WEBVTT 00:00:00.000 --> 00:00:01.960 align:middle line:90% [SQUEAKING] 00:00:01.960 --> 00:00:03.920 align:middle line:90% [RUSTLING] 00:00:03.920 --> 00:00:04.900 align:middle line:90% [CLICKING] 00:00:04.900 --> 00:00:11.262 align:middle line:90% 00:00:11.262 --> 00:00:12.220 align:middle line:90% GIAN PAOLO BERETTA: OK. 00:00:12.220 --> 00:00:13.960 align:middle line:90% Good afternoon. 00:00:13.960 --> 00:00:19.063 align:middle line:84% Here we are March 19, lecture 13. 00:00:19.063 --> 00:00:23.150 align:middle line:90% 00:00:23.150 --> 00:00:28.370 align:middle line:84% We are still in the process of studying 00:00:28.370 --> 00:00:34.550 align:middle line:84% mixtures, the wonderful world of mixtures. 00:00:34.550 --> 00:00:42.080 align:middle line:84% And so let me just briefly see where we are. 00:00:42.080 --> 00:00:46.820 align:middle line:84% You remember that we discussed the properties 00:00:46.820 --> 00:00:53.990 align:middle line:84% of isothermobaric mixing, which are measurable properties that 00:00:53.990 --> 00:01:01.010 align:middle line:84% allow us to infer the chemical potentials thanks 00:01:01.010 --> 00:01:04.165 align:middle line:90% to this expression here. 00:01:04.165 --> 00:01:09.190 align:middle line:90% 00:01:09.190 --> 00:01:20.430 align:middle line:84% We discussed the intermolecular forces between molecules 00:01:20.430 --> 00:01:27.240 align:middle line:84% and, for example, the fact that in the gas or ideal gas, regime, 00:01:27.240 --> 00:01:32.160 align:middle line:84% molecules spend most of their time 00:01:32.160 --> 00:01:34.470 align:middle line:84% far away from each other in spite 00:01:34.470 --> 00:01:41.400 align:middle line:84% of having an enormous amount of collisions per second, 00:01:41.400 --> 00:01:45.510 align:middle line:90% during which they do get close. 00:01:45.510 --> 00:01:50.940 align:middle line:84% But the time they spend close is small with respect 00:01:50.940 --> 00:01:54.730 align:middle line:90% to the average. 00:01:54.730 --> 00:02:05.640 align:middle line:84% And so since the properties are determined by the average, 00:02:05.640 --> 00:02:12.840 align:middle line:84% the effect of the intermolecular interactions 00:02:12.840 --> 00:02:21.390 align:middle line:84% is relatively negligible and that's the ideal gas regime. 00:02:21.390 --> 00:02:28.470 align:middle line:84% We will see more about that today 00:02:28.470 --> 00:02:32.075 align:middle line:84% from another point of view, which we will see in a moment. 00:02:32.075 --> 00:02:35.310 align:middle line:90% 00:02:35.310 --> 00:02:43.410 align:middle line:84% We introduced a first ideal behavior model 00:02:43.410 --> 00:02:51.330 align:middle line:84% based on this idea of particles not interacting, which 00:02:51.330 --> 00:02:56.430 align:middle line:84% is the ideal Gibbs-Dalton mixture model in which, as you 00:02:56.430 --> 00:02:59.790 align:middle line:84% remember, the idea is that we represent 00:02:59.790 --> 00:03:07.230 align:middle line:84% the properties of the mixture as equivalent to the properties 00:03:07.230 --> 00:03:12.960 align:middle line:84% of a composite system in which the various components are 00:03:12.960 --> 00:03:14.940 align:middle line:90% separate. 00:03:14.940 --> 00:03:16.690 align:middle line:90% They occupy the same volume. 00:03:16.690 --> 00:03:18.070 align:middle line:90% They are the same temperature. 00:03:18.070 --> 00:03:20.873 align:middle line:84% And they are at the partial pressure. 00:03:20.873 --> 00:03:24.840 align:middle line:90% 00:03:24.840 --> 00:03:32.160 align:middle line:84% According to this model, if each of the substances 00:03:32.160 --> 00:03:38.760 align:middle line:84% behaves as an ideal gas, so obeys the ideal gas pV equals 00:03:38.760 --> 00:03:47.820 align:middle line:84% RT equation of state, we found that the partial pressures 00:03:47.820 --> 00:03:51.480 align:middle line:84% add up to the pressure of the mixture. 00:03:51.480 --> 00:03:56.400 align:middle line:90% That's Dalton law, this here. 00:03:56.400 --> 00:03:58.230 align:middle line:84% And the partial pressures are given 00:03:58.230 --> 00:04:02.880 align:middle line:84% by the pressure of the mixture times the mole fraction. 00:04:02.880 --> 00:04:06.300 align:middle line:84% Plus, the chemical potential of the components 00:04:06.300 --> 00:04:08.970 align:middle line:84% in the mixture of each component is 00:04:08.970 --> 00:04:14.520 align:middle line:84% equal to this important expression that 00:04:14.520 --> 00:04:18.060 align:middle line:84% is the sum of the chemical potential of the pure component 00:04:18.060 --> 00:04:21.269 align:middle line:84% at the same temperature and pressure of the mixture 00:04:21.269 --> 00:04:23.910 align:middle line:84% plus RT, the log of a mole fraction. 00:04:23.910 --> 00:04:34.700 align:middle line:90% 00:04:34.700 --> 00:04:45.200 align:middle line:84% We then proved also the Amagat law of additive volumes 00:04:45.200 --> 00:04:48.320 align:middle line:84% whereby an ideal gas mixture is such 00:04:48.320 --> 00:04:54.140 align:middle line:84% that you can also view it as a composite 00:04:54.140 --> 00:04:59.780 align:middle line:84% of separate pure gases, all at the same temperature 00:04:59.780 --> 00:05:10.210 align:middle line:84% and pressure but each occupying the so-called proper volume, 00:05:10.210 --> 00:05:15.340 align:middle line:84% which is given at the volume that they would occupy 00:05:15.340 --> 00:05:22.243 align:middle line:84% when pure in the same amount at the same temperature 00:05:22.243 --> 00:05:22.785 align:middle line:90% and pressure. 00:05:22.785 --> 00:05:27.420 align:middle line:90% 00:05:27.420 --> 00:05:33.330 align:middle line:84% Then we discussed spontaneous mixing. 00:05:33.330 --> 00:05:37.530 align:middle line:84% What happens when we have separate gases 00:05:37.530 --> 00:05:39.510 align:middle line:84% kept separate by partitions that are 00:05:39.510 --> 00:05:44.790 align:middle line:84% movable so they are all at the same temperature and pressure. 00:05:44.790 --> 00:05:47.510 align:middle line:84% And without any external effect-- 00:05:47.510 --> 00:05:50.040 align:middle line:84% you see the weight is at the same level-- 00:05:50.040 --> 00:05:51.250 align:middle line:90% you just let them mix. 00:05:51.250 --> 00:05:56.190 align:middle line:84% You take the partitions away and let them mix. 00:05:56.190 --> 00:06:00.090 align:middle line:84% We found that the mixing under these conditions 00:06:00.090 --> 00:06:02.790 align:middle line:84% yields the final temperature and the final pressure 00:06:02.790 --> 00:06:05.730 align:middle line:90% equal to the initial ones. 00:06:05.730 --> 00:06:12.540 align:middle line:84% And the entropy increases by the so-called entropy of mixing, 00:06:12.540 --> 00:06:18.190 align:middle line:84% which is given by this term, minus the number of particles, 00:06:18.190 --> 00:06:24.060 align:middle line:84% total number of particles, the gas constant, and the sum of y 00:06:24.060 --> 00:06:29.160 align:middle line:84% log y, so the mole fractions times the logarithm 00:06:29.160 --> 00:06:32.360 align:middle line:90% of the mole fractions. 00:06:32.360 --> 00:06:37.220 align:middle line:84% We also noticed that we could have 00:06:37.220 --> 00:06:41.850 align:middle line:84% done that mixing in a better way, 00:06:41.850 --> 00:06:44.520 align:middle line:84% better from the thermodynamic point of view, in other words 00:06:44.520 --> 00:06:47.990 align:middle line:90% reversibly, in a weight process. 00:06:47.990 --> 00:06:54.110 align:middle line:84% And we computed how much work we could-- 00:06:54.110 --> 00:06:57.890 align:middle line:84% so therefore how much weight lifting we 00:06:57.890 --> 00:07:04.280 align:middle line:84% could get if we did that, indeed, reversibly. 00:07:04.280 --> 00:07:05.350 align:middle line:90% And here's the formula. 00:07:05.350 --> 00:07:12.080 align:middle line:90% 00:07:12.080 --> 00:07:18.110 align:middle line:84% And the last thing we discussed was a pictorial idea 00:07:18.110 --> 00:07:24.290 align:middle line:84% of how that extraction of the adiabatic availability 00:07:24.290 --> 00:07:33.270 align:middle line:84% could be done if we have semi-permeable membranes. 00:07:33.270 --> 00:07:37.360 align:middle line:84% So here we picture, first of all, 00:07:37.360 --> 00:07:43.440 align:middle line:84% the idea is that when we have a gas confined 00:07:43.440 --> 00:07:47.550 align:middle line:84% in a volume by a partition and on the other side 00:07:47.550 --> 00:07:49.678 align:middle line:84% of the partition there is volume-- 00:07:49.678 --> 00:07:50.720 align:middle line:90% I mean, there is vacuum-- 00:07:50.720 --> 00:07:53.430 align:middle line:90% 00:07:53.430 --> 00:07:58.230 align:middle line:84% the pressure of the molecules here on the piston 00:07:58.230 --> 00:08:05.520 align:middle line:84% will tend to push it so as to increase the volume. 00:08:05.520 --> 00:08:10.920 align:middle line:84% And that will lift the weight if properly connected. 00:08:10.920 --> 00:08:15.240 align:middle line:90% Similarly, the black particles. 00:08:15.240 --> 00:08:19.830 align:middle line:84% Now the idea here is that in an ideal Gibbs-Dalton mixture 00:08:19.830 --> 00:08:26.400 align:middle line:84% behavior, the particles essentially, even in a mixture, 00:08:26.400 --> 00:08:28.390 align:middle line:90% ignore each other. 00:08:28.390 --> 00:08:31.680 align:middle line:84% They live in the entire volume even as 00:08:31.680 --> 00:08:35.230 align:middle line:90% if the others weren't there. 00:08:35.230 --> 00:08:40.120 align:middle line:84% So for the gray particles to expand, to here whether there 00:08:40.120 --> 00:08:46.870 align:middle line:84% is the black particles or not, isn't really a difference. 00:08:46.870 --> 00:08:53.620 align:middle line:84% So that is the idea that we can achieve 00:08:53.620 --> 00:08:58.030 align:middle line:84% with these semi-permeable membranes. 00:08:58.030 --> 00:09:02.290 align:middle line:84% This one allows the black particles to go through 00:09:02.290 --> 00:09:03.470 align:middle line:90% but not the gray. 00:09:03.470 --> 00:09:08.560 align:middle line:84% So the gray will push it to the right and pull up this weight. 00:09:08.560 --> 00:09:14.410 align:middle line:84% This other membrane is permeable to the gray particles but not 00:09:14.410 --> 00:09:17.050 align:middle line:84% the black so the black will push it leftward 00:09:17.050 --> 00:09:19.100 align:middle line:90% and pull up this other weight. 00:09:19.100 --> 00:09:21.750 align:middle line:90% 00:09:21.750 --> 00:09:22.250 align:middle line:90% OK. 00:09:22.250 --> 00:09:29.210 align:middle line:90% 00:09:29.210 --> 00:09:34.580 align:middle line:84% You asked, do we have membranes in reality? 00:09:34.580 --> 00:09:35.785 align:middle line:90% And yes, we do. 00:09:35.785 --> 00:09:38.840 align:middle line:90% 00:09:38.840 --> 00:09:44.390 align:middle line:90% We can manufacture them. 00:09:44.390 --> 00:09:47.780 align:middle line:90% We engineer materials. 00:09:47.780 --> 00:09:50.060 align:middle line:84% These are some YouTube videos that you 00:09:50.060 --> 00:09:52.550 align:middle line:90% may want to google and watch. 00:09:52.550 --> 00:09:54.830 align:middle line:90% Not doing it here. 00:09:54.830 --> 00:10:04.490 align:middle line:84% And so you can, essentially, have a wall of material 00:10:04.490 --> 00:10:05.550 align:middle line:90% with holes. 00:10:05.550 --> 00:10:09.000 align:middle line:84% And if the holes are the proper size, 00:10:09.000 --> 00:10:12.620 align:middle line:84% they can allow the passage of some particles 00:10:12.620 --> 00:10:17.210 align:middle line:84% and prevent the other particles to pass. 00:10:17.210 --> 00:10:22.580 align:middle line:84% So that is the way we achieve a semi-permeable membrane. 00:10:22.580 --> 00:10:26.240 align:middle line:84% In nature, also in biology, this is 00:10:26.240 --> 00:10:37.960 align:middle line:84% very important on the surface of cells where you may have a-- 00:10:37.960 --> 00:10:42.130 align:middle line:84% you may have active and passive transport. 00:10:42.130 --> 00:10:44.080 align:middle line:84% Active transport is a different story 00:10:44.080 --> 00:10:52.840 align:middle line:84% because it requires some energy for the particles to move. 00:10:52.840 --> 00:10:56.620 align:middle line:84% But also there is passive transport and the fact 00:10:56.620 --> 00:11:04.150 align:middle line:84% that membranes can be changed by the cell depending on its needs. 00:11:04.150 --> 00:11:08.500 align:middle line:84% So for example, if the cell needs to eat, so to speak, 00:11:08.500 --> 00:11:15.220 align:middle line:84% it could open little holes so that the, say, oxygen comes in. 00:11:15.220 --> 00:11:20.450 align:middle line:84% If the cell needs to discharge maybe CO2, 00:11:20.450 --> 00:11:24.280 align:middle line:84% then it needs to open a little bigger holes 00:11:24.280 --> 00:11:27.085 align:middle line:84% so that these molecules can go out. 00:11:27.085 --> 00:11:30.140 align:middle line:90% 00:11:30.140 --> 00:11:35.930 align:middle line:84% I'll have another reference about biological structures 00:11:35.930 --> 00:11:38.660 align:middle line:84% but I don't want to talk about them because I'm not 00:11:38.660 --> 00:11:42.505 align:middle line:90% at all expert on this. 00:11:42.505 --> 00:11:45.750 align:middle line:90% 00:11:45.750 --> 00:11:49.330 align:middle line:84% But so the world is full of membranes. 00:11:49.330 --> 00:11:51.875 align:middle line:84% And we will see more in a few moments. 00:11:51.875 --> 00:11:55.620 align:middle line:90% 00:11:55.620 --> 00:12:02.530 align:middle line:84% Now there is one historical topic that is maybe interesting, 00:12:02.530 --> 00:12:04.173 align:middle line:90% just fun to discuss. 00:12:04.173 --> 00:12:08.230 align:middle line:90% 00:12:08.230 --> 00:12:09.765 align:middle line:90% It's called the Gibbs paradox. 