 --> 00:57:17,830 is the thing that will diverge. 896 00:57:17,830 --> 00:57:20,870 But the susceptibility per unit volume 897 00:57:20,870 --> 00:57:26,280 is the result of doing an integration of this covariance 898 00:57:26,280 --> 00:57:29,810 as a function of position. 899 00:57:29,810 --> 00:57:33,730 Let's see what we expect this covariance to do. 900 00:57:33,730 --> 00:57:38,990 So as a function of separation, if I 901 00:57:38,990 --> 00:57:45,640 look at the covariance between two spins-- OK, 902 00:57:45,640 --> 00:57:48,300 so there's this note that I have already 903 00:57:48,300 --> 00:57:50,330 subtracted out the average. 904 00:57:50,330 --> 00:57:52,470 So whether you are in the ferromagnetic phase 905 00:57:52,470 --> 00:57:55,120 or in the paramagnetic phase, the statement 906 00:57:55,120 --> 00:57:57,960 is how much is the fluctuation around the average 907 00:57:57,960 --> 00:58:01,100 at this point and this other point are related? 908 00:58:01,100 --> 00:58:03,620 Well, when the two points come together, what I'm looking 909 00:58:03,620 --> 00:58:09,680 is some kind of a variance of the randomness. 910 00:58:09,680 --> 00:58:12,610 Now, when I go further and further away, 911 00:58:12,610 --> 00:58:15,750 I expect that eventuality, what this 912 00:58:15,750 --> 00:58:18,595 does as far as a fluctuation around the averages 913 00:58:18,595 --> 00:58:24,910 is concerned, does not influence what is going on very far away. 914 00:58:24,910 --> 00:58:28,570 So I expect that as a function of going further and further 915 00:58:28,570 --> 00:58:33,050 away, this is something that will eventually 916 00:58:33,050 --> 00:58:34,345 die off and go to zero. 917 00:58:37,140 --> 00:58:40,650 Let's imagine that there is some kind 918 00:58:40,650 --> 00:58:44,660 of characteristic landscape below which 919 00:58:44,660 --> 00:58:48,930 the correlations have died off to 0. 920 00:58:48,930 --> 00:58:51,570 And I will say that this integral 921 00:58:51,570 --> 00:58:55,710 is less than or equal to essentially looking 922 00:58:55,710 --> 00:59:00,030 at the volume over which there are correlations. 923 00:59:00,030 --> 00:59:02,190 The correlations within this volume 924 00:59:02,190 --> 00:59:05,670 would typically be less than sigma squared. 925 00:59:05,670 --> 00:59:10,580 But let's bound it by sigma squared times the volume 926 00:59:10,580 --> 00:59:12,300 that we're dealing with, which is either 927 00:59:12,300 --> 00:59:15,400 four pi over 3c cubed. 928 00:59:15,400 --> 00:59:19,950 Coefficients here are not important. 929 00:59:19,950 --> 00:59:27,200 OK, so if I know that my response function 930 00:59:27,200 --> 00:59:30,930 per unit volume, like my compressibility 931 00:59:30,930 --> 00:59:34,700 or susceptibility, if I know that the left hand 932 00:59:34,700 --> 00:59:40,430 side as I go to TC is diverging and going to infinity, 933 00:59:40,430 --> 00:59:41,910 variance is bounded. 934 00:59:41,910 --> 00:59:43,190 I can't do anything with it. 935 00:59:43,190 --> 00:59:44,540 Data is bounded. 936 00:59:44,540 --> 00:59:47,260 I can't do anything with this. 937 00:59:47,260 --> 00:59:50,390 The only thing that I conclude, the only knob I have, 938 00:59:50,390 --> 00:59:54,040 is that C must go to infinity. 939 00:59:54,040 --> 00:59:59,600 So K over V going to infinity implies 940 00:59:59,600 --> 01:00:01,420 and is implied by this [INAUDIBLE]. 941 01:00:09,310 --> 01:00:11,740 So I was actually not quite truthful. 942 01:00:11,740 --> 01:00:15,580 Because you need to learn one other exponent. 