1 00:00:00,060 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high-quality educational resources for free. 5 00:00:10,730 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:20,540 --> 00:00:21,840 PROFESSOR: OK. 9 00:00:21,840 --> 00:00:22,463 Let's start. 10 00:00:26,420 --> 00:00:30,000 So last time, we started with kinetic theory. 11 00:00:36,120 --> 00:00:39,205 And we will focus for gas systems mostly. 12 00:00:43,890 --> 00:00:45,730 And the question that we would like 13 00:00:45,730 --> 00:00:50,140 to think about and answer somehow is the following. 14 00:00:50,140 --> 00:00:56,260 You start with a gas that is initially 15 00:00:56,260 --> 00:00:57,765 confined to one chamber. 16 00:01:01,400 --> 00:01:06,180 And you can calculate all of its thermodynamic properties. 17 00:01:06,180 --> 00:01:11,560 You open at time 0 a hole, allowing the gas 18 00:01:11,560 --> 00:01:17,290 to escape into a second initially empty chamber. 19 00:01:17,290 --> 00:01:19,400 And after some time, the whole system 20 00:01:19,400 --> 00:01:22,320 will come to a new equilibrium position. 21 00:01:22,320 --> 00:01:25,040 It's a pretty reversible thing. 22 00:01:25,040 --> 00:01:27,620 You can do this experiment many, many times. 23 00:01:27,620 --> 00:01:31,690 And you will always get roughly the same amount of time 24 00:01:31,690 --> 00:01:34,280 for the situation to start from one equilibrium 25 00:01:34,280 --> 00:01:36,820 and reach another equilibrium. 26 00:01:36,820 --> 00:01:38,250 So how do we describe that? 27 00:01:38,250 --> 00:01:41,010 It's slightly beyond what we did in thermodynamics, 28 00:01:41,010 --> 00:01:43,990 because we want to go from one equilibrium state 29 00:01:43,990 --> 00:01:46,910 to another equilibrium state. 30 00:01:46,910 --> 00:01:50,260 Now, we said, OK, we know the equations of motion 31 00:01:50,260 --> 00:01:55,200 that governs the particles that are described by this. 32 00:01:55,200 --> 00:02:00,970 So here we can say that we have, let's say, N particles. 33 00:02:00,970 --> 00:02:05,930 They have their own momenta and coordinates. 34 00:02:05,930 --> 00:02:09,160 And we know that these momenta and coordinates evolve 35 00:02:09,160 --> 00:02:12,230 in time, governed by some Hamiltonian, 36 00:02:12,230 --> 00:02:17,290 which is a function of all of these momenta and coordinates. 37 00:02:17,290 --> 00:02:17,880 OK? 38 00:02:17,880 --> 00:02:19,270 Fine. 39 00:02:19,270 --> 00:02:21,400 How do we go from a situation which 40 00:02:21,400 --> 00:02:23,100 describes a whole bunch of things 41 00:02:23,100 --> 00:02:25,580 and coordinates that are changing with time 42 00:02:25,580 --> 00:02:33,400 to some microscopic description of macroscopic variables going 43 00:02:33,400 --> 00:02:36,850 from one equilibrium state to another? 44 00:02:36,850 --> 00:02:39,740 So our first attempt in that direction 45 00:02:39,740 --> 00:02:46,390 was to say, well, I could start with many, many, many examples 46 00:02:46,390 --> 00:02:49,260 of the same situation. 47 00:02:49,260 --> 00:02:52,190 Each one of them would correspond to a different 48 00:02:52,190 --> 00:02:55,500 trajectory of p's and q's. 49 00:02:55,500 --> 00:02:59,300 And so what we can do is to construct 50 00:02:59,300 --> 00:03:05,720 some kind of an ensemble average, or ensemble density, 51 00:03:05,720 --> 00:03:08,800 first, which is what we did. 52 00:03:08,800 --> 00:03:15,255 We can say that the very, very many examples of this situation 53 00:03:15,255 --> 00:03:20,300 that I can have will correspond to different points 54 00:03:20,300 --> 00:03:23,890 at some particular instant of time, 55 00:03:23,890 --> 00:03:28,085 occupying this 6N-dimensional phase space, 56 00:03:28,085 --> 00:03:32,330 out of which I can construct some kind of a density in phase 57 00:03:32,330 --> 00:03:34,950 space. 58 00:03:34,950 --> 00:03:39,320 But then I realize that since each one of these trajectories 59 00:03:39,320 --> 00:03:42,490 is evolving according to this Hamiltonian, 60 00:03:42,490 --> 00:03:45,320 this density could potentially be a function of time. 61 00:03:48,030 --> 00:03:52,710 And we described the equation for the evolution 62 00:03:52,710 --> 00:03:54,850 of that density with time. 63 00:03:54,850 --> 00:03:59,980 And we could write it in this form-- V rho by dt 64 00:03:59,980 --> 00:04:05,350 is the Poisson bracket of the Hamiltonian with rho. 65 00:04:05,350 --> 00:04:10,120 And this Poisson bracket was defined 66 00:04:10,120 --> 00:04:13,650 as a sum over all of your coordinates. 67 00:04:16,339 --> 00:04:16,839 Oops. 68 00:04:22,490 --> 00:04:34,660 We had d rho by d vector qi, dot product with dH by dpi minus 69 00:04:34,660 --> 00:04:36,116 the other [? way. ?] 70 00:04:42,470 --> 00:04:47,740 So there are essentially three N terms-- well, 71 00:04:47,740 --> 00:04:52,920 actually, six N terms, but three N pairs of terms in this sum. 72 00:04:52,920 --> 00:04:55,890 I can either use some index alpha 73 00:04:55,890 --> 00:04:59,460 running from one to three N, or indicate them 74 00:04:59,460 --> 00:05:05,100 as contributions of things that come from individual particles 75 00:05:05,100 --> 00:05:08,210 and then use this notation with three vectors. 76 00:05:08,210 --> 00:05:11,280 So this is essentially a combination 77 00:05:11,280 --> 00:05:14,440 of sum of three terms. 78 00:05:14,440 --> 00:05:16,900 OK? 79 00:05:16,900 --> 00:05:22,910 So I hope that somehow this equation 80 00:05:22,910 --> 00:05:28,420 can describe the evolution that goes on over here. 81 00:05:28,420 --> 00:05:32,660 And ultimately, when I wait sufficiently long time, 82 00:05:32,660 --> 00:05:36,910 I will reach a situation where d rho by dt 83 00:05:36,910 --> 00:05:39,455 does not change anymore. 84 00:05:39,455 --> 00:05:43,830 And I will find some density that is invariant on time. 85 00:05:43,830 --> 00:05:46,650 I'll call that rho equilibrium. 86 00:05:46,650 --> 00:05:51,810 And so we saw that we could have our rho equilibrium, which then 87 00:05:51,810 --> 00:05:58,620 should have zero Poisson bracket with H to be a function of H 88 00:05:58,620 --> 00:06:00,140 and any other conserved quantity. 89 00:06:05,900 --> 00:06:08,780 OK? 90 00:06:08,780 --> 00:06:14,582 So in principle, we sort of thought 91 00:06:14,582 --> 00:06:21,030 of a way of describing how the system will evolve 92 00:06:21,030 --> 00:06:24,660 in equilibrium-- towards an equilibrium. 93 00:06:24,660 --> 00:06:30,290 And indeed, we will find that in statistical mechanics later on, 94 00:06:30,290 --> 00:06:35,340 we are going to rely heavily on these descriptions 95 00:06:35,340 --> 00:06:38,370 of equilibrium densities, which are 96 00:06:38,370 --> 00:06:41,090 governed by only the Hamiltonian. 97 00:06:41,090 --> 00:06:43,850 And if there are any other conserved quantities, 98 00:06:43,850 --> 00:06:45,150 typically there are not. 99 00:06:45,150 --> 00:06:49,650 So this is the general form that we'll ultimately 100 00:06:49,650 --> 00:06:52,630 be using a lot. 101 00:06:52,630 --> 00:06:56,730 But we started with a description 102 00:06:56,730 --> 00:06:58,820 of evolution of these coordinates 103 00:06:58,820 --> 00:07:01,420 in time, which is time-reversible. 104 00:07:05,870 --> 00:07:08,835 And we hope to end up with a situation that 105 00:07:08,835 --> 00:07:12,180 is analogous to what we describe thermodynamically, 106 00:07:12,180 --> 00:07:18,410 we describe something that goes to some particular state, 107 00:07:18,410 --> 00:07:20,230 and basically stays there, as far 108 00:07:20,230 --> 00:07:23,620 as the macroscopic description is concerned. 109 00:07:23,620 --> 00:07:29,550 So did we somehow manage to just look at something else, which 110 00:07:29,550 --> 00:07:32,470 is this density, and achieve this transition 111 00:07:32,470 --> 00:07:35,520 from reversibility to irreversibility? 112 00:07:35,520 --> 00:07:36,760 And the answer is no. 113 00:07:36,760 --> 00:07:40,610 This equation-- also, if you find, indeed, 114 00:07:40,610 --> 00:07:47,230 a rho that goes from here to here, you can set t to minus t 115 00:07:47,230 --> 00:07:49,705 and get a rho that goes from back here to here. 116 00:07:49,705 --> 00:07:52,500 It has the same time-reversibility. 117 00:07:52,500 --> 00:07:56,400 So somehow, we have to find another solution. 118 00:07:56,400 --> 00:08:00,180 And the solution that we will gradually build upon 119 00:08:00,180 --> 00:08:03,150 relies more on physics rather than 120 00:08:03,150 --> 00:08:06,850 the rigorous mathematical nature of this thing. 121 00:08:06,850 --> 00:08:09,310 Physically, we know that this happens. 122 00:08:09,310 --> 00:08:11,560 All of us have seen this. 123 00:08:11,560 --> 00:08:16,026 So somehow, we should be able to use physical approximation 124 00:08:16,026 --> 00:08:18,860 and assumptions that are compatible with what 125 00:08:18,860 --> 00:08:20,260 is observed. 126 00:08:20,260 --> 00:08:24,280 And if, during that, we have to sort of make 127 00:08:24,280 --> 00:08:27,450 mathematical approximations, so be it. 128 00:08:27,450 --> 00:08:30,330 In fact, we have to make mathematical approximations, 129 00:08:30,330 --> 00:08:35,620 because otherwise, there is this very strong reversibility 130 00:08:35,620 --> 00:08:37,000 condition. 131 00:08:37,000 --> 00:08:41,030 So let's see how we are about to proceed, thinking more 132 00:08:41,030 --> 00:08:43,820 in terms of physics. 133 00:08:43,820 --> 00:08:47,840 Now, the information that I have over here, 134 00:08:47,840 --> 00:08:50,335 either in this description of ensemble 135 00:08:50,335 --> 00:08:53,220 or in this description in terms of trajectories, 136 00:08:53,220 --> 00:08:55,960 is just enormous. 137 00:08:55,960 --> 00:09:01,090 I can't think of any physical process which 138 00:09:01,090 --> 00:09:05,990 would need to keep track of the joined positions of coordinates 139 00:09:05,990 --> 00:09:08,900 and momenta of 10 to the 23 particles, 140 00:09:08,900 --> 00:09:13,190 and know them with infinite precision, et cetera. 141 00:09:13,190 --> 00:09:15,280 Things that I make observations with 142 00:09:15,280 --> 00:09:18,790 and I say, we see this, well, what do we see? 143 00:09:18,790 --> 00:09:24,090 We see that something has some kind of a density over here. 144 00:09:24,090 --> 00:09:26,010 It has some kind of pressure. 145 00:09:26,010 --> 00:09:28,720 Maybe there is, in the middle of this process, 146 00:09:28,720 --> 00:09:30,530 some flow of gas particles. 147 00:09:30,530 --> 00:09:33,420 We can talk about the velocity of those things. 148 00:09:33,420 --> 00:09:35,130 And let's say even when we are sort 149 00:09:35,130 --> 00:09:39,460 of being very non-equilibrium and thinking about velocity 150 00:09:39,460 --> 00:09:43,550 of those particles going from one side to the other side, 151 00:09:43,550 --> 00:09:48,060 we really don't care which particle among the 10 to the 23 152 00:09:48,060 --> 00:09:50,520 is at some instant of time contributing 153 00:09:50,520 --> 00:09:55,130 to the velocity of the gas that is swishing past. 154 00:09:55,130 --> 00:09:59,720 So clearly, any physical observable that we make 155 00:09:59,720 --> 00:10:03,780 is something that does not require 156 00:10:03,780 --> 00:10:05,890 all of those degrees of freedom. 157 00:10:05,890 --> 00:10:08,060 So let's just construct some of those things 158 00:10:08,060 --> 00:10:10,480 that we may find useful and see how 159 00:10:10,480 --> 00:10:13,170 we would describe evolutions of them. 160 00:10:13,170 --> 00:10:16,020 I mean, the most useful thing, indeed, is the density. 161 00:10:16,020 --> 00:10:19,850 So what I could do is I can actually 162 00:10:19,850 --> 00:10:23,210 construct a one-particle density. 