1 00:00:00,070 --> 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 00:00:20,650 --> 00:00:21,830 PROFESSOR: OK, let's start. 9 00:00:24,530 --> 00:00:29,980 So last time, having dealt with the ideal gas 10 00:00:29,980 --> 00:00:33,225 as much as possible, we started with interacting systems. 11 00:00:39,420 --> 00:00:42,660 Let's say interacting gas. 12 00:00:42,660 --> 00:00:46,760 And the first topic that we did was 13 00:00:46,760 --> 00:00:51,860 what I called an approach towards interaction 14 00:00:51,860 --> 00:00:54,451 through the Cumulant expansion. 15 00:00:57,400 --> 00:01:03,720 The idea is that we certainly solved the problem where 16 00:01:03,720 --> 00:01:14,550 we had particles in a box and it was just a trivial system. 17 00:01:14,550 --> 00:01:20,690 Basically, the particles were independent of each other 18 00:01:20,690 --> 00:01:23,520 in the canonical ensemble. 19 00:01:23,520 --> 00:01:27,540 And things become interesting if you put interactions 20 00:01:27,540 --> 00:01:30,710 among all of the particles. 21 00:01:30,710 --> 00:01:33,480 Very soon, we will have a specific form 22 00:01:33,480 --> 00:01:35,520 of this interaction. 23 00:01:35,520 --> 00:01:37,270 But for the beginning, let's maintain 24 00:01:37,270 --> 00:01:40,190 it as general as possible. 25 00:01:40,190 --> 00:01:42,240 And the idea was that we are trying 26 00:01:42,240 --> 00:01:46,120 to calculate the partition function for a system that 27 00:01:46,120 --> 00:01:49,400 has a given temperature. 28 00:01:49,400 --> 00:01:54,410 The box has some volume V and there are N particles in it. 29 00:01:54,410 --> 00:01:58,250 And to do so, we have to integrate over 30 00:01:58,250 --> 00:02:02,140 the entirety of the phase space of N particles. 31 00:02:02,140 --> 00:02:09,100 So we have to integrate over all of the momenta and coordinates 32 00:02:09,100 --> 00:02:11,710 of e to the minus beta h. 33 00:02:11,710 --> 00:02:15,580 But we said that we are going to first of all, 34 00:02:15,580 --> 00:02:18,330 note that for identical particles, 35 00:02:18,330 --> 00:02:21,720 we can't tell apart the phase space 36 00:02:21,720 --> 00:02:25,340 if you were to make these in [? factorial ?] permutations. 37 00:02:25,340 --> 00:02:27,650 And we also made it dimensionless 38 00:02:27,650 --> 00:02:30,620 by dividing by h to the 3n. 39 00:02:30,620 --> 00:02:33,250 And then we have e to the minus beta 40 00:02:33,250 --> 00:02:39,340 h, which has a part that is e to the minus sum over i pi 41 00:02:39,340 --> 00:02:43,240 squared over 2m, that depends on the momenta. 42 00:02:43,240 --> 00:02:48,660 And a part that depends on the interaction U. 43 00:02:48,660 --> 00:02:52,390 The integrals over the momenta, we can easily perform. 44 00:02:52,390 --> 00:02:55,770 They give us the result of Gaussians 45 00:02:55,770 --> 00:03:03,240 that we can express as lambda to the power of 3n 46 00:03:03,240 --> 00:03:09,322 where we introduced this lambda to be h over root 2 pi m k t. 47 00:03:12,770 --> 00:03:18,170 And actually, sort of keeping in mind 48 00:03:18,170 --> 00:03:22,620 the result that we had for the case of the ideal gas, 49 00:03:22,620 --> 00:03:28,700 let's rewrite this as V over lambda cubed to the power of N. 50 00:03:28,700 --> 00:03:34,770 So I essentially divided and multiplied by V to the N 51 00:03:34,770 --> 00:03:37,200 so that this part becomes the partition 52 00:03:37,200 --> 00:03:42,010 function than I would have had if U was absent. 53 00:03:42,010 --> 00:03:47,650 And so then the remainder is the integral over all of the q's, 54 00:03:47,650 --> 00:03:50,580 but I divided by V. So basically, 55 00:03:50,580 --> 00:03:53,580 with each integration, I'm uniformly 56 00:03:53,580 --> 00:03:57,890 sampling all of the points inside the box with probability 57 00:03:57,890 --> 00:04:02,540 density 1/V e to the minus beta U. 58 00:04:02,540 --> 00:04:05,760 And we said that this quantity we 59 00:04:05,760 --> 00:04:10,470 can regard as sampling e to the minus beta U 60 00:04:10,470 --> 00:04:12,540 through 0 to all other probability 61 00:04:12,540 --> 00:04:17,149 distribution in which the particles ideal-like are 62 00:04:17,149 --> 00:04:19,160 uniformly distributed. 63 00:04:19,160 --> 00:04:23,870 And we call this e to the minus beta U. 64 00:04:23,870 --> 00:04:27,810 And the index 0 again to indicate 65 00:04:27,810 --> 00:04:31,265 that we are dealing with a uniform 0 66 00:04:31,265 --> 00:04:33,750 to order distribution with respect 67 00:04:33,750 --> 00:04:35,584 to which this average is taken. 68 00:04:38,190 --> 00:04:44,060 And then, this quantity we can write as Z0. 69 00:04:44,060 --> 00:04:46,760 And this exponential we can formally 70 00:04:46,760 --> 00:04:51,160 express as a sum over all, let's say, 71 00:04:51,160 --> 00:04:54,750 l running from 0 to infinity minus beta 72 00:04:54,750 --> 00:04:58,080 to the l divided by l factorial. 73 00:04:58,080 --> 00:05:02,290 Then, expanding this, a quantity that depends on the coordinates 74 00:05:02,290 --> 00:05:06,930 is U. So I could have U raised to the l power, this kind 75 00:05:06,930 --> 00:05:10,360 of average taken with respect to that. 76 00:05:10,360 --> 00:05:13,720 And immediately, my writing something like this 77 00:05:13,720 --> 00:05:17,110 suggests that what I'm after eventually 78 00:05:17,110 --> 00:05:20,120 is some kind of a perturbation theory. 79 00:05:20,120 --> 00:05:24,460 Because I know how to solve the problem where U is absent. 80 00:05:24,460 --> 00:05:26,370 That's the ideal gas problem. 81 00:05:26,370 --> 00:05:29,900 Maybe we can start to move away from the ideal gas 82 00:05:29,900 --> 00:05:33,160 and calculate things perturbatively 83 00:05:33,160 --> 00:05:34,850 in some quantity. 84 00:05:34,850 --> 00:05:36,910 And exactly what that quantity is 85 00:05:36,910 --> 00:05:39,090 will become apparent shortly. 86 00:05:39,090 --> 00:05:43,260 At this time, it looks like it's an expansion in beta U 87 00:05:43,260 --> 00:05:46,850 having to be a small quantity. 88 00:05:46,850 --> 00:05:51,050 Now, of course, for calculating thermodynamic functions 89 00:05:51,050 --> 00:05:55,520 and behaviors, we don't really rely usually on Z, 90 00:05:55,520 --> 00:05:59,120 but log Z, which can be related to the free energy, 91 00:05:59,120 --> 00:06:01,110 for example. 92 00:06:01,110 --> 00:06:04,230 So taking the log of that expression, 93 00:06:04,230 --> 00:06:07,490 I have a term that is log of Z0. 94 00:06:07,490 --> 00:06:11,190 And a term that is the log of that sum. 95 00:06:11,190 --> 00:06:14,530 But we recognize that that sum, in some sense, 96 00:06:14,530 --> 00:06:20,240 is a generator of moments of this quantity U. 97 00:06:20,240 --> 00:06:22,020 And we have experienced that when 98 00:06:22,020 --> 00:06:25,565 we take the logarithm of a generator of moments, what 99 00:06:25,565 --> 00:06:29,350 we will get is a generator of Cumulants, 100 00:06:29,350 --> 00:06:33,460 now starting with l equals to 1 minus beta 101 00:06:33,460 --> 00:06:38,180 to the power of l divided by l factorial U 102 00:06:38,180 --> 00:06:42,910 to the l0 with a subscript c. 103 00:06:42,910 --> 00:06:48,880 Of course, this subscript c sort of captures all of the various 104 00:06:48,880 --> 00:06:51,600 subtractions that you have to make in order 105 00:06:51,600 --> 00:06:54,120 to go from a moment to a Cumulant. 106 00:06:54,120 --> 00:06:56,070 But that's quite general and something 107 00:06:56,070 --> 00:07:00,720 that we know essentially is relating log of the expansion 108 00:07:00,720 --> 00:07:02,120 to the expansion itself. 109 00:07:02,120 --> 00:07:03,170 But that's well-known. 110 00:07:05,790 --> 00:07:10,230 So presumably, I can perturbatively calculate these. 111 00:07:10,230 --> 00:07:13,410 And I will have, therefore, progressively 112 00:07:13,410 --> 00:07:17,130 better approximations to log Z in the presence 113 00:07:17,130 --> 00:07:19,810 of this interaction. 114 00:07:19,810 --> 00:07:22,270 Now at this point, we have to start 115 00:07:22,270 --> 00:07:29,150 thinking about a particular expression for U. 116 00:07:29,150 --> 00:07:34,850 So let's imagine that our U is a sum over pairwise interactions. 117 00:07:34,850 --> 00:07:37,290 So if I think about the particles, 118 00:07:37,290 --> 00:07:41,030 there are molecules that are in the gas in this room. 119 00:07:41,030 --> 00:07:43,880 Basically, the most important thing is when the two of them 120 00:07:43,880 --> 00:07:47,960 approach each other and I have to put a pairwise interaction 121 00:07:47,960 --> 00:07:50,710 between them of this form. 122 00:07:50,710 --> 00:07:53,280 In principle, I could add three-point and higher-order 123 00:07:53,280 --> 00:07:54,530 interactions. 124 00:07:54,530 --> 00:07:57,290 But for all intents and purposes, 125 00:07:57,290 --> 00:07:59,140 this should be enough. 126 00:07:59,140 --> 00:08:04,190 And then again, what I have here to evaluate, 127 00:08:04,190 --> 00:08:07,735 the first term would be minus beta, 128 00:08:07,735 --> 00:08:14,730 the average of the first power, and the next one 129 00:08:14,730 --> 00:08:17,810 will be beta squared over 2. 130 00:08:17,810 --> 00:08:20,090 For the first Cumulant, it is the same thing 131 00:08:20,090 --> 00:08:21,870 as the first moment. 132 00:08:21,870 --> 00:08:24,550 The second Cumulant is the thing that is the variance. 133 00:08:24,550 --> 00:08:28,520 So this would have been U squared 0 minus average 134 00:08:28,520 --> 00:08:32,633 of U squared, and then there will be higher orders. 135 00:08:38,760 --> 00:08:43,669 So let's calculate these first two moments explicitly 136 00:08:43,669 --> 00:08:46,250 for this potential. 137 00:08:46,250 --> 00:08:54,110 So the first term U in this 0 to order average, what is it? 138 00:08:54,110 --> 00:08:59,270 Well, that's my U. So I have to do a sum over i 139 00:08:59,270 --> 00:09:06,570 and j of V of qi minus qj. 140 00:09:06,570 --> 00:09:08,380 What does the averaging mean? 141 00:09:08,380 --> 00:09:14,670 It means I have to integrate this over all qi divided 142 00:09:14,670 --> 00:09:19,030 by V. Let's call this qk divided V and this is [INAUDIBLE]. 143 00:09:22,958 --> 00:09:28,400 OK, so let's say we look at the first term in this series which 144 00:09:28,400 --> 00:09:33,390 involves V of q1 minus q2. 145 00:09:33,390 --> 00:09:36,800 Let's say we pick that pair, q1 and q2. 146 00:09:36,800 --> 00:09:45,940 And explicitly, what I have here is integrals over q1, q2, 147 00:09:45,940 --> 00:09:54,520 qn, each one of them coming with a factor of V. 148 00:09:54,520 --> 00:09:58,100 Now, of course, it doesn't matter which pair I pick. 149 00:09:58,100 --> 00:09:59,580 Everything is symmetric. 150 00:09:59,580 --> 00:10:03,090 The weight that I start with uniformly is symmetric. 151 00:10:03,090 --> 00:10:06,510 So clearly, my choice of this pair 152 00:10:06,510 --> 00:10:08,500 was arbitrary and the answer when 153 00:10:08,500 --> 00:10:12,950 I do the sum is the answer for 1 times the number of pairs, 154 00:10:12,950 --> 00:10:14,680 which is N N minus 1 over 2. 