1 00:00:00,060 --> 00:00:02,500 The following content is provided under a Creative 2 00:00:02,500 --> 00:00:04,019 Commons license. 3 00:00:04,019 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,730 continue 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 00:00:22,350 --> 00:00:25,440 PROFESSOR: OK, let's start. 9 00:00:25,440 --> 00:00:34,410 So we were talking about melting in two dimensions, 10 00:00:34,410 --> 00:00:38,750 and the picture that you had was something 11 00:00:38,750 --> 00:00:48,570 like a triangular lattice, which at zero temperature 12 00:00:48,570 --> 00:00:55,520 has particles sitting at precise sites-- 13 00:00:55,520 --> 00:00:59,190 let's say, on this triangular lattice-- 14 00:00:59,190 --> 00:01:01,135 but then at finite temperature, the particles 15 00:01:01,135 --> 00:01:05,370 will to start to deform. 16 00:01:05,370 --> 00:01:12,600 And the deformations were indicated by a vector u. 17 00:01:12,600 --> 00:01:19,630 And the idea was that this is like an elastic material, 18 00:01:19,630 --> 00:01:23,310 as long as we're thinking about these long wavelength 19 00:01:23,310 --> 00:01:24,090 deformations. 20 00:01:24,090 --> 00:01:28,140 u and the energy costs can be written 21 00:01:28,140 --> 00:01:32,830 for an isotropic material in two dimensions in terms 22 00:01:32,830 --> 00:01:34,955 of two invariants. 23 00:01:37,750 --> 00:01:40,460 And traditionally, it is written in terms 24 00:01:40,460 --> 00:01:46,855 of the so called lame coefficients, mu and lambda. 25 00:01:52,800 --> 00:02:03,010 Where this uij, which is the strain, 26 00:02:03,010 --> 00:02:08,800 is obtained by taking derivatives of the deformation, 27 00:02:08,800 --> 00:02:12,494 the iuj, and symmetrizing it. 28 00:02:12,494 --> 00:02:16,610 This symmetrization essentially eliminates an energy 29 00:02:16,610 --> 00:02:18,475 at a cost for rotations. 30 00:02:20,990 --> 00:02:25,650 And then because of this simple quadratic translation 31 00:02:25,650 --> 00:02:29,090 of invariant form, we could also express 32 00:02:29,090 --> 00:02:32,000 this in terms of fullier mode. 33 00:02:32,000 --> 00:02:37,670 And I'm going to write the fullier description slightly 34 00:02:37,670 --> 00:02:42,370 differently than last time. 35 00:02:42,370 --> 00:02:44,920 Basically, this whole form can be 36 00:02:44,920 --> 00:02:55,625 written as u plus 2 lambda over 2 q dot u tilde of q squared. 37 00:02:59,160 --> 00:03:04,410 And the other term-- other than previously I had written things 38 00:03:04,410 --> 00:03:06,630 in terms of q dot u and q squared 39 00:03:06,630 --> 00:03:14,320 u squared-- we write it in terms of q crossed with u tilde of q 40 00:03:14,320 --> 00:03:14,820 squared. 41 00:03:23,750 --> 00:03:27,790 Essentially, you can see that this ratifies that they're 42 00:03:27,790 --> 00:03:32,560 going to have modes that are in the direction of q, 43 00:03:32,560 --> 00:03:35,290 the longitudinal modes. 44 00:03:35,290 --> 00:03:38,160 Cost is nu plus 2 lambda, and those 45 00:03:38,160 --> 00:03:42,030 that are transfers or orthogonal to the direction of q, 46 00:03:42,030 --> 00:03:45,050 whose cost is just mu. 47 00:03:45,050 --> 00:03:49,030 And clearly if I were to go into real space, 48 00:03:49,030 --> 00:03:53,720 this is kind of related to a divergence of u. 49 00:03:53,720 --> 00:03:56,750 And the divergence of u corresponds 50 00:03:56,750 --> 00:04:02,780 to essentially squeezing or expanding this deformation. 51 00:04:02,780 --> 00:04:04,670 So what these measures is essentially 52 00:04:04,670 --> 00:04:08,410 the cost of changing the density. 53 00:04:08,410 --> 00:04:12,090 And this combination is related to the bark modulus. 54 00:04:12,090 --> 00:04:14,270 You have that even for a liquid. 55 00:04:14,270 --> 00:04:16,329 So if you have a liquid, you try to squeeze it. 56 00:04:16,329 --> 00:04:19,130 There will be a bulk energy cost. 57 00:04:19,130 --> 00:04:24,580 And this term, which in the real space is kind of related 58 00:04:24,580 --> 00:04:29,360 to kern u, you would say is corresponding to making 59 00:04:29,360 --> 00:04:31,630 the rotations. 60 00:04:31,630 --> 00:04:37,480 So if you try to rotate this material locally, 61 00:04:37,480 --> 00:04:41,530 then the corresponding sheer cost of the formation 62 00:04:41,530 --> 00:04:46,030 has a cost that is indicated by mu, the sheer modulus. 63 00:04:46,030 --> 00:04:50,590 And basically what really makes a solid is this term. 64 00:04:50,590 --> 00:04:55,050 Because as I said, a liquid also has the bark modulus, 65 00:04:55,050 --> 00:04:58,740 but lacks the resistance to try to sheer it, 66 00:04:58,740 --> 00:05:02,260 which is captured by this, that is unique and characteristic 67 00:05:02,260 --> 00:05:03,090 of a solid. 68 00:05:06,640 --> 00:05:10,640 So this is the energy cost. 69 00:05:10,640 --> 00:05:14,470 The other part of this whole story 70 00:05:14,470 --> 00:05:19,640 is that this structure has order. 71 00:05:19,640 --> 00:05:24,190 And we can characterize that order 72 00:05:24,190 --> 00:05:29,370 which makes it distinct from a liquid or gas a number of ways. 73 00:05:29,370 --> 00:05:32,390 One was to do an x-ray scattering, 74 00:05:32,390 --> 00:05:35,010 and then you would see the back peaks. 75 00:05:35,010 --> 00:05:38,870 And really that type of order is translational. 76 00:05:43,110 --> 00:05:45,961 And you characterize that by an order parameter. 77 00:05:49,050 --> 00:05:51,280 It's kind of like a spin that you 78 00:05:51,280 --> 00:05:56,130 have in the case of a magnet being up or down. 79 00:05:56,130 --> 00:06:01,990 In this case, this object was e to the i g 80 00:06:01,990 --> 00:06:05,597 dot u-- the deformation that you have that's on location r. 81 00:06:10,570 --> 00:06:18,520 And then these g's are chosen to be the inverse lattice vectors. 82 00:06:22,080 --> 00:06:24,410 It doesn't really matter whether I write here 83 00:06:24,410 --> 00:06:27,500 u of r or the actual position. 84 00:06:27,500 --> 00:06:31,550 Because the actual positions starts at zero temperature, 85 00:06:31,550 --> 00:06:36,530 we devalue r 0, such that the dot product of that g 86 00:06:36,530 --> 00:06:38,470 is a multiple of 2pi. 87 00:06:38,470 --> 00:06:43,990 And so essentially, that's what captures this. 88 00:06:43,990 --> 00:06:47,580 Clearly, if I start with a zero temperature picture 89 00:06:47,580 --> 00:06:51,890 and just move this around, the phase 90 00:06:51,890 --> 00:06:55,400 of this order parameter over here will change, 91 00:06:55,400 --> 00:06:58,640 but it will be the same across the system. 92 00:06:58,640 --> 00:07:02,200 And so this is long range correlation that 93 00:07:02,200 --> 00:07:04,580 is present at zero temperature, you 94 00:07:04,580 --> 00:07:08,350 can ask what happens to it at finite temperature. 95 00:07:08,350 --> 00:07:13,650 So we can look at the row g at some position, row g 96 00:07:13,650 --> 00:07:17,000 star at some other position. 97 00:07:17,000 --> 00:07:23,560 And so that was related to exponential of minus g squared 98 00:07:23,560 --> 00:07:27,380 over 2-- something like u squared 99 00:07:27,380 --> 00:07:31,475 x, or u of x minus 0 squared. 100 00:07:34,410 --> 00:07:39,260 And what we saw was that this thing 101 00:07:39,260 --> 00:07:44,570 had a characteristic that it was falling off with distance 102 00:07:44,570 --> 00:07:51,460 according to some kind of power law. 103 00:07:51,460 --> 00:07:55,830 The exponent of this power law, when calculated, 104 00:07:55,830 --> 00:07:58,500 clearly is related to this g squared. 105 00:07:58,500 --> 00:08:02,316 Because this is the quantity that goes logarithmically. 106 00:08:02,316 --> 00:08:07,660 And so the answer was g squared over 4 pi. 107 00:08:07,660 --> 00:08:12,390 Heat was dependent on these two modes being present. 108 00:08:12,390 --> 00:08:19,134 So you have nu 2 nu plus lambda, and then 2 nu plus lambda. 109 00:08:19,134 --> 00:08:20,300 You had a form such as this. 110 00:08:24,520 --> 00:08:30,450 Now this result was obtained as long 111 00:08:30,450 --> 00:08:36,150 as we were treating this field, u, as just the continuum 112 00:08:36,150 --> 00:08:40,799 field that satisfies this. 113 00:08:40,799 --> 00:08:45,190 And this result is really different, also, 114 00:08:45,190 --> 00:08:48,365 from the expectation that at very high temperature 115 00:08:48,365 --> 00:08:52,370 the particle in a liquid should not know anything 116 00:08:52,370 --> 00:08:54,870 about the particle further out in the liquid, 117 00:08:54,870 --> 00:09:00,050 as long as they're beyond some small correlation links. 118 00:09:00,050 --> 00:09:03,530 So we expect this to actually decay exponentially 119 00:09:03,530 --> 00:09:05,540 at high temperatures. 120 00:09:05,540 --> 00:09:09,570 And we found that we could account 121 00:09:09,570 --> 00:09:25,150 for that by addition of these locations, 122 00:09:25,150 --> 00:09:31,900 can cause a transition to a high temperature 123 00:09:31,900 --> 00:09:39,931 phase in which row g, row g star, between x and 0, 124 00:09:39,931 --> 00:09:40,806 decays exponentially. 125 00:09:45,160 --> 00:09:48,580 As opposed to this algebraic behavior, 126 00:09:48,580 --> 00:09:53,920 indicating that these locations-- once you 127 00:09:53,920 --> 00:09:57,320 go to sufficiently high temperature, 128 00:09:57,320 --> 00:10:00,700 such that the entropy of creating and rearranging 129 00:10:00,700 --> 00:10:05,655 these dislocations overcomes the large cost of creating them 130 00:10:05,655 --> 00:10:10,130 in the first place, then you'll have 131 00:10:10,130 --> 00:10:15,300 this absence of translational order, 132 00:10:15,300 --> 00:10:22,540 and some kind of exponential decay of this order parameter. 133 00:10:22,540 --> 00:10:27,370 So at this stage, you may feel comfortable enough 134 00:10:27,370 --> 00:10:30,760 to say that addition of these dislocation 135 00:10:30,760 --> 00:10:36,480 causes our solid to melt and become a liquid. 136 00:10:36,480 --> 00:10:43,970 Now, I indicated, however, that the sun also 137 00:10:43,970 --> 00:10:45,670 has an orientational role. 138 00:10:56,130 --> 00:11:03,970 What I could do is-- at each location in the solid, 139 00:11:03,970 --> 00:11:11,470 I can ask how much has the angle been deformed, 140 00:11:11,470 --> 00:11:13,320 and look at the bond angle. 