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 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:29,960 --> 00:00:31,640 PROFESSOR: Let's say that I tell you 9 00:00:31,640 --> 00:00:37,870 that I'm interested in a gas that has some temperature. 10 00:00:37,870 --> 00:00:39,910 I specify, let's say, temperature. 11 00:00:39,910 --> 00:00:42,560 Pressure is room temperature, room pressure. 12 00:00:42,560 --> 00:00:45,560 And I tell you how many particles I have. 13 00:00:45,560 --> 00:00:50,870 So that's [INAUDIBLE] for you what the macro state is. 14 00:00:50,870 --> 00:00:55,690 And I want to then see what the corresponding micro state is. 15 00:00:55,690 --> 00:01:01,010 So I take one of these boxes, whereas this is a box 16 00:01:01,010 --> 00:01:05,290 that I draw in three dimensions, I can make a correspondence 17 00:01:05,290 --> 00:01:09,060 and draw this is 6N-dimensional coordinate space, which 18 00:01:09,060 --> 00:01:10,960 would be hard for me to draw. 19 00:01:10,960 --> 00:01:13,850 But basically, a space of six n dimensions. 20 00:01:13,850 --> 00:01:17,030 I figure out where the position, and the particles, 21 00:01:17,030 --> 00:01:18,590 and the momenta are. 22 00:01:18,590 --> 00:01:22,270 And I sort of find that there is a corresponding micro state 23 00:01:22,270 --> 00:01:25,270 that corresponds to this macro state. 24 00:01:25,270 --> 00:01:26,180 OK, that's fine. 25 00:01:26,180 --> 00:01:28,500 I made the correspondence. 26 00:01:28,500 --> 00:01:37,410 But the thing is that I can imagine lots and lots of boxes 27 00:01:37,410 --> 00:01:40,910 that have exactly the same macroscopic properties. 28 00:01:40,910 --> 00:01:45,660 That is, I can imagine putting here side by side 29 00:01:45,660 --> 00:01:48,460 a huge number of these boxes. 30 00:01:48,460 --> 00:01:53,230 All of them are described by exactly the same volume, 31 00:01:53,230 --> 00:01:55,600 pressure, temperature, for example. 32 00:01:55,600 --> 00:01:57,760 The same macro state. 33 00:01:57,760 --> 00:02:02,260 But for each one of them, when I go and find the macro state, 34 00:02:02,260 --> 00:02:04,090 I find that it is something else. 35 00:02:06,880 --> 00:02:12,210 So I will be having different micro states. 36 00:02:12,210 --> 00:02:15,390 So this correspondence is certainly 37 00:02:15,390 --> 00:02:17,980 something where there should be many, 38 00:02:17,980 --> 00:02:25,160 many points here that correspond to the same thermodynamic 39 00:02:25,160 --> 00:02:27,510 representation. 40 00:02:27,510 --> 00:02:31,330 So faced with that, maybe it makes sense 41 00:02:31,330 --> 00:02:35,610 to follow Gibbs and define an ensemble. 42 00:02:40,250 --> 00:02:42,950 So what we say is, we are interested 43 00:02:42,950 --> 00:02:46,140 in some particular macro state. 44 00:02:46,140 --> 00:02:49,170 We know that they correspond to many, many, many 45 00:02:49,170 --> 00:02:52,070 different potential micro states. 46 00:02:52,070 --> 00:02:57,380 Let's try to make a map of as many micro states that 47 00:02:57,380 --> 00:03:00,190 correspond to the same macro state. 48 00:03:00,190 --> 00:03:13,620 So consider n copies of the same macro state. 49 00:03:13,620 --> 00:03:17,370 And this would correspond to n different points 50 00:03:17,370 --> 00:03:21,820 that I put in this 6N-dimensional phase space. 51 00:03:21,820 --> 00:03:24,940 And what I can do is I can define an ensemble density. 52 00:03:33,440 --> 00:03:40,160 I go to a particular point in this space. 53 00:03:40,160 --> 00:03:44,230 So let's say I pick some point that corresponds 54 00:03:44,230 --> 00:03:46,510 to some set of p's and q's here. 55 00:03:52,370 --> 00:03:55,007 And what I do is I draw a box that 56 00:03:55,007 --> 00:04:01,890 is 6N-dimensional around this point. 57 00:04:01,890 --> 00:04:06,100 And I define a density in the vicinity 58 00:04:06,100 --> 00:04:08,455 of that point, as follows. 59 00:04:12,520 --> 00:04:16,880 Actually, yeah. 60 00:04:16,880 --> 00:04:23,280 What I will do is I will count how many of these points 61 00:04:23,280 --> 00:04:33,200 that correspond to micro states fall within this box. 62 00:04:33,200 --> 00:04:45,165 So at the end is the number of mu points in this box. 63 00:04:49,010 --> 00:04:56,900 And what I do is I divide by the total number. 64 00:04:56,900 --> 00:05:00,220 I expect that the result will be proportional 65 00:05:00,220 --> 00:05:02,220 to the volume of the box. 66 00:05:02,220 --> 00:05:05,650 So if I make the box bigger, I will have more. 67 00:05:05,650 --> 00:05:09,950 So I divide by the volume of the box. 68 00:05:09,950 --> 00:05:14,280 So this is, let's call d gamma is the volume of box. 69 00:05:19,280 --> 00:05:22,190 Of course, I have to do this in order 70 00:05:22,190 --> 00:05:27,430 to get a nice result by taking the limit where 71 00:05:27,430 --> 00:05:32,330 the number of members of the ensemble becomes quite large. 72 00:05:32,330 --> 00:05:37,830 And then presumably, this will give me a well-behaved density. 73 00:05:37,830 --> 00:05:41,870 In this limit, I guess I want to also have the size of the box 74 00:05:41,870 --> 00:05:44,650 go to 0. 75 00:05:44,650 --> 00:05:47,600 OK? 76 00:05:47,600 --> 00:05:50,370 Now, clearly, with the definitions 77 00:05:50,370 --> 00:05:54,090 that I have made, if I were to integrate 78 00:05:54,090 --> 00:06:05,290 this quantity against the volume d gamma, what I would get 79 00:06:05,290 --> 00:06:09,990 is the integral dN over N. N is, of course, a constant. 80 00:06:09,990 --> 00:06:12,850 And the integral of dN is the total number. 81 00:06:12,850 --> 00:06:15,490 So this is 1. 82 00:06:15,490 --> 00:06:19,410 So we find that this quantity rho that I have constructed 83 00:06:19,410 --> 00:06:21,180 satisfies two properties. 84 00:06:21,180 --> 00:06:23,020 Certainly, it is positive, because I'm 85 00:06:23,020 --> 00:06:24,720 counting the number of points. 86 00:06:24,720 --> 00:06:27,030 Secondly, it's normalized to 1. 87 00:06:27,030 --> 00:06:29,970 So this is a nice probability density. 88 00:06:29,970 --> 00:06:37,250 So this ensemble density is a probability density function 89 00:06:37,250 --> 00:06:41,040 in this phase space that I have defined. 90 00:06:41,040 --> 00:06:43,490 OK? 91 00:06:43,490 --> 00:06:44,620 All right. 92 00:06:44,620 --> 00:06:48,850 So once I have a probability, then I 93 00:06:48,850 --> 00:06:52,230 can calculate various things according 94 00:06:52,230 --> 00:06:55,260 to the rules of probability that we defined before. 95 00:06:55,260 --> 00:06:59,185 So for example, I can define an ensemble average. 96 00:07:03,000 --> 00:07:05,560 Maybe I'm interested in the kinetic energy 97 00:07:05,560 --> 00:07:08,320 of the particles of the gas. 98 00:07:08,320 --> 00:07:12,090 So there is a function O that depends 99 00:07:12,090 --> 00:07:15,260 on the sum of all of the p squareds. 100 00:07:15,260 --> 00:07:20,570 In general, I have some function of O that depends on p and q. 101 00:07:20,570 --> 00:07:25,350 And what I defined the ensemble average would be the average 102 00:07:25,350 --> 00:07:28,010 that I would calculate with this probability. 103 00:07:28,010 --> 00:07:31,690 Because I go over all of the points in phase space. 104 00:07:31,690 --> 00:07:34,900 And let me again emphasize that what I call d gamma 105 00:07:34,900 --> 00:07:41,230 then really is the product over all points. 106 00:07:41,230 --> 00:07:46,290 For each point, I have to make a volume in both momentum 107 00:07:46,290 --> 00:07:48,090 and in coordinate. 108 00:07:48,090 --> 00:07:51,650 It's a 6N-dimensional volume element. 109 00:07:51,650 --> 00:07:55,380 I have to multiply the probability, which 110 00:07:55,380 --> 00:07:59,980 is a function of p and q against this O, which 111 00:07:59,980 --> 00:08:04,310 is another function of p and q. 112 00:08:04,310 --> 00:08:05,241 Yes? 113 00:08:05,241 --> 00:08:07,696 AUDIENCE: Is the division by M necessary 114 00:08:07,696 --> 00:08:09,958 to make it into a probability density? 115 00:08:09,958 --> 00:08:10,583 PROFESSOR: Yes. 116 00:08:10,583 --> 00:08:13,544 AUDIENCE: Otherwise, you would still have a density. 117 00:08:13,544 --> 00:08:14,210 PROFESSOR: Yeah. 118 00:08:14,210 --> 00:08:18,210 When I would integrate then, I would get the total number. 119 00:08:18,210 --> 00:08:20,380 But the total number is up to me, 120 00:08:20,380 --> 00:08:22,650 how many members of the ensemble I took, 121 00:08:22,650 --> 00:08:25,250 it's not a very well-defined quantity. 122 00:08:25,250 --> 00:08:26,710 It's an arbitrary quantity. 123 00:08:26,710 --> 00:08:29,940 If I set it to become very large and divide by it, 124 00:08:29,940 --> 00:08:33,100 then I will get something that is nicely a probability. 125 00:08:33,100 --> 00:08:35,590 And we've developed all of these tools 126 00:08:35,590 --> 00:08:37,299 for dealing with probabilities. 127 00:08:37,299 --> 00:08:40,982 So that would go to waste if I don't divide by. 128 00:08:40,982 --> 00:08:41,944 Yes? 129 00:08:41,944 --> 00:08:42,906 AUDIENCE: Question. 130 00:08:42,906 --> 00:08:45,792 When you say that you have set numbers, 131 00:08:45,792 --> 00:08:49,250 do you assume that you have any more informations than just 132 00:08:49,250 --> 00:08:52,070 the microscopic variables GP and-- 133 00:08:52,070 --> 00:08:53,390 PROFESSOR: No. 134 00:08:53,390 --> 00:08:57,735 AUDIENCE: So how can we put a micro state in correspondence 135 00:08:57,735 --> 00:09:00,310 with a macro state if there is-- on the-- like, 136 00:09:00,310 --> 00:09:01,920 with a few variables? 137 00:09:01,920 --> 00:09:04,930 And do you need to-- from-- there's 138 00:09:04,930 --> 00:09:07,057 like five variables, defined down 139 00:09:07,057 --> 00:09:09,655 to 22 variables for all the particles? 140 00:09:09,655 --> 00:09:11,280 PROFESSOR: So that's what I was saying. 141 00:09:11,280 --> 00:09:13,792 It is not a one-to-one correspondence. 142 00:09:13,792 --> 00:09:19,100 That is, once I specify temperature, pressure, 143 00:09:19,100 --> 00:09:21,080 and the number of particles. 144 00:09:21,080 --> 00:09:21,580 OK? 145 00:09:24,075 --> 00:09:24,575 Yes? 146 00:09:24,575 --> 00:09:27,535 AUDIENCE: My question is, if you generate identical macro 147 00:09:27,535 --> 00:09:33,004 states, and create-- which macro states-- 148 00:09:33,004 --> 00:09:34,828 PROFESSOR: Yes. 149 00:09:34,828 --> 00:09:37,820 AUDIENCE: Depending on some kind of a rule on how 150 00:09:37,820 --> 00:09:39,261 you make this correspondence, you 151 00:09:39,261 --> 00:09:42,870 can get different ensemble densities, right? 152 00:09:42,870 --> 00:09:44,270 PROFESSOR: No. 153 00:09:44,270 --> 00:09:48,551 That is, if I, in principle and theoretically, 154 00:09:48,551 --> 00:09:54,180 go over the entirety of all possible macroscopic boxes 155 00:09:54,180 --> 00:09:57,125 that have these properties, I will 156 00:09:57,125 --> 00:10:00,200 be putting infinite number of points in this. 157 00:10:00,200 --> 00:10:02,950 And I will get some kind of a density. 