WEBVTT 00:00:00.060 --> 00:00:01.780 The following content is provided 00:00:01.780 --> 00:00:04.019 under a Creative Commons license. 00:00:04.019 --> 00:00:06.870 Your support will help MIT OpenCourseWare continue 00:00:06.870 --> 00:00:10.730 to offer high-quality educational resources for free. 00:00:10.730 --> 00:00:13.330 To make a donation or view additional materials 00:00:13.330 --> 00:00:17.217 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.217 --> 00:00:17.842 at ocw.mit.edu. 00:00:29.960 --> 00:00:31.640 PROFESSOR: Let's say that I tell you 00:00:31.640 --> 00:00:37.870 that I'm interested in a gas that has some temperature. 00:00:37.870 --> 00:00:39.910 I specify, let's say, temperature. 00:00:39.910 --> 00:00:42.560 Pressure is room temperature, room pressure. 00:00:42.560 --> 00:00:45.560 And I tell you how many particles I have. 00:00:45.560 --> 00:00:50.870 So that's [INAUDIBLE] for you what the macro state is. 00:00:50.870 --> 00:00:55.690 And I want to then see what the corresponding micro state is. 00:00:55.690 --> 00:01:01.010 So I take one of these boxes, whereas this is a box 00:01:01.010 --> 00:01:05.290 that I draw in three dimensions, I can make a correspondence 00:01:05.290 --> 00:01:09.060 and draw this is 6N-dimensional coordinate space, which 00:01:09.060 --> 00:01:10.960 would be hard for me to draw. 00:01:10.960 --> 00:01:13.850 But basically, a space of six n dimensions. 00:01:13.850 --> 00:01:17.030 I figure out where the position, and the particles, 00:01:17.030 --> 00:01:18.590 and the momenta are. 00:01:18.590 --> 00:01:22.270 And I sort of find that there is a corresponding micro state 00:01:22.270 --> 00:01:25.270 that corresponds to this macro state. 00:01:25.270 --> 00:01:26.180 OK, that's fine. 00:01:26.180 --> 00:01:28.500 I made the correspondence. 00:01:28.500 --> 00:01:37.410 But the thing is that I can imagine lots and lots of boxes 00:01:37.410 --> 00:01:40.910 that have exactly the same macroscopic properties. 00:01:40.910 --> 00:01:45.660 That is, I can imagine putting here side by side 00:01:45.660 --> 00:01:48.460 a huge number of these boxes. 00:01:48.460 --> 00:01:53.230 All of them are described by exactly the same volume, 00:01:53.230 --> 00:01:55.600 pressure, temperature, for example. 00:01:55.600 --> 00:01:57.760 The same macro state. 00:01:57.760 --> 00:02:02.260 But for each one of them, when I go and find the macro state, 00:02:02.260 --> 00:02:04.090 I find that it is something else. 00:02:06.880 --> 00:02:12.210 So I will be having different micro states. 00:02:12.210 --> 00:02:15.390 So this correspondence is certainly 00:02:15.390 --> 00:02:17.980 something where there should be many, 00:02:17.980 --> 00:02:25.160 many points here that correspond to the same thermodynamic 00:02:25.160 --> 00:02:27.510 representation. 00:02:27.510 --> 00:02:31.330 So faced with that, maybe it makes sense 00:02:31.330 --> 00:02:35.610 to follow Gibbs and define an ensemble. 00:02:40.250 --> 00:02:42.950 So what we say is, we are interested 00:02:42.950 --> 00:02:46.140 in some particular macro state. 00:02:46.140 --> 00:02:49.170 We know that they correspond to many, many, many 00:02:49.170 --> 00:02:52.070 different potential micro states. 00:02:52.070 --> 00:02:57.380 Let's try to make a map of as many micro states that 00:02:57.380 --> 00:03:00.190 correspond to the same macro state. 00:03:00.190 --> 00:03:13.620 So consider n copies of the same macro state. 00:03:13.620 --> 00:03:17.370 And this would correspond to n different points 00:03:17.370 --> 00:03:21.820 that I put in this 6N-dimensional phase space. 00:03:21.820 --> 00:03:24.940 And what I can do is I can define an ensemble density. 00:03:33.440 --> 00:03:40.160 I go to a particular point in this space. 00:03:40.160 --> 00:03:44.230 So let's say I pick some point that corresponds 00:03:44.230 --> 00:03:46.510 to some set of p's and q's here. 00:03:52.370 --> 00:03:55.007 And what I do is I draw a box that 00:03:55.007 --> 00:04:01.890 is 6N-dimensional around this point. 00:04:01.890 --> 00:04:06.100 And I define a density in the vicinity 00:04:06.100 --> 00:04:08.455 of that point, as follows. 00:04:12.520 --> 00:04:16.880 Actually, yeah. 00:04:16.880 --> 00:04:23.280 What I will do is I will count how many of these points 00:04:23.280 --> 00:04:33.200 that correspond to micro states fall within this box. 00:04:33.200 --> 00:04:45.165 So at the end is the number of mu points in this box. 00:04:49.010 --> 00:04:56.900 And what I do is I divide by the total number. 00:04:56.900 --> 00:05:00.220 I expect that the result will be proportional 00:05:00.220 --> 00:05:02.220 to the volume of the box. 00:05:02.220 --> 00:05:05.650 So if I make the box bigger, I will have more. 00:05:05.650 --> 00:05:09.950 So I divide by the volume of the box. 00:05:09.950 --> 00:05:14.280 So this is, let's call d gamma is the volume of box. 00:05:19.280 --> 00:05:22.190 Of course, I have to do this in order 00:05:22.190 --> 00:05:27.430 to get a nice result by taking the limit where 00:05:27.430 --> 00:05:32.330 the number of members of the ensemble becomes quite large. 00:05:32.330 --> 00:05:37.830 And then presumably, this will give me a well-behaved density. 00:05:37.830 --> 00:05:41.870 In this limit, I guess I want to also have the size of the box 00:05:41.870 --> 00:05:44.650 go to 0. 00:05:44.650 --> 00:05:47.600 OK? 00:05:47.600 --> 00:05:50.370 Now, clearly, with the definitions 00:05:50.370 --> 00:05:54.090 that I have made, if I were to integrate 00:05:54.090 --> 00:06:05.290 this quantity against the volume d gamma, what I would get 00:06:05.290 --> 00:06:09.990 is the integral dN over N. N is, of course, a constant. 00:06:09.990 --> 00:06:12.850 And the integral of dN is the total number. 00:06:12.850 --> 00:06:15.490 So this is 1. 00:06:15.490 --> 00:06:19.410 So we find that this quantity rho that I have constructed 00:06:19.410 --> 00:06:21.180 satisfies two properties. 00:06:21.180 --> 00:06:23.020 Certainly, it is positive, because I'm 00:06:23.020 --> 00:06:24.720 counting the number of points. 00:06:24.720 --> 00:06:27.030 Secondly, it's normalized to 1. 00:06:27.030 --> 00:06:29.970 So this is a nice probability density. 00:06:29.970 --> 00:06:37.250 So this ensemble density is a probability density function 00:06:37.250 --> 00:06:41.040 in this phase space that I have defined. 00:06:41.040 --> 00:06:43.490 OK? 00:06:43.490 --> 00:06:44.620 All right. 00:06:44.620 --> 00:06:48.850 So once I have a probability, then I 00:06:48.850 --> 00:06:52.230 can calculate various things according 00:06:52.230 --> 00:06:55.260 to the rules of probability that we defined before. 00:06:55.260 --> 00:06:59.185 So for example, I can define an ensemble average. 00:07:03.000 --> 00:07:05.560 Maybe I'm interested in the kinetic energy 00:07:05.560 --> 00:07:08.320 of the particles of the gas. 00:07:08.320 --> 00:07:12.090 So there is a function O that depends 00:07:12.090 --> 00:07:15.260 on the sum of all of the p squareds. 00:07:15.260 --> 00:07:20.570 In general, I have some function of O that depends on p and q. 00:07:20.570 --> 00:07:25.350 And what I defined the ensemble average would be the average 00:07:25.350 --> 00:07:28.010 that I would calculate with this probability. 00:07:28.010 --> 00:07:31.690 Because I go over all of the points in phase space. 00:07:31.690 --> 00:07:34.900 And let me again emphasize that what I call d gamma 00:07:34.900 --> 00:07:41.230 then really is the product over all points. 00:07:41.230 --> 00:07:46.290 For each point, I have to make a volume in both momentum 00:07:46.290 --> 00:07:48.090 and in coordinate. 00:07:48.090 --> 00:07:51.650 It's a 6N-dimensional volume element. 00:07:51.650 --> 00:07:55.380 I have to multiply the probability, which 00:07:55.380 --> 00:07:59.980 is a function of p and q against this O, which 00:07:59.980 --> 00:08:04.310 is another function of p and q. 00:08:04.310 --> 00:08:05.241 Yes? 00:08:05.241 --> 00:08:07.696 AUDIENCE: Is the division by M necessary 00:08:07.696 --> 00:08:09.958 to make it into a probability density? 00:08:09.958 --> 00:08:10.583 PROFESSOR: Yes. 00:08:10.583 --> 00:08:13.544 AUDIENCE: Otherwise, you would still have a density. 00:08:13.544 --> 00:08:14.210 PROFESSOR: Yeah. 00:08:14.210 --> 00:08:18.210 When I would integrate then, I would get the total number. 00:08:18.210 --> 00:08:20.380 But the total number is up to me, 00:08:20.380 --> 00:08:22.650 how many members of the ensemble I took, 00:08:22.650 --> 00:08:25.250 it's not a very well-defined quantity. 00:08:25.250 --> 00:08:26.710 It's an arbitrary quantity. 00:08:26.710 --> 00:08:29.940 If I set it to become very large and divide by it, 00:08:29.940 --> 00:08:33.100 then I will get something that is nicely a probability. 00:08:33.100 --> 00:08:35.590 And we've developed all of these tools 00:08:35.590 --> 00:08:37.299 for dealing with probabilities. 00:08:37.299 --> 00:08:40.982 So that would go to waste if I don't divide by. 00:08:40.982 --> 00:08:41.944 Yes? 00:08:41.944 --> 00:08:42.906 AUDIENCE: Question. 00:08:42.906 --> 00:08:45.792 When you say that you have set numbers, 00:08:45.792 --> 00:08:49.250 do you assume that you have any more informations than just 00:08:49.250 --> 00:08:52.070 the microscopic variables GP and-- 00:08:52.070 --> 00:08:53.390 PROFESSOR: No. 00:08:53.390 --> 00:08:57.735 AUDIENCE: So how can we put a micro state in correspondence 00:08:57.735 --> 00:09:00.310 with a macro state if there is-- on the-- like, 00:09:00.310 --> 00:09:01.920 with a few variables? 00:09:01.920 --> 00:09:04.930 And do you need to-- from-- there's 00:09:04.930 --> 00:09:07.057 like five variables, defined down 00:09:07.057 --> 00:09:09.655 to 22 variables for all the particles? 00:09:09.655 --> 00:09:11.280 PROFESSOR: So that's what I was saying. 00:09:11.280 --> 00:09:13.792 It is not a one-to-one correspondence. 00:09:13.792 --> 00:09:19.100 That is, once I specify temperature, pressure, 00:09:19.100 --> 00:09:21.080 and the number of particles. 00:09:21.080 --> 00:09:21.580 OK? 00:09:24.075 --> 00:09:24.575 Yes? 00:09:24.575 --> 00:09:27.535 AUDIENCE: My question is, if you generate identical macro 00:09:27.535 --> 00:09:33.004 states, and create-- which macro states-- 00:09:33.004 --> 00:09:34.828 PROFESSOR: Yes. 00:09:34.828 --> 00:09:37.820 AUDIENCE: Depending on some kind of a rule on how 00:09:37.820 --> 00:09:39.261 you make this correspondence, you 00:09:39.261 --> 00:09:42.870 can get different ensemble densities, right? 