155 00:10:19,250 --> 00:10:23,510 Now, over here I have integrals over q3 or qn, 156 00:10:23,510 --> 00:10:26,260 et cetera, that don't appear in the function 157 00:10:26,260 --> 00:10:27,840 that I'm integrating. 158 00:10:27,840 --> 00:10:32,100 So all of those integrals will give me factors of 1. 159 00:10:32,100 --> 00:10:34,660 And really, the only thing that I am going to be left with 160 00:10:34,660 --> 00:10:39,020 is these two integrals over q1 and q2. 161 00:10:39,020 --> 00:10:42,310 But even then, the function that I'm integrating 162 00:10:42,310 --> 00:10:45,230 is only a function of the relative coordinate 163 00:10:45,230 --> 00:10:47,270 that I can call q. 164 00:10:47,270 --> 00:10:54,470 And then I can, if I want, integrate over q1 and q. 165 00:10:54,470 --> 00:10:56,970 The integral over q1 would also give me 166 00:10:56,970 --> 00:10:59,520 1 when I divide by the volume. 167 00:10:59,520 --> 00:11:04,280 So the answer is going to be N N minus 1 over 2, 168 00:11:04,280 --> 00:11:06,600 which is the number of pairs. 169 00:11:06,600 --> 00:11:11,420 The integral over the relative coordinate times 170 00:11:11,420 --> 00:11:14,330 the potential-- there is a factor of 1 over V 171 00:11:14,330 --> 00:11:17,780 that I can put out here. 172 00:11:17,780 --> 00:11:21,790 So you tell me what the first pair potential is 173 00:11:21,790 --> 00:11:25,790 and I tell you that this is the first correction that you will 174 00:11:25,790 --> 00:11:29,780 get once you multiply by minus beta 175 00:11:29,780 --> 00:11:32,890 to log Z from which you can calculate the partition 176 00:11:32,890 --> 00:11:35,450 functions. 177 00:11:35,450 --> 00:11:36,642 Yes. 178 00:11:36,642 --> 00:11:39,222 AUDIENCE: How did you turn the product of two differentials 179 00:11:39,222 --> 00:11:44,670 d q1 times d q2 into just one differential? 180 00:11:44,670 --> 00:11:49,590 PROFESSOR: OK, I have the integral dq q1 dq q2, 181 00:11:49,590 --> 00:11:53,000 some function of q1 minus q2. 182 00:11:53,000 --> 00:11:58,160 I change variables keeping q1 as one of my variables 183 00:11:58,160 --> 00:12:03,800 and q [INAUDIBLE] replace q2 with q minus q2. 184 00:12:03,800 --> 00:12:06,950 So then I have the integration over q1. 185 00:12:06,950 --> 00:12:10,200 I have the integration over q2. 186 00:12:10,200 --> 00:12:14,160 Not q2, but q f of q. 187 00:12:14,160 --> 00:12:15,520 And then everything's fine. 188 00:12:23,280 --> 00:12:25,950 All right, so that's the first term. 189 00:12:25,950 --> 00:12:29,140 Let's calculate the next term. 190 00:12:29,140 --> 00:12:33,540 Next term becomes somewhat interesting. 191 00:12:33,540 --> 00:12:39,520 I have to calculate this quantity as a Cumulant. 192 00:12:39,520 --> 00:12:41,785 It's the variance. 193 00:12:45,900 --> 00:12:49,480 V itself is a sum over pairs. 194 00:12:49,480 --> 00:12:53,200 So to get V squared to U squared, 195 00:12:53,200 --> 00:12:56,160 I have to essentially square this expression. 196 00:12:56,160 --> 00:12:59,780 So rather than having sum over one pair ij, 197 00:12:59,780 --> 00:13:02,690 I have the sum over two pairs ij and kl. 198 00:13:05,890 --> 00:13:09,120 And then what I need to calculate 199 00:13:09,120 --> 00:13:12,910 is an average that if I were to square this, 200 00:13:12,910 --> 00:13:22,230 I will get v of qi minus qj, v of qk minus ql 0. 201 00:13:22,230 --> 00:13:25,720 This will be the average of U squared. 202 00:13:25,720 --> 00:13:28,900 But when I calculate the variance, 203 00:13:28,900 --> 00:13:32,380 I have to subtract from it the average of U squared. 204 00:13:32,380 --> 00:13:40,502 So I have to subtract V of qi minus qj V of qk minus ql. 205 00:13:49,810 --> 00:13:52,510 Fine. 206 00:13:52,510 --> 00:13:57,190 So this is just rewriting of that expression 207 00:13:57,190 --> 00:14:00,000 in terms of the sum. 208 00:14:00,000 --> 00:14:06,590 Now, each one of these is N N minus 1 over two terms. 209 00:14:06,590 --> 00:14:10,016 And so this, when I sum over all possibility, 210 00:14:10,016 --> 00:14:15,640 is the square of N N minus 1 over two possible terms that 211 00:14:15,640 --> 00:14:20,220 will occur here in the series that I've written. 212 00:14:20,220 --> 00:14:26,190 Now, let's group those terms that can occur in this as 213 00:14:26,190 --> 00:14:27,620 follows. 214 00:14:27,620 --> 00:14:33,680 One class of terms will be when the pair 215 00:14:33,680 --> 00:14:45,260 ij that I pick for the first sum is distinct from the pair kl. 216 00:14:45,260 --> 00:14:49,140 So huge number of terms as I go through the pairs 217 00:14:49,140 --> 00:14:52,720 will have this possibility. 218 00:14:52,720 --> 00:14:55,400 OK, then what happens? 219 00:14:55,400 --> 00:15:02,640 Then when I calculate the average of qi minus qj qk minus 220 00:15:02,640 --> 00:15:09,580 ql, which is this part, what I need to do is to do all 221 00:15:09,580 --> 00:15:19,420 of these integral of this form over all q's. 222 00:15:19,420 --> 00:15:26,630 And then I have this V of qi minus qj V of qk minus ql. 223 00:15:29,745 --> 00:15:32,980 Now, my statement is that I have, let's say here, 224 00:15:32,980 --> 00:15:37,450 q1 and q2 and somewhere else q7 and q8. 225 00:15:37,450 --> 00:15:39,740 Or, q7 and q974. 226 00:15:39,740 --> 00:15:41,710 It doesn't matter. 227 00:15:41,710 --> 00:15:43,880 The point is that this integral that 228 00:15:43,880 --> 00:15:45,810 involves this pair of variables has 229 00:15:45,810 --> 00:15:47,310 nothing to do with the integral that 230 00:15:47,310 --> 00:15:49,650 involves the other pair of variables. 231 00:15:49,650 --> 00:15:54,460 So this answer is the same thing as the average of qi 232 00:15:54,460 --> 00:16:00,120 minus qj average of qk minus ql. 233 00:16:00,120 --> 00:16:04,000 Essentially, in the 0 order probability that we are using, 234 00:16:04,000 --> 00:16:06,620 the particles are completely independently 235 00:16:06,620 --> 00:16:11,040 exploring this space and these averages will independently 236 00:16:11,040 --> 00:16:13,160 rely on one pair and independently 237 00:16:13,160 --> 00:16:15,380 rely on another pair. 238 00:16:15,380 --> 00:16:21,060 And once this kind of factorization occurs, 239 00:16:21,060 --> 00:16:24,640 it's clear that this subtraction that I 240 00:16:24,640 --> 00:16:28,430 need to do for the variance will get rid of this term. 241 00:16:28,430 --> 00:16:33,090 So these pairs do not contribute. 242 00:16:33,090 --> 00:16:33,765 No contribution. 243 00:16:38,450 --> 00:16:41,510 If you like, this is this thing. 244 00:16:45,320 --> 00:16:53,678 All right, so let's do the next possible term, ij and ik, 245 00:16:53,678 --> 00:16:57,710 with k not equal to this k. 246 00:16:57,710 --> 00:17:02,720 So basically, I make the pairs have one point in common. 247 00:17:02,720 --> 00:17:05,460 So previously, I had a pair here, a pair here. 248 00:17:05,460 --> 00:17:08,390 Now I joined one of their points. 249 00:17:08,390 --> 00:17:11,760 So what do I get here? 250 00:17:11,760 --> 00:17:20,520 I will get for this average qi minus qj qi minus ql. 251 00:17:20,520 --> 00:17:24,470 You can see that essentially, the three integrals that I need 252 00:17:24,470 --> 00:17:27,060 to worry about are the integrals that 253 00:17:27,060 --> 00:17:28,510 involve these three indices. 254 00:17:28,510 --> 00:17:30,810 All the others will give me 0. 255 00:17:30,810 --> 00:17:40,370 So this is going to be the integral qi qj ql divided by V. 256 00:17:40,370 --> 00:17:47,170 I have V of qi minus qj V of qi minus ql. 257 00:17:50,026 --> 00:17:52,237 AUDIENCE: You switched [INAUDIBLE]. 258 00:17:52,237 --> 00:17:52,820 PROFESSOR: OK. 259 00:17:58,720 --> 00:18:00,050 I think that's fine. 260 00:18:04,440 --> 00:18:06,640 All right. 261 00:18:06,640 --> 00:18:11,150 Now, I can do the same thing that I did over here. 262 00:18:11,150 --> 00:18:13,760 I have three variables. 263 00:18:13,760 --> 00:18:17,370 I will pick qi to stay as one of my variables. 264 00:18:17,370 --> 00:18:23,920 I replace qj with this difference, q j i. 265 00:18:23,920 --> 00:18:29,440 I replace ql with its distance to i. 266 00:18:32,120 --> 00:18:34,610 And again, I can independently integrate 267 00:18:34,610 --> 00:18:37,810 over these three variables. 268 00:18:37,810 --> 00:18:41,520 So this again, becomes the same thing 269 00:18:41,520 --> 00:18:50,670 as V of qi minus qj average V of qi minus ql average. 270 00:18:50,670 --> 00:18:52,430 And again, there is no contribution. 271 00:19:02,430 --> 00:19:06,180 Now, the first class it was obvious because this pair 272 00:19:06,180 --> 00:19:09,800 and this pair were completely distinct. 273 00:19:09,800 --> 00:19:12,830 This class is a little bit more subtle 274 00:19:12,830 --> 00:19:17,300 because I joined one of the points together, 275 00:19:17,300 --> 00:19:23,090 and then I used this and this as well as this point 276 00:19:23,090 --> 00:19:25,280 as independent variables. 277 00:19:25,280 --> 00:19:29,450 And I saw that the sum breaks into pieces. 278 00:19:29,450 --> 00:19:32,680 And you won't be surprised if no matter 279 00:19:32,680 --> 00:19:37,870 how complicated I make various more interactions over here, 280 00:19:37,870 --> 00:19:40,780 I can measure all of the coordinates with respect 281 00:19:40,780 --> 00:19:44,150 to the single point and same thing would happen. 282 00:19:44,150 --> 00:19:52,548 This class of diagrams are called 1-particle reducible. 283 00:19:52,548 --> 00:19:56,160 And the first part had to do with distinct graphs. 284 00:19:56,160 --> 00:19:58,090 You didn't have to worry about them. 285 00:19:58,090 --> 00:20:01,170 But if I were to sort of convert this expression-- 286 00:20:01,170 --> 00:20:04,150 and we will do so shortly-- into graphs, 287 00:20:04,150 --> 00:20:06,575 it corresponds to graphs where there 288 00:20:06,575 --> 00:20:10,620 is a point from which everything else is hanging. 289 00:20:10,620 --> 00:20:12,760 And measuring coordinates with respect 290 00:20:12,760 --> 00:20:18,490 to that point will allow the average to break into pieces, 291 00:20:18,490 --> 00:20:23,520 and then to be removed through this subtraction. 292 00:20:23,520 --> 00:20:28,190 So this is simple example of something that is more general. 293 00:20:28,190 --> 00:20:31,560 So at the level of this second-order theorem, 294 00:20:31,560 --> 00:20:34,540 the only type of thing that will survive 295 00:20:34,540 --> 00:20:40,030 is if the pairs are identical, ij and other term 296 00:20:40,030 --> 00:20:41,780 is also the same pair ij. 297 00:20:44,380 --> 00:20:46,250 What do I get then? 298 00:20:46,250 --> 00:20:58,613 I get V of qi minus qj squared minus V 299 00:20:58,613 --> 00:21:04,740 of qi minus qj 0 order squared. 300 00:21:04,740 --> 00:21:09,260 So it's the variance of a single one of these bond 301 00:21:09,260 --> 00:21:11,060 contributions. 