141 00:11:13,320 --> 00:11:15,300 So maybe this particle moved here, 142 00:11:15,300 --> 00:11:17,270 and this particle moved here. 143 00:11:17,270 --> 00:11:19,700 Somewhere else, the particles may have 144 00:11:19,700 --> 00:11:21,900 moved in a different fashion. 145 00:11:21,900 --> 00:11:24,600 And the angle that was originally, 146 00:11:24,600 --> 00:11:28,630 say, along the x direction, had rotated somewhere else. 147 00:11:28,630 --> 00:11:32,630 And clearly, again, at zero temperature, 148 00:11:32,630 --> 00:11:39,040 I can look at the correlations of this angular order, 149 00:11:39,040 --> 00:11:42,660 and they would be the same across the system. 150 00:11:42,660 --> 00:11:46,510 I can ask what happens when I include these deformations 151 00:11:46,510 --> 00:11:49,330 and then the dislocations. 152 00:11:49,330 --> 00:11:53,480 So in the same way that we defined the translational order 153 00:11:53,480 --> 00:11:57,750 parameter, I can define an orientational order parameter. 154 00:12:07,310 --> 00:12:14,080 Let's call it sci at some location, r, 155 00:12:14,080 --> 00:12:16,965 which is e to the i. 156 00:12:16,965 --> 00:12:20,750 Theta at that location r-- 157 00:12:20,750 --> 00:12:24,780 Except that when I look at the triangular lattice, 158 00:12:24,780 --> 00:12:28,440 it may be that the triangles have actually rotated 159 00:12:28,440 --> 00:12:31,860 by 60 degrees or 120 degrees. 160 00:12:31,860 --> 00:12:36,250 And I can't really tell whether I clicked once, zero times, 161 00:12:36,250 --> 00:12:38,310 twice, et cetera. 162 00:12:38,310 --> 00:12:44,420 So because of this symmetry of the original lattice on their 163 00:12:44,420 --> 00:12:50,320 on their theta going to theta plus 2 pi over 6, 164 00:12:50,320 --> 00:12:53,570 I have to use something like this that will not 165 00:12:53,570 --> 00:12:57,480 be modified if I make this transformation, 166 00:12:57,480 --> 00:12:58,930 even at zero temperature. 167 00:12:58,930 --> 00:13:03,918 If I miscount some angle by 60 degrees, this will become fine. 168 00:13:07,670 --> 00:13:13,500 Now I want to calculate the correlations of this theta 169 00:13:13,500 --> 00:13:17,710 from one part of this system to another part of the system. 170 00:13:17,710 --> 00:13:20,440 So for that, what I need to do is 171 00:13:20,440 --> 00:13:27,330 to look at the relationship between theta 172 00:13:27,330 --> 00:13:32,470 and the distortion field, u, that I told you before. 173 00:13:32,470 --> 00:13:37,490 Now you can see that right on the top right corner 174 00:13:37,490 --> 00:13:41,790 I took the distortion field, and I took it's derivative, 175 00:13:41,790 --> 00:13:44,150 and then symmetrized the result in pencil. 176 00:13:44,150 --> 00:13:48,750 And that symmetrization actually removes any rotation 177 00:13:48,750 --> 00:13:50,430 that I would have. 178 00:13:50,430 --> 00:13:53,000 So in order to bring back the notation, 179 00:13:53,000 --> 00:13:55,390 I just have to put a minus sign. 180 00:13:55,390 --> 00:14:05,150 And indeed, one can show that the distortion or displacement 181 00:14:05,150 --> 00:14:09,450 u or r across my system-- let's call it 182 00:14:09,450 --> 00:14:16,210 u of x-- leads to a corresponding angular 183 00:14:16,210 --> 00:14:24,903 distortion, theta, at x, which is minus one half-- let's 184 00:14:24,903 --> 00:14:31,180 call it z hat dotted with curve of u. 185 00:14:31,180 --> 00:14:35,340 So if, rather than doing the i u j plus d j u i, 186 00:14:35,340 --> 00:14:38,210 if I put a minus sign, you can see that I 187 00:14:38,210 --> 00:14:40,600 have the structure of a curve. 188 00:14:40,600 --> 00:14:43,280 In two dimensions, actually curve 189 00:14:43,280 --> 00:14:45,640 would be something that would be pointing only 190 00:14:45,640 --> 00:14:48,710 along the z direction. 191 00:14:48,710 --> 00:14:51,830 And so I just make a scale on my dot, 192 00:14:51,830 --> 00:14:54,990 without taking that in the z direction. 193 00:14:54,990 --> 00:14:59,120 And so you can do some distortion, 194 00:14:59,120 --> 00:15:02,440 and convince yourself that for each distortion 195 00:15:02,440 --> 00:15:04,282 you will get an angle that is this. 196 00:15:04,282 --> 00:15:06,282 AUDIENCE: Do we need some kind of thermalization 197 00:15:06,282 --> 00:15:08,218 to fix the dimensions of this? 198 00:15:08,218 --> 00:15:12,100 Because that can go u has dimensions of fields, and u-- 199 00:15:12,100 --> 00:15:16,630 PROFESSOR: I'm only talking about two dimensions. 200 00:15:16,630 --> 00:15:20,860 And in any case, you can see that u 201 00:15:20,860 --> 00:15:23,530 is a distortion-- is a displacement-- 202 00:15:23,530 --> 00:15:26,730 the gradient is reduced by the displacement, 203 00:15:26,730 --> 00:15:29,830 so this thing is dimensionless as long as you 204 00:15:29,830 --> 00:15:31,924 have these dimensions. 205 00:15:31,924 --> 00:15:32,590 AUDIENCE: Sorry. 206 00:15:32,590 --> 00:15:33,464 PROFESSOR: Yes? 207 00:15:33,464 --> 00:15:34,630 AUDIENCE: That's a 2, right? 208 00:15:34,630 --> 00:15:35,244 Not a c? 209 00:15:35,244 --> 00:15:36,160 PROFESSOR: That's a 2. 210 00:15:38,760 --> 00:15:42,810 It's the same 2 that I have for the definition of the strain. 211 00:15:42,810 --> 00:15:44,575 Rather than a plus, you put a minus. 212 00:15:47,214 --> 00:15:48,630 AUDIENCE: So can we think of these 213 00:15:48,630 --> 00:15:51,405 as two sets of Goldstone modes, or is that not a way 214 00:15:51,405 --> 00:15:52,190 to interpret it? 215 00:15:52,190 --> 00:15:54,672 Is it like two order parameters? 216 00:15:54,672 --> 00:15:58,690 I mean, you have a think that has u dependence, but-- 217 00:15:58,690 --> 00:16:02,210 PROFESSOR: OK, so let's look at this picture over here. 218 00:16:02,210 --> 00:16:05,000 You do have two sets of Goldstone modes corresponding 219 00:16:05,000 --> 00:16:07,720 to longitudinal transfers. 220 00:16:07,720 --> 00:16:10,160 You can see that this curve is the thing 221 00:16:10,160 --> 00:16:12,540 is that I call the angle. 222 00:16:12,540 --> 00:16:16,360 So if you like, you can put the angle over here. 223 00:16:16,360 --> 00:16:20,170 But the difference between putting an angle here, 224 00:16:20,170 --> 00:16:23,420 and this term, is that in terms of the angle, 225 00:16:23,420 --> 00:16:26,000 there is no q dependence. 226 00:16:26,000 --> 00:16:28,200 So it is not a ghost. 227 00:16:28,200 --> 00:16:35,390 Because the cost of making a distortion of wave number q 228 00:16:35,390 --> 00:16:37,190 does not vanish as q squared works. 229 00:16:44,010 --> 00:16:48,850 All right, so then I can look at the correlation between, say, 230 00:16:48,850 --> 00:16:54,200 sci of x, sci star of zero. 231 00:16:54,200 --> 00:16:59,660 And what I will be calculating is expectation value of e 232 00:16:59,660 --> 00:17:03,350 to the i 6. 233 00:17:03,350 --> 00:17:05,680 And then I will have this factor of-- 234 00:17:13,040 --> 00:17:15,460 So let me write it in this fashion. 235 00:17:15,460 --> 00:17:18,990 Theta of x minus theta of 0. 236 00:17:23,750 --> 00:17:27,960 Since u is Gaussian distributed, theta 237 00:17:27,960 --> 00:17:30,680 in the Gaussian distributor. 238 00:17:30,680 --> 00:17:34,130 So for any Gaussian distributed entity, 239 00:17:34,130 --> 00:17:37,550 we can write the exponential of e 240 00:17:37,550 --> 00:17:45,300 to the something as its average as exponential of minus 1/2 241 00:17:45,300 --> 00:17:49,040 the average of whatever is in the exponent. 242 00:17:49,040 --> 00:17:54,380 So I will get 36 divided by 2. 243 00:17:54,380 --> 00:18:00,480 I do have the expectation value of delta theta squared. 244 00:18:00,480 --> 00:18:07,800 But delta theta is related up to this factor of 1/4 245 00:18:07,800 --> 00:18:10,900 to some expectation value of kern u. 246 00:18:10,900 --> 00:18:18,370 So I would need to calculate kern mu at x minus kern u 247 00:18:18,370 --> 00:18:23,870 at 0, the whole thing squared with the Gaussian average. 248 00:18:27,940 --> 00:18:35,580 Now, this entity-- clearly what I can do 249 00:18:35,580 --> 00:18:38,570 is to go back and look at these things 250 00:18:38,570 --> 00:18:45,820 in terms of Fourier space, rather than position space. 251 00:18:45,820 --> 00:18:54,830 So this becomes an integral d 2 q 2 pi to the d. 252 00:18:54,830 --> 00:19:01,280 I will get e to the i q dot x minus 1. 253 00:19:01,280 --> 00:19:07,090 And then I have something like q cross u tilde of q. 254 00:19:07,090 --> 00:19:09,620 And I have to do that twice. 255 00:19:09,620 --> 00:19:13,480 When I do that twice, I find that the different q's are 256 00:19:13,480 --> 00:19:14,530 uncorrelated. 257 00:19:14,530 --> 00:19:18,100 So I will get, rather than two of these integrals, one 258 00:19:18,100 --> 00:19:19,555 of these integrals. 259 00:19:19,555 --> 00:19:23,960 And because the q and q prime are said to be the same, 260 00:19:23,960 --> 00:19:28,430 the product of those two factors will be the integral 2 261 00:19:28,430 --> 00:19:33,010 minus 2 cosine of q dot x term that we are used to. 262 00:19:33,010 --> 00:19:36,440 And so that's where the x dependence appears. 263 00:19:36,440 --> 00:19:43,870 And then I need the average of q cross mu of q. 264 00:19:43,870 --> 00:19:47,752 And that I can read off the beta [INAUDIBLE], 265 00:19:47,752 --> 00:19:51,010 root the energy over here. 266 00:19:51,010 --> 00:19:53,360 You can see that there is a Gaussian cost 267 00:19:53,360 --> 00:19:56,770 for q plus u of q squared, which is simply 268 00:19:56,770 --> 00:19:59,240 1 of a lingering variance. 269 00:19:59,240 --> 00:20:03,141 So basically, this term you'll the sum of 1 over u. 270 00:20:05,787 --> 00:20:09,230 Now the difference between all of the calculations 271 00:20:09,230 --> 00:20:13,010 that we were doing previously, as was asked regarding 272 00:20:13,010 --> 00:20:17,220 Goldstone modes-- if I was just looking at u squared, which 273 00:20:17,220 --> 00:20:19,720 is what I was doing up here, I would 274 00:20:19,720 --> 00:20:23,670 need to put another factor of 1 over q squared [INAUDIBLE]. 275 00:20:23,670 --> 00:20:26,180 And then I would have the coulomb integral that 276 00:20:26,180 --> 00:20:27,960 would grow logarithmically. 277 00:20:27,960 --> 00:20:32,720 But here you can see that the whole thing-- the cosine 278 00:20:32,720 --> 00:20:34,620 integrated against the constant-- 279 00:20:34,620 --> 00:20:36,280 will average out to 0. 280 00:20:36,280 --> 00:20:39,250 So I will think you have 2 over u times 281 00:20:39,250 --> 00:20:41,440 this integral is a constant. 