00:12:09.765 --> 00:12:13.360 align:middle line:90% 00:12:13.360 --> 00:12:16.240 align:middle line:84% And we have the tools to now understand what it means 00:12:16.240 --> 00:12:19.390 align:middle line:90% and essentially to resolve it. 00:12:19.390 --> 00:12:21.250 align:middle line:90% The idea is this. 00:12:21.250 --> 00:12:28.660 align:middle line:84% We just mentioned what happens when we have particles that mix. 00:12:28.660 --> 00:12:32.050 align:middle line:84% So here suppose we have just a binary mixture, only 00:12:32.050 --> 00:12:35.080 align:middle line:84% two types of particles, the gray and the black. 00:12:35.080 --> 00:12:40.900 align:middle line:84% And the spontaneous mixing occurs this way. 00:12:40.900 --> 00:12:43.180 align:middle line:84% And if you compute the entropy of mixing, 00:12:43.180 --> 00:12:48.220 align:middle line:84% this summation, when you have only two types of particles, 00:12:48.220 --> 00:12:51.640 align:middle line:90% becomes a sum of just two terms. 00:12:51.640 --> 00:12:54.430 align:middle line:84% And of course, the mole fractions 00:12:54.430 --> 00:12:59.470 align:middle line:84% when you have two particles are not independent 00:12:59.470 --> 00:13:02.790 align:middle line:90% because they must add up to 1. 00:13:02.790 --> 00:13:06.970 align:middle line:84% So the mole fraction of second type 00:13:06.970 --> 00:13:09.310 align:middle line:84% must be equal to 1 minus the mole fraction 00:13:09.310 --> 00:13:12.360 align:middle line:90% of the first type of particles. 00:13:12.360 --> 00:13:16.820 align:middle line:84% In the particular case in which you have 50% of 1 and 50% 00:13:16.820 --> 00:13:22.310 align:middle line:84% of the other, so the mole fractions are 50% or 1/2, 00:13:22.310 --> 00:13:30.910 align:middle line:84% this entropy of mixing becomes nR log of 2. 00:13:30.910 --> 00:13:34.170 align:middle line:90% 00:13:34.170 --> 00:13:41.640 align:middle line:84% Now if we consider, for example, a case in which I have one 00:13:41.640 --> 00:13:51.290 align:middle line:84% particle, if we express one particle in terms of moles, 00:13:51.290 --> 00:14:07.690 align:middle line:84% that's 1 over the Avogadro's number mole because in one mole 00:14:07.690 --> 00:14:15.940 align:middle line:84% you have these many particles, 6.0 times 10 to the 23rd. 00:14:15.940 --> 00:14:18.838 align:middle line:90% Right? 00:14:18.838 --> 00:14:29.500 align:middle line:84% So nR in that case is equal to R over the Avogadro number. 00:14:29.500 --> 00:14:35.460 align:middle line:84% And this is called the Boltzmann constant 00:14:35.460 --> 00:14:39.270 align:middle line:84% which is one of the physical constants. 00:14:39.270 --> 00:14:53.370 align:middle line:84% So if we suppose here we have just one particle on one side 00:14:53.370 --> 00:15:01.900 align:middle line:84% and one particle on the other side so the total n is 2, 00:15:01.900 --> 00:15:06.310 align:middle line:84% it's actually 2 divided by Avogadro's number. 00:15:06.310 --> 00:15:13.530 align:middle line:84% So the entropy of mixing is twice, 00:15:13.530 --> 00:15:17.040 align:middle line:90% the Boltzmann constant log of 2. 00:15:17.040 --> 00:15:24.430 align:middle line:90% 00:15:24.430 --> 00:15:33.490 align:middle line:90% So what is the Gibbs paradox? 00:15:33.490 --> 00:15:36.315 align:middle line:84% The Gibbs paradox runs as follows. 00:15:36.315 --> 00:15:39.400 align:middle line:90% 00:15:39.400 --> 00:15:48.880 align:middle line:84% There is a saying that I know it in Latin. 00:15:48.880 --> 00:15:50.305 align:middle line:90% Natura non facit saltum. 00:15:50.305 --> 00:15:58.320 align:middle line:90% 00:15:58.320 --> 00:16:01.830 align:middle line:84% It means nature doesn't make jumps. 00:16:01.830 --> 00:16:08.130 align:middle line:84% And this I think it goes back to Aristotle or maybe Leibniz. 00:16:08.130 --> 00:16:15.305 align:middle line:84% I mean every philosopher has touched on this idea. 00:16:15.305 --> 00:16:19.110 align:middle line:90% 00:16:19.110 --> 00:16:21.550 align:middle line:90% So what Gibbs-- 00:16:21.550 --> 00:16:23.500 align:middle line:84% I think the Gibbs, they call it-- 00:16:23.500 --> 00:16:24.460 align:middle line:90% I really don't know. 00:16:24.460 --> 00:16:27.450 align:middle line:84% Maybe we should look it up about the history of these things 00:16:27.450 --> 00:16:31.720 align:middle line:84% is interesting as a chapter in itself. 00:16:31.720 --> 00:16:33.910 align:middle line:84% So I'm not an expert in that either. 00:16:33.910 --> 00:16:38.290 align:middle line:84% But suppose that it was Gibbs who realized that, all right, 00:16:38.290 --> 00:16:48.150 align:middle line:84% suppose I'm coloring these particles with a dye. 00:16:48.150 --> 00:16:59.070 align:middle line:84% And so suppose the amount of distance in color 00:16:59.070 --> 00:17:01.980 align:middle line:84% is measured by this parameter lambda. 00:17:01.980 --> 00:17:06.089 align:middle line:84% So I say 0 means the particles are the same color. 00:17:06.089 --> 00:17:10.770 align:middle line:84% Lambda greater than 0 means that the particles 00:17:10.770 --> 00:17:11.749 align:middle line:90% are different color. 00:17:11.749 --> 00:17:14.569 align:middle line:90% 00:17:14.569 --> 00:17:15.349 align:middle line:90% All right. 00:17:15.349 --> 00:17:17.920 align:middle line:84% So regardless of the color, here we 00:17:17.920 --> 00:17:22.750 align:middle line:84% see that the result, this result, the entropy of mixing, 00:17:22.750 --> 00:17:26.115 align:middle line:84% is the same regardless of how different the particles are. 00:17:26.115 --> 00:17:28.880 align:middle line:90% 00:17:28.880 --> 00:17:32.320 align:middle line:84% There is no detail about what the particles or the mass is. 00:17:32.320 --> 00:17:36.020 align:middle line:90% It's just how many there are. 00:17:36.020 --> 00:17:39.770 align:middle line:84% So the entropy of mixing is always 00:17:39.770 --> 00:17:51.380 align:middle line:84% given by this amount, nR log of 2, except-- 00:17:51.380 --> 00:17:54.660 align:middle line:84% and so you would say, well, as lambda goes to 0, 00:17:54.660 --> 00:18:00.610 align:middle line:84% I would expect that to also go continuously. 00:18:00.610 --> 00:18:06.400 align:middle line:84% Instead, as you go to 0, that jumps to 0 00:18:06.400 --> 00:18:13.820 align:middle line:84% because, if we think in terms of the simple-system model, 00:18:13.820 --> 00:18:20.920 align:middle line:84% you remember that when we have a pure substance, 00:18:20.920 --> 00:18:23.500 align:middle line:84% putting a partition or removing it 00:18:23.500 --> 00:18:25.270 align:middle line:90% doesn't change the properties. 00:18:25.270 --> 00:18:31.060 align:middle line:84% So that's one quick way, since we have seen a lot before, 00:18:31.060 --> 00:18:38.320 align:middle line:84% to show that if the particles are identical, 00:18:38.320 --> 00:18:41.400 align:middle line:84% removing the partition will not change any of the properties, 00:18:41.400 --> 00:18:42.150 align:middle line:90% including entropy. 00:18:42.150 --> 00:18:46.170 align:middle line:90% 00:18:46.170 --> 00:18:49.950 align:middle line:84% So there is no entropy of mixing in this case. 00:18:49.950 --> 00:18:55.950 align:middle line:84% If you mix the same substance, there is no entropy of mixing. 00:18:55.950 --> 00:18:57.610 align:middle line:84% As soon as they become different, 00:18:57.610 --> 00:19:01.500 align:middle line:84% there is entropy of mixing and it is discontinuous. 00:19:01.500 --> 00:19:06.275 align:middle line:84% So that's why the philosophers didn't like this idea. 00:19:06.275 --> 00:19:12.510 align:middle line:90% 00:19:12.510 --> 00:19:17.790 align:middle line:84% But in fact, it all has to do with the fact 00:19:17.790 --> 00:19:22.880 align:middle line:84% that we are doing modeling of reality. 00:19:22.880 --> 00:19:32.330 align:middle line:84% So Nature is the way we model it. 00:19:32.330 --> 00:19:37.850 align:middle line:84% So if we have the possibility to distinguish the particles 00:19:37.850 --> 00:19:46.130 align:middle line:84% and therefore they are different for us, then there is mixing. 00:19:46.130 --> 00:19:48.510 align:middle line:84% And if there is entropy of mixing, 00:19:48.510 --> 00:19:53.990 align:middle line:84% there is also the possibility to do work. 00:19:53.990 --> 00:19:59.630 align:middle line:84% As we have seen, we can do this work, this mixing, 00:19:59.630 --> 00:20:02.810 align:middle line:90% properly and extract some work. 00:20:02.810 --> 00:20:08.510 align:middle line:84% But if we don't have the possibility 00:20:08.510 --> 00:20:14.600 align:middle line:84% to recognize, to identify, to distinguish the particles, 00:20:14.600 --> 00:20:17.510 align:middle line:90% that work is impossible for us. 00:20:17.510 --> 00:20:22.370 align:middle line:84% So it has to do with the way we model it or, if you really want 00:20:22.370 --> 00:20:31.200 align:middle line:84% to go Nature, if the particles are truly indistinguishable 00:20:31.200 --> 00:20:33.870 align:middle line:84% in the sense of quantum mechanics, 00:20:33.870 --> 00:20:37.360 align:middle line:84% distinguishable/ indistinguishable, fermion, 00:20:37.360 --> 00:20:40.860 align:middle line:84% bosons, and that kind of discussion, 00:20:40.860 --> 00:20:45.820 align:middle line:84% then there is no way you can distinguish. 00:20:45.820 --> 00:20:51.330 align:middle line:84% There is no way you can extract any work. 00:20:51.330 --> 00:20:56.610 align:middle line:84% If the particles are distinguishable, 00:20:56.610 --> 00:21:00.250 align:middle line:84% then it all depends on your model, 00:21:00.250 --> 00:21:05.910 align:middle line:84% namely on your kind of tool that you 00:21:05.910 --> 00:21:07.265 align:middle line:90% are considering in your model. 00:21:07.265 --> 00:21:13.980 align:middle line:90% 00:21:13.980 --> 00:21:19.730 align:middle line:84% So that's essentially the resolution of the paradox. 00:21:19.730 --> 00:21:21.780 align:middle line:90% There's no paradox. 00:21:21.780 --> 00:21:24.930 align:middle line:90% It's true like this. 00:21:24.930 --> 00:21:29.280 align:middle line:84% And the discontinuity in going from here to there 00:21:29.280 --> 00:21:34.830 align:middle line:84% is essentially embedded in our modeling 00:21:34.830 --> 00:21:38.460 align:middle line:84% or, from the practical point of view, our ability 00:21:38.460 --> 00:21:42.330 align:middle line:84% to control the differences in particles 00:21:42.330 --> 00:21:44.370 align:middle line:84% because if the particles are different 00:21:44.370 --> 00:21:51.510 align:middle line:84% but we have no ways to control their differences 00:21:51.510 --> 00:21:59.610 align:middle line:84% and therefore build semi-permeable membranes that 00:21:59.610 --> 00:22:07.130 align:middle line:84% can select, then obtaining that work is out of our reach. 00:22:07.130 --> 00:22:10.590 align:middle line:90% 00:22:10.590 --> 00:22:12.450 align:middle line:90% OK. 00:22:12.450 --> 00:22:14.375 align:middle line:90% So that's the Gibbs paradox. 00:22:14.375 --> 00:22:19.460 align:middle line:90% 00:22:19.460 --> 00:22:26.390 align:middle line:84% And since we are at it, I want to take a little detour today 00:22:26.390 --> 00:22:35.540 align:middle line:84% also on some other topics that are important in statistics, 00:22:35.540 --> 00:22:40.360 align:middle line:84% in information theory, and also in quantum theory. 00:22:40.360 --> 00:22:43.600 align:middle line:90% 00:22:43.600 --> 00:22:48.800 align:middle line:84% So again, look at this box with particles, different particles. 00:22:48.800 --> 00:22:51.980 align:middle line:90% 00:22:51.980 --> 00:22:55.640 align:middle line:84% And suppose you make an experiment 00:22:55.640 --> 00:23:02.550 align:middle line:84% by having a little window someplace like a control volume 00:23:02.550 --> 00:23:09.180 align:middle line:84% somewhere in that container where you can look. 00:23:09.180 --> 00:23:13.200 align:middle line:84% If there is a particle in there, there can be only one at a time. 00:23:13.200 --> 00:23:16.460 align:middle line:90% It's a small control volume. 00:23:16.460 --> 00:23:24.320 align:middle line:84% You write it down in your experimental notebook and say, 00:23:24.320 --> 00:23:28.460 align:middle line:84% oh yeah, I saw a molecule of nitrogen, 00:23:28.460 --> 00:23:35.180 align:middle line:84% or I saw a molecule of oxygen, or I saw a molecule of argon, 00:23:35.180 --> 00:23:38.900 align:middle line:90% or I saw a molecule of CO2. 00:23:38.900 --> 00:23:46.040 align:middle line:90% If that's air, this is the idea. 00:23:46.040 --> 00:23:56.200 align:middle line:84% And if you are in a stable equilibrium state and therefore 00:23:56.200 --> 00:24:02.830 align:middle line:84% the probabilities that you observe one of these molecules 00:24:02.830 --> 00:24:09.040 align:middle line:84% is proportional to the fraction of particles 00:24:09.040 --> 00:24:12.130 align:middle line:84% overall so it's the mole fraction. 00:24:12.130 --> 00:24:15.200 align:middle line:84% So the mole fraction, which by the way 00:24:15.200 --> 00:24:16.840 align:middle line:90% is a number like probability. 00:24:16.840 --> 00:24:21.610 align:middle line:84% It's a number between 0 and 1 and it adds up to 1. 00:24:21.610 --> 00:24:24.700 align:middle line:84% So the mole fraction is the probability 00:24:24.700 --> 00:24:27.450 align:middle line:90% to get a particle of that type. 00:24:27.450 --> 00:24:35.930 align:middle line:90% 00:24:35.930 --> 00:24:43.260 align:middle line:90% So there is a measure-- 00:24:43.260 --> 00:24:50.330 align:middle line:84% if you are facing this situation with probabilities, 00:24:50.330 --> 00:24:54.920 align:middle line:84% you are in an uncertainty situation. 00:24:54.920 --> 00:24:59.540 align:middle line:84% You are uncertain about what will 00:24:59.540 --> 00:25:03.350 align:middle line:84% be the outcome of the next observation. 00:25:03.350 --> 00:25:10.010 align:middle line:84% You cannot predict the next time that's what I will get. 00:25:10.010 --> 00:25:14.330 align:middle line:84% But you can predict, because the mole fractions, 00:25:14.330 --> 00:25:18.740 align:middle line:84% with what frequency you will get these results upon repeated 00:25:18.740 --> 00:25:19.740 align:middle line:90% measurement. 