302 00:21:11,060 --> 00:21:14,390 If I write that in terms of integrals, 303 00:21:14,390 --> 00:21:19,200 this becomes the integral d cubed q1, 304 00:21:19,200 --> 00:21:26,580 d cubed q2-- well, qi qj. 305 00:21:26,580 --> 00:21:30,210 I will have factors of V. Then, I 306 00:21:30,210 --> 00:21:39,250 would have V squared qi minus qj for the first term 307 00:21:39,250 --> 00:21:51,360 and then the square of something like this for the second term. 308 00:21:58,680 --> 00:22:04,560 So putting everything together up to this order, 309 00:22:04,560 --> 00:22:05,580 what do I get? 310 00:22:05,580 --> 00:22:09,530 I will get log Z, which is log of Z0. 311 00:22:09,530 --> 00:22:13,560 Let me remind you, log of Z0 was N log 312 00:22:13,560 --> 00:22:24,400 of V over lambda cubed N. That's on form such as this. 313 00:22:24,400 --> 00:22:26,865 And then I have these corrections. 314 00:22:29,490 --> 00:22:33,520 Note that both of the terms that have survived 315 00:22:33,520 --> 00:22:37,240 correspond to looking at one pair. 316 00:22:37,240 --> 00:22:42,620 So the corrections will be of the order of N N minus 1 317 00:22:42,620 --> 00:22:43,850 over 2. 318 00:22:43,850 --> 00:22:48,000 Because in both cases, I really look at one pair. 319 00:22:48,000 --> 00:22:55,230 The first contribution was minus beta integral d cubed q v of q. 320 00:22:55,230 --> 00:22:57,600 There was a factor of 1 over V. Actually, 321 00:22:57,600 --> 00:23:01,270 I can take-- well, let's put the factor of 1 over V 322 00:23:01,270 --> 00:23:03,970 here for the time being. 323 00:23:03,970 --> 00:23:10,290 The next term will be beta squared over 2 324 00:23:10,290 --> 00:23:12,740 because I am looking at the second-order term. 325 00:23:12,740 --> 00:23:14,701 And what do I have? 326 00:23:14,701 --> 00:23:19,610 I have the difference of integral d cubed q 327 00:23:19,610 --> 00:23:21,240 over V V squared. 328 00:23:25,010 --> 00:23:32,210 And the square of the integral of d cubed q over V V. 329 00:23:32,210 --> 00:23:34,910 And presumably, there will be higher-order terms 330 00:23:34,910 --> 00:23:36,367 as we will discuss. 331 00:23:40,510 --> 00:23:45,760 Now, I'm interested in the limit of thermodynamics 332 00:23:45,760 --> 00:23:47,800 where N and V are large. 333 00:23:51,480 --> 00:23:53,250 And in that limit, what do I get? 334 00:23:53,250 --> 00:23:56,800 I will get log Z is log of Z0. 335 00:24:00,520 --> 00:24:10,340 And then I can see that these terms I can write as beta. 336 00:24:10,340 --> 00:24:14,900 N N minus 1 I will replace will N squared. 337 00:24:14,900 --> 00:24:18,250 The factors-- here I have a 1 over V. Here, 338 00:24:18,250 --> 00:24:20,310 I have 1 over V squared. 339 00:24:20,310 --> 00:24:23,920 So this factor is smaller by an amount that 340 00:24:23,920 --> 00:24:28,560 is order of V in the large V-limit. 341 00:24:28,560 --> 00:24:32,770 I will bring the factor of V outside, 342 00:24:32,770 --> 00:24:35,310 so I have something like this. 343 00:24:35,310 --> 00:24:38,090 And then, what do I have here? 344 00:24:38,090 --> 00:24:43,530 Actually, let's keep this in this form, 345 00:24:43,530 --> 00:24:50,455 minus beta integral d cubed q V plus beta 346 00:24:50,455 --> 00:24:57,460 squared over 2 integral d cubed q V squared [INAUDIBLE]. 347 00:25:04,590 --> 00:25:06,690 So the kinds of things that we are interested 348 00:25:06,690 --> 00:25:09,980 and we can measure are the energy of the system, 349 00:25:09,980 --> 00:25:11,585 but let's say we focus on pressure. 350 00:25:15,700 --> 00:25:21,140 And if you look through various things that we derived before, 351 00:25:21,140 --> 00:25:23,510 beta times the pressure you can get 352 00:25:23,510 --> 00:25:35,220 from taking a derivative of log Z with respect to V. 353 00:25:35,220 --> 00:25:39,160 We can express the log Z in terms of the free energy, 354 00:25:39,160 --> 00:25:41,940 and then the derivative of free energy with respect to volume 355 00:25:41,940 --> 00:25:44,830 will give you the pressure of the various factors of beta 356 00:25:44,830 --> 00:25:47,660 [? and signs, ?] we will come up with this. 357 00:25:47,660 --> 00:25:53,360 So from the first term, N long V, we get our ideal gas result. 358 00:25:53,360 --> 00:25:57,920 Beta p is the same thing as density. 359 00:25:57,920 --> 00:26:01,830 And we see that we get a correction from here when 360 00:26:01,830 --> 00:26:05,420 I divide by-- when I take a derivative with respect 361 00:26:05,420 --> 00:26:09,760 to V of 1 over V, I will get minus 1 over V squared. 362 00:26:09,760 --> 00:26:13,040 So the next term would be the square of the density. 363 00:26:16,260 --> 00:26:18,310 And then, there is a series that we 364 00:26:18,310 --> 00:26:27,650 will encounter which depends on the interaction potential. 365 00:26:32,960 --> 00:26:37,070 Now, it turns out that so far I have calculated 366 00:26:37,070 --> 00:26:41,400 terms that relied on only two points, ij. 367 00:26:41,400 --> 00:26:43,740 And hence, they become at the end 368 00:26:43,740 --> 00:26:46,750 proportional to density squared. 369 00:26:46,750 --> 00:26:48,930 If I go further in my expansion, I 370 00:26:48,930 --> 00:26:52,770 will encounter things that I will need triangular points. 371 00:26:52,770 --> 00:26:56,470 For example, i j k forming a triangle. 372 00:26:56,470 --> 00:26:58,490 And then, that would give me something 373 00:26:58,490 --> 00:27:00,750 that would be of the order of the density cubed. 374 00:27:03,300 --> 00:27:11,030 So somehow, I can also see that ultimately this series 375 00:27:11,030 --> 00:27:14,080 in perturbation theory, as we shall see, 376 00:27:14,080 --> 00:27:18,980 also more precisely can be organized in powers of density. 377 00:27:18,980 --> 00:27:21,210 Why is that important? 378 00:27:21,210 --> 00:27:23,930 Because typically, as we discussed right 379 00:27:23,930 --> 00:27:27,580 at the beginning, when you look at the pressure of a gas 380 00:27:27,580 --> 00:27:31,050 when it is dilute and over V is very small, 381 00:27:31,050 --> 00:27:34,140 it is always ideal gas-like. 382 00:27:34,140 --> 00:27:36,620 As you make it more dense, you start 383 00:27:36,620 --> 00:27:41,300 to get corrections that you can express in powers of density. 384 00:27:41,300 --> 00:27:45,020 And the coefficients of that are called Virial coefficients. 385 00:27:45,020 --> 00:27:46,810 So in some sense, we have already 386 00:27:46,810 --> 00:27:50,850 started in calculating the second Virial coefficient, 387 00:27:50,850 --> 00:27:53,620 and there are higher-order Virial coefficients, 388 00:27:53,620 --> 00:27:56,960 which together give you the equation of states relating 389 00:27:56,960 --> 00:28:01,059 pressure to density by a power series. 390 00:28:01,059 --> 00:28:01,850 AUDIENCE: Question. 391 00:28:01,850 --> 00:28:02,632 PROFESSOR: Yes. 392 00:28:02,632 --> 00:28:04,048 AUDIENCE: The Virial coefficients, 393 00:28:04,048 --> 00:28:06,670 if they have integral over the whole volume, 394 00:28:06,670 --> 00:28:09,956 then they will be functions of volume, right? 395 00:28:09,956 --> 00:28:10,580 PROFESSOR: Yes. 396 00:28:10,580 --> 00:28:14,170 But imagine that I'm thinking about the gas in this room 397 00:28:14,170 --> 00:28:16,590 and the interaction of two oxygen molecules. 398 00:28:16,590 --> 00:28:20,290 So what I am saying is you pick an oxygen molecule. 399 00:28:20,290 --> 00:28:22,940 There is interaction and I have to integrate over 400 00:28:22,940 --> 00:28:25,530 where the other oxygen molecule is. 401 00:28:25,530 --> 00:28:28,250 And by the time it is tiny, tiny bit 402 00:28:28,250 --> 00:28:30,800 [? away, ?] the interaction is 0. 403 00:28:30,800 --> 00:28:33,886 It doesn't really matter. 404 00:28:33,886 --> 00:28:36,353 AUDIENCE: If the characteristic volume of interaction 405 00:28:36,353 --> 00:28:37,561 is much smaller than volume-- 406 00:28:37,561 --> 00:28:38,741 PROFESSOR: Of the space. 407 00:28:38,741 --> 00:28:41,032 AUDIENCE: [INAUDIBLE], then the integrals just converge 408 00:28:41,032 --> 00:28:41,660 to [? constant. ?] 409 00:28:41,660 --> 00:28:42,660 PROFESSOR: That's right. 410 00:28:42,660 --> 00:28:45,170 So if you want to be even more precise, 411 00:28:45,170 --> 00:28:48,610 you will get corrections when your particles are 412 00:28:48,610 --> 00:28:50,840 close to the wall and uniformity, et cetera, 413 00:28:50,840 --> 00:28:52,790 is violated. 414 00:28:52,790 --> 00:28:55,740 So there will, actually, be corrections to, say, log 415 00:28:55,740 --> 00:29:00,800 Z that are not only proportional ultimately to number volume, 416 00:29:00,800 --> 00:29:03,620 but the area of the enclosure and other [? subleading ?] 417 00:29:03,620 --> 00:29:04,270 factors. 418 00:29:04,270 --> 00:29:07,740 But in the thermodynamic limit, we ignore all of that. 419 00:29:07,740 --> 00:29:11,860 So a lot of that is resolved by this statement here. 420 00:29:23,240 --> 00:29:28,510 Again, you won't be surprised that if I were to go ahead, 421 00:29:28,510 --> 00:29:30,740 then there will be a term in this series that 422 00:29:30,740 --> 00:29:34,280 would be beta cubed over 3 factorial integral over q 423 00:29:34,280 --> 00:29:36,740 of vq, et cetera. 424 00:29:36,740 --> 00:29:43,290 And that's actually very good because as I have written 425 00:29:43,290 --> 00:29:48,850 for you currently, this expression is totally useless. 426 00:29:48,850 --> 00:29:50,580 Why is it useless? 427 00:29:50,580 --> 00:29:55,820 Because let's think back about these two oxygen 428 00:29:55,820 --> 00:29:58,130 molecules in the gas in this room 429 00:29:58,130 --> 00:30:01,700 and what the potential of interaction between them 430 00:30:01,700 --> 00:30:04,960 would look like. 431 00:30:04,960 --> 00:30:08,340 Presumably, it is a function of the relative separation. 432 00:30:08,340 --> 00:30:12,650 We can call it r or q, it doesn't matter. 433 00:30:12,650 --> 00:30:19,260 And if you bring them closer than the typical size 434 00:30:19,260 --> 00:30:24,610 of these molecules, their potential will go to infinity. 435 00:30:24,610 --> 00:30:29,440 So basically, they don't want to come close to each other. 436 00:30:29,440 --> 00:30:31,820 If you go very far away, typically you 437 00:30:31,820 --> 00:30:33,660 have the van der Waals attraction 438 00:30:33,660 --> 00:30:35,440 between the particles. 439 00:30:35,440 --> 00:30:37,280 So out here, it is attractive. 440 00:30:37,280 --> 00:30:39,820 The potential is negative and falls off, 441 00:30:39,820 --> 00:30:42,980 typically, as 1 over r to the 6 related 442 00:30:42,980 --> 00:30:46,360 to the polarizabilities of the particles. 443 00:30:46,360 --> 00:30:51,570 And these, when you come very short distances, 444 00:30:51,570 --> 00:30:54,830 the electronic clouds will overlap 445 00:30:54,830 --> 00:30:57,910 and the potential goes to infinity. 446 00:30:57,910 --> 00:31:02,310 And if you want to sort of think about numbers, typical scales 447 00:31:02,310 --> 00:31:05,120 that we have here-- let's say here, 448 00:31:05,120 --> 00:31:08,730 are presumably of the order of angstroms. 449 00:31:08,730 --> 00:31:11,900 And the typical depths of these potentials 450 00:31:11,900 --> 00:31:16,720 in units that make sense to statistical physics 451 00:31:16,720 --> 00:31:20,370 are of the order of 100 degrees Kelvin. 