158 00:10:02,950 --> 00:10:06,540 AUDIENCE: What if you, say, generate infinite number 159 00:10:06,540 --> 00:10:10,560 of points, but all in the case when, like, 160 00:10:10,560 --> 00:10:13,960 all molecules of gas are in right half of the box? 161 00:10:13,960 --> 00:10:15,282 PROFESSOR: OK. 162 00:10:15,282 --> 00:10:17,602 Is that a thermodynamically equilibrium state? 163 00:10:17,602 --> 00:10:19,352 AUDIENCE: Did you mention it needed to be? 164 00:10:19,352 --> 00:10:20,990 PROFESSOR: Yes. 165 00:10:20,990 --> 00:10:23,500 I said that-- I'm talking about things 166 00:10:23,500 --> 00:10:27,186 that can be described macroscopically. 167 00:10:27,186 --> 00:10:28,560 Now, the thing that you mentioned 168 00:10:28,560 --> 00:10:32,300 is actually something that I would like to work with, 169 00:10:32,300 --> 00:10:36,450 because ultimately, my goal is not only 170 00:10:36,450 --> 00:10:40,540 to describe equilibrium, but how to reach equilibrium. 171 00:10:40,540 --> 00:10:45,520 That is, I would like precisely to answer the question, what 172 00:10:45,520 --> 00:10:49,670 happens if you start in a situation where all of the gas 173 00:10:49,670 --> 00:10:53,650 is initially in one half of the room? 174 00:10:53,650 --> 00:10:55,920 And as long as there is a partition, 175 00:10:55,920 --> 00:10:59,170 that's a well-defined macroscopic state. 176 00:10:59,170 --> 00:11:01,930 And then I remove the partition. 177 00:11:01,930 --> 00:11:05,050 And suddenly, it is a non-equilibrium state. 178 00:11:05,050 --> 00:11:10,620 And presumably, over time, this gas will occupy that. 179 00:11:10,620 --> 00:11:16,200 So there is a physical process that we know happens in nature. 180 00:11:16,200 --> 00:11:19,030 And what I would like eventually to do 181 00:11:19,030 --> 00:11:22,660 is to also describe that physical process. 182 00:11:22,660 --> 00:11:24,690 So what I will do is I will start 183 00:11:24,690 --> 00:11:28,300 with the initial configuration with everybody 184 00:11:28,300 --> 00:11:30,400 in the half space. 185 00:11:30,400 --> 00:11:33,360 And I will calculate the ensemble 186 00:11:33,360 --> 00:11:35,400 that corresponds to that. 187 00:11:35,400 --> 00:11:37,270 And that's unique. 188 00:11:37,270 --> 00:11:39,670 Then I remove the partition. 189 00:11:39,670 --> 00:11:44,690 Then each member of the ensemble will follow some trajectory 190 00:11:44,690 --> 00:11:48,010 as it occupies eventually the entire box. 191 00:11:48,010 --> 00:11:52,020 And we would like to follow how that evolution takes place 192 00:11:52,020 --> 00:11:55,720 and hopefully show that you will always have, 193 00:11:55,720 --> 00:11:59,060 eventually, at the end of the day, the gas occupying 194 00:11:59,060 --> 00:12:02,080 the system uniform. 195 00:12:02,080 --> 00:12:04,080 AUDIENCE: Yeah, but just would it 196 00:12:04,080 --> 00:12:08,195 be more correct to generate many, many different micro 197 00:12:08,195 --> 00:12:12,908 states as the macro states which correspond to them? 198 00:12:12,908 --> 00:12:15,350 And how many different-- 199 00:12:15,350 --> 00:12:17,702 PROFESSOR: What rule do you use for generating 200 00:12:17,702 --> 00:12:21,040 many, many micro states? 201 00:12:21,040 --> 00:12:26,050 AUDIENCE: Like, all uniformly arbitrary perturbations 202 00:12:26,050 --> 00:12:30,610 of particles always to put them in phase space. 203 00:12:30,610 --> 00:12:39,020 And look to-- like, how many different micro states 204 00:12:39,020 --> 00:12:40,562 give rise to the same macro state? 205 00:12:40,562 --> 00:12:42,520 PROFESSOR: Oh, but you are already then talking 206 00:12:42,520 --> 00:12:45,018 about the macro state? 207 00:12:45,018 --> 00:12:48,120 AUDIENCE: A portion-- which description 208 00:12:48,120 --> 00:12:51,720 do you use as the first one to generate the second one? 209 00:12:51,720 --> 00:12:55,967 So in your point of view, let's do-- [INAUDIBLE] 210 00:12:55,967 --> 00:12:59,320 to macro states, and go to micro state. 211 00:12:59,320 --> 00:13:00,980 But can you reverse? 212 00:13:03,765 --> 00:13:04,650 PROFESSOR: OK. 213 00:13:04,650 --> 00:13:09,360 I know that the way that I'm presenting things 214 00:13:09,360 --> 00:13:14,980 will lead ultimately to a useful description of this procedure. 215 00:13:14,980 --> 00:13:16,720 You are welcome to try to come up 216 00:13:16,720 --> 00:13:18,710 with a different prescription. 217 00:13:18,710 --> 00:13:22,430 But the thing that I want to ensure that you agree 218 00:13:22,430 --> 00:13:26,710 is that the procedure that I'm describing here 219 00:13:26,710 --> 00:13:30,120 has no logical inconsistencies. 220 00:13:30,120 --> 00:13:32,450 I want to convince you of that. 221 00:13:32,450 --> 00:13:35,420 I am not saying that this is necessarily the only one. 222 00:13:35,420 --> 00:13:37,350 As far as I know, this is the only one 223 00:13:37,350 --> 00:13:38,640 that people have worked with. 224 00:13:38,640 --> 00:13:43,770 But maybe somebody can come up with a different prescription. 225 00:13:43,770 --> 00:13:45,710 So maybe there is another one. 226 00:13:45,710 --> 00:13:47,430 Maybe you can work on it. 227 00:13:47,430 --> 00:13:49,050 But I want you to, at this point, 228 00:13:49,050 --> 00:13:51,440 be convinced that this is a well-defined procedure. 229 00:13:56,180 --> 00:13:58,550 OK? 230 00:13:58,550 --> 00:14:01,646 AUDIENCE: But because it's a well-defined procedure, 231 00:14:01,646 --> 00:14:03,986 if you did exist on another planet 232 00:14:03,986 --> 00:14:07,024 or in some universe where the physics were different, 233 00:14:07,024 --> 00:14:08,619 the point is, you can use this. 234 00:14:08,619 --> 00:14:10,660 But can't you use this for information in general 235 00:14:10,660 --> 00:14:14,220 when you want to-- like, if you have-- the only requirement 236 00:14:14,220 --> 00:14:18,370 is that at a fine scale, you have 237 00:14:18,370 --> 00:14:21,640 a consistent way of describing things; and at a large scale, 238 00:14:21,640 --> 00:14:25,092 you have a way of making sense of generalizing. 239 00:14:25,092 --> 00:14:25,800 PROFESSOR: Right. 240 00:14:25,800 --> 00:14:28,070 AUDIENCE: So it's sort of like a compression 241 00:14:28,070 --> 00:14:31,420 of data, or I use [INAUDIBLE]. 242 00:14:31,420 --> 00:14:33,200 PROFESSOR: Yeah. 243 00:14:33,200 --> 00:14:38,040 Except that part of this was starting with some physics 244 00:14:38,040 --> 00:14:39,550 that we know. 245 00:14:39,550 --> 00:14:42,980 So indeed, if you were in a different universe-- 246 00:14:42,980 --> 00:14:46,013 and later on in the course, we will be in a different universe 247 00:14:46,013 --> 00:14:47,512 where the rules are not classical or 248 00:14:47,512 --> 00:14:48,303 quantum-mechanical. 249 00:14:50,289 --> 00:14:52,080 And you have to throw away this description 250 00:14:52,080 --> 00:14:53,850 of what a micro state is. 251 00:14:53,850 --> 00:14:57,150 And you can still go through the entire procedure. 252 00:14:57,150 --> 00:15:00,250 But I want to do is to follow these set of equations 253 00:15:00,250 --> 00:15:02,610 of motion and this description of micro state, 254 00:15:02,610 --> 00:15:04,600 and see where it leads us. 255 00:15:04,600 --> 00:15:08,220 And for the gas in this room, it is a perfectly good description 256 00:15:08,220 --> 00:15:10,170 of what's happened. 257 00:15:10,170 --> 00:15:10,670 Yes? 258 00:15:10,670 --> 00:15:12,542 AUDIENCE: Maybe a simpler question. 259 00:15:12,542 --> 00:15:15,023 Is big rho defined only in spaces 260 00:15:15,023 --> 00:15:17,327 where there are micro states? 261 00:15:17,327 --> 00:15:19,660 Like, is there anywhere where there isn't a micro state? 262 00:15:19,660 --> 00:15:20,890 PROFESSOR: Yes, of course. 263 00:15:20,890 --> 00:15:23,420 So if I have-- thinking about a box, 264 00:15:23,420 --> 00:15:28,100 and if I ask what is rho out here, 265 00:15:28,100 --> 00:15:30,180 I would say the answer is rho is 0. 266 00:15:30,180 --> 00:15:33,270 But if you like, you can say rho is defined only 267 00:15:33,270 --> 00:15:35,170 within this space of the box. 268 00:15:35,170 --> 00:15:38,090 So the description of the macro state 269 00:15:38,090 --> 00:15:39,980 which has something to do with the box, 270 00:15:39,980 --> 00:15:42,010 over which I am considering, will also 271 00:15:42,010 --> 00:15:45,280 limit what I can describe. 272 00:15:45,280 --> 00:15:47,290 Yes. 273 00:15:47,290 --> 00:15:54,320 And certainly, as far as if I were to change p with velocity, 274 00:15:54,320 --> 00:15:56,800 let's say, then you would say, a space 275 00:15:56,800 --> 00:16:00,580 where V is greater than speed of light is not possible. 276 00:16:00,580 --> 00:16:02,150 That's the point. 277 00:16:02,150 --> 00:16:06,580 So your rules of physics will also define implicitly 278 00:16:06,580 --> 00:16:10,270 the domain over which this is there. 279 00:16:10,270 --> 00:16:12,590 But that's all part of mechanics. 280 00:16:12,590 --> 00:16:14,980 So I'm going to assume that the mechanics part of it 281 00:16:14,980 --> 00:16:16,110 is you are comfortable. 282 00:16:16,110 --> 00:16:17,452 Yes? 283 00:16:17,452 --> 00:16:20,060 AUDIENCE: In your definition of the ensemble average, 284 00:16:20,060 --> 00:16:23,790 are you integrating over all 6N dimensions of phase space? 285 00:16:23,790 --> 00:16:24,530 PROFESSOR: Yes. 286 00:16:24,530 --> 00:16:29,530 AUDIENCE: So why would your average depend on p and q? 287 00:16:32,480 --> 00:16:33,300 If you integrate? 288 00:16:33,300 --> 00:16:36,600 PROFESSOR: The average is of a function of p and q. 289 00:16:36,600 --> 00:16:38,650 So in the same sense that, let's say, 290 00:16:38,650 --> 00:16:41,520 I have a particle of gas that is moving on, 291 00:16:41,520 --> 00:16:45,070 and I can write the symbol, p squared over 2m, 292 00:16:45,070 --> 00:16:47,100 what is this average? 293 00:16:47,100 --> 00:16:49,890 The answer will be KT over 2, for example. 294 00:16:49,890 --> 00:16:51,380 It will not depend on p. 295 00:16:51,380 --> 00:16:56,530 But the quantity that I'm averaging inside the triangle 296 00:16:56,530 --> 00:16:58,580 is a function of p and q. 297 00:17:01,144 --> 00:17:01,644 Yes? 298 00:17:01,644 --> 00:17:03,685 AUDIENCE: So if it's an integration, or basically 299 00:17:03,685 --> 00:17:04,579 the--? 300 00:17:04,579 --> 00:17:06,970 PROFESSOR: The physical limit of the problem. 301 00:17:06,970 --> 00:17:10,281 AUDIENCE: Given a macro state? 302 00:17:10,281 --> 00:17:13,130 PROFESSOR: Yes. 303 00:17:13,130 --> 00:17:16,430 So typically, we will be integrating q 304 00:17:16,430 --> 00:17:20,619 over the volume of a box, and p from minus infinity 305 00:17:20,619 --> 00:17:22,740 to infinity, because classically, 306 00:17:22,740 --> 00:17:26,814 without relativity, this is a lot. 307 00:17:26,814 --> 00:17:27,314 Yes? 308 00:17:27,314 --> 00:17:28,103 AUDIENCE: Sorry. 