163 00:10:23,210 --> 00:10:26,455 So I want to look at some position in space 164 00:10:26,455 --> 00:10:30,530 at some time t and ask whether or not 165 00:10:30,530 --> 00:10:34,960 there are particles there. 166 00:10:34,960 --> 00:10:37,220 I don't care which one of the particles. 167 00:10:37,220 --> 00:10:40,300 Actually, let's put in a little bit more information; 168 00:10:40,300 --> 00:10:44,090 also, keep track of whether, at some instant of time, 169 00:10:44,090 --> 00:10:47,350 I see at this location particles. 170 00:10:47,350 --> 00:10:49,370 And these particles, I will also ask 171 00:10:49,370 --> 00:10:51,020 which direction they are moving. 172 00:10:51,020 --> 00:10:53,650 Maybe I'm also going to think about the case 173 00:10:53,650 --> 00:10:57,640 where, in the intermediate, I am flowing from one side 174 00:10:57,640 --> 00:10:58,270 to another. 175 00:10:58,270 --> 00:11:01,840 And keeping track of both of these is important. 176 00:11:01,840 --> 00:11:06,510 I said I don't care which one of these particles 177 00:11:06,510 --> 00:11:09,340 is contributing. 178 00:11:09,340 --> 00:11:18,180 So that means that I have to sum over all particles 179 00:11:18,180 --> 00:11:23,345 and ask whether or not, at time t, 180 00:11:23,345 --> 00:11:29,818 there is a particle at location q with momentum p. 181 00:11:32,890 --> 00:11:37,490 So these delta functions are supposed to enforce that. 182 00:11:37,490 --> 00:11:41,860 And again, I'm not thinking about an individual trajectory, 183 00:11:41,860 --> 00:11:44,750 because I think, more or less, all of the trajectories 184 00:11:44,750 --> 00:11:46,970 are going to behave the same way. 185 00:11:46,970 --> 00:11:50,940 And so what I will do is I take an ensemble 186 00:11:50,940 --> 00:11:53,161 average of this quantity. 187 00:11:53,161 --> 00:11:53,660 OK? 188 00:11:53,660 --> 00:11:55,300 So what does that mean? 189 00:11:55,300 --> 00:11:57,650 To do an ensemble average, we said 190 00:11:57,650 --> 00:12:04,200 I have to integrate the density function, rho, 191 00:12:04,200 --> 00:12:06,970 which depends on all of the coordinates. 192 00:12:06,970 --> 00:12:09,910 So I have coordinate for particle one, 193 00:12:09,910 --> 00:12:14,780 coordinate-- particle and momentum, particle and momentum 194 00:12:14,780 --> 00:12:17,690 for particle two, for particle three, 195 00:12:17,690 --> 00:12:22,750 all the way to particle number N. 196 00:12:22,750 --> 00:12:24,970 And this is something that depends on time. 197 00:12:24,970 --> 00:12:26,920 That's where the time dependence comes from. 198 00:12:26,920 --> 00:12:29,030 And again, let's think about the time dependence 199 00:12:29,030 --> 00:12:31,620 where 0 is you lift the partition 200 00:12:31,620 --> 00:12:34,750 and you allow things to go from one to the other. 201 00:12:34,750 --> 00:12:38,950 So there is a non-equilibrium process that you are following. 202 00:12:38,950 --> 00:12:44,120 What I need to do is I have to multiply this density 203 00:12:44,120 --> 00:12:50,130 with the function that I have to consider the average. 204 00:12:50,130 --> 00:12:53,790 Well, the function being a delta function, it's very nice. 205 00:12:53,790 --> 00:12:59,980 The first term in the sum, we'll set simply q1 equals to q. 206 00:12:59,980 --> 00:13:04,440 And it will set p1 equals to p when 207 00:13:04,440 --> 00:13:08,490 I do the integration over p1 and q1. 208 00:13:08,490 --> 00:13:12,940 But then I'm left to have to do the integration over particle 209 00:13:12,940 --> 00:13:20,155 two, particle three, particle four, and so forth. 210 00:13:20,155 --> 00:13:21,580 OK? 211 00:13:21,580 --> 00:13:24,930 But this is only the first term in the sum. 212 00:13:24,930 --> 00:13:27,480 Then I have to write the sum for particle number two, 213 00:13:27,480 --> 00:13:30,220 particle number three, et cetera. 214 00:13:30,220 --> 00:13:33,070 But all of the particles, as far as I'm concerned, 215 00:13:33,070 --> 00:13:34,720 are behaving the same way. 216 00:13:34,720 --> 00:13:44,510 So I just need to multiply this by a factor of N. 217 00:13:44,510 --> 00:13:49,700 Now, this is something that actually we encountered before. 218 00:13:49,700 --> 00:13:54,500 Recall that this rho is a joint probability distribution. 219 00:13:54,500 --> 00:13:56,600 It's a joint probability distribution 220 00:13:56,600 --> 00:14:01,010 of all N particles, their locations, and momentum. 221 00:14:01,010 --> 00:14:10,510 And we gave a name to taking a joint probability density 222 00:14:10,510 --> 00:14:13,180 function, which this is, and integrating 223 00:14:13,180 --> 00:14:16,266 over some subset of variables. 224 00:14:16,266 --> 00:14:21,150 The answer is an unconditional probability distribution. 225 00:14:21,150 --> 00:14:24,380 So essentially, the end of this story 226 00:14:24,380 --> 00:14:27,630 is if I integrate over coordinates 227 00:14:27,630 --> 00:14:32,060 of two all the way to N, I will get an unconditional result 228 00:14:32,060 --> 00:14:34,840 pertaining to the first set of coordinate. 229 00:14:34,840 --> 00:14:38,470 So this is the same thing up to a factor of N-- 230 00:14:38,470 --> 00:14:43,110 the unconditional probability that I will call rho one. 231 00:14:43,110 --> 00:14:44,885 Actually, I should have called this f1. 232 00:14:47,830 --> 00:14:50,570 So this is something that is defined 233 00:14:50,570 --> 00:14:52,470 to be the one particle density. 234 00:14:58,540 --> 00:15:04,430 I don't know why this name stuck to it, because this one, that 235 00:15:04,430 --> 00:15:07,220 is up to a factor of N different from it, 236 00:15:07,220 --> 00:15:13,340 is our usual unconditional probability for one particle. 237 00:15:13,340 --> 00:15:28,944 So this is rho of p1 being p, q1 being q at time. 238 00:15:28,944 --> 00:15:30,850 OK? 239 00:15:30,850 --> 00:15:35,930 So essentially, what I've said is I have a joint probability. 240 00:15:35,930 --> 00:15:37,990 I'm really interested in one of the particles, 241 00:15:37,990 --> 00:15:41,890 so I just integrate over all the others. 242 00:15:41,890 --> 00:15:47,190 And this is kind of not very elegant, 243 00:15:47,190 --> 00:15:49,100 but that's somehow the way that it 244 00:15:49,100 --> 00:15:51,090 has appeared in the literature. 245 00:15:51,090 --> 00:15:53,790 The entity that I have on the right 246 00:15:53,790 --> 00:15:59,730 is the properly normalized probability. 247 00:15:59,730 --> 00:16:05,960 Once I multiply by N, it's a quantity this is called f1. 248 00:16:05,960 --> 00:16:10,130 And it's called a density and used very much 249 00:16:10,130 --> 00:16:12,170 in the literature. 250 00:16:12,170 --> 00:16:13,236 OK? 251 00:16:13,236 --> 00:16:16,240 And that's the thing that I think will be useful. 252 00:16:16,240 --> 00:16:20,420 Essentially, all of the things that I 253 00:16:20,420 --> 00:16:23,020 know about observations of a system, 254 00:16:23,020 --> 00:16:26,870 such as its density, its velocity, et cetera, 255 00:16:26,870 --> 00:16:30,080 I should be able to get from this. 256 00:16:30,080 --> 00:16:33,800 Sometimes we need a little bit more information, 257 00:16:33,800 --> 00:16:36,610 so I allow for the possibility that I may also 258 00:16:36,610 --> 00:16:41,570 need to focus on a two-particle density, where 259 00:16:41,570 --> 00:16:48,710 I keep information pertaining to two particles at time t 260 00:16:48,710 --> 00:16:52,570 and I integrate over everybody else that I'm not interested. 261 00:16:52,570 --> 00:16:58,180 So I introduce an integration over coordinates number 262 00:16:58,180 --> 00:17:03,720 two-- sorry, number three all the way to N 263 00:17:03,720 --> 00:17:09,280 to not have to repeat this combination all the time. 264 00:17:09,280 --> 00:17:12,520 The 6N-dimensional phase space for particle 265 00:17:12,520 --> 00:17:15,839 i I will indicate by dVi. 266 00:17:15,839 --> 00:17:24,329 So I take the full N particle density, if you like, 267 00:17:24,329 --> 00:17:28,160 integrate over everybody except 2. 268 00:17:28,160 --> 00:17:32,950 And up to a normalization of NN minus 1, 269 00:17:32,950 --> 00:17:35,940 this is called a two-particle density. 270 00:17:35,940 --> 00:17:41,160 And again, absent this normalization, 271 00:17:41,160 --> 00:17:44,750 this is simply the unconditional probability 272 00:17:44,750 --> 00:17:47,592 that I would construct out of this joint probability 273 00:17:47,592 --> 00:17:48,092 [INAUDIBLE]. 274 00:17:55,970 --> 00:18:00,340 And just for some mathematical purposes, the generalization 275 00:18:00,340 --> 00:18:05,370 of this to S particles, we will call fs. 276 00:18:05,370 --> 00:18:11,510 So this depends on p1 through qs at time t. 277 00:18:11,510 --> 00:18:14,740 And this is going to be N factorial divided 278 00:18:14,740 --> 00:18:18,910 by N minus S factorial in general. 279 00:18:18,910 --> 00:18:24,110 And they join the unconditional probability that depends on S 280 00:18:24,110 --> 00:18:29,466 coordinates p1 through qs at time t. 281 00:18:34,312 --> 00:18:34,812 OK? 282 00:18:37,740 --> 00:18:40,860 You'll see why I need these higher ones 283 00:18:40,860 --> 00:18:47,620 although, in reality, all my interest is on this first one, 284 00:18:47,620 --> 00:18:50,120 because practically everything that I need 285 00:18:50,120 --> 00:18:54,190 to know about this initial experiment that I set up 286 00:18:54,190 --> 00:18:57,550 and how a gas expands I should be able to extract 287 00:18:57,550 --> 00:19:00,004 from this one-particle density. 288 00:19:00,004 --> 00:19:02,220 OK? 289 00:19:02,220 --> 00:19:05,350 So how do I calculate this? 290 00:19:05,350 --> 00:19:08,770 Well, what I would need to do is to look 291 00:19:08,770 --> 00:19:15,150 at the time variation of fs with t 292 00:19:15,150 --> 00:19:20,290 to calculate the time dependence of any one of these quantities. 293 00:19:20,290 --> 00:19:24,040 Ultimately, I want to have an equation for df1 by dt. 294 00:19:24,040 --> 00:19:26,330 But let's write the general one. 295 00:19:26,330 --> 00:19:33,850 So the general one is up to, again, this N factorial, 296 00:19:33,850 --> 00:19:44,510 N minus s factorial and integral over 297 00:19:44,510 --> 00:19:50,740 coordinates that I'm not interested, of d rho by dt. 298 00:19:55,830 --> 00:19:56,380 OK? 299 00:19:56,380 --> 00:20:01,080 So I just take the time derivative inside the integral, 300 00:20:01,080 --> 00:20:04,580 because I know how d rho by dt evolves. 301 00:20:04,580 --> 00:20:11,900 And so this is going to be simply the Poisson bracket of H 302 00:20:11,900 --> 00:20:13,190 and rho. 303 00:20:13,190 --> 00:20:14,540 And I would proceed from there. 304 00:20:17,980 --> 00:20:20,150 Actually, to proceed further and in order 305 00:20:20,150 --> 00:20:25,870 to be able to say things that are related to physics, 306 00:20:25,870 --> 00:20:28,410 I need to say something about the Hamiltonian. 307 00:20:28,410 --> 00:20:32,940 I can't do so in the most general case. 308 00:20:32,940 --> 00:20:35,030 So let's write the kind of Hamiltonian 309 00:20:35,030 --> 00:20:39,930 that we are interested and describes the gas-- so 310 00:20:39,930 --> 00:20:44,530 the Hamiltonian for the gas in this room. 311 00:20:44,530 --> 00:20:46,640 One term is simply the kinetic energy. 312 00:20:52,360 --> 00:20:55,420 Another term is that gas particles 313 00:20:55,420 --> 00:20:57,520 are confined by some potential. 314 00:20:57,520 --> 00:21:01,740 The potential could be as easy as the walls of this room. 315 00:21:01,740 --> 00:21:05,370 Or it could be some more general potential 316 00:21:05,370 --> 00:21:09,660 that could include gravity, whatever else you want. 317 00:21:09,660 --> 00:21:14,850 So let's include that possibility. 