452 00:31:20,370 --> 00:31:26,370 So it's why typical gases liquidy 453 00:31:26,370 --> 00:31:29,020 at range of temperatures that is of the order of 100 454 00:31:29,020 --> 00:31:31,790 degrees Kelvin. 455 00:31:31,790 --> 00:31:37,020 Now, if I want to take this potential and calculate this, 456 00:31:37,020 --> 00:31:38,920 I have trouble. 457 00:31:38,920 --> 00:31:44,000 Because the integral over here will give me infinity. 458 00:31:44,000 --> 00:31:47,620 So this expression is, in general, fine. 459 00:31:47,620 --> 00:31:49,780 But whenever you have perturbation theory, 460 00:31:49,780 --> 00:31:51,670 you start to evaluate things and you 461 00:31:51,670 --> 00:31:54,430 have to see whether the correction indeed 462 00:31:54,430 --> 00:31:56,100 is sufficiently small. 463 00:31:56,100 --> 00:31:58,500 And it seems we can't make it sufficiently small. 464 00:31:58,500 --> 00:31:59,384 Yes. 465 00:31:59,384 --> 00:32:05,312 AUDIENCE: If you're modeling interatomic or intermolecular 466 00:32:05,312 --> 00:32:07,782 potential as a Lennard-Jones potential, 467 00:32:07,782 --> 00:32:11,960 isn't the [INAUDIBLE] term at the-- 468 00:32:11,960 --> 00:32:17,150 or near the origin, a phenomenological choice? 469 00:32:17,150 --> 00:32:20,210 PROFESSOR: If you are using as a formula 470 00:32:20,210 --> 00:32:22,390 for this the Lennard-Jones potential, 471 00:32:22,390 --> 00:32:24,820 people phenomenologically write something 472 00:32:24,820 --> 00:32:27,780 like 1 over r to the 12th power, et cetera. 473 00:32:27,780 --> 00:32:30,153 But I don't have to assume that. 474 00:32:30,153 --> 00:32:30,980 AUDIENCE: Right. 475 00:32:30,980 --> 00:32:33,842 Because I've seen forms of the Lennard-Jones potential 476 00:32:33,842 --> 00:32:37,500 which is exponentially decaying at the origin. 477 00:32:37,500 --> 00:32:41,890 So it reaches a finite, albeit probably high value 478 00:32:41,890 --> 00:32:42,840 at the origin. 479 00:32:42,840 --> 00:32:46,620 Wouldn't that take care of the issue? 480 00:32:46,620 --> 00:32:49,740 PROFESSOR: If you believe in that kind of potential, yes. 481 00:32:49,740 --> 00:32:52,650 So what you are saying is that if I 482 00:32:52,650 --> 00:32:55,000 make this come to a finite value, 483 00:32:55,000 --> 00:32:58,250 I will be able to find the temperature that 484 00:32:58,250 --> 00:33:01,170 is sufficiently high compared to this, 485 00:33:01,170 --> 00:33:02,610 and then things would be fine. 486 00:33:06,820 --> 00:33:09,750 If we had that, then we would have fusion right here going 487 00:33:09,750 --> 00:33:10,930 on, right? 488 00:33:10,930 --> 00:33:14,950 So clearly, there is some better truth 489 00:33:14,950 --> 00:33:18,520 to a potential that is really very, very high compared 490 00:33:18,520 --> 00:33:21,940 to temperatures that we are looking at. 491 00:33:21,940 --> 00:33:25,790 So we don't have to worry about that. 492 00:33:25,790 --> 00:33:28,256 Well, we have to worry about this issue. 493 00:33:28,256 --> 00:33:30,570 OK? 494 00:33:30,570 --> 00:33:31,930 All right. 495 00:33:31,930 --> 00:33:36,880 So you say, well, how about the following-- let's say 496 00:33:36,880 --> 00:33:39,940 that I am something like the gas in this room 497 00:33:39,940 --> 00:33:42,700 and I am sufficiently dilute. 498 00:33:42,700 --> 00:33:46,010 I'm going to forget about these terms. 499 00:33:46,010 --> 00:33:50,880 But since this potential becomes essentially large, 500 00:33:50,880 --> 00:33:55,550 I can never ignore a particular term in this series. 501 00:33:55,550 --> 00:33:59,710 And I keep going adding more and more terms in the series. 502 00:33:59,710 --> 00:34:02,660 If I sort of think back about the picture 503 00:34:02,660 --> 00:34:07,930 that I was generating before diagrammatically-- which again, 504 00:34:07,930 --> 00:34:11,780 we will clarify later on-- this term 505 00:34:11,780 --> 00:34:14,800 corresponds to taking a pair of points and a V 506 00:34:14,800 --> 00:34:17,449 that connects them. 507 00:34:17,449 --> 00:34:21,080 And this term corresponds to two V's 508 00:34:21,080 --> 00:34:24,090 going between the same pair of points. 509 00:34:24,090 --> 00:34:26,020 And you say, well, there will be terms 510 00:34:26,020 --> 00:34:30,000 that will involve three V's, four V's, and they 511 00:34:30,000 --> 00:34:35,949 will contribute in the series in a manner that I can recognize. 512 00:34:35,949 --> 00:34:40,350 So why don't I add all of those terms together? 513 00:34:40,350 --> 00:34:46,699 And if you like, we can call the resulting object something 514 00:34:46,699 --> 00:34:49,340 else. 515 00:34:49,340 --> 00:34:51,219 What is that something else? 516 00:34:51,219 --> 00:34:55,585 Well, what I have done is I have minus beta V. 517 00:34:55,585 --> 00:34:58,430 Well, all of them-- so what is this? 518 00:34:58,430 --> 00:35:03,120 It is the integral d cubed q. 519 00:35:03,120 --> 00:35:11,740 I have minus beta V plus 1/2 beta V squared. 520 00:35:11,740 --> 00:35:13,840 The next term is going to be minus 1 521 00:35:13,840 --> 00:35:18,240 over 3 factorial beta V cubed because it will come 522 00:35:18,240 --> 00:35:21,050 from the third-order term in the expansion. 523 00:35:21,050 --> 00:35:23,740 And I can go on. 524 00:35:23,740 --> 00:35:27,850 And you say, well, obviously this whole thing 525 00:35:27,850 --> 00:35:31,820 came from the expansion of e to the minus beta V 526 00:35:31,820 --> 00:35:35,960 except that I don't have the 0 order term, 1. 527 00:35:35,960 --> 00:35:44,070 So this quantity that I will call f of q rather than V of q 528 00:35:44,070 --> 00:35:49,740 is obtained as e to the minus beta V of q minus 1. 529 00:35:56,220 --> 00:36:00,420 So if I were to, in fact, add up an infinite number 530 00:36:00,420 --> 00:36:03,500 of terms in the series rather than having 531 00:36:03,500 --> 00:36:07,770 to integrate V V squared V cubed, each one of them 532 00:36:07,770 --> 00:36:09,390 is divergence. 533 00:36:09,390 --> 00:36:12,600 I have to integrate this f of q. 534 00:36:12,600 --> 00:36:17,980 So let's see what this f of q-- or correspondingly, f of r 535 00:36:17,980 --> 00:36:19,670 looks like as a function of r. 536 00:36:22,910 --> 00:36:28,970 So in the range that I have my hard core 537 00:36:28,970 --> 00:36:35,650 and V is large and positive, this is going to give me 0. 538 00:36:35,650 --> 00:36:41,898 So the function is minus 1 down here. 539 00:36:41,898 --> 00:36:49,480 At exactly whatever point it is that the potential is 0, 540 00:36:49,480 --> 00:36:51,200 then f is also 0. 541 00:36:51,200 --> 00:36:54,505 So basically, I will come to the same 0. 542 00:36:57,180 --> 00:37:03,570 When the potential is negative, this will be larger than 1. 543 00:37:03,570 --> 00:37:05,530 My f will be positive. 544 00:37:05,530 --> 00:37:07,340 So presumably, it will have some kind 545 00:37:07,340 --> 00:37:11,400 of a peak around where this peak is. 546 00:37:11,400 --> 00:37:15,690 And then at large distances, it goes back towards 0. 547 00:37:15,690 --> 00:37:21,065 So this is going to go back towards 1 minus 1, which is 0. 548 00:37:21,065 --> 00:37:23,100 So actually, this end of the potential 549 00:37:23,100 --> 00:37:25,243 is the same thing as minus beta. 550 00:37:30,800 --> 00:37:36,330 And whereas I couldn't integrate any one term in this series, 551 00:37:36,330 --> 00:37:40,460 I can certainly integrate the sum over all positions 552 00:37:40,460 --> 00:37:43,320 and it will give me a number, which 553 00:37:43,320 --> 00:37:45,500 will depend on temperature, the properties 554 00:37:45,500 --> 00:37:47,790 of the potential, et cetera. 555 00:37:47,790 --> 00:37:51,080 And it is that number that will tell me 556 00:37:51,080 --> 00:37:54,660 what the density squared term in the expression for the pressure 557 00:37:54,660 --> 00:37:59,735 is and what the second Virial coefficient is. 558 00:37:59,735 --> 00:38:00,235 Yes. 559 00:38:00,235 --> 00:38:03,894 AUDIENCE: Is this basically an excluded volume of sorts? 560 00:38:03,894 --> 00:38:04,560 PROFESSOR: Yeah. 561 00:38:04,560 --> 00:38:06,870 So this part if I sent it to infinity, 562 00:38:06,870 --> 00:38:09,040 corresponds to an excluded volume. 563 00:38:09,040 --> 00:38:12,670 So if you want really excluded volume, this would be minus 1 564 00:38:12,670 --> 00:38:14,309 up to some point. 565 00:38:14,309 --> 00:38:15,225 AUDIENCE: [INAUDIBLE]. 566 00:38:15,225 --> 00:38:15,891 PROFESSOR: Yeah. 567 00:38:15,891 --> 00:38:18,800 But I'm talking about very general, 568 00:38:18,800 --> 00:38:21,730 so I would say that excluded volume can 569 00:38:21,730 --> 00:38:24,095 be very easily captured within this [INAUDIBLE]. 570 00:38:31,720 --> 00:38:34,400 All right. 571 00:38:34,400 --> 00:38:40,340 So we want to sort of follow-up from here. 572 00:38:40,340 --> 00:38:42,370 But since you will have a problem set, 573 00:38:42,370 --> 00:38:46,150 I will mention one other thing. 574 00:38:46,150 --> 00:38:50,640 That this reorganization of the series that I made here 575 00:38:50,640 --> 00:38:54,220 is appropriate to the limit of low densities, where 576 00:38:54,220 --> 00:38:58,950 I would have a nice expansion in powers of density. 577 00:38:58,950 --> 00:39:01,100 The problem that you will deal with 578 00:39:01,100 --> 00:39:04,370 has to do with plasmas, where the interaction 579 00:39:04,370 --> 00:39:06,350 range is very large. 580 00:39:06,350 --> 00:39:09,710 And you already saw something along those lines 581 00:39:09,710 --> 00:39:12,350 when we had the Vlasov equation as opposed 582 00:39:12,350 --> 00:39:13,900 to Boltzmann equation. 583 00:39:13,900 --> 00:39:16,450 There was a regime where you had to reorganize 584 00:39:16,450 --> 00:39:17,970 the series different ways. 585 00:39:17,970 --> 00:39:21,720 In that case, it was the BBGKY hierarchy 586 00:39:21,720 --> 00:39:25,205 whether you were looking at the dense limit or a dilute limit. 587 00:39:25,205 --> 00:39:28,890 So this is the analog of where the Boltzmann equation would 588 00:39:28,890 --> 00:39:30,640 have been inappropriate. 589 00:39:30,640 --> 00:39:32,550 The analog of the regime where you 590 00:39:32,550 --> 00:39:43,020 are dense and something like the Vlasov equation 591 00:39:43,020 --> 00:39:44,550 would be appropriate. 592 00:39:44,550 --> 00:39:47,150 So there is some kind of interaction range, 593 00:39:47,150 --> 00:39:50,770 n d cubed if it is much larger than 1. 594 00:39:50,770 --> 00:39:55,100 Then it turns out that rather than looking at diagrams that 595 00:39:55,100 --> 00:39:58,630 have the fewest number of points-- in this case, 2-- 596 00:39:58,630 --> 00:40:00,420 you have to look at diagrams that 597 00:40:00,420 --> 00:40:02,570 have the largest number of points. 598 00:40:02,570 --> 00:40:05,200 Because each additional point will give you 599 00:40:05,200 --> 00:40:07,960 an additional factor of N over V. 600 00:40:07,960 --> 00:40:13,140 You can see that these factors of N N minus 1 over 2 601 00:40:13,140 --> 00:40:15,810 came from the number of points that I had selected. 