1 00:00:00,090 --> 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 00:00:20,550 --> 00:00:23,340 PROFESSOR: OK, let's start. 9 00:00:23,340 --> 00:00:27,160 So we said that the task of statistical mechanics 10 00:00:27,160 --> 00:00:31,810 is to assign probabilities to different microstates given 11 00:00:31,810 --> 00:00:33,620 that we have knowledge of the microstate. 12 00:00:36,320 --> 00:00:43,310 And the most kind of simple logical place to start 13 00:00:43,310 --> 00:00:51,470 was in the microcanonical ensemble 14 00:00:51,470 --> 00:00:54,850 where the microstate that we specified 15 00:00:54,850 --> 00:00:58,760 was one in which there was no exchange of work or heat 16 00:00:58,760 --> 00:01:02,660 with the surroundings so that the energy was constant, 17 00:01:02,660 --> 00:01:05,580 and the parameters, such as x and N, 18 00:01:05,580 --> 00:01:08,270 that account for chemical and mechanical work 19 00:01:08,270 --> 00:01:11,000 were fixed also. 20 00:01:11,000 --> 00:01:14,820 Then this assignment was, we said, 21 00:01:14,820 --> 00:01:17,870 like counting how many faces the dice has, 22 00:01:17,870 --> 00:01:21,110 and saying all of them are equally likely. 23 00:01:21,110 --> 00:01:24,030 So we would say that the probability here 24 00:01:24,030 --> 00:01:31,340 of a microstate is of the form 0 or 1 depending 25 00:01:31,340 --> 00:01:36,450 on whether the energy of that microstate 26 00:01:36,450 --> 00:01:46,280 is or is not the right energy that is listed over here. 27 00:01:46,280 --> 00:01:50,660 And as any probability, it has to be normalized. 28 00:01:50,660 --> 00:01:53,170 So we had this 1 over omega. 29 00:01:59,860 --> 00:02:06,080 And we also had a rule for converting probabilities 30 00:02:06,080 --> 00:02:12,370 to entropies, which in dimensionless form-- that 31 00:02:12,370 --> 00:02:18,370 is, if we divide by kB-- was simply minus the expectation 32 00:02:18,370 --> 00:02:22,090 value of the log of the probability, this probability 33 00:02:22,090 --> 00:02:24,520 being uniformly 1 over omega. 34 00:02:24,520 --> 00:02:32,000 This simply gave us log of omega E, x, N. 35 00:02:32,000 --> 00:02:36,990 Once we had the entropy as a function of E, x, N, 36 00:02:36,990 --> 00:02:41,870 we also identified the derivatives of this quantity 37 00:02:41,870 --> 00:02:43,180 in equilibrium. 38 00:02:43,180 --> 00:02:46,540 And partial derivative with respect to energy 39 00:02:46,540 --> 00:02:50,010 was identified as 1 So systems that 40 00:02:50,010 --> 00:02:55,580 were in equilibrium with each other, this derivative dS by dE 41 00:02:55,580 --> 00:02:58,190 had to be the same for all of them. 42 00:02:58,190 --> 00:03:03,280 And because of mechanical stability, 43 00:03:03,280 --> 00:03:07,750 we could identify the next derivative with respect to x. 44 00:03:07,750 --> 00:03:11,460 That was minus J/T. And with respect to N 45 00:03:11,460 --> 00:03:20,020 by identifying a corresponding thing, we would get this. 46 00:03:20,020 --> 00:03:22,810 So from here, we can proceed and calculate 47 00:03:22,810 --> 00:03:28,830 thermodynamic properties using these microscopic rules, as 48 00:03:28,830 --> 00:03:37,620 well as probabilities in the entire space of microstates 49 00:03:37,620 --> 00:03:39,770 that you have. 50 00:03:39,770 --> 00:03:42,210 Now, the next thing that we did was 51 00:03:42,210 --> 00:03:45,320 to go and look at the different ensemble, 52 00:03:45,320 --> 00:03:53,240 the canonical ensemble, in which we said that, 53 00:03:53,240 --> 00:03:56,385 again, thermodynamically, I choose 54 00:03:56,385 --> 00:03:58,410 a different set of variables. 55 00:03:58,410 --> 00:04:02,630 For example, I can replace the energy with temperature. 56 00:04:02,630 --> 00:04:06,215 And indeed, the characteristic of the canonical ensemble 57 00:04:06,215 --> 00:04:09,190 is that rather than specifying the energy, 58 00:04:09,190 --> 00:04:11,680 you specify the temperature. 59 00:04:11,680 --> 00:04:15,120 But there's still no chemical or mechanical work. 60 00:04:15,120 --> 00:04:19,610 So the other two parameters are kept fixed also. 61 00:04:19,610 --> 00:04:23,640 And then the statement that we had 62 00:04:23,640 --> 00:04:29,810 was that by putting a system in contact with a huge reservoir, 63 00:04:29,810 --> 00:04:33,480 we could ensure that it is maintained at some temperature. 64 00:04:33,480 --> 00:04:40,130 And since system and reservoir were jointly microcanonical, 65 00:04:40,130 --> 00:04:44,460 we could use the probabilities that we had microcanonically. 66 00:04:44,460 --> 00:04:47,650 We integrated over the degrees of freedom of the reservoir 67 00:04:47,650 --> 00:04:49,220 that we didn't care about. 68 00:04:49,220 --> 00:04:52,630 And we ended up with the probability 69 00:04:52,630 --> 00:04:58,750 of a microstate in the canonical ensemble, which 70 00:04:58,750 --> 00:05:02,000 was related to the energy of that 71 00:05:02,000 --> 00:05:10,430 by the form exponential of minus beta h where we introduced beta 72 00:05:10,430 --> 00:05:13,940 to be 1 over kT. 73 00:05:13,940 --> 00:05:18,670 And this probably, again, had to be normalized. 74 00:05:18,670 --> 00:05:23,280 So the normalization we called Z. And Z 75 00:05:23,280 --> 00:05:27,770 is attained by summing over all of the microstates E 76 00:05:27,770 --> 00:05:32,190 to the minus beta H of the microstate where this could 77 00:05:32,190 --> 00:05:35,120 be an integration over the entire phase space 78 00:05:35,120 --> 00:05:39,520 if you're dealing with continuous variables. 79 00:05:39,520 --> 00:05:44,920 OK, now the question was, thermodynamically, 80 00:05:44,920 --> 00:05:49,790 these quantities T and E are completely 81 00:05:49,790 --> 00:05:54,540 exchangeable ways of identifying the same equilibrium 82 00:05:54,540 --> 00:05:58,160 state, whereas what I have done now 83 00:05:58,160 --> 00:06:01,135 is I have told you what the temperature is. 84 00:06:01,135 --> 00:06:05,090 But the energy of the system is a random variable. 85 00:06:05,090 --> 00:06:08,270 And so I can ask, what is the probability 86 00:06:08,270 --> 00:06:14,510 that I look at my system and I find a particular energy E? 87 00:06:14,510 --> 00:06:17,006 So how can I get that? 88 00:06:17,006 --> 00:06:20,390 Well, that probability, first of all, 89 00:06:20,390 --> 00:06:24,370 has to come from a microstate that has the right energy. 90 00:06:24,370 --> 00:06:26,730 And so I will get e to the minus beta 91 00:06:26,730 --> 00:06:31,960 E divided by Z, which is the probability of getting 92 00:06:31,960 --> 00:06:35,040 that microstate that has the right energy. 93 00:06:35,040 --> 00:06:37,740 But I only said something about the energy. 94 00:06:37,740 --> 00:06:41,310 And there are a huge number of microstates, as we've seen, 95 00:06:41,310 --> 00:06:43,150 that have the same energy. 96 00:06:43,150 --> 00:06:46,390 So I could have picked from any one of those microstates, 97 00:06:46,390 --> 00:06:49,910 their number being omega of E, the omega that we 98 00:06:49,910 --> 00:06:51,210 had identified before. 99 00:06:53,770 --> 00:07:00,120 So since omega can be written as exponential of s 100 00:07:00,120 --> 00:07:04,350 over kB, this you can also think of as being 101 00:07:04,350 --> 00:07:09,650 proportional to exponential of minus beta E minus T, 102 00:07:09,650 --> 00:07:12,790 the entropy that I would get for that energy 103 00:07:12,790 --> 00:07:19,564 according to the formula above, divided by Z. 104 00:07:19,564 --> 00:07:22,600 Now, we said that the quantity that I'm 105 00:07:22,600 --> 00:07:27,370 looking at here in the numerator has 106 00:07:27,370 --> 00:07:34,470 a dependence on the size of the system that is extensive. 107 00:07:34,470 --> 00:07:36,700 Both the entropy and energy we expect 108 00:07:36,700 --> 00:07:40,670 to be growing proportionately to the size of the system. 109 00:07:40,670 --> 00:07:45,750 And hence, this exponent also grows proportionately 110 00:07:45,750 --> 00:07:47,620 to the size of the system. 111 00:07:47,620 --> 00:07:51,520 So I expect that if I plot this probability 112 00:07:51,520 --> 00:07:57,100 as a function of energy, this probability to get energy E, 113 00:07:57,100 --> 00:08:00,470 it would be one of those functions. 114 00:08:00,470 --> 00:08:02,260 It's certainly positive. 115 00:08:02,260 --> 00:08:03,540 It's a probability. 116 00:08:03,540 --> 00:08:05,550 It has an exponential dependence. 117 00:08:05,550 --> 00:08:09,280 So there's a part that maybe from the density of states 118 00:08:09,280 --> 00:08:10,840 grows exponentially. 119 00:08:10,840 --> 00:08:14,060 e to the minus beta E will exponential kill it. 120 00:08:14,060 --> 00:08:17,500 And maybe there is, because of this competition, 121 00:08:17,500 --> 00:08:20,440 one or potentially more maxima. 122 00:08:20,440 --> 00:08:24,360 But if there are more maxima locally, I don't really care. 123 00:08:24,360 --> 00:08:26,760 Because one will be exponentially larger 124 00:08:26,760 --> 00:08:27,870 than the other. 125 00:08:27,870 --> 00:08:32,590 So presumably, there is some location corresponding 126 00:08:32,590 --> 00:08:37,810 to some E star where the probability is maximized. 127 00:08:42,900 --> 00:08:45,920 Well, how should I characterize the energy of the system? 128 00:08:45,920 --> 00:08:49,900 Should I pick E star, or given this probability, 129 00:08:49,900 --> 00:08:58,040 should I look at maybe the mean value, the average of H? 130 00:08:58,040 --> 00:09:00,600 How can I get the average of H? 131 00:09:00,600 --> 00:09:05,210 Well, what I need to do is to sum over all the microstates H 132 00:09:05,210 --> 00:09:07,460 of the microstate e to the minus beta 133 00:09:07,460 --> 00:09:11,626 H divided by sum over microstates e to the minus beta 134 00:09:11,626 --> 00:09:12,876 H, which is the normalization. 135 00:09:16,150 --> 00:09:19,880 The denominator is of course the partition function. 136 00:09:19,880 --> 00:09:23,900 The numerator can be obtained by taking a derivative of this 137 00:09:23,900 --> 00:09:27,060 with respect to beta of the minus sign. 138 00:09:27,060 --> 00:09:32,060 So what this is is minus the log Z by d beta. 139 00:09:34,900 --> 00:09:42,020 So if I have calculated H, I can potentially 140 00:09:42,020 --> 00:09:45,720 go through this procedure and maybe calculate 141 00:09:45,720 --> 00:09:48,880 where the mean value is. 142 00:09:48,880 --> 00:09:51,810 And is that a better representation 143 00:09:51,810 --> 00:09:55,280 of the energy of the system than the most likely value, which 144 00:09:55,280 --> 00:09:56,605 was E star? 145 00:09:56,605 --> 00:10:00,590 Well, let's see how much energy fluctuates. 146 00:10:00,590 --> 00:10:04,050 Basically, we saw that if I repeat this procedure 147 00:10:04,050 --> 00:10:10,120 many times, I can see that the n-th moment is minus 1 148 00:10:10,120 --> 00:10:15,340 to the n 1 over Z the n-th derivative of Z 149 00:10:15,340 --> 00:10:18,620 with respect to beta. 150 00:10:18,620 --> 00:10:22,000 And hence the partition function, 151 00:10:22,000 --> 00:10:24,850 by expanding it in powers of beta, 152 00:10:24,850 --> 00:10:28,540 will generate for me higher and higher moments. 