1 00:00:00,070 --> 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 --> 00:00:17,260 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,260 --> 00:00:21,730 at ocw.mit.edu 8 00:00:21,730 --> 00:00:25,160 MEHRAN KARDAR: OK, let's start. 9 00:00:25,160 --> 00:00:31,730 So in this class, we focused mostly 10 00:00:31,730 --> 00:00:41,150 on having some slab of material and having some configuration 11 00:00:41,150 --> 00:00:45,310 of some kind of a field inside. 12 00:00:45,310 --> 00:00:49,720 And we said that, basically, we are 13 00:00:49,720 --> 00:00:52,810 going to be interested close to, let's say, 14 00:00:52,810 --> 00:00:56,950 phase transition and some quantity that changes 15 00:00:56,950 --> 00:00:57,950 at the phase transition. 16 00:00:57,950 --> 00:01:01,330 We are interested in figuring out the singularities 17 00:01:01,330 --> 00:01:03,350 associated with that. 18 00:01:03,350 --> 00:01:06,710 And we can coarse grain. 19 00:01:06,710 --> 00:01:12,470 Once we have coarse grained, we have the field m, potentially 20 00:01:12,470 --> 00:01:16,950 a vector, that is characterized throughout this material. 21 00:01:16,950 --> 00:01:20,960 So it's a field that's function of x. 22 00:01:20,960 --> 00:01:28,780 And by integrating out over a lot of degrees of freedom, 23 00:01:28,780 --> 00:01:32,170 we can focus on the probability of finding 24 00:01:32,170 --> 00:01:37,080 different configurations of this field. 25 00:01:37,080 --> 00:01:40,150 And this probability we constructed 26 00:01:40,150 --> 00:01:46,010 on the basis of a number of simple assumptions 27 00:01:46,010 --> 00:01:50,780 such as locality, which implied that we would write 28 00:01:50,780 --> 00:01:56,360 this probability as a product of contributions 29 00:01:56,360 --> 00:02:01,590 of different parts, which in the exponent becomes an integral. 30 00:02:01,590 --> 00:02:05,520 And then we would put within this all kinds 31 00:02:05,520 --> 00:02:09,560 of things that are consistent with the symmetries 32 00:02:09,560 --> 00:02:10,599 of the problem. 33 00:02:10,599 --> 00:02:13,370 So if, for example, this is a field that 34 00:02:13,370 --> 00:02:16,090 is invariant on the rotations, we 35 00:02:16,090 --> 00:02:22,735 would be having terms such as m squared, m to the fourth, 36 00:02:22,735 --> 00:02:23,580 and so forth. 37 00:02:26,630 --> 00:02:28,620 But the interesting thing was that, of course, 38 00:02:28,620 --> 00:02:32,380 there is some interaction with the neighborhoods. 39 00:02:32,380 --> 00:02:36,520 Those neighborhood interactions we can, in the continuum limit, 40 00:02:36,520 --> 00:02:40,670 inclement by putting terms that are proportional to radiant 41 00:02:40,670 --> 00:02:42,540 of m and so forth. 42 00:02:50,240 --> 00:02:52,280 So there is a lot of things that you 43 00:02:52,280 --> 00:02:55,620 could put consistent with symmetry 44 00:02:55,620 --> 00:02:58,550 and presumably be as general as possible. 45 00:02:58,550 --> 00:03:00,880 You could have this and the terms 46 00:03:00,880 --> 00:03:04,620 in this coefficients of this expansion would 47 00:03:04,620 --> 00:03:08,260 be these phenomenological parameters characterizing 48 00:03:08,260 --> 00:03:11,230 this probability, function of all kinds 49 00:03:11,230 --> 00:03:13,090 of microscopic degrees of freedom, 50 00:03:13,090 --> 00:03:15,310 as well as microscopic constraints 51 00:03:15,310 --> 00:03:19,140 such as temperature, pressures, et cetera. 52 00:03:19,140 --> 00:03:27,390 Now the question that we have is if I start with a system, 53 00:03:27,390 --> 00:03:30,190 let's say a configuration where everybody's pointing up 54 00:03:30,190 --> 00:03:33,990 or some other configuration that is not the equilibrium 55 00:03:33,990 --> 00:03:36,930 configuration, how does the probability 56 00:03:36,930 --> 00:03:41,290 evolve to become something like this? 57 00:03:41,290 --> 00:03:44,970 Now we are interested therefore in m 58 00:03:44,970 --> 00:03:49,030 that is a function of position and time. 59 00:03:49,030 --> 00:03:54,180 And since I want to use t for time, this coefficient of m 60 00:03:54,180 --> 00:03:57,366 squared that we were previously calling t, 61 00:03:57,366 --> 00:04:02,250 I will indicate by r, OK? 62 00:04:02,250 --> 00:04:04,470 There are various types of dynamics that you 63 00:04:04,470 --> 00:04:07,020 can look at for this problem. 64 00:04:07,020 --> 00:04:09,562 I will look at the class that is dissipative. 65 00:04:15,880 --> 00:04:20,100 And its inspiration is the Brownian motion 66 00:04:20,100 --> 00:04:22,980 that we discussed last time where 67 00:04:22,980 --> 00:04:27,850 we saw that when you put a particle in a fluid 68 00:04:27,850 --> 00:04:31,910 to a very good approximation when the fluid is viscous, 69 00:04:31,910 --> 00:04:34,880 it is the velocity that is proportional to the force 70 00:04:34,880 --> 00:04:38,600 and you can't ignore inertial effects such as mass times 71 00:04:38,600 --> 00:04:41,330 acceleration and you write an equation 72 00:04:41,330 --> 00:04:45,930 that is linear in position. 73 00:04:45,930 --> 00:04:48,210 Velocity is the linear derivative 74 00:04:48,210 --> 00:04:50,180 of position, which is the variable that 75 00:04:50,180 --> 00:04:52,130 is of interest to you. 76 00:04:52,130 --> 00:04:55,000 So here the variable that is of interest to us, 77 00:04:55,000 --> 00:04:59,010 is this magnetization that is changing as a function of time. 78 00:04:59,010 --> 00:05:02,100 And the equation that we write down 79 00:05:02,100 --> 00:05:09,400 is the derivative of this field with respect to time. 80 00:05:09,400 --> 00:05:14,650 And again, for the Brownian motion, 81 00:05:14,650 --> 00:05:18,400 the velocity was proportional to the force. 82 00:05:18,400 --> 00:05:22,380 The constant of proportionality was some kind of a mobility 83 00:05:22,380 --> 00:05:24,340 that you would have in the fluid. 84 00:05:24,340 --> 00:05:27,530 So continuing with that inspiration, 85 00:05:27,530 --> 00:05:29,740 let's put some kind of a mobility here. 86 00:05:29,740 --> 00:05:32,830 I should really put a vector field here, but just 87 00:05:32,830 --> 00:05:38,420 for convenience, let's just focus on one component 88 00:05:38,420 --> 00:05:40,830 and see what happens. 89 00:05:40,830 --> 00:05:44,730 Now what is the force? 90 00:05:44,730 --> 00:05:49,670 Presumably, each location individually 91 00:05:49,670 --> 00:05:52,570 feels some kind of a force. 92 00:05:52,570 --> 00:05:56,660 And typically when we had the Brownian particle, the force 93 00:05:56,660 --> 00:05:58,420 we were getting from the derivative 94 00:05:58,420 --> 00:06:02,125 of the potential with respect to the [? aviation ?] 95 00:06:02,125 --> 00:06:06,390 of the position, the field that we are interested. 96 00:06:06,390 --> 00:06:13,030 So the analog of our potential is the energy 97 00:06:13,030 --> 00:06:17,280 that we have over here, and in the same sense 98 00:06:17,280 --> 00:06:21,430 that we want this effective potential, if you like, 99 00:06:21,430 --> 00:06:24,760 to govern the equilibrium behavior-- 100 00:06:24,760 --> 00:06:28,050 and again, recall that for the case of the Brownian particle, 101 00:06:28,050 --> 00:06:31,670 eventually the probability was related to the potential by e 102 00:06:31,670 --> 00:06:36,450 to the minus beta v. This is the analog of beta v, now 103 00:06:36,450 --> 00:06:38,770 considered in the entire field. 104 00:06:38,770 --> 00:06:45,740 So the analog of the force is a derivative of this beta H, 105 00:06:45,740 --> 00:06:53,600 so this would be our beta H with respect to the variable 106 00:06:53,600 --> 00:06:55,590 that I'm trying to change. 107 00:06:58,930 --> 00:07:05,325 And since I don't have just one variable, but a field, 108 00:07:05,325 --> 00:07:08,350 the analog of the derivative becomes this functional 109 00:07:08,350 --> 00:07:09,980 derivative. 110 00:07:09,980 --> 00:07:14,160 And in the same sense that the Brownian particle, 111 00:07:14,160 --> 00:07:17,540 the Brownian motion, is trying to pull you 112 00:07:17,540 --> 00:07:20,470 towards the minimum of the potential, 113 00:07:20,470 --> 00:07:24,450 this is an equation that, if I kind of stop over here, 114 00:07:24,450 --> 00:07:28,090 tries to put the particle towards the minimum 115 00:07:28,090 --> 00:07:33,000 of this beta H. 116 00:07:33,000 --> 00:07:36,530 Now the reason that the Brownian particle didn't go and stick 117 00:07:36,530 --> 00:07:40,220 at one position, which was the minimum, but fluctuated, 118 00:07:40,220 --> 00:07:43,310 was of course, that we added the random force. 119 00:07:43,310 --> 00:07:46,970 So there is some kind of an analog of a random force that 120 00:07:46,970 --> 00:07:51,157 we can put either in front of [? mu ?] or imagine that if we 121 00:07:51,157 --> 00:07:56,100 added it to [? mu, ?] and we put it over here. 122 00:07:56,100 --> 00:07:59,740 Now for the case of the Brownian particle, 123 00:07:59,740 --> 00:08:04,930 the assumption was that if I evaluate eta at some time, 124 00:08:04,930 --> 00:08:08,860 and eta at another time t2 and t1, 125 00:08:08,860 --> 00:08:17,520 that this was related to 2D delta function t1 minus t2. 126 00:08:17,520 --> 00:08:23,840 Now of course here, at each location, I have a noise term. 127 00:08:23,840 --> 00:08:28,400 So this noise carries an index, which 128 00:08:28,400 --> 00:08:31,980 indicates the position, which is, of course, a vector in D 129 00:08:31,980 --> 00:08:34,309 dimension in principal. 130 00:08:34,309 --> 00:08:38,610 And there is no reason to imagine that the noise here 131 00:08:38,610 --> 00:08:42,210 that comes from all kinds of microscopic degrees of freedom 132 00:08:42,210 --> 00:08:45,310 that we have integrated out should 133 00:08:45,310 --> 00:08:48,650 have correlation with the noise at some other point. 134 00:08:48,650 --> 00:08:54,520 So the simplest assumption is to also put the delta function 135 00:08:54,520 --> 00:08:58,710 in the positions, OK? 136 00:08:58,710 --> 00:09:02,340 So if I take this beta H that I have over there 137 00:09:02,340 --> 00:09:05,530 and take the functional derivative, what do I get? 138 00:09:05,530 --> 00:09:17,110 I will get that the m of xt by dt is the function of-- oops, 139 00:09:17,110 --> 00:09:20,650 and I forgot to put a the minus sign here. 140 00:09:20,650 --> 00:09:23,290 The force is minus the potential, 141 00:09:23,290 --> 00:09:26,100 and this is basically going down the gradient, 142 00:09:26,100 --> 00:09:28,060 so I need to put that. 143 00:09:28,060 --> 00:09:31,530 So I have to take a derivative of this. 144 00:09:31,530 --> 00:09:33,765 First of all I can take a derivative with respect 145 00:09:33,765 --> 00:09:35,780 to m itself. 146 00:09:35,780 --> 00:09:40,055 I will get minus rm. 147 00:09:44,640 --> 00:09:48,840 Well, actually, let's put the minus out front. 148 00:09:48,840 --> 00:09:52,580 And then I will get the derivative of m 149 00:09:52,580 --> 00:09:56,280 to the fourth, which is [? 4umq, ?] all kinds of terms 150 00:09:56,280 --> 00:09:57,790 like this. 151 00:09:57,790 --> 00:10:01,110 Then the terms that come from the derivatives 152 00:10:01,110 --> 00:10:05,650 and taking the derivative with respect to gradient 153 00:10:05,650 --> 00:10:08,400 will give me K times the gradient. 154 00:10:08,400 --> 00:10:10,780 But then in the functional derivative, 155 00:10:10,780 --> 00:10:13,370 I have to take another derivative, 156 00:10:13,370 --> 00:10:17,320 converting this to minus K log Gaussian of m. 157 00:10:17,320 --> 00:10:20,050 And then I would have L the fourth derivative 158 00:10:20,050 --> 00:10:24,310 from the next term and so forth. 159 00:10:24,310 --> 00:10:26,330 And on top of everything else, I will 160 00:10:26,330 --> 00:10:31,080 have the noise who's statistics I have indicated above. 