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. 00:00:20.540 --> 00:00:21.840 PROFESSOR: OK. 00:00:21.840 --> 00:00:22.463 Let's start. 00:00:26.420 --> 00:00:30.000 So last time, we started with kinetic theory. 00:00:36.120 --> 00:00:39.205 And we will focus for gas systems mostly. 00:00:43.890 --> 00:00:45.730 And the question that we would like 00:00:45.730 --> 00:00:50.140 to think about and answer somehow is the following. 00:00:50.140 --> 00:00:56.260 You start with a gas that is initially 00:00:56.260 --> 00:00:57.765 confined to one chamber. 00:01:01.400 --> 00:01:06.180 And you can calculate all of its thermodynamic properties. 00:01:06.180 --> 00:01:11.560 You open at time 0 a hole, allowing the gas 00:01:11.560 --> 00:01:17.290 to escape into a second initially empty chamber. 00:01:17.290 --> 00:01:19.400 And after some time, the whole system 00:01:19.400 --> 00:01:22.320 will come to a new equilibrium position. 00:01:22.320 --> 00:01:25.040 It's a pretty reversible thing. 00:01:25.040 --> 00:01:27.620 You can do this experiment many, many times. 00:01:27.620 --> 00:01:31.690 And you will always get roughly the same amount of time 00:01:31.690 --> 00:01:34.280 for the situation to start from one equilibrium 00:01:34.280 --> 00:01:36.820 and reach another equilibrium. 00:01:36.820 --> 00:01:38.250 So how do we describe that? 00:01:38.250 --> 00:01:41.010 It's slightly beyond what we did in thermodynamics, 00:01:41.010 --> 00:01:43.990 because we want to go from one equilibrium state 00:01:43.990 --> 00:01:46.910 to another equilibrium state. 00:01:46.910 --> 00:01:50.260 Now, we said, OK, we know the equations of motion 00:01:50.260 --> 00:01:55.200 that governs the particles that are described by this. 00:01:55.200 --> 00:02:00.970 So here we can say that we have, let's say, N particles. 00:02:00.970 --> 00:02:05.930 They have their own momenta and coordinates. 00:02:05.930 --> 00:02:09.160 And we know that these momenta and coordinates evolve 00:02:09.160 --> 00:02:12.230 in time, governed by some Hamiltonian, 00:02:12.230 --> 00:02:17.290 which is a function of all of these momenta and coordinates. 00:02:17.290 --> 00:02:17.880 OK? 00:02:17.880 --> 00:02:19.270 Fine. 00:02:19.270 --> 00:02:21.400 How do we go from a situation which 00:02:21.400 --> 00:02:23.100 describes a whole bunch of things 00:02:23.100 --> 00:02:25.580 and coordinates that are changing with time 00:02:25.580 --> 00:02:33.400 to some microscopic description of macroscopic variables going 00:02:33.400 --> 00:02:36.850 from one equilibrium state to another? 00:02:36.850 --> 00:02:39.740 So our first attempt in that direction 00:02:39.740 --> 00:02:46.390 was to say, well, I could start with many, many, many examples 00:02:46.390 --> 00:02:49.260 of the same situation. 00:02:49.260 --> 00:02:52.190 Each one of them would correspond to a different 00:02:52.190 --> 00:02:55.500 trajectory of p's and q's. 00:02:55.500 --> 00:02:59.300 And so what we can do is to construct 00:02:59.300 --> 00:03:05.720 some kind of an ensemble average, or ensemble density, 00:03:05.720 --> 00:03:08.800 first, which is what we did. 00:03:08.800 --> 00:03:15.255 We can say that the very, very many examples of this situation 00:03:15.255 --> 00:03:20.300 that I can have will correspond to different points 00:03:20.300 --> 00:03:23.890 at some particular instant of time, 00:03:23.890 --> 00:03:28.085 occupying this 6N-dimensional phase space, 00:03:28.085 --> 00:03:32.330 out of which I can construct some kind of a density in phase 00:03:32.330 --> 00:03:34.950 space. 00:03:34.950 --> 00:03:39.320 But then I realize that since each one of these trajectories 00:03:39.320 --> 00:03:42.490 is evolving according to this Hamiltonian, 00:03:42.490 --> 00:03:45.320 this density could potentially be a function of time. 00:03:48.030 --> 00:03:52.710 And we described the equation for the evolution 00:03:52.710 --> 00:03:54.850 of that density with time. 00:03:54.850 --> 00:03:59.980 And we could write it in this form-- V rho by dt 00:03:59.980 --> 00:04:05.350 is the Poisson bracket of the Hamiltonian with rho. 00:04:05.350 --> 00:04:10.120 And this Poisson bracket was defined 00:04:10.120 --> 00:04:13.650 as a sum over all of your coordinates. 00:04:16.339 --> 00:04:16.839 Oops. 00:04:22.490 --> 00:04:34.660 We had d rho by d vector qi, dot product with dH by dpi minus 00:04:34.660 --> 00:04:36.116 the other [? way. ?] 00:04:42.470 --> 00:04:47.740 So there are essentially three N terms-- well, 00:04:47.740 --> 00:04:52.920 actually, six N terms, but three N pairs of terms in this sum. 00:04:52.920 --> 00:04:55.890 I can either use some index alpha 00:04:55.890 --> 00:04:59.460 running from one to three N, or indicate them 00:04:59.460 --> 00:05:05.100 as contributions of things that come from individual particles 00:05:05.100 --> 00:05:08.210 and then use this notation with three vectors. 00:05:08.210 --> 00:05:11.280 So this is essentially a combination 00:05:11.280 --> 00:05:14.440 of sum of three terms. 00:05:14.440 --> 00:05:16.900 OK? 00:05:16.900 --> 00:05:22.910 So I hope that somehow this equation 00:05:22.910 --> 00:05:28.420 can describe the evolution that goes on over here. 00:05:28.420 --> 00:05:32.660 And ultimately, when I wait sufficiently long time, 00:05:32.660 --> 00:05:36.910 I will reach a situation where d rho by dt 00:05:36.910 --> 00:05:39.455 does not change anymore. 00:05:39.455 --> 00:05:43.830 And I will find some density that is invariant on time. 00:05:43.830 --> 00:05:46.650 I'll call that rho equilibrium. 00:05:46.650 --> 00:05:51.810 And so we saw that we could have our rho equilibrium, which then 00:05:51.810 --> 00:05:58.620 should have zero Poisson bracket with H to be a function of H 00:05:58.620 --> 00:06:00.140 and any other conserved quantity. 00:06:05.900 --> 00:06:08.780 OK? 00:06:08.780 --> 00:06:14.582 So in principle, we sort of thought 00:06:14.582 --> 00:06:21.030 of a way of describing how the system will evolve 00:06:21.030 --> 00:06:24.660 in equilibrium-- towards an equilibrium. 00:06:24.660 --> 00:06:30.290 And indeed, we will find that in statistical mechanics later on, 00:06:30.290 --> 00:06:35.340 we are going to rely heavily on these descriptions 00:06:35.340 --> 00:06:38.370 of equilibrium densities, which are 00:06:38.370 --> 00:06:41.090 governed by only the Hamiltonian. 00:06:41.090 --> 00:06:43.850 And if there are any other conserved quantities, 00:06:43.850 --> 00:06:45.150 typically there are not. 00:06:45.150 --> 00:06:49.650 So this is the general form that we'll ultimately 00:06:49.650 --> 00:06:52.630 be using a lot. 00:06:52.630 --> 00:06:56.730 But we started with a description 00:06:56.730 --> 00:06:58.820 of evolution of these coordinates 00:06:58.820 --> 00:07:01.420 in time, which is time-reversible. 00:07:05.870 --> 00:07:08.835 And we hope to end up with a situation that 00:07:08.835 --> 00:07:12.180 is analogous to what we describe thermodynamically, 00:07:12.180 --> 00:07:18.410 we describe something that goes to some particular state, 00:07:18.410 --> 00:07:20.230 and basically stays there, as far 00:07:20.230 --> 00:07:23.620 as the macroscopic description is concerned. 00:07:23.620 --> 00:07:29.550 So did we somehow manage to just look at something else, which 00:07:29.550 --> 00:07:32.470 is this density, and achieve this transition 00:07:32.470 --> 00:07:35.520 from reversibility to irreversibility? 00:07:35.520 --> 00:07:36.760 And the answer is no. 00:07:36.760 --> 00:07:40.610 This equation-- also, if you find, indeed, 00:07:40.610 --> 00:07:47.230 a rho that goes from here to here, you can set t to minus t 00:07:47.230 --> 00:07:49.705 and get a rho that goes from back here to here. 00:07:49.705 --> 00:07:52.500 It has the same time-reversibility. 00:07:52.500 --> 00:07:56.400 So somehow, we have to find another solution. 00:07:56.400 --> 00:08:00.180 And the solution that we will gradually build upon 00:08:00.180 --> 00:08:03.150 relies more on physics rather than 00:08:03.150 --> 00:08:06.850 the rigorous mathematical nature of this thing. 00:08:06.850 --> 00:08:09.310 Physically, we know that this happens. 00:08:09.310 --> 00:08:11.560 All of us have seen this. 00:08:11.560 --> 00:08:16.026 So somehow, we should be able to use physical approximation 00:08:16.026 --> 00:08:18.860 and assumptions that are compatible with what 00:08:18.860 --> 00:08:20.260 is observed. 00:08:20.260 --> 00:08:24.280 And if, during that, we have to sort of make 00:08:24.280 --> 00:08:27.450 mathematical approximations, so be it. 00:08:27.450 --> 00:08:30.330 In fact, we have to make mathematical approximations, 00:08:30.330 --> 00:08:35.620 because otherwise, there is this very strong reversibility 00:08:35.620 --> 00:08:37.000 condition. 00:08:37.000 --> 00:08:41.030 So let's see how we are about to proceed, thinking more 00:08:41.030 --> 00:08:43.820 in terms of physics. 00:08:43.820 --> 00:08:47.840 Now, the information that I have over here, 00:08:47.840 --> 00:08:50.335 either in this description of ensemble 00:08:50.335 --> 00:08:53.220 or in this description in terms of trajectories, 00:08:53.220 --> 00:08:55.960 is just enormous. 00:08:55.960 --> 00:09:01.090 I can't think of any physical process which 00:09:01.090 --> 00:09:05.990 would need to keep track of the joined positions of coordinates 00:09:05.990 --> 00:09:08.900 and momenta of 10 to the 23 particles, 00:09:08.900 --> 00:09:13.190 and know them with infinite precision, et cetera. 00:09:13.190 --> 00:09:15.280 Things that I make observations with 00:09:15.280 --> 00:09:18.790 and I say, we see this, well, what do we see? 00:09:18.790 --> 00:09:24.090 We see that something has some kind of a density over here. 00:09:24.090 --> 00:09:26.010 It has some kind of pressure. 00:09:26.010 --> 00:09:28.720 Maybe there is, in the middle of this process, 00:09:28.720 --> 00:09:30.530 some flow of gas particles. 00:09:30.530 --> 00:09:33.420 We can talk about the velocity of those things. 00:09:33.420 --> 00:09:35.130 And let's say even when we are sort 00:09:35.130 --> 00:09:39.460 of being very non-equilibrium and thinking about velocity 00:09:39.460 --> 00:09:43.550 of those particles going from one side to the other side, 00:09:43.550 --> 00:09:48.060 we really don't care which particle among the 10 to the 23 00:09:48.060 --> 00:09:50.520 is at some instant of time contributing 00:09:50.520 --> 00:09:55.130 to the velocity of the gas that is swishing past. 00:09:55.130 --> 00:09:59.720 So clearly, any physical observable that we make 00:09:59.720 --> 00:10:03.780 is something that does not require 00:10:03.780 --> 00:10:05.890 all of those degrees of freedom. 00:10:05.890 --> 00:10:08.060 So let's just construct some of those things 00:10:08.060 --> 00:10:10.480 that we may find useful and see how 00:10:10.480 --> 00:10:13.170 we would describe evolutions of them. 00:10:13.170 --> 00:10:16.020 I mean, the most useful thing, indeed, is the density. 