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 --> 00:00:13,340 To make a donation or view additional materials 6 00:00:13,340 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:21,100 --> 00:00:22,500 PROFESSOR: OK, let's start. 9 00:00:25,880 --> 00:00:29,825 So we've been trying to understand critical points. 10 00:00:36,570 --> 00:00:41,080 And this refers to the experimental observation 11 00:00:41,080 --> 00:00:44,570 that in a number of systems we can 12 00:00:44,570 --> 00:00:48,570 be changing some parameters, such as temperature, 13 00:00:48,570 --> 00:00:53,610 and you encounter a transition to some other type of behavior 14 00:00:53,610 --> 00:00:54,760 at some point. 15 00:00:54,760 --> 00:00:58,950 So the temperature, let's say, in this behavior 16 00:00:58,950 --> 00:01:00,070 is the control parameter. 17 00:01:00,070 --> 00:01:02,330 And you have to see, for example, 18 00:01:02,330 --> 00:01:05,390 this will be normal to superfluid transition. 19 00:01:05,390 --> 00:01:08,336 You have one now [INAUDIBLE] change temperature and going 20 00:01:08,336 --> 00:01:10,210 through this point. 21 00:01:10,210 --> 00:01:12,500 For other systems, such as magnets, 22 00:01:12,500 --> 00:01:14,730 you actually have two knobs. 23 00:01:14,730 --> 00:01:18,120 There is also the magnetic field. 24 00:01:18,120 --> 00:01:21,360 And there, you have to turn two knobs 25 00:01:21,360 --> 00:01:25,630 in order to end at this critical point, 26 00:01:25,630 --> 00:01:29,060 also in the case of the liquid gas system in the pressure 27 00:01:29,060 --> 00:01:33,553 temperature plane, you have to tune two things 28 00:01:33,553 --> 00:01:35,420 to get this point. 29 00:01:35,420 --> 00:01:38,630 And the interesting thing was that in the vicinity 30 00:01:38,630 --> 00:01:43,420 of his point, the singular parts of various thermodynamic 31 00:01:43,420 --> 00:01:48,710 quantities are interestingly independent of the type 32 00:01:48,710 --> 00:01:50,210 of material. 33 00:01:50,210 --> 00:01:55,320 So if we, for example, establish a coordinate at t and h 34 00:01:55,320 --> 00:01:59,400 describing deviations from this critical point, 35 00:01:59,400 --> 00:02:02,880 we have, let's say, the singular part 36 00:02:02,880 --> 00:02:06,820 of free energy as a function of t and h 37 00:02:06,820 --> 00:02:10,710 has a form like t to the 2 minus alpha, 38 00:02:10,710 --> 00:02:15,440 some scaling function ht to the delta, 39 00:02:15,440 --> 00:02:18,856 and these exponents, alpha and delta, other things 40 00:02:18,856 --> 00:02:21,700 that are universe. 41 00:02:21,700 --> 00:02:25,080 For example, we could get from that 42 00:02:25,080 --> 00:02:28,660 by taking two derivatives with respect to h, the singularity 43 00:02:28,660 --> 00:02:33,120 and the divergence of the susceptibility. 44 00:02:33,120 --> 00:02:35,776 And we said that the diverging susceptibility also 45 00:02:35,776 --> 00:02:39,660 immediately tells you that there is a correlation then 46 00:02:39,660 --> 00:02:44,790 that diverges, and in particular, we 47 00:02:44,790 --> 00:02:48,410 indicated its divergence to an exponent u. 48 00:02:48,410 --> 00:02:53,210 WE could for that also establish a scaling form 49 00:02:53,210 --> 00:02:56,140 on how the correlation then diverges 50 00:02:56,140 --> 00:03:01,980 on approaching this point generally in the ht plane. 51 00:03:01,980 --> 00:03:08,820 So this was the general picture. 52 00:03:08,820 --> 00:03:14,340 And building on that, we made one observation last time, 53 00:03:14,340 --> 00:03:18,370 which is that any point when you are away from h and t 54 00:03:18,370 --> 00:03:23,130 equals to 0, you have a correlation length. 55 00:03:23,130 --> 00:03:28,760 And then we concluded that if you are at t and h equals to 0, 56 00:03:28,760 --> 00:03:30,460 you have a form of scaling vertices. 57 00:03:38,560 --> 00:03:45,470 And basically what that means is that when 58 00:03:45,470 --> 00:03:48,150 you are at that point, you look at your system, 59 00:03:48,150 --> 00:03:52,510 it's a fluctuating system, and the fluctuations are 60 00:03:52,510 --> 00:03:56,035 such that you can't associate a scale with them. 61 00:03:56,035 --> 00:03:59,400 The scale has already gone into the correlation 62 00:03:59,400 --> 00:04:02,180 length that is infinite. 63 00:04:02,180 --> 00:04:03,990 And we said that therefore, if I were 64 00:04:03,990 --> 00:04:06,295 to look at some kind of a correlation function, 65 00:04:06,295 --> 00:04:12,270 such as a magnetization in the case of a magnet, 66 00:04:12,270 --> 00:04:17,700 that the only way that it became its separation 67 00:04:17,700 --> 00:04:23,550 is as a power of a distance. 68 00:04:23,550 --> 00:04:26,160 And this clearly has a property that if we 69 00:04:26,160 --> 00:04:30,990 were to rescale x and y by a certain amount, 70 00:04:30,990 --> 00:04:34,090 this correlation function nearly gets multiplied 71 00:04:34,090 --> 00:04:38,610 by a factor that is dependent on this rescale. 72 00:04:38,610 --> 00:04:41,810 And this is after we do the averaging, 73 00:04:41,810 --> 00:04:47,200 so it's a kind of statistical self-singularity, 74 00:04:47,200 --> 00:04:52,715 as opposed to some factor such as Sierpinkski gasket, which 75 00:04:52,715 --> 00:04:54,840 are identically and deterministically 76 00:04:54,840 --> 00:04:58,000 self-similar in that each piece, if you blow it up, 77 00:04:58,000 --> 00:05:01,320 looks like the entire thing. 78 00:05:01,320 --> 00:05:07,900 So what we have in our system is that if we have, let's say, 79 00:05:07,900 --> 00:05:14,245 a box which could be containing our liquid gas 80 00:05:14,245 --> 00:05:20,400 system at its critical point, or maybe a magnet 81 00:05:20,400 --> 00:05:23,980 at its critical point, will have a statistical field, 82 00:05:23,980 --> 00:05:26,327 this m of x. 83 00:05:26,327 --> 00:05:31,120 And it will fluctuate across the system. 84 00:05:31,120 --> 00:05:36,490 So maybe this would be a picture of the density fluctuation. 85 00:05:36,490 --> 00:05:43,966 What I can do is to take a scan along some particular axis-- 86 00:05:43,966 --> 00:05:49,860 let's call it x-- and plot what the fluctuations are 87 00:05:49,860 --> 00:05:51,600 of this magnetization. 88 00:05:51,600 --> 00:05:54,660 Let's say m of x. 89 00:05:54,660 --> 00:05:56,780 Now the average will be 0, but it 90 00:05:56,780 --> 00:05:59,700 will have fluctuations around the average. 91 00:05:59,700 --> 00:06:01,615 And so maybe it will look something 92 00:06:01,615 --> 00:06:12,540 like this-- kind of like a picture of a mountain, 93 00:06:12,540 --> 00:06:13,040 for example. 94 00:06:16,910 --> 00:06:20,140 Now one thing that we should remember 95 00:06:20,140 --> 00:06:25,420 is that this object would be piece of iron or nickle, 96 00:06:25,420 --> 00:06:29,160 and clearly I don't really mean that this 97 00:06:29,160 --> 00:06:33,220 is what is going on at the scale of a single atom 98 00:06:33,220 --> 00:06:34,920 or molecule of my substance. 99 00:06:34,920 --> 00:06:38,930 I had to do some kind of averaging in order 100 00:06:38,930 --> 00:06:42,300 to get the statistical field that I'm presenting here. 101 00:06:42,300 --> 00:06:45,470 So let's keep in mind that there is, 102 00:06:45,470 --> 00:06:54,360 in fact, some implicit analog of lattice size 103 00:06:54,360 --> 00:06:57,630 or some implicit shortest distance, 104 00:06:57,630 --> 00:07:00,810 shortest wavelength that I allow for my frustrations. 105 00:07:06,270 --> 00:07:14,035 Now I can sort of make this idea of scale invariance 106 00:07:14,035 --> 00:07:19,910 of a set of pictures, such as this one, more precise, 107 00:07:19,910 --> 00:07:26,470 as follows, by going through a procedure 108 00:07:26,470 --> 00:07:30,676 that I will call renormalization that has the following three 109 00:07:30,676 --> 00:07:31,175 steps. 110 00:07:38,040 --> 00:07:45,575 So the first step, what I will do is to coarse-grain further. 111 00:07:51,490 --> 00:08:10,630 And by this, I mean averaging m of x over a scale ta. 112 00:08:10,630 --> 00:08:15,950 So previously, I had done my averaging of whatever means, 113 00:08:15,950 --> 00:08:17,360 et cetera. 114 00:08:17,360 --> 00:08:21,260 We're giving contribution to the overall magnetization 115 00:08:21,260 --> 00:08:25,320 over some number, let's 100 by 100 by 100 116 00:08:25,320 --> 00:08:30,180 spins and a was my scaling distance. 117 00:08:30,180 --> 00:08:31,880 Why should I choose 100? 118 00:08:31,880 --> 00:08:36,880 Why not choose 200, some factor of what I had originally? 119 00:08:36,880 --> 00:08:40,929 So coarse-graining means increasing this minimum length 120 00:08:40,929 --> 00:08:43,620 scale from a to ba. 121 00:08:43,620 --> 00:08:46,850 And then I define a coarse-grained version 122 00:08:46,850 --> 00:08:48,690 of my field. 123 00:08:48,690 --> 00:08:50,760 So previously, I had m of x. 124 00:08:50,760 --> 00:08:57,630 Now I have m tilda of x, which is obtained by averaging, 125 00:08:57,630 --> 00:09:06,400 let's say, over volume around the point x that I had before. 126 00:09:06,400 --> 00:09:16,970 And this volume is a box of dimension ba to the d. 127 00:09:16,970 --> 00:09:24,220 And then I basically average over that. 128 00:09:24,220 --> 00:09:28,920 I guess let's call it original distance a equals 1, 129 00:09:28,920 --> 00:09:34,988 so I don't really have to bother by the dimensionality of y, 130 00:09:34,988 --> 00:09:35,487 et cetera. 