1 00:00:00,000 --> 00:00:03,860 GERBRAND CEDER: So what I want to do today is finish up 2 00:00:03,860 --> 00:00:10,850 Monte Carlo simulation and go into a little more advanced 3 00:00:10,850 --> 00:00:13,250 things in Monte Carlo simulation and then 4 00:00:13,250 --> 00:00:16,100 also go into free energy integrations 5 00:00:16,100 --> 00:00:17,860 and other coarse-graining methods 6 00:00:17,860 --> 00:00:20,210 so when direct simulation won't work. 7 00:00:25,700 --> 00:00:28,070 If you remember, I introduced Monte Carlo 8 00:00:28,070 --> 00:00:30,230 simulation as a way of importance sampling 9 00:00:30,230 --> 00:00:32,607 and contrasted it against simple sampling. 10 00:00:32,607 --> 00:00:34,190 What I want to show you today, there's 11 00:00:34,190 --> 00:00:37,100 actually quite a few things in between that, in some cases, 12 00:00:37,100 --> 00:00:38,270 can be practical. 13 00:00:38,270 --> 00:00:40,520 Remember the idea of simple sampling 14 00:00:40,520 --> 00:00:44,270 is that you sample randomly and then 15 00:00:44,270 --> 00:00:47,120 you weigh with a probability that's 16 00:00:47,120 --> 00:00:50,180 correct relatively, where the states have 17 00:00:50,180 --> 00:00:53,120 their correct relative probability to one another. 18 00:00:53,120 --> 00:00:56,150 Importance sampling is sampling with the correct probability 19 00:00:56,150 --> 00:01:01,010 and then just averaging the quantity you sample, 20 00:01:01,010 --> 00:01:04,980 but you can actually sample with things in between. 21 00:01:04,980 --> 00:01:09,920 So here you're sampling with a probability proportional 22 00:01:09,920 --> 00:01:14,150 to that exponential of that Hamiltonian. 23 00:01:14,150 --> 00:01:16,160 Here you're sampling with no Hamiltonian 24 00:01:16,160 --> 00:01:19,550 and therefore, you have to correct the probability later. 25 00:01:19,550 --> 00:01:23,210 You can sample with any Hamiltonian H0, which is not 26 00:01:23,210 --> 00:01:25,520 the true Hamiltonian of your system, 27 00:01:25,520 --> 00:01:28,380 and then correct for it in the probabilities. 28 00:01:28,380 --> 00:01:35,550 So if you sample with H0, then the states you sample 29 00:01:35,550 --> 00:01:37,920 have to be corrected with the relative probability 30 00:01:37,920 --> 00:01:41,270 of that state, the relative probability 31 00:01:41,270 --> 00:01:44,220 that you would get with the correct Hamiltonian versus what 32 00:01:44,220 --> 00:01:47,670 you get with the Hamiltonian which you decided to sample. 33 00:01:47,670 --> 00:01:48,600 OK? 34 00:01:48,600 --> 00:01:50,880 So this is the Hamiltonian which we decided to pick, 35 00:01:50,880 --> 00:01:53,230 the states that go in your sample, 36 00:01:53,230 --> 00:01:55,420 and then this is the probability correction. 37 00:01:55,420 --> 00:01:55,920 Sorry. 38 00:02:02,350 --> 00:02:04,630 And so whenever you sample with what's 39 00:02:04,630 --> 00:02:07,150 not the proper Hamiltonian of your system, 40 00:02:07,150 --> 00:02:08,965 people call it non-Boltzmann sampling. 41 00:02:15,290 --> 00:02:18,860 Why would you want to do non-Boltzmann sampling? 42 00:02:18,860 --> 00:02:24,810 Well, it can be relevant if the particular quantity you're 43 00:02:24,810 --> 00:02:28,290 after in your system is essentially not 44 00:02:28,290 --> 00:02:31,410 determined by the relevant thermodynamics states. 45 00:02:31,410 --> 00:02:35,340 Remember, I introduced this by say, 46 00:02:35,340 --> 00:02:39,030 what if you're looking at, say, average energy 47 00:02:39,030 --> 00:02:41,705 or average volume or average magnetization, 48 00:02:41,705 --> 00:02:43,080 then what you really need to know 49 00:02:43,080 --> 00:02:46,500 is what's the energy of the states 50 00:02:46,500 --> 00:02:49,470 that the system spent most of its time in? 51 00:02:49,470 --> 00:02:52,740 But what if a system spends only a small amount 52 00:02:52,740 --> 00:02:54,720 of its time in certain states, but those 53 00:02:54,720 --> 00:02:57,780 have the relevant property that you want to sample? 54 00:02:57,780 --> 00:03:03,840 For example, let's say that this is a phase space, this 55 00:03:03,840 --> 00:03:04,920 [INAUDIBLE]. 56 00:03:04,920 --> 00:03:06,990 So with importance sampling, you'd 57 00:03:06,990 --> 00:03:09,170 be drawn towards these states. 58 00:03:09,170 --> 00:03:09,870 OK? 59 00:03:09,870 --> 00:03:12,270 I don't know, maybe the ones that live here 60 00:03:12,270 --> 00:03:14,760 are optically active or something 61 00:03:14,760 --> 00:03:16,770 like that and the other ones aren't and you 62 00:03:16,770 --> 00:03:19,410 want to somehow get a lot of information 63 00:03:19,410 --> 00:03:21,630 of the optical activity of the material, 64 00:03:21,630 --> 00:03:24,570 you may want to build a Hamiltonian that drives you 65 00:03:24,570 --> 00:03:26,540 towards these states. 66 00:03:26,540 --> 00:03:27,210 OK? 67 00:03:27,210 --> 00:03:30,240 And as long as you then correct for the proper probability, 68 00:03:30,240 --> 00:03:32,610 you'll get a proper ensemble. 69 00:03:32,610 --> 00:03:35,970 It's one that's just much more efficiently biased 70 00:03:35,970 --> 00:03:38,700 towards the regional phase space where you want to get. 71 00:03:38,700 --> 00:03:41,440 So that's non-Boltzmann sampling. 72 00:03:41,440 --> 00:03:45,810 Another obvious way is you can use it to sample phase space 73 00:03:45,810 --> 00:03:48,250 more efficiently. 74 00:03:48,250 --> 00:03:56,280 If you had a phase base with a lot of local minima, 75 00:03:56,280 --> 00:04:01,260 you may want to define a new Hamiltonian that 76 00:04:01,260 --> 00:04:02,070 looks like this. 77 00:04:07,120 --> 00:04:09,550 And this is especially relevant if your Monte Carlo has 78 00:04:09,550 --> 00:04:11,290 some form of dynamics in it. 79 00:04:11,290 --> 00:04:14,230 So in the blue Hamiltonian, the true Hamiltonian, 80 00:04:14,230 --> 00:04:16,540 it may be very hard to get out of this minimum 81 00:04:16,540 --> 00:04:18,190 into the next one. 82 00:04:18,190 --> 00:04:20,649 Whereas if you raise the potential well, 83 00:04:20,649 --> 00:04:23,320 OK, in the red Hamiltonian it's going 84 00:04:23,320 --> 00:04:26,140 to be much easier to get out of it. 85 00:04:26,140 --> 00:04:26,680 OK? 86 00:04:26,680 --> 00:04:30,280 So essentially your flattening your phase space 87 00:04:30,280 --> 00:04:31,870 with the new Hamiltonian, so it's 88 00:04:31,870 --> 00:04:34,570 going to be much easier to get out of local minima 89 00:04:34,570 --> 00:04:37,640 and then you can correct for that probability. 90 00:04:37,640 --> 00:04:42,280 There's a lot of algorithms these days built on this idea, 91 00:04:42,280 --> 00:04:44,860 not just in Monte Carlo, but there 92 00:04:44,860 --> 00:04:46,815 are molecular dynamic schemes, there 93 00:04:46,815 --> 00:04:48,190 are all kinds of schemes that are 94 00:04:48,190 --> 00:04:52,420 built on this idea of lifting up the potential wells 95 00:04:52,420 --> 00:04:55,060 and then correcting the relative probability 96 00:04:55,060 --> 00:04:57,610 or the relative vibrational frequency 97 00:04:57,610 --> 00:05:00,700 or the relative time you spend in each potential well. 98 00:05:07,060 --> 00:05:09,380 OK. 99 00:05:09,380 --> 00:05:11,840 You can also do non-Metropolis Monte Carlo. 100 00:05:11,840 --> 00:05:14,330 This gets even a little more odd. 101 00:05:14,330 --> 00:05:21,200 Remember that we defined an acceptable Metropolis as one 102 00:05:21,200 --> 00:05:25,280 where the a priori probabilities were equal. 103 00:05:28,150 --> 00:05:30,160 So basically in the Markov chain, 104 00:05:30,160 --> 00:05:33,940 the rate at which I pick the i state from the j one 105 00:05:33,940 --> 00:05:37,982 as a potential next step in the Markov chain is symmetric. 106 00:05:37,982 --> 00:05:39,940 It's the same as the rate at which I pick the j 107 00:05:39,940 --> 00:05:41,365 state from the i state. 108 00:05:41,365 --> 00:05:44,290 j to i and i to j is the same. 109 00:05:44,290 --> 00:05:46,970 You don't necessarily have to do that. 110 00:05:46,970 --> 00:05:50,040 You can actually make these rates-- 111 00:05:50,040 --> 00:05:52,300 so these are the picking rates, the rates at which I 112 00:05:52,300 --> 00:05:56,890 try one state from the other, non-symmetric, and even more, 113 00:05:56,890 --> 00:06:01,030 you can make them dependent on the Hamiltonian. 114 00:06:01,030 --> 00:06:02,560 And why would you want to do that? 115 00:06:02,560 --> 00:06:04,300 I'll show you an example in a second. 116 00:06:04,300 --> 00:06:07,570 It's a way of forcing systems downhill in energy space 117 00:06:07,570 --> 00:06:09,740 much faster. 118 00:06:09,740 --> 00:06:14,110 So as long as you're correct in your detailed balance argument, 119 00:06:14,110 --> 00:06:15,250 you'll be OK. 120 00:06:15,250 --> 00:06:18,670 So the important thing is the detailed balance criteria 121 00:06:18,670 --> 00:06:21,730 because the detailed balance criteria is the one that 122 00:06:21,730 --> 00:06:26,033 ultimately ensures that you sample phase space 123 00:06:26,033 --> 00:06:27,700 or that you weigh states and phase space 124 00:06:27,700 --> 00:06:29,620 with the correct probability. 125 00:06:29,620 --> 00:06:31,790 And so as long as you correct here-- 126 00:06:31,790 --> 00:06:37,270 so if you changed the W0, you can still 127 00:06:37,270 --> 00:06:40,480 get satisfied detail balanced by essentially changing 128 00:06:40,480 --> 00:06:42,760 what the acceptance rates are. 129 00:06:42,760 --> 00:06:46,420 PI to J and PJ to I. You can write out what they are. 130 00:06:46,420 --> 00:06:50,440 Essentially the ratio of PIJ to PJI 131 00:06:50,440 --> 00:06:55,330 is the factor we had before, but now you 132 00:06:55,330 --> 00:06:58,540 correct by the relative weight, which we 133 00:06:58,540 --> 00:07:00,050 did the picking of the states. 134 00:07:00,050 --> 00:07:00,550 OK? 135 00:07:00,550 --> 00:07:04,227 So in the end it's a fairly trivial correction. 136 00:07:09,840 --> 00:07:13,380 I haven't seen too many examples of this, but one is this one. 137 00:07:13,380 --> 00:07:15,720 It's force-bias Monte Carlo, which 138 00:07:15,720 --> 00:07:18,157 almost looks like a hybrid between Monte 139 00:07:18,157 --> 00:07:19,365 Carlo and molecular dynamics. 140 00:07:23,993 --> 00:07:26,660 You know, I've given you several times this example of a liquid. 141 00:07:26,660 --> 00:07:30,190 How would you sample a liquid? 142 00:07:30,190 --> 00:07:32,980 In Monte Carlo, you just could pick random displacements 143 00:07:32,980 --> 00:07:34,435 of an atom within a given range. 144 00:07:38,020 --> 00:07:41,500 One way to actually drive the system to low energy 145 00:07:41,500 --> 00:07:45,430 faster is not take a random displacement of the atoms, 146 00:07:45,430 --> 00:07:49,260 but take one that's biased along the force on that atom. 147 00:07:49,260 --> 00:07:50,950 If you move an atom down with force, 148 00:07:50,950 --> 00:07:54,370 you're obviously going to lower the energy mathematically 149 00:07:54,370 --> 00:07:57,400 because the force is the gradient of the energy. 150 00:07:57,400 --> 00:08:03,150 So you could take a displacement factor 151 00:08:03,150 --> 00:08:06,930 that has a random component and that has some component that 152 00:08:06,930 --> 00:08:09,720 depends on the force. 153 00:08:09,720 --> 00:08:14,060 Actually if you only do this, that's 154 00:08:14,060 --> 00:08:18,492 actually how a lot of static relaxation schemes work. 155 00:08:18,492 --> 00:08:20,200 For example, in density functional theory 156 00:08:20,200 --> 00:08:21,980 you calculate the forces on atoms 157 00:08:21,980 --> 00:08:25,160 and then you relax them assuming a certain spring constant 158 00:08:25,160 --> 00:08:26,930 typically. 159 00:08:26,930 --> 00:08:29,720 But because you still have a random element, 160 00:08:29,720 --> 00:08:34,130 this is probably somewhat better described as mixed dynamics 161 00:08:34,130 --> 00:08:36,027 Monte Carlo methods. 162 00:08:36,027 --> 00:08:37,860 And if you would like to read more about it, 163 00:08:37,860 --> 00:08:39,527 I think this was one of the first papers 164 00:08:39,527 --> 00:08:41,600 that kind of introduced them. 165 00:08:46,100 --> 00:08:49,700 So what I want to do is a couple more case studies 166 00:08:49,700 --> 00:08:53,060 and then show you some of the complications that 167 00:08:53,060 --> 00:08:56,450 can arise when you do difficult systems with Monte Carlo 168 00:08:56,450 --> 00:08:59,750 and see how we solve them, and that'll lead us 169 00:08:59,750 --> 00:09:02,660 into free energy integration. 170 00:09:02,660 --> 00:09:05,990 So this is sort of a classic paper on-- 171 00:09:05,990 --> 00:09:09,530 it's from 1985-- using Monte Carlo to study surface 172 00:09:09,530 --> 00:09:12,140 segregation in copper nickel. 173 00:09:12,140 --> 00:09:14,872 It uses the embedded atom method as a potential so, 174 00:09:14,872 --> 00:09:16,580 one of the reasons I picked it because it 175 00:09:16,580 --> 00:09:19,950 integrates a lot of the things you've seen in class. 176 00:09:19,950 --> 00:09:23,660 So the idea was to take random copper nickel alloys 177 00:09:23,660 --> 00:09:26,540 and see which element segregates to the surface 178 00:09:26,540 --> 00:09:30,227 and then see how the segregation pattern away from the surface 179 00:09:30,227 --> 00:09:32,060 is, because there's been a lot of discussion 180 00:09:32,060 --> 00:09:36,570 about that in the literature, especially around that time. 181 00:09:36,570 --> 00:09:40,520 So what they do is they set up, just like you would probably, 182 00:09:40,520 --> 00:09:43,190 supercells. 183 00:09:43,190 --> 00:09:52,700 So I think they either had 24 or 24 or 48 layers, 184 00:09:52,700 --> 00:09:55,070 and then with a certain width. 185 00:09:55,070 --> 00:09:56,990 So here's your supercell. 186 00:09:59,870 --> 00:10:04,560 Again, you can define your Hamiltonian multiple ways. 187 00:10:04,560 --> 00:10:08,090 You could just seed the system with a particular concentration 188 00:10:08,090 --> 00:10:11,030 and then you have to decide which kind of Monte Carlo 189 00:10:11,030 --> 00:10:13,580 moves to allow. 