309 00:17:28,103 --> 00:17:36,007 So why is the [INAUDIBLE] from one end for every particle, 310 00:17:36,007 --> 00:17:37,983 instead of just scattering space, 311 00:17:37,983 --> 00:17:40,453 would you have a [INAUDIBLE]? 312 00:17:40,453 --> 00:17:42,923 Or is that the same thing? 313 00:17:42,923 --> 00:17:45,630 PROFESSOR: I am not sure I understand the question. 314 00:17:45,630 --> 00:17:50,560 So if I want to, let's say, find out just one particle that 315 00:17:50,560 --> 00:17:55,600 is somewhere in this box, so there is a probability that it 316 00:17:55,600 --> 00:17:58,180 is here, there is a probability that it is there, 317 00:17:58,180 --> 00:18:00,930 there is a probability that it is there. 318 00:18:00,930 --> 00:18:06,070 The integral of that probability over the volume of the room 319 00:18:06,070 --> 00:18:06,850 is one. 320 00:18:06,850 --> 00:18:08,150 So how do I do that? 321 00:18:08,150 --> 00:18:12,640 I have to do an integral over dx, dy, dz, probability 322 00:18:12,640 --> 00:18:14,690 as a function of x, y, and z. 323 00:18:14,690 --> 00:18:16,590 Now I just repeat that 6N times. 324 00:18:20,890 --> 00:18:21,390 OK? 325 00:18:25,390 --> 00:18:26,020 All right. 326 00:18:28,990 --> 00:18:31,440 So that's the description. 327 00:18:31,440 --> 00:18:35,660 But the first question to sort of consider 328 00:18:35,660 --> 00:18:44,940 is, what is equilibrium in this perspective? 329 00:18:55,730 --> 00:19:00,100 Now, we can even be generous, although it's 330 00:19:00,100 --> 00:19:04,370 a very questionable thing, to say that, really, 331 00:19:04,370 --> 00:19:08,320 when I sort of talk about the kinetic energy of the gas, 332 00:19:08,320 --> 00:19:14,230 maybe I can replace that by this ensemble average. 333 00:19:14,230 --> 00:19:17,240 Now, if I'm in equilibrium, the results 334 00:19:17,240 --> 00:19:20,640 should not depend as a function of time. 335 00:19:20,640 --> 00:19:24,020 So I expect that if I'm calculating things 336 00:19:24,020 --> 00:19:28,530 in equilibrium, the result of equations such as this 337 00:19:28,530 --> 00:19:33,200 should not depend on time, which is actually a problem. 338 00:19:33,200 --> 00:19:39,740 Because we know that if I take a picture of all of these things 339 00:19:39,740 --> 00:19:42,440 that I am constructing my ensemble with 340 00:19:42,440 --> 00:19:48,670 and this picture is at time t, at time t plus dt, 341 00:19:48,670 --> 00:19:51,420 all of the particles have moved around. 342 00:19:51,420 --> 00:19:56,150 And so the point that was here, the next instant of time 343 00:19:56,150 --> 00:19:58,020 is going to be somewhere else. 344 00:19:58,020 --> 00:19:59,670 This is going to be somewhere else. 345 00:19:59,670 --> 00:20:04,392 Each one of these is flowing around as a function of time. 346 00:20:04,392 --> 00:20:05,770 OK? 347 00:20:05,770 --> 00:20:08,175 So the picture that I would like you to imagine 348 00:20:08,175 --> 00:20:12,260 is you have a box, and there's a huge number of bees or flies 349 00:20:12,260 --> 00:20:14,615 or whatever your preferred insect 350 00:20:14,615 --> 00:20:17,010 is, are just moving around. 351 00:20:17,010 --> 00:20:18,860 OK? 352 00:20:18,860 --> 00:20:22,610 Now, you can sort of then take pictures of this cluster. 353 00:20:22,610 --> 00:20:25,535 And it's changing potentially as a function of time. 354 00:20:25,535 --> 00:20:28,490 And therefore, this density should potentially 355 00:20:28,490 --> 00:20:30,090 change as a function of time. 356 00:20:33,510 --> 00:20:38,590 And then this answer could potentially depend on time. 357 00:20:38,590 --> 00:20:42,540 So let's figure out how is this density changing 358 00:20:42,540 --> 00:20:44,762 as a function of time. 359 00:20:44,762 --> 00:20:49,100 And hope that ultimately, we can construct a solution 360 00:20:49,100 --> 00:20:52,970 for the equation that governs the change in density 361 00:20:52,970 --> 00:20:58,700 as a function of time that is in fact invariant in time. 362 00:20:58,700 --> 00:21:02,040 It is going back to my flies or bees or whatever, 363 00:21:02,040 --> 00:21:04,330 you can imagine a circumstance in which 364 00:21:04,330 --> 00:21:07,710 the bees are constantly moving around. 365 00:21:07,710 --> 00:21:12,020 Each individual bee is now here, then somewhere else. 366 00:21:12,020 --> 00:21:15,810 But all of your pictures have the same density of bees, 367 00:21:15,810 --> 00:21:18,160 because for every bee that left the box, 368 00:21:18,160 --> 00:21:21,220 there was another bee that came in its place. 369 00:21:21,220 --> 00:21:24,290 So one can imagine a kind of situation 370 00:21:24,290 --> 00:21:26,300 where all of these points are moving around, 371 00:21:26,300 --> 00:21:29,730 yet the density is left invariant. 372 00:21:29,730 --> 00:21:33,030 And in order to find whether such a density is possible, 373 00:21:33,030 --> 00:21:36,010 we have to first know what is the equation that 374 00:21:36,010 --> 00:21:38,494 governs the evolution of that density. 375 00:21:38,494 --> 00:21:40,410 And that is given by the Liouville's equation. 376 00:21:50,180 --> 00:22:01,420 So this governs evolution of rho with time. 377 00:22:01,420 --> 00:22:01,920 OK. 378 00:22:07,430 --> 00:22:12,060 So let's kind of blow off the picture that we had over there. 379 00:22:12,060 --> 00:22:15,310 Previously, they're all of these coordinates. 380 00:22:15,310 --> 00:22:20,260 There is some point in coordinate space 381 00:22:20,260 --> 00:22:22,140 that I am looking at. 382 00:22:22,140 --> 00:22:26,560 Let's say that the point that I am looking at is here. 383 00:22:26,560 --> 00:22:30,480 And I have constructed a box around it 384 00:22:30,480 --> 00:22:35,510 like this in the 6N-dimensional space. 385 00:22:35,510 --> 00:22:39,540 But just to be precise, I will be looking at some particular 386 00:22:39,540 --> 00:22:45,280 coordinate q alpha and the conjugate momentum p alpha. 387 00:22:45,280 --> 00:22:48,820 So this is my original point corresponds to, say, 388 00:22:48,820 --> 00:22:53,620 some specific version of q alpha p alpha. 389 00:22:53,620 --> 00:22:59,260 And that I have, in this pair of dimensions, 390 00:22:59,260 --> 00:23:02,570 created a box that in this direction 391 00:23:02,570 --> 00:23:09,110 has size dq alpha, in this direction has size dp alpha. 392 00:23:09,110 --> 00:23:11,420 OK? 393 00:23:11,420 --> 00:23:17,602 And this is the picture that I have at some time t. 394 00:23:17,602 --> 00:23:18,640 OK? 395 00:23:18,640 --> 00:23:24,030 Then I look at an instant of time that is slightly later. 396 00:23:24,030 --> 00:23:29,360 So I go to a time that is t plus dt. 397 00:23:29,360 --> 00:23:33,430 I find that the point that I had initially over here 398 00:23:33,430 --> 00:23:37,350 as the center of this box has moved around 399 00:23:37,350 --> 00:23:39,890 to some other location that I will 400 00:23:39,890 --> 00:23:42,610 call q alpha prime, p alpha prime. 401 00:23:45,560 --> 00:23:48,700 If you ask me what is q alpha prime and p alpha prime, 402 00:23:48,700 --> 00:23:50,330 I say, OK, I know that, because I 403 00:23:50,330 --> 00:23:51,990 know with equations of motion. 404 00:23:51,990 --> 00:23:54,220 If I was running this on a computer, 405 00:23:54,220 --> 00:24:03,090 I would say that q alpha prime is q alpha plus the velocity 406 00:24:03,090 --> 00:24:09,730 q alpha dot dt, plus order of dt squared, 407 00:24:09,730 --> 00:24:12,480 which hopefully, I will choose a sufficient and small time 408 00:24:12,480 --> 00:24:14,876 interval I can ignore. 409 00:24:14,876 --> 00:24:19,580 And similarly, p alpha prime would 410 00:24:19,580 --> 00:24:26,460 be p alpha plus p alpha dot dt order of dt squared. 411 00:24:29,446 --> 00:24:29,946 OK? 412 00:24:35,440 --> 00:24:39,885 So any point that was in this box will also move. 413 00:24:39,885 --> 00:24:42,015 And presumably, close-by points will 414 00:24:42,015 --> 00:24:45,140 be moving to close-by points. 415 00:24:45,140 --> 00:24:47,200 And overall, anything that was originally 416 00:24:47,200 --> 00:24:52,210 in this square actually projected from a larger 417 00:24:52,210 --> 00:24:57,830 dimensional cube will be part of a slightly distorted entity 418 00:24:57,830 --> 00:24:58,330 here. 419 00:24:58,330 --> 00:25:03,421 So everything that was here is now somewhere here. 420 00:25:03,421 --> 00:25:06,410 OK? 421 00:25:06,410 --> 00:25:14,850 I can ask, well, how wide is this new distance that I have? 422 00:25:14,850 --> 00:25:19,120 So originally, the two endpoints of the square 423 00:25:19,120 --> 00:25:22,020 were a distance dq alpha apart. 424 00:25:22,020 --> 00:25:26,430 Now they're going to be a distance dq alpha prime apart. 425 00:25:26,430 --> 00:25:28,740 What is dq alpha prime? 426 00:25:28,740 --> 00:25:40,220 I claim that dq alpha prime is whatever I had originally, 427 00:25:40,220 --> 00:25:43,930 but then the two N's are moving at slightly 428 00:25:43,930 --> 00:25:48,080 different velocities, because the velocity depends 429 00:25:48,080 --> 00:25:50,700 on where you are in phase space. 430 00:25:50,700 --> 00:25:53,590 And so the difference in velocity between these two 431 00:25:53,590 --> 00:26:01,900 points is really the derivative of the velocity with respect 432 00:26:01,900 --> 00:26:06,880 to the separation that I have between those two points, which 433 00:26:06,880 --> 00:26:10,715 is dq times dq alpha. 434 00:26:13,950 --> 00:26:16,490 And this is how much I would have 435 00:26:16,490 --> 00:26:18,910 expanded plus higher order. 436 00:26:21,694 --> 00:26:25,010 And I can apply the same thing in the momentum direction. 437 00:26:28,330 --> 00:26:40,860 The new vertical separation dp alpha prime 438 00:26:40,860 --> 00:26:44,230 is different from what it was originally, 439 00:26:44,230 --> 00:26:47,710 because the two endpoints got stretched. 440 00:26:47,710 --> 00:26:49,330 The reason they got stretched was 441 00:26:49,330 --> 00:26:53,210 because their velocities were different. 442 00:26:53,210 --> 00:26:55,270 And their difference is just the derivative 443 00:26:55,270 --> 00:26:57,640 of velocity with respect to separation 444 00:26:57,640 --> 00:26:59,800 times their small separation. 445 00:26:59,800 --> 00:27:02,650 And if I make everything small, I 446 00:27:02,650 --> 00:27:05,092 can, in principle, write higher order terms. 447 00:27:05,092 --> 00:27:06,633 But I don't have to worry about that. 448 00:27:13,378 --> 00:27:13,878 OK? 449 00:27:16,790 --> 00:27:24,940 So I can ask, what is the area of this slightly distorted 450 00:27:24,940 --> 00:27:26,440 square? 451 00:27:26,440 --> 00:27:30,680 As long as the dt is sufficiently small, 452 00:27:30,680 --> 00:27:34,010 all of the distortions, et cetera, will be small enough. 