1 00:00:00,060 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high-quality educational resources for free. 5 00:00:10,730 --> 00:00:13,340 To make a donation or view additional materials 6 00:00:13,340 --> 00:00:17,236 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,236 --> 00:00:17,861 at ocw.mit.edu. 8 00:00:20,980 --> 00:00:28,555 PROFESSOR: Last time, we looked at the case of a Bose gas, 9 00:00:28,555 --> 00:00:31,240 and we found that there was this condensation 10 00:00:31,240 --> 00:00:32,895 phenomena into the ground state. 11 00:00:36,710 --> 00:00:39,850 Let's go over it one more time. 12 00:00:39,850 --> 00:00:43,420 So the idea was that when you have 13 00:00:43,420 --> 00:00:48,350 a Bose system, the occupation of the different one-particle 14 00:00:48,350 --> 00:00:54,210 states, which you can specify for a non-interacting system 15 00:00:54,210 --> 00:00:58,990 in the grand canonical ensemble, these entities 16 00:00:58,990 --> 00:01:03,240 are independent for each one of the one-particle states 17 00:01:03,240 --> 00:01:07,340 and have a form that is related to the chemical potential 18 00:01:07,340 --> 00:01:16,890 and the energy of these states through this form, epsilon 19 00:01:16,890 --> 00:01:19,880 [? kd ?] in the one-particle state, 20 00:01:19,880 --> 00:01:25,100 and z standing for e to the beta mu, chemical potential 21 00:01:25,100 --> 00:01:27,950 divided by temperature. 22 00:01:27,950 --> 00:01:30,110 OK? 23 00:01:30,110 --> 00:01:35,140 Now, if we go and look at this in the limit 24 00:01:35,140 --> 00:01:39,110 where z is much less than 1, which 25 00:01:39,110 --> 00:01:41,530 is appropriate to high temperatures 26 00:01:41,530 --> 00:01:45,390 or when the case where the gas is mostly classical, 27 00:01:45,390 --> 00:01:46,680 you can expand this. 28 00:01:46,680 --> 00:01:49,160 You can see that in the lowest order, 29 00:01:49,160 --> 00:01:53,270 it is z e to the minus beta epsilon of k. 30 00:01:53,270 --> 00:01:57,840 So it looks like a standard Boltzmann weight 31 00:01:57,840 --> 00:02:01,540 that we would use in classical statistical mechanics. 32 00:02:01,540 --> 00:02:03,440 But then there are higher order terms. 33 00:02:03,440 --> 00:02:05,415 And in particular there would be a correction 34 00:02:05,415 --> 00:02:10,130 that is e to the minus beta epsilon of k squared 35 00:02:10,130 --> 00:02:15,270 and so forth that would modify this result. 36 00:02:15,270 --> 00:02:17,910 But I think this is something that I mentioned. 37 00:02:17,910 --> 00:02:22,790 If you look at a form such as this at high temperatures, 38 00:02:22,790 --> 00:02:24,970 it's like a Boltzmann weight. 39 00:02:24,970 --> 00:02:29,470 So the state that has this lowest energy, 40 00:02:29,470 --> 00:02:32,040 in this case if we say epsilon of k 41 00:02:32,040 --> 00:02:37,640 is h bar squared k squared over 2m corresponding to k 42 00:02:37,640 --> 00:02:40,360 equals to 0 for the lowest energy, 43 00:02:40,360 --> 00:02:46,244 has the highest mean occupation number of weight. 44 00:02:46,244 --> 00:02:47,690 OK? 45 00:02:47,690 --> 00:02:50,060 Now that is always the case. 46 00:02:50,060 --> 00:02:54,310 This is a function that, as epsilon of k 47 00:02:54,310 --> 00:02:57,610 becomes larger, no matter whether you expand it or not, 48 00:02:57,610 --> 00:03:00,790 becomes less and less. 49 00:03:00,790 --> 00:03:04,310 The other limit that is of interest to us 50 00:03:04,310 --> 00:03:06,370 is how big can z be? 51 00:03:06,370 --> 00:03:12,890 We said that the largest it can be is when it approaches 1. 52 00:03:12,890 --> 00:03:17,340 And when it is 1, this quantity, the occupation number, 53 00:03:17,340 --> 00:03:22,750 is e to the beta epsilon of k minus 1. 54 00:03:22,750 --> 00:03:27,650 Again, something that is largest as k becomes smaller, 55 00:03:27,650 --> 00:03:31,050 and you go to the lowest energy. 56 00:03:31,050 --> 00:03:34,560 But there is the difficulty that the one energy that corresponds 57 00:03:34,560 --> 00:03:38,450 to k equals to 0, if I look at this formula, 58 00:03:38,450 --> 00:03:41,300 it is 1 divided by 0. 59 00:03:41,300 --> 00:03:45,970 It just doesn't make too much sense. 60 00:03:45,970 --> 00:03:49,130 Now, of course, we said that typically we 61 00:03:49,130 --> 00:03:54,490 don't want to work in this variable. 62 00:03:54,490 --> 00:03:56,730 We want to figure out things as a function 63 00:03:56,730 --> 00:04:00,580 of the number of particles or the number of particles divided 64 00:04:00,580 --> 00:04:06,220 by volume density, which is a nice intensive quantity. 65 00:04:06,220 --> 00:04:06,720 OK? 66 00:04:06,720 --> 00:04:09,010 So typically, what should we do? 67 00:04:09,010 --> 00:04:11,650 We said that in the grand canonical ensemble, 68 00:04:11,650 --> 00:04:19,930 N is the sum over k these expectation values. 69 00:04:19,930 --> 00:04:22,060 I should really put an average here, 70 00:04:22,060 --> 00:04:24,820 but we said that in the thermodynamic sense, 71 00:04:24,820 --> 00:04:27,580 fluctuations being of the order of square root of N, 72 00:04:27,580 --> 00:04:29,074 we are going to ignore. 73 00:04:29,074 --> 00:04:30,490 So this is what you would normally 74 00:04:30,490 --> 00:04:34,530 call the number that we have in the system. 75 00:04:34,530 --> 00:04:41,240 And then you have to replace this sum in the limit 76 00:04:41,240 --> 00:04:44,420 where we have a large system with an integral. 77 00:04:44,420 --> 00:04:51,540 So we have the integral d cubed k 2 pi cubed V. 78 00:04:51,540 --> 00:05:00,230 And then we have this N of k, which is 1 over z inverse e 79 00:05:00,230 --> 00:05:03,710 to the beta epsilon of k minus 1. 80 00:05:07,790 --> 00:05:13,040 Now if I go like I did in the line above to the limit 81 00:05:13,040 --> 00:05:17,650 where z is much less than 1, then the integral 82 00:05:17,650 --> 00:05:25,670 that I have to do is simply the integral of z e 83 00:05:25,670 --> 00:05:29,840 to the minus beta epsilon k h bar squared k squared 2m. 84 00:05:33,220 --> 00:05:34,290 We know what this is. 85 00:05:34,290 --> 00:05:37,520 This is simply, up to a factor of z, 86 00:05:37,520 --> 00:05:40,680 these are the integrals that have given me 87 00:05:40,680 --> 00:05:42,745 these factors of 1 over lambda cubed. 88 00:05:47,130 --> 00:05:48,890 Actually, there's the factor of V 89 00:05:48,890 --> 00:05:51,004 here, so it is V over lambda cubed. 90 00:05:53,730 --> 00:05:57,790 So normally, you would then solve for z, 91 00:05:57,790 --> 00:06:04,760 and you would find that z is the density times lambda cubed. 92 00:06:04,760 --> 00:06:08,450 And this is a formula that we have seen a number of times, 93 00:06:08,450 --> 00:06:14,356 that the chemical potential is kt log of n lambda cubed. 94 00:06:14,356 --> 00:06:16,690 OK? 95 00:06:16,690 --> 00:06:17,190 Fine. 96 00:06:17,190 --> 00:06:19,140 But now we are interested in the other limit. 97 00:06:22,210 --> 00:06:26,791 Now in the other limit, we have this difficulty 98 00:06:26,791 --> 00:06:32,430 that this quantity, if I look at it as a function of z, 99 00:06:32,430 --> 00:06:34,140 this is the quantity that we would 100 00:06:34,140 --> 00:06:40,420 call f-- it is related to f 3/2 of z. 101 00:06:40,420 --> 00:06:48,080 So this is going to be 1 over lambda cubed f 3/2 plus of z. 102 00:06:48,080 --> 00:07:00,560 It's a function that increases with z up to the limiting value 103 00:07:00,560 --> 00:07:04,240 that we are allowed to have, which is at z equals to 1. 104 00:07:04,240 --> 00:07:08,380 It's a function that starts linearly, as we have discussed, 105 00:07:08,380 --> 00:07:14,360 but then up here, it comes to a finite value of zeta 3/2. 106 00:07:17,490 --> 00:07:17,990 OK? 107 00:07:20,540 --> 00:07:26,030 So basically, we know that when I evaluate this for finite z, 108 00:07:26,030 --> 00:07:28,730 I'm less than where I was here. 109 00:07:28,730 --> 00:07:33,650 So this is certainly less than 1 over lambda cubed zeta 110 00:07:33,650 --> 00:07:36,970 of 3/2, the limiting value that we have over here. 111 00:07:39,880 --> 00:07:42,730 So now I have a difficulty. 112 00:07:42,730 --> 00:07:50,710 This function over here, I can only make it so big. 113 00:07:50,710 --> 00:07:53,670 But I need to make it big enough so 114 00:07:53,670 --> 00:07:57,740 that I can put enough particles, the total number of particles 115 00:07:57,740 --> 00:08:00,200 that I have in the system. 116 00:08:00,200 --> 00:08:02,720 So what is happening over here? 117 00:08:02,720 --> 00:08:09,490 What is happening is that when I make this density larger 118 00:08:09,490 --> 00:08:13,730 and larger, ultimately when I hit this point at z 119 00:08:13,730 --> 00:08:17,470 equals to 1, as we have discussed 120 00:08:17,470 --> 00:08:23,160 there is an uncertainty as to what happens to this expression 121 00:08:23,160 --> 00:08:26,600 when z is equal to 1, and you are at this state 122 00:08:26,600 --> 00:08:30,020 that corresponds to k equals to 0. 123 00:08:30,020 --> 00:08:30,520 Right? 124 00:08:30,520 --> 00:08:33,559 I don't know how many things are over there. 125 00:08:33,559 --> 00:08:36,400 So I can have, in principle, the freedom 126 00:08:36,400 --> 00:08:41,380 to take advantage of the fact that I don't know what 1 over 0 127 00:08:41,380 --> 00:08:47,950 is to call that the number that I need to make up 128 00:08:47,950 --> 00:08:51,342 the difference between here and there. 129 00:08:51,342 --> 00:08:53,440 OK? 130 00:08:53,440 --> 00:09:02,460 And so, in principle, when z goes to 1, 131 00:09:02,460 --> 00:09:09,670 then this becomes 1 over lambda cubed, or V over lambda cubed, 132 00:09:09,670 --> 00:09:14,130 zeta of 3/2, obtained by essentially 133 00:09:14,130 --> 00:09:17,290 doing the integration with z equals to 1, 134 00:09:17,290 --> 00:09:21,280 evaluating it at this possible limit. 135 00:09:21,280 --> 00:09:25,580 And then whatever is left, I will 136 00:09:25,580 --> 00:09:29,560 put in the k equals to 0 state. 137 00:09:29,560 --> 00:09:30,610 OK? 138 00:09:30,610 --> 00:09:35,070 And you will say, well, what does that explicitly mean? 139 00:09:38,790 --> 00:09:44,910 Well, what it means is that I'm really evaluating things 140 00:09:44,910 --> 00:09:50,620 at the value of z that is extremely close to 1 141 00:09:50,620 --> 00:09:54,290 so that in the thermodynamic limit where N goes to infinity 142 00:09:54,290 --> 00:09:57,240 I wouldn't know the difference. 143 00:09:57,240 --> 00:10:01,240 But it is really something that vanishes only 144 00:10:01,240 --> 00:10:05,120 in that thermodynamic limit of N going to infinity. 