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 00:00:21,680 --> 00:00:22,380 PROFESSOR: OK. 9 00:00:22,380 --> 00:00:25,250 Let's start. 10 00:00:25,250 --> 00:00:30,090 So if we have been thinking about critical points. 11 00:00:35,050 --> 00:00:40,140 And these arise in many phased diagrams 12 00:00:40,140 --> 00:00:47,440 such as that we have for the liquid gas system where there's 13 00:00:47,440 --> 00:00:52,846 a coexistence line, let's say, between the gas and the liquid 14 00:00:52,846 --> 00:01:00,480 that terminate, or we looked in the case of a magnet 15 00:01:00,480 --> 00:01:07,930 where as a function of [INAUDIBLE] temperature 16 00:01:07,930 --> 00:01:10,920 there was in some sense coexistence 17 00:01:10,920 --> 00:01:15,200 between magnetizations in different directions 18 00:01:15,200 --> 00:01:16,655 terminating at the critical point. 19 00:01:20,290 --> 00:01:25,180 So why is it interesting to take it a whole phase 20 00:01:25,180 --> 00:01:27,540 diagram that we have over here? 21 00:01:27,540 --> 00:01:33,132 For example, for this system, we can also have solid, et cetera. 22 00:01:33,132 --> 00:01:35,090 AUDIENCE: So isn't this [INAUDIBLE] [INAUDIBLE] 23 00:01:35,090 --> 00:01:35,480 PROFESSOR: [INAUDIBLE]. 24 00:01:35,480 --> 00:01:35,740 Yes. 25 00:01:35,740 --> 00:01:36,440 Thank you. 26 00:01:39,540 --> 00:01:43,154 And focus on just the one point. 27 00:01:43,154 --> 00:01:44,570 In the vicinity of this one point. 28 00:01:47,380 --> 00:01:52,920 And the reason for that was this idea of universality. 29 00:01:59,090 --> 00:02:02,430 There many things that are happening 30 00:02:02,430 --> 00:02:04,660 in the vicinity of this point as far 31 00:02:04,660 --> 00:02:07,435 as singularitities, correlations, et cetera, 32 00:02:07,435 --> 00:02:10,840 are concerned that are independent of whatever 33 00:02:10,840 --> 00:02:14,530 the consequence of the system are. 34 00:02:14,530 --> 00:02:20,460 And these singularities, we try to capture through some scaling 35 00:02:20,460 --> 00:02:23,730 laws for the singularities. 36 00:02:23,730 --> 00:02:28,210 And I've been kind of constructing 37 00:02:28,210 --> 00:02:31,090 a table of your singularities. 38 00:02:31,090 --> 00:02:34,340 Let's do it one more time here. 39 00:02:34,340 --> 00:02:39,180 So we could look at system such as the liquid gas-- 40 00:02:39,180 --> 00:02:44,079 so let's have here system-- and then we 41 00:02:44,079 --> 00:02:46,230 could look at the liquid gas. 42 00:02:49,100 --> 00:02:54,514 And for that, we can look at a variety of exponents. 43 00:02:54,514 --> 00:03:02,380 We have alpha, beta, gamma, delta, mu, theta. 44 00:03:05,900 --> 00:03:11,390 And for the liquid gas, I write you some numbers. 45 00:03:11,390 --> 00:03:14,955 The heat capacity diverges with an exponent 46 00:03:14,955 --> 00:03:18,905 that is 0.11-- slightly more accurate 47 00:03:18,905 --> 00:03:21,360 than I had given you before. 48 00:03:21,360 --> 00:03:29,675 The case for beta is 0.33 gamma. 49 00:03:29,675 --> 00:03:30,175 OK. 50 00:03:30,175 --> 00:03:34,020 I will give you a little bit more digits just 51 00:03:34,020 --> 00:03:36,856 to indicate the accuracy of experiments. 52 00:03:36,856 --> 00:03:47,390 This is 1.238 minus plus 0.012. 53 00:03:47,390 --> 00:03:52,250 So these exponents are obtained by looking at the fluid system 54 00:03:52,250 --> 00:03:56,920 with light scattering-- doing this critical opalescence 55 00:03:56,920 --> 00:03:59,615 that we were talking about in more detail 56 00:03:59,615 --> 00:04:05,430 and accurately, Delta is 4.8. 57 00:04:05,430 --> 00:04:14,726 The mu is, again from light scattering 0.629 minus plus 58 00:04:14,726 --> 00:04:18,822 0.003. 59 00:04:18,822 --> 00:04:33,970 Theta is 0.032 0 minus plus 0.013. 60 00:04:33,970 --> 00:04:39,350 And essentially these three are [INAUDIBLE] light scattered. 61 00:04:48,780 --> 00:04:49,570 Sorry. 62 00:04:49,570 --> 00:04:55,010 Another case that I mentioned is that of the super fluid. 63 00:05:00,300 --> 00:05:04,270 And in this general construction of the lambda 64 00:05:04,270 --> 00:05:07,436 gives [INAUDIBLE] theories that we had, 65 00:05:07,436 --> 00:05:10,525 liquid gas would be in question one. 66 00:05:10,525 --> 00:05:14,980 Superfluid would be in question two. 67 00:05:14,980 --> 00:05:21,690 And I just want to mention that actually the most 68 00:05:21,690 --> 00:05:25,746 experimentally accurate exponent that has been determined 69 00:05:25,746 --> 00:05:30,660 is the heat capacity for dry superfluid helium transition. 70 00:05:30,660 --> 00:05:34,720 I had said that it kind of looks like a logarithmic divergence. 71 00:05:34,720 --> 00:05:36,880 You look at it very closely. 72 00:05:36,880 --> 00:05:41,850 And it is in fact a cusp, and does not diverge all the way 73 00:05:41,850 --> 00:05:45,740 to infinity, so it corresponds to a slightly negative value 74 00:05:45,740 --> 00:05:58,346 of alpha, which is the 0.0127 minus plus 0.0003. 75 00:05:58,346 --> 00:06:01,950 And the way that this has been data-mined 76 00:06:01,950 --> 00:06:06,604 is they took superfluid helium to the space shuttle, 77 00:06:06,604 --> 00:06:09,508 and this experiments were done away 78 00:06:09,508 --> 00:06:12,160 from the gravity of the earth in order 79 00:06:12,160 --> 00:06:15,600 to not to have to worry about the density difference 80 00:06:15,600 --> 00:06:19,470 that we would have across the system. 81 00:06:19,470 --> 00:06:23,133 Other exponents that you have for this system-- 82 00:06:23,133 --> 00:06:27,882 let me write down-- beta is around 0.35. 83 00:06:27,882 --> 00:06:32,896 Gamma is 1.32. 84 00:06:32,896 --> 00:06:37,520 Delta is 4.79. 85 00:06:37,520 --> 00:06:40,920 Mu is is 0.67. 86 00:06:40,920 --> 00:06:46,990 Theta is 0.04. 87 00:06:46,990 --> 00:06:52,470 And we don't need this for system. 88 00:06:56,250 --> 00:07:03,580 Any questions we could do players kind of 89 00:07:03,580 --> 00:07:12,010 add the exponents here I've booked usability 90 00:07:12,010 --> 00:07:18,820 even if it's minus 1 is a research data 91 00:07:18,820 --> 00:07:32,650 0.7 down all those are long this is more to say about new ideas 92 00:07:32,650 --> 00:07:49,650 and so on I think is that these numbers aren't you 93 00:07:49,650 --> 00:08:02,930 think that is simplest way for us is net 94 00:08:02,930 --> 00:08:10,480 my position and the question is why these numbers are all 95 00:08:10,480 --> 00:08:14,160 of the same as all the systems is therefore profound. 96 00:08:14,160 --> 00:08:16,200 These are dimensionless numbers. 97 00:08:16,200 --> 00:08:20,240 So in some sense, it is a little bit of mathematics. 98 00:08:20,240 --> 00:08:22,775 It's not like you calculate the charge of the electron 99 00:08:22,775 --> 00:08:25,080 and you get a number. 100 00:08:25,080 --> 00:08:29,240 These don't depend on a specific material. 101 00:08:29,240 --> 00:08:31,200 Therefore, what is important about them 102 00:08:31,200 --> 00:08:34,640 is that they must somehow be capturing 103 00:08:34,640 --> 00:08:37,440 some aspect of the collective behavior of all 104 00:08:37,440 --> 00:08:40,880 of these degrees of freedom, in which the details of what 105 00:08:40,880 --> 00:08:44,090 the degrees of freedom are is not that important. 106 00:08:44,090 --> 00:08:47,810 Maybe the type of synergy rating is important. 107 00:08:47,810 --> 00:08:52,860 So unless we understand and derive these numbers, 108 00:08:52,860 --> 00:08:55,950 there is something important about the collective behavior 109 00:08:55,950 --> 00:09:00,440 of many degrees of freedom that we have not understood. 110 00:09:00,440 --> 00:09:03,711 And it is somehow a different question 111 00:09:03,711 --> 00:09:08,160 if you are thinking about phase transitions. 112 00:09:08,160 --> 00:09:11,660 So let's say you're thinking about superconductors. 113 00:09:11,660 --> 00:09:14,220 There's a lot of interest in making high temperature 114 00:09:14,220 --> 00:09:18,600 superconductor pushing TC further and further up. 115 00:09:18,600 --> 00:09:21,240 So that's certainly a material problem. 116 00:09:21,240 --> 00:09:23,050 We are asking a different problem. 117 00:09:23,050 --> 00:09:26,320 Why is it, whether you have a high temperature superconductor 118 00:09:26,320 --> 00:09:30,590 or any other type of system, the collective behavior 119 00:09:30,590 --> 00:09:34,940 is captured by the same set of exponents. 120 00:09:34,940 --> 00:09:39,690 So in an attempt to try to answer that, 121 00:09:39,690 --> 00:09:49,450 we did this Landau-Ginzburg and try to calculate its singular 122 00:09:49,450 --> 00:09:53,720 behavior using this other point of approximation. 123 00:09:53,720 --> 00:09:56,150 And the numbers that we got, alpha 124 00:09:56,150 --> 00:10:00,680 was 0, meaning that there was discontinuity. 125 00:10:00,680 --> 00:10:09,483 Beta was 1/2, gamma was 1, delta was 3, my was 1/2, 126 00:10:09,483 --> 00:10:15,170 theta was 0, which don't quite match with these numbers 127 00:10:15,170 --> 00:10:18,070 that we have up there. 128 00:10:18,070 --> 00:10:22,660 So question is, what should you do? 129 00:10:22,660 --> 00:10:28,000 We've made an attempt and that attempt was not successful. 130 00:10:28,000 --> 00:10:33,480 So we are going to completely for a while forget about that 131 00:10:33,480 --> 00:10:36,440 and try to approach the problem from a different perspective 132 00:10:36,440 --> 00:10:38,770 and see how far we can go, whether we 133 00:10:38,770 --> 00:10:43,800 can gain any new insights. 134 00:10:43,800 --> 00:10:49,805 So that new approach I put on there 135 00:10:49,805 --> 00:10:51,847 the name of the scaling hypothesis. 136 00:10:59,640 --> 00:11:06,090 And the reason for that will become apparent shortly. 137 00:11:06,090 --> 00:11:12,550 So what we have in common in both of these examples 138 00:11:12,550 --> 00:11:17,570 is that there is a line where there 139 00:11:17,570 --> 00:11:21,700 are discontinuities in calculating 140 00:11:21,700 --> 00:11:24,630 some thermodynamic function that terminates 141 00:11:24,630 --> 00:11:26,730 at a particular point. 142 00:11:26,730 --> 00:11:30,030 And in the case of the magnetic system, 143 00:11:30,030 --> 00:11:35,980 we can look at the singularities approaching that point either 144 00:11:35,980 --> 00:11:38,580 along the direction that corresponds 145 00:11:38,580 --> 00:11:44,040 to change in temperature and parametrize that through heat, 146 00:11:44,040 --> 00:11:48,280 or we can change the magnetic field 147 00:11:48,280 --> 00:11:52,030 and approach the problem from this other direction. 