1 00:00:00,060 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high quality educational resources for free. 5 00:00:10,730 --> 00:00:13,330 To make a donation, or view additional materials 6 00:00:13,330 --> 00:00:17,236 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,236 --> 00:00:17,861 at ocw.mit.edu. 8 00:00:20,840 --> 00:00:23,166 PROFESSOR: Let's start. 9 00:00:23,166 --> 00:00:24,255 Are there any questions? 10 00:00:29,290 --> 00:00:34,220 We would like to have a perspective for this really 11 00:00:34,220 --> 00:00:41,350 common observation that if you have a gas that is initially 12 00:00:41,350 --> 00:00:45,730 in one half of a box, and the other half is empty, 13 00:00:45,730 --> 00:00:50,900 and some kind of a partition is removed so that the gas can 14 00:00:50,900 --> 00:00:54,620 expand, and it can flow, and eventually we 15 00:00:54,620 --> 00:00:57,400 will reach another equilibrium state where 16 00:00:57,400 --> 00:01:00,950 the gas occupies more chambers. 17 00:01:00,950 --> 00:01:04,450 How do we describe this observation? 18 00:01:04,450 --> 00:01:07,720 We can certainly characterize it thermodynamically 19 00:01:07,720 --> 00:01:11,940 from the perspectives of atoms and molecules. 20 00:01:11,940 --> 00:01:16,510 We said that if I want to describe 21 00:01:16,510 --> 00:01:22,410 the configuration of the gas before it starts, and also 22 00:01:22,410 --> 00:01:25,240 throughout the expansion, I would basically 23 00:01:25,240 --> 00:01:30,200 have to look at all sets of coordinates and momenta 24 00:01:30,200 --> 00:01:32,790 that make up this particle. 25 00:01:32,790 --> 00:01:34,710 There would be some point in this [? six ?], 26 00:01:34,710 --> 00:01:36,910 and I mention our phase space, that 27 00:01:36,910 --> 00:01:39,710 would correspond to where this particle was originally. 28 00:01:42,690 --> 00:01:46,110 We can certainly follow the dynamics of this point, 29 00:01:46,110 --> 00:01:48,480 but is that useful? 30 00:01:48,480 --> 00:01:50,730 Normally, I could start with billions 31 00:01:50,730 --> 00:01:54,690 of different types of boxes, or the same box 32 00:01:54,690 --> 00:01:56,780 in a different instance of time, and I 33 00:01:56,780 --> 00:02:01,130 would have totally different initial conditions. 34 00:02:01,130 --> 00:02:03,730 The initial conditions presumably 35 00:02:03,730 --> 00:02:08,150 can be characterized to a density in this phase space. 36 00:02:08,150 --> 00:02:11,435 You can look at some volume and see how it changes, 37 00:02:11,435 --> 00:02:13,790 and how many points you have there, 38 00:02:13,790 --> 00:02:16,610 and define this phase space density 39 00:02:16,610 --> 00:02:21,330 row of all of the Q's and P's, and it 40 00:02:21,330 --> 00:02:24,790 works as a function of time. 41 00:02:24,790 --> 00:02:28,400 One way of looking at how it works as a function of time 42 00:02:28,400 --> 00:02:32,580 is to look at this box and where this box will 43 00:02:32,580 --> 00:02:34,560 be in some other instance of time. 44 00:02:38,800 --> 00:02:45,250 Essentially then, we are following a kind of evolution 45 00:02:45,250 --> 00:02:48,540 that goes along this streamline. 46 00:02:48,540 --> 00:02:54,340 Basically, the derivative that we are going to look at 47 00:02:54,340 --> 00:03:00,150 involves changes both explicitly in the time variable, 48 00:03:00,150 --> 00:03:03,020 and also increasingly to the changes 49 00:03:03,020 --> 00:03:05,640 of all of the coordinates and momenta, 50 00:03:05,640 --> 00:03:09,360 according to the Hamiltonian that governs the system. 51 00:03:09,360 --> 00:03:13,700 I have to do, essentially, a sum over all coordinates. 52 00:03:13,700 --> 00:03:21,080 I would have the change in coordinate i, 53 00:03:21,080 --> 00:03:27,420 Qi dot, dot, d row by dQi. 54 00:03:27,420 --> 00:03:35,500 Then I would have Pi, dot-- I guess these are all vectors-- 55 00:03:35,500 --> 00:03:39,150 d row by dPi. 56 00:03:39,150 --> 00:03:44,010 There are six end coordinates that implicitly depend on time. 57 00:03:44,010 --> 00:03:46,500 In principle, if I am following along the streamline, 58 00:03:46,500 --> 00:03:48,325 I have to look at all of these things. 59 00:03:51,080 --> 00:03:56,240 The characteristic of evolution, according to some Hamiltonian, 60 00:03:56,240 --> 00:04:01,540 was that this volume of phase space does not change. 61 00:04:01,540 --> 00:04:05,370 Secondly, we could characterize, once we 62 00:04:05,370 --> 00:04:20,740 wrote Qi dot, as dH by dP, and the i dot as the H by dQ. 63 00:04:20,740 --> 00:04:24,560 This combination of derivatives essentially could be captured, 64 00:04:24,560 --> 00:04:30,230 and be written as 0 by dt is the Poisson bracket 65 00:04:30,230 --> 00:04:31,870 of H and [? P. ?] 66 00:04:35,270 --> 00:04:39,110 One of the things, however, that we emphasize 67 00:04:39,110 --> 00:04:44,050 is that as far as evolution according to a Hamiltonian 68 00:04:44,050 --> 00:04:50,740 and this set of dynamics is concerned, 69 00:04:50,740 --> 00:04:54,480 the situation is completely reversible in time 70 00:04:54,480 --> 00:04:57,020 so that some intermediate process, 71 00:04:57,020 --> 00:05:01,170 if I were to reverse all of the momenta, then the gas 72 00:05:01,170 --> 00:05:07,130 would basically come back to the initial position. 73 00:05:07,130 --> 00:05:08,040 That's true. 74 00:05:08,040 --> 00:05:10,170 There is nothing to do about it. 75 00:05:10,170 --> 00:05:13,410 That kind of seems to go against the intuition 76 00:05:13,410 --> 00:05:16,556 that we have from thermodynamics. 77 00:05:19,210 --> 00:05:23,120 We said, well, in practical situations, 78 00:05:23,120 --> 00:05:27,460 I really don't care about all the six end pieces 79 00:05:27,460 --> 00:05:30,270 of information that are embedded currently 80 00:05:30,270 --> 00:05:33,370 in this full phase space density. 81 00:05:33,370 --> 00:05:36,340 If I'm really trying to physically describe 82 00:05:36,340 --> 00:05:39,470 this gas expanding, typically the things 83 00:05:39,470 --> 00:05:43,170 that I'm interested in are that at some intermediate time, 84 00:05:43,170 --> 00:05:45,190 whether the particles have reached 85 00:05:45,190 --> 00:05:47,820 this point or that point, and what 86 00:05:47,820 --> 00:05:51,090 is this streamline velocity that I'm seeing before the thing 87 00:05:51,090 --> 00:05:54,570 relaxes, presumably, eventually into zero velocity? 88 00:05:54,570 --> 00:05:57,200 There's a lot of things that I would 89 00:05:57,200 --> 00:06:00,980 need to characterize this relaxation process, 90 00:06:00,980 --> 00:06:03,560 but that is still much, much, much 91 00:06:03,560 --> 00:06:06,600 less than all of the information that is currently encoded 92 00:06:06,600 --> 00:06:10,050 in all of these six end coordinates and momenta. 93 00:06:10,050 --> 00:06:14,880 We said that for things that I'm really interested in, 94 00:06:14,880 --> 00:06:17,610 what I could, for example, look at, 95 00:06:17,610 --> 00:06:20,960 is a density that involves only one particle. 96 00:06:25,650 --> 00:06:29,820 What I can do is to then integrate over 97 00:06:29,820 --> 00:06:35,000 all of the positions and coordinates of particles 98 00:06:35,000 --> 00:06:36,746 that I'm not interested in. 99 00:06:50,230 --> 00:06:55,020 I'm sort of repeating this to introduce some notation so as 100 00:06:55,020 --> 00:06:58,120 to not to repeat all of these integration variables, 101 00:06:58,120 --> 00:07:02,085 so I will call dVi the phase place contribution of particle 102 00:07:02,085 --> 00:07:02,585 i. 103 00:07:06,090 --> 00:07:10,320 What I may be interested in is that this is something 104 00:07:10,320 --> 00:07:12,790 that, if I integrate over P1 and Q1, 105 00:07:12,790 --> 00:07:17,390 it is clearly normalized to unity 106 00:07:17,390 --> 00:07:22,160 because my row, by definition, was normalized to unity. 107 00:07:22,160 --> 00:07:24,690 Typically we may be interested in something else 108 00:07:24,690 --> 00:07:32,180 that I call F1, P1 Q1 P, which is simply 109 00:07:32,180 --> 00:07:37,510 n times this-- n times the integral product 110 00:07:37,510 --> 00:07:47,620 out i2 to n, dVi, the full row. 111 00:07:50,470 --> 00:07:52,740 Why we do that is because typically you 112 00:07:52,740 --> 00:07:56,110 are interested or used to calculating things 113 00:07:56,110 --> 00:07:58,330 in [? terms ?] of a number density, 114 00:07:58,330 --> 00:08:02,190 like how many particles are within some small volume here, 115 00:08:02,190 --> 00:08:06,660 defining the density so that when I integrate over 116 00:08:06,660 --> 00:08:11,080 the entire volume of f1, I would get 117 00:08:11,080 --> 00:08:13,990 the total number of particles, for example. 118 00:08:13,990 --> 00:08:19,270 That's the kind of normalization that people have used for f. 119 00:08:19,270 --> 00:08:22,340 More generally, we also introduced 120 00:08:22,340 --> 00:08:31,600 fs, which depended on coordinates representing 121 00:08:31,600 --> 00:08:35,299 s sets of points, or s particles, if you like, 122 00:08:35,299 --> 00:08:37,059 that was normalized to be-- 123 00:08:56,710 --> 00:09:00,710 We said, OK, what I'm really interested in, in order 124 00:09:00,710 --> 00:09:05,220 to calculate the properties of the gases it expands in terms 125 00:09:05,220 --> 00:09:08,530 of things that I'm able to measure, is f1. 126 00:09:11,550 --> 00:09:14,860 Let's write down the time evolution of f1. 127 00:09:14,860 --> 00:09:16,500 Actually, we said, let's write down 128 00:09:16,500 --> 00:09:19,430 the time evolution of fs, along with it. 129 00:09:19,430 --> 00:09:22,520 So there's the time evolution of fs. 130 00:09:22,520 --> 00:09:26,740 If I were to go along this stream, 131 00:09:26,740 --> 00:09:32,900 it would be the fs by dt, and then I 132 00:09:32,900 --> 00:09:37,855 would have contributions that would correspond 133 00:09:37,855 --> 00:09:43,760 to a the changes in coordinates of these particles. 134 00:09:43,760 --> 00:09:47,930 In order to progress along this direction, 135 00:09:47,930 --> 00:09:51,732 we said, let's define the total Hamiltonian. 136 00:09:51,732 --> 00:09:55,995 We will have a simple form, and certainly for the gas, 137 00:09:55,995 --> 00:09:59,050 it would be a good representation. 138 00:09:59,050 --> 00:10:03,260 I have the kinetic energies of all of the particles. 139 00:10:03,260 --> 00:10:07,840 I have the box that confines the particles, or some other one 140 00:10:07,840 --> 00:10:12,450 particle potential, if you like, but I will write in this much. 141 00:10:12,450 --> 00:10:16,720 Then you have the interactions between all pairs of particles. 142 00:10:16,720 --> 00:10:22,669 Let's write it as sum over i, less than j, V of Qi minus Qj. 143 00:10:31,520 --> 00:10:36,840 This depends on n set of particles, coordinates, 144 00:10:36,840 --> 00:10:39,170 and momenta. 145 00:10:39,170 --> 00:10:42,480 Then we said that for purposes of manipulations that you have 146 00:10:42,480 --> 00:10:45,950 to deal with, since there are s coordinates 147 00:10:45,950 --> 00:10:48,880 that are appearing here whose time derivatives I have 148 00:10:48,880 --> 00:10:52,600 to look at, I'm going to simply rewrite 149 00:10:52,600 --> 00:10:54,590 this as the contribution that comes 150 00:10:54,590 --> 00:10:58,310 from those s particles, the contribution that comes 151 00:10:58,310 --> 00:11:01,310 from the remaining n minus s particles, 152 00:11:01,310 --> 00:11:04,571 and some kind of [? term ?] that covers the two 153 00:11:04,571 --> 00:11:05,320 sets of particles. 