943 01:00:15,580 --> 01:00:18,520 So then how the correlation then divergence 944 01:00:18,520 --> 01:00:23,710 as a function of temperature is also important 945 01:00:23,710 --> 01:00:31,700 and is indicated by an exponent that is called nu. 946 01:00:31,700 --> 01:00:39,430 So C diverges as T to some x point under this nu. 947 01:00:39,430 --> 01:00:42,860 So this is what you were seeing when I was showing you 948 01:00:42,860 --> 01:00:44,870 the critical opalescence. 949 01:00:44,870 --> 01:00:47,970 The size of these fluctuations became so large 950 01:00:47,970 --> 01:00:50,690 that you couldn't see through the sample. 951 01:00:50,690 --> 01:00:54,570 All kinds of wavelengths were taking place in the system. 952 01:00:54,570 --> 01:00:59,590 And if I had presented things as square, 953 01:00:59,590 --> 01:01:01,500 I could have given you that as a prediction. 954 01:01:01,500 --> 01:01:04,320 I should have shown you, say, the critical isoterm 955 01:01:04,320 --> 01:01:06,080 looks like this. 956 01:01:06,080 --> 01:01:08,220 Therefore, if you look at it at TC, 957 01:01:08,220 --> 01:01:10,090 you shouldn't be able to see through it. 958 01:01:10,090 --> 01:01:11,450 This is [INAUDIBLE]. 959 01:01:15,860 --> 01:01:22,560 All right, so those are the phenomena 960 01:01:22,560 --> 01:01:26,810 that we would like to now explain, 961 01:01:26,810 --> 01:01:29,670 phenomena being these critical exponent 962 01:01:29,670 --> 01:01:33,450 alpha, beta, gamma, and nu, et cetera, being universal 963 01:01:33,450 --> 01:01:36,880 and the same across very many different systems. 964 01:01:39,826 --> 01:01:40,808 AUDIENCE: Question. 965 01:01:40,808 --> 01:01:41,790 PROFESSOR: Yes. 966 01:01:41,790 --> 01:01:47,682 AUDIENCE: Is [INAUDIBLE] in elementary fundamental field 967 01:01:47,682 --> 01:01:50,873 theory, like-- I mean, like, in the standard model 968 01:01:50,873 --> 01:01:53,910 or in quantum field theory where something like this happens 969 01:01:53,910 --> 01:01:57,100 where there's-- I mean, it's not a thermodynamical system. 970 01:01:57,100 --> 01:01:58,400 It's like elementary theory. 971 01:01:58,400 --> 01:01:59,770 And yet-- 972 01:01:59,770 --> 01:02:02,200 PROFESSOR: Yep. 973 01:02:02,200 --> 01:02:07,800 In some sense, the masses of the different particles 974 01:02:07,800 --> 01:02:10,900 are like correlation lengths. 975 01:02:10,900 --> 01:02:13,280 Because it is the mass of the particles 976 01:02:13,280 --> 01:02:15,680 like in the nuclear potential that 977 01:02:15,680 --> 01:02:19,040 describes the range of the interactions. 978 01:02:19,040 --> 01:02:22,270 So there are phenomena such as the case 979 01:02:22,270 --> 01:02:25,910 of [INAUDIBLE] or whatever where the range is infinite. 980 01:02:25,910 --> 01:02:28,220 So in some sense, those phenomena 981 01:02:28,220 --> 01:02:30,254 are sitting at the critical point. 982 01:02:50,200 --> 01:02:56,020 OK, so the name of the statistical field theory 983 01:02:56,020 --> 01:02:59,760 that we will construct is Landau Ginzburg 984 01:02:59,760 --> 01:03:03,025 and originally constructed by Landau in connection 985 01:03:03,025 --> 01:03:05,350 to super fluidity. 986 01:03:05,350 --> 01:03:16,510 But it can describe a lot of different phase transitions. 987 01:03:23,310 --> 01:03:26,340 Let's roughly introduce it in the context 988 01:03:26,340 --> 01:03:28,880 of these magnetic systems. 989 01:03:28,880 --> 01:03:32,380 So basically, I have my magnet. 990 01:03:36,290 --> 01:03:41,340 And again, in principle, I have all kinds 991 01:03:41,340 --> 01:03:43,500 of complicated degrees of freedom 992 01:03:43,500 --> 01:03:45,530 which are the spins that have quantum mechanical 993 01:03:45,530 --> 01:03:49,420 interactions, the exchange interactions, whatever 994 01:03:49,420 --> 01:03:51,610 the result of the behavior of electrons 995 01:03:51,610 --> 01:03:54,230 and their common interactions with ions 996 01:03:54,230 --> 01:03:58,040 is that eventually, something like nickel becomes 997 01:03:58,040 --> 01:04:01,760 a ferromagnet at low temperatures. 