161 00:10:34,650 --> 00:10:37,880 So this entity, this equation, came 162 00:10:37,880 --> 00:10:42,820 from the Landau-Ginzburg model, and it 163 00:10:42,820 --> 00:10:51,325 is called a time dependent Landau-Ginzburg equation. 164 00:10:54,540 --> 00:11:02,120 And so would be the analog of the Brownian type of equation, 165 00:11:02,120 --> 00:11:05,810 now for an entire field. 166 00:11:05,810 --> 00:11:08,820 Now we're going to have a lot of difficulty 167 00:11:08,820 --> 00:11:12,750 in the short amount of time that is left to us to deal 168 00:11:12,750 --> 00:11:15,320 with this nonlinear equation, so we 169 00:11:15,320 --> 00:11:18,160 are going to do the same thing that we did in order 170 00:11:18,160 --> 00:11:22,300 to capture the properties of the Landau-Ginzburg model 171 00:11:22,300 --> 00:11:23,720 qualitatively. 172 00:11:23,720 --> 00:11:26,410 Which is to ignore nonlinearities, 173 00:11:26,410 --> 00:11:30,000 so it's a kind of Gaussian version of the model. 174 00:11:30,000 --> 00:11:33,810 So what we did was we linearized. 175 00:11:33,810 --> 00:11:34,720 AUDIENCE: Question. 176 00:11:34,720 --> 00:11:35,960 MEHRAN KARDAR: Yes? 177 00:11:35,960 --> 00:11:38,690 AUDIENCE: Why does the sign in front of the K term 178 00:11:38,690 --> 00:11:40,830 change relative to the others? 179 00:11:40,830 --> 00:11:48,140 MEHRAN KARDAR: OK, so when you have the function 0 of a field 180 00:11:48,140 --> 00:11:56,010 gradient, et cetera, you can show 181 00:11:56,010 --> 00:12:01,510 that the functional derivative, the first term 182 00:12:01,510 --> 00:12:06,940 is the ordinary type of derivative. 183 00:12:06,940 --> 00:12:12,152 And then if you think about the variations that are carrying m 184 00:12:12,152 --> 00:12:17,870 and with the gradient and you write it as m plus delta m 185 00:12:17,870 --> 00:12:21,660 and then make sure that you take the delta m outside, 186 00:12:21,660 --> 00:12:26,280 you need to do an integration by part that changes the sign. 187 00:12:26,280 --> 00:12:28,566 And the next term would be the gradient 188 00:12:28,566 --> 00:12:32,760 of the phi by the [? grad ?] m, and the next term 189 00:12:32,760 --> 00:12:35,550 would be log [INAUDIBLE] of [INAUDIBLE] gradient 190 00:12:35,550 --> 00:12:38,176 of m squared and so forth. 191 00:12:38,176 --> 00:12:43,436 It alternates and so by explicitly calculating 192 00:12:43,436 --> 00:12:45,580 the difference of two functionals 193 00:12:45,580 --> 00:12:51,320 with this integration evaluated at m and m plus delta m 194 00:12:51,320 --> 00:12:55,102 and pulling out everything that is proportional to delta m 195 00:12:55,102 --> 00:12:56,910 you can prove these expressions. 196 00:13:00,920 --> 00:13:05,880 So we linearize this equation, which I've already done that. 197 00:13:05,880 --> 00:13:09,980 I cross out the mq term and any other nonlinear terms, 198 00:13:09,980 --> 00:13:12,630 so I only keep the linear term. 199 00:13:12,630 --> 00:13:16,065 And then I do a Fourier transform. 200 00:13:21,330 --> 00:13:27,660 So basically I switch from the position representation 201 00:13:27,660 --> 00:13:32,620 to the Fourier transform that's called m tilde of q, I think. 202 00:13:32,620 --> 00:13:35,770 x is replaced with q. 203 00:13:35,770 --> 00:13:42,220 And then the equation for the field m linears. 204 00:13:42,220 --> 00:13:46,900 The form separates out into independent equations 205 00:13:46,900 --> 00:13:51,660 for the components that are characterized by q. 206 00:13:51,660 --> 00:13:55,200 So the Fourier transform of the left hand side is just this. 207 00:13:55,200 --> 00:13:58,460 Fourier transform of the right hand side 208 00:13:58,460 --> 00:14:05,660 will give me minus mu r plus K q squared. 209 00:14:05,660 --> 00:14:10,140 The derivative of [INAUDIBLE] will give me a minus q squared. 210 00:14:10,140 --> 00:14:15,290 The derivative of the next term L q to the forth, et cetera. 211 00:14:15,290 --> 00:14:17,776 And I've only kept the linear term, 212 00:14:17,776 --> 00:14:21,510 so I have m tilde of q and t. 213 00:14:21,510 --> 00:14:23,580 And then I have the Fourier transforms 214 00:14:23,580 --> 00:14:27,000 of the noise eta of x and t. 215 00:14:27,000 --> 00:14:29,820 They [? call it ?] eta tilde of q [? m. ?] 216 00:14:33,590 --> 00:14:39,870 So you can see that each mode satisfy 217 00:14:39,870 --> 00:14:42,940 a separate linear equation. 218 00:14:45,570 --> 00:14:49,290 So this equation is actually very easy 219 00:14:49,290 --> 00:14:56,960 to solve for any linear equation m tilde of q and t. 220 00:14:56,960 --> 00:15:00,230 If I didn't have the noise, so I would 221 00:15:00,230 --> 00:15:05,360 start with some value at t [? close ?] to 0, 222 00:15:05,360 --> 00:15:09,940 and that value would decay exponentially 223 00:15:09,940 --> 00:15:14,660 in the characteristic time that I will call tao of q. 224 00:15:14,660 --> 00:15:26,980 And 1 over tao of q is simply this mu r plus K q2 225 00:15:26,980 --> 00:15:30,860 and so forth. 226 00:15:30,860 --> 00:15:34,920 Now once you have noise, essentially each one 227 00:15:34,920 --> 00:15:39,180 of these noises acts like an initial condition. 228 00:15:39,180 --> 00:15:43,300 And so the full answer is an integral 229 00:15:43,300 --> 00:15:48,470 over all of these noises from 0 tilde time t of interest. 230 00:15:48,470 --> 00:15:54,260 Dt prime the noise that occurs at a time t prime, 231 00:15:54,260 --> 00:15:57,515 and the noise that occurs at time t [? trime ?] relaxes 232 00:15:57,515 --> 00:16:06,350 with this into the minus t minus t prime tao of q. 233 00:16:06,350 --> 00:16:12,420 So that's the solution to that linear noisy equations, 234 00:16:12,420 --> 00:16:14,320 basically [? sequencing ?] [INAUDIBLE]. 235 00:16:17,970 --> 00:16:24,330 So one of the things that we now see 236 00:16:24,330 --> 00:16:33,630 is that essentially the different Fourier components 237 00:16:33,630 --> 00:16:35,080 of the field. 238 00:16:35,080 --> 00:16:38,150 Each one of them is independently 239 00:16:38,150 --> 00:16:41,710 relaxing to something, and each one of them 240 00:16:41,710 --> 00:16:46,690 has a characteristic relaxation time. 241 00:16:46,690 --> 00:16:54,360 As I go towards smaller and smaller values of q, 242 00:16:54,360 --> 00:16:59,670 this rate becomes smaller, and the relaxation time 243 00:16:59,670 --> 00:17:01,580 becomes larger. 244 00:17:01,580 --> 00:17:03,880 So essentially shortwave wavelength modes 245 00:17:03,880 --> 00:17:07,630 that correspond to large q, they relax first. 246 00:17:07,630 --> 00:17:11,280 Longer wavelength modes will relax later on. 247 00:17:11,280 --> 00:17:19,480 And you can see that the largest relaxation time, tao max, 248 00:17:19,480 --> 00:17:25,243 corresponds to q equals to 0 is simply 1 over [? nu ?] r. 249 00:17:28,349 --> 00:17:37,270 So I can pluck this to a max as a function of r, 250 00:17:37,270 --> 00:17:41,200 and again in this theory, the Gaussian theory, 251 00:17:41,200 --> 00:17:45,410 we saw only makes sense as long as all is positive. 252 00:17:45,410 --> 00:17:49,410 So I have to look only on the positive axis. 253 00:17:49,410 --> 00:17:53,060 And I find that the relaxation time 254 00:17:53,060 --> 00:17:57,060 for the entire system for the longest wavelength 255 00:17:57,060 --> 00:18:02,310 actually diverges as r goes to 0. 256 00:18:02,310 --> 00:18:05,560 And recall that r, in our perspective, 257 00:18:05,560 --> 00:18:09,320 is really something that is proportional to T minus Tc. 258 00:18:12,260 --> 00:18:16,800 So basically we find that as we are approaching 259 00:18:16,800 --> 00:18:20,810 the critical point, the time it takes 260 00:18:20,810 --> 00:18:23,850 for the entirety of the system or the longest 261 00:18:23,850 --> 00:18:30,820 wavelength [? modes ?] to relax diverges as 1 over T minus Tc. 262 00:18:30,820 --> 00:18:35,010 There's an exponent that shows this, so called, 263 00:18:35,010 --> 00:18:36,660 critical slowing down. 264 00:18:43,950 --> 00:18:46,400 Yes? 265 00:18:46,400 --> 00:18:48,200 AUDIENCE: In principal, why couldn't you 266 00:18:48,200 --> 00:18:53,930 have that r becomes negative if you restrict your range of q 267 00:18:53,930 --> 00:18:58,930 to be outside of some value and not go arbitrarily 268 00:18:58,930 --> 00:19:01,380 close to the origin. 269 00:19:01,380 --> 00:19:05,230 MEHRAN KARDAR: You could, but what's the physics of that? 270 00:19:05,230 --> 00:19:06,056 See. 271 00:19:06,056 --> 00:19:08,550 AUDIENCE: I'm wondering if there is or is that not needed. 272 00:19:08,550 --> 00:19:09,300 MEHRAN KARDAR: No. 273 00:19:09,300 --> 00:19:14,360 OK, so the physics of that could be that you have a system that 274 00:19:14,360 --> 00:19:21,950 has some finite size, [? l. ?] Then the largest q that you 275 00:19:21,950 --> 00:19:27,030 could have [? or ?] the smallest q that you could have would be 276 00:19:27,030 --> 00:19:31,680 the [? 1/l. ?] So in principal, for that you can go slightly 277 00:19:31,680 --> 00:19:32,610 negative. 278 00:19:32,610 --> 00:19:37,180 You still cannot go too negative because ultimately this will 279 00:19:37,180 --> 00:19:39,490 overcome that. 280 00:19:39,490 --> 00:19:42,020 But again, we are interested in singularities 281 00:19:42,020 --> 00:19:46,884 that we kind of know arise in the limit of [INAUDIBLE]. 282 00:19:52,850 --> 00:19:56,150 I also recall, there is a time that we see as diverging 283 00:19:56,150 --> 00:19:57,810 as r goes to 0. 284 00:19:57,810 --> 00:20:02,060 Of course, we identified before a correlation length 285 00:20:02,060 --> 00:20:06,380 from balancing these two terms, and the correlation length 286 00:20:06,380 --> 00:20:10,540 is square root of K over r, which is again proportional 287 00:20:10,540 --> 00:20:15,500 to T minus Tc and diverges with a square root singularity. 288 00:20:15,500 --> 00:20:23,640 So we can see that this tao max is actually related 289 00:20:23,640 --> 00:20:30,622 to the psi squared over Nu K. 290 00:20:30,622 --> 00:20:34,150 And it is also related towards this. 291 00:20:34,150 --> 00:20:47,610 We can see that our tao of q, basically, if q is large 292 00:20:47,610 --> 00:20:54,980 such that qc is larger than 1 and the characteristic time 293 00:20:54,980 --> 00:21:03,340 is going to be 1 over [? nu ?] K times the inverse of q squared. 294 00:21:03,340 --> 00:21:07,390 And the inverse of q is something like a wavelength. 295 00:21:07,390 --> 00:21:12,290 Whereas, ultimately, this saturates for qc what is less 296 00:21:12,290 --> 00:21:18,440 than 1 [? long ?] wavelengths to c squared over [? nu ?] K. 297 00:21:18,440 --> 00:21:26,230 So basically you see things at very short range, 298 00:21:26,230 --> 00:21:29,040 at length scales that are much less than the correlation 299 00:21:29,040 --> 00:21:33,760 length of the system, that the characteristic time will 300 00:21:33,760 --> 00:21:37,475 depend on the length scale that you are looking at squared. 301 00:21:41,130 --> 00:21:48,070 Now you have seen times scaling as length squared 302 00:21:48,070 --> 00:21:51,870 from diffusion, so essentially this 303 00:21:51,870 --> 00:21:56,256 is some kind of a manifestation of diffusion, 304 00:21:56,256 --> 00:22:01,620 but as you perturb the system, let's say at short distances, 305 00:22:01,620 --> 00:22:03,930 there's some equilibrium system. 