00:09:42.870 --> 00:09:44.270 PROFESSOR: No. 00:09:44.270 --> 00:09:48.551 That is, if I, in principle and theoretically, 00:09:48.551 --> 00:09:54.180 go over the entirety of all possible macroscopic boxes 00:09:54.180 --> 00:09:57.125 that have these properties, I will 00:09:57.125 --> 00:10:00.200 be putting infinite number of points in this. 00:10:00.200 --> 00:10:02.950 And I will get some kind of a density. 00:10:02.950 --> 00:10:06.540 AUDIENCE: What if you, say, generate infinite number 00:10:06.540 --> 00:10:10.560 of points, but all in the case when, like, 00:10:10.560 --> 00:10:13.960 all molecules of gas are in right half of the box? 00:10:13.960 --> 00:10:15.282 PROFESSOR: OK. 00:10:15.282 --> 00:10:17.602 Is that a thermodynamically equilibrium state? 00:10:17.602 --> 00:10:19.352 AUDIENCE: Did you mention it needed to be? 00:10:19.352 --> 00:10:20.990 PROFESSOR: Yes. 00:10:20.990 --> 00:10:23.500 I said that-- I'm talking about things 00:10:23.500 --> 00:10:27.186 that can be described macroscopically. 00:10:27.186 --> 00:10:28.560 Now, the thing that you mentioned 00:10:28.560 --> 00:10:32.300 is actually something that I would like to work with, 00:10:32.300 --> 00:10:36.450 because ultimately, my goal is not only 00:10:36.450 --> 00:10:40.540 to describe equilibrium, but how to reach equilibrium. 00:10:40.540 --> 00:10:45.520 That is, I would like precisely to answer the question, what 00:10:45.520 --> 00:10:49.670 happens if you start in a situation where all of the gas 00:10:49.670 --> 00:10:53.650 is initially in one half of the room? 00:10:53.650 --> 00:10:55.920 And as long as there is a partition, 00:10:55.920 --> 00:10:59.170 that's a well-defined macroscopic state. 00:10:59.170 --> 00:11:01.930 And then I remove the partition. 00:11:01.930 --> 00:11:05.050 And suddenly, it is a non-equilibrium state. 00:11:05.050 --> 00:11:10.620 And presumably, over time, this gas will occupy that. 00:11:10.620 --> 00:11:16.200 So there is a physical process that we know happens in nature. 00:11:16.200 --> 00:11:19.030 And what I would like eventually to do 00:11:19.030 --> 00:11:22.660 is to also describe that physical process. 00:11:22.660 --> 00:11:24.690 So what I will do is I will start 00:11:24.690 --> 00:11:28.300 with the initial configuration with everybody 00:11:28.300 --> 00:11:30.400 in the half space. 00:11:30.400 --> 00:11:33.360 And I will calculate the ensemble 00:11:33.360 --> 00:11:35.400 that corresponds to that. 00:11:35.400 --> 00:11:37.270 And that's unique. 00:11:37.270 --> 00:11:39.670 Then I remove the partition. 00:11:39.670 --> 00:11:44.690 Then each member of the ensemble will follow some trajectory 00:11:44.690 --> 00:11:48.010 as it occupies eventually the entire box. 00:11:48.010 --> 00:11:52.020 And we would like to follow how that evolution takes place 00:11:52.020 --> 00:11:55.720 and hopefully show that you will always have, 00:11:55.720 --> 00:11:59.060 eventually, at the end of the day, the gas occupying 00:11:59.060 --> 00:12:02.080 the system uniform. 00:12:02.080 --> 00:12:04.080 AUDIENCE: Yeah, but just would it 00:12:04.080 --> 00:12:08.195 be more correct to generate many, many different micro 00:12:08.195 --> 00:12:12.908 states as the macro states which correspond to them? 00:12:12.908 --> 00:12:15.350 And how many different-- 00:12:15.350 --> 00:12:17.702 PROFESSOR: What rule do you use for generating 00:12:17.702 --> 00:12:21.040 many, many micro states? 00:12:21.040 --> 00:12:26.050 AUDIENCE: Like, all uniformly arbitrary perturbations 00:12:26.050 --> 00:12:30.610 of particles always to put them in phase space. 00:12:30.610 --> 00:12:39.020 And look to-- like, how many different micro states 00:12:39.020 --> 00:12:40.562 give rise to the same macro state? 00:12:40.562 --> 00:12:42.520 PROFESSOR: Oh, but you are already then talking 00:12:42.520 --> 00:12:45.018 about the macro state? 00:12:45.018 --> 00:12:48.120 AUDIENCE: A portion-- which description 00:12:48.120 --> 00:12:51.720 do you use as the first one to generate the second one? 00:12:51.720 --> 00:12:55.967 So in your point of view, let's do-- [INAUDIBLE] 00:12:55.967 --> 00:12:59.320 to macro states, and go to micro state. 00:12:59.320 --> 00:13:00.980 But can you reverse? 00:13:03.765 --> 00:13:04.650 PROFESSOR: OK. 00:13:04.650 --> 00:13:09.360 I know that the way that I'm presenting things 00:13:09.360 --> 00:13:14.980 will lead ultimately to a useful description of this procedure. 00:13:14.980 --> 00:13:16.720 You are welcome to try to come up 00:13:16.720 --> 00:13:18.710 with a different prescription. 00:13:18.710 --> 00:13:22.430 But the thing that I want to ensure that you agree 00:13:22.430 --> 00:13:26.710 is that the procedure that I'm describing here 00:13:26.710 --> 00:13:30.120 has no logical inconsistencies. 00:13:30.120 --> 00:13:32.450 I want to convince you of that. 00:13:32.450 --> 00:13:35.420 I am not saying that this is necessarily the only one. 00:13:35.420 --> 00:13:37.350 As far as I know, this is the only one 00:13:37.350 --> 00:13:38.640 that people have worked with. 00:13:38.640 --> 00:13:43.770 But maybe somebody can come up with a different prescription. 00:13:43.770 --> 00:13:45.710 So maybe there is another one. 00:13:45.710 --> 00:13:47.430 Maybe you can work on it. 00:13:47.430 --> 00:13:49.050 But I want you to, at this point, 00:13:49.050 --> 00:13:51.440 be convinced that this is a well-defined procedure. 00:13:56.180 --> 00:13:58.550 OK? 00:13:58.550 --> 00:14:01.646 AUDIENCE: But because it's a well-defined procedure, 00:14:01.646 --> 00:14:03.986 if you did exist on another planet 00:14:03.986 --> 00:14:07.024 or in some universe where the physics were different, 00:14:07.024 --> 00:14:08.619 the point is, you can use this. 00:14:08.619 --> 00:14:10.660 But can't you use this for information in general 00:14:10.660 --> 00:14:14.220 when you want to-- like, if you have-- the only requirement 00:14:14.220 --> 00:14:18.370 is that at a fine scale, you have 00:14:18.370 --> 00:14:21.640 a consistent way of describing things; and at a large scale, 00:14:21.640 --> 00:14:25.092 you have a way of making sense of generalizing. 00:14:25.092 --> 00:14:25.800 PROFESSOR: Right. 00:14:25.800 --> 00:14:28.070 AUDIENCE: So it's sort of like a compression 00:14:28.070 --> 00:14:31.420 of data, or I use [INAUDIBLE]. 00:14:31.420 --> 00:14:33.200 PROFESSOR: Yeah. 00:14:33.200 --> 00:14:38.040 Except that part of this was starting with some physics 00:14:38.040 --> 00:14:39.550 that we know. 00:14:39.550 --> 00:14:42.980 So indeed, if you were in a different universe-- 00:14:42.980 --> 00:14:46.013 and later on in the course, we will be in a different universe 00:14:46.013 --> 00:14:47.512 where the rules are not classical or 00:14:47.512 --> 00:14:48.303 quantum-mechanical. 00:14:50.289 --> 00:14:52.080 And you have to throw away this description 00:14:52.080 --> 00:14:53.850 of what a micro state is. 00:14:53.850 --> 00:14:57.150 And you can still go through the entire procedure. 00:14:57.150 --> 00:15:00.250 But I want to do is to follow these set of equations 00:15:00.250 --> 00:15:02.610 of motion and this description of micro state, 00:15:02.610 --> 00:15:04.600 and see where it leads us. 00:15:04.600 --> 00:15:08.220 And for the gas in this room, it is a perfectly good description 00:15:08.220 --> 00:15:10.170 of what's happened. 00:15:10.170 --> 00:15:10.670 Yes? 00:15:10.670 --> 00:15:12.542 AUDIENCE: Maybe a simpler question. 00:15:12.542 --> 00:15:15.023 Is big rho defined only in spaces 00:15:15.023 --> 00:15:17.327 where there are micro states? 00:15:17.327 --> 00:15:19.660 Like, is there anywhere where there isn't a micro state? 00:15:19.660 --> 00:15:20.890 PROFESSOR: Yes, of course. 00:15:20.890 --> 00:15:23.420 So if I have-- thinking about a box, 00:15:23.420 --> 00:15:28.100 and if I ask what is rho out here, 00:15:28.100 --> 00:15:30.180 I would say the answer is rho is 0. 00:15:30.180 --> 00:15:33.270 But if you like, you can say rho is defined only 00:15:33.270 --> 00:15:35.170 within this space of the box. 00:15:35.170 --> 00:15:38.090 So the description of the macro state 00:15:38.090 --> 00:15:39.980 which has something to do with the box, 00:15:39.980 --> 00:15:42.010 over which I am considering, will also 00:15:42.010 --> 00:15:45.280 limit what I can describe. 00:15:45.280 --> 00:15:47.290 Yes. 00:15:47.290 --> 00:15:54.320 And certainly, as far as if I were to change p with velocity, 00:15:54.320 --> 00:15:56.800 let's say, then you would say, a space 00:15:56.800 --> 00:16:00.580 where V is greater than speed of light is not possible. 00:16:00.580 --> 00:16:02.150 That's the point. 00:16:02.150 --> 00:16:06.580 So your rules of physics will also define implicitly 00:16:06.580 --> 00:16:10.270 the domain over which this is there. 00:16:10.270 --> 00:16:12.590 But that's all part of mechanics. 00:16:12.590 --> 00:16:14.980 So I'm going to assume that the mechanics part of it 00:16:14.980 --> 00:16:16.110 is you are comfortable. 00:16:16.110 --> 00:16:17.452 Yes? 00:16:17.452 --> 00:16:20.060 AUDIENCE: In your definition of the ensemble average, 00:16:20.060 --> 00:16:23.790 are you integrating over all 6N dimensions of phase space? 00:16:23.790 --> 00:16:24.530 PROFESSOR: Yes. 00:16:24.530 --> 00:16:29.530 AUDIENCE: So why would your average depend on p and q? 00:16:32.480 --> 00:16:33.300 If you integrate? 00:16:33.300 --> 00:16:36.600 PROFESSOR: The average is of a function of p and q. 00:16:36.600 --> 00:16:38.650 So in the same sense that, let's say, 00:16:38.650 --> 00:16:41.520 I have a particle of gas that is moving on, 00:16:41.520 --> 00:16:45.070 and I can write the symbol, p squared over 2m, 00:16:45.070 --> 00:16:47.100 what is this average? 00:16:47.100 --> 00:16:49.890 The answer will be KT over 2, for example. 00:16:49.890 --> 00:16:51.380 It will not depend on p. 00:16:51.380 --> 00:16:56.530 But the quantity that I'm averaging inside the triangle 00:16:56.530 --> 00:16:58.580 is a function of p and q. 00:17:01.144 --> 00:17:01.644 Yes? 00:17:01.644 --> 00:17:03.685 AUDIENCE: So if it's an integration, or basically 00:17:03.685 --> 00:17:04.579 the--? 