453 00:27:34,010 --> 00:27:37,930 And you can convince yourself of this. 454 00:27:37,930 --> 00:27:41,530 And what you will find is that the product 455 00:27:41,530 --> 00:27:46,250 of dq alpha prime, dp alpha prime, 456 00:27:46,250 --> 00:27:48,580 if I were to multiply these things, 457 00:27:48,580 --> 00:27:51,340 you can see that dq alpha and dp alpha 458 00:27:51,340 --> 00:27:53,125 is common to the two of them. 459 00:27:53,125 --> 00:27:57,200 So I have dq alpha, dp alpha. 460 00:27:57,200 --> 00:28:01,620 From multiplying these two terms, I will get one. 461 00:28:01,620 --> 00:28:06,610 And then I will get two terms that are order of dt, 462 00:28:06,610 --> 00:28:15,682 that I will get from dq alpha dot with respect to dq alpha, 463 00:28:15,682 --> 00:28:19,750 plus dp alpha dot with respect to dp alpha. 464 00:28:19,750 --> 00:28:23,900 And then there will be terms that are order of dt squared 465 00:28:23,900 --> 00:28:25,562 and higher order types of things. 466 00:28:28,640 --> 00:28:29,140 OK? 467 00:28:32,090 --> 00:28:38,680 So the distortion in the area of this particle, 468 00:28:38,680 --> 00:28:44,410 or cross section, is governed by something 469 00:28:44,410 --> 00:28:46,680 that is proportional to dt. 470 00:28:46,680 --> 00:28:53,170 And dq alpha dot over dq alpha plus dp alpha dot dp alpha. 471 00:28:53,170 --> 00:29:02,060 But I have formally here for what P i dot and qi dot are. 472 00:29:02,060 --> 00:29:06,460 So this is the dot notation for the time derivative 473 00:29:06,460 --> 00:29:08,580 other than the one that I was using before. 474 00:29:08,580 --> 00:29:11,435 So q alpha dot, what do I have for it? 475 00:29:11,435 --> 00:29:14,490 It is dh by dp alpha. 476 00:29:14,490 --> 00:29:20,205 So this is d by dq alpha of the H by dp alpha. 477 00:29:22,880 --> 00:29:28,620 Whereas p alpha dot, from which I have to evaluate d by dp 478 00:29:28,620 --> 00:29:35,780 alpha, p alpha dot is minus dH by dq alpha. 479 00:29:35,780 --> 00:29:37,620 So what do I have? 480 00:29:37,620 --> 00:29:43,360 I have two second derivatives that appear with opposite sign 481 00:29:43,360 --> 00:29:45,285 and hence cancel each other out. 482 00:29:49,085 --> 00:29:50,520 OK? 483 00:29:50,520 --> 00:29:57,190 So essentially, what we find is that the volume element 484 00:29:57,190 --> 00:30:01,590 is preserved under this process. 485 00:30:01,590 --> 00:30:05,970 And I can apply this to all of my directions. 486 00:30:11,240 --> 00:30:18,050 And hence, conclude that the initial volume that 487 00:30:18,050 --> 00:30:24,039 was surrounding my point is going to be preserved. 488 00:30:24,039 --> 00:30:24,538 OK? 489 00:30:27,820 --> 00:30:34,200 So what that means is that these classical equations of motion, 490 00:30:34,200 --> 00:30:40,030 the Hamiltonian equations, for this description of micro state 491 00:30:40,030 --> 00:30:43,410 that involves the coordinates and momenta 492 00:30:43,410 --> 00:30:47,330 have this nice property that they preserve volume 493 00:30:47,330 --> 00:30:50,071 of phase space as they move around. 494 00:30:50,071 --> 00:30:50,570 Yes? 495 00:30:50,570 --> 00:30:53,741 AUDIENCE: If the Hamiltonian has expontential time dependence, 496 00:30:53,741 --> 00:30:55,100 that doesn't work anymore. 497 00:30:55,100 --> 00:30:55,710 PROFESSOR: No. 498 00:30:55,710 --> 00:30:58,459 So that's why I did not put that over there. 499 00:30:58,459 --> 00:30:58,959 Yes. 500 00:31:06,070 --> 00:31:10,840 And actually, this is sometimes referred to 501 00:31:10,840 --> 00:31:14,980 as being something like an incompressible fluid. 502 00:31:24,770 --> 00:31:31,680 Because if you kind of deliver hydrodynamics for something 503 00:31:31,680 --> 00:31:34,910 like water if you regard it as incompressible, 504 00:31:34,910 --> 00:31:37,810 the velocity field has the condition 505 00:31:37,810 --> 00:31:42,050 that the divergence of the velocity is 0. 506 00:31:42,050 --> 00:31:46,440 Here, we are looking at a 6N-dimensional mention velocity 507 00:31:46,440 --> 00:31:51,790 field that is composed of q alpha dot and p alpha dot. 508 00:31:51,790 --> 00:31:55,170 And this being 0 is really the same thing 509 00:31:55,170 --> 00:32:00,190 as the divergence in this 6N-dimensional space is 0. 510 00:32:00,190 --> 00:32:04,230 And that's a property of the Hamiltonian dynamics 511 00:32:04,230 --> 00:32:04,960 that one has. 512 00:32:07,540 --> 00:32:08,040 Yes? 513 00:32:08,040 --> 00:32:09,960 AUDIENCE: Could you briefly go over 514 00:32:09,960 --> 00:32:15,240 why you have to divide by the separation 515 00:32:15,240 --> 00:32:20,126 when you expand the times between the displacement? 516 00:32:20,126 --> 00:32:23,620 PROFESSOR: Why do I have to multiply by the separation? 517 00:32:23,620 --> 00:32:24,991 AUDIENCE: Divide by. 518 00:32:24,991 --> 00:32:28,280 PROFESSOR: Where do I divide? 519 00:32:28,280 --> 00:32:30,420 AUDIENCE: dq alpha by-- 520 00:32:30,420 --> 00:32:32,180 PROFESSOR: Oh, this. 521 00:32:32,180 --> 00:32:35,804 Why do I have to take a derivative. 522 00:32:35,804 --> 00:32:39,740 So I have two points here. 523 00:32:39,740 --> 00:32:43,060 All of my points are moving in time. 524 00:32:43,060 --> 00:32:47,200 So if these things were moving with uniform velocity, 525 00:32:47,200 --> 00:32:50,360 one second later, this would have moved here, 526 00:32:50,360 --> 00:32:52,620 this would have moved the same distance, 527 00:32:52,620 --> 00:32:54,470 so that the separation between them 528 00:32:54,470 --> 00:32:56,640 would have been maintained if they 529 00:32:56,640 --> 00:32:59,070 were moving with the same velocity. 530 00:32:59,070 --> 00:33:00,935 So if you are following somebody and you 531 00:33:00,935 --> 00:33:03,650 are moving with the same velocity as them, 532 00:33:03,650 --> 00:33:06,210 thus, your separation does not change. 533 00:33:06,210 --> 00:33:11,500 But if one of you is going faster than the other one, 534 00:33:11,500 --> 00:33:14,750 then the difference in velocity will 535 00:33:14,750 --> 00:33:17,400 determine how you separate. 536 00:33:17,400 --> 00:33:20,390 And what is the difference in velocity? 537 00:33:20,390 --> 00:33:23,200 The difference in velocity depends, in this case, 538 00:33:23,200 --> 00:33:26,210 to how far apart the points are. 539 00:33:26,210 --> 00:33:29,300 So the difference between velocity here and velocity 540 00:33:29,300 --> 00:33:32,530 here is the derivative of velocity 541 00:33:32,530 --> 00:33:34,900 as a function of this coordinate. 542 00:33:34,900 --> 00:33:38,560 Derivative of velocity as a function of that coordinate 543 00:33:38,560 --> 00:33:40,145 multiplied by the separation. 544 00:33:50,520 --> 00:33:52,530 OK? 545 00:33:52,530 --> 00:33:54,960 So what does this incompressibility condition 546 00:33:54,960 --> 00:33:56,250 mean? 547 00:33:56,250 --> 00:34:01,310 It means that however many points I had over here, 548 00:34:01,310 --> 00:34:06,990 they end up in a box that has exactly the same volume, which 549 00:34:06,990 --> 00:34:13,040 means that the density is going to be the same around here 550 00:34:13,040 --> 00:34:16,050 and around this new point. 551 00:34:16,050 --> 00:34:25,420 So essentially, what we have is that the rho at the new point, 552 00:34:25,420 --> 00:34:32,630 p prime, q prime, and time, t, plus dt, is 553 00:34:32,630 --> 00:34:41,670 the same thing as the rho at the old point, p, q, at time, t. 554 00:34:41,670 --> 00:34:46,310 Again, this is the incompressibility condition. 555 00:34:46,310 --> 00:34:47,234 Now we do mathematics. 556 00:34:51,690 --> 00:34:53,510 So let's write it again. 557 00:34:53,510 --> 00:34:59,380 So I've said, in other words, that rho p, q, t 558 00:34:59,380 --> 00:35:02,510 is the same as the rho at the new point. 559 00:35:02,510 --> 00:35:05,270 What's the momentum at the new point? 560 00:35:05,270 --> 00:35:10,180 It is p plus. 561 00:35:10,180 --> 00:35:16,330 For each component, it is p alpha plus p alpha dot. 562 00:35:16,330 --> 00:35:19,810 Let's put it this way. p plus p dot 563 00:35:19,810 --> 00:35:30,118 dt, q plus q dot dt, and t plus dt. 564 00:35:30,118 --> 00:35:35,440 That is, if I look at the new location compared 565 00:35:35,440 --> 00:35:40,040 to the old location, the time changed, the position changed, 566 00:35:40,040 --> 00:35:42,870 the momentum changed. 567 00:35:42,870 --> 00:35:45,810 They all changed-- in each arguments 568 00:35:45,810 --> 00:35:47,880 changed infinitesimally by an amount that 569 00:35:47,880 --> 00:35:50,660 is proportional to dt. 570 00:35:50,660 --> 00:35:54,320 And so what I can do is I can expand this function 571 00:35:54,320 --> 00:35:56,320 to order of dt. 572 00:35:56,320 --> 00:36:01,390 So I have rho at the original point. 573 00:36:01,390 --> 00:36:03,590 So this is all mathematics. 574 00:36:03,590 --> 00:36:06,310 I just look at variation with respect 575 00:36:06,310 --> 00:36:07,930 to each one of these arguments. 576 00:36:07,930 --> 00:36:12,470 So I have a sum over alpha, p alpha 577 00:36:12,470 --> 00:36:20,910 dot d rho by dp alpha plus q alpha 578 00:36:20,910 --> 00:36:28,720 dot d rho by dq alpha plus the explicit derivative, 579 00:36:28,720 --> 00:36:30,400 d rho by dt. 580 00:36:30,400 --> 00:36:35,050 This entirety is going to be multiplied by dt. 581 00:36:35,050 --> 00:36:37,760 And then, in principle, the expansion 582 00:36:37,760 --> 00:36:39,574 would have higher order terms. 583 00:36:43,350 --> 00:36:45,260 OK? 584 00:36:45,260 --> 00:36:48,242 Now, of course, the first term vanishes. 585 00:36:48,242 --> 00:36:51,050 It is the same on both times. 586 00:36:51,050 --> 00:36:53,780 So the thing that I will have to set to 0 587 00:36:53,780 --> 00:36:55,510 is this entity over here. 588 00:36:58,270 --> 00:37:05,785 Now, quite generally, if you have a function of p, 589 00:37:05,785 --> 00:37:11,710 q, and t, you evaluate it at the old point 590 00:37:11,710 --> 00:37:14,860 and then evaluate at the new point. 591 00:37:14,860 --> 00:37:21,110 One can define what is called a total derivative. 592 00:37:21,110 --> 00:37:24,640 And just like here, the total derivative 593 00:37:24,640 --> 00:37:28,000 will come from variations of all of these arguments. 594 00:37:28,000 --> 00:37:30,470 We'll have a partial derivative with respect 595 00:37:30,470 --> 00:37:33,970 to time and partial derivatives with respect 596 00:37:33,970 --> 00:37:35,395 to any of the other arguments. 597 00:37:46,940 --> 00:37:50,320 So I wrote this to sort of make a distinction 598 00:37:50,320 --> 00:37:53,950 between the symbol that is commonly used, 599 00:37:53,950 --> 00:37:58,360 sometimes d by dt, which is straight, sometimes Df by Dt. 600 00:38:02,620 --> 00:38:09,120 And this is either a total derivative 601 00:38:09,120 --> 00:38:13,470 or a streamline derivative. 