318 00:21:14,850 --> 00:21:17,680 So these are so-called one-body terms, 319 00:21:17,680 --> 00:21:21,670 because they pertain to the coordinates of one particle. 320 00:21:21,670 --> 00:21:24,225 And then there are two-body terms. 321 00:21:24,225 --> 00:21:27,840 So for example, I could look at all pairs of particles. 322 00:21:27,840 --> 00:21:33,480 Let's say i not equal to j to avoid self-interaction. 323 00:21:33,480 --> 00:21:36,730 V of qi minus qj. 324 00:21:36,730 --> 00:21:40,910 So certainly, two particles in this gas in this room, 325 00:21:40,910 --> 00:21:44,760 when they get close enough, they certainly 326 00:21:44,760 --> 00:21:46,290 can pass through each other. 327 00:21:46,290 --> 00:21:47,840 They each have a size. 328 00:21:47,840 --> 00:21:51,970 But even when they are a few times their sizes, 329 00:21:51,970 --> 00:21:54,120 they start to feel some interaction, which 330 00:21:54,120 --> 00:21:55,270 causes them to collide. 331 00:21:55,270 --> 00:21:56,269 Yes. 332 00:21:56,269 --> 00:21:58,145 AUDIENCE: The whole series of arguments 333 00:21:58,145 --> 00:22:00,021 that we're developing, do these only hold 334 00:22:00,021 --> 00:22:01,897 for time-independent potentials? 335 00:22:01,897 --> 00:22:04,620 PROFESSOR: Yes. 336 00:22:04,620 --> 00:22:07,820 Although, it is not that difficult to sort of 337 00:22:07,820 --> 00:22:10,200 go through all of these arguments 338 00:22:10,200 --> 00:22:13,940 and see where the corresponding modifications are going to be. 339 00:22:13,940 --> 00:22:15,860 But certainly, the very first thing 340 00:22:15,860 --> 00:22:21,030 that we are using, which is this one, we kind of implicitly 341 00:22:21,030 --> 00:22:23,810 assume the time-independent Hamiltonian. 342 00:22:23,810 --> 00:22:27,370 So you have to start changing things from that point. 343 00:22:30,615 --> 00:22:31,115 OK? 344 00:22:34,090 --> 00:22:37,071 So in principle, you could have three-body and higher body 345 00:22:37,071 --> 00:22:37,570 terms. 346 00:22:37,570 --> 00:22:40,990 But essentially, most of the relevant physics 347 00:22:40,990 --> 00:22:42,330 is captured by this. 348 00:22:42,330 --> 00:22:44,970 Even for things like plasmas, this 349 00:22:44,970 --> 00:22:48,400 would be a reasonably good approximation, 350 00:22:48,400 --> 00:22:51,630 with the Coulomb interaction appearing here. 351 00:22:51,630 --> 00:22:53,860 OK? 352 00:22:53,860 --> 00:23:00,500 Now, what I note is that what I have to calculate over here 353 00:23:00,500 --> 00:23:04,710 is a Poisson bracket. 354 00:23:04,710 --> 00:23:08,760 And then I have to integrate that Poisson bracket. 355 00:23:08,760 --> 00:23:10,860 Poisson bracket involves a whole bunch 356 00:23:10,860 --> 00:23:14,620 of derivatives over all set of quantities. 357 00:23:14,620 --> 00:23:20,200 And I realize that when I'm integrating over derivatives, 358 00:23:20,200 --> 00:23:22,280 there are simplifications that can 359 00:23:22,280 --> 00:23:25,410 take place, such as integration by parts. 360 00:23:25,410 --> 00:23:30,260 But that only will take place when the derivative 361 00:23:30,260 --> 00:23:34,590 is one of the variables that is being integrated. 362 00:23:34,590 --> 00:23:41,370 Now, this whole process has separated out arguments of rho, 363 00:23:41,370 --> 00:23:44,100 as far as this expression is concerned, 364 00:23:44,100 --> 00:23:48,980 into two sets-- one set, or the set that is appearing out here, 365 00:23:48,980 --> 00:23:53,820 the first s ones that don't undergo the integration, 366 00:23:53,820 --> 00:23:58,250 and then the remainder that do undergo the integration. 367 00:23:58,250 --> 00:24:02,020 So this is going to be relevant when we do our manipulations. 368 00:24:02,020 --> 00:24:06,400 And therefore, it is useful to rewrite this Hamiltonian 369 00:24:06,400 --> 00:24:07,725 in terms of three entities. 370 00:24:12,560 --> 00:24:15,660 The first one that I call H sub s-- so if you like, 371 00:24:15,660 --> 00:24:18,550 this is an end particle Hamiltonian. 372 00:24:18,550 --> 00:24:22,720 I can write an H sub s, which is just exactly the same thing, 373 00:24:22,720 --> 00:24:25,560 except that it applies to coordinates 374 00:24:25,560 --> 00:24:27,920 that I'm not integrating over. 375 00:24:27,920 --> 00:24:30,390 And to sort of make a distinction, 376 00:24:30,390 --> 00:24:35,870 I will label them by N. And it includes the interaction 377 00:24:35,870 --> 00:24:37,770 among those particles. 378 00:24:45,900 --> 00:24:48,930 And I can similarly write something 379 00:24:48,930 --> 00:24:55,780 that pertains to coordinates that I am integrating over. 380 00:24:55,780 --> 00:24:58,930 I will label them by j and k. 381 00:24:58,930 --> 00:25:10,010 So this is s plus 1 to N, pj squared over 2m plus u of qj 382 00:25:10,010 --> 00:25:17,820 plus 1/2 sum over j and k, V of qj minus qk. 383 00:25:20,858 --> 00:25:23,790 OK? 384 00:25:23,790 --> 00:25:27,420 So everything that involves one set of coordinates, everything 385 00:25:27,420 --> 00:25:29,310 that involves the other set of coordinates. 386 00:25:29,310 --> 00:25:35,090 So what is left are terms that [? copy ?] one set of 387 00:25:35,090 --> 00:25:37,090 coordinates to another. 388 00:25:37,090 --> 00:25:41,680 So N running from 1 to s, j running from s plus 1 389 00:25:41,680 --> 00:25:48,340 to N, V between qm and qj. 390 00:25:48,340 --> 00:25:48,840 OK? 391 00:25:57,970 --> 00:26:01,980 So let me rewrite this equation rather than in terms 392 00:26:01,980 --> 00:26:09,450 of-- f in terms of the probabilities' rhos. 393 00:26:09,450 --> 00:26:13,890 So the difference only is that I don't 394 00:26:13,890 --> 00:26:17,210 have to include this factor out front. 395 00:26:17,210 --> 00:26:19,610 So I have dVi. 396 00:26:19,610 --> 00:26:27,150 I have the d rho by dt, which is the commutator of H with rho, 397 00:26:27,150 --> 00:26:33,636 which I have been writing as Hs plus HN minus s. 398 00:26:33,636 --> 00:26:35,920 I'm changing the way I write s. 399 00:26:35,920 --> 00:26:37,080 Sorry. 400 00:26:37,080 --> 00:26:41,376 Plus H prime and rho. 401 00:26:41,376 --> 00:26:44,250 OK? 402 00:26:44,250 --> 00:26:46,650 So there is a bit of mathematics to be 403 00:26:46,650 --> 00:26:50,160 performed to analyze this. 404 00:26:50,160 --> 00:26:53,400 There are three terms that I will label a, b, 405 00:26:53,400 --> 00:26:59,160 and c, which are the three Poisson 406 00:26:59,160 --> 00:27:02,450 brackets that I have to evaluate. 407 00:27:02,450 --> 00:27:06,310 So the contribution that I call a 408 00:27:06,310 --> 00:27:15,430 is the integral over coordinates s 409 00:27:15,430 --> 00:27:24,020 plus 1 to N, of the Poisson bracket of Hs with rho. 410 00:27:24,020 --> 00:27:29,130 Now, the Poisson bracket is given up here. 411 00:27:29,130 --> 00:27:35,650 It is a sum that involves N terms over all N particles. 412 00:27:35,650 --> 00:27:42,540 But since, if I'm evaluating it for Hs and Hs 413 00:27:42,540 --> 00:27:46,270 will only give nonzero derivatives with respect 414 00:27:46,270 --> 00:27:49,400 to the coordinates that are present in it, 415 00:27:49,400 --> 00:27:51,755 this sum of N terms actually becomes simply 416 00:27:51,755 --> 00:27:54,850 a sum of small N terms. 417 00:27:54,850 --> 00:28:01,860 So I will get a sum over N running from 1 to s. 418 00:28:01,860 --> 00:28:03,570 These are the only terms. 419 00:28:03,570 --> 00:28:11,715 And I will get the things that I have for d rho by dpN-- sorry, 420 00:28:11,715 --> 00:28:25,496 rho by dqN, dHs by dpN minus d rho by dpN, dHs by dqN. 421 00:28:33,207 --> 00:28:33,707 OK? 422 00:28:39,040 --> 00:28:40,634 Did I make a mistake? 423 00:28:40,634 --> 00:28:46,418 AUDIENCE: Isn't it the Poisson bracket of rho H, not H rho? 424 00:28:46,418 --> 00:28:47,382 PROFESSOR: Rho. 425 00:28:51,250 --> 00:28:51,970 Oh, yes. 426 00:28:51,970 --> 00:28:54,720 So I have to put a minus sign here. 427 00:28:54,720 --> 00:28:55,220 Right. 428 00:28:55,220 --> 00:28:56,200 Good. 429 00:28:56,200 --> 00:28:57,180 Thank you. 430 00:29:01,590 --> 00:29:02,580 OK. 431 00:29:02,580 --> 00:29:08,740 Now, note that these derivatives and operations that 432 00:29:08,740 --> 00:29:13,800 involve Hs involve coordinates that are not 433 00:29:13,800 --> 00:29:18,460 appearing in the integration process. 434 00:29:18,460 --> 00:29:20,770 So I can take these entities that 435 00:29:20,770 --> 00:29:23,190 do not depend on variables that are part 436 00:29:23,190 --> 00:29:30,080 of the integration outside the integration, which means that I 437 00:29:30,080 --> 00:29:35,590 can then write the result as being 438 00:29:35,590 --> 00:29:44,480 an exchange of the order of the Poisson bracket 439 00:29:44,480 --> 00:29:47,850 and the integration. 440 00:29:47,850 --> 00:30:04,750 So I would essentially have the integration only appear here, 441 00:30:04,750 --> 00:30:17,940 which is the same thing as Hs and rho s. 442 00:30:17,940 --> 00:30:19,350 This should be rho s. 443 00:30:24,800 --> 00:30:28,200 This is the definition of rho s, which is the same thing as rho. 444 00:30:31,492 --> 00:30:31,992 OK. 445 00:30:34,807 --> 00:30:36,015 What does it mean physically? 446 00:30:36,015 --> 00:30:40,430 So it's actually much easier to tell you 447 00:30:40,430 --> 00:30:43,810 what it means physically than to do the math. 448 00:30:43,810 --> 00:30:48,500 So what we saw was happening was that if I 449 00:30:48,500 --> 00:30:52,960 have a Hamiltonian that describes N particles, then 450 00:30:52,960 --> 00:30:58,370 for that Hamiltonian, the corresponding density 451 00:30:58,370 --> 00:30:59,980 satisfies this Liouville equation. 452 00:30:59,980 --> 00:31:02,810 D rho by dt is HN rho. 453 00:31:02,810 --> 00:31:07,620 And this was a consequence of this divergence-less character 454 00:31:07,620 --> 00:31:09,810 of the flow that we have in this space, 455 00:31:09,810 --> 00:31:13,620 that the equations that we write down over here 456 00:31:13,620 --> 00:31:17,220 for p dot and q dot in terms of H 457 00:31:17,220 --> 00:31:21,820 had this character that the divergence was 0. 458 00:31:21,820 --> 00:31:23,410 OK? 459 00:31:23,410 --> 00:31:28,230 Now, this is true if I have any number of particles. 460 00:31:28,230 --> 00:31:34,410 So if I focus simply on s of the particles, 461 00:31:34,410 --> 00:31:38,416 and they are governed by this Hamiltonian, 462 00:31:38,416 --> 00:31:41,640 and I don't have anything else in the universe, 463 00:31:41,640 --> 00:31:44,400 as far as this Hamiltonian is concerned, 464 00:31:44,400 --> 00:31:48,740 I should have the analog of a Liouville equation. 465 00:31:48,740 --> 00:31:52,720 So the term that I have obtained over there from this first term 466 00:31:52,720 --> 00:31:56,430 is simply stating that d rho s by dt, 467 00:31:56,430 --> 00:31:59,620 if there was no other interaction with anybody else, 468 00:31:59,620 --> 00:32:02,960 would simply satisfy the corresponding Liouville 469 00:32:02,960 --> 00:32:06,170 equation for s particles. 470 00:32:06,170 --> 00:32:09,620 And because of that, we also expect and anticipate-- 471 00:32:09,620 --> 00:32:11,990 and I'll show that mathematically-- 472 00:32:11,990 --> 00:32:13,890 that the next term in the series, 473 00:32:13,890 --> 00:32:18,800 that is the Poisson bracket of N minus s and rho, 474 00:32:18,800 --> 00:32:23,760 should be 0, because as far as this s particles 475 00:32:23,760 --> 00:32:28,780 that I'm focusing on and how they evolve, 476 00:32:28,780 --> 00:32:33,290 they really don't care about what all the other particles 477 00:32:33,290 --> 00:32:35,970 are doing if they are not [INAUDIBLE]. 