00:10:16.020 --> 00:10:19.850 So what I could do is I can actually 00:10:19.850 --> 00:10:23.210 construct a one-particle density. 00:10:23.210 --> 00:10:26.455 So I want to look at some position in space 00:10:26.455 --> 00:10:30.530 at some time t and ask whether or not 00:10:30.530 --> 00:10:34.960 there are particles there. 00:10:34.960 --> 00:10:37.220 I don't care which one of the particles. 00:10:37.220 --> 00:10:40.300 Actually, let's put in a little bit more information; 00:10:40.300 --> 00:10:44.090 also, keep track of whether, at some instant of time, 00:10:44.090 --> 00:10:47.350 I see at this location particles. 00:10:47.350 --> 00:10:49.370 And these particles, I will also ask 00:10:49.370 --> 00:10:51.020 which direction they are moving. 00:10:51.020 --> 00:10:53.650 Maybe I'm also going to think about the case 00:10:53.650 --> 00:10:57.640 where, in the intermediate, I am flowing from one side 00:10:57.640 --> 00:10:58.270 to another. 00:10:58.270 --> 00:11:01.840 And keeping track of both of these is important. 00:11:01.840 --> 00:11:06.510 I said I don't care which one of these particles 00:11:06.510 --> 00:11:09.340 is contributing. 00:11:09.340 --> 00:11:18.180 So that means that I have to sum over all particles 00:11:18.180 --> 00:11:23.345 and ask whether or not, at time t, 00:11:23.345 --> 00:11:29.818 there is a particle at location q with momentum p. 00:11:32.890 --> 00:11:37.490 So these delta functions are supposed to enforce that. 00:11:37.490 --> 00:11:41.860 And again, I'm not thinking about an individual trajectory, 00:11:41.860 --> 00:11:44.750 because I think, more or less, all of the trajectories 00:11:44.750 --> 00:11:46.970 are going to behave the same way. 00:11:46.970 --> 00:11:50.940 And so what I will do is I take an ensemble 00:11:50.940 --> 00:11:53.161 average of this quantity. 00:11:53.161 --> 00:11:53.660 OK? 00:11:53.660 --> 00:11:55.300 So what does that mean? 00:11:55.300 --> 00:11:57.650 To do an ensemble average, we said 00:11:57.650 --> 00:12:04.200 I have to integrate the density function, rho, 00:12:04.200 --> 00:12:06.970 which depends on all of the coordinates. 00:12:06.970 --> 00:12:09.910 So I have coordinate for particle one, 00:12:09.910 --> 00:12:14.780 coordinate-- particle and momentum, particle and momentum 00:12:14.780 --> 00:12:17.690 for particle two, for particle three, 00:12:17.690 --> 00:12:22.750 all the way to particle number N. 00:12:22.750 --> 00:12:24.970 And this is something that depends on time. 00:12:24.970 --> 00:12:26.920 That's where the time dependence comes from. 00:12:26.920 --> 00:12:29.030 And again, let's think about the time dependence 00:12:29.030 --> 00:12:31.620 where 0 is you lift the partition 00:12:31.620 --> 00:12:34.750 and you allow things to go from one to the other. 00:12:34.750 --> 00:12:38.950 So there is a non-equilibrium process that you are following. 00:12:38.950 --> 00:12:44.120 What I need to do is I have to multiply this density 00:12:44.120 --> 00:12:50.130 with the function that I have to consider the average. 00:12:50.130 --> 00:12:53.790 Well, the function being a delta function, it's very nice. 00:12:53.790 --> 00:12:59.980 The first term in the sum, we'll set simply q1 equals to q. 00:12:59.980 --> 00:13:04.440 And it will set p1 equals to p when 00:13:04.440 --> 00:13:08.490 I do the integration over p1 and q1. 00:13:08.490 --> 00:13:12.940 But then I'm left to have to do the integration over particle 00:13:12.940 --> 00:13:20.155 two, particle three, particle four, and so forth. 00:13:20.155 --> 00:13:21.580 OK? 00:13:21.580 --> 00:13:24.930 But this is only the first term in the sum. 00:13:24.930 --> 00:13:27.480 Then I have to write the sum for particle number two, 00:13:27.480 --> 00:13:30.220 particle number three, et cetera. 00:13:30.220 --> 00:13:33.070 But all of the particles, as far as I'm concerned, 00:13:33.070 --> 00:13:34.720 are behaving the same way. 00:13:34.720 --> 00:13:44.510 So I just need to multiply this by a factor of N. 00:13:44.510 --> 00:13:49.700 Now, this is something that actually we encountered before. 00:13:49.700 --> 00:13:54.500 Recall that this rho is a joint probability distribution. 00:13:54.500 --> 00:13:56.600 It's a joint probability distribution 00:13:56.600 --> 00:14:01.010 of all N particles, their locations, and momentum. 00:14:01.010 --> 00:14:10.510 And we gave a name to taking a joint probability density 00:14:10.510 --> 00:14:13.180 function, which this is, and integrating 00:14:13.180 --> 00:14:16.266 over some subset of variables. 00:14:16.266 --> 00:14:21.150 The answer is an unconditional probability distribution. 00:14:21.150 --> 00:14:24.380 So essentially, the end of this story 00:14:24.380 --> 00:14:27.630 is if I integrate over coordinates 00:14:27.630 --> 00:14:32.060 of two all the way to N, I will get an unconditional result 00:14:32.060 --> 00:14:34.840 pertaining to the first set of coordinate. 00:14:34.840 --> 00:14:38.470 So this is the same thing up to a factor of N-- 00:14:38.470 --> 00:14:43.110 the unconditional probability that I will call rho one. 00:14:43.110 --> 00:14:44.885 Actually, I should have called this f1. 00:14:47.830 --> 00:14:50.570 So this is something that is defined 00:14:50.570 --> 00:14:52.470 to be the one particle density. 00:14:58.540 --> 00:15:04.430 I don't know why this name stuck to it, because this one, that 00:15:04.430 --> 00:15:07.220 is up to a factor of N different from it, 00:15:07.220 --> 00:15:13.340 is our usual unconditional probability for one particle. 00:15:13.340 --> 00:15:28.944 So this is rho of p1 being p, q1 being q at time. 00:15:28.944 --> 00:15:30.850 OK? 00:15:30.850 --> 00:15:35.930 So essentially, what I've said is I have a joint probability. 00:15:35.930 --> 00:15:37.990 I'm really interested in one of the particles, 00:15:37.990 --> 00:15:41.890 so I just integrate over all the others. 00:15:41.890 --> 00:15:47.190 And this is kind of not very elegant, 00:15:47.190 --> 00:15:49.100 but that's somehow the way that it 00:15:49.100 --> 00:15:51.090 has appeared in the literature. 00:15:51.090 --> 00:15:53.790 The entity that I have on the right 00:15:53.790 --> 00:15:59.730 is the properly normalized probability. 00:15:59.730 --> 00:16:05.960 Once I multiply by N, it's a quantity this is called f1. 00:16:05.960 --> 00:16:10.130 And it's called a density and used very much 00:16:10.130 --> 00:16:12.170 in the literature. 00:16:12.170 --> 00:16:13.236 OK? 00:16:13.236 --> 00:16:16.240 And that's the thing that I think will be useful. 00:16:16.240 --> 00:16:20.420 Essentially, all of the things that I 00:16:20.420 --> 00:16:23.020 know about observations of a system, 00:16:23.020 --> 00:16:26.870 such as its density, its velocity, et cetera, 00:16:26.870 --> 00:16:30.080 I should be able to get from this. 00:16:30.080 --> 00:16:33.800 Sometimes we need a little bit more information, 00:16:33.800 --> 00:16:36.610 so I allow for the possibility that I may also 00:16:36.610 --> 00:16:41.570 need to focus on a two-particle density, where 00:16:41.570 --> 00:16:48.710 I keep information pertaining to two particles at time t 00:16:48.710 --> 00:16:52.570 and I integrate over everybody else that I'm not interested. 00:16:52.570 --> 00:16:58.180 So I introduce an integration over coordinates number 00:16:58.180 --> 00:17:03.720 two-- sorry, number three all the way to N 00:17:03.720 --> 00:17:09.280 to not have to repeat this combination all the time. 00:17:09.280 --> 00:17:12.520 The 6N-dimensional phase space for particle 00:17:12.520 --> 00:17:15.839 i I will indicate by dVi. 00:17:15.839 --> 00:17:24.329 So I take the full N particle density, if you like, 00:17:24.329 --> 00:17:28.160 integrate over everybody except 2. 00:17:28.160 --> 00:17:32.950 And up to a normalization of NN minus 1, 00:17:32.950 --> 00:17:35.940 this is called a two-particle density. 00:17:35.940 --> 00:17:41.160 And again, absent this normalization, 00:17:41.160 --> 00:17:44.750 this is simply the unconditional probability 00:17:44.750 --> 00:17:47.592 that I would construct out of this joint probability 00:17:47.592 --> 00:17:48.092 [INAUDIBLE]. 00:17:55.970 --> 00:18:00.340 And just for some mathematical purposes, the generalization 00:18:00.340 --> 00:18:05.370 of this to S particles, we will call fs. 00:18:05.370 --> 00:18:11.510 So this depends on p1 through qs at time t. 00:18:11.510 --> 00:18:14.740 And this is going to be N factorial divided 00:18:14.740 --> 00:18:18.910 by N minus S factorial in general. 00:18:18.910 --> 00:18:24.110 And they join the unconditional probability that depends on S 00:18:24.110 --> 00:18:29.466 coordinates p1 through qs at time t. 00:18:34.312 --> 00:18:34.812 OK? 00:18:37.740 --> 00:18:40.860 You'll see why I need these higher ones 00:18:40.860 --> 00:18:47.620 although, in reality, all my interest is on this first one, 00:18:47.620 --> 00:18:50.120 because practically everything that I need 00:18:50.120 --> 00:18:54.190 to know about this initial experiment that I set up 00:18:54.190 --> 00:18:57.550 and how a gas expands I should be able to extract 00:18:57.550 --> 00:19:00.004 from this one-particle density. 00:19:00.004 --> 00:19:02.220 OK? 00:19:02.220 --> 00:19:05.350 So how do I calculate this? 00:19:05.350 --> 00:19:08.770 Well, what I would need to do is to look 00:19:08.770 --> 00:19:15.150 at the time variation of fs with t 00:19:15.150 --> 00:19:20.290 to calculate the time dependence of any one of these quantities. 00:19:20.290 --> 00:19:24.040 Ultimately, I want to have an equation for df1 by dt. 00:19:24.040 --> 00:19:26.330 But let's write the general one. 00:19:26.330 --> 00:19:33.850 So the general one is up to, again, this N factorial, 00:19:33.850 --> 00:19:44.510 N minus s factorial and integral over 00:19:44.510 --> 00:19:50.740 coordinates that I'm not interested, of d rho by dt. 00:19:55.830 --> 00:19:56.380 OK? 00:19:56.380 --> 00:20:01.080 So I just take the time derivative inside the integral, 00:20:01.080 --> 00:20:04.580 because I know how d rho by dt evolves. 00:20:04.580 --> 00:20:11.900 And so this is going to be simply the Poisson bracket of H 00:20:11.900 --> 00:20:13.190 and rho. 00:20:13.190 --> 00:20:14.540 And I would proceed from there. 00:20:17.980 --> 00:20:20.150 Actually, to proceed further and in order 00:20:20.150 --> 00:20:25.870 to be able to say things that are related to physics, 00:20:25.870 --> 00:20:28.410 I need to say something about the Hamiltonian. 00:20:28.410 --> 00:20:32.940 I can't do so in the most general case. 00:20:32.940 --> 00:20:35.030 So let's write the kind of Hamiltonian 00:20:35.030 --> 00:20:39.930 that we are interested and describes the gas-- so 00:20:39.930 --> 00:20:44.530 the Hamiltonian for the gas in this room. 00:20:44.530 --> 00:20:46.640 One term is simply the kinetic energy. 00:20:52.360 --> 00:20:55.420 Another term is that gas particles 00:20:55.420 --> 00:20:57.520 are confined by some potential. 