00:17:04.579 --> 00:17:06.970 PROFESSOR: The physical limit of the problem. 00:17:06.970 --> 00:17:10.281 AUDIENCE: Given a macro state? 00:17:10.281 --> 00:17:13.130 PROFESSOR: Yes. 00:17:13.130 --> 00:17:16.430 So typically, we will be integrating q 00:17:16.430 --> 00:17:20.619 over the volume of a box, and p from minus infinity 00:17:20.619 --> 00:17:22.740 to infinity, because classically, 00:17:22.740 --> 00:17:26.814 without relativity, this is a lot. 00:17:26.814 --> 00:17:27.314 Yes? 00:17:27.314 --> 00:17:28.103 AUDIENCE: Sorry. 00:17:28.103 --> 00:17:36.007 So why is the [INAUDIBLE] from one end for every particle, 00:17:36.007 --> 00:17:37.983 instead of just scattering space, 00:17:37.983 --> 00:17:40.453 would you have a [INAUDIBLE]? 00:17:40.453 --> 00:17:42.923 Or is that the same thing? 00:17:42.923 --> 00:17:45.630 PROFESSOR: I am not sure I understand the question. 00:17:45.630 --> 00:17:50.560 So if I want to, let's say, find out just one particle that 00:17:50.560 --> 00:17:55.600 is somewhere in this box, so there is a probability that it 00:17:55.600 --> 00:17:58.180 is here, there is a probability that it is there, 00:17:58.180 --> 00:18:00.930 there is a probability that it is there. 00:18:00.930 --> 00:18:06.070 The integral of that probability over the volume of the room 00:18:06.070 --> 00:18:06.850 is one. 00:18:06.850 --> 00:18:08.150 So how do I do that? 00:18:08.150 --> 00:18:12.640 I have to do an integral over dx, dy, dz, probability 00:18:12.640 --> 00:18:14.690 as a function of x, y, and z. 00:18:14.690 --> 00:18:16.590 Now I just repeat that 6N times. 00:18:20.890 --> 00:18:21.390 OK? 00:18:25.390 --> 00:18:26.020 All right. 00:18:28.990 --> 00:18:31.440 So that's the description. 00:18:31.440 --> 00:18:35.660 But the first question to sort of consider 00:18:35.660 --> 00:18:44.940 is, what is equilibrium in this perspective? 00:18:55.730 --> 00:19:00.100 Now, we can even be generous, although it's 00:19:00.100 --> 00:19:04.370 a very questionable thing, to say that, really, 00:19:04.370 --> 00:19:08.320 when I sort of talk about the kinetic energy of the gas, 00:19:08.320 --> 00:19:14.230 maybe I can replace that by this ensemble average. 00:19:14.230 --> 00:19:17.240 Now, if I'm in equilibrium, the results 00:19:17.240 --> 00:19:20.640 should not depend as a function of time. 00:19:20.640 --> 00:19:24.020 So I expect that if I'm calculating things 00:19:24.020 --> 00:19:28.530 in equilibrium, the result of equations such as this 00:19:28.530 --> 00:19:33.200 should not depend on time, which is actually a problem. 00:19:33.200 --> 00:19:39.740 Because we know that if I take a picture of all of these things 00:19:39.740 --> 00:19:42.440 that I am constructing my ensemble with 00:19:42.440 --> 00:19:48.670 and this picture is at time t, at time t plus dt, 00:19:48.670 --> 00:19:51.420 all of the particles have moved around. 00:19:51.420 --> 00:19:56.150 And so the point that was here, the next instant of time 00:19:56.150 --> 00:19:58.020 is going to be somewhere else. 00:19:58.020 --> 00:19:59.670 This is going to be somewhere else. 00:19:59.670 --> 00:20:04.392 Each one of these is flowing around as a function of time. 00:20:04.392 --> 00:20:05.770 OK? 00:20:05.770 --> 00:20:08.175 So the picture that I would like you to imagine 00:20:08.175 --> 00:20:12.260 is you have a box, and there's a huge number of bees or flies 00:20:12.260 --> 00:20:14.615 or whatever your preferred insect 00:20:14.615 --> 00:20:17.010 is, are just moving around. 00:20:17.010 --> 00:20:18.860 OK? 00:20:18.860 --> 00:20:22.610 Now, you can sort of then take pictures of this cluster. 00:20:22.610 --> 00:20:25.535 And it's changing potentially as a function of time. 00:20:25.535 --> 00:20:28.490 And therefore, this density should potentially 00:20:28.490 --> 00:20:30.090 change as a function of time. 00:20:33.510 --> 00:20:38.590 And then this answer could potentially depend on time. 00:20:38.590 --> 00:20:42.540 So let's figure out how is this density changing 00:20:42.540 --> 00:20:44.762 as a function of time. 00:20:44.762 --> 00:20:49.100 And hope that ultimately, we can construct a solution 00:20:49.100 --> 00:20:52.970 for the equation that governs the change in density 00:20:52.970 --> 00:20:58.700 as a function of time that is in fact invariant in time. 00:20:58.700 --> 00:21:02.040 It is going back to my flies or bees or whatever, 00:21:02.040 --> 00:21:04.330 you can imagine a circumstance in which 00:21:04.330 --> 00:21:07.710 the bees are constantly moving around. 00:21:07.710 --> 00:21:12.020 Each individual bee is now here, then somewhere else. 00:21:12.020 --> 00:21:15.810 But all of your pictures have the same density of bees, 00:21:15.810 --> 00:21:18.160 because for every bee that left the box, 00:21:18.160 --> 00:21:21.220 there was another bee that came in its place. 00:21:21.220 --> 00:21:24.290 So one can imagine a kind of situation 00:21:24.290 --> 00:21:26.300 where all of these points are moving around, 00:21:26.300 --> 00:21:29.730 yet the density is left invariant. 00:21:29.730 --> 00:21:33.030 And in order to find whether such a density is possible, 00:21:33.030 --> 00:21:36.010 we have to first know what is the equation that 00:21:36.010 --> 00:21:38.494 governs the evolution of that density. 00:21:38.494 --> 00:21:40.410 And that is given by the Liouville's equation. 00:21:50.180 --> 00:22:01.420 So this governs evolution of rho with time. 00:22:01.420 --> 00:22:01.920 OK. 00:22:07.430 --> 00:22:12.060 So let's kind of blow off the picture that we had over there. 00:22:12.060 --> 00:22:15.310 Previously, they're all of these coordinates. 00:22:15.310 --> 00:22:20.260 There is some point in coordinate space 00:22:20.260 --> 00:22:22.140 that I am looking at. 00:22:22.140 --> 00:22:26.560 Let's say that the point that I am looking at is here. 00:22:26.560 --> 00:22:30.480 And I have constructed a box around it 00:22:30.480 --> 00:22:35.510 like this in the 6N-dimensional space. 00:22:35.510 --> 00:22:39.540 But just to be precise, I will be looking at some particular 00:22:39.540 --> 00:22:45.280 coordinate q alpha and the conjugate momentum p alpha. 00:22:45.280 --> 00:22:48.820 So this is my original point corresponds to, say, 00:22:48.820 --> 00:22:53.620 some specific version of q alpha p alpha. 00:22:53.620 --> 00:22:59.260 And that I have, in this pair of dimensions, 00:22:59.260 --> 00:23:02.570 created a box that in this direction 00:23:02.570 --> 00:23:09.110 has size dq alpha, in this direction has size dp alpha. 00:23:09.110 --> 00:23:11.420 OK? 00:23:11.420 --> 00:23:17.602 And this is the picture that I have at some time t. 00:23:17.602 --> 00:23:18.640 OK? 00:23:18.640 --> 00:23:24.030 Then I look at an instant of time that is slightly later. 00:23:24.030 --> 00:23:29.360 So I go to a time that is t plus dt. 00:23:29.360 --> 00:23:33.430 I find that the point that I had initially over here 00:23:33.430 --> 00:23:37.350 as the center of this box has moved around 00:23:37.350 --> 00:23:39.890 to some other location that I will 00:23:39.890 --> 00:23:42.610 call q alpha prime, p alpha prime. 00:23:45.560 --> 00:23:48.700 If you ask me what is q alpha prime and p alpha prime, 00:23:48.700 --> 00:23:50.330 I say, OK, I know that, because I 00:23:50.330 --> 00:23:51.990 know with equations of motion. 00:23:51.990 --> 00:23:54.220 If I was running this on a computer, 00:23:54.220 --> 00:24:03.090 I would say that q alpha prime is q alpha plus the velocity 00:24:03.090 --> 00:24:09.730 q alpha dot dt, plus order of dt squared, 00:24:09.730 --> 00:24:12.480 which hopefully, I will choose a sufficient and small time 00:24:12.480 --> 00:24:14.876 interval I can ignore. 00:24:14.876 --> 00:24:19.580 And similarly, p alpha prime would 00:24:19.580 --> 00:24:26.460 be p alpha plus p alpha dot dt order of dt squared. 00:24:29.446 --> 00:24:29.946 OK? 00:24:35.440 --> 00:24:39.885 So any point that was in this box will also move. 00:24:39.885 --> 00:24:42.015 And presumably, close-by points will 00:24:42.015 --> 00:24:45.140 be moving to close-by points. 00:24:45.140 --> 00:24:47.200 And overall, anything that was originally 00:24:47.200 --> 00:24:52.210 in this square actually projected from a larger 00:24:52.210 --> 00:24:57.830 dimensional cube will be part of a slightly distorted entity 00:24:57.830 --> 00:24:58.330 here. 00:24:58.330 --> 00:25:03.421 So everything that was here is now somewhere here. 00:25:03.421 --> 00:25:06.410 OK? 00:25:06.410 --> 00:25:14.850 I can ask, well, how wide is this new distance that I have? 00:25:14.850 --> 00:25:19.120 So originally, the two endpoints of the square 00:25:19.120 --> 00:25:22.020 were a distance dq alpha apart. 00:25:22.020 --> 00:25:26.430 Now they're going to be a distance dq alpha prime apart. 00:25:26.430 --> 00:25:28.740 What is dq alpha prime? 00:25:28.740 --> 00:25:40.220 I claim that dq alpha prime is whatever I had originally, 00:25:40.220 --> 00:25:43.930 but then the two N's are moving at slightly 00:25:43.930 --> 00:25:48.080 different velocities, because the velocity depends 00:25:48.080 --> 00:25:50.700 on where you are in phase space. 00:25:50.700 --> 00:25:53.590 And so the difference in velocity between these two 00:25:53.590 --> 00:26:01.900 points is really the derivative of the velocity with respect 00:26:01.900 --> 00:26:06.880 to the separation that I have between those two points, which 00:26:06.880 --> 00:26:10.715 is dq times dq alpha. 00:26:13.950 --> 00:26:16.490 And this is how much I would have 00:26:16.490 --> 00:26:18.910 expanded plus higher order. 00:26:21.694 --> 00:26:25.010 And I can apply the same thing in the momentum direction. 00:26:28.330 --> 00:26:40.860 The new vertical separation dp alpha prime 00:26:40.860 --> 00:26:44.230 is different from what it was originally, 00:26:44.230 --> 00:26:47.710 because the two endpoints got stretched. 00:26:47.710 --> 00:26:49.330 The reason they got stretched was 00:26:49.330 --> 00:26:53.210 because their velocities were different. 00:26:53.210 --> 00:26:55.270 And their difference is just the derivative 00:26:55.270 --> 00:26:57.640 of velocity with respect to separation 00:26:57.640 --> 00:26:59.800 times their small separation. 00:26:59.800 --> 00:27:02.650 And if I make everything small, I 00:27:02.650 --> 00:27:05.092 can, in principle, write higher order terms. 