148 00:11:52,030 --> 00:11:55,220 And we saw that there were analogs for doing 149 00:11:55,220 --> 00:11:59,195 so in the liquid gas system also. 150 00:11:59,195 --> 00:12:03,830 And in particular, let's say we calculated a magnetization, 151 00:12:03,830 --> 00:12:07,520 we found that there was one form of singularity coming this way, 152 00:12:07,520 --> 00:12:10,430 one form of singularity coming that way. 153 00:12:10,430 --> 00:12:14,005 We look at the picture for the liquid gas system 154 00:12:14,005 --> 00:12:18,400 that I have up there, and it's not necessarily clear 155 00:12:18,400 --> 00:12:24,310 which direction would correspond to this nice symmetry 156 00:12:24,310 --> 00:12:26,780 breaking or non-symmetry breaking 157 00:12:26,780 --> 00:12:29,740 that you have for the magnetic system. 158 00:12:29,740 --> 00:12:33,610 So you may well ask, suppose I approach the critical point 159 00:12:33,610 --> 00:12:35,440 along some other direction. 160 00:12:35,440 --> 00:12:38,950 Maybe I come in along the path such as this. 161 00:12:38,950 --> 00:12:41,120 I still go to the critical point. 162 00:12:41,120 --> 00:12:44,170 We can imagine that for the liquid gas system. 163 00:12:44,170 --> 00:12:46,955 And what's the structure of the singularities? 164 00:12:46,955 --> 00:12:51,221 I know that there are different singularities in the t and h 165 00:12:51,221 --> 00:12:51,720 direction. 166 00:12:51,720 --> 00:12:53,802 What is it if I come and approach 167 00:12:53,802 --> 00:12:55,670 the system along a different direction, 168 00:12:55,670 --> 00:13:00,400 which we may well do for a liquid gas system? 169 00:13:00,400 --> 00:13:03,270 Well, we could actually answer that if we go back 170 00:13:03,270 --> 00:13:07,680 to our graph saddlepoint approximation. 171 00:13:07,680 --> 00:13:10,050 In the saddlepoint approximation, 172 00:13:10,050 --> 00:13:15,134 we said that ultimately, the singularities in terms 173 00:13:15,134 --> 00:13:19,090 of these two parameters t and h-- 174 00:13:19,090 --> 00:13:20,400 so this is in the saddlepoint. 175 00:13:26,050 --> 00:13:33,185 Part obtained by minimizing this function that 176 00:13:33,185 --> 00:13:37,405 was appearing in the expansion in the exponent. 177 00:13:37,405 --> 00:13:39,470 There was a t over 2m squared. 178 00:13:39,470 --> 00:13:42,378 There was a mu n to the 4th. 179 00:13:42,378 --> 00:13:43,770 And there is an hm. 180 00:13:46,554 --> 00:13:52,325 So we had to minimize this with respect to m. 181 00:13:52,325 --> 00:13:56,840 And clearly, what that gives us is m. 182 00:13:56,840 --> 00:14:00,280 If I really solve the equation, that 183 00:14:00,280 --> 00:14:03,220 corresponds to this minimization, 184 00:14:03,220 --> 00:14:05,820 which is a function of t and h. 185 00:14:05,820 --> 00:14:10,970 And in particular, approaching two directions that's 186 00:14:10,970 --> 00:14:16,000 indicated, if I'm along the direction where h equals 0, 187 00:14:16,000 --> 00:14:18,869 I essentially balance these two terms. 188 00:14:18,869 --> 00:14:20,660 Let's just write this as a proportionality. 189 00:14:20,660 --> 00:14:23,515 I don't really care about the numbers. 190 00:14:26,580 --> 00:14:30,110 Along the direction where h equals 0, 191 00:14:30,110 --> 00:14:36,120 I have to balance m to the 4th and tm squared. 192 00:14:36,120 --> 00:14:41,150 So m squared will scale like e. 193 00:14:41,150 --> 00:14:47,048 m will scale like square root of t. 194 00:14:47,048 --> 00:14:49,495 And more precisely, we calculated 195 00:14:49,495 --> 00:14:53,480 this formula for t negative and h equals to 0. 196 00:14:56,220 --> 00:14:59,300 If I, on the other hand, come along the direction 197 00:14:59,300 --> 00:15:08,050 that corresponds to t equals to 0, along that direction 198 00:15:08,050 --> 00:15:09,510 I don't have a first term. 199 00:15:09,510 --> 00:15:14,440 I have to balance um the 4th and hm. 200 00:15:14,440 --> 00:15:20,143 So we immediately see that m will scale like h over u. 201 00:15:20,143 --> 00:15:25,008 In fact, more correctly h over 4u to the power of one third. 202 00:15:28,950 --> 00:15:31,030 You substitute this in the free energy 203 00:15:31,030 --> 00:15:34,464 and you find that the singular part of the free energy 204 00:15:34,464 --> 00:15:41,590 as a function of t and h in this saddlepoint approximation 205 00:15:41,590 --> 00:15:45,960 has the [INAUDIBLE] to the form of proportionality. 206 00:15:45,960 --> 00:15:49,740 If I substitute this in the formula for t negative, 207 00:15:49,740 --> 00:15:52,905 I will get something like minus t squared 208 00:15:52,905 --> 00:15:58,400 over 4 we have-- forget about the number t squared over u. 209 00:15:58,400 --> 00:16:02,430 If I go along the t equals to 0 direction, 210 00:16:02,430 --> 00:16:08,500 substitute that over there, I will get n to the 4th. 211 00:16:08,500 --> 00:16:15,570 I will get h to the 4 thirds divided by mu to the one third. 212 00:16:15,570 --> 00:16:18,170 Even the mu dependence I'm not interested. 213 00:16:18,170 --> 00:16:22,850 I'm really interested in the behavior close to t and h 214 00:16:22,850 --> 00:16:24,930 as a function of t and h. 215 00:16:24,930 --> 00:16:27,820 Mu is basically some non-universal number 216 00:16:27,820 --> 00:16:29,960 that doesn't go to 0. 217 00:16:29,960 --> 00:16:33,980 I could in some sense capture these two expressions 218 00:16:33,980 --> 00:16:39,050 by a form that is t squared and then 219 00:16:39,050 --> 00:16:41,430 some function-- let's call it g sub 220 00:16:41,430 --> 00:16:47,055 f which is a function of-- let's see 221 00:16:47,055 --> 00:16:52,338 how I define the delta h over t to the delta. 222 00:16:58,510 --> 00:17:03,610 So my claim is that I toyed with the behavior coming 223 00:17:03,610 --> 00:17:07,190 across these two different special direction. 224 00:17:07,190 --> 00:17:13,420 In general, anywhere else where t and h are both nonzero, 225 00:17:13,420 --> 00:17:17,980 the answer for m will be some solution of a cubic equation, 226 00:17:17,980 --> 00:17:21,099 but we can arrange it to only be a function of h 227 00:17:21,099 --> 00:17:25,410 over [INAUDIBLE] and have this form. 228 00:17:25,410 --> 00:17:30,880 Now I could maybe rather than explicitly 229 00:17:30,880 --> 00:17:33,210 show you how that arises, which is not 230 00:17:33,210 --> 00:17:36,580 difficult-- you can do that-- since there's something 231 00:17:36,580 --> 00:17:39,460 that we need to do later on, I'll 232 00:17:39,460 --> 00:17:43,030 show it in the following manner. 233 00:17:43,030 --> 00:17:48,980 I have not specified what this function g sub f is. 234 00:17:48,980 --> 00:17:53,710 But I know its behavior along h equals to 0 here. 235 00:17:53,710 --> 00:17:57,238 And so if I put h equals to 0, the argument of the function 236 00:17:57,238 --> 00:17:59,580 goes to 0. 237 00:17:59,580 --> 00:18:02,810 So if I say that the argument of the function 238 00:18:02,810 --> 00:18:06,950 is a constant-- the constant let's say is minus 1 239 00:18:06,950 --> 00:18:10,270 over u on one side, 0 on the other side, 240 00:18:10,270 --> 00:18:12,170 then everything's fine. 241 00:18:12,170 --> 00:18:20,060 So I have is that the limit as its argument goes to 0 242 00:18:20,060 --> 00:18:21,330 should be some constant. 243 00:18:26,890 --> 00:18:29,020 Well, what about the other direction? 244 00:18:29,020 --> 00:18:32,520 How can I reproduce from a form such as this 245 00:18:32,520 --> 00:18:37,510 the behavior when t equals to 0? 246 00:18:37,510 --> 00:18:40,400 Because I see that when t equals to 0, 247 00:18:40,400 --> 00:18:43,560 the answer of course cannot depend on t itself, 248 00:18:43,560 --> 00:18:47,840 but as a power law as a function of h. 249 00:18:47,840 --> 00:18:51,790 Is it consistent with this form? 250 00:18:51,790 --> 00:18:54,010 Well, as t goes to 0 in this form, 251 00:18:54,010 --> 00:18:58,580 the numerator here goes to 0, the argument of the function 252 00:18:58,580 --> 00:19:01,660 goes to infinity, I need to know something 253 00:19:01,660 --> 00:19:05,480 about the behavior of the function of infinity. 254 00:19:05,480 --> 00:19:09,800 So let's say that the limiting behavior as the argument 255 00:19:09,800 --> 00:19:13,400 of the function goes to infinity of gf 256 00:19:13,400 --> 00:19:21,438 of x is proportional to the argument to some other peak. 257 00:19:21,438 --> 00:19:24,432 And I don't know where that power is. 258 00:19:24,432 --> 00:19:28,230 Then if I look at this function, the whole function 259 00:19:28,230 --> 00:19:32,775 in this limit where t goes to 0 will behave. 260 00:19:32,775 --> 00:19:37,530 There's a t squared out front the goes to 0, 261 00:19:37,530 --> 00:19:39,890 the argument of the function goes to infinity. 262 00:19:39,890 --> 00:19:44,860 So the function will go like the argument to some power. 263 00:19:44,860 --> 00:19:49,313 So I go like h t to the delta to some other peak. 264 00:19:56,720 --> 00:20:00,190 So what do I know? 265 00:20:00,190 --> 00:20:02,320 I know that the answer should really 266 00:20:02,320 --> 00:20:06,360 be proportional to h to the four thirds. 267 00:20:09,210 --> 00:20:15,496 So I immediately know that my t should be four thirds. 268 00:20:20,480 --> 00:20:21,810 But what about this delta? 269 00:20:21,810 --> 00:20:24,660 I never told you what delta was. 270 00:20:24,660 --> 00:20:28,100 Now I can figure out what delta is, because the answer should 271 00:20:28,100 --> 00:20:31,860 not depend on t. t has gone to 0. 272 00:20:31,860 --> 00:20:34,460 And so what power of t do I have? 273 00:20:34,460 --> 00:20:42,630 I have 2 minus delta p should be 0. 274 00:20:42,630 --> 00:20:50,950 So my delta should be 2 over p, 2 over four thirds, 275 00:20:50,950 --> 00:20:52,365 so it should be three halves. 276 00:21:01,610 --> 00:21:08,920 Why is this exponent relevant to the question that I had before? 277 00:21:08,920 --> 00:21:13,840 You can see that the function that describes the free energy 278 00:21:13,840 --> 00:21:16,580 as a function of these two coordinates. 279 00:21:16,580 --> 00:21:21,600 If I look at the combination where h and t are non-zero, 280 00:21:21,600 --> 00:21:28,625 is very much dependent on this h divided by t to the delta, 281 00:21:28,625 --> 00:21:31,650 and that delta is three halves. 282 00:21:31,650 --> 00:21:34,420 So, for example, if I were to draw here 283 00:21:34,420 --> 00:21:43,630 curves where h goes like 3 to the three halves-- 284 00:21:43,630 --> 00:21:48,770 it's some coefficient, I don't know what that coefficient is-- 285 00:21:48,770 --> 00:21:55,320 then essentially, everything that is on the side 286 00:21:55,320 --> 00:22:01,160 that hogs the vertical axis behaves like the h singularity. 