190 00:10:13,580 --> 00:10:16,980 A copper nickel form [INAUDIBLE] solid solutions. 191 00:10:16,980 --> 00:10:18,590 So what kind of moves would you do 192 00:10:18,590 --> 00:10:23,620 to equilibrate the system if you had to set this up yourself? 193 00:10:30,770 --> 00:10:34,240 So again, essentially you would seed the system, probably 194 00:10:34,240 --> 00:10:41,500 with some concentration of copper and nickel 195 00:10:41,500 --> 00:10:43,740 except that there's a lot more atoms. 196 00:10:43,740 --> 00:10:46,830 So how would you do a Monte Carlo on this? 197 00:10:46,830 --> 00:10:49,550 Remember that you're going to have an embedded atom 198 00:10:49,550 --> 00:10:51,440 Hamiltonian essentially, so it's one 199 00:10:51,440 --> 00:10:53,680 that you can fairly rapidly evaluate. 200 00:10:53,680 --> 00:10:56,630 So in the end, it's a very simple energy function. 201 00:10:56,630 --> 00:10:59,450 It's a bunch of pairwise sums and then 202 00:10:59,450 --> 00:11:02,550 you just stick the result in a function and you're done. 203 00:11:02,550 --> 00:11:04,460 It's not like quantum mechanics. 204 00:11:04,460 --> 00:11:07,395 You have, essentially, a Hamiltonian you can abuse, 205 00:11:07,395 --> 00:11:09,890 so it's important to realize. 206 00:11:09,890 --> 00:11:11,580 So how would you do Monte Carlo on this? 207 00:11:18,262 --> 00:11:19,220 This is the one thing-- 208 00:11:19,220 --> 00:11:21,980 Monte Carlo is about choosing your perturbations, 209 00:11:21,980 --> 00:11:25,710 the rest is automatic after that essentially. 210 00:11:25,710 --> 00:11:29,213 It's just doing it a lot of times. 211 00:11:29,213 --> 00:11:30,880 So how would you equilibrate the system? 212 00:11:36,208 --> 00:11:38,750 It's too early in the morning, you're all still equilibrated. 213 00:11:44,140 --> 00:11:47,320 So first of all, you need to segregate some species 214 00:11:47,320 --> 00:11:49,730 to the surface assuming there will be segregation. 215 00:11:49,730 --> 00:11:51,980 So how do you do that? 216 00:11:51,980 --> 00:11:55,810 Again, you could do diffusive hops. 217 00:11:55,810 --> 00:12:00,590 Diffusive like hops, so do canonical like interchanges. 218 00:12:00,590 --> 00:12:03,400 You could interchange them nearby 219 00:12:03,400 --> 00:12:05,770 or you could interchange them far away. 220 00:12:05,770 --> 00:12:07,990 I got to stick to my colors here. 221 00:12:07,990 --> 00:12:11,920 So you could interchange positions of copper and nickel, 222 00:12:11,920 --> 00:12:14,170 but again, the slightly faster way to do it 223 00:12:14,170 --> 00:12:17,030 is to do it grand canonically, to define actually 224 00:12:17,030 --> 00:12:25,780 a Hamiltonian that's the energy minus mu copper, n copper 225 00:12:25,780 --> 00:12:31,130 minus from mu nickel, n nickel. 226 00:12:31,130 --> 00:12:31,940 OK? 227 00:12:31,940 --> 00:12:35,150 And since you're actually going to be interchanging 228 00:12:35,150 --> 00:12:40,520 the identity of atoms, n copper and n nickel 229 00:12:40,520 --> 00:12:42,320 are not independent variables. 230 00:12:42,320 --> 00:12:46,650 You keep total number of atoms constant. 231 00:12:46,650 --> 00:12:52,410 So you can actually write this as E minus mu copper minus mu 232 00:12:52,410 --> 00:12:57,830 nickle times n copper, since any change of a nickel 233 00:12:57,830 --> 00:13:00,140 is compensated by a change of copper. 234 00:13:00,140 --> 00:13:02,990 So we have essentially only one controlling chemical potential, 235 00:13:02,990 --> 00:13:06,140 which is the difference of the two species chemical 236 00:13:06,140 --> 00:13:07,170 potentials. 237 00:13:07,170 --> 00:13:08,090 OK. 238 00:13:08,090 --> 00:13:09,058 Yes, sir? 239 00:13:09,058 --> 00:13:11,498 AUDIENCE: If a [INAUDIBLE] in a specific ratio, nickel 240 00:13:11,498 --> 00:13:14,015 or copper, [INAUDIBLE] very long to find the correct value 241 00:13:14,015 --> 00:13:15,131 of [INAUDIBLE]? 242 00:13:19,280 --> 00:13:21,240 GERBRAND CEDER: Yes. 243 00:13:21,240 --> 00:13:23,490 AUDIENCE: [INAUDIBLE] 244 00:13:23,490 --> 00:13:24,570 GERBRAND CEDER: Yes. 245 00:13:24,570 --> 00:13:30,920 Yes, so you're controlling the external chemical potential, 246 00:13:30,920 --> 00:13:32,770 so you don't control the composition. 247 00:13:32,770 --> 00:13:33,270 True. 248 00:13:36,135 --> 00:13:39,900 What people sometimes do is that-- 249 00:13:39,900 --> 00:13:45,150 what's of interest to you is the bulk chemical potential, that's 250 00:13:45,150 --> 00:13:47,250 the source for the surface. 251 00:13:47,250 --> 00:13:49,500 You first do bulk simulations and then 252 00:13:49,500 --> 00:13:52,710 you roughly know what chemical potential corresponds 253 00:13:52,710 --> 00:13:57,540 to what compositions, because it's actually a little tricky. 254 00:13:57,540 --> 00:14:00,990 While energetically often, the center of the system 255 00:14:00,990 --> 00:14:02,610 does not influence the surface much 256 00:14:02,610 --> 00:14:05,220 so you think you have convergence. 257 00:14:05,220 --> 00:14:07,620 You still deplete it as a reservoir 258 00:14:07,620 --> 00:14:09,870 as you segregate one species to the surface, 259 00:14:09,870 --> 00:14:11,370 so that's sometimes why you actually 260 00:14:11,370 --> 00:14:13,230 need more bulk in the system. 261 00:14:13,230 --> 00:14:15,480 But yes, what you say is true. 262 00:14:15,480 --> 00:14:17,730 If you're only interested in one composition 263 00:14:17,730 --> 00:14:20,540 it can be better to stick there with that. 264 00:14:20,540 --> 00:14:21,120 OK. 265 00:14:21,120 --> 00:14:22,875 But let's say you do these exchange moves. 266 00:14:26,000 --> 00:14:27,380 Do you do it on a lattice? 267 00:14:27,380 --> 00:14:30,233 Do you do it in sort of a continuous space? 268 00:14:30,233 --> 00:14:31,400 I mean, how would you do it? 269 00:14:34,630 --> 00:14:35,870 This is an FCC solid. 270 00:14:35,870 --> 00:14:38,540 This is an FCC solid solution. 271 00:14:38,540 --> 00:14:42,585 So you could if you want to do it on a lattice, but would you? 272 00:14:42,585 --> 00:14:44,210 So you know what I mean with a lattice? 273 00:14:44,210 --> 00:14:46,910 You have pre-defined positions where the atoms can sit. 274 00:14:54,512 --> 00:14:56,220 Somebody has to be able to say something. 275 00:14:56,220 --> 00:14:59,200 I'm just going to stand here. 276 00:14:59,200 --> 00:15:02,512 And it's not good because you're being taped for OCW, 277 00:15:02,512 --> 00:15:03,970 so they're going to think these MIT 278 00:15:03,970 --> 00:15:05,230 students have nothing to say. 279 00:15:08,830 --> 00:15:10,600 Somebody in Western Africa sees this 280 00:15:10,600 --> 00:15:12,650 they're going to go like, why did they 281 00:15:12,650 --> 00:15:13,940 let these people in at MIT? 282 00:15:17,930 --> 00:15:19,883 So what else would you add? 283 00:15:19,883 --> 00:15:21,800 First of all, if you do it on a fixed lattice, 284 00:15:21,800 --> 00:15:23,990 think of what you end up with thermodynamically? 285 00:15:23,990 --> 00:15:25,970 What have you sampled? 286 00:15:25,970 --> 00:15:27,470 If you're on a fixed lattice, you've 287 00:15:27,470 --> 00:15:31,850 sampled the configurational entropy, essentially 288 00:15:31,850 --> 00:15:34,010 the possible distributions of copper and nickel 289 00:15:34,010 --> 00:15:36,770 on these fixed lattice sites. 290 00:15:36,770 --> 00:15:39,140 But if one atom is bigger than the other, 291 00:15:39,140 --> 00:15:43,130 for example, then they don't sit on a fixed lattice, 292 00:15:43,130 --> 00:15:46,520 then atoms will relax away from lattice positions. 293 00:15:46,520 --> 00:15:49,760 You will also not sample vibrational excitation 294 00:15:49,760 --> 00:15:52,790 because you never go away from your lattice position. 295 00:15:52,790 --> 00:15:56,140 So you could add small displacements 296 00:15:56,140 --> 00:15:57,560 to the perturbations. 297 00:15:57,560 --> 00:15:59,410 So remember, you do the sort of big 298 00:15:59,410 --> 00:16:02,350 into the chemical interchange as you interchange copper 299 00:16:02,350 --> 00:16:05,200 to nickel, but then to that you could add 300 00:16:05,200 --> 00:16:08,590 small displacements of the atoms so you not sample deviations 301 00:16:08,590 --> 00:16:10,780 from their equilibrium positions. 302 00:16:10,780 --> 00:16:12,430 And if you're done them, then you'll 303 00:16:12,430 --> 00:16:16,360 have both configurational and vibrational entropy. 304 00:16:16,360 --> 00:16:21,440 So remember that by the possible perturbations 305 00:16:21,440 --> 00:16:24,740 you pick for your system, you are essentially defining 306 00:16:24,740 --> 00:16:27,220 the phase space you sample. 307 00:16:27,220 --> 00:16:30,920 So you're defining this phase space you integrate over 308 00:16:30,920 --> 00:16:34,570 and that's essentially telling you what forms of entropy 309 00:16:34,570 --> 00:16:37,950 you allow in your system. 310 00:16:37,950 --> 00:16:40,710 Because, you know, what forms of entropy is the same as what 311 00:16:40,710 --> 00:16:45,090 forms of disorder or excitations I allow in your system. 312 00:16:45,090 --> 00:16:45,900 OK? 313 00:16:45,900 --> 00:16:48,960 So the cool thing about it is that you 314 00:16:48,960 --> 00:16:52,025 can look at the effects of both, and I'll show that in a bit. 315 00:16:52,025 --> 00:16:53,400 You could actually say, first I'm 316 00:16:53,400 --> 00:16:56,285 going to do it on a rigid lattice 317 00:16:56,285 --> 00:16:57,660 and I'll see the result, and then 318 00:16:57,660 --> 00:17:01,212 I'll also include displacement away from the rigid lattice 319 00:17:01,212 --> 00:17:02,670 and you can look at the difference, 320 00:17:02,670 --> 00:17:04,829 and that's essentially telling you 321 00:17:04,829 --> 00:17:07,680 what the effect of vibrational excitations 322 00:17:07,680 --> 00:17:11,069 are on your thermodynamics. 323 00:17:11,069 --> 00:17:25,079 OK, so here's the result. You know this was 1985? 324 00:17:25,079 --> 00:17:27,660 You'd be amazed how easy this is to do now 325 00:17:27,660 --> 00:17:29,850 if you wanted to do something like this yourself. 326 00:17:29,850 --> 00:17:33,270 With a Hamiltonian-like embedded atom it really goes very fast 327 00:17:33,270 --> 00:17:36,580 and you could equilibrate systems like this in no time. 328 00:17:36,580 --> 00:17:38,070 But here's the result. So this is 329 00:17:38,070 --> 00:17:42,690 a function of the bulk concentration of copper. 330 00:17:42,690 --> 00:17:45,870 This is the amount of copper in the first 331 00:17:45,870 --> 00:17:47,380 and second and third layer. 332 00:17:47,380 --> 00:17:49,570 So the green curve is the first layer. 333 00:17:49,570 --> 00:17:53,230 So clearly, copper segregates to the surface in this material. 334 00:17:53,230 --> 00:17:58,050 So as you can see, even for 20% copper in the bulk 335 00:17:58,050 --> 00:18:01,440 you have an almost perfect copper layer on the surface. 336 00:18:01,440 --> 00:18:04,830 I think this was actually a 111 surface. 337 00:18:04,830 --> 00:18:07,665 Copper in the second layer, interestingly, is depleted. 338 00:18:11,960 --> 00:18:16,070 So it's actually below the average bulk concentration. 339 00:18:16,070 --> 00:18:18,350 And then copper in the third layer 340 00:18:18,350 --> 00:18:22,890 essentially starts tracking the bulk concentration. 341 00:18:22,890 --> 00:18:29,750 So why do you think the copper in the second layer 342 00:18:29,750 --> 00:18:30,530 is depleted? 343 00:18:37,130 --> 00:18:40,490 Because first of all, why does copper go to the surface? 344 00:18:40,490 --> 00:18:45,277 It has a lower surface energy, so this is not rocket science. 345 00:18:45,277 --> 00:18:47,610 Most of the time the thing with the lower surface energy 346 00:18:47,610 --> 00:18:50,040 goes to the surface. 347 00:18:50,040 --> 00:18:50,940 OK? 348 00:18:50,940 --> 00:18:53,460 But why is depleted in the second layer? 349 00:18:53,460 --> 00:18:55,440 Because I sort of think, well, this 350 00:18:55,440 --> 00:18:56,790 is the lower surface energy. 351 00:18:56,790 --> 00:18:59,820 Even in the second layer it's not perfectly bonded, 352 00:18:59,820 --> 00:19:03,300 so maybe it still wants to enhance its concentration there 353 00:19:03,300 --> 00:19:05,290 but actually depletes it. 354 00:19:05,290 --> 00:19:10,600 This is not atypical, it's very common surface concentration. 355 00:19:13,780 --> 00:19:16,150 It's actually because copper and nickel in this system 356 00:19:16,150 --> 00:19:18,360 have an ordering interaction. 357 00:19:18,360 --> 00:19:24,260 So it's the fact that the first layer is almost pure copper, 358 00:19:24,260 --> 00:19:27,050 but the second layer, for chemical reasons, 359 00:19:27,050 --> 00:19:29,500 really wants to be nickel. 360 00:19:29,500 --> 00:19:30,430 OK? 361 00:19:30,430 --> 00:19:31,960 And this is a pure surface effect. 362 00:19:31,960 --> 00:19:33,970 Copper and nickel in the bulk are actually 363 00:19:33,970 --> 00:19:36,130 at low temperature phase separating, 364 00:19:36,130 --> 00:19:38,080 but in the surface because of strain effects 365 00:19:38,080 --> 00:19:40,180 they form an ordering interaction. 366 00:19:40,180 --> 00:19:42,370 And so the second layer wants to be nickel 367 00:19:42,370 --> 00:19:44,980 because the first layer is so much copper. 368 00:19:44,980 --> 00:19:49,060 And so you often see this in compound forming systems 369 00:19:49,060 --> 00:19:52,840 that you have damped oscillations 370 00:19:52,840 --> 00:19:55,240 in the concentration away from the surface. 371 00:20:05,070 --> 00:20:07,445 OK. 372 00:20:07,445 --> 00:20:09,570 So I want to show you one example of something that 373 00:20:09,570 --> 00:20:12,630 gets much more complicated, and you'll 374 00:20:12,630 --> 00:20:16,620 see how you start running into problems with Monte Carlo. 375 00:20:16,620 --> 00:20:20,460 So I've shown you a little bit how you would detect phase 376 00:20:20,460 --> 00:20:22,230 transitions in Monte Carlo. 