1 00:00:00,000 --> 00:00:01,620 PROFESSOR: So what we'll do today 2 00:00:01,620 --> 00:00:07,020 is really switching to the computational description 3 00:00:07,020 --> 00:00:10,560 and characterization of thermodynamic properties. 4 00:00:10,560 --> 00:00:13,320 As we have seen in a lot of the previous lectures 5 00:00:13,320 --> 00:00:16,620 that we have been focusing on finding out good energy 6 00:00:16,620 --> 00:00:19,170 models, that is being able to calculate, 7 00:00:19,170 --> 00:00:21,960 what is the energy of a system given 8 00:00:21,960 --> 00:00:24,600 the coordinates of its ions? 9 00:00:24,600 --> 00:00:26,400 And now what we want to understand, really, 10 00:00:26,400 --> 00:00:30,690 is, what is the evolution of the system in the presence 11 00:00:30,690 --> 00:00:32,352 of a finite temperature? 12 00:00:32,352 --> 00:00:33,810 Temperature really means that there 13 00:00:33,810 --> 00:00:37,140 is an average kinetic energy available for all 14 00:00:37,140 --> 00:00:39,720 the atoms and the molecules in your system. 15 00:00:39,720 --> 00:00:43,080 That means that there is really a microscopic dynamics. 16 00:00:43,080 --> 00:00:46,050 And we want to follow that. 17 00:00:46,050 --> 00:00:47,610 One of the most important things-- 18 00:00:47,610 --> 00:00:50,520 and this goes back to, really, the first lecture that you have 19 00:00:50,520 --> 00:00:54,030 seen in this class for the quantum part-- 20 00:00:54,030 --> 00:00:59,130 is that there is the fundamental de Broglie relation that 21 00:00:59,130 --> 00:01:03,450 gives us an estimate of what is the quantum wavelength 22 00:01:03,450 --> 00:01:05,040 of any kind of object. 23 00:01:05,040 --> 00:01:09,600 Remember, we have said that the wavelength times the momentum 24 00:01:09,600 --> 00:01:12,750 should be equal to the Planck constant. 25 00:01:12,750 --> 00:01:17,190 And that really means that for an object like an electron, 26 00:01:17,190 --> 00:01:20,340 the de Broglie wavelength that comes out from that relation 27 00:01:20,340 --> 00:01:24,070 is comparable to the distance between the atoms. 28 00:01:24,070 --> 00:01:26,190 So if the wavelength of an electron 29 00:01:26,190 --> 00:01:29,610 is of the order of the angstrom, 10 to the minus 10 meters, 30 00:01:29,610 --> 00:01:33,720 it means that the wavelike properties of the electrons 31 00:01:33,720 --> 00:01:34,830 needs to be studied. 32 00:01:34,830 --> 00:01:37,260 And the electron will behave as a wave. 33 00:01:37,260 --> 00:01:39,360 And that's why we use the Schrodinger 34 00:01:39,360 --> 00:01:42,570 equation to describe electrons. 35 00:01:42,570 --> 00:01:46,800 The wavelength of the nuclei is much, much smaller. 36 00:01:46,800 --> 00:01:50,160 That means that nuclei really didn't-- 37 00:01:50,160 --> 00:01:54,120 we don't really need to take into account 38 00:01:54,120 --> 00:01:58,320 to a good approximation the quantum nature of the nuclei. 39 00:01:58,320 --> 00:02:00,960 The wavelength is so small that they 40 00:02:00,960 --> 00:02:05,070 can be treated practically, correctly, 41 00:02:05,070 --> 00:02:07,320 as classical particles. 42 00:02:07,320 --> 00:02:10,050 And that's why, actually, nuclei, in a way, 43 00:02:10,050 --> 00:02:13,680 still follow the rules of classical dynamics. 44 00:02:13,680 --> 00:02:16,830 That is, they evolve in time following 45 00:02:16,830 --> 00:02:18,570 Newton's equation of motion. 46 00:02:18,570 --> 00:02:20,610 Again, this is an approximation that 47 00:02:20,610 --> 00:02:23,730 involves treating the nuclei as classical particles 48 00:02:23,730 --> 00:02:26,040 and not as quantum particles. 49 00:02:26,040 --> 00:02:28,290 But especially at high temperature, 50 00:02:28,290 --> 00:02:30,610 this is a very good approximation. 51 00:02:30,610 --> 00:02:34,930 And so a lot of approaches in computational and material 52 00:02:34,930 --> 00:02:39,840 science have to do with solving Newton's equation of motion 53 00:02:39,840 --> 00:02:42,300 for a system of interacting nuclei 54 00:02:42,300 --> 00:02:44,580 and evolving those nuclei in time 55 00:02:44,580 --> 00:02:47,590 to study the thermodynamic properties of the system. 56 00:02:47,590 --> 00:02:51,000 And so that's why I actually started with a reminder 57 00:02:51,000 --> 00:02:52,980 of Newton's equation of motion. 58 00:02:52,980 --> 00:02:56,730 That is really an ordinary second-order differential 59 00:02:56,730 --> 00:02:57,570 equation. 60 00:02:57,570 --> 00:03:01,020 That is, when you have a particle of mass m in a force 61 00:03:01,020 --> 00:03:04,020 field that is this F of r, well, we 62 00:03:04,020 --> 00:03:07,200 can calculate its trajectory-- that 63 00:03:07,200 --> 00:03:09,930 is, its position-- as a function of time 64 00:03:09,930 --> 00:03:13,080 and its velocity as a function of time 65 00:03:13,080 --> 00:03:16,320 by integrating this differential equation. 66 00:03:16,320 --> 00:03:20,190 And in the classical case that is always presented, 67 00:03:20,190 --> 00:03:25,380 say, is the evolution of a particle in a constant force 68 00:03:25,380 --> 00:03:30,000 field, like a particle under the gravitational law, that, 69 00:03:30,000 --> 00:03:33,870 to a large extent, is constant if you don't move very much. 70 00:03:33,870 --> 00:03:39,060 And because, in that case, the force is really a constant, 71 00:03:39,060 --> 00:03:40,970 the integration of this becomes trivial. 72 00:03:40,970 --> 00:03:43,890 It's really a second derivative that 73 00:03:43,890 --> 00:03:45,180 needs to be equal a constant. 74 00:03:45,180 --> 00:03:48,060 And integrating that gives us a parabola. 75 00:03:48,060 --> 00:03:51,960 But the reason why I'm focusing on this 76 00:03:51,960 --> 00:03:57,480 is to remind you of the initial conditions in our differential 77 00:03:57,480 --> 00:03:57,990 equations. 78 00:03:57,990 --> 00:04:00,420 That is, if we have a second-order differential 79 00:04:00,420 --> 00:04:04,980 equation, the general solution-- let's say in this case 80 00:04:04,980 --> 00:04:08,700 for this force field would be a parabola-- is uniquely 81 00:04:08,700 --> 00:04:14,160 identified once we specify the position at a certain instant 82 00:04:14,160 --> 00:04:17,040 and the velocity at a certain instant. 83 00:04:17,040 --> 00:04:20,399 That is, among all the possible parabolas in the world, 84 00:04:20,399 --> 00:04:23,340 you selected the one that your projectile 85 00:04:23,340 --> 00:04:26,730 is going to follow once you have the position 86 00:04:26,730 --> 00:04:29,130 and the velocity at a certain instant. 87 00:04:29,130 --> 00:04:33,420 And that's why actually dynamics and molecular dynamics 88 00:04:33,420 --> 00:04:36,360 and other physical properties basically depend essentially 89 00:04:36,360 --> 00:04:39,540 on the position of the particle in your system 90 00:04:39,540 --> 00:04:43,210 and the velocity of the particle in your system. 91 00:04:43,210 --> 00:04:46,860 That's why there is no physical observable that depends, say, 92 00:04:46,860 --> 00:04:49,050 on the fifth derivative with respect to time. 93 00:04:49,050 --> 00:04:51,480 It's really because the dynamics of that system 94 00:04:51,480 --> 00:04:54,500 follows Newton's equation of motion. 95 00:04:54,500 --> 00:04:59,040 OK, with this, what we really need to deal in general 96 00:04:59,040 --> 00:05:03,402 is with a system of interacting particles. 97 00:05:03,402 --> 00:05:04,860 And for a moment you could actually 98 00:05:04,860 --> 00:05:09,390 think that this i, interacting particles, 99 00:05:09,390 --> 00:05:12,670 could be the planets orbiting around the sun. 100 00:05:12,670 --> 00:05:16,770 So each of the planet i will have its own trajectory. 101 00:05:16,770 --> 00:05:18,990 Our r of t will have its own mass. 102 00:05:18,990 --> 00:05:21,930 And they all live in a force field. 103 00:05:21,930 --> 00:05:24,510 That's really where the many-body complexity, again, 104 00:05:24,510 --> 00:05:25,440 comes out. 105 00:05:25,440 --> 00:05:27,480 So many-body complexity, that is much easier 106 00:05:27,480 --> 00:05:31,140 to deal with than the many-body complexity of the electron 107 00:05:31,140 --> 00:05:35,730 problem because, again, now we describe each object just 108 00:05:35,730 --> 00:05:39,420 with a set of six variables, so its 3-space coordinate 109 00:05:39,420 --> 00:05:41,580 and its 3-momentum coordinate. 110 00:05:41,580 --> 00:05:45,720 But still, each and every object interacts with each other. 111 00:05:45,720 --> 00:05:48,270 When you look at the planets orbiting around the sun, 112 00:05:48,270 --> 00:05:50,640 you, at first approximation, could only 113 00:05:50,640 --> 00:05:53,220 consider the gravitational attraction to the sun. 114 00:05:53,220 --> 00:05:57,400 But in reality, also the planets interact with each other. 115 00:05:57,400 --> 00:06:00,480 And so, again, even if this is much simpler 116 00:06:00,480 --> 00:06:02,310 than the Schrodinger equation, as soon 117 00:06:02,310 --> 00:06:05,130 as you start to have more than two bodies interacting, 118 00:06:05,130 --> 00:06:08,760 it really becomes too complex to solve analytically. 119 00:06:08,760 --> 00:06:11,610 And so all the molecular dynamic techniques 120 00:06:11,610 --> 00:06:14,250 have, as their only object, basically, 121 00:06:14,250 --> 00:06:19,170 the integration in time of this equation of motion 122 00:06:19,170 --> 00:06:22,020 in an efficient form. 123 00:06:22,020 --> 00:06:25,740 There are a couple of comments that one wants to make. 124 00:06:30,130 --> 00:06:33,390 In general, we deal with conservative fields. 125 00:06:33,390 --> 00:06:37,480 That is, the force depends on position only. 126 00:06:37,480 --> 00:06:40,150 That also means that the work that you 127 00:06:40,150 --> 00:06:44,800 make in going from one point to another point, the integral 128 00:06:44,800 --> 00:06:47,380 of the force times the displacement 129 00:06:47,380 --> 00:06:51,140 is going to be independent from the trajectory. 130 00:06:51,140 --> 00:06:53,590 So the energy, if you want, is just 131 00:06:53,590 --> 00:06:56,660 a function of the position. 132 00:06:56,660 --> 00:06:59,980 The other thing that descends immediately 133 00:06:59,980 --> 00:07:04,360 from this equation of motion is that the total energy 134 00:07:04,360 --> 00:07:06,830 of the system is conserved. 135 00:07:06,830 --> 00:07:08,810 And that's very easy to prove. 136 00:07:08,810 --> 00:07:10,100 And I'll do it in a moment. 137 00:07:10,100 --> 00:07:14,590 And that's what's often called a microcanonical evolution. 138 00:07:14,590 --> 00:07:19,270 When you read about microcanonical ensembles, 139 00:07:19,270 --> 00:07:22,420 it means that you are actually under the condition 140 00:07:22,420 --> 00:07:26,560 in which the total energy of your system is conserved. 141 00:07:26,560 --> 00:07:29,570 And again, what is the total energy of your system? 142 00:07:29,570 --> 00:07:33,520 Well, if you have one particle, it's just, trivially, 143 00:07:33,520 --> 00:07:38,050 the kinetic energy 1/2 mv squared 144 00:07:38,050 --> 00:07:42,640 plus the potential energy V of r of that particle, 145 00:07:42,640 --> 00:07:45,460 and just doing everything in one dimension. 146 00:07:45,460 --> 00:07:48,670 And you ask, why is this going to be conserved? 147 00:07:48,670 --> 00:07:53,110 Well, you can just prove it by taking the time derivative 148 00:07:53,110 --> 00:07:55,060 of this quantity, OK? 149 00:07:55,060 --> 00:07:58,630 So we have taken kinetic energy plus potential energy. 150 00:07:58,630 --> 00:08:00,460 We look at the time derivative. 151 00:08:00,460 --> 00:08:03,700 And this is just simple algebra. 152 00:08:03,700 --> 00:08:10,840 What you get is 1/2 m times 2 times v times the derivative 153 00:08:10,840 --> 00:08:13,960 with respect to time, that is the acceleration, 154 00:08:13,960 --> 00:08:20,210 plus the derivative of the potential with respect to time. 155 00:08:20,210 --> 00:08:26,440 And this can just be written as mass times velocity times 156 00:08:26,440 --> 00:08:28,420 acceleration plus-- 157 00:08:28,420 --> 00:08:30,190 and we can rewrite the derivative 158 00:08:30,190 --> 00:08:32,830 of the potential with time as the derivative 159 00:08:32,830 --> 00:08:35,679 of the potential with respect to position 160 00:08:35,679 --> 00:08:39,892 times the derivative of the position with respect to time. 161 00:08:39,892 --> 00:08:41,559 And now you see where we are getting to, 162 00:08:41,559 --> 00:08:48,550 because this term here is equal to minus the force, 163 00:08:48,550 --> 00:08:52,460 while these terms here is equal to the velocity. 164 00:08:52,460 --> 00:08:56,030 And so it goes away with here. 165 00:08:56,030 --> 00:09:00,510 And so what we have is that m, the derivative 166 00:09:00,510 --> 00:09:04,080 of the total energy with respect to time, 167 00:09:04,080 --> 00:09:08,190 is equal to the mass times acceleration minus the force. 168 00:09:08,190 --> 00:09:12,930 That is 0 because the particle follows 169 00:09:12,930 --> 00:09:14,650 Newton's equation of motion. 170 00:09:14,650 --> 00:09:16,630 So because of Newton's equation of motion 171 00:09:16,630 --> 00:09:20,190 and because the particle follows exactly that evolution, 172 00:09:20,190 --> 00:09:22,740 we have that the sum of the kinetic energy 173 00:09:22,740 --> 00:09:24,600 plus the potential energy needs to be 174 00:09:24,600 --> 00:09:27,540 conserved during the evolution. 175 00:09:27,540 --> 00:09:31,950 And that's actually the fundamental sanity check 176 00:09:31,950 --> 00:09:34,620 that you will always have and you can always 177 00:09:34,620 --> 00:09:37,350 do during a molecular dynamics simulation. 178 00:09:37,350 --> 00:09:40,350 That is, you can check that your integration is 179 00:09:40,350 --> 00:09:44,250 correct by making sure that the total energy of your system 180 00:09:44,250 --> 00:09:46,440 is concerned. 181 00:09:46,440 --> 00:09:51,000 OK, so what do we do in a practical molecular dynamic 182 00:09:51,000 --> 00:09:51,810 simulation? 183 00:09:51,810 --> 00:09:54,960 Well, the idea is that, as in the motion of the planets, 184 00:09:54,960 --> 00:09:58,710 we want to integrate the equation of motions 185 00:09:58,710 --> 00:10:02,880 with this caveat that, again, each particle interacts 186 00:10:02,880 --> 00:10:05,200 with each other particle. 