282 00:20:41,440 --> 00:20:43,750 So the whole thing, at the end of the day, 283 00:20:43,750 --> 00:20:51,050 is exponential of-- that becomes a 9 divided by 2. 284 00:20:56,190 --> 00:21:00,830 There's a factor of 1 over the mu, 285 00:21:00,830 --> 00:21:06,260 and then I have twice the integral of d 2 q over 2 pi 286 00:21:06,260 --> 00:21:06,960 squared. 287 00:21:06,960 --> 00:21:08,730 Which is-- you can convince yourself 288 00:21:08,730 --> 00:21:13,695 simply the density of the system a number of times. 289 00:21:16,940 --> 00:21:25,490 So as opposed to the translational order, which 290 00:21:25,490 --> 00:21:29,420 was decaying as above our lot, then we 291 00:21:29,420 --> 00:21:31,460 include the phonon modes. 292 00:21:31,460 --> 00:21:34,450 When we include these phonon modes, 293 00:21:34,450 --> 00:21:40,940 we find that the orientational order decays much more weakly. 294 00:21:40,940 --> 00:21:43,980 So that was falling off as I went further and further. 295 00:21:43,980 --> 00:21:46,360 This, as I go further and further, 296 00:21:46,360 --> 00:21:49,310 eventually reaches a constant that is less than 1, 297 00:21:49,310 --> 00:21:51,020 but it is something. 298 00:21:51,020 --> 00:21:53,690 Using conversely proportional to temperature-- 299 00:21:53,690 --> 00:21:59,500 so as I go to 0 temperature, these go to 1. 300 00:21:59,500 --> 00:22:04,280 And basically because this order parameter, 301 00:22:04,280 --> 00:22:06,745 with respect to-- well, this measure 302 00:22:06,745 --> 00:22:10,080 of distortion with respect to that measure of distortion 303 00:22:10,080 --> 00:22:12,940 has an additional factor of gradient. 304 00:22:12,940 --> 00:22:16,810 I will get an additional factor of q squared, and then 305 00:22:16,810 --> 00:22:19,200 everything changes accordingly. 306 00:22:19,200 --> 00:22:26,650 So orientational order is much more robust. 307 00:22:26,650 --> 00:22:29,010 This phase that we were calling the analogue of a two 308 00:22:29,010 --> 00:22:33,570 dimensional solid had only quasi long range order. 309 00:22:33,570 --> 00:22:36,890 The long range order was decaying as a power law. 310 00:22:39,510 --> 00:22:40,980 Yes? 311 00:22:40,980 --> 00:22:44,620 AUDIENCE: Is n dependent on the position, or-- 312 00:22:44,620 --> 00:22:46,340 PROFESSOR: No. 313 00:22:46,340 --> 00:22:50,370 So basically, if you were to remember the number of points 314 00:22:50,370 --> 00:22:53,650 it should be the same as the number of allowed fullier 315 00:22:53,650 --> 00:22:55,150 modes. 316 00:22:55,150 --> 00:22:57,790 And this goes to an integral-- 2 q 317 00:22:57,790 --> 00:23:04,610 over 2 pi squared-- when I put the area in two dimensions. 318 00:23:04,610 --> 00:23:06,540 So the integral over whatever [INAUDIBLE] 319 00:23:06,540 --> 00:23:09,460 zone you have over the fullier modes 320 00:23:09,460 --> 00:23:12,800 is the same thing as the number of points 321 00:23:12,800 --> 00:23:17,050 that you have in the original lattice, divided by area, 322 00:23:17,050 --> 00:23:23,125 or 1 over the size of one of those triangles squared. 323 00:23:23,125 --> 00:23:23,625 Yes? 324 00:23:23,625 --> 00:23:26,940 AUDIENCE: Where is the x dependent in that expression? 325 00:23:26,940 --> 00:23:28,810 PROFESSOR: OK, the x dependence basically 326 00:23:28,810 --> 00:23:33,940 disappears because you integrate over the cosine of q x. 327 00:23:33,940 --> 00:23:38,297 And if x is sufficiently large, those fluctuations disappear. 328 00:23:38,297 --> 00:23:40,672 AUDIENCE: Oh, so we're really looking at the [INAUDIBLE]. 329 00:23:40,672 --> 00:23:41,660 PROFESSOR: Yes, that's right. 330 00:23:41,660 --> 00:23:43,256 So at short distances, there are going 331 00:23:43,256 --> 00:23:45,610 to be some oscillations or whatever. 332 00:23:45,610 --> 00:23:48,530 But it gradually-- we are interested in the long distance 333 00:23:48,530 --> 00:23:50,300 behavior. 334 00:23:50,300 --> 00:23:52,330 At very short distances, I can't even 335 00:23:52,330 --> 00:23:54,930 use the continuum description for things 336 00:23:54,930 --> 00:23:57,451 that are three or lattice spacings apart. 337 00:24:05,310 --> 00:24:10,790 So maybe I should explicitly say that this is usually 338 00:24:10,790 --> 00:24:19,430 called quasi long range order, versus this dependence, which 339 00:24:19,430 --> 00:24:20,640 is two long ranges. 340 00:24:32,120 --> 00:24:40,050 So given that this is more robust than these forum-like 341 00:24:40,050 --> 00:24:43,370 fluctuations, the next question is, well 342 00:24:43,370 --> 00:24:49,314 does it completely disappear when I include these locations. 343 00:24:49,314 --> 00:24:54,930 So again, this calculation, based on Gaussian's, relies 344 00:24:54,930 --> 00:25:01,130 on just the fullier modes of that line that I have up there. 345 00:25:01,130 --> 00:25:05,090 It does not include the dislocations, which, in order 346 00:25:05,090 --> 00:25:07,530 to properly account, you saw that we 347 00:25:07,530 --> 00:25:13,760 need to look at a collections of these locations appearing 348 00:25:13,760 --> 00:25:16,120 at different positions on the lattice. 349 00:25:16,120 --> 00:25:19,850 And they had these vectorial nature of the fullier 350 00:25:19,850 --> 00:25:22,410 interactions among them. 351 00:25:22,410 --> 00:25:30,566 So presumably, when I go into the base where 352 00:25:30,566 --> 00:25:45,150 these locations unbind-- and by unbinding-- 353 00:25:45,150 --> 00:25:47,342 as I said, in the low temperature 354 00:25:47,342 --> 00:25:48,716 picture of the dislocations, they 355 00:25:48,716 --> 00:25:51,950 should appear very close to each other 356 00:25:51,950 --> 00:25:55,920 because it is costly to separate them by an amount that grows 357 00:25:55,920 --> 00:25:57,870 invariably in the separation. 358 00:25:57,870 --> 00:26:02,220 In the unbound phase, you have essentially a gas 359 00:26:02,220 --> 00:26:06,160 of dislocations that can be anywhere. 360 00:26:06,160 --> 00:26:11,860 So the picture here now is that indeed this 361 00:26:11,860 --> 00:26:15,680 is a phase, that if I just focus on the dislocations, 362 00:26:15,680 --> 00:26:17,230 there is a whole bunch of them. 363 00:26:17,230 --> 00:26:19,910 In a triangular lattice, they could 364 00:26:19,910 --> 00:26:26,240 be pointing in any one of three directions, plus or minus. 365 00:26:26,240 --> 00:26:34,190 And then there is certainly an additional contribution 366 00:26:34,190 --> 00:26:41,550 to the angle that comes from the presence of these dislocations. 367 00:26:41,550 --> 00:26:46,830 So you calculate-- if you have a dislocation that 368 00:26:46,830 --> 00:26:54,323 has inverse b, let's say at the origin, what kind of angular 369 00:26:54,323 --> 00:26:56,630 distortion does it cause. 370 00:26:56,630 --> 00:27:01,530 And you find that it goes like v dot x, divided 371 00:27:01,530 --> 00:27:03,650 by the absolute value of x. 372 00:27:07,700 --> 00:27:10,120 This is for one dislocation. 373 00:27:10,120 --> 00:27:15,690 This is the theta that you would get for that dislocation 374 00:27:15,690 --> 00:27:17,580 at location x. 375 00:27:23,400 --> 00:27:27,270 Essentially, you can see that if I 376 00:27:27,270 --> 00:27:34,320 were to replace the u that I have here with the u that 377 00:27:34,320 --> 00:27:39,060 was caused by dislocation, you would get something 378 00:27:39,060 --> 00:27:40,880 like this formula. 379 00:27:40,880 --> 00:27:46,050 Because remember the u that was caused by dislocation 380 00:27:46,050 --> 00:27:50,410 was something like the gradient of the log potential. 381 00:28:07,120 --> 00:28:10,070 It's kind of hard to work, but maybe I'll 382 00:28:10,070 --> 00:28:12,070 make an attempt to write it. 383 00:28:15,540 --> 00:28:19,230 So let's take a gradient of theta. 384 00:28:21,840 --> 00:28:25,360 Gradient of theta, if I use that formula, 385 00:28:25,360 --> 00:28:33,640 you would say, OK, I have minus 1/2 z hat dot 386 00:28:33,640 --> 00:28:38,520 kern of something. 387 00:28:38,520 --> 00:28:44,410 And if I take a gradient of the kern, the answer should be 0. 388 00:28:44,410 --> 00:28:49,020 But that's as long as this u is a well defined object. 389 00:28:49,020 --> 00:28:54,100 And our task was to say that this u, then you 390 00:28:54,100 --> 00:28:57,530 have these dislocations, is not a well defined 391 00:28:57,530 --> 00:28:59,890 object, in the sense that you take the kern, 392 00:28:59,890 --> 00:29:02,350 and the gradient then you would get 0. 393 00:29:02,350 --> 00:29:05,260 So essentially, I will transport the gradient 394 00:29:05,260 --> 00:29:08,210 all the way over here, and the part of u 395 00:29:08,210 --> 00:29:12,880 that will survive that is the one that is characterized 396 00:29:12,880 --> 00:29:14,220 by this dislocation feat. 397 00:29:18,060 --> 00:29:24,540 Now, you can see that this object 398 00:29:24,540 --> 00:29:30,510 kind of looks like a Laplacian of this distortion. 399 00:29:30,510 --> 00:29:34,090 It's two the derivatives of this distortion field 400 00:29:34,090 --> 00:29:36,720 that had this logarithm in it. 401 00:29:36,720 --> 00:29:39,480 And when you take two derivatives of a logarithm, 402 00:29:39,480 --> 00:29:41,530 you get the delta function. 403 00:29:41,530 --> 00:29:43,250 So if you do things correctly, you 404 00:29:43,250 --> 00:29:48,940 will find that this answer here becomes a sum over i 405 00:29:48,940 --> 00:29:57,870 v i delta function of x minus xi. 406 00:29:57,870 --> 00:30:05,010 So basically, each dislocation at location x i-- again, 407 00:30:05,010 --> 00:30:07,630 depending on its v being in each direction-- 408 00:30:07,630 --> 00:30:11,950 gives a contribution to the gradient of theta. 409 00:30:11,950 --> 00:30:17,540 And if I were to take the gradient of the expression 410 00:30:17,540 --> 00:30:23,350 that I have over here, the gradient of this object 411 00:30:23,350 --> 00:30:25,450 is also-- this is like the field that you 412 00:30:25,450 --> 00:30:27,790 have for the logarithmic potential-- 413 00:30:27,790 --> 00:30:29,490 will give you the data function. 414 00:30:29,490 --> 00:30:33,380 So that's where the similarity comes. 415 00:30:33,380 --> 00:30:36,470 So the full answer comes out to be-- 416 00:30:36,470 --> 00:30:40,270 if you have a sum over the dislocations, 417 00:30:40,270 --> 00:30:44,700 the sum over the distortion fields that each one of them 418 00:30:44,700 --> 00:30:56,910 is causing-- and you will have a form such as this. 419 00:30:56,910 --> 00:30:57,410 Yes? 420 00:30:57,410 --> 00:30:59,243 AUDIENCE: Should the denominator be squared? 