131 00:09:38,760 --> 00:09:40,240 OK? 132 00:09:40,240 --> 00:09:42,980 So if I were to apply that to the picture 133 00:09:42,980 --> 00:09:45,670 that I have up there, what do I get? 134 00:09:45,670 --> 00:09:49,435 I will get an m tilda as a function of x. 135 00:09:55,500 --> 00:09:58,770 And essentially, let's say if I were 136 00:09:58,770 --> 00:10:04,250 to choose a factor of b that was like 2, 137 00:10:04,250 --> 00:10:07,970 I would take the average of the fluctuations 138 00:10:07,970 --> 00:10:11,270 that they have over 2 of those of those intervals. 139 00:10:11,270 --> 00:10:14,190 And so the picture that I would get it 140 00:10:14,190 --> 00:10:17,660 would be kind of a smoothened out version 141 00:10:17,660 --> 00:10:20,690 of what I have before over there. 142 00:10:23,910 --> 00:10:27,080 I will still have some fluctuations, 143 00:10:27,080 --> 00:10:31,480 but kind of ironed out. 144 00:10:31,480 --> 00:10:35,070 And basically, essentially, it means 145 00:10:35,070 --> 00:10:40,640 that if you were to imagine having taken a photograph, 146 00:10:40,640 --> 00:10:45,830 previously you had the pixel size that was 1. 147 00:10:45,830 --> 00:10:47,830 Now your pixel size is larger. 148 00:10:47,830 --> 00:10:51,510 It is factor of b. 149 00:10:51,510 --> 00:10:55,790 So it's this kind of detuning and averaging 150 00:10:55,790 --> 00:10:58,170 of the fluctuations that has gone. 151 00:10:58,170 --> 00:11:00,378 And so you have here now b. 152 00:11:08,690 --> 00:11:12,650 Now if I were to give you a photograph like that 153 00:11:12,650 --> 00:11:15,430 and a photograph like this, you would 154 00:11:15,430 --> 00:11:18,370 say that they are not identical. 155 00:11:18,370 --> 00:11:22,796 One of them is clearly much grainier than the other. 156 00:11:22,796 --> 00:11:27,010 So I say, OK, I can restore some amount of similarity 157 00:11:27,010 --> 00:11:30,530 between them by doing a rescaling. 158 00:11:34,070 --> 00:11:41,520 So I call a new variable x prime to be my old variable x divided 159 00:11:41,520 --> 00:11:44,220 by a factor of b. 160 00:11:44,220 --> 00:11:47,871 So when I do that to this picture, 161 00:11:47,871 --> 00:11:53,710 I will get m tilda as a function of x prime. 162 00:11:53,710 --> 00:12:00,120 x prime can go in further less, because all I do 163 00:12:00,120 --> 00:12:03,900 is I take this and squeeze it by a factor of b. 164 00:12:03,900 --> 00:12:05,840 So I will get a picture that maybe 165 00:12:05,840 --> 00:12:08,500 looks something like this. 166 00:12:19,140 --> 00:12:27,630 Now if I were to look at this picture and this picture, 167 00:12:27,630 --> 00:12:30,240 you would also see a difference. 168 00:12:30,240 --> 00:12:31,920 That is, there is a contrast. 169 00:12:31,920 --> 00:12:34,890 So here, there would be, let's say, black and white. 170 00:12:34,890 --> 00:12:37,340 And as you scan the picture, you sort of see 171 00:12:37,340 --> 00:12:39,840 some variation of black and white. 172 00:12:39,840 --> 00:12:44,010 If you look at this, you say the contrast is just too big. 173 00:12:44,010 --> 00:12:46,890 You have big fluctuations as you go across 174 00:12:46,890 --> 00:12:49,750 compared to what I had over there. 175 00:12:49,750 --> 00:12:58,620 So there's another step, which is called renormalize, which 176 00:12:58,620 --> 00:13:05,590 is that you define m prime to be m 177 00:13:05,590 --> 00:13:09,040 by a change of contrast factor zeta. 178 00:13:09,040 --> 00:13:13,240 So you take a knob that corresponds to contrast 179 00:13:13,240 --> 00:13:21,480 and you reduce it until you see pictures 180 00:13:21,480 --> 00:13:29,270 that kind of statistically look like what you started with. 181 00:13:29,270 --> 00:13:36,910 So in order to sort of generate pictures that are self-similar, 182 00:13:36,910 --> 00:13:38,980 you have this one knob. 183 00:13:38,980 --> 00:13:45,030 Basically, scaling variance means the change of size. 184 00:13:45,030 --> 00:13:48,350 But there is associated with change of size 185 00:13:48,350 --> 00:13:51,290 a change of contrast for whatever variable 186 00:13:51,290 --> 00:13:52,660 you are looking at. 187 00:13:52,660 --> 00:13:55,500 It turns out that that change of contrast 188 00:13:55,500 --> 00:13:58,140 would eventually map to one of these exponents 189 00:13:58,140 --> 00:13:59,320 that we have over there. 190 00:13:59,320 --> 00:14:00,530 Yes. 191 00:14:00,530 --> 00:14:03,200 STUDENT: Are you using m or n tilda? 192 00:14:03,200 --> 00:14:04,702 PROFESSOR: m tilda, thank you. 193 00:14:08,080 --> 00:14:14,530 So I guess the green is m tilda of x prime, 194 00:14:14,530 --> 00:14:18,900 and the pink is m prime of x prime. 195 00:14:30,560 --> 00:14:41,940 So what I have done mathematically is as follows. 196 00:14:41,940 --> 00:14:47,230 I have defined an m prime of x prime, 197 00:14:47,230 --> 00:14:51,740 which is 1 over zeta, this contrast factor 198 00:14:51,740 --> 00:14:56,506 b to the d because of the averaging over a volume 199 00:14:56,506 --> 00:15:06,230 that involved b to the d pixels of the original field centered 200 00:15:06,230 --> 00:15:09,300 at a location that was bx prime plus y. 201 00:15:13,750 --> 00:15:23,030 So in principle, I can go and generate 202 00:15:23,030 --> 00:15:27,686 lots and lots of configurations of my magnetization, 203 00:15:27,686 --> 00:15:32,480 or lots and lots of pictures of a system at the liquid gas 204 00:15:32,480 --> 00:15:35,760 critical point, or magnetic systems 205 00:15:35,760 --> 00:15:37,010 at their critical point. 206 00:15:37,010 --> 00:15:39,920 I can generate lots and lots of these pictures 207 00:15:39,920 --> 00:15:42,410 and construct this transformation. 208 00:15:45,080 --> 00:15:48,990 And associated with this transformation 209 00:15:48,990 --> 00:15:52,520 is a change of probability, because there 210 00:15:52,520 --> 00:15:57,200 was some probability-- let's call it P old, 211 00:15:57,200 --> 00:16:02,980 that was describing my original configurations m of x. 212 00:16:02,980 --> 00:16:07,470 Let's forget the vector notation for the time being. 213 00:16:07,470 --> 00:16:15,930 Then there will be, after this transformation, probability 214 00:16:15,930 --> 00:16:18,212 that describes these configurations m 215 00:16:18,212 --> 00:16:18,920 prime of x prime. 216 00:16:23,230 --> 00:16:26,730 Now you know that averaging is not something 217 00:16:26,730 --> 00:16:29,080 that you can reverse. 218 00:16:29,080 --> 00:16:32,490 So this transformation going from here and here, 219 00:16:32,490 --> 00:16:35,010 I cannot go back. 220 00:16:35,010 --> 00:16:39,270 There are many configurations over here 221 00:16:39,270 --> 00:16:41,690 that would correspond to the same average, 222 00:16:41,690 --> 00:16:43,810 like up, down or down, up would give you 223 00:16:43,810 --> 00:16:45,900 the same average, right? 224 00:16:45,900 --> 00:16:50,420 So a number of possibilities here 225 00:16:50,420 --> 00:16:54,340 have to be summed up to generate for you this object. 226 00:16:59,100 --> 00:17:02,540 Now the statement of self-similarity presumably 227 00:17:02,540 --> 00:17:08,380 is that this weight is the same as this weight. 228 00:17:08,380 --> 00:17:12,170 You can't tell apart that you generated configurations 229 00:17:12,170 --> 00:17:14,020 before or after that scaling. 230 00:17:16,970 --> 00:17:26,310 So this is same at critical point. 231 00:17:35,610 --> 00:17:37,880 I've not constructed either weight, 232 00:17:37,880 --> 00:17:42,420 so it really doesn't amount to much. 233 00:17:42,420 --> 00:17:53,390 But Kadanoff introduced this concept 234 00:17:53,390 --> 00:17:59,730 of doing this and thinking of it as a kind of group operation 235 00:17:59,730 --> 00:18:03,200 called renormalization group that I 236 00:18:03,200 --> 00:18:08,380 describe a little bit better and evolve the description as we 237 00:18:08,380 --> 00:18:10,890 go along. 238 00:18:10,890 --> 00:18:19,610 So if I look at my original system, 239 00:18:19,610 --> 00:18:22,560 I said that self-similarity occurs, 240 00:18:22,560 --> 00:18:25,720 let's say, exactly at this point that corresponds to t 241 00:18:25,720 --> 00:18:26,820 and h equals to 0. 242 00:18:35,090 --> 00:18:40,200 Now presumably, I can, in some sense, 243 00:18:40,200 --> 00:18:45,240 force these things, if I were to take its log, for example. 244 00:18:45,240 --> 00:18:49,110 I can construct some kind of a weight that 245 00:18:49,110 --> 00:18:52,620 is associated with m, and this would 246 00:18:52,620 --> 00:18:59,720 be a new weight that is associated with m prime. 247 00:19:05,090 --> 00:19:07,750 Presumably, right at the critical point, 248 00:19:07,750 --> 00:19:09,749 these two would be the same weight, 249 00:19:09,749 --> 00:19:11,290 and it would be the same Hamiltonian. 250 00:19:14,050 --> 00:19:16,980 What happens, if I do this procedure, 251 00:19:16,980 --> 00:19:23,460 to a system that is initially away from the critical point? 252 00:19:23,460 --> 00:19:28,690 So my initial system is characterized by deviations t 253 00:19:28,690 --> 00:19:34,660 and h from this scale in variant ways, which 254 00:19:34,660 --> 00:19:38,850 means that over here I have a correlation length. 255 00:19:44,834 --> 00:19:46,750 Now I go through all of these transformations. 