287 00:22:01,160 --> 00:22:06,510 Everything that is over here depends like a t singularity. 288 00:22:06,510 --> 00:22:12,230 So a path that, for example, comes along a straight line, 289 00:22:12,230 --> 00:22:16,960 if I, let's say, call the distance that I have 290 00:22:16,960 --> 00:22:21,500 to the critical point s, then t is something 291 00:22:21,500 --> 00:22:27,220 like s cosine of theta. h is something like s sine theta. 292 00:22:27,220 --> 00:22:29,630 You can see however that the information 293 00:22:29,630 --> 00:22:36,840 h over t to the delta as s goes to 0 will diverge, 294 00:22:36,840 --> 00:22:39,745 because I have other three halves down here 295 00:22:39,745 --> 00:22:44,550 for s that will overcome the linear cover I have over there. 296 00:22:44,550 --> 00:22:49,570 So for any linear path that goes through the critical point, 297 00:22:49,570 --> 00:22:53,760 eventually for small s I will see the type of singularity 298 00:22:53,760 --> 00:22:57,770 that is characteristic of the magnetic field 299 00:22:57,770 --> 00:23:03,050 if the exponents are according to this other point. 300 00:23:03,050 --> 00:23:05,230 We have this assumption, of course. 301 00:23:05,230 --> 00:23:10,040 But if I therefore knew the correct delta 302 00:23:10,040 --> 00:23:14,070 for all of those systems, I would be also able to answer, 303 00:23:14,070 --> 00:23:17,530 let's say for the liquid gas, whether if I 304 00:23:17,530 --> 00:23:20,260 take a linear path that goes through the critical point 305 00:23:20,260 --> 00:23:22,920 I would see one set of singularities or deltas 306 00:23:22,920 --> 00:23:25,980 that have singularities. 307 00:23:25,980 --> 00:23:34,000 So this delta which is called a gap exponent, 308 00:23:34,000 --> 00:23:35,290 gives you the answer to that. 309 00:23:38,420 --> 00:23:43,530 But of course I don't know the other exponents. 310 00:23:43,530 --> 00:23:46,995 There is no reason for me to trust the gap exponent 311 00:23:46,995 --> 00:23:51,010 that I obtained in this fashion. 312 00:23:51,010 --> 00:24:02,740 So what I say is let's assume that for any critical point, 313 00:24:02,740 --> 00:24:06,620 the singular part of the free energy 314 00:24:06,620 --> 00:24:10,080 on approaching the critical point which 315 00:24:10,080 --> 00:24:14,710 depends on this pair of coordinates 316 00:24:14,710 --> 00:24:18,730 has a singular behavior that is similar to what 317 00:24:18,730 --> 00:24:22,460 we had over here, except that I don't know the exponent. 318 00:24:22,460 --> 00:24:26,620 So rather than putting 2 t squared, 319 00:24:26,620 --> 00:24:30,070 I write t to the 2 minus alpha for reason 320 00:24:30,070 --> 00:24:32,736 that will become apparent shortly, 321 00:24:32,736 --> 00:24:39,820 and some function of h t to the delta and for some 322 00:24:39,820 --> 00:24:41,227 alpha and delta. 323 00:24:47,580 --> 00:24:49,700 So this is certainly already an assumption. 324 00:24:52,680 --> 00:24:57,730 This mathematically corresponds to having 325 00:24:57,730 --> 00:24:59,015 homogeneous functions. 326 00:25:06,190 --> 00:25:09,490 Because if I have a function of x and y, 327 00:25:09,490 --> 00:25:14,930 I can certainly write lots of functions such as x squared 328 00:25:14,930 --> 00:25:19,110 plus y squared plus a constant plus x cubed y cubed that I 329 00:25:19,110 --> 00:25:22,800 cannot rearrange into this form. 330 00:25:22,800 --> 00:25:25,560 But there are certain functions of x and y 331 00:25:25,560 --> 00:25:29,150 that I can rearrange so that I can pull out 332 00:25:29,150 --> 00:25:32,370 some factor of let's say x squared out front, 333 00:25:32,370 --> 00:25:34,480 and everything that is then in a series 334 00:25:34,480 --> 00:25:37,870 is a function of let's say y over x cubed. 335 00:25:37,870 --> 00:25:39,910 Something like that. 336 00:25:39,910 --> 00:25:42,440 So there's some class of functions 337 00:25:42,440 --> 00:25:46,074 of two arguments that have this homogeneity. 338 00:25:46,074 --> 00:25:50,400 So we are going to assume that the singular behavior close 339 00:25:50,400 --> 00:25:55,732 to the critical point is described by such a function. 340 00:25:55,732 --> 00:25:57,181 That's an assumption. 341 00:26:00,080 --> 00:26:05,190 But having made that assumption, let's follow its consequence 342 00:26:05,190 --> 00:26:07,120 and let's see if we learned something 343 00:26:07,120 --> 00:26:10,520 about that table of exponents. 344 00:26:10,520 --> 00:26:13,570 Now the first thing to note is clearly 345 00:26:13,570 --> 00:26:16,455 I chose this alpha over here so that when 346 00:26:16,455 --> 00:26:23,820 I take two derivatives with respect to t, 347 00:26:23,820 --> 00:26:27,050 I would get something like a heat capacity, 348 00:26:27,050 --> 00:26:32,550 for which I know what the divergence is. 349 00:26:32,550 --> 00:26:36,580 That's a divergence called alpha. 350 00:26:36,580 --> 00:26:37,980 But there's one thing that I have 351 00:26:37,980 --> 00:26:42,100 to show you is that when I take a derivative of one 352 00:26:42,100 --> 00:26:48,032 of these homogeneous functions, with respect 353 00:26:48,032 --> 00:26:52,270 to one of its arguments, I will generate 354 00:26:52,270 --> 00:26:53,850 another homogeneous function. 355 00:26:53,850 --> 00:26:57,080 If I take one derivative with respect to t, 356 00:26:57,080 --> 00:27:03,770 that derivative can either act on this, 357 00:27:03,770 --> 00:27:08,370 leaving the function unchanged, or it 358 00:27:08,370 --> 00:27:11,600 can act on the argument of the function 359 00:27:11,600 --> 00:27:16,740 and give me d to the 2 minus alpha. 360 00:27:16,740 --> 00:27:21,950 I will have minus h t to the power of delta plus 1. 361 00:27:21,950 --> 00:27:24,366 There will be a factor of delta and then 362 00:27:24,366 --> 00:27:28,713 I will have the derivative function ht to the delta. 363 00:27:32,530 --> 00:27:34,630 So I just took derivatives. 364 00:27:34,630 --> 00:27:37,687 I can certainly pull out a factor of t 365 00:27:37,687 --> 00:27:40,750 to the 1 minus alpha. 366 00:27:40,750 --> 00:27:46,420 Then the first term is just 2 minus alpha times 367 00:27:46,420 --> 00:27:47,365 the original function. 368 00:27:50,820 --> 00:27:59,240 The second term is minus delta h divided by t to the delta. 369 00:27:59,240 --> 00:28:01,500 Because I pulled out the 1 minus alpha, 370 00:28:01,500 --> 00:28:05,310 this t gets rid of the factor of 1 there. 371 00:28:05,310 --> 00:28:06,400 And I have the derivative. 372 00:28:12,000 --> 00:28:13,990 So this is completely different function. 373 00:28:13,990 --> 00:28:16,825 It's not the derivative of the original function. 374 00:28:16,825 --> 00:28:20,015 But whatever it is it is still only a function 375 00:28:20,015 --> 00:28:23,984 of the combination h over t to the delta. 376 00:28:23,984 --> 00:28:26,650 So the derivative of a homogeneous function 377 00:28:26,650 --> 00:28:28,973 is some other homogeneous function. 378 00:28:28,973 --> 00:28:30,452 Let's call it g2. 379 00:28:30,452 --> 00:28:31,438 It doesn't matter. 380 00:28:31,438 --> 00:28:36,370 Let's call it g1 ht to the delta. 381 00:28:36,370 --> 00:28:39,560 And this will happen if I take a second derivative. 382 00:28:39,560 --> 00:28:41,920 So I know that if I take two derivatives, 383 00:28:41,920 --> 00:28:44,960 I will get t to the minus alpha. 384 00:28:44,960 --> 00:28:49,100 I will basically drop two factors over there. 385 00:28:49,100 --> 00:28:56,730 And then some other function, ht to the delta. 386 00:28:56,730 --> 00:29:00,150 Clearly again, if I say that I'm looking at the line 387 00:29:00,150 --> 00:29:05,020 where h equals to zero for a magnet, 388 00:29:05,020 --> 00:29:08,300 then the argument of the function goes to 0. 389 00:29:08,300 --> 00:29:12,030 If I say that the function of the argument goes to 0 390 00:29:12,030 --> 00:29:15,790 is a constant, like we had over here, 391 00:29:15,790 --> 00:29:19,840 then I will have the singularity t to the minus alpha. 392 00:29:19,840 --> 00:29:24,160 So I've clearly engineered whatever the value of alpha 393 00:29:24,160 --> 00:29:28,410 is in this table, I can put over here 394 00:29:28,410 --> 00:29:33,630 and I have the right singularity for the heat capacity. 395 00:29:33,630 --> 00:29:36,660 Essentially I've put it there by hand. 396 00:29:36,660 --> 00:29:40,810 Let me comment on one other thing, which 397 00:29:40,810 --> 00:29:47,869 is when we are looking at just the temperature, 398 00:29:47,869 --> 00:29:49,410 let's say we are looking at something 399 00:29:49,410 --> 00:29:51,360 like a superfluid, the only parameter 400 00:29:51,360 --> 00:29:58,060 that we have at our disposal is temperature and tens of ITC. 401 00:29:58,060 --> 00:30:03,050 Let's say we plug the heat capacity 402 00:30:03,050 --> 00:30:06,631 and then we see divergence of the heat capacity on the two 403 00:30:06,631 --> 00:30:07,130 sides. 404 00:30:09,910 --> 00:30:15,430 Who said that I should have the same exponent on this side 405 00:30:15,430 --> 00:30:16,865 and on this side? 406 00:30:19,380 --> 00:30:22,230 So we said that generally, in principle, 407 00:30:22,230 --> 00:30:26,370 I could say I would do that. 408 00:30:26,370 --> 00:30:31,090 And in principle, there is no problem with that. 409 00:30:31,090 --> 00:30:33,980 If there is function that has one behavior here, 410 00:30:33,980 --> 00:30:38,820 another behavior there, who says that two exponents 411 00:30:38,820 --> 00:30:41,710 have to be the same? 412 00:30:41,710 --> 00:30:44,550 But I have said something more. 413 00:30:44,550 --> 00:30:49,310 I have said that in all of the cases that I'm looking at, 414 00:30:49,310 --> 00:30:54,440 I know that there is some other axis. 415 00:30:57,820 --> 00:31:02,850 And for example, if I am in the liquid gas system, 416 00:31:02,850 --> 00:31:06,760 I can start from down here, go all the way around 417 00:31:06,760 --> 00:31:10,940 back here without encountering a singularity. 418 00:31:10,940 --> 00:31:15,120 I can go from the liquid all the way to gas 419 00:31:15,120 --> 00:31:16,650 without encountering a singularity. 420 00:31:19,450 --> 00:31:24,630 So that says that the system is different from a system 421 00:31:24,630 --> 00:31:29,120 that, let's say, has a line of singularities. 