153 00:10:28,540 --> 00:10:35,620 And we therefore concluded that the cumulant would be obtained 154 00:10:35,620 --> 00:10:39,650 by the same procedure, except that I will replace this 155 00:10:39,650 --> 00:10:42,320 by log Z. So this will be the n-th derivative 156 00:10:42,320 --> 00:10:43,570 of log Z with respect to beta. 157 00:10:49,090 --> 00:10:57,910 So the variance H squared c is a second derivative 158 00:10:57,910 --> 00:11:00,800 of log Z. The first derivative gives me 159 00:11:00,800 --> 00:11:03,870 the first cumulant, which was the mean. 160 00:11:03,870 --> 00:11:08,616 So this is going to give me d by d beta 161 00:11:08,616 --> 00:11:15,140 with a minus sign of the expectation value of H, 162 00:11:15,140 --> 00:11:19,980 which is the same thing as kBT squared, 163 00:11:19,980 --> 00:11:23,600 because the derivative of 1 over kT 164 00:11:23,600 --> 00:11:28,670 is 1 over T squared times derivative with respect to T. 165 00:11:28,670 --> 00:11:31,740 And it goes to the other side, to the numerator. 166 00:11:31,740 --> 00:11:34,380 And so then I have the derivative 167 00:11:34,380 --> 00:11:38,450 of this object with respect to temperature, 168 00:11:38,450 --> 00:11:40,550 which is something like a heat capacity. 169 00:11:40,550 --> 00:11:43,820 But most importantly, the statement 170 00:11:43,820 --> 00:11:46,600 is that all of these quantities are 171 00:11:46,600 --> 00:11:51,310 things that are of the order of the size of the system. 172 00:11:51,310 --> 00:11:54,710 And just like we did in the central limit theorem, 173 00:11:54,710 --> 00:11:57,040 with the addition of random variables, 174 00:11:57,040 --> 00:12:00,220 you have a situation that is very much like that. 175 00:12:00,220 --> 00:12:02,690 The distribution, in some sense, is 176 00:12:02,690 --> 00:12:07,830 going to converge more and more towards the Gaussian dominated 177 00:12:07,830 --> 00:12:11,770 by the first and second cumulant. 178 00:12:11,770 --> 00:12:15,130 And in fact, even the second cumulant we see 179 00:12:15,130 --> 00:12:17,980 is of the order of square root of n. 180 00:12:17,980 --> 00:12:21,370 And hence the fluctuations between these two quantities, 181 00:12:21,370 --> 00:12:22,970 which are each one of them order of n, 182 00:12:22,970 --> 00:12:25,700 is only of the order of square root of n. 183 00:12:25,700 --> 00:12:27,700 And when the limit of n goes to infinity, 184 00:12:27,700 --> 00:12:30,150 we can ignore any such difference. 185 00:12:30,150 --> 00:12:33,690 We can essentially identify either one of them 186 00:12:33,690 --> 00:12:37,430 with the energy of the system thermodynamically. 187 00:12:37,430 --> 00:12:40,260 And then this quantity, if I identify 188 00:12:40,260 --> 00:12:42,850 this with energy of the system, is simply 189 00:12:42,850 --> 00:12:47,520 the usual heat capacity of constant x. 190 00:12:47,520 --> 00:12:51,090 So the scale of the fluctuations over here 191 00:12:51,090 --> 00:12:56,926 is set by square root of kBT squared the heat capacity. 192 00:12:59,890 --> 00:13:02,800 By the way, which is also the reason 193 00:13:02,800 --> 00:13:06,540 that heat capacities must be positive. 194 00:13:06,540 --> 00:13:09,750 Because statistically, the variances 195 00:13:09,750 --> 00:13:12,410 are certainly positive quantities. 196 00:13:12,410 --> 00:13:16,060 So you have a constraint that we had 197 00:13:16,060 --> 00:13:18,760 seen before on the sign emerging in 198 00:13:18,760 --> 00:13:20,570 its statistical interpretation. 199 00:13:25,350 --> 00:13:33,030 What I did over here was to identify an entropy associated 200 00:13:33,030 --> 00:13:36,120 with a particular form of the probability. 201 00:13:36,120 --> 00:13:41,160 What happens if I look at an entropy with this probability 202 00:13:41,160 --> 00:13:42,500 that I have over here? 203 00:13:42,500 --> 00:13:44,870 OK, so this is a probability phase space. 204 00:13:44,870 --> 00:13:46,860 What is its entropy? 205 00:13:46,860 --> 00:13:55,450 So S over k is expectation value of log p. 206 00:13:55,450 --> 00:13:57,645 And what is log of p? 207 00:13:57,645 --> 00:14:02,920 Well, there is log of this minus beta H. So what I will have, 208 00:14:02,920 --> 00:14:08,880 because of the change in sign-- beta expectation value of H. 209 00:14:08,880 --> 00:14:13,050 And then I have minus log Z here. 210 00:14:13,050 --> 00:14:21,930 The sign changes, and I will get plus log Z. 211 00:14:21,930 --> 00:14:27,030 So yeah, that's correct. 212 00:14:31,150 --> 00:14:34,090 If I were to rearrange this, what do I get? 213 00:14:34,090 --> 00:14:41,311 I can take this minus-- let's see. 214 00:14:45,207 --> 00:14:50,900 Yeah, OK, so if I take this to the other side, 215 00:14:50,900 --> 00:14:53,730 then I will get that log Z equals-- 216 00:14:53,730 --> 00:14:57,090 this goes to the other side-- minus beta. 217 00:14:57,090 --> 00:15:01,740 Expectation value of H we are calling E. 218 00:15:01,740 --> 00:15:07,585 And then I have multiplying this by beta. 219 00:15:07,585 --> 00:15:10,790 The kB's disappear, and I will get minus TS. 220 00:15:15,100 --> 00:15:21,620 So we see that I can identify log Z with the combination E 221 00:15:21,620 --> 00:15:33,480 minus TS, which is the Helmholtz free energy. 222 00:15:37,900 --> 00:15:46,880 So log Z, in the same way that the normalization here that 223 00:15:46,880 --> 00:15:51,470 was omega, gave us the entropy. 224 00:15:51,470 --> 00:15:55,110 The normalization of the probability 225 00:15:55,110 --> 00:15:57,940 in the canonical ensemble, which is also called the partition 226 00:15:57,940 --> 00:16:02,390 function if I take its log, will give me the free energy. 227 00:16:02,390 --> 00:16:05,600 And I can then go and compute various thermodynamic 228 00:16:05,600 --> 00:16:08,460 quantities based on that. 229 00:16:08,460 --> 00:16:13,640 There's an alternative way of getting the same result, which 230 00:16:13,640 --> 00:16:18,450 is to note that actually the same quantity is appearing over 231 00:16:18,450 --> 00:16:24,032 here, right? 232 00:16:24,032 --> 00:16:28,940 And really, I should be evaluating the probability here 233 00:16:28,940 --> 00:16:31,550 at the maximum. 234 00:16:31,550 --> 00:16:33,930 Maximum is the energy of the system. 235 00:16:33,930 --> 00:16:39,730 So if you like, I can call this variable over here epsilon. 236 00:16:39,730 --> 00:16:42,360 This is the probability of epsilon. 237 00:16:42,360 --> 00:16:45,290 And it is only when I'm at the maximum 238 00:16:45,290 --> 00:16:55,030 that I can replace this epsilon with E. Then since with almost 239 00:16:55,030 --> 00:17:00,129 probability 1, I'm going to see this state 240 00:17:00,129 --> 00:17:01,670 and none of the other states, because 241 00:17:01,670 --> 00:17:05,250 of this exponential dependence. 242 00:17:05,250 --> 00:17:09,950 When this expression is evaluated at this energy, 243 00:17:09,950 --> 00:17:12,680 it should give me probability 1. 244 00:17:12,680 --> 00:17:20,270 And so you can see again that Z has to be-- so basically what 245 00:17:20,270 --> 00:17:22,880 I'm saying is that this quantity has 246 00:17:22,880 --> 00:17:27,849 to be 1 so that you get this relationship back 247 00:17:27,849 --> 00:17:29,250 from that perspective also. 248 00:17:36,437 --> 00:17:37,260 AUDIENCE: Question. 249 00:17:37,260 --> 00:17:38,960 PROFESSOR: Yes. 250 00:17:38,960 --> 00:17:42,560 AUDIENCE: So I agree that if in order 251 00:17:42,560 --> 00:17:45,440 to plug in d, mean energy, into that expression, 252 00:17:45,440 --> 00:17:50,140 you would get a probability of 1 at that point. 253 00:17:50,140 --> 00:17:54,820 But because even though the other energies are 254 00:17:54,820 --> 00:17:59,320 exponentially less probable, they strictly speaking 255 00:17:59,320 --> 00:18:01,567 aren't 0 probability, are they? 256 00:18:01,567 --> 00:18:02,150 PROFESSOR: No. 257 00:18:02,150 --> 00:18:04,460 AUDIENCE: So how does this get normalized? 258 00:18:04,460 --> 00:18:06,920 How does this probability expression get normalized? 259 00:18:06,920 --> 00:18:09,336 PROFESSOR: OK, so what are we doing? 260 00:18:09,336 --> 00:18:14,920 So Z, I can also write it as the normalization of the energy. 261 00:18:14,920 --> 00:18:20,580 So rather than picking the one energy that maximizes things, 262 00:18:20,580 --> 00:18:26,530 you say, you should really do this, right? 263 00:18:26,530 --> 00:18:30,522 Now, I will evaluate this by the saddle point method. 264 00:18:30,522 --> 00:18:35,090 The saddle point method says, pick the maximum of this. 265 00:18:35,090 --> 00:18:40,680 So I have e to the minus beta F evaluated at the maximum. 266 00:18:40,680 --> 00:18:43,850 And then I have to integrate over 267 00:18:43,850 --> 00:18:46,840 variations around that maximum. 268 00:18:46,840 --> 00:18:49,320 Variations around that maximum I have 269 00:18:49,320 --> 00:18:51,870 to expand this to second order. 270 00:18:51,870 --> 00:18:55,960 And if I really correctly expand this to second order, 271 00:18:55,960 --> 00:19:03,220 I will get delta E squared divided by this 2kTCx, 272 00:19:03,220 --> 00:19:08,440 because we already established what this variance is. 273 00:19:08,440 --> 00:19:10,840 I can do this Gaussian integration. 274 00:19:10,840 --> 00:19:15,170 I get e to the minus beta F of e star. 275 00:19:15,170 --> 00:19:17,810 And then the variance of this object 276 00:19:17,810 --> 00:19:26,565 is going to give me root of 2 pi kTC of x-- T-square. 277 00:19:31,590 --> 00:19:39,190 So what do I mean when I say that this quantity is Z? 278 00:19:39,190 --> 00:19:42,810 Really, the thing that we are calculating always in order 279 00:19:42,810 --> 00:19:47,070 to make computation is something like a free energy, 280 00:19:47,070 --> 00:19:48,800 which is log Z. 281 00:19:48,800 --> 00:19:52,260 So when I take the log, log of Z is 282 00:19:52,260 --> 00:19:58,870 going to be this minus beta F star S. But you're right. 283 00:19:58,870 --> 00:19:59,870 There was a weight. 284 00:19:59,870 --> 00:20:01,720 All of those things are contributing. 285 00:20:01,720 --> 00:20:03,640 How much are they contributing? 286 00:20:03,640 --> 00:20:12,300 1/2 log of 2 pi kBT squared C. 287 00:20:12,300 --> 00:20:15,340 Now, in the limit of large N, this 288 00:20:15,340 --> 00:20:23,550 is order of N. This is order of log N. And I [INAUDIBLE]. 289 00:20:23,550 --> 00:20:25,510 So it's the same saddle point. 290 00:20:25,510 --> 00:20:29,040 The idea of saddle point was that there is a weight. 291 00:20:29,040 --> 00:20:31,940 But you can ignore it. 292 00:20:31,940 --> 00:20:33,690 There maybe another maximum here. 293 00:20:33,690 --> 00:20:34,795 You say, what about that? 294 00:20:34,795 --> 00:20:37,990 Well, that will be exponentially small. 295 00:20:37,990 --> 00:20:43,050 So everything I keep emphasizing only works because of this N 296 00:20:43,050 --> 00:20:44,240 goes to infinity limit. 297 00:20:46,990 --> 00:20:50,130 And it's magical, you see? 298 00:20:50,130 --> 00:20:53,440 You can replace this sum with just one, the maximum. 299 00:20:53,440 --> 00:20:55,910 And everything is fine and consistent. 300 00:20:55,910 --> 00:20:57,950 Yes. 301 00:20:57,950 --> 00:20:59,480 AUDIENCE: Not to belabor this point, 302 00:20:59,480 --> 00:21:06,570 but if you have an expected return for catastrophe where 303 00:21:06,570 --> 00:21:11,630 this outlier causes an event that brings the system down, 304 00:21:11,630 --> 00:21:16,020 couldn't that chase this limit in the sense that as that goes 305 00:21:16,020 --> 00:21:18,500 to 0, that still goes to infinity, 306 00:21:18,500 --> 00:21:21,570 and thus you're-- you understand what I'm saying. 