602 00:38:13,470 --> 00:38:16,580 That is, you are taking derivatives 603 00:38:16,580 --> 00:38:20,560 as you are moving along with the flow. 604 00:38:20,560 --> 00:38:25,860 And that is to be distinguished from these partial derivatives, 605 00:38:25,860 --> 00:38:29,650 which is really sitting at some point in space 606 00:38:29,650 --> 00:38:33,360 and following how, from one time instant to another time 607 00:38:33,360 --> 00:38:36,930 instant, let's say the density changes. 608 00:38:36,930 --> 00:38:40,190 So Df by Dt with a partial really 609 00:38:40,190 --> 00:38:43,240 means sit at the same point. 610 00:38:43,240 --> 00:38:48,060 Whereas this big Df by Dt means, go along with the flow 611 00:38:48,060 --> 00:38:50,880 and look at the changes. 612 00:38:50,880 --> 00:38:56,955 Now, what we have established here is that for the density, 613 00:38:56,955 --> 00:38:59,990 the density has some special character because 614 00:38:59,990 --> 00:39:03,125 of this Liouville's theorem, that the streamlined derivative 615 00:39:03,125 --> 00:39:04,500 is 0. 616 00:39:04,500 --> 00:39:09,330 So what we have is that d rho by dt is 0. 617 00:39:11,990 --> 00:39:16,530 And this d rho by dt I can also write down 618 00:39:16,530 --> 00:39:25,445 as d rho by dt plus sum over all 6N directions, d 619 00:39:25,445 --> 00:39:28,550 rho by dp alpha. 620 00:39:28,550 --> 00:39:34,870 But then I substitute for p alpha dot from here. 621 00:39:34,870 --> 00:39:36,880 p dot is minus dH by dq. 622 00:39:44,010 --> 00:39:49,460 And then I add d rho by dq alpha. 623 00:39:49,460 --> 00:39:52,740 q alpha dot is dH by dp r. 624 00:39:56,556 --> 00:39:57,056 OK? 625 00:40:01,120 --> 00:40:07,110 So then I can take this combination with the minus sign 626 00:40:07,110 --> 00:40:14,520 to the other side and write it as d rho by dt 627 00:40:14,520 --> 00:40:20,780 is something that I will call the Poisson bracket of H 628 00:40:20,780 --> 00:40:27,680 and rho, where, quite generally, if I 629 00:40:27,680 --> 00:40:31,480 have two functions in phase space 630 00:40:31,480 --> 00:40:36,530 that is defending on B and q, this scalary derivative 631 00:40:36,530 --> 00:40:42,700 the Poisson bracket is defined as the sum over all 6N 632 00:40:42,700 --> 00:40:44,710 possible variation. 633 00:40:44,710 --> 00:40:49,740 The first one with respect to q, the second one with respect 634 00:40:49,740 --> 00:40:51,100 to p. 635 00:40:51,100 --> 00:40:55,190 And then the whole thing with the opposite sign. 636 00:40:55,190 --> 00:40:58,950 dA, dP, dB, dq. 637 00:41:03,270 --> 00:41:04,895 So this is the Poisson bracket. 638 00:41:10,450 --> 00:41:12,110 And again, from the definition, you 639 00:41:12,110 --> 00:41:14,540 should be able to see immediately 640 00:41:14,540 --> 00:41:16,870 that Poisson bracket of A and B is 641 00:41:16,870 --> 00:41:24,279 minus the Poisson bracket of B and A. OK? 642 00:42:01,440 --> 00:42:06,290 Again, we ask the question that in general, I 643 00:42:06,290 --> 00:42:14,410 can construct in principle a rho of p, q, let's say, 644 00:42:14,410 --> 00:42:16,280 for an equilibrium ensemble. 645 00:42:16,280 --> 00:42:20,780 But then I did something, like I removed a partition 646 00:42:20,780 --> 00:42:23,940 in the middle of the gas, and the gas is expanding. 647 00:42:23,940 --> 00:42:28,980 And then presumably, this becomes a function of time. 648 00:42:28,980 --> 00:42:33,810 And since I know exactly how each one of the particles, 649 00:42:33,810 --> 00:42:35,680 and hence each one of the micro states 650 00:42:35,680 --> 00:42:37,740 is evolving as a function of time, 651 00:42:37,740 --> 00:42:41,200 I should be able to tell how this density in phase space 652 00:42:41,200 --> 00:42:42,430 is changing. 653 00:42:42,430 --> 00:42:46,970 So this perspective is, again, this perspective 654 00:42:46,970 --> 00:42:50,100 of looking at all of these bees that are buzzing around 655 00:42:50,100 --> 00:42:54,190 in this 6N-dimensional space, and asking the question, if I 656 00:42:54,190 --> 00:42:58,350 look at the particular point in this 6N-dimensional space, what 657 00:42:58,350 --> 00:43:00,600 is the density of bees? 658 00:43:00,600 --> 00:43:03,890 And the answer is that it is given 659 00:43:03,890 --> 00:43:08,820 by the Poisson bracket of the Hamiltonian 660 00:43:08,820 --> 00:43:11,709 that governs the evolution of each micro state 661 00:43:11,709 --> 00:43:12,750 and the density function. 662 00:43:20,940 --> 00:43:21,470 All right. 663 00:43:21,470 --> 00:43:23,880 So what does it mean? 664 00:43:23,880 --> 00:43:25,320 What can we do with this? 665 00:43:29,520 --> 00:43:32,170 Let's play around with it and look at some consequences. 666 00:43:37,870 --> 00:43:43,402 But before that, does anybody have any questions? 667 00:43:43,402 --> 00:43:44,840 OK. 668 00:43:44,840 --> 00:43:45,340 All right. 669 00:43:49,350 --> 00:43:51,300 We had something that I just erased. 670 00:43:51,300 --> 00:43:55,730 That is, if I have a function of p and q, 671 00:43:55,730 --> 00:43:58,210 let's say it's not a function of time. 672 00:43:58,210 --> 00:44:01,360 Let's say it's the kinetic energy for this system where, 673 00:44:01,360 --> 00:44:04,290 at t equals to 0, I remove the partition, 674 00:44:04,290 --> 00:44:06,280 and the particles are expanding. 675 00:44:06,280 --> 00:44:08,462 And let's say the other place you have a potential, 676 00:44:08,462 --> 00:44:11,600 so your kinetic energy on average is going to change. 677 00:44:11,600 --> 00:44:14,380 You want to know what's happening to that. 678 00:44:14,380 --> 00:44:17,980 So you calculate at each instant of time 679 00:44:17,980 --> 00:44:21,380 an ensemble average of the kinetic energy 680 00:44:21,380 --> 00:44:24,340 or any other quantity that is interesting to you. 681 00:44:24,340 --> 00:44:29,460 And your prescription for calculating an ensemble average 682 00:44:29,460 --> 00:44:36,440 is that you integrate against the density the function 683 00:44:36,440 --> 00:44:42,260 that you are proposing to look at. 684 00:44:42,260 --> 00:44:45,580 Now, in principle, we said that there 685 00:44:45,580 --> 00:44:48,620 could be situations where this is dependent on time, 686 00:44:48,620 --> 00:44:53,040 in which case, your average will also depend on time. 687 00:44:53,040 --> 00:44:57,130 And maybe you want to know how this time dependence occurs. 688 00:44:57,130 --> 00:44:59,720 How does the kinetic energy of a gas that 689 00:44:59,720 --> 00:45:04,100 is expanding into some potential change on average? 690 00:45:04,100 --> 00:45:04,600 OK. 691 00:45:04,600 --> 00:45:05,810 So let's take a look. 692 00:45:05,810 --> 00:45:07,990 This is a function of time, because, as we said, 693 00:45:07,990 --> 00:45:09,220 these go inside the average. 694 00:45:09,220 --> 00:45:12,710 So really, the only explicit variable that we have here 695 00:45:12,710 --> 00:45:13,680 is time. 696 00:45:13,680 --> 00:45:15,430 And you can ask, what is the time 697 00:45:15,430 --> 00:45:16,765 dependence of this quantity? 698 00:45:22,000 --> 00:45:24,480 OK? 699 00:45:24,480 --> 00:45:35,380 So the time dependence is obtained by doing this, 700 00:45:35,380 --> 00:45:39,950 because essentially, you would be adding things 701 00:45:39,950 --> 00:45:43,210 at different points in p, q. 702 00:45:43,210 --> 00:45:45,830 And at each point, there is a time dependence. 703 00:45:45,830 --> 00:45:49,290 And you take the derivative in time with respect 704 00:45:49,290 --> 00:45:51,860 to the contribution of that point. 705 00:45:51,860 --> 00:45:54,940 So we get something like this. 706 00:45:54,940 --> 00:45:58,420 Now you say, OK, I know what d rho by dt is. 707 00:45:58,420 --> 00:46:05,080 So this is my integration over all of the phase space. 708 00:46:05,080 --> 00:46:09,400 d rho by dt is this Poisson bracket of H and rho. 709 00:46:09,400 --> 00:46:14,400 And then I have O. OK? 710 00:46:14,400 --> 00:46:16,930 Let's write this explicitly. 711 00:46:16,930 --> 00:46:19,340 This is an integral over a whole bunch 712 00:46:19,340 --> 00:46:20,530 of coordinates and momenta. 713 00:46:26,340 --> 00:46:30,725 There is, for the Poisson bracket, 714 00:46:30,725 --> 00:46:32,690 a sum over derivatives. 715 00:46:32,690 --> 00:46:39,390 So I have a sum over alpha-- dH by dq alpha, 716 00:46:39,390 --> 00:46:48,030 d rho by dp alpha minus dH by dp alpha, d rho by dq alpha. 717 00:46:48,030 --> 00:46:50,970 And that Poisson bracket in its entirety 718 00:46:50,970 --> 00:46:57,210 then gets multiplied by this function of phase space. 719 00:46:57,210 --> 00:47:00,080 OK. 720 00:47:00,080 --> 00:47:04,475 Now, one of the mathematical manipulations 721 00:47:04,475 --> 00:47:07,760 that we will do a lot in this class. 722 00:47:07,760 --> 00:47:11,160 And that's why I do this particular step, although it's 723 00:47:11,160 --> 00:47:13,795 not really necessary to the logical progression 724 00:47:13,795 --> 00:47:17,470 that I'm following, is to remind you 725 00:47:17,470 --> 00:47:21,260 how you would do an integration by parts when you're 726 00:47:21,260 --> 00:47:23,290 faced with something like this. 727 00:47:23,290 --> 00:47:26,700 An integration by parts is applicable 728 00:47:26,700 --> 00:47:31,660 when you have variables that you are integrating that are also 729 00:47:31,660 --> 00:47:35,340 appearing as derivatives. 730 00:47:35,340 --> 00:47:39,740 And whenever you are integrating Poisson brackets, 731 00:47:39,740 --> 00:47:43,130 you will have derivatives for the Poisson bracket. 732 00:47:43,130 --> 00:47:45,360 And the integration would allow you 733 00:47:45,360 --> 00:47:48,530 to use integration by parts. 734 00:47:48,530 --> 00:47:52,130 And in particular, what I would like to do 735 00:47:52,130 --> 00:47:55,510 is to remove the derivative that acts on the densities. 736 00:47:58,060 --> 00:48:05,730 So I'm going to essentially rewrite that as 737 00:48:05,730 --> 00:48:11,090 minus an integral that involves-- again. 738 00:48:11,090 --> 00:48:14,190 I don't want to keep rewriting that thing. 739 00:48:14,190 --> 00:48:20,340 I want to basically take the density out and then 740 00:48:20,340 --> 00:48:24,070 have the derivative, which is this d by dp in the first term 741 00:48:24,070 --> 00:48:28,790 and d by dq in the second term, act on everything else. 742 00:48:28,790 --> 00:48:36,890 So in the first case, d by dp alpha will act on O 743 00:48:36,890 --> 00:48:38,540 and dH by dq alpha. 744 00:48:41,150 --> 00:48:48,260 And in the second case, d by dq alpha will act on O 745 00:48:48,260 --> 00:48:49,691 and dH by dp alpha. 746 00:48:53,459 --> 00:48:57,000 Again, there is a sum over alpha that is implicit. 747 00:48:59,916 --> 00:49:01,380 OK? 748 00:49:01,380 --> 00:49:03,080 Again, there is a minus sign. 749 00:49:03,080 --> 00:49:07,230 So every time you do this procedure, there is this. 750 00:49:07,230 --> 00:49:11,000 But every time, you also have to worry about surface terms. 