998 01:04:01,760 --> 01:04:06,390 Now, hopefully and actually precisely in the context 999 01:04:06,390 --> 01:04:08,350 that we are dealing with, I don't 1000 01:04:08,350 --> 01:04:11,480 need to know any of all those details. 1001 01:04:11,480 --> 01:04:14,045 I will just focus on the phenomena 1002 01:04:14,045 --> 01:04:16,720 that there is a system that undergoes 1003 01:04:16,720 --> 01:04:20,600 a transition between ferromagnetic and paramagnetic 1004 01:04:20,600 --> 01:04:24,755 behavior and focus on calculating 1005 01:04:24,755 --> 01:04:28,380 an appropriate partition function for the degrees 1006 01:04:28,380 --> 01:04:32,080 of freedom that change their behavior ongoing through TC. 1007 01:04:34,770 --> 01:04:39,140 So what I expect is that again, just like we 1008 01:04:39,140 --> 01:04:42,380 saw for the case of liquid gas system, 1009 01:04:42,380 --> 01:04:45,490 on one side for the magnet, there 1010 01:04:45,490 --> 01:04:48,825 will be an average zero magnetization 1011 01:04:48,825 --> 01:04:50,160 in the paramagnet. 1012 01:04:50,160 --> 01:04:53,690 But there will be fluctuations of magnetization, 1013 01:04:53,690 --> 01:04:56,280 presumably with long wavelengths. 1014 01:04:56,280 --> 01:04:59,840 On the other side, there will be these fluctuations 1015 01:04:59,840 --> 01:05:02,430 on top of some average magnetization that 1016 01:05:02,430 --> 01:05:04,790 has formed in the system. 1017 01:05:04,790 --> 01:05:08,870 And if I stick sufficiently close to TC, 1018 01:05:08,870 --> 01:05:11,210 I expect that that magnetization is small. 1019 01:05:11,210 --> 01:05:13,290 That's where the exponent beta comes from. 1020 01:05:13,290 --> 01:05:15,820 So if I go sufficiently close to TC, 1021 01:05:15,820 --> 01:05:19,870 these new Ms that I have at each location hopefully 1022 01:05:19,870 --> 01:05:21,310 will not be very big. 1023 01:05:21,310 --> 01:05:25,360 So maybe in the same sense that when I was doing elasticity 1024 01:05:25,360 --> 01:05:27,380 by going to low temperature, I could 1025 01:05:27,380 --> 01:05:29,500 look at small deformations. 1026 01:05:29,500 --> 01:05:31,932 By sticking in the vicinity of TC, 1027 01:05:31,932 --> 01:05:36,430 I can look at small magnetization fluctuations. 1028 01:05:36,430 --> 01:05:39,990 So what I want to do is to imagine 1029 01:05:39,990 --> 01:05:46,500 that within my system that has lots and lots of electrons 1030 01:05:46,500 --> 01:05:49,860 and other microscopic degrees of freedom, 1031 01:05:49,860 --> 01:05:55,350 I can focus on regions that I average. 1032 01:05:55,350 --> 01:05:57,275 And with each region that I average, 1033 01:05:57,275 --> 01:06:00,940 I associate magnetization as a function 1034 01:06:00,940 --> 01:06:05,360 of position, which is a statistical field 1035 01:06:05,360 --> 01:06:14,740 in the same sense that displacement was. 1036 01:06:14,740 --> 01:06:26,370 See, over here, I will write this quantity-- 1037 01:06:26,370 --> 01:06:29,800 the analog of the U-- in fact by using 1038 01:06:29,800 --> 01:06:34,930 two different vectorial symbols for reasons 1039 01:06:34,930 --> 01:06:39,390 that will become obvious shortly, hopefully. 1040 01:06:39,390 --> 01:06:44,380 Because I would like to have the possibility of having 1041 01:06:44,380 --> 01:06:52,730 R to describe systems that live in D dimensions 1042 01:06:52,730 --> 01:06:56,380 where D would be one if I'm dealing with a wire. 1043 01:06:56,380 --> 01:06:59,750 D would be two if I'm dealing with a flat plane. 1044 01:06:59,750 --> 01:07:02,160 D equals to three in three dimensional space. 