306 00:22:03,930 --> 00:22:07,120 Let's say we do some perturbation to it 307 00:22:07,120 --> 00:22:09,790 at some point, and that perturbation 308 00:22:09,790 --> 00:22:14,940 will start to expand diffusively until it reaches 309 00:22:14,940 --> 00:22:18,050 the size of the correlation length, at which point 310 00:22:18,050 --> 00:22:21,050 it stops because essentially correlation length is 311 00:22:21,050 --> 00:22:25,390 an individual block that doesn't know about individual blocks, 312 00:22:25,390 --> 00:22:26,900 so the influence does not last. 313 00:22:30,480 --> 00:22:33,380 So quite generally, what you find-- 314 00:22:33,380 --> 00:22:36,480 so we solved the linearized version 315 00:22:36,480 --> 00:22:38,450 of the Landau-Ginzburg model, but we 316 00:22:38,450 --> 00:22:43,960 know that, say, the critical behaviors for the divergence 317 00:22:43,960 --> 00:22:47,040 of the correlation length that is predicted here 318 00:22:47,040 --> 00:22:49,030 is not correct in three dimensions, 319 00:22:49,030 --> 00:22:50,830 things get modified. 320 00:22:50,830 --> 00:22:54,580 So these kind of exponents that come from diffusion also 321 00:22:54,580 --> 00:22:56,690 gets modified. 322 00:22:56,690 --> 00:23:01,640 And quite generally, you find that the relaxation time 323 00:23:01,640 --> 00:23:09,610 of a mode of wavelength q is going to behave something 324 00:23:09,610 --> 00:23:21,140 like wavelength, which is 1 over q, 325 00:23:21,140 --> 00:23:27,710 rather than squared as some exponent z. 326 00:23:27,710 --> 00:23:33,190 And then there is some function of the product 327 00:23:33,190 --> 00:23:34,990 of the wavelength you are looking at 328 00:23:34,990 --> 00:23:39,020 and the correlation length so that you will cross over 329 00:23:39,020 --> 00:23:42,260 from one behavior to another behavior 330 00:23:42,260 --> 00:23:44,280 as you are looking at length scales that 331 00:23:44,280 --> 00:23:46,050 are smaller than the correlation length 332 00:23:46,050 --> 00:23:49,090 or larger than the correlation length. 333 00:23:49,090 --> 00:23:52,600 And to get what this exponent z is, 334 00:23:52,600 --> 00:23:56,850 you have to do study of the nonlinear model 335 00:23:56,850 --> 00:23:58,950 in the same sense that, in order to get 336 00:23:58,950 --> 00:24:01,350 the correction to the exponent [? nu ?], 337 00:24:01,350 --> 00:24:03,500 we had to do epsilon expansion. 338 00:24:03,500 --> 00:24:05,115 You have to do something similar, 339 00:24:05,115 --> 00:24:10,970 and you'll fine that at higher changes, 340 00:24:10,970 --> 00:24:15,240 it goes like some correction that does not actually 341 00:24:15,240 --> 00:24:18,580 start at order of epsilon but at order of epsilon squared. 342 00:24:18,580 --> 00:24:24,520 But there's essentially some modification 343 00:24:24,520 --> 00:24:27,920 of the qualitative behavior that we 344 00:24:27,920 --> 00:24:33,180 can ascribe to the fusion of independent modes 345 00:24:33,180 --> 00:24:37,190 exists quite generally and universal exponents 346 00:24:37,190 --> 00:24:40,700 different from [? to ?] will emerge from that. 347 00:24:45,600 --> 00:24:51,900 Now it turns out that this is not the end of the story 348 00:24:51,900 --> 00:24:54,820 because we have seen that the same probability 349 00:24:54,820 --> 00:25:00,650 distribution can describe a lot of different systems. 350 00:25:00,650 --> 00:25:04,910 Let's say the focus on the case of n equals to 1. 351 00:25:08,680 --> 00:25:11,760 So then this Landau-Ginzburg that I described 352 00:25:11,760 --> 00:25:15,680 for you can describe, let's say, the Ising model, 353 00:25:15,680 --> 00:25:24,040 which describes magnetizations that lie 354 00:25:24,040 --> 00:25:26,700 along the particular direction. 355 00:25:26,700 --> 00:25:31,040 So that it can also describe liquid gas phenomena 356 00:25:31,040 --> 00:25:34,480 where the order parameter is the difference in density, 357 00:25:34,480 --> 00:25:37,760 if you like, between the liquid and the gas. 358 00:25:37,760 --> 00:25:40,970 Yet another example that it describes 359 00:25:40,970 --> 00:25:43,650 is the mixing of an alloy. 360 00:25:50,110 --> 00:25:58,380 So let's, for example, imagine brass that has a composition 361 00:25:58,380 --> 00:26:01,550 x that goes between 0 and 1. 362 00:26:01,550 --> 00:26:04,345 On one end, let's say you have entirely copper 363 00:26:04,345 --> 00:26:08,250 and on the other and you have entirely zinc. 364 00:26:08,250 --> 00:26:11,890 And so this is how you make brass as an alloy. 365 00:26:11,890 --> 00:26:16,580 And what my other axis is is the temperature. 366 00:26:16,580 --> 00:26:21,690 What you find is that there is some kind of phase diagram 367 00:26:21,690 --> 00:26:29,920 such that you get a nice mixture of copper and zinc 368 00:26:29,920 --> 00:26:32,180 only if you are at high temperatures, 369 00:26:32,180 --> 00:26:34,070 whereas if you are at low temperature, 370 00:26:34,070 --> 00:26:37,660 you basically will separate into chunks 371 00:26:37,660 --> 00:26:40,950 that are rich copper and chunks that are rich in zinc. 372 00:26:40,950 --> 00:26:44,710 And you'll have a critical demixing 373 00:26:44,710 --> 00:26:50,560 point, which has exactly the same properties as the Ising 374 00:26:50,560 --> 00:26:51,500 model. 375 00:26:51,500 --> 00:26:54,760 For example, this curve will be characterized 376 00:26:54,760 --> 00:26:56,610 with an exponent beta, which would 377 00:26:56,610 --> 00:26:59,811 be the beta of the Ising mode. 378 00:26:59,811 --> 00:27:05,410 And in particular, if I were to take someplace 379 00:27:05,410 --> 00:27:10,080 in the vicinity of this and try to write down a probability 380 00:27:10,080 --> 00:27:13,200 distribution, that probability distribution 381 00:27:13,200 --> 00:27:15,960 would be exactly what I have over there where 382 00:27:15,960 --> 00:27:22,470 m is, let's say, the difference between the two types of alloys 383 00:27:22,470 --> 00:27:28,420 that I have compared to each other over here. 384 00:27:28,420 --> 00:27:32,060 So this is related to 2x minus 1 or something like that. 385 00:27:34,880 --> 00:27:40,280 So as you go across your piece of material close to here, 386 00:27:40,280 --> 00:27:42,190 there will be compositional variations 387 00:27:42,190 --> 00:27:45,540 that are described by that. 388 00:27:45,540 --> 00:27:51,620 So the question is, I know exactly what the probability 389 00:27:51,620 --> 00:27:56,120 distribution is for this system to be an equilibrium given 390 00:27:56,120 --> 00:27:58,910 this choice of m. 391 00:27:58,910 --> 00:28:03,020 Again, with some set of parameters, R, U, et cetera. 392 00:28:03,020 --> 00:28:07,730 The question is-- is the dynamics again described 393 00:28:07,730 --> 00:28:10,350 by the same equation? 394 00:28:10,350 --> 00:28:12,810 And the answer is no. 395 00:28:12,810 --> 00:28:15,640 The same probability of distribution 396 00:28:15,640 --> 00:28:19,430 can describe-- or can be obtained 397 00:28:19,430 --> 00:28:21,250 with very different dynamics. 398 00:28:21,250 --> 00:28:23,640 And in particular, what is happening 399 00:28:23,640 --> 00:28:28,970 in the system is that, if I integrate this quantity m 400 00:28:28,970 --> 00:28:34,920 across the system, I will get the total number of 1 401 00:28:34,920 --> 00:28:38,750 minus the other, which is what is given to you, 402 00:28:38,750 --> 00:28:41,530 and it does not change as a function of time. 403 00:28:41,530 --> 00:28:50,585 d by dt of this quantity is 0. 404 00:28:50,585 --> 00:28:53,058 It cannot change. 405 00:28:53,058 --> 00:28:53,970 OK. 406 00:28:53,970 --> 00:28:59,300 Whereas the equation that I have written over here, 407 00:28:59,300 --> 00:29:03,630 in principle, locally, I can, by adding the noise 408 00:29:03,630 --> 00:29:05,740 or by bringing things from the neighborhood, 409 00:29:05,740 --> 00:29:08,135 I can change the value of m. 410 00:29:08,135 --> 00:29:09,320 I cannot do that. 411 00:29:09,320 --> 00:29:15,000 So this process of the relaxation that would go 412 00:29:15,000 --> 00:29:19,100 on in data graphs cannot be described by the time dependent 413 00:29:19,100 --> 00:29:22,270 on the Landau-Ginzburg equation because you have this 414 00:29:22,270 --> 00:29:23,652 conservation here. 415 00:29:28,080 --> 00:29:30,260 OK. 416 00:29:30,260 --> 00:29:34,050 So what should we do? 417 00:29:34,050 --> 00:29:39,270 Well, when things are conserved, like, say, 418 00:29:39,270 --> 00:29:42,580 as the gas particles move in this fluid, 419 00:29:42,580 --> 00:29:45,460 and if I'm interested in the number of particles 420 00:29:45,460 --> 00:29:48,390 in some cube, then the change in the number 421 00:29:48,390 --> 00:29:51,050 of particles in some cube in this room 422 00:29:51,050 --> 00:29:54,310 is related to the gradient of the current that 423 00:29:54,310 --> 00:29:56,260 goes into that place. 424 00:29:56,260 --> 00:30:01,380 So the appropriate way of writing an equation that 425 00:30:01,380 --> 00:30:04,710 describes, let's say, the magnetization changing 426 00:30:04,710 --> 00:30:07,150 as a function of time. 427 00:30:07,150 --> 00:30:09,410 Given that you have a conservation, 428 00:30:09,410 --> 00:30:14,340 though, is to write it as minus the gradient 429 00:30:14,340 --> 00:30:15,670 of some kind of a current. 430 00:30:19,895 --> 00:30:22,980 So this j is some kind of a current, 431 00:30:22,980 --> 00:30:25,045 and these would be vectors. 432 00:30:29,770 --> 00:30:32,720 This is a current of the particles 433 00:30:32,720 --> 00:30:36,760 moving into the system. 434 00:30:36,760 --> 00:30:42,150 Now, in systems that are dissipative, 435 00:30:42,150 --> 00:30:46,260 currents are related to the gradient of some density 436 00:30:46,260 --> 00:30:49,210 through the diffusion constant, et cetera. 437 00:30:49,210 --> 00:30:56,360 So it kind of makes sense to imagine that this current is 438 00:30:56,360 --> 00:31:03,840 the gradient of something that is trying-- or more precisely, 439 00:31:03,840 --> 00:31:06,780 minus the gradient of something that 440 00:31:06,780 --> 00:31:10,790 tries to bring the system to be, more or less, 441 00:31:10,790 --> 00:31:13,330 in its equilibrium state. 442 00:31:13,330 --> 00:31:18,740 Equilibrium state, as we said, is determined by this data H, 443 00:31:18,740 --> 00:31:21,890 and we want to push it in that direction. 444 00:31:21,890 --> 00:31:29,250 So we put our data H by dm over here, 445 00:31:29,250 --> 00:31:32,283 and we put some kind of a U over here. 446 00:31:35,670 --> 00:31:39,370 Of course, I would have to add some kind 447 00:31:39,370 --> 00:31:43,180 of a conserved random current also, 448 00:31:43,180 --> 00:31:47,640 which is the analog of this non-conserved noise 449 00:31:47,640 --> 00:31:50,507 that I add over initially. 450 00:31:50,507 --> 00:31:51,007 OK. 451 00:31:55,350 --> 00:32:00,450 Now, the conservative version of the equation, you can see, 452 00:32:00,450 --> 00:32:04,870 you have two more derivatives with respect 453 00:32:04,870 --> 00:32:07,340 to what we had before. 454 00:32:07,340 --> 00:32:10,590 And so once I-- OK. 455 00:32:10,590 --> 00:32:19,530 So if we do something like this, dm by dt is mu C. 456 00:32:19,530 --> 00:32:22,000 And then I would have [INAUDIBLE] 457 00:32:22,000 --> 00:32:30,170 plus of R, rather than by itself. 458 00:32:30,170 --> 00:32:35,880 Actually, it would be the plus then of something like R m 459 00:32:35,880 --> 00:32:41,701 plus 4 U m cubed, and so forth. 460 00:32:41,701 --> 00:32:47,520 And we are going to ignore this kind of term. 461 00:32:47,520 --> 00:32:50,150 And then there would be high order terms 462 00:32:50,150 --> 00:32:53,735 that would show up minus k the fourth derivative, 463 00:32:53,735 --> 00:32:55,130 and so forth. 464 00:32:59,320 --> 00:33:04,380 And then there may be some kind of a conserved noise 465 00:33:04,380 --> 00:33:06,220 that I have to put outside. 466 00:33:11,055 --> 00:33:11,555 OK. 467 00:33:14,960 --> 00:33:18,890 So when I fully transform this equation, what do I get? 468 00:33:18,890 --> 00:33:27,530 I will get that dm by dt is-- let's say in the full space, 469 00:33:27,530 --> 00:33:34,520 until there is a function of qnt is minus 470 00:33:34,520 --> 00:33:39,770 U C. Because of this plus, then there's an additional factor 471 00:33:39,770 --> 00:33:42,040 of q squared. 