307 00:21:21,570 --> 00:21:24,770 If an outlier causes this simulation-- that's my word-- 308 00:21:24,770 --> 00:21:30,307 causes this system to crumble, then-- so is there a paradox 309 00:21:30,307 --> 00:21:30,806 there? 310 00:21:30,806 --> 00:21:33,170 PROFESSOR: No, there is here the possibility 311 00:21:33,170 --> 00:21:36,345 of a catastrophe in the sense that all the oxygen 312 00:21:36,345 --> 00:21:38,420 in this room could go over there, 313 00:21:38,420 --> 00:21:42,870 and you and I will suffocate after a few minutes. 314 00:21:42,870 --> 00:21:44,940 It's possible, that's true. 315 00:21:44,940 --> 00:21:47,040 You just have to wait many, many ages 316 00:21:47,040 --> 00:21:49,040 of the universe for that to happen. 317 00:21:52,760 --> 00:21:53,643 Yes. 318 00:21:53,643 --> 00:21:56,058 AUDIENCE: So when you're integrating, 319 00:21:56,058 --> 00:21:58,956 that introduces units into the problem. 320 00:21:58,956 --> 00:22:02,938 So we have to divide by something to [INAUDIBLE]. 321 00:22:08,850 --> 00:22:10,650 PROFESSOR: Yes, and when we are doing 322 00:22:10,650 --> 00:22:12,440 this for the case of the ideal gas, 323 00:22:12,440 --> 00:22:15,020 I will be very careful to do that. 324 00:22:15,020 --> 00:22:17,980 But it turns out that those issues, 325 00:22:17,980 --> 00:22:20,435 as far as various derivatives are concerned, 326 00:22:20,435 --> 00:22:23,220 will not make too much difference. 327 00:22:23,220 --> 00:22:26,050 But when we are looking at the specific example, 328 00:22:26,050 --> 00:22:29,120 such as the ideal gas, I will be careful about that. 329 00:22:37,130 --> 00:22:41,070 So we saw how this transition occurs. 330 00:22:41,070 --> 00:22:44,420 But when we were looking at the case 331 00:22:44,420 --> 00:22:46,750 of thermodynamical descriptions, we 332 00:22:46,750 --> 00:22:51,410 looked at a couple of other microstates. 333 00:22:51,410 --> 00:22:53,340 One of them was the Gibbs canonical. 334 00:22:58,830 --> 00:23:00,690 And what we did was we said, well, 335 00:23:00,690 --> 00:23:03,900 let's allow now some work to take place 336 00:23:03,900 --> 00:23:06,450 on the system, mechanical work. 337 00:23:06,450 --> 00:23:11,170 And rather than saying that, say, the displacement 338 00:23:11,170 --> 00:23:14,500 x is fixed, I allow it to vary. 339 00:23:14,500 --> 00:23:20,100 But I will say what the corresponding force is. 340 00:23:20,100 --> 00:23:22,435 But let's keep the number of particles fixed. 341 00:23:25,430 --> 00:23:28,040 So essentially, the picture that we have 342 00:23:28,040 --> 00:23:33,060 is that somehow my system is parametrized by some quantity 343 00:23:33,060 --> 00:23:34,370 x. 344 00:23:34,370 --> 00:23:46,380 And I'm maintaining the system at fixed value of some force J. 345 00:23:46,380 --> 00:23:51,300 So x is allowed potentially to find the value that 346 00:23:51,300 --> 00:23:54,380 is consistent with this particular J 347 00:23:54,380 --> 00:23:56,320 that I impose on the system. 348 00:23:56,320 --> 00:23:59,900 And again, I have put the whole system in contact 349 00:23:59,900 --> 00:24:03,630 with the reservoir temperature T so 350 00:24:03,630 --> 00:24:08,630 that if I now say that I really maintain 351 00:24:08,630 --> 00:24:16,550 the system at variable x, but fix J, 352 00:24:16,550 --> 00:24:20,800 I have to put this spring over there. 353 00:24:20,800 --> 00:24:25,630 And then this spring plus the system 354 00:24:25,630 --> 00:24:32,320 is jointly in a canonical perspective. 355 00:24:32,320 --> 00:24:34,310 And what that really means is that you 356 00:24:34,310 --> 00:24:41,950 have to keep track of the energy of the microstate, 357 00:24:41,950 --> 00:24:49,830 as well as the energy that you extract from the spring, 358 00:24:49,830 --> 00:24:54,150 which is something like J dot x. 359 00:24:54,150 --> 00:24:59,170 And since the joint system is in the canonical state, 360 00:24:59,170 --> 00:25:00,940 it would say that the probability 361 00:25:00,940 --> 00:25:07,370 for the joint system, which has T and J and N specified, 362 00:25:07,370 --> 00:25:09,240 but I don't know the microstate, and I 363 00:25:09,240 --> 00:25:13,090 don't know the actual displacement x, 364 00:25:13,090 --> 00:25:18,270 this joint probability is canonical. 365 00:25:18,270 --> 00:25:22,790 And so it is proportional to e to the minus beta 366 00:25:22,790 --> 00:25:27,680 H of the microstate plus this contribution Jx 367 00:25:27,680 --> 00:25:31,530 that I'm getting from the other degree of freedom, which 368 00:25:31,530 --> 00:25:35,490 is the spring, which keeps the whole thing at fixed J. 369 00:25:35,490 --> 00:25:39,560 And I have to divide by some normalization that 370 00:25:39,560 --> 00:25:42,990 in addition includes integration over x. 371 00:25:42,990 --> 00:25:50,610 And this will depend on T, J, N. 372 00:25:50,610 --> 00:25:56,320 So in this system, both x and the energy of the system 373 00:25:56,320 --> 00:25:57,230 are variables. 374 00:25:57,230 --> 00:26:00,880 They can potentially change. 375 00:26:00,880 --> 00:26:06,980 What I can now do is to either characterize like I did over 376 00:26:06,980 --> 00:26:11,310 here what the probability of x is, or go by this other route 377 00:26:11,310 --> 00:26:14,120 and calculate what the entropy of this probability 378 00:26:14,120 --> 00:26:19,250 is, which would be minus the log of the corresponding 379 00:26:19,250 --> 00:26:21,080 probability. 380 00:26:21,080 --> 00:26:23,840 And what is the log in this case? 381 00:26:23,840 --> 00:26:30,870 I will get beta expectation value of H. 382 00:26:30,870 --> 00:26:38,330 And I will get minus beta J expectation value of x. 383 00:26:38,330 --> 00:26:44,274 And then I will get plus log of this Z tilde. 384 00:26:48,630 --> 00:26:55,360 So rearranging things, what I get is that log of this Z tilde 385 00:26:55,360 --> 00:27:00,200 is minus beta. 386 00:27:00,200 --> 00:27:04,580 I will call this E like I did over there. 387 00:27:04,580 --> 00:27:09,920 Minus J-- I would call this the actual thermodynamic 388 00:27:09,920 --> 00:27:12,530 displacement x. 389 00:27:12,530 --> 00:27:15,400 And from down here, the other side, 390 00:27:15,400 --> 00:27:16,775 I will get a factor of TS. 391 00:27:22,370 --> 00:27:25,000 So this should remind you that we 392 00:27:25,000 --> 00:27:30,270 had called a Gibbs free energy the combination 393 00:27:30,270 --> 00:27:37,990 of E minus TS minus Jx. 394 00:27:37,990 --> 00:27:40,770 And the natural variables for this G 395 00:27:40,770 --> 00:27:43,800 were indeed, once we looked at the variation 396 00:27:43,800 --> 00:27:54,180 DE, T. They were J, and N that we did not do anything with. 397 00:27:54,180 --> 00:28:03,870 And to the question of, what is the displacement, since it's 398 00:28:03,870 --> 00:28:06,900 a random variable now, I can again 399 00:28:06,900 --> 00:28:11,480 try to go to the same procedure as I did over here. 400 00:28:11,480 --> 00:28:15,110 I can calculate the expectation value 401 00:28:15,110 --> 00:28:18,390 of x by noting that this exponent here 402 00:28:18,390 --> 00:28:21,160 has a factor of beta Jx. 403 00:28:21,160 --> 00:28:25,895 So if I take a derivative with respect to beta J, 404 00:28:25,895 --> 00:28:29,840 I will bring down a factor of x. 405 00:28:29,840 --> 00:28:35,182 So if I take the derivative of log of this Z tilde 406 00:28:35,182 --> 00:28:39,680 with respect to beta J, I would generate 407 00:28:39,680 --> 00:28:42,370 what the mean value is. 408 00:28:42,370 --> 00:28:49,490 And actually, maybe here let me show you 409 00:28:49,490 --> 00:28:52,440 that dG was indeed what I expected. 410 00:28:52,440 --> 00:28:54,990 So dG would be dE. 411 00:28:54,990 --> 00:28:56,860 dE has a TdS. 412 00:28:56,860 --> 00:29:02,160 This will make that into a minus SdT. 413 00:29:02,160 --> 00:29:04,110 dE has a Jdx. 414 00:29:04,110 --> 00:29:08,110 This will make it minus xdJ. 415 00:29:08,110 --> 00:29:12,320 And then I have mu dN. 416 00:29:12,320 --> 00:29:14,050 So thermodynamically, you would have 417 00:29:14,050 --> 00:29:17,630 said that x is obtained as a derivative of G 418 00:29:17,630 --> 00:29:23,040 with respect to J. 419 00:29:23,040 --> 00:29:26,670 Now, log Z is something that is like beta 420 00:29:26,670 --> 00:29:33,410 G. At fixed temperature, I can remove these betas. 421 00:29:33,410 --> 00:29:34,150 What do I get? 422 00:29:34,150 --> 00:29:39,380 This is the same thing as dG by dJ. 423 00:29:39,380 --> 00:29:41,740 And I seem to have lost a sign somewhere. 424 00:29:45,310 --> 00:29:51,250 OK, log Z was minus G. So there's a minus here. 425 00:29:51,250 --> 00:29:52,320 Everything is consistent. 426 00:29:56,310 --> 00:29:59,780 So you can play around with things in multiple ways, 427 00:29:59,780 --> 00:30:05,040 convince yourself, again, that in this ensemble, 428 00:30:05,040 --> 00:30:08,810 I have fixed the force, x's variable. 429 00:30:08,810 --> 00:30:12,850 But just like the energy here was well defined up 430 00:30:12,850 --> 00:30:15,300 to something that was square root of N, 431 00:30:15,300 --> 00:30:17,670 this x is well defined up to something 432 00:30:17,670 --> 00:30:19,750 that is of the order square root of N. 433 00:30:19,750 --> 00:30:25,280 Because again, you can second look at the variance. 434 00:30:25,280 --> 00:30:30,150 It will be related to two derivatives of log G, which 435 00:30:30,150 --> 00:30:33,970 would be related to one derivative of x. 436 00:30:33,970 --> 00:30:35,960 And ultimately, that will give you 437 00:30:35,960 --> 00:30:43,561 something like, again, kBT squared d of x by dJ. 438 00:30:43,561 --> 00:30:44,060 Yes. 439 00:30:44,060 --> 00:30:46,483 AUDIENCE: Can you mention again, regarding the probability 440 00:30:46,483 --> 00:30:47,024 distribution? 441 00:30:47,024 --> 00:30:48,510 What was the idea? 442 00:30:48,510 --> 00:30:50,360 PROFESSOR: How did I get this probability? 443 00:30:50,360 --> 00:30:51,562 AUDIENCE: Yes. 444 00:30:51,562 --> 00:30:57,660 PROFESSOR: OK, so I said that canonically, I 445 00:30:57,660 --> 00:31:02,260 was looking at a system where x was fixed. 446 00:31:02,260 --> 00:31:04,950 But now I have told you that J-- I 447 00:31:04,950 --> 00:31:07,540 know what J is, what the force is. 448 00:31:07,540 --> 00:31:12,600 And so how can I make sure that the system 449 00:31:12,600 --> 00:31:15,020 is maintained at a fixed J? 450 00:31:15,020 --> 00:31:18,860 I go back and say, how did I ensure 451 00:31:18,860 --> 00:31:23,120 that my microcanonical system was at a fixed temperature? 452 00:31:23,120 --> 00:31:25,740 I put it in contact with a huge bath 453 00:31:25,740 --> 00:31:27,750 that had the right temperature. 454 00:31:27,750 --> 00:31:30,000 So here, what I will do is I will 455 00:31:30,000 --> 00:31:35,090 connect the wall of my system to a huge spring that 456 00:31:35,090 --> 00:31:37,690 will maintain a particle [? effect ?] force 457 00:31:37,690 --> 00:31:39,570 J on the system. 458 00:31:39,570 --> 00:31:41,710 When we do it for the case of the gas, 459 00:31:41,710 --> 00:31:44,510 I will imagine that I have a box. 460 00:31:44,510 --> 00:31:47,980 I have a piston on top that can move up and down. 461 00:31:47,980 --> 00:31:49,610 I put a weight on top of it. 462 00:31:49,610 --> 00:31:51,820 And that weight will ensure that the pressure 463 00:31:51,820 --> 00:31:54,850 is at some particular value inside the gas. 464 00:31:54,850 --> 00:31:58,270 But then the piston can slide up and down. 465 00:31:58,270 --> 00:32:01,040 So a particular state of the system 466 00:32:01,040 --> 00:32:05,850 I have to specify where the piston is, how big x is, 467 00:32:05,850 --> 00:32:08,820 and what is the microstate of the system. 