154 00:11:11,720 --> 00:11:15,430 This, actually, I didn't quite need here until the next stage 155 00:11:15,430 --> 00:11:18,910 because what I write here could, presumably, 156 00:11:18,910 --> 00:11:23,460 be sufficiently general, like we have here some n 157 00:11:23,460 --> 00:11:26,670 running from 1 to s. 158 00:11:26,670 --> 00:11:30,260 Let me be consistent with my S's. 159 00:11:30,260 --> 00:11:48,795 Then I have Qn, dot, dFs by dQn, plus Pn, dot, dFs by dPn. 160 00:11:58,240 --> 00:12:05,780 If I just look at the coordinates that appear here, 161 00:12:05,780 --> 00:12:10,470 and say, following this as they move in time, 162 00:12:10,470 --> 00:12:13,380 there is the explicit time dependence 163 00:12:13,380 --> 00:12:15,490 on all of the implicit time dependence, 164 00:12:15,490 --> 00:12:17,790 this would be the total derivative moving 165 00:12:17,790 --> 00:12:20,600 along the streamline. 166 00:12:20,600 --> 00:12:26,410 Qn dot I know is simply the momentum. 167 00:12:26,410 --> 00:12:29,192 It is the H by dPn. 168 00:12:29,192 --> 00:12:33,720 The H by dPn I have from this formula over here. 169 00:12:33,720 --> 00:12:37,470 It is simply Pn divided by m. 170 00:12:37,470 --> 00:12:40,300 It's the velocity-- momentum divided by mass. 171 00:12:40,300 --> 00:12:43,010 This is the velocity of the particle. 172 00:12:43,010 --> 00:12:45,460 Pn dot, the rate of change of momentum 173 00:12:45,460 --> 00:12:47,350 is the force that is acting on the particle. 174 00:12:51,020 --> 00:12:54,720 What I need to do is to take the derivatives of various terms 175 00:12:54,720 --> 00:12:55,680 here. 176 00:12:55,680 --> 00:12:59,550 So I have minus dU by dPn. 177 00:13:03,120 --> 00:13:03,782 What is this? 178 00:13:03,782 --> 00:13:05,740 This is essentially the force that the particle 179 00:13:05,740 --> 00:13:07,990 feels from the external potential. 180 00:13:07,990 --> 00:13:09,620 If you are in the box in this room, 181 00:13:09,620 --> 00:13:13,820 It is zero until you hit the edge of the box. 182 00:13:13,820 --> 00:13:20,230 I will call this Fn to represent external potential that 183 00:13:20,230 --> 00:13:22,570 is acting on the system. 184 00:13:22,570 --> 00:13:24,100 What else is there? 185 00:13:24,100 --> 00:13:28,270 I have the force that will come from the interaction 186 00:13:28,270 --> 00:13:35,150 with all other [? guys. ?] I will write here a sum over m, 187 00:13:35,150 --> 00:13:44,675 dV of Qm minus Qn, by dQn-- dU by dQm. 188 00:13:44,675 --> 00:13:45,375 I'm sorry. 189 00:13:49,920 --> 00:13:50,795 What is this? 190 00:13:50,795 --> 00:13:58,460 This Is basically the sum of the forces that 191 00:13:58,460 --> 00:14:03,900 is exerted by the n particle on the m particle. 192 00:14:03,900 --> 00:14:05,655 Define it in this fashion. 193 00:14:10,790 --> 00:14:15,060 If this was the entire story, what I would have had 194 00:14:15,060 --> 00:14:21,610 here is a group of s particles that 195 00:14:21,610 --> 00:14:26,070 are dominated by their own dynamics. 196 00:14:26,070 --> 00:14:29,450 If there is no other particle involved, 197 00:14:29,450 --> 00:14:33,760 they basically have to satisfy the Liouville equation 198 00:14:33,760 --> 00:14:39,070 that I have written, now appropriate to s particles. 199 00:14:39,070 --> 00:14:42,810 Of course, we know that that's not the entire story 200 00:14:42,810 --> 00:14:46,430 because there are all these other terms 201 00:14:46,430 --> 00:14:50,700 involving the interactions with particles that I have not 202 00:14:50,700 --> 00:14:51,800 included. 203 00:14:51,800 --> 00:14:54,360 That's the whole essence of the story. 204 00:14:54,360 --> 00:14:57,610 Let's say I want to think about one or two particles. 205 00:14:57,610 --> 00:15:00,510 There is the interaction between the two particles, 206 00:15:00,510 --> 00:15:03,670 and they would be evolving according to some trajectories. 207 00:15:03,670 --> 00:15:05,670 But there are all of these other particles 208 00:15:05,670 --> 00:15:09,830 in the gas in this room that will collide with them. 209 00:15:09,830 --> 00:15:13,390 So those conditions are not something 210 00:15:13,390 --> 00:15:17,980 that we had in the Liouville equation, 211 00:15:17,980 --> 00:15:19,600 with everything considered. 212 00:15:19,600 --> 00:15:21,400 Here, I have to include the effect 213 00:15:21,400 --> 00:15:23,600 of all of those other particles. 214 00:15:23,600 --> 00:15:25,210 We saw that the way that it appears 215 00:15:25,210 --> 00:15:32,270 is that I have to imagine that there's another particle whose 216 00:15:32,270 --> 00:15:36,390 coordinates and momenta are captured 217 00:15:36,390 --> 00:15:41,340 through some volume for the s plus 1 particle. 218 00:15:41,340 --> 00:15:45,350 This s plus 1 particle can interact 219 00:15:45,350 --> 00:15:47,870 with any of the particles that are in the set 220 00:15:47,870 --> 00:15:50,710 that I have on the other side. 221 00:15:50,710 --> 00:15:55,870 There is an index that runs from 1 to s. 222 00:15:55,870 --> 00:16:03,470 What I would have here is the force that will come from this 223 00:16:03,470 --> 00:16:10,180 s plus 1 particle, acting on particle n the same way that 224 00:16:10,180 --> 00:16:13,650 this force was deriving the change of the momentum, 225 00:16:13,650 --> 00:16:19,390 this force will derive the change of the momentum of-- I 226 00:16:19,390 --> 00:16:20,730 guess I put an m here-- 227 00:16:23,410 --> 00:16:26,230 The thing that I have to put here 228 00:16:26,230 --> 00:16:32,960 is now a density that also keeps track of the probability 229 00:16:32,960 --> 00:16:38,282 to find the s plus 1 particle in the location in phase place 230 00:16:38,282 --> 00:16:39,740 that I need to integrate with both. 231 00:16:39,740 --> 00:16:44,030 I have to integrate over all positions. 232 00:16:44,030 --> 00:16:48,170 One particle is moving along a straight line by itself, 233 00:16:48,170 --> 00:16:50,000 let's say. 234 00:16:50,000 --> 00:16:53,670 Then there are all of the other particles in the system. 235 00:16:53,670 --> 00:16:56,280 I have to ask, what is the possibility that there 236 00:16:56,280 --> 00:16:59,690 is a second particle with some particular momentum 237 00:16:59,690 --> 00:17:02,310 and coordinate that I will be interacting with. 238 00:17:07,180 --> 00:17:14,989 This is the general set up of these D-B-G-K-Y hierarchy 239 00:17:14,989 --> 00:17:15,530 of equations. 240 00:17:19,829 --> 00:17:23,130 At this stage, we really have just 241 00:17:23,130 --> 00:17:28,640 rewritten what we had for the Liouville equation. 242 00:17:28,640 --> 00:17:34,750 We said, I'm really, really interested only one particle 243 00:17:34,750 --> 00:17:36,730 [? thing, ?] row one and F1. 244 00:17:36,730 --> 00:17:37,860 Let's focus on that. 245 00:17:37,860 --> 00:17:41,210 Let's write those equations in more detail 246 00:17:41,210 --> 00:17:46,450 In the first equation, I have that the explicit time 247 00:17:46,450 --> 00:17:52,920 dependence, plus the time dependence of the position 248 00:17:52,920 --> 00:17:59,510 coordinate, plus the time dependence of the momentum 249 00:17:59,510 --> 00:18:05,210 coordinate, which is driven by the external force, 250 00:18:05,210 --> 00:18:07,850 acting on this one particle density, which 251 00:18:07,850 --> 00:18:10,900 is dependent on p1, q1 at time t. 252 00:18:14,420 --> 00:18:17,590 On the right hand side of the equation. 253 00:18:17,590 --> 00:18:24,890 I need to worry about a second particle with momenta P2 254 00:18:24,890 --> 00:18:28,130 at position Q2 that will, therefore, 255 00:18:28,130 --> 00:18:29,910 be able to exert a force. 256 00:18:29,910 --> 00:18:33,820 Once I know the position, I can calculate the force 257 00:18:33,820 --> 00:18:36,580 that particle exerts. 258 00:18:36,580 --> 00:18:38,160 What was my notation? 259 00:18:38,160 --> 00:18:45,785 The order was 2 and 1, dotted by d by dP1. 260 00:18:45,785 --> 00:18:54,220 I need now f2, p1, q2 at time t. 261 00:18:54,220 --> 00:18:58,390 We say, well, this is unfortunate. 262 00:18:58,390 --> 00:19:03,300 I have to worry about dependence on F2, 263 00:19:03,300 --> 00:19:05,710 but maybe I can get away with things 264 00:19:05,710 --> 00:19:08,260 by estimating order of magnitudes 265 00:19:08,260 --> 00:19:10,450 of the various terms. 266 00:19:10,450 --> 00:19:13,620 What is the left hand side set of operations? 267 00:19:13,620 --> 00:19:15,550 The left hand side set of operations 268 00:19:15,550 --> 00:19:22,310 describes essentially one particle moving by itself. 269 00:19:22,310 --> 00:19:27,690 If that particle has to cross a distance of this order of L, 270 00:19:27,690 --> 00:19:32,710 and I tell you that the typical velocity of these particles 271 00:19:32,710 --> 00:19:36,980 is off the order of V, then that time scale is going 272 00:19:36,980 --> 00:19:41,550 to be of the order of L over V. The operations 273 00:19:41,550 --> 00:19:53,190 here will give me a V over L, which 274 00:19:53,190 --> 00:19:56,660 is what we call the inverse of Tau u. 275 00:19:59,910 --> 00:20:04,750 This is a reasonably long macroscopic time. 276 00:20:04,750 --> 00:20:06,850 OK, that's fine. 277 00:20:06,850 --> 00:20:09,500 How big is the right hand side? 278 00:20:09,500 --> 00:20:12,050 We said that the right hand side has 279 00:20:12,050 --> 00:20:15,430 something to do with collisions. 280 00:20:15,430 --> 00:20:19,030 I have a particle in my system. 281 00:20:19,030 --> 00:20:24,080 Let's say that particle has some characteristic dimension 282 00:20:24,080 --> 00:20:25,530 that we call d. 283 00:20:28,720 --> 00:20:33,520 This particle is moving with velocity V. Alternatively, 284 00:20:33,520 --> 00:20:36,820 you can think of this particle as being stationary, 285 00:20:36,820 --> 00:20:39,895 and all the other particles are coming at it 286 00:20:39,895 --> 00:20:42,970 with some velocity V. 287 00:20:42,970 --> 00:20:47,340 If I say that the density of these particles is n, 288 00:20:47,340 --> 00:20:52,150 then the typical time for which, as I shoot these particles, 289 00:20:52,150 --> 00:20:56,130 they will hit this target is related 290 00:20:56,130 --> 00:21:00,800 to V squared and V, the volume of particles. 291 00:21:03,330 --> 00:21:09,570 Over time t, I have to consider this times V tau x. 292 00:21:09,570 --> 00:21:15,640 V tau xn V squared should be of the order of one. 293 00:21:15,640 --> 00:21:21,240 This gave us a formula for tau x. 294 00:21:21,240 --> 00:21:23,480 The inverse of tau x that controls 295 00:21:23,480 --> 00:21:35,100 what's happening on this side is n V squared V. 296 00:21:35,100 --> 00:21:39,450 Is the term on the right hand side more important, 297 00:21:39,450 --> 00:21:41,920 or the term on the left hand side? 298 00:21:41,920 --> 00:21:44,340 The term on the right hand side has 299 00:21:44,340 --> 00:21:47,050 to do with the two body term. 300 00:21:47,050 --> 00:21:49,050 There's a particle that is moving, 301 00:21:49,050 --> 00:21:50,710 and then there's another particle 302 00:21:50,710 --> 00:21:56,190 with a slightly different velocity that it is behind it. 303 00:21:56,190 --> 00:21:58,670 In the absence of collisions, these particles 304 00:21:58,670 --> 00:22:01,410 would just go along a straight line. 305 00:22:01,410 --> 00:22:04,040 They would bounce off the walls, but the magnitude 306 00:22:04,040 --> 00:22:05,930 of their energy, and hence, velocity, 307 00:22:05,930 --> 00:22:09,590 would not change from these elastic collisions. 