472 00:33:42,040 --> 00:33:48,540 And then I have R plus kq squared plus L 473 00:33:48,540 --> 00:33:51,740 cubed to the fourth, et cetera. 474 00:33:51,740 --> 00:33:54,960 And then I will have a fully transformed version 475 00:33:54,960 --> 00:33:58,529 of this conserved noise. 476 00:33:58,529 --> 00:33:59,028 OK. 477 00:34:02,280 --> 00:34:08,060 You can see that the difference between this equation 478 00:34:08,060 --> 00:34:12,480 and the previous equation is that all of the relaxation 479 00:34:12,480 --> 00:34:18,639 times will have an additional factor of q squared. 480 00:34:18,639 --> 00:34:25,635 And so eventually, this shortest relaxation time actually 481 00:34:25,635 --> 00:34:29,790 will glow like the size of the system squared. 482 00:34:29,790 --> 00:34:31,522 Whereas previously, it was saturated 483 00:34:31,522 --> 00:34:34,620 at the correlation length. 484 00:34:34,620 --> 00:34:37,639 And because you will have this conservation, 485 00:34:37,639 --> 00:34:42,280 though, you have to rearrange a lot of particles 486 00:34:42,280 --> 00:34:44,580 keeping their numbers constant. 487 00:34:44,580 --> 00:34:49,015 You have a much harder time of relaxing the system. 488 00:34:49,015 --> 00:34:53,389 All of the relaxation times, as we see, 489 00:34:53,389 --> 00:34:57,426 grow correspondingly and become higher. 490 00:34:57,426 --> 00:34:58,310 OK. 491 00:34:58,310 --> 00:35:04,940 So indeed, for this class, one can show that z starts with 4, 492 00:35:04,940 --> 00:35:07,940 and then there will be corrections 493 00:35:07,940 --> 00:35:11,110 that would modify that. 494 00:35:11,110 --> 00:35:16,298 So the-- yes? 495 00:35:16,298 --> 00:35:19,376 AUDIENCE: How do we define or how do we 496 00:35:19,376 --> 00:35:22,322 do a realization of the conserved noise, conserved 497 00:35:22,322 --> 00:35:25,396 current-- conserved noise in the room? 498 00:35:25,396 --> 00:35:26,146 MEHRAN KARDAR: OK. 499 00:35:26,146 --> 00:35:29,700 AUDIENCE: So it has some kind of like correlation-- 500 00:35:29,700 --> 00:35:32,790 self-correlation properties, I suppose, 501 00:35:32,790 --> 00:35:36,080 because, if current flowing out of some region, 502 00:35:36,080 --> 00:35:39,430 doesn't it want to go in? 503 00:35:39,430 --> 00:35:41,600 MEHRAN KARDAR: If I go back here, 504 00:35:41,600 --> 00:35:46,270 I have a good idea of what is happening because all I need, 505 00:35:46,270 --> 00:35:50,930 in order to ensure conservation, is that the m by dt 506 00:35:50,930 --> 00:35:53,053 is the gradient of something. 507 00:35:53,053 --> 00:35:53,920 AUDIENCE: OK. 508 00:35:53,920 --> 00:35:58,015 MEHRAN KARDAR: So I can put whatever I want over here. 509 00:35:58,015 --> 00:36:00,619 AUDIENCE: So if it's a scalar or an [INAUDIBLE] field? 510 00:36:00,619 --> 00:36:01,410 MEHRAN KARDAR: Yes. 511 00:36:01,410 --> 00:36:04,019 As long as it is sitting under the gradient-- 512 00:36:04,019 --> 00:36:04,560 AUDIENCE: OK. 513 00:36:04,560 --> 00:36:07,610 MEHRAN KARDAR: --it will be OK, which means this quantity here 514 00:36:07,610 --> 00:36:12,730 that I'm calling a to z has a gradient in it. 515 00:36:12,730 --> 00:36:15,640 And if you wait for about five minutes, 516 00:36:15,640 --> 00:36:19,030 we'll show that, because of that in full space, 517 00:36:19,030 --> 00:36:22,590 rather than having-- well, I'll describe 518 00:36:22,590 --> 00:36:25,060 the difference between non-conserved and conserved 519 00:36:25,060 --> 00:36:26,770 noise in fully space. 520 00:36:26,770 --> 00:36:30,046 It's much easier. 521 00:36:30,046 --> 00:36:30,546 OK. 522 00:36:35,010 --> 00:36:40,750 So actually, as far as what I have discussed so far, which 523 00:36:40,750 --> 00:36:45,340 is relaxation, I don't really need the noise 524 00:36:45,340 --> 00:36:47,790 because I can forget the noise. 525 00:36:47,790 --> 00:36:49,360 And all I have said-- and I forgot 526 00:36:49,360 --> 00:36:54,170 the n tilde-- is that I have a linear equation that 527 00:36:54,170 --> 00:36:56,230 relaxes your variable to 0. 528 00:36:56,230 --> 00:36:59,175 I can immediately read off for the correlation, 529 00:36:59,175 --> 00:37:05,920 then this-- a correlation times what I need the noise for it 530 00:37:05,920 --> 00:37:11,410 so that, ultimately, I don't go to the medium is the potential, 531 00:37:11,410 --> 00:37:15,050 but I go to this pro-rated distribution. 532 00:37:15,050 --> 00:37:20,880 So let's see what we have to do in order to achieve that. 533 00:37:20,880 --> 00:37:24,235 For simplicity, let's take this equation. 534 00:37:28,380 --> 00:37:34,940 Although I can take the corresponding one for that. 535 00:37:34,940 --> 00:37:39,650 And let's calculate-- because of the presence of this noise, 536 00:37:39,650 --> 00:37:44,260 if I run the same system at different times, 537 00:37:44,260 --> 00:37:46,930 I will have different realizations of the noise 538 00:37:46,930 --> 00:37:49,930 than if I had run many versions of the system because 539 00:37:49,930 --> 00:37:52,020 of the realizations of noise. 540 00:37:52,020 --> 00:37:55,315 It's quantity and tilde would be different. 541 00:37:55,315 --> 00:37:59,540 It would satisfy some kind of a pro-rated distribution. 542 00:37:59,540 --> 00:38:02,480 So what I want to do is to calculate averages, 543 00:38:02,480 --> 00:38:05,280 such as the average of m tilde. 544 00:38:05,280 --> 00:38:11,920 Let's say, q1 at time T with m tilde q2 at time t. 545 00:38:15,340 --> 00:38:21,870 And you can see already from this equation 546 00:38:21,870 --> 00:38:26,250 that, if I forget the part that comes from the noise, 547 00:38:26,250 --> 00:38:28,755 whatever initial condition that I have 548 00:38:28,755 --> 00:38:31,440 will eventually decay to 0. 549 00:38:31,440 --> 00:38:33,640 So the thing that agitates and gives 550 00:38:33,640 --> 00:38:38,680 some kind of a randomness to this really comes from this. 551 00:38:38,680 --> 00:38:40,620 So let's imagine that we have looked 552 00:38:40,620 --> 00:38:43,620 at times that are sufficiently long so 553 00:38:43,620 --> 00:38:45,880 that the influence of the initial condition 554 00:38:45,880 --> 00:38:46,950 has died down. 555 00:38:46,950 --> 00:38:48,720 I don't want to write the other term. 556 00:38:48,720 --> 00:38:53,570 I could do it, but it's kind of boring to include it. 557 00:38:53,570 --> 00:38:57,340 So let's forget that and focus on the integral. 558 00:38:57,340 --> 00:39:00,140 0 to t. 559 00:39:00,140 --> 00:39:04,230 Now, if I multiply two of these quantities, 560 00:39:04,230 --> 00:39:08,318 I will have two integrals over t prime. 561 00:39:08,318 --> 00:39:10,290 All right. 562 00:39:10,290 --> 00:39:16,625 Each one of them would decay with the corresponding tau 563 00:39:16,625 --> 00:39:18,170 of q. 564 00:39:18,170 --> 00:39:20,260 In one case, tau of q1. 565 00:39:20,260 --> 00:39:27,080 In the other case, tau of q2. 566 00:39:27,080 --> 00:39:29,430 Coming from these things. 567 00:39:29,430 --> 00:39:35,780 And the noise, q1 at time to 1 prime, and noise q2 568 00:39:35,780 --> 00:39:39,764 at time [? t2 prime. ?] 569 00:39:39,764 --> 00:39:41,760 OK. 570 00:39:41,760 --> 00:39:48,050 Now if I average over the noise, then I 571 00:39:48,050 --> 00:39:51,414 have to do an average over here. 572 00:39:51,414 --> 00:39:51,913 OK. 573 00:39:55,140 --> 00:40:01,310 Now, one thing that I forgot to mention right at the beginning 574 00:40:01,310 --> 00:40:04,146 is that, of course, the average of this 575 00:40:04,146 --> 00:40:05,880 we are going to set to 0. 576 00:40:05,880 --> 00:40:08,020 It's the very least that is important. 577 00:40:08,020 --> 00:40:09,520 Right? 578 00:40:09,520 --> 00:40:11,550 So if I do that, clearly, the average 579 00:40:11,550 --> 00:40:14,280 of one of these in full space would 580 00:40:14,280 --> 00:40:21,950 be 0 also because the full q is related to the real space delta 581 00:40:21,950 --> 00:40:24,750 just by an integral. 582 00:40:24,750 --> 00:40:26,375 So if the average of the integral is 0, 583 00:40:26,375 --> 00:40:29,390 the average of this is 0. 584 00:40:29,390 --> 00:40:33,200 So it turns out that, when you look at the average of two 585 00:40:33,200 --> 00:40:36,010 of them-- and it's a very simple exercise 586 00:40:36,010 --> 00:40:43,115 to just rewrite these things in terms of 8 of x and t. 587 00:40:43,115 --> 00:40:47,490 8 of x and t applied average that you have. 588 00:40:47,490 --> 00:40:52,840 And we find that the things that are uncorrelated in real space 589 00:40:52,840 --> 00:40:56,280 also are uncoordinated in full space. 590 00:40:56,280 --> 00:40:58,970 And so the variance of this quantity 591 00:40:58,970 --> 00:41:05,000 is 2d delta 1 prime minus [? d2 prime. ?] 592 00:41:05,000 --> 00:41:09,950 And then you have the analog of the function in full space, 593 00:41:09,950 --> 00:41:16,330 which always carries the solution of a factor of 2 pi 594 00:41:16,330 --> 00:41:17,370 to the d. 595 00:41:17,370 --> 00:41:20,000 And it becomes a sum of the q's, as we've 596 00:41:20,000 --> 00:41:22,860 seen many times before. 597 00:41:22,860 --> 00:41:25,060 OK. 598 00:41:25,060 --> 00:41:28,350 So because of this delta function, 599 00:41:28,350 --> 00:41:30,920 this delta integral becomes one integral. 600 00:41:30,920 --> 00:41:34,480 So I have the integral 0 to t. 601 00:41:34,480 --> 00:41:39,170 The 2t prime I can write as just 1t prime, 602 00:41:39,170 --> 00:41:42,990 and these two factors merge into one. 603 00:41:42,990 --> 00:41:49,680 It comes into minus t minus t prime over tau of q, 604 00:41:49,680 --> 00:41:53,650 except that I get multiplied by a factor 2 605 00:41:53,650 --> 00:41:56,900 since I have two of them. 606 00:41:56,900 --> 00:42:05,640 And outside of the integral, I will have this factor of 2d, 607 00:42:05,640 --> 00:42:10,875 and then 2 pi to the d delta function with 1 plus 2. 608 00:42:15,125 --> 00:42:15,625 OK. 609 00:42:19,430 --> 00:42:21,880 Now, we are really interested-- and I already 610 00:42:21,880 --> 00:42:24,825 kind of hinted at that in the limit 611 00:42:24,825 --> 00:42:29,130 where time becomes very large. 612 00:42:29,130 --> 00:42:32,930 In the limit, where time becomes a very large, 613 00:42:32,930 --> 00:42:36,630 essentially, I need to calculate the limit of this integral 614 00:42:36,630 --> 00:42:38,308 as time becomes very large. 615 00:42:42,132 --> 00:42:44,680 And as time grows to very large, this 616 00:42:44,680 --> 00:42:47,470 is just the integral from 0 to infinity. 617 00:42:47,470 --> 00:42:55,280 And the integral is going to give me 2 over tau of q. 618 00:42:55,280 --> 00:42:59,255 Essentially, you can see that integrating at the upper end 619 00:42:59,255 --> 00:43:02,810 will give me just 0. 620 00:43:02,810 --> 00:43:05,445 Integrating at the smaller end, it 621 00:43:05,445 --> 00:43:07,120 will be exponentially small as it 622 00:43:07,120 --> 00:43:11,855 goes to infinity as a factor of 2 over tau. 623 00:43:11,855 --> 00:43:12,750 OK. 624 00:43:12,750 --> 00:43:14,202 Yes? 625 00:43:14,202 --> 00:43:18,460 AUDIENCE: Are you assuming or-- yeah. 626 00:43:18,460 --> 00:43:22,230 Are you assuming that tau of q is even in q 627 00:43:22,230 --> 00:43:24,039 to be able to combine to-- 628 00:43:24,039 --> 00:43:24,830 MEHRAN KARDAR: Yes. 629 00:43:24,830 --> 00:43:25,320 AUDIENCE: --2 tau? 630 00:43:25,320 --> 00:43:25,850 So-- OK. 631 00:43:25,850 --> 00:43:28,183 MEHRAN KARDAR: I'm thinking of the tau of q's that we've 632 00:43:28,183 --> 00:43:29,201 calculated over here. 633 00:43:29,201 --> 00:43:29,742 AUDIENCE: OK. 634 00:43:29,742 --> 00:43:30,990 MEHRAN KARDAR: OK. 635 00:43:30,990 --> 00:43:31,820 Yes? 636 00:43:31,820 --> 00:43:34,850 AUDIENCE: The e times the [INAUDIBLE] 637 00:43:34,850 --> 00:43:36,370 do the equation of something? 