468 00:32:08,820 --> 00:32:15,020 So the variables that I'm not sure of are mu and x. 469 00:32:15,020 --> 00:32:18,640 Now, once I have said what mu and x is, and I've 470 00:32:18,640 --> 00:32:21,570 put the system in contact with a bath at temperature T, 471 00:32:21,570 --> 00:32:26,370 I can say, OK, the whole thing is canonical, 472 00:32:26,370 --> 00:32:31,110 the energy of the entire system composed of the energy of this, 473 00:32:31,110 --> 00:32:36,850 which is H of mu, and the energy of the spring, which is Jx. 474 00:32:36,850 --> 00:32:41,880 And so the canonical probability for the joint system 475 00:32:41,880 --> 00:32:44,630 is composed of the net energy of the two. 476 00:32:44,630 --> 00:32:47,910 And I have to normalize it. 477 00:32:47,910 --> 00:32:50,180 Yes. 478 00:32:50,180 --> 00:32:52,080 AUDIENCE: Are you taking x squared 479 00:32:52,080 --> 00:32:58,535 to be a scalar, so that way the dx over dJ 480 00:32:58,535 --> 00:32:59,410 is like a divergence? 481 00:32:59,410 --> 00:33:03,483 Or are you taking it to be a covariance tensor? 482 00:33:03,483 --> 00:33:06,290 PROFESSOR: Here, I have assumed that it is a scalar. 483 00:33:06,290 --> 00:33:12,840 But we can certainly do-- if you want to do it with a vector, 484 00:33:12,840 --> 00:33:16,660 then I can say something about positivity. 485 00:33:16,660 --> 00:33:20,460 So I really wanted this to be a variance and positive. 486 00:33:20,460 --> 00:33:23,650 And so if you like, then it would be the diagonal terms 487 00:33:23,650 --> 00:33:28,900 of whatever compressibility you have. 488 00:33:28,900 --> 00:33:34,030 So you certainly can generalize this expression to be a vector. 489 00:33:34,030 --> 00:33:37,830 You can have xI xJ, and you would have d of xI with respect 490 00:33:37,830 --> 00:33:40,120 to JJ or something like this. 491 00:33:40,120 --> 00:33:42,900 But here, I really wanted the scalar case. 492 00:33:42,900 --> 00:33:45,410 Everything I did was scalar manipulation. 493 00:33:45,410 --> 00:33:48,696 And this is now the variance of something and is positive. 494 00:33:58,420 --> 00:34:01,260 OK, there was one other ensemble, 495 00:34:01,260 --> 00:34:02,510 which was the grand canonical. 496 00:34:09,139 --> 00:34:16,739 So here, I went from the energy to temperature. 497 00:34:16,739 --> 00:34:18,800 I kept N fixed. 498 00:34:18,800 --> 00:34:21,940 I didn't make it into a chemical potential. 499 00:34:21,940 --> 00:34:23,760 But I can do that. 500 00:34:23,760 --> 00:34:25,480 Rather than having fixed number, I 501 00:34:25,480 --> 00:34:27,610 can have fixed chemical potential. 502 00:34:27,610 --> 00:34:33,280 But then I can't allow the other variable 503 00:34:33,280 --> 00:34:35,250 to be J. It has to be x. 504 00:34:35,250 --> 00:34:37,788 Because as we have discussed many times, at least one 505 00:34:37,788 --> 00:34:38,954 of them has to be extensive. 506 00:34:41,780 --> 00:34:45,679 And then you can follow this procedure that you had here. 507 00:34:45,679 --> 00:34:50,020 Chemical work is just an analog of mechanical work 508 00:34:50,020 --> 00:34:51,510 mathematically. 509 00:34:51,510 --> 00:34:53,560 So now I would have the probability 510 00:34:53,560 --> 00:34:58,550 that I have specified this set of variables. 511 00:34:58,550 --> 00:35:01,640 But now I don't know what my microstate is. 512 00:35:01,640 --> 00:35:03,550 Actually, let me put mu s. 513 00:35:03,550 --> 00:35:07,020 Because I introduced a chemical potential mu. 514 00:35:07,020 --> 00:35:10,460 So mu sub s is the microstate of the system. 515 00:35:10,460 --> 00:35:15,510 I don't know how many particles are in that microstate. 516 00:35:15,510 --> 00:35:18,670 The definition of mu s would potentially cover N. 517 00:35:18,670 --> 00:35:21,110 Or I can write it explicitly. 518 00:35:21,110 --> 00:35:29,920 And then, what I will have is e to the beta mu N minus beta 519 00:35:29,920 --> 00:35:34,260 H of the microstate's energy divided 520 00:35:34,260 --> 00:35:38,870 by the normalization, which is the grand partition function 521 00:35:38,870 --> 00:35:42,440 Q. So this was the Gibbs partition function. 522 00:35:42,440 --> 00:35:48,930 This is a function of T, x at the chemical potential. 523 00:35:48,930 --> 00:35:55,240 And the analog of all of these expressions here would exist. 524 00:35:55,240 --> 00:35:58,370 So the average number in the system, 525 00:35:58,370 --> 00:36:01,160 which now we can use as the thermodynamic number, 526 00:36:01,160 --> 00:36:09,530 as we've seen, would be d of log of Q with respect to d beta mu. 527 00:36:09,530 --> 00:36:12,430 And you can look at the variances, et cetera. 528 00:36:12,430 --> 00:36:14,450 Yes. 529 00:36:14,450 --> 00:36:16,450 AUDIENCE: So do you define, then, 530 00:36:16,450 --> 00:36:19,180 what the microstate of the system 531 00:36:19,180 --> 00:36:24,010 really is if the particle number is [INAUDIBLE]? 532 00:36:24,010 --> 00:36:27,760 PROFESSOR: OK, so let's say I have five particles or six 533 00:36:27,760 --> 00:36:29,530 particles in the system. 534 00:36:29,530 --> 00:36:33,070 It would be, in the first case, the momentum 535 00:36:33,070 --> 00:36:35,380 coordinates of five particles. 536 00:36:35,380 --> 00:36:37,095 In the next case, in the momentum, 537 00:36:37,095 --> 00:36:39,750 I have coordinates of six particles. 538 00:36:39,750 --> 00:36:43,780 So these are spaces, as it has changed, 539 00:36:43,780 --> 00:36:46,410 where the dimensionality of the phase space 540 00:36:46,410 --> 00:36:49,990 is also being modified. 541 00:36:49,990 --> 00:36:53,316 And again, for the ideal gas, we will explicitly calculate that. 542 00:36:57,880 --> 00:37:04,940 And again, you can identify what the log of this grand partition 543 00:37:04,940 --> 00:37:06,220 is going to be. 544 00:37:06,220 --> 00:37:12,590 It is going to be minus beta E minus TS. 545 00:37:12,590 --> 00:37:16,660 Rather than minus Jx, it will be minus mu N. 546 00:37:16,660 --> 00:37:24,400 And this combination is what we call the grand potential g, 547 00:37:24,400 --> 00:37:34,850 which is E minus TS minus mu N. You can do thermodynamics. 548 00:37:34,850 --> 00:37:36,580 You can do probability. 549 00:37:36,580 --> 00:37:41,520 Essentially, in the large end limit, and only 550 00:37:41,520 --> 00:37:46,670 in the large end limit, there's a consistent identification 551 00:37:46,670 --> 00:37:52,320 of most likely states according to the statistical description 552 00:37:52,320 --> 00:37:53,835 and thermodynamic parameters. 553 00:38:05,631 --> 00:38:06,130 Yes. 554 00:38:06,130 --> 00:38:11,210 AUDIENCE: Can one build a more general ensemble or description 555 00:38:11,210 --> 00:38:14,640 where, say, J is not a fixed number, 556 00:38:14,640 --> 00:38:17,430 but I know how it varies? 557 00:38:17,430 --> 00:38:21,340 I know that it's subject to-- say the spring is not exactly 558 00:38:21,340 --> 00:38:22,753 a Hookean spring. 559 00:38:22,753 --> 00:38:23,695 It's not linear. 560 00:38:23,695 --> 00:38:25,810 The proportion is not linear. 561 00:38:25,810 --> 00:38:28,310 PROFESSOR: I think you are then describing a [? composing ?] 562 00:38:28,310 --> 00:38:30,040 system, right? 563 00:38:30,040 --> 00:38:33,820 Because it may be that your system 564 00:38:33,820 --> 00:38:37,470 itself has some nonlinear properties 565 00:38:37,470 --> 00:38:40,090 for its force as a function of displacement. 566 00:38:40,090 --> 00:38:42,130 And most generally, it will. 567 00:38:42,130 --> 00:38:43,980 Very soon, we will look at the pressure 568 00:38:43,980 --> 00:38:47,380 of a gas, which will be some complicated function 569 00:38:47,380 --> 00:38:50,700 of its density-- does not necessarily have to be linear. 570 00:38:50,700 --> 00:38:53,240 So the system itself could have all kinds 571 00:38:53,240 --> 00:38:55,720 of nonlinear dependencies. 572 00:38:55,720 --> 00:38:59,190 But you calculate the corresponding force 573 00:38:59,190 --> 00:39:00,390 through this procedure. 574 00:39:00,390 --> 00:39:05,270 And it will vary in some nonlinear function of density. 575 00:39:05,270 --> 00:39:08,260 If you're asking, can I do something 576 00:39:08,260 --> 00:39:11,950 in which I couple this to another system, which 577 00:39:11,950 --> 00:39:14,720 has some nonlinear properties, then I 578 00:39:14,720 --> 00:39:16,960 would say that really what you're doing 579 00:39:16,960 --> 00:39:18,980 is you're putting two systems together. 580 00:39:18,980 --> 00:39:23,290 And you should be doing separate thermodynamical calculations 581 00:39:23,290 --> 00:39:25,730 for each system, and then put the constraint 582 00:39:25,730 --> 00:39:27,430 that the J of this system should equal 583 00:39:27,430 --> 00:39:31,280 the J of the other system. 584 00:39:31,280 --> 00:39:35,186 AUDIENCE: I don't think that's quite what I'm getting at. 585 00:39:35,186 --> 00:39:41,260 I'm thinking here, the J is not given 586 00:39:41,260 --> 00:39:42,721 by another thermodynamical system. 587 00:39:42,721 --> 00:39:45,230 We're just applying it-- like we're not 588 00:39:45,230 --> 00:39:49,072 applying thermodynamics to whatever is asserting to J. 589 00:39:49,072 --> 00:39:54,000 PROFESSOR: No, I'm trying to mimic thermodynamics. 590 00:39:54,000 --> 00:39:56,805 So in thermodynamics, I have a way 591 00:39:56,805 --> 00:39:59,070 of describing equilibrium systems 592 00:39:59,070 --> 00:40:02,080 in terms of a certain set of variables. 593 00:40:02,080 --> 00:40:06,420 And given that set of variables, there are conjugate variables. 594 00:40:06,420 --> 00:40:08,420 So I'm constructing something that 595 00:40:08,420 --> 00:40:11,230 is analogous to thermodynamics. 596 00:40:11,230 --> 00:40:14,270 It may be that you want to do something else. 597 00:40:14,270 --> 00:40:17,210 And my perspective is that the best way 598 00:40:17,210 --> 00:40:20,560 to do something else is to sort of imagine 599 00:40:20,560 --> 00:40:23,380 different systems that are in contact with each other. 600 00:40:23,380 --> 00:40:25,360 Each one of them you do thermodynamics, 601 00:40:25,360 --> 00:40:28,130 and then you put equilibrium between them. 602 00:40:28,130 --> 00:40:30,570 If I want to solve some complicated system 603 00:40:30,570 --> 00:40:36,000 mechanically, then I sort of break it down into the forces 604 00:40:36,000 --> 00:40:38,780 that are acting on one, forces that are acting on the other, 605 00:40:38,780 --> 00:40:40,790 and then how this one is responding, 606 00:40:40,790 --> 00:40:42,750 how that one is responding. 607 00:40:42,750 --> 00:40:45,600 I don't see any advantage to having 608 00:40:45,600 --> 00:40:48,850 a more complicated mechanical description 609 00:40:48,850 --> 00:40:52,780 of an individual system. 610 00:40:52,780 --> 00:40:54,990 AUDIENCE: It's not like in reality. 611 00:40:54,990 --> 00:40:57,245 It's hard to obtain situations where 612 00:40:57,245 --> 00:40:58,720 the external force is fixed. 613 00:40:58,720 --> 00:41:00,470 Really, if we're doing an experiment here, 614 00:41:00,470 --> 00:41:04,466 the pressure in the atmosphere is a fixed number. 