377 00:20:22,230 --> 00:20:24,270 You'd look at things like the heat capacity 378 00:20:24,270 --> 00:20:26,730 especially, particularly powerful for detecting 379 00:20:26,730 --> 00:20:28,350 second order transitions. 380 00:20:28,350 --> 00:20:32,280 For first order transitions, you look at discontinuities 381 00:20:32,280 --> 00:20:34,950 in things like the energy, things 382 00:20:34,950 --> 00:20:37,650 like the relation between mu and see chemical potential 383 00:20:37,650 --> 00:20:40,650 and concentration, and you'll pick that up. 384 00:20:40,650 --> 00:20:43,770 The problem you'll face is often that Monte Carlo systems, 385 00:20:43,770 --> 00:20:48,660 just like real systems, often face significant hysteresis. 386 00:20:48,660 --> 00:20:52,770 So here's an example from quite a few years ago 387 00:20:52,770 --> 00:20:55,470 of a fairly complicated lattice model Hamiltonian. 388 00:20:55,470 --> 00:20:57,750 This, again, was on an FCC lattice. 389 00:20:57,750 --> 00:20:59,820 This is the palladium vanadium system, OK? 390 00:20:59,820 --> 00:21:00,450 Phase diagram. 391 00:21:00,450 --> 00:21:04,440 This is palladium on this side, this side is vanadium, 392 00:21:04,440 --> 00:21:08,220 and this is a lattice model with a bunch of interactions 393 00:21:08,220 --> 00:21:09,420 in the Hamiltonian. 394 00:21:09,420 --> 00:21:12,150 A nearest neighbor pair, second neighbor pair, 395 00:21:12,150 --> 00:21:13,680 third neighbor pair, and so on. 396 00:21:13,680 --> 00:21:18,130 Four body interactions, four body here, three body here. 397 00:21:18,130 --> 00:21:19,800 It's a fairly complicated Hamiltonian 398 00:21:19,800 --> 00:21:22,860 that tends to give you a lot of local minima 399 00:21:22,860 --> 00:21:24,660 and how would you get this phase diagram? 400 00:21:24,660 --> 00:21:26,670 Well, what would you do? 401 00:21:26,670 --> 00:21:30,960 Again, to equilibrate faster you wouldn't keep the concentration 402 00:21:30,960 --> 00:21:33,420 fixed, you would scan the chemical potential 403 00:21:33,420 --> 00:21:38,380 and evolve in concentration from one end to the other. 404 00:21:38,380 --> 00:21:38,880 OK? 405 00:21:38,880 --> 00:21:40,630 Let me actually do that on the next slide. 406 00:21:45,412 --> 00:21:46,870 So you have some chemical potential 407 00:21:46,870 --> 00:21:50,080 that gives you the difference in palladium and vanadium 408 00:21:50,080 --> 00:21:51,610 amounts or the concentration. 409 00:21:55,160 --> 00:21:55,970 OK. 410 00:21:55,970 --> 00:21:58,190 And so you would plot something like-- 411 00:22:01,395 --> 00:22:03,420 so you would have a driving chemical potential 412 00:22:03,420 --> 00:22:04,920 and you would plot the average spin, 413 00:22:04,920 --> 00:22:06,587 we'd say is a concentration of vanadium. 414 00:22:11,660 --> 00:22:14,040 And let's say you scan at this temperature, 415 00:22:14,040 --> 00:22:15,800 so what would you see? 416 00:22:15,800 --> 00:22:18,890 If you start here, you would go through a solid solution 417 00:22:18,890 --> 00:22:20,150 regime. 418 00:22:20,150 --> 00:22:24,350 So you would see your concentration go up when 419 00:22:24,350 --> 00:22:28,070 you hit the two phase region. 420 00:22:28,070 --> 00:22:28,570 OK? 421 00:22:28,570 --> 00:22:30,278 The chemical potential is constant there, 422 00:22:30,278 --> 00:22:32,530 so that means at that chemical potential you 423 00:22:32,530 --> 00:22:36,170 have a discontinuity in the concentration, 424 00:22:36,170 --> 00:22:37,400 so this would go straight up. 425 00:22:40,120 --> 00:22:42,240 Then you would go in a single phase region. 426 00:22:45,580 --> 00:22:46,360 OK? 427 00:22:46,360 --> 00:22:49,180 So in a single phase region, the chemical potential 428 00:22:49,180 --> 00:22:51,860 changes rapidly with concentration. 429 00:22:51,860 --> 00:22:56,090 So there, you actually tend to be kind of flat. 430 00:22:56,090 --> 00:22:58,130 It's not quite flat, but it tends 431 00:22:58,130 --> 00:23:01,280 to just be sloped a lot less. 432 00:23:01,280 --> 00:23:04,460 Then you would, again, go in to a two phase region. 433 00:23:04,460 --> 00:23:08,420 So you would, again, form a step. 434 00:23:08,420 --> 00:23:12,332 You would go into a single phase. 435 00:23:12,332 --> 00:23:15,100 So we basically see this kind of behavior, 436 00:23:15,100 --> 00:23:18,010 and where you have these discontinuities 437 00:23:18,010 --> 00:23:21,730 in concentration, you would know that there's a first order 438 00:23:21,730 --> 00:23:23,650 transition. 439 00:23:23,650 --> 00:23:26,440 You could also plot the energy, and the energy 440 00:23:26,440 --> 00:23:28,480 is discontinuous as well. 441 00:23:28,480 --> 00:23:31,570 The internal energy system is discontinuous at a first order 442 00:23:31,570 --> 00:23:33,680 transition. 443 00:23:33,680 --> 00:23:36,610 You don't necessarily see things in the heat capacity. 444 00:23:36,610 --> 00:23:37,940 Now what happens in reality? 445 00:23:40,490 --> 00:23:44,350 If you think about it, if you come from this end 446 00:23:44,350 --> 00:23:46,450 so you have a solid solution, you 447 00:23:46,450 --> 00:23:47,950 go through two phase region, and you 448 00:23:47,950 --> 00:23:49,810 have to form this compound, which 449 00:23:49,810 --> 00:23:51,580 is called nickel [INAUDIBLE]. 450 00:23:51,580 --> 00:23:55,070 So often in Monte Carlo, just like in a real system, 451 00:23:55,070 --> 00:23:56,770 you have nucleation problems. 452 00:23:56,770 --> 00:24:00,010 If you have to form a phase or an arrangement that's 453 00:24:00,010 --> 00:24:03,340 extremely different from the host from which it forms, 454 00:24:03,340 --> 00:24:05,950 it will often just not nucleate, and well, what happens 455 00:24:05,950 --> 00:24:08,980 is that you will overshoot. 456 00:24:08,980 --> 00:24:13,690 So you will actually push the disordered phase way too far 457 00:24:13,690 --> 00:24:16,330 out of equilibrium and then at some point, 458 00:24:16,330 --> 00:24:19,060 you will be so far out of equilibrium 459 00:24:19,060 --> 00:24:22,450 that you literally shoot into the ordered phase. 460 00:24:22,450 --> 00:24:25,630 And then what happens when you come back? 461 00:24:25,630 --> 00:24:27,270 You have hysteresis the other way. 462 00:24:29,880 --> 00:24:32,520 So you'll overshoot the order phase sometimes and disorder, 463 00:24:32,520 --> 00:24:34,900 although disordering is a little easier to do. 464 00:24:34,900 --> 00:24:37,860 So just like in real systems at strong first order conditions 465 00:24:37,860 --> 00:24:44,060 you'll see a lot of hysteresis, and it's often very hard 466 00:24:44,060 --> 00:24:46,950 to get rid of. 467 00:24:46,950 --> 00:24:48,950 I think some of you may have noticed that if you 468 00:24:48,950 --> 00:24:51,560 did the molecular dynamics lab. 469 00:24:51,560 --> 00:24:56,270 Seeing first order transitions in a molecular dynamics lab, 470 00:24:56,270 --> 00:24:58,830 you also are often plagued with hysteresis. 471 00:24:58,830 --> 00:25:01,940 I mean, if you study melting, for example, 472 00:25:01,940 --> 00:25:03,830 it's very asymmetrical hysteresis. 473 00:25:03,830 --> 00:25:07,280 If you heat up a solid, it's pretty easy to melt it. 474 00:25:07,280 --> 00:25:10,370 If you cool down a liquid, it's about impossible 475 00:25:10,370 --> 00:25:13,670 to nucleate the solid unless you force it. 476 00:25:13,670 --> 00:25:17,360 So there you have enormous amounts of hysteresis. 477 00:25:17,360 --> 00:25:22,370 So the point of this is that to study strong first order 478 00:25:22,370 --> 00:25:25,070 transitions, sometimes the best way 479 00:25:25,070 --> 00:25:28,940 to find their transition temperatures or concentrations 480 00:25:28,940 --> 00:25:31,370 is not by direct simulation. 481 00:25:31,370 --> 00:25:33,880 Essentially, what you need to do is go 482 00:25:33,880 --> 00:25:35,130 through a thermodynamic route. 483 00:25:35,130 --> 00:25:39,718 Try to extract free energies and find where they cross, 484 00:25:39,718 --> 00:25:41,510 because first order transitions are defined 485 00:25:41,510 --> 00:25:45,830 by where free energies of the two phases intersect, 486 00:25:45,830 --> 00:25:48,650 and that's by far the most accurate way of detecting 487 00:25:48,650 --> 00:25:50,870 phase transitions. 488 00:25:50,870 --> 00:25:53,930 There's a real problem here is that, how 489 00:25:53,930 --> 00:25:55,130 do you get free energy? 490 00:26:00,320 --> 00:26:03,920 The fundamental difference between free energy and energy 491 00:26:03,920 --> 00:26:06,920 is that energy is an average quantity 492 00:26:06,920 --> 00:26:08,930 and free energy is not. 493 00:26:08,930 --> 00:26:11,660 See, energy is the average of something 494 00:26:11,660 --> 00:26:15,200 that is defined in the microscopic states. 495 00:26:15,200 --> 00:26:17,660 For every microscopic state I go through, 496 00:26:17,660 --> 00:26:21,087 I can define an energy, and then the internal energy, 497 00:26:21,087 --> 00:26:22,670 the thermodynamic energy of the system 498 00:26:22,670 --> 00:26:25,370 is just the average of that quantity. 499 00:26:25,370 --> 00:26:27,590 For free energy or entropy-- 500 00:26:27,590 --> 00:26:30,980 which are essentially the same because if I know the entropy, 501 00:26:30,980 --> 00:26:33,080 I know the free energy-- 502 00:26:33,080 --> 00:26:36,350 it is not the average of a quantity that's defined 503 00:26:36,350 --> 00:26:38,235 in the microscopic state. 504 00:26:38,235 --> 00:26:38,735 You know? 505 00:26:38,735 --> 00:26:41,600 If you put the atoms in a fixed position 506 00:26:41,600 --> 00:26:46,940 somewhere with some velocity, that's a microscopic state. 507 00:26:46,940 --> 00:26:48,980 You cannot define the entropy of that. 508 00:26:48,980 --> 00:26:52,310 The entropy is a property of the ensemble as a whole, 509 00:26:52,310 --> 00:26:54,660 not of a given macroscopic state. 510 00:26:54,660 --> 00:26:59,120 So entropy and free energy cannot be obtained as averages. 511 00:26:59,120 --> 00:27:01,730 They're actually integrals. 512 00:27:01,730 --> 00:27:07,730 You can see that the entropy is a sum over all phase 513 00:27:07,730 --> 00:27:12,340 space of this quantity, P log P. You can write 514 00:27:12,340 --> 00:27:17,850 the free energy same way as an integrated quantity over all 515 00:27:17,850 --> 00:27:21,570 of phase space of this quantity. 516 00:27:21,570 --> 00:27:24,540 If you rewrite it, you sort of see it better. 517 00:27:24,540 --> 00:27:26,820 You are averaging something. 518 00:27:26,820 --> 00:27:29,700 If you look at this, you are averaging something 519 00:27:29,700 --> 00:27:31,230 because you have a probability here, 520 00:27:31,230 --> 00:27:34,000 but what's the quantity you're averaging? 521 00:27:34,000 --> 00:27:36,670 It's essentially the free energy itself. 522 00:27:36,670 --> 00:27:40,460 So it's a quantity that's flat in phase space. 523 00:27:40,460 --> 00:27:41,250 OK? 524 00:27:41,250 --> 00:27:46,020 So that makes it extremely difficult to sample. 525 00:27:46,020 --> 00:27:49,095 Actually this is not a mathematical problem, 526 00:27:49,095 --> 00:27:50,220 this is a physical problem. 527 00:27:50,220 --> 00:27:52,920 It's just the same as what happens in nature. 528 00:27:52,920 --> 00:27:55,530 You can measure energy, you can measure volume, 529 00:27:55,530 --> 00:27:57,120 you cannot measure free energy. 530 00:27:57,120 --> 00:28:00,900 There are no free energy meters because it's 531 00:28:00,900 --> 00:28:04,230 an extensive quantity that that is determined 532 00:28:04,230 --> 00:28:05,650 by the whole ensemble. 533 00:28:05,650 --> 00:28:08,642 You only really get free energies ever indirectly. 534 00:28:08,642 --> 00:28:10,100 You can get free energy difference, 535 00:28:10,100 --> 00:28:13,320 but you get free energy by integrating lower order 536 00:28:13,320 --> 00:28:14,040 quantities. 537 00:28:21,400 --> 00:28:27,430 OK, so you can write it as an average 538 00:28:27,430 --> 00:28:30,130 but it's a little misleading. 539 00:28:30,130 --> 00:28:35,620 You can actually write it as the average of this thing here. 540 00:28:35,620 --> 00:28:39,710 If you can calculate the exponential 541 00:28:39,710 --> 00:28:42,303 of beta the Hamiltonian and average that over phase space-- 542 00:28:42,303 --> 00:28:43,970 and notice I didn't make an error there. 543 00:28:43,970 --> 00:28:45,980 I did not drop a minus sign. 544 00:28:45,980 --> 00:28:48,320 It's the exponential of the positive beta 545 00:28:48,320 --> 00:28:50,000 times the Hamiltonian. 546 00:28:50,000 --> 00:28:51,650 If you can average that quantity, 547 00:28:51,650 --> 00:28:54,770 you can show that you can actually have the free energy. 548 00:28:54,770 --> 00:28:56,390 And the proof is given here, it's 549 00:28:56,390 --> 00:28:59,870 sort of an almost trivial proof. 550 00:28:59,870 --> 00:29:03,050 But you see that's kind of problematic. 551 00:29:03,050 --> 00:29:06,110 The Hamiltonian is an extensive quantity, 552 00:29:06,110 --> 00:29:09,420 so it scales with the size of the system. 553 00:29:09,420 --> 00:29:13,790 So you're taking the exponential of something that's extensive, 554 00:29:13,790 --> 00:29:15,477 so that gets very big. 555 00:29:15,477 --> 00:29:17,810 So first of all, you can only do this for finite systems 556 00:29:17,810 --> 00:29:20,480 and for systems that are really, really small. 557 00:29:20,480 --> 00:29:23,810 Because otherwise, essentially, that quantity, 558 00:29:23,810 --> 00:29:26,660 the exponential of beta the Hamiltonian, the difference 559 00:29:26,660 --> 00:29:31,670 between two states becomes excessively large 560 00:29:31,670 --> 00:29:34,490 as your system size gets bigger because that quantity 561 00:29:34,490 --> 00:29:37,250 in the exponential is extensive. 562 00:29:37,250 --> 00:29:38,010 OK? 563 00:29:38,010 --> 00:29:41,570 You see, if I'm calculating the Hamiltonian difference, say, 564 00:29:41,570 --> 00:29:45,800 between two phases, the Hamiltonian value goes in there 565 00:29:45,800 --> 00:29:47,570 as the extensive quantity. 