00:20:57.520 --> 00:21:01.740 The potential could be as easy as the walls of this room. 00:21:01.740 --> 00:21:05.370 Or it could be some more general potential 00:21:05.370 --> 00:21:09.660 that could include gravity, whatever else you want. 00:21:09.660 --> 00:21:14.850 So let's include that possibility. 00:21:14.850 --> 00:21:17.680 So these are so-called one-body terms, 00:21:17.680 --> 00:21:21.670 because they pertain to the coordinates of one particle. 00:21:21.670 --> 00:21:24.225 And then there are two-body terms. 00:21:24.225 --> 00:21:27.840 So for example, I could look at all pairs of particles. 00:21:27.840 --> 00:21:33.480 Let's say i not equal to j to avoid self-interaction. 00:21:33.480 --> 00:21:36.730 V of qi minus qj. 00:21:36.730 --> 00:21:40.910 So certainly, two particles in this gas in this room, 00:21:40.910 --> 00:21:44.760 when they get close enough, they certainly 00:21:44.760 --> 00:21:46.290 can pass through each other. 00:21:46.290 --> 00:21:47.840 They each have a size. 00:21:47.840 --> 00:21:51.970 But even when they are a few times their sizes, 00:21:51.970 --> 00:21:54.120 they start to feel some interaction, which 00:21:54.120 --> 00:21:55.270 causes them to collide. 00:21:55.270 --> 00:21:56.269 Yes. 00:21:56.269 --> 00:21:58.145 AUDIENCE: The whole series of arguments 00:21:58.145 --> 00:22:00.021 that we're developing, do these only hold 00:22:00.021 --> 00:22:01.897 for time-independent potentials? 00:22:01.897 --> 00:22:04.620 PROFESSOR: Yes. 00:22:04.620 --> 00:22:07.820 Although, it is not that difficult to sort of 00:22:07.820 --> 00:22:10.200 go through all of these arguments 00:22:10.200 --> 00:22:13.940 and see where the corresponding modifications are going to be. 00:22:13.940 --> 00:22:15.860 But certainly, the very first thing 00:22:15.860 --> 00:22:21.030 that we are using, which is this one, we kind of implicitly 00:22:21.030 --> 00:22:23.810 assume the time-independent Hamiltonian. 00:22:23.810 --> 00:22:27.370 So you have to start changing things from that point. 00:22:30.615 --> 00:22:31.115 OK? 00:22:34.090 --> 00:22:37.071 So in principle, you could have three-body and higher body 00:22:37.071 --> 00:22:37.570 terms. 00:22:37.570 --> 00:22:40.990 But essentially, most of the relevant physics 00:22:40.990 --> 00:22:42.330 is captured by this. 00:22:42.330 --> 00:22:44.970 Even for things like plasmas, this 00:22:44.970 --> 00:22:48.400 would be a reasonably good approximation, 00:22:48.400 --> 00:22:51.630 with the Coulomb interaction appearing here. 00:22:51.630 --> 00:22:53.860 OK? 00:22:53.860 --> 00:23:00.500 Now, what I note is that what I have to calculate over here 00:23:00.500 --> 00:23:04.710 is a Poisson bracket. 00:23:04.710 --> 00:23:08.760 And then I have to integrate that Poisson bracket. 00:23:08.760 --> 00:23:10.860 Poisson bracket involves a whole bunch 00:23:10.860 --> 00:23:14.620 of derivatives over all set of quantities. 00:23:14.620 --> 00:23:20.200 And I realize that when I'm integrating over derivatives, 00:23:20.200 --> 00:23:22.280 there are simplifications that can 00:23:22.280 --> 00:23:25.410 take place, such as integration by parts. 00:23:25.410 --> 00:23:30.260 But that only will take place when the derivative 00:23:30.260 --> 00:23:34.590 is one of the variables that is being integrated. 00:23:34.590 --> 00:23:41.370 Now, this whole process has separated out arguments of rho, 00:23:41.370 --> 00:23:44.100 as far as this expression is concerned, 00:23:44.100 --> 00:23:48.980 into two sets-- one set, or the set that is appearing out here, 00:23:48.980 --> 00:23:53.820 the first s ones that don't undergo the integration, 00:23:53.820 --> 00:23:58.250 and then the remainder that do undergo the integration. 00:23:58.250 --> 00:24:02.020 So this is going to be relevant when we do our manipulations. 00:24:02.020 --> 00:24:06.400 And therefore, it is useful to rewrite this Hamiltonian 00:24:06.400 --> 00:24:07.725 in terms of three entities. 00:24:12.560 --> 00:24:15.660 The first one that I call H sub s-- so if you like, 00:24:15.660 --> 00:24:18.550 this is an end particle Hamiltonian. 00:24:18.550 --> 00:24:22.720 I can write an H sub s, which is just exactly the same thing, 00:24:22.720 --> 00:24:25.560 except that it applies to coordinates 00:24:25.560 --> 00:24:27.920 that I'm not integrating over. 00:24:27.920 --> 00:24:30.390 And to sort of make a distinction, 00:24:30.390 --> 00:24:35.870 I will label them by N. And it includes the interaction 00:24:35.870 --> 00:24:37.770 among those particles. 00:24:45.900 --> 00:24:48.930 And I can similarly write something 00:24:48.930 --> 00:24:55.780 that pertains to coordinates that I am integrating over. 00:24:55.780 --> 00:24:58.930 I will label them by j and k. 00:24:58.930 --> 00:25:10.010 So this is s plus 1 to N, pj squared over 2m plus u of qj 00:25:10.010 --> 00:25:17.820 plus 1/2 sum over j and k, V of qj minus qk. 00:25:20.858 --> 00:25:23.790 OK? 00:25:23.790 --> 00:25:27.420 So everything that involves one set of coordinates, everything 00:25:27.420 --> 00:25:29.310 that involves the other set of coordinates. 00:25:29.310 --> 00:25:35.090 So what is left are terms that [? copy ?] one set of 00:25:35.090 --> 00:25:37.090 coordinates to another. 00:25:37.090 --> 00:25:41.680 So N running from 1 to s, j running from s plus 1 00:25:41.680 --> 00:25:48.340 to N, V between qm and qj. 00:25:48.340 --> 00:25:48.840 OK? 00:25:57.970 --> 00:26:01.980 So let me rewrite this equation rather than in terms 00:26:01.980 --> 00:26:09.450 of-- f in terms of the probabilities' rhos. 00:26:09.450 --> 00:26:13.890 So the difference only is that I don't 00:26:13.890 --> 00:26:17.210 have to include this factor out front. 00:26:17.210 --> 00:26:19.610 So I have dVi. 00:26:19.610 --> 00:26:27.150 I have the d rho by dt, which is the commutator of H with rho, 00:26:27.150 --> 00:26:33.636 which I have been writing as Hs plus HN minus s. 00:26:33.636 --> 00:26:35.920 I'm changing the way I write s. 00:26:35.920 --> 00:26:37.080 Sorry. 00:26:37.080 --> 00:26:41.376 Plus H prime and rho. 00:26:41.376 --> 00:26:44.250 OK? 00:26:44.250 --> 00:26:46.650 So there is a bit of mathematics to be 00:26:46.650 --> 00:26:50.160 performed to analyze this. 00:26:50.160 --> 00:26:53.400 There are three terms that I will label a, b, 00:26:53.400 --> 00:26:59.160 and c, which are the three Poisson 00:26:59.160 --> 00:27:02.450 brackets that I have to evaluate. 00:27:02.450 --> 00:27:06.310 So the contribution that I call a 00:27:06.310 --> 00:27:15.430 is the integral over coordinates s 00:27:15.430 --> 00:27:24.020 plus 1 to N, of the Poisson bracket of Hs with rho. 00:27:24.020 --> 00:27:29.130 Now, the Poisson bracket is given up here. 00:27:29.130 --> 00:27:35.650 It is a sum that involves N terms over all N particles. 00:27:35.650 --> 00:27:42.540 But since, if I'm evaluating it for Hs and Hs 00:27:42.540 --> 00:27:46.270 will only give nonzero derivatives with respect 00:27:46.270 --> 00:27:49.400 to the coordinates that are present in it, 00:27:49.400 --> 00:27:51.755 this sum of N terms actually becomes simply 00:27:51.755 --> 00:27:54.850 a sum of small N terms. 00:27:54.850 --> 00:28:01.860 So I will get a sum over N running from 1 to s. 00:28:01.860 --> 00:28:03.570 These are the only terms. 00:28:03.570 --> 00:28:11.715 And I will get the things that I have for d rho by dpN-- sorry, 00:28:11.715 --> 00:28:25.496 rho by dqN, dHs by dpN minus d rho by dpN, dHs by dqN. 00:28:33.207 --> 00:28:33.707 OK? 00:28:39.040 --> 00:28:40.634 Did I make a mistake? 00:28:40.634 --> 00:28:46.418 AUDIENCE: Isn't it the Poisson bracket of rho H, not H rho? 00:28:46.418 --> 00:28:47.382 PROFESSOR: Rho. 00:28:51.250 --> 00:28:51.970 Oh, yes. 00:28:51.970 --> 00:28:54.720 So I have to put a minus sign here. 00:28:54.720 --> 00:28:55.220 Right. 00:28:55.220 --> 00:28:56.200 Good. 00:28:56.200 --> 00:28:57.180 Thank you. 00:29:01.590 --> 00:29:02.580 OK. 00:29:02.580 --> 00:29:08.740 Now, note that these derivatives and operations that 00:29:08.740 --> 00:29:13.800 involve Hs involve coordinates that are not 00:29:13.800 --> 00:29:18.460 appearing in the integration process. 00:29:18.460 --> 00:29:20.770 So I can take these entities that 00:29:20.770 --> 00:29:23.190 do not depend on variables that are part 00:29:23.190 --> 00:29:30.080 of the integration outside the integration, which means that I 00:29:30.080 --> 00:29:35.590 can then write the result as being 00:29:35.590 --> 00:29:44.480 an exchange of the order of the Poisson bracket 00:29:44.480 --> 00:29:47.850 and the integration. 00:29:47.850 --> 00:30:04.750 So I would essentially have the integration only appear here, 00:30:04.750 --> 00:30:17.940 which is the same thing as Hs and rho s. 00:30:17.940 --> 00:30:19.350 This should be rho s. 00:30:24.800 --> 00:30:28.200 This is the definition of rho s, which is the same thing as rho. 00:30:31.492 --> 00:30:31.992 OK. 00:30:34.807 --> 00:30:36.015 What does it mean physically? 00:30:36.015 --> 00:30:40.430 So it's actually much easier to tell you 00:30:40.430 --> 00:30:43.810 what it means physically than to do the math. 00:30:43.810 --> 00:30:48.500 So what we saw was happening was that if I 00:30:48.500 --> 00:30:52.960 have a Hamiltonian that describes N particles, then 00:30:52.960 --> 00:30:58.370 for that Hamiltonian, the corresponding density 00:30:58.370 --> 00:30:59.980 satisfies this Liouville equation. 00:30:59.980 --> 00:31:02.810 D rho by dt is HN rho. 00:31:02.810 --> 00:31:07.620 And this was a consequence of this divergence-less character 00:31:07.620 --> 00:31:09.810 of the flow that we have in this space, 00:31:09.810 --> 00:31:13.620 that the equations that we write down over here 00:31:13.620 --> 00:31:17.220 for p dot and q dot in terms of H 00:31:17.220 --> 00:31:21.820 had this character that the divergence was 0. 00:31:21.820 --> 00:31:23.410 OK? 00:31:23.410 --> 00:31:28.230 Now, this is true if I have any number of particles. 00:31:28.230 --> 00:31:34.410 So if I focus simply on s of the particles, 00:31:34.410 --> 00:31:38.416 and they are governed by this Hamiltonian, 00:31:38.416 --> 00:31:41.640 and I don't have anything else in the universe, 00:31:41.640 --> 00:31:44.400 as far as this Hamiltonian is concerned, 00:31:44.400 --> 00:31:48.740 I should have the analog of a Liouville equation. 00:31:48.740 --> 00:31:52.720 So the term that I have obtained over there from this first term 00:31:52.720 --> 00:31:56.430 is simply stating that d rho s by dt, 00:31:56.430 --> 00:31:59.620 if there was no other interaction with anybody else, 00:31:59.620 --> 00:32:02.960 would simply satisfy the corresponding Liouville 00:32:02.960 --> 00:32:06.170 equation for s particles. 00:32:06.170 --> 00:32:09.620 And because of that, we also expect and anticipate-- 00:32:09.620 --> 00:32:11.990 and I'll show that mathematically-- 00:32:11.990 --> 00:32:13.890 that the next term in the series, 00:32:13.890 --> 00:32:18.800 that is the Poisson bracket of N minus s and rho, 00:32:18.800 --> 00:32:23.760 should be 0, because as far as this s particles 00:32:23.760 --> 00:32:28.780 that I'm focusing on and how they evolve, 00:32:28.780 --> 00:32:33.290 they really don't care about what all the other particles 00:32:33.290 --> 00:32:35.970 are doing if they are not [INAUDIBLE]. 