422 00:31:29,120 --> 00:31:33,550 So if I now take the functions that in principle 423 00:31:33,550 --> 00:31:37,014 have two different singularities, 424 00:31:37,014 --> 00:31:43,740 t to the minus alpha minus t to the minus alpha plus on the h 425 00:31:43,740 --> 00:31:48,950 equals to 0 axis and try to elevate them 426 00:31:48,950 --> 00:31:54,930 into the entire space by putting this homogeneous functions 427 00:31:54,930 --> 00:32:00,430 in front of them, there is one and only one way 428 00:32:00,430 --> 00:32:05,405 in which the two functions can match exactly on this t 429 00:32:05,405 --> 00:32:09,680 equals to 0 line, and that's if the two exponents are the same 430 00:32:09,680 --> 00:32:12,518 and you are dealing with the same function. 431 00:32:12,518 --> 00:32:15,830 So that we put in a bit of physics. 432 00:32:15,830 --> 00:32:21,266 So in principle, mathematically if you don't have the h axis 433 00:32:21,266 --> 00:32:25,839 and you look at the one line and there's a singularity, 434 00:32:25,839 --> 00:32:27,630 there's no reason why the two singularities 435 00:32:27,630 --> 00:32:29,700 should be the same. 436 00:32:29,700 --> 00:32:31,920 But we know that we are looking at the class 437 00:32:31,920 --> 00:32:35,430 of physical systems where there is the possibility 438 00:32:35,430 --> 00:32:39,200 to analytically go from one side to the other side. 439 00:32:39,200 --> 00:32:42,270 And that immediately imposes this constraint 440 00:32:42,270 --> 00:32:47,590 that alpha plus should be alpha minus, and one alpha 441 00:32:47,590 --> 00:32:49,205 is in fact sufficient. 442 00:32:49,205 --> 00:32:54,850 And I gave you the correct answer for why that is. 443 00:32:54,850 --> 00:32:57,930 If you want to see the precise mathematical details 444 00:32:57,930 --> 00:33:00,130 step by step, then that's in the notes. 445 00:33:04,330 --> 00:33:06,020 So fine. 446 00:33:06,020 --> 00:33:08,360 So far we haven't learned much. 447 00:33:08,360 --> 00:33:10,790 We've justified why the two alphas should 448 00:33:10,790 --> 00:33:13,480 be the same above and below, but we 449 00:33:13,480 --> 00:33:17,770 put the alpha, the one alpha, then by hand. 450 00:33:17,770 --> 00:33:19,945 And then we have this unknown delta also. 451 00:33:19,945 --> 00:33:23,046 But let's proceed. 452 00:33:23,046 --> 00:33:26,170 Let's see what other consequence emerge, because now we 453 00:33:26,170 --> 00:33:27,970 have a function of two variables. 454 00:33:27,970 --> 00:33:30,490 I took derivatives in respect to t. 455 00:33:30,490 --> 00:33:33,546 I can take derivatives with respect to m. 456 00:33:33,546 --> 00:33:39,910 And in particular, the magnetization 457 00:33:39,910 --> 00:33:46,340 m as a function of t and h is obtained 458 00:33:46,340 --> 00:33:52,800 from a derivative of the free energy with respect to h. 459 00:33:52,800 --> 00:33:55,230 There's potential. 460 00:33:55,230 --> 00:33:59,140 It's the response to adding a field could 461 00:33:59,140 --> 00:34:01,250 be some factor of beta c or whatever. 462 00:34:01,250 --> 00:34:03,440 It's not important. 463 00:34:03,440 --> 00:34:06,460 The singular part will come from this. 464 00:34:06,460 --> 00:34:10,690 And so taking a derivative of this function 465 00:34:10,690 --> 00:34:14,510 I will get t this to the 2 minus alpha. 466 00:34:14,510 --> 00:34:17,159 The derivative of a can be respect to h, 467 00:34:17,159 --> 00:34:19,849 but h comes in the combination h over t 468 00:34:19,849 --> 00:34:24,610 to the delta will bring down a factor of minus delta up front. 469 00:34:24,610 --> 00:34:29,058 Then the derivative function-- let's call it gf1, for example. 470 00:34:36,210 --> 00:34:41,906 So now I can look at this function in the limit 471 00:34:41,906 --> 00:34:49,400 where h goes to 0, climb along the coexistence line, 472 00:34:49,400 --> 00:34:51,280 h2 goes to 0. 473 00:34:51,280 --> 00:34:55,550 The argument of the function has gone to 0. 474 00:34:55,550 --> 00:34:58,150 Makes sense that the function should be constant 475 00:34:58,150 --> 00:35:00,140 when its argument goes to 0. 476 00:35:00,140 --> 00:35:03,192 So the answer is going to be proportional to t 477 00:35:03,192 --> 00:35:06,108 to the 2 minus alpha minus delta. 478 00:35:09,940 --> 00:35:13,720 But that's how beta was defined. 479 00:35:13,720 --> 00:35:21,400 So if I know my beta and alpha, then I 480 00:35:21,400 --> 00:35:25,988 can calculate my delta from this exponent identity. 481 00:35:28,550 --> 00:35:31,760 Again, so far you haven't done much. 482 00:35:31,760 --> 00:35:37,075 You have translated two unknown exponents, this singular form, 483 00:35:37,075 --> 00:35:40,540 this gap exponent that we don't know. 484 00:35:40,540 --> 00:35:44,630 I can also look at the other limit 485 00:35:44,630 --> 00:35:48,280 where t goes to 0 that is calculating 486 00:35:48,280 --> 00:35:51,550 the magnetization along the critical isotherm. 487 00:35:55,280 --> 00:36:01,140 So then the argument of the function has gone to infinity. 488 00:36:01,140 --> 00:36:03,990 And whatever the answer is should not depend on t, 489 00:36:03,990 --> 00:36:06,470 because I have said t goes to 0. 490 00:36:06,470 --> 00:36:09,970 So I apply the same trick that I did over here. 491 00:36:09,970 --> 00:36:13,420 I say that when the argument goes to infinity, 492 00:36:13,420 --> 00:36:19,330 the function goes like some power of its argument. 493 00:36:24,191 --> 00:36:29,430 And clearly I have to choose that power such that the t 494 00:36:29,430 --> 00:36:34,130 dependence, since t is going to 0, I have to get rid of it. 495 00:36:34,130 --> 00:36:40,060 The only way that I can do that is if p is 2 minus alpha 496 00:36:40,060 --> 00:36:45,720 minus delta divided by that. 497 00:36:52,710 --> 00:36:56,980 So having done that, the whole thing will then 498 00:36:56,980 --> 00:36:59,866 be a function of h to the p. 499 00:37:02,610 --> 00:37:05,146 But the shape of the magnetization 500 00:37:05,146 --> 00:37:08,520 along the critical isotherm, which was also 501 00:37:08,520 --> 00:37:13,590 the shape of the isotherm of the liquid gas system, 502 00:37:13,590 --> 00:37:16,200 we were characterizing by an exponent 503 00:37:16,200 --> 00:37:18,242 that we were calling 1 over delta. 504 00:37:22,100 --> 00:37:25,930 So we have now a formula that says 505 00:37:25,930 --> 00:37:31,540 my delta shouldn't in fact be the inverse of p. 506 00:37:31,540 --> 00:37:39,170 It should be the delta 2 minus alpha minus delta. 507 00:37:39,170 --> 00:37:39,670 Yes? 508 00:37:39,670 --> 00:37:42,309 AUDIENCE: Why isn't the exponent t minus 1 509 00:37:42,309 --> 00:37:43,975 after you've differentiated [INAUDIBLE]? 510 00:37:47,090 --> 00:37:51,115 Because g originally was defined as [INAUDIBLE]. 511 00:37:51,115 --> 00:37:54,070 PROFESSOR: Let's call it pr. 512 00:37:58,010 --> 00:38:01,440 Because actually, you're right. 513 00:38:01,440 --> 00:38:05,300 If this is the same g and this has particular singularity 514 00:38:05,300 --> 00:38:07,220 [INAUDIBLE]. 515 00:38:07,220 --> 00:38:09,446 But at the end of the day, it doesn't matter. 516 00:38:15,550 --> 00:38:18,770 So now I have gained something that I didn't have before. 517 00:38:18,770 --> 00:38:21,060 That is, in principle I hit alpha 518 00:38:21,060 --> 00:38:25,320 and beta, my two exponents, I'm able to figure out 519 00:38:25,320 --> 00:38:27,190 what delta is. 520 00:38:27,190 --> 00:38:30,320 And actually I can also figure out what gamma is , 521 00:38:30,320 --> 00:38:34,714 because gamma describes the divergence 522 00:38:34,714 --> 00:38:35,630 of the susceptibility. 523 00:38:43,560 --> 00:38:48,215 [INAUDIBLE] which is the derivative of magnetization 524 00:38:48,215 --> 00:38:51,980 with respect to field, I have to take 525 00:38:51,980 --> 00:38:55,680 another derivative of this function. 526 00:38:55,680 --> 00:38:58,270 Taking another derivative with respect to h 527 00:38:58,270 --> 00:39:00,774 will bring down another factor of delta. 528 00:39:00,774 --> 00:39:04,390 So this becomes minus 2 delta. 529 00:39:04,390 --> 00:39:10,850 Some other double derivative function h 2 to the delta. 530 00:39:10,850 --> 00:39:12,660 And susceptibilities, we are typically 531 00:39:12,660 --> 00:39:18,110 interested in the limit where the field goes to 0. 532 00:39:18,110 --> 00:39:23,420 And we define them to diverge with exponent gamma. 533 00:39:23,420 --> 00:39:32,640 So we have identified gamma to be 2 delta plus alpha minus 2. 534 00:39:40,990 --> 00:39:42,860 So we have learned something. 535 00:39:42,860 --> 00:39:45,126 Let's summarize it. 536 00:39:45,126 --> 00:39:55,400 So the consequences-- one is we established 537 00:39:55,400 --> 00:39:59,854 that same critical exponents above and below. 538 00:40:12,400 --> 00:40:15,800 Now since various quantities of interest 539 00:40:15,800 --> 00:40:18,730 are obtained by taking derivatives 540 00:40:18,730 --> 00:40:21,216 of our homogeneous function and they 541 00:40:21,216 --> 00:40:25,530 turn into homogeneous functions, we 542 00:40:25,530 --> 00:40:40,500 conclude that all quantities are homogeneous functions 543 00:40:40,500 --> 00:40:45,270 of the same combination ht to the delta. 544 00:40:45,270 --> 00:40:46,687 Same delta governs it. 545 00:40:53,160 --> 00:40:58,020 And thirdly, once we make this answer our assumption 546 00:40:58,020 --> 00:41:02,600 for the free energy, we can calculate the other exponents 547 00:41:02,600 --> 00:41:04,170 on the table. 548 00:41:04,170 --> 00:41:22,920 So all, of almost all other exponents related to 2, 549 00:41:22,920 --> 00:41:27,070 in this case alpha and delta. 550 00:41:27,070 --> 00:41:30,490 Which means that if you have a number of different exponents 551 00:41:30,490 --> 00:41:34,440 that all depend on 2, there should 552 00:41:34,440 --> 00:41:38,680 be some identities, exponent identities. 553 00:41:47,670 --> 00:41:51,620 It's these numbers in the table, we predict if all of this 554 00:41:51,620 --> 00:41:56,390 is varied have some relationships with t. 555 00:41:56,390 --> 00:42:00,700 So let's show a couple of these relationships. 556 00:42:00,700 --> 00:42:07,330 So let's look at the combination alpha plus 2 beta plus gamma. 557 00:42:07,330 --> 00:42:10,930 Measurement of heat capacity, magnetization, susceptibility. 558 00:42:10,930 --> 00:42:13,140 Three different things. 559 00:42:13,140 --> 00:42:16,550 So alpha is alpha 2. 560 00:42:16,550 --> 00:42:21,530 My beta up there is 2 minus alpha minus delta. 561 00:42:21,530 --> 00:42:28,140 My gamma is 2 delta plus alpha minus 2. 562 00:42:28,140 --> 00:42:30,350 We got algebra. 563 00:42:30,350 --> 00:42:33,370 There's one alpha minus 2 alpha plus alpha. 