256 00:19:49,460 --> 00:19:53,080 I can do those transformations also 257 00:19:53,080 --> 00:19:56,880 for a point that is not at the critical point. 258 00:19:56,880 --> 00:19:59,260 But at the end of the day, I certainly 259 00:19:59,260 --> 00:20:02,020 will not get back my original weight, 260 00:20:02,020 --> 00:20:05,420 because I look at the picture after this transformation. 261 00:20:05,420 --> 00:20:09,300 Before the transformation, I had a long correlation length, 262 00:20:09,300 --> 00:20:11,910 let's say a mile. 263 00:20:11,910 --> 00:20:15,640 When I do this transformation, that correlation length 264 00:20:15,640 --> 00:20:19,600 is reduced by a factor of b. 265 00:20:19,600 --> 00:20:26,605 So the new system has deviated more from the critical point. 266 00:20:30,080 --> 00:20:33,630 Because the further you go away from the critical point, 267 00:20:33,630 --> 00:20:37,540 you have a larger correlation length. 268 00:20:37,540 --> 00:20:42,320 So the idea is that right at the critical point, 269 00:20:42,320 --> 00:20:45,180 the two weights are the same. 270 00:20:45,180 --> 00:20:47,350 Deviation from the critical point 271 00:20:47,350 --> 00:20:52,430 is described by these two parameters, t and h. 272 00:20:52,430 --> 00:20:56,890 And if you do the renormalization procedure 273 00:20:56,890 --> 00:20:59,720 on a Hamiltonian that deviates, you 274 00:20:59,720 --> 00:21:03,420 will get a Hamiltonian that more deviates, still 275 00:21:03,420 --> 00:21:09,420 describable by parameters t and h that have changed. 276 00:21:09,420 --> 00:21:16,045 So again, this says that c was, in fact, 277 00:21:16,045 --> 00:21:23,380 b times c of t prime and h prime, 278 00:21:23,380 --> 00:21:28,070 and t prime and h prime are further away. 279 00:21:32,610 --> 00:21:39,870 Now the next thing that Kadanoff said was, OK, therefore there 280 00:21:39,870 --> 00:21:43,820 is a transformation that tells me 281 00:21:43,820 --> 00:21:48,400 after I do a rescaling by a factor of b how 282 00:21:48,400 --> 00:21:55,390 the new t and the new h depend on the old t and the old h. 283 00:22:03,060 --> 00:22:06,130 So there is a mapping in this space. 284 00:22:06,130 --> 00:22:08,510 So a point that was here will go over there. 285 00:22:08,510 --> 00:22:11,240 Maybe a point that is here will map over there. 286 00:22:11,240 --> 00:22:14,200 A point that is here will map over here. 287 00:22:14,200 --> 00:22:17,550 So there is a mapping that tells you 288 00:22:17,550 --> 00:22:23,140 how th get transformed under this procedure. 289 00:22:23,140 --> 00:22:29,765 Actually the reason this is called a renormalization group, 290 00:22:29,765 --> 00:22:32,260 groups we are really thinking usually 291 00:22:32,260 --> 00:22:36,520 in terms of operations that are invertible. 292 00:22:36,520 --> 00:22:39,940 This transformation is not invertible. 293 00:22:39,940 --> 00:22:41,110 But this is a mapping. 294 00:22:41,110 --> 00:22:44,090 So potentially this mapping is invertible. 295 00:22:44,090 --> 00:22:48,220 You can say that if this point came from this point 296 00:22:48,220 --> 00:22:51,781 under inversion, it will go back to the original point, 297 00:22:51,781 --> 00:22:52,703 and so forth. 298 00:22:55,540 --> 00:23:02,740 The next part of the argument is what did we do over here? 299 00:23:02,740 --> 00:23:07,230 We got rid of some short wavelength fluctuations. 300 00:23:07,230 --> 00:23:10,740 Now one of the things that I said right at the beginning 301 00:23:10,740 --> 00:23:14,510 was that as long as you are getting rid of short scale 302 00:23:14,510 --> 00:23:18,720 fluctuations, you are summing over a cube that his 100 303 00:23:18,720 --> 00:23:20,230 square, 200 cube. 304 00:23:20,230 --> 00:23:23,350 It doesn't matter, 100 cube, 200 cube-- 305 00:23:23,350 --> 00:23:27,130 you are doing some analytical function. 306 00:23:27,130 --> 00:23:30,087 So the transformation that relates these 307 00:23:30,087 --> 00:23:34,460 to these, the old to new, should be analytical, 308 00:23:34,460 --> 00:23:39,050 and hence you should be able to write a Taylor series for it. 309 00:23:39,050 --> 00:23:42,690 So let's try to make a Taylor series for this. 310 00:23:42,690 --> 00:23:45,480 Taylor series start with a constant. 311 00:23:45,480 --> 00:23:49,150 But we know that the constant has to be 0 in both cases 312 00:23:49,150 --> 00:23:52,070 because the starting point was the point that 313 00:23:52,070 --> 00:23:56,350 was scale invariant and was mapping onto itself. 314 00:23:56,350 --> 00:23:59,120 So the first thing that I can write down are linear terms. 315 00:23:59,120 --> 00:24:04,870 So there could be a term that is proportional to t. 316 00:24:04,870 --> 00:24:08,262 There could be a term that is proportional to h. 317 00:24:08,262 --> 00:24:13,690 There could be a term here that is proportional to h. 318 00:24:13,690 --> 00:24:16,550 There could be a term that is proportional-- 319 00:24:16,550 --> 00:24:18,880 well, let's call this t. 320 00:24:18,880 --> 00:24:21,810 Let's call this h. 321 00:24:21,810 --> 00:24:23,420 And then there will be terms that 322 00:24:23,420 --> 00:24:26,150 will be order of t squared and higher. 323 00:24:32,710 --> 00:24:36,610 So I just did an analytical expansion, 324 00:24:36,610 --> 00:24:41,670 justified by this summing over just finite degrees of freedom 325 00:24:41,670 --> 00:24:44,990 at short scale. 326 00:24:44,990 --> 00:24:47,440 Now if I have a structure, such as the one that I 327 00:24:47,440 --> 00:24:51,460 have over there, I also know some things 328 00:24:51,460 --> 00:24:53,590 on the basis of symmetry. 329 00:24:53,590 --> 00:24:59,540 Like if I'm on the line that corresponds to h equals to 0, 330 00:24:59,540 --> 00:25:02,410 there is no difference between up and down. 331 00:25:02,410 --> 00:25:04,910 Under rescaling, I still don't know the difference 332 00:25:04,910 --> 00:25:06,620 between up and down. 333 00:25:06,620 --> 00:25:11,810 So I should not generate an h if h was originally 0 334 00:25:11,810 --> 00:25:14,160 just because t deviated from 0. 335 00:25:14,160 --> 00:25:18,610 So by symmetry, that has to be absent. 336 00:25:18,610 --> 00:25:20,740 And similarly, by symmetry, there 337 00:25:20,740 --> 00:25:24,790 is no difference between h positive and h negative. 338 00:25:24,790 --> 00:25:29,310 As far as t is concerned, h and minus h should behave the same. 339 00:25:29,310 --> 00:25:32,530 So this series should start at order of h squared and not h, 340 00:25:32,530 --> 00:25:34,330 so that term should be absent. 341 00:25:38,230 --> 00:25:44,990 So at this level, we have a nice separation into t prime is at 342 00:25:44,990 --> 00:25:45,900 and h Prime. 343 00:25:45,900 --> 00:25:46,470 Is dh. 344 00:25:49,332 --> 00:25:53,080 Now we know something more, which 345 00:25:53,080 --> 00:25:55,930 is that the procedure that we are doing 346 00:25:55,930 --> 00:26:00,360 has some kind of a group character, in that if I, 347 00:26:00,360 --> 00:26:06,180 let's say, originally transform by some factor b1, 348 00:26:06,180 --> 00:26:14,170 change by a factor of 2, then change by a factor of 3, 349 00:26:14,170 --> 00:26:18,580 the answer is equivalent to changing by a factor of 2 350 00:26:18,580 --> 00:26:20,960 times 3, or 3 times 2. 351 00:26:20,960 --> 00:26:23,990 Doesn't matter in which order I do them. 352 00:26:23,990 --> 00:26:28,010 So also, I would get, if I were to do b1 first 353 00:26:28,010 --> 00:26:30,364 and b2 later, it would be the same thing. 354 00:26:33,330 --> 00:26:35,590 So what does that imply? 355 00:26:35,590 --> 00:26:39,930 That if I do two of these transformation, 356 00:26:39,930 --> 00:26:47,930 I find that my new t is obtained in one case by the product, 357 00:26:47,930 --> 00:26:52,848 in the other case by the product of the two a's. 358 00:26:58,116 --> 00:27:02,880 So that's, again, some kind of a group character. 359 00:27:02,880 --> 00:27:08,100 And furthermore, if I don't change the length scale, 360 00:27:08,100 --> 00:27:10,300 everything should stay where it is. 361 00:27:13,500 --> 00:27:16,650 So you glance at those, and you find 362 00:27:16,650 --> 00:27:19,390 that there is only one possibility, 363 00:27:19,390 --> 00:27:23,350 that a as a function of b should be b to some power. 364 00:27:30,090 --> 00:27:35,720 So you know therefore that at the lowest 365 00:27:35,720 --> 00:27:40,670 order under rescaling by a factor of b, 366 00:27:40,670 --> 00:27:43,670 t prime should be b to some y-- I called 367 00:27:43,670 --> 00:27:48,490 it yt-- times t plus higher orders, 368 00:27:48,490 --> 00:27:54,770 while h prime is b to some other power of yh times h plus higher 369 00:27:54,770 --> 00:27:55,270 orders. 370 00:28:00,940 --> 00:28:04,976 And you say, OK, fine. 371 00:28:08,640 --> 00:28:09,730 What's this good for? 372 00:28:12,300 --> 00:28:18,460 Well, let's take a look at what we did over there. 373 00:28:18,460 --> 00:28:22,360 We said that I take some bunch of initial configurations, 374 00:28:22,360 --> 00:28:24,450 sum their weights to get the weight 375 00:28:24,450 --> 00:28:27,530 of the new configuration. 376 00:28:27,530 --> 00:28:31,970 What happens if I sum over all initial configurations? 377 00:28:31,970 --> 00:28:35,860 Well, if I sum over all initial configuration, 378 00:28:35,860 --> 00:28:40,560 I will get the partition function. 