145 00:10:05,120 --> 00:10:09,510 And we saw that if I kind of look at the occupation 146 00:10:09,510 --> 00:10:15,050 number of the k equals to 0 state, what it is 147 00:10:15,050 --> 00:10:27,220 is z inverse, which is e to the minus beta mu minus 1. 148 00:10:27,220 --> 00:10:31,250 And if I'm in the limit where z is very small, 149 00:10:31,250 --> 00:10:36,650 or beta mu is-- sorry, z is very much close to 1 150 00:10:36,650 --> 00:10:41,430 or beta mu is very small, then I can expand this. 151 00:10:41,430 --> 00:10:47,930 And this becomes approximately, when mu goes to 0, 152 00:10:47,930 --> 00:10:49,718 minus 1 over beta mu. 153 00:10:52,530 --> 00:10:55,230 OK? 154 00:10:55,230 --> 00:11:00,130 And I want this to be the N0, what 155 00:11:00,130 --> 00:11:05,040 is left over when I subtract from the total number 156 00:11:05,040 --> 00:11:11,520 the amount that I can put in to the ground state. 157 00:11:11,520 --> 00:11:13,010 OK? 158 00:11:13,010 --> 00:11:17,820 And so you can see that this is a quantity that is extensive. 159 00:11:17,820 --> 00:11:21,130 It is proportional to the number of particles. 160 00:11:21,130 --> 00:11:26,670 And mu is the inverse of that, so it is infinitesimal. 161 00:11:26,670 --> 00:11:31,480 And if I choose that value of mu that is, essentially, 162 00:11:31,480 --> 00:11:35,540 pushing z infinitesimally close to 1, 163 00:11:35,540 --> 00:11:41,088 then I can ensure that I have a finite occupation over there. 164 00:11:41,088 --> 00:11:43,020 OK? 165 00:11:43,020 --> 00:11:44,150 So what does it mean? 166 00:11:44,150 --> 00:11:50,220 If I could actually look at these occupation numbers, 167 00:11:50,220 --> 00:11:58,070 then I am at temperatures less than Tc of n. 168 00:11:58,070 --> 00:12:00,010 And we saw that basically we could 169 00:12:00,010 --> 00:12:03,300 get what the temperature is when this happens 170 00:12:03,300 --> 00:12:11,180 by equating this combination g over lambda cubed, 171 00:12:11,180 --> 00:12:15,090 in principle, over-- I had said g equals to 1. 172 00:12:15,090 --> 00:12:18,220 I could repeat it with g not equal to 1. 173 00:12:18,220 --> 00:12:26,210 And n, which is g over lambda cubed zeta of 3/2, 174 00:12:26,210 --> 00:12:29,640 rather than f 3/2 plus, this defines for me 175 00:12:29,640 --> 00:12:33,345 this temperature Tc of n. 176 00:12:33,345 --> 00:12:36,180 OK? 177 00:12:36,180 --> 00:12:42,060 Now, if I could look at the occupation numbers 178 00:12:42,060 --> 00:12:46,070 when I was at temperature less than Tc, what would I see? 179 00:12:46,070 --> 00:12:52,520 I would see that the occupation number as a function of k 180 00:12:52,520 --> 00:12:56,230 is exactly the form that we have over here with z equals 181 00:12:56,230 --> 00:13:07,830 to 1, 1 over e to the beta epsilon of k minus 1. 182 00:13:07,830 --> 00:13:11,870 But then at k equals to 0, which I 183 00:13:11,870 --> 00:13:16,370 can indicate by some kind of delta function, 184 00:13:16,370 --> 00:13:20,360 I have everybody else, which is this N minus 185 00:13:20,360 --> 00:13:22,910 V over lambda cubed zeta of 3/2. 186 00:13:25,670 --> 00:13:28,090 OK? 187 00:13:28,090 --> 00:13:34,200 Actually, let's remember that our k and momentum 188 00:13:34,200 --> 00:13:36,650 are related to each other. 189 00:13:36,650 --> 00:13:42,620 So rather than asking what the density is 190 00:13:42,620 --> 00:13:46,360 as a function of this label k, I can maybe 191 00:13:46,360 --> 00:13:49,350 ask what the density is for things 192 00:13:49,350 --> 00:13:52,390 that have different momentum P. 193 00:13:52,390 --> 00:13:54,250 And according to this formula, it 194 00:13:54,250 --> 00:14:00,000 is going to be beta P squared over 2m, this kind of modified 195 00:14:00,000 --> 00:14:05,140 version of the Boltzmann weight, and a delta function at P 196 00:14:05,140 --> 00:14:08,220 equals to 0 there. 197 00:14:08,220 --> 00:14:12,340 The rest of the particles are up here. 198 00:14:12,340 --> 00:14:17,070 Now, when people saw the signature 199 00:14:17,070 --> 00:14:22,080 of Bose-Einstein condensation, it was precisely this. 200 00:14:22,080 --> 00:14:29,910 So this is the figure that shows this 201 00:14:29,910 --> 00:14:34,420 from the experiment of Eric Cornell and group, 202 00:14:34,420 --> 00:14:39,920 and you see two thermometers. 203 00:14:39,920 --> 00:14:43,420 They are cooling the temperature of this gas 204 00:14:43,420 --> 00:14:46,740 of atoms that is confined in a trap. 205 00:14:46,740 --> 00:14:48,390 The way that they are cooling it is 206 00:14:48,390 --> 00:14:52,570 that they are evaporating away the high energy particles. 207 00:14:52,570 --> 00:14:55,460 And then the rest of the particles 208 00:14:55,460 --> 00:14:58,780 are cooled down because they thermalize, and the highest 209 00:14:58,780 --> 00:15:00,970 energy is always removed. 210 00:15:00,970 --> 00:15:03,285 But in the process, the density in the trap 211 00:15:03,285 --> 00:15:05,760 is also being reduced. 212 00:15:05,760 --> 00:15:12,660 And if the density goes down, then Tc of n will also go down. 213 00:15:12,660 --> 00:15:16,780 So what is plotted over here on the right thermometer 214 00:15:16,780 --> 00:15:21,450 is the Tc of n that you would calculate from that formula. 215 00:15:21,450 --> 00:15:24,160 And on the right thermometer is actually 216 00:15:24,160 --> 00:15:27,000 the temperature of the particles in the gas. 217 00:15:27,000 --> 00:15:30,660 And you can see that right at some point, then 218 00:15:30,660 --> 00:15:35,390 the Tc of the particles in the gas goes below Tc of n. 219 00:15:35,390 --> 00:15:38,220 And what is being plotted over here is as follows. 220 00:15:38,220 --> 00:15:39,740 So you have the particles that are 221 00:15:39,740 --> 00:15:44,180 in the trap at some temperature. 222 00:15:44,180 --> 00:15:49,250 We remove the trap, and then the particles are going to escape. 223 00:15:49,250 --> 00:15:51,330 The particles that have zero momentum 224 00:15:51,330 --> 00:15:53,410 are going to stay where they are. 225 00:15:53,410 --> 00:15:55,480 The particles that have large momentum 226 00:15:55,480 --> 00:15:57,370 are going to go further away. 227 00:15:57,370 --> 00:16:00,570 So let's say after one second you take a picture. 228 00:16:00,570 --> 00:16:03,910 And what you see over here from the picture 229 00:16:03,910 --> 00:16:06,930 is over here there are the particles that 230 00:16:06,930 --> 00:16:09,655 had most momentum and ran further away. 231 00:16:09,655 --> 00:16:12,000 While at the center, you see the particles 232 00:16:12,000 --> 00:16:14,640 that had small momentum. 233 00:16:14,640 --> 00:16:17,270 You can see that always, even at high temperature, 234 00:16:17,270 --> 00:16:19,910 the zero momentum is the peak. 235 00:16:19,910 --> 00:16:24,190 But once you go below this condition of Tc of n, 236 00:16:24,190 --> 00:16:28,650 in addition to the normal peak that is over here, 237 00:16:28,650 --> 00:16:32,210 you get some additional peak that is appearing at P 238 00:16:32,210 --> 00:16:35,130 equals to 0, which is the condensation that you 239 00:16:35,130 --> 00:16:39,005 have into that one single state. 240 00:16:39,005 --> 00:16:39,505 OK? 241 00:16:45,460 --> 00:16:50,560 The other thing that I started to do last time but let's 242 00:16:50,560 --> 00:16:54,090 complete and finish today is I wanted 243 00:16:54,090 --> 00:17:00,080 to show you the heat capacity of this system. 244 00:17:00,080 --> 00:17:00,580 OK. 245 00:17:00,580 --> 00:17:01,690 So how do we do that? 246 00:17:01,690 --> 00:17:05,240 So the first formula that we had, 247 00:17:05,240 --> 00:17:09,869 the one that I used over here if I were to divide by V, 248 00:17:09,869 --> 00:17:15,050 is that the density, in general, is given by g over lambda 249 00:17:15,050 --> 00:17:18,970 cubed f 3/2 plus of z. 250 00:17:22,710 --> 00:17:24,609 OK? 251 00:17:24,609 --> 00:17:29,480 And we said that if z goes to 1, then I 252 00:17:29,480 --> 00:17:32,770 have to interpret that appropriately that this 253 00:17:32,770 --> 00:17:37,570 is really just excited states, and there's 254 00:17:37,570 --> 00:17:41,020 going to be a separation into the ground 255 00:17:41,020 --> 00:17:43,340 state and the excited states. 256 00:17:43,340 --> 00:17:48,660 So there is really two portions of n when we are below Tc. 257 00:17:48,660 --> 00:17:49,210 OK? 258 00:17:49,210 --> 00:17:52,010 But then there is also a formula for pressure. 259 00:17:52,010 --> 00:18:00,480 Beta P was g over lambda cubed f 5/2 of z. 260 00:18:00,480 --> 00:18:06,680 Again, these f's we had defined-- f plus m minus 1 of z 261 00:18:06,680 --> 00:18:11,120 was 1 over m minus 1 factorial integral 0 262 00:18:11,120 --> 00:18:17,470 to infinity dx x to the m minus 1 z inverse e to the x 263 00:18:17,470 --> 00:18:20,000 plus 1-- minus 1. 264 00:18:24,240 --> 00:18:24,740 OK? 265 00:18:27,560 --> 00:18:29,730 Fine. 266 00:18:29,730 --> 00:18:33,920 But in order to calculate the heat capacity, 267 00:18:33,920 --> 00:18:36,520 I need to work with the energy. 268 00:18:36,520 --> 00:18:38,560 And we said that, quite generally, 269 00:18:38,560 --> 00:18:41,610 whether you have classical system, 270 00:18:41,610 --> 00:18:47,000 quantum gas, bosons, fermions, the energy and pressure 271 00:18:47,000 --> 00:18:52,680 are related simply by this formula, which is ultimately 272 00:18:52,680 --> 00:18:58,470 a consequence of the scaling of the energy being proportional 273 00:18:58,470 --> 00:19:03,230 to momentum squared that we have over there. 274 00:19:03,230 --> 00:19:05,280 OK? 275 00:19:05,280 --> 00:19:10,900 So let's look at this formula. 276 00:19:10,900 --> 00:19:11,620 What do I have? 277 00:19:11,620 --> 00:19:19,020 For T that is less than Tc of n, my z is equal to 1. 278 00:19:19,020 --> 00:19:22,820 Pressure comes entirely from particles 279 00:19:22,820 --> 00:19:24,900 that are moving around. 280 00:19:24,900 --> 00:19:29,040 So although the density breaks into two components, 281 00:19:29,040 --> 00:19:31,970 in principle I guess I would say that the pressure also 282 00:19:31,970 --> 00:19:34,190 breaks into two components. 283 00:19:34,190 --> 00:19:38,230 But the component that is at k equals to 0 284 00:19:38,230 --> 00:19:39,410 is not moving around. 285 00:19:39,410 --> 00:19:43,180 It is not giving you any contribution to the pressure. 286 00:19:43,180 --> 00:19:44,035 So what do we? 287 00:19:44,035 --> 00:19:50,460 We have P is kt, taking the beta to the other side. 288 00:19:50,460 --> 00:19:56,860 I have g over lambda cubed, and then this f plus evaluated at z 289 00:19:56,860 --> 00:20:02,860 equals to 1, which is this zeta function of 5/2. 290 00:20:02,860 --> 00:20:03,470 OK? 291 00:20:03,470 --> 00:20:05,210 So fine. 292 00:20:05,210 --> 00:20:08,490 So what do I have for energy? 