00:25:19.740 --> 00:25:23.680 align:middle line:90% 00:25:23.680 --> 00:25:27.310 align:middle line:84% But when you are facing the next observation 00:25:27.310 --> 00:25:30.820 align:middle line:84% and you have a big question mark in your mind 00:25:30.820 --> 00:25:33.640 align:middle line:84% and ask, all right, what will be the next outcome, 00:25:33.640 --> 00:25:36.190 align:middle line:84% you are in a state of uncertainty. 00:25:36.190 --> 00:25:42.850 align:middle line:84% And the uncertainty can be measured 00:25:42.850 --> 00:25:49.240 align:middle line:84% by this measure of information, which is called the Shannon 00:25:49.240 --> 00:25:50.950 align:middle line:90% information entropy. 00:25:50.950 --> 00:25:58.690 align:middle line:84% Shannon, 1948, said, well, if I look at a situation described 00:25:58.690 --> 00:26:00.910 align:middle line:84% by probabilities like these events, 00:26:00.910 --> 00:26:11.800 align:middle line:84% and I want to find the most logical mathematical expression 00:26:11.800 --> 00:26:13.780 align:middle line:84% or function of these probabilities 00:26:13.780 --> 00:26:18.006 align:middle line:84% that expresses my state of uncertainty. 00:26:18.006 --> 00:26:20.860 align:middle line:90% 00:26:20.860 --> 00:26:24.380 align:middle line:84% He set up some postulates, some desiderata 00:26:24.380 --> 00:26:31.460 align:middle line:84% about this measurement function of probabilities. 00:26:31.460 --> 00:26:36.320 align:middle line:84% And the only function that satisfies those criteria 00:26:36.320 --> 00:26:37.150 align:middle line:90% is this one. 00:26:37.150 --> 00:26:41.890 align:middle line:90% 00:26:41.890 --> 00:26:45.700 align:middle line:90% For example, see if you are-- 00:26:45.700 --> 00:26:49.190 align:middle line:84% that is equal to 0 if you have certainty. 00:26:49.190 --> 00:26:53.260 align:middle line:84% So if the probabilities are all zeros 00:26:53.260 --> 00:26:58.870 align:middle line:84% except one event that is equal to 1, so you have certainty. 00:26:58.870 --> 00:27:02.090 align:middle line:84% So the particles are only one type. 00:27:02.090 --> 00:27:04.610 align:middle line:90% The mole fraction is 1. 00:27:04.610 --> 00:27:05.110 align:middle line:90% OK. 00:27:05.110 --> 00:27:07.840 align:middle line:90% 00:27:07.840 --> 00:27:13.240 align:middle line:90% Log of 1 is 0. 00:27:13.240 --> 00:27:19.670 align:middle line:84% And x log x for those that are 0 is again 0 00:27:19.670 --> 00:27:28.360 align:middle line:84% because you remember that these functions p log p look this way. 00:27:28.360 --> 00:27:38.260 align:middle line:84% Between 0 and 1 is the function x log x. 00:27:38.260 --> 00:27:40.450 align:middle line:90% Right? 00:27:40.450 --> 00:27:43.840 align:middle line:84% So in that case of complete certainty, 00:27:43.840 --> 00:27:51.020 align:middle line:84% the information entropy is 0, or on the opposite side, 00:27:51.020 --> 00:27:56.450 align:middle line:84% if all particles have equal probability 00:27:56.450 --> 00:28:05.120 align:middle line:84% so all the mole fractions are equal, and in that case, 00:28:05.120 --> 00:28:16.640 align:middle line:84% the entropy is equal to log of n of the number of-- 00:28:16.640 --> 00:28:18.990 align:middle line:90% log of r in this case. 00:28:18.990 --> 00:28:23.820 align:middle line:84% It's the number of different particles. 00:28:23.820 --> 00:28:24.880 align:middle line:90% You can work it out. 00:28:24.880 --> 00:28:27.120 align:middle line:84% You substitute each probability is 00:28:27.120 --> 00:28:36.660 align:middle line:84% 1 over r and that this summation gives log of r. 00:28:36.660 --> 00:28:41.490 align:middle line:84% That's the maximum, the most uncertain situation 00:28:41.490 --> 00:28:43.770 align:middle line:90% when they are all alike. 00:28:43.770 --> 00:28:49.860 align:middle line:84% If you have instead a situation like for the molecules of air, 00:28:49.860 --> 00:28:58.230 align:middle line:84% it's not even and therefore some are more frequent than others 00:28:58.230 --> 00:29:06.083 align:middle line:84% so you could guess some better than if everything was just 50%. 00:29:06.083 --> 00:29:09.200 align:middle line:90% 00:29:09.200 --> 00:29:12.350 align:middle line:90% You can also view-- 00:29:12.350 --> 00:29:15.890 align:middle line:84% this is interesting also for us for the next step 00:29:15.890 --> 00:29:21.620 align:middle line:84% in the next definition of ideal solution. 00:29:21.620 --> 00:29:27.470 align:middle line:84% You can also view this as a weighted sum 00:29:27.470 --> 00:29:30.880 align:middle line:90% weighted on the mole fractions. 00:29:30.880 --> 00:29:36.140 align:middle line:84% So I can view this as a summation. 00:29:36.140 --> 00:29:43.360 align:middle line:84% The mole fraction times minus log of y. 00:29:43.360 --> 00:29:46.500 align:middle line:90% 00:29:46.500 --> 00:29:53.130 align:middle line:84% Now this term in parentheses, we could call it the surprise. 00:29:53.130 --> 00:29:56.000 align:middle line:90% The surprise. 00:29:56.000 --> 00:29:57.460 align:middle line:90% The surprise is a measure. 00:29:57.460 --> 00:30:01.700 align:middle line:90% 00:30:01.700 --> 00:30:08.720 align:middle line:84% It's high for the least probable ones. 00:30:08.720 --> 00:30:11.570 align:middle line:90% So if I get a-- 00:30:11.570 --> 00:30:13.040 align:middle line:90% say that if I get-- 00:30:13.040 --> 00:30:16.670 align:middle line:84% if I see a molecule of CO2 there and I 00:30:16.670 --> 00:30:20.730 align:middle line:84% know that there are only 500 PPMs, parts per million, 00:30:20.730 --> 00:30:22.910 align:middle line:84% so there is very low probability, 00:30:22.910 --> 00:30:26.810 align:middle line:84% so if I see that one, I'm more surprised 00:30:26.810 --> 00:30:32.210 align:middle line:84% than if I see a nitrogen or an oxygen. 00:30:32.210 --> 00:30:35.360 align:middle line:84% Argon also is a bit of a surprise 00:30:35.360 --> 00:30:38.420 align:middle line:90% because argon is less than 1%. 00:30:38.420 --> 00:30:47.120 align:middle line:84% So I need to do 100 trials to have, on average, one argon. 00:30:47.120 --> 00:30:57.680 align:middle line:84% And here I need 2,000 trials to get one CO2. 00:30:57.680 --> 00:30:59.440 align:middle line:90% So that's the surprise. 00:30:59.440 --> 00:31:04.140 align:middle line:90% 00:31:04.140 --> 00:31:07.200 align:middle line:84% Now, in information theory, there 00:31:07.200 --> 00:31:12.690 align:middle line:84% is also this problem that is also relevant for measurements. 00:31:12.690 --> 00:31:22.110 align:middle line:84% For example, suppose that your device finds the particle 00:31:22.110 --> 00:31:26.760 align:middle line:84% and characterizes the particle by the mass. 00:31:26.760 --> 00:31:35.520 align:middle line:84% So when the particle is in your window, your device, maybe 00:31:35.520 --> 00:31:38.850 align:middle line:84% a spectrometer or whatever, it tells you, OK, it 00:31:38.850 --> 00:31:40.970 align:middle line:90% was a particle with this mass. 00:31:40.970 --> 00:31:45.880 align:middle line:90% 00:31:45.880 --> 00:31:51.790 align:middle line:84% So in the end, when you do a measurement, like a repeated 00:31:51.790 --> 00:31:54.560 align:middle line:84% measurement procedure, in the end, 00:31:54.560 --> 00:32:04.210 align:middle line:84% you will find the average or the mean mass that shows up. 00:32:04.210 --> 00:32:06.970 align:middle line:84% And suppose that that's all that you 00:32:06.970 --> 00:32:10.810 align:middle line:90% can measure of your substance. 00:32:10.810 --> 00:32:14.310 align:middle line:84% So all you can measure is the average value of the mass. 00:32:14.310 --> 00:32:19.970 align:middle line:90% 00:32:19.970 --> 00:32:23.600 align:middle line:84% If that's the only information I have about my system, 00:32:23.600 --> 00:32:28.250 align:middle line:84% could I make an educated guess of what the probability 00:32:28.250 --> 00:32:31.150 align:middle line:84% distribution, so therefore what the mole fractions are? 00:32:31.150 --> 00:32:34.410 align:middle line:90% 00:32:34.410 --> 00:32:40.290 align:middle line:84% And here comes information theory. 00:32:40.290 --> 00:32:45.360 align:middle line:84% And again, based on Shannon information entropy, 00:32:45.360 --> 00:32:52.570 align:middle line:84% you say, all right, in order to make the educated guess, 00:32:52.570 --> 00:32:58.000 align:middle line:84% I should avoid any bias that is not 00:32:58.000 --> 00:33:02.230 align:middle line:84% contained in the information I have about the system. 00:33:02.230 --> 00:33:07.550 align:middle line:84% So I should use the information I have, the measured values, 00:33:07.550 --> 00:33:11.110 align:middle line:90% but not extra stuff. 00:33:11.110 --> 00:33:16.420 align:middle line:84% So I want to make the least biased guess. 00:33:16.420 --> 00:33:19.810 align:middle line:84% And the least biased guess is obtained 00:33:19.810 --> 00:33:22.780 align:middle line:84% by saying, all right, what is the probability 00:33:22.780 --> 00:33:28.540 align:middle line:84% distribution that maximizes the Shannon information? 00:33:28.540 --> 00:33:31.270 align:middle line:84% In other words, it leaves me with the highest 00:33:31.270 --> 00:33:35.140 align:middle line:84% possible uncertainty compatible with that mean value 00:33:35.140 --> 00:33:39.010 align:middle line:90% that I have observed. 00:33:39.010 --> 00:33:45.890 align:middle line:84% In this way, I have not built in the probability distribution, 00:33:45.890 --> 00:33:47.236 align:middle line:90% any extra information. 00:33:47.236 --> 00:33:51.890 align:middle line:90% 00:33:51.890 --> 00:33:54.980 align:middle line:84% And so, mathematically, this means 00:33:54.980 --> 00:33:59.300 align:middle line:84% you have to maximize the Shannon entropy 00:33:59.300 --> 00:34:05.120 align:middle line:84% subject to the constraint that your mean value be 00:34:05.120 --> 00:34:07.700 align:middle line:90% the known or measured value. 00:34:07.700 --> 00:34:11.389 align:middle line:84% Of course, you also have to put the auxiliary constraint 00:34:11.389 --> 00:34:16.699 align:middle line:84% that probabilities must add up to 1. 00:34:16.699 --> 00:34:21.139 align:middle line:84% And if you solve this problem with the method 00:34:21.139 --> 00:34:24.080 align:middle line:90% of Lagrange multipliers-- 00:34:24.080 --> 00:34:27.650 align:middle line:84% I don't want to just go over that today, but maybe later 00:34:27.650 --> 00:34:34.429 align:middle line:90% on, since we will review it. 00:34:34.429 --> 00:34:39.230 align:middle line:84% In any case, I presume that you have seen it somewhere else. 00:34:39.230 --> 00:34:44.520 align:middle line:84% So the method simply says, all right, 00:34:44.520 --> 00:34:46.860 align:middle line:84% if you have a constraint maximization, 00:34:46.860 --> 00:34:48.980 align:middle line:84% you must assign to each of the constraints 00:34:48.980 --> 00:34:56.456 align:middle line:84% a Lagrange multiplier, an undetermined multiplier, 00:34:56.456 --> 00:35:03.100 align:middle line:84% and then do an unconstrained maximization of the optimizing 00:35:03.100 --> 00:35:11.350 align:middle line:84% function minus the constraints multiplied by these multipliers. 00:35:11.350 --> 00:35:14.080 align:middle line:84% And then once you obtain the solution 00:35:14.080 --> 00:35:17.180 align:middle line:84% of the unconstrained maximum in terms of these multipliers, 00:35:17.180 --> 00:35:19.540 align:middle line:84% you substitute back into the constraints 00:35:19.540 --> 00:35:23.160 align:middle line:84% and get the values for the multipliers. 00:35:23.160 --> 00:35:23.660 align:middle line:90% All right. 00:35:23.660 --> 00:35:28.630 align:middle line:84% That's a quick so I don't expect you to have learned it 00:35:28.630 --> 00:35:29.600 align:middle line:90% from these words. 00:35:29.600 --> 00:35:31.940 align:middle line:84% But in any case, if you do the job, 00:35:31.940 --> 00:35:35.470 align:middle line:84% here's the probability distribution that you get. 00:35:35.470 --> 00:35:41.680 align:middle line:84% And if you substitute this probability back 00:35:41.680 --> 00:35:48.700 align:middle line:84% into the constraint, this one that gives you the average M, 00:35:48.700 --> 00:35:55.450 align:middle line:84% it gives you a relation between the lambda 00:35:55.450 --> 00:35:59.110 align:middle line:84% M, which is the Lagrange multiplier, and the mean value. 00:35:59.110 --> 00:36:02.410 align:middle line:84% So you get that the Lagrange multiplier depends 00:36:02.410 --> 00:36:06.160 align:middle line:90% on that mean value of the mass. 00:36:06.160 --> 00:36:13.000 align:middle line:84% So this is the best or at least the least biased assignment 00:36:13.000 --> 00:36:18.730 align:middle line:84% of mole fractions that you can do upon a measurement of only 00:36:18.730 --> 00:36:19.410 align:middle line:90% the mass. 00:36:19.410 --> 00:36:25.150 align:middle line:90% 00:36:25.150 --> 00:36:32.320 align:middle line:84% We will see in a while that this probability distribution 00:36:32.320 --> 00:36:39.670 align:middle line:84% is very similar to what you have for the description 00:36:39.670 --> 00:36:43.570 align:middle line:84% of stable equilibrium states within statistical mechanics 00:36:43.570 --> 00:36:46.720 align:middle line:90% and quantum theory. 00:36:46.720 --> 00:36:48.680 align:middle line:84% It's called the canonical distribution 00:36:48.680 --> 00:36:52.420 align:middle line:84% if instead of mass you put energy. 00:36:52.420 --> 00:37:01.750 align:middle line:84% And that allows or allowed researchers like Jaynes-- 00:37:01.750 --> 00:37:10.750 align:middle line:84% it's a big name who made these very neat connections 00:37:10.750 --> 00:37:13.640 align:middle line:84% between information theory and statistical mechanics. 00:37:13.640 --> 00:37:17.730 align:middle line:90% 00:37:17.730 --> 00:37:25.500 align:middle line:84% Or also, if you still go to the library here at MIT, 00:37:25.500 --> 00:37:32.990 align:middle line:84% there is a wonderful set of video lectures by Professor 00:37:32.990 --> 00:37:42.110 align:middle line:84% Myron Tribus back in, I think, the '60s or the '70s where he 00:37:42.110 --> 00:37:47.493 align:middle line:84% does the information theory point of view of thermodynamics. 