478 00:32:35,970 --> 00:32:37,685 So anything interesting should ultimately 479 00:32:37,685 --> 00:32:39,940 come from this third term. 480 00:32:39,940 --> 00:32:44,530 But let's actually go and do the calculation for the second term 481 00:32:44,530 --> 00:32:49,510 to show that this anticipation that the answer should be 0 482 00:32:49,510 --> 00:32:51,190 does hold up and why. 483 00:32:55,100 --> 00:33:02,410 So for the second term, I need to calculate a similar Poisson 484 00:33:02,410 --> 00:33:05,690 bracket, except that this second Poisson 485 00:33:05,690 --> 00:33:09,290 bracket involves H of N minus s. 486 00:33:09,290 --> 00:33:14,790 And H of N minus s, when I put in the full sum, 487 00:33:14,790 --> 00:33:19,530 will only get contribution from terms that start from s plus 1. 488 00:33:19,530 --> 00:33:22,100 So the same way that that started from N, 489 00:33:22,100 --> 00:33:27,840 this contribution starts from s plus 1 to N. 490 00:33:27,840 --> 00:33:31,360 And actually, I can just write the whole thing as above, d 491 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, 492 00:33:50,570 --> 00:34:01,990 this is the rho. d rho by d pj dotted by dHN minus s by dqj. 493 00:34:11,340 --> 00:34:16,010 So now I have a totally different situation 494 00:34:16,010 --> 00:34:19,010 from the previous case, because the previous case, 495 00:34:19,010 --> 00:34:22,630 the derivatives were over things I was not integrating. 496 00:34:22,630 --> 00:34:26,500 I could take outside the integral. 497 00:34:26,500 --> 00:34:29,010 Now all of the derivatives involve things 498 00:34:29,010 --> 00:34:31,252 that I'm integrating over. 499 00:34:31,252 --> 00:34:36,090 Now, when that happens, then you do integration by parts. 500 00:34:36,090 --> 00:34:42,620 So what you do is you take rho outside 501 00:34:42,620 --> 00:34:47,400 and let the derivative act on everything else. 502 00:34:47,400 --> 00:34:49,080 OK? 503 00:34:49,080 --> 00:34:54,219 So what do we end up with if we do integration by parts? 504 00:35:02,640 --> 00:35:06,140 I will get surface terms. 505 00:35:06,140 --> 00:35:09,130 Surface terms are essentially rho-evaluated when 506 00:35:09,130 --> 00:35:11,660 the coordinates are at infinity or at the edge 507 00:35:11,660 --> 00:35:14,360 of your space, where rho is 0. 508 00:35:14,360 --> 00:35:17,080 So there is no surface term. 509 00:35:17,080 --> 00:35:19,140 There is an overall change in sign, 510 00:35:19,140 --> 00:35:23,540 so I will get a product i running from s plus 1 511 00:35:23,540 --> 00:35:27,420 to N, dVi. 512 00:35:27,420 --> 00:35:30,640 Now the rho comes outside. 513 00:35:30,640 --> 00:35:34,840 And the derivative acts on everything that is left. 514 00:35:34,840 --> 00:35:37,470 So the first term will give me a second derivative 515 00:35:37,470 --> 00:35:45,930 of HN minus s, with respect to p, with respect to q. 516 00:35:45,930 --> 00:35:49,560 And the second term will be essentially the opposite way 517 00:35:49,560 --> 00:35:50,560 of doing the derivative. 518 00:35:57,130 --> 00:35:58,787 And these two are, of course, the same. 519 00:35:58,787 --> 00:35:59,620 And the answer is 0. 520 00:36:03,106 --> 00:36:04,600 OK? 521 00:36:04,600 --> 00:36:10,845 So we do expect that the evolution of all 522 00:36:10,845 --> 00:36:13,650 the other particles should not affect 523 00:36:13,650 --> 00:36:15,920 the subset that we are looking at. 524 00:36:15,920 --> 00:36:18,550 And that's worn out also. 525 00:36:18,550 --> 00:36:20,710 So the only thing that potentially 526 00:36:20,710 --> 00:36:26,470 will be relevant and exciting is the last term, number c. 527 00:36:26,470 --> 00:36:29,030 So let's take a look at that. 528 00:36:29,030 --> 00:36:34,170 So here, I have to do an integration over variables 529 00:36:34,170 --> 00:36:38,580 that I am not interested. 530 00:36:38,580 --> 00:36:49,470 And then I need now, however, to do a full Poisson 531 00:36:49,470 --> 00:36:53,080 bracket of a whole bunch of terms, 532 00:36:53,080 --> 00:36:55,900 because now the terms that I'm looking at 533 00:36:55,900 --> 00:36:58,510 have coordinates from both sets. 534 00:36:58,510 --> 00:37:01,200 So I have to be a little bit careful. 535 00:37:01,200 --> 00:37:10,160 So let me just make sure that I follow the notes that I 536 00:37:10,160 --> 00:37:11,720 have here and don't make mistakes. 537 00:37:15,620 --> 00:37:17,140 OK. 538 00:37:17,140 --> 00:37:19,935 So this H prime involves two sums. 539 00:37:23,260 --> 00:37:31,730 So I will write the first sum, N running from 1 to s. 540 00:37:31,730 --> 00:37:37,480 And then I have the second sum, j running from s 541 00:37:37,480 --> 00:37:42,890 plus 1 to N. What do I need? 542 00:37:42,890 --> 00:38:01,774 I need the-- OK. 543 00:38:04,740 --> 00:38:07,490 Let's do it the following way. 544 00:38:07,490 --> 00:38:12,910 So what I have to do for the Poisson bracket 545 00:38:12,910 --> 00:38:16,690 is a sum that involves all coordinates. 546 00:38:16,690 --> 00:38:20,070 So let's just write this whole expression. 547 00:38:20,070 --> 00:38:23,700 But first, for coordinates 1 through s. 548 00:38:23,700 --> 00:38:27,500 So I have a sum N running from 1 to s. 549 00:38:27,500 --> 00:38:29,360 And then I will write the term that 550 00:38:29,360 --> 00:38:32,456 corresponds to coordinates s plus 1 to m. 551 00:38:32,456 --> 00:38:37,400 For the first set of coordinates, what do I have? 552 00:38:37,400 --> 00:38:43,520 I have d rho by dqn. 553 00:38:43,520 --> 00:38:47,510 And then I have d H prime by dpn. 554 00:38:47,510 --> 00:38:50,410 So I didn't write H prime explicitly. 555 00:38:50,410 --> 00:38:54,150 I'm just breaking the sum over here. 556 00:38:54,150 --> 00:39:04,590 And then I have sum j running from s plus 1 to N, 557 00:39:04,590 --> 00:39:14,381 d rho by dqj times d H prime by dpj. 558 00:39:14,381 --> 00:39:20,690 And again, my H prime is this entity over here 559 00:39:20,690 --> 00:39:25,100 that [? copies ?] coordinates from both sets. 560 00:39:25,100 --> 00:39:26,780 OK. 561 00:39:26,780 --> 00:39:32,970 First thing is I claim that one of these two sets of sums is 0. 562 00:39:32,970 --> 00:39:34,926 You tell me which. 563 00:39:34,926 --> 00:39:36,300 AUDIENCE: The first. 564 00:39:36,300 --> 00:39:37,830 PROFESSOR: Why first? 565 00:39:37,830 --> 00:39:42,764 AUDIENCE: Because H prime is independent of p [? dot. ?] 566 00:39:48,416 --> 00:39:50,200 PROFESSOR: That's true. 567 00:39:50,200 --> 00:39:50,700 OK. 568 00:39:50,700 --> 00:39:52,170 That's very good. 569 00:39:52,170 --> 00:39:55,580 And then it sort of brings up a very important question, 570 00:39:55,580 --> 00:39:57,710 which is, I forgot to write two more terms. 571 00:39:57,710 --> 00:40:01,787 [LAUGHTER] 572 00:40:01,787 --> 00:40:19,630 Running to s of d rho by dpn, d H prime by dqn minus sum j 573 00:40:19,630 --> 00:40:32,320 s plus 1 to N of d rho by dpj dot dH prime by dqj. 574 00:40:32,320 --> 00:40:35,760 So indeed, both answers now were correct. 575 00:40:35,760 --> 00:40:37,620 Somebody said that the first term 576 00:40:37,620 --> 00:40:41,770 is a 0, because H prime does not depend on pn. 577 00:40:41,770 --> 00:40:45,310 And somebody over here said that this term is 0. 578 00:40:45,310 --> 00:40:48,844 And maybe they can explain why. 579 00:40:48,844 --> 00:40:51,149 AUDIENCE: [INAUDIBLE]. 580 00:40:51,149 --> 00:40:53,250 PROFESSOR: Same reason as up here. 581 00:40:53,250 --> 00:40:55,460 That is, I can do integration by parts. 582 00:40:55,460 --> 00:40:57,196 AUDIENCE: [INAUDIBLE]. 583 00:40:57,196 --> 00:41:03,890 PROFESSOR: To get rid of this term plus this term together. 584 00:41:03,890 --> 00:41:08,310 So it's actually by itself is not 0. 585 00:41:08,310 --> 00:41:11,710 But if I do integration by parts, 586 00:41:11,710 --> 00:41:14,550 I will have-- actually, even by itself, it is 0, 587 00:41:14,550 --> 00:41:19,183 because I would have d by dqj, d by pj, H prime. 588 00:41:19,183 --> 00:41:22,900 And H prime, you cannot have a double derivative pj. 589 00:41:22,900 --> 00:41:25,740 So each one of them, actually, by itself is 0. 590 00:41:25,740 --> 00:41:28,480 But in general, they would also cancel each other 591 00:41:28,480 --> 00:41:30,621 through their single process. 592 00:41:30,621 --> 00:41:31,120 Yes. 593 00:41:31,120 --> 00:41:34,100 AUDIENCE: Do you have the sign of [? dH ?] of [? H prime? ?] 594 00:41:34,100 --> 00:41:35,820 PROFESSOR: Did I have the sign incorrect? 595 00:41:35,820 --> 00:41:36,570 Yes. 596 00:41:36,570 --> 00:41:39,120 Because for some reason or other, 597 00:41:39,120 --> 00:41:46,048 I keep reading from here, which is rho and H. So let's do this. 598 00:41:49,476 --> 00:41:49,976 OK? 599 00:41:49,976 --> 00:41:50,824 AUDIENCE: Excuse me. 600 00:41:50,824 --> 00:41:51,449 PROFESSOR: Yes? 601 00:41:51,449 --> 00:41:54,886 AUDIENCE: [INAUDIBLE]. 602 00:41:54,886 --> 00:41:55,920 PROFESSOR: OK. 603 00:41:55,920 --> 00:41:57,170 Yes, it is different. 604 00:41:57,170 --> 00:41:57,670 Yes. 605 00:41:57,670 --> 00:42:03,430 So what I said, if I had a more general Hamiltonian 606 00:42:03,430 --> 00:42:07,200 that also depended on momentum, then 607 00:42:07,200 --> 00:42:11,020 this term would, by itself not 0, but would cancel, 608 00:42:11,020 --> 00:42:13,370 be the corresponding term from here. 609 00:42:13,370 --> 00:42:15,290 But the way that I have for H prime, 610 00:42:15,290 --> 00:42:19,010 indeed, each term by itself would be 0. 611 00:42:19,010 --> 00:42:19,510 OK. 612 00:42:19,510 --> 00:42:24,330 So hopefully-- then what do we have? 613 00:42:27,150 --> 00:42:34,240 So actually, let's keep the sign correct and do this, 614 00:42:34,240 --> 00:42:37,290 because I need this right sign for the one term 615 00:42:37,290 --> 00:42:38,040 that is preserved. 616 00:42:38,040 --> 00:42:39,330 So what does that say? 617 00:42:39,330 --> 00:42:47,870 It is a sum, n running from 1 to s. 618 00:42:51,030 --> 00:42:52,330 OK? 619 00:42:52,330 --> 00:42:56,940 I have d H prime by dqn. 620 00:42:59,500 --> 00:43:01,100 And I have this integration. 621 00:43:01,100 --> 00:43:10,150 I have the integration i running from s plus 1 to N dV of i. 622 00:43:10,150 --> 00:43:16,480 I have d rho by dpn dot producted 623 00:43:16,480 --> 00:43:19,410 with d H prime by dqN. 624 00:43:19,410 --> 00:43:23,760 d H prime by dqn I can calculate from here, 625 00:43:23,760 --> 00:43:31,940 is a sum over terms j running from s plus 1 626 00:43:31,940 --> 00:43:42,280 to N of V of qn minus qj. 627 00:43:52,440 --> 00:43:53,010 All right. 628 00:43:56,433 --> 00:43:57,411 AUDIENCE: Question. 629 00:43:57,411 --> 00:43:58,878 PROFESSOR: Yes. 630 00:43:58,878 --> 00:44:02,045 AUDIENCE: Why aren't you differentiating [? me ?] 631 00:44:02,045 --> 00:44:04,095 if you're differentiating H prime? 632 00:44:07,884 --> 00:44:10,200 PROFESSOR: d by dqj. 633 00:44:13,400 --> 00:44:14,442 All right? 634 00:44:14,442 --> 00:44:17,025 AUDIENCE: Where is the qn? 635 00:44:17,025 --> 00:44:19,161 PROFESSOR: d by dqn. 636 00:44:19,161 --> 00:44:20,604 Thank you. 637 00:44:20,604 --> 00:44:22,050 Right. 638 00:44:22,050 --> 00:44:25,390 Because always, pn of qn would go together. 639 00:44:25,390 --> 00:44:26,764 Thank you. 640 00:44:26,764 --> 00:44:27,264 OK. 641 00:44:31,121 --> 00:44:31,620 All right. 642 00:44:31,620 --> 00:44:34,870 So we have to slog through these derivations. 643 00:44:34,870 --> 00:44:38,980 And then I'll give you the physical meaning. 