00:32:35.970 --> 00:32:37.685 So anything interesting should ultimately 00:32:37.685 --> 00:32:39.940 come from this third term. 00:32:39.940 --> 00:32:44.530 But let's actually go and do the calculation for the second term 00:32:44.530 --> 00:32:49.510 to show that this anticipation that the answer should be 0 00:32:49.510 --> 00:32:51.190 does hold up and why. 00:32:55.100 --> 00:33:02.410 So for the second term, I need to calculate a similar Poisson 00:33:02.410 --> 00:33:05.690 bracket, except that this second Poisson 00:33:05.690 --> 00:33:09.290 bracket involves H of N minus s. 00:33:09.290 --> 00:33:14.790 And H of N minus s, when I put in the full sum, 00:33:14.790 --> 00:33:19.530 will only get contribution from terms that start from s plus 1. 00:33:19.530 --> 00:33:22.100 So the same way that that started from N, 00:33:22.100 --> 00:33:27.840 this contribution starts from s plus 1 to N. 00:33:27.840 --> 00:33:31.360 And actually, I can just write the whole thing as above, d 00:33:31.360 --> 00:33:50.570 rho by d qj dotted by d HN minus s by dpj, plus d rho by-- no, 00:33:50.570 --> 00:34:01.990 this is the rho. d rho by d pj dotted by dHN minus s by dqj. 00:34:11.340 --> 00:34:16.010 So now I have a totally different situation 00:34:16.010 --> 00:34:19.010 from the previous case, because the previous case, 00:34:19.010 --> 00:34:22.630 the derivatives were over things I was not integrating. 00:34:22.630 --> 00:34:26.500 I could take outside the integral. 00:34:26.500 --> 00:34:29.010 Now all of the derivatives involve things 00:34:29.010 --> 00:34:31.252 that I'm integrating over. 00:34:31.252 --> 00:34:36.090 Now, when that happens, then you do integration by parts. 00:34:36.090 --> 00:34:42.620 So what you do is you take rho outside 00:34:42.620 --> 00:34:47.400 and let the derivative act on everything else. 00:34:47.400 --> 00:34:49.080 OK? 00:34:49.080 --> 00:34:54.219 So what do we end up with if we do integration by parts? 00:35:02.640 --> 00:35:06.140 I will get surface terms. 00:35:06.140 --> 00:35:09.130 Surface terms are essentially rho-evaluated when 00:35:09.130 --> 00:35:11.660 the coordinates are at infinity or at the edge 00:35:11.660 --> 00:35:14.360 of your space, where rho is 0. 00:35:14.360 --> 00:35:17.080 So there is no surface term. 00:35:17.080 --> 00:35:19.140 There is an overall change in sign, 00:35:19.140 --> 00:35:23.540 so I will get a product i running from s plus 1 00:35:23.540 --> 00:35:27.420 to N, dVi. 00:35:27.420 --> 00:35:30.640 Now the rho comes outside. 00:35:30.640 --> 00:35:34.840 And the derivative acts on everything that is left. 00:35:34.840 --> 00:35:37.470 So the first term will give me a second derivative 00:35:37.470 --> 00:35:45.930 of HN minus s, with respect to p, with respect to q. 00:35:45.930 --> 00:35:49.560 And the second term will be essentially the opposite way 00:35:49.560 --> 00:35:50.560 of doing the derivative. 00:35:57.130 --> 00:35:58.787 And these two are, of course, the same. 00:35:58.787 --> 00:35:59.620 And the answer is 0. 00:36:03.106 --> 00:36:04.600 OK? 00:36:04.600 --> 00:36:10.845 So we do expect that the evolution of all 00:36:10.845 --> 00:36:13.650 the other particles should not affect 00:36:13.650 --> 00:36:15.920 the subset that we are looking at. 00:36:15.920 --> 00:36:18.550 And that's worn out also. 00:36:18.550 --> 00:36:20.710 So the only thing that potentially 00:36:20.710 --> 00:36:26.470 will be relevant and exciting is the last term, number c. 00:36:26.470 --> 00:36:29.030 So let's take a look at that. 00:36:29.030 --> 00:36:34.170 So here, I have to do an integration over variables 00:36:34.170 --> 00:36:38.580 that I am not interested. 00:36:38.580 --> 00:36:49.470 And then I need now, however, to do a full Poisson 00:36:49.470 --> 00:36:53.080 bracket of a whole bunch of terms, 00:36:53.080 --> 00:36:55.900 because now the terms that I'm looking at 00:36:55.900 --> 00:36:58.510 have coordinates from both sets. 00:36:58.510 --> 00:37:01.200 So I have to be a little bit careful. 00:37:01.200 --> 00:37:10.160 So let me just make sure that I follow the notes that I 00:37:10.160 --> 00:37:11.720 have here and don't make mistakes. 00:37:15.620 --> 00:37:17.140 OK. 00:37:17.140 --> 00:37:19.935 So this H prime involves two sums. 00:37:23.260 --> 00:37:31.730 So I will write the first sum, N running from 1 to s. 00:37:31.730 --> 00:37:37.480 And then I have the second sum, j running from s 00:37:37.480 --> 00:37:42.890 plus 1 to N. What do I need? 00:37:42.890 --> 00:38:01.774 I need the-- OK. 00:38:04.740 --> 00:38:07.490 Let's do it the following way. 00:38:07.490 --> 00:38:12.910 So what I have to do for the Poisson bracket 00:38:12.910 --> 00:38:16.690 is a sum that involves all coordinates. 00:38:16.690 --> 00:38:20.070 So let's just write this whole expression. 00:38:20.070 --> 00:38:23.700 But first, for coordinates 1 through s. 00:38:23.700 --> 00:38:27.500 So I have a sum N running from 1 to s. 00:38:27.500 --> 00:38:29.360 And then I will write the term that 00:38:29.360 --> 00:38:32.456 corresponds to coordinates s plus 1 to m. 00:38:32.456 --> 00:38:37.400 For the first set of coordinates, what do I have? 00:38:37.400 --> 00:38:43.520 I have d rho by dqn. 00:38:43.520 --> 00:38:47.510 And then I have d H prime by dpn. 00:38:47.510 --> 00:38:50.410 So I didn't write H prime explicitly. 00:38:50.410 --> 00:38:54.150 I'm just breaking the sum over here. 00:38:54.150 --> 00:39:04.590 And then I have sum j running from s plus 1 to N, 00:39:04.590 --> 00:39:14.381 d rho by dqj times d H prime by dpj. 00:39:14.381 --> 00:39:20.690 And again, my H prime is this entity over here 00:39:20.690 --> 00:39:25.100 that [? copies ?] coordinates from both sets. 00:39:25.100 --> 00:39:26.780 OK. 00:39:26.780 --> 00:39:32.970 First thing is I claim that one of these two sets of sums is 0. 00:39:32.970 --> 00:39:34.926 You tell me which. 00:39:34.926 --> 00:39:36.300 AUDIENCE: The first. 00:39:36.300 --> 00:39:37.830 PROFESSOR: Why first? 00:39:37.830 --> 00:39:42.764 AUDIENCE: Because H prime is independent of p [? dot. ?] 00:39:48.416 --> 00:39:50.200 PROFESSOR: That's true. 00:39:50.200 --> 00:39:50.700 OK. 00:39:50.700 --> 00:39:52.170 That's very good. 00:39:52.170 --> 00:39:55.580 And then it sort of brings up a very important question, 00:39:55.580 --> 00:39:57.710 which is, I forgot to write two more terms. 00:39:57.710 --> 00:40:01.787 [LAUGHTER] 00:40:01.787 --> 00:40:19.630 Running to s of d rho by dpn, d H prime by dqn minus sum j 00:40:19.630 --> 00:40:32.320 s plus 1 to N of d rho by dpj dot dH prime by dqj. 00:40:32.320 --> 00:40:35.760 So indeed, both answers now were correct. 00:40:35.760 --> 00:40:37.620 Somebody said that the first term 00:40:37.620 --> 00:40:41.770 is a 0, because H prime does not depend on pn. 00:40:41.770 --> 00:40:45.310 And somebody over here said that this term is 0. 00:40:45.310 --> 00:40:48.844 And maybe they can explain why. 00:40:48.844 --> 00:40:51.149 AUDIENCE: [INAUDIBLE]. 00:40:51.149 --> 00:40:53.250 PROFESSOR: Same reason as up here. 00:40:53.250 --> 00:40:55.460 That is, I can do integration by parts. 00:40:55.460 --> 00:40:57.196 AUDIENCE: [INAUDIBLE]. 00:40:57.196 --> 00:41:03.890 PROFESSOR: To get rid of this term plus this term together. 00:41:03.890 --> 00:41:08.310 So it's actually by itself is not 0. 00:41:08.310 --> 00:41:11.710 But if I do integration by parts, 00:41:11.710 --> 00:41:14.550 I will have-- actually, even by itself, it is 0, 00:41:14.550 --> 00:41:19.183 because I would have d by dqj, d by pj, H prime. 00:41:19.183 --> 00:41:22.900 And H prime, you cannot have a double derivative pj. 00:41:22.900 --> 00:41:25.740 So each one of them, actually, by itself is 0. 00:41:25.740 --> 00:41:28.480 But in general, they would also cancel each other 00:41:28.480 --> 00:41:30.621 through their single process. 00:41:30.621 --> 00:41:31.120 Yes. 00:41:31.120 --> 00:41:34.100 AUDIENCE: Do you have the sign of [? dH ?] of [? H prime? ?] 00:41:34.100 --> 00:41:35.820 PROFESSOR: Did I have the sign incorrect? 00:41:35.820 --> 00:41:36.570 Yes. 00:41:36.570 --> 00:41:39.120 Because for some reason or other, 00:41:39.120 --> 00:41:46.048 I keep reading from here, which is rho and H. So let's do this. 00:41:49.476 --> 00:41:49.976 OK? 00:41:49.976 --> 00:41:50.824 AUDIENCE: Excuse me. 00:41:50.824 --> 00:41:51.449 PROFESSOR: Yes? 00:41:51.449 --> 00:41:54.886 AUDIENCE: [INAUDIBLE]. 00:41:54.886 --> 00:41:55.920 PROFESSOR: OK. 00:41:55.920 --> 00:41:57.170 Yes, it is different. 00:41:57.170 --> 00:41:57.670 Yes. 00:41:57.670 --> 00:42:03.430 So what I said, if I had a more general Hamiltonian 00:42:03.430 --> 00:42:07.200 that also depended on momentum, then 00:42:07.200 --> 00:42:11.020 this term would, by itself not 0, but would cancel, 00:42:11.020 --> 00:42:13.370 be the corresponding term from here. 00:42:13.370 --> 00:42:15.290 But the way that I have for H prime, 00:42:15.290 --> 00:42:19.010 indeed, each term by itself would be 0. 00:42:19.010 --> 00:42:19.510 OK. 00:42:19.510 --> 00:42:24.330 So hopefully-- then what do we have? 00:42:27.150 --> 00:42:34.240 So actually, let's keep the sign correct and do this, 00:42:34.240 --> 00:42:37.290 because I need this right sign for the one term 00:42:37.290 --> 00:42:38.040 that is preserved. 00:42:38.040 --> 00:42:39.330 So what does that say? 00:42:39.330 --> 00:42:47.870 It is a sum, n running from 1 to s. 00:42:51.030 --> 00:42:52.330 OK? 00:42:52.330 --> 00:42:56.940 I have d H prime by dqn. 00:42:59.500 --> 00:43:01.100 And I have this integration. 00:43:01.100 --> 00:43:10.150 I have the integration i running from s plus 1 to N dV of i. 00:43:10.150 --> 00:43:16.480 I have d rho by dpn dot producted 00:43:16.480 --> 00:43:19.410 with d H prime by dqN. 00:43:19.410 --> 00:43:23.760 d H prime by dqn I can calculate from here, 00:43:23.760 --> 00:43:31.940 is a sum over terms j running from s plus 1 00:43:31.940 --> 00:43:42.280 to N of V of qn minus qj. 00:43:52.440 --> 00:43:53.010 All right. 00:43:56.433 --> 00:43:57.411 AUDIENCE: Question. 00:43:57.411 --> 00:43:58.878 PROFESSOR: Yes. 00:43:58.878 --> 00:44:02.045 AUDIENCE: Why aren't you differentiating [? me ?] 00:44:02.045 --> 00:44:04.095 if you're differentiating H prime? 00:44:07.884 --> 00:44:10.200 PROFESSOR: d by dqj. 00:44:13.400 --> 00:44:14.442 All right? 00:44:14.442 --> 00:44:17.025 AUDIENCE: Where is the qn? 00:44:17.025 --> 00:44:19.161 PROFESSOR: d by dqn. 00:44:19.161 --> 00:44:20.604 Thank you. 00:44:20.604 --> 00:44:22.050 Right. 00:44:22.050 --> 00:44:25.390 Because always, pn of qn would go together. 00:44:25.390 --> 00:44:26.764 Thank you. 00:44:26.764 --> 00:44:27.264 OK. 00:44:31.121 --> 00:44:31.620 All right. 00:44:31.620 --> 00:44:34.870 So we have to slog through these derivations. 00:44:34.870 --> 00:44:38.980 And then I'll give you the physical meaning. 