00:37:47.493 --> 00:37:49.160 align:middle line:84% So he develops the entire thermodynamics 00:37:49.160 --> 00:37:55.410 align:middle line:84% just by building up on information theory concepts. 00:37:55.410 --> 00:38:01.720 align:middle line:84% And indeed there is a ground for doing it 00:38:01.720 --> 00:38:06.450 align:middle line:84% because the molecular view that we are discussing here 00:38:06.450 --> 00:38:09.750 align:middle line:84% and the tools developed by Shannon 00:38:09.750 --> 00:38:12.160 align:middle line:90% do provide that possibility. 00:38:12.160 --> 00:38:17.480 align:middle line:90% 00:38:17.480 --> 00:38:23.830 align:middle line:84% So I said that I wanted to make a little digression today. 00:38:23.830 --> 00:38:29.630 align:middle line:84% So here is a digression in the digression. 00:38:29.630 --> 00:38:39.740 align:middle line:84% And it has to do with the treatment 00:38:39.740 --> 00:38:43.555 align:middle line:90% of a particle in a box. 00:38:43.555 --> 00:38:50.890 align:middle line:90% 00:38:50.890 --> 00:38:51.470 align:middle line:90% OK. 00:38:51.470 --> 00:38:59.680 align:middle line:84% So we take a box of sides l1, l2, and l3 00:38:59.680 --> 00:39:01.390 align:middle line:90% in the three directions. 00:39:01.390 --> 00:39:06.010 align:middle line:84% And we consider only one particle inside the box. 00:39:06.010 --> 00:39:18.490 align:middle line:84% That's the idea of this little survey with some equations 00:39:18.490 --> 00:39:26.770 align:middle line:84% that are borrowed from quantum mechanics, which 00:39:26.770 --> 00:39:28.820 align:middle line:90% you may be familiar or not. 00:39:28.820 --> 00:39:33.520 align:middle line:84% But even if you are not, it's OK. 00:39:33.520 --> 00:39:38.290 align:middle line:84% This is just a quick view that tells you how 00:39:38.290 --> 00:39:40.620 align:middle line:90% all these things are related. 00:39:40.620 --> 00:39:44.680 align:middle line:90% So it might be interesting. 00:39:44.680 --> 00:39:53.590 align:middle line:84% I will not ask for these in the oral exam, of course. 00:39:53.590 --> 00:39:56.710 align:middle line:84% And actually, one day I will let you 00:39:56.710 --> 00:40:04.720 align:middle line:84% know which of the many things that we cover, the slides that 00:40:04.720 --> 00:40:10.420 align:middle line:84% are going to be part of the exam so that you 00:40:10.420 --> 00:40:17.350 align:middle line:84% can focus because otherwise it's too much, I perfectly realize. 00:40:17.350 --> 00:40:18.110 align:middle line:90% All right. 00:40:18.110 --> 00:40:27.280 align:middle line:84% So the idea in quantum mechanics is that in order to describe 00:40:27.280 --> 00:40:32.050 align:middle line:84% a molecule like that, or even simpler, 00:40:32.050 --> 00:40:35.920 align:middle line:84% if you have a molecule on a one-dimensional box that we 00:40:35.920 --> 00:40:37.160 align:middle line:90% already mentioned-- 00:40:37.160 --> 00:40:41.980 align:middle line:90% 00:40:41.980 --> 00:40:46.000 align:middle line:84% so suppose we have just one-dimensional box. 00:40:46.000 --> 00:40:49.930 align:middle line:84% In classical mechanics, the particle is here 00:40:49.930 --> 00:40:52.850 align:middle line:90% and it goes back and forth. 00:40:52.850 --> 00:40:58.300 align:middle line:84% It's perfectly situated somewhere 00:40:58.300 --> 00:40:59.890 align:middle line:90% at every instant of time. 00:40:59.890 --> 00:41:04.010 align:middle line:84% In quantum mechanics, the story is different. 00:41:04.010 --> 00:41:06.700 align:middle line:84% Particle cannot be-- is not here-- 00:41:06.700 --> 00:41:09.310 align:middle line:90% 00:41:09.310 --> 00:41:13.100 align:middle line:84% viewing the particle as localized is wrong. 00:41:13.100 --> 00:41:18.000 align:middle line:84% It gives you some results that are wrong. 00:41:18.000 --> 00:41:22.680 align:middle line:84% So the idea of quantum mechanics is that you start from-- 00:41:22.680 --> 00:41:35.520 align:middle line:84% you have to find a probability distribution which 00:41:35.520 --> 00:41:39.670 align:middle line:84% can be obtained by a function, let's say phi. 00:41:39.670 --> 00:41:43.640 align:middle line:90% 00:41:43.640 --> 00:41:52.880 align:middle line:84% And this function is the solution 00:41:52.880 --> 00:41:55.120 align:middle line:90% of an eigenvalue problem. 00:41:55.120 --> 00:41:59.630 align:middle line:90% 00:41:59.630 --> 00:42:04.280 align:middle line:84% And for example, first of all, this function 00:42:04.280 --> 00:42:07.970 align:middle line:90% can be also written as-- 00:42:07.970 --> 00:42:09.650 align:middle line:90% it factors. 00:42:09.650 --> 00:42:17.600 align:middle line:84% So it's a product of a function of x and a function of time. 00:42:17.600 --> 00:42:27.770 align:middle line:84% And the function of time, if I remember, it's minus h bar times 00:42:27.770 --> 00:42:33.926 align:middle line:84% the energy times time over something. 00:42:33.926 --> 00:42:39.930 align:middle line:90% 00:42:39.930 --> 00:42:44.490 align:middle line:90% It will come up somewhere-- 00:42:44.490 --> 00:42:45.650 align:middle line:90% over m probably. 00:42:45.650 --> 00:42:51.300 align:middle line:90% 00:42:51.300 --> 00:42:52.620 align:middle line:90% Not sure. 00:42:52.620 --> 00:42:57.630 align:middle line:84% So the idea is that this function here, 00:42:57.630 --> 00:43:02.040 align:middle line:84% the one that depends on x, is the solution 00:43:02.040 --> 00:43:07.230 align:middle line:90% of an eigenvalue problem. 00:43:07.230 --> 00:43:13.200 align:middle line:90% And these functions are-- 00:43:13.200 --> 00:43:21.610 align:middle line:84% you can plot them as follows, as we know. 00:43:21.610 --> 00:43:25.270 align:middle line:90% 00:43:25.270 --> 00:43:29.080 align:middle line:84% Actually, I think we have the expression that's 00:43:29.080 --> 00:43:40.520 align:middle line:84% something like square root of 2 over l where l is the side-- 00:43:40.520 --> 00:43:44.110 align:middle line:90% so this is from 0 to l-- 00:43:44.110 --> 00:43:55.953 align:middle line:84% times sine of some j, some maybe x. 00:43:55.953 --> 00:44:00.340 align:middle line:90% 00:44:00.340 --> 00:44:02.920 align:middle line:84% And there are some other constants here 00:44:02.920 --> 00:44:09.310 align:middle line:90% that now slips my memory. 00:44:09.310 --> 00:44:23.170 align:middle line:84% It's maybe h over 2m or something like that or maybe 00:44:23.170 --> 00:44:25.780 align:middle line:90% also an l. 00:44:25.780 --> 00:44:29.250 align:middle line:90% There is also an l here. 00:44:29.250 --> 00:44:38.940 align:middle line:84% So the idea is that this j is the integer numbers. 00:44:38.940 --> 00:44:42.750 align:middle line:84% So there are many possible contributions. 00:44:42.750 --> 00:44:48.840 align:middle line:84% Each one of these for each j corresponds to a solution 00:44:48.840 --> 00:44:55.590 align:middle line:84% and to that solution corresponds also an energy level. 00:44:55.590 --> 00:45:00.270 align:middle line:84% So for example, the lowest energy level corresponding to j 00:45:00.270 --> 00:45:04.325 align:middle line:84% is equal to 1 is a function like this. 00:45:04.325 --> 00:45:07.320 align:middle line:90% 00:45:07.320 --> 00:45:09.450 align:middle line:84% Then you may have the second energy 00:45:09.450 --> 00:45:22.225 align:middle line:84% level in which the function is something like this. 00:45:22.225 --> 00:45:26.640 align:middle line:90% 00:45:26.640 --> 00:45:44.360 align:middle line:84% And then j equals to 3 and so on. 00:45:44.360 --> 00:45:50.630 align:middle line:84% Corresponding to each of these j's, there are energies, 00:45:50.630 --> 00:45:54.660 align:middle line:84% or actually on the slide, I'm calling them epsilon j. 00:45:54.660 --> 00:45:57.790 align:middle line:90% 00:45:57.790 --> 00:46:07.760 align:middle line:84% And these energies are h squared over 2m multiplied by j squared 00:46:07.760 --> 00:46:09.010 align:middle line:90% over l squared. 00:46:09.010 --> 00:46:13.020 align:middle line:90% 00:46:13.020 --> 00:46:21.260 align:middle line:84% Maybe there's a pi here because it needs to go to 0 at 0 00:46:21.260 --> 00:46:22.430 align:middle line:90% and at l. 00:46:22.430 --> 00:46:25.820 align:middle line:90% So OK. 00:46:25.820 --> 00:46:34.040 align:middle line:84% So that means that if this epsilon 1 is the lowest energy 00:46:34.040 --> 00:46:41.555 align:middle line:90% level, epsilon 2 is equal to-- 00:46:41.555 --> 00:46:44.330 align:middle line:90% 00:46:44.330 --> 00:46:47.300 align:middle line:84% that corresponds to j equals 2 so it equals-- 00:46:47.300 --> 00:46:49.600 align:middle line:90% j squared is 4 times epsilon 1. 00:46:49.600 --> 00:46:52.540 align:middle line:90% 00:46:52.540 --> 00:46:58.480 align:middle line:84% And epsilon 3 is 3 to the square so it's 00:46:58.480 --> 00:47:03.350 align:middle line:90% 9 times epsilon 1 and so on. 00:47:03.350 --> 00:47:09.215 align:middle line:90% 00:47:09.215 --> 00:47:15.270 align:middle line:84% So already this gives you the energy levels. 00:47:15.270 --> 00:47:20.320 align:middle line:84% Now in an experiment that you do, for example, 00:47:20.320 --> 00:47:26.550 align:middle line:84% with spectroscopy on the particle, 00:47:26.550 --> 00:47:28.730 align:middle line:84% you can measure the energy of the particle. 00:47:28.730 --> 00:47:31.560 align:middle line:90% 00:47:31.560 --> 00:47:34.830 align:middle line:84% And you're facing a situation like the one 00:47:34.830 --> 00:47:39.000 align:middle line:84% that we've just done with the mass of the particles 00:47:39.000 --> 00:47:40.630 align:middle line:90% that you don't know which one. 00:47:40.630 --> 00:47:41.130 align:middle line:90% OK. 00:47:41.130 --> 00:47:43.670 align:middle line:90% Here it's a similar situation. 00:47:43.670 --> 00:47:46.680 align:middle line:90% You do a measurement of energy. 00:47:46.680 --> 00:47:54.240 align:middle line:84% And you don't know which one of the possible energies 00:47:54.240 --> 00:47:57.880 align:middle line:84% you get because the particle is not definitely on this, 00:47:57.880 --> 00:47:58.480 align:middle line:90% not always. 00:47:58.480 --> 00:48:02.730 align:middle line:84% I mean, most of the times, it's in a superposition 00:48:02.730 --> 00:48:04.590 align:middle line:90% of energy levels. 00:48:04.590 --> 00:48:08.850 align:middle line:84% So each one of them has a probability. 00:48:08.850 --> 00:48:15.200 align:middle line:84% And the energy of the particle is the weighted sum 00:48:15.200 --> 00:48:18.365 align:middle line:84% with these probabilities of these energy levels. 00:48:18.365 --> 00:48:25.190 align:middle line:90% 00:48:25.190 --> 00:48:30.230 align:middle line:84% So the state of the particles is represented by a probability 00:48:30.230 --> 00:48:36.800 align:middle line:84% distribution, which, at least in the stable equilibrium states, 00:48:36.800 --> 00:48:44.930 align:middle line:84% can be viewed as the probability of finding the energy levels 00:48:44.930 --> 00:48:48.460 align:middle line:90% among these possibilities. 00:48:48.460 --> 00:48:52.610 align:middle line:84% That is for the particle in one-dimensional box. 00:48:52.610 --> 00:48:54.650 align:middle line:84% If you have a three-dimensional box, 00:48:54.650 --> 00:48:59.990 align:middle line:84% each direction has its own contribution. 00:48:59.990 --> 00:49:04.840 align:middle line:84% And actually, these functions factor. 00:49:04.840 --> 00:49:10.210 align:middle line:84% So the three orthogonal translational degrees 00:49:10.210 --> 00:49:15.710 align:middle line:84% of freedom of the particle are superposed in each other. 00:49:15.710 --> 00:49:23.770 align:middle line:84% So it's like having three kind of subparticles 00:49:23.770 --> 00:49:28.420 align:middle line:84% like in a perfect gas that ignore each other. 00:49:28.420 --> 00:49:32.050 align:middle line:84% Each degree of freedom lives by itself 00:49:32.050 --> 00:49:37.120 align:middle line:84% occupies its directional modes by itself 00:49:37.120 --> 00:49:40.630 align:middle line:84% without interacting with the other modes. 00:49:40.630 --> 00:49:47.990 align:middle line:84% That's why this is such a simple extension of that. 00:49:47.990 --> 00:49:55.360 align:middle line:84% Just multiply, so to speak, the various functions. 00:49:55.360 --> 00:50:00.020 align:middle line:84% Not only, but from this one, from these results, 00:50:00.020 --> 00:50:06.220 align:middle line:84% you can also obtain the probability of finding-- 00:50:06.220 --> 00:50:14.390 align:middle line:84% the probability density of finding these particle 00:50:14.390 --> 00:50:17.410 align:middle line:84% if you make a position measurement, finding it 00:50:17.410 --> 00:50:21.070 align:middle line:90% between x and x plus dx. 00:50:21.070 --> 00:50:24.080 align:middle line:84% That's the PDF, the probability density function. 00:50:24.080 --> 00:50:32.770 align:middle line:84% It's given by this psi, the modulus of this one or also 00:50:32.770 --> 00:50:34.360 align:middle line:90% that one. 00:50:34.360 --> 00:50:40.040 align:middle line:84% And I think there is an i here that I'm 00:50:40.040 --> 00:50:44.930 align:middle line:84% missing because this is supposed to be i 00:50:44.930 --> 00:50:46.730 align:middle line:90% is the square root of minus 1. 00:50:46.730 --> 00:50:48.602 align:middle line:90% OK. 00:50:48.602 --> 00:50:53.460 align:middle line:84% So that means that when I do the complex conjugate of this one, 00:50:53.460 --> 00:50:57.470 align:middle line:90% that i becomes minus i. 00:50:57.470 --> 00:51:04.190 align:middle line:84% And so the modulus of phi is equal to-- this one cancels 00:51:04.190 --> 00:51:17.180 align:middle line:84% and you're left with this, which is the modulus of psi squared. 00:51:17.180 --> 00:51:20.430 align:middle line:84% And that is the probability density function. 00:51:20.430 --> 00:51:23.360 align:middle line:84% So if you want to know how much is 00:51:23.360 --> 00:51:25.850 align:middle line:84% the probability in that interval, you multiply by dx. 00:51:25.850 --> 00:51:29.090 align:middle line:90% 00:51:29.090 --> 00:51:35.930 align:middle line:84% And if you want to know the probability to go from, 00:51:35.930 --> 00:51:37.940 align:middle line:84% say, from here to there, you just 00:51:37.940 --> 00:51:44.090 align:middle line:84% do the integral from whatever x, say x1 to x2, 00:51:44.090 --> 00:51:46.550 align:middle line:90% to get the overall distribution. 