644 00:44:38,980 --> 00:44:43,075 So I can rearrange this. 645 00:44:46,750 --> 00:44:49,910 Let's see what's happening here. 646 00:44:49,910 --> 00:44:55,180 I have here a sum over particles that 647 00:44:55,180 --> 00:44:58,980 are not listed on the left-hand side. 648 00:44:58,980 --> 00:45:02,660 So when I wrote this d rho by dt, 649 00:45:02,660 --> 00:45:07,090 I had listed coordinates going from p1 through qs that 650 00:45:07,090 --> 00:45:09,940 were s coordinates that were listed. 651 00:45:09,940 --> 00:45:12,520 If you like, you can think of them as s particles. 652 00:45:17,030 --> 00:45:21,460 Now, this sum involves the remaining particles. 653 00:45:21,460 --> 00:45:22,540 What is this? 654 00:45:22,540 --> 00:45:23,690 Up to a sign. 655 00:45:23,690 --> 00:45:28,000 This is the force that is exerted by particle j 656 00:45:28,000 --> 00:45:30,150 from the list of particles that I'm not 657 00:45:30,150 --> 00:45:33,410 interested on one of the particles on the list 658 00:45:33,410 --> 00:45:35,262 that I am interested. 659 00:45:35,262 --> 00:45:38,090 OK? 660 00:45:38,090 --> 00:45:42,140 Now, I expect that at the end of the day, all of the particles 661 00:45:42,140 --> 00:45:46,950 that I am not interested I can treat equivalently, 662 00:45:46,950 --> 00:45:49,460 like everything that we had before, 663 00:45:49,460 --> 00:45:55,140 like how I got this factor of N or N minus 1 over there. 664 00:45:55,140 --> 00:46:01,700 I expect that all of these will give me the same result, which 665 00:46:01,700 --> 00:46:06,410 is proportional to the number of these particles, which 666 00:46:06,410 --> 00:46:08,462 is N minus s. 667 00:46:08,462 --> 00:46:11,110 OK? 668 00:46:11,110 --> 00:46:14,950 And then I can focus on just one of the terms in this sum. 669 00:46:14,950 --> 00:46:17,495 Let's say the term that corresponds to j, 670 00:46:17,495 --> 00:46:18,450 being s plus 1. 671 00:46:23,480 --> 00:46:28,870 Now, having done that, I have to be careful. 672 00:46:28,870 --> 00:46:32,210 I can do separately the integration 673 00:46:32,210 --> 00:46:37,660 over the volume of this one coordinate that I'm keeping, 674 00:46:37,660 --> 00:46:39,930 V of s plus 1. 675 00:46:39,930 --> 00:46:42,270 And what do I have here? 676 00:46:42,270 --> 00:46:49,180 I have the force that exerted on particle number N 677 00:46:49,180 --> 00:46:52,520 by the particle that is labelled s plus 1. 678 00:46:57,880 --> 00:47:03,540 And this force is dot producted with a gradient 679 00:47:03,540 --> 00:47:12,390 along the momentum in direction of particle N of its density. 680 00:47:12,390 --> 00:47:15,820 Actually, this is the density of all particles. 681 00:47:15,820 --> 00:47:20,510 This is the rho that corresponds to the joint. 682 00:47:20,510 --> 00:47:24,950 But I had here s plus 1 integrations. 683 00:47:24,950 --> 00:47:27,860 One of them I wrote down explicitly. 684 00:47:27,860 --> 00:47:30,028 All the others I do over here. 685 00:47:36,860 --> 00:47:38,540 Of the density. 686 00:47:38,540 --> 00:47:43,770 So basically, I change the order of the derivative 687 00:47:43,770 --> 00:47:46,890 and the integrations over the variables not involved 688 00:47:46,890 --> 00:47:48,670 in the remainder. 689 00:47:48,670 --> 00:47:51,450 And the reason I did that, of course, 690 00:47:51,450 --> 00:47:54,875 is that then this is my rho s plus 1. 691 00:47:58,590 --> 00:48:00,930 OK? 692 00:48:00,930 --> 00:48:05,160 So what we have at the end of the day 693 00:48:05,160 --> 00:48:13,130 is that if I take the time variation of an s particle 694 00:48:13,130 --> 00:48:21,380 density, I will get one term that I expected, 695 00:48:21,380 --> 00:48:26,060 which is if those s particles were interacting only 696 00:48:26,060 --> 00:48:30,620 with themselves, I would write the Liouville equation 697 00:48:30,620 --> 00:48:34,460 that would be appropriate to them. 698 00:48:34,460 --> 00:48:36,510 But because of the collisions that I 699 00:48:36,510 --> 00:48:41,930 can have with particles that are not over here, 700 00:48:41,930 --> 00:48:46,010 suddenly, the momenta that I'm looking at could change. 701 00:48:46,010 --> 00:48:50,400 And because of that, I have a correction term here 702 00:48:50,400 --> 00:48:53,170 that really describes the collisions. 703 00:48:53,170 --> 00:48:57,980 It says here that these s particles 704 00:48:57,980 --> 00:49:00,870 were following the trajectory that 705 00:49:00,870 --> 00:49:03,400 was governed by the Hamiltonian that 706 00:49:03,400 --> 00:49:06,330 was peculiar to the s particles. 707 00:49:06,330 --> 00:49:11,210 But suddenly, one of them had a collision with somebody else. 708 00:49:11,210 --> 00:49:12,800 So which one of them? 709 00:49:12,800 --> 00:49:14,200 Well, any one of them. 710 00:49:14,200 --> 00:49:20,160 So I could get a contribution from any one of the s particles 711 00:49:20,160 --> 00:49:24,970 that is listed over here, having a collision with somebody else. 712 00:49:24,970 --> 00:49:27,970 How do I describe the effect of that? 713 00:49:27,970 --> 00:49:37,020 I have to do an integration over where this new particle that I 714 00:49:37,020 --> 00:49:39,160 am colliding with could be. 715 00:49:39,160 --> 00:49:43,090 I have to specify both where the particle is that I am colliding 716 00:49:43,090 --> 00:49:47,750 with, as well as its momentum. 717 00:49:47,750 --> 00:49:50,100 So that's this. 718 00:49:50,100 --> 00:49:55,190 Then I need to know the force that this particle is 719 00:49:55,190 --> 00:49:56,420 exerting on me. 720 00:49:56,420 --> 00:50:05,680 So that's the V of qs plus 1 minus qN divided by dqN. 721 00:50:05,680 --> 00:50:10,240 This is the force that is exerted by this particle 722 00:50:10,240 --> 00:50:13,980 that I don't see on myself. 723 00:50:13,980 --> 00:50:18,990 Then I have to multiply this, or a dot product of this, 724 00:50:18,990 --> 00:50:23,840 with d by dpN, because what happens in the process, 725 00:50:23,840 --> 00:50:28,150 because of this force, the momentum of the N particle 726 00:50:28,150 --> 00:50:29,880 is changing. 727 00:50:29,880 --> 00:50:34,300 The variation of that is captured 728 00:50:34,300 --> 00:50:38,080 through looking at the density that 729 00:50:38,080 --> 00:50:44,370 has all of these particles in addition to this new particle 730 00:50:44,370 --> 00:50:46,250 that I am colliding with. 731 00:50:46,250 --> 00:50:50,270 But, of course, I am not really interested in the coordinate 732 00:50:50,270 --> 00:50:54,970 of this new particle, so I integrate over it. 733 00:50:54,970 --> 00:50:57,670 There are N minus s such particles. 734 00:50:57,670 --> 00:51:00,280 So I really have to put a factor of N minus s 735 00:51:00,280 --> 00:51:04,640 here for all of potential collisions. 736 00:51:04,640 --> 00:51:05,775 And so that's the equation. 737 00:51:09,010 --> 00:51:13,900 Again, it is more common, rather than to write 738 00:51:13,900 --> 00:51:20,150 the equation for rho, to write the equation for f. 739 00:51:20,150 --> 00:51:22,730 And the f's and the rhos where simply 740 00:51:22,730 --> 00:51:27,270 related by these factors of N factorial over N minus 1 741 00:51:27,270 --> 00:51:28,880 s factorial. 742 00:51:28,880 --> 00:51:36,550 And the outcome of that is that the equation for f 743 00:51:36,550 --> 00:51:41,160 simply does not have this additional factor of N minus s, 744 00:51:41,160 --> 00:51:44,190 because that disappears in the ratio of rho 745 00:51:44,190 --> 00:51:46,230 of s plus 1 and rho s. 746 00:51:46,230 --> 00:51:53,690 And it becomes a sum over N running from 1 to s. 747 00:51:53,690 --> 00:52:01,790 Integral over coordinates and momenta of a particle s plus 1. 748 00:52:04,620 --> 00:52:08,660 The force exerted by particle s plus 1 749 00:52:08,660 --> 00:52:19,540 on particle N used to vary the momentum of the N particle. 750 00:52:19,540 --> 00:52:25,040 And the whole thing would depend on the density that includes, 751 00:52:25,040 --> 00:52:30,850 in addition to the s particles that I had before, 752 00:52:30,850 --> 00:52:33,810 the new particle that I am colliding with. 753 00:52:36,642 --> 00:52:39,010 OK? 754 00:52:39,010 --> 00:52:42,190 So there is a set of equations that 755 00:52:42,190 --> 00:52:48,730 relates the different densities and how they evolve in time. 756 00:52:48,730 --> 00:52:53,660 The evolution of f1, which is the thing that I am interested, 757 00:52:53,660 --> 00:52:55,420 will have, on the right-hand side, 758 00:52:55,420 --> 00:52:57,850 something that involves f2. 759 00:52:57,850 --> 00:53:01,100 The evolution of f2 will involve f3. 760 00:53:01,100 --> 00:53:05,910 And this whole thing is called a BBGKY hierarchy, 761 00:53:05,910 --> 00:53:09,760 after people whose names I have in the notes. 762 00:53:09,760 --> 00:53:11,500 [SOFT LAUGHTER] 763 00:53:13,620 --> 00:53:19,450 But again, what have we learned beyond what 764 00:53:19,450 --> 00:53:21,315 we had in the original case? 765 00:53:21,315 --> 00:53:23,720 And originally, we had an equation 766 00:53:23,720 --> 00:53:29,370 that was governing a function in 6N-dimensional space, which 767 00:53:29,370 --> 00:53:30,770 we really don't need. 768 00:53:30,770 --> 00:53:33,500 So we tried our best to avoid that. 769 00:53:33,500 --> 00:53:36,700 We said that all of the physics is in one particle, maybe 770 00:53:36,700 --> 00:53:38,560 two particle densities. 771 00:53:38,560 --> 00:53:41,770 Let's calculate the evolution of one-particle and two-particle 772 00:53:41,770 --> 00:53:42,800 densities. 773 00:53:42,800 --> 00:53:45,980 Maybe they will tell us about this non-equilibrium situation 774 00:53:45,980 --> 00:53:47,340 that we set up. 775 00:53:47,340 --> 00:53:50,860 But we see that the time evolution of the first particle 776 00:53:50,860 --> 00:53:53,410 density requires two-particle density. 777 00:53:53,410 --> 00:53:56,700 Two-particle density requires three-particle densities. 778 00:53:56,700 --> 00:54:01,250 So we sort of made this ladder, which ultimately will 779 00:54:01,250 --> 00:54:03,740 [? terminate ?] at the Nth particle densities. 780 00:54:03,740 --> 00:54:08,040 And so we have not really gained much. 781 00:54:08,040 --> 00:54:13,050 So we have to now look at these equations a little bit more 782 00:54:13,050 --> 00:54:17,150 and try to inject more physics. 783 00:54:17,150 --> 00:54:20,755 So let's write down the first two terms explicitly. 784 00:54:23,340 --> 00:54:28,580 So what I will do is I will take this Poisson bracket of H 785 00:54:28,580 --> 00:54:32,850 and f, to the left-hand side, and use the Hamiltonian 786 00:54:32,850 --> 00:54:37,760 that we have over here to write the terms. 787 00:54:37,760 --> 00:54:41,680 So the equation that we have for f1-- and I'm 788 00:54:41,680 --> 00:54:44,360 going to write it as a whole bunch of derivatives 789 00:54:44,360 --> 00:54:46,980 acting on f1. 790 00:54:46,980 --> 00:54:52,930 f1 is a function of p1 q1 t. 791 00:54:52,930 --> 00:54:57,100 And essentially, what Liouville's theorem 792 00:54:57,100 --> 00:55:01,100 says is that as you move along the trajectory, 793 00:55:01,100 --> 00:55:06,290 the total derivative is 0, because the expansion 794 00:55:06,290 --> 00:55:08,720 of the flows is incompressible. 795 00:55:08,720 --> 00:55:10,670 So what does that mean? 796 00:55:10,670 --> 00:55:17,870 It means that d by dt, which is this argument, plus q1 dot 797 00:55:17,870 --> 00:55:25,126 times d by dq1 plus p1 dot by d by dp1. 798 00:55:27,960 --> 00:55:31,775 So here, I should write p1 dot and q1 dot. 799 00:55:34,670 --> 00:55:37,340 In the absence of everything else is 0. 