602 00:40:15,810 --> 00:40:17,850 So if I had to select three points, 603 00:40:17,850 --> 00:40:23,100 I would have N cubed and the corresponding V. So the more 604 00:40:23,100 --> 00:40:25,010 points that I have in my series, I 605 00:40:25,010 --> 00:40:27,540 will have more powers of the density. 606 00:40:27,540 --> 00:40:29,870 So in that case, it turns out that rather 607 00:40:29,870 --> 00:40:33,910 than looking at two points and all lines between them, 608 00:40:33,910 --> 00:40:36,040 you organize things in terms of what 609 00:40:36,040 --> 00:40:40,910 are called ring diagrams, which are 610 00:40:40,910 --> 00:40:46,330 things such as this, this, this. 611 00:40:46,330 --> 00:40:54,090 So basically, for a given number of lines, 612 00:40:54,090 --> 00:40:59,270 the most number of points is obtained by creating your ring. 613 00:40:59,270 --> 00:41:03,980 And so one of the problems that you have 614 00:41:03,980 --> 00:41:07,510 is to sort of sum these ring diagrams and see what happens. 615 00:41:13,900 --> 00:41:17,150 But it seems like what I'm telling you here 616 00:41:17,150 --> 00:41:20,100 is while we calculated order of density squared, 617 00:41:20,100 --> 00:41:23,370 but maybe I want to calculate order of density cubed. 618 00:41:23,370 --> 00:41:27,180 And it makes much more sense rather than when 619 00:41:27,180 --> 00:41:30,910 faced with potentials of this form arranging 620 00:41:30,910 --> 00:41:34,310 the series in powers of the potential to arrange it 621 00:41:34,310 --> 00:41:40,490 in powers of this quantity e to the minus beta V minus 1. 622 00:41:40,490 --> 00:41:45,640 So let's go through a route in which I directly 623 00:41:45,640 --> 00:41:49,850 expand everything in terms of this quantity. 624 00:41:49,850 --> 00:41:55,440 And that's the second type of expansion 625 00:41:55,440 --> 00:41:58,596 that I will call the cluster expansion. 626 00:42:05,750 --> 00:42:09,470 So once more, what we want to calculate 627 00:42:09,470 --> 00:42:11,110 is the partition function. 628 00:42:11,110 --> 00:42:14,030 It depends on temperature, volume, number 629 00:42:14,030 --> 00:42:19,200 of particles obtained by integrating over 630 00:42:19,200 --> 00:42:21,580 all degrees of freedom. 631 00:42:21,580 --> 00:42:25,190 The integration over momenta we saw is very simple. 632 00:42:25,190 --> 00:42:30,710 Ultimately, will give us this N factorial lambda to the 3N. 633 00:42:30,710 --> 00:42:33,530 This time, I won't pull out the factor of V 634 00:42:33,530 --> 00:42:36,380 to the N at this point. 635 00:42:36,380 --> 00:42:43,130 And then I have the integration over all of these q's. 636 00:42:43,130 --> 00:42:46,180 I did not divide by V because I did not 637 00:42:46,180 --> 00:42:51,030 multiply by V to the N. OK. 638 00:42:51,030 --> 00:42:54,170 And then I have e to the minus beta U, 639 00:42:54,170 --> 00:43:00,190 but this is my U. U is the sum of various terms. 640 00:43:00,190 --> 00:43:06,630 So e to the minus beta U will be a product over all, 641 00:43:06,630 --> 00:43:15,068 let's say-- let's call this k pairs ij. 642 00:43:15,068 --> 00:43:20,480 e to the minus beta V of qi minus qj. 643 00:43:23,380 --> 00:43:27,020 So basically, the only thing that I did 644 00:43:27,020 --> 00:43:29,860 was e to the minus beta-- this quantity 645 00:43:29,860 --> 00:43:32,220 I wrote as a product of contributions 646 00:43:32,220 --> 00:43:33,400 from the different pairs. 647 00:43:38,490 --> 00:43:42,810 Now, this quantity I have it in the line above. 648 00:43:42,810 --> 00:43:45,560 This is also a relative position. 649 00:43:45,560 --> 00:43:50,650 Clearly, I can think of this as 1 650 00:43:50,650 --> 00:43:55,420 plus the f that would correspond to the distance between qi 651 00:43:55,420 --> 00:43:57,310 and qj. 652 00:43:57,310 --> 00:44:02,510 And again, I can either write it as Vf of qi minus qj, 653 00:44:02,510 --> 00:44:07,630 or simplify of i-- my notation-- and write it as f i j. 654 00:44:07,630 --> 00:44:13,900 So f i j means-- so let me maybe write it here. 655 00:44:13,900 --> 00:44:18,020 f i j I have defined to be e to the minus beta 656 00:44:18,020 --> 00:44:21,518 V of qi minus qj minus 1. 657 00:44:28,190 --> 00:44:31,210 So what do I expect? 658 00:44:31,210 --> 00:44:33,760 I saw that the first correction to something 659 00:44:33,760 --> 00:44:39,150 that was interesting to me had one power of f in it 660 00:44:39,150 --> 00:44:41,480 that I had to ultimately integrate. 661 00:44:41,480 --> 00:44:46,410 So maybe what I should do is I should start organizing things 662 00:44:46,410 --> 00:44:49,520 in terms of how many f's I have. 663 00:44:49,520 --> 00:44:52,970 So an expansion in powers of f. 664 00:44:52,970 --> 00:44:54,610 So what's going to happen here? 665 00:44:54,610 --> 00:44:59,100 I would have 1 over N factorial lambda to the 3N. 666 00:44:59,100 --> 00:45:05,460 I have a product of all of the integrations. 667 00:45:05,460 --> 00:45:10,010 And I have all of these factors of 1 plus f1 2 times 1. 668 00:45:10,010 --> 00:45:15,240 So basically, this is maybe-- it's really a product of 1 669 00:45:15,240 --> 00:45:20,460 plus f1 2, 1 plus f1 3, 1 plus all of one of these things. 670 00:45:20,460 --> 00:45:23,690 So the thing that has least number of f's is 671 00:45:23,690 --> 00:45:26,570 when I pick the 1 from all of these brackets 672 00:45:26,570 --> 00:45:29,240 that are multiplying each other. 673 00:45:29,240 --> 00:45:37,690 The next term is to pick 1f from one pair 674 00:45:37,690 --> 00:45:40,580 and all the others would be 1. 675 00:45:40,580 --> 00:45:49,840 The next term would be sum over ij kl f i j f k l, 676 00:45:49,840 --> 00:45:53,270 and then there will be diagrams that would progressively 677 00:45:53,270 --> 00:45:55,790 have more and more factors of f. 678 00:46:05,670 --> 00:46:09,350 Now, what I will do is to represent the various terms 679 00:46:09,350 --> 00:46:11,546 that I generate in this series diagrammatically. 680 00:46:25,440 --> 00:46:30,790 So first of all, I have to integrate over N points. 681 00:46:30,790 --> 00:46:43,925 So I put points 1 through N. So I have 1, 2, 3, 4. 682 00:46:43,925 --> 00:46:48,980 It doesn't matter how I put them-- N. 683 00:46:48,980 --> 00:46:53,840 And then, this 1 would correspond 684 00:46:53,840 --> 00:46:56,970 to just this diagram. 685 00:46:56,970 --> 00:47:03,330 The next thing is I put a line for f i j. 686 00:47:03,330 --> 00:47:05,700 So let's say that I picked in this series 687 00:47:05,700 --> 00:47:07,400 the term that was f 2 3. 688 00:47:07,400 --> 00:47:13,180 I will represent that by a line that goes between 2 and 3. 689 00:47:13,180 --> 00:47:16,930 Later on in the picture, maybe I will 690 00:47:16,930 --> 00:47:21,580 pick a term that is f 2 3 f 4 5. 691 00:47:21,580 --> 00:47:26,130 So some second-order term could be something like this. 692 00:47:26,130 --> 00:47:28,510 Maybe later on in the series, I will 693 00:47:28,510 --> 00:47:31,120 pick a term that is connecting f3 and f4. 694 00:47:31,120 --> 00:47:34,760 So this would be a third-order term in the series. 695 00:47:34,760 --> 00:47:40,540 Maybe later on the series, I have some other pair over here. 696 00:47:40,540 --> 00:47:44,870 So any one of the huge number of terms 697 00:47:44,870 --> 00:47:48,460 that I have generated here has one diagram 698 00:47:48,460 --> 00:47:50,020 that is appearing over here. 699 00:47:53,120 --> 00:47:56,630 Now, the next step is, of course I have to integrate over all 700 00:47:56,630 --> 00:47:57,440 of the q's. 701 00:47:57,440 --> 00:48:00,520 I mean, these diagrams, what they represent is essentially 702 00:48:00,520 --> 00:48:04,130 some f of q1 minus q3, some f of whatever. 703 00:48:04,130 --> 00:48:07,570 And I have to integrate over all of these q's. 704 00:48:07,570 --> 00:48:11,290 So the contribution to the partition function 705 00:48:11,290 --> 00:48:14,770 that I will get would be some value associated 706 00:48:14,770 --> 00:48:17,140 with one of these graphs obtained by doing 707 00:48:17,140 --> 00:48:18,420 the integrations over q. 708 00:48:21,560 --> 00:48:24,870 Now, the thing that I want you to notice 709 00:48:24,870 --> 00:48:38,760 is that contribution of one of these graphs or diagrams 710 00:48:38,760 --> 00:48:50,640 is the product of contributions of its linked clusters. 711 00:48:59,190 --> 00:49:02,910 So here I have one particular diagram 712 00:49:02,910 --> 00:49:04,350 that is represented here. 713 00:49:04,350 --> 00:49:10,390 Well, let's say 2, 3, 4, 5 are linked together 714 00:49:10,390 --> 00:49:12,460 and separate from whatever this is. 715 00:49:12,460 --> 00:49:13,950 Let's say this is 7, 9. 716 00:49:16,640 --> 00:49:21,520 So when I do the integrations over q2, q3, q4, q5, 717 00:49:21,520 --> 00:49:25,530 I don't really rely on any of the other things. 718 00:49:25,530 --> 00:49:28,120 So my integration here, we break off 719 00:49:28,120 --> 00:49:29,760 into the integration that involves 720 00:49:29,760 --> 00:49:32,690 this, the integration that involves this, as well 721 00:49:32,690 --> 00:49:35,060 as integrations over all of these points 722 00:49:35,060 --> 00:49:36,650 that are not connected to anybody. 723 00:49:36,650 --> 00:49:41,520 All of those will simply give me a factor of v. 724 00:49:41,520 --> 00:49:45,110 So is everybody happy with this simple statement 725 00:49:45,110 --> 00:49:49,070 that the contribution-- if I think of these linked clusters 726 00:49:49,070 --> 00:49:52,940 as some collection of points that are linked together, 727 00:49:52,940 --> 00:49:55,100 the value that I would get for this diagram 728 00:49:55,100 --> 00:49:58,365 would be the contribution-- product of the contributions 729 00:49:58,365 --> 00:49:59,490 that I would have for this. 730 00:50:04,060 --> 00:50:05,590 So yes? 731 00:50:05,590 --> 00:50:07,417 AUDIENCE: Can you explain [INAUDIBLE]? 732 00:50:07,417 --> 00:50:08,000 PROFESSOR: OK. 733 00:50:08,000 --> 00:50:13,060 So let's pick a particular thing. 734 00:50:13,060 --> 00:50:16,510 Let's say we do something like this, 735 00:50:16,510 --> 00:50:25,795 where this is number 7, 245, 6, 5, 4. 736 00:50:25,795 --> 00:50:26,920 This is a particular thing. 737 00:50:26,920 --> 00:50:28,230 So what do I have to do? 738 00:50:28,230 --> 00:50:33,020 There's also a whole bunch of points-- 1, 2, 3. 739 00:50:33,020 --> 00:50:36,980 What I am instructed to do is to integrate over 740 00:50:36,980 --> 00:50:44,490 q1, q2, q3, et cetera. 741 00:50:44,490 --> 00:50:49,240 Now, the integral over q1 is just the integral of q1. 742 00:50:49,240 --> 00:50:52,110 So this integral by itself will give me 743 00:50:52,110 --> 00:50:57,190 V. Any number of points that are by themselves 744 00:50:57,190 --> 00:51:02,950 will also give me V. So the first term 745 00:51:02,950 --> 00:51:04,870 that becomes nontrivial is when I 746 00:51:04,870 --> 00:51:23,020 have to do the integration over q4, q5, q6 of f 4 5, f 5 6, 747 00:51:23,020 --> 00:51:24,010 f 6 4. 