00:51:46.550 --> 00:51:51.610 align:middle line:84% So these functions are these ones that I've plotted. 00:51:51.610 --> 00:51:54.850 align:middle line:90% 00:51:54.850 --> 00:51:57.210 align:middle line:90% I wish I had more space. 00:51:57.210 --> 00:51:57.800 align:middle line:90% OK. 00:51:57.800 --> 00:51:58.680 align:middle line:90% Do it this way. 00:51:58.680 --> 00:52:05.510 align:middle line:90% 00:52:05.510 --> 00:52:14.860 align:middle line:84% So the probabilities are the square of these functions. 00:52:14.860 --> 00:52:17.290 align:middle line:84% The wave functions can go negative. 00:52:17.290 --> 00:52:22.030 align:middle line:84% But the probability is essentially the square. 00:52:22.030 --> 00:52:26.500 align:middle line:84% So take the square of this sine and it 00:52:26.500 --> 00:52:29.680 align:middle line:90% becomes something like this. 00:52:29.680 --> 00:52:40.480 align:middle line:84% The square of this other sine, it becomes something like this 00:52:40.480 --> 00:52:41.730 align:middle line:90% and so on. 00:52:41.730 --> 00:52:49.980 align:middle line:90% 00:52:49.980 --> 00:52:51.010 align:middle line:90% And so on. 00:52:51.010 --> 00:52:54.570 align:middle line:90% 00:52:54.570 --> 00:53:01.260 align:middle line:84% When the particle is in a given state, 00:53:01.260 --> 00:53:06.570 align:middle line:84% it is not typically in just one of these energy levels. 00:53:06.570 --> 00:53:09.220 align:middle line:84% It is a probability distribution. 00:53:09.220 --> 00:53:14.790 align:middle line:84% So each one of these distributions in space 00:53:14.790 --> 00:53:19.090 align:middle line:84% is weighted by a probability distribution. 00:53:19.090 --> 00:53:20.790 align:middle line:90% So there will be a p1. 00:53:20.790 --> 00:53:24.540 align:middle line:84% There will be a p2, p3, and so on. 00:53:24.540 --> 00:53:31.650 align:middle line:84% So you have a weighted sum of these probabilities with which 00:53:31.650 --> 00:53:38.310 align:middle line:84% you can compute, for example, the probability 00:53:38.310 --> 00:53:44.220 align:middle line:84% that your particle is between x and x plus dx, 00:53:44.220 --> 00:53:53.930 align:middle line:84% it's this area times p1 plus this area times p2 plus 00:53:53.930 --> 00:53:58.730 align:middle line:84% that area times p3 up to infinity 00:53:58.730 --> 00:54:01.070 align:middle line:90% because you have to go up. 00:54:01.070 --> 00:54:03.390 align:middle line:90% This j goes up to infinity. 00:54:03.390 --> 00:54:08.290 align:middle line:90% 00:54:08.290 --> 00:54:12.940 align:middle line:84% And that's how you compute the actual stable equilibrium state 00:54:12.940 --> 00:54:14.630 align:middle line:90% probability distribution. 00:54:14.630 --> 00:54:17.700 align:middle line:90% 00:54:17.700 --> 00:54:21.930 align:middle line:84% We already mentioned the fact that when you do this job, 00:54:21.930 --> 00:54:27.090 align:middle line:84% you can compute therefore the probability density. 00:54:27.090 --> 00:54:31.500 align:middle line:84% For every x, you will find that it will always 00:54:31.500 --> 00:54:36.660 align:middle line:84% go to 0 at the edges because at the edges, 00:54:36.660 --> 00:54:40.060 align:middle line:84% no matter how you weight it, it's always 0. 00:54:40.060 --> 00:54:48.810 align:middle line:84% So the sum of 0 is 0 whereas, in the center, 00:54:48.810 --> 00:54:58.500 align:middle line:84% you will have some wiggles, especially if the low energy 00:54:58.500 --> 00:54:59.890 align:middle line:90% levels are important. 00:54:59.890 --> 00:55:05.070 align:middle line:84% As you go upwards, this becomes so you may have, 00:55:05.070 --> 00:55:08.250 align:middle line:90% if the quantum number goes-- 00:55:08.250 --> 00:55:13.740 align:middle line:84% as it goes higher, these wiggles become more intense 00:55:13.740 --> 00:55:15.820 align:middle line:90% and they add up. 00:55:15.820 --> 00:55:17.710 align:middle line:84% And so the one cancels the others. 00:55:17.710 --> 00:55:21.760 align:middle line:84% And so the 0, you'll get only 0 at the walls 00:55:21.760 --> 00:55:24.130 align:middle line:84% but not in the middle, not in the bulk. 00:55:24.130 --> 00:55:27.620 align:middle line:90% 00:55:27.620 --> 00:55:29.330 align:middle line:90% OK. 00:55:29.330 --> 00:55:33.110 align:middle line:90% So these are the energy levels. 00:55:33.110 --> 00:55:35.330 align:middle line:84% Here is the name for the probabilities 00:55:35.330 --> 00:55:39.680 align:middle line:90% that we have simplified here. 00:55:39.680 --> 00:55:42.830 align:middle line:84% In three dimensions, you have three labels 00:55:42.830 --> 00:55:45.886 align:middle line:84% because you have one quantum number per direction. 00:55:45.886 --> 00:55:49.440 align:middle line:90% 00:55:49.440 --> 00:55:54.900 align:middle line:84% And the typical problem that you face 00:55:54.900 --> 00:56:01.350 align:middle line:84% if you do energy measurements is, well, how do I-- 00:56:01.350 --> 00:56:05.380 align:middle line:84% what's my best guess for this probability distribution? 00:56:05.380 --> 00:56:08.530 align:middle line:90% 00:56:08.530 --> 00:56:10.470 align:middle line:84% That's the information theory point of view. 00:56:10.470 --> 00:56:15.210 align:middle line:90% 00:56:15.210 --> 00:56:18.615 align:middle line:84% The thermodynamics point of view is, all right, 00:56:18.615 --> 00:56:23.640 align:middle line:84% the probabilities if we accept that the probabilities represent 00:56:23.640 --> 00:56:31.860 align:middle line:84% the state, and if we accept that entropy is measured 00:56:31.860 --> 00:56:37.850 align:middle line:84% by a function like this, then we can use the maximum entropy 00:56:37.850 --> 00:56:45.780 align:middle line:84% principle which says, for a given energy, 00:56:45.780 --> 00:57:01.920 align:middle line:84% my system has an energy and entropy diagram. 00:57:01.920 --> 00:57:10.230 align:middle line:84% And for a given energy, I may have many different states. 00:57:10.230 --> 00:57:13.830 align:middle line:84% But if I'm looking for the stable equilibrium state, 00:57:13.830 --> 00:57:17.490 align:middle line:84% I know that it has to have the maximum entropy for that given 00:57:17.490 --> 00:57:18.930 align:middle line:90% energy. 00:57:18.930 --> 00:57:21.090 align:middle line:84% So that's precisely the same problem 00:57:21.090 --> 00:57:22.260 align:middle line:90% that we have seen before. 00:57:22.260 --> 00:57:25.770 align:middle line:84% Maximize entropy provided you accept 00:57:25.770 --> 00:57:29.953 align:middle line:84% that formula for the entropy of the quantum particle. 00:57:29.953 --> 00:57:34.450 align:middle line:90% 00:57:34.450 --> 00:57:37.780 align:middle line:84% Then the mathematics of maximization 00:57:37.780 --> 00:57:39.926 align:middle line:84% is the same using Lagrange multipliers. 00:57:39.926 --> 00:57:43.180 align:middle line:90% 00:57:43.180 --> 00:57:47.800 align:middle line:84% Now since we know already the answer, 00:57:47.800 --> 00:57:57.190 align:middle line:84% instead of calling the Lagrange multiplier lambda E, which since 00:57:57.190 --> 00:57:59.440 align:middle line:84% later on we recognize that that lambda is 00:57:59.440 --> 00:58:03.340 align:middle line:84% the inverse of the temperature divided by the Boltzmann 00:58:03.340 --> 00:58:06.200 align:middle line:84% constant, we already call it 1 over kT. 00:58:06.200 --> 00:58:09.510 align:middle line:90% 00:58:09.510 --> 00:58:12.360 align:middle line:84% And so you find this distribution. 00:58:12.360 --> 00:58:16.830 align:middle line:84% So the probabilities are given by e to the minus 00:58:16.830 --> 00:58:24.270 align:middle line:84% the energy, these energy values, divided by-- 00:58:24.270 --> 00:58:31.830 align:middle line:84% or multiplied by the Lagrange multiplier divided by this Q. 00:58:31.830 --> 00:58:35.230 align:middle line:84% You can view it as a normalization factor. 00:58:35.230 --> 00:58:38.162 align:middle line:84% It's essentially the summation of all these things 00:58:38.162 --> 00:58:50.370 align:middle line:84% at the numerator, this Q. I also see that since the energy 00:58:50.370 --> 00:58:52.160 align:middle line:90% eigenvalues add up. 00:58:52.160 --> 00:58:54.870 align:middle line:90% 00:58:54.870 --> 00:58:57.780 align:middle line:84% The exponential of a sum is equal to the product 00:58:57.780 --> 00:58:58.810 align:middle line:90% of the exponentials. 00:58:58.810 --> 00:59:04.380 align:middle line:84% So these probability distributions factor 00:59:04.380 --> 00:59:09.060 align:middle line:84% into separate probability distributions 00:59:09.060 --> 00:59:13.690 align:middle line:84% for each of the directions of the box. 00:59:13.690 --> 00:59:18.730 align:middle line:84% So you can view the problem also just as-- 00:59:18.730 --> 00:59:22.630 align:middle line:84% you can solve it in one direction per time 00:59:22.630 --> 00:59:24.660 align:middle line:84% and then simply multiply the results. 00:59:24.660 --> 00:59:27.580 align:middle line:90% 00:59:27.580 --> 00:59:31.630 align:middle line:84% These denominators or denominator for the overall 00:59:31.630 --> 00:59:33.915 align:middle line:84% is called the partition function. 00:59:33.915 --> 00:59:38.640 align:middle line:90% 00:59:38.640 --> 00:59:39.160 align:middle line:90% Yes. 00:59:39.160 --> 00:59:43.570 align:middle line:84% It plays the role of the normalization factor 00:59:43.570 --> 00:59:49.500 align:middle line:84% so that when you sum the probabilities, they add up to 1. 00:59:49.500 --> 00:59:52.440 align:middle line:84% But it also plays an interesting role 00:59:52.440 --> 00:59:58.740 align:middle line:84% because, essentially, you can extract everything if you 00:59:58.740 --> 01:00:00.010 align:middle line:90% know this partition function. 01:00:00.010 --> 01:00:04.890 align:middle line:84% If you know this summation, by playing with it in terms 01:00:04.890 --> 01:00:08.490 align:middle line:84% of taking some derivatives, you can 01:00:08.490 --> 01:00:12.790 align:middle line:84% get everything of the distribution, 01:00:12.790 --> 01:00:16.650 align:middle line:84% including the probabilities themselves. 01:00:16.650 --> 01:00:20.130 align:middle line:84% So if you take the partial derivative 01:00:20.130 --> 01:00:24.510 align:middle line:84% of the logarithm of the partition function as written 01:00:24.510 --> 01:00:28.200 align:middle line:84% here and take the derivative with respect 01:00:28.200 --> 01:00:34.950 align:middle line:84% to the energy level, you get the probability and multiply by kT 01:00:34.950 --> 01:00:40.570 align:middle line:84% and minus, you get the probability 01:00:40.570 --> 01:00:42.840 align:middle line:90% of observing that energy level. 01:00:42.840 --> 01:00:47.420 align:middle line:90% 01:00:47.420 --> 01:00:51.260 align:middle line:84% Or you take this other derivative. 01:00:51.260 --> 01:00:55.100 align:middle line:84% You take the derivative with respect to temperature 01:00:55.100 --> 01:01:06.410 align:middle line:84% because, yes, in a moment we'll recognize that this T is also-- 01:01:06.410 --> 01:01:11.480 align:middle line:84% this Lagrange multiplier, satisfies 01:01:11.480 --> 01:01:13.430 align:middle line:84% our definition of temperature-- is 01:01:13.430 --> 01:01:17.180 align:middle line:84% the partial of E with respect to S. 01:01:17.180 --> 01:01:21.980 align:middle line:84% And so you identify it with the temperature. 01:01:21.980 --> 01:01:25.430 align:middle line:84% So if you take the log of-- the partial of log of Q 01:01:25.430 --> 01:01:34.010 align:middle line:84% with respect to T, you get the energy, the mean energy. 01:01:34.010 --> 01:01:37.100 align:middle line:84% The mean energy is the average value of the energy. 01:01:37.100 --> 01:01:40.550 align:middle line:84% So it's the weighted sum of these energy 01:01:40.550 --> 01:01:43.070 align:middle line:84% levels weighted on the probability that 01:01:43.070 --> 01:01:44.510 align:middle line:90% represents that state. 01:01:44.510 --> 01:01:47.930 align:middle line:90% 01:01:47.930 --> 01:01:50.140 align:middle line:90% And also, the entropy. 01:01:50.140 --> 01:01:53.240 align:middle line:90% 01:01:53.240 --> 01:01:57.140 align:middle line:84% Also the entropy because the probabilities factor, 01:01:57.140 --> 01:02:00.380 align:middle line:84% the entropies add, because the logarithm-- remember, 01:02:00.380 --> 01:02:03.650 align:middle line:84% the logarithm of the product is equal to the sum 01:02:03.650 --> 01:02:06.590 align:middle line:90% of the logarithms. 01:02:06.590 --> 01:02:09.680 align:middle line:84% So you can write the overall entropy 01:02:09.680 --> 01:02:12.680 align:middle line:84% and split it into the entropies of the three 01:02:12.680 --> 01:02:15.830 align:middle line:90% directions of the box. 01:02:15.830 --> 01:02:20.300 align:middle line:84% And each one of them can be obtained as a proper derivative 01:02:20.300 --> 01:02:26.030 align:middle line:84% of Q or if you've already taken the derivative that gives you 01:02:26.030 --> 01:02:33.170 align:middle line:84% the energy, you can write S as E over T plus Boltzmann 01:02:33.170 --> 01:02:37.160 align:middle line:84% constant times the logarithm of the partition function. 01:02:37.160 --> 01:02:44.750 align:middle line:84% This is somehow as close as you can get to the Euler 01:02:44.750 --> 01:02:46.250 align:middle line:90% free energy or the free energy. 01:02:46.250 --> 01:02:51.780 align:middle line:90% 01:02:51.780 --> 01:02:54.450 align:middle line:90% In fact, some people call it-- 01:02:54.450 --> 01:03:01.800 align:middle line:84% you could also rewrite that as Ei minus TSi 01:03:01.800 --> 01:03:12.920 align:middle line:84% is equal to kB T log of Q, probably with a minus sign. 01:03:12.920 --> 01:03:13.420 align:middle line:90% Yeah. 01:03:13.420 --> 01:03:19.840 align:middle line:90% 01:03:19.840 --> 01:03:20.340 align:middle line:90% OK. 01:03:20.340 --> 01:03:36.060 align:middle line:90% 01:03:36.060 --> 01:03:44.380 align:middle line:84% Now going back to these probabilities, 01:03:44.380 --> 01:03:47.710 align:middle line:84% this probability distribution is the one 01:03:47.710 --> 01:03:50.680 align:middle line:84% that characterizes the stable equilibrium state. 