293 00:20:08,490 --> 00:20:15,260 Energy is going to be at 3/2 V times this formula, 294 00:20:15,260 --> 00:20:21,870 kd T g over lambda cubed zeta 5/2. 295 00:20:21,870 --> 00:20:30,590 And I noticed that lambda, being h square root of 2 pi mk T, 296 00:20:30,590 --> 00:20:33,950 scales like T to the minus 1/2. 297 00:20:33,950 --> 00:20:37,870 So 1 over lambda cubed scales like T to the 3/2 plus 298 00:20:37,870 --> 00:20:40,890 this is something that's scaling like T to the 5/2. 299 00:20:45,120 --> 00:20:49,030 And this is simply proportional to the volume and temperature 300 00:20:49,030 --> 00:20:50,780 to the [INAUDIBLE] fifth power. 301 00:20:50,780 --> 00:20:54,460 It knows nothing about the density. 302 00:20:54,460 --> 00:20:55,010 Why? 303 00:20:55,010 --> 00:20:59,650 Because if you were at the same temperature and volume, 304 00:20:59,650 --> 00:21:02,020 put more particle in the system, all 305 00:21:02,020 --> 00:21:04,630 of those additional particles would go over here, 306 00:21:04,630 --> 00:21:10,370 would make no contribution to pressure or to energy. 307 00:21:10,370 --> 00:21:12,420 You wouldn't see them. 308 00:21:12,420 --> 00:21:16,450 So once you have this, the heat capacity at constant volume, 309 00:21:16,450 --> 00:21:24,860 dE by dT, is simply 5 over 2T of whatever energy is. 310 00:21:35,960 --> 00:21:41,300 And there is a reason that I write it in this fashion. 311 00:21:41,300 --> 00:21:43,810 So what you would say is that if I 312 00:21:43,810 --> 00:21:50,030 were to plot the heat capacity at constant volume 313 00:21:50,030 --> 00:21:56,040 as a function of temperature, well, 314 00:21:56,040 --> 00:22:01,646 it's natural units are of the order of kB. 315 00:22:01,646 --> 00:22:05,800 You can see that this T and this T I can cancel out. 316 00:22:05,800 --> 00:22:08,280 So I have something that has units of kB. 317 00:22:08,280 --> 00:22:11,940 It is extensive proportional to volume. 318 00:22:11,940 --> 00:22:17,600 I can really certainly write it in this fashion. 319 00:22:17,600 --> 00:22:24,090 And what I have is that the curve 320 00:22:24,090 --> 00:22:27,881 behaves like T to the 3/2. 321 00:22:27,881 --> 00:22:31,130 So this is proportional to T to the 3/2. 322 00:22:34,020 --> 00:22:41,530 And presumably, this behavior continues all the way 323 00:22:41,530 --> 00:22:48,820 until the temperature at which this system ceases 324 00:22:48,820 --> 00:22:51,610 to be a Bose-Einstein condensate. 325 00:22:51,610 --> 00:22:55,530 So the low temperature form of the heat capacity's calculation 326 00:22:55,530 --> 00:22:58,840 is very simple. 327 00:22:58,840 --> 00:23:01,530 Now, at very high temperatures, we 328 00:23:01,530 --> 00:23:09,090 know that the heat capacity is going to be 3/2 N kT, right? 329 00:23:09,090 --> 00:23:16,190 So ultimately, I know that if I go to very high temperatures, 330 00:23:16,190 --> 00:23:23,300 my heat capacity is going to be 3/2 N, 331 00:23:23,300 --> 00:23:28,530 if I divide it by [? kB. ?] OK? 332 00:23:28,530 --> 00:23:32,080 Now, there is a reason that I drew this line 333 00:23:32,080 --> 00:23:35,160 over here less than the peak here. 334 00:23:38,840 --> 00:23:44,030 Well, in order to calculate the heat capacity on this side, 335 00:23:44,030 --> 00:23:46,860 we'll have to do a little bit more work. 336 00:23:46,860 --> 00:23:54,980 You can no longer assume that z equals to 1. 337 00:23:54,980 --> 00:23:58,580 z is something that is less than 1 338 00:23:58,580 --> 00:24:02,790 that is obtained by solving N equals to g over 339 00:24:02,790 --> 00:24:07,050 lambda cubed f 3/2 plus of z. 340 00:24:07,050 --> 00:24:08,770 OK? 341 00:24:08,770 --> 00:24:12,200 But I can still presumably use these formulae. 342 00:24:12,200 --> 00:24:17,670 I have that the energy is 3/2 PV. 343 00:24:17,670 --> 00:24:25,760 So I have V kT times-- what is the thing 344 00:24:25,760 --> 00:24:27,400 that I have on the right-hand side? 345 00:24:27,400 --> 00:24:31,940 It is g over lambda cubed, just as I have over here 346 00:24:31,940 --> 00:24:38,130 except that I have to put f 5/2 plus of z. 347 00:24:38,130 --> 00:24:41,100 OK? 348 00:24:41,100 --> 00:24:44,800 And if I want to calculate the heat capacity, what 349 00:24:44,800 --> 00:24:46,700 I need to do is to take a derivative 350 00:24:46,700 --> 00:24:51,020 of the energy with respect to temperature. 351 00:24:51,020 --> 00:24:55,790 And then I notice, well, again, right like I had before, 352 00:24:55,790 --> 00:24:58,996 whatever appears here is proportional to T to the 5/2. 353 00:25:02,090 --> 00:25:04,950 So when I take a derivative with respect to temperature, 354 00:25:04,950 --> 00:25:10,850 I essentially get 5/2 this whole quantity divided by T. 355 00:25:10,850 --> 00:25:14,300 So just like I had over here, I will write the result 356 00:25:14,300 --> 00:25:22,220 as 3/2 V kT g over lambda cubed. 357 00:25:22,220 --> 00:25:31,290 And then I have the 5/2 1 over T of f 5/2 plus of z. 358 00:25:35,320 --> 00:25:39,380 But you say that's the wrong way of doing things 359 00:25:39,380 --> 00:25:45,680 because in addition to this combination out front be 360 00:25:45,680 --> 00:25:50,300 explicitly proportional to T to the 5/2 power, 361 00:25:50,300 --> 00:25:55,230 this factor z here is implicitly a function of temperature 362 00:25:55,230 --> 00:25:59,380 because it is related to density through that formula N equals 363 00:25:59,380 --> 00:26:04,450 to g over lambda cubed f 5/2 plus of z. 364 00:26:04,450 --> 00:26:07,760 So you should not forget to take the derivative of what 365 00:26:07,760 --> 00:26:11,660 is over here with respect to T. OK? 366 00:26:11,660 --> 00:26:15,110 So what do I get when I take a derivative of this with respect 367 00:26:15,110 --> 00:26:16,110 to T? 368 00:26:16,110 --> 00:26:22,230 Well, the argument is not T, it is z. 369 00:26:22,230 --> 00:26:25,740 So I will have to take a dz by dT, 370 00:26:25,740 --> 00:26:31,100 and then the derivative of f 5/2 with respect to z. 371 00:26:31,100 --> 00:26:33,500 And we saw that one of the characteristics 372 00:26:33,500 --> 00:26:35,860 of these functions was that when we 373 00:26:35,860 --> 00:26:39,270 took the derivative of each one of them, 374 00:26:39,270 --> 00:26:42,170 we generated one with a lower argument. 375 00:26:42,170 --> 00:26:46,270 So the derivative would be f 3/2 plus of z. 376 00:26:46,270 --> 00:26:49,200 But in order to have this [? latter ?] property, 377 00:26:49,200 --> 00:26:54,900 I had to do a z d by dz, and so I essentially 378 00:26:54,900 --> 00:26:59,160 need to restore this 1 over z here. 379 00:26:59,160 --> 00:27:05,140 So you can see that, really, the difference between what 380 00:27:05,140 --> 00:27:10,830 I had at low temperatures and high temperatures, 381 00:27:10,830 --> 00:27:15,110 at low temperatures this was just a number that I was using 382 00:27:15,110 --> 00:27:17,400 over here, zeta of 5/2, and I didn't 383 00:27:17,400 --> 00:27:19,407 have to bother about this other term. 384 00:27:22,270 --> 00:27:27,000 Now, we can figure out what the behavior of this term 385 00:27:27,000 --> 00:27:31,350 is certainly at small z because we 386 00:27:31,350 --> 00:27:34,920 have the expansion of these functions at small z. 387 00:27:34,920 --> 00:27:40,000 And if you do sufficient work, you will find-- well, 388 00:27:40,000 --> 00:27:44,280 first of all, at the largest values of z-- sorry, 389 00:27:44,280 --> 00:27:46,790 at the smallest value, z going to 0, 390 00:27:46,790 --> 00:27:52,150 you can confirm that the heat capacity in units of N kB 391 00:27:52,150 --> 00:27:54,060 is simply 3/2. 392 00:27:54,060 --> 00:27:57,900 So you will come to this line over here, 393 00:27:57,900 --> 00:28:02,610 and then the first correction that you calculate you'll find 394 00:28:02,610 --> 00:28:06,220 is going to take you in that direction. 395 00:28:06,220 --> 00:28:09,730 Actually, if you do the calculation for bosons 396 00:28:09,730 --> 00:28:11,960 and fermions at high temperatures, 397 00:28:11,960 --> 00:28:14,890 you can convince yourself that while bosons 398 00:28:14,890 --> 00:28:18,615 will go in this direction, fermions, the heat capacity, 399 00:28:18,615 --> 00:28:22,610 will start to go in the other direction. 400 00:28:22,610 --> 00:28:25,950 But OK, so you have this behavior. 401 00:28:25,950 --> 00:28:28,080 And then what we haven't established 402 00:28:28,080 --> 00:28:31,440 and the reason that I'm still doing all of this algebra 403 00:28:31,440 --> 00:28:34,850 is that, OK, this curve is going up, 404 00:28:34,850 --> 00:28:37,990 where is it going to end up? 405 00:28:37,990 --> 00:28:40,540 Is it going to end up here, here, 406 00:28:40,540 --> 00:28:43,790 or right where this curve hits? 407 00:28:43,790 --> 00:28:45,080 OK? 408 00:28:45,080 --> 00:28:48,180 And what I will show you is that this 409 00:28:48,180 --> 00:28:54,290 has to end up right at this point. 410 00:28:54,290 --> 00:29:00,440 And actually, in order to show that, all we need to do 411 00:29:00,440 --> 00:29:06,540 is to figure out what these terms do when I evaluate them 412 00:29:06,540 --> 00:29:09,660 at z equals to 1. 413 00:29:09,660 --> 00:29:12,690 And because these things are telling 414 00:29:12,690 --> 00:29:16,105 me what is happening as I am approaching to this point, 415 00:29:16,105 --> 00:29:19,360 the question that I ask is, what is the limiting behavior 416 00:29:19,360 --> 00:29:23,800 of this function as I get to z equals to 1? 417 00:29:23,800 --> 00:29:26,070 But the limiting value of this one I know. 418 00:29:26,070 --> 00:29:28,340 It is precisely what I want over here. 419 00:29:28,340 --> 00:29:31,110 It is zeta of 5/2. 420 00:29:31,110 --> 00:29:36,572 So I have to somehow show you that this next term is 0. 421 00:29:36,572 --> 00:29:41,580 Well, this next term, however, depends on dz by dT. 422 00:29:41,580 --> 00:29:44,110 How can I get that? 423 00:29:44,110 --> 00:29:47,170 Well, all of these things are done under the conditions 424 00:29:47,170 --> 00:29:50,040 where the volume is fixed, or the number of particles 425 00:29:50,040 --> 00:29:52,390 is fixed, if the density is fixed. 426 00:29:52,390 --> 00:29:58,090 So z and T are related through the formula that 427 00:29:58,090 --> 00:30:03,450 says N over V, which is the density, 428 00:30:03,450 --> 00:30:09,260 is g over lambda cubed f 3/2 plus of z. 429 00:30:09,260 --> 00:30:10,520 OK? 430 00:30:10,520 --> 00:30:14,450 So if I take the system and I change its temperature, 431 00:30:14,450 --> 00:30:17,540 while keeping the number of particles, volume, 432 00:30:17,540 --> 00:30:20,300 fixed so that the density is fixed, 433 00:30:20,300 --> 00:30:25,270 if I take d by dT on both sides, on the other side 434 00:30:25,270 --> 00:30:29,550 I'm assuming that I'm doing the calculations under conditions 435 00:30:29,550 --> 00:30:31,970 where the density does not change. 