308 00:22:09,590 --> 00:22:12,450 But if the particles can catch up 309 00:22:12,450 --> 00:22:19,040 and interact, which is governed by V2, V on the other side, 310 00:22:19,040 --> 00:22:22,700 then what happens is that the particles, when they interact, 311 00:22:22,700 --> 00:22:25,140 would collide and go different ways. 312 00:22:25,140 --> 00:22:29,020 Quickly, their velocities, and momenta, and everything 313 00:22:29,020 --> 00:22:31,760 would get mixed up. 314 00:22:31,760 --> 00:22:36,130 How rapidly that happens depends on this collision distance, 315 00:22:36,130 --> 00:22:41,430 which is much less than the size of the system, 316 00:22:41,430 --> 00:22:44,670 and, therefore, the term that you have on the right hand 317 00:22:44,670 --> 00:22:48,770 side in magnitude is much larger than what 318 00:22:48,770 --> 00:22:50,910 is happening on the left hand side. 319 00:22:50,910 --> 00:22:55,230 There is no way in order to describe 320 00:22:55,230 --> 00:22:59,590 the relaxation of the gas that I can neglect collisions 321 00:22:59,590 --> 00:23:01,010 between gas particles. 322 00:23:01,010 --> 00:23:04,060 If I neglect collisions between gas particles, 323 00:23:04,060 --> 00:23:06,530 there is no reason why the kinetic energies 324 00:23:06,530 --> 00:23:08,380 of individual particles should change. 325 00:23:08,380 --> 00:23:11,690 They would stay the same forever. 326 00:23:11,690 --> 00:23:13,330 I have to keep this. 327 00:23:13,330 --> 00:23:16,305 Let's go and look at the second equation in the hierarchy. 328 00:23:16,305 --> 00:23:17,140 What do you have? 329 00:23:17,140 --> 00:23:30,200 You have d by dT, P1 over m d by d Q1, P2 over m, P d by d Q2. 330 00:23:30,200 --> 00:23:36,650 Then we have F1 d by d Q1, plus F2, d 331 00:23:36,650 --> 00:23:42,180 by d Q2 coming from the external potential. 332 00:23:42,180 --> 00:23:44,770 Then we have the force that the involves 333 00:23:44,770 --> 00:23:47,320 the collision between particles one and two. 334 00:23:50,600 --> 00:23:55,750 When I write down the Hamiltonian for two particles, 335 00:23:55,750 --> 00:23:58,310 there is going to be already for two particles 336 00:23:58,310 --> 00:24:00,640 and interactions between them. 337 00:24:00,640 --> 00:24:03,940 That's where the F1 2 comes from. 338 00:24:03,940 --> 00:24:09,850 F1 2 changes d by the momentum of particle one. 339 00:24:09,850 --> 00:24:11,910 I should write, it's 2 1 that changes 340 00:24:11,910 --> 00:24:13,960 momentum of particle two. 341 00:24:13,960 --> 00:24:19,500 But as 2 1 is simply minus F1 2, I can put the two of them 342 00:24:19,500 --> 00:24:22,220 together in this fashion. 343 00:24:22,220 --> 00:24:28,900 This acting on F2 is then equal to something 344 00:24:28,900 --> 00:24:43,370 like integral over V3, F3 1, d by dP1, plus F3 2, d by dP2. 345 00:24:43,370 --> 00:24:49,694 [INAUDIBLE] on F3 P1 and Q3 [INAUDIBLE]. 346 00:24:53,890 --> 00:24:56,510 Are we going to do this forever? 347 00:24:56,510 --> 00:25:00,120 Well, we said, let's take another look 348 00:25:00,120 --> 00:25:04,010 at the magnitude of the various terms. 349 00:25:04,010 --> 00:25:06,770 This term on the right hand side still 350 00:25:06,770 --> 00:25:10,660 involves a collision that involves a third particle. 351 00:25:10,660 --> 00:25:13,450 I have to find that third particle, 352 00:25:13,450 --> 00:25:17,120 so I need to have, essentially, a third particle 353 00:25:17,120 --> 00:25:19,150 within some characteristic volume, 354 00:25:19,150 --> 00:25:22,740 so I have something that is of that order. 355 00:25:22,740 --> 00:25:26,790 Whereas on the left hand side now, 356 00:25:26,790 --> 00:25:30,590 I have a term that from all perspectives, 357 00:25:30,590 --> 00:25:34,580 looks like the kinds of terms that I had before except 358 00:25:34,580 --> 00:25:39,730 that it involves the collision between two particles. 359 00:25:39,730 --> 00:25:43,770 What it describes is the duration that collision. 360 00:25:43,770 --> 00:25:45,890 We said this is of the order of 1 over tau 361 00:25:45,890 --> 00:25:50,100 c, which replaces the n over there 362 00:25:50,100 --> 00:25:52,940 with some characteristic dimension. 363 00:25:52,940 --> 00:25:56,040 Suddenly, this term is very big. 364 00:25:58,570 --> 00:25:59,916 We should be able to use that. 365 00:25:59,916 --> 00:26:00,790 There was a question. 366 00:26:00,790 --> 00:26:04,513 AUDIENCE: On the left hand side of both of your equations, 367 00:26:04,513 --> 00:26:07,892 for F1 and F2, shouldn't all the derivatives that 368 00:26:07,892 --> 00:26:11,328 are multiplied by your forces be derivatives 369 00:26:11,328 --> 00:26:13,958 of the effects of momentum? [INAUDIBLE] the coordinates? 370 00:26:13,958 --> 00:26:15,986 [INAUDIBLE] reasons? 371 00:26:15,986 --> 00:26:20,170 PROFESSOR: Let's go back here. 372 00:26:20,170 --> 00:26:27,640 I have a function that depends on P, Q, and t. 373 00:26:27,640 --> 00:26:31,750 Then there's the explicit time derivative, d by dt. 374 00:26:31,750 --> 00:26:38,370 Then there is the Q dot here, which will go by d by dQ. 375 00:26:38,370 --> 00:26:42,610 Then there's the P dot term that will go by d by dP. 376 00:26:42,610 --> 00:26:45,560 All of things have to be there. 377 00:26:45,560 --> 00:26:48,270 I should have derivatives in respect to momenta, 378 00:26:48,270 --> 00:26:50,990 and derivatives with respect to coordinate. 379 00:26:50,990 --> 00:26:53,530 Dimensions are, of course, important. 380 00:26:53,530 --> 00:26:57,070 Somewhat, what I write for this and for this 381 00:26:57,070 --> 00:26:59,080 should make up for that. 382 00:26:59,080 --> 00:27:02,630 As I have written it now, it's obvious, of course. 383 00:27:02,630 --> 00:27:06,080 This has dimensions of Q over T. The Q's cancel. 384 00:27:06,080 --> 00:27:09,170 I would have one over T. D over Dps cancel. 385 00:27:09,170 --> 00:27:12,750 I have 1 over P. Here, dimensionality is correct. 386 00:27:12,750 --> 00:27:15,610 I have to just make sure I haven't made a mistake. 387 00:27:15,610 --> 00:27:17,830 Q dot is a velocity. 388 00:27:17,830 --> 00:27:21,200 Velocity is momentum divided by mass. 389 00:27:21,200 --> 00:27:23,730 So that should dimensionally work out. 390 00:27:23,730 --> 00:27:26,500 P dot is a force. 391 00:27:26,500 --> 00:27:27,665 Everything here is force. 392 00:27:30,579 --> 00:27:31,745 In a reasonable coordinate-- 393 00:27:31,745 --> 00:27:33,266 AUDIENCE: [INAUDIBLE] 394 00:27:33,266 --> 00:27:35,101 PROFESSOR: What did I do here? 395 00:27:35,101 --> 00:27:36,100 I made mistakes? 396 00:27:36,100 --> 00:27:37,100 AUDIENCE: [INAUDIBLE] 397 00:27:49,119 --> 00:27:50,910 PROFESSOR: Why didn't you say that in a way 398 00:27:50,910 --> 00:27:57,800 that-- If I don't understand the question, 399 00:27:57,800 --> 00:28:01,730 please correct me before I spend another five minutes. 400 00:28:05,450 --> 00:28:10,950 Hopefully, this is now free of these deficiencies. 401 00:28:10,950 --> 00:28:12,223 This there is very big. 402 00:28:15,150 --> 00:28:17,660 Now, compared to the right hand side in fact, 403 00:28:17,660 --> 00:28:20,140 we said that the right hand side is 404 00:28:20,140 --> 00:28:23,650 smaller by a factor that measures how many particles 405 00:28:23,650 --> 00:28:26,060 are within an interaction volume. 406 00:28:26,060 --> 00:28:27,950 And for a typical gas, this would 407 00:28:27,950 --> 00:28:31,810 be a number that's of the order of 10 to the minus 4. 408 00:28:31,810 --> 00:28:34,560 Using 10 to the minus 4 being this small, 409 00:28:34,560 --> 00:28:38,270 we are going to set the right hand side to zero. 410 00:28:38,270 --> 00:28:40,410 Now, I don't have to write the equation for F2. 411 00:28:45,770 --> 00:28:51,550 I'll answer a question here that may arise, which is ultimately, 412 00:28:51,550 --> 00:28:54,700 we will do sufficient manipulations 413 00:28:54,700 --> 00:28:58,650 so that we end up with a particular equation, known 414 00:28:58,650 --> 00:29:00,685 as the Boltzmann Equation, that we 415 00:29:00,685 --> 00:29:06,660 will show does not obey the time reversibility 416 00:29:06,660 --> 00:29:08,890 that we wrote over here. 417 00:29:08,890 --> 00:29:14,520 Clearly, that is built in to the various approximations I make. 418 00:29:14,520 --> 00:29:17,140 The first question is, the approximation 419 00:29:17,140 --> 00:29:21,950 that I've made here, did I destroy this time 420 00:29:21,950 --> 00:29:23,310 reversibility? 421 00:29:23,310 --> 00:29:25,110 The answer is no. 422 00:29:25,110 --> 00:29:28,600 You can look at this set of equations, 423 00:29:28,600 --> 00:29:31,390 and do the manipulations necessary to see 424 00:29:31,390 --> 00:29:35,110 what happens if P goes to minus P. You will find that you will 425 00:29:35,110 --> 00:29:38,520 be able to reverse your trajectory without any problem. 426 00:29:38,520 --> 00:29:39,020 Yes? 427 00:29:41,750 --> 00:29:44,144 AUDIENCE: Given that it is only an interaction 428 00:29:44,144 --> 00:29:46,096 from our left side that's very big, 429 00:29:46,096 --> 00:29:48,980 that's the reason why we can ignore the stuff on the right. 430 00:29:48,980 --> 00:29:50,810 Why is it that we are then keeping 431 00:29:50,810 --> 00:29:55,820 all of the other terms that were even smaller before? 432 00:29:55,820 --> 00:29:57,300 PROFESSOR: I will ignore them. 433 00:29:57,300 --> 00:29:57,800 Sure. 434 00:29:57,800 --> 00:30:00,290 AUDIENCE: [LAUGHTER] 435 00:30:10,285 --> 00:30:14,570 PROFESSOR: There was the question of time reversibility. 436 00:30:14,570 --> 00:30:19,050 This term here has to do with three particles coming 437 00:30:19,050 --> 00:30:23,330 together, and how that would modify what 438 00:30:23,330 --> 00:30:26,590 we have for just two-body collisions. 439 00:30:26,590 --> 00:30:29,140 In principle, there is some probability 440 00:30:29,140 --> 00:30:31,380 to have three particles coming together 441 00:30:31,380 --> 00:30:34,400 and some combined interactions. 442 00:30:34,400 --> 00:30:37,050 You can imagine some fictitious model, 443 00:30:37,050 --> 00:30:40,810 which in addition to these two-body interactions, 444 00:30:40,810 --> 00:30:43,810 you cook up some body interaction so that it 445 00:30:43,810 --> 00:30:47,190 precisely cancels what would have happened 446 00:30:47,190 --> 00:30:48,910 when three particles come together. 447 00:30:48,910 --> 00:30:51,110 We can write a computer program in which 448 00:30:51,110 --> 00:30:52,890 we have two body conditions. 449 00:30:52,890 --> 00:30:56,360 But if three bodies come close enough to each other, 450 00:30:56,360 --> 00:30:59,440 they essentially become ghosts and pass through each other. 