1045 01:07:02,160 --> 01:07:06,260 Maybe there's some relationship to relativistic field theories 1046 01:07:06,260 --> 01:07:10,520 where we would be four for space time. 1047 01:07:10,520 --> 01:07:15,480 But M I will allowed to be something else that 1048 01:07:15,480 --> 01:07:20,030 has components M1, M2, Mn. 1049 01:07:23,620 --> 01:07:29,120 And we've already seen two cases where n is either 1 or 3. 1050 01:07:29,120 --> 01:07:34,120 Clearly, if I'm thinking about the case of a ferromagnet, 1051 01:07:34,120 --> 01:07:36,850 then there are three components of the magnetization. 1052 01:07:36,850 --> 01:07:38,640 N has to three. 1053 01:07:38,640 --> 01:07:40,400 But then if I'm looking at the analogous 1054 01:07:40,400 --> 01:07:44,300 thing for the liquid gas system, the thing 1055 01:07:44,300 --> 01:07:46,460 that distinguishes different locations 1056 01:07:46,460 --> 01:07:48,990 is the density-- density fluctuations. 1057 01:07:48,990 --> 01:07:51,060 And that's a scalar quantity. 1058 01:07:51,060 --> 01:07:54,520 So that corresponds to N equals to 1. 1059 01:07:54,520 --> 01:07:58,610 There's other-- then we are dealing with super fluidity 1060 01:07:58,610 --> 01:08:04,230 where what we are leaving it is a quantum mechanical object 1061 01:08:04,230 --> 01:08:07,230 that has a phase and an amplitude. 1062 01:08:07,230 --> 01:08:10,510 It has an x component-- real and imaginary component that 1063 01:08:10,510 --> 01:08:13,130 corresponds to n equals to 2. 1064 01:08:13,130 --> 01:08:17,020 Again, n equals to 1 would describe something 1065 01:08:17,020 --> 01:08:19,180 like liquid gas. 1066 01:08:19,180 --> 01:08:24,770 N equals to 3 would correspond to something like a magnet. 1067 01:08:24,770 --> 01:08:27,310 And there's actually no relationship between n 1068 01:08:27,310 --> 01:08:30,130 and v. N could be larger than 3. 1069 01:08:30,130 --> 01:08:33,340 So imagine that you take a wire. 1070 01:08:33,340 --> 01:08:36,960 So x is clearly one-dimensional. 1071 01:08:36,960 --> 01:08:40,160 But along the wire, you put three component spins. 1072 01:08:40,160 --> 01:08:42,770 So N could still be true. 1073 01:08:42,770 --> 01:08:49,350 So we can discuss a whole bunch of different quantities 1074 01:08:49,350 --> 01:08:55,056 at the same time by generalizing our picture of the magnet 1075 01:08:55,056 --> 01:09:00,479 to have n component and existing D dimensions. 1076 01:09:00,479 --> 01:09:02,370 OK? 1077 01:09:02,370 --> 01:09:06,529 Now, the thing that I would like to construct 1078 01:09:06,529 --> 01:09:10,640 is that I look at my system. 1079 01:09:10,640 --> 01:09:13,510 And I characterize it by different configurations 1080 01:09:13,510 --> 01:09:17,960 of this field M of R. And if I have 1081 01:09:17,960 --> 01:09:22,250 many examples of the same system at the same temperature, 1082 01:09:22,250 --> 01:09:24,850 I will have different realizations 1083 01:09:24,850 --> 01:09:27,210 of that statistical field. 1084 01:09:27,210 --> 01:09:29,550 I can assign the different realizations 1085 01:09:29,550 --> 01:09:35,040 some kind of weight of probability, if you like. 1086 01:09:35,040 --> 01:09:38,840 And what I would like to do is to have 1087 01:09:38,840 --> 01:09:42,580 an idea of what the weight of probability 1088 01:09:42,580 --> 01:09:46,479 of different configurations is. 1089 01:09:46,479 --> 01:09:50,900 Just because I have a background in statistical mechanics, 1090 01:09:50,900 --> 01:09:54,680 in statistical mechanics, we are used to [INAUDIBLE] weight. 1091 01:09:54,680 --> 01:09:58,744 So we take the log of weight of probabilities 1092 01:09:58,744 --> 01:10:00,860 and call them some kind of effective Hamiltonian. 1093 01:10:07,200 --> 01:10:12,850 And this effective Hamiltonian is 1094 01:10:12,850 --> 01:10:16,520 distinct from a true microscoping Hamiltonian 1095 01:10:16,520 --> 01:10:19,900 that describes the system. 