564 00:42:33,370 --> 00:42:36,120 Alpha is cancelled. 565 00:42:36,120 --> 00:42:40,410 Minus 2 deltas plus 2 deltas then it does cancel. 566 00:42:40,410 --> 00:42:45,342 I have 2 times 2 minus 2, so that 2. 567 00:42:45,342 --> 00:42:51,250 So the prediction is that you take some line on the table, 568 00:42:51,250 --> 00:42:55,000 add alpha, beta, 2 beta plus gamma, they 569 00:42:55,000 --> 00:42:56,390 should add up to one. 570 00:42:56,390 --> 00:42:59,820 So let's pick something. 571 00:42:59,820 --> 00:43:04,500 Let's pick a first-- actually, let's 572 00:43:04,500 --> 00:43:08,070 pick the last line that has a negative alpha. 573 00:43:08,070 --> 00:43:11,270 So let's do n equals to 3. 574 00:43:11,270 --> 00:43:17,160 For n equals to 3 I have alpha which is minus .12. 575 00:43:17,160 --> 00:43:25,800 I have twice beta, that is .37, so that becomes 74. 576 00:43:25,800 --> 00:43:33,860 And then I have gamma, which is 1.39. 577 00:43:33,860 --> 00:43:34,856 So this is 74. 578 00:43:37,880 --> 00:43:43,990 I have 9 plus 413 minus 2, which is 1. 579 00:43:43,990 --> 00:43:52,880 I have 3 plus 7, which is 10, minus 1, which is 9. 580 00:43:52,880 --> 00:43:56,625 But then I had a 1 that was carried over, so I will have 0. 581 00:43:56,625 --> 00:44:00,160 So then I have 1, so 201. 582 00:44:00,160 --> 00:44:02,650 Not bad. 583 00:44:02,650 --> 00:44:07,705 Now this goes by the name of the Rushbrooke identity. 584 00:44:16,010 --> 00:44:19,830 The Rushbrooke made a simple manipulation 585 00:44:19,830 --> 00:44:24,960 based on thermodynamics and you have a relationship with these. 586 00:44:24,960 --> 00:44:27,980 Let's do another one. 587 00:44:27,980 --> 00:44:33,450 Let's do delta and subtract 1 from it. 588 00:44:33,450 --> 00:44:34,820 What is my delta? 589 00:44:34,820 --> 00:44:39,360 I have delta to the delta 2 plus alpha minus delta. 590 00:44:41,864 --> 00:44:45,550 This is small delta versus big delta. 591 00:44:45,550 --> 00:44:47,358 And then I have minus 1. 592 00:44:50,350 --> 00:44:55,184 Taking that into the numerator with the common denominator 593 00:44:55,184 --> 00:44:59,880 of 2 plus alpha minus delta, this minus delta 594 00:44:59,880 --> 00:45:04,300 becomes plus delta, which this becomes 2 delta minus 595 00:45:04,300 --> 00:45:07,730 alpha minus 2. 596 00:45:07,730 --> 00:45:15,362 2 delta 597 00:45:15,362 --> 00:45:17,860 AUDIENCE: Should that be a minus alpha in the denominator? 598 00:45:17,860 --> 00:45:19,696 PROFESSOR: It better be. 599 00:45:19,696 --> 00:45:20,195 Yes. 600 00:45:25,430 --> 00:45:28,060 2 delta plus alpha minus 2. 601 00:45:28,060 --> 00:45:30,952 Then we can read off the gamma. 602 00:45:30,952 --> 00:45:33,097 So this is gamma over beta. 603 00:45:35,720 --> 00:45:39,740 And let's check this, let's say for m equals to 2. 604 00:45:42,770 --> 00:45:46,140 No, let's check it for m plus 21, for the following 605 00:45:46,140 --> 00:45:53,870 reason, that for n equals to 1, what we have for delta is 606 00:45:53,870 --> 00:46:01,500 4.8 minus 1, which would be 3.8. 607 00:46:01,500 --> 00:46:06,170 And on the other side, we have gamma over beta. 608 00:46:06,170 --> 00:46:12,150 Gamma is 1.24, roughly, divided by beta .33, 609 00:46:12,150 --> 00:46:14,870 which is roughly one third. 610 00:46:14,870 --> 00:46:17,520 So I multiply this by 3. 611 00:46:17,520 --> 00:46:27,420 And that becomes 3.72. 612 00:46:27,420 --> 00:46:34,438 This one is known after another famous physicist, Ben Widom, 613 00:46:34,438 --> 00:46:35,935 as the Widom identity. 614 00:46:39,930 --> 00:46:41,690 So that's nice. 615 00:46:41,690 --> 00:46:48,620 We can start learning that although we don't know anything 616 00:46:48,620 --> 00:46:52,910 about this table, these are not independent numbers. 617 00:46:52,910 --> 00:46:55,060 There's relationship between them. 618 00:46:55,060 --> 00:46:58,975 And they're named after famous physicists. 619 00:46:58,975 --> 00:46:59,475 Yes? 620 00:46:59,475 --> 00:47:03,075 AUDIENCE: Can we briefly go over again what extra assumption 621 00:47:03,075 --> 00:47:06,024 we had put in to get these in and these out? 622 00:47:08,690 --> 00:47:11,642 Is it just that we have this homogeneous function 623 00:47:11,642 --> 00:47:12,415 [INAUDIBLE]? 624 00:47:12,415 --> 00:47:13,980 PROFESSOR: That's right. 625 00:47:13,980 --> 00:47:18,230 So you assume that the singularity 626 00:47:18,230 --> 00:47:22,600 in the vicinity of the critical point as a function 627 00:47:22,600 --> 00:47:26,300 of deviations from that critical point 628 00:47:26,300 --> 00:47:29,302 can be expressed as a homogeneous function. 629 00:47:29,302 --> 00:47:33,900 The homogeneous function, you can rearrange any way you like. 630 00:47:33,900 --> 00:47:37,610 One nice way to rearrange it is in this fashion. 631 00:47:37,610 --> 00:47:41,940 It will depend, the homogeneous function on two exponents. 632 00:47:41,940 --> 00:47:44,540 I chose to write it as 2 minus alpha 633 00:47:44,540 --> 00:47:48,290 so that one of the exponents would immediately be alpha. 634 00:47:48,290 --> 00:47:50,530 The other one I couldn't immediately 635 00:47:50,530 --> 00:47:52,770 write in terms of beta or gamma. 636 00:47:52,770 --> 00:47:55,740 I had to do these manipulations to find out 637 00:47:55,740 --> 00:47:59,290 what the relationship [INAUDIBLE]. 638 00:47:59,290 --> 00:48:03,700 But the physics of it is simple. 639 00:48:03,700 --> 00:48:09,000 That is, once you know the singularity of a free energy, 640 00:48:09,000 --> 00:48:11,110 various other quantities you obtain 641 00:48:11,110 --> 00:48:13,090 by taking derivatives of the free energy. 642 00:48:13,090 --> 00:48:17,460 That's [INAUDIBLE] And so then you 643 00:48:17,460 --> 00:48:19,862 would have the singular behavior of [INAUDIBLE]. 644 00:48:27,110 --> 00:48:34,530 So I started by saying that all other exponents, 645 00:48:34,530 --> 00:48:39,640 but then I realized we have nothing so far that tells us 646 00:48:39,640 --> 00:48:43,840 anything about mu and eta. 647 00:48:43,840 --> 00:48:48,490 Because mu and eta relate to correlations. 648 00:48:48,490 --> 00:48:51,190 They are in microscopic quantities. 649 00:48:51,190 --> 00:48:55,740 Alpha, beta, gamma depend on macroscopic thermodynamic 650 00:48:55,740 --> 00:48:59,140 quantities, magnetization susceptibility. 651 00:48:59,140 --> 00:49:03,440 So there's no way that I will be able to get information, 652 00:49:03,440 --> 00:49:04,400 almost. 653 00:49:04,400 --> 00:49:07,290 No easy way or no direct way to get information 654 00:49:07,290 --> 00:49:09,080 about mu and eta. 655 00:49:11,680 --> 00:49:19,630 So I will go to assumption 2.0. 656 00:49:19,630 --> 00:49:25,720 Go to the next version of the homogeneity assumption, which 657 00:49:25,720 --> 00:49:29,580 is to emphasize that we certainly 658 00:49:29,580 --> 00:49:32,930 know, again from physics and the relationship 659 00:49:32,930 --> 00:49:35,330 between susceptibility and correlations, 660 00:49:35,330 --> 00:49:37,810 that the reason for the divergence 661 00:49:37,810 --> 00:49:42,200 of the susceptibility is that the correlations become large. 662 00:49:42,200 --> 00:49:45,740 So we'll emphasize that. 663 00:49:45,740 --> 00:49:51,170 So let's write our ansatz not about the free energy, 664 00:49:51,170 --> 00:49:54,730 but about the correlation length. 665 00:49:54,730 --> 00:50:02,044 So let's replace that ansatz with homogeneity 666 00:50:02,044 --> 00:50:04,474 of correlation length. 667 00:50:13,240 --> 00:50:17,680 So once more, we have a structure 668 00:50:17,680 --> 00:50:21,784 where is a line that is terminate 669 00:50:21,784 --> 00:50:25,780 when two parameters, t and h go to 0. 670 00:50:25,780 --> 00:50:30,650 And we know that on approaching this point, 671 00:50:30,650 --> 00:50:33,260 the system will become cloudy. 672 00:50:33,260 --> 00:50:36,880 There's a correlation length that 673 00:50:36,880 --> 00:50:41,430 diverges on approaching that point a function of these two 674 00:50:41,430 --> 00:50:42,640 arguments. 675 00:50:42,640 --> 00:50:45,280 I'm going to make the same homogeneity assumption 676 00:50:45,280 --> 00:50:46,520 for the correlation length. 677 00:50:46,520 --> 00:50:48,460 And again, this is an assumption. 678 00:50:48,460 --> 00:50:52,805 I say that this is a to the minus mu. 679 00:50:52,805 --> 00:50:56,770 The exponent mu was a divergence of the correlation length. 680 00:50:56,770 --> 00:51:00,830 Some other function, it's not that first g that we wrote. 681 00:51:00,830 --> 00:51:06,610 Let's call it g psi of ht to the delta. 682 00:51:12,380 --> 00:51:17,650 So we never discussed it, but this function immediately 683 00:51:17,650 --> 00:51:22,170 also tells me if you approach the critical point 684 00:51:22,170 --> 00:51:25,820 along the criticalizer term, how does the correlation length 685 00:51:25,820 --> 00:51:31,540 diverge through the various tricks that we have discussed? 686 00:51:31,540 --> 00:51:35,976 But this is going to be telling me something more 687 00:51:35,976 --> 00:51:44,670 if from here, I can reproduce my scaling assumption 1.0. 688 00:51:44,670 --> 00:51:49,860 So there is one other step that I can make. 689 00:51:49,860 --> 00:52:02,670 Assume divergence of c is responsible-- 690 00:52:02,670 --> 00:52:11,275 let's call it even solely responsible-- for singular 691 00:52:11,275 --> 00:52:11,775 behavior. 692 00:52:17,475 --> 00:52:21,670 And you say, what does all of this mean? 693 00:52:21,670 --> 00:52:27,150 So let's say that I have a system could be my magnet, 694 00:52:27,150 --> 00:52:33,014 could be my liquid gas that has size l on each search. 695 00:52:36,170 --> 00:52:44,830 And I calculate the partition function log z. 696 00:52:44,830 --> 00:52:49,935 Log z will certainly have the part that is regular. 697 00:52:49,935 --> 00:52:55,280 Well-- log z will have a part that is certainly-- let's 698 00:52:55,280 --> 00:52:57,620 say the contribution phonons, all kinds 699 00:52:57,620 --> 00:53:00,880 of other regular things that don't have anything 700 00:53:00,880 --> 00:53:03,570 to do with singularity of the system. 701 00:53:03,570 --> 00:53:09,550 Those things will give you some regular function. 702 00:53:09,550 --> 00:53:11,580 But one thing that I know for sure 703 00:53:11,580 --> 00:53:14,120 is that the answer is going to be extensive. 