379 00:28:45,330 --> 00:28:47,880 Now essentially, all the original 380 00:28:47,880 --> 00:28:52,630 configurations I regrouped and put 381 00:28:52,630 --> 00:29:00,530 into these coarse-grained configurations that 382 00:29:00,530 --> 00:29:01,760 are weighted this way. 383 00:29:04,670 --> 00:29:06,780 So there could be an overall constant 384 00:29:06,780 --> 00:29:09,250 that emerges from this. 385 00:29:09,250 --> 00:29:16,330 But this really implies that the singular part of log z, 386 00:29:16,330 --> 00:29:19,790 and presumably this depends on how far 387 00:29:19,790 --> 00:29:24,660 away I am from the critical point, 388 00:29:24,660 --> 00:29:32,060 is the same as log z that singular after I do this t 389 00:29:32,060 --> 00:29:32,957 prime and h prime. 390 00:29:41,730 --> 00:29:44,073 Now there is one other issue, which is extensivity. 391 00:29:46,620 --> 00:29:49,470 Up to signs, factors of beta, et cetera, 392 00:29:49,470 --> 00:29:55,520 this is b times an intensive free energy, 393 00:29:55,520 --> 00:29:57,260 which is a function of t and h. 394 00:30:01,880 --> 00:30:07,200 So this is the same as v prime, because the volume shrunk. 395 00:30:07,200 --> 00:30:11,610 I took all of my scales and shrunk it by a factor of v, 396 00:30:11,610 --> 00:30:15,717 v prime, f of t prime and h prime. 397 00:30:21,460 --> 00:30:25,080 So now let's go this way. 398 00:30:25,080 --> 00:30:30,300 Note that v prime is the original v divided 399 00:30:30,300 --> 00:30:35,450 by b to the d scaling factor. 400 00:30:35,450 --> 00:30:37,570 So you do the divisions here, and you 401 00:30:37,570 --> 00:30:43,630 find that f as a function of t and h 402 00:30:43,630 --> 00:30:48,736 is the ratio of v prime to v, which is b to the minus d, 403 00:30:48,736 --> 00:30:52,530 f as a function of t prime and h prime. 404 00:30:52,530 --> 00:30:56,720 But t prime we said to lowest order is b to the yt t. 405 00:30:56,720 --> 00:30:59,140 h prime is b to the yh h. 406 00:31:08,100 --> 00:31:10,520 This is actually the more correct form 407 00:31:10,520 --> 00:31:12,250 of writing a homogeneous function. 408 00:31:19,950 --> 00:31:23,390 So previously in last lecture, we 409 00:31:23,390 --> 00:31:29,350 assumed that the free energy had a homogeneous form. 410 00:31:29,350 --> 00:31:32,900 Now subject to these conditions and assumptions 411 00:31:32,900 --> 00:31:35,340 of renormalization group, we have 412 00:31:35,340 --> 00:31:39,850 concluded that it should have that homogeneous form. 413 00:31:39,850 --> 00:31:42,880 Now you say this homogeneous form does not 414 00:31:42,880 --> 00:31:44,770 look like the homogeneous forms that I 415 00:31:44,770 --> 00:31:48,310 had written for you before. 416 00:31:48,310 --> 00:31:50,080 I say, OK. 417 00:31:50,080 --> 00:31:53,350 Presumably this is true for any factor of b 418 00:31:53,350 --> 00:31:56,540 that I want to choose. 419 00:31:56,540 --> 00:32:09,820 Let me choose a b, a rescaling factor such that v to the yt t 420 00:32:09,820 --> 00:32:11,940 is of the order of 1. 421 00:32:11,940 --> 00:32:13,320 Could be 1, could be pi. 422 00:32:13,320 --> 00:32:14,020 I don't care. 423 00:32:16,660 --> 00:32:19,050 Which means that I chose a factor 424 00:32:19,050 --> 00:32:24,816 of b that will scale with t as t to the minus 1 over yt. 425 00:32:28,008 --> 00:32:34,540 I put this b-- this expression is true for all choices of b. 426 00:32:34,540 --> 00:32:39,750 If I chose that particular value, what I get 427 00:32:39,750 --> 00:32:46,570 is t to the d over yt, some function. 428 00:32:46,570 --> 00:32:50,080 First argument has now become 1 or some constant. 429 00:32:50,080 --> 00:32:53,820 Really it only depends on the second argument 430 00:32:53,820 --> 00:33:00,520 in the combination h and t to the power of yh over yt. 431 00:33:06,200 --> 00:33:10,090 So you can see that this is, in fact, 432 00:33:10,090 --> 00:33:13,650 the same as the first line that I have above. 433 00:33:13,650 --> 00:33:17,730 And I have identified that 2 minus alpha 434 00:33:17,730 --> 00:33:22,130 would be related to this factor of yt, which 435 00:33:22,130 --> 00:33:27,230 is how you would scale under renormalization, 436 00:33:27,230 --> 00:33:30,700 the parameters t and h. 437 00:33:30,700 --> 00:33:34,935 And the gap exponent is related to the ratio of yh over yt. 438 00:33:45,350 --> 00:33:49,657 Similarly, we had that the correlation length-- 439 00:33:49,657 --> 00:33:50,490 I have a line there. 440 00:33:50,490 --> 00:33:56,860 Psi of t and h is b psi of t prime and h prime. 441 00:33:56,860 --> 00:34:02,706 So I have that psi as a function of t and h 442 00:34:02,706 --> 00:34:10,750 is b times psi as a function of b to the yt t, b to the yh h. 443 00:34:10,750 --> 00:34:16,520 So that's also correct. 444 00:34:16,520 --> 00:34:24,389 I can again choose this value of v, substitute it over there. 445 00:34:24,389 --> 00:34:25,699 What do I get? 446 00:34:25,699 --> 00:34:29,239 I get that psi as a function of t and h 447 00:34:29,239 --> 00:34:34,159 would be t to the minus 1 over yt, 448 00:34:34,159 --> 00:34:40,139 some scaling function-- let's call it g psi-- of, again, h 449 00:34:40,139 --> 00:34:41,810 to the power of yh over yt. 450 00:34:48,969 --> 00:34:56,100 So I have got an answer that nu should be 1 over yt. 451 00:34:56,100 --> 00:35:00,800 I can get the scaling form for the correlation length. 452 00:35:00,800 --> 00:35:04,170 I identify the divergence of correlation length 453 00:35:04,170 --> 00:35:05,680 with inverse of this. 454 00:35:05,680 --> 00:35:10,140 And by the way, I get, if I substitute nu as 1 over yt 455 00:35:10,140 --> 00:35:14,260 here, the Josephson hyperscale in relation to minus alpha 456 00:35:14,260 --> 00:35:14,860 equals to b. 457 00:35:27,540 --> 00:35:31,040 I can go further if I want. 458 00:35:31,040 --> 00:35:36,440 I can calculate magnetization as a function of t and h, 459 00:35:36,440 --> 00:35:38,880 would correspond to basically the behaviors 460 00:35:38,880 --> 00:35:46,710 that we identify with exponents beta or delta as d log z. 461 00:35:55,074 --> 00:36:01,500 Yeah, let's say f by dh. 462 00:36:01,500 --> 00:36:05,180 If I take a derivative over there, 463 00:36:05,180 --> 00:36:07,000 you can immediately see that what 464 00:36:07,000 --> 00:36:13,510 that gives me is b to the power of yh minus d, and then 465 00:36:13,510 --> 00:36:16,800 some scaling function which is the derivative of this scaling 466 00:36:16,800 --> 00:36:22,510 function b the yt t, b to the yh h. 467 00:36:25,030 --> 00:36:29,090 And again, if I make this choice, 468 00:36:29,090 --> 00:36:39,980 then this goes over to t to the power of d minus yh over yt, 469 00:36:39,980 --> 00:36:44,755 and then some scaling function of h t to the delta. 470 00:36:47,790 --> 00:36:50,240 So I can continue with my table. 471 00:36:50,240 --> 00:36:52,833 And for example, I will have beta 472 00:36:52,833 --> 00:36:58,980 to be d minus yh divided by yt. 473 00:36:58,980 --> 00:37:01,420 I can go and calculate delta, et cetera. 474 00:37:06,700 --> 00:37:11,730 Actually I was a little bit careless 475 00:37:11,730 --> 00:37:18,770 with this factor zeta, which presumably 476 00:37:18,770 --> 00:37:24,270 is implicit in all of these transformations that I have. 477 00:37:24,270 --> 00:37:26,480 And I have to do special things to figure out 478 00:37:26,480 --> 00:37:30,180 what zeta is so that I will get self-similarity right 479 00:37:30,180 --> 00:37:32,780 at the critical point. 480 00:37:32,780 --> 00:37:36,202 But we can see that already we have 481 00:37:36,202 --> 00:37:41,970 the analog of a rescaling for m. 482 00:37:41,970 --> 00:37:47,530 And so it is easy to sort of look at those two equations 483 00:37:47,530 --> 00:37:54,420 and identify that my zeta should be precisely this one. 484 00:37:54,420 --> 00:38:01,560 So the zeta is not independent of the relevance 485 00:38:01,560 --> 00:38:04,220 of the magnetic field. 486 00:38:04,220 --> 00:38:09,850 And if you think about it, the field and the magnetization 487 00:38:09,850 --> 00:38:14,710 are conjugate variables in the sense that in the weight here, 488 00:38:14,710 --> 00:38:20,360 I will have a term that is like hm-- integrated, of course. 489 00:38:20,360 --> 00:38:23,150 And so hm integrated, you can see 490 00:38:23,150 --> 00:38:27,830 that up to a factor of b to the d from integration, 491 00:38:27,830 --> 00:38:31,930 the dimensionality that I assign to h and the dimensionality 492 00:38:31,930 --> 00:38:37,700 that I assign to m should be related. 493 00:38:37,700 --> 00:38:40,770 And not only for the magnetization, 494 00:38:40,770 --> 00:38:45,145 but for any pair of variables that are so conjugate-- 495 00:38:45,145 --> 00:38:47,970 there's some f, and there's some x-- 496 00:38:47,970 --> 00:38:51,760 there will be a corresponding relation between what 497 00:38:51,760 --> 00:38:58,740 would happen to this x at the critical point and this factor 498 00:38:58,740 --> 00:39:01,501 f when I deviate from the critical point. 499 00:39:11,730 --> 00:39:17,800 So all of this is kind of nice, but it's 500 00:39:17,800 --> 00:39:20,310 a little bit hand waving. 501 00:39:20,310 --> 00:39:30,450 I essentially traded one set of assumptions about homogeneity 502 00:39:30,450 --> 00:39:34,345 and scaling of free energy correlation length 503 00:39:34,345 --> 00:39:40,380 to some other set of assumptions about two parameters moving 504 00:39:40,380 --> 00:39:45,040 away from a scale invariant critical point. 