1096 01:10:19,900 --> 01:10:21,820 And it's just a way of describing 1097 01:10:21,820 --> 01:10:24,620 what the logarithm of the probability 1098 01:10:24,620 --> 01:10:28,560 for the different configurations is [INAUDIBLE]. 1099 01:10:28,560 --> 01:10:29,310 Say, well, OK. 1100 01:10:29,310 --> 01:10:32,440 How do you go and construct this? 1101 01:10:32,440 --> 01:10:33,360 Well, I say, OK. 1102 01:10:33,360 --> 01:10:41,820 Presumably, there is really some true microscopic Hamiltonia. 1103 01:10:41,820 --> 01:10:44,800 And I can take whatever that microscopic Hamiltonia 1104 01:10:44,800 --> 01:10:48,880 is that has all of my degrees of freedom. 1105 01:10:48,880 --> 01:10:52,110 And then for a particular configuration, 1106 01:10:52,110 --> 01:10:58,106 I know what the probability of a true microscopic configuration 1107 01:10:58,106 --> 01:10:58,606 is. 1108 01:11:01,320 --> 01:11:05,780 Presumably, what I did was to obtain my M of R 1109 01:11:05,780 --> 01:11:13,980 by somehow averaging over these other two degrees of freedom. 1110 01:11:13,980 --> 01:11:18,960 So the construction to go from the true microscopic 1111 01:11:18,960 --> 01:11:22,510 probabilities to this effective weight 1112 01:11:22,510 --> 01:11:25,300 is just a change of variable. 1113 01:11:25,300 --> 01:11:29,825 I have to specify some configuration M of R 1114 01:11:29,825 --> 01:11:32,060 in my system. 1115 01:11:32,060 --> 01:11:35,400 That configuration of M of R will 1116 01:11:35,400 --> 01:11:40,780 be consistent with a huge number of microscopic configurations. 1117 01:11:40,780 --> 01:11:42,560 I know what the weight of each one 1118 01:11:42,560 --> 01:11:45,390 of those microscopic configurations is. 1119 01:11:45,390 --> 01:11:47,890 I sum over all of them. 1120 01:11:47,890 --> 01:11:50,370 And I have this. 1121 01:11:50,370 --> 01:11:52,840 Now, of course, if I could do that, 1122 01:11:52,840 --> 01:11:54,880 I would immediately solve the full problem 1123 01:11:54,880 --> 01:11:57,880 and I wouldn't need to have to deal with this. 1124 01:11:57,880 --> 01:12:00,720 Clearly, I can't do that. 1125 01:12:00,720 --> 01:12:04,250 But I can guess what the eventual form of this 1126 01:12:04,250 --> 01:12:07,920 is in the same way that I guessed 1127 01:12:07,920 --> 01:12:11,150 what the form of the statistical field theory for elasticity 1128 01:12:11,150 --> 01:12:14,450 was just by looking at symmetries and things 1129 01:12:14,450 --> 01:12:17,220 like that, OK? 1130 01:12:17,220 --> 01:12:34,750 So in principle, W from change of variables starting 1131 01:12:34,750 --> 01:12:48,989 from into the minus beta H in practice from symmetries 1132 01:12:48,989 --> 01:12:52,210 and a variety of other statements and constraints 1133 01:12:52,210 --> 01:12:55,516 that I would tell you about. 1134 01:12:55,516 --> 01:12:58,462 Actually, let's keep this. 1135 01:12:58,462 --> 01:13:00,426 And let's illuminate this. 1136 01:13:16,140 --> 01:13:17,860 OK. 1137 01:13:17,860 --> 01:13:25,090 So there is beta H, which is a function of M 1138 01:13:25,090 --> 01:13:27,060 as a function of this vector x. 1139 01:13:30,340 --> 01:13:33,000 First thing that I will do is what 1140 01:13:33,000 --> 01:13:39,600 I did for the case of elasticity, which 1141 01:13:39,600 --> 01:13:44,730 is to write the answer as an integral in D 1142 01:13:44,730 --> 01:13:52,450 dimensions of some kind of a density at location x. 1143 01:13:52,450 --> 01:13:59,700 So this is the same locality type of constraint 1144 01:13:59,700 --> 01:14:06,920 that we were discussing before and has some caveats associated 1145 01:14:06,920 --> 01:14:08,910 with that. 1146 01:14:08,910 --> 01:14:13,250 This is going to be a function of the variable 1147 01:14:13,250 --> 01:14:16,290 at that location x. 1148 01:14:16,290 --> 01:14:18,540 So that's the field. 