00:44:38.980 --> 00:44:43.075 So I can rearrange this. 00:44:46.750 --> 00:44:49.910 Let's see what's happening here. 00:44:49.910 --> 00:44:55.180 I have here a sum over particles that 00:44:55.180 --> 00:44:58.980 are not listed on the left-hand side. 00:44:58.980 --> 00:45:02.660 So when I wrote this d rho by dt, 00:45:02.660 --> 00:45:07.090 I had listed coordinates going from p1 through qs that 00:45:07.090 --> 00:45:09.940 were s coordinates that were listed. 00:45:09.940 --> 00:45:12.520 If you like, you can think of them as s particles. 00:45:17.030 --> 00:45:21.460 Now, this sum involves the remaining particles. 00:45:21.460 --> 00:45:22.540 What is this? 00:45:22.540 --> 00:45:23.690 Up to a sign. 00:45:23.690 --> 00:45:28.000 This is the force that is exerted by particle j 00:45:28.000 --> 00:45:30.150 from the list of particles that I'm not 00:45:30.150 --> 00:45:33.410 interested on one of the particles on the list 00:45:33.410 --> 00:45:35.262 that I am interested. 00:45:35.262 --> 00:45:38.090 OK? 00:45:38.090 --> 00:45:42.140 Now, I expect that at the end of the day, all of the particles 00:45:42.140 --> 00:45:46.950 that I am not interested I can treat equivalently, 00:45:46.950 --> 00:45:49.460 like everything that we had before, 00:45:49.460 --> 00:45:55.140 like how I got this factor of N or N minus 1 over there. 00:45:55.140 --> 00:46:01.700 I expect that all of these will give me the same result, which 00:46:01.700 --> 00:46:06.410 is proportional to the number of these particles, which 00:46:06.410 --> 00:46:08.462 is N minus s. 00:46:08.462 --> 00:46:11.110 OK? 00:46:11.110 --> 00:46:14.950 And then I can focus on just one of the terms in this sum. 00:46:14.950 --> 00:46:17.495 Let's say the term that corresponds to j, 00:46:17.495 --> 00:46:18.450 being s plus 1. 00:46:23.480 --> 00:46:28.870 Now, having done that, I have to be careful. 00:46:28.870 --> 00:46:32.210 I can do separately the integration 00:46:32.210 --> 00:46:37.660 over the volume of this one coordinate that I'm keeping, 00:46:37.660 --> 00:46:39.930 V of s plus 1. 00:46:39.930 --> 00:46:42.270 And what do I have here? 00:46:42.270 --> 00:46:49.180 I have the force that exerted on particle number N 00:46:49.180 --> 00:46:52.520 by the particle that is labelled s plus 1. 00:46:57.880 --> 00:47:03.540 And this force is dot producted with a gradient 00:47:03.540 --> 00:47:12.390 along the momentum in direction of particle N of its density. 00:47:12.390 --> 00:47:15.820 Actually, this is the density of all particles. 00:47:15.820 --> 00:47:20.510 This is the rho that corresponds to the joint. 00:47:20.510 --> 00:47:24.950 But I had here s plus 1 integrations. 00:47:24.950 --> 00:47:27.860 One of them I wrote down explicitly. 00:47:27.860 --> 00:47:30.028 All the others I do over here. 00:47:36.860 --> 00:47:38.540 Of the density. 00:47:38.540 --> 00:47:43.770 So basically, I change the order of the derivative 00:47:43.770 --> 00:47:46.890 and the integrations over the variables not involved 00:47:46.890 --> 00:47:48.670 in the remainder. 00:47:48.670 --> 00:47:51.450 And the reason I did that, of course, 00:47:51.450 --> 00:47:54.875 is that then this is my rho s plus 1. 00:47:58.590 --> 00:48:00.930 OK? 00:48:00.930 --> 00:48:05.160 So what we have at the end of the day 00:48:05.160 --> 00:48:13.130 is that if I take the time variation of an s particle 00:48:13.130 --> 00:48:21.380 density, I will get one term that I expected, 00:48:21.380 --> 00:48:26.060 which is if those s particles were interacting only 00:48:26.060 --> 00:48:30.620 with themselves, I would write the Liouville equation 00:48:30.620 --> 00:48:34.460 that would be appropriate to them. 00:48:34.460 --> 00:48:36.510 But because of the collisions that I 00:48:36.510 --> 00:48:41.930 can have with particles that are not over here, 00:48:41.930 --> 00:48:46.010 suddenly, the momenta that I'm looking at could change. 00:48:46.010 --> 00:48:50.400 And because of that, I have a correction term here 00:48:50.400 --> 00:48:53.170 that really describes the collisions. 00:48:53.170 --> 00:48:57.980 It says here that these s particles 00:48:57.980 --> 00:49:00.870 were following the trajectory that 00:49:00.870 --> 00:49:03.400 was governed by the Hamiltonian that 00:49:03.400 --> 00:49:06.330 was peculiar to the s particles. 00:49:06.330 --> 00:49:11.210 But suddenly, one of them had a collision with somebody else. 00:49:11.210 --> 00:49:12.800 So which one of them? 00:49:12.800 --> 00:49:14.200 Well, any one of them. 00:49:14.200 --> 00:49:20.160 So I could get a contribution from any one of the s particles 00:49:20.160 --> 00:49:24.970 that is listed over here, having a collision with somebody else. 00:49:24.970 --> 00:49:27.970 How do I describe the effect of that? 00:49:27.970 --> 00:49:37.020 I have to do an integration over where this new particle that I 00:49:37.020 --> 00:49:39.160 am colliding with could be. 00:49:39.160 --> 00:49:43.090 I have to specify both where the particle is that I am colliding 00:49:43.090 --> 00:49:47.750 with, as well as its momentum. 00:49:47.750 --> 00:49:50.100 So that's this. 00:49:50.100 --> 00:49:55.190 Then I need to know the force that this particle is 00:49:55.190 --> 00:49:56.420 exerting on me. 00:49:56.420 --> 00:50:05.680 So that's the V of qs plus 1 minus qN divided by dqN. 00:50:05.680 --> 00:50:10.240 This is the force that is exerted by this particle 00:50:10.240 --> 00:50:13.980 that I don't see on myself. 00:50:13.980 --> 00:50:18.990 Then I have to multiply this, or a dot product of this, 00:50:18.990 --> 00:50:23.840 with d by dpN, because what happens in the process, 00:50:23.840 --> 00:50:28.150 because of this force, the momentum of the N particle 00:50:28.150 --> 00:50:29.880 is changing. 00:50:29.880 --> 00:50:34.300 The variation of that is captured 00:50:34.300 --> 00:50:38.080 through looking at the density that 00:50:38.080 --> 00:50:44.370 has all of these particles in addition to this new particle 00:50:44.370 --> 00:50:46.250 that I am colliding with. 00:50:46.250 --> 00:50:50.270 But, of course, I am not really interested in the coordinate 00:50:50.270 --> 00:50:54.970 of this new particle, so I integrate over it. 00:50:54.970 --> 00:50:57.670 There are N minus s such particles. 00:50:57.670 --> 00:51:00.280 So I really have to put a factor of N minus s 00:51:00.280 --> 00:51:04.640 here for all of potential collisions. 00:51:04.640 --> 00:51:05.775 And so that's the equation. 00:51:09.010 --> 00:51:13.900 Again, it is more common, rather than to write 00:51:13.900 --> 00:51:20.150 the equation for rho, to write the equation for f. 00:51:20.150 --> 00:51:22.730 And the f's and the rhos where simply 00:51:22.730 --> 00:51:27.270 related by these factors of N factorial over N minus 1 00:51:27.270 --> 00:51:28.880 s factorial. 00:51:28.880 --> 00:51:36.550 And the outcome of that is that the equation for f 00:51:36.550 --> 00:51:41.160 simply does not have this additional factor of N minus s, 00:51:41.160 --> 00:51:44.190 because that disappears in the ratio of rho 00:51:44.190 --> 00:51:46.230 of s plus 1 and rho s. 00:51:46.230 --> 00:51:53.690 And it becomes a sum over N running from 1 to s. 00:51:53.690 --> 00:52:01.790 Integral over coordinates and momenta of a particle s plus 1. 00:52:04.620 --> 00:52:08.660 The force exerted by particle s plus 1 00:52:08.660 --> 00:52:19.540 on particle N used to vary the momentum of the N particle. 00:52:19.540 --> 00:52:25.040 And the whole thing would depend on the density that includes, 00:52:25.040 --> 00:52:30.850 in addition to the s particles that I had before, 00:52:30.850 --> 00:52:33.810 the new particle that I am colliding with. 00:52:36.642 --> 00:52:39.010 OK? 00:52:39.010 --> 00:52:42.190 So there is a set of equations that 00:52:42.190 --> 00:52:48.730 relates the different densities and how they evolve in time. 00:52:48.730 --> 00:52:53.660 The evolution of f1, which is the thing that I am interested, 00:52:53.660 --> 00:52:55.420 will have, on the right-hand side, 00:52:55.420 --> 00:52:57.850 something that involves f2. 00:52:57.850 --> 00:53:01.100 The evolution of f2 will involve f3. 00:53:01.100 --> 00:53:05.910 And this whole thing is called a BBGKY hierarchy, 00:53:05.910 --> 00:53:09.760 after people whose names I have in the notes. 00:53:09.760 --> 00:53:11.500 [SOFT LAUGHTER] 00:53:13.620 --> 00:53:19.450 But again, what have we learned beyond what 00:53:19.450 --> 00:53:21.315 we had in the original case? 00:53:21.315 --> 00:53:23.720 And originally, we had an equation 00:53:23.720 --> 00:53:29.370 that was governing a function in 6N-dimensional space, which 00:53:29.370 --> 00:53:30.770 we really don't need. 00:53:30.770 --> 00:53:33.500 So we tried our best to avoid that. 00:53:33.500 --> 00:53:36.700 We said that all of the physics is in one particle, maybe 00:53:36.700 --> 00:53:38.560 two particle densities. 00:53:38.560 --> 00:53:41.770 Let's calculate the evolution of one-particle and two-particle 00:53:41.770 --> 00:53:42.800 densities. 00:53:42.800 --> 00:53:45.980 Maybe they will tell us about this non-equilibrium situation 00:53:45.980 --> 00:53:47.340 that we set up. 00:53:47.340 --> 00:53:50.860 But we see that the time evolution of the first particle 00:53:50.860 --> 00:53:53.410 density requires two-particle density. 00:53:53.410 --> 00:53:56.700 Two-particle density requires three-particle densities. 00:53:56.700 --> 00:54:01.250 So we sort of made this ladder, which ultimately will 00:54:01.250 --> 00:54:03.740 [? terminate ?] at the Nth particle densities. 00:54:03.740 --> 00:54:08.040 And so we have not really gained much. 00:54:08.040 --> 00:54:13.050 So we have to now look at these equations a little bit more 00:54:13.050 --> 00:54:17.150 and try to inject more physics. 00:54:17.150 --> 00:54:20.755 So let's write down the first two terms explicitly. 00:54:23.340 --> 00:54:28.580 So what I will do is I will take this Poisson bracket of H 00:54:28.580 --> 00:54:32.850 and f, to the left-hand side, and use the Hamiltonian 00:54:32.850 --> 00:54:37.760 that we have over here to write the terms. 00:54:37.760 --> 00:54:41.680 So the equation that we have for f1-- and I'm 00:54:41.680 --> 00:54:44.360 going to write it as a whole bunch of derivatives 00:54:44.360 --> 00:54:46.980 acting on f1. 00:54:46.980 --> 00:54:52.930 f1 is a function of p1 q1 t. 00:54:52.930 --> 00:54:57.100 And essentially, what Liouville's theorem 00:54:57.100 --> 00:55:01.100 says is that as you move along the trajectory, 00:55:01.100 --> 00:55:06.290 the total derivative is 0, because the expansion 00:55:06.290 --> 00:55:08.720 of the flows is incompressible. 00:55:08.720 --> 00:55:10.670 So what does that mean? 00:55:10.670 --> 00:55:17.870 It means that d by dt, which is this argument, plus q1 dot 00:55:17.870 --> 00:55:25.126 times d by dq1 plus p1 dot by d by dp1. 00:55:27.960 --> 00:55:31.775 So here, I should write p1 dot and q1 dot. 00:55:34.670 --> 00:55:37.340 In the absence of everything else is 0. 