751 00:49:11,000 --> 00:49:14,050 So on the surface, you would potentially 752 00:49:14,050 --> 00:49:19,850 have to evaluate things that involve rho, O, and these d 753 00:49:19,850 --> 00:49:21,305 by d derivatives. 754 00:49:25,020 --> 00:49:28,290 But let's say we are integrating over momentum 755 00:49:28,290 --> 00:49:30,650 from minus infinity to infinity. 756 00:49:30,650 --> 00:49:35,240 Then the density evaluated at infinity momenta would be 0. 757 00:49:35,240 --> 00:49:38,760 So practicality, in all cases that I can think of, 758 00:49:38,760 --> 00:49:43,280 you don't have to worry about the boundary terms. 759 00:49:43,280 --> 00:49:47,200 So then when you look at these kinds of terms, 760 00:49:47,200 --> 00:49:51,380 this d by dp alpha can either act on O. 761 00:49:51,380 --> 00:49:58,000 So I will get dO by dp alpha, dH by dq alpha. 762 00:49:58,000 --> 00:50:03,070 Or it can act on dH by d alpha. 763 00:50:03,070 --> 00:50:08,530 So I will get plus O d2 H, dp alpha dq alpha. 764 00:50:08,530 --> 00:50:11,950 And similarly, in this term, either I 765 00:50:11,950 --> 00:50:20,720 will have dO by dq alpha, dH by dp alpha, or O d2 H, dq alpha 766 00:50:20,720 --> 00:50:23,740 dp alpha. 767 00:50:23,740 --> 00:50:25,770 Once more, the second derivative terms 768 00:50:25,770 --> 00:50:31,620 of the Hamiltonian, the order is not important. 769 00:50:31,620 --> 00:50:34,600 And what is left here is this set 770 00:50:34,600 --> 00:50:39,590 of objects, which is none other than the Poisson bracket. 771 00:50:39,590 --> 00:50:44,440 So I can rewrite the whole thing as d by dt of the expectation 772 00:50:44,440 --> 00:50:47,630 value of this quantity, which potentially 773 00:50:47,630 --> 00:50:49,860 is a function of time because of the time 774 00:50:49,860 --> 00:50:53,410 dependence of my density is the same thing 775 00:50:53,410 --> 00:51:00,090 as minus an integration over the entire phase space of rho 776 00:51:00,090 --> 00:51:03,170 against this entity, which is none 777 00:51:03,170 --> 00:51:09,200 other than the Poisson bracket of H with O. 778 00:51:09,200 --> 00:51:13,830 And this integration over density of this entire space 779 00:51:13,830 --> 00:51:17,710 is just our definition of the expectation value. 780 00:51:17,710 --> 00:51:27,650 So we get that the time derivative of any quantity 781 00:51:27,650 --> 00:51:32,370 is related to the average of its Poisson bracket 782 00:51:32,370 --> 00:51:36,110 with the Hamiltonian, which is the quantity that is really 783 00:51:36,110 --> 00:51:37,437 governing time dependences. 784 00:51:40,913 --> 00:51:41,413 Yes? 785 00:51:41,413 --> 00:51:43,350 AUDIENCE: Could you explain again 786 00:51:43,350 --> 00:51:46,420 why the time derivative when N is the integral, 787 00:51:46,420 --> 00:51:48,688 it's rho as a partial derivative? 788 00:51:48,688 --> 00:51:50,140 PROFESSOR: OK. 789 00:51:50,140 --> 00:51:53,190 So suppose I'm doing a two-dimensional integral 790 00:51:53,190 --> 00:51:56,060 over p and q. 791 00:51:56,060 --> 00:52:01,010 So I have some contribution from each point in this p and q. 792 00:52:01,010 --> 00:52:05,220 And so my integral is an integral dpdq, 793 00:52:05,220 --> 00:52:08,770 something evaluated at each point in p, 794 00:52:08,770 --> 00:52:12,420 q that could potentially depend on time. 795 00:52:12,420 --> 00:52:14,600 Imagine that I discretize this. 796 00:52:14,600 --> 00:52:17,230 So I really-- if you are more comfortable, 797 00:52:17,230 --> 00:52:20,310 you can think of this as a limit of a sum. 798 00:52:20,310 --> 00:52:23,170 So this is my integral. 799 00:52:23,170 --> 00:52:26,800 If I'm interested in the time dependence of this quantity-- 800 00:52:26,800 --> 00:52:29,130 and I really depends only on time, 801 00:52:29,130 --> 00:52:32,290 because I integrated over p and q. 802 00:52:32,290 --> 00:52:34,970 So if I'm interested in something 803 00:52:34,970 --> 00:52:38,990 that is a sum of various terms, each term 804 00:52:38,990 --> 00:52:41,760 is a function of time. 805 00:52:41,760 --> 00:52:43,880 Where do I put the time dependence? 806 00:52:43,880 --> 00:52:49,990 For each term in this sum, I look at how it depends on time. 807 00:52:49,990 --> 00:52:53,805 I don't care on its points to the left and to the right. 808 00:52:59,030 --> 00:52:59,990 OK? 809 00:52:59,990 --> 00:53:04,420 Because the big D by Dt involves moving with the streamline. 810 00:53:04,420 --> 00:53:07,050 I'm not doing any moving with the streamline. 811 00:53:07,050 --> 00:53:12,350 I'm looking at each point in this two-dimensional space. 812 00:53:12,350 --> 00:53:15,730 Each point gives a contribution at that point 813 00:53:15,730 --> 00:53:17,380 that is time-dependent. 814 00:53:17,380 --> 00:53:20,580 And I take the derivative with respect to time at that point. 815 00:53:23,200 --> 00:53:23,720 Yes? 816 00:53:23,720 --> 00:53:27,672 AUDIENCE: Couldn't you say that you have function O 817 00:53:27,672 --> 00:53:30,636 as just some function of p and q, 818 00:53:30,636 --> 00:53:34,457 and its time derivative would be Poisson bracket? 819 00:53:34,457 --> 00:53:35,082 PROFESSOR: Yes. 820 00:53:35,082 --> 00:53:37,291 AUDIENCE: And does the average of the time derivative 821 00:53:37,291 --> 00:53:38,957 would be the average of Poisson bracket, 822 00:53:38,957 --> 00:53:40,826 and you don't have to go through all the-- 823 00:53:40,826 --> 00:53:41,570 PROFESSOR: No. 824 00:53:41,570 --> 00:53:43,280 But you can see the sign doesn't work. 825 00:53:45,942 --> 00:53:47,304 AUDIENCE: How come? 826 00:53:47,304 --> 00:53:48,720 PROFESSOR: [LAUGHS] Because of all 827 00:53:48,720 --> 00:53:50,390 of these manipulations, et cetera. 828 00:53:50,390 --> 00:53:54,030 So the statement that you made is manifestly incorrect. 829 00:53:54,030 --> 00:53:58,190 You can't say that the time dependence of this thing 830 00:53:58,190 --> 00:54:03,366 is the-- whatever you were saying. [LAUGHS] 831 00:54:03,866 --> 00:54:06,794 AUDIENCE: [INAUDIBLE]. 832 00:54:06,794 --> 00:54:07,510 PROFESSOR: OK. 833 00:54:07,510 --> 00:54:08,760 Let's see what you are saying. 834 00:54:08,760 --> 00:54:11,090 AUDIENCE: So Poisson bracket only 835 00:54:11,090 --> 00:54:14,402 counts for in place for averages, right? 836 00:54:14,402 --> 00:54:15,360 PROFESSOR: OK. 837 00:54:15,360 --> 00:54:17,560 So what do we have here? 838 00:54:17,560 --> 00:54:25,630 We have that dp by dt is the Poisson bracket of H and rho. 839 00:54:25,630 --> 00:54:26,630 OK? 840 00:54:26,630 --> 00:54:33,730 And we have that O is an integral of rho O. Now, 841 00:54:33,730 --> 00:54:37,460 from where do you conclude from this set of results 842 00:54:37,460 --> 00:54:44,900 that d O by dt is the average of a Poisson bracket 843 00:54:44,900 --> 00:54:47,390 that involves O and H, irrespective 844 00:54:47,390 --> 00:54:48,670 of what we do with the sign? 845 00:54:48,670 --> 00:54:53,230 AUDIENCE: Or if you look not at the average failure of O 846 00:54:53,230 --> 00:54:57,745 but at the value of O and point, and take-- I 847 00:54:57,745 --> 00:55:01,050 guess it would be streamline derivative of it. 848 00:55:01,050 --> 00:55:07,214 So that's assuming that you're just like assigning value of O 849 00:55:07,214 --> 00:55:10,570 to each point, and making power changes with time 850 00:55:10,570 --> 00:55:13,076 as this point moves across the phase space. 851 00:55:13,076 --> 00:55:13,810 PROFESSOR: OK. 852 00:55:13,810 --> 00:55:20,230 But you still have to do some bit of derivatives, et cetera, 853 00:55:20,230 --> 00:55:21,652 because-- 854 00:55:21,652 --> 00:55:24,520 AUDIENCE: But if you know that the volume of the 855 00:55:24,520 --> 00:55:28,756 in phase space is conserved, then we basically 856 00:55:28,756 --> 00:55:30,980 don't care much that the function 857 00:55:30,980 --> 00:55:34,340 O is any much different from probability density. 858 00:55:34,340 --> 00:55:35,080 PROFESSOR: OK. 859 00:55:35,080 --> 00:55:38,070 If I understand correctly, this is what you are saying. 860 00:55:38,070 --> 00:55:41,150 Is that for each representative point, 861 00:55:41,150 --> 00:55:46,190 I have an O alpha, which is a function of time. 862 00:55:46,190 --> 00:55:51,880 And then you want to say that the average of O 863 00:55:51,880 --> 00:55:57,370 is the same thing as the sum over alpha of O alpha of t's 864 00:55:57,370 --> 00:55:59,690 divided by N, something like this. 865 00:55:59,690 --> 00:56:02,150 AUDIENCE: Eh. 866 00:56:02,150 --> 00:56:08,020 Uh, I want to first calculate what does time derivative of O? 867 00:56:08,020 --> 00:56:12,406 O remains in a function of time and q and p. 868 00:56:12,406 --> 00:56:13,390 So I can calculate-- 869 00:56:13,390 --> 00:56:14,380 PROFESSOR: Yes. 870 00:56:14,380 --> 00:56:20,834 So this O alpha is a function of 871 00:56:20,834 --> 00:56:22,176 AUDIENCE: So if I said-- 872 00:56:22,176 --> 00:56:22,800 PROFESSOR: Yes. 873 00:56:22,800 --> 00:56:23,140 OK. 874 00:56:23,140 --> 00:56:23,640 Fine. 875 00:56:23,640 --> 00:56:27,292 AUDIENCE: O is a function of q and p and t, 876 00:56:27,292 --> 00:56:29,890 and I take a streamline derivative of it. 877 00:56:29,890 --> 00:56:31,616 So filter it with respect to t. 878 00:56:31,616 --> 00:56:37,500 And then I average that thing over phase space. 879 00:56:37,500 --> 00:56:39,994 And then I should get the same version-- 880 00:56:39,994 --> 00:56:40,910 PROFESSOR: You should. 881 00:56:40,910 --> 00:56:42,185 AUDIENCE: --perfectly. 882 00:56:42,185 --> 00:56:43,035 But-- 883 00:56:43,035 --> 00:56:46,890 PROFESSOR: Very quickly, I don't think so. 884 00:56:46,890 --> 00:56:49,150 Because you are already explaining things a bit 885 00:56:49,150 --> 00:56:51,455 longer than I think I went through my derivation. 886 00:56:51,455 --> 00:56:54,096 But that's fine. [LAUGHS] 887 00:56:55,004 --> 00:56:57,290 AUDIENCE: Is there any special-- 888 00:56:57,290 --> 00:56:59,600 PROFESSOR: But I agree in spirit, yes. 889 00:56:59,600 --> 00:57:02,920 That each one of these will go along its streamline, 890 00:57:02,920 --> 00:57:06,280 and you can calculate the change for each one of them. 891 00:57:06,280 --> 00:57:08,810 And then you have to do an average of this variety. 892 00:57:08,810 --> 00:57:09,705 Yes. 893 00:57:09,705 --> 00:57:15,254 AUDIENCE: [INAUDIBLE] when you talk about time derivative 894 00:57:15,254 --> 00:57:19,300 of probability density, it's Poisson bracket of H and rho. 895 00:57:19,300 --> 00:57:23,156 But when you talk about time derivative of average, 896 00:57:23,156 --> 00:57:27,330 you have to add the minus sign. 897 00:57:27,330 --> 00:57:29,840 PROFESSOR: And if you do this correctly here, 898 00:57:29,840 --> 00:57:31,134 you should get the same result. 