421 00:31:08,750 --> 00:31:12,990 PROFESSOR: Yes, that's right. 422 00:31:12,990 --> 00:31:16,430 The potential goes logarithmically. 423 00:31:16,430 --> 00:31:19,090 The field, which is the gradient of the potential, 424 00:31:19,090 --> 00:31:21,910 falls off as 1 over separation. 425 00:31:21,910 --> 00:31:23,930 So since I put the separation out there, 426 00:31:23,930 --> 00:31:25,625 I have to put the separation squared. 427 00:31:34,300 --> 00:31:39,520 So you can see that the singular part, the part that 428 00:31:39,520 --> 00:31:44,200 arises from dislocations-- if I have a soup of dislocations, 429 00:31:44,200 --> 00:31:45,770 I can figure out what theta is. 430 00:31:49,720 --> 00:31:55,220 Now what I did look for-- actually, I 431 00:31:55,220 --> 00:32:02,680 was kind of hinting at that-- if I take the gradient of theta-- 432 00:32:02,680 --> 00:32:05,800 and I forgot to put the factor of 1/2pi 433 00:32:05,800 --> 00:32:15,770 here-- does the 4 vertices that had charged to pi-- 434 00:32:15,770 --> 00:32:16,900 I had the potential. 435 00:32:16,900 --> 00:32:18,220 That was 1/r. 436 00:32:18,220 --> 00:32:21,440 So for dislocations it becomes d/2pi. 437 00:32:21,440 --> 00:32:25,970 If I take the gradient, then the gradient translates 438 00:32:25,970 --> 00:32:34,630 to sum over pi, the i data function of x minus x 439 00:32:34,630 --> 00:32:39,750 i-- the expression that I have written over there. 440 00:32:39,750 --> 00:32:44,810 And if I do the fullier transform, 441 00:32:44,810 --> 00:32:47,445 you see what I did over here was essentially 442 00:32:47,445 --> 00:32:49,980 to look at theta in fullier space. 443 00:32:49,980 --> 00:32:53,559 So let's do something similar here. 444 00:32:53,559 --> 00:32:55,350 So when I do the fullier transform of this, 445 00:32:55,350 --> 00:33:04,110 I will get pi q-- the fullier transform of this angular feat. 446 00:33:04,110 --> 00:33:09,195 And on the right hand side, what I would get 447 00:33:09,195 --> 00:33:12,315 is essentially the fullier transform 448 00:33:12,315 --> 00:33:15,580 of the field of dislocations. 449 00:33:15,580 --> 00:33:28,080 So I have defined my v of q to be sum over i into the i q dot 450 00:33:28,080 --> 00:33:34,470 position of the i dislocation, the vector that 451 00:33:34,470 --> 00:33:37,776 characterizes the dislocation. 452 00:33:37,776 --> 00:33:41,240 And it would make sense to also tap 453 00:33:41,240 --> 00:33:43,480 into the normalization that gives 1 454 00:33:43,480 --> 00:33:45,640 over the square root of area. 455 00:33:45,640 --> 00:33:48,520 If you don't do that, then at some other point 456 00:33:48,520 --> 00:33:50,290 you have to worry about the normalization. 457 00:33:58,090 --> 00:34:03,720 So if I just multiply both sides by q-- 458 00:34:03,720 --> 00:34:08,050 and I think I forgot the minus sign throughout, 459 00:34:08,050 --> 00:34:17,260 which is not that important-- but theta tilde of q 460 00:34:17,260 --> 00:34:24,340 becomes i q dot b of q, divided-- maybe 461 00:34:24,340 --> 00:34:31,022 I should've been calling this b tilde-- divided by q squared. 462 00:34:37,288 --> 00:34:38,734 So this is important. 463 00:34:42,120 --> 00:34:49,580 Essentially, you take the collection of dislocations 464 00:34:49,580 --> 00:34:52,929 in this picture and you calculate 465 00:34:52,929 --> 00:34:56,790 what the fullier transform is, call that the tilde of q. 466 00:34:56,790 --> 00:34:59,010 Essentially, you divide by 1 factor of q, 467 00:34:59,010 --> 00:35:02,490 and you can get the corresponding angle of feat. 468 00:35:07,300 --> 00:35:12,720 Now what I needed to evaluate for here 469 00:35:12,720 --> 00:35:18,790 was the average of theta tilde of q squared. 470 00:35:23,190 --> 00:35:28,050 And you can see that if I write this explicitly, let's 471 00:35:28,050 --> 00:35:36,440 say, q i for be tilde i 4, then the two of them I will 472 00:35:36,440 --> 00:35:40,152 get q beta b tilde of beta. 473 00:35:40,152 --> 00:35:42,562 And then I would have a q to the side. 474 00:35:45,460 --> 00:35:50,840 And the average over here becomes the average 475 00:35:50,840 --> 00:35:55,200 over all contributions of these dislocations 476 00:35:55,200 --> 00:35:57,360 that I can put across my system. 477 00:36:00,990 --> 00:36:06,772 Now, explicitly I'm interested in the limit where q goes to 0. 478 00:36:06,772 --> 00:36:10,710 So these things depend on q. 479 00:36:10,710 --> 00:36:14,520 What I'm interested in is the limit 480 00:36:14,520 --> 00:36:20,270 as q goes to 0, especially what happens to this average. 481 00:36:28,280 --> 00:36:36,040 It becomes-- multiplying two of these things together-- 482 00:36:36,040 --> 00:36:41,070 actually, in the limit where q goes to 0, 483 00:36:41,070 --> 00:36:46,930 what I have is the sum over all of the b's. 484 00:36:46,930 --> 00:36:49,030 So in the limit where q goes to 0, 485 00:36:49,030 --> 00:36:53,740 this becomes an integral or sum. 486 00:36:53,740 --> 00:36:56,770 It doesn't matter which one of them I write. 487 00:36:56,770 --> 00:36:59,820 q has gone to 0, so I basically need 488 00:36:59,820 --> 00:37:06,620 to look at the average of the alpha of x, the beta of x 489 00:37:06,620 --> 00:37:09,170 [INAUDIBLE], divided by area. 490 00:37:12,730 --> 00:37:17,390 So what is there in the numerator? 491 00:37:21,160 --> 00:37:26,410 We can see that in the numerator, sq goes to 0. 492 00:37:26,410 --> 00:37:32,780 What I'm looking at is the sum of all of these dislocations 493 00:37:32,780 --> 00:37:35,940 that I have in the system. 494 00:37:35,940 --> 00:37:39,275 Now the average up the sum is 0, because in all 495 00:37:39,275 --> 00:37:42,710 of our calculations, we've been restricting the configurations 496 00:37:42,710 --> 00:37:44,550 that we moved from. 497 00:37:44,550 --> 00:37:46,465 Because if I go beyond that strategy, 498 00:37:46,465 --> 00:37:48,115 it's going to cost too much. 499 00:37:50,920 --> 00:37:52,965 But what I'm looking at is not the average 500 00:37:52,965 --> 00:37:57,460 of b, which is 0, but the average of b squared, 501 00:37:57,460 --> 00:37:59,770 which is the variance. 502 00:37:59,770 --> 00:38:06,540 So essentially I have a system that has a large area, a. 503 00:38:06,540 --> 00:38:09,540 It is on average neutral. 504 00:38:09,540 --> 00:38:12,530 And the question is, what is the variance of the net charge. 505 00:38:15,040 --> 00:38:19,120 And my claim is that the variance of the net charge 506 00:38:19,120 --> 00:38:25,870 is, by central limit theorem, proportional to the area-- 507 00:38:25,870 --> 00:38:28,710 actually, it is proportional to the units that are 508 00:38:28,710 --> 00:38:30,950 independent from each other. 509 00:38:30,950 --> 00:38:39,450 So roughly I would expect that in this high temperature phase, 510 00:38:39,450 --> 00:38:42,520 I have a correlation that is c. 511 00:38:42,520 --> 00:38:48,570 And within each portion of side c, will be neutral. 512 00:38:48,570 --> 00:38:53,540 But when I go within things that are more than c apart, 513 00:38:53,540 --> 00:38:57,120 there's no reason to maintain the strategy. 514 00:38:57,120 --> 00:39:02,560 So overall I have something like throwing coins, 515 00:39:02,560 --> 00:39:05,060 but at each one of them, the average 516 00:39:05,060 --> 00:39:09,040 is 0, with probability being up or down. 517 00:39:09,040 --> 00:39:12,550 But when I look at the variance for the entire thing, 518 00:39:12,550 --> 00:39:16,301 the average will be proportional to the area 519 00:39:16,301 --> 00:39:22,280 in units of these things that are independent of each other. 520 00:39:22,280 --> 00:39:26,090 It was from the normalization factor of 1 over area. 521 00:39:26,090 --> 00:39:29,120 And these, really, I should write as a proportionality, 522 00:39:29,120 --> 00:39:32,030 because I don't know precisely what 523 00:39:32,030 --> 00:39:35,550 the relationship between these independent sides 524 00:39:35,550 --> 00:39:37,000 that correlation [INAUDIBLE]. 525 00:39:37,000 --> 00:39:39,500 But they have to be roughly proportional. 526 00:39:43,500 --> 00:39:45,318 So what do you compute? 527 00:39:45,318 --> 00:39:50,670 You compute that the limit as q goes 528 00:39:50,670 --> 00:40:02,580 to 0, of the average of my theta tilde of q squared 529 00:40:02,580 --> 00:40:04,350 is a structure such as this. 530 00:40:04,350 --> 00:40:07,650 I forgot to put one more thing here. 531 00:40:07,650 --> 00:40:10,970 I don't expect to be any correlations 532 00:40:10,970 --> 00:40:13,780 between the x component and the y 533 00:40:13,780 --> 00:40:16,390 component of this answer-- the variance, 534 00:40:16,390 --> 00:40:18,940 the covariance of the dislocations in one 535 00:40:18,940 --> 00:40:20,990 direction and the other direction-- 536 00:40:20,990 --> 00:40:23,570 so I put the delta function there. 537 00:40:23,570 --> 00:40:26,820 If I put this over here, I would get the q squared 538 00:40:26,820 --> 00:40:28,700 divided by q to the 4th. 539 00:40:28,700 --> 00:40:34,160 So I will get a 1 over4 q squared. 540 00:40:34,160 --> 00:40:37,931 And I have the c squared and then some unknown coefficient 541 00:40:37,931 --> 00:40:38,431 up here. 542 00:40:48,270 --> 00:40:55,740 So it's interesting, because we started 543 00:40:55,740 --> 00:41:00,470 without thinking about dislocations, just in terms 544 00:41:00,470 --> 00:41:01,983 of the distortion field. 545 00:41:01,983 --> 00:41:09,540 And we said that this object is related to the angle. 546 00:41:09,540 --> 00:41:12,640 And indeed, we had this distortion, 547 00:41:12,640 --> 00:41:16,270 that energy cost of distortions is proportional to angle 548 00:41:16,270 --> 00:41:17,950 squared. 549 00:41:17,950 --> 00:41:21,940 And that angle, therefore, is not the Goldstone mode 550 00:41:21,940 --> 00:41:24,050 because it doesn't go like q squared. 551 00:41:27,050 --> 00:41:29,180 Now we go to this other phase now, 552 00:41:29,180 --> 00:41:32,530 with dislocations all over the place, 553 00:41:32,530 --> 00:41:38,210 and we calculate the expectation value of theta squared. 554 00:41:38,210 --> 00:41:40,790 And it looks like it came from a theory that 555 00:41:40,790 --> 00:41:42,016 was like Goldstone modes. 