187 00:10:05,200 --> 00:10:08,160 And so it's really a many-body problem, 188 00:10:08,160 --> 00:10:11,200 even if it is a classical many-body problem. 189 00:10:11,200 --> 00:10:13,980 And so it's really easier to integrate 190 00:10:13,980 --> 00:10:19,620 than a many-body Schrodinger equation. 191 00:10:19,620 --> 00:10:23,580 And in general, the way particles interact 192 00:10:23,580 --> 00:10:28,170 with each other can be described by empirical force fields 193 00:10:28,170 --> 00:10:32,160 or, in the best-case scenario, can be described 194 00:10:32,160 --> 00:10:34,650 using quantum mechanics. 195 00:10:34,650 --> 00:10:37,260 That is the [? functional ?] theory [INAUDIBLE].. 196 00:10:37,260 --> 00:10:40,290 And so what you will have to do in that case 197 00:10:40,290 --> 00:10:44,790 is that for every configuration of your ionic system, 198 00:10:44,790 --> 00:10:48,870 you need to solve an entire total energy self-consistent 199 00:10:48,870 --> 00:10:49,470 problem. 200 00:10:49,470 --> 00:10:51,262 And that's why actually the first principle 201 00:10:51,262 --> 00:10:55,920 in molecular dynamics becomes a very, very expensive challenge. 202 00:10:55,920 --> 00:11:00,330 And we'll see in one of the later classes 203 00:11:00,330 --> 00:11:03,070 how this is solved. 204 00:11:03,070 --> 00:11:06,810 There are a lot of systems that can be described accurately, 205 00:11:06,810 --> 00:11:10,140 even with very simple potential. 206 00:11:10,140 --> 00:11:13,620 Even more than in the case of the total energy, 207 00:11:13,620 --> 00:11:15,900 certain thermodynamic transitions, 208 00:11:15,900 --> 00:11:19,290 like the transition from a solid to a liquid, 209 00:11:19,290 --> 00:11:23,490 can be thought of as broadly independent of the details 210 00:11:23,490 --> 00:11:25,650 of the interacting potentials. 211 00:11:25,650 --> 00:11:30,480 So even studying a system with a Lennard-Jones potential that, 212 00:11:30,480 --> 00:11:34,080 after all, was developed especially to do molecular 213 00:11:34,080 --> 00:11:37,290 dynamic simulations of rare gas atoms, 214 00:11:37,290 --> 00:11:43,650 or even just studying a system that is made by hard spheres 215 00:11:43,650 --> 00:11:47,730 that don't really interact when they travel in space, 216 00:11:47,730 --> 00:11:49,990 apart when they really hit each other-- 217 00:11:49,990 --> 00:11:53,100 so what is called a contact interaction that is 0 all 218 00:11:53,100 --> 00:11:56,670 the time apart from the instant in which the two spheres 219 00:11:56,670 --> 00:11:57,420 collide-- 220 00:11:57,420 --> 00:12:01,080 can sometimes give you a very accurate qualitative 221 00:12:01,080 --> 00:12:05,400 description of the thermodynamic transition in a problem. 222 00:12:05,400 --> 00:12:13,270 And molecular dynamics actually was first introduced really 223 00:12:13,270 --> 00:12:18,700 to study systems, like liquids, for which it's very difficult 224 00:12:18,700 --> 00:12:21,160 to develop analytic techniques. 225 00:12:21,160 --> 00:12:25,900 You see, if you want to study a solid, 226 00:12:25,900 --> 00:12:29,590 you can actually choose a potential. 227 00:12:29,590 --> 00:12:31,810 And if the potential is simple enough, 228 00:12:31,810 --> 00:12:35,950 you can analytically calculate the potential energy 229 00:12:35,950 --> 00:12:40,060 of a periodic repetition of all your atoms. 230 00:12:40,060 --> 00:12:44,050 But if you study liquid, disorder 231 00:12:44,050 --> 00:12:48,830 is really an essential part of the state of the system. 232 00:12:48,830 --> 00:12:51,880 And so the entropic contribution to the free energy 233 00:12:51,880 --> 00:12:54,040 of the system is fundamental. 234 00:12:54,040 --> 00:12:58,600 And really there are no reasonable analytic ways 235 00:12:58,600 --> 00:13:01,610 to address a lot of the questions that you have. 236 00:13:01,610 --> 00:13:04,270 And that's why people really moved 237 00:13:04,270 --> 00:13:08,750 to solving on the computer the Newton equation of motion. 238 00:13:08,750 --> 00:13:13,750 And this is a short history, really, of molecular dynamics, 239 00:13:13,750 --> 00:13:17,260 reminding you what is the history of computers, really. 240 00:13:17,260 --> 00:13:21,580 In 1952, one of the first computers, MANIAC, 241 00:13:21,580 --> 00:13:23,990 was operational in Los Alamos. 242 00:13:23,990 --> 00:13:27,130 And that's when Metropolis, Nicholas Metropolis, 243 00:13:27,130 --> 00:13:31,657 in collaboration with actually a husband and wife 244 00:13:31,657 --> 00:13:33,490 couple, Rosenbluth and Rosenbluth and Teller 245 00:13:33,490 --> 00:13:37,540 and Teller, developed what is called the Monte Carlo method. 246 00:13:37,540 --> 00:13:40,930 That was a very powerful method to address 247 00:13:40,930 --> 00:13:44,260 exactly this entropic free energy problem. 248 00:13:44,260 --> 00:13:47,770 And that you'll see in one of the later classes. 249 00:13:47,770 --> 00:13:54,010 But really, 1956 sees the first molecular dynamic simulation 250 00:13:54,010 --> 00:13:56,020 by Alder and Wainwright. 251 00:13:56,020 --> 00:14:00,880 And as I said, in looking at the dynamics of hard spheres, 252 00:14:00,880 --> 00:14:03,100 these are the milestone papers. 253 00:14:03,100 --> 00:14:05,710 Vineyard is studying radiation damage 254 00:14:05,710 --> 00:14:10,180 in copper, what happens when you have a copper atom that 255 00:14:10,180 --> 00:14:12,760 is really put in motion by a very 256 00:14:12,760 --> 00:14:17,200 energetic radiation kicking it and starting running around. 257 00:14:17,200 --> 00:14:20,530 And really this is the classic paper 258 00:14:20,530 --> 00:14:24,880 of Aneesur Rahman studying, with a Lennard-Jones potential, 259 00:14:24,880 --> 00:14:26,860 the dynamics of liquid argon. 260 00:14:26,860 --> 00:14:29,710 Again, a Lennard-Jones potential is actually very accurate 261 00:14:29,710 --> 00:14:33,460 to describe rare gases that ultimately 262 00:14:33,460 --> 00:14:35,680 repel each other at very close distance 263 00:14:35,680 --> 00:14:38,710 and, at large distances, attract each other 264 00:14:38,710 --> 00:14:44,170 with this weak dipole-dipole interaction that goes as 1 265 00:14:44,170 --> 00:14:45,820 over r to the 6th. 266 00:14:45,820 --> 00:14:49,660 And then really the first ab-initio molecular dynamics 267 00:14:49,660 --> 00:14:51,910 and the first theory for ab-initio molecular dynamics 268 00:14:51,910 --> 00:14:58,140 was developed in the mid '80s by Car and Parrinello. 269 00:14:58,140 --> 00:15:03,120 OK, now, there is a universal way 270 00:15:03,120 --> 00:15:08,400 to think of these dynamics of your classical system composed 271 00:15:08,400 --> 00:15:09,910 of N particles. 272 00:15:09,910 --> 00:15:12,570 And as we have said, since they follow 273 00:15:12,570 --> 00:15:15,870 Newton's equation of motion, what we really need to track, 274 00:15:15,870 --> 00:15:18,060 the only quantities that we need to track, 275 00:15:18,060 --> 00:15:21,120 is their position as a function of time 276 00:15:21,120 --> 00:15:23,850 and their velocity as a function of time. 277 00:15:23,850 --> 00:15:26,070 And generally speaking, the kinetic energy 278 00:15:26,070 --> 00:15:28,500 is just a function of the velocities. 279 00:15:28,500 --> 00:15:32,500 And the potential energy is just a function of the positions. 280 00:15:32,500 --> 00:15:34,980 So if you have N particles, you really 281 00:15:34,980 --> 00:15:40,390 have to follow in time six n variables. 282 00:15:40,390 --> 00:15:47,280 And if we define for a moment a six-dimensional space, what 283 00:15:47,280 --> 00:15:51,420 we really have is that an instantaneous state 284 00:15:51,420 --> 00:15:55,170 of your dynamical system, that is all its position and all 285 00:15:55,170 --> 00:16:01,440 its velocity, can be represented by a single point in this 6N 286 00:16:01,440 --> 00:16:03,010 dimension of space. 287 00:16:03,010 --> 00:16:05,190 So if you have got just a hydrogen molecule 288 00:16:05,190 --> 00:16:07,740 with two nuclei, what you really need 289 00:16:07,740 --> 00:16:10,265 is a point in six dimensions. 290 00:16:10,265 --> 00:16:11,640 If you have a water molecule, you 291 00:16:11,640 --> 00:16:13,920 need a point in nine dimensions. 292 00:16:13,920 --> 00:16:18,140 And the evolution-- sorry. 293 00:16:18,140 --> 00:16:20,500 If you have a water molecule, you 294 00:16:20,500 --> 00:16:24,290 need a point in 18 dimensions. 295 00:16:24,290 --> 00:16:26,110 And if you have a hydrogen molecule, 296 00:16:26,110 --> 00:16:28,300 you need a point in 12 dimensions. 297 00:16:28,300 --> 00:16:30,760 And the dynamical evolution of your system 298 00:16:30,760 --> 00:16:35,860 is described exactly by the evolution of this point. 299 00:16:35,860 --> 00:16:41,720 So we often think at this phase space that we'll have, 300 00:16:41,720 --> 00:16:47,760 for an N particle, 6N dimension, of which-- 301 00:16:47,760 --> 00:16:51,530 let's try to draw a bit dimension-- 302 00:16:51,530 --> 00:16:54,680 3N of them, those represented in green, 303 00:16:54,680 --> 00:16:57,620 will really represent the positions. 304 00:16:57,620 --> 00:17:01,470 And the other 3N will represent velocities. 305 00:17:01,470 --> 00:17:02,180 OK. 306 00:17:02,180 --> 00:17:05,060 And obviously this is all orthogonal dimension. 307 00:17:05,060 --> 00:17:09,589 And your point will be somewhere. 308 00:17:09,589 --> 00:17:13,970 The certain instant in time T will have a set of velocities, 309 00:17:13,970 --> 00:17:15,470 will have a set of positions. 310 00:17:15,470 --> 00:17:17,599 And the Newton equation of motion 311 00:17:17,599 --> 00:17:22,260 will make it evolve in this phase space. 312 00:17:22,260 --> 00:17:25,910 So really your molecular dynamic integration 313 00:17:25,910 --> 00:17:32,180 consists in following in time this evolution. 314 00:17:32,180 --> 00:17:34,940 And that evolution is, in principle, 315 00:17:34,940 --> 00:17:38,180 analytically and uniquely defined 316 00:17:38,180 --> 00:17:42,650 once you know the position and the velocities 317 00:17:42,650 --> 00:17:45,020 at every instant. 318 00:17:45,020 --> 00:17:49,430 And-- well, we'll see that later. 319 00:17:49,430 --> 00:17:53,900 OK, so what do we do once we have 320 00:17:53,900 --> 00:17:57,740 a computational algorithm that allows us to evolve 321 00:17:57,740 --> 00:18:00,320 this point in phase space? 322 00:18:00,320 --> 00:18:05,150 And I sort of summarized here the three main goals 323 00:18:05,150 --> 00:18:09,230 that could follow from a molecular dynamic simulation 324 00:18:09,230 --> 00:18:13,430 in the order that we'll see them. 325 00:18:13,430 --> 00:18:16,070 Really, one of the most common ones 326 00:18:16,070 --> 00:18:19,670 is to use molecular dynamic simulations 327 00:18:19,670 --> 00:18:23,180 to calculate thermodynamic properties, that 328 00:18:23,180 --> 00:18:28,070 is to calculate what we call ensemble averages. 329 00:18:28,070 --> 00:18:32,430 Suppose that you have a system like a solid. 330 00:18:32,430 --> 00:18:35,130 And you have calculated, say, its lattice parameter, 331 00:18:35,130 --> 00:18:37,860 like you're doing in lab 2 or lab 3. 332 00:18:37,860 --> 00:18:41,790 And suppose that you start heating up this solid. 333 00:18:41,790 --> 00:18:42,930 What happens? 334 00:18:42,930 --> 00:18:45,900 Well, the atoms, in the classical sense, 335 00:18:45,900 --> 00:18:48,720 start oscillating more and more. 336 00:18:48,720 --> 00:18:52,410 And you know from experience that most solids 337 00:18:52,410 --> 00:18:56,290 start expanding when you warm them up. 338 00:18:56,290 --> 00:18:58,530 So a molecular dynamic simulation 339 00:18:58,530 --> 00:19:03,640 could tell you how much a solid expands when you heat it up 340 00:19:03,640 --> 00:19:07,230 because basically you start evolving your system from 0 341 00:19:07,230 --> 00:19:08,160 temperature. 342 00:19:08,160 --> 00:19:10,410 And then you increase its temperature. 343 00:19:10,410 --> 00:19:12,070 And really, temperature, remember, 344 00:19:12,070 --> 00:19:15,580 is just the average kinetic energy of the ions. 345 00:19:15,580 --> 00:19:17,500 And as you increase the temperature, 346 00:19:17,500 --> 00:19:19,600 the ions oscillate more and more. 347 00:19:19,600 --> 00:19:25,530 And if you let your unit cell expand or contract 348 00:19:25,530 --> 00:19:29,580 in time following what is the stress overall 349 00:19:29,580 --> 00:19:32,370 of all the atoms in your unit cell, 350 00:19:32,370 --> 00:19:35,850 you see that your system increases 351 00:19:35,850 --> 00:19:40,110 the dimension of the unit cell and starts oscillating around 352 00:19:40,110 --> 00:19:43,240 a value that's really an average, 353 00:19:43,240 --> 00:19:45,480 depending on temperature. 354 00:19:45,480 --> 00:19:50,500 And all these sort of statements can be actually made formally 355 00:19:50,500 --> 00:19:52,590 in a way that we'll see in a moment that is 356 00:19:52,590 --> 00:19:54,480 the language of thermodynamics. 357 00:19:54,480 --> 00:19:59,220 That is, if we have a finite temperature, 358 00:19:59,220 --> 00:20:03,660 we'll have a certain probability associated 359 00:20:03,660 --> 00:20:06,300 with every microscopic configuration. 360 00:20:06,300 --> 00:20:09,810 OK, so if you are at 0 temperature, 361 00:20:09,810 --> 00:20:12,450 really what you have is that you have a probability 362 00:20:12,450 --> 00:20:17,240 1 to be in the lowest energy state possible and probability 363 00:20:17,240 --> 00:20:21,190 0 of being in any other energy state. 364 00:20:21,190 --> 00:20:24,630 If you start increasing your temperature, 365 00:20:24,630 --> 00:20:28,980 states that are energetically similar to the ground 366 00:20:28,980 --> 00:20:32,040 state but not really the ground state 367 00:20:32,040 --> 00:20:37,020 start to have a certain probability of being occupied. 368 00:20:37,020 --> 00:20:39,580 And the more temperature you have, 369 00:20:39,580 --> 00:20:44,190 the more very costly configurations 370 00:20:44,190 --> 00:20:46,830 can be accessed by your system. 