566 00:29:47,570 --> 00:29:49,670 It's not the normalized one. 567 00:29:49,670 --> 00:29:53,160 It's not the one, say, per unit cell or something like that. 568 00:29:53,160 --> 00:29:56,120 So that energy difference is infinite 569 00:29:56,120 --> 00:29:58,370 in the extensive limit, OK? 570 00:29:58,370 --> 00:30:03,020 So even if two phases only are one joule a part per molecule, 571 00:30:03,020 --> 00:30:05,450 per mol set, one joule per mol. 572 00:30:05,450 --> 00:30:09,040 In the extensive limit, they're still an infinite energy apart. 573 00:30:09,040 --> 00:30:10,040 OK? 574 00:30:10,040 --> 00:30:15,250 So you can't practically actually sample that, so 575 00:30:15,250 --> 00:30:17,080 how do people get free energies? 576 00:30:17,080 --> 00:30:21,790 Well, there are hundreds of papers 577 00:30:21,790 --> 00:30:23,943 on free energy integration and the reason 578 00:30:23,943 --> 00:30:25,360 that there's hundreds of papers is 579 00:30:25,360 --> 00:30:28,990 that it's such a difficult thing and everybody claims to have 580 00:30:28,990 --> 00:30:32,810 the magic potion to do it. 581 00:30:32,810 --> 00:30:35,590 The first thing to realize is that you almost never need 582 00:30:35,590 --> 00:30:38,170 free energy, you always need free energy differences 583 00:30:38,170 --> 00:30:41,290 between two things, and that's a powerful statement 584 00:30:41,290 --> 00:30:44,098 because that's a lot easier to do as you'll see. 585 00:30:44,098 --> 00:30:45,640 The second thing is that you probably 586 00:30:45,640 --> 00:30:49,240 shouldn't believe half of what you read in papers. 587 00:30:49,240 --> 00:30:51,340 I used to track that field and everybody said, 588 00:30:51,340 --> 00:30:55,390 oh, I have a great method to get free energies out 589 00:30:55,390 --> 00:30:58,090 of a single simulation, because that's sort of the problem. 590 00:30:58,090 --> 00:31:00,490 You'll see in a second that the way we get free energy 591 00:31:00,490 --> 00:31:02,513 is that we have to do Monte Carlo 592 00:31:02,513 --> 00:31:04,930 at a lot of different points to get the free energy at one 593 00:31:04,930 --> 00:31:07,060 point. 594 00:31:07,060 --> 00:31:09,190 There's tons of papers that write that you 595 00:31:09,190 --> 00:31:10,730 can do it with one simulation. 596 00:31:10,730 --> 00:31:13,450 Well, either implicitly they do a lot of simulations 597 00:31:13,450 --> 00:31:16,570 within that one and just call it one, 598 00:31:16,570 --> 00:31:19,060 or you can do it in limited cases 599 00:31:19,060 --> 00:31:23,090 if you know a lot about the form of your phase space. 600 00:31:23,090 --> 00:31:26,080 So if you have extremely simple Hamiltonians-- 601 00:31:26,080 --> 00:31:30,680 if I give you a nearest neighbor Ising model, the magnetic model 602 00:31:30,680 --> 00:31:32,830 which is the nearest neighbor interaction, 603 00:31:32,830 --> 00:31:35,680 essentially the amount of excitations 604 00:31:35,680 --> 00:31:38,470 out of the low energy states there is very finite. 605 00:31:38,470 --> 00:31:40,450 Like I said, you could flip one isolate, spin, 606 00:31:40,450 --> 00:31:42,700 you get a times j. 607 00:31:42,700 --> 00:31:44,822 So you could almost numerically start writing out 608 00:31:44,822 --> 00:31:46,030 what the free energy becomes. 609 00:31:46,030 --> 00:31:50,925 So in very simple models, if you know the form of the phase 610 00:31:50,925 --> 00:31:53,980 space, you can actually get towards free energy models. 611 00:31:53,980 --> 00:31:57,280 In general, it's pretty much an unsolved problem. 612 00:31:57,280 --> 00:31:58,900 And the way we practically get it 613 00:31:58,900 --> 00:32:01,490 is with three types of methods. 614 00:32:01,490 --> 00:32:05,010 One is free energy integration, and I put lambda integration 615 00:32:05,010 --> 00:32:05,510 under there. 616 00:32:05,510 --> 00:32:08,970 I'll show in a second what it is. 617 00:32:08,970 --> 00:32:11,130 And the second one is overlapping distribution 618 00:32:11,130 --> 00:32:14,070 methods, which is slightly less important, especially 619 00:32:14,070 --> 00:32:15,480 for solids. 620 00:32:15,480 --> 00:32:20,940 So it's really only 2 previous others, which I used to cover 621 00:32:20,940 --> 00:32:22,650 and I don't even do anymore now. 622 00:32:22,650 --> 00:32:23,640 OK. 623 00:32:23,640 --> 00:32:25,770 Let me first show you overlapping distribution 624 00:32:25,770 --> 00:32:30,180 methods, which I'm less familiar with because I never use it, 625 00:32:30,180 --> 00:32:33,580 but the idea of it is quite simple. 626 00:32:33,580 --> 00:32:36,660 So if you want to know the free energy difference between two 627 00:32:36,660 --> 00:32:39,300 states, remember that the free energy is KT 628 00:32:39,300 --> 00:32:41,730 log the partition function q. 629 00:32:41,730 --> 00:32:45,120 So the delta, the difference is the log 630 00:32:45,120 --> 00:32:48,020 of the ratio of the partition function, OK? 631 00:32:48,020 --> 00:32:49,340 You are with us? 632 00:32:49,340 --> 00:32:51,010 So you can write out what that is. 633 00:32:51,010 --> 00:32:54,750 The partition function is the sum 634 00:32:54,750 --> 00:32:57,020 over all the states of the exponential 635 00:32:57,020 --> 00:32:59,760 of minus beta the Hamiltonian. 636 00:32:59,760 --> 00:33:02,220 If I multiply this by 1-- and I'm 637 00:33:02,220 --> 00:33:04,980 going to multiply this by 1, then I'm 638 00:33:04,980 --> 00:33:11,190 going to write 1 as the exponential of beta H1. 639 00:33:11,190 --> 00:33:12,040 Sorry. 640 00:33:12,040 --> 00:33:13,260 Got to be consistent here. 641 00:33:13,260 --> 00:33:20,460 H1 nu times exponential minus beta H1. 642 00:33:20,460 --> 00:33:23,400 OK, so that's one. 643 00:33:23,400 --> 00:33:26,550 So then I can collect the terms here, and what I get 644 00:33:26,550 --> 00:33:29,100 is that I get I still sum over all the states. 645 00:33:29,100 --> 00:33:33,180 I get the exponential, the Hamiltonian difference 646 00:33:33,180 --> 00:33:37,050 to n1 weighted by the probability 647 00:33:37,050 --> 00:33:41,100 of this state in the ensemble of Hamiltonian one. 648 00:33:41,100 --> 00:33:43,830 This is essentially the probability 649 00:33:43,830 --> 00:33:46,800 of that state is the exponential minus beta H 650 00:33:46,800 --> 00:33:51,000 over the partition function taken with Hamiltonian 1. 651 00:33:51,000 --> 00:33:52,740 So what have I written here? 652 00:33:52,740 --> 00:33:56,340 The exponential of the probability of that state 653 00:33:56,340 --> 00:34:01,540 weighted in Hamiltonian 1 of this quantity. 654 00:34:01,540 --> 00:34:03,960 So essentially, what I'm averaging 655 00:34:03,960 --> 00:34:07,650 is the exponential of minus beta the Hamiltonian difference 656 00:34:07,650 --> 00:34:10,469 between state two and one, but I average it 657 00:34:10,469 --> 00:34:13,800 in the ensemble of one and that gives me the free energy 658 00:34:13,800 --> 00:34:15,960 difference. 659 00:34:15,960 --> 00:34:22,130 The reason that's called overlapping distribution 660 00:34:22,130 --> 00:34:24,250 methods is that-- 661 00:34:24,250 --> 00:34:27,100 let me show you that in a second-- 662 00:34:27,100 --> 00:34:30,670 you're trying to say something about state two, 663 00:34:30,670 --> 00:34:34,570 but you're sampling in the ensemble of one. 664 00:34:34,570 --> 00:34:38,980 So the only way this is ever going to work 665 00:34:38,980 --> 00:34:43,300 is if one and 2 are not too far apart so that the states that 666 00:34:43,300 --> 00:34:47,920 are relevant for two are also sampled to some extent when 667 00:34:47,920 --> 00:34:50,739 you're in one, and that's why it's called overlapping 668 00:34:50,739 --> 00:34:51,980 distribution method. 669 00:34:51,980 --> 00:34:57,280 So if I sample in one, essentially 670 00:34:57,280 --> 00:34:58,330 let's look at the energy. 671 00:34:58,330 --> 00:35:02,875 I sample energy states around the average with some spread, 672 00:35:02,875 --> 00:35:05,170 spread could be the heat capacity. 673 00:35:05,170 --> 00:35:06,940 If I'm in two I do the same thing 674 00:35:06,940 --> 00:35:09,870 around the average energy of two. 675 00:35:09,870 --> 00:35:11,580 And essentially what I'm doing is 676 00:35:11,580 --> 00:35:15,900 I'm integrating things for ensemble two 677 00:35:15,900 --> 00:35:20,690 just by the way I walk through ensemble one. 678 00:35:20,690 --> 00:35:25,040 And so it's a relatively elegant way of doing it, 679 00:35:25,040 --> 00:35:28,670 and the key aspect is that the states in the two ensembles 680 00:35:28,670 --> 00:35:29,270 are the same. 681 00:35:32,210 --> 00:35:35,440 The ensembles are the same, so the accessible states 682 00:35:35,440 --> 00:35:38,390 are the same, it's just that you weigh them differently. 683 00:35:38,390 --> 00:35:42,220 So this is almost just like non-Boltzmann sampling. 684 00:35:42,220 --> 00:35:45,250 I Boltzmann sample for ensemble one, 685 00:35:45,250 --> 00:35:47,350 but you could say I non-Boltzmann sample 686 00:35:47,350 --> 00:35:50,920 for ensemble two and I correct the probability and this is 687 00:35:50,920 --> 00:35:52,240 how I get the free energy. 688 00:35:55,180 --> 00:35:57,430 So you can already see when this is not going to work. 689 00:35:57,430 --> 00:36:00,910 This is not going to work when your states are too far apart. 690 00:36:04,680 --> 00:36:06,810 OK. 691 00:36:06,810 --> 00:36:10,200 By far the most used method to get 692 00:36:10,200 --> 00:36:12,630 free energy is sort of a trivial one in some sense, 693 00:36:12,630 --> 00:36:16,100 it's free energy integration. 694 00:36:16,100 --> 00:36:19,120 But it's the one that always works 695 00:36:19,120 --> 00:36:20,790 if you put enough time in it and it 696 00:36:20,790 --> 00:36:23,520 starts from this kind of trivial idea 697 00:36:23,520 --> 00:36:26,820 that the difference in a quantity 698 00:36:26,820 --> 00:36:30,000 is the integral of the differential, which this is why 699 00:36:30,000 --> 00:36:33,100 you come to MIT to learn this. 700 00:36:33,100 --> 00:36:39,330 So if I can actually sample this, 701 00:36:39,330 --> 00:36:41,070 I may be able to get at the quantity 702 00:36:41,070 --> 00:36:43,380 simply by integrating that. 703 00:36:43,380 --> 00:36:44,650 And why is that important? 704 00:36:44,650 --> 00:36:48,000 Because if A is a free energy or an entropy, 705 00:36:48,000 --> 00:36:52,110 the derivatives of free energies and entropies 706 00:36:52,110 --> 00:36:54,090 are things that can be sampled because they 707 00:36:54,090 --> 00:36:57,000 tend to be either averages or fluctuations. 708 00:36:57,000 --> 00:36:59,580 For example, for the entropy, you 709 00:36:59,580 --> 00:37:02,950 wanted the entropy difference between two states 710 00:37:02,950 --> 00:37:05,082 so you integrate the derivative of the entropy. 711 00:37:05,082 --> 00:37:06,540 Well, the derivative of the entropy 712 00:37:06,540 --> 00:37:09,600 is the heat capacity and the heat capacity 713 00:37:09,600 --> 00:37:11,430 you can get from Monte Carlo. 714 00:37:11,430 --> 00:37:14,430 The heat capacity is essentially the fluctuation of the energy. 715 00:37:22,250 --> 00:37:25,780 So if you want to know the entropy at a given temperature, 716 00:37:25,780 --> 00:37:28,900 you start from some reference temperature 717 00:37:28,900 --> 00:37:32,200 where you know the entropy or you just fix it to some value 718 00:37:32,200 --> 00:37:34,960 if you want to reference everything to the same thing 719 00:37:34,960 --> 00:37:38,560 and you integrate the heat capacity. 720 00:37:38,560 --> 00:37:40,480 What do you take as reference states? 721 00:37:40,480 --> 00:37:42,878 Well, again, you could just integrate between two states 722 00:37:42,878 --> 00:37:44,170 and get the entropy difference. 723 00:37:44,170 --> 00:37:47,350 Often you start from 0. 724 00:37:47,350 --> 00:37:50,920 If you start from 0 temperature in models 725 00:37:50,920 --> 00:37:53,440 with discrete degrees of freedoms like the Ising 726 00:37:53,440 --> 00:37:56,800 model, the spin model, the entropy is 0 at 0 Kelvin 727 00:37:56,800 --> 00:37:59,230 so that's an easy integration state. 728 00:37:59,230 --> 00:38:02,710 In some cases, you can also find the entropy at infinity 729 00:38:02,710 --> 00:38:05,390 because in infinity, your phase space is random, 730 00:38:05,390 --> 00:38:07,390 the probability distribution is flat, 731 00:38:07,390 --> 00:38:11,170 and so sometimes you can get analytically the entropy there. 732 00:38:11,170 --> 00:38:13,510 So these are all proper reference states. 733 00:38:22,040 --> 00:38:23,650 So here's an example. 734 00:38:23,650 --> 00:38:28,690 This is, again, our very simple 2D square magnetic Ising model. 735 00:38:28,690 --> 00:38:31,160 You would essentially integrate C over T, 736 00:38:31,160 --> 00:38:36,040 so you'd essentially integrate under this curve from 0 up. 737 00:38:36,040 --> 00:38:37,990 The way you practically do it is that you 738 00:38:37,990 --> 00:38:42,010 wouldn't start from 0, because the reason is you're 739 00:38:42,010 --> 00:38:43,370 integrating. 740 00:38:43,370 --> 00:38:52,860 So you're integrating C over T and that integral, 741 00:38:52,860 --> 00:38:56,700 in reality of course, converges as T goes to 0. 742 00:38:56,700 --> 00:38:59,100 But numerically, it will never in your simulation 743 00:38:59,100 --> 00:39:03,360 because T goes to 0 and you force T to 0 744 00:39:03,360 --> 00:39:04,260 in your simulation. 745 00:39:04,260 --> 00:39:07,920 That's a well defined number, but the heat capacity 746 00:39:07,920 --> 00:39:12,150 will not go to 0 just because of numerical noise. 747 00:39:12,150 --> 00:39:13,590 In reality, it should be really 0, 748 00:39:13,590 --> 00:39:17,070 but what will actually happen is that because 749 00:39:17,070 --> 00:39:19,110 of some minor noise and fluctuations, 750 00:39:19,110 --> 00:39:21,300 we will get non-zero heat capacity. 751 00:39:21,300 --> 00:39:24,510 So you divide by a number that gets exceedingly small 752 00:39:24,510 --> 00:39:26,910 as you go to 0, and so you're integral will blow up 753 00:39:26,910 --> 00:39:28,330 numerically. 