638 00:43:36,370 --> 00:43:37,980 Right? 639 00:43:37,980 --> 00:43:42,030 MEHRAN KARDAR: At this stage, I am focusing on this expression 640 00:43:42,030 --> 00:43:44,610 over here, where a is not. 641 00:43:44,610 --> 00:43:46,640 But I will come to that expression also. 642 00:43:49,900 --> 00:43:54,440 So for the time being, this d is just the same constant 643 00:43:54,440 --> 00:43:56,455 as we have over here. 644 00:43:56,455 --> 00:43:58,200 OK? 645 00:43:58,200 --> 00:44:01,310 So you can see that the final answer 646 00:44:01,310 --> 00:44:08,680 is going to be d over tau of q 2 pi 647 00:44:08,680 --> 00:44:12,314 to the d delta function q1 plus q2. 648 00:44:15,150 --> 00:44:21,700 And if I use the value of tau of q that I have, tau of q 649 00:44:21,700 --> 00:44:29,820 is mu R plus kq squared, which becomes d over mu R 650 00:44:29,820 --> 00:44:33,600 plus kq squared and so forth. 651 00:44:33,600 --> 00:44:38,310 2 pi to the d delta function q1 plus q2. 652 00:44:42,340 --> 00:44:49,510 So essentially, if I take this linearized time dependent line 653 00:44:49,510 --> 00:44:54,180 of Landau-Ginzburg equation, run it for a very long time, 654 00:44:54,180 --> 00:44:56,970 and look at the correlations of the field, 655 00:44:56,970 --> 00:44:59,340 I see that the correlations of the field, 656 00:44:59,340 --> 00:45:03,380 at the limit of long times, satisfy this expression. 657 00:45:06,220 --> 00:45:10,090 Now, what do I know if I look at the top line 658 00:45:10,090 --> 00:45:14,380 that I have for the probability of distribution? 659 00:45:14,380 --> 00:45:17,890 I can go and express that probability of distribution 660 00:45:17,890 --> 00:45:19,175 in full mode. 661 00:45:19,175 --> 00:45:22,160 In the linear version, I immediately 662 00:45:22,160 --> 00:45:27,580 get that the probability of m tilde of q 663 00:45:27,580 --> 00:45:31,960 is proportional to a product over different q's, 664 00:45:31,960 --> 00:45:36,890 e to the minus R plus kq squared, 665 00:45:36,890 --> 00:45:42,250 et cetera, and tilde of q squared over 2. 666 00:45:42,250 --> 00:45:44,600 All right. 667 00:45:44,600 --> 00:45:48,060 So when I look at the equilibrium linear 668 00:45:48,060 --> 00:45:51,730 as Landau-Ginzburg, I can see that, if I calculate 669 00:45:51,730 --> 00:45:57,560 the average of m of q1, m of q2, then 670 00:45:57,560 --> 00:46:01,950 this is an equilibrium average. 671 00:46:01,950 --> 00:46:06,980 What I would get is 2 pi to the d delta function 672 00:46:06,980 --> 00:46:10,870 of q1 plus q2 because, clearly, the different q's are recovered 673 00:46:10,870 --> 00:46:12,620 from each other. 674 00:46:12,620 --> 00:46:17,530 And for the particular value of q, what I will get is 1 over R 675 00:46:17,530 --> 00:46:21,030 plus kq squared and so forth. 676 00:46:21,030 --> 00:46:21,890 Yes? 677 00:46:21,890 --> 00:46:26,235 AUDIENCE: So isn't the 2 over tau-- [INAUDIBLE]? 678 00:46:26,235 --> 00:46:27,026 MEHRAN KARDAR: Yes. 679 00:46:36,132 --> 00:46:37,124 Over 2. 680 00:46:43,072 --> 00:46:43,572 Yeah. 681 00:46:54,976 --> 00:46:55,476 OK. 682 00:47:02,420 --> 00:47:03,412 OK. 683 00:47:03,412 --> 00:47:07,975 I changed because I had 2d cancels the 2. 684 00:47:07,975 --> 00:47:10,600 I would have to put here tau of q over 2. 685 00:47:10,600 --> 00:47:13,360 I had 2d times tau of q over 2. 686 00:47:13,360 --> 00:47:15,100 So it's d tau. 687 00:47:15,100 --> 00:47:18,618 And inverse of tau, I have everything correct. 688 00:47:18,618 --> 00:47:21,330 OK. 689 00:47:21,330 --> 00:47:28,031 So if you make an even number of errors, the answer comes up. 690 00:47:28,031 --> 00:47:28,530 OK. 691 00:47:28,530 --> 00:47:31,700 But you can now compare this expression 692 00:47:31,700 --> 00:47:36,940 that comes from equilibrium and this expression that 693 00:47:36,940 --> 00:47:42,760 comes from the long time limit of this noisy equation. 694 00:47:42,760 --> 00:47:48,120 OK So we want to choose our noise 695 00:47:48,120 --> 00:47:55,590 so that the stochastic dynamics gives the same value 696 00:47:55,590 --> 00:48:01,100 as equilibrium, just like we did for the case of a Brownian 697 00:48:01,100 --> 00:48:06,340 particular where you have some kind of an Einstein equation 698 00:48:06,340 --> 00:48:15,125 that was relating the strength of the noise and the mobility. 699 00:48:15,125 --> 00:48:19,120 And we see that here all I need to do 700 00:48:19,120 --> 00:48:27,198 is to ensure that d over mu should be equal to 1. 701 00:48:27,198 --> 00:48:27,698 OK. 702 00:48:33,011 --> 00:48:41,810 Now, the thing is that, if I am doing this, 703 00:48:41,810 --> 00:48:48,430 I, in principle, can have a different noise for each q, 704 00:48:48,430 --> 00:48:52,110 and compensate by different mobility for each q. 705 00:48:52,110 --> 00:48:55,350 And I would get the same answer. 706 00:48:55,350 --> 00:49:01,690 So in the non-conserved version of this time dependent dynamics 707 00:49:01,690 --> 00:49:07,300 that you wrote down, the d was a constant 708 00:49:07,300 --> 00:49:11,280 and the mu was a constant. 709 00:49:11,280 --> 00:49:14,940 Whereas, if you want to get the same equilibrium result out 710 00:49:14,940 --> 00:49:19,300 of the conserved dynamics, you can see that, essentially, 711 00:49:19,300 --> 00:49:23,650 what we previously had as mu became something that 712 00:49:23,650 --> 00:49:26,540 is proportional to q squared. 713 00:49:26,540 --> 00:49:31,540 So essentially, here, this becomes mu C q squared. 714 00:49:31,540 --> 00:49:35,070 So clearly, in order to get the same answer, 715 00:49:35,070 --> 00:49:42,750 I have to put my noise to be proportional to q squared also. 716 00:49:42,750 --> 00:49:46,570 And we can see that this kind of conserved noise that I put over 717 00:49:46,570 --> 00:49:49,990 here achieves that because, as I said, 718 00:49:49,990 --> 00:49:54,870 this conserved noise is the gradient of something, which 719 00:49:54,870 --> 00:49:58,090 means that, when I go to fully space, if it be q, 720 00:49:58,090 --> 00:50:00,740 it will be proportional to q. 721 00:50:00,740 --> 00:50:02,620 And when I take its variants, it's variants 722 00:50:02,620 --> 00:50:05,800 will be proportional to q squared. 723 00:50:05,800 --> 00:50:09,580 Anything precisely canceled. 724 00:50:09,580 --> 00:50:12,580 But you can see that you also-- this 725 00:50:12,580 --> 00:50:16,390 had a physical explanation in terms of a conservation law. 726 00:50:16,390 --> 00:50:20,450 In principle, you can cook up all kinds of b of q and mu 727 00:50:20,450 --> 00:50:22,070 of q. 728 00:50:22,070 --> 00:50:26,780 As long as this equality is satisfied, you will have, 729 00:50:26,780 --> 00:50:31,010 for these linear stochastic equations, 730 00:50:31,010 --> 00:50:35,381 the guarantee that you would always get the same equilibrium 731 00:50:35,381 --> 00:50:35,880 result. 732 00:50:35,880 --> 00:50:38,700 Because if you wait for this dynamics 733 00:50:38,700 --> 00:50:43,060 to settle down after long times, you will get to the answer. 734 00:50:43,060 --> 00:50:43,842 Yes? 735 00:50:43,842 --> 00:50:48,312 AUDIENCE: I wonder how general is this result for stochastic 736 00:50:48,312 --> 00:50:50,070 [INAUDIBLE]? 737 00:50:50,070 --> 00:50:50,820 MEHRAN KARDAR: OK. 738 00:50:50,820 --> 00:50:51,736 AUDIENCE: But what I-- 739 00:50:51,736 --> 00:50:54,380 MEHRAN KARDAR: So what I showed you 740 00:50:54,380 --> 00:50:58,570 was with for linearized version, and the only thing that I 741 00:50:58,570 --> 00:51:01,220 calculated was the variance. 742 00:51:01,220 --> 00:51:04,090 And I showed that the variances were the same. 743 00:51:04,090 --> 00:51:06,310 And if I have a Gaussian problem of distribution, 744 00:51:06,310 --> 00:51:08,920 the variance is completely categorizable 745 00:51:08,920 --> 00:51:10,360 with distribution. 746 00:51:10,360 --> 00:51:11,880 So this is safe. 747 00:51:11,880 --> 00:51:15,580 But we are truly interested in the more general 748 00:51:15,580 --> 00:51:18,740 non-Gaussian probability of distribution. 749 00:51:18,740 --> 00:51:24,440 So the question really is-- if I keep the full non-linearity 750 00:51:24,440 --> 00:51:29,730 in this story, would I be able to show 751 00:51:29,730 --> 00:51:31,830 that the probability of distribution that 752 00:51:31,830 --> 00:51:36,140 will be characterized by all kinds of moments 753 00:51:36,140 --> 00:51:38,472 eventually has the same behavior as that. 754 00:51:38,472 --> 00:51:39,180 AUDIENCE: Mm-hmm. 755 00:51:39,180 --> 00:51:41,460 MEHRAN KARDAR: And the answer is, in fact, yes. 756 00:51:41,460 --> 00:51:46,310 There's a procedure that relies on converting 757 00:51:46,310 --> 00:51:48,940 this equation-- sorry. 758 00:51:48,940 --> 00:51:51,660 One equation that governs the evolution 759 00:51:51,660 --> 00:51:55,421 of the full probability as a function of time. 760 00:51:55,421 --> 00:51:55,920 Right? 761 00:51:55,920 --> 00:52:00,930 So basically, I can start with a an initial probability 762 00:52:00,930 --> 00:52:04,430 and see how this probability evolves as a function of time. 763 00:52:04,430 --> 00:52:08,220 And this is sometimes called a master equation. 764 00:52:08,220 --> 00:52:10,255 Sometimes, called a [INAUDIBLE] equation. 765 00:52:13,000 --> 00:52:20,790 And we did cover, in fact, this in-- next spring 766 00:52:20,790 --> 00:52:23,370 in the statistical physics and biology, 767 00:52:23,370 --> 00:52:25,630 we spent some time talking about these things. 768 00:52:25,630 --> 00:52:29,390 So you can come back to the third version of this class. 769 00:52:29,390 --> 00:52:35,015 And one can ensure that, with appropriate choice 770 00:52:35,015 --> 00:52:40,270 of the noise, the asymptotic solution for this probability 771 00:52:40,270 --> 00:52:43,880 distribution is whatever Landau-Ginzburg 772 00:52:43,880 --> 00:52:45,600 or other probability distribution 773 00:52:45,600 --> 00:52:47,458 that you need most. 774 00:52:47,458 --> 00:52:51,130 AUDIENCE: So is this true if we assume 775 00:52:51,130 --> 00:52:54,140 Landau-Ginzburg potential for how resistant? 776 00:52:54,140 --> 00:52:54,975 MEHRAN KARDAR: Yes. 777 00:52:54,975 --> 00:52:55,965 AUDIENCE: OK. 778 00:52:55,965 --> 00:52:59,200 Maybe this is not a very good-stated question, 779 00:52:59,200 --> 00:53:01,705 but is there kind of like an even more general level? 780 00:53:01,705 --> 00:53:03,330 MEHRAN KARDAR: I'll come to that, sure. 781 00:53:03,330 --> 00:53:06,390 But currently, the way that I set up the problem 782 00:53:06,390 --> 00:53:09,110 was that we know some complicated equilibrium 783 00:53:09,110 --> 00:53:13,250 force that exist-- form of the probability that exist. 784 00:53:13,250 --> 00:53:16,510 And these kinds of linear-- these kinds 785 00:53:16,510 --> 00:53:20,090 of stochastic linear or non-linear evolution 786 00:53:20,090 --> 00:53:22,340 equations-- generally called non- [INAUDIBLE] 787 00:53:22,340 --> 00:53:26,150 equations-- one can show that, with the appropriate choice 788 00:53:26,150 --> 00:53:30,690 of the noise, we'll be able to asymptotically reproduce 789 00:53:30,690 --> 00:53:34,192 the probability of distribution that we knew. 790 00:53:34,192 --> 00:53:35,894 But now, the question is, of course, you 791 00:53:35,894 --> 00:53:37,685 don't know the probability of distribution. 792 00:53:37,685 --> 00:53:40,113 And I'll say a few words about that. 793 00:53:40,113 --> 00:53:40,654 AUDIENCE: OK. 794 00:53:40,654 --> 00:53:41,570 Thank you. 795 00:53:41,570 --> 00:53:44,540 MEHRAN KARDAR: Anything else? 796 00:53:44,540 --> 00:53:45,420 OK. 797 00:53:45,420 --> 00:53:52,130 So the lesson of this part is that the field 798 00:53:52,130 --> 00:53:58,070 of dynamic or critical phenomena is quite rich, much richer 799 00:53:58,070 --> 00:54:01,450 than the corresponding equilibrium critical phenomena 800 00:54:01,450 --> 00:54:04,610 because the same equilibrium state can 801 00:54:04,610 --> 00:54:08,360 be obtained by various different types of dynamics. 