556 00:41:45,490 --> 00:41:50,750 So you would say that once I am in this phase, where 557 00:41:50,750 --> 00:41:55,970 the dislocations are unbound, there is an effective energy 558 00:41:55,970 --> 00:42:03,880 cost for these changes in angle that 559 00:42:03,880 --> 00:42:11,307 is proportional to the radiant of the angle squared. 560 00:42:15,710 --> 00:42:20,680 So that means fullier space, this would go to k a over 2, 561 00:42:20,680 --> 00:42:28,222 integral into q 2 pi squared, q squared theta tilde of q 562 00:42:28,222 --> 00:42:28,721 squared. 563 00:42:31,500 --> 00:42:36,170 So that if you had this theory, you would definitely 564 00:42:36,170 --> 00:42:41,130 say that the expectation value of theta tilde of q squared 565 00:42:41,130 --> 00:42:45,245 is 1 over k a q squared. 566 00:42:49,160 --> 00:42:53,950 The variance is k a q squared invers. 567 00:42:53,950 --> 00:43:00,290 You compare those two things and you find that once 568 00:43:00,290 --> 00:43:03,960 the dislocations have unbound, and there 569 00:43:03,960 --> 00:43:07,710 is a correlation lend that essentially tells you 570 00:43:07,710 --> 00:43:11,280 how far the dislocations are talking to each other 571 00:43:11,280 --> 00:43:15,060 and maintaining neutrality, that there is exactly 572 00:43:15,060 --> 00:43:19,290 an effective stiffness, like a Goldstone note, 573 00:43:19,290 --> 00:43:23,410 for angular distortions, that is proportional to c squared. 574 00:43:29,760 --> 00:43:33,750 And hence, if I were to look at the orientation of all 575 00:43:33,750 --> 00:43:45,530 their correlations, I would essentially 576 00:43:45,530 --> 00:43:51,150 have something like expectation value of theta q squared, 577 00:43:51,150 --> 00:43:52,590 which is 1 over q squared. 578 00:43:52,590 --> 00:43:55,385 If I fully transform that, I get the log. 579 00:43:55,385 --> 00:43:57,890 And so I will get something that falls off 580 00:43:57,890 --> 00:44:02,340 in the distance to some other exponent, if I recall 581 00:44:02,340 --> 00:44:02,840 [INAUDIBLE]. 582 00:44:07,720 --> 00:44:11,940 If I have a true liquid-- in a liquid, 583 00:44:11,940 --> 00:44:14,700 again, maybe in a neighborhood of seven 584 00:44:14,700 --> 00:44:17,240 or eight particles, neighbors, et cetera, 585 00:44:17,240 --> 00:44:20,830 they talk to each other and the orientations are correlated. 586 00:44:20,830 --> 00:44:22,830 But then I go from one part of the liquid 587 00:44:22,830 --> 00:44:24,810 to another part of the liquid, there 588 00:44:24,810 --> 00:44:27,630 is no correlation between bond angles. 589 00:44:27,630 --> 00:44:32,070 I expect these things to decay exponentially. 590 00:44:32,070 --> 00:44:39,750 So what we've established is that neither the phonon 591 00:44:39,750 --> 00:44:45,160 nor the dislocations are sufficient to give 592 00:44:45,160 --> 00:44:51,250 the exponential decay that you expect for the bond. 593 00:44:51,250 --> 00:44:57,690 So this object has quasi long range order, 594 00:44:57,690 --> 00:45:00,756 versus what I expect to happen in the liquid, which 595 00:45:00,756 --> 00:45:03,468 is exponential of minus x over psi. 596 00:45:08,900 --> 00:45:15,650 So the unbinding of dislocations gives rise 597 00:45:15,650 --> 00:45:20,120 to the new phase of matter that has this quasi long range 598 00:45:20,120 --> 00:45:22,810 order in the orientations. 599 00:45:22,810 --> 00:45:25,060 It has no positional order. 600 00:45:25,060 --> 00:45:28,260 It's a kind of a liquid crystal that is called a hexatic. 601 00:45:43,790 --> 00:45:45,060 Yes? 602 00:45:45,060 --> 00:45:47,992 AUDIENCE: So your correlation where you got 1 over k q 603 00:45:47,992 --> 00:45:51,980 squared, doesn't that assume that you're allowing the angle 604 00:45:51,980 --> 00:45:55,480 to vary in minus [INAUDIBLE] when you do your averaging? 605 00:45:55,480 --> 00:45:57,300 What about the restriction-- 606 00:45:57,300 --> 00:46:04,030 PROFESSOR: OK, so what is the variance of the angle here? 607 00:46:04,030 --> 00:46:07,630 There's a variance of the angle that is controlled by this 1 608 00:46:07,630 --> 00:46:09,430 over k a. 609 00:46:09,430 --> 00:46:13,830 So if I go back and calculate these in real space, 610 00:46:13,830 --> 00:46:19,013 I will find that if I look at theta at location x minus theta 611 00:46:19,013 --> 00:46:22,570 at location 0, the answer is going to go 612 00:46:22,570 --> 00:46:28,210 like 1 over k logarithm x. 613 00:46:28,210 --> 00:46:30,690 So what it says is that if things 614 00:46:30,690 --> 00:46:35,270 are close enough to each other-- and this is in units of 1/a-- 615 00:46:35,270 --> 00:46:38,170 up to some factor, let's say log 5, et cetera. 616 00:46:38,170 --> 00:46:40,480 So I don't go all the way to infinity. 617 00:46:40,480 --> 00:46:46,840 The fluctuations in angle are inversely set by a parameter 618 00:46:46,840 --> 00:46:52,440 that we see as I approach right after the transition is 619 00:46:52,440 --> 00:46:54,310 very large. 620 00:46:54,310 --> 00:46:57,120 So in the same sense that previously 621 00:46:57,120 --> 00:47:00,210 for the positional correlations I 622 00:47:00,210 --> 00:47:04,400 had the temperature being small and inverse temperature being 623 00:47:04,400 --> 00:47:08,880 large, limiting the size of the translational fluctuations, 624 00:47:08,880 --> 00:47:13,930 here the same thing happens for the bond angle fluctuations. 625 00:47:13,930 --> 00:47:19,080 Close to the transitions, they are actually small. 626 00:47:19,080 --> 00:47:21,190 So the question that you asked, you 627 00:47:21,190 --> 00:47:24,820 could have certainly also asked over here. 628 00:47:24,820 --> 00:47:29,500 That is, when I'm thinking about the distortion field, 629 00:47:29,500 --> 00:47:33,125 the distortion field is certainly going to be limited. 630 00:47:33,125 --> 00:47:38,120 If it becomes as big as this, then it doesn't make sense. 631 00:47:38,120 --> 00:47:41,280 So given that, what sense or what 632 00:47:41,280 --> 00:47:44,830 justification do I have in making these Gaussian 633 00:47:44,830 --> 00:47:46,210 integrals? 634 00:47:46,210 --> 00:47:48,205 And the answer is that while it is true 635 00:47:48,205 --> 00:47:53,140 that it is fluctuating, as I go to low temperature, 636 00:47:53,140 --> 00:47:56,270 the degree of fluctuations is very small. 637 00:47:56,270 --> 00:48:01,580 So effectively what I have is that 638 00:48:01,580 --> 00:48:05,190 I have to integrate over some finite interval 639 00:48:05,190 --> 00:48:09,510 a function that kind of looks like this. 640 00:48:09,510 --> 00:48:14,120 And the fact that I replaced that with an integration 641 00:48:14,120 --> 00:48:17,340 from minus infinity to infinity rather than from minus a 642 00:48:17,340 --> 00:48:19,130 to a just doesn't matter. 643 00:48:30,460 --> 00:48:36,410 So we know that ultimately we should get this, 644 00:48:36,410 --> 00:48:42,730 but so far we've only got this, so what should we do? 645 00:48:42,730 --> 00:48:45,810 Well, we say OK, we encountered this difficulty 646 00:48:45,810 --> 00:48:52,550 before in something that looked like an angle in the xy model-- 647 00:48:52,550 --> 00:48:55,740 that low temperature had power-law decay, 648 00:48:55,740 --> 00:48:57,860 whereas we knew that at high temperatures 649 00:48:57,860 --> 00:49:00,430 they would have to have exponential decay. 650 00:49:00,430 --> 00:49:03,620 And what we said was that we need these topological defects 651 00:49:03,620 --> 00:49:06,060 in angle. 652 00:49:06,060 --> 00:49:18,950 So what you need-- topological defects-- or in our case, 653 00:49:18,950 --> 00:49:20,505 theta is a bond angle. 654 00:49:27,640 --> 00:49:32,101 And these topological defects in the bond angle have a name. 655 00:49:32,101 --> 00:49:33,350 They're called disconnections. 656 00:49:43,180 --> 00:49:47,950 And very roughly they correspond to something like this. 657 00:49:47,950 --> 00:49:53,180 Suppose this is the center one of these discriminations, 658 00:49:53,180 --> 00:49:57,490 and then maybe next to this, here 659 00:49:57,490 --> 00:50:03,820 I have locally at the distance r-- if I look at a point, 660 00:50:03,820 --> 00:50:07,840 I would see that the bonds that connect it to its neighbor 661 00:50:07,840 --> 00:50:11,875 have an orientation such as the one that I have indicated over 662 00:50:11,875 --> 00:50:13,590 here. 663 00:50:13,590 --> 00:50:21,210 Now what I want to do is, as I go around and make a circuit, 664 00:50:21,210 --> 00:50:26,320 that this angle theta that I have here to be 0, 665 00:50:26,320 --> 00:50:31,700 rotates and comes back up to 60 degrees. 666 00:50:31,700 --> 00:50:36,140 So essentially what I do is I take this line 667 00:50:36,140 --> 00:50:43,190 and I gradually shift it around so that by the time 668 00:50:43,190 --> 00:50:47,517 I come back, I have rotated by 60 degrees. 669 00:50:47,517 --> 00:50:49,100 It's kind of hard for me to draw that, 670 00:50:49,100 --> 00:50:52,980 but you can imagine what I have to do. 671 00:50:52,980 --> 00:50:57,200 So what I need to do is to have the integral 672 00:50:57,200 --> 00:51:02,860 over a circuit that encloses this discrimination such 673 00:51:02,860 --> 00:51:07,805 that when I do a d s dotted with the gradient of the bond 674 00:51:07,805 --> 00:51:13,000 orientational angle, I come back to pi over 675 00:51:13,000 --> 00:51:16,440 6 times some integer. 676 00:51:16,440 --> 00:51:21,590 And again, I expect the [INAUDIBLE] dislocations that 677 00:51:21,590 --> 00:51:28,600 correspond to minus plus 1. 678 00:51:28,600 --> 00:51:32,320 Then the cost of these is obtained 679 00:51:32,320 --> 00:51:36,220 by taking this distortion fee, gradient of theta, 680 00:51:36,220 --> 00:51:40,910 whose magnitude at a distance r from the center of this object 681 00:51:40,910 --> 00:51:52,580 is going to be 1 over 2 pi r times pi over 6 times 682 00:51:52,580 --> 00:51:55,710 whatever this integer n is. 683 00:51:55,710 --> 00:52:00,570 And then if I substitute this 1 over r behavior 684 00:52:00,570 --> 00:52:04,330 in this expression, which is the effective energy 685 00:52:04,330 --> 00:52:08,230 of this entity, I will get the logarithmic cost 686 00:52:08,230 --> 00:52:11,550 for making a single disclination. 687 00:52:11,550 --> 00:52:13,220 Which means that at low temperature, 688 00:52:13,220 --> 00:52:15,940 I have to create disclination pairs. 689 00:52:15,940 --> 00:52:18,430 And then there will be an effective interaction 690 00:52:18,430 --> 00:52:21,230 between disclination pairs, that is 691 00:52:21,230 --> 00:52:24,510 [INAUDIBLE] in exactly the same way that we calculated 692 00:52:24,510 --> 00:52:26,150 for the x y model. 