754 00:39:28,330 --> 00:39:31,770 So what you typically do is you look at your heat capacity 755 00:39:31,770 --> 00:39:36,810 and say, well, below 0.5 or 0.75 I essentially 756 00:39:36,810 --> 00:39:38,460 have no heat capacity. 757 00:39:38,460 --> 00:39:40,860 That means all the way from 0 to there 758 00:39:40,860 --> 00:39:43,830 I have essentially no entropy, and you just 759 00:39:43,830 --> 00:39:47,400 start integrating from 0.5 or 0.75. 760 00:39:47,400 --> 00:39:49,470 If you want to be more accurate, there 761 00:39:49,470 --> 00:39:54,990 are ways of analytically writing the heat capacity from 0 762 00:39:54,990 --> 00:39:57,330 to low temperature by things like low temperature 763 00:39:57,330 --> 00:39:59,070 expansions. 764 00:39:59,070 --> 00:40:01,410 Essentially from here to about here, 765 00:40:01,410 --> 00:40:04,800 all that would ever happen is single spin flips. 766 00:40:04,800 --> 00:40:07,320 So you'd have this ferromagnet sitting there, 767 00:40:07,320 --> 00:40:09,900 once in a while one spin would go, [FAST-SOUNDING WHISTLE].. 768 00:40:09,900 --> 00:40:11,520 So they're single excitation so you 769 00:40:11,520 --> 00:40:14,290 can write out what the partition function pretty much looks 770 00:40:14,290 --> 00:40:14,790 like. 771 00:40:14,790 --> 00:40:19,320 It's the groundstate energy plus just the first excited states. 772 00:40:19,320 --> 00:40:21,450 So you can write out analytically 773 00:40:21,450 --> 00:40:24,490 what the entropy is up to that point 774 00:40:24,490 --> 00:40:25,740 and then integrate from there. 775 00:40:33,260 --> 00:40:37,760 Practically you can integrate a whole bunch of other variables. 776 00:40:37,760 --> 00:40:40,190 The one that if you integrate from infinity 777 00:40:40,190 --> 00:40:45,140 that's very practical is to use the Gibbs-Helmholtz relation. 778 00:40:45,140 --> 00:40:46,820 Gibbs-Helmholtz relation essentially 779 00:40:46,820 --> 00:40:49,790 tells you that the average energy 780 00:40:49,790 --> 00:40:52,580 is the derivative of the free energy over T 781 00:40:52,580 --> 00:40:56,840 with respect to 1 over T. This is actually a generic relation. 782 00:40:56,840 --> 00:41:02,140 If you think here the average of any Hamiltonian 783 00:41:02,140 --> 00:41:09,350 is actually the derivative of its corresponding free energy 784 00:41:09,350 --> 00:41:14,600 over T with respect to 1 over T. 785 00:41:14,600 --> 00:41:15,470 OK. 786 00:41:15,470 --> 00:41:19,432 So if you do this in a canonical system, 787 00:41:19,432 --> 00:41:21,140 then you're Hamiltonian's like the energy 788 00:41:21,140 --> 00:41:24,300 minus mu times some concentration. 789 00:41:24,300 --> 00:41:27,470 So then that's the average quantity you would get here. 790 00:41:27,470 --> 00:41:28,520 Why is this useful? 791 00:41:28,520 --> 00:41:32,300 Because essentially now you're integrating in 1 over T, 792 00:41:32,300 --> 00:41:36,020 so in beta, and that quantity is 0 when 793 00:41:36,020 --> 00:41:37,470 you're at infinite temperature. 794 00:41:37,470 --> 00:41:39,660 So beta is 0 and T equals infinity. 795 00:41:39,660 --> 00:41:44,150 So now this is an easy way to integrate from infinity. 796 00:41:44,150 --> 00:41:46,160 Why do you want integrate from infinity? 797 00:41:46,160 --> 00:41:48,830 Like I said before, sometimes that infinite temperature, 798 00:41:48,830 --> 00:41:50,510 you know the free energy analytically, 799 00:41:50,510 --> 00:41:54,260 because everything is totally random and so something that's 800 00:41:54,260 --> 00:41:56,420 a useful integration state. 801 00:41:56,420 --> 00:41:58,250 Another one that's quite practical 802 00:41:58,250 --> 00:42:02,698 is simply integrating in composition, 803 00:42:02,698 --> 00:42:04,740 and that's why I sort of shown this on this phase 804 00:42:04,740 --> 00:42:09,450 because it shows the three major ways of integration. 805 00:42:09,450 --> 00:42:11,750 If you come from low temperature, 806 00:42:11,750 --> 00:42:13,430 you integrate the heat capacity. 807 00:42:13,430 --> 00:42:15,110 If you come from high temperature, 808 00:42:15,110 --> 00:42:17,180 you use the Gibbs-Helmholtz relation 809 00:42:17,180 --> 00:42:20,610 to integrate the average Hamiltonian, so the energy 810 00:42:20,610 --> 00:42:22,280 most cases. 811 00:42:22,280 --> 00:42:26,760 If you come from the sides, you integrate in composition space. 812 00:42:26,760 --> 00:42:29,750 So you integrate essentially sigma d mu, 813 00:42:29,750 --> 00:42:32,825 and the reason is that the composition-- 814 00:42:32,825 --> 00:42:36,430 let me write in regular thermodynamic variables-- 815 00:42:36,430 --> 00:42:39,670 is essentially derivative of free energy with respect 816 00:42:39,670 --> 00:42:41,810 to the chemical potential. 817 00:42:41,810 --> 00:42:43,632 So when you integrate that-- 818 00:42:43,632 --> 00:42:46,090 I've written composition here as the average spin in a spin 819 00:42:46,090 --> 00:42:52,240 model, so you integrate Cd mu, and that gives you dF 820 00:42:52,240 --> 00:42:56,020 and so that's essentially what this here is. 821 00:42:56,020 --> 00:42:59,080 Why can you integrate from the sides? 822 00:42:59,080 --> 00:43:01,180 Well, if you have pure 1-- 823 00:43:01,180 --> 00:43:05,020 so you call this A or B. If you have pure A, 824 00:43:05,020 --> 00:43:07,630 you know the free energy in many cases 825 00:43:07,630 --> 00:43:11,490 because if you have a model with only configuration entropy when 826 00:43:11,490 --> 00:43:13,240 you have pure A, you have no configuration 827 00:43:13,240 --> 00:43:16,700 entropy so the free energy there is just the energy. 828 00:43:16,700 --> 00:43:17,200 OK? 829 00:43:17,200 --> 00:43:21,190 So in some sense, if you think of this phase diagram 830 00:43:21,190 --> 00:43:23,530 as this line going to infinity, you essentially 831 00:43:23,530 --> 00:43:29,840 know the thermodynamic properties at all four edges 832 00:43:29,840 --> 00:43:32,630 and then you can integrate. 833 00:43:32,630 --> 00:43:36,440 This is by far the most accurate way of determining first order 834 00:43:36,440 --> 00:43:38,480 transitions, by far. 835 00:43:38,480 --> 00:43:40,340 The reason is that you can get the answer 836 00:43:40,340 --> 00:43:42,590 as accurately as you want it. 837 00:43:42,590 --> 00:43:45,290 So you have to integrate now along a path, that's 838 00:43:45,290 --> 00:43:47,130 the painful thing. 839 00:43:47,130 --> 00:43:50,150 So if I like the free energy here, 840 00:43:50,150 --> 00:43:51,680 I don't have to just simulate here, 841 00:43:51,680 --> 00:43:55,715 I have to simulate pretty much all the way up from here. 842 00:43:55,715 --> 00:43:57,840 So rather than simulating at one set of conditions, 843 00:43:57,840 --> 00:44:00,490 I got to simulate along a whole path. 844 00:44:00,490 --> 00:44:01,890 But the nice thing is that-- 845 00:44:01,890 --> 00:44:04,830 so where does your error come from? 846 00:44:04,830 --> 00:44:08,000 It comes from things you all control. 847 00:44:08,000 --> 00:44:10,710 You know, how long I've sampled at each state 848 00:44:10,710 --> 00:44:13,350 to get the quantity I'm integrating, like heat capacity 849 00:44:13,350 --> 00:44:13,860 or energies. 850 00:44:13,860 --> 00:44:14,790 You can control that. 851 00:44:14,790 --> 00:44:17,220 If you want it better, you sample longer. 852 00:44:17,220 --> 00:44:19,560 How many steps I take along the integration path, 853 00:44:19,560 --> 00:44:22,140 so now you're numerically integrating along a part. 854 00:44:22,140 --> 00:44:24,880 If I want that more accurate, I take more steps. 855 00:44:24,880 --> 00:44:28,350 So while it looks difficult, you have all the properties 856 00:44:28,350 --> 00:44:29,013 under control. 857 00:44:29,013 --> 00:44:30,430 So that's the nice thing, that you 858 00:44:30,430 --> 00:44:33,080 know how to make it better if you don't like the answer. 859 00:44:35,772 --> 00:44:37,230 The only thing to keep in mind when 860 00:44:37,230 --> 00:44:40,260 you free energy integration, that you have to iterate 861 00:44:40,260 --> 00:44:42,660 through equilibrium states. 862 00:44:42,660 --> 00:44:44,640 These thermodynamic relations do not 863 00:44:44,640 --> 00:44:48,520 hold when you're away from equilibrium states. 864 00:44:48,520 --> 00:44:52,350 So if you integrate through transitions, 865 00:44:52,350 --> 00:44:57,060 then you want to be absolutely sure that that transition is 866 00:44:57,060 --> 00:45:00,070 occurring in equilibrium, and that's often the problem, 867 00:45:00,070 --> 00:45:02,640 and that's why we combine all these schemes. 868 00:45:02,640 --> 00:45:07,440 Like if I want to get, say, this phase boundary, 869 00:45:07,440 --> 00:45:12,270 I would probably integrate one phase up from here to here 870 00:45:12,270 --> 00:45:16,140 and then I would integrate either the solid solution 871 00:45:16,140 --> 00:45:19,860 from the side or from infinity to that point. 872 00:45:19,860 --> 00:45:20,940 OK? 873 00:45:20,940 --> 00:45:24,840 So I would never try to get the solid solution by integrating 874 00:45:24,840 --> 00:45:27,510 from the ordered phase through the transition 875 00:45:27,510 --> 00:45:29,550 into the solid solution because I get way 876 00:45:29,550 --> 00:45:32,340 too much error from non-equilibrium phenomena 877 00:45:32,340 --> 00:45:35,140 at the transition. 878 00:45:35,140 --> 00:45:35,640 OK. 879 00:45:43,940 --> 00:45:48,860 Then we come to the last form of free energy integration, which 880 00:45:48,860 --> 00:45:52,070 is a form of thermodynamic integration 881 00:45:52,070 --> 00:45:54,590 that has a bit of a science fiction component to it. 882 00:45:57,870 --> 00:45:58,620 Come on. 883 00:45:58,620 --> 00:45:59,800 OK. 884 00:45:59,800 --> 00:46:02,790 Anyway, I think I've said all these things. 885 00:46:02,790 --> 00:46:05,100 What the advantages and disadvantages are, 886 00:46:05,100 --> 00:46:08,880 thermodynamic integration. 887 00:46:08,880 --> 00:46:14,030 You can actually do something a little more fancy or esoteric 888 00:46:14,030 --> 00:46:15,720 in thermodynamic integration, which 889 00:46:15,720 --> 00:46:19,660 tends to go by the name of lambda integration. 890 00:46:19,660 --> 00:46:24,700 What I showed you before was you were integrating with respect 891 00:46:24,700 --> 00:46:26,710 to derivatives of physical parameters, 892 00:46:26,710 --> 00:46:29,380 like temperature or concentration 893 00:46:29,380 --> 00:46:32,380 or 1 over temperature. 894 00:46:32,380 --> 00:46:34,540 You can actually integrate with respect 895 00:46:34,540 --> 00:46:36,190 to nonphysical parameters. 896 00:46:36,190 --> 00:46:39,250 For example, I may want to get the free energy 897 00:46:39,250 --> 00:46:42,420 difference between systems that have 898 00:46:42,420 --> 00:46:46,230 two different Hamiltonians. 899 00:46:46,230 --> 00:46:47,190 What would that mean? 900 00:46:47,190 --> 00:46:48,930 Maybe I want to know how to free energy 901 00:46:48,930 --> 00:46:52,350 change if I turn on a certain interaction in the Hamiltonian. 902 00:46:52,350 --> 00:46:55,342 Like maybe I want to know, I turn on Coulombic interactions 903 00:46:55,342 --> 00:46:57,300 and I want to know, how does this really affect 904 00:46:57,300 --> 00:46:59,620 the free energy of my system? 905 00:46:59,620 --> 00:47:02,003 Maybe I want to add a particle. 906 00:47:02,003 --> 00:47:04,170 You could think of actually changing the temperature 907 00:47:04,170 --> 00:47:05,907 as a way of changing your Hamiltonian, 908 00:47:05,907 --> 00:47:07,740 because the thing you put in the exponential 909 00:47:07,740 --> 00:47:11,430 is beta times Hamiltonian, so it's really kind of one unit. 910 00:47:11,430 --> 00:47:13,230 It's when you change that product 911 00:47:13,230 --> 00:47:16,950 that you're changing something to the probability density. 912 00:47:16,950 --> 00:47:20,710 I'll show you some cool examples of this in a second, 913 00:47:20,710 --> 00:47:23,620 but let me show you how it works. 914 00:47:23,620 --> 00:47:27,060 So now you're going to integrate along a path of lambda that 915 00:47:27,060 --> 00:47:30,990 essentially describes how you go from Hamiltonian 1 916 00:47:30,990 --> 00:47:33,270 to Hamiltonian 2. 917 00:47:33,270 --> 00:47:35,700 And Hamiltonian 2 could be totally different physics, 918 00:47:35,700 --> 00:47:38,380 it could be different chemistry, whatever. 919 00:47:38,380 --> 00:47:40,680 So I'm going to write the Hamiltonian 920 00:47:40,680 --> 00:47:45,630 as a linear combination of the Hamiltonians of 1 and 2, OK? 921 00:47:45,630 --> 00:47:48,570 So now you see, as lambda goes from 0 to 1, 922 00:47:48,570 --> 00:47:50,910 if I'm 0 I have Hamiltonian 1, if I'm 923 00:47:50,910 --> 00:47:52,630 one I have Hamiltonian 2. 924 00:47:52,630 --> 00:47:53,130 OK? 925 00:47:53,130 --> 00:47:54,750 So the integral 0 to 1 for lambda 926 00:47:54,750 --> 00:47:58,368 defines the past one which I integrate. 927 00:47:58,368 --> 00:47:59,910 So what I want to know is essentially 928 00:47:59,910 --> 00:48:03,047 the free energy between lambda is 1 and 0-- 929 00:48:03,047 --> 00:48:05,130 but I'll get the free energy along the whole path, 930 00:48:05,130 --> 00:48:07,660 as you'll see in a second. 931 00:48:07,660 --> 00:48:09,090 You can see how you can get that. 932 00:48:09,090 --> 00:48:12,240 If you look at the derivative of the free energy with respect 933 00:48:12,240 --> 00:48:16,170 to lambda, well, the free energy is the log 934 00:48:16,170 --> 00:48:17,890 of the partition function. 935 00:48:17,890 --> 00:48:19,920 So if I take the derivative of that, 936 00:48:19,920 --> 00:48:21,867 you can sort of do the math here. 937 00:48:21,867 --> 00:48:23,700 If I take a derivative of these exponentials 938 00:48:23,700 --> 00:48:25,920 I get the exponential back, so that's 939 00:48:25,920 --> 00:48:28,200 going to give me a partition function. 