1149 01:14:18,540 --> 01:14:21,385 But I will also allow various derivatives 1150 01:14:21,385 --> 01:14:28,250 to appear so that I go beyond just single [INAUDIBLE] 1151 01:14:28,250 --> 01:14:30,730 will just give me independent things happening 1152 01:14:30,730 --> 01:14:35,250 at each location by allowing some kind of connections 1153 01:14:35,250 --> 01:14:36,860 in a neighborhood. 1154 01:14:36,860 --> 01:14:41,010 And if I go and recruit higher and higher order derivatives, 1155 01:14:41,010 --> 01:14:45,000 naturally I would have more pips. 1156 01:14:45,000 --> 01:14:48,600 Somebody was asking me-- you were asking me last time 1157 01:14:48,600 --> 01:14:53,080 in principle, if the system is something 1158 01:14:53,080 --> 01:14:56,450 that varies from one position to another position, 1159 01:14:56,450 --> 01:15:01,010 the very function itself would depend on x. 1160 01:15:01,010 --> 01:15:07,100 But if we assume that the system is uniform, 1161 01:15:07,100 --> 01:15:10,150 then we can drop that force. 1162 01:15:13,990 --> 01:15:25,720 So to be precise, let's do this for the case of five 0 field. 1163 01:15:25,720 --> 01:15:29,620 Because when you are at zero field in a magnet, 1164 01:15:29,620 --> 01:15:33,420 the different directions of space are all the same to you. 1165 01:15:33,420 --> 01:15:36,230 There's no reason to be pointing one direction as opposed 1166 01:15:36,230 --> 01:15:38,060 to another direction. 1167 01:15:38,060 --> 01:15:50,730 So because of this symmetry in rotations, in this function, 1168 01:15:50,730 --> 01:15:54,420 you cannot have M-- actually, you couldn't have M for other 1169 01:15:54,420 --> 01:15:55,290 reasons. 1170 01:15:55,290 --> 01:15:57,020 Well, OK. 1171 01:15:57,020 --> 01:16:00,170 You couldn't have M because it would break the directionality. 1172 01:16:03,780 --> 01:16:07,640 But you could have something that is M squared. 1173 01:16:07,640 --> 01:16:13,815 Again, to be precise, M squared is some, let's say, 1174 01:16:13,815 --> 01:16:18,250 alpha running from 1 to n and alpha of x and alpha of x. 1175 01:16:23,320 --> 01:16:28,880 Now, if the different directions in space are the same, 1176 01:16:28,880 --> 01:16:31,990 then I can't have a gradient appearing by itself 1177 01:16:31,990 --> 01:16:34,630 because it would pick a particular direction. 1178 01:16:34,630 --> 01:16:38,030 In the same sense that M squared, 1179 01:16:38,030 --> 01:16:40,745 two Ms have to be appearing together, 1180 01:16:40,745 --> 01:16:43,170 if the different directions in space 1181 01:16:43,170 --> 01:16:46,580 forward and backward are to be treated the same, 1182 01:16:46,580 --> 01:16:51,452 I have to have gradients appearing together 1183 01:16:51,452 --> 01:16:52,440 in powers of two. 1184 01:16:52,440 --> 01:16:56,190 So there's a term that, before doing that, 1185 01:16:56,190 --> 01:16:59,480 let's say also sometimes I will write something 1186 01:16:59,480 --> 01:17:02,880 that is M to the fourth. 1187 01:17:02,880 --> 01:17:10,280 M to the fourth is really this quantity M dot M squared. 1188 01:17:10,280 --> 01:17:16,340 If I write M to the sixth, it is M dot n. 1189 01:17:16,340 --> 01:17:18,780 Mute and so forth-- so there's all kinds 1190 01:17:18,780 --> 01:17:21,990 of terms such as that can be appearing in this series. 1191 01:17:21,990 --> 01:17:26,400 Gradients-- there's a term that I will write symbolically 1192 01:17:26,400 --> 01:17:29,770 as gradient of M square. 1193 01:17:29,770 --> 01:17:38,820 And by that, I mean to we take a derivative of n alpha 1194 01:17:38,820 --> 01:17:43,660 with respect to the alt component 1195 01:17:43,660 --> 01:17:51,200 and then repeat the same thing sum 1196 01:17:51,200 --> 01:17:54,310 wrote over both alpha and alt. 1197 01:17:54,310 --> 01:17:56,130 So that would be the gradient. 