748 00:51:26,990 --> 00:51:31,350 I don't know what this is, but there 749 00:51:31,350 --> 00:51:33,960 is something that will come from here. 750 00:51:33,960 --> 00:51:37,570 Then, later on I have to integrate over 7 751 00:51:37,570 --> 00:51:39,310 and something else. 752 00:51:39,310 --> 00:51:44,652 So then I have the integral over q7, q-- let's call this 8. 753 00:51:49,380 --> 00:51:50,760 f of 7 8. 754 00:51:50,760 --> 00:51:53,910 So this is something else. 755 00:51:53,910 --> 00:51:59,460 So the overall value of this term in the perturbation theory 756 00:51:59,460 --> 00:52:02,800 is the product of contributions from a huge number of one 757 00:52:02,800 --> 00:52:04,690 clusters. 758 00:52:04,690 --> 00:52:06,070 Here, I have a two cluster. 759 00:52:06,070 --> 00:52:07,740 Here, I have a three cluster. 760 00:52:07,740 --> 00:52:09,360 Maybe I will have more things. 761 00:52:13,410 --> 00:52:20,140 Also, notice that if I have more of these two clusters, 762 00:52:20,140 --> 00:52:24,350 the result for them would be exactly the same. 763 00:52:24,350 --> 00:52:27,550 So if I have lots of these pairs, the same way that I 764 00:52:27,550 --> 00:52:30,080 had lots of single points-- and it became V 765 00:52:30,080 --> 00:52:32,020 to the number of single points-- it 766 00:52:32,020 --> 00:52:34,820 will become whatever that y is to the number of pair 767 00:52:34,820 --> 00:52:38,530 points, x, whatever that is, to the number of triplets 768 00:52:38,530 --> 00:52:40,680 that I have in triangles. 769 00:52:40,680 --> 00:52:44,970 So this is how this object is built up. 770 00:52:44,970 --> 00:52:45,680 Yes? 771 00:52:45,680 --> 00:52:48,580 AUDIENCE: How are we deciding how many triplets 772 00:52:48,580 --> 00:52:51,226 and how many pairs we actually have? 773 00:52:51,226 --> 00:52:53,100 PROFESSOR: At this point, I haven't told you. 774 00:52:53,100 --> 00:52:55,940 So there is a multiplicity that we have to calculate. 775 00:52:55,940 --> 00:52:57,500 Yes. 776 00:52:57,500 --> 00:53:00,080 So at this point, all I'm saying is given 777 00:53:00,080 --> 00:53:03,099 this, the answer is the product of contribution. 778 00:53:03,099 --> 00:53:05,015 There is a multiplicity factor-- you're right. 779 00:53:05,015 --> 00:53:07,185 We have to calculate that. 780 00:53:09,980 --> 00:53:11,780 OK? 781 00:53:11,780 --> 00:53:14,715 Anything else? 782 00:53:14,715 --> 00:53:16,040 All right. 783 00:53:16,040 --> 00:53:21,300 So I mean, think of this as taking a number 784 00:53:21,300 --> 00:53:24,850 and writing it as the product of its prime factors, 785 00:53:24,850 --> 00:53:27,500 factorizing into primes. 786 00:53:27,500 --> 00:53:31,770 So you sort of immediately know that somehow prime factors, 787 00:53:31,770 --> 00:53:34,730 the prime numbers, are more important. 788 00:53:34,730 --> 00:53:37,920 Because everything else you can write as prime numbers. 789 00:53:37,920 --> 00:53:40,530 So clearly, what is also buried in heart 790 00:53:40,530 --> 00:53:46,610 of calculating this partition function is these clusters. 791 00:53:46,610 --> 00:53:56,950 So let me define the analog of prime numbers as follows. 792 00:53:56,950 --> 00:54:08,870 I define bl to be the sum over contributions 793 00:54:08,870 --> 00:54:17,430 of all linked clusters of l-points. 794 00:54:39,350 --> 00:54:42,050 So let's go term by term. 795 00:54:42,050 --> 00:54:46,480 b1 corresponds essentially to the one point 796 00:54:46,480 --> 00:54:50,460 by itself in the diagram that I was drawing down here. 797 00:54:50,460 --> 00:54:53,970 And corresponds to integrating over 798 00:54:53,970 --> 00:54:56,890 the coordinate that goes all over the space. 799 00:54:56,890 --> 00:55:03,330 And hence, the volume is the same thing as the volume V. 800 00:55:03,330 --> 00:55:08,810 b2 is a cluster of two points. 801 00:55:08,810 --> 00:55:11,160 So it is this. 802 00:55:11,160 --> 00:55:16,300 And so it is the integral over q1 and q2, 803 00:55:16,300 --> 00:55:24,910 which are two endpoints of this, of f of q1 minus q2. 804 00:55:24,910 --> 00:55:27,430 And this we have seen already many times. 805 00:55:27,430 --> 00:55:33,550 It's the same thing as volume, the integral over q f of q. 806 00:55:33,550 --> 00:55:36,270 It is the thing that was related to our second Virial 807 00:55:36,270 --> 00:55:36,770 coefficient. 808 00:55:40,040 --> 00:55:43,600 Now, this is very important for b3. 809 00:55:46,640 --> 00:55:52,040 I note that I underlined here "all." 810 00:55:52,040 --> 00:55:54,580 And this is for later convenience. 811 00:55:54,580 --> 00:55:57,610 It seems that I'm essentially pushing complexity 812 00:55:57,610 --> 00:56:00,080 from one place to another. 813 00:56:00,080 --> 00:56:03,090 So there are a number of diagrams 814 00:56:03,090 --> 00:56:08,130 that have three things linked together. 815 00:56:08,130 --> 00:56:09,440 This is one of them. 816 00:56:09,440 --> 00:56:14,380 Let's say, think of 1, 2, 3 connected to each other. 817 00:56:14,380 --> 00:56:16,920 But then, I have diagrams. 818 00:56:16,920 --> 00:56:19,380 At this stage, I don't really care 819 00:56:19,380 --> 00:56:22,430 that this is one particle irreducible. 820 00:56:22,430 --> 00:56:26,800 Here, I have no constraints on one particle irreducibility. 821 00:56:26,800 --> 00:56:29,710 And in fact, this comes in three varieties 822 00:56:29,710 --> 00:56:33,640 because the bond that is missing can be one of three. 823 00:56:33,640 --> 00:56:38,040 So once I pick the triplets of three points, sum 824 00:56:38,040 --> 00:56:42,360 over all clusters that involve these three points is this, 825 00:56:42,360 --> 00:56:45,560 in particular with this one, appearing three times. 826 00:56:45,560 --> 00:56:46,283 Yes. 827 00:56:46,283 --> 00:56:48,116 AUDIENCE: Do we still count all the diagrams 828 00:56:48,116 --> 00:56:49,990 when particles are identical? 829 00:56:49,990 --> 00:56:51,220 PROFESSOR: Yes. 830 00:56:51,220 --> 00:56:56,230 Because here, at this stage, there 831 00:56:56,230 --> 00:56:57,830 is an N factorial out here. 832 00:56:57,830 --> 00:57:00,410 All I'm really doing is transcribing 833 00:57:00,410 --> 00:57:02,700 a mathematical formula. 834 00:57:02,700 --> 00:57:04,600 The mathematical formula says you 835 00:57:04,600 --> 00:57:06,840 have to do a bunch of integrations. 836 00:57:06,840 --> 00:57:09,580 And I am just following translating 837 00:57:09,580 --> 00:57:11,730 that mathematical formula to diagrams. 838 00:57:11,730 --> 00:57:13,600 At this stage, we forget. 839 00:57:13,600 --> 00:57:17,510 This has nothing to do with this N factorial or identity. 840 00:57:17,510 --> 00:57:24,990 It's just a transcription of the mathematics 841 00:57:24,990 --> 00:57:31,710 So at this stage, don't try to correct my conceptual parts 842 00:57:31,710 --> 00:57:37,090 but try to make sure that my mathematical steps are correct. 843 00:57:37,090 --> 00:57:39,115 OK? 844 00:57:39,115 --> 00:57:39,615 Yes. 845 00:57:39,615 --> 00:57:41,070 AUDIENCE: I didn't understand what 846 00:57:41,070 --> 00:57:43,690 that missing links represents. 847 00:57:43,690 --> 00:57:45,040 PROFESSOR: OK. 848 00:57:45,040 --> 00:57:53,340 So at the end of this story, what I want to write down 849 00:57:53,340 --> 00:57:57,390 is that the value of some particular term in this series 850 00:57:57,390 --> 00:58:00,610 is related to product of contributions. 851 00:58:00,610 --> 00:58:03,360 I described to you how productive contributions 852 00:58:03,360 --> 00:58:05,450 comes along. 853 00:58:05,450 --> 00:58:08,880 And each contribution is a bunch of points 854 00:58:08,880 --> 00:58:12,400 that are connected together. 855 00:58:12,400 --> 00:58:16,360 So what I want to do, in order to make my algebra easier 856 00:58:16,360 --> 00:58:19,360 later on, is to say that-- let's say 857 00:58:19,360 --> 00:58:22,760 these three points are part of a cluster. 858 00:58:22,760 --> 00:58:25,320 They are connected together somehow. 859 00:58:25,320 --> 00:58:28,950 So they will be giving me some factor. 860 00:58:28,950 --> 00:58:32,030 Now, they are connected, if they are connected like this, 861 00:58:32,030 --> 00:58:35,360 like this, like this, like that. 862 00:58:35,360 --> 00:58:38,080 So basically, any one of these is 863 00:58:38,080 --> 00:58:41,690 a way of connecting these three points so that they 864 00:58:41,690 --> 00:58:43,250 will make a contribution together. 865 00:59:00,960 --> 00:59:01,510 Yes. 866 00:59:01,510 --> 00:59:03,764 AUDIENCE: So the one with all three bonds 867 00:59:03,764 --> 00:59:05,252 comes with a different [INAUDIBLE] 868 00:59:05,252 --> 00:59:08,200 in the series than the ones with only two bonds, right? 869 00:59:08,200 --> 00:59:09,030 PROFESSOR: Yes. 870 00:59:09,030 --> 00:59:11,020 Now, there is reason, ultimately, 871 00:59:11,020 --> 00:59:15,300 why I want to group all of them together. 872 00:59:15,300 --> 00:59:17,760 Because there was a question before about 873 00:59:17,760 --> 00:59:19,980 multiplicity factors. 874 00:59:19,980 --> 00:59:23,190 I can write down a closed form, nice expression 875 00:59:23,190 --> 00:59:27,320 for the multiplicity factor if I group them together. 876 00:59:27,320 --> 00:59:30,660 Otherwise, I have to also worry that there are three of these 877 00:59:30,660 --> 00:59:32,650 and there is one of them, et cetera. 878 00:59:32,650 --> 00:59:36,220 There is an additional layer of multiplicity. 879 00:59:36,220 --> 00:59:40,140 And I want to separate those two layers of multiplicity. 880 00:59:40,140 --> 00:59:46,140 So here, the thing that I have is I have these endpoints. 881 00:59:46,140 --> 00:59:49,870 I want to say that the eventual result for endpoints 882 00:59:49,870 --> 00:59:55,630 comes from clusters that involves 1's, clusters that 883 00:59:55,630 --> 01:00:01,350 involve pairs, and cluster that involve triplets. 884 01:00:01,350 --> 01:00:05,010 Now, I realize suddenly that if I made the multiplicity count 885 01:00:05,010 --> 01:00:10,140 here, then for this triplet, I could have put all of them 886 01:00:10,140 --> 01:00:12,440 together like this or like that. 887 01:00:12,440 --> 01:00:14,190 And the next order of business when 888 01:00:14,190 --> 01:00:17,330 I am thinking about four objects, 889 01:00:17,330 --> 01:00:21,440 I can connect them together into a cluster in multiple ways. 890 01:00:21,440 --> 01:00:25,350 And there is a separate way of calculating 891 01:00:25,350 --> 01:00:27,370 the relative multiplicity of what 892 01:00:27,370 --> 01:00:31,650 comes outside and the multiplicity of this quartic 893 01:00:31,650 --> 01:00:33,040 with respect to entirety. 894 01:00:36,690 --> 01:00:39,350 So you have to think about this a little bit. 895 01:00:39,350 --> 01:00:44,240 So I'll proceed with this. 896 01:00:44,240 --> 01:00:53,640 So this mathematically would be the integral q1, q2, q3, f 897 01:00:53,640 --> 01:01:05,310 1 2, f 2 3, f 3 1 plus, say, f 1 2, f 2 3, plus f 1 3 f 3 2 898 01:01:05,310 --> 01:01:11,250 plus f 2 3, f 3 1, something like this. 899 01:01:14,080 --> 01:01:17,230 Well, 1, 2. 900 01:01:17,230 --> 01:01:17,945 2 is repeated. 901 01:01:17,945 --> 01:01:18,750 3 is repeated. 902 01:01:18,750 --> 01:01:20,026 I should repeat 1. 