371 00:20:46,830 --> 00:20:48,990 And really, the average properties 372 00:20:48,990 --> 00:20:51,360 at a finite temperature need to be 373 00:20:51,360 --> 00:20:55,410 an average over all these possible states weighed 374 00:20:55,410 --> 00:20:59,160 with their appropriate thermodynamic weight, weighed 375 00:20:59,160 --> 00:21:02,740 with the probability of being in that state. 376 00:21:02,740 --> 00:21:05,880 And so this is what an ensemble average does really. 377 00:21:05,880 --> 00:21:10,440 It tries to span all the possible states in a system 378 00:21:10,440 --> 00:21:13,110 with a probability that is proportional to 379 00:21:13,110 --> 00:21:15,030 their thermodynamical weight. 380 00:21:15,030 --> 00:21:19,770 And then for each state, you can calculate a certain property, 381 00:21:19,770 --> 00:21:28,000 like what is its average potential energy, what 382 00:21:28,000 --> 00:21:32,710 is its lattice parameter, and so on. 383 00:21:32,710 --> 00:21:35,290 And out of that, you can calculate 384 00:21:35,290 --> 00:21:37,150 the average thermodynamic property. 385 00:21:37,150 --> 00:21:40,040 And we'll see, actually, an example in a moment. 386 00:21:40,040 --> 00:21:45,290 So we'll see a practical application of this. 387 00:21:45,290 --> 00:21:48,520 The other thing that you can do with molecular dynamic 388 00:21:48,520 --> 00:21:52,240 simulator is actually study the evolution 389 00:21:52,240 --> 00:21:54,710 in real time of your system. 390 00:21:54,710 --> 00:21:57,790 So you could actually study a chemical reaction. 391 00:21:57,790 --> 00:22:03,460 What happens if you have, say, an explosive molecule that 392 00:22:03,460 --> 00:22:05,530 is decomposing, OK? 393 00:22:05,530 --> 00:22:09,130 And in particular, whenever you have a chemical reaction that 394 00:22:09,130 --> 00:22:12,520 tends to be a bond-breaking and bond-forming reaction, 395 00:22:12,520 --> 00:22:15,790 you'll probably need to use a quantum mechanical approach. 396 00:22:15,790 --> 00:22:18,400 But in principle, you can set up your system 397 00:22:18,400 --> 00:22:21,130 in an initial configuration and see what 398 00:22:21,130 --> 00:22:23,300 happens as you evolve in time. 399 00:22:23,300 --> 00:22:25,270 You could look at a catalytic reaction. 400 00:22:25,270 --> 00:22:28,690 Say, maybe you are interested in studying fuel cells. 401 00:22:28,690 --> 00:22:32,740 And you want to see how a hydrogen molecule decomposes 402 00:22:32,740 --> 00:22:34,930 when it arrives on a platinum surface. 403 00:22:34,930 --> 00:22:37,300 Well, you can take your hydrogen molecule 404 00:22:37,300 --> 00:22:41,530 and project it delicately towards the platinum surface. 405 00:22:41,530 --> 00:22:44,710 And follow, with your molecular dynamic techniques, 406 00:22:44,710 --> 00:22:49,390 the dynamics of all these nuclei as the molecule chemisorbs 407 00:22:49,390 --> 00:22:52,240 and dissociates in time. 408 00:22:52,240 --> 00:22:57,190 And then really the last application 409 00:22:57,190 --> 00:23:05,630 of molecular dynamics is more of an optimization algorithm. 410 00:23:05,630 --> 00:23:10,810 There are problems in which it's very complex 411 00:23:10,810 --> 00:23:15,850 to find what is the lowest possible solution, what 412 00:23:15,850 --> 00:23:17,710 is the optimal solution. 413 00:23:17,710 --> 00:23:20,590 That is, you could try to find out 414 00:23:20,590 --> 00:23:29,050 what is the schedule of all your flights in an airplane company. 415 00:23:29,050 --> 00:23:31,900 And obviously that's a complex optimization problem 416 00:23:31,900 --> 00:23:35,470 because you can't move one plane at a time 417 00:23:35,470 --> 00:23:39,220 and figure out what is the best possible solution because there 418 00:23:39,220 --> 00:23:43,460 might be a completely different choice of planes 419 00:23:43,460 --> 00:23:46,420 and itineraries that actually gives you a best 420 00:23:46,420 --> 00:23:48,560 performance overall on the net. 421 00:23:48,560 --> 00:23:53,170 And so you can always take one optimization problem 422 00:23:53,170 --> 00:23:56,320 and express a cost function. 423 00:23:56,320 --> 00:23:59,860 That is depending on what is your interest. 424 00:23:59,860 --> 00:24:03,310 Minimize the amount of oil, let's say, used 425 00:24:03,310 --> 00:24:05,500 in running all these routes. 426 00:24:05,500 --> 00:24:08,800 So when you have a cost function of a very complex problem, 427 00:24:08,800 --> 00:24:14,320 you have really an object that depends on many variables 428 00:24:14,320 --> 00:24:19,370 and that has a lot of possible minima. 429 00:24:19,370 --> 00:24:21,650 OK, so this would be, really, the problem 430 00:24:21,650 --> 00:24:27,320 of optimizing the air routes of an airline. 431 00:24:27,320 --> 00:24:30,650 What you have is a lot of possible variables. 432 00:24:30,650 --> 00:24:33,830 And there are a lot of reasonable solutions. 433 00:24:33,830 --> 00:24:38,930 But you really want to find the one that is the best for you, 434 00:24:38,930 --> 00:24:41,090 something like this. 435 00:24:41,090 --> 00:24:44,030 And molecular dynamics can actually 436 00:24:44,030 --> 00:24:46,730 help to solve this problem because it's 437 00:24:46,730 --> 00:24:50,930 a problem that is absolutely analogous of having 438 00:24:50,930 --> 00:24:53,690 a system of interacting particles that 439 00:24:53,690 --> 00:24:56,810 has a certain potential energy surface 440 00:24:56,810 --> 00:25:00,320 and trying to explore that potential energy surface, 441 00:25:00,320 --> 00:25:04,970 moving around, until you find the lowest minimum. 442 00:25:04,970 --> 00:25:09,000 You could think of this as a mountain range. 443 00:25:09,000 --> 00:25:11,090 And you have your skiers that are really 444 00:25:11,090 --> 00:25:13,130 your particles moving around. 445 00:25:13,130 --> 00:25:16,520 And you want to sort of let the skiers ski 446 00:25:16,520 --> 00:25:21,470 around as much as possible until they find the minimum energy 447 00:25:21,470 --> 00:25:22,440 state. 448 00:25:22,440 --> 00:25:25,610 And so you can use a thermodynamic analogy. 449 00:25:25,610 --> 00:25:27,950 You heat up your system. 450 00:25:27,950 --> 00:25:31,430 That really means that every particle 451 00:25:31,430 --> 00:25:33,810 has a lot of kinetic energy. 452 00:25:33,810 --> 00:25:36,920 So having a lot of kinetic energy 453 00:25:36,920 --> 00:25:40,550 can easily overcome any kind of potential barrier. 454 00:25:40,550 --> 00:25:43,550 And you have a lot of these particles moving around. 455 00:25:43,550 --> 00:25:46,980 And then you cool them down slowly. 456 00:25:46,980 --> 00:25:49,070 And as you cool them down, they'll 457 00:25:49,070 --> 00:25:55,220 start piling up in the lowest possible energy state. 458 00:25:55,220 --> 00:25:59,720 And once you have reached, very slowly, 0 temperature, 459 00:25:59,720 --> 00:26:03,090 you look at where all your particles have ended up. 460 00:26:03,090 --> 00:26:07,100 And so you have at least a good sampling 461 00:26:07,100 --> 00:26:10,770 of the relevant minima in your potential energy surface. 462 00:26:10,770 --> 00:26:13,850 So actually molecular dynamics-- and this is called simulated 463 00:26:13,850 --> 00:26:15,440 annealing, in this context-- 464 00:26:15,440 --> 00:26:20,869 can actually be used as an optimization technique. 465 00:26:25,000 --> 00:26:32,860 And we don't have to go as far as the case of an airline 466 00:26:32,860 --> 00:26:35,800 or what is called the traveling salesman problem. 467 00:26:35,800 --> 00:26:38,470 But often what we have is that we 468 00:26:38,470 --> 00:26:43,660 have a system that is fairly complicated to characterize. 469 00:26:43,660 --> 00:26:47,410 Say we want to describe maybe an interface 470 00:26:47,410 --> 00:26:53,770 between a perfect silicon crystal and the SiO2 substrate 471 00:26:53,770 --> 00:26:56,350 because we are in the electronic industry. 472 00:26:56,350 --> 00:26:58,570 Well, I mean, we have a problem because how do we 473 00:26:58,570 --> 00:27:02,740 construct the interface between a crystalline solid 474 00:27:02,740 --> 00:27:04,420 and an amorphous system? 475 00:27:04,420 --> 00:27:08,230 I mean, we can try putting atoms in a reasonable position. 476 00:27:08,230 --> 00:27:11,150 But it's never going to work very well. 477 00:27:11,150 --> 00:27:13,030 And so what we can do is actually 478 00:27:13,030 --> 00:27:17,200 use molecular dynamic techniques to let the atoms evolve 479 00:27:17,200 --> 00:27:19,630 according to their interaction and to the Newton 480 00:27:19,630 --> 00:27:23,335 equation of motion and somehow find, by themselves, 481 00:27:23,335 --> 00:27:27,430 a slightly more favorable green minima instead 482 00:27:27,430 --> 00:27:32,350 of the initial configuration in which we have put them. 483 00:27:32,350 --> 00:27:38,320 OK, so, in principle, if you had infinite computing power 484 00:27:38,320 --> 00:27:41,230 or infinite mathematical prowess, 485 00:27:41,230 --> 00:27:44,380 you would have really solved the problem 486 00:27:44,380 --> 00:27:50,920 of characterizing on a computer any material that you wanted. 487 00:27:50,920 --> 00:27:53,200 The reason why you can't really do 488 00:27:53,200 --> 00:27:57,730 that in general and indiscriminately 489 00:27:57,730 --> 00:28:03,460 has to do with four problems that arise in the approach 490 00:28:03,460 --> 00:28:04,660 that I've mentioned. 491 00:28:04,660 --> 00:28:11,110 And I've mentioned them here, I guess, in order of importance. 492 00:28:11,110 --> 00:28:15,430 And the first and most dramatic problem 493 00:28:15,430 --> 00:28:18,290 is the one of time scales. 494 00:28:18,290 --> 00:28:24,320 That is, atoms move around and vibrate at time scales that 495 00:28:24,320 --> 00:28:26,960 are characteristic of atoms. 496 00:28:26,960 --> 00:28:30,740 That is, say, in a molecule, it will 497 00:28:30,740 --> 00:28:35,480 take tens of femtoseconds or hundreds of femtoseconds 498 00:28:35,480 --> 00:28:37,430 to have a periodic oscillation. 499 00:28:37,430 --> 00:28:41,060 So these are femtoseconds. 500 00:28:41,060 --> 00:28:43,880 10 to the minus 15 seconds is the order 501 00:28:43,880 --> 00:28:51,090 of magnitude of the dynamics of the atoms at room temperature. 502 00:28:51,090 --> 00:28:54,380 Now, a lot of interesting problems 503 00:28:54,380 --> 00:29:00,410 take place in times of seconds, minutes, or hours. 504 00:29:00,410 --> 00:29:06,110 A classical problem is how a protein folds itself 505 00:29:06,110 --> 00:29:07,670 in its native state. 506 00:29:07,670 --> 00:29:11,150 And you know, this is a process that can take seconds. 507 00:29:11,150 --> 00:29:13,190 And so we have, really, the issue 508 00:29:13,190 --> 00:29:18,890 that all our calculation takes place on the timescales of 10 509 00:29:18,890 --> 00:29:20,930 to the minus 15. 510 00:29:20,930 --> 00:29:24,410 And often we need to reach physical properties that 511 00:29:24,410 --> 00:29:26,780 are of the order of seconds. 512 00:29:26,780 --> 00:29:30,650 So we need to span a 15 order of magnitude. 513 00:29:30,650 --> 00:29:36,890 And that means that if it takes a time x to perform 514 00:29:36,890 --> 00:29:39,540 one single molecular dynamic operation, 515 00:29:39,540 --> 00:29:43,535 we have a 10 to the 15 cost to get to the reasonable time 516 00:29:43,535 --> 00:29:46,160 scale for a lot of macroscopic processes. 517 00:29:46,160 --> 00:29:49,070 And that's really a scale that we can't bridge. 518 00:29:49,070 --> 00:29:54,980 If we do quantum mechanical simulations for a small system, 519 00:29:54,980 --> 00:29:58,130 we can easily reach the times of tens 520 00:29:58,130 --> 00:30:01,610 of picoseconds or hundreds of picoseconds. 521 00:30:01,610 --> 00:30:05,330 If we do classical molecular dynamic simulation, 522 00:30:05,330 --> 00:30:09,380 we can go maybe three, four, five orders of magnitude 523 00:30:09,380 --> 00:30:10,430 better. 524 00:30:10,430 --> 00:30:14,630 And so we can start accessing nanoseconds 525 00:30:14,630 --> 00:30:21,870 and maybe start getting closer towards the microsecond. 526 00:30:21,870 --> 00:30:24,960 But there is still an enormous gap to reach. 527 00:30:24,960 --> 00:30:27,780 And this is really a difficult problem 528 00:30:27,780 --> 00:30:32,460 to overcome because we have dynamics taking place 529 00:30:32,460 --> 00:30:34,410 at different levels. 530 00:30:34,410 --> 00:30:35,770 Say, think of a protein. 531 00:30:35,770 --> 00:30:40,740 We have the atomic dynamics of an atom vibrating around. 532 00:30:40,740 --> 00:30:46,050 And then we have dynamics of subunits maybe interacting 533 00:30:46,050 --> 00:30:49,290 with the different amino acids along the chain. 534 00:30:49,290 --> 00:30:53,220 And then we have structural motifs that want to move around 535 00:30:53,220 --> 00:30:57,900 and then maybe the first coil-up in a certain configuration. 536 00:30:57,900 --> 00:30:59,700 But that's not really very good. 537 00:30:59,700 --> 00:31:02,220 And then after a while they sort of open up 538 00:31:02,220 --> 00:31:04,680 and recoil in a different configuration. 539 00:31:04,680 --> 00:31:09,810 And so the more you look at your system on a larger and larger 540 00:31:09,810 --> 00:31:12,030 length scale, the more you discover 541 00:31:12,030 --> 00:31:16,680 that there are dynamics of more and more complex groups 542 00:31:16,680 --> 00:31:19,050 that are slower and slower. 543 00:31:19,050 --> 00:31:24,210 And integrating from the atomic motion up to the mesoscale 544 00:31:24,210 --> 00:31:26,610 and the macroscopic scale of these dynamics 545 00:31:26,610 --> 00:31:31,030 is really an impossible problem to do by brute force. 546 00:31:31,030 --> 00:31:35,790 And I would say that it's still the conceptual problem 547 00:31:35,790 --> 00:31:39,420 for which less progress has been made. 548 00:31:39,420 --> 00:31:42,030 It's really the most difficult one. 549 00:31:42,030 --> 00:31:45,000 And you'll hear a lot of talks nowadays 550 00:31:45,000 --> 00:31:48,150 in science about multiscale modeling 551 00:31:48,150 --> 00:31:50,850 and about bridging timescales. 552 00:31:50,850 --> 00:31:54,780 And it's obviously a very important challenge. 553 00:31:54,780 --> 00:31:57,060 But I still think we are at the stage in which there 554 00:31:57,060 --> 00:31:59,310 are very little solutions. 