505 00:39:45,040 --> 00:39:47,990 I didn't calculate anything about what the scale invariant 506 00:39:47,990 --> 00:39:49,640 probability is. 507 00:39:49,640 --> 00:39:53,830 I didn't show that, indeed, two parameters are sufficient, 508 00:39:53,830 --> 00:39:57,740 that this kind of scaling takes place, et cetera. 509 00:39:57,740 --> 00:40:01,750 So we need to be much more precise 510 00:40:01,750 --> 00:40:05,740 if we want to do, ultimately, calculations that give us 511 00:40:05,740 --> 00:40:09,530 what these numbers yt and yh are. 512 00:40:09,530 --> 00:40:13,770 So let's try to put this hand waving 513 00:40:13,770 --> 00:40:15,720 on a little bit more firm setting. 514 00:40:20,220 --> 00:40:24,600 So let's see how we should proceed. 515 00:40:27,620 --> 00:40:37,040 We start with some experimental system, critical point. 516 00:40:40,410 --> 00:40:46,420 So I tell you that somebody in the experiment, the liquid gas 517 00:40:46,420 --> 00:40:51,000 system, they saw a diverging correlation length, 518 00:40:51,000 --> 00:40:53,600 critical opalescence, et cetera. 519 00:40:53,600 --> 00:40:59,920 So then I associate with that some kind 520 00:40:59,920 --> 00:41:02,390 of a statistical field. 521 00:41:07,080 --> 00:41:11,160 And let's kind of stick with the notation 522 00:41:11,160 --> 00:41:12,520 that we have for the magnet. 523 00:41:12,520 --> 00:41:14,622 Let's call it m of x. 524 00:41:14,622 --> 00:41:18,800 And in general, this would be the part 525 00:41:18,800 --> 00:41:21,900 where one needs to put in a lot of thinking. 526 00:41:21,900 --> 00:41:24,830 That is, the experimentalist comes and tells you 527 00:41:24,830 --> 00:41:29,260 that I see a system that undergoes a phase transition. 528 00:41:29,260 --> 00:41:30,790 There are some response functions 529 00:41:30,790 --> 00:41:32,970 that are divergent, et cetera. 530 00:41:32,970 --> 00:41:36,110 You have to put in some thinking to think about what 531 00:41:36,110 --> 00:41:39,600 the appropriate order parameter is. 532 00:41:39,600 --> 00:41:47,510 And based on that order parameter or statistical field, 533 00:41:47,510 --> 00:42:00,610 you construct the most general weight consistent 534 00:42:00,610 --> 00:42:06,840 with symmetries, with not only asymmetries 535 00:42:06,840 --> 00:42:12,300 but the kind of assumptions that we have been putting in play. 536 00:42:12,300 --> 00:42:18,790 So we put in assumptions about locality, symmetry. 537 00:42:22,190 --> 00:42:24,040 Stability is, of course, paramount. 538 00:42:24,040 --> 00:42:28,120 But there is a list of things that you have to think about. 539 00:42:28,120 --> 00:42:32,770 So once you do that you say, OK, I 540 00:42:32,770 --> 00:42:38,602 associate with my configurations m of x some set 541 00:42:38,602 --> 00:42:39,310 of probabilities. 542 00:42:43,650 --> 00:42:46,390 Probabilities are certainly positive. 543 00:42:46,390 --> 00:42:50,830 So I can take its log, call its minus its log to be some kind 544 00:42:50,830 --> 00:42:56,325 of a weight, beta h, that governs these m of x's. 545 00:42:59,440 --> 00:43:03,010 If I say that I'm obeying locality, 546 00:43:03,010 --> 00:43:05,820 then I would write the answer, for example, like this. 547 00:43:05,820 --> 00:43:07,830 But it doesn't have to be. 548 00:43:07,830 --> 00:43:11,010 I have to write some particular example. 549 00:43:11,010 --> 00:43:13,530 But you may construct your example 550 00:43:13,530 --> 00:43:16,060 depending on the system of interest. 551 00:43:16,060 --> 00:43:19,750 And let's say we are looking at something like a superfluid, 552 00:43:19,750 --> 00:43:23,210 maybe, that we don't even have the analog of magnetic field, 553 00:43:23,210 --> 00:43:25,060 and we go and construct terms that 554 00:43:25,060 --> 00:43:29,880 are symmetric and made for a two component m. 555 00:43:29,880 --> 00:43:32,470 And I will write a few of these terms 556 00:43:32,470 --> 00:43:36,590 to emphasize that this is, in principle, a long list. 557 00:43:36,590 --> 00:43:40,290 There's coefficient of m to the sixth. 558 00:43:40,290 --> 00:43:44,440 We saw that the gradient terms could start with this k. 559 00:43:44,440 --> 00:43:47,670 But maybe there's a higher order gradient, 560 00:43:47,670 --> 00:43:53,080 and there's essentially and infinity of terms 561 00:43:53,080 --> 00:43:56,920 that you can write down that are consistent with these 562 00:43:56,920 --> 00:43:59,064 assumptions that you have made so far. 563 00:44:03,020 --> 00:44:05,940 So you say, OK. 564 00:44:05,940 --> 00:44:12,080 Now I take this, and implicit in all of these calculations 565 00:44:12,080 --> 00:44:16,470 is, indeed, some kind of a short scale cutoff. 566 00:44:16,470 --> 00:44:19,770 To construct the statistical field, 567 00:44:19,770 --> 00:44:29,440 I do apply the three steps of RG-- renormalization group, 568 00:44:29,440 --> 00:44:31,660 as I described before. 569 00:44:31,660 --> 00:44:38,130 And this will give me a new configuration 570 00:44:38,130 --> 00:44:40,570 for each of the old configurations 571 00:44:40,570 --> 00:44:43,205 through the formula that I gave you over there. 572 00:44:55,500 --> 00:44:59,740 So in principle, this is just a transformation from one 573 00:44:59,740 --> 00:45:03,630 set of variables to a new set of variables. 574 00:45:03,630 --> 00:45:05,810 So if I do this transformation, I 575 00:45:05,810 --> 00:45:13,206 can calculate the weight of the new configurations, 576 00:45:13,206 --> 00:45:14,930 m prime of x prime. 577 00:45:18,350 --> 00:45:20,830 I can take minus the log of that. 578 00:45:23,420 --> 00:45:27,510 And again, up to some constant, it 579 00:45:27,510 --> 00:45:30,090 will be the same as a probability. 580 00:45:30,090 --> 00:45:32,010 So there could be, in this procedure, 581 00:45:32,010 --> 00:45:34,780 some set of constants that are generated 582 00:45:34,780 --> 00:45:36,730 that don't depend on m. 583 00:45:39,800 --> 00:45:41,370 And then there will be a function 584 00:45:41,370 --> 00:45:45,700 that depends on m prime of x prime. 585 00:45:45,700 --> 00:45:48,220 Now the statement is that since I 586 00:45:48,220 --> 00:45:57,090 wrote the most general function over here, whatever I put here 587 00:45:57,090 --> 00:46:01,340 will have to have exactly the same form, because I said 588 00:46:01,340 --> 00:46:03,420 put anything that you can think of that 589 00:46:03,420 --> 00:46:06,120 is consistent with symmetries over here. 590 00:46:06,120 --> 00:46:08,240 So you put everything there. 591 00:46:08,240 --> 00:46:12,120 What I put here should have exactly the same functional 592 00:46:12,120 --> 00:46:15,080 form, but with coefficients that have changed. 593 00:46:28,550 --> 00:46:32,120 So you basically prime everything, 594 00:46:32,120 --> 00:46:36,380 but you have this whole thing. 595 00:46:36,380 --> 00:46:40,050 Now this may seem like truly difficult thing. 596 00:46:40,050 --> 00:46:42,170 But we will actually do this. 597 00:46:42,170 --> 00:46:46,410 We will carry out this transformation explicitly 598 00:46:46,410 --> 00:46:48,450 in particular cases. 599 00:46:48,450 --> 00:46:51,920 And we will show that this transformation amounts 600 00:46:51,920 --> 00:46:56,240 to constructing a rescaling of each one of these parameters-- 601 00:46:56,240 --> 00:47:05,010 t prime, u prime, v prime, k prime, l prime, and so forth-- 602 00:47:05,010 --> 00:47:07,987 as functions of the old parameters. 603 00:47:24,590 --> 00:47:28,730 So this is, if you like, a mapping. 604 00:47:32,360 --> 00:47:41,040 You take some set of parameters-- t, u, v, k, l, 605 00:47:41,040 --> 00:47:45,270 blah, blah, blah-- and you construct a mapping, 606 00:47:45,270 --> 00:47:51,877 s prime, which is some function of the original set 607 00:47:51,877 --> 00:47:52,460 of parameters. 608 00:47:56,090 --> 00:47:59,840 So this is a huge dimensional space. 609 00:47:59,840 --> 00:48:02,960 Any points that you start on the transformation 610 00:48:02,960 --> 00:48:05,210 will go to another point. 611 00:48:05,210 --> 00:48:08,750 But the key is that we wrote the most general form 612 00:48:08,750 --> 00:48:13,406 that we could, so we had to stay within this space. 613 00:48:22,770 --> 00:48:26,430 So why are you doing this? 614 00:48:26,430 --> 00:48:33,170 Well, I started by saying that the key to this whole thing 615 00:48:33,170 --> 00:48:39,710 is have to having a handle as to what this self-similar scale 616 00:48:39,710 --> 00:48:43,210 invariant probability is. 617 00:48:43,210 --> 00:48:47,470 I can't construct that just by guessing. 618 00:48:47,470 --> 00:48:49,750 But I can do what we usually do, let's say, 619 00:48:49,750 --> 00:48:53,070 in constructing wave functions in quantum mechanics that 620 00:48:53,070 --> 00:48:55,410 have some particular symmetry. 621 00:48:55,410 --> 00:48:57,880 Maybe you start with some wave function that 622 00:48:57,880 --> 00:48:59,830 doesn't have the full symmetry, and then 623 00:48:59,830 --> 00:49:02,050 you rotate it and rotate it again, 624 00:49:02,050 --> 00:49:03,880 and you average over all of them, 625 00:49:03,880 --> 00:49:05,710 and you end up with some function that 626 00:49:05,710 --> 00:49:07,830 has the right symmetry. 