01:03:50.680 --> 01:03:53.800 align:middle line:90% It characterizes this state. 01:03:53.800 --> 01:03:55.515 align:middle line:90% How about these other ones? 01:03:55.515 --> 01:04:06.030 align:middle line:90% 01:04:06.030 --> 01:04:12.450 align:middle line:84% So this canonical distribution, the maximum entropy one, 01:04:12.450 --> 01:04:15.430 align:middle line:90% looks like this. 01:04:15.430 --> 01:04:18.700 align:middle line:84% Here, for simplicity, I have drawn 01:04:18.700 --> 01:04:25.910 align:middle line:84% a system which doesn't have this eigenvalue structure 01:04:25.910 --> 01:04:29.470 align:middle line:84% but it has energy levels that are equally spaced 01:04:29.470 --> 01:04:32.710 align:middle line:90% and there is only seven of them. 01:04:32.710 --> 01:04:38.510 align:middle line:84% The lowest one, 1, 2, 3, 4, 5, 6, 7. 01:04:38.510 --> 01:04:39.110 align:middle line:90% OK. 01:04:39.110 --> 01:04:40.245 align:middle line:90% Equally spaced. 01:04:40.245 --> 01:04:45.150 align:middle line:90% 01:04:45.150 --> 01:04:50.670 align:middle line:84% We have already mentioned that for a system 01:04:50.670 --> 01:04:55.600 align:middle line:84% with an upper bound to the energy spectrum-- 01:04:55.600 --> 01:04:58.500 align:middle line:90% this need an r here-- 01:04:58.500 --> 01:05:01.950 align:middle line:90% that should be spectrum-- 01:05:01.950 --> 01:05:06.820 align:middle line:84% the E versus S diagram doesn't go up indefinitely 01:05:06.820 --> 01:05:09.660 align:middle line:90% but it bends and returns. 01:05:09.660 --> 01:05:11.615 align:middle line:90% So that's the case. 01:05:11.615 --> 01:05:14.250 align:middle line:90% 01:05:14.250 --> 01:05:17.110 align:middle line:84% So we have only seven energy levels. 01:05:17.110 --> 01:05:24.510 align:middle line:84% And we can picture these probabilities by a histogram. 01:05:24.510 --> 01:05:29.070 align:middle line:84% So this is the maximum entropy, the canonical distribution. 01:05:29.070 --> 01:05:32.670 align:middle line:84% For example, this is a zero entropy. 01:05:32.670 --> 01:05:38.640 align:middle line:84% A zero entropy state is one in which you are certain-- 01:05:38.640 --> 01:05:42.090 align:middle line:84% remember, the entropy in terms of probabilities 01:05:42.090 --> 01:05:44.460 align:middle line:84% represents your state of uncertainty. 01:05:44.460 --> 01:05:48.010 align:middle line:84% Well, if I know that every time I measure, 01:05:48.010 --> 01:05:51.720 align:middle line:84% I always find the particle in that energy level, 01:05:51.720 --> 01:05:53.020 align:middle line:90% I have no uncertainty. 01:05:53.020 --> 01:05:54.900 align:middle line:90% The entropy is zero. 01:05:54.900 --> 01:06:03.210 align:middle line:84% And that is when I have zeros for all probabilities except 1, 01:06:03.210 --> 01:06:04.250 align:middle line:90% which is equal to 1. 01:06:04.250 --> 01:06:07.610 align:middle line:90% 01:06:07.610 --> 01:06:14.480 align:middle line:84% A point like here is one that has an intermediate probability 01:06:14.480 --> 01:06:15.895 align:middle line:90% between 0 and the maximum. 01:06:15.895 --> 01:06:19.510 align:middle line:90% 01:06:19.510 --> 01:06:26.680 align:middle line:84% You can concoct an infinite number of possibilities 01:06:26.680 --> 01:06:32.710 align:middle line:84% of different probabilities, probability distributions, that 01:06:32.710 --> 01:06:37.060 align:middle line:84% have the same entropy and, of course, the same energy. 01:06:37.060 --> 01:06:40.260 align:middle line:90% Here are the ones. 01:06:40.260 --> 01:06:40.760 align:middle line:90% OK. 01:06:40.760 --> 01:06:45.230 align:middle line:90% 01:06:45.230 --> 01:06:48.072 align:middle line:90% Why is that important? 01:06:48.072 --> 01:06:52.930 align:middle line:90% 01:06:52.930 --> 01:06:55.390 align:middle line:84% When we are interested in describing 01:06:55.390 --> 01:06:59.830 align:middle line:84% non-equilibrium states, what is important 01:06:59.830 --> 01:07:05.860 align:middle line:84% is to be able to understand or model 01:07:05.860 --> 01:07:10.930 align:middle line:84% how these states or these probability distributions 01:07:10.930 --> 01:07:17.980 align:middle line:84% evolve towards equilibrium because Nature non facit saltum, 01:07:17.980 --> 01:07:27.220 align:middle line:84% but Nature usually likes to go towards equilibrium. 01:07:27.220 --> 01:07:30.560 align:middle line:84% And so there is a spontaneous tendency. 01:07:30.560 --> 01:07:38.350 align:middle line:84% So there is a natural law that pushes a system to even 01:07:38.350 --> 01:07:44.420 align:middle line:84% out or redistribute the energy load. 01:07:44.420 --> 01:07:47.720 align:middle line:84% You can interpret these histograms-- 01:07:47.720 --> 01:07:50.230 align:middle line:90% 01:07:50.230 --> 01:07:52.780 align:middle line:84% the probabilities-- you see as written here-- 01:07:52.780 --> 01:07:58.810 align:middle line:84% measure the degree of involvement of the level ej 01:07:58.810 --> 01:08:03.790 align:middle line:90% in sharing the mean energy load. 01:08:03.790 --> 01:08:09.220 align:middle line:84% You give some energy to the particle. 01:08:09.220 --> 01:08:15.700 align:middle line:84% And the particle has to decide where to put it. 01:08:15.700 --> 01:08:20.210 align:middle line:84% It can put it in translational mode. 01:08:20.210 --> 01:08:23.649 align:middle line:84% But as we've seen here, if the particle is confined, 01:08:23.649 --> 01:08:27.290 align:middle line:84% the translational modes have these kind of energy levels. 01:08:27.290 --> 01:08:29.470 align:middle line:84% So how much energy do I put in this mode 01:08:29.470 --> 01:08:32.470 align:middle line:84% and how much in that and how much in that? 01:08:32.470 --> 01:08:38.560 align:middle line:84% If the particle also has an internal structure, 01:08:38.560 --> 01:08:42.069 align:middle line:84% so for example, it could vibrate, 01:08:42.069 --> 01:08:47.899 align:middle line:84% or if it has a structure so that it can rotate, it can put also, 01:08:47.899 --> 01:08:54.319 align:middle line:84% you get additional modes, rotational modes, 01:08:54.319 --> 01:08:56.450 align:middle line:84% and then you have electronic modes. 01:08:56.450 --> 01:09:05.569 align:middle line:84% So the particle has to decide how to distribute and distribute 01:09:05.569 --> 01:09:07.609 align:middle line:90% evenly. 01:09:07.609 --> 01:09:17.840 align:middle line:84% So if it is unconstrained, the most even distribution 01:09:17.840 --> 01:09:21.380 align:middle line:84% is the maximum entropy, the equilibrium. 01:09:21.380 --> 01:09:24.810 align:middle line:84% At equilibrium, it will have distributed things 01:09:24.810 --> 01:09:26.430 align:middle line:90% so as to maximize the entropy. 01:09:26.430 --> 01:09:31.345 align:middle line:84% So entropy also measures the degree of sharing. 01:09:31.345 --> 01:09:35.660 align:middle line:90% 01:09:35.660 --> 01:09:39.380 align:middle line:84% The highest degree of sharing is achieved here. 01:09:39.380 --> 01:09:44.960 align:middle line:84% And the natural phenomena have in common 01:09:44.960 --> 01:09:48.620 align:middle line:84% the idea that there is a spontaneous tendency 01:09:48.620 --> 01:09:56.300 align:middle line:84% to share the energy as evenly as possible 01:09:56.300 --> 01:10:01.430 align:middle line:84% among the available energy levels. 01:10:01.430 --> 01:10:08.870 align:middle line:84% So the question about dynamics is how do these things 01:10:08.870 --> 01:10:10.460 align:middle line:90% evolve in time. 01:10:10.460 --> 01:10:15.540 align:middle line:84% And here I have a slide with some animation. 01:10:15.540 --> 01:10:16.710 align:middle line:90% Hopefully, it should work. 01:10:16.710 --> 01:10:17.250 align:middle line:90% Let's see. 01:10:17.250 --> 01:10:22.380 align:middle line:90% 01:10:22.380 --> 01:10:23.010 align:middle line:90% Yeah. 01:10:23.010 --> 01:10:28.860 align:middle line:84% So for example, I started from very close to zero entropy. 01:10:28.860 --> 01:10:33.150 align:middle line:84% And this is a smooth transition in which 01:10:33.150 --> 01:10:37.350 align:middle line:84% the particle, while keeping the energy constant, 01:10:37.350 --> 01:10:41.190 align:middle line:84% redistributes the probabilities until it eventually 01:10:41.190 --> 01:10:45.896 align:middle line:84% achieves the stable equilibrium canonical distribution. 01:10:45.896 --> 01:10:52.090 align:middle line:90% 01:10:52.090 --> 01:10:56.140 align:middle line:84% And also, this one is another example. 01:10:56.140 --> 01:11:02.470 align:middle line:84% Started from an intermediate state with two initially 01:11:02.470 --> 01:11:07.990 align:middle line:84% full energy levels and the other ones 01:11:07.990 --> 01:11:14.370 align:middle line:84% get filled until, in the end, you get to the same state. 01:11:14.370 --> 01:11:20.800 align:middle line:90% 01:11:20.800 --> 01:11:26.710 align:middle line:84% I rescued these viewgraphs from work 01:11:26.710 --> 01:11:38.240 align:middle line:84% that I had done for my PhD thesis here, 1981, 01:11:38.240 --> 01:11:45.110 align:middle line:84% to devise an equation of motion that would take any probability 01:11:45.110 --> 01:11:51.320 align:middle line:84% distribution of this sort and evolves it smoothly 01:11:51.320 --> 01:11:55.220 align:middle line:90% towards a stable equilibrium. 01:11:55.220 --> 01:11:58.700 align:middle line:84% And that's developed in the subject 01:11:58.700 --> 01:12:04.730 align:middle line:84% that I now call it steepest entropy ascent 01:12:04.730 --> 01:12:08.030 align:middle line:84% but also has to do with what I like 01:12:08.030 --> 01:12:11.220 align:middle line:84% to call the fourth law of thermodynamics 01:12:11.220 --> 01:12:17.240 align:middle line:84% because, essentially, this kind of dynamical description 01:12:17.240 --> 01:12:23.990 align:middle line:84% of non-equilibrium states is common of a large number 01:12:23.990 --> 01:12:27.170 align:middle line:90% of theories of non-equilibrium. 01:12:27.170 --> 01:12:35.420 align:middle line:84% There are some features and there 01:12:35.420 --> 01:12:44.180 align:middle line:84% are some aspects, some general aspects of this modeling ideas, 01:12:44.180 --> 01:12:50.210 align:middle line:84% that are so common in spite of the different fields in which 01:12:50.210 --> 01:12:56.180 align:middle line:84% you find them, that you can say that perhaps 01:12:56.180 --> 01:13:03.140 align:middle line:90% they represent another law of-- 01:13:03.140 --> 01:13:09.680 align:middle line:84% shouldn't say of Nature, but another law of modeling 01:13:09.680 --> 01:13:12.860 align:middle line:84% Nature correctly, like the second law 01:13:12.860 --> 01:13:18.110 align:middle line:84% we have described as something that your model should not 01:13:18.110 --> 01:13:18.810 align:middle line:90% violate. 01:13:18.810 --> 01:13:21.170 align:middle line:90% Otherwise, it is a bad model. 01:13:21.170 --> 01:13:24.410 align:middle line:84% Also, the fourth law will be something 01:13:24.410 --> 01:13:29.790 align:middle line:84% that is similarly to be satisfied 01:13:29.790 --> 01:13:34.920 align:middle line:84% if your model has to provide a compatible description of how 01:13:34.920 --> 01:13:38.700 align:middle line:84% states evolve towards equilibrium-- 01:13:38.700 --> 01:13:43.140 align:middle line:84% compatible with all we know about evidence 01:13:43.140 --> 01:13:44.765 align:middle line:90% of physical reality. 01:13:44.765 --> 01:13:49.670 align:middle line:90% 01:13:49.670 --> 01:13:55.270 align:middle line:84% Well, a little more because we haven't finished yet 01:13:55.270 --> 01:13:59.240 align:middle line:90% with the particle in the box. 01:13:59.240 --> 01:14:09.640 align:middle line:90% And let me take it-- 01:14:09.640 --> 01:14:12.310 align:middle line:84% start again from the partition function 01:14:12.310 --> 01:14:19.010 align:middle line:84% Q of the one dimensional case, which 01:14:19.010 --> 01:14:23.260 align:middle line:84% is therefore the summation of e to the energy levels over kT. 01:14:23.260 --> 01:14:29.550 align:middle line:84% And here are the energy levels substituted directly. 01:14:29.550 --> 01:14:32.550 align:middle line:84% So you see now in this expression, 01:14:32.550 --> 01:14:34.440 align:middle line:90% what are the fixed parameters. 01:14:34.440 --> 01:14:36.300 align:middle line:84% I mean, some are physical constants-- 01:14:36.300 --> 01:14:40.940 align:middle line:84% Planck constant h, Boltzmann constant kb. 01:14:40.940 --> 01:14:44.570 align:middle line:90% m is the mass of your particle. 01:14:44.570 --> 01:14:48.830 align:middle line:84% j is just the dummy variable in this summation. 01:14:48.830 --> 01:14:51.410 align:middle line:84% There are only two parameters in this formula-- 01:14:51.410 --> 01:14:58.970 align:middle line:84% T, which is the temperature, and l, which is the size of the box 01:14:58.970 --> 01:15:00.610 align:middle line:90% or the side of the box. 01:15:00.610 --> 01:15:04.100 align:middle line:90% 01:15:04.100 --> 01:15:07.970 align:middle line:84% So Q is a function of two variables. 01:15:07.970 --> 01:15:11.370 align:middle line:84% That means that, as we like to do when we have functions, 01:15:11.370 --> 01:15:14.720 align:middle line:90% we take the differentials. 01:15:14.720 --> 01:15:18.740 align:middle line:84% So if I take the differential of the logarithm of this summation, 01:15:18.740 --> 01:15:26.540 align:middle line:84% logarithm of Q, and it takes a little while, but you can do it. 01:15:26.540 --> 01:15:27.715 align:middle line:90% You can check. 