800 00:55:37,340 --> 00:55:44,790 Then, of course, for q1 dot, we use the momentum 801 00:55:44,790 --> 00:55:46,820 that we would get out of this. 802 00:55:46,820 --> 00:55:53,060 q1 dot is momentum divided by mass. 803 00:55:53,060 --> 00:55:55,790 So that's the velocity. 804 00:55:55,790 --> 00:55:59,740 And p1 dot, changing momentum, is 805 00:55:59,740 --> 00:56:02,905 the force, is minus dH by dq1. 806 00:56:02,905 --> 00:56:09,360 So this is minus d of this one particle potential divided 807 00:56:09,360 --> 00:56:14,630 by dq1 dotted by [? h ?] p1. 808 00:56:14,630 --> 00:56:21,240 So if you were asked to think about one particle in a box, 809 00:56:21,240 --> 00:56:24,400 then you know its equation of motion. 810 00:56:24,400 --> 00:56:27,470 If you have many, many realizations 811 00:56:27,470 --> 00:56:32,080 of that particle in a box, you can construct a density. 812 00:56:32,080 --> 00:56:34,186 Each one of the elements of the trajectory, 813 00:56:34,186 --> 00:56:36,250 you know how they evolve according 814 00:56:36,250 --> 00:56:38,010 to [? Newton's ?] equation. 815 00:56:38,010 --> 00:56:43,120 And you can see how the density would evolve. 816 00:56:43,120 --> 00:56:45,540 It would evolve according to this. 817 00:56:45,540 --> 00:56:48,280 I would have said, equal to 0. 818 00:56:48,280 --> 00:56:53,420 But I can't set it to 0 if I'm really thinking about a gas, 819 00:56:53,420 --> 00:56:56,360 because my particle can come and collide 820 00:56:56,360 --> 00:57:00,050 with a second particle in the gas. 821 00:57:00,050 --> 00:57:02,980 The second particle can be anywhere. 822 00:57:02,980 --> 00:57:11,570 And what it will do is that it will exert a force, which 823 00:57:11,570 --> 00:57:15,000 would be like this, on particle one. 824 00:57:15,000 --> 00:57:17,300 And this force will change the momentum. 825 00:57:17,300 --> 00:57:20,040 So my variation of the momentum will not 826 00:57:20,040 --> 00:57:22,410 come only from the external force, 827 00:57:22,410 --> 00:57:27,100 but also from the force that is coming 828 00:57:27,100 --> 00:57:30,380 from some other particle in the medium. 829 00:57:30,380 --> 00:57:33,640 So that's where this d by dp really 830 00:57:33,640 --> 00:57:37,080 gets not only the external force but also 831 00:57:37,080 --> 00:57:39,890 the force from somebody else. 832 00:57:39,890 --> 00:57:42,830 But then I need to know where this other particle is, 833 00:57:42,830 --> 00:57:45,830 given that I know where my first particle is. 834 00:57:45,830 --> 00:57:50,340 So I have to include here a two-particle density which 835 00:57:50,340 --> 00:57:56,740 depends on p1 as well as q2 at time t. 836 00:57:56,740 --> 00:57:58,230 OK. 837 00:57:58,230 --> 00:58:00,546 Fine. 838 00:58:00,546 --> 00:58:05,420 Now you say, OK, let's write down-- I need to know f2. 839 00:58:05,420 --> 00:58:07,220 Let's write down the equation for f2. 840 00:58:07,220 --> 00:58:12,110 So I will write it more rapidly. 841 00:58:12,110 --> 00:58:15,500 I have p1 over m, d by dq1. 842 00:58:15,500 --> 00:58:22,232 I have p2 over m, d by dq2. 843 00:58:22,232 --> 00:58:27,450 I will have dU by dq1, d by dp1. 844 00:58:27,450 --> 00:58:35,580 I will have dU by dq2, d by dp2. 845 00:58:35,580 --> 00:58:40,470 I will have also a term from the collision between q1 and q2. 846 00:58:44,500 --> 00:58:49,680 And once it will change the momentum of the first particle, 847 00:58:49,680 --> 00:58:52,490 but it will change the momentum of the second particle 848 00:58:52,490 --> 00:58:53,735 in the opposite direction. 849 00:58:53,735 --> 00:58:57,110 So I will put the two of them together. 850 00:58:57,110 --> 00:59:03,050 So this is all of the terms that I would get from H2, Poisson 851 00:59:03,050 --> 00:59:06,600 bracket with density acting on the two-particle density. 852 00:59:09,530 --> 00:59:13,220 And the answer would be 0 if the two particles 853 00:59:13,220 --> 00:59:15,490 were the only thing in the box. 854 00:59:15,490 --> 00:59:17,910 But there's also other particles. 855 00:59:17,910 --> 00:59:20,770 So there can be interactions and collisions 856 00:59:20,770 --> 00:59:22,700 with a third particle. 857 00:59:22,700 --> 00:59:25,920 And for that, I would need to know, 858 00:59:25,920 --> 00:59:29,680 let's actually try to simplify notation. 859 00:59:29,680 --> 00:59:35,470 This is the force that is exerted from two to one. 860 00:59:35,470 --> 00:59:41,822 So I will have here the force from three to one 861 00:59:41,822 --> 00:59:44,075 dotted by d by dp1. 862 00:59:48,380 --> 00:59:49,240 Right. 863 00:59:49,240 --> 00:59:52,600 And the force that is exerted from three to two 864 00:59:52,600 --> 00:59:57,910 dotted by d by dp2 acting on a three-particle density that 865 00:59:57,910 --> 01:00:02,870 involves everything up to three. 866 01:00:02,870 --> 01:00:04,880 And now let's write the third one. 867 01:00:04,880 --> 01:00:07,380 [LAUGHTER] 868 01:00:08,380 --> 01:00:10,390 So that, I will leave to next lecture. 869 01:00:10,390 --> 01:00:14,250 But anyway, so this is the structure. 870 01:00:14,250 --> 01:00:15,840 Now, this is the point at which we 871 01:00:15,840 --> 01:00:23,340 would like to inject some physics into the problem. 872 01:00:23,340 --> 01:00:30,080 So what we are going to do is to estimate the various terms that 873 01:00:30,080 --> 01:00:33,880 are appearing in this equation to see whether there 874 01:00:33,880 --> 01:00:37,582 is some approximation that we can make 875 01:00:37,582 --> 01:00:42,680 to make the equations more treatable and handle-able. 876 01:00:42,680 --> 01:00:45,090 All right? 877 01:00:45,090 --> 01:00:54,900 So let's try to look at the case of a gas-- 878 01:00:54,900 --> 01:00:58,160 let's say the gas in this room. 879 01:00:58,160 --> 01:01:01,790 A typical thing that is happening 880 01:01:01,790 --> 01:01:03,690 in the particles of the gas in this room 881 01:01:03,690 --> 01:01:06,240 is that they are zipping around. 882 01:01:06,240 --> 01:01:09,760 Their velocity is of the order-- again, 883 01:01:09,760 --> 01:01:14,991 just order of magnitude, hundreds of meters per second. 884 01:01:14,991 --> 01:01:15,490 OK? 885 01:01:15,490 --> 01:01:20,899 So we are going to, again, be very sort of limited 886 01:01:20,899 --> 01:01:22,315 in what we are trying to describe. 887 01:01:22,315 --> 01:01:24,110 There is this experiment. 888 01:01:24,110 --> 01:01:26,585 Gas expands into a chamber. 889 01:01:26,585 --> 01:01:30,802 In room temperature, typical velocities are of this order. 890 01:01:30,802 --> 01:01:34,620 Now we are going to use that to estimate 891 01:01:34,620 --> 01:01:36,710 the magnitude of the various terms that 892 01:01:36,710 --> 01:01:39,830 are appearing in this equation. 893 01:01:39,830 --> 01:01:42,200 Now, the whole thing about this equation 894 01:01:42,200 --> 01:01:44,870 is variation with time. 895 01:01:44,870 --> 01:01:46,920 So the entity that we are looking 896 01:01:46,920 --> 01:01:51,450 at in all of these brackets is this d by dt, 897 01:01:51,450 --> 01:01:55,830 which means that the various terms in this differential 898 01:01:55,830 --> 01:01:58,980 equation, apart from d by dt, have 899 01:01:58,980 --> 01:02:03,320 to have dimensions of inverse time. 900 01:02:03,320 --> 01:02:06,620 So we are going to try to characterize 901 01:02:06,620 --> 01:02:08,220 what those inverse times are. 902 01:02:12,430 --> 01:02:21,355 So what are the typical magnitudes of various terms? 903 01:02:25,640 --> 01:02:28,740 If I look at the first equation, I 904 01:02:28,740 --> 01:02:31,640 said what that first equation describes. 905 01:02:31,640 --> 01:02:35,180 That first equation describes for you a particle in a box. 906 01:02:35,180 --> 01:02:38,390 We've forgotten about everything else. 907 01:02:38,390 --> 01:02:41,710 So if I have a particle in a box, 908 01:02:41,710 --> 01:02:43,590 what is the characteristic time? 909 01:02:43,590 --> 01:02:46,730 It has to be set by the size of the box, given 910 01:02:46,730 --> 01:02:49,740 that I am moving with that velocity. 911 01:02:49,740 --> 01:02:52,550 So there is a timescale that I would 912 01:02:52,550 --> 01:02:57,215 call extrinsic in the sense that it is not really 913 01:02:57,215 --> 01:02:58,640 a property of the gas. 914 01:02:58,640 --> 01:03:02,020 It will be different if I make the box bigger. 915 01:03:02,020 --> 01:03:04,010 There's a timescale over which I would 916 01:03:04,010 --> 01:03:07,430 go from one side of the box to another side of the box. 917 01:03:07,430 --> 01:03:09,920 So this is kind of a timescale that 918 01:03:09,920 --> 01:03:13,100 is related to the term that knows something 919 01:03:13,100 --> 01:03:17,140 about the box, which is dU by dq, d by dp. 920 01:03:17,140 --> 01:03:21,480 I would say that if I were to assign some typical magnitude 921 01:03:21,480 --> 01:03:24,170 to this type of term, I would say 922 01:03:24,170 --> 01:03:31,980 that it is related to having to traverse a distance that 923 01:03:31,980 --> 01:03:39,630 is of the order of the size of the box, 924 01:03:39,630 --> 01:03:42,640 given the velocity that I have specified. 925 01:03:42,640 --> 01:03:44,465 This is an inverse timescale. 926 01:03:44,465 --> 01:03:45,950 Right? 927 01:03:45,950 --> 01:03:50,190 And let's sort of imagine that I have an-- 928 01:03:50,190 --> 01:03:55,620 and I will call this timescale 1 over tau U, 929 01:03:55,620 --> 01:04:00,610 because it is sort of determined by my external U. 930 01:04:00,610 --> 01:04:03,070 Let's say we have a typical size that 931 01:04:03,070 --> 01:04:04,760 is of the order of millimeter. 932 01:04:04,760 --> 01:04:06,890 If I make it larger, it will be larger. 933 01:04:06,890 --> 01:04:12,520 So actually, let's say we have 10 to the minus 3 meters. 934 01:04:12,520 --> 01:04:14,340 Actually, let's make it bigger. 935 01:04:14,340 --> 01:04:18,740 Let's make it of the order of 10 to the minus 1 meter. 936 01:04:18,740 --> 01:04:20,900 Kind of reasonable-sized box. 937 01:04:20,900 --> 01:04:24,270 Then you would say that this 1 over tau c 938 01:04:24,270 --> 01:04:29,010 is of the order of 10 to the 2 divided by 10 to the minus 1, 939 01:04:29,010 --> 01:04:31,110 which is of the order of 1,000. 940 01:04:31,110 --> 01:04:35,390 Basically, it takes a millisecond 941 01:04:35,390 --> 01:04:38,530 to traverse a box that is a fraction 942 01:04:38,530 --> 01:04:42,220 of a meter with these velocities. 943 01:04:42,220 --> 01:04:42,720 OK? 944 01:04:42,720 --> 01:04:44,450 You say fine. 945 01:04:44,450 --> 01:04:46,360 That is the kind of timescale that I 946 01:04:46,360 --> 01:04:50,100 have in the first equation that I have in my hierarchy. 947 01:04:50,100 --> 01:04:52,070 And that kind of term is certainly 948 01:04:52,070 --> 01:04:56,950 also present in the second equation for the hierarchy. 949 01:04:56,950 --> 01:04:59,860 If I have two particles, maybe these two particles 950 01:04:59,860 --> 01:05:01,990 are orbiting each other, et cetera. 951 01:05:01,990 --> 01:05:04,880 Still, their center of mass would move, typically, 952 01:05:04,880 --> 01:05:06,569 with this velocity. 953 01:05:06,569 --> 01:05:08,110 And it would take this amount of time 954 01:05:08,110 --> 01:05:11,010 to go across the size of the box. 955 01:05:11,010 --> 01:05:15,210 But there is another timescale inside there 956 01:05:15,210 --> 01:05:25,815 that I would call intrinsic, which involves dV/dq, d by dp. 957 01:05:31,380 --> 01:05:36,770 Now, if I was to see what the characteristic magnitude 958 01:05:36,770 --> 01:05:42,470 of this term is, it would have to be V divided 959 01:05:42,470 --> 01:05:47,730 by a lens scale that characterizes the potential. 