627 00:49:07,830 --> 00:49:11,940 So we start with a weight that I don't 628 00:49:11,940 --> 00:49:16,930 know whether it has the property that I want. 629 00:49:16,930 --> 00:49:19,530 And I apply the action of the group, which 630 00:49:19,530 --> 00:49:22,020 is this scaling variance, to see what 631 00:49:22,020 --> 00:49:23,780 happens to it under that transformation. 632 00:49:26,290 --> 00:49:29,060 But the point that I am interested, 633 00:49:29,060 --> 00:49:31,940 or the behavior that I am interested, 634 00:49:31,940 --> 00:49:37,030 is where I basically get the same probability back. 635 00:49:37,030 --> 00:49:41,250 So I'm very interested at the point where, 636 00:49:41,250 --> 00:49:45,770 under the transformation, I go back to myself. 637 00:49:45,770 --> 00:49:47,360 And that's called a fixed point. 638 00:49:52,670 --> 00:49:59,530 So S is a shorthand for this infinite vector of parameters. 639 00:49:59,530 --> 00:50:05,200 I want to find the point s star in this parameter space. 640 00:50:05,200 --> 00:50:10,051 Actually, let me call this transformation R 641 00:50:10,051 --> 00:50:14,990 and indicate that I'm renormalizing by a scale b, 642 00:50:14,990 --> 00:50:19,760 such that, when I renormalize by a scale 643 00:50:19,760 --> 00:50:23,390 b, my original set of parameters, 644 00:50:23,390 --> 00:50:29,440 if I am at this fixed point, I will end up at that point. 645 00:50:29,440 --> 00:50:32,790 So clearly, this is a system that 646 00:50:32,790 --> 00:50:38,160 has exactly these properties that I was harping in 647 00:50:38,160 --> 00:50:39,150 at the beginning. 648 00:50:39,150 --> 00:50:42,440 This is the point that is truly scale invariant. 649 00:50:42,440 --> 00:50:47,000 That's the point that I want to get at. 650 00:50:47,000 --> 00:50:49,436 So again, once we have done this transformation 651 00:50:49,436 --> 00:50:55,340 in a specific case, we'll figure out what this fixed point is. 652 00:50:55,340 --> 00:51:00,680 But for the time being, let's think a little bit away 653 00:51:00,680 --> 00:51:11,500 from this and deviate from fixed point. 654 00:51:17,460 --> 00:51:22,530 So I start with an initial point S 655 00:51:22,530 --> 00:51:30,690 that is, let's write it, S star plus a little bit away. 656 00:51:30,690 --> 00:51:32,820 Just like in the picture that I have 657 00:51:32,820 --> 00:51:34,450 here, I started with a fixed point, 658 00:51:34,450 --> 00:51:36,330 and I said I go away by an amount 659 00:51:36,330 --> 00:51:39,360 that I had parameterized by t and h. 660 00:51:39,360 --> 00:51:42,700 Now I have essentially a whole line 661 00:51:42,700 --> 00:51:46,070 of deviations forming a vector. 662 00:51:46,070 --> 00:51:55,710 I act with Rb on this, and I note that if delta S goes to 0, 663 00:51:55,710 --> 00:51:57,700 then I should go back to S star. 664 00:52:00,880 --> 00:52:08,460 But if delta S is small, maybe I can 665 00:52:08,460 --> 00:52:14,000 look at the delta S prime, which is 666 00:52:14,000 --> 00:52:16,800 a linearized version of these transformations. 667 00:52:16,800 --> 00:52:19,770 So basically these transformations 668 00:52:19,770 --> 00:52:26,140 are highly nonlinear just as the transformation over here, 669 00:52:26,140 --> 00:52:29,680 in principle, would have been highly nonlinear. 670 00:52:29,680 --> 00:52:34,940 But then I expanded it around the point t and h equals to 0. 671 00:52:34,940 --> 00:52:39,870 Similarly, I'm assuming that this delta S is small, 672 00:52:39,870 --> 00:52:47,300 and therefore delta S prime can be related to delta S 673 00:52:47,300 --> 00:52:52,772 through the action of a matrix that is a linearized version. 674 00:52:52,772 --> 00:52:54,770 Let's call it here RL of b. 675 00:52:54,770 --> 00:53:06,630 So this is a linearized transformation, 676 00:53:06,630 --> 00:53:08,544 which means that it's really a matrix. 677 00:53:12,260 --> 00:53:16,730 In this particular case, in principle, I started with a 2 678 00:53:16,730 --> 00:53:18,150 by 2 matrix. 679 00:53:18,150 --> 00:53:20,570 The off diagonal terms were 0, so it 680 00:53:20,570 --> 00:53:23,360 was only the diagonal terms that mattered. 681 00:53:23,360 --> 00:53:26,600 But in general, it would be a matrix, 682 00:53:26,600 --> 00:53:28,610 which would be the square of whatever 683 00:53:28,610 --> 00:53:31,632 the size of the parameter space is that I am looking at. 684 00:53:36,350 --> 00:53:41,590 Now then you have a matrix, it's good 685 00:53:41,590 --> 00:53:46,280 always to think in terms of its eigenvalues and eigendirection. 686 00:53:46,280 --> 00:53:48,620 In this problem that I had over here, 687 00:53:48,620 --> 00:53:51,540 symmetries had already diagonalized the matrix. 688 00:53:51,540 --> 00:53:53,730 I didn't have off diagonal terms. 689 00:53:53,730 --> 00:53:54,860 But I don't know here. 690 00:53:54,860 --> 00:53:57,880 It could be all kinds of off diagonal terms. 691 00:53:57,880 --> 00:54:08,740 So the properties are captured by diagonalize, RL, 692 00:54:08,740 --> 00:54:18,120 which means that I find a set of vectors in this space-- 693 00:54:18,120 --> 00:54:23,680 let's call them Oi-- such that under action of this, 694 00:54:23,680 --> 00:54:29,070 I will get lambda Oi, lambda i. 695 00:54:29,070 --> 00:54:31,770 Of course, the transformation depends on the rescaling 696 00:54:31,770 --> 00:54:34,856 parameter, so there should be a b here. 697 00:54:39,620 --> 00:54:42,600 Now of course, you will get a totally different matrix 698 00:54:42,600 --> 00:54:45,310 for each b. 699 00:54:45,310 --> 00:54:48,790 So is it really hopeless that for each b I have to look 700 00:54:48,790 --> 00:54:54,700 at a new matrix, new diagonalization, et cetera? 701 00:54:54,700 --> 00:54:58,270 Well, exactly this thing that we had over here 702 00:54:58,270 --> 00:55:01,230 now comes into play, because I know 703 00:55:01,230 --> 00:55:06,980 that if I make a transformation size b1 followed 704 00:55:06,980 --> 00:55:11,880 by a transformation size b2, the answer is a transformation 705 00:55:11,880 --> 00:55:13,750 size b1, b2. 706 00:55:13,750 --> 00:55:16,290 And it doesn't matter in which order I do it. 707 00:55:20,138 --> 00:55:23,790 AUDIENCE: Can't you just mix notation, because L used to be 708 00:55:23,790 --> 00:55:24,337 [INAUDIBLE]? 709 00:55:24,337 --> 00:55:25,045 PROFESSOR: Sorry. 710 00:55:46,200 --> 00:55:54,800 So in particular, I see that these linearized 711 00:55:54,800 --> 00:55:59,500 matrices commute with each other for different values of b. 712 00:55:59,500 --> 00:56:01,590 And again, from your quantum mechanics, 713 00:56:01,590 --> 00:56:05,410 you probably know that if matrices commute then 714 00:56:05,410 --> 00:56:08,880 they have the same eigenvectors. 715 00:56:08,880 --> 00:56:13,295 So essentially, I was correct here in putting no index b 716 00:56:13,295 --> 00:56:15,660 on these eigenvectors, because it's 717 00:56:15,660 --> 00:56:18,750 independent of eigenvector, whereas the eigenvalues, 718 00:56:18,750 --> 00:56:21,964 in principle, depend on b. 719 00:56:21,964 --> 00:56:25,000 And how they depend on b is also determined 720 00:56:25,000 --> 00:56:32,500 by this transformation, that is lambda i of b1, lambda i of b2 721 00:56:32,500 --> 00:56:35,880 should be the same thing as lambda b1, b2. 722 00:56:39,610 --> 00:56:44,490 And of course, lambda i of 1 should be 1. 723 00:56:44,490 --> 00:56:48,065 If you don't change scale, nothing should change. 724 00:56:48,065 --> 00:56:51,190 And this is exactly the same set of conditions 725 00:56:51,190 --> 00:56:54,110 as we have over here, which means 726 00:56:54,110 --> 00:56:59,875 that we know that the eigenvalue's lambda i can 727 00:56:59,875 --> 00:57:04,460 be written as b to the power of some set of yi. 728 00:57:08,330 --> 00:57:11,970 So we just generalized what we had done before, 729 00:57:11,970 --> 00:57:15,430 now to this space that includes many parameters. 730 00:57:24,940 --> 00:57:28,510 So the story is now something like this. 731 00:57:28,510 --> 00:57:32,090 There is this multi-dimensional space 732 00:57:32,090 --> 00:57:37,730 with lots and lots of parameters-- 733 00:57:37,730 --> 00:57:41,710 t, u, v, blah, blah, blah, many of them. 734 00:57:41,710 --> 00:57:46,095 And somewhere in this space of parameters, presumably 735 00:57:46,095 --> 00:57:49,460 there is a fixed point, S star. 736 00:57:55,110 --> 00:57:58,810 Now in the vicinity of that S star, 737 00:57:58,810 --> 00:58:03,710 I have established that there are some particular directions 738 00:58:03,710 --> 00:58:07,015 that I can obtain by diagonalizing this. 739 00:58:07,015 --> 00:58:10,090 So let's imagine that this is one direction, 740 00:58:10,090 --> 00:58:13,460 this is another direction, this is a third direction. 741 00:58:16,630 --> 00:58:24,480 And that if I start with a beta h-- well, 742 00:58:24,480 --> 00:58:26,085 actually, let's do this. 743 00:58:26,085 --> 00:58:34,430 That is, if I start with an S that is S star plus whatever 744 00:58:34,430 --> 00:58:38,640 is a projection of my components are 745 00:58:38,640 --> 00:58:42,480 along these different dimensions, so let's call them, 746 00:58:42,480 --> 00:58:47,480 let's say, ai along these Oi hat-- just 747 00:58:47,480 --> 00:58:51,940 make sure we kind of think of them as vectors-- 748 00:58:51,940 --> 00:58:55,520 that under rescaling, I will go to S 749 00:58:55,520 --> 00:59:03,560 prime, which is S star plus sum over i, ai b to the yi Oi. 750 00:59:08,240 --> 00:59:12,140 That is, some of these directions, 751 00:59:12,140 --> 00:59:15,215 the component will get stretched if yi is positive. 