00:55:37.340 --> 00:55:44.790 Then, of course, for q1 dot, we use the momentum 00:55:44.790 --> 00:55:46.820 that we would get out of this. 00:55:46.820 --> 00:55:53.060 q1 dot is momentum divided by mass. 00:55:53.060 --> 00:55:55.790 So that's the velocity. 00:55:55.790 --> 00:55:59.740 And p1 dot, changing momentum, is 00:55:59.740 --> 00:56:02.905 the force, is minus dH by dq1. 00:56:02.905 --> 00:56:09.360 So this is minus d of this one particle potential divided 00:56:09.360 --> 00:56:14.630 by dq1 dotted by [? h ?] p1. 00:56:14.630 --> 00:56:21.240 So if you were asked to think about one particle in a box, 00:56:21.240 --> 00:56:24.400 then you know its equation of motion. 00:56:24.400 --> 00:56:27.470 If you have many, many realizations 00:56:27.470 --> 00:56:32.080 of that particle in a box, you can construct a density. 00:56:32.080 --> 00:56:34.186 Each one of the elements of the trajectory, 00:56:34.186 --> 00:56:36.250 you know how they evolve according 00:56:36.250 --> 00:56:38.010 to [? Newton's ?] equation. 00:56:38.010 --> 00:56:43.120 And you can see how the density would evolve. 00:56:43.120 --> 00:56:45.540 It would evolve according to this. 00:56:45.540 --> 00:56:48.280 I would have said, equal to 0. 00:56:48.280 --> 00:56:53.420 But I can't set it to 0 if I'm really thinking about a gas, 00:56:53.420 --> 00:56:56.360 because my particle can come and collide 00:56:56.360 --> 00:57:00.050 with a second particle in the gas. 00:57:00.050 --> 00:57:02.980 The second particle can be anywhere. 00:57:02.980 --> 00:57:11.570 And what it will do is that it will exert a force, which 00:57:11.570 --> 00:57:15.000 would be like this, on particle one. 00:57:15.000 --> 00:57:17.300 And this force will change the momentum. 00:57:17.300 --> 00:57:20.040 So my variation of the momentum will not 00:57:20.040 --> 00:57:22.410 come only from the external force, 00:57:22.410 --> 00:57:27.100 but also from the force that is coming 00:57:27.100 --> 00:57:30.380 from some other particle in the medium. 00:57:30.380 --> 00:57:33.640 So that's where this d by dp really 00:57:33.640 --> 00:57:37.080 gets not only the external force but also 00:57:37.080 --> 00:57:39.890 the force from somebody else. 00:57:39.890 --> 00:57:42.830 But then I need to know where this other particle is, 00:57:42.830 --> 00:57:45.830 given that I know where my first particle is. 00:57:45.830 --> 00:57:50.340 So I have to include here a two-particle density which 00:57:50.340 --> 00:57:56.740 depends on p1 as well as q2 at time t. 00:57:56.740 --> 00:57:58.230 OK. 00:57:58.230 --> 00:58:00.546 Fine. 00:58:00.546 --> 00:58:05.420 Now you say, OK, let's write down-- I need to know f2. 00:58:05.420 --> 00:58:07.220 Let's write down the equation for f2. 00:58:07.220 --> 00:58:12.110 So I will write it more rapidly. 00:58:12.110 --> 00:58:15.500 I have p1 over m, d by dq1. 00:58:15.500 --> 00:58:22.232 I have p2 over m, d by dq2. 00:58:22.232 --> 00:58:27.450 I will have dU by dq1, d by dp1. 00:58:27.450 --> 00:58:35.580 I will have dU by dq2, d by dp2. 00:58:35.580 --> 00:58:40.470 I will have also a term from the collision between q1 and q2. 00:58:44.500 --> 00:58:49.680 And once it will change the momentum of the first particle, 00:58:49.680 --> 00:58:52.490 but it will change the momentum of the second particle 00:58:52.490 --> 00:58:53.735 in the opposite direction. 00:58:53.735 --> 00:58:57.110 So I will put the two of them together. 00:58:57.110 --> 00:59:03.050 So this is all of the terms that I would get from H2, Poisson 00:59:03.050 --> 00:59:06.600 bracket with density acting on the two-particle density. 00:59:09.530 --> 00:59:13.220 And the answer would be 0 if the two particles 00:59:13.220 --> 00:59:15.490 were the only thing in the box. 00:59:15.490 --> 00:59:17.910 But there's also other particles. 00:59:17.910 --> 00:59:20.770 So there can be interactions and collisions 00:59:20.770 --> 00:59:22.700 with a third particle. 00:59:22.700 --> 00:59:25.920 And for that, I would need to know, 00:59:25.920 --> 00:59:29.680 let's actually try to simplify notation. 00:59:29.680 --> 00:59:35.470 This is the force that is exerted from two to one. 00:59:35.470 --> 00:59:41.822 So I will have here the force from three to one 00:59:41.822 --> 00:59:44.075 dotted by d by dp1. 00:59:48.380 --> 00:59:49.240 Right. 00:59:49.240 --> 00:59:52.600 And the force that is exerted from three to two 00:59:52.600 --> 00:59:57.910 dotted by d by dp2 acting on a three-particle density that 00:59:57.910 --> 01:00:02.870 involves everything up to three. 01:00:02.870 --> 01:00:04.880 And now let's write the third one. 01:00:04.880 --> 01:00:07.380 [LAUGHTER] 01:00:08.380 --> 01:00:10.390 So that, I will leave to next lecture. 01:00:10.390 --> 01:00:14.250 But anyway, so this is the structure. 01:00:14.250 --> 01:00:15.840 Now, this is the point at which we 01:00:15.840 --> 01:00:23.340 would like to inject some physics into the problem. 01:00:23.340 --> 01:00:30.080 So what we are going to do is to estimate the various terms that 01:00:30.080 --> 01:00:33.880 are appearing in this equation to see whether there 01:00:33.880 --> 01:00:37.582 is some approximation that we can make 01:00:37.582 --> 01:00:42.680 to make the equations more treatable and handle-able. 01:00:42.680 --> 01:00:45.090 All right? 01:00:45.090 --> 01:00:54.900 So let's try to look at the case of a gas-- 01:00:54.900 --> 01:00:58.160 let's say the gas in this room. 01:00:58.160 --> 01:01:01.790 A typical thing that is happening 01:01:01.790 --> 01:01:03.690 in the particles of the gas in this room 01:01:03.690 --> 01:01:06.240 is that they are zipping around. 01:01:06.240 --> 01:01:09.760 Their velocity is of the order-- again, 01:01:09.760 --> 01:01:14.991 just order of magnitude, hundreds of meters per second. 01:01:14.991 --> 01:01:15.490 OK? 01:01:15.490 --> 01:01:20.899 So we are going to, again, be very sort of limited 01:01:20.899 --> 01:01:22.315 in what we are trying to describe. 01:01:22.315 --> 01:01:24.110 There is this experiment. 01:01:24.110 --> 01:01:26.585 Gas expands into a chamber. 01:01:26.585 --> 01:01:30.802 In room temperature, typical velocities are of this order. 01:01:30.802 --> 01:01:34.620 Now we are going to use that to estimate 01:01:34.620 --> 01:01:36.710 the magnitude of the various terms that 01:01:36.710 --> 01:01:39.830 are appearing in this equation. 01:01:39.830 --> 01:01:42.200 Now, the whole thing about this equation 01:01:42.200 --> 01:01:44.870 is variation with time. 01:01:44.870 --> 01:01:46.920 So the entity that we are looking 01:01:46.920 --> 01:01:51.450 at in all of these brackets is this d by dt, 01:01:51.450 --> 01:01:55.830 which means that the various terms in this differential 01:01:55.830 --> 01:01:58.980 equation, apart from d by dt, have 01:01:58.980 --> 01:02:03.320 to have dimensions of inverse time. 01:02:03.320 --> 01:02:06.620 So we are going to try to characterize 01:02:06.620 --> 01:02:08.220 what those inverse times are. 01:02:12.430 --> 01:02:21.355 So what are the typical magnitudes of various terms? 01:02:25.640 --> 01:02:28.740 If I look at the first equation, I 01:02:28.740 --> 01:02:31.640 said what that first equation describes. 01:02:31.640 --> 01:02:35.180 That first equation describes for you a particle in a box. 01:02:35.180 --> 01:02:38.390 We've forgotten about everything else. 01:02:38.390 --> 01:02:41.710 So if I have a particle in a box, 01:02:41.710 --> 01:02:43.590 what is the characteristic time? 01:02:43.590 --> 01:02:46.730 It has to be set by the size of the box, given 01:02:46.730 --> 01:02:49.740 that I am moving with that velocity. 01:02:49.740 --> 01:02:52.550 So there is a timescale that I would 01:02:52.550 --> 01:02:57.215 call extrinsic in the sense that it is not really 01:02:57.215 --> 01:02:58.640 a property of the gas. 01:02:58.640 --> 01:03:02.020 It will be different if I make the box bigger. 01:03:02.020 --> 01:03:04.010 There's a timescale over which I would 01:03:04.010 --> 01:03:07.430 go from one side of the box to another side of the box. 01:03:07.430 --> 01:03:09.920 So this is kind of a timescale that 01:03:09.920 --> 01:03:13.100 is related to the term that knows something 01:03:13.100 --> 01:03:17.140 about the box, which is dU by dq, d by dp. 01:03:17.140 --> 01:03:21.480 I would say that if I were to assign some typical magnitude 01:03:21.480 --> 01:03:24.170 to this type of term, I would say 01:03:24.170 --> 01:03:31.980 that it is related to having to traverse a distance that 01:03:31.980 --> 01:03:39.630 is of the order of the size of the box, 01:03:39.630 --> 01:03:42.640 given the velocity that I have specified. 01:03:42.640 --> 01:03:44.465 This is an inverse timescale. 01:03:44.465 --> 01:03:45.950 Right? 01:03:45.950 --> 01:03:50.190 And let's sort of imagine that I have an-- 01:03:50.190 --> 01:03:55.620 and I will call this timescale 1 over tau U, 01:03:55.620 --> 01:04:00.610 because it is sort of determined by my external U. 01:04:00.610 --> 01:04:03.070 Let's say we have a typical size that 01:04:03.070 --> 01:04:04.760 is of the order of millimeter. 01:04:04.760 --> 01:04:06.890 If I make it larger, it will be larger. 01:04:06.890 --> 01:04:12.520 So actually, let's say we have 10 to the minus 3 meters. 01:04:12.520 --> 01:04:14.340 Actually, let's make it bigger. 01:04:14.340 --> 01:04:18.740 Let's make it of the order of 10 to the minus 1 meter. 01:04:18.740 --> 01:04:20.900 Kind of reasonable-sized box. 01:04:20.900 --> 01:04:24.270 Then you would say that this 1 over tau c 01:04:24.270 --> 01:04:29.010 is of the order of 10 to the 2 divided by 10 to the minus 1, 01:04:29.010 --> 01:04:31.110 which is of the order of 1,000. 01:04:31.110 --> 01:04:35.390 Basically, it takes a millisecond 01:04:35.390 --> 01:04:38.530 to traverse a box that is a fraction 01:04:38.530 --> 01:04:42.220 of a meter with these velocities. 01:04:42.220 --> 01:04:42.720 OK? 01:04:42.720 --> 01:04:44.450 You say fine. 01:04:44.450 --> 01:04:46.360 That is the kind of timescale that I 01:04:46.360 --> 01:04:50.100 have in the first equation that I have in my hierarchy. 01:04:50.100 --> 01:04:52.070 And that kind of term is certainly 01:04:52.070 --> 01:04:56.950 also present in the second equation for the hierarchy. 01:04:56.950 --> 01:04:59.860 If I have two particles, maybe these two particles 01:04:59.860 --> 01:05:01.990 are orbiting each other, et cetera. 01:05:01.990 --> 01:05:04.880 Still, their center of mass would move, typically, 01:05:04.880 --> 01:05:06.569 with this velocity. 01:05:06.569 --> 01:05:08.110 And it would take this amount of time 01:05:08.110 --> 01:05:11.010 to go across the size of the box. 01:05:11.010 --> 01:05:15.210 But there is another timescale inside there 01:05:15.210 --> 01:05:25.815 that I would call intrinsic, which involves dV/dq, d by dp. 01:05:31.380 --> 01:05:36.770 Now, if I was to see what the characteristic magnitude 01:05:36.770 --> 01:05:42.470 of this term is, it would have to be V divided 01:05:42.470 --> 01:05:47.730 by a lens scale that characterizes the potential. 