704 00:53:14,120 --> 00:53:19,050 If I have any nice thermodynamic system 705 00:53:19,050 --> 00:53:24,810 and I am in v dimensions, then it 706 00:53:24,810 --> 00:53:28,686 will be proportional to the volume of that system 707 00:53:28,686 --> 00:53:29,618 that I have. 708 00:53:32,420 --> 00:53:38,120 Now the way that I have written it is not entirely nice, 709 00:53:38,120 --> 00:53:44,630 because log z is-- a log is a dimensionless quantity. 710 00:53:44,630 --> 00:53:47,940 Maybe I measured my length in meters or centimeters 711 00:53:47,940 --> 00:53:51,262 or whatever, so I have dimensions here. 712 00:53:51,262 --> 00:53:56,900 So it makes sense to pick some landscape to dimensionalize it 713 00:53:56,900 --> 00:54:00,470 before multiplying it by some kind of irregular function 714 00:54:00,470 --> 00:54:04,525 of whatever I have, t and h, for example. 715 00:54:10,370 --> 00:54:15,580 But what about the singular part? 716 00:54:15,580 --> 00:54:18,770 For the singular part, the statement 717 00:54:18,770 --> 00:54:21,270 was that somehow it was a connective behavior. 718 00:54:21,270 --> 00:54:23,350 It involved many, many degrees of freedom. 719 00:54:23,350 --> 00:54:27,665 We saw for the heat capacity of the solid at low temperatures, 720 00:54:27,665 --> 00:54:32,040 it came from long wavelength degrees of freedom. 721 00:54:32,040 --> 00:54:36,120 So no lattice parameter is going to be important. 722 00:54:36,120 --> 00:54:41,272 So one thing that I could do, maintaining extensivity, 723 00:54:41,272 --> 00:54:48,605 is to divide by l over c times something. 724 00:54:52,750 --> 00:54:55,030 So that's the only thing that I did 725 00:54:55,030 --> 00:55:00,100 to ensure that extensivity is maintained when 726 00:55:00,100 --> 00:55:05,025 I have kind of benign landscape, but in addition a landscape 727 00:55:05,025 --> 00:55:08,940 that is divergent. 728 00:55:08,940 --> 00:55:12,095 Now you can see that immediately that says that log 729 00:55:12,095 --> 00:55:18,330 z singular as a function of t and h, 730 00:55:18,330 --> 00:55:23,726 will be proportional to c to the minus t. 731 00:55:23,726 --> 00:55:26,000 And using that formula, it will be 732 00:55:26,000 --> 00:55:31,150 proportional to t to the du, some other scaling function. 733 00:55:31,150 --> 00:55:34,718 And it's go back to gf ht to the delta. 734 00:55:42,040 --> 00:55:46,650 Physically, what it's saying is that when 735 00:55:46,650 --> 00:55:52,550 I am very close, but not quite at the critical point, 736 00:55:52,550 --> 00:55:56,580 I have a long correlation length, much larger 737 00:55:56,580 --> 00:56:00,280 than microscopic length scale of my system. 738 00:56:00,280 --> 00:56:06,010 So what I can say is that within a correlation length, 739 00:56:06,010 --> 00:56:11,375 my degrees of freedom for magentization or whatever it is 740 00:56:11,375 --> 00:56:15,120 are very much coupled to each other. 741 00:56:15,120 --> 00:56:17,940 So maybe what I can do is I can regard 742 00:56:17,940 --> 00:56:21,480 this as an independent lock. 743 00:56:21,480 --> 00:56:25,130 And how many independent locks do I have? 744 00:56:25,130 --> 00:56:28,060 It is l over c to the d. 745 00:56:28,060 --> 00:56:30,710 So the statement roughly is a part 746 00:56:30,710 --> 00:56:34,390 of the assumption is that this correlation length that is 747 00:56:34,390 --> 00:56:36,340 getting bigger and bigger. 748 00:56:36,340 --> 00:56:38,700 Because things are correlated, the number 749 00:56:38,700 --> 00:56:40,670 of independent degrees of freedom that you 750 00:56:40,670 --> 00:56:43,980 are having gets smaller and smaller. 751 00:56:43,980 --> 00:56:47,550 And that's changing the number of degrees of freedom 752 00:56:47,550 --> 00:56:51,400 is responsible for the singular behavior of the free energy. 753 00:56:51,400 --> 00:56:56,520 If I make this assumption about this correlation then diverges, 754 00:56:56,520 --> 00:56:57,784 then I will get this form. 755 00:57:01,090 --> 00:57:05,920 So now my ansatz 2.0 matches my ansatz 1.0 756 00:57:05,920 --> 00:57:09,940 provided du is 2 minus alpha. 757 00:57:09,940 --> 00:57:16,950 So I have du2 plus 2 minus alpha which 758 00:57:16,950 --> 00:57:20,734 is known after Brian Josephson, so this 759 00:57:20,734 --> 00:57:26,662 is the Josephson relation. 760 00:57:26,662 --> 00:57:33,600 And it is different from the other exponent identities 761 00:57:33,600 --> 00:57:37,100 that we have because it explicitly 762 00:57:37,100 --> 00:57:39,790 depends on the dimensionality of space. 763 00:57:39,790 --> 00:57:42,060 d appears in the problem. 764 00:57:42,060 --> 00:57:45,947 It's called hyperscale for that reason. 765 00:57:50,420 --> 00:57:51,200 Yes? 766 00:57:51,200 --> 00:57:54,230 AUDIENCE: So does the assumption that the divergence in c 767 00:57:54,230 --> 00:57:56,479 is solely responsible for the singular behavior, what 768 00:57:56,479 --> 00:57:58,020 are we excluding when we assume that? 769 00:57:58,020 --> 00:58:02,066 What else could happen that would make that not true? 770 00:58:02,066 --> 00:58:05,440 PROFESSOR: Well, what is appearing here maybe 771 00:58:05,440 --> 00:58:10,401 will have some singular function of t and h. 772 00:58:10,401 --> 00:58:12,275 AUDIENCE: So this similar to what 773 00:58:12,275 --> 00:58:15,581 we were assuming before when we said that our free energy could 774 00:58:15,581 --> 00:58:17,890 have some regular part that depends 775 00:58:17,890 --> 00:58:21,874 on [INAUDIBLE] the part that [INAUDIBLE]. 776 00:58:21,874 --> 00:58:23,297 PROFESSOR: Yes, exactly. 777 00:58:26,430 --> 00:58:30,620 But once again, the truth is really whether or not 778 00:58:30,620 --> 00:58:33,120 this matches up with experiments. 779 00:58:33,120 --> 00:58:39,300 So let's, for example, pick anything in that 780 00:58:39,300 --> 00:58:41,040 table, v equals to t. 781 00:58:41,040 --> 00:58:45,660 Let's pick n goes to 2, which we haven't done so far. 782 00:58:45,660 --> 00:58:54,610 And so what the formula would say is 3 times mu. 783 00:58:54,610 --> 00:59:03,670 Mu for the superfluid is 67 is 2 minus-- well, 784 00:59:03,670 --> 00:59:07,990 alpha is almost 0 but slightly negative. 785 00:59:07,990 --> 00:59:16,240 So it is 0.01. 786 00:59:16,240 --> 00:59:17,270 And what do we have? 787 00:59:17,270 --> 00:59:26,440 3 times 67 is 2.01. 788 00:59:26,440 --> 00:59:27,750 So it matches. 789 00:59:27,750 --> 00:59:31,830 Actually, we say, well, why do you emphasize 790 00:59:31,830 --> 00:59:34,300 that it's the function of dimension? 791 00:59:34,300 --> 00:59:39,040 Well, a little bit later on in the course, 792 00:59:39,040 --> 00:59:44,990 we will do an exact solution of the so-called 2D Ising model. 793 00:59:47,670 --> 00:59:51,820 So this is a system that first wants to be close to 2, 794 00:59:51,820 --> 00:59:53,120 n equals to 1. 795 00:59:53,120 --> 00:59:56,780 And it was an important thing that people could actually 796 00:59:56,780 --> 01:00:00,860 solve an interacting problem, not in three dimensions 797 01:00:00,860 --> 01:00:01,760 but in two. 798 01:00:01,760 --> 01:00:05,780 And the exponents for that, alpha is 0, 799 01:00:05,780 --> 01:00:08,860 but it really is a logarithmic divergence. 800 01:00:08,860 --> 01:00:11,700 Beta is 1/8. 801 01:00:11,700 --> 01:00:20,060 Gamma is 7/4, delta is 15, mu is 1, and eta is 1/4. 802 01:00:20,060 --> 01:00:28,190 And we can check now for this v equals to 2 n 803 01:00:28,190 --> 01:00:32,980 equals to 1 that we have two times 804 01:00:32,980 --> 01:00:35,970 our mu, which exactly is known to be 805 01:00:35,970 --> 01:00:43,410 1 is 2 minus logarithmic divergence corresponding to 0. 806 01:00:43,410 --> 01:00:46,036 So again, there's something that works. 807 01:00:50,600 --> 01:00:56,150 One thing that you may want to see and look at 808 01:00:56,150 --> 01:01:01,500 is that the ansatz that we made first also 809 01:01:01,500 --> 01:01:05,820 works for the result of saddlepoint, 810 01:01:05,820 --> 01:01:09,930 not surprisingly because again in the saddlepoint 811 01:01:09,930 --> 01:01:14,020 we start with a singular free energy and go through all this. 812 01:01:14,020 --> 01:01:18,000 But it does not work for this type of scaling, 813 01:01:18,000 --> 01:01:23,690 because 2 minus alpha would be 0 is not 814 01:01:23,690 --> 01:01:29,925 equal to d times one half, except in the case of four 815 01:01:29,925 --> 01:01:30,425 dimensions. 816 01:01:33,060 --> 01:01:37,390 So somehow, this ansatz and this picture 817 01:01:37,390 --> 01:01:42,680 breaks down within the saddlepoint approximation. 818 01:01:42,680 --> 01:01:47,180 If you remember what we did when we calculated fluctuation 819 01:01:47,180 --> 01:01:49,815 corrections for the saddlepoint, you 820 01:01:49,815 --> 01:01:54,800 got actually an exponent alpha that was 2 minus mu over 2. 821 01:01:54,800 --> 01:02:00,400 So the fluctuating part that we get around the saddlepoint 822 01:02:00,400 --> 01:02:02,420 does satisfy this. 823 01:02:02,420 --> 01:02:04,995 But on top of that there's another part that 824 01:02:04,995 --> 01:02:07,730 is doe to the saddlepoint value itself 825 01:02:07,730 --> 01:02:10,720 that violates this hyperscaling solution. 826 01:02:10,720 --> 01:02:11,220 Yes? 827 01:02:11,220 --> 01:02:15,280 AUDIENCE: Empirically, how well can we probe the dependence 828 01:02:15,280 --> 01:02:19,192 on dimensionality that we're finding in these expressions? 829 01:02:19,192 --> 01:02:21,050 PROFESSOR: Experimentally, we can 830 01:02:21,050 --> 01:02:23,730 do d equals to 2 d equals to 3. 831 01:02:23,730 --> 01:02:27,948 And computer simulations we can also do d equals to 2 d 832 01:02:27,948 --> 01:02:28,840 equals to 3. 833 01:02:28,840 --> 01:02:31,695 Very soon, we will do analytical expressions 834 01:02:31,695 --> 01:02:35,310 where we will be in 3.99 dimensions. 835 01:02:35,310 --> 01:02:38,960 So we will be coming down conservatively around 4. 836 01:02:38,960 --> 01:02:42,260 So mathematically, we can play tricks such as that. 837 01:02:42,260 --> 01:02:45,983 But certainly empirically, in the sense of experimentally 838 01:02:45,983 --> 01:02:48,456 we are at a disadvantage in those languages. 839 01:02:51,490 --> 01:02:51,990 OK? 840 01:03:00,220 --> 01:03:03,190 So we are making progress. 841 01:03:03,190 --> 01:03:05,600 We have made our way across this table. 842 01:03:05,600 --> 01:03:09,110 We have also an identity that involves mu. 843 01:03:09,110 --> 01:03:11,320 But so far I haven't said anything about eta. 844 01:03:15,420 --> 01:03:21,950 I can say something about the eta reasonably simply, 845 01:03:21,950 --> 01:03:24,255 but then you try to build something profound 846 01:03:24,255 --> 01:03:27,290 based on that. 847 01:03:27,290 --> 01:03:35,046 So let's look at exactly at tc, at the critical point. 