752 00:59:15,215 --> 00:59:19,980 It will get diminished if yi is negative. 753 00:59:19,980 --> 00:59:25,480 And so now some notation comes into play. 754 00:59:51,970 --> 00:59:58,570 If yi is positive, the corresponding direction 755 00:59:58,570 --> 00:59:59,804 is called relevant. 756 01:00:04,890 --> 01:00:15,115 Eigendirection is relevant. 757 01:00:20,190 --> 01:00:27,390 If yi is negative, the corresponding eigendirection 758 01:00:27,390 --> 01:00:28,362 is irrelevant. 759 01:00:34,270 --> 01:00:41,770 And very occasionally, we may run into the case 760 01:00:41,770 --> 01:00:43,940 where yi is 0. 761 01:00:43,940 --> 01:00:45,590 And there is a terminology. 762 01:00:45,590 --> 01:00:48,680 The corresponding eigendirection is marginal. 763 01:00:51,980 --> 01:00:57,250 And what that means is that I need to resort to higher order 764 01:00:57,250 --> 01:01:00,840 terms to see whether it is attracted or repelled 765 01:01:00,840 --> 01:01:02,586 by the fixed point. 766 01:01:02,586 --> 01:01:04,580 So we need higher orders. 767 01:01:07,180 --> 01:01:11,105 After all, so far I have only linearized the transformation. 768 01:01:18,720 --> 01:01:42,967 Now the set of irrelevant directions to this particular S 769 01:01:42,967 --> 01:01:51,426 fixed point, S star, defines basing of attraction of S star. 770 01:02:06,010 --> 01:02:11,030 So let me go back to the picture that I have over here 771 01:02:11,030 --> 01:02:18,640 and be precise and use the arrow going away as an indication 772 01:02:18,640 --> 01:02:21,810 that the corresponding b is positive, 773 01:02:21,810 --> 01:02:25,330 and I'm forced out along this direction. 774 01:02:25,330 --> 01:02:31,640 Let me choose going in as an indicator 775 01:02:31,640 --> 01:02:34,920 that the corresponding y is negative. 776 01:02:34,920 --> 01:02:38,260 And as I make b larger and larger, 777 01:02:38,260 --> 01:02:41,230 I shrink along this axis. 778 01:02:41,230 --> 01:02:44,190 So in this three dimensional representation 779 01:02:44,190 --> 01:02:49,200 that I have over there, I have one relevant direction and two 780 01:02:49,200 --> 01:02:51,180 irrelevant directions. 781 01:02:51,180 --> 01:02:55,730 The two irrelevant directions will define the plane 782 01:02:55,730 --> 01:02:58,810 in this three dimensional space, which 783 01:02:58,810 --> 01:03:02,070 is the basing of attraction. 784 01:03:02,070 --> 01:03:09,460 So basically these two define a surface, 785 01:03:09,460 --> 01:03:15,400 and presumably any point that is in this surface in the three 786 01:03:15,400 --> 01:03:18,910 dimensional picture under looking 787 01:03:18,910 --> 01:03:21,960 at larger and larger things will get 788 01:03:21,960 --> 01:03:23,940 attracted to the fixed point. 789 01:03:23,940 --> 01:03:25,760 If you are away from the surface, 790 01:03:25,760 --> 01:03:27,820 maybe you will approach here, and then you 791 01:03:27,820 --> 01:03:28,830 will be pushed out. 792 01:03:32,910 --> 01:03:36,680 All right, fine. 793 01:03:36,680 --> 01:03:40,150 Now let's go and look at the following. 794 01:03:40,150 --> 01:03:44,300 We have a formula, psi of t and h. 795 01:03:44,300 --> 01:03:47,740 Or quite generally, psi under rescaling 796 01:03:47,740 --> 01:03:52,190 is b times the new psi. 797 01:03:52,190 --> 01:03:56,200 Or the new psi under any one of these transformation, psi 798 01:03:56,200 --> 01:04:01,500 prime, is the old psi divided by the old correlation 799 01:04:01,500 --> 01:04:03,605 length divided by a factor of b. 800 01:04:07,400 --> 01:04:11,100 So if I look at the fixed point-- 801 01:04:11,100 --> 01:04:16,140 so if I ask what is psi at the fixed point-- 802 01:04:16,140 --> 01:04:22,890 then under the transformation, I have the same parameters. 803 01:04:22,890 --> 01:04:24,660 So psi at the fixed point should be 804 01:04:24,660 --> 01:04:27,990 the psi of the fixed point divided by b. 805 01:04:27,990 --> 01:04:30,870 There are only two solutions to this. 806 01:04:30,870 --> 01:04:38,896 Either psi of S star is 0 or psi of S star is infinite. 807 01:04:45,530 --> 01:04:48,020 Now we introduce physics. 808 01:04:48,020 --> 01:04:51,860 Psi being 0 means that I have units 809 01:04:51,860 --> 01:04:54,860 that are completely uncorrelated to each other. 810 01:04:54,860 --> 01:04:58,010 Each one of them does whatever it wants. 811 01:04:58,010 --> 01:05:01,750 So this describes essentially, let's say, 812 01:05:01,750 --> 01:05:03,610 a system of infinite temperature. 813 01:05:03,610 --> 01:05:06,095 Every degree of freedom does whatever it wants. 814 01:05:10,520 --> 01:05:18,310 Well, I should say this corresponds to disordered 815 01:05:18,310 --> 01:05:22,390 or ordered phases. 816 01:05:25,420 --> 01:05:29,070 Because after all, we said that when 817 01:05:29,070 --> 01:05:32,210 we go to the ordered states also, 818 01:05:32,210 --> 01:05:34,590 there is an overall magnetization, 819 01:05:34,590 --> 01:05:38,550 but fluctuations around the overall magnetization 820 01:05:38,550 --> 01:05:41,520 have only a finite correlation length. 821 01:05:41,520 --> 01:05:44,780 And as you go further and further into the ordered phase, 822 01:05:44,780 --> 01:05:47,510 that correlation length shrinks to 0. 823 01:05:47,510 --> 01:05:50,400 So there is a similarity between what goes on 824 01:05:50,400 --> 01:05:52,730 at very high temperature and what 825 01:05:52,730 --> 01:05:55,050 goes on at very low temperature as 826 01:05:55,050 --> 01:05:58,340 far as the correlation of fluctuations is concerned. 827 01:05:58,340 --> 01:06:00,290 There is, of course, a long range order 828 01:06:00,290 --> 01:06:02,880 in one case that is absent in the other. 829 01:06:02,880 --> 01:06:05,630 But the correlation of fluctuations 830 01:06:05,630 --> 01:06:08,650 in both of those cases basically becomes 831 01:06:08,650 --> 01:06:13,920 finite, and under rescaling, goes all the way to 0. 832 01:06:13,920 --> 01:06:17,000 And clearly this is the interesting case, 833 01:06:17,000 --> 01:06:19,615 where it corresponds to critical point. 834 01:06:28,090 --> 01:06:37,150 So we've established that, once we found this fixed point, 835 01:06:37,150 --> 01:06:44,180 that those set of parameters are what can give us the scale 836 01:06:44,180 --> 01:06:45,960 invariant behavioral that we want. 837 01:06:49,580 --> 01:06:56,040 Now this list is hundreds of parameters. 838 01:06:56,040 --> 01:07:00,140 So this point corresponds to a very special point 839 01:07:00,140 --> 01:07:04,150 in this hundreds of parameter space. 840 01:07:04,150 --> 01:07:06,660 So let's say there is one point somewhere there which 841 01:07:06,660 --> 01:07:08,980 is the fixed point. 842 01:07:08,980 --> 01:07:11,170 And then you take your magnet and you 843 01:07:11,170 --> 01:07:13,630 change your critical temperature, 844 01:07:13,630 --> 01:07:15,710 are we going to hit that point? 845 01:07:15,710 --> 01:07:16,880 The answer is, no. 846 01:07:16,880 --> 01:07:20,000 Generically, you are not going to hit that point. 847 01:07:20,000 --> 01:07:22,010 But that's no problem. 848 01:07:22,010 --> 01:07:22,760 Why? 849 01:07:22,760 --> 01:07:24,395 Because if this basing of attraction. 850 01:07:27,340 --> 01:07:44,410 Because for any point on basing of attraction, I do rescaling, 851 01:07:44,410 --> 01:07:48,665 and I find that psi prime is psi over b. 852 01:07:48,665 --> 01:07:50,140 It becomes smaller. 853 01:07:50,140 --> 01:07:54,960 So you generically tend to become smaller. 854 01:07:54,960 --> 01:07:57,140 But ultimately, you end up at this point. 855 01:07:57,140 --> 01:08:01,730 And this point, the correlation length is infinite. 856 01:08:01,730 --> 01:08:06,540 So any point on this basing of attraction, in fact, 857 01:08:06,540 --> 01:08:09,840 has infinite correlation length. 858 01:08:09,840 --> 01:08:15,950 So every point on the basis of psi prime equals to psi, 859 01:08:15,950 --> 01:08:20,680 and hence psi has to be infinite. 860 01:08:25,670 --> 01:08:26,423 Yes. 861 01:08:26,423 --> 01:08:27,370 AUDIENCE: Question. 862 01:08:27,370 --> 01:08:31,590 Why should there be only one fixed point? 863 01:08:31,590 --> 01:08:32,906 PROFESSOR: There is no reason. 864 01:08:32,906 --> 01:08:33,899 AUDIENCE: OK. 865 01:08:33,899 --> 01:08:35,123 So this is just an example? 866 01:08:35,123 --> 01:08:35,789 PROFESSOR: Yeah. 867 01:08:35,789 --> 01:08:39,310 So locally, let's say that we found such a fixed point. 868 01:08:39,310 --> 01:08:41,819 Maybe globally, there is hundreds of them. 869 01:08:41,819 --> 01:08:42,990 I don't know. 870 01:08:42,990 --> 01:08:45,815 So that will always be a question in our minds. 871 01:08:45,815 --> 01:08:50,520 So if I just write down for you the most general set 872 01:08:50,520 --> 01:08:54,170 of transformations, who knows what's happening? 873 01:08:54,170 --> 01:08:57,890 Ultimately, we have to be guided by physics. 874 01:08:57,890 --> 01:09:02,560 We have to say that, if in the space of all parametrization, 875 01:09:02,560 --> 01:09:05,300 there are some that have no physical correspondence, 876 01:09:05,300 --> 01:09:07,710 we throw them out, we seek things 877 01:09:07,710 --> 01:09:09,830 that can be matched to our physical system. 