555 00:31:59,310 --> 00:32:03,270 Somehow related to the time-scale problem, 556 00:32:03,270 --> 00:32:06,525 there is the length scale problem. 557 00:32:09,250 --> 00:32:13,890 Suppose you want to study a system close to a phase 558 00:32:13,890 --> 00:32:14,490 transition. 559 00:32:14,490 --> 00:32:17,760 Suppose that you want to see a gas evaporate. 560 00:32:17,760 --> 00:32:22,350 You want to study a liquid-to-vapor transition. 561 00:32:22,350 --> 00:32:25,560 What happens exactly at the temperature 562 00:32:25,560 --> 00:32:29,100 where the liquid and the vapor are in equilibrium? 563 00:32:29,100 --> 00:32:32,490 Suppose that you are studying water boiling. 564 00:32:32,490 --> 00:32:39,120 At normal condition, it boils at 373 Kelvin. 565 00:32:39,120 --> 00:32:43,320 And at that temperature, why there is a phase transition? 566 00:32:43,320 --> 00:32:45,660 Basically because the free energy 567 00:32:45,660 --> 00:32:50,580 of the liquid and the free energy of the gas are the same. 568 00:32:50,580 --> 00:32:53,250 You go a little bit above the temperature, 569 00:32:53,250 --> 00:32:55,540 the free energy of the gas is lower. 570 00:32:55,540 --> 00:32:58,020 So the system wants to be all gas. 571 00:32:58,020 --> 00:33:00,570 You go a little bit below the boiling temperature, 572 00:33:00,570 --> 00:33:02,530 the free energy of the liquid is lower. 573 00:33:02,530 --> 00:33:04,800 So the system wants to be a liquid. 574 00:33:04,800 --> 00:33:08,190 But as you get closer and closer to the temperature, 575 00:33:08,190 --> 00:33:11,670 these two free energies becomes comparable. 576 00:33:11,670 --> 00:33:15,930 That means that the cost of transforming from liquid 577 00:33:15,930 --> 00:33:19,080 to the gas and from gas to liquid 578 00:33:19,080 --> 00:33:21,630 becomes smaller and smaller. 579 00:33:21,630 --> 00:33:25,650 And it is 0 at the transition temperature. 580 00:33:25,650 --> 00:33:30,900 That means, in practice, that since it doesn't cost anything 581 00:33:30,900 --> 00:33:34,620 to transform from one state to the other, 582 00:33:34,620 --> 00:33:37,560 you have lost all possible length 583 00:33:37,560 --> 00:33:45,330 scale because you can nucleate a bubble of gas of any size 584 00:33:45,330 --> 00:33:47,440 with 0 cost. 585 00:33:47,440 --> 00:33:49,320 So if you think of the transformation 586 00:33:49,320 --> 00:33:54,840 from a liquid to a vapor as the nucleation of bubble of gas 587 00:33:54,840 --> 00:33:58,680 exactly at the temperature of the transition, 588 00:33:58,680 --> 00:34:00,330 that cost has become 0. 589 00:34:00,330 --> 00:34:03,790 And we can nucleate bubbles of any size. 590 00:34:03,790 --> 00:34:06,060 So the phase transition is a moment 591 00:34:06,060 --> 00:34:08,159 where we lose length scales. 592 00:34:08,159 --> 00:34:11,100 The fluctuation can have any size. 593 00:34:11,100 --> 00:34:14,639 And so any kind of finite simulation 594 00:34:14,639 --> 00:34:17,159 will break down because it's really 595 00:34:17,159 --> 00:34:21,690 only in the infinite system that we represent appropriately all 596 00:34:21,690 --> 00:34:24,760 these sizes popping up in our system. 597 00:34:24,760 --> 00:34:27,060 And so that's why it becomes very 598 00:34:27,060 --> 00:34:30,199 difficult to study accurately, say, a boundary 599 00:34:30,199 --> 00:34:31,260 at a phase transition. 600 00:34:31,260 --> 00:34:34,770 We say that the length scale starts diverging. 601 00:34:34,770 --> 00:34:37,380 And that's why one needs to be careful. 602 00:34:37,380 --> 00:34:42,750 This problem is not as serious as the problem of time scales. 603 00:34:42,750 --> 00:34:48,929 But it's still one that you want to consider very carefully 604 00:34:48,929 --> 00:34:52,440 and one you need to spend some time thinking, actually. 605 00:34:52,440 --> 00:34:56,310 That is, often we use a periodic boundary condition 606 00:34:56,310 --> 00:34:58,720 to describe an extended system. 607 00:34:58,720 --> 00:35:00,810 And so the question that you always 608 00:35:00,810 --> 00:35:05,850 have to ask yourself is, how large should I make my unit 609 00:35:05,850 --> 00:35:09,870 system that is periodically repeated before it's really 610 00:35:09,870 --> 00:35:13,920 describing accurately the infinite system that 611 00:35:13,920 --> 00:35:15,630 is the goal of my simulation? 612 00:35:15,630 --> 00:35:18,660 Suppose that you want to study water, OK? 613 00:35:18,660 --> 00:35:21,990 Or suppose that you want to study any liquid. 614 00:35:21,990 --> 00:35:25,170 In principle you want an infinite system. 615 00:35:25,170 --> 00:35:27,990 In principle you can deal only with a finite number 616 00:35:27,990 --> 00:35:29,530 of degrees of freedom. 617 00:35:29,530 --> 00:35:31,950 So the two possibilities that you have, 618 00:35:31,950 --> 00:35:35,850 you could study a droplet of water, OK? 619 00:35:35,850 --> 00:35:38,910 But then obviously you encounter the problem 620 00:35:38,910 --> 00:35:41,100 that, when you study a droplet, you 621 00:35:41,100 --> 00:35:45,300 have a very large influence coming from the surface effect. 622 00:35:45,300 --> 00:35:48,360 A droplet is a droplet because of the surface tension. 623 00:35:48,360 --> 00:35:52,290 And before reaching the thermodynamic limit 624 00:35:52,290 --> 00:35:55,950 of that droplet behaving as a bulk liquid, 625 00:35:55,950 --> 00:35:59,130 you need really to go to enormous sizes. 626 00:35:59,130 --> 00:36:03,240 If instead you study, say, the same number of molecules 627 00:36:03,240 --> 00:36:05,700 in periodic boundary conditions, you 628 00:36:05,700 --> 00:36:08,170 eliminate the presence of the surface. 629 00:36:08,170 --> 00:36:11,340 And so you can reach the thermodynamic limit 630 00:36:11,340 --> 00:36:15,870 of the system behaving as a bulk much faster. 631 00:36:15,870 --> 00:36:18,510 But still you need to always ask your question, 632 00:36:18,510 --> 00:36:22,110 is your simulation still large enough? 633 00:36:22,110 --> 00:36:25,890 And there is a size that is large enough. 634 00:36:25,890 --> 00:36:29,130 And that's the other very important concept. 635 00:36:29,130 --> 00:36:35,250 That is, whenever your simulation cell is so large 636 00:36:35,250 --> 00:36:39,900 that the dynamics of one water molecule 637 00:36:39,900 --> 00:36:45,080 doesn't affect the dynamics of a water molecule 638 00:36:45,080 --> 00:36:48,570 that is far away from yours as much as it 639 00:36:48,570 --> 00:36:52,940 is its periodic image, your system has become infinite. 640 00:36:52,940 --> 00:36:56,870 That is, whenever you have a molecule in a liquid, 641 00:36:56,870 --> 00:37:01,640 you really dynamically interact not with every other molecule 642 00:37:01,640 --> 00:37:08,180 in the infinite liquid but only with a sphere of neighbors. 643 00:37:08,180 --> 00:37:10,650 And you need to understand how large 644 00:37:10,650 --> 00:37:15,290 is that sphere of neighbors because those and only those 645 00:37:15,290 --> 00:37:18,410 are the molecules that affect yourself 646 00:37:18,410 --> 00:37:20,390 in the center of that sphere. 647 00:37:20,390 --> 00:37:25,490 And if you decide that for a water molecule at room 648 00:37:25,490 --> 00:37:29,210 temperature the sphere of neighbors with which there 649 00:37:29,210 --> 00:37:32,720 is a dynamic interaction has only a radius of 10 650 00:37:32,720 --> 00:37:37,370 angstroms, whenever your unit cell is larger, is a cube, 651 00:37:37,370 --> 00:37:40,160 say, larger than 20 angstroms inside, 652 00:37:40,160 --> 00:37:43,670 you are studying, in practice, an infinite system 653 00:37:43,670 --> 00:37:46,850 because every molecule in that system 654 00:37:46,850 --> 00:37:51,940 sees all the independent neighbors that it needs to see, 655 00:37:51,940 --> 00:37:52,610 OK? 656 00:37:52,610 --> 00:37:56,750 So even a simulation with a finite number of molecules 657 00:37:56,750 --> 00:38:01,490 can reproduce exactly the behavior of an infinite system 658 00:38:01,490 --> 00:38:06,560 provided a molecule interacts with other independent 659 00:38:06,560 --> 00:38:12,590 molecules up to a maximum range of dynamical correlation. 660 00:38:12,590 --> 00:38:15,170 And we'll actually define this quantity 661 00:38:15,170 --> 00:38:17,850 in the lecture that follows. 662 00:38:17,850 --> 00:38:21,050 So again, length scales are another problem 663 00:38:21,050 --> 00:38:24,110 and a problem in which periodic boundary conditions tend 664 00:38:24,110 --> 00:38:25,500 to help a lot. 665 00:38:25,500 --> 00:38:28,370 But again, suppose that you want to study something 666 00:38:28,370 --> 00:38:32,300 like the strength of a metal. 667 00:38:32,300 --> 00:38:37,850 Well, you have now a problem in which a lot of length scales 668 00:38:37,850 --> 00:38:43,070 becomes intertwined because in the strength of a metal, 669 00:38:43,070 --> 00:38:48,440 you might want to consider what is the dynamics of impurities 670 00:38:48,440 --> 00:38:50,660 that strengthen that metal. 671 00:38:50,660 --> 00:38:53,330 Like, if you have steel, you have carbon atoms 672 00:38:53,330 --> 00:38:54,800 in between the ions. 673 00:38:54,800 --> 00:38:58,820 And so you need to understand what is the atomic dynamics. 674 00:38:58,820 --> 00:39:03,650 But then there will be dynamics of units on a larger length 675 00:39:03,650 --> 00:39:07,190 scale because in a metal there's going to be these location 676 00:39:07,190 --> 00:39:09,080 and slip and glide planes. 677 00:39:09,080 --> 00:39:14,570 And so there is coordinated motion of thousands of atoms 678 00:39:14,570 --> 00:39:16,500 that you want to consider. 679 00:39:16,500 --> 00:39:20,630 And then, in reality, a real metal is never a crystal. 680 00:39:20,630 --> 00:39:23,130 But it's always a polycrystalline material. 681 00:39:23,130 --> 00:39:24,770 So you have grains. 682 00:39:24,770 --> 00:39:28,220 And inside that grain you have a consistent ordering 683 00:39:28,220 --> 00:39:29,150 of the atoms. 684 00:39:29,150 --> 00:39:31,370 But then grains are mismatched. 685 00:39:31,370 --> 00:39:33,240 And there are grain boundaries. 686 00:39:33,240 --> 00:39:35,510 And so you have a lot of length scales 687 00:39:35,510 --> 00:39:38,420 to consider if you really want to predict 688 00:39:38,420 --> 00:39:44,020 what would be the mechanical response of a real system. 689 00:39:44,020 --> 00:39:47,680 OK, so this is the challenge, if you want, number 2. 690 00:39:47,680 --> 00:39:50,170 The challenge number 3 is the one 691 00:39:50,170 --> 00:39:54,850 that we have described more in detail up to now 692 00:39:54,850 --> 00:39:58,660 that is determining how accurate can 693 00:39:58,660 --> 00:40:01,930 be your prediction of the interaction between nuclei. 694 00:40:01,930 --> 00:40:04,510 And you have seen everything that we could tell you 695 00:40:04,510 --> 00:40:08,440 about empirical potential quantum mechanical simulations. 696 00:40:08,440 --> 00:40:12,850 And in principle, even if it's obviously very expensive 697 00:40:12,850 --> 00:40:15,520 to make very accurate predictions of the interaction 698 00:40:15,520 --> 00:40:19,600 between atoms, in principle it's the less conceptually 699 00:40:19,600 --> 00:40:21,230 challenging of the problems. 700 00:40:21,230 --> 00:40:25,990 I mean, we can just try all our electronic structure methods 701 00:40:25,990 --> 00:40:26,680 at will. 702 00:40:26,680 --> 00:40:30,920 And we can try to see how accurate we become. 703 00:40:30,920 --> 00:40:34,510 And the last challenge that is also not 704 00:40:34,510 --> 00:40:39,550 really a conceptual challenge and it can be solved 705 00:40:39,550 --> 00:40:43,480 but is just a very expensive one to overcome 706 00:40:43,480 --> 00:40:48,940 is this one in which we treat the nuclei 707 00:40:48,940 --> 00:40:50,650 as classical particles. 708 00:40:50,650 --> 00:40:53,950 And that tends to be, in general, very good. 709 00:40:53,950 --> 00:40:56,470 But it's sort of an approximation that 710 00:40:56,470 --> 00:40:59,920 breaks down the more you go towards lighter 711 00:40:59,920 --> 00:41:02,020 and lighter nuclei because, remember, 712 00:41:02,020 --> 00:41:03,490 the de Broglie relation. 713 00:41:03,490 --> 00:41:06,730 Your wavelength times your momentum 714 00:41:06,730 --> 00:41:08,890 is equal to the Planck constant. 715 00:41:08,890 --> 00:41:11,350 So the lighter you are, the smaller 716 00:41:11,350 --> 00:41:14,800 your momentum, the longer your wavelength, the more quantum 717 00:41:14,800 --> 00:41:15,970 you become. 718 00:41:15,970 --> 00:41:20,080 So something like hydrogen that has a nucleus that is just one 719 00:41:20,080 --> 00:41:23,470 proton, so it's fairly light-- or it's the lightest of all 720 00:41:23,470 --> 00:41:24,850 possible nuclei-- 721 00:41:24,850 --> 00:41:30,040 tend to still have significant quantum properties 722 00:41:30,040 --> 00:41:32,290 even at relatively high temperatures. 723 00:41:32,290 --> 00:41:34,670 You see, the more you increase the temperature, 724 00:41:34,670 --> 00:41:37,870 the more the average momentum of your particles become 725 00:41:37,870 --> 00:41:40,390 and the more classical they become. 726 00:41:40,390 --> 00:41:43,870 But the lower the temperature, the more quantum they become. 727 00:41:43,870 --> 00:41:47,170 And something like hydrogen is still significantly 728 00:41:47,170 --> 00:41:49,000 quantum at room temperature. 729 00:41:49,000 --> 00:41:50,480 And what does that mean? 730 00:41:50,480 --> 00:41:53,140 Well, it means that that particle, 731 00:41:53,140 --> 00:41:56,380 it's really a delocalized system that can actually 732 00:41:56,380 --> 00:41:58,000 tunnel through barriers. 733 00:41:58,000 --> 00:42:00,460 That's if you want one of the important differences 734 00:42:00,460 --> 00:42:02,710 between a particle that behaves according 735 00:42:02,710 --> 00:42:05,680 to classical mechanics and one that behaves according 736 00:42:05,680 --> 00:42:07,150 to quantum mechanics. 737 00:42:07,150 --> 00:42:09,580 If you are behaving as a classical particle, 738 00:42:09,580 --> 00:42:13,150 you need to have enough kinetic energy to overcome a barrier. 739 00:42:13,150 --> 00:42:15,520 If you are a skier and you are in a valley 740 00:42:15,520 --> 00:42:17,490 and you want to go in the other valley, 741 00:42:17,490 --> 00:42:20,200 you need to have enough velocity to be 742 00:42:20,200 --> 00:42:22,460 able to overcome the mountain. 