615 00:41:04,466 --> 00:41:06,314 But maybe in other circumstances-- 616 00:41:06,314 --> 00:41:09,680 PROFESSOR: Yes, so if you are in another circumstance, 617 00:41:09,680 --> 00:41:13,520 like you have one gas that is expanding in a nonlinear 618 00:41:13,520 --> 00:41:16,520 medium, then what I would do is I 619 00:41:16,520 --> 00:41:18,720 would calculate the pressure of gas 620 00:41:18,720 --> 00:41:21,270 as a function of its volume, et cetera. 621 00:41:21,270 --> 00:41:24,080 I would calculate the response of that medium 622 00:41:24,080 --> 00:41:25,840 as a function of stress, or whatever 623 00:41:25,840 --> 00:41:27,930 else I'm exerting on that. 624 00:41:27,930 --> 00:41:31,320 And I say that I am constrained to move around the trajectory 625 00:41:31,320 --> 00:41:33,450 where this force equals this force. 626 00:41:43,040 --> 00:41:46,970 So now let's carry out this procedure for the ideal gas. 627 00:41:51,500 --> 00:41:58,410 And the microscopic description of the ideal gas 628 00:41:58,410 --> 00:42:06,460 was that I have to sum over-- its Hamiltonian is composed 629 00:42:06,460 --> 00:42:09,750 of the kinetic energies of the particles. 630 00:42:09,750 --> 00:42:12,450 So I have their momentum. 631 00:42:12,450 --> 00:42:17,050 Plus the potential that we said I'm going to assume 632 00:42:17,050 --> 00:42:22,620 describes box of volume mu. 633 00:42:22,620 --> 00:42:27,290 And it is ideal, because we don't 634 00:42:27,290 --> 00:42:29,010 have interactions among particles. 635 00:42:29,010 --> 00:42:35,030 We will put that shortly for the next version. 636 00:42:35,030 --> 00:42:37,720 Now, given this, I can go through 637 00:42:37,720 --> 00:42:39,850 the various prescriptions. 638 00:42:39,850 --> 00:42:43,130 I can go microcanonically and say 639 00:42:43,130 --> 00:42:47,850 that I know the energy volume and the number of particles. 640 00:42:47,850 --> 00:42:52,870 And in this prescription, we said that the probability 641 00:42:52,870 --> 00:42:59,320 is either 0 or 1 divided by omega. 642 00:42:59,320 --> 00:43:08,510 And this is if qi not in box and sum over i Pi 643 00:43:08,510 --> 00:43:14,743 squared over 2m not equal to E and one otherwise. 644 00:43:14,743 --> 00:43:16,118 AUDIENCE: Is there any reason why 645 00:43:16,118 --> 00:43:19,494 we don't use a direct delta there? 646 00:43:19,494 --> 00:43:20,910 PROFESSOR: I don't know how to use 647 00:43:20,910 --> 00:43:24,100 the direct delta for reading a box. 648 00:43:24,100 --> 00:43:27,210 For the energy, the reason I don't 649 00:43:27,210 --> 00:43:30,960 like to do that is because direct deltas have dimensions 650 00:43:30,960 --> 00:43:34,220 of 1 over whatever thing is inside. 651 00:43:34,220 --> 00:43:35,950 Yeah, you could do that. 652 00:43:35,950 --> 00:43:41,010 It's just I prefer not to do it. 653 00:43:41,010 --> 00:43:42,640 But you certainly could do that. 654 00:43:42,640 --> 00:43:44,240 It is certainly something that is 655 00:43:44,240 --> 00:43:48,505 0 or 1 over omega on that surface. 656 00:43:53,950 --> 00:43:57,760 The S for this, what was it? 657 00:43:57,760 --> 00:44:00,350 It was obtained by integrating over 658 00:44:00,350 --> 00:44:04,270 the entirety of the 6N dimensional phase space. 659 00:44:04,270 --> 00:44:08,560 From the Q integrations, we got V to the N. 660 00:44:08,560 --> 00:44:11,030 From the momentum integrations, we 661 00:44:11,030 --> 00:44:15,580 got the surface area of this hypersphere, which 662 00:44:15,580 --> 00:44:22,110 was 2 pi to the 3N over 2, because it was 3N dimensional. 663 00:44:22,110 --> 00:44:25,410 And this was the solid angle. 664 00:44:25,410 --> 00:44:30,920 And then we got the radius, which 665 00:44:30,920 --> 00:44:36,660 is 2mE square root raised to the power of number 666 00:44:36,660 --> 00:44:37,840 of dimensions minus 1. 667 00:44:41,710 --> 00:44:45,730 Then we did something else. 668 00:44:45,730 --> 00:44:51,190 We said that the phase space is more appropriately divided 669 00:44:51,190 --> 00:44:55,250 by N factorial for identical particles. 670 00:44:55,250 --> 00:45:01,410 Here, I introduced some factor of H 671 00:45:01,410 --> 00:45:06,130 to dimensionalize the integrations over PQ space 672 00:45:06,130 --> 00:45:10,820 so that this quantity now did not carry any dimensions. 673 00:45:10,820 --> 00:45:15,535 And oops, this was not S. This was omega. 674 00:45:18,710 --> 00:45:27,900 And S/k, which was log of omega-- 675 00:45:27,900 --> 00:45:30,680 we took the log of that expression. 676 00:45:30,680 --> 00:45:36,890 We got N log of V. From the log of N factorial, 677 00:45:36,890 --> 00:45:40,500 Stirling's approximation gave us a factor of N over e. 678 00:45:44,570 --> 00:45:48,400 All of the other factors were proportional to 3N over 2. 679 00:45:48,400 --> 00:45:50,360 So the N is out front. 680 00:45:50,360 --> 00:45:53,850 I will have a factor of 3 over 2 here. 681 00:45:53,850 --> 00:46:03,990 I have 2 pi mE divided by 3N over 2. 682 00:46:03,990 --> 00:46:07,750 And then there's E from Stirling's approximation. 683 00:46:07,750 --> 00:46:10,640 And that was the entropy of the ideal gas. 684 00:46:10,640 --> 00:46:11,140 Yes. 685 00:46:11,140 --> 00:46:13,966 AUDIENCE: Does that N factorial change 686 00:46:13,966 --> 00:46:17,952 when describing the system with distinct particles? 687 00:46:17,952 --> 00:46:19,160 PROFESSOR: Yes, that's right. 688 00:46:19,160 --> 00:46:22,500 AUDIENCE: So that definition changes? 689 00:46:22,500 --> 00:46:25,300 PROFESSOR: Yes, so this factor of-- if I 690 00:46:25,300 --> 00:46:29,280 have a mixture of gas, and one of them are one kind, 691 00:46:29,280 --> 00:46:31,760 and two of them are another kind, 692 00:46:31,760 --> 00:46:35,760 I would be dividing by N1 factorial, N2 factorial, 693 00:46:35,760 --> 00:46:37,910 and not by N1 plus N2 factorial. 694 00:46:41,510 --> 00:46:46,780 AUDIENCE: So that is the correct definition for all the cases? 695 00:46:46,780 --> 00:46:49,420 PROFESSOR: That's the definition for phase space 696 00:46:49,420 --> 00:46:52,070 of identical particles. 697 00:46:52,070 --> 00:46:54,310 So all of them are identical particle, 698 00:46:54,310 --> 00:46:55,910 is what I have written. 699 00:46:55,910 --> 00:46:57,370 All of them are distinct. 700 00:46:57,370 --> 00:46:59,330 There's no such factor. 701 00:46:59,330 --> 00:47:02,990 If half of them are of one type and identical, half of them 702 00:47:02,990 --> 00:47:04,890 are another type of identical, then 703 00:47:04,890 --> 00:47:10,678 I will have N over 2 factorial, N over 2 factorial. 704 00:47:10,678 --> 00:47:11,634 AUDIENCE: Question. 705 00:47:11,634 --> 00:47:12,518 PROFESSOR: Yeah. 706 00:47:12,518 --> 00:47:14,809 AUDIENCE: So when you're writing the formula for number 707 00:47:14,809 --> 00:47:19,355 of microstates, just a question of dimensions. 708 00:47:19,355 --> 00:47:22,334 You write V to the N. It gives you 709 00:47:22,334 --> 00:47:25,610 something that mentions coordinate to the power 710 00:47:25,610 --> 00:47:29,031 3N times something of dimensions of [INAUDIBLE]. 711 00:47:29,031 --> 00:47:33,385 The power is 3N minus 1 divided by H to 3N. 712 00:47:33,385 --> 00:47:37,110 So overall, this gives you 1 over momentum. 713 00:47:37,110 --> 00:47:42,860 PROFESSOR: OK, let's put a delta E over here-- actually, not 714 00:47:42,860 --> 00:47:49,810 a delta E, but a delta R. OK, so now it's fine. 715 00:47:52,510 --> 00:47:56,080 It's really what I want to make sure is 716 00:47:56,080 --> 00:47:59,305 that the extensive part is correctly taken into account. 717 00:48:01,872 --> 00:48:03,455 I may be missing a factor of dimension 718 00:48:03,455 --> 00:48:06,800 that is of the order of 1 every now and then. 719 00:48:06,800 --> 00:48:10,440 And then you can put some [INAUDIBLE] to correct it. 720 00:48:10,440 --> 00:48:13,510 And it's, again, the same story of the orange. 721 00:48:13,510 --> 00:48:19,690 It's all in the skin, including-- 722 00:48:19,690 --> 00:48:22,330 so the volume is the same thing as the surface area, 723 00:48:22,330 --> 00:48:28,450 which is essentially what I'm-- OK? 724 00:48:28,450 --> 00:48:32,170 So once you have this, you can calculate various quantities. 725 00:48:32,170 --> 00:48:46,810 Again, the S is dE over T plus PdV over T minus mu dN over T. 726 00:48:46,810 --> 00:48:51,920 So you can immediately identify, for example, 727 00:48:51,920 --> 00:48:58,180 that 1 over T derivative with respect to energy 728 00:48:58,180 --> 00:49:02,880 would give me a factor of N/E. If I take 729 00:49:02,880 --> 00:49:08,060 a derivative with respect to volume, P over T, 730 00:49:08,060 --> 00:49:10,310 derivative of this object with respect to volume 731 00:49:10,310 --> 00:49:14,870 will give me N over V. 732 00:49:14,870 --> 00:49:20,430 And mu over T is-- oops, actually, 733 00:49:20,430 --> 00:49:23,620 in all of these cases, I forgot the kB. 734 00:49:23,620 --> 00:49:25,040 So I have to restore it. 735 00:49:29,650 --> 00:49:35,750 And here I would get kB log of the N out front. 736 00:49:35,750 --> 00:49:44,490 I can throw out V/N 4 pi m E over 3N. 737 00:49:44,490 --> 00:49:48,520 And I forgot the H's. 738 00:49:48,520 --> 00:49:51,490 So the H's would appear as an H squared here. 739 00:49:51,490 --> 00:49:55,370 They will appear as an H squared here raised to the 3/2 power. 740 00:50:05,440 --> 00:50:11,450 We also said that I can look at the probability 741 00:50:11,450 --> 00:50:13,790 of a single particle having momentum P1. 742 00:50:16,330 --> 00:50:18,520 And we showed that essentially I have 743 00:50:18,520 --> 00:50:22,280 to integrate over everything else if I'm not interested, 744 00:50:22,280 --> 00:50:27,110 such as the volume of the particle number one. 745 00:50:27,110 --> 00:50:34,520 The normalization was omega E, V, N. And integrating 746 00:50:34,520 --> 00:50:38,580 over particles numbers two to N, where 747 00:50:38,580 --> 00:50:41,350 the energy that is left to them is 748 00:50:41,350 --> 00:50:47,010 E minus the kinetic energy of the one particle, 749 00:50:47,010 --> 00:50:49,920 gave me this expression. 750 00:50:49,920 --> 00:50:52,100 And essentially, we found that this 751 00:50:52,100 --> 00:50:56,580 was proportional to one side divided by E. 1 752 00:50:56,580 --> 00:51:01,690 minus P1 squared over 2m raised to the power that was very 753 00:51:01,690 --> 00:51:04,875 close to 3N over 2 minus/plus something, 754 00:51:04,875 --> 00:51:08,170 and that this was proportional, therefore, 755 00:51:08,170 --> 00:51:15,430 to this P1 squared over 2m times-- P1 756 00:51:15,430 --> 00:51:20,135 squared 2mE-- 3N over 2E. 757 00:51:24,300 --> 00:51:32,242 And 3N over 2E from here we see is the same thing as 1 over kT. 758 00:51:36,400 --> 00:51:38,870 So we can, with this prescription, 759 00:51:38,870 --> 00:51:43,950 calculate all of the properties of the ideal gas, 760 00:51:43,950 --> 00:51:50,830 including probability to see one particle with some momentum. 761 00:51:50,830 --> 00:51:56,970 Now, if I do the same thing in the canonical form, in which 762 00:51:56,970 --> 00:52:00,750 the microstate that I'm looking at is temperature, 763 00:52:00,750 --> 00:52:04,950 volume, number of particles, then 764 00:52:04,950 --> 00:52:08,410 the probability of a microstate, given 765 00:52:08,410 --> 00:52:11,630 that I've specified now the temperature, 766 00:52:11,630 --> 00:52:15,980 is proportional to the energy of that microstate, which 767 00:52:15,980 --> 00:52:23,940 is e to the minus beta sum over i Pi squared over 2m. 768 00:52:23,940 --> 00:52:27,480 Of course I would have to have all of Qi's in box. 769 00:52:30,540 --> 00:52:33,720 Certainly they cannot go outside the box. 770 00:52:33,720 --> 00:52:35,120 And this has to be normalized. 