940 00:48:28,200 --> 00:48:30,330 Because remember, if I take derivative of the log 941 00:48:30,330 --> 00:48:32,130 I get 1 over this thing. 942 00:48:32,130 --> 00:48:33,990 So you know, I get log z. 943 00:48:33,990 --> 00:48:36,660 When you take the derivative of that, I'm going to get 1 over z 944 00:48:36,660 --> 00:48:39,420 and then times d, z, d, whatever I'm taking 945 00:48:39,420 --> 00:48:42,540 the derivative of with respect. 946 00:48:42,540 --> 00:48:45,310 So that gives me that partition function here. 947 00:48:45,310 --> 00:48:48,120 And then I take the derivative of what's inside the log 948 00:48:48,120 --> 00:48:49,530 and that gives me this. 949 00:48:49,530 --> 00:48:53,700 And essentially what shows up is the derivative 950 00:48:53,700 --> 00:48:56,190 of the Hamiltonian with respect to lambda, 951 00:48:56,190 --> 00:48:57,450 and this is actually classic. 952 00:48:57,450 --> 00:48:58,320 This always happens. 953 00:48:58,320 --> 00:49:01,650 If you take any derivative of the partition function, 954 00:49:01,650 --> 00:49:04,290 you essentially always end up with a weighted derivative 955 00:49:04,290 --> 00:49:05,850 of the Hamiltonian. 956 00:49:05,850 --> 00:49:08,790 And so what you see is this is the probability, 957 00:49:08,790 --> 00:49:12,090 this exponential weighted by q is the probability. 958 00:49:12,090 --> 00:49:16,820 So essentially what this free energy derivative is, 959 00:49:16,820 --> 00:49:20,120 it's the average of the derivative of the Hamiltonian 960 00:49:20,120 --> 00:49:21,770 with respect to lambda. 961 00:49:21,770 --> 00:49:22,310 OK? 962 00:49:22,310 --> 00:49:27,330 This is actually quite generic in statistical mechanics. 963 00:49:27,330 --> 00:49:32,930 So the quantity that you need to integrate is this derivative. 964 00:49:32,930 --> 00:49:35,600 Now if we've linearized our Hamiltonian, 965 00:49:35,600 --> 00:49:38,570 that derivative is just Hamiltonian difference. 966 00:49:38,570 --> 00:49:39,140 OK? 967 00:49:39,140 --> 00:49:42,690 But this is actually more generically true. 968 00:49:42,690 --> 00:49:44,720 But if you linearize it, all you need to do 969 00:49:44,720 --> 00:49:47,038 is average the Hamiltonian difference. 970 00:50:01,690 --> 00:50:04,270 So I'm going to show you an example which 971 00:50:04,270 --> 00:50:08,210 I got out of this paper, which is hidden here by-- 972 00:50:08,210 --> 00:50:09,210 kind of move this thing. 973 00:50:09,210 --> 00:50:09,878 There we go. 974 00:50:13,677 --> 00:50:15,260 And this was a study where they wanted 975 00:50:15,260 --> 00:50:20,270 to look at the effects of a water dipole 976 00:50:20,270 --> 00:50:22,620 on the free energy of water. 977 00:50:22,620 --> 00:50:26,077 And so the reason you have a dipole in water is, of course, 978 00:50:26,077 --> 00:50:26,660 you have H2O-- 979 00:50:29,210 --> 00:50:31,280 how does this work again? 980 00:50:31,280 --> 00:50:35,220 So the hydrogens are slightly positively charged 981 00:50:35,220 --> 00:50:39,260 and so this is then minus 2 times that quantity. 982 00:50:39,260 --> 00:50:44,250 And so because of that, you of course, have a dipole which-- 983 00:50:44,250 --> 00:50:47,090 does a dipole point from positive to negative 984 00:50:47,090 --> 00:50:50,785 or, yeah, from negative to positive dipoles? 985 00:50:50,785 --> 00:50:51,410 Well, whatever. 986 00:50:51,410 --> 00:50:53,660 We'll define it this way. 987 00:50:53,660 --> 00:50:59,330 So you have a dipole moment and in a simple simulation, 988 00:50:59,330 --> 00:51:03,410 you can essentially just represent the water 989 00:51:03,410 --> 00:51:05,510 by its dipole, nothing else. 990 00:51:05,510 --> 00:51:07,140 No atoms, no molecules. 991 00:51:07,140 --> 00:51:09,950 So you want to look at the dipole interactions 992 00:51:09,950 --> 00:51:11,030 between water. 993 00:51:11,030 --> 00:51:13,910 Now if you want to look at how does the free energies 994 00:51:13,910 --> 00:51:17,790 change with the strength of that dipole, for example, 995 00:51:17,790 --> 00:51:22,910 you could write dipolar strength in terms of some parameter 996 00:51:22,910 --> 00:51:25,290 lambda, and that's what I've done here. 997 00:51:25,290 --> 00:51:28,460 So the positive and the negative charge in the dipole 998 00:51:28,460 --> 00:51:31,392 depend on the parameter lambda, and so essentially 999 00:51:31,392 --> 00:51:33,350 what I'm going to do is look at the free energy 1000 00:51:33,350 --> 00:51:34,460 as a function of lambda. 1001 00:51:37,160 --> 00:51:40,400 And let me show you how it works. 1002 00:51:40,400 --> 00:51:43,400 The green line here, there's a few too many lines on this plot 1003 00:51:43,400 --> 00:51:44,240 unfortunately. 1004 00:51:44,240 --> 00:51:45,605 This is the exact result-- 1005 00:51:49,770 --> 00:51:52,350 and actually, this was well parameterized 1006 00:51:52,350 --> 00:51:55,230 because at 0 and 1 you should get the same answer 1007 00:51:55,230 --> 00:51:59,180 because when lambda is 1, the dipole is the same as at 0, 1008 00:51:59,180 --> 00:52:01,950 it's just inverted the plus n minus charge. 1009 00:52:01,950 --> 00:52:03,420 OK? 1010 00:52:03,420 --> 00:52:08,100 So the purple line is what they got with lambda integrations 1011 00:52:08,100 --> 00:52:09,750 paper, so it does pretty well. 1012 00:52:09,750 --> 00:52:12,432 I mean it's this line here, sorry. 1013 00:52:12,432 --> 00:52:14,640 But what you see is there's of course a bit of error. 1014 00:52:14,640 --> 00:52:18,360 This point should be the same as that point, OK? 1015 00:52:18,360 --> 00:52:22,650 And so you clearly see that they accumulated some error 1016 00:52:22,650 --> 00:52:24,810 along the integration path. 1017 00:52:24,810 --> 00:52:28,320 And if you're really hard core computational 1018 00:52:28,320 --> 00:52:31,973 and you think this is fun, this is one way 1019 00:52:31,973 --> 00:52:34,140 you can check your integration errors is essentially 1020 00:52:34,140 --> 00:52:38,170 do a circular integration in your phase space. 1021 00:52:38,170 --> 00:52:44,160 So not only go from Hamil state 1 to 2, but come back 1022 00:52:44,160 --> 00:52:45,690 and you essentially have some idea 1023 00:52:45,690 --> 00:52:48,180 of the error you've accumulated along the integration path. 1024 00:52:53,970 --> 00:52:57,600 This one, the next one I like as an example. 1025 00:52:57,600 --> 00:53:01,620 This is starting to get very close to alchemy. 1026 00:53:01,620 --> 00:53:04,980 You know, in the old days people tried to turn lead into gold. 1027 00:53:04,980 --> 00:53:07,530 This is getting pretty close. 1028 00:53:07,530 --> 00:53:10,540 You can look at three energy changes 1029 00:53:10,540 --> 00:53:14,880 literally by changing chemistry. 1030 00:53:14,880 --> 00:53:16,590 I know nothing about organic chemistry, 1031 00:53:16,590 --> 00:53:19,320 but I think that ring there is called the phenyl 1032 00:53:19,320 --> 00:53:21,690 and so you can put different groups on the phenyl 1033 00:53:21,690 --> 00:53:24,360 and if you put chlorine on it, it's chlorophenyl. 1034 00:53:24,360 --> 00:53:27,480 If you pull a methyl group on its methylphenyl, 1035 00:53:27,480 --> 00:53:29,760 and if you put a cyan group it's cyanophenyl. 1036 00:53:29,760 --> 00:53:35,670 So there's clearly some logic in chemical names. 1037 00:53:35,670 --> 00:53:38,880 But for example, you could write a Hamiltonian 1038 00:53:38,880 --> 00:53:43,225 that slowly changes one of these species into another one. 1039 00:53:43,225 --> 00:53:44,850 And what does that mean, slowly change? 1040 00:53:44,850 --> 00:53:46,830 That means that you mix the interaction. 1041 00:53:46,830 --> 00:53:48,630 Let's say you did this with potential, 1042 00:53:48,630 --> 00:53:52,170 you would essentially write a potential from that group. 1043 00:53:52,170 --> 00:53:55,950 Potential that comes from this group here that you attach 1044 00:53:55,950 --> 00:53:58,225 has a weighted average of-- 1045 00:53:58,225 --> 00:53:59,850 let's say we go from, like we did here, 1046 00:53:59,850 --> 00:54:02,730 methylphenyl to chlorophenyl. 1047 00:54:02,730 --> 00:54:06,030 So you would weigh the potential with the parameter lambda 1048 00:54:06,030 --> 00:54:07,860 and integrate in that space and you'd 1049 00:54:07,860 --> 00:54:10,320 get the free energy difference between these two 1050 00:54:10,320 --> 00:54:11,605 groups attached. 1051 00:54:17,660 --> 00:54:19,780 OK. 1052 00:54:19,780 --> 00:54:25,870 So here's my take on Monte Carlo. 1053 00:54:25,870 --> 00:54:30,532 What I really like about it, it's conceptually simple. 1054 00:54:30,532 --> 00:54:31,990 Of all the simulation methods, it's 1055 00:54:31,990 --> 00:54:37,630 probably the one that has the least amount of frills. 1056 00:54:37,630 --> 00:54:43,510 It tends to be very easy to implement. 1057 00:54:43,510 --> 00:54:46,820 And maybe the most important thing, 1058 00:54:46,820 --> 00:54:49,930 it's as accurate as your Hamiltonian 1059 00:54:49,930 --> 00:54:53,320 can be so you can push the sampling as far as you 1060 00:54:53,320 --> 00:54:54,200 want it to be. 1061 00:54:54,200 --> 00:54:57,400 So the sampling part can be done as accurately as you have time 1062 00:54:57,400 --> 00:54:58,210 for, essentially. 1063 00:54:58,210 --> 00:54:59,957 Time and money for. 1064 00:54:59,957 --> 00:55:01,540 Which is not always true about models. 1065 00:55:01,540 --> 00:55:05,290 If I set up a potential model, I can't necessarily 1066 00:55:05,290 --> 00:55:08,110 improve that infinitely better to model 1067 00:55:08,110 --> 00:55:09,520 the energetics of my system. 1068 00:55:09,520 --> 00:55:13,030 But at least with Monte Carlo, the sampling part, so 1069 00:55:13,030 --> 00:55:17,320 the finite temperature part can be done with as little error 1070 00:55:17,320 --> 00:55:20,440 as you'd like it to be. 1071 00:55:20,440 --> 00:55:22,630 I think there's sort of two or three 1072 00:55:22,630 --> 00:55:28,850 major disadvantages is that it's not a dynamical method, 1073 00:55:28,850 --> 00:55:31,090 so it doesn't give you any kinetic information 1074 00:55:31,090 --> 00:55:33,623 like molecular dynamics does. 1075 00:55:33,623 --> 00:55:35,290 Sometimes that's an advantage because it 1076 00:55:35,290 --> 00:55:38,170 means you don't need a kinetic mechanism 1077 00:55:38,170 --> 00:55:41,770 to study how the system goes through it's phase space. 1078 00:55:41,770 --> 00:55:45,400 But the second one is the major hit most of the time. 1079 00:55:45,400 --> 00:55:48,130 It's an extremely wasteful method 1080 00:55:48,130 --> 00:55:50,710 in terms of energy evaluations. 1081 00:55:50,710 --> 00:55:53,530 You do a lot of kind of random excursions in phase 1082 00:55:53,530 --> 00:55:56,960 space and every time you need to get the energy, 1083 00:55:56,960 --> 00:56:00,430 and so that's why it's really great to implement it 1084 00:56:00,430 --> 00:56:02,050 with fast energy methods. 1085 00:56:02,050 --> 00:56:04,827 You know, [INAUDIBLE] small Hamiltonian G. That's just 1086 00:56:04,827 --> 00:56:06,910 adding a few numbers and multiplying a few numbers 1087 00:56:06,910 --> 00:56:08,620 and you got the energy, bam! 1088 00:56:08,620 --> 00:56:10,690 Or something like potential models 1089 00:56:10,690 --> 00:56:14,440 if you do it in a continuous space or embedded atom 1090 00:56:14,440 --> 00:56:16,840 works great with those methods. 1091 00:56:16,840 --> 00:56:19,840 It's essentially impossible to implement it with quantum 1092 00:56:19,840 --> 00:56:21,670 mechanics directly, you know? 1093 00:56:21,670 --> 00:56:24,200 I mean, that embedded. 1094 00:56:24,200 --> 00:56:25,900 The method, the copper nickel one 1095 00:56:25,900 --> 00:56:28,420 I showed you did millions of energy evaluations 1096 00:56:28,420 --> 00:56:30,520 to equilibrate the system. 1097 00:56:30,520 --> 00:56:33,120 So can you imagine you're going to millions of direct DFT 1098 00:56:33,120 --> 00:56:34,740 calculations? 1099 00:56:34,740 --> 00:56:35,640 No. 1100 00:56:35,640 --> 00:56:38,010 So that the fact that you need typically 1101 00:56:38,010 --> 00:56:42,430 a fast energy method is sort of a limitation. 1102 00:56:42,430 --> 00:56:45,150 The stochastic nature of it can be a limitation. 1103 00:56:45,150 --> 00:56:49,710 It's less and less so, but when you run two Monte Carlo 1104 00:56:49,710 --> 00:56:53,380 simulations you don't get the same answer, 1105 00:56:53,380 --> 00:56:57,060 so because of that, there is noise in data. 1106 00:56:57,060 --> 00:57:00,780 It's a lot harder to write methods on top of it, 1107 00:57:00,780 --> 00:57:03,020 in some sense, drivers for it. 1108 00:57:03,020 --> 00:57:06,570 If you do DFT, if you converge, you 1109 00:57:06,570 --> 00:57:08,790 get the same answer every time. 1110 00:57:08,790 --> 00:57:10,220 So it's not stochastic. 1111 00:57:10,220 --> 00:57:14,790 So you can now develop algorithms that use that input 1112 00:57:14,790 --> 00:57:15,840 and do stuff with it. 1113 00:57:15,840 --> 00:57:18,510 It's a lot harder to work with stochastic input, 1114 00:57:18,510 --> 00:57:22,830 let me tell you, because you kind of have to average noise 1115 00:57:22,830 --> 00:57:24,570 away. 1116 00:57:24,570 --> 00:57:26,900 And I think the fourth one-- you know, I should update. 1117 00:57:26,900 --> 00:57:29,730 This is becoming less and less an issue. 1118 00:57:29,730 --> 00:57:33,540 It used to be, but as computers get faster it's really, 1119 00:57:33,540 --> 00:57:35,760 I would say we can get free energy when 1120 00:57:35,760 --> 00:57:39,870 we want it, especially on models with discrete degrees 1121 00:57:39,870 --> 00:57:40,388 of freedom. 1122 00:57:40,388 --> 00:57:42,680 Models with continuous degree of freedom you can do it. 1123 00:57:42,680 --> 00:57:46,320 There's still more work, but it's not much of a disadvantage 1124 00:57:46,320 --> 00:57:49,065 anymore. 1125 00:57:49,065 --> 00:57:53,260 OK, I've gone through the references Before. 1126 00:57:53,260 --> 00:57:54,070 OK. 1127 00:57:54,070 --> 00:57:55,785 So what I want to start maybe now-- 1128 00:57:55,785 --> 00:57:57,160 and I'm probably not going to get 1129 00:57:57,160 --> 00:58:05,800 this finished-- is start to talk about coarse-graining methods. 