743 00:42:22,460 --> 00:42:24,880 If you are a quantum skier, you can actually 744 00:42:24,880 --> 00:42:28,090 have some finite probability of tunneling through. 745 00:42:28,090 --> 00:42:34,600 And so there are tunneling problems that are important 746 00:42:34,600 --> 00:42:37,450 not only for hydrogen but sometimes 747 00:42:37,450 --> 00:42:40,930 even for much more massive particles 748 00:42:40,930 --> 00:42:44,530 or even for groups of particles, provided 749 00:42:44,530 --> 00:42:47,410 that the energy barrier that they need to overcome 750 00:42:47,410 --> 00:42:49,390 is small enough. 751 00:42:49,390 --> 00:42:53,410 And I've shown you in the first quantum class an example 752 00:42:53,410 --> 00:42:57,370 in a perovskite crystal in which we have this cubic structure 753 00:42:57,370 --> 00:42:59,860 with an octahedral cage of oxygen 754 00:42:59,860 --> 00:43:04,390 that can configure itself in a ground state that 755 00:43:04,390 --> 00:43:08,230 is ferroelectric, in which the oxygen cage displaces 756 00:43:08,230 --> 00:43:10,420 with respect to the cubic symmetry 757 00:43:10,420 --> 00:43:13,750 and induces a permanent electrical dipole 758 00:43:13,750 --> 00:43:14,805 in your system. 759 00:43:14,805 --> 00:43:16,180 Well, there are certain materials 760 00:43:16,180 --> 00:43:17,680 that have this structure. 761 00:43:17,680 --> 00:43:21,100 But since the octahedral cage can 762 00:43:21,100 --> 00:43:24,790 have roughly six possible choices on where 763 00:43:24,790 --> 00:43:31,030 to move offsite, if you decrease the temperature enough, 764 00:43:31,030 --> 00:43:33,190 the oxygen cage will start actually 765 00:43:33,190 --> 00:43:36,910 tunneling between all the six possibilities. 766 00:43:36,910 --> 00:43:40,420 Instead of finding itself at 0 temperature 767 00:43:40,420 --> 00:43:43,300 in a broken symmetry configuration in which 768 00:43:43,300 --> 00:43:48,010 the oxygen cage has chosen one of those six possibilities, 769 00:43:48,010 --> 00:43:50,470 in reality, because of quantum tunneling, 770 00:43:50,470 --> 00:43:53,620 it keeps tunneling between all the six possibilities. 771 00:43:53,620 --> 00:43:56,500 And on average your system looks cubic. 772 00:43:56,500 --> 00:44:00,580 And on average you don't have really any ferroelectricity. 773 00:44:00,580 --> 00:44:03,220 But the polarizability of the system 774 00:44:03,220 --> 00:44:06,430 is going to be different from that one in which 775 00:44:06,430 --> 00:44:10,420 the octahedral cage was sitting squarely in the cube. 776 00:44:10,420 --> 00:44:15,480 So there is a dielectric response that is different. 777 00:44:15,480 --> 00:44:21,860 So there are cases that are fairly exotic, I would say, 778 00:44:21,860 --> 00:44:26,150 for most applications, in which tunneling effects of the nuclei 779 00:44:26,150 --> 00:44:29,000 could become important. 780 00:44:29,000 --> 00:44:33,380 But there is one other important consequence 781 00:44:33,380 --> 00:44:35,750 that comes from quantum mechanics that 782 00:44:35,750 --> 00:44:39,260 tends to be important in a lot of cases. 783 00:44:39,260 --> 00:44:42,950 And that consequence has to do with the quantization 784 00:44:42,950 --> 00:44:45,590 of vibrational excitations. 785 00:44:45,590 --> 00:44:48,890 That is, if you look at the motion of any system-- 786 00:44:48,890 --> 00:44:50,660 a molecule, a solid-- 787 00:44:50,660 --> 00:44:54,770 in classical terms, well, that molecule or that solid 788 00:44:54,770 --> 00:44:58,280 can have any amount of kinetic energy. 789 00:44:58,280 --> 00:45:00,740 In a hydrogen molecule, the atoms 790 00:45:00,740 --> 00:45:04,160 can have 0 kinetic energy if we treat it classically. 791 00:45:04,160 --> 00:45:06,200 Or they can move just a little bit. 792 00:45:06,200 --> 00:45:08,960 So they have a very minute amount of kinetic energy. 793 00:45:08,960 --> 00:45:11,600 And we can just increase the kinetic energy 794 00:45:11,600 --> 00:45:14,800 that those atoms can have at will. 795 00:45:14,800 --> 00:45:20,380 In reality, because the system is a quantum system, 796 00:45:20,380 --> 00:45:24,190 the vibrational modes of a molecule 797 00:45:24,190 --> 00:45:29,320 or the vibrational modes of a solid are actually quantized. 798 00:45:29,320 --> 00:45:34,780 And you can't have arbitrary amounts of vibrational energy. 799 00:45:34,780 --> 00:45:38,470 Actually, in its quantum ground state, 800 00:45:38,470 --> 00:45:42,340 a molecule will not be at rest. 801 00:45:42,340 --> 00:45:46,280 But it will be in what is called the zero-point motion state. 802 00:45:46,280 --> 00:45:51,200 That is what is the quantum mechanical ground state. 803 00:45:51,200 --> 00:45:54,280 You can think of the analogy with the stability 804 00:45:54,280 --> 00:45:56,470 of the atoms. 805 00:45:56,470 --> 00:46:01,450 In its ground state, the electron around the proton 806 00:46:01,450 --> 00:46:04,960 is not collapsing onto the proton because 807 00:46:04,960 --> 00:46:07,060 of the Coulombic repulsion. 808 00:46:07,060 --> 00:46:09,490 If the electron was a classical particle, 809 00:46:09,490 --> 00:46:11,380 it would collapse on the nucleus. 810 00:46:11,380 --> 00:46:15,220 And the hydrogen atom would be just a proton and an electron 811 00:46:15,220 --> 00:46:17,000 sitting in the same place. 812 00:46:17,000 --> 00:46:19,330 But because the electron is quantum, 813 00:46:19,330 --> 00:46:21,340 it can't really collapse. 814 00:46:21,340 --> 00:46:24,580 And its lowest energy state is a state 815 00:46:24,580 --> 00:46:29,290 that is the 1s orbit for the hydrogen in which the electron 816 00:46:29,290 --> 00:46:32,740 is around the atom but doesn't collapse onto it. 817 00:46:32,740 --> 00:46:37,690 And the excitation of that electron to the next energy 818 00:46:37,690 --> 00:46:40,750 state is a quantized excitation. 819 00:46:40,750 --> 00:46:44,140 You can't go to a level of 1s that 820 00:46:44,140 --> 00:46:47,020 has an energy for the hydrogen of minus 1 821 00:46:47,020 --> 00:46:51,700 Rydberg to another state that has energy minus 0.99 822 00:46:51,700 --> 00:46:55,220 or minus by 0.98 or any kind of amount. 823 00:46:55,220 --> 00:46:58,240 The next state in which the electron can live 824 00:46:58,240 --> 00:47:03,820 is a state that has an energy of minus 1/4 and then minus 1/9 825 00:47:03,820 --> 00:47:05,890 and so on and so forth. 826 00:47:05,890 --> 00:47:10,440 The same thing happens for the vibrational level of molecules, 827 00:47:10,440 --> 00:47:11,170 OK? 828 00:47:11,170 --> 00:47:15,940 A quantum molecule in its ground state 829 00:47:15,940 --> 00:47:19,840 is not in a state where the atoms do not move. 830 00:47:19,840 --> 00:47:23,410 But it's in a state where the atoms have a certain amount 831 00:47:23,410 --> 00:47:25,300 of quantum kinetic energy. 832 00:47:25,300 --> 00:47:28,400 And it's in a state called zero-point motion. 833 00:47:28,400 --> 00:47:31,150 And if you want to increase the kinetic energy 834 00:47:31,150 --> 00:47:34,390 of this molecule, you can't do it continuously. 835 00:47:34,390 --> 00:47:35,980 There is a quantized cap. 836 00:47:35,980 --> 00:47:39,100 So you need to provide enough energy 837 00:47:39,100 --> 00:47:42,550 to heat up the system to the next state 838 00:47:42,550 --> 00:47:45,640 and then to the next state and then to the next state. 839 00:47:45,640 --> 00:47:50,740 And that, if you think, affects deeply how you 840 00:47:50,740 --> 00:47:53,150 are going to heat up a system. 841 00:47:53,150 --> 00:47:56,920 That is, if you look at the specific heat of a solid, 842 00:47:56,920 --> 00:47:59,440 that is basically the quantity that tells you 843 00:47:59,440 --> 00:48:03,850 how much heat the system is going to be able to take, 844 00:48:03,850 --> 00:48:07,720 and you look at what happens at very low temperature, 845 00:48:07,720 --> 00:48:13,150 you discover that actually the system is almost unable to take 846 00:48:13,150 --> 00:48:17,830 any heat because all the vibrational excitation are 847 00:48:17,830 --> 00:48:21,610 much larger than the average amount of kinetic energy 848 00:48:21,610 --> 00:48:24,490 that you have if you are looking at what 849 00:48:24,490 --> 00:48:28,600 it takes to bring something from 1 Kelvin to 2 Kelvin. 850 00:48:28,600 --> 00:48:33,640 OK, but this is still relevant if you have a system 851 00:48:33,640 --> 00:48:37,210 like water, even at 300 Kelvin. 852 00:48:37,210 --> 00:48:41,050 In water, say, at 300 Kelvin, what you have is 853 00:48:41,050 --> 00:48:42,490 you have these water molecules. 854 00:48:42,490 --> 00:48:46,030 Remember what you have-- your oxygen, hydrogen, and hydrogen. 855 00:48:46,030 --> 00:48:50,020 And the modes of vibration of these molecules 856 00:48:50,020 --> 00:48:53,800 are stretching modes of the hydrogens. 857 00:48:53,800 --> 00:48:56,830 And then there are scissor-like models 858 00:48:56,830 --> 00:48:59,110 in which the two bonds do this. 859 00:48:59,110 --> 00:49:04,300 And then these are intermolecular modes, just sort 860 00:49:04,300 --> 00:49:05,860 of relevant to one molecule. 861 00:49:05,860 --> 00:49:09,880 And then there are all the molecules interacting. 862 00:49:09,880 --> 00:49:14,020 Well, at room temperature, all the internal modes 863 00:49:14,020 --> 00:49:18,910 of the molecule, the stretching and the scissor modes, 864 00:49:18,910 --> 00:49:24,610 have so much vibrational energy in their ground state 865 00:49:24,610 --> 00:49:28,660 that it's almost impossible to excite them 866 00:49:28,660 --> 00:49:32,320 to the next vibrational state with the average temperature 867 00:49:32,320 --> 00:49:34,960 that you have at 300 Kelvin. 868 00:49:34,960 --> 00:49:38,530 So even a system as fundamental as water 869 00:49:38,530 --> 00:49:43,390 has really a hybrid behavior in which 870 00:49:43,390 --> 00:49:48,190 the vibrational interaction between molecules 871 00:49:48,190 --> 00:49:51,400 can actually be populated arbitrarily 872 00:49:51,400 --> 00:49:53,770 by the temperature that is surrounding you. 873 00:49:53,770 --> 00:49:58,300 But it's almost impossible to excite a molecule 874 00:49:58,300 --> 00:50:04,060 from a zero-point stretching motion to the next one. 875 00:50:04,060 --> 00:50:07,060 And so, say, the specific heat of water 876 00:50:07,060 --> 00:50:11,320 will be slightly exotic because really part of its vibration 877 00:50:11,320 --> 00:50:12,640 are quantized. 878 00:50:12,640 --> 00:50:15,070 And so there are a number of cases 879 00:50:15,070 --> 00:50:22,510 in which this classical nuclear approximation is not exact. 880 00:50:22,510 --> 00:50:27,610 And luckily they can be studied with specific techniques 881 00:50:27,610 --> 00:50:31,030 that we'll see in some of the later-application classes 882 00:50:31,030 --> 00:50:32,300 during this course. 883 00:50:32,300 --> 00:50:34,750 But remember this-- if there is one thing that you want 884 00:50:34,750 --> 00:50:37,660 to remember about the behavior of nuclei-- 885 00:50:37,660 --> 00:50:41,380 that this vibrational excitation are quantized. 886 00:50:41,380 --> 00:50:44,200 And we actually see this quantization 887 00:50:44,200 --> 00:50:50,130 in physical properties like the specific heat. 888 00:50:50,130 --> 00:50:57,900 OK, so let's discuss the first of the possible applications 889 00:50:57,900 --> 00:51:01,440 of molecular dynamics that I have mentioned. 890 00:51:01,440 --> 00:51:04,650 And that's using molecular dynamics 891 00:51:04,650 --> 00:51:08,580 to really calculate average properties of a system 892 00:51:08,580 --> 00:51:10,590 at finite temperatures. 893 00:51:10,590 --> 00:51:15,330 And what I have written here is really the summary 894 00:51:15,330 --> 00:51:17,320 of statistical mechanics. 895 00:51:17,320 --> 00:51:20,910 So that is what, if you want, Boltzmann would tell you. 896 00:51:20,910 --> 00:51:23,580 You have a dynamical system that is 897 00:51:23,580 --> 00:51:28,680 described by this point moving around in phase space, OK? 898 00:51:28,680 --> 00:51:31,920 So with a certain possible set of positions 899 00:51:31,920 --> 00:51:35,130 and with a certain possible set of momenta. 900 00:51:35,130 --> 00:51:38,100 And suppose that we want to calculate, say, 901 00:51:38,100 --> 00:51:42,030 what is an average property at a certain temperature. 902 00:51:42,030 --> 00:51:45,060 And we call that property A. It could be the lattice 903 00:51:45,060 --> 00:51:47,700 parameter of the system. 904 00:51:47,700 --> 00:51:50,640 Or it could be the diameter, say-- it's even simpler-- 905 00:51:50,640 --> 00:51:52,740 of your water bubble. 906 00:51:52,740 --> 00:51:54,540 OK, you have this water bubble. 907 00:51:54,540 --> 00:51:57,660 And you want to see what is the diameter of this water bubble 908 00:51:57,660 --> 00:51:59,580 at a certain temperature. 909 00:51:59,580 --> 00:52:02,950 And this diameter is going to increase with temperature. 910 00:52:02,950 --> 00:52:06,840 And what you do, you just let this system evolve. 911 00:52:06,840 --> 00:52:09,000 And during its evolution, you keep 912 00:52:09,000 --> 00:52:11,130 measuring what is its diameter. 913 00:52:11,130 --> 00:52:13,530 And you let this evolve a lot. 914 00:52:13,530 --> 00:52:16,320 And you measure what is its width. 915 00:52:16,320 --> 00:52:20,910 Well, the way Boltzmann would set this problem is say, 916 00:52:20,910 --> 00:52:26,850 we really need to integrate over all possible configurations 917 00:52:26,850 --> 00:52:27,840 of the system. 918 00:52:27,840 --> 00:52:31,620 That is, we need to go through all possible configurations 919 00:52:31,620 --> 00:52:32,490 of the system. 920 00:52:32,490 --> 00:52:38,130 That is, we need to integrate over the full phase space. 921 00:52:38,130 --> 00:52:41,970 And for every possible point in phase space, 922 00:52:41,970 --> 00:52:47,190 we need to calculate the diameter of my water bubble. 923 00:52:47,190 --> 00:52:51,180 And then we need to take into account 924 00:52:51,180 --> 00:52:55,450 what is the probability of being in that configuration. 