899 00:57:33,858 --> 00:57:34,766 AUDIENCE: Oh, OK. 900 00:57:34,766 --> 00:57:35,945 PROFESSOR: Yes. 901 00:57:35,945 --> 00:57:39,005 AUDIENCE: Well, along that line, though, 902 00:57:39,005 --> 00:57:43,455 are you using the fact that phase space volume is 903 00:57:43,455 --> 00:57:47,074 incompressible to then argue that the total time 904 00:57:47,074 --> 00:57:49,900 derivative of the ensemble average 905 00:57:49,900 --> 00:57:54,045 is the same as the ensemble average of the total time 906 00:57:54,045 --> 00:57:56,520 derivative of O, or not? 907 00:58:08,326 --> 00:58:09,700 PROFESSOR: Could you repeat that? 908 00:58:09,700 --> 00:58:11,810 Mathematically, you want me to show 909 00:58:11,810 --> 00:58:16,875 that the time derivative of what quantity? 910 00:58:16,875 --> 00:58:19,515 AUDIENCE: Of the average of O. 911 00:58:19,515 --> 00:58:21,970 PROFESSOR: Of the average of O. Yes. 912 00:58:21,970 --> 00:58:24,370 AUDIENCE: Is it in any way related to the average 913 00:58:24,370 --> 00:58:27,065 of dO over dt? 914 00:58:27,065 --> 00:58:29,520 PROFESSOR: No, it's not. 915 00:58:29,520 --> 00:58:34,660 Because O-- I mean, so what do you mean by that? 916 00:58:34,660 --> 00:58:37,850 You have to be careful, because the way that I'm defining this 917 00:58:37,850 --> 00:58:41,640 here, O is a function of p and q. 918 00:58:41,640 --> 00:58:46,250 And what you want to do is to write something 919 00:58:46,250 --> 00:58:56,990 that is a sum over all representative points, divided 920 00:58:56,990 --> 00:59:00,780 by the total number, some kind of an average like this. 921 00:59:00,780 --> 00:59:04,114 And then I can define a time derivative here. 922 00:59:04,114 --> 00:59:05,030 Is that what you are-- 923 00:59:05,030 --> 00:59:06,710 AUDIENCE: Well, I mean, I was thinking 924 00:59:06,710 --> 00:59:10,230 that even if you start out with your observable being defined 925 00:59:10,230 --> 00:59:14,080 for every point in phase space, then if you were to, 926 00:59:14,080 --> 00:59:16,150 before doing any ensemble averaging, 927 00:59:16,150 --> 00:59:19,970 if you were to take the total time derivative of that, 928 00:59:19,970 --> 00:59:25,030 then you would be accounted for p dot and q dot as well, right? 929 00:59:25,030 --> 00:59:26,960 And then if you were to take the ensemble 930 00:59:26,960 --> 00:59:29,710 average of that quantity, could you 931 00:59:29,710 --> 00:59:31,420 arrive at the same result for that? 932 00:59:31,420 --> 00:59:34,570 PROFESSOR: I'm pretty sure that if you do things consistently, 933 00:59:34,570 --> 00:59:35,320 yes. 934 00:59:35,320 --> 00:59:37,230 That is, what we have done is essentially 935 00:59:37,230 --> 00:59:41,830 we started with a collection of trajectories in phase space 936 00:59:41,830 --> 00:59:45,235 and recast the result in terms of density and variables 937 00:59:45,235 --> 00:59:49,950 that are defined only as positions in phase space. 938 00:59:49,950 --> 00:59:53,200 The two descriptions completely are equivalent. 939 00:59:53,200 --> 00:59:55,810 And as long as one doesn't make a mistake, 940 00:59:55,810 --> 00:59:58,022 one can get one or the other. 941 00:59:58,022 --> 01:00:01,635 This is actually a kind of a well-known thing 942 01:00:01,635 --> 01:00:05,490 in hydrodynamics, because typically, you write down, 943 01:00:05,490 --> 01:00:09,070 in hydrodynamics, equations for density and velocity 944 01:00:09,070 --> 01:00:11,850 at each point in phase space. 945 01:00:11,850 --> 01:00:14,220 But there is an alternative description 946 01:00:14,220 --> 01:00:18,400 which we can say that there's essentially particles that are 947 01:00:18,400 --> 01:00:19,700 flowing. 948 01:00:19,700 --> 01:00:22,800 And particle that was here at this location 949 01:00:22,800 --> 01:00:25,860 is now somewhere else at some later time. 950 01:00:25,860 --> 01:00:28,820 And people have tried hard. 951 01:00:28,820 --> 01:00:30,570 And there is a consistent definition 952 01:00:30,570 --> 01:00:34,620 of hydrodynamics that follows the second perspective. 953 01:00:34,620 --> 01:00:37,700 But I haven't seen it as being practical. 954 01:00:37,700 --> 01:00:41,440 So I'm sure that everything that you guys say is correct. 955 01:00:41,440 --> 01:00:45,250 But from the experience of what I know in hydrodynamics, 956 01:00:45,250 --> 01:00:47,450 I think this is the more practical description 957 01:00:47,450 --> 01:00:49,090 that people have been using. 958 01:00:53,170 --> 01:00:56,060 OK? 959 01:00:56,060 --> 01:00:58,810 So where were we? 960 01:00:58,810 --> 01:00:59,310 OK. 961 01:01:01,920 --> 01:01:05,810 So back to our buzzing bees. 962 01:01:09,960 --> 01:01:13,240 We now have a way of looking at how 963 01:01:13,240 --> 01:01:17,070 densities and various quantities that you can calculate, 964 01:01:17,070 --> 01:01:22,670 like ensemble averages, are changing as a function of time. 965 01:01:22,670 --> 01:01:24,145 But the question that I had before 966 01:01:24,145 --> 01:01:25,270 is, what about equilibrium? 967 01:01:31,700 --> 01:01:35,170 Because the thermodynamic definition of equilibrium, 968 01:01:35,170 --> 01:01:38,550 and my whole ensemble idea, was that essentially, I 969 01:01:38,550 --> 01:01:41,920 have all of these boxes and pressure, 970 01:01:41,920 --> 01:01:44,120 volume, everything that I can think of, 971 01:01:44,120 --> 01:01:46,510 as long as I'm not doing something that's 972 01:01:46,510 --> 01:01:52,230 like opening the box, is perfectly independent of time. 973 01:01:52,230 --> 01:01:56,260 So how can I ensure that various things that I calculate 974 01:01:56,260 --> 01:01:58,170 are independent of time? 975 01:01:58,170 --> 01:02:03,640 Clearly, I can do that by having this density not really depend 976 01:02:03,640 --> 01:02:05,950 on time. 977 01:02:05,950 --> 01:02:07,830 OK? 978 01:02:07,830 --> 01:02:10,180 Now, of course, each representative point 979 01:02:10,180 --> 01:02:11,370 is moving around. 980 01:02:11,370 --> 01:02:13,210 Each micro state is moving around. 981 01:02:13,210 --> 01:02:14,710 Each bee is moving around. 982 01:02:14,710 --> 01:02:21,100 But I want the density that is characteristic of equilibrium 983 01:02:21,100 --> 01:02:23,340 be something that does not change in time. 984 01:02:26,030 --> 01:02:26,560 It's 0. 985 01:02:29,220 --> 01:02:34,830 And so if I posit that this particular function of p and q 986 01:02:34,830 --> 01:02:38,150 is something that is not changing as a function of time, 987 01:02:38,150 --> 01:02:40,900 I have to require that the Poisson bracket 988 01:02:40,900 --> 01:02:44,310 of that function of p and q with the Hamiltonian, 989 01:02:44,310 --> 01:02:48,435 which is another function of p and q, is 0. 990 01:02:48,435 --> 01:02:49,330 OK? 991 01:02:49,330 --> 01:02:52,290 So in principle, I have to go back 992 01:02:52,290 --> 01:02:57,000 to this equation over here, which is a partial differential 993 01:02:57,000 --> 01:03:03,190 equation in 6N-dimensional space and solve with equal to 0. 994 01:03:03,190 --> 01:03:06,640 Of course, rather than doing that, we will guess the answer. 995 01:03:06,640 --> 01:03:12,390 And the guess is clear from here, because all I need to do 996 01:03:12,390 --> 01:03:18,500 is to make rho equilibrium depend on coordinates in phase 997 01:03:18,500 --> 01:03:24,210 space through some functional dependence on Hamiltonian, 998 01:03:24,210 --> 01:03:28,354 which depends on the coordinates in phase space. 999 01:03:28,354 --> 01:03:29,820 OK? 1000 01:03:29,820 --> 01:03:31,730 Why does this work? 1001 01:03:31,730 --> 01:03:36,580 Because then when I take the Poisson bracket of, let's 1002 01:03:36,580 --> 01:03:45,640 say, this function of H with H, what do I have to do? 1003 01:03:45,640 --> 01:03:50,590 I have to do a sum over alpha d rho with respect 1004 01:03:50,590 --> 01:03:55,510 to, let's say-- actually, let's write it in this way. 1005 01:03:55,510 --> 01:04:02,850 I will have the dH by dp alpha from here. 1006 01:04:02,850 --> 01:04:06,770 I have to multiply by d rho by dq alpha. 1007 01:04:06,770 --> 01:04:10,510 But rho is a function of only H. So I 1008 01:04:10,510 --> 01:04:14,500 have to take a derivative of rho with respect to its argument H. 1009 01:04:14,500 --> 01:04:16,270 And I'll call that rho prime. 1010 01:04:16,270 --> 01:04:18,610 And then the derivative of the argument, 1011 01:04:18,610 --> 01:04:22,090 which is H, with respect to q alpha. 1012 01:04:22,090 --> 01:04:24,580 That would be the first term. 1013 01:04:24,580 --> 01:04:29,755 The next term would be, with the minus sign, from the H here, 1014 01:04:29,755 --> 01:04:37,520 dH by dq alpha, the derivative of rho with respect to p alpha. 1015 01:04:37,520 --> 01:04:42,460 But rho only depends on H, so I will get the derivative of rho 1016 01:04:42,460 --> 01:04:44,650 with respect to its one argument. 1017 01:04:44,650 --> 01:04:49,610 The derivative of that argument with respect to p alpha. 1018 01:04:49,610 --> 01:04:51,410 OK? 1019 01:04:51,410 --> 01:04:54,470 So you can see that up to the order of the terms that 1020 01:04:54,470 --> 01:04:59,020 are multiplying here, this is 0. 1021 01:04:59,020 --> 01:05:02,020 OK? 1022 01:05:02,020 --> 01:05:10,650 So any function I choose of H, in principle, satisfies this. 1023 01:05:10,650 --> 01:05:14,710 And this is what we will use consistently 1024 01:05:14,710 --> 01:05:17,680 all the time in statistical mechanics, 1025 01:05:17,680 --> 01:05:20,560 in depending on the ensemble that we have. 1026 01:05:20,560 --> 01:05:22,275 Like you probably know that when we 1027 01:05:22,275 --> 01:05:25,000 are in the micro canonical ensemble, 1028 01:05:25,000 --> 01:05:28,110 we look at-- in the micro canonical ensemble, 1029 01:05:28,110 --> 01:05:31,630 we'd say what the energy is. 1030 01:05:31,630 --> 01:05:35,620 And then we say that the density is 1031 01:05:35,620 --> 01:05:41,100 a delta function-- essentially zero. 1032 01:05:41,100 --> 01:05:44,820 Except places, it's the surface that 1033 01:05:44,820 --> 01:05:47,410 corresponds to having the right energy. 1034 01:05:47,410 --> 01:05:50,160 So you sort of construct in phase space 1035 01:05:50,160 --> 01:05:53,100 the surface that has the right energy. 1036 01:05:53,100 --> 01:05:56,889 So it's a function of these four. 1037 01:05:56,889 --> 01:05:58,180 So this is the micro canonical. 1038 01:06:00,790 --> 01:06:02,750 When you are in the canonical, I use 1039 01:06:02,750 --> 01:06:05,822 a rho this is proportional to e to the minus beta H 1040 01:06:05,822 --> 01:06:07,610 and other functional features. 1041 01:06:07,610 --> 01:06:10,175 So it's, again, the same idea. 1042 01:06:13,140 --> 01:06:13,640 OK? 1043 01:06:21,570 --> 01:06:25,910 That's almost but not entirely true, 1044 01:06:25,910 --> 01:06:29,894 because sometimes there are also other conserved quantities. 