436 00:30:31,970 --> 00:30:34,740 And then what do I have over here? 437 00:30:34,740 --> 00:30:38,030 I have to take the temperature derivative of this. 438 00:30:38,030 --> 00:30:40,180 There's an implicit derivative here 439 00:30:40,180 --> 00:30:44,280 since this is T to the 3/2 proportionately. 440 00:30:44,280 --> 00:30:52,150 So when I take a derivative, I will get 3 over 2T, 441 00:30:52,150 --> 00:30:59,030 and then I will have f 3/2 plus of z. 442 00:30:59,030 --> 00:31:03,550 But then I have to take the derivative that 443 00:31:03,550 --> 00:31:06,565 is implicit in the dependence on z here. 444 00:31:06,565 --> 00:31:14,250 And that's going to be identical to this, 445 00:31:14,250 --> 00:31:17,960 except that when I take a derivative of f 3/2, 446 00:31:17,960 --> 00:31:23,660 I will generate f 1/2 of z. 447 00:31:23,660 --> 00:31:25,030 OK? 448 00:31:25,030 --> 00:31:29,690 So since the left-hand side is equal to 0, 449 00:31:29,690 --> 00:31:31,850 I can immediately solve for this, 450 00:31:31,850 --> 00:31:39,590 and I find that T dz by dT 1 over z 451 00:31:39,590 --> 00:31:51,920 is minus 3/2 f 3/2 plus of z divided by f 1/2 plus of z. 452 00:31:51,920 --> 00:31:54,360 OK? 453 00:31:54,360 --> 00:32:02,370 So essentially, what I can do is I can multiply through by T. 454 00:32:02,370 --> 00:32:07,740 So this T will get rid of this T. I can bring a T, therefore, 455 00:32:07,740 --> 00:32:09,320 here. 456 00:32:09,320 --> 00:32:12,930 And this combination over here I can 457 00:32:12,930 --> 00:32:21,510 replace from what I have there with minus 3/2 f 3/2 plus of z 458 00:32:21,510 --> 00:32:24,226 divided by f 1/2 half plus of z. 459 00:32:27,960 --> 00:32:30,990 Why is any of this useful? 460 00:32:30,990 --> 00:32:35,340 Well, it immediately gives the answer that I need. 461 00:32:35,340 --> 00:32:37,130 Why? 462 00:32:37,130 --> 00:32:44,290 Because f 1/2 is the derivative of this function. 463 00:32:44,290 --> 00:32:46,590 And you can see that I drew it so that it 464 00:32:46,590 --> 00:32:50,620 has infinite derivative when z hits 1. 465 00:32:50,620 --> 00:32:55,930 And we had indeed said that this function zeta of m, which 466 00:32:55,930 --> 00:32:59,640 is the limit of this when z goes to 1, 467 00:32:59,640 --> 00:33:03,160 only exists for m that is larger than 1, 468 00:33:03,160 --> 00:33:07,840 and it is divergent for m that is less than 1. 469 00:33:07,840 --> 00:33:15,590 So when z hits 1, this thing goes to infinity, 470 00:33:15,590 --> 00:33:18,720 and the fraction disappears. 471 00:33:18,720 --> 00:33:22,490 And so basically, the additional correction 472 00:33:22,490 --> 00:33:26,680 that you have beyond this becoming zeta of 5/2, 473 00:33:26,680 --> 00:33:30,290 which is what we have over here, vanishes. 474 00:33:30,290 --> 00:33:35,870 So I have shown that, indeed, whatever this curve is doing, 475 00:33:35,870 --> 00:33:43,080 it will hit that point exactly at Tc of n 476 00:33:43,080 --> 00:33:46,830 at the same height as the low temperature curve. 477 00:33:46,830 --> 00:33:51,090 And you can do a certain amount of work and algebra 478 00:33:51,090 --> 00:33:56,450 to show that it goes there with a finite slope. 479 00:33:56,450 --> 00:33:59,020 So that's the shape of the heat capacity 480 00:33:59,020 --> 00:34:02,317 curve for the Bose-Einstein condensate. 481 00:34:02,317 --> 00:34:03,150 AUDIENCE: Professor? 482 00:34:03,150 --> 00:34:03,850 PROFESSOR: Yes? 483 00:34:03,850 --> 00:34:08,929 AUDIENCE: So it seems that f 1/2 goes to infinity, 484 00:34:08,929 --> 00:34:10,967 but it's [? on ?] the half, so it is half 485 00:34:10,967 --> 00:34:12,950 comes from the [INAUDIBLE] 3. 486 00:34:12,950 --> 00:34:13,449 So-- 487 00:34:13,449 --> 00:34:14,240 PROFESSOR: Exactly. 488 00:34:14,240 --> 00:34:16,850 AUDIENCE: --you have higher dimensional system, then 489 00:34:16,850 --> 00:34:19,520 it seems that [INAUDIBLE] won't vanish there. [INAUDIBLE]. 490 00:34:22,739 --> 00:34:26,320 PROFESSOR: So if you have a higher dimensional system, 491 00:34:26,320 --> 00:34:28,320 indeed the situation would be different. 492 00:34:32,659 --> 00:34:36,170 And I guess the borderline dimension would need to be-- 493 00:34:36,170 --> 00:34:40,261 since this is a V over 2 minus 1-- 494 00:34:40,261 --> 00:34:41,909 AUDIENCE: [INAUDIBLE] 495 00:34:41,909 --> 00:34:44,239 PROFESSOR: V over 2 minus 1 being 1, 496 00:34:44,239 --> 00:34:46,775 which is the borderline, will give you a dc of 4. 497 00:34:46,775 --> 00:34:49,040 So something else happens above 4. 498 00:34:49,040 --> 00:34:49,830 Yes? 499 00:34:49,830 --> 00:34:53,870 AUDIENCE: So it seems that the nature of this phase 500 00:34:53,870 --> 00:34:56,400 transition, indeed it changes with the dimension. 501 00:34:56,400 --> 00:34:57,400 PROFESSOR: Exactly, yes. 502 00:34:57,400 --> 00:34:59,014 The nature of this phase transition 503 00:34:59,014 --> 00:35:00,138 does depend on [INAUDIBLE]. 504 00:35:11,360 --> 00:35:11,860 Good. 505 00:35:14,811 --> 00:35:15,310 OK? 506 00:35:18,460 --> 00:35:24,026 So we will come back to this heat capacity curve 507 00:35:24,026 --> 00:35:27,120 a little bit later because I want 508 00:35:27,120 --> 00:35:31,630 to now change directions and not talk about 509 00:35:31,630 --> 00:35:37,650 Bose-Einstein condensation anymore, but about a phenomena 510 00:35:37,650 --> 00:35:41,297 that is much more miraculous in my view, 511 00:35:41,297 --> 00:35:42,380 and that is superfluidity. 512 00:35:55,590 --> 00:36:02,350 So the first thing that is very interesting about helium 513 00:36:02,350 --> 00:36:05,110 is its phase diagram. 514 00:36:05,110 --> 00:36:11,460 So we are used to drawing phase diagrams 515 00:36:11,460 --> 00:36:15,700 as a function of pressure and temperature, 516 00:36:15,700 --> 00:36:21,020 which have a separation between a liquid and a gas 517 00:36:21,020 --> 00:36:23,780 that terminates at a critical point. 518 00:36:23,780 --> 00:36:29,780 So we have seen something like this many times, where 519 00:36:29,780 --> 00:36:33,355 you have on the low pressure/high temperature 520 00:36:33,355 --> 00:36:35,810 side a gas. 521 00:36:35,810 --> 00:36:39,186 On the higher pressure/lower temperature side, 522 00:36:39,186 --> 00:36:41,170 you have a liquid. 523 00:36:41,170 --> 00:36:46,390 And they terminate, this line of coexistence between the two 524 00:36:46,390 --> 00:36:48,690 of them, at a critical point. 525 00:36:48,690 --> 00:36:52,990 And for the case of helium, the critical point 526 00:36:52,990 --> 00:36:57,660 occurs roughly at 5.2 degrees Kelvin 527 00:36:57,660 --> 00:36:59,570 and something like 2.6 atmospheres. 528 00:37:03,784 --> 00:37:04,750 OK? 529 00:37:04,750 --> 00:37:11,650 So far, nothing different from, say, liquid water, for example, 530 00:37:11,650 --> 00:37:16,110 and steam, except that for all other substances, 531 00:37:16,110 --> 00:37:19,140 when we cool it at the low enough temperatures, 532 00:37:19,140 --> 00:37:24,140 we go past the triple point, and then we have a solid phase. 533 00:37:24,140 --> 00:37:28,450 For the case of helium, it stays liquid 534 00:37:28,450 --> 00:37:32,270 all the way down to 0 temperature. 535 00:37:32,270 --> 00:37:34,990 If you put sufficiently high pressure on it, 536 00:37:34,990 --> 00:37:37,300 ultimately it will become a solid. 537 00:37:41,570 --> 00:37:45,570 But at normal pressures-- so this maximal pressure 538 00:37:45,570 --> 00:37:50,210 is something of the order of 26 atmosphere. 539 00:37:50,210 --> 00:37:54,600 For pressures that are less than 26 atmosphere, 540 00:37:54,600 --> 00:37:58,530 you can cool the liquid all the way down to 0 temperature, 541 00:37:58,530 --> 00:38:02,100 and it stays happily a liquid. 542 00:38:02,100 --> 00:38:06,340 So let's think about why that is the case. 543 00:38:06,340 --> 00:38:11,510 Well, why do things typically become solids anyway? 544 00:38:11,510 --> 00:38:15,110 The reason is that if I think about the interaction 545 00:38:15,110 --> 00:38:20,310 that I have between two objects, two atoms and molecules, 546 00:38:20,310 --> 00:38:23,750 we have discussed this in connection 547 00:38:23,750 --> 00:38:28,470 with condensation before that what 548 00:38:28,470 --> 00:38:33,690 we have is that at very large separations 549 00:38:33,690 --> 00:38:36,450 there is a van der Waals' attraction. 550 00:38:36,450 --> 00:38:42,400 Typically, the dipole moments of these atoms 551 00:38:42,400 --> 00:38:45,900 would be fluctuating, and the fluctuations would somehow 552 00:38:45,900 --> 00:38:48,170 align them temporarily in the manner 553 00:38:48,170 --> 00:38:51,220 that the energy can be reduced. 554 00:38:51,220 --> 00:38:55,625 At very short distances, the electronic clouds will overlap, 555 00:38:55,625 --> 00:38:59,710 and so you would have something such as this. 556 00:38:59,710 --> 00:39:04,590 And then, typically, you will have the function that 557 00:39:04,590 --> 00:39:08,628 joins them and look something like this. 558 00:39:08,628 --> 00:39:10,970 OK? 559 00:39:10,970 --> 00:39:15,380 And this is no different for the case of helium. 560 00:39:15,380 --> 00:39:20,620 You have here essentially a hard core appearing 561 00:39:20,620 --> 00:39:25,220 at the distance that is of the order of 2.6 angstroms. 562 00:39:25,220 --> 00:39:28,240 There is a minimum that is of the order of 3 angstroms. 563 00:39:33,160 --> 00:39:36,685 Now, the depth of this potential is not very big for helium. 564 00:39:36,685 --> 00:39:43,270 It is only of the order of, say, 9 degrees Kelvin. 565 00:39:43,270 --> 00:39:49,930 And the reason is that the strength of this van der Waals' 566 00:39:49,930 --> 00:39:54,700 attraction is proportional to the polarizability that 567 00:39:54,700 --> 00:39:57,310 scales typically with the number of electrons 568 00:39:57,310 --> 00:39:59,080 that you have in your atom. 569 00:39:59,080 --> 00:40:01,730 And so it is, in fact, proportional 570 00:40:01,730 --> 00:40:06,010 to your atomic number squared. 571 00:40:06,010 --> 00:40:10,140 And with the exception of hydrogen, 572 00:40:10,140 --> 00:40:13,000 this is going to be the smallest value 573 00:40:13,000 --> 00:40:16,360 that you are going to get for the van der Waals' attraction. 574 00:40:16,360 --> 00:40:21,070 And hydrogen does not count because it forms a molecule. 575 00:40:21,070 --> 00:40:24,890 So helium does not form a molecule. 576 00:40:24,890 --> 00:40:27,685 You have these [? balls. ?] They have these attractions. 