925 00:52:55,450 --> 00:52:59,610 And if you are in a system at a constant temperature, 926 00:52:59,610 --> 00:53:04,470 the probability of finding yourself at a given temperature 927 00:53:04,470 --> 00:53:08,550 is given by exponential of minus beta is just 1 928 00:53:08,550 --> 00:53:10,410 over the Boltzmann constant times 929 00:53:10,410 --> 00:53:12,570 the temperature times the energy. 930 00:53:12,570 --> 00:53:15,470 That means that configurations that 931 00:53:15,470 --> 00:53:19,200 have a high internal energy are very unlikely. 932 00:53:19,200 --> 00:53:22,050 And configurations that have lower energy 933 00:53:22,050 --> 00:53:25,410 are more likely, to the point that if the temperature is 934 00:53:25,410 --> 00:53:29,190 0, that is this beta is diverging to infinity, 935 00:53:29,190 --> 00:53:33,000 only the lowest energy configuration possible is 936 00:53:33,000 --> 00:53:34,380 represented. 937 00:53:34,380 --> 00:53:36,840 So this is the statistical mechanic points. 938 00:53:36,840 --> 00:53:40,320 To calculate an average quantity at a finite temperature, 939 00:53:40,320 --> 00:53:45,030 we need to sum over all possible configurations 940 00:53:45,030 --> 00:53:46,920 in space and momentum. 941 00:53:46,920 --> 00:53:49,830 For each possible r and p configuration, 942 00:53:49,830 --> 00:53:53,490 we'll have a certain value of the diameter of that bubble. 943 00:53:53,490 --> 00:53:56,970 We weigh that with the statistical weight 944 00:53:56,970 --> 00:53:58,560 of that configuration. 945 00:53:58,560 --> 00:54:03,310 And we actually normalize this expectation value. 946 00:54:03,310 --> 00:54:06,150 And this is what we obtain. 947 00:54:06,150 --> 00:54:09,420 So if, for a moment, you think at a phase space that 948 00:54:09,420 --> 00:54:14,220 is two dimensional and in this square 949 00:54:14,220 --> 00:54:17,070 we represent all the possible positions and velocities 950 00:54:17,070 --> 00:54:19,470 of our system, what Boltzmann tells 951 00:54:19,470 --> 00:54:22,560 us is that for every possible point, 952 00:54:22,560 --> 00:54:25,110 we will have a certain value of A. 953 00:54:25,110 --> 00:54:27,960 And the sum of all the possible values 954 00:54:27,960 --> 00:54:32,700 of A weighted with the probability of finding them 955 00:54:32,700 --> 00:54:34,680 gives us the expectation value. 956 00:54:34,680 --> 00:54:36,630 And so what you really need to do 957 00:54:36,630 --> 00:54:41,580 is an integral over all space of this. 958 00:54:41,580 --> 00:54:47,040 And the cost of these integrals diverges 959 00:54:47,040 --> 00:54:49,650 as the number of dimensions diverge 960 00:54:49,650 --> 00:54:55,800 because, again, every time we increase the size of the system 961 00:54:55,800 --> 00:55:00,930 by one particle, we have six more dimensions over which 962 00:55:00,930 --> 00:55:02,040 to integrate. 963 00:55:02,040 --> 00:55:04,740 And again, if you think that we do 964 00:55:04,740 --> 00:55:07,020 this just by numerical integrations 965 00:55:07,020 --> 00:55:09,570 by taking points on a grid and maybe 966 00:55:09,570 --> 00:55:13,560 we just discretize our system with 10 points along one 967 00:55:13,560 --> 00:55:15,810 dimension, every time you add a particle, 968 00:55:15,810 --> 00:55:19,860 you have 10 to the 6 points that multiply your system. 969 00:55:19,860 --> 00:55:22,020 So this quantity explodes. 970 00:55:22,020 --> 00:55:25,830 The point of view of molecular dynamics is different. 971 00:55:25,830 --> 00:55:31,980 What we say is that, well, let's just let this system evolve. 972 00:55:31,980 --> 00:55:34,440 Let this system evolve in time. 973 00:55:34,440 --> 00:55:38,830 And let it evolve at a constant temperature. 974 00:55:38,830 --> 00:55:42,330 And so now our point in phase space 975 00:55:42,330 --> 00:55:47,950 is going to go around, following a trajectory. 976 00:55:47,950 --> 00:55:52,450 And if we let it go around long enough, 977 00:55:52,450 --> 00:55:56,850 it will spend more time in the parts of phase space that 978 00:55:56,850 --> 00:56:00,270 are more favorite, the ones that are more likely, 979 00:56:00,270 --> 00:56:02,490 according to the Boltzmann factor. 980 00:56:02,490 --> 00:56:07,050 And so if our evolution, really, is distributed 981 00:56:07,050 --> 00:56:13,590 in the trajectory according to this thermodynamic weight, 982 00:56:13,590 --> 00:56:17,850 thermodynamic factor, then our expectation value, 983 00:56:17,850 --> 00:56:21,390 the thermodynamic average for our observable A, 984 00:56:21,390 --> 00:56:25,560 is just trivial, obtained by integrating 985 00:56:25,560 --> 00:56:27,900 over the trajectories all the values 986 00:56:27,900 --> 00:56:29,890 of A. That is what I was saying before. 987 00:56:29,890 --> 00:56:33,360 If you want to calculate the diameter of your bubble, what 988 00:56:33,360 --> 00:56:35,880 you do in the molecular dynamic sense, 989 00:56:35,880 --> 00:56:39,180 you just let the bubble evolve according 990 00:56:39,180 --> 00:56:42,400 to the equation of motion at a constant temperature. 991 00:56:42,400 --> 00:56:44,850 And you keep monitoring its diameter. 992 00:56:44,850 --> 00:56:47,850 And you will see that the diameter will have an average 993 00:56:47,850 --> 00:56:53,670 and that this is a much more efficient way then considering 994 00:56:53,670 --> 00:56:56,460 all possible configurations of the bubble, 995 00:56:56,460 --> 00:57:00,600 weighing them with how expensive that configuration is, 996 00:57:00,600 --> 00:57:04,050 and doing it for all the possible configurations. 997 00:57:04,050 --> 00:57:07,380 The molecular dynamic algorithm automatically, if you want, 998 00:57:07,380 --> 00:57:11,340 keeps spanning the most likely configurations. 999 00:57:11,340 --> 00:57:16,260 And so it does, for you, the job of going in the place where 1000 00:57:16,260 --> 00:57:19,900 it's most likely to be. 1001 00:57:19,900 --> 00:57:22,930 So in order to calculate a sample average, 1002 00:57:22,930 --> 00:57:24,880 we can actually transform a problem 1003 00:57:24,880 --> 00:57:27,850 into integration over a trajectory. 1004 00:57:27,850 --> 00:57:32,300 And that is almost always doable under what is called 1005 00:57:32,300 --> 00:57:35,050 the criterion of ergodicity. 1006 00:57:35,050 --> 00:57:37,825 So the equivalence between the two formulations 1007 00:57:37,825 --> 00:57:39,700 that I'd written before-- that is calculating 1008 00:57:39,700 --> 00:57:43,960 a thermodynamic observable as an integral in phase space 1009 00:57:43,960 --> 00:57:47,620 or as an integral over trajectories-- are equivalent, 1010 00:57:47,620 --> 00:57:52,240 provided your trajectory is really able 1011 00:57:52,240 --> 00:57:56,450 to go all over the place in phase space. 1012 00:57:56,450 --> 00:57:58,840 So if, again, this were phase space, 1013 00:57:58,840 --> 00:58:01,390 you could have, in principle, the problem 1014 00:58:01,390 --> 00:58:04,420 of a known ergodic system if there 1015 00:58:04,420 --> 00:58:08,770 is something that prevents your trajectory 1016 00:58:08,770 --> 00:58:13,990 to reach certain so-called forbidden segments 1017 00:58:13,990 --> 00:58:16,450 in your phase space. 1018 00:58:16,450 --> 00:58:18,710 And I can give you an example. 1019 00:58:18,710 --> 00:58:23,590 Suppose that you are studying something like silicon 1020 00:58:23,590 --> 00:58:25,940 and you are in the crystalline state. 1021 00:58:25,940 --> 00:58:28,510 And now you heat it up. 1022 00:58:28,510 --> 00:58:32,290 And once you go above the melting temperature, 1023 00:58:32,290 --> 00:58:35,800 your silicon will become liquid, OK? 1024 00:58:35,800 --> 00:58:40,780 Now, suppose that you cool it down again below, now, 1025 00:58:40,780 --> 00:58:42,130 the melting temperature. 1026 00:58:42,130 --> 00:58:45,880 OK, in principle, the liquid silicon 1027 00:58:45,880 --> 00:58:48,850 should become again crystalline silicon 1028 00:58:48,850 --> 00:58:52,600 because, at that temperature in which we have brought it back 1029 00:58:52,600 --> 00:58:56,680 to, the thermodynamic stable state is really 1030 00:58:56,680 --> 00:58:58,360 the crystalline solid. 1031 00:58:58,360 --> 00:59:01,270 If you do that in practice with our molecular dynamic 1032 00:59:01,270 --> 00:59:04,420 simulation, you see that you start from the crystal. 1033 00:59:04,420 --> 00:59:05,930 You heat it up. 1034 00:59:05,930 --> 00:59:08,980 It becomes a liquid with no problems. 1035 00:59:08,980 --> 00:59:11,590 And then you cool it back down. 1036 00:59:11,590 --> 00:59:14,020 And instead of going back to a crystal, 1037 00:59:14,020 --> 00:59:18,160 it goes back to a glassy, amorphous state. 1038 00:59:18,160 --> 00:59:22,390 Silicon will find itself in a configuration 1039 00:59:22,390 --> 00:59:25,600 in which locally all atoms are happy to be 1040 00:59:25,600 --> 00:59:27,520 fourfold coordinated. 1041 00:59:27,520 --> 00:59:32,290 But if we look at the long-range order, it's completely lost. 1042 00:59:32,290 --> 00:59:34,310 The system is not crystalline anymore. 1043 00:59:34,310 --> 00:59:36,070 It's glassy. 1044 00:59:36,070 --> 00:59:37,340 What's happening there? 1045 00:59:37,340 --> 00:59:41,410 Well, it happens that it's just, for a liquid cooling 1046 00:59:41,410 --> 00:59:46,930 down, so much easier to trap itself 1047 00:59:46,930 --> 00:59:52,720 into a set of configurations in which atoms are really fairly 1048 00:59:52,720 --> 00:59:55,510 happy as they are, because what is important 1049 00:59:55,510 --> 00:59:59,470 for silicon atoms is to be fourfold coordinated. 1050 00:59:59,470 --> 01:00:03,280 But what is important to be in a crystal 1051 01:00:03,280 --> 01:00:07,030 is that this fourfold coordination extends 1052 01:00:07,030 --> 01:00:09,760 and becomes a long-range order. 1053 01:00:09,760 --> 01:00:12,820 But for silicon, it's very inexpensive to mess up 1054 01:00:12,820 --> 01:00:16,390 that long-range order, like for many other systems. 1055 01:00:16,390 --> 01:00:20,140 And in principle, same thing happens for polymers. 1056 01:00:20,140 --> 01:00:21,340 You lower it down. 1057 01:00:21,340 --> 01:00:24,520 And they sort of get trapped in a glassy configuration. 1058 01:00:24,520 --> 01:00:27,280 A lot of polymers might want to stay actually 1059 01:00:27,280 --> 01:00:29,020 in a crystalline state. 1060 01:00:29,020 --> 01:00:32,060 But really that is impossible to reach. 1061 01:00:32,060 --> 01:00:33,670 And so if you want to study, say, 1062 01:00:33,670 --> 01:00:35,830 what are the properties of your silicon 1063 01:00:35,830 --> 01:00:40,360 and you have obtained or you are trying to obtain your silicon 1064 01:00:40,360 --> 01:00:44,830 structure from a liquid that you cooled down, you'll actually-- 1065 01:00:44,830 --> 01:00:46,810 in your molecular dynamics, you'll 1066 01:00:46,810 --> 01:00:52,690 keep orbiting around in this glassy mass state 1067 01:00:52,690 --> 01:00:56,380 and will never reach that region of phase space 1068 01:00:56,380 --> 01:00:59,680 that is actually very, very small because it 1069 01:00:59,680 --> 01:01:03,340 needs to have all atoms long range in very 1070 01:01:03,340 --> 01:01:04,840 specific positions. 1071 01:01:04,840 --> 01:01:10,090 And so you are really spanning a system in a nonergodic way. 1072 01:01:10,090 --> 01:01:13,960 That is, you spend a lot of time in a region of phase space 1073 01:01:13,960 --> 01:01:17,950 that are not really representative but out of which 1074 01:01:17,950 --> 01:01:19,810 is very difficult to escape. 1075 01:01:19,810 --> 01:01:23,440 OK, so there are problems in, again, 1076 01:01:23,440 --> 01:01:27,070 brute force molecular dynamic approaches that you 1077 01:01:27,070 --> 01:01:30,340 have to be careful about. 1078 01:01:30,340 --> 01:01:35,630 OK, so let's see how we actually go 1079 01:01:35,630 --> 01:01:39,590 and try to do this on our computer experiment. 1080 01:01:39,590 --> 01:01:42,350 And I'll start with the simplest case 1081 01:01:42,350 --> 01:01:46,010 that is what is called the microcanonical case. 1082 01:01:46,010 --> 01:01:50,750 That is, we perform a simulation where the number of particles 1083 01:01:50,750 --> 01:01:54,650 is constant, the volume is constant, 1084 01:01:54,650 --> 01:01:57,920 and the total energy of the system is constant. 1085 01:01:57,920 --> 01:02:01,040 This is what is called a thermodynamic ensemble. 1086 01:02:01,040 --> 01:02:05,120 Again, these are all extensive variables. 1087 01:02:05,120 --> 01:02:07,910 And we are fixing them. 1088 01:02:07,910 --> 01:02:10,760 Sometimes it's appropriate, instead, 1089 01:02:10,760 --> 01:02:13,400 to do a Legendre transformation and go 1090 01:02:13,400 --> 01:02:15,950 into a different thermodynamic ensemble. 1091 01:02:15,950 --> 01:02:19,040 Say, you could want not to be at a constant volume. 1092 01:02:19,040 --> 01:02:22,820 But you might want to study a system at a constant pressure 1093 01:02:22,820 --> 01:02:24,830 where the volume changes. 1094 01:02:24,830 --> 01:02:27,530 And so we need to describe how we'll actually 1095 01:02:27,530 --> 01:02:28,580 be able to do that. 1096 01:02:28,580 --> 01:02:31,670 That is, how will we be able to move 1097 01:02:31,670 --> 01:02:35,990 to a system in which different intensive thermodynamic 1098 01:02:35,990 --> 01:02:37,520 variables are constant? 1099 01:02:37,520 --> 01:02:42,170 But if you take a system of particles in periodic boundary 1100 01:02:42,170 --> 01:02:47,180 conditions and you have 10 water molecules in a certain unit 1101 01:02:47,180 --> 01:02:50,990 cell and you evolve them with the Newton equation of motion, 1102 01:02:50,990 --> 01:02:54,980 you can obviously know that the number of molecules 1103 01:02:54,980 --> 01:02:57,450 and the volume of the unit cell are not going to change. 1104 01:02:57,450 --> 01:02:59,090 And if you integrate your equation 1105 01:02:59,090 --> 01:03:02,510 of motion appropriately, the total energy of the system, 1106 01:03:02,510 --> 01:03:05,750 that is the kinetic energy plus the potential energy, 1107 01:03:05,750 --> 01:03:08,190 is not going to change. 1108 01:03:08,190 --> 01:03:12,005 And so as we evolve this point in phase space, 1109 01:03:12,005 --> 01:03:15,680 its trajectory is going to span what we call 1110 01:03:15,680 --> 01:03:18,470 the microcanonical ensemble. 1111 01:03:18,470 --> 01:03:23,210 And you let this point evolve long enough. 