01:05:47.730 --> 01:05:50.040 And the potential, let's say, is of the order 01:05:50.040 --> 01:05:52.780 of atomic size or molecular size. 01:05:52.780 --> 01:05:54.230 Let's call it d. 01:05:54.230 --> 01:05:58.910 So this is an atomic size-- or molecular size. 01:06:03.740 --> 01:06:07.240 More correctly, really, it's the range of the interaction 01:06:07.240 --> 01:06:09.730 that you have between particles. 01:06:09.730 --> 01:06:13.280 And typical values of these [? numbers ?] 01:06:13.280 --> 01:06:17.320 are of the order of 10 angstroms, or angstroms, 01:06:17.320 --> 01:06:18.080 or whatever. 01:06:18.080 --> 01:06:19.905 Let's say 10 to the minus 10 meters. 01:06:22.920 --> 01:06:23.480 Sorry. 01:06:23.480 --> 01:06:27.250 The first one I would like to call tau U. 01:06:27.250 --> 01:06:30.070 This second time, that I will call 01:06:30.070 --> 01:06:36.414 1 over tau c for collisions, is going to be the ratio of 10 01:06:36.414 --> 01:06:39.950 to the 2 to 10 to the minus 10. 01:06:39.950 --> 01:06:45.010 It's of the order of 10 to the 12 in both seconds. 01:06:45.010 --> 01:06:47.340 OK? 01:06:47.340 --> 01:06:54.700 So you can see that this term is much, much larger in magnitude 01:06:54.700 --> 01:06:58.584 than the term that was governing the first equation. 01:06:58.584 --> 01:06:59.400 OK? 01:06:59.400 --> 01:07:03.730 And roughly, what you expect in a situation 01:07:03.730 --> 01:07:11.840 such as this-- let's imagine, rather than shooting particles 01:07:11.840 --> 01:07:17.990 from here, you are shooting bullets. 01:07:17.990 --> 01:07:20.370 And then the bullets would come and basically 01:07:20.370 --> 01:07:23.440 have some kind of trajectory, et cetera. 01:07:23.440 --> 01:07:27.360 The characteristic time for a single one of them 01:07:27.360 --> 01:07:31.150 would be basically something that 01:07:31.150 --> 01:07:33.230 is related to the size of the box. 01:07:33.230 --> 01:07:38.170 How long does it take a bullet to go over the size of the box? 01:07:38.170 --> 01:07:40.620 But if two of these bullets happen 01:07:40.620 --> 01:07:45.130 to come together and collide, then there's 01:07:45.130 --> 01:07:48.650 a very short period of time over which 01:07:48.650 --> 01:07:51.100 they would go in different directions. 01:07:51.100 --> 01:07:53.690 And the momenta would get displaced 01:07:53.690 --> 01:07:55.600 from what they were before. 01:07:55.600 --> 01:07:59.600 And that timescale is of the order of this. 01:08:02.910 --> 01:08:05.880 But in the situation that I set up, 01:08:05.880 --> 01:08:09.510 this particular time is too rapid. 01:08:09.510 --> 01:08:12.480 There is another more important time, 01:08:12.480 --> 01:08:16.880 which is, how long do I have to wait 01:08:16.880 --> 01:08:20.330 for two of these particles, or two of these bullets, 01:08:20.330 --> 01:08:23.550 to come and hit each other? 01:08:23.550 --> 01:08:28.080 So it's not the duration of the collision that is irrelevant, 01:08:28.080 --> 01:08:32.450 but how long it would be for me to find another particle 01:08:32.450 --> 01:08:34.500 to collide with. 01:08:34.500 --> 01:08:38.779 And actually, that is what is governed by the terms 01:08:38.779 --> 01:08:41.399 that I have on the other side. 01:08:41.399 --> 01:08:44.600 Because the terms on the other side, what they say is I 01:08:44.600 --> 01:08:47.270 have to find another particle. 01:08:47.270 --> 01:08:55.470 So if I look at the terms that I have on the right-hand side 01:08:55.470 --> 01:08:58.290 and try to construct a characteristic time 01:08:58.290 --> 01:09:02.279 out of them, I have to compare the probability that I will 01:09:02.279 --> 01:09:06.840 have, or the density that I will have for s plus 1, 01:09:06.840 --> 01:09:16.080 integrated over some volume, over which 01:09:16.080 --> 01:09:20.184 the force between these particles is non-zero. 01:09:23.770 --> 01:09:27.580 And then in order to construct a timescale for it, 01:09:27.580 --> 01:09:32.279 I know that the d by dt on the left-hand side acts on fs. 01:09:32.279 --> 01:09:35.250 On the right-hand side, I have fs plus 1. 01:09:35.250 --> 01:09:37.824 So again, just if I want to construct dimensionally, 01:09:37.824 --> 01:09:44.529 it's a ratio that involves s plus 1 to s. 01:09:44.529 --> 01:09:46.640 OK? 01:09:46.640 --> 01:09:52.569 So I have to do an additional integration 01:09:52.569 --> 01:09:59.410 over a volume in phase space over which two particles can 01:09:59.410 --> 01:10:01.820 have substantial interactions. 01:10:01.820 --> 01:10:03.710 Because that's where this [? dv ?] by dq 01:10:03.710 --> 01:10:09.820 would be non-zero, provided that there is a density for s 01:10:09.820 --> 01:10:13.300 plus 1 particle compared to s particles. 01:10:13.300 --> 01:10:15.000 If you think about it, that means 01:10:15.000 --> 01:10:18.390 that I have to look at the typical density 01:10:18.390 --> 01:10:26.770 or particles times d cubed for these additional operations 01:10:26.770 --> 01:10:32.480 multiplied by this collision time that I had before. 01:10:32.480 --> 01:10:34.300 OK? 01:10:34.300 --> 01:10:39.945 And this whole thing I will call 1 over tau collision. 01:10:42.620 --> 01:10:50.080 And another way of getting the same result is as follows. 01:10:50.080 --> 01:10:56.190 This is typically how you get collision times by pictorially. 01:10:56.190 --> 01:11:01.320 You say that I have something that can interact over 01:11:01.320 --> 01:11:05.530 some characteristic size d. 01:11:05.530 --> 01:11:11.640 It moves in space with velocity v 01:11:11.640 --> 01:11:17.050 so that if I wait a time that I will call tau, 01:11:17.050 --> 01:11:19.840 within that time, I'm essentially 01:11:19.840 --> 01:11:28.690 sweeping a volume of space that has volume d squared v tau. 01:11:28.690 --> 01:11:33.000 So my cross section, if my dimension is d, is d squared. 01:11:33.000 --> 01:11:36.680 I sweep in the other direction by [? aman ?] d tau. 01:11:36.680 --> 01:11:40.290 And how many particles will I encounter? 01:11:40.290 --> 01:11:42.130 Well, if I know the density, which 01:11:42.130 --> 01:11:45.020 is the number of particles per unit volume, 01:11:45.020 --> 01:11:48.450 I have to multiply this by n. 01:11:48.450 --> 01:11:51.930 So how far do I have to go until I hit 1? 01:11:51.930 --> 01:11:54.340 I'll call that tau x. 01:11:54.340 --> 01:11:57.910 Then my formula for tau x would be 01:11:57.910 --> 01:12:05.080 1 over nvd squared, which is exactly what I have here. 01:12:05.080 --> 01:12:12.280 1 over tau x is nd squared v. OK? 01:12:12.280 --> 01:12:19.020 So in order to compare the terms that I 01:12:19.020 --> 01:12:22.710 have on the right-hand side with the terms 01:12:22.710 --> 01:12:25.850 on the left-hand side, I notice that I 01:12:25.850 --> 01:12:31.210 need to know something about nd cubed. 01:12:31.210 --> 01:12:36.110 So nd cubed tells you if I have a particle 01:12:36.110 --> 01:12:40.000 here and this particle has a range of interactions 01:12:40.000 --> 01:12:44.950 that I call d, how many other particles fall 01:12:44.950 --> 01:12:46.370 within that range of interaction? 01:12:49.490 --> 01:12:53.750 Now, for the gas in this room, the range of the interaction 01:12:53.750 --> 01:12:55.680 is of the order of the size of the molecule. 01:12:55.680 --> 01:12:58.350 It is very small. 01:12:58.350 --> 01:13:01.180 And the distance between molecules in this room 01:13:01.180 --> 01:13:03.340 is far apart. 01:13:03.340 --> 01:13:10.770 And indeed, you can estimate that for gas, nd cubed 01:13:10.770 --> 01:13:14.480 has to be of the order of 10 to the minus 4. 01:13:14.480 --> 01:13:16.260 And how do I know that? 01:13:16.260 --> 01:13:20.610 Because if I were to take all of the gas particles in this room 01:13:20.610 --> 01:13:24.560 and put them together so that they are touching, 01:13:24.560 --> 01:13:26.690 then I would have a liquid. 01:13:26.690 --> 01:13:29.600 And the density of, say, typical liquid 01:13:29.600 --> 01:13:33.740 is of the order of 10,000 times larger than the density of air. 01:13:33.740 --> 01:13:37.516 So basically, it has to be a number of this order. 01:13:37.516 --> 01:13:39.960 OK? 01:13:39.960 --> 01:13:41.355 So fine. 01:13:44.180 --> 01:13:47.310 Let's look at our equations. 01:13:47.310 --> 01:13:53.170 So what I find is that in this equation for f2, 01:13:53.170 --> 01:13:56.570 on the left-hand side, I have a term 01:13:56.570 --> 01:14:00.850 that has magnitude that is very large-- 1 over tau c. 01:14:04.020 --> 01:14:07.280 Whereas the term on the right-hand side, 01:14:07.280 --> 01:14:12.210 in terms of magnitude, is something like this. 01:14:15.420 --> 01:14:17.920 And in fact, this will be true for every equation 01:14:17.920 --> 01:14:20.420 in the hierarchy. 01:14:20.420 --> 01:14:25.570 So maybe if I am in the limit where nd cubed is much, 01:14:25.570 --> 01:14:29.950 much less than 1-- such as the gas in this room, which 01:14:29.950 --> 01:14:34.490 is called the dilute limit-- I can ignore the right-hand side. 01:14:34.490 --> 01:14:38.511 I can set the right-hand side to 0. 01:14:38.511 --> 01:14:39.010 OK? 01:14:41.720 --> 01:14:45.380 Now, I can't do that for the first equation, 01:14:45.380 --> 01:14:47.320 because the first equation is really 01:14:47.320 --> 01:14:50.020 the only equation in the hierarchy that does not 01:14:50.020 --> 01:14:53.484 have the collision term on the left-hand side. 01:14:53.484 --> 01:14:54.950 Right? 01:14:54.950 --> 01:14:59.611 And so for the first equation, I really need to keep that. 01:14:59.611 --> 01:15:03.650 And actually, it goes back to all of the story 01:15:03.650 --> 01:15:07.340 that we've had over here. 01:15:07.340 --> 01:15:11.540 Remember that we said this rho equilibrium has 01:15:11.540 --> 01:15:16.600 to be a function of H and conserved quantities. 01:15:16.600 --> 01:15:20.170 Suppose I go to my Hamiltonian and I 01:15:20.170 --> 01:15:24.290 ignore all of these interactions, which 01:15:24.290 --> 01:15:26.710 is what I would have done if I just 01:15:26.710 --> 01:15:31.370 look at the first term over here and set the collision 01:15:31.370 --> 01:15:34.040 term on the right-hand side to 0. 01:15:34.040 --> 01:15:36.380 What would happen then? 01:15:36.380 --> 01:15:40.180 Then clearly, for each one of the particles, 01:15:40.180 --> 01:15:44.480 I have-- let's say it's energy is going to be conserved. 01:15:44.480 --> 01:15:46.330 Maybe the magnitude of its momentum 01:15:46.330 --> 01:15:49.450 is going to be conserved in the appropriate geometry. 01:15:49.450 --> 01:15:52.390 And so there will be a huge number 01:15:52.390 --> 01:15:55.870 of individual conserved quantities 01:15:55.870 --> 01:15:58.300 that I would have to put over there. 