577 00:40:27,685 --> 00:40:30,020 Now, typically things become solid 578 00:40:30,020 --> 00:40:32,750 as you go to 0 temperature is because you 579 00:40:32,750 --> 00:40:36,130 can find the minimum energy in which you put everybody 580 00:40:36,130 --> 00:40:39,240 at the minimum energy separation. 581 00:40:39,240 --> 00:40:40,450 So you form a lattice. 582 00:40:40,450 --> 00:40:43,430 Everybody is around this separation, 583 00:40:43,430 --> 00:40:47,680 and they're all happy and that solid minimizes the energy. 584 00:40:47,680 --> 00:40:52,530 So why doesn't helium do the same thing? 585 00:40:52,530 --> 00:40:55,200 Maybe it should just do it at a lower temperature 586 00:40:55,200 --> 00:40:57,680 because the scale of this is lower. 587 00:40:57,680 --> 00:40:59,350 Again, the reason it doesn't do that 588 00:40:59,350 --> 00:41:02,070 is because of quantum mechanics. 589 00:41:02,070 --> 00:41:05,510 Because you can say that, really, 590 00:41:05,510 --> 00:41:11,200 I cannot specify exactly where the particle are because that 591 00:41:11,200 --> 00:41:13,550 would violate quantum mechanics. 592 00:41:13,550 --> 00:41:16,650 There has to be some uncertainty delta x. 593 00:41:19,590 --> 00:41:22,380 And associated with delta x there 594 00:41:22,380 --> 00:41:25,950 is going to be some kind of a typical momentum, which 595 00:41:25,950 --> 00:41:28,490 is h bar over delta x. 596 00:41:28,490 --> 00:41:32,240 And there will be some contribution 597 00:41:32,240 --> 00:41:35,230 to the kinetic energy of particles 598 00:41:35,230 --> 00:41:41,140 due to this uncertainty in delta x that is of this form. 599 00:41:41,140 --> 00:41:47,540 And if I make the localization more precise, 600 00:41:47,540 --> 00:41:51,910 I have to pay a higher cost in kinetic energy. 601 00:41:51,910 --> 00:41:54,940 Well, presumably the scale of the energies 602 00:41:54,940 --> 00:41:59,960 that I can tolerate and still keep these things localized 603 00:41:59,960 --> 00:42:08,780 is of the order of this binding energy, which 604 00:42:08,780 --> 00:42:13,980 is of the order of, as we said, 9 or 10 degrees Kelvin. 605 00:42:13,980 --> 00:42:16,263 Let's say 10 degrees Kelvin. 606 00:42:16,263 --> 00:42:16,763 OK? 607 00:42:20,700 --> 00:42:24,260 Now, the other property that distinguishes helium 608 00:42:24,260 --> 00:42:29,870 from krypton and, let's say, all the other noble gases 609 00:42:29,870 --> 00:42:33,370 is that its mass is very small. 610 00:42:33,370 --> 00:42:37,880 So because this mass is small, this uncertainty in energy 611 00:42:37,880 --> 00:42:40,346 becomes comparatively larger. 612 00:42:40,346 --> 00:42:41,260 OK? 613 00:42:41,260 --> 00:42:44,590 And so if you put the mass of the helium here 614 00:42:44,590 --> 00:42:48,450 and you use this value of 10 degrees 615 00:42:48,450 --> 00:42:51,600 K converted to the appropriate energy units, 616 00:42:51,600 --> 00:42:55,280 you ask what kind of a delta x would 617 00:42:55,280 --> 00:42:58,090 be compatible with this kind of binding energy, 618 00:42:58,090 --> 00:43:00,230 you find that the delta x that you get 619 00:43:00,230 --> 00:43:08,140 is of the order of 0.5 angstroms. 620 00:43:08,140 --> 00:43:09,630 OK? 621 00:43:09,630 --> 00:43:13,470 So basically, you cannot keep this in the bottom 622 00:43:13,470 --> 00:43:17,410 of the potential if you have uncertainty that is of this 623 00:43:17,410 --> 00:43:19,330 order. 624 00:43:19,330 --> 00:43:27,220 So the atoms of helium cannot form a nice lattice and stay 625 00:43:27,220 --> 00:43:32,500 in order because that registry would violate the quantum 626 00:43:32,500 --> 00:43:33,860 uncertainty principle. 627 00:43:33,860 --> 00:43:35,350 They are moving around. 628 00:43:35,350 --> 00:43:37,860 And so you have a system that has 629 00:43:37,860 --> 00:43:41,790 been [? melted ?] due to quantum fluctuations 630 00:43:41,790 --> 00:43:45,050 rather than thermal fluctuations. 631 00:43:45,050 --> 00:43:47,060 OK? 632 00:43:47,060 --> 00:43:52,500 So that's all nice, except that something 633 00:43:52,500 --> 00:43:56,370 does happen when you cool helium. 634 00:43:56,370 --> 00:44:00,870 And actually, it is interesting that helium provides us 635 00:44:00,870 --> 00:44:05,060 not only with one liquid that persists all the way to 0 636 00:44:05,060 --> 00:44:09,480 temperature, but two liquids because helium 637 00:44:09,480 --> 00:44:10,370 has two isotopes. 638 00:44:14,978 --> 00:44:19,010 There is helium-4, which is the abundant one, 639 00:44:19,010 --> 00:44:27,304 and there is helium-3, which is a much less available isotope. 640 00:44:27,304 --> 00:44:31,470 But the number of protons being different 641 00:44:31,470 --> 00:44:37,290 in the nucleus of these means that helium-4 is a boson, 642 00:44:37,290 --> 00:44:38,870 while helium-3 is fermion. 643 00:44:42,730 --> 00:44:47,450 So we have the opportunity to observe both a Bose system that 644 00:44:47,450 --> 00:44:50,260 is liquid at low temperature and a Fermi 645 00:44:50,260 --> 00:44:53,210 liquid at [INAUDIBLE] temperatures. 646 00:44:53,210 --> 00:44:55,380 Now, the case of the Fermi liquid 647 00:44:55,380 --> 00:44:59,180 is not really that different in principle 648 00:44:59,180 --> 00:45:02,065 from the Fermi liquid of the electrons in copper 649 00:45:02,065 --> 00:45:03,630 that we already discussed. 650 00:45:03,630 --> 00:45:05,910 There are some subtleties. 651 00:45:05,910 --> 00:45:09,780 In the case of the bosons, [? well, ?] 652 00:45:09,780 --> 00:45:12,660 we saw that for bosons there is potentially 653 00:45:12,660 --> 00:45:15,970 this kind of phase transition. 654 00:45:15,970 --> 00:45:21,640 In reality, what happens is that helium 655 00:45:21,640 --> 00:45:25,090 has a transition between two forms 656 00:45:25,090 --> 00:45:27,900 of the liquid at a temperature that 657 00:45:27,900 --> 00:45:32,870 is of the order of 2.18 degrees Kelvin. 658 00:45:32,870 --> 00:45:39,795 So we can call this helium I and this helium II, 659 00:45:39,795 --> 00:45:46,990 OK, two different forms of the helium-4 liquid that you have. 660 00:45:46,990 --> 00:45:50,473 So how was this seen experimentally? 661 00:46:00,630 --> 00:46:04,410 I kind of told you how in these experiments 662 00:46:04,410 --> 00:46:07,110 of the Bose-Einstein condensation, 663 00:46:07,110 --> 00:46:10,510 part of the way that they cooled the system 664 00:46:10,510 --> 00:46:14,530 was to remove the high-energy particles, 665 00:46:14,530 --> 00:46:17,300 and the remaining particles cooled down. 666 00:46:17,300 --> 00:46:20,810 So that's the common method that actually people 667 00:46:20,810 --> 00:46:22,720 use for cooling various things. 668 00:46:22,720 --> 00:46:28,890 It's called evaporative-- evaporative cooling. 669 00:46:33,450 --> 00:46:38,285 And for the case of helium, it is actually reasonably simple 670 00:46:38,285 --> 00:46:39,720 to draw what it is. 671 00:46:42,690 --> 00:46:53,890 You have a bath that is full of helium that is reasonably 672 00:46:53,890 --> 00:46:59,000 nicely thermally isolated from the environment. 673 00:46:59,000 --> 00:47:03,560 So you want to cool it, remove the energy from it. 674 00:47:03,560 --> 00:47:10,760 And the thing that you do is you pump out the gas. 675 00:47:10,760 --> 00:47:13,240 OK? 676 00:47:13,240 --> 00:47:22,670 So this is a coexistence because this part 677 00:47:22,670 --> 00:47:26,760 up here on top of the liquid helium is not vacuum. 678 00:47:26,760 --> 00:47:34,050 What happens is that there is evaporation of helium, 679 00:47:34,050 --> 00:47:38,600 leaving the liquid and going into the gas. 680 00:47:38,600 --> 00:47:43,350 And once they are in the gas, they are pumped out. 681 00:47:43,350 --> 00:47:47,520 But every time you grab a helium from here 682 00:47:47,520 --> 00:47:51,530 to put it into the gas, you've essentially taken it 683 00:47:51,530 --> 00:47:55,780 from down here and moved it to infinity, 684 00:47:55,780 --> 00:47:59,460 so you lose this amount of binding energy. 685 00:47:59,460 --> 00:48:02,020 It's called latent heat. 686 00:48:02,020 --> 00:48:04,840 Essentially, this is the latent heat, 687 00:48:04,840 --> 00:48:08,440 and it is actually of the order of 7 degrees 688 00:48:08,440 --> 00:48:10,890 Kelvin because typically it has to be 689 00:48:10,890 --> 00:48:14,750 the depth of the typical positions of these particles. 690 00:48:14,750 --> 00:48:15,510 OK? 691 00:48:15,510 --> 00:48:20,080 So for each atom that you pump out, 692 00:48:20,080 --> 00:48:22,880 you have reduced the total energy 693 00:48:22,880 --> 00:48:27,040 that you have in the liquid, and the liquid cools down. 694 00:48:27,040 --> 00:48:29,500 And what is the trajectory? 695 00:48:29,500 --> 00:48:34,810 You always actually remain on this coexistence line. 696 00:48:34,810 --> 00:48:36,140 The pressure is changing. 697 00:48:36,140 --> 00:48:37,670 The temperature is lowering. 698 00:48:37,670 --> 00:48:40,840 But you're always remaining on the coexistence 699 00:48:40,840 --> 00:48:44,220 of the gas and liquid line. 700 00:48:44,220 --> 00:48:47,880 So you follow what is happening over here, 701 00:48:47,880 --> 00:48:51,560 and you find that as you are approaching here, 702 00:48:51,560 --> 00:48:57,320 suddenly this system starts to bubble up. 703 00:48:57,320 --> 00:49:02,670 It is just like having a kettle that is boiling. 704 00:49:02,670 --> 00:49:05,860 Indeed, it is, again, in like the case of kettle, 705 00:49:05,860 --> 00:49:09,650 there are these bubbles that are carrying the steam that 706 00:49:09,650 --> 00:49:12,140 is going to go over here and being pulled out, 707 00:49:12,140 --> 00:49:14,480 and so it is bubbling. 708 00:49:14,480 --> 00:49:19,100 And then suddenly, [INAUDIBLE] heat 2.18 degrees Kelvin, 709 00:49:19,100 --> 00:49:22,680 and the whole thing quiets down. 710 00:49:22,680 --> 00:49:26,242 And below that, it stays quiet. 711 00:49:26,242 --> 00:49:28,830 And so let's see if we can show that. 712 00:49:40,031 --> 00:49:41,760 We saw the phase diagram. 713 00:49:52,810 --> 00:49:54,605 So this is the liquid. 714 00:49:57,385 --> 00:49:59,486 Let's see if we can get the sound also. 715 00:49:59,486 --> 00:50:00,152 [VIDEO PLAYBACK] 716 00:50:00,152 --> 00:50:02,080 -[INAUDIBLE]. 717 00:50:02,080 --> 00:50:04,396 This rapid [INAUDIBLE] evaporates [INAUDIBLE] 718 00:50:04,396 --> 00:50:08,300 helium cools, until at 2 degrees above absolute 0, 719 00:50:08,300 --> 00:50:10,672 a dramatic transformation takes place. 720 00:50:13,636 --> 00:50:14,624 Suddenly, we see-- 721 00:50:14,972 --> 00:50:16,180 PROFESSOR: Everything quiets. 722 00:50:16,180 --> 00:50:19,391 --[INAUDIBLE] stops and that the surface of the liquid helium is 723 00:50:19,391 --> 00:50:21,194 completely still. 