1130 00:58:05,800 --> 00:58:07,750 And why do you need coarse-graining methods? 1131 00:58:07,750 --> 00:58:13,500 it's essentially a nice follow up on what we just discussed. 1132 00:58:13,500 --> 00:58:18,990 Monte Carlo allows you to get full phase space sampling, 1133 00:58:18,990 --> 00:58:22,740 but you cannot do it on an accurate Hamiltonian like 1134 00:58:22,740 --> 00:58:25,930 density functional theory. 1135 00:58:25,930 --> 00:58:30,780 So what if you actually need highly accurate energetics 1136 00:58:30,780 --> 00:58:32,760 and you need to sample phase space well? 1137 00:58:32,760 --> 00:58:35,520 Then you're in trouble because the two 1138 00:58:35,520 --> 00:58:38,880 are very hard to combine. 1139 00:58:38,880 --> 00:58:41,250 Sampling phase space, well, means 1140 00:58:41,250 --> 00:58:44,520 a lot of energy evaluations and a lot of energy evaluations 1141 00:58:44,520 --> 00:58:47,700 precludes using a very highly accurate Hamiltonian, 1142 00:58:47,700 --> 00:58:50,880 but there are problems for which you need both anyway. 1143 00:58:50,880 --> 00:58:54,270 And then you sort of need to go to coarse-graining methods. 1144 00:58:54,270 --> 00:58:56,100 And the idea in coarse-graining methods 1145 00:58:56,100 --> 00:59:01,290 is that you try to either remove spatial degrees of freedom 1146 00:59:01,290 --> 00:59:03,690 or temporal degrees of freedom or your system 1147 00:59:03,690 --> 00:59:06,510 very systematically so that your model becomes 1148 00:59:06,510 --> 00:59:11,470 simpler and simpler, giving up as little accuracy as possible. 1149 00:59:11,470 --> 00:59:11,970 OK? 1150 00:59:11,970 --> 00:59:16,110 So accumulating as little error on the way as possible. 1151 00:59:16,110 --> 00:59:19,090 I'll talk first about temporal coarse-graining 1152 00:59:19,090 --> 00:59:21,373 since it's easier, but if we have time in the end 1153 00:59:21,373 --> 00:59:23,790 I may say a little about spatial coarse-graining, which is 1154 00:59:23,790 --> 00:59:26,310 a much more difficult problem. 1155 00:59:29,790 --> 00:59:32,750 OK, let me skip this. 1156 00:59:32,750 --> 00:59:33,931 OK. 1157 00:59:33,931 --> 00:59:35,473 Well, actually let me show it to you. 1158 00:59:39,560 --> 00:59:41,060 Something wrong with the color here. 1159 00:59:45,160 --> 00:59:49,510 Here's an example of why you need coarse-graining. 1160 00:59:49,510 --> 00:59:51,550 This is the copper, aluminum phase diagram. 1161 00:59:54,310 --> 00:59:56,890 All the stable phases in here are 1162 00:59:56,890 --> 01:00:00,250 in many cases within something like 5 to 10 milli electron 1163 01:00:00,250 --> 01:00:03,250 volt of several other phases. 1164 01:00:03,250 --> 01:00:06,250 So that means that to actually know 1165 01:00:06,250 --> 01:00:08,470 that these are the stables facing that system, 1166 01:00:08,470 --> 01:00:11,480 you need highly accurate energetics. 1167 01:00:11,480 --> 01:00:14,880 You can't afford more than a few million electoral votes 1168 01:00:14,880 --> 01:00:15,990 to an error. 1169 01:00:15,990 --> 01:00:17,940 That's a very small error. 1170 01:00:17,940 --> 01:00:22,010 So an election volt is about 100 kilojoules, 1171 01:00:22,010 --> 01:00:25,770 so a million electron volt is about 100 joules. 1172 01:00:25,770 --> 01:00:26,840 All right. 1173 01:00:26,840 --> 01:00:27,822 Yeah. 1174 01:00:27,822 --> 01:00:29,530 So I don't know what that in calories is, 1175 01:00:29,530 --> 01:00:33,620 but anyway, so highly accurate energetics 1176 01:00:33,620 --> 01:00:36,700 and then to get the temperature behavior 1177 01:00:36,700 --> 01:00:38,510 you need to sample phase space because you 1178 01:00:38,510 --> 01:00:41,360 need to get the entropy and the excitations. 1179 01:00:46,100 --> 01:00:49,860 For some problems like these, essentially we 1180 01:00:49,860 --> 01:00:51,630 figured out how to do this. 1181 01:00:51,630 --> 01:00:54,650 It's still hard work, but essentially the road map 1182 01:00:54,650 --> 01:00:56,750 is laid out. 1183 01:00:56,750 --> 01:01:01,580 The idea is that you successively integrate over 1184 01:01:01,580 --> 01:01:04,160 slower and slower timescales. 1185 01:01:04,160 --> 01:01:07,520 So you look at what excursions, what excitations 1186 01:01:07,520 --> 01:01:11,090 occur in the system, first at the really fast time scales 1187 01:01:11,090 --> 01:01:15,110 and then you try to either variationally remove them 1188 01:01:15,110 --> 01:01:16,670 or you try to integrate over them, 1189 01:01:16,670 --> 01:01:18,140 and those two are different things. 1190 01:01:18,140 --> 01:01:21,843 If you variationally remove a degree of freedom, 1191 01:01:21,843 --> 01:01:23,510 then you're really finding, essentially, 1192 01:01:23,510 --> 01:01:27,290 what gives you the lowest energy for that degree of freedom. 1193 01:01:27,290 --> 01:01:29,990 If you integrate over the excursions 1194 01:01:29,990 --> 01:01:32,330 of that degree of freedom, then essentially you 1195 01:01:32,330 --> 01:01:34,430 capture its full entropic component, 1196 01:01:34,430 --> 01:01:36,680 and I'll come back to the distinction between the two. 1197 01:01:36,680 --> 01:01:39,170 But let's take, for example, a simple binary 1198 01:01:39,170 --> 01:01:42,170 solid, like that aluminum copper I showed you. 1199 01:01:42,170 --> 01:01:45,690 What are the excitations in that system? 1200 01:01:45,690 --> 01:01:49,670 Well, it's a metal so at the highest level there's probably 1201 01:01:49,670 --> 01:01:51,920 electronic excitations. 1202 01:01:51,920 --> 01:01:56,810 If the Fermi level of the system cuts through a band 1203 01:01:56,810 --> 01:01:59,810 so you have density of states at the Fermi level, 1204 01:01:59,810 --> 01:02:02,860 then electrons can get excited across the Fermi level. 1205 01:02:02,860 --> 01:02:03,360 OK? 1206 01:02:03,360 --> 01:02:05,900 So that's a form of entropy right there already. 1207 01:02:05,900 --> 01:02:07,610 Most of the time we don't worry about it. 1208 01:02:07,610 --> 01:02:10,190 That's one that we variationally remove. 1209 01:02:10,190 --> 01:02:13,310 You do the FT and you find the lowest energy state. 1210 01:02:13,310 --> 01:02:15,080 You don't say, I'm going to integrate 1211 01:02:15,080 --> 01:02:17,150 over all the accessible electronic states. 1212 01:02:17,150 --> 01:02:20,610 You find the lowest one. 1213 01:02:20,610 --> 01:02:23,293 Then you have vibrational excitations. 1214 01:02:23,293 --> 01:02:24,710 These are ones you can get around. 1215 01:02:24,710 --> 01:02:28,935 They're essentially present in every material by definition. 1216 01:02:28,935 --> 01:02:31,800 These live on timescale 10 to the minus 11, 1217 01:02:31,800 --> 01:02:35,970 10 to the minus 12, 10 to the minus 13 seconds. 1218 01:02:35,970 --> 01:02:40,050 And then typically the slower one is configurational ones. 1219 01:02:40,050 --> 01:02:45,690 If you have just A and B that they interchange positions, 1220 01:02:45,690 --> 01:02:47,175 they start giving you disorder. 1221 01:02:51,240 --> 01:02:54,390 Again, the reason you can sort of do this problem easily 1222 01:02:54,390 --> 01:02:57,240 is that this would be the perfect Monte Carlo 1223 01:02:57,240 --> 01:03:00,420 problem, just like in that copper nickel example 1224 01:03:00,420 --> 01:03:01,650 segregation. 1225 01:03:01,650 --> 01:03:03,600 If you could do fast energy evaluations, 1226 01:03:03,600 --> 01:03:05,340 you would just do small displacements 1227 01:03:05,340 --> 01:03:07,410 to capture the vibrations and then 1228 01:03:07,410 --> 01:03:09,900 you would do big exchanges to capture 1229 01:03:09,900 --> 01:03:11,640 the configurational excitations. 1230 01:03:11,640 --> 01:03:13,110 But because you need the accuracy, 1231 01:03:13,110 --> 01:03:16,973 you'd almost need to do it on a DFT Hamiltonian. 1232 01:03:16,973 --> 01:03:18,390 You could say the vibrational ones 1233 01:03:18,390 --> 01:03:21,990 you could capture very well with molecular dynamics. 1234 01:03:21,990 --> 01:03:25,530 You just track the displacements of the atoms. 1235 01:03:25,530 --> 01:03:28,770 Think about it, the sort of slower phonons 1236 01:03:28,770 --> 01:03:32,020 go on a timescale of maybe 10 to the minus 11. 1237 01:03:32,020 --> 01:03:32,910 So what is that? 1238 01:03:32,910 --> 01:03:35,040 That's 10 picoseconds. 1239 01:03:35,040 --> 01:03:38,130 So if you simulate 100 picoseconds in nanoseconds, 1240 01:03:38,130 --> 01:03:40,230 you're going to start fairly allegorically 1241 01:03:40,230 --> 01:03:41,410 sampling the vibration. 1242 01:03:41,410 --> 01:03:44,190 So you'd have a pretty good result 1243 01:03:44,190 --> 01:03:46,230 for vibrational free energy, but you'd never 1244 01:03:46,230 --> 01:03:48,598 get down to the configurational timescale. 1245 01:03:52,190 --> 01:03:53,600 So how do you solve that problem? 1246 01:03:59,990 --> 01:04:04,760 Again, the idea is that we integrate over 1247 01:04:04,760 --> 01:04:07,490 the fast degrees of freedom and try 1248 01:04:07,490 --> 01:04:10,460 to define a Hamiltonian that's only 1249 01:04:10,460 --> 01:04:13,740 defined in the phase space of the slow degrees of freedom. 1250 01:04:13,740 --> 01:04:14,240 OK? 1251 01:04:14,240 --> 01:04:18,350 And the question is how accurate can we do this? 1252 01:04:18,350 --> 01:04:21,050 So I'm going to focus on this alloy problem 1253 01:04:21,050 --> 01:04:22,500 just so that we keep our focus. 1254 01:04:22,500 --> 01:04:27,140 So what I want to get to is a Hamiltonian that has integrated 1255 01:04:27,140 --> 01:04:31,400 away the electronic excitations, the vibrational excitations, 1256 01:04:31,400 --> 01:04:34,280 and therefore that just lives in the phase space 1257 01:04:34,280 --> 01:04:36,320 of the substitutional excitations, which 1258 01:04:36,320 --> 01:04:37,760 is a much smaller phase space. 1259 01:04:44,790 --> 01:04:47,170 OK. 1260 01:04:47,170 --> 01:04:48,490 Let me show you the math. 1261 01:04:51,190 --> 01:04:56,873 If you think of a crystal of A and B atoms, 1262 01:04:56,873 --> 01:04:58,790 they may live on a lattice, but of course they 1263 01:04:58,790 --> 01:05:00,430 can be displaced from the lattice 1264 01:05:00,430 --> 01:05:03,520 just through static relaxation but also 1265 01:05:03,520 --> 01:05:05,410 through vibrational excursions. 1266 01:05:05,410 --> 01:05:10,090 So normally you could define that system 1267 01:05:10,090 --> 01:05:12,220 just by coordinate vectors. 1268 01:05:12,220 --> 01:05:15,050 If I have n atoms, I need n coordinate vectors. 1269 01:05:15,050 --> 01:05:17,980 I'm going to change the way I characterize that system 1270 01:05:17,980 --> 01:05:21,110 by first, a lattice index. 1271 01:05:21,110 --> 01:05:22,780 So this is a topological index. 1272 01:05:22,780 --> 01:05:26,660 This is essentially if I have a crystalline material, 1273 01:05:26,660 --> 01:05:28,900 I could start indexing the possible sites 1274 01:05:28,900 --> 01:05:30,680 in that material. 1275 01:05:30,680 --> 01:05:35,770 So I will be the index of these possible lattice sites 1276 01:05:35,770 --> 01:05:39,490 and delta r will be the displacement from these lattice 1277 01:05:39,490 --> 01:05:40,080 sites. 1278 01:05:40,080 --> 01:05:40,870 OK? 1279 01:05:40,870 --> 01:05:45,280 So do you agree that the combination of these two 1280 01:05:45,280 --> 01:05:48,220 is essentially the same as having a full coordinate? 1281 01:05:48,220 --> 01:05:49,510 OK. 1282 01:05:49,510 --> 01:05:58,040 Now the set of indices i I'm going to represent essentially 1283 01:05:58,040 --> 01:06:00,030 by a lattice model. 1284 01:06:00,030 --> 01:06:04,040 So the set of indices i is essentially saying at each 1285 01:06:04,040 --> 01:06:07,220 lattice point or around there-- because the atom 1286 01:06:07,220 --> 01:06:10,040 doesn't exactly have to sit there, but around there-- 1287 01:06:10,040 --> 01:06:12,380 is it an A or a B there? 1288 01:06:12,380 --> 01:06:13,880 So the variables to describe that 1289 01:06:13,880 --> 01:06:17,180 are the same variables as a spin model or lattice model, 1290 01:06:17,180 --> 01:06:19,160 it's a binary problem now. 1291 01:06:19,160 --> 01:06:19,940 OK? 1292 01:06:19,940 --> 01:06:24,380 So the question is, at I is it A or B? 1293 01:06:24,380 --> 01:06:27,200 But again, I don't need to specify that the atom exactly 1294 01:06:27,200 --> 01:06:31,460 sits at A or B. That I specify by the displacements, the delta 1295 01:06:31,460 --> 01:06:33,530 r's, OK? 1296 01:06:33,530 --> 01:06:35,810 So what I've done is I've separated variables 1297 01:06:35,810 --> 01:06:39,920 that give me the configurational topology from the variables 1298 01:06:39,920 --> 01:06:43,910 that gives me the excursions away from the ideal lattice 1299 01:06:43,910 --> 01:06:44,410 site. 1300 01:06:49,840 --> 01:06:51,270 And now you see where I'm going. 1301 01:06:51,270 --> 01:06:55,500 I'm going to integrate over the excursions 1302 01:06:55,500 --> 01:06:58,050 and then retain something that only 1303 01:06:58,050 --> 01:07:00,780 exists in the space of the configurational degrees 1304 01:07:00,780 --> 01:07:01,860 of freedom. 1305 01:07:01,860 --> 01:07:04,090 OK, here we go. 1306 01:07:04,090 --> 01:07:07,660 So the partition function is the sum over all states. 1307 01:07:07,660 --> 01:07:09,310 So again, the sum over all states 1308 01:07:09,310 --> 01:07:12,540 you would have normally be the sum over all vectors. 1309 01:07:12,540 --> 01:07:16,230 I'm going to write that as the sum over configurational states 1310 01:07:16,230 --> 01:07:19,690 and then the sum over all displacement states, which 1311 01:07:19,690 --> 01:07:20,910 have called nu. 1312 01:07:20,910 --> 01:07:31,520 So nu is the set of delta ri, and so the energy 1313 01:07:31,520 --> 01:07:33,890 depends on both variables. 1314 01:07:33,890 --> 01:07:35,290 OK? 1315 01:07:35,290 --> 01:07:38,770 Now what I'm going to do is I'm going to essentially assume 1316 01:07:38,770 --> 01:07:40,780 I can do this integration. 