848 01:03:37,860 --> 01:03:43,262 So let's say you are sitting at t and h equals to 0. 849 01:03:43,262 --> 01:03:45,980 You have to prepare your system at that point. 850 01:03:45,980 --> 01:03:49,450 There's nothing physically that says you can't. 851 01:03:49,450 --> 01:03:53,640 At that point, you can look at correlations. 852 01:03:53,640 --> 01:03:57,420 And the exponent eta for example is a characteristic 853 01:03:57,420 --> 01:03:59,150 of those correlations. 854 01:03:59,150 --> 01:04:05,490 And one of the things that we have is that m of x m of 0, 855 01:04:05,490 --> 01:04:10,966 the connected parts-- well, actually at the critical point 856 01:04:10,966 --> 01:04:13,340 we don't even have to put the connected part 857 01:04:13,340 --> 01:04:16,840 because the average of n is going to be 0. 858 01:04:16,840 --> 01:04:19,725 But this is a quantity that behaves 859 01:04:19,725 --> 01:04:24,570 as 1 over the separation that's actually 860 01:04:24,570 --> 01:04:28,560 include two possible points, x minus y. 861 01:04:28,560 --> 01:04:34,760 When we did the case of the fluctuations 862 01:04:34,760 --> 01:04:39,280 at the critical point within the saddlepoint method, 863 01:04:39,280 --> 01:04:42,445 we found that the behavior was like the Coulomb law. 864 01:04:42,445 --> 01:04:45,830 It was falling off as 1x to the d minus 2. 865 01:04:45,830 --> 01:04:48,320 But we said that experiment indicated 866 01:04:48,320 --> 01:04:52,940 that there is a small correction for this 867 01:04:52,940 --> 01:04:54,580 that we indicate with exponent eta. 868 01:04:54,580 --> 01:04:59,850 So that was how the exponent eta was defined. 869 01:04:59,850 --> 01:05:04,560 So can we have an identity that involves the exponent eta? 870 01:05:04,560 --> 01:05:08,330 We actually have seen how to do this already. 871 01:05:08,330 --> 01:05:12,050 Because we know that in general, the susceptibilities 872 01:05:12,050 --> 01:05:16,400 are related to integrals of the correlation functions. 873 01:05:20,860 --> 01:05:25,850 Now if I put this power law over here, 874 01:05:25,850 --> 01:05:28,110 you can see that the answer is like trying 875 01:05:28,110 --> 01:05:32,100 to be integrate x squared all the way to infinity down. 876 01:05:32,100 --> 01:05:35,512 So it will be divergent and that's no problem. 877 01:05:35,512 --> 01:05:37,720 At the critical point we know that the susceptibility 878 01:05:37,720 --> 01:05:40,010 is divergent. 879 01:05:40,010 --> 01:05:48,500 But you say, OK, if I'm away from the critical point, 880 01:05:48,500 --> 01:05:53,940 then I will use this formula, but only 881 01:05:53,940 --> 01:05:57,060 up to the correlation length. 882 01:05:57,060 --> 01:06:00,250 And I say that beyond the correlation length, 883 01:06:00,250 --> 01:06:03,010 then the correlations will decay exponentially. 884 01:06:03,010 --> 01:06:07,810 That's too rapid a falloff, and essentially the only part 885 01:06:07,810 --> 01:06:10,155 that's contributing is because what 886 01:06:10,155 --> 01:06:13,110 was happening at the critical point. 887 01:06:13,110 --> 01:06:20,345 Once I do that, I have to integrate ddx over x to the d 888 01:06:20,345 --> 01:06:24,290 minus 2 plus eta up to the correlation length. 889 01:06:24,290 --> 01:06:27,040 The answer will be proportional to the correlation length 890 01:06:27,040 --> 01:06:30,400 to the power of 2 minus eta. 891 01:06:30,400 --> 01:06:36,230 And this will be proportional to p to the power of c goes 892 01:06:36,230 --> 01:06:38,180 [INAUDIBLE] to the minus mu. 893 01:06:38,180 --> 01:06:39,698 2 minus eta times mu. 894 01:06:44,920 --> 01:06:46,696 But we know that the susceptibilities 895 01:06:46,696 --> 01:06:51,210 diverge as t to the minus gamma. 896 01:06:51,210 --> 01:06:55,320 So we have established an exponent identity 897 01:06:55,320 --> 01:07:04,210 that tells us that gamma is 2 minus eta times mu. 898 01:07:04,210 --> 01:07:09,170 And this is known as the Fisher identity, after Michael Fisher. 899 01:07:13,341 --> 01:07:16,500 Again, you can see that in all of the cases in three 900 01:07:16,500 --> 01:07:18,730 dimensions that we are dealing with, 901 01:07:18,730 --> 01:07:20,668 exponent eta is roughly 0. 902 01:07:20,668 --> 01:07:24,350 It's 0.04 And all of our gammas are 903 01:07:24,350 --> 01:07:27,650 roughly twice what our mus are in that table. 904 01:07:27,650 --> 01:07:30,630 It's time we get that table checked. 905 01:07:30,630 --> 01:07:34,790 The one case that I have on that table where eta is not 0 906 01:07:34,790 --> 01:07:41,162 is when I'm looking at v positive 2 where eta is 1/4. 907 01:07:41,162 --> 01:07:44,870 So I take 2 minus 1/4, multiply it 908 01:07:44,870 --> 01:07:47,640 by the mu that is one in two dimension, 909 01:07:47,640 --> 01:07:51,162 and the answer is the 7/4, which we 910 01:07:51,162 --> 01:07:53,830 have for the exponent gamma over there. 911 01:08:02,075 --> 01:08:05,459 So we have now the identity that is 912 01:08:05,459 --> 01:08:09,810 applicable to the last exponents. 913 01:08:09,810 --> 01:08:13,160 So all of this works. 914 01:08:13,160 --> 01:08:16,359 Let's now take the conceptual leap 915 01:08:16,359 --> 01:08:19,185 that then allows us to do what we 916 01:08:19,185 --> 01:08:22,890 will do later on to get the exponents. 917 01:08:22,890 --> 01:08:25,890 Basically, you can see that what we have imposed 918 01:08:25,890 --> 01:08:30,569 here conceptually is the following. 919 01:08:30,569 --> 01:08:35,109 That when I'm away from the critical point, 920 01:08:35,109 --> 01:08:39,000 I look at the correlations of this important statistical 921 01:08:39,000 --> 01:08:40,562 field. 922 01:08:40,562 --> 01:08:43,640 And I find that they fall off with separation, 923 01:08:43,640 --> 01:08:46,810 according to some power. 924 01:08:46,810 --> 01:08:52,580 And the reason is that at the critical point, 925 01:08:52,580 --> 01:08:54,970 the correlation length has gone to infinity. 926 01:08:54,970 --> 01:08:57,279 That's not the length scale that you have to play with. 927 01:08:57,279 --> 01:09:02,890 You can divide x minus y divided by c, which is what we do away 928 01:09:02,890 --> 01:09:06,765 from the critical point. c has gone to infinity. 929 01:09:06,765 --> 01:09:09,410 The other length scale that we are worried about 930 01:09:09,410 --> 01:09:12,689 are things that go into the microscopics. 931 01:09:12,689 --> 01:09:18,260 but we are assuming that microscopics is irrelevant. 932 01:09:18,260 --> 01:09:21,939 It has been washed out. 933 01:09:21,939 --> 01:09:24,810 So if we don't have a large length scale, 934 01:09:24,810 --> 01:09:27,670 if we don't have a short length scale, some function 935 01:09:27,670 --> 01:09:30,960 of distance, how can it decay? 936 01:09:30,960 --> 01:09:33,510 The only way it can decay is [INAUDIBLE]. 937 01:09:36,510 --> 01:09:45,010 So this statement is that when we are at a critical point, 938 01:09:45,010 --> 01:09:47,479 I look at some correlation. 939 01:09:47,479 --> 01:09:50,170 And this was the magnetization correlation. 940 01:09:50,170 --> 01:09:54,020 But I can look at correlation of anything else 941 01:09:54,020 --> 01:09:56,069 as a function of separation. 942 01:10:01,820 --> 01:10:06,990 And this will only fall off as some power of separation. 943 01:10:06,990 --> 01:10:09,400 Another way of writing it is that if I 944 01:10:09,400 --> 01:10:13,500 were to multiply this by some length scale, 945 01:10:13,500 --> 01:10:17,570 so rather than looking at things that are some distance apart 946 01:10:17,570 --> 01:10:20,030 at twice that distance apart or hundred times 947 01:10:20,030 --> 01:10:24,006 that distance apart, I will reproduce the correlation 948 01:10:24,006 --> 01:10:32,970 that I have up to some other of the scale factor. 949 01:10:32,970 --> 01:10:35,520 So the scale factor here we can read off 950 01:10:35,520 --> 01:10:38,730 has to be related to t minus 2 plus eta. 951 01:10:38,730 --> 01:10:41,450 But essentially, this is a statement again 952 01:10:41,450 --> 01:10:45,680 about homogeneity of correlation functions 953 01:10:45,680 --> 01:10:47,968 when you are at a critical point. 954 01:10:50,510 --> 01:10:54,496 So this is a symmetry here. 955 01:10:54,496 --> 01:11:00,260 It says you take your statistical correlations 956 01:11:00,260 --> 01:11:02,670 and you look at them at the larger scale 957 01:11:02,670 --> 01:11:04,860 or at the shorter scale. 958 01:11:04,860 --> 01:11:08,400 And up to some overall scale factor, 959 01:11:08,400 --> 01:11:10,280 you reproduce what you had before. 960 01:11:14,020 --> 01:11:20,190 So this is something to do with invariance on the scale. 961 01:11:28,290 --> 01:11:32,930 This scaling variance is some property 962 01:11:32,930 --> 01:11:37,190 that was popular a while ago as being associated 963 01:11:37,190 --> 01:11:38,772 with the kind of geometrical objects 964 01:11:38,772 --> 01:11:39,980 that you would call fractals. 965 01:11:46,100 --> 01:11:54,000 So the statement is that if I go across my system 966 01:11:54,000 --> 01:11:58,590 and there is some pattern of magnetization fluctuations, 967 01:11:58,590 --> 01:12:00,416 let's say I look at it. 968 01:12:00,416 --> 01:12:02,465 I'm going along this direction x. 969 01:12:04,980 --> 01:12:11,240 And I plot at some particular configuration 970 01:12:11,240 --> 01:12:13,580 that is dominant and is contributing 971 01:12:13,580 --> 01:12:17,092 to my free energy, the magnetization, 972 01:12:17,092 --> 01:12:21,490 that it has a shape that has this characteristic self 973 01:12:21,490 --> 01:12:25,620 similarity kind of maybe looking like a mountain landscape. 974 01:12:28,420 --> 01:12:31,510 And the statement is that if I were 975 01:12:31,510 --> 01:12:40,650 to take a part of that landscape and then blow it up, 976 01:12:40,650 --> 01:12:45,290 I will generate a pattern that is of course not the same 977 01:12:45,290 --> 01:12:46,210 as the first one. 978 01:12:46,210 --> 01:12:48,970 It is not exactly scale invariant. 979 01:12:48,970 --> 01:12:53,150 But it has the same kind of statistics as the one 980 01:12:53,150 --> 01:12:57,220 that I had originally after I multiplied this axis 981 01:12:57,220 --> 01:12:59,088 by some factor lambda. 982 01:13:03,770 --> 01:13:05,345 Yes? 983 01:13:05,345 --> 01:13:10,960 AUDIENCE: Under what length scales are those subsimilarity 984 01:13:10,960 --> 01:13:15,032 properties evident and how do they compare to the length 985 01:13:15,032 --> 01:13:16,740 scale over which you're doing your course 986 01:13:16,740 --> 01:13:18,356 grading for this field? 