724 00:50:21,194 --> 00:50:23,652 The temperature is actually being lowered even further now, 725 00:50:23,652 --> 00:50:25,460 but nothing is happening. 726 00:50:25,460 --> 00:50:28,582 Well, this is really one of the great phenomenon 727 00:50:28,582 --> 00:50:31,827 in 20th-century physics. 728 00:50:31,827 --> 00:50:32,410 PROFESSOR: OK. 729 00:50:32,410 --> 00:50:38,695 I'll show you this movie at the next lecture. 730 00:50:41,220 --> 00:50:43,970 But this is just a segment of it that 731 00:50:43,970 --> 00:50:50,040 shows the important property of this fluid, helium II, which 732 00:50:50,040 --> 00:50:51,270 is the superfluid. 733 00:50:51,270 --> 00:50:54,850 So now you're above the temperature it was boiling, 734 00:50:54,850 --> 00:50:56,560 but it quieted down. 735 00:50:56,560 --> 00:50:58,600 You can see that the fluid started 736 00:50:58,600 --> 00:51:01,390 to go through the capillaries that 737 00:51:01,390 --> 00:51:03,460 were at the bottom of this. 738 00:51:03,460 --> 00:51:07,510 So at the bottom of this glass there were fine capillaries. 739 00:51:07,510 --> 00:51:11,380 As long as you were up here, the capillaries 740 00:51:11,380 --> 00:51:14,690 were sufficient to keep the fluid above. 741 00:51:14,690 --> 00:51:18,370 Once you went below, the fluid went through 742 00:51:18,370 --> 00:51:21,060 as if there is no resistance. 743 00:51:21,060 --> 00:51:25,690 And that's the superfluid property. 744 00:51:25,690 --> 00:51:28,331 -Superfluidity and superconductivity were baffling 745 00:51:28,331 --> 00:51:29,656 concepts for scien-- 746 00:51:29,656 --> 00:51:30,239 [END PLAYBACK] 747 00:51:30,239 --> 00:51:34,010 PROFESSOR: OK, we'll go back to that later. 748 00:51:34,010 --> 00:51:38,090 So let's list some of the properties 749 00:51:38,090 --> 00:51:41,705 of this superfluid state. 750 00:52:03,020 --> 00:52:04,850 OK. 751 00:52:04,850 --> 00:52:12,140 So what this last experiment showed 752 00:52:12,140 --> 00:52:16,175 is that it appears to have zero viscosity. 753 00:52:21,350 --> 00:52:24,940 So what's the way that you would calculate 754 00:52:24,940 --> 00:52:27,770 the viscosity of a fluid? 755 00:52:27,770 --> 00:52:35,660 And one way to sort of do that relies on 756 00:52:35,660 --> 00:52:41,550 like what we had over here, passing the fluid 757 00:52:41,550 --> 00:52:44,720 through some pore or capillary. 758 00:52:44,720 --> 00:52:46,500 In the case of this experiment, you 759 00:52:46,500 --> 00:52:50,500 have to really make this capillary extremely fine 760 00:52:50,500 --> 00:52:53,990 so that practically nothing will go through it. 761 00:52:53,990 --> 00:52:55,770 But this could be a quite general way 762 00:52:55,770 --> 00:53:01,120 of measuring viscosity of some fluid. 763 00:53:01,120 --> 00:53:04,710 So you have some fluid. 764 00:53:04,710 --> 00:53:09,330 You can measure its viscosity by trying 765 00:53:09,330 --> 00:53:14,650 to pass it through a capillary by exerting pressure 766 00:53:14,650 --> 00:53:19,580 on one side and measuring, in essence, the difference 767 00:53:19,580 --> 00:53:23,340 between the two pressures that you have on the two sides 768 00:53:23,340 --> 00:53:27,005 as a measure of the force that you are exerting on this. 769 00:53:27,005 --> 00:53:30,790 And once you do that, then these pistons enhance 770 00:53:30,790 --> 00:53:33,660 the velocity-- sorry, the fluid, are 771 00:53:33,660 --> 00:53:37,430 moving through with some kind of velocity. 772 00:53:37,430 --> 00:53:43,100 And some measure of viscosity, after you 773 00:53:43,100 --> 00:53:46,780 divide by various things that have dimensions of length 774 00:53:46,780 --> 00:53:49,980 and pressure, et cetera, would be related 775 00:53:49,980 --> 00:53:54,275 to the velocity divided by delta P-- 776 00:53:54,275 --> 00:53:58,470 is to achieve the same velocity, how much pressure difference 777 00:53:58,470 --> 00:54:00,990 you have to put. 778 00:54:00,990 --> 00:54:05,680 Now, for this experiment of superfluid helium, 779 00:54:05,680 --> 00:54:10,500 you'll find that essentially you can get a finite velocity 780 00:54:10,500 --> 00:54:14,240 by typically infinitesimal pressure difference. 781 00:54:14,240 --> 00:54:19,520 So when you try to use this method to calculate viscosity, 782 00:54:19,520 --> 00:54:24,210 you find that, to whatever precision you can measure, 783 00:54:24,210 --> 00:54:28,130 essentially this value of eta is not distinguishable from 0. 784 00:54:30,660 --> 00:54:31,670 OK? 785 00:54:31,670 --> 00:54:32,521 Yes? 786 00:54:32,521 --> 00:54:36,369 AUDIENCE: The way you defined it, it would be infinity? 787 00:54:36,369 --> 00:54:39,480 PROFESSOR: Then I probably defined it-- 788 00:54:39,480 --> 00:54:47,730 [? no? ?] Delta P goes to 0 for a finite V. Yes, so viscosity-- 789 00:54:47,730 --> 00:54:49,230 AUDIENCE: So what you're essentially 790 00:54:49,230 --> 00:54:51,194 writing is the Reynolds number, yeah? 791 00:54:51,194 --> 00:54:52,126 PROFESSOR: Exactly. 792 00:54:52,126 --> 00:54:52,667 AUDIENCE: OK. 793 00:55:02,500 --> 00:55:04,820 PROFESSOR: All right? 794 00:55:04,820 --> 00:55:07,460 But there is another experiment that 795 00:55:07,460 --> 00:55:16,050 appears to indicate that there is a finite viscosity, 796 00:55:16,050 --> 00:55:20,070 and that is a famous experiment due to Andronikashvili. 797 00:55:25,760 --> 00:55:31,815 And basically, this is the other experiment. 798 00:55:31,815 --> 00:55:40,950 You imagine that you put your fluid inside a container. 799 00:55:40,950 --> 00:55:49,320 And then you insert in this container a set of plates, 800 00:55:49,320 --> 00:55:52,670 could be copper plates or something else, 801 00:55:52,670 --> 00:55:54,900 that are stacked on top of each other. 802 00:55:59,076 --> 00:56:06,366 So essentially, you have a stack of these plates that 803 00:56:06,366 --> 00:56:11,730 have a common axis, and there's a rod 804 00:56:11,730 --> 00:56:14,870 that is going through them. 805 00:56:14,870 --> 00:56:24,930 And what you created here is a torsional oscillator 806 00:56:24,930 --> 00:56:30,220 in the sense that this rod that holds all of these things 807 00:56:30,220 --> 00:56:35,780 together has a particular preferred orientation. 808 00:56:35,780 --> 00:56:45,670 And if you start, let's say-- well, it may or may not 809 00:56:45,670 --> 00:56:55,290 have a particular orientation-- but if you start to rotate it, 810 00:56:55,290 --> 00:57:00,050 it will certainly have a moment of inertia that 811 00:57:00,050 --> 00:57:08,830 resists the rotation that is coming from all of the plates 812 00:57:08,830 --> 00:57:11,550 if it was in vacuum. 813 00:57:11,550 --> 00:57:15,900 So if I could measure the moment of inertia of these objects, 814 00:57:15,900 --> 00:57:19,800 either through torsion or some other mechanism, 815 00:57:19,800 --> 00:57:21,840 if it is a torsional oscillator I 816 00:57:21,840 --> 00:57:24,530 could figure out what the frequency is. 817 00:57:24,530 --> 00:57:27,000 If I knew the [? spinning ?] constant 818 00:57:27,000 --> 00:57:30,920 that wants to bring the angle to some particular value, 819 00:57:30,920 --> 00:57:33,600 I would divide that by the moment of inertia. 820 00:57:33,600 --> 00:57:36,400 The square root of that would give me the frequency. 821 00:57:36,400 --> 00:57:40,390 So the frequency is a measure of the moment of inertia. 822 00:57:40,390 --> 00:57:42,380 So if I had this thing in vacuum, 823 00:57:42,380 --> 00:57:44,050 I could figure out what the moment 824 00:57:44,050 --> 00:57:47,310 of inertia of all of the plates was. 825 00:57:47,310 --> 00:57:55,340 Now, if I add the fluid to this, then as I start to rotate this, 826 00:57:55,340 --> 00:57:59,480 some of the fluid will certainly move with the plates. 827 00:57:59,480 --> 00:58:02,120 And so the moment of inertia that you would measure 828 00:58:02,120 --> 00:58:04,840 in the presence of the fluid is larger 829 00:58:04,840 --> 00:58:07,920 than it is in the absence of fluid. 830 00:58:07,920 --> 00:58:11,500 And that additional moment of inertia 831 00:58:11,500 --> 00:58:15,270 tells you about how much of the fluid is, in some sense, 832 00:58:15,270 --> 00:58:19,200 stuck to the plates, and that's another different measure 833 00:58:19,200 --> 00:58:21,650 of the viscosity. 834 00:58:21,650 --> 00:58:22,890 OK? 835 00:58:22,890 --> 00:58:29,560 So if we sort of look at this experiment, 836 00:58:29,560 --> 00:58:36,200 we would imagine that if I measure the moment of inertia 837 00:58:36,200 --> 00:58:44,130 as a function of temperature, something 838 00:58:44,130 --> 00:58:47,770 should happen when I hit Tc. 839 00:58:47,770 --> 00:58:54,420 Above Tc, I'm getting some value that is measuring, essentially, 840 00:58:54,420 --> 00:58:58,470 the moment of inertia of the plates plus the fluid 841 00:58:58,470 --> 00:59:01,580 that is moving around, [? we do. ?] 842 00:59:01,580 --> 00:59:05,400 But what happens when you hit Tc? 843 00:59:05,400 --> 00:59:11,050 If the superfluid doesn't care about things around it, 844 00:59:11,050 --> 00:59:14,020 you would imagine that this would drop down to the value 845 00:59:14,020 --> 00:59:17,270 that it has in vacuum. 846 00:59:17,270 --> 00:59:20,350 In reality, what it does is something like this. 847 00:59:23,090 --> 00:59:26,170 So it does go to the value that you 848 00:59:26,170 --> 00:59:31,750 would have in vacuum but only at 0 temperature. 849 00:59:31,750 --> 00:59:38,280 So at any finite temperature we know Tc. 850 00:59:38,280 --> 00:59:44,220 There is indeed some fraction of the superfluid 851 00:59:44,220 --> 00:59:48,660 that is trapped by these plates. 852 00:59:48,660 --> 00:59:50,472 So what's happening? 853 00:59:50,472 --> 00:59:52,600 OK, so this is one [INAUDIBLE]. 854 00:59:56,150 --> 01:00:02,330 The other property-- maybe I will squeeze it somewhere here. 855 01:00:27,590 --> 01:00:30,230 There are a whole bunch of things 856 01:00:30,230 --> 01:00:35,150 that indicate that there is a coupling between temperature 857 01:00:35,150 --> 01:00:42,471 and mechanical properties, so-called thermo-mechanical 858 01:00:42,471 --> 01:00:42,970 couplings. 859 01:00:49,520 --> 01:00:57,990 What you find in this experiment, where 860 01:00:57,990 --> 01:01:01,730 the fluid spontaneously goes, let's 861 01:01:01,730 --> 01:01:05,620 say, from this side to this side, 862 01:01:05,620 --> 01:01:09,080 so the direction is like this, is 863 01:01:09,080 --> 01:01:11,680 that when you measure the temperatures 864 01:01:11,680 --> 01:01:13,890 you will find that the temperature of this one 865 01:01:13,890 --> 01:01:19,540 goes up, and the temperature of this one goes down. 866 01:01:19,540 --> 01:01:28,380 So somehow the motion led to some temperature differences. 