451 00:30:59,440 --> 00:31:03,140 That computer program would be fully reversible. 452 00:31:03,140 --> 00:31:05,550 That's why sort of dropping this there 453 00:31:05,550 --> 00:31:07,817 is not causing any problems at this point. 454 00:31:12,990 --> 00:31:17,640 What is it that you have included so far? 455 00:31:17,640 --> 00:31:23,680 What we have is a situation where the change in F1 456 00:31:23,680 --> 00:31:26,330 is governed by a process in which I 457 00:31:26,330 --> 00:31:29,760 have a particle that I describe on the left hand 458 00:31:29,760 --> 00:31:34,930 side with momentum one, and it collides with some particle 459 00:31:34,930 --> 00:31:39,710 that I'm integrating over, but in some particular instance 460 00:31:39,710 --> 00:31:43,000 of integration, has momentum P2. 461 00:31:43,000 --> 00:31:46,690 Presumably they come close enough to each other 462 00:31:46,690 --> 00:31:50,990 so that afterwards, the momenta have changed over so 463 00:31:50,990 --> 00:31:55,165 that I have some P1 prime, and I have some P2 prime. 464 00:32:04,410 --> 00:32:10,270 We want to make sure that we characterize these correctly. 465 00:32:10,270 --> 00:32:15,330 There was a question about while this term is big, 466 00:32:15,330 --> 00:32:18,770 these kinds of terms are small. 467 00:32:18,770 --> 00:32:22,580 Why should I basically bother to keep them? 468 00:32:22,580 --> 00:32:25,040 It is reasonable. 469 00:32:25,040 --> 00:32:29,040 What we are following here are particles in my picture that 470 00:32:29,040 --> 00:32:31,550 were ejected by the first box, and they 471 00:32:31,550 --> 00:32:33,250 collide into each other, or they were 472 00:32:33,250 --> 00:32:35,320 colliding in the first box. 473 00:32:35,320 --> 00:32:41,725 As long as you are away from the [? vols ?] of the container, 474 00:32:41,725 --> 00:32:43,980 you really don't care about these terms. 475 00:32:43,980 --> 00:32:47,450 They don't really moved very rapidly. 476 00:32:47,450 --> 00:32:51,250 This is the process of collision of two particles, 477 00:32:51,250 --> 00:32:55,550 and it's also the same process that is described over here. 478 00:32:55,550 --> 00:32:59,150 Somehow, I should be able to simplify the collision 479 00:32:59,150 --> 00:33:02,720 process that is going on here with the knowledge 480 00:33:02,720 --> 00:33:05,650 that the evolution of two particles 481 00:33:05,650 --> 00:33:09,190 is now completely deterministic. 482 00:33:09,190 --> 00:33:12,680 This equation by itself says, take two particles 483 00:33:12,680 --> 00:33:14,800 as if they are the only thing in the universe, 484 00:33:14,800 --> 00:33:17,570 and they would follow some completely deterministic 485 00:33:17,570 --> 00:33:20,690 trajectory, that if you put lots of them together, 486 00:33:20,690 --> 00:33:24,420 is captured through this density. 487 00:33:24,420 --> 00:33:28,360 Let's see whether we can massage this equation 488 00:33:28,360 --> 00:33:30,440 to look like this equation. 489 00:33:30,440 --> 00:33:33,050 Well, the force term, we have, except that here we 490 00:33:33,050 --> 00:33:34,510 have dP by P1 here. 491 00:33:34,510 --> 00:33:39,240 We have d by dP 1 minus d by dP2. 492 00:33:39,240 --> 00:33:41,720 So let's do this. 493 00:33:41,720 --> 00:33:50,030 Minus d by dP2, acting on F2. 494 00:33:50,030 --> 00:33:52,050 Did I do something wrong? 495 00:33:52,050 --> 00:33:57,250 The answer is no, because I added the complete derivative 496 00:33:57,250 --> 00:33:59,070 over something that I'm integrating over. 497 00:34:01,610 --> 00:34:03,590 This is perfectly legitimate mathematics. 498 00:34:06,420 --> 00:34:08,880 This part now looks like this. 499 00:34:08,880 --> 00:34:12,500 I have to find what is the most important term that 500 00:34:12,500 --> 00:34:14,580 matches this. 501 00:34:14,580 --> 00:34:16,280 Again, let's think about this procedure. 502 00:34:19,610 --> 00:34:24,020 What I have to make sure of is what 503 00:34:24,020 --> 00:34:26,699 is the extent of the collision, and how important is 504 00:34:26,699 --> 00:34:27,650 the collision? 505 00:34:27,650 --> 00:34:29,900 If I have one particle moving here, 506 00:34:29,900 --> 00:34:32,960 and another particle off there, they will pass each other. 507 00:34:32,960 --> 00:34:35,469 Nothing interesting could happen. 508 00:34:35,469 --> 00:34:38,199 The important thing is how close they come together. 509 00:34:38,199 --> 00:34:41,260 It Is kind of important that I keep 510 00:34:41,260 --> 00:34:45,100 track of the relative coordinate, Q, 511 00:34:45,100 --> 00:34:49,570 which is Q2 minus Q1, as opposed to the center 512 00:34:49,570 --> 00:34:55,030 of mass coordinate, which is just Q1 plus Q2 over 2. 513 00:34:59,320 --> 00:35:01,860 That kind of also indicates maybe it's 514 00:35:01,860 --> 00:35:05,320 a good thing for me to look at this entire process 515 00:35:05,320 --> 00:35:06,930 in the center of mass frame. 516 00:35:06,930 --> 00:35:08,440 So this is the lab frame. 517 00:35:11,730 --> 00:35:15,770 If I were to look at this same picture in the center 518 00:35:15,770 --> 00:35:20,370 of mass frame, what would I have? 519 00:35:20,370 --> 00:35:25,880 In the center of mass frame, I would have the initial particle 520 00:35:25,880 --> 00:35:32,480 coming with P1 prime, P1 minus P center of mass. 521 00:35:35,340 --> 00:35:39,180 The other particle that you are interacting with 522 00:35:39,180 --> 00:35:47,520 comes with P2 minus P center of mass. 523 00:35:47,520 --> 00:35:53,110 I actually drew these vectors that are hopefully 524 00:35:53,110 --> 00:35:55,312 equal and opposite, because you know 525 00:35:55,312 --> 00:35:57,610 that in the center of mass, one of them, in fact, 526 00:35:57,610 --> 00:35:59,690 would be P1 minus P2 over 2. 527 00:35:59,690 --> 00:36:01,850 The other would be P2 minus P1 over 2. 528 00:36:01,850 --> 00:36:03,620 They would, indeed, in the center of mass 529 00:36:03,620 --> 00:36:05,610 be equal and opposite momenta. 530 00:36:08,310 --> 00:36:15,250 Along the direction of these objects, 531 00:36:15,250 --> 00:36:20,850 I can look at how close they come together. 532 00:36:20,850 --> 00:36:23,680 I can look at some coordinate that I 533 00:36:23,680 --> 00:36:32,860 will call A, which measures the separation between them 534 00:36:32,860 --> 00:36:35,000 at some instant of time. 535 00:36:35,000 --> 00:36:38,280 Then there's another pair of coordinates 536 00:36:38,280 --> 00:36:43,110 that I could put into a vector that tells me how head to head 537 00:36:43,110 --> 00:36:44,590 they are. 538 00:36:44,590 --> 00:36:48,120 If I think about they're being on the center of mass, 539 00:36:48,120 --> 00:36:50,690 two things that are approaching each other, 540 00:36:50,690 --> 00:36:52,950 they can either approach head on-- that 541 00:36:52,950 --> 00:36:55,200 would correspond to be equal to 0-- 542 00:36:55,200 --> 00:37:02,590 or they could be slightly off a head-on collision. 543 00:37:02,590 --> 00:37:11,940 There is a so-called impact parameter 544 00:37:11,940 --> 00:37:17,420 B, which is a measure of this addition fact. 545 00:37:20,030 --> 00:37:23,830 Why is that going to be relevant to us? 546 00:37:23,830 --> 00:37:29,100 Again, we said that there are parts of this expression 547 00:37:29,100 --> 00:37:31,900 that all of the order of this term, 548 00:37:31,900 --> 00:37:35,030 they're kind of not that important. 549 00:37:35,030 --> 00:37:39,530 If I think about the collision, and what the collision does, 550 00:37:39,530 --> 00:37:45,250 I will have forces that are significant when 551 00:37:45,250 --> 00:37:49,780 I am within this range of interactions, 552 00:37:49,780 --> 00:37:54,730 D. I really have to look at what happens when the two 553 00:37:54,730 --> 00:37:56,400 things come close to each other. 554 00:37:56,400 --> 00:38:01,736 It Is only when this relative parameter A has approached D 555 00:38:01,736 --> 00:38:05,110 that these particles will start to deviate 556 00:38:05,110 --> 00:38:07,840 from their straight line trajectory, 557 00:38:07,840 --> 00:38:12,775 and presumably go, to say in this case, P2 prime minus P 558 00:38:12,775 --> 00:38:14,580 center of mass. 559 00:38:14,580 --> 00:38:16,865 This one occurs [? and ?] will go, 560 00:38:16,865 --> 00:38:22,610 and eventually P1 prime minus P center of mass. 561 00:38:22,610 --> 00:38:25,545 These deviations will occur over a distance 562 00:38:25,545 --> 00:38:31,610 that is of the order of this collision and D. 563 00:38:31,610 --> 00:38:37,640 The important changes that occur in various densities, 564 00:38:37,640 --> 00:38:40,530 in various potentials, et cetera, 565 00:38:40,530 --> 00:38:46,210 are all taking place when this relative coordinate is small. 566 00:38:46,210 --> 00:38:50,570 Things become big when the relative coordinate is small. 567 00:38:50,570 --> 00:38:54,590 They are big as a function of the relative coordinate. 568 00:38:54,590 --> 00:38:59,140 In order to get big things, what I need to do 569 00:38:59,140 --> 00:39:03,130 is to replace these d by dQ's with 570 00:39:03,130 --> 00:39:06,120 the corresponding derivatives with respect 571 00:39:06,120 --> 00:39:07,990 to the center of mass. 572 00:39:07,990 --> 00:39:10,670 One of them would come be the minus sign. 573 00:39:10,670 --> 00:39:12,990 The other would come be the plus sign. 574 00:39:12,990 --> 00:39:15,195 It doesn't matter which is which. 575 00:39:15,195 --> 00:39:17,270 It depends on the definition, whether I 576 00:39:17,270 --> 00:39:19,980 make Q2 minus Q1, or Q1 minus Q2. 577 00:39:23,490 --> 00:39:28,190 We see that the big terms are the force that 578 00:39:28,190 --> 00:39:31,840 changes the momenta and the variations 579 00:39:31,840 --> 00:39:35,760 that you have over these relative coordinates. 580 00:39:35,760 --> 00:39:40,260 What I can do now is to replace this 581 00:39:40,260 --> 00:39:44,610 by equating the two big terms that I have over here. 582 00:39:44,610 --> 00:40:03,280 The two big terms are P2 minus P1 583 00:40:03,280 --> 00:40:08,240 over m, dotted by d by dQ of F2. 584 00:40:17,950 --> 00:40:23,750 There is some other approximation that I did. 585 00:40:23,750 --> 00:40:28,540 As was told to me before, this is the biggest term, 586 00:40:28,540 --> 00:40:30,155 and there is the part of this that 587 00:40:30,155 --> 00:40:32,380 is big and compensates for that. 588 00:40:32,380 --> 00:40:34,750 But there are all these other bunches of terms. 589 00:40:34,750 --> 00:40:38,260 There's also this d by dt. 590 00:40:38,260 --> 00:40:43,900 What I have done over here is to look 591 00:40:43,900 --> 00:40:49,250 at this slightly coarser perspective on time. 592 00:40:49,250 --> 00:40:53,130 Increasing all the equations that I have over there 593 00:40:53,130 --> 00:40:55,830 tells me everything about particles 594 00:40:55,830 --> 00:40:59,260 approaching each other and going away. 595 00:40:59,260 --> 00:41:03,240 I can follow through the mechanics 596 00:41:03,240 --> 00:41:06,190 precisely everything that is happening, even 597 00:41:06,190 --> 00:41:08,930 in the vicinity of this collision. 