00:27:05.092 --> 00:27:06.633 But I don't have to worry about that. 00:27:13.378 --> 00:27:13.878 OK? 00:27:16.790 --> 00:27:24.940 So I can ask, what is the area of this slightly distorted 00:27:24.940 --> 00:27:26.440 square? 00:27:26.440 --> 00:27:30.680 As long as the dt is sufficiently small, 00:27:30.680 --> 00:27:34.010 all of the distortions, et cetera, will be small enough. 00:27:34.010 --> 00:27:37.930 And you can convince yourself of this. 00:27:37.930 --> 00:27:41.530 And what you will find is that the product 00:27:41.530 --> 00:27:46.250 of dq alpha prime, dp alpha prime, 00:27:46.250 --> 00:27:48.580 if I were to multiply these things, 00:27:48.580 --> 00:27:51.340 you can see that dq alpha and dp alpha 00:27:51.340 --> 00:27:53.125 is common to the two of them. 00:27:53.125 --> 00:27:57.200 So I have dq alpha, dp alpha. 00:27:57.200 --> 00:28:01.620 From multiplying these two terms, I will get one. 00:28:01.620 --> 00:28:06.610 And then I will get two terms that are order of dt, 00:28:06.610 --> 00:28:15.682 that I will get from dq alpha dot with respect to dq alpha, 00:28:15.682 --> 00:28:19.750 plus dp alpha dot with respect to dp alpha. 00:28:19.750 --> 00:28:23.900 And then there will be terms that are order of dt squared 00:28:23.900 --> 00:28:25.562 and higher order types of things. 00:28:28.640 --> 00:28:29.140 OK? 00:28:32.090 --> 00:28:38.680 So the distortion in the area of this particle, 00:28:38.680 --> 00:28:44.410 or cross section, is governed by something 00:28:44.410 --> 00:28:46.680 that is proportional to dt. 00:28:46.680 --> 00:28:53.170 And dq alpha dot over dq alpha plus dp alpha dot dp alpha. 00:28:53.170 --> 00:29:02.060 But I have formally here for what P i dot and qi dot are. 00:29:02.060 --> 00:29:06.460 So this is the dot notation for the time derivative 00:29:06.460 --> 00:29:08.580 other than the one that I was using before. 00:29:08.580 --> 00:29:11.435 So q alpha dot, what do I have for it? 00:29:11.435 --> 00:29:14.490 It is dh by dp alpha. 00:29:14.490 --> 00:29:20.205 So this is d by dq alpha of the H by dp alpha. 00:29:22.880 --> 00:29:28.620 Whereas p alpha dot, from which I have to evaluate d by dp 00:29:28.620 --> 00:29:35.780 alpha, p alpha dot is minus dH by dq alpha. 00:29:35.780 --> 00:29:37.620 So what do I have? 00:29:37.620 --> 00:29:43.360 I have two second derivatives that appear with opposite sign 00:29:43.360 --> 00:29:45.285 and hence cancel each other out. 00:29:49.085 --> 00:29:50.520 OK? 00:29:50.520 --> 00:29:57.190 So essentially, what we find is that the volume element 00:29:57.190 --> 00:30:01.590 is preserved under this process. 00:30:01.590 --> 00:30:05.970 And I can apply this to all of my directions. 00:30:11.240 --> 00:30:18.050 And hence, conclude that the initial volume that 00:30:18.050 --> 00:30:24.039 was surrounding my point is going to be preserved. 00:30:24.039 --> 00:30:24.538 OK? 00:30:27.820 --> 00:30:34.200 So what that means is that these classical equations of motion, 00:30:34.200 --> 00:30:40.030 the Hamiltonian equations, for this description of micro state 00:30:40.030 --> 00:30:43.410 that involves the coordinates and momenta 00:30:43.410 --> 00:30:47.330 have this nice property that they preserve volume 00:30:47.330 --> 00:30:50.071 of phase space as they move around. 00:30:50.071 --> 00:30:50.570 Yes? 00:30:50.570 --> 00:30:53.741 AUDIENCE: If the Hamiltonian has expontential time dependence, 00:30:53.741 --> 00:30:55.100 that doesn't work anymore. 00:30:55.100 --> 00:30:55.710 PROFESSOR: No. 00:30:55.710 --> 00:30:58.459 So that's why I did not put that over there. 00:30:58.459 --> 00:30:58.959 Yes. 00:31:06.070 --> 00:31:10.840 And actually, this is sometimes referred to 00:31:10.840 --> 00:31:14.980 as being something like an incompressible fluid. 00:31:24.770 --> 00:31:31.680 Because if you kind of deliver hydrodynamics for something 00:31:31.680 --> 00:31:34.910 like water if you regard it as incompressible, 00:31:34.910 --> 00:31:37.810 the velocity field has the condition 00:31:37.810 --> 00:31:42.050 that the divergence of the velocity is 0. 00:31:42.050 --> 00:31:46.440 Here, we are looking at a 6N-dimensional mention velocity 00:31:46.440 --> 00:31:51.790 field that is composed of q alpha dot and p alpha dot. 00:31:51.790 --> 00:31:55.170 And this being 0 is really the same thing 00:31:55.170 --> 00:32:00.190 as the divergence in this 6N-dimensional space is 0. 00:32:00.190 --> 00:32:04.230 And that's a property of the Hamiltonian dynamics 00:32:04.230 --> 00:32:04.960 that one has. 00:32:07.540 --> 00:32:08.040 Yes? 00:32:08.040 --> 00:32:09.960 AUDIENCE: Could you briefly go over 00:32:09.960 --> 00:32:15.240 why you have to divide by the separation 00:32:15.240 --> 00:32:20.126 when you expand the times between the displacement? 00:32:20.126 --> 00:32:23.620 PROFESSOR: Why do I have to multiply by the separation? 00:32:23.620 --> 00:32:24.991 AUDIENCE: Divide by. 00:32:24.991 --> 00:32:28.280 PROFESSOR: Where do I divide? 00:32:28.280 --> 00:32:30.420 AUDIENCE: dq alpha by-- 00:32:30.420 --> 00:32:32.180 PROFESSOR: Oh, this. 00:32:32.180 --> 00:32:35.804 Why do I have to take a derivative. 00:32:35.804 --> 00:32:39.740 So I have two points here. 00:32:39.740 --> 00:32:43.060 All of my points are moving in time. 00:32:43.060 --> 00:32:47.200 So if these things were moving with uniform velocity, 00:32:47.200 --> 00:32:50.360 one second later, this would have moved here, 00:32:50.360 --> 00:32:52.620 this would have moved the same distance, 00:32:52.620 --> 00:32:54.470 so that the separation between them 00:32:54.470 --> 00:32:56.640 would have been maintained if they 00:32:56.640 --> 00:32:59.070 were moving with the same velocity. 00:32:59.070 --> 00:33:00.935 So if you are following somebody and you 00:33:00.935 --> 00:33:03.650 are moving with the same velocity as them, 00:33:03.650 --> 00:33:06.210 thus, your separation does not change. 00:33:06.210 --> 00:33:11.500 But if one of you is going faster than the other one, 00:33:11.500 --> 00:33:14.750 then the difference in velocity will 00:33:14.750 --> 00:33:17.400 determine how you separate. 00:33:17.400 --> 00:33:20.390 And what is the difference in velocity? 00:33:20.390 --> 00:33:23.200 The difference in velocity depends, in this case, 00:33:23.200 --> 00:33:26.210 to how far apart the points are. 00:33:26.210 --> 00:33:29.300 So the difference between velocity here and velocity 00:33:29.300 --> 00:33:32.530 here is the derivative of velocity 00:33:32.530 --> 00:33:34.900 as a function of this coordinate. 00:33:34.900 --> 00:33:38.560 Derivative of velocity as a function of that coordinate 00:33:38.560 --> 00:33:40.145 multiplied by the separation. 00:33:50.520 --> 00:33:52.530 OK? 00:33:52.530 --> 00:33:54.960 So what does this incompressibility condition 00:33:54.960 --> 00:33:56.250 mean? 00:33:56.250 --> 00:34:01.310 It means that however many points I had over here, 00:34:01.310 --> 00:34:06.990 they end up in a box that has exactly the same volume, which 00:34:06.990 --> 00:34:13.040 means that the density is going to be the same around here 00:34:13.040 --> 00:34:16.050 and around this new point. 00:34:16.050 --> 00:34:25.420 So essentially, what we have is that the rho at the new point, 00:34:25.420 --> 00:34:32.630 p prime, q prime, and time, t, plus dt, is 00:34:32.630 --> 00:34:41.670 the same thing as the rho at the old point, p, q, at time, t. 00:34:41.670 --> 00:34:46.310 Again, this is the incompressibility condition. 00:34:46.310 --> 00:34:47.234 Now we do mathematics. 00:34:51.690 --> 00:34:53.510 So let's write it again. 00:34:53.510 --> 00:34:59.380 So I've said, in other words, that rho p, q, t 00:34:59.380 --> 00:35:02.510 is the same as the rho at the new point. 00:35:02.510 --> 00:35:05.270 What's the momentum at the new point? 00:35:05.270 --> 00:35:10.180 It is p plus. 00:35:10.180 --> 00:35:16.330 For each component, it is p alpha plus p alpha dot. 00:35:16.330 --> 00:35:19.810 Let's put it this way. p plus p dot 00:35:19.810 --> 00:35:30.118 dt, q plus q dot dt, and t plus dt. 00:35:30.118 --> 00:35:35.440 That is, if I look at the new location compared 00:35:35.440 --> 00:35:40.040 to the old location, the time changed, the position changed, 00:35:40.040 --> 00:35:42.870 the momentum changed. 00:35:42.870 --> 00:35:45.810 They all changed-- in each arguments 00:35:45.810 --> 00:35:47.880 changed infinitesimally by an amount that 00:35:47.880 --> 00:35:50.660 is proportional to dt. 00:35:50.660 --> 00:35:54.320 And so what I can do is I can expand this function 00:35:54.320 --> 00:35:56.320 to order of dt. 00:35:56.320 --> 00:36:01.390 So I have rho at the original point. 00:36:01.390 --> 00:36:03.590 So this is all mathematics. 00:36:03.590 --> 00:36:06.310 I just look at variation with respect 00:36:06.310 --> 00:36:07.930 to each one of these arguments. 00:36:07.930 --> 00:36:12.470 So I have a sum over alpha, p alpha 00:36:12.470 --> 00:36:20.910 dot d rho by dp alpha plus q alpha 00:36:20.910 --> 00:36:28.720 dot d rho by dq alpha plus the explicit derivative, 00:36:28.720 --> 00:36:30.400 d rho by dt. 00:36:30.400 --> 00:36:35.050 This entirety is going to be multiplied by dt. 00:36:35.050 --> 00:36:37.760 And then, in principle, the expansion 00:36:37.760 --> 00:36:39.574 would have higher order terms. 00:36:43.350 --> 00:36:45.260 OK? 00:36:45.260 --> 00:36:48.242 Now, of course, the first term vanishes. 00:36:48.242 --> 00:36:51.050 It is the same on both times. 00:36:51.050 --> 00:36:53.780 So the thing that I will have to set to 0 00:36:53.780 --> 00:36:55.510 is this entity over here. 00:36:58.270 --> 00:37:05.785 Now, quite generally, if you have a function of p, 00:37:05.785 --> 00:37:11.710 q, and t, you evaluate it at the old point 00:37:11.710 --> 00:37:14.860 and then evaluate at the new point. 00:37:14.860 --> 00:37:21.110 One can define what is called a total derivative. 00:37:21.110 --> 00:37:24.640 And just like here, the total derivative 00:37:24.640 --> 00:37:28.000 will come from variations of all of these arguments. 00:37:28.000 --> 00:37:30.470 We'll have a partial derivative with respect 00:37:30.470 --> 00:37:33.970 to time and partial derivatives with respect 00:37:33.970 --> 00:37:35.395 to any of the other arguments. 