903 01:01:24,130 --> 01:01:24,780 OK. 904 01:01:24,780 --> 01:01:26,960 Of course, the last three are the same thing, 905 01:01:26,960 --> 01:01:29,630 but this is the expression for this. 906 01:01:29,630 --> 01:01:34,340 Now, before would be a huge complex of things. 907 01:01:34,340 --> 01:01:37,060 It would have a diagram that is of this form. 908 01:01:37,060 --> 01:01:40,720 It would have a number of diagrams that are of this form. 909 01:01:40,720 --> 01:01:44,540 It would have diagrams that are of this form, diagrams 910 01:01:44,540 --> 01:01:49,250 that are like this, and so forth. 911 01:01:49,250 --> 01:01:53,600 Basically, even within the choice 912 01:01:53,600 --> 01:01:55,470 of things that have four clusters, 913 01:01:55,470 --> 01:01:57,760 there is a huge number when you go 914 01:01:57,760 --> 01:01:59,975 to further and further [? down. ?] 915 01:02:12,870 --> 01:02:20,340 OK, so maybe this statement will now clarify things. 916 01:02:20,340 --> 01:02:26,510 So what I have is that the partition function 917 01:02:26,510 --> 01:02:30,650 is 1 over N factorial lambda to the 3N. 918 01:02:33,380 --> 01:02:42,440 And then it is all of the terms that 919 01:02:42,440 --> 01:02:44,890 are obtained by summing this series. 920 01:02:47,450 --> 01:02:54,170 And so I have to look at all possible ways 921 01:02:54,170 --> 01:02:59,235 that I can break N-points into these clusters. 922 01:03:01,880 --> 01:03:14,330 Suppose I created a situation where I have clusters of size l 923 01:03:14,330 --> 01:03:16,100 and then I have nl of them. 924 01:03:19,120 --> 01:03:26,120 So I saw that the contribution that I would get from the 1 925 01:03:26,120 --> 01:03:28,430 clusters was essentially the product 926 01:03:28,430 --> 01:03:29,650 of the number of 1 clusters. 927 01:03:29,650 --> 01:03:31,830 It was V to the number of points that 928 01:03:31,830 --> 01:03:34,250 are not connected together. 929 01:03:34,250 --> 01:03:38,290 So I have to take b1, raise it to the power of N1, 930 01:03:38,290 --> 01:03:41,040 which is the number of 1 cluster. 931 01:03:41,040 --> 01:03:47,240 Then, I have to do the same thing for b2, b3, et cetera. 932 01:03:47,240 --> 01:03:50,160 And then I have to make sure that I 933 01:03:50,160 --> 01:03:54,580 have looked at all ways of partitioning 934 01:03:54,580 --> 01:03:56,860 N the numbers into these clusters. 935 01:03:56,860 --> 01:04:02,470 So I have a sum over nl l has to add up to n. 936 01:04:02,470 --> 01:04:06,130 And I have to sum over all nl's that are 937 01:04:06,130 --> 01:04:09,060 consistent with this constraint. 938 01:04:11,650 --> 01:04:15,750 But then there is this issue of the multiplicity. 939 01:04:15,750 --> 01:04:21,166 Given that I chose some articular set of these, 940 01:04:21,166 --> 01:04:26,930 if I were to reorganize the numbers, 941 01:04:26,930 --> 01:04:29,130 I would get the same contribution. 942 01:04:29,130 --> 01:04:32,210 So let's say we pick this diagram that 943 01:04:32,210 --> 01:04:34,320 has precisely a contribution that 944 01:04:34,320 --> 01:04:39,580 is V, which is b1 to the number of these one points. 945 01:04:39,580 --> 01:04:43,520 This to the number of b2 to the power 946 01:04:43,520 --> 01:04:46,290 of the number of these, this, et cetera. 947 01:04:46,290 --> 01:04:50,100 But then I can take these labels-- 1, 2, 3, 4, 5, 948 01:04:50,100 --> 01:04:54,270 6-- that I assigned to these things and permute them. 949 01:04:54,270 --> 01:04:58,510 If I permute them, I will get exactly the same contribution. 950 01:04:58,510 --> 01:05:03,310 So there is a large number of diagrams 951 01:05:03,310 --> 01:05:05,990 that have exactly this same contribution because 952 01:05:05,990 --> 01:05:06,990 of this permutation. 953 01:05:06,990 --> 01:05:09,230 There was a question back there. 954 01:05:09,230 --> 01:05:11,086 AUDIENCE: You've already answered it. 955 01:05:15,165 --> 01:05:16,040 PROFESSOR: All right. 956 01:05:19,120 --> 01:05:21,060 Actually, this is an interesting thing-- 957 01:05:21,060 --> 01:05:23,170 I don't know how many of you recognize this. 958 01:05:23,170 --> 01:05:27,840 So essentially, I have to take a huge number, N, 959 01:05:27,840 --> 01:05:31,210 and break it into a number of 1 clusters, 2 clusters, et 960 01:05:31,210 --> 01:05:32,460 cetera. 961 01:05:32,460 --> 01:05:34,360 The number of ways of doing that is 962 01:05:34,360 --> 01:05:36,700 called a partition of an integer. 963 01:05:36,700 --> 01:05:41,520 And last century, the [INAUDIBLE] 964 01:05:41,520 --> 01:05:43,370 calculated what that number is. 965 01:05:43,370 --> 01:05:47,160 So there is a [INAUDIBLE] theorem associated with that. 966 01:05:47,160 --> 01:05:48,740 And actually, later in the course 967 01:05:48,740 --> 01:05:50,930 I will give you a problem to calculate 968 01:05:50,930 --> 01:05:53,445 the asymptotic version of the [INAUDIBLE]. 969 01:05:57,160 --> 01:05:58,390 But this is different story. 970 01:05:58,390 --> 01:06:01,830 So given that you had made one of these partitions 971 01:06:01,830 --> 01:06:05,950 of this integer, what is this degeneracy factor? 972 01:06:05,950 --> 01:06:09,090 So let me tell you what this degeneracy factor is. 973 01:06:09,090 --> 01:06:13,960 So given this choice of nl's, you 974 01:06:13,960 --> 01:06:18,550 say-- well, the first thing is what I told you. 975 01:06:18,550 --> 01:06:21,690 For a particular graph, the value 976 01:06:21,690 --> 01:06:26,320 is independent of how these numbers were assigned. 977 01:06:26,320 --> 01:06:29,540 So I will permute those numbers in all possible ways 978 01:06:29,540 --> 01:06:31,630 and I will get the same thing. 979 01:06:31,630 --> 01:06:33,272 So that will give me N factorial. 980 01:06:33,272 --> 01:06:34,605 It's the number of permutations. 981 01:06:37,130 --> 01:06:41,860 But then, I have over-counted things because within, 982 01:06:41,860 --> 01:06:45,850 let's say, a 2 cluster-- if I count 7, 8, 8, 983 01:06:45,850 --> 01:06:47,370 7 they are the same thing. 984 01:06:47,370 --> 01:06:50,520 And by multiplying by this N factorial, 985 01:06:50,520 --> 01:06:53,320 I have over-counted that. 986 01:06:53,320 --> 01:06:58,990 So I have to divide by 2 for every one of my 2 clusters. 987 01:06:58,990 --> 01:07:04,750 So I have to divide by 2 to the power of N2. 988 01:07:04,750 --> 01:07:07,700 For the 3 clusters, I have the permutation of everything 989 01:07:07,700 --> 01:07:08,330 that is inside. 990 01:07:08,330 --> 01:07:10,930 So it is 3 factor here. 991 01:07:10,930 --> 01:07:14,050 So what I have here is I have to divide 992 01:07:14,050 --> 01:07:21,020 by the over-counting, which is l factorial-- labels 993 01:07:21,020 --> 01:07:24,740 within a cluster-- to the number of clusters 994 01:07:24,740 --> 01:07:27,790 that I have that are subject to this. 995 01:07:30,830 --> 01:07:40,195 There is another thing that this pair is 100, 101 and this pair 996 01:07:40,195 --> 01:07:42,280 is 7, 8. 997 01:07:42,280 --> 01:07:46,310 Exchanging that pair of numbers with that pair of numbers 998 01:07:46,310 --> 01:07:50,960 is also part of the symmetries that I have now over-counted. 999 01:07:50,960 --> 01:07:54,210 So that has to be taken into account 1000 01:07:54,210 --> 01:07:59,630 because the number of these 2 clusters here is 2. 1001 01:07:59,630 --> 01:08:00,790 Here is actually 3. 1002 01:08:00,790 --> 01:08:03,050 I have to divide by 3 factorial. 1003 01:08:03,050 --> 01:08:06,406 In general, I have to divide by nl factorial. 1004 01:08:10,300 --> 01:08:11,490 OK. 1005 01:08:11,490 --> 01:08:14,810 Again, this is one of those things 1006 01:08:14,810 --> 01:08:19,580 that the best advice that I can give you is to draw five 1007 01:08:19,580 --> 01:08:22,890 or six points so your n is small. 1008 01:08:22,890 --> 01:08:25,430 And draw some diagrams. 1009 01:08:25,430 --> 01:08:27,640 And convince yourself that what I told you 1010 01:08:27,640 --> 01:08:30,510 here rapidly is correct. 1011 01:08:30,510 --> 01:08:32,590 And it is correct. 1012 01:08:35,192 --> 01:08:35,691 Yes. 1013 01:08:35,691 --> 01:08:38,517 AUDIENCE: Can you just clarify which is in the [INAUDIBLE]? 1014 01:08:38,517 --> 01:08:39,100 PROFESSOR: OK. 1015 01:08:39,100 --> 01:08:41,700 So the N factorial is outside. 1016 01:08:41,700 --> 01:08:44,290 And then-- so maybe write it better. 1017 01:08:44,290 --> 01:08:49,819 For each l, I have to multiply nl factorial and l 1018 01:08:49,819 --> 01:08:52,422 factorial to the power of nl. 1019 01:09:04,020 --> 01:09:07,954 So this is my partition function in terms of these clusters. 1020 01:09:10,760 --> 01:09:13,270 Even with all of these definitions 1021 01:09:13,270 --> 01:09:18,670 that I have up there, maybe not so obvious an answer. 1022 01:09:18,670 --> 01:09:22,590 And one of the reason it is kind of an obscure answer 1023 01:09:22,590 --> 01:09:28,240 is because here I have to do a constrained sum. 1024 01:09:28,240 --> 01:09:31,870 That is, I have all of these variables. 1025 01:09:31,870 --> 01:09:38,220 Let's call them n1, n2, n3, et cetera. 1026 01:09:38,220 --> 01:09:42,920 Each one of them can go from 0, 1, 2, 3. 1027 01:09:42,920 --> 01:09:44,890 But they are all linked to each other 1028 01:09:44,890 --> 01:09:49,399 because their sum is kind of restricted by some total value 1029 01:09:49,399 --> 01:09:52,500 l, by this constraint. 1030 01:09:52,500 --> 01:09:57,530 And constrained sums are hard to do. 1031 01:09:57,530 --> 01:10:00,310 But in statistical physics, we know 1032 01:10:00,310 --> 01:10:03,170 how to gets rid of constrained sums. 1033 01:10:03,170 --> 01:10:06,290 The way that we do that is we essentially 1034 01:10:06,290 --> 01:10:09,560 allow this N to go all the way. 1035 01:10:09,560 --> 01:10:13,420 So if I say, let's make-- it's very hard for me 1036 01:10:13,420 --> 01:10:15,560 to do this n1 from 0 to something, 1037 01:10:15,560 --> 01:10:19,760 n2 from 0 to something with all of this constrained. 1038 01:10:19,760 --> 01:10:22,800 My life would be very easier if I could independently 1039 01:10:22,800 --> 01:10:26,080 have n1 go take any value, n2 take any value, 1040 01:10:26,080 --> 01:10:28,450 n3 take any value. 1041 01:10:28,450 --> 01:10:32,570 But if I do that essentially for each choice of n1, n2, n3, 1042 01:10:32,570 --> 01:10:38,010 I have shifted the value of big N. 1043 01:10:38,010 --> 01:10:39,970 But there is an ensemble that I know 1044 01:10:39,970 --> 01:10:42,650 which has possibly any value of N, 1045 01:10:42,650 --> 01:10:44,044 and that's the grand canonical. 1046 01:10:50,820 --> 01:10:54,810 So rather than looking at the partition function, 1047 01:10:54,810 --> 01:10:57,840 I say I will look at the grand partition function 1048 01:10:57,840 --> 01:11:01,920 Q that is obtained by summing over 1049 01:11:01,920 --> 01:11:06,470 all N. Can take any value from 0 to infinity. 1050 01:11:06,470 --> 01:11:12,280 I have e to the beta mu N times the partition function that 1051 01:11:12,280 --> 01:11:13,395 is for N particles. 