598 00:41:08,930 --> 00:41:13,480 If I have two squishy balls, and I run my hand through them 599 00:41:13,480 --> 00:41:15,450 properly, I can see how the things 600 00:41:15,450 --> 00:41:17,670 get squished then released. 601 00:41:17,670 --> 00:41:20,350 There's a lot of information, but again, a lot 602 00:41:20,350 --> 00:41:22,140 of information that I don't really 603 00:41:22,140 --> 00:41:25,610 care to know as far as the properties of this gas 604 00:41:25,610 --> 00:41:29,650 expansion process is concerned. 605 00:41:29,650 --> 00:41:35,386 What you have done is to forget about the detailed variations 606 00:41:35,386 --> 00:41:39,130 in time and space that are taking place here. 607 00:41:39,130 --> 00:41:42,530 We're going to shortly make that even more explicit 608 00:41:42,530 --> 00:41:45,110 by noting the following. 609 00:41:45,110 --> 00:41:49,340 This integration over here is an integration 610 00:41:49,340 --> 00:41:52,750 over phase space of the second particle. 611 00:41:52,750 --> 00:41:57,440 I had written before d cubed, P2, d cubed, Q2, 612 00:41:57,440 --> 00:41:59,530 but I can change coordinates and look 613 00:41:59,530 --> 00:42:05,020 at the relative coordinate, Q, over here. 614 00:42:09,070 --> 00:42:12,300 What I'm asking is, I have one particle 615 00:42:12,300 --> 00:42:14,090 moving through the gas. 616 00:42:14,090 --> 00:42:17,230 What is the chance that the second particle comes 617 00:42:17,230 --> 00:42:23,510 with momentum P2, and the appropriate relative distance 618 00:42:23,510 --> 00:42:28,740 Q, and I integrate over both the P and the relative distance Q? 619 00:42:28,740 --> 00:42:31,400 This is the quantity that I have to integrate. 620 00:42:38,190 --> 00:42:41,280 Let's do one more calculation, and then we 621 00:42:41,280 --> 00:42:46,430 will try to give a physical perspective. 622 00:42:46,430 --> 00:42:50,670 In this picture of the center of mass, what did I do? 623 00:42:50,670 --> 00:43:02,010 I do replaced the coordinate, Q, with a part that was the impact 624 00:43:02,010 --> 00:43:05,320 parameter, which had two components, 625 00:43:05,320 --> 00:43:07,505 and a part that was the relative distance. 626 00:43:10,820 --> 00:43:14,100 What was this relative distance? 627 00:43:14,100 --> 00:43:17,360 The relative distance was measured 628 00:43:17,360 --> 00:43:23,640 along this line that was giving me the closest approach. 629 00:43:23,640 --> 00:43:26,270 What is the direction of this line? 630 00:43:26,270 --> 00:43:30,330 The direction of this line is P1 minus P2. 631 00:43:30,330 --> 00:43:32,150 This is P1 minus P2 over 2. 632 00:43:32,150 --> 00:43:32,900 It doesn't matter. 633 00:43:32,900 --> 00:43:36,380 The direction is P1 minus P2. 634 00:43:36,380 --> 00:43:43,860 What I'm doing here is I am taking the derivative precisely 635 00:43:43,860 --> 00:43:46,920 along this line of constant approach. 636 00:43:50,260 --> 00:43:54,410 I'm taking a derivative, and I'm integrating along that. 637 00:43:57,790 --> 00:44:03,310 If I were to rewrite the whole thing, what do I have? 638 00:44:03,310 --> 00:44:14,180 I have d by dt, plus P1 over m, d by dQ1, plus F1, d by dP1-- 639 00:44:14,180 --> 00:44:22,870 don't make a mistake-- acting on F1, P1, Q1, t. 640 00:44:22,870 --> 00:44:25,580 What do I have to write on the right hand side? 641 00:44:25,580 --> 00:44:29,500 I have an integral over the momentum 642 00:44:29,500 --> 00:44:33,620 of this particle with which I'm going to make a collision. 643 00:44:33,620 --> 00:44:37,220 I have an integral over the impact parameter 644 00:44:37,220 --> 00:44:41,910 that tells me the distance of closest approach. 645 00:44:41,910 --> 00:44:47,110 I have to do the magnitude of P2 minus P1 646 00:44:47,110 --> 00:44:50,180 over n, which is really the magnitude 647 00:44:50,180 --> 00:44:53,170 of the relative velocity of the two particles. 648 00:44:53,170 --> 00:44:57,710 I can write it as P2 minus P1, or P1 minus P2. 649 00:44:57,710 --> 00:44:59,070 These are, of course, vectors. 650 00:44:59,070 --> 00:45:02,070 and I look at the modulus. 651 00:45:02,070 --> 00:45:04,813 I have the integral of the derivative. 652 00:45:07,610 --> 00:45:11,700 Very simply, I will write the answer 653 00:45:11,700 --> 00:45:19,130 as F2 that is evaluated at some large distance, 654 00:45:19,130 --> 00:45:23,150 plus infinity minus F2 evaluated at minus infinity. 655 00:45:23,150 --> 00:45:24,240 I have infinity. 656 00:45:24,240 --> 00:45:26,730 In principle, I have to integrate over F2 657 00:45:26,730 --> 00:45:29,560 from minus infinity to plus infinity. 658 00:45:29,560 --> 00:45:34,250 But once I am beyond the range of where 659 00:45:34,250 --> 00:45:37,090 the interaction changes, then the two particles 660 00:45:37,090 --> 00:45:39,000 just move away forever. 661 00:45:39,000 --> 00:45:41,030 They will never see each other. 662 00:45:41,030 --> 00:45:47,180 Really, what I should write here is F2 of-- after the collision, 663 00:45:47,180 --> 00:45:54,990 I have P1 prime, P2 prime, at some Q plus, 664 00:45:54,990 --> 00:46:04,280 minus F2, P1, P2, at some position minus. 665 00:46:04,280 --> 00:46:08,210 What I need to do is to do the integration when 666 00:46:08,210 --> 00:46:12,420 I'm far away from the collision, or wait 667 00:46:12,420 --> 00:46:16,070 until I am far after the collision. 668 00:46:16,070 --> 00:46:21,630 Really, I have to just integrate slightly below, after, 669 00:46:21,630 --> 00:46:25,270 and before the collision occurs. 670 00:46:25,270 --> 00:46:28,430 In principle, if I just go a few d's 671 00:46:28,430 --> 00:46:31,080 in one direction or the other direction, 672 00:46:31,080 --> 00:46:32,050 this should be enough. 673 00:46:34,560 --> 00:46:38,670 Let's see physically what this describes. 674 00:46:38,670 --> 00:46:41,390 There is a connection between this 675 00:46:41,390 --> 00:46:43,930 and this thing that I had over here, in fact. 676 00:46:46,680 --> 00:46:51,420 This equation on the left hand side, if it was zero, 677 00:46:51,420 --> 00:46:53,600 it would describe one particle that 678 00:46:53,600 --> 00:46:57,840 is just moving by itself until it hits the wall, at which 679 00:46:57,840 --> 00:47:01,160 point it basically reverses its trajectory, 680 00:47:01,160 --> 00:47:04,100 and otherwise goes forward. 681 00:47:04,100 --> 00:47:06,080 But what you have on the right hand side 682 00:47:06,080 --> 00:47:07,980 says that suddenly there could be 683 00:47:07,980 --> 00:47:10,830 another particle with which I interact. 684 00:47:10,830 --> 00:47:13,350 Then I change my direction. 685 00:47:13,350 --> 00:47:16,830 I need to know the probability, given 686 00:47:16,830 --> 00:47:21,200 that I'm moving with velocity P1, 687 00:47:21,200 --> 00:47:24,400 that there is a second particle with P2 that 688 00:47:24,400 --> 00:47:25,500 comes close enough. 689 00:47:30,010 --> 00:47:32,540 There is this additional factor. 690 00:47:32,540 --> 00:47:35,320 From what does this additional factor come? 691 00:47:35,320 --> 00:47:38,330 It's the same factor that we have over here. 692 00:47:38,330 --> 00:47:42,140 It is, if you have a target of size d squared, 693 00:47:42,140 --> 00:47:48,780 and we have a set of bullets with a density of n, 694 00:47:48,780 --> 00:47:52,650 the number of collisions that I get depends both on density 695 00:47:52,650 --> 00:47:55,720 and how fast these things go. 696 00:47:55,720 --> 00:48:01,020 The time between collisions, if you like, is proportional to n, 697 00:48:01,020 --> 00:48:07,320 and it is also related to V. That's what this is. 698 00:48:07,320 --> 00:48:10,760 I need some kind of a time between the collisions 699 00:48:10,760 --> 00:48:12,750 that I make. 700 00:48:12,750 --> 00:48:15,120 I have already specified that I'm 701 00:48:15,120 --> 00:48:17,580 only interested in the set of particles that 702 00:48:17,580 --> 00:48:20,450 have momentum P2 for this particular [? point in ?] 703 00:48:20,450 --> 00:48:26,270 integration, and that they have this kind of area or cross 704 00:48:26,270 --> 00:48:27,500 section. 705 00:48:27,500 --> 00:48:31,930 So I replace this V squared and V 706 00:48:31,930 --> 00:48:33,570 with the relative coordinates. 707 00:48:33,570 --> 00:48:37,790 This is the corresponding thing to V squared, 708 00:48:37,790 --> 00:48:41,150 and this is really a two particle density. 709 00:48:41,150 --> 00:48:43,850 This is a subtraction. 710 00:48:43,850 --> 00:48:46,480 The addition is because it is true 711 00:48:46,480 --> 00:48:51,370 that I'm going with velocity P1, and practically, any collisions 712 00:48:51,370 --> 00:48:54,830 that are significant will move me off kilter. 713 00:48:54,830 --> 00:48:58,480 So there has to be a subtraction for the channel that 714 00:48:58,480 --> 00:49:02,630 was described by P1 because of this collision. 715 00:49:02,630 --> 00:49:05,230 This then, is the addition, because it 716 00:49:05,230 --> 00:49:09,240 says that it could be that there is no particle going 717 00:49:09,240 --> 00:49:10,620 in the horizontal direction. 718 00:49:10,620 --> 00:49:15,140 I was actually coming along the vertical direction. 719 00:49:15,140 --> 00:49:17,650 Because of the collision, I suddenly 720 00:49:17,650 --> 00:49:21,150 was shifted to move along this direction. 721 00:49:21,150 --> 00:49:28,560 The addition comes from having particles that would correspond 722 00:49:28,560 --> 00:49:33,550 to momenta that somehow, if I were in some sense 723 00:49:33,550 --> 00:49:35,850 to reverse this, and then put a minus sign, 724 00:49:35,850 --> 00:49:38,990 a reverse collision would create something 725 00:49:38,990 --> 00:49:40,860 that was along the direction of P1. 726 00:49:46,110 --> 00:49:49,440 Here I also made several approximations. 727 00:49:49,440 --> 00:49:56,570 I said, what is chief among them is that basically I 728 00:49:56,570 --> 00:50:01,470 ignored the details of the process that is taking place 729 00:50:01,470 --> 00:50:05,520 at scale the order of d, so I have thrown away 730 00:50:05,520 --> 00:50:08,490 some amount of detail and information. 731 00:50:08,490 --> 00:50:12,540 It is, again, legitimate to say, is this 732 00:50:12,540 --> 00:50:15,970 the stage at which you made an approximation 733 00:50:15,970 --> 00:50:19,790 so that the time reversibility was lost? 734 00:50:19,790 --> 00:50:21,920 The answer is still no. 735 00:50:21,920 --> 00:50:26,890 If you are careful enough with making precise definitions 736 00:50:26,890 --> 00:50:30,520 of what these Q's are before and after the collision, 737 00:50:30,520 --> 00:50:35,220 and follow what happens if you were to reverse everything, 738 00:50:35,220 --> 00:50:38,640 you'll find that the equations is fully reversible. 739 00:50:38,640 --> 00:50:43,890 Even at this stage, I have not made any transition. 740 00:50:43,890 --> 00:50:47,560 I have made approximations, but I haven't made something 741 00:50:47,560 --> 00:50:49,920 to be time irreversible. 742 00:50:49,920 --> 00:50:52,490 That comes at the next stage where 743 00:50:52,490 --> 00:50:55,660 we make the so-called assumption of molecular chaos. 744 00:51:08,390 --> 00:51:12,990 The assumption is that what's the chance 745 00:51:12,990 --> 00:51:16,512 that I have a particle here and a particle there? 