987 01:13:18,356 --> 01:13:25,476 PROFESSOR: OK, so basically we expect this to be applicable 988 01:13:25,476 --> 01:13:27,230 presumably at length scales that are 989 01:13:27,230 --> 01:13:28,780 less than the size of your system 990 01:13:28,780 --> 01:13:31,330 because once I get to the size of the system 991 01:13:31,330 --> 01:13:33,950 I can't blow it up further or whatever. 992 01:13:33,950 --> 01:13:37,560 It has to certainly be larger than whatever 993 01:13:37,560 --> 01:13:41,870 the coarse-graining length is, or the length scale at which 994 01:13:41,870 --> 01:13:45,825 I have confidence that I have washed out the microscoping 995 01:13:45,825 --> 01:13:48,340 details. 996 01:13:48,340 --> 01:13:50,400 Now that depends on the system in question, 997 01:13:50,400 --> 01:13:53,630 so I can't really give you an answer for that. 998 01:13:53,630 --> 01:13:56,140 The answer will depend on the system. 999 01:13:56,140 --> 01:13:58,040 But the point is that I'm looking 1000 01:13:58,040 --> 01:14:01,720 in the vicinity of a point where mathematically I'm 1001 01:14:01,720 --> 01:14:04,250 assured that there's a correlation length that 1002 01:14:04,250 --> 01:14:05,580 goes to infinity. 1003 01:14:05,580 --> 01:14:09,580 So maybe there is some system number 1 that average out 1004 01:14:09,580 --> 01:14:12,390 very easily, and after a distance of 10 1005 01:14:12,390 --> 01:14:14,890 I can start applying this. 1006 01:14:14,890 --> 01:14:16,730 But maybe there's some other system 1007 01:14:16,730 --> 01:14:19,520 where the microscopic degrees of freedom are very problematic 1008 01:14:19,520 --> 01:14:22,950 and I have to go further and further out before they average 1009 01:14:22,950 --> 01:14:23,730 out. 1010 01:14:23,730 --> 01:14:27,200 But in principle, since my c has gone to infinity, 1011 01:14:27,200 --> 01:14:31,240 I can just pick a bigger and bigger piece of my system 1012 01:14:31,240 --> 01:14:33,170 until that has happened. 1013 01:14:33,170 --> 01:14:36,280 So I can't tell you what the short distance length 1014 01:14:36,280 --> 01:14:39,880 scale is in the same sense that when [INAUDIBLE] says 1015 01:14:39,880 --> 01:14:44,230 that coast of Britain is fractal, well, 1016 01:14:44,230 --> 01:14:47,040 I can't tell you whether the short distance is 1017 01:14:47,040 --> 01:14:50,840 the size of a sand particle, or is it 1018 01:14:50,840 --> 01:14:54,648 the size of, I don't know, a tree or something like that. 1019 01:14:54,648 --> 01:14:55,624 I don't know. 1020 01:15:03,920 --> 01:15:13,740 So we started thinking about our original problem. 1021 01:15:13,740 --> 01:15:19,360 And constructing this Landau-Ginzburg, 1022 01:15:19,360 --> 01:15:25,270 [INAUDIBLE] that we worked with on the basis of symmetries 1023 01:15:25,270 --> 01:15:30,010 such as invariance on the rotation, et cetera. 1024 01:15:30,010 --> 01:15:32,895 Somehow we've discovered that the point that we 1025 01:15:32,895 --> 01:15:37,700 are interested has an additional symmetry that maybe we 1026 01:15:37,700 --> 01:15:41,706 didn't anticipate, which is this self-similarity and scale 1027 01:15:41,706 --> 01:15:42,205 invariance. 1028 01:15:45,290 --> 01:15:49,240 So you say, OK, that's the solution to the problem. 1029 01:15:49,240 --> 01:15:52,190 Let's go back to our construction 1030 01:15:52,190 --> 01:15:55,930 of the Landau-Ginsburg theory and add 1031 01:15:55,930 --> 01:15:58,010 to the list of symmetries that have 1032 01:15:58,010 --> 01:16:03,250 to be obeyed, this additional self-similarity of scaling. 1033 01:16:03,250 --> 01:16:07,990 And that will put us at t equals to 0, h equals to 0. 1034 01:16:07,990 --> 01:16:10,880 And for example, we should be able to calculate 1035 01:16:10,880 --> 01:16:13,530 this correlation. 1036 01:16:13,530 --> 01:16:15,987 Let me expand a little bit on that because we 1037 01:16:15,987 --> 01:16:17,320 will need one other correlation. 1038 01:16:17,320 --> 01:16:19,550 Because we've said that essentially, 1039 01:16:19,550 --> 01:16:22,530 all of the properties of the system 1040 01:16:22,530 --> 01:16:25,760 I can get from two independent exponents. 1041 01:16:25,760 --> 01:16:29,690 So suppose I constructed this scale invariant theory 1042 01:16:29,690 --> 01:16:32,470 and I calculated this. 1043 01:16:32,470 --> 01:16:33,770 That would be on exponent. 1044 01:16:33,770 --> 01:16:36,420 I need another one. 1045 01:16:36,420 --> 01:16:39,560 Well, we had here a statement about alpha. 1046 01:16:39,560 --> 01:16:43,830 We made the statement that heat capacity diverges. 1047 01:16:43,830 --> 01:16:48,690 Now in the same sense that the susceptibility is a response-- 1048 01:16:48,690 --> 01:16:52,470 it came from two derivatives of the free energy with respect 1049 01:16:52,470 --> 01:16:54,000 to the field. 1050 01:16:54,000 --> 01:16:56,540 The derivative of magnetization with respect 1051 01:16:56,540 --> 01:16:59,400 to field magnetization is one derivative [INAUDIBLE]. 1052 01:16:59,400 --> 01:17:01,810 The heat capacity is also two derivatives 1053 01:17:01,810 --> 01:17:05,400 of free energy with respect to some other variable. 1054 01:17:08,400 --> 01:17:10,700 So in the same sense that there is 1055 01:17:10,700 --> 01:17:14,011 a relationship between the susceptibility 1056 01:17:14,011 --> 01:17:16,770 and an integrated correlation function, 1057 01:17:16,770 --> 01:17:21,800 there is a relationship that says that the heat capacity is 1058 01:17:21,800 --> 01:17:26,000 related to an integrated correlation function. 1059 01:17:26,000 --> 01:17:31,860 So c as a function of say t and h, let's say the singular part, 1060 01:17:31,860 --> 01:17:36,115 is going to be related to an integral of something. 1061 01:17:40,080 --> 01:17:45,420 And again, we've already seen this. 1062 01:17:45,420 --> 01:17:48,512 Essentially, you take one derivative of the free energy 1063 01:17:48,512 --> 01:17:50,470 let's say with respect the beta or temperature, 1064 01:17:50,470 --> 01:17:53,060 you get the energy. 1065 01:17:53,060 --> 01:17:55,860 And you take another derivative of the energy 1066 01:17:55,860 --> 01:17:59,300 you will get the heat capacity. 1067 01:17:59,300 --> 01:18:02,120 And then that derivative, if we write 1068 01:18:02,120 --> 01:18:05,320 in terms of the first derivative of the partition function 1069 01:18:05,320 --> 01:18:09,780 becomes converted to the variance in energy. 1070 01:18:09,780 --> 01:18:12,750 So in the same way that the susceptibility was 1071 01:18:12,750 --> 01:18:16,220 the variance of the net magnetization, 1072 01:18:16,220 --> 01:18:19,555 the heat capacity is related to the variance 1073 01:18:19,555 --> 01:18:23,580 of the net energy of the system at an even temperature. 1074 01:18:23,580 --> 01:18:26,090 The net energy of the system we can write this 1075 01:18:26,090 --> 01:18:29,400 as an integral of an energy density, 1076 01:18:29,400 --> 01:18:31,450 just as we wrote the magnetization 1077 01:18:31,450 --> 01:18:34,050 as an integral of magnetization density. 1078 01:18:34,050 --> 01:18:40,105 And then the heat capacity will be related to the correlation 1079 01:18:40,105 --> 01:18:42,486 functions of the energy density. 1080 01:18:47,226 --> 01:18:53,380 Now once more, you say that I'm at the critical point. 1081 01:18:53,380 --> 01:18:57,210 At the critical point there is no length scale. 1082 01:18:57,210 --> 01:19:02,360 So any correlation function, not only that of the magnetization, 1083 01:19:02,360 --> 01:19:08,580 should fall off as some power of separation. 1084 01:19:08,580 --> 01:19:12,920 And you can call that exponent whatever you like. 1085 01:19:12,920 --> 01:19:16,772 There is no definition for it in the literature. 1086 01:19:16,772 --> 01:19:20,300 Let me write it in the same way as magnetization 1087 01:19:20,300 --> 01:19:23,320 as d minus 2 plus eta prime. 1088 01:19:23,320 --> 01:19:28,320 So then when I go and say let's terminate it at the correlation 1089 01:19:28,320 --> 01:19:30,800 length, the answer is going to be 1090 01:19:30,800 --> 01:19:35,330 proportional to c to the 2 minus eta prime, 1091 01:19:35,330 --> 01:19:38,792 which would be t to the minus mu. 1092 01:19:38,792 --> 01:19:41,150 2 minus eta prime. 1093 01:19:41,150 --> 01:19:46,590 So then I would have alpha being mu 2 minus eta. 1094 01:19:51,550 --> 01:19:56,100 So all I need to do in principle is 1095 01:19:56,100 --> 01:20:00,160 to construct a theory, which in addition 1096 01:20:00,160 --> 01:20:02,980 to rotational invariance or there's 1097 01:20:02,980 --> 01:20:05,720 whatever is appropriate to the system in question, 1098 01:20:05,720 --> 01:20:09,860 has this statistical scale invariance. 1099 01:20:09,860 --> 01:20:13,710 Within that theory, calculate the correlation functions 1100 01:20:13,710 --> 01:20:17,740 of two quantities, such as magnetization and energy. 1101 01:20:17,740 --> 01:20:20,520 Extract two exponents. 1102 01:20:20,520 --> 01:20:22,936 Once we have two exponents, then we 1103 01:20:22,936 --> 01:20:24,920 know why your manipulations will be 1104 01:20:24,920 --> 01:20:26,597 able to calculate all the exponents. 1105 01:20:29,520 --> 01:20:32,850 So why doesn't this solve the problem? 1106 01:20:32,850 --> 01:20:36,270 The answer is that whereas I can write immediately for you 1107 01:20:36,270 --> 01:20:40,800 a term such as m squared, that is rotational invariant, 1108 01:20:40,800 --> 01:20:46,725 I don't know how to write down a theory that is scale invariant. 1109 01:20:46,725 --> 01:20:50,180 The one case where people have succeeded to do that 1110 01:20:50,180 --> 01:20:52,590 is actually two dimensions. 1111 01:20:52,590 --> 01:20:54,890 So in two dimensions, one can show 1112 01:20:54,890 --> 01:20:56,835 that this kind of scale invariance 1113 01:20:56,835 --> 01:21:00,615 is related to conformal invariance 1114 01:21:00,615 --> 01:21:04,070 and that one can explicitly write down conformal invariant 1115 01:21:04,070 --> 01:21:08,500 theories, extract exponents et cetera out of those. 1116 01:21:08,500 --> 01:21:12,860 But say in three dimensions, we don't know how to do that. 1117 01:21:12,860 --> 01:21:16,460 So we will still, with that concept 1118 01:21:16,460 --> 01:21:19,020 in the back of our mind, approach it 1119 01:21:19,020 --> 01:21:24,810 slightly differently by looking at the effects of the scale 1120 01:21:24,810 --> 01:21:27,190 transformation on the system. 1121 01:21:27,190 --> 01:21:31,140 And that's the beginning of this concept of normalization.