693 00:52:26,150 --> 00:52:33,040 So up to just this minor change that the charge of a refect 694 00:52:33,040 --> 00:52:36,850 is reduced by a factor of six, this theory 695 00:52:36,850 --> 00:52:41,680 is identical the theory of the unbinding of the x y 696 00:52:41,680 --> 00:52:43,020 model [INAUDIBLE] defects. 697 00:52:43,020 --> 00:52:43,950 Yes? 698 00:52:43,950 --> 00:52:46,079 AUDIENCE: Why is it pi over 6 and not 2 pi over 6? 699 00:52:46,079 --> 00:52:47,120 PROFESSOR: You are right. 700 00:52:47,120 --> 00:52:49,646 It should be 2 pi over 6. 701 00:52:49,646 --> 00:52:50,602 Thank you. 702 00:52:55,870 --> 00:52:57,520 Yes? 703 00:52:57,520 --> 00:52:59,918 AUDIENCE: So when you were saying that 704 00:52:59,918 --> 00:53:05,960 the-- so the distance of this hexatic phase would require 705 00:53:05,960 --> 00:53:12,880 the dislocations to occur to [INAUDIBLE] [INAUDIBLE] 706 00:53:12,880 --> 00:53:15,420 orientational defects. 707 00:53:15,420 --> 00:53:18,200 Is there an analogous case where-- 708 00:53:18,200 --> 00:53:23,589 I guess you can't have dislocations in the orientation 709 00:53:23,589 --> 00:53:24,630 without the dislocation-- 710 00:53:24,630 --> 00:53:28,480 PROFESSOR: So if you try to make these objects 711 00:53:28,480 --> 00:53:32,050 in the original case, in the origin of lattice, 712 00:53:32,050 --> 00:53:34,540 you will find that their cost grows actually 713 00:53:34,540 --> 00:53:39,640 like l squared log l, as opposed to dislocations, 714 00:53:39,640 --> 00:53:43,450 whose cost only grows as log l. 715 00:53:43,450 --> 00:53:48,830 So these entities are extremely unlikely to occur 716 00:53:48,830 --> 00:53:50,970 in the original system. 717 00:53:50,970 --> 00:53:54,590 If you sort of go back and ask what they actually 718 00:53:54,590 --> 00:53:59,640 correspond to, if you have a picture that you have generated 719 00:53:59,640 --> 00:54:01,290 on the computer, they're actually 720 00:54:01,290 --> 00:54:04,530 reasonably easy to identify. 721 00:54:04,530 --> 00:54:07,200 Because the centers of these disclinations 722 00:54:07,200 --> 00:54:09,250 correspond to having points, that 723 00:54:09,250 --> 00:54:14,780 have, rather than 6 neighbors, 5 or 7 neighbors. 724 00:54:14,780 --> 00:54:18,800 So you generate the picture, and you find mostly you have 725 00:54:18,800 --> 00:54:23,690 neighborhoods with 6 neighbors, and then there's a site where 726 00:54:23,690 --> 00:54:26,147 there's 5 neighbours, and another site that's 7 727 00:54:26,147 --> 00:54:27,550 neighbors. 728 00:54:27,550 --> 00:54:30,635 5 and 7 come more or less in pairs, 729 00:54:30,635 --> 00:54:34,916 and you can identify these disclination pairs reasonably 730 00:54:34,916 --> 00:54:35,416 easy. 731 00:54:39,250 --> 00:54:43,730 So at the end of the day, the picture that we have 732 00:54:43,730 --> 00:54:46,400 is something like this. 733 00:54:46,400 --> 00:54:51,670 We are starting with the triangular lattice 734 00:54:51,670 --> 00:54:55,890 that I drew at the beginning, and you're 735 00:54:55,890 --> 00:54:58,130 increasing temperature. 736 00:54:58,130 --> 00:55:00,970 We're asking what happens. 737 00:55:00,970 --> 00:55:03,940 So this is 0 temperature. 738 00:55:03,940 --> 00:55:07,160 Close to 0 temperature, what we have 739 00:55:07,160 --> 00:55:14,930 is an entity that has translational quasi 740 00:55:14,930 --> 00:55:16,366 long range order. 741 00:55:16,366 --> 00:55:24,140 So this quantity goes like 1 over x to this power a to g. 742 00:55:27,010 --> 00:55:37,166 Whereas the orientations go to a constant. 743 00:55:42,100 --> 00:55:48,340 Now, this a to g is there because there's 744 00:55:48,340 --> 00:55:50,590 a shear modulus. 745 00:55:50,590 --> 00:55:55,430 And so throughout this phase, I have a shear modulus. 746 00:55:55,430 --> 00:55:58,070 The parameter that I'm calling u, 747 00:55:58,070 --> 00:56:00,730 I had scaled inversely with temperature. 748 00:56:00,730 --> 00:56:04,590 So I have this shear modulus u that 749 00:56:04,590 --> 00:56:08,440 diverges once we scale by temperature as 1 750 00:56:08,440 --> 00:56:09,910 over temperature. 751 00:56:09,910 --> 00:56:16,160 But then as I come down, the reduction is more than one 752 00:56:16,160 --> 00:56:19,870 over temperature because I will have 753 00:56:19,870 --> 00:56:23,350 this effect of dislocations appearing in pairs, 754 00:56:23,350 --> 00:56:26,290 and the system becomes softer. 755 00:56:26,290 --> 00:56:28,010 And eventually you will find that there's 756 00:56:28,010 --> 00:56:32,030 a transition temperature at which the shear modulus drops 757 00:56:32,030 --> 00:56:34,770 down to 0. 758 00:56:34,770 --> 00:56:39,180 And we said that near this transition, 759 00:56:39,180 --> 00:56:42,790 there is this behavior that mu approaches mu 760 00:56:42,790 --> 00:56:48,500 c, whatever it is, with something-- let's 761 00:56:48,500 --> 00:56:51,240 call this t 1. 762 00:56:51,240 --> 00:56:56,590 T 1 minus t to this exponent mu bar 763 00:56:56,590 --> 00:56:58,831 which was planned to be 6963. 764 00:57:05,800 --> 00:57:12,630 Now, once we are beyond this temperature t 1, then 765 00:57:12,630 --> 00:57:22,940 our positional correlations decay exponentially 766 00:57:22,940 --> 00:57:24,890 at some correlation, like c. 767 00:57:27,900 --> 00:57:32,480 And this c is something that diverges 768 00:57:32,480 --> 00:57:34,090 on approaching this transition. 769 00:57:34,090 --> 00:57:40,450 So basically I have a c that goes up here to infinity. 770 00:57:40,450 --> 00:57:46,230 And the fact that if we calculate the c, 771 00:57:46,230 --> 00:57:50,190 it diverges according to this strange formula that 772 00:57:50,190 --> 00:57:54,705 was 1 over t minus t 1 to this exponent mu 773 00:57:54,705 --> 00:57:58,530 bar Very strange type of divergence. 774 00:58:03,850 --> 00:58:08,860 But then, associated with the presence of this c 775 00:58:08,860 --> 00:58:12,160 is the fact that when you look at the orientational 776 00:58:12,160 --> 00:58:22,380 correlations, they don't decay as an exponential 777 00:58:22,380 --> 00:58:25,750 but as a power-law 8 of c. 778 00:58:28,870 --> 00:58:37,490 And this 8 of c is related to this k a, and falls off as 1 779 00:58:37,490 --> 00:58:41,180 over c squared. 780 00:58:41,180 --> 00:58:45,870 So here it diverges as you approach this transition. 781 00:58:45,870 --> 00:58:49,890 Now, as we go further and further on, 782 00:58:49,890 --> 00:58:53,518 the disclinations will appear-- disclinations 783 00:58:53,518 --> 00:58:56,990 with [INAUDIBLE] resolve of the angles 784 00:58:56,990 --> 00:58:59,140 to be parallel to each other. 785 00:58:59,140 --> 00:59:01,150 And there's another transition that 786 00:59:01,150 --> 00:59:08,820 is [INAUDIBLE], at which this is going to suddenly go down to 0. 787 00:59:08,820 --> 00:59:13,096 And close to here, we have that a to c 788 00:59:13,096 --> 00:59:18,820 reaches the critical value of 1/4 v 789 00:59:18,820 --> 00:59:24,210 to square root of-- let's call it 790 00:59:24,210 --> 00:59:32,320 t 2-- v the square root singularity. 791 00:59:35,887 --> 00:59:42,560 And then finally we have the ordinary liquid phase, 792 00:59:42,560 --> 00:59:53,790 where additionally I will find that psi 6 of x psi star 6 of 0 793 00:59:53,790 --> 00:59:57,260 decays exponentially. 794 00:59:57,260 --> 01:00:00,520 Let's call it psi 6. 795 01:00:00,520 --> 01:00:04,100 And this psi 6 is something that will diverge 796 01:00:04,100 --> 01:00:12,638 of this transition as an exponential of minus 1 797 01:00:12,638 --> 01:00:15,584 over square root of t minus t2. 798 01:00:19,530 --> 01:00:27,020 So this is the current scenario of how 799 01:00:27,020 --> 01:00:35,010 melting could occur for a system of particles in two dimensions. 800 01:00:35,010 --> 01:00:37,376 If it is a continuous phase transition, 801 01:00:37,376 --> 01:00:40,770 it has to go through these two transitions 802 01:00:40,770 --> 01:00:44,850 with the intermediate exotic phase. 803 01:00:44,850 --> 01:00:48,840 Of course, it is also possible-- and typically people 804 01:00:48,840 --> 01:00:53,530 were seeing numerically when they were doing hearts, 805 01:00:53,530 --> 01:00:55,580 spheres, et cetera, that there is 806 01:00:55,580 --> 01:01:02,360 a direct transition from here to here, which is discontinuous, 807 01:01:02,360 --> 01:01:03,960 like you have in three dimensions. 808 01:01:03,960 --> 01:01:07,260 So that's an area of a discontinuous transition 809 01:01:07,260 --> 01:01:09,270 that is not ruled out. 810 01:01:09,270 --> 01:01:11,305 But if you have continuous transitions, 811 01:01:11,305 --> 01:01:15,524 it has to have this intermediate phase in [INAUDIBLE]. 812 01:01:25,444 --> 01:01:26,436 Any questions? 813 01:01:31,888 --> 01:01:32,388 Yes? 814 01:01:32,388 --> 01:01:36,356 AUDIENCE: [INAUDIBLE] so the red one is mu. 815 01:01:36,356 --> 01:01:40,910 The yellow one is theta psi, and the purple one is [INAUDIBLE]. 816 01:01:40,910 --> 01:01:44,600 PROFESSOR: The correlation, then, that I would put here. 817 01:01:44,600 --> 01:01:46,196 So they are three different entities. 818 01:02:27,470 --> 01:02:30,490 So throughout the course, we have 819 01:02:30,490 --> 01:02:33,550 been thinking about systems that are 820 01:02:33,550 --> 01:02:36,800 described by some kind of an equilibrium probability 821 01:02:36,800 --> 01:02:39,050 distribution. 822 01:02:39,050 --> 01:02:43,260 So what we did not discuss is how the system 823 01:02:43,260 --> 01:02:46,190 comes to that equilibrium. 824 01:02:46,190 --> 01:02:52,750 So we're going to now very briefly talk about dynamics, 825 01:02:52,750 --> 01:02:56,010 and the specific type of dynamics 826 01:02:56,010 --> 01:02:58,670 that is common to condensed matter 827 01:02:58,670 --> 01:03:01,785 systems at finite temperature, which I 828 01:03:01,785 --> 01:03:03,790 will call precipative dynamics. 829 01:03:06,510 --> 01:03:16,084 And the prototype of this is a Brownian particle 830 01:03:16,084 --> 01:03:22,300 that I will briefly review for you. 831 01:03:22,300 --> 01:03:31,230 So what you have is that you have a particle that 832 01:03:31,230 --> 01:03:36,510 is within some kind of a solvent, 833 01:03:36,510 --> 01:03:40,060 and this particle is moving around. 