1045 01:06:39,640 --> 01:06:43,620 Let's say that, for example, we have a collection of particles 1046 01:06:43,620 --> 01:06:48,100 in space in a cavity that has the shape of a sphere. 1047 01:06:48,100 --> 01:06:50,740 Because of the symmetry of the problem, 1048 01:06:50,740 --> 01:06:56,300 angular momentum is going to be a conserved quantity. 1049 01:06:56,300 --> 01:06:58,570 Angular momentum you can also write 1050 01:06:58,570 --> 01:07:02,030 as some complicated function of p and q. 1051 01:07:02,030 --> 01:07:05,340 For example, p cross q summed over all of the particles. 1052 01:07:05,340 --> 01:07:10,130 But it could be some other conserved quantity. 1053 01:07:10,130 --> 01:07:13,680 So what does this conservation law mean? 1054 01:07:13,680 --> 01:07:20,410 It means that if you evaluate L for some time, 1055 01:07:20,410 --> 01:07:23,920 it is going to be the same L for the coordinates 1056 01:07:23,920 --> 01:07:26,010 and momenta at some other time. 1057 01:07:26,010 --> 01:07:29,700 Or in other words, dL by dt, which 1058 01:07:29,700 --> 01:07:34,470 you obtained by summing over all coordinates dL 1059 01:07:34,470 --> 01:07:44,370 by dp alpha, p alpha dot, plus dL by dq alpha q alpha dot. 1060 01:07:44,370 --> 01:07:46,440 Essentially, taking time derivatives 1061 01:07:46,440 --> 01:07:48,150 of all of the arguments. 1062 01:07:48,150 --> 01:07:51,780 And I did not put any explicit time dependence here. 1063 01:07:51,780 --> 01:07:57,440 And this is again sum over alpha dL by dp alpha. 1064 01:07:57,440 --> 01:08:04,670 p alpha dot is minus dH by dq alpha. 1065 01:08:04,670 --> 01:08:14,570 And dL by dq alpha, q alpha dot is dH by dp alpha. 1066 01:08:14,570 --> 01:08:17,834 So you are seeing that this is the same thing 1067 01:08:17,834 --> 01:08:24,540 as the Poisson bracket of L and H. 1068 01:08:24,540 --> 01:08:29,830 So essentially, conserved quantities which 1069 01:08:29,830 --> 01:08:33,990 are essentially functions of coordinates and momenta 1070 01:08:33,990 --> 01:08:37,680 that you calculate that don't change as a function of time 1071 01:08:37,680 --> 01:08:40,420 are also quantities that have zero Poisson 1072 01:08:40,420 --> 01:08:43,700 brackets with the Hamiltonian. 1073 01:08:43,700 --> 01:08:47,569 So if I have a conserved quantity, 1074 01:08:47,569 --> 01:08:49,414 then I have a more general solution. 1075 01:08:55,890 --> 01:09:02,140 To my d rho by dt equals to 0 requirement. 1076 01:09:02,140 --> 01:09:05,410 I could make a rho equilibrium which 1077 01:09:05,410 --> 01:09:10,563 is a function of H of p and q, as well as, say, L of p and q. 1078 01:09:14,649 --> 01:09:20,979 And when you go through the Poisson bracket process, 1079 01:09:20,979 --> 01:09:34,120 you will either be taking derivatives 1080 01:09:34,120 --> 01:09:37,870 with respect to the first argument here. 1081 01:09:37,870 --> 01:09:43,319 So you would get rho prime with respect to the first argument. 1082 01:09:43,319 --> 01:09:47,859 And then you would get the Poisson bracket of H and H. 1083 01:09:47,859 --> 01:09:50,810 Or you would be getting derivatives with respect 1084 01:09:50,810 --> 01:09:53,819 to the second argument. 1085 01:09:53,819 --> 01:09:59,280 And then you would be getting the Poisson bracket of L and H. 1086 01:09:59,280 --> 01:10:01,540 And both of them are 0. 1087 01:10:01,540 --> 01:10:04,850 By definition, L is a conserved quantity. 1088 01:10:04,850 --> 01:10:10,440 So any solution that's a function of Hamiltonian, 1089 01:10:10,440 --> 01:10:12,340 the energy of the system is conserved 1090 01:10:12,340 --> 01:10:14,570 as a function of time, as well as 1091 01:10:14,570 --> 01:10:18,190 any other conserved quantities, such as angular momentum, 1092 01:10:18,190 --> 01:10:22,200 et cetera, is certainly a valid thing. 1093 01:10:22,200 --> 01:10:28,680 So indeed, when I drew here, in the micro canonical ensemble, 1094 01:10:28,680 --> 01:10:33,020 a surface that corresponds to a constant energy, 1095 01:10:33,020 --> 01:10:36,450 well, if I am in a spherical cavity, 1096 01:10:36,450 --> 01:10:39,270 only part of that surface that corresponds 1097 01:10:39,270 --> 01:10:43,490 to the right angular momentum is going to be accessible. 1098 01:10:43,490 --> 01:10:48,870 So essentially, what I know is that if I have conserved 1099 01:10:48,870 --> 01:10:55,050 quantities, my trajectories will explore the subspace 1100 01:10:55,050 --> 01:10:59,070 that is consistent with those conservation laws. 1101 01:10:59,070 --> 01:11:05,720 And this statement here is that ultimately, those spaces that 1102 01:11:05,720 --> 01:11:09,720 correspond to the appropriate conservation law 1103 01:11:09,720 --> 01:11:11,750 are equally populated. 1104 01:11:11,750 --> 01:11:15,400 Rho is constant around. 1105 01:11:15,400 --> 01:11:20,960 So in some sense, we started with the definition of rho 1106 01:11:20,960 --> 01:11:24,460 by putting these points around and calculating 1107 01:11:24,460 --> 01:11:28,410 probability that way, which was my objective definition 1108 01:11:28,410 --> 01:11:30,100 of probability. 1109 01:11:30,100 --> 01:11:32,320 And through this Liouville theorem, 1110 01:11:32,320 --> 01:11:33,920 we have arrived at something that 1111 01:11:33,920 --> 01:11:38,770 is more consistent with the subjective assignment 1112 01:11:38,770 --> 01:11:40,040 of probability. 1113 01:11:40,040 --> 01:11:42,790 That is, the only thing that I know, 1114 01:11:42,790 --> 01:11:44,360 forgetting about the dynamics, is 1115 01:11:44,360 --> 01:11:46,240 that there are some conserved quantities, 1116 01:11:46,240 --> 01:11:49,060 such as H, angular momentum, et cetera. 1117 01:11:49,060 --> 01:11:51,990 And I say that any point in phase space that does not 1118 01:11:51,990 --> 01:11:56,600 violate those constants in this, say, micro canonical ensemble 1119 01:11:56,600 --> 01:11:58,520 would be equally populated. 1120 01:11:58,520 --> 01:12:00,860 There was a question somewhere. 1121 01:12:00,860 --> 01:12:02,314 Yes? 1122 01:12:02,314 --> 01:12:11,324 AUDIENCE: So I almost feel like the statement that the rho has 1123 01:12:11,324 --> 01:12:15,967 to not change in time is too strong, because if you go over 1124 01:12:15,967 --> 01:12:20,458 to the equation that says the rate of change 1125 01:12:20,458 --> 01:12:24,574 is observable is equal to the integral that 1126 01:12:24,574 --> 01:12:28,795 was with a Poisson bracket of rho and H, then 1127 01:12:28,795 --> 01:12:30,450 it means that for any observable, 1128 01:12:30,450 --> 01:12:33,138 it's constant in time, right? 1129 01:12:33,138 --> 01:12:34,040 PROFESSOR: Yes. 1130 01:12:34,040 --> 01:12:38,330 AUDIENCE: So rho of q means any observable we can think of, 1131 01:12:38,330 --> 01:12:40,232 its function of p and q is constant? 1132 01:12:40,232 --> 01:12:42,640 PROFESSOR: Yep. 1133 01:12:42,640 --> 01:12:43,370 Yep. 1134 01:12:43,370 --> 01:12:49,720 Because-- and that's the best thing that we 1135 01:12:49,720 --> 01:12:53,730 can think of in terms of-- because if there 1136 01:12:53,730 --> 01:12:56,660 is some observable that is time-dependent-- 1137 01:12:56,660 --> 01:12:59,655 let's say 99 observables are time-independent, 1138 01:12:59,655 --> 01:13:03,830 but one is time-dependent, and you can measure that, 1139 01:13:03,830 --> 01:13:06,140 would you say your system is in equilibrium? 1140 01:13:06,140 --> 01:13:07,104 Probably not. 1141 01:13:11,856 --> 01:13:12,356 OK? 1142 01:13:12,356 --> 01:13:14,022 AUDIENCE: It seemed like the same method 1143 01:13:14,022 --> 01:13:17,155 that you did to show that the density is a function of 1144 01:13:17,155 --> 01:13:20,850 [INAUDIBLE], or [INAUDIBLE] that it's 1145 01:13:20,850 --> 01:13:27,860 the function of any observable that's a function of q. 1146 01:13:27,860 --> 01:13:28,360 Right? 1147 01:13:28,360 --> 01:13:29,500 PROFESSOR: Observables? 1148 01:13:29,500 --> 01:13:30,530 No. 1149 01:13:30,530 --> 01:13:36,000 I mean, here, if this answer is 0, 1150 01:13:36,000 --> 01:13:39,580 it states something about this quantity averaged. 1151 01:13:39,580 --> 01:13:43,720 So if this quantity does not change as a function of time, 1152 01:13:43,720 --> 01:13:48,410 it is not a statement that H and 0 is 0. 1153 01:13:48,410 --> 01:13:52,700 A statement that H and O is 0 is different from its ensemble 1154 01:13:52,700 --> 01:13:55,330 average being 0. 1155 01:13:55,330 --> 01:13:59,450 What you can show-- and I think I have a problem set for that-- 1156 01:13:59,450 --> 01:14:04,280 is that if this statement is correct for every O 1157 01:14:04,280 --> 01:14:08,870 that the average is 0, then your rho 1158 01:14:08,870 --> 01:14:11,940 has to satisfy this theorem equals 1159 01:14:11,940 --> 01:14:15,992 to-- Poisson bracket of rho and H is equal to 0. 1160 01:14:23,860 --> 01:14:24,420 OK. 1161 01:14:24,420 --> 01:14:28,030 So now the big question is the following. 1162 01:14:28,030 --> 01:14:31,390 We arrived that the way of thinking about equilibrium 1163 01:14:31,390 --> 01:14:34,610 in a system of particles and things 1164 01:14:34,610 --> 01:14:38,000 that are this many-to-one mapping, et cetera, in terms 1165 01:14:38,000 --> 01:14:42,030 of the densities-- we arrived that the definition of what 1166 01:14:42,030 --> 01:14:44,990 the density is going to be in equilibrium. 1167 01:14:44,990 --> 01:14:49,730 But the thermodynamic statement is much, much more severe. 1168 01:14:49,730 --> 01:14:52,770 The statement, again, is that if I have a box 1169 01:14:52,770 --> 01:14:56,320 and I open the door of the box, the gas 1170 01:14:56,320 --> 01:15:01,300 expands to fill the empty space or the other part of the box. 1171 01:15:01,300 --> 01:15:04,850 And it will do so all the time. 1172 01:15:04,850 --> 01:15:07,790 Yet the equations of motion that we have over here 1173 01:15:07,790 --> 01:15:10,090 are time reversal invariant. 1174 01:15:10,090 --> 01:15:12,710 And we did not manage to remove that. 1175 01:15:12,710 --> 01:15:15,920 We can show that this Liouville equation, et cetera, 1176 01:15:15,920 --> 01:15:19,030 is also time reversal invariant. 1177 01:15:19,030 --> 01:15:21,340 So for every case, if you succeed 1178 01:15:21,340 --> 01:15:24,340 to show that there is a density that is in half of the box 1179 01:15:24,340 --> 01:15:27,610 and it expands to fill the entire box, 1180 01:15:27,610 --> 01:15:30,940 there will be a density that presumably goes the other way. 1181 01:15:30,940 --> 01:15:34,760 Because that will be also a solution of this equation. 1182 01:15:34,760 --> 01:15:39,430 So when we sort of go back from the statement of what 1183 01:15:39,430 --> 01:15:42,670 is the equilibrium solution and ask, 1184 01:15:42,670 --> 01:15:46,720 do I know that I will eventually reach this equilibrium 1185 01:15:46,720 --> 01:15:50,210 solution as a function of time, we have not shown that. 1186 01:15:50,210 --> 01:15:55,040 And we will attempt to do so next time.