867 01:01:28,380 --> 01:01:31,830 A dramatic one that, again, maybe we'll show in the movie 868 01:01:31,830 --> 01:01:34,540 next time is the fountain effect. 869 01:01:38,540 --> 01:01:42,300 What you do is you-- let's imagine that, 870 01:01:42,300 --> 01:01:49,010 again, you have a vat of helium, and inside of it 871 01:01:49,010 --> 01:01:55,010 you put a tube that is, let's say, shaped like this. 872 01:01:55,010 --> 01:02:03,930 And you have the fluid that is occupying everything here. 873 01:02:03,930 --> 01:02:10,160 Then inside this tube, you put some material 874 01:02:10,160 --> 01:02:13,280 that is a good absorber of heat, such as, let's say, 875 01:02:13,280 --> 01:02:18,770 a piece of carbon, and then you shine laser 876 01:02:18,770 --> 01:02:24,380 on it so that this thing becomes hot. 877 01:02:24,380 --> 01:02:28,380 So in this case, the temperature goes up, 878 01:02:28,380 --> 01:02:34,740 and what you find happens is that the fluid will 879 01:02:34,740 --> 01:02:37,500 start to shoot up here like a fountain. 880 01:02:37,500 --> 01:02:40,270 It's called the fountain effect because of that. 881 01:02:40,270 --> 01:02:45,640 And essentially, whereas here pressure was converted 882 01:02:45,640 --> 01:02:50,130 to temperature, here temperature, the heating, 883 01:02:50,130 --> 01:02:52,460 gets converted to pressure. 884 01:02:52,460 --> 01:02:54,440 So this is like-- I don't know if you've ever 885 01:02:54,440 --> 01:02:57,610 done this, where you put your hand under water, 886 01:02:57,610 --> 01:03:01,160 and if you squeeze your hand and the water shoots up. 887 01:03:01,160 --> 01:03:05,080 So it's kind of putting pressure here, causing a fountain that 888 01:03:05,080 --> 01:03:07,860 is shooting up like that. 889 01:03:07,860 --> 01:03:08,360 OK? 890 01:03:08,360 --> 01:03:13,890 So there is this coupling between temperature 891 01:03:13,890 --> 01:03:19,000 and-- actually, this issue of boiling-not boiling 892 01:03:19,000 --> 01:03:22,910 is also a manifestation of a coupling between temperature 893 01:03:22,910 --> 01:03:26,200 and mechanical properties, which is whether the fluid is boiling 894 01:03:26,200 --> 01:03:28,160 or not boiling. 895 01:03:28,160 --> 01:03:30,470 OK? 896 01:03:30,470 --> 01:03:36,820 So given everything that we have been talking about so far, 897 01:03:36,820 --> 01:03:40,755 it is tempting to imagine that there 898 01:03:40,755 --> 01:03:44,600 is a connection between all of this 899 01:03:44,600 --> 01:03:51,010 and the Bose-Einstein condensation. 900 01:03:51,010 --> 01:03:52,010 AUDIENCE: [INAUDIBLE]. 901 01:03:52,010 --> 01:03:54,885 PROFESSOR: Yes? 902 01:03:54,885 --> 01:03:59,998 AUDIENCE: Do superfluids have no vorticity or can they? 903 01:04:04,390 --> 01:04:06,710 PROFESSOR: Yes, they can have vorticity, 904 01:04:06,710 --> 01:04:08,520 but it is of a special kind. 905 01:04:08,520 --> 01:04:11,210 It is quantized vorticity. 906 01:04:11,210 --> 01:04:16,950 And so what you can certainly do is you can take a vat of helium 907 01:04:16,950 --> 01:04:20,040 and set it into rotation, and then you 908 01:04:20,040 --> 01:04:23,679 will create vortex lines inside this superfluid. 909 01:04:23,679 --> 01:04:24,220 AUDIENCE: OK. 910 01:04:24,220 --> 01:04:26,370 I was wondering if there was an analogy 911 01:04:26,370 --> 01:04:30,280 to superconductivity and magnetic fields. 912 01:04:30,280 --> 01:04:33,480 PROFESSOR: The analog of the magnetic field is the rotation. 913 01:04:33,480 --> 01:04:34,195 AUDIENCE: Right. 914 01:04:34,195 --> 01:04:34,820 PROFESSOR: Yes. 915 01:04:34,820 --> 01:04:35,361 AUDIENCE: OK. 916 01:04:35,361 --> 01:04:36,824 PROFESSOR: So in that sense, yes. 917 01:04:40,993 --> 01:04:41,493 OK? 918 01:04:44,950 --> 01:04:47,130 OK. 919 01:04:47,130 --> 01:04:59,520 So similarities between BEC, Bose-Einstein Condensation, 920 01:04:59,520 --> 01:05:00,780 and superfluidity. 921 01:05:11,130 --> 01:05:19,350 I would say that perhaps the most important one, although I 922 01:05:19,350 --> 01:05:21,850 guess at the beginning of the story they didn't know 923 01:05:21,850 --> 01:05:31,410 this one, is that helium-3, which is a fermion, 924 01:05:31,410 --> 01:05:43,030 does not become superfluid around this same temperature 925 01:05:43,030 --> 01:05:46,060 of 2 degrees Kelvin, which is where 926 01:05:46,060 --> 01:05:47,570 helium-4 becomes superfluid. 927 01:05:50,800 --> 01:05:52,860 What is special about this is, well, you 928 01:05:52,860 --> 01:06:00,990 would say that once you have mass and density, 929 01:06:00,990 --> 01:06:04,660 you can figure out what the Bose-Einstein temperature is. 930 01:06:04,660 --> 01:06:09,620 You had this formula that was n lambda 931 01:06:09,620 --> 01:06:15,350 cubed is equal to zeta of 3/2. 932 01:06:15,350 --> 01:06:23,820 So that gives you some value of kB Tc being h 933 01:06:23,820 --> 01:06:32,320 squared 2 pi mass of helium, and then you have a zeta of 3/2 934 01:06:32,320 --> 01:06:33,520 and n somewhere. 935 01:06:36,390 --> 01:06:55,990 3-- I guess this n-- oops, n-- yeah, it's correct-- n h 936 01:06:55,990 --> 01:07:05,510 cubed 2 pi m kB Tc to the 3/2 being zeta of 3/2. 937 01:07:05,510 --> 01:07:16,874 So 2 pi m kB Tc is proportional to n zeta of 3/2 938 01:07:16,874 --> 01:07:19,530 to the 2/3 power. 939 01:07:19,530 --> 01:07:21,230 OK. 940 01:07:21,230 --> 01:07:28,570 So I had a picture here, which was the typical attraction 941 01:07:28,570 --> 01:07:30,000 potential. 942 01:07:30,000 --> 01:07:33,330 The typical separation of helium atoms 943 01:07:33,330 --> 01:07:36,700 is something like 3.6 angstroms. 944 01:07:36,700 --> 01:07:41,050 How do I know that is because the density I can figure out 945 01:07:41,050 --> 01:07:45,390 is typically 1 over the volume per particle, 946 01:07:45,390 --> 01:07:51,460 and this volume per particle is roughly 3.6 angstroms cubed. 947 01:07:51,460 --> 01:07:55,050 It's something like 46.2 angstrom cubed. 948 01:07:55,050 --> 01:07:58,660 So I can put that value of n over here. 949 01:07:58,660 --> 01:08:01,280 I know what the mass of helium is. 950 01:08:01,280 --> 01:08:05,010 And I can figure out what this Tc is, 951 01:08:05,010 --> 01:08:11,490 and I find that Tc is 3.13 degrees K. You say, well, it's 952 01:08:11,490 --> 01:08:17,970 not exactly the 2 pi and something that we had. 953 01:08:17,970 --> 01:08:22,520 But given that we completely ignored the interactions 954 01:08:22,520 --> 01:08:25,660 that we have between this, and this is the result 955 01:08:25,660 --> 01:08:27,930 that we have for [? point ?] particles 956 01:08:27,930 --> 01:08:32,319 of mass m with no interactions, this is actually very good. 957 01:08:32,319 --> 01:08:35,660 You would say, OK, this is roughly in the right range. 958 01:08:35,660 --> 01:08:38,340 And presumably if I were to include 959 01:08:38,340 --> 01:08:42,620 the effects of interactions, potentially I 960 01:08:42,620 --> 01:08:46,204 will get an interaction, temperature, 961 01:08:46,204 --> 01:08:48,819 that is roughly correct. 962 01:08:48,819 --> 01:08:53,540 So this is another reason. 963 01:08:53,540 --> 01:09:00,430 The third reason is that you have automatically, 964 01:09:00,430 --> 01:09:04,330 when you are talking about the Bose-Einstein condensation, 965 01:09:04,330 --> 01:09:07,910 a relationship between pressure and temperature. 966 01:09:07,910 --> 01:09:09,950 So remember the formula that we had 967 01:09:09,950 --> 01:09:11,790 for the Bose-Einstein condensation, 968 01:09:11,790 --> 01:09:16,189 beta P was something, where P is kT 969 01:09:16,189 --> 01:09:22,470 1 over lambda cubed zeta of 5/2, right? 970 01:09:22,470 --> 01:09:25,479 So once you specify the temperature, 971 01:09:25,479 --> 01:09:28,970 you've also specified the pressure. 972 01:09:28,970 --> 01:09:31,410 They go hand in hand. 973 01:09:31,410 --> 01:09:35,370 So then you can start to explain a lot of these phenomena, 974 01:09:35,370 --> 01:09:39,109 such as here you increase the temperature, 975 01:09:39,109 --> 01:09:42,359 pressure had to go up, and it was the increased pressure 976 01:09:42,359 --> 01:09:43,569 than did that. 977 01:09:43,569 --> 01:09:46,304 It kind of tells you something about what 978 01:09:46,304 --> 01:09:49,729 is going on in this experiment over here, 979 01:09:49,729 --> 01:09:54,310 where the temperature goes up where you have higher pressure. 980 01:09:54,310 --> 01:09:58,120 It actually also explains why things 981 01:09:58,120 --> 01:10:02,250 stop bubbling when you go into the superfluid. 982 01:10:02,250 --> 01:10:04,960 Because let's back out and think of why 983 01:10:04,960 --> 01:10:09,515 things are bubbling when you put your kettle on top of the oven. 984 01:10:09,515 --> 01:10:14,000 And the reason is that there are variations in temperature. 985 01:10:14,000 --> 01:10:17,600 There could be a local hotspot over here, 986 01:10:17,600 --> 01:10:20,770 and in this local hotspot, eventually there 987 01:10:20,770 --> 01:10:25,580 will a bubble that forms, and the bubble goes away. 988 01:10:25,580 --> 01:10:32,400 Why is it that it forms here is because heat gets spread out 989 01:10:32,400 --> 01:10:34,370 through diffusion. 990 01:10:34,370 --> 01:10:40,180 If I strike a match or even if I have a blow torch here, 991 01:10:40,180 --> 01:10:43,750 you won't feel the heat for some time. 992 01:10:43,750 --> 01:10:49,410 But if I do this, you hear my hand clapping 993 01:10:49,410 --> 01:10:51,060 much more rapidly. 994 01:10:51,060 --> 01:10:57,920 So pressure gets transmitted very efficiently through sound, 995 01:10:57,920 --> 01:11:01,870 whereas heat goes very slowly through diffusion. 996 01:11:01,870 --> 01:11:06,710 And that's why when you boil something, essentially 997 01:11:06,710 --> 01:11:11,880 you will have the diffusion of heat, local hotspots, bubbles. 998 01:11:11,880 --> 01:11:18,740 But if temperature changes are connected to pressure changes, 999 01:11:18,740 --> 01:11:21,760 then for the superfluid, having a local hotspot 1000 01:11:21,760 --> 01:11:23,770 is like doing this. 1001 01:11:23,770 --> 01:11:26,800 It creates immediately a pressure change, 1002 01:11:26,800 --> 01:11:29,926 and pressure changes are rapidly taken away. 1003 01:11:29,926 --> 01:11:33,380 And that's why this could also explain 1004 01:11:33,380 --> 01:11:34,610 what's going on over here. 1005 01:11:38,370 --> 01:11:45,350 So I think this number two was first noted by Fritz London. 1006 01:11:45,350 --> 01:11:49,330 Maybe we'll talk about that a bit more next time around. 1007 01:11:49,330 --> 01:11:55,630 Based on all of these similarities, 1008 01:11:55,630 --> 01:12:02,200 Laszlo Tisza, who spent many years at MIT 1009 01:12:02,200 --> 01:12:08,100 after being in Europe, proposed the two-fluid model. 