834 01:03:40,060 --> 01:03:43,580 So you would say, let's for simplicity 835 01:03:43,580 --> 01:03:47,730 actually focus on the one direction, x. 836 01:03:47,730 --> 01:03:50,830 And you would say that the mass of the particle 837 01:03:50,830 --> 01:03:56,450 times its acceleration is equal to the forces 838 01:03:56,450 --> 01:03:58,515 that it's experiencing. 839 01:04:02,280 --> 01:04:10,880 The forces-- well, if you are moving in a fluid, 840 01:04:10,880 --> 01:04:14,780 you are going to be subject to some kind 841 01:04:14,780 --> 01:04:21,490 of a dissipative force which is typically 842 01:04:21,490 --> 01:04:24,800 portional to your velocity. 843 01:04:24,800 --> 01:04:27,670 If you, for example, solve for the hydrodynamic 844 01:04:27,670 --> 01:04:31,470 of a sphere in a fluid, you find that mu 845 01:04:31,470 --> 01:04:33,945 is related to viscosity inversely 846 01:04:33,945 --> 01:04:36,760 to the size of the particle, et cetera. 847 01:04:36,760 --> 01:04:38,410 But that behavior is generic. 848 01:04:38,410 --> 01:04:42,730 You're not going be thinking about that. 849 01:04:42,730 --> 01:04:46,210 Now suppose that additionally I put 850 01:04:46,210 --> 01:04:50,470 some kind of an optical trap, or something that tries 851 01:04:50,470 --> 01:04:54,060 to localize this potential. 852 01:04:54,060 --> 01:04:58,680 So then there would be an additional force v 853 01:04:58,680 --> 01:05:04,230 2, the derivative of the potential with respect to x. 854 01:05:07,086 --> 01:05:10,950 And then we are talking about Brownian particles. 855 01:05:10,950 --> 01:05:13,730 Brownian particles are constantly jiggling. 856 01:05:13,730 --> 01:05:18,804 So there is also a random force that is a function of time. 857 01:05:26,932 --> 01:05:34,370 Now we are going to be interested in the dynamics that 858 01:05:34,370 --> 01:05:38,780 is very much controlled by the dissipation term. 859 01:05:38,780 --> 01:05:44,770 And acceleration we can forget. 860 01:05:44,770 --> 01:05:50,930 And if we are in that limit, we can write the equation 861 01:05:50,930 --> 01:06:00,005 as mu-- I can sort of rearrange it slightly as-- actually, 862 01:06:00,005 --> 01:06:02,910 let me change location to this. 863 01:06:06,230 --> 01:06:13,210 So that the eventual velocity x dot 864 01:06:13,210 --> 01:06:19,026 is going to be proportional to the external force. 865 01:06:21,840 --> 01:06:26,950 mu the coefficient that is the mobility. 866 01:06:26,950 --> 01:06:34,580 So mu essentially relates the force to the velocity. 867 01:06:34,580 --> 01:06:38,100 Of course, this is the average force. 868 01:06:38,100 --> 01:06:43,300 And there is a fluctuating part, so essentially, I 869 01:06:43,300 --> 01:06:47,535 call mu times this to be the a times function of t. 870 01:06:51,720 --> 01:06:57,360 Now, if I didn't have this external force, 871 01:06:57,360 --> 01:07:02,130 the fluctuations of the particles would be diffusive. 872 01:07:02,130 --> 01:07:04,700 And you can convince yourself that you 873 01:07:04,700 --> 01:07:13,090 can get the diffusive result provided that you relate 874 01:07:13,090 --> 01:07:18,230 the correlations of this force that fluctuating 875 01:07:18,230 --> 01:07:23,440 and have 0 average, the diffusion constant d 876 01:07:23,440 --> 01:07:28,372 of the particle in the medium through delta of t minus t. 877 01:07:32,660 --> 01:07:38,170 So if their track was not there, you solve of this equation 878 01:07:38,170 --> 01:07:42,200 without the track and find that the prohibitive distribution 879 01:07:42,200 --> 01:07:50,590 for x grows as a Gaussian whose width grows with time, as d t. 880 01:07:50,590 --> 01:07:55,090 d therefore must be the diffusion constant. 881 01:07:55,090 --> 01:07:59,700 Now, in the presence of the potential, 882 01:07:59,700 --> 01:08:02,930 this particle will start to fluctuate. 883 01:08:02,930 --> 01:08:05,600 Eventually if you wait long enough, 884 01:08:05,600 --> 01:08:07,600 there is a probability that it will be here, 885 01:08:07,600 --> 01:08:10,100 a probability to be somewhere else. 886 01:08:10,100 --> 01:08:13,870 So at long enough times, there's a probability p 887 01:08:13,870 --> 01:08:17,340 of x to find the particle. 888 01:08:17,340 --> 01:08:23,120 And you expect that t of x will be 889 01:08:23,120 --> 01:08:28,430 proportional to exponential of minus v of x divided 890 01:08:28,430 --> 01:08:30,149 by whatever the temperature is. 891 01:08:33,109 --> 01:08:38,439 And you can show that in order to have this occur, 892 01:08:38,439 --> 01:08:44,863 you need to relate mu and d through the so called Einstein 893 01:08:44,863 --> 01:08:45,362 relation. 894 01:08:48,609 --> 01:08:52,540 So this is a brief review of Brownian particles. 895 01:08:52,540 --> 01:08:54,420 Yes? 896 01:08:54,420 --> 01:09:01,960 AUDIENCE: The average and time correlation of eta 897 01:09:01,960 --> 01:09:06,044 can be found by saying the potential is 0, right? 898 01:09:06,044 --> 01:09:08,160 PROFESSOR: Mm hmm. 899 01:09:08,160 --> 01:09:09,729 AUDIENCE: Those will still be true 900 01:09:09,729 --> 01:09:12,880 even if the potential is not 0, right? 901 01:09:12,880 --> 01:09:14,779 PROFESSOR: Yes. 902 01:09:14,779 --> 01:09:22,120 So I just wanted to have an idea of where this d comes from. 903 01:09:22,120 --> 01:09:27,200 But more specifically, this is the important thing. 904 01:09:27,200 --> 01:09:32,430 That if at very long times you want 905 01:09:32,430 --> 01:09:35,760 to have a probability distribution coming 906 01:09:35,760 --> 01:09:41,660 from this equation, that has the Boltzmann form 907 01:09:41,660 --> 01:09:46,210 with k t, the coefficients of mu and the noise, 908 01:09:46,210 --> 01:09:49,830 you have to relate through the so called Einstein relation. 909 01:09:49,830 --> 01:09:53,109 And once you do that, this result 910 01:09:53,109 --> 01:09:57,990 is true no matter how complicated this v of x is. 911 01:10:00,920 --> 01:10:03,960 So in general, for a complicated v of x, 912 01:10:03,960 --> 01:10:08,020 you won't be able to solve this equation analytically. 913 01:10:08,020 --> 01:10:10,390 You can only do it numerically. 914 01:10:10,390 --> 01:10:14,650 Yet you are guaranteed that this equation with this noise 915 01:10:14,650 --> 01:10:18,080 correlator will have asymptotically a probability 916 01:10:18,080 --> 01:10:19,423 distribution of [INAUDIBLE]. 917 01:10:42,780 --> 01:10:45,580 The problem that we have been looking at all 918 01:10:45,580 --> 01:10:50,285 along is something different. 919 01:10:50,285 --> 01:10:53,960 Let's say you have, let's say, a piece of magnet 920 01:10:53,960 --> 01:10:59,400 or some other system that we characterize, 921 01:10:59,400 --> 01:11:04,020 let's say, by something m of x. 922 01:11:04,020 --> 01:11:07,640 Again, you can do it for vector, but for simplicity, let's do it 923 01:11:07,640 --> 01:11:10,110 for the scalar case. 924 01:11:10,110 --> 01:11:16,420 So we know, or we have stated, that subject to the symmetries 925 01:11:16,420 --> 01:11:20,250 of the system, I know the probability. 926 01:11:20,250 --> 01:11:27,890 For some configuration of this field is governed by a form, 927 01:11:27,890 --> 01:11:30,990 let's say, that has Landau Ginzburg character. 928 01:11:52,070 --> 01:11:54,500 So that has been our starting point. 929 01:11:54,500 --> 01:11:58,350 We have said that I have a prohibitive distribution that 930 01:11:58,350 --> 01:12:01,120 is of this form. 931 01:12:01,120 --> 01:12:05,025 So that statement is kind of like this statement. 932 01:12:08,080 --> 01:12:12,050 But the way that I came to that statement 933 01:12:12,050 --> 01:12:15,640 was to say that there was a degree of freedom 934 01:12:15,640 --> 01:12:20,430 x, the position of the particle, that was fluctuating subject 935 01:12:20,430 --> 01:12:24,790 to forces and external variables from the particles 936 01:12:24,790 --> 01:12:30,645 of the fluid, that was given by this so called Langevin 937 01:12:30,645 --> 01:12:31,145 equation. 938 01:12:37,770 --> 01:12:41,350 So I had a time dependent prescription 939 01:12:41,350 --> 01:12:46,170 that eventually went to the Boltzmann way that I wanted. 940 01:12:46,170 --> 01:12:51,910 Here I have started with the final Boltzmann weight. 941 01:12:51,910 --> 01:12:57,620 And the question is, can I think about a dynamics 942 01:12:57,620 --> 01:13:05,130 for a field that will eventually give this state. 943 01:13:05,130 --> 01:13:11,680 So there are lots and lots of different dynamics 944 01:13:11,680 --> 01:13:14,300 that I can impose. 945 01:13:14,300 --> 01:13:17,060 But I want to look at the dynamics that 946 01:13:17,060 --> 01:13:21,370 is closest to the Brownian particle that I wrote, 947 01:13:21,370 --> 01:13:25,240 and that's where this word dissipative comes. 948 01:13:25,240 --> 01:13:30,390 So among the universe of all possible dynamics, 949 01:13:30,390 --> 01:13:37,840 I'm going to look at one that has a linear time 950 01:13:37,840 --> 01:13:40,378 derivative for the field n. 951 01:13:44,940 --> 01:13:48,850 So this is the analog of the x dot. 952 01:13:48,850 --> 01:13:54,340 And so I write that it is equal to some coefficient, 953 01:13:54,340 --> 01:13:58,230 that with their minds, the ease with which 954 01:13:58,230 --> 01:14:02,020 that particle-- well the field of that location x 955 01:14:02,020 --> 01:14:07,040 changes as a function of the forces that is exerted on it. 956 01:14:07,040 --> 01:14:10,790 I assume that mu is the same across my system. 957 01:14:10,790 --> 01:14:13,910 So here I'm already assuming there's no x dependence. 958 01:14:13,910 --> 01:14:15,900 This system is uniform. 959 01:14:15,900 --> 01:14:19,130 And then there was a d v by d x. 960 01:14:19,130 --> 01:14:22,460 So v was ultimately the thing that was appearing 961 01:14:22,460 --> 01:14:24,200 in the Boltzmann weight. 962 01:14:24,200 --> 01:14:27,590 So clearly the analog of the v that I have 963 01:14:27,590 --> 01:14:30,190 is this Landau Ginzburg. 964 01:14:30,190 --> 01:14:37,070 So I will do a function of the derivative of this quantity 965 01:14:37,070 --> 01:14:44,425 that I will call beta h, with respect to m of x. 966 01:14:48,520 --> 01:14:53,640 Again, over there, I had one variable, x. 967 01:14:53,640 --> 01:14:56,220 You can imagine that I could have had a system where 968 01:14:56,220 --> 01:15:02,120 two particles, x 1 and x 2, also have an interaction among them. 969 01:15:02,120 --> 01:15:05,780 Then the equation that I would have had over there 970 01:15:05,780 --> 01:15:09,740 would be the force that is acting on particle 1, 971 01:15:09,740 --> 01:15:11,910 by taking the total potential-- which 972 01:15:11,910 --> 01:15:17,350 is the external potential plus the potential that 973 01:15:17,350 --> 01:15:19,440 comes from the inter particle interaction. 