1317 01:07:40,780 --> 01:07:43,310 We'll talk in a second about how you can do that. 1318 01:07:43,310 --> 01:07:46,510 So I'm only going to do the sum over the displacement, 1319 01:07:46,510 --> 01:07:49,880 but for a given configurational state. 1320 01:07:49,880 --> 01:07:51,650 So what does that practically mean? 1321 01:07:51,650 --> 01:07:56,030 That means that I'm integrating the phase space of a fixed 1322 01:07:56,030 --> 01:07:59,500 topology, but allowing the atoms to sort of vibrate 1323 01:07:59,500 --> 01:08:02,770 around their average positions. 1324 01:08:02,770 --> 01:08:04,510 That's essentially what I'm doing. 1325 01:08:04,510 --> 01:08:07,120 So essentially I'm capturing, in that integral, 1326 01:08:07,120 --> 01:08:09,610 the vibrational free energy component 1327 01:08:09,610 --> 01:08:13,570 for a given configuration. 1328 01:08:13,570 --> 01:08:16,970 And so I'm going to define. 1329 01:08:16,970 --> 01:08:20,203 So this is that integral we talked about. 1330 01:08:20,203 --> 01:08:22,120 So I'm going to give that a free energy, which 1331 01:08:22,120 --> 01:08:24,050 is just the logarithm of that. 1332 01:08:24,050 --> 01:08:29,300 And then just if I substitute that in here, 1333 01:08:29,300 --> 01:08:35,069 you see that what I end up with is this 1334 01:08:35,069 --> 01:08:36,450 and this is kind of important. 1335 01:08:36,450 --> 01:08:37,270 What do I have? 1336 01:08:37,270 --> 01:08:41,130 I have the partition function of a lattice model. 1337 01:08:41,130 --> 01:08:44,160 I have a partition function that only 1338 01:08:44,160 --> 01:08:46,770 sums over different topologies, it only 1339 01:08:46,770 --> 01:08:52,670 sums over different configurational states, OK? 1340 01:08:52,670 --> 01:08:55,370 So I've reduced the phase space. 1341 01:08:55,370 --> 01:08:57,189 I'm not summing over vibrational states. 1342 01:08:57,189 --> 01:08:58,609 Remember, I've integrated them. 1343 01:08:58,609 --> 01:09:00,713 But what is the quantity I'm summing? 1344 01:09:00,713 --> 01:09:01,880 This is the important thing. 1345 01:09:01,880 --> 01:09:04,729 The quantity I'm summing is not the energy, 1346 01:09:04,729 --> 01:09:08,420 it's the free energy of the vibrations essentially. 1347 01:09:08,420 --> 01:09:09,740 OK? 1348 01:09:09,740 --> 01:09:14,300 So as you coarse-grain in time, your Hamiltonian 1349 01:09:14,300 --> 01:09:17,450 at the slower timescale is essentially the free energy 1350 01:09:17,450 --> 01:09:19,640 of the faster timescale. 1351 01:09:19,640 --> 01:09:20,720 OK? 1352 01:09:20,720 --> 01:09:21,290 Why? 1353 01:09:21,290 --> 01:09:23,359 Because you successively integrate. 1354 01:09:23,359 --> 01:09:27,350 So in essence, if I sum over lattice model stage 1355 01:09:27,350 --> 01:09:30,260 and I put A's and B's on different lattice positions, 1356 01:09:30,260 --> 01:09:32,630 what this is telling you is that the quantity-- 1357 01:09:32,630 --> 01:09:35,600 that's my Hamiltonian-- is not the energy of that state. 1358 01:09:35,600 --> 01:09:39,210 It's the vibrational free energy of that state. 1359 01:09:39,210 --> 01:09:42,140 So if I take that as my Hamiltonian 1360 01:09:42,140 --> 01:09:44,960 and then I do this partition function, 1361 01:09:44,960 --> 01:09:47,359 I will essentially have an exact result. 1362 01:09:47,359 --> 01:09:51,260 I will have the exact partition function of the system. 1363 01:09:51,260 --> 01:09:52,310 OK? 1364 01:09:52,310 --> 01:09:54,590 And that's pretty amazing because I've really 1365 01:09:54,590 --> 01:09:58,460 sort of separated the timescale, integrate over them separately, 1366 01:09:58,460 --> 01:10:01,730 but I get what's essentially still-- 1367 01:10:01,730 --> 01:10:05,840 it's an almost exact result, because let's say 1368 01:10:05,840 --> 01:10:07,590 I'm going to do it this way. 1369 01:10:07,590 --> 01:10:09,530 So let's say on this I do Monte Carlo now. 1370 01:10:12,260 --> 01:10:15,170 There is a small assumption that I've made in all of this. 1371 01:10:18,530 --> 01:10:20,360 Think of the physical picture here. 1372 01:10:23,390 --> 01:10:25,970 Essentially what I'm saying is that I got 1373 01:10:25,970 --> 01:10:28,160 these A's and B's sitting on-- 1374 01:10:28,160 --> 01:10:30,018 you can't call them exactly lattice site, 1375 01:10:30,018 --> 01:10:32,060 but they're associated with a given lattice site. 1376 01:10:32,060 --> 01:10:35,090 They may be displaced from it, and they sort of vibrate around 1377 01:10:35,090 --> 01:10:37,640 and I integrate that those vibrations to get 1378 01:10:37,640 --> 01:10:38,870 the vibrational free energy. 1379 01:10:38,870 --> 01:10:42,320 And then once in a while they hop, exchange, 1380 01:10:42,320 --> 01:10:46,490 and if I sample that that gives me the free energy coming 1381 01:10:46,490 --> 01:10:49,200 from that slower timescale. 1382 01:10:49,200 --> 01:10:51,240 There's one assumption I've made in all of this. 1383 01:10:54,700 --> 01:10:59,070 It's sort of a subtle one, but I've essentially 1384 01:10:59,070 --> 01:11:03,460 assumed that the time scales are uncoupled, 1385 01:11:03,460 --> 01:11:05,572 and here you have to be careful with what I say. 1386 01:11:05,572 --> 01:11:07,030 I say the timescales are uncoupled. 1387 01:11:07,030 --> 01:11:09,070 I don't mean the energetics is uncoupled. 1388 01:11:09,070 --> 01:11:12,790 Obviously the vibrational free energy 1389 01:11:12,790 --> 01:11:14,510 depends on the configuration. 1390 01:11:14,510 --> 01:11:16,390 If I arrange the atoms differently 1391 01:11:16,390 --> 01:11:18,250 over the lattice sites, I get a different vibrational free 1392 01:11:18,250 --> 01:11:18,750 energy. 1393 01:11:18,750 --> 01:11:20,020 That's not the problem. 1394 01:11:20,020 --> 01:11:25,100 I've assumed that you can define a free energy. 1395 01:11:25,100 --> 01:11:25,880 OK? 1396 01:11:25,880 --> 01:11:28,050 That this thing exists. 1397 01:11:28,050 --> 01:11:30,290 So what does that physically mean? 1398 01:11:30,290 --> 01:11:35,870 What I assume is that for a given lattice model state, 1399 01:11:35,870 --> 01:11:37,640 the system actually waits long enough 1400 01:11:37,640 --> 01:11:39,740 before it goes to the next one. 1401 01:11:39,740 --> 01:11:42,920 If it actually only did one vibration and bam, 1402 01:11:42,920 --> 01:11:45,860 it goes to the other one, then the system is not 1403 01:11:45,860 --> 01:11:47,353 ergodic in its vibration. 1404 01:11:47,353 --> 01:11:48,770 And what that means is essentially 1405 01:11:48,770 --> 01:11:50,390 it doesn't sample all its vibrations 1406 01:11:50,390 --> 01:11:53,490 before it goes onto the next one. 1407 01:11:53,490 --> 01:11:57,420 So it's the fact that I can separate the excitation that 1408 01:11:57,420 --> 01:11:59,070 really allows me to do this. 1409 01:11:59,070 --> 01:12:04,110 Now in most materials, this is no problem whatsoever. 1410 01:12:04,110 --> 01:12:08,160 So vibrations, again, these are timescale 10 to minus 11, 10 1411 01:12:08,160 --> 01:12:12,090 to the minus 13 kind of range. 1412 01:12:12,090 --> 01:12:14,610 The exchanges between atoms on lattices 1413 01:12:14,610 --> 01:12:17,130 depends on the diffusion constant, 1414 01:12:17,130 --> 01:12:20,310 but you're going to be hard pressed 1415 01:12:20,310 --> 01:12:22,950 to find any solid where that happens faster 1416 01:12:22,950 --> 01:12:26,950 than of a rate of, say, 10 to the 4 per second, for example. 1417 01:12:26,950 --> 01:12:28,590 10 to the 4, 10 to the 5 per second. 1418 01:12:28,590 --> 01:12:30,765 That's really fast diffusion. 1419 01:12:30,765 --> 01:12:32,640 That gives you diffusion constants of like 10 1420 01:12:32,640 --> 01:12:35,400 to the minus 7, 10 to minus 8, which are very high. 1421 01:12:35,400 --> 01:12:37,550 Extremely high. 1422 01:12:37,550 --> 01:12:40,110 So at room temperature, for a lot of-- 1423 01:12:40,110 --> 01:12:44,980 say for metals, this is well below one hop per second. 1424 01:12:44,980 --> 01:12:49,230 Fast conductors you start to get to a few hops per second 1425 01:12:49,230 --> 01:12:50,130 at room temperature. 1426 01:12:50,130 --> 01:12:51,505 Then of course, if you go higher, 1427 01:12:51,505 --> 01:12:53,560 temperature goes faster, but almost always are 1428 01:12:53,560 --> 01:12:57,050 these timescales extremely well separated. 1429 01:12:57,050 --> 01:13:00,720 Places where they might not be like fast proton motion. 1430 01:13:00,720 --> 01:13:02,760 They might not be very well separated 1431 01:13:02,760 --> 01:13:04,500 and then this stuff breaks down. 1432 01:13:04,500 --> 01:13:07,050 But remember, if they move that fast, you should just 1433 01:13:07,050 --> 01:13:10,050 do molecular dynamics because then it's within the range. 1434 01:13:10,050 --> 01:13:13,080 It's well within the scope of molecular dynamics, 1435 01:13:13,080 --> 01:13:17,380 and that's the perfect approach at that point. 1436 01:13:17,380 --> 01:13:17,880 OK. 1437 01:13:42,680 --> 01:13:49,160 OK, let me do one more thing and then we'll stop. 1438 01:13:49,160 --> 01:13:51,680 There's a variety of approximations you can do. 1439 01:13:55,810 --> 01:13:59,140 Remember that your Hamiltonian in your lattice model 1440 01:13:59,140 --> 01:14:03,580 should be the free energy of the higher order states, which 1441 01:14:03,580 --> 01:14:07,000 are essentially the vibrational and the electronic excitations 1442 01:14:07,000 --> 01:14:09,640 that you've removed. 1443 01:14:09,640 --> 01:14:13,600 Now in some cases people say, I don't 1444 01:14:13,600 --> 01:14:16,960 want to do all that work of integrating 1445 01:14:16,960 --> 01:14:18,850 over the electronic states and integrating 1446 01:14:18,850 --> 01:14:20,230 over the vibrational states. 1447 01:14:20,230 --> 01:14:22,060 I'm going to variationally remove 1448 01:14:22,060 --> 01:14:25,030 them, which means I'm going to find the lowest 1449 01:14:25,030 --> 01:14:28,630 energy electronic states and the lowest energy displacement 1450 01:14:28,630 --> 01:14:30,770 state, delta ri state. 1451 01:14:30,770 --> 01:14:31,390 OK? 1452 01:14:31,390 --> 01:14:33,310 So I do that practically while you 1453 01:14:33,310 --> 01:14:35,350 take an arrangement of atoms and you just 1454 01:14:35,350 --> 01:14:38,830 relax them both the electronic states and the positions 1455 01:14:38,830 --> 01:14:43,220 to the minimum energy, and that gives you some E value. 1456 01:14:43,220 --> 01:14:44,960 So what it essentially is is if you 1457 01:14:44,960 --> 01:14:48,410 think of the F as the free energy of an ensemble, 1458 01:14:48,410 --> 01:14:52,820 the E you take is the lowest energy value in that ensemble. 1459 01:14:52,820 --> 01:14:55,280 So if you do that and then you stick that in Monte Carlo, 1460 01:14:55,280 --> 01:14:56,600 what do you have? 1461 01:14:56,600 --> 01:14:58,610 You have proper energetics and you only 1462 01:14:58,610 --> 01:15:00,980 have configurational entropy because rather 1463 01:15:00,980 --> 01:15:03,560 than integrating all the vibrations and the electronics 1464 01:15:03,560 --> 01:15:08,950 states, you've taken the minimal state out of that sub-ensemble. 1465 01:15:08,950 --> 01:15:10,630 You'll integrate over-- that means 1466 01:15:10,630 --> 01:15:14,140 essentially your system hasn't sampled excursions 1467 01:15:14,140 --> 01:15:15,880 for those variables, so you don't have 1468 01:15:15,880 --> 01:15:17,960 entropy from those variables. 1469 01:15:17,960 --> 01:15:20,270 OK? 1470 01:15:20,270 --> 01:15:22,580 And then you can do all kinds of approximation. 1471 01:15:22,580 --> 01:15:25,940 You can say, well, I don't care about the vibrations. 1472 01:15:25,940 --> 01:15:27,650 I think that's too much work. 1473 01:15:27,650 --> 01:15:29,743 I'm going to just get the electronic entropy. 1474 01:15:29,743 --> 01:15:31,910 And one of the reasons electronic entropy and metals 1475 01:15:31,910 --> 01:15:34,580 is easy, any time you have delocalized states, 1476 01:15:34,580 --> 01:15:36,028 you can write as a simple integral 1477 01:15:36,028 --> 01:15:37,820 over the density of states, which you often 1478 01:15:37,820 --> 01:15:40,470 have [INAUDIBLE] density functional theory. 1479 01:15:40,470 --> 01:15:43,040 So if you say I'm going to take the minimal energy 1480 01:15:43,040 --> 01:15:44,762 and the electronic entropy, well then 1481 01:15:44,762 --> 01:15:46,220 after you've done your Monte Carlo, 1482 01:15:46,220 --> 01:15:50,270 you have configurational entropy and electronic entropy. 1483 01:15:50,270 --> 01:15:54,230 And then if you want to go all the way, 1484 01:15:54,230 --> 01:15:56,653 rather than minimizing the energy you integrate over 1485 01:15:56,653 --> 01:15:58,070 to displacement, then you're going 1486 01:15:58,070 --> 01:16:00,910 to get the vibrational entropy component as well. 1487 01:16:07,180 --> 01:16:08,420 OK. 1488 01:16:08,420 --> 01:16:10,420 I'm going to stop here because the rest is going 1489 01:16:10,420 --> 01:16:11,860 to take us a little too long. 1490 01:16:11,860 --> 01:16:16,240 And so I'll pick some of this up again. 1491 01:16:16,240 --> 01:16:18,640 It's unfortunately almost two weeks away from now, 1492 01:16:18,640 --> 01:16:22,420 I think, because like I said, so Tuesday's no lecture. 1493 01:16:22,420 --> 01:16:25,592 Thursday-- oh, wait Thursday's you, no? 1494 01:16:25,592 --> 01:16:28,050 Yeah, so actually Thursday's a lecture by Professor Marzari 1495 01:16:28,050 --> 01:16:31,650 and then it's the next Tuesday's the lab and then Thursday 1496 01:16:31,650 --> 01:16:34,085 after that I pick this up again and then 1497 01:16:34,085 --> 01:16:35,460 I think we're in May or something 1498 01:16:35,460 --> 01:16:37,740 and we're almost over. 1499 01:16:37,740 --> 01:16:40,530 Anyway, so have a good-- 1500 01:16:40,530 --> 01:16:42,030 what is this holiday again? 1501 01:16:42,030 --> 01:16:42,770 [INAUDIBLE] Day? 1502 01:16:42,770 --> 01:16:44,060 No? 1503 01:16:44,060 --> 01:16:44,970 [INAUDIBLE] Day? 1504 01:16:44,970 --> 01:16:45,540 OK. 1505 01:16:45,540 --> 01:16:47,510 Watch the marathon.