771 00:52:38,660 --> 00:52:41,480 Now we can see that in this canonical ensemble, 772 00:52:41,480 --> 00:52:45,670 the result that we had to do a couple of lines of algebra 773 00:52:45,670 --> 00:52:50,030 to get, which is that the momentum of a particle 774 00:52:50,030 --> 00:52:54,510 is Gaussian distributed, is automatically satisfied. 775 00:52:54,510 --> 00:52:58,260 And in this ensemble, each one of the momenta you can see 776 00:52:58,260 --> 00:53:02,370 is independently distributed according to this probability 777 00:53:02,370 --> 00:53:05,090 distribution. 778 00:53:05,090 --> 00:53:09,060 So somethings clearly emerge much easier 779 00:53:09,060 --> 00:53:11,690 in this perspective. 780 00:53:11,690 --> 00:53:16,290 And if I were to look at the normalization Z, 781 00:53:16,290 --> 00:53:21,875 the normalization Z I obtained by integrating over 782 00:53:21,875 --> 00:53:23,830 the entirety of the phase space. 783 00:53:23,830 --> 00:53:31,720 So I have to do the integration over d cubed Pi d cubed qi. 784 00:53:31,720 --> 00:53:35,240 Since this is the phase space of identical particles, 785 00:53:35,240 --> 00:53:39,370 we said we have to normalize it by N factorial. 786 00:53:39,370 --> 00:53:42,950 And I had this factor of H to the 3N 787 00:53:42,950 --> 00:53:45,810 to make things dimensionless. 788 00:53:45,810 --> 00:53:52,400 I have to exponentiate this energy sum over i Pi squared 789 00:53:52,400 --> 00:53:58,730 over 2m and just ensure that the qi are inside the box. 790 00:53:58,730 --> 00:54:02,340 So if I integrate the qi, what do I get? 791 00:54:02,340 --> 00:54:03,810 I will get V per particle. 792 00:54:03,810 --> 00:54:07,650 So I have V to the N divided by N factorial. 793 00:54:10,320 --> 00:54:13,100 So that's the volume contribution. 794 00:54:13,100 --> 00:54:15,000 And then what are the P integrations? 795 00:54:15,000 --> 00:54:19,490 Each one of the P integrations is an independent Gaussian. 796 00:54:19,490 --> 00:54:21,680 In fact, each component is independent. 797 00:54:21,680 --> 00:54:24,780 So I have 3N Gaussian integrations. 798 00:54:24,780 --> 00:54:27,310 And I can do Gaussian integrations. 799 00:54:27,310 --> 00:54:31,420 I will get root 2 pi m inverse of beta, 800 00:54:31,420 --> 00:54:36,030 which is kT per Gaussian integration. 801 00:54:36,030 --> 00:54:37,820 And there are 3N of them. 802 00:54:42,950 --> 00:54:47,060 And, oh, I had the factor of h to the 3N. 803 00:54:47,060 --> 00:54:47,870 I will put it here. 804 00:54:54,930 --> 00:55:00,630 So I chose these H's in order to make this phase space 805 00:55:00,630 --> 00:55:02,060 dimensionless. 806 00:55:02,060 --> 00:55:06,680 So the Z that I have now is dimensionless. 807 00:55:06,680 --> 00:55:10,100 So the dimensions of V must be made up 808 00:55:10,100 --> 00:55:13,840 by dimensions of all of these things that are left. 809 00:55:13,840 --> 00:55:17,550 So I'm going to make that explicitly clear by writing it 810 00:55:17,550 --> 00:55:22,470 as 1 over N factorial V over some characteristic volume 811 00:55:22,470 --> 00:55:25,070 raised to the N-th power. 812 00:55:25,070 --> 00:55:29,130 The characteristic volume comes entirely from these factors. 813 00:55:29,130 --> 00:55:32,100 So I have introduced lambda of T, 814 00:55:32,100 --> 00:55:41,740 which is h over root 2 pi mkT, which is the thermal de Broglie 815 00:55:41,740 --> 00:55:42,240 wavelength. 816 00:55:45,960 --> 00:55:49,750 At this stage, this h is just anything 817 00:55:49,750 --> 00:55:52,370 to make things dimensionally work. 818 00:55:52,370 --> 00:55:54,530 When we do quantum mechanics, we will 819 00:55:54,530 --> 00:55:57,920 see that this lens scale has a very important 820 00:55:57,920 --> 00:55:59,260 physical meaning. 821 00:55:59,260 --> 00:56:03,310 As long as the separations of particles on average 822 00:56:03,310 --> 00:56:06,380 is larger than this, you can ignore quantum mechanics. 823 00:56:06,380 --> 00:56:08,000 When it becomes less than this, you 824 00:56:08,000 --> 00:56:09,570 have to include quantum factors. 825 00:56:14,070 --> 00:56:17,540 So then what do we have? 826 00:56:17,540 --> 00:56:23,200 We have that the free energy is minus kT log Z. 827 00:56:23,200 --> 00:56:31,440 So F is minus kT log of this partition function. 828 00:56:31,440 --> 00:56:40,590 Log of that quantity will give me N log V over lambda cubed. 829 00:56:40,590 --> 00:56:43,610 Stirling's formula, log of N factorial, 830 00:56:43,610 --> 00:56:46,880 will give me N log N over e. 831 00:56:46,880 --> 00:56:48,105 So that's the free energy. 832 00:56:50,950 --> 00:56:55,520 Once I have the free energy, I can calculate, let's say, 833 00:56:55,520 --> 00:56:57,090 the volume. 834 00:56:57,090 --> 00:57:02,040 Let's see, dF, which is d of E minus TS, 835 00:57:02,040 --> 00:57:08,930 is minus SdT minus PdV, because work 836 00:57:08,930 --> 00:57:11,800 of the gas we had identified as minus PdV. 837 00:57:11,800 --> 00:57:14,820 I have mu dN. 838 00:57:14,820 --> 00:57:16,180 What do we have, therefore? 839 00:57:16,180 --> 00:57:27,780 We have that, for example, P is minus dF by dV at constant T 840 00:57:27,780 --> 00:57:32,300 and N. So I have to go and look at this expression 841 00:57:32,300 --> 00:57:35,880 where the V up here, it just appears in log V. 842 00:57:35,880 --> 00:57:45,820 So the answer is going to be NkT over V. 843 00:57:45,820 --> 00:57:49,740 I can calculate the chemical potential. 844 00:57:49,740 --> 00:57:56,630 Mu will be dF by dN at constant T and V, 845 00:57:56,630 --> 00:58:01,210 so just take the derivative with respect to N. So what do I get? 846 00:58:01,210 --> 00:58:10,930 I will get minus kT log of V over N lambda cubed. 847 00:58:10,930 --> 00:58:13,420 And this E will disappear when I take 848 00:58:13,420 --> 00:58:16,920 the derivative with respected to that log inside. 849 00:58:16,920 --> 00:58:22,340 So that's the formula for my mu. 850 00:58:22,340 --> 00:58:25,520 I won't calculate entropy. 851 00:58:25,520 --> 00:58:33,830 But I will calculate the energy, noting that in this ensemble, 852 00:58:33,830 --> 00:58:39,350 a nice way of calculating energy is minus d log Z by d beta. 853 00:58:39,350 --> 00:58:44,900 So this is minus d log Z by d beta. 854 00:58:44,900 --> 00:58:59,170 And my Z, you can see, has a bunch of things-- V, N, et 855 00:58:59,170 --> 00:58:59,690 cetera. 856 00:58:59,690 --> 00:59:04,140 But if I focus on temperature, which is the inverse beta, 857 00:59:04,140 --> 00:59:07,610 you can see it appears with a factor of 3N over 2. 858 00:59:07,610 --> 00:59:12,710 So this is beta to the minus 3N over 2. 859 00:59:12,710 --> 00:59:18,050 So this is going to be 3N over 2 derivative of log beta 860 00:59:18,050 --> 00:59:21,430 with respect to beta, which is 1 over beta, 861 00:59:21,430 --> 00:59:24,250 which is the same thing as 3 over 2 NkT. 862 00:59:58,080 --> 01:00:05,460 So what if I wanted to maintain the system at some fixed 863 01:00:05,460 --> 01:00:07,820 temperature, but rather than telling you 864 01:00:07,820 --> 01:00:10,200 what the volume of the box is, I will tell you 865 01:00:10,200 --> 01:00:15,050 what its pressure is and how many particles I have? 866 01:00:15,050 --> 01:00:18,690 How can I ensure that I have a particular pressure? 867 01:00:18,690 --> 01:00:22,740 You can imagine that this is the box that contains my gas. 868 01:00:22,740 --> 01:00:25,920 And I put a weight on top of some kind 869 01:00:25,920 --> 01:00:32,300 of a piston that can move over the gas. 870 01:00:32,300 --> 01:00:38,230 So then you would say that the net energy that I 871 01:00:38,230 --> 01:00:42,820 have to look at is the kinetic energy of the gas 872 01:00:42,820 --> 01:00:46,050 particles here plus the potential energy 873 01:00:46,050 --> 01:00:48,910 of this weight that is going up and down. 874 01:00:48,910 --> 01:00:50,680 If you like, that potential energy 875 01:00:50,680 --> 01:00:58,310 is going to be mass times delta H. Delta H times area gives you 876 01:00:58,310 --> 01:00:59,500 volume. 877 01:00:59,500 --> 01:01:04,320 Mass times G divided by area will give you pressure. 878 01:01:04,320 --> 01:01:06,900 So the combination of those two is the same thing 879 01:01:06,900 --> 01:01:08,830 as pressure times volume. 880 01:01:08,830 --> 01:01:10,690 So this is going to be the same thing 881 01:01:10,690 --> 01:01:17,310 as minus sum over i Pi squared over 2m. 882 01:01:17,310 --> 01:01:21,940 For all of the gas particles for this additional weight that 883 01:01:21,940 --> 01:01:24,420 is going up and down, it will give me 884 01:01:24,420 --> 01:01:27,910 a contribution that is minus beta PV. 885 01:01:33,380 --> 01:01:40,090 And so this is the probability in this state. 886 01:01:40,090 --> 01:01:43,660 It is going to be the same as this provided 887 01:01:43,660 --> 01:01:49,240 that I divide by some Gibbs partition function. 888 01:01:52,410 --> 01:01:55,840 So this is the probability of the microstate. 889 01:01:55,840 --> 01:02:00,480 So what is the Gibbs partition function? 890 01:02:00,480 --> 01:02:02,350 Well, what is the normalization? 891 01:02:02,350 --> 01:02:05,540 I now have one additional variable, 892 01:02:05,540 --> 01:02:10,440 which is where this piston is located in order to ensure 893 01:02:10,440 --> 01:02:12,950 that it is at the right pressure. 894 01:02:12,950 --> 01:02:18,255 So I have to integrate also over the additional volume. 895 01:02:21,680 --> 01:02:30,020 This additional factor only depends on PV. 896 01:02:30,020 --> 01:02:33,280 And then I have to integrate given 897 01:02:33,280 --> 01:02:35,640 that I have some particular V that I then 898 01:02:35,640 --> 01:02:40,870 have to integrate over all of the microstates that 899 01:02:40,870 --> 01:02:43,720 are confined within this volume. 900 01:02:43,720 --> 01:02:50,420 Their momenta and their coordinates-- what is that? 901 01:02:50,420 --> 01:02:51,720 I just calculated that. 902 01:02:51,720 --> 01:02:57,280 That's the partition function as a function of T, V, and N. 903 01:02:57,280 --> 01:03:01,700 So for a fixed V, I already did the integration 904 01:03:01,700 --> 01:03:05,425 over all microscopic degrees of freedom. 905 01:03:05,425 --> 01:03:08,850 I have one more integration to do over the volume. 906 01:03:08,850 --> 01:03:10,240 And that's it. 907 01:03:10,240 --> 01:03:13,370 So if you like, this is like doing a Laplace transform. 908 01:03:13,370 --> 01:03:20,750 To go from Z of T, V, and N, to this Z tilde of T, P, and N 909 01:03:20,750 --> 01:03:23,700 is making some kind of a Laplace transformation 910 01:03:23,700 --> 01:03:27,280 from one variable to another variable. 911 01:03:27,280 --> 01:03:30,990 And now I know actually what my answer 912 01:03:30,990 --> 01:03:34,330 was for the partition function. 