802 00:54:08,360 --> 00:54:12,400 And I explained to you just one conservation law, 803 00:54:12,400 --> 00:54:14,090 but there could be some combination 804 00:54:14,090 --> 00:54:17,020 of conservation of energy, conservation of something. 805 00:54:17,020 --> 00:54:21,360 So there is a whole listing of different universality classes 806 00:54:21,360 --> 00:54:26,254 that people have targeted for the dynamics. 807 00:54:26,254 --> 00:54:30,620 But not all of this was assuming that you 808 00:54:30,620 --> 00:54:35,800 know what the ultimate answer is because, in all cases, 809 00:54:35,800 --> 00:54:38,870 the equations that we're writing are 810 00:54:38,870 --> 00:54:43,910 dependent on some kind of a gradient descent conserved 811 00:54:43,910 --> 00:54:47,100 or non-conserved around something that corresponded 812 00:54:47,100 --> 00:54:49,460 to the log of probability of distribution 813 00:54:49,460 --> 00:54:52,190 that we eventually want to put. 814 00:54:52,190 --> 00:54:55,770 And maybe you don't know that, and so 815 00:54:55,770 --> 00:55:04,000 let me give you a particular example in the context of, 816 00:55:04,000 --> 00:55:06,464 let's say, surface interface fluctuations. 817 00:55:12,160 --> 00:55:15,160 Starting from things that you know and then building 818 00:55:15,160 --> 00:55:18,780 to something that maybe you don't. 819 00:55:18,780 --> 00:55:21,888 Let's first with the case of a soap bubble. 820 00:55:25,720 --> 00:55:30,680 So we take some kind of a circle or whatever, 821 00:55:30,680 --> 00:55:34,980 and we put a soap bubble on top of it. 822 00:55:34,980 --> 00:55:41,080 And in this case, the energy of the formation-- 823 00:55:41,080 --> 00:55:43,780 the cost of the formation comes from surface tension. 824 00:55:50,020 --> 00:55:54,900 And let's say, the cost of the deformation 825 00:55:54,900 --> 00:55:58,830 is the changing area times some sigma. 826 00:56:04,450 --> 00:56:10,160 So I neglect the contribution that 827 00:56:10,160 --> 00:56:13,870 comes from the flat surface, and see if I make a deformation. 828 00:56:13,870 --> 00:56:17,460 If I make a deformation, I have changed the area of the spin. 829 00:56:17,460 --> 00:56:20,220 So there is a cost that is proportion to the surface 830 00:56:20,220 --> 00:56:22,930 tension times the change in area. 831 00:56:22,930 --> 00:56:26,230 Change in area locally is the square root of 1 832 00:56:26,230 --> 00:56:29,240 plus the gradient of a height profile. 833 00:56:29,240 --> 00:56:34,750 So what I can do is I can define at each point on the surface 834 00:56:34,750 --> 00:56:39,420 how much it has changed its height from being perfectly 835 00:56:39,420 --> 00:56:39,920 flat. 836 00:56:39,920 --> 00:56:42,320 So h equals to 0 is flat. 837 00:56:42,320 --> 00:56:47,770 Local area is the integral dx dy of square root of 1 838 00:56:47,770 --> 00:56:50,790 plus gradient of h squared, minus 1. 839 00:56:50,790 --> 00:56:53,060 That corresponds to the flat. 840 00:56:53,060 --> 00:56:54,420 And so then you expand that. 841 00:56:54,420 --> 00:56:58,360 The first term is going to be the integral gradient of h 842 00:56:58,360 --> 00:56:58,860 squared. 843 00:57:01,500 --> 00:57:06,490 So this is the analog of what we had over there, only 844 00:57:06,490 --> 00:57:07,640 the first term. 845 00:57:07,640 --> 00:57:10,440 So you would say that the equation that you would write 846 00:57:10,440 --> 00:57:16,210 down for this would be all to some constant mu 847 00:57:16,210 --> 00:57:18,690 proportional to the variations of this, 848 00:57:18,690 --> 00:57:24,260 which will give me something like sigma Laplacian of h. 849 00:57:24,260 --> 00:57:27,970 But because of the particles from the air constantly 850 00:57:27,970 --> 00:57:30,380 bombarding the surface, there will 851 00:57:30,380 --> 00:57:32,897 be some noise that depends on where 852 00:57:32,897 --> 00:57:36,108 you are on the surface in time. 853 00:57:36,108 --> 00:57:40,335 And this is the non-conserved version. 854 00:57:44,810 --> 00:57:48,030 And you can from this very quickly get 855 00:57:48,030 --> 00:57:53,610 that the expectation value of h tilde of q squared 856 00:57:53,610 --> 00:57:58,730 is going to be proportional to something like D 857 00:57:58,730 --> 00:58:03,840 over mu sigma q squared, because of this q squared. 858 00:58:03,840 --> 00:58:06,990 And if you ask how much fluctuations you 859 00:58:06,990 --> 00:58:10,170 have in real space-- so that typical scale 860 00:58:10,170 --> 00:58:13,620 of the fluctuations in real space-- 861 00:58:13,620 --> 00:58:18,010 will come from integrating 1 over q squared. 862 00:58:18,010 --> 00:58:21,110 And it's going to be our usual things that 863 00:58:21,110 --> 00:58:23,330 have this logarithmic dependence, so there will 864 00:58:23,330 --> 00:58:26,850 be something that ultimately will go logarithmically 865 00:58:26,850 --> 00:58:28,960 with the size of the system. 866 00:58:28,960 --> 00:58:31,440 The constant of proportionality will be 867 00:58:31,440 --> 00:58:34,080 proportional to kt over sigma. 868 00:58:34,080 --> 00:58:38,900 So you have to choose your D and mu to correspond to this. 869 00:58:38,900 --> 00:58:42,750 But basically, a soap film, as an example 870 00:58:42,750 --> 00:58:45,970 of all kinds of Goldstone mode-like things 871 00:58:45,970 --> 00:58:47,410 that we have seen. 872 00:58:47,410 --> 00:58:49,890 It's a 2-dimensional entity. 873 00:58:49,890 --> 00:58:53,120 We will have logarithmic fluctuations-- not very big, 874 00:58:53,120 --> 00:58:56,560 but ultimately, at large enough distances, 875 00:58:56,560 --> 00:58:59,110 it will have fluctuations. 876 00:58:59,110 --> 00:59:02,230 So that was non-conserved. 877 00:59:02,230 --> 00:59:04,750 I can imagine that, rather than this, 878 00:59:04,750 --> 00:59:12,150 I have the case of a surface of a pool. 879 00:59:12,150 --> 00:59:17,890 So here I have some depth of water, 880 00:59:17,890 --> 00:59:21,930 and then there's the surface of the pool of water. 881 00:59:21,930 --> 00:59:26,730 And the difference between this case and the previous case-- 882 00:59:26,730 --> 00:59:29,530 both of them can be described by a height function. 883 00:59:33,130 --> 00:59:36,280 The difference is that if I ignore evaporation 884 00:59:36,280 --> 00:59:41,020 and condensation, the total mass of water 885 00:59:41,020 --> 00:59:42,800 is going to be conserved. 886 00:59:42,800 --> 00:59:49,420 So I would need to have divided t of the integral dx d qx 887 00:59:49,420 --> 00:59:55,060 h of x and t to be 0. 888 00:59:55,060 --> 00:59:57,746 So this would go into the conserved variety. 889 01:00:00,500 --> 01:00:05,120 And while, if I create a ripple on the surface of this 890 01:00:05,120 --> 01:00:10,770 compared to the surface of that, the relaxation time 891 01:00:10,770 --> 01:00:13,320 through this dissipative dynamics 892 01:00:13,320 --> 01:00:17,280 would be much longer in this case as opposed to that case. 893 01:00:17,280 --> 01:00:20,390 Ultimately, if I wait sufficiently long time, 894 01:00:20,390 --> 01:00:24,640 both of them would have exactly the same fluctuations. 895 01:00:24,640 --> 01:00:27,710 That is, you would go logarithmically with the length 896 01:00:27,710 --> 01:00:31,050 scale over which [INAUDIBLE]. 897 01:00:31,050 --> 01:00:37,170 OK, so now let's look at another system that fluctuates. 898 01:00:37,170 --> 01:00:40,620 And I don't know what the final answer is. 899 01:00:40,620 --> 01:00:43,240 That was the question, maybe, that you asked. 900 01:00:43,240 --> 01:00:46,620 The example that I will give is the following-- 901 01:00:46,620 --> 01:00:50,480 so suppose that you have a surface. 902 01:00:50,480 --> 01:00:56,330 And then you have a rain of sticky materials 903 01:00:56,330 --> 01:00:59,670 that falls down on top of it. 904 01:00:59,670 --> 01:01:03,500 So this material will come down. 905 01:01:03,500 --> 01:01:05,400 You'll have something like this. 906 01:01:05,400 --> 01:01:07,860 And then as time goes on, there will 907 01:01:07,860 --> 01:01:12,090 be more material that will come, more material that will come, 908 01:01:12,090 --> 01:01:14,210 more material that will come. 909 01:01:14,210 --> 01:01:20,840 So there, because the particles are raining down randomly 910 01:01:20,840 --> 01:01:25,810 at different points, there will be a stochastic process 911 01:01:25,810 --> 01:01:27,450 that is going on. 912 01:01:27,450 --> 01:01:32,230 So you can try to characterize the system 913 01:01:32,230 --> 01:01:35,360 in terms of a height that changes as a function of t 914 01:01:35,360 --> 01:01:38,890 and as a function of position. 915 01:01:38,890 --> 01:01:44,930 And there could be all kinds of microscopic things going on, 916 01:01:44,930 --> 01:01:46,930 like maybe these are particles that 917 01:01:46,930 --> 01:01:51,260 are representing some kind of a deposition process. 918 01:01:51,260 --> 01:01:54,390 And then they come, they stick in a particular way. 919 01:01:54,390 --> 01:01:56,580 Maybe they can slide on the surface. 920 01:01:56,580 --> 01:02:01,360 We can imagine all kinds of microscopic degrees of freedom 921 01:02:01,360 --> 01:02:04,200 and things that we can put. 922 01:02:04,200 --> 01:02:08,660 But you say, well, can I change my perspective, 923 01:02:08,660 --> 01:02:12,500 and try to describe the system the same way that we 924 01:02:12,500 --> 01:02:16,180 did for the case of coarse grading and going 925 01:02:16,180 --> 01:02:18,010 from without the microscopic details 926 01:02:18,010 --> 01:02:22,930 to describe the phenomenological Landau-Ginzburg equation? 927 01:02:22,930 --> 01:02:26,910 And so you say, OK, there is a height that is growing. 928 01:02:26,910 --> 01:02:30,660 And what I will write down is an equation 929 01:02:30,660 --> 01:02:32,510 that is very similar to the equations 930 01:02:32,510 --> 01:02:36,090 that I had written before. 931 01:02:36,090 --> 01:02:40,340 Now I'm going to follow the same kind of reasoning 932 01:02:40,340 --> 01:02:44,570 that we did in the construction of this Landau-Ginzburg model, 933 01:02:44,570 --> 01:02:49,350 is we said that this weight is going to depend on all 934 01:02:49,350 --> 01:02:52,660 kinds of things that relate to this height 935 01:02:52,660 --> 01:02:53,870 that I don't quite know. 936 01:02:53,870 --> 01:02:58,230 So let's imagine that there is some kind of a function 937 01:02:58,230 --> 01:03:01,570 of the height itself. 938 01:03:01,570 --> 01:03:04,420 And potentially, just like we did over there, 939 01:03:04,420 --> 01:03:08,176 the gradient of the height, five derivatives of the height, 940 01:03:08,176 --> 01:03:08,675 et cetera. 941 01:03:12,460 --> 01:03:16,280 And then I will start to make an expansion of this 942 01:03:16,280 --> 01:03:20,750 in the same spirit that I did for the Landau-Ginzburg Model, 943 01:03:20,750 --> 01:03:23,540 except that when I was doing the Landau-Ginzburg Model, 944 01:03:23,540 --> 01:03:26,840 I was doing the expansion at the level of looking 945 01:03:26,840 --> 01:03:30,450 at the probability distribution and the log of the probability. 946 01:03:30,450 --> 01:03:34,625 Here I'm making the expansion at the level of an equation that 947 01:03:34,625 --> 01:03:35,500 governs the dynamics. 948 01:03:38,580 --> 01:03:41,250 Of course, in this particular system, 949 01:03:41,250 --> 01:03:44,620 that's not the end of story, because the change in height 950 01:03:44,620 --> 01:03:49,600 is also governed by this random addition of the particles. 