00:37:46.940 --> 00:37:50.320 So I wrote this to sort of make a distinction 00:37:50.320 --> 00:37:53.950 between the symbol that is commonly used, 00:37:53.950 --> 00:37:58.360 sometimes d by dt, which is straight, sometimes Df by Dt. 00:38:02.620 --> 00:38:09.120 And this is either a total derivative 00:38:09.120 --> 00:38:13.470 or a streamline derivative. 00:38:13.470 --> 00:38:16.580 That is, you are taking derivatives 00:38:16.580 --> 00:38:20.560 as you are moving along with the flow. 00:38:20.560 --> 00:38:25.860 And that is to be distinguished from these partial derivatives, 00:38:25.860 --> 00:38:29.650 which is really sitting at some point in space 00:38:29.650 --> 00:38:33.360 and following how, from one time instant to another time 00:38:33.360 --> 00:38:36.930 instant, let's say the density changes. 00:38:36.930 --> 00:38:40.190 So Df by Dt with a partial really 00:38:40.190 --> 00:38:43.240 means sit at the same point. 00:38:43.240 --> 00:38:48.060 Whereas this big Df by Dt means, go along with the flow 00:38:48.060 --> 00:38:50.880 and look at the changes. 00:38:50.880 --> 00:38:56.955 Now, what we have established here is that for the density, 00:38:56.955 --> 00:38:59.990 the density has some special character because 00:38:59.990 --> 00:39:03.125 of this Liouville's theorem, that the streamlined derivative 00:39:03.125 --> 00:39:04.500 is 0. 00:39:04.500 --> 00:39:09.330 So what we have is that d rho by dt is 0. 00:39:11.990 --> 00:39:16.530 And this d rho by dt I can also write down 00:39:16.530 --> 00:39:25.445 as d rho by dt plus sum over all 6N directions, d 00:39:25.445 --> 00:39:28.550 rho by dp alpha. 00:39:28.550 --> 00:39:34.870 But then I substitute for p alpha dot from here. 00:39:34.870 --> 00:39:36.880 p dot is minus dH by dq. 00:39:44.010 --> 00:39:49.460 And then I add d rho by dq alpha. 00:39:49.460 --> 00:39:52.740 q alpha dot is dH by dp r. 00:39:56.556 --> 00:39:57.056 OK? 00:40:01.120 --> 00:40:07.110 So then I can take this combination with the minus sign 00:40:07.110 --> 00:40:14.520 to the other side and write it as d rho by dt 00:40:14.520 --> 00:40:20.780 is something that I will call the Poisson bracket of H 00:40:20.780 --> 00:40:27.680 and rho, where, quite generally, if I 00:40:27.680 --> 00:40:31.480 have two functions in phase space 00:40:31.480 --> 00:40:36.530 that is defending on B and q, this scalary derivative 00:40:36.530 --> 00:40:42.700 the Poisson bracket is defined as the sum over all 6N 00:40:42.700 --> 00:40:44.710 possible variation. 00:40:44.710 --> 00:40:49.740 The first one with respect to q, the second one with respect 00:40:49.740 --> 00:40:51.100 to p. 00:40:51.100 --> 00:40:55.190 And then the whole thing with the opposite sign. 00:40:55.190 --> 00:40:58.950 dA, dP, dB, dq. 00:41:03.270 --> 00:41:04.895 So this is the Poisson bracket. 00:41:10.450 --> 00:41:12.110 And again, from the definition, you 00:41:12.110 --> 00:41:14.540 should be able to see immediately 00:41:14.540 --> 00:41:16.870 that Poisson bracket of A and B is 00:41:16.870 --> 00:41:24.279 minus the Poisson bracket of B and A. OK? 00:42:01.440 --> 00:42:06.290 Again, we ask the question that in general, I 00:42:06.290 --> 00:42:14.410 can construct in principle a rho of p, q, let's say, 00:42:14.410 --> 00:42:16.280 for an equilibrium ensemble. 00:42:16.280 --> 00:42:20.780 But then I did something, like I removed a partition 00:42:20.780 --> 00:42:23.940 in the middle of the gas, and the gas is expanding. 00:42:23.940 --> 00:42:28.980 And then presumably, this becomes a function of time. 00:42:28.980 --> 00:42:33.810 And since I know exactly how each one of the particles, 00:42:33.810 --> 00:42:35.680 and hence each one of the micro states 00:42:35.680 --> 00:42:37.740 is evolving as a function of time, 00:42:37.740 --> 00:42:41.200 I should be able to tell how this density in phase space 00:42:41.200 --> 00:42:42.430 is changing. 00:42:42.430 --> 00:42:46.970 So this perspective is, again, this perspective 00:42:46.970 --> 00:42:50.100 of looking at all of these bees that are buzzing around 00:42:50.100 --> 00:42:54.190 in this 6N-dimensional space, and asking the question, if I 00:42:54.190 --> 00:42:58.350 look at the particular point in this 6N-dimensional space, what 00:42:58.350 --> 00:43:00.600 is the density of bees? 00:43:00.600 --> 00:43:03.890 And the answer is that it is given 00:43:03.890 --> 00:43:08.820 by the Poisson bracket of the Hamiltonian 00:43:08.820 --> 00:43:11.709 that governs the evolution of each micro state 00:43:11.709 --> 00:43:12.750 and the density function. 00:43:20.940 --> 00:43:21.470 All right. 00:43:21.470 --> 00:43:23.880 So what does it mean? 00:43:23.880 --> 00:43:25.320 What can we do with this? 00:43:29.520 --> 00:43:32.170 Let's play around with it and look at some consequences. 00:43:37.870 --> 00:43:43.402 But before that, does anybody have any questions? 00:43:43.402 --> 00:43:44.840 OK. 00:43:44.840 --> 00:43:45.340 All right. 00:43:49.350 --> 00:43:51.300 We had something that I just erased. 00:43:51.300 --> 00:43:55.730 That is, if I have a function of p and q, 00:43:55.730 --> 00:43:58.210 let's say it's not a function of time. 00:43:58.210 --> 00:44:01.360 Let's say it's the kinetic energy for this system where, 00:44:01.360 --> 00:44:04.290 at t equals to 0, I remove the partition, 00:44:04.290 --> 00:44:06.280 and the particles are expanding. 00:44:06.280 --> 00:44:08.462 And let's say the other place you have a potential, 00:44:08.462 --> 00:44:11.600 so your kinetic energy on average is going to change. 00:44:11.600 --> 00:44:14.380 You want to know what's happening to that. 00:44:14.380 --> 00:44:17.980 So you calculate at each instant of time 00:44:17.980 --> 00:44:21.380 an ensemble average of the kinetic energy 00:44:21.380 --> 00:44:24.340 or any other quantity that is interesting to you. 00:44:24.340 --> 00:44:29.460 And your prescription for calculating an ensemble average 00:44:29.460 --> 00:44:36.440 is that you integrate against the density the function 00:44:36.440 --> 00:44:42.260 that you are proposing to look at. 00:44:42.260 --> 00:44:45.580 Now, in principle, we said that there 00:44:45.580 --> 00:44:48.620 could be situations where this is dependent on time, 00:44:48.620 --> 00:44:53.040 in which case, your average will also depend on time. 00:44:53.040 --> 00:44:57.130 And maybe you want to know how this time dependence occurs. 00:44:57.130 --> 00:44:59.720 How does the kinetic energy of a gas that 00:44:59.720 --> 00:45:04.100 is expanding into some potential change on average? 00:45:04.100 --> 00:45:04.600 OK. 00:45:04.600 --> 00:45:05.810 So let's take a look. 00:45:05.810 --> 00:45:07.990 This is a function of time, because, as we said, 00:45:07.990 --> 00:45:09.220 these go inside the average. 00:45:09.220 --> 00:45:12.710 So really, the only explicit variable that we have here 00:45:12.710 --> 00:45:13.680 is time. 00:45:13.680 --> 00:45:15.430 And you can ask, what is the time 00:45:15.430 --> 00:45:16.765 dependence of this quantity? 00:45:22.000 --> 00:45:24.480 OK? 00:45:24.480 --> 00:45:35.380 So the time dependence is obtained by doing this, 00:45:35.380 --> 00:45:39.950 because essentially, you would be adding things 00:45:39.950 --> 00:45:43.210 at different points in p, q. 00:45:43.210 --> 00:45:45.830 And at each point, there is a time dependence. 00:45:45.830 --> 00:45:49.290 And you take the derivative in time with respect 00:45:49.290 --> 00:45:51.860 to the contribution of that point. 00:45:51.860 --> 00:45:54.940 So we get something like this. 00:45:54.940 --> 00:45:58.420 Now you say, OK, I know what d rho by dt is. 00:45:58.420 --> 00:46:05.080 So this is my integration over all of the phase space. 00:46:05.080 --> 00:46:09.400 d rho by dt is this Poisson bracket of H and rho. 00:46:09.400 --> 00:46:14.400 And then I have O. OK? 00:46:14.400 --> 00:46:16.930 Let's write this explicitly. 00:46:16.930 --> 00:46:19.340 This is an integral over a whole bunch 00:46:19.340 --> 00:46:20.530 of coordinates and momenta. 00:46:26.340 --> 00:46:30.725 There is, for the Poisson bracket, 00:46:30.725 --> 00:46:32.690 a sum over derivatives. 00:46:32.690 --> 00:46:39.390 So I have a sum over alpha-- dH by dq alpha, 00:46:39.390 --> 00:46:48.030 d rho by dp alpha minus dH by dp alpha, d rho by dq alpha. 00:46:48.030 --> 00:46:50.970 And that Poisson bracket in its entirety 00:46:50.970 --> 00:46:57.210 then gets multiplied by this function of phase space. 00:46:57.210 --> 00:47:00.080 OK. 00:47:00.080 --> 00:47:04.475 Now, one of the mathematical manipulations 00:47:04.475 --> 00:47:07.760 that we will do a lot in this class. 00:47:07.760 --> 00:47:11.160 And that's why I do this particular step, although it's 00:47:11.160 --> 00:47:13.795 not really necessary to the logical progression 00:47:13.795 --> 00:47:17.470 that I'm following, is to remind you 00:47:17.470 --> 00:47:21.260 how you would do an integration by parts when you're 00:47:21.260 --> 00:47:23.290 faced with something like this. 00:47:23.290 --> 00:47:26.700 An integration by parts is applicable 00:47:26.700 --> 00:47:31.660 when you have variables that you are integrating that are also 00:47:31.660 --> 00:47:35.340 appearing as derivatives. 00:47:35.340 --> 00:47:39.740 And whenever you are integrating Poisson brackets, 00:47:39.740 --> 00:47:43.130 you will have derivatives for the Poisson bracket. 00:47:43.130 --> 00:47:45.360 And the integration would allow you 00:47:45.360 --> 00:47:48.530 to use integration by parts. 00:47:48.530 --> 00:47:52.130 And in particular, what I would like to do 00:47:52.130 --> 00:47:55.510 is to remove the derivative that acts on the densities. 00:47:58.060 --> 00:48:05.730 So I'm going to essentially rewrite that as 00:48:05.730 --> 00:48:11.090 minus an integral that involves-- again. 00:48:11.090 --> 00:48:14.190 I don't want to keep rewriting that thing. 00:48:14.190 --> 00:48:20.340 I want to basically take the density out and then 00:48:20.340 --> 00:48:24.070 have the derivative, which is this d by dp in the first term 00:48:24.070 --> 00:48:28.790 and d by dq in the second term, act on everything else. 00:48:28.790 --> 00:48:36.890 So in the first case, d by dp alpha will act on O 00:48:36.890 --> 00:48:38.540 and dH by dq alpha. 00:48:41.150 --> 00:48:48.260 And in the second case, d by dq alpha will act on O 00:48:48.260 --> 00:48:49.691 and dH by dp alpha. 