913 01:03:34,330 --> 01:03:41,890 It was 1 over N factorial V over lambda cubed raised 914 01:03:41,890 --> 01:03:50,900 to the power of N. So I have 1 over N factorial lambda 915 01:03:50,900 --> 01:03:52,450 to the power of 3N. 916 01:03:52,450 --> 01:03:55,430 And then I have to do one of these integrals 917 01:03:55,430 --> 01:04:01,550 that we have seen many times, the integral of V to the N 918 01:04:01,550 --> 01:04:04,540 against an exponential. 919 01:04:04,540 --> 01:04:06,050 That's something that actually we 920 01:04:06,050 --> 01:04:12,180 used in order to define N factorial, except that I have 921 01:04:12,180 --> 01:04:17,010 to dimensionalize this V. So I will get a factor of beta P 922 01:04:17,010 --> 01:04:19,260 to the power of N plus 1. 923 01:04:22,040 --> 01:04:24,330 The N factorials cancel. 924 01:04:24,330 --> 01:04:28,250 And the answer is beta P to the power of N 925 01:04:28,250 --> 01:04:44,410 plus 1 divided by lambda cubed to the power of N. 926 01:04:44,410 --> 01:04:52,060 So my Gibbs free energy, which is minus 927 01:04:52,060 --> 01:04:57,430 kT log of the Gibbs partition function, 928 01:04:57,430 --> 01:05:10,100 is going to be minus NkT log of this object. 929 01:05:10,100 --> 01:05:14,380 I have ignored the difference between N and N plus 1. 930 01:05:14,380 --> 01:05:16,960 And what I will get here is the combination beta 931 01:05:16,960 --> 01:05:20,490 P lambda cubed. 932 01:05:20,490 --> 01:05:21,760 Yes. 933 01:05:21,760 --> 01:05:24,170 AUDIENCE: Is your beta P to the N plus 1 934 01:05:24,170 --> 01:05:26,150 in the numerator or the denominator? 935 01:05:26,150 --> 01:05:31,341 PROFESSOR: Thank you, it should be in the denominator, which 936 01:05:31,341 --> 01:05:41,220 means that-- OK, this one is correct. 937 01:05:41,220 --> 01:05:43,750 I guess this one I was going by dimensions. 938 01:05:43,750 --> 01:05:46,420 Because beta PV is dimensionless. 939 01:05:49,120 --> 01:05:51,340 All right, yes. 940 01:05:51,340 --> 01:05:53,680 AUDIENCE: One other thing, it seems 941 01:05:53,680 --> 01:05:57,386 like there's a dimensional mismatch in your expression 942 01:05:57,386 --> 01:06:03,599 for Z. Because you have an extra factor of beta P-- 943 01:06:03,599 --> 01:06:09,480 PROFESSOR: Exactly right, because this object 944 01:06:09,480 --> 01:06:15,110 is a probability density that involves a factor of volume. 945 01:06:15,110 --> 01:06:18,220 And as a probability density, the dimension of this 946 01:06:18,220 --> 01:06:20,060 will carry an extra factor of volume. 947 01:06:22,780 --> 01:06:27,565 So if I really wanted to make this quantity dimensionless 948 01:06:27,565 --> 01:06:31,900 also, I would need to divide by something 949 01:06:31,900 --> 01:06:35,850 that has some dimension of volume. 950 01:06:35,850 --> 01:06:41,100 But alternatively, I can recognize that indeed this 951 01:06:41,100 --> 01:06:43,330 is a probability density in volume. 952 01:06:43,330 --> 01:06:46,180 So it will have the dimensions of volume. 953 01:06:46,180 --> 01:06:49,130 And again, as I said, what I'm really always 954 01:06:49,130 --> 01:06:52,750 careful to make sure that is dimensionless is the thing that 955 01:06:52,750 --> 01:06:57,280 is proportional to N. If there's a log of a single dimension 956 01:06:57,280 --> 01:07:00,310 out here, typically we don't have to worry about it. 957 01:07:00,310 --> 01:07:02,085 But if you think about its origin, 958 01:07:02,085 --> 01:07:04,750 the origin is indeed that this quantity 959 01:07:04,750 --> 01:07:06,034 is a probability density. 960 01:07:13,450 --> 01:07:17,110 But it will, believe me, not change anything 961 01:07:17,110 --> 01:07:18,944 in your life to ignore that. 962 01:07:23,330 --> 01:07:27,960 All right, so once we have this G, then we recognize-- again, 963 01:07:27,960 --> 01:07:29,920 hopefully I didn't make any mistakes. 964 01:07:29,920 --> 01:07:35,210 G is E plus PV minus TS. 965 01:07:35,210 --> 01:07:46,952 So dG should be minus SdT plus VdP plus mu dN. 966 01:07:52,710 --> 01:07:56,730 So that, for example, in this ensemble I can ask-- well, 967 01:07:56,730 --> 01:07:58,170 I told you what the pressure is. 968 01:07:58,170 --> 01:07:59,820 What's the volume? 969 01:07:59,820 --> 01:08:04,356 Volume is going to be obtained as dG by dP 970 01:08:04,356 --> 01:08:07,210 at constant temperature and number. 971 01:08:07,210 --> 01:08:10,740 So these two are constant. 972 01:08:10,740 --> 01:08:15,420 Log P, its derivative is going to give me NkT over P. 973 01:08:15,420 --> 01:08:18,439 So again, I get another form of the ideal gas 974 01:08:18,439 --> 01:08:20,090 equation of state. 975 01:08:20,090 --> 01:08:22,700 I can ask, what's the chemical potential? 976 01:08:22,700 --> 01:08:28,720 It is going to be dG by dN at constant T and P. 977 01:08:28,720 --> 01:08:31,090 So I go and look at the N dependence. 978 01:08:31,090 --> 01:08:34,609 And I notice that there's just an N dependence out front. 979 01:08:34,609 --> 01:08:41,300 So what I will get is kT log of beta P lambda cubed. 980 01:08:41,300 --> 01:08:44,020 And if you like, you can check that, say, 981 01:08:44,020 --> 01:08:48,200 this expression for the chemical potential 982 01:08:48,200 --> 01:08:54,399 and-- did we derive it somewhere else? 983 01:08:54,399 --> 01:08:56,620 Yes, we derived it over here. 984 01:08:56,620 --> 01:08:59,790 This expression for the chemical potential 985 01:08:59,790 --> 01:09:04,390 are identical once you take advantage of the ideal gas 986 01:09:04,390 --> 01:09:06,590 equation of state to convert the V 987 01:09:06,590 --> 01:09:16,069 over N in that expression to beta P. 988 01:09:16,069 --> 01:09:23,420 And finally, we have the grand canonical. 989 01:09:23,420 --> 01:09:24,697 Let's do that also. 990 01:09:38,140 --> 01:09:43,220 So now we are going to look at an ensemble where I tell you 991 01:09:43,220 --> 01:09:46,470 what the temperature is and the chemical potential. 992 01:09:46,470 --> 01:09:49,700 But I have to tell you what the volume is. 993 01:09:49,700 --> 01:09:53,324 And then the statement is that the probability 994 01:09:53,324 --> 01:09:57,640 of a particular microstate that I will now indicate mu 995 01:09:57,640 --> 01:10:03,700 s force system-- not to be confused with the chemical 996 01:10:03,700 --> 01:10:09,310 potential-- is proportional to e to the beta mu 997 01:10:09,310 --> 01:10:16,890 N minus beta H of the microstate energy. 998 01:10:16,890 --> 01:10:22,770 And the normalization is this Q, which will be a grand partition 999 01:10:22,770 --> 01:10:28,160 function that is function of T, V, and mu. 1000 01:10:28,160 --> 01:10:31,330 How is this probability normalized? 1001 01:10:31,330 --> 01:10:38,330 Well, I'm spanning over a space of microstates 1002 01:10:38,330 --> 01:10:41,030 that have indefinite number. 1003 01:10:41,030 --> 01:10:47,450 Their number runs, presumably, all the way from 0 to infinity. 1004 01:10:47,450 --> 01:10:53,280 And I have to multiply each particular segment that 1005 01:10:53,280 --> 01:10:58,090 has N of these present with e to the beta mu N. 1006 01:10:58,090 --> 01:11:02,930 Now, once I have that segment, what I need to do 1007 01:11:02,930 --> 01:11:06,820 is to sum over all coordinates and momenta 1008 01:11:06,820 --> 01:11:10,100 as appropriate to a system of N particles. 1009 01:11:10,100 --> 01:11:13,690 And that, once more, is my partition function 1010 01:11:13,690 --> 01:11:23,590 Z of T, V, and N. And so since I know what that expression is, 1011 01:11:23,590 --> 01:11:27,637 I can substitute it in some e to the beta mu 1012 01:11:27,637 --> 01:11:34,840 N. And Z is 1 over N factorial V over lambda cubed raised 1013 01:11:34,840 --> 01:11:41,240 to the power of N. 1014 01:11:41,240 --> 01:11:45,530 Now fortunately, that's a sum that I recognize. 1015 01:11:45,530 --> 01:11:48,530 It is 1 over N factorial something raised 1016 01:11:48,530 --> 01:11:51,190 to the N-th power summed over all N, 1017 01:11:51,190 --> 01:11:53,900 which is the summation for the exponential. 1018 01:11:53,900 --> 01:11:58,770 So this is the exponential of e to the beta mu 1019 01:11:58,770 --> 01:11:59,850 V over lambda cubed. 1020 01:12:08,910 --> 01:12:13,190 So once I have this, I can construct 1021 01:12:13,190 --> 01:12:24,150 my G, which is minus kT log of Q, which is minus 1022 01:12:24,150 --> 01:12:32,380 kT e to the beta mu V divided by lambda cubed. 1023 01:12:38,320 --> 01:12:42,615 Now, note that in all of the other expressions 1024 01:12:42,615 --> 01:12:47,030 that I had all of these logs of something, they were extensive. 1025 01:12:47,030 --> 01:12:50,680 And the extensivity was ensured by having results 1026 01:12:50,680 --> 01:12:52,570 that were ultimately proportional 1027 01:12:52,570 --> 01:12:55,205 to N for these logs. 1028 01:12:55,205 --> 01:12:58,410 Let's say I have here, I have an N here. 1029 01:12:58,410 --> 01:13:00,380 For the s, I have an N there. 1030 01:13:00,380 --> 01:13:04,770 Previously I had also an N here. 1031 01:13:04,770 --> 01:13:06,850 Now, in this ensemble, I don't have 1032 01:13:06,850 --> 01:13:10,660 N. Extensivity is insured by this thing being proportional 1033 01:13:10,660 --> 01:13:22,770 to G. Now also remember that G was E minus TS minus mu N. 1034 01:13:22,770 --> 01:13:26,540 But we had another result for extensivity, 1035 01:13:26,540 --> 01:13:33,130 that for extensive systems, E was TS plus mu N minus PV. 1036 01:13:33,130 --> 01:13:38,930 So this is in fact because of extensivity, we 1037 01:13:38,930 --> 01:13:43,600 had expected it to be proportional to the volume. 1038 01:13:43,600 --> 01:13:47,270 And so this combination should end up 1039 01:13:47,270 --> 01:13:50,170 being in fact the pressure. 1040 01:13:50,170 --> 01:13:54,070 I can see what the pressure is in different ways. 1041 01:13:54,070 --> 01:13:57,422 I can, for example, look at what this dG is. 1042 01:13:57,422 --> 01:13:59,780 It is minus SdT. 1043 01:13:59,780 --> 01:14:04,243 It is minus PdV minus Nd mu. 1044 01:14:07,530 --> 01:14:11,860 I could, for example, identify the pressure 1045 01:14:11,860 --> 01:14:15,390 by taking a derivative of this with respect to volume. 1046 01:14:15,390 --> 01:14:17,100 But it is proportional to volume. 1047 01:14:17,100 --> 01:14:21,090 So I again get that this combination really 1048 01:14:21,090 --> 01:14:23,760 should be pressure. 1049 01:14:23,760 --> 01:14:26,530 You say, I don't recognize that as a pressure. 1050 01:14:26,530 --> 01:14:29,460 You say, well, it's because the formula for pressure 1051 01:14:29,460 --> 01:14:33,430 that we have been using is always in terms of N. So let's 1052 01:14:33,430 --> 01:14:34,030 check that. 1053 01:14:34,030 --> 01:14:35,840 So what is N? 1054 01:14:35,840 --> 01:14:42,560 I can get N from minus dG by d mu. 1055 01:14:42,560 --> 01:14:46,980 And what happens if I do that? 1056 01:14:46,980 --> 01:14:49,030 When I take a derivative with respect to mu, 1057 01:14:49,030 --> 01:14:52,030 I will bring down a factor of beta. 1058 01:14:52,030 --> 01:14:53,480 Beta will kill the kT. 1059 01:14:53,480 --> 01:15:00,640 I will get e to the beta mu V over lambda cubed, which 1060 01:15:00,640 --> 01:15:03,620 by the way is also these expressions 1061 01:15:03,620 --> 01:15:07,840 that we had previously for the relationship between mu 1062 01:15:07,840 --> 01:15:10,480 and N over V lambda cubed if I just 1063 01:15:10,480 --> 01:15:12,810 take the log of this expression. 