1010 01:12:14,390 --> 01:12:16,790 So again, roughly inspired by what 1011 01:12:16,790 --> 01:12:21,570 we saw for the case of the Bose-Einstein condensate, 1012 01:12:21,570 --> 01:12:23,950 we saw that for the Bose-Einstein condensate, 1013 01:12:23,950 --> 01:12:27,940 below Tc you had to make a separation 1014 01:12:27,940 --> 01:12:33,970 for the number of things that were in the ground state, 1015 01:12:33,970 --> 01:12:37,280 k equals to 0, and all of the others 1016 01:12:37,280 --> 01:12:40,180 that were in the higher excited states. 1017 01:12:40,180 --> 01:12:44,130 Pressure, heat capacity, everything 1018 01:12:44,130 --> 01:12:48,950 came from the excited component, whereas the component that 1019 01:12:48,950 --> 01:12:53,230 was at the k equals to 0 state was not making any contribution 1020 01:12:53,230 --> 01:12:55,300 to these phenomena. 1021 01:12:55,300 --> 01:12:58,860 So what he proposed was that somehow, 1022 01:12:58,860 --> 01:13:03,870 again, similarly when you are in the superfluid, 1023 01:13:03,870 --> 01:13:06,670 you really have two components, two fluids 1024 01:13:06,670 --> 01:13:08,790 that are in coexistence. 1025 01:13:08,790 --> 01:13:11,800 There is a superfluid component, and there 1026 01:13:11,800 --> 01:13:15,130 is a normal component. 1027 01:13:15,130 --> 01:13:22,030 And as you change temperature, these two components 1028 01:13:22,030 --> 01:13:24,570 get converted to each other. 1029 01:13:24,570 --> 01:13:28,000 So just like the case of the Bose-Einstein condensate, 1030 01:13:28,000 --> 01:13:31,800 it is not like you have so many of this fluid, so 1031 01:13:31,800 --> 01:13:33,460 many of that fluid. 1032 01:13:33,460 --> 01:13:37,170 These two things are always interchanging. 1033 01:13:37,170 --> 01:13:40,920 And then what is happening in this experiment, 1034 01:13:40,920 --> 01:13:45,600 presumably, is that the superfluid component 1035 01:13:45,600 --> 01:13:50,450 goes through, the normal component leaves behind. 1036 01:13:50,450 --> 01:13:54,140 The superfluid component is the component that, in some sense, 1037 01:13:54,140 --> 01:13:57,550 has less entropy, is at lower temperature. 1038 01:13:57,550 --> 01:14:00,990 So when it goes here, it cools down this part. 1039 01:14:00,990 --> 01:14:04,890 This part then goes up in temperature. 1040 01:14:04,890 --> 01:14:09,900 Again, there is no way that you can push all of the superfluid 1041 01:14:09,900 --> 01:14:13,410 here and have normal here because these things get 1042 01:14:13,410 --> 01:14:14,040 converted. 1043 01:14:14,040 --> 01:14:17,520 So some amount of superfluid goes through here. 1044 01:14:17,520 --> 01:14:20,160 Some amount gets converted from the normal back 1045 01:14:20,160 --> 01:14:21,000 to the superfluid. 1046 01:14:21,000 --> 01:14:24,120 So there is this exchange that is constantly 1047 01:14:24,120 --> 01:14:26,170 taking place in the system. 1048 01:14:26,170 --> 01:14:32,260 And so on the basis of this model and having 1049 01:14:32,260 --> 01:14:36,070 different velocities for the normal and superfluid 1050 01:14:36,070 --> 01:14:41,940 component, Tisza could explain a number of these things. 1051 01:14:41,940 --> 01:14:44,750 OK? 1052 01:14:44,750 --> 01:14:59,240 So I will mention some important differences 1053 01:14:59,240 --> 01:15:03,750 at the end of this lecture and leave their resolution 1054 01:15:03,750 --> 01:15:05,030 for next. 1055 01:15:05,030 --> 01:15:10,470 So while those were similarities between BEC and superfluidity, 1056 01:15:10,470 --> 01:15:12,280 there are very important differences. 1057 01:15:31,780 --> 01:15:36,080 One important observation immediately 1058 01:15:36,080 --> 01:15:46,880 is that if I try to compress superfluid helium or normal 1059 01:15:46,880 --> 01:15:48,230 helium, it doesn't matter. 1060 01:15:48,230 --> 01:15:50,810 It is just like trying to compress 1061 01:15:50,810 --> 01:15:53,222 any other fluid such as water. 1062 01:15:53,222 --> 01:15:58,300 It does not like to be squeezed. 1063 01:15:58,300 --> 01:16:05,900 So helium liquid is approximately, of course 1064 01:16:05,900 --> 01:16:20,710 not entirely, incompressible, while BEC, we 1065 01:16:20,710 --> 01:16:24,770 saw that the pressure is only a function of temperature. 1066 01:16:24,770 --> 01:16:28,020 It doesn't know anything about the density. 1067 01:16:28,020 --> 01:16:35,610 So you can change the density. 1068 01:16:35,610 --> 01:16:37,870 All of the additional particles will 1069 01:16:37,870 --> 01:16:39,740 go to go to the k equals to 0 state. 1070 01:16:39,740 --> 01:16:41,780 It doesn't really care. 1071 01:16:41,780 --> 01:16:44,110 So BEC is infinitely compressible. 1072 01:16:52,880 --> 01:16:54,490 And of course, this is immediately 1073 01:16:54,490 --> 01:16:56,830 a manifestation of the interactions. 1074 01:17:00,990 --> 01:17:04,880 In the case of the BEC, we didn't put any interactions 1075 01:17:04,880 --> 01:17:05,920 among the particles. 1076 01:17:05,920 --> 01:17:09,840 We can put as many of them as we like into the single state k 1077 01:17:09,840 --> 01:17:11,210 equals to 0. 1078 01:17:11,210 --> 01:17:14,370 Of course, real heliums, they have the self-avoiding-- 1079 01:17:14,370 --> 01:17:16,464 the interaction between them. 1080 01:17:16,464 --> 01:17:18,130 They cannot go [? through ?] each other. 1081 01:17:18,130 --> 01:17:20,760 So that's an important difference. 1082 01:17:24,830 --> 01:17:41,400 Number 2 is that detailed shapes/ T dependences of heat 1083 01:17:41,400 --> 01:17:47,970 capacity and density of the superfluid part are different. 1084 01:17:51,260 --> 01:17:53,150 OK. 1085 01:17:53,150 --> 01:17:58,640 So I have up there what the heat capacity of a BEC 1086 01:17:58,640 --> 01:18:01,100 should look like. 1087 01:18:01,100 --> 01:18:07,720 Now, people have measured the heat capacity of helium 1088 01:18:07,720 --> 01:18:12,410 as a function of temperature, and something does indeed 1089 01:18:12,410 --> 01:18:16,350 happen at Tc. 1090 01:18:16,350 --> 01:18:21,310 And what you find the shape is is something like this. 1091 01:18:25,500 --> 01:18:33,300 It goes to 0, and then it does something like this. 1092 01:18:33,300 --> 01:18:37,780 And so while it doesn't look that different from what 1093 01:18:37,780 --> 01:18:42,400 I have up there, except that when you look at the behavior 1094 01:18:42,400 --> 01:18:45,770 down here, at the transition, over there 1095 01:18:45,770 --> 01:18:47,790 it went to a finite value. 1096 01:18:47,790 --> 01:18:49,995 Here it diverges as a log. 1097 01:18:56,700 --> 01:19:01,460 So when you go to Tc, it actually goes to infinity. 1098 01:19:01,460 --> 01:19:03,760 And because of the shape that this has, 1099 01:19:03,760 --> 01:19:06,130 this is sometimes called the lambda transition. 1100 01:19:12,760 --> 01:19:17,560 Now, there's lots of issues about interesting things 1101 01:19:17,560 --> 01:19:20,090 happening at the transition, and we 1102 01:19:20,090 --> 01:19:23,050 won't be able to resolve that. 1103 01:19:23,050 --> 01:19:25,910 But one thing that we should be able to figure out, 1104 01:19:25,910 --> 01:19:29,150 based on what we know already, is the behavior 1105 01:19:29,150 --> 01:19:31,900 as things go to 0 temperature. 1106 01:19:31,900 --> 01:19:34,950 And I said that for the Bose-Einstein condensate, 1107 01:19:34,950 --> 01:19:38,680 the temperature dependence is T to the 3/2. 1108 01:19:38,680 --> 01:19:43,050 What is important is that for the superfluid, 1109 01:19:43,050 --> 01:19:45,750 it is actually proportional to T cubed. 1110 01:19:50,020 --> 01:19:53,930 And T cubed, if you go back in the course, 1111 01:19:53,930 --> 01:19:57,890 is a signature of what was happening for phonons. 1112 01:19:57,890 --> 01:19:59,860 So Landau looked at this and said 1113 01:19:59,860 --> 01:20:03,430 there must be some kind of a phonon-like excitation going 1114 01:20:03,430 --> 01:20:09,085 on, which is different from this case [INAUDIBLE] excitations 1115 01:20:09,085 --> 01:20:12,710 that we have. 1116 01:20:12,710 --> 01:20:17,000 The other thing that we had was that if we 1117 01:20:17,000 --> 01:20:23,560 look at the fraction that is in-- let's say the density that 1118 01:20:23,560 --> 01:20:30,910 is in the k equals to 0 state as a function of temperature, what 1119 01:20:30,910 --> 01:20:38,850 happens for the superfluid-- sorry, 1120 01:20:38,850 --> 01:20:41,370 what happens for the Bose-Einstein condensate-- 1121 01:20:41,370 --> 01:20:43,110 for the Bose-Einstein condensate, 1122 01:20:43,110 --> 01:20:47,250 let's use the green-- essentially, 1123 01:20:47,250 --> 01:20:56,380 the curve will come down to Tc linearly, and then it goes up, 1124 01:20:56,380 --> 01:21:01,580 and it reaches its asymptotic value. 1125 01:21:01,580 --> 01:21:06,050 From this result, the fraction that is excited, 1126 01:21:06,050 --> 01:21:09,130 the difference to 1 is proportional to 1 1127 01:21:09,130 --> 01:21:10,130 over lambda cubed. 1128 01:21:10,130 --> 01:21:12,060 So this goes like T to the 3/2. 1129 01:21:16,100 --> 01:21:20,820 You can ask whether what was observed in the Andronikashvili 1130 01:21:20,820 --> 01:21:24,870 experiment, eventually this measure 1131 01:21:24,870 --> 01:21:29,930 also, the fraction that is in the superfluid component 1132 01:21:29,930 --> 01:21:34,290 or excited states, does this curve match that curve? 1133 01:21:34,290 --> 01:21:38,350 But if I take this curve and put it over here, first of all, 1134 01:21:38,350 --> 01:21:40,770 down here it is different. 1135 01:21:40,770 --> 01:21:45,980 It goes like Tc minus T to the 2/3 power rather than linear 1136 01:21:45,980 --> 01:21:48,600 that you have over here. 1137 01:21:48,600 --> 01:21:50,790 That, again, is less [? worry ?] than the fact 1138 01:21:50,790 --> 01:21:54,265 that over here it goes to 0 proportional to T 1139 01:21:54,265 --> 01:21:56,380 to the fourth. 1140 01:21:56,380 --> 01:21:58,564 Again, something that needs explanation. 1141 01:22:03,410 --> 01:22:07,500 And finally, I will write down one statement, 1142 01:22:07,500 --> 01:22:11,130 and then maybe we'll explain next time, 1143 01:22:11,130 --> 01:22:20,800 is that BEC with the spectrum of excitations, 1144 01:22:20,800 --> 01:22:29,020 which is k squared over 2m, cannot be superfluid. 1145 01:22:32,020 --> 01:22:38,592 There is an interesting reason for that that we will explain. 1146 01:22:38,592 --> 01:22:43,190 And indeed, we'll see that having excitations 1147 01:22:43,190 --> 01:22:47,180 that are [INAUDIBLE] like is compatible with superfluidity. 1148 01:22:47,180 --> 01:22:49,890 This kind of excitation is not. 1149 01:22:49,890 --> 01:22:55,100 So why is that, et cetera, we'll talk about next time.