00:48:53.459 --> 00:48:57.000 Again, there is a sum over alpha that is implicit. 00:48:59.916 --> 00:49:01.380 OK? 00:49:01.380 --> 00:49:03.080 Again, there is a minus sign. 00:49:03.080 --> 00:49:07.230 So every time you do this procedure, there is this. 00:49:07.230 --> 00:49:11.000 But every time, you also have to worry about surface terms. 00:49:11.000 --> 00:49:14.050 So on the surface, you would potentially 00:49:14.050 --> 00:49:19.850 have to evaluate things that involve rho, O, and these d 00:49:19.850 --> 00:49:21.305 by d derivatives. 00:49:25.020 --> 00:49:28.290 But let's say we are integrating over momentum 00:49:28.290 --> 00:49:30.650 from minus infinity to infinity. 00:49:30.650 --> 00:49:35.240 Then the density evaluated at infinity momenta would be 0. 00:49:35.240 --> 00:49:38.760 So practicality, in all cases that I can think of, 00:49:38.760 --> 00:49:43.280 you don't have to worry about the boundary terms. 00:49:43.280 --> 00:49:47.200 So then when you look at these kinds of terms, 00:49:47.200 --> 00:49:51.380 this d by dp alpha can either act on O. 00:49:51.380 --> 00:49:58.000 So I will get dO by dp alpha, dH by dq alpha. 00:49:58.000 --> 00:50:03.070 Or it can act on dH by d alpha. 00:50:03.070 --> 00:50:08.530 So I will get plus O d2 H, dp alpha dq alpha. 00:50:08.530 --> 00:50:11.950 And similarly, in this term, either I 00:50:11.950 --> 00:50:20.720 will have dO by dq alpha, dH by dp alpha, or O d2 H, dq alpha 00:50:20.720 --> 00:50:23.740 dp alpha. 00:50:23.740 --> 00:50:25.770 Once more, the second derivative terms 00:50:25.770 --> 00:50:31.620 of the Hamiltonian, the order is not important. 00:50:31.620 --> 00:50:34.600 And what is left here is this set 00:50:34.600 --> 00:50:39.590 of objects, which is none other than the Poisson bracket. 00:50:39.590 --> 00:50:44.440 So I can rewrite the whole thing as d by dt of the expectation 00:50:44.440 --> 00:50:47.630 value of this quantity, which potentially 00:50:47.630 --> 00:50:49.860 is a function of time because of the time 00:50:49.860 --> 00:50:53.410 dependence of my density is the same thing 00:50:53.410 --> 00:51:00.090 as minus an integration over the entire phase space of rho 00:51:00.090 --> 00:51:03.170 against this entity, which is none 00:51:03.170 --> 00:51:09.200 other than the Poisson bracket of H with O. 00:51:09.200 --> 00:51:13.830 And this integration over density of this entire space 00:51:13.830 --> 00:51:17.710 is just our definition of the expectation value. 00:51:17.710 --> 00:51:27.650 So we get that the time derivative of any quantity 00:51:27.650 --> 00:51:32.370 is related to the average of its Poisson bracket 00:51:32.370 --> 00:51:36.110 with the Hamiltonian, which is the quantity that is really 00:51:36.110 --> 00:51:37.437 governing time dependences. 00:51:40.913 --> 00:51:41.413 Yes? 00:51:41.413 --> 00:51:43.350 AUDIENCE: Could you explain again 00:51:43.350 --> 00:51:46.420 why the time derivative when N is the integral, 00:51:46.420 --> 00:51:48.688 it's rho as a partial derivative? 00:51:48.688 --> 00:51:50.140 PROFESSOR: OK. 00:51:50.140 --> 00:51:53.190 So suppose I'm doing a two-dimensional integral 00:51:53.190 --> 00:51:56.060 over p and q. 00:51:56.060 --> 00:52:01.010 So I have some contribution from each point in this p and q. 00:52:01.010 --> 00:52:05.220 And so my integral is an integral dpdq, 00:52:05.220 --> 00:52:08.770 something evaluated at each point in p, 00:52:08.770 --> 00:52:12.420 q that could potentially depend on time. 00:52:12.420 --> 00:52:14.600 Imagine that I discretize this. 00:52:14.600 --> 00:52:17.230 So I really-- if you are more comfortable, 00:52:17.230 --> 00:52:20.310 you can think of this as a limit of a sum. 00:52:20.310 --> 00:52:23.170 So this is my integral. 00:52:23.170 --> 00:52:26.800 If I'm interested in the time dependence of this quantity-- 00:52:26.800 --> 00:52:29.130 and I really depends only on time, 00:52:29.130 --> 00:52:32.290 because I integrated over p and q. 00:52:32.290 --> 00:52:34.970 So if I'm interested in something 00:52:34.970 --> 00:52:38.990 that is a sum of various terms, each term 00:52:38.990 --> 00:52:41.760 is a function of time. 00:52:41.760 --> 00:52:43.880 Where do I put the time dependence? 00:52:43.880 --> 00:52:49.990 For each term in this sum, I look at how it depends on time. 00:52:49.990 --> 00:52:53.805 I don't care on its points to the left and to the right. 00:52:59.030 --> 00:52:59.990 OK? 00:52:59.990 --> 00:53:04.420 Because the big D by Dt involves moving with the streamline. 00:53:04.420 --> 00:53:07.050 I'm not doing any moving with the streamline. 00:53:07.050 --> 00:53:12.350 I'm looking at each point in this two-dimensional space. 00:53:12.350 --> 00:53:15.730 Each point gives a contribution at that point 00:53:15.730 --> 00:53:17.380 that is time-dependent. 00:53:17.380 --> 00:53:20.580 And I take the derivative with respect to time at that point. 00:53:23.200 --> 00:53:23.720 Yes? 00:53:23.720 --> 00:53:27.672 AUDIENCE: Couldn't you say that you have function O 00:53:27.672 --> 00:53:30.636 as just some function of p and q, 00:53:30.636 --> 00:53:34.457 and its time derivative would be Poisson bracket? 00:53:34.457 --> 00:53:35.082 PROFESSOR: Yes. 00:53:35.082 --> 00:53:37.291 AUDIENCE: And does the average of the time derivative 00:53:37.291 --> 00:53:38.957 would be the average of Poisson bracket, 00:53:38.957 --> 00:53:40.826 and you don't have to go through all the-- 00:53:40.826 --> 00:53:41.570 PROFESSOR: No. 00:53:41.570 --> 00:53:43.280 But you can see the sign doesn't work. 00:53:45.942 --> 00:53:47.304 AUDIENCE: How come? 00:53:47.304 --> 00:53:48.720 PROFESSOR: [LAUGHS] Because of all 00:53:48.720 --> 00:53:50.390 of these manipulations, et cetera. 00:53:50.390 --> 00:53:54.030 So the statement that you made is manifestly incorrect. 00:53:54.030 --> 00:53:58.190 You can't say that the time dependence of this thing 00:53:58.190 --> 00:54:03.366 is the-- whatever you were saying. [LAUGHS] 00:54:03.866 --> 00:54:06.794 AUDIENCE: [INAUDIBLE]. 00:54:06.794 --> 00:54:07.510 PROFESSOR: OK. 00:54:07.510 --> 00:54:08.760 Let's see what you are saying. 00:54:08.760 --> 00:54:11.090 AUDIENCE: So Poisson bracket only 00:54:11.090 --> 00:54:14.402 counts for in place for averages, right? 00:54:14.402 --> 00:54:15.360 PROFESSOR: OK. 00:54:15.360 --> 00:54:17.560 So what do we have here? 00:54:17.560 --> 00:54:25.630 We have that dp by dt is the Poisson bracket of H and rho. 00:54:25.630 --> 00:54:26.630 OK? 00:54:26.630 --> 00:54:33.730 And we have that O is an integral of rho O. Now, 00:54:33.730 --> 00:54:37.460 from where do you conclude from this set of results 00:54:37.460 --> 00:54:44.900 that d O by dt is the average of a Poisson bracket 00:54:44.900 --> 00:54:47.390 that involves O and H, irrespective 00:54:47.390 --> 00:54:48.670 of what we do with the sign? 00:54:48.670 --> 00:54:53.230 AUDIENCE: Or if you look not at the average failure of O 00:54:53.230 --> 00:54:57.745 but at the value of O and point, and take-- I 00:54:57.745 --> 00:55:01.050 guess it would be streamline derivative of it. 00:55:01.050 --> 00:55:07.214 So that's assuming that you're just like assigning value of O 00:55:07.214 --> 00:55:10.570 to each point, and making power changes with time 00:55:10.570 --> 00:55:13.076 as this point moves across the phase space. 00:55:13.076 --> 00:55:13.810 PROFESSOR: OK. 00:55:13.810 --> 00:55:20.230 But you still have to do some bit of derivatives, et cetera, 00:55:20.230 --> 00:55:21.652 because-- 00:55:21.652 --> 00:55:24.520 AUDIENCE: But if you know that the volume of the 00:55:24.520 --> 00:55:28.756 in phase space is conserved, then we basically 00:55:28.756 --> 00:55:30.980 don't care much that the function 00:55:30.980 --> 00:55:34.340 O is any much different from probability density. 00:55:34.340 --> 00:55:35.080 PROFESSOR: OK. 00:55:35.080 --> 00:55:38.070 If I understand correctly, this is what you are saying. 00:55:38.070 --> 00:55:41.150 Is that for each representative point, 00:55:41.150 --> 00:55:46.190 I have an O alpha, which is a function of time. 00:55:46.190 --> 00:55:51.880 And then you want to say that the average of O 00:55:51.880 --> 00:55:57.370 is the same thing as the sum over alpha of O alpha of t's 00:55:57.370 --> 00:55:59.690 divided by N, something like this. 00:55:59.690 --> 00:56:02.150 AUDIENCE: Eh. 00:56:02.150 --> 00:56:08.020 Uh, I want to first calculate what does time derivative of O? 00:56:08.020 --> 00:56:12.406 O remains in a function of time and q and p. 00:56:12.406 --> 00:56:13.390 So I can calculate-- 00:56:13.390 --> 00:56:14.380 PROFESSOR: Yes. 00:56:14.380 --> 00:56:20.834 So this O alpha is a function of 00:56:20.834 --> 00:56:22.176 AUDIENCE: So if I said-- 00:56:22.176 --> 00:56:22.800 PROFESSOR: Yes. 00:56:22.800 --> 00:56:23.140 OK. 00:56:23.140 --> 00:56:23.640 Fine. 00:56:23.640 --> 00:56:27.292 AUDIENCE: O is a function of q and p and t, 00:56:27.292 --> 00:56:29.890 and I take a streamline derivative of it. 00:56:29.890 --> 00:56:31.616 So filter it with respect to t. 00:56:31.616 --> 00:56:37.500 And then I average that thing over phase space. 00:56:37.500 --> 00:56:39.994 And then I should get the same version-- 00:56:39.994 --> 00:56:40.910 PROFESSOR: You should. 00:56:40.910 --> 00:56:42.185 AUDIENCE: --perfectly. 00:56:42.185 --> 00:56:43.035 But-- 00:56:43.035 --> 00:56:46.890 PROFESSOR: Very quickly, I don't think so. 00:56:46.890 --> 00:56:49.150 Because you are already explaining things a bit 00:56:49.150 --> 00:56:51.455 longer than I think I went through my derivation. 00:56:51.455 --> 00:56:54.096 But that's fine. [LAUGHS] 00:56:55.004 --> 00:56:57.290 AUDIENCE: Is there any special-- 00:56:57.290 --> 00:56:59.600 PROFESSOR: But I agree in spirit, yes. 00:56:59.600 --> 00:57:02.920 That each one of these will go along its streamline, 00:57:02.920 --> 00:57:06.280 and you can calculate the change for each one of them. 00:57:06.280 --> 00:57:08.810 And then you have to do an average of this variety. 00:57:08.810 --> 00:57:09.705 Yes. 00:57:09.705 --> 00:57:15.254 AUDIENCE: [INAUDIBLE] when you talk about time derivative 00:57:15.254 --> 00:57:19.300 of probability density, it's Poisson bracket of H and rho. 00:57:19.300 --> 00:57:23.156 But when you talk about time derivative of average, 00:57:23.156 --> 00:57:27.330 you have to add the minus sign. 00:57:27.330 --> 00:57:29.840 PROFESSOR: And if you do this correctly here, 00:57:29.840 --> 00:57:31.134 you should get the same result. 