974 01:15:19,440 --> 01:15:21,120 So I would have to take a derivative 975 01:15:21,120 --> 01:15:26,630 of the net potential energy v, divided with respect 976 01:15:26,630 --> 01:15:29,430 to either x 1 or x 2 to calculate the force 977 01:15:29,430 --> 01:15:31,470 on the first one or the second one. 978 01:15:31,470 --> 01:15:34,856 So here for a particular configuration 979 01:15:34,856 --> 01:15:38,120 m of the field across the system, 980 01:15:38,120 --> 01:15:42,690 if I'm interested in the dynamics of this position x, 981 01:15:42,690 --> 01:15:49,930 I have to take this total internal potential energy, 982 01:15:49,930 --> 01:15:51,790 and take the derivative with respect 983 01:15:51,790 --> 01:15:54,830 to the variable that is sitting on that side. 984 01:15:54,830 --> 01:15:57,510 So that's why this is a functional 985 01:15:57,510 --> 01:16:00,130 derivative of this end. 986 01:16:00,130 --> 01:16:06,950 And then I will have to put a noise, eta. 987 01:16:06,950 --> 01:16:10,010 Well, again, if I had multiple particles, 988 01:16:10,010 --> 01:16:14,520 I would subject each one of them to an independent noise. 989 01:16:14,520 --> 01:16:20,710 So at each location, I have an independent noise. 990 01:16:20,710 --> 01:16:24,490 So the noise is a function of time, 991 01:16:24,490 --> 01:16:26,910 but it is also wearing across my system. 992 01:16:30,140 --> 01:16:37,300 So if I take that form and do the functional derivative-- 993 01:16:37,300 --> 01:16:40,680 so if I take the derivative with respect to m of x, 994 01:16:40,680 --> 01:16:43,055 I have to take the derivative of these objects. 995 01:16:43,055 --> 01:16:49,550 So I will have minus derivative of t m squared is t m. 996 01:16:49,550 --> 01:16:56,250 The u m to the 4th is 4 u m q, and so forth. 997 01:16:56,250 --> 01:17:00,500 Once I have gotten rid of these terms, 998 01:17:00,500 --> 01:17:04,150 then I would have terms that depend on the gradient. 999 01:17:04,150 --> 01:17:08,450 So I would have minus the derivative of this object 1000 01:17:08,450 --> 01:17:10,670 with respect to the gradient. 1001 01:17:10,670 --> 01:17:14,300 So here I would get k gradient of m. 1002 01:17:14,300 --> 01:17:17,500 And then the next term would be Laplacian derivative, 1003 01:17:17,500 --> 01:17:18,660 with respect to Laplacian. 1004 01:17:18,660 --> 01:17:23,480 So I would put l Laplacian of m, and so forth 1005 01:17:23,480 --> 01:17:28,822 with the methodologies of taking function derivatives. 1006 01:17:28,822 --> 01:17:31,690 And then I have the noise. 1007 01:17:35,530 --> 01:17:41,110 So this leads to an equation which 1008 01:17:41,110 --> 01:17:48,589 is called a time dependent, Landau Ginzberg. 1009 01:17:55,491 --> 01:18:01,390 Because we started with the Landau Ginzberg weight, 1010 01:18:01,390 --> 01:18:04,200 and this equation, as we see shortly, 1011 01:18:04,200 --> 01:18:10,510 subject to similar restrictions as we had before, will give us, 1012 01:18:10,510 --> 01:18:12,503 eventually, this probability distribution. 1013 01:18:16,130 --> 01:18:19,410 This is a difficult equation in the same sense 1014 01:18:19,410 --> 01:18:21,605 that the original Landau Ginzberg 1015 01:18:21,605 --> 01:18:24,820 is difficult to look at correlations, et cetera. 1016 01:18:24,820 --> 01:18:28,100 This is a nonlinear equation, causes various difficulties, 1017 01:18:28,100 --> 01:18:30,655 and we need approaches to be able to deal 1018 01:18:30,655 --> 01:18:34,190 with the difficult, non linealities. 1019 01:18:34,190 --> 01:18:38,790 So what we did for the Landau Ginzburg 1020 01:18:38,790 --> 01:18:43,430 was to initially get insights and simplify the system 1021 01:18:43,430 --> 01:18:47,910 by focusing on the linearized or Gaussian version. 1022 01:18:47,910 --> 01:18:53,453 So let's look at the version of this equation that 1023 01:18:53,453 --> 01:18:54,439 is linearized. 1024 01:18:58,390 --> 01:19:01,650 And when it is linearized, what I have on the left hand side 1025 01:19:01,650 --> 01:19:05,020 is d m by d t. 1026 01:19:05,020 --> 01:19:09,930 What I have on the right hand side is mu. 1027 01:19:09,930 --> 01:19:12,060 I have t m. 1028 01:19:12,060 --> 01:19:15,430 I got rid of the non linear term, 1029 01:19:15,430 --> 01:19:21,180 so the next term that I will have will be k Laplacian of m, 1030 01:19:21,180 --> 01:19:27,216 and then will be minus l 4th derivative of m, and so forth. 1031 01:19:27,216 --> 01:19:29,918 And then there will be a noise [INAUDIBLE]. 1032 01:19:41,560 --> 01:19:46,040 One thing that I can immediately do 1033 01:19:46,040 --> 01:19:49,000 is to go to fullier transform. 1034 01:19:49,000 --> 01:19:55,440 So m of x goes to m theta of q. 1035 01:19:55,440 --> 01:20:00,530 And if I do that, but not fullier transform in time, 1036 01:20:00,530 --> 01:20:04,820 I will get that the time derivative of m tilde of q 1037 01:20:04,820 --> 01:20:11,020 is essentially what I have here. 1038 01:20:11,020 --> 01:20:14,880 And I forgot the minus that I have here. 1039 01:20:14,880 --> 01:20:16,420 So this minus is important. 1040 01:20:24,640 --> 01:20:30,970 And then this becomes negative, this becomes positive. 1041 01:20:30,970 --> 01:20:34,980 So that when I go fullier transform, what I will get 1042 01:20:34,980 --> 01:20:43,780 is minus t plus k q squared plus l q to the 4th, and so forth. 1043 01:20:43,780 --> 01:20:50,400 And tilde of q with this mu out front. 1044 01:20:50,400 --> 01:20:55,266 And then the fullier transform of what my noise is. 1045 01:21:00,150 --> 01:21:06,840 First thing to note is that even in the absence of noise, 1046 01:21:06,840 --> 01:21:11,180 there is a set of relaxation times. 1047 01:21:11,180 --> 01:21:14,360 That is, for eta it was to 0. 1048 01:21:14,360 --> 01:21:18,660 Or in general, I would have n tilde of q and p. 1049 01:21:18,660 --> 01:21:24,220 I can solve this equation kind of simply. 1050 01:21:24,220 --> 01:21:27,910 It is the m by d t is some constant times 1051 01:21:27,910 --> 01:21:31,735 n-- let's call it gama of q-- which 1052 01:21:31,735 --> 01:21:33,880 has dimensions of 1 over time. 1053 01:21:33,880 --> 01:21:38,160 So I can call that 1 over tau of q. 1054 01:21:38,160 --> 01:21:40,020 If I didn't have noise, if I started 1055 01:21:40,020 --> 01:21:45,160 with some original value at time 0, 1056 01:21:45,160 --> 01:21:50,360 it is going to decay exponentially 1057 01:21:50,360 --> 01:21:54,270 with this characteristic time. 1058 01:21:54,270 --> 01:21:56,770 And once I have noise, it is actually 1059 01:21:56,770 --> 01:21:59,940 easy to convince yourself that the answer is 1060 01:21:59,940 --> 01:22:07,590 0 2 t d t prime e to the minus this gamma of q or inversify. 1061 01:22:10,800 --> 01:22:17,799 Tau of q times eta of q i t prime. 1062 01:22:21,210 --> 01:22:30,970 So you see that you have a hierarchy of relaxation times, 1063 01:22:30,970 --> 01:22:40,900 ta of q, which are 1 over u t plus k q squared, and so forth, 1064 01:22:40,900 --> 01:22:46,800 which scale in two limits. 1065 01:22:46,800 --> 01:22:52,890 Either the wavelength lambda, which is the inverse of q 1066 01:22:52,890 --> 01:22:57,630 is much larger than the correlation length-- 1067 01:22:57,630 --> 01:23:00,260 and the correlation length of this model 1068 01:23:00,260 --> 01:23:04,980 you have seen to be the square root of t over k, 1069 01:23:04,980 --> 01:23:08,600 square root of k over t-- or the other limit, 1070 01:23:08,600 --> 01:23:13,470 where lambda is much less than c. 1071 01:23:13,470 --> 01:23:15,640 In this limit, where we are looking 1072 01:23:15,640 --> 01:23:20,640 at modes that are much shorter than the correlation length, 1073 01:23:20,640 --> 01:23:22,130 this term is dominant. 1074 01:23:22,130 --> 01:23:25,996 This becomes 1 over mu k q squared. 1075 01:23:25,996 --> 01:23:32,340 In the other limit, it goes to a constant 1 over u t. 1076 01:23:32,340 --> 01:23:36,850 So this linear equation has a whole bunch 1077 01:23:36,850 --> 01:23:39,360 of modes that can be characterized 1078 01:23:39,360 --> 01:23:42,310 by their wavelength or their wave number. 1079 01:23:42,310 --> 01:23:45,680 You find that the short wavelength modes 1080 01:23:45,680 --> 01:23:50,060 have this characteristic, time, that becomes longer and longer 1081 01:23:50,060 --> 01:23:52,580 as the wavelength increases. 1082 01:23:52,580 --> 01:23:57,280 So if you make the wavelength twice as large, 1083 01:23:57,280 --> 01:24:00,640 and you want to relax a system that is linearly 1084 01:24:00,640 --> 01:24:05,430 twice as large, this says that it will take 4 times longer. 1085 01:24:05,430 --> 01:24:09,840 Because the answer goes like lambda squared. 1086 01:24:09,840 --> 01:24:14,720 Whereas eventually you reach the size of the correlation length. 1087 01:24:14,720 --> 01:24:17,670 Once you are beyond the size of the correlation length, 1088 01:24:17,670 --> 01:24:18,680 it doesn't matter. 1089 01:24:18,680 --> 01:24:21,030 It's the same time. 1090 01:24:21,030 --> 01:24:24,280 But the interesting thing, of course, to us 1091 01:24:24,280 --> 01:24:26,750 is that there are phase transitions that 1092 01:24:26,750 --> 01:24:27,560 are continuous. 1093 01:24:27,560 --> 01:24:29,330 And close to that phase transition, 1094 01:24:29,330 --> 01:24:32,250 the correlation length goes to infinity. 1095 01:24:32,250 --> 01:24:37,610 Which means that the relaxation time also will go to infinity. 1096 01:24:37,610 --> 01:24:39,520 So according to this theory, there's 1097 01:24:39,520 --> 01:24:44,080 a particular divergence as 1 over t minus t c. 1098 01:24:44,080 --> 01:24:49,560 But it will be modified, and as I will discuss next time, 1099 01:24:49,560 --> 01:24:53,770 this is only-- even within to dissipative class-- 1100 01:24:53,770 --> 01:24:56,800 one type of dynamics that you can have. 1101 01:24:56,800 --> 01:24:59,310 And there are additional dynamics, 1102 01:24:59,310 --> 01:25:05,020 and this system characterizes criticality 1103 01:25:05,020 --> 01:25:08,630 as single universality class in statics. 1104 01:25:08,630 --> 01:25:13,160 There are many dynamic universality classes that 1105 01:25:13,160 --> 01:25:16,100 correspond to this same static.