1052 01:11:16,750 --> 01:11:21,130 So that's the definition of how you would go from the canonical 1053 01:11:21,130 --> 01:11:23,260 where you have fixed N to grand canonical 1054 01:11:23,260 --> 01:11:27,630 where you have fixed the chemical potential mu. 1055 01:11:27,630 --> 01:11:31,620 So let's apply this sum over there. 1056 01:11:31,620 --> 01:11:36,690 I have a sum over N, but I said that if I allowed 1057 01:11:36,690 --> 01:11:40,940 the nl's to vary independently, it 1058 01:11:40,940 --> 01:11:45,480 is equivalent to varying that N, recognizing 1059 01:11:45,480 --> 01:11:50,019 that this n is sum over l l nl. 1060 01:11:50,019 --> 01:11:50,935 That's the constraint. 1061 01:11:54,140 --> 01:11:55,800 So that's just the first term. 1062 01:11:55,800 --> 01:12:01,000 I have rewritten this sum that was constrained and this sum 1063 01:12:01,000 --> 01:12:05,370 over the total number as independent sums over the nl's. 1064 01:12:05,370 --> 01:12:06,640 Got rid of the constraint. 1065 01:12:09,570 --> 01:12:12,290 Now, I write W. Oh, OK. 1066 01:12:12,290 --> 01:12:13,404 The partition function. 1067 01:12:13,404 --> 01:12:14,820 Now, write the partition function. 1068 01:12:14,820 --> 01:12:18,680 I have 1 over N factorial. 1069 01:12:18,680 --> 01:12:21,240 I have lambda cubed raised to this power. 1070 01:12:21,240 --> 01:12:25,980 So actually, let me put this in this fashion. 1071 01:12:25,980 --> 01:12:28,470 So both e to the beta mu over lambda 1072 01:12:28,470 --> 01:12:31,090 cubed that is raised to that power. 1073 01:12:31,090 --> 01:12:33,310 OK, so that's that part. 1074 01:12:33,310 --> 01:12:36,140 The sum I have gotten rid of. 1075 01:12:36,140 --> 01:12:40,350 I notice that my W has an N factorial. 1076 01:12:40,350 --> 01:12:43,160 So this is the N factorial that came from here. 1077 01:12:45,670 --> 01:12:49,790 But then there is an N factorial that is up here from the W. 1078 01:12:49,790 --> 01:12:54,440 So the two of them will cancel each other, N factorial. 1079 01:12:54,440 --> 01:13:00,770 And then I have a product over all clusters. 1080 01:13:00,770 --> 01:13:04,980 Part of it is this bl to the power of nl, 1081 01:13:04,980 --> 01:13:08,790 and then I have the contribution that is nl factorial l 1082 01:13:08,790 --> 01:13:10,236 factorial to the power of nl. 1083 01:13:15,930 --> 01:13:17,150 So this part cancels. 1084 01:13:19,840 --> 01:13:25,360 For each l, I can now independently 1085 01:13:25,360 --> 01:13:31,910 sum over the values of nl can be anything, 0 to infinity. 1086 01:13:31,910 --> 01:13:33,762 And what do I have? 1087 01:13:33,762 --> 01:13:39,650 I have bl to the power of nl. 1088 01:13:39,650 --> 01:13:43,790 I have division by nl factorial. 1089 01:13:43,790 --> 01:13:48,630 I have l factorial to the power of nl. 1090 01:13:48,630 --> 01:13:55,920 And then I have e to the beta mu over lambda cubed raised 1091 01:13:55,920 --> 01:13:57,431 to the power of l nl. 1092 01:14:08,500 --> 01:14:12,650 I recognize that each one of these terms in the sum 1093 01:14:12,650 --> 01:14:19,150 is 1 over nl factorial something raised to the power of nl, 1094 01:14:19,150 --> 01:14:22,240 which is the definition of the exponential. 1095 01:14:22,240 --> 01:14:27,880 So my q is a product over all l's. 1096 01:14:27,880 --> 01:14:35,690 Once I exponentiate, I have e to the beta mu 1097 01:14:35,690 --> 01:14:42,380 over lambda cubed raised to the power of l. 1098 01:14:42,380 --> 01:14:46,890 And then I have bl divided by l factorial. 1099 01:14:53,080 --> 01:14:59,770 Or, log of Q, which is the quantity that I'm interested, 1100 01:14:59,770 --> 01:15:04,240 is obtained by summing over all clusters 1 1101 01:15:04,240 --> 01:15:11,290 to infinity e to the beta mu over lambda cubed raised 1102 01:15:11,290 --> 01:15:16,366 to the power of l bl divided by l factorial. 1103 01:15:26,840 --> 01:15:30,330 So what does this tell us, which is 1104 01:15:30,330 --> 01:15:34,050 kind of nice and fundamentally important? 1105 01:15:34,050 --> 01:15:38,110 You see, we started at the beginning over here 1106 01:15:38,110 --> 01:15:40,510 with a huge number of graphs. 1107 01:15:43,280 --> 01:15:49,240 These graphs could be organizing all kinds of clusters. 1108 01:15:49,240 --> 01:15:51,980 And they would give us either the partition function 1109 01:15:51,980 --> 01:15:56,700 or summing over N, the grand partition function. 1110 01:15:56,700 --> 01:16:05,230 But when we take the log, I get only single connect objects. 1111 01:16:05,230 --> 01:16:07,510 And this is something that you had already 1112 01:16:07,510 --> 01:16:10,440 seen as the connection that we have 1113 01:16:10,440 --> 01:16:13,750 between moments and Cumulants. 1114 01:16:13,750 --> 01:16:17,010 So the way that we got Cumulants was 1115 01:16:17,010 --> 01:16:19,860 to look at the expansion of the log. 1116 01:16:19,860 --> 01:16:23,250 So the function itself was the generator of all moments 1117 01:16:23,250 --> 01:16:24,830 and we took the log. 1118 01:16:24,830 --> 01:16:26,990 And graphically, we presented that 1119 01:16:26,990 --> 01:16:29,480 as getting the moments by putting together 1120 01:16:29,480 --> 01:16:34,920 all kinds of clusters of points that corresponded to Cumulants. 1121 01:16:34,920 --> 01:16:37,300 And this is again, a representation 1122 01:16:37,300 --> 01:16:38,660 of the same thing. 1123 01:16:38,660 --> 01:16:41,490 That is, in the log you have essentially 1124 01:16:41,490 --> 01:16:44,290 individual contributions. 1125 01:16:44,290 --> 01:16:49,790 Once you exponentiated, you get multiple contributions. 1126 01:16:49,790 --> 01:16:54,870 Now, the other thing is, that if I am thinking about the gas, 1127 01:16:54,870 --> 01:17:00,730 then log of Q is e to the minus beta g. 1128 01:17:00,730 --> 01:17:11,270 And g is E minus TS minus mu N. Bur for a gas that 1129 01:17:11,270 --> 01:17:16,840 has extensive properties, E is TS mu N minus PV. 1130 01:17:16,840 --> 01:17:19,210 So this is the same thing as beta PV. 1131 01:17:22,320 --> 01:17:25,730 So this quantity that we have calculated here 1132 01:17:25,730 --> 01:17:27,830 is, in fact, related to the pressure 1133 01:17:27,830 --> 01:17:30,001 directly through this formula. 1134 01:17:32,810 --> 01:17:35,770 And the important part or the important observation 1135 01:17:35,770 --> 01:17:40,800 is that it says that it should be proportional to volume. 1136 01:17:40,800 --> 01:17:44,170 You can see that each one of my b's 1137 01:17:44,170 --> 01:17:48,500 that I calculated here will have one free integral. 1138 01:17:48,500 --> 01:17:50,010 If you like, its center of mass. 1139 01:17:50,010 --> 01:17:52,310 You can go over the entire volume. 1140 01:17:52,310 --> 01:17:58,560 So all of my b's are indeed proportionality to volume. 1141 01:17:58,560 --> 01:18:00,440 And you can see what disaster it would 1142 01:18:00,440 --> 01:18:05,310 have been if there was a term here that was not just a linked 1143 01:18:05,310 --> 01:18:07,510 cluster but product of two linked clusters. 1144 01:18:07,510 --> 01:18:10,240 Then I would have something that would go like V squared. 1145 01:18:10,240 --> 01:18:12,010 It's not allowed. 1146 01:18:12,010 --> 01:18:16,970 So essentially, this linked cluster nature is also 1147 01:18:16,970 --> 01:18:19,850 related to extensivity in this sense. 1148 01:18:23,350 --> 01:18:28,620 So in some sense, what I have established here-- again, 1149 01:18:28,620 --> 01:18:35,170 related to that-- is that clearly all of my bl's 1150 01:18:35,170 --> 01:18:38,650 are proportional to volume. 1151 01:18:38,650 --> 01:18:40,510 And I can define something that I 1152 01:18:40,510 --> 01:18:48,540 will call bl bar, which is divide the thing by the volume. 1153 01:18:48,540 --> 01:18:54,770 And then what we have established 1154 01:18:54,770 --> 01:19:03,400 is that the pressure of this interacting gas as an expansion 1155 01:19:03,400 --> 01:19:09,142 beta p, which takes this nice, simple form, sum over l 1156 01:19:09,142 --> 01:19:14,770 e to the beta mu divided by lambda cubed bl 1157 01:19:14,770 --> 01:19:21,580 bar-- the intensive part of the contribution of these cluster-- 1158 01:19:21,580 --> 01:19:23,300 divided by l factorial. 1159 01:19:23,300 --> 01:19:26,540 This, of course, runs from 1 to [INAUDIBLE]. 1160 01:19:29,510 --> 01:19:30,070 Yes. 1161 01:19:30,070 --> 01:19:33,592 AUDIENCE: [INAUDIBLE] is raised to the l? 1162 01:19:33,592 --> 01:19:35,133 PROFESSOR: Who's asking the question? 1163 01:19:35,133 --> 01:19:37,999 AUDIENCE: The term in the parentheses is raised to the l? 1164 01:19:37,999 --> 01:19:38,790 PROFESSOR: Exactly. 1165 01:19:38,790 --> 01:19:42,120 Thank you very much. 1166 01:19:42,120 --> 01:19:43,534 Yes. 1167 01:19:43,534 --> 01:19:44,450 AUDIENCE: [INAUDIBLE]. 1168 01:19:53,575 --> 01:19:55,930 But they don't, actually. 1169 01:19:55,930 --> 01:19:58,870 Could that be true, for example [INAUDIBLE]? 1170 01:20:04,667 --> 01:20:05,250 PROFESSOR: OK. 1171 01:20:05,250 --> 01:20:07,750 So the triangle, I have it here. 1172 01:20:07,750 --> 01:20:12,320 Integral d q1, d q2, d Q3, f 1 2, f 2 3, f 3. 1173 01:20:12,320 --> 01:20:14,370 Let's write it explicitly. 1174 01:20:14,370 --> 01:20:23,210 It is f of q1 minus q2, f of q2 minus q3, f of q3 minus q1. 1175 01:20:23,210 --> 01:20:25,725 I call this vector x. 1176 01:20:25,725 --> 01:20:28,500 I call that vector y. 1177 01:20:28,500 --> 01:20:30,700 This vector is x plus y. 1178 01:20:35,310 --> 01:20:36,109 Yes. 1179 01:20:36,109 --> 01:20:38,504 AUDIENCE: Does the contribution from [INAUDIBLE]? 1180 01:20:44,260 --> 01:20:45,290 PROFESSOR: OK. 1181 01:20:45,290 --> 01:20:52,720 So you say that I have here an expansion for the pressure. 1182 01:20:52,720 --> 01:20:55,930 I had given you previously an expansion for pressure 1183 01:20:55,930 --> 01:20:59,240 that I said is sensible, which is this Virial expansion, which 1184 01:20:59,240 --> 01:21:01,870 is powers of density. 1185 01:21:01,870 --> 01:21:06,160 This is not an expansion in powers of density. 1186 01:21:06,160 --> 01:21:10,320 So whether or not the terms in this expansion 1187 01:21:10,320 --> 01:21:13,230 becomes smaller or larger will depend 1188 01:21:13,230 --> 01:21:16,490 on density in some indirect way. 1189 01:21:16,490 --> 01:21:21,100 So my next task, which I will do next lecture, 1190 01:21:21,100 --> 01:21:23,540 is within this ensemble I have told you 1191 01:21:23,540 --> 01:21:25,740 what the chemical potential is. 1192 01:21:25,740 --> 01:21:28,160 Once I know what the chemical potential is, 1193 01:21:28,160 --> 01:21:30,770 I can calculate the number of particles 1194 01:21:30,770 --> 01:21:36,380 as d log Q by d beta mu. 1195 01:21:36,380 --> 01:21:41,640 And hence, I can calculate what the density is. 1196 01:21:41,640 --> 01:21:46,390 And so the answer for this will also be a series in powers of e 1197 01:21:46,390 --> 01:21:47,980 to the beta mu. 1198 01:21:47,980 --> 01:21:51,280 And what then I will do is I will combine these two 1199 01:21:51,280 --> 01:21:56,620 series to get an expansion for pressure in powers of density, 1200 01:21:56,620 --> 01:22:00,410 and then identify the convergence of this series 1201 01:22:00,410 --> 01:22:03,320 and all of that via that procedure.