00:57:33.858 --> 00:57:34.766 AUDIENCE: Oh, OK. 00:57:34.766 --> 00:57:35.945 PROFESSOR: Yes. 00:57:35.945 --> 00:57:39.005 AUDIENCE: Well, along that line, though, 00:57:39.005 --> 00:57:43.455 are you using the fact that phase space volume is 00:57:43.455 --> 00:57:47.074 incompressible to then argue that the total time 00:57:47.074 --> 00:57:49.900 derivative of the ensemble average 00:57:49.900 --> 00:57:54.045 is the same as the ensemble average of the total time 00:57:54.045 --> 00:57:56.520 derivative of O, or not? 00:58:08.326 --> 00:58:09.700 PROFESSOR: Could you repeat that? 00:58:09.700 --> 00:58:11.810 Mathematically, you want me to show 00:58:11.810 --> 00:58:16.875 that the time derivative of what quantity? 00:58:16.875 --> 00:58:19.515 AUDIENCE: Of the average of O. 00:58:19.515 --> 00:58:21.970 PROFESSOR: Of the average of O. Yes. 00:58:21.970 --> 00:58:24.370 AUDIENCE: Is it in any way related to the average 00:58:24.370 --> 00:58:27.065 of dO over dt? 00:58:27.065 --> 00:58:29.520 PROFESSOR: No, it's not. 00:58:29.520 --> 00:58:34.660 Because O-- I mean, so what do you mean by that? 00:58:34.660 --> 00:58:37.850 You have to be careful, because the way that I'm defining this 00:58:37.850 --> 00:58:41.640 here, O is a function of p and q. 00:58:41.640 --> 00:58:46.250 And what you want to do is to write something 00:58:46.250 --> 00:58:56.990 that is a sum over all representative points, divided 00:58:56.990 --> 00:59:00.780 by the total number, some kind of an average like this. 00:59:00.780 --> 00:59:04.114 And then I can define a time derivative here. 00:59:04.114 --> 00:59:05.030 Is that what you are-- 00:59:05.030 --> 00:59:06.710 AUDIENCE: Well, I mean, I was thinking 00:59:06.710 --> 00:59:10.230 that even if you start out with your observable being defined 00:59:10.230 --> 00:59:14.080 for every point in phase space, then if you were to, 00:59:14.080 --> 00:59:16.150 before doing any ensemble averaging, 00:59:16.150 --> 00:59:19.970 if you were to take the total time derivative of that, 00:59:19.970 --> 00:59:25.030 then you would be accounted for p dot and q dot as well, right? 00:59:25.030 --> 00:59:26.960 And then if you were to take the ensemble 00:59:26.960 --> 00:59:29.710 average of that quantity, could you 00:59:29.710 --> 00:59:31.420 arrive at the same result for that? 00:59:31.420 --> 00:59:34.570 PROFESSOR: I'm pretty sure that if you do things consistently, 00:59:34.570 --> 00:59:35.320 yes. 00:59:35.320 --> 00:59:37.230 That is, what we have done is essentially 00:59:37.230 --> 00:59:41.830 we started with a collection of trajectories in phase space 00:59:41.830 --> 00:59:45.235 and recast the result in terms of density and variables 00:59:45.235 --> 00:59:49.950 that are defined only as positions in phase space. 00:59:49.950 --> 00:59:53.200 The two descriptions completely are equivalent. 00:59:53.200 --> 00:59:55.810 And as long as one doesn't make a mistake, 00:59:55.810 --> 00:59:58.022 one can get one or the other. 00:59:58.022 --> 01:00:01.635 This is actually a kind of a well-known thing 01:00:01.635 --> 01:00:05.490 in hydrodynamics, because typically, you write down, 01:00:05.490 --> 01:00:09.070 in hydrodynamics, equations for density and velocity 01:00:09.070 --> 01:00:11.850 at each point in phase space. 01:00:11.850 --> 01:00:14.220 But there is an alternative description 01:00:14.220 --> 01:00:18.400 which we can say that there's essentially particles that are 01:00:18.400 --> 01:00:19.700 flowing. 01:00:19.700 --> 01:00:22.800 And particle that was here at this location 01:00:22.800 --> 01:00:25.860 is now somewhere else at some later time. 01:00:25.860 --> 01:00:28.820 And people have tried hard. 01:00:28.820 --> 01:00:30.570 And there is a consistent definition 01:00:30.570 --> 01:00:34.620 of hydrodynamics that follows the second perspective. 01:00:34.620 --> 01:00:37.700 But I haven't seen it as being practical. 01:00:37.700 --> 01:00:41.440 So I'm sure that everything that you guys say is correct. 01:00:41.440 --> 01:00:45.250 But from the experience of what I know in hydrodynamics, 01:00:45.250 --> 01:00:47.450 I think this is the more practical description 01:00:47.450 --> 01:00:49.090 that people have been using. 01:00:53.170 --> 01:00:56.060 OK? 01:00:56.060 --> 01:00:58.810 So where were we? 01:00:58.810 --> 01:00:59.310 OK. 01:01:01.920 --> 01:01:05.810 So back to our buzzing bees. 01:01:09.960 --> 01:01:13.240 We now have a way of looking at how 01:01:13.240 --> 01:01:17.070 densities and various quantities that you can calculate, 01:01:17.070 --> 01:01:22.670 like ensemble averages, are changing as a function of time. 01:01:22.670 --> 01:01:24.145 But the question that I had before 01:01:24.145 --> 01:01:25.270 is, what about equilibrium? 01:01:31.700 --> 01:01:35.170 Because the thermodynamic definition of equilibrium, 01:01:35.170 --> 01:01:38.550 and my whole ensemble idea, was that essentially, I 01:01:38.550 --> 01:01:41.920 have all of these boxes and pressure, 01:01:41.920 --> 01:01:44.120 volume, everything that I can think of, 01:01:44.120 --> 01:01:46.510 as long as I'm not doing something that's 01:01:46.510 --> 01:01:52.230 like opening the box, is perfectly independent of time. 01:01:52.230 --> 01:01:56.260 So how can I ensure that various things that I calculate 01:01:56.260 --> 01:01:58.170 are independent of time? 01:01:58.170 --> 01:02:03.640 Clearly, I can do that by having this density not really depend 01:02:03.640 --> 01:02:05.950 on time. 01:02:05.950 --> 01:02:07.830 OK? 01:02:07.830 --> 01:02:10.180 Now, of course, each representative point 01:02:10.180 --> 01:02:11.370 is moving around. 01:02:11.370 --> 01:02:13.210 Each micro state is moving around. 01:02:13.210 --> 01:02:14.710 Each bee is moving around. 01:02:14.710 --> 01:02:21.100 But I want the density that is characteristic of equilibrium 01:02:21.100 --> 01:02:23.340 be something that does not change in time. 01:02:26.030 --> 01:02:26.560 It's 0. 01:02:29.220 --> 01:02:34.830 And so if I posit that this particular function of p and q 01:02:34.830 --> 01:02:38.150 is something that is not changing as a function of time, 01:02:38.150 --> 01:02:40.900 I have to require that the Poisson bracket 01:02:40.900 --> 01:02:44.310 of that function of p and q with the Hamiltonian, 01:02:44.310 --> 01:02:48.435 which is another function of p and q, is 0. 01:02:48.435 --> 01:02:49.330 OK? 01:02:49.330 --> 01:02:52.290 So in principle, I have to go back 01:02:52.290 --> 01:02:57.000 to this equation over here, which is a partial differential 01:02:57.000 --> 01:03:03.190 equation in 6N-dimensional space and solve with equal to 0. 01:03:03.190 --> 01:03:06.640 Of course, rather than doing that, we will guess the answer. 01:03:06.640 --> 01:03:12.390 And the guess is clear from here, because all I need to do 01:03:12.390 --> 01:03:18.500 is to make rho equilibrium depend on coordinates in phase 01:03:18.500 --> 01:03:24.210 space through some functional dependence on Hamiltonian, 01:03:24.210 --> 01:03:28.354 which depends on the coordinates in phase space. 01:03:28.354 --> 01:03:29.820 OK? 01:03:29.820 --> 01:03:31.730 Why does this work? 01:03:31.730 --> 01:03:36.580 Because then when I take the Poisson bracket of, let's 01:03:36.580 --> 01:03:45.640 say, this function of H with H, what do I have to do? 01:03:45.640 --> 01:03:50.590 I have to do a sum over alpha d rho with respect 01:03:50.590 --> 01:03:55.510 to, let's say-- actually, let's write it in this way. 01:03:55.510 --> 01:04:02.850 I will have the dH by dp alpha from here. 01:04:02.850 --> 01:04:06.770 I have to multiply by d rho by dq alpha. 01:04:06.770 --> 01:04:10.510 But rho is a function of only H. So I 01:04:10.510 --> 01:04:14.500 have to take a derivative of rho with respect to its argument H. 01:04:14.500 --> 01:04:16.270 And I'll call that rho prime. 01:04:16.270 --> 01:04:18.610 And then the derivative of the argument, 01:04:18.610 --> 01:04:22.090 which is H, with respect to q alpha. 01:04:22.090 --> 01:04:24.580 That would be the first term. 01:04:24.580 --> 01:04:29.755 The next term would be, with the minus sign, from the H here, 01:04:29.755 --> 01:04:37.520 dH by dq alpha, the derivative of rho with respect to p alpha. 01:04:37.520 --> 01:04:42.460 But rho only depends on H, so I will get the derivative of rho 01:04:42.460 --> 01:04:44.650 with respect to its one argument. 01:04:44.650 --> 01:04:49.610 The derivative of that argument with respect to p alpha. 01:04:49.610 --> 01:04:51.410 OK? 01:04:51.410 --> 01:04:54.470 So you can see that up to the order of the terms that 01:04:54.470 --> 01:04:59.020 are multiplying here, this is 0. 01:04:59.020 --> 01:05:02.020 OK? 01:05:02.020 --> 01:05:10.650 So any function I choose of H, in principle, satisfies this. 01:05:10.650 --> 01:05:14.710 And this is what we will use consistently 01:05:14.710 --> 01:05:17.680 all the time in statistical mechanics, 01:05:17.680 --> 01:05:20.560 in depending on the ensemble that we have. 01:05:20.560 --> 01:05:22.275 Like you probably know that when we 01:05:22.275 --> 01:05:25.000 are in the micro canonical ensemble, 01:05:25.000 --> 01:05:28.110 we look at-- in the micro canonical ensemble, 01:05:28.110 --> 01:05:31.630 we'd say what the energy is. 01:05:31.630 --> 01:05:35.620 And then we say that the density is 01:05:35.620 --> 01:05:41.100 a delta function-- essentially zero. 01:05:41.100 --> 01:05:44.820 Except places, it's the surface that 01:05:44.820 --> 01:05:47.410 corresponds to having the right energy. 01:05:47.410 --> 01:05:50.160 So you sort of construct in phase space 01:05:50.160 --> 01:05:53.100 the surface that has the right energy. 01:05:53.100 --> 01:05:56.889 So it's a function of these four. 01:05:56.889 --> 01:05:58.180 So this is the micro canonical. 01:06:00.790 --> 01:06:02.750 When you are in the canonical, I use 01:06:02.750 --> 01:06:05.822 a rho this is proportional to e to the minus beta H 01:06:05.822 --> 01:06:07.610 and other functional features. 