951 01:03:49,600 --> 01:03:53,080 So there is some function that changes 952 01:03:53,080 --> 01:03:55,070 as a function of position and time, 953 01:03:55,070 --> 01:03:57,040 depending on whether, at that time, 954 01:03:57,040 --> 01:03:59,370 a particle was dropped down. 955 01:03:59,370 --> 01:04:01,920 I can always take the average of this 956 01:04:01,920 --> 01:04:07,740 to be 0, and put that average into the expansion of this 957 01:04:07,740 --> 01:04:10,220 starting from a constant. 958 01:04:10,220 --> 01:04:13,200 Basically, if I just have a single point 959 01:04:13,200 --> 01:04:17,010 and I randomly drop particles at that single point, 960 01:04:17,010 --> 01:04:19,560 there will be an average growth velocity, 961 01:04:19,560 --> 01:04:21,850 an average addition to the height, 962 01:04:21,850 --> 01:04:23,530 that averages over here. 963 01:04:23,530 --> 01:04:27,580 But there will be fluctuations that are going [INAUDIBLE]. 964 01:04:27,580 --> 01:04:31,360 OK but the constant is the first term 965 01:04:31,360 --> 01:04:34,330 in an expansion such as this. 966 01:04:34,330 --> 01:04:38,220 And you can start thinking, OK, what next order? 967 01:04:38,220 --> 01:04:41,300 Can I put something like alpha h? 968 01:04:41,300 --> 01:04:43,840 Potentially-- depends on your system-- 969 01:04:43,840 --> 01:04:49,440 but if the system is invariant whether you started from here 970 01:04:49,440 --> 01:04:52,490 or whether you started from there-- something like gravity, 971 01:04:52,490 --> 01:04:55,540 for example, is not important-- you say, OK, 972 01:04:55,540 --> 01:05:00,110 I cannot have any function of h if my dynamics will proceed 973 01:05:00,110 --> 01:05:05,330 exactly the same way if I were to translate this surface 974 01:05:05,330 --> 01:05:07,590 to some further up or further down. 975 01:05:07,590 --> 01:05:10,760 If I see that there's no change in future dynamics on average, 976 01:05:10,760 --> 01:05:14,660 then the dynamic cannot depend on this. 977 01:05:14,660 --> 01:05:16,970 OK, so we've got rid of that. 978 01:05:16,970 --> 01:05:19,370 And any function of h-- can I put 979 01:05:19,370 --> 01:05:23,640 something that is proportional to gradient of h? 980 01:05:23,640 --> 01:05:28,070 Maybe for something I can, but for h itself I cannot, 981 01:05:28,070 --> 01:05:30,510 because h is a scalar gradient. 982 01:05:30,510 --> 01:05:33,990 If h is a vector, I can't set something 983 01:05:33,990 --> 01:05:40,620 that is a scalar equal to a vector, so I can't have this. 984 01:05:40,620 --> 01:05:41,914 Yes? 985 01:05:41,914 --> 01:05:43,455 AUDIENCE: Couldn't you, in principle, 986 01:05:43,455 --> 01:05:47,130 make your constant term in front of the gradient also 987 01:05:47,130 --> 01:05:48,570 a vector and [INAUDIBLE]? 988 01:05:50,659 --> 01:05:51,700 MEHRAN KARDAR: You could. 989 01:05:51,700 --> 01:05:54,690 So there's a whole set of different systems 990 01:05:54,690 --> 01:05:56,630 that you can be thinking about. 991 01:05:56,630 --> 01:06:00,220 Right now, I want to focus on the simplest system, which 992 01:06:00,220 --> 01:06:02,770 is a scalar field, so that my equation can 993 01:06:02,770 --> 01:06:05,404 be as simple as possible, but still we 994 01:06:05,404 --> 01:06:07,070 will see it has sufficient complication. 995 01:06:10,640 --> 01:06:12,940 So you can see that if I don't have them, 996 01:06:12,940 --> 01:06:15,770 the next order term that I can have 997 01:06:15,770 --> 01:06:17,480 would be something like a Laplacian. 998 01:06:20,560 --> 01:06:24,440 So this kind of diffusion equation, you can see, 999 01:06:24,440 --> 01:06:28,780 has to emerge as a low-order expansion of something 1000 01:06:28,780 --> 01:06:29,280 like this. 1001 01:06:29,280 --> 01:06:32,140 And this is the ubiquity of the diffusion equation appearing 1002 01:06:32,140 --> 01:06:34,380 all over the place. 1003 01:06:34,380 --> 01:06:36,220 And then you could have terms that 1004 01:06:36,220 --> 01:06:38,890 would be of the order of the fourth derivative, 1005 01:06:38,890 --> 01:06:40,520 and so forth. 1006 01:06:40,520 --> 01:06:42,480 There's nothing wrong with that. 1007 01:06:46,410 --> 01:06:49,940 And then, if you think about it, you'll 1008 01:06:49,940 --> 01:06:55,420 see that there is one interesting possibility that 1009 01:06:55,420 --> 01:06:58,980 is not allowed for that system, but is 1010 01:06:58,980 --> 01:07:03,950 allowed for this system, which is something that is a scalar. 1011 01:07:03,950 --> 01:07:06,366 It's the gradient of h squared. 1012 01:07:09,320 --> 01:07:15,100 Now I could not have added this term for the case of the soap 1013 01:07:15,100 --> 01:07:17,960 bubble for the following reason-- 1014 01:07:17,960 --> 01:07:22,760 that if I reverse the soap bubble so that h 1015 01:07:22,760 --> 01:07:27,050 becomes minus h, the dynamics would proceed exactly 1016 01:07:27,050 --> 01:07:28,960 as before. 1017 01:07:28,960 --> 01:07:34,020 So the soap bubble has a symmetry of h going to minus h, 1018 01:07:34,020 --> 01:07:38,160 and so that symmetry should be preserved in the equation. 1019 01:07:38,160 --> 01:07:41,820 This term breaks that symmetry because the left-hand side 1020 01:07:41,820 --> 01:07:45,760 is odd in h, whereas the right-hand side of this term 1021 01:07:45,760 --> 01:07:48,080 would be even in h. 1022 01:07:48,080 --> 01:07:51,990 But for the case of the growing surface-- and you've seen 1023 01:07:51,990 --> 01:07:53,140 things that are growing. 1024 01:07:53,140 --> 01:07:55,520 And typically, if I give you something 1025 01:07:55,520 --> 01:07:59,270 that has grown like the tree trunk, for example, 1026 01:07:59,270 --> 01:08:02,400 and if I take the picture of a part of it, 1027 01:08:02,400 --> 01:08:05,370 and you don't see where the center is, where the end is, 1028 01:08:05,370 --> 01:08:08,040 you can immediately tell from the way 1029 01:08:08,040 --> 01:08:11,760 that the shape of this object is, that it is growing 1030 01:08:11,760 --> 01:08:13,960 in some particular direction. 1031 01:08:13,960 --> 01:08:17,590 So for growth systems, that symmetry does not exist. 1032 01:08:17,590 --> 01:08:22,220 You are allowed to have this term, and so forth. 1033 01:08:22,220 --> 01:08:26,300 Now the interesting thing about this term 1034 01:08:26,300 --> 01:08:31,465 is that there is no beta h that you can write down 1035 01:08:31,465 --> 01:08:36,040 that is local-- some function of h such 1036 01:08:36,040 --> 01:08:40,229 that if you take a functional derivative with respect to h, 1037 01:08:40,229 --> 01:08:46,310 it will reproduce that term-- just does not exist. 1038 01:08:46,310 --> 01:08:51,750 So you can see that somehow immediately, 1039 01:08:51,750 --> 01:08:59,380 as soon as we liberate ourselves from writing equations that 1040 01:08:59,380 --> 01:09:03,899 came from functional derivative of something, 1041 01:09:03,899 --> 01:09:07,310 but potentially have physical significance, 1042 01:09:07,310 --> 01:09:10,810 we can write down new terms. 1043 01:09:10,810 --> 01:09:14,000 So this is actually-- also, you can do 1044 01:09:14,000 --> 01:09:17,020 this, even for two particles. 1045 01:09:17,020 --> 01:09:22,770 A potential v of x1 and x2 will have some kind of derivatives. 1046 01:09:22,770 --> 01:09:24,505 But if you write dynamical equations, 1047 01:09:24,505 --> 01:09:26,600 there are dynamical equations that 1048 01:09:26,600 --> 01:09:31,080 allow you to rotate 1 x from x1 to x2. 1049 01:09:31,080 --> 01:09:32,759 That kind of term will never come 1050 01:09:32,759 --> 01:09:34,369 from taking the derivative. 1051 01:09:37,130 --> 01:09:38,330 So fine. 1052 01:09:41,370 --> 01:09:43,490 So this is a candidate equation that 1053 01:09:43,490 --> 01:09:47,420 is obtained in this context-- something that is grown. 1054 01:09:47,420 --> 01:09:49,380 We say we are not interested in it's 1055 01:09:49,380 --> 01:09:51,850 coming from some underlying weight, 1056 01:09:51,850 --> 01:09:55,060 but presumably, this system still, 1057 01:09:55,060 --> 01:09:58,270 if I look at it at long times, will 1058 01:09:58,270 --> 01:10:01,450 have some kind of fluctuations. 1059 01:10:01,450 --> 01:10:04,450 All the fluctuations of this growing surface, 1060 01:10:04,450 --> 01:10:07,340 like the fluctuations of the soap bubble, 1061 01:10:07,340 --> 01:10:10,986 and they have this logarithmic dependence. 1062 01:10:10,986 --> 01:10:13,194 You have a question? 1063 01:10:13,194 --> 01:10:17,400 AUDIENCE: So why doesn't that term-- what if I put in h times 1064 01:10:17,400 --> 01:10:19,968 that term that we want to appear and then I 1065 01:10:19,968 --> 01:10:21,200 vary with respect to h? 1066 01:10:21,200 --> 01:10:24,019 A term like what we want to pop out together with other terms? 1067 01:10:24,019 --> 01:10:25,810 MEHRAN KARDAR: Yeah, but those other terms, 1068 01:10:25,810 --> 01:10:27,660 what do you want to do with them? 1069 01:10:27,660 --> 01:10:32,700 AUDIENCE: Well, maybe they're not acceptable [INAUDIBLE]? 1070 01:10:32,700 --> 01:10:35,360 MEHRAN KARDAR: So you're saying why not have a term that 1071 01:10:35,360 --> 01:10:38,500 is h gradient of h squared? 1072 01:10:38,500 --> 01:10:42,690 Functional derivative of that is gradient of h squared. 1073 01:10:42,690 --> 01:10:47,390 And then you have a term that is h Laplacian-- it's 1074 01:10:47,390 --> 01:10:50,785 a gradient of, sorry, gradient of h. 1075 01:10:57,816 --> 01:11:00,690 And then you expand this. 1076 01:11:00,690 --> 01:11:02,820 Among the terms that you would generate 1077 01:11:02,820 --> 01:11:07,560 would be a term that is h, a Laplacian of h. 1078 01:11:07,560 --> 01:11:09,745 This term violates this condition 1079 01:11:09,745 --> 01:11:13,490 that we had over here. 1080 01:11:13,490 --> 01:11:18,000 And you cannot separate this term from that term. 1081 01:11:18,000 --> 01:11:20,960 So what you describe, you already 1082 01:11:20,960 --> 01:11:22,940 see at the level over here. 1083 01:11:22,940 --> 01:11:27,380 It violates translation of symmetry in [? nature ?]. 1084 01:11:27,380 --> 01:11:29,775 And you can play around with other functions. 1085 01:11:29,775 --> 01:11:31,488 You come to the same conclusion. 1086 01:11:35,300 --> 01:11:39,660 OK, so the question is, well, you added some term here. 1087 01:11:39,660 --> 01:11:43,750 If I look at this surface that has grown at large time, 1088 01:11:43,750 --> 01:11:46,720 does it have the same fluctuations as we had before? 1089 01:11:49,330 --> 01:11:54,820 So a simple way to ascertain that 1090 01:11:54,820 --> 01:11:58,780 is to do the same kind of dimensional analysis 1091 01:11:58,780 --> 01:12:02,570 which, for the Landau-Ginzburg, was a prelude to doing 1092 01:12:02,570 --> 01:12:04,810 renormalization. 1093 01:12:04,810 --> 01:12:08,550 So we did things like epsilon expansion, et cetera. 1094 01:12:08,550 --> 01:12:12,960 But to calculate that there was a critical dimension of 4, 1095 01:12:12,960 --> 01:12:16,300 all we needed to do was to rescale x and m, 1096 01:12:16,300 --> 01:12:18,480 and we would immediately see that mu 1097 01:12:18,480 --> 01:12:21,770 goes to mu, b to the 4 minus D or something-- 1098 01:12:21,770 --> 01:12:24,760 D minus 4, for example. 1099 01:12:24,760 --> 01:12:28,970 So we can do the same thing here. 1100 01:12:28,970 --> 01:12:33,130 We can always move to a frame that 1101 01:12:33,130 --> 01:12:35,250 is moving with the average velocity, 1102 01:12:35,250 --> 01:12:37,910 so that we are focusing on the fluctuations. 1103 01:12:37,910 --> 01:12:40,570 So we can basically ignore this term. 1104 01:12:40,570 --> 01:12:45,620 I'm going to rescale x by a factor of b. 1105 01:12:45,620 --> 01:12:50,980 I'm going to rescale time by a factor of b to something 1106 01:12:50,980 --> 01:12:52,290 to the z. 1107 01:12:52,290 --> 01:12:57,360 And this z is kind of indicative of what 1108 01:12:57,360 --> 01:13:02,430 we've seen before-- that somehow in these dynamical phenomena, 1109 01:13:02,430 --> 01:13:08,620 the scaling of time and space are related to some exponent. 