746 00:51:16,512 --> 00:51:17,970 You would say, it's a chance that I 747 00:51:17,970 --> 00:51:20,985 have one here and one there. 748 00:51:20,985 --> 00:51:34,070 You say that if two of any P1, P2, Q1, Q2, t 749 00:51:34,070 --> 00:51:45,760 is the same thing as the product of F1, P1, Q1, t, F1, P2, Q2, 750 00:51:45,760 --> 00:51:46,260 t. 751 00:51:50,981 --> 00:51:55,940 Of course, this assumption is generally varied. 752 00:51:55,940 --> 00:51:58,700 If I were to look at the probability 753 00:51:58,700 --> 00:52:02,750 that I have two particles as a function of, let's say, 754 00:52:02,750 --> 00:52:08,140 the relative separation, I certainly 755 00:52:08,140 --> 00:52:12,040 expect that if they are far away, 756 00:52:12,040 --> 00:52:16,290 the density should be the product of the one particle 757 00:52:16,290 --> 00:52:16,790 densities. 758 00:52:19,990 --> 00:52:24,500 But you would say that if the two particles come to distances 759 00:52:24,500 --> 00:52:29,510 that are closer than their separation d, 760 00:52:29,510 --> 00:52:32,800 then the probability and the range of interaction d-- 761 00:52:32,800 --> 00:52:34,490 and let's say the interaction is highly 762 00:52:34,490 --> 00:52:40,660 repulsive like hardcore-- then the probability should go to 0. 763 00:52:40,660 --> 00:52:44,360 Clearly, you can make this assumption, 764 00:52:44,360 --> 00:52:47,560 but up to some degree. 765 00:52:47,560 --> 00:52:51,220 Part of the reason we went through this process 766 00:52:51,220 --> 00:52:57,520 was to indeed make sure that we are integrating things 767 00:52:57,520 --> 00:53:00,500 at the locations where the particles are 768 00:53:00,500 --> 00:53:03,000 far away from each other. 769 00:53:03,000 --> 00:53:06,630 I said that the range of that integration over A 770 00:53:06,630 --> 00:53:08,640 would be someplace where they are 771 00:53:08,640 --> 00:53:11,720 far apart after the collision, and far apart 772 00:53:11,720 --> 00:53:12,680 before the collision. 773 00:53:15,290 --> 00:53:19,300 You have an assumption like that, 774 00:53:19,300 --> 00:53:21,700 which is, in principle, something 775 00:53:21,700 --> 00:53:23,670 that I can insert into that. 776 00:53:27,140 --> 00:53:30,940 Having to make a distinction between the arguments that 777 00:53:30,940 --> 00:53:37,570 are appearing in this equation is kind of not so pleasant. 778 00:53:37,570 --> 00:53:41,070 What you are going to do is to make another assumption. 779 00:53:41,070 --> 00:53:45,296 Make sure that everything is evaluated at the same point. 780 00:53:53,610 --> 00:53:59,850 What we will eventually now have is the equation that d by dt, 781 00:53:59,850 --> 00:54:11,110 plus P1 over n, d by dQ1, plus F1, dot, d by dP1, 782 00:54:11,110 --> 00:54:19,010 acting on F1, on the left hand side, 783 00:54:19,010 --> 00:54:24,380 is, on the right hand side, equal to all collisions 784 00:54:24,380 --> 00:54:29,245 in the particle of momentum P2, approaching 785 00:54:29,245 --> 00:54:34,350 at all possible cross sections, calculating 786 00:54:34,350 --> 00:54:36,870 the flux of the incoming particle 787 00:54:36,870 --> 00:54:38,880 that corresponds to that channel, which 788 00:54:38,880 --> 00:54:42,550 is proportional to V2 minus V1. 789 00:54:42,550 --> 00:54:49,460 Then here, we subtract the collision of the two particles. 790 00:54:49,460 --> 00:54:55,050 We write that as F1 of P1 at this location, 791 00:54:55,050 --> 00:55:06,390 Q1, t, F1 of t2 at the same location Q1, t. 792 00:55:06,390 --> 00:55:21,610 Then add F1 prime, P1 prime, Q1 t, F1 prime, P2 prime, Q2, t. 793 00:55:24,730 --> 00:55:27,850 In order to make the equation eventually 794 00:55:27,850 --> 00:55:38,070 manageable, what you did is to evaluate all 795 00:55:38,070 --> 00:55:41,200 off the coordinates that we have on the right hand 796 00:55:41,200 --> 00:55:46,680 side at the same location, which is the same Q1 that you specify 797 00:55:46,680 --> 00:55:48,950 on the left hand side. 798 00:55:48,950 --> 00:55:53,260 That immediately means that what you have done 799 00:55:53,260 --> 00:55:56,280 is you have changed the resolution with which you 800 00:55:56,280 --> 00:55:58,060 are looking at space. 801 00:55:58,060 --> 00:56:00,250 You have kind of washed out the difference 802 00:56:00,250 --> 00:56:03,080 between here and here. 803 00:56:03,080 --> 00:56:08,100 Your resolution has to put this whole area that 804 00:56:08,100 --> 00:56:12,740 is of the order of d squared or d cubed in three dimensions 805 00:56:12,740 --> 00:56:14,320 into one pixel. 806 00:56:14,320 --> 00:56:17,630 You have changed the resolution that you have. 807 00:56:17,630 --> 00:56:20,340 You are not looking at things at this [? fine ?] [? state. ?] 808 00:56:23,330 --> 00:56:27,370 You are losing additional information here 809 00:56:27,370 --> 00:56:31,670 through this change of the resolution in space. 810 00:56:31,670 --> 00:56:34,980 You have also lost some information 811 00:56:34,980 --> 00:56:39,960 in making the assumption that the two [? point ?] densities 812 00:56:39,960 --> 00:56:44,090 are completely within always as the product one particle 813 00:56:44,090 --> 00:56:45,530 densities. 814 00:56:45,530 --> 00:56:49,680 Both of those things correspond to taking something 815 00:56:49,680 --> 00:56:52,730 that is very precise and deterministic, 816 00:56:52,730 --> 00:56:57,380 and making it kind of vague and a little undefined. 817 00:56:57,380 --> 00:57:01,930 It's not surprising then, that if you have in some sense 818 00:57:01,930 --> 00:57:05,450 changed the precision of your computer-- let's say, 819 00:57:05,450 --> 00:57:08,450 that is running the particles forward-- at some point, 820 00:57:08,450 --> 00:57:10,920 you've changed the resolution. 821 00:57:10,920 --> 00:57:13,320 Then you can't really run backward. 822 00:57:13,320 --> 00:57:17,180 In fact, to sort of precisely be able to run the equations 823 00:57:17,180 --> 00:57:19,430 forward and backward, you would need 824 00:57:19,430 --> 00:57:22,460 to keep resolution at all levels. 825 00:57:22,460 --> 00:57:26,270 Here, we have sort of removed some amount of resolution. 826 00:57:26,270 --> 00:57:29,000 We have a very good guess that the equation that you 827 00:57:29,000 --> 00:57:33,510 have over here no longer respects time reversal 828 00:57:33,510 --> 00:57:37,680 inversions that you had originally posed. 829 00:57:37,680 --> 00:57:42,840 Our next task is to prove that you need this equation. 830 00:57:42,840 --> 00:57:46,000 It goes in one particular direction in time, 831 00:57:46,000 --> 00:57:49,730 and cannot be drawn backward, as opposed to all 832 00:57:49,730 --> 00:57:55,340 of the predecessors that I had written up to this point. 833 00:57:55,340 --> 00:57:56,480 Are there any questions? 834 00:57:59,714 --> 00:58:00,682 AUDIENCE: [INAUDIBLE] 835 00:58:08,105 --> 00:58:13,532 PROFESSOR: Yes, Q prime and Q1, not Q1 prime. 836 00:58:13,532 --> 00:58:14,240 There is no dash. 837 00:58:14,240 --> 00:58:15,170 AUDIENCE: Oh, I see. 838 00:58:15,170 --> 00:58:16,425 It is Q1. 839 00:58:16,425 --> 00:58:18,750 PROFESSOR: Yes, it is. 840 00:58:18,750 --> 00:58:20,060 Look at this equation. 841 00:58:20,060 --> 00:58:22,460 On the left hand side, what are the arguments? 842 00:58:22,460 --> 00:58:28,110 The arguments are P1 and Q1. 843 00:58:28,110 --> 00:58:30,730 What is it that I have on the other side? 844 00:58:30,730 --> 00:58:32,800 I still have P1 and Q1. 845 00:58:32,800 --> 00:58:36,350 I have introduced P1 and b, which 846 00:58:36,350 --> 00:58:40,490 is simply an impact parameter. 847 00:58:40,490 --> 00:58:42,590 What I will do is I will evaluate 848 00:58:42,590 --> 00:58:48,110 all of these things, always at the same location, Q1. 849 00:58:48,110 --> 00:58:50,650 Then I have P1 and P2. 850 00:58:50,650 --> 00:58:56,460 That's part of my story of the change in resolution. 851 00:58:56,460 --> 00:58:59,910 When I write here Q1, and you say Q1 prime, 852 00:58:59,910 --> 00:59:01,330 but what is Q1 prime? 853 00:59:01,330 --> 00:59:02,745 Is it Q1 plus b? 854 00:59:02,745 --> 00:59:03,860 Is it Q1 minus b? 855 00:59:03,860 --> 00:59:06,020 Something like this I'm going to ignore. 856 00:59:09,800 --> 00:59:11,520 It's also legitimate, and you should 857 00:59:11,520 --> 00:59:16,027 ask, what is P1 prime and Q2 prime? 858 00:59:16,027 --> 00:59:16,610 What are they? 859 00:59:20,120 --> 00:59:26,780 What I have to do, is I have to run on the computer 860 00:59:26,780 --> 00:59:29,300 or otherwise, the equations for what 861 00:59:29,300 --> 00:59:35,430 happens if I have P1 and P2 come together 862 00:59:35,430 --> 00:59:38,604 at an impact parameter that is set by me. 863 00:59:38,604 --> 00:59:41,430 I then integrate the equations, and I 864 00:59:41,430 --> 00:59:44,760 find that deterministically, that collision will 865 00:59:44,760 --> 00:59:48,230 lead to some P1 prime and P2 prime. 866 00:59:48,230 --> 00:59:59,590 P1 prime and P2 prime are some complicated functions 867 00:59:59,590 --> 01:00:02,760 of P1, P2, and b. 868 01:00:05,560 --> 01:00:10,040 Given that you know two particles are approaching 869 01:00:10,040 --> 01:00:16,940 each other at distance d with momenta P1 P2, in principle, 870 01:00:16,940 --> 01:00:20,300 you can integrate Newton's equations, 871 01:00:20,300 --> 01:00:22,790 and figure out with what momenta they end up. 872 01:00:25,600 --> 01:00:30,700 This equation, in fact, hides a very, very complicated function 873 01:00:30,700 --> 01:00:33,790 here, which describes P1 prime and P2 prime 874 01:00:33,790 --> 01:00:36,760 as a function of P1 and P2. 875 01:00:36,760 --> 01:00:40,815 If you really needed all of the details of that function, 876 01:00:40,815 --> 01:00:44,360 you would surely be in trouble. 877 01:00:44,360 --> 01:00:45,620 Fortunately, we don't. 878 01:00:45,620 --> 01:00:50,700 As we shall see shortly, you can kind of get a lot of mileage 879 01:00:50,700 --> 01:00:51,630 without knowing that. 880 01:00:51,630 --> 01:00:52,755 Yes, what is your question? 881 01:00:52,755 --> 01:00:54,622 AUDIENCE: There was an assumption 882 01:00:54,622 --> 01:00:57,210 that all the interactions between different molecules 883 01:00:57,210 --> 01:00:59,770 are central potentials [INAUDIBLE]. 884 01:00:59,770 --> 01:01:02,850 Does the force of the direction between two particles 885 01:01:02,850 --> 01:01:04,990 lie along the [INAUDIBLE]? 886 01:01:04,990 --> 01:01:07,760 PROFESSOR: For the things that I have written, yes it does. 887 01:01:07,760 --> 01:01:10,160 I should have been more precise. 888 01:01:10,160 --> 01:01:14,950 I should have put absolute value here. 889 01:01:19,054 --> 01:01:20,740 AUDIENCE: You have particles moving 890 01:01:20,740 --> 01:01:23,368 along one line towards each other, 891 01:01:23,368 --> 01:01:25,326 and b is some arbitrary vector. 892 01:01:25,326 --> 01:01:27,990 You have two directions, so you define a plane. 893 01:01:27,990 --> 01:01:31,460 Opposite direction particles stay at the same plane. 894 01:01:31,460 --> 01:01:34,040 Have you reduced-- 895 01:01:34,040 --> 01:01:36,426 PROFESSOR: Particles stay in the same plane? 