01:15:27.715 --> 01:15:30.740 align:middle line:90% 01:15:30.740 --> 01:15:34.340 align:middle line:90% It can be written this way. 01:15:34.340 --> 01:15:37.730 align:middle line:84% So of course, it has a dT and a dl 01:15:37.730 --> 01:15:39.810 align:middle line:84% because the two variables are T and l. 01:15:39.810 --> 01:15:44.310 align:middle line:90% 01:15:44.310 --> 01:15:47.100 align:middle line:84% So once you have done it, you may 01:15:47.100 --> 01:15:52.410 align:middle line:84% say that the coefficient that multiplies dT 01:15:52.410 --> 01:16:00.440 align:middle line:84% is equal to the derivative of the log of Q with respect to T. 01:16:00.440 --> 01:16:04.500 align:middle line:90% And so you obtain this relation. 01:16:04.500 --> 01:16:07.710 align:middle line:84% And this other one is equal to the derivative 01:16:07.710 --> 01:16:14.670 align:middle line:90% of log of Q with respect to l. 01:16:14.670 --> 01:16:19.380 align:middle line:84% Actually, notice that since it's dl/l, this is also 01:16:19.380 --> 01:16:21.210 align:middle line:90% the derivative of the log of l. 01:16:21.210 --> 01:16:24.020 align:middle line:90% 01:16:24.020 --> 01:16:30.210 align:middle line:84% So you obtain that derivative of log of Q over log of l 01:16:30.210 --> 01:16:34.080 align:middle line:90% is equal to E sub i over kT. 01:16:34.080 --> 01:16:53.960 align:middle line:84% Now, E sub i, I don't find it in this slide, but it is E sub i. 01:16:53.960 --> 01:16:58.340 align:middle line:84% If you go a couple of slides earlier, you find it. 01:16:58.340 --> 01:17:01.630 align:middle line:84% And you should correct me if I'm writing it wrong. 01:17:01.630 --> 01:17:07.560 align:middle line:90% 01:17:07.560 --> 01:17:09.810 align:middle line:84% E sub i should be the weighted sum 01:17:09.810 --> 01:17:13.440 align:middle line:84% of the probabilities of the energy 01:17:13.440 --> 01:17:20.550 align:middle line:90% levels of the single direction. 01:17:20.550 --> 01:17:23.140 align:middle line:90% 01:17:23.140 --> 01:17:28.580 align:middle line:84% So essentially, it's the partial energy. 01:17:28.580 --> 01:17:33.880 align:middle line:84% You see, the overall energy is equal to the sum of the energies 01:17:33.880 --> 01:17:38.980 align:middle line:84% that the particle has in the three directions. 01:17:38.980 --> 01:17:42.100 align:middle line:90% It's like additive. 01:17:42.100 --> 01:17:47.890 align:middle line:84% They are independent modes, independent lives 01:17:47.890 --> 01:17:50.590 align:middle line:84% that the particle has in the three directions. 01:17:50.590 --> 01:17:55.900 align:middle line:84% So here we are watching it in just one direction. 01:17:55.900 --> 01:18:02.060 align:middle line:84% Then, another formula that we had a couple of slides ago 01:18:02.060 --> 01:18:08.595 align:middle line:84% is this one, the one that I also wrote here for the free energy. 01:18:08.595 --> 01:18:12.600 align:middle line:90% 01:18:12.600 --> 01:18:20.100 align:middle line:84% So if I take the differential of this one, here is what I get. 01:18:20.100 --> 01:18:21.810 align:middle line:84% Again, it takes a little bit of doing 01:18:21.810 --> 01:18:27.030 align:middle line:84% because I have to take the differential of log of Q. 01:18:27.030 --> 01:18:29.860 align:middle line:84% But the differential of log of Q, we just take it. 01:18:29.860 --> 01:18:32.980 align:middle line:90% 01:18:32.980 --> 01:18:33.890 align:middle line:90% We just took it. 01:18:33.890 --> 01:18:36.220 align:middle line:90% And so here it is. 01:18:36.220 --> 01:18:37.790 align:middle line:90% We substitute it there. 01:18:37.790 --> 01:18:42.340 align:middle line:84% This one cancels with the minus 1/T squared that comes from 01:18:42.340 --> 01:18:43.760 align:middle line:90% the derivative of this. 01:18:43.760 --> 01:18:45.670 align:middle line:84% And so here is what I'm left with-- 01:18:45.670 --> 01:18:50.920 align:middle line:90% dS 1/T dE plus something dl. 01:18:50.920 --> 01:18:54.130 align:middle line:90% 01:18:54.130 --> 01:18:57.100 align:middle line:84% This is an entropy form of the fundamental relation 01:18:57.100 --> 01:18:58.605 align:middle line:84% for the stable equilibrium state. 01:18:58.605 --> 01:19:01.460 align:middle line:90% 01:19:01.460 --> 01:19:03.770 align:middle line:90% So this is the Gibbs relation. 01:19:03.770 --> 01:19:05.790 align:middle line:84% You like to put it in energy form. 01:19:05.790 --> 01:19:07.340 align:middle line:90% You just solve for dE. 01:19:07.340 --> 01:19:08.540 align:middle line:90% And here is it. 01:19:08.540 --> 01:19:13.400 align:middle line:84% It's dE is minus TdS minus something dl. 01:19:13.400 --> 01:19:18.140 align:middle line:90% 01:19:18.140 --> 01:19:28.360 align:middle line:84% Now, if you remember, when we have 01:19:28.360 --> 01:19:35.770 align:middle line:84% a substance in the simple-system model limit, 01:19:35.770 --> 01:19:42.640 align:middle line:84% the Gibbs relation is that dE is TdS minus pdV. 01:19:42.640 --> 01:19:44.890 align:middle line:90% pdV, OK? 01:19:44.890 --> 01:19:47.950 align:middle line:84% Volume is the parameter, and pressure 01:19:47.950 --> 01:19:50.860 align:middle line:84% is the coefficient in the differential that 01:19:50.860 --> 01:19:55.480 align:middle line:84% measures the partial of the energy with respect 01:19:55.480 --> 01:19:58.330 align:middle line:90% to volume at constant S. 01:19:58.330 --> 01:20:02.410 align:middle line:84% All right, so that prompts us by just looking at this. 01:20:02.410 --> 01:20:04.960 align:middle line:84% Here, we don't have volume because there 01:20:04.960 --> 01:20:06.470 align:middle line:90% is no volume for this particle. 01:20:06.470 --> 01:20:08.210 align:middle line:84% It is only the length of the box. 01:20:08.210 --> 01:20:13.480 align:middle line:84% The size of the box is the equivalent of the volume. 01:20:13.480 --> 01:20:17.610 align:middle line:84% So we can define the equivalent of a pressure. 01:20:17.610 --> 01:20:20.650 align:middle line:90% 01:20:20.650 --> 01:20:21.380 align:middle line:90% I will call it-- 01:20:21.380 --> 01:20:24.340 align:middle line:84% I don't know what name to give, but I will 01:20:24.340 --> 01:20:28.120 align:middle line:90% call it directional pressure. 01:20:28.120 --> 01:20:29.790 align:middle line:90% It has dimensions of pressure. 01:20:29.790 --> 01:20:32.740 align:middle line:90% 01:20:32.740 --> 01:20:39.400 align:middle line:84% And I say-- and I define it as the negative of the partial 01:20:39.400 --> 01:20:47.290 align:middle line:84% of the energy with respect to volume but done while changing 01:20:47.290 --> 01:20:51.190 align:middle line:90% only the-- 01:20:51.190 --> 01:20:54.100 align:middle line:90% one direction. 01:20:54.100 --> 01:20:57.110 align:middle line:84% So you see, since I have a box now this size, 01:20:57.110 --> 01:21:01.270 align:middle line:84% this shape, there are three different ways in which I 01:21:01.270 --> 01:21:04.000 align:middle line:90% can change the volume. 01:21:04.000 --> 01:21:12.130 align:middle line:84% One is to keep l2 and l3 fixed and make l1 longer. 01:21:12.130 --> 01:21:15.700 align:middle line:84% And then there are the other two, OK? 01:21:15.700 --> 01:21:19.940 align:middle line:84% So I'm changing the volume just by changing l1. 01:21:19.940 --> 01:21:26.510 align:middle line:84% That's the pressure assigned to direction one. 01:21:26.510 --> 01:21:29.960 align:middle line:84% How much the energy changes in the stable equilibrium 01:21:29.960 --> 01:21:34.405 align:middle line:84% at fixed S, fixed l2 and l3 when I change l1. 01:21:34.405 --> 01:21:38.680 align:middle line:90% 01:21:38.680 --> 01:21:41.195 align:middle line:84% So you can also rewrite it this way. 01:21:41.195 --> 01:21:43.960 align:middle line:90% 01:21:43.960 --> 01:21:47.290 align:middle line:84% And if you rewrite it this way, because the volume-- 01:21:47.290 --> 01:21:49.490 align:middle line:84% I mean, yeah, I'll let you think about this. 01:21:49.490 --> 01:21:50.940 align:middle line:90% This is very simple. 01:21:50.940 --> 01:21:53.770 align:middle line:90% 01:21:53.770 --> 01:21:57.460 align:middle line:84% All right, so the partial of E with respect to the logarithm 01:21:57.460 --> 01:22:02.350 align:middle line:90% of l1 is obtained here. 01:22:02.350 --> 01:22:10.390 align:middle line:84% It goes from this differential, this coefficient minus 2E is 01:22:10.390 --> 01:22:13.400 align:middle line:84% exactly the partial of E with respect to the logarithm 01:22:13.400 --> 01:22:15.830 align:middle line:84% of the dl/l is the logarithm of l. 01:22:15.830 --> 01:22:19.210 align:middle line:90% 01:22:19.210 --> 01:22:32.670 align:middle line:84% So here, what we find is that pressure is 2E/V. 01:22:32.670 --> 01:22:36.550 align:middle line:84% That is the closest we can get for the moment without 01:22:36.550 --> 01:22:38.050 align:middle line:90% additional assumption. 01:22:38.050 --> 01:22:43.380 align:middle line:90% So we found that pi-- 01:22:43.380 --> 01:22:49.410 align:middle line:90% let's say-- 1 times the volume-- 01:22:49.410 --> 01:22:52.530 align:middle line:90% is that the volume-- 01:22:52.530 --> 01:22:55.080 align:middle line:90% is equal to 2Ei. 01:22:55.080 --> 01:22:59.160 align:middle line:90% 01:22:59.160 --> 01:23:00.241 align:middle line:90% This is pi i. 01:23:00.241 --> 01:23:07.790 align:middle line:90% 01:23:07.790 --> 01:23:13.790 align:middle line:84% In a moment, if you add the information that eventually we 01:23:13.790 --> 01:23:24.360 align:middle line:84% will have that E sub i is equal to 1/2 of kT, 01:23:24.360 --> 01:23:27.120 align:middle line:90% you substitute up there. 01:23:27.120 --> 01:23:31.980 align:middle line:90% You find that pi V equals-- 01:23:31.980 --> 01:23:36.860 align:middle line:90% the 2 cancels, equals kT. 01:23:36.860 --> 01:23:41.880 align:middle line:84% k, remember, is the gas constant n R. So 01:23:41.880 --> 01:23:44.820 align:middle line:90% you find that pi V equals nRT. 01:23:44.820 --> 01:23:47.340 align:middle line:90% 01:23:47.340 --> 01:23:51.960 align:middle line:84% That's the ideal gas equation of state. 01:23:51.960 --> 01:23:56.170 align:middle line:84% It requires, though, this step, which we don't get yet. 01:23:56.170 --> 01:23:58.200 align:middle line:90% And it's not always. 01:23:58.200 --> 01:24:03.420 align:middle line:84% But at this stage, without assumptions about how large 01:24:03.420 --> 01:24:06.510 align:middle line:84% the temperature is, which is something 01:24:06.510 --> 01:24:10.300 align:middle line:84% that we require to discuss to get to this result, 01:24:10.300 --> 01:24:16.740 align:middle line:84% we have obtained essentially the equation of state. 01:24:16.740 --> 01:24:20.760 align:middle line:84% And of course, you have one such equation 01:24:20.760 --> 01:24:24.850 align:middle line:90% of state for each direction. 01:24:24.850 --> 01:24:29.530 align:middle line:84% You may have energy in the x, y, and z direction, 01:24:29.530 --> 01:24:32.220 align:middle line:84% and so you have these directional pressures. 01:24:32.220 --> 01:24:34.865 align:middle line:84% And so you have three such relations here. 01:24:34.865 --> 01:24:46.060 align:middle line:90% 01:24:46.060 --> 01:24:51.610 align:middle line:84% You also find that since the energy sums-- 01:24:51.610 --> 01:24:56.800 align:middle line:84% I mean is additive, the energy of the three directions 01:24:56.800 --> 01:25:00.040 align:middle line:84% adds up to the overall energy of your particle, 01:25:00.040 --> 01:25:03.140 align:middle line:84% this relation between energy and pressure. 01:25:03.140 --> 01:25:07.510 align:middle line:84% So you can also be written as the sum of these three 01:25:07.510 --> 01:25:12.680 align:middle line:90% pressures times V/2. 01:25:12.680 --> 01:25:17.780 align:middle line:90% 01:25:17.780 --> 01:25:24.110 align:middle line:84% And from the fact that the energy sums, 01:25:24.110 --> 01:25:29.060 align:middle line:90% I can also take the-- 01:25:29.060 --> 01:25:34.480 align:middle line:84% if I take the differential of this, 01:25:34.480 --> 01:25:38.560 align:middle line:84% so dE is equal to the sum of the dEi's but the expressions 01:25:38.560 --> 01:25:40.490 align:middle line:90% for the dEi's is here. 01:25:40.490 --> 01:25:48.440 align:middle line:84% So the Gibbs relations for the various directions are here. 01:25:48.440 --> 01:25:51.670 align:middle line:84% I can sum them up to get the overall Gibbs relation 01:25:51.670 --> 01:25:54.400 align:middle line:90% for my particle in the box. 01:25:54.400 --> 01:25:57.700 align:middle line:90% And so I have dE equal TdS. 01:25:57.700 --> 01:26:01.670 align:middle line:84% And instead of pdV, we have three contributions, 01:26:01.670 --> 01:26:03.020 align:middle line:90% one for each direction. 01:26:03.020 --> 01:26:07.070 align:middle line:90% 01:26:07.070 --> 01:26:11.960 align:middle line:84% So this is, in general, the Gibbs relation 01:26:11.960 --> 01:26:13.690 align:middle line:90% for the particle in the box. 01:26:13.690 --> 01:26:16.320 align:middle line:90% 01:26:16.320 --> 01:26:19.720 align:middle line:90% And it requires three pressures. 01:26:19.720 --> 01:26:24.700 align:middle line:90% 01:26:24.700 --> 01:26:37.071 align:middle line:84% Now, we can make an observation that the partition function 01:26:37.071 --> 01:26:45.570 align:middle line:84% that we now recognize the role is a summation. 01:26:45.570 --> 01:26:50.880 align:middle line:84% So is a summation of various terms. 01:26:50.880 --> 01:27:06.360 align:middle line:84% You can think of a summation as the various terms. 01:27:06.360 --> 01:27:10.050 align:middle line:84% Consider them as the terms in a histogram. 01:27:10.050 --> 01:27:13.280 align:middle line:84% And then the summation is the area. 01:27:13.280 --> 01:27:18.410 align:middle line:84% So this is like the first term, say, from 0 to 1 and from here 01:27:18.410 --> 01:27:20.065 align:middle line:90% to 2 and 3 and so on. 01:27:20.065 --> 01:27:22.690 align:middle line:90% 01:27:22.690 --> 01:27:27.770 align:middle line:84% And so the summation gives you essentially the area. 01:27:27.770 --> 01:27:31.640 align:middle line:84% And this is the value of the element in the sum. 01:27:31.640 --> 01:27:35.630 align:middle line:84% The idea here is that you sometimes 01:27:35.630 --> 01:27:43.290 align:middle line:84% can approximate this summation as the integral of a curve. 