1064 01:15:12,810 --> 01:15:17,640 And then if I substitute e to the beta mu V over lambda cubed 1065 01:15:17,640 --> 01:15:20,860 to BN, you can see that I have the thing that I was calling 1066 01:15:20,860 --> 01:15:33,080 pressure is indeed NkT over V. So everything 1067 01:15:33,080 --> 01:15:35,070 is consistent with the ideal gas law. 1068 01:15:35,070 --> 01:15:39,370 The chemical potential comes out consistently-- extensivity, 1069 01:15:39,370 --> 01:15:44,300 everything is correctly identified. 1070 01:15:44,300 --> 01:15:47,390 Maybe one more thing that I note here 1071 01:15:47,390 --> 01:15:52,430 is that, for this ideal gas and this particular form that I 1072 01:15:52,430 --> 01:16:01,670 have for this object-- so let's maybe do something here. 1073 01:16:01,670 --> 01:16:06,960 Note that the N appears in an exponential with e 1074 01:16:06,960 --> 01:16:08,400 to the beta mu. 1075 01:16:08,400 --> 01:16:11,790 So another way that I could have gotten my N 1076 01:16:11,790 --> 01:16:18,157 would have been d log Q with respect to beta mu. 1077 01:16:20,840 --> 01:16:29,030 And again, my log Q is simply V over lambda cubed 1078 01:16:29,030 --> 01:16:31,820 e to the beta mu. 1079 01:16:31,820 --> 01:16:34,300 And you can check that if I do that, I 1080 01:16:34,300 --> 01:16:39,060 will get this formula that I had for N. Well, 1081 01:16:39,060 --> 01:16:44,380 the thing is that I can get various cumulants 1082 01:16:44,380 --> 01:16:49,200 of this object by continuing to take derivatives. 1083 01:16:49,200 --> 01:16:58,430 So I take m derivatives of log of Q with respect to beta mu. 1084 01:17:02,580 --> 01:17:05,245 So I have to keep taking derivatives of the exponential. 1085 01:17:08,080 --> 01:17:12,560 And as long as I keep taking the derivative of the exponential, 1086 01:17:12,560 --> 01:17:15,310 I will get the exponential back. 1087 01:17:15,310 --> 01:17:17,820 So all of these things are really the same thing. 1088 01:17:20,380 --> 01:17:24,640 So all cumulants of the number fluctuations of the gas 1089 01:17:24,640 --> 01:17:27,550 are really the same thing as a number. 1090 01:17:27,550 --> 01:17:30,250 Can you remember what that says? 1091 01:17:30,250 --> 01:17:33,140 What's the distribution? 1092 01:17:33,140 --> 01:17:35,610 AUDIENCE: [INAUDIBLE] 1093 01:17:35,610 --> 01:17:37,230 PROFESSOR: Poisson, very good. 1094 01:17:37,230 --> 01:17:41,430 The distribution where all of the cumulants were the same 1095 01:17:41,430 --> 01:17:43,530 is Poisson distribution. 1096 01:17:43,530 --> 01:17:46,550 Essentially it says that if I take a box, 1097 01:17:46,550 --> 01:17:52,070 or if I just look at that imaginary volume in this room, 1098 01:17:52,070 --> 01:17:54,150 and count the number of particles, 1099 01:17:54,150 --> 01:17:56,820 as long as it is almost identical, 1100 01:17:56,820 --> 01:18:00,060 the distribution of the number of particles within the volume 1101 01:18:00,060 --> 01:18:03,970 is Poisson. 1102 01:18:03,970 --> 01:18:09,170 You know all of the fluctuations, et cetera. 1103 01:18:09,170 --> 01:18:09,875 Yes. 1104 01:18:09,875 --> 01:18:11,250 AUDIENCE: So this expression, you 1105 01:18:11,250 --> 01:18:17,060 have N equals e to the beta mu V divided by lambda to the third. 1106 01:18:17,060 --> 01:18:19,550 Considering that expression, could you then 1107 01:18:19,550 --> 01:18:22,040 say that the exponential quantity is proportional 1108 01:18:22,040 --> 01:18:26,024 to the phase space density [INAUDIBLE]? 1109 01:18:26,024 --> 01:18:28,016 PROFESSOR: Let's rearrange this. 1110 01:18:28,016 --> 01:18:35,150 Beta mu is log of N over V lambda cubed. 1111 01:18:35,150 --> 01:18:39,780 This is a single particle density. 1112 01:18:39,780 --> 01:18:44,780 So beta mu, or the chemical potential up to a factor of kT, 1113 01:18:44,780 --> 01:18:52,350 is the log of how many particles fit within one de Broglie 1114 01:18:52,350 --> 01:18:54,520 volume. 1115 01:18:54,520 --> 01:18:58,240 And this expression is in fact correct only in the limit 1116 01:18:58,240 --> 01:18:59,880 where this is small. 1117 01:18:59,880 --> 01:19:01,543 And as we shall see later on, there 1118 01:19:01,543 --> 01:19:03,560 will be quantum mechanical corrections 1119 01:19:03,560 --> 01:19:05,635 when this combination is large. 1120 01:19:05,635 --> 01:19:10,490 So this combination is very important in identifying 1121 01:19:10,490 --> 01:19:12,362 when things become quantum mechanic. 1122 01:19:21,210 --> 01:19:25,220 All right, so let's just give a preamble 1123 01:19:25,220 --> 01:19:27,390 of what we will be doing next time. 1124 01:19:34,352 --> 01:19:38,570 I should have erased the [INAUDIBLE]. 1125 01:19:38,570 --> 01:19:46,610 So we want to now do interacting systems. 1126 01:19:46,610 --> 01:19:51,160 So this one example of the ideal gas I 1127 01:19:51,160 --> 01:19:54,070 did for you to [INAUDIBLE] in all possible ensembles. 1128 01:19:54,070 --> 01:19:56,980 And I could do that, because it was 1129 01:19:56,980 --> 01:20:00,920 a collection of non-interacting degrees of freedom. 1130 01:20:00,920 --> 01:20:06,140 As soon as I have interactions among my huge number of degrees 1131 01:20:06,140 --> 01:20:08,860 of freedom, the story changes. 1132 01:20:08,860 --> 01:20:13,270 So let's, for example, look at a generalization of what 1133 01:20:13,270 --> 01:20:24,680 I had written before-- a one particle description which 1134 01:20:24,680 --> 01:20:30,310 if I stop here gives me an ideal system, and potentially 1135 01:20:30,310 --> 01:20:35,640 some complicated interaction among all of these coordinates. 1136 01:20:35,640 --> 01:20:37,380 This could be a pairwise interaction. 1137 01:20:37,380 --> 01:20:39,880 It could have three particles. 1138 01:20:39,880 --> 01:20:42,560 It could potentially-- at this stage 1139 01:20:42,560 --> 01:20:44,710 I want to write down the most general form. 1140 01:20:49,120 --> 01:20:52,410 I want to see what I can learn about the properties 1141 01:20:52,410 --> 01:20:56,100 of this modified version, or non-ideal gas, 1142 01:20:56,100 --> 01:20:59,460 and the ensemble that I will choose initially. 1143 01:20:59,460 --> 01:21:01,910 Microcanonical is typically difficult. 1144 01:21:01,910 --> 01:21:05,700 I will go and do things canonically, 1145 01:21:05,700 --> 01:21:07,190 which is somewhat easier. 1146 01:21:07,190 --> 01:21:11,310 And later on, maybe even grand canonical is easier. 1147 01:21:11,310 --> 01:21:14,860 So what do I have to do? 1148 01:21:14,860 --> 01:21:19,580 I would say that the partition function is obtained 1149 01:21:19,580 --> 01:21:26,340 by integrating over the entirety of this phase space-- product 1150 01:21:26,340 --> 01:21:30,950 d cubed Pi d cubed qi. 1151 01:21:30,950 --> 01:21:35,010 I will normalize things by N factorial, 1152 01:21:35,010 --> 01:21:38,830 dimensionalize them by h to the 3N. 1153 01:21:38,830 --> 01:21:42,840 And I have exponential of minus beta 1154 01:21:42,840 --> 01:21:46,360 sum over i Pi squared over 2m. 1155 01:21:46,360 --> 01:21:56,120 And then I have the exponential of minus U. 1156 01:21:56,120 --> 01:22:00,300 Now, this time around, the P integrals are [INAUDIBLE]. 1157 01:22:00,300 --> 01:22:02,590 I can do them immediately. 1158 01:22:02,590 --> 01:22:07,200 Because typically the momenta don't interact with each other. 1159 01:22:07,200 --> 01:22:09,900 And practicality, no matter how complicated 1160 01:22:09,900 --> 01:22:12,260 a system of interactions is, you will 1161 01:22:12,260 --> 01:22:16,310 be able to integrate over the momentum degrees of freedom. 1162 01:22:16,310 --> 01:22:20,660 And what you get, you will get this factor of 1 1163 01:22:20,660 --> 01:22:25,512 over lambda to the power of 3N from the 3N momentum 1164 01:22:25,512 --> 01:22:26,053 integrations. 1165 01:22:29,320 --> 01:22:32,610 The thing that is hard-- there will be this factor of 1 over N 1166 01:22:32,610 --> 01:22:38,280 factorial-- is the integration over all of the coordinates, 1167 01:22:38,280 --> 01:22:42,060 d cubed qi of this factor of e to the minus beta 1168 01:22:42,060 --> 01:22:45,680 U. I gave you the most general possible U. There's no way 1169 01:22:45,680 --> 01:22:48,930 that I can do this integration. 1170 01:22:48,930 --> 01:22:53,300 What I will do is I will divide each one of these integrations 1171 01:22:53,300 --> 01:22:57,480 over coordinate of particle by its volume. 1172 01:22:57,480 --> 01:23:01,040 I will therefore have V to the N here. 1173 01:23:01,040 --> 01:23:05,400 And V to the N divided by lambda to the 3N N factorial 1174 01:23:05,400 --> 01:23:07,760 is none other than the partition function 1175 01:23:07,760 --> 01:23:11,490 that we had calculated before for the ideal gas. 1176 01:23:11,490 --> 01:23:15,830 And I will call Z0 for the ideal gas. 1177 01:23:15,830 --> 01:23:21,410 And I claim that this object I can interpret 1178 01:23:21,410 --> 01:23:26,030 as a kind of average of a function e 1179 01:23:26,030 --> 01:23:30,470 to the beta U defined over the phase space of all 1180 01:23:30,470 --> 01:23:33,420 of these particles where the probability 1181 01:23:33,420 --> 01:23:38,360 to find each particle is uniform in the space of the box, 1182 01:23:38,360 --> 01:23:40,520 let's say. 1183 01:23:40,520 --> 01:23:45,920 So what this says is for the 0-th order case, 1184 01:23:45,920 --> 01:23:49,490 for the ideal case that we have discussed, once I have set 1185 01:23:49,490 --> 01:23:53,970 the box, the particle can be anywhere in the box uniformly. 1186 01:23:53,970 --> 01:23:57,540 For that uniform description probability, 1187 01:23:57,540 --> 01:23:59,820 calculate what the average of this quantity is. 1188 01:24:02,630 --> 01:24:06,530 So what we have is that Z is in fact Z0, that 1189 01:24:06,530 --> 01:24:09,360 average that I can expand. 1190 01:24:09,360 --> 01:24:11,920 And what we will be doing henceforth 1191 01:24:11,920 --> 01:24:15,420 is a perturbation theory in powers of U. 1192 01:24:15,420 --> 01:24:20,150 Because I know how to do things for the case of U equals 0. 1193 01:24:20,150 --> 01:24:26,530 And then I hope to calculate things in various powers of U. 1194 01:24:26,530 --> 01:24:30,460 So I will do that expansion. 1195 01:24:30,460 --> 01:24:34,030 And then I say, no really, what I'm 1196 01:24:34,030 --> 01:24:37,860 interested in is something like a free energy, which is log Z. 1197 01:24:37,860 --> 01:24:43,540 And so for that, I will need log of Z0 plus log of that series. 1198 01:24:43,540 --> 01:24:45,540 But the log of these kinds of series 1199 01:24:45,540 --> 01:24:50,200 I know I can write as minus beta to the l over l 1200 01:24:50,200 --> 01:24:55,120 factorial, replacing, when I go to the log, moments 1201 01:24:55,120 --> 01:24:58,660 by corresponding cumulants. 1202 01:24:58,660 --> 01:25:02,290 So this is called the cumulant expansion, 1203 01:25:02,290 --> 01:25:07,410 which we will carry out next time around. 1204 01:25:07,410 --> 01:25:07,980 Yes. 1205 01:25:07,980 --> 01:25:09,916 AUDIENCE: In general, [INAUDIBLE]. 1206 01:25:21,190 --> 01:25:24,210 PROFESSOR: For some cases, you can. 1207 01:25:24,210 --> 01:25:26,680 For many cases, you find that that 1208 01:25:26,680 --> 01:25:28,160 will give you wrong answers. 1209 01:25:28,160 --> 01:25:31,900 Because the phase space around which you're expanding 1210 01:25:31,900 --> 01:25:33,570 is so broad. 1211 01:25:33,570 --> 01:25:35,760 It is not like a saddle point where 1212 01:25:35,760 --> 01:25:38,880 you have one variable you are expanding 1213 01:25:38,880 --> 01:25:41,860 in a huge number of variables.