896 01:01:36,426 --> 01:01:42,280 AUDIENCE: If the two particles were moving towards each other, 897 01:01:42,280 --> 01:01:45,080 and also you have in the integral 898 01:01:45,080 --> 01:01:50,040 your input parameter, which one is [INAUDIBLE]. 899 01:01:50,040 --> 01:01:54,000 There's two directions. 900 01:01:54,000 --> 01:01:55,880 All particles align, and all b's align. 901 01:01:55,880 --> 01:01:57,039 They form a plane. 902 01:01:57,039 --> 01:01:59,330 [? Opposite ?] direction particles [? stand ?] in the-- 903 01:01:59,330 --> 01:02:02,690 PROFESSOR: Yes, they stand in the same plane. 904 01:02:02,690 --> 01:02:04,362 AUDIENCE: My question is, what is 905 01:02:04,362 --> 01:02:06,070 [INAUDIBLE] use the integral on the right 906 01:02:06,070 --> 01:02:08,930 from a two-dimensional integral [? in v ?] 907 01:02:08,930 --> 01:02:12,170 into employing central symmetry? 908 01:02:12,170 --> 01:02:14,170 PROFESSOR: Yes, you could. 909 01:02:14,170 --> 01:02:19,901 You could, in principle, write this as b db, if you like, 910 01:02:19,901 --> 01:02:20,900 if that's what you want. 911 01:02:24,390 --> 01:02:27,414 AUDIENCE: [INAUDIBLE] 912 01:02:27,414 --> 01:02:29,080 PROFESSOR: Yes, you could do that if you 913 01:02:29,080 --> 01:02:30,945 have simple enough potential. 914 01:02:39,050 --> 01:02:47,180 Let's show that this equation leads to irreversibility. 915 01:02:47,180 --> 01:02:49,020 That you are going to do here. 916 01:03:09,440 --> 01:03:12,835 This, by the way, is called the Boltzmann equation. 917 01:03:18,880 --> 01:03:25,064 There's an associated Boltzmann H-Theorem, 918 01:03:25,064 --> 01:03:38,400 which restates the following-- If F of P1, Q1, and t 919 01:03:38,400 --> 01:03:53,200 satisfies the above Boltzmann equation, 920 01:03:53,200 --> 01:04:08,540 then there is a quantity H that always decreases in time, where 921 01:04:08,540 --> 01:04:20,350 H is the integral over P and Q of F1, log of F1. 922 01:04:26,340 --> 01:04:28,330 The composition of irreversibility, 923 01:04:28,330 --> 01:04:30,940 as we saw in thermal dynamics, was 924 01:04:30,940 --> 01:04:33,490 that there was a quantity entropy that 925 01:04:33,490 --> 01:04:34,980 was always increasing. 926 01:04:34,980 --> 01:04:37,560 If you have calculated for this system, 927 01:04:37,560 --> 01:04:41,700 entropy before for the half box, and entropy afterwards 928 01:04:41,700 --> 01:04:45,490 for the space both boxes occupy, the second one 929 01:04:45,490 --> 01:04:47,066 would certainly be larger. 930 01:04:50,030 --> 01:04:52,900 This H is a quantity like that, except that when 931 01:04:52,900 --> 01:04:55,720 it is defined this way, it always 932 01:04:55,720 --> 01:04:58,500 decreases as a function of time. 933 01:04:58,500 --> 01:05:01,605 But it certainly is very much related to entropy. 934 01:05:04,330 --> 01:05:09,080 You may have asked, why did Boltzmann 935 01:05:09,080 --> 01:05:11,110 come across such a function, which 936 01:05:11,110 --> 01:05:15,790 is F log F, except that actually right now, 937 01:05:15,790 --> 01:05:19,860 you should know why you write this. 938 01:05:19,860 --> 01:05:22,020 When we were dealing with probabilities, 939 01:05:22,020 --> 01:05:26,060 we introduced the entropy of the probability distribution, which 940 01:05:26,060 --> 01:05:31,460 was related to something like sum over iPi, log of Pi, 941 01:05:31,460 --> 01:05:34,900 with a minus sign. 942 01:05:34,900 --> 01:05:38,170 Up to this factor of normalization N, 943 01:05:38,170 --> 01:05:42,610 this F1 really is a one-particle probability. 944 01:05:42,610 --> 01:05:45,600 After this normalization N, you have 945 01:05:45,600 --> 01:05:49,600 a one-particle probability, the probability 946 01:05:49,600 --> 01:05:54,150 that you have occupation of one-particle free space. 947 01:05:54,150 --> 01:05:57,380 This occupation of one-particle phase space 948 01:05:57,380 --> 01:06:01,550 is changing as a function of time. 949 01:06:01,550 --> 01:06:06,990 What this statement says is that if the one-particle density 950 01:06:06,990 --> 01:06:09,790 evolves in time according to this equation, 951 01:06:09,790 --> 01:06:12,880 the corresponding minus entropy decreases 952 01:06:12,880 --> 01:06:15,970 as a function of time. 953 01:06:15,970 --> 01:06:19,230 Let's see if that's the case. 954 01:06:19,230 --> 01:06:24,340 To prove that, let's do this. 955 01:06:24,340 --> 01:06:30,810 We have the formula for H, so let's calculate the H by dt. 956 01:06:30,810 --> 01:06:34,500 I have an integral over the phase space 957 01:06:34,500 --> 01:06:44,520 of particle one, the particle that I just called one. 958 01:06:44,520 --> 01:06:47,570 I could have labeled it anything. 959 01:06:47,570 --> 01:06:52,060 After integration, H is only a function of time. 960 01:06:52,060 --> 01:06:54,050 I have to take the time derivative. 961 01:06:54,050 --> 01:06:57,950 The time derivative can act on F1. 962 01:06:57,950 --> 01:07:04,010 Then I will get the F1 by dt, times log F1. 963 01:07:04,010 --> 01:07:10,870 Or I will have F1 times the derivative of log F1. 964 01:07:10,870 --> 01:07:15,870 The derivative of log F1 would be dF1 by dt, and then 1 965 01:07:15,870 --> 01:07:16,720 over F1. 966 01:07:16,720 --> 01:07:18,420 Then I multiply by F1. 967 01:07:21,260 --> 01:07:22,910 This term is simply 1. 968 01:07:26,815 --> 01:07:29,180 AUDIENCE: Don't you want to write the full derivative, 969 01:07:29,180 --> 01:07:31,630 F1 with respect [INAUDIBLE]? 970 01:07:31,630 --> 01:07:35,060 PROFESSOR: I thought we did that with this before. 971 01:07:35,060 --> 01:07:40,560 If you have something that I am summing over lots 972 01:07:40,560 --> 01:07:46,420 of [? points, ?] and these [? points ?] can be positioned, 973 01:07:46,420 --> 01:07:50,040 then I have S at location one, S at location two, 974 01:07:50,040 --> 01:07:56,230 S at location three, discretized versions of x. 975 01:07:56,230 --> 01:07:58,460 If I take the time derivative, I take 976 01:07:58,460 --> 01:08:00,480 the time derivative of this, plus this, 977 01:08:00,480 --> 01:08:03,285 plus this, which are partial derivatives. 978 01:08:14,940 --> 01:08:18,069 If I actually take the time derivative here, 979 01:08:18,069 --> 01:08:24,950 I get the integral d cubed P1, d cubed Q1, the time derivative. 980 01:08:24,950 --> 01:08:28,310 This would be that partial dF1 by dt 981 01:08:28,310 --> 01:08:31,430 is the time derivative of n, which is 0. 982 01:08:31,430 --> 01:08:34,479 The number of particles does not change. 983 01:08:34,479 --> 01:08:38,970 Indeed, I realize that 1 integrated against dF1 by dt 984 01:08:38,970 --> 01:08:41,640 is the same thing that's here. 985 01:08:41,640 --> 01:08:43,890 This term gives you 0. 986 01:08:43,890 --> 01:08:47,149 All I need to worry about is integrating 987 01:08:47,149 --> 01:08:51,460 log F against the Fydt. 988 01:08:51,460 --> 01:09:01,520 I have an integral over P1 and Q1 of log F against the Fydt. 989 01:09:01,520 --> 01:09:10,389 We have said that F1 satisfies the Boltzmann equation. 990 01:09:14,844 --> 01:09:21,149 So the F1 by dt, if I were to rearrange it, 991 01:09:21,149 --> 01:09:25,370 I have the F1 by dt. 992 01:09:25,370 --> 01:09:29,899 I take this part to the other side of the equation. 993 01:09:29,899 --> 01:09:34,830 This part is also the Poisson bracket 994 01:09:34,830 --> 01:09:40,300 of a one-particle H with F1. 995 01:09:40,300 --> 01:09:42,620 If I take it to the other side, it 996 01:09:42,620 --> 01:09:47,399 will be the Poisson bracket of H with F1. 997 01:09:47,399 --> 01:09:49,080 Then there is this whole thing that 998 01:09:49,080 --> 01:09:52,050 involves the collision of two particles. 999 01:09:52,050 --> 01:09:56,550 So I define whatever is on the right hand side 1000 01:09:56,550 --> 01:10:02,060 to be some collision operator that acts on two 1001 01:10:02,060 --> 01:10:04,160 [? powers ?] of F1. 1002 01:10:04,160 --> 01:10:11,060 This is plus a collision operator, F1, F1. 1003 01:10:11,060 --> 01:10:16,130 What I do is I replace this dF1 by dt 1004 01:10:16,130 --> 01:10:24,850 with the Poisson bracket of H, or H1, if you like, with F1. 1005 01:10:24,850 --> 01:10:28,340 The collision operator I will shortly write explicitly. 1006 01:10:28,340 --> 01:10:32,502 But for the time being, let me just write it as C of F1. 1007 01:10:42,460 --> 01:10:44,450 There is a first term in this sum-- 1008 01:10:44,450 --> 01:10:50,860 let's call it number one-- which I claim to be 0. 1009 01:10:50,860 --> 01:10:53,590 Typically, when you get these integrations 1010 01:10:53,590 --> 01:10:56,070 with Poisson brackets, you would get 0. 1011 01:10:56,070 --> 01:10:58,190 Let's explicitly show that. 1012 01:10:58,190 --> 01:11:07,240 I have an integral over P1 and Q1 of log of F1, 1013 01:11:07,240 --> 01:11:11,330 and this Poisson bracket of H1 and F1, 1014 01:11:11,330 --> 01:11:14,220 which is essentially these terms. 1015 01:11:14,220 --> 01:11:27,110 Alternatively, I could write it as dH1 by dQ1, dF1 by dt1, 1016 01:11:27,110 --> 01:11:34,310 minus the H1 by dt1, dF1, by dQ1. 1017 01:11:37,770 --> 01:11:43,090 I've explicitly written this form for the one-particle 1018 01:11:43,090 --> 01:11:44,380 in terms of the Hamiltonian. 1019 01:11:47,230 --> 01:11:49,030 The advantage of that is that now I 1020 01:11:49,030 --> 01:11:51,803 can start doing integrations by parts. 1021 01:11:55,830 --> 01:12:01,550 I'm taking derivatives with respect to P, 1022 01:12:01,550 --> 01:12:05,680 but I have integrations with respect to P here. 1023 01:12:05,680 --> 01:12:08,930 I could take the F1 out. 1024 01:12:08,930 --> 01:12:10,820 I will have a minus. 1025 01:12:10,820 --> 01:12:15,530 I have an integral, P1, Q1. 1026 01:12:15,530 --> 01:12:18,100 I took F1 out. 1027 01:12:18,100 --> 01:12:22,940 Then this d by dP1 acts on everything that came before it. 1028 01:12:22,940 --> 01:12:25,610 It can act on the H1. 1029 01:12:25,610 --> 01:12:33,500 I would get d2 H1 with respect to dP1, dQ1. 1030 01:12:33,500 --> 01:12:38,490 Or it could act on the log of F1, in which case 1031 01:12:38,490 --> 01:12:43,380 I will get set dH1 by dQ1. 1032 01:12:47,890 --> 01:12:51,900 Then I would have d by dP acting on log 1033 01:12:51,900 --> 01:13:00,990 of F, which would give me dF1 by dP1, 1034 01:13:00,990 --> 01:13:04,018 then the derivative of the log, which is 1 over F1. 1035 01:13:07,810 --> 01:13:10,300 This is only the first term. 1036 01:13:10,300 --> 01:13:16,822 I also have this term, with which I will do the same thing. 1037 01:13:16,822 --> 01:13:22,553 AUDIENCE: [INAUDIBLE] The second derivative [INAUDIBLE] 1038 01:13:22,553 --> 01:13:27,598 should be multiplied by log of F. 1039 01:13:27,598 --> 01:13:28,806 PROFESSOR: Yes, it should be. 1040 01:13:33,505 --> 01:13:34,540 It is Log F1. 1041 01:13:38,180 --> 01:13:38,920 Thank you. 1042 01:13:41,780 --> 01:13:46,160 For the next term, I have F1. 1043 01:13:46,160 --> 01:13:53,620 I have d2 H1, and the other order of derivatives, dQ1, dP1. 1044 01:13:53,620 --> 01:13:57,720 Now I'll make sure I write down the log of F1. 1045 01:13:57,720 --> 01:14:04,510 Then I have dH1 with respect to dQ1. 1046 01:14:13,110 --> 01:14:18,100 Then I have a dot product with the derivative of log F, which 1047 01:14:18,100 --> 01:14:24,008 is the derivative of F1 with respect to Q1 and 1 over F1. 