1198 01:17:56,130 --> 01:18:01,050 So basically, the xi appears twice. 1199 01:18:01,050 --> 01:18:03,830 M alpha appears twice. 1200 01:18:03,830 --> 01:18:07,660 And you can go and construct higher and higher order terms 1201 01:18:07,660 --> 01:18:12,380 and derivatives, again ensuring that each index 1202 01:18:12,380 --> 01:18:18,240 both on the side of the position x and the side of the fields M 1203 01:18:18,240 --> 01:18:21,199 is a repeated index to respect the symmetries that 1204 01:18:21,199 --> 01:18:21,740 are involved. 1205 01:18:24,740 --> 01:18:34,310 So if we-- there is actually maybe one other thing 1206 01:18:34,310 --> 01:18:43,200 to think about, which is that again like 1207 01:18:43,200 --> 01:18:48,460 before, I assume that I can make an analytical expansion in M. 1208 01:18:48,460 --> 01:18:50,852 And who says you are allowed to do 1209 01:18:50,852 --> 01:18:53,480 an analytical expansion in M? 1210 01:18:53,480 --> 01:18:56,260 Again, the key to that is the averaging 1211 01:18:56,260 --> 01:18:59,110 that we have to do in the process. 1212 01:18:59,110 --> 01:19:05,260 And I want you to think back to the central limit theorem. 1213 01:19:05,260 --> 01:19:09,380 Let's imagine that there is a variable x where 1214 01:19:09,380 --> 01:19:15,430 the probability of selecting that variable x 1215 01:19:15,430 --> 01:19:18,220 is actually kind of singular. 1216 01:19:18,220 --> 01:19:21,130 Maybe it is something like e to the minus x 1217 01:19:21,130 --> 01:19:23,550 that is a discontinuity. 1218 01:19:23,550 --> 01:19:28,340 Maybe it even has an integrable divergence at some point. 1219 01:19:28,340 --> 01:19:30,642 Maybe it has additional delta function. 1220 01:19:30,642 --> 01:19:35,480 All kinds of-- it's a very complicated, singular type 1221 01:19:35,480 --> 01:19:38,160 of a function. 1222 01:19:38,160 --> 01:19:41,590 Now, I tell you-- add thousands of these variables 1223 01:19:41,590 --> 01:19:46,390 together and tell me what the distribution of the sum is. 1224 01:19:46,390 --> 01:19:49,550 And the distribution of the sum, because of the central limit 1225 01:19:49,550 --> 01:19:53,580 theorem, you know has to look Gaussian. 1226 01:19:53,580 --> 01:19:59,200 So then take its log, it has a nice analytical expansion. 1227 01:19:59,200 --> 01:20:03,660 So the point is that again, part of the course 1228 01:20:03,660 --> 01:20:07,920 grading that we did to get from whatever microscopic degrees 1229 01:20:07,920 --> 01:20:10,680 of freedom that we have, to reaching 1230 01:20:10,680 --> 01:20:13,350 the level of this effective field theory, 1231 01:20:13,350 --> 01:20:15,870 we added many variables together. 1232 01:20:15,870 --> 01:20:18,320 And because of the central limit theorem, 1233 01:20:18,320 --> 01:20:21,615 I have a lot of confidence that quite generically, 1234 01:20:21,615 --> 01:20:25,410 I can make an analytical expansion such as this. 1235 01:20:25,410 --> 01:20:27,590 But I have not. 1236 01:20:27,590 --> 01:20:31,540 So having done all of this, you would 1237 01:20:31,540 --> 01:20:40,550 say that in order to describe this magnet, what we need 1238 01:20:40,550 --> 01:20:45,150 is to evaluate the partition function, 1239 01:20:45,150 --> 01:20:50,970 let's say, as a function of temperature, 1240 01:20:50,970 --> 01:20:55,430 which if I had enormous power, I would 1241 01:20:55,430 --> 01:21:01,190 do by trace of into the beta H microscopic. 1242 01:21:01,190 --> 01:21:04,950 But what I have done is I have subdivided 1243 01:21:04,950 --> 01:21:09,690 different configurations of microscopic degrees of freedom 1244 01:21:09,690 --> 01:21:13,590 to different configurations of this effective magnetization. 