01:06:07.610 --> 01:06:10.175 So it's, again, the same idea. 01:06:13.140 --> 01:06:13.640 OK? 01:06:21.570 --> 01:06:25.910 That's almost but not entirely true, 01:06:25.910 --> 01:06:29.894 because sometimes there are also other conserved quantities. 01:06:39.640 --> 01:06:43.620 Let's say that, for example, we have a collection of particles 01:06:43.620 --> 01:06:48.100 in space in a cavity that has the shape of a sphere. 01:06:48.100 --> 01:06:50.740 Because of the symmetry of the problem, 01:06:50.740 --> 01:06:56.300 angular momentum is going to be a conserved quantity. 01:06:56.300 --> 01:06:58.570 Angular momentum you can also write 01:06:58.570 --> 01:07:02.030 as some complicated function of p and q. 01:07:02.030 --> 01:07:05.340 For example, p cross q summed over all of the particles. 01:07:05.340 --> 01:07:10.130 But it could be some other conserved quantity. 01:07:10.130 --> 01:07:13.680 So what does this conservation law mean? 01:07:13.680 --> 01:07:20.410 It means that if you evaluate L for some time, 01:07:20.410 --> 01:07:23.920 it is going to be the same L for the coordinates 01:07:23.920 --> 01:07:26.010 and momenta at some other time. 01:07:26.010 --> 01:07:29.700 Or in other words, dL by dt, which 01:07:29.700 --> 01:07:34.470 you obtained by summing over all coordinates dL 01:07:34.470 --> 01:07:44.370 by dp alpha, p alpha dot, plus dL by dq alpha q alpha dot. 01:07:44.370 --> 01:07:46.440 Essentially, taking time derivatives 01:07:46.440 --> 01:07:48.150 of all of the arguments. 01:07:48.150 --> 01:07:51.780 And I did not put any explicit time dependence here. 01:07:51.780 --> 01:07:57.440 And this is again sum over alpha dL by dp alpha. 01:07:57.440 --> 01:08:04.670 p alpha dot is minus dH by dq alpha. 01:08:04.670 --> 01:08:14.570 And dL by dq alpha, q alpha dot is dH by dp alpha. 01:08:14.570 --> 01:08:17.834 So you are seeing that this is the same thing 01:08:17.834 --> 01:08:24.540 as the Poisson bracket of L and H. 01:08:24.540 --> 01:08:29.830 So essentially, conserved quantities which 01:08:29.830 --> 01:08:33.990 are essentially functions of coordinates and momenta 01:08:33.990 --> 01:08:37.680 that you calculate that don't change as a function of time 01:08:37.680 --> 01:08:40.420 are also quantities that have zero Poisson 01:08:40.420 --> 01:08:43.700 brackets with the Hamiltonian. 01:08:43.700 --> 01:08:47.569 So if I have a conserved quantity, 01:08:47.569 --> 01:08:49.414 then I have a more general solution. 01:08:55.890 --> 01:09:02.140 To my d rho by dt equals to 0 requirement. 01:09:02.140 --> 01:09:05.410 I could make a rho equilibrium which 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. 01:09:14.649 --> 01:09:20.979 And when you go through the Poisson bracket process, 01:09:20.979 --> 01:09:34.120 you will either be taking derivatives 01:09:34.120 --> 01:09:37.870 with respect to the first argument here. 01:09:37.870 --> 01:09:43.319 So you would get rho prime with respect to the first argument. 01:09:43.319 --> 01:09:47.859 And then you would get the Poisson bracket of H and H. 01:09:47.859 --> 01:09:50.810 Or you would be getting derivatives with respect 01:09:50.810 --> 01:09:53.819 to the second argument. 01:09:53.819 --> 01:09:59.280 And then you would be getting the Poisson bracket of L and H. 01:09:59.280 --> 01:10:01.540 And both of them are 0. 01:10:01.540 --> 01:10:04.850 By definition, L is a conserved quantity. 01:10:04.850 --> 01:10:10.440 So any solution that's a function of Hamiltonian, 01:10:10.440 --> 01:10:12.340 the energy of the system is conserved 01:10:12.340 --> 01:10:14.570 as a function of time, as well as 01:10:14.570 --> 01:10:18.190 any other conserved quantities, such as angular momentum, 01:10:18.190 --> 01:10:22.200 et cetera, is certainly a valid thing. 01:10:22.200 --> 01:10:28.680 So indeed, when I drew here, in the micro canonical ensemble, 01:10:28.680 --> 01:10:33.020 a surface that corresponds to a constant energy, 01:10:33.020 --> 01:10:36.450 well, if I am in a spherical cavity, 01:10:36.450 --> 01:10:39.270 only part of that surface that corresponds 01:10:39.270 --> 01:10:43.490 to the right angular momentum is going to be accessible. 01:10:43.490 --> 01:10:48.870 So essentially, what I know is that if I have conserved 01:10:48.870 --> 01:10:55.050 quantities, my trajectories will explore the subspace 01:10:55.050 --> 01:10:59.070 that is consistent with those conservation laws. 01:10:59.070 --> 01:11:05.720 And this statement here is that ultimately, those spaces that 01:11:05.720 --> 01:11:09.720 correspond to the appropriate conservation law 01:11:09.720 --> 01:11:11.750 are equally populated. 01:11:11.750 --> 01:11:15.400 Rho is constant around. 01:11:15.400 --> 01:11:20.960 So in some sense, we started with the definition of rho 01:11:20.960 --> 01:11:24.460 by putting these points around and calculating 01:11:24.460 --> 01:11:28.410 probability that way, which was my objective definition 01:11:28.410 --> 01:11:30.100 of probability. 01:11:30.100 --> 01:11:32.320 And through this Liouville theorem, 01:11:32.320 --> 01:11:33.920 we have arrived at something that 01:11:33.920 --> 01:11:38.770 is more consistent with the subjective assignment 01:11:38.770 --> 01:11:40.040 of probability. 01:11:40.040 --> 01:11:42.790 That is, the only thing that I know, 01:11:42.790 --> 01:11:44.360 forgetting about the dynamics, is 01:11:44.360 --> 01:11:46.240 that there are some conserved quantities, 01:11:46.240 --> 01:11:49.060 such as H, angular momentum, et cetera. 01:11:49.060 --> 01:11:51.990 And I say that any point in phase space that does not 01:11:51.990 --> 01:11:56.600 violate those constants in this, say, micro canonical ensemble 01:11:56.600 --> 01:11:58.520 would be equally populated. 01:11:58.520 --> 01:12:00.860 There was a question somewhere. 01:12:00.860 --> 01:12:02.314 Yes? 01:12:02.314 --> 01:12:11.324 AUDIENCE: So I almost feel like the statement that the rho has 01:12:11.324 --> 01:12:15.967 to not change in time is too strong, because if you go over 01:12:15.967 --> 01:12:20.458 to the equation that says the rate of change 01:12:20.458 --> 01:12:24.574 is observable is equal to the integral that 01:12:24.574 --> 01:12:28.795 was with a Poisson bracket of rho and H, then 01:12:28.795 --> 01:12:30.450 it means that for any observable, 01:12:30.450 --> 01:12:33.138 it's constant in time, right? 01:12:33.138 --> 01:12:34.040 PROFESSOR: Yes. 01:12:34.040 --> 01:12:38.330 AUDIENCE: So rho of q means any observable we can think of, 01:12:38.330 --> 01:12:40.232 its function of p and q is constant? 01:12:40.232 --> 01:12:42.640 PROFESSOR: Yep. 01:12:42.640 --> 01:12:43.370 Yep. 01:12:43.370 --> 01:12:49.720 Because-- and that's the best thing that we 01:12:49.720 --> 01:12:53.730 can think of in terms of-- because if there 01:12:53.730 --> 01:12:56.660 is some observable that is time-dependent-- 01:12:56.660 --> 01:12:59.655 let's say 99 observables are time-independent, 01:12:59.655 --> 01:13:03.830 but one is time-dependent, and you can measure that, 01:13:03.830 --> 01:13:06.140 would you say your system is in equilibrium? 01:13:06.140 --> 01:13:07.104 Probably not. 01:13:11.856 --> 01:13:12.356 OK? 01:13:12.356 --> 01:13:14.022 AUDIENCE: It seemed like the same method 01:13:14.022 --> 01:13:17.155 that you did to show that the density is a function of 01:13:17.155 --> 01:13:20.850 [INAUDIBLE], or [INAUDIBLE] that it's 01:13:20.850 --> 01:13:27.860 the function of any observable that's a function of q. 01:13:27.860 --> 01:13:28.360 Right? 01:13:28.360 --> 01:13:29.500 PROFESSOR: Observables? 01:13:29.500 --> 01:13:30.530 No. 01:13:30.530 --> 01:13:36.000 I mean, here, if this answer is 0, 01:13:36.000 --> 01:13:39.580 it states something about this quantity averaged. 01:13:39.580 --> 01:13:43.720 So if this quantity does not change as a function of time, 01:13:43.720 --> 01:13:48.410 it is not a statement that H and 0 is 0. 01:13:48.410 --> 01:13:52.700 A statement that H and O is 0 is different from its ensemble 01:13:52.700 --> 01:13:55.330 average being 0. 01:13:55.330 --> 01:13:59.450 What you can show-- and I think I have a problem set for that-- 01:13:59.450 --> 01:14:04.280 is that if this statement is correct for every O 01:14:04.280 --> 01:14:08.870 that the average is 0, then your rho 01:14:08.870 --> 01:14:11.940 has to satisfy this theorem equals 01:14:11.940 --> 01:14:15.992 to-- Poisson bracket of rho and H is equal to 0. 01:14:23.860 --> 01:14:24.420 OK. 01:14:24.420 --> 01:14:28.030 So now the big question is the following. 01:14:28.030 --> 01:14:31.390 We arrived that the way of thinking about equilibrium 01:14:31.390 --> 01:14:34.610 in a system of particles and things 01:14:34.610 --> 01:14:38.000 that are this many-to-one mapping, et cetera, in terms 01:14:38.000 --> 01:14:42.030 of the densities-- we arrived that the definition of what 01:14:42.030 --> 01:14:44.990 the density is going to be in equilibrium. 01:14:44.990 --> 01:14:49.730 But the thermodynamic statement is much, much more severe. 01:14:49.730 --> 01:14:52.770 The statement, again, is that if I have a box 01:14:52.770 --> 01:14:56.320 and I open the door of the box, the gas 01:14:56.320 --> 01:15:01.300 expands to fill the empty space or the other part of the box. 01:15:01.300 --> 01:15:04.850 And it will do so all the time. 01:15:04.850 --> 01:15:07.790 Yet the equations of motion that we have over here 01:15:07.790 --> 01:15:10.090 are time reversal invariant. 01:15:10.090 --> 01:15:12.710 And we did not manage to remove that. 01:15:12.710 --> 01:15:15.920 We can show that this Liouville equation, et cetera, 01:15:15.920 --> 01:15:19.030 is also time reversal invariant. 01:15:19.030 --> 01:15:21.340 So for every case, if you succeed 01:15:21.340 --> 01:15:24.340 to show that there is a density that is in half of the box 01:15:24.340 --> 01:15:27.610 and it expands to fill the entire box, 01:15:27.610 --> 01:15:30.940 there will be a density that presumably goes the other way. 01:15:30.940 --> 01:15:34.760 Because that will be also a solution of this equation. 01:15:34.760 --> 01:15:39.430 So when we sort of go back from the statement of what 01:15:39.430 --> 01:15:42.670 is the equilibrium solution and ask, 01:15:42.670 --> 01:15:46.720 do I know that I will eventually reach this equilibrium 01:15:46.720 --> 01:15:50.210 solution as a function of time, we have not shown that. 01:15:50.210 --> 01:15:55.040 And we will attempt to do so next time.