1112 01:03:23,210 --> 01:03:28,220 And all of a sudden, you will have a meaningful trajectory, 1113 01:03:28,220 --> 01:03:33,710 that is a meaningful assembly collection of microstates 1114 01:03:33,710 --> 01:03:38,900 of representative configurations in space and in velocities 1115 01:03:38,900 --> 01:03:42,770 of your system, out of which you can calculate averages 1116 01:03:42,770 --> 01:03:47,820 of any of the observables that you want to calculate 1117 01:03:47,820 --> 01:03:50,190 and that you want to characterize. 1118 01:03:50,190 --> 01:03:55,170 So what do you do in these molecular dynamic simulations? 1119 01:03:55,170 --> 01:03:58,710 Well, there are really four important steps 1120 01:03:58,710 --> 01:04:00,390 that you need to take into account. 1121 01:04:00,390 --> 01:04:05,640 And we keep thinking of water, liquid water, as an example. 1122 01:04:05,640 --> 01:04:07,932 That is, you need to start from something. 1123 01:04:07,932 --> 01:04:09,390 That is, you need to put your water 1124 01:04:09,390 --> 01:04:10,950 molecule in a certain position. 1125 01:04:10,950 --> 01:04:13,980 And you need to give them initial velocities. 1126 01:04:13,980 --> 01:04:18,210 And you are almost never going to be good enough 1127 01:04:18,210 --> 01:04:22,710 in choosing a state of the system that is somehow 1128 01:04:22,710 --> 01:04:23,670 meaningful. 1129 01:04:23,670 --> 01:04:26,670 You'll always, when you start building something complex, 1130 01:04:26,670 --> 01:04:30,420 put molecules in places that they don't want to stay. 1131 01:04:30,420 --> 01:04:34,410 And one needs to be very careful, then, in all of this. 1132 01:04:34,410 --> 01:04:38,190 But suppose that we have, in any case, an initial configuration 1133 01:04:38,190 --> 01:04:39,900 of position and velocity. 1134 01:04:39,900 --> 01:04:44,680 Then what we need to do is, over and over again, integrate 1135 01:04:44,680 --> 01:04:47,460 the equation of motion, that is, from the current position 1136 01:04:47,460 --> 01:04:52,020 and the current velocities, predict the new position 1137 01:04:52,020 --> 01:04:54,780 after a certain time. 1138 01:04:54,780 --> 01:04:57,570 And once you have the new position at a certain time, 1139 01:04:57,570 --> 01:05:00,450 you can recalculate what are the forces acting 1140 01:05:00,450 --> 01:05:02,760 on the particle in the new position. 1141 01:05:02,760 --> 01:05:05,190 And with these forces, you can, again, 1142 01:05:05,190 --> 01:05:09,610 move the particles by a certain amount and so on and so forth. 1143 01:05:09,610 --> 01:05:11,940 And that is where the molecular dynamics 1144 01:05:11,940 --> 01:05:13,600 algorithm comes into play. 1145 01:05:13,600 --> 01:05:16,530 We need to integrate our equation of motion. 1146 01:05:16,530 --> 01:05:20,320 And with this, our system will start to evolve. 1147 01:05:20,320 --> 01:05:23,280 We have put that point in phase space in motion. 1148 01:05:23,280 --> 01:05:25,540 And it'll start moving around. 1149 01:05:25,540 --> 01:05:27,720 And if we are integrating correctly, 1150 01:05:27,720 --> 01:05:34,090 the total energy of the system is going to be conserved. 1151 01:05:34,090 --> 01:05:36,930 Now what we need to do is we need 1152 01:05:36,930 --> 01:05:41,460 to make sure that our characterization, 1153 01:05:41,460 --> 01:05:44,640 our thermodynamic averages, do not depend 1154 01:05:44,640 --> 01:05:46,260 on our initial condition. 1155 01:05:46,260 --> 01:05:49,890 Again, we have started from a configuration 1156 01:05:49,890 --> 01:05:51,900 that we have built by hand. 1157 01:05:51,900 --> 01:05:55,230 And because of that, it's most likely 1158 01:05:55,230 --> 01:05:59,160 a very unfavorable configuration of the system. 1159 01:05:59,160 --> 01:06:05,190 OK, so we need to let the system evolve and somehow start 1160 01:06:05,190 --> 01:06:08,610 figuring out really in which condition 1161 01:06:08,610 --> 01:06:10,060 it would want to stay. 1162 01:06:10,060 --> 01:06:15,030 Think of the previous example of silicon that is glassifying. 1163 01:06:15,030 --> 01:06:16,620 Well, we had liquid silicon. 1164 01:06:16,620 --> 01:06:18,120 We have to cool it down. 1165 01:06:18,120 --> 01:06:20,970 In a way, what's happening is that we are never 1166 01:06:20,970 --> 01:06:25,770 equilibrating it because the system goes into a glass phase. 1167 01:06:25,770 --> 01:06:30,370 And so it never loses memory of its initial condition. 1168 01:06:30,370 --> 01:06:34,510 It's always sort of trapped in this pseudo liquid, now glass, 1169 01:06:34,510 --> 01:06:35,280 state. 1170 01:06:35,280 --> 01:06:37,710 While in reality, at a temperature 1171 01:06:37,710 --> 01:06:40,680 below the freezing or the melting point, 1172 01:06:40,680 --> 01:06:43,320 equilibrium would be given by a crystalline state 1173 01:06:43,320 --> 01:06:45,030 with the ions vibrating. 1174 01:06:45,030 --> 01:06:47,610 So there are cases in which equilibration 1175 01:06:47,610 --> 01:06:50,580 is so difficult that we actually never reach it. 1176 01:06:50,580 --> 01:06:53,010 Silicon never goes to a crystal. 1177 01:06:53,010 --> 01:06:54,430 That's why we need to be careful. 1178 01:06:54,430 --> 01:06:57,210 But in most cases awfully we are lucky. 1179 01:06:57,210 --> 01:06:59,940 And we lose memory of initial condition. 1180 01:06:59,940 --> 01:07:03,360 Say, the opposite is actually very easy to achieve. 1181 01:07:03,360 --> 01:07:06,420 If we start from the solid and we warm it up, 1182 01:07:06,420 --> 01:07:08,140 it becomes a liquid. 1183 01:07:08,140 --> 01:07:11,400 And if we wait long enough, that liquid 1184 01:07:11,400 --> 01:07:15,420 will have lost all memory of having come from a solid. 1185 01:07:15,420 --> 01:07:18,210 OK, so it's actually very easy in one direction 1186 01:07:18,210 --> 01:07:21,670 and very difficult in the other direction. 1187 01:07:21,670 --> 01:07:28,090 But once this equilibration time has passed, well, at that point 1188 01:07:28,090 --> 01:07:30,270 we have truly everything in place. 1189 01:07:30,270 --> 01:07:33,120 Our point in phase space is moving around. 1190 01:07:33,120 --> 01:07:38,040 And it's really going into the regions of high probability 1191 01:07:38,040 --> 01:07:41,710 and distributed, according to Boltzmann distribution. 1192 01:07:41,710 --> 01:07:45,600 And so at that point, we can start an average 1193 01:07:45,600 --> 01:07:50,340 among any of the physical observables 1194 01:07:50,340 --> 01:07:52,230 that we want to calculate. 1195 01:07:52,230 --> 01:07:54,810 And so there are these four phases. 1196 01:07:54,810 --> 01:08:02,220 And I'll briefly mention how we actually choose them. 1197 01:08:02,220 --> 01:08:03,630 Initialization. 1198 01:08:03,630 --> 01:08:07,020 Well, again, we need position and velocities. 1199 01:08:07,020 --> 01:08:11,040 You need to be careful, in particular, 1200 01:08:11,040 --> 01:08:15,150 to avoid overlap between your molecules. 1201 01:08:15,150 --> 01:08:18,420 If you put two atoms very close, they 1202 01:08:18,420 --> 01:08:22,630 are going to repel each other with an enormous strength. 1203 01:08:22,630 --> 01:08:25,229 And if you put them close enough, 1204 01:08:25,229 --> 01:08:27,899 they are going to shoot in opposite directions 1205 01:08:27,899 --> 01:08:31,649 very quickly and completely kick off and mess up 1206 01:08:31,649 --> 01:08:34,630 everything on the side. 1207 01:08:34,630 --> 01:08:40,680 So what is difficult is really, always in configuration space, 1208 01:08:40,680 --> 01:08:43,770 finding the right structures. 1209 01:08:43,770 --> 01:08:47,640 This is the difficult part of choosing initial condition. 1210 01:08:47,640 --> 01:08:51,000 Choosing velocities is much easier. 1211 01:08:51,000 --> 01:08:53,830 One possibility is just giving zero velocity 1212 01:08:53,830 --> 01:08:57,270 to the system or a very small velocity 1213 01:08:57,270 --> 01:09:00,180 and then slowly adding things. 1214 01:09:00,180 --> 01:09:05,430 And what we usually see is that configuration in velocity space 1215 01:09:05,430 --> 01:09:07,050 is much faster. 1216 01:09:07,050 --> 01:09:08,399 And this is the reason. 1217 01:09:08,399 --> 01:09:12,029 Think of that trajectory for a moment of one particle. 1218 01:09:12,029 --> 01:09:15,245 And think of what happens when you have a collision. 1219 01:09:15,245 --> 01:09:17,370 Suppose that you have particles, for a moment, that 1220 01:09:17,370 --> 01:09:18,870 are hard spheres. 1221 01:09:18,870 --> 01:09:23,890 In a collision, the velocity of the particle 1222 01:09:23,890 --> 01:09:27,149 all of a sudden changes sign. 1223 01:09:27,149 --> 01:09:31,020 So in the velocity part of the phase space, 1224 01:09:31,020 --> 01:09:35,760 your point makes a sudden jump, OK, 1225 01:09:35,760 --> 01:09:39,300 because the velocity that was plus 10 has become 1226 01:09:39,300 --> 01:09:41,819 all of a sudden minus 10. 1227 01:09:41,819 --> 01:09:45,899 So in velocity space, for the limit case 1228 01:09:45,899 --> 01:09:50,250 of a system of hard spheres, your representative point 1229 01:09:50,250 --> 01:09:54,060 hops everywhere continuously. 1230 01:09:54,060 --> 01:09:57,870 In reality, I mean, you have interaction at all times. 1231 01:09:57,870 --> 01:10:00,060 And you don't really hop. 1232 01:10:00,060 --> 01:10:03,030 But you can sort of move very quickly 1233 01:10:03,030 --> 01:10:05,550 from one region to the other region. 1234 01:10:05,550 --> 01:10:09,420 There is not anything similar to the glass entanglement 1235 01:10:09,420 --> 01:10:11,200 in configuration space. 1236 01:10:11,200 --> 01:10:16,140 So it's very easy to thermalize velocity. 1237 01:10:16,140 --> 01:10:20,130 It's very difficult to thermalize and equilibrate 1238 01:10:20,130 --> 01:10:20,910 position. 1239 01:10:20,910 --> 01:10:25,880 And that's where, all, you need to be careful. 1240 01:10:25,880 --> 01:10:29,140 And if you really want to be accurate, 1241 01:10:29,140 --> 01:10:31,540 what you can also do instead of giving zero 1242 01:10:31,540 --> 01:10:35,620 velocity to your system, you can use your concept 1243 01:10:35,620 --> 01:10:37,070 of statistical mechanics. 1244 01:10:37,070 --> 01:10:40,300 That is, a system of classical particles 1245 01:10:40,300 --> 01:10:44,560 will have a distribution of velocities 1246 01:10:44,560 --> 01:10:48,340 that depends on the temperature at which it is. 1247 01:10:48,340 --> 01:10:49,960 And this is 0 in Celsius. 1248 01:10:49,960 --> 01:10:52,390 So it's 273 Kelvin. 1249 01:10:52,390 --> 01:10:57,640 And the distribution of velocity in a classical system 1250 01:10:57,640 --> 01:11:00,970 is actually given by the Maxwell-Boltzmann distribution 1251 01:11:00,970 --> 01:11:04,910 written above that has basically this shape. 1252 01:11:04,910 --> 01:11:10,330 So it means that if we are, say, at 0 Celsius, 273 Kelvin, 1253 01:11:10,330 --> 01:11:16,570 we'll have a lot of molecules with a certain velocity. 1254 01:11:16,570 --> 01:11:19,060 And then, in principle, we can have 1255 01:11:19,060 --> 01:11:21,670 molecules with higher and higher velocity. 1256 01:11:21,670 --> 01:11:24,560 But the probability of finding them decreases. 1257 01:11:24,560 --> 01:11:26,540 And as we increase the temperature, 1258 01:11:26,540 --> 01:11:28,870 the probability distribution of the velocities 1259 01:11:28,870 --> 01:11:31,270 for the molecules changes. 1260 01:11:31,270 --> 01:11:36,730 And so as usual, you can define both a median or an average 1261 01:11:36,730 --> 01:11:37,360 in your system. 1262 01:11:37,360 --> 01:11:42,160 That is, you can look at what is the most likely velocity, that 1263 01:11:42,160 --> 01:11:45,310 is the top of this curve, at 0 Celsius. 1264 01:11:45,310 --> 01:11:49,660 Or you could also define what is the average velocity. 1265 01:11:49,660 --> 01:11:51,550 And since that this curve is asymmetric, 1266 01:11:51,550 --> 01:11:55,310 it's going to be really shifted a little bit. 1267 01:11:55,310 --> 01:11:57,670 And these are actually the numerical expressions 1268 01:11:57,670 --> 01:12:00,700 for these two velocities, the most likely 1269 01:12:00,700 --> 01:12:03,350 and the average velocity. 1270 01:12:03,350 --> 01:12:06,970 And so I've given you the example of an oxygen molecule 1271 01:12:06,970 --> 01:12:08,320 at room temperature. 1272 01:12:08,320 --> 01:12:13,550 We are talking about 1,000 meters per second. 1273 01:12:13,550 --> 01:12:15,640 So these velocities, if you want, 1274 01:12:15,640 --> 01:12:20,620 are similar to the speed of sound in your material 1275 01:12:20,620 --> 01:12:24,370 because, at the end, sound propagates at a certain speed 1276 01:12:24,370 --> 01:12:27,520 because that's the velocity of the collision from one 1277 01:12:27,520 --> 01:12:31,370 molecule to the next that are really carrying sound away. 1278 01:12:31,370 --> 01:12:33,520 So if you want to be sophisticated, 1279 01:12:33,520 --> 01:12:36,430 you give to your system a set of velocities 1280 01:12:36,430 --> 01:12:40,990 that could be, say, all identical to the most likely 1281 01:12:40,990 --> 01:12:41,590 velocity. 1282 01:12:41,590 --> 01:12:47,170 But again, equilibration and thermalization 1283 01:12:47,170 --> 01:12:49,450 in velocity space is very easy. 1284 01:12:49,450 --> 01:12:52,600 So once you have this, that is, once you 1285 01:12:52,600 --> 01:12:55,690 have a set of positions and the set of velocities, 1286 01:12:55,690 --> 01:13:00,010 you need to use Newton's equation of motion of saying, 1287 01:13:00,010 --> 01:13:06,170 of predicting, where your system is going to go. 1288 01:13:06,170 --> 01:13:11,110 And so that's when the molecular dynamic algorithm 1289 01:13:11,110 --> 01:13:12,500 comes into play. 1290 01:13:12,500 --> 01:13:14,830 So what you need to do is, as we say, 1291 01:13:14,830 --> 01:13:20,020 you need to use an integrator, the algorithm that tells you 1292 01:13:20,020 --> 01:13:23,470 what your next position and what your next velocities 1293 01:13:23,470 --> 01:13:24,820 are going to be. 1294 01:13:24,820 --> 01:13:28,540 And there are a number of integrators. 1295 01:13:28,540 --> 01:13:29,890 There are subtleties. 1296 01:13:29,890 --> 01:13:33,790 But often you find the name of Verlet algorithm. 