960 01:05:47,730 --> 01:05:50,040 And the potential, let's say, is of the order 961 01:05:50,040 --> 01:05:52,780 of atomic size or molecular size. 962 01:05:52,780 --> 01:05:54,230 Let's call it d. 963 01:05:54,230 --> 01:05:58,910 So this is an atomic size-- or molecular size. 964 01:06:03,740 --> 01:06:07,240 More correctly, really, it's the range of the interaction 965 01:06:07,240 --> 01:06:09,730 that you have between particles. 966 01:06:09,730 --> 01:06:13,280 And typical values of these [? numbers ?] 967 01:06:13,280 --> 01:06:17,320 are of the order of 10 angstroms, or angstroms, 968 01:06:17,320 --> 01:06:18,080 or whatever. 969 01:06:18,080 --> 01:06:19,905 Let's say 10 to the minus 10 meters. 970 01:06:22,920 --> 01:06:23,480 Sorry. 971 01:06:23,480 --> 01:06:27,250 The first one I would like to call tau U. 972 01:06:27,250 --> 01:06:30,070 This second time, that I will call 973 01:06:30,070 --> 01:06:36,414 1 over tau c for collisions, is going to be the ratio of 10 974 01:06:36,414 --> 01:06:39,950 to the 2 to 10 to the minus 10. 975 01:06:39,950 --> 01:06:45,010 It's of the order of 10 to the 12 in both seconds. 976 01:06:45,010 --> 01:06:47,340 OK? 977 01:06:47,340 --> 01:06:54,700 So you can see that this term is much, much larger in magnitude 978 01:06:54,700 --> 01:06:58,584 than the term that was governing the first equation. 979 01:06:58,584 --> 01:06:59,400 OK? 980 01:06:59,400 --> 01:07:03,730 And roughly, what you expect in a situation 981 01:07:03,730 --> 01:07:11,840 such as this-- let's imagine, rather than shooting particles 982 01:07:11,840 --> 01:07:17,990 from here, you are shooting bullets. 983 01:07:17,990 --> 01:07:20,370 And then the bullets would come and basically 984 01:07:20,370 --> 01:07:23,440 have some kind of trajectory, et cetera. 985 01:07:23,440 --> 01:07:27,360 The characteristic time for a single one of them 986 01:07:27,360 --> 01:07:31,150 would be basically something that 987 01:07:31,150 --> 01:07:33,230 is related to the size of the box. 988 01:07:33,230 --> 01:07:38,170 How long does it take a bullet to go over the size of the box? 989 01:07:38,170 --> 01:07:40,620 But if two of these bullets happen 990 01:07:40,620 --> 01:07:45,130 to come together and collide, then there's 991 01:07:45,130 --> 01:07:48,650 a very short period of time over which 992 01:07:48,650 --> 01:07:51,100 they would go in different directions. 993 01:07:51,100 --> 01:07:53,690 And the momenta would get displaced 994 01:07:53,690 --> 01:07:55,600 from what they were before. 995 01:07:55,600 --> 01:07:59,600 And that timescale is of the order of this. 996 01:08:02,910 --> 01:08:05,880 But in the situation that I set up, 997 01:08:05,880 --> 01:08:09,510 this particular time is too rapid. 998 01:08:09,510 --> 01:08:12,480 There is another more important time, 999 01:08:12,480 --> 01:08:16,880 which is, how long do I have to wait 1000 01:08:16,880 --> 01:08:20,330 for two of these particles, or two of these bullets, 1001 01:08:20,330 --> 01:08:23,550 to come and hit each other? 1002 01:08:23,550 --> 01:08:28,080 So it's not the duration of the collision that is irrelevant, 1003 01:08:28,080 --> 01:08:32,450 but how long it would be for me to find another particle 1004 01:08:32,450 --> 01:08:34,500 to collide with. 1005 01:08:34,500 --> 01:08:38,779 And actually, that is what is governed by the terms 1006 01:08:38,779 --> 01:08:41,399 that I have on the other side. 1007 01:08:41,399 --> 01:08:44,600 Because the terms on the other side, what they say is I 1008 01:08:44,600 --> 01:08:47,270 have to find another particle. 1009 01:08:47,270 --> 01:08:55,470 So if I look at the terms that I have on the right-hand side 1010 01:08:55,470 --> 01:08:58,290 and try to construct a characteristic time 1011 01:08:58,290 --> 01:09:02,279 out of them, I have to compare the probability that I will 1012 01:09:02,279 --> 01:09:06,840 have, or the density that I will have for s plus 1, 1013 01:09:06,840 --> 01:09:16,080 integrated over some volume, over which 1014 01:09:16,080 --> 01:09:20,184 the force between these particles is non-zero. 1015 01:09:23,770 --> 01:09:27,580 And then in order to construct a timescale for it, 1016 01:09:27,580 --> 01:09:32,279 I know that the d by dt on the left-hand side acts on fs. 1017 01:09:32,279 --> 01:09:35,250 On the right-hand side, I have fs plus 1. 1018 01:09:35,250 --> 01:09:37,824 So again, just if I want to construct dimensionally, 1019 01:09:37,824 --> 01:09:44,529 it's a ratio that involves s plus 1 to s. 1020 01:09:44,529 --> 01:09:46,640 OK? 1021 01:09:46,640 --> 01:09:52,569 So I have to do an additional integration 1022 01:09:52,569 --> 01:09:59,410 over a volume in phase space over which two particles can 1023 01:09:59,410 --> 01:10:01,820 have substantial interactions. 1024 01:10:01,820 --> 01:10:03,710 Because that's where this [? dv ?] by dq 1025 01:10:03,710 --> 01:10:09,820 would be non-zero, provided that there is a density for s 1026 01:10:09,820 --> 01:10:13,300 plus 1 particle compared to s particles. 1027 01:10:13,300 --> 01:10:15,000 If you think about it, that means 1028 01:10:15,000 --> 01:10:18,390 that I have to look at the typical density 1029 01:10:18,390 --> 01:10:26,770 or particles times d cubed for these additional operations 1030 01:10:26,770 --> 01:10:32,480 multiplied by this collision time that I had before. 1031 01:10:32,480 --> 01:10:34,300 OK? 1032 01:10:34,300 --> 01:10:39,945 And this whole thing I will call 1 over tau collision. 1033 01:10:42,620 --> 01:10:50,080 And another way of getting the same result is as follows. 1034 01:10:50,080 --> 01:10:56,190 This is typically how you get collision times by pictorially. 1035 01:10:56,190 --> 01:11:01,320 You say that I have something that can interact over 1036 01:11:01,320 --> 01:11:05,530 some characteristic size d. 1037 01:11:05,530 --> 01:11:11,640 It moves in space with velocity v 1038 01:11:11,640 --> 01:11:17,050 so that if I wait a time that I will call tau, 1039 01:11:17,050 --> 01:11:19,840 within that time, I'm essentially 1040 01:11:19,840 --> 01:11:28,690 sweeping a volume of space that has volume d squared v tau. 1041 01:11:28,690 --> 01:11:33,000 So my cross section, if my dimension is d, is d squared. 1042 01:11:33,000 --> 01:11:36,680 I sweep in the other direction by [? aman ?] d tau. 1043 01:11:36,680 --> 01:11:40,290 And how many particles will I encounter? 1044 01:11:40,290 --> 01:11:42,130 Well, if I know the density, which 1045 01:11:42,130 --> 01:11:45,020 is the number of particles per unit volume, 1046 01:11:45,020 --> 01:11:48,450 I have to multiply this by n. 1047 01:11:48,450 --> 01:11:51,930 So how far do I have to go until I hit 1? 1048 01:11:51,930 --> 01:11:54,340 I'll call that tau x. 1049 01:11:54,340 --> 01:11:57,910 Then my formula for tau x would be 1050 01:11:57,910 --> 01:12:05,080 1 over nvd squared, which is exactly what I have here. 1051 01:12:05,080 --> 01:12:12,280 1 over tau x is nd squared v. OK? 1052 01:12:12,280 --> 01:12:19,020 So in order to compare the terms that I 1053 01:12:19,020 --> 01:12:22,710 have on the right-hand side with the terms 1054 01:12:22,710 --> 01:12:25,850 on the left-hand side, I notice that I 1055 01:12:25,850 --> 01:12:31,210 need to know something about nd cubed. 1056 01:12:31,210 --> 01:12:36,110 So nd cubed tells you if I have a particle 1057 01:12:36,110 --> 01:12:40,000 here and this particle has a range of interactions 1058 01:12:40,000 --> 01:12:44,950 that I call d, how many other particles fall 1059 01:12:44,950 --> 01:12:46,370 within that range of interaction? 1060 01:12:49,490 --> 01:12:53,750 Now, for the gas in this room, the range of the interaction 1061 01:12:53,750 --> 01:12:55,680 is of the order of the size of the molecule. 1062 01:12:55,680 --> 01:12:58,350 It is very small. 1063 01:12:58,350 --> 01:13:01,180 And the distance between molecules in this room 1064 01:13:01,180 --> 01:13:03,340 is far apart. 1065 01:13:03,340 --> 01:13:10,770 And indeed, you can estimate that for gas, nd cubed 1066 01:13:10,770 --> 01:13:14,480 has to be of the order of 10 to the minus 4. 1067 01:13:14,480 --> 01:13:16,260 And how do I know that? 1068 01:13:16,260 --> 01:13:20,610 Because if I were to take all of the gas particles in this room 1069 01:13:20,610 --> 01:13:24,560 and put them together so that they are touching, 1070 01:13:24,560 --> 01:13:26,690 then I would have a liquid. 1071 01:13:26,690 --> 01:13:29,600 And the density of, say, typical liquid 1072 01:13:29,600 --> 01:13:33,740 is of the order of 10,000 times larger than the density of air. 1073 01:13:33,740 --> 01:13:37,516 So basically, it has to be a number of this order. 1074 01:13:37,516 --> 01:13:39,960 OK? 1075 01:13:39,960 --> 01:13:41,355 So fine. 1076 01:13:44,180 --> 01:13:47,310 Let's look at our equations. 1077 01:13:47,310 --> 01:13:53,170 So what I find is that in this equation for f2, 1078 01:13:53,170 --> 01:13:56,570 on the left-hand side, I have a term 1079 01:13:56,570 --> 01:14:00,850 that has magnitude that is very large-- 1 over tau c. 1080 01:14:04,020 --> 01:14:07,280 Whereas the term on the right-hand side, 1081 01:14:07,280 --> 01:14:12,210 in terms of magnitude, is something like this. 1082 01:14:15,420 --> 01:14:17,920 And in fact, this will be true for every equation 1083 01:14:17,920 --> 01:14:20,420 in the hierarchy. 1084 01:14:20,420 --> 01:14:25,570 So maybe if I am in the limit where nd cubed is much, 1085 01:14:25,570 --> 01:14:29,950 much less than 1-- such as the gas in this room, which 1086 01:14:29,950 --> 01:14:34,490 is called the dilute limit-- I can ignore the right-hand side. 1087 01:14:34,490 --> 01:14:38,511 I can set the right-hand side to 0. 1088 01:14:38,511 --> 01:14:39,010 OK? 1089 01:14:41,720 --> 01:14:45,380 Now, I can't do that for the first equation, 1090 01:14:45,380 --> 01:14:47,320 because the first equation is really 1091 01:14:47,320 --> 01:14:50,020 the only equation in the hierarchy that does not 1092 01:14:50,020 --> 01:14:53,484 have the collision term on the left-hand side. 1093 01:14:53,484 --> 01:14:54,950 Right? 1094 01:14:54,950 --> 01:14:59,611 And so for the first equation, I really need to keep that. 1095 01:14:59,611 --> 01:15:03,650 And actually, it goes back to all of the story 1096 01:15:03,650 --> 01:15:07,340 that we've had over here. 1097 01:15:07,340 --> 01:15:11,540 Remember that we said this rho equilibrium has 1098 01:15:11,540 --> 01:15:16,600 to be a function of H and conserved quantities. 1099 01:15:16,600 --> 01:15:20,170 Suppose I go to my Hamiltonian and I 1100 01:15:20,170 --> 01:15:24,290 ignore all of these interactions, which 1101 01:15:24,290 --> 01:15:26,710 is what I would have done if I just 1102 01:15:26,710 --> 01:15:31,370 look at the first term over here and set the collision 1103 01:15:31,370 --> 01:15:34,040 term on the right-hand side to 0. 1104 01:15:34,040 --> 01:15:36,380 What would happen then? 1105 01:15:36,380 --> 01:15:40,180 Then clearly, for each one of the particles, 1106 01:15:40,180 --> 01:15:44,480 I have-- let's say it's energy is going to be conserved. 1107 01:15:44,480 --> 01:15:46,330 Maybe the magnitude of its momentum 1108 01:15:46,330 --> 01:15:49,450 is going to be conserved in the appropriate geometry. 1109 01:15:49,450 --> 01:15:52,390 And so there will be a huge number 1110 01:15:52,390 --> 01:15:55,870 of individual conserved quantities 1111 01:15:55,870 --> 01:15:58,300 that I would have to put over there. 1112 01:15:58,300 --> 01:16:00,560 Indeed, if I sort of go back to the picture 1113 01:16:00,560 --> 01:16:04,860 that I was drawing over here, if I ignore collisions 1114 01:16:04,860 --> 01:16:08,810 between the particles, then the bullets that I send 1115 01:16:08,810 --> 01:16:13,740 will always be following a trajectory such as this 1116 01:16:13,740 --> 01:16:17,200 forever, because momentum will be conserved. 