1048 01:14:31,660 --> 01:14:33,400 Here are the terms that are proportional 1049 01:14:33,400 --> 01:14:34,680 to the second derivative. 1050 01:14:34,680 --> 01:14:37,310 The order of the derivatives does not matter. 1051 01:14:37,310 --> 01:14:38,760 One often is positive. 1052 01:14:38,760 --> 01:14:43,010 One often is negative, so they cancel out. 1053 01:14:43,010 --> 01:14:44,910 Then I have these additional terms. 1054 01:14:44,910 --> 01:14:47,190 For the additional terms, you'll note 1055 01:14:47,190 --> 01:14:50,480 that the F1 and the 1 over F1 cancels. 1056 01:14:54,770 --> 01:14:58,960 These are just a product of two first derivatives. 1057 01:14:58,960 --> 01:15:06,180 I will apply the five parts process one more time 1058 01:15:06,180 --> 01:15:11,760 to get rid of the derivative that is acting on F1. 1059 01:15:11,760 --> 01:15:17,150 The answer becomes plus d cubed P1, d cubed Q1. 1060 01:15:17,150 --> 01:15:29,160 Then I have F1, d2 H1, dP1, dQ1, minus d2 H1, dQ1, dP1. 1061 01:15:29,160 --> 01:15:34,202 These two cancel each other out, and the answer is 0. 1062 01:15:40,110 --> 01:15:43,560 So that first term vanishes. 1063 01:15:43,560 --> 01:15:55,816 Now for the second term, number two, what I have 1064 01:15:55,816 --> 01:15:56,940 is the first term vanished. 1065 01:15:56,940 --> 01:16:00,080 So I have the H by dt. 1066 01:16:00,080 --> 01:16:06,260 It is the integral over P1 and Q1. 1067 01:16:09,490 --> 01:16:17,660 I have log of F1. 1068 01:16:17,660 --> 01:16:22,060 F1 is a function of P1, and Q1, and t. 1069 01:16:22,060 --> 01:16:26,760 I will focus, and make sure I write the argument of momentum, 1070 01:16:26,760 --> 01:16:31,710 for reasons that will become shortly apparent. 1071 01:16:31,710 --> 01:16:35,180 I have to multiply with the collision term. 1072 01:16:35,180 --> 01:16:39,520 The collision term involves integrations 1073 01:16:39,520 --> 01:16:46,360 over a second particle, over an impact parameter, 1074 01:16:46,360 --> 01:16:51,450 a relative velocity, once I have defined what P2 and P1 are. 1075 01:16:51,450 --> 01:16:57,340 I have a subtraction of F evaluated at P1, 1076 01:16:57,340 --> 01:17:01,760 F evaluated at P2, plus addition, 1077 01:17:01,760 --> 01:17:06,705 F evaluated at P1 prime, F evaluated at P2 prime. 1078 01:17:14,430 --> 01:17:18,700 Eventually, this whole thing is only a function of time. 1079 01:17:18,700 --> 01:17:22,310 There are a whole bunch of arguments appearing here, 1080 01:17:22,310 --> 01:17:25,430 but all of those arguments are being integrated over. 1081 01:17:29,230 --> 01:17:35,500 In particular, I have arguments that are indexed by P1 and P2. 1082 01:17:35,500 --> 01:17:38,930 These are dummy variables of integration. 1083 01:17:38,930 --> 01:17:41,750 If I have a function of x and y that I'm 1084 01:17:41,750 --> 01:17:45,110 integrating over x and y, I can call x "z." 1085 01:17:45,110 --> 01:17:47,060 I can call y "t." 1086 01:17:47,060 --> 01:17:48,990 I would integrate over z and t, and I 1087 01:17:48,990 --> 01:17:51,310 would have the same answer. 1088 01:17:51,310 --> 01:17:54,280 I would have exactly the same answer 1089 01:17:54,280 --> 01:17:58,650 if I were to call all of the dummy integration variable that 1090 01:17:58,650 --> 01:18:01,700 is indexed 1, "2." 1091 01:18:01,700 --> 01:18:04,420 Any dummy variable that is indexed 2, 1092 01:18:04,420 --> 01:18:10,130 if I rename it and call it 1, the integral would not change. 1093 01:18:10,130 --> 01:18:11,900 If I do that, what do I have? 1094 01:18:11,900 --> 01:18:14,920 I have integral over Q-- actually, 1095 01:18:14,920 --> 01:18:19,030 let's get of the integration number on Q. 1096 01:18:19,030 --> 01:18:21,350 It really doesn't matter. 1097 01:18:21,350 --> 01:18:24,840 I have the integrals over P1 and P1. 1098 01:18:24,840 --> 01:18:29,560 I have to integrate over both sets of momenta. 1099 01:18:29,560 --> 01:18:33,730 I have to integrate over the cross section, which 1100 01:18:33,730 --> 01:18:37,490 is relative between 1 and 2. 1101 01:18:37,490 --> 01:18:43,840 I have V2 minus V1, rather than V1 minus V2, rather than 1102 01:18:43,840 --> 01:18:45,550 V2 minus V1. 1103 01:18:45,550 --> 01:18:47,620 The absolute value doesn't matter. 1104 01:18:47,620 --> 01:18:52,970 If I were to replace these indices with an absolute value, 1105 01:18:52,970 --> 01:18:56,750 [? or do a ?] V2 minus V1 goes to minus V1 minus V2. 1106 01:18:56,750 --> 01:18:59,400 The absolute value does not change. 1107 01:18:59,400 --> 01:19:01,250 Here, what do I have? 1108 01:19:01,250 --> 01:19:04,720 I have minus F of P1. 1109 01:19:04,720 --> 01:19:11,550 It becomes F of P2, F of P1, plus F of P2 prime, 1110 01:19:11,550 --> 01:19:13,640 f of P1 prime. 1111 01:19:13,640 --> 01:19:14,530 They are a product. 1112 01:19:14,530 --> 01:19:18,450 It doesn't really matter in which order I write them. 1113 01:19:18,450 --> 01:19:20,080 The only thing that really matters 1114 01:19:20,080 --> 01:19:26,930 is that the argument was previously called F1 of P1 1115 01:19:26,930 --> 01:19:31,870 for the log, and now it will be called F1 of P2. 1116 01:19:31,870 --> 01:19:32,965 Just its name changed. 1117 01:19:37,150 --> 01:19:39,980 If I take this, and the first way of writing things, 1118 01:19:39,980 --> 01:19:43,350 which are really two ways of writing the same integral, 1119 01:19:43,350 --> 01:19:49,044 and just average them, I will get 1/2 an integral d cubed Q, 1120 01:19:49,044 --> 01:19:56,350 d cubed P1, d cubed P2, d2 b, and V2 minus V1. 1121 01:19:56,350 --> 01:20:09,100 I will have F1 of P1, F1 of P2, plus F1 of P1 prime, F1 1122 01:20:09,100 --> 01:20:11,630 of P2 prime. 1123 01:20:11,630 --> 01:20:17,360 Then in one term, I had log of F1 of P1, 1124 01:20:17,360 --> 01:20:19,180 and I averaged it with the other way 1125 01:20:19,180 --> 01:20:24,630 of writing things, which was log of F-- let's put the two 1126 01:20:24,630 --> 01:20:28,320 logs together, multiplied by F1. 1127 01:20:28,320 --> 01:20:30,400 So the sum of the two logs I wrote, 1128 01:20:30,400 --> 01:20:32,540 that's a log of the product. 1129 01:20:32,540 --> 01:20:34,770 I just rewrote that equation. 1130 01:20:34,770 --> 01:20:39,100 If you like, I symmetrized It with respect to index 1 and 2. 1131 01:20:39,100 --> 01:20:42,222 So the log of 1, that previously had 1132 01:20:42,222 --> 01:20:43,930 one argument through this symmetrization, 1133 01:20:43,930 --> 01:20:48,860 became one half of the sum of it. 1134 01:20:48,860 --> 01:20:55,790 The next thing one has to think about, what I want to do, 1135 01:20:55,790 --> 01:20:59,280 is to replace primed and unprimed coordinates. 1136 01:21:04,280 --> 01:21:07,570 What I would eventually write down 1137 01:21:07,570 --> 01:21:22,290 is d cubed P1 prime, d cubed P2 prime, d2 b, V2 prime minus V1 1138 01:21:22,290 --> 01:21:33,760 prime, minus F1 of P1 prime, F1 of P2 prime, plus F1 of P1, 1139 01:21:33,760 --> 01:21:35,790 F1 of P2. 1140 01:21:35,790 --> 01:21:42,308 Then log of F1 of P1 prime, F1 of P2 prime. 1141 01:21:47,100 --> 01:21:52,050 I've symmetrized originally the indices 1 and 2 1142 01:21:52,050 --> 01:21:55,420 that were not quite symmetric, and I end up 1143 01:21:55,420 --> 01:22:00,210 with an expression that has variables P1, P2, and functions 1144 01:22:00,210 --> 01:22:04,170 P1 prime and P2 prime, which are not quite symmetric again, 1145 01:22:04,170 --> 01:22:11,360 because I have F's evaluated for P's, but not for P primes. 1146 01:22:11,360 --> 01:22:14,260 What does this mean? 1147 01:22:14,260 --> 01:22:17,620 This mathematical expression that I have written down here 1148 01:22:17,620 --> 01:22:24,010 actually is not correct, because what this amounts to, 1149 01:22:24,010 --> 01:22:30,520 is to change variables of integration. 1150 01:22:30,520 --> 01:22:33,340 In the expression that I have up here, 1151 01:22:33,340 --> 01:22:37,910 P1 and P2 are variables of integration. 1152 01:22:37,910 --> 01:22:41,700 P1 prime and P2 prime are some complicated functions 1153 01:22:41,700 --> 01:22:43,770 of P1 and P2. 1154 01:22:43,770 --> 01:22:47,730 P1 prime is some complicated function that I don't know. 1155 01:22:47,730 --> 01:22:52,360 P1, P2, and V, for which I need to solve in principle, 1156 01:22:52,360 --> 01:22:55,060 is Newton's equation. 1157 01:22:55,060 --> 01:22:58,250 This is similarly for P2 prime. 1158 01:22:58,250 --> 01:23:00,380 What I have done is I have changed 1159 01:23:00,380 --> 01:23:06,000 from my original variables to these functions. 1160 01:23:06,000 --> 01:23:11,620 When I write things over here, now P1 prime and P2 prime 1161 01:23:11,620 --> 01:23:14,365 are the integration variables. 1162 01:23:14,365 --> 01:23:16,680 P1 and P2 are supposed to be regarded 1163 01:23:16,680 --> 01:23:20,810 as functions of P1 prime and P2 prime. 1164 01:23:20,810 --> 01:23:22,770 You say, well, what does that mean? 1165 01:23:22,770 --> 01:23:26,790 You can't simply take an integral dx, 1166 01:23:26,790 --> 01:23:34,096 let's say F of some function of x, and replace this function. 1167 01:23:34,096 --> 01:23:35,470 You can't call it a new variable, 1168 01:23:35,470 --> 01:23:38,100 and do integral dx prime. 1169 01:23:38,100 --> 01:23:43,020 You have to multiply with the Jacobian of the transformation 1170 01:23:43,020 --> 01:23:49,140 that takes you from the P variables to the new variables. 1171 01:23:55,680 --> 01:23:58,550 My claim is that this Jacobian of the integration 1172 01:23:58,550 --> 01:23:59,917 is, in fact, the unit. 1173 01:24:03,110 --> 01:24:06,310 The reason is as follows. 1174 01:24:06,310 --> 01:24:11,010 These equations that have to be integrated to give me 1175 01:24:11,010 --> 01:24:15,260 the correlation are time reversible. 1176 01:24:15,260 --> 01:24:20,020 If I give you two momenta, and I know what the outcomes are, 1177 01:24:20,020 --> 01:24:22,350 I can write the equations backward, 1178 01:24:22,350 --> 01:24:25,530 and I will have the opposite momenta go back 1179 01:24:25,530 --> 01:24:28,510 to minus the original momenta. 1180 01:24:28,510 --> 01:24:33,400 Up to a factor of minus, you can see that this equation has 1181 01:24:33,400 --> 01:24:38,660 this character, that P1, P2 go to P1 prime, P2 prime, then 1182 01:24:38,660 --> 01:24:44,850 minus P1 prime, minus P2 prime, go to P1, and P2. 1183 01:24:44,850 --> 01:24:47,480 If you sort of follow that, and say 1184 01:24:47,480 --> 01:24:50,770 that you do the transformation twice, 1185 01:24:50,770 --> 01:24:53,840 you have to get back up to where a sign actually 1186 01:24:53,840 --> 01:24:55,800 disappears to where you want. 1187 01:24:55,800 --> 01:24:58,820 You have to multiply by two Jacobians, 1188 01:24:58,820 --> 01:25:01,560 and you get the same unit. 1189 01:25:01,560 --> 01:25:05,730 You can convince yourself that this Jacobian has to be unit. 1190 01:25:05,730 --> 01:25:08,340 Next time, I guess we'll take it from there. 1191 01:25:08,340 --> 01:25:11,180 I will explain this stuff a little bit more, 1192 01:25:11,180 --> 01:25:15,810 and show that this implies what we had said about the Boltzmann 1193 01:25:15,810 --> 01:25:17,084 equation.