1245 01:21:13,590 --> 01:21:19,965 And so essentially, that sum is the same as integrating over 1246 01:21:19,965 --> 01:21:23,231 all configurations of this constrained magnetization 1247 01:21:23,231 --> 01:21:23,730 field. 1248 01:21:26,720 --> 01:21:30,810 And the probabilities of these configurations 1249 01:21:30,810 --> 01:21:35,400 of the magnetization field are exponential of this minus data 1250 01:21:35,400 --> 01:21:39,290 H that I'm constructing on the basis of principles 1251 01:21:39,290 --> 01:21:43,180 that I told you-- principles where that I first of all 1252 01:21:43,180 --> 01:21:45,270 have a locality. 1253 01:21:45,270 --> 01:21:49,330 I have an integrality of x. 1254 01:21:49,330 --> 01:21:51,320 And then I have to write terms that 1255 01:21:51,320 --> 01:21:54,270 are consistent with the symmetry. 1256 01:21:54,270 --> 01:21:57,140 First term that we saw that is consistent with the symmetry 1257 01:21:57,140 --> 01:22:00,050 is this M squared. 1258 01:22:00,050 --> 01:22:02,290 And let's give a name to its coefficient. 1259 01:22:02,290 --> 01:22:06,490 Just like the coefficient of elasticity, I put it nu over 2. 1260 01:22:06,490 --> 01:22:10,050 Let's get here a T over 2. 1261 01:22:10,050 --> 01:22:11,840 There will be higher order terms. 1262 01:22:11,840 --> 01:22:13,580 There will be M to the fourth. 1263 01:22:13,580 --> 01:22:15,046 There will be M to the sixth. 1264 01:22:15,046 --> 01:22:17,170 There will be a whole bunch of things in principle. 1265 01:22:20,220 --> 01:22:22,280 There will be gradient types of terms. 1266 01:22:22,280 --> 01:22:27,079 So there will be K over 2, gradient of M squared. 1267 01:22:27,079 --> 01:22:31,160 There will be L over 2 Laplacian of M squared. 1268 01:22:31,160 --> 01:22:33,600 There would be higher order terms that 1269 01:22:33,600 --> 01:22:37,070 involve multiplying M's and various gradient 1270 01:22:37,070 --> 01:22:40,460 of M, et cetera. 1271 01:22:40,460 --> 01:22:42,590 And actually, in principle, there 1272 01:22:42,590 --> 01:22:44,240 could be an overall constant. 1273 01:22:44,240 --> 01:22:48,280 So there could be up here some overall constant 1274 01:22:48,280 --> 01:22:56,790 of integration-- let's call it beta F0-- which depends 1275 01:22:56,790 --> 01:23:00,530 on temperature and so forth but has nothing 1276 01:23:00,530 --> 01:23:04,970 to do with the ordering of these magnetization type of degrees 1277 01:23:04,970 --> 01:23:07,140 of freedom. 1278 01:23:07,140 --> 01:23:13,230 Actually, if I imagine that I go slightly away from the H equals 1279 01:23:13,230 --> 01:23:17,610 to 0 axis, just I did before, I will add here 1280 01:23:17,610 --> 01:23:27,260 a term minus beta H of M and evaluate this whole thing. 1281 01:23:27,260 --> 01:23:30,638 So this is the Landau-Ginzberg theory. 1282 01:23:36,570 --> 01:23:40,600 In principle, you have to put a lot of terms in the series. 1283 01:23:40,600 --> 01:23:45,360 Our task would be to show you quite rigorously 1284 01:23:45,360 --> 01:23:47,730 that just a few terms in the series 1285 01:23:47,730 --> 01:23:50,730 aren't enough just as in the case of theory of vibration 1286 01:23:50,730 --> 01:23:52,790 for small enough vibration. 1287 01:23:52,790 --> 01:23:55,680 In this case, the analog of small enough vibration 1288 01:23:55,680 --> 01:23:58,910 is to be close enough to the critical point, exactly 1289 01:23:58,910 --> 01:24:02,720 where we want to calculate these universal exponents 1290 01:24:02,720 --> 01:24:06,030 and to then calculate the universal exponents, which 1291 01:24:06,030 --> 01:24:07,990 will turn out to be a hard problem 1292 01:24:07,990 --> 01:24:11,836 that we will struggle with in the next many lectures.