1110 01:13:08,620 --> 01:13:10,690 But there's also an exponent that 1111 01:13:10,690 --> 01:13:14,690 characterizes how the fluctuations in h 1112 01:13:14,690 --> 01:13:18,070 grow if I look at systems that are larger and larger. 1113 01:13:18,070 --> 01:13:20,990 In particular, if I had solved that equation, 1114 01:13:20,990 --> 01:13:24,210 rather than for a soap bubble in two dimensions, 1115 01:13:24,210 --> 01:13:28,650 for a line-- for a string that I was pulling so that I had 1116 01:13:28,650 --> 01:13:31,280 line tension-- the one-dimensional version of it, 1117 01:13:31,280 --> 01:13:35,860 the one-dimensional version of an integral of 1 over q squared 1118 01:13:35,860 --> 01:13:37,940 would be something that would grow 1119 01:13:37,940 --> 01:13:40,100 with the size of the system. 1120 01:13:40,100 --> 01:13:43,100 So there I would have a chi of 1/2, for example, 1121 01:13:43,100 --> 01:13:44,850 in one dimension. 1122 01:13:44,850 --> 01:13:47,250 So this is the general thing. 1123 01:13:47,250 --> 01:13:51,800 And then I would say that the first equation, dhy dt, 1124 01:13:51,800 --> 01:13:54,290 gets a factor of b to the chi minus 1125 01:13:54,290 --> 01:13:58,930 z, because h scaled by a factor of chi, 1126 01:13:58,930 --> 01:14:06,280 t scaled by a factor of z, the term sigma Laplacian of h 1127 01:14:06,280 --> 01:14:10,090 gets a factor of b to the chi minus 2 1128 01:14:10,090 --> 01:14:13,350 from the two derivatives here-- sorry, the z and the 2 look 1129 01:14:13,350 --> 01:14:15,380 kind of the same. 1130 01:14:15,380 --> 01:14:17,412 This is a z. 1131 01:14:17,412 --> 01:14:20,640 This is a 2. 1132 01:14:20,640 --> 01:14:23,830 And then the term that is proportional 1133 01:14:23,830 --> 01:14:27,740 to this non-linearity that I wrote down-- actually, it 1134 01:14:27,740 --> 01:14:32,270 is very easy, maybe worthwhile, to show that sigma to the 4th 1135 01:14:32,270 --> 01:14:36,290 goes with a factor of b to the chi minus 4. 1136 01:14:36,290 --> 01:14:41,100 It is always down by a factor of two scalings in b with respect 1137 01:14:41,100 --> 01:14:44,470 to a Laplacian-- the same reason that when 1138 01:14:44,470 --> 01:14:47,360 we were doing the Landau-Ginzburg. 1139 01:14:47,360 --> 01:14:50,790 We could terminate the series at order of gradient squared, 1140 01:14:50,790 --> 01:14:53,610 because higher-order derivatives were irrelevant. 1141 01:14:53,610 --> 01:14:55,790 They were scaling to 0. 1142 01:14:55,790 --> 01:15:02,890 But this term grows like b to the 2 chi, 1143 01:15:02,890 --> 01:15:08,820 because it's h squared, minus 2, because there's two gradients. 1144 01:15:08,820 --> 01:15:11,950 Now thinking about the scaling of eta 1145 01:15:11,950 --> 01:15:17,670 takes a little bit of thought, because what we have-- we 1146 01:15:17,670 --> 01:15:21,330 said that the average of eta goes to 0. 1147 01:15:21,330 --> 01:15:26,290 The average of eta at two different locations 1148 01:15:26,290 --> 01:15:29,380 and two different times-- it is these particles that 1149 01:15:29,380 --> 01:15:32,460 are raining down-- they're uncorrelated 1150 01:15:32,460 --> 01:15:33,800 at different times. 1151 01:15:33,800 --> 01:15:36,026 They're uncorrelated at different positions. 1152 01:15:38,600 --> 01:15:40,190 There's some kind of variance here, 1153 01:15:40,190 --> 01:15:43,110 but that's not important to us. 1154 01:15:43,110 --> 01:15:48,840 If I rescale t by a factor of b, delta of bt-- sorry, 1155 01:15:48,840 --> 01:15:54,230 if I scale t by a factor of b to the z, delta of b to the zx 1156 01:15:54,230 --> 01:15:57,550 will get a factor of b to the minus z. 1157 01:15:57,550 --> 01:16:02,110 This will get a factor of b to the minus d. 1158 01:16:02,110 --> 01:16:06,240 But the noise, eta, is half of that. 1159 01:16:06,240 --> 01:16:14,210 So what I will have is b to the minus z plus d over 2 times 1160 01:16:14,210 --> 01:16:18,180 eta on the rescalings that I have indicated. 1161 01:16:18,180 --> 01:16:23,780 I get rid of this term. 1162 01:16:23,780 --> 01:16:28,960 So this-- divide by b to the chi minus z. 1163 01:16:28,960 --> 01:16:37,760 So then this becomes bh y dt is sigma b to the z minus 2. 1164 01:16:37,760 --> 01:16:44,550 Maybe I'll write it in red-- b to the z minus 2. 1165 01:16:44,550 --> 01:16:52,470 This becomes sigma to the 4, b to the z minus 4. 1166 01:16:52,470 --> 01:16:54,850 And then lambda over 2. 1167 01:16:54,850 --> 01:16:57,450 This is Laplacian of h. 1168 01:16:57,450 --> 01:17:02,580 This, for derivative of h, this is Laplacian of h. 1169 01:17:02,580 --> 01:17:09,900 This term becomes b to the chi plus z minus 2, gradient of h 1170 01:17:09,900 --> 01:17:12,290 squared. 1171 01:17:12,290 --> 01:17:24,804 And the final term becomes b to the chi minus d minus z over 2. 1172 01:17:36,005 --> 01:17:38,950 AUDIENCE: [INAUDIBLE]? 1173 01:17:38,950 --> 01:17:42,990 MEHRAN KARDAR: b to the minus chi-- you're right. 1174 01:17:42,990 --> 01:17:51,424 And then, actually, this I-- no? 1175 01:17:54,370 --> 01:18:09,944 Minus chi minus d over 2 minus d over 2 [INAUDIBLE], 1176 01:18:09,944 --> 01:18:10,942 That's fine. 1177 01:18:14,940 --> 01:18:21,011 So I can make this equation to be invariant. 1178 01:18:31,813 --> 01:18:34,970 So I want to find out what happens 1179 01:18:34,970 --> 01:18:38,180 to this system if I find some kind of an equation, 1180 01:18:38,180 --> 01:18:41,155 or some kind of behavior that is scale invariant. 1181 01:18:41,155 --> 01:18:45,690 You can see that immediately, my choice for the first term 1182 01:18:45,690 --> 01:18:49,730 has to be z equals to 2. 1183 01:18:49,730 --> 01:18:52,100 So basically, it says that as long 1184 01:18:52,100 --> 01:18:57,060 as you're governed by something that is diffusive, 1185 01:18:57,060 --> 01:19:00,700 so that when you go to Fourier space, you have q squared, 1186 01:19:00,700 --> 01:19:03,430 your relaxation times are going to have 1187 01:19:03,430 --> 01:19:07,140 this diffusive character, where time is distance squared. 1188 01:19:07,140 --> 01:19:10,480 Actually, you can see that immediately from the equation 1189 01:19:10,480 --> 01:19:13,700 that this diffusion time goes like distance squared. 1190 01:19:13,700 --> 01:19:17,380 So this is just a statement of that. 1191 01:19:17,380 --> 01:19:21,940 Now, it is the noise that causes the fluctuations. 1192 01:19:21,940 --> 01:19:29,830 And if I haven't made some simple error, 1193 01:19:29,830 --> 01:19:33,600 you will find that the coefficient of the noise term 1194 01:19:33,600 --> 01:19:36,980 becomes scale invariant, provided that I choose it 1195 01:19:36,980 --> 01:19:41,910 to be z minus d over 2 for chi. 1196 01:19:41,910 --> 01:19:45,610 And since my z was 2, I'm forced to have 1197 01:19:45,610 --> 01:19:48,490 chi to be 2 minus d over 2. 1198 01:19:48,490 --> 01:19:51,770 And let's see if it makes sense to us. 1199 01:19:51,770 --> 01:20:00,170 So if I have a surface such as the case of the soap bubble 1200 01:20:00,170 --> 01:20:04,210 in two dimensions, chi is 0. 1201 01:20:04,210 --> 01:20:06,880 And 0 is actually this limiting case 1202 01:20:06,880 --> 01:20:10,020 that would also be a logarithm. 1203 01:20:10,020 --> 01:20:14,140 If I go to the case of d equals to 1-- like pulling a line 1204 01:20:14,140 --> 01:20:18,090 and having the line fluctuate-- then I have 2 minus 1 1205 01:20:18,090 --> 01:20:20,550 over 2, which is 1/2, which means 1206 01:20:20,550 --> 01:20:22,700 that, because of thermal fluctuations, 1207 01:20:22,700 --> 01:20:26,290 this line will look like it a random walk. 1208 01:20:26,290 --> 01:20:27,710 You go a distance x. 1209 01:20:27,710 --> 01:20:30,420 The fluctuations in height will go 1210 01:20:30,420 --> 01:20:33,290 like the square root of that. 1211 01:20:33,290 --> 01:20:35,000 OK? 1212 01:20:35,000 --> 01:20:37,220 So all of that is fine. 1213 01:20:37,220 --> 01:20:39,660 You would have done exactly the same answer 1214 01:20:39,660 --> 01:20:44,210 if you had just gotten a kind of scaling such as this 1215 01:20:44,210 --> 01:20:46,850 for the case of the Gaussian Model 1216 01:20:46,850 --> 01:20:49,310 without the nonlinearities. 1217 01:20:49,310 --> 01:20:51,850 But for the Gaussian Model with nonlinearities, 1218 01:20:51,850 --> 01:20:55,460 we could also then estimate whether the nonlinearity u 1219 01:20:55,460 --> 01:20:57,210 is relevant. 1220 01:20:57,210 --> 01:21:00,470 So here we see that the coefficient of our nonlinearity 1221 01:21:00,470 --> 01:21:07,690 is lambda, is governed by something 1222 01:21:07,690 --> 01:21:11,260 that is chi plus z minus 2. 1223 01:21:11,260 --> 01:21:17,270 And our chi is 2 minus z over 2. 1224 01:21:17,270 --> 01:21:19,720 z minus 2 is 0. 1225 01:21:19,720 --> 01:21:25,720 So whether or not this nonlinearity is relevant, 1226 01:21:25,720 --> 01:21:28,590 we can see depends on whether you're 1227 01:21:28,590 --> 01:21:31,940 above or below two dimensions. 1228 01:21:31,940 --> 01:21:35,900 So when you are below two dimensions, 1229 01:21:35,900 --> 01:21:39,840 this nonlinearity is relevant. 1230 01:21:39,840 --> 01:21:42,720 And you will certainly have different types 1231 01:21:42,720 --> 01:21:45,810 of scaling phenomena then what you 1232 01:21:45,810 --> 01:21:52,620 predict by the case of the diffusion equation plus noise. 1233 01:21:52,620 --> 01:21:56,300 Of course, the interesting case is 1234 01:21:56,300 --> 01:22:02,120 when you are at the marginal dimension of 2. 1235 01:22:05,840 --> 01:22:12,030 Now, in terms of when you do proper renormalization group 1236 01:22:12,030 --> 01:22:15,820 with this nonlinearity, you will find that, 1237 01:22:15,820 --> 01:22:20,350 unlike the nonlinearity of the Landau-Ginzburg, which 1238 01:22:20,350 --> 01:22:23,800 is marginally irrelevant in four dimensions. 1239 01:22:23,800 --> 01:22:29,250 du by dl was minus u squared, this lambda 1240 01:22:29,250 --> 01:22:31,670 is marginally relevant. 1241 01:22:31,670 --> 01:22:36,320 d lambda by dl is proportional to plus lambda squared. 1242 01:22:36,320 --> 01:22:44,040 It is relevant marginality-- marginally relevant. 1243 01:22:44,040 --> 01:22:46,340 And actually, the epsilon expansion 1244 01:22:46,340 --> 01:22:51,250 gives you no information about what's happening in the system. 1245 01:22:51,250 --> 01:22:55,050 So people have then done numerical simulations. 1246 01:22:55,050 --> 01:22:57,845 And they find that there is a roughness that 1247 01:22:57,845 --> 01:23:02,640 is characterized by an exponent, say something like 0.4. 1248 01:23:02,640 --> 01:23:06,780 So that when you look at some surface that is grown, 1249 01:23:06,780 --> 01:23:12,130 is much, much rougher than the surface of a soap bubble 1250 01:23:12,130 --> 01:23:15,950 or what's happening on the surface of the pond. 1251 01:23:15,950 --> 01:23:21,820 And the key to all of this is that we wrote down equations 1252 01:23:21,820 --> 01:23:24,410 on the basis of this generalization 1253 01:23:24,410 --> 01:23:27,760 on symmetry that we had learned, now applied 1254 01:23:27,760 --> 01:23:31,310 to this dynamical system, did an expansion, 1255 01:23:31,310 --> 01:23:34,640 found one first term. 1256 01:23:34,640 --> 01:23:37,330 And we found it to be relevant. 1257 01:23:37,330 --> 01:23:39,980 And is actually not that often that you 1258 01:23:39,980 --> 01:23:42,460 find something that is relevant, because then it 1259 01:23:42,460 --> 01:23:44,990 is a reason to celebrate. 1260 01:23:44,990 --> 01:23:47,150 Because most of the time, things are irrelevant, 1261 01:23:47,150 --> 01:23:50,260 and you end up with boring diffusion equations. 1262 01:23:50,260 --> 01:23:53,170 So find something that is relevant. 1263 01:23:53,170 --> 01:23:56,010 And that's my last message to you.