1297 01:13:33,790 --> 01:13:36,220 This is one class of algorithms. 1298 01:13:36,220 --> 01:13:40,710 And the other common class is the Gear predictor-corrector, 1299 01:13:40,710 --> 01:13:42,310 just to give you the terminology. 1300 01:13:42,310 --> 01:13:45,080 And we look, in a moment, at the Verlet algorithm. 1301 01:13:45,080 --> 01:13:48,100 They're all fairly simple. 1302 01:13:48,100 --> 01:13:52,360 But what is really important is that this algorithm 1303 01:13:52,360 --> 01:13:53,930 should be robust. 1304 01:13:53,930 --> 01:13:58,360 That is, they should give you a trajectory that 1305 01:13:58,360 --> 01:14:02,020 is very close to the trajectory that you would 1306 01:14:02,020 --> 01:14:05,950 get from the perfect analytical integration 1307 01:14:05,950 --> 01:14:07,600 of the equation of motion. 1308 01:14:07,600 --> 01:14:12,220 And in particular, what is the most important thing 1309 01:14:12,220 --> 01:14:17,830 is that they should be accurate in, say, conserving 1310 01:14:17,830 --> 01:14:19,510 the constant of motion. 1311 01:14:19,510 --> 01:14:22,060 As we have seen from the Newton's equation of motion, 1312 01:14:22,060 --> 01:14:24,730 the sum of the kinetic energy and the potential energy 1313 01:14:24,730 --> 01:14:26,560 should be a constant. 1314 01:14:26,560 --> 01:14:29,800 If your integration, if your numerical evaluation, 1315 01:14:29,800 --> 01:14:32,080 of that equation is poor, you will 1316 01:14:32,080 --> 01:14:36,290 see that the constant of motion is not any more constant. 1317 01:14:36,290 --> 01:14:38,890 So again, this is the sanity check 1318 01:14:38,890 --> 01:14:42,220 on the accuracy of your integration. 1319 01:14:42,220 --> 01:14:45,190 And there are more subtle elements 1320 01:14:45,190 --> 01:14:49,000 of the integration algorithm that can become important. 1321 01:14:49,000 --> 01:14:51,770 But we won't go into that. 1322 01:14:51,770 --> 01:14:55,120 And then as I said, straightforward integration 1323 01:14:55,120 --> 01:14:56,710 of the Newton's equation of motion 1324 01:14:56,710 --> 01:14:59,650 gives you the microcanonical ensemble in which 1325 01:14:59,650 --> 01:15:01,570 the total energy's conserved. 1326 01:15:01,570 --> 01:15:04,960 But sometimes you want to study systems in which 1327 01:15:04,960 --> 01:15:07,270 the temperature is fixed. 1328 01:15:07,270 --> 01:15:08,890 Or you could want to study systems 1329 01:15:08,890 --> 01:15:11,360 in which the pressure is fixed. 1330 01:15:11,360 --> 01:15:16,480 And so you need to augment your molecular dynamic integrator 1331 01:15:16,480 --> 01:15:20,740 with some feature that allows you to control 1332 01:15:20,740 --> 01:15:22,880 those intensive variables. 1333 01:15:22,880 --> 01:15:26,050 So if you want, say, to control the temperature of the system, 1334 01:15:26,050 --> 01:15:29,950 what we say usually is that we couple the system 1335 01:15:29,950 --> 01:15:31,300 to a thermostat. 1336 01:15:31,300 --> 01:15:35,050 That is, we make sure that the temperature in your evolution 1337 01:15:35,050 --> 01:15:36,010 doesn't change. 1338 01:15:36,010 --> 01:15:39,710 And you'll see examples in the next class of this. 1339 01:15:39,710 --> 01:15:43,540 And in the case, in particular, of something, 1340 01:15:43,540 --> 01:15:47,740 say, like the temperature, you have a whole set 1341 01:15:47,740 --> 01:15:50,800 of possible thermostats. 1342 01:15:50,800 --> 01:15:54,340 That is, you could use stochastic approaches. 1343 01:15:54,340 --> 01:15:56,950 That is, you have your evolution of your system. 1344 01:15:56,950 --> 01:15:59,260 And your system doesn't conserve, 1345 01:15:59,260 --> 01:16:01,390 if it's microcanonical, the temperature. 1346 01:16:01,390 --> 01:16:03,910 Sometimes it goes towards higher temperature. 1347 01:16:03,910 --> 01:16:06,620 Sometimes it go towards lower temperature. 1348 01:16:06,620 --> 01:16:12,160 And what you could do is either every now and then 1349 01:16:12,160 --> 01:16:16,310 absorb some energy or give back some energy in your system-- 1350 01:16:16,310 --> 01:16:18,370 and that would be a Langevin dynamics-- 1351 01:16:18,370 --> 01:16:22,300 in order to control the average kinetic energy of your system. 1352 01:16:22,300 --> 01:16:27,910 Or you could somehow forbid, in the microcanonical evolution, 1353 01:16:27,910 --> 01:16:32,780 your system to choose a state in which the temperature changes. 1354 01:16:32,780 --> 01:16:38,140 So if your new positions are such that your velocity gives 1355 01:16:38,140 --> 01:16:41,890 an increase in temperature, you just 1356 01:16:41,890 --> 01:16:45,430 shorten how much your atoms move in order 1357 01:16:45,430 --> 01:16:47,650 to make the velocity smaller and in order 1358 01:16:47,650 --> 01:16:49,720 to make the kinetic energy conserved. 1359 01:16:49,720 --> 01:16:52,720 Or there are more complex dynamical ways 1360 01:16:52,720 --> 01:16:54,940 of imposing temperature that goes 1361 01:16:54,940 --> 01:16:57,670 under the name of extended Hamiltonian, 1362 01:16:57,670 --> 01:16:59,290 or extended system, of which we'll see 1363 01:16:59,290 --> 01:17:02,500 an example that is [INAUDIBLE]. 1364 01:17:02,500 --> 01:17:07,810 But let me give you really what is the simplest and still 1365 01:17:07,810 --> 01:17:12,520 one of the most used numerical integrators. 1366 01:17:12,520 --> 01:17:15,650 And it tends to be very robust and very accurate. 1367 01:17:15,650 --> 01:17:20,030 So it's still used in a lot of molecular dynamic simulations. 1368 01:17:20,030 --> 01:17:23,890 And there is often no reason to do anything better than this. 1369 01:17:23,890 --> 01:17:25,930 So in a way, we are in a much luckier situation 1370 01:17:25,930 --> 01:17:28,140 than in quantum mechanics. 1371 01:17:28,140 --> 01:17:31,750 This is good enough in almost all cases. 1372 01:17:31,750 --> 01:17:34,930 And this is how we look at the problem. 1373 01:17:34,930 --> 01:17:38,620 That is, what we need to do always 1374 01:17:38,620 --> 01:17:45,940 is we need to calculate the position of each particle 1375 01:17:45,940 --> 01:17:51,970 at a time t plus delta t once we know 1376 01:17:51,970 --> 01:17:55,780 the position, the velocity, and the forces 1377 01:17:55,780 --> 01:17:58,030 on that particle at time t. 1378 01:17:58,030 --> 01:18:01,360 So suppose I'm at a certain instant in time that I call t. 1379 01:18:01,360 --> 01:18:03,980 I know everything about my system-- position, velocity, 1380 01:18:03,980 --> 01:18:04,750 and forces. 1381 01:18:04,750 --> 01:18:06,970 What I really need to do is predict 1382 01:18:06,970 --> 01:18:08,860 what are the new positions. 1383 01:18:08,860 --> 01:18:10,870 And when I'm in the new positions, 1384 01:18:10,870 --> 01:18:16,090 I want to calculate the forces again and evolve the system. 1385 01:18:16,090 --> 01:18:17,150 And now, what do we do? 1386 01:18:17,150 --> 01:18:20,470 Well, let's do a simple Taylor expansion. 1387 01:18:20,470 --> 01:18:27,130 That is, this is r of t is a function of the variable t. 1388 01:18:27,130 --> 01:18:29,080 So we can use a Taylor expansion. 1389 01:18:29,080 --> 01:18:33,160 And r of t plus delta t is going to be r of t 1390 01:18:33,160 --> 01:18:37,510 plus the derivative of r with respect to t times delta t. 1391 01:18:37,510 --> 01:18:38,980 And that's the velocity. 1392 01:18:38,980 --> 01:18:41,590 Plus 1/2 of the second derivative 1393 01:18:41,590 --> 01:18:43,693 of the position with respect to time. 1394 01:18:43,693 --> 01:18:44,860 And that's the acceleration. 1395 01:18:44,860 --> 01:18:48,310 Times delta t squared, and so on for the third derivatives 1396 01:18:48,310 --> 01:18:51,400 and so on for the fourth, fifth, ad infinitum. 1397 01:18:51,400 --> 01:18:54,550 This is just the Taylor expansion as a function, OK? 1398 01:18:54,550 --> 01:18:58,900 And so it gives us the position of t plus delta t, 1399 01:18:58,900 --> 01:19:03,890 knowing all the derivatives-- first, second, third, fourth, 1400 01:19:03,890 --> 01:19:04,450 nth-- 1401 01:19:04,450 --> 01:19:05,890 at times t. 1402 01:19:05,890 --> 01:19:07,970 And once we have the derivatives, 1403 01:19:07,970 --> 01:19:11,470 we can not only predict what the position would 1404 01:19:11,470 --> 01:19:14,380 be at t plus delta t, but also what the position would 1405 01:19:14,380 --> 01:19:16,900 be at t minus delta t. 1406 01:19:16,900 --> 01:19:18,190 And it's written here. 1407 01:19:18,190 --> 01:19:23,110 And if you think, the terms with an odd power 1408 01:19:23,110 --> 01:19:25,720 change sign in delta t. 1409 01:19:25,720 --> 01:19:29,440 And the terms with an even power in delta t do not change sign. 1410 01:19:29,440 --> 01:19:31,780 So we have just done a Taylor expansion. 1411 01:19:31,780 --> 01:19:36,370 And now what we do is we actually sum these two 1412 01:19:36,370 --> 01:19:38,660 quantities together, OK? 1413 01:19:38,660 --> 01:19:43,360 And this is what we obtain, OK? 1414 01:19:43,360 --> 01:19:45,970 We sum the two quantities together. 1415 01:19:45,970 --> 01:19:50,020 All the odd terms disappear. 1416 01:19:50,020 --> 01:19:51,550 This sums to 0. 1417 01:19:51,550 --> 01:19:52,960 This sums to 0. 1418 01:19:52,960 --> 01:19:56,140 The fifth order, seventh order, and so on sum to 0. 1419 01:19:56,140 --> 01:20:00,700 So what we have is that we have rt plus delta t plus rt 1420 01:20:00,700 --> 01:20:06,920 minus delta t is equal to 2 times rt plus alpha-- 1421 01:20:06,920 --> 01:20:09,130 sorry-- acceleration times delta t squared. 1422 01:20:09,130 --> 01:20:10,540 It's just written here. 1423 01:20:10,540 --> 01:20:12,820 And then rearrange the terms. 1424 01:20:12,820 --> 01:20:15,430 But you see, this is the Verlet algorithm. 1425 01:20:15,430 --> 01:20:18,740 Now what we have when we look at this expression, 1426 01:20:18,740 --> 01:20:21,850 we have an expression that tells us 1427 01:20:21,850 --> 01:20:26,050 what is the position at my new time step, 1428 01:20:26,050 --> 01:20:30,100 knowing the position at my current instant, 1429 01:20:30,100 --> 01:20:33,220 knowing the position at the previous instant. 1430 01:20:33,220 --> 01:20:36,820 We are keeping configurations from our trajectories. 1431 01:20:36,820 --> 01:20:38,800 So we know where we were before. 1432 01:20:38,800 --> 01:20:40,450 We know where we are now. 1433 01:20:40,450 --> 01:20:42,670 And we know where we are going to be 1434 01:20:42,670 --> 01:20:46,150 just by adding to this the term acceleration 1435 01:20:46,150 --> 01:20:49,610 at the present time times delta t squared. 1436 01:20:49,610 --> 01:20:53,210 We make an error because we are not 1437 01:20:53,210 --> 01:20:56,300 introducing terms that are higher 1438 01:20:56,300 --> 01:20:58,730 order in delta t squared. 1439 01:20:58,730 --> 01:21:02,940 But you see, the term in delta t cubed has canceled out. 1440 01:21:02,940 --> 01:21:07,520 So the error that we make is just of the order delta t 1441 01:21:07,520 --> 01:21:08,420 fourth. 1442 01:21:08,420 --> 01:21:11,900 That is, that means that the smaller we make the delta t, 1443 01:21:11,900 --> 01:21:15,620 the smaller we make the time step, the more accurate 1444 01:21:15,620 --> 01:21:17,420 our integration is. 1445 01:21:17,420 --> 01:21:20,205 And so this is, if you want the most critical parameter 1446 01:21:20,205 --> 01:21:22,160 in your molecular dynamic simulation, 1447 01:21:22,160 --> 01:21:26,030 you need to make sure that the steps that you take 1448 01:21:26,030 --> 01:21:30,230 are small enough, considering what is really 1449 01:21:30,230 --> 01:21:34,880 the variation of your force as a function of time 1450 01:21:34,880 --> 01:21:37,910 because remember that now that we use 1451 01:21:37,910 --> 01:21:42,830 Newton's equation of motion, the acceleration at a time t 1452 01:21:42,830 --> 01:21:48,860 is just given by the force divided by the mass. 1453 01:21:48,860 --> 01:21:51,680 And because the force is just the gradient of the potential, 1454 01:21:51,680 --> 01:21:53,520 it's just given by this. 1455 01:21:53,520 --> 01:21:55,980 So at a certain instant in time t, 1456 01:21:55,980 --> 01:21:58,130 you have the acceleration acting in your particles 1457 01:21:58,130 --> 01:22:01,100 just by taking the force and dividing by the mass. 1458 01:22:01,100 --> 01:22:03,470 So at every instant, you calculate force. 1459 01:22:03,470 --> 01:22:06,590 And the force lets you evolve the system. 1460 01:22:06,590 --> 01:22:10,100 And it keeps giving you the new positions. 1461 01:22:10,100 --> 01:22:13,550 And so you know what is the trajectory and configuration 1462 01:22:13,550 --> 01:22:14,420 space. 1463 01:22:14,420 --> 01:22:17,630 And you can very simply find out what 1464 01:22:17,630 --> 01:22:20,150 is your velocity at every instant t, 1465 01:22:20,150 --> 01:22:23,300 if you need to calculate what is the velocity just 1466 01:22:23,300 --> 01:22:28,500 by doing a final difference calculation of the tangent 1467 01:22:28,500 --> 01:22:29,000 basically. 1468 01:22:29,000 --> 01:22:31,520 The velocity is the tangent of the trajectory 1469 01:22:31,520 --> 01:22:32,570 with respect to t. 1470 01:22:32,570 --> 01:22:35,990 OK, this concludes class. 1471 01:22:35,990 --> 01:22:38,810 What we'll see in the next lecture 1472 01:22:38,810 --> 01:22:43,460 is how we actually take this Verlet integration 1473 01:22:43,460 --> 01:22:46,520 and how we use it to do the characterization 1474 01:22:46,520 --> 01:22:50,040 of the thermodynamic properties of a real system. 1475 01:22:50,040 --> 01:22:54,610 So today is Thursday. 1476 01:22:54,610 --> 01:22:56,810 Next week is spring break. 1477 01:22:56,810 --> 01:23:03,280 So we'll see each other again on Tuesday, 29th. 1478 01:23:03,280 --> 01:23:07,500 And I will be at a conference. 1479 01:23:07,500 --> 01:23:11,290 And all the TAs will be at a conference next week. 1480 01:23:11,290 --> 01:23:15,420 So it will be slightly difficult communicating via email. 1481 01:23:15,420 --> 01:23:19,560 So try to ask a lot of questions between now 1482 01:23:19,560 --> 01:23:23,610 and Sunday for your problem set that is due the Tuesday 1483 01:23:23,610 --> 01:23:24,690 after spring break. 1484 01:23:24,690 --> 01:23:27,990 And that will be problem set 2. 1485 01:23:27,990 --> 01:23:30,500 Otherwise, enjoy spring break.