1117 01:16:17,200 --> 01:16:19,990 You will always-- I mean, except up to reflection. 1118 01:16:19,990 --> 01:16:21,760 And say the magnitude of velocity 1119 01:16:21,760 --> 01:16:25,030 would be always following the same thing. 1120 01:16:25,030 --> 01:16:26,380 OK? 1121 01:16:26,380 --> 01:16:30,050 So however, if there is a collision between two 1122 01:16:30,050 --> 01:16:34,090 of the particles-- so the particles that come in here, 1123 01:16:34,090 --> 01:16:37,290 they have different velocities, they will hit each other. 1124 01:16:37,290 --> 01:16:38,810 The moment they hit each other, they 1125 01:16:38,810 --> 01:16:41,080 go off different directions. 1126 01:16:41,080 --> 01:16:43,400 And after a certain number of hits, 1127 01:16:43,400 --> 01:16:45,460 then I will lose all of the regularity 1128 01:16:45,460 --> 01:16:48,490 of what I had in the beginning. 1129 01:16:48,490 --> 01:16:54,390 And so essentially, this second term 1130 01:16:54,390 --> 01:16:57,680 on the right-hand with the collisions 1131 01:16:57,680 --> 01:17:00,550 is the thing that is necessary for me 1132 01:17:00,550 --> 01:17:06,200 to ensure that my gas does come to equilibrium in the sense 1133 01:17:06,200 --> 01:17:09,870 that their momenta get distributed and reversed. 1134 01:17:09,870 --> 01:17:12,890 I really need to keep track of that. 1135 01:17:12,890 --> 01:17:16,350 And also, you can see that the timescales for which 1136 01:17:16,350 --> 01:17:18,800 this kind of equilibration takes place 1137 01:17:18,800 --> 01:17:22,770 has to do with this collision time. 1138 01:17:22,770 --> 01:17:28,030 But as far as this term is concerned, for the gas 1139 01:17:28,030 --> 01:17:31,570 or for that system of bullets, it doesn't really matter. 1140 01:17:31,570 --> 01:17:35,030 Because for this term to have been important, 1141 01:17:35,030 --> 01:17:37,510 it would have been necessary for something 1142 01:17:37,510 --> 01:17:41,470 interesting to physically occur should three particles come 1143 01:17:41,470 --> 01:17:44,520 together simultaneously. 1144 01:17:44,520 --> 01:17:47,450 And if I complete the-- say that never 1145 01:17:47,450 --> 01:17:51,350 in the history of this system three particles will 1146 01:17:51,350 --> 01:17:55,900 come together, they do come together in reality. 1147 01:17:55,900 --> 01:17:58,800 It's not that big a difference. 1148 01:17:58,800 --> 01:18:00,780 It's only a factor of 10 to the 4 difference 1149 01:18:00,780 --> 01:18:03,290 between the right-hand side and the left-hand side. 1150 01:18:03,290 --> 01:18:05,290 But still, even if they didn't, there 1151 01:18:05,290 --> 01:18:07,990 was nothing about equilibration of the gas that 1152 01:18:07,990 --> 01:18:10,910 would be missed by this. 1153 01:18:10,910 --> 01:18:15,710 So it's a perfectly reasonable approximation and assumption, 1154 01:18:15,710 --> 01:18:20,040 therefore, for us to drop this term. 1155 01:18:20,040 --> 01:18:23,020 And we'll see that although that is physically motivated, 1156 01:18:23,020 --> 01:18:26,980 it actually doesn't resolve this question of irreversibility 1157 01:18:26,980 --> 01:18:31,650 yet, because that's also potentially a system 1158 01:18:31,650 --> 01:18:33,130 that you could set up. 1159 01:18:33,130 --> 01:18:35,880 You just eliminate all of the three-body interactions 1160 01:18:35,880 --> 01:18:37,030 from the problem. 1161 01:18:37,030 --> 01:18:39,330 Still, you could have a very reversible set 1162 01:18:39,330 --> 01:18:42,570 of conditions and deterministic process 1163 01:18:42,570 --> 01:18:45,850 that you could reverse in time. 1164 01:18:45,850 --> 01:18:48,180 But still, it's sort of allows us 1165 01:18:48,180 --> 01:18:51,250 to have something that is more manageable, 1166 01:18:51,250 --> 01:18:54,340 which is what we will be looking at next. 1167 01:18:54,340 --> 01:18:57,270 Before I go to what is next, I also 1168 01:18:57,270 --> 01:19:01,450 mentioned that there is one other limit where 1169 01:19:01,450 --> 01:19:04,420 one can do things, which is when, 1170 01:19:04,420 --> 01:19:07,580 within the range of interaction of one particle, 1171 01:19:07,580 --> 01:19:09,930 there are many other particles. 1172 01:19:09,930 --> 01:19:15,180 So you are in the dense limit, nd cubed greater than 1. 1173 01:19:15,180 --> 01:19:18,000 This does not happen for a liquid, because for a liquid, 1174 01:19:18,000 --> 01:19:22,230 the range does not allow many particles to come within it. 1175 01:19:22,230 --> 01:19:24,100 But it happens for a plasma where 1176 01:19:24,100 --> 01:19:26,800 you have long-range Coulomb interaction. 1177 01:19:26,800 --> 01:19:29,780 And within the range of Coulomb interaction, 1178 01:19:29,780 --> 01:19:32,140 you could have many other interactions. 1179 01:19:32,140 --> 01:19:36,620 And so that limit you will explore in the problem set 1180 01:19:36,620 --> 01:19:41,290 leads to a different description of approach to equilibrium. 1181 01:19:41,290 --> 01:19:44,210 It's called the Vlasov equation. 1182 01:19:44,210 --> 01:19:48,140 What we are going to proceed with now will lead to something 1183 01:19:48,140 --> 01:19:52,310 else, which is called-- in the dilute limit, 1184 01:19:52,310 --> 01:19:54,306 it will get the Boltzmann equation. 1185 01:19:58,766 --> 01:19:59,266 OK? 1186 01:20:03,250 --> 01:20:06,925 So let's see what we have. 1187 01:20:19,200 --> 01:20:26,210 So currently, we achieved something. 1188 01:20:26,210 --> 01:20:34,140 We want to describe properties of a few particles 1189 01:20:34,140 --> 01:20:37,450 in the system-- densities that describe only, 1190 01:20:37,450 --> 01:20:41,990 say, one particle by itself if one was not enough. 1191 01:20:41,990 --> 01:20:44,040 But I can terminate the equations. 1192 01:20:44,040 --> 01:20:48,740 And with one-particle density and two-particle density, 1193 01:20:48,740 --> 01:20:51,240 I should have an appropriate description 1194 01:20:51,240 --> 01:20:54,730 of how the system evolves. 1195 01:20:54,730 --> 01:20:57,310 Let's think about it one more time. 1196 01:20:57,310 --> 01:20:59,115 So what is happening here? 1197 01:20:59,115 --> 01:21:02,450 There is the one-particle description 1198 01:21:02,450 --> 01:21:06,580 that tells you how the density for one particle, 1199 01:21:06,580 --> 01:21:10,430 or an ensemble, the probability for one particle, its position 1200 01:21:10,430 --> 01:21:12,740 and momentum evolves. 1201 01:21:12,740 --> 01:21:15,300 But it requires knowledge of what 1202 01:21:15,300 --> 01:21:19,150 would happen with a second particle present. 1203 01:21:19,150 --> 01:21:23,220 But the equations that we have for the density that 1204 01:21:23,220 --> 01:21:28,020 involves two particles is simply a description of things 1205 01:21:28,020 --> 01:21:32,390 that you would do if you had deterministic trajectories. 1206 01:21:32,390 --> 01:21:35,130 There is nothing else on the right-hand side. 1207 01:21:35,130 --> 01:21:38,930 So basically, all you need to do is in order 1208 01:21:38,930 --> 01:21:43,240 to determine this, is to have full knowledge of what happens 1209 01:21:43,240 --> 01:21:46,810 if two particles come together, collide together, go 1210 01:21:46,810 --> 01:21:48,880 away, all kinds of things. 1211 01:21:48,880 --> 01:21:52,410 So if you have those trajectories for two particles, 1212 01:21:52,410 --> 01:21:55,540 you can, in principle, build this density. 1213 01:21:55,540 --> 01:21:57,480 It's still not an easy task. 1214 01:21:57,480 --> 01:22:00,050 But in principle, one could do that. 1215 01:22:00,050 --> 01:22:03,840 And so this is the description of f2. 1216 01:22:03,840 --> 01:22:09,050 And we expect f2 to describe processes 1217 01:22:09,050 --> 01:22:12,960 in which, over a very rapid timescale, say, 1218 01:22:12,960 --> 01:22:18,200 momenta gets shifted from one direction to another direction. 1219 01:22:18,200 --> 01:22:23,270 But then there is something about the overall behavior that 1220 01:22:23,270 --> 01:22:26,030 should follow, more or less, f1. 1221 01:22:26,030 --> 01:22:27,920 Again, what do I mean? 1222 01:22:27,920 --> 01:22:33,400 What I mean is the following, that if I open this box, 1223 01:22:33,400 --> 01:22:35,720 there is what you would observe. 1224 01:22:35,720 --> 01:22:39,060 The density would kind of gush through here. 1225 01:22:39,060 --> 01:22:40,860 And so you can have a description 1226 01:22:40,860 --> 01:22:44,480 for how the density, let's say in coordinate space, 1227 01:22:44,480 --> 01:22:47,590 would be evolving as a function of time. 1228 01:22:47,590 --> 01:22:51,630 If I ask how does the two-particle prescription 1229 01:22:51,630 --> 01:22:55,780 evolve, well, the two-particle prescription, part of it 1230 01:22:55,780 --> 01:22:57,450 is what's the probability that I have 1231 01:22:57,450 --> 01:22:59,890 a particle here and a particle there? 1232 01:22:59,890 --> 01:23:04,120 And if the two particles, if the separations are far apart, 1233 01:23:04,120 --> 01:23:06,880 you would be justified to say that that is roughly 1234 01:23:06,880 --> 01:23:09,920 the product of the probabilities that I have something here 1235 01:23:09,920 --> 01:23:11,710 and something there. 1236 01:23:11,710 --> 01:23:14,960 When you become very close to each other, however, 1237 01:23:14,960 --> 01:23:17,900 over the range of interactions and collisions, 1238 01:23:17,900 --> 01:23:21,510 that will have to be modified, because at those descriptions 1239 01:23:21,510 --> 01:23:24,760 from here, you would have to worry 1240 01:23:24,760 --> 01:23:28,830 about the collisions, and the exchange of momenta, et cetera. 1241 01:23:28,830 --> 01:23:35,610 So in that sense, part of f2 is simply following f1 slowly. 1242 01:23:35,610 --> 01:23:42,340 And part of f1 captures all of the collisions that you have. 1243 01:23:42,340 --> 01:23:45,600 In fact, that part of f2 that captures in the collisions, 1244 01:23:45,600 --> 01:23:49,930 we would like to simplify as much as possible. 1245 01:23:49,930 --> 01:23:52,960 And that's the next task that we do. 1246 01:23:52,960 --> 01:24:02,340 So what I need to do is to somehow express 1247 01:24:02,340 --> 01:24:06,630 the f2 that appears in the first equation 1248 01:24:06,630 --> 01:24:09,690 while solving this equation that is the second one. 1249 01:24:12,870 --> 01:24:15,790 I will write the answer that we will eventually 1250 01:24:15,790 --> 01:24:18,740 deal with and explain it next time. 1251 01:24:18,740 --> 01:24:21,760 So the ultimate result would be that the left-hand side, 1252 01:24:21,760 --> 01:24:23,910 we will have the terms that we have currently. 1253 01:24:33,800 --> 01:24:36,280 f1. 1254 01:24:36,280 --> 01:24:39,660 On the right-hand side, what we find 1255 01:24:39,660 --> 01:24:44,240 is that I need to integrate over all momenta 1256 01:24:44,240 --> 01:24:47,320 of a second particle and something that 1257 01:24:47,320 --> 01:24:53,940 is like a distance to the target-- one term that 1258 01:24:53,940 --> 01:24:56,700 is the flux of incoming particles. 1259 01:24:59,580 --> 01:25:11,930 And then we would have f2 after collision minus f2 1260 01:25:11,930 --> 01:25:12,880 before collision. 1261 01:25:18,580 --> 01:25:21,390 And this is really the Boltzmann equation 1262 01:25:21,390 --> 01:25:26,890 after one more approximation, where we replace f2 with f1. 1263 01:25:26,890 --> 01:25:28,420 f1. 1264 01:25:28,420 --> 01:25:30,730 But what all of that means symbolically 1265 01:25:30,730 --> 01:25:34,780 and what it is we'll have to explain next time.