01:27:43.290 --> 01:27:48.210 align:middle line:84% And that approximation is good when 01:27:48.210 --> 01:27:55.110 align:middle line:84% the error that you make, these differences here become small. 01:27:55.110 --> 01:27:56.350 align:middle line:90% So this is the idea. 01:27:56.350 --> 01:28:00.040 align:middle line:90% 01:28:00.040 --> 01:28:07.760 align:middle line:84% It requires that these things be small somehow. 01:28:07.760 --> 01:28:11.200 align:middle line:84% And that condition is written here. 01:28:11.200 --> 01:28:19.600 align:middle line:84% It means that this combination of terms must be smaller than 1. 01:28:19.600 --> 01:28:25.420 align:middle line:84% This is a dimensionless number, which contains temperature 01:28:25.420 --> 01:28:29.110 align:middle line:90% and the square of the side. 01:28:29.110 --> 01:28:31.430 align:middle line:90% And it has to be small. 01:28:31.430 --> 01:28:35.380 align:middle line:84% If this is small, you can approximate the summation 01:28:35.380 --> 01:28:36.470 align:middle line:90% with an integral. 01:28:36.470 --> 01:28:38.380 align:middle line:84% And then this is a kind of integral 01:28:38.380 --> 01:28:44.320 align:middle line:84% that can be done because it's an integral from 0 to infinity. 01:28:44.320 --> 01:28:51.630 align:middle line:90% And here is the result. 01:28:51.630 --> 01:28:55.120 align:middle line:84% And this is the partition function for one direction. 01:28:55.120 --> 01:28:57.120 align:middle line:84% If you multiply the partition functions 01:28:57.120 --> 01:29:05.050 align:middle line:84% for the various directions, so you might apply the three. 01:29:05.050 --> 01:29:07.470 align:middle line:84% You get the partition function for the overall particle 01:29:07.470 --> 01:29:08.970 align:middle line:90% in the box. 01:29:08.970 --> 01:29:20.180 align:middle line:90% And you see here, I have an li. 01:29:20.180 --> 01:29:26.090 align:middle line:84% It's squared, but actually, it's to the 1/2. 01:29:26.090 --> 01:29:29.150 align:middle line:90% So you can take it outside also. 01:29:29.150 --> 01:29:31.530 align:middle line:84% And when you multiply the three of them, 01:29:31.530 --> 01:29:38.960 align:middle line:84% it becomes l1 times l2 times l3, which is the volume of the box. 01:29:38.960 --> 01:29:43.250 align:middle line:84% And so the partition function also here is Q to the 2/3, 01:29:43.250 --> 01:29:44.760 align:middle line:90% but it's to the 3/2. 01:29:44.760 --> 01:29:47.690 align:middle line:90% So you can take the V outside. 01:29:47.690 --> 01:29:51.890 align:middle line:84% So essentially, this says that the overall partition function 01:29:51.890 --> 01:29:57.020 align:middle line:90% depends on the volume only. 01:29:57.020 --> 01:30:00.050 align:middle line:84% Essentially in this limit, you are 01:30:00.050 --> 01:30:07.410 align:middle line:84% losing the effect of the details of the shape of the box. 01:30:07.410 --> 01:30:12.452 align:middle line:84% What counts is just the product of the three, 01:30:12.452 --> 01:30:13.285 align:middle line:90% which is the volume. 01:30:13.285 --> 01:30:18.970 align:middle line:90% 01:30:18.970 --> 01:30:27.250 align:middle line:84% In fact, in that limit, if you take the formulas we have up 01:30:27.250 --> 01:30:31.630 align:middle line:84% here for, say, for the energy E sub i, 01:30:31.630 --> 01:30:36.640 align:middle line:84% you have to take the partial of the logarithm with respect to T. 01:30:36.640 --> 01:30:43.930 align:middle line:84% Well, here is Q. Take the logarithm with respect to T. 01:30:43.930 --> 01:30:47.980 align:middle line:90% It's just 1/2 times kT. 01:30:47.980 --> 01:30:51.570 align:middle line:90% That's the result. 01:30:51.570 --> 01:30:57.000 align:middle line:84% And therefore-- and it's equal in all directions. 01:30:57.000 --> 01:31:02.700 align:middle line:84% This is so-called equipartition theorem. 01:31:02.700 --> 01:31:10.950 align:middle line:84% The particle has decided in this limit 01:31:10.950 --> 01:31:16.770 align:middle line:84% to distribute equally the energy in the three 01:31:16.770 --> 01:31:20.180 align:middle line:90% translational modes. 01:31:20.180 --> 01:31:25.450 align:middle line:90% And each one gets a half of kT. 01:31:25.450 --> 01:31:29.200 align:middle line:84% Meaning that since the energy of the particle is the sum 01:31:29.200 --> 01:31:34.360 align:middle line:84% of the energies of the three directions, 01:31:34.360 --> 01:31:37.330 align:middle line:90% E is equal to 3/2 of kT. 01:31:37.330 --> 01:31:40.466 align:middle line:90% 01:31:40.466 --> 01:31:43.080 align:middle line:84% This is a structureless particle. 01:31:43.080 --> 01:31:49.020 align:middle line:84% And so the specific heat, which is the partial of E with respect 01:31:49.020 --> 01:31:53.450 align:middle line:84% to T is 3/2 of the Boltzmann constant. 01:31:53.450 --> 01:31:59.480 align:middle line:90% 01:31:59.480 --> 01:32:03.595 align:middle line:84% Also, these pressures become all equal to one another. 01:32:03.595 --> 01:32:07.740 align:middle line:90% 01:32:07.740 --> 01:32:12.150 align:middle line:84% So if I take the Gibbs relation and these pi's are 01:32:12.150 --> 01:32:21.330 align:middle line:84% equal to one another, I can write it this way. 01:32:21.330 --> 01:32:24.600 align:middle line:90% And so I get-- 01:32:24.600 --> 01:32:29.700 align:middle line:84% so these three terms become kT, which is here, 01:32:29.700 --> 01:32:33.090 align:middle line:90% times the sum of the dl/l. 01:32:33.090 --> 01:32:36.520 align:middle line:84% This is the sum of the logarithm. 01:32:36.520 --> 01:32:41.480 align:middle line:84% So it's the derivative of the volume over the volume. 01:32:41.480 --> 01:32:44.070 align:middle line:90% 01:32:44.070 --> 01:32:50.410 align:middle line:90% It's d of log of V. 01:32:50.410 --> 01:32:53.200 align:middle line:84% And so we have the Gibbs relation. 01:32:53.200 --> 01:32:55.510 align:middle line:84% But now, if we take the Gibbs relation, 01:32:55.510 --> 01:32:58.420 align:middle line:84% and remember that what multiplies 01:32:58.420 --> 01:33:02.470 align:middle line:84% dV is what we call pressure, here we 01:33:02.470 --> 01:33:04.870 align:middle line:84% find that, therefore, the pressure 01:33:04.870 --> 01:33:08.620 align:middle line:84% of the particle in the box is kT/V. 01:33:08.620 --> 01:33:14.300 align:middle line:84% So we get pV is equal to kT, which is the ideal gas law. 01:33:14.300 --> 01:33:18.090 align:middle line:90% 01:33:18.090 --> 01:33:19.770 align:middle line:84% We also get, of course, the expression 01:33:19.770 --> 01:33:27.000 align:middle line:84% for the entropy, which can be written 01:33:27.000 --> 01:33:30.840 align:middle line:84% as a function of temperature and volume only. 01:33:30.840 --> 01:33:34.050 align:middle line:84% So we lost the information about-- the need 01:33:34.050 --> 01:33:36.760 align:middle line:90% to know the details of the box. 01:33:36.760 --> 01:33:39.600 align:middle line:84% So that means that in that limit-- 01:33:39.600 --> 01:33:44.670 align:middle line:84% and this limit, this condition that this is small can also be 01:33:44.670 --> 01:33:49.060 align:middle line:84% written this way, that the temperature, 01:33:49.060 --> 01:33:54.570 align:middle line:84% if you fix the volume, it means that the temperature must be big 01:33:54.570 --> 01:33:59.430 align:middle line:84% enough so that these constants divided by the volume 01:33:59.430 --> 01:34:00.125 align:middle line:90% to the 2/3-- 01:34:00.125 --> 01:34:04.640 align:middle line:90% 01:34:04.640 --> 01:34:07.990 align:middle line:84% so that the temperature is much greater than this number. 01:34:07.990 --> 01:34:12.810 align:middle line:90% 01:34:12.810 --> 01:34:15.460 align:middle line:84% Or you could also put it differently. 01:34:15.460 --> 01:34:18.030 align:middle line:84% If you fix the temperature, you need 01:34:18.030 --> 01:34:21.170 align:middle line:90% the volume to be big enough. 01:34:21.170 --> 01:34:25.500 align:middle line:84% The system must be diluted enough. 01:34:25.500 --> 01:34:33.990 align:middle line:84% So in these limits, we find the ideal gas law. 01:34:33.990 --> 01:34:44.670 align:middle line:84% And we find that a box of the same volume, it could be very-- 01:34:44.670 --> 01:34:49.170 align:middle line:84% many different shapes can have the same volume. 01:34:49.170 --> 01:34:51.960 align:middle line:84% They all have the same properties. 01:34:51.960 --> 01:35:00.690 align:middle line:84% That's why in most practical situations, since this 01:35:00.690 --> 01:35:07.290 align:middle line:84% is satisfied except when you go to very low temperatures, that's 01:35:07.290 --> 01:35:10.592 align:middle line:84% why we typically use just the volume as parameter 01:35:10.592 --> 01:35:12.550 align:middle line:84% because otherwise you would say, all right, you 01:35:12.550 --> 01:35:15.340 align:middle line:84% said at the beginning you need to put the parameters 01:35:15.340 --> 01:35:16.580 align:middle line:90% in the fundamental relation. 01:35:16.580 --> 01:35:18.650 align:middle line:84% But then you end up always putting only volume. 01:35:18.650 --> 01:35:19.150 align:middle line:90% Why? 01:35:19.150 --> 01:35:24.170 align:middle line:84% Well, here is the reason because for most practical situations, 01:35:24.170 --> 01:35:30.420 align:middle line:84% but not for small systems, volume is all that counts. 01:35:30.420 --> 01:35:36.800 align:middle line:90% 01:35:36.800 --> 01:35:46.790 align:middle line:84% OK, so just a couple of minutes we are left. 01:35:46.790 --> 01:35:49.520 align:middle line:84% I want to introduce this idea, which 01:35:49.520 --> 01:35:52.360 align:middle line:90% is going to be our next step. 01:35:52.360 --> 01:35:57.920 align:middle line:90% 01:35:57.920 --> 01:36:06.050 align:middle line:84% Remember that we talked about the gas state 01:36:06.050 --> 01:36:09.320 align:middle line:84% as one in which the particles spend most of their time distant 01:36:09.320 --> 01:36:13.200 align:middle line:84% from one another, in spite of their colliding, OK? 01:36:13.200 --> 01:36:22.150 align:middle line:90% 01:36:22.150 --> 01:36:26.980 align:middle line:90% So in the ideal gas-- 01:36:26.980 --> 01:36:31.480 align:middle line:84% Gibbs-Dalton mixture of ideal gases, 01:36:31.480 --> 01:36:35.410 align:middle line:84% we found that essentially the particles live in the space 01:36:35.410 --> 01:36:38.355 align:middle line:84% without seeing each other except when they collide. 01:36:38.355 --> 01:36:45.880 align:middle line:90% 01:36:45.880 --> 01:36:55.630 align:middle line:84% And that results to the fact that the entropy of mixing 01:36:55.630 --> 01:37:01.720 align:middle line:84% is the only effect that results upon mixing. 01:37:01.720 --> 01:37:09.430 align:middle line:84% You mix gases at a given spontaneous process. 01:37:09.430 --> 01:37:11.860 align:middle line:90% The temperature doesn't change. 01:37:11.860 --> 01:37:13.080 align:middle line:90% The pressure doesn't change. 01:37:13.080 --> 01:37:15.960 align:middle line:90% 01:37:15.960 --> 01:37:18.868 align:middle line:84% And the only thing that changes is the entropy. 01:37:18.868 --> 01:37:19.535 align:middle line:90% It's the mixing. 01:37:19.535 --> 01:37:25.550 align:middle line:90% 01:37:25.550 --> 01:37:31.220 align:middle line:84% Now, we need to define also an ideal behavior 01:37:31.220 --> 01:37:32.810 align:middle line:90% for the liquid phase. 01:37:32.810 --> 01:37:37.740 align:middle line:84% Now, in liquid phase, the particles do see each other. 01:37:37.740 --> 01:37:42.380 align:middle line:84% Actually, they live most of the times close to one another, 01:37:42.380 --> 01:37:53.090 align:middle line:84% so close to where this intermolecular potential has 01:37:53.090 --> 01:37:55.010 align:middle line:90% a minimum. 01:37:55.010 --> 01:37:58.006 align:middle line:90% So for them, that is important. 01:37:58.006 --> 01:38:01.220 align:middle line:90% 01:38:01.220 --> 01:38:05.840 align:middle line:84% However, we are going to define an ideal solution 01:38:05.840 --> 01:38:15.750 align:middle line:84% behavior, one in which the curves for the interaction 01:38:15.750 --> 01:38:22.890 align:middle line:84% between equal particles or different particles 01:38:22.890 --> 01:38:26.940 align:middle line:90% are all very similar. 01:38:26.940 --> 01:38:31.920 align:middle line:84% So if they are all very similar, the particle 01:38:31.920 --> 01:38:35.580 align:middle line:84% doesn't care what the neighbor is. 01:38:35.580 --> 01:38:43.350 align:middle line:84% In this limit, again, when you mix stuff, the only effect-- 01:38:43.350 --> 01:38:48.540 align:middle line:84% you don't get any energy of mixing or enthalpy of mixing. 01:38:48.540 --> 01:38:50.370 align:middle line:84% You don't get any change of volume 01:38:50.370 --> 01:38:53.760 align:middle line:84% of mixing because the particle doesn't want 01:38:53.760 --> 01:38:56.260 align:middle line:84% to change their preferred distances, 01:38:56.260 --> 01:39:02.650 align:middle line:84% which would imply a enthalpy of mixing or volume of mixing, 01:39:02.650 --> 01:39:06.570 align:middle line:84% which would imply changing the distance. 01:39:06.570 --> 01:39:15.360 align:middle line:84% Because essentially they like each other in the same way. 01:39:15.360 --> 01:39:22.410 align:middle line:84% The only effect upon mixing is the randomness, 01:39:22.410 --> 01:39:23.855 align:middle line:90% so the entropy of mixing. 01:39:23.855 --> 01:39:26.576 align:middle line:90% 01:39:26.576 --> 01:39:30.370 align:middle line:84% And therefore, in this model that we define, 01:39:30.370 --> 01:39:34.920 align:middle line:84% we will assume that the entropy of mixing of the single particle 01:39:34.920 --> 01:39:38.160 align:middle line:90% is related to the surprise. 01:39:38.160 --> 01:39:44.640 align:middle line:84% And that is the only effect of this ideal situation 01:39:44.640 --> 01:39:46.740 align:middle line:90% that we're going to define. 01:39:46.740 --> 01:39:48.370 align:middle line:90% OK? 01:39:48.370 --> 01:39:53.960 align:middle line:84% So with that, we've set the ground for the next lecture. 01:39:53.960 --> 01:40:01.000 align:middle line:90%