878 01:09:15,330 --> 01:09:16,224 Yes? 879 01:09:16,224 --> 01:09:18,099 AUDIENCE: If there are multiple fixed points, 880 01:09:18,099 --> 01:09:21,013 do the planes of the basing of attraction 881 01:09:21,013 --> 01:09:22,429 have to be parallel to each other? 882 01:09:26,160 --> 01:09:28,750 PROFESSOR: They may have to have some conditions 883 01:09:28,750 --> 01:09:31,380 on non-intersecting or whatever. 884 01:09:31,380 --> 01:09:35,710 These are only linear in the vicinity of the fixed point. 885 01:09:35,710 --> 01:09:39,210 So in principle, they could be highly curved surfaces 886 01:09:39,210 --> 01:09:44,569 with all kinds of structures and things that I don't know. 887 01:09:44,569 --> 01:09:46,004 Yes? 888 01:09:46,004 --> 01:09:50,260 AUDIENCE: Is there any reason why you might or might not 889 01:09:50,260 --> 01:09:54,940 have attracting point that is actually 890 01:09:54,940 --> 01:09:57,630 a more complicated structure, like say, 891 01:09:57,630 --> 01:10:00,534 a limit cycle or even a [INAUDIBLE]? 892 01:10:00,534 --> 01:10:01,200 PROFESSOR: Yeah. 893 01:10:01,200 --> 01:10:06,950 So again, we are governed ultimately by physics. 894 01:10:06,950 --> 01:10:09,230 When I write these equations, they 895 01:10:09,230 --> 01:10:13,310 are as general as equations as the people in dynamical systems 896 01:10:13,310 --> 01:10:19,270 use that also includes cycles, chaotic attractors, 897 01:10:19,270 --> 01:10:21,740 all kinds of strange things. 898 01:10:21,740 --> 01:10:27,450 And we have to hope that when we apply this procedure 899 01:10:27,450 --> 01:10:30,450 to an appropriate physical system, 900 01:10:30,450 --> 01:10:33,015 the kind of equations that we get 901 01:10:33,015 --> 01:10:37,870 are such that their behavior is indicative of the physics. 902 01:10:37,870 --> 01:10:43,960 So there is one case I know where people sort of found 903 01:10:43,960 --> 01:10:47,190 chaotic renormalization group trajectories 904 01:10:47,190 --> 01:10:49,640 for some kind of a [INAUDIBLE] system. 905 01:10:49,640 --> 01:10:53,900 But always, again, this is a very general procedure. 906 01:10:53,900 --> 01:10:57,900 We have to limit mathematics, ultimately, 907 01:10:57,900 --> 01:11:00,450 by what the physical process is. 908 01:11:00,450 --> 01:11:03,880 So it's good that you know that these equations can 909 01:11:03,880 --> 01:11:06,140 do all kinds of strange things. 910 01:11:06,140 --> 01:11:09,510 But then we take a particular physical system, 911 01:11:09,510 --> 01:11:12,904 we have to beat on them until they behave properly. 912 01:11:17,380 --> 01:11:23,270 So let's imagine that we have a situation, such as this, 913 01:11:23,270 --> 01:11:27,330 where we have three parameters. 914 01:11:27,330 --> 01:11:28,880 Two of them are irrelevant. 915 01:11:28,880 --> 01:11:31,700 One of them is relevant. 916 01:11:31,700 --> 01:11:35,020 Then presumably, I take my physical system 917 01:11:35,020 --> 01:11:37,810 at some temperature and it would correspond 918 01:11:37,810 --> 01:11:41,220 to being on some point in this phase diagram. 919 01:11:44,290 --> 01:11:47,070 Some color that we don't have. 920 01:11:47,070 --> 01:11:48,890 Let's say over here. 921 01:11:48,890 --> 01:11:51,030 And I change the temperature. 922 01:11:51,030 --> 01:11:56,490 And I will take some trajectory-- 923 01:11:56,490 --> 01:11:59,440 in this case, three dimensional space. 924 01:11:59,440 --> 01:12:04,850 And this is a line in this three dimensional space. 925 01:12:04,850 --> 01:12:08,750 And experimentally, I've been told that if I take, 926 01:12:08,750 --> 01:12:12,170 let's say, my piece of iron and I change temperature, 927 01:12:12,170 --> 01:12:18,270 at some point I go through a point that 928 01:12:18,270 --> 01:12:20,300 has infinite correlations. 929 01:12:20,300 --> 01:12:25,990 So I have to conclude that my trajectory for iron 930 01:12:25,990 --> 01:12:31,670 will intersect with surface at some point. 931 01:12:31,670 --> 01:12:34,210 And I'll say, OK, I take nickel. 932 01:12:34,210 --> 01:12:37,430 Nickel would be something else. 933 01:12:37,430 --> 01:12:39,570 And I change temperature of nickel, 934 01:12:39,570 --> 01:12:43,070 and I will be doing something completely different. 935 01:12:43,070 --> 01:12:46,300 But that experimentalist also has a point 936 01:12:46,300 --> 01:12:48,490 where you have ferromagnetic transition, 937 01:12:48,490 --> 01:12:51,210 so it must hit this surface. 938 01:12:51,210 --> 01:12:54,960 Then you do cobalt, where some other trajectory 939 01:12:54,960 --> 01:12:58,590 comes and hits off the surface. 940 01:12:58,590 --> 01:13:02,510 Now what we now know is that when 941 01:13:02,510 --> 01:13:06,760 we rescale the system sufficiently, all of them 942 01:13:06,760 --> 01:13:11,030 ultimately are described at the point where 943 01:13:11,030 --> 01:13:13,225 they have infinite correlation length by what 944 01:13:13,225 --> 01:13:15,810 is going on over here. 945 01:13:15,810 --> 01:13:18,930 So if I take iron, nickel, cobalt, clearly 946 01:13:18,930 --> 01:13:20,570 at the level of atoms and molecules, 947 01:13:20,570 --> 01:13:24,550 they are very different from each other. 948 01:13:24,550 --> 01:13:28,170 And the difference between ironness, nickelness, 949 01:13:28,170 --> 01:13:34,090 cobaltness is really in all of these irrelevant parameters. 950 01:13:34,090 --> 01:13:36,810 And as I go and look at larger and larger scale, 951 01:13:36,810 --> 01:13:40,170 they all diminish and go away. 952 01:13:40,170 --> 01:13:42,970 And at large scale, I see the same thing, 953 01:13:42,970 --> 01:13:47,610 where all of the individual details has been washed out. 954 01:13:47,610 --> 01:13:51,650 So this is able to capture the idea of universality. 955 01:13:54,610 --> 01:13:56,860 But there is a very important caveat 956 01:13:56,860 --> 01:14:02,970 to this, which is that the experimental system, 957 01:14:02,970 --> 01:14:05,920 whether you take iron or cobalt or some mixture 958 01:14:05,920 --> 01:14:09,940 of these different elements, you change one parameter 959 01:14:09,940 --> 01:14:14,610 temperature, and you always see a transition from, let's say, 960 01:14:14,610 --> 01:14:16,720 paramagnetic to ferromagnetic behavior. 961 01:14:19,240 --> 01:14:24,640 Now if I have, say, a line here in three dimensional space 962 01:14:24,640 --> 01:14:26,770 and I draw another line that corresponds 963 01:14:26,770 --> 01:14:30,220 to change in temperature, I will not intersect it. 964 01:14:30,220 --> 01:14:36,200 I have to do something very special to intersect that line. 965 01:14:36,200 --> 01:14:41,180 So in order that genetically I have a phase transition-- which 966 01:14:41,180 --> 01:14:44,960 is what my experimentalist friends tell me-- 967 01:14:44,960 --> 01:14:51,260 I know that I can only have one relevant direction, 968 01:14:51,260 --> 01:14:55,880 because the dimensionality of the basing of attraction 969 01:14:55,880 --> 01:15:00,200 is the dimensionality of the space minus however 970 01:15:00,200 --> 01:15:02,570 many relevant directions I have. 971 01:15:02,570 --> 01:15:04,790 And I've been told by experimentalists 972 01:15:04,790 --> 01:15:07,290 that they exchange one parameters, 973 01:15:07,290 --> 01:15:10,310 and generically they hit the surface. 974 01:15:10,310 --> 01:15:12,220 So that's part of the story. 975 01:15:12,220 --> 01:15:16,490 I better find a theory that, at the end of the day, when 976 01:15:16,490 --> 01:15:20,790 I do all of this, I find a fixed point that 977 01:15:20,790 --> 01:15:24,320 not only is well-behaved and is not a limit cycle, 978 01:15:24,320 --> 01:15:28,470 but also a fixed point that has one and only one 979 01:15:28,470 --> 01:15:31,316 relevant direction, if that's the physical system 980 01:15:31,316 --> 01:15:34,460 that I'm describing. 981 01:15:34,460 --> 01:15:37,400 Now of course, maybe that was for the superfluid, where 982 01:15:37,400 --> 01:15:39,360 they could only change temperature, 983 01:15:39,360 --> 01:15:42,630 and you have a situation where the magnet comes into play 984 01:15:42,630 --> 01:15:46,270 and they say, oh, actually we also have the magnetic field. 985 01:15:46,270 --> 01:15:50,810 And we really have to go to the space of zero field. 986 01:15:50,810 --> 01:15:54,800 And then if I expand my space of parameters 987 01:15:54,800 --> 01:15:59,720 here to include terms that break the symmetry, 988 01:15:59,720 --> 01:16:02,360 in that generalized space, I should only 989 01:16:02,360 --> 01:16:05,870 have two relevant directions. 990 01:16:05,870 --> 01:16:10,480 So it is kind of strange story, that all we are doing here 991 01:16:10,480 --> 01:16:11,650 is mathematics. 992 01:16:11,650 --> 01:16:13,300 But at the end of the day, we have 993 01:16:13,300 --> 01:16:16,925 to get the mathematics to have very specific properties that 994 01:16:16,925 --> 01:16:20,250 are dictated by very rough things about experiments. 995 01:16:23,770 --> 01:16:28,650 So this was kind of conceptually rich. 996 01:16:28,650 --> 01:16:31,770 So I'll let you digest that for a while. 997 01:16:31,770 --> 01:16:37,140 And next lecture, we will start actually doing this procedure 998 01:16:37,140 --> 01:16:40,060 and finding these kinds of [INAUDIBLE] relations.