01:15:58.300 --> 01:16:00.560 Indeed, if I sort of go back to the picture 01:16:00.560 --> 01:16:04.860 that I was drawing over here, if I ignore collisions 01:16:04.860 --> 01:16:08.810 between the particles, then the bullets that I send 01:16:08.810 --> 01:16:13.740 will always be following a trajectory such as this 01:16:13.740 --> 01:16:17.200 forever, because momentum will be conserved. 01:16:17.200 --> 01:16:19.990 You will always-- I mean, except up to reflection. 01:16:19.990 --> 01:16:21.760 And say the magnitude of velocity 01:16:21.760 --> 01:16:25.030 would be always following the same thing. 01:16:25.030 --> 01:16:26.380 OK? 01:16:26.380 --> 01:16:30.050 So however, if there is a collision between two 01:16:30.050 --> 01:16:34.090 of the particles-- so the particles that come in here, 01:16:34.090 --> 01:16:37.290 they have different velocities, they will hit each other. 01:16:37.290 --> 01:16:38.810 The moment they hit each other, they 01:16:38.810 --> 01:16:41.080 go off different directions. 01:16:41.080 --> 01:16:43.400 And after a certain number of hits, 01:16:43.400 --> 01:16:45.460 then I will lose all of the regularity 01:16:45.460 --> 01:16:48.490 of what I had in the beginning. 01:16:48.490 --> 01:16:54.390 And so essentially, this second term 01:16:54.390 --> 01:16:57.680 on the right-hand with the collisions 01:16:57.680 --> 01:17:00.550 is the thing that is necessary for me 01:17:00.550 --> 01:17:06.200 to ensure that my gas does come to equilibrium in the sense 01:17:06.200 --> 01:17:09.870 that their momenta get distributed and reversed. 01:17:09.870 --> 01:17:12.890 I really need to keep track of that. 01:17:12.890 --> 01:17:16.350 And also, you can see that the timescales for which 01:17:16.350 --> 01:17:18.800 this kind of equilibration takes place 01:17:18.800 --> 01:17:22.770 has to do with this collision time. 01:17:22.770 --> 01:17:28.030 But as far as this term is concerned, for the gas 01:17:28.030 --> 01:17:31.570 or for that system of bullets, it doesn't really matter. 01:17:31.570 --> 01:17:35.030 Because for this term to have been important, 01:17:35.030 --> 01:17:37.510 it would have been necessary for something 01:17:37.510 --> 01:17:41.470 interesting to physically occur should three particles come 01:17:41.470 --> 01:17:44.520 together simultaneously. 01:17:44.520 --> 01:17:47.450 And if I complete the-- say that never 01:17:47.450 --> 01:17:51.350 in the history of this system three particles will 01:17:51.350 --> 01:17:55.900 come together, they do come together in reality. 01:17:55.900 --> 01:17:58.800 It's not that big a difference. 01:17:58.800 --> 01:18:00.780 It's only a factor of 10 to the 4 difference 01:18:00.780 --> 01:18:03.290 between the right-hand side and the left-hand side. 01:18:03.290 --> 01:18:05.290 But still, even if they didn't, there 01:18:05.290 --> 01:18:07.990 was nothing about equilibration of the gas that 01:18:07.990 --> 01:18:10.910 would be missed by this. 01:18:10.910 --> 01:18:15.710 So it's a perfectly reasonable approximation and assumption, 01:18:15.710 --> 01:18:20.040 therefore, for us to drop this term. 01:18:20.040 --> 01:18:23.020 And we'll see that although that is physically motivated, 01:18:23.020 --> 01:18:26.980 it actually doesn't resolve this question of irreversibility 01:18:26.980 --> 01:18:31.650 yet, because that's also potentially a system 01:18:31.650 --> 01:18:33.130 that you could set up. 01:18:33.130 --> 01:18:35.880 You just eliminate all of the three-body interactions 01:18:35.880 --> 01:18:37.030 from the problem. 01:18:37.030 --> 01:18:39.330 Still, you could have a very reversible set 01:18:39.330 --> 01:18:42.570 of conditions and deterministic process 01:18:42.570 --> 01:18:45.850 that you could reverse in time. 01:18:45.850 --> 01:18:48.180 But still, it's sort of allows us 01:18:48.180 --> 01:18:51.250 to have something that is more manageable, 01:18:51.250 --> 01:18:54.340 which is what we will be looking at next. 01:18:54.340 --> 01:18:57.270 Before I go to what is next, I also 01:18:57.270 --> 01:19:01.450 mentioned that there is one other limit where 01:19:01.450 --> 01:19:04.420 one can do things, which is when, 01:19:04.420 --> 01:19:07.580 within the range of interaction of one particle, 01:19:07.580 --> 01:19:09.930 there are many other particles. 01:19:09.930 --> 01:19:15.180 So you are in the dense limit, nd cubed greater than 1. 01:19:15.180 --> 01:19:18.000 This does not happen for a liquid, because for a liquid, 01:19:18.000 --> 01:19:22.230 the range does not allow many particles to come within it. 01:19:22.230 --> 01:19:24.100 But it happens for a plasma where 01:19:24.100 --> 01:19:26.800 you have long-range Coulomb interaction. 01:19:26.800 --> 01:19:29.780 And within the range of Coulomb interaction, 01:19:29.780 --> 01:19:32.140 you could have many other interactions. 01:19:32.140 --> 01:19:36.620 And so that limit you will explore in the problem set 01:19:36.620 --> 01:19:41.290 leads to a different description of approach to equilibrium. 01:19:41.290 --> 01:19:44.210 It's called the Vlasov equation. 01:19:44.210 --> 01:19:48.140 What we are going to proceed with now will lead to something 01:19:48.140 --> 01:19:52.310 else, which is called-- in the dilute limit, 01:19:52.310 --> 01:19:54.306 it will get the Boltzmann equation. 01:19:58.766 --> 01:19:59.266 OK? 01:20:03.250 --> 01:20:06.925 So let's see what we have. 01:20:19.200 --> 01:20:26.210 So currently, we achieved something. 01:20:26.210 --> 01:20:34.140 We want to describe properties of a few particles 01:20:34.140 --> 01:20:37.450 in the system-- densities that describe only, 01:20:37.450 --> 01:20:41.990 say, one particle by itself if one was not enough. 01:20:41.990 --> 01:20:44.040 But I can terminate the equations. 01:20:44.040 --> 01:20:48.740 And with one-particle density and two-particle density, 01:20:48.740 --> 01:20:51.240 I should have an appropriate description 01:20:51.240 --> 01:20:54.730 of how the system evolves. 01:20:54.730 --> 01:20:57.310 Let's think about it one more time. 01:20:57.310 --> 01:20:59.115 So what is happening here? 01:20:59.115 --> 01:21:02.450 There is the one-particle description 01:21:02.450 --> 01:21:06.580 that tells you how the density for one particle, 01:21:06.580 --> 01:21:10.430 or an ensemble, the probability for one particle, its position 01:21:10.430 --> 01:21:12.740 and momentum evolves. 01:21:12.740 --> 01:21:15.300 But it requires knowledge of what 01:21:15.300 --> 01:21:19.150 would happen with a second particle present. 01:21:19.150 --> 01:21:23.220 But the equations that we have for the density that 01:21:23.220 --> 01:21:28.020 involves two particles is simply a description of things 01:21:28.020 --> 01:21:32.390 that you would do if you had deterministic trajectories. 01:21:32.390 --> 01:21:35.130 There is nothing else on the right-hand side. 01:21:35.130 --> 01:21:38.930 So basically, all you need to do is in order 01:21:38.930 --> 01:21:43.240 to determine this, is to have full knowledge of what happens 01:21:43.240 --> 01:21:46.810 if two particles come together, collide together, go 01:21:46.810 --> 01:21:48.880 away, all kinds of things. 01:21:48.880 --> 01:21:52.410 So if you have those trajectories for two particles, 01:21:52.410 --> 01:21:55.540 you can, in principle, build this density. 01:21:55.540 --> 01:21:57.480 It's still not an easy task. 01:21:57.480 --> 01:22:00.050 But in principle, one could do that. 01:22:00.050 --> 01:22:03.840 And so this is the description of f2. 01:22:03.840 --> 01:22:09.050 And we expect f2 to describe processes 01:22:09.050 --> 01:22:12.960 in which, over a very rapid timescale, say, 01:22:12.960 --> 01:22:18.200 momenta gets shifted from one direction to another direction. 01:22:18.200 --> 01:22:23.270 But then there is something about the overall behavior that 01:22:23.270 --> 01:22:26.030 should follow, more or less, f1. 01:22:26.030 --> 01:22:27.920 Again, what do I mean? 01:22:27.920 --> 01:22:33.400 What I mean is the following, that if I open this box, 01:22:33.400 --> 01:22:35.720 there is what you would observe. 01:22:35.720 --> 01:22:39.060 The density would kind of gush through here. 01:22:39.060 --> 01:22:40.860 And so you can have a description 01:22:40.860 --> 01:22:44.480 for how the density, let's say in coordinate space, 01:22:44.480 --> 01:22:47.590 would be evolving as a function of time. 01:22:47.590 --> 01:22:51.630 If I ask how does the two-particle prescription 01:22:51.630 --> 01:22:55.780 evolve, well, the two-particle prescription, part of it 01:22:55.780 --> 01:22:57.450 is what's the probability that I have 01:22:57.450 --> 01:22:59.890 a particle here and a particle there? 01:22:59.890 --> 01:23:04.120 And if the two particles, if the separations are far apart, 01:23:04.120 --> 01:23:06.880 you would be justified to say that that is roughly 01:23:06.880 --> 01:23:09.920 the product of the probabilities that I have something here 01:23:09.920 --> 01:23:11.710 and something there. 01:23:11.710 --> 01:23:14.960 When you become very close to each other, however, 01:23:14.960 --> 01:23:17.900 over the range of interactions and collisions, 01:23:17.900 --> 01:23:21.510 that will have to be modified, because at those descriptions 01:23:21.510 --> 01:23:24.760 from here, you would have to worry 01:23:24.760 --> 01:23:28.830 about the collisions, and the exchange of momenta, et cetera. 01:23:28.830 --> 01:23:35.610 So in that sense, part of f2 is simply following f1 slowly. 01:23:35.610 --> 01:23:42.340 And part of f1 captures all of the collisions that you have. 01:23:42.340 --> 01:23:45.600 In fact, that part of f2 that captures in the collisions, 01:23:45.600 --> 01:23:49.930 we would like to simplify as much as possible. 01:23:49.930 --> 01:23:52.960 And that's the next task that we do. 01:23:52.960 --> 01:24:02.340 So what I need to do is to somehow express 01:24:02.340 --> 01:24:06.630 the f2 that appears in the first equation 01:24:06.630 --> 01:24:09.690 while solving this equation that is the second one. 01:24:12.870 --> 01:24:15.790 I will write the answer that we will eventually 01:24:15.790 --> 01:24:18.740 deal with and explain it next time. 01:24:18.740 --> 01:24:21.760 So the ultimate result would be that the left-hand side, 01:24:21.760 --> 01:24:23.910 we will have the terms that we have currently. 01:24:33.800 --> 01:24:36.280 f1. 01:24:36.280 --> 01:24:39.660 On the right-hand side, what we find 01:24:39.660 --> 01:24:44.240 is that I need to integrate over all momenta 01:24:44.240 --> 01:24:47.320 of a second particle and something that 01:24:47.320 --> 01:24:53.940 is like a distance to the target-- one term that 01:24:53.940 --> 01:24:56.700 is the flux of incoming particles. 01:24:59.580 --> 01:25:11.930 And then we would have f2 after collision minus f2 01:25:11.930 --> 01:25:12.880 before collision. 01:25:18.580 --> 01:25:21.390 And this is really the Boltzmann equation 01:25:21.390 --> 01:25:26.890 after one more approximation, where we replace f2 with f1. 01:25:26.890 --> 01:25:28.420 f1. 01:25:28.420 --> 01:25:30.730 But what all of that means symbolically 01:25:30.730 --> 01:25:34.780 and what it is we'll have to explain next time.